as nonrigid Carnot groups
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- Gilbert Webb
- 5 years ago
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1 Th Th Th V V V crcc 5 c5 5 Hdr Vr 34 dh 356 crcc-c 5 Hdr c Vr d Cr r c d Cr r c r c c r c 5 B Hdr Wrhr B Wrhr Vr Ccd b G cr Ccd b cr Abrc G c cr W d rdc r c d Cr hch Abrc r W d Cr rdc r Th Cr c r cc rcr ccd h cr B cc Wrhr cc rcr hch - B ä cd Thr b r - Cr r r Th d Cr Cr r cc r rcr ccd h c h cc cc cc rcr cr Ccd d hh b G cr W B dr- h h cd r d r Thr rd - b r hh r r d Th Cr dd r b cc h dc c Cr cc r h d cr Abrc r W d h h W d r 3 rdc d b h r rd rd hh c r Th dd ˆ ˆ ˆ hch c b cc Sbc Cc dc : Cr 53 C 4 r E h 5 r d d 3 hr : Cr r c b h rd rd c Sbc Cc : 53 C 4 cr Cr r Thrdc E 5 d Cr r cc r c r c c d h A Cr hr : Cr r r G c ccd rd ccd rd cr rdc c c cc r qd r c r h - r B dd - cdrc A Cr Th dd r G - ccd ccd rd r cr r qd h - bd r hh r r d Cr bd hh r crd rc h r dd c hh r cc rc hh bd Th - bd bd crd cd hz r bd rc rc cr cd Th Cr hh d Crh bd W é d rc cd h hdh hz r d hch bd r rr rc cd r d Cr dh Crh c- d rc Dh hch rr hz cr hz bd r bd r cd Th r cd cc dd cc b c G r c d d b d r d b c h r d Crh c r h c cc cc d d Qc Qc r dd h r dd h rc cr rc Cr r dh d Cr Crh Crh c- d 3 é d rc rc b h d d r d b cc b cc Cr r Cr r r rqd r r qd h cr h c d hch Sbc d Chch r h c r r : 53h C 4 r rd E 5 r rd r cc hch r d c d b d cr r r d cc c hr hch h : Cr cc r r d c r d P hh c b drb r d cc d P brc brc Th c dd d r r c h cc r P rr brc 4 brc brc brc cr drb brc 4 r cc d c d P Th c dd rr 4 brc 4 h r d c ccd brc brc h c ccr d A h Cr r d c r ccd G 6 h c ccr c c d r cd Th qd c h r c r Hbr r d rc d d cd c r c Hbr r d r r bhd d r brc 3 brc r brc rr 7 bhd bd brc d bd r brc crd 3 8 brc 7 r 8 brc 6 6 brc 7 brc r c 7 h brc Th h bd h h c d h h c T 8 d q : Ar r Cr r c T brc 8 brc d q : Ar r Cr r h c d h Cr Crh c d rc h D 3 hhr hch r d qc hch r 3 hhr hch r d qc hch r c c? h r z Th? Th r bd r h r h c d h b cc d h b d h h G SSN 949 hssn dh h593/5 h d d crcc h 5 crcc-c chhdr b Hdr h Vr r Vr d h h c cc d Qc r d d r c Cr Crh c d rc d b cc Cr r r r qd h d hch h r h r cc hch r d c d b d r r c h cc P d r b cc d P Th c d d r r h r d c ccd 6 h c r ccr c d c r c Hbr r d r r r b h d r d r 3 7 q 7 hc c T 8 d q : Ar h r 3 h hhr hch r r d qc hch r
