Segmentation de séquences d'images en vue du codage
191
0
0
Texte intégral
(2)
(3) ! " . #%$'& ()$+*-,/.0,01324* 5$ #$0647$+*98:$0#;5=<>1?().@& $0# $+* A 7B$ 57 8C245D.@& $. EFGIH3G JLKNMPORQSUTNMQ VWQXZY[S\T ]_^ `"ab\] QcdYZTNe b SfY ] IQ gWhLJiMPKRejQhfKRQVWQkOmlejSLQPOVLQonpY[KReqO JiYrK sutwvWxyZz|{}:~% EG J b Lh K [b TNQSLejK ] QITReTRKNQIVLQ bWa TNQhLKQSl b KNJL b\]b\ ejQolYrTRLMPYZTReqUhLQ. g b Wh TRQPSUhfQ ] Qu3XwKNe ]pk[\Lb SUTNYrejSLQ L] QkYrhVWQXZYrSUT ] QhLKR a b u J b ORM VWQ[ ll[QkYrS g `" ¡ R¢¤£¤¥|¦\§¨i© ª b hLK«Y[V¬ `"¡ d®¯l¯ °d±N²\²L³r ¤©´§µW ¶ MK«YrK«V `"· ` ¸º¹ ±r» ¥¼¨?±Z©>§µW K«YrSf½ b eqO ®o¾ ªmª ¯ `¿ ¸º¹ ±r» ¥¼¨?±Z©>§µW QPKRK«ÀrS l mÁo·` g °d±N²\²L³r ¤©´§µW QPKRSfY[SfV l `pÂo`" ¸º¹ ±r» ¥¼¨?±Z©>§µW .
(4) »%±±Z»u¥§ µW¥q£ ¸
(5) § ¥|± M a § £ w£ ¨f©>³Z¨i¥|³ »§¤£²f±Z R§¨f©_£±[¦[³«² © ¥ N£ ¥ ¥|±r¨ §§©¢ R±Z R¦. §© § [³rµ© ¥¼¨ [³Zµ «±Z¨ ³r [ ³Zµ «±Z¨ ©
(6) !"[³Zµ ±Z N§ ¤¥$# © % ¥$# % ¥$#&±r .
(7) \QPY[S gwQKNKNY \hfe K a Q O b S YrSLSfMQ ONY L YZTReqUhLQ Y[K a Q ]jb SLQ
(8) m ^ Y V b SLSfM ] ^ bWaa Y[ORe b SQT ] Q J ] Y[ejORejKdVWQ KRQPS\TNKRQPKdVLY[SfO ] ^ eSUTRMPKRQkORONYrSUTdV b YrejSLQ VLQ ] Yl b KNJL b\]b\ ejQ lYrTRLMP%YrTReqUhLQ[lQPK a epJ b hLK T b hWTRQkO ] QkO ab SLSfYreqONORY[S a QkOoY[JLJLKNejORQPOoQTJ b hLK ]_^ Y aPa hLQPe ] Q a QJWTNe b SLSLQ ] VfYrSfO¡hLS a QSUTRKNQ Y a TNe SLejX[QPY[hO a ejQSUTNeiUhLQ
(9) WYreqOmYrhfONO e whLY[eS u QKNSfYrSiVlQP[QKmV ^ Y¬X b eK hfejVWMIQTmORhLejXwe VWQIJLKkO a QoTRK«Y¬XZYre ] lQK a e J b hfK¡V ^ Y¬X b ejK ORh ^ QS ab hfKNY QK PuQ \hiYrSfV ] QPO ab SiVWe TNe b SfOS ^ MTNYrejQSUTJiY[O ] QPOJ ] hfO|Y¬X b K«Y L] QPOP lQPK a e J b hLKORYuJ ] QejSLQVWeqORJ b SLe e ] e TNM[ QPKRK«ÀrSl YrK«\hfMPOP b SKRQkO J b SfONY f] Q d YrK a Q ]jb SfQ a TRM"VLQUhLeUe ] QPO T e%J b ONO e L] Q VWQ SLQIJfY\O TNKRQ¡ b \Qh nºfe ] eJfJiQ gLY ] Q eQPKdQTl b SUTNORQun Y[KNV O?UhLeºY¬X\Q a ] QPhLK«O b SiOdKNMPORh ] TNYrTNOo ^ b S\T T b hZ b hfKNO¡J b hfONORM Y[uM ] e b KNQK ] QPO¡%ejQSfOP b ORM ¿KRQkO J b Y¬X[Q a UhLeW ^ Yre QPhVLQPO KNh a TNhLQhiO QkOmVWejO a hiORORe b SfOP ` TRejQSLSLQ Q a QS a e KNQoJ b hfK¡T b hLTP ¿LKNejO T b JLLQ ¶ K«YZTRejS mJ KNQ VWQ%QkO ab SLSfY[ejONORY[S a QkO%QS ejS b KNYZTNejUhLQ
(10) J b hfK ONY JLKNM a eQPhfO Q Y[ejVWQIJLK b QkORORe b SLSLQ ]] QIYrejSfO e UhLQ ] ^ Yr%e TNeMUh ^ e ] ^ Y "b ! QKRTRQ\ le a LQ ]3 e ]jb VWQPY[h ] ^ QS a a]b JiMkVWejQoXwejXZYrSUTRQ
(11) wT b hr b hLK«O b hLX\QKRTRQ u] Y b SLSfQoJiY Q\ mb\] Y[SfV KNM b SfVJ b hLK ORQPOoS b KNQhfORQPOdKNQ ] Q a TNhLKNQPOIVWQ%%QPOoKRMkVLY a TRe b SfOoQT ORQPO ab SfORQe ] OmY¬XweqO MkO npY[O a Y $] # YrhLKNQS Q ] YTRK«YrSfUhLe ]] eTRMuJiQPKNO b SLSfe fMPQ ?UhLe ^ YY[JLJLKNejOdJfY[K ] ^ %Q wQP%J ] Q Uh ^ e ] |Y[hWTXweXwKNQ ] Y%XweQ ONYrSfOmO TRKNQPONOP u Y KNe a Q # QP b SLSLejQKmJ b hLKO b S a & hfK b hLX\QKRTP '( # # YrbhLK«Yu] Sfb VWKNaeqYrY[OReS b K b J +a b *+hLK ORQP[O b SiO ab b SiO QPe ] O e SLQ h iYrK«)V npYZTNKRe KRe Q,K O QPJ Y[' -\. S Y\ORY\%O ¯¤/ b KNORQ,T ª b SLe ¶ Y[ORh ]j] / ¿whfY[S ¶ h dgW ] XweQ [QkYrSLSLejS QPKRK«ÀrS+l YrK«\hfMPO b ORQJ Y[-[S l b KNK b O l b S\T«O Q npY[KNV %O 10oO a Y[K e QPO fn"Le ] ejJLJiQ%gWY ] Q ejQKmQT # ] hLejO ª b KRKNQPOJ b hLK ] Y b SLSLQ Yr eqYrS a QIUh ^ e ] O b S\TORh a KNMQK u] ^ ejSUTRMKNejQhLKVWhJLK b QTl b KNJLLQ ab le a LQ ] ¶ YrhLTRLejQKmQT l YrK /a 2 YrK b UhLeQPK¡J b hLK ] QPhLK b SZ b hLKVWhYZTRejS 3 # e ] eqYrSfQQT ¶ MK«YrK«VunºeJiYrh ] TJ b hfK4TNKRQ K«YrejSLQVWQ ib SLLQhfK¿QTVWQ ib SLSLQwhL%QPhLKP lQPK a e V ^ Y¬X b ejK¡KRQPuJ ] e ] Q K \] QIVWQ JfY[KRQPSUTNOmY[V b JWTNe |OP Y5|Y[%e ]j] Q76 J KNQ?KNQ T«YrSUTRQuQT8 KPKRQ:9Y[h O QPeS VWQ ] Y[UhLQ ]] Q ^ Yre QPh ] Y b SLSLQ a fY[S a Q VWQISf<Y ;jTNKRQIQT ] Q b SLLQPhLKVWQ XwejXUKNQ[ UYrejuQ
(12) pJ b hLK%Y¬X b ejK YrK«VWM ] Y a] MVWh b SLLQPhLK%JiQPSfVLY[S\T%VWQh= YrSfOQS TNLM b KRejQ bTRK eqOmYrSiO¡QTVWQPue3QSJLKNYrTReqUhLQ LONYrSfOmY¬X b ejKVLMPORQPORJiMPKRMIhLSLQORQh ] QI%eSwhWTNQ[ # b TRfY[K QK QPSQT>Ie * e 2 e eSUT b S WgwQK Q QPh a LQK,w\Y a Q * ¿e a b ? # h a dQ +a * QPK ] YrejKRQp®Q ] PSLQ"dQYrKRT ¬ ] e a Q ¶ YrhLTRLejQ,K < ¿Q ? QPK ¶ b KNQ S kej%e TNKRoQT ¯>KRQPSfY ¶ b K b * b Xwe * ce ab\] Q ¶ hfQ Y[S knºLe ] ejJLJ Q ¶ hLe L] ejS ZlY[K a eSI¯A@¡YrS b @O * e b %ejSLeq\hfQ"[QPh ] eS ,B ] Y[VLe%ejK >nIQfwY[K«KRYreTK«bV S Qb T CX ` X\\Q QP] Y[wS SLDEQ ¶ ] YrY hi VWSfFQ Y[e> KNQ] QPe GS Yr JL# fYrY[hfQ KR] QgWY[lORJ QkbORORK eqT«YrY[QOGS # lYrhLYrKNKNQeSUhiTO ? gWY¬l XZY[] KRw
(13) SfrYrnºK aejQ? KNh KR* QI g b e M]SL] QM ` V b hfYrK«VgWUhLe ]] Y a e U®h hLQPO ª Y ] b ,T ¶ e ] QKRT b ª e ]] ÀrGS b KRejSLSLQ BºY a feQPK UlYrK a BºYrS D VWK b[ QS K b Q +a * \hfe b SUT ab SUTRKNe hfM MSfMKNQhfORQ%QSUT /]_^ Yr eqYrS a QORw%JfYZTNLejUhLQVLh a QPSUTRKNQoQTUhLe b SUTKNQSfVWh b SORM b hfK %fb SUTNY[eSLQ L] QPYrhejS b h f] eqY L] Q[. H. .
(14) .
(15) xPy# arb xkz
(16) b ^ . Y %JLKNQPONO e SV ejY QLVWM%fSLeTRe b SVWhTNY[h VWQ ab %JLKNQPONO e b S I dQPOm%MTN b VWQPO a] Y[ONO eqUhLQPO¡Yrh= %MTR b VLQPO b KRejQSUTRMPQPO b[ QT4 I I H lMTR b VWQkO b KNeQPSUTRMQkO b[ QTo I I I I o I I I I H ) n ] YrSVWQ a QT TNQITR=kO Q!I o I I I I o I I I I H 1 t !r ªyZz" xkz b
(17) t# $b % xPz'& )b (*f b[y]j",b[+ -&-. z/0] t W KNY[SfOE KRYrTRe SfO¡l KRJL eqUhLQPO¿M M%QPS\T«YrejKRQkO I I I I WW #"# Yu^ 2 dK b e OR] Yre b TNSYZTNe b S I oo II II II II oo II II II II W H # Q ¶ K«Y[VLeQPS\Tl b KRJf b[]jb[ ejUhLQ0I I o I I I I 3 WW #"# Y^ ¾d hLQX\KNQ%KRTRQhLTRKNhfQ%KRQ!I oo II II II II oo II II II II 43 W ¯>SUTRQPKRJfKRMTNYZTNe b S VWQ ] ^ b hLX\QKRTRhLKNQdQTVWQ ] Y QPKR%QTRhLKNQ I I 4 W 53 # QPO a fY[JiQkYrh iYrhWTVWQ b KR%Q 6¼T b JD´fYZT 9 I I I I 4 # W QkO¡TRK«YrSiOE b KRYZTNe b SfO M b VWMkO eqUhLQPOI I I o I I I I W W # YuVLe ] YZT«YZTRe b S M b VWMPORejUhLQ/I I I o I I I I WW WW H #"# Y^ 2 K b Q ORae b b SfS O TR KNM h b a VWTNMke bO eqS UhLQ II II II II oo II II II II !!66 W H e ] TRKNQPO ab SfSLQ WQPOdI I o I I I I o I I I I !6 WW HH ee ]] TNTNKRKRQkQkO¡OmJfYrKNYrM Kmb[KN] YrQ aejKNb QPSfO O TRKNoh a TNIe b S I II II oo II II II II \ W # Y ] e SLQVWQIJfY[K T«Y QIVWQkOmQPY[h I I I o I I I I H W # Y%VWwSfY[%ejUhLQ% I I o I I I I o I I I I 7 H98#
(18) 8 yZv:! _& z x 8 ) ; y<_& t $bW a 8 /0 t
(19) b [ t# ] # = b b j ] ] de MKNQSUTRQkOYrJfJLK LQkOJ hLK¡TRK«YreTRQPK Y%VWej%QSfORe STRQ%J KNQ Q I 4 HH [ ?>@bw[a ª I o b I I]] I I o I I I I k4 JLJLK fQ KNejVWej%QSfORe SLSLQ QI I o I I I I bwa a H 6 HH H ## QkO H VWb e ! MPKRJLQPJLS Ka QkO¡fQQoSUTRKNKNM Q hLVWK«QO h=ejX[ QJ ] YrISfO ORIh aPa IQPON ORe |O¡IV ^ hLoSLQ ORIMP UhL QPSIa Q I II H H gUTNQ KRh a TRhLhLKNX[Q QPu MPQPSLSUMT K«Y ] IQI V I^ hf S ORWoOT %IQ VWIQ aIb VL Y QoIKR M oa hL K«ORIe b KNeQPIS\ TNM Ib\ IQ T A B kH zt /0.-( t t
(20) xPvLxPz
(21) : DC v: 8 t* ; yE_& vF_& z.
