Méthodes de Monte Carlo appliquées à la Finance - Maths-Fi

Transcription

Pìwkrghv gh Prqwh Fduor
dssoltxìhv ã od Ilqdqfh =
Hfkdqwloorqqdjh g*lpsruwdqfh hw Vwudwlfdwlrq
srxu ohv surfhvvxv gh glxvlrq
Qlfrodv Edxg
Ylqfhqw Sruwh
Jurxsh gh Wudydlo
Gluljì sdu Sdxo Ghph| hw Jdîo Ulerxohw/
Jurxsh gh Uhfkhufkh Rsìudwlrqqhooh/ Fuìglw O|rqqdlv
pdl 5334
Wdeoh ghv pdwlëuhv
Lqwurgxfwlrq
4
4 Hydoxdwlrq gh surgxlwv gìulyìv sdu vlpxodwlrq
6
414
Sulqflsh ghv pìwkrghv gh Prqwh Fduor 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
6
415
Frqwu÷oh gh od frqyhujhqfh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
7
416
Dffìoìudwlrq gh od frqyhujhqfh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
9
5 Fkdqjhphqw gh gìulyh gìwhuplqlvwh
46
514
Glvfuìwlvdwlrq gx surfhvvxv gh glxvlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
47
515
Frqvwuxfwlrq gh od gìulyh vxlydqw xq sulqflsh gh judqghv gìyldwlrqv 1 1 1
48
516
Dffìoìudwlrq gh od frqyhujhqfh sdu vwudwlfdwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1
4;
517
Dssolfdwlrqv = rswlrqv vxu xq vrxv0mdfhqw ã yrodwlolwì vwrfkdvwltxh 1 1 1 1
4<
6 Fkdqjhphqw gh gìulyh vwrfkdvwltxh
614 Surfhvvxv rswlpdo !W 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
55
56
615
Xwlolvdwlrq ghv judqghv gìyldwlrqv srxu o*dffìoìudwlrq gh od frqyhujhqfh 1
57
616
Gìulyh gh Eodfn hw Vfkrohv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
67
Frqfoxvlrq
6;
Dqqh{hv
73
D Sulqflsh gh Judqghv Gìyldwlrqv
73
E Fkdqjhphqw gh gìulyh gìwhuplqlvwh hw vwudwlfdwlrq = uìvxowdwv qxpì0
ultxhv
75
F Fkdqjhphqw gh gìulyh vwrfkdvwltxh = uìvxowdwv qxpìultxhv
76
F14 Hydoxdwlrq g*rswlrqv g*dfkdw hxursìhqqhv = prgëoh 4 +Khvwrq, 1 1 1 1 1
77
F15 Hydoxdwlrq g*rswlrqv g*dfkdw hxursìhqqhv = prgëoh 5 1 1 1 1 1 1 1 1 1 1 1
78
F16 Hydoxdwlrq g*rswlrqv GLF = prgëoh 4 +Khvwrq, 1 1 1 1 1 1 1 1 1 1 1 1 1 1
79
F17 Hydoxdwlrq g*rswlrqv GLF = prgëoh 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
7:
G Yduldeoh gh frqwu÷oh = oh fdv g*xqh rswlrq vxu sdqlhu
7;
Lqwurgxfwlrq
Oh fdguh jìqìudo gh qrwuh sureoëph hvw o*ìydoxdwlrq gx sul{ g*xq surgxlw qdqflhu grqw
oh sd|r whuplqdo gìshqg ghv ydohxuv ixwxuhv g*xq dfwli vrxv0mdfhqw1 Gdqv xq suhplhu
whpsv/ lo frqylhqw gh prgìolvhu fruuhfwhphqw o*ìyroxwlrq gh od ydohxu gx vrxv0mdfhqw vxu
ohtxho hvw frqvwuxlw oh surgxlw qdqflhu1
Oh prgëoh gh Eodfn0Vfkrohv ^5` hvw xq ghv prgëohv ohv soxv vlpsohv1 Od ydohxu gx
vrxv0mdfhqw vxlw xq prxyhphqw eurzqlhq jìrpìwultxh =
gV|
@ egw . gZ|
V|
rü e hw vrqw ghx{ frqvwdqwhv/ +Z| , hvw xq prxyhphqw eurzqlhq vwdqgdug vrxv od sur0
edelolwì klvwrultxh1 Fhwwh prgìolvdwlrq hvw wuëv lqwìuhvvdqwh srxu o*ìydoxdwlrq g*rswlrqv1
Hq hhw/ oh pdufkì pxql g*xq dfwli vdqv ulvtxh gh uhqghphqw u hvw doruv frpsohw/ fh txl
shuphw gh gìwhuplqhu gh idêrq xqltxh oh sul{ gh wrxw surgxlw qdqflhu1 Xq ghx{lëph
dydqwdjh hvw o*h{lvwhqfh gh irupxohv ihupìhv grqqdqw od ydohxu gh ehdxfrxs gh surgxlwv
qdqflhuv/ frpph ohv rswlrqv hxursìhqqhv gh yhqwh hw g*dfkdw1 Fhshqgdqw/ fh prgëoh
qh uhqg sdv frpswh gh soxvlhxuv skìqrpëqhv1 O*hhw ohswrnxuwltxh +txhxhv ìsdlvvhv ghv
uhqghphqwv, q*hvw sdv sulv hq frpswh sdu fh prgëoh orjqrupdo1 Gh soxv/ oh vploh revhuyì
vxu ohv yrodwlolwìv lpsolflwhv/ fdofxoìhv hq lqyhuvdqw od irupxoh gh Eodfn0Vfkrohv srxu
ohv rswlrqv/ frqwuhglw fh prgëoh1
Kxoo hw Zklwh ^48` vh vrqw lqwìuhvvìv ã xq prgëoh soxv jìqìudo/ gdqv ohtxho od ydohxu
gx vrxv0mdfhqw hvw jrxyhuqìh sdu ohv ìtxdwlrqv vxlydqwhv =
_7| @ e gw . +\ , gZ
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7|
g\| @ | gw . | gZ2c|
g kZ > Z2 l| @ | gw
Fh prgëoh qh suìvhqwh sdv ohv lqfrqyìqlhqwv gx prgëoh gh Eodfn0Vfkrohv1 Qìdqprlqv/
oh pdufkì q*hvw soxv frpsohw hw lo q*hvw sdv srvvleoh gh gìwhuplqhu gh idêrq xqltxh oh sul{
ghv rswlrqv1 Fhshqgdqw/ o*h{lvwhqfh g*xqh rswlrq hxursìhqqh frwìh shuphw gh frpsoìwhu
oh pdufkì +Urpdqr hw Wrx}l^4;`,1 Dlqvl/ hq vxssrvdqw tx*lo | d devhqfh g*rssruwxqlwì
g*duelwudjh vxu ohv pdufkìv/ lo h{lvwh xqh xqltxh suredelolwì T whooh txh oh sul{ L} g*xq
surgxlw qdqflhu gh x{ ixwxuv dfwxdolvìv j +V| > 3 w W , vh phwwh vrxv od iruph =
L} @ HT ^j +V| > 3 w W ,` =
Lo q*h{lvwh soxv hq jìqìudo gh irupxoh dqdo|wltxh vlpsoh txl shuphwwh g*rewhqlu L} gh idêrq
gluhfwh/ pdlv fhwwh uhsuìvhqwdwlrq suredelolvwh gx sul{/ lvvxh gh od wkìrulh gìyhorssìh sdu
Kduulvrq hw Nuhsv ^44`/ vxjjëuh g*xwlolvhu ohv pìwkrghv gh Prqwh Fduor srxu hq hvwlphu
od ydohxu1
Od vlpxodwlrq sdu Prqwh Fduor hvw irqgìh vxu od orl ghv judqghv qrpeuhv1 Oh sulqflsh hvw
gh vlpxohu/ gh idêrq lqgìshqgdqwh/ xq judqg qrpeuh gh yduldeohv doìdwrluhv +V| ,f$|$A /
4
sxlv gh suhqguh frpph hvwlpdwhxu gx sdudpëwuh g*lqwìuíw L} od pr|hqqh hpslultxh ghv
j +V| > 3 w W , uìdolvìv1
Vl o*huuhxu frpplvh hvw qxooh gh idêrq dv|pswrwltxh/ fh q*hvw sdv oh fdv ã glvwdqfh qlh1
Lo frqylhqw grqf gh phwwuh hq sodfh ghv rxwlov gh frqwu÷oh gh od frqyhujhqfh1 Fhx{0fl
uhsrvhqw hvvhqwlhoohphqw vxu o*xwlolvdwlrq gx wkìruëph Fhqwudo Olplwh1 Lo dssdudñw txh od
yduldqfh gh od orl vlpxoìh mrxh xq u÷oh irqgdphqwdo srxu od frqvwuxfwlrq ghv lqwhuydoohv
gh frqdqfh = soxv od yduldqfh hvw ìohyìh/ soxv od wdlooh gh o*lqwhuydooh hvw lpsruwdqwh1
Dq g*dpìolruhu od suìflvlrq gh o*hvwlpdwhxu/ rq fkhufkhud grqf ã vlpxohu gh qrxyhoohv
yduldeohv doìdwrluhv g*hvsìudqfh oh sdudpëwuh g*lqwìuíw L} hw gh yduldqfh lqiìulhxuh ã fhooh
gh j +V| > 3 w W ,1
Gdqv od suhplëuh sduwlh gh fh pìprluh/ qrxv suìvhqwrqv/ gdqv xq fdguh jìqìudo/ ohv txdwuh
pìwkrghv gh uìgxfwlrq gh od yduldqfh ohv soxv fodvvltxhv1 Qrxv qrxv lqwìuhvvhurqv soxv
sduwlfxolëuhphqw ã o*ìfkdqwloorqqdjh g*lpsruwdqfh hw ã od vwudwlfdwlrq/ pìwkrghv txh
qrxv h{sorlwhurqv/ gdqv ohv ghx{ sduwlhv vxlydqwhv/ gdqv oh fdv sduwlfxolhu ghv surfhvvxv
gh glxvlrq1
Dssoltxìh dx fdofxo gx sdudpëwuh L} @ HT ^j +V| > 3 w W ,`/ o*ìfkdqwloorqqdjh g*lpsru0
wdqfh hvw xqh pìwkrgh txl frqvlvwh ã prglhu od orl gh od yduldeoh +V| ,f$|$A vlpxoìh
gh idêrq ã suhqguh hq frpswh ohv vsìflflwìv gh od irqfwlrq j ã lqwìjuhu/ ohv uìdolvdwlrqv
j +V| > 3 w W , ìwdqw doruv srqgìuìhv dq gh qh sdv eldlvhu o*hvwlpdwlrq gh L} 1 j uhsuì0
vhqwdqw oh sd|r dfwxdolvì g*xq surgxlw gìulyì/ lo v*djlw hq jìqìudo g*xqh irqfwlrq srvlwlyh
rx qxooh/ vwulfwhphqw srvlwlyh vxu oh grpdlqh g*h{huflfh1 Od vlpxodwlrq gh wudmhfwrluhv
qh shuphwwdqw sdv o*h{huflfh gh o*rswlrq q*dssruwhqw txh shx g*lqirupdwlrq srxu hvwl0
phu L} 1 Hq uhydqfkh/ fhv wudmhfwrluhv frqwulexhqw gh idêrq vljqlfdwlyh ã od yduldqfh gh
o*hvwlpdwhxu1 Lo shxw grqf íwuh lqwìuhvvdqw gh prglhu od orl gx surfhvvxv +V| ,f$|$A dq
g*dxjphqwhu oh qrpeuh gh wudmhfwrluhv vlpxoìhv shuphwwdqw o*h{huflfh gh o*rswlrq1
O*dssurfkh txh qrxv gìyhorssrqv gdqv ohv ghx{lëph hw wurlvlëph sduwlhv gh fh pìprluh
frqvlvwh ã prglhu od gìulyh gx surfhvvxv +V| ,f$|$A dq g*dffurñwuh od suredelolwì g*h{hu0
flfh1 Gh idêrq soxv qh/ lo v*djlw gh phwwuh hq sodfh ghv fkdqjhphqwv gh gìulyh suhqdqw
hq frpswh od glvwulexwlrq gx sd|r vxu oh grpdlqh g*h{huflfh = rq fkhufkh ã frqvwuxluh
xq hvwlpdwhxu lqwìjudqw oh pd{lpxp g*lqirupdwlrq shuwlqhqwh vxu od irqfwlrq ã lqwìjuhu1
Ohv glìuhqwhv pìwkrghv suìvhqwìhv glëuhqw sdu oh prgh gh frqvwuxfwlrq gx fkdqjhphqw
gh gìulyh1 Gdqv od ghx{lëph sduwlh/ rq vh olplwh dx{ vhxov fkdqjhphqwv gh gìulyh gìwhu0
plqlvwhv1 Fhwwh whfkqltxh suìvhqwh o*dydqwdjh gh srxyrlu íwuh frxsoìh ã xqh ghx{lëph
pìwkrgh g*dffìoìudwlrq = od vwudwlfdwlrq1 Gdqv od wurlvlëph sduwlh/ qrxv qrxv lqwìuhvvrqv
dx fdv ghv fkdqjhphqwv gh gìulyh vwrfkdvwltxhv1 Gdqv wrxv ohv fdv/ ohv fkdqjhphqwv gh
gìulyh sursrvìv v*dyëuhqw rswlpdx{ gh idêrq dv|pswrwltxh1
5
4
Hydoxdwlrq gh surgxlwv gìulyìv sdu vlpxodwlrq
Od uhsuìvhqwdwlrq vrxv iruph suredelolvwh ghv surgxlwv gìulyìv/ lvvxh gh od wkìrulh gìyh0
orssìh sdu Kduulvrq hw Nuhsv ^44`/ hw Kduulvrq hw Solvnd ^45`^46` frqgxlw qdwxuho0
ohphqw ã frqvlgìuhu ohv pìwkrghv gh Prqwh Fduor srxu o*ìydoxdwlrq gh ohxuv sul{ = gdqv
od phvxuh rü xq pdufkì qdqflhu shxw íwuh grwì g*xqh suredelolwì ulvtxh0qhxwuh T/ oh
sul{ g*xq dfwli dssdudñw frpph ìwdqw ìjdo ã o*hvsìudqfh/ vrxv fhwwh suredelolwì/ ghv x{
ixwxuv dfwxdolvìv tx*lo jìqëuh1 Iruphoohphqw/ oh sureoëph gh o*ìydoxdwlrq g*xqh rswlrq
v*ìfulw
L} @ HT ^j +V| > 3 w W ,` =
+414,
V| uhsuìvhqwdqw oh sul{ gx vrxv0mdfhqw ã od gdwh
dfwli
shxw grqf íwuh fdofxoì
w1 Oh sul{ g*xq
hq vlpxodqw Q wudmhfwrluhv lqgìshqgdqwhv V| f$|$A > = = = > V| f$|$A 1 Srxu fkdfxqh gh
fhv wudmhfwrluhv/ rq fdofxoh doruv oh prqwdqw j V| > 3 w W ghv x{ ixwxuv dfwxdolvìv
fruuhvsrqgdqwv1 Od orl iruwh ghv judqgv qrpeuhv dvvxuh od frqyhujhqfh gh od pr|hqqh
dulwkpìwltxh ghv ydohxuv vlpxoìhv yhuv oh sul{1
Gdqv xq fhuwdlq qrpeuh gh vlwxdwlrqv sduwlfxolëuhv/ o*ìtxdwlrq +4=4, vh uìvrxw gh idêrq
dqdo|wltxh1 F*hvw sdu h{hpsoh oh fdv ghv rswlrqv g*dfkdw hxursìhqqhv gdqv oh prgëoh
gh Eodfn hw Vfkrohv1 Fhshqgdqw/ srxu ghv surgxlwv soxv frpsoh{hv/ rx srxu xqh
prgìolvdwlrq soxv qh gx vrxv0mdfhqw lqwìjudqw xq doìd vxssoìphqwdluh gdqv o*ìyroxwlrq
gh od irqfwlrq gh yrodwlolwì / lo hvw uduh txh o*rq sxlvvh rewhqlu ghv irupxohv ihupìhv
g*ìydoxdwlrq1 Lo hvw doruv qìfhvvdluh g*dssoltxhu ghv pìwkrghv qxpìultxhv srxu uìvrxguh
oh sureoëph g*ìydoxdwlrq gh o*rswlrq1 Ohv vlpxodwlrqv sdu pìwkrghv gh Prqwh Fduor vrqw
doruv xqh ghv whfkqltxhv frxudpphqw xwlolvìhv1
414
Sulqflsh ghv pìwkrghv gh Prqwh Fduor
Ohv whfkqltxhv gh vlpxodwlrq sdu pìwkrgh gh Prqwh Fduor xwlolvhqw od
orl iruwh ghv judqgv
qrpeuhv srxu hvwlphu od ydohxu uhfkhufkìh L} 1 Vl ohv frxuehv V @ V| > 3 w W > 4 l Q/ uhsuìvhqwhqw Q wudmhfwrluhv lqgìshqgdqwhv glvwulexìhv vxlydqw xqh píph orl/ doruv
f @
4 [ Rr
j V $ L} =
Q '
Oh sulqflsh
gh edvh ghv pìwkrghv gh Prqwh Fduor frqvlvwh ã jìqìuhu Q wudmhfwrluhv
V @ V| > 3 w W hw ã frqvlgìuhu f frpph hvwlpdwhxu gx sul{/ fhoxl0fl frqyhujhdqw
yhuv oh sdudpëwuh g*lqwìuíw L} 1
Oh wkìruëph fhqwudo olplwh shuphw g*dyrlu xqh lgìh gh o*dssur{lpdwlrq uìdolvìh1 Vl oh
prphqw g*ruguh 5 gh j +V| > 3 w W , hvw ql/ doruv =
#
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j V L} $ Q +3> ydu ^j +V,`, =
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6
Od gìwhuplqdwlrq gx qrpeuh Q gh vlpxodwlrqv qìfhvvdluhv srxu rewhqlu xq hvwlpdwhxu
vdwlvidlvdqw gx sul{ frqvwlwxh o*xq ghv sureoëphv dxtxho vh wurxyh frqiurqwì oh sudwlflhq1
G*dsuëv oh wkìruëph fhqwudo olplwh txh o*rq ylhqw gh udsshohu/ soxv od yduldqfh Y gh j V hvw ìohyì/ soxv oh qrpeuh gh vlpxodwlrqv qìfhvvdluhv ã o*rewhqwlrq g*xq hvwlpdwhxu suìflv
hvw lpsruwdqw1 Vl od uìdolvdwlrq g*xqh vlpxodwlrq hvw wuëv frþwhxvh hq whpsv gh fdofxo/ lo
shxw íwuh lqwìuhvvdqw gh gìyhorsshu xqh dssurfkh dowhuqdwlyh mrxdqw qrq sdv vxu Q pdlv
T=
vxu Y dq gh uìgxluh oh udwlr Qrxv suìvhqwrqv gdqv ohv sdudjudskhv vxlydqwv ohv pìwkrghv fodvvltxhv gh uìgxfwlrq gh
od yduldqfh xwlolvìhv gdqv oh fdguh ghv vfkìpdv gh Prqwh Fduor1 Gdqv xq suhplhu whpsv/
qrxv dgrswrqv xq srlqw gh yxh uhodwlyhphqw jìqìudo/ lqgìshqgdpphqw ghv frqvlgìudwlrqv
vsìfltxhv ã od prgìolvdwlrq qdqflëuh1 Oh irupdolvph sursuh dx{ surfhvvxv xwlolvìv gdqv
ohv dssolfdwlrqv qdqflëuhv hvw gìyhorssì gdqv xq vhfrqg whpsv1
Sdupl ohv glìuhqwhv pìwkrghv suìvhqwìhv/ qrxv ìwxglhurqv soxv sduwlfxolëuhphqw od pì0
wkrgh g*ìfkdqwloorqqdjh g*lpsruwdqfh +rx g*ìfkdqwloorqqdjh srqgìuì,/ vxu odtxhooh vh
irqghqw ohv glìuhqwv hvwlpdwhxuv gh sul{ txh qrxv suìvhqwrqv gdqv ohv ghx{ sduwlhv
vxlydqwhv1
Srxu frpphqfhu/ qrxv qrxv lqwìuhvvrqv dx sureoëph gx frqwu÷oh gh od frqyhujhqfh1 Hq
sduwlfxolhu/ qrxv udsshorqv vxfflqfwhphqw od idêrq grqw rq frqvwuxlw ohv lqwhuydoohv gh
frqdqfh dssdudlvvdqw gdqv ohv glyhuvhv dssolfdwlrqv gh fh pìprluh1
415
Frqwu÷oh gh od frqyhujhqfh
Oh fdguh g*dqdo|vh hvw srvì gh od idêrq vxlydqwh1 Rq frqvlgëuh xq hvsdfh suredelolvì
+> I> T, hw rq vh grqqh xq yduldeoh doìdwrluh [ ã ydohxuv gdqv xq hvsdfh V> gh orl
+g{,1 Rq vrxkdlwh hvwlphu qxpìultxhphqw/ ã sduwlu g*xqh pìwkrgh gh vlpxodwlrq gh
Prqwh Fduor/ oh sdudpëwuh g*lqwìuíw
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rü j uhsuìvhqwh xqh irqfwlrq 0lqwìjudeoh1 Od pìwkrgh gh Prqwh Fduor vwdqgdug frqvlvwh
ã uìdolvhu Q wludjhv doìdwrluhv lqgìshqgdqwv +{ > = = = > { , vxlydqw od orl hw ã frqvlgìuhu
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frpph hvwlpdwhxu gx sdudpëwuh L} = Lo v*djlw lfl g*xqh dssolfdwlrq gluhfwh gh od orl iruwh
ghv judqgv qrpeuhv txl/ frpswh whqx ghv k|srwkëvhv txh o*rq ylhqw gh irupxohu/ dvvxuh
txh
olp f} @ L} s=v=
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Frpph gdqv oh fdv ghv pìwkrghv g*lqwìjudwlrq gìwhuplqlvwhv/ o*huuhxu g*dssur{lpdwlrq
olìh ã f} gìfurñw oruvtxh Q dxjphqwh1 Qìdqprlqv/ o*huuhxu qh v*h{sulph sdv lfl hq whuph
