• Aucun résultat trouvé

Periodic solutions of piecewise affine gene network models: the case of a negative feedback loop

N/A
N/A
Protected

Academic year: 2021

Partager "Periodic solutions of piecewise affine gene network models: the case of a negative feedback loop"

Copied!
26
0
0

Texte intégral

(1)Periodic solutions of piecewise affine gene network models: the case of a negative feedback loop Etienne Farcot, Jean-Luc Gouzé. To cite this version: Etienne Farcot, Jean-Luc Gouzé. Periodic solutions of piecewise affine gene network models: the case of a negative feedback loop. [Research Report] RR-6018, INRIA. 2006, pp.22. �inria-00112195v5�. HAL Id: inria-00112195 https://hal.inria.fr/inria-00112195v5 Submitted on 14 Nov 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. Periodic solutions of piecewise affine gene network models: the case of a negative feedback loop Etienne Farcot — Jean-Luc Gouzé. N° 6018 Novembre 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6018--FR+ENG. Thème BIO.

(3)

(4)  

(5)   

(6)  "!$#% &  '()+*-,./  0+1 2

(7) '!  3!45&!6+87"9 5:!

(8) *- $ ;6<>=@?BAAC?EDGF(H>IKJ(< LNM ?BF(A+OPQCISRTJUQCVKW XY[Z]\_^`'acbed-fhgjilkcZ \/^>im[nporqporsKnptBu[^>i vNwcoyxz^k|{or\/oKwc^ }~y€[€(oKwlk‚j^$wc^]ƒ„Y[^]w ƒ„Y[^‡†Uˆ‰KŠ[‹]ŒSdŽ‚o9r^ \m[w ^yŠrŠK‰$dK€‘~rsr^>i ∗. †. ’ “C”]• –9—[˜y•K™ aš†›k Y[nœi€‘~y€(^]wCkcY[^^ žjnpilkc^]†‘ƒ^~r†‘›u[†‘npƒ nŸkzgor U~6ilk ~rm[q¡^€(^ w n¡oj[npƒ'orw m[n¡knpi€[w o¢K^ †¤£9 ¥oKwG~6ƒqœ~ri iCoy (€[np^]ƒ ^ ¦nœil^ ~9§ †[^'jn¡¨(^]wc^]†Kkcnœ~rqB^]tBu‘~9k n¡oK†‘inp†›jn¡\/^]†‘ilnpor†›©‚oKw+\_oKwc^K£¢€[w o¢hnœj^]6kcY[^]n¡wnp†Kk ^ w„~rƒkcnpor†ilkcw u‘ƒ8k u[wc^Nnœi~‚†‘^ sK~ykcnpr^G ¥^ ^]jmª~rƒ„« qpohor€¤¬a­kNnœiN~yqœiloicY[o¢¦†/kcY‘~ykGkcY‘^i ~y\/^‚icgjizk ^ \ iGƒ or†Br^]wcsK^'kco9¦'~yw„/~6u[†[nœtBu[^ilk ~rm[q¡^‚^]tBu[npqpn¡m[w npu[\®€(oKn¡†BkNn¡†[n¡\/^ †ªilnpor† j¬XY[nœi'^ žhkc^ †ª[i~kcY‘^ orw ^ \or Cfh†[oru‘i icn¯£h¦Y‘npƒ„Y icY[o9¦^]kcY[^6^žjnœizk ^ †‘ƒ ^°or kcY[^>il^|oKwcm‘nŸk„ioK†[q¡gK¬XY[^6ƒ or†‘icnp[^ w ^]±ƒqœ~ri i oy ²^>tKuª~9kcnpor†ªinœiGu‘icu‘~yqpqpg_izk u‘jnp^]~riN~6\/oj[^ q‘or UsK^ †[^‚w ^ sru‘qp~ykcoKwcg†[^ kš¦'orw «hi]¬+a­knpiN†[oyk'~ri ilu[\/^>›kcY‘~yk~rq¡q(j^]ƒ]~>g›w„~9kc^>i ~yw ^np[^ †Kk npƒ]~yq@£r¦Y‘npƒ„Y/nœim[n¡oKq¡oKsrnœƒ ~rq¡qpg$npwcw ^ qp^ y~y†Kk>£ym[ujkNY‘~rim(^ ^]† jor†[^np†_kcY‘^9~KizkG\/~9xzoKwcn¡kzgor (€[w ^ hnporu‘iilkcuªjn¡^>i ¬Gb°u[w ¦oKwc«$wc^ qpnp^]ior†_ƒqœ~ri ilnœƒ ~rqKw ^]icu[q¡k iC~ym(oKujk³[žj^]€(ornp†Kk„ioy ª\/or†‘oykcoK†[^r£yƒ or†‘ƒ]~>r^or€(^]w ~ykcoKw i+~rƒkcnp†[s°oK†€(oBiln¡kcnpr^'9~yw nœ~ym[qp^]i]¬ ´ToKwc^]o¢r^]w]£¢kcY[^u‘ic^]›kc^>ƒ Y[†‘nptBu[^>i~rwc^r^ w gq¡np«r^ qpgkco~y€[€[qpgn¡†/\/orw ^'sK^ †[^]w ~rqjƒoK†Kkc^ žhk i]£yoK€ª^]†[np†[s$jnpwc^>ƒ8kcnpor†ªi µoKwC ¥ujk u[wc^ ¦oKwc«U¬ ¶T·B¸U¹cº›» –¢¼”K™ €(^ w n¡oj[npƒ6iloKq¡u[kcnpor†‘i]£j€[n¡^>ƒ^ ¦nœic^½­q¡np†[^]~rw]£h\/oK†[oyk or†[^6or€(^ w„~9k orw„i. ∗ †. ̾„¿¢ÇcÍÀlξ Á8¾8ÁÈ ¾jɄÇcœÊ8ÃËyÄ8Å]ÀlÄjƄDŽ­¿ÀšÁÈ Å9É8ÇcÀlÊ8ÄjËyµÃ8ÀlÅÄj­ÀšÁÅ9ÀlÄjœÃÅ 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.

(9) ±+  / /5 5 ,./  !$#%   !$ ,.$C

(10) !- 2  !  &  '+ /  0 

(11) '! 

(12) l5 2:S

(13)  &!6+87 2C'C!

(14) N+ ™  ~y†ªi5ƒ ^Ew„~y€[€(oKwlk±q  ^žjnpilkc^]†‘ƒ^S^k q  u‘†[npƒ nŸk    u[†[^SoKwcm‘nŸk ^E€ w n¡oj[nptBu[^Sizk„~ym[qp^E^>izk \/oK†Kkcw ^K£‚€(oru[w  ”   u[†[^5ƒqœ~ri il^  ]tBu‘~9k n¡oK†‘ijn¡¨  w ^ †Kk n¡^]q¡qp^]i›~9§ †[^]i€‘~yw\_oKw ƒ ^]~rujžE^]† jn¡\/^]†‘ilnpor† ©^ k€[qpu‘i]£€(oru[wq¡^>iicgjizk Z \/^]ijor†Bk qœ~ilkcw u‘ƒkcu[w ^   np†Kk ^ w„~rƒ8k npor† ^>izku[†[^ m(oru‘ƒ q¡^S† ]sK~9k n¡K^r¬ ašq6^]ilk ]sK~yqp^ \/^]†Kk  \/or†Bkcw  tKu‘^ ƒ^]i !\ ]\/^]i icgjizk Z \/^]i ƒoK†Br^]wcsK^ †Kkr^]w iu[†±u[†‘nptBu["^ ]tBu[npq¡npm[w ^|izk„~ym[qp^°^ †5[n¡\/^ †ªilnpor† [¬G{^>ƒn  kc^ †ªu‘†±w ]icu[qŸk„~9k #$›fj†[oru‘i icn¯£jtKu‘nU~Kicicu[w„~yn¡k q ^ žhnœilkc^ †ªƒ^‚ic^ u[qp^‚j^ƒ^>iGiloKq¡u[kcnpor†‘i]&¬ %~$ƒqœ~Kicic^  ]tBu‘~9k npor†‘iGƒoK†‘ilnœ ]'w  ^^>izkƒoKu[w ~r\/\_^]†KkGujkcnpqpnp'i  ^ƒoK\_\/^\/ojjZ]q¡^j^ w]ic^]~yu[ž j^6'w ]sru[qœ~9k n¡oK†(s ]† k nptBu[^K)¬ %¤^]ik„~yujž±[^6  sKw ~K[~9k n¡oK†‘iN†[^6icor†Bk'€ª~riilu[€‘€(oK'i ]*i ]sK~yujž^ †Kk wc^|^]ujž²£hYhgB€(orkcY[Z>il^ m[nporqporsKnptBu[^]\_^]†KkGnp†9xzu‘ilkcn¡+³  ^r£B\ ~rnpiG^¨U^]ƒ8k u  ^°[~y†‘iGqp~$€[qpu[€‘~ywckj^]ikcw„~¢9~yujž_ilu‘wGqp^‚icu9xz^ k]¬N{^kcw„~>y~ynpq[,i  ~r€[€[u[np^‚icu[wNj^>i kc+Y  orw Z \/^>iƒqœ~ri ilnœtBu[^]ij^|€(ornp†Kk³‘žh^|-  oK€  w„~9kc^]u[w„iG\/or†[orkcoK†[^]iN^ kƒ or†‘ƒ ~¢r^>i £B~¢r^]ƒ9~yw nœ~ym[qp^]iG€(oBiln¡kcnpr^]i]¬  ^°€[qpu‘i]£hqp^]i kc^>ƒ Y‘†[nptBu[^>i  r^ qpor€‘€  ^]iiloK†Kk‚k wcZ>iƒ^]wlk„~ynp†[^ \/^ †Bk°~y€[€‘q¡nœƒ ~rm[q¡^>.i $j^>iicn¡kcu‘~ykcnpor†‘i‚€[qpu‘i(s  +†  w„~yqp^]i]£[oru[hw„~y†Kku[†[^rornp^ €(oru‘w[^| ¥ujk u[w ^]iwc^>ƒ Y‘^ w„ƒ Y[^>i ¬ /‡» •]” ¹ 1˜ 0  ”±™ icorqpujk n¡oK†‘i€  w n¡oj[nptBu[^>i £jq¡np† ]~rn¡w ^|€ª~yw\/orw„ƒ^]~rujž²£hor€  w„~9k ^ u[w„i'\_oK†[oyk or†[^>i.

(15) ©.  

(16)    

