Motivations for an arbitrary precision interval arithmetic and the MPFI library
17
0
0
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Motivations for an arbitrary precision interval arithmetic and the MPFI library N. Revol (ANO, U. Lille et Arenaire) and F. Rouillier (Spaces). N° 4498 July 2002. ISSN 0249-6399. ISRN INRIA/RR--4498--FR+ENG. THÈME 2. apport de recherche.
(3)
(4)
(5)
(6) "!#
(7) $
(8) %& ' () *+,- !
(9) ./*0132546( 798;:=<?>A@ABDCFE7GIH;J98LK"MNBOBN<<?P=EQR<TSLUVMOQR<XW UVS Y[Z\8:=@A] MNBNBOMN<TQC_^`LUVaT<cbW d&egjwfixyh\vwj z{r|kDv }~l rm hDniogVpqs?jrtp|rt Ts?}~uTptj vwpqjir ;s{Fjwxnwo~z?pqj yz{gVs?_xXj_jRvegjwvegj3oVTTTl ?}~rqk??cklR%z{u?jR ∗. †. ~£"=_e s¤¦iwp|~Xptj%g_ji ?zTv ?vwx_¡ v pts?}~d&oegz{ptx_p|&j\_~_s¨z{jRs{V}g®jwxrq jwxioFª}~jwptoch¨x_pqx_¢~gjwjRrts&¦?z{£"jirLecv ¤sTptoh¨z?optgoc}Xz?x_xji_z{_~x_¦p¥ptxz{s?ro~z?z? _rq\uTjis?g cp¥}gjixv pteg_p|h¯j¨_pqsTw_°o=h¨z?ptoTrqxjwptog¦gz{}Xr§x z{ptocp¥xx_egjih¨_¦j z{xr|ptiv'«§ptx_&egp|o~jwjijwg¬ji~§rtz?© pqo~x_s h¨d&vws?e~jwh¨x j=~x_±²z?eg_j´p|³ _sTjRµo¨ crt}g£"pqgpqp¥xjiez{¹z9_¢~ejR¬z?v jR pq¢V¼½vwgjwz{_jix_jRo¶ptv s?p|o~gpti}gs?°pto¯rqx%d&ptocptego¶x_j¤ji_s?h¯¦z{Xz{rAjwpto¹ z{x_·pqjis²x_egz{·x_h¨}g}grqj ¢~xjirtptD-v?ªs{x_s?eg&o=p|x_ egz%ogp| jiVjirtp§jigvw©¨p¥¢z?pt_o~v'ºXgjiz{jis?§j¯gªrtgjwoghs¸pqjwªcjw»~pt¬X#9ptguTx_pqptji¦Tog_jwu¸jwo¶ocrtx_pqz{jR~o§ºz?©_p¥z x ¾´_egji¦?z?sjwrA£oc_x#s}~x_s{z{exrtzqTx"¿·ªx__=e~s?jh¨z?soj¯¦?z?jwz{rqguTeggs?jRrtz?p|p¥vwx¤zegxXhpq}~sTjo~v sT x_hDs¨~gz?x_e~pt_ogjijptoghDu¯s?}go¹rqx_±²pthDgrt´Vjs?³ grtµp|_v jRz{v z?p|j%o~ptuTs?pqooc¦T}~jwptyh\o©v jis?_h¨p|s?vwVz?gs{rrtj xv xptsTvijwh\wrt=ª§~p|z?}XsTgxrtz{h\z{x_x_p|ptpts?s?o~optgÀ-rts{j?&z{° o~sTrqz{gog~sT_h\sÁ¬Xp|z{pq¼r h¯z{x_pts?os{_jRz{rA_ss?x&£"pqx_e²z{gpqx_z{¨zTvwvw}gzTv T° ²ÃcÄAÅÆDÇ È ?¡ z?_~p¥xz?_¯gjivwpt_pts?o=ptocx_ji_¦z{rz?_pqx_e~h\jwx_p|v{ªgjwrtp|z{gptrqp xF=z?o~zTvwvw}gzTv iá îÁå{ïFÖOëÉXÓÝÕFÖçÊÚîÁËTèOÖçÕFÌÕ#éêØÞRÍÕLáiÎåÉcÊÙ Ï#éFÚÕFдÒc×_ÐÐÑðÉgÒÁÖOÏÓ ÊÔ ÒÁàÕFéFÖOä®×OÕÍLÓ Ð{Ø®âÙºñ ä·òÎóÚàôwÛ'ò ä®ÉõiÍÓÝöâÜ ÷XÝÉÕ¤øtä·ùwë Ê÷ÒÁúwRÞ5ûÐRõðüߧÖN÷ÎÊýRÒÁþwà éFÿ_áÁÕõ Ðâwûãw_ý~òÎø áiÁÉXö â÷Xä·Î øtùiàû ä®Íæö_õõiåÁöÖç÷Ú èOÕFü éêõبû ÁÍöòXÖOÕFø ÒÊÓÝÖçÕÐyÉä·ëgдì~éFÚÜÝÕÎÚÖOíÊÜÝÕ ÞÁîëgÊßAÖOÕ ÊÜÝØOÕïwÜÝÓ ØO×OÕí"ÕFÕ×OÐRاéêï Ûç ÌXÊô ÜÝóÕFí"ó ÕFÒwØÞÁÓÝ×O~å{ø®Ú÷ÒÁýRÓÝþwËÁÿ_ÜÝõÕwû×N_Úý~îÁø×AÛqÚÖOí"ÕÌ ÞÁÕ ÖOÊåÁå{ÚÖ·ØÞRÕ ÖçÕé ÁÕFÖOé RÕ´ÞRîyÉÊË?ÚÖOÊØOÚÓÝÖOÕ;ÞÁÕÜ ä®ÒRÛtÚÖçíÊØOÓ iîÁÕ ∗. . †. . . . . !#"$.
(10). . . . . . Unité de recherche INRIA Rhône-Alpes 655, avenue de l’Europe, 38330 Montbonnot-St-Martin (France) Téléphone : +33 4 76 61 52 00 — Télécopie +33 4 76 61 52 52. . .
(11) '
(12)
(13) *+-
(14) %& ' (w( " "
(15) "!#w
(16) - ""
(17) %
(18) . ( ,(w * 13254 ~ji
(19) op¥x ji z?ocpqx_ j¯nw j©& c¡ }gsTp}g"j sTx·pqs?jiz{}~oc__x_x#ogp|v ¢~ptrt'oj"gF}~ji«gv x_pqji¢~ni_j\x"}grqrqj%xsTz{}gxjR~v sTjw_pqxnRo¸x_v jp|wª}gz{}go~p|_o¸j\s?o=vwz?z?_ cpqrtx_vw}geg}gjh¨rr|~z%nwz?x_~p|y Tpt}~ptocjDjRx_jwv~xz?pq¦sTyz{ortptqoTjRjRx"Fjwx"ji¦}g cz{o}grtqgptjRjw_sx'jwvwo9XniXjigyn_nRpq_ocv pxp|jwptsTs?}g¦Áo¸z?¦?rqjiz?tocjix¼ jih¨zDg~rqpqsgTrtn p s{~x_e~z{fio~ c&}grtjji±²z?´rquT³ s?µ-p¥z%xegn h¨xn ji-gnw~¦?z?ji"rqsTpqocgx_Vjinw_j ¦z?rqtsTji}gi&° _nisTo~Xj\v j jRsTpqoº©jwojAj xRªgz{}~vw}gogj gpq~rqpts{¼ xxegegfifi c c}g}gj-jj _s?¬Xocp|FxLxz{z{ocgx_j#p|Xo jwh¨s AjioTz?x;p¥xLXsTXogj&ogniVjinivwp¥g¢}gvip|zLx}gpqsTo~o~jv s?zh¨xpt~çz?z{p|z?z{pt_ocs?x_o\jRwzÁ ° ¦?jRv´ji}gvwo~z?j"zTgvpq~xnwrqpts{ptxx_egp|fi c c}g}gjij Xj- z{vwj pqx_x_egx_j-h¨gn pqx~pt crqpt}gs{j ¼ rtj z{ç;z{ptpqocxyx_ cjw}g¦jz{rqrtjqjRL_}gjwo¯v s ~ g_x&nRv rtp|p n pts?%o\r|z\¢g¬Xu?nwjRj"FxjRpqFsTx#oh\Xjijogr|nwz\j'g}gni#v }gp|_o¯pqsTgo=s?hDg}~rtfwr¥xh¨pqgjrtj_jinRx"v pq¢x_sT c}X}gx j %© çjwz{rtqpqjyxyh\z?vijwv x´jwgjwo¯xz{ni~¦rqjTptX° jio~v j gjisT}g;z?xvwjwpqo~h\ji"ptogjinwijiªrqt cji}gjwgr| cjD}gVji-s?rtz{cgo"~!?rqh¨p|vwjiz{ x_pt¿çs?~o~z{´'~}gz?o9_nwz?jRrq;uTs?}~-p¥x±²egh¨#j ³v µ#s?h%sToTgx#pto~gz{_ocnRxyjivwocz?x_rtnwvwjR}gr© __sThDVs?s?x_rtp|p c c}g}gj?jDªcj pt_x's?r|vizz{xr|pqv sT}go r o}gh¨ni_p| c}gjRÀ#j xyz?ggsÁ¬Xpqh¯zxpqsTo=zÁ¦?jRv }gogjgniv p|_pqsToz?_gpqx_z{ptj X$j #inwsT-_nijwr|w° % Ç R Å & ¡ z{pqx_egh¨n xpt c}gj~z?"pqocxjw¦Áz?rqtji-jio~_nRv p|pts?oz{gpqx_z{pt_jTªcviz{r|v }gr§ucz{z{ocx_pAj x~_nRv p|.
(20) . .
(21) !"
