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A Time Domain Characterization of 2-microlocal Spaces

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(1)A Time Domain Characterization of 2-microlocal Spaces Stéphane Seuret, Jacques Lévy Véhel. To cite this version: Stéphane Seuret, Jacques Lévy Véhel. A Time Domain Characterization of 2-microlocal Spaces. [Research Report] RR-4545, INRIA. 2002. �inria-00072043�. HAL Id: inria-00072043 https://hal.inria.fr/inria-00072043 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A Time Domain Characterization of 2-microlocal Spaces Stéphane Seuret — Jacques Lévy Véhel. N° 4545 Septembre 2002. ISSN 0249-6399. ISRN INRIA/RR--4545--FR+ENG. THÈME 4. apport de recherche.

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(161) cmf „V-aBlmpIU,U,Vcevm\bsn–„T_Tc©\gcY\blhTfBc š }0Vœ : U(Q RTna V‹ve•Ive\^\^hTˆBˆYRIccmRcmadVrc vmU E y †  yˆ†€y${ J 0 E yˆ†  t†<yb{ J|–kceRT\bl0vm\^ˆBRIccmVrvmU¨fn—5' J *sranhi„V vmVœ°”0ve\gcqcmVrhµaBl0f7hTVfB—ceRTVcy”f(—™f7`g`^fd”0\gh&ˆ(VtkjTveVrlelq\^fBh&l#'™cmRTV©`^a7lyc¬fBhTV„<V\^hTˆ‹lmp]U,UYVcmve\^s\gh anh o a* 6. . . . y. . yˆ† ;  . '. yˆ† ;  ' G; ?TE y †  t† y { J ";G? 5 ;7 '  G ;G?"E€E yˆ†  yˆ†€y${ J E yˆ†  >†€y${ J]J & ";G? 5 ; 7 '. . '$ /. : ’ h ec RTV-,š —™fB’ `^h`gfdo ”0V\^V hTo ˆ–J–:cm”.R&VV-pD”0”\g`^fB`0_&U,` o f7lqvmcYVA­7fB_&—~\gvecmVYR&VcmRTceV,\gU,_&V¡lmV(a5fn B—f7\ <po ]ˆBcmRTU

(162) Vi_&hsveo \£celm\^srj&ana7`0sVrsl¢a7lq\^VAh&l%lqcm#Vra o fB —.8 ceRT9V,; _&lman_&h ano ` 8w9 ;. RTfBU,f7ˆBVh&VfB_l¤%¥B` o Vv~lmj&a7sVrlL E 7 9 r J š : ’ — ETy J o VrhTfnceVrl›ceRTVw—™_&h&s|ce\gf7h  A &A-A B '  

(163) &  A ' –dfBh&VwVra7lq\^`gp lmVVrl3cmRadc b\ lanc`^VraBlqcsfBhIcm\^h]_TfB_&l adc~VAaBszR-jf7\ghIcr–kVAlqj<Vrs\^aB`g`^p‹an7c y {Bš:’ h o VV o –T\g—†+"† ƒ  „" D T–kf7hTV 0” vm\gcmVAl !. . %. . %. . . E aJ 0  y † ; ;@?  ! š QR]_&l 8  ; ;@? aBvmf7_Th. ”0\£ceRww.  ETy JІ. $.    † 0 T –kfBh&V¢RaBl. †+a†6ƒ  „).

(164) . . . '. .

(165)  . '.  . E "J 0  yˆ† ; 5 ;7 yˆ† yˆ  † yˆ †€y${  ”0\£ceR&6L†ƒ  „=!7–]cmR]_&l 8 ; 5 ;7anvefB_&h o y {7š  E y^J†. . . . . . . o yb{ Mš w h«cmR&VfBcmRTVrv

(166) R&aBh o –:\g— ;G? ' 5 ; 7  . 0 yˆ† ;  5 ; 7. . w hTV‹\bl

(167) cmR]_&l,an`^`gfd”V o cefDczan•BV 2 y${-\gh' J *fBv©\gh' D †[*|–waBh o an`blmf¡cmf§sfBhlq\ o Vv