(17) !"# %$&'() *+ ,- ..   ,. !$ . /"01/. 24357698:<;>=@?A:B3C=. D \/oK†[s±wc^>ƒ^]†Kk°m[nporqporsKnpƒ]~yq+ƒoK†‘ƒ^]€jk i]7£ EFGHEIJ$,*K(L!M HN+ nœi|~€‘~ywckcnœƒu[qœ~yw qpg5n¡†Bkcw n¡sKu[np†[s±or†[^r¬${oK~rw ic^ qpg5i ~ynœ²£ n¡k°wc^  ¥^ w„ikco5~il^ kor sr^ †‘^]iƒoj[n¡†[s  ¥orw€‘wcorkc^ np†‘i‚m(^ np†[s±~rm[qp^$kco±~Kƒ8k n¡y~9kc^orwnp†[Y[npm[nŸk°kcY‘^^ žj€[wc^>icicnpor† or oyk Y[^ wsK^ †[^]i ¦n¡kcY[np† k Y[^i ~y\/^ ic^k>¬5fhnp†‘ƒ ^iluªƒ Y ilgjilkc^]\/i~yw ^/q¡np«r^ qpg k onp†BroKq¡K^/qp~rwcsK^_†Bu[\m(^]w i$or 'sK^ †[^>i £+~r†‘ icnp†‘ƒ^ \/orw ^ o9r^ w sr^]†[^^žj€[w ^]i ilnpor†nœi«h†[o9¦†k o±m(^›~ †‘or†j½­q¡np†[^>~yw\/^]ƒ„Y‘~y†[nœic\ £(\ ~9k Y[^ \ ~ykcnœƒ ~yq+\/ojj^]qpi°~rwc^K£[k Y[oru‘srYE~ w ^]tBu[npw ^] kcohoKq¯£   ~rƒ n¡†‘s~/\/~9xzoKw‚ƒ„Y‘~rq¡qp^ †[sK^r¬XY[nœi  ~rƒ8kY‘~Kin¡\/€(^ qpqp^]T~/qœ~yw sr^$~y\/oru‘†Kk‚or C¦orw «U£jsKn¡hnp†[s/m‘n¡wckcYkco†Bu[\/^ w oru‘i\/oj[^ qœi £ ic^ )^ O + Pª ¥orw~$w ^ hnp^ ¦$¬ D \/or†[s›jnŸ¨U^ w ^ †BkNkzgB€(^>iGor ²\/ojj^]qpi]£B~ic€(^]ƒnœ~yqªƒ qp~KiciGoy ¤€[n¡^>ƒ^]¦npic^‚q¡np†[^>~ywjn¡¨(^]wc^]†Kkcnœ~yqª^>tBu‘~9k n¡oK†‘i Y‘~¢r^±m(^]^ † €[wcoK€(oKic^] n¡†kcY[^ ^]~rwcqpg3R‹ QTS9+Š  Ui O V+P­¬ XY[nœi_ƒ qp~Kici›Y‘~rik Y[^~rjy~y†Kk„~ysK^oy m(^ np†[sS\ ~ykcY[^]\/~ykcnœƒ ~rq¡qpg \u‘ƒ Y \/orw ^6k w ~Kƒ8k ~rm[qp^6kcYª~y†T\/orw ^ƒ qp~Kicicnœƒ ~yq@£‘ic\/ohoyk Y†[oK†j½@qpnp†[^]~rw‚^]tBu‘~ykcnpor†‘i]£‘†[or†‘^kcY[^]q¡^>ici‚or¨(^]wcnp†[s±~ ƒ or\/€‘~rw ~rm[q¡^$9~rwcnp^kzg oy m(^]Y‘~¢Bnporu‘w i]¬ fhnp†‘ƒ^n¡k iCnp†Kk wco[ju‘ƒ8k n¡oK†¤£9kcY[^‚ƒqœ~ri i+or ª€[np^]ƒ ^ ¦nœil^ ½@qpn¡†‘^]~yw\/oj[^ qœiY‘~Ki+kcYhu‘iCm(^ ^]†/izk u‘jnp^]›np†›y~yw n¡oKu‘i  ~ricY[n¡oK†‘i]¬vC^ w n¡o[jnpƒ kcw„~9xz^]ƒkcorw np^]i]£ªorw°q¡np\/nŸk$ƒgjƒqp^]i]£(oy k Y[^]ic^_ilgjilkc^]\/iYª~>r^m(^]^ †E^]ic€(^]ƒnœ~yqpqpg n¡†hr^]ilkcnpsK~9k ^]²¬|XY[^³‘w ilk°w ^]icu[q¡k i°ƒ or†‘ƒ ^ w †[^] ~ €‘~ywckcnœƒu[qœ~yw›ƒ ~Kil^K£Cƒ„Y‘~yw„~rƒkc^]wcXn W ^] mhgSk Y[^±^>tKuª~yqpnŸkzg oy ~yqpq'j^]ƒ]~>g w„~9kc^>ioy kcY[^ icghilkc^]\Z¬ Y‚†‘[^ wk Y[npi_~ri icu[\_€[kcnpor†¤£ kcw„~9xz^]ƒkcorw np^]i6~rwc^ izk w„~ynpsrYKk$qpn¡†[^>i$n¡† w ^ sKn¡oK†‘i|¦Y[^ w ^_kcY‘^ilgjilkc^]\npi~9§ †[^K£+~r†‘Ek Y[^ ¦Y[orqp^ ~r†‘~yqpgjilnœi|m(^>ƒor\/^>i6\u‘ƒ Y icn¡\/€[qp^ w>¬‚ork ~rm[q¡gK£(~r†T^ žj€[q¡nœƒn¡k°vCornp†‘ƒ ~r'w \ ~r€Tnœi€(oKi icn¡m[qpgj^ w n¡r^>²£ª¦Y‘npƒ„Y nœi°~/ ¥w„~rƒkcnpor†‘~rqq¡np†[^>~yw\ ~y€¬‚a­k|npikcY[^ † €(oKi icn¡m[qp^‚kco›wc^>ju‘ƒ^kcY[^°€‘wcoKm[q¡^]\ or ¤^žjnœizk ^ †‘ƒ ^°~y†‘±izk„~ym[npqpnŸkzg/oy €ª^]wcnpojjnœƒ‚orw m[n¡k iGkco_~y†±^ npsr^ †ª~y†‘~yqpgjicnpi]¬+XY[nœi\/^kcY‘oj ¦'~Ki'np†Kk wco[ju‘ƒ^>5n¡A† O ‰j(£ SRP­£‘~y†‘ icu[m‘ic^]tBu[^]†Kkcqpg±n¡\/€[w o¢K^]±mhg5jn¡¨(^ w ^ †Bk~yujk Y[orw„[i Op‹>j£[Œ[\£ QP­¬ X+o$oru[wG«h†[o9¦q¡^>jsr^r£ror†‘q¡g›or†[^‚ilkcu‘[g_jnp/†[orkGw ^]tBu[npwc^'k Y[nœiGu[†[n¡ ¥orw \®[^]ƒ ~¢gw„~9k ^]i~Kicicu[\/€jk n¡oK†¤£rm[ujkƒ or†‘ƒ ^ w †[^]›oK†_kcY[^ oyk Y[^ wY‘~r†‘5icghilk ^ \ i'¦Y[oKic^°np†Kk ^ w„~rƒ8k npor†5ilkcw u‘ƒ8k u[w ^|ƒ or†‘icnpilk i'np† ~_ilnp†[sKq¡^K£h†[^ sB~9k n¡r^|q¡ohoKZ€ Op‹ ]P­¬G`nporqporsKnpƒ]~yqª^ ž[~y\/€[qp^]i £ icu‘ƒ Y~rikcY[^ H^H _J$,9O ]P(\ ~y«r^k Y[npiGkšgh€(^‚or Unp†Kk ^ w„~rƒ8k npor† izk wcuªƒ8kcu[w ^w ^ qp^ y~y†Kk>a¬ `‘orwNicu‘ƒ„Y icgjizk ^ \ i]£ykcY[^^žjnœizk ^ †‘ƒ ^r£ m[ujk/†[oyk_u[†[nœƒn¡kšgK£or ~Eizk„~ym[qp^±qpn¡\/n¡k/ƒgjƒ q¡^±np†[n¡\/^ †ªilnpor† > 3 £G~r†‘ oy ~Sizk„~ym[qp^ ¥ojƒuªinp†jnp\/^ †‘icnpor† 2 nœi€[w o9r^ † np@† Op‹ ]P­¬GXY[nœi€[wcohor +wc^]q¡np^]ioK†`w oruh¦^ w  iN³[žj^] €(ornp†KkkcY[^]orw ^ \£[~r†‘5npi†[ork^ †Bkcnpwc^]q¡g5ic^ q¡ CƒoK†Kk ~rn¡†‘^]²¬ b°u[w\ ~rn¡† w ^]icu[qŸknœi~Tƒ or\/€[qp^kc^€[w ohor 'or 'k Y[^^ žhnœilkc^ †ªƒ^±~y†‘ u[†[nœƒn¡kšg or 'k Y[^\/^]†Kkcnpor†‘^] ~9kckcw„~rƒkcoKw i6np†  ¥^ ^]jmª~rƒ„« qpohor€Ticghilk ^ \ i ¬´Torw ^ o9r^ w>£hkcY‘^\ ~9k Y[^ \ ~9k npƒ]~yq¤kcohoKqpinp†BroKq¡r^>5kco€[w o9r^|k Y[npiw ^]ilu‘qŸk°ic^ ^ \ k oY‘~>r^m(^ ^ †Tnpsr†[oKwc^>5np† kcY‘^ƒor†Bkc^žhk‚oy €[np^]ƒ ^ ¦nœil^6qpnp†[^]~ywsK^ †[^6†[^ kš¦oKwc«\/oj[^ qœi £[~r†‘5oy¨U^ w‚~_r^]wcg€[w or\/nœicn¡†[s› ¥w„~y\/^]¦orw «U¬ ´ToKwc^‚€[wc^>ƒnœil^]q¡gK£yk Y[^wc^>ilu[q¡kNoK† ¦Y[npƒ„Y¦^‚\ ~ynp†[qpg›w ^ qpg›nœiN~6³[žj^>/€(ornp†KkNkcY‘^ orw ^ \® ¥orwN\/or†[orkcoK†[^r£Bƒor†ªƒ ~¢r^or€(^ w„~9k orw„i £ ¦Y[^]wc^$\/or†[orkcoK†[npƒ nŸkzg5~r†‘ƒ or†‘ƒ ~¢hn¡kšg5~yw ^$j^ ³‘†[^>¦nŸk YTwc^]ic€(^]ƒkkco±~/€‘~ywckcnœ~yq¤orw„j^]w]¬XY[^ w ^$^žjnœizk„i‚~rƒkcu‘~rq¡qpg5~/¦^]q¡q¡½ j^]r^ qpor€(^] ƒoKwc€[uªi~ym(oKujk°or€(^ w„~9k orw„i‚ic~ykcnœiz ¥ghnp†[s±icu‘ƒ YE€[wcoK€(^ wckcnp^]i]£‘^>il€(^>ƒnœ~yqpq¡g5 ¥orw|icgjizk ^ \ i¦nŸk Y €(oBiln¡kcnpr^9~yw nœ~ym[qp^]i]¬ XY[^>il^w ^]icu[qŸk„i‚Y‘~¢r^6k Y[^›~rj9~r†Kk„~ysr^$oy m(^ np†[ssK^ †[^ w nœƒy£ª~y†‘ k Y[^YBgh€(oyk Y[^]ic^]i‚or†T¦Y[nœƒ„YkcY‘^ g wc^]q¡g ~rwc^$oK†kcY[^oyk Y[^ w Y‘~r†‘ †‘~9k u[w„~yqpq¡g5i ~9kcnœil³‘^] np† k Y[^ƒor†Bkc^ žBkoy C€[np^]ƒ ^ ¦nœil^~y§/†‘^$\/ojj^ qœior Csr^ †[^$†‘^kz¦orw «ji ¬Naš†oKw j^]wkco sKn¡r^~/\/orw ^ €[w ^]ƒ npic^ ¥oKwc^ k ~Kizk ^‚oy ¤kcY[^°w ^]ilkoy ²k Y[nœi€‘~y€(^]wq¡^ k'u‘i'izk„~9kc^°†[o9¦3k Y[^°\ ~ynp†w ^]icu[q¡k]¬CX+o\ ~y«r^n¡k'u[†ªj^ w„izk„~y†‘[~rm[qp^r£Bq¡^ k'u‘i xzu‘ilk‚i ~>g kcY‘~yk C nœi~›€(^]wcnpojjnœƒ|il^>tBu[^ †‘ƒ ^|oy wc^]srnpor†‘i]£hƒw oKi ic^]mBgkcY[^|k w„~¢xz^>ƒ8k orw n¡^>ioy C~y†Bg€[np^]ƒ ^ ¦nœil^$~9§ †[^|\/oj[^ q²oy  sr^]†[^†[^kz¦oKwc«jiƒor†‘icnœizk n¡†‘snp†±~†[^]sK~ykcnpr^ ¥^ ^>jm‘~Kƒ «/q¡ojor€¤¬ D qpico‘£ Z nœi'~€‘~ywckcnœƒu[qœ~yw'ilu[m‘ic^k'oy ¤oK†[^w ^ srnpoK†¤£roKw  W oK†[(^ p£ ¦Y[nœƒ Ynœi~_¦^]q¡qj^ ³‘†[^] vCornp†‘ƒ ~r'w $il^>ƒ8kcnpor†£[~y†‘ T nœi~›³ªw ilkw ^kcu‘wc† \ ~y€ oK† Z ¬ b)c ·h» – · edgfih  T : Z → Z j lk[9_(

(18) $Hm-%$n1 C o 1. 1. 1. 1. pZq . lk ∀x ∈ Z1 , Tm x → 0 !sk m → ∞ o n = 2r pZq  n > 2 r lktlkHGHuF#  )$U\v\)(wRH>xsuy [ _\ q = Tq o zZH { r q ∈ ˚Z1 $ ( >{* x ∈ Z1 \ {0} r Tm x → q $ m → ∞ o. D w ~r€[np o9r^ w hn¡^]¦oy CkcY[^kcohoKqpi‚~y†ª w ^ ¥^ w ^ †‘ƒ ^]i¦^wc^]q¡g5or†nœisKn¡K^ †n¡† ic^]ƒ8k npor† ‹r¬ h¬XY[^ †¤£‘kcY‘^ƒqœ~ri ioy C€[np^]ƒ ^½ ¦nœil^6~9§ †[^|\/ojj^ qœi'¦^° ¥ojƒuªioK†±nœi'€[w ^]ic^ †Kk ^]±n¡† ic^]ƒkcnpor† j¬Nfhor\/^|sr^ †‘^ w„~yq(orm‘ic^ w 9~9k npor†‘i~ym(oru[k'\/or†[orkcoK†[npƒ nŸkzg~y†‘ ƒoK†‘ƒ ~¢hnŸkzg  ¥orwkcY‘npiƒqœ~ri ioy Cicgjizk ^ \ i~rwc^6\ ~r[^6np†Til^>ƒ8k n¡oK†©[¬NXY[^]ic^$wc^]\/~rwc«ji'qp^]~K±kco/ ¥ojƒ u‘ior† †[^]sK~ykcnpr^| ¥^ ^]jmª~rƒ„« qpohor€ilgjilkc^ \ inp†ic^]ƒkcnporK† ]‘£[¦Y[^ w ^|kcY[^$\ ~ynp† w ^]icu[q¡k i~yw ^6ilk ~ykc^> ~y†‘5€[w o9r^ †¤¬ /"0L|. }~=T5€ ;‚„ƒ† a‡ˆ>5€ƒ† a‡8H‰ ")35€=y;‚,‡=. XY[^$\ ~ynp†wc^>ilu[q¡k‚oy kcY[nœi€‘~r€(^ wizk wcoK†[srqpg w ^ qpg±or† oK†[^6kcY[^ oKwc^]\£hk Y‘~9k¦^$wc^>ƒ ~rq¡q²Y[^]wc^K¬a­knœi‚~›³[žj^] €(ornp†BkkcY[^]orw ^ \ oy ²\/oK†[oyk or†[^r£BƒoK†‘ƒ ~¢r^or€(^]w ~ykcorw„i ¬+`'orkcY \/or†[orkcor†‘npƒ nŸkzg_~y†‘ ƒ or†‘ƒ]~>hn¡kšg_icY‘~yqpq‘m(^‚u‘†‘j^ w„ilkcohoj_¦n¡kcYw ^]ic€(^]ƒ8kGk oiloK\/^ €‘~rwlk np~rqGoKw j^]w]£ic^ ^m(^]q¡o9¦® ¥oKw›j^³ª†[nŸk n¡oK†‘i›~y†‘ †‘oyk ~ykcnpor†‘i]¬Eb°€(^ w„~9kcoKw i6¦n¡kcY m(oyk Y k Y[^]ic^€[w or€(^]wlk n¡^>i$Y‘~¢r^ qp^]~K kco 9~rwcnporuªi°w ^]icu[q¡k i6oK† ^žjnœizk ^ †‘ƒ ^ ~r†‘Su[†[nœƒn¡kzgEor ³[žj^>S€(ornp†Kk„i ¬ D \/oK†[s^]~rwcqpgEizk u‘jnp^]i]s£ Š$w ~Kil†[oKic^ q  ic«BnpnCY‘~ri$€[wco9hnp[^] ‹€‹Œ ŽR‘1’.