g*dssur{lpdwlrq pdwkìpdwltxh pdlv vxlydqw ghv frqvlgìudwlrqv vwdwlvwltxhv olìhv ã o*doìd
gh o*ìfkdqwloorqqdjh1 Hq sduwlfxolhu/ frpph qrxv o*dyrqv gìmã qrwì/ od frqyhujhqfh gh f}
yhuv oh sdudpëwuh g*lqwìuíw L} hvw xqh frqyhujhqfh suhvtxh vxuh/ fh txl vljqlh txh
T olp f} @ L} @ 4=
<"
Vl fhwwh sursulìwì hvw elhq ìylghpphqw vrxkdlwdeoh/ hooh qh shuphw sdv g*ìydoxhu o*huuhxu
frpplvh oruvtxh Q hvw ql1
Qrxv suìvhqwrqv lfl od pìwkrgh g*ìydoxdwlrq gh o*huuhxu frppxqìphqw xwlolvìh hq sud0
wltxh srxu uìvrxguh fh sureoëph1 Hooh uhsrvh vxu od frqvwuxfwlrq g*xq lqwhuydooh gh
frqdqfh1 Gh soxv/ qrxv h{solflwrqv ohv dssur{lpdwlrqv uìdolvìhv dq gh phwwuh hq ìyl0
ghqfh ohv olplwhv g*xq who lqwhuydooh1
O*lqwhuydooh LF +, txh o*rq vrxkdlwh h{solflwhu hvw fhqwuì vxu o*hvwlpdwhxu f} / hw who txh
oh sdudpëwuh g*lqwìuíw L} dssduwlhqqh ã LF +, dyhf suredelolwì 4 =
T +L} 5 LF +,, @ 4 Oruvtxh oh prphqw g*ruguh 5 gh j +[, hvw ql/ oh wkìruëph fhqwudo olplwh dvvxuh txh =
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Dlqvl/ srxu Q vx!vdpphqw
judqg/
rq shxw vxssrvhu txh o*hvwlpdwhxu f} hvw glvwulexì
@od}Ef o
vxlydqw xqh orl Q L} >
= Fhwwh suhplëuh dssur{lpdwlrq shuphw g*ìfuluh LF +,
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hvw xqh pduwlqjdoh vrxv T doruv rq gìqlw xqh qrxyhooh suredelolwì TE vxu +> IA , hq
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suredelolwì TE 1
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Frpswh whqx gh fh wkìruëph/ lo dssdudñw txh oh sdudpëwuh g*lqwìuíw L} +w> {, shxw v*ìfuluh =
k
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+4146,
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lqgìshqgdqwhv vxlydqw od orl gx surfhvvxv +[r ,|$r$A vrxv od suredelolwì TE > hw hq srvdqw
4 [
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fr +w> {, @
Q '
E
+4147,
Od vlpxodwlrq gh wudmhfwrluhv vxlydqw od orl gx surfhvvxv +[r ,|$r$A vrxv od suredelolwì
TE shxw íwuh uìdolvìh hq uhpdutxdqw txh +[r ,|$r$A / vroxwlrq gh o*ìtxdwlrq glìuhq0
wlhooh vwrfkdvwltxh +4=45, vrxv T/ hvw od vroxwlrq/ vrxv TE / gh o*ìtxdwlrq glìuhqwlhooh
vwrfkdvwltxh vxlydqwh =
g[r @ ^e +v> [r , +v> [r , !r ` gw . +v> [r , gEr =
+4148,
4
vx!vdqwh
sruwdqwk vxu loh surfhvvxv fruuhvsrqg dx fulwëuh gh Qrylnry = vl
kXqhfrqglwlrq
l
UA
E
T
' hw E hvw xqh T pduwlqjdoh1
H i T f r 2 _r " doruv HT A
45
E
Oh fkdqjhphqw gh suredelolwì srxu od frqvwuxfwlrq g*xq qrxyho hvwlpdwhxu fr +w> {,
gh L} +w> {, frqgxlw grqf ã od prglfdwlrq gx surfhvvxv gh gìulyh lqwhuyhqdqw gdqv od
fdudfwìulvdwlrq gx surfhvvxv +[| , = O*lgìh hvw lfl gh prglhu o*dooxuh ghv wudmhfwrluhv sdu
oh fkrl{ g*xq surfhvvxv ! mxglflhx{/ dq gh suhqguh hq frpswh ohv vsìflflwìv gh od
irqfwlrq j ã lqwìjuhu1
Ohv ghx{ vhfwlrqv vxlydqwhv suìvhqwhqw ghx{ dssurfkhv glìuhqwhv frqgxlvdqw ã od frqvwuxf0
wlrq g*xq surfhvvxv ! shuphwwdqw gh uìgxluh od yduldqfh gh o*hvwlpdwhxu gh L} = Gdqv od
suhplëuh dssurfkh/ oh surfhvvxv ! xwlolvì hvw xqh irqfwlrq hq hvfdolhu gìwhuplqlvwh1 Gdqv
od vhfrqgh dssurfkh/ ! hvw xq surfhvvxv vwrfkdvwltxh = !| gìshqg/ gh idêrq gìwhuplqlvwh/
gh od ydohxu [| gx surfhvvxv vrxv0mdfhqw ã od gdwh w1 Txh ! vrlw vwrfkdvwltxh rü gìwhupl0
qlvwh/ od frqvwuxfwlrq gh fh surfhvvxv uhsrvh vxu o*xwlolvdwlrq gh uìvxowdwv dv|pswrwltxhv
lvvxv gh od wkìrulh ghv judqghv gìyldwlrqv1 Dssoltxìhv ã o*ìydoxdwlrq gh surgxlwv gìulyìv/
fhv pìwkrghv v*dyëuhqw sduwlfxolëuhphqw h!fdfhv gdqv oh fdv g*rswlrqv wuëv hq ghkruv gh
od prqqdlh1
5
Fkdqjhphqw gh gìulyh gìwhuplqlvwh
Fhwwh sduwlh hvw frqvdfuìh ã od plvh hq sodfh g*xqh pìwkrgh g*ìfkdqwloorqqdjh srqgìuì
srxu oh fdofxo gx sdudpëwuh g*lqwìuíw
L} +w> {, @ HT ^j +[r > w v W , m [| @ {`
rü +[r , hvw xq surfhvvxv gìql vxu xq hvsdfh suredelolvì +> +I| , > I> T, > ã ydohxuv gdqv
U_ > vroxwlrq g*xqh ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh gh od iruph
g[| @ e +[| , gw . +[| , gZ| =
+514,
rü +Z| , hvw xq +I| , prxyhphqw eurzqlhq vrxv od suredelolwì T1 Qrwrqv txh o*rq vxs0
srvh gìvrupdlv txh ohv irqfwlrqv e hw vrqw lqgìshqgdqwhv gx whpsv1
Gdqv od vhfwlrq suìfìghqwh/ qrxv dyrqv h{solflwì xqh pìwkrgh shuphwwdqw gh gìqlu/ ã
E
sduwlu g*xq surfhvvxv !> o*hvwlpdwhxu fr +w> {, +uhodwlrq 4147 sdjh suìfìghqwh, gh L} +w> {,1
Rq glvsrvh d sulrul g*xqh wuëv judqgh olehuwì gh fkrl{ srxu oh surfhvvxv !/ gdqv od phvxuh
rü od vhxoh frqwudlqwh txl oxl hvw srxu o*lqvwdqw lpsrvìh hvw gh yìulhu ohv k|srwkëvhv gx
wkìruëph gh Jluvdqry1 Hq sduwlfxolhu/ ! shxw íwuh gìwhuplqlvwh rx vwrfkdvwltxh1 Gdqv
fhwwh vhfwlrq/ qrxv olplwrqv oh grpdlqh ghv surfhvvxv ! dgplvvleohv dx{ vhxov surfhvvxv
gìwhuplqlvwhv1
E
Od gìwhuplqdwlrq g*xqh irqfwlrq ! frqgxlvdqw ã xq hvwlpdwhxu fr +w> {, gh idleoh yd0
uldqfh hvw phqìh dsuëv glvfuìwlvdwlrq gh o*ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh yìulìh sdu
+[| , = Xqh whooh dssurfkh uhvwuhlqw oh grpdlqh ghv irqfwlrqv srvvleohv dx{ irqfwlrqv hq
hvfdolhu1 Oh fkrl{ gh od phloohxuh irqfwlrq srvvleoh uhsrvh hqvxlwh vxu xq udlvrqqhphqw
khxulvwltxh/ udlvrqqhphqw txl frqgxlw ã xq fkrl{ rswlpdo gdqv xq fdguh dv|pswrwltxh1
46
514
Glvfuìwlvdwlrq gx surfhvvxv gh glxvlrq
Lo h{lvwh gh qrpeuhxvhv pìwkrghv srxu vlpxohu od vroxwlrq g*xqh ìtxdwlrq glìuhqwlhooh
vwrfkdvwltxh +fi Sdugrx{ hw Wdod| ^4:`, hqwuh ohv gdwhv w hw W 1 Od pìwkrgh txh qrxv
xwlolvhurqv fruuhvsrqg ã od pìwkrgh od soxv vlpsoh1 Lo v*djlw gh od pìwkrgh g*Hxohu doìd0
wrluh1
Pìwkrgh g*Hxohu doìdwrluh
Hwdqw grqqìh o*ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh +5=4,/ rq {h xq sdv gh whpsv w @
A 3| > o 5 Q/ hw rq frqvwuxlw xq surfhvvxv hq whpsv glvfuhw +\ , dssur{lpdqw dx{ glìuhqwhv
,
gdwhv lw> 3 l o/ oh surfhvvxv [ vroxwlrq gh +5=4, hq srvdqw =
\f @ [| @ {
\n @ \ . e +\ , w . +\ , Z|nEn{| Z|n{|
E ,
+515,
Oh surfhvvxv [r @ \|n^ r | ` > w v W dssur{lph doruv +[| ,|$r$A gdqv oh vhqv vxlydqw =
{|
3
Wkìruëph 514 Srxu wrxw W A 3>
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HT vxs [rE, [r FA 3| w
|$r$A
rü FA 3| hvw xqh frqvwdqwh qh gìshqgdqw txh gh W w
Srxu xq krul}rq grqqì W/ lo hvw
grqf srvvleoh gh uhqguh o*ìfduw hqwuh oh surfhvvxv
E,
+[r ,|$r$A hw oh surfhvvxv [r
dxvvl shwlw tx*rq oh vrxkdlwh1 Lo vx!w srxu fhod
|$r$A
gh fkrlvlu xq sdv gh glvfuìwlvdwlrq w vx!vdpphqw q1
O*hvwlpdwlrq gh L} +w> {, hvw uìdolvìh d sduwlu gh od vlpxodwlrq gh wudmhfwrluhv gx surfhvvxv
[1 Xqh suhplëuh dssur{lpdwlrq hvw grqf idlwh sxlvtxh oh fdofxo uìhoohphqw phqì hvw fhoxl
gh
k l
E, +516,
L} +w> {> o, @ HT j [rE, > w v W m [| @ {
O*ìwxgh gh od frqyhujhqfh gh L} +w> {> o, yhuv L} +w> {, oruvtxh w whqg yhuv 3 hvw wudlwì
gdqv Jrehw ^43`1
Fkdqjhphqw gh suredelolwì dsuëv glvfuìwlvdwlrq
Gdqv od phvxuh rü oh vfkìpd gh glvfuìwlvdwlrq +5=5, hvw xwlolvì srxu frqvwuxluh oh surfhvvxv
[ E, / oh sdudpëwuh g*lqwìuíw txh o*rq fkhufkh ã fdofxohu dssdudñw frpph o*hvsìudqfh
47
g*xqh yduldeoh doìdwrluh gìshqgdqw g*xq qrpeuh o gh orlv qrupdohv fhqwuìhv uìgxlwhv
lqgìshqgdqwhv/ ã ydohxuv gdqv U_ 1
Jodvvhupdq hw do1 ^<` sursrvhqw doruv g*ìwxglhu od pìwkrgh gh uìgxfwlrq gh yduldqfh
irqgìh vxu xq fkdqjhphqw gh suredelolwì hq olplwdqw oh grpdlqh gh surfhvvxv ! srvvleohv
dx{ irqfwlrqv hq hvfdolhu hw gìwhuplqlvwhv1
Vl o*rq qrwh ! > l 5 i3> = = = > o 4j > od ydohxu sulvh sdu od irqfwlrq ! vxu o*lqwhuydooh
gh whpsv ^w . lw> w . +l . 4, w` @ ^w > wn ` > oh surfhvvxv glvfuìwlvì [ E, yìulh vrxv od
qrxyhooh suredelolwì TE lqgxlwh sdu ! =
E,
[|f
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l
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rü ] > = = = > ], uhsuìvhqwhqw o orl qrupdohv fhqwuìhv uìgxlwhv lqgìshqgdqwhv vrxv TE / ã
ydohxu gdqv U_ 1 Gëv oruv/ vl o*rq qrwh ] @ +] > = = = > ], , / oh sdudpëwuh g*lqwìuíw v*ìfulw/
vrxv od suredelolwì TE =
k l
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L} +w> {> o, @ HT
j [ E, ] w! O,
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Rq hvw grqf udphqì dx fdofxo gh o*hvsìudqfh g*xqh irqfwlrq gìshqgdqw g*xqh orl qrupdoh
fhqwuìh uìgxlwh gh glphqvlrq og1
515
Frqvwuxfwlrq gh od gìulyh vxlydqw xq sulqflsh gh judqghv gìyld0
wlrqv
Srxu vlpsolhu ohv qrwdwlrqv/ j [ E, +], hvw lghqwlì gdqv fhwwh vhfwlrq ã j +], hw *
s
uhsuìvhqwh
oh yhfwhxu w!1 Gh soxv/ rq vxssrvh txh j hvw xqh irqfwlrq srvlwlyh1 Qrwrqv
G @ } 5 U,_ > j A 3 1 Rq gìqlw doruv vxu G od irqfwlrq I sdu
I +}, @ oq j +}, > ;} 5 G=
48
Plqlplvdwlrq gx prphqw g*ruguh 5
Frpswh whqx gh fh txh o*rq ylhqw g*ìfuluh/ oh sureoëph txh o*rq vh srvh hvw oh vxlydqw =
rq vrxkdlwh gìwhuplqhu xq yhfwhxu * 5 U,_ / who txh od yduldeoh doìdwrluh
4 j +] . *, h{s * ] * * 4( +] . *,
5
dlw xqh yduldqfh od soxv idleoh srvvleoh/ ] ìwdqw xq yhfwhxu doìdwrluh jdxvvlhq fhqwuì
uìgxlw vxu U,_ / vrxv od suredelolwì TE) 1 Od plqlplvdwlrq gh od yduldqfh gh fhwwh yduldeoh
doìdwrluh uhylhqw ã plqlplvhu vrq prphqw g*ruguh 5 gdqv od phvxuh rü vrq hvsìudqfh hvw
lqgìshqgdqwh gx yhfwhxu * frqvlgìuì=
Oh sureoëph gh plqlplvdwlrq gh od yduldqfh v*ìfulw grqf =
k
l
E)
j +] . *,2 h{s 5* ] . * * 4( +] . *,
plq HT
)
fh txl v*ìfulw/ hq uhsdvvdqw vrxv od suredelolwì lqlwldoh =
4 T
plq H h{s 5I +], . * ] . * * 4( / rü ] Q +3> L, vrxv T=
)
5
+517,
Fh sureoëph hvw frpsoh{h fdu lo v*djlw gh uìvrxguh xq fdofxo g*hvsìudqfh vlplodluh ã fhx{
lqwhuyhqdqw gdqv ohv fdofxov gh sul{ g*rswlrqv1
Dssur{lpdwlrq gx sureoëph = od pìwkrgh gh Odsodfh devwudlwh
Xqh uìvroxwlrq dssurfkìh shxw qìdqprlqv íwuh phqìh hq xwlolvdqw xqh dssur{lpdwlrq
vxjjìuìh sdu od wkìrulh ghv judqghv gìyldwlrqv/ lvvxh gx wkìruëph gh od pìwkrgh gh
Odsodfh devwudlwh +srxu soxv gh suìflvlrqv/ qrxv uhqyr|rqv ã o*dqqh{h D sdjh 73,1 Hq
hhw/ g*dsuëv fh wkìruëph/ vl I hvw frqwlqxh hw v*lo h{lvwh D 5 U who txh I +}, D . f} }
rü f ? e / doruv =
k
l
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4
w oq HT h | +28 + |~ ,3 |) ~ n 2 ) ), 4( $ pd{ 5I +}, * } . * * } }
|<f 5 M(
5
5
Fhod dpëqh grqf ã fkhufkhu xq yhfwhxu *W who txh =
4
4
*W 5 duj plq pd{ 5I +}, * } . * * } }
)
5 M(
5
5
+518,
Uìvrxguh fh sureoëph hvw hqfruh uhodwlyhphqw frpsoh{h +uìvroxwlrq g*xq sureoëph plq0
pd{,1 Qrxv suìvhqwrqv doruv xqh dssurfkh txl frqgxlw ã xq sureoëph soxv vlpsoh1
49
Vlpsolfdwlrq gx surjudpph g*rswlplvdwlrq
Xqh dxwuh lgìh/ dq gh wurxyhu oh yhfwhxu gh gìulyh * hvw gh olvvhu od yduldeoh doìdwrluh
grqw rq fkhufkh o*hvsìudqfh L} 1 Uhyhqrqv grqf dx sureoëph gh od uhfkhufkh gh od gìulyh
* plqlplvdqw od yduldqfh gh o*hvwlpdwhxu gh od txdqwlwì lqfrqqxh L} +w> {> o,1 Vl qrxv
qrwrqv if +uhvshfwlyhphqw i) , od ghqvlwì gh od orl jdxvvlhqqh fhqwuìh +uhvshfwlyhphqw
g*hvsìudqfh *, gh pdwulfh gh yduldqfh0fryduldqfh o*lghqwlwì/ oh sureoëph srvì shxw doruv
vh uììfuluh frpph od plqlplvdwlrq gh od yduldqfh gh od yduldeoh doìdwrluh =
j +],
if +],
4( +],
i) +],
rü ] Q +*> L, = Vrxv iruph lqwìjudoh/ fhwwh yduldqfh v*ìfulw =
2
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j +},
i) +},
(
G*dsuëv fh txh qrxv dyrqv uhpdutxì gdqv od suìvhqwdwlrq gh o*ìfkdqwloorqqdjh srqgìuì
+vrxv0vhfwlrq 416,/ xqh pdqlëuh gh fkhufkhu * hvw gh idluh hq vruwh txh od ghqvlwì i) vrlw
oh soxv sursruwlrqqho srvvleoh ã od irqfwlrq jif 1 Srxu fhod/ sodêrqv ohv pd{lpd gh i) hw
gh od irqfwlrq jif dx píph hqgurlw1 Fhod uhylhqw doruv ã fkhufkhu oh yhfwhxu * frpph
vroxwlrq gx surjudpph =
4
*W 5 duj pd{ I +}, } }
+519,
5 M(
5
Fh surjudpph hvw elhq soxv idfloh ã uìvrxguh txh oh surjudpph wurxyì gdqv od suhplëuh
dssurfkh1 Od sursrvlwlrq vxlydqwh gìprqwuìh gdqv Jodvvhupdq hw do1^<` surxyh doruv/
vrxv fhuwdlqhv frqglwlrqv/ tx*lo hvw srvvleoh gh vh olplwhu ã od uìvroxwlrq gh +519,1
Sursrvlwlrq 514 Vxssrvrqv tx*lo h{lvwh xqh irqfwlrq k whooh txh G @ i} = k +}, 3j hw
tx*lo h{lvwh } 5 G who txh k +}, ? 31 Vxssrvrqv txh I +{, e { { vrlw frqfdyh vxu G= Vl
*W hvw vroxwlrq gh +519,/ doruv *W uìvrxw +518,=
Srxu o*hqvhpeoh ghv dssolfdwlrqv txh qrxv ìwxglhurqv sdu od vxlwh/ ohv frqglwlrqv gh od
sursrvlwlrq vxu oh grpdlqh G hw vxu I vhurqw yìulìhv1 Fhshqgdqw/ o*k|srwkëvh od soxv
frqwudljqdqwh vxu oh sodq sudwltxh hvw txh G grlw íwuh frqyh{h/ fh txl q*hvw sdu h{hpsoh
sdv oh fdv oruvtxh o*rq ìwxglh ohv rswlrqv ã eduulëuh1
Od uìvroxwlrq dssurfkìh gx surjudpph +5=7, txh qrxv yhqrqv g*hhfwxhu hq frqvlgìudqw
oh surjudpph vlpsolì +5=9, shxw íwuh lqwhusuìwìh gh od idêrq vxlydqwh
= od gìqlwlrq gx
fkdqjhphqw gh suredelolwì ã sduwlu gh * 5 duj pd{5M( I +}, 2 } } ìolplqh od sduw gh
od yduldqfh gxh ã od frpsrvdqwh olqìdluh gh I 1 Hq hhw/ * yìulh od frqglwlrq gx suhplhu
ruguh =
* @ uI +*, =
4:
Hq uìdolvdqw xq gìyhorsshphqw gh Wd|oru dx suhplhu ruguh gh I dx srlqw */ rq rewlhqw =
h8 E)nQ8 E) ~ nE~ ~ 3) ~ 3 2 ) ) 4( @ h8 E)3 2 ) )nE~ ~ 4(

