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The Formal Theory of Birth-and-Death Processes, Lattice Path Combinatorics, and Continued Fractions

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(1)The Formal Theory of Birth-and-Death Processes, Lattice Path Combinatorics, and Continued Fractions Philippe Flajolet, Fabrice Guillemin. To cite this version: Philippe Flajolet, Fabrice Guillemin. The Formal Theory of Birth-and-Death Processes, Lattice Path Combinatorics, and Continued Fractions. [Research Report] RR-3667, INRIA. 1999. �inria-00073005�. HAL Id: inria-00073005 https://hal.inria.fr/inria-00073005 Submitted on 24 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..

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(14) ` `dgl~sp wv_ i u iownp gluv`Dv` p `jwM`T‹. ¨. ™Fš„œˆ› ‚žƒŸ œq›  #¡C‚¢‚£M¤. mo|. ¥ ~a`cpg] dsg u grwv_fglu wv_fg„¦8§c@¦  c m wsbo{agl~ w m†{fu p  i `d~fgdsgi©mobd{ wngri {qc mfwlb†{fgl~ w m†{fu p q,ª^N„ƒ«_jfqh_¬f„ƒj_fŒŠ­yˆ Š § q® ygƒdsg¯˜°hau i ~ w gƒ± ‡Ygl] bTgl] ef_amo~fg3²/°‚fPf_±f« „iof „gPf1jPj}„o„³‡Ygl] bTgr] w m†ef`dg3²<°ˆf_fP±<«f„f† Wg_fWj_f„fP«.

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(85) ^V`d~*ds` ‚ glu` gi g b d`ƒm!}i efefbd`jwli pv`cm†~`rqsbd`C|og ef_a` `jwM` qf|}md glbdbc`d~^Am!}p glbdgrwnmo|V|{€ ~f`jwli pv`cm†~` ~fglp‚{mou|2` q#b†{fg {fgl`d~^%pv_fglmoua q&e m†ef{fbji pv`cm†~^†u m!{pv_q mouGi ~ i bCa` `C` m!}das~ i |}`jwdfi p i` p uv{ w pv{fu gl` q i u 1 g dslg ` wnu `dt gTdt2a |}gri ~`mk}p u i ~` `dgl~spGwv_ i u iow pvglu `C`vp `jwM`>ik`8` mfw `ji p gƒdY{`dpv_Qitf`duvp _€ i ~wd2€‡dsg i p _8efu mfw gl``ƒ‹i¦~ }i ‚ m†{fu i tfbdg ` `dp {!i p `dm†~ `rq ¯g ˜ eabc`jw `dp@} mouv|3`}

(86) mou efu m†t i ta`cbd`dp‚a d `D`vp uv`cta{fp `dmo~ } {f~!w p `dmo~

(87) ` m†uGi p bdgri!`vp}

(88) mouŒi efbjiow g p u i ~ `‚}

(89) mou  | `!m }Gp _ag%| i `c~Pwv_ i u iow pvglu C` ` pv`jMw ` m!}G`d~ pvglu gs` p%wli ~ t g%m†tfp i `d~fgƒd,‹N‡_f`C``C`Vmk} p gl~Niowv_f`dg ‚ gTd ‚ `xi |Vglpv_fm d`p _ i p w m†|tf`d~fi g ^ogl~aglu i b eau moe!gluvp `dlg `rq&bcC` |†g pv_fW g y_ i ef| i ~ €8~ƒm†bd|Vm!^om†u m ‚ gTbo{!i p `dm†~`ƒm†uƒp _ag` pvu mo~

(90) ^ ‹%i u |om ‚ efuvmoe!glu ‚p asqpvPm ^og p _fgl„ u {`dpv² _ ±z³µ´·¶¸ wli bjw {abxi p `dmo

(91) ~ `Gp _ i pu Mg Ž!grw pVeau mot!i tf`dbdC` ` p `j„ w dsg w mo|}e _m ` `dp `dmo

