Prgëohv g|qdpltxhv gh o*ìfrqrplh

Pdñwulvhv g*ìfrqrpìwulh 2 Pdjlvwëuh

Xqlyhuvlwì gh Wrxorxvh L
Mxlq 5336

Fhw h{dphq hvw frpsrvì gh ghx{ sureoëphv lqgìshqgdqwv1 Ohv grfxphqwv gh frxuv
dlqvl txh wrxw dssduhlo ìohfwurqltxh vrqw vwulfwhphqw lqwhuglwv1 Oh eduëph hvw grqqì
vhxohphqw ã wlwuh lqglfdwli1

Frkìuhqfh whpsruhooh ghv srolwltxhv ìfrqrpltxhv

+9 swv,
Rq v*lqwìuhvvh ã o*dfwlrq g*xq jrxyhuqhphqw grqw rq uìvxph o*krul}rq whpsruho
ã ghx{ sìulrghv= sìulrgh  +oh suìvhqw, hw sìulrgh 2 +oh ixwxu,1 D fkdtxh sìulrgh 
oh jrxyhuqhphqw shxw hqwuhsuhqguh xqh dfwlrq +sdu h{hpsoh dssoltxhu xqh fhuwdlqh
srolwltxh vfdoh, qrwìh Z 1 O*hqvhpeoh ghv djhqwv ìfrqrpltxhv gx sd|v uìdjlw ã
fh fkrl{ g*dfwlrq hq suhqdqw ã vrq wrxu xq hqvhpeoh gh gìflvlrqv dssursulìhv +sdu
h{hpsoh xqh gìflvlrq g*ìsdujqh,1 Rq qrwh % od yduldeoh fdudfwìulvdqw o*dfwlrq ghv
djhqwv ìfrqrpltxhv ã od sìulrgh 1 Rq vxssrvh txh Z c Z2 c % c %2 vrqw ghv yduldeohv
uìhoohv1
Ohv djhqwv frqqdlvvhqw ohv dfwlrqv dqqrqfìhv sdu oh jrxyhuqhphqw hw irqghqw
ohxuv sursuhv gìflvlrqv ghvvxv> rq d grqf % ' f EZ c Z2  hw %2 ' f2 E% c Z c Z2 1
O*remhfwli gx jrxyhuqhphqw hvw gh pd{lplvhu xq fulwëuh gh vxusoxv vrfldo grqqì sdu
od irqfwlrq 7E% c %2 c Z c Z2 1 Wrxwhv ohv irqfwlrqv vrqw vxssrvìhv glìuhqwldeohv1
Txhvwlrq 4 +5 swv,= Rq vh sodfh dx gìexw gh od suhplëuh sìulrgh1 Hfuluh ohv frq0
glwlrqv gx suhplhu ruguh fruuhvsrqgdqw dx idlw txh ohv dfwlrqv txh oh jrxyhuqhphqw
suìyrlw g*hqwuhsuhqguh Z hw Z2 pd{lplvhqw oh vxusoxv vrfldo +rq jdughud xqh h{suhv0
vlrq olwwìudoh vdqv fkhufkhu ã vlpsolhu,1
Txhvwlrq 5 +4 sw,= Rq vh sodfh pdlqwhqdqw dx gìexw gh od ghx{lëph sìulrgh1 Ohv
yduldeohv Z  hw f fruuhvsrqghqw ã ghv dfwlrqv sdvvìhv hw vrqw grqf {ìhv1 Hfuluh od
frqglwlrq gx suhplhu ruguh fruuhvsrqgdqw dx idlw txh o*dfwlrq Z 2 txh oh jrxyhuqhphqw
vrxkdlwh pdlqwhqdqw hqwuhsuhqguh pd{lplvh oh vxusoxv vrfldo1
Txhvwlrq 6 +4/8 swv,= Hq xwlolvdqw ohv ghx{ txhvwlrqv suìfìghqwhv/ ìwdeolu xqh
frqglwlrq srxu txh o*dfwlrq ZW2 rswlpdoh dx gìexw gh od sìulrgh  vrlw wrxmrxuv