(168) cmRTV-veVran` h]_TU

(169) „<Vv E yb{•J š : fnQ—0RTceV RT`gV V— oc~\gR&¶FanVhveVo hIœªclm\ o cmVrV¬vmfBU(—l©' J \^l

(170) *RTanhfd”o Vcm BR&VrV¢vVVrtka7jlqf7pDhTcmVf hIce_Tlh ”o V¢Vrve_&lqlqceV aBh U(o a5FYpŸfBlmhTVV(VU cevm\^sVrfBlU,cmfjT`^Vvmt V o š _&QsRTVŸV cmhTRTVAV-slyVrcele_ lm\£o cypp fnn— $DcmfŸcmRTV©fBhTVfB—a(hTVr” —™_&h&s|ce\gf7h o Vve\^ BV o —™vefBU $DcmR&anc¬”0\^`g`„<V`^fBhTˆ(cef‹lmfBU,V :A =B  ? ”0\gcmR  0  X =!€aBh o  0  ! š ’ hŸa¢”a5p7–n”.V0janvmcm\gcm\^fBh©cmRTV%lm_T„&lmVcwfB— 9 r ‘ EFHG€KJ “^A .  GD  0 ,anh o  D  ” \^hIcmf ce\g`^Vrl fn—lmaBUYV,lm\grVB–3aBh o ”V©cmvzanh&lm`^ancmV©cmRTV,jTvefB„&`gVrU cmfcmRTV

(171) ²q\^hT\£ce\^aB` ´©cm\^`gV ‘ E‚HG€ J -CL n !anh o #! ” ' lmVV žˆB_TveV D * š {Tf7v©VtkaBU,jT`gV7–:\£—%aµ—™_Th&scm\^fBh $ „<V`^fBh&ˆ7l

(172) cmf : A ;>B= ;G? ”0\£ceR 0 =   !7D– $*a o U,\gcel anvefB_Th o yb{%a o Vrvm\^ dadcm\^ BVfB—Ff7v o Vv š QRTV—™f7vmU©_T`ba€lq\^U,jT`gpYlma5pklceR&adc $ 8 : A ;>B= ;G? U,VAanh&lceR&adc . . . /. . . –˜— ‚/–u™.

(173) J.  

(174)  

(175) 

(176)   "!#$

(177) %'&. 1. 0. −1. −2. −3. −4. −5. −6 −1. {\^ˆB_&vmV D F:u:a5 ]\gh&ˆ©fB—ceRTV€R&an`g—jT`^aBhTV 0. 1. 2. 3. 4. . 5. D . š. 6. 7. : ;  = ;G? š QRTVrhJ–&\g—D$ 8 < : ;>= ;G? ”0\gcmR-ˆ w  }!€”0\gcmR D !B–fBhTV€veVjT`baBsVrlL$ $ 8 € AB —™fBveU

(178) _T`ba1' D †H*.„]p % & A A  ' A B % & A ' –&aBh o cmA RTB \blhTV” —™_Th&scm\^fBhi”0\^`g`J„<V`^fBh&ˆ©cef : HA ;B 5 ;97t= ;@? ' 5 ;7 š  . w. hTVsaBh‹—™f7vmU(aB`g\^V¬cmRTVAlqV ceRTfB_&ˆBRIcel„]p‹lea5p]\ghTˆ©cmRadc0cmR&V€fBj<Vvzadcmf7v $ y. Q5R. \^h. . $ETy J†:‡E y^J y < † y { 65 ;97 . ' —™f7v0cmRTVsfBvevmVAs|cjf7`gp]hTf7U,\^aB`/ *U(aBj&l : A ;%B= ;G? ‡ZT_ YHe E_alDk#`/_"eaZX`Y. \ghIcmf. : A ; B 5 ; 7n= ;G? ' 5 ; 7  . š. ¦§V¬lqcmveVrlellmV 7Vvzan`T\^hIcmVrvmVAlyce\ghTˆ©jTvefBj<Vvmcm\^VrlwfB—›cmRTVAlqV¢lqjaBsVAl : ;>= ;G? š uwvefBj<f7lm\£ce\gf7h&l' štD *cmfA'° š W[*anveV xy_lyc~VtIceVh&lm\^fBh&lfB—3cmR&VsfBveveVrlmjf7h o \^hTˆ©f7hTVrl\^h€e J fJcmfYcmR&V€ˆBVh&A B Vvzan`›sa7lqV š QRTV žvelqc0jTvmf7jfIlq\gcm\^fBh-\bl0anh-VU