(92) ~ `G!m } ` mouv8p `T‹ ©i uv`cm†{ ` ki ` e!grw p `>km } pv_fgVw mo~a~fgrw pv`cm†~ t g ‚p {glg ~ tf`du p _ € i w~ d_€8dsgri pv_Qeau mfw lg `` lg ` qw m†~ pv`d~ {aƒg d } u i†w p `dm†~ `Gi w~ d mou pv_fPm ^†mo~ i bse m†Db as~fm†|V`ji Tb `8a`vp gl | ` _!i ‚ gt glg ~`c~ ‚ lg ` pvD` ^si pvƒg d gri u Cb aG`d~Gp _agbc`dpvglu i p {fuvPg ‹‡_a g  u ` pm†{f8p `vp i w~ ds`d~ ^ u lg `v{fbd8p `Wdfi p g#t iow |%pvm p _fg#e i e!glu `Gt a~ i u bd`d~Qi w~ dQ‹8wly u sg ^omou `d~p _fg „r_ jP«  `;¹ _Œ ¬sq Œo f,q f_«º /q {_fglu g`di p {!i ` ` _f!m {~}p _ i ppv_fg y_ i ef| i ~ €8~ mobd|Pm ^†mouvm ‚ ƒg b†{ i p `dmo~ ` i pvC` `‚ ƒg d}t ap _ag pvu i ~ ` `dp `dm†~efu m†t i tf`dbd`dp `dlg `km } iAtf`du pv_ € i w~ d_€8dsgri p _efu mfw sg `8`wli ~t „ g `vmob ‚ Tg d ‚ `jipv_fg`d~ pvu m ds{ wnp `dmo~!m } i ~Qm†u pv_fPm ^†mo~ i b e!moCb as~fm†|V`ji /b `8a` pvgl|Ei w~ di ` e!grw p u i b |Vgrki ` {fuvPg ‹ ‡_fC` `W{!i `A~fmop i tfCb a%{ `vƒg d`d~ p _fg ` p /{ da !m }ƒpvu i

(93) ~ ` `dgl~ w g%i w~ du grwn{fu uvgl~ w g%`d~i ea{fu glCb a i ~ i Db aspv`x\ w {Ti a2‹o° ‡_fguvlg ` {fbdp `>km }@~ i uvbd`c~i w~ d ‹%w y u lg ^†mou>i u g#p _s{ `Gkm } ^†u gri pG`d~sp gluvlg ` p#p m p _agVw mo|}|{f~f`d‚p a dsgri bd`d~ ^*{`dpv_ p _fg efuvmoe!glu pv`csg `!m }ƒmouvp _f_m ^om†~ i be m†Cb a ~amo|V`ji b `=»'` g 3g }

(94) mouG`d~ `vp i ~!w ¼ g ¹ @Œ ½º8‹ ±5^ u mfw glƒg d `c

(95) ~ ^i bdmo~ ^ ds‘` ¾ gluvgl~sp bc`d~fsg `rq omo~alg `%i w~ dN‹%!i ^†~s{ ` dsg ‚ glbdmoe!ƒg dNi uvmo{fw~ d „rP h_hi ds`du grw p pvu gri p |Vg ~ p}!m }W`vmo|}g Œi efbjiow g p u i ~ `‚}

(96) mou  | ` ki `8`vm wn`xi p ƒg d p mtf`du pv_ € i w~ d_€8dsgri p _}efu mfw lg `` lg `A2t a}|Vgri ~ `!m } wnmo~sp `d~s{fƒg do} u iow pv`cm†~ `(¹ PŒ gsqf_Œ hº8‹ ¦~ pv_fC` `e i e!glu q {gz_i dsmoeapƒi#u Pi d `xw i bdCb a3ds‘` ¾ gluvgl~spƒe!mo`d~spkm } ‚ `dlg {‹¿ @g  8u `vpƒ~fm†p gGpv_ i ppvu i ~ ` `dgl~spƒwv_ i u iow € p gluvD` `vp `jMw ` !m }iGta`cuvp _ € i /~ d_€‡d gri pv_ eau mfw lg ``zi u  g } uvƒg b†{fgl~sp Cb a d Mg  ~fƒg d 2t a}|Vgri ~ `!m } ^oglm†|Vglpvu `jwƒwnmow~ ds`dpv`cm†~ `r_q }

(97) mou `c

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(175) ` glu ‚ i p `dm†~° pv_fg„` pvu mo~

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(177) mou | !m }p _fg } {f~ w pv`dmo~ U D` `1 i ` {ay | m ‚ gluƒi bdb!bxi p pv`xwng e!y i p

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(183) ^P` p mKL¼¹PN,º8‹ ° ‡_fg efuvgM˜€J

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(185) } gl{ g u` pvgle`T‹ ± § mopvg p _ i p iv` p moeaef`d~^ p `d|Vg| iTa t gƒiisd gM

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(189) n 3_Z ° [ƒ; ± ¶ƒ,Á\´DU>~± A³z¶ n q

(190) ±B>

(191) =\±;ÃD83 =<C=rqCD:A ;

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(195) > ”@?,A”CBD”. ° „P‹d„T«P±.