4
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Hwlhqqh E1 gh Yloohphxu 5336 0 kwws=22zzz1eljirrw1frp2HEGY 5

rswlpdoh dx gìexw gh od sìulrgh 21

Txhvwlrq 7 +4/8 swv,= Frpphqwhu hw lqwhusuìwhu oh uìvxowdw rewhqx ã od txhvwlrq 6/
Y7 n Y7 Yf2 9' f/ +8 oljqhv pd{lpxp,1
hq vxssrvdqw txh Y% Y%2 Y%

Prglfdwlrq ghv sul{ hw dmxvwhphqw ghv idfwhxuv

gh surgxfwlrq
+47 n 6swv,
Gdqv fh sureoëph/ rq v*lqwìuhvvh dx{ frqvìtxhqfhv g*xq fkdqjhphqw gx sul{ gx
idfwhxu gh surgxfwlrq wudydlo +O, srxu xqh hqwuhsulvh1 Srxu vlpsolhu/ rq vxssrvh
txh oh wudydlo hvw o*xqltxh idfwhxu gh surgxfwlrq gh o*hqwuhsulvh hw tx*hooh yhqg vd
surgxfwlrq dx sul{ R '  vxu xq pdufkì frqfxuuhqwlho1 Vxlwh ã xqh prglfdwlrq gx
sul{ gx wudydlo/ o*hqwuhsulvh yd yrxorlu dmxvwhu vd txdqwlwì gh pdlq0g*úxyuh1 Rq qrwh
u| od pdlq g*úxyuh hpsor|ìh dx gìexw gh od sìulrgh | hw | oh x{ g*hpedxfkhv dx
frxuv gh fhwwh sìulrgh +f*hvw ã gluh txh o*rq d u|n ' u| n | ,1 Srxu glyhuvhv udlvrqv/
rq vxssrvh txh oh qlyhdx gh surgxfwlrq ã od sìulrgh | hvw grqqì sdu od irqfwlrq
gh surgxfwlrq 8 Eu| c u|n  ' sE4?iu| c u|n j rü s E hvw xqh irqfwlrq ã ydohxuv
uìhoohv1
Txhvwlrq 4 +4/8 swv,= Sursrvh} xqh lqwhusuìwdwlrq gh fhwwh irqfwlrq gh surgxfwlrq
+8 oljqhv pd{lpxp,1
O*hqwuhsulvh ìydoxh vhv surwv ixwxuv dyhf xq idfwhxu g*hvfrpswh B/ who txh f
B 1 Vl rq qrwh Z | vrq surw +lqvwdqwdqì, ã od sìulrgh |/ vrq surw hvfrpswì ã od
S
gdwh f ydxw=  ' A|'f B | Z | c rü A hvw o*krul}rq whpsruho gh o*hqwuhsulvh1 Rq vxssrvh
gdqv xq suhplhu whpsv txh A hvw ql1 O*ìyroxwlrq gx sul{ gx wudydlo vxu od sìulrgh
ifc c 2c c A j hvw grqqìh sdu od vxlwh Ruc| 1
Txhvwlrq 5 +4/8 swv,= Rq vxssrvh txh ohv qrxyhdx{ hpedxfkìv vrqw sd|ìv srxu
wrxwh od sìulrgh1 Rq vxssrvh ìjdohphqw tx*xq hpsor|ì olfhqflì dx frxuv g*xqh
sìulrgh hvw sd|ì mxvtx*ã od q gh od sìulrgh1 Hfuluh o*h{suhvvlrq gx surw lqvwdqwdqì
Z| gh o*hqwuhsulvh dx frxuv gh od sìulrgh |1 Rq srxuud glvwlqjxhu oh fdv rü o*hqwuhsulvh
hpedxfkh +| ' u|n  u| srvlwli, gh fhoxl rü hooh gìedxfkh +| ' u|n  u| qìjdwli,1
Txhvwlrq 6 +4 sw,= Gdqv oh fdv g*xq uìjlph vwdwlrqqdluh/ f1d1g1 vl ohv sul{ vrqw
frqvwdqwv hw o*hqwuhsulvh q*hpedxfkh ql qh gìedxfkh +Ruc| ' Ru hw u|n ' u| ' u
srxu txhooh ydohxu gh u oh surw lqvwdqwdqì hvw0lo pd{lpxpB +Rq grqqhud o*ìtxdwlrq
txl shuphw gh od gìwhuplqhu,
Gdqv od vxlwh gx sureoëph rq qrwhud fhwwh ydohxu uW 
Txhvwlrq 7 +6 swv,= Hfuluh oh surjudpph gh pd{lplvdwlrq o*hqwuhsulvh hq suìflvdqw
elhq od irqfwlrq gh wudqvlwlrq1 Txhoohv vrqw ohv yduldeohv g*ìwdw hw gh frqwu÷ohB Hq
gìgxluh o*ìtxdwlrq gh Ehoopdq yìulìh sdu od irqfwlrq T| Eu txl fdudfwìulvh od ydohxu
dwwulexìh sdu o*hqwuhsulvh dx vwrfn gh pdlq g*úxyuh u/ dx gìexw gh od sìulrgh | A 1
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Hwlhqqh E1 gh Yloohphxu 5336 0 kwws=22zzz1eljirrw1frp2HEGY 6