(22) tG tG" vWy¬xPv:. tF t# twv:HIJ|& v: LK %M f gwej%J ] e a YZTRe b SVWQ ] ^ eY Q/o I I I I o I I I I 4 f ` wTRK«Y a TRe b SVWQIYrK«UhLQhLK«Odo I I I I o I I I I H ¯.
(23) ff H ¿ KMkbO heqOR] ONTNYrYrS TNOa QoVLVLQIQ ORQ % %Y[KNQUSUhLTNQPYrhLTRK«e bOISfoO [ I II II II oo II II II II H
(24) H H f nºf K Wb[ L ] %ORhLQKVWD M QIa fKNMPYrO SUb[TR] e hW]j]TRb e bSLS SfY QdoVW Q I] ^ ejIY QI II oo II II II II HH f W VWQPh SfeX\QPYrh= VWQ KRMkO b\] hLTRe b S I I o I I I I H f nºK b[ L] %Q ] eM %] Y KNY e ] eTRM VWQ ] YVWwSfYr%eq\hfQ o I I I I H f53 b S a] hfO e b SiO I I I o I I I I o I I I I J 1 H t .-( t
(25) xPb vLxPza
(26) b xPyZz z( t
(27) P z'
(28)
(29) t#&'|& t C v: 8 t ; y<|& v
(30) v:( z'/0 t BM W ¯>SUTRK VWh TNe S I I I o I I I I o I I I I 3 W ª KNY[SfO %eqORORe b S VLQ ]_^ eS= b KN%YrTRe b SVWhJfY[ONORM I o I I I I 3 W H ¯>S a] hfORe b SVWQIS b hfX[Q ]j] QPO¡KNM e b SiO I I I o I I I I W b S a] hfO e b SiO I I I o I I I I o I I I I A 1M z ! ;k z
(31) ¯>S ab SwX[MPSLeQPSUTNOmVWQ ] Y%uMTR b VWQ Y\O MPQ O hLK ] Q K«Y[VLeQPS\T I I I 3 hLTRKNQPOm%MTN b VWQPOVWQ O Q %QSUTNYrTRe b S I I o I I I I W ¿K b eqORONYrS a QIVWQIKRM e b SfO u]_^ M a fQ ]j] QVWhJLe WQ ] I I I I W ¿K b eqORONYrS a QIVWQIKRM e b SfO u]_^ M a fQ ]j] QVWQkO ?b SLQPO¡J ] YZTNQPO I I W H hfO e b SVWQ KNM e b SfOo I I I I o I I I I H dejO a hfONORe b S! I I I o I I I I o I I I I H MkO b\] hLTRe b S I o I I I I o I I I I f H nºK b[ f] PuQkOmVWQ hLeTRQkO¿ I I I I o I I I I f H H dhJLe WQ ]3u] Y ?b SLQIJ ] YZTRQ I I I o I I I I f H gwM ] Q a TNe b SVWQ Y[KNUhLQPhLKNO I I I o I I I I H ¶ MPSLMK«Y ] ejONYZTNe b S VWQ ] YuKRMPMXZY ] hfYrTRe b S VLQ ] Y%VWwSfY[ueqUhLQ; I b S a] hfO e b S I I I o I I I I o I I I I H M ~<&'.%i yZz x!+b ( t9a btP z'
(32) t 8 . z
(33) B w3 ¯>SUTRK VWh TNe S I I I o I I I I o I I I I 3ww3 H dlQPeqOO Q a KNQPeJWS TNe& b ShLXwVWKRQ Q]_^ Y I]j[ b IKReTR L %oQ VWQ8I hfIORe b SIVW Q KRM I e b oSfO II II II II \ w3 H QPJLKNMPORQSUTNYrTRe b S VLQ ]_^ eS= b KN%YrTRe b S I o I I I I \ 3ww3 HH H ¶¶ QkQkOOTNTNee bb SSVWVWQ Q YY aa KK bb D´D>KRYrM K TRe b QkSfO O I II II oo II II II II 43 w3 H ] Y[ONO QP%QSUTVWwSfYr%eq\hfQIVLQPOmYrK TNQPO¡JfY[K b K«VWKNQoVWQIXZY ] hfYrTRe b S \ w H n"ORQhiV b D ab VWQ VWQ ] ^ Y ]\b KNe TNL%Q[ I o I I I I D3 3 ` w3 WQ%J ] Q J MPVLY [b[ ejUhLQ o I I I I o I I I I D3 H = H t .-( t
(34) xP] vLa xPz
(35) b " a vWy P ]_z'^ ]j
(36) [b ty 8 . z
(37) b M= L4 JfJ e YrTRe SVWejKRQ TNQIVLQ Y KReTRL%QIVWQ hfORe S hfSLQIeY QIKRMPQ ]j] Q D3A4 4LL4 hfhfOO ee bb SSVWVWQ Q KNKNMM ee bb SfSfO¡O¡ORY¬Y[X[SfQ aOmYY aa TRTRhfhfYY ]] eqeqORONYrYZTRTRee bb SS oo II II II II D 3A4 4 L4 nºKNMTNKNY[e TNQ%QSUT I I o I I I I o I I I I 4 4LL4 WW 2ºº2 ]] ejejuuejejSfSfYrYrTRTRee bb SSVWVWhQPO¡ KNKNM hL e e T b SfIO¡ VWQIITR K«YrISi O e TRe Ib S! oo II II II II 44 H H # ab b 4L H L4L4 ¿ KNHW e T KNQPOm ¿VWejKNONeQ ORT Y hfKNQ QIORe VWbVWQSQPO a b SUISUTN KNT Y\ OhLTNoK«Q O III III III III ooo III III III III 44#4#33 L4 H ¿KNe T KNQ VWQ ab SUTNKNY\OTNQ ]jbWa Y ] I I o I I I I J6 ¯R¯.
(38) L4 H H dejO a hiORORe b S I o I I I I o I I I I f L H ¿ KNe T KNQ VWQoTRQ wTNhLKRQ I I I I o I I I I f 4 4L b S a] hfO e b S I I I o I I I I o I I I I ;6 H t .-( t
(39) xPvLxPz
(40) t 8 /0 t
(41) [t# $jz( v:. t# " vWy Pz'
(42) ty 8 . z
(43) B ]j[b KReTRfuQ VWQ hfORe b S VWQKNM e b SfOPeSUTNK b VWh a TRe b S VWQ%Y[KNUhLQPhLK«OIQT TRK«YreTRQP%QSUTVWQIS b hfX[Q ]j] QPO¡KNM e b SiODI I I o I I I I ;6 nºKNMTNKNY[e TNQ%QSUT I I o I I I I o I I I I ;6 3 W e ] TNKNY Q YrKNM b\] Y[eKNQ o I I I I o I I I I ;6 3 W · TRe ] eqORYrTRe b S%VWQ ]_^ ejS b KNYZTRe b S%VWQ¡ b hLX\Q%QSUTpJ b hLKºY[uM ] e b KNQK ] Y ab SUTNeSwhLeTRMITNQ%J b KRQ ]] Q I I I o I I I I ;6 4 W H `º] ejuejSfYrTRe b SVWQPO¡KNM e b SfO¡VWQITRK«YrSiO eTRe b S!o I I I I [ H H ¿KNe T KNQPOmVWQ hfORe b S I o I I I I o I I I I [ H ¿KNe T KNQ VWQ ab SUTNKNY\OTNQ I I I I o I I I I [ H ¿KNe T KNQ VWQoTRQ wTNhLKRQ I I I I o I I I I [ H H ¿KNe T KNQ VWQI b hLX[QP%QSUT! I I I o I I I I [ 3 b S a] hfO e b S I I I o I I I I o I I I I
(44) xPyr xkz
(45) tF& jv:&.-i y¬z|x;+( t t k z
(46) t y 8 i. z'
(47) " vWy ( % itJK (;6 t
(48) x b v
(49) a |& v b C -J|& t ] t v:. t b : 1 l VWe YrTRe SfOmVLQ Y%ORQ %QPS\T«YZTNe S9 I I o I I I I ;6 ¶ MPSLMK«YZTNe b SVWQ ] YO Q %QSUT«YZTRe b S fSLQ! o I I I I ;6 ¶ MPSLMK«YZTNe b SVWQ ] YO Q %QSUT«YZTRe b S K b ONO e KNQDI I I I 3 ;6 l b VWe a YrTRe b SfOmVLh ab VLY QIVLQ ab SUT b hLKmI I o I I I I H ;6 H l b VWe a YrTRe b SVWQ ] Y ab %J QSfONYZTNe b SVWQ b hLX[QPuQPSUT%I I I I H
(50) H ;6 MkO h ] TNYrTNO; I I I o I I I I o I I I I H 3 ;6 i gwMkUhLQS a Q fb KNQYrS BI I I I o I I I I H ;6 i gwMkUhLQS a Q cQ%@O Do I I I I o I I I I ;6 i H gwMkUhLQS a Q Y[KRJL b SLQ
(51) I I I I o I I I I # ;6 b S a] hfO e b S I I I o I I I I o I I I I P 4 - [
(52) r xPz'
(53) b
(54) v:|&a z|x 8b ¯>SUTRK VWh TNe S I I I o I I I I o I I I I P\ [ gwhLejXwe3V ^ b[ QTNO I I o I I I I o I I I I 3 6 [ L ¶ MPSLMK«YZTNe b SVWhY[ONUhLQ I I I o I I I I 3 6 [ L gwhfeXwe3VWhY[ONUhLQ o I I I I o I I I I 3 [ L H hfO e b SiO¡JfYrKm b hfX[Q%QPS\TmQT b S a TRe b SLSfY ] eTRM I I I I 3 [ H MkO h ] TNYrTNO; I I I o I I I I o I I I I 3 [ b S a] hfO e b S I I I o I I I I o I I I I 3 :
(55) &k z
(56) a :M 1 L n QK«O J Q TRejX[QkOI I I o I I I I o I I I I 3J3. ¯ ¯R¯.
