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RelatPeedr gamoTni tloefsI nterest Books CONSTANTINESCU: Distributainodtn hse iArp plicatiionPn hsy sics COOPER COOPER: & IntroducttioDo ynn amiPcr ogramming KURATOWSKI: A HalCfe nturoyf P olisMha thematics LAKSHMIKANTHAM LEELA: & An IntroducttioNo onn lineDairf ferenEtqiuaalt ioinnsS pace SINAI: RigorouRse sulitnst heT heoroyf P haseT ransitions Journals AnalysMiast hematica ProblemosfC ontroaln dI nformatTihoeno ry FunctioAnnaall ysis by L.V .K ANTOROVICH NobelP rizWei nner Membero ft heA cademyo fS ciences oft heU SSR and G.P .A KILOV Translabtye d HOWARDL .S ILCOCK SECONDE DITION PERGAMONP RESS OXFORD NEW YORK TORONTO SYDNEY PARIS FRANKFURT · · · · · U.K. PergamoPnr esLst d.H,e adingtHoinl Hla ll, OxforOdX 3 OBW,E ngland U.S.A. PergamoPnr esIsn c.M,a xwelHlo useF,a irviPeawr k, ElmsforNde,w York1 0523U,. S.A. CANADA PergamoPnr esCsa nadaL td.S,u it1e0 4, 150C onsumerRsd .W,i llowdaOlnet,a riMo2 J 1P 9,C anada AUSTRALIA PergamoPnr es(sA ustP.t)y L.t d.P,. O.B ox5 44, PottPso inNt. ,S .W. 20A1u1s,t ralia FRANCE PergamoPnr esSsA RL,2 4 rued esE coles, 75240P ariCse,d ex0 5,F rance FEDERALR EPUBLIC PergamoPnr esGsm bH,6 242K ronberg-Taunus, OF GERMANY Hammerweg6 ,F ederRaelp ubloifcG ermany Copyrig©h t1 982P ergamoPnr esLst d. "Nauka" & Publishers RightRse serveNd.o parto f thipsu blicatmiaoyn b e Arellp roduced, isnta o rreetdr ieval osryt srtaenms mititnae ndy formo rb ya nym eanse:l ectroenliecc,t rostmaatgince,t tiacp e, mechanicpahlo,t ocopyirnegc,o rdionrgo therwiwsiet,h out permissiionwn r itifnrgo mt hep ublishers. Firesdti tipounb lishbeyd" NaukaP"u blishe1r9s5,9 Seconde diti1o9n8 2 LibraorfCy o ngreCsast alogiinPn ugb licaDtaitoan KantoroviLceho,ni Vdi tal'e.v ich Functioannaall ysis. TranslatoifFo unn ktsionaaln'anlyiiz . Bibliograpp.h y: Includiensd ex. 1.F unctioannaall ysIis.A. k iloGvl,e bP avlovich, joinatu thor.T itle. 11. QA320.K283 1982 515.7 80-21734 BritiLsihb raCrayt aloguiinnP gu blicatDiaotna KantoroviLceho,n iVdi talevich Functioannaall ys-i2sn.d e d. Functioannaall ysis 1I..T itle AkiloGv ,P 515.7 1O1A.3 20 80-41734 ISBN0 -08-023036H-a9r dcover ISBN0 -08-026486F-l7e xicover PrintiendG reaBtr itabiyn Wheaton Co.L td.,E xeter A. & Deditcota htmeee dm oorofyu d re taera chers, GrigMoirkihia Fiilkohvtiaecnnhdg ol'ts VladviamniorSv miicrhn ov I PREAFCE TOT HE SECOND EDITOIN yeahrasvp ea ssseidnt chaeep pearoaftn hficere es dti toifto hnbi oso ukn der TtWhtEetiN lTFeYu c-nOotDniDAa nlya sliiNnso redmS paIcnte hsat.tie mr adicchaaln hgaevste a ken plabceo,tw hi tmhaitnh emiattsaiencltdsfoi tsst aitntu hsse y stoefmm o descrni enitdieafisc. Onei mporatsapnoetfct th ese ccohanncgeterhpsnel s ao cfuefn ctiaonnaalwlyi stithshi en mathecmaadlti isciWphleirnefueanssc. t niaaoln alwyasssie sea ntt, h aep pearaonftc hee firesdt itaisaoc no,ma pratinveewl pya rnodm ipsairontfag n alnyosawidsa,ty hste e rm "funcntaialon