(19) €$H

(20) 9FwRm ]. np\_€(oKwlk„~y†Bk$wc^>ilu‘qŸk„„i O¡‹K‹ P­¬ ^ w ^r£¦^ u‘ic^ ~ \/oKwc^/wc^>ƒ^]†Kk$€‘wc^>il^ †Bk ~ykcnpor† oy icu‘ƒ„Y w ^]icu[qŸk„i £+ju‘^/kco fj\_n¡k Y Op‹]©+P­£+¦Y[oKic^  ¥orw \u[qœ~9k n¡oK†5nœi¦^]q¡q~K[~y€jk ^]±kco oru[w€[u[w €(oKic^6~r†‘ ƒoK†Kk ^žhk]¬G~r\/^ qpgr£[¦^6u‘ic^|k Y[^ oKwc^]\ j¬ or O¡‹>©+P­¬ `'^ ¥oKwc^ izk„~9k n¡†[skcY[^ qœ~9kckc^]w]£qp^ku‘i$np†Kk wco[ju‘ƒ^iloK\_^/†[ork ~ykcnpor†‘i  ¦Y[npƒ„Y ~rwc^ ilnp\/n¡qœ~yw6k o k Y[oKic^/np† Op‹]©P ¬ S^[^ †[oyk ^ ~r†‘ n¡ ¤kcY[^>il^°np†[^]tBu‘~rq¡n¡kcnp^]iY[oKqp  ¥orw^>~rƒ„Y±ƒohoKw [n¡†‘~ykc

(21) ^  wc^>il€¤¬C^ †Bkcw g or ¤r^]ƒkcoKw i¥w ^]ic€¤¬\ ~ykcw npƒ ^]i ~r†‘ x<y ¬y XY‘^ †¤£9¦x^'j6^]†[yoyk ^ x

(22) y n¡  x 6 y ~y†ª x 6= y a¬ `‘orw x 6 y £ [x, y] = {z | x 6 z 6 y} £y~y†‘ (x, y) = {z | x < z x< y} ¬ `‘orw‚~y†hg±il^ k A £ A˚ j^ †‘oykc^>i'kcY‘^$n¡†Bkc^ w nporwor  A £[~r†‘ c`(A) nŸk„i‚ƒqpoKicu[wc^K¬Co¢¦$£[¦^ƒ ~y†ilk ~9k ^|k Y[^6 ¥orqpq¡o9¦np†[s‘£ b)c ·h» – · (fih  p ∈ R ˚ r $ ( T : [0, p] → [0, p]

(23)  L_\\ y r C _ (0, p) o  RF+ . n +. DT (0) = lim DT (x) x→0 x>0. 1. HuF#  o  -%.  z   x > 0r x < p o DT (x) > 0     0 < x < y < po DT (y)

(24) DT (x) $   R F +-%  lk$ Tp!<"p o r $ FxM( λ = ρ(DT (0)) r _k\i 

(25) H$€H$I _y 1 DT (0) o  k r !sk\ n → ∞ o λ 6 1 =⇒ ∀x ∈ [0, p] r T n x → 0 k1„uI# L )$% v\)ywRH9xsuy ‚\_  zt H {+ r q ∈ (0, p) λ > 1 =⇒ qo $ 9 {* x ∈ [0, p] \ {0} r T n x → q $+ qn=→T ∞ o XY‘^ƒ]~ric^ T 0 6= 0 qp^]~K[ikco ^]nŸk Y[^ w°~/u[†[nœtBu[^6³‘žh^> €(ornp†Kk>£‘orwkco±jn¡r^]wcsKn¡†‘s_oKwcm‘nŸk„i ¬#‚ o9¦^ K^ w]£jkcY‘npiƒ]~ric^6¦npq¡q†[ork. y~ €‘€(^]~ywnp†oru[w‚ƒoK†Kk ^žhk]£‘~y†‘5kcYBuªinŸk‚nœi†‘oykj^k ~rnpq¡^>5n¡† kcY‘^6kcY[^ oKwc^]\ ~rm(o¢r^K¬ b°†[^$\ ~¢g5wc^]\/~rwc«±†[o¢¦ek Y‘~9knp†kcY‘^ƒ ~ric^$¦Y[^ † npikz¦npƒ ^jnŸ¨U^ w ^ †Bkcnœ~ym[qp^r£‘kcY[^ƒ or†‘ƒ]~>hn¡kšg5ƒ or†‘[nŸk n¡oK† ~rj\/n¡k i~ icn¡\/€[qp^ilu[§ ƒ n¡^]†KkCƒor†ªjnŸk n¡oK†¤¬+XY‘^N\/~r€ T : [0, p]T→ R j^]†[oyk ^]i¤kcY[^ ikcY›ƒohoKw [n¡†‘~ykc^C ¥u[†ªƒ8kcnpor†oy  T : [0,(C)p] → [0, p] ¬ $ – &» %N» ” ' • ' )» ( (f  RI lk$ ($ _ i, j, k ∈ {1 · · · n} r $ ( >$l 0 < x < p r i. +. ∂ 2 Ti (x) 6 0, ∂xk ∂xj. $ 9 $_ i, j _k\HGuFX L )$ k \

(26) k lk$ <lk_(v\$ _L*U# L

(27)  o k T $# lx< )

(28)  _ (C) ilk H-+* o D ƒ8k u‘~yqpqpgr£nŸk±npiƒ q¡^>~yw/kcY‘~yku[†‘[^ w kcY‘npi±ƒor†ªjnŸk npor†3^>~rƒ„Y kc^]wc\ ∂T. ro   DT nœi~ j^>ƒw ^]~ricnp†[sS ¥u[†‘ƒ8k n¡oK† or |^]~Kƒ Y ƒojorw„jn¡†ª~9kc^ x ¬‚a­k|npi\/oKwc^]o¢r^]w‚ilkcw npƒkcqpgj^]ƒ wc^>~ricn¡†[s np†E~9k°qp^]~rilkoK†[^or Ck Y[^]ic^›ƒohoKw jnp†‘~ykc^]i]£ª~y†‘ (C) kcYhu‘i‚ ¥oKq¡qpo9¦‚i ¬ b°m‘ic^ w r^›mBgTk Y[^/¦'~¢gk Y‘~9k6k Y[^/†[oyk n¡oK†Soy 'ƒ or†‘ƒ]~>hn¡kšg, ¦$¬ w>¬ k$~ €‘~ywckcnœ~yqCorw„j^]w ¦^/[^]~yq¦nŸk Y Y[^ w ^_nœi|¦'^]~y«r^]†[^]EmBg kcY‘^6  ~rƒ8kkcYª~9k‚nŸkƒ or†‘ƒ ^ w †‘i'oK†[q¡g±orw„j^ w ^]±€‘~rn¡w„i (x, y) or 9~yw nœ~ym[qp^]i]¬ }^]\/~rwc«y~ym[qpgr£rkcY[^]orw ^ \ ‹‚w ^ qpnp^]iGoK† vC^ w wcoK†j½^`‘wcoKm(^ †[npu‘iCk Y[^ oKwc^]\ £B¦Y[npƒ„Y¦npqpqªm(^u‘ic^]~9k'icor\/^‚€(oKn¡†Bknp† kcY‘^€[w ohoy ²oy  oru‘w\ ~ynp†Tw ^]icu[q¡k]¬‚a­k°uªilu‘~rq¡qpgic~¢gjikcY‘~yk|~y†hg5\ ~ykcw nŸž¦nŸk Y €(oBiln¡kcnpr^^]†Kkcw np^]i°~rj\/n¡k i°~€(oKicnŸk n¡r^K£(icn¡\/€[qp^^ npsr^]†B9~rq¡u[^K£ ~ri icojƒnœ~9kc^> k o ~T€(oKicnŸk n¡r^ ^]n¡sK^ †hr^]ƒ8k orw>¬-^ w ^r£¦^ ¦npqpqNor†[qpgS†[^ ^] k Y[^  ¥orqpqpo¢¦np†[s ƒorw orqpqœ~ywc/g .$nŸ  A nœi~wc^>~yq n × n \ ~9k wcn¡ž5icu‘ƒ Y kcY‘~yk A > 0 £hk Y[^ † A ~Kj\/nŸk„i~_€(oBiln¡kcnpr^6wc^>~yq²^ npsr^ †h9~rq¡u[^K¬ i. ∂xj. k. 0. 

(29)   !$#%  ,./ . | 01/. 1:B3\ƒ ;‚ a‡8:#2. XY[^$sr^]†[^ w„~yq( ¥orw \ oy +kcY‘^6€‘n¡^>ƒ^ ¦nœic^$~9§ †[^6\/ojj^]qpi¦^$ƒ or†‘icnp[^ wY‘^ w ^|\ ~¢gm(^$¦w nŸkckc^ †T~ri. dx = κ(x) − Γx dt. l‹3. XY[^$y~yw np~rm[qp^]i (x . . . x ) w ^ €[w ^]ic^ †Kkƒor†ªƒ^ †Bkcw„~9k n¡oK†‘in¡†T€[wcorkc^ np†‘iorw‚\/} D €[w ojjuªƒ^]5 ¥wcoK\ n n¡†Bkc^ w„~rƒkcnp†[s/sK^ †[^>i ¬ fhnp†‘ƒ^°sK^ †[^‚k w„~y†‘i ƒw n¡€jk npor†‘~yq‘w ^ sKu[qœ~9kcnpor† nœi¦npj^]q¡gilu[€‘€(oKic^]/kco ¥orqpq¡o9¦ ~ilkc^]^ €5ilnpsr\/ornœ/qœ~¢¦$£BnŸkY‘~rim(^ ^ †5icu[srsK^]ilkc^] 1. n. 465a‹74 8.

(30)  

(31)    