(57) . #Q. VWMX\Q ]jb JLJ Q%QPS\T V ^ Y[JLJ ] e a YZTNe b SfOTNQ ]j] QkO UhLQ ] Y+XweqVWM b TRM ] MPJL b SfeQ
(58) ] Y+XUeqVWM bwab SD MKNTRQK S POa Q bY[hOIVW] Q M eTPh ] TR# ej%QPOIQPVWO eqWY O Tb PSUuT Qka OIKNMVWM Q ] aYb %SfM JLa KNQkQPORONOROeTRe bMS VWQV ^ aejb VWY QP KoQ VWb QkKNOejQORSUMPTRUMkhLO QPb\S a QkQOT,V Yr^ eJLJ Y Q ] QPMPOO YrhiORORewuMTR b VWQkO VWQVLQh We %Q MSfMK«YZTRe b SG b! KNQSUT hLSLQ¡Y ] TRQPKRSfYrTRejX[Q¿JLK b %QT TNQhfORQ¡Yrh %MTN b VWQPO a] Y\OROReq\hfQPOP5 ¿QkOI%MTR b VLQPOORQ%KNY[JLJLK bwa fQSUTIVLh ab %J b KRTRQPuQPSUT VLQ ] ^ & e ] whLYrej S pVWQPO TRejSfYrTNYrejKN Q fSfY ] VWQPO O WO T PuQkOVWQ ab VLY QQT VLM a KNejX[QSUThLSfQej%Y QQPS ^ [ b TRQPKR%QkO¿V QTNO¿JLKNMPORQSUTNOVLY[SfO ] YO a PSLQ[peSfORe UhfS ORwO TP%QdVWQ ab VLY Q b KRejQSUTRM b[ D QT b J KNQ QSVWQPh MTNY[JiQkO¿hLSLQJLKNQ% e KNQIMTNY[JiQ V ^ Y[SfY ] WO
(59) Q fYrJfJiQ ] MPQO Q %QSUTNYrTRe b S UhLemVWM ab %J b ORQ ] ^ ejY Q QS b\ QT«7 O 6 QSiO QP L] QPOuVWQJLe WQ ] O X[MPKRe iYrSUTuhLSLQ ab SiVWe TNe b S ^V b b\ MSfMeTR,M 9 QT hfSLQ ORQ ab SfVWQMTNY[JiQUhLe¿ b VWM ] eqO Q ] Y b KNuQQT ] Q ab SUTRQPSwh VWQPO b[ QTNOmORQ %QSUTRMkO # Q¿TRK«Y¬X¬Y[e ] VW]_^Q b[ a QT TNa Q¿TN=PORQ¡] QPO T ab SfONY]_^ a KRM ] YoO Q %QSUTNYrTRe b SVLQ¡ORMPUhLQPaS b a QkO V ^ ejbY QkO% a ONY fY[S\To\hfQ Q TNe iSfY QPO ToVWQ ejS\TNM KRQPKdVLY[SfOhLS O WO T %QuVWQ VfY Q KRejQSUTNM b[ QTP. c b hiO Y¬X b SfO ab %%QS a M S b TRKNQTNKNY¬XZYre ] QS+S b hfO b SfVLY[SUTORhLK ] Q JiYrK«Y[VWe %Q VWQ ORQ %QPS\T«YZTNe b SVWQ ] Y b KNJL b\]b\ ejQmYZTRfMYZTNejUhLQ\ ] Y ] e SLQdVLQJfY[K T«Y QdVWQPO¿QPY[h %VLh K«Y[VLeQPS\T JfYrKRTRejKdVLQPOY[KNUhLQPhLKNOdVWQPOKRM e b SfOdUhLQ ] ^ b SX[QPhWTQ wTNKNY[eKNQ[8 ¿Q O a LMPY YV b SLSLM ] ejQh VWQS b KNQhfORQPOYrJLJ ] e a YZTNe b SfOY[ejOpVLYrSfO ] QV b Y[eSLQ VWh ab VLY Q
(60) a Q ] Y ab SfVWhLeT VWQkOIVWe a h ] TNMPOoKNMQ ]j] QPOP ¿QPOIVWe a h ] TRMPO O b SUTIVWhLQPO ] Y JLKNMPORQS a Q VWQ VWMTNYre ] O iSf%O b h a Q%UhLe KRQPXweQPS\TIYrh ' P%Q ] Y JLKNMPORQS a Q VWQu b hLX\Q%QSUTNOdK«YrJLeqVWQPO UhLe MS KNQSUTdVWQkO b[ QTNO fSfOdVLYrSiO ] ^ "Y WQ TRQPuJ b KNQ ] ` S Q ! Q,T ?VfYrSfO ] QuV b YrejSLQVLh ab VLY
(61) Q ] QPO ab SfVLe TNe b SfOV ^ M a fY[S\TNe ]j]jb SLSfY Q b KNJL b\]b\ eq\hfQ SLQO b SUTJfY\OX[MPKRe fMPQPOP #º^ YrSiY ] wORQVWb QkO VL e a h a ] TNMPO KRQPb S ab a S\TNKRMPQPO] S b hfOub Y ab SfVWhf e Tb hLS YrhWTNKRQ JfYrK«Y[VWe %QVWQ ORQ %QPS\T«YZTNe S Y[ORM QTRTRQ ejO D e O hfK Y hfORe SVWQ KRM e SfOP c b hiOY¬X b SfOY[eSfORe VWMX\Q ]jb JLJ MhLS Y ]\b KNe TNL%QVWQ3 hiO e b S VLQKRM e b SfOPm¯>%J ] M%QSUTRM ORhLK hLSLQºO TRKNh a TRhLKNQºVWQ KNY[JLLQpV b SUT ] QPO S & hfVLO QT ] QkO Yr K TNQPO O b SUT MXZY ] hLMk%O ] ^ Y ]\b KNe TNL%Q QPO T ] Y b ejOQ a Y a Q QTIO h ONYr%%QSUT
(62) iQ We f] QJ b hfKJLKNQSfVLKRQ QPS ab %JWTNQhLSLQ ] Y[K Q XZYrKNeMTRMuVWQ a KReT PKRQkO ` S JiYrKRTRe a h ] eQPK ?VLY[SfO ] Q hWT VWQ KNQSiVWKRQ ] Q%O WO T PuQuVWQ ab VLY Q []jb[ Y ] Q%QSUT ab LMKNQSUTWS b hfOY¬X b SfOJLKNeqO¡QS ab uJLTRQVLY[SfO ] QPO a KReTPKRQkOmVWQ hfORe b S ] QPO a Y[JfY a e TNMPOmVWQkOmY ]j[b KReTRL%QkOVLQ ab VLY QoY[JLJ ] ejUhLMkOmYrJL K PO ] YO Q %QSUT«YZTRe b S3 ` SfS S b hfOoY¬X b SfOme ]] hfO TRKNM ] Q J b TNQSUTRejQ ] Uh ^ b ! KNQ ] YORQ %QSUTNYrTRe b S hfS ORwO TP%Q VWQ ab VLY Q[ ` SQ ! Q,T ] Y O Q %QSUTNYrTRe b SejSUTRK b VWhLeT ] YS b TRe b S V ^ b\ QTmVLY[SfO ] Q ab VWQPhLKP ejSfORe ] Q%VLMX[Q ]b JLJ Q%QSUTVWQ b S a TRe b SfSfY ] e TNMPO TRQ ]] QPO\hfQ ] QORhLeXweºV ^ b[ QTNO3VWQPXweQPS\T.
(63) hLSLQoQ wTNQSfORe b SSfYZTNhLKNQ ]j] QoVWQPOY ]j[b KReTRL%QkO"VWQ ab VLY Q[ ¿QPO4 b S VLY[SfO ] Q a Y\VWKNQºVWQPO3S b hLX\QPY[h b\ Q a TRe |O)= WMPO3JfYrK ] YmS b KNuQºVLQ ab. a TRe b SLSiY ] e TNMPO¿KRQPS\TNKRQPSUT VLY QpXweqVWM b ln ` ¶ f. .
(64)
(65) ? "!$# %'&)(*(,+- . / < +!01&23/54768.5+:9*+- . / ; 9*;=< /5&> "!$# %'&)(*(,+- . M a Q%%QSUT ] Y ab %JLKRQkORORe b SdV ^ ejY Q
(66) PQT QS JfY[K TNe a h ] eQPK ] Y ab %JLKNQPONO e b SIVWQºORMPUhLQS a QPO V ^ eY QPOQkOTVWQX\QSwhLQ hLSORhZQT VLQ K«YrSfVoejS\TNMK T K a Q Yrh VWMX\Q ]jb JLJ Q%QPS\T3V ^ Y[JLJ ] e a YZTRe b SiO TRQ ]] QPO UhLQ ] Q XwejVLM b TNM ] MJL b SLQ
(67) ] Y+XUeqVWM bwab S MPKRQPS a Q
(68) QT ] QkO h ] TNe%QkVWejY+J b hLK SLQ a eTRQPKUhLQ ] QkO%J ] hiO%e%J b KRTNYrSUTNQPOP # QkO%eY QPO ab SUTRejQSLSfQSUThLSLQMS b KR%QUhfYrSUTNe TNM V ^ eS= b KN%YrTRe b S a QUhLemKNQSfV ab@? TRQPh ] QhLKO T bWa+* Y Q b h ] QhLK%TNKNY[SfORueqONO e b S O eQ ]] QPO SLQO b SUTJfY\O%JLKRMkY ] Y L] Q%QPS\T ab %JLKNQPONO MPQPOP npY[KQW Q%J ] Q J b hLKTNKNY[SfORuQT TNKRQhLSLQ ORMPUhLQS a QVWQ [IejY QkOpJfY[K¿ORQ ab SfVWQVWQ [ × \ JLe WQ ] O"Y¬X\Q a [ SLejX[QPY[h uVWQ KNejO 6 4 eTNOºJfYrKpJLe W Q ] 9 QU W ORJ Y[]jSfb OeTR]QPY K a] b QP%OJL KRQkb OR[ORQPQSfK,OZ e VW] Q| YrhfTRK«VWYrK«SiYrOeTp%hLeqORSOReVWb M S e T"UVWhLQ e O Hb SUl T JLe T«wOBWORdA ejUO QhLaQPb uSfQPVWSUQ[T i S!VWQ %eQPh+ ] e%eTRMPO b S QkOT b[ f] e M%VWQuKRQ a LQK a LQKIVWQkOIKRQkV b SfVfYrS a QPOoVWQ%%Y[SLe KNQ TRK«YrSfOR%QTRTRKNQ ] ^ ejS b KNYZTNe b S Y¬X\Q a hLS S b KNQVWQ eTNOJ ] hfO | Y[e L] Q[ # Q hWT VWQ ] Y ab %JLKRQkORORe b S V ^ eY QQkOT%VWQKNMPVLhLeKNQY[hWTNY[S\T%UhLQJ b ONO e L] Q ] QS b KNQ VLQ eTNO SLM a QPONORY[eKNQJ b hLK KNQ ab SfO TRKNhLeKNQ hLSLQ ab JLejQ3i V ] QuVWQ ] ^ eY Q b KRe ejSfY ] Q[ # YKNQJfKRMkO QPS\T«YZTNe b S a YrS b SLeq\hfQ V ^ hLSLQeY Q¿VLe e T«Y ] Q¿KNQPUhLejQKRTDCFE G × H × I e T«O% b@J N ×M O b SUT ] QPO VWej%QSfORe b SfO VWQ ] ^ ejY QoQT ] QIS b KNQoVLQ eTNO¡J b hfKmKRQPJLKRMkO QPSUTRQK¿hLSJ b eSUT ´D ej%Y Q6 hLSM a iYrSUTRe ] D ]jb SV ^ eY QO QPbKNY%Y[JLJiQ ] M JfeW Q ] \hfe QPO ThLSLQY KNMXwejYrTRe b SVWhTRQPKR%QYrS [] YreqO JLe a TNhLKNQ Q ] Q%QPS\T E9¡gWe b SQkOT a YrJfY L] QVLQ KRQ ab SiOTNKRhLejKNQohfSLQ ab JLejQVWQ ]_^ ej%Y Q JfY[K TNeKdVWQ eTNO b SVWejKNY%UhLQ ] QoTNYrh= VWQ ab %JLKRQkORORe b S b\ TRQPSwh QkOTVWQ T =. B C. n b hLKdhLS ORWOT %QVWQ ab %JLKNQPONORe b SV b SLSLM
(69) iJ ] hfO ] Q TNY[h VWQ ab %JLKRQkORORe b SQPO TdM ] QPX[M J ] hfO ]_^ ej%Y Q ab VWMQ O ^ M ]jb e SLQ VWQ ] ^ eY Q b KRe ejSfY ] Q[ gwQ ]jb S ] QkO Y[JLJ ] e a YZTRe b SiO% ] QPO TNY[h VLQ ab uJfKRQkORORe b S KNQ a fQK a LMPOmO b SUTVWe ! MPKRQPSUTNOP · SLQJLKNQ%e KNQ a] Y\ORORe a YrTRe b SVWQPO %MTN b VWQPOIVWQ ab VLY QJ QhWTTRKNQQSwXweqORY MQ JfY[K TNeKoVWQPO b\ Q a TRe |OoUhLQ ]_^ b S O Q = WQ[ %¿QQKRSUTNT Y[e]_Sf^ ejQP%OdY u Q MTRb KNbe VWeSfQkO Y ] b 8Q SU6 Td%MMTNTNMu bVLVWMfQPOpSLejONQPYrOdSiJO3b J hLQKKRTRQkTNO KR9Q #a Y[QPJfO Y TN LY[]hQP OdVWVWQ QuaKRb Q uab JfSiKROQkTNORKRORhLe bejKNSQ YZQTR LTRYQeja SUTRTNQOD SLQuVWMJfY\ORORQSUTJfY\O b h H / ¿QT TNQ ab SUTRK«YrejS\TNQ QPO ToO b hLX\QSUTmTRK b JO TRKNe a TRQ\ # QTNYrh= VWQ ab %JLKRQkORORe b S V b SUT b S Y QPO b ejS/VLYrSfO ] YJ ] hLJfY[K T VWQkOoY[JLJ ] e a YZTNe b SfO ab S a QKNSfY[S\TIVWQPO ej%Y Qk%O rQkOTO hLJ MKNejQhLK"V ^ Y[h b eSiOphLS b K«VWKNQVWQ K«YrSiVWQhLK a Q ] hLefUh ^ b ShLTRe ] eqO QdVLYrSfO ] Y ab uJfKRQkORORe b SVWQdV b SLSLMPQPO [YreqO"QPS %QTNQ%JfO¿e ] JiQPKR%QT"hLSLQdJiQTReTRQIVWejO T b KNORe b S. . .