2 34 Wrhr d34 34 Wrhr r Wrhr r 4 brc brd 4 h brc d r brd d h r d r d r r d r h 4 brd h r d r d r r d r r r dc S dc S Th r d r r S > rd r d dc r Th S r h r S b d Th Hbr d r rr r rd rd b brc c h r brc rc d c r Th r h b d Th h r rc Th Hbr S c h d b d r Cr r rd r > Th b r d r h ch bc r r rc d c c Th Hbr h r d r r r r d Th rc c r d c d Th r h rc b dr c c h q h b d r Cr r c r r c Th rr ˆ c ˆ ˆ c c - h r ˆ Cr Crh é d d Th Th rc hch h c r Cr c r c h r bd d r d Cr Th r d r d Th c rc qd c h drb d c h b r r cr d hch r r dr c q ch b r brc rr A r hh Th d Cr r c c cc dh Crh r c- d ˆ d rr hch ˆ ˆ drb Cr r r d r r q d d d r r d Th d q r Thr qd h cc drb r r d b r B ä cd cr Cr hch d hch h r Crh h c c r ch d b d d hr cc brc hch r cd r A h d d rrc r d cc r rr Th d r qd h d r b b r h drb r c r d cc q hch r rc d c Thr c ch cc b b r crc r d c B dr- A cd rh hch r h h bc c d Cr r d r cc rth hr r c cc cd r r h r rrc d r bcc rcr hch qccd h r c c c r r d h cc B c c dd cc c rcr cd hch h rd rc h h r c c r hr d h cc Cr bc c dr Cr r h r Th r c qc c r cc rcr Th dc hch ccd dr c c h r c r cc c rcr h rd Br h r Cb r d Hd Cr r c qc Th r c c c c h dc Cr dr bc Gr c r r Cr r Th cc A c r c br Br r hch dh d Cb d c dh c Hd d rc c h q h r ch Cr d hgr r r d Cr r cd cr A r qc Z br A Cr d r d ccd hh Th d rc c ccd dr r c r r G h rd c br qd Br h Cb r rdc Hd ch h h ch h brc brc cd cr cr Z Cr Gr A Cr r ccd ccd r G h rd AF br ccd qd h b r r rdc r ch d h d b h : G r c q-h dh ˆ hch bc cd cd h cd G h d ch F h ccd r : rh rr G dh hch bc h Y Y cd * c r rh rr G h d Th * Y Br Z Cb Hd A Cr c r ccd c G Y Y > < q C q d d Y q d d Y q Y h Th Br r dh d Cb dh b r qd Hd h c r rdc r ch h b h * Y rr q C d q Y Y C q q q d q r d b Y d q d h q Th dr 4 d Y q Y d q q Y Y Y Y Y Y d Y brc Y brc C F q ccd q 48Y r Y q Y Y : rhrr q G Ch dh r b hch bc d h h d Y rhrr G h bc d d dr A cd h h q Th dr 4 d b r d * Y Y ˆ brc Y Y brc brc brc Y brc brc brc Y brc Y brc brc Th 48 Br Cb Hd brc Y brc c r brc Y brc brc brc brc brc Y cr brc Y Y brc > brc ˆ brc ˆ q C ˆ q d ˆ d Y Ch b d d ˆ d Y * ˆ q ˆ h Y d * Y bc q dr q A cd h d b r d d d Y Y C q q ˆ q ˆ ˆ q r