(32) !"# %$&'() *+ ,-. V. kcYª~9kNnœj^>~yqpnJW]^]²£Bjnœi ƒor†Bkcnp†Bu[oKu‘iNil¦n¡k ƒ„Y[^]iN\/~¢g›m(^‚u‘ic^]/np†‘ilkc^]~K_kco\/ojj^]q[kcY[^>il^ƒ or\/€[qp^ž/icgjizk ^ \ iO V+P­¬ D ƒ ƒ orw„jn¡†‘srqpgr£ j^ †‘oykc^6kcY[^$np†‘ƒ wc^>~ricn¡†[s/ilkc^ €  ¥u[†‘ƒkcnpor†¤£‘orw ‚^]~¢hnœilnœj^| ¥u[†‘ƒ8k n¡oK7† . s (· , θ) : R → {0, 1}  nŸ  x < θ, s (x, θ) = 0 nŸ  x > θ, s (x, θ) = 1 ¦Y[nœƒ Y±wc^ €‘wc^>il^]†Kk~y†^¨U^]ƒkor ¤~Kƒ8k n¡y~9kcnpor†¬ D qœico‘£ s (x, θ) = 1 − s (x, θ) £BnœiNn¡k i[^]ƒw ^]~Kilnp†[s$r^]w icnpor†¤£B~y†‘ wc^]€[w ^]ic^ †Kk„i np†[Y[npm[nŸk n¡oK†¤¬ XY[^]†¤£ κ : R → R nœi~|€[n¡^>ƒ^ ¦nœic^ƒ or†‘ilk ~r†Kk€[w ojju‘ƒ8k npor†kc^]wc\‡kcY‘~ykGƒ]~y†_m(^^žj€[w ^]i il^>np†›k ^ w \/iCor (ilkc^]€› ¥u[†‘ƒkcnpor†‘i ¬ npi~jnœ~ysKor†‘~rq‘\ ~9k wcn¡ž ¦Y[oKic^°jnœ~ysKor†‘~rq‘^ †Bkcw np^]i Γ = γ £j~yw ^°j^ sKw ~K[~9k n¡oK†/w„~9k ^]iNor ¤y~yw np~rm[q¡^>i s (x , θ ) Γ ∈ R np†5k Y[^ilgjilkc^]\ ¬ fhnp†‘ƒ^°^>~rƒ„Y9~yw nœ~ym[qp^ x nœi'~ƒoK†‘ƒ^]†Kkcw„~9k n¡oK† ¥or ¤\ }‚ D oKwoy ¤€‘wcorkc^ np)† £Bn¡k'w ~r†[sr^>iGn¡†5icor\/^np†Kkc^]wcy~yqªor ¤†[or†[½@†[^]sK~9k npr^ 9~rq¡u‘^]ij^ †[orkc^> [0,  ] ¬ Y[^ †/kcY[nœiGƒoK†‘ƒ^]†Kk w ~ykcnpor†›w ^]~rƒ„Y[^>iC~°k Y[wc^>ilY[oKqp›y~yqpu[^r£KiloK\/^'orkcY[^]wsr^]†[^np†_kcY[^†[^ kš¦'orw «U£ i ~>gEsr^]†[^/†hu[\m(^ w £+nœiicu‘[j^]†[q¡gS€[w ojju‘ƒ ^]S¦n¡kcY ~[nŸ¨U^ w ^ †Bk€[w ojju‘ƒkcnpor† w„~9k ^ .$k Y[^ y~yqpu[^/or  ƒ Y‘~r†[sr^>i ¬ `[oKw ^]~Kƒ Y i ∈ {1 · · · n} kcjY[^]wc^6nœikcYBuªi‚~›³‘†[n¡kc^ic^k‚oy +k Y[wc^>ilY‘orqœ±9~rqpu[^]i]£h¦Y‘npƒ„Yic^ w r^6~Ki'€ª~yw„~y\/^k ^ woy Cilκkc^]€5 ¥u[†ªƒ8kcnpor†ªi . ¯ }, Θ = {θ , . . . , θ ¦Y[^]wc^_kcY‘^_k Y[w ^]icY[orqœ[i$~yw ^_orw„j^ w ^] . ¬XY[^ ^žhkcw ^ \/^/9~yqpu[^>i ~r†‘ ~yw ^_†[ork kcY‘wc^>ilY[oKqp‘i £yicn¡†ªƒ^kcY‘^ gm(oru‘†‘kcY[^y~yqp0u[^]<ioyθ  x<£r~r·†‘· · <kcYhu‘θiC\/~¢g$<†‘oykCm( ^ƒw oKi il^>²¬ o¢¦'^ r^ w>£y~|ƒor0†hr^ †Bkcnpor†‘~r qB†[ork ~9k n¡oK† ¦npq¡q¤m(^ . θ = 0 £‘~r†‘ θ = max ¬ ‚o9¦$£h~9k'~6kcnp\/^ ilu‘ƒ„Y/k Y‘~9k £KkcY[^]wc^npiicor\/^ icu‘ƒ Y k Y‘~9k ¬Ca­kN ¥oKq¡qpo9¦‚i kcYª~9k°^]~Kƒ Y ~yžjnpi‚ort CkcY‘^›izk„~9k ^xic€‘(t)~rƒ ^$∈¦Θnpqpq+m(^u‘il^  ¥u[qpq¡g€‘~ywcjkc∈n¡kcnp{1or†‘^]· · · n¡n}†Bkco±or€(^]† ic^ srκ\/^](x(t †Kk im())^kz¦6=^ κ^]†T(x(t kcY[w ^]ic))Y[orqœ[i]¬‚fhnp†‘ƒ ^ kcY‘^^ žhkcw ^ \/^9~rq¡u[^>iG¦npqpqª†‘oyk'mª^|ƒw oKi ic^]/mBg_kcY[^ ªo¢¦  ic^ ^°qp~ykc^ w 8£Kk Y[^³‘w„izk'~r†‘ qœ~rilk'il^]sr\/^ †Bk iNnp†‘ƒqpu‘j^°oK†[^oy ²k Y[^ npw ^ †ªj€(ornp†Kk„i . o n o n © , θ ] ∪ (θ , θ ) | j ∈ {1 · · · q − 2} ∪ Θ D ∈ [θ , θ ), (θ N~Kƒ Y€[w ojju‘ƒk D = Q D j^³‘†[^>i~$w ^]ƒ8k„~y†[sKu[qœ~y‚w I-%$_ª£h¦Y[oBil^jnp\/^ †‘icnpor† nœiGkcY[^°†Bu[\m(^]woy  D kcY‘~yk'~yw ^†[ork icn¡†[sKq¡^ kcoK†‘i ¬ Y[^ † dim D = n £ªor†[^u‘icu‘~yqpqpg5i ~¢ghik Y‘~9k°nŸk°nœi~ H^E J$ ^ *K I -„$ _‘£Uor)w H^E J$ G I-%$_ª£U~r†‘5k Y[oKic^ joK\/~rn¡†T¦nŸk YTqpo9¦^ wjnp\/^ †‘icnpor† ~yw ^ƒ ~yqpqp^]4 !"_,

(33) k_TE F-%$_ )O © P­¬ ^u‘il^$k Y[^†‘oyk ~ykcnpor† kcow ^ €[w ^]ic^ †BkkcY[^›il^ k oy ~yqpq¤joK\ ~ynp†‘ioy kcY[^6 ¥oKwc\ ~rm(o¢r^K¬XY[^ †£ D ¦n¡qpqj^]†[oyk ^|k Y[^ic^k‚oy ~yqpq²w ^ sKu[qp~ykcoKwcgjor\ ~rDn¡†‘i]£[~r†‘ D k Y[^ic^k‚oy  ~yqpq¤ic¦n¡k ƒ„Y[n¡†‘s jor\ ~rn¡†‘i]¬ D ƒoK†Br^ †‘n¡^]†Kk¦~>g k o/w ^ €[w ^]ic^ †Bkw ^ sru‘qp~rwjoK\/~rn¡†ªi\/~r«r^]iu‘ic^6oy C~/[npi ƒw ^k ^|\ ~r€¤£[¦Y[oKic^|w„~y†[sK^|¦^6¦w n¡kc^$~Ki . +. +. +. −. ±. n +. i. i. +. n + n×n +. ii. i. i. i. j. qi −1 i. 1 i. 0 i. qi −1 i. 1 i. i. i. i. i. qi i. i. i. i. 0 i. i. n i=1. j. qi −1 i. 1 i. j i. qi i. j+1 i. i. i. +. j. −. i. i. r. s. A=. n Y. {0 · · · qi − 1}.. l] . XY[^6\ ~r€ n¡†TtKu‘^]izk npor†5nœi'k Y[^ †Tj^³ª†[^]~ri £‘~y†‘ ic^ †‘‘i^]~Kƒ Y wc^]sru[qœ~yw[or\ ~ynp†±k o_kcY[^icu[€(^ w„icƒ wcnp€jkor +n¡k i q¡o9¦^ wc½­q¡^  µ k ƒorw †[^ 3w . d Q (θ , θ ) =d (a: D −→1 .A. . a − 1) ¬|aš†Tk Y[^ ¥orqpqpo¢¦np†[sª£ª¦'^¦npq¡qor µkc^]† nœj^]†Kkcn¡ ¥gTw ^ sru‘qp~rw joK\/~rn¡†ªi~r†‘±kcY‘^ npwnp\ ~ysK^|np† A £jk ~rq¡«hnp†[s_ ¥orw^ž[~r\_€‘q¡^$~rm(orujk‚icor\/^  joK\ ~ynp† a p¬ XY[^jgh†‘~r\_nœƒ i_or†w ^ sKu[qp~rw_jor\ ~ynp†‘i]£Gƒ]~yqpq¡^> Y[^ w 4^ H^E J$  *($ -

(34)  _ƒ ~y†m(^[^³‘†[^> tKu‘nŸk ^ilnp\/€[q¡gK£N[u[^5kcoSkcY[^ icn¡\/€[qp^›^ žh€‘wc^>icicn¡oK†Toy Gk Y[

(35) ^ ªo¢¦‡n¡†S^]~rƒ„Y ¬b°†Sil^ k i°or  or† kcY‘^›orkcY[^]w°Yª~y†‘²£UkcY[^ ‘o9¦ nœi°np†EsK^ †[^ w„~yq+†[ork u[†[nœtBu[^ qpg_j^ ³‘†[^>²¬Ca­kNnœiG~r†Bgh¦'~¢g€(oKi ilnpm[qp^Dkco∈jD^³ª†[^icorqpujkcnpor†‘iGn¡†D~6w npsrorw oru‘iC¦'~¢gr£rghnp^ qœjnp†[s$¦Y‘~ykN¦n¡qpq‘m(^‚\/^ †Bkcnpor†[^> ~ri6k Y[^ _yE X$ *($ -

(36)  8£~9k$k Y[^€[wcnœƒ^ or ƒor†ªilnœj^ w np†[sic^kc½@y~yqpu[^] icorqpujk n¡oK†‘i]¬±XY[^qp~yklkc^],w  ij^³‘†‘nŸk n¡oK† w ^]ilk i$or† `n¡qpnp€[€(o¢  i°k Y[^ oKwcgTor 'jn¡¨(^]wc^]†Kk np~rqC^>tBu‘~9k n¡oK†‘i|¦nŸk Y ~jnpi ƒoK†Kk n¡†hu[oru‘i6wcnpsrYBkl½­Y‘~y†‘Silnœj^r£~r†‘En¡k i| ¥oKwc\u[qœ~9k npor†Enp†SkcY[^ €‘~rwlk npƒ u[qp~rwƒ ~ric^°or +^]tBu‘~9k n¡oK†‘i z3‹ 'ƒ ~r†±m(^| ¥oru[†‘5np4† Op‹r£U‹>+Š P@£[~r\/or†[s›orkcY[^]w i]¬fhnp†‘ƒ ^|¦^6¦n¡qpq²or†[qpg ^ †ªƒoru‘†Kkc^]wilgjilkc^]\/i  ¥orwC¦Y[nœƒ Y›kcY[^ `n¡qpnp€[€(o¢iloKq¡u[kcnpor†‘iq¡^>~r$k o6ilnp†[sKq¡^'9~yqpu[^>$kcw„~9xz^]ƒkcorw np^]i]£¢¦^icY‘~yqpqhor†[qpgj^]i ƒw npm(^NkcY‘^'w ^ sru‘qp~rwjgh†‘~y\/nœƒ i ¦n¡kcYTiloK\/^6[^k ~rn¡q@¬ i=1. i. ai −1 i. ai i. r. 1. n. r. | 0L|. s.  5M;‚ 329 "ƒ 81‰7=. }^]sru[qœ~9k orw gjoK\/~rn¡†ªi~yw ^oy (€‘~rwlk nœƒu[qœ~ywn¡\/€(orwck ~r†‘ƒ^K¬XY[^ g ¥oKwc\‡k Y[^\ ~ynp†/€‘~ywckor Uilk ~9k ^ic€‘~Kƒ^r£K~y†‘›kcY[^jgh†‘~y\/nœƒ i or† k Y[^ \ƒ ~r†Em(^›^ žj€[wc^>icic^] tBu[n¡kc^/icn¡\/€[qpgK¬ D ƒkcu‘~rq¡qpgr£²oK†Siluªƒ Y ~5jor\ ~ynp† D £UkcY[^_€[w ojju‘ƒkcnpor†Ew ~ykc^ κ npi6ƒoK†‘izk„~y†Bk]£ ~y†ª±kcYBuªi^]tBu‘~ykcnpor† z‹3 'nœi~9§ †[^K¬Ca­k„i‚iloKq¡ujk n¡oK†5nœi^žj€[qpnpƒ nŸk q¡g±«h†[o9¦†¤£h ¥oKw^]~Kƒ YƒohoKw [n¡†‘~ykc^ i . ‹€‹Œ ŽR‘1’.   κi κi −γi t −e xi − , ϕi (x, t) = xi (t) = γi γi. LV .

(37) ‰. €$H

(38) 9FwRm. ~y†ª5npi9~rq¡nœ± µoKw‚~yqpq t ∈ R icu‘ƒ„Y5kcYª~9k +. x(t) ∈ D. ¬a­k ¥oKq¡qpo¢¦‚inp\/\_^>jnœ~9kc^]q¡gk Y‘~9k. φ(D) = (φ1 · · · φn ) =. . κn κ1 ··· γ1 γn. . œn i6~y† ~9kckcw„~rƒkcnpr^›^]tBu[npq¡npm[w n¡u‘\ €(ornp†Kk6 ¥orw|k Y[^ ªo¢¦ LV ¬ ‚^]†‘ƒ^K£¤nŸ 'nŸk$q¡np^]i6n¡†ªilnœj^ D £¤nŸk$nœi6~y† ~rƒkcu‘~rq^>tBu[n¡qpnpm[w n¡u[\ oy  icghilkc^]\ l‹3 8¬$bk Y[^ w ¦npic^r£ªkcY[^

(39) ‘o9¦ ¦n¡qpqwc^>~rƒ„YkcY[^›m(oru‘†‘[~yw g n¡† ³‘†[n¡kc^k n¡\/^K£Uu[†[qp^]i i φ(D) qpnp^]i^ ž[~rƒ8k q¡g oK†TkcY[^ m(oru‘†‘[~yw gor  D ¬ ‚o9¦^ r^]wCkcY‘npiNicnŸk u‘~9k n¡oK†_nœiGƒ q¡^>~yw q¡g†[oykNsr^ †‘^ w ∂D npƒr£r~r†‘_¦^‚~Kicicu[\/^np†_k Y[^‚wc^>izkGoy (k Y[nœi€ª~y€(^ wGn¡kG†‘^ r^ w ojƒ]ƒu[w„i ¬ D k|kcY[^k n¡\/^›¦Y‘^ † kcY[^

(40) ªo¢¦ w ^]~Kƒ Y[^>i ∂D kcYhu‘i £UkcY[^›y~yqpu[^oy  κ ¦npqpqCƒ Yª~y†[sr^K£U~y†‘TkcY‘~yk|oy  φ ~Kƒ ƒoKw [n¡†[sKq¡gK¬ XY[^€(ornp†Kk φ(D) npi‚oy µk ^ †Tƒ]~yqpq¡^>GR

