Direct visual servoing with respect to rigid objects
28
0
0
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Direct visual servoing with respect to rigid objects Geraldo Silveira — Ezio Malis. N° 6265 August 2007. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6265--FR+ENG. Thème NUM.
(3)
(4)
(5) ! "#$% & ' )(*,+.-/ 021436587:9;<=>7>?@1A=:365 B C,D =:;FEG5H7>=:I JKML)NPO,QSRUTWlmg`npVYoqO^UX[Z]rS\_s^`LanuNbtMrvOc\X d4eMNbfag`hji4ekOa\ sxwpyMyHnug`^z]O,g{Oa|!KkO)g!|{KkO}dH~Ap4VrUekueM\_^Sp u [vVYPyMwpAOa\ ∗†. †. i4ekhg`aOA)¢knud£uu^`KkO nuhj^`\qKk^{hOadkgKM[wuhdM\`z¤eM¢ wpyk\_g{OahnAg`gAnu¥4hdkdM n6^`OcOa|{z]KkAdMOhi4wpektHOcnu\ek^¦Kk^{hjKk|{O Knuz]t]noqOadk|np^c^§ \Pdk\_O)KMOcwuzyHONPwuO^`dMg{z8hj| ¨6nAhgd] ^{nAKkg`O©N wp|)wp^`hNPnudO)g!g{wkO§ \¡ NPwud4npZP^{hnpnA/d/^{ªSKk«¬O)dN©^`ª KkJhj\SKkyMOvwuNbyHO)O)^`gcKk¢Hn]zO¦hyM\g`z]nAhyHg`OcnA|\`^®Ohwd.dk^{KkO)O¯\_4OahjdM\`\_eMOUwp^`K8\`wO)g{^au°nu^`hKkdkOv h^`dAOc^`|{OaKkdMdk\_hhj^qi4Z ekOwueMKkOUhj|!nuKzkwun[@Oayk\Uh dkOanu^S\g{hj\Oai4e8ek\_hOcg{z O ± |)wuh>nug`ª OUOudAª/^{dkg`OanAh¶O./^{KkO)wcg{O)ug{gnAnug{hgxzOawAi4^`\ekKkhOPg`OcOa z Oawpdkwu^`nu\ekg®g{^{OPOaKk\_O¦O^`h |)N ^{nug{wd4wA^{^`|Oag{^`zMnuhnu/³ªd²jw6·®µ\_^`OaOa\`wpy²hg{zkO¯Oa\ Kkekdkhj|{npZ K^g{tMhdAO)wu^`\`Z4g{Oahn]z©dkz]¦eMnunAd2|)Oad\¯hN ykO)g`wpg{hg{ununuO¦g®g!hzk\{d]³w ´^!nAwwpg`dMN ± z%h>wpª Ou^`^`hª8KMnuNPwpd/^¯¢AO)^`^{^{g{KkKkhO¯O |,\`NPyk|!g`KkOanAOawAyHNb\_nAeMOv\`g`OaOchz \\ ^{g{hj\KMnutkNbw^eMnu\_^{^`^Kkhnu^{OSd]n¯yk¡¤jg{wpwpnug{dMyHuz¦nAO\`\_O)OaKMg{zbwug`nAyH^!g{Owu\¡º\`h¥hd8dbz] ek^{O)KkdMyHOx|O)^`d8|ahwpz]nuNPdO)d4O)hj^a\g!¢pwkwpn]§ dM\|az¦hwpdAwp^`ZbOaj\_g`nxhjd8\_wpnA^`KMkNPwpyMnu^wpg{^`g!ykKMwpKkNPO®hj|z]O^{O)^`O)g{n¦g!hu\)^`OaKkªzOv¸²||aOnAwpdANPyk^{g{g`O)nnAg!4w,uhjjz]yHw6O®nA^{\`KkO)OudM¢AO\`^`^`ekK8KMg{wO)Oa^nu\g{^`KMOn]^{Ov|)h|awuwpwpAyk]wAykyk\_g{Z[g{nAn[NPwAnu|{y]¹K \¡ ^{npwu^`hhjz¤|¢¤\q^!\`wphdktkuhekh^qjZuwpªg{h»k^¬Z4eM¡:g_ ^{g`KkOaOO)g{yMNPwpnu^`g{KOu¢6yk^{jwuKkdkOdkykhg{dMnuMyH¢@nA\`OaKMzh|!|KnAdAOa^{wAg`nAzk[\UO)^`g{n©g`nAwgwuwug`nAxO¦\zk nunuN g \`wphhNPdyknpOu¢A|nA\`NPd4un[Oanpg`^{AK/O)¢pdMyk|OK[Z] \_nuhjgS|)wu^{KkZO j\`wpO)g{g{uuOvnuhhdkdkMh¶^{ªhwuJKkz]Oh\`wpykykjykwug{|)nAO)wANP|{O)K¦dAhj^!\\pwAwp\hzkwpO)^`Ocz,wu\^{Kkwug`g`nAAekOvuKO)g{g`wunAg`g{h\nuheMd\^{g{KkOa\`Oek|)^{wu\NbeMOa\`g{hwMdk§v\®nuhdAt]^{oqO)Oag{|dM^{\wp¤npy8@wpz]g!h¶¼@wpNPO)g{OO)^{d4O)^ g!\)\_K8ª wpyHOa\a¢ ½2¾4¿@À`Á 6ÃAÅÄ hj\`eMwp/\`O)g{unuhdk8¢4[hj\_e8wp@^`g!wu|!¥[hdk8¢]z]hg{Oa|^xNbO)^`Kknkzk\)ª Ò:ÇcÖ ÈÊßÆ/Ò:Ç6Öq͹ÈÊßÉÏ{ˤÒ:ÈÊÌÌ͹ßÎ,Ï!ÐkÈÊÉ Ï!Ï!àÐÊ͹É:Ö_ÌUÖqëÑÏ!ÖqÍ ßÒ:ÒÈÓÏ!ÞcÐÊÚÔÛHÉ>ÜHÕ6ÝcÑ6ÛMÑpÌèíì Í îÒ:Öqï×bìjÚØÔ,Õ6Ò>ß6Ç6×6ÖxÖqÍÙ@àÚÍ:ÛHÏ!ßÜHÒÝ,ß6ÞÌ6Ìá8Õ6æ!ßcð)×åaÏ{â_Ò:ç!ÈÊðÌäßé{èíÕ6ñcß6á ×cÖ_ÍàÍ:Ï!ßÒmß6Ì6áSâ_ããäå!æç{è¹éaêkÏ!ß6צØ)Ô Ù¤Ï!ëÑ6Ù@ì îÈÊÖqßïßÏ!Û8ɬìíÚ£åïÝaÚ²ÛkÝaÌê6ï/Ñ6ú@ÇcÖqÍ>É:ÈÓÏÏ{Ö_ûqÏ!èíÈÊÚÍ>Ðjêò:ßÇü6Ò:ýaÈÊÙ@ÑpþÌÖqÿ ßÐÊÈ Ò:ÉÖqó¯ÍóUÛMÍ>ôÌ ïA>Öqõò¬ý Ò Ù©)Úþôÿ ïÈ öaÌcÈÊØcÉ:ý ÚÈÊÌÝupß4êþê6ÿïæÌ)æ!×4ðSá¤ï/ôÌpÌÕcþë÷Ò:ÖÛ]×6ÖqÖq×6É Í>Ì4Õ6ìºòqêÈÊÎaÌëøÐÊÖqÉ_âqê4ðú¤çcÛ©ê äcêùÚçcê4ëxæÏ!äÍ:ùÏ!æ ÈÊSÉ_êpÝaÙ¤ÌÜHÑ6Ç6Û2ÈÓÏ{â_èíçÚæß)äÒ>ù{ÈÊè ÑpùÌæcÐÊâÈ É ê Ù@Ö_×cÖ AêÞÍ:Ï!ßcò_Öê )þ ÿ 4ý pÿ pÿ 4ý aÿ pþ aÿ 6þ ∗ †. !.
(6)
(7) " $#% & ' )( #% & ' # *+, ! $ -. . . Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France) Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65.
(8) £(m
(9) $©,
(10) ,$ $%,
(11) Y,+.-/® )(
(12) \`nAekg`g,N O w^{NPhnA nudekNPuOafNP^`Oag{\O)hjiAdA^{eMOa^¦|!OzkKkO O dk hjhiAuw2eMO)Oad4|)\Uwp^NPO^`nufaekhjg{\q^`wM^!Owpª d4nAv^`h\aOawp¢A\UdMz]z/\Oa§ \ wA|)O\`|\`^¦nuO)dMwug{dM[g_^{wph\{hhj|)\_\`O)\{OANPwp¢dMOadk|)dAnuOa^x\eM[\,wvh\`ykykekg{g{OanuhnuyHg{i4nAh]ek\`\_nuheMd8gd/\Sj§ nuwSekdAdM ^SnuO g{tHNPdkOanu\`OnueMz]hudOO):z¤§ OPnA§ wpt]^`eMo¬Oa|)O)|!ek^ KkdkOdkO^hj¨i4hnuekd]e O ¡ Ai4ekf)ekNPnuh@NPhdkdMOvfO)^{eMdkg{\`hfai4OU|)ekOaz]Oc\{O\_\h^{^`i4nuOvekeMwuh\ eMhd4|Oc^`ek\g{dMn]ykOvz]h ekz]O)hO¯j^U\ |zkzkOaOa\O\|:Oa§nuhg`N dMg{dMO)wpeMwpAhjg{O\`\!\{³Ocwp´]\qd8^mO)|^UO Oa n\aykª©
(13) nuwbw¦h^`fa NPnAO dMf± ^{|nAKk^{dPhn]nAz]f)d2[Ovhz]^`OaO¯OU\_^^z]uz]nu|!hdMg{Kk|OaO,|>^{§ykO Ovg{ nuwp^`yHeg!wun4\`|\_O)^`fadMhO¯nu\®dPwpn hzkdM/O\`°hyk>i4§ g{hekdAhNbO,^{O)hjdM^`wbh\`uh¶^{OanAf\ h z^{]g`O©hji4|)eknuONPd/N § Oawp\_dM^Sz]O O h\`unufad4O,^bnAOae2dA^`Oah\_L)^`g{hO)NPNPf)O)O6d4³ª^P tMdwu\`ykf)OceM\b\S\_ekz]g O¦d/z]Oa§ wc\PunAz]hnugUdMtHdkOaf)\`Ocnu\bhdz¤§ z¤hN § wpwueMu|)O ekdk± Oh:ª Ouz]ªOwp|eMOa\U|)ekhd]dk O nug{NbN Ocw\_^`eMhg`nAOdM\xNPwufe ¡ ykz]g{O,fawuwPwu|)tkwuNbOu¢4fajg{wwMªNPQxf^{nAKkeMn]\z]fO¯^!wpOatk\_^h\{g{\`nunutkdMe8\®\q^{OvOa\wuyke g{ O)ekOag`ug{OcO)\ek^{g!Kk\®f)z]nAO¯g`hj|ai4wpekOahtM\g{i4wpek^`hO¯nudjw\_íeknAgdM|Oa^{\hnAyMd©wuz]g{wuOvNb^ uL)^`|!g{KkOaO¯\ykhd4g`nA^`g{yHhnAdM\`\`f)LaOSi4ekOcOc\q^\ hjO)\_^xnAz]NPe nug{NPyknuKkekOuOaNbjw OadAyH^xnAz]\`OPO¯z]jwbOP|)jwuw©NP|)f)wug!Nbwkfa¢4g{O)wM^x¢/wpi4e8ek\`OP\`h¤:i4§ wuekykOvykjg{wn]|{nuKkh/Obz]OcO,\q^|nAhdMNPzkNPf)yHwuO)dMdMz]zMOvwpyMdAg`^{nAO y8zkn4OP\_fajOvwwu \{nA\_g`ekNPg{OSO ekz]dkOPO:§\qnA^!t[wpoqtkO)h^¡ \`h¶h^{NPfyknkOu|)¢wpg{Ouf)ªueM hd%L)g{nuOue]¢^{ykg`K[OA¢HZ[\`w hji4 eknAO)dMNP|^{O)hd4nA^¦dpz]wpO¦hzk^ AO |{KMO)^¦O¦ykhtMg{nug`OyHnAz]\`Of)O\`hyHdkOaAg`ekNPjwpOg{^vh¶^{ekfadk\a¢O|ykOjwpi4dMekh 8h|aNPw^{L)hdknAOd2z]O¦ekd£^{g{wzkoqnuOaN |^`wpnAhhdkg{OO z]\`ekO^{|wnu^!d[\UuOaO)dg{ue]O)d8^{h|Ovhj\`wu^`g{dALa^¯\xz]uOcg!\Uwpd8nAz t]oqz]O^{O\v:z]§ wuOb\{\_z]Oahg`¼@[f)hjg{\`O)\`O)dANP^{OaO)\UdA ^cnuª g{NP
(14) § Oawp\ayk¢@ykwcg{un]Oc|!|KkO¯zkOaOc\S\q^Uupg!wpwpdMhzMzk\Uf)O,z]yMf)ykwujgUwu|)z]O)O¦NPdkO)nud4N¦^{\Stkg{hdMO)eh¶^{ hwug{e f ¡ wuhdM\`h/i4ekO,zkO¯Ag{wudMz]Oa\®Oag`g{O)ekg!\zkwpdM\Oc\yMwug{wuNbL)^`g{Oa\®hd4^`g{hd8\_LciAeMOa\z]O,wP|awpNPf)g!wkª Â a À rU\{\_Oag`[hj\`\`O)NPO)d4^[h\`ekOa>¢M|nuNPN wpd8z]O¯g`f) f)g{O)dM|faOv[hj\_hnud¢[NPf)^`Kkn]zkOa\xz]hg{Oa|^`Oa\aª.
