• Aucun résultat trouvé

Duality and Separation Theorems in Idempotent Semimodules

N/A
N/A
Protected

Academic year: 2021

Partager "Duality and Separation Theorems in Idempotent Semimodules"

Copied!
30
0
0

Texte intégral

(1)Duality and Separation Theorems in Idempotent Semimodules Guy Cohen, Stéphane Gaubert, Jean-Pierre Quadrat. To cite this version: Guy Cohen, Stéphane Gaubert, Jean-Pierre Quadrat. Duality and Separation Theorems in Idempotent Semimodules. [Research Report] RR-4668, INRIA. 2002. �inria-00071917�. HAL Id: inria-00071917 https://hal.inria.fr/inria-00071917 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Duality and Separation Theorems in Idempotent Semimodules Guy Cohen — Stéphane Gaubert — Jean-Pierre Quadrat. N° 4668 December 2002. ISSN 0249-6399. ISRN INRIA/RR--4668--FR+ENG. THÈME 4. apport de recherche.

(3)

(4) 

(5)    !"!#%$&')(*!#+, -!. /!#%#+0!$ 13254&687:9;=<">@?BADC=E59!F*<;G1HF*25I;=JKA

(6) >L;

(7) F*<NMOPQ;=JKJD;GRS2!F:TNJUF*A VWXUY[Z]\3^`_:abYdcegfihjalknmoZDh0kqp-hjalY[agr'f

(8) h'abkqm s Ztr'u:rh'XvY[Zvr0wDknY[pelZUx-Zvr y"z'kq{ ZDh|}ZUhjf

(9) egf

(10) c ~fqppBknzh s Z,zjZvwW€ZUzwWZmƒ‚\*„q„q †^ˆ‡‰ZvwUZUYSŠBZvz‰‹

(11) ŒnŒn‹8^‹

(12) „3p€f

(13) ŽnZvr ‘ ’v“U”i•–q“q—8˜ Z]wDknm€r'a s Zvz.r'cŠ€r'ZUY[abY3k s c€elZ™r.fqm s wDknm:šqZDx[rc€Š€rZUhjr.k

(14) ›!r'ZUY[abY3k s c€elZ™rkišnZUzr'ZUY[abz'abmŽnr alh'Wf

(15) mža s ZUY[pƒk

(16) h'Zvm*hGf s€s aŸhjalknm  ˜ Z¡alm*hjz'k s c€wUZ¢f£mknmebalmZ™f

(17) z¤pz'kq{ ZvwDh'abkqm/kqmr'cŠ€r'ZUY[abY3k s c€elZ™rU¥ hjœ WZ¡pzjk

(18) { Z™wKhjalknm¦kq›tf£pBknalm*h‘agrHhjWZ¢YHf

(19) x:abYHf

(20) efqppzjk=x-alYHfihjalknm§›QzjkqY¨ŠƒZUebk œ kq›]h'WZ¡pƒkqabm*h‘abmžh'WZ r'cŠ€r'ZUY[abY[k s celZn  ˜ Z¤c€r'Z¤h'Wagr[pzjk

(21) { ZvwDh'abkqm£hjk r'ZUpBf

(22) zfih'ZGf¢pƒkqabm*hS›QzjkqY©f wDknm:šqZDx&r'ZDh™  ˜ Z}f

(23) egr'k r'Wk hjW€fihªh'WZ8pzjk

(24) { Z™wKh'abkqm‘Y3abmabY[al«vZvr.h'WZ]f

(25) mBf

(26) ebkqŽqc€Zk

(27) ›@¬alebŠƒZUz'hv­ r.p€z'kq{ ZvwKhjalšnZY[ZDhjz'agw

(28)   ˜ Z s ZUšqZvelknp Y[knz'œ ZŽnZUmZvzjfqelebuSf,h'WZvkqzjuSkq› s c€f

(29) eƒp€f

(30) abzr›Qkqz.a s ZvY[pBkqh'ZUm*hrZvY3abY[k s cebZvrv  ˜ ZknŠ-hjfqalm®fnrfSwDkqzjkqebegf

(31) zju s c€f

(32) ebaŸh uozjZvr'celhjr8ŠBZUh œ ZvZUm¢h'WZ3z'k œ fqm s wUkqebcY[m†r'p€fqwUZvr‰k

(33) ›YHfihjz'agwDZ™r œ alh'W†ZUm*h'zjabZvr‰alm†a s ZvY[pBkqh'ZUm*h r'ZUY[abz'abmŽ*rU  ˜ Z‰alebelcBr hjzjf

(34) h'Zh'W€Zz'Z™rc€eŸhr"Š:uSr'Wk œ abmŽ,pBknelu:WZ s zf,f

(35) m s W€fqeŸ›°¯±r'p€fqwUZvrkišqZvzh'WZ‰YHfix:¯²p€elc€r r'ZUY[abz'abmŽB  ³}´*µN¶j·d¸ ”i¹’q— |¡fix:¯ºpelcBrtrZvY[alzjalm€Ž€»@r'ZUY[abY[k s celZ™rU»¬fqWm-¯±¼f

(36) m€fnwW¢hjWZUknz'ZvYG»ebalmZ™f

(37) z,ZUx*hjZUm€r'abkqm» s c€f

(38) ebaŸh un» s cBf

(39) e!pBf

(40) abzjrv»€pzjk

(41) { Z™wKhjalknm»zjZvr'a s cBfih'abkqm!»B½]f

(42) ebkqagrwUkqmmZ™wKhjalknm»ŽqZvmZUzf

(43) ebab«UZ s wUkqmi{ cŽ*fqwUalZ™rU»zjk œ r'p€fnwDZq»wUkqebcY[mGr'p€fnwDZq . ʲâU¿=͙ÐÁÏʲÅÏРоÖ0ÁÇп=ÁQÈçÇÀÂÁÅÇè±Ýºé ¿tÃ5ÄKÔ=ÅÇбƑʰÃ5ÃÄKÈÅÏÁ]Æ]ÉiÞ@Èß:ÅQÞÊÇÀËÈàBÌÂÓƒÌ ÍGáÁÇӃÎ=âdÉ=É

(44) ãÂäÏÄK¾!ÅÏÊÇ¿™Ð ÐцÞ@ÒDÌÂÍGÛKÐ Ò=ÊÇ¿=ŲÈÐSÀÂÝÓBÞÎ=ÅÏəÄKÉ=É

(45) ÅÇÐÄDÈÈÔ Ý²¿tÕNÊÇÄKÄÖ0à-Ö.Ð ÅÏÎ=ålÔ=ÄKÀÅÇÊ°Ö͑È×iԙŲÝÈЪÖ0ӀбæÃ5ÈÄKÌÂÎiÅÏÆGÈjʲÀÂØgÄKÙÚÔ,É=ÄåBÅÇÄKáÛKÀÂÅÇÁÇÈÝ ÖÜÅÇбÊÇʲп™Ó€ÅÇÄKæDÎ=Ð ÛKÔD¿ Ê. Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France) Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30.

(46)   5    !#+!$&     .$  !$¦!#%#+0‰ !$ (n!#+, -!. $  ’. . —

(47).  s.   kqcBrdwDknm€r'a z'knm€r]ebZvrtr'kqc€r¯±rZvY3al¯ºY3k s cebZvr]ZUhSrknc€r¯²Zvm€rZvYdŠebZvr,wUkqm:šqZUx-Zvr s Z™rtrZvY[aŸ¯ s s Y[k cebZvr0r'cz Z™rªr'ZUY[al¯ºfqmmZ™f

(48) c-x s knm*hªe²­ f ss alh'abkqmGZvrh0a s ZvY[pBkqh'ZUm*hjZq  knc€rªalm*hjz'k s cagr'kqm€rªcmZ]pzjk

(49) ¯ { Z™wKhjalknm mknm/elabm fqalzjZorczHc€m/r'kqcBr ¯±rZvY[aŸ¯ºY[k s celZn¥¤egf pzjk

(50) { ZvwDh'abkqm s ­ c€m¦pƒkqabm*h®ZvrhHelZ}pebc€r[Žnzjfqm s e Y[ZUm*h s c¤rknc€r ¯±r'ZUY[aŸ¯ºY[k s ceb Z nc€a€Y[alm€kqzjZwDZ‰pBknalm*hv  kqc€rc-h'abelagr'kqm€r.elZ‰pzjk

(51) { ZDh pBknczr p€f

(52) zjZUz"c€m pƒkqabm*h s ­ cmowDknm:šqZDx-Zn  kqc€rªY[kqm*h'zjkqmBrªfqc€rjra *cZ]elZ8pzjk

(53) { ZDh Y[abmabY3agr'Z8e²­ f

(54) m€fqelknŽqcZ s Z8egfSY h'zja *cZ p€z'kq{ ZvwKhjalšnZ s Z¬abebŠBZvzh™  kqc€r s šnZUebkqppƒkqm€rp€elc€rŽ m zf

(55) ebZUY[ZUm*hcmZh'W kqzjalZ s Z™r"r'ZUY[aŸ¯ºY[k s cebZvrZvm s c€f

(56) ebaŸh   knc€r.kqŠ-hjZUmknm€rwUkqY[Y[Z‰wDknz'knelegf

(57) abzjZ s Zvr.z rcelhjf

(58) hjr s Z s cBf

(59) ebaŸh Zvmnhjz'ZeÇ­ Zvr'p€fqwUZ s Z™rebalŽnmZvr.ZDh e²­ Zvr'p€fqwUZ s Z™r]wDkqebkqm€mZvr s Z[YHfih'zjagwDZvr GwDk:Z ®wUalZvmnhr s f

(60) m€r s Z™r]rZvY3al¯±f

(61) mmZ™f

(62) c-x}a s ZUY[pƒk

(63) hjZUm*hjrv  kqc€r abebelc€rh'zjkqmBr‰ebZvr8z r'celhjfihrZvm†Y[knmnhjzjfqm*h *cZUe *cZvr8pƒkqebuqX s z'Z™r8ZDh s ZUY[al¯²Z™rp€fnwDZ™r8r'cz]ebZ3r'ZUY[al¯ºfqmmZvfqc YHf

(64) x:p€elc€rv   ¸ “™’ ¶ –   ’ — _-ZUY[abfqmmZ™f

(65) c YHfix:¯²p€elc€rv»@r'ZUY[abY[k s celZ™rU»!h'W kqzjXUY[Z s Z‘¬‰f

(66) Wm¯º¼f

(67) mBfqwW»p€z'knelknmŽqZU¯ Y[Zvmnh.elabm fqalzjZq» s c€f

(68) ebalh »*pzjk

(69) { Z™wKh'abkqm!»qz r'a s cBfih'abkqm!»qwDknz'zjZvr'pBknm s fqm€wDZ s Z‰½]fqelknabrv»nwUkqmi{ cŽ*f

(70) agrknm€rŽ m ¯ zf

(71) ebagr Zvrv»*Z™rpBfqwDZ s Z™relabŽqm€Zvrv»-Zvr'p€fqwUZ s Zvr0wDknelknmmZ™rU .