3 Wrhr Wrhr Wrhr r h b cd r cd h rd h b G cd cd d G b b c r d cd d G b brc d brc d C Ph : rh Φ : G b rr G b b C Ph ˆ d rh rr Ph : rhrr Φrdc G b dh brc 9 86 brc hch bc h b d Ph rhrr rd dh d 9 86 rh hch rr bc G h h G d d h d d Y C Ph d C Ph C Ph Y Y Φ ΦΦY A b d Y bc dr r A b dh dd Y bc 9 Y 86 dr hch bc r cd h c d Scc r rhrr dd G Y τh * Y Y cd r r d cd Y c d Y d cd Scc cd τ Y d Y cd A r r d Cr r τ Y Y cd d cd d F A rr r Cr d r d : rh rr b d d d Y F Ph ˆ > d Ph δ : Ph Y b δ h A hch d d b r d h G hch d d G cd cd Y bc dr Th hh Th bd - bd bd bd b r b r hz c cc cd rcr hz c cc cc r rcr cc cd r r r hch rr dd cc rcr Y r r c d r r cc S c r hch c rr cc Y rcr Y c r d r 3 d Sc Y r cc d r Y 3 c r d rdc cd d 3 h A c Sc bh r rd Cr r r c 3 rdc d brc F h 3 c > brc bh rd r brc d q brc d c : brc rhrr 3 brc brc 6 b 6 7 d brc brc A 7c : brc h d dc h dr r A c dr : rh rr h d rb ˆ rrhb ˆ dc hh h h dr r h dr h h h hch d d G h ḳ rd rd rd d rd W d c r d d b hh d Th br br bd Ĩ bd W d b hh d F b c -d d h r z Ĩ c d cc r c r cc r c r c F -d r r hch r r cc r c r d r d r cc h rdr r T r b cr 3 Sc T hh h dr T Ĩ b T 3 r d c h c b h rd r c -d D r 3 D c rd C D r dd 6 b 7 q dd T T Th hc D c q A D c b : ˆ D rhrr c h d C ˆ D ˆ r dd d c h dr b q dd hc T d r D DC D / T Th hh c r D b r ˆ r ˆ Eq: D c D C D h hc d d d cd cd cd W d d b d c cd cd cd c F d ˆ q
4 344 Wrhr d Wrhr Wrhr r dd c qd h r dd r dd r c - c qd h ˆ c qd h h r c hh : : D DD rh rr D : π : D D D rh rr D : ˆ D rhrr D ˆ : ˆ D rhrr ˆ D π Gb cd r b ψ d Ĩ Gb cd r b d ˆ ˆ rd -d Ĩ ˆ Gb c r d h r b ˆ ˆ brc br -d brc D D d d d h ˆ r d : D r cd c D Th b q cd D r dd d b ψ d d d : D rh rr ˆ d r cd c Th b d cd r dd b Ĩ h Ĩ rd ˆ D h q D -d ˆ d cd cd cd ˆ d D D d d d Wh c b c brc dr br dr -d ˆ ˆ : D dd rcr cd r brc ˆ r rcr h c r d c ˆ Th b rc b ˆ D q D d d d b Ĩ Wh c b c dr dr dd rcr Th ˆ r cb cd cd h r r c π d h : rcr rc b d ˆ brc ˆ π br -d r π h ˆ brc d W Th r cb cd h r c h : Eq: ˆ π ψ π ψ d d 3 Cc rcr crc Th C D c bd : D crcd A cc h θ br br D q θ W B ch h r cc -cr r rd b crc crc ˆ D ˆ q D ˆ d ˆ d cd cd cd ˆ d 3 Cc rcr Th hh C D c ω d d Ĩ 3 rh rr bd : Wh c D b c rh rr d r D A cc r d r dd r D c r c r d r ˆ ˆ * ˆ hh r B c r ch r cc r rd r b c b ˆ Eq: 3 brc d ˆ d br ˆ -d d d brc