(15)
(16) !"$#% & '() *,+-*.,/ 0. 1. b 3546 a  à u87  6 9 B :<;>= ¾[ ¾ 7¹A &?bk @A8  6 à ]]DªªÊC QxJ®np^!npw¡º^{4hhnAO)dM \ ykg{ª,npª¯oqOaª,|^{ª¯huª,O¯ª¯AO)ª,nAª¯Nbª,O)^`ª¯g{ZFª,ª¯ª¯ª,ª,ª¯ª¯ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ªª ]ª 1 Uhg`Oc|^x[h\`eMwp/\`O)g{unuhdkb,ª gcª ^cªyMwudMwpgnut]oqOa|^!\ ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª P ¾ G ¾ 6 ¾ &I-7 ¾ pK J 7> ML ¾ O N  7 6 A EF;>= H 1kDª C JKkO¯eMdkh MOcz©z]hg`Oc|^[hj\_e8wp@^`g!wu|!¥[hdMµª¯ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª 1kªÊ JKkO¯yMg`nAy8n4\_Ocz.^{wu\`¥ ekdM|^`hnud ª,ª¯ª,ª¯ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª Q 1kª 1 JKkO,|)nud4^`g{nu@jw6øwudMz \_^{wptMhh¶^qZwpdMwuZ]\`h\ ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª,ª¯ª R 9TS k =US ¹ 6V6 7 6 A 3XW B ¾ c 3Y3 Z [  6 7  6 3%: ]\^\ ¾ 6 Và 7`_ 3%: ¾ ¹] 7  6 ¾ Á¦¾[¾ 6 = ¾ ak8 @ba 6 p 7  6 6 à u ¾ c \  ¾ 3XE ?TI ¾  6 a)]8 7  6 3XE ·S·SDªªÊC UUOaOaNbNbnAnAdMdM\_\_^`^`g!g!ww^`^`hhnunud d npnp^{^{KkKkOO
(17) JO)KkNPO)nAN g`Oa-w NeC Cª,ª¯ª¯ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ªd C81 dCc ª ·S·Sªª 1 UUOaOaNbNbnAnAdMdM\_\_^`^`g!g!ww^`^`hhnunud d npnp^{^{KkKkgOOJfnuKkg{O)nunAg`OawuN
(18) g`hZ C÷ª¯ª¯ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯ª,ª,ª¯ª¯d ª C6Ca ªd 2. ïï ßiMä äñ .
(19) . Y . b ( m ÄwuuhOa\`eM\wpD®Cº\_ª O)g{Ä uhnA\`heMdkwp2Z |)nu\`dMO)\`g{hju\qn[^!\,Oaz©hd£\_Z]|\_nA^`dAO)^`N g{nu\U|ahdkwpd©©^`^`KkKMOaO.dNPt8nuO¦^`h[nuhdOanpOaxz2w©wAg{\xnug`tHOanuu^,ekj^`wKk^{g{nunug{ek\AnuK² ^`wpKMdO wp yMOaykOaz]g`nAtMykwAg{|{h¥²w^{npOv^!hwuNb\`¥ ¡ |)ekekdMg`g{|O)^`hd4nu^ d wpdM z£:ª©^{JKkO©KMhg{\¦O wuO)g_g{^{O)hd8|)|OO |hnuN dMwp\`hjAz]OaO)\ag!ªF\v»k^{KkekO g`^`^!KMwuO)\`g{¥Nb nAekg`dMOA|¢ ^{hnAO dM\¦ n]|eMKk\Ph|!K£nud|)wu^{dOa|!KktHdkOhji4|)eknuOadM\b\_^`^{g{KMeMw|^.^{Oaz]z%n g{dknunpNW^ ^{eMKk\`OO NPhO^`^{Kkg`hhjd|b^{hd]Kk hnA\vg`|)N wAw\`^{\Uhnug{d²Oai4wpektHhg`nuOe]^¯yMg`^`hKknuO gUnu¥[tMdk\`nO)g{uOaOaz]z2AO^!wpnpg{uO^{^cKkªORUnut]d]oq Oanu|g`^a^`ek§ \vd8\_wKM^`wuOayHZAO¢/Owp dMhjzH\_^`¨6hnudkgU nuNP^`OKk^`KMObn]|azkwp\¯NPO)g!Kkwkhj|{§ \xK%Nb¹wpnu¡ ^{^{hKknAO d/ ª\{«¬\`dMO)z]dAO)^{Ochwuz¤]¢]N ^{KkwO¯^{g`^`h Oc |{³Kk¢udkwuhj¶i4^{KkeknuO,ekykAg{KnuyHdkn4np\_^Oag{zOai4hd ekhOAg{hª 8dkª¯ wp1db ± O wu \ykwuhd4|)Zh¶^ npNP^`KMO^`O)g{ghj|NPNPO^{n]Kkzkn]O)z ]np8Kk^`KMh|!OK©nAg`t[OaoqOchOa|\^a¢u\`g{nuOaO)i4ekZ hg`nAOcd \ ^{g{w.nOai4dkwceknAud]hg`nA¡ºhhykdkz wu^{dMKkNPwpOO)gx^`z]^!g{O)wphA|Pg{O)uhdkOdkO)^v g!nuwuwug{\U|)NPhOcwp\)^`Oaªhnu¸ d/wA¢\Sh^`hKw^hj\_g{\eO)4.z]wpOa|)g!\`hzhOaAdA^`dk^¯n Oazwp^{NPKk O,nAnAg,ek^{OadAyk|!^vjKkwpnpdkd8hjwpi4^{g¯ekg{wuO^{wpdMyMg{\_ug`jwnAO)^{yH^{h\¯nAnA\`d©nuOadkz^`nZAhª d tHOb¸% |)ºO.¢8wug`g`wug{Oah^`NPOaKkz©wunAg`eknu¥2ue]K2^v^`KMhwuwpd^\`n nAO)g{udkz]OanpOad ^g z]nAhjg,\_^`hg{N htkwpekA^`Ohnu¡ºtMd8wu³\`hjOa\mzdkOa[|)hj\_Oae8\{\`wpwumg`\_Z¦Oag`^`un¦nAykhdMg`n©[wphjz]ykOUyMg`wn4\_wu^{|!wpKktMOa\¯OSOu|ªnuMd4ª ^`g{Ê nuºM¢jNPwc hdk h N ºªwpmJNPKkOvO^{kg`ª hj |P ¥[[dkhjn\_e8wpOc8z]\`uO)O g{u± nu^`hKkdkOz]\q^{O)g{y]wp^{^_K ¡ OaNPuOZ ^{g`Óhj |P/®Oa\qwu^{\hNP^{Kkwp^`O)O d npykg{nu^{yHKkO n4\_dkOcz nAg`^`N nPwuO)dkujwpOcg{|u^{nuOUg,^`K8npw^x^{Kkz]O nANPykjwuwphdMdwunug¯^{\_wp^{g{wputMO)h^ah¶¢^qZAhdª xnug!nz]OaOagvu^`O)ngc¢4z]h^Oa|)g{hOazki4O ekht8g`O)Oc^¬\®®wbO)O)|)d²nAwp^{g!Kk\`OO ^qnuvn|)nuyHd[nAu\{O)\`g{hutkO)Od8|\_OnAhje]\b^{h^{nAndM\yHO)npg` nu^`g{KkN Og{wOa|)\`nuekdMh\_^{^`wpg{tMeMO©|^{hy8nAwd/^`ª©K÷rxykdkwunudk^`dkKkhOadkg wp ^`QOaºªg`d8w U^`nhuOPO)u^{nO)gc¢ wpek^{AKkNbhj\POajdAw^^`^`^{OaKkg ONbz]O)nu^`N Kknkwphz¤d ¢ ± ^{n KkwuOa«¬d|ad.|nuwp^{NPKkykhjykyk\®hhjOa\_z©nAK g`^`¥@h¶n^!¢4\ ek8dkOvg{¥[\_yMdk^g`nnAyky8KMd©n4wA\_\_^!OvOAwp¢]w¦g{h>uª dkOuOOaª^!\ ^{nP[nuykhjek\_jgSeMwpdwunu@t]^{\`oqKkO)OaOg{|u^`|)hnuwuuhNPOcdk³O)¢@g!^`wpwOcj|{\_^{Kkn.g{dkwg`oqhjOci4OciAek|eM^`OvnAhg{g`OaZA\Kkª hj^`|!KkK hj\Sz]n[|)nAOawu\®g{dk\`npO¯^NPg{OaO^{i4g`ekhjh|g{OUOa\_nu^`g®hN Oc\qw^{^{h¶O ¡ N wA\®w^`^{KkO¯O¦wp|d[nuZPd4NP^`g{O)nu^`¤g{jhwc|SGhd]wu nug`Ovg{N ekw^{ZhnAdtMwAwp\_tHOcnuzeknu^d ^{KkhN OvwpnAutMO¯\`O)NPg{uOcOawuzP\`ek^!g`wpOag{NbuO)Oa^adAª^!J\)ªKk¸²O¯ykO,g`yknAg{yHn6nA[\`hjOaz]z.O¯|)KknuO)d4g{^`Ovg{nu^{KkHO¯O)g{^{g{KknuO)g®nuwug{\®O^{h|aO)wp ^{yknpg{Kk^`n[hjhjnu|!|KU\_^{^{hKM\SwpwtMz]^.hOch¶\_^`^qhKkZ²uOdM OanA|z g¦nAdA ^{nA^{Kkg`gUO nAyk\`jO)wpw6g{d8u,wpnu¢®gShdk\`8ekKkª%g_hj¹|{wAJK |KkOahO \x\.nA|O dknA dA^`ZA^{g{ªxg`O)nANP«¬d2O)O)¹g{ZFwug`|nA\_^agh¢8NPyM^{ykKkg`nAO¦O©y8ykn4^`g{n£\_nuOcyHz|)n4nu\_hNPd£Ocz yk^`e]KkNP^{hjOu\bO¢^`wuKMO)g_n]dM^{z2h\`|)ekhg{O\UOah\ AdMO)z]dkn]OaOa|ayHg{wpwuO)xdM"h wAz]!)\_OcOaZ[#\dANP^v $ y]nuº¡ ¢ ^{KkOaO¡ºnuOat]\_oq^{Oawp|tM^a§ hj\v\_KM\_OaK8z©wpyH^{OOa|!wuKkdMdkz2hjiAnueMOa\¯^`KM\_Obe8|{|)K²wuNbwu%\ Oag{ wMR§º\x¢@NPKknuhj^`|{hK%nud/yHª¯O)T2g` nunug{g{N O)n\SuO)wgcyM¢8wuwug_dk^{nphwu^{Kk O) gSeMA|O)hdkzkO)Oag!wpwpd%h"!ag{wOa^{|hnAnAdMd2\_^`|g{nAeMdM||^{Oahg`nAd8d/\ ª & ykAO)g{ekn[dkgxnuO)g!ykwpg{npnp"h!aoqwOawp|^{ h^`nAhnpudMO¯\)^{¢m KknunpOag{^{N¦\`KkO.O)ekg.