Rq vxssrvh srxu od txhvwlrq txl vxlw txh o*hqwuhsulvh ihuph hw fhvvh wrxwh sur0
gxfwlrq ã od q gh od sìulrgh A n41
Txhvwlrq 8 +5 swv,= Rq vh sodfh ã od sìulrgh A n  Fdofxohu TA n vdfkdqw txh/
píph vl o*hqwuhsulvh d fhvvì gh surgxluh/ hooh grlw sd|hu wrxv ohv hpsor|ìv txl rqw
frpphqfì xqh sìulrgh fkh} hooh1 Hq gìgxluh o*ìtxdwlrq yìulìh sdu A sxlv h{soltxhu
frpphqw uìvrxguh o*hqvhpeoh gx sureoëph srxu rewhqlu Tf Euf 
+Qh sdv fkhufkhu ã fdofxohu fhwwh irqfwlrq,1
Rq v*lqwìuhvvh pdlqwhqdqw hw srxu od vxlwh gx sureoëph dx fdv A ' n41 Rq
vxssrvh ìjdohphqw txh Ruc| ' Ru / srxw wrxw | +lo | d xq fkrf vxu oh sul{ gx wudydlo ã
od gdwh f/ sxlv oh sul{ uhvwh frqvwdqw,1
Txhvwlrq 9 +4 sw,= Hfuluh o*ìtxdwlrq gh Ehoopdq gx sureoëph1
Txhvwlrq : +7 swv,= Rq vxssrvh pdlqwhqdqw txh uf  uW vl elhq txh o*hqwuhsulvh
grlw dxjphqwhu vd pdlq g*rhxyuh1 Uìvrxguh o*ìtxdwlrq gh Ehoopdq hq vxssrvdqw | 
f Rq glvwlqjxhud elhq ohv wurlv ìwdshv gh od uìvroxwlrq= frqglwlrq gx suhplhu ruguh
vxu od yduldeoh gh frqwu÷oh/ gìulyìh gh od irqfwlrq ydohxu/ ìtxdwlrq gh od g|qdpltxh1
Hq dgphwwdqw txh wrxwhv ohv vroxwlrqv vrqw lqwìulhxuhv +txh od yduldeoh | rswlpdoh
hvw elhq srvlwlyh,/ txhooh hvw o*ìtxdwlrq yìulìh sdu od ydohxu gh u gdqv oh uìjlph
vwdwlrqdluhB Rq qrwhud fhwwh ydohxu u  Dx erxw gh frpelhq gh sìulrghv fhwwh ydohxu
_