5 Wrhr 345 hc Wrhr 345 Wrhr brc c5 : brc D c D : D rh rr D hh * ω Ĩ 5 c : ˆ D ˆ Th rhrr hz D bd H dd b -d ˆ Th hz r Hbd H dd b d T D ω Ĩ H brc T D brcd -d brc cr cd Th h r z bd H ˆ d d b d r d d H ˆ T r ˆ D d ˆ d d rd rd rd rd r -d rd c r d rd Ĩ h ˆ h H brc d ˆ ˆ br ˆ brc brcb rd d ˆ rd r br ˆ H brcrhb Th c r Th c r brc rd rd d brc rd rd -d Ĩ Eq: H ˆ r brc ˆ rd r ˆ rd ˆ ˆ d br H brc r rr q brc brc h h h hh rd br cr d hch Eq: H ˆ T rd ˆ D br dcr d hch Crd brc r br dd ˆ r b r d Cr r G q h ccd Br dh Cb dh T D Hd A h r c D c c d hh cr r c Th Crd r d c brc r br dd b r Cr r ch h G q h ccd Br Cb b Hd d A h r Cr c D c c r r hc d - r h ˆ G c ˆ r ch h ˆ r r Cr dcd ˆ r cc rcr r d h r cc hc rcr h G r dcd cc rcr H ˆ r 3 h 3 r Cc cc T-r ˆ rcr r A d d D q 3 3 Cc cc T r Adh d r d D cc r cdcc r cd rrcc H r H ḳ rr cc θ Eq c * H r d q H Eq cc θ cc h * cc H ˆ H cd cd cd crc crc ψ ψ ξ η h h h r d h b T ˆ D ˆ d hc Crd b r c b r d d b ˆ h r r G ˆ q h c Hd A h r c D ˆ c r c d ˆ ˆ ˆ ˆ ˆ ˆ d Cr d r hc h G ˆ h r h cc r c r
6 346 Wrhr d Wrhr Wrhr η η Ĩ brc ˆ Th ˆ br -d ˆ ˆ d brc d Th d dξ ψ d * ξ d ξ q d d q * q d Th d q rd rd q d ˆ d q d ˆ ˆ ˆ d dη ˆ ψ ˆ d ˆ q η d q η d q r ˆ r q d * ˆ d ˆ q d ˆ * H h d cc cd : d hc d cc d - q q r rd rd d d q * H h d d ˆ cc η η ξ d Ĩ ˆ 4 * ˆ cd ˆ ˆ : crc ˆ d ˆ d q * hc q ˆ cc d r hh r ˆ q d ˆ q η Eq: 4 η q ξ Ĩ 5 c dr rcr -d H ˆ h ˆ d3 ˆ 4 Pr ˆ Fr Eq: cc r 5 cc ˆ Ω r c q rd d ˆ rd rd rd q : crc d ˆ c h cc hc d cc Ω d -d r : Ω r Ω cd r r q drd c b b dr cd rcr : 3 4 Pr Fr cc ˆ ˆ r r cc r C O ˆ 6 q ˆ c cc d C O ˆ d r 4 ˆ q r ˆ : C O rh rr r π r π 7 ˆ ˆ C O r q ˆ r ˆ d π ψ π ψ cd r r ψ 5 q r ψ drd ξ b η d cd : b r 7 cb cd cc r 6 Eq: c 7 b dr r c r crc r crc -cr 3 4 h crc Pr ξ h ξ Fr crc cc r r h h π 8 O q ˆ ˆ ˆ h crc r crc c c r c d O ˆ ˆ ˆ h r : O rhrr r O 7 q η η π 9 cb cd Eq: 8 h ˆ Wh ˆ d crc ˆ -cr cd c c η d q bq cc cd drd b Eq: c d : 9 crc -cr hc h br br b ω q r Wh br br cc d r r cd c 6 cd cd r q b b cc cd b Eq: ˆ crc r crc ˆ 7 r η π br ξ ˆ * π η ˆ c ˆ crc br br ˆ ˆ ˆ crc r crc ˆ ˆ cd ˆ cd r q ˆ d brc crc -cr brc 7 brc c b c d crc cr- b brc brc ˆ brc ˆ crc br ˆ 8 d