alyisusis sead"l moitsnetr chanwgiet"ahmb altyh emaantailcyWashlia st" . imso rfuen,c tiaonnaallny oswpi rso vaic doemsm loangn uafogrea lalr eoafms a thecmsa ti involtvhcieon ngc oefcp otn tiNnous ieitroyui.sn vestiintg hateth ieooonrffu y n ctions, diffeiraeelnq tuatoirom nast haetmipchayls iincn sum,e rimceatlh omdast,h ecmaalt i ecnoomiocrcs o nttrhoeloo rriy nn, u merootuhsfie erl tdask,e s plcaocutela-d-ok re placwei-theoxunttse eiu vsoef t hlea nguaangdre e suolffut nsc tiaonnaallyI stii ss . precitshefailcsytt haetxa pilnosnt, h oen hea ntdh,re a pdiedv pemleonotffu nctional analayssa mi ast mhaetidciaslc iapnldoi,nnt eh,oe t hhearn tdh,ee v enrc-ariseirnogl e playbeyid tt sce hniqiunae pspa ltiicons. Thea uthortsh encsoheta en wgietsbh o th apnradin dxei Tehtepy r.i idsaen atural manifetsitooanfa s enosfpe a rctiipaitnis oingc nainhfiti srtioceaevln tTsh.ea nxiety, howeviepsrr ,o vobkyie mdai gnitnhfaget oeft hree atdahetwr e e nvaigfoser t hbioso k, for it iinfas c,tn ,o wn ol oenrpg osstiopb rloeudc aec omephresnitveebx otookffu nctional anal(yesviaestan n incttroloredylvu )e. eCqounesnatlhltoyu,wg ehh avmea dsei gcnainfit rveisiiopnnr se pnagtr hpier eseednittw ieto hno,u igethx tpi eedtnort e ttahioenv erpallaln andt,oal aregxeetn tth,se e clteiaonnad r rangoeftm oecpnsait d opfotret dhfi er esdti tion. Howevetrhe,ea rraen umboefrt p oicfosrw hicha ctcuhoneht a bse esnu bsatlalnyt i changpeadr,t iciuntl htaehr eloyr typo ooglfioc vaelc stpoarac netdsh teh eoorfiy n tegral opaetroWrhse.r efoarsm erlayc cutonhwtea bsa soendt hteh eoorfny o rmed spaces, and tpoogliocvaelc tsopra cweesrc eo vesreepda r(alattehloyfau igrhful ly)l a yso ptional materiinta helidi,st iwoehn a vtea ktehnte h eoorfty po ologvieoccrast lp aacset shb ea sis ofo uerx psoitiicnoo nn,fo rwmiittthyhl ego icdaelv eloopffum necntti aonnaallhy esnicse: thceh anignte h beo okt'lisetW .e haadveda ec dh apdteearlw iinttghh e el emeonftt hse theoorfpy a rtoidraelrlseypd ea scO.u rd eveloopfmt ehtneht e oorfiy n treaoglpe rators antdh erierep srentaitbsia osonensdi desaaplc eosfm easurfuanbcltei ons. Asb eformeo,so tft hbeo oiksd evottoet dha ep paltiicoofnfu sn ctiaonnaallty os is applaineyadsl iwshi,wc ehr ade i nscttifeiavteue or ft hfier esdti tTihopenr .e soefnt chee se scetiiontn hsbe o oskt ilmauttehddee vpemleoonftt hree letvpoaincbtso ,ti hnt hUeSS R anadb roIantd h.pe r eseednittt ihaoecn cu onotft hehsaebs e esno mwehaetx tended and modeirzneAdn.o tehsesrae lfen attieuo rf tehdiisti istoh inen cluosfsi oomtnepo icosf functniaaoln alcyosnniesc tweidta hp paltiictoonm sa themaetcnioomciacals n cdo ntrol theoarlyt,h owuehg ahv bee eunn abtloge i vteh etshese p atcheed ye seSrovmele.e ss topical hmabaset eeenrxl icuadlTe hde.b ibliohgarbsae pehsnyu bsatlaclnhyta inged. ChapIti eisrn trcotdoIurnyi .wt e p restehneetl emeonftt htseh eoorfty po ogliocal xi xii Prefatcote h e SEedictoinodn spactehtseh, e oorfmy