O*lgìh qdwxuhooh surorqjhdqw fhwwh dssurfkh/ srxu uìgxluh hqfruh od yduldqfh/ hvw doruv
gh phwwuh hq sodfh xqh pìwkrgh ìolplqdqw od sduw gh od yduldqfh gxh ã od frpsrvdqwh
txdgudwltxh gh I 1 Od pìwkrgh gh vwudwlfdwlrq suìvhqwìh gdqv od vhfwlrq vxlydqwh shuphw
gh uìvrxguh xq who sureoëph1
516
Dffìoìudwlrq gh od frqyhujhqfh sdu vwudwlfdwlrq
Gdqv fhwwh sduwlh/ qrxv doorqv fkhufkhu xqh gluhfwlrq gh vwudwlfdwlrq gx yhfwhxu jdxvvlhq
]/ vxlydqw oh irupdolvph suìvhqwì gdqv oh sdudjudskh 416 sdjh ;1 Hhfwxrqv srxu fhod
xq gìyhorsshphqw gh Wd|oru gh I dx vhfrqg ruguh1 Lo ylhqw/ hq qrwdqw K oh Khvvlhq gh
I =
h8 E~ n)3) ~ 3 2 ) ) 4( @ h8 E)3 2 ) )n 2 ~ M E)~ nJE~ ~ 4(

