ΓΡΑΜΜΙΚΗ ΑΛΓΕΒΡΑ

- i / , i r r ' '! l ' l

rVhllllifit PrllOIE

And6oon oro dllnvrKql

' l l l r l r

,il;l ll. ir,trii

,,1{Ftj r;

n. t

NANINIITHMIAKII EKAOIIII KPHTHI

GILBERT STRANGMassachusetts Institute of Technology

fpauurrH AAIEBPA

ral EoAPl,tofE:

rpkq M@a

IIANEIrtrHMrAKE> EKAo>Err KpHTHxI60!rrri 6oesd na11rplrrrris Ev6osos Apep',x11s

llPA( EIO 1999

Atui6oorl oatdinau.6" a--

--\

nA"D IIAM@rAot

NANEnDTHMIAKE> EKAoxEI' KPHTHII^PwA TEXNO^OTIA, KAI EPEYN^'

Hedl|}'lro Koirrc, T.O. 1527, 7 I 1 10. Tt),. (081) 394235, Fax: 394236A€riva: M'ials5,10681. Tt),. (01) 38183?2,Fd: (01) 3301581

e-m!il: [email protected] xa! [email protected]: hnp://www.p€k.uch.gr.

tEIPAT nANEnltft IMlAKll BIBAIOoHKH eETlKaN Eru:THMaN

fftfug re@otirou: Cilb€n Strutr Lircd Algcbn and itsapplications, lrd cdition

@ /98a Harcoun Bracc Jovlnovich, PublhheB@ 1993 yt.Iill'.1 ny*i yL(rddar n^NEruriHMrAxEr EK^o:E|) KplrrHt

nu.nq txdoot Orrd6pros l9e5Zti$e4 tfioon Oxrd'6ercs1996

Teit,t t"6ottt: M&ercs 1999Alt.;doott otu dJ,trt ti: n.ieE ndpq os

Ka0rnfi riE Mdonf arLr.onTprl.L fla!/Iio! KeirlN

Telr"i Enryllut $tuqeaor(r ArlFirens Koeoy!awd,{$, NixoETlavdart<, Mdqo Tee6iEou

*oqeoe$ia z 6oroi,t.r't: >oqia >a66dxt (n.E.K.)A {oaoilorl-Mowie Erairo{4j rynorsr^xAzAN^(E.flnEurax^do!

trttidor .EaqiLloir Arp4lQfi Tldvfi

Al€lJewriE o$adrr :idqoloE TqoXolds

rsBN 960-7309-70-7

IIEPIEXOMENA

nPoAoro: METAoPATTHnPoAoro> :YrrPAoEA

I nNAKE KAI AnAAoloH cAuss

1.1 Eloqyo}ll1.2 H f€o perpio tov lQappud)v €Erodosov

1.3 nqad6€ryla AnqlolqrlE Gauss

1.4 :up6ohop6E rar Iloltrantrooloop6E nrvdxorvAoxioEtc

L5 Teryuvrxoi IoQdyovrfcror Evoltrayds feaupdv

1.6 AvriorQoqor kor AvdorQopor.Aalt1oeG

1.? Er6r,{oi I]ivdxeq rar Eqaapo'ydE

Aorrto€rE yro Enav' 4!n

3l l131821J13545,a5160

69

G. STRANG

, AIANy>MA]'IKoI Xopol KAI IIaMMIKE EEDQ:ED

2.1 Alovuopoilr(oi XdQo! rcr !r6xoQo!

2.2 H lioiln €E|o6oE0rv oiE nQoE n ayv6ororjE

2.3 fQdpt-r!l{{ A$EdQrIold, 6do€rg ||or 6rdoroolAoxioeLa

2,4 Or x€oo€qlE oepel,rddelE 1Jr[61oqor

2.5 fqdqipdtd ){(tt 6[,(r!a

2.6 lQspplxoi p€raolnparlol.roi

Aoxrjoe6 yr,o urcvti),qrp1

7380829092

1001031 1 5t t813I135t46

1533 OPooroNtolHTA

3.1 Kde€ra 6!o!!oFara xar, oqooytirlr,ol r.rnolopolAoxiaete

3.2 EourreQrxd "yr!6pEvo xcr ,[Qo6oME o€ aroEiEEAoxiaetE

3.3 nQo6otrig xdL nQoo€yyto€rs Etraxioroi! r€rpay6vovAaxfoetg

3.4 Oo0oy6v|3E 6dos!9, oQoo"fdwror, ryhorcErcr, 1 ogooxovovuonoirpl Tov Cram-Schmidi

Aaxfioe63.5 O rd)01E p€raofl[aflop6E Fourier

Aax1o€tE3,6 AvaoxofiIon xar nQoofirrriri

AaxiozrEAorio€rE yta €rrqvdLq{I

153

1A115111188

\922@

2t32212282412M

fPAMMIKH AAfEBPA

t474 OPrzoYrE:

4.1 ErooyoTri,1.2 Ibrdrlroq Tltq ooiloljoaq

,1.3 Tllnor 1,(a rlvoailolloo

,1.4 Equato-y6q rt0! oarlolodrv

Aoxio€[e 1ro €n(l!d),i!rI

251256260

214218281

c IAro l rME: KAI IAToATANYTMAI 'A

5.1 Er,ocl1,oyrl

5.2 II bLotdrvrcr lroQqli €v6E rriv(rxu

5.3 Or FELodoFlE 6raq,d0(01 rar oL 6!vci[€!s Ar

5,4 AlapoQ[xaE €E[o6oeLS 7a! ro ono€rDio d^'

5.5 Mr,/abu{ol nivarcE: :![perorxoi avsvrL rpplrlqv6rxc! oQ0o1d)\,!or 61'cvlr oQ0opova6t.rio!

5.6 Mtr(oIItl(rrLopo| oiror{irl!(g

Ao'tlioo'E 1,'o rndvci),rtl,r

2842932963033051t1321

:l:llt351

372

6 OETTK,\ OPDMENoT flrN^KE:

6.1 Elirrroxcl, t#yloxo rdr ocl fuaT'xd olll8iaAoxiarLs

6.2 KQrrliAr(l 0ttrxrtq oarorporlroE

6.3 HproproFlror ]{cL a6aroroL rivarcE,,4r =,[Mr

3833rJ5393396

C. STRANG

6.4AqldE e}'qroonoi,rplE rol to nllirov ro! Rayleigh

H pE0o6oE rol n€n€Qoo@vov oioll€iovAaxriaus

403,105

411

419

IYno^orrtMor ME IINAKE:

7.2 H ordopl *cr! o aQropdc t{dldmaorE evoE trildxaAaxioe4

7.3 O onoloylop6E io! r6ioxtl.r.(i1

7.4 E rovs)"qnrlrlE [60o6ot'ylo 10 Ar =,

42r423430432441443450

8 lpAMMrRot nporpAMMATrtMoi KAI @EoPlA nAIfNIaN

8.1 f pcppr,xlg Avro6tlregAoxia€rE

8.2 H pd0o6oE Simplex xd{ I F€0o6oE Kamarkar

8.3 O€oQi,q Adotrou

8.4 Movrtl's Alxr{ov

8.5 O€0JQia narl,\,iov xqr ro OEdStpq MinimaxAeaio€rE

NAPAPTEMA AIIAPAPTI\,{A BflAPAPTHMA f

BrBArortAoIAAY'ED Enr rrMENoN AtKE EoNEwETgtto

45245E4(/)47848t492495503506515

51853153S554555585

IIPOAOTOI TOY METAOPA'TH

Touto ro 6!6r'io lAcqrtrlxfu {itry€6aaq €ivor, o?16 no}.IaE on6qr€K, Eexoalor6.Aur6 og€il€ror oro 6r| €aval yQallavo and 6vov "ForrQ" ror.r €[6o!q, rounaQozotrouo€i qv el6l'Gq rou or,rlx€4rivou xaq €v nol,Loag, rrlv xaooQiE€r.Al/dF€o( orq oQ€rd! ro! rfrft I onl,6qro rar q dv€lll Qori rorv !6€6v. qroooQoria ll€raEli oE(,)QiaE ror nQdElE, ro knr6raro xm{[oQ. rl €u)|ollo rurl,cno6€iE€o1, ,(qr I €oQiTrFo rou uvoyv(oorLrori )colvori oro onoto on€o06\,rrar,

AprOp1uz4 ov(i}.uo4, brc9oqrx6g alrodrorq, 6ir o. yQagil[dra, yQdpprrkTtQolQolrpaltoFoS, e€oQicr ttsrY\,iorl/, t(rXl'q pricoxqparLopoE Fouier, €ivat l€Qlxi]scro rLg eqcppoy€5 rrK lQapllx{E dly€GQaE rrou outlr(owaL oro x€iF€vo. O0u,-rplrrxti :tpooavcro),ropevog ovolvriorlg 0c 6qer rnio6 6l.t tt 0e(,rpic (cx6Ftlxar io 0€(l)Qrua qE [oQqrk ro! Jordan) rou oulrri0orq naQouordB€lor oE 6riorEapIlloio pa0iparcl orcr Jtdvenloripro. flQ6xrltoL yta 6ra np<iypcrtr, 1pr]oLpo6r6l,io rrou Eivol oQloror€lvrxa lQa[pavo.

:ru rQuroruno, o ouyyQacl€ae XQnoLFonorfi FrxQaE, ouvtoFrE nQotdoftc xatfi).dott }.6yo y},qe!p6. (riJd rcur6XQovc nux.r6 o€ fi€Qlel6p€vo. To x€ip€vomaLrri 4oo[l{rLzrl ovdyvcuotl nQoond0toc va percqapro our6 xo oxu),, robrriorcpro pe loupop xcr! oin! €Ll,I!!xi Fsrdqeaol. All.od ouro eflar euxolo(1..I. orrlv dorano+ov6,.6oro 3.3.26) oLl,ou (6noir ro aot€io 6rd6dE€rqr avdf€ouor4 .ypcgpiq) eivcL ro).u 6uonoLo.

:Itr(rlrr:.{ 6oti0€|d €ila cn6 ro owd6€Ieo Arlp{rel KoQqtrolv(ixl t(ou€iU€ lll l€vr}.ll €1rq,t6l€ro ,{sr €lrrlourro€ ro x€iF€vo $€ n}.yl6o< noQ'at'lQ{o€(l)v,nou 6ot0otlv rov (rvdyvtirrll. Erycwrui rriolg irdv rt oqr6o;.il xou :r6q@onTaoXovd, 6L€u0tn'ni tu1, IlcvenLor4pLoxriv Ex66os{Dv Ka'ylqE, L6ioE ornv oaoia:r66om1 rov reprxriw 6gw. Me 6Lz{ rou rporpomj I.1. uLo0erdr rov 6po't6L6poaqoq :rivcatrq" cvri rou roxd rraQd6oorv lploqonoro4evou

't6Ldlov

nivczcg". O npriroE enLrpd:rrl ro o1r rorLop6 rou ouoLaonxon "L6ropogpitr"rou cro6i6er ooord tor 6po "singulffily", rvd ar6 ro bmrepo 6ev rcptiyrrcrcvcLolo ouolcorrx6. Bo{o€Lcr €riolE €i1a afi6 rot\ ouva66Lqol,s K6fio All'

IIPO OTO> TOY METAOPA>TH

FiFQror.6riorrtro t(a! XoedLqrro Kop'rdqo, nou &d6ooor rqooemDui to rcl4evo'iaL ot,\,eraE(rv to €rrQ€oiQNo. Oloug outorb toljs eqqe[or6 0€4pd. Elfilqo &r F€tl 6ori0eui toug naQ,oliioro orlegc otov dl*rpc cvcprirorl 6va rcigevo lqqoqro,caL 6@orcG6ao0ri6 orrvd:ua.

Iorlvtog 1994n'iet5 ndlr{r1o6

AvqnLmook Ko0ryqnkiou ndv€nr.oqplo! KQrlrqg

IIPOAOI OI TOY IYIIPAOEA

H faqFFLrirt i\ly€6Qo €ivsr €va pova6rr6 pdonpa. An6 q $!o nl€uad €taorlexd0aqo rct 6FoEpo. Av tlere rpic 8rrvuopcrrcr oro! 12-6rdoraro X6Qo,fnoQ€i1€ oxe66v va ro deize. 'EvoE ow6ucopog 6r(l)g ro nQ6io or,l,/ ro b€rireQopeiov 6io tpoq6E ro rQiro dLdwopa siv{lr oljorol,orsQos oc4 qcvrcoic, ol,i,<iioa4 tivcr cx6po 6woroE. Aev l6p<o cv xdnotos propei rc 6rr. d,lou< ourorgtolE ow6uaouo14, cL}.<i x<brr,x (o' a1jl6 ro Fd&Uo) oAXilftE v( lo r.d!€r€.:iyouQo o! ow6lqopoi rxsivo.rv rorv rQldrv 6rovuoldrov 6€v 0a yqrloouv 6r!0rov l2-6Ld('roio t6eo. (Oo6dpo! 6rr ro pdoqpd 61er. 11611 cqliocr" ow{004 onp6l,o1o6 6r.cr6dlercr, euxolql). Aur6 fiou sxeivor, or, ow6uoopoi 7rpl(ar;r, eivar,idfl olpa\rtlx6 -r(ar 6Xr Fovo yLq xa0cQorjE Faor JarlxouE,

Auoi eiva! I tiLlrl rl,flqci r1E fqcpprxrlg ALye6pa6. X,otulltrut xlJ;t ypI-orponrkar, A6xq lQ6lro nerv. 6!6oor6iov rol.u cqrplp6vcr xaL I lorlnliorlFdoid rou FsorlFqros XCrv6Tov, Mro rtiorc xcrdoraor| 6€v prroeolio€ va ow€-Iror€i. H faoFFo.ri .i\)'y€6Qo it€L liv€r rooo 600014 xar 1600 €qoQF6oLFl ooo oAr€rQoorrxo( Ao./!opos. xar €urvldx €ivo! €ur.ol6cQq. To nQ6yQo![o onor6d)v6raeff vo crLldE€L ror 1 faagulxri A).ye6pn eircr n;.6ov €ua6o-E anob€xni o)!r1a onuQairytro ldonlc.yro ro 6etrtpo rl rpiro 6ro9 ivc xpoarcLroripevo yc0€ir,{6( onou6d(, xat 6vc xevrqrx6 qr{pc rrov Ma0rltrcrrrx6v.

O ozonoq curori rou 6r,6liou €ivor vo 6€iE6L xar rK 6io r)€uQdE -rlv ol.roQ-qr{i qE fQappLlolE A}.tE6QoE. xdr rnv aEtcr qE. H rQoonde€ra 6ev orlxevrqdr-vslot oL6xlqQn o€ oeoqrparc n<rL anodeil€q, ctv 1{o! id ouotlQd MdoqrorLlidundelow. H dprpcoq eivcr l,ry6r€Qo orlv auoqQ6qrot riar noLlJ nsQroodr€Qooilv x(rrol6lol. naooraeA n €E4yio@ raei va atro'eiEa. Xto &6).io,onotg xat ofi1l rdh, oL r6dr_< dqlowot pe roQa6€iyparo. 'Ordv douMt€K B€uno16QorE, tolE rardld6oiveq. H rxovonFo yra Fdoryoltxrl oxari4 ea aranlu-I0ci, qv tt1g 6o0€i aaxsrrl 6oul,erd. Kcr. oL orpcwrx6repeg !66€E qE fqogfrxr6A]"'leBQaS 6t1, tit& noli 6uoxo),eE.

NPO OTO: TOY >YT|PAOFA

Oa rl0Elq vo fio rqooQd 6tr (rur6 rivcr €va Fo0qrorlr6 6!6),i,o. Aev€l€L onlouorslje€i 16oo :roLu r6ore vq tdoel ro orioTr6 rou. A€v 7ilorEllc)orr ot qowldg i or zc0ryr1rfE Em0Wolw 6aa ?.€v6 Bd0t11.o [€ taEK drQ€E xlv€66oud6a Fnoa€i va €mr€vl0ti xdtt oEt6i'o1o, uat€i ro 6t6l,to vtr 6ol0d. Elnfqt,lxot morento r(r 6sift. nioo) on6 ro ov€donto z(rt rrQoool Lxo orutr dlnori rou6r6i.iou. 61r €i1,o! tpqrua\,o yro vo 6r6ciEer rrQo$crLxd Mco4crrLxrl. Y:rriplouvncQrryQaqor no! l)a aoQcLeiu€r€, nar iorog u:rdpXow rneElytloe6 nou 6€v 0(rtQrrcorsi.r€, aLld b€v proQ€i. !o ocg brarprileL q bwapl nol aror€L€i nt 6dolrou p(roiFaros. flqoloqri an]"d xor qlol'(d cln6 pLo €r0€ia { €vc efiinroo orov,r-6rdor.rro XdQo R'r. Aur6 ro 6tjFc tivcr 6,rt;.oLr.n€eo ota Mo0lpatlra, i{atxdO€ qorrnxrli F roQ€i v(l xo xdv€r.

E6tb ocr ;rpfret va liver I €QdJqoI: n6E 0(1 anQ€r€ vc crQxio€! ro Fdoura;ot :rrproo6rclx)l {polfiFac 6Qxowar oro nQ6ro pdor]F(r l,v(,QiBowo( z&r yLayQopfrx6( €ELorio€rE. IIse' 6].(r durd :rr.orcrx'r 6rr :rptnrt vc cqliloopr Fr ll€Ero(ror1t fr nyvchorcov, /r = 6, xar. Ftr trlv nrrLorlol€pl xar lQ,qor[61€pl pa0o6ouni).uory up cnul.oupri causs (6xr $: oQigolo€El). nivor pLc r6l,€ta Eroay(Dlltorov noLtrdnlsorootld filv(ixcov. Kor €!rruX6E. av 1(or" I p60o6os €i\'cr 1600€!lxo),q. urloQloriv oqlEi(r €sp€trL(i)8l yro rylv xarov6qori rqg xcr rda o1e66r 1Lur.de€ qolrqq. nQin€! rov€iE va xardlci6€r. xao(i)E I anai'orqtl o6ly€i aridrov aQt(fro nllaxo A o' aro d\,o, rQlyrDvrx6 U, 6rt o A nvdnieroL oe lv6pevo6io rQr./ol,rlov nrv(i1((l)1/: 4 = LU. ALtrit rl aqQorrtplort 6rv eivsr 6c0rd. xcrrivcr ErhoLo v( rnoLne€w€i. rivor 6Fog rqoFepd olprcr.rLxri oqv nadE{. fispivc cud t:ivcL dva tieiTro ev6g oo6cqoi po0rlpcrroE, rort io E€IrlQi!€r atr6 FLan(goudicrol crolo).oulr1at |rovo [r'trQ,rErK yQqpprc"v 11 rov ,1 I.

'Evc dll.o 06Fc eilcL va 6Q€0€i n o('Jori iolurrFo. Av rrrdQle! EEolxEf(,)olpe :rpdf€( nlvdrolv, 16rz o KttpdA<tLo 1 6* rpin€t ut f i1'tL ueyi. To Klq'rl,olo2 o:rcnri neEoo6trpl 6oul€ta, ,.or (ro16 o+roiv€r boukd br(rqoQ€rrro{ €i6orE

olr aQrulrlrrxoi unoLoyroFoi nou F roQ€i va xciver dva6 u:rololtorrl;, nLl'ci ptuxorav6rlort lrtq .'l.r: = il nou crpXiler pe rry arcLorprj xcL rlyciver 6c0utepc.H rdEn trQan€L vo yvtrqiler drL dD.alr q rcxirrpo dp'lovrar r6ar<. A!'tt yrqor|,,xflrp4rivc 6tcv!opc1a, xat:ral6paor€ bmvuopcrrrxor,s 1ri-rpotq.

'EI(|) n€lso€ioir o! x6oo€Qr( 0rprL(i)6e6 lrr6lc0QoL --o I6g9E r(l)v orll6v rou A, o irqogyoappcbv, xsr oL l-q6ev61opor tou ,4 i{or" rou r4r €ilct o nlo orior€tr€oFcltx6(rp6:rq va ellylurt 1 ypclqrrxl elriqtipt xcr ov€EaQrloio. ,ror va zoravotoouvoL 6woLrg '6cioq'xar "6ram(ol ro! roEl"- AUIoi or XdrQoL ovorruooovrororo6rara d,id oro0t-Qd, raL noQdyow naQo6€i$trrc rcrrti 6vnv eweLri4 tpuotx6rQono. Arrtol €niolE onolstroi,v ro d€LOi yLo qv ,4Ji = /).

Mrop6 vcr 616oo €va :raad6ery[o, 10 v([ 6€iEo n6E pnoQouFr v(r 6oqrs prar.56c pe 6LcaogerLxorig rp6row; Eildr ro 0€F€).rrir6eg 6rua tou ;roLl,or).croLaolror),4 uri. r, n6q ri.vcxa eni 6vc 8rdvuopcl I' rvc re(i)xo otd6to to /r €ivar afildEaaropoi. :ro o:r6pevo ortidro €ilar dvq ouv6lJooF6E r(,)v ortldv ror ,4. :rorpiro nd6!o eiror 6ao 6rdvu.4a orov I6Qo ro)v olt).61'. (Or(,,Qoip€ 6vav I6Qobravuogdr(ov, nou neQl€/,€r 6).oq rot'< owbuaoFoug xor 6Xr g6vo aur6v.) l't(r

nPoAofo: >Yrf!Aq'EA

avov ol,l,€6Qrorrt, o,4 avaroerord 6aav yaotrtlrx6 rtsrcoxrttlaxropo xar Al: €ivaLro orordleopc qg eqapponq ror oro r. Ksr ttr t6ooeQo eivaL o:roU6cic, ncLro 6r6Lio rpin€L vo 6do€L I[s 6raouvtjto€r,(.

Tc K€qdl,crc 1 5 €iv(lL nQdwarr I xaabLci €v6s FoeitoloE lac$puinE Al.1€6acr. I]€Q!€Xouv 6vav $€ydlo apL6$o €qaa[oyd]v oq OxoLrtt, 11l MltcvLrt, rrEflr0aloxnx€! riuL rI :rarroT|rri, ro OlliovoFuld ?ro[ q Blo],olio. Or €qap$oy€guureE 6ev d1orru rapcpeqrmri oro r€IoE cl,I<i t:ivtrr 6ep6vtg pre ra Fao1FarLx{i.Ta Sirrua rar rc lpcqrpcrc, €xt( tou 6r! ei.vct ororbcrieg nr116q rtvdxorv,:roilor.rv orl$cvrLx6 p6)"o or1 Mrycvrarl nol ori]v Efilortilrl Yno].oyLortbv ]tolbivouv eriolq :rapaaeiypcrra yro rI 6tbcorctric. Auto nou rc MaEFarrxtipnoporiv vcr xtirou', xsr ro ,.idl€L iooo xctld rt faappLrrt Aly€644, €iva! va!e1ropi.!ow povt6l.c ;tou €t ''at E! paracL xQr'rgtr€vrr a1E erqp1oyig Aur.j 'ar lva6r6),io nor atupitw oE xaeaQotia pt191porz"o1i< 1tc \tdiEouv ecpappoop&aMag\lauvA. fl!o1r:!(o oxr or xa0ryrlr6g 61ouv €I€ue€Qio zrvrlo€o),' 1,c brbdEowo.r( XQEldEovrar or ctorqrdE. H fiOoonde€t(l ea owapct{p0€! r}.{Qoq.

Av 6XEie 6€r nQonlot|l.reveq cx66oe6 ro! 6r6),iou. 0o 6rarirctdo€tE alJ'aylq.H ,.oQ(iyQ(1qoq 1.1 eivor orxeicr, cl"ld 61t q 1.2. :iyouQa ro rrv€{tto 6€vaxs! GtrldEsu suro ro pooqrs €iaar ax6[a !cDvto!6 brorl ro slllltcilrcv6 ror€iva[. ALbdoxovrdq ro orrv€zdF, 6aiko pro ol.6tr].lol o€tQd otro 6€trrtdo€tE-ot1v op.Tcivtoorl xcrr omE uoxrior:Lg (cxatottib€E €i.\rcr! \rd€q. oE [!a nol.u €uQeiqrrprolj), zcL erio6 ora nseL616[61o. OL fieQloo6r€Qe( atr'clJrdq rL5 6€].rL(bo€K0o qrcvouv p6vo orrtv ra0ll€Qlvrl nQoondo€Lq q6 br6aoxali{(q xtlr Fxpd0ryry,<a1]Torr 1('! po0ripcrloE- 6rav I rqrd).l.IU enrlrlytoyl 1i doqoq 0o 6EiE€! rrl6rd(poQd. Avoq€e{, 6uo a}.}.al€E no]J qrclirowcl! oto"r fiivclxc xov I€Qr€ZoFElt'rv:Ot ypct l tytxot t t< raonytn t tut t ' t r 'vn' rvodparDFrvor oro xfrFts!o. xar uncruxfrpra 'via (rcL :rpocftpnrxl) ;trpdypcrqoE yc to\ Tar:D MErdoxtlturnp' Fo ,iet.Aur6< rivar io0rE o cLy6pL0poE nor E€IoeiEeL ora ouytQova MaoTrarl&d. xatix€r qdQe! €fi(\rdoraon m1v q4prcxrl rnoleqlcoio. Asv €ivd! xinol€ ql),o fiaQd€v(rE yQnyoQoi xQofioE nol.).anl.aoLaopou $€ 6vov ouyx€xpttl€vo fiil,orol 'looq

p6vo vc 1vroqilere (6norE €ydr yvd,arqa) 6tL r !6io urdrqzet rcr eivnL orycvrLxtl.Efaar €uxaaimlol vc crvoxnl.ri;rteq n6oo orevci ouvb€€iar F€ rI lpopplxrtAl"yr6Qo (ror .r.ody€r rorq [ryd5LxonE apropoiE).

Au6 eircr 6vc nori)lo trdo+n radFpul4E AIy€6eaE. H e€toaio noQaxr-v€ir(!, rc! €1,roxuexor, drr6 p{(JL€q arpcqFoyiq. Tcur6lpor(, o oxonoE Ei.lrar nroio.,'6n('t xdr ro uLLrc €tvar xali{ 0€F€}.t(|lB€ao. Aqoll io K€qrilolo 2 fiQolt0'prloer nepo a:to qv crc).ortpl rcL roa ,4 ',

Fara! rlv 6€rr rotl 8lavropdrrxol

Idao!, ro K€qdlaro 3 enrxevrpdrvercr orqv opqoyaudr4ttL. f€oF€rerxci, atl[ocivnL rcrcrvolr6 :rpLv crn6.ro :rpdrro pir0rpc. Al"y€6aurd. ra Bruaro €i.var ou{€ioror ouoLcb6'4 vn 1'vnpiler xaveiq a6re xLirrord 6tavtiopaxa €i\roL rde€rd, rt noLoLr:toltopoL eivcr. oq0oydivLoL, r1 nd4 vc rpo6ciltr rrivto o' avov ul6y,oeo. I trcijEvc xcnnoreuciEa prc opeorcvovrx{ 6dot. Mtp, urorrpdrE axr6 ro x€qd}'orc.Kcr6rw, ro Keqdl.cro 4 rcporoldL€t xr.E tpllouoe<. tov ow6erln6 rpixo uercElqg,4r = tJ xar qE -4r =.\J. Awsr 6irox! €va tQrrripLo ar'flore€qJruorrtroE, xo

NPO O|O: TOY IYITPAAEA

onoio avad€rxv{€r ro Q6}.o ro,\r rolorrrrdrv. ,{1116 €roriy€r ro l€}.€uraio p€yd}.o6t1rcr oro pci0r1;rc.

)ro K!'t.il.cra .r ;r.rrrouUroLrrat rt br oyovr o:ror toyl :rqrv ,rro r4 gopqjJordan. Or r6LorLir6_< xorT<r L6lo6ruvuo}ora paq o611our xc-r' €uU€iav o1i6 €.va1':rivozo A orq btwtiFe|c rou ,4r- A[vou1, r:lrorioeK nou rle],ioooYrar lpovrrri

6wc4[xti rpo6Lrpcrc. o€ avri.oeol ir€ xo oxcr[xo ap66].Ipo ,4c = 6. ncp€1ouvtnioqg ;rLqpoqopieg nou 5ev rivot aqoqcretg ruo rov i6ro rov :tivcxc 6raqri.voxoq Markov rz€r rSro-rryr] ),,,"- = 1, dvcrq op0oy6vros ni,vczcg 6X€r 6La ra,\ = 1. 1((rr 6vsc orrFtr€rpLz6,< nivariog €X€r rQ(y[(rrxdg i6rcrqr6E. A.,, ro ldenrd

oo( q)rdoE! orrlv oaxl rou K€qo).oiou 6, oi oldocK tl!:raEl 6rorLurb\'. o6lydra1..(r[ oQtlouodrv or4qr€rprrcbr :rr"vrixrrv 0c cvc6€iEo1jv rI owol! o].6xh1por ror]avrLx€+6vor. (H rrL€licrtc rapcilpcrtpoq rou xdo€ x€qrctr(liou fivar nQooraErlx{.)Karo,.Irv. ro K€$dlalo 7 rnrx€liQd^,€roL orry a0L0FIrLxrl fQqpprxrl Al,yr6pa,;ror eirnr 16dq yro ||(l0nttorl1{olis' rjnoLoyropotE. Ka! flol€uo 6It ox6pl xaL[La o!.,'topl lolrd oro K€qd],oro 8 irdv€! 6lv(rrrl pre €rooy0ryl oror 1qc14rLx6:r0o1s4punouo. rou ol t tcr rnv xo:ro xor ervar. rcuroxQovc qvcr l oUpoLr4rfEtror-r X(liaovro' rou 6QiorErc! o1o i6),og. xcr 6ev elerdlr:ror.

Oo rle€lo '" crva(p60trl ro EyXttpiiro rou laaxl,lou, xcr dvcr riLlo 6r6lio.To eyl€tpi.iiro n€ed1€r 16luo€LE 6].cov rov aoxio€orv (olyn€Orl.o[6qvotdv0rv rcovAoxno€ov lta Enavd),Iln oxo ralo( rq0€v6E Gfi6 ia K€qd).aL(r l-5), xor €nionqtra oultroyf r6€dr! rcr reorcio€ov 1!o irlv €qoQpoop6\rn fQclpprri ALye6po.El.riE(l) 6rl:0[ 66ooro''rrE 0(r !ry1oorv dvo ovriruno ofio rov sx66fi] (HBJCc,lleBe Depa.t enr,7555 Caldwell Ave'rue, Chicaso IL 60618). Errionq el.'rt!o ox!oI c\rcy1,6oi€E auroli rotl 6r6Lio1j Oa ouv€Zioollv oro en6pevo. Atrr6 61eL ti.r).oEnayuylj ora EEezttoopitu Ma04puttxci,x(l'r ouvtild!€L rn fQqpFlnn ALyr6aa[€ rq AraqoeL)|iq EErodoorg o€ 6v<r xripovo yLcr oiuqovc EqoqFooFi\ro Ma-olfarrxc! rar MaE[.rrLXd ^/!o 1q1, r€tvo],oyia. Il€p ap6dveL crr .rorl Fourier.1rrycr6rrig porc6Ur€E, [epr.xd6 dLcqopudq €Ero{i)orLE, aQL0FIrirdS [€0660u5 xcrLlitLttoronoilorl allil ro olFei.o erxi\,JtonE €ivd! 1 fpapgLxrl Al.y€6Qa. 'EI€l

6npoolcrr0ei (ln6 qv Wellesley-Cambridgc Prcss (Box 157, Wellesley MA 02181)raL €1\r€ EEarQ€xL,rd <rrobtrr6 no),Iir T$ruara Ertrlonv pra ovoo€orQlpdvt' :26()0I. yra !a brddEouv o.rr r rvnr : tQracnrrpo (noaarrr l ro.

Auro ro 6r6).i.o, r.006! xo! €rsi\,o rou aro).ouolo€, otolerler. vn 6ei[eL tr,[tioQ€i !c xdv€r €vaE urolo./Loriiq O(l)Qf< a1Jr6 vc rrpLcpXei). H ].uol €\r6Erpo6l. tporoE 5r v orupcrc :rrc oro ypcarpo plrc drrrp4c ocrQdg. i ortv ruarolEvoE xxtrort onog o xnv6vcE ror Crarner, dl.Id nQoI(,)Q{i oillv zaraor€lrt evoqarorElsouorr)ion cl"yopi0pol. Au6 anrarei xal.€E |,8s€E: 610! ro MaolForuid€nLqoiv! H A),^/€60(l :tcpcldver oaqrjq, onLrt roL oraoean. l-r.n rry crcl.oqrt,ro [6ielpa 'iAdE€orv oro Keqci],aro 1 iler xcrL be&epo onor6 vc evr.opoeL pe). , nropt1' r t : t4v zuruvo4o4 r1c :rrpinruo4< n t : r i , . t , f rQt. lvroc ra raayparxci6rtparo. 'O$r,)s 6€v ltl rirvo 6Lc orrlv rcifq. To xEi.[Evo 'rQi]i€r vo or.pr].rpcirve r.crl.Lti xcL vc 6i'ver ra xuEn oryeic.

ME 6uo L6ylo, Tprrcilacr €rc 6rdlio rrou va ErLrQ6n€! \rc 6r6o10ouv pe

nro^olo> :YfrPAoEA

€tim,tio or sqaQFoy6s, o€ ow6udop6 F€ ro Mooqparlxd rou ofiorerjnrovrdr.'Evs r€roro 6[6l,to nQoordoryo va yQdrlxn.T€L€r6vovtaE, qvni €i1,or Flo xdl,ri €$xaleia ln p6vc vc :ro rc er4opLmrir

Fou. Eipa[ €EotQ€rr i €ul.vciF.Dv olol8 ovcrw6o[€s nou rouE dQ€o€ ro 6r6lio,xa! 6t(ow riorcrld6€0 awlrrQoo(orlEri€l. Ilol,]'oi gou q(ow owil€L 166€E rarF€ €Iouv evoaQQuv€!, xm cvaqdqro p6vo rlivr€ ov6paio: Dan Drucker, VinceCiambalvo, Steve Kleiman, Beresford Parlett, xqt Jim Simmonds. Kor orol.ou€ei Fra7r),I06eo qtl,(ov ftal xQLxr,xdv no! €hror ud€qylqcvoE 611 610r. H l<rpouoc rpl,Tl1dx6ooq ixer 6€Lilo0€i lipLE o' autd nou ovrol 6low 6rOdE€! -oe eolrqriE ra!oto ovY'YQaqds, 'Hxav rotui !€^ldln eulqQioryprl I ow€Qyqoio F€ r$, SophiaKoulouras, nou 6cxrul,cryEiqrpe ro X€rQ6yQaqo, xar rov Micha€l Michaud, rouq(€6iao€ io 6t6).io xcL ro €E6pulLo. fldvrrl an' 6].o, euywopovdr tr1 oriluyo,ra nor6ld xcr roL( yovetg pou. To 6r6Lio ovrix€! rar o' autor.E, xaedE xar ionv€upc rou -Jrou ierllxd eivo! ro ficv. AE po! €rfiQqn€i va qqlEQ6o(l) nd],rour6 ro 6r6lio orov nat6pa xcL rr1 prp6pc pou, nou nQdogeQqv 1600 fio).ld: :aEflrxcQroro licr, Touq o!o,

Cilb€n Srang

IIINAKE' KAIA|IAAOIOH GAUSS

1.1 r f,t'Ararl'

To nevrgrx6 irg66l,1pc -qc yptr[prrtjq d],ye6aaq eivor I Luoq olor4rdnD!ypcrpprnti:v elrodroro.rv. H crr.louoicQl xqL n).iov €vltroqr6poroa :repi:rrtoorl eirnLrzetvrl orlv o:roiu to:r1.10o5 rorv cpdrmov roodlar $€ ro jrli0oE ro)! €Ero(i)o€o)!.oc Eer{ruiootlpe trorn6! [e cr!r6 1() 6colro fiQ66trn1cr: n eEn@o E ott nQo€ n

Yni{QIo!\r 6!1C) r{a0reQ(U6vo! rQ6nor }.uo€1Dq yQ(luBr,xdrv EE(odro€ov. H nQdrxrleilnL 4 gd6obo6 qg axd"otEris. aary onoLa rol,lcrnl,irorcr q< rptirtnE rEioroonqror uro;lpatoE crgrcrLpoivrar. on6 rq cil.Leq sEro6orjL< €ror d;mr vcr urc erq0ticr:r'aui< o:rptiro< ci.lvororog. Arx6 o6ryri or 6va prrp6repo otior4rcr n- ItiLorlota-rv Fc rr I clvd;morq. H 6ra6r1{aoio €fio\r(rl(4r6(ivEr(u pdxQlg orou(nop€iv€L llic €Eioorort tt' 6vcr oylrroro. q ofioic1 Luv€I(! qidooE. K(lrofi!!.6€!€i!tl|. bnozo).o va rdgE 'roog rtl niorD rar !c 6pon[€ rorq dlloq q!6orors [€crvtiorporpl oepir. 6cr ereErpycoroipe oe lilo evc o1eru6 :rcrpdberlpc. Mirbrirr:Qrl 1t(r ]rro trrq6euFdq p60o6oE lQno!|lonorei xt1't t6te NaN oetaouo6e.YriraX€L €\raq .lxp|,6rt6 r!'106, nor I€1€r( rnnoq xou Craner, o oroioq 6iv€r rI),nor (6 .abt rrE orrorrq tqr6q ftdv cryvcomoiv) ros n{,ixo buo oprlororirv r enirr. An6 rcr :rcpa8einrcna ror nvclnolrcr o' ivcr rypLpi6ro 6ev brcqriretcL:rcivroft roLcr p(0o6oq rivtrr" xoL{r€al (ro rt, = 3 ri r = 4 6Qtoxolror n€Qino! oroavdrrrpo 6pLo rm uopovlg pLcrg ouvl0r"opdvf d\'0Q(irn!vlE lraQEilq).

Ilpdlpcrr, 1 :rLo erLtrl8erp€vr1 pt0o6oq nou ;rrp .crp6cirer opilororg eivctorlv apirEl prc orprpopci. evch 1crcl'orrprj eivar 11 [€0o6og nou xplo4ronoreircL:rirv.rort yra q Luoq peld),ov ouryirrov eErodroecuv. Ipdroq rcq or61oq eivarva xcrrc),ir6oqre crn6v rov trl.y6arelo, ro! ovopci(erar yn'rxci nta,l,orgri Gauss.

H rd;u I ivor nupantravtrrru o:rl4. xcr or zti:rorc ltopqrl r4c Uroqri vc rivcr161 orrcic orov crvclvdrmrl. )'cur6 ro rerpirlcro 0a e[ryrlootrpe, err6< an6

I I INAKF ]J KA I A I IA O IOH ( IA I ]SS

rov i.6Lo ror ci./6Qr0[o, r€ooEpc qorv6trsva, 6oeir€Ao cn6 rov cn]"6 l|qt({!roFociicrio|,(tlig. r(r onoi.a €ivaL:

(l) H yEaFEraia rorv yatlfqltzu)! .-Erod)ot(o!. AEv siacr Enxolo va qa-vrcoeortr€ tvo 10'brdolaro Fnirit6o o dv(rv l1-6Ldor(1ro 160{). Eivtlr cltoBllrczo),6trpo ra 6ot4rr 1l rllr'rrlxd r( Fnil€6o aft rttrvovrar o'6va Fovabtrcoqprio clrou rolr X(;)Qor-r ztlrh zd:roro rp6ro 6fKrE €ivcr o/€bo! bucr6v. Mr:rQi( fnin06( ('1i! rQ€L! 6r0othoaq ofloupc eivcrL lwcrrov. Kcrro:rLv, l lpultptxti.;1.i,€6Q& tr€raqr6p€r ro rp66lrua orrq ttootptg 11 ttg irlx€r(l 6(cor(iorLE, ororr I6l(io0lol r[Q6n€L vcl f.irtcorri r1yrqrerpiu (zcl[ 10 )iu1oQ0(;)vt[).

(2) H cQlr]vsi.c rlq $€uobou oncd.orq{< (|Jq ncQclorrorohoq< ro! niwz(louv-releorriw. Oa eLorq(ilorpe -rov orp6o),ro[6 ]r[,'(i,a('rv 1L(l ro oum!l((l , IEFotboetuv.:teprypitgorrcrq rouq o1v(irorou< trru t\( bdluoBa.rr rcl rt! F,Ero(iro€r:.olvo:rr tzJ. pr : r tL.r :r tc. ' | r = l . lorr ' , t . ta; , '&I d\ tytr ' ! t ' r t t t iuClyorro-roi4o4 rot A at in ynr;1ttut Lt.t nri< xitro rpryLuttxoi' trirattL L xar tv<)<tit,ot tpLyun,Lxol U.

l'lQdn€r no{;xc !c1 €rocyilyolrlrr !e orrori}ra(r6 rpo:ro :riv(xeg z(L 6Lcrvl-ofLclro. xeedE,ior rou; uitro!( ro),l,ur),oor(lo[ou ro!5. Oc oeioorltc e;rto{; to!avaolQoqo At 1!.a 1.'\ |rtioreoq)o ,4- €voE nivalis i.

(3) :rrE nrQroo6r€Qs! reQ[rrrrio€lE I cuicri.orq]{ neol!)eci y,o)QiE 6uo{o),1(1.:e x(ifioLe< €[b!xi< nrQrjrrdoeq 6|r:uit 1cL anotuyxtivEr -rflc 6161t o! sEro6oelgrlptirploov oe ),(v0ooli:vrl ou0(i, rQ('lylLC fiou b[oQ06r'€r(tr crxolc E!(tr].dooo-wrig rLg. tirr:6tot! ot tqro6oclE 6e\,610!,'tritl ).rcl trovov liiol. :rlv rg).€ur(lklnagimtrol p:ropet 'ta y4t uztiq7tr ).ia1{ vl uniQIEr dnuQo i1it0o6 lioEorr.)r pia riroLcr :repi:rtoorl a;rorrlicrq, 0al"oUr€ lrr pclcr)ioo![e rov rQoro [e rovoroio rt tr60o6oc cnc),oLqniq tE(rad(i)\'€( ,{(0d!c (r' (rrlt ra €vb€I6trcv.r.

(4) Eivcr on[aw!r{6 \'(r y\,(r)Qiloule N9\6eha xo nh1ooE rar aQtepryftt;wxodExt'tv ttor un(trron\'r.rr yK \,(1 Luooutr€ 6'va oiuitB(l [r- rt p60odo (ln(li.or.{)i;. To x6otoq tc'rv I:ro},o"/|oBti-rv npoo6ro0iE€! out,'(i rrt\r arpi6Er(t roxtrovriior. O tlno),o./LoritE tmoQ€i !(l,ah''cL cr.itot+uara nedE€o! a),l.ir oxr roLl.ilrArotrrrrot4ruara.'Hbn 6r, gcrh crr6 dvcr rzcrrol4mpro 611r(rtr, ro o(tril+r( maoli,r),onoirtolq [roa€i. vc eircL or1;rcrvrrx6. (M€Qrtiil n0o6],rt[.rr.r ci!.11 €uaio0lra.lliLa 6xr.) Xoai: vc proqrt or ),sn.ropaQtrpE, 06)"o!Fs !a 5on$E r.l oxorriFar(nor orvuvrti*r(lr" orlv na(iln 11(L rov rQono ),!olq rouq.

To rrl,116 orordl.€otL(l alroU tou req(kriou 0c eivcr 6vcrg,6oo ro 6rvcrov crrorairogcrrz6r€poq (iy6ar0[oq nncd.orrir]<. Auto< €ivcrl nQoy$orltrdo dyoQropos nor lqito4roroLeircr oe picr xcrrccr),1xruri irorxrLir €9cQpoy6''.TelrdtQovo, I zaravolori ror ml y),rboocr r(,v ftrchLrDv (o nivoxcq o!!i€).€ordv. or. niloz€q ror ertel.onv 6rlpcra cral.oqnc ri r:vcrl?'o?6_( ?Qcrpp(i)! xar oLreLzoi rpr.lovrzoi :rcrpirlovreq Z xar Ii) uottl'ti ololcorrrd [i.r ot[E].itrol llq

1 2 H TEAMETPIA TAN IPATIIVIIKON ESDQ:EQN

I.2 I H IEQMETPIA TAN TPAMMIKAN EEI'AIEAN

O rp6noE Y|c fllv ,{olov6qon oljxou xou 06uaroq €i.var ro roaci6€r1tro.Sezrvoupe pr brlo rl7iEQ6ollxo x(lnetv€g EErodro€rq. avq{DailoardE 611 0o Fno-QolJoox€ vc xls Luo€r€ Xoaig ro Frioryo rtq ypaFFrzrlE ri]'1e6pag. Ilcp' 6L' autdr lniE,, or H,r 6J,orr. mov c!Lr* tud ruxorpro:

2 x u = 7' - l J = 5

Yndrplow 6uo rp6fior !o 6ou[€ dur6 1() o]JorlFo, xcl svbl(rqEQdlr(lor€ vcrelerdooqre xcr to4 buo.

o npcir,roE rQ6rog €rnx€vrQdv€ro! 1,!QCD (lJ16 rK Fqrovo)Fdvss elrodtoeLg, peirLlc tr6yla rls ypapl!4. A1J16 sivo! ro rlro ouvl0roF6vo }iar yiv€rar lollyooa.H r:Etfloq 2r - , = 1 ropr"otriverar F€ FIa ere€icr oro €fii7iE6o r. y. H elJo€ioatpvd nr6 rn oIBtLc{ c = 1, 1J = | xctL:t = +. y = 0 (zcr06E xar rs (0, -1) &q!(2.3) xcrr 6La i(l e.''b!ci[Eoa o4lEis). H beuregq e!i,ocuo1, rr: + y = 5, fiqQdy€r Frobrrntprl el]otia (:1. 1.1o). H kl,iol fltq ei\c\" duldr = l rar r6F1sI ilv rrQd lFu0€fcr orl ),!ol. To olllsi,o roFl( €ivc! ro p6vo xolv6 x(rl oxq 6Jo ElJoeies rclouvrndg ei!c! n [ovo6[ri ].uon rio! r(|)v 6rio c[rotiroecr.rv. OL ouvrer<rlpiveg tou€i,\,(! 'l- = 2 xcrr l/ = 3 -xc or]lroFo ea 6Qe0Ei tl€ o]JornrcrIl&i (lro),o!qrl,

H b€ur€Qn nQoo6./1!oll 6ev eivqL 16oo oovl0ropivr1. Enr€vro(i)vs.icr orlgonileg rou .Tpapprxo! o!or!$oroE. O! 6io E€IoQror€E €ELoci)o€r,i Eivcr( ofil!nQ(yparlx6qr0 !!o 6rdwopalwil €Eiottto\

^ , , . , l (d r i rn l l

ri +r ,dnjr]] 2l

(d)

lX. l.l H ye.+rnaio ftw yaq+t6r zdr tuv mt]..irv-

.l?1.,1-il=l1l

G)

To rp66Lrya ei\n'" va 6qoipz tov arv6uaop6 tat 6nwopdrav-aq)"1ov m\vapotegn n),Eupti, trou ttaedyEr to trdvuopa oq 6E€,1t ,t]"EuAd. Altrd aa6r6rdorcrtcr 6Lcrvrogcrru ruprordvovtar pr tr rctetd< lQcppa( rou :X. 1.16. Ordf{')oror eivol o! a0r0poi, rj 11c! y no! rioi.),cfi],aordlori! To 6(rvrjotloi(-ofill€(.H oLl r6€a an€Lxoail0r(lL o orir6To oX{[(. 6'rou 2 (poaiq I ortU-1 rooorfo8rcL3 qoaiq o.rl orrtU-2. r€arlsrarrd, rrllT6 naai(y€r ro r€aifllo raaa],It].6yaaFFo.

l r lAlvloorxLi. nuoavrL ro ouoro bLavuoua. | : orn orElu n}'tuoa r-v rEro,i,oru,v

Lr lpc_(. H Err6v( ro)v orlL0)! €r{[6€6c[0)ve[ 6rr I Lrlol €ivcr r = 2. y = 3.

Oa lfioQouo€'v(l Eo6rur€i fi€Qlooor€Qo( XQovoi o'(ur6 To rrcQc6€[1!( (ttrdfipor41d) !c nQol(')Qtioo olo fr = 3. TQ€lq eElo(ioerg €iv(r (lxo[n 6r.Nolov !aB€L€nl0oiv zcr rrQooq)!ieo1r'', riol.u trt./ol. rsat norLd.i.(. @€(r0lorr o(r !:16rz0naadasryBa ro

2 u + r , + t t j = 5

411 6r, = -2 ( I)- 2 u + 7 1 : + 2 t t ) = t )

Mnooo![r ndLr lcr B€L€rliooltr€ tlE 1pof416( ll rlE orll],eE, xa! Esxrvi(F€ cn6 irElpcrpp6-<. Kri0e elio{'rot ;repL"lpdqer 6rcr eain:6o orrE rO€lE 6roor('1osL,(. To np(i)roerin€bo €t.f'(r 2u + r,+ 1/, = 5. rcl|" 61er ole6roo0€i or.o 2X. 1.2. Avr6 n€Odlelr( or l l€ i( l ( ; .0,0) x(rL (0.5.1)) 1{Cr (0.( ] .5) x(! nQoo6roQi!€1ar ar ' cur ir ra rot(rolp€ia rl aro ono!(6li;rore rQia o4r€lcr rou -€g 600r, 6€\, n€prdxowcl oE Urcrc!o€i.c. :ilF€rdloqr€ ev n(Q6bo 6rL ro €ri €bo 2x + u + |l, = l0 €i!q! nc0dl,),Il,o(]ro nQortyoiFrlr,o. T(r cvri{norxo ol[eic ei1,(r (5.0.0) 1{Cr (0. 10,0) x(ll (0.0. l0).o€ bLn),holrl afidoroon (rrro rrlv aQXn -fio! ei!a! xo zevrQlrd orl$Eio L = 0. | = 0.r, = (1. Al.).dqovraq q 6sErd n).Ered [€raroniqowE ro cnirsbo naQZd,].I].c mo.vr-curo rou, T o 6s €'rinrbo 2'l + u + u, = 0 nEpld Llro q! aexri.*

To brnrrao srinsbo €iv(r ro 4u 6r = 2 rar iz€r q(l)yaaqror€i liararopuqo,6rorr ro r,pnopt:i vu:rhpn ororu6rpotr r'[t. O orlrr:],rorri< ror u, oq6civer\r.r pl6€!il€r.rL. rouro 6po)E rra$4riler €nin€6o rolJ rQr8lhor(ror Xri)Qori.

(Eit/ n €qfolljol nra! au = 3 ri oTll\, a1{eai.l rEpinrorort nor u = 0. ortli 0anrpdlpcrrle :rch),r €vcr erirsbo.) To oxripa bsixv€r r1v rop{ ror brurdpor urr6borpt to nprirro. H ioll ti1,(r irft e.r'0ti..r. :rrq rptrg 6rcortiotrg (nurroulr.r 6uoelrooloeLq, oTq n 6nordo€L( anarxouvraL n - 1.

Td.oq, ro rairo €nir€8o ripvEr aurn q! sx0€i.l o'€!a olu€io. To €rfnEao(bf l dx€r oxEbrcoe€i) rnpror i(veL qv tpiq ef iorool 2r+7r,+2a,=9,r l1rrd[v€rrlv r-uo€i.(l oro = 1, u = 1, r.J = 2. To onr€io am6 livrL ro lpr4rpr.ro oroqpcr.

IIdq €,-r€xr€iv€rar a!r'i I €'rova mLs ?r brflorciorLsr Er€i ecr €roul€ nelLo6oeLq rou:repr.e1ou,n ttyvdororE. H npdq efiooorl en opileL rhIL evc"€riir€bo", ron bfl 0o €ivaL 'rlaov 6va otlvlero$Evo b'brdoraro Enir€bo ro! rAF

nrNAKt! KAI Al lA^Ol . t H GArSS

+ L.i! or dno rori)ftr e:ro(;)oe( rir(! 2r +r +u =s xcr 2u+!+ u = 10. rire rd.nireS( 6rv

0d ritrvolrdr acL ber 0c uiaTe iiort.

1 2 H TEAMETPIT\ TAN IP MMIKAN [:DA:EAN

dnim6o 2,+u t r '= 5

.nim6o 4x 6u = -2

I,-'" l= ii1ill,llXli"r"

I

Ir. 1.2 H srxolcr ro! qllLlirv: rF|vop€lo €nine6a.

6Lcrm<rrou Xrirporr. Kqril lidnoLo rQ6no I "b!i(otclo{ ro! ec( €ivo! n - 1".Ao16. :r(p6Lo nou pag qxriroror ocrv o1EQE6, ne6r€r vcr €i!c.r €Es(e€xrxq l,€nx6z(r €fiin€6o p6ocr orov n-bLirotcrro zdeo. Edv 1 rdr<rqtr1 6roorcoq eivuL oXpovo<. rors xo €rrin€bo I = 0 r6[ver rov rerea6rciororo I(i)Oo li(r n(Qcr"l,E!ror rpLbrrimuro rirpo pdou oror onoio lowe ({ zdrix€Qa, ro! x6o[o ofi(.l!ril{n 61(11 t' = 0). Evc (i}J,o rnifi€6o ei\,co ro z = 0, ro oTroio Eival Erior]srt)r6liror(ro. Am6 €i,'(L 1() ouvn0lo[dvo Enirr8o r.9 oeoqoopevo./Lc ltd0e qylrox L Arrzr ra rqr6Lrio,ctc €rfnEbo ea r{pvowurl Aur6 ;ror ixow xorr6.i\,.r' ro ouvl0ro[dvo €lrfrebo z,y yLa t =0 Kor€6rixalrt: or( 6!o 6ruorclor|c,xcrL ro eropno e:rine6o lcrg uqnilu pr pia ru0eiu. Te),oE €\,c ldtoQlo €nin€6ooqr{v€r dvc trovo6rro oq$€io. Eivc! ro oq[eio orr]\, rotrt r{rv 4 €nur6&Dv o{|q 4Srdordofrq ) . l1r ) . i ! r r f l t 4 nQohrl l |vrc rErouofrc.

o(1 dlar prrelir6tE ('a dvro ro 7ioQd6€lypc (1fi6 q (Jz€xr}i6qxa rQa6rlE€rnecniQ.r.. To oE|oortueill)ro €ivor 6rr n./QolFrxrt d).y€6pa $ropEi va boul.€U€rtL€ oriorobtrorE rLtooq sqrodoE(,v. H rpdq €lioooq opileL evc n 1-6rdmcro€rincbo orrE n 6LaordocL(. H bcUr€pr €Etocoot opiler tvc hL),o tlri.ne8o rar crrciritrvovrcL Gl,nilouUs) o' 6vc pnaq6tepo orvolo "brdoroor5 n 2'. Yno0€rolrcEorL 6Lc :rdve xr dr, rcioe v6o eri.nedo (rirBr vdc EEioorot) [€rd)\r€r rI 6rdotcottxarir dvc. :ro rdloq, 6rcv xcL rc n tlrire8c 61ouv ?.1q0ei ur' 6qLv, q ropi6X€r rI SLirorrrol FI6€r'. EivaL rvc orpeio, nou:rrpLdlercL os 6lrr icr €rin€6cr.xrrl or ou'rflcypdv:g ror urcvo:rorotv xar-rLg rr rlrotirorLg. Eivor q ).1lotl! Hrr"r6vc crrnl rivor 6LnLo0Trxf Il€op€rQic €( XQ€LoooEi. rn\r uroor{arEl rIEili,y€6QoE ouo!(rfir,io opcx €ir'clr o(Liolyl.