e tsrpiacc aenstd,h t eh eoorfay b satcmrte assupraescH .ee rmea ny resuarlsett sa wtietdh poruostoT. fh ree aidasesr s eudtm ob efa milwiiattrhh t eh eoorfy functioofarn esva alr iaanbdl ett oohpleo ogfny - dimeanlEs uicolnisdpeaarcnoe u,g thol y thlee voefal g eneurniavle rcsoiutriysn me a thecmaaaltn liaysSiubst.l aenrdm ore specimaaltiezireinbda o ltt,hht eh eaonrtdyh aep lpaictiaornmesa ,r koeffbd y b eisnegt in smatlylp aen,md a yb eo mittaefi dr srateta dTihnerg e.a dpearr ticiunltaerrilenys ted appaltiicooffun nsc tiaonnaallmy asayi lsos moi ant um beorfo thmeorera bsatcsrte icotns deianlwgi ttphoo gliocsapla acnedsp otgloiocvaeolc rst pacoe-rsih,fe i asl refaamdiyl iar witthhb ea siidce oafts h teh eoorfny o rmsepda cesm-ahyte u rinm meadtietloty h e relecvhaanptot enar psp lications. An umboefpr e opwleero efg r ehaetli pnt h weo rokfp repatrhpiern egns tee dition. First ofa lwlem usmte ntAi.oV n.B ukvhaltoovw ,h ocmr eiddsiu ten ,oo tn lfoyre dittiehn g enttierxet boofo bktu,ht e alrseow rfoiCrth ianpXgt eoorndr erneodmr eds pacetsh ea nd related im§na ot3feC rhiaapIltV e§,r 1 o fC hapVtIe§,r 1 o fC hapXtIea,rn cde rtain othseetrci oHneas l.ss uob satlarlneytv iitsheeexd ps oitiinCo hna ptIe-rItsVh ;ui snp arts oft hbeo ohkea ctteoda ,l l ianntdpe unrtpso saec so,- aaust hor. V. Dem'yaannodAv .M .R ubinmoavds eo mceo rcrteiionnts he ex opsitiino n ChaFp.Xt Veo rf t hmee thods teoefdp eessctae nndat d detdhn ee §w§ 4a nd5 I..K . Daugamvaedtse u sbtanatdidailta inocdnor sr eicotnsC hianp tXeIrVXs,V a nXdV III. V.P .I lm'aidnse o msei figcnaiinmtp roveimnte henext sp osoif§t §ia3on n4d o fC hapter XIp,a rticiuntl haperr losyoo ffL emm2aa nTdh eor1ei m§n o3fC hapXtIe.r Rubinsrhetwer§io 8not feC hapItXeb,ra siiontng w orokfL .KVa.n tvoiraconhGd . SS hh.. G. Rubinshstoemaiedn d;i ttioto hnwises r mea dbeyV .L .L evTihnre.e vioefwt ehbreo ok, ProsfeosBr. Vulimkahd,se o mvea luraebmlraek ass,d iYdu A.. A bramoAv.iM c.h , VersSh.iZ Kk.Vi, .s lySa.kKSouv.ta, t elGa.Yd azL.eo ,za novAs.k MiAei.k, l Be.r P,To .l yak and PV.K. h vain. Thaeu theoxrpsrs eniscset rhea ntkoas lt lh omseen tiaobnoeavdne ad,l tsoot howsheo heeldp rienia ndtgh mea nuscarnitdph pteor osf. Thbei bglriaopchoyn sitswtposa t rsoal:f i osfmt o nroagpohnfus n ctiaonnaallay nsdi s relattpoeidca sna,dl iosflitt ercaittucerodem, p rimsoislnjtygo n uaralr tiAcr leefesr.e nce oft hfoer m Vuli-k1i1hn cdtaietsh meo ngoraphB .b Vyu liokchc ururnidnegr the Z. aut'hssou rarmnew itthhn eu mbeirnt hIleIi o sfmt o nroagpwhhsi,al r ee feresncuche as Lev[i3]ni ncdtaieasp apbeyrV .L .L eviintn h lei osflt i tercaittTuehrdbee.i bliography makensop retenccoemle ptaetn Iencs sase.ws h earr ee uslhta asl ready ianpa pe ared publimsohnerodag pwheh avaesa,r ulper,e fertroge idv ree fear etnotc hbee o orka ther thatnot hoeir gipnaaeplr . Theb oocko nsoifes itgsh teen choafwp htieicrsdhs i ,v ieidanectsdhoet cioannsd sbusteicoAnr se.fe