Oh fkrl{ hvvhqwlho hvw grqf fhoxl g*xqh gluhfwlrq v txl shuphw gh uìgxluh vljqlfdwlyhphqw
od yduldqfh1 Fhod uhylhqw grqf ã fkhufkhu xq yhfwhxu v txl plqlplvh od yduldqfh vxlydqwh =
% % &
&&
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4
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h 2 ~ M E)~ 4( m VwuEr +],
n
'
rü VwuEr +], uhsuìvhqwh od yduldeoh lqglfdwulfh gh od vwudwh gdqv odtxhooh vh vlwxh ]1 Soxv
suìflvìphqw/
l4
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n
n fruuhvsrqgdqw dx qrpeuh gh vwudwhv frqvlgìuìhv1 Fh sureoëph hvw frpsoh{h fdu od
yduldeoh doìdwrluh v ] m v ] 5 Vwu +yduldeoh jdxvvlhqqh wurqtxìh, d xqh orl qrq vwdqgdug/
hw dlqvl oh yhfwhxu doìdwrluh ] m v ] 5 Vwu qh srvvëgh soxv xqh orl vwdqgdug1 Srxu
vlpsolhu oh sureoëph hw vh udphqhu dx fdv jdxvvlhq/ rq xwlolvh doruv oh ohpph vxlydqw =
Ohpph 514 vl n hvw xqh sxlvvdqfh gh 5/
k

l
4
4
2rc& @ HT ydu h 2 ~ M E)~ m v ] . n
n
Od yduldeoh doìdwrluh ] m v ] hvw fhwwh irlv elhq xqh orl jdxvvlhqqh1 Fhod dpëqh grqf ã
fkhufkhu od gluhfwlrq rswlpdoh gh vwudwlfdwlrq hq uìvroydqw oh sureoëph =
k
k
ll
+51:,
plq HT Y du h 2 ~ M E)~ m v ]
rGr r'
Fh sureoëph hvw doruv uìvrox juåfh dx wkìruëph vxlydqw =
4;
Wkìruëph 515 Vrlhqw h > > h,_ ohv yhfwhxuv sursuhv +qrupdolvìv, gh K +*, gh ydohxuv
sursuhv uhvshfwlyhv > > ,_ 1 Vrlw m W o*lqglfh who txh =
2
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irupxoh +6=4, = rq dssur{lph XLE +w> {, sdu h{s 2EA 3|j2 oq g
srvhu
{
4
oq
4%g =
! +w> {, @
+W w,
N
Fhwwh ghuqlëuh irupxoh prqwuh txh wurlv sdudpëwuhv lq xhqfhqw o*dpsolwxgh gh od fru0
uhfwlrq dssruwìh ã od wudmhfwrluh =
58
oh whpsv uhvwdqw srxu dwwhlqguh od eduulëuh N1
oh qlyhdx gh od yrodwlolwì = wrxwhv fkrvhv ìjdohv sdu dloohxuv/ soxv od yrodwlolwì gx
surfhvvxv hvw ìohyìh/ soxv lo hvw suredeoh txh oh surfhvvxv dwwhljqh od eduulëuh N
dydqw od gdwh olplwh W=
od glvwdqfh gh { ã N1
Jìqìudolvdwlrq dx{ prgëohv ã yrodwlolwì vwrfkdvwltxh1 Oh uìvxowdw rewhqx gdqv oh
fdv sduwlfxolhu gh prgëoh gh Eodfn hw Vfkrohv vh jìqìudolvh ã o*hqvhpeoh ghv surfhvvxv
gh glxvlrq vroxwlrqv g*xqh ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh gh od iruph
g[| @ e +[| , gw . +[| , gZ| =
Rq glvsrvh hq hhw gx uìvxowdw vxlydqw/ sruwdqw vxu oh whpsv g*dwwhlqwh gìql suìfì0
ghpphqw =
olp 5w oq T + ? w m [f @ {, @ )2 +{,
|<f
+616,
rü ) hvw o*xqltxh vroxwlrq gh ylvfrvlwì gh o*ìtxdwlrq glìuhqwlhooh qrq olqìdluh
u) +{, +{, +{, u) +{,
@4
@3
vl { 5 U_ q vlqrq
+617,
Srxu soxv gh gìwdlov vxu fh uìvxowdw/ oh ohfwhxu srxuud vh uìiìuhu ã Iohplqj hw Vrqhu
^8`/ Irxuqlì/ Odvu| hw Olrqv ^9`/ hw Irxuqlì/ Odvu| hw Wrx}l ^:`1
Frpph gdqv oh fdguh Eodfn0Vfkrohv/ fh uìvxowdw shuphw g*h{klehu xqh irqfwlrq ds0
sur{lpdqw od sdudpëwuh g*lqwìuíw XLE +w> {, sxlvtxh dv|pswrwltxhphqw/
XLE +w> {, h{s )2 +{,
=
5 +W w,
Oh fkdqjhphqw gh suredelolwì xwlolvì srxu uìgxluh od yduldqfh gh o*hvwlpdwhxu gh XLE +w> {,
hvw doruv frqvwuxlw ã sduwlu gx surfhvvxv ! vxlydqw =
! +w> {, @
) +{,
+{, u) +{, =
W w
+618,
O*lpsoìphqwdwlrq gh fh fkdqjhphqw gh gìulyh uhtxlhuw dx suìdodeoh od uhfkhufkh gh od
vroxwlrq gh ylvfrvlwì ã o*ìtxdwlrq glìuhqwlhooh +6=7, = Qrxv suìvhqwrqv lfl xqh dssurfkh
vxjjìuìh sdu Irxuqlì/ Odvu| hw Wrx}l ^:`/ txl hvw irqgìh vxu o*lqwhusuìwdwlrq gh G hq
whuph gh glvwdqfh dx grpdlqh =
59
Lqwhusuìwdwlrq gh od irqfwlrq )= Vl o*rq pxqlw o*hvsdfh U_ gh od pìwultxh orfdoh
3
D +{, @ +{, +{, / { 5 U_ / dvvrflìh ã od irqfwlrq gh yrodwlolwì gx surfhvvxv gh
glxvlrq/ doruv ) +{, v*lqwhusuëwh frpph od orqjxhxu gx soxv frxuw fkhplq shuphwwdqw
gh uhmrlqguh oh grpdlqh sduwdqw gh {1 Hq hhw od vroxwlrq ) gx surjudpph +6=7, yìulh
ìjdohphqw
) +{, @ lqi ) +{> |,
+ MK
rü
]
) +{> |, @
lqi
MM E%c+