(70) . WV Q ] ^ eS b KRYrTRe b SVLYrSiO ] YuQkO hfKRQ b@Ju] ^ & e ] Uhf%Y[eSSfQ ] YIJ QK«½ b e TºJiY[OP #"^ b\ Q a TRe QkOT VWQ%KNQPO TReTRhLQPK hLSLQ ab JLeQVWQ ] ^ b KNe eSfY ] Y¬X\Q a hLSLQ a QKRTNYrejSLQ%VWeqOT b K«O e b S Y[ejOIXweqO hLQ ]] QD %QSUTJLK bWa LQVWQ ] ^ ejY QVWQVLMJfY[K Tk¾dS/JiQPhWT b[ TRQSfeKVLQPO ejY QkOIQSYrJfJfYrKNQS a Q eqVWQSUTReqUhLQPO ] ^ b KRe ejSfY ] Y¬X\Q a hLS TNYrh= VWQ ab uJfKRQkORORe b S V ^ QSwXwejK b S ucMPY[SL b ejSfO% a QPK T«YrejSLQPOmY[JLJ ] e a YZTNe b SfO¡KRQkUhLe PKRQPSUT¡hfS T«Yrh VWQ ab %JLKRQkORORe b SJ ] hfOM ] QX\M[pn b hLK a Q a e e ] QkOTuSLM a QPONORY[eKNQV ^ Q wTRK«YrejKRQ ]_^ e=S b KN%YrTRe b S UhLeS b hiOJ QKNuQT TNKNY VWQKNM a hLJiMPKRQPKhLSLQ ej%Y Q ORe%J ] e iMQ YreqO¡KRQPJLKNMPORQSUTNYrTRejX[QdVLQ ]_^ eY Q b KNe eSiY ] Q[. ;6. . &)( !04 9 / &)($ -(*(,+ ; &)( ;=< !04 9 5 / &)( % + &). 9*4)&)( & 9 # QkOIJLKN Q%b eQPKNOIJiY[OIQSa ab %JLKNQPONO e babS V ^ ejb Y QuhLTRe ] eqO QPSUT]b ] Y TRfM b KNejQ%VLQ ] ]_^ eS= b KN%YrTRbe b S dQh K hLJ QPOVWQ"TNQ LSfejUhLQkOVLQ VfY Q SUT MTRMmVWMPX[Q JLJiMkO VLQJLhLeqO QPO YrSLSfMQPO O e D YrSUTNQdVLY[SfO a Q a Y[VWKNQ[ ] QPO¿TRQ a LSLeq\hfQPOO JfYrTReqY ] QPOU UhLe TNKNY¬XZYre ]j] QPS\T¡VWejKRQ a TRQP%QSUTY¬X\Q a] QPO XZY ] QhLK«O VWQkOM a iYrSUTRe ]]jb SiO VWQ ]_^ ej%Y QQT ] QkO %MTR b VLQPO TNKNY[SfO b KNuMPQPO3 UhLeºhWTNe ] eqO QPS\T ] YKNQJLKNMPORQSUT«YZTRe b S O J Q a TRK«Y ] QuVWQ ] ^ ejY QuQT ] Y ORejuJ ] e fQSUTk ¿QPOo%MTN b VWQPO3VWM ab D J b ONYrSUT ]_^ ej%Y QQPS L]jbwa OKNM h ] ejQK«O%QTQW J ]jb eTNY[S\T ] Y ab KNKRM ] YrTRe b S ]_^ ejS\TNMKNeQPhLKVWQ a fY\\hfQ L]bWa \ VLY[SfO ] QV b YrejSLQORJfYZTNejY ]Lb hORJiQ a TNKNY ] b SUTºJfYrKRTRejQVWQ a QUh ^ b SYrJLJ Q ]j] Q ² N§»u¥ N§ #U¢¨?¢ R±r©_¥|³r¨!¦[§££ ¬£«© »%§£¦\§ «³Z»² R§£«£¤¥|³Z¨ ¦ ¥¼»%± #\§ # QPO L]jbwa O%hWTRe ] ejORMPO J b hLK¿QW J ]jb eTRQK ] Y ab KNKRM ] YrTRe b S%VWQ ] ^ ejY QSLQO b SUTºJfY[O ] ejMPO O b S ab S\TNQSwh # QkO TRK«YrSD ORe TNe b SfOQPS\TNKRQ L]bWa OVWQPXUejQSfSLQSUTXweqO e L] QPOUhfY[SfV b SQkORONY¬[QV ^ YrT TRQPeSiVWKRQ VWQPOTNYrh= VWQ ab %JLKRQkORORe b SM ] QPX[MkO # QkOuMTR b VWQkO b KNaeb QPSUTRMQkO b\ QT,L ] YrJLJ Q ] MQkOmYrhfONORe %MTR b VWb QkOmVWQ¦[§µ ¹ ¥ »bW§a #U¢¨?¢ R±&! © ¥|³Z¨ Y[JLJfY[KNY[ejONO QPSUT %%QhLSLQ Y TNQKNSfYZTNeX\Q[ ¿ QkO%MTN VWQPO O QK«YrJfJLK LQPSUTVLh ab %J b K TNQ%QSUTVWQ ]_^ & e ] whLY[eS VWQPO TRejSfYrTNYrejKNQ fSiY ] VWQPOORWOT %QPO%VWQ ab VfY Q
(71) ºQT VWM a KNeX\QSUT hLSfQ ej%Y Q QS+TNQKN%QPOV ^ b\ QT«O "# ^ ejY Q S ^ QkOTJ ] hfOVWM ab %J b ORMQQPS L]jbWa OmYrK eTRK«YrejKRQkO¿VLh J b ejSUTmVLQ XUhfQIVLQIO b S ab S\TNQSwh W YreqO¡QS b[ QTNO ab KNKRQkO J b SiVLYrSUT VWQkOQPS\TNe TNMPO JLKNMPORQSUTRQkOVLYrSiO ] Y O a SLQ\ #º^ eSUTNMK TVLh ab VLY Q b KNejQSUTRM b\ QTQkOT V ^ b[ TNQSLejK¿VLQPOºejY QPO"ORejuJ ] e f MQkOºYreqO ab SUTRQPSfYrSUTºhLSfQejS b KNYZTRe b S ] ejMQY[h ab SUTNQSwh VWQ ] YO a SLQ\¾dS/J QhWTQkO J MKNQKUh ^ hLSLQY[SfY ] WO QVWQ ]_^ eY Q TRK«YreTRQPK b JWTNe%eqO Q ] QPO J QK b KN%Y[S a QkO¡VWh ab VWQhfKP # QkO%MTR b VWQkO VWQ ] YJLKNQ%e KNQ MPSLMK«YZTNe b S ab VWQPS\TORe%J ] Q%QSUT ] Q ab S\TNQSwh VWQ a fY\\hfQ L]jbwa 6 ] YJ b OReTRe b SQT ] Y b KN%Q VWQkO f]bWa OdQPO T ab SLSwhLQuYJLKNe b KNe 9 JiYrK ab SUTRKNQ ] QPO¿uMTR b VWQkO b KRejQSUTRMPQPO b[ QT¿V b ejX[QPSUT ab VWQPK ] YJ b ORe TNe b S ] Y b KR%QQT ] Q ab SUTRQSwhVWQ a fY\\hfQ b[ QTPdgWe hLSfQej%Y QQkOToKRQPJLKNMPORQSUTRMPQ QS VWMTNY[e ] QPY[h ab hLJV ^ eS= b KN%YrTRe b SfO X b SUT5 TRKNQSLM a QkORONYrejKRQkOJ b hLK ab VWQK ] QkO ab SUT b hfKNOVLQPO b\ QT«O% a Q UhLeIVWMPONY¬XZYrSUTNY Q ] QPO%MTR b VWQkO b KNejQSUTRMPQPO b\ QTJiYrKK«YrJfJ b K TY[h %MTR b VWQkO a] Y[ONORejUhLQkO¡npYrK ab SUTRKNQ ] QPO¡%MTN b VWQPO b KRejQSUTRMPQPO b[ QTV b SLSfQSUTVWQIuQPe ]j] QhLK«OKRMkO h ] TNYrTNO ]jb K«ORUhLQ ] Y ab %JLKNQPO D ORe b SQPO TTNKk O b KRTRQ\mQ ]j] QkOJ QKN%QT TNQSUTIhLSLQ KNY[SfVWQORejuJ ] e a YrTRe b S VLQ ] ^ ejY QQSSLQ KNQTRQPSfYrSUT%VWQ ] ^ ejY QUhLQ ] QPOTRK«YreTNO M b %MTNKReqUhLQPO ORY[e ]j] Y[SUTNO Y ]b KNO UhLQ ] QPO fYrhLTRQPO KNMPU# hLQkQOpS a%QPMOmTNY[ hb L VWUQPhLO Q b ]jKR] QkejQO SU]_TN^ M& QPe O ] b[U hf%QTºY[JLeSKRMkQPO O QPTSUTRQbSUejT"SfY[OmhfORORQORSfeLORhLe fS%] Q ejSUO TRb MSUK TmTSLVWM h\] J e b MPeQPSUOPT" VWQXUhfQVWQPO YrJfJ ] e a YrTRe b SfO ] YK b[ b TNejUhLQ\ # YVWM ab %J b ORe TNe b SVWQ ]_^ eY QQPS b[ QTNO¿QPO T¡hLSJ b ejS\T VWQVWMPJfYrKRTJ b hLKORY ab %JLKRMPLQSiO e b S a QUhLe b hLXwKNQ ] QPO%J b KRTRQkO VLQPOYrJfJ ] e a YrTRe b SfO V ^ YrhWT b YZTNejOR%Q[ `º]j] Q b! KNQY[hfONO e¡hLS K«YrSiV J b TRQPSUTRejQ ] J b hLKVWMPX[Q ]b JLJiQPK ] QkO%S b hD X[Q ]] QPO b S a TRe b SLSfY ] eTRMkOºVWQln ` ¶ D rTNQ ]j] QkO"UhLQ ] Y YrSLejJLh ] YZTNe b SV ^ b\ QT«O% ] Q ab VLY Q ORM ] Q a TNe 6 ab VLQK Y¬X[Q a J ] hfO VWQuJLKRM a eqO e b S ] QuJLKNQ%ejQKIJ ] YrS JfY[KdK«YrJfJ b K T Yrh b SfV 9 ] Q ORhLeXwe V ^ b[ QTNOojj # YuOTNKRh a TRhfKRQIV ^ hLS ORWOT %QoVLQ ab VLY Q b KNejQSUTRM b[ QTQkOTmhLS ib S 5. . . . . . .
(72). . . .
(73) H ORhLJLJ b KRT¡J b hfKVWMX\Q ]jb JLJ QKmVWQkOYrJfJ ] e a YrTRe b SfO a b \S TNQSUT Y[ORQPV. . 4 9 /5&)( % +-&). 9*4)&)( & 9. (. . .