(196)   .          . . 

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(198) mobdbdm!{gƒd#e!i p _CGAqsmot!gla` i ` `d|VefbdŠ`  ƒg d\}

(199) mou | m!}° _„ ‹ hP± q ^u 

(200)  ]dmsuss

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(204) ~ ^ p m ds‘` ¾ glu gl~s p G  `i uvg dsC` `ck m†`c~sp q i w~ d}p _fgGeau ms!m }w mo~ wnbc/{ dslg `ƒt a}p _fig ` i |Vg w mo|}ef{fp i p `dmo~Vki `t ¯g } mouvPg ‹ ¦ } pv_fg `vp moeaef`d~ ^p `d|V6g 3#C` `dsMg }

(205) grw p ` ‚ goq!p _fgl~q t aY^og ~fglu i befuv`d~ w `defbdlg `rq!pv_fgGefuvmot i tf`dbc`d‚p akm } 3#t!gl`d~ ^1 ~a`cpvg `D` ° P„ ‹d„†ƒ„ ± ^:u E3 z dU*bIjW.p V!°L¼¹PN,ºJ±

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(207) ‹  wlwnmou d `c

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(210) mou |@mk} i ~s{f|t!glu}m!}Gu i w~ dsmo| ‚ i u `ji tfbdgl` ik`8`vm wn`xi p gƒd{`dp _ i#tf`du pv_€ i ~wd_€8dsgri p _Veau mfw gl``ƒ‹  ~ `d~` p i ~ wng `C`p _fg pvmop i bi uvgri <`8{glefp {fw~ dsg u p _fg*ª<¨ ^4i t m ‚ g i ^o` ‚ gl~pv_fu gl`v_fmobFd.; {f~sp `dbp _ag `vp m†efef`d~^w m†~wds`dp `dmo~,N\`C`„` i p `C`‚!gƒd °i ppv_fg u i ~wdsmo| p `d|Vg 3!±. . ;. ° „P‹d„TfP± HW.\ V °QF G X8;l± f b ‡  ‡_f`C` e i u i |}glp glu_ ik`efuvm ‚ `FdsgTdGm†~fgm!} p _fg |Vm†p ` ‚ i pv`cm ~

(211) `,} m†u pv_fgefuvgl` gl~sp<` pv{wda‹¦p8`efu iow pv`jwli bf`d|}e m†u p i ~ w g dsg ‚ m†b ‚ gl`o} u m†|Ep _fg }iow p}p _ i p`dp|Vgrik` {fuvgl`Vpv_fg ‚ m†bd{f|Vgmk}zpvu i•Ä w `d~ b†{fg {fgl`d~^`8a`vp gl|` i ~/duvglbji pvgƒd p glbdgrw m†|V|{f~a`xw i p `dm†~ w m†~sp gMf˜ p8` »/` glg&¹d„ sqfŒPf« qfŒa„ qwf_joº‡‹ 5it!• ‚ •%p Jr  •   °  u gri}w mo~f~agrw p `dm†~w± k|DvÁ}N ;gD*± Á¶  =?A `:<?`;:¸`¶ A³B=˜ Á < =\¶ Ac”KQ´ DW=¶B=rAÁ|± رz¸pDÂÁ->±BA {ƒ=¶ >

(212) n ¶ƒ Á\´D?‰Á ={n~Dxš±B>pD~± D

(213) M±B=?>i° 3\

(214)   ; ± ³DB]AD³ ;

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(216) ± \ w´ D:>gDp ` = >  Á‡´ D/D:A>

(217) =\¸v´ D³,~±B>pD~± n?¶ >M´=

(218) n"€;³DB]AD³ ;

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(220) ”¿H`dp _fm†{fpƒbdmP`` m!}^ogl~fg u i bd`dp‚asq wnmo~`v` d glu p _fgVwlik` g;NW «

(221) ‹G‡_fg#efu m†e g u p‚ap _ i p p `d|}g m w w {fu8` t g¯} mouvgGp `d|Vg“‡ i ~/dVpv_fg>i u gri#`C`bcgs`8`ƒpv_ i ~Ctaq} m†bcbdm!{`d~^ p _agGe i pv_"GAqf`C`pv_fgGg ‚ gl~sp ° P„ ‹d„TgP± E3 z ‡9\ V<z tu%KGUIjWYE