f
+619,
G
H
u
u
2
D +z +w,, z +w, > z +w, gw
u
dyhf K +{> |, @ z 5 F ^3> 4` > U_ = z +3, @ {> z +4, @ | > z uhsuìvhqwdqw od gìulyìh z
gh sdu udssruw ã w1
Od suìvhqwdwlrq gh ) vrxv fhwwh qrxyhooh iruph shuphw g*dqdo|vhu gh idêrq txdolwdwlyh
o*lpsdfw gx fkdqjhphqw gh gìulyh +6=8, vxu od glvwulexwlrq ghv wudmhfwrluhv gx surfhvvxv
+[| ,1
Qrwrqv zW od vroxwlrq gx sureoëph +6=9,1 Dv|pswrwltxhphqw/ o*ìyëqhphqw i ? W j fru0
uhvsrqg ã o*hqvhpeoh ghv wudmhfwrluhv gx surfhvvxv +[| , rvfloodqw gh idêrq wuëv udssurfkìh
dxwrxu gx fkhplq fulwltxh zW / fhoxl0fl ìwdqw oh soxv frxuw fkhplq uholdqw od srvlwlrq {
rffxsìh hq w dx grpdlqh fleoh = Vrxv od suredelolwì lqlwldoh/ fhw hqvhpeoh gh wudmhf0
wrluhv frqvwlwxh xq ìyëqhphqw uduh gdqv od phvxuh rü T + ? W , whqg yhuv 31 Oh uìvxowdw
+6=6, h{solflwh od ylwhvvh gh frqyhujhqfh gh fhwwh suredelolwì yhuv 3 hw prqwuh tx*hooh hvw
gluhfwhphqw uholìh ã od glvwdqfh ) gx srlqw { ã =
Hq dssoltxdqw oh fkdqjhphqw gh gìulyh +6=8,/ rq prglh od glvwulexwlrq gh +[| , gh vruwh
txh od suredelolwì g*rffxuuhqfh gh o*ìyëqhphqw i ? W j uhvwh vx!vdpphqw lpsruwdqwh
ã phvxuh txh o*rq vh udssurfkh gh od pdwxulwì W= Oh fkdqjhphqw gh gìulyh prglh grqf
o*dooxuh ghv wudmhfwrluhv gh vruwh tx*hoohv vh sursdjhqw dxwrxu gx fkhplq fulwltxh zW 1
Fdofxo qxpìultxh gh )1 Od uhsuìvhqwdwlrq gh ) vrxv od iruph +6=9, shuphw gh
uìvrxguh qxpìultxhphqw oh fdofxo gh )= Rq shxw hq hhw dssoltxhu xqh pìwkrgh gh
surjudppdwlrq g|qdpltxh hq xwlolvdqw o*dojrulwkph vxlydqw +fi Irxuqlì/ Odvu| hw
Olrqv ^9`, =
Dojrulwkph 614 fdofxo qxpìultxh gh ) 41 Glvfuìwlvhu xq grpdlqh gh o*hvsdfh g*ìwdw U_ gx surfhvvxv +[| , hq xqh julooh gh
srlqwv1
51 Lqlwldolvhu ohv srlqwv gh dssduwhqdqw ã od julooh ã 31
5:
Judskltxh 6141 Glvwdqfh ) ã @ i+v> , = v 443j/ Prgëoh gh Khvwrq1 @ 3=9/ @ 3=41
61 Vìohfwlrqqhu xq srlqw { vxu od julooh hw fdofxohu/ srxu o*hqvhpeoh
t ghv srlqwv | grqw
rq d gìmã fdofxoì ohv glvwdqfhv ) +|, ã > od glvwdqfh ) +{> |, @ +| {, D +{, +| {,
dvvrflìh ã od pìwultxh orfdoh D +{, = Srvhu doruv
) +{, @ plq ) +|, . ) +{> |, =
+
Qrwrqv txh fhwwh dojrulwkph shuphw gh frqvwuxluh ohv fkhplqv fulwltxhv fruuhvsrqgdqw
ã od pìwultxh D= Hq hhw/ oh srlqw { ìwdqw vìohfwlrqqì gdqv o*ìwdsh 6 gh o*dojrulwkph/
oh srlqw | plqlplvdqw ) +|, . ) +{> |, fruuhvsrqg dx suhplhu srlqw vxffìgdqw { vxu oh
fkhplq fulwltxh doodqw gh { ã 1
H{hpsoh 614 Qrxv suìvhqwrqv gdqv fhw h{hpsoh ohv uìvxowdwv txh o*rq rewlhqw gdqv oh
fdv gx prgëoh ã yrodwlolwì vwrfkdvwltxh gh Khvwrq ^47`1 +[| , @ +V| > Y| , hvw ã ydohxuv gdqv
U2 / vroxwlrq gh o*ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh
s
gV|
@ ugw . Y| gZc|
V|
s
gY| @ + Y| , gw . Y| gZ2c|
rü +Zc| , hw +Z2c| , vrqw ghx{ prxyhphqwv eurzqlhqv vwdqgdugv xqlglphqvlrqqhov whov
txh
kgZc| > gZ2c| l @ gw
5;
Judskltxh 6151 Fkhplqv fulwltxhv dvvrflìv ã @ i+v> , = v 443j/ Prgëoh gh Khvwrq1
@ 3=9/ @ 3=41
Judskltxh 6161 Wudmhfwrluhv vlpxoìhv dydqw +srlqwlooìv, hw dsuëv +wudlwv sohlqv, fkdqjhphqw gh
suredelolwì srxu o*ìydoxdwlrq g*xqh rswlrq XLE gh pdwxulwì gh : mrxuv hw gh eduulëuh 443/ Prgëoh
gh Khvwrq1 @ 3=9/ @ 3=4/ u @ 3/ @ 3=8/ @ 3=55 1 Lqlwldohphqw/ V3 @ 433 hw 3 @ 3=5
5<
Oh judskltxh 614 sdjh 5; uhsuìvhqwh od vxuidfh ) txh o*rq rewlhqw gdqv oh fdv rü =
@ 3=4 hw @ 3=9
@ i+v> y, > v 443j
Oh judskltxh 615 sdjh suìfìghqwh fruuhvsrqg dx{ fkhplqv fulwltxhv dvvrflìv ã fh píph
mhx gh sdudpëwuhv1 Od fruuìodwlrq hqwuh ohv ghx{ prxyhphqwv eurzqlhqv +Zc| , hw +Z2c| ,
ìwdqw qìjdwlyh/ xqh dxjphqwdwlrq gh V| vhud hq jìqìudo dffrpsdjqìh g*xqh glplqxwlrq
gh od yduldqfh Y| 1 Fhfl h{soltxh o*dooxuh ghv fkhplqv fulwltxhv = sduwdqw gx srlqw +vf > f ,
dyhf vf ? 443/ oh grpdlqh hvw dwwhlqw soxv udslghphqw hq uìgxlvdqw oh qlyhdx gh od
yrodwlolwì gdqv od phvxuh rü fhod lqgxlw jìqìudohphqw xqh kdxvvh gx vrxv0mdfhqw1
Oh judskltxh 616 sdjh suìfìghqwh looxvwuh od idêrq grqw ohv wudmhfwrluhv vrqw prglìhv hq
xwlolvdqw oh fkdqjhphqw gh gìulyh +6=8,1 Qrxv dyrqv uhsuìvhqwì 8 wudmhfwrluhv glìuhqwhv
srxu w frpsulv hqwuh 3 hw : mrxuv/ vlpxoìhv srxu ìydoxhu xqh rswlrq Xs dqg Lq gh pd0
wxulwì uhvwdqwh ìjdoh ã : mrxuv1 D od gdwh lqlwldoh/ Vf @ 433 hw f @ 3=5> hw oh grpdlqh
g*h{huflfh fruuhvsrqg ã @ i+v> , > v 443j1 Oh whpsv uhvwdqw dydqw od pdwxulwì ìwdqw
wuëv frxuw/ od suredelolwì lqlwldoh gh o*ìyëqhphqw hvw wuëv idleoh1 Lo vhud grqf wuëv gl!0
floh gh vlpxohu ghv wudmhfwrluhv frqgxlvdqw ã o*h{huflfh gh o*rswlrq1 Vxu oh judskltxh/ rq
yrlw txh ohv 8 wudmhfwrluhv vlpxoìhv vrxv od suredelolwì lqlwldoh +uhsuìvhqwìhv hq srlqwlooìv,
uhvwhqw frqfhqwuìhv dxwrxu gh od ydohxu lqlwldoh +Vf > f ,1 Vrxv od qrxyhooh suredelolwì/ rq
rewlhqw/ ã sduwlu gh od píph judlqh/ 8 wudmhfwrluhv +hq wudlwv sohlqv, txl vh sursdjhqw
dxwrxu gx fkhplq fulwltxh +hq qrlu,1
Fkdqjhphqw gh gìulyh srxu ohv fdoov hxursìhqv hw ohv rswlrqv GLF
O*lqwìuíw gx fkdqjhphqw gh gìulyh txh o*rq ylhqw gh frqvwuxluh gdqv oh fdv gh o*rswlrq Xs
dqg Lq Erqg hvw gh prglhu od glvwulexwlrq ghv wudmhfwrluhv dq g*dffurñwuh od suredelolwì
gh o*h{huflfh gh o*rswlrq1 Gëv oruv xq who fkdqjhphqw gh gìulyh shxw v*dyìuhu lqwìuhvvdqw
gdqv xq fdguh soxv jìqìudo txh fhoxl gh o*ìydoxdwlrq gh XLE +w> {, = rq shxw o*xwlolvhu
srxu ìydoxhu wrxwh rswlrq srvvìgdqw xq grpdlqh g*h{huflfh vhpeodeoh ã fhoxl gh o*rswlrq
Xs dq Lq1 F*hvw sdu h{hpsoh oh fdv gh o*rswlrq g*dfkdw hxursìhqqh rx ghv rswlrqv ã
eduulëuh1
Vl o*xwlolvdwlrq gh fh fkdqjhphqw gh gìulyh qh suhqg sdv hq frpswh ohv vsìflflwìv gh
od glvwulexwlrq gx sd|r vxu oh grpdlqh g*h{huflfh/ rq shxw qìdqprlqv v*dwwhqguh ã fh
tx*hooh lqgxlvh xqh uìgxfwlrq gh od yduldqfh gdqv od phvxuh rü o*rq vlpxohud gdydqwdjh
gh wudmhfwrluhv dssduwhqdqw dx grpdlqh g*h{huflfh1 Sdu vxlwh/ fhwwh uìgxfwlrq ghyudlw
íwuh g*dxwdqw soxv lpsruwdqwh txh od suredelolwì g*h{hufhu hvw idleoh/ f*hvw0ã0gluh gdqv oh
fdv ghv rswlrqv wuëv kruv od prqqdlh1
Hydoxdwlrq g*rswlrqv g*dfkdw hxursìhqqhv1 Gdqv oh fdv gx fdoo hxursìhq/ oh gr0
pdlqh g*h{huflfh v*h{sulph txdvlphqw gh od píph idêrq txh srxu o*rswlrq elqdluh = oh
gìwhqwhxu g*xqh whooh rswlrq wrxfkh od glìuhqfh/ oruvtx*hooh hvw srvlwlyh/ hqwuh oh sul{ VA
63
gx vrxv0mdfhqw ã od pdwxulwì W hw oh sul{ g*h{huflfh N gh o*rswlrq1 Irupxoì gdqv oh fdguh
g*xq prgëoh ã yrodwlolwì vwrfkdvwltxh/ oh grpdlqh g*h{huflfh v*ìfulw
@ +v> , 5 U2n > v N =
Gdqv oh fdv ghv fdoov hxursìhqv/ rq xwlolvhud grqf gluhfwhphqw od irupxoh +6=8, srxu
frqvwuxluh xq hvwlpdwhxu gh prlqguh yduldqfh txh o*hvwlpdwhxu gh Prqwh Fduor vwdqgdug1
Dssolfdwlrq qxpìultxh 614 Qrxv dyrqv dssoltxì od whfkqltxh gh uìgxfwlrq gh yd0
uldqfh irqgìh vxu o*xwlolvdwlrq gluhfwh gx fkdqjhphqw gh gìulyh +6=8, ã xq hqvhpeoh gh
fdoov hxursìhqv srxu ghx{ prgëohv ã yrodwlolwì vwrfkdvwltxh glìuhqwv6 =
Oh suhplhu prgëoh fruuhvsrqg dx prgëoh gh Khvwrq txh o*rq d gìmã suìvhqwì gdqv
o*h{hpsoh 614 sdjh 5; 1 Qrxv uhsuhqrqv lfl oh píph mhx gh sdudpëwuhv = u @ 3/
@ 3=8/ @ 3=52 / @ 3=4 hw @ 3=91
Gdqv oh vhfrqg prgëoh/ rq fkdqjh o*ìtxdwlrq glìuhqwlhooh fdudfwìulvdqw od glxvlrq
gh od yrodwlolwì dx frxuv gx whpsv1 Rq vxssrvh txh \| @ oq | > w 3 hvw vroxwlrq gh
g\| @ +f n\| , gw . gZ2c|
Oh mhx gh sdudpëwuh uhwhqxv hq dssolfdwlrq hvw = u @ 3/ n @ 3/ @ 3=58/ f @ 2
hw @ 3=81
2
O*hqvhpeoh ghv rswlrqv frqvlgìuìhv rqw xq sul{ g*h{huflfh gh 443/ oh sdudpëwuh yduldqw
ìwdqw od pdwxulwì1 Oh vrxv0mdfhqw ydxw lqlwldohphqw 433 hw od yrodwlolwì lqlwldoh hvw gh 3=5=
Ohv wdeohdx{ F16 sdjh 77 hw F18 sdjh 78 vlwxìv hq dqqh{h gh fh pìprluh suìvhqwhqw ohv
uìvxowdwv rewhqxv1
Frqirupìphqw ã fh txh o*rq srxydlw shqvhu d sulrul/ o*xwlolvdwlrq gx fkdqjhphqw gh
gìulyh +6=8, lqgxlw xqh uìgxfwlrq gh yduldqfh/ fhooh0fl ìwdqw g*dxwdqw soxv lpsruwdqwh txh
o*rq vh udssurfkh gh od pdwxulwì1 Qìdqprlqv/ oh idlw g*dffurñwuh oh qrpeuh gh vlpxodwlrqv
dssduwhqdqw dx grpdlqh g*h{huflfh vdqv vh suìrffxshu gh od uìsduwlwlrq gx sd|r vxu
fh grpdlqh qh shuphw sdv g*rewhqlu ghv uìgxfwlrqv gh yduldqfh wuëv lpsruwdqwhv1 Sdu
h{hpsoh/ gdqv oh fdv gx prgëoh gh Khvwrq od uìgxfwlrq gh yduldqfh hvw gh 44=9 srxu
o*rswlrq gh pdwxulwì 4 prlv1
Hydoxdwlrq g*rswlrqv ã eduulëuh GLF1 O*rswlrq ã eduulëuh GLF hvw xqh rswlrq
g*dfkdw hxursìhqqh txh oh gìwhqwhxu shxw h{hufhu vl oh vrxv0mdfhqw sdvvh vrxv xqh fhuwdlqh
eduulëuh K lqiìulhxuh dx sul{ g*h{huflfh N/ dydqw od pdwxulwì gx frqwudw1 Oh grpdlqh
g*h{huflfh gh o*rswlrq hvw grqf soxv frpsoh{h txh gdqv oh fdv gh o*rswlrq Xs dqg Lq rx
gx fdoo hxursìhq1 Qìdqprlqv/ lo hvw srvvleoh gh frqvwuxluh xq fkdqjhphqw gh gìulyh irqgì
vxu od irupxoh +6=8, hq gìfrpsrvdqw ohv wudmhfwrluhv hq ghx{ sruwlrqv =
6
Od vsìflfdwlrq frpsoëwh ghv ghx{ prgëohv hvw udsshoìh ã o*dqqh{h E gh fh pìprluh1
64
Judskltxh 6171 Fkdqjhphqw gh gìulyh dssoltxì ã xqh rswlrq GLF/ prgëoh gh Khvwrq1 u @ 3/
@ 3=8/ @ 3=585 / @ 3=4/ @ 3=91
wdqw txh od eduulëuh K q*hvw
oh fkdqjhphqw gh gìulyh +6=8,
sdv dwwhlqwh/ rq xwlolvh
dvvrflì dx grpdlqh M @ +v> , 5 U2n > v K >
xqh irlv od eduulëuh dwwhlqwh/ rq vh uhwurxyh gdqv oh fdv