(74). # QkOouMTR b VWa QkO b KNeQPSUaTRb MQkO b[ ] QTIVWb M a KNeb X\QSUTI] hLSLb Q%eY ] QuQaS b TRQbKN%QPOIV ^ b[ ] QaTNb OoJLKNMPORQSUTN] O VLY[SfOIhLSLQO SfQuQT VWQSUT Y J OReTRe S Y KN%Q 6 QkO S\T hLK«O+9mQT Q S\TNQSwh(6 Y TRQ%w TRhLKNQ,u 9 VWQ a fY a hLS VWQPO b[ QTNO%VWMTRQ a TRMkO #º^ b[ Q a TNedS ^ QkOTJfY\O%VWQKRQkOTNe TNhLQK%hLSLQ ej%Y QuejVLQSUTReqUhLQ ] ^ b KRe ejSfY ] ?YreqOoV ^ Q% UTNKNY[eKNQ ] ^ eS b KRYrTRe b SJ QKRTRejSLQSUTNQ%VLM a KNejX¬Y[SUT ] YuO a P SLQ[ # QPO¡JLK b JfKRejMTNMPO M b %MTNKReqUhLQPO¡VWQkO b[ QTNOmO b SUTmJLKRMkO QPKRX\MQkO% a QUhLe M ] ej%eSLQ ] QPOQ ! QT«OmVWQ L]bWa V ^ hLSLQ YrJLJfK bWa LQ a] Y[ONO eqUhLQ[ · SORwO TP%QoVWQ ab VfY Q b KNeQPSUTRM b[ QT b J KNQIQS H MT«YrJ QPOP # YJLKNQ%e P KRQpMTNY[JiQpQkOT ] YmORQ %QPS\T«YZTNe b S a fY[K MQ V ^ ejVWQPSUTRe f QK ] QPO b[ QTNO JfKRMkO QPS\T«O VLYrSfO ] YO a SLQ\ · S b\ QTdQPO ToVWM%fSLe ab %%QhLS QSfORQ L] Q VWQJLe WQ ] O 6|hLSLQKNMD e b S19 X[MKNe i YrSUTdhfSLQ ab SfVLe TNe b S V ^ b b\ MSfMeTRM%¾dS ] hfeºY[ONO e SLQhLSLQ%MTNejUhLQT TRQ 6V b ^ ejh ]Y Y QuQ ] OR%9 Q'\ bwhfaeY O ] QPejORK QT SUT OR] hL^ eqK VWQhLSUS TNefa fQPKPYr%J/# Y V ^ JYr]JLhLJ Jf] eYra KRYZTTNe VWb QPS OJfYrY[JLK J TNe] ae a h YZ] TNeQPe b KSf Oa QV \^ YrhfSfeºY Ji] WQPORK Q D uQT¿V ^ Y[VLY[JWTRQPK ] QkO"Y ]\b KNe TNL%QPO a fY\\hfQ a Y[OºJfYrKRTRe a h ] ejQKk3npYrKºQ%W Q%J ] Qw VLYrSfO hLSLQYrJLJ ] e a YZTNe b S %MPVLe a Y ] Q b S J QhWTY¬X b ejK a LQK a LQPKuVWQkOuJfYrKRTRe a h ] QPO ab KNKRQ%D O J b SfVfYrSUT VWQkO a Q ]j] h ] QkO% b h VWQkOoTRhL%QPhLKNO3Y¬X\Q a VLQPO a KNeTKNQPOoVWQuTNY[e ]j] Q3VWQ O b b KNS\uTQ
(75) VWi M%VWf QSLejTRQPQ%OU TNVfhLYrKNSfQ O b ]hQPOVWQORJ aM b a h e ] QPa hLYZKkTRe b # SiQPOO VWa Q YrK«]_Y ^ Yra JfTRMPJ KR] eqe Oa TNYrejTRUe hLb QkS O VW#QkO Q b[a b VLQYTNO Qa LVWQQK XZa Y[LS\MkTO O ^ YrJLJ ] eqUhLQK hLSLQ KNY[SfVWQ XZYrKNejMTRMV ^ ejY QkO% ] Q a KNe T KNQVWQ ORM ] Q a TRe b SV ^ b[ QTNO SLQuV b eTIJiY[OohWTRe ] ejORQKVWQkOoeS b KRYrTRe b SfOIORhLK ]_^ eY Q%YJLKNe b KNe_ "# ^ ejSUTRK b VWh a TNe b S VWQ ] YORQ %QPS\T«YZTNe b S VLYrSiOIhLSLQYrJLJ ] e a YrTRe b S/VWQ ab VLY Q S b hfO b[ f] e Q VWMf SfeK VWQPO a KNe T KNQPO V ^ b b[ MSLMPe TNM MPSLMK«Yrh )mXZY ] Y f] QkOJ b hLKhLSLQ eY Q V ^ QSUTNKRMPQ YrK e TNKNY[eKNQ[ L · SLQ b eqO UhLQ ] QkO b\ QT«O b S\TMTNM VWMTRQ a TRMkO VLY[SfO ]_^ MT«YrJ Q VWQ O Q %QSUTNYrTRe b S ] QPhLKNOoJ b ORe TNe b S QT b KNuQV b eX\QSUT TRKNQ ab VWMPQPOP # Q ab VLY QuVWQ ab SUT b hLKNO ab VLQ ]_^ e=S b KN%YrTRe b SKRQ ] YrTRejX[Q u] YuJfY[K TNe TNe b SVWQ ] ^ QPORJfY a Q[ H ejSfY ] QPuQPSUT ] Q ab VLY QIVWQITRQ% wTRhLKNQ ab VWQ ] Q ab SUTRQPSwhVWQkO b[ QTNOP. . . -. /5&> & 9 9*&9 )(,& # QJLK b J b O VWQ a QTRTRQ TR=k O QQPO T ] QVLMX[Q ]b JLJ Q%QSUTuV ^ hLS Y ]j[b KReTRfuQ VWQORQ %QSUTNY"D TRe b SVWQ ORMPUhLQPS a QkOmV ^ ejY QPOmONY a fY[S\TmUhLQ ]_^ b[ Q a TNe$fSfY ] QPO TVWQ ] ^ ejSUTRM KRQPKmVLYrSfO¡hfS ORwO TP%QIVLQ ab VLY Q b KRejQSUTNM b\ QTk c b hfO%Y¬X b SfO ab %%QS a M S b TRKNQ TRK«Y¬X¬Y[e ] QS S b hfO b SfVLY[SUTORhLK ] Q JfY[KNY\VWe %Q VWQ ORQ %QPS\T«YZTNe b SVWQ ] Y b KNJL b\]b\ ejQmYZTRfMYZTNejUhLQ\ ] Y ] e SLQdVLQJfY[K T«Y QdVWQPO¿QPY[h %VLh K«Y[VLeQPS\T JfYrKRTRejKdVLQPOY[KNUhLQPhLKNOdVWQPOKRM e b SfOdUhLQ ] ^ b SX[QPhWTQw TNKNY[eKNQ[8¿ Q O a LMPY YV b SLSLM ] ejQh VWQS b KNQhfORQPOYrJLJ ] e a YZTNe b SfOY[ejOp VLYrSfO ] QV b Y[eSLQ VWh ab VLY Q
(76) a Q ] Y SLQ VWQPO¡VWe a h ] TRMkO¿KNMQ ]] QPOP # Y O b hLK a QdVWQ a QPOVWe a h ] TNMPO¡QkOT ] Y JLKNMPORQS a QdVWQ VWMTNYre ] O f SiO% b h a Q%\hfe KNQXwejQSUToY[h %Q ] YJfKRMkO QPS a QuVWQ b hfX[Q%QPS\ToKNY[JLeqVWQ UhLe MPS= KNQVWQkO b[ QTNO i SfO"VLY[SfO ] ^ Y"W QmTRQPuJ b KNQ ] ` SQ ! QT,U VLY[SfO ] QV b %Y[eSfQVWh ab VLY Q
(77) ] QPO ab SfVWeTRe b SiOV ^ M a fY[S\TNe ]j]jb SLSfY QI b KNJL b\]b\ eq\hfQdSLQO b SUTJfY\OmX[MPKRe fMPQPOP #º^ YrSfY ] WORQ VWQkOºVWe a h ] TRMPOºKRQPS ab S\TNKRMPQPOS b hfOpY ab SiVWhLeT hfS%Y[hWTRKNQmJfY[KNY\VWe %QVWQORQ %QSUTNYrTRe b S JfY[K hfO e b SVWQ KNM e b SfO . . . .
(78) c b hfOY¬X b SfOYrejSfO eVLMX[Q ]b JLJ M hLS'Y ]\b KNe TNL%Q VWQ hfORe b S0VWQ/KRM e b SfOPD¯>%J ] MD %QSUTRMO hfKhLSLQOTNKRh a TRhfKRQVLQ K«YrJfLQV b SUT ] QkOS & hfVfOQT ] QPOY[KTRQkOO b SUTMXZY ] hLMkO% ] ^ Y ]\b KNe TNL%Q"QPO T ] Y b ejO Q a Y a Q¿QTpO h ONYr%%QSUT fQ We L] QJ b hLKJLKNQSfVWKNQ"QPS ab %JWTRQ hLSLQ ] Y[K Q XZYrKNeMTRM VLQ a KNeTKNQPOPc b TNY[u%QPS\T, VfYrSfO ] Q hWTIVWQKNQSfVLKRQ ] Q ORWOT %QVWQ ab VLY Q []jb[ Y ] QPuQPSUT ab LMKNQSUT"S b hfOuY¬X b SfOJfKReqOQPS ab %JWTRQVLY[SfO ] QkO a KReTPKRQkOVWQ %hfQOReSUb TNS YrTR] e QPb O S a Y[JfY a e TNMPOVLQPOIY ]j[b KReTRL%QkOVWQ ab VLY QUhLepO b SUToYrJfJ ] eqUhLMPOoYrJfKkO ] YORQ D eSiY ] Q%QSUTS b hfOY¬X b SfOIe ]] hfO TRKNM ] QJ b TRQPS\TNeQ ] UhLQ ] YORQ uQPSUTNYZTNe b S b"! KRQ hfS ORwO TP%QdVWQ ab VLY Q[ # Y O Q %QSUTNYrTRe b SejSUTRK b VLhLe T ] YS b TNe b SV ^ b[ QT¡VfYrSfO ] Q ab VWQPhLKP ejSfORe ] Q%VLMX[Q ]b JLJ Q%QSUTVW Q b S a TRe b SfSfY ] e TNMPO TRQ ]] QPO\hfQ ] QORhLeXweºV ^ b[ QTNO 3VWQPXweQPS\T b j ] ] hLSLQoQ wTNQSfORe SSfYZTNhLKNQ QoVWQPOY ]j[b KReTRL%QkO"VWQ ab VLY Q[ ¿QPO4 b S a TRe b SLSiY ] e TNMPO¿KRQPS\TNKRQPSUT VLY[SfO ] Q a Y\VWKNQºVWQPO3S b hLX\QPY[h b\ Q a TRe |)O = WMPO3JfYrK ] YmS b KNuQºVLQ ab VLY QpXweqVWM b ln ` ¶ f. .