(41) $\ _\oy k Y[^jor\ ~ynp† D ¬XY‘^ †¤£[k Y[^ƒ or†Bkcnp†Bu[oKu‘i ‘o9¦ ƒ]~y†m(^$w ^]ju‘ƒ ^]5kco~ jnœicƒ wc^ kc^ ½¯k n¡\/^$jgh†‘~y\/nœƒ ~rq¤ilgjilkc^ \£[¦nŸk Y~/ilk ~ykc^$ic€‘~Kƒ^$ilu[€‘€(orwckc^]5mBgk Y[^$m(oru[†ª[~yw n¡^>i'oy Cmªo¢žj^]inp† D ¬ ‚^]†‘ƒ^K£hkcY[^$izk„~9k ^|ic€‘~rƒ ^°oy kcY‘npijnœicƒw ^k ^½@kcnp\/^|icgjizk ^ \nœi'€‘~ywck'or  £j¦Y[nœƒ Y5\ ~¢gil^]^ \€‘wcoKm[q¡^]\ ~9kcnœƒ°~9k'³‘w„ilkicnpsrYKk>¬ aš†   ~rƒ8k>£BnŸknœi~rƒ8k u‘~yqpqpg›†[oyk'~yqp¦'~¢gji€(oKi ilnpm[qp^kco[^³‘†[^°[npi ƒw ^k ^½@kcnp\_D^k w„~¢xz^>ƒ8k orw n¡^>ior†joK\/~rn¡†ªiGoy  D Y‘~¢hn¡†‘sjnp\/^ †j½ icn¡oK† oKw|qp^]i i ¬_b|†S~r†Bg joK\ ~ynp† £¤or†EkcY[^/oyk Y[^ w6Y‘~y†ª²£¤icu‘ƒ„Y ~5[^³‘†[n¡kcnpor† \ ~>g ~rq¡¦~>gji°m(^_€[w or€(^ w qpg €[w o¢hnœnj^]−¤£2np† k ^ w \/ior kcY[^ ‘no9¦ −qp1n¡†‘^]inp† kcY‘^Dkz¦oEw ^ sKu[qp~rwjoK\/~rn¡†ªiic^ €‘~rw ~ykc^] mBg D ¬Ea­ kcY‘^]ic^ ‘o9¦ qpn¡†[^>im(oyk Y €(ornp†Kk6k o¢¦~yw„[i £²oKw6~¢¦'~¢g ¥wcoK\ D ¥¦Y[nœƒ Y nœi6^žj€[w ^]i il^>T ¥orw \ ~yqpq¡gTu‘icnp†[s5k Y[^ ‘o9¦®ƒohoKw jnp†‘~ykc^›¦Y[nœƒ Y nœi6†[orw \ ~yqkco 8£jkcY[^6qœ~9kckc^]wnpiƒ]~yqpq¡^>±wc^>il€(^]ƒkcnpr^ qpg X$I

(42) N%!M$ _horw !sky_,)!M$ _ ¬'bk Y[^ w ¦npic^r£   ¦Y[^]† ªo¢¦qpn¡†‘^]im(oyk Y ƒw oKi i np† D kcY‘^$i ~y\/^$jnpwc^>ƒ8kcnpor†£jor†[^6u‘icu‘~yqpqpgwc^  ¥^ w„j ik o D ~KiK~ LH$\ $H\<!M$ _ ¬Gaš† kcY[^6kzo ¦o o_³‘w„izk‚ƒ]~ric^]i]£BkcY[)^ `npq¡np€[€(o9 k Y[D^ oKwcg nœiwc^>tBu[n¡w ^]¤£ª€[wco9hnp[n¡†[s±ic^kc½@y~yqpu[^] k w ~9xz^]ƒkcoKwcnp^]i‚or†TkcY[^›¦'~rq¡qœi]£‘kcY‘^›icoy½šƒ ~rq¡qp^]icqpnpjnp†[s±\/ojj^>i O¡‹>+Š P@¬$b°†k w ~r†‘il€ª~yw ^ †Kk ¦'~rq¡qœi £²k w ~9xz^>ƒ8kcoKwcnp^]i6~yw ^/ilnp\/€[q¡gSj^³‘†‘^]SmhgEƒoK†Kk n¡†hu[nŸkzgK£¤ ¥wcoK\ kcY[^ ‘o¢¦®q¡np†[^]i6oK†Sm(orkcY icnpj^>i ¬ ^ icY‘~yqpq^ †‘ƒoKu[†Kk ^ w or†‘q¡gkcY[nœikcY[npw„5«hn¡†ª5or ¦'~yqpq²np†5k Y[^$wc^>izkor +kcY[nœi€‘~r€(^ w>¬ npiwc^]€[wc^>il^]†Kk ^]±mhg 7£   £[oK†[^$ilYª~yqpq²\/oKilkoy µk ^ †[^ †[orkc^6mhg φ(a) £[orw φ £jkcY[^ ‚orkc^6kcY‘~ykn¡   ¥ojƒ]~yq²€(ornp†Kk~rDi icoj∈ƒnœD~9kc^> k o_kcY‘npi‚joK\ ~ynp†¤¬ a ∈ A o o d(D) = a b°†ªƒ^k Y[^ ‘o9¦ LV Nnœi'srnpr^ †±np†5~›m(o>ž £hn¡knpi'^>~ricg_kco_ƒ or\/€[ujk ^°kcY[^°k n¡\/^$~y†‘€(oBiln¡kcnpor†5~yk'¦Y[nœƒ Y5n¡knp†Kkc^]w ic^]ƒk i kcY‘^m(oru[†ª[~yw g±oy  D £‘n¡ ^ r^]w]¬'XY[^€(oBicicnpDm[n¡qpn¡kšg± ¥orw^]~Kƒ Y   ~Kƒ^k‚k om(^^ †‘ƒoKu[†Kk ^ w ^] mhg±k Y[^ ‘o9¦ j^ €(^]†‘[i‚u‘†[nptBu[^]q¡g or†kcY[^ €(oBiln¡kcnpor†or ‚kcY‘^5 ¥ojƒ ~rq'€(ornp†Kk . {x | x = θ } ¥w ^]ic€¤¬ {x | x = θ } ›ƒ]~y†m(^ƒw oKi ic^] nŸ |~r†‘ oK†[q¡g n¡  ¥w ^]ic€¤¬ φ > θ 8¬ D ƒ]ƒorw„jnp†[s±k o5k Y[nœi6orm‘ic^ w 9~9k n¡oK†¤£U¦^ j^ †[orkc^ I (a) = {i ∈ {1 · · · n}| φ > θ } £ φ <θ ~y†ª ¬ XY[^]†¤I£ I (a)(a)= ={i I∈ {1(a)· · ·∪n}|I φ(a)< npθikcY[^}ic^k‚oy ^>icƒ]~y€[np†[s/jnpw ^]ƒ8k n¡oK†‘ioy  D ¬ D qœico‘£hnŸk¦npqpqUmª^|€[w„~rƒkcnœƒ ~rq‘kco_€(ornp†Kkorujk'kcY[^|qpo9¦^ w~y†‘u‘€[€(^ w'kcY[w ^]icY[oKqp[im(oru‘†‘jnp†[s_~m(o¢žU£BkcYª~y†[«jiNk okcY‘^|€‘~ynpw ioy   ¥u[†‘ƒkcnpor†‘i θ : A → Θ £ θ (a) = θ ~r†‘ θ (a) = θ ¬CXY[^]ic^' ¥u[†‘ƒ8k n¡oK†‘i¦n¡qpq‘ƒqp^]~yw qpg$or†‘q¡gm(^u‘il^  ¥u[q[†[ork ~9k n¡oK†‘i]£ ~y†ª5m[wcnp†[s/†[o_ ¥u‘wlk Y[^ w‚n¡†[ µoKwc\ ~ykcnpor† ~rm(orujkk Y[^icghilkc^]\¬ Y[^]†nŸk‚nœiu[†ª~y\m[npsru[oKu‘i]£h¦'^6¦npq¡q¤oK\_n¡kkcY[^j^]€(^ †‘j^]†‘ƒ^$or† a np†5k Y[^ic^]tBu[^ q@¬ ‚o9¦$£hn¡†5^]~rƒ„Y±[n¡w ^]ƒkcnpor† i ∈ I kcY[^°kcnp\/^|~yk'¦Y[nœƒ Y x(t) ^ †‘ƒ oru[†Bkc^ w„iGk Y[^6ƒorw w ^]ic€(or†‘jnp†[sYBgh€(^ w €[qp~r†[^r£K ¥oKw x ∈ D £ ƒ ~r† ^]~ricnpq¡gm(^ilY[o9¦†5kco/m(^ .    ‰ φ − θ (a) φ − θ (a) −1 ln min , . τ (x) = γ φ −x φ −x X~y«hnp†[skcY[^_\/np†[n¡\u‘\ τ (x) = min τ (x) £¤~y†‘ w ^ np†9xz^]ƒ8k n¡†‘s5nŸk6np†E^]tBu‘~ykcnpor† LV £²¦^›sr^ k°kcY[^_^žjn¡kcnp†[s5€(ornp†Kk6oy  ¦Y[^]† ilk ~rwlk n¡†[s±~9k ¬fhnp†‘ƒ^kcY‘npi€[w ojƒ^>icinœi‚np†Kk ^ †‘[^]kco±m(^w ^ €(^]~ykc^>~rq¡oK†[s/kcw„~9xz^]ƒ8k orw n¡^>i £ ¦n¡qpq+sr^]†[^ w„~yqpq¡g±qpnp^ D or†/kcY[^m(oru[†‘[~rwcgor (k Y[x^ƒ u[wcw ^ †BkGm(o>ž²£K^ž[ƒ^]€jkG ¥orwGkcY[^‚np†[n¡kcnœ~yq(ƒor†‘[nŸk n¡oK†¤£K¦Y[npƒ„Y \ ~>g›Y[o9¦^ K^ wGm(x^ƒ Y[oBil^]†/or† ~6  ~rƒ^ k ¦n¡kcY[oKujk‚q¡oBicioy sK^ †[^]w ~rq¡n¡kzgr¬ ^6kcY‘^ †sr^ k‚~ H$\ #4-„$  T : ∂D → ∂D . r. s. s. a. r. a. a. i. ai −1 i − out. i. + out. out. ± i. − out. i. ai −1 i. i. ai i. ai i + out. i. ai −1 i. i. i. ai i. a. ai −1 i. − i. + i. ai i. out. a. − i. i. i. i. i∈Iout. i. + i. i. i. i. i. i. a. a. Tax. a. a. S . = ϕ (x, τ (x)) = φ + α(x)(x − φ).. ¦Y[^]wc^ α(x) = exp(−τ (x)Γ) ¬GXY[^$qœ~9klk ^ w‚[^ €(^ †‘‘ioK† a £‘~Kiil^]^ †5 ¥wcoK\ @‰ 8¬ ‚o9¦$£hkcY‘^6np†[nŸk nœ~yq¤icgjizk ^ \ l‹3 \ ~>gm(^]^ † w ^]ju‘ƒ ^]k o ~_jnœicƒ wc^ kc^|kcnp\_^jgh†‘~y\/nœƒ ~rq²icghilk ^ \£[~ri'\/^]†Kkcnpor†‘^]5^]~rwcqpnp^ w>¬+a­k ƒoK†‘icnpilk i'n¡† nŸk ^ w„~9kc^>ior +~›sKq¡oKm‘~yq(\ ~y€ or† ~_jor\ ~ynp† ¬ D i~rq¡w ^]~Kjg_\/^]†KkcnpoK†[^]²£ \u‘izkm(^6ƒ or†Bk ~ynp†[^> np†Tk Y[^_ic^k|oy  n − 1 [or\ ~ynp†‘ioy  D ¬6Tfhnp†‘ƒ ^ T Y‘~ri‚k o±Dom m(^›n¡kc^ w„~9k ^]ToK†TkcY[nœi|joK\ ~ynp†¤£U~yqpqDom €(ornp†Kk i°¦Y[nœƒ„Y w ^]~Kƒ YEiloK\/^ jn¡\/^ †ªilnpor†‘~rq  orwqp^]i i Njor\ ~rn¡†±~y µkc^ w~y€[€‘q¡ghnp†[s ~$³‘†[n¡kc^|†Bu[\m(^]w'oy ²k n¡\/^>i £h\u‘ilk'm(^°^ž[ƒqpu‘j^>/ ¥w or\ ¬ n−2 `‘u[wlk Y[^ w \/orw ^r£B¦Y[^ w ^]~ri T npi'wc^]~Kjnpq¡g j^³‘†‘^]±or†kcw„~y†‘icT€‘~yw ^ †BkN¦~yqpqpi]£Bm[qœ~rƒ„« ~y†‘¦Y[n¡kc^|¦'~rq¡qœi~rwc^\uªƒ Y5q¡^>icioKDom mBhn¡oKu‘i kco j^>~yq¦nŸk Y¤¬ S^5k YBu‘i icu[€[€(oKic^± ¥w or\ †[o9¦or† k Y‘~9k Dom ^]†Kkcnpw ^ qpg ƒor†ªilnœizk„i›oy kcw„~y†‘ic€‘~yw ^ †Bk›¦'~rq¡qœi]¬ D \/orw ^ j^ k ~ynpqp^]5jnœi ƒu‘i ilnpor† ~ym(oKujk'kcY[^|k or€(oKq¡oKsrg/oy k Y[nœijor\ ~ynp† ƒ ~r†±m(^| ¥oru‘†‘5n¡@† O QP­¬ D ilnp\/€[q¡^$ƒw n¡kc^ w npor†kco/^ †‘icu[w ^°kcY‘~yk T. T. s. T. T. 465a‹74 8.

