Bi-invariant Means in Lie Groups. Application to Left-invariant Polyaffine Transformations
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Bi-invariant Means in Lie Groups. Application to Left-invariant Polyaffine Transformations. Vincent Arsigny — Xavier Pennec — Nicholas Ayache. N° 5885 April, 2006.. ISSN 0249-6399. ISRN INRIA/RR--5885--FR+ENG. Thème BIO. apport de recherche.
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(33) . ¶¸·¥¶¸¹º>¼>¾8¶¿¼¹À¼>¹wÇ ¶¸¹©Û!¶¿Â-¾Æ>Ü3Å»Ç. yv q-scwg ²{g38skxIqtIg.x«!gNz n × n qtpWfafagskv~ faskvygNqN gNwgz\áEuqtsk{sâ-³Ìvyg |{x{w;wxs>sv°xIqË §cwg G v~qóÌvyg|{x{ÌrvÏsqwgNrskzgzygfagHs§vyz°z/u»g§v°stskgN e »% spWwv~zgzygfagHsx{« §vyz°zu»g m \bdcwg³Ìvyg zy|{gNuwx« G §vyz°zu»g §vÏskskgN g G skcwgYs{w|{gNHsdqk{gx{« G >sd»xIv°Hs m §vyzyz»u`g.kg«¬gkg3 skxa{q T G r§cwvyc{´u»g vyIswv°skvy²{gzyp skcwxIw|{cHsx{«{q-scwg zyv°wg3qk{g u`gNqtswkx½Órvyfaskvyw| G {kxIw m §'grgwx{skg L á¬gNqkÌ R â-scwgzyg«¸skÉ á¬gNqkÌdv°|IcIskÉ¥âfÒwz°skvywzyvyN>sv°xIuWp©gzygfagNIs m ∈ G wwkskcgf xIkgIr§'g.§vyz°zz°gs Inv : G → G u`g.skcg v°W²{gNqkvyx{$xI»gNskxIN vÏ«QfÃ{wwvyw| Φ : G → G v~qwv -QggHskv~uz°gI>§'g-§kv°skg D Φ v°sq-Àʼ¹HÔH¹»ÀÌÁüŠá¬xIrv -QggHskv~z fûâs vyqafa{wwvyw| «¬kxIf skx »§cwv~cfag3qdsc>sskx´s{w|{gNIs ²{g38skxIÃz°xrmN>skg3D¦>Φs m ῧcwvyc¦v~qÃuIqtv~{Tz°zyp±G¦vyrT®wv°skg3GqtvyfÃzwvyqkwz~{gNfagHsâ v°s;{qqkxWvyskg3q ©sw|IgHs²IgN8sx{ >s á¿xskcg v°r®»wvÏsgNqkv°fÃ{z-rv~qkwzyIgNf gNHsN z°vy²Wv°w| /vy±Ñrv -QggHs ²{g38skxIqt»{g½â8 Φ(m) Sym(n). m. m. . m. m. m. . m. Φ(m).