Arsvrrolcrc - :rtl,EE

E'ftoradqouF€ orLE ort],€E. Aud q qopd I bLuvuopcrrx{ sEioool (I i.bra

$E rrv (1)) €ivcr

To rcecnirv(D eL,rcn qtdtcimrna \rctiarnra-oriLeE. To 6ldvoo[( 6 oryl 6eErd:r) . fuc( rxFr f lq ouvrmuofs 5.-2.q. ad uura( or nwrnr ' ;or: F.rc r : f l rainouv'va ox€bd(ooutr€ ro b!civ!o[rc. To ,rdruopq b tautilxtar. pE 10 olpeio nuoxoio! ot ouw*aypl*g Efuar 5, -2,9. Kdoe orpeio rolj rorbldordtou xdpol]o\4!sro[X!:i o 6!c 6ldlrioB(l z<rL crvri.otporpc. A! i rtcv I [66d ro! Descafies.nori frlrirQ€liJ€ rrl ./€(l)l.r€roia o€ d).y€6Qo €Ql,(lqot€\rog p€ xlE ouvreru./pf.v€E rouorpeior. Mnopoupe vc( yQduo]Jtrs ro 6!ch,!o[o o€ $Lc or4l.l 4 F ioQoi$E vdnorcxoQisolr[s i[q owr€rc riv€S xou o€ [Icr liotcr oclv 11"'6 = (5,-2,9) rlFnooolrlre !o 10 n(rQco1nooupe pe irtr 6ilo5 ro! E€rrvd on6 q! coz,li.* >€6Lo ro 6r6trio 1Qrlo4roTiolo]JpE noQ€v06or(q xq! ).6$[cxc. orc! or owlotcr"|,B6v€gx0raxDQo!\'r(r oQr!6v1[o. ricr! r€1Qclollxas (ryxu]'ss (l{,)aiE x6FBatd) 61(,! 6vcl6ldr1,uo[o-ofllll r]Jn6vfi car xco€T(r,

Ilpct4rcruti. aur6 rou dl6r onFooio tLtat q np6o0eaq ,tawapdtav xut o,tdJarIco.aop6E pe oQ.e/d. :ro :X. 1.3cr 6L6fier€ rylv nQoo0€oq vc €{1€tr€i,ro!owlorrDoq rpoq ouvrorri)oqi

:ro oppn 6t:!rri r:rdqt:r dvcs no),Ia'r).ootcrotr6q F€ 2 (ror av tra\' 2, ro6!d1,ud$o 0o 6r[Qcne !l1 6EixvEr oq! alrio€q kar€uouvol):

NINAKE' KAI AIIA O]OH CAISS

* MEataoi ollyaqEi< neonllotv rc iivc or ro diLo! €i!( nadytofl to 6tritota. No[ilo

6t bEr ude/fl 6rc{op.i. Nlnoarir! yc 6ruiiEft ro 6iios. r. o+r.io ri rol': 1o.r< datoFon<. (oL(

lEarvora G6 rrr i6,a (a/]:r (0.0.0)). :rc iEt 6,dorion< tiw, ioo< Elaol6cpo rc b,dlilst ro!<

.til.til.-,lilt;l

lil.l;l.lil[;]

'iil [i] ,|ll l_il

(2)

I,] H I'EANfETPI,\ TON 'PAII'IIKON E:I:Q:EON

!x. r.J H rzdv( tuv onli,i)!: o yeulllrzG orvbradpd< rov oqldJv roolh(L fF l).

)to oXqpct bulLd ncpr6ltta! sniolq ttLa ar6 rLE xevrpLarg L66eE q<.1qcrpprxr1Ehlyr6paq. Xalo+roroLouvrclL or biiro 6(orxiE nQi(E€K: bLuvuopurrr .rro,trla;r,1a-utdio1,rut lr tet9lloi,a xar xar6ntr nQootietltat. To (urord),€o$r ),ilErurlAqrt[zo; oulbuaoFoq. rcr o' curtiv rlv n€Qinrcool o yQ(4rllltoE owbuuo]r6s

Av(lropilere tq olpcroicr cruron ro]J eLbrrou owbu(loBo l Au\rEr q\, EEio0)oI(:). H €qiociol !ryoror touE ouvt€I€oidG rr.f.u nou Ocl ncp{lqcv rqv 6rlLcix).€uph 6. Auroi or (rar0troi €irc(r u = 1. r,= l, ,.= 2 xar bivouv ro\' ('{')('16

lacF[Lzo owbucotr(i r{,)v Srrwuotrdr0r!-mnld!. zae6! eniorK zar ro olp€io( l. 1.2) oqv €u(ov( r(')r ypaFFdiv (ono! rapvom(rt rs rQio Enin€6a).

IloLl,afiLootcrop6E xar rp6o0to1 erreloUvrcL XcrEm(i 1,|( rdo€ ouvLordoc.ft o!ro. ot yQq4rutoi olr'brlaoloi €LvcI enLrpenroi p6vov 6rcr rn 6r.a',uogcrtrixorl ro i6ro nLrl0og ouvLorroociv. IrpeLrirore 6rL 61,a rtr 6r(lrjopcxa mo o1npollt(v rQr6rdorat(. nrre6l,o :rou grpr.xd6 crr6 rq otvlordocE rouE {rtrv pq66v.

Mry EEIrdt€ rov or6/p- nou rivcr va 6oupr rdect or.d rE 6rio I rQrr5bL(lorao€K. mL! r br(.ordo€!<. Me n elrodroerg u ;rpog n cprimorg, 0aunrtpXav n €rin€bq o1nv €rx6vc yr.r lQuBtr6!. :qr erx6vcr 1tn or{,eE u:rdrprlow

lll.til.,lill;l

NINAKE: KAI ANAAOIOH 6AI]SS

r 6rar,1jo$ora orv dvo brcivuoua b orlv 6EErd rl,€rsd. Or €Ao(i)o€Iq litrorv 6vdvy0aFBrz6 ouvbuao$6 r(lJv n 6rovtlo[i{TtDv riou rooljrc! tl€ 10 b. :ro o!./}.r€xpL[6voraph6€L.4ro 6plxcpr 6vcv rtroro owiucop6 (6cv urriplolv dIIo| aIId ylarci;roreg efrooroeiq aut6 0c Eivar o6tr"'arov. Ioac66q(oq, o tp6noq 1Lc lcxcrovorlooufe qv xolrl rrpi:rttLro4 eivol va F€).€r4oortr€ rlv rt1xl. ff alr6rL'r i toupr r t 1u-,Ur rUitr . rot" , ( t<C.6u< oro! o 'rr ' t xaraAp, ' r . or lv )rrAinr,uolnou orotrttilolpe rdrri,uopg4.

no(i)r( o! vorlriqoriFs:

Eu6va tav yqayp(bv:'fopri n entn66rorEu6ra uw ot4litr". H 6rlLd rLerpit b eivc{ xdriolog yQctltll1.ioq o1J"'6rioo[6irov 6LcwoFdrorv-orll.drvAio4 tav elnrhoear. To oql€io io[ri( i(l)v €fi(rr66cDv = or o!we].€or66 orovowbuco$o rorv orl),ci)v

H 16uiFoQ{tI ftaint(rdrl

Ynoedor€ 611 60ro{6Bco1€ fi..}i or[E rQ€lE orsoroo€Ls zqL 6rr rc rgiosrir€6o. oryl! e$i6vs (rv yQCl46!, 6Er rilrortet. Tr tmoQei va fid€l orQq6s:Mrc 6w(lr6qrc. rou i6I onleld)o(rp€. €i\rcrl rc efiir€6o vu rlvtA ncrpd].1.1].tr.MroQei 6!0 cr' 1r,( sElo(i)o€G n.I. 2?r + fr + 11, = 5 xar 4& + 2r, + 2u = ll, vcr €ivcrl(loup6i6aor€E -on6c dEv ura(QZEr ),uon (To:1. 1.4(1 6iver plc dro!'I). :roblEldorcxo fiQo6),lpc, u€ €oOei€s q!d. pc! €rtre6tr. a! i €tlal rl p6vl n€QirirootlzqrdQQ€rlolE. To n0o6L:lt(r eivar rbr6pogpo 6rcv ot E!9€i.€E Ei!c[ fisQd].I,n],€<rar Fl !6r6poQ(po 61(v orr1,c1,rdrvr.rl. TQia 0[(j)6 efiitredcr orrg rQ€6 broor{io€r:[noQol .,'( ilow rrQ66),IFo X0rQiq va Eivo! ]rdQd].111,q.

H vaa buoxol. ic : rcpLorhvetcr oro:X. 1.16. Kt l r rc rpic enir t6a ei !(r lzdo€r.l ornv odiito. ,4.]16 qv zdrollJl o ltrcliEow dlc rpiy(l)lo. Kdo€ leryosE rrnibc,rv rfu!€icr a),Ih 6ev urtiplcr onr0io xo!','6 'toL o1o tQi(i.*

{r) {6)

Ir. 1.4 Ibrdpoaq.q n.arnlrlo.t!: adF[i.1].nor i afteic ].nosov

'r To rAirl, .nii.6o 6.! .i!ar nnaililtlo or( .itLa enirEb( d).il eilcr ncpiri.irlio orry €loric

H TEANfETI'IA TON I PAN,IMIKON E:I>O:EAN

A!r4 €ivo( ou\rl06orcel r€Ainrtrrol €r€ivlE irDv rapaL],{I(Dv €?rL7r66()v xcuovrroroLt€i o' sru r6r6poprp orior4rcr 6rcDg ro

1 t + r + u = 2

2 u + 3 ) = s ( 3 )

3 1 1 + u + 4 r l r = 6

OL 5ro :rpdrreg trQr.otugtg jrLFUp6g oOporlopevrg bil'ouv rtv rpiq. Aur6 681,roxu€r yra rI 6€Erd rl€uad. E!i.o!;o1 1 ouv rEio()ot 2 F€iov eliocool 3 obryeiorov ;rnpcrlo^/.qr6 0 = l. AQC. ofiorg 0c( 6larirorore€t oroqpn-ftad xrn pc qvcral.orrtrt Gauss. or rEro<iotti tivcLL aotp6i6am4.

Yndpl€r rar 6v(l dlLo rordpoagro o1)orllro, or,yt€v6( nQoq ro npolyoilpevod.},n. $ cnLei lli(lrnr, (vri r{(4rtriq }.uortg. Eirv to 6 rr1; rr}.eurcri.cr-c eli.oarorlglivet 7. t6te oL rpeLg rlLotioeq ouv6uiqovrul x(r|, 6ivow 0 = 0, ]ro! qniv€raLo(')o1o. b!61! I rAiq €Eior)ol rtvar ro d0QoloF( rt')v b]Jo nQotu,'. 2- crurlvr1p ruQinrclon 1(l tQic( sntn€lo dlouv roLrrl trra ol6riLt0rl !:!0r:ie (:2. 1.41).H ouvdn€ra rIE ol.Lcyrl< rc,)v 6eErd! nLs!Q6\r eivor peror6:tLo1 tow cnL:r66r,rvn(QdlJ.ll.c otov ecrr6 rorE xar./rc rrt b€Erd n),r:uad, = (2.5.7) I Erx6\ro aLld!€!:og',,'!,rd. To xirr(r) €nifi€bo per(x.rwei.rcu" rrQos iq fiirvo. o!!c!rd rc( dLl.cr xo!ozlfLcrtiitlcrr $Icl etlo€ia (1JI6 l,!oE!q. To rQ66),rlu( €ivo! noLL lbro[oQclo. (ll.].crr<ip<L nae,, un6 tapa toA6q ),iae6 u,ni" an6 xo't,i

't,i1eE.

YndOZet qtloLx(i I arpotq r€Qirrool xQlci)v naQa),LrlLc,)v En|n€b(,)1,. fro rrc'rrQLooorEQ€< belLdg nLeupi< 6rv unhpler. Luoq (:X. i.4d). fLa Elbrxdg b€t!€Eiri€lpiE (o'i@E q z| = (0.{).0)l) lrn(iQle! dra ol.6x},r1po uri.:re6o }.uoec'rv -br.orL rcrrpic tni.rc6c rcr,"rilovrcr o' 6vc.

Ti. oup6(i.1,eL pt trlv t:rxr5rc rov orlldrv 6rcv ro olorrpa Ei\rcL r6!6poQ,qro:Arirrl rrpEnEL !c loldor. io 06[c 6p{DE €t\,dL ntb<. Ynhplorr nri},r rqeLE orq},Egm1v rermeqi d.erpri rnrv EEroriraeo^, xar rpoo:rc0oupr va rq otvblcioor4eriort vu :rcrpd.1o4r ro b:

.tij.lil..iil., (4)

r'(1 ro i, = (2,5,7) aur6 {r(l! 6x\,.1ro1'. l-!(l xo lJ = (2,5.6) 6ev !rcv. H crricf_ivor otl. rn,tdE ot t-ouE mfiAe6 ntptyorr(u o' tvt efiifie,o. Apa raL 6).oL ol./Q.lF$Ixoi auv6uco1roi rotrq reprlovrar t:rto$ o' .ruro ro f]rinr:6o (rou r€Avilffi'rlv.eln). Edv ro 6rciv|opc b 6ev :rep|d1ercu" o'aur6 ro €ni.nE6o 16r€ 6€vurdQlrr l.uort.

Arro ox[6ai\'€r. rL< n€aloo6r€A€E qoaiE: €v y€ver" 6vc r.6L6poqqo oiorlpcr 6ev6X€L ],ljol. YrdQI€! 6u0rq ncr evc ev6et6$evo to 6 vc:reqr6lcrcr oro mfnebo trDvoql6! zaL o' .1utv tr1l nepi:rrcuol udplow :rcia( no).I6q ),!o€L!. : o!ui1, r:rvrepi.:rrcoorl or. rptL; orlie< proqorlv vc ouvbrca9oi,t rc rinetpotq eittorq 6o\tva raair"/orN ro 6- Auro 5i\rrr nt! €!x6vo-r(|)! orltuirv 1.56 :ror crvrroroL1si m4,€rrov(l r(|Jv ypatB6v 1.4y.

l u NI \A IL ! KA I ANAAOI4 IH ( JALSS

rr. r.5 ltrcIoqg)t:E n€arnrdorrqr ro /J fxrdq ri F!ro< tolj etr|Ii6o! rov orr).drv

O).c (u1d eivqr qLl0ri( r.oxuaro[oi ,ro!' 6€\, €Xouv 6lroE 6rx(rloLoyloei.. n(;<11rDaiionFr'6r! o! rO€rE orrtl€E nEpLdXovrcr o'iv(l srin€bol Mi(1 cr.ivrnol €ivolvo 6Qo1,l€ avov ouvbuooF6 r(']v oq).(iJv nou 6iver depoLqr( g1bdv. Merd <r,[6zrirorov uroLolLop6 crnog eiv(L u =3. r = -1. u = -2. TqerS rpopig q onil,q Irooitu 1t rqv ortii,q 2 onv 6no gopdq rlv mil.I 3. H otrih I elvrrL oto eaineborarv bto tiLhorv zcrt p6vov ot btio onil,q tivut (rveldBrlreg. 'Orcv e\,c brhvvqrcr{trrrE ro , = (2.5.7) n€Qrd1€rar sniofi o cr-'r6 ro sriaebo -ro bLrivropn nwocivcL rl orri),n I ow I orni.I 3- ror€ xo o!oq[s proQEi vo Luo€i. Ee iooubuvatoi 6ptu; €iv(r r(cr roll,oi riIIoL oUvbr(ogoi. To i6ro brirvuolo , €ivcrtniolK 4(oIrttrq l)-(ottlLt 2)-(oq),t 3). rou nao'innr€r r0oo0drovrcq ort!nqrixl Lrioq ro\r ouv6uc1oB6 (3, -1, -2) nou €66€ Fn66v. EreLbrl gnopoqre vanaoo06oowr onorobinor€ rrolJ,(tn)do(o xor (3, -1, -2). uridQIE! I(!( o)'61a1.rcI€uo€i(l liro€rov 6nco; yvoqi.loqre crd rn! elxola ro\, yQo[t16v.

4r.16 ril(rr ufioloyrorlxcr aerotrx6. 6uv eivu 6pog o rprryFcrLx66 Loyoq ynror onofo nrA+tdvouF€ or oril,€( !c Jrteraxolr(r o 6v( €ntr€bo. H d.io€rceircrr 6rt Tvopi(cpe rto< or odl.rE 0rr ouv6udqovrttv ciore vc bci.roouv ro pr16iv.bt6rL r(L oL yO4rF€< dz(r'\,c! to ibto. Aurrl eiva ptn pn04ycrrz{ clti0€L(. oXrlt'ro),oyrorL?ri zcL :rcqcpdveL ul,10r1E r(|L mLg n 6Lofido€lE. Etiy t.r n Efiinx .t6er iTotv xou6 oqyxio, t6tc ,tdt ot diX.ES t cQt$oy.at oro i6to ttratttDo.Eitr xutapp€rr q erdvcr tov lpcpprirv. ro i6Lo 0o ooF6civ€L rc& F€ rrtv l.!x6!cicov oq)dJ1,. Avr6 €ilat €va 0epelL66eE olgndpcopc q; fgr4rpr,xrl< AL1e6p<rg.xe[ q)(vtQ{i\rfl xl 6rcqoQd ro! K€eal.aior I cn6 ro KcrptiLaro 2. Ar[6 roxEqdl,cLo p€I€rir ro rrlo ev6lcqaQov nQ66l.qBa q pq 6rrjpoqEq rie()lrr(l)ol-6rrou lndQX€[ [ia Luol rou nQ6n€r v.r 6Q€0€i. To en6pevo x€qd],c!o [€],eri! rIlr'!!xll r[€Qinloorl onori pnoQei vcr ufidQxow no],].eq rl zcr zqrpic ],iol. Kar orrEbuo n€Qlrwdo€E 6€! [roQoul€ vc ouv€Ii.oorF€ T,topig xcrtri].],qLo oqr6o].top6(xu\ixtot) y.cL Trrpiq )i(r.iD,tlo (1tr6e$Vo (analoLqi). Merd rq ao'rio€lq,E€xrvdqr€ tr€ rov (llyopr6Fo.

12 H '€OMEI?IA ToN fPAMMIXqN EIEO'EON

A>XIDEE

lrJ >tcordoft qv cr 6va rov ruq$6v ylo rK 6d,o P+.w6ocB c +! = a, b -2! = 4(6io I€p.|d$e€E suoei?s) ra! tlp srx6vo ro)v sql6v (ow6uootG ior 6'no oqld/vrootrd Fe io 6dwaFa-odll,rl (4 4) orqv 6sE d 'rtsued).

La2 Aiore ro rQi1ovr,x6 gllo'rlrd

u + u + u = 6 1

a + u = h

A.tEt. dr! q liloq oaE 6i1€r Cvfi, o!\,6!do!d rov oql.tlr nou worlto pe tq otrjtrrloirl SEErd nt€led.

LzJ n€erled{tr€ r tv roFl i iov iQ[6v r- ' ru &Dvt +u+u+z-6xar,u+u+z=4v' . \{, + u = 2 (610 otov r.iQa6ldorato ,(6Qo). Etldr pld ?uosiq, 6va onr€io rl io xrv6o$volo; no|,a €tvor I iqd edv oqfi€ed,qeoel to t&aolo €rii.ris6o r = -1i

lr.il :t€6.dor€ 14 €1,orlEx+2u =2

' 2 - ! = 2

U ' r

I\,hoeoi'v vo llooi'v nvr4(eovd or rQ€E €EIo6osE: Tt ow6olv€r sro q0fua &av6I€s or 6€hrs nl€!a4 etllor Fq664 Ynd€(€r Fq pn6evlxri srlt ofri r()v 6€Er6v n).!'JedrvnoD srnednsr.oq re€l5 slloct€E va dFvovrqr qro 1610 otreto ror otE tqeq €ho6oe6la s'IolJv Fra ̂ !oI;

1,2.s Be€lft 6io oru€to oinv €ljo€td rotriE rov 1016r sr!fid6ov t = 0 xc! z = 0 xdlr + V + z + t = 1 orov rered6dororo X60o,

'.2'j Oflv ao b = Q,517) 6q€tt€ pld }\rjoq (u, !, rr) rorr oloritrdros (4) olaqoesrflrl ari6rrs lus€r{ (1,0,1) 1{or, (4, -1, -1) riou pq|rorslrovtdr, oro r.€L|r€vo.

r2,7 A6oft 6rio axoprl 6ptrdE 'il,s'Jc€a, s'{r4 qE b = (2,s,7), y',a i6 olot€g ro o',oqtra(4) Un€ial" A6ote 6!0 arigt Oe6r& nleuedr, €ttd< qs [ = @,5,6), yw ira orols{

tll EEryIcE rmli ro otoqrau + o + u = 2

tt+2u +3u =l

u + 2 u = O

€il,dr tbdpoaqo, 6Aaorcvror 6r,av ow6rJool6 ro)v redr 4nix'E@ ln 6iv€r. d00or-old 0 = 1. M€ rola qri r€€:r€',lu avnxorosroo€t rc pr|6fv trF &!d5 rrstryiE,'n@e or eEwdcK va sxol,v Xig8E -'io', 'roto silar Na an' n4 hio i

ITINAI(E XA! AJIAAOIOIT GAUS,S

L2.9 H €r)i6\,o r{l)! ort),6v qs reoryotpsvlE dorrlo4s ei!a!

A€ttse 6t! or tQ€a sriLss ora seloteed '*aLsrowo', olo i6rc €niff6o, e4edgovrosrlv iQirl onllt os o'6!aqr6 iov 6to dLl{Dv. no€s €tadt or hrocrS (r, o, u) e&v to6 €[ro! ro pt6svu{6 &dvlloto (0,0,0);

1.a10 Y'r6 not€s oworlxes .rw w vt,92,h r.Qttxdwo.r to orftio (0, fi ), (l, !r), (2, g, o!Ft(t etoEta;

r2.rl OL €Ewdo€€au +2U =0

2 o + a U = 0

Clow ol.yorgc qv ),r,1oq c = !, = 0. frd nolq rIICg ro! a lJrdaxsr oLddlqQl ttioeta

12t2 z4?fitAotE ao uri,rztE E +! + z = 1 { ro #eoE rou Erur66ou fion daQdtelot otoeet11.6 dl6o{r6eF r > 0, y > 0,, > 0. Kdvrs ro t6to .|ra rc r + y + z = 2 oro [6@oxtFd. noro 6dvuqra €tvur xdoero o' arrd ra c rc6a:

Lzl:| E€xrv(irrros on6 qv s!8€to c +4y - 7,6oltre qv e€to(Dott rrE roedl"ldlqg aJortqsdn6 ro t = 0, c =0. Be€tc llp 4iooo4 Fldg dl-}.rE €uostos nou smuwd qr re{tt.la1o a =3,9 =1.

"[i].,[i].,lil=,

I . ] I IAPAAL I I MA A I IAAOIOH! CAT ]5S

L3 T IIAPAAE|ri'A AIIAAOIOHE GAUSS

O reono< ncrla\,6lof curod rou oiB(1roq ri,var xo n(ad6rr1Fcr. ApXiEolFsorq rp€|g bLqo1i(o€t< tr€ ro ororltrc

To n066),ql( €tvcL vc 6A,orF€ rL< dlv(,)orE< rttrr-E rtrrv u.r xttr rt.. zsr Otr€f(lAFooo4rr rurd.oqll Gauss. (O Gruss ctvs./vll)Qiq8rcr o(lv o UdlLoroE dLovr,.,v turHr rrrr').'uv. oiyouau dtru: oXr lL uur4v rlv iu:rvruorl nou rou zoorroFio(or bar(l ),rnrd. Ev ror.rorg, xurd rrprcwrxf riXI. crr' 6L€E rLE !64€E nou qeQorv1o ololir iou, ullr1i rliv(lL n oux!6r€$r 1Qlorponotou*svl.) H Fd0obo( E€nrvi(nquLubttuq tuM.a :iaru r\q nQtbr4< tEiaoo4< atu) tq nn6lorrE<, irot (boltut axcAtt(tti to u uttj uq [tio rc).tnttit(.{rodoslS. AlJro cn(lr€i:

(() vtr cq.crrpiooltr€ 2 qioQ6E rlv na(;)q rEior)o4 c:r6 q brrirrpq.(6) !( (lq(lLAiooqre -l r1,op6q* l4v nAd)rrt €,lio(|)ol (rn6 rlv rQirrt.

To ftrort),roF(r rivaL 6vo Loobuvqo oUoqpc tlLodroccDv

2 u + r ' + l l r : 5

4 u 6 f = 2

2 u + ' l r + 2 u ' = 9 .

2r l+ r ,+ ru = 5-8 r 2 r , = 12

8 r , + 3 r , = 14 .

(1.)

(2')

O ouvttl,rorri< 2:ror:ro).1o ,(oi(!r rov:tprirro riprooro u orqv:rptirq eliooorlriv(r y!(')oro< 0r< rQdro5 o641riE frc v&;rpoobLoeio€r I cdrcl,oq)li rotK o(Dorol<ro).trstr),(or(lor€(. bLCLp€i ,.[(i}"rore !r rov o6476 ror5 cpt0poriE :rou 6ei('xovr(r

:1o FnoF0ao 6rtlc anol.ot(prlq (lloorlpe qv rpritrl €Eio1l)ort. OL bro i.1.L€qaqLodo€[E neprixorr Fovov rLg Ber(6].rF(q r, aar llJ xar lr]roQou[€ vo rq](QFodoljF€o culg qr ibtu brabLxaoi.a &'rol,o!qrtE. O o6ryyd.. o' uur6 tu 6ipu Eilu -8x(l| 0 ag,urQ(oor8e 6r'cr nol,l.arLriuo r$ bnlrEQnE clJ1rlg eEioorolq cr6 rL<uno).oLne< €Ero(losK (o aurj r1v n€eir()orl cnopdv€! [i(r fft! F6vov, I rQirl)dor€ vc on(irirl,oritr€ ro r,. IlAooodroxt€ Lomov q 5r[repr1 elio0roq orlv rpirrlri t€ (iL),c )oyL<r

( /r dq(rQoutrr I qoAi< nt b.L\sQl €Erouo4 u:r<i r4r rpiTr1.

* : r.E.: Arti(lrl!( naoooroortr n|v nLr;rrt rlidtrnn] mrll tAir'l

2+ r ,+ r ' = 58 1 , 2 1 1 ' = 1 2

t = L

FI r(xrrr{ri €ni},uoqg cruori rou ouoriF(ro< €iv(L :rqorpcr\,lig. H-re)'eut(i(r elioaxnl6i6er u = 2. AvtrrooLcrx6l't(lE orl beln€Qtl 6pioxoqrt r' = 1. T€]'o!. I radxleEioool btber u = 1 . H ct:tLri (rurri 6lcl6rtooi & l'a'ttrrur ardfieoq.l (i avntQolt.xi)

Eriovolo$6dvo]Jpe: H 6La6oltxrl (?rolo[qt 'IsQnYCrYs ro!( oblYous 2, 13, ].Aqr(lio€rrr: roLLcn).dor( xile€ ya(r[FriE (r'16 rL6 lptrppig (lJro xdt(,) rls. 'Eqr(ro€

oro "TQLIU)\,u4o ' ouorilFtl (3). Ko16rLv, ro oiorrl[(l (.u16 i'rioqre pe crwiorpoqlnoQ€i(. (no xd.r(o neoE ro /rd1'o. ctrrtxagtotd!'to< ff10€ trQ6oq,arc uto]oI'oe€iocr rqrri ortv €Eio(')on (n6 rov(o ryl<.

Euteir|,o11, 'E\al xql.6E rQ6fios xcrr(ryQ(rqls r{')! tiq1d10)v ura)'o|(prlC €ivol !coul:rtpJ<i6orpe flt b€qLir nL€uQil oov Frc €ni r)'iov m{l'l. A€v €ivaL avdYxlvq ovrLyeaqolp€ ro ouFdokl ri x(r| r'x(! {r'ti(rL = o€ kdreE 6llttcl. 'EroL }o!n6!

trlivo|trr !r' €\,c( curoyuFvr)F;vo €l,irxl(Tto:

Iro rilo< uaoF6ver dva rQ!'fo)\,td orlorru(, droqto yto (\dbQoFtl ((vr!].arclorcloYl.

MnoQ€irr vo nQorrtrioEt€ curi q pd0o6o rou €Eaoq,d'iEsr oft or nQdE€ri orlvcQlol€Qrl ,r)!r'uQd nov €ftotloeov yivoltcrL uri<rqq xor oq 6eErd 6!6t! o' a!"r{vrnporrorrilovro xar or 6do nlnQiq ! .t1.

: (!cr !€./al,o ;rp66L4r(. r1 gtltthntp4 :tpooad€etcl ctrarr€i I6K6oxtx{r.ncl,oq ti. Aurtl zngopilstnL cr:r6 rq optotepe< tr).€upd< (o! €Elodo€ov €v6 rlcv.i6eo'rl (vtlxotdor(or1 elnptirtar enio6 cro rLE belLeE ll'€rrQdE :xo tradJro6{ga lpqorporoLoUpc roi.},cr:r}.dorc qg :rpcilrlq eEiocootlg Y|d va FYlbeviooupt6Lorq tolE owrcLeord( r({r(') cn rov np6ro o6416. Kcro:rrv xcrortoflowerq 6rr'rrat orri).t xirr(r) Gro xoa berirtpo o64yo. Ta)'oE, I r€l€r'rdfo €Ei'J{DolmpLiXt:L p6vov rov rri.Eurclio .iy1,{Doro r',). noltrofilcorsouavo I€ tov t€l€uraioo6ry6. H crvir6aoFt bloblzcoi.r rEl.s(i)l€! 6r(v xo oiorlFa €ivat tQl.fovtt 6. H(!d6Qo$I trvrrrcrriotcoq :rcrpd1,er. rr1 Luoq oe nvriorQoq,I brdrcEq -crplitowaq

cifio rov r€L€ljIaio ityv(1)oro, ).rivo1'xal,< 1c16fit! uX jrQo( rov rQot€).€ur.1i,o lialr€L€rdvovrcE xdrorE tt€ rov 'Ip6ro.

E! oQtoFoi. ot o,qtoi 6tr Ei|Itpit Etnt vtr p46*iEottu. @€}'oupe vabrcra'orlr€ [' .rmou!.

IIINAKE: KAI ANAAOIOH GAUSS

H 6rd6r)iqoia anal.olqig €ivor, xd)aq n)'{QIE, lo]Jldtto.rov ]{ord qv "fiQoE rarprp6<" xcrruouvo1, x(rt, o6T/€i oro cnLonor.ttrrivo orrorlpc

(3)

I z r r 5 l 12 r r s l [2 I I s l4 r r o 2 l -10 -8 2 - l2 l r r 8 2 t z l

| : 1 2 , l Lo x .1 l4 l l 0 0 I 2 l

H onolulio ilg qral,o.qig

Od.outr€ vc :iirvox$€ 6uo rQl:'rrior6. Mrop€i vo q(vouv rirroq nQ60)Q€!-(r1,ot! p6ir; rou eiSap€ rov .rl16ar0Fo vu boul€u€L 6t'o\ I ondrvrryi rolE 0oq!)lirttr n€erooor€eo rI trigo6o. H np{irrrl tptinlo4 rivcr: Yn6 noizg otv0fixeEE|Wt'.dt vd aaoriya 4 6&6txd.oi.t, H €Q(;tlpl 6€v eivcrt yer4etprr{ rd,}rir()../eilaLzri.

H ar.iwtot eivur: Edr o d'.y6Qrelo: ncQdyEr n o6tyoiq. roi€ undrQl€! tklz(lt Fovcl6rLrl luol rou orlorllllsrq. ro ollorlF(t eiv.lr Fl !6l6uoQ9o ac(t ),u\,€I(|tr{' b|(1boxrxi (lrcloq,rl xcr! clilbQoll svrlr.crdor(ol. Edrv oploi oq e6ol €v6<<t6f|'!oi Eytpauogti wa lr4r€rlrri. iorE I cn(rtro(fill 7rQ€n€! v(r orc4{ctrrlo€r -ri.T€,,1\roT|zu frr f : r t 'oar"qrv(. To otor4prt prol 'n i vrr ! iv ' { r l rul v( t r4v r iv, !

Edv f tor! i i r r , f l o rplrru: ob11oc. ortv ruv(, i , rC.f l rLpft lurvru. rorr 1,r:rci,or{'i ror u ulro tq ril},rE tlrorirot4 0( utlcrt crbivcrq. To i6Lo ul.l0rirrrz(r 1r( x(i0r srbldF€oo 6nlrc. :I!r((;)or€ ox( 6r(( slbri{rEooq o6lly6g luroQri!( trrlb€vLoo€i. x(i.llll br(b[].ctoi(l.rrc),o(frlq, (xopr1 xar orcv o owre].em{g o',(r'r'l rtv oiot. oto (pxLx6 ouorrlpcr. rivut pt Ft6$tx6q. MLI{i)vrqS 're6xse(:hrt, yvropiloupt av 0a epqavrc0ti xdnoro p46tvrxlt ttprv va rpoonaeioouyzrrrqri p6vov crpou ezre),6ooupr 11 brubLxaoic cna),otr1,r6.

> (ezrr6E n€QrrTrdlo€r< (turo ro nao6),nr(. Fnoo€i vcr ((vr+€r(l)nrooEi 'r.rL

rl rr;TaLorqi v( ouwxroO€i. 'Evcr rtroro olotrpc ouvuroioli(trrL or( pI Lbr6-rk)trl( r-iv(t p6voa o aiy6prQto< rou Xptrtir(rr(! €nrorcrin.: citrL€q nreurrio€rGrl (( orrixi( eav(l! clcrn6qErxrll. ArJl(i r( (0r'Qdn€!I( ouolrlF(rla ri\,(r r rj.rt,pqc. eirt 6€v rxouv ),i,o€q rire 61orw nripc:roL)dc xcrt bev p:ropolgr vcolxrtyr: trct :tltlpeE otiro),o o6rytbv.

llH IAIOMOPOO IIAPAAEIIMA (OrQCncu€r(tr €\,((Ll.riooo1,ro< rlr uEro(iJoElq: z(r 3)

],3 IIAPAAEIT!1l\ AIIAAOIOH> OAIJSS

2r + 2 r '+ 5 r , =. l u+6 r+ i l r ' =

'li) o\rorrllc €i\,.lL n),aov tptltorz6 zut Lwercrr lt€ o1,(i6QopI (lnclorqri.

IAIOMOPOO IIAPA EIIMA ((OrQdn€uro)

L5

. 3 l ' = - 2r +zlu, =

J , ' J r ' = _ l l r '

2r +2 r +5u , =,ll, +,11, + sttj = 4\t, =

-li;rQc be.,' lln(iQI€L €\rdL),(yi ypallrdrl, no! cfio(periy€r ro tlb€vrx6 oq 6airrQ{

ollyrzii 0dol. Or f6r8< or €Erodro€ls BrroQ€i .'rc ),fvo\,rar rl vtr g1v luvovtcr.

nN KE: KAI AnAr\OloH CA!ss

Edv or flto rrLeuoieg rELoootq €ivclL 3rr = 6 t(rL 1u,=1. ror€ b€v trrvolrct.Eirr oq6ei or 6!() ul)Id< €Eto(l)o€rq rrr €ival ouf6r6aoriq onorg or 3{r = 6zor 4il, = I 161F o (rE{! qv L6r.6poprp1 :repin-roro1 67oqrr tireqsg }.rioeLg.fva;piloqre Urr otr u, = 2. a).Id I nadry €Eiotuol 6rv gropci v(l zll0oaiorL $ovlTlq x(( ro ll i{(t|, ro ?,.

:qv noahyaaqo 1.5 ecl oulitrrtoorut' rI< ri'v(liL(ry6! y0ctrtrcbv. orav rootrmllcr eircL 91 Lbr6goprlo. Totr or Evctrl.(r1dq ncrpdlouv €vu ;r),Iqeg onro),oobrydv. To Kerldtrcuo 2 rElr(iqu rrtv r6roFo0q n€pir.rtDol z raor,coairxomoaivowag oq\r LrnlrLor,qt. To 3u lnoQ€i crz6p1 vcr cr:rn)"tir1tr ro 4u,zttt9( ovoliroortr€ ro 3 beurpo o6ryo. (A€v rlnirel€L rei.roq obityo-c.) n0og 1()rcp6v u€o)porp€ otr ruL ot r o6q1oi eilcL pr1 pq6eueoi. 7,opig vcr lpeLdltroLv cr),)"irto4e 1ll otlQ(i r()v r:ELo(;)oe(')v. H rEoirrri)o)l otlrri, tr€ rI! ono[oor\rfliEoutrr, ri1,(r r1 xrrhirrpl 6wn-r1.

To x6oro6 rq6 ano,,orqtq

H (iLtr! [os eQdqol rivur irolu rQ(rrixrt. nth :< Eryaonttq eA$L< atu -

rct"vru xou tut tnc ,oLtpfi, o lw or\or\ltu tk tt tfuxiau.q xu tt ttyularoxa,E(iv ro ,' eivar lr€y(iLo. 1611 0s b6oouBe q oaorl lrrlE o 6vcr! ljno)'oytofil, fio!0( crr€Lio€L rl broblr{coi(l (r[o].orqic (lrnoor'i v( br(lodtere xafioto ouExrr6fiQolQqruc rl vu lprprBo:rorrjoort rouE x(r)6rx0E Fonrun 1()! ,raQcQrll[(ros r).Aqrou lvtLrpi(or.pt: (ii.c rc {irll{lra. 0( d;rQ€rre vo eitr(lorr oe oiol v(l lroar6'ryorpe to:rLrl0oE tr,r'r ;rQdEeur\r:ro! 0c exrL'l.tosL o tlno).oyLoriq. Ayvoriote nQo(otr./tLril rr( nphEsrE ort brEd r €lAd rt,-rv €9t.t{i)ot0)\r z(t pcxQ)iore povov rt(npcilaLg orlv cr.oLor8prj ir).!'!pri.. Arrd< tiv(L 6r() ct6dr!. H Ufc €iv.l|, bt(i.etolpr tov obr116 i,rtl vo i;Qou!€ noLo no).),(ntriror.o (ioro l) qq obryo! eELorirot0-rc:rpi:rrr.v cqrcLpidorrut trn6 nl\r ttiope,'{ iI! EEiotlo1. Kcrortv, xdvoar( cult1!rrlv crl.cipeol rrt( ttL.( eEiolDott oro rlv dl.).I. orvttvrdt€ o!a€td)( a!c! o1J,'-6uraF6 noLl.(ri),(ol.rdgon-aqcripeol<" . Or ouw!-).eora,< rIE o6qyor cqlod)oeoqro),l..1]r),&orhlovrclr pr I xcr zcri,:rrv cquL$r1r1,r(r a:r6 rou: or','r€LEor6-< xIErnotravlc.

Aq outr{rll,itoo4r€ vc 0eopotrpe xir0r 6rrriQran rrrL x.i0t roD.a:rl.ooraotrori cr.{.ripeor oav picr 6rcrzrrprs(v1 ra('1EI. :rn\r apz'i. .rv I rpdrrt €EiotDoI6xrL grjro< rr, r6rt yrc rci0r ul6€,'(opo lou r€txXcivorF€ oxlv racirq oltll,r-oerdlovrur r na(iE€L< lLi( 1(t 'r(1 6Qon[s ro ro].],an].horo I, xar rL( u6Lorrtqy|( !c 6Qolrtt€ rorE \riorE orrltal.eordq x.rril $rjro( rlc yactr$nc. Yn(iQXor! rl - llai4+liq ,(.itrrr .ri6 q! ripdrrrl. irpcr mo :rpdrro rtE ord6ro I cncr),orrtlt xotdE€r.rtn(r] l) = r: '] na(.rE€rE. (ML.r (i).ll Zcrorl'r1 ror rrr rr t:ircrr r rEriq: zprrri(cr.lrr' ul).a[orpr xu ory n1 at't,r tori;, ext6s Dv n qq rctirr4; ypnyptq.):lF€r(i)or€ rd)Q.r orr r(1 rEl.€urai.a 6rt$ara btoaxouv LLy6r€ao. uqroll oI EELodroEtqFrle.rilolv 6raplidrq. :ro orcibro 6notr (loqrr A elLoriroeL-<. ilper.cl6ptrore Ar Aapirleq lrclvc prlbrviooqe rrl otrl),1 ztirc, ar'tov o5ry6 (ott1tqt,l'vc [e tov

or,iiol,ro[6 rou €(lcrQuoocrp€ i{(lr fio ]iQ(bxo 6ri1(r, orc! to ir ixav ioo F€ n).:r!.vo).Lxir ro nLtoos r(Dv fiQdq€()v orillr aQrfi€Qi rr),€uQd rorv eEro(i)o€u)! €ivc!ro ri0poropcr rr,rv ,4: Li t(r ol.0! flE rr$6-< ro! A @ro 1 6(DE ni

I ] NAPAAEIII\JA ANAAOIOH: (iAI]SS

( 1 r+ . . +n r ) ( 1+ . . + n ) =r0 r+1 ) (2 "+1 ) n ( n + t )

2

; NIr iioriofle do rf ZrQit|t !roi( ilc a.irn) (16 ro 1.176 lb zAr0r (Qi1y[a 5(4..ri!.rd !(

i\(r ro l. qoi zorEi< cno ro!< Erbrd4r€oor! darolon< &v q(ivEl(L rd q{ xd(tr ro rbdtco.. Ar{rt: Yin ,,rt,i,. tr! nlooornr4 Gollt.

Ar,roi €ivoL owleroF6loL runor ltc ro troporoFo rorv nporrov n a0L0trcbv riar.ro(i0aofolr( ft)v tportc|rv rr rerpcrycbvov. Avtr"xcror"or6vtag n = | nor n = 2 xu,, = 100 oror tnro l(rrr n) 6),uroupe otr 1 6Lu6onzr1 aral"orqf pnopei vcrliQrr zcrlivu, buo ! arpi:rol 6vcr.rpi.ro rou ez(topprqio! fiipcrrc (:rou,1n elcrz(),o n-o61-o(ptLu o avu PC. olpaiv€r 4l beurp6),rrrc). To oqndpcoFcr ri.var

t\ct pxyiLo n . pia xa).ri czrip4q roli .tiqots ra| npirleav eivu \n' .

(hirv ro t(61cfbE rou uu(rrurcroc 6|nloor(1oeri x Li1or. privov r:x ro'v ('uvrrl.t:-orrirr eivcrr prlbtv, t6-re to x6oro; orEdv€ror o-ro olrrcir).dolo.

ll u,l/dDQol.n (1vru{fti(orool Dtlc(r oiluc(!r|}ri( IaXUl€Ql. O r€),€uroiog i.f!o)'oroi 6Qiozero! [s tll( xc(t [ovov nQdrEI (ric 6rciQsor] Ir€ ro\'r€L€uroio obllo),o :r.x)reklr(rioE cl.Tvootoq XpeLirletclL 6uo x.o.,a. To fiLrl0os 1(,).,' nodrEe(,)v llg(^llbQopyl( (vr!xsriror(rorl< et!(r

1 + 2 + . . . + , , - r r ( r r + 1 ) ̂ r r :1 * t '

Oc 6oupr.6r! ntl2 u,<.6pn 6lprcrc:rpoerorpLrilouv q 6eErri nl.eupd yrc( A(ibQolLI c\rx!z(rlr.rx( a\, (te( 4 6410 fi,,erre euqit|tNtlt !ft i fteaE g -nol,i;.Ltor€pcq an6 rq nr/3 116 apLorrprlE.

nAr"! cro ll€err{d lp6vra. oleb6v xir0€ [.r6ruqrri{6< 0.r !fi€oerE oxr, .l!roi. oL((pr0troi Ei.vcr.. olor(mrzir. 66),rr0r(Jr.. M€.iuo Io./La. 6!c ./s\{xo o[m4[( D eEr-il;rortLrv;rr- l' clvcimoUq, bfl 0& turopolo€ va ],!0ri gr:ro],n L.T6reqoriq .lno nrl3:roi.).(nl"cdr(lopo!g. (Ynlar(v $riLora r(1r Or-ola{!ara ro! ro arob€ixv!cv. X(])-,.li; otrlr< !c l.a[6d!ouv ur'6vrv rou! o].aq rrq $E066ou!-) Araoo66rnr.r, (urnrt fzritrlol cnob€ixrlrr €oqa)+rtvll ,a({ )aa iot, DxaAl€L ttlo !iqo6o< itotrarurti pthu' CntE.t ndi,a nonotrorlqt Aur6 e,Eclprdrcr t1n6 i:vcl (n1,6 yr:',ln,oE: 6no o]lr,8llcouoi 6{11 6r(tvroudra)\, a' i!a\, 6!6rdororo lri)(xr q(rivo\,r(lr va(:r(rronv 8 ro).),ad.aor.1oFotE. $nooo!\,6[(.! \,.1 ̂ /i!olrv xcL Fe 7. Arlro ,ioiio(o€ rov €roaril cno los,s. ror afv.l|, 3 orov log,7 = 2.8. Auni n .lr,.r,{ri}.uutrcrpoxirrloe $La zlrrarl.lzr.zri SeaoqpL6rnra y.c r1v erprorl ro1l gLirp6reqorbr\'(ror n. T€),rzil o €z0drr6 d:reoe (oro epervTr.z6 rdvrpo r1s IBM) ,.idro.rn6ro 1.5 onor zaL r&patrsvEL rI orL.4rt ro! yadqcd.* Exrx1rb(./La rn\r anc).oLq)i, I

llt NINAKE> XAI ANAAOIOH G/\USS

arao€Ei C €iva[, onyroltutd, 1600 F€ydr,$ aralro wo! 1600 7ro),I4 ]iQooodo€qxat o rA|oyQsIIorroUG ei1r'dL 1600 nohid.oros, 60r€ I v6q p!0o6oE vs 4(€Lt'l)pi(,E (ri t€l€i(l)E), 0eo4rqtr*6 w6rcq6pov. To rro reooqqro nq66Lrpcr eivartroto to x6o[os 6T<rv lqlo4ronor,orivtar, ndJoi aapdltq|ot enefepyao <, xatt"oxL cxopl Ylooro,

A>KIDEIT

1.3.1 Eqoegdor€ oJral.oLpt xgt avd6popt ovrtxoidordol .yro va ),uo€r€ ro orroiqtq

2t - 3L' =3

4u -5 r+ u=121t - | -3u =5.

flotor €ttoL or otqtoi: Kdi.r.rEiqE. rrE resr4 ]redE.re rdtd lK onoi€E sao no]J.ffiId-g|o yeq$1is a9oeelror on6 1t5 Jrs,

1.3.2 l'ta io ouarllrat r + v + w = 2

u + 3 ! + 3 u - 0

1 t + 3 r + S u = 2 1

6qlile ro rary(Dv|-l6 onoqFa p€rd rn tnabotn ri (rJral,orqli, xae(ix xa! q MoI.

1.33 A"ctF ro ori.rqFo xaL desirF rorE obTrous 6rav

2 u - L i = 0

- u + z D u = 0

- 1 t + 2 b z = O

Mrce.ire va 6('.rr€ nt 6€ELd 'rl,€ued oav pro p,rtl otrtl,I (6ix('{ ro ledqEr€ ror,

". rr, z F*e( vq Wdo!-r€ orc Los qs Ln ).

1,3,4 E.raQpooiE oral.oLpi mo o!o?)F.l

u+ r+u= -231t + 3x u = 6

u ' t ] + t , = _ ] '

M6I< €teauo0€i Fl6€vor6 o€ o6lyLr.i 060r arr4r€raodore qv €Ei@oI pe rnv (!-16'i&or rrE ridr (Drxior€. floLo< ol)vi€I€oEqs rou ? oqv laiq EEi@ot avri ro!

l.l frAIAA!trtlA AnAAoI@H: c trSs

1 tror undax€r r(laa, e( zo€Ftonos c6rvoq q o!!dI!ol qs SLabr"doicq xai 0oc!dyia!. q pieobo dnoloqriE vo oiorlattl

r.3.5 AlorE trr ult{foodo ard}.otl1tr- to onoqtLa

r : l = al r +6! = l8

:Xobrdo1€ ro oxrtpa no! rcqrotd 7d0€ !Eio(.,ol o(! ptd €leEio lor Enrt6o! I:, gor rllori€! rilloaroL mI lnor. naoo0aote rnioLua caopt EUeEi(1. ro ledqi]tto nl(!ro! 6uir€als €Eio.)o[ rox raoxnrrst Brtd rI] ano].ory1].

1.3.6 Baeitr flc 416r ror.r o yra rs oroiFE atrorlry1d!& rt ordoLqri. naooo,aLld i oFetd-rU1(1, yr(l ro

a l r + r ' = l

411+ tLli =2.

H ltnorulia oro rredro 6rltl(l broaeda€rd( elaudoool{u( teut{ris oi(t dFoE zdt 1larorltic1 mo relfurdio 6rlrL.r.

I.3.7 ((r)E(i! oL hrro trO(;)i€< raollli ro! ll silur or abteq. noi€ eo c1!attcr],!+'sL I onu].oryl]01 o /:l Fi!( tbr6toa(rc<i OrlciErs aa( r(lQ{ibFr?lc 3 eJti 3 ,!o! uTLreirut svatrlaYas

(d) Eda or bro nQ(i,rFs ofltLE< rou ,4 ria(L ibLrE, r(ii€ e(l (1!axd.n+8t )t c(nd.oLqrldlL o jl Fi!(r tbropoa(fio(i

1.3.E nodorl< noD.dilldor.odBo1lq'dqoqrloFLs 0(! 78!r(!dr.1! )t rriorl lvos qlo itlcros rd-ieox r = 600i nod( d€tll€aolenro Terrdqovror d dvo Pc nou 0lirf).€f 8.000 nqdEeqro 6ntr€pol0nro it o 6!c! vAX ro! s,rs).ri 80.000 rl o a!(,r CRAY X-MP/2 roud,rri.i 12 er(trolliar(r (Ot nedEu tiraL bttr].iq (!taL60i.( loBilo ott o CRAYfilct oLliovoprr6r€qoq. s.i! rov brc0iftrt.)

1.3.e Atrief(( ri {4r(: t4 E.i! l reinj eEioool EEt(vd f. tL$€!rl6 olt.ieori (dq1il,n

[. 0L) to€ 6Ea oqdreeiraL rol],cr]"ddro rt. €Eiooo1ls I do rr! sEiooor 3.(6) E.i! llraiot €liooot arsr rr1l5r,lta6 btslxrQo du!t.]..m'l (tr€edtd ro 0u) iord 6.vq,oLerircd aoiLcrltdLo rrt< .qioooF 2 on6 qv EEiolrol 3.(y) Ecia r teiq €liooot nrair(fl rc 0u raL 0r, to€ 5€! dqcoqcird nollcn)".ioLoqc rgioooL- r i qE .Eio(,,,tl. 2 dno qv Eqioool 3.

1.3.10 ([o].n reoaesnzri) KovolL"A o noi].oft).dolaoBoq 6no Btyabr"tb! aaL0fdv

la + ib)(. + id) = (u( l)d) + i(h. + ad)

xuar,af6dlfr r€oo€aq noiiatr].ooKorlouq ac, Lc, b.1..,.1. Atvo.bl1ds to ;, lmoqtirs!a rnoiotio.r€ iLE rooo F€c a. ,.1a(' b. +a.l F€ teets uot Fovov roil.crlaoro-oFonsi (Eflrear€td !d aaluft ipooodo.E.6106I a+b, /.oeiE,6moq. tnt€tdrloll'.oorooo orr r ra6o0to! xa.(d!.tar iEt oqolorLotonq ridolE o avdr CRAY XMP/,18 rar o nol].olrldoraogo( F6rov 6!av Eni trlio!.)

20 IlINAKE: KAI AIIAAO]OH C USS

13.11 Xeryuotrortcfte anclotqri yro vd ].ioEr€ rc

6 u + ! + u = 7

u + 2 1 ) + 2 u i = 1 1 u + 2 u + 2 u = l t )

2 u + 3 \ t 4 u = 3 2 u + 3 ! 4 u = l .

l t l r u . r tast ,vr - i , ,x i4 t t . 'doxtdr r to t t tx t tp t :o at r t v- t ' t ' ; , Iu ,u ' t t t /OaruP;v.grdddr.,'. Ynoe€or. ofl(a) an' ooorJs cqtilouv pra lqolrd oqv Kol|(,oqvra, 80% [6!o!v zdt 20% qerh/ot r.(6) ar' ooo\ oarilouv t{.1 lQoud extdE Katrrq6a'rss. 90% Fdvow 6Eo xor 10%enaraqtol'rar.Ed! yaoaiLolls rry .taXtxli rcndoraol,I.X. 200 eraroFptaro 6Eo)tor30 0l1oroppnara

F!oa, 16rr rilor euxol,o vo 6qo1|ps rou (arepols 11 rar I al|tdr! ro! rcL'id ldvorv6q(D .lc(

triool0, 9(200.000.000) + 0,2(30.1U0.0ff)) =d

0, l(200.000.000) + 0,8(30.1U0.000) = rl

To rpclllarod na66lruo 0ilor ac1 ndls naos r(l rioo xtu !a 1Jnoloyiooutlt rI!oqxrrrt rodortra p6oo) rlE re).o{<.

1,3.12 E(ia r = 200.000.000 ,tor r = 30.000.000 oro id).oE ro! Xeovo!, brur!ftdarF (Xo)Oisva rrs.Tf) rs rE,o-rp,s r,r' rcodhLociqou ro aqXu'r 1 xu. .

1.3.13 Ed' rs u rar I mo r!).os filaL r(l ilta p€ ra u rar I sot! cqlrt, noLEs €i!(tl or Flt-qdrqr6: noroE €i\,({ o L6yo< io! u neoi ro u, n ouri q "ora0.Od xaldo'isol :

1 .1 :YNII]OAI:MO: KAI NOAA,\NT\,\:IA:MO: IIINAKON

1,4 T IYMBOADMOT KAI NOAAAIIAA:L{IMOT NINAKAN

M€ ro 3 €ri 3 nopdb€rTlh $q, B]roaouoatl€. B(tar rdao, vc YadqoriF€ oI€Errq €Erorbo€K o!cr,ui!)id. Oo lnoaouoa$€ rrz6gq vG ltcx(YQdqoul€ lenropepo;gzcr0€vcr cn6 tu 6rlp<rttr q-< traal.oqrtg, rror crqutpolv 6vcr :rolJ'a:r)'cior"o t1g prr<rlioororr uno qr rlLl.I 1|r1 rc qrdroow oc prcr rpL.lovt"ztl poprp{. aot6oo. o' i:rtt

lr€ydLo olJorltr(l, n1JT6g o rponoq rclrcrlqcqllq rI! arftIorqiq 0a oblYoroc o€(b(iqo6o. Xa€rdiErar €vag roln owonrLl6reQoq raoroE. Oo €rooyn'rowt olJtr-i,o),[opo rrvdr(,ra yLa v.r ;repLypcirpqre ro cp1tn6 ouoqfa zcrl no)'lofi)'ooLoo[6rrrdr(rv Il(1 ra rsaryaZ(lrorpe tLg npriEeLg nou tr:r).onorouv ro ouotlpu.

}perd;ore 6rt xaL oto ;rdQd6€[1ltd $(rq

ftrqc!i.qo\,1o! Ip€r( braqoQ€rrroi ru:roL leleocirv. >rI 6eEId rrl.€uQd eivcr! 10r\trivro[c-or1lq 17. :qv oQlor€Qrl rl.euQd ei!o! o! dllcoorol u,tJ,u. :q!(pLort:p1 ;ritrpri €fiiolE Eilar El,vi(l oovrrLeol6g (6voi ofi' arjIou; or+r6cri.vel\ ' . t r4nrvr: .r(u). E:voL grroLxu vu nu-Drornvou.rf tor! rQrr( uTvuoruLE Ir dv'r

2 t + 1 t + u t = )

4u 6 r = -2-2u+7 t t +2u r = 9

o Zrlv(Doro< oilur o ,ri = rl Luorl L'i,vrlr jr =

(1 )

(loo yr(r xouq e\r!6c ou'x€),roriq, rorl xo1or6[ovror oE xQelg yQa$[69 xuL 18€tEorrl),rg, to or,ro16 o lpc ei,tnt &a< nivaxag qia ati eia. A1rr.6< Lt'lEror nitarxtl<

O ,.1 eivcr evag rrrpay.rrrrdE niv(rx.lE. 3L6rL ro r).rteoE 1or! GYvdmov orFrirr€L

$r ro :rLrl0og rorv r!Lori,o€{,rv. f€vrx6r€p.l, €ilv ro oioxilpo fi€Qrdtet n EErodro€rEzclL n ayvdrorouE. 6Xor+€ €\ral' rarpc1(ouz6 rilaxo orJlreLsord!. ?.i€..r( n.Ax6pI y€!126ft:gr. $ropti va dlorge ororrpc n €E(o(irot(,)v [E n d./vtbororq.:'crrr{v qr ;rr:pin-nr>or1 o ni,'alraS sivar rzapd,tr)"4)'6ypupyoq tx m leagF6q xar norriie;. Mr drLlc l6yr.c. drcrE 'nivcrzcs nr eni n'.