resnuccaehsI X.i4n.cd2ai t§4e s,su bse2co tfCih oanp ItXe-rh owever, inc roesfesr-ernwcietsha i cnh apttehcreh, a pntuemrb iesro mittTehde.o raermes numebrecdos necuttihvreoluuygte h aocshce tiTohne,o rXeImV .b3e.i2tn hgse e cond theoir§ne mo3 fC hapXtIe-Vrt houwgiht thhisanet ic otint uwlbeodr efertrose idm ply asT heor2e.m L. KATNOVRIOCGH.P, .A KILOV v. FROMT HE PREFACE TOT HE FIRST EDITOIN FUNCITONaAnLa liyasss icsi endticisifipclo ifcn oepm aratrieevcneotlr yi gIihtna t.sa ken shaapse iannd epebnrdaenoncftmh a thaetmical oanlnyia ntl hylesa i2ss0t -- 30 years thoutghhih sa sn otp reenvtietdfr omo ccupoynineog ft he aclep notsritiino ns conteamrpmyoa rthematics. Fucntiloa nnaaliyassm iossb tr inltml ainaifestoaftt hrieao dnic chaaelnt ghiatsta king plaicnme a themaattp ircetss- eanc haenc gopmaraibnil mep rotantcote h oen teh at occurwrhee(dni tnh seee vnteceenntthvu arryi)aw belrieens t urcoeiddn mtaot hematics, laeditnotg h dee veloopfmt ehdneit ff ereanntidin atlec garlaclu lus. Thicsh aenh gabse eenx preasbsoeavdlei l na n ewa pprcohta ot hset uodfvy a rious prolbemosfm athetmicaaanlya sliTsh.ei nvestoifgi antdiiaovlfuni n dcutiaonndts h e reilaoatnnsed q uantsci oonnetchteihmna gbs e erne plbaycac e odl leicntvievset igation thaits ,i navne stoifgs aptaicooeffuns n ctiaonndts h etirra nsform(afutnicnotanilso operatTohrusas)d .i ffeenartloi peartora nio nrtr eatglr ansforremg airasdsa e pdp lying nottoa s ilnefug nctbiuottnoa w hocllea osffs nu ctsi-oannodn set udtiheeeffs e ocfat tarnsformaont iao noc fful nacsotsni asn,d ctohneut iitnyo,ns eei nnos rae n otohfte hre, operataonrd, o sno. Anotihmepro rfetaatneuot rffu nctiaonnaalliy tssh giees n earbasaltca rtp protaotc hhe stuodfpy r olbemosfa nyasliwsh,imc ahk ieptso slseti ocb ombainnsdeu bjteoacs til neg investpirgoabtlitehomaanstt fi rssit gahpetpa qrui tdei veFroserex .a mpasl teu,od fty h e fnuctieoqnuaalFt (ixo=n) w hexra en da roeb jeocfatm so roerl easrsb itdroaamirny , makeipsto isbstloeb rityn,og g estuhcedyhr i veprrbsoleme sa st hseo luotfdi ioffne rential equiaotnisn,tr eaeglq uatbiooudnnasr,y -pvrabollueeam nsid n finsiytseto efam lsg ebraic equatainodtn hspe,r boleomfm omenTthset. r ansfriotmii onnd ivfuindcutailto on s spacoeffus n ctitohnosui,gmt ha y taitme evseb neh artdop erceiisnv eev,e rtahse less imporitnpa rnitnl cea iswp ast hter ainosfrnio tma lbgreaeiqcu atainodrne aslt itoon s variaabnlfuden sc tniadole pendence. Thinse pwo ionfvt i edwin do atr ifrsoema s impslter itvogi ennge rNaelwip zrbeol.e ms arisiintn hgde e lvoepmoefna tn alnyastiusrc aalllfolyrea dt arnsittoai n oenwl evoefl abscttriaTohne.is nec lutdhpeerd bo leomfc omlpetefonrea ss sy stoeffum n ctitohnes , soluboifbl oiutnydv aarlyu e pwriotbahlg ieinmv sce lnao sffus n ctiaonnsdsi ,ml utaneous inveasttiioogfwn hso cllea sosfpe rso blaesfom rsi ,na sntcient hset uodfty h e depee ndenc oft hseo luotfib oaon u anrdy-vparloubeol net mhr ei