(43)  

(44)    

(45) !"# %$&'() *+ ,-. S. ~yqpq²¦'~rq¡qœi~yw ^°kcw„~y†‘ic€‘~rwc^]†KknpikcY[^~ym‘ic^ †ªƒ^6oy C~yujk oy½­wc^]sru[qœ~9kcnpor†¤£hn¡† kcY[^ic^ †‘ic^|kcY‘~yk†[o/€[w ojjuªƒ8kcnpor†5kc^ w \ κ j^]€(^ †‘[i or† x ¬ D k|^>~rƒ„Y €(ornp†Kk6oy ilu‘ƒ„YS~±joK\ ~ynp†¤£(n¡k6npi°€(oBicicn¡m‘q¡^k o ƒor†ªilnœizk ^ †Bkcqpg~r€[€[qpgT~u‘†[nptBu[^_q¡ojƒ]~yq+\ ~y€ ¬ D ƒ8k u‘~yqpqpgr£U^>~rƒ„Y ¦'~rq¡qnpi‚^]nŸk Y[^ wor† k Y[^m(oKu[†‘[~rwcg±oy CkcY[^¦Y[orqp^joK\/~rn¡† S D £(~y†‘ kcYBuªi‚oy G~ ilnp†[sKq¡^$w ^ sKu[qœ~yw‚jTor\ ~ynp† D £ªorwƒ ~y† m(^¦wcn¡klk ^ †E~ri~y† n¡†Bkc^ w„ic^]ƒ8k n¡oK† ∂D ∩ ∂D oy Ckz¦o±wc^]sru[qœ~ywjor\ ~rn¡†Tm(oru‘†‘[~yw np^]i]¬'a­k°nœi°ƒqp^]~yw‚kcY‘~yk°or†k w ~r†‘il€ª~yw ^ †Kk ¦'~rq¡qœi'kcY[^]wc^6nœi‚~yqp¦'~>gji'^ žj~Kƒ8k q¡goK†[^6oy +k Y[^6kš¦'o jor\ ~rn¡†‘i]£[i ~>g D £‘ilu‘ƒ„Y5k Y‘~9k T nœi†[orkkcY‘^$npj^]†Kk nŸkzgr¬ D €[€‘q¡ghnp†[s T nœi|kcY[^]† ƒ or†‘icnœizk ^ †Kk$¦n¡kcY kcY‘^/oKwcnp^ †Bk ~9k n¡oK†Soy kcY[^ ‘o9¦)qpn¡†‘^]i]¬ N^k]£+¦^/oK†[q¡gSƒ ~yw ^_Y[^ w ^/¦nŸk YS ¥oKwc¦'~rw  kcw„~9xz^]ƒ8k orw n¡^>i £ icn¡†‘ƒ ^ T npi†[oyk‚€[w or€(^ w qpgj^³ª†[^] or†5k Y[^6 ¥u[qpqjoK\/~rnp† Dom ¬ fhu[€‘€ªoBil^ †[o9¦ k Y‘~9k k Y[^ w ^ npi~ ¦'~yqpq W ⊂ Dom £6~y†‘e~ ic^]tBu[^]†‘ƒ^ a . . . a or w ^ sru‘qp~rw jor\ ~ynp†‘i icu‘ƒ„YkcY‘~yk ¬Go¢¦$£[n¡k‚npikc^]\/€jkcnp†[s/kcoizk u‘jgk Y[^jgh†‘~r\_nœƒ ioy +kcY‘npin¡kc^]w ~ykc^>±\ ~r€ or† ¬ (T ◦ T aš† €‘~ywckcnœƒu[qœ~yw>·£r·~y· †h◦gT³[žj)(W ^>_€()ornp∩†KkNWoy U6=kcY[∅nœiG\ ~y€ j^ kc^]wc\/np†[^]iN~6€(^ w n¡oj[npƒ'kcw„~9xz^]ƒ8k orw gor (kcY[^ƒ or†Kk n¡†hu[oruªiCkcnp\/^ilgjilkc^]\ lW3‹ 8¬   ,.56+5

(46) '„  !5

(47) '

(48) !7 „    &5  aš† oKw j^]wCk o~y€[€‘q¡gk Y[^ oKwc^]\ ‹k o~y† n¡kc^ w„~9k ^or (k Y[^‚kcw„~y†‘icnŸk n¡oK†/\ ~y€¤£B~KiG[^³‘†[^>/np† €[w ^ hnporu‘iGic^]ƒ8k npor†¤£Kn¡kNnœiG†[^>ƒ^]i i ~yw g kcoSƒ„Y[^]ƒ„«S¦Y‘^kcY‘^ w›nŸk›nœi\/or†[orkcor†‘^~r†‘ ƒ or†‘ƒ ~¢r^K£+np† k Y[^5ic^ †‘ic^oy ‚€[wcoK€(^ wckcnp^]i (M ) ~y†‘ (C) ¬ aš† k Y[nœi›il^>ƒ8k n¡oK† nŸk nœiicY[o9¦† kcY‘~ykkcY[^>il^±ƒ or†‘jn¡kcnpor†ªiƒ ~r†[†[oykmª^±i ~9k npil³‘^> mBgEk Y[^ kcw„~y†‘icn¡kcnpor† \ ~y€ oK† ~¦Y[oKq¡^¦'~yqpq¯£C~y†‘ kcY‘~yk›iloK\/^ w ^³‘†[^]\_^]†KkilY‘oru[qœ5m(^$joK†[^r¬GX+o›k Y[nœi‚~ynp\ £[or†[^$icY[oKu[qp ƒ or\/€[ujk ^|k Y[^xl~rƒoKm[nœ~y†5or ~/ƒ or\/€(oKicn¡kc^|\ ~r€¤£jor +kcY[^6 ¥oKwc\ . ¬GXY[nœiw ^]tBu[npw ^]ikcY‘^ƒor\/€[ujk„~9k n¡oK† oy C~y†~rwcm[n¡k w ~rwcg ¬ D(T ◦ T %¤^ k W m(^T~r†3·~r· wc·m‘◦nŸk Tw ~rwc)g ¦~yqpq¯¬b°†[^Y‘~ri ic^ ^]† np†3€[w ^ hnporu‘i/ic^]ƒkcnpor† kcY‘~yk TDT= T £N ¥orwicor\/^u[†[nœtKu‘^ qpgj^ ³‘†[^] ¬ %^k›\/orw ^ o9r^ w x ∈ W icu‘ƒ„Y k Y‘~9kk Y[^ w ^^ žjnpilk i›~Tu[†[nœtBu[^ s ∈ I (a) icu‘ƒ Y kcY‘~yk τ (x) = τ (x) £~r†‘ j^]†[oyk ^ a k Y[^|¦'~rq¡q²icu‘ƒ Y±k Y‘~9k Tx ∈ W ¬GXY[^ †±k Y[^ w ^°^ žjnpilk i~y†±oK€(^ †±†‘^ npsrYBm(oKu[wcY‘ohoj oy  x np†±¦Y‘npƒ„Y±kcY[^ W ⊂ {x | x = θ }  ¥orqpqpo¢¦np†[s/Y[orqœ[i . Œ T x = φ + (x − φ ) α (x), ¦Y[^]wc^ T nœi/kcY[^ ikcY ƒ ohorw„jnp†‘~9k ^5 ¥u[†‘ƒ8k n¡oK† oy  T ¬X+o ~ym[m‘wc^ hnœ~9k ^ƒoK\/€[ujk ~ykcnpor†‘i]£¦^TY‘~¢r^Tj^]†[oyk ^] α kcY‘^ ikcY jnœ~ysKor†‘~rq(^]†Kk wcg±oy +k Y[^jnœ~ysroK†‘~yqU\ ~9k wcn¡ž α(x) j^³‘†[^> n¡†€[w ^ hn¡oKu‘iic^]ƒkcnpor†7. i. i. a. a. a. a. a. b. a. a. −1. T. 1. T. a`. `. a1. a`−1. a`. a. a1. a`−1. ai. a. 0. s. out. 0. s. a i. i. a i. i. s. i. i. i. αi (x) =. . φas − θs φas − xs.  γγi. Q . s. .. XY[^]† n¡k‚npiw ~ykcY[^]wilkcw„~ynpsrYBkl ¥orw ¦'~yw kco ƒ or\/€[ujk ^|k Y[^$€‘~ywckcnœ~yq¤j^ w np9~9k npr^]i~9k~_€(ornp†Kk x .  nŸ  i = j  (α (x))    l‹]Š ∂T nŸ  j = s γ φ −x = − (α (x))  ∂x   oykcY‘^ wc¦nœic^ .  γ φ −x 0 a­k~y€[€(^>~yw„i ¥wcoK\ z‹>Š k Y‘~9k6kcY‘^_€ª~ywckcnœ~yqj^ w n¡y~9k n¡r^_~rwc^_oy 'ƒoK†‘ilk ~y†Bk$ilnpsr†Sn¡ z£~r†‘Eor†[qpgTn¡ z£ sign (φ − x ) nœi°³[žj^]²¬ D ƒ8kcuª~yqpq¡gK£h~yqpq(j^]srw„~r[~ykcnpor†/w„~9k ^]i γ ~rwc^‚€(oBiln¡kcnpr^r£h~y†ª α (x) = exp(−γ τ (x)) ∈ (0, 1) icnp†‘ƒ^ τ (x) nœiN†‘or†j½­†[^ sB~9k n¡r^K¬ o¢¦ \ ~y«r^kz¦ooKm‘il^]wcy~9kcnpor†‘iCoK†_k Y[^icn¡sK†/€‘~yklkc^]wc† oy  φ − x np† sr^]†[^ w„~yq@¬+XY[^‚³‘w ilkNor†[^ƒoK†‘ƒ^]wc†‘in¡k iNn¡†h9~yw nœ~y†‘ƒ^ ¦n¡kcYwc^>il€(^>ƒ8kkco/kcnp\/^$^ rorqpujk n¡oK7† . $ – »&%N» ” ' • ' »)( d7f 7H- lk)Hu(H      r  ()

(49) $Z j {i_k\$"lk)v\$ \_L sign (φ − x ) $ H<(H { j * γi γs. s. i. j. i. s. a i a s. i. s. s. γi γs. a i. i. s. i. s. a. i i lk  ! r _Zlk) \ _k\$ sign (φa − x ) = sign (φa − ϕ (x, t)) $ * t > 0 o i i i i q $

(50) J$ r _(

(51) ‚lk‚!M$ _ W

(52) k+  $  _k\ j ^EI# \_yE„19lky# 9 

(53)  # 9_

(54) \ F _ $ k*J$ ()19lka lkG$ j  {+„ j {+$ L 4*J  a a {x j x) = sign (φj − θj ).  _-G| x_Jj$ =* rθj}r$ * x !sk

(55) k4 

(56) $  ){ $Ulksign !M$l (φWj 0 −⊂T{x [k$ sign (φa − x ) = sign (φa − θ ). | x s = θs } r s s s s a. D ic^]ƒ or†‘±  ~rƒk‚ƒ or†‘ƒ ^ w †‘ik Y[^$n¡† ªu[^ †‘ƒ ^$oy +kcY‘^jnŸ¨U^ w ^ †BkƒohoKw jnp†‘~ykc^]i'or† ¬ sign (φa − x) $ – &» %N» ”' •' »)(

(57) f € >$\* 

(58) $#yEG _H

(59) L  i ∈ I (a) r k$+ >__k\ x < θ+ (a) < φa r   φa < θ− (a) < x r i i i $ _ j lk

(60) $+  sign (φa − x ) # ‚

(61) \ ^$ \a_K_k\>out !sk\J)

(62) X+ H c`(D ) 19 _k\ i I-%$_ o iq  θ ∈i{θ− (a), + θ (a)} i a i i i i # i_k\ 

(63) $ (_TE lkyH HkX r lky#  E Av\$ sign (φa − θ ) o i. ‹€‹Œ ŽR‘1’. i.