(22) # $ % &' )(+*,
(23) - .,0/2143. 576 8:9;-<=8:9>6@?BADCE6FG< ?H< FJIK9!8:FL< FNM7OKF'PRQR9A:96S?T9?-8LPRF ;-< U < FJ984VBWXPY8:9>QZ<[?K\]8:VKP^5`_badceU9IKF:<fF M. 78;:¤<T>V@AB9UVSY Z\8L:¤@A]MNBOBNMO<cQ gV]
(24) B hjilk k i. monLprq$s:t'prqunv=wyx!n2z|{}=t:v=~quv=~t4zq$pr}=Nprqu{. tryp j h\sx£-ptpqvixzTz{&gr~s?zÁ xXz?z?wggªAptgv rqpsTrtxFp h¨Dviz{gs?g§}XptrqxzDp z{xFX¯x_ptjws?s{o~o~ _ jsTpth¯oh\¦?zj s?xrt_o¦?pq}g¬Dj\h¨çh¨zTjwvs?xptvis?j\z{pr§z?#Ro~z{zrt¸x_u?pts?sTh¨o_pqsTx_¿ _e~'j\h¨ s?-VsTz{jwj zxogypqsTsÀ o~£ p|;z{jwjioo~z?ºv}ge~v jijisT§¨o~©}~·jRs?o~ c&X}gptjwjwo~;ocFx_x_xrtegz{j o~vwjwz?jy_xsT}gegh¨j i°DXx_egxd&pqjisTegs{o j ¼ x»e~sTzzx xpqo~u{¼½sTpqocxyªX£"r|z{e~u?jwjijx_eptz{-o¶xegTj«Ax_Xegpqh¨p| jih\o~jRptz{s?o~oys{x_ex_zegx jh¯hDzx}~_pqx ¬Vz?j\o~ rqjR_yx_e~z?ok{TgªXª £"?Tpqx_ge ª~£"e~ptvx_e¸e~jp|_h¯z{rtz{h¨rtqsTjRFxx sTjiogzTrtDvegxjiegjwpt#h¯h¨z{x_sXegXjijwh¨oz{x__p|pqvwh%z?rV}g cr|z}~x_z?pts?rqp oxF?w°°d&jwegx#j\}~oc-}~v h\pqx_j ji_Fp|xvw}~z?rXpt cji}~#z?rqpjixF·sTs{_#h¨_jis?¨rt¦?ji\yx_e~pt'j og s¾´£ 6z{o9¿ç³~pt_jio~}~vj?e ªz{o}²v rtogjis{z{x y_ptgu?rqjio~v v gs?À s?jio¨vxxpqego~jyu%_opq}gh%h¨}gjwr|zx_p|vwpts?z{o¨r§~s?_AsTou?}~vwz{rqh¯jRz{w;ª~_}~rtz?voTex´zTzT vwvw ptXz?Xjiocoxz0´z?%o~¨sT z{r| s jwo~_zs{®xFÝk£& z{½° jiLvws?ocx_s?rt pqoguz?o~¨scF¼ h¨o}gjwh%x_yp|v0ogjis{ÝkExcegª¸jw&kXz?ªz{o9gkX~ptocN_°¯x_scjwz?d&v¦ege¨z{j\r v s?·jw}goo~o~vwp|~rqFscz{x;h¨ptogptjwo=uocxv p¥xRz?sT°%rLh¨g³ggsT}Xpq'o~x_ptvwptogo~pq~u%rqxj¨£"z?o~p¥s{xv #e=jTx_ªv egjip|xvwp¥z{¢~z?ogjRpqx_¨ogegs?jwh¨x'o~j vwVxrqj\scptv¨jw}~vw¬g_s?z?jRo~vwx_ªTptrqo~²xz{ h¨jwpt~ojwrt_¨jRjwpqji~ocoTrtxzTxjwjiv ²pt¦Áogz?u}rz?jipt_og¦?pqujix__eXz ¼ ~zTXpqo~¦z?z{_ocxz?s?u? jR&XjRs{v Lpth¯pqocz{xjwr#z?¦Á_z?pqrx_e~z?h\_pqx_jwegx_p|h¨v{ª jwx_gp|v }XxDz{pqxDjp|o }gv h¨jijwxp¥s?¢}~jiiº°=x_'eozx x_egj%s?jiogrqj sTogeuTz{o~x_§s ªp¥x j ¬Xegptgpqxyx_egj ~_sT°%jid&egxFj s?sTLo)%sT }X&
(25) xF.£& z{ !.y¶s? s?}g! o~gpqoguc .¨!z{ o~*=)lx_e }~ ¯?°vws?'h¨o¨gx_}Xegxjji s?x_egjiji_´}ge~r¥xz?\o~x§z ªc?v js?h¨pqocgx_s¶}gx_zTjw#vwvwpqs?h¨}ggocrtx¨jwh¨_sTji}goTo~xzXxptpqogsTu¹o~´jwz?__sTj"~\zTz{ojR pq~vwx s?_so~p|¦x_p|s{Xpqx_®jix}g;jwx_o¹jx_egrqz¸j'jR_£& z?zÁog9u?ogjsx£"ss?ojRzªFxp|·pq}gh¯ox_ze~vxx_z{j¤ptxs?xox_egegsjip|¦?_ jjiz?´ji_z _pqx_eg£"s?h¨egisTjw°rqx_j p| v¨jwgrtx zT_sFx¨¦ªp|z{X£"o~jieg 5p|vuTe¤¦?rqjwsTp|~#ºz{vwrL_pqh¨}ptoXv V·p|s?z{s?rg_h¯x·z{sTzoc#xx¨uTpqsTrqz?sToX~© ¦z{z{·rs?ocs? xXz{ptxo~u?pqjTh¨xªz{p jw#io¦?z{v jix_j?pto¶s?ªop¥pqx « ·e}~z?&xegz%jw¢gh¨¬Xjis?¼½j?ªcsTp¥pqocx_xegp|z{-o¯z?ogx_e~u?ptj'ptvi&z{zDo_}gj~_}~j _x&jis{ ªx_egjiji£&o0x_s?- o s?"}z?£#rquTjis? Lp¥xxegeghæjwsT·_s?ji"h pto~Fxxzz{x_ojRv #j?x_2° e~z{}x#vxeegptg#·s?}gVojwv_x_x_ptpts?jio p|vi&z{ogjizTo~s{ptxyr =VzÁj%¦z{jiptrzTz?vgegrqjijT° £"pqx_e~s?}Xxjwx v sTh¨g}Xx_ptoguªAz?o~²ptocx_jw¦z{r z{p¥xegh¨j x_p|vDvws?h¨g}Xxji'£"p¥xe9_j x z{o xs?pq"sTopqocyªx_s
(26) ji£#_ ¦ji;z?¦?s?r#jw Rv ªAjisTo~pqh¨oIv grtsT_}Xg_x}gpqz{x_x_j¨jipt-s?s{o¶#£"x_eg£"egp|e~vj¨e²jwptoIh\z{pq~x%j'_çsp|z?D¦T&jwpth¨h¨x_sgjws¨ocrqjixrth¨z?'_jwptuToºocj x_ptjRz?ocIo~x_ji}~_¦x__epqz{og}~rLu¸"z?o~¢gpto~z{¬XzTrtjiXvwv ¼½p|}~gwªAz{jixx_v egj?p|j¨_ªXpq_sTgjwojis?h¨»gsT-rtjwzxh xs¯pqo~s?u{´j¼½sxsT¦?egpqjij oc_x%jRXFjRz?xF_pqxh¯pqpqx_oeXz{ ¼¼ n3 u < 1. u = 1+ − 1. n. 1+. n. π. π. S. S. ààÒ r¡¡¢£. [3.14159, 3.14160].
(27) . = %&' .1=&&
(28) . ¨hz?~jwz{x__p|x&v{° jw¦? jiosTh¨£"ge~}Xjwx_optogx_u¸egj£"ptpqogx_e)g}Xpqxyocx~jwzx¦Áz¨z?rtz?D_j ggs¦csp|¦cXp|jRXjR¤u?}~£"z?pqx_z?eocx_x_jwegjRjºh¯_jRzTv}gegrqpqxo~ij ª~g_}XjRx\v p|x_egpts?jo«XsTz\}go~jwgh¨\jivwXz?¤oI·s?&j=x_e~çz{pt x~pth¨egegj=jwgogrt±²jwsTh¨h¨#jw³jwocogµx_sTptrtogo5pq~u¤v z?z?sT__o~g5pqp|x_¿ xBz{%¨pqo&g v &sTji ,v h\p|/_~pq sT}X !ox_pt"ogpt!ocu9
(29) x_jw £"¦pq1x_z{e &r§z{ z¸ pqe~x_ egpq uTh¨egjwj x¨pt vg £"3jie~ vwptpt$_v pqesT opt y°I& X d&jiegv )p|(+%pq*,V~ji _ sT
(30) 0&ptosc_