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(35) . Y¾Ç8¶ÏÔ{¹Ø´ÂÀ¼>Ú. . qtxIfagYvyr®wv°skgÉÊwv°fag»qtvyx{{zÛ!¶¿ÂÔ{¾kÆÜ3Å»ÇÆÖ Õ>¶ ÂÆÁaƾPÅ W¶~Ç8ÁÇ-c»½²{gY{zyqkxÒu`ggN;gNgHskzypa|I{v°rÉ °v |$fÒ»c¯v°fa`x{ks{gvyÑx{fawrs>sv°xIzskx{fp ÊYäYwgskxÃscwgcv°|Ic¯z°gN²{gNzx«@sgNcwvÏÉ {z°v°sp©wgg3rgN¯skx%rgN{z!§vÏsc qkc¯vyr®wv°skgÉÊwv°fag»qtvyx{{z@qtIg3qQ§-g§vyz°z!wx{sx{»qtv~rgYskcvyq spW»g x{«!³v°g.|{x{q-v°skcvyq§-x{¨Q 4 ' /¡>¢a¢ ¯ ¡ N¢r Þ gЧvyz°zHwgNgN Wwfu»gNx{«wwxsv°xIqzyIqkqkvyNzyz°p »qtg3´sx´rgNqv°u`g skcwg kxI»gNtsv°g3q'x{«!³v°g.|IkxIwqN!bdcgp${kg skcg.«¬x{zyz°x>§vyw|Ë bx {Hp ²IgNskx{ v ∈ T G {vP gIsx Wp.s{w|{gNIs@²IgN8sx{skx G s!skcwgv~rgHsvÏspÒN u»gIqkqkxrv~>skg3 v°© x{gÉPskxÉÊx{gYfÃwgÚ°ÂEÖÀE·¥¶¸¹»º½¼¾8¶¿¼>¹»À²{g38sx{-®gz~ X »rg®gNuWp X (m) = D L .v vE g{\qkvyf z°p;uHp$z°g«¸stÉÊfÒwz°skvywzypHvyw| v §'g©NÎ|{vy²{g (T G, +, .) -§cwv~cZv~qauWp±xIqtsk8sv°xIÎκ>ÂÄÀ¥Æ>¾ÇPÅw¼IÄÂd qtskskwg%x« Û@¶¿Â ¼>Ú ÔHÂ3µ¾k¼4vE g{ |{vy²{gÒvÏs.¯gÓHs$zy|{gu{vy.xI»gNskvyx{©§cwv~c¯v~qY{{qqtxrvyskvy²{g»©uwv°ÉPzyv°gN v°wgYwkxrr»8sYNzyz°g3$scwg Û@¶¿Â µ¾k¼IÄ IÂÀPQrgwx{skg3%cwgg [ , ] dbdcwv~qx{`g>sv°xI©zyxIqkgzyp$g g38sq skcwg-fÒzÏsv°wzyv~>sv°²IgÐwx{`gkskvygNq/x«rscwg-|{x{w G i¥kskv~zy{N3«¬x{!xIfafÒrs>skvy²{g'³ÌvygÐ|{x{q skcwg³v°g uI¨{gsdv~qzy§-½prq-Wwz°zP bdcwv~q!v°wg!wxWw8s!vyq!I8sk»zyz°prgNkvy²{g3.«¬x{fÔH¹Q¾k¼Ú3³v°g-uw{¨Igs/xIqkfaxHx{skc²{gNskxI®gNzyq {©u»g xIqtv~rgNkg3Iq;Û@¶¿ÂÇܵ¼Ú ÔIÂ3µ¾k¼ x{«\²{gNskxI®gz~wqx{ G `qkvyg qtvyg §'gN(TvywG,gH+,skv°«¬p . ,T[ , G]) §v°skcscwgqtgsx«!z°g«¸stÉÊv°W²>vy{Hsd²IgNskx{-®gNzyq q.fagHsv°xIwgNwkgN²Wv°xIqtzyp{»skcg xsskvyx{©«¬xI skcwga³Ìvyg {z°|Iguw´x{« G §vyz°z!u`g vy¯scwv~q kskv~zyg \iÊsc»{qaWwfÒu`gx{«gfÃ{k¨>uz°g.zy|{gNuw{vyYwx{`gkskvygNq á¿v°©{rvÏsv°xIskxÃv°sqIqkqkxrv~>skvy²WvÏsp g uwv°ÉÊz°vywgN{kv°spwâ\§cvyc©g skcwg «¬xIz°zyx>§v°|Ë v¸â ∀a, b ∈ g, [a, b] = −[b, a]