Or ri.vcrrq rpoori.6€\'rcr gerc[ri torg tj ro}.}.n:rlcorttlovra pe otcr0tpdg.6fur< arpr66s rtl br.cv6oycr-rc xtiot oulterc.4rivrl lqcoprorir. Ilpdnrnt, pno-por;re rc O:roporipe tu 6rnuroFnta ocv tLbu€E rr€emr{i)o!:rq nLva,r,tv dvat

lilttl| 2 I r' l

A= l 4 6 o| 2 7 2 )

22 TIINAKE: KAI ATIAAOI@H GAI]SS

nilaxEq p€ pi!1 Nvo! ariLry. 'Oftog ltc! ro bLovtopcro, 6rio riivax€E €ivar6uraT6v .r'a fiQoot€Ool^,, 0o!o! oTo1, €louv ro i6to ox{[a:

Ilo]"i,an],sorsoF6g nilaxa Fs 6rdvootrc

AQxilolJtlr nl6ov vo xalolpo7rolo![r sur6\, rov oup6o].[o[6. A( yQd!Jo![€1!' oumlpc (t) r()! 1e[6v rElodor{Dv tl€ rQE!E olv(i)oroug oqv a7r}.oroqF6vlFo(xpt tlr rivoli€q ,4;r = L O rrrog cur6E oe n).rlp1 cvrimr!1 eivcr

t; il I i il li il'l: il li il

li i,)t;l l ;l

t,tr , 1 l t

f | ! ] = [2u+, i+ . r '1 .

,,,u[j]

H bcEld n),r!od rt eivar cro,ae-ri{ ooqrlg eivor, ro 6r,crvuopo-ooj},1 r(ov "[I ouo'yevrirv 6guv". H aaror€arl rtr€rlAd a;roreleirar an6 ro 6td1,!oltc1 ,r fiol,l.a7d,cr'oroolr6vo pe tov nlvcxc,4. O noltrcnl.noLcop6g (1!16g ocr oQtoeei &a, 0016 r,.rttwtraprtX1 n qyrxti otirrr4prt. :uvsn(i)E. ! nQd)q owte rrypeq tou yLvopd-\ro! Ao fiQd7i€! vo Tiooxurir€! "no),l,anLoora(!,o!rag" tr1v rpdrq 1pc4rp1 ror,4!e ro brcivuopcr-oxliLl ?:

(2)

Arn6 rivcrL ioo pe r1v :rprirrl ourmoou rou b 2, + 1r + rrr = 5, €ilaL I nQ(ixl €Ei'oo)oil -rorJ ouofliporog, H d€rJi€QJl ouvror(i)o( TolJ yl\rotd\rou /r fiQooaloQi(ero!diro rt 6ffrrpt l(',.r,U)i ror 4 -eivor 4u 6L) aat n reirl orv.orJoa neolrnnr.lcr6 qv tpitrl ypcpprl. 'Erfl

I rEioool nvdxrtlv ,4.r = ll r"oo6uvapri. cxpr6tiq !tro oiorrpo -rov rpLtirv t€rod)or:o! F€ ro onoio E€d!,liooBe.

H rie.rqn orn\r €Eio(|lon (2) Eilar oa[€].rdb4( 1,|.l 6l.016 roxE 'ro].1.a'r].oord-ogorq nrvcizcrv. E€zrvh F' 6va 6Ldvlopc-"1qcgpt ltl 6va 6rcivuosc-mrjl.1, tovo:roitov ra pryrl rcrQrdloriv, x.u" :rcpciyer 6vcv xcr povc6rx6 cpL0f6. Aur6g oorl,nurpqrivog aareLL6E ).tlcraL eootrprz6 yrvdprvo rov 6tio 6ruvuoptirov. Mtd),l.c l.6yLc, ro yrv6pevo ev6c rivazc 1 rri n. nou eivar evcr 6rtivxopc ypapptjxur tr'69 rivcrz( rr eni, 1. i1 \rat'noparcq arilqq tLy,ll,6vug nivuxug I eni 1(:.r.E.: nou rov.rauiloupe pe ro pova6rr6 ororpio rou):

=[2 1+1 .1+1 2 ] = is l .

r.r !YMBo,\l:}jo: KAI nor\ An,\A:lA!lvot nINAKQN

Aur6 rfl6€6c16vn 6rL 1 :rporrLv6pevl ),nol r = (1. 1.2) Lxcvonorei r1v 4tlq

Erir' eierdoouyt tov ol,o uno).oyto[6, no]J.an].(otcofoi ;rircz<r pe btirvr-(]F.l, 8a lorpE orr urciplouv 6uo rQofiot !a ro 1a.iao!p€. O €vu,< tiv(lt ypctrttlnpo< yQoul.t. K.ior yQolrBrl ro! fiivaaa owbudE€r.rr [€ ro llirlroF.r ri(! bi.\,fl Fr(ou\ ' rrr{ .?tr f ! I rorr l , rvot\rvuu. Ynrlcxolrv roirr ro"rrrprzci l rvoucv,r orrrv uauuluuvrptr< lpcppiq: 1c :rupt'r6cr..1gcr

[F Yac4rtr€i ju = ti i rllil [i,;1i,:1:,i] lilEror ro eET/o1il orviHoq. d.),(i o bEUrrQoE lQonoq civor €_E i,oor [\,61(rqdQ(lJ!,Io(Dq Fci],lorcr rcL rro s!brqq,6g'rv. xcL €vrrl.€i rov no).].(d.(or(&tFo .rflii,t Jrao(d i;,/. n) lrlotr€vo rUAioz€r(1r 6rd [r(l< z(r €iarJr ouybtaoF66 rotv reuitv ornAitr

fr l fr l f . l f ' lt ' f . n l ) . f ( . 4 . . = 2 | 3 | - s I 0 | - u I I | = | o { r l

Lr l ! l Lr . l t1 )H .lnrivrnol €ir(lr 2 qoa€S I mil.t I ouv 5 .loaF( I rrrriit 2. Auro .rvrLorolx€iorry "rrzdvc rr'rl orrl).ti.rv' €!o< yaqrtrLr.on ouorrnrcrq -{r = 1,. E(iv I b€Eldn),€L'ad l, at€r rLE ouvLortioes 7. 6. ?. r6rF I Luo't.r dxrr rL< ouvLofli)o€q 2.5.0.Orurxd rl Frzovo roJv y(rqf(i)v outrq(l)lri. tr' (uro (x(( orrtv ouok1. rpanrr !cz(i!oL'llr. rouq iblottq no),1(rli),coroollotlq).

O xov6vu< y( rq oriile< 0( Xprlo4ro:roLrir(r a(lr' ercrvirLrlrrl trro 6L6l.ioz(r ror In,n1ri . ,4r i iuvor1rf yrr ! , r 6, ;ooutrr r ' t t ' t , r l t :

lA To ylvqr€vo ,4r Fnoe€i va uno].oyLoo€i XQloqrofiordwsE o].oxl,IQ€Sorrll.eg. 6ru[ o"rlv (3). ff au6 ro ,4:r rLvat" ivag ouviuaopig m4hbv touA. O! ouw€I.€otd( nou rrorll,orrlaot{ilouv rlE ort]l,eg eivcr oL ourtercrnr6veg

E(il rpoor(0rloor,B€ ri:r 6r(rund)oorp. ror l€uxo iuvov.r oxLq n brcordo€!:.7pr.ra!6pcrorc oqr6ol,ro[6 yr.l xr0qrLri owrorrior rorJ A. AljTo eiv(( €ino].o-H or,vrordo( orl Toctr[4 t xt1r o t orrt].I j oqdo],ilEraL $€ o,j, O n06roqbrizlrl: qu.,'€Q(i)v€t to1' (lprotro yQ(4qtnq rcr o b€nrtoo: ro\' cpL0Fo orrll.lq (orlvrqolloiyt-v4 .iozrtol ro drr ry(lv 3 zaL ro oF rila\'6). Al o ,4 eivcrt:rilazugrr uri rr. rore o 6eizqg i ptrc6ci).Lerar aro I dt,r; nr u:rirpXouv rrr 1pc44r6;ztLL o .j rLn6l drrg n. )wo),rzir l;rt'rp'1olv mn owtottiroe<;ror oXqpcrilotv prartpaLiq)"6yp<44r1 btrita!1 rcr ro o,,,,, rivar oq 6elrir xdno 1t.rvitr.

fr( ro 6rd!ro[c.r cQxet dvc; bein-rtq. H ] ou\rKrdrou or+r6olii,erar pe r;.(:ro\' iol.ltln)-(1orcrotro nQoruor+Evoq eix(4r€. .r1 = 2. ir = 5 z(L rr = 0.) To r'

NIN_AKE: KAI AIIAAOIOH CAUSS

yede€r(! oulr]o(DE oclv 6rdvuoy<r-onllq -dvcg ni.v<rxrg I eri n- lrrpr,zig qop6E6tlos ond1,€rc! oe p',a yac4rtui, 6ncos ro r = (2,5.0). Or fi(lQ€!06osrE z((|" rclr6gFcl1c( divouv egqtroq oro 6rr, owo bev Eivor dvo< rtvoxoE 1 eri 3. EivcL(rxQ!6dE av( brirvuoF.r-myll,I rou rrQoocuQrvli yp{iq€rar oarl6vrlo.

fLo !s r€Qryedvou[€ rollvop€vo A.r XQqoL[oroLoqre xo oup6o].o na6o0€off

la;;:r; eilcrL I i-orrvrm(;,oo rou Ar'

Arir6 ro dgeow[cr $q o6n/ei x(lrd [ftroq qE i-ortg 'l,Qopltrl( ro! -4. oro(rynr(floll(t iou €oo1€pLlrou "ywopdlof io! [r ro i.. bidovrcE fio\, 6rixrl] jo),€E TlE rudq otr6 I los n rior aQoo00Tovr(E r(r ofior€laol(rrcr -To (ioQoroBc€iv(L i r ,L.r ' r +.1,r .h+.. + rr ; , , : r ' , , . BL6roqe : tr i l . r 6rL ro $rpoq turv lpcppdlv ( tor),{0o5 rorv oqL6v) rou ,4 :rpirrt va roArdlE! [E 10 pJlttoq ro! .ri. 'E!aq ?ne:ri rr :rivcxuq ;ro1,)rcr:rl,crorciler ivu r-bLdoroto btd6 roBa ('|c[ n(rpirl€r tvcl m-brhoraro bravuoFo). Awi vcr ypcirjotpe rc :rdwcr ov<rl.urrxci. eivcl cr:rlouotepovu 6ou),euovpe pe ot4r6olcr s0ooropirrrl;v. Kt crut6 6pox 6Ev €ivc 1600 onl6 60cro ot,1r6oLlotos n!vdrx()v.*

Exrppcoq er6g 6ripcnog analorerig trs nilaro

M6IQL r(irQa lQlor[otroLl]oalf yL(l ro cQXLxo ouoqFa €ELodro€{Dv rI otvro$yl.lqarpri .1.r' = 6. Ti yivcr( pr t6 :rqrilrLg nor ettel.otivrar" xcrrci rrp' ancl,oLrprll Irorcrpciber"ypti pcg. ro rpdrro 6r1gc rlrcrv v crrpcrLpdoot4re 2 rpoprg r1v nptbq elioarolcr:r' q betrepl. frcr qv 6rlrd n}'ergir, arrt6 o4pni.r,tr 6tt :rpirr:r v trqolQ6ooupe2 rpoptE qv fiQ6rq o1Jv!or6o(. rou , ano q bElr€Qq r{(u roxuerq6[(rot€ ofl ro[6u, o.ttori^tota itttuT.ut*t4tt lir.tr fid.la aoftoovtt ro b g tot' ttuJptt'omotT.tttir\4 :rh,uxa:

Arir6 ro blunloid]voxF€ lQnoqoTro(6vro( rov xs1'61'o ro].].atr],rlorco[ou fiivor{ar:ri 6Lrivloptl

Eh=

* O Finsr.in {oi/qE ror'r($oui o\rooirotro_ zcrd ro} onoio aw. .r(rda4r6u!qrro:

d€iz.F o,l(,i\., (rroldro. zd, ap6o0sm} Erc.(r. drjlj. ,iuy,ol,r dlrj /.oai< io t.

"= l-l i 3lL 0 0 1.1

I I o ol I s l | 5 ll2 l0 l l -2 1= l r2 l .Loorl lel Lel

1.1 :YIIBOT\t:IlO: KAI nO,\AATAA!tA)trlO: IIlNAK!lN

H nal,rrf zli' rpirll o!{ordrou 5 xnr 9. rcpfu€nov or. i.6'sq (ioyo rrlc €L6rztqtroaqlq rI< n0(trl! x(lr-raiqc lar4qrtq ror -E). H v6rr Senr€Qtt o!'\,rord)o( 6Xrlrt oc'rorrirrloi 12 xar €t,qGviorrlLr- u€rir ro:rpri,ro 61p( qq cir.rioL(1,rtg.

EircrL nrzoio vc r€0r./oirllJoutr€ ni!(it E( nor rzre)"orv grgovrr4revc 6r1g(rnclrciorqitE. on!r< o '. :ltrar(i)1ortr. raioqc ror "tcrrrrorLz6 rivftxu" :ror, be.vzrirtr uro),lrog riio.rr.

18 O rilarcq rox oqryel zciot 6Ldvropcr c1'al,).oi(trio €ivcr. o rauroflr(iEnivaraE | , p.E I oll 6(cy6vro x clL 0 ncl.vrou nllou. O rivcrrcq fiou cqcirQ€i6vc notr),(1nidolo I rrlg 1p(pprtq j a:r6 q lqcpprt i eLrar o atoqttb64gflivalass Et $c I oll bltlydrlo *ar-rov crpL0p6 ) orit lQouui i xcft orlorrl),Ll.

I IAPAAEITMA

E(irl ;rol.i.ffrl.(ot(iorr!: onoLo6inorr. br(ivuo$( Ir frf rov rclrrotlx6 fii.'\,oli(r.n(i0!r.rr. rd}'|, ro brdll]otlc 1r. AlJIoE o :ri.vtrxa< rivur ro cad],oyo ro! I ra)!,ror{\u;!: /, = 1,. Edv o[(); noiiruriuor(irrtrs pt rov Ell :ruipvert I11ir =(, . rr. rr )rr). A!r)i riv(r trlo xrnrzit na.ilq o1n 6r!r(i ,-rifr,0{i rrDv r.E'odrofr,rvzr1r ro orll.i'\,rrx(j r.tvirL 1 dltr liler o tv (prorLai nfuuA.i.

l-r( !c b|,Crlprioortrc rrp Loorrpc. ;rpi:rrL vt1 cq(A[ooor]rr rill' i.bLa nphElztu orrg 6rio:r),flrpiq r1q,{.r'= 6. Me (i},},( l"6yL.r. x06ntr v( no},Itl]Iicorhoortraro brdvlotr( .1.r' pr tov:rivuzcr -8. Kar r(i),l 6roL 0r|(LpeitaL 2 qropiq I rpd;Tltoljvlordrorr.rno tt1 6trireqrl rr uqrjrovtcL 11 raorn 1.11 rQilt cuvrorrirou crcr)-i,oi(,)r€<. M€rd .1r' .11116 ro 6rl|tu. ro vt'o zcu cr:r),oictepo ouotrlp( (Loo6lvupo pcro rcri.r6) €fvar cxardrir< E(.4.r) = Elt. Allro rivar arl.onorrao 1,61a, ror p16rvL,zcrl n{ir(|) cln rov npdrro o6ty(t. rn)6riv(l1o 6€. 616r' trroaoqrs 1,.l Er(r,i).uoLj|lroto (Azrrd oiiorltrc (rpooo6rorra< (tlrioiao(tra 2 qop6,< r1v :rprinl rii.orDol or1bsir€ptl). 'Eror x.ll rir 6rro ororlpcrrcr i:xorv r1'v i.6ro crar6d)E l" t .r.

nol.lnr .aoroop66 nrvriz(Dv

EpXotrrrorr rirQc orlv mo evbLcrpdpoUoa epcirrlo\: miq tu $ @ftloutttbtto ril,arLs; MiG a!6€LErt lloqe cra'r1v tlnal,orqit Gauss. fropi(or4rr rov.ra)(rxo rilar.r A. yrt,piioqre r rcoatveL perh ro:rprjjIo 6!pa (lrlc),orgrt: zartroiLq ./va)Ai.oqLr ro! nivcrz( E ror exre),ei criro ro 6rjptr.'EtoL d.:ri[orpe orL

f l r ) r ' l [ | o o lr=10 r U l z , r r f r l = r ) I r l

Lo o t l L ) ( ) 1 l

il| 2 1 1 l I4 6 0l 6iv[r -D/l = |

12 72 ) L

i 1 o o lE=l 2 101

L 0 0 1 l

210 t327

NINAXF' (r\I r\IIAAOIOH GAUSS

H necirlit 'ior lpiq 10a[Fi rorl ,4 6ev {aivsral v' trl.Ldlotlv orov t]4, €vd)bio VoA€< I rQ(;rrl lQapgri crpcrp60lxe cln6 rt b€urr:At. Krrt' our6v rov ra6jroo nolJ,onloo(ropoi rirvdn\r qoi1,€rcr vd outrqo)\,€i. pr rq ;rpti[eL5 cnc].orqljgFtrqEn ro)v yeq{rdv rou /. MroQoulr€ \,o yeduoup€ ro (nor€L€qro €irs o(vE(,4.r) = Eb. €q,s$r6-tovraE rov E orL< 6uo rr).€uQ6( flt: sEio(l)olE, €irr ocv(Z.,{).r = Eb. O v6o! rivrrrr4 f-,1 r<rr<rozeur'rleraL droL dorr or 6uo errta(rlLoo;ortq v(l oupq.r,rvow. Me drl).cr ),o11(l. or :rcrpooioerq €[vcL ri€Qr.rr6g zcllnoporiFe vo tpirlDv[e 0'1:J.6;. EAr = Eb.*

Yntiplt:r piu ur6BI onaiqol 1L<r rov noLl.a:rLcrorcopo nLvdx(Dv: fvrrlQi,-lot[€ rrbg va dpioiolp€ ro 1Lv6[€vo,4r. €v6q nivaxa eni.brcivuopc, rcl o v6o;tl(g oaloFoq otl iretn€ \'a €ivar o4r6r6(fi6g F axr61' roa no),],or.),cot(lo[o.Otcrv o:rivcrztrq B :reprdlrL pin xcr lovo ortl,rt x. 16r€ ocr;rpircL ro lv6gevoitld.zrr;v ,48 vc tcmiierct gr ro nv6paro aivcrxcr rri bLcivuopcr -4r. Ax6FIrol,u1rQu 0o €avft vo rolu€r yE!rr6rrQ(: 'Olcr! o B n€pril€L 6rrkpoprE orrj),r;.) , .X. . rr , . r ' : . . r ' r , EIniEo![s o! or l ] ,e! ror i ,4,a rs civ(r cxQL6(;)E or,4rr , ,4r ' r . i . i1.T6te o roL),cnLaoraop6E :rlvcirrov 6cr rivcrr crlorimcro5. @cr onegr6p<rorr rovB cr;roe),ouprro tor6 olilLe(. rI picl birrLa otry dLLq. xoL ea rol.Larl.aoLdqoupr',(ci€r gio EtX(')A!o!.i. O xo\r6voq cur6q €(p(Qllo!€1or orouG ni!c1{€G no! roL}.a-xl,crolcroclll€ ,laolyouFdlq. H rgorq miLl rou EA roourcr gr l'gopd-< rrlvn0;tn oriLn rou ,,1.

rcr ro i6ro LnXurr xnr 1Lcr rL< {i},}.r< orrt}.r(.:ryFr(iorr orL I nedTll FaS {oi(irlorl €iX€ l'Cl xd\'€r" Fe nQriE€LE ll€raEu xrrv

'ypuspdrv tvti 4 brurrprl ge orrl).€S. Micl rqirt nqoodlloq, xcL eLrfloqre rlxutrtn€Ql. €ilct rcr arpLlpcirpoupe rd.0r oroL1€io tou ,4,B. flpri4art, u:rciq;1erivta zor tovor xcv6vcg rou 6ev popilto:roLq rov avrxdr).ur1rr. rriwc4 acivarro :rilvra vc 6oulE1iou'' ra),a. ,4.!.16< 6€\, EnlrQ€/rr! rov ro),1or),(oruop6o:roror6rFort Lrifouq nLvilzov. E(iv riv.{. r€xQ(./o!!xoi, 6roE oro ficlQirb€!1,pdpt4, r6x€ nQane! vcr sivcL rou L6iou pey€6oug. Eirv r:i.vo! naaallr1}.6lAsg$or,ror€ 5€v nQ€ncr vo 6xouv ro i6!o lllyrOo<. al,l,s to frIiqog toy oiql6r ro', Arqhta ta rcoirat rc n Ai0o5 nv ypappar orl B. Totu trovov prloQ€i lcrnoiJ,cn).(ro!ao0€i o A |r€ xhor mrl).q rou B. M€ ddL( liiyr(l. 6rav o ;1 eivaL 2€ni ,r zct o B eivct nr sri fr. 16r€ o no ,cr;rl.oortropog eivcL 6war6< xcrr ro'!06tEao AB 0a dwn p ai. n.

n€QlyQd(poup€ rdp( ro1, rQ6no .IQ€oIE rou oroLl€io! rou r4B rrou 6pioxexaLor! yQdFF{ i xci orl o{Iq j rou AA.

li:illillil

* ANo rnar r( ro $rtu(( ror '-f,Cod.r(ralGraor a6uof .n! ro! I. (l .1) = (2 i). r. ror

todin ooo iE{('i!. {;rc 6rioToi( q(uu:or((d€ 6tr uf,oa.i !( rirur idoo!. qtu\ rn iho (kr

uoaotuer€ u:o!rf lkr lo! "!ftrom,o !6!10 z l=1 2. f,or yr( iikzra ri!( o\r.x iriUo..

I,4 >YMBO DMO> KAI NO AAJI A>IA>MO> ruNAKON 27

axI

3l 17 8 l8 l=12 3 l

AB

rr. 1.6. na0d6€r, lro rol),artraorsopoi nrvdxov.

1c To oiorxelo (i, j) ro! .48 el,\,or, io eooreQr*6 "/h/6pwo ttE.yQdppik irou ,4 pe q on1l,l j rot B. fla ro naQdd€rypo rou >r(. 1.6,

(AB)32 = a31b12 + a32fo2 + a$b32 + aub42. (4)

,,,1

241t€into4t 'Orav ypdqoop€ ,4.B, oL filvox€E 6€v 61ow crapolrrptog xdnor,crox6ol [e rlv qriqlol9rl causs. >ro fiQonyoul.{€vo r[qQqo€rytls FsS, ro llvq€1/otra! -84, 6no! .E 6vdg "olo!X€!d)6lE' nlvdroE. AQy6ftQo ea'!L\er PA i LA'i1 ctx6pl xal LDU. 2e xdo€ n€Qlfii(Doq xQloq.lonoloriF€ rov i6!o ^/Ev[r6 rov6vcvlc rov nol,lqflLsolcou6 fiw(irio1,,

IIAPAAEITMA 1

12 3lL4 0 l

f l 2 o l i17 1o lls -r ol =l+ x ol

lt;lle

fla roQd6c!"/po, ro olorleio 1? eivor (2)(1) + (3X5), 6t1". to rorrlreqrx6 y',v6pevoqs ne6rqE ]aapF{E p€ rlv nQ6q o4}un. To 8 etvar (4X2) + (0)( 1) ori6 1rlaernepq lpq4rri x<u. oerheprl orrlLl. H rairrl mril,q rou B Eivc!! 0 rio! ^ft oordrov 1,61o ro i6ro rolier *ar oxo,t AB.

IIAPA EITMA 2

o riva'idE oe|,Orced neoEE\eL &d)ay4 yao#p61, xor irL\&x& 6EELCL.

NINAKE> KAI ANAAOIOH CAUSS

IIAPAAEIIMA 3 O rnurorr'{6E rivarclq eqiv€r rdo€ bdwoBcr avall.oiorox(! xdo€ rivarq €Jriolq ova),l,oi,urto:

IIAPAAElrMA 4

I A = A ? t l L B I = 8 .

l . / , I= l20 + l . 2 i '+3d l,,=li !l l: ll

At.r6 qc.v€Q{i\,€r rrtv !b!6qt<r :rou rL:rilope 6tL to1u€. ytc orri).€E. O B ulrord'ef-lclr o,1o 6uo orrll€S. Irl trl|cr 6i:rl,n orlv rirl).1. xcrL o ,4 rol').an)'croLrileL xtroe ptcrou( E€7,o)Qr('Td. Erot, ,4i08 aiil4 tou AB dvar ouv,utopiS tav o141ov tuu,4.'O;rtog axpL6tig orov nol,tlunLrrot(roBo r vsitcr ari bLirwopc, oL or4),eE tou,4nol.l(lnl.(loLdqo1ia! F6 roq ael0[onE (ri yeciFtiora) ro! B. H nQ6q oll)'t lou,41-i eivoL "a qrop.E' q oojl,n I ouv "r qoalg" rl ofll),rl 2.

Pordp€ r{i)Qo yl(l rLE ?papptE tou AlJ. O no},}'ar}.r.oL(ro[6s FroQei €niolEvn livrr y(r+?/oj 4oE y11,4pa. H 6€rirtQyl yQcrg$rt rou ]rloga.Jou Xptorpotror€irouE (lQrolous 2 v"qL 3 uro q bsur€Ql lQqui rot, A. Autoi o! (rQloFoinoi,),c(nloo!(iqouv rq yQCrFFds to! B yro 1'a b6oow 2lo bl + 3[. dl. OltoioS.r1 :rpclrrl lqqprl rou lLvoFdvou XQlortroTtotei rors cer0FouE I xtlr 0 1!(| v(r b(oo€toE 1rlo oee(i |[o l,] +0[c d]. 'O;rr:.r< crpr6riS oqv cncr),ot<1ti, un rtlv onoicErrivqo(rv 6),cr. rir0r '1pcpy{ tou ,48 riv(r ow6uaop6g itoyyQdplttir ou B.

:uvolrilo1JtlE rd)ao rouE rQe[c 6((poe€rr]iouE iQ6fiouE y((r va 6l6tre! x(vei(roa fioD,c(n),uo( coF6 nLvd1{{ov:

lD (i) Kdo€ mor)(€i.o rou -48 etl(rt to yw6pEvo prag 1pa14rrlE xcuploE orrlL'|E:

(,44),, = yqoFprl i too ,,1 €fli orrlLt j rou .B

(ii) Kde€ orql.ylTou .43 Etva! ro yr\,6p€vo Ev6g '!h.rxdxcrr lrrdE orttr4E:

ori1?J., j rou AB = A eni onitrl j iolj .B(iii) Kdo€ yeagl1li rou ,48 €ivdr to'flv6Fsvo LrlctE lQaltrlE

x(rr, €v(rg rivoea:

lQogpri i xou ,48 = yQdFpr{ i tot -4 eri 8

H ropdryletol durri €ivor tQ{o+rl yro rytv €ra}.r]O€!ol ttldq arro rlE olps-vrLx6repeg L6r6tqteg to! nollotr),qo[soFox nl'cirlDv. Yrio0aotE 6r[ A.B xor" Ceivnr ncrpcL),r1),6ypc4r1ror riil'.rr{€q zqr 6r ro o"/pn rot8 €ttlrQ€jr€r no}.}.at}.c-orcrop6 p'crr,rr1 q 6!draql: ro d.rioog t('v oqL6v r(,)v .4 rot B raerd!€!.

I J :YMBOAI'NIO)- KAI IIOA AII A)IA:MO: NNAKAN

(lriororrc. rr€ ro rLileo: r0)\' yacffdrv rrLlv B ,r.at" C. T6re 11 l6r6qrc eivcr 1€EiIE:

lE o nol),orLaoraop6q nLv{ixorv eivcr rpooetolaLor.Lxbs: @BtC = A(BC).

Eciv o C etvar 6v<r 6rrivuopa. btlctui, o riiloitr< {u picr 1r6vo oujL1, crn6Ei\'cL r"oo6nr(4ro F€ rlv criroixlort (rAh = t(,'l,r:) rou cvcq6Qcp€ riQorllo![t-vorE. AlJro [ivcr ro o€tli],ro otrr)v rO)a \16110)'\r ro! no]J,(1ntrL1orsotlolj rrvi(xorv.tdv o a FXrr nF0rooor€!)€t or l ) . r : . (o uuvu ̂ uL' urvrr r ivur \(1 rrc ozr{routrr r lFia 6i.nlr1 ('Tl1, dll.l x(r rc eq)aQ[6oou[e rov vo[o ncQloo6lEQEq 9op6q.

'Etor

oF €vo yrvotr€vo n€aloooriEnv fllcir{Dv 6€v ta€rdrlovraL roQsvo€oelq. ,4!16btorirotdrvFtc(I €ni,olq, ol,yliQivoli(1( ndo€ ororlaio rnrv (,41)C xcr,4(BC).0ubtirt o!()q rc(Qctxiltor (Aox. 1.4.20) yldlt nQot+rolF€ -rov ouF6ol.roFo rirvdx(,)a.

OtLouf(r va rQoxoQrloorrlr oq ol(on p€rulo no),lan),colaoflou nrvdxo]v^(t! cnatrorq,rlE Causs. aL)d r0olloqr6'vo: ripinuL va pvlporeuoor4e brio or6p1rbrolllEq -pLrr tb!6-rlr{r nou o :ro},l"cnl.aoLcoptiq :rrvriltlov 6Xer" xtrt pr,u cil,},r1roubtv d1.r. H [br61lt( rorJ ro1rr€! €ivar, li

lF O noll,ux).(oLuotloE ftvq1{Cuv €ivqr EnrFEQrcnrxdE:

A(B + Cl = AD + AC xctt (B + C)D = BD + C D.

Orlorrci. r( olril.rr( (1116! r(')v nr!.iki.,' nQirrel !c l.rLQ!dlo!! or , xc(C dXow ro ibro op;rcr, cipc rpomi0*rttt ,r.at ot A xer D 6zouv ro r(1illl.Ln),oItl€{lo( yrc no}.Lanl(1orcolo cro lreoorcir zaL aro rioo. H rlxI66EGn .rlJ.ortou xcrdrc pt Ld.1cr ciwL :ro),r 6cpot{.

H L6r6xrTr( rio! 6€v roxtit:r eivcl zdna;g nro rv6rcrgipouoc:

lG O no)llan),ooraoF6q nrvdrov Aev €ivcr. F€raoErrroE: :u,io(,),< ,F.4 l.tF.

nAPAAE|f l lA 5 YroHfnrr of l E f iv, i r o ] iv, , r , (E :ro, r ,o(] | l7uFf nAot lor ' -st-vcLrq. tou oroior 16qdo4 rFrrv lc arpoeei 3uo rpopiq rnl iAdlrt €Eioool ar'rI! b€ur€pl xar uro06orE orr o F civar o fiLvor(tE.1!cr ro e:r6urvo 6r1rc, r1gr0oo0€ol< rlq yActrl{s I oTry 1,0.rFU{ 3:

01 ti:ilill_j

l.

IIINAKE: KAI ANA OIOH GA(jSS

Auroi or 6to frivaxe? ttetatl0.Evrar:

i 3l =.,0 1 l

f r u 0l I r0 ol | | 0 ol I r 0 0lr ;F=10 ro l l2 l " l=1.2 t0 l o ) . ) .d EG=l -2 l0 l .

L0 l r l L o o l l L -2 | l l L 0 r rJ

.*=liii]..,"".1iii]

Toiro to yrv6g€\,o errelei zct ru 6no 6tpnrc pt o:rotabr];rort otrpri. I1(vrolQ6vo\ zrrr lc 6{(,. 6rofl o' cuoil' rlv €rbr i r€Qlar{Dol I brdi(rEl €irolIr'rQK onpclora.

IIAPAAEIIMA 6 Ynoo€ore or! o .4 €Lvctr o i6rog ulLri o C eivcL o;rivaxr4'yt{r 10 rd,E!.roio 6ip( -ttpoogirtt 14 ypupptj 2 ot4 ypuppfl 3. T6acr I 6(iroEn;rciter Qrjlo. )r1v ;rpdrrrl :repimr,n4. 6aou eq,sQF6(ouFr tol E aur n(rroarvrov C. I6eu€al sEio(l)oq l€xo6(il.),fraL rp./ rrtv rairrt. A!5i €ivaL rt 6rdr((Enrou 6vrrrrg owawrlocrpt- oqv cr:rcrl.otr14. H:rpriq rliotrrl err.6pd oq bnneprlrorr rrir6Qi( oq! rQirq. Edv o t aLo€r Ferd rov G, xdr. q rQirl €Eioooq 6e1trli.otrlraL rrlv enl6pcorl r4g nQ{i}rfi. Oc b€ir€ dva gl6rvLx6 orn odorl (3,1) ro!EG. rvti: un<ipXer dvc -2 orqv crvriororXrl 0!or1 tou GE:

Apu DG I GE. 'Ev( rulai.o ,rqQdbr:ry|( 0o e6er7,vt: rrltbov oilolQs ro i6Lo:rprlptr or:reqroo6tapoL:rivcxe< 6ev percrri0evrcrr- e6ri 69o6 or:riv<rxq dXouvx<i:roro v6rptr. Y:rrjplc ivnE l,61o9 1cr r1v EF - FE xrLt rlcrg l61os yLo rtvEG I GE. AEi\er,'tc xrivoupe dlc cx6pr1 6iUC, yrc !a borp€ tL oq6criver ge'rouj tgrtc tivrtztc unuloq ig lart.roX26ruc

To ytrfurvo GIn eirnt 11 riecygcrlltrn 6rirrc!4-r4 fficd.oLgiE. Aurft.1yar o,ti1'axuq itor gtpvtL ror ttpyLxri A mov tiva qryat,rxd U. Ocl to EoroboriFEornv €fi op€vrl rraQc{Ypaqro,

O irllog rivaxng,Ll'G eivar rLo 6poqrpoE 6rorL q 6Ldrcrlrl rcov agr0pciv 2lou E. I rou tF z(t 1 tor G 6ev 6t(rr4ci101x€ (utoi Unn,,ic! zcr' €u0ei(.! oroT!'t6trs!o. Auorul(i)S 4 bLriralq eivtu ..xcrddLd.t 1L<r tlv arn),orrprl. EutuXrico1r<t4 titrn 4 oami \uirafu yrur1v u,rrceoEf ruv 6apdrut attdotEtis ttot:€niolq out6dv€r ollv &r6pern :taptiypaqo.

:+r€L6or€ or, xo lrvEr€vo rd{(D xQryovrr6v nLvdzov €ivdL rrd}.! xdr(DrQLyovrx6g nivaxcq.

1,4 IYMBO ItMOt XAI nO AIl AtlAtMOt nINA(ON 3t

A>XIIEEIE

r.4r YroLctior€ ro yLrdpsvd

tls6rdors 6vo qe1ryoS xoorrow aE6ruv rdl ta 6dh4 ,rQoE ro orgeio c = 2, y = I xaro = 0, I = 3. neooolote ro 6to 6lavrroFora ouF llQ6tovro5 ro rloQdtAq)"EQdIFo.

1.4.2 Aoul€$owdg onlu riQog oflilq, lnol,o,ytois rd ,tw6fs!o

1.43 BA€lft 6io €qor€alid $rdglva xa! eva tr\d$svo ,$rdr(t)r:

[ : i] til .- ti : ll [-i] *, r rrnr

li iltlr .- ti : ill:l .- fi iltlt

,' -, ,[-]] ML r, -2 r[i] .- [-i]o ' ',

f3 -6 0 l |2 l,a j=10 2 -2 l l r l ,

L1 -1 -11 L 1 .1

To ne{nrc 6iv.r ro prixoE rou &ov$qarog (qro terad}Dllo).

L4.4 E. 6lds m €ni n ri\uiros .4 noD.qnlaodlll dvo n-6rdororo bdv$qra r, 'i600!p€pory(orlro iou.artao|0oFol :(o€rdlovtor n ow6abrr 6rov o.4 'roLl,o'rl.oodge!dwv a srl ? ]dvaxa Bi

1.45 Ynol,ollote ro yrl6levo

Be€tft pta ) io{ , rou sljor{Faros ,{, = 0, y!' 0!16! tov ntvuxa .4 xar F€ pl6svu{doq 6€Ed nfapd xa0 ruv rQrd,v €Erddro€(ov. MrloQ€tc va 6e€li€ ,i€ewodr€Qsg dr6pta ''issr4:

la.6 rQdtlne rorE 3 enL 2 ,rLro&4 A xsr B tls oto|lelc, cvrioror.tc, aii = i+ j ao]"Dij =(-1)i*i.

Ld7 fed$te io €ftn€{o6 T1,6psvo rou 6rdvriotrdrog-yaoa{ri V = [Vt!A . . . g,] xa! ro!6rovljolaroE-onilq r, Xaloq.oriordMas io ofu6olo tou aoeoiqraroE.

L4.t A60r€ 'iaer6Einrcra 3 sni 3 ('ir 666sis ro ,4 - 0)a) 6lcry6 o riwra: oir =0rltrilj.6) oulareleu.on trirat c ;ij = dji, yra x'ios i xor j.y) dv@ tery@'a@ nivarar aij = 0. yLd rd!€e i > j.6) owtolrtrFEqlxon ftlvara: oii = -aji, yrd ,idoe i |.dr j.

NINAKE: KAI ANA OIOH CAUSS

1.4.9 Mc roro o€ed nqAE€ov noffarlqg|dEou! or €r6F€ve< eo!.riv€s rov / sni ro ri

D O l 0 I = t . ND O I 0 J = l . N

DO l0 J = l.ND O t 0 I = l . N

10 8(r) = B(I) + A(l.J) * X(I) 10 B(I) = B(I) + A(l.J) * x(J)

Air6 pd0flrdrorn oxonrdr. ta o]roff)dopdra riv(rr roodnvotra. q ooo! rJnoodoolusdr oaxrxd B(l)=0. 'O!aJ<

r 6opi q( FORTRAN x(l0rmii rov bFtliFAo r.66Lxa'tdn06 onor€Ifqrottx6rceo (Il(l0daqllo r). Aur6 .ivar no),n onorrleoloru.or€aoor $a brorDolarrxri lD(aui onos I CRAY. br6rL o co(l}repuor xtl'.),o< lnoe€i vcu.d!$ no),l(i B(I) ra1lroxa6vrE, !a6 o nq6ro( x,t6'xcrE rdr$ nol).as d),1.q4( o'ilu t{ttov,'trivo B(I) xor bev br(vuotsrorvu i.

1.4.10 Eav ru mori(€Lc ro!,,1sivoL ri; rqitoLl|orooim€ o+6o1,rqr6 bsdtdv yr( va rqd!,€r.(l) rov ngiro ob'Tr6(2) rov noD,cnrooL(oni .\,r uts ysdtrfis I nou aqdre€ir(tl an6 tt yeq$i i.(l) ro v6o morxsio nou sflxaerdri ro oij rtFi ntv oqqie.oil(4) ror b.u€ao o5'tyri.

t.4.ll AUoas ! !r€!6asi Adole olrlfioadbrryl{r 6rav civor !rF$d(.c) Eliv it no(i,q xoL rqirt odll ro! I ei,',(ll i6Lr<. ro ib(o oqr{tuivrL rur tr rrl!rqdrnt tar roiq @i\ rou ,4 D.6) Eda I n06nt x(r reirl yAnNIi rou lJ rir(r i6r€<. ro i6ro o+6(ivfl xcr F€ r'tvtredJrr ad roint req$i rou AB.y) Edr! I nQ(irflt L( reiq T Oo$ot ro! A Fi'vor iblt<, ro i6Lo ollrdclv{ aqL FF flt!nadrirt xar '!eiq 'yaq$I io! ,4.B.b) ( i la): = r{rrr .

1.4.12 H redq yqaFfi rou ,1-d eilcL yo.4rrLxos sw6uffI& o)ov r()v yQdrrrrdrv ro! B.noroL rivar ot ouEE),roiE cwon rox lm6uaoFof xor troLa rt trO(irq yqotFri ior)

' 1 2 I 1 f l l r4-; _i i l 8-lY 1l, L I O I

r.4.13 Ib lvoF.vo 6r\) xdft' raLpvr"6v mvir'iov riaqr nd].r "dro

re(ro!'xG (6ra r(rororx€ia uftAd'wo nts 6Layoviou €ir.lL rrl16av). Etral.nosnoE ro pe 6v.l rae'i6€L'yFa 3cni 3 xor fEryiiora n6s ou6 dnerar dr'6 rolr v6fou( nollar).aoroolton ftrvdfi)!.

l.{.14 Bqstrr ruad6.illcto rwan.,v 2 cfli 2. rfrou',v,;,drrlet A: - L $ .l nou qn trolo reoyForrxd ororXric.6) B1 = 0, e!6 B 10.y) C, = -rC. xoais va oqr6(ivet CD = u.d) tl' =0. roQis taliva morr€io t()v t zar F !(l €irar 0.

l .4. ls f lpqredlar rc lcntFrcrou 1,4 x.( . t :miLstof AE ""

E = t l ] lt U j I

1.4.16 Er.r.N€nm€ tov rAooEareLmLxo roFo (.sF)C = li(ac) yLa rotE rivarcs mo

I J :\'NIBOAI:IVIO: K]\I NOAAANAA:IA:IIO! NINAKAN

1.4.17 Ynootor. 6rr o ,1 $roiio€ror rrr xrior' 2 eni 2 ni\axa (AB = BA) xot t 6oad o

. t , r l . . - 0 t . . l 0 I l. r - | . , / | r r rorru l , ( t r . ror< ur .

[o o] "* " . -

to u,

AriErf orr o = d "(

6 =. = 0. Av,.l/i = .lA yra aAee nivorc B. ror. o .,l €ircLno].Id7L(ioroE io! tourorrror,.

l .4 . lE Eot . ) r , ro Srdvuqro mi) , r l l | f orvmr i ro€< 1.0. . . . .0 . ̂ l iEre 6rr o rovo!(C (AB) ' =-1(Br) oadtzdi.r q! reilq mill ror rlB !c riv(r A .ri rlv na.;q {niil to! B.

1.4.19 loroL .l,1 rorK uiorr[lor( nivazr! .iv(r oilol,qo ioor Ft (.1+ B)r:

( , +-1)r . , , l r +2?rB + ar , ,1 lA + B)+ B\A+ BJ,e l+ B) lB + A) , .41+ Ar i + DA+ 82.

1,4,20 :E olruo),totr6 o0aorofdro!. ro ;. J-oroLxr.io ro! A, Ei!!lr

(.1/r),J = tn,/,bar.

Ed! or ..{, /t .i!or niw*..- n €ni ?Lu( oro r( ororrFi( iolr( €ivoL l. daeiru ro(.1.i1)ir. M€ rov ihro o+L0ol"totro, o raoo€r.(0Lo1a6s !6po< (,lr)t' = A(rC.')

\ - l \ - , , . , . r , 1,, t" . , / t r , , , \z - \ z - - t t - t r I t ' - ^ | z t " r | |

r \ r . , / I \ ; /

Y,To),oyi,rr.ls hio n)..!oi( or(v o (r !tv(' I sri , NA xt\a? .rt =2.

l.:1.2t Yr.iat.r aw. ral(atoc rEiro; v(l boutr€ rov nd;(Jt].aorcof6 rLvu,a0,!. d(v mf].rg€:ri yaq'f6.. E.iv or oii].r! ior .l .iv( .1.. . . .." t.or oL .reqtris ror B ri!(t rc6,u l i ( ' r ruru-yqcr ' t t r : , r . . . . r , . . rorr . r r r r ivar ivac n iv ,s ' ,c z , ' r

- 1 - B - . 1 r t + r r f : + + . u t " ,

() A(i,mE dla naQdber?fc I sri 2 o!ro! rou,t.r!dv( rol).otri\(o(aqroli.d) Elrty'ior. ./(lri 't 6€ird r). q.i 6i6rt nt o,iooilp'i Il r o,/i6/,, .rts ro oorxrio(.{a),;.

l . { . 2 2 o r r i ! u 4 ( n o , , r d t o l v r o , r , n c 6 o r . v v u l

-r ,p, = [ ' ! '9 ' ' i l .' ' t s i t t I cos Pl

c) Etr(i,te€im. nl! ..1(rr)rl(dr) = -.I(dr + 9r) (sro rl: rcwonFEs yLa ro cos{dr + 0:),{cr sin(rr + dr ).6) noro €iva' ro l,L!o|c!o ,'l(r) td ,4(-rl)r

1.4.23 f(r ro( rilca€(

I r l. i l l i l . . B- l : ' j l * , , r -=. . ra lT

- i lt r r l 1 , , - , r L i . i J

IIINAKE> KAI AIIAAOIOH CAUSS

6Q€ile rE 6uvdp€E A'?, A3 ('to\ ei\dt A2 tltL A), ... xat 82,83, ...1taL C2,C3, ...

'.4.A fito^tE\Lx6s cn6 aov ro)..l.dftldoraop6 p€ orrll,Es €tlal o rondrl,aotaol6s p€ |rr)'oa.Ed! oi navdftE 1oQroootv o€ pn].or (!'ionlvdx€s) r{r! i{r ollpod iol]s €nnA6no1jvrov roi.LanLaoroqrd p€ p'rlox, rd,r€ €rrratn€rsr:

t . . .

d) Awtlraraoflioff avid ia c p€ dqr0tro!6 lrar, €r!6€6ar6ot€ dir o rol,)'or)'oo|aorosrard p? o,{ ei,vc! sooros.6) Aciro!€ 6io ol{dtrl naQadeiypora (pe ,) pe ,4 ra eh,or 3 erii 4 xar B !s eivor4 €ni 2. or rdo€ro! 6rqoaroroi to! ,4 na6ri€r va iaedlou, r€ rolJs oatdw|olls6ror{r)qrolro1JE rou B.

I f ' ' li t l ) s E l

t t - l. ')l: ")l '1"

I .5 ' IPTANIKOI I IAPATONTE> KAIF,NAAAATEI |PAMMON

L5 T TPITANIKOI NAPAIONTEI KAI ENAAAATET TPAMMAN

O( €n(\,eE€t(iooul€ rla onalorqrl .fro vcr 6oiF€ nd)< yi!€r(tt I n€QryAoqirM€ na1,o2€<. To ollr€io €zzivrloyl_c llrcv ro ouorrlF(l ,4r = r:

( 1 )l2 " l f , l f i l| 4 -6 0l , l= l -2 1=r,12 7 2 .1 L t | l Le l

^ li i il lil til .

Tpiu {rnv trr 6fuuru a:rc7'orq{E:

l) AqdQ€o€ 2 qroQ6; qr, ;tptltl eliorxr4 ua6 n] beutepq.2) Aq,crip666 -l {pad qv ^adrl rEio('}ot (Ino q! rairl..1) Aqaiecor I q opd 14 6rlrrpl lEiourol 'oro r4r rqirt.

To c;toritrr.opu rjrav evc v6o ztrL or).ororepo orrorrpcr pe vdo nivulrtr owtel.e-orr;1 no! orjlldoLiqoufr€ p€ A:

Aut6; o:rivczcrg t/ eivu rfua qtyav*iE -6Lc icr oloL1€ic x(iro (n fll, r!Q!cbr(ydvro €iv.r( pI6€!.

H b€Er(i nLuu()(i fio! Eivc! 41,( v(o br(ivuopo ri, rrAoirltul,s cn6 ro cpl,robrrivuopc b pe rc i6L( 6li[crra nou o6rllqocrv tov / olov I/. AOC l cna].oL<prj

Srziv4oe pe rc ,,1 zcr i.E$ciaFoo€ r(. l),2) x(( 3) F arrDt q! osrQd.Tel.€i(l)o€ Fr roa f,I xu! ro ..

To tt),trrcio orci6ro eivcr q Luol ror l/r = r pe avdEQoFI crrltlxardr:r(ot. npo<ot|.ytr{! 6€v rloto},olproa $r: aur6. Eorr(i!o![€ ryl' rpooolq pcE oq oI6oI ro]v.4. zcLL U.

O:rivaxa; E nn ro 6rpc 1), o t yrc ro 64pc 2) xal o G 1ro ro 6{pc 3).opioo!7(rv rt6l orl\, nQolyoru€1n nt.a(iyQcrgo. Auroi ovoF(iloEc4 ororya.idsrg:rivax$. t<tr tivc( €i?m),o ltr 6otr€ nd,i 6ol).euow. fLn vrl ar{,cuQio€r rlo!€i<€\'c noLl..1n1,(iro(o ) rIE eEi.o(l)oK j on6 q! €Eioool i. oiter rov cqt6p6 -,\ or4v0iol (i,.i). I.( ro !;r6),orncr zparaer rov rurrorLx6 aivmc. 1rr 1 orn 6Lo./(;uo2ar 0 nalrorl oLiou. TorE o noLlcirLao!(lolr6E mvciro]v €rr€I€i ro 6iFd.

To oftoti),€o[c! tt,rr tprrirr' 6lpritorv eivar GF-4,4 = t/. !4reLcborr orr, o-l roL).arl.(old!€rrlr rptittn pe tov E. petr'r pc rov -F xcr perdt gr rov d.Oft gropoior4te v(l nol.).crieor(iooltr€ ror\ GFt Fcrli rcrL vq 6aotF€ 6v.rv

(2)

ouyz€zprB6vo nivtlxcl nou qeQl'€t tov ,4 mov U (zae6! Enio$ z(t! to 6 olo c)-

n(raol€ino}.roE ra Fl6€\,rz.i. curo yiv€roL ors €.Eri,(:

Aur6 eirru xcl6 cr),Ld 11 nro €v6lorpiQouoq €pd)llol €ildr ctiQl66g rl awioreoql:Ilrir p;ropoqrr vc ;rtipr tlrro rov [/ rioo mo! ,41 llitt pnoQol,pE ra araqiaouyEiq. iiyata t4g uxtthotqfig Gausx

'E.,'d [ovo6rr6 6iua, Lt. ro 6rltro ] ) (l(tQ€ilot euxolc. Avri v rl(lalaioou[t

nQooearolp€ bro qop6g r1v 'ladq yQapFt oq 6€!rcQI. (OI! 6no qoaic fll!bsuftel ye,.FFylor lv nadrl l ) To (xor i t reogcr rou ve xf tvortu zcLt4vulr t iptrqa(r rnv ne6o0eol €tvc(L vu lrpiooupe rtoco orov rcu[ot!)46 nivc(xo:

NINAKE> (AI AI1AAOIOH GAUSS

(4)

E t I . - 1 G - 1 L I A . (s)

* OL nEarooit.ql riraa.< b.v awrdraa{r owtu r6do Elzoiq! Iloaowtriiorllr tol! awdldq ots

nLo oufrlldrL,.i mry .10[6! i(arilrx(to. nilir t(o! q(i!{( 6n' .i!d a@o t( bon!€ rbrn tl!

GiouEtal r.eirtoot.

""-l' i lli llj,l=lir,l'''

l;:illi:rlliiilH lia:rpirl4 (lr,(rQ€i fitv dl,Lt. Ie opo).o{u nLvrix'rv. o avcq trlv(r(s tiv(! oariarpoqog rou irlJ,ou. Ed! o norxer6brF nivuxag tr 61et ro\, cqt0p6 -) otqv0€ort (i.j), 1or€ o ((vriorpog,o5 dleL rov +) firlv i6Lt( 0€oI.* O trivcraE (ut6Eor'!6o),iq€1(rL u€ E 1. Eror E I rni /l rivcL o rcrnotrx6g :rivcxcr;. Arrri eivaLr €Eior'jor (4).

Mr[oQoiB€ v aworQduor4tE xd0t 6r]tr( tqs aralorqrlc xolotFoTrotdw(qrouE E I xcr F I rtrL G '. To teLLx6 rqd6).qpc eivcr vc crvcrLgaooul€ 1l! o]'r1bL(bLxcoicr 6rti lrt<rq r(r vo bolpr roLoE :rircrcrE rlipveL-ror U nioo oio! ,4.Irlper.rirora 6-ft cqod ro 6ipd.3) irav tu r€Aet,i.tio xaNd nt tx'i6aoq drt5 tuv Aoaov U, o ftivct cti Nou, G, neimt va dvar o xQ(i)toi ttou 0d Nrrctpaq€i allravriqsq xn€louron. Alrioreoqol €llgcliLovtor tr€ (t!rt0Eq 8lcircEyl. (iQc to6rnrrpo cruimporpo 6{uc si\r(I I-r rta! ro xslolrdo Eivar .E '.

Mnop€ir€ va cl.rr'rarclorioet€ oIt ozal,ll o(rE rcv CFEA mt 0€ol lou U. Y!(!c 6rire n6g or crwiotpo(po! (1,c1rQouv ra c(Q1rxd 61gcrc.

Td4n cvcryvr:pi(or4re rov ;riv(rx( :rou g€pveL tov U ttioo) olov A. Aurog:rpiaer vc eilaL o E rI rG-r. zrrt am6 eivar ro .tt6i qs tr:rol'ottpi1q. z(0ti)s;.oL o ouv6€ofo! tr€r(Eri rou rl rou le*Lr'riouyr ztr tou {/ rou zartti'ylEq€. O

nivrrxaE aur6q o![6olifrrcr pe I brorr rircr xdrta rpryaux6q u],]'a iTtt xat yLLcter6rzrl L6r6rlr( :ro! qcfverar p6vov prrci r1v rxr(l'eo1 luv no),1.u:r)'coLcoptix:

f r l l I l l . | | | l2 | | | | I | | 2 I | . r b r

L r l l I r lL r r l L I l l l

Bi6(1ro o , circL rirtt,r rpL1(,r'vr"rog, f€ I orlv zupr"tr 6rcy('rro zct 0 o16nrivrr. To nfLoorlpatrmo eivar 6n ta moqeia xdta sn't\ 6&y61ao Eirclarlpr66E ot ,ro .q.nhsotsoli; ) = 2, I nar 1. 'OrGv 7iol,trcnl,(roldlovrolniacilr:(, r{caoaLzd neprpilorpr 6rr brv r:tripler tp6nog vcr borlpe cn' €!0€io!ro (urord),eo[(. Ebd or iri.v(xa< epqavi.lovtoL cxA!6d]E $E ri o0rori) o€[Qd.rroL rirorc ro 1v6per6 rouq vc yropri vcr ypctrtri c4rtoo;q. Ddr o onoJ'.ot (orriqcrrolLvlgorrrrr zitoe noLiar),ctotuotl .\;; -rov (Qt0[6 nou no].Larl(rodq€t rlvob116 lpcrppl ne!\r n1.,, (lrf(rQ(o!x (o16 rI fa(l$[rt i, rcL:rcrpdler dvu plbevrroorl 0(on (i.J)- rift .tuoi oL,tou,a .aoftrn7iE 6ivor| iw ,iptE toroptt tir],; ans|otgtiE. Or Fioxr( roljg orov 7rilcxc /,. o r.rrokrE ViQver. tov U ;rloo otor-'1. aiaftr or o(1)or6E.

LH \TQty@vtlti ittQtyononohot A = LU\ EIl:v 6s\r craclto!\5dr Evstr-tratE yecr4rl)v, o ctOXUr6E tri\rattqg A F ioQ€L va lQrrq€i oov 1lvotr€voA = LU. O nivclxc{g , €ivsr xsro rQrl0rr!1a6E, p€ 1 oq brslti)vlo xotrou: ?roLtranLoolaordq .\;, (norr opilovtut oIry at qlolqrl) xdroJ (lno tl6!(r./d![o. O U eivar o dvor -rQrylrllw6q nivqxqc trou 6[9ct!i!€1st FfidxI brdtjoxrxyl o:rrd,orgrl xor. aQ[v afi6 xlv avd6Qo[n cllirxarqorqor]. Tqliraldrrc ororl€i( xo! €i!q! oL oolyoi.