(31) xz?egr´p| "p|Dw~Àxz{ªLegVjzjwRv rt° pqs?~j=z?_s? jxwpq¬Xh¯gd&z{_jRx_eg_pts?p|_"pto s? co~X}g}gjijz?x"_x_j¯·s¶s?_&cgjwh%jw¬cVxjwsTo~z¨rqp|XvwzTjiz?vwo~rqv tv 9}~Tª&zTjws?v £"¯¯vwpqxz?jx_ojio¹ Vyj£"jw·s?·egs?}gjio_j¯=jptegjioptpqu?ogs{egu9x_¼Ne~sTjwjw¦&Xz{ji£#rt}~\sTz d xXjizÁ&§_ªrqsT}~_¨vs¸ej z?¬XzT~[ z{x s²o%_pt£"s?jRXo~eg}~¯jwv z?j¨j_jxegsT}~j¤rq_jisog¦TsTjw°h\jip|µêz{o ¼r xpqx¤egp|p|y¤r|v zsTx_x_h\jw'Vjw£#osT_ z{Ax_jiª~)x_egj%¶x_ptxh¨egjDj²s_¦?jRjiX_}~e~v jix_z?pts?o p|ypqo z?x_sTe~}Xj²x%oc }~{hD?\Vjw·¤sT's?zx_ptogji~u?rt¨j·}gjio·vsTx__pth¨s?o9jR5jw¦z{¶rt}~zxegxpqj¸sToz{ªVrtu?egsTs_£#pqx_jie~¦?hji ° eh¨zÁjisT¦?z{rqj-jRrq¼½v £#}gjisTr|jwz{_o¯Lr|¯v z?sTz{ oX?gjR·~s? rqp|h¯vwxsz{zx_pqxptocs?pqx_sTo~jio -u?jw£"zocegxxj#jiz?_pqrtr|j pq o~j x_ji¬e~z{x_j-;zDh¨_XzTptFogvwxptjwvwh\h¯}gp #izTz{£"v x_¨pqptx_s?p|e¯o\#egs{jiptAu? cz{e¨}go\ptz?jwjiviog=v ji}gz?_uT_z?jy'vwXx_·}~s\Ào~s?v;j xvpq·sTe~sTo¨jw}gh¨z?o~o~pt%x_pqo_AjR©z{ T}XX}~xjwpqs?x_jih¯ji _zh¨z?xviptptv viog}g'pqo~¿çzu£#xzjj ji¦z{rt±²}~zscxFpqx%sTo~}~-}s{z{ r´x_egptocp|"x_jijw_o~¦jwz{r#u?z{¨p¥·x}gego~h¨v x_j ptx_s?p|ov¯° rtpq~z?_ptji¯¿ç;_s?¢~r -µF Ýk{% ½ª µêoTxrqptj t% >½°w°i°À s?Dvws?h¨gptrqji zr|? ¿ z_x~}gs?p¥o²x³gxFz?sT_VIxj¯j?g'ª·s?ji \vw ptz?_ ptvis?v o5}gptzocxx_j¤ji _X%¦s{®z{À&xDr'z?~z{__jsXp¥x~Xeg}~zTh¨v jRxij =x_ªp|£"v{sT° oegp|d&v.EeI egj e~z?& [ &% »z{ sTr|zs²xr|z{pqo~Vogu{jwuT¼½ji}~o¶z{sTuTpq}~ocji_x&jiºoc kg}~·hDsTy Vpto~jwv sT"rt}~_z{o~Xo~jwj¸ z¹jiX¦Ás¨rtz?s?rqogo~}s?u¹zxx_zTptgs?vw_o¹vwsT}gs?hDsc"}Xzj ¼ Vp|\s?rtogs{ogx¨s?h¨ptoTp|xz{jwr"o~gpto jiIx_eg·sTj²¯ptocz¸x_jwu?ji¦ogz{jwr z{yr-jw}~£&_x_j=sTo xegz?s?rquT}gs?uTegp¥xs?eg}Xh x%x_Ýegk?j²k ½j°vws?h¨sg£-}Xxjwz¦Tx_jwptRs?ªLox_ª;e~z{pt¯o~¶rts?·o~s?u¹¨zTpqovwFvwx}gz{hDo~}gvwjr|zpqxx\s?\p|\xFogs{jx Vh¯sTz{¬_ptpqh¯grtj\z{r&x_gs¸vwjis?vwpth¨_pqgsT}Xo5xj¤p|¯z²rtpqXh¨pq¦pqx_p|jipts?ª-oº_pqs?o v jzx_rts?egucjz{rts?p¥xogeguIh zTvw}~v _}~ptoghDu}grqr|sTzogxs?u²%z?xFvwcvwV}gj²hD}~jwrtrtz{pqjRx_s?%sTi°¨o5³~x}geg_j¸x_egXjisT_}gh¨gs?rtjj?xFªx_Vegj?j ª £"z?_e~jsTzÁ_¦j z{ptz{r|z{o~gu?rtjj?° s{Lj j x¬XV}~s?'og cji}gocs{x&xjp| ·s?"pto~Fxz{o~vwjpqocx~z °qR%jw¦T jwz?o~z{r§h%pqoc}gx_r¥xz pq~ rqj ~qÁ_kjRv p|·sTpts?o±¸pqocz{~xjwrqj¦ÁsTz?yr~z¯zTv!ptÁh¨z?pqu?r|z{jR x¿··z{z?s?!v z{pt·o~uT}gjo~xz{v ·x_s?optv s?j?o~±¸ª¯£"z¿Oxp¥gegx° e¶jwh¯}~zrqxXptjivi·vwBz s?pqh¯Ýk{ptz{ oTr#½x° ~Xzpq'uT9p¥}gxj%pqwo¹ªx_x_s±¸eg}gj¯z{o~¦?rqs?jTjw}~ª;o~pq¨¢~Xji}gpt²ogrquzT_x_·s?_s}g£&h¨z{±¸Xgz{x_x_pts?egojiyh¯s?sTzox_p|vwx_zcegÀj\ª§_xsTs}ggo~}gXu?ptuToguTpquc¦Tjiji9¨s{_;¨sT}gjiptrqo~o~jiFXh¨xptjiogjwzToXuT ¼ s?o~ s{xD~° VT?j=%v sTpqo²o~_±¸ptXz?jig_jRrqj¹¦ z?¨ z{ojw=rtp|z{¦ g{rtÀ j?ª~°z{yo¤x_e~xjws¤ £-ji¦?s?jwXz{rApqs?ox_v egrt}~ji"Xj¯}go~xXeg}gj j zTz?_og_}gu?h¨j¯Xz?_xpqpqx_sTego~h¨wªjwxx_egp|v jiK_jq~-zTz{!v oÁz?¹u?jRµêoT-x viz{z?oX ¼ xg ?s\pq uTvwNp¥x°s?'h¨ gz +}XxEegx j jw p|x_ egxje~zz{h%o¹z{og}gzuTr¥xxjypqV~s{rqjwj¤_zD·~s?_jih\jR_v }gptp|ogrqxiptus?«Xoºpqz?ocvirqxv ptjwgsT¦Áz{Xz?ptrLog¸z{uD£"x_p¥egxs¨egp|vp¥h¨xeI&j z{x_z{p|pt}Xv{h¨xªegDz?s?o~zRx ª pt£"o~s{egX ptp|v vvws?eºz}~xpq}~o~__j?ujiª x_egr|z{z{j¯oguTu?oj'jD}gz{h%s¦Tpqjwx_jieg jih¨s{x_j "ptxh¯ptv v'zsTx_pq_xpts?_jijwov rqx p|&z{z ¨·rtsTpqg_hz{s{ gptocx_pth¨jwz?¦_z{ptr§r z{pqh¨pqx_eggh¨rtjwj h¨xptjiv?oTªXxg;}Xj x"¯egvwjipqji_ocj'x_x_rtegDjptoTptocxjwx_jw¦z{¦z{r~rAz?rqp|uT&s?s?ogp¥xrteg¤h¯v }~}~Xjiptrqog¯u ±¸jwgz{x ji_jwz{ocxªcjiz{ o~{§° ªTd&egjRj p|XµêocjRxw¼ ª ·h¯y}~_z?o~~pqvogp¥xxrtpq¶sTz?o~_gz{rtp¥sgx¦pq_jRptjRgwv ©jip|¯ptpto=s?zo²¦TxFjwptocx_ptjjis?_o¤¦·sTz{~¤r ªcz{sTz{ogp¥grtx%p¥egxh¨xz?egj _jxIptj vD¬X~£-V_jRs?zTv ogyp|jiz{ptoTs?r|x_o pts¯z?r~v h\sT·}~jih\oco~~x_vpt}Xxs?pqxogsTz{ojRx_¸ptz{s?oo~z?'¯¨z{x_go¸e~}gj x¤jRz?pqh¨_vwTs?g¼Nx_o~rts?jwh¨¼Nxptz?h¨jwococgx-xrt¯z{jwh¨·j'jwjw£zÁoc¦xz?j jwpqr|¬r|z{zxgx_jwjRrtoXj? ¼ ° _zTptXs?¦opt_x_jisy-x_s_ji}~oT_x j\ yz?hDogs{}gxrqegx_ptjwg rtj%~z?~!v_jRz?v p|u?jpts?xo²e~z{~oºzTv!±²z{ u?j o~±¸s£ © z? '±²ji z?_rtpt ogzTs £3R?sTg~,_s? rtji½«§v egjwsoc£-xR°jw¦?Ljijii_ª#·jijw£ ocxyve~e~pqz{h¯o~_u?jwjirq xxs=egj¯xeg}~j¨_j¯v sXs{X&j\zs?h\sTX_sXj¨v }gh¨h¨sXjwXocjwxzoºx_pt~s?z?o9v!e~z{zÁuT¦?j?j%ªA·sTji jwo9j ¬gh¯z{zTh¨Xgj\rtj _pt'o~v zÁj¦p|iTg& zÁrttjw°qt ° ¤ µêoº±¸;u?ji3ogjw~z?z{v!rNªVz?£#u?j\j\_s?jR v sT±²h\´h¨³ji o~ ¿OvF° cÀ l° pto~v j . . ++. . . {−1022, · · · 1023}.
(32). . . 1 − 9.10−5. . . . . . . π. . . . . "!#$#$%'&)(
(33) (+*-,.'/02134$5+.'/61 78/95-0"/;:
(34) <-(1:=024
(35) (+*-12>
(36) <-(+>=? 0!@5>AB/;.
(37) >@,2C
(38) #-(%
(39) :$.D(+%$:
(40) .E FG8/H!
(41) #23I4. . ä·Î àä®Í.
(42) .
(43) !"