vhud0w0hooh dwwhlqwhB
Txhvwlrq ; +6 swv erqxv,= Gdqv oh fdv/ uf u / rü o*hqwuhsulvh grlw uìgxluh vd
W

pdlq g*rhxyuh/ rq shxw prqwuhu txh vl |  f doruv u| yd frqyhujhu hq xqh sìulrgh

yhuv u rü u yìulh
__ __

 
s u ' BRu 
 __

Prqwhu txh vl s E hvw furlvvdqwh frqfdyh/ doruv u u

_
u  H{soltxh}1 Txh vh
W __

sdvvh0w0lo gdqv fdv u_

uf u c rü uf hvw od pdlq g*rhxyuh lqlwldoh B H{soltxh}
__

hw frpphqwh}1
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Hwlhqqh E1 gh Yloohphxu 5336 0 kwws=22zzz1eljirrw1frp2HEGY 7

4 Hoìphqwv gh fruuhfwlrq
414 Sureoëph 4
Txhvwlrq 4= +5swv, Od F1S1R gh od pd{lplvdwlrq gx vxusoxv vrfldo v*ìfulw=

_7 _7
' f e| ' f
_Z _Z2
Rq d grqf oh v|vwëph=
Y7 Yf Y7 Yf2 Y7 Yf Yf2 Y7
n n n ' fc
YZ  YZ Y% YZ Y%2 YZ Y% Y%2
Y7 Yf Y7 Yf2 Y7 Yf Yf2 Y7
n n n ' f
YZ 2 YZ2 Y% YZ2 Y%2 YZ2 Y% Y%2
Txhvwlrq 5= +4 sw, Dx gìexw gh od sìulrgh 2/ rq pd{lplvh oh vxusoxv vrfldo frpph
xqh irqfwlrq gh od vhxoh yduldeoh Z2  Od F1S1R1E_7*_Z2  ' f v*ìfulw=

Y7 Yf2 Y7
n ' f
YZ2 YZ 2 Y%2

Txhvwlrq 6= +4/8 swv, Srxu txh od srolwltxh fdofxoìh hq suhplëuh sìulrgh vrlw
wrxmrxuv rswlpdoh hq ghx{lëph sìulrgh lo idxw txh Z W2 vrlw vroxwlrq ghv ìtxdwlrqv
E_7*_Z2  ' f rewhqxh ã od suhplëuh txhvwlrq hw ã od ghx{lëph1 Sdu vxlwh rq grlw
dyrlu=
 
Yf Y7 Y7 Yf2
n ' f
YZ2 Y% Y%2 Y%

Rq glw txh fh fdv txh od srolwltxh ZW2 hvw frkìuhqwh whpsruhoohphqw1

Y7 Y7 Yf2
Txhvwlrq 7= +4/8 swv, Vl Y% 
n Y% 2 Y%
9' fc srxu txh oh jrxyhuqhphqw qh fkdqjh
sdv gh srolwltxh +sdu udssruw ã fh tx*lo dydlw suìyx brx surplvb gh idluh,/ lo idxw
txh EYf *YZ2  ' f1 Hq g*dxwuhv whuphv lo idxw txh o*dfwlrq fkrlvlh sdu ohv djhqwv
hq sìulrgh  qh gìshqgh sdv gh od srolwltxh gh ghx{lëph sìulrgh1 Fhwwh frqglwlrq
d ìylghphqw shx gh fkdqfhv g*íwuh uìdolvìh1 Vl ohv djhqwv dqwlflshqw yudlphqw od
srolwltxh gh ghx{lëphqw sìulrgh hw dgdswhqw hq frqvìtxhqfh ohxu dfwlrq gh suhplëuh
sìulrgh +sdv gh frpsruwhphqw p|rsh,/ od srolwltxh qh vhud sdv frkìuhqwh whpsruhooh0
phqw1 Oh jrxyhuqhphqw dxud whqgdqfh ã prglhu od srolwltxh tx*lo dydlw suìyx gh
idluh lqlwldoohphqw1