(179) „<V o&o \gh&ˆ(jTvmf7jVrvqcyp(„<Vcy”VVrh-cmRTVlmj&aBsVrl : A ;%B= ;G? š [ Ž ¼ 8 ¼ HŒ , H , ¼ + /"_`( ƒ'y { 8i9 rts

(180)  ^s:HG€FG  G  ‡v% '  <%

(181)  v%%& &  >L$

(182)  :0 

(183)  ^s     0 w  ^$' : -A ;%B = ;G? :  =  ?   -A B. QRTV€j&vmf]fn—fn—uwvmf7jfIlq\gcm\^fBh¡ štD \^l0lmjT`^\£cmcmV o \ghIcmfYcy”f(lq\^U,jT`gVrv0`gVrU,U,a7l .¸ “ /9_T( |y { 819 rts

(184) ^sHG€FG  G  /v> '   %

(185)   v%%&m&  %r$

(186)  0 . $' : HA ;>B= ;G? [ ^] . . : =  ?  A-B. Ž ¼J¼ ‘ 3Vc%_&l%cmveVranc~cmR&V©sa7lqV>5  B! ' \ š V š veVj&`^a7s\^hTˆ$µ„]p $ anh o €„]pAL† š . ‚ƒ‚ „ †‡'ˆ%‡'ˆ. . 3

(187)  ^s     2 h#l. 2 [*|–cmRTV©ˆBVrhTVvzan`3sraBlmV\bl0cmR&Vh. o V o _&sV o „]p.

(188) D. †. !    

(189) >$'. ’ n— ƒ  „ ƒ  „ª–BceRTV¢vmVAlq_T`gc.\blwf7„] I\^fB_l š JVc._&lcmveVranccmR&V¬sa7lqVtƒ  „ 2 ƒ  „H†@!7–7ceRTV%ˆ7VhTVrveaB`&veVrlm_T`gc.”0\^`g` ceRTVhiVAaBlm\g2 `^p(—™fB`^`^fd” „]p‹\gcmVvzadce\gf7h š 3Vc~_&l~a7lmlm_TU,V¢cmR&anc $ : ;>= ;G? –y { ,anh o cmRadc 0 y –T”0\gcmRTf7_kc%`^f7lelfn—ˆ7VhTVrveaB`g\gcyp š w hTV aB`^lmf‹a7lmlm_TU,Vrl~cmR&ancy D &–8 „]pivmA-VrB jT`baBs\^hTˆ2 y $E[J„]p y $EX†)J š QR&VveV

(190) Vtk\blyczl%a‹jf7`gp]hTf7UY\ban`  lq_szR‹ceR&ad> c ' J *RTf7` o l Nš w hTV \blhTfd” `^f]fB•]\gh&ˆ©—™f7v~aYjf7`gp]hTf7UY\ban/`   ceR&adc%leadce\^lqž&VAl 44 $E y J<†4  E y^J † $E aJ†4  E aJ 44 0  yˆ†   ? E yA†  y J  ? 5  7 ' D  * 44 y 65 ;7 44 65 ;97 3Vc_&l o VhTfBcmV„]# p   cmRTV.jf7`gp]hTf7UY\ban`Ifn— o VˆBveVV ƒ  „ 2 ƒ  „€†%!”0\£ce7 R€cmRTVleanU,V.sfIV³Ÿs\^VhIcel7 aBl *_Tj€cmf o VˆBveV‡V ƒ  „ š Q3fŸlm\gU,jT`^\g—™pŸcmR&VhTfnczadce\gf7h&l–k`^Vc¬_&l o Vž&hTV E y J 2 & A ' A    & A ' aBh o  ETy J 2 & A A '  

(191)  & A ' š QR&VhJ– . .