(72) 

(73)   !#"$% '&#(#")* # + #", "-./01$. 2. 3547698 :689;.   =±m*h'zjk s c€wKhjalknm <> ‹  y"zjZUebalY[abm€f

(74) zjalZ™r ‹  <  A@.kqY[pebZDhjZ%B‰z s Zvz'Z s _:ZUhjrfqm s ~Zvr'a s cBfih'Z s |¡f

(75) p€r ‹  ‹ _:ZUY[abY[k s celZ™rkišqZvzC= s ZUY[pBkqh'Zvmnh‰_-ZUY[alzjabmŽnr ‹  2€A   B‰ppƒknr'aŸhjZ,_-ZUY[alY[k s cebZvr 2€  knmelabmZ™f

(76) zoy"zjk

(77) { ZvwDh'knzjrv)» Dm€alšnZUzr'fqe]_:Zvp€f

(78) zfihjalknm VWZvkqzjZUY f

(79) m s ¬abebŠBZvzh}pzjk

(80) { Z™wKh'abšqZ Y[ZDh'zjagw 2€  <   knmelabmZ™f

(81) zy"zjk

(82) { ZvwDh'knz 2€  ‹F   DmabšqZvzjrjf

(83) e_:Zvp€f

(84) zfihjalknm¤VWZvkqzjZUY 2€  2€ ½8ZUmZvzjfqelab«UZ s ¬alebŠBZvzhp€z'kq{ ZvwKhjalšnZ]Y3ZUh'zjabw 2€  \BF   Gh k s abY[ZUm€r'alknm€f

(85) e5ZUxf

(86) Y[pebZ \B  ‡cBf

(87) e@œ _:ZvY[alY[k s cebZvr0fqm s ¬f

(88) W€m-¯º¼fqm€fqwWGVWZUknz'ZvY[r \B  <   ‡cBf

(89) ef

(90) m s yz'Z s c€f

(91) e5yfqalzr \B  ‹>   =±m:šqkqebc-hjalknm€r \B  2€  ~01Z IBZDx-alšnZ]_-ZUY[alzjabmŽnr \B  \B  ~0ZUp€z'Z™rZvmnhfihjalknm‘k

(92) K› JabmZvfqMz Lknz'YHr \B  ?F   Gpp€elagwUf

(93) h'abkqm¥ s c€fqelalh u‘ŠƒZDh œ ZUZvm¤zjk œ fqm s wDkqebcY[morp€fnwDZ™r ~0ZU›QZUzjZUm€wUZvr < %  O 68P4RQ7S7TU8VW4R6. 2 ? ?. „ E. E <Œ. <2 <? <H <H <E. ‹qŒ ‹n‹ ‹

(94) \ ‹N?. =±m¢h'Wagr8p€f

(95) pƒZUz™» œ ZSrh'c s u¡rZvY[alY[k s cebZvr‰kišqZvz‰r'ZUY[alzjabmŽnr œ Wknr'ZSf ss alh'abkqm¡agra s ZvY3pƒk

(96) hjZUm*hv»NhjW€fih œ ,Z V wUWfqelZ†eXh Nu:1p")agwU*f

(97) e‰ #ZDx +f

(98) Y[$#1p"YebZ¢ "-kqZ›dr/Zv0Y[[$Kal  zjalm€Ž aŸhjWfqma s ZUY[pBkqh'Zvmnhof ss aŸhjalknmžh'W€f

(99) h Z¢r'W€f

(100) ebe]wDkqmBra s ZUz agrdh'WZ\"Z]N^_`a$-$##", [ ab

(101) » R » œ Wagwœ W&agrthjWZGr'ZDh R ∪ {−∞} »Z *calp€pBZ s œ œ aŸhjW&hjWZGf ss alh'abkqm f

(102) m s h'WZ.Ydcelh'abpebabwvfihjalknm   ˜ ZrWBf

(103) ebe*fqebr'kwDknm€r'a s Zvz!hjWZwDkqY[pebZDhjZ s (a, b) 7→ max(a, b) YHf

(104) x*¯ºpebc€r®r'ZUY[abz'abmŽ€» R » œ W€abwWžagr®knŠ-hjfq(a,almZ b)s 7→Š*u/a+b f s { kqabmalm€Ž h'k R f +∞ ZvelZvY[ZUm*hv VWZ cd.0e\$##", [ ab abr‰fHrc€Š€rZvY[alzjalm€ŽHk

(105) › f

(106) m s R f kqŠ-hf

(107) abmZ s Š:hu gqZUZvpabmŽHkqmebu®hjWZt«vZUzjk ZvelZvY[ZUm*hv» −∞ »€f

(108) m s Bh'W€Z,cmalhZUebZUY[ZUm*hv» 0i   R = s ZvY[pBkqh'ZUm*h[rZvY3abY[k s cebZvrtabm€wUelc s Z‘f}m:cYdŠƒZUzSk

(109) ››Ïf

(110) Y[abelagf

(111) zdZDxf

(112) Y[pebZvrvj   Lkqzdalm€rhjfqm€wDZn»h'WZ‘r'ZDh kq›dwDknm*šnZDx¦›Qcm€wDh'abkqm€r s #Z k€mZ s kqm f§šqZ™wKh'knzGr'p€fnwDZ¢wvf

(113) mŠƒZ¢hjWkqc€ŽqW*hGkq›dfqrGf§r'ZUY[abY3k s c€elZ kišqZvz hjWZ3Y[f

(114) x:¯²pebc€r8rZvY[alzjalm€Ž€l  Gmkqh'WZvz›ÏfqY[alebabfqz8wUebfnr'r‰k

(115) ›.a s ZvY[pBkqh'ZUm*h,rZvY[alY[k s cebZvr]wDknm€r'abrhjrk

(116) ›.r'cp-¯ r'ZUY[abebf

(117) hhjabwUZvr œ alh'Wf£ŠBkqhhjkqY ZUebZUY[ZUm*h™» œ WabwWwUkqabm€wDa s Z œ aŸhjWrZvY3abY[k s cebZvr¤kišqZUz[hjWZ†¼ªk:kqebZvfqm r'ZUY[abz'abmŽB VWZ8abmnhjZUzjZvrh.›Qknz.a s ZvY3pƒk

(118) hjZUm*h0rZvY3abY[k s cebZvrf

(119) zjknr'Zr'pBZ™wDagf

(120) ebelu[›QzjkqY+h'W€Z s ZUšqZvelknpY[ZUm*h.kq› hjWZ3Y[f

(121) x:¯²pebc€r8f

(122) ebŽqZvŠzjfqabwdf

(123) ppzjknfnwWoh'k¤kqp-hjalYHf

(124) ewUkqm*h'zjkqe f |¡fnrebkiš < E H 2»€|¢fqr'elkiš}f

(125) m s _-f

(126) YSŠBknzj!r g:na mo š # r0s < ENEq €a» J!aŸhjš:almkitš 1 u_s5‹

(127) ŒqŒ < f

(128) m sos agrjwDzjZDh'Z < E(E*‹-/» pknelNk gqkneŸhrkišHf

(129) m s |¢fqr'elkiš < E(E H *» Jalh'š:abmkiq ZvšqZvmnhru-rh'ZvY[r f @.kqWZv\m # +0s < EnN ?»n¼fqwvwDZUebebXa # +0s < E(En‹/» @.knWZU\m 1 r0s < E(En„/» @.kqWZvi \m # +0s < ENE H » ½]fqcŠƒZUz'h@f

(130) m s yelc€r < ENE H Z» @.kqWZv m 1 70s < ENE(E i  !_:ZvZf

(131) egrk f @.cmabmŽnW€f

(132) Y[ZD¯ ½8zjZUZUm < E H E€Z» vƒabY3Y[Zvz'YHf

(133) m€m k # X0s < En

(134) \B» ˜ fqŽqmZvcz < ENE < i»n½8cmBf œ fqz s Zvm€f < ENEq »q½8knm s zf

(135) mHf

(136) m s |¡almknc-x < Eq < » p]abY < En n‹w» @ªf

(137) x ‹qŒqŒ*‹ i ›Qkqz0Y[kqzjZ‰ŠBfqew g:Žqzjkqcm s   max. max. max. max. yRy Ô z*{1|[|[}. max.