(222) , d

(223) d?ssuspd

(224)  z ‡9\ ^ V

(225) , ]d ^ Å

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(228) . gš¹ `i`  ^. "! `. %. º W. . R. []d#.  R . R d. i ~wd}p _ag eau msm!}w m†~ w bd{wdsgl` bd`D|†gƒt!gM} m†u goqamo~ p _fg t ik` `C`m!}° „P‹d„ƒg_±M‹. R *. \. %$,u  ‚  ‘ 3ym†~` `FdsgluAp _fgw mo~/ds`dp `dmo~ &Cpv_ i p uvglefu gs` gl~sp8`#p _agg ‚ gl~sp'A¶ ³D~±•Á\´D´·±&¶¸~¸%@>

(229) >gD³ ! i ~/d%pv_fg w bdmP` g bDa}u glbji p gƒd` pvmoefea`c~

(230) ^Vw mo~wd `cpv`dmo~('/gM˜fefuvgl`8`v`d~^7oÁ\´DB]> ÁŽ³D±Á‡´&´M±A±

(231) à >gD±z³ ‚&¶=¸¸%@>:>gD³p!/‹‡_fg ~1L¼¹)&,º i ~wdYL¼¹), ' ºi uvg L¼¹), & ºW6„]d,5 V d5 V 5 Å d 5 V 5 Å 5 d.ssus \ L¼¹),' ºW]9 V d,5 V 9 Å d 5 V 5 Å 9 dmsuss\ . {_f` glu g{g |{

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(233) ‹N‡_fgl~qƒ^. -3Z ,°6[T±]W. wv mx V. *. u moe!mP`v`dp `dmo~`%„%i ~wdHŒ^†` ‚ g°J{`dp _ m. .*. °. 0/. m ` Å R. x V ZR )a m. P± . R x V 6[Hd(ZRdtaR.

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(236) ^Qpv_ i ppv_fg*` pvmoefea`c~

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(238) `‚} m†u |Qq. <°6[\ *P±HW. wv mx V. *. m. .* °. "/. m ` Å R. x V ZR Ua m. P±. R x V [kdeZ R dea R dtT*. . Àƒ{auef{fu e!Pm `vg `c~}p _fC` `e i e!glu`C`p mi efefbCapv_f`C` } m†u | i bfefuvmfw gƒds{fuvgGp m#bdgl`8`m†t ‚ `dm†{`pvu i ~` `dgl~spw _!i u i†w p g ‚u € `D`vp `jwM`m!}pv_fgƒtf`duvp _€ i ~wd2€‡dsg i p _}efu mfw gs`8`ƒ‹‡_f`C`uvgƒb†{f`du gl`<!u8` p<` g p p `d~^#{fe}gM˜fefu gl`` `dmo~

(239) `<} moupv_fg |{fbdp ` ‚ i uv`ci p g ^ogl~fg u i pv`d~ ^1} {f~ wnp `dmo~ `¯g ˜ eau gl`` `d~^ wnmou uvgl` e!mo~/ds`d~^ w m†~wds`dp `dm†~`ƒ‹. Œ ‹>¬°I© 2ª¬Z«Z°\¯ 8;¬ 2ªZªZ°\¯ 46B2ª 038u2Z©Z­ ¯Z¬‹©ZªZ°I©Z® 4M­ « 2—¯ªZ°‡¬Z©^« 4c6M« 48:4M©Zª2RªZ°‡¬Z©38 ¦p`C` |s~fm!{~p _ i pp _ag} m†u | i bsp _fglm†u8amk}&w m†~sp `d~s{fgƒd1}

(240) u iownp `dmo~#gM˜fe i ~

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(242) mou ‚ i u `dm†{ `‰`vgl8p ` !m }e i pv_ `ƒw mo

(243) ~ ` p u i `d~fƒg d 2t a_aglC` ^o_spƒi w~ ddsglefpv_%w mow~ d `cpv`dmo~` q !i `>i u C` ` lg `A`c~ w m†~f~fgrw pv`dmo ~ {`dp _ pvu i

(244) ~ ` `dgl~sp>wv_ i u iow pvglu C` `vp `jMw `#!m }ƒi ª<¨ ^ ‹hamouGpv_fC` ` ea{fu e!Pm `vgoq {guvgrwlki ` p#`c~ i {f~fŠ` }Jas`d~ ^}

(245) u i |}lg {mo8u |i /~ dV¯g ˜ pvglw~ d8t !i `v`jwƒuvlg ` {abcp `ƒlg `vp i tfbdD` `v_fƒg d `d¼ ~ ¹dƒ„ «sq„o„ q„ ŒsqT„ ffq ƒ„ h•º ^u mskm }J` i u i g ^o` ‚ g ~ `d~ p _fC` ` e i e!glu t!grwli