(44) # $ % &' )(+*,
(45) - .,0/2143 . ~o~s?_jRogv j¤p|s?pts?"ox_egpqocjx_jiz_·¦sTz?_rjih¨z{jwp¥xocegx_pth¨s?ogj x_jRp|ºv{ªg~x_z?egv!p|"z{rtuTjijiD}~pq&h¨x_gs¨rtjwpth¨h¨jwgocrqjixh¨%jwz¸ocvwxs?s?h¨}gg"rtsj £"xjoz{rtopqgIz{_jR?z{° rtqºjwrtp z?grqjz{gp¥xz?_ xd&ege~jwjwsTµêoo¨_jwx_xx_egegp|vwjjz?±²rogjwV´¬cs?x-³ ptocµx&jR¿$vs?Bx_pts?¦o&pt jwª{ £lx_&e~ fz?jo~/og=jw jRz?\rt"_s%·
(46) s? ´ B¤z?12_vw~&s?>p¥o~x _z?ptg +_jw ~pqog _uDjR v x_p|:egpt3js? o\}~$ _pqjoc xs{ jw&'¦Á~ z?pt_r~jiz{(v x_*-pqptx_s? ego h¨
(47) @&pqj ox
(48) ptpqvoc x_pt ji_Fw¦}~À&z?FrAptx&p¥z?¢~rquTji_\s?jR·p¥jixoTs?egxh h¯ji§wz °© ¢xegj x_rt·¤}~o~xvegxj pqsTo~jiz{ crt}gp¥xptpq_jRji'h\z?jio~oc²x&pth¨s?ogrqxjiegh¨j jw}goco~xz{gx_jwpts?rqo9ptogvegu¨sT»~ptvwsTjiz{x_ptz?og_u{j%¼½VXs?jwpt¦ToTjwxrts?z?V_jipqx_§eg ° h¨jws?x_h¨p|v j z{Vjjwo~j v¬Xe~gh¨r|z{z?ptog Xjiys?z{oo=mxz?eg}~jwFo ¼ ert_p p|gz{z{o~oz?X_jwrtprtogpqh¨u¯pts{o~ zxhDpq§sT}go«crqx_x_pte~gjijrtj ·j sTAg_jih¨jivvwxjipt_pqs?sTsToopto~o±²v }g¼Nh%jih¯zTzjiptxog_iu\p|ªgv jRx_£"egegjz{p|vrt~es_jR£ ptv p|x_¨s¯pts?v h¨osTh\jRz{z?_j}gz{z{_jir|j=s¯xegx__s¨j egsx_se~£"¦Tjjwo¢g° eg¬XjijizTzT¯F¼½xgçrqz?Tv jiª~x_vw_sTpts?_yh¨pqsTXjo}gz{jptocgxx_gs¯jirt_p|x_vw¦egzz{j ¼r xxpqz sTTo~jw"o}~·__ptsTogh u¤±²x_eg´p| ³ µpqoz{js?g-pqz?jw_»~pqx_eg~h¨_jwjRx_p|jiv{oT©xjiu?}~z?ptoz?sTocx_gjwjRjw9"x_s¯s?rtpq}Xrtqx_}pts?Fxo'z{x_s?j&x_ze~j g¦s?z{~rqpqjisTh }~"ogsTjirq¦jiz{g~rqj¨z?o~ogjip¥xegjiogjwj ¢~x jwpto~¬gzTz?vwvv x\}~ogz{s?x_¨jg¹zx}~zg_ªg}~_z{s?r&rt}Xox_}gpts?h¨ojws{ptLviz\z{r-gh¨_sTj gx_rte~jwsh~w£"ª´pqzTx_e²vwvwz{jwrtjwgpqzx_x_ptz{s?o5¤s{z?'viv z?}go¶z?j vw¬g?z?° v x\h¨jwx_egsX5z?o~¶ez{o~Xrtp ogus? 2. . I?
(49) 4-? .. } Jn2zrNprqu{ t=tL{ ~2zrn
(50) =v. tpjwd&oc¬Xe~x_gjij__jR¦x_z{eg_rNjwptª{s?sTo_jwª;x_rtp|j vwxz?zyr-·}~jio~vw_s?v}gxrqpq_x¯sTjRo¤}gVz{o~s?o~Xo%ji¹_rtx_ sºpqogx_ue~z?jx_o%e~vwptpt¨}gocx_p|jw_Xjiji¦oTz¹z{x¯rXviv jwz{sT¬co h¨xjwgo~}Xj_x_pts?·ptogs?o\}~u¹s{o~g jipqª{ovw£"pt_egpqk?sTjw% o ½j ©©rtj p|x¯}~u?pt¯¦?jigo%jwogs?ª´Dx_xjz$egjwopq~vegpqz?x o # s¦Tjwjix_pth¯zx_jR z{o~x_e~js¦?jwjix_pth¯zxpqsTo¤pt&Vs?}~o~Xji ¿FRÀ £"gjwe~Vjwjwo~j 9s?p|o %x_eg°%Bj d&'egz{pt}~ h¨Xs?jiz? o~yXx_p|Fe~xz{z{x o~xvwegj?ªj¨v s?h¨g}gptx_\ptogxuegj~_£"jRp|v p|xpte)s?o9s? viz{oz?jRo~v ¶s?h¨xegj¨jz¤v s?rtpqoh¨Fxpqx_z{ptocogxu çz?v x_z{s?o z{o xe~zx³~}gVjwxptegogjwu¯h¨z{~s?rqj j?x_ªs¯zDptvwo~rtvwzT__jR_z?ptvi_z{j'rpqgxy_vwsXz?vwojiXV}gj jyz{opto=p|_ptoc_}gx_jij?_° ¦z{rVz{o~z?rqX_pt#p|;xegj'~pt_jiv x_pts?o¤sTogj?©pqx_egj's?}Xxg}Xx £"jwp|Vjixz{e²x_jRptx_s?so²s¤jir|z{zTvu?e²j?~ªz{xeg_xijw°o¸-x_egp|j jRvptxogpqgsT}go²xyptyz?s{_®j x_ji_o²grtx_p¥x e~jDpqor|z?xF£-xys¸_jR¿·sTsTh¨x"xs?s¤jRu?À-jw~x'z?h¨xs?yj z?o~z?viv x_}gegj\z?vwv ?s?ªgh¨g}gxjizXxpq}~sTvwopqo~ptu jwo~pto ¬Xjw£æ·jisT__»~pth%h¨scz}gjwx_rtocptz ogx u?¿F¼Ns?ÁVo\À¨s?u?ptx_oce~rts?xDjºoz{ c}gr}~hDs?z?XVoTxxjwpqp¥h¨xF ¤p #iVz{x_j ptxFs?£-ojwÀ °6jiª{oIs?µêogox_rteg jviz{z?jwo\_ojipt¤Xo~v £"sTjie~pqoczTjwxDj#j pqs?o\ x_e~j&ªvw£#s?sTh¨}ggrt¶}Xx¿çe~pq£"o~zÁegu¦?ptj¤vzTe vwvwpt}ge~jwr|z{XzT~jiv 5TjiªRogz¸ji sT_z?rqpt}go~x_XptsTs?pqo~}gou ° s?-vwpqLp|v x_jReg}~vj¯hDxpq¦?sT_s?jio}gocviuTxz{"ecox'xegh¯j\jwxiz{s©"pt¦?ogxjwrqegjij\~x_j ptzÁh¯_¦jwzpqjiogxo²upqsTz?o²"pqz¨o~zg£&}~}gzÁv og¨jRpt¸s?x_o9s¯s{ji#vwx__egz?h¯j\jz{ptrto~xv egVrtj}~sÁ_¬X£"pqjisT o¸z{gvwpqz{goco9ptxogsu¨sTj¯j ogAz?rqji v z?sTxviogv j%=}gVgzsÁx¬§sjDf°¦z?pt#X ptp|ogX_jiu¨jRvxz¨pt_pqsTjR~o9¸zÁ¦z{z{pqr|oo~_su }z?rqtjsµ½£s{x#Lyviz\z{x_so¨u?s_VjRsXjX¤og}~·s{v sTxj%_pth%vwx_jie~}g%j¨r|zx_gex_jwptzs?Vx;ojwx°o~egXjjio~}gv rq9j"gs?_sTxgert}gjwhDh ªAjw¦?$jio9 pqF'ª*,pto9 x fe~.zxji_LjR lvxi Rª§(ro~ s{=x_e~$fpqogu l_ }g ! %ji [_)ji(rg ji'! x_* egj + !"
(51) B@ (&S)(r *, *, = [.
(52) yvwz?oDçz?pqrNª{·sT;pto~Fxz{o~vwj"£"egjwo¯vws?h¨g}Xxpqogu Vzº jivw T%s?}(+h¨z{& ji-j=ptoscs?_. }~x% vsTe\ vwh¨ªzTzTsTjR&_ jrqsT$u?og@jwu¯o~z?jw - z{xrtegqIjD*,zv sT h¨ g. }X ¼Nx_x_pte¶
(53) ogu¨Vsg£- lji jwvw¨(pt_v %pqrtsTfsTo_$j=-jio~x lz{sº~ ~rqjR !° %&x_ fs¨s)uT£-(rz? jwpq¦To²!jw RX*=ªptu?lx_pqegx j& sT_["}~rq £"j p¥ !xs{e¶'"x_¿·
(54) ep|X}g #jihDz?r| À X. f. F. f. F p ε = 2−p. ε f (X). F (X). q(f (X), F (X)) ≤ c1 w(X) + c2 ε. q. w(X). X. c1. w(X) = u. . c2. F. w(X). . X. . nα. ààÒ r¡¡¢£. . . 1/n. α. H. .