(192) . .  #.  #. . '. %. ^yG8  E y^J M. . fd”€–.  TE y J †  E

(193) aJ U p“sfBh&lqcmve_&s|ce\gf7hJ–D‡ETy J+†6 ‡ETy J†:  E y J 2 9 y 5 ;7 –Tanh o  TE y J† 

(194) E aJ 0 Q. _lqV—™_T`JveVU(anve•(\^lceR&adc ' hTfnceV¢cmR&anc !5† . 0. D † ƒ  „˜* š. 0. 44  y 4.  E y^JY\^l(a. 0 0. 0. 0. %. %. . ..  ETy J†:  ETy J G ‡ 2 y ^ETy J y 5 ;7  #. & '  & A '  †   ^EaJ/  & %. ' %   

(195)  .

(196) E y J  & % A ' %   

(197)  . A. 44 4. jf7`gp]hTf7U,\^aB`”0\gcmR fBh&`gp“f7hTV-hTfBhkœªVrvmf§sf]V³Ÿs\gVrh7c š QR]_&l 5 4. yˆ†  '  ?] yˆ†   ? ' #   yˆ †   '  ? E y †  y J   ? ' #   yˆ †   '  ? E y †  y J   ?  5  7‚‰. o \^vmVAs|c0_TjTj<Vv%„f7_Th o —™fBv Ÿ \bl0fB„Tcean\^hTV o „]p(czan•]\gh&ˆ. =. yG E y^J† ^ EFHJ 0  y ; 5 ; 7f‰. Q%jTjT`^p]\ghTˆ&' D ‡H*.anh o ' J *|–]ceRTV `^a7lyccmVrvmU —™f7`g`^fd”~l. 0. %. 4 5. M. E J 0. . . y E y^J<†  ^E"J  9 44 y 85 ;7 †  85 ;7 44 4 y ;7  #  ;7  # 4 y E ^ETy J<† E HJ]J†  E ^EaJ† E [JTJ ŽE 9  ^EFHJ J yˆ† . yˆ† 2 Q. . . E J 2 y E E y^J†. 2 ,\^h4' J *. E HJ]J,† E E aJ† E [JTJ \bl.cmveVrancmV o .

(198) . y †  ^ETy J† ^EFHJ   E E y^J† ^EFHJTJ<†  E ^E"J† yˆ† ^ETy J† E HJ   E y^J† E

(199) aJ  yˆ †  y ; 5 ;7   yˆ† ; ' ;G? E y  yˆ† J  ;G? 5 ;7  yˆ †   '  ? y‡†A #  &  '  ?

(200) ' y ; 5 ;97   y‡ †  ; ' ;@? E y  yˆ† J  ;G? 5 ;7. ' D H‡ *. aBl. E HJ]J . –˜— ‚/–u™.

(201) DD.  

(202)  

(203) 

(204)   "!#$

(205) %'&. Zk\gh&stV +†6ƒ  „aBh o !L† veVrlmj<Vrs|ce\g 7V`^p©„]p1E y  ƒ  „ 2 ƒ  „ †!B–. E    uJ0anveV€j<f7lm\£ce\g 7VB–&f7hTV_Tj&jVrvqœª„f7_Th o l y ;  5 ; 7 – y †A #  &  '  ?

(206) 'aBh o  y#† J; 5 ;97y– E y  y#†  J #  &  '  ? ':anh o E y  y#†  J š QRT\^l.ˆB\^ BVAl–7_&lm\ghTˆ E J.  yˆ†  '  ? E y  yˆ† J   ?

(207)  5  7 ' ;  '  yˆ† ; ' G; ? E y  yˆ† J  G; ? 5  7 ' / ‰. 0. ±  BVhIce_&an`^`gp7–veVU,VrU

(208) „<Vve\ghTˆŸcmR&anc     2 56 –Ff7hTVR&a7l>†+d†6ƒ  „  w h&V sfBh&s`g_ o Vrlr–k_&lm\ghTˆ wL†  D T–]ceR&adc. 2 †   † ƒ „  6. 6.  E y^J†. .¸ “  : >; =  ?  AB [ ^]. 6. w † |š.  E aJ 0  yA†   ?XE y  yˆ† J  ? 5  7 . . ”0R&\^szR-ˆB\^ BVAlVtTaBscm`^p,cmRTV€veVr­I_T\^vmV o veVrlm_T`gc š . /. . '. . . /9_n/ 9y { 8 9 rts