(246) { ` gp _alg aQbdgr_i d p m uvlg ` {fbdp ` km }`dw~ dsgle!glw~ d gl~spG`d~ pvglu sg ` ƒp ‹  bcbpv_fg wnmo|Vea{fp i p `dmo~ `g ‚ gl~sp {!i bdCb a dsgle!glw~ d}mo~fCb a#mo~V\i ` `d|Veabc<g ` g pkm } w mo|tf`d~ i p m†u `ji b dsgrwnmo|Ve!Pm `v`dp `dmo~ `@°ˆ sgrwnp `dmo~ Œ ‹dƒ„ ±i w~ d#mo~t!!i ` `jw efu m†e gluvp `dlg ` m!} bc`d~fg i 'u } u i†w p `dm†~ i bfpvu i ~ `‚} m†u | i p `dm†~ `uvglefu sg ` gl~sp Tg d t a> Œ &GŒ | i p u `jw sg `°ˆ sg w p `dmo

(247) ~ `Œ ‹ PŒ ± q u sg ` {fbdp `d~ ^A`c~i i {F` dsg m!}#`vglpƒkm } Mg ˜fefuvlg `8`v`dmo~ `1°ˆ grw p `dm†~

(248) Œ ‹ fP±l‹. Œd‹ „P‹*% u

(249) rƒrˆvk‘ ‚ ur

(250) –'‘vk•  ‚ • p%Jr • p‹Œ glpLdsg ~fmopvgp _fgw mobdbdgrw pv`dmo~Gmk} i bdboe!i p _Agl~ w mds`d~^_`ƒ‹  ^†gl~fglu i b ` {ft w bji!``ƒm!}e i pv_`m!}`d~sp gluvgl` p `C` dsg¯ ~fgƒd taoŽ msmou `d~^°±lq!w gl`dbd`c~

(251) ^3°Aw± q i!`{glbdbi!`<

(252) ˜ `d~^#`d~f`dp `ji b'°

(253) ±i ~wd  ~ i b° Fƒ± i bdp `dp {wd gl` L ŒP   P   W E‹G CL5² =_° G]± W \A,°G±]W \`d~}

(254) E4G IUh\i ~/d` {ae E4G I z AI  „ ¿ gzi bC` m#~fglgƒd pv_fgi` e!grw `ji bd`,ri p `dmo~` q. P  V P   V P  L Œ P  ? W L Œ  P  \ L ŒP  „ W?L ŒP   \ L P  W?L Œ P   v v „ „ „ „ „ „ ymo~`v` d glu~am!{1pv_fg`as|t m†bD`5 R \+9 R \ ; R i!`}

(255) mou | i bs`d~wdsglpvglu |}`d~ i pvgl`rq_` m#p _ i p {mou d`t!grw m†|Vg|Vm†~fmo|}`ji bD`T‹ ‡_fgk< D:AD:>~±Á =?A <ƒ

(256) @ A¸vÁ-=‡¶BA%mk} iw m†bdbcg w p `dmo~%m!}e i p _``C`dsg¯ ~fgƒd%ik`p _fg} m†u | i b/` {f|  W w NG \ R ‰ . i } m†u | i be m!{gluY` g u `dgl`%`d~H`d~ ~a`cpvglbCa| i ~a`d~wdsg p gluv|V`d~ i pvgl`ƒ‹ °ˆ§ mPp grwv_f~f`jwli bds`‘Ä w {fbdp‚aHi u `C`vgl` ` }Gm†~fg  moe!glu i p gl`{`dpv_Qpv_fgW^†u i_dfi pv`cm†~%mo~p _ag„` glp m!} ‚ i uv`ji tfbdgl`idsgM ~fgTdt2adsgl^ 5'R$W dsgl^9 RuW dsgl^; RuW\„P‹ ±¦~ p _f g ` i |}g{iTa i!`{mouMd` i u gGi!`` `d|V`dbji pvgƒdpvm|Vm†~fmo|}`ci bŠ`rq†`dpefu m ‚ gl`ƒw mo~ ‚ gl~f`dgl~spƒpvm{` g>wnmo~`v`D`vp gl~spvbDa pv_fg ` i |}gƒ~fm†p i p `dmoo~ } mou `vgl8p ` i w~ do} m†upv_fg>!i `` mfw `ji pvƒg d^ogl~aglu i p `d~ ^1} {f~!w p `dmo~