(64) Œ. €$H

(65) 9FwRm. €  _k\i4 

(66) $ (_yE _H

(67) L   t_k\lk>k\$( r θi+ (a) o. sign (φai − xi ). # 

(68)   ^$ \. Wr. _(

(69) . θi− (a) < φai <. XYhu‘i]£[ƒ or†‘jn¡k n¡oK† (M ) ƒ ~y†‘†[oykm(^$9~yqpnœ±oK†5kcY[^$¦Y[oKq¡^6¦~yqpq W £[u[†[qp^]i i~rq¡q¤[n¡w ^]ƒkcnpor†‘i~yw ^|^>icƒ]~y€[np†[s‘¬ fjn¡†‘ƒ ^6~_jn¡¨(^]wc^]†Kk np~rqUoy k Y[^° ¥oKwc\ nœi~›€[w ojju‘ƒkoy jnŸ¨U^ w ^ †Bkcnœ~yqœi'oy +qpojƒ ~rq(\ ~y€‘i £ ~y†ªEicn¡†‘ƒ ^k Y[^_qœ~9kckc^]w|\ ~¢goK†[qpgm(^_€(D(T oBiln¡kcnpr^›◦ npT† ~±w ^ · sK· n¡·oK◦† T¦Y[)^ w ^k Y[^_r^>ƒ8kcoKw φ − x Y‘~ri6~³[žj^>Eicnpsr†E€‘~yklD(T kc^]wc†£²)nŸk ic^ ^ \ iw ^ qp^ y~y†Kkkco wc^>izk wcnœƒ8kkcY[^ilkcuªjgkcoilu‘ƒ„Y w ^ sKn¡oK†‘i ¬ ‚o9¦$£rƒoK†‘jn¡kcnpor† ƒor†ªƒ^ w †‘i+kcY[^9~yw nœ~9kcnpor†ªioy ªkcY[^kc^ w \ inp† z‹>Š ¬ D iicY[o9¦†›mBg€[w or€(oBiln¡kcnpor†±‹r£9k Y[^ic^]ƒ or†‘h½­orw„j^ w j^]wcnp9~ykcnpr^]i~yw ^|np(C) †j ¥orw \ ~9k n¡r^6np†5k Y[npi‚ƒ]~ric^r¬XY[^]g±ƒ]~y† m(^6^ žj€[wc^>icic^]5~Ki .   n¡  i = j, m = s oKw j = s, m = i γ (α (x))     γ φ −x  l‹r3‹ ∂ T   = ¡ n   γ γ φ −x  ∂x ∂x m=j=s 1+ (α (x))   γ γ (φ − x )   y o k [ Y ^ w ¦npic^ .  0 ‚^]wc^|~rsK~ynp†¤£KkcY[^>il^6j^ w n¡y~9kcnpr^]i~rwc^or ¤³[žj^]±icn¡sK†or†‘q¡g/n¡  φ − x Yª~ri'~³[žj^]±icn¡sK†€‘~yklkc^]wc†¬ ^ †‘ƒ ^r£hm(orkcY³‘w„ilk'orw„j^ w ~y†ª ic^]ƒoK†‘5orw„j^]wj^]wcnp9~ykcnpr^]i~rwc^6or CƒoK†‘ilk ~y†Bk‚icn¡sK† np† kcY[^ic~r\_^6w ^ sKn¡oK†‘i]¬Nb°†[^6nœik YBu‘i‚†‘~9k u[w ~rq¡qpgqp^]5kcoƒ or†‘icnpj^]w w ^]ƒ8k„~y†[sKu[qœ~ywGwc^]srnpor†‘iG€‘~ywckcn¡kcnpor†[np†[s$k Y[^¦'~rq¡q W £h[^³‘†[^> mBg_kcY[^|ilnpsr† or  φ − x £BoKwNnp†oyk Y[^ wN¦'orw„[iGmBg_kcY[^°€(oKicnŸk n¡oK† ¥np† kc^]wc\ ior +kcY[^$€ª~ywckcnœ~yq²orw„j^ w Noy €(oKn¡†Bk i¦nŸk Ywc^>il€(^>ƒ8kkco/kcY[^6 ¥ojƒ]~yq²€(ornp†Kk>¬ a­k¦npq¡qUm(^|icY[o9¦†np†k Y[^° ¥orqpq¡o9¦np†[s›ic^]ƒ8k n¡oK†‘iNkcY‘~yk'np†k Y[^6il€(^>ƒnœ~yqUƒ ~Kil^or †[^ sB~9kcnpr^‚ ¥^]^]jm‘~Kƒ « qpohor€±icgjizk ^ \ i £hor†[qpg or†‘^/icu‘ƒ Y wc^]srnpor†Sƒ or†Kk„~ynp†‘i°k Y[^/qpor†[sr½¯k ^ w \ [gB†ª~y\/npƒ]i £¤or† ^]~rƒ„YS¦'~rq¡qoy ~5€(^ w n¡o[jnpƒ_ic^]tBu[^]†‘ƒ^_or 'joK\ ~ynp†‘i|n¡† ilk ~ykc^ ic€‘~rƒ ^r¬C`'^ ¥oKwc^€[w o¢hnp†[s$k Y[npi]£hqp^k'u‘i'[npi ƒu‘i i~›ilqpnpsrYKk qpg_\/oKwc^sr^]†[^ w„~yq(ƒ ~Kil^‚kcYª~y† ¥^ ^][m‘~rƒ„«/q¡ojor€±ilgjilkc^]\/i]¬Gb°u[w'Y[or€(^ Y[^]wc^$npik Y‘~9kk Y[nœi‚jnœicƒ u‘i ilnpor† srnpr^$icor\/^6np†Kk u[nŸk npor†or +k Y[^$m‘~Kilnœƒ°  ~Kƒ8k ikcY‘~yk‚¦n¡qpqic^ w r^$~9 µkc^]wc¦~yw„[i ¬a­k~rqpico/icu[sKsr^]ilk i kcYª~9k'kcohoKqpiN ¥wcoK\ €(oBiln¡kcnpr^°or€(^ w„~9k orw„ikcY[^]orw g_\/npsrYBk'm(^|^§ƒnp^ †Bkn¡† ~m[wcoB~rj^]wƒ or†Kk ^žhkk Y‘~y†±kcY[^6icorqp^‚ ¥^ ^>jm‘~Kƒ « q¡ojor€ icghilkc^]\ i ¬ ~r\/^ qpgr£(icu[€[€(oKic^srnpr^ † ~ €(^ w npojjnœƒ$ic^]tBu[^]†‘ƒ^oy Njor\ ~ynp†‘i £(icu‘ƒ YTkcY‘~ykkcY[^]wc^^žjnpilk i°~ u[†[nœtKu‘^ ^]i ƒ ~r€[np†[s6jnpw ^]ƒ8k n¡oK†› ¥orw^>~rƒ„Y_m(o>ž a ¬CXY[npiGƒ]~ric^'nœiG~rƒ8k u‘~yqpqpg$kcaY‘^‚, .il.np\/. a€[qp,^]aizkGoK†[=^¦aY[^]†/ilkcuªjghn¡†[s$€(^ w npojjnœƒoKwcm[n¡k i]£ricnp†‘ƒ^ n¡ Ck Y[^ w ^~yw ^il^]r^ w„~yq¤^]i ƒ ~r€[np†[sjnpwc^>ƒ8kcnpor†ªi ¥oKw‚oK†[^m(o>žn¡†TkcY[^›ic^]tBu[^ †ªƒ^r£‘kcY[^]†EiloK\_^$kcw„~9xz^]ƒkcorw np^]i^]i ƒ ~y€(^$kcY[^›ƒgjƒ q¡^ ~9k|k Y[npi6m(o¢žU¬XY‘^/il^ k6oy €(ornp†Kk i6wc^]\/~rn¡†‘n¡†[s5np†EkcY‘^_€(^ w npojjnœƒ›il^>tBu[^ †‘ƒ ^ npi|†‘oyk6^]~KilnpqpgTj^ ³‘†[^]Sn¡†SsK^ †[^]w ~rq¯£ kcY‘oru[sKY5u‘†‘j^ wkcY‘^$u[†[n¡ µoKwc\ j^>ƒ ~¢gw„~9kc^>i~Kicicu[\/€jkcnpoK†5n¡k‚\ ~>g±m(^6^žj€[w ^]i aic^]±. .mB.g5a ~_qpnœizk‚oy C~y§/†‘^6n¡†‘^]tBu‘~yqpn¡kcnp^]i O Œ[\£ Q+P­¬ XY[^]†¤£j^]~Kƒ Y5¦'~rq¡q²wc^>~rƒ„Y[^]mhg±iluªƒ ƒ^>icicnpr^°nŸk ^ w„~9k ^]i'oy k Y[^|kcw„~y†‘icn¡kcnpor†±\ ~r€5Y‘~rik o_m(^6€‘~ywckcn¡kcnpoK†[^]±n¡†Bkco w ^ srnpor†ªi np†Tsr^ †‘^ w„~yq@¬ D ƒ8k u‘~yqpqpgr£ªor†Ticu‘ƒ„Y ~/¦'~rq¡q€(oKn¡†Bk i‚Y‘~¢r^ n − 1 ƒojorw„jn¡†ª~9kc^>i¦Y[nœƒ Y ~rwc^$†[ork°ƒor†‘ilk„~y†K3k .Nnp†Tor2†[^jnpwc^>ƒ8kcnpor† n¡k6^]tBu‘~rqpi|~±kcY[w ^]icY[oKqp y~yqpu[^r¬ D \/or†[s5kcY[^>il^ n − 1 £¤or†[^›^ ž[~rƒ8k qpgnœi°kcY[^_^>icƒ]~y€[np†[s jnpwc^>ƒ8kcnpor† s or GkcY‘^_€[w ^ hn¡oKu‘i m(o¢žU£ a ¬GXYhu‘i‚~rƒ ƒ orw„jnp†[sk o/€[w or€(oKicn¡kcnpor†h£jk Y[^6 ¥orqpq¡o9¦np†[s_nœi³[žj^] a1. a`−1. a`. ai. a. 2. m. i. i. s. s. s. γi γs. s. i. j. i. a i a s. i. s. s. i. s. 2. γi γs. s. a. i. a. 1. `. `+1. 1. 1. `. i. n−2. i−1. i−1. i. i. sign (φasi−1 − xsi−1 ) = sign (φasi−1 − θsmi−1 ),.  ¥orw‚icor\/^ m £[~r†‘ ~y†hg€(ornp†Kk x oK†~›kcw„~9xz^]ƒ8k orw gƒw oKi icn¡†[s›kcY[^ƒ or†‘icnp[^ w ^]5ic^]tBu[^ †‘ƒ ^6oy CjoK\/~rnp†‘i ¬ XYBuªin¡ Cor†[^~ri icu[\/^]i £jk Y[^ † jnpw ^]ƒ8k npor†‘iwc^]\/~rn¡†5 ¥oKw¦Y‘npƒ„Y kz¦oilnpsr†‘i‚~yw ^6€(oKi ilnpm[qp^r¬ D k^>~rƒ„Y ¦'~rq¡q@£ kcY‘^ np\ ~ysK^]i6or 'kcY[^>il^ 2s 6= sw ^ srnpor† \ ~>gSn n¡−†Bkc2^ w„ic^]ƒ8k›il^]r^ w„~yqwc^]srnpor†‘i$~r\_oK†[s kcY‘^ 2 €‘~ywckcn¡kcnpor†[np†[skcY[^†[^ žBk¦'~rq¡q@¬ ‚^]†‘ƒ^K£hoK†[^°\ ~>g wc^ ³‘†[^k Y[^|€‘~ywckcn¡k n¡oK†or ¤kcY[^|qœ~9kckc^]w]£j~y†ªnŸk ^ w„~9k n¡†[skcY‘npi'€‘wcojƒ ^]i i £Bormjk„~ynp†5~€‘~ywckcn¡kcnpoK†or +~¦'~yqpq(np†Kk o w ^ srnpor†ªiCicu‘ƒ„Y_k Y‘~9k~yqpq[€ª~ywckcnœ~yq‘j^ w np9~9k n¡r^>iCoy Un¡kc^]w ~ykc^>i £roy (k Y[^ ¥orw \ £r ¥orw k ∈ {1 · · · `} ~yw ^or (³[žj^] ilnpsr†¤£B~y†‘›k Y[^ ic^]ƒoK†‘ orw„j^ w|j^]wcnp9~ykcnpr^]i°~Ki°¦^ qpq@¬XY[nœi°€‘wcojƒ ^]i i¦oru[qœ qp^]~KTkco ƒ or†‘icnœj^ w6~±Y[npsrYE†hu[\m(^ w6oy wc^]srnpor†‘inp†EsK^ †[^]w ~rq¯£ ^]~Kƒ Y or ¦Y[nœƒ„Y \ ~¢g ic~ykcnœiz ¥gEkcY‘^YBgh€(oyk Y[^]ic^]ioy 6‹r¬XY[nœiil^]^ \ i6  ~rw$ ¥w or\kcw nphnp~rqGn¡† k Y[^\/oKilksK^ †[^]w ~rqGƒ ~Kil^K£+m[ujk ¦oKu[qpƒ^ wck ~rn¡†[qpgm(^6 ¥wcu[n¡kl ¥u‘q¤np†TiloK\_^6\/orw ^|€ª~ywckcnœƒu[qœ~ywƒ]~ric^]i]¬ ‚^]†‘ƒ^K£‘q¡^ k‚u‘i‚q¡ohoK«  ¥orwicu‘ƒ YTic€(^]ƒ np~rq¤ƒ]~ric^]ior ^>tBu‘~9k n¡oK†‘ i z3‹ <¬ `‘orwkš¦oicu‘ƒ ƒ ^]i ilnpr^6m(o¢žh^>i a ~y†‘ a £‘or†[^$Y‘~ri]£‘~Ki ic^ ^ †5 ¥w or\  Œ . z‹¢ − x ) α (x), + (φ φ −T x=φ −φ ~y†ª5n¡ CkcY[^$ ¥ojƒ ~rq¤€(ornp†Kk i~yw ^ ~rq¡npsr†‘^] œ£[n¡†kcY[^ic^ †‘ic^6kcY‘~yk £[kcY[^ilnpsK† oy k Y[^tKuª~y†Kk nŸkzg wc^]\ ~ynp†‘i u[†‘ƒ„Y‘~y†‘sr^] ~9 µk ^ w|~y€[€‘q¡ghnp†[s T ¬ D ƒkcu‘~rq¡qpg oK†[^›~riic^ ^]† ~rq¡w ^]φ~Kjg±=kcY‘φ~yk α (x) ∈ (0, 1) £U~r†‘ ~y€[€[qpghnp†[s φ T−np\/x€[qpn¡^>i‚k Y‘~9k i n−2. i−1. n−2. ∂Tik ∂xj. i−1. ai j. j. ai j. ai−1 j. ai−1 j. a j. i. a j. j. i. j. i−1. j. j. s. 465a‹74 8.