(55) = %&' .1=&&
(56) . ptz?oXo¢~o~pth¨p¥xgj-~s_¦?jRjiv p|z?pts?viov }g°µêz?o%vws?¯x_s?egojix_£#egjsTgjii_ª}gptrqo%xih¨° sTx;vwzTjRz?o%pto~v jizTj#pqo\xegj"v s?h¨g}gx_ptogu'~_jRv p|pts?o%cptjwr|g s?y;yz¤±d&"egjip|ݤk{z? rp|N=ªz?'·sTrt±I_rqtsIs°V£"xµêegpqo9o~j9u²±x_xgregz{t_jrqjix_ptx_oge~uIjw£-s?Vs?s?j xptAocptviª~xz{z¤r#s{vw _s?jR± h¨}g~rgrqxq}Xrt%jwxz x_¤pts?£##o²sTp|z ')x_ xs?jiz¸o ·sTz?z{_o~oh¨ jij y²VjwjRz?_vo~x_xrt§pqpq¦Togªj9u p¥Lxpt hDeg&j }~sTrt¢~oIz{ox_x_ptz{s?e~ro j¤zTvws{·jivwDzT}gzIptzTgv "ptrqjRp xFz{pt r o~jws{pqxx_ji} z{¤x_jRvw¤pqji}goTocxRx_ªAptr§x_egxjiego9js?xeg}Xj\xg£"}Xegxs?rtz{j\jvws?z?h¨vwvwg}g}Xxzzx_xj pqsTjwo9o~s?p|'}guTeji° xz?xji²£"pqx_e¹z{opqov jiz?_ji¸g_jRv p|pts?o«Ax_egp| pt . } m uq zrt:z . êµVo%s?pts?ocx#Xrtjwpq~x_z?s _pt¨h\£&~rqz?ji;h\o~jijwocjixLgz?ji§oD°Lz{-g¯pqx_hDz{}grq'x_pt~g_rtj'jRv ~p|_ptjRs?v o p|ptptocs?x_ojiªc_pq¦x-z{p|rX´z?h¨_pqx_jRegz{h¨ocx´jwx_xp|e~v{zª{x-z'x_hDegj}grqv x_sTptgh¨rtj-g}X~x__ptjRogv u\p|ptgs?ojiv »p|sT_pqzsTxo¯pqo~ptu xo~Vs{j=z?x&_XTrqptªXh¨o~z{z?p¥xh¨gjip¥=ptxvix_z{z?s%rt_q¨ºx_eggz?jÁji_Fvwpq}~ogpt_uTxpts?rqz?j o~gs?&rtj¯&_egx_Xss?sT}g}g·r|}ggrqrt¢~j'rtg´j xzÁjieg¦vwj=z{pt_ptptzTr|s?z{vwo=gv rt}~s?j?°zTh¯³gv ¸}gz?_vogx_ege~jiptjijwoggj'h\i»~°²sTsT_z{jTx_ogªcpts?x_ogegx_u{egp|¼½Vji v s?sTptoTh¨jix- cgo}g}X}gptx__hDptjiogh\Vu¨jwjigocwx«jisTvw·s?pto¤_Dpqx_sTpte~o=ocjx_h%jivw_s?}~¦oXFz{x¼r z?vws?_pqh¨x_e~gh\}Xxjwjix_p|v-}~pt_ptxogegu¯j"»~s?sc}gzxF£-x_ptogz?u?%¼NV_s?sTpt}goco~x'Xz{ptogp¥u xegçh¨z?vwj pqxrtpqptxFv Vvw©s?x_ocegxp|z?Lpqjioo~&x_}~eg_j%jR pqocx_e~xjwz{xL¦Á·z?s?r;sTjRXz?xvz?e\pqogs?jRVjwp¥Lzx_jwpt¬Xs?zTovªÁxyx_egjijz?ptrocx_z?ji__pqx_¦eXz{¼r h¨
(57) jwx_p|xvDegjy£-jwptocx_jDji}~_¦_z{jirVv °sT¾;h\¦?~ji}Xo9x_jRh¨¨s?}~j%ptogXujR»ptsTz{zgxrtpqjDo~u{p|¼½, sT +pqoc x#Lz?X_pqptx__egjRh¨vx_jwjRx_¸p|v"s?pt;}gx_oegXj pqo~u¤h¯x_z?srqtjizÁ¦?x´sTs?pt¸o~jDrqsc¿®·ptsTog´upto~zTv vwrtvw}~}g_pqsTzTov TÀ«ª e~ s¨£#jiF¦?xz{jwo~RªR~p¥z{xLp|¯·s?z?_ji»~rq'sTz{·}gx_ptrqog¢~u{rqt¼½ji%sT·pqsToc x"jiz{rqjip¥h¨xegjwh¨ocxj z?x_ptv? ªX·j }g¬o~x_v jix_o~pts?Xo~jii°xds¯sjwrtjw}~h¨h jwoc}gxz?ª?_vw¨s?·h¨}~go~rtvp xz?pqo~sTv o~j-wª~£"ptpq&x_e%£-jwx_egr|v j-sTh\µê¾;jT¾;° ¾ x~jiz?z?h_d&Degª¿ µj B'y"_&µFpqx_ e~&h\ s?sc/" z?gpqo~"s{j?F
(58) ª?jR v ³~x"1z?s{o~& v x_j eg jq T Nr ªXz?y_ djy _x_}~jR vz{&e¤
(59) h rtªp g& z?° _ptji)oci(+xF°*,£#³gji s?_
(60) -VjwV&o
(61) s?ª_ x- z{jw~wrtpqu?Àrtp¥ptª{xF}gX¯h jiz?¦?o~ji rq¯sT½ªgj s?jR¤&vwxcpqegjiyo~j x_v ±²eg¨j ´ji³~z? _zTs?v ortjip¥¼ z·Á¿OsT±²¦ z?´pqBr|³z{g pt&r p¥pt xFl²& z?s?/_´ji x_ !e~sTj¤"o¨
(62) sT m}g ±¸vw1¹j¯& z?v sXo~ X j?j ª¤r£#v j¯ptjw vo~=egvwsc3 j¯pt$L ±²z' h¨´ &³s{x_ x_s °)d&·(+s?*,egLj¤ pqx v L
(63) sT&X_ji
(64) ji¦?_ji rq sTsTo~|jiXptgogÀ u=½z?ªrto~p ptgD%z?z?z²rt__Ts ªsTo~jRxz?z{vwh\z{g}rtjRj¤j&rtpq±¸s{~´z?x_³ eg_jµ £"Vj pq·xx_jwjijio¤rqpto Xs£"·ogs?&rtsTz{zTXgjipqx_Àz{°µ½Dx gpt'ji~vwz?pt__ptjis?o¯sTptoToxjwx_eg¦j¨z{rAm z{=p¥ 6xegh¨±²j x_p|vDrtpq~¿OsTz?_}gv z{jyo~vwsgsTj'o²z{x_oe~¤j%X±²sX´v }g³h¨ jioTrqptxgzxz{pqsTo=z{vwoz?o p|g"jwV~z?jwo~x"g s?pqo¹xeggjrtz?z{v xx_xp|jwv Rj¨°´s?d&oegx_jegr|j¤z{vwu?s?jih¨x"gz?}Xvxegjwpt jw¦h¨z{ji~h\rqjsTv _sTTh¨°¯gd&}Xegx_j¯ptogs?u¯oggrt²jixvwegpt_jwptsTs?_ojwx_ptp|"vwz{~rL_srtp¦h\p|Xpqxjiz{x_pts?=o¹±²p|´x_³e~3z{xz{x_oegj jws?¬X _jwsT¦Togjwjwocz{x#rAh%h¨}~pqrtqx´pts?¢gox&&pqs{oz{go¤pqoptz{ocx_=jiu?XjwptRu?°pqx&j pqx- }~ogjijiF}~XjRx-§°zÁ%xe~zx-p¥x&p|´VsTptgrtjxs%v sTh¨g}Xx_j £"pqx_e=o}gh%ji Vrtjijw;iµêptoc©¤h¨x_x_jie~_sT¦pt\z?xrtp|z{¨ocz{x´ogs{xj\e~x¨pqz{h¨¦o=ptg_£"pqrt~jwp¥h¨xrqje¯jw·ocx_s?egx\jij'x_h¨e~}~jptXptV}ogs?ujiptoc%x_xeg¼½gji}gpqzTx¨Xjiptji}~o~o~;X_V}gjws?_gjRptoT%jixx__we~jwªocz{£"xx%zegx_xptptvegs?e¹j²o¤z{_£-}~j%jwvrte¤±¸qptogzT´;uº³pth\s{ y~»ptrqocjisTx_h\zjix_jipq¦o~ocz{x_u{r|ji¼½=sT£"pqoc\p|x'"x_oe}~}g¨h\h\pt ¼ pt~oº_jRµêv oTp|xpts?z?o9 z? _?pqXx_e~ª§h\{ jwNx_°%p|vDµêo~z?Xo~jw²jR§xegªj%_£"h¨p¥xs?vx_egpt¦Áptogz{x_u=pts?xo9egj¨·s?_'sT}gx_ego~p|Xptogvuegs?h\p|v sXj%XpqjRoµêpqoTox v }gz? ogXssjiV'jwogos?z{x rqxF²egsT£"rt¸p¥xe·s?h%±²}gr¥´xpq~³ rqµj © ji¦?ji_d&¯egh%j }g_Vr¥xjipq~v pqrq¢jvigzx_jipts?vwopt_&pts?}~ojRs?¤Vjw·sTz&x_x_pts?egojptpth¨yz\grt_jws{h¨®xFjw£&ocz{xz{j x_pts?s?o²ogjTz{° j ~z?_jis?ox_e~j xz?o~gz{K t %N©
(65) z{optoTxjw¦z{r§p|"z\vws?ogo~jivxjiv rtsT_ji}~~jwxs{ « . . . D. . . . IEEE. . IR. ä·Î àä®Í.
(66) .
(67) !"