IIAPAAEITMA I

1.5 TPlfar*IKOI n]\P TONTE: KAt E\A ATE: TPAMNION

o ,, = ll ;] *", -.' 21 .. F e r o v , = rl

IAPAAEIIMA 2 (ro! ta€ririEr(lt e!d'loy{ yQaF$6\').

f13" = l,j

[ r ] 1 l [ r o oJ f r r r l11 = l 2 r = r I 0 l0 t 1 l

L l 2 3 l L ] r r l L l l 0 1 l

6* ;rcpc^1orro:ror.eitcL ar A = LU

TIAPAAEIIMA 3 (ono! o),or. or obnloi zct trol"ldrLoorcor6E sivaL ioot [€ l)

o .4 = 19 ?lt r + l

= LU,

IIINA(E> KA] ANAAOIOH CAIISS

Atro rov,,l rQo-< rov [/ {vol,rar aqse€o€q, an6 xo1, U nAoS ro.\, A.,[Qoo0€o€K.

IAPAAEIIMA 4 (6rrw o Y €iv(r o ra!'rorrzoE xnL o I i6to_< !r rov A)

f r o ol14= l )1 1 o l .

L ) , r )1 r 1 l

Ts 6iFcrrrr an(l,orq,tk eivcr 1) o E nou rrqcoe€i )11 qop6< q ypupgrl I o,roItl TQaFu'l 2. 2) o F nou uqureri ,\,1 gopig 14 y0'rFtui I u:ro rI yUuUUrj 3 rurJ. l , ' ( , nol , , ( t r( lefr )r : qoAis r t yeqrt l ' i 2 ono rI yrxrHrl J. TL' undratrroFarrv'r. o r4rrroflr{oc nr!qru<: o auro ro ncpribrrTp( U = L :rlv (vriorQoq,lHar€!'0u\rol, e(iv o r.icv6vtrq fog eiv.rt oom6E. or" cvriotporlor 0c irpene va pugqiQvou! nfo(, orov ,l

f I t,lurrroiorrroq or"r (F | .,r.rl.,i1,."G o,o, (c | .(rqa,oiorNo. or"! /)) =,1.

f 1 I f 1l ) : r I I t : r iI t I L),,'

irl,i

,l*,f'r 1.,,i r=,r.r l L ) : : l l

H brdr(E1 sivar I orrrorrl 6ore v urorpnryowcrr dl.ll),0nr6pAo€Lq yttulri rcovtrt!dxr')v. :el€trixrr€ 6r[ oL rqe€v06orK b€v rirov arclenieq. L61trr rou :tpooercrr_

A = LU: H n€QirE()oq r €ni ,

. H. aupclovrorotlol A = LU eivut rooo ev6rtlpdpouocr rou crlilrr vc qolllrloo!'lr€ /r€Olooor€po cr rcr, owrjOroq cirololdrl€L on6 ra pd0ipcr( feag[Fx{c AIys6AoE. L6r"(1irttrxr 6rcv awir elLxevrqcirvovtaL otrlv (rpr1p4[rvr1 0rdrpr1o1.flL0uv6v vt. €O€coQei.ro x.t! no),rl 6us,roLl d.).d Eoei.q ibl rI l!(,)Qi.l€r€. Eovto r€letJtuio ll(rodb€ryla 4 o6ly$)€i 6v( 6rllrcr :rcpcldpa. d)ms vs €nrrQ(rn€ionoroo6rinor€ U (vri rla er6Lzrfu rsptmrioecr4 U = ,f, r6t€ .yiv€ror 6uvcr6v( boqr€ n6q 6our!€i€L o zc1,6va< y€vlrd.. Ksronr! nedrr€r vc ''oiF€ ,tdJEIQrtortonor€irar s!'rt I raQ(ryovroroirtoq LII oqv rqtiEl.

O. r<rv6vcrg eiluL 6rt" o taxag L, tqappo( pwog mov LI, o6qyti nioa

ttl frocrrrr4 t ror U l{ r l l yoqF ' i2 rouUl=4.l l LypqFFrt I rou t/ l

(7)

H (1]106€r€I yiv€rar 6qaqp6low4q n 6qyaru t4E atd.oLqi€. >xl|6tErd nXl.lupa6c o6lIjoow rov ,4 mol' U. Oa 6eiloupe 6r zdrolv q! i6La 6oul,s!d rdL oxn!uerotc0rl n).rupd. rEsL€iqovrcc rov L ,q.q or xo, or Oro x).rupic rI< (7, oolyorv

l.i lPlfANlKOl IIAPATONTE' KAI ENr\AA]\|E: |PAMN{aN

oiov i6ro L-. zor r(l 6iBlrr( nor tlo! ndv€ o! atl1o\'€ivar GvrLoraiVtUG. ol,'tdE oIo u o f / . c u Q r c , r v u i o ' c z . r I ( 7 ) | \ ' , n o , " o i .

To rptbro 6rluc uq,urpei to rv6povo 111 e;rf r1v (1.0.0) crro q Edrtpqlpcrppi, ;rQdypc nou €Ed.0if€r ro 111. H b€ttrcp4 r{iqort ortclQ€i roy"v6pcro /31 €:ri rt "1pr44rt 1 crno q 1pc4rp1 3. To tpiro 6tlpo tivcr va(q,arQiooqr€ ro yrvotrrlo ror lrr €ni q rid ./QcrU[{ 2 (rou €i.v(t I 0, L.0) (iro rl TQctrtri 3. H orrptr nou crxol,or.r0rl0rlre eivrA q or,rortl. xar 6toLiTorlrr rot r , rurorrzo : f l vL/vu O rvuorT' f l t r .nTr?o( votroc r 'cc r)rrrpi : f l 1l(i '"o!6(),rioortrc" poli 1rq tov nivcr(i {,/ zard q 6rirpxetc 6Lorrom(;v r(,)v nouaniaor(opri-rv. )ro tt'),oc. orav rLc o , 6X€r $€rcrpsn€ioc .1. otcr crpLorcpir 61rL gei.vcr poro o ti. Auro €rt6€6or6v0t ro y€yoloE6rL xaL or bno n)*up€; rq< (7) xrtrtrlrlyouv vc eivcrt ioes naor ror tr

drerr rdrQcr fiQanut !c €ivc{ iorq.

H rcpqovroroiloq ,4 = /,t' t:ir(L rrioo (rotp(oLotrzrlg otpdoicq, xc(trooo lrQolc. ci.rotr orrq crozrioer: ufio6cu{au€1o! zctt p[c( beurtQl ,r0ooi./1toil,XplortrcroLori[r,3 r:ri 3;rivcne; a].),4( [noQtlix€ vtt dlirti r(;)g or ou]"],oYrof(oir{t,uQtt6loarc{ zar y(r $Eya},rji€QouE rrivcx€E. Mlu rQirrl clrob€Gl q,r(i}Er orov;1 = I[,r lle €r(l,(')l,ri. ov(i./ovrc: x(ioe ?) €ni 4 nQd6].nFu oTllv rrro[€vl Xop]-Ioreprl tciE4 r - 1. To 6r6iio llolJ Etoc)tufti ors EEeQttoott(rt Mtl|Itltlrtxtibei/rer;rtlq I crno).orrprl Etqrlo!6i.(€f dvsv airLo trivczq oe rci0€ 6rila x(rL 1rl(i{)0oLolr( (!r(irv 1(1)! nLlirxo! Ei!c! @rQr6d( A = Ltl. E6tLt 6irorfto €''( (rxollrln(Q(i6cr./$r x(L r({ronr! 6Zrqouls xo'v A = ,LI o€ XQrloll.

IIAPAAEITMA (A = Itl, pt: p16r:vrxri otls rEr,6E Odo€'E)

A1rTo 6€it!€L ndrq €laq rivcrx(q Ie rprLg [n pq6evlz6< 6n^/0r\rioug 6XsL ]raatiyovrt-Epe buo pq pr16tvLx6< 6(ryu)vio!E. To nupciSerl,;rc crt6 :tpoipx*cr rcr6 dva€v6raq,(ao! n066l,n$o 6r.qroArxdx €Erociro€cDv (Ilnpirlpcrrtoq 1.7).

Erc yaoFF'1a6 oumlFo = Auo tlrrylDrtxri .tarorrlFcrc

YirdQler 6!c erbrnqeqov iprlxtrx6 lrlrrpn olerr,nri pe rrlv,'{ = -LU. Au6€i.v(L xhrr n€Qroo6rEpo cn6 prc rml.1 zarayat'qi r0)'/ 6nFdxo/ d'r.lLotq4Excrl 6i , '€r rorE:r i rcrzeq:ror:rpmer l tcr vo otroxr ' IQ(Do€l q Luorl toU, ' lz = 6.nad"/For.. o 1 0c gnopotroe v cworOei, €d'\, lv(Dpiqdp€ rolq -L

"cL t/. Me

braSoxLxrt ana),orqri rraoxopl4rr on6 to b oto r (xcrit qv o:roi.ct livctcr ilptlol

.l' i i ,l [" 1,] [' i ' r]

NNAKE: KA] AIlAAO]OH (;AL]SS

ror ,) xar $€ cvd6Qouq olrlrcrdorcol nlycivoqre on6 to c oro t (]iaxd qvo;roic livetoL lEloq ro! ai). Orav €XouF€ 6a€r roug ;ioeiryo!'t€E. ttrioQoiJF€, rareo €nQ€ri€. vo 6o1jl6vouF€ lDaiq rov ,4.

:E ./Ld)oo.r rrrvcirov, 1 cnclorqrl xtr;ail€! xo ,4r = b 6e 6io rqlyavlxti

npd;ro Lc = b zco F€rd Ux = c. (8)

Aur6 eivnr, r,ooduvopo pElo At =b. nd.^a .aouiorE rn 6EiTEaq EEioooq p€ LyLa va ndptr LUt = Lc, not tila Ar = b. Kde€ rpryovlx6 ouoqp( }.rlvt:ro!yprlyopo. npdrutr rio! ridver oxQL6cbS 1 cncLotrpf, xa! €vo ra),6 riQ6yQop0cboul.euer pe tlv i,6to oetpti:

1. ns.Qd.yorfinoi4oE (afi6 rov ,4 6Q€g rouq tr xar U)2. Aloe lo.!t61o.j5 L xal LI )id[ b 6aeE ro r)

O 6Lsxcualqrdg qur{i}v r{Dv 6lpdtcDv o+taivel 6r[ pltr o].6xllQrl o€ed 6oE[d','rLsriQci)v [roQ€i vcr Tiael crwlz€ff€vo €fi€EEQlsoi(r(. H tlrloQoorivq Arios rt(c.!o-no!€i qv cEio(Doll (8) xql ],tvet ta 6uo tpLlcorr,z<i ouotlilaro, xq06vcr o€ n2/26q[ara. II hlorl yra xdee vld itlu:t)ed b' pnogei ra 6Qe0d Fe n2 xQdEetttrdvo. A1,'ro €ivul nol,u LryotcQo .tfio rc nr/3 0rlucrtcr :tou onrrlrouvrqr yLCl vcrrroQ(yolrorroqoei o A o"ril 6EE!d r[lE!Qd.

nAPAAEIrMA (ro rrQortyorip€vo n(lpo6cryF( I€ 6€Etd nLElad b)

lu t - ; l r t =1

, , - n r + 2 : L t - : i i 3 = l/t:1, - t)

r r + 2 r i r = l

0vol.usror or(r otlort[ffi (1

Ur=c

ff aljrouq roljE €r6rxorg6aiv€L oro 2n. BI€r€r€

{ , r = l

c t + f 2 = 1

-c2+ ca = I- . 3 + c 4 = 1

: l t , t2 =I

.42 ia i =2

rr za =3

z t = 4

"rarvlo€r6€i!" Trt!(r€! ro]I,i,s \ Lc= b l,[vErar l€

'r),{ooE rov nadEsrDl' raft-6la6oyx6 xQ6no xu EL\&r

L c = b rilf'fl

rou6rv i l?=l i ]

(uro (1916(;r: nor o1u6(i'\'€t ,r.lrd rrl! (ncd,oql. Ktrr6nLr r1 tl.r = c l.uvmcrL pe(adbQolll canzcrdfi((ol. ono)( ,r.iarc.

nqmitAqat r H [oatp'i IU rivcrL "crorlpmp4 zcrrir giav draoy4: O Lr iTrrottr oi; ztrrdr rrlzog rrlq zuai(< 6rcy(Dvi(,r.6nou o I iTrr yovdrbq. A!.16 6roro(;)\'€rcrr €izo),tr. Eld7oqr cr6 ror {.r i|el 6q,6vo rh,ara D cnor€}.ortI€lorr:io 1or,\ oltyoi\ dr. .t. . . . ./,,.

I . 5 1 I ) I |ON IKOI NAPATONTE: KA I E : ! , \AA I ' I ] I I ' I ' AMMON

I r r 1 1 l d 1 t 1 r l r l 1 . 1' ""10' ,l

(e).'=l'',,]:ro rrl ,r(io rra0(!6trLytr.r oIoL or obrtyoi rir(a r/i = l. : (urlr rI1 r€airrroolo./J oi!(t o tcluloltz6: ;rivulaq xur I n(Lxnloc iroQ(tllplorl brv iXtr o41uoitr.A1lTo ol((0< ilr(r! notu frbr)i6 x(r ol'vriH')< o lDl,r efvut br(1q'opt.TrxoE (ljro ro!L L i .

H rqtyr,alnxil tapayonottoi4al yprirytar atyd ,t = LDLI, 6,tot Lxat Lt *our povti6r5 atq Snythwo xat o D dtat o 6nyhuoE triraxaEfl,'y o6ttyt;'v.

Yn(iQXer ll drrvlitJ€K. (v x(r nQord,ri ptlri),r1 oir11uo1. 1'( ou!6o),(!€1u[ c!ro<o'\io< datJ rpryrovLx6c :rirarcq pr to i6Lo .lphppcr U. 'Orcv 6),r:rers ,l-DUu:rovotirrr rirL o I/ d1tr povribeq or1 br(Iivro yt ritrl,( tro1Lc. ort eiroe ypcrpptla7€( 6L(t|Q€0et pc 1('! oblyo x(r rolr l(rrclrlQrEop(ore rouE I rcL fI pc rov ibrorLxiiio. E!( rr(QdbEL.4re Ei,!c! ro

. l r 2 l r l f l : l l r ' r ' lr Li;l Li ll _.1-Li ll _,ll i] '-,,'

,4l,16 ro yLv6[€vo 6X€r Fo!l(bs: orrl b!(ll,drrc r(Dv r r(L U, rcr rolE obl./oriq IzcL 2 oror D.

Ilapatilptlorl 2 n€QrlQd(ovr(; ru 6rilatcr rq< (n(Ao(0]a. [no0.i v.r bdoqrerrl\'€\'rijr('orl 6rL Srv L"rdQl€r rLs!0rei(( olo v{l yivouv or u;ro).o1Lo1toi pe6((to0€rrzri 6r.iraE t- Aur6 €il(L ).('10o<. YrAatxr zAlo& €),ruo€Qic, x(lLu:rcipXer zaL ivtlc ai.l6er0FoS rou Cmuf nor' 6r€u0rref rou; u:roloyro|oi,; privcv ri..l(ppdg 6Lnrpop€rrz6 ra6ro. :i./ouQ( o[o< 6rv ur.lpx€r ;rh]artq €i.€x6caia.rulou :tptiErL< gcrcrln ypcpudv oe prc rrry(ic 6Ldr(Et, €uroi.a ocr ardoresq(llo 6va 6{pc rcr l16evLzh:rol :rcrp1101oav oe xhnoLo :rpolyoupevo drlgrcr. E:rio1<.brv u;rlrpler €L€u0€aic mc rctrll.n L.D xcL U. To:rro crEroolleiorro eivcl ro€tI!:

-12

n EiLv A = LtDtt\ xar A = L7D2U2,6r.o\i or , €ivol rrdlo lQrYrovllcoi, F€

FovdoES orrl fua16vm, o[ U eivot dvo rQry(D1,l,Loi Ue Fovdo€E orl6|,ay6vlor.!o! or D €ivdl bLcytbvrol ntvaxeE p€ pl Fi6€vrt{d otolxeis qrr] 6ldY6vto,

x6xe Lt = L2, Dr = Dz, t\ = U2. H ,roQrrtovrorolrpl ,DU ltoOdls xd! 1,U oQilovr(|t FovooilpqYrq c!n6 iov A.

H ffi66ettl eivct pLe xcL{ irorlol yra avliorQoqolq trivoxe5, nou €E€to'

Eo{c.r orqv €trop€q ncrQdYQ(9o

EvaLl,oytg TQaFF6v 'rar nirat sg Fsrooao€(ov

nQ6tr0r ro)OC vo awL[€ft)]tioorlp€ ro nQ66lrlpa t[ou tliXa! tdQa aroqet,Yap€r

o crpL0p6g nor tt€Q+dvaBs v(l lqrp4rorotrlooupt: o(tv 06l"/6, [noQ€i !a €ivttl

lNiv. Awo ga pnopouo€ va trQo'|uli€L oro p6oov rou utrorlolLolou i ax6Fn xal

m1v trpXl (oqv:tepimool nou drr =0).'Evct anl6 1(rQ'i6€!'fF( ril'(r| ro

H 6uoro),icr eivar guveQi. Kd!6vo fiot'lonldoto t$ nQdlrYlq YQc+prl( 6ev [/roQ€i1cl .E(),€lV€r rov ouvi€)'€oni 3.

H 0€Qor€t(. eivc! €E ioou qol,tQli. Eral tiErE 1$ 6{0 EErodtocti xctr ilor

mtyciv*e to 3 nirlo. orrl 06oq lou o6lYoi. t'owrlv llv ar).rl tt€Qatri(|)ol o

nivcxrrc ftr yrv6tcv apiotog civor rElurvr,x6g rc! ro orLor1go

3u + 4t' = b:

Z r = b l

ec1 lw6tov c.pdoo)E P€ avdoQopl altulotdoioon.fLcl vc drcrrunt,rooug€ cruq qv rpaEq onl Y)'drooc rr'rv :rwr'rxorv rpdncL ltr

6eouFe rov Ilil,(|ti(r F€idoeoqg nou rpoln'ei tqv evclLali 1pr4qtlv. Aw6E

ei,vcr o f^n=t l l' -Lr 0 l 'r&r 61roE, o no),LotrlaotnoF6g ge rov P evc).ldooe! rLE YQa[p6c:

4l2l '

O P iler rlv i6r,cr eniapaol zor. oro li, €lalldooot'tq xo a)r x(rl ,]. To v6o

ouorara ei.v.lr P,4r = P6. Or dTvootor u xa! o 6ev percr6tiL)o1'[gt r(otd rlv

€vc).Lcyri yQ{rp[6v,

NNAK!: KAI AJI,\AOI@H OAUSS

lslllrl=li:l

,,=lt ll ll il =[;

'Errr lrq6trtz6 or 06o1 o6ryol qtiptr 6to 6r'vn6r4req: H 6uoxd.ia FnoaEi vl

f€p..t€riFrdf cttzoAa, i pnoqei ro tirat oo6aei. Auro crroqcroiietcr rottrirlo-vrrL< xdrot az rc y4btt,rzri. Eciv u:rripXtr" ivr gr1 p4bevLz6 otoLXrio tr:ro xtirt'rtr(l( olrlv i6rtr orl),1. 16r€ cxlc)"orille pLu rvcl./,u1tl lpallrirv. To 1u] ylbtrrxorrrotXrio livraL o crvapcrioE obll6q tr(( I u;ral,otrprl propet !o orJle1r 0r'i.i':rlv rFQirrr0nrll pe< oLu €Eo$6vr(tL (1n6 ro! aoloFo /. Edv (? = 0. ro nQ66Lru.rr,i.vcr. (o0Q.i7rsrrro xor o niv(l'ioE r,iv(r t.t.dpoQqog.'* Aev lJndQ1€r [),rrib( 1.1tlov(lbLxrl ),no1. E(iv ro ./ 6€r rivcr trlb6v, ttLcl svallcyi ro)v lpotrtt{l! 1 nar 3 0opercxr"vtloet to d oq 0dor1obryor rcL ro nqdro 6rl1cr eiv(r n),r1ptg.:rt brnrtprt0io1 obryorr rv rornoq u;rrip1.r n<iLr lvrr p16tvLzo. O aptep6g a tfvuL rriqxrldft, (1n (ur6 (ro ( ano ncrvo eiv.lL .i1Qloro), x(rr €irr ro (1 b€v etv(r Fl6[1UQrrriior(lL trL<r szotrq ercdJ,a?i yA(trtr(;v. H na(;)rrt tr€rho€ol Prr znL q brirepl'rt'rri lrr(i0€ol 4i trQo?unlouv (.,'r' rou! ria(}r€q

Ilporrrphpe rrirpo oe !rcr irro brloltol,I nepintnxnl. Yno0torr 6rr o A eiv.r!ni'v(7.(r! 3 Eni 3 qE FoAqltS

' { I P I T O \ I K O I f i \ P q T O \ I I : I A I I \ A \ \ \ T E : T P \ \ I \ I O \

i;l

* hf i!{lirt ztrorrE rret.iqri to(tq(;\ ,(r otct o (uradq oiryi,E €ir(r o/..dd, rupi\("(itrt z(r nr!! b rirdr caEddr lrlhi\. Anirryovi(( |.tdirfro o6ry6 ftaropiiorrr€ ro drtit]4(r

** !.r,E,r H '6!iqov. A' z(r .Iup(€i o b!!r!oo< ooo<, .rd .tit!r".i.a.iitrr o na(iroq i( !(

u4rNvilmar !. tu! tja. \inculanry(=rior(oqri() rn; Arniroti.

lrlil

fo o r l [ rP| l=10 I 0 l x { { & :=10

l1 0 0 l L0

^."'lii:llt:il[:ii]"

AlJroi tjQiuzovr(r tr'6v.rtDaaio rdxv.lo$rl: tr(foaF6oTg Tlt lr!rchO€oyloTov r(lllrorLxonlvcz(l, fL( :rtrpir6eL.lpc P|qI =Pl npdret vc eircL o rauorrx6g nivcxcg pe€vrll.L(ytrtvrq rG ya([F6q 1 xcL 3.

K(! x(ifl (1{6$l oErooll€iorro: YndQX€L alrcE ni.v(rzc! p€roodo€0rq nor] rxrs-),ri. r(r rrq bio €v(),}"(yiE 6Lri Frcq, ncr fircrL ro ltlopelo ror 6Uo peporropivu'v

Frr(0aorcov (onox o Prr bp ra(i)roEl):

Eriv iipcpt rr vcr zirvo4rr, 0o $roAorro(.p€ vc :rolJ'cr),croLdroo4re rov ;rivataFcg $o rov P a:r' r1v cp1{. T6re 1 rcrtr},ou1,ti pr rov P,4 6ev Ocr ouvavtouoe

NIN , \ I I ! l \A I ATA \O IOH G? \L ISS

br,oxo),i€!.2(r ro 16!0 z(zo u( tlr(r 4 upTrzrj rrl rl bLdnctEl ft)! tEtooroEUn'.M€ r(. 10([[6! oT't or,)oT'i 6dr(ql.

€iv(r rtfttt rQry(ruzo:.

' or (/.rna,nri: !r.ri iiu on o lJ irr r'iL-rrrf r(( n{ar rrr(:r ror Z zor ror t,I, z'r ',r

: r+ (e i . r ' r ) , r = l . n i to rndr x i rd ro i | rs / , i

{lll'li) yt-yovi)( 6fl siv(r rprltDlrzo: f-i!{( ortrnr!4r(flro. :rrytlo('); c,-rotrivtL v( Yia€Irt (n(io(qit. To c:roor k-i(Jro 6r(')( fiv(r roirro: Ehv l cljlcr),orq11nopri rnor,vrriro0ri $s rt 6oi0rrtr ev(),Lr.yr;r'v lpt4qtdrr. ror. llnoaoitr€ r(( q (\'rco0o+tt'(ir! (1j1a< or €vciL(ry6q yilolrcr {trrll (QIi] rrl< 6r(6rxcoi(lq. Atro n(a(ilrL ro'r/'.1 rrrr o rr, lciyr :tiutxu; ieL ypt uritrru t n))t4,i< ypayluit'. H (n(l,o(0iborLfirfr tr! oior)"itpEE ya(FFiq, ot onoifq nltoa ri!(t oF tu( ij'(1l(E'l rot,nrl1xi1.L g1 prlbevLzti ororlfi( orrq udoflc rr)\'obly(;)l. Me rii)'u ltilttr. o P-'1r:rrrpr!:rr.r rq owrl0 r o! ivrl :rr rp(l1ovro:roi1orl or' /- fni tr. H 0r0)0i( rltq (1ntrlon{IcOr!ss FrToafi r( or,1'orl,ro0fi ( ): t:ri(:

1.f lrrl 1r4 6rl4neq\ :tteixronrl, urcipxrr fvu< nil,czoi |rro06o€0)q Prol, (1cr6tcr(ioos! llE yeq4ri-i rou I (;)orr v(l uno'trqeoi.\, tllb€llx(i ofl!udo&a r0w obrly(;)v. :' crtrl trl! ;r€0itrr(r)ol

(l ) I -.lr' = , €ter Fr(1 [ov(brzri ),uort

(2) (r'ni 6eior€r{lr pr a:r(),ot{t z(r rv(r,},cyri yea$t6v

(3) cvcbL(1r(io(x)1,rq EE (Qxq< lL! yaqrtrfq, Inoeoitu' !c

ncQaYolT onolioorJl(rl to! P,.l oll Foa.{,rl ,u.

ltry bbpoqq4 tttgirturrrrT. xtrppitr cvcrbLhrcil 6tv p;ropei vc acrpirltr 6va:r\prq onvolo obtltij!.

)nrErd)orf orL i r t l - . t i , t txr t< t t l t t ( 'oro< P 4tt uq i [Lt : yaamiq tu n'rre,rofuxtj. tE zhnor( rbrxri bLrirrr\rt. flpriltLcrt. P = 1 eivtu o c'riororfpo(x(lr:rro cr|rq0ropivo; rtv(Ic. (bfv (i),(iiEl ri;rort). To tt,'optro bi'o rrvi'{t)'\

trt-r(0(oft,)c €[!(L trt.r vi( ]rfr(i(ltorl. A|Ioi bt! t'il(L (f i1"/2t !c trtrcriotlrcr:4. ' 1r P. ro; . ; . , ( : r ; .uor, r . r ; ,^r r nr o|rr , , i , r rr i , t rL,o,tr t f2frut: ru, i l ! 'nrt7. fui ! , ' trrlqlSFit,lrcrq. 0c i6r\(rv i\'(\' b(q oafrlzo P.

d{'drri'Orroq I]oi;rrL v( eiorr:rqooFzrrzoi trtro\' /-. Y:rooior€ orL r1 u:rtdor<1tl(gcllqei t4 1g(ttttrt 1 a:ro rtt lp(trtrt 1.:r(ptiTovru: lr = l. Kar6nrl txo{)aoltor r !u, . / . , ,dLr I rc 7,. t , , r t r i : I l rur 1. E,( ! ror"nl 4 r f r ; r . , ,7r1 7ir ' , orrN ' r1t7t1. , ,;ro).i,cmLl:lorcorrE 0.1 lrrc6),t0t-i o! l.r = I orrtv I'r1 = Zt'.:

15 TPlrANl(OI I1A?]\fO|*TE>: KAI ENAr\,^,\rE): l'I'AMIVAN

11 r 1 l f1 1 r l 11 1 l l,APAAETTMA ,,=

]; I ;] _

13 : :] _

13 ; t] ="

Me t:ruL),c1pive-< -r( yqc$$€E 6aiorout€ rov ,U rcbpc 6pru,< )r1 = 1 xal .l:r = 2r

Ioror 0a iildl o' ontloi.: MIno< @('lrirar .!d').(./ri llnt{t,i,vi

r.5.:l no].Icni( rddt. ior rirurc l = -D r l' r (: onl! feidoort (6) frt 1o! CFt orlv,,iid.rot (3):

| 1 0 0 ll 2 10 t r iL I l r l

1 0- 2 1

1 1

t1i1n1litoutlE: E\u v,,Jl,o ;rp6lpcrpptr (rcr)'oL<p{g Gauss n<rtclpciqer ror I, rov U)i(r ro! P. Or fill(r€< alrroi TrEQt6Xola qv n)'lpoq,oqic 7ro1j aoIlxi( nt-Q[€iXeo ,4 x.rl lld!),[orc o€ rrro €uxQqori] iroptprl Erfoq< t:flrp6,1ouv ril ]uorl rotr.4.r: = 6 pi:oru bro rpryawrrtiv olorl$(ir(or ,1clt !:ivc! ro r0art!)to ctv(il'oYo r{DvunololLopt,-'r aor-r xitvolr[ts rl(Q(z('lror y& 1,4 6aotfu nr at'rioraoEo niwlx(xat I ' ) ^ t o t l t = A o .

A'KHEEI,

1,5.1 noru il(l9 (i!o lartolr,io< ii!(a.$ €ivor tr! dotroo(toql

1.5.2 oo() noii(n].i(sro rlls laotrfris 2 0c q,aLqioFr 1 cn(r.o(ill uio q lAaBBri 3r

f r r ) 0 l f r 0 0 lP lu o t l z ,u r 2 I n , tu r P4 I r .

| {_r I 0l L1 0 r.l

il

13 I r l l r I l J/ ( . t = I I l l 2 & t - 1 = I r l l

L ] r l l L l I 8 l

f2 r r l f u l f2 l-'=l::lll;l=L3l

I r r ) o l l r 7 r ] l'=li i l'lLi i ll

I]oii(r).dorim. rriolq auL d€ amimqoq'll n tL at intrrrlaEL< th,at

1.s.4 Eq(atom. md).otqritto !(1 n.ra4.ft ro!-q rusilolff( I aar l,'1( ror\

, f 2 t l" =

l s 7 l

r.s.s loeqowotrorlor. tov -,1 ot l-ll, xoL ]eii1l{t ro du,) rar ,ryta6 ororlF( arr =.

nou trrq(rai!.1o1ttEi.i t1t! crd,o(f11, YLo 10

TIINAI(E: KAI IIAAOIOH GAUSS

15.6 Balit€ rous E? xa! En zor E-r,6rav

E=l l 9 l- L 6 l j

1.5.7 Bqeiru rs yrvop€va acd rot fiGF dra! (rdqalsiro\,rcs 1( ?i'ivo tayyollxdprtoe!)

f r ' l

f r I 11 I. 12 r | ^_10r | ._10 I I'= luor l "=lnzr l " - loor I

Lr , o o l . l [o o o r l L0 0 2 l . l

r3.8 ( .w€Qn anobrGr r{s ,4 = rU) H relrr yaqlll'l rou U rQoialffct and Tltv rei:lrllAapti iol, .,1 cq{ edrvras no},lon}'ciota tov yeapp6a 1 xar 2 (ro! Ul):

yaqqlt 3 rol I/ = r0aru{ 3 rou / -.\i1 (yedt4i 1to! LD-)1? (]a.Ipttl2 to! U)

(, rL(ld oq,dreonwar ol.yeqllpts ro$ U tft O(r ot to.44rds tor Ar Atuivqot:dr6rL &ov 6x€r i6yl xatouonoLN,i F (l obryog yadluli ...

(6) H treo\yolttEv'l €Eiorrol ciror ibLa fr rrtv

laqqr{ 3 roo ,4 =)lr (yqouri I rou U)+}r: (yqaFsrl2 rou U) + l(yodFFli 3 torJ U).

noq xcldlas ndlLqnLsoroqroi rL\,ritov (a''d to lD) xdlEr d!r6 dxat66s i6o trr14lpcpgrj 3 tor I eai to! Urncaoto|(l x(rr or dl).€g yeup$6< ror -LU otpnlttou, p' (l!'rds tolJ A.

r.s.9 a) Yno ro(rs ouv0ir€s €ilar o .4 u L6dloa.pos. <itcrv o / eivat ro yLvdptvo

I r 0 0 l f l r l f r r 0 l^= l - r ro l l d : l l0 r r l r

L o I I IL dr l Lo o I l

6) Auorr ro onoqFd .4' = b E€xra(i,ras ar6 to t. = 6:

l | 0 ( r l fc i l fo ll - l I 0 l c r l= 0 l= r 'Ln I r l 1 ,1 .1 L r . l

rs.r0 d) rrdti tQerdlo!'rct rrpiror nrl2 6ripma nolratrtaoKorrorr-oediasolg yrd vdl.$ei .,ifled'o an6 r(! ,c = b xt Uc=ct.

6) n6o.r 6riparo XaryUo,ror€i rt snol,orqi lLa va Ilo€L l0 ouotrltffa p€ lov i6Lo60 €J 60 niva-ro or,\.r€l€md,u

f r 0 0 l f2 a a l fu l f2 lr 5 . l r A i o l € t r ! l t r 0 l l 0 | 2 | | , l = | t r I n "e i c vu noL lan loo ,don . t ouc

L I 0 r l L0 0 l l L r l L2 l,a/ yn !(l 6A€itf rov ,4.

I,5 TPTONIKOI NAPATONTE: KAI ENAAAATE! TPAMMAN

r-!i.u ndE oa l'oQoiodle ra traQapvroronlost€ tov .4 s' sva Yrv6Fvo tt-L, til{oiery{nurc$ r.ar dflo reryo'flot Oa €tI€ tor'Io touE ldtorE riagiYori€t F€ tovA = L U ;

1.5.13 Ator€ lre snoloM, x&oma! ev ovdy,rt e1/oi.loyfg 'yaolltul

u + 4 u + h t t = - 2 o + w = o-2r-ae+3u=32 rut 1,+u =0

u + t - I u + + w - 1

noro! riLvorrE! pelogdo€os drloltoimcli

15.14 Karoyed{ne 6}'oq iol8 aEi 3 €tit 3 r vdr€< pftageoeog, 1o! P = I ouptieQ ql-6avop&voo. ne,oo6roatolr rol.s awLor06eo1jg rous. o! otto{ot etaar €rriols ,tltcx€g$etaoto.os -u.dvonororiry rav PP-t = f t dt ttt{ts)(ovKt otov t&o |.(rdloYo

lJ.U; BQ€Ite fl! naQayowororio€E P,4 = tDU (*ar €taqeeiote rd y',o

r0 1 1 l,4=11 011

t2 3 4)

f l 2 1 lrc l A= z 4 21 .

t1 1 1 .1

f516 Baein sru" prl 6d$oQqo 4 €ri 4 ,rt1,d,(o, trou lQ€uiEetar tQ.5 el,o)..ltry& flq+r$6vytd !a @Q€r €16 rdQsg rlv arqLqi. El 51^u16vr 05 €lvd| to tlqQd6€lyFa evqgnlw|lag peraodosq.

13.17 :q 6droEr ,PU, rorr ripoqrortv o! d\'yE68roidg,1 atd)'oqi svdAdoo€r 'fcappdg

ldvov oio 1dlosl

f l I l lA= l l 1 3 l -

1258J

f l l l l [ 1 0 0 ] [ l I l ll 0 0 2 l=PU= l0 0 l l l 0 3 61 .Lo 3 5 l to 1o . l [o o 2 .1

f r20 . ]A= la 8 3 l

Lob5 l

no|og €ivor o I o' armiv tlfv fi€Q{rw(oo$ Avriosla pt nV PA = LU rd,r londed6€r1pa pgld to 1J, o! no ,ant'oqiaotq 6eioxovrd| oqv oCoI ro!5 (o )2i €tvar1 xor o )31 €t10I 2).

1J.lt Anoqaoiore dv ta €Jt4r€vd oootrlpotd ei.,at l6Ld$oa{a i 6rr xat €d' *o1l1| Fto,*q{rla ri d!r€e€{ }'i|o!r€:

u - b = z r - u = O r + u = 7

=2 1tdt, u-v =0 , !ar u+t) =1

1 t - u - 2 u - t = 0 u + u = l

15,19 nor€E tLFds tuv 4,6, c o6ryoi'v oe Evulaay& laaHrdv xot notlE tirivow iolls nlvot €st6Lfuoeqoq;

."' o=[z 1].

NINAKE: X I ATIAAOIOHG USS

1.6 I ANTDTPOOOI KAI ANAETPOOOI

O (vtiorpoqog ev6_< n eni n niloxa €i.var 6vag dl,),og n uti tt nivoriog. Eo,o nQd)roq ni!(t?.(rq €i.vc! o J4. o avriorQop6s iou yQde€rfl -4 I (ruL i.€y*cu ",4(nri('xaoqos ). H 0€tr€Lr66qE r6L6qro €ivar !nl,i. Etit':to ..r .aoftont ttxo! A xat xatd:t r ttt rov A t, qeioxEot nia@, Ex€i nov E*tiiocttE.

t a v , 4 r = | ' r 6 r e , 4 r l ; = t .

Apc ,4-1,,1.r = .r. O ;ri.voz(rE ,4 €rli rov fiivqlia ,4-l rivcr o rcruroru6E :rivtrxcE.A€v d1ow 6F!rS 6),oL or niaori€6 rrvriorQoqo. To lp66l,4ro €Fqcryi(€roL 6xovro ,4.r €i,vor pq6iv d,ld( ro r 6ev €ivs! p16dv. O crwimpogoE 0a 6rrQ€r[€vc nqlcirrr, nionr, cn6 ro .4.r: oto ;r;. Kcrv6vcg fiivaxo( 6$oq 6€v FnoQ€i,!a nol,),arrLoodo€r ro pr16errx6 brrivuopc xar vc :raptiyer 6va 91 prl6evrxobtdr, uqra o cw4v v1v n€Oinrtool o ,4 r 6w e( r;:r(iQ1€t. Or or61oL gcg eivurv(t oQioouF€ lov olriorQoqo. !o Tov unol,oyaooqr€ xft 1(r lov XQrprForolriooule.orrs nrQmrdo€q r!o! rndQx€r -r{.rr6rrv 6€ v(r xcrn}dr6ooF€ noLoL nlv(rrigs q(ow

lX O nh,azsg ,4 eircu <rwrmqilqro6 €d'\, unaQD(€! 6vaE rivdr(dE .B ero!titott BA = I xa! AB = f. Ynriplel ro roh, ,vaE rEroros B, },ey€ro[ovrifiQoqoE iou .4 roL oup6oli!€ial p€ ,4-r.

A r A = I x q l A A - t = L o)

2tt1t€itnol I AEv FnoQ€i vc urldQxo]Jv 6uo brsqoQcrlr(oi avriorQoqo!. 6!0r! €(iv8.4 = I xru" AC = I, tdre i1olLt

B = B(ACI =(BAIC =C.

Eiw unhqler ivcg crqrorepoq-cvriorpoqoq rq! dvcs 6ell69-trvriorqo{poq (o Ilno),l.cn),cord!€r u:r6 r' aprmeph xcu" o C an6 tc beErci) r6re rlrrroi eivcu ot

tryteiotttl20 ctmiaqorpoq rou ,4 I eirm o i6Log o ,4 zcL rxqvonoLsi rlv (1)!

z\ltEitt orl 3'E\sa n[!ux(q I erri 1 eiva! avr|orQdl]ltrog 6xqv 6ev €ivar Frl66!:eirv,{=Ial x6:€ A-t =l l la l . O cvt i ,orqoqoE E\6<2ff iL2 r ivolro $rroQ€i vslt(rr((yQoqr€i (l[€o{DE xor t€l,rrd (€q' 6oov I dd - ,( 6€v plSsviqero!):

f " a l ' 1 l , r b ll . d l =a,r

i "L . " l

6 A\TI:TIO'I OI (AI ANA:TrOir'OI

t\p€iooq 4 Elc< 6rq(')\r!oq ni\,clll(l( Eilcr. c11.{rrQirljL[og 6rcv nc!61( 6r(r],d)lro ororX€io rox 6E! !:ir(r( U466',:

l' , I r6re '4 = l"'' ,r,,]

f r r ' lz r y t oo t t so : , : r 2 i ! ' , x , , : . 1 l ; ; r r v r r ru r ' , v r ro ru t r , , r roc . o r u r l l . r c

t , ,tor B uri ,1 eircL oi.1or.pc i6r€.E r(iL 6sv trropci vc civcL or.6ro nrrjl.eq rou 1,nol, r'il(r 6n(foQ{rru{6!. (Oc 6rirn d}.ior)< l{.r xa),tr€aoq Loyonq "/r(ri (lrlloq o]1 5€! ri.v(lr avrLorp6vrtroq).

Orsv €X0WE vft i.arvorF€ l€ buo nivcrz€C, lEv trnoao1)lre vc nouB€ rtnot€' ' r , r rov uvr iorpoq o rou J - B. To t i0pororrr t pruqtr r , r t .v ' r . r t z ' u ru U nv r i rur(arorpa1(po. crvelriprlra cr;r6 rlv trvrrorQerlrL+r6rryo -rt,rv pepovcrg6vt'rv ,4 r<lt. An6 r'lv rii),q. o avrirxporpo; ror yrvopdvor,,11J c:to.rrlei rov r no rilcrbiorou: lxo).oyrolroi( xr!iln!)v. fru ourqurolla'\,o!q cpreuottE rl ,r(ra(orool ut!(rl ibra: ro(f l+/J) Ia;r lo:roLcir(1r bioroi f t . r ! ( i ) -rC) 1/.7, bL(ron(i1({ oc l /o €r i i| / . . f r , r n r \ { a " r c , , , r , , : . r 1 r , t E r 1 r , , u n , n . L , , : r } . , r " . , 1 1 , r o u n C r r r r v , , l ! ' o a ' i n r l f , ( v, lB.r =u.drc lJr = A t l t x<Lr. t = lJ tA t l t , f iou 6etz!0t f i ( , )E ot (rwaoloo(fo!r11<rvi.lowcL or: crriotporll bxhru!1.

lL To ),[\,opevo ,4! nvtLotperpipcov nLvirio! 6x€[ (vti.orgoqo. Au6E6Qioli€rar Trol,tranLqordBolxcrg ro]Jq qvri.o,Qoqous nivax€S 06: qvriorQoqrlbrdrc1EI:

@41 , -u 'n ' (2)

Ar{its'E4 frc va b€iEorpe 6rr o B ,{ eivcr o crvriorpoq,o< ror ,4 B, rou :rol-i(nL(or(ilortrr: Zaloqtoro|il!i(,( rov ntx]o|-r(1tQl(xLno !o[o y|lr \r' c(firlQaoorJttp

( -1 r ) ( , r - 1 t )=Auu \A t -A1A t -AA t - I

tB t . 4 t ) i !B )=B) .1 tAB=B \ rB=B tB = r .

n' ,C^tr . , . - ] , , iv^v, ic , .X,1 ' y , i t , , rnHC 4 r f0rnnorfOoui r iv ' ,z fc :

6BC) | =C tB )A t .

TIv cllrrorpoqi qq 6rrircrh_( q ollcrvrto,atre rpollorgivroE, orcv crvi'ora€lJau€rorg oroLlsr0;6e6 iivc"eg ,4.I, G iror obrlouv m1v uncdoL<pl. cr;r6 .rov I/ :ri,or,;

NINAXF : | .A I AT IA , \O I4 'H UA I ] !S

orov A. Iaox(oadrvra< {qr cncloryyl o C-FEA ilrav o U. :rI! (vriorQoqlrcr€rouvol. L = E tF tG t rlrcv to yrv6prvo-rorv crvrrorprigov. Errrbrl o Grq |1!'trolor(\ rr )rrrrrrioc. o r; | fl ,,r(!itr rar nAUTuc.

Mlu r-Eio(,)olt :rrvdzriv 6'rt'rc n GFEA = U FnoQti ,/(1 ro).l(ljr)..rorao0€irord [d].t rrr rov G r (cr:t ro uprmqrd) { u€ rov '4 I kln ra 6€ELci) i lt€tov U r (zr ffi rl"q 6uo peprtg). Edr Xe€Lcr(6paor(v rov ,'l r rrlltoq 0a rpcr!I/ rGFE. ncrpcxoirir vo ro €nd,tOlrloer€ 6L6rt erxo)'c livovtcrt iirOr1.

O uroi,olop6g tou ,4 |

Oroplorr -qv FEi,oo)orl ;1A r = I Edv i)€trlpt0€i rrril4 naoE ani14.aurrl 1 rltocrl nQoo6roQiqr:r rLq or{Ieq rou A I. H naortl olrt),q tcttr / L

noJ,),cn),oord!r:i(lr [€ ror A 1l(l va 6riroer qv nAdrq oIn),q 10! l(lrxorrltoli:-.tr'1 = r',. n(rnoUn,,, 1"1 = c1 xcr ,4.rr = c:r. 6:rou tc ri uiv(I or or{}!:q lou .1. :'ivc 3 c:ri 3 ncpri6etlpcr. ,,{ €nt A-r tiv(!

'EroL dtotJtr€ 10k1 ouolnpcrc EErodoecov (rl /r ouorriltara) fiou dlou,olcl rov ibtorircx(l orrteleord'v ,{. OL 6rEr6< :r}.trrpi:q €ivcr 6r,rrq,oQet[x6E ctrr,cl gnoQor:le rqrzrr).loo4rt u:rcr),orr1,rj ot 6,c n oturlyura ruurolqdlrD<. Alrrt I 6L(6rxcoic,:rou liyr.rnr ti0o6oE txtr Gauss-Jordan. crvti 'vcr oropcrnjoeL olo! a/ rot ,'(cp7ttrr! rrt\r (,'ri6Qol1 altrrcrhotcol, ouvtxi!tL rrrlcrQ(;)wcE nollcn)..iotct [ra6yp.4rtL{( rrn' tq tln6 rcivor yp(frtr6(. EroL noQ.il€L lqbcvLxil 16l]0 zdrcD cr' rtbruT, , \ .o ooo

To :rcrpcibrrlgc u:rrour?ni€L xrE rQcLE orritr€q er,.r. cr xcr bol),erEr trE yOattd;ptjxo!,< i!r:

IIAPAAEIIMA

|2]r ' l [ - l

t I l rool| 4 -6 0 l r , . r : , r . = ( ' ( r c i l= 0101 . (3 )L-z 7 2) l I t . l Lo o 1l

| 2 r 1 r o o l6 c . l = l , l 6 0 0 1 0'

lz 72oor;[-a "'

f2 1 I r o o l-10 8 2 2 l0 l

L0 8 3 l 0 l l

12 1 1 1 o o l-10 8 2 2 r 0 l= lu

L0 0 1 -1 I t lL ' ) .

6 ANTI I l lL ) l ,o l 1 \A l A- \A :TPOOOI 51

Aur6 o4rnL4prirver rI lLaSoXLXrt crd.oL{)i. Or rQ€r< nQo)rr( otlirg ror rr}.er-r(1iou ni,v(xu bivou -rov irvt,) rory!)vrlo U- OL ro€L! ai),I,€q ornL€<. nor €ivr{lot 6€Er€E ,r)"n,a(( rraorito4ruop6vrq 1cr avdSpopt rrvrrrrrrriorcot, riv(r ot i6regr. f f l : ortL( ro, I (TL'uo r ivur ro crorr i rr olru rpq,1qro?t5 ru! orr , llrrcob6v npdlaarv orol r<ruorLr4ci ni.vaxa L = GFE.) To naoro $roo rtEcrrcl),orqltq nriyr ano ror,4 oror aI xoL r(Da( ro ir ,o pr"oo ecr rrier" nr6 rov I/orov 1. @rrcilvovrcq ptbrrrx(i ririyo uno rorE o6ryor.r< orov rrl"eu(rio ni.v(lr(.qrrdvoL,tt€ oro! A r:

[ . r r r ] - 10 280 40 I - l

0 0 l2l8I t 0 - 40 l l

0 0 12116l0 4 /80 l - l

i'l#l

iltrllti

I3I

) / d

ll

s / 16,3 / i j

I

= [1 ;1 '1 .

:ro r!).e!r(io tltip(l btru06ocfLf lrr' rotrc obT/orq. O ri.v(x(< or!r'rei!:ord)\' oror4rrorrp6 pLoo ayLv! o r(!rorrx6(. Aqo! ro,4 mjlt orr-r 1. or ibre( rrQdEelEdrlt br!(i n}'t uA(i :rgirir|vcr €qrp(v ro\,.1 oro\,,4 '. Eror r:ro).o1iorpe tovavrioraoq o. Mr( (iru!io,(Ii] yr( ro tr[).],oa: Mroa€irF .,'c bairc fil! oQi.lo!rr{1 16tgt,trvr!6pevit mouE naQo.,'op(o16g ror rl r. :ror roorrrcr. pr ro yLv6pevo ttorobry,;,! (2)( 8)(l) x.'r €trqalflEr.rr oro rdl.oE.6ra1 or lix4rpic 6ruqpiucrL pe

IIcA(i ri]\, rao$cQit .11l1r1 tmruXi( oro.v r;rol.o.1Lop6 roU,4 L. ero b* ro!orvrord. Icp.rbcXotrar 6tL o urtirlrrxrtoc LuvEr ro .l/ = l, o' ivc 6rjgc cA,ri 1Ln

.r' =,1 rb avri rrlr Lt = b xol, tl.t = t:.

Oa trnoponoafLr ! . r r( i [c n(o pczprd rcl l rc lphlroqr c = L\ l : yat r =r l ' L / . . :nr, ;or ' or. , , - or d, v i ,7tp,,r , ! ,p, . x, , , L ' , inr\ : raurraoruo).o1Logd 6er 0c irpe:re rc oTrpattoorpr. roq nivcxeq I 'zur" Lr r. At"r6u( ra(r(t,'11),,'rvr- riozoncr xp6r,o, c.tor otl }e€ral6\'rouoav (xtiF1 rr/2 :rpirlers1.tl \,ct rioD.(irl,uo('toor.U€ rov .r tr€ rov U ' rt ro l, Ua ro! Z '. IlcrpoBorcnapartiaton roluer zur lrcr rov,.1 pLcr rcrL o notrLcn),ooL(ot16; A 'l Ocr7p€rcq6r(.y :idl.r nr 6r1p(1tc. At,rri rou l4rupe dwt 4 ),i'o4 zdt oy o),u tatft orx€ra rot' at|t ror etj.lov,

ndau4ato\ I Ano Lc0c1ait n€pL6a./€La. aE llno],oyi.ooxtrr ro rl"ytooE r0)! rAlrEEov nor a'r.(ronl,r1[ ./r( n1v $rpeo1 .ror ,4 '. :uvrlecoq. ro n).ioo< yr.t rdo€brlrir rierpch ai.vor. nr. to gLo6 npoXt-rprirv-rc< orry nrcrlorrp{ zaL ro Froo aqv

NIN KE: (AI AIIAAOIOH C USS

Ctvd6eoFl (1"rlxclrdortlol. fla n 6(.qoQErrz6( 6€Et€< rl,EUQsg. sr,.ro 6iv€r nr. vcro!!rr€00,sF6dvovlo< rr< nr/3 nQ.iE€r< oro! i6ro rov,4 dXolF€ 6\'s onvolo 4nr/3;rPG€ll)v.

To a;roril.ropc cm6 eivcr ev toriroq :rol.i pelcrl,rhrqo (rr' ro rpclypatrx6,;ror eivcr pLxp6repo l6yat rorv 6!:Er{i)v nLeupd;v r:; (:roLlri glberrrd). :r1SrcrbolLxrl rmJ.orrp1 yivovtoL crl,trcrlig p6vo rirrr) on to I oqv j e€ol. Toxupudrrr uurr i iXpL trovo , r ouvrotr ' 'orq. t ieu ortv nt(r l l r (( t r t {oI t l Io ro i } , i { i , 'srurr ru, iEcr, ' r 1r(( ro , , u).Lr i l r r oro t , r - j t : 2. Au0oitow.rc yrcl oi .u Tlr . / .ixorlrr'ro trl,loLlzo /ri.n0oE yr( rn bKbo1rxll ofic),oqri FivrrL xr/6. Auro irQi,rcrva o!'\,6u(o0€i I. r< ouv{oFrE ,)r/3 nQdEsK no! €{{,(QF6Eovr(lr oro\,,4 z(! r(alar/ l t , rv,r l r1r ' | rza ol trur( nou rr) .r?.r :ruQdyor.v r(1.r , -r i rF ur"rd yivow(l l[e[ov(,)[6v.i. eir€ rcrvr6XQovc, orr{'E orl! fi0o6o Gauss-Jordrn. fo ft).trti:i.tiqoar'ur :tu,i!r dr ; ta ror rtrtiol.trtro rot A-t tivrLt

O (pL0g6q aurdE eivcr alroolprit'-rtcr grzp6g. crrpri o roL),cr:rlcorcrop6q :n-rclz.c'rv xcll go\'o Tprtciler(r nr ,rA.iErr<. XQ€L(iqor'r(lr Lolnd! yrtl ror uno},oyroF6ror -,1 | r6orc :rpd!eL<. 60€5 XplL.iiowor zcrr 1c rov u;ro).oytop6 rov ;lr. Aur6ro Teyov6q qrivrrcrL otc66v c:rioterro (xcr o t':ro),oyLoF6E rou ,4' xp€(i_Lsrcr ro6r:r),iroro. cor' 6,11 xcrcl(dciv{t. llca ol. trr'rd. o A-r br! Xpr-ril(€Icr t or bt!0( €nQ€fir v(l unoLoyilercr.

nagariettott 2l1ei,v eQxiooupe vc q)rL(if,or+re p1b*r,rri nirvor cr' q br,c.7civro.otoa (1l,ldAr0fo Causs-Jordan. nqol('rprtoc4re, 6:ro; z(r m1v (lrc],orqri Gauss.piXpr ro ri),og rr5 zrrtcrorernlq rou U. Ocr Fropoio(l'"^ vc xirroqe xcrL 6rcrp-ptrrxri. Oc p:ropouocpc va Xpqo4ro:roLrjooqe tor 6cirepo obry6 vopiq. p6).Lgq0irvt4rr o cw6v. drmt rc rcpcilorla Fl6flrr{d r(ioo ar6 rdrvor tou 6oo zcra;ro xrirror ror. Ar,16 brl eivcr diu:rvo. Exeirl rrt ott.4ui I b€ur€al ya(44ri 0c€ivcr rQ(xrlx(i yl}rarrl, €rdJ x(r(t ro xiLo! 0(t EZ€r no),Lri ll&vLa.i, ).6yo r(Dvnv, i ! , , "v n,r f f i , r rXou! :rQorytol l olc ( :ro nLivrd . /Autqr ic.

A!flmQ€UrFoa = MI r6r6poQqoE

Od)'oqrs. rt_),og. !(. yvDoiEorill€ ,roror ni\,a).e< ri\,.l! .11'noreivrFor i(C! noror6tL. Auoi €iv(r 16l:'l) €v6raq'6Qouo( t-adrlol. rou 6trr xo1,),6( (rno1.rtorrs.np.iyFcrrr. z(O€vc cr6 r.r rQrbr( nal'l€ z€q.D,.rt.r 6r( b6oouv 6v( 6r(q,[email protected] Loobnv(lo) xQrfliQro c11'r!orQ€r]J4(6rrt1(!- M€QLzig rtoQ€( r( zQr1tor.1 €n€-xrrivor"rcr oe lapcLl,1).6./pr4ruoui rivcx€S,tcr. $ovotrL€uQor( (1'dorpoqoxE: roK€qdLoro 2 €r(Lr€i w€EdarIrFE ./aq4ri- ] clEEi{aqrsg or4},€S rar ro K€qir}.dLo3 cvrLo-rpdqer rov /r4r'rl roa,,1",1. Ta rr6lomc x€{{)(i}.(!cr €rLrAdnouv U6\'orapayt'xuou; :rivaxq zcr elerrilouv p4 p4Sevrzds oQi€ouaEs i pr1 pq6xwxiE

I r ANTI :TPO' i r ) l N , \ l A : !A : l l ,O4 'OI il

dtortpii i 1nl 1[ttErtxoiE o6'4fois. ]'o rr),rur(io (xro xarrrlQro tia(r €z€ivono| ouv(1,r1io.19[ zord ri]! (l:rcl,oq,il Grlss. O(r,ou$r v.r b€iEoufrr (o€ ],il/€Ezii:r",r . o rJru',.!r|rrzd: :rirA(tr',i't or,:l orr rxrrr-/Z,iy .