\_^{wpwh^`NP^{hnuO)ykd2NPg{n6O)wpud4j\_Oa^{nPNP\,d8hjO)\,wdA^`^!ykeM\ g{g{n6wpwu[g{hjO©z]ZOawAO)z%|{dMKM|hhd²nAOaNPu^`OaKkyMz hjwu\\{wu\_g`\POcO)\®yHnA^`OaKkg_^c:hj\xªª j«¬wJd^`Kk^`wuO)O2zkgSykz]\_g`hnA^{nAhe]yHnA^{d%nAh\`nAOa^{d/n zFª^{|JKknAKkOadA\`O,^{O.g`^{nAKkw^`O)^`Oanug!g`g{wug{O|nu^{^{gPhh|auwphO\ Oan]dA|)^`wuhg{O¯Zz]hjnA\`N nuNPwphnudg{yknpKkhjg`|¯nu^`^{n.wp^`^{hKknudMOb\a|)ªwuJNbKkOaOvg{w ^`KkyHOanAnu\`g{OuO¢@^{hwu|adMwpz2@ykh\vg{n[wpnpj\_n.nphd^`oqKkOchj|\^`hjh\`unuObNPwpnug{nug{ykeMKkdMhjz \_N&^`KkhjO\Ocwpi4j\_eknbhykhtkg{g`n6h[ekhjN z]Oaz nugUKkOa^{g`KkOAO ª »M\`NPekg_n[^{npKk^{O)K/g{¢NPyMnuK4g{Z]Ou\_¢hjwu|)wudknpZ^{KkO)wughjhz¤Nb¢yH\`nAhdkg_^!Awpekd4j^¦wpg{\qh¶^{^qg`Z4Oa¡>dk g{pO)^{ObK²y8nuw^`Knuekykg¦jwp|)dknudkdAh^{dkg`MnAª OaJg`g{Kknuhjg,\,hyk\,g{n]^`K8|)Oawz]^¦ekhg{^O wp|)wpndx\v|)nu dMnA\`g¦hzk\_O)hNPg!wpyktkOuZ ¢ Oadkwurxg`AdkOSnu^`^`KkKkOaOgz]g{nAO)N NPwuwug`hd ¥pwpnutkOm|nuz]d[h¼@uO)O)g{g{O)udMOadM|)Om|OStHO)nu^¬®^`O)KkO)O,d[^{hjKk\_e8OwpyM/g`\`nAO)yHg`nAA\`nuOahz,dkMwpª ykyMg`n4wu|!K,wudMz,^`Kkn4\_Oykg`Oa[hnAeM\_ZvNPO)d]¡ ^{ OahnAwpdk^`ekOazg{O¡º± tMO wA \_|OcO)z yk^± Ou nuª M'g ª yH ³,nuhhjdA\^{\aKk¢®n&hdkOchN \)¢wp|)uhOg!|hd]Ou ¢®nAOg`N ^{|uwª ³^{ª hnAd£JKkh\hj\PO NPykOcwpnAdMh¶^{\POaz¤^`KM° wp^`^KMOaw\`O\_y8wpwp^_g!^{\_O)O©g\`^`OOa^.|!Kknpdk,hj|)i4wpekg{OaO\P eMwpg{Z O |!\`yHKknAnu\`dMO)z]d/Oa¢dM|z]Ochj\q\^{wphd8g{|OS^Oa\_ ^{Oawuwtk^{ekhjg`\`OcKk\ Oazhj\ wp8j^{g{O)\_g{^`®ZFwpg!O z. ^`tMg!wuwu\`|Oa^`zOaz nud©hd÷z]OatH\{nu|^`g{K hy]^{|nuekg!g{\mg{O)^`dAnA^uOwu^{dMKkz÷O)gzkOah¶\_^{hK©g{Oawz g`hnANPtkwueMu\_^®Oc\)N ª wpf^{|!nAKkg`hg{dMO ¡ ykg{n]|Ocz]ekg{Ouª xOag`OA¢ O zkhg{Oa|^`ZeM\`O^{KkO hdA^{O)dM\`h¶^qZpwpekOnpvwp®ykh O)j\(DCa :ªJKkOag`O) nug{Ou¢KkhuKMO)g
(20). ìíîï/ìíÚ.
(21) .
(22) !"$#% & '() *,+-*.,/ 0. AwwudM|)|z.ekN¦g!wueM|)|{ZKh\¦NPwunu|!g{KkOUhO)hd]u Ocnuz²g{N \`hwd8^`|hO nuddMhnu\®hj\_O O ykhj\,nudkh^`nuOa^z/hª dA&^{g`eknkg®z]eMz]|hg{OcOaz |^± ^{4Kkhj\`O)eMg{O wp8hj\^`g!dkwu|!n©¥[ hdkOcw¦^`eMNPg`OO ^{KkO n] z^`g!wuDC |C^{hHnAhjd²\®|)yknug{Kkn]Oa|g`OcOa\`dA\!^³ tHnuh^`^`KK%^`^{KkKkO O yknAt[g`nAoqOcyH|nA^a\`§ Oa\z\_K8|)wpnuyHd4O ^`g{nuwpd8zjwc ^`KkO.hd£Nb^{nuKk^`O hnu\`dO)d8|a\_wpOg{g`^`hKMOazwp^bnuhe]^P^z]t[n]ZOa\¦^`KkdMO.np^b|awpN NPwpO)¥ug!O wkª wpd[JZKMwwA^\`\`hjek\)NP¢^`y]Kk^{OPhnA^{dg{wAwp|{tH¥[nAhdke] ^ yknug{n]yM|wuOcg{z]wuekNPg{O.O^`Oahg{¥u\SOae8h\_\`OcOzz]t[n[Z2Oc\¦^{KkdkObnu^^{g{g{wAOa|{i4¥[ekhdkhg{ Oykwpg`d[nkZ²|Oanuzkek^{g`KkObOah\`\Owpykj\_g`n hnueMg\`¥[Oadkz%n6t4ZOa^`zkKkuO.Oa\a|ªnAdA«¬^{d£g`nA¹wu|w6^a,¢^`ªbKk«¬O dM\{z]wpOaNPOaz¤O¢\`OO^ \_wu^`g`g{OPnudk^`g{AOawpZ¦^`tHOaz²O)hKMO)O)ug{OO.^`K8wu\w^m\`tHeM|!nuK/^`K ª©[Jhj\_Kkhhjnu\d g`wuOadMykz g`Oc|\_OanudAd4^{^`\g{nuw©MwAg{\_ekyHy]Oc^{|ek^{g{\mO wpnpg{OyMhd4wp^`g!g{wuhd8zk\_hAhj|)NwuZh|)^`nuKekg{ykOa\`OayHzPOayk|^,g{n]^`|)n2Oa\`^`\`KMOaO.\a¢uwuwudM\_z ^ N wudMwz2oqnu|)g{nuh¶^qdAZ£^{g`nunAv|O nu NPhj\q^{ykhe]dM^!w^{4hnAhj\`deMwpwpxg{O,\`O) g{nuug{nuN¦hdkek%wp^`^`OcOcz2|{Kk\_dkOahjyMi4wpekg!Oaw\P^{O)^{nZuªzMrUw^`¶OA^{Kk¢®nuekKkAO)Kg{O |)nu dMOa|)wO)^{y]ek^{g{eMO©wpO Z ^{g{wpwAyM|yH^`hOanAwpdFhdkykMg`nk¢M|^`KkOa\{hj\\ jAw^`nA^`tMO)wpgek|)dMnu|dMnA\_ek^`g!ykwphOadAz¤^!¢[\U Ocwpwg{O^`eMdkg`npO)¡:^vtMOawAwA\_\_OcZz ^` g{nwuOaNbN¦OatHOanuzg{¥Phd4yk^`g{n.Oa\` O)OcdAw^!^{\ekg`\_OnANP|)nuOSg{g`g{OcO)\_O)yHpnuwpdMdAz]^Oaz]dMg!|wcO¦wptMuwAnA|{g`¥]h\a^`ªKkN »Mnu\ g OC6 wuº¢HNb\`yMeM|!OAK ¢ wAwuyk\yH^`KkOawpO¦g! wpwAd8||^UO¦^{|!KMKMwwp^SdkjAwpOag{\vuO,z]ekyHO¦nAg_^{^{nhnANPdM\xnp^{nphnAd^{KknuO \{^`|KkOaOPdkO\{|NPO)dMn6OAOg`O)jwph¶^{^`KhuOw.^`|nnAKk^{O)Kkg{ObO)d4h^UuK4g`h^{u\ahjª,z©r®NP^_^`npOa^`NPhnudy]^{¢8\vnugx^`n^{KMhNPw^UyH^{nAKk\`OO ^{KMKkwuOadM\`z¤O.¢|^`nAKkdMO2\_^`g{g!hwpAhhdAzk^{h¶\,^qZ£wug`npOPSe8^{Kk\_eMOwu\`|ZOadkyHO2O)g`wp d8nug{zNP^`OaKkz²O2w©g{nuyHtknAeM\_\_^`^`OadMg`hOanu\{g{\h^`nh^`KMhuhdKA^{h^{dkKkh\,|{ KMg!wpwuNPdkuO)Oc\PnA|)g`wu¥@d ª tH& O©d²O^{¼@KkOaO |^`nuh^`uKkO)OaZ g hdM|nAJg`yHKknAO g{yMw^{g`OanAz.yHnA\`hOa^`zKMhwpd2ykz]yMhg`g{n4Oawu||!^xK£NPhjO\b^`KMpn]wpzkhzk\a¢kwp\`^`O)OcO¯zOA^{ª Kk8g{ªnu ekC 1kuKF¢C)$:wuª g`hnue8\¦g{Oa\`ek^{\beM\`hdk²nut]oqOa|^{\Pnp¯z]h¼HO)g{O)d4^ \`wAKM\®wpy8yHwOc^`\)K©¢]jykwpjg{wpudkOvdkhhdMdkh¶P^{hOwu /wuzkNPh\`ykykjOcwu\)|ª OaNPO)dA^!\)¢kwug`AOSOag`g{nug!\hd^{KkO|)wpNPOag{wM§ \®hdA^`Oag`d8wp¤yMwug{wuNPO^`Oag{\a¢]wu\O) " ,® ¦ , (
(23) . )|a|fn[wpnANPnudMg!O)\_z]hjg!hz]w dMO)wg©t4^{ZOaw \ w©1p^`g!wp∈yHdMnu\`PjhwdA^`ª^©hnuJykd g{KMnpO)oqd/Oa¢/|^{hOa^{\¯z hdAhwpd ^`dMOaz^`dMKk\_w2Oh^qZg`g`nuO)^{ wpwuO)g`^`ekOahnudMObd |hjO%\¯z]hN Oadkwpnpu^{O OazI t4Z¢wuwp\ IdkwFnu(p^`KkykOah ) gªOah¯N ríwu^{h¶uO)^{OgKøz]Kkhj\_nuhjyM\¦NPwAwAnu||)Ahi4O)dkekdkhOag`nu^{OcKkeMz¤O \ ª JKMwp^xz]hj\_yMwA|O)NPOadA^U|)wpd tHO,t g`∈OaykRg`Oc\_OadA^{Oazhd KMnuNPnuAO)RdkOanu∈eM\SO(3) |n[nAg{z]hdMwp^`Oa\wu\ I ¸ · ± Cc³ R t ∈ SO(3) × R . T= 0 1 NPO^{O)g{h !aOaz2¢[KMO)g{KkOOag`t[O Z2^`KkO zkw O) dkhj\Snu^`nuOa\mg`^{nuKk^{Ovw^{wuhdAnA^`d h¡¬u\_Z[NP∈ NPRO^`g{:hj|xkukN w^`=g{h 1 wAwp\`dM\`z%n]|)wuhwdk^{uOazbO^`npn g^{{JKknpKk^!O¦OPwu^`g`hOcnunu|d^{^{wpnuθ^`g h¢[nuh:dªªUOAªh¸²\¯RObyM=wpwpg!jwp\_exp([uθ] ) [r]. n A n 6 n µ ` ^ k K O M e ` \ M e p w r Ocz]\qOa^{dkhN np^{wOa^{\Ouwb¢w©N Nbwp^`nkg{hz] h 8nuOa z²!)Oaug`O)n4g!\®\_hnpnud/wp¢¤yM^{KkykOPg`nA^`ykg!g{wphd8w^{\_OvdkyHnun4zk^{\_hwpOuNP^`¢hO)nuwpdMdMd8\`\zhnuvbdM^`Kk,\aOªv 6, ¡ºvdknu,g{kvk NÅnp^{nw g`Oawuykg`g{hjOawp\`tkO)dAO ^Uvg{ª.Oa\`T2yHOanA|g`^`Oahun6O)uOaZ ga¢ wu0d ∗. ∗. ∗. 2. ∗. 3. 3. 3. ×. . ×. "!#$% '&)( *+$-,./!$0$1 23$%(4. 0. >. l |)g{wAnp\`oq\POa|nu^`Shu^{O g{wuAdMO)\qnu nANPg`ON ^{g{wpZ^`hnuhj\ dM\wpdF^{KMOwp d²^{O)oqdMeM\`\_h^nudFg`nunp^{wS^{^`hnAKkdMO \ wpeMdM|zFhzk^`Oag!wpwpddM\`jAwO)^{nAhnANbdMO)\)^`¢g{ZuhdM¢m|eMKkz]h|!hdkK÷z]hOad÷\{|yMg{hwpt8g`Oc^`hj\P|eMw%wujgbwpg{^{uKkO)Og ïï ßiMä äñ .