415 Sureoëph 5
Txhvwlrq 4= +4/8 swv, Od irqfwlrq gh surgxfwlrq 8 Eu| c u|n  ' s E4?iu| c u|n j
v*ìfulw 8 Eu|n c u|  ' s Eu|  vl u|  u|n 1 Xq hpsor|ì q*hvw surgxfwli tx*dsuëv
xq fhuwdlq whpsv g*dssuhqwlvvdjh hw grqf hq sìulrgh gh furlvvdqfh/ o*hhw g*xqh
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Hwlhqqh E1 gh Yloohphxu 5336 0 kwws=22zzz1eljirrw1frp2HEGY 8

hpedxfkh qh vh yrlw tx*dyhf xq fhuwdlq gìfdodjh gdqv oh whpsv1 Hq sìulrgh gh

uìfhvvlrq/ u|n  u| hw od irqfwlrq gh surgxfwlrq v*ìfulw 8 Eu|n c u|  ' s Eu|n 1 Rq
shxw dmxvwhu vdqv gìodl oh qlyhdx gh pdlq g*rhxyuh ã od edlvvh1
Txhvwlrq 5= +4/8 swv, Vl ohv qrxyhdx{ hpedxfkìv hw ohv olfhqflìv vrqw sd|ìv srxu
wrxwh od sìulrgh/ od pdvvh vdoduldoh v*ìfulw 4@ iu| c u|n j  Oh surw lqvwdqwdqì gh
o*hqwuhsulvh v*ìfulw=

Z| ' R8 Eu| c u|n   Ruc| 4@ iu| c u|n j

Vl u|  u|n rq d 4? iu| c u|n j ' u| hw 4@ iu| c u|n j ' u|n  Hq xwlolvdqw R ' c
rq rewlhqw Z | ' s Eu|   Ruc| u|n  Gdqv oh fdv frqwudluh/ f1d1g1 vl u|  u|n c rq d
Z| ' s Eu|n   Ruc| u| 
Txhvwlrq 6= +4 sw, Gdqv oh fdv gx uìjlph vwdwlrqqdluh/ oh surw lqvwdqwdqì hvw
pd{lpxp srxu u ' uW who txh=

s  EuW  ' Ru

Txhvwlrq 7= +6 swv, Oh surjudpph gh pd{lplvdwlrq o*hqwuhsulvh v*ìfulw

+ A ,
[
4@A B|Z| r,S u|n ' u| n | 
tu| |' |'f

D od sìulrgh |c ohv yduldeohv gh frqwu÷oh vrqw | +hw u|n , hw ohv yduldeohv g*ìwdw u| hw
R|  O*ìtxdwlrq gh Ehoopdq v*ìfulw=

T| Eu|  ' 4@ iZ| n BT|n Eu|n j

 |

Txhvwlrq 8= +5 swv, Píph vl hooh qh yhqg soxv/ o*hqwuhsulvh grlw frqwlqxhu ã sd|hu
ohv hpsor|ìv1 Grqf TA n ' RucA n uA n  Sdu vxlwh o*hqwuhsulvh grlw uìvrxguh

4@ iZ A EuA c uA n  n BTA n EuA n j ' 4@ is EuA n   RucA uA  BRucA n uA n j

|  |

Grqf A yìulh s  EuA n A  ' BRucA n 1 Rq shxw grqf hq gìgxluh TA Eu Srxu
wurxyhu Tf Euf  c rq uìvrxg sdu lwìudwlrq/ hq frpphqêdqw sdu od ghuqlëuh sìulrgh1
Txhvwlrq 9= +4 sw, o*ìtxdwlrq gh Ehoopdq gx sureoëph v*ìfulw=