(36) á {Hskv°ÉÊx{fafÒrs>sv°²Wv°sp â8r§cvycv°fawzyvygNq [a, a] = 0 vyv¸â ∀a, b, c ∈ g, [a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0 á IIx{uv4vyrgNHskv°sprâ nWvyf z°g-gÓwfawzygNqx«³v°g'uwI¨{gsqÌv~qg\|{vy²{gN uHp { v°sq/fÒwz°skvywzyvyN>skvy²{g-qtuw|{x{w»q z°vy¨{g SL(n) x{ SO(n) i¥.skcwg3qtg-Iqtg3qskcwg-³ÌvygÐzy|{gNGL(n) uwdv~qº½ÂÄÀÊƾ\ÇPÅw¼IÄÂÆÖÐÇÝܼ¾kÂ-Áa¼ÀE¾8¶¿ÄÂÇ8 Ãskcwg ³Ìvyguw{¨Igs\u`gs§-gg´s§'xgNz°gNfagHsq x{«4skcwv~q-zy|{gu.v~q\scwg ÄÆÁÁ ÜrÀʼÀÊƾ x«-skcwg3qtgas§-x%fÃ>skv~gNqNÌvP gI [M, N ] = M.NM− N.MNÃi¥1kskv~zy{NQscwg$³ÌvygÃzy|{gNuw;x« vyq Isc>s-x« v~q\scwgYqkwuW²{g38sx{Ðqk{gx« x«ÌfÃskvygNq\§v°skc; sk{g GL(n) M (n) gN}H{zÐsxÑÈNgx@ªskcwg%SL(n) ³Ìvyg{z°|Iguw%x« SO(n) vyqÒskcwg²{g38Msx{(n) aqtIg´x{«Yqk¨{g§qkpWf fagskvy fÃ>skv~gNqN wxI x{fawzygsg{x{wHsx{%³Ìvyg zy|{gNuwIqwqkg g Ê G Nau`g gkg3qtgNIsgN! >uWpa.|{x{w x«QfÃ>skv~g3q\{skvyw| x{ g I²Hv~.§cs\vyq'zyzygN vÏsq¼{ Õ Æ>¶¸¹»À ¾kÂ¥Å`¾kÂǹ»ÀʼÀE¶¿Æ¹ ÞÑg§v°zyz!qtgNg skcsscwgÒwx{`gkskvygNqx«!skcvyqYkgNwgNqkgHsskvyx{©©skcwg gÓrv~qsgg x«/uvÏÉÊv°Ad(G) W²>v~Hs-fagskvyNq-«¬x{scwg.|{x{w G {kg cwvy|{cwzyp$zyv°¨{gNÌ bdcwv~qdf g3q-sc>sxIwg.fÃgNIc;gNz°gNf gNHsdx{«scwg |{x{w;v°Hsxa;Ú ¶¸¹Q¼¾ÒÆž¼>À¥Æ>¾ á¬vP gI°w fÃ>skv°Ówâ§cwvyc¯I8sqx{%scwgÒ³ÌvygÒ{z°|Iguww ©x{g wgNv~qkgzyp{{©gzygfagHs m x« G I8sqx{©{ . . . . . . . . . . . e. v. v. e. m. e. e. e. . .
(37) ¶¸·¥¶¸¹º>¼>¾8¶¿¼¹À¼>¹wÇ ¶¸¹©Û!¶¿Â-¾Æ>Ü3Å»Ç. . g zygfagHs v x« g uWp Ad(m).v = m.v.m >bdcvyq x{`g>sv°xI¯v~q Nzyz°g3;kgNwgNqkgHsskvyx{ v°;skcg.qkg»qtg x«-¾kÂÊÅQ¾kÂǹÀ¥¼>ÀP¶¿Æ>¹À Âƾ8ØáEqtgNg 3 `«¬xI x{fawzygsgskg3>skfagNIs8â8r§cwv~c$fag3q skcsdskcwv~qdfÃwwvyw|av~q.ÄÆÁż>ÀP¶¿µÚyÂd§vÏscskcwg ³Ìvyg |{x{wqskskwg x« G \bdcwv~qdxIf »>skvyuwvyz°v°sp xIqtv~qtsqdx«skcg.«¬x{zyz°x>§vyw| wkxI»gNtsv°g3qË v¸â Ad(e) = Id vyv¸â ∀m ∈ G, Ad(m ) = Ad(m) v°vyv¸â ∀m, n ∈ G, Ad(m.n) = Ad(m).Ad(n) vy²wâ Ad : G → GL(g) vyqqkfaxHx{skcÌ bdcwv~q {fax{wHsqskx%qk½pWvyw|Ãsc>s Ad vyq.©Ç8ÁÃÆNÆÀ Ô{¾kÆ>Ü3Å Æ>ÁÃÆ>ÁÃÆ>¾PÅ W¶~Ç8Á á¬xI ³v°g|{x{wcwxÉ fax{fax{wcwv~qtf´â −1. . −1. −1. . . . .