Yno0ior€ or o xilrczc< 6ZrL 6vo Jrl.rta€< onvo),o (l'rd n obryoi!'€: opL-{'lon (l,16; rivcL trl rbL(ttroaqo!. (Yn€loutriEoutr€ rir)'t 6rr or obryoi sivcr ['trrtaEvrrioi.) H €giooJorl -,1.'l | = 1 bivEr rr lr;rlprotdr otoTrirrcrc rl.r, = e, lLcr rrgori).eq ror,.1 nor rivrA buvu.r6v vtr rrtr0oiv tr€ cmclorq,Jt I $r 6rcbr,'coi.GGAuss-Jordan. Mropti va lper"cro0ouv r:v(),1"<tyeq 1pc$pr;).,'. ci,Lri ot ottl),eg ror.4 | t:ir,<r poroo{1rc.vr<1 oQrotrtr€!.

Mrl'(;r!r(l( (r'rtrrlQ(i. nar_nrr ,\'( 6fi.Eo\rp€ olr o niach(a J I p ur,tig tr;ortl).e: ritaL rnioJl; x(r (erorred (a orQo(ol. Ar,l,oar(. rtlt' .'1.,1 L - 16yoqri.uorr rcn'roTporrr zru fl]\, .{ r..l = /. ().i(i yr(d: Atrrrl r-iv(r l|(l l(n tsrertqra\ilr,qr'rrr1oq. Mrrr 6Lzrlto).oyio neozll:rl€r (;y6 ro rnolrrvo 7eq(il(no: 6v(1( lro\'o;r),r1"()o< (lrioroo(lo< rro< rsrQ(1()vrlroi niv(x(l r'ivcr rc!r(il0ov( x(r qLq,inL€upo<<rvriorpoq,oq. Mnoaoiitrr otr.)E vtr ro rlaYltooqrr x(l| to)0iq ro Krtd),(Lo 2, trrt"'1 cr),ri n(tacTlipryIr Ktiq{ ititta rr< rtqidt'r ca$t..hr\k t €i u tuIlcxia.q@yLiq a7; qnrtqri 1tr trc.r arotltrritr\4 tii(ra. Enrlpdt(olJirc lQet( ottorr((trorz€rrronrr fl \|(1).r('r\,:

l) ll,,. 7(r !( cq(rpoltre :ro),itr:rl.cioru ,\ rq: 1pa3pl< j a;r' rr1 11x.trtLn ;l l l ' , , .7ur vr( fv( l ) i . , iooor ' l l r rr :71rr4r1u:: r zrrr 73) , 0i , r). yLc wr rlr(rpoiJtr€ oLr. rL< yg4rtrdg F€ lot\ ob1loi5 to4.

H 6r, ,brz ' ,or,{ Cru.. Jordun r ivar ortv nlrrr lprtr tzorqru Irr t yry,rvrr ,( i ' , (1ro-),ori0ia (lno ro).),l1nL(o[(1ououq ir rv(irit 'r;

(D r . . . C . . r . . . E ) , t = I

0 , 1

(4)

O iruzu< orry n(rAd\'0rol. carorcp(i rou -'1. rivar:rpo<lcvrirg ilcq cprmtp6;c\'rioraqo;. Yn(iQ7€r. z(.r rn(i rrt orltr€io)ot l. nQ€n€r !c ri\'(ll ioo: tr€ rovbrEd .l\'ridreo(to. Erov xtiot lttl t6t6poaqot ttiraLaa tirat alnfireitl\poa.

To aarimpo(rro rivcr €riofi c),{6r: Xrid€ awtoqitl't1toE ftiyaxo.s tir,alp4 ddpo(ltpoE. Eirr o ,'1 eivar rrvrrotp6rlrLpoc. ax€r r) o6Jlyotq. l oirro rivcLrlrA'epo or pLtr crpui( ne.Qiff(l](Jilr Eciv o A iX avriorQoqo. rdr€ b€! lrnoQ€i !ci.Xrr Frc o),oxl,rtpit oni],l [nbEvrxljv. (nori 6€! 0t1 F roponof o (vriorpor{iog v(xol.lcnl(ordorr tr( oriiLt [tb€vL'{6v x(' vc 'raady€r t'rc orrj},lro1l rarrorL'{on.):€ lrc iLy6rtao crpaicr :rrpi:rrt.ro1. unooior€ 6rr I cncloq rt E€zrvd [€ €lcvaurmq€1'r1o nivcz( -{ rdJir zcrrcppirr orcv :rpozirlrr o

0 r/.000000 rl

Aur6E o nilazag, ororn xt av €ivsr xcr r, 6ev gnopei vc 6I€! ovrlorQo(po.Mlo a'i66€GI €lvor va XQrp$onornooupe :rpdleq gcraln mv oql,(iv (yranedq Aoad:) <iore vc xrivoupe o)'r; rry tafq otiltl Fl6€vr)iri. AgalQxortaE toooord roltrorldoro lg ofllL$ 2 xctl tot6nn, flls onil4E 1. qtdvoutl€ o€ 6vc.vaivolra:rou oiyoupu 6ev eiv(rr or"rLorQ€lJ4loE. Apa zar o oQ,xrlq ,4 6€v eivqrcrr"nmpirpLgog anr €1oqru trrav av qoon. 'Orcw

E€ri|,v(ip€ ard 6vav a]rrme6rl'lllo:rivuxu I u:rclLotqri 6tv fnoQ€i v(r r{clraQQ€uoer.

H afiorlorqri 6t1€r 6vo cnor€tr€oForlx6 xoLrrlQro yra rlv u;rcp[r1 tou,4-r:np€a€r v(r undQxow n o6rlyoi..

' Et'u( x€rQay o1'H6e niuaxu( Eivcl artloreiv ryo<xiv xat pr;vov uiv ri:lrr y4 dbpopqo<.

To KEqd),alo 2 0o XQyloqronor.rjoer nqtile6 yq<4rFri)! orov ,,1 yra vtt ,t attrtlrlEcl ogprc( lQcrppri pl6€vrxdx xcr oro tbLo crnoral,€opa. firr opIoF6vE( bcELiq x).t!Qdg,ro oufil[c ,4r = b 6ev eXE! )'uo€K. ft(( 6vav qwlorQ6qrryo niv(lra. ufidpxeln(i!.rotE tlolobr,{li iluorl ir = ,4_rr.

XQ€rsl6F(lore lvcrv ar6p1 nivo.o nou, €nrl,xdx, eiaar nol,i an;.odotsQos.rno xov or.lifieoqo, Towog ovopdtlcruL cvdorQogos roo ,4 ltcr 01416()r!l!,t'(rr FrAr . OL onil'trE rou otrlForilovtol on' r!0ri(q ruro 116 lQ'aFl6l ro'.J A -q i-@nyeqqoi iolr A fi!€rat 4 toxrl oril\ roj AI - xo:t trot xlnro,oxeurilercL lopig

NAAKE: (AI AIIAAOI+H (J USS

Er" ,4 = 13 ,1 1] "u* ,' =

(A1\ i j = A j i . (s)

O ovciorporpog €v6q xdroJ rQryol,lxou nivoxc €ivar dvo rQ!^/clv$.ioE. O qvdoreo-rpog rou A? o8qyei niot,r otov A.

liilTcu6lpovc or. oxril,e< rou -4 yilovr(r| yQoFF€s rou r". Edv o,4 eivc( rn eJn rivaxng. r6te o.AT €i1,(rr lt €ni rn. Aur6 nou ouu6oivsr, rEllxs. €ivq! 6vqExctonrprogo< rou rivcucr xcrrd FnxoE rlE xupicg brrloviou rou, xarci rov o:toioro morxeio rou -4" oII loq{ri t rar oq orii,t j nQodQxerar (rn6 ro orolx€iorou ,4 :rou 6pioxernL onl lQcBpil j xo! otl otil'I i:

l b \N l l : lP r . r l ' ( ,1 | {A l \NA! l l ,o4o l

Ynoodolt i6QC otr iTorpr 6io :Iia(z€< ,{ 1.(1r B. Ed! 1or,( ;rooofj6oou$€l'l(t tr€r(i tou< qruorpFl'cxlr. ro.no16),€ot(c rivct ro i6ro tlt ro vc rouqcrvcrorgil,orlpr' )t(r $€r(i r(r ror< rrQoouaool,lr€r (.4 + B)T ei,!.Mo ibro $€,'lT + B? . Arv tivltr 6l(r! q(v€06 rroroq ei,\,c[ o cldolQoq]o< croq 1!.,'olra.,'o! .48r1 rroq cvrror06q ou / r rcL crrroi ei.vcr" or 6coo{oi rrlnor criniq rltg r(Q(yairqori

lM (l) O av{iorpoqoq rcr Al} eLvu (AB\r = Br Ar.(2) O (vdoraoq'o! ro1,,4 ' € i ! (L ( /-r)r =(r l r) '

: , l t r t -r{ ; ,orr 6r i , ' r r i ro:1L.r rov (- lA)/ t rodi tr t ({ ,€l i€ ir 'ov ror (-{B) I . K(( orr(6io nf lnm(i)ofL< ( l , rrorpigol le q r \rr ! rrr !1 :rc ipvorrcr; Br, . l / xaL B r; l - r . H(n66.rErl./r(i rol c(\,lioraoqo rirc! nrxoLrt. (L).ri ou6g tbrir c;rru"rti uouvl0rorrlunot(ovrl oro!<;rol.icnL(or{()trori< rir\,(!xov. Fl ;rp<bq yqcfrllit ror (,,18)" fi!ftrq:rpt i ; r1 orr j ) .1 ror '1B.:rou orpaivrr 6tL or or l i€q ror / i lorJl orrb!.rot l f tr ! f i ' Trt ; rrr ' r r ' ; , , r4) r1: rnu l i Mrr,{ ' tua:ovr,r : ur ' /C,( tr t rrc. rolro,r l turr lu i r lor.1gr4rLrir ror -.1/ iyoul ollbluo0ti yiorrr r1q ned)r1q lpaplnjc rol A?. Atrrjuzirrtilr: tilrl 1 ;r1xixrl ypcruLrri rot, lr'-{r.

Krrr ' r rrrr) .oyirrv o|p;r i : rrotr z(r or r?.) . tc Tg4rrr i : ran' (--1. /J)/ zar l l l - '17.

NAPAAEIIMA

A]J =

l lT.r1 =

f r o l f r r - r l f - r - r r lI r r l l : : r l =ls s slf : r l l , , , , , l r s l

ltill;'l=li:lIi( v(( {rnobFilor.|lr rot riro ror, (.{ L)7 lrzruorrl€ (:rri rrq -.1-{. I r , l = / zru r( iavoqrE cvcoraort or,q. A:r ' r r l l ru ptpL(i .17 =./ . e;Ttv!,Qi:ol,It- (;ro ro (l)orr rt (!cor.-xnt rj lt votrivor' 6iv€L

( . 1 r ) r . 1 " = . / z t u - { / t - { I ) / = / .

Auro z.i1fr ro\'(--l r)r cvri.orltqo ror' -{/ zcrL arobr-rzvntr ro (2).Florr ' r : , rr , r6r iErr ' , ,nolE rnfc ) , t (rorf : . Fnupo, lr t , ( f r ! ( t , ryor lrr t r Lr

Erbtxrl x(rili,ol,i( nrr.ixnr!, :Itu(!ir< try rro t:v6rc<1,epoLro( (m 6lrq. Evagm'plttfetxis ttirrttr.cs tivftt ivft; ttiraxaj nou rcoitar K tov atdmqoqo rctr-'1r =.1. Avro/z(orLxri o xri!(7r( u( €i'wr r[rr.xryro!L:{6;,,i.lr x.iu€ ororzEio rou0( Lool'I(r l(r ro r(TonrQrxo rorr (n{j ro dl),o lrqo; trl,c &.r.7oviot,: o,, = a,;.Aio (lni.(i n{ra(16Fii,tr.rra Eia(L Ia

rrt! ci7.i.r.

(6)

'=ll;l'.","=l;!l

NINAKE: KAI ATIAAOIOH CAIJSS

'Evcr< oulperprx6q ni\'a1rcq 5ev eivcrL cvrLorqclrqrog, [d]'.ror( 0o pnoQo{o€ voeivor (vag ri.vrrxcq Ulb€arrdrv. Eci\' 6pnq udpleL o ,{ I, rorixoq ri.r,ar urioqgolJtrp€1eLx6E. A116 ro (2) nLo ;rtivr'r. o avtrorpoqoq rI_( ,4 I rooura! rdlror€pt: tov (,4"; I nor yrc orpprtqLxorg aivcxeq eivcr" orpr6dlq o,4 '. Aprr o'4l , ry lv r4v nrurxruorl o A I roo. i ror pr rov uvt iorporpo rorr ?ur rroluvu,c,rvf t louF[€rQr1{6E oo1, rov ,4.

X4perprxoi. :rfvnxeE etrupcvilowcrr oe rdo€ lirlpu rou oroiou or xnv6veqeivar bi.xcLor. "K{i0r 6AdoI iler trrLtr i.or1 z(! ovri0grrl GvlibQool", ?ior roororz€io nor 6ivrL 11 6pdo1 rou i :rrivar oro j raLordqer [€ rr] 6Adol rou jnalor'oto i. Oc Soqre rotnrl rr1 oqrperqi<r orla €7[6pevn ,roQdyQ(gro. mlg6raqoQrx6E €Ero6osL_(. E6(i) ][FQLoQLEo$oor€ oqv croLoLgrl Gauss xcrL lqtirpr vatrdeow€ nolEE €ivar o! Erl"rrdro€rq rlq ouFFerQiag. O,|oLeoxr:,.d auti ttEeroeitErtI dodtd oro 1rari. Or rno),o1opoi ndvar an rrl 6trr.1iwro trupoulrrrL tougunololro$orJq ,t(1(J) on rl btdyci)vlo xcr! o dvo rprytrvLx6< U :rpoo6Lopi.lmurrl,YlQoS (rro rov ridlo iQlyollro Z:

lN Edv o ,4 €ivql oup[erQrx6s 1(Cr[ ncQc'/ovronoleilar oE ,4 = rru X(I)Q{Eevcd.lcyas yQop[6v rou raxaolp6qo]Jv fil or]lr|lerQicr 1o!, 16is o dvo roryo-vrr6E [/ ei1ror o aldoiQocpo( rou lidror rQlyov!11ou -L. H doQqyo\.roTioiqonyi. ' ter.u,A=LDLI.

flc rI1, (r]r6dfLErl, oeorEjore rov crvdorQoqo ror.r A = LDU. Ot ar(icr-rQo{lole[qc!i!o1,ta pt rrvriorporpq 0LdrcrErl xcrL bivow ,4l = U1 D1 L1 . Extt6il oA €i\raL o4rF€roLX6E. lrlorlrc! [t6 rov ,4". dpc ilorpo 6uo naqcryovtonor.rjoeqror ,4 or rhro rpr,^7rrrvr,z6 eni. 6(1"/(i)vro €ni irrto tpryovrr6. (O -L" etvcr irvr,;rpLIDvr.r6q pe $ov.i6eq oq 6!cydr\r!o. 6ri(,)E arQ!6dE aor o t/.) Kard ro 1I aljTrtq :rupclovroroirlorl eivc{ [ovctiLxri. AQu o Z] nQdrr€r \ro rcuriq,€i(r pe rov U,nQdypa nou olozlla6v€L Tltv a'rob€rfu.

IIAPAAEIIMA (pt: ouyperprr6 A xu L1 =U)

f r 2 l f r o l f r o l l r 2 ll : * l l ' r l lo +l l rr t l

M€rci ano zdo€ 6rtFa xrl< aral.o|,(p4E, o nil,crxcg orq 6eqrd xdi(D y(I)vta naAa[6v€Lotltrttr:ror.roE orrorE zaL F€rd ro nAd)ro 6ytp.1:

ttl;ll;b

. b J ;j;l

I.6 NTI>ITO@OI XAI ANA>IPOOOI

H eqyaoia rrE orrd,orqrk ger.rirvercu xor' ouoiov oro Fto6, ),oy{r otlqrcraios, ari6n313 ot nt 16. Arv undaln cvripq vo mropv4roverioor4r ro oroLlriu xar u;r6rrE 6uo pcprdE qg brqoviou, i rs qrooqx.rroouF€ xcr rov tr xal rov a/.

AIKH'EIE

1.6.1 BAeit€ ro1,S (l'rtorqoqo1'r (6!1 (rn(rrffiror erirx6 olonlrd) rol

, l 0 2 L 1 2 0 1 , f c o s 9 - . i n , l^r - [ j o ] . " ,

= l , r :1 . , ' r =

[ , i "0 - . r ]

1.6.2 o) Ba€ift rolJg dwrorAot ol4 nuv flrvdx(')v !era86o€o!

6).EE{yiot€ ytart y',a nivoftc rftroodoEov I', o P I iv nAnoft o i6os le turl'r . rro qv and&rEr xaro'Foro'imr 6fl o! povd6€E €ivor oq o0roni 0don 6ore vnrooxumtr n PPj =/.

l.6J A;d qv A D = C 6apir. n'ro y(n lov,4 r. Kdvt€ ro i6,o yLq tov PA = ZU.

1,6/ c) Ed! o ,4 eivor owLote€qqro( '.dt AA = ,4C. b€fEr€ yQriyoao 6rr B = C.

6) Edv / = [; ;]. 6e€ift roadr,4wd ddt A B = AC o't'\s B t C.

f.65 Ed! B €ivdr o ovltorqoqos ro! ,4'?. bdft. dfl o .|\.ri('tooqos ro! A ei\ar AB.(:uv€rliJs o A eivdr oErdroavqros, 6tov o /2 .ivdr ovfloradvlllos.)

1.6.6 Xelsrponouiors q !d0o6o Gruss'Jordun yta v'olrlorAtrl,€r€ ror4

f r 0 0 l, 4 , = l l I l l . , 4 ' -

lo o rJ

2, r o l. t 2 11, Ar=0 I 2 l

1.6.7 Bq.ire iorE niva,{e( 2 erri 2, ro! rilar 6mrpo0.r.xol It' A = I rdt A = -Ltsrrcuilorct fe tols o\.iiciOo{ok rorr: r{2 = L

L6.E ariErF orL o l; ;]

bpv ixft ovt{otaoqo. raoon(oJ,uds vo luoHF ,o

11 l l t a b l_ t1 0 l13 3 l t c d l - t0 1 l '

r.6.9 A€ih€ dn &ov orooyxdv€r rl ara),or(fri .tra Ev.n dtoFoapo rivaM otuE o

f0 0 r l11 0 01.L 0 1 0 1

ro o 1ll 0 r 0 lL l 0 0 l

l 0 0 1 l

fill

[ 2 14 6 l

^=13 33;lto00e l

NNAKE: K I AIIA.AO]OH OAUSS

16r. o.4 bfl piioari va omotqiq€ral H rqiq -yeaFFi ro! ,1 r. ftol).oft),aord(olroE10! A. 0o inatn. va 6dJo.L tv rpiq yqotqrrl ro! ,,1 r,4 = 1. fnri .ir(t1 a1ir6

1.6.10 t tA. . , , r t . , u ,dudi ron vot , , t ro rqoror roLS r .vr^roo ' ro ' ( r '^

l.6.ll A(i)orr noa(16tittrcto rrvdxov,4 ror -B €tor oorl) O ,1 + B !o pltr dltloraaq€ror raq 610 tro! or A a( , ornoraa{{,owal2) o i +, v (lnora6q,€ror r{ra o},o no! or ,,1 zc(r B l€! avrrma6qrolrcr3) or /1., xd9(;)s adL o .4 + /J clrroradqrovror.:ot l t0 i t r tc l to doqor Xaro+onorr tms or l l - r ( i+ B)B 1 = B t+A1 l . t tb.iErs drL r(| o]J 1+;1 1 oiraL anotpdpqroi -tjaFirF dro|rl z(l( rino yro ro!(tliioreofd rou.

1,6,12 oo'rs rbLoTltr|s eao( n|1,axa brftnAoiviaL rcr oror (lvriorpoq,o rou (lrro0iroaroc()1 lndexsL o A-r)r (I) o r{ eilc| tol.row'.loq (2) o ,,1 eilrl' onB[ErEa6s (3) o A uivorra,b,(yr;v'os (1) ols ra srotxcio ro! €i!a' (rda((or aareloi (5) 6r{l ru morxrio ro!€ilcr al(i('f(ro (o!fnr.-or.af6dro!ro< oy€eoiol< 6rros o 3i 1).

f . 6 . f : t F , , v - l - I x o r 8 = i . r ' r o ; o t r d r ! T . $ . l r D . R t A . A t J t v u t t l 4 t .

1.6.14 (Evb(q6eo!ou) Ano60iEre orr aro[! xaryr( naad,h).oy0a!+o1( nivca!! oL A-.1?uaL ir .1 efv(lr ri(i\rore olprt€rarxoi. driErr $ naedlbrryt.a 6xr crJroi BnoA€i var i !d 6 'd ' l oer f lxo . , ,xopr t ?nr I . , ( r , rC. , t "v ,xurrg r rv , ,xr 9

L6.rs A.iEre orr 1(1 xdor rerqclorrro rirorc B, o,1 = B+Br eilclr r(i.\'loro oL,ppftaLaosao o Ii = B 3? eiroL ;rriltore cvtroupfstqrros ro! otfrdi!.r 611 1i7 = -/i.

^ l l IB! ' . r , uuru 'sroNr, !dr ,corn! a

i i 1 lo yr ' . i ' , r u \ / i du! d0eorq.d fvoc

d' t ' t ' , r0 xra r ( . r !n : d l rouUU,_e,"ou r i ldzd

r.6.16 c) noo( mo'xoi( rildr 6!vd6!aa oaroeon! o$lripnlrc ct: tivcv or||l(Er6 rivcra

6) oooc oro'r€id €i!( buldtov !d oArdeou! (rrlriqqrc or iaor om64qL€ieL!6ir!(a( raqrlr( xi

1,6.17 aJ Ea\ A = LDU rr€ Foliiij€q mr! 6L{rro!io!q tuv , "dr

t,I, roK Eivar r dvriototrlnaad'rovtotroitof ror -,17; !4LeLrirore oiL o n rdL o -,1" (rttqcpvrzoi nivcreqloQiE tvd).loltq 7ad|llld!) itouv aoLvouq obl oiq.6) no'a raryolLai. omiFmo ec b6oou1q IuoI rori l"g = tr

L6.rE Etir I = lr Dr Ur rar,.l = rrrrur. bEiErE oat L | = Li. D1 = D2 xcr U j = U2. Eq\o A €ildl (wmaav+Lor, xors r rogayowonoilol rilor rrovcS'zi.o) :urtrsa.ivor8 flc EEro6o.r! L1'L.tD\ = AUjLt.

' "ar

sEll}norE yLori r Fiar /.' uu r vo, xd-u rp yuvrzl ' v; i1 nI^1 or, , '(.)'7, ,v' i4

tlt:ll

;l3llt

0 0o 23 00 0

.1 00 a

[ 1 . 0 0 0 l+100

ic i i roLo o j r . l

]6 ANIT'TIOOOI KAI ANA'IPOOOI

6) >1Jytaivar€ rrE ftiat€r 6royorfolE o' aurtv q\, €qiol])or 'rdr roi6'n, orryrelvaftxar ra exrdE ira,yoviolr oiorl€la.

1.6.Ilt Yn6 'roi€s ouvotr€E yld rd oroq€la io! Eav{r! o A ovlrrretv4ros, 6iar

f a D c l f d 8 o l,a= ld e 0 l , i , 4= lc d 0 l ;

L.f 0 0.1 L0 0 e.l

1.6.20 Edv o 3 €d 3 drsxaE,4 6xer yeappi 1 + yaotllli 2 = yedu}i 3, 6dEiE 6ir€ivdL d6tvaro'', ro ?ru0ei. ro or:orqpo Ar =11 2 41r. Mnoe€i o A vd €tlord1,trorQd{qros:

1.6.21 Yroloyiors qv olrrperprrri :rcqo"yowo:roLry4 ,LD.L" ruv

1 1 3 5 lr=l i r i r i l ta b1--L; i;];l

1.6.22 Be€ir€ rov awi.oreoqo ro!

f+ 0 I 9tt+ r 0 0 l'=l i + 'ol 't i j + 1l

r.6.23 E<iv ,4 xcr B eivcr r€rQsrovrxoi niror€s, 6ei€rs 6rr o 1-,4.8 sLvar awrorelv4-ros,6rov o 1- 8,4 €Lvar qvilolQCliqros. E€&vloff an6 Tf,t',t B(I - AB) = (I - BA)8,

NINAKE: KAI ANA,\O]OH 'JAI]SS

I.7 I EIAIKOI TIINAKf,T KAI E@APMOTET

I cRrtr} rry ncqrrilp(lqo iXorfrE Srio orozor<. O 6vo! €ilaL va €Etyloort'r-alol re6t(o tr€ ror orroio €trq(ai!o\,r.A ortv rpir!1$elirkr o|mllcrrcr ypcr|urz0rrliror;oF.i,a. MaX0L r{ira( 6.1 trvltrov€noqr€ o ftrt1o ro 6'6},io z(no'(l €{Gauoln.zcr rilrrr cirios'rr 6.rr i,rc vcr repr.Tpirrl,oryc tva yeliio rcr tc),t-ir'.'g pr-trirorL16nooililtrc rrt! z.rraoL €r(roru.ri( trtxalrrriq I rt)v orxolotlulir! 0( tnQtnlr ac rr(itr0:roir pcrxpLci. Y;rcipler oor6oo pLcr rporoioyrxrl xrrr r:r6Lrr(t,opor)oc *prrepoyt 'rorlev c:rcrLrei :ro),r rpoero4ruoiu.

O tiLioq oroloc t-ivct vtl br|!oqrr'. tru rrtl i6I(1 tlr,rrt c(f'.r0ttoyri. rt; €LbuitqLt\r6rprq ro! ouxvcr ixolv or,-riluxt:< ourrl,uort;)v. EiwL rtoLrltl0tmo va^ r ' ( rr ,qr tu /u/ .o r-rr'14. Xxrc\ov ;ruiuott ur'411tr irrr r;rorr1;ro+rc, opcrti pri),rorc xcu. pc r1vnQ(;rq !(lfld o|xv(l ii'( unorr')n(,rtr(i ortqrrrakr( ri r(ia(l no7,L6! ptb€vrx(irv()ro,/fi!)a. :111rlLfrrr(i(l nfairr(,,o11. .n€rb)i fv(t: (a.rroc rivcrz(lc rEadzEL noi,i,fu.16repouc cno n: rpoprig:r).rpoqoqtrri. ot rnoloyrolroi nQifirl l'tr borll,€t,oulnoiir (1r),oi,orrpcr trn 6.rr or ivav y€tr(iro niv(x(. O( fFrr(ioori!{, Lbr(irra(ratnot ti< ttuur<, rLov onoirrv ro $t $tbtvLx(i ororxrt(l ouyzFlre(bvovr(t1ir12r,r (ro rn xlprc brcrltirvro.1[<r v<r bor[a:rrir< ctlrriq rlltoqr( (lltclcxL(r-ralont btabL,((roio (n(iorqn(. nce(r1drcr 0c bori6rl,()Lrttr tL{ 6v1lv f'bLxo rc'!'oc'bt

O ibro< o niloz.rE q(irerrr. orlv rEiooorl (6). Or r.noltrrq nco(iyo((to!eiryoiw t1v tq,cppoyj.

'lb ne(ibr.Lytr(i pu; :rpori;rrtL 6tir',r prl((kiloutre cr:r6 !r'c oNeXiq rpo-ii).rF.i d dv.1 6rozprr6. To (]ulE (a( ne66Littr( itL'r (inreo|c (./!,;)ororc (laott lIr|o r(.f) i,[(l xd0r' .r) r{or 6er g:ropt:i r( },!oti pc ffIo},lJTn .rzoi6er.r o 6vavr:ro),o1orr1. Ilpi:rtr ioL:r6v va ;rqrorrtylrr0.i rrn6 ilrr btrrrt+ro nA66iilgrr -ooont0r(Jl'6rr()ouq cyv(iroror,( iToL,tL€ 16iJ() lsy.1l.urEQi] orpi6sra 0a nsrlX(ivor]rrc),l.i rcr rdoo lfy(LiroQo ro 1{ooro(. :c! €\ra .rr}.o c},iri zaL ro}.r rrrrzotrrwrTig npti6l.rp( rnLl,iyollttr rt br(1(pprrri EEiooof

d ' u , .. r L r

= / ( r ) . 0 < . t ! l . ( l )

\ . T r l , , \ d r / p l r l r l , , z r t r E ' o i ' o l d c n v u : r r t v , r / v i u o r r t o r J v ' e r r o r t , r . U r t I o J o

1ev{ 6po I. Yrrha1cnhroLa €},rir0raic oro nQ6iih (r, c(poi x.i00 o1r\,5r,(opo6C + D.r 0a p:roqorou vo neoorE0ti o. onol.rErinorF ).riort. To doaorotra 0a 4r('vIrlil ir"il Lr'ot. (qroi 11 6€rir€Qrt n(phy{')./oq litq C + Dr bev oweLoqepeL ri:rore.fr' (116 rt .r6F6ftLornra rou ano$d'veL 1.6"1o ttov c[0crLp6rrov or.toeptiv C r.rcl|,, o€Qctr i.r(r €rrrorr',,.irrrolro( Ft "orl,oanzti oL1rqt-I a€ xdo€ fizao rou

r4t)) = 0. u(1) = 0. 12)To (xo-ri),eo$cr €i\ror 6!(l rQo6i,I[( o]JlopLax(.rv rL[0)v [€ 6no o)t[€i.a, nou'io0ryQi1qcr 6XL i1.r [rrc6(rrx6 dJd dra qcrLv6$€'vo orcoepdq zcrr,]orcioe04

I.7 EIAIKOI IIIN,\KE> KAI EiDAPMOTE:

-y!o riaQ{i6€rytlo, fllv rioxovoprl SeqpoxqooloE oe pi,o pti66o fiou ld dxQa qEzQcrrorrwar o€ ors0€Qyi O€p$oraoois 0" xcrL Oeqrciverol nr6 aq/i 0€ap6qtagI@r.

or4rloeire 6rr o m61og Fas €iv(r0 vcr fi(rQdlo I€ €v[( 6!co.Qw6 rl n€Jt€eaold-rto! brnortiororv np66l,ipo p€ irl?,o l,6yrc., 6va ,rQ66lrXrd 'fQ(lllrxri6 aL't€6aos.ff atr6v ro ),6yo 6E1 lnoQoulr€ vq 6€x0oiF€ ncqri 6vc :ro:reqcroFivo nl,r1-eoE irl,nQoeoQLdv 1Lc trp J, 1.1. rrs rqr6s rIE o€ rornialolto olpsia rr = h,x1 = 2h.,. . . .r, = rrr xsL qur6 rou 0<l urol,oyioorrF€ 0o uivcl! tiQoo€yyloftxi.EI$aq ?rL, . . . j 2,, ilE rlQaltlonrrk truoqs o' quid la i6!o otlp€iq. >Id dxpq ;u0 = 0z(rr ;un,I = 1 = (D + 1)[. ilow riq 6o8€i. or ooor€E rqr€E u0 = 0, r"-r = 1.

H nad)q eEirlol €ivtrl, F€ r! crr.u.q€rororjFE q\ lrl.edtc)\o eufdrliAqou xd€€ raQdl(l)yos eivcrL 6pLo q).ixtov 6LcrqoQ6v. ItoQ€i \,o 'rQooay1to0€iorr4rnrrirwag oe xrinoLo n€n€Ql(rofalo 6iFd x(rr Fi Er[rlQ€rowoE oro /r (i Ar) v.rriooo€yyioEr ro Bq6dv. ftn t1v rfu/d'. oadqlow 6rdVoQsE €vr.),laxrrri6s ).io€LS:

To l€l€ilrolo, €nr!6ri €i1,(o oulrgErQrxd n€Qi ro r, €ivor rrQd1rafl To nlo cxQ!6i<,fL(l q b€ur€Qn rl(Qi.loyo unaQx€! 6vcE CtLQr6G ol,v6ucoFq nou xQrylorponotsip6vov rrE r!p(S ma r zc{ o Lr:

tlzu u(r + t) - 2r(r) + r(.r ,)

du

(4)

Avr6E q(€! ror ro nQo16QIps vo €ivar oullErQr'.os n€Qi xo r:. Enaval.aF6dvowe:r1 6rjlr<i nler.rpdr apootllileL r1v apcrrucrrLxri rrpri 16 Xuldrz xrr06E ro I - 0.al,ld ne6r€[ !a mopodoouF€ o' 6vo 0€rr1(o h.

:'6v(r rurrlxd svtudrF€oo oyl.tcio J =jh, t 6logoQlrrl €Ei,ooot -tliuldt =

l(x ) ovrlr{(0iororo! nl,to! oft6 1C) bi(r,{Q!io cvdi'oy6 rllq (4) ror rro ,onl,aord-lolrcq pe ro /r1,

ur . t +2ur - u t I t i tuh t . (s )

YrrdQ1ow n €q!o6our6 ounig rIE FoEprig, Fia yrc xdeerrptij = 1...., z. H:tpt6t4zco yl r€).€v[c.iq rsQdorool oLF r€Qrl(rF6irvow 16 116 ,|c.r u,,r. nou 6.v €ivcrrityvuoror .or rrpig roq rivcr or orrvoQLdxig orw01xrg ?ar F€taqfQowar ofi11,6e(Lri nl.euqri rov elrodo€{Dv ouv€loq6QolraE orouE Fn opo"fer€iE 6aorlg ({ iou-l,ri}orov 0c owerorp6pow edv 6ev eircr pq66v). Eivdl srrxolo !d xar(rlorloouB€qv (5) oov Fr.( ricrrdmcot rooppo:rfcrg, orrlv onoi.o q Ao{ (uj - uj+r) €Qtop€vlcr:r'rc 6rlrci nar 4 fuj uj t\ €Qt6[Evl arr6 ra aQrm€Qd €ElooQQoro rrar Fet4v c:rtir)'a<r lrlfh) oto x6rtpo.

NIN KE: KAI AN AOIOH CALiSS

H bo1ri] nov n €Eroo)o€rDv (5) F roa€i va roaaora€€i xrr.tr€pa [€ ni,l,ox€,.,4x =6. a!c).4ow€ i =t /e r1 n =s:

2 1

7 2 17 2 - 1

I 2

(6)

Ano rcbQa x(! oro €EtE 0a \oiLitltorltt g try t{/ooo4 (6), raL 6rv €|vor avdyrlru xorrdrpe:rior'; oqv nlyj tou apo6l.rjparoq. Arlt6 nor; dx€r onpcottr ei.v.rl orr6ZouU€ jriv(li€< olw€L€ordrv rlrv o:toi.cuv 1 rd!r1 r trraopei vc ai.r<l :roLu peyill,(trLri ot onoior nL)ogc(v(irg rro),! oni1o!\r on6 ro vc €i,vo! nr1ci.oL. O ;rtvcrcrE6X€r TroL),as €rbrxiq !6r6fi1€E rli rorv ofioi,rr\r rper4 ci..v<rL Ottrrel.Lriberg:

(I) O ,tivcr<nE ei|st rararcyrirvron OLn rc pr1 pt6evrxd ororlsi( 6Qioro.|'rariTclv{') mI b[o./d)vro rar xlq b]Jo yer"rovLxt:q brolovi.ouq. Exr6; cn (rrrjv 11Itupibct 6ev o'!dpl€! rLnor€: (1ji = 0 6rov j -jl > L Auri( t(1 pr1{iavrxticn.{ofioLouv TQolcrxltrd rlv ('ra],or$i Ottss.

(2) O ,rivaxqE €fist olppcrprxig. Kiroe otorluio (,j looljr(r [€ D]v ricrlo-;npLx1 eLrovo ro! dji. t1c(r etor ,4 = A?, EroL o civc0 iqLlr,rvtx6; U Ocr eiv(Lo (vrlmpoqoq ror xhror rQ!y(,)!txoi /,, ral l r€i.rxl n(ec1/o1,ro]ro|Iol 0( €i,vcllA = LDLI H ow[6r0io (rrrt rou r{ cvrqvq}r}.cl rq or4perqic q; cpTr.xfEbloqroprzrig tEiotlorts. E(iv I rroad(lcoyoq rra\r:rrprmri; trilear 6roE 11 d3r/tJtl du ld:L:. tort o A 6tr 0a rpcv orrpptrpr.zoq.

(3) O r&axag dvar 6€rud ograFh'o6. Aurl tivur. irLc eri:t).dov r.6L6qrcrfiou 0t1 Enal,il0€r0ei. ]aa0t.]q 6(, 1]Jlo),o./iqovr.,| or o6lyoi xcu. o1[aivar 6tr oror\4yoi thaL Hsotoi. 21o K€{td}"a!o 6 0.r booouv .rolirroi roobrtlcFor oQroBoirou 0r:rrxd oQLoui\ro!. oL :repr.oo6repor cr!' roxg orrotolq 6ev 61ow ri;rora vcrxirvor! pr rI! arr(),oqrJl. Oor6oo olt4leraia ror oet|,roi o61^1oi eXouv gLo dprorlo!v€r€r.r: Evull,ayiq lqa|l|rdv €iv.ft neqLrtdq 16oo oq 0soQicr 6oo rcr. m4v'IQ(iEr. Auro, o€ ovri0€or1 pe rov rirnxa ,4 mo rdl.oE.rtq n.ec1pd9ou. ro! 6ev€i\raL 0rr|rir opLoF€\roG r.rcr. 1a{fq errr.}"(yiq lpctrtrdir, €i1,cL r€l.€i(l)q E}irro€4rdvocoro orpd)+{c {xAoyp}.onof qol]c.

Enrdra(qoqrs oro rsvrpLx6 y€yo\r6q 6T|" o ,4 eirm rpr,6LclcirvLoE. Ilor.aortFaol.a tx0r auro yL(1 rn.,'ffirlloqrt: K(t' cpXriE, crg unoodooqr€ 6rL Exr€l.orilEro 1106r() 6{Fa rlq ffio},oLqrJtq r{or rrrlpdyoup€ prl8evLxri rrirto (1JI6 rov rrQ(i)ro

|;r]I J(i) I| ! (2h\

I . r (3 f t ) | .t$h) |

L/(5/ i ) l

2 -7t21

12-1

, , ]

1

i -rI 2 i

t 21

2

I2

1.? EIdIKOI IIINAKE> KA] EOAPMOT€>

:I. 1,7 EvaE rdrlrff|6iE rrivaxca ror or noodyovlds rou.

:vyxQhowoE ps 6vav l€vr,xd rihcrto 5 €ri 5, 6l6nor.rpe 6rio peldXq onlouoreri-oeIE:

a) \ndggo in pdvo pq pqdetwd moryeio xdro dti6 tov o61ryd.6) To{ro ro 6lFCr sxre}dolnr€ o€ plo riotld pr,xpri lpcppf. Merri rov

rrQoo6[oQl,og6 ro1j rotrtldfi],ooLor )21 = -], lper,rio€rpe pia xar, pdvo ttgrTl4tt o 1 left 1 ao rco poi - aq a ie € oq E,

To pq6evr.xd lolr6v oqv rlQdxrl lQdpp{ l(ar rrQ6al ofli}ul qnloJroi,qoqlnol{ ro nQ(hlo 6rlpo. Ed r}.dov q zgr'raydvLa pogqfi 6rat4pfi0t1xe xa*i tqvafidlo tEi (1,,oei\ ctfi ouoi,clcv evc1,l,a16g lqqrptlvl),

1) To dewepo 6r]pc cnalolprlE, xqo(l)E xar rq en6peld, 6dlowa[ €nloqE tKqnl.oriofiloeq t(l)! o) xol 6).

MrioQoiF vo olvoulooq$ ro ret!!r{6 oror6},€oFa F€ 6rdgoQouE rQ6no!s. Ofir"o oriolidl,rirlrn{ds €i!q! !q 6oupe qv ,ru-riaQayovlorioilorl ioo ,4:

I- i1

- i 1- 5 r

_z3

r - 3I

1 - ]1

5

oL naQdqQrio€q a) - y) unoQow vq 6!qrur[ll)0olJ1/ (l)g €EfE: Or napdyot'reg Lxar. U utjg tpt6taydvrou trivaxa eit'at 6r6rayrivlot xar ixoxv liyo-tu|i qv i6la6opfi p46et'writ' pt tov -4. lqper,riroft enio4g 6tr o! , ror L/ €tvdr droorpoqor o6vcq ror d;.l,ou, 6noq cvapev6rav on6 rq ou14lerQio, lio! 6-ir or o8qyoi di eilc{tOeflxoi.* Or o&ryoi rqoqcvdrq ouyxl"ivouv oe prn opnxtl tr4.ni +1 ro06c 10 nFeyqldv€r. T€roror fiivsxcq xdl,orw rov rlrto),oylo.rri riol"d €ntuxrop6vo.

AurdE or cnLonoqoerq oiqyoriv oe prro :rl'{p1 o .crti iou ri},riOols iolnpdleov. :e xdoe 6rlpo qE orrololqk onoftotwdr 6to p6vo rrQdEeq rs!

* ro lrvtuso xov obrydrv Eiva, n ooiloroa io ,4 i det A = 6.

NINAKF! KAI AIIA,\O]OH Cr\LSS

liidQ1onv n rdroro 6rtfaro. Aptt uvri. yLa n1l3 eitla4 xoeuj:Eovutt pth,o2n xrrr" o rlnol.oyrop6s- eivc rcT,rnepoq rarci o)'6rl,1pe< tirEt:r,E F€Y€0ouq. ToiSLo a),I6€uEL ,iar yro lqv a'vdbpoprl crvtLxartiotooq, ono! TrdL! lQerillovrar 2nnpirErrg cvti ftrv nrl2. Aarr ro rlieoq rav naAEE(D! Eivel atriioyo nx n, aa''6Xr Brcg 6wirp€o( rou ?r. T(, rpL6ra161!o olrotriu(lro l.wovrar ox€dov o[rllrcio.

Yroetotts ltslu{orsoc 6rr. o A eivar €voE ro!"'rcr[6]( fii.voxcq ror onoior6Lo rc olo(rric €i!(lr tr166v cxr6E cnndrv ror r€ar1olxor orl tu,rai.6a i i < u(:2. 1.7). To 'rrrrvrcra6 (ru1)r!pog' rivaL tt,= I ./m 6vcv 6r(./tilvto nivoxo,u = 2 ylo 6vcv rqbL(tyri/vro xcr u = rt 1|o 6vcv fil,llQl rivoxa. To n0tDro 6liB(1tllE onuLoLqrtE c(nc(rsi rr(r1' - 1) :rpdleLE 1r(1r !oI!:p' cn (rrro ro 6t$a €IoutrEnd{),r 10r.rl!0r(6 !:{Qo< r1,. Aqrou ixo![e n 1fita:nit, 11an ,o81:i1 tvrj< tuu'nedoiqni.wxa rplrn tn uturti ultt troctEae.

To d),llgoi ro! fiQdEro)a €taar aldloyo 10! rr xcl rd)Qo 6]'6rouF€ orl €i"rolzur u.vtil,olo rou rerpcydrvou ro! u'. Ka06E ro r| rLlordl€t ro ?r o fii1,(lxo!lrliLrr rcL ro r),n0oE-rrD.'r nQcEco!./h€rar nafu,Io!6Qrta(. rlr. Mrtr (itQl6iox€elrxtipr4ol rou ;rl,rl0o4 rpd(eov r:laptrirrt crn6 to ye'/ovoq 6tr orrtv xdr{,r 6eEriry(1)!kl ro r]JQoq xlq r(trvicrE 6er eircrt rLdov u. trgoli 6ev rirdQx€r ldrQoq YlC 1600

lolrprl ),tnpibcr. O oxQr6ig (rQr0[6s ro(iE€ow no! noQdlouv tous I',D xtL U(X(Doiq !a l),1oe€rol)pe rov ,4 outrrperprx6) ei.vaL P = u(u'- 1)(3r - 21r, + 1)/3.fr( i\,o\r Trl.IQl /iivcrxa no! ixs! r, = rr Edvdr(riQvow€ P = rr(n - l)(tr + l)/3.*:lrlolfiiop€: Elcq xql!L.oer.6{g ;rivru(g ,4 e2et tpr.lcovLnoiE :r<rptQovreg I xot [/no! dxow ro ibro €ipo( ruL 16oo 11 cacrlotgrj 6o0 kor yl oair6Qo[l .lvtt fficrotaol]€ivcr nol.i ./Q{yoQfr.

E6(;) rrLc(i)oape xo [6rQI[d rov:rpdletw,06],ou$e oFt,rE vrr rovioorFsro ztgo ollL€i.o. fla 6\,av rivazo rsn€Qdopd\r(1)\, b!aqoQ6\,, 0r(l)6 o ,4, oaltimporpoE eivdL dvcq :rLr1p1g rivaxcrg. 'Etot, yLa !a Luoou[€ ro ,4ri = l. €ir(,rd.ti :tto tix o vn 6Qolp rcuE L xut U. at' d,tr nv A L. IloLlcnLcoxroltoEror A I pe To b n:rarrei rrr na(iE€l,<. c1,d aQt{oxr, 4rr 1c vc },rloolpt to Ic = bzc1| rat6:rrv ro tt;. = r:, 6ql,(5i yL(r vc exrelrooq€ &a6otlnri ondol{pri xaruvtr6popLxq cvtLxnrr imnorl nou:rcrpclyow to I =U tc =U tL tb=A \b

E),ni.Eoue orr to rcrphberlpc (uro rEurinQirrtoE 6uo oxo'Iou:: aq' €1,6,(ioxro€ r1v rcrcv61o1 ex laQou! rotr .rvoyvd]orl r$ bLc6rxcoicrc oTio].orqlig

(nou rL6o\' 0ecoporpe tLrq n),iao6 epnE6rDtriql) xnr cr1' eripor 66ort dvc

lvrloro ncrpci8rLnrcr rotr ei6ouE rr,rv gelcitrtov lpap[rxdv o!finpdr<ov, fiou ovroqowcrrrdrpe oalv na(iEI. :ro €rogEvo x€qtr).orc ttrrorQdqroul€ oq "08oarFrrri"6o1ri] €\r6q olorrt$rrro! ./qc$Lxrbv elrodroeov ,,tr = b. 6q].abt1 rlv n:tcpl1 rcL to

Fovooilr('l1o lou r].

+ Eiftoa frrud< !d o(( iiE&ndroortr 6n o -Ir €i!(r (?irxro<. r\fon or n 1. n 7(r r - 1

fir(r 6trrl.traoi (an)toi. ivo! (n alronq 0c bror!,Eird t'. r. l

l 7 E la l (O l l l lN . \KE: } l l I n ,AP\ IO[E:

lqdil(r Erao11r,1-ruort;

O0{'}grlrLz<i. l pl rbLolroa(tl n€aixrrDort a/€L oi.oz).IED{)ri. fr(( at{qrdoultr o'ia( nirjAt-( olvoi,o ob)tydr turopci lc xpr(iiontr rrc),1a.76< ypcrypd-rv. K<rtrrlrLv.ro otorl1rrr ,4.r = b ),rrvrrrl ge cvhbpoll ar-nrcrrchorlotl. :rry rpdEl [troQ€|rcr TgrrriiovruL r! (ooLt ecr trLirg tn(),]'rtyi< lqutrtr(;a br({oQtrLXri i] )'rort rorr,:roioTiionre yr:ropt-i r.izo).c lu livtr rr),tit'rc d/otorrt. O( (qrspdoortrt briootii6t< (tri.eit';: nQo((orflzi: y'( ro tr(i0nr( orrt! rri:rt) 7(r \'rr zirror|r rrtv(cr(ioll{ri nro ttorl10ri 2(( r(r boitrr yr(ri TprLdirrtu zur :rrn:lilrrar.

O+rrlefir! 6rl yL(r ia( ol,oril(1 [0rQior t(f./i0oriq i.I. 100 rni l(X). i] (;rdor(fri

Tprrrii|t(L tvtr iQiro rolr rxcrotrfr,oiox rQdE€K. Ano xi0r' jrailEl ra6nfr vft(r'r4riT oupr ircr or ir)1t( (rlaoyyrl,r'rxrlt. )r1\,r10(,r< ka.rri4rs !v( or.10Eoo nLl0oqrl,,lq irn' (i./. tqi(.. 1(r ir( no).li (6r'\(11() !,no).ol,ror{).

'l-lrin I n06o0ro't 6r(,

(0rOtr('y br(qoQrnzon trrya0o|q 6ivfl

{) .3.15 +0.00123 - 0. :1.16

x(r rtr rfir.rr(ic l,t(iiu rrro! fuzAo ctnqxi Xl1!o!1(r olndrrzh. To t0(;)rrlttc rir(rroirror rL olrrroqiLxnry til,rr (tr<i rtr trr.frt)1ofrir,( cr(f,(il.tr(rrl crrpo.lyi),rror6 ort)rri.rzo oq (iiu( rI. ),r,orl(i

Arri lta rir( t-izo).o iraoij).r r(. I](x l\ri0tzt (^d rov John von Neun nn:ror,r1r(v r1 ryerrztl pa0rltrrrLzt qrroroyv')tri( rov z(ra(l jtor' :(.t1'rz(ir or l,no),o-Tori; ir<rvcr eqLrril rrt\'frldirort f!(i< r'r(rrotqrllaiori n$i:€r'x'. fi!'riyp(rr. ir(lp'.'1.r, irof z(L a11l)l 't f'{ritrlo)t roLr oq ri\r(lro< rxg,iryr,tronoi1or6 rm6 rov vonNcLrnr{nfl ir(v;ro)"li nfoinlLrrl. o olivlr((olr6< r(1r! Cruss x(tt vDD NcnDr nn 6i!florry rrr).,1 urrliorqrl lrrrr rrELooryrior( nionorc rororri(. Til o('Jorri (1ni(l'rrto'l' r r i r rrn inlu ' udlzr , , \ \ ' i lkrn\on.:rol r ' i { ' roi . i r r r , r I r i r , rr : rr . io! z i . ' , , r ia, i

Aio (cTt.ri ir(rxrbriy|(r( bc!r.rotu'\(r (tno rov Noble z(r rot< FoA)1hez , U 1 4 , , l u r . U , ( , 1 , ( ! f u , ' o o r ' ! r , I , , r \ A u r t f , r N l , , t r 1 r i , r 7 r ? " , a o r o i r r . u . ldrrx)i,yi,i€!o)1.. T{1 ;r(0(1br-iltrarc cit(r

, f l I I"= l t t .ooot ]

H aq;rl 0i0rt lrai €i\(!:

. , I o . ooot l lrr l

1O Mt:pr.zoi rivaxr< tsi."vcr EEaLQ€r|lailci),l,oL 6* etrcr. O niv(xag -,1 €ilql or'roi). O -.1' EiarrL o€ zd,rt acrdorr(ort.

Ano nooorrzit dr:ro1,4. o -{ €iv(r ozr6(ivtD.rilol|re ro re),trrttrio ororz€io rorl .,1 or:

emrio0nror o0 [rx06g a].].ay6( cvd)KzI zcrhor(o1 (bllabrt €r|rlio0I-

Lbrqoglog nr ir o -{ '6tr ni laL. Er iva:: = 1 cur4 livtruL LbL6popqoq r<tr

IIINAKE: K,\I NA OIOH CAIJSS

or 6to ori),€< yivovror i6rE6. OeoQtlor€ 6!0 ro),rl rovrrv6E osErdE rrL€uQes rou

u + t = 2 u +

u+ l . t xn l l r=2 r + 1,{XX}lu = 2.0001.

H liol ro! nQd)rou €ivur u =2. | =0,1),601 rou 6€nrlQou eivar u = u = 1.MLU a ,ayf otu nipnto 1lneio to1' h peye9ir94xe ot gra a).tuyi ror qtt@utlnEnr r4E 7r1o4g. Kaplia apL9g4trxil pi9oboE 6tv propei va unqiy€taur41' x41' Eueoqnokt o€ prltQi| dKtztedYi(. H i.clxri xoiilolool FnoQei vqFsraroTrLoo€i, alo 6vc oqpei,o o' 6vcr riLLo al,l.d 6fl pnoQri, va €Ea),€rqosL. H:rprrypcrrLrrl Lrlol eivaL nol ElJaioerFl1{o! I l]no).oylqop€vl LuoI 6sv pnoQei !(rsivar lry6r€€|o €uaio0rFi.

H 6€r.rr€Qt oion F(rs €ivc0:

1P Axopq xar 6vcg nivcxog oe xcrLri xordoraol LrfioQel vo ,.clclmQtrq€idn6 61d1, qioxliE ,rordrrFsg olydQlo[o.

Me Lti;q orpeil,oqre va rrsQab€Xroqrr 6fl I afl' €u0€ia6 oJrd.oqri causs €ivar.yt(. rov nivoxq,4/, (rvdp€o(. orcu( qr(l)xriE nor6qr(rE alloQiErous. YnooaorE6tt ro 0.0001 yiv€rctl srob€xro ocv nQ6ros obrTyoE x(.! 10000 qoe€s I np6rrlypcrpp{ crrlctgeirnr cr6 q 6erireprl. To xrino 6eErd aror1€to }iv€rar -9999 a}.}dro oqd}.|ta oteowul€uolg yla roia !4qtd 6l!er -10000. Kdoe ilvoq cn6 ra ITOU qQXLXO! JriV(|riC €E(rgslio0ll{€,

O€@q4or€ ro €r6rx6 nqQdb€ly[(

0 , 0 0 0 1 & + u = 1

u + ! = 2 .

Metti qv nrnl,oLgti. 11 6€ir€al €Eio(,)c'll 0s inesrE va €lvor

-99991. = 9998. i u=0,99990.

H arpoyyi),tro1 ocr bdroer. 10000o = -10000, { u = t. MdIQ6 e66q l{0xooxeoqirrlg 6€uree$ €Eio(ooIE 6w c.wovorldrcr o€ Flc avarer6ll iliol, 616 ro u €ivaL0(|)016 xard nQoodwLcr4 rQL{ba 6€xo6rx6v Vrpi{Dv. K(I06E 6pog ouvexilernr rqcvdr6popl av'rr.xctirorcrorl, r1rp<in| elioooq pr ro 0(0016 u e(l 6nQ€r6 va €ircr

0 , 0 0 0 1 u + 0 , 9 9 9 9 = 1 , { 1 l = 1 .

Avti, yf o1116, belopevor, r1v rqlj a = 1 nou eivci ld9oE povo oro rdxaQro V4qi.o.6Iow€

0 , 0 0 0 1 r + l = 1 , t i u = 0 .

I7 f r / \ rKot l l lNA l i [ : KAI E@ PNtorE:

To r ;ror r,;roLo.Tiocrte riv(r rd.rir,rq irigo<. fl(1p' riio roU o ,.1 r:ivcr o€ xcirtxcrtdotcrol. Ira xcrr' €rl{)€icv rrn(riorqri siror 6r(i{')! (orc0rlq. Or r(qiqovreg1,. D aLu U. eite crzpr6eiq eire xcrci :rpooiy"1o1, o[a(r rti€io]q ti),l,ru 2l"(rcza,<(n d.rr o (lQtr^o: n|ruxrl::

, , I l o l fo .ooo l o l f r roooo l- - 110000 r l I o eeee lLo r l '

O t{zAog oblyog 0.0l){)l irao!€vto8 (orri0€L(i z(ir I 0€A(orri({ r'ivc(r qclrarit r ' ) : u ) ' i ) o , l , t u ; . A , r r t , r v , r I r f i r t H , n t 1 , , ( :

lQ 'Or(,r( oxpr66< 6!c tLtbsllxo or 0(o1 oblyo! €n66a}'[ $r(r 0€(oaryryri(fL.ryrt orlv n:r(r).orrpr1. 6tor xcr alrclq tl|neoq obl16i €]rr6d)"1€r $L(l nQo-ztLxrt a),Icr1i. O t;;roloyrorlE, €1ir6q rli neQrrrx6orcoq nou ulirQ1€t €rblrirE(roqd).[ort yrcr 10 clr|e€ro,0c nAinr v( orryltQivtr y.dor: ()6rl./() p€ oLousrouq.il.).ouc 6uv(rouq obryouE orry ib!({ orJlLl. H €n!},oyriro! [ryu].orirQo!, .r 'o lou: roLr u:rn\ ' t \ rot( zu. 4 rvutr) . t r74 tov cvttoro.T,,r y1'up,r t i ,v. i ro,rirort rrrrrl r1 pd.1Lorr1 r4rr1 vtr 1irer, o o6ry6E, Ld\etlr pEOrxi oiiy1ott.