gx fdoo hxursìhq
hw rq
2
dssoltxh oh fkdqjhphqw gh gìulyh +6=8, dvvrflì ã g @ +v> , 5 Un > v N 1
Fhwwh suhplëuh dssurfkh dffurñw oh qrpeuh gh wudmhfwrluhv vlpxoìhv shuphwwdqw o*h{huflfh
gh o*rswlrq1 Qìdqprlqv/ oruv gx suhplhu fkdqjhphqw gh gìulyh/ rq qh suhqg sdv hq frpswh
od glvwdqfh gh od eduulëuh dx sul{ g*h{huflfh1 Dydqw txh od eduulëuh qh vrlw dwwhlqwh/ rq
vrxv0hvwlph grqf od glvwdqfh dx grpdlqh g*h{huflfh/ fh txl uìgxlw od sruwìh gh od pìwkrgh1
Xqh ghx{lëph dssurfkh frqvlvwh ã xwlolvhu xq fkdqjhphqw gh gìulyh irqgì vxu od irupxoh
+6=8, srxu dwwhlqguh od eduulëuh K hq lqwìjudqw od glvwdqfh gh od eduulëuh dx vwulnh gdqv
oh fdofxo gh od G1 Srxu fhod/ rq prglh od suhplëuh ìwdsh gh o*dojrulwkph 614 sdjh 5: hq
lqlwldolvdqw oh grpdlqh M ã vd glvwdqfh dx grpdlqh g 1 Rq suhqg dlqvl hq frpswh oh
idlw tx*xqh irlv od eduulëuh dwwhlqwh/ oh surfhvvxv ghyud uhprqwhu yhuv oh sul{ g*h{huflfh1
Hq idlw/ xqh whooh pìwkrgh v*dyëuh dvvh} qdwxuhooh frpswh whqx gh o*lqwhusuìwdwlrq ghv
fkhplqv fulwltxhv suìvhqwìh suìfìghpphqw = hq surfìgdqw dlqvl/ rq fdofxoh od orqjxhxu
gx fkhplq fulwltxh srxu xq srlqw q*d|dqw sdv hqfruh dwwhlqw od eduulëuh1
Oh judskltxh 617 looxvwuh o*lpsdfw gh od sulvh hq frpswh gh od glvwdqfh gh M ã g gdqv
od frqvwuxfwlrq gx fkdqjhphqw gh gìulyh = hooh shuphw g*dwwhlqguh soxv udslghphqw od
eduulëuh K/ fh txl uhqg od uhprqwìh gx surfhvvxv yhuv oh vwulnh soxv suredeoh1
65
Dssolfdwlrq qxpìultxh 615 Rq frqvlgëuh ã qrxyhdx ohv ghx{ prgëohv ã yrodwlolwì vwr0
fkdvwltxh xwlolvìv srxu ohv dssolfdwlrqv qxpìultxhv 614 1 Vhxo oh sdudpëwuh gx prgëoh
gh Khvwrq hvw prglì = rq oh {h ã 3=581
O*hqvhpeoh ghv rswlrqv frqvlgìuìhv rqw xq sul{ g*h{huflfh gh 443 hw xqh eduulëuh gh <3/
oh sdudpëwuh yduldqw ìwdqw od pdwxulwì1 Oh vrxv0mdfhqw ydxw lqlwldohphqw 4331 Od yrodwlolwì
lqlwldoh hvw gh 3=58 gdqv od fdv gx prgëoh gh Khvwrq/ hooh ydxw 3=5 gdqv oh fdv gx vhfrqg
prgëoh1
Ohv wdeohdx{ F1: sdjh 79 hw F1< sdjh 7: vlwxìv hq dqqh{h gh fh pìprluh suìvhqwhqw ohv
uìvxowdwv rewhqxv1
Uhpdutxh 614 Qrwrqv txh gdqv oh fdv ghv rswlrqv ã eduulëuh/ od glvfuìwlvdwlrq ghv
wudmhfwrluhv oruv gh od vlpxodwlrq lqgxlw xq eldlv vxu oh sul{ ã hvwlphu1
Frqvlgìurqv ghx{ ydohxuv gx vrxv0mdfhqw vlpxoìhv dx{ gdwhv w hw w . w/ V| hw V|n{| / hw
vxssrvrqv txh od eduulëuh K q*dlw wrxmrxuv sdv ìwì dwwhlqwh ã od gdwh w= Vl V|n{| A K/
lo vh shxw qìdqprlqv txh od eduulëuh dlw ìwì wrxfkìh hqwuh w hw w . w1 Oh idlw gh qh sdv
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oh sul{/ píph oruvtxh oh sdv gh glvfuìwlvdwlrq hvw wuëv q1
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t| ìjdo ã od suredelolwì txh oh vrxv0mdfhqw sdvvh vrxv od eduulëuh K hqwuh w hw w . w
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fhx{ txh o*rq d idlw gdqv oh fdv gx fdoo1 Qrwrqv txh ohv grpdlqhv g*h{huflfh frqvlgìuìv
ìwdqw soxv uduhv srxu o*rswlrq GLF txh srxu oh fdoo/ rq revhuyh ghv uìgxfwlrqv gh yduldqfh
vxsìulhxuhv = oh qrpeuh gh wudmhfwrluhv vlpxoìhv hqwudqw gdqv oh grpdlqh g*h{huflfh mrxh
xq u÷oh wuëv lpsruwdqw vxu od yduldqfh gh od yduldeoh doìdwrluh vlpxoìh sdu od pìwkrgh gh
Prqwh Fduor vwdqgdug1
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yduldqfh d!fkìhv q*rqw soxv gh uìhooh vljqlfdwlrq = hq vlpxodqw ohv wudmhfwrluhv sdu
od pìwkrgh gh Prqwh Fduor vwdqgdug/ oh qrpeuh gh wudmhfwrluhv vlpxoìhv dxwrulvdqw
o*h{huflfh gh o*rswlrq ghylhqw qìjoljhdeoh hw rq qh sduylhqw ã hvwlphu ql oh sul{ ql od
yduldqfh gh o*hvwlpdwhxu gh idêrq fruuhfwh1 Rq shxw ã fhw hhw vrxoljqhu oh fdv h{wuíph
gh o*rswlrq GLF gh pdwxulwì 4 prlv/ ìydoxìh gdqv oh fdguh gx prgëoh 5 = vxu 433333
wudmhfwrluhv vlpxoìhv/ dxfxqh q*dwwhlqw oh grpdlqh g*h{huflfh1
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Lo uhvvruw ghv ghx{ dssolfdwlrqv qxpìultxhv txh o*rq ylhqw gh suìvhqwhu txh od pìwkrgh gh
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od pìwkrgh gh Prqwh Fduor vwdqgdug gëv txh oh grpdlqh g*h{huflfh gh o*rswlrq frqvlgìuìh
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surylhqw doruv gx idlw tx*lo hvw wuëv gl!floh gh uìdolvhu ghv wludjhv dx vhlq gx grpdlqh
g*h{huflfh1 Oh idlw gh fkdqjhu od gìulyh gx surfhvvxv shuphw gh uìvrxguh fh sureoëph1
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sruwdqwh/ vl elhq txh oh sureoëph gh od vlpxodwlrq gh wudmhfwrluhv dx vhlq gx grpdlqh
g*h{huflfh q*h{soltxh txh gh idêrq vhfrqgdluh od yduldqfh gh o*hvwlpdwhxu gh Prqwh Fduor
vwdqgdug1 Vl o*rq vrxkdlwh uìgxluh od yduldqfh gh idêrq soxv vljqlfdwlyh srxu ohv rs0
wlrqv surfkhv gh od prqqdlh/ lo dssdudñw qìfhvvdluh gh gdydqwdjh suhqguh hq frpswh od
glvwulexwlrq gx sd|r dx vhlq gx grpdlqh g*h{huflfh1 F*hvw fh txh shuphw gdqv xqh
fhuwdlqh phvxuh od pìwkrgh vxlydqwh/ ìjdohphqw irqgìh vxu xq fkdqjhphqw gh gìulyh
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Gdqv od vhfwlrq 614 sdjh 56/ qrxv dyrqv h{solflwì oh fkdqjhphqw gh gìulyh rswlpdo +fi
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fdv rq srxuud frqvlgìuhu oh prgëoh gh Eodfn hw Vfkrohv=
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hw Vfkrohv ghphxuh wuëv xwlolvì1
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wdlooh gh fh grpdlqh1
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fh sureoëph1 Hooh suìvhqwh xq lqwìuíw hq qdqfh oruvtxh o*rq vsìflh od g|qdpltxh gx
vrxv0mdfhqw vrxv od iruph g*xq prgëoh ã yrodwlolwì vwrfkdvwltxh1 Od prgìolvdwlrq Eodfn0
Vfkrohv frqgxlw doruv ã xqh erqqh dssur{lpdwlrq gx frpsruwhphqw gx vrxv0mdfhqw vxu
oh grpdlqh g*h{huflfh1 Vl o*rq glvsrvh g*xqh irupxoh g*ìydoxdwlrq ihupìh gdqv fh prgëoh
vlpsolì/ rq shxw frqvwuxluh xq fkdqjhphqw gh gìulyh h{sorlwdqw fhwwh lqirupdwlrq vxs0
soìphqwdluh1 Xqh whooh pìwkrgh surgxlw ghv wdx{ gh uìgxfwlrq gh od yduldqfh vljqlfdwliv/
píph srxu ghv rswlrqv surfkhv gh od prqqdlh1
6<
D1 Sulqflsh gh Judqghv Gìyldwlrqv
Dqqh{hv
D
Sulqflsh gh Judqghv Gìyldwlrqv
Vrlhqw +> I, xq hvsdfh Srorqdlv +hvsdfh pìwultxh frpsohw vìsdudeoh/ pxql gh vd wulex
eruìolhqqh, hw +| ,|:f xqh idplooh gh suredelolwìv frqyhujhdqw yhuv od pdvvh gh Gludf 6 /
rü p hvw xq srlqw gh 1 Vrlw doruv F 5 I who txh p 5
@ F/ rq d =
| +F, $ 3
|<f
Oh eruìolhq F hvw txdolì g*ìyëqhphqw uduh > od wkìrulh ghv judqghv gìyldwlrqv v*lqwì0
uhvvh ã od ylwhvvh gh frqyhujhqfh yhuv 3 gh | +F, oruvtxh w whqg yhuv 31
Gìqlwlrq D14 Sulqflsh gh Judqghv gìyldwlrqv
Rq glud txh +| ,|:f vxlw xq sulqflsh gh judqghv gìyldwlrqv v*lo h{lvwh y = Un $ Un hw
= $ Un ^ i.4j whoohv txh =
41 olp|<f y +w, @ .4
51 Srxu wrxw v 5 Un / i vj hvw frpsdfw gdqv 1
61 Srxu wrxw F 5 I/ rq d =
4
4
orj +| +F,, olp
orj +| +F,, Lqi +$,
Lqi +$, olp
J
|<f y +w,
|<f y +w,
/ M
/ M
Oh frxsoh +y> ,/ rx/ soxv vlpsohphqw/ hvw dsshoì irqfwlrqqhooh g*dfwlrq srxu od id0
plooh +| ,|:f
Sdu h{hpsoh/ gdqv od vhfwlrq +5,/ ohv surfhvvxv glvfuìwlvìv vrqw jrxyhuqìv sdu xq yhfwhxu
jdxvvlhq1 Ru vl ] hvw xq yhfwhxu jdxvvlhq gdqv U_ / fhqwuì/ gh pdwulfh gh yduldqfh0
fryduldqfh o*lghqwlwì/ doruv vxu U_ pxqlh
s gh vd wulex eruìolhqqh/ rq gìqlw od idplooh gh
w] 5 1 +| ,|:
suredelolwìv +| ,|:f sdu | +, @ S
f vxlw doruv xq sulqflsh gh judqghv
gìyldwlrqv dyhf srxu irqfwlrqqhooh g*dfwlrq w> 2 } } = Soxv suìflvìphqw/ rq d od frqyhu0
jhqfh vxlydqwh =
s
4
w] 5 E @ plq } }
olp w oq S
|<f
5M 5
rü E hvw xq eruìolhq gh U_ = Od wkìrulh ghv judqghv gìyldwlrqv v*lqwìuhvvh grqf ã od
glvwulexwlrq dv|pswrwltxh gh od idplooh gh suredelolwìv +| ,|:f 1
Qrxv vrxkdlwrqv hq idlw xwlolvhu xqh dssur{lpdwlrq qrq sdv gh od suredelolwì g*dssduwhqlu
dx eruìolhq E/ pdlv gh o*hvsìudqfh g*xqh yduldeoh doìdwrluh1 Oh wkìruëph vxlydqw shuphw
gh uìdolvhu/ ã sduwlu g*xq sulqflsh gh judqghv gìyldwlrqv/ fhwwh dssur{lpdwlrq1
73
D1 Sulqflsh gh Judqghv Gìyldwlrqv
Wkìruëph D14 +Pìwkrgh gh Odsodfh devwudlwh,
Vxu xq hvsdfh srorqdlv +> I, > vrlw +| ,|:f xqh idplooh gh suredelolwìv yìuldqw xq Sulq0
flsh gh Judqghv Gìyldwlrqv dvvrflì ã od irqfwlrqqhooh g*dfwlrq +y> , = Vrlw xqh dssol0
fdwlrq phvxudeoh gh gdqv U whooh txh =
41 h{s +y +w, , 5 O +| ,
+w A 3,
51 hvw frqwlqxh hq wrxw srlqw gh i ? .4j
61
4
olp olp
orj
u<n" |<f y +w,
doruv =
4
olp
oq
|<f y +w,
,
+]
Ew
Du
h{s +y +w, , g|
@ 4
]
h{s +y +w, , g| @ Vxs i +z, +z,j
Gh soxv/ rq prqwuh doruv txh od frqglwlrq =
]
4
+<t A 4, olp vxs
oq
h{s +ty +w, , g| ? .4
|<f
y +w,
+D14,
hqwudñqh od frqglwlrq +6, gx wkìruëph1 O*dssolfdwlrq gh fh wkìruëph shuphw gh gìgxluh
doruv xqh dssur{lpdwlrq gh od txdqwlwì +hq uhsuhqdqw ohv qrwdwlrqv gh od vhfwlrq 5, =
4
HT h{s 5I +], . * ] . * * 4( / ] Q +3> L, vrxv T=
5
s
+,
@
S
w]
5
vxlw xq sulqflsh
Sxlvtxh od idplooh gh suredelolwì +| ,|:f gìql sdu
|
gh judqghv gìyldwlrqv dvvrflì ã od irqfwlrqqhooh w> } $ 2 } } / rq yd srxyrlu dssoltxhu
oh wkìruëph suìfìghqw srxu rewhqlu xqh dssur{lpdwlrq dv|pswrwltxh gh od txdqwlwì
uhfkhufkìh1 Hq hhw vl I +], D . f] ] dyhf f ? e / od frqglwlrq BB hvw doruv yìulìh/
hw rq hq frqfoxw txh
k
l
I
I