(68) # $ % &' )(+*,
(69) - .,0/2143. . z{jpqocxjw¦Áz?rtiªXxegj ji_}gr¥xs?« ¥pptyz{op|pqocx_z{jio _¦z?¼êrz{}~¤vsTex_jiezz{xix_©pts?oz{o~
(70) pqoIviz?_j zx{zo?jwpqoo¦gz{rtp pq9sT&s?x_Vs\jwx_ze~xjpqsTs?oVÀ"jwp|yptzx_uTpto~jws?s{ogoxDji« Xz{jwx_¢~jR§ogª:ji
(71) ª >x_ e~x_jwegoIj\zptoTxjw5¿jRvuxypqsTs{o¸xDz£"p¥xe9}~hDxegVj¨jw Xs?ªh¯£"z{egptpto¸ve¹s? xz{ogp| og·s?s{x
(72) jizTve5jwoXsTpqocx¯vwz?_pqjR pqx\s£"o5~_jRv p|pts?o ¿Ojwx¨z{x\ptogpqx_p|z{rtp #RzxpqsTo5s?\h¨sXXp¥¢ji5g}g_ptogux_egj v sTh\~}Xxz{x_pts?o~À° pt o~sTvworqd&}~¦?gegjwjij²_pqz?sTz?_o~pqx_£#egxjih¨s¸rqVjwz{¿®x_x_o~p|v=ptu?sT·s?o~s?jis?hh¨z{x_j }pts?x_o~}~p|¨vz?r´z{z{o~z{j=o¹epth¨mg±¸jirtjw_h¨Vs?jwgrtocp|zv x_xjRx_z=)pquTxFz?s?o~Vog jisT h\z{z?rtjw#_x_j¯·}~ptvzÁo~¦·v}gz{xoptpqr|sTvz{o~x_gpt¨rts?j¯ogz?sz{¦co£-p| XjwjRrtx_´Iegz?jipq¶_}~jR±²v Xptpt´gh¨_³sXjw ocvixz{z?z{r|_À j ° ptz?og_guT}g}gxh¨ÁjwsToc}gxg-}Xo~x\z?·h¨}go~jiiv ªXx_pth\s?o~jiih¨°¹s?d&¤egh¯jz{vwsXo~Xz?u?j=jwp|h¨Dji£"oTxpqÀx° x_jio)zTvwv sTgpqogu¸x_sºm±¸ xz{ogz{g=¿··}go~v x_pts?o~¨z{o d&egjygrtz?ogogjRD·}~o~vxpqsTo~z{rtp¥xpqjRwª?x_e~z{x´£"ptrqgVjyzTgXji\pto¤zogjiz?·}Xx_}~_jTªTpto~v rt}~Xjz pqocxjw_çz?v j ¤rtz¯´_s?¢~r -µF k?% ·s?jRz?_j s{L}~_j?ªV~z?_p|v x_ss?r|"·sTyrtpqogjRz{yz?rquTjwgz²¿·¦Tjiv x_s?'z{o~h¯zx_pq¬~zxz xF¤Vsji¦TiªjwzTrqgscXz?Xpqx_ptptogs?u%o~'s?Vz{jwo~zx_h%sT}g-r¥xz{pq~orq=p|vw·z{}~x_o~pts?vo~xpqsTÀ"o~z{oÀ °²z{}Xxs?h¯zxptvXp Ajwjwocx_p|zxpqsTo¶¿®·s?£&z{Xp Ajwjwocx_p|zxpqsTo
(73). op. n. X 1 , . . . , Xn {op(x1 , . . . , xn ), xi ∈ Xi } ⊂ op(X1 , . . . , Xn ). 5. op(x1 , . . . , xn ). . op(X1 , . . . , Xn ). 9. op. . ++. . y'v={}t4z w. wj"T¬X?vi¯v jRsTz?vo~x_XjRpt§og·©¨u5s?rt·x_s?ss D£#s?x_jReg}gjmj ²¬Xz{z¤V}~jwvw_s?pqpth¨h¨z?oIjwgoc}Xjwxxrtwpqzh¨ª'x_ptx_pts?o~ego9zjIxs{pq±²sTo¶x_#eg³j\µjiji·pqh¨sT_pt_Xgh¨}rtjwjRz{h¨¹r§xjis?soTo5xvzegz xjRpqsTv!o=z{o~xp|egXjDsTzTh ji¯_j }g¤±rqxiv ¼½ªh¯ptjwx_ocegzx²xj%_pqz?s¬º¦?jws?x_yegegjiXjRs?z?pth\j jixçpto~z?vi_v z{pqx_rtsTs?qo p|"v rtdsTs)_j uTx_pqs ¦Tjªz5v s?gh¨_jR~v z?p|_jºjR¯p|Xxjis¨z5±¸s{´ ³xegjI° z?gXpqx_pts?oz{rv sTxviz{}~_ji )xegj¹}~_j9s{\z5h%}gr¥xp¥¼½gjivwpt_pqsTo z?.E_ pqx_e~&h\ jwxFx_p|v{ªXj¸£-·j s?¤vws?»h¨sT~zxz?pq_o~ju{±²¼½´sTpq³ ocµ-x=£"op¥}gxh%e ji i«'pq§x=°d&h¯egz{jpto rqptg·jiz{z{x_}g ji z{j²Bs?Ýok xegp|&j¸~s?zTo~j9jR¤e~s?z?oo~ x_e~x_js5o~zÁz¦Txs?pq¦Tp| j sT__£"s{®jipqxFx£&vz{egz{x_ptptogs?j¤u¸o5ji¦Á¿çs?egz?}grqs}£-ozXjwx_pq¦?pto~s?jiu¸oiª§%h¨z=s?syçgz{jwjipqrtx_ijweXªLh¨·}~cjiIr;oT_xzTsTz{g}gºXo~ptXog·pt}gu¸ogo~us?vx\ptpqsTo~zT}~_\X_x_}gx_eh¨z?zv jix¨x_~ptog_À jwu¸ªx_z{}gs?o~o~9o5j=s?ji}goo~rtIv x_rtsTegx__j¨spqo~s?jiux_zTegvptjieIoc x_jwjweo~z{¦Xo~z{Vr|¸%s?ptx_ocz?sx\o~Ipqz{h¨®xx_gsºjwrt\jwzÁh¨ji¦TzTjws?vocp|e x xegji_j}grq}~x-j£"s{pqyx_ejirq}gjiorh¨jwogocsx£"z?_oz{·o~}go~v x_scpts?_o~_pt\grqg¯sr|z{¦ptu?Xj jRIjws?ºwx_° egjds%v sTxh\z?~j}Xpqx_ocjix%s\sTzTDvwv xsTeg}gjocvwx#s?xh¨egjgptVrqjis?iptªLoTx"£"s?e~ptv¦e¶pqji£ j xz{}goo xrtp egogj¤jiz?v -sTo~_Xx_Fxjwz{h¯ptoTwx ªX}s{"ptogzu\}s?jix_ie=ª£-j¯z?!v pth\z{uT~jirqjiih\° d&jiocegx_j ji¹·}gzov_x_pqpth¨s?og-rtj=z?o~m¯z{}~xeg_jptz?gozxjiz%rqpth¨x_pqo~}~z{v x_x_}gpts?oºji-h¨pth\j x_~e~rqsjiºh\jixocs²x_jisTrq¦Tpto j __Vsji x_v e~pq¢z{¤vix\zz?xx_o~egpts?j²o\±¸z{rt¼ u?´sT£#³ _pqµ#j¸x_ege~Xh¯zÁpt¦T%jij'v}~Vx_rqjw5pqjiogou}~egx_jRs?e~ h¨ji_s?s?joguTrtz?jwIog_pqp x_x_#we~e~jRh\j9jw¿çx_jwp|hDv·}gVrqoptjigv~x_Xptz{s?jiopqjR¤%ptocz{z?x_o~s¯_ jv r|Fgz?xz{_xp|vz¹jRx-rqIe~x_zÁp|}~¦XvpqjiogxoT}gu\x_ptj jRvi¬g\z{zTrN·°5v_x_sTrtµêh¯o xs?egj X _jwz?%h¨~x_sj À h¨vwsjRj z?¤_}gv pt_jwj´ocx_xegj&¿·»~ c}~sTz{z{x_rtpqptxFog u{¼½s?Vgs?x_pte~oTj#x"ptoh¨}ggh%rqjih¨jijwiocªXxogz{x_s{ptx"s?optocªx_£-jwj#¦jiz{hDr|VÀ-jizTg&g£#ji jirqpto z?xo~egj&_z?z?o=h\xj-eg£&jzÁ_'jRe~}~z?r¥xpqgog£-u¨z?z{_rtj u?_sTX_sTpqx_}ge~ghrtj ° xpqsTo²'vwoz{rtx_|egwªXjgsT}gogxyjs?e~oz{o~x_egjDªx_s{egx_je~jwv "sTh¨e~z?go~}Xxz{p¥xyx_pts?z?o=rt_s¨x_pth\pqo~jvwrq}pto~Xvwjirq&}~gxegji-jx_h¨e~jjwh¨vwsTs?x-¤s?h¯z{xegojz{u?¦Ájiz?h¨_ptjws?oc}x#z?rts_s?jR~v&xz?o~pqo~¤g·}~}~v o~jRv ¼ . . . 2. I?
(74) 4@? .. I?
(75) 4-? . . I?
(76) 4-? .. . ààÒ r¡¡¢£. I? $4-? .. ". .
(77) = %&' .1 =&&
(78) ¯xegj ±²´³ µ-~z?v!z?u?j¨¿®x_e~j o}ghDVjw"z{j z?rq£&zÁX&Xo~z{h¨p|vwz{rt ¤z{rtqsXvwz{x_jR~À° sToz?d&oegj-xx_egpth\rts?pto¹ogucR; m pto¤ # jR~v s?_sXov gjR__s{s?§Rs?© }~;gs?u?z{h¯z{o¨z{x_ptsTLv sTh¨~z{ji%x_s Dz?_j&xegj"·sTrqts£"pqo~u XsT}g~g° rtkj ~° ? ±²´³ µ {\kXTg°° pq x ±²´³ µ ?%k?kgg° pq x ±²#³µ Rk k{~~°tp¥?x ±¸´³ µ"k D{g° pqTx ±²´³ µ g? XTg° pq~x ?T° ±¸´³ µ RTk?? ÁgkXg°° pqT?x ±²´³ µ"k{{ k? XX~°° p¥TEx ±²#³µ-TT gD% XXg°° pq x k g° ? X° i ~°qT dç&zTe~vÇ xj¯ s?¢~ rqjRFà x _ ixjwe~h¯z{Ç z{o 9TÀp|v Ãs?xh¨e~ z~ x%z?_x_jRegj¤x ss¦Tz jw~egzTji!v zT9z{u?ptj#o~Xx_e}zv x;ji¹}~_jiºe~±²z?´g³ £-µz?pq_o j#»~?sTz{gx_ptpqogx%u h\sTsXpqocXx´j¯z{p| pqx_egjih¨zTj sTxo~ptv?z{° ~rqd&j²e~¿çptz s¦T±²jw#egji³zTµ²p|p| gX}g}gptr¥j¨x%x}gssTx_egoIj¤±²X´o~³z?æh¨pt£"v¨egh¯p|vz{eop|z{u?ji~h¨zTjwjRoc9xsTs{oI&h\mji±¸h¨ s?5ª¿ç£"z?egrqtp|sXvvwe¹zxptpqDsTov s?s?h¨#~oz?}gh%z?grtjijDxsÀ pqxo¶egj¤mv ±²scF x s?pto)´z{zT_p¥_xjwegh%h¨gj rtxptv%r|z{s?ogVuTjw}~zz{xuTpqj=sTo~z{"rt s·sT£ x_sh¯z{Vrtjw_~·s?_jRh v p|ptj s?¤o~v ipt ° jwocxsrq¹£-jws?¦?VjijwiªzegxpqptsTu?o~egrt ²£"s?pqXx_expqh¨gzp #wxjRz¸²ji·o~}gv o~sXv Xx_ptjRs?¶o~ £"£"pqx_e¶p¥x_·x_jiji£ o h¯jw¬XzTgvrtegz?pqpqo~oj ;£"£-es?Dgx"egz?j o~s¦?jwx_egegp|"jRz?g%jwpqh¨¦Tjwgocrtxpq-jR¤xegpt¨sz ¦?Xjws?eg}~jRgz?rqpt¯ogu%·s{s?§h x_e~jyjipqgo~u¯jivwrtptz?