(38) . . ³v°g.|IkxIwqg ¹`Æ>À²{gNskxIdqkIgNqdvy%|{gNwgzQuwrsc»½²{g.af xIkg x{fawzyvyN>skg3;qtskskwg{Ë!vyqsgNI {x «Ò6á¿x{fafÒws>sv°²Ig3â${wrv°skvyx{ + d6±qkNz~´fwzÏsv°z°v~skvyx{ . yscwgp¦x{z°pÎc½²{gÑ6á¬wxIrÉ x{fafÒws>sv°²Igv°$|{ggz¸âfÒwz°skvywzyvyN>skvyx{ HÃ{´v°W²Igqtvyx{ x{`g>sx{ᬧcvyc$x{kg3qt`x{»wqskx scwgqz~dfwzÏsv°z°v~skvyx{uHp −1 «¬xI²{gNskxI×dqkIgNqâ8 +x>§qtcx{wz~¯xIwg|IgwgN{z°vyÈgÒskcwgawx{skvyx{Ñx«Áü>¹;skx;skcwv~q spW»gax«'wx{rÉÊzyv°wg3.qt»{g Íb/xrx qkxrxIwg {;gzypÃx{;skcwgö¸¹»º½¼¾8¶¿¼>¹QÄÂÅQ¾kÆž8ÀP¶¿ÂÇ'sc>sdskcg f g3%qtcx{wz~ ¼.Å`¾8¶¿Æ¾8¶Ìqkskv~q«¬pIrv°x{rg sxa|{ggzyv°ÈNgscwg v°W²>v~gYwx{`gkskvygNqdx{«scwgvÏscwfagsvy.fagN{;vy%²{g38skxIqtIg3q i¥wggNÌ'vyscwg¯Iqtg;x«Y²IgNskx{aqk{gNqN@skcwg©{kv°skcwfagskv~;fag3q wgNqkgHsqaqtskx{w|Ñv°W²>v~g kxI»gNtsv°g3qËÌvyW²>vy{g-§vÏscakg3qt`gNs@skx Wp s{qtz~>sv°xI{Ò§v°skcagNqk»g38s@sx.{WpfÒwz°skvywzyvyN>sv°xI uWp.qz~3bdcwv~qÌfag3qÌskcsskcg'v°skcwfagsvyÐfagN v~qÌvyW²>vy{Is/§vÏsckg3qt`gNsÌskx zyzscwg'{z°|Iguwv~ xI»gNskvyx{q vyrgNÑuWp©skcwga²IgN8sx{ qk{gaqtskskwg{ iÊs fa{¨{g3qY|IxWxW qkgqkgÒsc>s scwgawxsv°xIÑx{« fag3%;skcg.{z°|Iguwv~ qskskwg qtcx{wz~$u`g$ÄÆ>ÁÅw¼ÀE¶¿µÚ°Â i¥%scwgÒIqtg x{«@|{x{wqNrscwgvyH²>{kv~g §vÏsc©kg3qt`gNsskx´zyg«¸skÉ-{%kvy|{cHskÉPfwzÏsv°z°v~skvyx{q á¬skcwg |IkxIw$N$u`g.wx{wÉÊx{fafÒrs>sv°²Ig3â\{Ãscwg.v°W²Igqtvyx{´xI»gNskxI-{kgskcwg.gN}Hwvy²>zygHsdx«Ìscwg.v°W²>vÏÉ {gÒwkxI»gNtsv°g3qIqkqkxrv~>skg3skx$skcwgf g3¯vy¯²IgN8sx{qkIgNqNÞÍcwgN§-g sk»qtz~>sgÒ´|Iv°²Ig¯qkgsYx{« qfawzygNqx{ ¯wx{u»uwvyz°v°sp1fagN{qkwg{!v°s v~q kg3{qkx{{uwz°gasxѧv~qkcªskcsscwgvyafagN±u»g;skqkz~>skg3 gÓw{8sz°p©vyÑskcwg´qfag §-½pIÌskcwgÃqfag wkxI»gNtsp©v~q rg3qtvy{uwzyg§cwgNѧ-gs¨Ig skcwgavyH²Igqtg3qx{« scwgqfawzygNqN #&% ¢ 4 E" 4 (¡ ¿£ 3 ¢ ! ¬ 2