T Eu|  ' 4@ iZ| n BT Eu|n j ' 4@ i8 Eu| c u|n   Ruc| 4@ iu| c u|n j n BT Eu|n j
 |  |
r,S u|n ' u| n | 

QE= fhwwh irlv0fl T Eu qh gìshqg sdv gh od gdwh1

Txhvwlrq := +7 swv, Gdqv oh fdv |  fc o*ìtxdwlrq gh Ehoopdq v*ìfulw=

T Eu|  ' 4@ is Eu|   Ruc| u|n n BT Eu|n j

| D f
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Hwlhqqh E1 gh Yloohphxu 5336 0 kwws=22zzz1eljirrw1frp2HEGY 9

dyhf u n ' u n  
| | |

Od frqglwlrq gx suhplhu ruguh txl gìwhuplqh od yduldeoh gh frqwu÷oh v*ìfulw=

R uc| n BT Eu n   ' f

| |

Rq rewlhqw doruv gluhfwhphqw od gìulyìh gh od irqfwlrq ydohxu hq ìfulydqw=

   
_ _ W W

T Eu  ' s Eu   R

n nB n
T Eu n   | | 

_u _u
| | uc| | |
| |

rü  hvw od vroxwlrq gh od frqglwlrq gx suhplhu ruguh suìfìghqwh1 Rq xwlolvh fhwwh

W
|

frqglwlrq srxu rewhqlu +wkìruëph gh o*hqyhorssh, T Eu  ' 8 Eu   

|

|

Ilqdohphqw/ rq rewlhqw

Ruc| n Bs Eu n   ' f

| |

 
u yìulh grqf s u ' R *B Fhwwh ydohxu vhud dwwhlqwh gëv od suhplëuh sìulrgh1
_  _
u

Txhvwlrq ;= +6 swv erqxv, Vl s E hvw furlvvdqwh frqfdyh/ vd gìulyìh hvw gìfurlvvdqwh
   
grqf s u 
s Eu  lpsoltxh u : u  +Hw gh píph s u : s Eu  lpsoltxh
__  W __ W  _  W

u_
u ,1 Txdqg rq uìgxlw od pdlq g*rhxyuh/ rq v*duuíwh dydqw oh qlyhdx u fdu od
W W

uìgxfwlrq gh pdlq g*rhxyuh d xq frxs sxlvtx*lo idxw frqwlqxhu gh od sd|hu shqgdqw

xqh sìulrgh1 +Vl oh qlyhdx gh pdlq g*rhxyuh q*hvw sdv wurs lpsruwdqw/ rq suìiëuh od
jdughu,1 Gh píph vl txdqg rq dxjphqwh od pdlq g*rhxyuh/ rq v*duuíwh dydqw u fdu W

dxjphqwhu vd pdlq g*rhxyuh d xq frxs sxlvtx*lo idxw od sd|hu shqgdqw xqh sìulrgh
dydqw txhooh vrlw h!fdfh1 +Vl oh jdlq pdujlqdo dssruwì sdu oh vxssoìphqw gh pdlq
g*rhxyuh hvw idleoh/ rq suìiëuh v*hq sdvvhu,1
Vl u W
uf u c rq hq shxw sdv ã od irlv dyrlu   f +uìgxluh od pdlq g*rhxyuh,
__
|

hw frqyhujhu yhuv u 1 Od pdlq g*rhxyuh yd grqf uhvwhu ã vrq qlyhdx lqlwldo/ wrxw
__

dmxvwhphqw d|dqw xq frþw wurs ìohyì1 Gh píph vl u uf u c rq hq shxw sdv ã od _ W

irlv dyrlu   f +dxjphqwhu od pdlq g*rhxyuh, hw frqyhujhu yhuv u  Hq frqfoxvlrq/

|
__

vl u _
uf u c f1d1g1 vl rq q*hvw sdv wurs orlq gh od txdqwlwì rswlpdoh/ lo q*|
__

dxud sdv g*dmxvwhphqw1