: ror nt \ I rz(. .1 o f !br7( j | ! \ ,o; ob'1loa 0,0001 0( o!yz$ror(\ , f r . ro oroL/r: iorlno r.rlit) rol ro1, Loorjrcr trr. I rur Otr yvoruv crp6ot{ trra e!ui),c./rl ./a(44r{i)\r.Mr ni\(zf<. (rro. onrrc r(tL rQoityorlirvr)q. orlL{1iafr noi},(ni(lor(otro !ni 6!(niv(,..i lLrr.ior(rtq. O vioq iTi\,(lzoc A" = P!' aTrL rlv rca.ryo\,ronotrlorl

, I r r l=f r ul f r , ' l i r I' l ' , ' , r x ' t t l

- [4 .400 t I I l u u , ,oJ ' ) | l L r I

ot 6r() otu[oi tivtlr rdra( I zcL 0,9999 zar oqrq,(r)!on.', rdiF|.l orlv rr'r!r1 1rt1i0ourorq. nAorl"/oLrtr[lo,! )tr{1v 0,0001 r(r q999.

IL,, r , rxrt u6rt /4ot brurC vr l i r , , r , r ' , v , rz, !rr , r lvrrqqrrajrrr) l oraurty zlrt rtiaary oh)jiuotl..;ror t-!rrh!€r ox' lrovov rlr A-orrl orrll11rrD,ri r(r ci),rrrrcfxotL€lEq orrii.0! yr(1 !a fiaHL rov ftf./o),!r€po buyor6 obr 6. :rlv n).'tpll o6iyrto'lc\r-\' /arcilovraL tr6ao rlrlii.(i,i! yocutLrirv cLl"(i r(r orlLo;r :ror rlipvou' cru414 1tilor1rrpr1 or4 oiot obrl"/or. (Mr dii( L6trd xar(ii€r(, puc e;rcrvcrpioplorlrorv petu6iqriv rt irclq roD.cnl.ooLc(otroq (1no 8rlLri !t| i\,( ni.,'Gr{(l trrdo€ol:.)H 6loxo)"1(t roiJ !c ritrcorr owrlplrLxoi eivcrL to r6oroq. To lrdE+o 6),r)!.r!)vontidrr ro! arc4Livorr riv(L 7_oovo669o rrd 1 peprz{ obrf11or1 aiv.rl (QX€r('1

o1.id,4,! rr,,Lr(i orov 0r-fr€)r0)6rt ci"/6pretro rlq.eretrryrz{( racgt,Lxlql\),./E6po!: rrtdr,ot.r'i px ptptxl o6fiyr1o4. Eivar buvcrol r'a yilorrl tLrQ(rd<ozolLl 6El,rLdoEL!. tlitrr( I r:itr(o1 ror crr prcr ot{),1 /a€L/r!€.rcr ala6h01Lorlrritr-(ri. Ildvrorq, o (!a"/v6ml! ./r(l)oiq€L n),iov ti ncivrL o uroi.olLorrlq o'

IIINA(E: KAI ANA OIOH GAUSS

6vo oioqpo lpoppxdv €Erodro€ov, >WxOW6F€vt Fs rI "O€rDQrFuiri" n€pryQsg{-6pdre tov.4 I xat iro),Lcnlaorriore ,4 .. t 6- q nepr.lpaqrlgag xqravdLro€ nol,uXQovo (zat ufloltovi) lou cvc1vt6otr1. Ocr eul6gow vtr urtiple dvag ernol6regoEraonos €qrlyllorlE rou nG 6Qioxer(t rQd"nrorlxd ro ir. NoFilo) 6ll(14 6rr. 6evun.iQx€r.

AEKHTElE

1.7.1 Tqororoui('rc ro,Toe{ib€ryFo oro xsilsvo. d,ldlotras ro a = 2 sr' (1 = l. x(lr6a€its rrv noeqovronoirtort ,DU rou a€ou qwori ra6rqr(i)vlou r raru.

17.2 redl|'I€ rov 3 €,ri 3 rivo'ro fl0ntQuold\a0! braqoed'v (l = 1/4) y( qv

u ( 0 ) = u ( l ) = 0 .

1.7.3 BQ€ir0 tor ?iivol|( 5 enL 5 no! riAooly"fl!,er q!

9ror = ?rrr =0,

ovn rOtor(l\.ro! rr! owoproxiE ow0itx€E p€ rrg &0 = ur r(t 116 = ris. B€6oro0eiftorr o nivdxri< oar. eqa$ol6lrios oro .'ta0sE 6rdwot( (1,I.l.I,l) roe(it$ tou66v, dao o / dvu| rbr6rro(x|o<. AsL&s, dwimoLro,6 eda r u(r) sivgt Inort rouswrxorE nrJo6l.l[eIos. i6ie ro i6rc o![6oi!€, x.lr !€ rry u(i)+1. or br|o ouvoQi(rd<dw6rl{eq b€v oraeolv rlv d6e6o(5DFo oro! oeo (.' + Di da.r rl ).!oI tjer silot

r.1a f( h = | /4 xlt J@) = a/ 'in 2'r, r .Eio{ddr 6Eqoed'v (s) fiv(ll

Aumc qv0rE rr0o< ro Lr.ur,11r rar 6g!:ii€ro l"doos ouyteivolrdg Ftr rlv nAoyp.lrrl{liLnor | =sin2'rJ 6rdv ' = r la, r =|12,ur =3/a.

1.7.s Iloio olionlfro 5 .ni 5 dlrrxa0rord qv (6), .dv o! owoeraxds milEr.eg .yirolJv

u(0) = I , l l ( l ) = 0 i

r.7.6 (>wrmdrs0 Ynoloyim€ rov ovrioreoqo ro! 3 €nf 3,tivsr.o ro! Hilben

F. 6no reono1jE. xaloqrono(;lrds rI ou1'Iororc\,l] bloalrooio onoloryis tu\, c!u$Jordm: (i) fF axqr6€ir uroLolrolous (ii) maol.yxLfnovtor 1t(io€ aaL0p6 oe iQia

t i .i ill,ll +l il

t1 1 + t

liiil

A:KII:ED nA EII^NA H!,Hr Keqr. I

laqia. >Ir€trDory lle6x€riar .ylo ptd r€at''torot 6:io! I o6{notl 6€1, 6ot0d. O -4eiva! oe l.ql.ri ruriisiosr '{or aocEjllt€wos.

L7.7 tqr.sitdi€ its 6eEt4 Jtls'Jefs rE ,4c = 6, yra to! neoTyoitrsvo rdvaxa r(or .tro tl4I1Jo€E o = (1,1,1) '.ar c = (0,6, -3.6).

1.7.8 Ators ro ar = b = (1,0,...,0) ym iov 10 €nt 10 nirono ro! Hilbcn |16 air =1/(t +j - l), re'prfonodwas xd'ror,o fle6yeoFFa yra yeoH,Lu{ds €Ero6oe6. Kli.liexor6n$ pld Wxqi dl,l,oyl o€ 6v( drolI.l,o ro! A ri rou b xar suyrelvaie flr }.io€€.

1.7.9 >vrxqi\,ors rol,5 oqyorb qg xavovook aaaloL{4E F auioq q< $serriE otiyprts,

,_ to,oor o l^=L r roool(Avr6 eivft rQd$afl svo naEi6€r.flrq noo XQ€dtsrol ovoordopwl nQn, rqv ono-rotrpt)

1.7.10 EErnrTors lLsit oq peaw4 o6h,Iot d)o! o! nollorlloolooi4 ,\;1 orov L wcvo-noroi! rI! )ijl < 1. >W€edvdrs d edv ro oioq€Ld rcr cQilxorl ntvoru ,4rrdvoJrorct! qv loijl < 1, dr€ peid qv naqayoryi ruv Frp€vu{6v oirp re6dlorit\[ 6],a rs orolt€lo eQddoovla! dn6 io 2, FsId 6e u:rd ,t 6rlptno ppciooowcrqr6 ro 2r. Mro{€ir€ vo r(ardox€lJdaere d1(r trdQd6enlo 3 ert 3 lr! 6Ia ro laiil < f|.ar l,\ijl < I, ror onoi('r o t€fflnatoE o6qyq 1lq €avdt 4;

AEKIIEEIE IIA EnANAAIIIPH: K8qdlorc 1

L1 a) rodUw 6l,o!E rols rl.vaffg 3 €rit 3 |rc orolleta

6) Ynol,oyiore ra yriqreva AB xat BA xat A2.

r.2 Yno),oyiors olE ni1,u1t4 AB ltsL BA*ar A-r rr B-1 mr (18)-r, &av

Ba€iff naecSctrFora mv'itu\ 2 E ri 2 $e on = 1 /2, Td as. onoLos ) A 2 = I 6 \ A - t = A ' I . r \ A 2 = A .

taw p€ aral.ol9li xor o1/risaqrl ovt|'(oidomorl:

, 1,tttr oi.i = j.

,=[] il *,r=l; ?]13

! + r = 3

IA

NINAKE: KAI ANAAOIOH GAUSS

r.s naAayovronoDiore rols naoryo1]pelolJE rivas€E otl loeqi A = rar i P,1 = tU

1.6 o) Yndaxow 16rivaatE2 tni 2 rov oroiov rc oro(Xcis €ivaL l ri0 n6sor.t dEd)v€rvar awLme€u'trorl6) (noln 5!oxo).6ftAll) Ed! 6dl.Err orlv d.ll I xor 0 mc moq€ia €v6i 10 €ri l0nivaro. fl €ivdr fl(kvorsao vc r.ia€lF, aad! artmA6vrlo rl fvcr tbt6poeqo rivoxor

1.7 Ynderolv 16 nivaxeg 2 €ri 2 r(,)v onoi(iv ta mortfic filat 1 rot I n6oot rEcrrdrv ri!( awrsrqalq'ot;

lE noLrs Eivat ot leqqris rou E,,l o€ dtr/aqLoit tr€ trs YA.{4fs iou,'1.6r(v

r.9 r0d!r€ dvo o!mlfo 2 rri 2 no1, ix$ dfttaEr ).no€K.

l.l0 B0€ire p. ro s(ifli tre rI $d0o6o Gsuss-Jordan rotE svtLme6po1\. €(p 6oov lrdalol,v,

l,ll E,t! d /t rivur 2 tni 2 xu eosotr!! itv tredrrl 0qloordl otrl b€ut€al t' etlar o! ,2.EB xcL 88r

r.U Ah{.la( { +€!6as: Mr 6Lt{ctoloyi.r 'tl.r .r'tl€€s, 'ror cwlnaod6€L1la Yro 1ru6as:l) E.iv o ,'t rtvar (ELorQirl,rpoE r(r o( yA(lFp€s tou €ivct o€ owi0rq 6Ldtah movB. tore tta o B Eiv0L owroradq'uos.2) Edv or .4 aar B aL',(lt outl€rau{ot, rotP rot o .4, sivat olpB€1q(x6E3) Et(v or ,4 !d' B €Lvcr o\flotat1,4to', 16r. xat o ,A eivar amtGqil4ros.{) Kdo€ Ft r6L6Fo$pos Jrild"ds FftoA.i !o ndaorotoroLloti o als llvot'tlo--l = /,tr rlq xtft, rqryovLzon t xor t!& u(tu leLYovrtoi t/-

Lx! Anotr ro -'lx = b, ).ivor(ls rc blo rary(,vrxd @qrtlct(r ./-. = 6 xor t/3 = c:

noto rrl'ip( ror .4-' 6aixars on ow6 ro ti6do l,i

r.r4 BapirF, Fr 6!varo', oLous tous 3 lnt 3 trildrEE B itol 6orea) DA =2Ay4xaIE A6) B A =28 ra x(r01 Ay) o BA ru erai.).doo€L qv rdt'l IL€ r'l!l€).€Eaid Ya4lllri rox -'rb) o BA va €vc)'Ldoo€t ttv ftedr4 pE tlv cL€uoi(r ori).I rou -4.

, [i : i] i, [i i rr',-li i i]

f r o r l [ : ro ] | I I : ll= l r I o l . .4= l r 2 r l .

" . r ' l= l I 2 r l .

Lo r l l Lo r2 l L -2 | l l

f r 0 0 l f2 2 4 ' l f01A=t ,u=11 r i i L lo t 31 , l ,= lo l .

l r i J l l l o 011 L l . l

A:I(IDED rlA E[ANA.4FI,H: K@, I

rJt Be€iI€ rtv iqni rou c drav

| | - r -1 . ] f c I r . lr ra rovnedn , r= l -1 "

- : I e i rqL A- , = - - .1 - | 1 c 1 l

| _ r l n r l t . r tL - r _ l _ l n l [1 l l c l

1.16

Lt7

fla nol€g rqrdg m|J ,t dt$ ro cnJorqro

kx +u = l

' +kU =1

xoF.pio, pld i dnEeq l,1Jde6;BA€lre rt oq4rsreu.i ra{dyovtorotqorl .4 = IDt T ruv

f l 2 0 lA=12 6 4 l na r e=19 o l

[o + t t l Lo c )

Ynoodsr€ 6r! o .4 raorairwer o'l' rov rdmofid 4 lri 4 ,rl,vars, pe dvru.ardoraolqs onl',rE 2 on' ro 6{dvuolrd u:

f l u r00 . l. l o u , o o l^= lo 1 , ; I o l .

Lo0 .o r lo) naqayorrofiorior! io! o. ,tr,1,o$e1o Zy, tJrioodroviqs t,2 I 0.6) Bapli€ tov ,4-r, rro! 6Xer qv i6la tolai 11€ iov ,4. tcr€ I€ orofot{i { 6eiEE 6l! 66v lmipx€r tior r@:

u + o + u = o u + r r + r r = 0u + 2 r + 3 w = 0 x c r l r + u + 3 r ! = 0

3 u + 5 u + 7 w = l 3 . d + 5 u + 7 u = 1 .

Or z €rd n rihdx€S F ta0dssos si'qt svSrqgae€v raed6€0Tra .'otddas,'. To w,4.svrirolE noQqravoll! orqv old6o. Fxot' dvrLore6polE no! avirolJl' o!I! oF,i6a, oldForrrcs dqxrr oqv opd6d '.o, rxsvo'roL€Lra, o xr,j6\ds 4(p24) = etp)p3-616tl lr.avonoEirar yla 61,ous toq ntvax!5.o) n6oa orcrrEta r€€r€Xovtor on5 oFd6es 1tl}v a €ni 4 ,(aL tory r GJri 'l nwir.t v;6) Bpei:re pta 6rlvqrq t stor droI€ 6l.0r or 3 €ni 3 nivar(eE tretoffo€os vd lrlovoororow

nrsLygivrs 16 yqqal.€E rou D.{ ,.oL q onil& rou A D,rA" D -lt ?l

rlINAKE> KAI ANAAOIOI] GAUSS

1.22 o) Edv o A rivdt dl'ttoreilltrlo: r.ros rivor o owimeoqix rou '47: - ,6, Edv o A fivdr Fnicnl! @|Fr rqrx6S notog eivut o

"t(imeoqoE lor' A

, z 1 l1) E0oatr6m€ 'ot! 6!0 il]norls olo! ,4 =

I I r I

L2l n€tqar(rtrlosevo! l€ n =2 lrar n =3 68€it€ rolls

12 31 " f 2 31 " t 2 3 l '

[o o ] ' lo 1 l ' lo r l

1.24 Erzrvd)l'ld< F ala radro €rine6o u +2ti u =6.6seirE rrlv 'tior"'I lou

a) rcqdl.?'i).o! srLrdbo! 6id qE oqxrlsOja1'OE l,.ntq-r 'ioqc)'l',il,o! e'llnibolinor nequtx'j|rot rc 'nl[ei(l (6 0'0)xar (2'2'0)'

ri wos toitor :tcrparitror trinitrol rou ouvsEdr to na(irro xar to 6t1rr€eo olo

otlL€io (4,1.0).

l.25 floro ,tonan].doro qs

n';c x(rulo6oivFr (x ais v.l TtoD,utlootdoct€l ritr o,4 rivdr oq4Pratxo<. dvtF

rroi!'qro: xor Tqb,ry,i,!,os: noror tiv(t or oqYoi tou:

c) norc 6tdv1Jouo . 0( rd!€' ,'tr = (st'l)'I 1) + 2(otiln 3). Y|o llov 'iLvaxd 3 lnt 3:

el r,rr(to,nriicm l.,av ntlaxd ror 4(Ft (oflih 1) + 2(otilrl 3) = 0 Enal40€lorP

6rL o -4 €i'dL L6ronoo@< (i.rY6t.aot ono 3 obnYoi) tat sEryiotf.vrari ow6dirtL

Aln8fc i1 u€ubi<, Lu brxuro).olio Y,r. d.flJds xnr dwrn'roabnYFa yra vFUo6<lrr i"" i.t = /.2U: (FF itvu rqtlorvtxuus I xualg. urbFvrrid

":l 6''1":9

lilk i r U t p l Y ! d v ' x o L 5 L F F F o v d 6 | s o n l b L ' q d u o ) . t d 1 r ' ' | = / . .rap(yolroroiryI €ivot Fo!46'xi2) Eriv ,4: +.4 = J. rritt l-' = ,4 + Ill Edv ol,a ra n'(y'i'wr otorxrio lor' ,:l rivor Fnba!.IotF o 4 fivd rbroFoeq(r

laauFig 2 dqcQsild ('t.i til! Y€qqri 3 nl dtoboxtxl

L28 netao|urrlopevol tr€ tl tLd0o6o GlussJordan !ro^otrdtr IolE

1,26

r1 o 0 l r r 2 0 l]1=12 r 0 l l 0 I s l l

L r , s l l L0 o l l

t l r . , o l " t l o o l I f l o o l - l

l r r o l l i l o l l l I o ll . r o t l L ln o t . l Lo m r l

129 ledvrs rot! ni!(ra.s 2 €ni 2 troric) ortrorotqow rtv tiartn0wol 1td0€ blovnolalos6) naooaJow xdo€ 6tdvuola oro! dEola zr

ri moiq,ouv rcioe 6uivlou( ttard 90' cwio€rc [€ 10 ao]616) dlo 'onv xdot dtd!1]otto os rAoE rI! etloeia (x)'iolE 45") rr = ,,2

AIANY>MATIKOI XAPOIKAI fPAMMIKE> EEDQ>EI:

2.1 T AIANY):MATTKOI XSTPOI KAI YNOXAPOI

H (ncaor(f rt rl:rl.oxorri ro yactrttro oi'or4rc ..1.r' = /, ororZtio xOo: ororxrio.K(1r(i r1'rrZ'i orirxro,o't. (n)-o,'Torci fniortq r(r q (lrl')ai(1. Or t'(orzi,. ra{Drriorr.lrrr rrlr' r1:a1€4 *t to govoatisuttt, r,:rtipltr pttr. zr4rpir r1 ri:rerpq ),r'oercrrrrcrrtlvruL noi,i rL,t{oL6rra( tu.r(i rrtv cn(),or(t.n. Xot-rriitrur r' (.ft€OoooutLr

|iu rur6g1 r(priypculo o (uldq tLq ra('rri(n,q. U€rd iI! oroi( (uro< o xllx).oqrbr;x 0.r i}fr ol'ox)'.trx,)0!i. Oorooo. o prlTcrropo< 11< tr:rcrlorqjc 6ivr,L ir(lr, r , ' r 'o rgrrr ' ?,rr ,r !ortort : ru! ?u(tr tr '4ou ouL'rr I , ((roc fv 'n u ?aLrr i 'c dl . )7oc , rcr i rar ru:r t t i loryt-yr( l i ( lo r€Qr) z((r( \o1r l . To zeq(i l (o (1ur6 n. icr ( t rr l1 'z(abr(i rI; fQ.4rIrzri< A).y€dQ(( zo( int'r< riv(L 6rozoirlrtptr rlro ro npdrro.

K(r' aezri! Ze{,((6t(lorE rrlv a1,vo,(. rorr Statuoparxoi' xtiteor. fru \uur(](yriloLtr€ or,lriv rrll rbi(l ieznArr (!.16 roLE nro !v6L((i60ovr€; j(drao|( nouori l tu i i ioar( l r pr B.Rr.R', . . . r ,n{ i0lrr ivc: yrc } i ( i {h 0€r '16 crzipuro. Oxti;poq R" urorc),eirar cl:16 <il"tr rc bLcvi,qrrrru-ottl),eg l(r- r (rrvlordor<. (OLor,!Lor{;oc< Ei!(L n(xrlF(rrzoi (eL0troi.) o Trlpo< I3: :rtrpLorhvstar pe rool'vrt0Lotr6vo Enijrrbo r..v: or 610 orryror6oF< rolr 6(rv!otr(ro( yivovr(! orolvrmcllilr:.r zcr r rou alnoroiTou oryriou. O R' eilrl rE ioor otzeio< zuror rgrr: ot'vrnr;)otg bilorl ivc orlptio tor rgrbrrirrt.lrou Z6A0r,. O $olobLtior(ro<7(;raoi Rr eir,.rr Fi( rr'ori(. To or}r(!rrz(i 1r( rrt laqrtrrroi (iiy€6a( €iv(r (irLrl r-,rilr(oil orLq a brcotriorLq €iv(r r: ioor cnLrl: lL( 6v.r 6rdvuofr( orovurru6rrirxato ltirgo -R7..1Ltr ror o:roio biroxolc 61oryt pa4rmpLxrj ero:rrtic, roLrtivo ;ror TperdlrruL vc lipoqe eivcrL or snrh oNrordror'< ror.

:rou! Zdporli cl11rori<.z(udrE )rcr or,{(ior brcruotrtlfll{6 xrirpo. btro:rptiEfl<

MnoaoirtlE va naoo0iooultt 6io 6nit6lttnoxu r.a ft .],anr.aot(iooupe inviopata pt apfiyo{g

fro rorg zciraor: R" or rad(q€rE cnrr6-< yivorrcr otvtordro(r nQo< olilrnrclo.r.Eirv.r' civcl ro 6Ldrvr.,oyu ror Rr fLa otrvtottboeq 1. 0, 0. 3, rore.2r sivcL 1obrirrlolcr ps or,vrrxrirot! 2,0,0.6. Mnoa€i vcr 6tc;(to1tLr0€i prc oi.ozl1p1 oetpira;r6 r6r<ir1rr:q 6roE: o [rio0€rrx6: rouo; .r + y = 1/ +.r], I nnaQEl "llbEvtzorbru\'1rou(1ro!' :ror rzcvonorei rlv0+i =ri, I ln(rQql svoq btcvnoparoE " :r"roLr rx(lvoTror€i rlv x+r = 0. An oior GrJr6q rrq Lbrorll-r€q, orrdr, ;(orr€ar).c$6hvor! xar -rLg ;rporyoigtrt:< rar:L!, triv.r O€tt€)"Io)btrg rcu o xotdloyri<roria rr.lo(rrioErar orlv AoxIoI 2.1.5. TrJnLxtl, npaypanxal 6rcruotturtt,,o;

TtipoEzivar iva dvo),o 'fruvuotrdl(,r1" €qoStdotliro 1tE zov6vEg iruruopcaxit;,rQ6o0EaI_< la ftd,l,.rtrhuorcopoit pt reayyartoriE dQr0ltoJ6. H nAooolol z(lLo ;ro),icn),(ror.rolroc neir€r v( n(Qdi/oLiv 6r(!nolt.1rc Et,r.j< ror Xteort tt(t\ !(lrx(vonorouv rLq ozrr'; (rJTig 16roTrF€q.

K(i(l z(vo!(, r( &ovlid$(r(i lag €i.v(rr otjvlerotralc brcllop(rrc-o ]].4( riat, .vt?o.v o r ! ' rv { . j r ' r , ,uc l , , r \o, : R . Qorooo. .r rL 'nrx, ' ( our, , t ro: Fnrrtr )rr . v ' roreq,rd!(or0 xt1[ ri)"I( npdyFatu oc(l bL(liollcrc r(fi oooa I jTQooo€oil x(to :ro)"i(:r).(otcropo< pe (IQrotroiE eir(r n().ri opropdvor. Ai''outr€ let(l ncQ(tbtil-

(l) O tr aodtloturo< /ritoo< R-. Tcr b[(lvtloir(lrd ro! ixouv (intL0eEaurroroo€S, 6noE ro .ri = (1.2, 1,2, l. . . .) 01,o o! volloL t[Q6o0rorF r(lL roL],(jrlo-oraotroir ncQ(4l6aoN or ibroL.

(1) o y,ti?o< tav ni\ix,rr,,l r212. : (ur1v r1v nseinro)ol 1(r blcllrotrcltc"r.iv(L ri.v(xql MroQortrE 1cr nQoooiooupr. 6uo ;riv0re;. A + B = B + '4. !|'Idpxerd\, q trlb€vrr{o; :ri.vuxu< nri. O Trirqro; ottog eilcrL ibro< pe rov R". (Or 6tLoNrottiocq 6rcu0ero1Jvrcr o(! oQooylblro naQall.IL6ypctrto. c!li. o0luo orl)'Yl.)Kri0c r:rr),o71 r,rv ]tt v.t n HG dbLbe ivc :rup6poro :ruQ(i6nruo. to,'XdrQo rtov

13) O T;aoq oL)raenia€tuv JQ). A€x6$.1ort sbL| o).eq rrq oll!(lorriortq/( . f ) nox oeiqovrar o' ivu or lxt :rprpero 6L. iot4tLa ), .X. ro 0 < r '< 1. O ldaocariroc rrpr],q6dv€r nc /(.f) = .r:, a(r) = sin r. ro dropoLolir ro!-( (/ + a)(,r) =r'r+sin .r z(0rig xaL o).a r.l roD,a: .dor.r, onr|)c ro 3rir r(1| sin i. To bLalnotrcrari''cL olvaprtiocLE zar I6rdorcol €iva! n.il.L drneLprl xct pcitrlora pLcr crtrpic

tr€i,aln1€al* tln6 crlTlt! ror R-.

Enrn).6o! n(1pcbEii,t(arc bibovrar m|E uokriocr.g. (i).Li oL bL(rlrloBarrroi 16-por. ior lpeLc!6g.rore r€eroo6r€po Eilct rarror ().l.ox r€aL61ovrcr Fdod orott<R''. Auouq roq ltbpo4 0iloqLt rc neprlphrpqe zcL va e!11,{oorpe lutiri'vcrL 16oo evbLcqdpovr€E. f€rDlsrQuto. oi{€qoeirE rov ouvl€LoFtvo retbrdotoro

Trbpo Rr znL bLc).6!re dvcr e:rirr6o 6rt:p16pevo a:r6 rlr cppl. Aw6 ro enineiotiwr an6 p6vo rou irdE 'raruopctddq XbAoE. EA,' 'Io).),&n).aoriroorpt ala

A]ANY:N'\TIKOI XAPOI KA] I'I'AI\,II\'IIKE: E:I:O>EI:

* > r.E.i I rn.la rt.llarl "((rolot

(!.i<.ta oiyaa'orl. tlr.(iiovr(t Wtocr< cno rrt oeopi( r0rr

: ] AIANY:III,\TIKO] XAPOI Kr\I YNOXOPOI

6rd\roFc ror rnrr66or F0 3 rt 3 rt x.iroro riL),o aaL0[6. raipvoljlrs ira 6Ldvro$an€pLEzotrEvo oro i.bro €ninE6o. Edv rpoooioorl8 6r(r brovno$.r.ro rori €rtnabo!,ro rieporotrdr roltq napl1[6v€L oro €ni;r€bo. To €;rir€bo crr6 €ilaL 6vo! u'rdroAoglor caXrzor Xoraor fi'rcr awrrraoocdrr€r€r [rcr ro16 rLq e€B€),'0rbfor€a€: Lbf€ETlt( OF0rai(q rI: fa(4rurzlq A),y€i;acc.

OPlzMOz Yn.;^pao< €vo< b(illJol.rrrzou Z(baou ].6yrrcr 6va Ft zavo riiroon-r . c b d n r n o t , r ! , : ' ( n ' o j o , , : :

(i) Etir npooOiooqre 6no SLuvnopcrru:r zur 9 rol uoli.rpou. ro ir0goropcirorc .r + ,/ n€0r€I€r.rr orov rirozoao.

(ii) E(i1, r(oi).(ljrl.(ordooure (rc 6((ivuotr(.r'ror !..royLirqor pr iv(v trqL0po. . To i rn; . I , r , i l ' ( i n ! , / nrr ,rr l r r , . r , r , {} . r oru! ' nn/ r- 'n.

Mt- clLltr iolLcr, u6lorpog etr(r (va r oolvolo no! €iv(t z),€tor6 frE nQo! rlvrpdoofol x(! xo'r' iro),.1,((7ri(o!copo ll€ cQr00orq. OL rp(ilrq (!ra: (xol,orJooi,'ror)q x(a(i1,rq ro1, nfrn6(il.Lovroq X(;)oor Xo)otq !(l [(lE oblyofi, 6Eo, (m rovtjnoxr)po. 'Eror bcv ci.r(r (lrir"/,nt ao €n(1),I0cuoo!tr€ rt: oliror rblo F€C, brorr(lqrou (),I(lEuouv oroa repr6hD,orc lrirpo. (ruo$(i-r(,)E (kr (ril0rrjouv r(rr o. nir0tu;r61r'rpo. )11Lr.rrirrxr: 6rL t:rbrrri fu yqntrnr) 'u'nvolrt (rr itteu:r t 01 ruL(,r;rrj7"opo. Toriro o<lciLcrcr orov

"c1olc (ii): AL(ldltc rov apr0p6 vtr r:ir(r r - 0.

H.-, , \ , .a!- ! . iu blvnro,, t ro 7r( rvuv ur.Jlu 'Qo rr \u. rc, ,rrQri7t , rvcr 2.r l ForobLrlruopu. ro Blbevrzo. Tolroq si!(! avc( lldsvoblcioruroq brllvuotlclrria6(xrirpog :ror:rrpL!:1rL ptivo ro q$tio orlv u0xr1. Or rrrvrivrq (i) '{(r (ii) oi(vo-nolorvr(r. c(fou nQ6o0roJl x(1! no),),ufiL.rolcoBoE siv.1! 6!v.11oi: 1() (i0Qo!o[a0 + 0 rrerdX€r.rr o .rrro\r rol Xdao sv6q ol[Eior xar To ibrc or$6.ri.,'sr rar tr€o).cr rtl ro).],cnidoft 4. Ar . €i1,at o #xQ6rLQo< 6i,')'(nt;< dtu,liott(ntx6</.iiQoi: ro xsro tlrooirvoLo 6€v eivLft tlro}(])po<. :rtv dL),I azoci( rr{pirrrDol. oleTrLrircpoq bruvL,opurrxo; r:r61arpo: civur olor),1poq o rglzcig lipo< rd0r6(i!1'd$( neQldxerc( orov rv ),61o !x6x(l)0o. O1a\r o cQXLx6,( XriJ(roq ei\,oL o-Rr, rorE 6).()r or 6 ,(toi lri6xrlJQor neQryori(polr.rr triroLa: o Lbloi o .Rr, x.iore:riaebo brri rrE .rQpiq, zhOe $6eicr brci qq aqxri_( x.r r ibla 1 reTri [6vr] rilq(ro |lr6evL116 6r(ilrouc).

\4, przLr rru"ubf l / t .uru :rou brvor 'pr roAu "dr

oprotr iv,( LrQ/orfuu. zuvorr\q,cllrqi Tlt 6rc19opir lcrcln ev6E r:rooUvoLor ncrL t:v6g unoTd;por. :€ xdo!: r€-arn-rtl)ort I €a(|JTltort ror na€rr€r vc anclvrtooutri(llr(Ir'louG (i) x(L (ii)r Mroarir€ v(l rpooodo€r€ 61ull'o$uru 1tcu" vc no)"Lur)"c'oL.ioEr€ 6r(r\'r-,of(rro U€ cpr0po. T,rLrpi< rcr elxarcrL€i!,€r€ To\r urioxorQol

nAPAAfII \ ' I4 | O'uqrlorr o).Lr r l r brulUot,(ru o.vrru7t, , ! , ;eircr 0erLxdq )i [I6dv. Ehl o cppoq 16qoq ei!.r' ro r.9 Enir€bo Rr, r6rt crr6

,, lar ro rrporo rr rupr4pripro r l r . o, oL,vr r( ' t t , rv, c.r a 0 ,tr ! > 0. Aur6, rap6io nou :reprexer .ro ir16tv rcr r1 rp6o0eot llirgaq,frer" p6oc. 6ev eircr ri6ltllpoq. ncoc6dlrrcr o rav6vsc (1i). cqor 1c rov

AIANYIMATIKOI XAPOI KAI TPAMMIKE> E=DA:EI'

aplep6 1 x(l! ro bldvuoUa [1 1], xo ro].ldrrldoLo ffi = [-1 1] €ivar ororQiro r€rcQrqpoQro cvri yr,o io nQdro.

Edv ouful€Q!]d6outr€ ,icr io rQiro t€iaQr4[6Qlo [aEi tt€ ro ?re6ro, 16r€o nol.Ianlaoraolos €t1'( €rdE€0: edo€ rotrlarldo|o ar ocr noQqr61,€! orovur6l(|)Q,o rar o xsvdvss (ii) 0a rriovoro!€irsr. T6pa ql(Dg 0s aQo6rdEera0 ozov6voE (i), oqou rl rra6o€€ort rou [1 2] f€ ro [ 2 - 1] 6iv€r ro 8rdvuoFo[-1 1] noo 6w ,i€ptdl€rtrr o' avrd id ietaQrqF6era. O UlrtqdrsaoE un6Xo]QoEJrou risQ!61€[ 10 ,rQ(DTo reloQqp6Qro €lvor ol,6xlqpo to -R2,

IIAPAAEIrMA 2 Av E€z0vtioow€ c.n6 rov 6lcl.uoporrxd )(6Qo rov nrvdxov3 sri 3. r6re dvcg 6woro9 u:r61,rpoE eivcc ro orivolo ro)v xdro rQryovlr.d,vrwdrl((])v. Evas cii,Los eivar ro orivol,o rov or4rperpLxtirv nrv<ixtuv. Kcr ortEblo trEQrri6orr4. ro creoofoFolc ,4 + B ,iar to ro).ldriLdols c/ rlrlooloFo$vrL6 [brdTr]TeE ut"t A xeu B, ztrr eivru x<irt.r iQrycDvllto| c! o! ,4,, €ir,o[ rdmDrprlrrrvr.xo{ nnt ouppergixoi qv or ,4, B €lvg|, soFl.l€rQDiol. Ououid, o [l]6evLr6Efii1,Clr(l< crvix€r xu! orol]E 6rJo unoldeoug,

EQI6p(ots t6Qq ors tloQd6€tyFarq-rt!€!6ri u:roXrirprrv. Awoi ouv66ow(rdpcon pe 6vcrv nircorcr ,4 zgt 6ilow n},r1p'ogoqieE 1r,c ro oriorryo ,4c = l. !e!€Qrxag tsQtm6o€K rr€pr61ow 6tcvtioyurcr pe 'r owrmdroeg, 6:ro19 or, ooiLeE rouA, rc(! €ivor un6lo)Qo! roo R"r. le d).l,rg neEnrriroeE ta 6|0violrsxo how nowLor6o€S. 6rr(l)6 or yQcrpBag iou ,4 (ri io i6ro io a), xar rivsr, un6x0)Qor rou1l?r. A[€lxQlviEou[€ rs no rlqvoJ I'tvn roqd6er,lpc tgrdlv eEiodo€rov 1.ta! 6do

Etrv unrlplav fieqLoo6reqot dlvororor, <rl' 6,rr, e!Lo{i}o€r,S €a ri€Qqrarop va 6Qodlr€fiol,l.6E l{o€tE (rtaQ' 6}.0 nou 6€a €ivclr" ndmore droL). !r4r ncrporoc repi,ntcr-ror1uri{iQ{ouv n€Qloo6r€QEE €EroooelE (rfi6 rous a}'v(iorouE lcL rpener, vc nepqrd-'to4Le or" out'tiqaE 6tv Oa undaY€t ),loq, 'Evo ouorqgq f€ rn > rz luv€ro! y|,aopwy6ve5 p6vo 6eldg rl.er,peg nar oqv lp'qpaxn{oqra yrs 6va fiohi "l€rrro"unool'votlo 6l,ll)v tov bwnrdrv rpr6uimcrov bLcwoprirarv ir. @6l,or4e vc 6poriyr0ur6 io uriooi,volo fiDv 6,

'EvaS rQ6roE v(r n€QryQdq,ow€ dur610 unooivo),o sa!o[ 1600 arr].66, fiol)€riliol.o FroQ€i, xo! vo iov naQa6l6uo1JtrE.

2A To orloapo -4c = b lw€io! 16r€ xol Udvov rdft, 6iov ro travuopaD €xqadlerar oav .yQ,o{+rxog orw6uco}r6g x<ov orll6v rou ,4,

li llt;t lfrl ( l )

: I AI]\NY:I\I 1'IIiOI XOI'OI (AI YNOXAPOI

!I.2.1 O }(i)aoq r.lv orrti(i)l: rv( Inir$bo rorJ rL,u(marou XrrQou.

A!{Jl I r€CLyQ.rqti b€v n€QLaX€L r(oir FL.l crvrrbL(r{muorl 1or, ouorlFciog A,. = ,o11l tloQrt,lii

,frl.fll=f'rlL2l l r l L/ , ,1

12)

Auric r iv(r or io,rq r tnr( rErourorr< r ,rc jTtoc r( ,uc oL.o n7!d,orou(. T,1,xrot,x,: 61.€iro,'ltr rifl ro r0d6t'Ir( ri!(lr r,, rE,lE: N.. dar0on! (0iOyui ,

",. ,

"ouno^Ir 'n/ . , (r . , ,Corrrvo. r n. r Iv nr,u,r t . lur hfurco, l orr)) . , t . vLl ' I r (A{ i7r)r .v ro hrr.vuolrdo. lo oudrlllr /.].v€lrlt rorr uxuti,,: or..v u;rdexotv rr:ror(,r ouvrr),(oric ?ctl ro0k.! ,1rr ' ( . (r , . ' ) r ivur r1 ) .Uo4..

Exoltiv(j)< ro ouvolo r0)! €(u.r6v bsEL6v n),ruptfi b dvat n oitotro .rovrtLt, o,1) tttaotet,,rut, otllit,ror A. A€Erir rl,€uad trir.o btjvcrrdv v.r €ivur ri i&aI iAolrt ori),I. 6r(1, €Xoqe ouvrrl.torrg a - I xuL r. = 0. Mr.r {!l}.n 6r,var6rnr(til(r r] brnrEarl ord,I: l1 =0 r{(lt r = L MLa iairrl Ei!(lr rt 6€q(i n},supirl) =0,or ' ,v , :0 r .0 t t r i rn\ , r tv rrrQltr t , iua ." ,1, , , .74 n.r , ,7,u,nut, . to A = O. ;rroro.zL (v rL \ ' ( t o j r l v (zar . l

Oro.pr l 'c r , ,p,r olou rouc ouvbu(ator lE rr , ! buo or l) . t i ,v xur rcprypri ,q ouyrr u , . r . t r t / . f o F . . 7 r u F F r p t v . r t : r < t . l . t = L 2 t : t t r t r t i r x a t g o n , L i , t i h t ! ! t t : x t r u toro t,tlnto nou iraQdyEret exi tu 6iro 6&lrj6lakr-orilE< 1:1. Z.ti. ir,.;rn(r ro ),€116 orvo).o rov eqrzrrirv 6_ Eciv ro 6 eivuL exr6< rorr i:rurdbou. r6reo€v €racr yecr$uroq ow6uaopoq r(,)v 6rjo orddrv xnL o. urn{v rrlv :repi.;rrr,ro1 ror1J = 6 6sv 6X€L ),iol.

Arlro nou 6Xs_r o4rcoiu eivut 61 10 €v ).oy(,) €Jrine6o 6ev eivaL ercr craL6u r oouvoio rou. R'. a),}.d iv,r( u:rn1t'lAo( xor Ou z{r}.Fr I ur ,.;pog o.r ntunv rcu i.u Torpo: orffun tyo: .Iy,rxu dnorr].rir(r lao 61o.,E touE .ruy'uaopoiE rrov

\IANY:MATIKO] XAPOI KAI |PAMIUIKI: E=E!2:EI:

oltlri)v tor zur org6oLil€ror p€ R(.4). H I:Eioaot Ar = b ).th,trur rltr xutydror &v o b:reptiTerar oror lriao ori)ltu rtr n. frrr 6'va'v m €ni n nivrrxcr(uIo( 0(l €iloL 61uq ur6/,o)!ro! rox -R"', 616T| or ottl)€q dloriv nr orvtotootq actoL or(Lxrio€r! (i) r{cr (ii) €ra},Io0uovraL €ircIa:

(i) Ynoe6orr: otr tu i, aclr. i,'neqrdlovror oror, 160cl orl),(;a xtt e:topivtrl<.4.r' = tJ ./lu xrinoro.? xar ,4.r'' = 6'"^.r xri:roro.r',6aor rc I rttrr .r' bivor.v rollEorrr ,btrcolrorrE f io! 7(uQdlotv rc b rcr b/ . T6rr A(r+xj /) =b+h' uL," r .o b+b'tivtlr" rni.orF orrv6urou6< r0)1 ortidra. Ed1, ro 1) LooiraL [s q (orrti,I 1) I'si.ov rI(orlt),rl 2) xor ro tl' toolt(t tr€ 2 q,oQd< rrl (or1L1 2), r6te to 6 + 6' rooura po -rr1

(olli,I l) ouv lil (oritU 2). T( F(tDird bLcvnotrara €ilcL ri)"€rord {,r: rQoE rlvr4, ' ' r t t t r4 r ,u I :TC, ' ,nt , , : r , rr tat 7, ' ( urnXd'pouc rxovonorr.rur.

(ii) Ei(v ro b ri.vt( oto xi-rpo otrlLr'rv. ro i6ro oop6(1irfr r(L [rr' r(i0€noLiarLcroro (b. A,' zdnoroq ouvbu(ollo( r(')v (rillL(;)v n(lo(ilsr ro 6 ()..2..1.r = ,). rarf rTolL(1'Ii(otdlovr(( x(ioF ouvral€o'irt rolj otlvbucotrot Ir€ c 0(lr(eiiloritr€ ro r1). MF ir),Io l6yrc A(c.r) = .r.

f€r'4rrrpLxri, I yr_vrlill n€oinr(i)ol .i\r0r ofi(,)< crulrl ro! :T,Iprtros 2.1 -carlurpfoou|t rLE d(ror(iftng ,_rorJ pnoQ€i !c ti\,(( nol.ir brcqroab'rL#q. A€v riacrc,'(ii,zn !( dlo![e i!( bLbr.ioraro eri]iebo rvrog ro1, rpLbr(or('rro! Xi)Qou.Il(poporc. o xri0ero< ;rpoc rov t(i)ix, orrtl,(iJv tdrQoE, norJ €nians oXEbtdor4r€oro:X. 2.1, FnooFt !(l [1]! ei!.r| $io e!o€i.l. H gra (}raci( nr.oijTT0Jol, roufrlxQorcQo! 61r\r(10r tdqor orrl).rijr, nqoiplrrcrL rn6 rov prlbevLx6 :rivaxa ,1 = 0.To tLolo Er.i\ruo[.r orov ydgo oq].riv tor (o p6voq orv6luopri< rorv or1].rin)aiv.r ro l, = 0 xcrL porov cmtrl 1 r:rJ,o1l ror I r:rL.rpdrtr rq Liol io! 0.r = r.trlv.!1.).I cra(ic :reqt:rrtoorl. uo06orr 611 ,4 t:ivuL o rcurorrroq 5 t:ri 5:rvuzuc. l -orr u 7r i .oo: or1).J,v r"u r ' .v,r . uLori . r iu ' ro R'z,rr nr ,mvrr or4l .Lcror-rcrrrorrxon t:ivcL 6wcror ',tc orJ,'Er(o0ox! ,{ctcl},inlrr zaL va rapiryotlvonorob| l .1orr o Livuo,rLr ro, n ' . Auro d' \ ' I r ! (r X(v' ,xr ' lL,r ,rrrx, , 1! , 'L,r , ,J ' , u, 'v, ,ror ruuorrzon :rircrcr. Kti1t y4 bipopc2oq 5 *i 5 :tiv x q lFL o16214Qo roR' oev rino arrtui)r.h {r dvcv rixoro nir.i)i.r u:rhqXouv 5 obilloi x.r. liopoupe!( ),iJoorFe ro /.r: = b pe .rnrrLoLrg{ causs. :w€rd_< xciee D crvrirL orov xti-rpoorrl).(i! s.,'o! [n L6r.6lLoQ(por ]rilaxtl.

Alrd,cFdirvroo€ $€ rior.o r06rio ro KEqirLcl|o 1 n€pL6X€raL o€ rouro €bd rox.(I]li].(Lo. Er€L F€],rx)iaouc rn! (nlouorepl (zaL nro ourloroldvt) n€pinrrool,tvoq nivcz( , €ni , xo1-,ofioiorJ o ZdpoE otrll6v aivcrL ro 1?". Tdpa 0cto-QorllL€ €riorl! rbrotloQqoua ni.!(1,i8! z.lL xraQ.1L).I),o"/Aa$Follg rii'cx€q ororibrlnol€olll[(lro!, r0)! ofioi(D! o lri)po! orlidv 6pi.or{€raL xi{rolj [€roEn rou |I6€rutonrro7rbpou zcL ol.o'r).{po! ro! ltirpou. Au6g scli pe rov xh0er6 ror u:r6xt.rpo6i3€L Fia ar6 rLq 6110 rpoorllioeq pcg yLc rtv xcrcvoqon ror -4rt = b.

,2.1 AIANY:MATIKOT XOPOI KAI YIIOXAPOI

O Mq6D?dl0aog ror, ,4

H 6srft€a,q naoo6fnot rou ,&r = D s[vc! "6!ir1i" qE ra(irqg. Mtrg 6\,6t-rr-qiQow 6X[ F6vo ronE 6€Er"€g cuq6g 6 elvol eglxrdE, cd,Ld xa! xo odvolo r(Dv)r.rioaov z nou rq emtulxdvouv. H 6€€td fi},€uQd t = 0 4€r nd}'toxE irlv 606u.ri}'rioq r = g. otrtui F-noQet vd undexouv xar dr€rQEE tilJ.4 }'r!oet{. (Kcr rmrig-Iou1, rrd\|Iox€ 6rsv o! dlltDoroL elvd! ri€QloodreQo! cn' tq elrorireE, 6r1},c6rj6$t n > m.) To airolnt rar lloeaN btt Ac = 0 Efua. dfi6 ltitro iou lraEdrawopatrx6E Xbpoq -o Fq6e 'X.neo€ rov A.

O pt16ev61oqog evdg ntlorid olp6olttfior ps,A/(r4) Nd! orroreLri,rar on' dl,aro olcyliouara , ytc ic oriold .Aa a 0. Ant4 sl,var Avag lridxoQos rou .Rn,6noE rcr o 1ci4oE oql,tix rlrcv utdlorpoE tol Rn.

IoIJi€l I oJtoitlol (i): Edv,4c =0 raar Aa' =0 r6rs,4(r+r') =0. Iqtrler,enlorE xo! rt onoirlol (ii): E6v .4c = 0 rdte ,4(cr) = 0. Kor or 6rio owgTi€E,raterrv 1,o rq6row 6(n, q 6€Etd rl€lQd elaau F l l-r46€vn ti. M6vov o! Ito€qe\i\ olroyE|oie olroaip'oioE (6 = 0) olotelow uorr4o. O pr16ev61oqog rou,rQortyowEvou ndQc6€l.T1otos 6Q'l,or€id eruold:

H nQ6tI sEiooorl 6ivel u = 0 xq! q 0errreQrl eEloool ovdyxdEer r6i€ to u = O. >'aunlv l,o!fi6v div rlsQirlroroq o pq6ev6logog leQdt(€! pdlo io l.rq6€vw6 bkivuopq.O Fdvog ouv6uaou6g noo ndQdy€r io FI6dv orl 6e!rd ll'euqd eivar u = o = 0.

H xardorooq o},-)"<iler. p6trr5 nQooodoooFs pLc rpirq otqLl nou €ivc! o1)v-6udo!d6 rov &io riQ6ioli

O I6eoE orlL6v iolJ B etvcr i6lo6 p'cr.ndv rou,4. H 1,da orntrI fi€e|,€leiaroro €nhr€6o rou :X. 2.1 r{ft ehcr driQ[6dg ro dopolopo io\, 6io oqld,v Ue 1|6onoieg lexrvfoqre. O Fl6sv6xoaog 6goE cworl rou v6ou dvcrc B *rpG1er,ro 6(iwopr(! pe ouwordro6E 1,1,-1 *aodE lial rd0s fiollqnl{iqto auroir tou6Lov|iouaro<:

[il]t;tlil

f1 o 1la= ls 4 el ,

L2 4 6 )

[ r 0 1l I c l [01l5 4 el l c l=101.L2 a 6J L-c) Lol

^IANY:MATXOI XATOIKAI TPAMMIKE:E:I:OIET'

O pl6ev61t.rpog rou B siv(rr I €rOsio ,ro! n€Qlix€! olo ro olll€io ri = c. y = c,; = c.6:rou to c perc6irll,etnr rur6 -oc 6u\ o€. (H €u0€itt r€Qvd air'rrtv spxri,o;rr4 aqr6(et oe zdroe u7r6x(J)ao.) Aur6E o povoiiLciororoE Ftb€v6x(,)a,oE i(€L Eva1/zir0e-ro xrirpo (evcr eri:r€bo) jrou ox€i[!€ra! aF€ocr Fe rLg orrll€S rou nivqra xar6l€[ [6(aix€Qo €!6ro(p6Qov,

:llvouiLoups @e},oqre. na rdo€ oimlllro ,4:r = 6, vu elttcrorE lriavoi v(6aioxor$€ 6),€E rLq eqLxrig 6elr6E nlerp€g 6 xct0<iE xur 6Leq rK Luoeq tou,,lr: = 0. T(r bL(lllioFaro b mprdlowcu orov ldrpo oqL6v r(t ra 6la\,io[cfic rfiov Fq6s!61(,)Qo. Torro olp(i!€r 6rr nQ6neL vo ]Jnol,oyioouBe xls 6Lqordo€Ls(ru(bv rcDv ufiolrbpoJ! x00(i)E lar €v( riordl,Lnl,o o!orrl$( 6r(v!o$drov nor ious:rrrtrriyouv. E).riloupr orr 0rr xarul,{lot1tr iXovrc< zutuvorioet 1(or turg 7!ootpr(un6lolQor( nou owb€ovr(r or€vir Frr(|Eu rcrE x(! F€ rov fiivcxo ,4. 6llcl6rl rovX(i)Qo oriL(iJv rolj A. iov [nb€v6X()Qo rou -4, rae6q ror xol\ 6110 rd0€ro!\ o'(rlrotE ur6xoQo!8.

ATKH'ET'

2.1.1 Aeiftf 6xr or orrclntofrs (i) xcr (ii) r'c1 av.r\' 6LovuoB.rtLro i(dJqo Filal re!4farra!€EdqrTps. r.1r.roxtli(iiovroElq) [!( rwoonlo],o ror 6r6tdm(rou xd]poo, xlflmo (d< raos ritv n06o0ror z(r

F(i).Lolc zoL rK raoi rlv cq,oiqrol dt.rvuo|drolv. o)!L 6NDE 1(.lL r,E neos to!ioi0,unldo(rqr6 trE aareFouq.

6) lvc rnoonvolo ior brbdor(rou xdqo! (dr((rcaet|r6 cr6 rd 6rJl) alriecrcrrrrerrtFoerc) x)..Lmo {os reor tov ro}.icrt(ordoFo c}.ld o;(r o; rqos fllvrQoo0rot d(rluott(rov.

2.12 nor(l (nd ra rn6leva unoonvo),(1 ftr I?r silcr nahrFarL u6rrpor:o) To 0niroijo rov browdpirro! ||r ra6ut oulrn6od ,r =0.6) To rrir.bo ro! bnvloudro! i) f€ ,r = ]y) r( dr.rvnolora l' !€ 611," = 0 Gt[i €il(t tt alour bio tnoX6sr!, to! tntnabolll,r =0 ao tou €mn6nou rr =0).b) To !r!o!o{ra!o 6rdvu'Fa b = (tJ.0,0).u) 'O].ot or olv6ldopoL tov 6no bLcvlorrdrur! lr = (1,1,0) 1|Cr y = (2,0, l).! )Tc bruvr ioparr l (br . r r , r r ) no! t ) r ( lor roro lv q! l r br+3rr =0.

2.r.3 fi€eryecilr€ rov xdreo on]l6v xsr rov Fl6€voi(r)ao tuv [v(x(,v

, - l - l ' n _ f 0 0 0 l

" o | | l xa r "=10 0 o l

2.1.4 loroE €ivar o Ft,E t€Ao( uiroX()Ao( rov mvdxo! 3 €ni 3 rro! n.ardXfl ol,ot( rol5orr!€torlions rivorFq lior 6).0l"r rolc xdro) reryolr,ror]s nivo?fri Ilorog €ilaL op€yclu0ao! un6)(.)Aos no! n|ardXt1ar noL orouE 6!0 duol|< rinot.beols!

2.1.5 A16 rt! na6oe€q rar rov rolianlaoLaouo F€ aaLoFoL\ o' 6vo 6LavxducrLao rdqourdrro\r v,, rzuvoro'ouEur or uoFr vor xav6v.s:

2.I AIANY'M,\TTKOI XAPOI KAI YNOXOPOI

2 . r+ (u+ , ) =@+u \+ z3. Yrdax€r dva $o'a6r't6 "prDe!u.6 6r&!opa" 0 dlor rlore r+0 =.r,

4. rra zd0E r utr(iar€L sa FovaoLxo -r EoL 6cr r + (-r) = o

6. (crcr)J =. r (c2")7 . ( ( x + ! ) = $ + c !8. k l + c2) I = l : t r + ( "0.

ollldri(otJ! enift6o. €!0€ia, qtr€i.o. !n6toqo, FrpG!6X(')Oo rox A, rdreo oflti\(hr

2,1.9 Acilt€ orL ro ouvolo to\, Fr r6LoFoAq(ov 2 €ri 2 rrvdao)v 6€v €ivar 6Lawol(lux6sI(i)aos. AEiqr€ €triofi 6fl ro ovolo r()v r6Losoqqrv 2 €rri 2 rLvdxov tuv .ivolbrdwofanxo< xinaos.

() Yro06c€ (nl I neoooccrl co,6P reooosr.r Frd Fovd6a os,ido€ oufld,ss.oftie (3,1) + (s,0) Loo$ror Fs (e.2) alri y|a (8, l). noror Tdvoa€s xdtanq1oi-wor cv aqaoouf€ rov t6ro noD,arl,aorooto $ ooreiro',r.

6) dsiEr. 6rL ro ouvoLo d),o! ro)t srrrx6v sqr0p{i}! F. rLs rredEc6 ,: + j/ xft crcl.Ioyldvr( 6oif \,( (ooi!1.1! dvrioto(Io !e ?y xdr rN'. drqr dltrs bm\ooldrrrost(i)ao!.