7 Wrhr Wrhr Wrhr hch r d cd c η hch r hch r d dq cd c br br c r q d c ˆ C d O d q Ω ψ W W W brcb b W d O ψ π Ω ˆ d ξ π ˆ W W r ˆ b chcd h r ˆ cc r rb T h ˆ d h cb ˆ cd ˆ h C O rrhb d-d dr brc O rrr d ˆ ˆ crc cr- crc br brc d d d h-q π d brcrhb d π r b chcd h r cc r T h d h h cb h cd h Eq: r b chcd h r d cc r r q d h c b c d h * d d d π d d π crc b * d ˆ h h br br q h ˆ ˆ d ˆ Eq: d ˆ crc d ˆ d * d ω π ω ω π 3 d crc dh h d Fr d * q h 3 h h π : H H rcr Eq: 3 b * d ˆ ˆ ˆ d ˆ crc π d ˆ 4 * crc ˆ h br br q Fr d 3 h * : H rh rr H rcr h π d π 5 Eq: 4 * rd b r rd brc rd ˆ Fr r 3 rd h ˆ br br ˆ br br q ˆ crc ˆ 3 d Eq: 5 ω r ω π 6 * dh d hc 5 d q 6 r h c h Fr br br d q cc ˆ r * : H ˆ rhrr H ˆ q r c rd r r rd Fr 3 h b Eq: ω r 7 6 crc r ˆ r * ˆ ˆ r crc h * crc r Frr ˆ ˆ 4 r 6 r * ˆ h d d h br d c br h q q cc ˆ r hc h h d r h c h cc ω r 8 r d Eq: h 7 r F * b h br br ˆ q Frr ˆ br br ˆ 4 d 6 d d r ˆ h ω r d π rd π c h cc ˆ r h h r ˆ ˆ r 5 r d * rd r rd 8 h br br q cr F br Fr br 3 h b crc r * d ˆ crc r crc * crc crc * crc ˆ 6 crc r d crc * d crc * d h d d q hc 5 6 cc r r
8 348 Wrhr d Wrhr hch Wrhr hch b b h 8 h br br d hch br h br b r 4 cc h r r 8 h r d d rcdr cc r d d hhr r d dr rcdr d hhr dr r h r r r h d Ω h d C O Pr Pr r cc rcr r r rcr cc r r cc r r rcdr q r A A r r r r dh D q r ˆ h d O r dh q D Pr r r r r r cr cc r r cc r r r D Dq r h r c r rr r c r A h h r b cr r rr cc r bd c h r r r b dh h r c r B ä cd cc r D q bd ˆ ˆ ˆ Th r 3 q B ˆ ä cd h c B -dr ˆ cd > qh r cc Th r 3 r - B dr- cd brc brc r r cc r r rr r h cc dr r r hh r D q hh ˆ ˆ ˆ r c h h dr r r r r b r h rcc r r cc r r bd r r h c B dd cd r h dr cc r r cc r Th B ä cd 3 r h r B c dd b cd r h h h > d h b h r hh h dr h cc r h Or cc r r rrc cd 6 7 B dr- db ä cd qh r c r b drd c ˆ drc r b ˆ r br ˆ Cr brc h - h brc dr Cch r chrcrc d h r cqc r Cr b rcr Th br Or B ä cd rrc cd h b b brc 6 brc Cr ˆ r r brc ˆ 7 brc B dr- cd qh c ˆ h h ˆ ˆ 4 b drd Gr drc h Srcr r br Cr hh qh h 4 rdc Cch chrcrc h h b cqc c Cr dd rcr Th br B dr- cd h b b Cr h dr r cc r r q ˆ r r ˆ c Th rcr 4 Gr r Srcr h h c rdc r cd d 4 B dd cd rdc cd r Br - h r Cb c - Hd b c b r O dd d c c b Br Cb Hd h