(138) \. r/  Z1 U  **Me1[  /e^

(139) r1[! r(!. Z0ŽqabšqZ0¬‰f

(140) Wm-¯±¼f

(141) m€fnwWdh u:pƒZh'WZvkqzjZUYHr›QknzwDknY3p€elZUh'Z0a s ZUY[pƒk

(142) h'Zvm*h.rZvY[alY[k s cebZvr œ h'WZ8mkqh'abkqmHk

(143) ›wDkqY[pebZDhjZUmZ™r'rabr s Z#k€mZ s alm®h'ZUzjYHrkq›5h'WZ‰m€fihjczf

(144) eƒkqz s ZUz.k

(145) ›5h'W€Z‰r'ZUY[alY[k s cebZ   ˜ Z r'f Wk œ h'W€f

(146) hSf}cm€alšnZUzr'fqe"rZvp€f

(147) zfihjalknm†zjZvr'celhdWkqe s r f VWZvkqzjZUY ¡ŠBZvelk œ i » œ alh'Wknc-hSfqm*u f s€s aŸhji alknm€f

(148) e fnr'r'cY[p-hjalknm€rknm¡hjWZ3r'ZUY[abY[k s celZ3kqz8kqm¢h'WZ[r'ZUY[alzjabmŽ€»Nal›.kqm€Zthj(f gqZ™r‰fnr‰f¤mkqm€elabmZvfqz s c€f

(149) er'p€fqwUZ fqm&kqp€pBk*ralh'ZorZvY[alY[k s cebZq &V@k†zjZvwUkišqZvzdf†r'ZUpBf

(150) zfih'abkqm hjWZUknz'ZvY alm:šnkqebš*abmŽ¢f†ebalmZ™f

(151) z s c€fqer'p€fnwDZq» œkq›‰ZShjrkqh'pƒc kqs ebu}kqŽnY[abwvkqf

(152) zjeZdšqŽqZ™ZvwKmh'knZUz®zf

(153) rebebpBu fqwDs Z™cBrUf

(154) ¥¡eqf p€f

(155)  abz!rUe»5r'alY[Kab*ebfqz ®hjk®wDknh'm€W€rZ3agrknhjr[mZvk

(156) r›8h'h WBfik h,f

(157) wUzjkqabY[r'ZdpwDebZDegfqhjZ¢rjr'abrwvZvf

(158) Y[ebelaluoY[abk m¡s h'cW€ebZvZdr hjWZUknz'u » Z *calp€pBZ s œ alh'W†f‘ŠabelabmZvfqz8wUkqm*h'abm:ckqcBrp€fqalzjalm€Ž h· | ·i »fqm s ftœ* ‰abr8f®pzjZ s c€f

(159) ep€f

(160) abz œ X,WabYwW r'ZUpBf

(161) zfih'Z™rpBknalm*hr f r'ZUZ j\ i   ˜ Ztabm*h'zjk s c€wDZtf‘½]fqelknabrwDkqm€mZvwDh'abkqm X → Y, x 7→ x » Y → X, y 7→ » WabwWou:abZUe s rf

(162) m*hjaŸ¯ºabr'kqY[knz'pW€abr'YHr.ŠƒZDh œ ZUZvmGh'WZtegfih'h'agwDZ™r0k

(163) ›h'WZdZUebZUY[ZUm*hjrk

(164) › X fqm s Y œ WagwW y œ fqz'ZGwDebknr'Z s ›Qkqz3h'WagrHwDknz'zjZvr'pƒkqm s Zvm€wDZn  Lknz3abm€rhjfqm€wDZn» œ WZvm X = R abrShjWZ}rZvY3abY[k s cebZGkq› ¯ s abY[ZUm€r'alknm€f

(165) e5wUkqebcY[mGšnZvwDh'kqzrªkišqZvz.hjWZdwUkqY[pebZDh'Z s YHf

(166) x*¯ºpebc€r0r'ZUY[alzjabmŽ R » Y = R »€f

(167) m s n »f

(168) ebeZUebZUY[ZUm*hrHk

(169) › X f

(170) m s Y f

(171) zjZ¡wDebknr'Z s »ªfqm s h'WZ¢wUkqmi{ c€Žnfihjalknm hy | xi = max x) W€ZUzjZ s Zvmk

(172) hjZvr‰h'W€ZShjzjfqm€r'pBk*ralh'abkqm,  Lkqz knpBZvzjf

(173) h'abkqm¡agr,r'alY[pebu (yx =+(−x) fqm s f wUebfnr'r3k

(174) ›8a s ZUY[pBkqh'Zvmnh‘r'ZUY[abz'abmŽ*rdhjW€fih yœ Z}=wUf

(175) (−y) ebe0zj1Z I€ZUx:œ abšqZn» œ Zor'Wk œ hjW€fih s cBf

(176) e0p€fqalzrHr'f

(177) h'agr ›Qu&f Y[knz'ZS›Ïf

(178) Y[abelagf

(179) z™»ebalmZ™f

(180) z™»ŽqZvkqY[ZDhjz'agw[¬fqWm-¯±¼ªfqm€fqwW¡h'WZvkqzjZUYo» œ WagwW W€fqr8hjWZ[›Qkqebelk œ abmŽ¤›QkqzjY f r'ZUZ VW€ZUkqzjZUY 2

(181) \tŠƒZUebk ¥al› abr.ftwDknY[pelZUh'Z8rcŠBrZvY3abY[k s cebZk

(182) › X »*aŸ› x ∈ X Šc-h x 6∈ V »qhjWZUm®h'WZvz'Z ZUx-abrh0ZUebZUY[ZUm*hjr œ i Vr'c€wW‘hjW€fih =±m[hjWabrp€f

(183) pƒZUz™». −. −. n×1 max. 1×n max. max. 1≤i≤n i −. f<i. i. −. >. >. >. y, z ∈ Y. VW€Z‰r'ZUp€fqzjf

(184) h'abmŽtp€fqalz. and hy | xi 6= hz | xi . g a  r  m q k ' h  W b a  m S Ž  Š c  h j h  W 8 Z € p

(185) f abzkq›!wDkqm

(186) { cŽnf

(187) h'Zvr ( x, P (y, z). hy | vi = hz | vi, ∀v ∈ V,. −. −. V. (x)). » œ W€ZUzjZ. PV (x). C B. H. M. P. A.  VRS9P: _-ZUp€fqzjf

(188) h'abkqm k

(189) ›h'WZ®wUkqm:šqZUx ABC f

(190) m s h'W€Z®pBknalm*h M Š:u¡hjWZ®W€fqeŸ›°¯ 'r p€fqwUZ H   agrh'WZSŠBZ™r h,f

(191) pp€z'k=x-abY[f

(192) h'abkqmG›QzjkqY ŠBZvelk œ kq› x Š:uof

(193) m¡ZvelZvY[ZUm*h8kq› V  ]_:almBwDZ P (x) Y[alm€alY[ab«UZvr8f

(194) m fqm€f

(195) ebkqŽncZkq›ƒ¬alebŠBZvzh™­ r"pz'kq{ ZvwDh'abšqZY[ZUh'zjabwq» f < i abrr'alY[abebfqzh'k8h'WZ0r'ZUp€fqzjf

(196) h'abkqmtp€z'knpBZvzh u,abm"c€wDeba s Zvfqm r'p€fnwDZvrv» œ WZUzjZ P agrHhjWZ¢knzhjWkqŽnkqm€fqe0pzjk

(197) { ZvwDh'knz®kqm V f

(198) m s hjWZ¢šnZvwKhjkqz (x, P (x)) ŽnalšnZvr®h'WZ V. V. V. Ø y Ø°Þ.

(199) 

(200)   !#"$% '&#(#")* # + #", "-./01$. ?. s alzjZvwDh'abkqmokqz'h'WknŽqknm€f

(201) eƒh'k®fHr'ZUpBf

(202) zfih'abmŽ[W:u:pBZvz'pegf

(203) m€Zq .VW€ZlgqZUu s agr'wUz'Zvp€f

(204) m€wUuq»Š:uGwDkqY[p€fqz'agr'kqm œ alh'W šnZvwDh'kqz.rp€fnwDZ™rU»*abrh'WBfih.knmZm€ZUZ s rK e$"k

(205) ›ebabmZvfqz›Qknz'YHrhjkSr'ZUp€fqzjf

(206) h'ZfdpBknalm*h›QzjkqYÜftr'cŠ€r'p€fnwDZq»*kqz Y[knz'Z,ŽnZUmZvzjfqelebuq»-›QzjkqY f®wUkqm:šqZUxGr'ZDhv VW€Zdf HmZ,›Qknz'Y k

(207) ›hjWZSr'ZUp€fqzjf

(208) h'abkqm¤h'WZvkqzjZUY abralebebc€r hjzjf

(209) h'Z s ab'm L@abŽ€  < » œ WagwWGr'Wk œ rf[YHfix:¯ºpelcBr.pƒkqebu:WZ s z'knm¤ŽnZUmZvzjf

(210) h'Z s Š:uHh'WzjZUZ]ZUx*hjz'ZvYHf

(211) e5pƒkqabmnhrU» A, B, C » f3pƒkqabm*h WagwW s k*Z™rmk

(212) h0ŠƒZUebkqmŽ3hjk[h'WZ,pƒkqebu:WZ s zjkqm!»nhjkqŽnZDh'W€ZUz alh'W}f3W€f

(213) el›r'p€fqwUZ almoebalŽnWnh Žnzjf=u i œ WMagwW¤œ wUkqm*hjfqalmBrh'W€Zpƒkqebu*W€Z s zjkqm»*Šc-hmkqhh'W€Z‰pƒkqabm*h M  VWœZWBf

(214) el›°¯ºr'p€fqwUZagrknŠ-hjHfqalmf Z s ›QzjkqY hjWZ,pzjk

(215) { Z™wKh'abkqm P kq› M  _:ZvZ e2  \3›Qkqz s ZUhjf

(216) abegrU  VWZopzjZvr'ZUm*h[a s ZUY[pBkqh'Zvmnh®¬‰f

(217) Wm-¯±¼f

(218) m€fnwW£h'W€ZUkqzjZUY W€fnr[f†ebkqm€Ž¢Wagr hjkqzjuq §VW€hZ kBzjrh3zjZvr'celh3kq› hjWagr g*abm s rZvZUYHr‰h'koW€f=šqZSŠBZvZUm pzjkišqZ s Š*qu vNalY[Y[ZUzjYHf

(219) mm f < E H(H i »ƒ›Qkqz,wUelk*rZ s wDknm:šqZDx¡r'cŠ€r'ZDhr‰kq› » WZvz'Z K agr0fSr'ZUY[abz'abmŽ œ aŸhjWGfqm‘a s ZUY[pƒk