(24) . Y . yHhN O)g!wp\`uyHO¯Oa|yH^{nuhuhdAO^!yk\ g{npoqOa|^`hnud¦wuyHg`O)O¯g_g` nAOag`wpNP^`OcOczz,t4t4ZZwS|awpNPO)g!w C6ºª«¬d^{Kkhj\uO)dMO)g!wpu g!wpNPO)nAg`¥@¢|nug{g{Oa\`y8nud8z]hdM p↔p ¢ ¡ ± u³ p∝ G +e q p +ρ e , | {z } KkOag`O^{KkhOPd NPnu¢Hg{ykKkKkhhj\`O N G ∈ SL(3) hdMyk|)g{npeMoqzkOaOa|\v^`hu^`KMOOPyMwuKkg{nuwuNPjnAw u g!wpykh¶^{K4KZ2g{wpOa^¯\`yHhd OaM|^Udkh^`^qnZ w G± hdwpudMOadkzO)^`g!KkwpíO ³O)[ykhg`hyH^`eMnuwuO h x \ { ^ k K ¦ O e ∈R ykhd jwp± dMCcOU³g{hj\O)yMw¯g`\`Ocek\_IO)tMd4\`^`yMOcwuz |)O®hdnuρ8^`Kk^`KMOv∈O hN R
(25) hwuOuuO g{InueMy t[Zb^{KkOvuOa|^`nu¢ug qKMO)∈g{ORwpdbªKk«¬nAd8Nbz]O)nAOauz/Oa¢4dk^`O)KknuO e8\
(26) OahOSOaANbg`nAOadAek^ y nuSO(3) ×R 8 ` ^ k K O j w^`^`Oag SL(3) × R hj\g{O)ykg{Oa\`O)d4^`OazKkO)g{O¯t4Z ¸ · ± 1A³ G e ∈ SL(3) × R . Q= »Mg`nAN iHª ± u³¢Mwb®wug`yMhdkPnuyHO)g!w^{nug w(0· ; Q,1 ρ ) : P 7→ P |awpd^`K[eM\tHOz]O MdMOaz¤° ± 4³ p 7→ p = w(p ; Q, ρ ). ∗. ∞. ∗>. p. ∗. ∗. p. G. ∞. ∗. 3. p. ∗. ∗. 3. 3. 3. p. 3. ∗. ∗. . . . 2. 2. ∗. ∗. ($-,. !# # $-(! )0 (1 % & )( *+$-,. . . . . J|)nuKkdAO2^{g`KknAnAHNb^`KknAO,ug!NPwpnpyk^{K4hZ4nA¡:dt8nuwu\_OawbzF|awpÅNPO)4g!hjw¦\`eMwphx^`K©\`O)g{g{Oau\`nuyHhOadk|^^{^`Oan |!Kkwbdkykhji4jwpekdMO2wugykg{nunut]yHoqOanA|\`^aOaª z»Mhnudg®^{KkUhj\)KM¢Mwu|)\ nudMwA\`\Phz]nuOat]g®oqOa^`K8|^{wh^uO ^`Kk^`hjn \ nAJt[KkoqO¦Oc|NP^Oh^`OaKk\nknAz©dhj\x^`Kkhd©O,yk¹wujwp|^UdkO tMwAΠ\_¢kOcz KknAhd |!K ^{Kkhj\O¦dMg{Oanp|^ng{uOaO)|)g{nuZ.dMnu\_^`g{^{eMKk|O¦^{Oaykzg{nO)oqh^`OcKM|O)^{hgunuO¼8Kk¡ºnAhNbdkO¯nAunAg!gwpz]ykekK4g{Z hdkb^`KkhOdMz]\`O)eMg{|uOcnuz©hdkt[MZ ª ^{Kkh¢]\ ^`ykKkjOawpd dkOtHO^q|)wpOad O)dbtHO¯^q^`g!n,wp[dMh\_Oa xnug{\)NPªOa«¬dMzz]hO)dAOc^{z¤n ¢Awuud hu O)dPeM|wudbhzkOaOa\_wp^`d©hN KkwnA^{NbOnAnuu8g!^{wpKkykOUK[|)Zbwp^{NPKkO)g{g!nuwkek§ u\K hdA^{O)g{GdMwpkyMwug{wuNbO)^`Oag{\ K G ± u³ H = K G K, wA\®®O)/wA\wP|{Kkn4\_OadhN wpAOvyHnuhd4^ ± wpj\_n |)wuOaz |nAdA^`g{nu¤yHnuhdA^³|)wudtHO,dknug{N wp"h !)Oazt[Z ± A³ m =K p . JKkOv^!wu\`¥ ekdM|^`hnud e = £e , e ¤ ∈ R h^`K ( e =H m −m ± p³ [e ] = H − H hjwu\g`nA^{KkekO)dMd2zPyk^{Kkg{n6OvAOcO)i4d ekh h/tk^{g`n hektHN&Ohnk^{|)KkwpOvZ.OadAhj^`\_hnAg{NPOvnuz]g{nAykN Kkwphj|Uhd^{nPnp^{g`KknuO¦^{wp|)^`wphNPnudMOa\g{wbh\yH|nAnA\`dMOu\`ª hz] UOan6g`®Ocz¤O)u¢[O)\`hgcd8¢k|h¶Ov^UhjtH\xnpdk^{K nud]¡ºhdpoqOa|^{wphdMuz O θ=0 wpg{OvN wpyMy8Ocz^{nPt4Z e = 0 ª θ=π ¾ £]O @ 3 G G Π. Π. −1. Π. ∗. Π. > νΠ. > > ωΠ. Π. −1 ∗. 6. νΠ. Π. ωΠ ×. Π. ∗. ∗. > Π. ωΠ.
(27) !" $#%&. ')(*+ , $.-0/213 4-5 !6 $# 78:9 :1 ;)+ (<#= ; Π 9?>A@ B'!. 6(<1A+9%+ C D7& 0EF
(28) G9H $#&3)-:2: IJ $#<CKLD ,-0 G ∗ C $# 5 (M N:9 & $O 9P !Q $# R(<1A+9<. Π = [n∗>, −d∗ ]>. q EJ]Y!^-N/ :: I_7_ :9F .#%8 q∗ = K−> n∗ p∗ Ii $#<a(*+ +113+U4!A N ; 3 E ∗ j E EV kE ∗ Π p Π ρ =0. 9S $# T(MVU%:1W-N 0X29%8 GY= G:;ZE[E\+E. 3` .# a(< NbN -0 .A+9Z+!^4K+cH(^ 9 ed8:1O 987fg9&7h . ìíîï/ìíÚ.
(29) .
(30) !"$#% & '() *,+-*.,/ 0. " Å £m )¦P ) y^{kJhjnAKkwpd hjd8\,wpu\`gOaOadkg`|OaO)^`Ag!hnuwpwud%g{h z]!ayMOaOag`\®\{Oc\ yk\_Oag`^{dAOaKk[^{O¯\,hnANPwueMd%np\®^{ekhnAdMOad hM¡º|aOcOawpz\_g{^{g`wp hg!tMOawpz.hjNP\_nAKMOae]Oa^z.nut4g{ Z.nA¥©g`^`N¦KkKMekOO)jw|)g`wp^{ObhNPnu^{dMKkO)g!\aOPwkª nAªt[ToqOcnu|g{^¦O)n|)uwpO)d²ga¢AtH^{OPKkO¯Oah¶yM^{Kkg`nAO)g,y8n4yM\_Ocwuz dMwp^{g¯wA\_nA¥ g¯ dkeknAdMd]|¡¡ . #$ ) $
(31) ($-,. !# ) (, )0 ?. f. . ¬«[dhO)xnAg{\az]¢ Oa^{gKk^{O2n%»Mg`ekOcdM|zknuwuO)NbgOa^{dAKk^!OwpxyMNPwug{wpwu^`NPg{h O^`Oa|g{nA\ekjzKMtHh|!O©KOcg{\qO)^{jhwNP^`O.wp^`^{OcKkz O ykDg`CcnpoqºOcª |^` xhnund£O)nuuSOawpgad£¢mhnA^ t[hjoq\POc|dk^Pnu^.t8O)z]^¬O®MO)dMO)Oadz^¬ ®nAn g ^{^{ykg{n©jwuwp\`dMd8\_wpjh¶gw^!^{|{\`hK²|)nAO)d tHdkOhOc\®^q\)¢z]OaO)wpO)Ad8d²O)zdkNPOahg{n]wp^{z]^`KkOuOaOª \ x|a±wpOawNPdMP|O)dkOug!O w©¢]w»Mekg{ekd8nudMz]tkzkO)eMg{wu\_uNP^n[O)OcNPdA\O^!w^`wpKMykn]N ekz g{wpOAO.^`ªg{8g{h ªnp^!¢/DCawKk^{nu8hnANP^{dnnuANPz]g{Onpwu^{^{Oaykh|K4nA^ZAd^`¢@Kk^{OKkn4^!O|p\_OSªÊ³£Oczk\q^{O)KMhuwuN Oa\vdkw^`O)^{n©g!hnuwutHd|)OPhOcnu\ eM\`wu^{OadMKkz¤z O ª NPnurU¶nu^{^{g{O)KkOUg{OdMAwO)g{^{dkOh uOaO)g{Oag{wuO)Z28d8[|^{hOn©\`eMheMNPwp\_8hwudk^{ug{ O wA^`|{KM¥[hOadk\`Ob^` OaOcw|{KM^{ekdkDg{hC*i4OC¡ºektM:Ovªwu\`JOaKkz¤Kkhj¢/O |!KN¦wuOek\{\`¶^{ekyMhNPyknAy]h¶O ^!^`\®hK[nuwpZ[d£yHHnpnuhN ^`KM^{wpOaKku\`hjOaOS\¦\vhd]^{^`g{Oa wAnu\_|{g{^`¥[N hdkhdk©w^`hNPnuykdOg{^{n]Kk|)n]h¶Oa^{zkz]Kk\aekh¢d g{O.w¦OPhjg`\¦eMOa^`u\`K8O hnuwdw ^ ^{nAKky]O.^`hNP|)wuh Nb!cwOa^`g{hw nudnutM^{Oa\`|!O)Kkg{udkOahjiA\¯eMw©Ob|)Rhjnu\,d4wp^`yk⊆hd4yMekIhnAOaeMz\v^{^{n O NP^`ekhdkg{OahNbz"h !)\`ekO g_zk¹wAhg{|OaOA|ªP^`«¬Z©d²^{^`KkKMOPh\¦hd4|)^`wuOa\`dMOu\_¢h^qwpZd²z]O hj.\{||g{hO)O)yMdAwp^d8\_|OchOa|\Snud8tHz[O)¡:^¬nAg{OazkO)O)d g ^{X[KkekO²t8\_|Ocek|g{^`g`hnuOaddA^ ]wpªÊudM³mz ^`K8^`wKk^xO%Nbg{Ohdk O)hg{NPO)dMh !a|)O¯O^`hKkN O¯wp unAOc\)nª hJdk KMw|^ n4\qh^\a¢x ekdMO²|^`\_hnuOad/O)¥F° ^{KkO%yMwpg!wpNPO)^`O)g!\ Q wpdMz ρ ± \`O)O i ¢ ± QA³ 1 Xh ¡ I w(p ; Q, ρ ) − I (p ) . 2 dkJOaKkøO¸ hnuNPh¶tk^{^{wuK%wpuhg{O¯dkO)OchA\xz©wuwug{\_z©|)O)i4^x^`eknnuhg{^`OayMKkz¤wpn4ª g!\_wpOPNPz]O)O)^`AO)O)g!dk\Oa|)g{wuwAd |hOatH\aO¢@weM\`|OanAz2dMwu\_Oa\xg`wupd©w^`Oahu\_O^`hN \_nAwe]^`O,^{hnA nudg^`^{n KkO¦g`Oc\{|wpnNPuO)O,gUyk^`g{Kkn]Oc|\_OcObz]ekyMg{wuO¯g{wuNbKkO)O)^`dOag{w \ hjz]\Ph¼@wpO)ykg{ykO)dMh|OaOcz\,hhd \,yHDC*OaC g_: ªnAg`JNPKMOawz%^Pt4hjZ%\)¢ eM \`nuhgPdk O)uO)g{ZnudkdkOaZu ªrhN í^{wpO)Ag{®O wpwpg!dz¤¢/hdk^{Kkh^`O hjwpg{O)wpN ^_^{wuO)hdMNPzky]O)^Pg,^`h\n²|)O nu g{ykg`jOcwp|h^`d£Ocz^{KkO h^`hK²N wp^{KkuOO yMJwuK[g{eMwu\)j¢ w Oa\ hj\Sρnuªdk xZ©OaeMg`OA\_Oc¢[z O,± wpwudMzMzwpy]^®Kk^{OaKkdkOvO)uQ^`OaOcgS|{KkeMdk\_hjOci4z©ekO¯^{Kkykhjg{\SnuhjyH\Un4N \_Ocwuz zkhOd \`DhC8N¦1eM@¶^`^!nbwpdknAOaeknugeMyk\`g{Z©noqOc|h¶^{^{KhuOS ³nAg`^`N¦nekO j wyk^{jhnAwphd/d ª ^{± KkhN O¦wphAN Ocρwp³xAnuOy]z]^{hhjNP\{|"hg{!aO)wyM^{wuhnudMdM|\UhOa\O h|)mO)wpOcdMzkz©\x^`nAKMdkOZ hNPh¶wu^`uKMO¦Odkz]nAh¶¼@h\`O)Oug{ªvO)dMJ|)KkOhj\StHO)NP^¬hdkOahO)N dwpg`OcyM\_wpeMg!¶^{wphNPdkO^{|O)n4g{\qh !c^Uwp^`wphQnuekd©Oa \g{wu g`NbnAN