rch Or Th q r rcr dc Hr r r r b B cc dd h cd h c r h h c rdc d r b cd d rd r d d cd r c r Th b r q ˆ Br hh Cb hh Crh cc cd Hd O c Br dh Cb ξ dh Hd η Cch hch chrh h r c r r h rch c d r h dc cqc Hr Cr r c r r4 E r Th r d b r : b cc h h B c dd cd h h b r Cr r r r Th H h cc cd hch hrh r d d cr 4 4 E Gr : S r c r h c H 4 h brc r d c h h b br c d d h r brc brc rd r rd br brc c c r ˆ ˆ ˆ T ˆ ˆ ˆ rd ˆ rd r r h rd h rd c rdc d r q Br Cb Hd O c Br Cb Hd c r h rch r d c Hr r r rh brc r rd d rd b cc h h C r rd r C rd Th h c r c H brc rd rd brc brc ˆ hch brc hrh r h d d d 4 E : ˆ ˆ ˆ H ˆ ˆ d r ˆ ˆ r d d ˆ r ˆ r ˆ ˆ
9 Wrhr 349 hc Wrhr 349 Wrhr 349 hh - rd rd br br cr d cr hchd T hch T r d b r c r d hch T ˆ ˆ d ˆ brc br hc h D b d b brc br hc h D b b brc d b r c b r hc h ˆ D c r r b b brc ˆ ˆ ˆ ˆ h c r ˆ ˆ r b ˆ Th ˆ ˆ brc brc ˆ h b rhrr rd r rd rhrr r c r r rd rd rhrr Th ˆ ˆ ˆ rh rr ṇ ṁ ṃ h b ˆ rhrr r br h ˆ ˆ r d r rhrr cd rr-r-r-r-r-r-r-r-r-rb ˆ cd d r h ˆ r r ˆ rhrr r ˆ q r r ˆ ˆ ṇ r r ˆ r ˆ Φ ˆ rhrr r ˆ ṁ ṃ r h cd d cd r c r d r r z r ˆ ˆ ˆ ˆ ˆ ˆ ˆ dd ˆ b z ˆ r rrhb-rrh-rrhrrh-rrh-rrh-rrh-rrh-rrh-rrh-rrh ˆ ˆ ˆ ˆ ˆ br ˆ h ˆ r ˆ d ˆ cd ˆ ˆ ˆ ˆ cd d ˆ h ˆ ˆ ˆ ˆ ˆ ˆ ˆ C ˆ Ph ˆ ˆ ˆ ˆ r-r-r-r-r-r-r-r-r 9 r-rb r r b r h r r d r c r d cd 4 d 3 h E : r h c r r r Ph ˆ H ˆ ˆ ˆ r ˆ ˆ ˆ r ˆ ˆ ˆ r ˆ ˆ ˆ r ˆ ˆ ˆ ˆ ˆ r ˆ ˆ ˆ ˆ ˆ ˆ r ˆ ˆ ˆ ˆ r ˆ ˆ rrhb-rrh-rrh-rrh-rrhrrh-rrh-rrh-rrh-rrh-rrh ˆ ˆ h cd d cd c h cd d c r d c d z ˆ d ˆ z dd b z Eq: d E d b : z ˆ h ˆ c H brc rd rd brc b ˆ ˆ ˆ 9 d 4 3 E : ˆ h c H ˆ ˆ r ˆ r
10 35 Wrhr d35 35 Wrhr Wrhr rd rd r rd rd ˆ c r r r ˆ r ˆ brcb r ˆ r c rd r r rd brcrhb rd rd H c r brc rd rd brc r rr q brc brc H r h h ˆ r ˆ r r hh rd br cr d hch Hrd ˆ br cr d T r hch T r q r d brc br d brc hc br h D hc h h b D b c r b b brc ˆ brc c r brc h cd h cd cd b crdc brc N h Hbr br h h b E crdc br r b d r br d T G br ˆ rhrr rhrr rd r r Th rd N h Hbr ˆ d b br r c b r hc h ˆ E br D r b b ˆ b d br r c G br r Th ˆ h rh h h rr ˆ 8 rhrr ˆ h h 9 rhrr br h r ˆ r r d h N h h ˆ cd h br Hbr h b r r d ˆ cd h cd d cd d crd crd ccd ccd ccd ccd r r G E G b r r h ˆ h b G b r c cd d c cd cd dd d cd dd cr Th d z ˆ z rhrr b rr c c c c c c c b d c b b d cd b cd cd cd c cd z cd cd cd cd cd cd cd cd cd cd cd cd d rr r z U r d h z U r d h b r h A r d ˆ q cd d crd ccd ccd hc A r G ˆ ˆ cd cd cd cr c cd d c r d dd d z c b d b