(220) h'Zvm*h0f ss alh'abkqm»rjfihjabr›Qu:alm€Ž3r'kqY[Z‰f

(221) x-alknY[r œ WagwW‘W€kqe s KW€ZUm œ œŽnZUmZvzjfqKe€fq=rjr'cRY[p-h'abd  kqm€G rrkqabmHY[hjalegWf

(222) Z‰z‰r'z'ZUZ™Y[rc€abz'eŸh abmœ ŽBfn»nr0ŠcphªzjkifqšqrjZ r'scY[Š:alu}mŽt_-f

(223) h'YSW€f

(224) ŠBh.knhjzjW!r g:Zna mpƒo!kqf

(225) abm m*s hh'_-k3WprZvab« p€f

(226) f z< fiE(hjE*Z0‹ W€i fqcr.m abs m*ZUšnz8ZUz'Y[h'abkqŠzjebZZ wUk:kqz s almBfih'Z™rU VWZpzjZvr'ZUm*h"¬f

(227) W€m-¯º¼fqm€fqwWSh'WZvkqzjZUY W€kqe s rcm s Zvz"Y[kqzjZŽqZUm€ZUzf

(228) e-fqrjrc€Y3ph'abkqm€rv»qf

(229) m s u:abZUe s r s abzjZvwKh"ZUx-pelagwDalh›QkqzjYdc€e  ›Qkqz"r'ZUp€fqzjf

(230) h'abmŽ]W:u*pƒZUzjpegf

(231) mZ™rU VWagrŽqZvmZUzf

(232) ebaŸh udabrpƒknrjr'alŠebZŠBZ™wUfqc€rZ Z kqz gabm e")0# !kqz s Zvz'Z s rh'zjc€wDh'czjZvrv9  =±m,hjWZwUfnrZk

(233) ›:h'WZ.YHfix:¯²p€elc€r!rZvY3abzjalmŽB»UhjWagr!Y[Zvfqm€r5hjW€fih hjœ WZSœ wUk:Z®wDabZUm*hjrk

(234) ›h'W€ZSr'ZUp€fqzjf

(235) h'abmŽHW€f

(236) el›°¯±rp€fnwDZ™rh'W€f

(237) h œ ZtŠcabe s wUfqmohjf(gqZ,hjWZ +∞ šifqelcZn»ƒr'kHhjW€fih hjWZvr'ZW€fqeŸ›°¯±r'p€fqwUZvrmZvZ s mkqhŠƒZwDebknr'Z s ›Qknzh'WZ‰c€r'c€f

(238) e€h'kqpƒkqebkqŽnuq ¬ZUmBwDZq»*kqc€z"z'Z™rc€eŸhrf

(239) p€pelu3ZUšnZUm3h'k r'kqY[Z,wDknm:šqZDx®r'cŠ€r'ZDhr œ WabwW}fqz'Z8mk

(240) hwUelk*rZ s »rZvZ8h'WZ,ZUx-fqY[pelZ,f

(241) hªhjWZ,ZUm s kq›~0ZvY[fq!z g < „SŠBZvelk œ   =±m @.kqWZvm 1 M0s5‹

(242) ŒnŒn‹ i » œ Zdf

(243) pp€elu¤h'WZdpz'Z™rZvm*hz'Z™rc€eŸhrh'k‘wUkqm:šqZUx‘›QcmBwKh'abkqmBrkišqZvzh'WZSYHfix:¯²p€elc€r r'ZUY[f abz'abmŽB»:fqm s zjZvwUkišqZvzabm¤p€fqzhjabwUcegf

(244) z0fSr'ZUpBf

(245) zfih'abkqm®h'WZvkqzjZUY Sebf-vƒabY3Y[Zvz'YHf

(246) m‘›QkqzwUelk*rZ s wDknm*šnZDx r'ZDhrU GVWZ‘r'pabz'alhtk

(247) ›ªh'WZ‘pz'Z™rZvm*h œ kn!z g¡agrtfqebr'k}šqZvz'u¢wUelk*rZ3hjk}h'W€f

(248) htk

(249) ›h'W€ZHh'WZvkqzju s ZvšqZUebkqpƒZ s Š:u J!aŸhjš:almkišN»*|¡fqr'elkišN»qfqm s _:Wpab« ‹qŒqŒ < » Wk,ZvrhjfqŠebabr'WHa s ZUY[pBkqh'Zvmnhªf

(250) m€fqelknŽqcZ™rk

(251) ›5rZvšqZUzf

(252) ewUebfnr'r'agwUf

(253) e hjWZUknz'ZvYHr0k

(254) ››Qcm€wDh'abkqm€fqe@fqm€f

(255) ebu-rf agrU ªVWi Ztœ zjZUpzjZvr'ZUm*hfih'abkqm¤h'W€ZUkqzjZUY ›Qkqz‰elabmZvfqz0›QkqzjYHr f @.knz'knelegf

(256) zj u 2NE ŠƒZUebk œ i f

(257) m s h'WZ®zjZUegfihjZ s f

(258) m€fqelu*hjabw3›Qknz'Yh'WZ¤¬f

(259) W€m-¯º¼fqm€fqwW¢h'WZvkqzjZUY f @.kqzjkqebebfqz'u¢\nŒoŠƒZUebk œ i f

(260) zjZ ZUx:h'ZUmBrabkqm€r[kq›]h'WZ¢wUkqzjz'Z™rpƒkqm s alm€Ž†zjZvr'celhjrHkq› f Jalh'š:alm€kiš # Y_s0‹

(261) ŒqŒ < i    d  p‰kqeb(k gnkqelhjr'kiš < ENE(E r'cŽnŽqZvrh'Z s h'k¤h'W€Z[r'ZvwUkqm s f

(262) c-hjWkqz8h'W€Z3abm*h'Zvz'Z™r h8k

(263) ›.zjZUš:agralh'abmŽ¤Y[f

(264) x:¯²pebc€r8zjZvr'a s cBfih'abkqm¡h'WZvkqzju f œ alh'W i f®½]fqelknabr0wUkqmm€ZvwKhjalknm¤pƒkqabm*hk

(265) ›š:alZ »€f

(266) m s hjWZtpzjZvr'ZUm*h kqz g®alebebc€r hjzjf

(267) h'Z™rªh'WZ,›Qzjcalh›QcewW€f

(268) zfqwDh'Zvzªkq› hjWagr0a s Zvf» œ W€abwWGabr0fqebr'kHf

(269) pp€elabZ s h'œ k s aNZUzjZUm*h0pz'knŠebZUYHrœ alm f G)g:abfqm # C0sN‹

(270) ŒnŒn‹ i   ˜ Z]h'WBf

(271) m*g®W€alYo» fqm s œ Z]hjW€f

(272) *m gGf

(273) egrk[|£  G)g:abfqm»y  Jkqh'alh'kB»  €|¡fqm€wDabmZvelebaÇ» =K B_:abmŽqZvzf

(274) m s   ˜ f

(275) ŽnmZUc€zv»:›Qknzc€r'ZD›Qce s abrjwDc€rjr'alknm€rU  Labm€f

(276) ebebuq» Z}m€k

(277) h'ZohjW€fihGf†p€z'ZvelabY[alm€fqz'u§šqZUzr'alknm kq›‰hjWZ¡pzjZvr'ZUm*hHz'Z™rc€eŸhr®f

(278) ppƒZvfqz'Z s abm f @.knWZUm 1 u_sB‹qŒqŒqŒ i œ   n. max. -‹    P: wV

(279) )V 67PV :; ‹  <  S¸   ´ “ ´ ”i¹ ´ ” ´ ¹ ´ “v’&•@¹ ‘´ ’Ϲ @•-“ ´ ¹  •’8˜ Zjk€zr h®zjZvwvf

(280) eberknY[ZowUebfnr'r'agwUf

(281) e m€k

(282) h'abkqmBrªfqŠBknc-hkqz s ZUzjZ s r'ZDhr0f

(283) m s z'Z™ra s c€f

(284) h'Z s YHf

(285) p€rv  ˜ Z,rjf=u3h'W€f

(286) h0fSp€fqzhjabfqelebuHkqz s ZUzjZ s rZUh (S, ≤) agre"M 01 aŸ›"f

(287) m:uorc€Š€rZUh X ⊂ S W€fnrfHebZvfqrhcp€pBZvzŠƒkqcm s f s ZUm€k

(288) h'Z s Š:u ∨ X i r  =±m}p€f

(289) z'h'agwDc€ebfqzv» WBfqr®Šƒk

(290) hjW f£Y[abmalYHfqe ŠBkqhh'knY ZUebZUY[Zvmnh s ZUmkqh'Z s β Z *c€f

(291) eh'k ∨ ∅ »fqm s f£YHf

(292) x:abYHf

(293) e S hf 'knp i ZUebZUY[ZUm*h >S s ZUmkqh'fZ s τ œ Wi agwW abr,Z *c€⊥S f

(294) ehjk  ‘_:abm€wDZ[hjWZ®ŽqzjZvf

(295) h'Z™r h]elk ZUz]Šƒkqcm s k

(296) ›ªf r'ZDh X wUfqm ŠƒZ s Z1kBmZ s Š:u ∧ X = ∨{y ∈ S | y ∨≤Sx, ∀x ∈ X} » S abr®f wUkqœ Y[pebZDh'Zoegfih'h'agwDZq  _-ZUZ ¼ªalzg:Wk < E

(297) \*Œ»€‡‰cŠzjZUabel!¯ *fqwUk

(298) h'abm # )0s < EN?(2»€¼ªebunhjW¢f

(299) m s nfqmk œ alh'« < E H ‹ i ›Qkqz‰Y3knz'Z s ZDhf

(300) abebr knm¤knfz s ZUzjZ s fqelŽnZUŠzf

(301) agw8r hjz'c€wDh'czjZvr0fqm s z'Z™ra s c€f

(302) h'abkqm  S. S. yRy Ô z*{1|[|[}.