(209)  ^s>HGI FG  v% |%>>

(210) "  ‡v%'%&&  %  

(211)    0 .  : -A ;>B= ;G? . Ž ¼J¼ ‘ ’ 5— ƒ ? „ ƒ    „°–TcmRTV€veVrlm_T`gc\blfB„] ]\^fB_&l š ¬lmlm_TU,V cmRadc 2 2ƒ   „ 2 ƒ    „" !7–TcmR&VˆBVrhTVvzan`›veVrlm_T`gc~”0\g`^`JceRTVh¡VAaBlm\g`^pŸ—™f7`g`^fd” š Q¬l0_&lm_&an` h&fd”€–I”V¢”0\g`^`JaBlelq_&UYV¢”0\gcmRTf7_kc0`gfIlmlfn—ˆBVrhTVvzan`^\£cypYceR&adc}y { 2 Yanh o 0 y š U pŸa7lmlm_TU,jkcm\^fBh3–/' J * R&fB` o l–TaBh o fBhTV€”anhIcel.cef(jTvmfd 7V Q. . $E y J<†4‡ETy J †   # $E"J†:‡E

(212) aJ 44 0 y 65 ;7  &   # '  65 ;97 &   # ' 44  yˆ† ;  &   # ' '  ? E yˆ †   y J   ? 5 ;7 ' &   # '€‰. 44 44. Qan•]\^hTˆ. =. 2 ©\^h ' J *pI\^V` o. .  #. l.   $ E y^J†:‡E y J 0  y ; . . —™f7v¢a-sVrvqczan\^hµj<fB`^pIh&fBU,\^aB` fn— o VrˆBveVVYadc UYfIlycƒ  „ † š€’ h7ceVˆ7veancm\^hTˆŸž&vzlqc¬ceRT\^l¢`baBlqc¬\^hTVr­I_&aB`g\gcyp „<Vcy”VVrh ,aBh o y3–&aBh o ceRTVhi„<Vcy”VVrh anh o y3–&fBhTV fB„Tcean\^h&l $E y^J†4  ? ETy J 0  y ; ' D [W * aBh o E $E y^J†4  ? ETy J]J<†6E $E aJ†4  ? E "J]J 0  yˆ† y ; ' D ƒ * —™f7v¢aj<fB`^pIh&fBU,\^aB`   ? fn— o VrˆBveVVYadc UYfIly‡c ƒ  „^† q!A'™cmRTV(leanU,V

(213) f7hTV©—™fBv¢„fBcm6R ' D )W *~aBh o ' D ƒ *]* š w h&V¬žvelqc0”0vm\gcmVAl $. .  .  .  .  #.  .  #.  #. . . ' #. . 44 44. ‚ƒ‚ „ †‡'ˆ%‡'ˆ. .  #. $E y^J†4  ? ETy J † y 8 5 ;7   '9#. . $EaJ†4  ? E"J 44 0 E JŽE J G 44 9 9H9  8 5 ;7   ' #.  #.  .

(214) D. . !    

(215) >$'. ”0R&VveV 44 44. E9 J 2. aBh o. E 9H9 J 2.  w. $E y^J†4  ETy J y 5 ;97  ' # ? †. T.  #. .  #. 44 44. ' #. 44. .  #. 4 44 † 65 ! $E

(216) aJ†:  ?

(217) E aJ 44 y 65 ! 4 ;7   ' #  ;7   ' # 44 5 5    # $

(218) E aJ†:  ?

(219) E aJ y ;97  ' # 65 †  ;7   ' # y? ;7   ' # †  y 5 ;97     # $

(220) E aJ†:  ?