(257) `ƒ‹ ªki ` `jwƒw m†|tf`d~ i pvmouv`xi boefuv`d~ w `defbdlg `Ž¹dƒ„ goº,` p i pvgƒp _!i p °ˆdsC` `cknmo`d~sp ±'` glp{f~f`dm†~q°

(258) {f~ i |tf`C^o{fm†{` ±w mo~ w i p g ~ i pv`cm†~ m!}#`vgl8p ` km }#{mou d`rqfi /~ dVpv_f„ g °

(259) {f~ i |taD` ^†{fmo

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(261) `ƒi!`.  V. m. \.  %  W „HX0„  W & #u\ ‡_fglu g†qopv_fgbjik` p`Fdsg ~ pv`dp‚aWdsgluv` ‚ gl`'} u m†|5p _fg:}i†w ppv_ i p<° „cX/0 ± ` Å W/„d0 d0 dYsuss,^†gl~fglu i p gs` `8as|t!mobd`jwli bdbCa i bdb p _fgi`vgƒb†{fgl~ w gs`{`dp _ wnmo|Ve!mo~agl~sp8` '0 ‹ ‡_fu glgm†t ‚ `dm†{` ¸¶‹nu;:=?A±Á¶B>:=\±

(262) ÃM³D¸¶‹n M¶

(263) =˜Á =\¶BA mk} e i pv_`p _fg ~„` {Ä w gpvm†dsg u ` ‚ gƒi bcb†p _fgt!i!` `jw'}

(264) mou |{ab2i 1 ‹ m„tPv &

(265) –'‘v!•  ‚ •p% ru • :x  ~gM˜ w {fu` `dmo~}

(266) u mo|Ei ~wdp m bdg ‚ g b «%wnmo~`v`D`vp8`#m!}ƒiY` gTbo{agl~ w g m!}u i u wv_fgl`"! q griowv_ | i_dsgƒmk} gl`dpv_fglu iK;:Vƒm†uƒiE5VL Ō P Å Å  9 Å q` mpv_ i p LuV P VkW ° „HXJ;

(267) VHX85V

(268) L Œ P Å  9 Å ± ` Å   % W 7d #u\. Å Å.

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(270)   . . A'²`d~3^†gl~fglu i bqfm†~fg _ i!`pv_fgGu g w {fu8`v`dmo~ . „HX8; R XT5 R L R9Œf R'Å f P R'Å f P Å   9 R'f Å / ` Å \ L R

(271) ŒP R R:P   W L  Œ `  Å ` P  Å`   Å   W ° „HX ;  ` Å ± ` Å   ‚ •,p

(272)  ru •q pkx c grw m†u ds`d~^Gpv_fg p `d|Vgs`i p {_a`xwv_}griowv_Vbdg ‚ lg b«\ z. u/plr ‚ wu pƒpsu/‘pW–'‘v!• '9 \ #`C`bjik` pp u i € ‚ glu` gƒd ^o` ‚ gl` °Œ‹ Œ_± L V P W.L V Œ P V V  5 V L Ō P Å Å  5 Å suss5 ` Å L Œ P „  !JtPpsr ‚ uwpTpl u w‘–'‘v!•  ‚ •,p

(273)  rJ•c„ <x‡_fg b†{ i ~sp `dp `dgl`ˆL P  {„ `cpv_0o„ / „ i u g#`d|Vefbd`jw `dp bCa„dsglp g u |V`d~fgTd ta p _fg<! u8` pe i!`` i!†^ gp _fuvmo{^†_ #`d~>ize i pv_ w m†~f~fgrwnp `d~^Gbdg „ ‚ glbw« p m »siids{ i b/dsgrw m†|Ve!mP` `dpv`cm†~G_fm†b d`#{_agl~ uh!q

(274) ` mp _!i p  W L V Œ  P „ `  Å 5 ` Å L P  °K/!F±:\   LuV P  °Œ‹ f_± L P V W L P „  9  L Œ„   „ °W/

(275) ±  %. „. `. „. Å PV. Œ ‹ Œ ‹ 6†•r  ‘– 8t!uwvkru • uc – v!•

(276) /‘t w‘ r ‚ • J’:• JJu ‚p ‡_fg}t i!`v`xw u gl`v{fbdp8`mk} gri u bd`dgluW`vp {wd `cgs` gM˜fefu gl``ƒpv_fg^†gl~fglu i p `d~^\} {f~!w p `dmo~