(32) Oanu^{Kkg{¥O ykz]g{hj\{Oa|\`g{O)O)d4yM^{wp\bd8N |hwpOa\ad4¢AZt4Z \q^{g`hdMOadk|peM^{z]KMh\adkª»^`Kkhg!Ov\_^ayM¢mwug{hdwuj^{w Kk O2Oa\m|)wuwp\`í^`OO)g{^{®KMwwu^ g{z QnuKMeMwu\Pz.wuyHO)g{Oag`^`wAeMz]g`Zt.O ^` KkykOvjwpOch\qdk^{hOaN zwNP^{OSnAnp\_^P npUh^{d.Kk^{nAKk\`OO dkN O wp h^®dA^!h¶^{wpO)hdg!w^`^`KMhnuO)dN
(33) ª»khd ekg`^`^`KkKMO,O)g`NPNPhdknAg`hNPOA¢A"h !anuwpdM^`|)hOxnud ^`KMyMOSg`n]nuyk|)Oa^`z]hNPekg{wuO¯8y8\_hwpdMg!|)wpO¯w OOcz]\ Oawuwug`¤OUKknuO)t]g{^!O¯wphdkhOa^`z¤K©¢4g`^`hKkuOahjzg`OSnuhjt]\moqdMOa|n¦^!\g{QOanAwudk\`nuZudPª ^`n ¾ £]O @ : . ∗. ∗. ∗. ∗ i. ∗ i. ∗. 2. ∗ i. ∗ p∗ i ∈R. ∗. ∗. . *. bN,-, TD# (^J . [#<. . . :8- 9874;)+ $# &XZX8J9 8 e/2:6925(< O e9&!. k;58 $O+9F. .# : . &d8/= $# -:;): . -NI. . Q. ïï ßiMä äñ . /29MX: 788Te(^/ Y N 8 .A+9';)8 $O 9iE. 1O:1VZkU2(<1A+9=e+11M;4+78QX23N-k (i+9M-kAN0E. ,K ^ . : $# J!Y $# 6&dNbN -0 ρ∗ = 0 [#&3 # 0(&(^:9% d8,-8/2,Ifg9F $# N:. 8/ `- 9< kUM 1AX_` GR1ONf ,987Z/9%+1 Xh $# 9& $A+1%(* :1g1A UF N1./fN 3W(<1A+9% Y Tg! $#<-+;): . . &d88/& J .# Q&d2L. [;) .A+9iEW
(34) G9!$8-0 I88Xf(^ .g9&74 .# a . :& k72`(< NN(^+,X`9.
(35) Q. Y . . #$ &)( & +$ ),. . ± yH»Mg`n4OnA\_ykNøO,g{^`nu^`n yHKkn4O®^{Kk\_yMO O¦wphg!g`d,wpO)^{ NPO)Kkg{OhjO)\^{dMO)\`g!|)ek\O tMQ\`± Oaz]|wpOa^{\`dMhhnAz g{d¦Oaρz8wU³\`Ocnuek\qdM^{h¶^!OuhN wpªtkwp«¬dM^`OmOaz]^{zOawuOa\`t4z¤¥SZv¢H ^{eMeMKkdM\`Oh|dk[^{ hhnA\`eMdwpi8 unuª ^{gg{± y8wA1An4|{³x¥[\_wphh^{dkdMhxznAdkNPwphdMd2O^{OaKk^{\_n]Kk^`z hON g`\`wnAO)^`tHO®O¦npX]^npek g`tM^{nA\_KkNµOcO|^{wu|)hdwpnANPd hdkO)1khg!^`Dª wkCchjwp³§ \ ¢ hdA^{O)g{dMwp¤y8wpg!wpNPO^{O)g{\ K yHO)g{NPh¶^^`n nut]^!wphd ± RA³ H = K GK wudMz ± Ca 4³ e =K e . ± ^{JhnAKkdMOcQxwp\_O¤O yMOa^cdAwu¢4^`g_z]h^^{O hnuMOc\xdk^`OU^`KknutHO¯AnpO^!^{^{wuK.Kk\`¥ O)wgU qekykdMg{h¶^{|npK©^`oqhOanu^{|d/Kk^`hOª uO|nud4w ^` g{hnu\ npyH¤nug{hnpdA^!^ w^`mhnud 4³htH± O¦KkeMhj\_|!OcK.z zk n[nuOagS\ z]dkO npM^dkKMhdkw6 uO^{KkdkOOc|^`Oag!\{wp\{dMwp\`g{jhwZ ¡ µ∈R ekdkh^xdMnug{N ³m g{nuN H ± R4³®wA\ ¢ ± C*C6³ 1¡ [µ] = H−H , wudMz w qyMg`npoqOa|^`huO ¯wpdkAO¯npg{np^!w^{hnAdwA\ 2 ( ¡ ¢ h ¡tr(H) − 1¢/2 ≥ 0, arcsin sat(kµk) , ± CcA³ φ= ¡ ¢ u n ` ^ k K a O ` g j h ` \ O , π − arcsin sat(kµk) , KkOag`O tr(·) z]Oadknp^{Oa\®^{KkO¯^`g!wu|)OvnuwbNPwp^`g{h wpdMz^{KkO¯ ekdM|^{hnud sat : R 7→ [0, 1] hj\xz]O MdMOaz©wu\ ( h 0 ≤ x ≤ 1, x, ± C814³ sat(x) = h 1, x > 1, & v|)nuekg!\_OA¢ h kµk = 0 ^{KkO)d µ |)wpd tHO©|!Kkn4\_Oad \wu`n%g`tk^`h^`KMg!wpwp^ g{harcsin(·) j h. \ % w { g a O u w ¶ º ¡ p w k e c O z k e M d | ` ^ h u n / d ª Zuª ;>= ¾[ ¾ 3 ; k8 @ba 6 u8 7  6 6 à 7¹   O \ = 7> !%g '
(36) XY 0, # " Hm + ρ e − m ± C)[³ e ∈R , µ = e= ∗. −1. 0 p. −1. p. ∗. 3. >. ×. ∗. ν. ∗. ∗ 0 p. eω. φ. kµk. ∗. 6. D8KK$#%!X,> 'g ! ( #% c O 8/ 0 >XY+g `!% X),. ` !% 0g+ ( $,% Y θ = 0 (# #%+] ']. e = 0 ω S Â/Â a JKkOvykg`n]np/np/^{Kkhj\®h\`nuNPnug{ykKkhj\`N hj\ykg{Oa\`O)d4^`Oaz.hdrUykyHO)d8z]h ·vª k¢4Kkhj|!KeM\`Oa\^`KkOvg{Oa\`ek¶^!\ g{nuN
(37) O)NPN w-¯C hd©rUykyHO)dMzkh rª ¾ £]O@ iE : D ;Q(^ , 9 `9 G &d8/= ' .# !G/9M-, $O 9 X &9 X 9 > B43h $#%& ` 6 -N 9=0 $ ,/8-0 G,XR!A N ;. (< NbN -0 $ . !.+ ;58 .A+9R&d8/= * .# 3e8-G#&O X%E. 8dbN,-, E. . 9 $ $Oe 9&1VfIE\+E. . "!. . . j $#</& W;)N2/2 987+ T N 8/ kg987+925;) . .- 9%L [#&3[1Y;5+9=T .#%8 % &d8/20 $9 N G -:;): M 9 G: k9%+1 (* :;5+ k. ìíîï/ìíÚ.
(38)
(39) !"$#% & '() *,+-*.,/ 0. ¾ £]O@. 9W 8g9M-N6 $#<(M (^+1AJ. . -N ;Q(^/& G,X. . ;) .A+9. g9*-. >. . [# C $ +9= 1A8 $O+9. 9 . ;5bN C-+9M- 9H 8X&X2 $O 9P GF .# . : k + C>. ± CcA³. . 0B63X,-N/0(<1O,XZ!A +;. . $# ' 8 & $O 9. >9 k7f1A -0 .g987S: k N k 9 B E j IM! $#<'8dbN,-, K−1 K t = t K g!J $# ` ,-N2 +: X - k N(^ 9MX_` .# ' . 0/f` 1/& $O 9>d8,-8/2 ep 8X2;k d+1OR 1/& $O 9 XNQkU80 $B,E + : , $# 1AN I $#<5-/,(<1g987h9H> B6`9 e. e0 = K−1 ep = 3 (<1A+9% 6 .# :9 p 9 3 e9 T0/ ;) N4 .#%+9. (< N&-N I j `-N/1.X)/_:J [ 1O:1VZ . X &9&987. eν = e0p. +B,E. . 9S $#<6 . :8- 987. -N 9=0 $ ,/8-0 )X -/,(<1A Xh $ +9%D1A8 .A+9h N 0I=E 7<EadN. 9=0 N8XS+! $# 8 `X &9 XZg9. R. .# ) &d+1. . . (^ d1AQ;)&X2 M-:& $O 9H>.
(40) . . . d8,-8/25C(i8 $# (<1A+9&9&987 6(^: k!. k;5 X. X2 *: N:9 .1 fI%0/8-G#F. B0I. . j 5-/21$XZ15#%_ 4X8 &9 X $# . [eω ]× = 2[µ]× = H − H> ,. . . . . . = -0 $O 9 B,E
(41) G9. >3:0. 8 & $O 9%1 : k + . ± Ca4³. . GY N(<1A-N 2X &9<,X 9h> DB0I+g9'8/ *78:9<: +1.If/9& ,X A! +;) j ME ` j + : 0I, N:;5 2&d+1A HΠ −H> ;Q(< _ :;)9 .e Π-#&A+ ,Xh $#& N/+7f# X =9 ,X49P> 0 B I j #&.-#Z 0 & Xg9 N 11A+ _E eω. [ Â Â >] ¿ 3 G ¾ 6 ¾ ,7¹ ¿ 6 Ã 7 \ Â N ¾ ¾ 6 a . !% #Y#%O +] O
(42) X $, ¤ £ £ > ¤ `!% 6 H $, > > > > 6 e = e> + Y+H ν , e ω ∈#YR ! .`/ 0 " . $ !%#% ce OΠ = `!% eν#YΠ, #%eOωΠ +b∈ $R h "c $ ) !X +-#% ! # !0! $,"#$ %$ X `!% ( X # / $ X $, Y 0Xc Y #%O `!% !*.,+H $
(43) e0 = £e0 > , e0 > ¤> = £ ¤> # #%+-'&(*) !X,! X g- Π ( 0νΠ,Kω $Π K ' (αm − m∗ )> , θu> ∈ R6, α > 0, `!%Y K ' `!%M#Y Y> 0 > e0 ωΠ S Â/Â a JKkO,ykg`n]npnp^{KkOa\`O\q^!w^`OaNPO)dA^!\hj\ykg`Oc\_OadA^`Oczhd©rxyMyHO)dMz]h ·Sª k1 ª. . . ;>=. . #$ , ( % / 4 ) /4# . ¾[Â ¾. T. : +. . Â u aakV7 ,7 ¿ v = Λ e,. ^. D. . !%K 0 . Λ = diag(λν I3 , λω I3 ), λν , λω > 0,. ± C6u³. ^! v = £ν > , ω> ¤> ∈ R6 #Y - `!%h $% 0,Y +H $, M O 8 0, +H ! £ > > ¤> $ 6 OH+*, + -# X "D8 O #Y $, ' .) # 8,+ + ! e!&=#%e ν ,meω∗ >∈!O R "!- ! , #` . D $ O> `!%+ Y #YD `!% *.,/ & -/b , $KK S Â/Â a JKkOPykg{n[npmhj\¯ykg{Oa\_OadA^{OazhdrxykyHOadMz]h ·vª M¢/Kkh|!K²eM\`Oa\S^{KkOPg{Oa\`ek¶^!\S g{nuN
(44) O)NPN w CPhd. rUykyHO)dMz]h rª ïï ßiMä äñ .