(39) Ñ ¡½ Â$ļ¹ Ô{¶¸º½Â$ÀÊƯÀ ÂÃÇÂÀ ÆtÖ ÅƽÇ8¶¸ÀP¶¸º½Â;¹»ÜrÁaµÂ¾Ç´¼©Ç8ÀE¾8ÜÄÀEÜr¾Â$ÆÖ;ÄÆÁÁ ÜrÀ¥¼>ÀP¶¸º½Â Ô¾Æ>Ü3Å @¶¸À À Â$ÜWÇ8ܼÚ@ÁÜrÚ ÀP¶ Å`Ú϶¿Ä¼ÀE¶¿Æ¹¦¶¸¹ R !8¹ À W¶~ÇÒÄÆ>¹»ÀÊÂãI#À "@Ú°ÂÀ (x ) µÂ N ÅwÆ>Ƕ¸ÀP¶¸º>ÂÒ¹»ÜrÁaµÂ¾Ç ¼¹`Õ (w ) µÂ N ¹QÆ>¹·¥¹QÂÊÔI¼ÀE¶¸º>¹QÆ>¾8Áü>Ú %¶ $3ÂÕ Ð¶ÏÔ WÀEÇ . . . . . . . . . . . . . . . . i. i. . &')(+*-,/.(1032 45,768,4 9;:8<5=1(1:#(1> ?A@B( ,/C DE?6F(:8,?8(1:#668(68C ,HG%03(1:#,.(1032 45<5.DI68,JLKJ < MN,:#, 68<ODE45?(EPQ45,7PR68SD JT:#<5=1CU68S 0/> 4 68<52 45<5.DI6F<5( V 'C <5?39W<5,45J ?AX *U9 .v> 4OD`X DE?F?F([.<ODI6F< \W< 6]9^(EP68C ,+=1:#(1> 2_0/>`4Ad(m).v 6F<52 45<5.DI68<5( B=V Y m.v.m 68C ,+0+−1DI6F:#Z< aT=.ADED?8,1m@c−1 bQ,eLd[mf+.DCLDe\UR,3m6F−1 C ,e.vG%?F<5=032 D45,m6FRC <5m?g−1 68<503.D,IKe LP%(1m:80 @WbC <5.hCH( 4 9 >`?F,?i6]bi(30+DI6F:#< a+0/>`4 6F<52 45<5.DI68<5( ?D J;( ,0+DI6F:#< a+< \U,:#?F<5( BV −1. Ad(R).M = R.M.R. . .
(40) 3. Y¾Ç8¶ÏÔ{¹Ø´ÂÀ¼>Ú. . P w¼>¾8¶¸À WÁÃÂÀP¶¿Ä;Áa¼¹ZÆÖaÀ wÂ;Õ{¼>À¥¼©¶~Ça¶¸¹º>¼>¾8¶¿¼¹À @¶¸À ª¾kÂÇEÅÂÄÀYÀÊÆ©ÁÜrÚ ÀP¶ Å`Ú϶¿Ä¼ÀE¶¿Æ¹" µÜrÀйQwÆ>À =@¶¸À1© ¾kÂ8ÇPÅÂÄÀ-ÀÊÆ´¶¸¹º>¾Ç8¶¿Æ¹ ¶¸¹%ÔI¹`¾k¼Ú á 3â 1 1 Σw 6= , x Σwx rÜWÇÒÀ Âa¼>¾8¶¸À WÁÃÂÀP¶¿ÄaÁü¹Ñ¶~ǹ`Æ>À4ÖÜrÚ¸Ú Ø©¼{ÕI¼ÅQÀÊÂÕÀ¥Æ´À W¶~ÇÒÁ ÜrÚ ÀP¶ Å`Ú ¶¿Ä¼ÀE¶¸º> Ç8ÀE¾8ÜÄÀPÜr¾k d¹À wÂÃÄÆ>¹»· ÀP¾k¼¾8 Ø "\À ÂYÔHÂÆ>ÁÃÂÀP¾8¶¿ÄÁü>c¹ " @¾8¶¸ÀEÀ¥Â¹ w¾k E((x ), (w )) "\¶~Ç\ÖÜrÚ¸Ú Ø;¼IÕ{¼Å`À¥ÂÕ 38À-¶~ÇÔ¶¸º>¹1µØ áE{â E((x ), (w )) = exp(Σ w log(x )).  ¾kÂļ>Ú¸Ú/À ÂÒÄÚ°¼½ÇÇ8¶¿Ä¼>Ú@ÄÆ>¹º>Âã{¶¸ÀPØö¸¹`ÂÝܼ>Ú ¶¸ÀPØ$µÂÀ йÑÀ  À ÐÆÃÁü>¹w Ç á Iâ exp(Σ w log(x )) < Σ w x , ¹Qº>¾ À ÂÕI¼>À¥¼´¶~Ç ¹`ÆÀоkÂÕÜÄÂÕ´ÀÊÆ;¼aÇ8¶¸¹HÔ{Ú°ÂÅwƶ¸¹À. @ ¢
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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