ghts-ihdaen deo qfu attohireoo nn thbeo undcaornyd nistI.itw o apsr eciisnte hlfoeyr mluatainoidnn vestoifg acshtu ion probltehmatsth mee thoodffus n ctiaonnaallpy rsvoiesdt ob ep articfruuliatrfully. Moreoivnme arn,cy sa eiswt a sr,ea mrkatbhlgeye ,n eroaftl hiaetp rypo atchhaa ltwl eod more gaennaedttr hasela ,mt ei mdee epaenmrdo erc oncrreetgaeur,li atnicdeo sn nections tob er evebaelcedai,un ssneii afigncdte toaifil nsd iuvaplir dolbemwse rber uesdah side and nol oenrog bscurtehedes seeo nftc hmea ttIeitris .nt hwiasty h tahtre ea ltionbsehtiwpe en prolbemosfd iffeenrfotr masn odir gibnescm oescl reear. Thewa yw apsa vfoerdt hcer teiaoonffu nctniaaoln albyysi insv easttiiiognn ss everal xiii Frotmh e Ptroet fahFceie rE sdti tion XIV aroecfal sia cmsaaslt ahitecamanla -ltyhcsecai lusolvfua sar tiiioenngtser,qa ulna stt,hi eo theooofrrh yto gofnuianocltnt hsae,p proxtihmeoaoCftrh ibyeoy ns thhepevr ,o bolfe m mome-naltolsfw hincaht ucraalfolllraey n d e awp prIonafa ccthth. ie n dailv idu probolffeu cmntsi aonnaaallry osfrsioeswm i tthheiae nrsa esa-sfo,re xamwpilttehh,e conoctfeh fupent c titohcneaa llco vufial naru tisiO ontn hoshet.e h rna dt,hd eee vlopment oft hsee tr-etthdiecicoisapl-l tihtnehe esooffurn yc toifaor nevsaa alrb itloeop,l ogy, abascattlrb grea-prteenpciahqrfoueraeds s y stematiican b dasecfotvtrre mol,fo pment, thnee twr .eI npnda rttihcteuh leoaoafrrb ,yas ctstrp awcaeestsx r esmieglnyfoi rfi cant functainolanyasli s. Funcatniaolnayblsre i esc ktcohoana nefiv drec s otmi eni tnod epeexneicdnsweeti ntth thsey stceomnacstttii(rcobu Dny.H iblearntod te hrosft) h teh eoofor pyae troorns infien-idtimaeuln nasiristyaop csnae,n tdh dee vel(oip1nm9- e112n89t3o )fa g eneral theoonfrro ym elndie saaprc ietnsh w eo ortkfh Heu anrgmiaatnah teimFcR.iei asanzn d, moreese paclioltfyh ,Pe o lmiasthh emSaB.ta incaicahn. Ienrtiefunsn tc tionianlti efiafunenrsdat lwhyheseiritnt rsu n oeudt i ttthehsacn ti ques (tthhee ooofrp yae troirHnsi e lrsbtapc eet,c .i)mo ptrhaaandpt pa ltiiicoqnnu sa ntum mechTahnleia2 csy0ste . a rsse tehhanaep ve pea roanfne ctwer einfundn sc tional analysis, praticiutnlh waeor orlSkfyo vmiaetth eamnasmt;ei tcihiatonsrdd es s hualhvtaemsd o st impoarptpaaltniiticot nnh rsee otpihcyamslai tchsce,apm lha ytasipipcalsni,ae ladyn sdi s, othaerreo amfsa thematics. Thbiosod ko neosct l atioemm b raacltelhd eee vlotprienangnda dsp paltiiocofn s funcntaailno alIiytpss r iism.da erviotlttoyh et edh eoonfro myre sdap ecsa,ni dn cludes thmeo ismtp ofarctotatfsnh ttih sea oosrir yg,id neavpleelbldyyRo iaensBdza nwahcihl,e altsaoki inantcgouc nsoto lmaewt oerIrkts .sujc bemta tittseht reh eoonfrro ym sepda ces, thteh eooofrep yr aatnotdrht seh eofurnyc toeifqo unaaCtloi inodsnesar.ta btleiens t ion patinodo ne-aalrsi nawlsnei eloalepr r aatnoedrq su aatnisdop nesfcnu,ic fitscip oanc es anodep raatroderi ss caultse sneIgdnptr aht.i ctuhleiaarsd er e ,t aileodtf h ter