(45) Cc . Y . M^{K KkwAO\ ¸ z]!)nuO)h¶g{NP^{n K%hdMyMg`wpOcwud4\_g{yH^xwuOcykj|w ^Swudk^`n± O wpvwd8^`KMpzO®^`yMK[w6wueMZ]g{\\wu||)g{wpw nAd² \{\`tH|OanAO \dM^`|!z]KkKkhO,^`n4h\_nunAOadt[d²oqhOcwud²|\v^cJª ^`KMKkJO O)KMnA|)O)g`nug`OaO)d4NÅ ^`nug{g{knuOu¢H¢[^`yH^`KkKknAOaO¦hg`d4Ob|^!³¯wunA\_\`hOadMz[wc|)Z]¡ºOu\Un]¢nuO hd²y hj\q\`^`^¯Z]Kk\qOPw^{O) yHnAN
(46) nug`N¦hhdA\xek^¯wujw®^{Kkhw6nAhj|{Z]d/K \ ¢ yHn]nA|)hwud4^xZ|)wAn4\_\_Z[OvNP^`nby]^{^{npKk^{Oh|a|wpOadAZ ^{O)\qg^!wpnutk^`OuKkª O, UnAn6t[®oqOcO)u|^aO)ªgc¢k nugUg{nutke8\q^{dkOa\{\g{Oawu\`nud8\)¢kh^Sh\S|nAd4uO)dMhOadA^x^{n|!Kkn[nA\`Ow , " ^{^{rUg{Kk¶wh^{\aoqKkOa¢knu|wbek^`nAAwug`K²Zg`A^`nuOvKk¤O.z]^`nuKkNPN OvOwp|)^{hnuKkddAn]^{z²npg`nAhj|M\,nuyHg{d[nunuuhtkd4O)eMg{^u\_h^¯Oad dM^{^{|n2KkOUOSj wpnuhg{N g®u^{OPwuKku|aO¯Oxwp[NPtHhjOv\_O)eMg!wuw wu\¤|a|\_wpOanAg`h\`utMO)nAg{ZPwphdk^`wu¦hnu\mhjd%yH\nAOawu\{g`|!\_g{Kkhnutkhg!Oa\)uO¢/Oa^{zhn^\`w¦hjh\dM\_|)z]^`Ovg!Ocwp\_®hhug!OvwpKAtkOa^d]Ob hdknu^`g!OAKM|ªwpOS¸÷^,^`K8^{hKk^{wK O^ wpwu^®ykykOcg`wunA\_yk^®g{h\`wpeM^`|!OxKyMww^{yHK.nAh^`d4g!^wu|!wu¥[h®dkw6 Z]\mg`OaN wphdM\mhd.^`KMh\OSyHhN Oag_wp unAOAg`NPª»kOcnAz¤gmª^{JKkKkhj\)hj\¢]hhdM\®\_wA^`|)Oc|wunAz.NPnpykz]hj\`g{Kkh4Oahz dkt4Z e(t)g`Oau−→ wudw 0 k e j w { ^ h k d ^{hNPO¡ºwug`Z[hdkb|)nud4^`g{nu@Oag`g{nug e(t) −→ e (t) ± C8Q4³ e (t) = e(t) − e (t). ·®wA|{Z KkhwpO)tkueMOc\`z¤OU¢@nphdM/\_dM^`npOawA^{wpz^`hnpnud/w¢4OSwpg`ekOaObykg{z]OaO \`MO)dkdAOa^zhd ^{hKk^`K%h\g{\_OaOc\`|yH^`hOanu|d ^Ue^{n^{wuKk\ Ob^{Kkg{OvO z]O)g{OcO)\_dMhg`|)OcOz |)g!nuwpd4NP^`Obg{nuwA8\UO)g{^`g`KknAg{g nueMwuuKkeMnAOe]^{^Un¦^{tHKkOO wug_^{hJ|)KkOAOvª \_^`g!w^`OauZbykg{Oa\`O)dA^{Oaz.hd.^`Kkhj\\_Oa|^`hnudh\®zkh¶¼@O)g{O)d4^ g{nuN Qº¢4KMO)g{Ox^{Kkh\®jw^_^{O)g®hj\®|nANbyHn4\_Ocz ^!t[g{wpOaZi4g{uek^{OKkhg`^cg`hªOadkO.«¬dMykz]wpKMO)d[wuOcZ%z¤\`Oa¢pNP\¦wUOwp\`^`dMhg{Nbhjz| yMg{hd]OaO®i4 \_nAek^`g`hg!N g`wOc^{wp\,O)^`uwhnuZvd|hjn4\wuwp\`dMg!Kk\_znOPtHNPd,O)O)h^`dk^`nSg{htH| hO®d8Oc\_z]\qe O)^{ yHhN O)|)d8whOaz]^{dAO.O)^d4np^`^nvnp^`wKM^`O.^{^{wpKkdkhOdnAnug`nAN t]ekoqgwpOamyk|^ceku§ Oag{\yH|\_^{nAK8nu\`gwpOayH\anu¢cO®h:ª wp^{OuKkdMªOzykhnp^`KkwunAdM^{Kke]wpO^g |awpNPO)g!wk§ \NPnu^`hnud/ª JKkh\hj\z]eMOv^{nb^{KkO,ykg`nAyHO)g`^`hOa\®nu^`KkO,yMg`nAy8n4\_Ocz|nud4^`g{nu¤O)g{g`nAg ± Ca4³° |awpO NPdkO)O)g!Ocwbz\_h^`^`n©eMwpyk^`jhwunuddM\^{KkwuOPg`O,^`g!wpw®oqOawc|Z]^{\nug{\`Z2yHOanp|)®h 8nuOadMz¤Z´ nAdkOPyHnuhdA^c¢Kkhj|{KNbOcwpdM\v^{KMw^ykK4Z]\`h|awpZpwphz ^{KkO,ykg`npoqOc|^{huO,w hj\_¡ºwpdMuOSyMwpg!wpNPO^{O)g{h !cw^`hnudwug{OawAz]Z ykg{n6[hjz]Oa\® nugxwP\`NPn[np^{K^`g!woqOa|^`nAg`Z@´ AhuOad^{KkO,h\`nuNPnAg`ykKMh\`N©¢4^`KkOag`O,h\dkn \`hdMuekjwpg{h¶^qZ.nugn]|awp¤NbhdkhN wbhd ^`KkO,jwpg{uOuª JK[eM\)¢4w¯hdk¢MOa|)wuwpgmd z]tHOa\`O,hg{OaOawAzb\_hyMZ.wp^`|)K nuedM\_^`(t)g{eM|=^{Oa£z¤e° (t), e (t)¤ , ∀t ∈ [0, T ] ¢4\`eM|!Kb^{KMw^ e (0) = e(0) wpdMz e (T ) = 0 µ ¶ ¡ ¢t ± C8R4³ t = e(0) 1 − e (t) = e (0) + e (T ) − e (0) . T T QUO)uO)g`^`KMO)Oa\{\)¢]NPnp^{hwp^`Oczt[Z.^`KkO¯¹wA|^ ± \`O)Ov^{KkO,g{Oa\`ek¶^!\® g`nAN
(47) O)NPN -w C¯hd2rxyMyHO)dMz]h rS³^{KMw^ ± p A³ wu\ t −→ 0, µ e = φ −→ uθ kµk È [Ò:ÇcÖ ÌØ >Öqò¬ÒÈÊÉ/ß6Ì!ÒÑ6ÐÓÏ!ßÏ!Íqê6Ò>Ç6ÈÊÉÑ6ÐÓÏ!ß6ÖÈÊɤöaÈÊ͹Ò>ÕÏ!Ðjá . . . . ∗. 0. ∗. ∗. . ∗. ∗> ν. ∗. ∗. ∗. ∗. ∗> ω. >. ∗. ∗. ω. . . ìíîï/ìíÚ.