(79) . . . $ 0
(80) . . oYrSfO ] Q a Y\VWKNQIV ^ hLSORWO T %QVWQ ab VLY Q b KNejQSUTRM b[ QT ] Y%l b KRJL b[]jb[ ejQIlYZTNLMY<D TReqUhLQdY[JLJfY[KNY<;jTºTNKkO"YrT TRK«Y a TRejX[Q a YrK a[^ QkOThLSLQoYrJLJLK bWa LQ M b %MTRKNejUhLQVWhTRK«YreTRQPuQPSUT VWh/O e SiY ] QT QS TNY[S\TUhLQuTRQ ]] Q3Q ]j] Q%J QKNuQTVWQ%%Y[SLejJLh ] QKYreqORM%QSUT VWQPO a KReTPKRQkO VWQ b KR%
(81) Q VWQTNYre ]j]
(82) Q VLQ ab SLSLQ WeTRMj # QPOJfKRQPuejQK«O ab S a QPJWTNO VWQ ] Yl b KNJL b\]b\ ejQ lYrTRLMP%YrTReqUhLQ hLKRQPSUT MPS b S a MPOQPS k L JfYrKºlejS *\b @O * e gwhLK a QT TNQ Y[ORQrlYrTRLQPK b S QT gWQKNKNY QS MT«Y L] eKNQSUT ] QPO8 b SfVWQ%QPS\T«OVLY[SfO ] QkOYrSfSLMQkO O b e LYrSUTNQ[QPJLhLeqO% a QT TRQ TRfM b KNejQO ^ QkOT ] YrK QPuQPSUTVWMPX[Q ]jb JfJiMPQ¿QTpYoMTRM¡VWe ! hfORMQmVLY[SfO ] Y ab %hLSfY[hWTRMejS\TNQKNSfY<D TRe b SiY ] Q[ # Y JLK b[] e MK«YZTRe b SVWQ ab S MKNQS a QPO\V ^ YrKRTRe a] QPOºQTV ^ Y[JLJ ] e a YZTRe b SiOºejSfVWhfO TRKNeQ ]] QPO QS/TNM b e SfQSUTP # YJLhLeqONORY[S a QVWQORQPO b hWTNe ] O ] ^ b SUT |YreTQSUTRKNQK VLYrSfOVWQS b KNQh a fY[uJiOV ^ YrJLJ ] e a YZTNe b SfOPpeY QKNeQI%MPVLe a Y ] QfO a ejQS a QPOVLQPOmYZTNMKNejY[h )WXwejORe b SK b\ ib D TReqUhLQ [TRM ] MkVWMTNQ a TNe b S \Y[SfY ] WO QT b J b[ K«YrJfLejUhL
(83) Q ZKRQ ab SLSfYreqONORY[S a Q¡VWQ a Y[KNY a T KNQPO¿j # QPO ] Q a TNQhLK«O¡ejS\TNMKNQPONO MkO¡JfYrKmhfSLQ ab SfSfYreqORONYrS a QIYrJLJLK b b SfVWejQ VWQ ] Y%l b KRJf b[]jb[ eQIlYZTNLMD YZTNejUhLQ J QhLX\QSUTORQ KR%M MKNQKYr=h b hfXUK«Y QPO ´¿oYrSfO a Q a fY[JLe TNKRQIS b hfOY ]]jb SfO ORQh ] Q%QSUTVWM a KNeKNQ ] QkOY[ORJ Q a T«OIJLK«YZTReqUhLQPO VLQ a QTRTRQTNLM b KRejQ[¯ ] O S b hfO J QKN%QT TNK b SUT VWQ %ejQh ab %JLKNQSfVLKRQ ] QIVLMX[Q ]b JLJ Q%QSUTVWQkO a fYrJLeTRKNQPO¡UhLe3ORhLejX[QPS\Tk # Yl b KRJL b[]jb[ ejQlYrTRLMPYZTReqUhLQ%QPO T hLSLQ K«YrS a LQ S b S ] eSLMkYrejKRQ%VWh TRK«YreTRQPuQPSUT VWhORe SfY ] UhLewYrJLJ ] eq\hfQ ] YmTRfM b KNejQ"VWQPO QSiO QP L] QPO ] ^ Y[SfY ] WO QºV ^ ej%Y Q\ # QkO b\ QT«O3VWQ ] ^ hLSfeX\QK«OO b SUT MPSLMK«Y ] Q%QSUTTRKNejVLe%QSiO e b SLSfQ ] O%wYreqO¡e ] O¡O b SUT¡J QK«½hiO TNKNY¬X\QK«OºhLSLQ JLK b Q a TRe b S ejVWej%QSfORe b SLSLQ ]] Q%O hLK ] YKNMTRejSLQ\ ¿QT TRQuJLK b Q a TRe b SS ^ QkOTIJfY[O ] YO b %%Q VWQkO ] hL%eSfY[S a QkOVWQT b hiO ] QPOJ b ejSUTNO ] Q ]jb S V ^ hfSLQ ] e SLQVWQ XUeqORe b S a Q \hfe QK«YreTdVWQ ] YXwejORe b S/hLSJLK bwa QPONO hiO ] ejSLMPY[eKNQ[ ] YJ ] hLJiYrKRT VWQPO b[ QTNOO b SUT b JfY[UhLQkO QT a Y a LQPSUT YrejSfORe a Qh UhLe O b SUToVWQPKRKNe PKRQ\8 ¿QTRTRQuJLK b JLKNeMTRMmhiOTNefQuhLSLQuYrJLJfK bWa LQS b S ] eSfMPYrejKNQ VWQ ] ^ Y[SfY ] WO Q%V ^ eY Q[ # Yl b KNJL b\]b\ ejQ lYrTRLMPYZTReqUhLQ TRLM b KNeQ%ejS\TNKRejSfO PUhLQPuQPSUT M b %MTNKReqUhLQ?ejSUTRK b VLhLe TIS b TNYr%%QSUT ] Y S b TRe b S V ^ M ] M%QPS\TO TRKNh a TNhLK«YrSUTP #º^ M ] M%QPS\T O TRKNh a TNhLKNY[SUT QPO T hLS b[ QT3VWQ b KR%Q ab SLSwhLQ ¬UhLerQkOT a b eqO eZQPS b S a TRe b S VWQPO3JLK b JLKNeMTRMkO M b %MTNKReqUhLQPOdUh ^ b SX[QPhWTdQ% wTRK«YrejKRQVLQ ] ^ ejY Q[ #"^ ejY Q Y[SfY ] WO QPKQPO T ab %JfY[KRMPQ ]_^ M ] M%QPS\TO TRKNh a TNhLK«YrSUTP # Yl b KNJL b\]b\ ejQ lYrTRLMP%YrTReqUhLQ%MTNY L] e T hLS b KNY ] ejOR%Q YZTNLMYZTNejUhLQ%J b hLK VWM a KNeKNQ%\hiYrSUTReTNYrTRejX[Q%QPS\T ] QPOIJLK b JLKNejMTRMkO M b %MTRKNeq\hfQPOoV ^ hfS b[ QTP. 6. . %#3.
(84) .
(85) !"$# &%' ". ,. . %'".5("( % !0 9*+- . (. %'# 5 1+ ; &)( 4 -4 !0&). 9*+:% &)( hiYrSfV S b hfO%JfYrK ]jb SfOV ^ QSiO QP L] QPOS b hfO'| Y[ejO b SfO%KNM MPKRQPS a Q VWQkO%eY QPO # QPOdej%Y QkO TRQPeSUTRQkOIVWQ KNejOoO b SUToVWM a KNe TNQPO ab u%QuMT«YrSUTIVLQPO/ b S a TRe b SfO (. . . . . ) ixP+t * Á je SfYrejKNQPOP VWQ IR2 → IR ,-/.0-/.. 132547698 :<;!8=. g b e T hLS O b hiOED´QSfORQ L] QºVWQ 2 6|hLSLQeY Q · S M X] M%QSUT OTNKRh a TRhfKNY[S\TdQPO ToIRhLS O b hfO D QPSfO QP S b TRM B LQkOTVWM%fSLe JiYrKk. je SfY[eKNQ,9 QT B hLS M ] M%QSUTO TRKNh a TRhLK«YrSUTP L] QVWQ IR2 # Q TRK«YrSfO ] YrTRMVLQ B JfYrK p~ . p. #º^ MK Lb ]ORe b S. ] ^ QPSfO QP. WV Q Q VWQPOJ. ] ^ QPSfO QP L] Q JfY[K ]_^ M ] M%QSUTO TRKNh a TNhLKNY[SUT QPO TVWMfSfeQ ab %uQ b eSUTNO p TNQ ] OmUXhLQ B 6¼TRK«YrSiO ] YZTNMIVLQ B JiYrK p~9BQkOT ab SUTRQSwhVLYrSiO X p Bp = {p + q / q ∈ B}. B (X) = {p ∈ IR2 , Bp ⊆ X} =. # QkO¡Q ! QT«O M b %MTNKReqUhLQPO¡VWQ ] ^ MPK b O e b SO b S\Tk ab b[ QTNOmY[h MTRK«YrS \] QPuQPSUTNOP • hfJLhLKNQoVWQkO. \. X−b. b∈B. ]u KRMTRKNM a eqONO QPuQPSUT¡VWQkO b[ QTNOmVWQIT«Yre ]] QORhLJiMPKRejQhfKRQ u Y T«Yre ]] Q VWQ B b ^ b\ QT«OVWQoTNY[e ]j] Q eS MPKRejQhfKRQ B • VWejORJfY[KReTRe SV #º^ Q wTN] QSfORe b S] VW^ b Q a QTRTRQdVLM fSLeTRe b S ] ^ b ] ^ QkO JfY a QVLQPO4 b S a TRe b fS O86 IR2 → IR9"QPO T \b RT QPSwh QSKNQ%J Y[½PYrSUT J MK«YZTRQPhLK ∩ JiYrK JiMPKNYrTRQhfK inf •. ∀p ∈ IR2. ,-/.0->,. B (f )(p) = {inf (f (x)), x ∈ Bp }. 1@?BAC;/D!? EF? E9;!8 =. `º]j] Q QPO TVLMfSLejQ ab %%Q ] ^ QPSfORQ f] Q WV QPO J b e SUT«O p TRQ ] O \hfQ ] ^ ejSUTRQK«ORQ a TNe b S VWQ ] _ ] ^ ] a a 6¼TNKNY[SfO YrTRM VWQ M M%QPS\TO TRKNh TRhLK«YrSUT B fJ YrK p~9¡Y¬X\Q X fS QO b e TmJfY\O¡SUh ]] Q[ δB (X) = {p ∈ IR2 , Bp ∩ X 6= ∅} =. [. X−b =. [. ˇx B. # YuVWe ] YrTNYrTRe b SQkOT ] YTNKNY[SfOE b KRYrTRe b SVWhiY ] QIVLQ ] ^ PM K b O e b 3S de ] YZTNQK ] QSUT M K b VLQKmO b S ab %J ] M%QSUT«YrejKRQ X c b∈B. #º^ Q wTN] QSfORe b S] VL^ b Q. QSKNQ%J Y[½PYrSUT. a QTRTRQVWM%fSLeTRe b S u ]_^ PQ ORJfY a Q VWQPO b S a NT e b SfO ] ^ b J MK«YZTRQPhLK ∪ JiYrK JiMPNK YrTRQhfK sup . Bp. X. QkOTmMk\hfeXZY<D. IR2 → IR. QPO T [b RT QSwhLQ. x∈X. δB (X) = (B (X c ))c. #º^ MK b ORe b S/QT ] YVLe ] YZT«YZTRe b S O b SUT ] QPO KNejUhLQkOVWQ Y[ORQJ b hLK ] Y ab SiOTNKRh a TRe b S VWQ S b KNQhfORQPO TNKNY[SfO b KN%YrTRe b SfO b KNJL b\]b\ eqUhLQPOP c b hfO Y ]j]jb SfO?JLKNMPORQSUTNQK VLY[SfO ] YmO hLeTRQ a Q ]] QPOmV b SUTS b hiOmYrhLK b SfO QPO b ejST b hWTY[h ]jb S VLQPOmJLK bWa fYrejSfO a fY[JLe TNKRQkO ∀p ∈ IR2. ˇp } δB (f )(p) = {sup(f (x)), x ∈ B.
(86)
(87) !"$# &%' " ,-/.0-. 3. 1 69?; =E 8 6
(88) 8 D!8;. # YO Q %QSUTNYrTRe b SV ^ eY QPOºSLM a QPONO eTRQ ] YUhfY[S\TNe a YZTNe b SVWQPO"XZYrKNejYrTRe b SfOpVWQ ab S\TNKNY\OTNQ QS a fY[UhLQJ b eSUTmVWQ ] ^ eY QJ b hLK%QTRTRKNQdQPSMXweqVWQS a Q ] QPO ab SUT b hfKNO¿VWQPO b[ QTNO b h a QUhLe"KRQPXUejQSUTYrh/%Q
(89) KNQJ MKNQK ] QPO ?Pb SLQkO b b\ SfQPOIVWQ ] ^ ejY Q\ # Q KNY\VWejQSUT b KNJL b\]b\ eq\hfQ WVWM iSLe JfY[K QPh a LQK ab %%Q ] QVWe ] YZTRMI b ejSfO ] ^ MK b VLM[ S b hfO/ b hLKNSLeTohLSfQ %QPORhLKNQ VWh b VWh ] QuVWh K«Y[VLeQPS\ToVWQ ] Y5 b S a RT e b S(6_O hfJLJ b O MPQ a b S D TRejS ? %QSUTVWe ! MPKRQPS a eqY f] Q:9 a] Y\ORORejUhLQP%QSUT¡VLMfSLe ab %%Q[ Γ(f ) = δB (f ) − B (f ).