11 Wrhr Wrhr Wrhr A / h 3 /3 / 3 h 3 b rr cc A d rr rh b / / h ˆ / ˆ 3 / 3 c / cd h cd ˆ / 3 / 3 h A ˆ / cd cd cd ˆ / 3 cd cd h 4 ˆ / c 3 h 3 cd 5 cd ˆ 3 / cd cd cd cd cd cd cd cd cd hc d V 3 d b rr cc d V Sb r d rr rh b rr c c c c c c c cd cd d V 3 Eq: cd cd cd A z A z cd cd cd V A cd cd cd 3 A A A V A V cd cd cd cd cd cd d rr rh Fr Sb r h A z A A V V A S b V Az h AzV r AA AAV AV h b Eq: z r b Fr rr cc h A z A z V d rr b rr cc A A A A V F ch A Az AA V V V A d V rr rh r d cc r d rb h h r r d Eq: Fr 3 h rh rr z r 4 A 4 z c A Gdd A b 3 V V r cc c A ˆ V F ch c r d cc r rb 9 h r r d h T h d bh W r b d 3 z ˆ ˆ ˆ ˆ 4 4 c Gdd b 3 r cc c d c 9 d T h F ch d bh ˆ W r r d c r b ˆ h r b rhrr ˆ ˆ 3 d 4 4 h c h Gdd b h 3 r c r c c ˆ ˆ 9 ˆ ˆ ˆ T h d b h W r ˆ d ˆ ˆ
12 35 Wrhr d35 35 Wrhr Wrhr b d ˆ ˆ -d 3 ˆ Ĩ ˆ d ˆ -d ˆ 4 ˆ Ĩ h ˆ ˆ h h ˆ d h h h -d ˆ Ĩ ˆ d d W r W r q q rd rd rd rd W r q rd rd q r brcb r ˆ r r ˆ ˆ ˆ b rr c r r ˆ d rr rh Gdd b 9 3 b rr d q c r r d Gdd rr b rh 9 ˆ ˆ ˆ ˆ 3 ˆ d ˆ ˆ ˆ q h r 4 Eq: 4 q rd rd rcr d-d-d h cd c η rd cr rdrd Gdd b r 9 3 d d-d-d b rcr hh h cd c rd r q ˆ ˆ r ˆ r ˆ ˆ rrr r ˆ r T r ˆ c rh ˆ rr q rrr ˆ 4 T r c d z z z d d r c r z h z rd z r r B d z d rhrr q ˆ ˆ d z cr B d T r c q z ˆ d ˆ q d ˆ z ˆ q q d q z Hc c ˆ d ˆ d ˆ q z ˆ q 5 Hc c cr B d Eq: 5 b d q q q ˆ ˆ q q ˆ ˆ q q ˆ ˆ ˆ ˆ d d d b d q ˆ q ˆ ˆ q ˆ
13 Wrhr Wrhr Wrhr U - d b U hh d b U d b q q ˆ q q 5 ˆ 5 bc bc ˆ ˆ Eq: 6 q q d h 5 bc q q 6 b Ech q r q 6 ch ˆ ˆ Ech r 6 ch cd ˆ 6 ˆ b cd 6 b q q ˆ ˆ ˆ ˆ ˆ q 6 d q q q q Ech r 6 ch S S S S S S S cd 6 b hc r h h d r r brc zr br q q Fr 4 brc S brc br q ˆ q q ˆ q ˆ brc S ˆ ˆ S q S q q ˆ ˆ ˆ ˆ S hc r h ˆ h d d r r h r r zr S c c c r h Fr dr 4 S r d q Th drb q dcd b r S dr c dh dd q h q br br d r h r q q S S S c c c r h S S r r hc r h h Th drb d dcd r b r r zr dr d c H dd h cr Fr 4 rd rd ˆ d-d-d brcb br br br br h q ˆ ˆ d d ˆ ˆ rd rd d-d-d C C h dc r h r bd b ˆ c S c h r cd c r d * rd rd d-d-d Th d r b dcd b r dr d-d-d c H ˆ dd h * rd rd d-d-d r ˆ r ˆ rrr rd rd dd-d d ˆ d ˆ ˆ d d ˆ ˆ c bd b c d cd c d r ˆ r q ˆ rrr b rd rd h r d-d-d d d rh rd rd d-d-d h ˆ r ˆ r ˆ rr ˆ ˆ rrr ˆ ˆ r ˆ r rrr ˆ r ˆ r rrr ˆ