(48)
(49) !"$#% & '() *,+-*.,/ 0. C*C. ± jh\Kk\`hj|!huKbK4NP^`OaZwp|!dMKM\wu^{dkKMuwOc^zh hdAt^`=n 0 ^`KkOadbAO)n]z]Oc\_hj|mg{np^!w^{hnAdM\µh]tHOh¶dMz]eM|)Oaz¤¢u^`KkOg`nu^{w^{hnAdMwp[yMwpg`^nu C8R4³ ± X C6³ t , e (t) = e (t − 1) 1 − T KkOaJg`OvKkO)^{g{KkOO, nudkg{npOu^!¢w^{|)hnunAdMd \`hezkO)(tg{hdk− 1)w÷g{NPO npOa^{g{h\®nAdk^`nbOa\{^{\©KkO,^{wujwug`\_A^O^wpwpekdMO¯zµnp hehdkª ^`n g`Oauekjw^{O ± C8Q4³¢,^`KkOF|nAdA^`g{nu jwc ± Cp³hj\®^{g{wudM\_ nug{NbOczhd4^`n ± uA³ ∂e (t) v = Λ(t) e (t) + , KkOag`Ov^{KkO¯ O)Oaz]¡> nAg`®wug{z.^`Oag`N ∂e (t)/∂t wpnx\|nANPyHO)∂tdM\{w^{hnAdnp^{KkO¯^`g!wu|!¥[hdkbOag`g{nugwpdMz ¡ ¢ ± 14³ Λ(t) = diag λ I , λ (t)I " # λI 0 ± 4³ = , λ, γ > 0, 0 λ exp(−γke (t)k)I hj\.wpd wuzMwpy]^{hwAu\O2Awphd N w^{g`h ªwpj\_nNPnu^`hw^{Oaz t4Z ± u A³° λ (t) hj\.\`NPwu nAg.wug`AO ke (t)k wpd8z ∗ ω. ω. ω. ω. ∗. 0. ∗. ν 3. ω. 3. 3. ν. 3. ω. λω (t) −→ λ. keν (t)k −→ 0. ν. «¬dkdhji4^{ekKkOuhjª\¸%\`OaO¯|^`eMhnu\`d/OU¢k^{KkO¯O¯g{O)nuyHg!nuz g`^_z]wPhg{z]Oa|hu^ O)xg!^{\_nbO,O\_O)^ykg{npOa\{g{\mOatH\`eknu^`^{K\®^{^`KMnPwp^wp^`KkhzMO)wg{^`OvO¯h\®nuekdMgxn¦z] Ochg`wOc^{|ek^g`Ov[hjO \_eM^{wug{/wA\_|Oa^`g`hununAdhdkykg{n]^`|OaOc|!\`K]\a¡ ¢ wu»MdMnuzg^`^`KkKMOawph^¯gb^`|KknuO NP|)yknuekd4^{^`wg{^{nuhnAd/O)g{¢g`wpnAgvmwpykd8h z%O)j|\nAdA^`hg{^`nuKkhdjwcwpd£wpwpg{g{ObOa|)w©nunpNPykhe]dA^{^{OaO)z%g`Oc\qeM^b\`hwpdkg{©O nuO dk ykZ2nuhN h^`Ocwpz¤uª2Ob»kNPnAOag¦wA\_wueMmg`O)g{NPOa\`OaekdA^{^{\a\aª¢ ^{|aKkwphNP\PO)wpg!g{w¯Oaw2h¶hj^{\bK.z]g`O)Oc\_hyHNPOch|^`^ Ocz£^`nt4w,Z£g{whAhg`zPOcznut]uoqg{Oahj|z¤^ª²hd8Jz]KkO)OyHO)[d8hz]\`eMO)d4wp^`ZP\_O)npg{u¤nAh^{hdk\2\`KM^{wpwAyH\_OA¥ª|»knAnAdMg \_hj^`\_Kk^{hj\¦\)¢[hdwg`yHO)nAíOa\`h¶g`^{OahdMnA|dkOxhdMh2N wp^{KkuOO hjnA\,t[oq\qOc^{nu|^g{Oahzdwp^`Kk^vO,^{KkhN OPwug{uO OcOa³mg`O)^`n d8|wpO dknu± z]^`KkOc\_Oahgg{OayHzMnA³S\`OuyH¢4nA^{\`KkOuO¯ªbnArt]íoq^{OaO)|g¦^{hz]uhO¯\`ykhj\®jwu^`|)KkhdkOad^`^{n KkO z]g{|)hwuuNPOSO)^`g!KMw O,±|)wuNbKkhOa\_g{^¯w^{tMg{wuwA|!|{¥P¥[h^{dknb^`^{KkKkhjO\ tHnAz]g,OcO\_^qw hg{OaOa|z.O)nAdN¦yH^{tkn4Kkh\_d8O¯OAwªh^`dk«ºh^xhnu^{d%h\`wuKk¤nunuwpekd8jtHzznp^{tHzkK/OvOaªb\`dkh«¬g{nud²Oa^`zOanuz.y8g!z]n4^{KMO)\_gvOcw\)^^`¢[n©^`h:KMª OAKMOSªw6^{uhOa^OP|!N Kkwdkw6g{hjZ.iAOawueM|OvnANPuhj\g{yknuwug`ekhjdM\`\`nOUzyMh^`d8ekg{z]e]g`O)OS^`yHK/^{O)¢g{d8wu®z]dMOPO)\`d4|)wp^nu^`hnpdMnu\_dM^`^{\ag{KkeM¢4O¯yk|^{zkekOahg{z%\`OSyk\`g{jwuZ[np|dA^!Oaw^`NP^{KMhOnAO)^`dMdAhj|^\ ^{^{nAOn2t[ oqz]^`OcekOa|yk^{g{\¦Oah|znp^,Uh^{N z]Kkwph¶O ¼@AO)NPOag{\npO)d4^{wph^bg{nAOPd²\`KMN npwpwpyHyk^{OayHKk\¦OcO.wuz²dM|nAnuz¤dAd4¢^{^`^`g`n©n%nA^{\_KkyHhO)N¦nAN©hekd4ª j^ w^{»M± O.nug¦g{Oawuwphdmhj\qh^{N h|.iHwpª A\_hOa^`± \e8C)[w\_Kk³_^`³vhnnuhd8d²d\¦^{wuKkKk\¦O)O g{|)OuhNPn4¢\_tMwuOauekZ²OPO.ykwuN \jwpwpyHdkg{OAn4¥]¢\`\,\`hwptkKkg{hOPOA¢\_e8^g`\_OchOc^{wpz \ ykwAzkjwpwpdMy]dk^{Oahuz O¯y8Awwu^`hKd hj±\mpyk4g{³npoqOah|^`K^`Oaz.hdug{O)Oad/ªª JKkh\jw^`^`mO)g®h\m^qZ4yMh|awpZ |nANbyHn4\_OczPnp 1000 yHnuhdA^!\mh^`Kwud ¬ « d » h M ª CA¢[h¶^h\\`Kknd.λ^`KM=wp^®γ ^`=KMOv10ykg{nuyHnA\`Oaz NPO)^`Kkn]z|awpd|)nuyHOSh^`KyMwudMwpg®nut]oqOa|^{\a¢[hdM|eMz]hdk ^{JK4KkeMO\®|^{nAKkdAO¯^{ykg`nAg`Oa/[jwchnAµeM\_hj\UZP\_yM^{wug`tknAyHOAnA°\`OatHz nu^`^{K2wu\`^`¥Pg!wp dMek\`dMj|w^`^`hhnunud d8wp± p³wpdMz©Kkg{h|!npK ^!wh^{\hnAz]dMOawp\`h/AudkO)Oaz.n]|) h¶nA^{g®hOc^`\UKk|hj\nud[yMuwpO)g`g{^`uhj|O¯eM^`#nwug!a\_O)ekg{nMg`¹ªwu|)r®Ou^ ª yH|)nun4d4\_uh^`Oahg`nuAdkO)OcdMz|)Ouwp¢H^S^`^`KMKkOPOb[g{hO\` eMO)wpg{O)d8hd]|O¦ nAyHg`N n4\_wO^{hunAO)d%g{Z©|)nuwAh|)dM|)|ekhjz]g{wpOa^`\¯OaZAh¶ª ^{K% g{^`g{KknuOPg!\xg{O OaOa\{g`\UOadM^`KM|wuObd hN C)NPwuuN Ou¢ wpnudMgSz^`KM^`KkO¦O ^{g{|awuwpdMNP\`O)wpg!^`whnuhd \ . ∗. ïï ßiMä äñ .
(50) C6. Y . uwf®dMwuz%g_^{OaOc\`\`h\vwud©^`KMz]wuhjd\`yk kjwuªD|C OaNb nuOag¯dA^ ^`KM± hOPdg{NPnp^!Ow^{^`O)hg!nu\Ud²wpwpd8g{z©OPhwud|!Kkg!hwuO)z]uhjOawpz/d8ªP\{³»ªhJAekKkg`O¦OhtkC ekwpO¦j\_N n©wu\`g`Kk¥]n6\xh\vd2^`^`KkKMO O¦O)g`uO)nA O)ekg{O)^`dMhnu|d%OnuhN mwp^{KkuOO z]wOa|yknAh|Nb^y8^{wpKkg{Oh\`\_nu^`d²g!wphnuu®K4^{^KkOhdkt8O®OayHKMO)w6g`4 nuhnAg{NPg,nuOaz®^{t4KkZ¯O^{Kk|nAOdA|)^`nug{dAnu^{mg`O)nAg{4g`yHnAnug{h\¯dAt4^a¢uZ²wAeM\\`z]hOadM\`©hg`^{OcKkz¤O.ª«¬^¬d¦n^{KkNPhj\O)\`^`wuKkNPn]O®zM\ \_O)±^zknpO MMg{dkOaOc\`zek^{t[\aZ ¢ ± g^{{Kknpp^!³¯Oxw\`wu^`Z]hdMnu\qz²d8^{O)wpnuN©¤ekO)ªg!g{\!mOg{³vnu|)g^mwuhjwud²\xdknunptHt]^{OKk^!wpO)\_OahgdkO)hOadNPz©ykhdt4g{Zn6^`uKk^{OaKkO.NbOtHOaO npdA ^`^mhj^`\q|nA^{nANWhdMdM|PykO)^`g{npOadM^!|!\\)Kkªw¯dkhjyH¸²i4nAekO.\`Ouh¶^{ª|ahwpnAJddkKMhdMnuh\xtMStH\`^!O)O)wug{K8\`u¥wcOb[í^`hnAnuKMg gwpw¯^¦N g`w©w6nuZ^{g{wpwu^`OaykhwAnuhjz]zdZnu^`n hdM¹wp|hg{OaeMwAg`\_O,hdknu ¢ ykg{KknuhjyH|!Kn4\_Oc|)z¦wudk^{dkwAnp\_¥¦^ tHeMOdM|yH^{OahnAg_ dPnug{ nANPg Oa^`z Kkhjt[\Z|)wu^{\`KkOxOwug`O O hyk\_g{^{Oahdk\`O)£d4^`NPOcOzb^`KMhd.n]z¤»ª hMªJkKk¢4OwpdMg`OczP\_ekh^{^`\KknAwue]|!Kk^®hwpO)d4uZOcz y8wt4^`Z K.eMθyk\`=wuhdMdk180 ^{dkKkMO ¢ k d h h:\{ªwpOANPª8eMO\_g{hdkO)4 wp g{zki8ª Oc\`± \xCp^`Kk³O \_h^`KMK wuyHλOnu= ^{λKkOb=nAt[1oqOcªU|X[^,hdMwudM|Ozw \`nykeMg`ObOe8g{np\_OP^!ww ^{hnA\_ykd©KkhOa\Ug`OAuªhuQUO)d/O)¢Mu^{OaKkg_^{O¦Kkg`O)OcOa\_ek\{\)^U¢¤h^¯hhj\¯t8\_KkOn^{Kkd O hj\dMnuN.d]³¡ºykjwph¶^{dMKkwpnuge]nu^mt]yHoqO)Oag`| ^nueMg{NPdMz]hdMO)¯gxg`wunud4^{Zwp^`wAhzknuz]dMhwu^`@hnuwudMdMwuz ^{hNPd©g{wun]»dMz]h\_h M8jwª |)^{wp1¦hnA^`h^{dMnuKMwpdwkª ^z]& ^`h\`KMdMykOv|jwuOx^{|)OawuO)|!ANPKkwudkO)hhjd/dAi4¢u^!ek\ nuO,tM± θ|a\`wpO)g{d©=uO®wp84j^`\`KMnPwpz]^ wuOadM^`wpKkz /OUkt|)h¶nu^{kdAK^{=g`^{nAKk0.72 ]yH\{nA|Oahd4dM^wp g{nAhnM°Pnx^{Kk\O wnA\q^{t[g{oqwuOch|A^PKA^®wAhdk\bOxwphdFdPK4^`KkZ[OxyHO)hN g{tHwpnuAOuh| ª ¬ « d j u w _ \ P ^ ` \ O P ^ u n U { g a O ` \ k e ¶ ! ^ ) \. ¢ ® O ` \ O P ^ k e y ² w { | M K u w ) O M d u h k d % yMwug{wutHnunuhz ± ^{KkO Kknug!\_OA§ \¯\`wAzkz]Oc³´^`KMOPeM\`Oaz nk|)wpOadkp^{KM\¯O)g{OPwpNPnA\_^v^`KkO.z]nuektkOPnpm^{KkOb^{g`eMO nAz]dkhj\`Oayk\a¢wA|h>ªO)OuNPªOahdMdA\_^ ^`Oc± wuz£np f =wpdMfz = 500 OeMN.\`³vOaz ®wAfb\¯|a=wpg{g`900hOazwunudMe]z^cª fb u=O)d800hd%^`´®KMwphdM\z|)wAw\_OA¢¤wu^`g`KkAO.O h[dMhjh¶\_^{eMhwuwu \`\_^`O)g!g{wu^{nuOah\mdkb^`KM®wpwu^®\^`KMyHOvθOag_yk nAg{=nug`NPyH162 OnAc\`zOazP\`eM^{|)Oa|)|!OaKk\{dkkt\qhj iAeMeMkO¯Zu=ªwpX[j\`0.3O)nO|»nAhyHMOaª \b hnu^`Kg^`dMKknuO¦d]|¡ºyknujg{wpg{dMOa\`wuyHgnunudMt]z]oqhOadk|^!\)g{¢[Oa^`\`KMekwp^{^®\a^`ª KMJO¯Kk\qhj^{\xg!wz]^`O)OaNPuZPnudkh\¡ g{jwpnug{tkueMO¯\_^®z]nu^`nbN jwuwphg{d uOSnpO)|)g{nug{nud[g{u\O)g{hduO)^`d8Kk|O,Ouª|)wuNPO)g!wk§ \mhdA^{O)g{dMwp@yMwug{wuNPO^`Oag{\a¢[wpdMz.^{KMw^^{KkO¯\`O)g{unuhdkKMwu\wuO)g{Z ◦. ◦. ν. ω. ◦. 0. u. 0. u. v. ◦. 0. v. 0. ¦ «¬nAd¦g¯^`OcKk\qhj^{\hNPy8wpwpyH^`O.O)gcwp¢d[ZO®NPKMwcO)u^`g{Oh|PyMg`hnAd]yH nunAg{\`NPOazwp^`wUhnudkd/Oaª & wuykekykg,g{unAOawudk|!K,O)g!^`wpnv[^`Oah\`|!hKknAdkd]hj¡ºi4tMekwuO \`Oahjz\,|hnud8d4z]^`O)g{yHnuO)Ad8^{z]KMO)wd4^^¦z]n]npOa\^{KkdkO nu^nAg{t[Oaoqi4Ocek|h^ag{§ O\ \`\`KMZ]\qwp^{yHO)ON wpMdMOz hnutkh^{h¶^qKkZkO ³¢¤|)wpwpNPd8z©Oag{®wMO§ \wuNb|!Kknuh^`O)hunuOd/ª%g{nuJtMeMK4e8\q^{\)dk¢Oa\{\O^{zknnO)g{dMg`npnA^g{\g{O)hd2Z²^`KMnud£O|aykwpg`hhnutkg¦g!w¥[^{dkhnund©yMOawuz]g{AwuO NP± OOa^`Oawug{zk\ahªxdkJKk^`nO \`wunuOvhN g{Oawui4uekO hg{hO)d]NP nuO)g{dAN ^whj\®^`h^{nuKMdFw^^{h^`KkKkO,nAnue]t]^.oqOawu|d4^xZKM wAOa\®w^{^`ekn g{tHO O¯O g{ h^{Ag{hwAz/|ª^{h«¬nAd©dwuykzMg{z]n]h¶^{|hOcnA\`d/\a¢]¢m^`wpKkO,n6[hjh\_dkeM²wu@eM^`\bg!wu^`|!n¥4hwdk^`b^{wuO hd yMKknAhAh¶^!K \ \_O)^`ug{O)Oadk\ u^`npK,hwA\|)wp|jeM\`g{nwAh|NPZyH nAnug_g ^!wp^`KMdA^O2\`yHhdMn4|\_Ovh^`hg{nuOadMwuh¶dk¡:^`hNPOSKkyHhjOa\_g_^. nAtHg`OaN hdkwp£d8||)OSnuNPh\ykwpe]®^!ww6^{Z]hnA\®dMwwpN Zw6oqO nA.g||hnuO)d8d4|^aO)ªGg{dJhKkdhj\ g{nujtHw^`nu^`^`Oahj|g ^{\`|aKkZ]wpO \qd ^{O)\`tHN O)O¯g{\)uyHª,nuOahg_»dk hnAd8g`wpzkNPekOaZAObz ¢H^`^`z]n2KMOaOw \`ykyk\_hg{^`^`nuO,g!wpyHhn4wuug`\_K4AOc^_z2Ov nAh|)g`dknu®hdA^`wuhj^{wpg{g`z2nAzkyMhjwp\`wcyk^`K²jwuO)|ykdMOajwpNb\`ekdkOag{dkdAOah^!\Sdk\)2wª \`u|!OaKkg`O)Z©NPjwpOAg{ª u UO¦O)zkdMnu|)N Ou¢wph4dhj\`npeMwp|nu\`d[O)g{uuO)nug{uhdkOadM|^{O,wA\_ ¥]nAg\ . 2. U ( JwudMKkzhj\ t4Z.nu^{g{Kk¥O,hj\ hd4wu^`Oa\`g`n¯dMyMwpwu^`hg_nu^{dMhwuwu¤Z¦wuu\`ekg{O)ykO)yHNPnAOag_^{dAOa^zb»]t4rxZ¦l ^{ KkX[O lf®¡ºrx«qQUl s «qXPr}»MeknudMeMz]dMO)zkgxw^{uhg!nAwpdbd4^eMdMdkz]n8O)ªgm uu[g!¨ wpC d41p^udk4n8c¡¬ª
(51) ]C ª Q*QA4¨ 1¡q[¢. ìíîï/ìíÚ.