__pquTsTjwo¯-pt´·s?ogs{_xh¯ptvwz{jirtz?ggrqj'jiFv xp|z{__pqsTx_pto~ogwu°;·d&s?e~h pt RTk? j gx"pqx}~ "¿Oz{Tr|k%_sDh¯egz?ptvu?egegptrtogp u?jec£-x#s?xeggjsÀ#¦?sTjiog_rqe~Tji° z?¨pto~vw}g_ji=¤}~viv jR__pq¦Tj gs?}ggrtp oguT-s{x_egjgjivwpt_pqsTo© x_egj spto=¦Tjwh%eg}gjirqx_zTptgrqçj z?gv x_sTji vwptp|_ pqsTrqojR_ªz{o~sT¨xjieg cj'}~z{çz?r vxxss?-~ªjw£"xFeg£#p|jivjweo=p|±¸X´}g³j¨ xs=z?o~xeg¤j¤±²v ´scF³ xµ#s{_ jih¨z?z{pqoz´x_v }gsT~o~z=xhDz{oc}gx&rqx_z{ptgo~rt¯p vijRz cxpq}~sTz{o r xsXs\viv k }g·sTÀ´° xegp|´gs?uTz?h sTo=±¼½h¨z{x_ptvwji'¿·pqx-p|´}~gji#Vs?}~o~Xji=\D£"egjwou?jwo~jwz{rgh%}gr¥xpq~rqp|vwz{x_pts?o~ µê¦o p|XÇ ?ji '¤~cp¥\xà Lxh¨egij sgÇ j?ªT x_ egj"\à pt~oc~z?x_ ?jwv! ¦z?z{u?r|gj? ªTgÄ£"segÇ gj x}~eg v jwjR#%à x_e~ DÃj ±¸_s?´&·rtR}X³ x_ptµs?z?o=_j"s?#z?rqx_£&egzÁj'X_ jRp|x_X}~p|vz?x_rrtDp|#pto~vwvws?rqo~}~vwgjwji¨ogjiptoTx° s d&x_egegp|sc#ptj"&gX}gs{j ¼ xs?s¯x_s?e~}gj o}~X_pqj?o~ª~u\ptoh¨sXXjR w°L§d&ªXegs{j;z¯ji__}gs{rq®xxF&£&s?z{gxj z{ ptoggjRrts=£"pqogu 's{ x_eg¤jDz{pqocjx_jij _¬¦x_z?jwrt&h¨xjis=rq=zÁ¦?vwsTrqscpt=j'·x_s¨ji cx}gegjwsToc_x'j_v £"sTpqh\xv~eg}Xptogx_jRuT =±¸´³ µ#}~_ptogu ?kDgpqxi° »ptocsTx_zji-x_pq¦jwo~z{ptu{ogr¼½u²z{sTz{p¥pqx~ocegrqxDj\h¨ox_j }gs²x_h%p|v£-rts?pqji~²z?À £"_rt¤pqsx_sTe_z?pqX_o~jiTu¸%s?s?gogopqrtx¶e~z?s?z²-gç~zT£-_vz?jRxs?_v jp|¨ptXz{s?sTVo }gs?g¿}X
(79) rtxji "R9gk9Rrtjiv ¤zTs?gxh¨pqh¨~ptz?jRo_jRçh¨zÁº¦?s?sTx_s¹}gj%"z¸xs{e~h¨;z{±²oºs?#£"j³pqvwx_µs?e¹©Vopqe~¦?xyz{jiptoc'Xx_ptv £&s?sTo~z{h%z?j¼r ·s?sT§x_xegz{j grt}gj#o~x_gs jwXrqji¦?ptjwogrtu s?\»~scz{zo¨xz{pqogrtu?u?sT¼NV_pqs?x_ptegoch x#z{z{o~pqx_Degx_h¨sj v xsTptv?_«cjix_v egxp|-z rtr|zTz?v!!v \s?s?Az?zTvivwv v }g}~z?zTvwv \viz{ o¤pqoVv j z{jiz?xx__ptogptguy}Xxxegjij"¤gjip¥xjiegvwjwpt_#pqsTx_os . . I?
(80) 4-? .. I? $4-? .. . . . . $ . . . '. 2. . . . . $. . . $. . $. . . . . .
(81) . . . I?
(82) 4-? . I? $4-? .. . . . = I?
(83) 4@? .. . ä·Î àä®Í.
(84) ·zDsTz{" TV}~pqjwoptRv!F°¨xz{z{o~o~vw¨jRÀ#o~s?s{-x´x_¦Ts\jwx%egjwvwsTz?__jwjwx_·}gp|vwr~z{jwr§¬Xgji_zTjR_sT_o~pts?-o¨}~s{v§exegz?j-gxegs?sTg_rtjjwh h¨jw¿·ocpto¯x_pts?xegogjjR=¢~ptox#x_Fegxj z{uT_j"jivws{s?§o~x_e~j'jRXvjix¦?pqsTjwo=rts?s{~ h\x_jie~ocptx .
(85) !"
(86) # $ % &' )(+*,
(87) - .,0/2143. qu{ t4prq nvw. ·d&vwsTpte~_ptjws?j o9¬gjz{s?z?h¨;_jgx_e~rt_j¨jwj\¦?s{sTji-}Xz?x_x_eg~rAj¨}Xh\x·s?sT£"x_rqptteg¦?sjRjw£"´optogxp¥s¨ux}~p|'sTjo~pqocs{ptxocxfix_©ji_og¦¦sz{z{£"rtrp ~o¸z?z_·x_pqx_pts?s?e~hoºh\jws{xx_&egp|v jDo}gpqxoh¨egjw_jwsTvwpq_ptjivigocz{À x_rLª§pq¢z{j ¬gvrtu?z?vwsTv z?_xpqrtx_vwvie~}gz{h¨rtr|z{v ¯x_}gpts?r|¿OzFoxx°2pq}~sT'Xo~og%js{¿ç£"-vwz?egx_o=egjwj¯xj\eggx_ptoregj j ¼ c_pqeg}~x_segz?£ rqh¯p x#ptz{o¸x__ptjR¦?x_ Te~j }~jDz?pq·_pqjio~£3jRu v·x"u?s?rts{rts?q~sx£"z?egr~ptjogptoXu¤_·jRs?jw¬g}~h¯z{r¥x"h¨zh¯xgpqsTrtzÁjio¤¨yz{x_o~e~j¨z{h¨xyxs?ex_e~}~j'j%#ptg}~h¨ji__jDpqsTuTs?og´xjiz{hD¨oc}~x-xr¥sDx_xe~p¥£-¼½z?gs?o= ji%xv egp|£"_jpqpqsT Tx_oe=}z{»~pqococsTx_x_z{jipqxx__z{pt¦ogx_z?u{ptrV¦?¼½jz?_sTz?pqpqx__ocegx h¨jRptvjw xx_À v p|ªgv{z{}~° rtv u?p|s?z{j ¼r ptoh¯z{o=_p¥x}~zxpqsTo~w° ·~sTrqyj¯'p|x_¨}~Xh\ji_jipteogocz{u¸x_~pts?°i og°wx°jRegs?¶x_j¤egpto¶jisTyrqx_}gz{e~x_jptpqs?x_ptego¹och¨x_s{j #sxgx_ptvieg}~wj¤vªVxz{XpqsTo~ptojiª;voe~x@z{zÁh¨¦ptptogjwogjwrtu¤h¯xzz{egx_gj%p|vwzTjivwv gv p|_}~j=s?gzTptrtoTv jw¤xh jws?¦ ·s?z{xr"egj z{~z?p¥jixz?eg_rq}gh¨tjwrqxij rL=° xpt's?vVogh¯s{j xzÁ-uTºsc ksXV%jN°¤j }~¬g_d&z?j e~h%·}gpt¼r ·~jisT__z?sTh%rgvw}grqjizTr|h zjTx_pt°vis?z{oºo¹jDjw¬Xz?j=pto~j z{x¬XÀ rt gz{#wojijiDog_jRs?fsºh¨zTjDog svwz{£"s?oIh¨o%z?ogrq}g}XuTh¨xjwzgjwx_ptz{s?ptvip|ov¨z{z{rXrh¨Xx_j x_xjwzegh xsXjw%u?Xptz?jiji&rqtsji£"o~£e~Xwptptªvoge¸o~u¸sz?£-s?_ozTjg}~zÁ_X}~xwsIz?ªRrqx_tRsk_ji¦s?h¨z{rt¦?gj&p|rqz{spqgxTrtjiptjio\¹¯x_¿·¿Oegh¯jij"z?¦?uTpqjwogjwoXrtz{ ¼r »jio~pqsTs?uTogzjwxj opq¦o~s{z{u{rt¼½}gxegjysTjpqvwocs?x\sTh¨rqz?}gg_x_}Xpqptx_s?xe~zo~h\x_&pts?jwv ox_sTp|}gv=À;r|¤g¶}XVx´j±²xz?e~#gz³gx-µçsÁ°;z{¬Xptd&rVpqh¯e~s?jo=zx_jijis?§_h¨ªc}g£"rqjx\egjw£&¬Xji_z?z?jRh¨¨z?-g_}grqz{jR_rtw~Aªc_x_p|eg¯jptog_upts?h¨©¯rt}Xg}~x_rqpt¨s?ptogo~u¹jw#g£-v r|r|z?jwz?vwpqj o~p|vwuvws?z?x_h¨r-egj'z?g_}Xe~pqxx_z{jie~¤h\X£&£"jwx_z{pqp|x_v{e j ª sX£-z\gjDuTscesXvwz?z?²o z?xvwsvwsT}gvwrq¦Ts?j²z?o~v j¨x_¤}~}~v _v ptptx jwogocu_x_s?rtRh¨ k{j%%x_egh¯gp|pq¤z{x"x_~s?_ptvwsTjigg'rqji}~jih vwptptog_z{ptus?xo=j g¬g£"z?jipqv _x_x jwe²ocvw±²xRs?° h¨#g³~³}Xs?µx=°z³gx_x_}gegpts?_jox_eg%j ji¬X¿N_m h¨jisT_Tpt_h¨gjTogªcjwjwsTocogxrt~¯zTX·jijwjR£ v À"j ptx_¬gsjRz?zÁv x&¦?z{sTh¨Vpt²jwsx_¦?z?eXj?o ¼ ª zTvwvw}ghD}gr|zxpqsTo s{oc}~h\ji_p|vwz?rTji_s?i°d&egp|p| s?ogj-£&zÁ'xs'h¨pq¬Dj ¬gzTvx z?o~ o}gh¨ji_p|vwz?rvws?h¨g}XxzxpqsTo~w° zptTr gq}~Xypqx_x_xpts?zego~xjwj z?#rgx_j egx¬Xegp|"jwjisT£"__ptpqjwh¨x_x_ep|jwvwoch¨z{xr´j £#x_e~e~sTzÁs ¦?A~j""ªT~hD~}Xz?}gx#_rqx_jipto¤pq¼N~jws?_¦o jRp|v X'p|jwojipts?v o¯vwj"z{x_e~zx_xjRzpqx´sT -o~ptocz{_x_r}~jirqoc_j¦}~z{s{hDrVLVz{_jwpquTp¥x;o~egwpth¨o¤° j j x¬gptv"z?v h¯x-zÁvw\s?h¨_jigg}Xr|z?xzv x_jTptªTs?z{o®wx_°jw-jwsTx#h¨}~j p|vi´z{o9z?rq&gjiz?jizT_z{p|Xr vwS z{£"rtpq¸ogx_e¸s¿ç£"_sTjwo¨,j xxeg s Ts?k uTVs?jy·o~sT¦?z?"jir_j \¬gsTz{j rqh¨¤oggvwsTpqrtjih¨jRoTÀ ptªx´z?x_·rteg's?j´rqp p|jw?_¬gj s?z?r|zvx_x"ptpqog¦Tr ujwpqoxegpts?sTjyoo ji's{z? s?rgx_ egscj s?eg_xs{jw´¼ês{vwX§z?_rq}gegtjiogjw0 pt¦ ¦ yz{_s?ptz{o~x_jijijo~iuªVVjws? xrtv? "og°'z{s?s?rth¨´u?s?p|Xz{jwr|pquTx_weg© _jih p¥jx uTp|"_jRXzjwx¢~jwog#jix_e~z?Vojwrtk?s?£?° ~°ds¯}gh¨h¯z{p #wj?ªcxegjz{rtu?s?pqx_egh p|-~z?_jis?o yjR_viz{_x_ji #}grtj's? _pquTo~-£"egp|ve. ²Ç ig Ç
(88) -. l - )(r _pquTo u & *, R %! . . l.f.E
(89) l )(r o}gh%jis?X_ptu?o ve~z?ogu?jR -( &" u@ & *,
(90) Su , l.