(303) „. r/  Z1 U  **Me1[  /e^

(304) r1[! r(! =º›. f

(305) m s. fqz'Z,knz s Zvz'Z s r'ZDhrU» œ Ztrjf=uHhjW€fih‰f[YHfqp r'c€wW¤h'W€f

(306) h. aglr !1$e e]aŸ›hjWZUzjZ f :S→T ZUx-abrhjr0f3YHf

(307) p ‹ f (s) ≤ t ⇐⇒ s ≤ f (t) , f i Wœ €abwW¡Y[Z™f

(308) m€rh'W€f

(309) h‰›Qknz8fqele t ∈ T »Bh'WZ[r'ZDh]k

(310) ›r'cŠ€r'kqebc-h'abkqm€rv» {s ∈ S | f (s) ≤ t} W€fnr‰f‘YHfix-alYHfqe ZvelZvY[ZUm*hv» f (t)  %=ºh,agr‰m€k

(311) h s a ®wUceŸh]hjkGr'ZUZSh'W€f

(312) h f abr8zjZvr'a s cBfih'Z s al› »!f

(313) m s kqm€eluoal› »aŸh,agr8Y3knmk

(314) hjkqmZ aÇ  Zq  »Bf

(315) m s f s ≤ s =⇒ f (s) ≤ f (s )i (S, ≤). (T, ≤) f] : T → S. ]. ]. 0. 2 f i. 0. f ◦ f ] ≤ IT ,. f ] ◦ f ≥ IS ,. œ €W ZUzjZ I s ZUmkqh'Zvr‰hjWZ[a s Zvm*h'alh u¡YHf

(316) p¢knm†fGrZUh X  %=º› (X, ≤) abr,fqm¢knz s Zvz'Z s r'ZDhv» œ Z s ZUm€k

(317) h'Z3Š:u h'WZxe( $[ 0kqz s ZUzjZ s r'ZDh™»:›Qkqz WagwW  "‡‰cZ8h'k3h'WZ,r'u:Y3Y[ZUh'zjuHk

(318) ›!hjWZ (X , ) s 1Z k€mabmŽ[pzjkqpƒZUz'h u f ‹ i »-alh0abr0wUelZ™f

(319) z.h'W€f

(320) h fœ : S →xT agyrzj⇐⇒ Zvr'a s cBxfih'≥Z s yal› »€f

(321) m s kqm€elu®al› » f : T → S agrtzjZvr'a s c€fihjZ s   ˜ WZvm S, T fqz'Z‘wUkqY[pebZDhjZ®kqz s ZUzjZ s r'ZDhjrv»h'WZvz'Z‘abrSf}rabY[pelZGwW€f

(322) zfqwDh'Zvz'ab«vf

(323) h'abkqm¢kq› zjZvr'a s c€fihjZ s YHfqp€rU  ˜ Z[r'f=uGh'WBfih,f‘Y[kqm€k

(324) h'knmZdYHfqp f : S → T agr   a$aŸ›"›Qknz,f

(325) ebe U ⊂ S » » WZvz'Z f (U ) = {f (x) | x ∈ U }   =±m§p€fqzhjabwUcebfqzv» œ WZUm U = ∅ » œ ZGŽqZUh f (∨ U ) = ∨ f (U ) œ   f (β ) = β .´ &•   & (S, ≤)  (T, ≤) !e")0#  !(#!e $#1

(326) $  W*#h "- f : S → T  $ [$[ ex  C  U` x    + $e  a$#s ¼ªu¤r'u:Y[Y3ZUh'zjuq» f : T → S agr0wDknmnhjalm:cknc€rv» œ WabwWGY[Zvfqm€rh'WBfihv¥ \ f (∧ U ) = ∧ f (U ) , ∀U ⊂ T . f i D‰rabmŽ3h'W€Z,Y3knmk

(327) hjkqmagwDalh u®kq› f fqm s f »-h'knŽqZUh'WZvz œ aŸhjW 2 i » œ Z]Z™fqr'alebu®ŽqZUhh'W€f

(328) h f ?qf i f ◦f ◦f =f , f ?iŠ i f ◦f ◦f =f , f ?

(329) w i f ◦ g = (g ◦ f ) , f Wœ €ZUzjZ g : T → W abrf

(330) mkqh'WZvzªzjZvr'a s c€fihjZ s YHf

(331) pX  =ºhagrmk

(332) h s a ®wDc€eŸh0hjk®wWZvew g[hjW€fih „qf abrabmi{ ZvwDh'abšqZ ⇔ f ◦ f = I ⇔ f abr0r'czÇ{ ZvwDh'abšqZq» f f i „

(333) Š abr0r'czÇ{ ZvwDh'abšqZ ⇔ f ◦ f = I ⇔ f agralm

(334) { ZvwKhjalšnZq  f f i |¡kqzjZUkišqZvzv»*al› {f } agr0f

(335) m}f

(336) zjŠalh'zf

(337) zju3›Ïf

(338) Y[alebu‘kq›@zjZvr'a s c€fihjZ s Y[fqp€r S → T » X. op. op. op. ≤. ≤. ]. S. op. op. T. ]. op. op ]. ]. ]. ]. ]. ]. ]. ]. ]. ]. S. ]. fH i. ]. T. ]. ]. i i∈I. ( ∨ f i )] =. ∧ fi] .. ‹  ‹   ´  ¸ ¹   ´ ’ ¸B´ ” D¹ ´  ¸ “ ´ N“  ´ ϔ   B’  =±m®hjWZ8r'Z*cZveÇ» ⊕, ⊗, ε, e) s Zvmk

(339) hjZvrªf 'r ZUY[abz'abmŽ œ Wk*rZ‰f ss alh'abkqmHagr"a s ZvY3pƒk

(340) hjZUm*h f aÇ  Zq  a ⊕ a = ai »:f

(341) m s ε f

(342) m s e fq(K, z'Z0hjWZmZvc-h'zf

(343) eBZUebZUY[ZUm*hr ›Qknz ⊕ f

(344) m s ⊗ »Bz'Z™rpƒZvwDh'abšqZUebuq  ˜ Zdr'W€f

(345) ebe@f s kqp-h0hjWZdcBrc€fqe@wUkqm:šqZvmnhjalknm€rv»f

(346) m s œ zjaŸhjZ,›Qkqz‰almBr hf

(347) m€wUZ ab abm€rh'Z™f s kq›  Gm a s ZUY[pƒk

(348) h'Zvm*htY3knmkqa s wUfqm¢ŠƒZHZ*cabppƒZ s aŸhjW¢hjWZj

(349) /!€knz s Zvz zjZUegfihjalknm» a a≤⊗bb ⇔ a ⊕ b = b »›Qkqz œ WagwW a(S,⊕⊕,b ε)= ∨{a, b} fqm s ε = œ β   ˜ Z}r'f=u h'WBfih[h'WZ r'ZUY[abz'abmŽ abr e")0# al›alh‰agrwUkqY[pebZDhjZtfqrf[mBfih'c€zjfqelebu‘knz s Zvz'Z s rZUhv»Bfqm s al›hjWZtebZD›°h]f

(350) m s zjalŽnW*h YScelh'abpelagwUf

(351) Kh'abkqm€rv» L , R : K → K » L (x) = ax » R (x) = xa »€fqz'Z,wUkqm*h'abm:ckqc€rv  i∈I. i∈I. K. K a. K a. K a. K a. Ø y Ø°Þ.

(352) 

(353)   !#"$% '&#(#")* # + #", "-./01$ H. z'abŽqW*h i K!¯ $##", "-Z/0 X agr‘f£wUkqY[Ydc-hfihjalšnZ}Y3knmkqa s (X, ⊕, ε) »Z*cabppƒZ s œ alh'Wf YHf

(354) p » z'abŽqW*hfqwDh'abkqm i »-h'W€f

(355) hr'f

(356) h'agr k€Z™r X × K → X (x, λ) → xλ f nf x(λµ) = (xλ)µ , f i f

(357) Š i (x ⊕ y)λ = xλ ⊕ yλ , x(λ ⊕ µ) = xλ ⊕ xµ , f qw i xε = ε , ελ = ε , xe = x , ›Qknz3fqele »  &_:abm€wUZ ⊕) agrda s ZvY3pƒk

(358) hjZUm*hv» (X, ⊕) abr3a s ZUY[pƒk

(359) hjZUm*h f alm s ZUZ s »alh ›Qknelebk œ r,x,›QzjkqyY ∈ f X

(360) Š i λ,f

(361) m µs ∈f qKw i hjW€fih x (K,  &VW€Z‘mkqh'abkqm&kq› ebZD›°h K¯±r'ZUY[alY[k s cebZ®agr s #Z k€mZ s£s c€f

(362) ebebuq= oVxeWz'=kncŽqx(eW€kqc-⊕h,e)hjWZ®=p€fqxepBZv⊕zv»xef

(363) ebe=h'W€Z¤x r'⊕ZUY[xali Y[k s cebZvr,h'W€f

(364) h œ Z r'W€fqelewDknm€r'a s Zvz œ abebeŠƒZ[kišqZvz‰a s ZUY[pƒk

(365) hjZUm*h]r'ZUY[abz'abmŽ*rU  ˜ ZHrW€fqelefqebr'k¤wUkqm€r'a s ZUz K¯ # $1#"Y "./01$K¥ f¤ŠagrZvY[alY[k s cebZ[abr,for'ZDh,Z *cabppƒZ s aŸhjW¢h k€»zjabŽqW*h]fqm s ebZD›°h™» ¯±rZvY3abY[k s cebZ[rh'zjc€wDh'czjZvrv»rcBwW hjW€fih‘h'WZ†z'abŽqW*h¤fqm s elZU›°hGfnwKh'abkqmBr‘wDknœ Y[Ydc-hjZqœ   =±mp€fqzhjabwUcegf

(366) z™»fqKmža s ZUY[pBkqh'ZvmnhorZvY[alzjalm€Ž K agr‘f ¯ºŠagrZvY3abY[k s cebZ,aŸ›kqmZ8hf gqZ,fnrelZU›°hf