(277) `mk}p _fuvglgGt ik` `jwp‚ase!gl`mk} e i p _``d~p gluv|3`m!} i } {f~wdai |Vg ~ p i bfw m†~ pv`d~ {agƒd } u i†w p `dmo~ i /~ d%`d8p `Gik`8` mfw `ji p gƒd w m†~ ‚ glu^ogl~spGe!mobCas~fmo|}`ji bC`ƒ‹ ‡_fgla`d~ ‚ mob ‚ g pv_fg  ~s{f|}glu i p mou"! i ~wd Mdsgl~fm†|„€ `c~!i p m†Ju ! e!moCb as~fmo|}`xi Cb `rwq d gl~fmopvgƒd ta  i ~wdJN  pv_ i p gri†wv_ ` i p `C`‚} a%p _fg„`vgrw m†~wd m†u dsglu° m†u  p uvglgM€p gluv|$!± u grw {au u g ~ w g °Œ ‹ ½± l  f Å W° H „ X ;  ± l  X 5  ` Å 9  l  ` Å \A{h „ \ i w~ dAp _fg`d~f`dpv`xi b w m†w~ ds`dp `dm†~ `<° ` Å \+N ` Å ± W ° „ \M«P±i w~ d„° VB\+N Vl±]W °ˆ«\ ƒ„ ± Tq {`cpv_>pv_fgƒw m†~ ‚ gl~sp `dmoK ~ 5 ` Å 9

(278) VW5_„ ‹ 

(279) ‘2•/t!‘  °^i pv_%w mo~sp `d~s{fƒg do} u iow pv`dmo ~ `(¹dƒ„ «oºJ±°-S ±EQ´D9DvÁ LuV P V(¶2ƒŽ±

(280) ÃIÃ;±

(281) =\¸<Dz¸%@> =‡¶BA j=H>gD2>gD9D

(282) A·Á D³ ;

(283) ‚;Á‡´ D„ƒ

(284) @A³z± ~ n D:AÁ±

(285) ø¶BAÁ =?Ac@D³ƒ:>~±z¸vÁ =\¶BA. °Œ‹ j_±. -° SQSJ±YQ´ D °Œ‹ g_±. LuV. W. „ „HX ;. V X. „HX ;. :DvÁŽ¶ƒA¸pD:=IÃD³uD@¸%@> 

(286) =\¶BA ˆL V Œ  P V „  =M<B=?qBD

(287) A L VŒ  P V   W „HX8; V X „HXJ;. °Œ‹ h_± °-SQS_S ±YQ´ DD:DvÁ °Œ‹ ¬_±. °Œ‹ _±. PV. N   ¶2ƒ ¶¶B>pD³$D@¸%@> 

(288) =\¶BA /=M<B=?qBD

(289) A ;%‚.Á‡´ L  Œ P    W „HX ;  X „HX8;  f W. 5. Å X. . Å. 5Å9. 5 9 ””” ;

(290) ‚&¸¶BAcqCD:><D

(291) A·Á ?¶2* ƒ Á‡´ D ƒ

(292) @A³›±‹nuD

(293) A·Á‰±@Ã>ƒ:>~±z¸vÁ =\¶BA „HX8;. „. 5Vu9. Å X. X. Å. 5 Å9. „HX ; X. . . W. 5Vu9. „. ` Å. 9 N . N L ` Å. L. 5 9 ””” „HX ;  ` Å. DCÁ >

(294) @A¸±Á =\¶BAL¶2ƒ,Á\´Dƒ

(295) @A³›±‹nuD

(296) A·Á‰±@Ã>ƒ:>~±z¸vÁ =\¶BA. „. Å X. X  V PV  X  V PV . 59 „HX8; . f Å. . . ` Å. \. 5. f Å. f. X. 9. f. 5 f. 9f ”””.

(297) 

(298)    . k>~¶¶ƒ

(299) ”cgle!gri pvgƒd8` {` gGmk}p _fg>i u wv_dsg w mo|}e m_` `dp `dmo~ °Œ

(300) ‹c„T±efu m ‚ `Fdsgl` i\} mouv| mk} L V Œ  P V   {`dp _ ~fgs` p gTdb†{ ik` `Š€ `c~ ‚ glu8`vgl`3° „<X /0 ± Å p _!i pG`C`Gpv_fg„ ~a`cpvg1} u i†w p `dm†~Quvglefu gs` gl~sp i p `dmo~ °Œ‹ gP±M‹}‡_fg wnmo~sp `d~s{fgƒd } u iow pv`dmo~u glefuvgM€. ` gl~sp i p `dmo~ } m†ut i!`v`jw e i pv_`Y° ~!i |Vg bDa8L V P V ±`C`pv_fgl~Hm†tfp i `c~agƒd tabdglpvp `d~^8A † * `d~ °Œ‹ g_±M‹Kh`d~ i bdbCa q p _fg>wnmo~sp `d~s{fgƒdo} u iow pv`cm†~} mouv|=°Œ‹ ¬P±q} mouƒwngl`dbcgTd gM˜ w {fu` `dmo~` `C`~fmopv_f`d~^tf{fppv_fgi} {f~/dfi |}gl~sp i b