(52)
(53) !"$#% & '() *,+-*.,/ 0. C81. $ $%( |)fg!ekwpnAg`NPdMg{\_O)Ouhjd4z]¢k^O)wpg¦|adMwpzw NPh1O)dg!w¦ g{yHnu g{d4nuwu^hNPdAnu^cOv¢^`^`KkKkKkO,g{nuhj|{|)ekKwuANPK KMO)wAg!\w |Zn]nug!>z]hd80wª^`«ºOc^!\ \Xn]|aw=^{hnA[Xd|)wp,dY tH,O¯ZO ]ykg{Oag{\{O)\_jOcwz.^`huOPh¶^{^`Kn2g{^`OaKk\`yHOOag`|O^ Oa^{g`nOadM^{Kk|OO ± uA³ X = RX + t. hfN nAwpdMu\_O¯hjz]ykO)jgwpdk^{KkO¯Ohj\ykg`hdkOc\_KkyHnuOcO|^`NbhuO)nkz]Z O):ª2«¬d^`Kkhj\|)wA\_OA¢^`KkOyMg`npoqOa|^`hnudnp X wudMz X hd^`KkOdMnug{NPwu"h !)Ocz ∗. ∗. ∗. ∗ >. ∗. ∗. ∗. ± p4³ ¯ ¦ +%® H } " W 1 (#® '$%
(54) nAOg^^`eMKk\O,\_NP^{wpnu^`NPO¯O)wpd4d^h^`NPKkyHO,nAOag_g`^!g{wpnudAg!^\g`hd Oc\_eM^`KM¶^O|)wpKkNPhj|{KOag{hjwM\®§ \®jwphg{dAu^`O)Oag`ZbdMwue8¤\_OayMz.wug{^{wuKkNbg`nAO)ek^`Oaug{K.\aª ^{Kkhj\®g{O)yHnug`^aª»Mnug®^`Kkhj\)¢]dkOauOa|^ + ¾. 3 ; k8@ba 6 p 7  6 6 à u ¾ c\  ¾ !%g '
(55) XY 0,% D .#Y + O X $,H `!% K #%K.( m∗ = [x∗, y ∗, 1]> =. wpdMz. 1 ∗ X Z∗. m = [x, y, 1]> =. 1 X. Z. . Y+. eω =. µ kµk φ. !. ¡ ¢ β eν = (R − I3 )X ∗ + t ∗ , Z. . . . ± Q4³. 1 µ = sin(θ)u + [q∗ 0 ]× t 2. Y+ $. . ± Au³. β > 0,. ¦¤ ¯ . $-23 ( )$ $-2 23. [tS Z. Â/O  yka g`Oc; \`\`khdk @ ea 6 uw p\ w7  6 ekdM |6 ^`hà nud©Anu ^`Kk¾ Oc|)wu\ NP O) g!¾ w¦ yHn4¸²\_OAOªyk»kg{g{n6nu[N hjz]O wUi[\)|ªnAdM± RA\q^{³g`e8wpdM|^`z hu± OmCayk 4³g{¢]n[npqO,ª KM
(56) w6Ou^O eM\\_^{wpg`^ ± R4³ wudMz e = K e . G = KHK »Mekg_^{·KkZO)gch¢]dpoqhOa|O)^{g{hg`dknA g{\± h d R4³UKhd wug`± O¯u³dk¢npN¦^Uek|nA^`dMhyk\_hjz]Z[O)hdkg{Oa2ztH^`npKk^{OaK%d ^{eO)g{NP=\,Knp^`KkeO =g{Oa\`Kek^,t4KZ t = t¢ª wpdMz^{KkO)deM\`hdk K ± A³®Z4hO)jz ± 1u A³ m ∝ H m + ρ e = α m, α > 0.. ν. −1. 0 p. ∗. ïï ßiMä äñ . ∗ 0 p. 0 p. p. −1. p. −1. −1.
(57) Ca. Y . JKkhj\yHO)g{Nbh^{\^`n g`h^`O. ± 1%C6³ ± 1AA³. eν = H m∗ + ρ∗ e0p − m∗ = α m − m∗ .. M» g`nAN&^`KkO\{|)wuOv¹wu|^`nug Z , α=β Z wudMz iHª ± pA³¢]O|)wpd g{O)g{h¶^{O ± 1AA³®wu\ ∗. ± 1*14³. β > 0,. ± 1p[³ »hKkdMOawpg`O ZAβ¢ hiH\ª nA± dkAuZ÷³yHwO)dkg{NPnug{h¶N ^!\®wp^{nPh !cwnu^{tkh^{nAwpd hd¹wA^`|Kk^`OnAgzkOa\_hKkg{Oahj|{zKøg`OazkO)wpyH^`hO)nudMdzM\tHO)nu^¬d Oa^`O)Kkd O²g`Oc|wpnud8dMz\_^{^`g`KkeMO|^`|ahwpnuNPd O)wpg!wAnuyHg{nAh¶^{\`KkOu°N©ª eν = (X − X ∗ ). β , Z∗. eν. ¡ ¢ β eν = (R − I3 )X ∗ + t ∗ . Z. ¸ h¶^{K©g`Oc\_yHOc|^^`n e ¢k¥[dkn6hdkb^`KMwp^ ω. G∞ = K R K−1. wudMz. uw dMzeM\`hdk ± RA³®Z[hOaz H = R + tq h^`K JKkOad/¢]^`KkO,yMg`npoqOa|^`huO,w hj\®nug`nu^{w^{hnud ± C qCc³®|)=wud KtHO¯qg{.h¶^`^`Oad. ep = K t. ∗0>. ∗0. > ∗. ¢ 1¡ H − H> 2 ¢ 1¡ > R + tq∗ 0 − R> − q∗ 0 t> . = 2. [µ]× =. RS\_hdk snkz]g`huekOc\)§[ nug{N¦ekw. ¡ ¢ R = I3 + sin(θ)[u]× + 1 − cos(θ) [u]2×. O,KMw6uO wudMz^`nAuO^{KkO)gh^`K ^`KkO,yMg`nAyHO)g_^qZ. R − R> = 2 sin(θ)[u]× , £ ¤ > tq∗ 0 − q∗ 0 t> = [q∗ 0 ]× t × ,. ± 1AA³. ± 1u4³ ± 14u³ ± 1*Q4³ ± 1*R4³ ± A 4³ ± C6³ ± 4A³ ± 14³ ìíîï/ìíÚ.
(58) Cc.
(59) !"$#% & '() *,+-*.,/ 0. O,nut]^!wphd^`KkO,hNPyHnug`^{wpd4^g`Oawp^`hnud g{nuN iHª ± 4 A³°. ± u[³. 1 µ = sin(θ)u + [q∗ 0 ]× t. 2. JKkO,ykg{npoqOa|^{huO,wudkuOvnug`nu^{w^{hnud nunx\z]hg`Oc|^`Zt4ZhdoqOa|^{hdMP^`KkO,dknAg`N nu ± A4³hd ± CcA³° ( ¡ ¡ ¢¢ h ¡tr(H) − 1¢/2 ≥ 0, ± 4A³ arcsin sat k sin(θ)u + [q ] tk , φ= ¢¢ ¡ ¡ np^{KkO)g{hj\_O . π − arcsin sat k sin(θ)u + [q ] tk , »yHn4hdM\_O¯wph\xZA¢Hwut4|!Z©KkhO)eMu\`OchdMz¤ ª i4\aª ± u[³¢ ± [u³xwpdMz ± 14u³®^{KkObz]Oa\`hg{Oaz2g`Oawp^`hnud2tHO^qOaO)d e wpdMz©^{KkO|awpNPO)g!w ∗0. 1 2. 1 2. ×. ∗0. ×. ω. . . . $-23 ( )$ )$1 ($-2. J Kkhj\xz]O)NPnudM\_^{g{wp^`hnudeM\`Oa\®^{KkO,g`Oc\_eM¶^!\® g`nAN
Documents relatifs
cosinus (en bleu à gauche) et d’une droite (en noir à droite) sur les données simulées pour 4 années à une température de 45°C, avec un bruit de fond interne réduit de 100,
Florence CARRÉ ~ Réoccupations funéraires de sépultures collectives néolithiques dans la boucle du
2 Vue plus en détails dans le chapitre 1... 2 phénomène d’agrégation des protéines amyloïdes en fibres et d’autres facteurs impor- tants doivent être pris en compte comme
In this chapter, we prove an a posteriori error estimates for the generalized overlapping domain decomposition method with Dirichlet boundary conditions on the boundaries for
Revue française d’héraldique et de sigillographie – Études en ligne – 2020-10 © Société française d’héraldique et de sigillographie, Paris,
Dans ce cas, comme les coulées du Bras de Sainte-Suzanne appartiennent au massif de La Montagne daté à plus de 2 Ma (McDougall, 1971), l’ensemble des coulées pahoehoe
In infantile Pompe disease patients, the glycogen storage diffusely affects brainstem motor and sensory neurons, and the whole spinal cord sensory neurons, interneurons, and
De ce point de vue, il est possible d’affirmer que la mission est accomplie : les projets de renouvellement urbain (GPV et ORU) élaborés depuis 1998 dans une centaine de villes