(70)  

(71)    

(72) !"# %$&'() *+ ,-. Q. kcY‘^‚ ¥ojƒ ~rq‘€(ornp†Kk\u‘izk'mª^w ^ €[qœ~rƒ ^]_mhg›kcY‘~ykoy Uk Y[^‚ ¥orqpq¡o9¦np†[s$m(o¢ž²¬CXY[npiNsK^ or\/^ kcw npƒƒ or†‘jn¡kcnpor†icY[oKu[qp_k YBu‘i'~>roKnp/u‘i kco/wc^ ³‘†[^°kcY‘^6€‘~ywckcn¡kcnpor†‘i'or +icu‘ƒ]ƒ^]i icn¡r^°¦~yqpqpi]£j~ri'\/^ †Bkcnpor†[^>±np†±k Y[^$~ym(o9r^|jnœicƒ u‘i ilnpor†¤£jicnp†‘ƒ^|n¡knp\_€(oBil^>ikcY‘~yk'k Y[^]ic^ w ^ srnpor†ªi'[o/†[oyk‚o¢r^]wcqœ~y€± ¥wcoK\oK†[^6¦'~rq¡qUk o_k Y[^6 ¥orqpq¡o9¦np†[s‘¬ a­kG\/~¢gm(^ilk ~ykc^]›\/orw ^ ¥oKwc\ ~yqpqpgr¬}^]ƒ]~yqpqBk Y‘~9k a wc^  ¥^ w„i+kco|k Y[^ ikcY ƒw oKi il^>m(o>ž²£r~r†‘k Y‘~9k s npinŸk„iCu[†[nœtBu[^^>icƒ]~y€[np†[s jnpwc^>ƒ8k n¡oK†¤¬ D i icu[\/^$~/ilkcw or†[s_r^]w icnpor†5oy +k Y[^$€[wc^]hn¡oKu‘i~yqpnpsr†[\/^ †Bkor + ¥ojƒ ~yq¤€ªoKnp†Kk i . z‹>© ∀i ∈ {1 · · · `}, ∀j ∈ {1 · · · n} \ {s }, φ = φ . XY[nœi~rq¡npsr†‘\_^]†Kk‚~Kicicu[\/€jk n¡oK†5¦npq¡q~rq¡qpo9¦u‘i'k o/u‘ic^|k Y[^ oKwc^]\ ‹6n¡† kcY‘^6w ^ sKn¡oK†5¦Y[^ w ^|kcY[^$qpor†[sr½¯k ^ w \jgh†‘~r\/npƒ]i'k ~y«r^ €[qœ~rƒ ^r¬ `[u‘wlk Y[^ w \/orw ^r£rn¡kN¦npqpq‘m(^°ilY[o9¦†/kcY‘~ykGk Y[^sr^ oK\_^ kcw nœƒ'€‘wcoK€(^ wckš!g l‹]© Gƒ ~y†~yqœico$m(^‚ormjk„~ynp†[^] ~KiG~ƒ or†‘ic^]tBu[^ †ªƒ^ oy C ¥u[wckcY‘^ w°~Kicicu[\/€jk n¡oK†‘ioK† k Y[^np†Kkc^]w ~Kƒ8kcnpoK†Tizk wcuªƒ8kcu[w ^oy CkcY‘^icgjizk ^ \ .n¡ Ck Y[^qp~yklk ^ w°ƒoK†‘ilnœilk i‚n¡†E~ icnp†[srqp^$†‘^ sK~ykcnpr^  ¥^ ^>jm‘~rƒ„«5qpohoK€¤£‘nŸk ¥orqpqpo¢¦‚ik Y‘~9kkcY‘^ wc^^ žjnpilk i~ €(^ w n¡oj[npƒ$ic^]tBu[^]†‘ƒ^oy m(o>žj^>i‚i ~9kcnœil ¥gBnp†[s l‹]© 8£(~Ki¦'^¦n¡qpq+ic^ ^np†kcY[^ †[^ žBkic^]ƒ8k n¡oK†¤¬ i. i. ai j. i+1. &!$+87 9 ±:!

(73) *- /$  ²+‚,. . 901/. ai+1 j. 2435€181ƒ 8 29 3  >5 2‚8,‡8:#2>=. aš†5k Y[nœi‚il^>ƒ8kcnpor†£j¦^6 ¥ojƒu‘ior†icghilk ^ \ ioy +k Y[^6 ¥orw \  dx1   dt   dxi dt. . l‹] . = κ01 + κ11 s− (xn , θn1 ) − γ1 x1 1 = κ0i + κ1i s+ (xi−1 , θi−1 ) − γ i xi ,. i = 2 . . . n.. ša †/ilu‘ƒ„Y_icgjizk ^ \ i]£9^>~rƒ„Yy~yw np~rm[qp^ x ~rƒkcnp9~9k ^]i+kcY[^€‘wcojjuªƒ8kcnpor†›oy ªkcY[^†[^ žhkCy~yw np~rm[qp^r£ x £r^ž[ƒ^]€jk x ¦Y[nœƒ Y_np†[Y[npm[n¡k i ¬ ^ †‘ƒ ^r£(k Y[^ npw|n¡†Bkc^ w„~rƒkcnpor†Eilkcw u‘ƒ8k u[wc^nœikcY‘~yk|oy ~†[^ sB~9k n¡r^ ¥^ ^>jm‘~rƒ„«qpohor€¤¬[Y€ kco ~ kcw„~y†‘icqœ~9k n¡oK†¤£‘kcY‘^ gT~rwc^oy  x kcY‘^° ¥oKwc\ j^]~rqŸk¦nŸk Y mBg5fh†[oKu‘icicnUnp†4Op‹]P­¬aš†±k Y[^$ƒ ~ric^ n = 3 or†[^°³‘†ª[i'kcY[^6¦^]q¡q¡½­«B†‘o¢¦†ZH,(H _X$,[O ] P­¬}^]\ ~yw « †[o9¦ kcY‘~yk‚~y†hg €‘wcoj[u‘ƒ8k n¡oK†k ^ w \ κ nœin¡†‘[^ €(^ †‘[^ †Kk‚oK†±kcY‘^69~rwcnœ~ym[qp^ x £[~r†‘k YBu‘i‚~yqpqU¦~yqpqpi\u‘izk‚m(^|kcw„~y†‘ic€‘~yw ^ †Bk]¬ n¡kcY[oKujk'n¡†Bkcw ojju‘ƒ n¡†‘su[†[†[^>ƒ^>ici ~yw g ¥orw \/~rq¡nœic\ £Kqp^k'u‘iNoK†[qpg/ic~¢gk Y‘~9k'¦^°ƒ ~rqp€q _ ^H$I

(74) L  E H$\kh£ IG nœi~y†orw n¡^]†Kk ^]²£ qœ~ym(^ qp^] sKw ~r€[Y¤£+¦Y[oBil^r^ wckcnœƒ^]iwc^]€[wc^>il^]†Kk9~yw np~rm[qp^]i6or kcY[^5ilgjilkc^]\ £~y†‘ ^][sr^]iw ^ €[w ^]ic^ †Kkn¡†Bkc^ w„~rƒkcnpor†‘im(^kz¦^ ^]† kcY‘^ \¬ `‘orw6^ž[~y\/€[qp^r£ x −→ x n¡ '~r†Enp†‘ƒw ^]~Kil^›oy  x \ ~¢gTq¡^>~r kco~ j^]ƒ wc^>~ric^or  κ ~9k6qp^]~rilk$~9k$icor\/^›€(oKn¡†Bk x np† ilk ~9k ^il€ª~rƒ^K¬±XY[^]†¤£~†[^]sK~ykcnpr^/qpohor€ nœi~Tƒgjƒ q¡^ np† IG ¦n¡kcY ~y† oj[ †hu[\m(^ woy †[^]sK~9k n¡K^/^]jsK^]i]¬a­knpi^]~Kilnpq¡g icY[o¢¦†±k Y‘~9k‚~r†Bg±ilgjilkc^ \¦Y[oBil^|np†Kkc^]w ~Kƒ8k n¡oK†±sKw ~r€[Y5ƒoK†‘icnpilk i'np† ~/ilnp†[srqp^|†[^ sB~9k n¡r^|qpohor€ ¥np†BroKq¡hnp†[s/~yqpqUy~yw np~rm[q¡^>i Gnœi ^]tBu[np9~rq¡^]†Kkk o z ‹ ] £[u[€ kco ~/icgh\/\_^ kcw gnp†ilk ~ykc^$ic€‘~rƒ ^r¬ XY[^'€‘~rwlk nœƒu[qœ~yw+ ¥orw \ oy  z ‹ ] +n¡\/€[qpn¡^>i+k Y‘~9kCkcY[^]wc^'nœior†[qpg$or†[^kcY‘wc^>ilY[oKqp θ np†›^>~rƒ„Y›jnpwc^>ƒ8kcnpoK† i £r~y†ª θ £ θ w ^ €‘wc^>il^ †Bk kcY‘^$mªoKu[†‘[ioy +k Y[^$w ~r†[sr^|or  x £[n¡†T~rƒ]ƒorw„[~r†‘ƒ^°¦n¡kcY @ 8¬ aš†5k Y[^6 ¥orqpqpo¢¦np†[s/¦^$~rq¡¦'~¢gji~ri icu[\/^°k Y‘~9k ~y†‘ κ < γ θ lR‹ V ∀i ∈ {1 · · · n} κ +κ >γ θ XY[nœi‚n¡\/€[qpn¡^>ikcY‘~ykY[npsrY[^]w‚9~rq¡u[^>ioy  ¥ojƒ]~yq¤€(oKn¡†Bk i~yw ^$~rm(o¢r^6kcY[w ^]icY[oKqp[i]£‘~r†‘ qpo¢¦'^ w‚9~yqpu[^>i‚~yw ^6m(^ qpo9¦k Y[w ^]icY[orqœ[i]¬ bkcY‘^ w ¦npic^r£[~yqpqUk w ~9xz^]ƒkcoKwcnp^]i'¦oKu[qœ ƒor†hr^ w sr^°kco ~_u‘†[nptBu[^6³[žj^>5€(ornp†Kk>£‘il^]„^ O¡ ‹ ]Pq¡^]\/\/~‹r¬ `'^  ¥orw ^ ^]†Kkc^]wcnp†[s np†Kk o \/oKwc^Tj^k„~ynpq@£'q¡^ ku‘i or€(^ w„~9k ^~S ¥^]¦ icnp\_€‘q¡n¡ ¥ghn¡†[s oK€(^ w„~9kcnpor†ªi ¬ `+npw„izk>£¦'^Tƒ„Y[oKic^ kco il^ k ¬GXY[^]†¤£ w =x −θ i. i+1. n. 1. i. i. −. i. j. i. j. 1 i. 0 i. 2 i. i. 0 i. j. j. 1 j. 1 i. 1 i i. 0 i. 1 i i. w˙ j = κ0j + κ1j h± (wj−1 , 0) − γj θj1 − γj wj ,. ~y†ª kcYhu‘i$~yqpqCkcY[w ^]icY[orqœE9~yqpu[^>i|~rwc^GW ^ w o‘¬ D ƒ ƒ orw„jnp†[srqpgr£¤~yqpqwc^ sKu[qœ~yw6jor\ ~rn¡†‘i6~rwc^_†[o9¦‡nœj^]†Kkcn¡³‘^>S¦n¡kcY ¥m(oKu[†‘j^> w ^]ƒ8k„~y†[sKu[qœ~yw/w ^ sKn¡oK†‘i/oy  _kcY[^ 2 orwckcY‘~r†Kk„i/or  R ¬ XY[^]g3~yw ^\ ~y€[€(^> ~Ki ^žj€[qœ~ynp†[^]3np† ic^]ƒkcnpor† j¬p‹y£or†Kk o kcY[^ jnœicƒ wc^ kc^ic^k A = {0, 1} ¬‚XY[^›ƒoKwcw ^]ic€(or†‘jnp†[s jnœicƒ wc^ kc^\ ~y€[€‘n¡†[s d : D → A ic^ †‘[i†[^ sB~9k n¡r^ƒohoKw jnp†‘~ykc^]ikco 0 £ ~y†ª5€(oKicnŸk n¡r^6oK†[^]i'k o ¬ XY[^$m(oru[†‘‘ior +kcY[^$¦1Y[oKq¡^$[or\ ~ynp† ~rwc^6†[o¢¦ j^]†[oyk ^] ~y†‘ θ = θ − θ . θ = θ − θ = −θ n. n. n. r. − i. ‹€‹Œ ŽR‘1’. 0 i. 1 i. 1 i. + i. 2 i. 1 i.

(75) ‹]Š. €$H

(76) 9FwRm. b°m‘ic^ w r^°kcY‘~yky~yw np~rm[qp^]i'\/~¢g†[o9¦em(^6†‘^ sK~ykcnpr^r£[~y†‘jo/wc^]€[w ^]ic^ †Kk‚ƒoK†‘ƒ^]†Kkcw„~9k n¡oK†‘i'or†‘q¡g±u[€5k o ~›k w ~r†‘ilqœ~9k n¡oK†¤¬ X Y[^E†[^ ¦ jgh†‘~r\/npƒ]~yq°ilgjilkc^]\ m(^]Y‘~>r^>i nœj^]†Kkcnœƒ ~rq¡qpg kco l‹] 8£‚or†‘ƒ ^TkcY[w ^]icY[orqœ[iY‘~¢r^Tm(^]^ † ic^k±kco W ^ w o‘£~r†‘3kcY[^  ¥orqpqpo¢¦np†[s/wc^]†‘~y\/np†[sKi'Y‘~¢r^6m(^ ^]†joK†[^ . ~y†‘ φ ← φ − θ . κ ←κ −γ θ , XY[^]†¤£B ¥oKw'^]~Kƒ Y i ∈ {1 · · · n} £Bk Y[^° ¥ojƒ]~yqU€ªoKnp†KkƒohoKw [n¡†‘~ykc^ φ \ ~>g or†[qpg/k ~r«r^kš¦'o›y~yqpu[^]i]¬ ^6~ym‘m[wc^]Bnœ~ykc^‚k Y[^ \ ~Ki  ¥orqpqpo¢¦‚i . ~y†ª φ = κ + κ . z‹>‰ κ φ = γ γ b°†[^$icY‘~yqpq¤~rqpico_†[orkc^ ¬ }^]\/~rwc«k Y‘~9k φ npiN†[κ^ sB~9=k n¡rγ^φ~y†‘ φ €(oBiln¡kcnpr

Références

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