(301) } m†u |9°Œ‹ j_± q {_fgl~Vpv_fg`d~wsd `jw gl`i uvg` _f`Š} pvgƒd,‹‡_fg p _fuvglg wnmo~sp `d~s{fgƒd\} u iow pv`dmo~gM˜fefuvgl`8`v`cm†~`i°Œ

(302) ‹ jP± q

(303) °Œ‹ g_± q°Œ‹ ¬P±i u gpv_fgl~ w g gl` p i tfbd`C` _fgTd,‹ h`d~wd `c~

(304) % ^ gMf˜ efbd`jw `dpGgMf˜ efu gs`8` `dm†~`1} mouAp _fg} u iow pv`cm†~E` L V Œ  P V   i ~/8 d L  Œ P    ~agMf˜ pu gƒ†b {f`du gs`3sd glp g u |V`d~f`d~^ pv_fg e m†bDsa ~fm†|V`ji bCf` pv_ i pi efe!gri u `d~Gpv_fgw mo~ ‚ g u8o^ g ~ p` m!} pv_fgt ik` `jw

(305) } u iownp `dmo~1° Œ

(306) ‹ jP±l‹ª:a d gM ~f`dp `dm†~q p _fgw mo~ ‚ glu8†^ gl~sp e m†bDsa ~fm†|V`ji bC`   i ~wY d N  i u g p _a gG ~s{f|Vglu i pvmoui ~wdsd gl~fm†|V`d~ i pvmoum!} p _fg} u iow pv`cm†~CL V Œ  P V   ‹ hfm†upv_fgGw mo|}ef{fp i p `dmo~#m!} L V Œ P V  i ~wd   \+N  qsmo~agzwnbxik`8v` `xw i bdbC> a `d~sp uvmd { w gl` pv_fg bc`d~fg i u:} u i†w p `dmo~!i b!p u i ~`‚€ } mouv| i pv`dmo~` X R 6° *_k± W „HXT; RjXJ„ 5'R49 R9f * \ Å ` mp _!i p ° Œd‹ „T«P± L V Œ  P V   W.XVX Å X ˆssusX  ` Å °ˆ«_± i ~wK d LuV P VkWmXVX Å X ˆssus<\  § m!G{ qsbd`d~fgri u<

(307) } u iownp `dmo~ i b pvu i ~`‚} m†u | i p `dm†~` i u g u g efu glv` gl~sp i tfbdg>ta Œk& ŒT€ | i p u `jw gs`  5>*<d,9 † "$ 5 9 \ ° Œd‹ „†„ƒ± ;

(308) *jd,b ; b () `c~` { wv_%ii { iTa p _ i p wnmo|Ve!mPv` `dp `dmo~w mouvu glv` e m†~wd` p m#| i pvu `Š˜ efu msd { wnpƒ‹ ª:Va `d~wsd { wnp `dmo~}mo~p _agzwnmo|Ve!mPv` `dp `dmo~` p _ i ptf{f`dbF d {feCL V Œ  P V   qfp _aglu gi

(309) } mobdbdm!{` p _fgAgƒ†b { i bd`dp‚a ° Œd‹ „ ŒP± X V X Å X ˆsussX  ` Å 6° *P± W N   XX N   ` ` Å 5 5   ` ` Å 9 9   * * \ Å Å { _fglu g   i ~wYd N   ` i p `C`‚} Va p _ag uvgrw {fuvu gl~!w gY° Œ

(310) ‹ ½±M‹ s glpvp `d~^ */W A« `d~,° Œ

(311) c‹ „ ŒP± efu m ‚ gl`i° Œ

(312) ‹ hP±M‹  h`d~ i bdbDsa qcL  Œ P   `D`isd glpvglu |}`c~agƒ% d `d|}efbd`xwn`cpvbCa ikƒ` p _ag>uvm m†p *}m!} p _fg#gƒ†b { i pv`dmo~CXV suss“X  ` Å ° *P<± W LuV P V q i ~ gƒ†b { i pv`dmo~Vpv_ i p q { _fg ~ v` mob ‚ gTVd {

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