(90)
(91)
(92) ~
(93)
(94) grad(f )
(95) =. #Q. ∂f ∂x. 2. +. . ∂f ∂y. 2 1/2. «K Y[VWejQSUT b KNJL b\]b\ eq\hfQ Y[ONORe SLQ a iY[UhLQuJ b eSUT ] Y J ] hfO K«YrSfVWQVWe ! MKNQS a Q QSUTNKRQ ] QPO ] hL%eSfY[S a QkOVWQ VWQPh J b ejSUTNOmVWQO b SX b ejOReSiY Q\ ,-/.0-. 132 6FE
(96) 6
(97) . S ^ b QPSiO T O eqJfVWY[MKOIb JSibO ON] O ^ eMP LK ]b QVWMoVWQ%VWQ KRM ]_a ^ QhLSiJ O MQPKN QK L]] Q ^ QPXSfO QPJf Yr LK ] ]Q%^ M e] SfMPue TNejQPY SU] Tm O TRb KN\h Qa SLTRSihLYrK«SUYrToSUT hfBSLQ VW` e S ] YrTNYZMTNSLe MPb KNS Y ] VWe Q ] ] ^ QPSfO QP L] QMPK b VWM\ "# ^ MPK b O e b S QPO ThLSLQ YrJLJ ] e a YZTNe b S/S b SejSZQ a TRejX[Q
(98) a[^ QkOTD D´VWejKNQ%UhLQ J ] hfOReQPhLKNOIQPSfORQ f] QkO 6 X, Y, ...9oJ QhfX[QSUTY¬X b ejK ] Q%QMK b VWM 6 X B 9 # Q%J ] hfO J QTReT QSfORQ L] QTNQ ] UhLQuO b S MK b VWMQkOT X B QPO TIY[JLJ Q ] M ] ^ b hLX[QPK TIVWQ X QTIe ] QkOT b[ TRQSwh QSVWe ] YrTNY[S\TO b S MK b VWM[ # Y ab eSfY[ejO b S MK b ORe b S ORhLejXUejQVWQVWe ] YrTNYZTNe b S QkOT b hLSLQ hfX[QKRTRhfKRQ
(99) « b. #ºJ ^ Bˇ QPO T L]_] ^ QSib O QP. X ◦ B = γB (X) = δBˇ (B (X)). L ] QITRK«YrSfORJ b ORMoVWQ B Bˇ = {−x / x ∈ B} #º^ Qb SihfOX[^ bQPQ KRTRhfKRQ Q hLX\b QKNKR^JLT Qkb\O]T]b\ hLeqSfUQhLQ O ej%QPO ^ JT ] e hL Sfa QYZTRbe b hfSX[QVWKRQ TRbhf] ^KRQPQSfO Y QP] b M L ] KNQ ejUejSLhLe^Q TNejYJf]Yr KRTRe ba h ] b e KNQ[ # Q TRQPKR%QV hLX[QPK TNhLKRQO Y[JLJ ejUhLQ V YrhWTNKRQkOITRK«YrSfO KNYZTNe SfO Uh hfSLQMK ORe SO hfeXwejQ V ^ hLSLQVLe ] YZT«YZTRe b S3¯ ] QS \]b\ Q¡T b hWTNQPO ] QkOpTRK«YrSiEO b KRYZTNe b SuUhLe X[MPKRe fQPS\T ] QPO"JLK b JLKNeMTRMkO Y ]j M KNejUhLQkOORhLejXZYrSUTRQkO b b a b ]] ]_^ a] b ^ L] QPOP • TRK«YrSfO KNYZTRe S K eqORONYrSUTRQ\ Q Q JLKNMPORQKNX[Q eS hfO e SV QSiO QP X ⊆ Y ⇒ (X ◦ B) ⊆ (Y ◦ B). •. •. TRK«YrSfO b KNYZTRe b S YrSUTRe D´Q wTRQPSfO ejX[Q\ ] QKRMkO h ] TNYrTQPO TeS a ] fh OVLY[SfO ] ^ PQ SfORQ f ] Q VWQ VWMJfY[K Tk TRK«YrSfO b KNYZTRe b SejVWQP%J b TRQSUTNQ[. X ◦B ⊆X. (X ◦ B) ◦ B = X ◦ B.
(100)
(101) !"$# &%' " ,-/.0-. 1@?6. 4. E 6
(102) .
(103) . ` S ejSwX[QK«ONYrSUT _] ^ b K«VWKNQ VLQPO b JiMPKNYrTRe b SfO?hWTNe ] eqO MPQPO J b hLK VWMfSfeK ] ^ b hLX\QKRTRhLKNQPS b hfO b[ TNQS b SfO hLSLQ S b hLX\Q ]j] Q b J MK«YZTNe b S ] Y' QPKR%QTNhLKNQ[ = (δ (X)) # bY aoQKN] uQTRhLKNQQkOT ]_^ a b JX] MK«•YZBTNe b =]_S ^ b φVWhfY (X) ] QVWQ ]_^ b hfX[QKRTRhfKRQ\ #º^ Q ! QTV ^ hfSLQ5 QPKR%QTNhLKNQ QPO TV S QI%QUhLQ Q hfe VWQ hfX[QKRTRhfKRQ
(104) UY[ejOmY[JLJ ] ejUhLMY[h ab %J ] M%QSUTNY[eKNQ[ ˇ B. B. B. # QkO¡JLK b JLKNeMTRMkOY ] M KNeq\hfQPO¡VWQ ] Y' QPKR%QTRhLKNQ O b S\Tk b b a b ]] ]_^ a] b ^ • TRK«YrSfO KNYZTRe S K eqORONYrSUTRQ\ Q Q JLKNMPORQKNX[Q eS hfO e SV QSiO QP (X c • B)c = X ◦ B. L] QPOP. Si X ⊆ Y, alors (X • B) ⊆ (Y • B). •. •. TRK«YrSfO b KNYZTRe b S%Q% UTNQSfORejX[Q[ ] ^ QPSfO QP L] QdVLQdVWMPJfYrKRT¿QkOTeS a] hiOVfYrSfO ] QdKRMkO h ] T«YZTP TRK«YrSfO b KNYZTRe b SejVWQP%J b TRQSUTNQ[. X ⊆X •B. (X • B) • B = X • B. ,-/.0-. . =E
(105) 6 6 E<? E9;!8=. . D258 6FE
(106) 6. . . . E. . D!? 6. E 6
(107) .
(108) . #º^ b hfX[QKRTRhfKRQ¡V ^ hLS%QSfORQ L] Q JfY[KhfSuM ] MP%QSUT"OTNKRh a TRhfKNY[S\T QkOT ] YoORhLK|Y a QKNQ ab hD X[QPK TNQJfY[K B ]jb K«OVWQ O b SVWMJ X] Y a Q%QSUT ]_^ ejS\TNMKNeQPhLKVWQ ]_^ QSiO BQP L] Q[ · SLQJLK b JfKRejMTNM ORe%e ] Y[eKNQdO ^ Y[JLJ ] ejUhLQ ] Y QKNuQTRhLKNQwYreqO a QT TRQ/ b eqO D a e ] ^ M ] MPuQPSUT¡O TRKNh a TNhLK«YrSUT¿JfY[K D ab hLKRT ] Q ab %J ] MP%QSUTNY[eKNQoVLQ ]_^ QSfORQ L] Q[ npYrK ab SfO TRKNh a TNe b S ]_^ b hLX\QKRTRhLKNQoQT ] Y QPKR%QTNhLKNQoKNQPORJ Q a TNQSUT ] QkOm\hiYZTRKNQoJLKRejS a eJ QPO VWQT b hLTRQdTRK«YrSfO b KNYZTRe b S% b KNJL b\]b\ eqUhLQ[ ejSwXZYrKNejY[S a QJiYrKºTNKNY[SfO ] YrTRe b S ab uJiYZTRe e ] D eTRMY¬X[Q a] QPO¡ b b TNLMTNeQk%O ab SLSiYreqORONYrS a Q ]jbWa Y ] QIQT ab SUTRejSwhLe TNM[ ,-/.0-
(109). 1 :. ?. ?. . ?. . E 806. E<8 ? E. . # Q a fYrJ QPY[huiYrhWT¿VLQ b KN%QQPO T"hLSLQTRK«YrSfO b KNYZTRe b S%UhLefJ QKN%QT¡VWQVWMTRQ a TRQPK a QUhLQ ] ^ b Lh X[QPK TNhLKRQ86 a fY[JiQkYrh L] Y[S a 9 b h ] Y QKN%QTNhLKRQ 6 a iYrJ QPYrh S b ejK 9 b SUT$|YreTºVWejORJfY[KNY<;jTNKRQ\ ` T«YrSUTºV b SLSfM"hLSuM ] M%QSUTpOTNKRh a TRhfKNY[S\T B b S VWM%fSLeT Y[eSiO e ] Q a fY[JiQkYrhiYrhWTºVWQ b KN%Q V ^ hLSLQ b S a TNe b S f JfYrKk chapeau blanc : W T HB (f ) = f − γB (f ) chapeau noir : BT HB (f ) = φB (f ) − f.
(110)
(111) !"$# &%' ". . . &)( 9,% .5("( %D!0 9*+- . ( 14) / 4)(,+ ; &)( b SUTNKNY[eKNQ%QSUTY[h TNKNY[SfO b KN%YrTRe b SfOQh a] eqVWejQSLSLQkOdUhLeY ejONO QPSUTdO hfK ] ^ QPSfO QP L] QVWQ ] ^ ejY Q
(112) ] QkO TRK«YrSfO b KNYZTRe b SiO M b VWMPORejUhLQkO3SfQO ^ Y[JLJ ] ejUhLQPSUT Uh ^ ] ^ ejSUTRMPKRejQhLKpVWQ ?b SLQPO JLKNMPVWMTRQPKR%ejSLMQkO% YrJLJ Q ] MQkO Y[ON\hfQPO M b VWMPOReq\hfQPOPejSfORe VLYrSiOIT b hWTRQ%TNKNY[SfOE b KRY"D TRe b S M b VWMkO eqUhLQ eSUTRQPKRXwejQSLSfQSUT3VWQh=d ej%Y QkO ] ^ ejY Q TNKNY[SfO b KNuQPK X QT ] QpY[ON\hfQ M b VWMPORejUhLQ M gwhLJfJ b O b SfOpUhLQ¡S b hfOpTRK«Y¬X¬Y[e ]j]jb SfOQS ab SLSLQ%W e TNM 1 ar^ QPO T >D Y<>D VWejKRQ
(113) ] QmX b eqO ejSfY Q¡V ^ hfS JLe W Q ] |6 e 9 QPO TP . . n b hLKKNQPO TRKNQejSfVWKNQ ] ^ Y a TRe b S VWQ ] YTRK«YrSfO b KNYZTRe b SY[hY[ONUhLQ M b VWMkO eqUhLQ ] Y ab SD SLQ% weTRM \hfe O ^ Y[JLJ ] ejUhLQ a iY[UhLQ JLe WQ ] S ^ QkOTJiY[O V4 (i, j) %Y[ejO V4 (i, j) ∩ M ar^ QPO T D D>VWejKRQ%Uh ^ hLS/JLe WQ ] S ^ Y[JLJfY[K TNQSfY[S\T JfY\OIYrh %Y\ORUhLQ%SLQORQK«Y JfY[O ab SfORejVWMPKRM ab %%Q X b ejORejSV ^ Yrh a hLSJLe wQ ] ¿QPOTRK«YrSfO b KNYZTNe b SfOO b SUT ] Y Y[ORQ VWQkO b JiMPKNYrTRe b SfO ] QkOJ ] hfO JiQPK b KRYrSUTNQPO ab %uQIJfY[KmQ WQ%J ] Q ] Y KNQ ab SfO TRKNh a TRe b S V4 (i, j) = {(i − 1, j), (i, j − 1), (i + 1, j), (i, j + 1)}. ,->,-/.. 1@? ;/D/?"EF? E9;!8 = . 8
(114) z xPz'
(115) . 8. . ": ; . . I. 3± ¦Z¥±r©´±r©_¥|³r¨ #U¢«³P¦\¢¤£¤¥µf§/¦[§ ´© ±r¥§ ¦\§ §¨L£§» § X ¦[±r¨L£ § »±¬£µf§ M º ¨ ³r©´¢«§ δM(1) (X) ¿ §¤£¤©¦\¢ "¨f¥|§ «³Z»u»%§ ¥¼¨f©>§ «£§ ©_¥|³r¨ ¦rµ ¦Z¥±r©´¢%¦[§ ©>±Z¥§ . §©¦[§. M. (1). (1). ³
(116) 5C §£«© § ³r¥q£«¥¼¨?± #U§¦ ±\¦± «§¨ «§²f³ZµW ± N³r¨f¨?§ ¹ ¥¼©>¢ ³r¥q£«¥|§ # Y VWe ] YZT«YZTNe b S M b VWMkO eqUhLQVWQTNYre ]j] QS VWQ ] ^ QPSfO QP L] Q X ab SfVWeTRe b SfSLMQY[hY[ON\hfQ b[ TNQSwhLQIQSe TNMK«YrSUT n b ejO ] YVWe ] YrTNYZTNe b S M b VWMkO eqUhLQoVLQITNYre ]j] Q (n) M δM (X) fQPO T δM (X) = (δB ) ∩ M. (n). (1). (1). (1). δM (X) = δM (δM (...δM (X))) | {z } nf ois. PQ %Y[KNUh b SfOUhLQ ] YVWe ] YZT«YZTNe b S M b VLMPORejUhLQVLQTNY[e ]j] Q ] YuVLe ] YZT«YZTRe b S b KNJL b\]b\ eq\hfQoVWQIT«Yre ]] Q n Y¬X\Q a M . n. (n). S ^ QPO TdJfY\O ] ^ e SUTRQPKNORQ a TRe b SVWQ. b %%Q ab SUTRKNQ D´Q WQ%J ] Q S b hfO J b hLX b SfO?eY ejSLQK hLS QPSfORQ f] Q b KN%MºVWQpJ ] hfO ejQhfKNO ab %J b ORY[S\TNQPO ab SLSLQ% WQPO3V b SUT ] ^ hfSLQQPO T ] ^ QPSfO QP L] Q X # YVWe M] YZT«YZTNe b S M b VWMPOReq\hfQ VWQS ^ e%J b KRTRQUhLQ ]j] Q T«Yre ]] QVWQ X VLY[SfO M ORQK«YT b hZ b hLK«O ] Y ab uJ b ONYrSUTNQejSLe TNejY ] Q 6J ] Q^ QhWSfT ORaQb SU LTR]QPQSL8eK9 VLT«QPYrO¡SfJfVLY[ejO¿K TN\eQkhfOmQ V ]_^ ^ ejY[S\hWTNTRQKNK«QPO OQ a aTNb e b%SJ VWb Q ONYr] SUYTRVWQkOe ] YraTNb YZSLTNSfe b QS WQPVWOPQm T«Yre ]] Q n VWQ X Y¬X\Q a M #º^ Q wTNQSfORe b S VWQ a QT TNQTRK«YrSfO b KNYZTNe b S J b hLK ] QkOej%Y QkO TRQPeSUTNQPOVLQ KNejO%QkOT VWejKRQ a TNQ[ # YVLe ] YZT«YZTRe b S M b VWMkO eqUhLQVWQ ] 3Y b S a TRe b S f VLY[SfO ] Y b S a TNe b SY[ON\hfQ g QkOT VWM%fSLejQVWQ ] Y |Y\½ b SORhLeXZY[S\TNQ[ δM (X) 6= (δ n (X)) ∩ M. δg1 (f ) = inf (δ(f ), g) δgn+1 = inf (δ(δgn (f )), g).