(367) m s z'abŽqW*hfqwDh'abkqm€rªhjWZtr'ZUY[alzjabmŽ[pz'k s cBwKh (a, b) 7→ a ⊗ b   K ˜ Zdrjf=u[h'W€f

(368) hf[z'abŽqW*h ¯ºr'ZUY[abY3k s c€elZ abr "M 0# aŸ›aŸhagrwDknY[pelZUh'Ztfnr0fSmBfih'c€zjfqelebu®kqz s Zvz'Z s r'ZDh™»"f

(369) m s al› »›QknzSf

(370) ebe x ∈KX fqm s λ ∈ KX»"h'WZ¤elZU›°h[fqm s z'abŽqW*hdYScelh'abpelagwUf

(371) h'abkqm€rv» R : X → X » f

(372) m s » »:fqz'Z0Šƒk

(373) hjW¤wUkqm*h'abm:ckqc€rv  @.kqY[pebZDh'Z‰elZU›°h ¯±rZvY3abY[k s cebZvrªf

(374) m s x 7→ xλ wUkqY[pebZDhjZ K¯ºŠLagrZvY3: abKY[k →s cXebZvrµfqz'7→Z s vµ#Z k€mZ s abm}f3r'abY3abegf

(375) z œ f=uqK  =±mGh'WZtr'Z *cZveÇ»€fqKeler'ZUY[abY[k s celZ™r œ abele ŠƒZ,zjalŽnWnhr'ZUY[abY3k s c€elZ™rU»-cm€elZ™r'rkqh'WZvz œ agrZ,r'pBZ™wD0a k€Z s    •    ´  f L€z'Zv,Z @.kqY[pebZDhjZ,_-ZUY[alY[k s cebZvrf

(376) m s _:ZvY3abY[k s cebZvrk

(377) X› Lc€m€wKhjalknm€r i  J!ZDh K s ZUm€k

(378) h'Zf wUkqY[pebZDhjZHa s ZvY[pBkqh'ZUm*hdr'ZUY[abz'abmŽ€x  G e e")0#  [b a ^$#1"Y "./_,agr,k

(379) ›ªh'WZ[›Qknz'Y ›Qkqz r'kqY[Z]f

(380) zjŠalh'zf

(381) zju[r'ZDh I ¥h'WZ]ZUebZUY[Zvmnhrk

(382) › K fqz'Z‰›Qcm€wDh'abkqm€r KI → K »fqm s K abrZ nc€alppƒZ s œ alKh'WGh'WZ f ss alh'abkqm »-f

(383) m s h'WZ]fqwDh'abkqm λ) 7→ aλ, (aλ)(i) = a(i)λ » ›Qknzf

(384) ebe a, b(a,∈b)K7→, λa∈⊕Kb, .(a¼ªuG⊕wDb)(i) kqmBra s =ZUzja(i) abmŽ3h'⊕WZdb(i)fnwKhjalknm (a, λ) 7→ λa,(a,(λa)(i) »€knmZtwUfqmGr'ZUZ = λa(i) s n f 0 r 3 f l e U Z ° ›  h  r v Z 3 Y b a [ Y k  c b e q Z   K VWZ8rZvY[alY[k s cebZ R »*ZUšn(k gqZ s alm®h'WZ‰alm*h'zjk s c€wDh'abkqm»*abr.f

(385) m®ZUxf

(386) Y[pebZk

(387) ›f,›QzjZUZ‰wDkqY[pebZDhjZzjalŽnW*h ¯ºr'ZUY[abY3k s c€elZn  Gmk

(388) hjWZUzZDxfqY3p€elZªalmSh'W€Zrjf

(389) Y[ZwUf

(390) h'ZUŽnkqzjuq»™h'k œ WagwW œ Z œ alebe-z'ZUh'czjmt›QzjkqYÚhjalY[Z R hjkdhjalY[Z,abm‘hjWagrp€f

(391) pƒZUz™»-abr.hjWZ,rZUh R kq›!›Qcm€wDh'abkqm€rª›QzjkqY f[r'ZDh U hjk R » œ aŸhjW‘hjWZ]pBknalm*h œ agrZ r'cpzjZUYScY fqr ⊕ knpBZvzjf

(392) h'abkqmGf

(393) m s h'WZ,wDknm:šqZUm*hjalknm€f

(394) eNf ss aŸhjalknm¤kq›@f3z'Z™f

(395) eNwDkqmBr hf

(396) m*hfnr f ebZD›°hk

(397) ›!zjabŽqW*h i fnwKhjalknm VWabrrZvY3abY[k s cebZ f h'W€f

(398) h œ Z,zjZD›QZvz0fqr F ›QknzrWknzh0abmGh'WZtr'Z *cZve i agr0wDkqY[pebZDhjZq  =±m}f[wDknY3p€elZUh'Ztr'ZUY[alY[k s cebZ X » œ Z s 1Z k€m€Zq»:›Qkqz‰f

(399) ebe x, y ∈ X f

(400) m s λ ∈ K » Eqf x\y = (L ) (y) = >{λ ∈ K | xλ ≤ y} , f i E

(401) Š x/λ = (R ) (x) = >{y ∈ X | yλ ≤ x} . f i yfqzjfqpWzfqr'almŽdh'WZ s 1Z k€malh'abkqmokq›@zjZvr'a s cBfih'Z s YHfqp€rU» Œ xλ ≤ y ⇐⇒ λ ≤ x\y ⇐⇒ x ≤ y/λ . f< i G. f. X λ. X v. I. I. I. I. I. n×1 max. max. yRy Ô z*{1|[|[}. U max. def. X ] x. def. X ] λ. max.

(402) . r/  Z1 U  **Me1[  /e^

(403) r1[! r(!. VW€Z8zjZvr'a s c€fihjalknm‘›Qknz'YSce  f 2 i » f ? i f

(404) m s f H i u:abZUe s f x(x\y) ≤ y , (x/λ)λ ≤ x , f <(< i <f (< Š i (x\y)λ ≤ x\(yλ) , x(λ/µ) ≤ (xλ)/µ , <f N< w i x\(xλ) ≥ λ , (xλ)/λ ≥ x , <f N< s i x\(∧ Λ) = ∧(x\Λ) , (∧ U )/λ = ∧(U/λ) , <f (< Z i x(x\(xy)) = xy , ((xλ)/λ)λ = xλ , <f N< › i x\(x(x\y)) = x\y , ((x/λ)λ)/λ = x/λ , <f N< Ž i λ\(x\z) = (xλ)\z , (x/µ)/λ = x/(λµ) , <f N< W i (∨ U )\y = ∧(U \y) , x/(∨ Λ) = ∧(x/Λ) , ›Qknzf

(405) ebe x, y, z ∈ X » U ⊂ X » Λ ⊂ K » œ WZUzjZ (U \y) = {u\y | u ∈ U } »qf

(406) m s x/Λ = {x/λ | λ ∈ Λ}   Labm€f

(407) ebebuq»ial› agrfŠ€abr'ZUY[abY3k s c€elZf

(408) m s »

(409) h'WZYHf

(410) p€r f

(411) m s wDknY[Ydc-hjZq»iWZvm€wDZn» Š:u f ?

(412) w i »-hjWXZUabz0zjZvr'a s cBfih'Z s YHfqp€r0wDknY3µ,YSc-νh'Zn∈» Kœ WagwW¤Y[Z™f

(413) m€rªxhjW€7→fih xλ x 7→ νx ‹ (ν\x)/µ = ν\(x/µ) . f< i _-alm€wUZ8h'WZvz'Z,agrmk®f

(414) YSŠabŽqcalh uq» œ ZtrW€fqeler'abY3p€elu œ zjalh'Z ν\x/µ ›Qknz f < ‹ i   ¤´ &•-”   kqh'Zh'W€f

(415) h Enf agr s c€f

(416) enk

(417) ›-hjWZ s #Z k€malh'abkqm ?-  < ab m J!aŸhjš*abmki-š 1 70sihjWZegfih'h'ZUz@z'Z *calzjZvr hjWZ,fqrjrc€Y3ph'abkqm¤h'W€f

(418) h0h'WZ,f fqwDi h'abkqmGk

(419) ›šqZvwDh'knzjrªkqmorjwUf

(420) egf

(421) zfrªr'f

(422) ih'agr k€Z™r x(∧ λ) = ∧ (xλ) ^ˆr'ZUZ J!aŸhjš:almkiš # C0s  ƒ  \€  H WabwW¤abr zjaŸh'h'Zvm‘›QknzªzjabŽqW*hªfnwKh'abkqm¤k

(423) ›!šnZvwDh'kqzr.kqm¤rjwUfqebfqzjr"^ WZUzjZvfnrU» abm h'Wagr[p€f

(424) pƒZUz™» fœ Zor hjf abweg i!hji k†œ hjWZoY[kqzjœ Z¤m€f

(425) h'czf

(426) e0fnr'r'cY[p-h'abkqm§h'W€f

(427) h[h'Wagr[pzjkqpƒZUz'h u£Wkqe s r œ œ aŸhjW ∨ abm€rh'Z™f s k

(428) › ∧ ¥hjWabrabrhjWZtwUfnrZ]›Qknz0almBr hf

(429) m€wUZ,aŸ›h'WZdcm s Zvz'ebu:almŽHr'ZUY[abz'abmŽ[abrfHrZvY3abzjalmŽ[kq›@›Qknz'YHf

(430) e r'ZUzjabZvrv»*knz0k

(431) ›YHfih'zjagwDZvrv»:kišqZvzf[wUkqY[pebZDh'Z,a s ZvY3pƒk

(432) hjZUm*hr'ZUY[alzjabmŽ€  ‹  2€     ¸ ’ ϓ ´  ´  ¸ ¹   ´ ’  =º› agr3f†wDknY3p€elZUh'ZGz'abŽqW*h ¯±rZvY3abY[k s cebZq» ZGwvf