2"16 'Edo, l ' ro€nin.dorourqL6rdmsroUx( i ,ao! !ceEio(osrr+2y+. t = 6. nora s ivo! r rtliooq rou Frud6oD l'0, 6L€e')(ofdvot' (no qv sgxri x.l. rded,Ii)@u mo Pr Eivorra /' rcL ff1 u6xtopoL rou R3;

2.L7 noror otr' rot5 €rdlltvous Eiacr uidto)Ao! ro! R-rc) O),Fs or oro) .or ,0 i ts , dno)< n (1,0,1,0,1,0, . . . ) . no! nrq ap6dlouv dr i€LqorEpl6€vLXouS ri(torE.6) Ohs ot arclotjoi€g (..r. r::....) p€ rj = 0 qr6 rdtoto o4rEio xoL n6Q.l.y) oL.s or e€irousrs (lxolouotcg .rj.' < :rj 1La iof j.b)'ol€s or ourTlivouorg axol.ou0is6: ra r:r 6)(ow 6aro n(0dE ro j - -.€) O),8< oL (IeroF4rLxds ndo6or: i.l,I - rj .ivdr ro i6ro ym xder j.

t i oLfc o, y fd |srerx iE naonlror ( r '1 .1. r1. , ( : . r . . . . )s f lLre inovlKoloro l roL.r r .

2.r.8 tloL8s drr616 snouslK n€0ryga.pls €ilar u0rords: Or llJoeq i! ro1,

*=[i ; i] =t3lf",tL";t

IANYIMA IKOI XAPOI (AI fDAMMIKE: E:D9>EI'

2.2 r H AI2H m EEITOIEAN ME r ATNA'TOYE

H 6rc6tzcroia ofld,o(plig €ivor ;rl,!ov :rol.i oLxei( 1( rrrpoprvrxor5 :rivcrragzcL upzti ilo rcea6€L1rc 1<l vtr eqv€QrD0oiv or vdeq 6wut6rqteg aou nn-eouortrtowsr 6rcrv o riva;raq €ivco ncaaLl,4.6yaq+ro(. H i6ra 4 n;rai'otgt1:rpolarpei Xtoqi< oqpcr'tutic alL(../t_<. 6r(v 61oq grdv€t oq1, €uQ€{tI rIE },rlolt !t€crcbpoBrxrl (!r'Lurdmclorl uir€to6Qlovr(! oQlolrh€E fu dqoQ6i.

'loroE eira ot-ror6, nprv ro fi(ro(i6€LTra. vo 6[oooqlliooritr€ rL6 8wororrFrq1.(orr{irrciE lJl! (rQro[N!xr] rEi,ooron ar = 6. flQdx€lrqr yra "o!orn$a" $[cls€EioororlG r{cA .v6g o/!(i)fio!. MnoQ[i vo €ival 3r = 4 rl 0r = 0 ri 0r = 4.nupubrintrt,r n,ru qcvrpo'vow ttg tptrc 6wrrr<ir4rt<:

(i) Eil1, d 10. ror€ yrcr zcie€ D. l'n(iQxer Br(r ),ior1 r = 6/a za r1 Inol cr,r{Fiva! lro1,c(b!zrl. AtJq €ilsr n pq!'661topqn xeeixrtttol (€v6q .r}'rr('Ia6rlilgou Irni I nivctxa n).

(ii) Eirv o = 0 x(r 6 = (). r6rc unirplouv dtrElQrs Lllor:li xc(r xdeE r:rxovofiofai rrl! 0ll = 0. Aml tL\@ \ atip@iq neQinrool, 6fio! wdaX€r luoq(rL),.( 6xr pov(r&ri.

(iii) E(i! rl = 0 x(r| i) 10. 16r€ b€'r' ur/rQx€r liort q< 0.r = .,. Auri eivar I Flor4t6t6 aodl nqiLnlli,al.

O),eq (uri< o! ,rsQlrwdo€K lroQouv vu orjF6oiry xcr .[|(1 T€TocrycovtaoiqnivctzE<. A!flx(0rmoill€ q! 'd

l {)' ge rqv o A eivcrL cvrlorQ6lrLFos ', c'lo!o ,4-r iler :rcil.L v6rp( frq rraQr(LuL6yQc44roljg rivar4. r1 bwar6qrc (i)eEc1qa\,i!€ror xo! bev luroQ€t !(l aloljlr8 ia1J16xeolc ljncQ€n xa! povoorlpcvro

!uct( ),!ong .r tL( xdo€ b. MnoQEi vc nrdQtouv ir:reqrE luoe6 1c xtior I rlcirrLqeE ).uoeLg 1c xrirorcr 6 xcr ecppic ),6o1 yrtr ri),Ic ri yia Ltlol yur rdcroro lxoL zq+ri.r./L( iriJ,(.

3ezLvripr gr dvcv :rivcza 3 sni 4, c.ylotottclE ortv op1ri tq 6sE!(i tri'€uQd ,:

O obly6E ar = 1 6€v lr]b€vi!€roL x(r or ornlOrqrdveq ororx€r66tlc iraAE€Ls ea:rcrpdryow pl6evLxh oqv :rpdrq onil'1 xcito cor6 cur6v rov ob{y6:

I r 3 3 2lA=l 2 6 e s l .

L l -330.1

| - r 3 3 2 ll0 0 3 r l .L0 0 6 2 l

A.

o xroliqLo( 6€ur€Qo6 oBnyos 6^/l!e pn6€v rar yr'dnro xorrd[€ aro lidro ylopq pl6evrro oroLXeio -€Xovroq raxd vou rylv €vo).Lqrt ya.4rF(;)v. :qa raaorioorrpiruan ro rrrotTtio ut6 xnru t tuu tnioq< lq6iv itut !\ o (rQXrrdS nivuxu€

I I H r\1:H d E:-I:A:EAN NIE r ]\fN!2:lOY>

Ir(r rr:xacr1()vr.rog. .1116 0a €ori$ar\'E 6rr o :rivnrt6 Eilar Latottopqoq. M€rcQd.).Iio./Q.1!r$o nivcrxa. onoq zar l(l z!,r! nQirisr !a r€ptlevowE F,.I€lhb€,<xcr 68 rn.ialrL l.6.7oq !c reppcrrioor4re q! cnc)'o[(prj. To lt6to nou lroaorJp€rn zcivorpe ei.rcr. rc rci7rc ar4r t pn'1 ar/1).4, 6:to! o obl16g bev ltlbevii€tar.A, l . r |a ' rruc b"o q oaic rr 6r t r t r pr 79' ,114, tro rr tv rQr r t . qr ' l !L,uf I Lrr , \

[] 3 1 ?lu= { ) 0 3 Il { )oool

Mri,(irvrq (lroTltad. nQoxo)Q(if€ orltv r6tttpttl or1l1. orot owcrtr'rpt tii'Loprl5evLz6 orry obityrxl 0aoit r(r 6ur ttro0orft€ jri€o! va ou!t7(oo!tr!-. HbL(boTrzti Gncr),otq,rt i:X8r o).ozl.tQ(')0€i.

H reilll{ goprll ror t,I Ft,rcu irdir dv(,) rpl./o)vr'n1.611r'r; or obryoi* btr tivcrxur' cvtilxl orlv zrprcr br<yrirvro. To r,v6r"tr<1ipor etrcrL 6rL rct trl ytbtlvrr.i(iororz€ic ouyiic f,xodaovra( o iv(r (norrrnr4ru anai,ut i lu(t xL4taxar\ poQrpil:ror oro:X. 2.2 rcrrLroriirrnL yt(l trts r€Qirircroit 5 t'ri 9. Or obrlyoi 6xow ftnFtor zuxirinra. rvri; rru irlLa otor,Teic propti va t:ivttt t1611 p1b6v.

0 t * *

0 0 0 ! ' *

00000

00000

0 t )

00

0 t

0 0

!r.2.2. Ik r|ltrltb€ura ,rrortei( .atic tliL"on +rcrorroi aircrt Ii

Mnorx)nFr 1a ol,loUioo4ra $E lo"/r.r arir(i no! ({(lradrvEl to oXilt'a:

(i) Ot lrt $16 'rdq yacFpd! +(r1,'i!o!rar rpdr€: 6rt1q]oa€ru{(i 0u,{.1!.+e€v()J,(yi:q 11x44rri)v- r1(tr or obn./oi €ircrl rc npdrc frl u6t!(zl1otolX€irr (mcbv

(ii) Kd-rtl) cir6 z(i00 obnlo rndaX€L trrcr onj],1 ;rq6evLzdrv rrou nQotiuntou!tr€rt (1n6 Grc),or(t{.

(iii) Kde€ o6lyoc 6QiozDr(r or( 8c[r.ci ror ob41or rrlg cl'I6 rirvor 1p(ppti;zct crm6 rcphytr to uorllrr ,tt(r or.i)..G.

* Or(rt0.ir. dr or ohryoi rirar pI pl|6nrzoi. K(r.i lt 6L.ia".ra lrtq oroiorqrj!. lroar r(

;q.ntc lrtb.vr"o oc trtrr obllzrt oiot. (it.,i flud Fllat ae.oo)!avo. Eldi.iooolr< ipqrtra! rl

d(riorki trc oniirt ror nrydyolr(q urrt fi6l,*rt. a(rd,.Iioltre oe lrLd cad6( (ot r|16!!,,i)!)oblirt! 7(r trtbrlLzdr (ro ?,i(' ror\

AIANY>MATIKOI XAPOI KAI TPAMMIKE> E!I>A:EI'

Aqori E€xnnoop€ Ie rov r{ xor }i(ra},iEcFE ('xov U, o avoyvr;onK oilorQaq.l ftv.rQ(l'rtori. nwbiovr$ uuroi or niv.rx|C UF ivov x.iru rar./rDvrxo :rivdza.6nr,E xcr nqotloutrdvoEi IoXU€I ,4 = rU; Aqou ro 6yl$axo rl_c oiraloqrtq 6svcil),aEa\', 5€v undpr€r L6yoq va Fry Lolier. KcL ncil"L oe zciee 6rlpc aqcLpouFe 6\'cno),Lcn),iroro Frag ypcpprlE c;t6 pLov ci),1.1 xcitco ot6 nu{. To ovriotpoqo xti0c6{pc.rog 1ive.r(! rnio6 6:torg ncrr rpollorp€var!. raooOftol.rc{s €vo Trol},on}.dorc:r ,- 'u;rpo4yor4r ivu; ug0rpioutrr . Aurc ro ovI iol-uoq u tniof i y ivuvlur ur lQonofiou pcs- €1r[1Q6rr€[ vc 10 €YYQdqoup€ xoT' e!0€[ov olov t:

O cvc./v(i)orrlq [roQsi vo 6[on!or(i)o€! 6rr ,4 = /-U xal tkr:rQiner rio or][€r{i)oer6r o -L 6ev €ivor" n<raol,Ltl6yaa$FoE dl.).d r€rpcryowlt6E. Aur6g eircr rivcrncgrdlEeog '7) = 3 rrou €ivar ro ?rlioog r(rw lQopldv flDv A xdr LI.

H F6vil irQ(iErl no! b€v XQ€r..oory1€ oro nc,rQ(rb[lytlci [oE, o2,]d XQ€ld(sxor, s!ldv€[. eivc& Uro €vol,],ayll yQoup6v.'Ofio( itcr( oro KEqq],sLo 1. auro 0q Erolly(l./€ivav riralta [Elo06o€og P, aou Faoeei va xdv€l evcL],oliE yQ<rgpdv, ,[Qr!clQzioEr I crficrl,oreri. Aqo! nooXorodB€ ofllv el6pevrl o{Lq. 6T crv 6ev l]flarpzowoblyoi oE Ulo 6Ebo[6q orlil,l. b€! unitQx€! xa! svclllo]vc( !fiooaoorJt€ orl o ll€i.vcr( Fl r6!o[oQ'qog. Ibo{ ro ituQ[o 0€6Qrlpqi

28 >E &d0€ nr €fii rr riivsxs,4 cwroror,Xei dvsq nfrqxqE Ferq06o€04 Pxcr 6vcg xdro rpryovr,xog niloxcE .L ps povd6sE o,rl 6loydwro, 1.idod)E ,io!aloE rn sfii ?r zhgoxor6E nLvdrrrE [/. 610! d)o[e PA = LU.

o fiozoE p(lE ro)Qd €i!(rL !c 6Qo!l.l€ rlE lio€rE (€(p' ooov lTidQxour) rouollor4$oroE ,4.r = 6.

Yro06ote 6tr lenvcipr pe rlv opolr'v{ nepi:rroor1, ll = 0. T6r€, srsLbi ornQllE€[E or|E ypappdg 6€v 6xo!v €ni6Qaol fic1 Fn6e1,Lriir q! 6€E!i1E ri),€uQdE, qAf, = 0. a\,dylrdr .l'i)dE l'tlv Ur = 0:

[ 1 o o lL= 21 01 .

[ 1 2 1J

liiiillllillU.t =

Or d^/v()oror r, r, ar xo! , IoQiqovroL o€ 6rio o[d5E<. H pi.o oFd6a .rrrorE].€ir.rrar6 atq 6aou.iE pna61nr4, oUr€E rou avrLororxoi\ a€ orilxE 1tE o,|yonq.H :raurt rur rqir l ort l4 nfpr i louv o5ryorlc. c ipa ra u xur u r ivar or ruprrcll€rc6lrF€q. H tiL},1 opci6a ororrleircL cn6 rr5 el,eJ0eqeg p*a "qtis no.l

l.l ll ,\\:11r'J E:EA>EaN IIE 4 ArNO:TOY:

alrfoiorXonv or onl,xE nou 6tr ttEprqov o6\yois. A|rtg rivar I 6euep4 zcrI rtrftArl orriil, cip( or r,. rtrL, eilcr s).er0eQ€q [€r(61'11(<.

frc vrl baortr€ rlv rrro lavLxit inol ror a,I./ = 0 (it rco6nrqrc rou Ar = 0)pnoQo!$e vu bd)oottr: a!0oiprre< r+16_< orLq rl€u0sps< [sTa6l.tr6(. Yno0aor€ 6rLolotL(iEottrr- oir:: autg rrg rrlL€q. o:rlo;q. r' xcr 17. T6-r€ or 6cirLxic pt:tu6lry(::rpoobropilovrur <rr(otllc ztrr or r4rdq rollq rrno).oyiiovrcfl pdorrr rarr r:).rr0ipr';vtrrrc6ilr(i)v rcr ptr roubpolLzrl crvrrzarciorcrol. Arvovrug apoq ra arivrLr.

3 r , + ! = 0 i T a a € X € r u ' = ( 1 / 3 ) t /

u + 31, + 3ir ' + 2y - 0 rcp6X€I rr = 3r ' , .

Yraotrr "5rni)i rurrrlliu ).|or0rr ror ouorlfr(ro: !E bi,o €).€riJ€a€q n(0o-t,*raorq r xcL i/. H lFluil inof ri.vo' o ouv6Lr{iot6(

llil.'[,i]Kolrdirr., ;roa(1]{(id, ndl,L ritlr rrLulJTokl troQql qE ),ioIE rorr ,,1.r' = 0. To\. i l r ,otr( ( 3. l . { ) ,0) c\ i .vfr r i t i . !ort , or(v or s l ,uu0rpr. ; pf t<rt i )"r1r ig t louv rLq rrpiqt = | ter u ={). 'fo rs}.r'rr(i.o bdvuotrcl rilcr il 1,!oll. olcv r, -0 }r(lL ?/ = L'O),tq oL ),tiorL; r'ircL lecppLzoi oLwbrrcoloi alr6r' rtov bro. Xve:rtb;. drcg xc),6gra6no( l,[( 1( LiporfLL^ o).e< tL< Lrlo(c ror /l.r = 0 r'ivcL

r. A(pn rl rrnd.orqri qrriotr orrlv ti.r = 0, rla F!(xa'6drooritra 'rore< fil,rrlor 6asrza< r.1| xolFq ot €:),u]{)rA€< [€rotj],Iriq.

2. Na bdoouur or trr( 0i,ni0!rrt trrr(rijhFrirlv fltlli I, oT'q (iils( r).€norergirfra6ilr6q rlv r+r4 0,{(! \,c L ft)utLr- oJq neoq rq ijoorxfq [er.16ir]riq.

3. Kdo€ el.sr0€al [Er tii,ryi ,.r.rQdyrL frL( ).riol, dot'r rou 6rlBcrtoq 2. r(r oLd,,!6r(otroi oLtov audv tor, Lnoeurv crrotrJ.o[v rov prlStvriltupo rovTdrpo o).1l)! tr)\' ),io€r)! rlq ,'1.f = 0.

ftr4tnprxr?. 1rrr6rc tivcr it FEiq: Miorl oror rmpabLirorcto xrirpo 6l.ovrfrl bu,ardr! 8L(vroltriror :r:, or ltrotLq r1c l/ = 0 a:rore).ouv d1'.1 SLbri(]rcrofni;r€bo ro! y46tvir1,topo ron ?,1. :ro ncAdbrryBcl. cru6< :rcrphyetcrL ano radno b'clnolclrcr ( 3,10.0) z(L ( 1.0, (1/3).1). or uwbucropoi crut lv rowdr,o brovrotltirrDv clnorEiouy €\ro o[\,oio, a].Etoro o; lpo;-r1v :rq6o0tor1 rcrL rov:ro1,).rrnLuorcot6 pe crprOloq. OL rp.iErL-< ur€r tl6ryoiv cn),(ir( or nrAroo6r€pcgi.r,oEr: rrlc ,.I:t = 0 raL oioL .1rrol oL ou,6r,c1oFoi crlotel.o& tor g16ev61orpo.

Arj{o r-ivrlr ro oltr€io oro onoi.o ava.rurpiiorpe rr( eE(lrprrLxir rvbr.cqdpo.v(l!rirQr}L(. Yro0aore 6rr lexLvci$r pr dvcrr:rlvrrr riolr il€r n€Qroo6r6pe< orrl),eq(cr' 6.rt 1!x4rtr€<. /, > ?r Tor€. aqon lnoa€i. !a u]rdaxoN ro no),r! rrr o6ryoi(nEv undplor\, (Q,.!ri< ypr4rp(g Ic vtl ollzpcrrjooN n€aloo6rrpols). 9d nain

A]AN12MATIKO1 XOPOI KAI IPAMMIKE> E:I:O'EI'

va utaaYnv, rouAdxlo'tov, n. rr'.deEA€Ep€ra6,l4riE.Mril'tota0cuttiplor.tvox6pl r€aroo6r€a€s €l.€ue€e€E p€rq6trqr6g €dv oup6oiv€r vo Fl6aviqovror p€8!rc8yQ<44r6gTor Lr. flciwoE 6,rr xol aa otrp6aive; toulriloror pic at6 rqperc6l.qreg rp€ner vu ei.vcr eler!0ep1. :irl p€rtr61'Itr1 arltri eivar 6rvtrr6v va6o0ouv crocipeteg rr.p66 acL cur6 o611ei oro en6$€vo oljF l6Qaopo:

2C Edv 6!d opof€vgE orior4po -4o = 0 61el neQroo6teQols d'ylci)otolE ct['6,rr eE[o6oel,( (D > m), r6re *Is! ro! pla Fn r€'teqldq l{ol: Y,rdexe! }.uqrtr, il(lqoeflrri ar' itlv r€rQrUpdq ,r = 0.

>rlv 7[QcryFarll6r$d ed rJnaQlo v d.]r€!Qsg lio€tg, (rqoll xdo€ nol.]'crfiLdotoer 0s lxqvonorei €riiolE rlv ,4(cn) = 0. o $16€vol(,)Qo( fi€Qt6x€[ try €ue€ia 6dtou :r: xc1! etiv uairplow r[EQroo6r€QsE €L€ue€QEg p€ra6].qr6E, t&s o ttlb€!61c0QoEytrerdr l€y(r].ur€Qos oTrd F[a sueEl(r torJ z-6rriotcfior x,jspou. o pl'euiTapogiytt t4 t' i6La " 6nmao4 " ye o .fi 0og t@v d.eu9lpat pen6)'4tti;.

H xewgrz{ r6ad qE ttdotao4s e16E !fio1dQou oqi.(er<t cxqL6cbg ottlr€n6F€vn ,rqQdyQoqo tio! €i!a[ 6!aq sQ!0[6s ]iou petpd rorLE 6c0por,5 eleuoepicE.

H pn o[ol€vris nsQi,]rtool. b 10, €ivor dol€td brdqoQentri. EnrotQdqoupeoro oQXlr6 TrqQd6e["/l1o ,4c = 6, nal errqQp6Eoutl€ lt6 tlQsE€tg, Trou o6riYloc(! afi'rov ,4 orov L/, rru ot6 bio nlflqig xIE €Elo(l)oIE. To qror6Leops 0a eiv(rt ivddvor -rpr1ovr,x6 ouotrpa LI;r = r::

To 6rdvuopo cno! +qqviorlxe oq 6e!Ld rl,etpd F€rd ra 6ylpaic. i{s crnal,oLqrig,eircr. ro -L rb. 6nog arpr6dtg tot. oto rporlyoupr€vo x€Qalalo.

A€v €ivor ooadg ov olrr6q ot €Erodo€6 6tor,r! Luort. H TairYl €Eiotl-ron €iva!rolu olqi6o]'I. H deroxse{ qq n}'er.rpti eivnr. gr16dv xcr ouvutrirg ot EEtobttxr1tiv.u ctoll!6i6aateE, an6t edr h - 2b, + sbr = 0. M€ dlLo 1'6Yrd, ro dtirolorat' ryLxrtitt' \Lauuap&au b 6 lar 61oq o taduiotuxo< XdQoE. llrua' ol'o:rou urciplow mploooreQot lry\'ooroL e?i' 6,1! €ELod)o€rg, F rop€i va FYN uridQlet).uo4. An6 r1v Ilnphlpoqo 2.1, yvoai,qoups 1,ic! 6vov d 'o xQ6tro vd €E€xdEooF€ro i6ro lrprpn: 11 ,4e = b }'t!ercr, r6te xnr. govov r6rc, 6rc! ro I rtptdlercrr orov

Xdpo ontLd\' lou A. O ev ).6yo ur6lropoE ncgriyercr" a?i6 its l6ooeQr( orrt).Egror ,4 (xcr, 61. ror Ul):

(2)li iiilfi] 1,,r,-,,,1

I r l t I l f r l 121

lill!ll:lL;l

flcp' 6),o nor t:rdpXouv r6oosQ( 6Lqvlopara, oL ouv6ucopoi. rouq y$iEouv

[o!o! tvc Enin€bo rolj rQrbLdoraxou X!)Qou. H b€ur€pl oril.I €ilal r0sK qiop6qr1 rqdrq xcr 1rdrcpq or1l1 efvcr io1 pu tlv rpr,;tq ouv r{i6ro!o xt!(iol.ro trtErpirl<. (Xrpcrtbote 6r cmdq oL elapqp6rq ornltsE. I 6€ur€ert 1{ct 16rctQfl1. €ivc!rxeirq crgrr6ri-rE ;rou 6rv rlow o6ryo1E.) O XrlQoE mlild)v pnop€i nl,6ov vrrnEorlQ(rq,€i l|€ 6uo crtLdr; brcqoQ€lrzo!! rQonor!. Afi' r1 l(rc tl€Qlct €iv(r roE;rin€bo nou ncQo./€ror rl'r.i rri ooiJ$( I z(r 3. Or (iD,€< orrll.rs ,rrQrilorror o'(r'ro ro €rin€6o zcL 6€v olrl€LoqtQouv rinors v6o. Ioobivctr(r. uurd €ivuL rosnix€bo rou (noral.€ir(rr cn6 6).( r( bravuolcrrc (6r.6r.6r) nou rx((ronorouvr1r l , - 2/ , . - 5/ ,1 = t) ?, .r i ruroc r i \ ' ( o nFgropropo: :rou rcirrr !u ryuvo:rorr i rur,:,ort ro oriinltr(r \'rr iXrr ).!o1. Kdgr oriU rx,rvororn aurov rov xrQroQrotrdno! iror €nr6ril.).srar ,iar oro /,. frolsrprxd. 0c bontrs orr ro 6'(ia!ott( (5. 2.1)€iv(l| rdo€ro oE ridur o il.n.

Edv ro 6 repL6letcL o' curri ro rninebo. cipo crtlzeL otov l{i)po orl}.dr.rrirr. oL LuorrE ror ,4r = 6 6eioxolirt! r:iJXoIc. H t€Ltrltoic( EEi,o(,)orl xouoljofil[(rro< (v(iyrrc! fil]l, 0 = 0. :1r< c),[rot0cE $cr(t6trIriE r, x(rr ]/ ](no0ouB€ v(lbdooutrr cr€rri.a€r€-q r4dq, 6iorE x(L npo11o!p6{,[. T6re or 6corxi< grr(6],qrig:rpoo6ropilovtcrL nirh gr trvtr6pogrzti (lrr1{ctdolrrol. E,E€tiriolpr ala *6rxo:rugcibtrypa, oro onoio oL owrm6o€q ro! 6 61ou'e;rL),e7ei vo €ivur l. 5. 5(naooizortr€ v( r)|cvonoL€iruL rt ,r - 2rr + 5br = 0). To oior ra A.r = , yiv€rcL

I I H \')ll, E:I:a:EAN N{E a ArNO:TOY:

H (rJr(),orq)rl ro [€rcrQdner oro

H rr),rurcic eliornl rilcr 0 = 0. ontrri crvcrtr€r6tar. ev6 oL r?.i.r; birour

3L '+ r=3 '1 '= r ] i ;u + 3 ? r + 3 r ' + 2 y = l i t u = - 2 3 L l t .

K(lr ndl! ]]naQxcr FL(r bffiLrl (nnaiti l!|oel,)v. Korrd)!.(,< xor lrE r6oo€Q!!ou,'roi(i)os6 |r.r!,i.. rt y€,'rx{ trilorl [iroari vc( lQc(qei orl $oQq)i

lili;llfllil

lii;illllt;l

|'] l:l."lil"'lil(3)

AIANY:NlA'I'I KOI XAPOI KA PAMMIKE: E!I!O'EI:

:r./dir = Jr,)[i] + rotuFroi"

Aw4 €fvar I i.6!a u€ rq l,riol rou /xr = 0 oqv €Eiooorl (l), €xr6E rou 6rr undQ,X€rz(.! €vo( \,609 6Qog. AutoE riva ro (-2.0. 1,0) nou .t''t.n Vw tr'txi ^{ort rouAf = 0. Aqoi aur6E tr €r ro orimnro. zqr6fiLv or 6{0 r€},€vI(rioL 6QoL ncQdyowrsQroo6rEQ€_c tujo€16 (616r! rrnvonor,oriv qv ,4r = 0), Kdge hio4 iov Ar. = bdvar d0goroya praE etirxtlg ),io4E xar png 7ia4g tou Al = 0:

To opolevdg zoppdrr rpo6pxetcrr an' ro1, |q6€1,6xoQo. H €rbu(rl hiol rou (3);(Qo€QXer(r ofi6 q ),uol ro! ouoflll(Ios 61av o' 6),€E rLE €l€ior'QrS F€!(6I'I16E6X€r bo0€i I rqrytFn6aa. Auro €ivcr rrlr ro F6vo v6o zopp<irL, aqou o p16ev61o4og6xer r]b1 unol'oyroeet. Orav nol,Lan).corrioere qv eEioool olo nlaioro F€ rov,4. nciplrre ,4ir- , . , ,* i =[+0.

feogerprzri, or levrriE ).rioeq lepilouv rcil,r prr blbLirororl €nlqdv€ltl -irotl6ptg brv etvcl un6XDQoS. 6!6r! 6ev lepLiler qv crpp1. H ev ).6yrr enrtpcivcrcrivit ntrQdIULI :ruo; rov p46rvoX"rqo ;rou Fi1(rtrr nQortyor+ivr'4. xur dXetpcrcro:noOei xurh qv er,brnri l,rio1. 'EtoL or unoLoyropoi :reprLr4t6irvow irc

l. Avrlrrye ro ,4.r' = 6 oro U.r' = c.

2. Mi66vro€ olr< lrs €lerio€Q€E prrlr6Ll]1€E zor 6Q€r [L.r €rbrri ].iort.

3, Kcl!€ rry 6eE!d ri)"euQd [lba! x(l! 6d)o€, brd6oxrxo, o€ xd0E €Luuo€QrlFtT(r6Llfil rlr, 1rl1ll d1,(. Olrol,ra< x.r!'r6lQovc oxli dll.€q €},€i0€QeE[rra6].triq rlv xrtot FI66v, 60€5 Fra oFoy!:}'lt ].not (6v(1 drhvuo[a x rouB&€aoxd)oou).

Ilgollorpdvr,4 qrrousioE€ ro 6rtpc 2, Ornv q e(ioorol rjrav I ,4r = 0, I €r6rxi).roq rpcv ro Ft6€vrx6 6 ivxopa! Aw6 rclEdler oro 1err6 oXripq, 6Fnq ro.r"io,i = {) 6€v €txe rurra)o€i mry €Efoool (1). Ttirga ;rpoorioera orr5 ogoleveiqlLtiot6, 6n4 oro (3).

H (|noLot,(pi cror(r),inrer rov qQr0!r6 r(Dv odnyd! xa066 rt(! rov oQrEr6r('v ek!0€QrD.f'[Exa6],l1ti,/. Etiy ut tiq,Xouv r o6qyot, t6ra undQ,Xour | 6d.ouiqFxrsqAqrig xar n - r Ettit0tpE6 pEtatlqrtE.2' nrr6v rov nqr0p6 r 0c 6riroo4rcrvcr 6vo1rc eivcr r1 rrif4 rox nivaxa- xat 161,I ara).oLtp'j $noa€i vc orrvol'roe€i(|)c tEliE:

2D Yno0sor€ 6rr I ara),or.rr,rt urr;rttr to Ar = b ato U t: = c 'EorrLr otrrncipxou! r o5ryoi xcr or rr r 1pc4r[€c-iolr tI tivcL p1b6v. Otl undaxtrgrcr Lrio1, r6re xcA 1t6vor. otuv or tt)'euoitg tl| r ouvror(.]o€E rou c tr,vl1r.eriorlq gr156v. Edr r' = rrr, 1jrdpler ndvrott piu Ltrotl.

H TevLxt )"uol ri'vtrL ro hgporolLcr pto< eLbr"rriq )'uo)t: (u€ oL€< rL!lirru€a€q tr€rft6hri< [I6€vtotriv€g) zar pl(1g otro./€votr: )"uoi]! (r xq n rcleioepeq F€rc6),tri< otrv crveltipr{rrq :r(qopirpotlc). Edv r = n, t6ro titvr;rirplorw e),tr0rpt:< pero6).tr€g I cr, o pl6€v6x(,rQoq nEqttxtr lLovo ro 7 = 0.

O cpLtUoq r ),iyrtc{! rd$r1ror nivaru ,4.

:rp0r(i)orE rr: lro (,ta(1ifc nf0Ljrrdo€G. or(r rlr(ih Fi!(L rt f($ rlrrrfpt1 5r!(1ri:

( I) Or(! | = ?r. 161r b!,,' un(i0xolv r).€rJolafc trfr(16)"tr6( oro .f.(2) Ot<n r = rr. rrirr'6ev rj;rriitXo]'lr $il6rvu{ig 1a(4(tra( (tro! {/.

fr( i = /r, o l1<\eroTroixrg neutlztL tr61'o ro / = 0. ll tdvt ),iol ei1(r 1lf , i \ , , r . I i ( , = rrr . 6n r ;rc i t tou ntpropLot lot Y(r 10 , . o X{i)po( orr lL( l ) , ' s iv( lol,6ri.rtpo ro -R"' rcr-ro ol,orip( ],tlafrcr i/Lu 2(i0f beFr(ir n)'tua(ir.

Mur ttpoaqnul naqartp4q :€ :toLi.d 516)"i( rl bl(brxaoic (na)'oqiiE baloraparti orov fr. (Ai,('t duvt7i.qrl trlxpr< (irol o;ii\r(lrac aL0!-r o. it(r.lxofrrl rirou:rl{ avqlpi:vr1xi.qrcxorrritrop{t,ri . H bLllr{.opti 6pior*cr oto 6tL o}'or oL o6ryoixrlyovrTono(oilrcr orrtv r4rt +1, 6(Qt0vraq xri0c 1p(pLltri pe ILcr orcr0tpri. rtrL;roa('tlo,'r(r Urtb!',rzch 6/r tri)vo x.lI(,) al.iir rcu:rrivto (1ir6 la.i0r o6il1o. fr( ro'tr r r ' , r : , , I r o ' a l , ' r t u , ' J L ; ^ 7 " , r ' o A ! r 1 0 , ( r r , , !

'Ortlv o -{ r'ilcr rrr0(./1D1lr6! xa yrl Lbr6poqq'oE rcrra)'11otpt oto'v tcturotrzoni.\'.rzc. |Iaozur.rr 1Lrr gLcr :reqi:rro-ro1 at< ffialot.tiq Gdrrl'./,r/d, clvri lrl:owlOropivl<, zarci crus. clvuytLryrt< oril! ,{ = Ztr. O:ra4 .1rar6dq I GaussJordan rilcr :tLo crpl{ o: xQrl',rrxolr< rrololtolror( $. rnoai,c'rwaoN triv(lxt;,(r ,.410F r(Ll,Lcrzri boprj Tcirt:ror orov I r, i:roL zL tnlrtl 11 eLbrzrl z)'qr(xor{

!roaq.rt .lnorr0l :rcipa :roli€< nQ.iE€L< ./Icl rcr tiv(lr 1 rpdrrtl €rL)'o./li o €!(r','raoiolLorrl. EX€r 11016(]0 r(inoro 0o(,)pttrt;46 r\6rcl.'d0ov o.1\' $Lcl

i l1ovo!rl1)i

IoQ(tit'rou,4: A\'€gaarltox r1,< r:rr).oy1E ttor oro,xtr.(Dbdr! natlE€tD!. cj1( onoioqottuirQrl.r4L6avolTrlr €voii(liq yai44r6! r(1r 6l(rlp6orLq t0ctrf(bv u!: ox(lo€adq. IrJ.rz4 crvrllp€v1z],r;rriz(rt'i !toa.{)i ror .{ !'i\,.rlrdrror€ I ibrcl.

lii::l

AIANY:MATIKO] XOIOI KAI I'PAMMIKEI E=I>O>EI>

AEKHEEII

2.2.r flooc drvord oxE6rdqro@ 2 €ni 3 xlqroxord,! mvdrov (6no5 aur6 rou :X. 2-2)

Broa.ir€ !a 6adrE: To GoLX€io or( l€Erd r.)v o6{ydv rivfl a6dqoea.

2.2.2 Kor(oxFr(me ro lurporFqo ouorruf, ro! Froarir'. Fr nsqlosor.qolc (ylri'oolrdn 6fl FqLo(lo€L< d,Id }oqis Liorl.

2.2.3 Yno).oyiore qv ndao,o1'roroilq ,a/Tn rov

f r 2 0 r l. {=10 r r0 l .

L r 2 u r l

llqoobLoqim€ d!.r onvoio 6(oLXdJv rrrtu6T'ttdr! xat dlc orvol,o rlnlelsr! rr11t16).I-16! 1.(L deritt rr yslrzi ),ro1 to| .4i = 0. rad$s rtv o€ tLr( Foa0{ ncao8ors qs( l ) . nor( f i l l l l 1 t rdEr l ror Ai

2.2.4 nqoobroal.ore tv alqororrl poe$i U, rls 6doltiq p€i(6),iFtq. lt( p).t!0€e€g p€r(1-tjlrl1ts xcod5 xat rlv 1€a(a4 Mor lo1| 4,. =0, yto ioa

, , f0 r 4 0 l"= lo z r o l '

K(iorLv rqqy6orr crcl.oLi{at urrv rlr = r. !e o!orotd,o.! ,r xat |l: oq 6e(dr).tlleti xar 6pdre rLs ouleixes d,urr ro /r: = b !s rivoL ouF6r6coro (6IIa61 vddX€r ).!dI) x(l06s xc| I1l yrv|tl luot mtv i soA<{,ri f' auri trF EEioo(nts (3). noKcivdr rt tdLft tolr Ai

2.2.5 Kdrr to i6r(1 6ifsr( yta fl br , l. , ,r . /)r orrl 6cqd al.led ro! dvdoreoqo! rivoK

l? i1,{_t i ; lLu ur

2.2.6 red!'it, trl ./evlzn I'tidltor

oov i'oAoro!.1 Nos €r6rx'ts Li'or5 tor .4r = b zor trp 1€!r'dk lnorl: rorl ,,lr = 0. 6ni'\oqu (31

2.2.7 l€eryQd{ne ro orilolo rov Eqrztdv 6€ld! nleladr! 6 yra ro

[i ?l t:r = l3]llz :l t'r L i,,,1

tr : :ti;l tlr

,2.2 H AYtH r EEDQ:EON ME i AINOtTOYI

6eioxovlqs roljg r€etoA|.ouoG ylo ro 6 rioo ,idvow rlv rQiq r€iooot 0 = 0 (ptdrlv uqlougil). nota €ltar n dErli lIdo€s €tvo! o! sl€iose€E trsra6l4!p{: 1160€6 €|l,dro! hiocK;

2a8 B!,eir€ Tr|' rrtrt qS c no! xaotod 6r'vooi t{ },{o't ro!

u + v + 2 u = 2

2u+3u - 1t =S

3u+41)+ u =c.

2.2.9 Yn6 ,rol€s oworix€s ro rd q ra! b2 (sq'doo\, ,(eeldgovEd, 3xE! to .4o =, ltoi,

^ t t 2 0 3 1 , I D , lo= Lz t o z l , '= lh l 'Beeils 6'io 6laviqaro r orov llrl6svtloeo rc! A ,.ood5 rur q ywur{ MoI io!

2.2.10 a) Be€Lie 6),€s rK Ldo8q rou

6) notls Et!a! or Lio€K dv I 6sEL.d nlelQd sndE& ond (0, 0, 0) oE (a,6,0)i

Zz.U Ynootor€ ofl rl lova6u4 liu4rcu/4r=0 (m eEwdoe6!E n oyv6croq) €tvarr = 0. rbla €ivar t rdE'| iou ,4i

2.2.12 Bpelle tao.2 e'''i 3 orloqpc ,4o = b roo oaoiou q tsvrxrl liiol 1,a €ivdr

22,13 Be€h€ dd q{onrro 3 s'rt 3 tl€ irlv t5lo l€uxri ricrl 6,u r reo'rwoqrdvl,)s, io o'loio&v )'tvftdl 6idv bl + b2 I bj.

2.2,14 1e&VlE &a 2 eni,2 orrorlpa ,4r = b oio oJroto vd l]ridflotlv no),).!S ).iosr4 ooudw.kaud l.q4do I ot rs6si. Aea rc s{or4ra aur6 6€v 0a 4€! l$o+

r r z 3 41 [" , ] ror^=LB B r al li;l =tsl

f r l l r la=l r l+u l2 l .L0 . l L l l

AIANY:MAT]KOI XOPOI KAI IPAMNIIKE> E:D9:EII

2.3 I IPAMMIKH ANEEAPTHUA, BA'EI' KAI AIAETA'H

Or aprotroi ,t 11(r n. [ovo! ro!9. 6ivou\, pro oi€tri €rftvt{ nDv nAayFoT|"r6vbrcrordoeolv ev6q lpcgprzou omtrignro;. O nil,crlio< olo ropirbrrypci paE eilerA€K yQo[Fiq x({r r6oo€arq orrtl€S, a},Id I raiq./ac$$r] itror 6vug ouv6uco[6gtrx ci),1,0;r dno xcl perir r1v arololtpt lcra6lyl0lhc or B(a plbulrzr] yAogtri,1n,0i.q la(4lgiu nqalptrrrzrl eni6paol o.ro oyoyrv€q np66},IBc ,4:r = 0. Or o11t},€(criorl( 6€v {ra! (reEdatrFeg, zaOoq I bertrprl rar" q tdrcpq orrl},r1 frtrv cr},<ir<olvbucopoi 116 nQ(itt! zcr rp|r$, cvcyxdlowcq 6-ror rov Xtirlp orll,rirv vofxqrul,rorri o a\,o 6!6Kimaro e;ri.ne5o.

O €vbra{€Qorv aar0[o! rou aQtiE!r lrc TroQolio(d!€ror €i.,'or { r.i€rr 7 . H rdErloai,oelx€ tl 6\'a\, xagaad unoloyrorrLo rQoTro, o(lv cQr,o[oq 10)! oblyd]! orlv6rc6(r(oic ond,otg)iE. 1 roobrvc4c, ocv cpr0[6q r{')v [I [I6evur{;'r' yQctrtr{l\rrxor rt:),rro n|voxu U. Touloi o oQroF6i €ivclt 1600 lrlxcvLx6( rorr ftr $ropo!o!vc rcr(lvol0!i cfio ir!(l,' !fioloy,or)i. Oc ilav 6pco; orptil4rcr vc tov ;rnp(lnloorpco crrr6 ro olpeio.6rorL eXer" xal pt<r c:rb1 6rcrro0lrlxrl oltlcloio: H rdEU rcrciro ,Aiieo< !1)i')tLtoia< uttqtfnn, i'payptbl] rolt irilnxa,4. @!l,o4e )'.otnovvo bdool jpE o'a1j l t "v rr} : roo6r) lT<r. z<ror iE r(r o 'dl ,Leq raQd[olsE l l 'a]J1iv,aro\r oQloft6 ficoroo6rseo Bq0I[qrLXo fiuQd uro],o1[or!r6.

O or61oq (lu lq Tllq fiuQ(lQsqo! ei,,'a[ vc sEl./toor{€ xo! va Xolorlororl-oolue l6oorgq lbde<:

L fpnpgLxq rweluprrpicr r1 tEdprrlol2. n(o(ly(l)yri !vo< uroldQo!3. B,iort r vo( u'roxieou4. Arhoraol ev6q ufiold)Qo1J.

To riodro 64rc si.lcL r,( oQi.oolr[r rn ),OdltIrri drs€aqttoid. Ao0ivroq ev6Gouvci),or bravrroghrorv r,r. r,r. . . . .1,a. sE€r.iqorp€ xorJg ypr4Lprxol< ouv8rcropougrolr(. r rL + rrfr + . +.),tr,. O r€rQLt4r61,oE o!v6ucoF6( pc oLo rG "tidQl" ri = 0,r0o(p.r\,6E n(Qdyer ro Ub€vrxo 6dv!otrt1:our +.. + 0i-,/i =0. To aQlbr4[a ei.vo!crv cruoq eivaL o povcSr.x6E tp6noE nupcyrollg rou p1b*6g. Edv roito cl.toeller1or€ r(r b|avuo}lara €iv(|, av€Edarlra- Edv rrrdp7er ,{d]ioro! d],Log ouv6uno[6<rox rapirl,€f ro tLt66\,. rdre rc bLcvnopcrrc rircL tEnpt4ph,a.

2E Erlr o terpr"py&og ow6uaop6q eivcL o p6vo6 :rou n<rprl^7er, to p166v,5ILc6{.

ct t , t+ctr2+ cA., i =0otp6{ r€! p6vov 6rdv ct = c2 = . . = ct = 0,

x6r€ 1{r | ovlioporo ur...., ur eivcr. ypcppr.r"ri4 cvelriqrrpo. Av clu16 6€!outr6t{iv€1, rox€ eflor yeL4rpul6q elcprrp€vc xar xirnor.o c:r' cud eivcLypaFurx6g ox1'3uoo[6q rov !r[ol"oin0Jv.

:.] fPAIINIIIiH ANE:AT'I'H>]A. tsA:EI> KAI AIA:TA:H

H lqcr3prz)i 'rldrprlol ;ruprorcivtrur ttrzol.r otor 1Qr6[cor(1o I(iQo. orrll

6),c rc bLcrvirolcrra lezLrorrv an rqv up14. At-o 6Luruolcru rirtlr €Eetp.''c.6rcrv :ttpr(Tovtur orrv ib(r r1,0riu. Tai( 6r(vri+.1ro €iaor EE(al+riv.r. 6r(vnt-eri7olrcrr oro ibrc rftin€bo. Mrtl rlrX(i( €n).o}ri -rprrll SLnvm$rir(')1. tl'tatEr5ftirlQc Zcp(rirtLndrrzdr. 0( €ra€riF va :r(pd1er lpcrpBrz{ av€qoarloil:r. Ar'rrl! hi.il tr€prri. rioospc Srcvtotcrc .rol Rr rivcrL :rirvrore eEcrqq$6va.

IIAPAAEIIMA I Erirr i:ru (:rti rc 6rrrr (]$(lr{1 i.X. ro ' r Fiv(r rtbq ro Flb€v|r{o6rd!r,otrc. rors oitouQ( ro orlao],o €ivcr./0.4+urdq €E.rprl$6vo. Mnoaoutr€ !.rbruidEoutrt rr = I rc{ ri),u r(1 (ill'( ., = 0. Arroq €i!(r 6vaq Fl r€rartqrfvoqoLr\,6wof(oq nou ri(Q(i./r'r ro ll116i1.

llAl'A^tjllMA 2 Or ori),rg rol rirc(x(l

tivaL lprqprxrirc rl{1prrltivrc, ftq'ori 1 t)rurrpl or)il.l Ei!(! ratntr(io!( 1I; trQ(;)nlg.O rJuvbuoloq rol l< tre i j ( ia l -3.1.0.0 bi lFL trLf t or l ) ,1 | Ibt !o{( i ! .

Or "/0(lrtr6( r'ialr e:riorlq r:cpr4rilrq. II 1g4qLr1 3 r'ivcr iorl pt bio (foQds rt1pl1pgrr1 2 |rior:rivrr <lo0ic rI yocftri l. (Atro( ei!(lt o ibloq ol)lb!(rotLoq noui:rqrr:;rl vc Lrcro:roror! r( ,r. rr. rr rt; brELriS jrirrpci;. tirotr to ,,1.r' = , \,.r ti.!(r.outLbltjlto1o. Xootc trtl ,r - 2l: + 5tr = 0. rt rpirrt tEiduoil bt! 0o ibt'v€ 0 = 0.)

I lAtAAElf l {A 3 Or nnl; . ' r rou rsrT' , \

tivar reagLrrrirq artlriprlrtq. ALrrj r ;r6rrivfl urtoltrito\, or.1v t(t orol)(€t.i rtlE6|a./r1r!iou 6a\, [i]5[vilo1r.1r. To yLtlri. qa.',€adarrc{' trol.L( ovoqt ioorirr f!(lvowbnrl6 rtw orlLtirv :rorr :rapci"Trr ro p16er:

qira w |tiforyt 6 ta t1,t:j, xsr t:t dvar xar' crdyxty p46dt H trinrr.rl.tcEiot-ro1 6ivtr (r = 0. Kcrdrrv I rrliu€ul EEiooort 6ivtL c: = 0 xoL (1vrlr.r0Lorrirttrg otlv nedrtl eEioljol, (royniritl rll .r = 0. O Foro< orJv8nootLoE rorr(0.rlEr ro lrl6Evrr6 6(ivrou.1 Eil.1L o rrrart4Ldvo;, daa ra bLclrnotrara €ircLi,aaFFLr1cd< (vEEapqra.

[ 1 3 3 2 l,,= ,? _: : ;]

f r I 2 l, {= l0 I 5 l

l o o 2 j

.[i]...[j].".[j]til

AIANY:MATIKOI XA?OI KAI TPAMMIKE: EN>A:ED

M€ ottl6oLrcr6 rwdr{ov, cvl6 ro trqed6€tyFo 0o lQoq6ra!

AriEqrs 6tr o /46€vri.teog ncqeiTe pivo to y46*wb 0rtittto4at q = t:! = c. =

0. A|'16 .i1/o .rxar6d)g ro i6ro lr€ ro vc( norip€ 6rr or otikS sivar yqarynxiL

aveEdQtqrE€.nqeotroro: outrl'otloFq €qaQ$6!,€ra! x(lL orK ypopy6E rou -4. nor.r eivct

eniols c(v€EdQqte!. Yfio06ore ot|

cr (3,4,2) + cr(o. l . s) + .3(0,0,2) = (0,0,0).

An6 i(s itQ(i)r€s ouvlordosE 6Qioliorlpo 3cL = 0 ll (r = 0 Kcrl6rilv o[ 6€urep€goollor(i)oFE 6ivolv c2 = 0 xo! 16),oE .: = 0.

H ibL(r Lbda €nEtttEi\,€ro! oc ororov6rinox€ rl,LFaxor6 nivcxc [/. dQu ol ttlFrDevrxis ypcFBi( rou irQir€l v(r €ilcl av€qdQrryeg. EftlorE, €dv E€xoQioow€ ltqmrii'eE rrou n€QLdIou\, totg obrryorg. arnlg tivo qv€EdQfiF€s tlo trQorryoilt€lor(Qcro€ryF(a F€ lov

.. tr 1 i ilu=10 0 I r l .L0 0 0 0 l

or onltrrE I nc(! 3 civcri ovsEdQrilxog, Ko!€!a utrooilol'o pe tpeq oql'tg 6evsi.,,(l[ clr€(iQirFo x.$ oiyol)Qo or r6oo€OK 6ev ei!q! qv€EdQlll€q. Etvat a].toelc,6tr, ovelriqrrpeg €ilo! €niorls xcrt ot oqleg 1 xtrl 4 aa sv loiiloLg al.J'hlap€ tore),€rl[aio 1 o. 0. 0a rixov t{r" dvriG €EaQrnF6v€(. Auti< ltot tivtt tyyt\ltivaut'tllg)ztttE, civ t o! ori),€€ rou neetil.ow o6,tyoiE. O \t"ttt(ax x.rv6lcrE €ilc.t:

2F O! r pI pl6Evud6 'yQ'q4rri4 evog xlqrcxrorori nivat a U €ivat lQ'ogtlrxdrgav€EdQflF€E r{ar ro i6lo oljlr6aiv€r 1.a! F€ tq r orri}'€6 nou n€elq(ow oOlYorS.

IAPAAEIIMA 4 Or mrj].rc rou tcutorrxoi n .Jri n nivaxu

eivcr lpcypra6g cve[dra,Trpeg. Avtd rn eL6r,zd 6tcvriopcrc 4ow tov st6!x6outr6oLrol6 cl. e, . . . , en rat, trsQlordlow ltovaa|ctia 5lav6oFarcr t{ard rK 6t€u-

13 4 2l f .L l fo llu 151 . : l=101.L0 0 2.1 L(r l Lol

'[:;,i]

I P \ I l l l -ad . \ : , rE :AI ' - 9 ) tA . r r \ :L r : ( l l l \ :T \ tH

0rvoeLq owrerclpivorr. :rov R{ crud rivl ra

til|;]t:lt'l

, f l 2 l l'= i r 3 2 l '

Tu repLoo6rcprr oiroicr reooirpolr brc','rJotL(iltLrv rol .Rr riv<rL lpr44lrrir< ovr-i<ipqrcr. To;r0oqprilttvo 6po;g eivuL ro rLo rrlxo),o z(r 1() nro oilorQo.

f t . r rrr r l r t rorrr : t rnoto ou*o). , ,6t ,r l r 'qrt i r ,"v rr , . {" : n(uc rr tvlpcrpprerl cvrlrrprqoiu. oTrycrtiore ror:rivuzr I tou oroiol ot r otiti,r< r'ilcrtu 6o96rra brrrvtioluru. K(r6i(! i.riort ro oiorllo ..1. = 0. E(i1, tn(iAztt).iol br(iqoQFrlzri rr;rri rlv c = 0. 16rF r( 6(Ar!('t(rr( fia(L F:(prrFiv(. E(i1'6f! ur(iQtorl1' e),.roFaF. trrlGii),Iri< (l rciErt aivar n) rorf 6L-v rndozrr .il.lo(tril60ao/,{')ao. r'zro: rori . = 0 r{(! r( br.1\,!otrcr( €iv(r (vr:darrtra. Edv Ircilr1 eivcr grxpi,rtpl rol n, 16rr lirr rorrLdxrorov petc6),qrl tivcr tienerpl vctrrr,ey€i. [l t|l]br'\,(,{ri 1t(l|" or olll},€E fiv(r yln4ll(lri(i( €EcrQfillli''rg.

Mf(l r€ai:rr(Ddil ilfr erbr'.ro elbrr(tdQov. 'Eoftr (irr r( bL(rirotr(r(r 61ou!/, oL,vrordou<. irQ( o -.1 eiru( al'(rs riv(lx(r< ,ll s;ri ,. Y:rot)aorf orr r > ,r.Tole o( orrli,r. 6er lnoQri !o €ia(r (vr:(iQrrFrcl A€1 |xoari v( un(lo7ou1, tlo$17oi. bL6rL br.r' r'rriqToi'v (Qz€rd< tQ(![6< i(r !( ror'. brz0our. H r(irll ;ra€rcr\,(r fi\'or [rz0orr0]l rlnd r zo( 6!(r ()B()?1\,6( oiml[c -.1. = {) trr nFrn o&ueolc{r11'(;oro( (r' 6.rr !Erir(;('fl< iXrr n(h'rorr ltro€rq . 10.

2C Evo otlvolo ,r 6r.(vropt'rrcDv ror R"' €ilol xct' uvdyxl! rEaOqF€1,o.

O (rvqldorrt: e (v(lv()aiorr o (ltro tllc;rca(r)i.qri rot' 2C.

IAPAAEIIMA 5 OF('prlorr rL; tptrl orrlir< ror

:rov -R 60r p;rr-rpri vc r:rripXotv toitr aveEhprlra 6L.rauotr(r( ltcr yLa v(ldaontr€ roa o!r6r,{(rtro ro}v orrti,urv iTor :re{i./tr ro rD6v. inlot'tu. ro ,1. = 0:

r ' : t lt - r '= l , lL l l l

Erhv bruoor;re r1v tLp{ I m1v €),€ueEAl [ex(6),trn r]. rorE rl cv(5ootuxJt.rurr(r(io{col orlv U. = 0 6irtr r': = -1. .r = 1. M (urd r( rpi.r 6dQl.rl nadrq oTll),t lei.ol rI 6€1]T€Qrt ('ur, rllv rQirlt, roorlTcr F€ Ft66v.

A!ANY:NIA'I]KO] XOPOI KAI TPAMMIKE: E:I:O!EI:

IlaQdTo?roE 6aor ofi O((l'Qo

To €n6F€vo 6{!rs oq! ouqtqol lts iox 6t(t\,uoF(rtu(oi,s X6eou! 0s etvc!v(r ooioouF€. rL orlfo(!€r'y(r dvc( ouvotro 6LovloFrinov vc. naQdy€r rov X6Qo.Xpryrtronoriocrtrr' our6a rov 6ao orrp oQlrt rou 1|rqal'rriou, 6rov FL).ylor+€ yr.rro r:rinlbo rr-r u :r, rt'rilr tuL u:rd rrc 6uo onlLr c rvo<:rivuxu xur ovopuior4r urnr-rro r*iru6o yrirpo m1L6v rou :tivox<r. O levtz6E oprop6g rirnt nn),6g o e$1q:

2H Eriv evog 6r<rvuoporrr6E l6QoE y ajrorEl€ixa! ar'610!5 roug yQoI-plriol5 ow6uqol@riE ouyxExQ4ravo)v brawoBdx@v u|,. . . , u1, t6re ru 6rcrru-oFard aurd raQdyow rov lirpo. Me rid).c ld.yro, x.igE d[irwop<r u rou y

F oQ€i vo €r.qQqoo€i 016 ow6rnopo5 rarv tu:

= clwt + + qu\ '/!o rdnororis owr€trsor6E ci,

EnrrQdnsr(r ro i6ro b!(ivuofo | !( n(rQ(iy€r(rt on6 t\,(l i n€Qroo6r€pd ouvol,(louw€),t(r16v .i. Koo(i)( ro ori!o),o nou ro noQdry€! tl^oeti v(. rivd ,1016 p€yd].o-lcli,rors tlxoQri !( n€Q!6xsr z(rr ro lrl6svrx6 bLdvlolr(l or ornrdlsorac brv€ivaL ?rsr' cv(iyxrlv lLovc(brxol.

nAPAAEIIMA 6 T(l 6Lo\rlrotr(trcr {ljr = (1. 0. 0). u,r = (0, l. 0) r{cr uh = (-2. 0. 0):ragdlow ro antfisbo (ro €fiifir:bo.rj,l/) [doo olov r(n6rdoloxo ldrQo. To 1610r{('lvouv xcr! rcr buo npdrrcr 6ruvtlogcrc F6\'o ro!l(. €v{i ra u'| rnr" u,1 ruprilouv

nAPAAEIrMA 7 O XdQoE orll,(;rv €v6( niv(zc eiv((t uxQr6drg o X.beo1 trourapdynar afi q ..riX.ES. O oQrol66 LxcvoroL€ir(. uqroii ro v(r o€oQriooulrtoiol5 tor5 owbucopor,5 ruv mr11'riv rivcrL ro i6ro F! ro v( 0€{Dqniooufr roaXdQo rrou ncQdyow. O noD,antraorclopd; 1or 14 F€ ro .r bi,l,€r 6vo\r {Jr.16!(l0!6rrDv ortL(i)a x(r ro brdvuoFa ,4r: rr:Qrdt€r(L orov XdQo onttr(i)v.