(117)
(118) !"$# &%' " ,->,->,. 132547698 :<;!8=. . 8. . . :<;. !6 . . #ºb ^ MK b ORe b S M bb VWMkO eqb UhLQ] QPO T ] Y TRK«YrSi OE b KRYZTNe b S VWhiY ] QVLQ g S Q%w JfKRQkORORe SJ hLK QkO¡eY QPO ejSfY[eKNQPOQPO TP. ] Y/VWe ] YrTNYrTRe b S M b VWMkO eqUhLQ[. n nM (X) = M − δM (M − X). QTJ b hfK ] QkO¡eY QPO TRQPeSUTNQPOVWQ KReqO ,->,-. . 1@?. n ng (X) = −δ−g (−f ) . 8=: EF6. . +E9;!8= . pn YrK ab SfO TRKNh a TRe b S ] QpKNMPORh ] TNYZT3VLQ ] YmVLe ] YZT«YZTRe b S M b VWMPOReq\hfQQkOT ejS a] hfO VLYrSiO ] QºY\ORUhLQ\ npYrKY[e ]j] QPhLK«O% ] Y%VWe ] YZT«YZTNe b S QkOThLSfQ b JiMPKNYrTRe b SQ wTRQPSfOReX\Q[p b S a JfYrKRTRejKmV ^ hLSLQ a QPK D TNY[eSfQeTRMK«YZTNe b S ] YVWe ] YrTNYrTRe b S M b VWMPORejUhLQSLQ b VWe fQJiY[O ]_^ eY Q[ # Y O hLeTRQuVWQ VLeD ] YZTNYrTRe b SfO M b VLMPORejUhLQkOºVWQmT«Yre ]] Q a K b eqONORY[S\TNQ36 δ(n) 9 ab SwX[QK Q¡X\QK«OphLSLQdeY QYrJfJiQ ] MPQ ej%Y QKNQ ab SfO TRKNhLeTRQ[ # YKNQ ab SfO TRKNh a TNe b S VLQ MX 6|YrK«\hfQhLK 9 ab SfVWeTRe b SLSLQ ]] Q%QPS\T ] ] b b ^ eqVWQ%J b TRQPS a Q\ M 6|# Y[ON\hf,Q ] 9QPO T Y] VLe YZaT«b YZTRe S a M b VWMPORejUhLQIVW] Q a X VL YrSiO M hfON\] h ^ QKRMkO h TNYrTdVWQ YKNQ SfO TRKNh TRe SVLY[SfO Q Y\O ejSfY[eKNQQPO T QSfORQ L] QuVWQPO ab D bJ ONYrSUTRQkO ab SLSLQ WQkO"VWhY\ORUhLQUhLe ab hLJiQPSUT ]_^ QSiO QP L] QVWQoVWMJfY[K T \Q ]] QKNQ ab SfO TRKNhLeT ] QPO¡JfY[K TNe a h ] QkOmVWQ M UhLe O b S\TmYrK«UhLMQkOJfYrK X oYrSfO ] Q a Y\OpSwhL%MKNejUhL
(119) Q ] YIKRQ ab SiOTNKRh a TRe b S%VWQ f VLYrSfO g Y ab %%QORQh ] Op<Y Wej%Y ] QPO¡JLe a OVLQ g \hfe O b S\T marqu´es JfYrK f c b hfO¡X[QPKRK b SiO¿J ] hfO ]b eSUhLQ a QT TNQITRK«YrSfO b KNYZTRe b S S b hiOmJiQPKR%QT TRK«Y%VWQ a fY[S QPK ] ^ b b T b JLejQ VWQ ] ^ ejY Q K«Y[VWejQSUTJ b hLK ] hLeue%J b ORQK ] QPO Q wTRKNQY O b hLiYreTRMPO QT M ] ejuejSLQPKYrejSfO e ] Y/O hfKNORQ uQPSUTNYZTNe b S QS QSiVWKRMPQ ]b KNO%V ^ hLSLQY[JLJ ] e a YZTRe b S VWejKRQ a TRQVWQ ] ^ Y ]\b KNe TNL%QVWQ ] e SLQVWQJfYrKRTNY QVWQkOQPY[h ` S b hWTRKNQ a QT TRQTRK«YrSfO b KNYZTNe b S QkOT ] Y Y[ORQVWQkO ] TRKNQPO ab SLSfQ WQPOmUhLe S b hfOJiQPKR%QT TRK b SUTVWQ O ej%J ] e fQPK ] ^ ejY QONYrSiOVWM D K«Y[VLQK ] Q ab SUT b hLK, ab SfVWeTRe b S SLM a QPONORY[eKNQJ b hLK¡Y b hLTRejK hLSLQ ib SLSLQoO Q %QSUTNYrTRe b S . !. . . + 9,% &)( .5. & < &)( . oYrSfO ] Y ORQ a TRe b SJfKRM a MPVLQSUTRQoS b hiOY¬X b SfO"XwhUhLQ ] QkO ab ejSfYreqO b SfOV ^ MPK b O e b SiO"QT¡VWQ VWe ] YrTNYrTRe b SfO ab SfO TReTRhLQPSUT"VWQPO ] TNKRQkOp b KNJL b[]jb[ ejUhLQkO6 b hfX[QKRTRhfKRQkO QT QKN%QTNhLKRQkO 9UhLe M ] ejuejSLQPS\T ] QPO ab %J b ORY[SUTRQPO ab SLSLQ% wQkO VWQTNYre ]j] Q¡ejS MKNeQPhLKNQ oa Q ]] QmVWQ ]_^ M ] M%QPS\TºO TRKNh a D TRhfKNY[S\Tk # QPO b hfX[QKRTRhfKRQkOºQT QKN%QTRhfKRQkO"J QhLX\QSUTmYrhfONO e TNKRQ ab ejSLMQkO"J b hfK b\ TRQPSLeK VWQkO ] TRKNQPOdJ ] hfOoMX b[] hLMPOP ¿QPJiQPSfVLYrSUToT b hfO ] QPO ] TRKNQPO ab SfOTNKRhfe T«OdVWQ a QT TRQuYrSLe KNQ QS' P%QTNQ%JfO\h ^ e ] O¿M ] ejuejSLQPS\T ] QkO"J QTNe TNQPO ab %J b ORY[S\TNQPO\VWM K«Y[VWQPSUT ] QkO ab SUT b hLK«O VWQkO b[ QTNO KNQPO TNY[SUTNOP #º^ ejS\TNK b VWh a TRe b SVWQ ] YKNQ ab SfO TRKNh a TNe b S V b SLSLQhfSLQVWej%QSfORe b S ] ] b b[]jb[ ]^ ] b ab b ejORSfhL] TRY JLKN]mJ MQkM%O K % a] QQPQ SUO T« b\Yr ej]KRY QQT«Y[KRO¡Qh aQb S SfaTRObK«TNYKNKRQdh QJfa TRKRe Mkb O S3QPKRS\JfT« Om ¿YrQJLT TR K QejPUO hLeqVW]Q\Q MP 7Q ] TRJfhWK«K TY k O eQISUTNKNM K Qb TNej%VWK bhLehLSfeTRX[YrQQPTRS\e QPT S S] QVWkhL QkK O ab SU%T bJ ´hLK ONYrYrb JfSUKNTRK e QkkOOD hLSLQMK b ORe b S ] QPO ab %J b ORY[S\TNQPOuUhLemS ^ Y¬X¬Y[eQPSUT JfY\O ab %J ] TNQ%QSUTVWejORJfY[KRh MTNY[eQPSUT hWTNe ] ejORMQkO ab %uQY[KNUhLQPhLKNO%J b hfKKRQ ab SfOTNKRhfeKNQ ] QPOJfYrKRTRe a h ] QPO b KRe ejSfY ] QkO ¿QT TRQ b J MK«YZTNe b S fY[JLJiQ ] MPQ b hfX[QKRTRhfKRQ JfY[KKNQ ab SfO TRKNh a TNe b S LQPO T ] QJLe b SLSfeQPKdVWQkO ] TRKNQPO ab S D SL%Q wQkO4 ¿QkO ] TRKNQPOJ b ONO kVWQSUT¡V ^ %Q a Q ]j] QSUTRQkO¿J QK b KNYrS a QPO¿\hfe?O b SUT ] ^ b KRe ejSLQoV ^ hfS ejuJ b KRTNY[SUTmVLMX[Q ]b JLJ Q%QSUTVWhTRK«YreTRQPuQPSUTV ^ ejY QPOP b %%QSi½ b SiOdJiYrKVWM a KNeKNQ ] Q a Y[O ejSfYrejKNQ[ ª b hWTORQuJiY[ONO Q ab %uQORe ] QPO ] TRKNQPO ab SLSL%Q wQkO ab SfORej=V KNQSUT a fY\\hfQ ab uJ b ONYrSUTNQ ab SLSLQ WQ eSfVLeXweqVWhLQ ]j] Q%QSUTP ¿fY\\hfQ. 3 6.
(120)
(121) !"$# &%' ". \. a b %J b ORY[S\TNQ ab SLSLQ% wQ 6|O b eTuVWQ ] ^ QPSfO QP L] Q X ºO b e T%VWQO b S ab %J ] M%QPS\T«YrejKRQ X c 9 QPO TuO b eT JfKRMkO QPKRX\MQejS\TNM KNY ] QPuQPSUTpO b e T ab uJ ] TRQ%QPS\T%Q ! Y a MQ\ # QkO ab %J b ONYrSUTRQkO ab SLSLQ% wQkO O b SUTV b S a VWQkO JfY[K TNe a h ] QkOIeSiVWeXweqO e L] QPOP7 ¿QT TRQ a YrK«Y a TRMKNeqOTNejUhLQ%Y[ONO hfKRQ ] Y ab SfORQKNX¬YrTRe b SVWh ab SUT b hLKmVWQkO b[ QTNOmUhLe b S\TORhLKNX[M a h Y[h ] TNKNY Q[ z xPz ³r¥¼© E µW¨ §¨f£§» I §u§© P (E) ¹ §¨L£§» I §%¦[§£o£³rµw£ ! §¨L£§» I §£¦[§ ¨ N ³ f ² ¢ R Z ± > © § W µ E (E) → P (E) §£¤© «³r¨f¨ § § µf±r¨ ¦ ∀A ⊆ E¹ ±/¦Z¥ d¢ R§¨ «§ £ Z»%¢© ¤¥ µf§§¨f© R§ AΨ§ P© Ψ(A) §£«©pµW¨?§dµW¨f¥|³Z¨¦\§ «³Z»²L³Z£±r¨f©>§¤£ «³Z¨f¨?§ §£o¦[§ A ³rµ¦\§. 8
(122) '
Documents relatifs