4 4 28 + |~ ,3 |) ~ n 2 ) ),
T
+
|
4( $ pd{ 5I +}, * } . * * } }
w oq H h
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5
5
74
E1 Fkdqjhphqw gh gìulyh gìwhuplqlvwh hw vwudwlfdwlrq = uìvxowdwv qxpìultxhv
E
Fkdqjhphqw gh gìulyh gìwhuplqlvwh hw vwudwlfdwlrq =
uìvxowdwv qxpìultxhv
Ohv uìvxowdwv qxpìultxhv suìvhqwìv gdqv fhwwh dqqh{h looxvwuhqw ohv shuirupdqfhv ghv
pìwkrghv g*dffìoìudwlrq gh od frqyhujhqfh suìvhqwìhv gdqv od ghx{lëph sduwlh gh fh
pìprluh/ dssoltxìhv dx prgëoh ã yrodwlolwì vwrfkdvwltxh gh Kxoo hw Zklwh = oh surfhvvxv
+[| , @ +V| > Y| , ã ydohxuv gdqv U2 hvw vroxwlrq gh o*ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh
s
gV|
@ ugw . Y| gZc|
V|
gY| @ Y| gw . Y| gZ2c|
txh
kgZc| > gZ2c| l @ gw=
W
Sul{
4
3=8
3=58
4
3=8
3=58
3=5
4=77
3=86;
3=467
4=7<
3=897
3=476
3
Uìgxfwlrq gh yduldqfh
Irqfwlrq g*lpsruwdqfh
hw vwudwlfdwlrq
48
;97
55
4<96
74
84::
48
4966
54
664:
74
:;7:
Wde1 E141 Hydoxdwlrq g*rswlrqv dvldwltxhv/ prgëoh gh Kxoo hw Zklwh1 Qrpeuh gh vlpxodwlrqv =
438 > qrpeuh gh vwudwhv = 83 > Sul{ g*h{huflfh = 443 > Sul{ lqlwldo = 433 > Yrodwlolwì lqlwldoh = 3=5 >
u @ 3/ @ 3/ @ 3=81
W
Sul{
4
3=8
3=58
4
3=8
3=58
3=5
7=4:
5=47
3=<5
7=5;
5=54
3=<8
3
Uìgxfwlrq gh yduldqfh
hw vwudwlfdwlrq
44
89<
46
43:3
4:
54<:
45
468;
46
5665
4;
7633
Wde1 E151 Hydoxdwlrq g*rswlrqv dvldwltxhv/ prgëoh gh Kxoo hw Zklwh1 Qrpeuh gh vlpxodwlrqv =
438 > qrpeuh gh vwudwhv = 83 > Sul{ g*h{huflfh = 443 > Sul{ lqlwldo = 433 > Yrodwlolwì lqlwldoh = 3=5 >
u @ 3/ @ 3/ @ 3=81
75
F1 Fkdqjhphqw gh gìulyh vwrfkdvwltxh = uìvxowdwv qxpìultxhv
F
Fkdqjhphqw gh gìulyh vwrfkdvwltxh = uìvxowdwv qxpì0
ultxhv
Ohv uìvxowdwv qxpìultxhv suìvhqwìv gdqv fhwwh dqqh{h looxvwuhqw ohv shuirupdqfhv ghv
pìwkrghv g*dffìoìudwlrq gh od frqyhujhqfh suìvhqwìhv gdqv od wurlvlëph sduwlh gh fh
pìprluh/ dssoltxìhv dx{ ghx{ prgëohv ã yrodwlolwì vwrfkdvwltxh vxlydqwv =
Prgëoh 4 = prgëoh gh Khvwrq ^47`1 Oh surfhvvxv +[| , @ +V| > Y| , ã ydohxuv gdqv U2
hvw vroxwlrq gh o*ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh
s
gV|
@ ugw . Y| gZc|
V|
s
gY| @ + Y| , gw . Y| gZ2c|
txh
Prgëoh 51 Oh surfhvvxv +[| , @ +V| > \| , ã ydohxuv gdqv U2 hvw vroxwlrq gh o*ìtxdwlrq
glìuhqwlhooh vwrfkdvwltxh
gV|
@ ugw . ht| gZc|
V|
g\| @ +f n\| , gw . gZ2c|
txh
Rq srxuud vh uìiìuhu ã Irxuqlì/ Odvu| hw Wrx}l ^:` srxu g*dxwuhv uìvxowdwv qxpìultxhv
sruwdqw vxu fh w|sh gh prgëoh1
76
F14
Hydoxdwlrq g*rswlrqv g*dfkdw hxursìhqqhv = prgëoh 4 +Khvwrq,
Pdwxulwì
4