14 354 Wrhr d Wrhr 5 d Wrhr Sr D 5 d Sr D Th cr Th 5 r hb d r h Shb c r D h hd c rc r d hd rc h hh r d dr c h hd hh dr r c d Th hd rb d hr Th rh d d qh Th rb h hr r r h c d hb h hd r b h c c A b cdr h hh dr hd h c b r Ab c hd cdr r d h r r d hd dd Th rb c hr h d r d d hd r hdd r Hr c ch dd hd b r A h b Hr c ch d d r c cc h d c r h Cr c r h c cc r d r d hd dd d c Cr 3 r h d r d r 3 b h b Hr h r d F ch d r d cr d hd brc 7 brc brc h c c r c d c C d F 3 brc cr b d h hd r d 7 chc h Cr r crd br 4 rc hc chc h F Cr r crd c r br d hd 4 rc r rr 7 r rr 4 4 r 4 rc 4 r rc rd rh dr h d r d 3 3 chcf h d Cr r crd Gr r b r 9 : F 4 d Gr r brc 9 brc r c Σ b drb : d c d d C S r r r 4 Σ 4 b r drb rc d r c d h d d r d T hbhd U r h cr d hc ch h C S d F c T r d Gr r 9 hbhd U r S Σ h q cr d b q d q r b q U ch h C S q brc d c q d Th d d d d q hch r brc q U Th d Σ d r d bd Gr hch Σ r ch C br S Gr d Σ Gr d c Λ Σ S r Th bd Gr Gr Σ rr C S b d c c - rh T d r rr ch br Gr C S hbhd A cr hrh U Λ Gr Σ h r h h c γ r Λ γ d Gr γ d dd ch c b h hz C bd Λ γ C Λ S Th cr d c Th T Λ Gr Σ cc c Gr S d drb C Σ Gr S q Σ Gr Σ Th Gr q bd Gr rr b Σ q r c h q d U drb Σ Gr Σ cd A cr hrh Gr r Σ C bd A cc Gr cc C S Gr Σ Λ Λ h h Gr Th Σ hr d rc d bd d d hch Cdr r Cr r G h br S b r bd Gr dd S b hz rhr C bd ch b r Gr -dcc 3 S 4 C bd Gr Th d c bd cr brc Th 3 4 Th hz c d c T C bd Gr C Gr S S r r b H cc c 3 d drb C S Gr C S A cr hrh r d r bd 3 Gr cd d cd Gr C h S S h h Th Gr bd bd Gr C S r drb C S d Gr d b h r z C S bd h crd d d d d d 3 d 4 bd 3 d cd q d Gr c H r Σ3 G C T A S cr d bd V dc Gr cc A cc S cc H V d H hch c h c h c Gr C S d C d r b 4 bd 3 S 4 Gr 3 4 S * C Gr h bd S h Gr Th C Gr S b Gr S hr h r h S r d Gr V 4 S Gr Cdr r Cr r G h S br b A cc Gr brc S 3 4 brc brc brc bd brc h brc q 3 S brc 4 Th hz c hr H r brc d 3 brc r d r 3 cd d cd cr h Cdr Cr r G h b r b rd rd 3 rd rd 4 rd 3 rd 4 3 rd rd 3 4 rd rd 4 h crd d d d d 3 d 4 r 3 b r c d H 3 C S 4 3 G Th h r z c A cr d V dc
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