eatment spaocdfei esffr aebnfultneiic otonsfse vvearraiilan btulrceobesydSd L . S. o boTlheevs.e tpoiacrsre e ltaaotg eedn ienrvaaelts iotoifinn gr taoelepg rrast.o Thbeo oibksa soenadc o uorlfse ec turaLtee sng girUianndvi evnfoe rrs stiutdye nts speciiamnla itzhcieanamlgna atlaiyncsdoi msp utmaattihoenmaalt ics. TOPOLOGICALA ND METRCI SPACES IMNA TMHAETaICniS m porrotilaasencc rtod tetodh ceo ncoaefs p pctae -tihasas ,te t betwweheons e ceelrertmetaeliinaanotr nspes r esbcyrm iebaoenfds aTxhiseoe mts .i s saiitndh s,ii st utahota ibvoeeeen nn,d owwiettdhshe t c rtuoutfrhe re e levaIntnth issp ace. chatphtbeea rssb uijceo ciftn sva etsiwtoiibnlget l o pioclaaonlgd smaepctterhsiae:crs ee sefotrws h oeslee mace onntcsce lpots hoeabfnse e pesonss tuTloapgtoielscdopa.al c es weirnet creiod1dn 9u b1yH0 a duos(rsffHe aeu srfdfa)on mde tsrpiaaccl eiesta trllei er byFe rch[]e1.t § Geneirnafolr matiseotnOs rd.oe nr seedt s 1. In stuhbissew cert eaiclsolon me el eamrecyno tncaenpndto st afrtoimo nt he g1e.n1te.h reaoolsfr e *ytI stn.h ciosn wteme axitn atnia nrifomnpa oli nt oft hvei ew, taking conocsfee pottcr, o lltebocei t niitotunic,vl eaelnanydro tn eioepnfdr edceifisneBi yt ion. theel emeans tewtsem eoatfnho ejb ects ocfo mwphoiscehd .i t is Wed entohsteeoe taf lnla tnuurmabbley Nr t,sh s eeo taf lr len aulm bbeyr asnt dh e seotaf l clo mpnluemxbb eyr s IR, LeAta nBdb es eTthsne.o C t. aaEt Aiio ncnda itthetashot ejb eacb te ltotonh gseseA t; thneoa ttaioA¢tn h aad ton eosbtl e otnAogW. e s ayA i tassh uasbteo atfs eBat,n wdr ite Ac B( oBr A ,)ie fv eerlyeo mAfea nltbs eol toBonI. gAf sc BnadBc At,h weesn a y => thAaa tn Bda reeq ,ua anwldr iAt Be=.T h eem pste(ytat ithst h,se ec to ntnaoi ning elemiedsne tnsob)tyt e hsdey m0b-Io(fl)P i asn syt atperemteantiteon l ienmgoe afn ts sett hAte,hns e u bcsoents oifs tEaiAlnfol grw hai (cPih)ss a tiisdsfie tenedobd y {EaA(:)aP}o,rb ri{e:a(fl)aPy} . LeAta nBdb see Itwfsi .eta hec lhe amE eAtn htie asrse s obcsyioa dmteeefi drn,ui alt ee, unieqluee/m (eEaBn )t,th aem na pp/frionmAgt Bos i staboied d e fiannewddew , r ite fA: -+B F.o era Xc chA t,h iem afg(eXo X)fi dse fitnboeet d h see{b tE Bt:h eexries ts aE Xw ibt= h/( )aF}o.er a Ycc h Bt,h ieen rvismea( goperr eim-1(a Ygoe)fi) Ydsfen fied tboet hsee{ tEa Af(: aE Y )} m.ap ipn:g -+/ABi ssa tibodeo ne-it!f(o ia-=)of ( n 2a)e impal1=i a 2e•s i sIsat ti bodAae m apnpogin Btfio(f A=)B .m app/iA:n -+Bg si said to beab jeictiifo bnio tot nhie s- taon"-do onnIteffoi a"sbA . j i ectthiteohnmne a, p pgi ng defibnyge( df )( aa=a),E A, ciaslt lhieevnd e rmsaep ptif,on gai ndsde nobt1y-e 1•d Wen ogwi vdee fitnhoeif t ibtoahssneeis tc - thoeeprooarnetIswitf.ii ectaahecl lh e ment osfo mneoe nm-pitnyds eextt hieasrse is aoatcs eeXd"t t hween tshaaafy mat i olfy ' oc A, *T hfoeu dnatoitfoh gnees n tehreaoolsfr e wyte slr aebi ytd hG ee rmmaant hemGaC.t aincittinoha lrena tter haolftfh nei necteetneunrtyh. 3