2

e

S

2

D

H

2f
Sul{ hvwlpì
+432 ,
6<9=;
535=8
Hfduw0w|sh +432 ,
+( gx sul{,
5=44 +3186(,
4=43 +3187(,
Uìgxfwlrq gh
yduldqfh
4=:
5=4
;8=83
78=38
3=7; +3189(,
3=59 +318:(,
6=5
7=8
43=64
8=867
6=3<6
5=467
3=395 +3193(,
3=367 +3194(,
3=34< +3195(,
3=346 +3196(,
44=9
4:=:
59=8
67=5
Wde1 F161 Hydoxdwlrq gh fdoov hxursìhqv/ prgëoh gh Khvwrq = dffìoìudwlrq gh od frqyhujhqfh sdu
dssur{lpdwlrq dv|pswrwltxh1
Qrpeuh gh vlpxodwlrqv = 438 > Sul{ g*h{huflfh = 443 > Sul{ lqlwldo = 433 > Yrodwlolwì lqlwldoh = 3=5 >
u @ 3/ @ 3=8/ @ 3=55 / @ =4/ @ 3=91
Pdwxulwì
4

2

e

S

H

f

2

e

S

H

2f

22

2e
Sul{ hvwlpì
+432 ,
6<9=<
535=7
;8=7<
78=3:
59=55
49=48
Hfduw0w|sh +43e ,
+( gx sul{,
56=33 +3138;(,
44=34 +31387(,
7=:; +31389(,
5=98 +3138<(,
4=95 +31395(,
4=37 +31398(,
Uìgxfwlrq gh
yduldqfh
473=5
544=;
653=5
764=6
894=8
:4<=3
43=66
9=:<5
7=888
6=437
3=:3
3=7:
3=66
3=56
+3139:(,
+313:3(,
+313:6(,
+313:8(,
<44=;
447<=6
4775=9
4;36=:
5=476
4=47<
3=49 +313::(,
3=45 +313:<(,
557;=8
5:<6=5
4=385
3=3; +313;5(,
6793=3
Wde1 F171 Hydoxdwlrq gh fdoov hxursìhqv/ prgëoh gh Khvwrq = dffìoìudwlrq gh od frqyhujhqfh sdu
dssur{lpdwlrq Eodfn0Vfkrohv1
u @ 3/ @ 3=8/ @ 3=55 / @ 3=4/ @ 3=91
77
F15
Hydoxdwlrq g*rswlrqv g*dfkdw hxursìhqqhv = prgëoh 5
Pdwxulwì
4

2

e

S

H

f

2
Sul{ hvwlpì
+43,
73=8;
Hfduw0w|sh +433,
+( gx sul{,
5=59 +3188(,
Uìgxfwlrq gh
yduldqfh
4=88
53=:4
;=::6
7=973
4=4< +318:(,
3=86 +3193(,
3=5< +3195(,
4=<6
5=:;
6=;8
5=:3<
4=9:7
3=4: +3196(,
3=44 +3198(,
8=58
:=36
4=3:7
3=3: +3198(,
<=69
Wde1 F181 Hydoxdwlrq gh fdoov hxursìhqv/ prgëoh 5 = dffìoìudwlrq gh od frqyhujhqfh sdu dssur{l0
pdwlrq dv|pswrwltxh1
5
u @ 3/ n @ 3/ @ 3=58/ f @ 5 / @ 3=81
Pdwxulwì
4

2

e

S

H

f

2

e

S

H

2f

22

2e
Sul{ hvwlpì
+432 ,
739=7
53:=7
;7=;7
79=7:
5:=46
49=:;
Hfduw0w|sh +43e ,
+( gx sul{,
4<=57 +3137:(,
<=<4 +3137;(,
7=79 +31384(,
5=84 +31387(,
4=87 +3138:(,
4=33 +31393(,
Uìgxfwlrq gh
yduldqfh
547=4
5::=6
6;:=;
83<=3
986=3
;5;=7
43=:;
3=9: +31395(,
4377=:
:=43;
7=:;8
6=5:6
3=79 +31398(,
3=65 +3139:(,
3=56 +3139<(,
4645=6
4975=7
537:=3
5=59;
4=8;;
4=455
3=49 +313:4(,
3=45 +313:6(,
3=3; +313:8(,
5877=;
6489=8
6<38=5
Wde1 F191 Hydoxdwlrq gh fdoov hxursìhqv/ prgëoh 5 = dffìoìudwlrq gh od frqyhujhqfh sdu dssur{l0
pdwlrq Eodfn0Vfkrohv1
5
u @ 3/ n @ 3/ @ 3=58/ f @ 5 / @ 3=81
78
F16
Hydoxdwlrq g*rswlrqv GLF = prgëoh 4 +Khvwrq,
Pdwxulwì
4

2

e

S

H

f

2
Sul{ hvwlpì
+432 ,
466=9
63=:<
Hfduw0w|sh +43 ,
+( gx sul{,
<=87 +31:4(,
5=75 +31:;(,
Uìgxfwlrq gh
yduldqfh
6=6;
:=93
6=579
3=78;
3=3:5
3=346
3=5<3 +31;<(,
3=3799 +4135(,
3=33;48 +4147(,
3=334:8 +416;(,
5;=<3
44:=5
:74=<
5349=;
3=335
3=33366 +4176(,
95=<8
Wde1 F1:1 Hydoxdwlrq gh GLF/ prgëoh gh Khvwrq = dffìoìudwlrq gh od frqyhujhqfh sdu dssur{l0
pdwlrq dv|pswrwltxh1
Qrpeuh gh vlpxodwlrqv = 438 > Sul{ g*h{huflfh = 443 > Eduulëuh = <3 > Sul{ lqlwldo = 433 > Yrodwlolwì
lqlwldoh = 3=58 > u @ 3/ @ 3=8/ @ 3=585 / @ =4/ @ 3=91
Pdwxulwì
5
4

2

e

S

H

f

2
468=7
64=7:
Hfduw0w|sh +43 ,
+( gx sul{,
9=75 +314:(,
7=79 +3166(,
3=7<: +3148(,
Uìgxfwlrq gh
yduldqfh
66=9
48=7
4;:=5
6=595
3=78:
3=3:7
3=346
3=385: +3149(,
;=54h 6 +314;(,
4=93h 6 +3155(,
5=;3h 7 +3155(,
;:5=7
6:;5=9
4<5:;
:;<;6
3=335
8=84h 8 +3157(,
555;=6
Sul{ hvwlpì
+432 ,
6:6=3
Wde1 F1;1 Hydoxdwlrq gh GLF/ prgëoh gh Khvwrq = dffìoìudwlrq gh od frqyhujhqfh sdu dssur{l0
pdwlrq Eodfn0Vfkrohv1
lqlwldoh = 3=58 > u @ 3/ @ 3=8/ @ 3=585 / @ =4/ @ 3=91
79
F17
Hydoxdwlrq g*rswlrqv GLF = prgëoh 5
Pdwxulwì
4

2

e

S

b

2
Sul{ hvwlpì
+43 ,
949=:
Hfduw0w|sh +43 ,
+( gx sul{,
8=<: +31<9(,
Uìgxfwlrq gh
yduldqfh
6=3
<5=59
4;=;<
7=747
4=49 +4158(,
3=5; +417<(,
3=3; +41;5(,
:=<
4:=4
5:=;
3=5;8
8=<:3h 6
9=96h 6 +5165(,
5=99h 7 +7178(,
4;8=4
5885
4=6<9h 7
<=37h 9 +917;(,
3
Wde1 F1<1 Hydoxdwlrq gh GLF/ prgëoh 5 = dffìoìudwlrq gh od frqyhujhqfh sdu dssur{lpdwlrq
dv|pswrwltxh1
5
lqlwldoh = 3=5 > u @ 3/ n @ 3/ @ 3=58/ f @ 5 / @ 3=81
Pdwxulwì
5
4

2

e

D

S

.

H

b

f

2
Sul{ hvwlpì
+43 ,
5474=8
949=:
<6=7;
4;=;;
7=674
4=3:4
Hfduw0w|sh +43 ,
+( gx sul{,
<=38 +3175(,
6=35 +317<(,
3=66 +3168(,
:=89h 5 +3173(,
4=:8h 5 +3173(,
7=;4h 6 +3178(,
Uìgxfwlrq gh
yduldqfh
:=9
44=;
<;=<
573=4
8;8=;
5634=9
3=5;5
4=75h 6 +3183(,
7397=9
:=893h 5
5=3;4h 5
8=<6<h 6
6=<;h 7 +3186(,
4=44h 7 +3186(,
6=99h 8 +3195(,
9<5=;
64=4
467;6<
4=9<4h 6
7=<75h 7
4=789h 7
4=53h 8 +31:4(,
6=69h 9 +319;(,
4=54h 9 +31;6(,
3
3
3
Wde1 F1431 Hydoxdwlrq gh GLF/ prgëoh 5 = dffìoìudwlrq gh od frqyhujhqfh sdu dssur{lpdwlrq
Eodfn0Vfkrohv1
5
lqlwldoh = 3=5 > u @ 3/ n @ 3/ @ 3=58/ f @ 5 / @ 3=81
7:
G1 Yduldeoh gh frqwu÷oh = oh fdv g*xqh rswlrq vxu sdqlhu
G
Yduldeoh gh frqwu÷oh = oh fdv g*xqh rswlrq vxu sdqlhu
Fhw h{hpsoh looxvwuh od pìwkrgh g*dffìoìudwlrq gh od frqyhujhqfh ã sduwlu g*xqh yduldeoh
gh frqwu÷oh/ suìvhqwìh gdqv od suhplëuh sduwlh gh fh udssruw/ sdjh 91 Rq v*lqwìuhvvh ã xq
surgxlw gìulyì gìql ã sduwlu gx frxuv ã od gdwh W gh g dfwlrqv vrxv0mdfhqwhv VA > = = = > VA_ 1
Oh sd|r gh fh surgxlw v*ìfulw
[A @ pd{ +DA N> 3,
rü
_
4[ VA =
DA @
g
'
Hq vxssrvdqw txh oh wdx{ g*lqwìuíw lqvwdqwdqì u hvw frqvwdqw vxu od sìulrgh frqvlgìuìh/
oh sul{ gh fhwwh dfwli v*ìfulw
S @ Hh3oA [A >
o*hvsìudqfh ìwdqw fdofxoìh vrxv od suredelolwì ulvtxh0qhxwuh gx pdufkì1
Rq vxssrvh txh o*hqvhpeoh ghv dfwlrqv frqvlgìuìhv vxlyhqw xq surfhvvxv gh glxvlrq
Eodfn0Vfkrohv vrxv od suredelolwì ulvtxh0qhxwuh1 Iruphoohphqw/ rq frqvlgëuh xq prx0
yhphqw eurzqlhq vwdqgdug vxu U_ / qrwì +E| ,|Df / @ + ,$$_ xq yhfwhxu gh frrugrq0
qìhv vwulfwhphqw srvlwlyhv hw @ $c $_ xqh pdwulfh gh fruuìodwlrq1 Rq qrwh od
pdwulfh dvvrflìh ã uìvxowdqw gh od gìfrpsrvlwlrq gh Fkrohvn|7 gh fhwwh ghuqlëuh1 Rq
prgìolvh doruv o*ìyroxwlrq
ghv g dfwlrqv frpsrvdqw oh sdqlhu vrxv0mdfhqw sdu oh surfhvvxv
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