(433) ebe e( $[  r'ZUY[abY[k s celZ¢k

(434) › X h'WZ 0 e K¯±rZvY3abY[Xk s cebZ X œ aŸhjWcm s Zvz'Kebu:alm€Ž§rZUh X »f sœ s alh'abkqm (x, y) 7→ h'WZ agr›Qkqzh'WZ‰m€fihjczf

(435) e€knz s ZUz"kq› f

(436) m s elZU›°hªfqwDh'abkqm »   Lkqz ∧ wUebfq{x,z'alh uqy}» œ f Zr'W€f

(437) ∧ebe€r'kqY[ZDhjalY[Z™r s ZUm€k

(438) h'Z0Š:u (λ, x)X7→i λ x = x/λ h'WKZ×ebZD›°Xhªfq→wKhjalXknmSkq(λ,› Xx) → Vx/λ W€f

(439) h X agr0f[rZvY[alY[k s cebZ]›Qkqebelk œ rª›QzjkqY ›QkqzjYdce  f <(< s i » f <N< Ž i »€f

(440) m s f <(< W i X  =±mop€f

(441) z'h'agwDc€ebfqzv» f <N< Ž i u:abZUe s r λ∈Λ. λ∈Λ. op. op. ·. (λµ). 2 f< i. op. ·. op. op. x = x/(λµ) = (x/µ)/λ = λ op· (µ op· x) ,. ›Qknzf

(442) ebe λ, µ ∈ K fqm s x ∈ X » œ W€abwW®r'Wk œ r œ W:u X YSc€r h.ŠBZ8wDkqmBra s ZUzjZ s fnr"f-0 e BzfihjWZUz"h'W€fqm f [b a r'ZUY[abY3k s c€elZn ,=±m s ZUZ s »@wDknm€r'a s Zvz'abmŽ q fnr,f‘zjabŽqW*htfqwKhjalknm kqce s z'Z nc€alzjZdhjWZ p€z'knpBZvzh u[ru:Y[Y[ZDhjz'agwUfqeBhjk f < 2 i h'k3Wkne s »*hjW€fih(x,abrv»:λ)Š:u 7→f nx/λ f i » x/(λµ) = (x/λ)/µ »:œ Šc-hh'WagrªpzjkqpƒZUz'h u m€ZUZ s mkqhWkqe s ›QknzfHrZvY[alY[k s cebZ X kišnZUz0f  e"Y",/   rZvY3abzjalmŽ K   op. op. Ø y Ø°Þ.

(443) 

(444)   !#"$% '&#(#")* # + #", "-./01$. ‡‰ZUmkqh'abmŽHŠ:u. op. \. fqm s. \nf f< i. op. E. h'WZ,zjZvr'a s c€fihjZ s kqpƒZUzfih'abkqmBr.Š€calelh›QzjkqY » œ Z]ŽqZUhª›QzjkqY. Œ » f< i. op. ·. /. λ. op. x. op. \. op. ] x = (LX λ ) (x) = ⊥{y ∈ X | y/λ ≥ x} ,. = xλ ,. \qŠ f< i. /. y=. op (RyX )] (x). = >{λ ∈ K | y/λ ≥ x} , = x\y ..  nm f < \nf i abrf

(445) mGalm:šqknelch'abš*alh u‘p€z'knpBZvzh uN¥h'WZdz'Z™ra s c€f

(446) h'Z s egf œ kq›@hjWZ,z'Z™ra s c€f

(447) h'Z s ge f œ kq›@hjWZtzjalŽnW*h fnwKhjalknm¤qk › X abrhjWZ,zjalŽnWnh0fqwDh'abkqmok

(448) › X alhjr'ZUel›  "VWZvz'ZU›QkqzjZq» ” ¸  ¸ ’ ϓ  ¸     e"M 01  K ^$1#"Y "./01$ X  (X ) = X s op op. 2  4R6 wV 69:7P  P4[:TU8947P9;

(449) Y69V R:P9;  : RP w8VW4R69:U47P:

(450) 767QYV. :*PN8 *P

(451) 4 [:UT8V R:

(452) ):*8 PVWT 2€  <  }¸     ´ •-” ” ¸™´ –q“ ¸ ”  !J ZDh s ZUmkqh'ZdfHwUkqY[pebZDh'Ztr'cŠ€r'ZUY[abY[k s celZdk

(453) ›fHwDkqY[pebZDhjZtrZvY[aŸ¯ Y[k s cebZ X kišqZvz]f}wUkqY[pebZDhjZHa s ZvY[pBVkqh'ZUm*hdr'ZUY[abz'abmŽ K   ˜ Z‘wUf

(454) ebe e  e # 1 , knm V h'WZ. YHfqp. PV : X → V,. PV (x) = >{v ∈ V | v ≤ x}. h'WZ‰elZ™fqrh.cp€pBZvz.Šƒkqcm s k

(455) › ∈ V | v ≤ x} ŠƒZUebkqm€Žnrh'k,hjWZ8r'ZDhªŠBZ™wUfqc€rZ V agr.wUkqY[pebZDh'Z i  VW:c€rv» f agrªh'WZ]zjZvr'a s cBfih'Z s YHf

(456) pG{v k

(457) ›@h'WZtwvf

(458) mknmagwUf

(459) eƒabmi{ ZvwDh'abkqm  "_:abm€wDZ »-c€r'alm€Ž ?

(460) w » P Zœ ®ŽnZDhthjW€fih P = P »"f

(461) m s r'abm€wDZ i abrSalm

(462) { ZvwKhjalšnZq»c€r'ialm€Ž : Vf „qf→i » Xœ Z‘r'ZUZ®h'W€if

(463) h =P i : X → fV agi r r'czÇ{ ZvwDh'abšqZq  ˜ Z¤rjf=u†h'W€f

(464) h agrdqf bw##!  wb 1"Y ` ¤k

(465) ›f¢wUkqY[pebZDh'Z¤rcŠBrZvY3abY[k s cebZ V al›f

(466) m:u ZvelZvY[ZUm*h v ∈ V wvf

(467) mGŠBZ œ zjalWhh'Zvm}fqr v = ∨{wλ | w ∈ W } »:›Qkqzr'kqY[Z λ ∈ K     ´:¸ ” ´ "! y"zjk

(468) { ZvwDh'knrz L€kqzjYdcegf i  & V  $Y e")0# ,$[1$#1"Y "./_-  X # . 'bw##!  wb 1",   1f  V. 2 V. 2 V. V. V. V. w. ? f< i

(469) r!Z 1s. V. V. w. W. PV (x) =. ∨ w(w\x) .. ˜ ZowUf

(470) m œ z'alh'Z P (x) = ∨ wλ »›Qkqz[rknY[Z λ ∈ K   LzjkqY P (x) ≤ x » œ Z¤ŽnZDh »ªkqz™»Z*cabšif

(471) ebZUm*h'ebuq» λ ≤ w\x  VWagrHrWk œ rShjW€fih P (x) ≤ ∨ w(w\x)   ¼ªc-h™» wλ ≤ x abrf

(472) mGZUebZUY[ZUm*hk

(473) › V » œ W€abwW»-Š:u f <N< f i »abrebZvrjrªh'WBf

(474) m x  "VWagr0pzjkišqZ™r f < ? i   ∨ w(w\x) ˜ Z}YHf=u&zjZ œ zjaŸhjZ f < ? i fqr P = ∨ P , œ WZvz'Z P s Zvmk

(475) hjZvr3hjWZ¡pzjk

(476) { Z™wKh'knz[kqm h'WZ%$kqm€Z s alY[ZUmBrabkqm€fqe'&]rp€fnwDZ wK  ._:abY[alegf

(477) z›Qknz'YSce  ›QkqzhjWZ,pzjk

(478) { ZvwDh'knzfqppƒZvf

(479) zjZ s abm f |}kqebebZUz < En q i   ” ¸  ¸ ’ ϓ  ¸ )( ‡cBf

(480) e!wW€fqzjfnwKhjZUzjal«™fih'abkqm‘k

(481) ›h'WZ,pzjk

(482) { Z™wKhjkqz i +* 1 V ⊂ X (# u e")0# +$[#^ $##", "-./0 # . hf bw1 1!

(483)  ab 1",  W s-1 „ P (x) = ⊥{z ∈ X | w\z ≥ w\x, ∀w ∈ W } . f< i

(484) r!Z 1s‰_-alm€wUZ w\z ≥ w\x ⇐⇒ z ≥ w(w\x) »€h'Wagr›Qknelebk œ rª›Qz'knY f < ? i   V. w∈W. w. w∈W. w. w. w. w∈W. V. V. yRy Ô z*{1|[|[}. V. V. w∈W. w. w. w∈W.