(221) E aJ y  yR 6 5 ;97  ' # †  y  #  ; 5 ;7   ;  '9# yˆ  6 5 ;97  ' # 0  y y † ; 5 ;97 ; 5 ;7  #   y  y yˆ †  0  yˆ† y ;  5 ;7  # G . 0. E 9[9 J. 44. $E

(222) aJ†4  EaJ y 5 ;97  ' # ?.  #. $E aJ†4  ? E"J 44 ‰ 44  8 5 ; 7   ' # „]pA yˆ† y ;  5 ;7  # š. $E"J<†4  ? E"J † y 8 5 ; 7   ' #. \^ I\ o \^hTˆ-' D ƒ*.„]p y 5 ;97 –/E 9 J0\^l„<fB_Th o V o hcmR&V€fncmR&Vv0R&aBh o –T_&lm\^hTˆ&' D W)*|–kf7hTV R&aBl  . . 0 0 0 0. .  #. ml \^h&sV A \bl„f7_Th o V o „]p-! š QRTVleanU,V€•]\^h o fn—U,aBhT\^jT_T`badcm\^fBhlfn—Vtkjf7hTVhIczl ™' „T_Tc%Vra7lq\^Vv *aBl0adc0ceRTVVrh o fn—cmRTV€jTveV ]\^fB_&l j&vmf7jfIlq\gcm\^fBh sranh „<V j<Vvm—™fBveU,V o›š }0VrU,aBvm•]\^hTˆ«ceR&adc y ";G? 0 E y †  y J ;G? anh o _&lm\ghTˆ % n  a†6E P†w!•J !B–k\gc~\bl0Vra7lq\^`gp‹ BVve\gž&V o cmR&anc %. %. %. %. . . .

(223) . y‡†A y ; 5 ; 7 . #. 2 0. 0. yˆ† ; '  ?

(224)  yˆ †  ; '  ?

(225)  yˆ †  ; '  ?

(226) . &.  # .  #. &. .  #. &. . ':'I y ; 5 ;97 ' E yˆ†  y J & ;  ? & ' ' ' E yˆ†  y J  ? & ' 5 ;7 ' yˆ† & ;  ? & . # . '. . . # . . . .  #. . . '. '. .  #. .  #.  #. ; 5 ; 7 . ' . . ”0R&\^szR-ˆB\^ BVAlcmRTV€veVrlm_T`gc š  f7U

(227) „T\^hT\^hT‡ ˆ JVU,U(aBl~ šED aBh o šED –kuwvefBj<f7lm\£ce\gf7hi šED \bljTvmfd 7V o›š ¦§V€hTfd” sfBU,j&aBvmV¬cmRTVAlqV : AH;>B= ;G? ”0\gcmR-cmRTVs`^a7lmlm\bsan`Fj<fB\^hIcy”0\^lmV€¤~¥B` o Vrv~lqj&a7sVAl š [ Ž ¼ 8 ¼ HŒ , H , ¼ + /"_/ ƒ' y { 8 9 rts

(228)  ^s-wv>u

(229) %

(230) }  v%j&  > $

(231)   D  s 68 7 A ; B 2 : A ;>B=  ; . #. 9; . $'. –˜— ‚/–u™.

(232) D.  

(233)  

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(235)   "!#$

(236) %'&. ‡. [ Ž ¼J^¼ ] ‘ JVct$ „<V‹a-—™_Th&scm\^fBh«\^h?AH; B š5w hTVŸ”0ve\gcmVrlr–3_lq\^hTˆ¡ceRTV‹anj&jTvmf5tk\^U(adcm\^hTˆ-jf7`gp]hTf7U,\^aB`D —™f7_Th o \^h4' D *|–. 44 $E y^J†4‡ETy J 44 yˆ†€y { 85 ;7. † $ E

(237) aJ†:{ ‡ 65 E

(238) aJ >†€y ;97. 44. Q R&\^ljTvefd BVAl5$ 8 : A-;>B= "; š 7 w h©cmRTVfnceRTVvR&aBh o –n`^VcD$,„<V0a~—™_&h&s|ce\gf7h©\^h ceR&adc E y^J\ oio Vž&hTV o „]p A &A A B '   

(239) & A ' * . . %.  E y^J†. ceR]_&l. ƒ  „ †. %. . 44 $E y^J†4‡ETy J 44 yˆ†€y { 85 ;7. 44. . 0. E yA†<yb{ ; E yA †   0 0. : >; =  ; AHB. š Qan•]\gh&ˆ . 44  44 $E aJ†:‡E aJ 44 44 44 >†€y { 65 ;97 44 5 ;7  >†€y${ ; 5 ;7 J y †<y { J ; 5 ;97 ‰

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