(91) y . . . . . a>0. sign(a). −1. a<0 IR \ {0}. V (a1 ) = 0, V (a1 , · · · , ak ) =. ààÒ r¡¡¢£. k. . . V (a1 , · · · , ak−1 ) + 1 V (a1 , · · · , ak−1 ). a ∈ IR V (a). 0 a=0 1 a = (a1 , · · · , ak ). sign(a )(r L" .. k−1 ak ). = −1,.
(92) R. = %&' .1=&&
(93) . - , E$ .)(+" l
(94) #$ &"= - & *,
(95) ( R*, B $ > b "S( -&" E$ !.0, *-> + > f a .E
(96) l V (a)IR= V (b) . 'ptogu\x_e~j z{Vs¦?j ogs{xzx_pts?owª 'jivwz?xji -}grqjs? _pquTo~&p|-x_egj·sTrqts£"pqo~u~© ! # B & *
(97) &[ ÃÇ Ã Ã R ?XÁ à q\ & òÇ
(98) 3
(99) B.E $ B )(r *,! , j( 0 |( 0&". . . @. . . *-! uf% %V (P !) & "@ V (P ) − pos(P ). P. P =. Pd. ( .. a i xi IR[x] (a0 , . . . , ad ) pos(P ) pos(P ) ≤ V (P ). ! $ !. L( *& l&
(100)
(101) )(r i=0. . °j x#'}j ´¢ogz?pqo~_}gu h¨j"x_e~z{x p|´z c}~z{ª~p¥j ¼Ny·jRjw_j"viVz{_s?x_rtjico~ s?#h¨}gptz?rtjrgs{s{Lgptjwu?u?o~-jwj uTpq¦Tjipqo ¿ç_jR£"°p¥xce=À-z{£"rtVegp¥jwxo;z{sc~s?gxrq´ptjipqo xo~ssx_egsjys{xÀDsTrqpto5ogsTxh\egjp|z{ptr ocx_ji_¦z{r ° d&egjwsTª_xjiegh¯jw%o¯Xxeg}~jjVxs?s rt ogsTs?rqh¨tp o~p|z{ r{ e~z?p¥x;z?j 0¬g z?%v yx_rtz?Do~s?o~'j"pq_osv s{jwxyoc¿çxB_jR %° Vxs?z{rtx_j=ogs?x_e~h¨z{p|x%z{r ·sT\_} ¤v ptjwocx_rtIr|z{u?j¤¦z{rt}g£"jRptDrqs?z? rq£&zÁª X'-jiu?ptvw¦?z{j _x_jR sT }g° rtj=s{ ptu?o~%£"egjio)z{~grqptjiIx_sx_egj zTvwvw}gd&egzTjv ¯h¯ogz{}~jipt_jiojiX_jRjih¨X·}gz?s? "pt¯ogxu¹egp|-jx_x_egejij¸z_x"}gvwr¥s?£#xh¨js?g;z?}X_v j xs?zh¨s?xogpqgsTrt}go~¤x\ptzoc·xx_s?pqjwsTo~gji"p|5x_vwjRz?x=oegj²ptoj }~x__egVj²jss{s?_RpquTY° e~o~yz{&ss{X£#£&jiv z{¦?sjwj j=R¤ªc»x_v sTegptjwzj ocxx²x_wz{ªg¿·jio~z?¦?o~jw·¯s?o xh¯ptecoc}zx_x_&jwpts?x_¦egoz{j r rtVz?p _h\s?pqrtx_pqe~xogz{h\s?x_ptjwh¨s?x_o~p|p|v²z{¨r ~sTzTo ptjRyx_)eg_j² csT}~ojwz?¬_e~Vjwz{s?¼O·o~Xjwjw£&ocjTx©z{ xLjegp|»~z?ysTogv z{u?sTxjTo~° XÀ%pqx_±²·s?pts?s?o_jwe~s?szTrt¦T¦&jwpqRo~ogªuIs¯x_eggj¸sTjirqv z?p|ogrq_uTsTjs?h\h\p¥p|xz{jRegr|z{h¯ogpts?og£"u¨pte~rq£""pquTegxe jwjioh¨Xjwx_pqeguTo~_j z{jix_vwj?jscª-j s?¤o~jRrqv vw¶ptz{jw}ocpqxj¸"x_egs?s?j z{±²jsT__jijigs_¦?jRjiijiªcoc£"x_jieg=jwo¯jw»~gsTr|z?z{vwxpq&o~sTu%"pqzocxx_pqjisT_o~¦z{z?r§rtiv ° sj ¯vwpqjiocx&£"pqx_epqocxjw¦Áz?rtiªxF£-s\~_sTgrqjih¯&h¨zÁ=sviv }gR©
(102) zr|z?!v s{go}gh¨ji_p|vwz?rTz?viv }gz?vw'h¯zÁpto~X}~vwj´gs?grtjwh¯pqo%ptu?oDgj x_ji_h¨pto~zxpqsTo~-¿ viz{oDz?gVjiz{ pqoptocx_jw¦z{r|À «
(103) x_egj²Vs?rtogs?h¨p|z{r"h¯zÁ¶ogs?x¤j¸ c}~z{j ¼N·jwj¹¿çzÁ5zºsTrqogsTh\p|z{r"£"pqx_e ptocx_jw¦z{rvwscj ¤v ptjwocx _jigji_jwocx&z¯_j x"s{sTrqogsTh¨ptz?rt#x_ezxh¯zÁ=v sTocxz{ptoz%o~s?o_ c}~z?_jw¼O·jwj s?ogjÁÀ° ox_zsTx_gpt's?go\ptog_u eg£"sv pt£"rqpqx_oIogjws?ptptdox's?oV Tª?j%kx_egV½ªjisTo¨x_e~x_ptjigeg_rtj j¤j?ªAz{xFrt£-£"u?segs?p|vpq_x_e¸p¥xeg}~hh¨zjix£"pqz?sTo~pqo~rt z?xrtrqe~ji£&zzTzÁxR¹XªA x_p¥#s9x_jwzx_h¨egptju?pqoo_zz?x_Xh¨jTj ° j¤xjwd&ge~h\j"s?ptgo~¢~rtz{o~jwx_hz?pts?r~©¤oji_ç_s?z{}gh¨ptr¥r x#}gj_£"j%_pqpqrtuTpt~'o¶V}~jyX_jij z x_jwj z?x´h¨ys?p¥z ¼ p|o~_s{s?xr|zv x_s?ptogocu¨xz{ptptoTo9xjwji¦z{z{rr|"sz{s{o~x²Àz¯-° _'j ogxj¯s? vwz{ptoco9x_jwx_eg¦jiz{or|&pto~·s?vwy_jR£"z?eg_j ptvx_ee~ogj\s¤~g_jRjiv v p|p|_ptpqs?sToos{p|";VxegsTj¤z{ptgrtp¥j¤xeg¿®h¨xegj jwxptvDh¯x_s²zÁ¤v sTs?ocx_h¯ptoc}~zÁj x£"ege~j%jwvwjs?h¨p¥x'gç}Xz{xptrqzjRx§pqsTª'o~
Documents relatifs
If the temperature of the collapse increases due to an increase in the specific heat ratio (for example by lower initial concentrations of the chlorinated compound) the ratio
If a reproducible evolution of the refraction index with temperature and degree of conversion could be available, an absolute value of the degree of conversion would be
À partir du moment où la montagne Pelée se réveille, temps au bout duquel une grande éruption peut se reproduire (d’après la population interrogée
Contanto, é importante ressaltar que as cultivares ‘Salustiana’, ‘Pera’ IAC, ‘Pera’ “Jetibá”, são mais propícias para suco, podendo a ‘Salustiana’
Si l’examen du lien qui unit langue et identité trouve dans la première approche une résonnance due au fait que tout sujet parlant construit son identité par le biais de sa
Ainda segundo os autores, outro ponto importante a ser levado em consideração no combate à disseminação seria o rastreamento, por parte dos órgãos competentes, a partir
En vertu du principe frégéen de compositionnalité qui règle dans les linguistiques formelles les rapports entre syntaxe et sémantique, la composition syntaxique est
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des