(485) <Œ. r/  Z1 U  **Me1[  /e^

(486) r1[! r(!.  • . ´ ‰˜ Z[z'ZUh'czjm†h'k¤h'WZ R ¯±r'ZUY[alY[k s cebZ F la m*hjz'k s c€wUZ s fih xf

(487) Y[pebZ®‹Gf

(488) m s†s abrjwDcBr'r hjWZtf

(489) p€pelagwUf

(490) h'abkqmGk

(491) ›pz'Zvš:alknc€rªz'Z™rcelhjrabmGh'Wagrr'ZvwDh'abkqm  Labzjr0h k

(492) ›f

(493) ebeÇ»-knŠ€rZvz'šnZhjW€fih . max.  ∀f, g ∈ F, f \g = inf g(u) − f (u) ,. œ alh'Wdh'WZwDknm:šqZUm*hjalknmtWZU  zjZ"hjW€fih +∞−∞ = +∞ f rabm€wUZ.abmSfqm*u,wUkqY[pebZDh'Zªa s ZUY[pƒk

(494) h'Zvm*hr'ZUY[abz'abmŽ S » β \β = τ \τ = τ i G‰r'r'cY[Zmk hjW€fih abr.f,h'knpBknelknŽqagwUfqešqZ™wKh'knzr'p€fqwUZf

(495) m s wUkqm€r'a s ZUzh'WZ8wDkqY[pebZDhjZr'cŠ€r'ZUY[alY[k s ¯ c€elZ V ŽqZvmZUzfihjœ Z s Š:u Uh'WZGrZUh W k

(496) ›wUkqm*h'abm:ckqcBrtebalmZ™f

(497) zd›Qcm€wKhjalknm€rSkišqZUz U   @.elZ™f

(498) zjelun»h'Wagrdagrdh'WZ r'cŠ€r'ZUY[abY[k s celZ,kq›e²  rU  w

(499)  ƒwDkqm:šnZDxH›Qcm€wDh'abkqm€rkišqZvz C  Lknzf

(500) m:u f ∈ F » P (f ) »Bfqr s Z1k€m€Z s alm e2  < » agrªh'W€ZtwDegfqrjragwUfqeƒe²  rU  w

(501)  €wUkqm:šqZUx®W:cele!k

(502) › f   L€kqz w ∈ UW » u∈U. S. S. S. S. S. 1. V.   w\f = inf f (u) − w(u) = − sup w(u) − f (u) ,. W€abwW[agr"mkqh'WabmŽ,Šc-h.h'WZkqppƒknr'aŸhjZ0k

(503) ›ƒh'WZdLZUmBwWZUe€h'zf

(504) m€r›QkqzjY hjœ WZUmou:abZUe s r u∈U. u∈U. PV (f )(·) =. ∨ w(w\f ) = ∨. f∗. kq› f Zvš=fqelcBfih'Z s fih w    * m. ? f< i.  w(·) − f ∗ (w) ,. hjW€fih8abrh'korjf=uq»Bh'WZSeÇ  rU  w

(505)  NwDknm:šqZDxoW:cebekq› f abrh'W€ZYL€ZUm€wWZve!hjzjfqm€r›QkqzjY)kq›h'WZLZvm€wWZUe@h'zf

(506) mBr ›Qknz'Y kq› f   Labm€f

(507) ebelun»  nm < „ rjf=u-r8h'W€f

(508) h,h'WZHe²  rv  w

(509)  !wUkqm:šqZUx¡W*c€ele"kq› f abr,h'WZHebZvfnr h,›QcmBwKh'abkqm g abm F rc€wW hjW€fih g agrelZ™r'rhjW€f f

(510) m!»-i kqz0Z *c€fqeNh'kB» f f pƒkqabm*h œ abr'Z i   2€  ‹     ƒ´ ”=’™•   ´ •-”i•-“  ¸   ´:¸ ” ´    ´:¸ ” ´  f DmabšqZvzjrjf

(511) e_:Zvp€f

(512) zfihjalknm¤VWZvkqzjZUY i * 1 V ⊂ X N1  - e")0# $e1$#1"Y "-Z^ /_ +-_1 x ∈ X sY 1 f ∀v ∈ V, v\P (x) = v\x , f <H i w∈W. ∗. w∈W. ∗. V.  Š f <H i. x ∈ V ⇐⇒ x\PV (x) = x\x .. :_ ZvZUabmŽ fnrf $'rjwUf

(513) egf

(514) zpzjk s c€wKh &»  nm f < H f i rjf=u-r!hjW€fihhjWZšqZvwDh'knz (x, P (x)) agr $ knzhjWkqŽnkqm€fqe'& hjkhjWZ0rZvY[y\x alY[k s cebZ »qfqm s < H Š r'Wk r@h'W€f

(515) hh'WZ $W*u:pƒZUzjpegf

(516) mZ & rZvp€f

(517) zfihjZvr ›x QzjkqY V »al›fqm s kqmVebu®al› x 6∈f V  "i VWagœ rhjZUzjY3abmknelknŽqu œ abele5ŠƒZ{ c€rh'{ya0k€Z |s y\P abm '\€(x)   = y\x}

(518) r!Z 1s‰_-alm€wUZq»ƒŠ:u s #Z k€malh'abkqm»Bh'W€Z alm < „ ŠBZvelknmŽnr0h'k®hjWZdr'ZDh™» ZtW€f=šqZ]hjW€fih » ›QknzHf

(519) ebe v ∈ V   Dr'abmŽ P (x) ≤⊥x f

(520) m f s h'i W€Z}Y3knmk

(521) hjkqmagwDalh u£k

(522) › yœ 7→ v\y » œ Zov\P ŽqZUh[h'(x) WZ}zj≥ZUšnv\x ZUzrZ abmZ ncBf

(523) ebaŸh un» œ WagwW&rW€k œ r f < H f i   =º› x ∈ V »@h'WZvm P (x) = x »fqm s x\P (x) = x\x »h'zjabš*agf

(524) ebebuq  @.knm:šqZUzr'ZUebuq»-al› x\P (x) = x\x » œ ZtWBf=šqZq»€Š*u < Œ i »hjW€fih P (x) ≥ x(x\x) »5f

(525) m s »BŠ:u <(< Z i »Bh'WBfih » W€abwW}r'Wk œ r.h'W€f

(526) h P (x) ≥ x  ._:fabm€wDZ P (x) ≤ x » œ Z,W€f=šnZ x = P (x) ∈f V   x(x\x) = x œ Ã@ÄKÉqÅÏÐ À ÊÏÅÇʲÈj РʲÔdÒ=ÀÂÄKÁ²ÀÂԙРÔdÅQÁæDÝ ¿=Ð"ÄKÀÂÔDÑ=¿=æDљÄ'Ð Ã5ÐÔUԉбÊÇæDÀÂÒ

(527) ÄKÐ ÐÅÔi¿™ÈʲÀÂÌN¿iÔ=ÈjÑ8Ô=Ê ÄÊÇʲ¿=ÈjÐÊÇÀÂÝ ÄKÄKÔ3ÔUæDÀÂÁÐ ÔDÈʲÖ.ÀÂÄKÒ=ÔiÀÂÈÛKÌÎ=Ã@ÄKÔ=¿™Î=ÄÀÂݺÁʲ¿‰Èj ÊÇÈÊÇÀÂÔ=ÄKŲÈÔ8ÑdԙʲÄKÁ²ÄÌËÔ=ÈjÐÈÊÇÐÉ=Ö.Á É= Î=Ì ÍÀÂÁÇÔ Êʲ¿™ÆKÐÐ ÐÅÏÉdÎ=Ì ÀÂÐÔtÈÝ Á Ö0ÄKÅÏÀÂÅÇÔ=Ð Ñ]ݱʲÃ!Ì ÍD¿ié ÈjÊÈÅÏЪʲ¿™ ÐÁ²Ä.Ý ÄKʲÅÇ¿iÅÏÈjÐ Ý Ê!Ê.ÊÇ¿=ÈÀÂÌÂÁBÛKÐ äÇÒ=ÅÇΙÅÇÈÌÂбÀÂÝç V. V. V. V. V. V. V. 1. V. V. βS ⊗ τ S = βS. V. V. Rmax. −∞ + ∞ = −∞. Ø y Ø°Þ.

(528) 

(529)   !#"$% '&#(#")* # + #", "-./01$. <N<. ¤´ &•-”  VWZ[rZvp€f

(530) zfihjalm€Ž¤r'ZDh H = {v ∈ V | v\P (x) = v\x} abr]fGrZvY[alY[k s cebZq %=±m s ZUZ s » Š:u f <(< W i aŸhagrrhjf

(531) Š€elZ]Š:u‘f s€s aŸhjalknm}f

(532) m s f <N< Ž i r'Wk œ r.h'W€f

(533) haŸhagrrhjfqŠelZ]Š:u‘rjwUfqebfqzfnwKhjalknm  ¤´ &•-”  G0wUwDknz s almŽ¡h'k}hjWZ¤p€z'Zvš*abkqcBr,z'ZvYHf

(534) zgƒ»alhSagrSr'c®wDabZUm*hSh'k†wW€Zvweg < H f knmebu¢›Qknz v f i zf

(535) m€ŽqabmŽSabm}f3ŽqZUm€ZUzfih'abmŽ[rc€Š€rZUh W kq› V    •   ´    Lknzh'WZ0r'ZUY[alY[k s cebZ alm*hjz'k s c€wUZ s fih xf

(536) Y[pebZ‹fqm s hjWZ0rc€Š€rZvY[alY[k s cebZ kq› e²  rU  w

(537)  qwUkqm:šqZUxt›Qcm€wDh'abkqm€rŽqZvmZUzfihjZ s Š*udFh'WZr'cŠ€r'ZDh W kq›NwUkqm*h'abm:ckqc€r"elabmZvfqz›Qcm€wDh'abkqm€r.fqr s agr'wUc€VrjrZ s f

(538) h xf

(539) Y[pebZ H »-h'WZSZ *c€fqelalh u < H f zjZvrh'zjagwKh'Z s h'k fqr0knŠ€r'ZUzjšqZ s abmoh'WZtp€z'Zvš*abkqcBr0z'ZvYHf

(540) z g kq›Nh'WZ]_:ZUp€fqzjf

(541) h'abkqm®VWZvkqzjZUY+r'f f=u-rhji W€f fih.h'W5Z LZvm€wWZUve€h'∈zf

(542) Wm€r›QkqzjY+k

(543) ›f

Références

Documents relatifs

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des

[…] Chez nous, le parterre […] est composé communément des citoyens les moins riches, les moins maniérés, les moins raffinés dans leurs mœurs ; de ceux

But on another hand, increasing q at constant operative pressure results in decreasing the equilibrium pressure, by decreasing the operative temperature (Figure 9), and finally, it

R´ esum´ e : On pr´esente des r´esultats autour du probl`eme suivant: pour un point mat´eriel dont l’acc´el´eration est born´ee par une constante donn´ee, et pour un

Ensuite, pour déterminer lesquelles des voix créées pouvaient être qualifiées d’« androgynes », nous avons fait un test de perception dans lequel nous avons demandé

Toutefois, des améliorations sont à prévoir car bien que la protection induite par ces glycoconjugués soit significative, elle n’est cependant pas optimale pour mettre

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des

Venant compléter les nombreux guides spécialisés qui informent le touriste gay sur les différents lieux et structures qu’il pourra fréquenter s’il recherche son