E,,v oL otn),rc r ivur rLr 6rrrvt 'oprrru ouvrrrupivorv r1. r . . . . . e, , rou :rsoi tlo'r,r.rt .1n6 rov t(urorrz6:ri.vc*a. t6tr ar,.rd rcQ(ryow tov ,6|". Kdg€ 6r(ivuolrab = (br . . . . .1. ) cirsr owbucoFdE aut6v rolv orrt),16v. : om6 ro nae/r6e!t|r(. ra6ripr1 eivcu or i6rr:g oL ouvLmriroeg: I = ljc1 + . + 6,,e,,. Aev €i!(t! o[q tl6vov oror{}.eg r1E rcworLxrlg :rou rraQi.yow roa Il" !

fra as (1no{r(1oioow€ av xo , €ivcl! owSraoFog ro! orl}d)v, fiQoonsooipevcr Truooqre ro A.r = b. fLct vcr @roq(oioor4r€ av oL orriLEq rivor cv€EdQqrEE.],tlvoulrs ro /jrj = 0. H naQey(Dyi oXfitEtxar p€ ror ld)eo nnld)1' ,'.ar 4awlcpqa t oXni(rtar pe rot p46ttiy1tpo. O rQcbro< 0o 6npure vn efvcr€raQ,{(lE tt€"/ir},oE. o 6€ur€eo< 0a dnpe?rs va rr€er61€r F6vo ro Ft;evrr6 6Lci1'uogtr.Ttr 6trr![qrcrrs er. . . . . cn tr€Qvoiw rar rtq 6uo 6oxLptg: Ilcrpiryouv rov -Iln xcu

' : r P \ \ t \ f l t i i t , \ : , r t : \ t ' t H ) t A F 4 : f l : x \ t . ! t . \ : t A : | l

€ia(L lpqrlrtzdri (r€:drarlr(. Mr).r;r!tr15 IO6ZEqo: l..r'tiya Duivtlopa avtolt rcuourdi.or' 6Er rtitt Xapiao. Toi"ro o6rf/ri ori\' rbds !tc< 6dorF

2l Btit''i. E\oE btovuoFolxori X(lQo! €ivar 6va od!o),o 6lulrolrdrol, fiouil€r xaxr61pov(r xlq 6!!) (6r6flFeE:

(l) Ei!.o yQaftrrir(ig ov€Ed0lrtro.(2) n.rQdry€! rov I{iQo.

Auto< o or,vbl(op6; rbtor4rt,}\' ti\(lr tarri.(i)q t)€tr€i.(;brtq 1r( r't faqrtuzrlAtl€i,ac z(( o'I((irrL ofl ziror bL(i\r,opu rtrov 7r;po rivcL orvbrcutro< rr,,\'6cor-z,;,1 6L(n rxiftrirf,v. (rfoi truci ror mrgilow. !13ctvrr. caol1 orr o ouvbltropog(r,16( ci \ rr t rol(bLrd<: E(iv 1' = dr r r +. + r^.r) , xcr €xiol< r ' =, r '1 + + lJ/ .r 'p.rorF rl c(f,(riarort birrr 0 = t(d, /,,)r,,. K(li x(lr)u q (a€,E(prlloi( n.ti(r! ro QoLorrl:. r(0rir rri0r duvrFkorrj( ((1, l') npinfl v{l trtbFli.t€ra'. Aoa a, =ri r(rr.rtipllrt r'ttt< zat trid' i: a. rpi:ro< Lrr 1'pritlrnyt ro I oat'ot'tt\rtk\tj kr1,( ; f l e . ; t ' i tu ' t \4 t . i r tD1, .

O(1 :rp6:rfl (:r' rrtr (a7rt r( rorioorpr orr rq br(\'ioB(lr( or'\'rr'r(./tri\'(oarr. . . . . . , , &-v ( ;rort i .or\ '14 I 'o\(brzr i t i r lot ro1' ,R". Mrol7( i no( iytr( l rc orr lylx4rtuz'j (iiyrbac riwr rorcbLz(i. (),iri o7L rr,16. E!(( 6r(vl'otr(rt7oq t6Qo:tfr afteia brcqoar ruaov tirioco^, tlrLN ^rr: rrrerctr,rvtao: :rlrcr,r< r'irur(\,rloroirl'rtro<. r(jrf oI ooiLeE tot riv(L (vrldp lre; zcr cnorriouv [r( d(lolro| R". or biro mrt).€< 116( ororor'brtnorf tril rbr6t{opqo! rri!(,{( 6n!)q o

,1= [] 1l| .: :'l

cnorti.on\' ?otl ro', Rr. Kti0e btbLriorrrro brrilloprr tiraL Tpcpprzci: or,r'6r,rolo:(r'r{;\'r,,v rrrrli,(;)l' nor €i\'.[ z(r (11,f!(iel|trr<.

IIAPAAEII'MA 8 Or-oariort ro ouvrl0rol(ivo eni:rrbo .r.r (:)r. l.l) irou €ivr1rczar6dr< ro -n:. To 6r(i,'rrol.r fr cno t(ovo ro! t-ivur 11x4tpLno; avt-ltipqro ciLd6rv:r(pri.1tr ro R'. T(1 tpi.r brovrotr(r( r'1. r':, r,i (]i.youQo n(Qhi,olr! ro ,Rr c),],.ieivcr rltrprqlrdv(. KA* kiyoE Lrlr' uurti t( 6r.r1vrqr.lr.l, 1..X. ra I x(r r,,, dx[LxcL rLg 6no LbLorrFeg :rcrptrltL ro t:ri:rlbo xcL etva crveldpt1ro. ripc <r:rorei,et6doft . : f t l tdorr ndir orL ivu;6umurytnx6< TLbpog 6tr i r t t t i . t ! .ho 6lar.

nAPAAtrtINIA 9 O€orp4ort ror z).rll(zr'116 3 eai .l ;rilcrzc I::

f l i r r lt '= l0 o 3 11.

L0 0 0 0 l

AIANY:MATIKOI XOI'OI KAI IPi\MMIKE> E=I>O)I'I>

rx. 2.3. 'Evs s1llolo nol raady€L ro € ft6o x(r Fl( 6dol to! R?.

'Oar,r; rirlra. o! rioo€Qq rnii,e! rcrqdlow ror Xd)ao otd,6\,. b€! €[vo! opo(oleldpopeg. Yadplorv no).16< enLlolig 1c pLc 6d.ot. toor6oo €F€iE nQoftfvotltrtl||(| e!6!zr] €rri,olli: O. oril€g ftou neglfl our o64'tois (o' dfiliv fll! n€Qlrro.,orl IrlQorrl xrlr I rQiTn, no! cwLmoLlow ortq 6corxtg yrtc6).F6g) dror)air 6doqtou Ttipot nq),tbv, IrpeL6oc4re oro 2F 6tL aur€g eivcr crvelripqr4 ruL tfv<truxo).o va 6oqre 6tt rnQiryouv xcrL tov x(;)Qo. nQdlrsrr. o x6aog mql.())v rou U€il'(L ro €nin€6or,i/ piou orov Rr. Aw6 6Et' KtutiEt:tu Bs 1o! I6Qo otllLdv

:wouiEor4r€: Or onl).cg t:v6E rtvoxcr napciyow tov ldrpo otrll.d)v rou. EirveivcL nvelhpqreE, clr[or€lotv 6dor1 rou yipou mql,6v 6ft€ o ttivcxo( eivolretpcrycovLr66 eite :rcgcL1,r1l.61qc44rog. Edw 0ftrpnoout€ o16,.),rlQo to! R't xolrrnrrrrlooqrr c:r'rrc m{l.rc !a crnotttlouv 6dol cturoi 10u XtrlQo!. ror€ o t(ivctro(rp€n€r v(t €ivor r€ra(rf(t!l1{os xa| ovtlo[Qau[oE.

Ardoraort 6tovDo||atD.oir 1d)Oou

IICIQ' 6].0 irou 6ev urcipleL Fovoorllro!.q €lrJ.oy4 6rior15 i((! irr€rQeE b!o-qopenz6-r 6wtrr6rwrg r-EuntoEtow €E ioou xd'd, unriQ,t€r LdrL xotv6 os 6leEarneE rrg en0.o.rdq. Auni €ir,o! [rcl rbr6qru nou IoQCrl,.rlQiE€L rov i6Lo rov x6Qo:

2J Ario 6do€6 10! 6rqwolcrlxori xirpou V regr.61ow rov i6ro aQrqr66lowopriror,. O oaieF6g su!6E, xoi!6E yrd 6lss rrE 6{ioet6, €xqQdl€! touE

"6q€For,! El,€lre€aia(" rou XdQorr x(,r, ovopdE€raL iui:otdoq ro:l V.'

Ilpiirrr v curo6eiiorpr ro rlrjc: oie< or bu(ri< 6<iot:r< ;reprdTow ror tbLo(er9lr6 |(rlr.otr(irt)r. Zlr.4rr ne6r(r (lJr' ro1' (\'(ryli)orl rc 6€L r( r(e(16€iy[crr(lzcr lrr o4rt(irotr rtl 6r(ior(orj rorq. To ctrirr6o r., mo:x. 2.3 6/rL bnoi(rvi(ttr(r( o€ zdoa 6dotl. cipcr I brdoraori ro!' €il(lr 2. :rrq rp€rq braorciorr:Zqfrriotrcor€ raic bLcurorar(. zcrri lrjzo: rl' trlov,rr .r. y. : rl rEorv ciiJ,or(ylxart(zci c!€qeT'ir(,)r!) 6rtt'0nrotrn'. H 6uiatao4 rov R Eivat . OTrirqro; or4i,ti;v rou t'. oro Ilcpri6rLll(r 9, (ilr btciotcol 2. rtrc'\'6ni(6ri alrl;''bLlLclor(ro< u,-ro7r'reoq rou -Rr". o Fr)trFar]ao<;rivce(( r-ir(( trd).Lo! !:qaLo€rdo:.br(;rr o l(iao( orili(;)v ror,:rrpLi/rL F6vo ro lulb€vLzo 6Lira1JoFc. :rJtq(,)voi$r vtl{)foQoirfr€ ro r{sro o1iroio (](r dr'r04 dwori toll X<ir12ou x(r rI &(ior oi ror iorlIr lLlllt'\'.

O roXrQlolroq iou 2J. (r( ror onoio riaprrircl[ I rbir( Iq< 6((iorcorl< cir(rrrrxx\ i ,v(4roi t rr rov

: ] I IAN I \ I IKH ANE:APTH: I , \ . B , \ :E I : KA I6 IA ! IA :H

2K Yro06mr 6( r,r ... . ri,,i eivcr 6rlo 6cioeLE rou L6ioubrrrvu4rartzoti lrirpor V. T6rr r?r = r.

An{tbr'€'t Yio{)adrr (;r ro i!(1 oivo}.o ti\(lt trrzaorfao ror'(t.i.ou. )..2. ,r < ,r.Orr zlrli.rilorlc oe crri{col. A(J ori r( r', (jrorrr.oril ti(i.rl. :rQd;r€L v( r(A('ryouvrr)v /('r(!) z(tr z(r{ls ri., fkr y(x({ r1(r o(v or\'dr!(lro< ro^, ? ,:

n , i - t i t t t + . . . + , t , , , ) r . , " = I , , " , . ,

Mf iril.lrr<. ronro 1_odq,['T(rr ll' = 1, ,'1. 6rori r( rt, t'i.r,(lr. or mjl.oc ror II z(rrrr r,, circrL oL orlltq ror l,'. A€v:irxn,trr noror eirar or owre),roric a,, , tiLri1'opi(oL1Le xhrr erbrcq,dpov: o ;1 rivrrL rr e:ri rr zuL rrr < rr. }i3r1.i,rva pr rolC. Otr r,n(ia7r tut rtrpLuptvl ),rb1 rorr -{r =0. Iio}.),cn}.ird.iiovrcq Br ro! l,'!7or!r l -{r = {) )j II'. = 0. nou or (illL (;r, r(1 6L(1not[rrtl & r oulbro!6tre!(l1rr fl( orcr0€eac.r bi.\,or!! rioAorotr( trilf,av. fi.v€rcr qc\'tao ),orno! orL r(l r,, bflrircr creEriQr4rrr zrn rouro rvrrlriuztL or4v ln60ror 6r, dortionv 6.iol. AA(:r$n{:r 1,( .rnoQprl+,ortrr r)t or\,(llorrlr( /,, <

".H r, ,0, ' :Tuy,, , uiTobulr t l i r ,A i .1", r , , i . ' nt i rof T0rtor lro:ror lor. ' , , : r tur l /ofr ' f -ro:. Trrr rcr brilorye orr zriOt or\r,i.o rl + I 6r(rvfotrrirr,)! rof R"' 'rpi:rer r( tir(1r

r Il,-rrrr' \( ol!r(Do.r nl 'tt.f:'t ?!!i,rr'tro: rr-nt'rtrLlxoLlr(r !E dro b(ro!.nzol!: r'_r.tr,r.:

\ l ' i , r ! r '1 ( .v ( r . ra { r l t r id r ( ro b rdwoFu rv \uovr r : f ! ( d (Aro t r ( t r€ reooear< or \ ro r l r ' r : t I, , , , , r io1( i \d L t ro rTro ro r Rr Aro r r ! , i ) . i . r t . n ia ( o t -nn( f r rc rp (bdm(ror 'a rna /u l r t r i . , r , (

r ' R ' r r ' r ' .

r vrL urph. T( urorT€iu (roi lif rr ru1l\ iiLrnrrof fno/,r,!o! !i1(r r:drdn(ro ||(riq',d,1.,jto:

AIAI\'Y'MATIKOI XOPOI X-{I TPAMMIK!:: F!I'O>EI:

eEapr+rdvo. Mrc aap6porc an66€tEt el€xr€ia€raL xar ot xd0e 6rovuoporlxdyirpo. 61" xcr' cvcipqv ltbpou< 6r.nvur4rcirov-otr1l.cbv. Iledy1ror!, ttrioQoip€ va6oupe 6rr ro yevrx6 oupntpoopo ei'vcL to tlrjq: )' &'ar bnvuopa xi Kieo 6rGmtioetr< k, xd9t oirc),o pt nqnoircpa an6 k bnvtiapara.ilol EE a(1t1!voxu xdee aivo),o yt: lrydrt:ptt atui k hLariapatu 6eL ptropet ut raanr tuv'/.aao

YfidQXow rr dll.o 'Suixci' e€cDet[o1o, EX x(l)v ofloi(Dv Fa4rov€uoup€ $olovilc. nou pcq urLrp€neL vcr acrtcl{1orpt ot: pxr 6dq, EerlvdraroE an6 6vo oulolo6rowopifinrv, eire noLu flraQ6 eir0 rotr! FEyir),o:

2L KdOe lpopllx(bE (v€EdoflFo oi!o].o tou y Bt[oQ€i va €'[€rixoo€lo€ pto 6doyl, €rLolvilr'lolrds €! c!!oyr(q ri€Q[oo6r€Qo olavriopoxd.

Kso€ ouvolo no! fiqQdy€r rov y cvdy€tdr o€ lr[(1 6dorl, (rwocbvrog €v(rv(rYzn o8!op6va 6ldvJopclrd.

To <rfLoorpeir,no eircrL 6r! [ro 6dol €ivs! ivc /iyroTo avEgiQnFo olrolo.nol bev yir,trcu vcr pe.lcrl.rireL nx6prrl ;reproooreQo XoQiE v<r Xa0ei q a!€EoQrloic.Mlc 6ooq €ivclf criols t\d EldILdo lrdQtiyor otivoro. rori 6ev BroQsi !c li!€lcraop4 praporrpo rE'eoloufld)trcq vo trsQaltr ro! X{;po.

KcL piu rel.rnrj o4r€t(1)ol ofllv oQololiq rrF fQa[[LiilE A].y€6Qo(. fiori6t'.! Xorlolpojroroli$E rollE 6qouE "6hcq !v6! rii!oz(' rl "xciEl E!o< xdraou" rl''bLcrorcol pL<rg 6dols". Or q,a(ioeE (ur€E b€v izouv v6lla. Eiv(! I 6(iolcolror ltirpou oql.rirv:ror-r roorircr [e n],'rirEl ro! rivtuc.6nog ()(l 6ci!o4e oolv€ro$cvl rcQalyQ(1q,o.

Ano.{'aoinf €triolF llv drir.i raad'l,o!! iov R*. rqoorood,raq !a ]"no€rf ro.rlL' ' ' + r r q = ( 0 , 0 , 0 . 1 ) .

2.3.2 Aroqo!0rift ]ta qv cv8Eaa loi(l 1i oIL lorva ) ( 1 . 1 , 2 ) . ( 1 , 2 , 1 ) , ( 3 . 1 . l ) .6) rr rr . ! r' { ) (1.1,0),(1,0.0),(0.1,1),( , .y,z), r i ( l of toLouobif toc oaLoFons..y.z.

A'XII'EI'

2.3.1 Aroqaoiors (v ra tn,Jt€lc btdwoFarc tivdr aldEciaqrc rl oxr i!lom(< ro. r r r l +r t ' r t + . i r t r +. ' . tq = 0:

tiltiltillil

].:] TPANIIUIKH,\NE=APTII:IA B,\:ED KA] AIA:T]\:II

2.1.3 A€iEt. oiL .(i! orotobrinorc brat(;!Lo olorl€io ror

livd trt56l. rore or lAqqliq €ivartactFrxoc rlcprlfiluq.

2.3.4 Eia(r (/,ile.L( 01, (ot(alc rrr = .r = rr + rrr €ivoL Eniots tlcBfra65 c!€E{iaqlar(Ynoberlrt Yro€;m€ orwdnoLos olv6rdorroE c I u, r + .r ur + .r u\ = 0 raL 6qEire trdion .j 'o)(ii.L )

2.3.s Yro0iorE orl rc |cvropora trol 06].o!p€ vo eq€rcoolp€ roio0lxowrdL orK tqolrra.-' d r . / l u r R r ' 1 I ' : r J , . 1 n , , : u j r o t u . ! r r , { I , r r , r . o . \ t . r r < a I r u r ( n -(!.Eaaflloi(Ki E(trca|6ore (l]Td m(l brovrlot'arc F Aox.2.:.1.

2.3.6 nuprlp{il,x! ylortrelprr(i ro! r,noX()ao ro! Rr rror iToa(ly0rar dno raG ) ( 0 . 0 , 0 ) . ( 0 , 1 . 0 ) , ( 0 . 2 , 0 ) .6 ) ( 0 .0 . r ) , ( 0 ,1 ,1 ) . ( 0 ,2 . r ) .y) xar r(! iEt nAoryolfrcla n|aurofftr.1. noLo lrnrt (liro aurir clrorrtonl 6aort:l) oi(1 x(1 lr(vno!(ltft !0 0citra; oulroldori.

2 .3.7 Aru,uJ ldf t a \ ro, nr€rz i rur oro! unu)(uqo ror 'ndqd?rrur L.nuro . . . . r^L(jroalnf ro

", d(! oTrtirc rou xir(ra / zoL naoon(0isrf !( iioL.if ro /1], = l).

Ook) fildL ro @orLrii.otr( orav( ) x , r : ( 1 , 1 , 0 ) . ! . 1 = ( 2 . 2 . l ) . ! ' r = ( 0 . 0 . 2 ) . b = ( 3 , 4 . 5 ) .r r ) u i r = (1.2,0) , r ' r = (2, s .0) , r r = (0,0.2) , r , . r - (0,0 0) , ̂ / ( ono'obuors 1, .

2.3.{t rlrerlq.il],ft fe IDyLa 1$ o]i61o oro €rin06o ".

!, -r 1,1,a. -'i,1," -" I = l1 :]

k ' . _ru l ' .A ' .u1 t , 'duuotru ' IUqu' dr l iu \ .

2.3.9 Evrorti(olxc( ro',E ontlor<.6e€ixf [r(1 6.io1] 10! tdrao! drlr.6! ror

T = o d cL0 0 " i I

[ 0 | 4 3 l. , 0 0 2 2 l"= o o o o l

L0 0 0 0 l

Erqiaatoxf ri(ief dnilrlroli be! (u}.r mq d(iml oa! o'r!h'{idFo r.)! 6odtrrl! ml).d)!.Ba.it€ €rior ivaa rilduo -'l I dxrlir rry r].+rdzoili loqqi U e].ld u€ dLdq:oq€rLzo)(.irao orti6a.

1.1.10 Ynoodorf orr 0rlraonr(r r.ior ri!a?( 2 €ri: oa! ila *brd!uorl(". nop' o).o ror' (rrdb.! filtlL btc!iop(lra tr€ xrt oL,ltoLofdr'! d!!o(c, aroutl€ rtl!o!f! raooeeqq rr!(irnnror nol,i(1ni(oraopo! f0 uplotroi! r.ft ro oilol,o i(,,! riwdri.)! €fv(1r

"ieror6 r)! nao!

crtiq tL: rpdlrL_c. B0€ir. [(,6(iof dritou ro! td,ao!. Ioros !r6)(.)qos ftoaAl€r(rcr6 ro onlolo oto! r.,v zi+Ldror(;N ftrldzo! ari

l.l.ll Ba.irt blo bluqoQErrxl! 6doflq ror uoxdaol oio! r(')! brou,oldro! rot, Rr. r(,r!oaoi.)! or bi,o xalr€q o1Jlrm(bo€q filar io.q.

AIANY:IIATIKOI XOPO] KA] TPAMMIKE> I]3I:O:EI:

zJ.U aprire (!lrnaad6{}irc mo! 0n6p€'o toxlqrofo: Ed! ro rr.!riur.!r arort}.otv6drort ro! I'Lalloucrrzon xorpor R{ xct o ll rivor iaaq MoIoqoF tou. rotu rdnoLo, rn.u!o;o ruv , , o^o, , ̂ , | ;d{ r l rou I I

2.1.13 Ba.ir€ q bld.mool rorra) x(;aor rlra SroluottdrN tou -Ra rov oroior ot olvrortboEs ixow ugaoruBd 0t i l , ,16, vo lu,Au' rou n.dolxoi 4 'n , d n iv , , ra.y) rou r6aor oi,orr to)v rtvdx(,! I cni 4.

.z . r . r4 r 'o ,uv , 'v&, I L ; ;

j l ,^ ,* ' - * ro nJvolo 'u \ lqaFuu,v n ' r 'a 6dorr

ror Rr rcrL (EqorpLorir) 6(,rire ono to or:vol,o ot1i(irv, lro 6dol rou Ar'

2,J ,15 YrJerm, of l o I ixr buiL ' r , , . t l . ^ r : : t . or 'd) onoto6inoft I aaeEdarrFo 6rdr!o!(t(l aftoft].onv 6dol.6) ono'abrirrore I br.lvnorroia raqcilorv tor V, orois)'o!! 6ciorl.M0 dl).o ).oIn, ot.1! o {lar0ttoE ()v btcrllottd'io! tf'ar soqrq, I p(a an t|srbrontr€s rl]E 6dorlq owerdrrd! fl]v ritU.

2.3.16 BaFirF I brdotctort ro! tdraou tu)v or$!r€lart{6! 3 srri 3 nNdxo!, lta0tbs ,luL ptc

2.3,17 A.iEr€ ofl ov or v x(lr I4' Fi!(tr iar5rd('ToroL r:rolorOot ror ,R5. t6rt ot V rcL it'txo ilo

"or!o Brl $I60'rao btd!!o[o. vtl6&6t: EEr'lt]or€ crno 6ds3[ ./ru rorE

bno !roxo)ao\. no! sljvo),rad onlcltiiol]v sivo)'o 6Et 6L{'uro}(i1(0!.

23,1a A1\0iqI tar)/qi c) Eda ot mrtrEE iou "l

sivar Yaa[pt169 a!€EdaflF€S, lore roA, =, fxer lio sre|66s )'1]ol yLC u(ioe b.tj) Evoq rilu"os 5 fri 7 bw ex€r nore leatprrr;( aleldefltrFs srrt)'sg

2,3.19 Yrooemr o1 raiQlolls n bLsvl|ottora lorr -R"' o.r' drrl),rq ro! riaaac 1. E(i!(ud riac, c!€EdanFa, ro'a €iroL I ldfu lot1 ,4i Edv noe(i,Iou! tor R"' no!( eildL

r rdEtiTL o!p6.1irsr otdv dlrd ariord,ouv 6dot tou R'?'i

2.3.20 BaEilr prc 6(il'lt ror uroxdreo! ro! 2 orf 2 nrldzov to! onoio' 1(r aeeoid[(td 10'oqld)!

".rt ro doqoidFdto (Dv ]aapl(;! .i!dt o].d iod. (rK E€x('r$dtri ent6q.i6rot:

Ba.tr. niu€ rqcuurad ov.Ealaqlols 3 .fti 3 rilouts p' (rxtiv tl! tbLoqto.)

2.3.21 Ed! o ,4 .i!d iaas 64 rri 17 rilotda riieEoq 11, 161J(l clrE(iaflF( 8uvnoForcudvororclv io ,4, = 0i n6od av€edeqta 6Lc'l'o!(rd r.Tdvororolv ro Ar , =0:

2.3.22 Yno06m8 orr o brllluottona6s X6eo( v axft bdm(ot 7 kot o ll/ oi!( 1,noloao5nteorid.lt 1. A|'lei< i l'e.|diq:(1) K.i0€ 6.iot to! l,l' uttrfiv.ror o. lltd 6dol to! f, €flolliitrrolIds tqio azoBrl

(2) KdoE 6d!ot rorr v r€aIEIEt 6dol ro! lI/, ror tpornnret f,aqa)'Eino'ro! taid

2.3.23 Yro0dote 6rr |1, 1)2,..., rq €ivoL Evlia broltropnra tor R7.c) rd bLollopora atrd (€i.!d!) (8Ev €iva,) (Froaei !( €ivcr) 'feoFptz6s aloEdaxlF(6) Aud (raa(iloll!) (b€! ftoe4ouv) (proaoi !a ndqhyoN) tov -R7.

t) Ed! xd bLdvnoFard dlrL sivc' or ont].€q to! n. t6ft to At = l, (alft) (it€! aXEo(pnoaet !d Frt! 4€t l.nor.

] ] OI TE:>EPI: OET'IEAIOAEI: YNOXAPOI

l. O I(bQoE oqLr|v r.or;1, riou outdol"i.icr(I ttE ?(,4).2. O p46xvo7r'tpoE ron A. nor oq6olilsrc' tr€ ,\(,'1).3. O XthQoq yQq+(6v r.ou ,.1. :ror rivcrL o ltirpo< ot4),rir tou A/. Aur6g

103

2.4 I OI TETIEPI! OEMEAIOAEI' YIOXAPOI

H r(QorlYor]$€1'l1 n.rad"/a.qoE co){oilelx€ n€arooortQo tlE o0IoForE trced $r xtl-r{ior!:u6r. f!(,railoritr€ rr eivcl' Ircr 6doit (l"iir oXL r(irs 0(i q 6pow€. Tdracl.leznducrq cr6IrG avd"lrrzrl nupLyec(pi ev6-< rro1o)ao!,0o 0iiatr€ !a uroLo-yioortr€ tra orJtzExaqrall 6dol.

Or ur6lopor ;rrqLyA(i9olr(I. t:a./iatr. ltr rotq €rotr€lottg bro rAotror\. :rolrptb-ro bibercrl &c o!1o),o 6@\,!opdr(D! no! rcrpirlouv-rov lti-r-oo. Attl tivur r1arpinr0roq tor,Tr,rpo! oqi(;l. 6r(a dlolv zcreoqrogri or orrlLt-r-. ltor 8cuttqtobibovrcr opr"olttvor r-rleroQrolloi rou :rpdncr vc Lzavo;rorei o rlno1o-'Qo<. Mo(brarzprvi.lercl or noru brclropcx( TrtQrilortat mov zcboo. sL).ir;rorsq ouv0qxtq:rpirrr v( Lx(roroLouv. o fl6!:v6x{'0oq (rrore},rirur ar6 6La ru 6((!1,(J$(lr(

r , , . l . f . r r " , (r

z, iHr r l io"of u ' , r r(o ro or.rorr t l r , . n ' t f .nr,1v, lir<rr:ftproprop6. :rrlv ro(i)rrt rrEQryauqit Unoeri 1,c u(iQXola (itolor€( orrii€c.:rrlv 60ir00rt trnoart v(r r:rcipTorw tn(v(l([ij(votLFvor rrotooLotroi. :€ xcfttliucno r|c nEQt]rl(boEtq l!1 riv(( b1,,/(rrdv. tur $r(1 [crrir. !(1 YQ(uouu! trrLt 6 oq.d),(i )(Qr(i(r'r(r tll( ouorl}Lr1rrzi 6|r16rr(oi(1.

O cv(r./1(;orilc !noQri !(1 fcrrrdlJer fiotcr 0o ti,,ut il bLqbrr(oic. O(l 6€i-Eorl€nd)q 0( ciQo]Jle pro oriorl 1L( xr106!ft\' (1n louE rrfioXdQous ro! ox€riqovrcr [s1o,' .4. IQloqrorordrvr(: rotlq /- xcL a/ (r(lr P) r.0Lr n(QaYovxqL x(r(i rlvtr:rcr),oLqrri. KcrrdnLr, :rup ti),o :ror cu16 u( xcrvt! I)tv l(QdyQoq,o 1t./(l.rire0tdn,r f l : , i ) . ) . ic. : rgrnrr \ , i rLrr( !o| t( rr< . .zeu,r: : r .Q.:rr , i ,u irc:

f ) t , , ) r t t a l r t t r " ' t 1 y y r ) ' : ' t . , t J ' r u , t i . ' , , , i ' - , " , j , = , , = " .tujrt o:tit,ttxtts lysL itut apLmtpri ra,rirrrpoqo B i drik: rioeoqoC tj bHrupo utia4oqo A '.

frc vc opluvdroo4r r1v d),t1 oult1t4otl. escoponB€, frt- rrt oeLa.i. 2o061,(1,cr:r6 rolq rtlooL_Qrq 0tyt),Lrir6trg 2rirpou<. Auo a:r6 auo!< r:ircL our€i.or 1{(1! 61l()

ai\'.ll o R(,.t'/) xtlL :raptr"/erar cr:r' r|,q yQaI$6E rolJ ,'1.4. O aorm€aoq lrtb€vo]rDpo( ror -4. nou !i,' I o trlb€\roXc'raoq rou

Atl16-( r€arix€r 6).cr rc 8rcl,roporcl , yLa rc onoia ATg = 0oq6o),i!erc1t pe,\r(.1r).

A T .

To crlroo1$siorro €ilaL orL oL 6!cl re),€!'tciot lnoloqoL :tpodqlovrcr crr6 tov,{7. Oruv o ,'1 sircr ivcq rivcncr'( rr e:ri r, pnopei.rs rtr 6ei.rr roror Xtipor "rpt-ioleronv ' (lron_( rorq lla6y,r.rpouq. eE$dEovta( ro n).t0oE rt,;v ouvr€rayravorv:

O prl6rv6xtDpog ,\r(r1 ) rcr o xdJQo! ya(1[Fd! R(i]') €ivor lln6ttj)aor .ror R''.

l 0 l AIANY:MATIKOI XAPOI KAI |PAMMIKE' ES]'O:EI>

o (lprorep6g Fl6evdlolpoE nr(,4r) rcr o rbpog onlL6v R(A) €ivor ljn6loQo!rou ,R'r.

O|yQapp6E €Xouv n owr.ortioeg xor or or{)'q 61ouv m. Ir(r dvav anl6 nivexa6n(DE o

o r6eoE orlldrv sivcr I e!0eio 7ro! r€Ql'd an6 to I I l. o lApog 1pola,,lu .iuot

I {u0eio nou rrQvd ono ro [ 0 0l/ . :ror.r r€Qr€x;r;L oro F]. O lqb€voxoJooEcivaL 6vc urine6o tou -d xdl o oQlor€Q6s pI6€v6XcDQoE etvcrl $ra €u0€icr oro -R'?l

o ir(A) neQ!6X€! ro o y'/(A' ) :teprileL ro

11 o olL0 0 0 l '

ltl[:].-lil

f r 3 3 2 l I r 3 3 2ft /=10 0 3 l l xa rA= l 2 6 9 s l .

Lo o o ol L-r -3 3 0.1

Xrurelr|ore 611 6t.c ro 6laviopdra €ilcr til(rv1Jop(tr(t onileE. Axo[n t{d! ol'ypc4rp6g 61ow qvqoxQcqol Hor o X6QoE yQdppcbv tou ,4 Ei,vsr, o xdQos ot4l.6vrou,4?. >!r{OoE, o qr}.6g ni,voxcg eivnL o Ll, p€rd qv sra}'orgli. To fiQ661.!pdpcrg eivcr vc oov66oouF€ rorJE uri6xoQoug rou [/ [€ roirE rJJI6l(0QooE rolr /. ffcr!x6, rqvr6lQovd p€ rov dQxlx6 tiivoxo,4, e€opoqe xcrr. ror xl.qruxr;r6 U:

fLd va xarvoroFlioo]JFe, oEeidEorp€ ,ro]JE r€so€pLg lifioloQoos $€ $q nLo €!6!(r-

3. O rirpoE lpapprirv ror A O x6aoE lAap[d)1, €v6< n].+amotot :rivcxc t/eiva|" qcvep6g. ITepreler, 6noE lido€ I6QOE ̂/ea$dr', 6l.oxE xouE ol'vbuoopo{ErrDv yQa[F{i).,' -rDor6oo I rgirr] 1qa$n b€v orw€roedQ€! -rirot€. Or 6rio nQ6t€q^lqal$dg cnorcl'ouv 6cioq tou lirpou yaoppdtl. naQ6FomE xdvovd6 r.otn€i ral.7Lc xd0e rl4rcxor6 nivoxo pe r obqloriE xcr. r $I Fqasvrx6E yeo$F6<: oryq y46rwxi< yQult l ic nu va nv,t , iq4nc xdt o XLLaoc yaatu,;v rcu r t6noT ao4 r . Kar,a ant lrj oriprroroq, ro i6r.o ruxo1"c oupnepcivoupe ro crvtiLolonpripnrcr xcr yr"o ro\r aQtlx6 ni""rtxa A. KcLL nuou 1 rpirl lQo!+{ 6evouv€rog€Q€r 1rfio1€.

2M. O ltirpog ypcppdJv tou ,4 6ter fltv i6ra braoxsot r Fe rov 1tbpo yQdF-

Fr6v rou U xa0tirg xar qv i,6!c! 6dorl, 6!oxr or. Xdqor lqcprpriw T cr,rilovtar,.

tL I f0 I. i =.- r=/ ' o I

t ) 6= .1 I ILL . r L l t rl = I

t r u, , ! r , i , roacr)(o,\rgtr i ror. UoDq Drl mr' l i l l )z.r'trldi).lrrr tr Dr tld343e U rD^rj nor j,r.[t(-rrD3l1 5risgnqr ro rrny t D)x I :iltr]o5r1(| jnol,[qo ]r7,r Dn^r3gpnrr.j L'I jlr[y{]Dri l :rllr0iri./r llr rDz 5rrrrt)o 5r1:7)r^r,DrorfuUdi( 0 = xrl or d ')on1 orolo ôr ri Souor']1 o n)^rr 5(!(lrD"t iornv

(f ).\'ior t oDg ̂ no13rorr or.igi)r ̂or ̂ 9!nD .rl ̂ ,loulzoir nor t)rt)rlo,t,rtre.r ,/ D.l. :rltr'/gD'3rl (:gzroDo) Jrrrolgtln jrl rnl LtoDroDrDzD,rD tolodqD^r)}l 0 = .rr) o|Drr+t'r3 rDL 0 trlrr lr -'i.rt|t9,rr3tl iidrgni {3 :inx)Iorntr )r19.o rDz I fnu U hu(gr)Dl [ar0n3.{3 r0nx ro DkrToqDrQ D)r]^lQ'^r,rorD)l :Ro/, trgo^l lol i r ( : l . tx^,rgftor i lno1^( l t t ro,r{or^(n^l3xr5r;(roroF^r)'j,uulqDl3rl j3d3gfl31i| .r - r/ ll){r no! I) = .f,') D tuo+o om q/,a)/.r^t) .r'lrD1?lDorxonrDz lrcriq Drhl'.] r/ [oDro?e rrXt (t )]\'Sod(Dl9^3qtlll O NZ

:oonl.)){o^]Q[rl nor Io])iJ r)rn,rlilodq 7r^ o!9alr o(lno/,rt'^D^,r J l l roh:rD) r ' { i r : rr rnioi i { rne ^D-:t - / nor j3l ibd/. jJ,ar^.reLr i l r i . r j r r

olj|r (r)(lj1oj(ln)zD) [ f-,ror.:IilrJdl jr)lu(]Dir,\t i(Dzrnnr]l ./ JortrQorx)loo!'r r"r^{':(lo0Dz r(rr.1v xulu(iD:r^)) D)^ll ,\oion .r x!,ro(,o':r ri ,v"r (,)o:rlrflfô//r,ol!r t)zr,\: o,\D)i nox inorlor_torl]r ra :nor oljv -),rot onoxo,\rQhd.rot td to|g| o tn,rt! y trot toneflo,rtghd o dy trtr'hi(lror1^D D).\'.r ))xD)z'(I)'qu l L N ' ' l z 0 = , , ' . ) o ' o D J ! t l D ^ l , l ) = ' | l j ) ' l j t l r o L U l 0 f . j l . r q f { j l 1

rj:))1{r t'r jrdr,)l rnDo(j)oi:,r ̂rr"rnrhd/, ̂ o)r rrJtuono or |lotuouoy!)) r,r ,v)rtrj l rbro,{Dur Jt(r :o;(9ro 59"r) f i r ) o l ]o,rr l l0tolro V ,ror:odolo^rqhd O 'Z

) 1ror J]lilr)rlL iixr^rqtul Irl :r 'jrtqr(]zr) ,rlllooturrl.l ))^ odrto[oi,!]) ^r)r[ rl)rg.Jrlno r3vDLiD! rJ^ JJDn D)i{ iriftoohrr)dz ))^ l,rrhl1)dl, jrlo! rrinoor(l1),)o1rt D^

olor{orq ^))rL r^ jnnlo jrLDL }iro^r)z or ))^ rn))oLrodorlri ))0 'ta or.rn )r,(,)')nn3'UoD9 'Jnj ro ilno:lvD).Dz D^ ti/. 5,]rn]) .rID .i 'rr )7D^(2o^/',]) ,v,'rnb('i/, oa(,)Iôr ,!,rol,DiiDr ,rorl; nor i]ll'b(l/, dr ju rrl llbotl^rz3: ^rQ r1o lr1'!,HrlUI

'^(tnriDd,', od(r7 ,rol ,\ro /rrox))rorloDng,\ro x) j('qrlzD rornv rtJr,v):rNDr :nor ^(,)riri')d/, ,([ ror'r,r,rQ,r,rr ro)) fr) : i r f i i r ] l j rzr l rdobDrQ ^notr , t , rz t ; ro uo Dr3etr ' / r )

"r \ 'A I nor a D)zI j i ldrnl l " j r r ox)), l rot [odr i f 'nor j . rnn)]dt rO r nor Z r ]z I : j r lnD(l / , : rr ou]rreluo(l! rl nol Z qrlriDd/, H ,l .ror jD,rD .ljD ̂ ,)oeUrz?r^?rl;l r)^ ̂ or?r^ne rD^rr

I/ ror j,rlrl))d/, rO mrr^?l îq ,t1o!ll tr bDdrorr/lrr )r^ r.raoxrl )rn[t r0Dz tLgr.Djr'jn)ôdlonDJ or^ irorrr lrXrrd,tu ̂ tq /l nor ^c) riDd/, :od(,)x o nd)) lr llor ,!!)rl'ri))il/, ̂ .Drir'ldD ô)r rorlonng^fto roxn irrlll r1r^l? l) Lror 5.riln))dr'. rO or(il()1?)îr^{')rjlixl/, oi(nl \or ^no^[ir jr.r!ri]! SriQ(J)rr'Ao)o ro rro oro Dj]3rlol(lq DllD IJ

l uda5\ ( , l t ( - r .1 l i t \ ' l l , { l ( : ) i l d l : : l I l i ) r :

f IA : ! \ ' \ , qT ' } :O I \O IO I K A I TTA \ f \ I IK | : f : I :A : | I :

:E rorro ro lrapirb€r1lc l zar levLz6repcr. t:iroL uixo).o vn 6otpe orL tcrbLcrvtrolLcrcr auir eivcr cveidprryc. KcrOL- o!!6urottoE .rrr + cr.rr 6lsr rlvrrgrj c1 toq r, orryrordrocr zcl r1v rqll c3 cr< y ol,vlord)o.r. O [ovoq iaoroq vaurcrvo:rorr i ror I (r : {r+(r:r ' r =0. t i r<r rct qto!$r cr = cr =0. Ta buo srr i rbLuvroycrtu ncrpirlow cnio4 ioa FlbFvot(nao ?,.rr I y€vLzit ),rol €i.vor tvr{(ou,'6lJ(o[oq r,.rir + r.ri:. Aoa i(l n r = 4 2 bravuotrsra clrorrLow grcr 6cio1.

o trl6r'!o7,arQo! oao$(iittar urirlq xtr ;zrplln< rot A xcL 1 brdotaotl torn - r ),;.^t€.I(Lt tt\ ;t 1, Dtirt tl.

1. O Io)Aog otll,cirv tort A fle(irr( r{(irL qiuflxo [t rllv opo].oyic: oulvir oXd)Qo< orl),(irv ro! A oaofdl!ru|,rra(o (rrtr(i)v) ro! A klr'€xei npo€axFrar royadtrlrc R). Auro riv(( oitrqnLrao xcr tr[ rI ouvlul rb€.r rou r€5iou r4r0]v Uroqorwriqrrryrl; /, r\ oUv6IoLr 6),r'rv ran,bw(rrox r+(tv./(.r). To / cl,rJxr( oro:rebio oprqori n(lr ro.l(.r) oro,-[€6io r4(!)v. :trlv fieeinlorol Uoq q ouvooqorlri!(r ll/(.r) = A.r. To r€bio oQrotrorJ lrN cnor€ltix(r (lnd 6)'a r( r' llro R" 11(l10 xrbao r4r!')1, rtq (l]r'oi(l r( blJacrd braviot(rra,4.r. (Mc rilla I61c.6)"a rcr| 1Lu ra o:roiu 1 ,{r = ir p:ropti vtr ),r0ri.) h{,)ailorlLe 6rr at;to €t,'crt crxQ!66qro oivo),o 6loa rov ouv6!co[(i)v r0)\'oq)"(;)!. (iQcr to nubio ttuor si\,(l! o ZdQoE011161. Oc rE(xo),o!0rloo!F€ v(l Zelorforrorou$r ro! oeo Xti)Oo< (ft\Ld1' ti\Lcl0c 11[oot.Trtooqu z(r rov ollarotro o$6o),roFo r?(',1).+

'I o nQotjLrttrcl uG< €i!.i! !u 6Qoutr€ tlr(1 ii{iol ror z(irpou orlLtiN R(Lt) ctr),itzcrL pLcr 6cio1 rou )(d'Qo! orll.drv 1Z(At. Or nAor cvtoi tbat iraEoetrtxoi(apxti vcr xotrrilore ror( nir(rtc!) (),tr(i dlow trll ib!( blcioraol. Alro €iltrl1(11 1() cELoorl[rilDro 8blir.

H nod)rl x(r. rQirrt or{},it ror L'crorolonr 6cio1 ro! xti-rpou orl}drv ror.Aurir €ilol oL orrli.e; ror ntpiyouv o64yoiE rcrr xti0u {i),Lr oiiLl €i''(lo! !b]J.rot.oc u"rdv ruv bl ' r . To ib.u .uxL,, | Zur yrr rov ,r l ,x r{ , ' : r .v, ,?, , -(vzcrL or o6ryoi btv eiv(r o-ouroi xar or ofilI€E civc! 6rc(roQ0r!xd(. H fiQdryl zaLrpiq or1L1 rou,4 rcrorEl.orjv 6(idl rorJ IriJQou orn),Lir! ror. :ilouacl rl bcnr€aloI]tLn 6er, €i\,.r( .1!€E(i0q1n, .nLd Eivar rQrrl.doLa rnq npdrlq. H riroprl oni),Iroonr(r tra (orytl.n 1) + (1/3)(oritU 3).'Oi(v oarol6a€c oirllrq rou U rmolel"oli','6dorl ror Ttirpor mrlLriv ror t/. rrirt: or. uvrioroLTeE ottil.Es tou I dnort),on\rii.iorl rou X6Qo! oq),(;v rorJ ,.1.

H uLrirr 1L' (lur6 ..i.,'.rr 6rr ,4r = 0 LoIiEr 6rav czpr5tirq t/r = 0. Ta 6iroororripatc tircu. 'oo6n1o$c xrrr dlouv tq ibreq Luoeq. H rirap.rl orlLl rou L'rirc^ eriorE ior pe (orrjit 1) + (1/3)(mrih 3). To yrv6pr:vo nivaxa eri 6(ivroFa,'1r = 0 ExqphlEL lpcrp$Lxrl e(cipr1o1 rtDv orll.tirv ror A, pe ouvttLtottq norza0opiqowaL ar' ro r. Arlti I rlhprrlorl cn'.nororxei or: pLcr r:qdprrot Ur = 0.perclr rov or1).cbv ror U, [e rolq i6rorq czsr6i\ aurrcl.tar..1. EdL iw otrotroat416t, nu A tilaL toeAiqtato. r6re ro i6Lo ouu6aiytr /.aL rc rLq uyrimoLxeq

* E !d .nrlrltot ditnrrrf dr dr( A./t'lta.i o l.irao<./ad!+Ld,! (driqer xiiir rL€ ro ibro tpdr$c.: n " r , ' \ ' t . , : r . | \ ' , : , i t n F o ' ' , ' , u " y ' .

ortj)j. tu1, Li xtt rflore6qr4.*

frc'\'( tlpor,ft r(rAu tu( 6.ioil ro! )f(;Jaou onti(ll'R(-.1). fploLtrororotittt<r.,to ro| i1tlr rlbq 1Lu rov fr. Or I orl),r( no1,;(€Qr6IoLry oblloi: (nor€io|rotlorl ror t'. n(ia1'ortrr i.or;rov rr; ilrr: orrji.r! r'ro!,1ir(z( .l oi!r{{(')1( !r€ r(1rqrl::

20 H brl{orcrorl rou toQou oq}.drv R(,4) roonror [s xr]v rciErt r. rroljLoorrr(lr zcL [r 11] blalor(o1rou ldpor 1pcrgtr(irv: O a0reF6q rt0v uv€E(Qxrift)vorrll,ti^, toorr(r F€ ro! uQl0p6 rr,w nvelcrprrircov yQ.4+rdv. Mr(1 6dol ronn(,4) ffiorEl.eircr (no rrc r ori).Es rou.4. (1,ri('Toq[< rov 0rll,6r rou a/rou x€arslouv obryorK.

Atl16 ro./fi,oloq.6rr o /(;)ooq orrt)dv r(r o X(;po< "/a&trtr6r ilollv rltr i6(r6rrirrr(or] r'. eivar (arri ro nro rvbr({Iipovro 0l.i,)otttorcr rrF l-Qqr!rlii< AUr60(lE.:r'Ilil ouvol,(it.r(r (tlrl qtQaat "ldE\ y?uppiv = rdl4 rrtlrir"xra !xqai.!€L av((lno16i,€oI(! nor,I( i\rr\'tri\'crzu 10 r'rri l2 bev tf.r(r n(t0dlor, noo([(!dq. Erioil(iitr zcirr 1rr ror': rtrqtrlr,l'rzor,::rivrzt:: Eriy or I'prrutti< twi; rrpui'rurrzor\:rinnt tirtt i'panlrtxti< ottriprr1rc1 ro t:6r, nry(tuiva tut yt tr< orilit< rot(z(L (tlrorao(Jor<). Krrr :rriri,L. rou),riTroror utov ot"71Q((tirr. cl'rri btt rp(rirt(r

frc v( brirr'trr1 crotrt tpopri <irL o Xrbpoq ypcrsprirv zcL o Xdpoq m1l.6v rolU iTouv r1v ibrc brdor(ot f, oropliorr sru ru:rrxrj z,rrirorrror1 [r rriEl r = 3. t)r ),1r1(zorro( ni!(r(( a.r d/fr oii,orQ( rar'( (\,fE.iQ1I1eq yQ(l(lri(:

I(]7rprioFtlore on rrr{ia1ow uriol< rgrr; rrvr(riQrrtr€! orrt),€q i( 6/L jr.ro.rn.ilo.Or orI ; . 'c r1 'ur ' \ lu ' \1 ' ruf l - t r I t r lot \ ' rx. : . or. \ ' r f rul t ' r l r 1. 'qrr ,N lr . roqiooltry ' . b ' i :o,1,, uf i rr r . -xr: 6dorzi : onl)r : ' t : raunl. r , ' r (ant. z, .r iznt ' i !d,1pa$Lzdr; trvtlriprrFrq. r6re :rpi:rrt vtr taorrLonv dciol (ror, 7,;,pot' mrl)drr rou{i.6rL ror..ll). Y:ro06orc orr

"dnoro< orw6lcrqr6; crtrdrr rox 6crorzrir or1}.dx

ot TE: :EPE OLi t t , \ t ! r \ t t : 1 r loxaPol

' Nlor q(trrr(L {i iti".rrar t,ro roa rr irird.riiotrofro xo! r|,1,r! d(}rla. lriloL riirp! dro

nLniro. fr( zd.r rd: rrrlt. hN aaler /(!\o:. 1(' dfllrio(otd lO mo ox.ir' oirlt'.i. rndr .niort: ri

xro i.rtrrn zlr 1i trr, drB(urzo.

f .r , -- -

. . lo o o , i ,"l::::

108 AIANY:MATIKOI XAPO] KAI fPAMMI(I) E€I)!)>EI:

rcadl€r ro trlb€v:

Aotllsuo.!.Ioq rraos ra rr(ilurj rim'4 ouvri0o4, 6),crouue 6-rr ro r. npdner. vcrFlb€vil€r(A, l.o./(D rou otutlorl d3 10. Ktrr6*w ncr to t|r rp6;teL vcr prl6evile-rar",brort dr I0. xat r6l,oq rr = 0. Auo uro6r:rzvleL t1v ypapprzrj crvalcprloicx(,r o!tm).tQd)v€! rn! ano6rlEq. E:tetbrj ,4r = 0. or xcrL p6vov cv t/;?r = 0.nuLrrr Crr ivortrr , l r t q, tpa'r4. r i rupr4 zor exr4 or i l ' t rou A oro.n: xr , rv trrrvo cp1a6g,4, rov o:roio ldfu.or(l our€ fio! yr,(rQiqor4re. o' cnx6 to nc0d6eLy[aunor€),oih 6iror ron R(,4).

Atboupe epquol oro 6rr" o Xrirpog .7p<rpprirv rtrL o lrirgoq ortldv !€roeoQi-o0l}u! xu. o. buo Lrrtc( 14! (rr i ( Ior ' f ' l nrov A. Arv Ftry,ronixqr t l l rov Ar.{rv xar oiloupu 0u proqouoope \rcr clcolQ6uoljl(€ rov ,,1. rv(lll.dooovrug trqYoclrlliE to! $ rrg or4).€i ro! ()r(lL rov-XdQo orl)d! io! [r ro! IdOo yQa[4rd!1o!). xrfl x(lrorLl, v( uvcilor4rE rov Ar oq\r xl,+crz(,)rl Foprpj tou (aor eiv(r6LcrpoqrrLxyl <r:16 rov U7). Ku! clJro o6lyei ororg o(,)otou: X(i)po!<,6€v ei.vcu,69t,4 1 or,rorl r6ic. YnciQXorJ! r[ol,]'.ag XerloEls 10! crrlcrorQogou. al"]d rotrrl 6rvei.lcf trt(r cn'cur6(. To yr'./ovoq €i.vcr orr o! R(A) &ctrR(At) 61(J!r, rq1, i.6(rbrcrotclol x([ .i!c! 6!!tlr6v vtr 6pe0ouv o!./Xpovo.rE crn6 ror [/.

'EeX€Io! t6Q( I ocLQir rou rrriraior 0tr$str|{;)bo1jF rrrrol@Qou. fior.r ficQ4tE!€FiXAr r60a qq.lvrlE. A(po! o! rp€rE nQolo! Z(i)Qo[ €i.\,ol oL t?(,4). ,/V(,4), rq!R(/1). o rdi&proq 0u rpirer vcr eivar o "\r(A1). ArLtoE fi.vo! o p16ev61t'rpo<rou,,v, i , rr4 'r1r,u A. i1 aptmtp6E pf infu.,rAos rou 1 d'oT| r4l , / = n d4ru:vfry' A = 0 t(.rt 1cr 6rd\aropu tpqlavi.leroL or1'v cpLorepl prqLd ro! ,,1.

4. O aQror€Qdg FIbcliiXo,QoE roo A l= o [IbrvdIDAoE roD A7 ) F.iv o .4t:ivcl :rivcna< rrr sri. n. 16rr o,4" rivnr,. siii ,r, o F16t:v61tLrp6g rou aivcLuiroX'r0n; r . f R . ,{ur roor. tvuo/a j / iXf l / , ouvrordofc. Iqc, lovrrrE1rI 0.or otnrordioaq ar]Tiq noL),anLcorir(ouv rLg 1pr+[dq -ro! ,4 xc{ ncrpdryouv rr]trlo€vu{I YaaFtrt:

= [0 0].

l;llrl.' Ii] ."'lf l

_lu'd=Ia' u, , , lL A

'Ero retoto 6Ldvuola-ypapp{ yT ovopriterar. aprcr€Q6 Fl6€!o6!dvto$d rou,4.ErzoLc 6pioroqre q Srdorcol .rl'rot ron Ft3€\,o1r,)Aou ,V(A"). I'a x.ids

rl'\,o)r.r. ro nLt0oE ro\'6aorxdv F€ra6).Irdrv ouv ro n),16o9 trov e),r!0epurvF€ra6).Ir6v tiQdnc! lcl rcrprciler ge ro ouvolLz6 nl.rteoq orit).6v. flcr xov ,4 olnotrcv ]" + (n r) = ,. Mr: d),Lu L61Lcr.

2.J Ot t€trEPtt oEMEAtOAEtt YIOXAPOI

6rdoroorl 1or IdQo! orlL6v + bldoxaol T ou lrlaev6Kogou = d.rioog orll"dJv.

109

o xov6va! avr6E eqaQu6lercr. eE i.oou rcr orov ,4"EE ioou xclo< :rivcxa< gr rov A xcr 1 6Ldorcorl rou€nioff r. 'Ero[ €Ioup€

t - d i r r ' N \ A r ) - n .

riou 6X€! r. o-rnl'€s, eivalX6Qou oqldv rolj ef1/ar

(1)

2P o dQro'r€Q6E prl6€v6xoQo9 y'/(,4") !1er, 6idotcroq m - r.

Ts 6Bvriopqra U xot6o1.,id[ {dfiou orqv ofic}.orqrl, 6r(rv or ^fpopfaE roo Aouv6rtilornr yrd v.r ?i.rQdyouv iE m - r lnbtrtxl€ yeapptE rou U. lLa \c6 Q o { $ € r d , E E x r v d F E a n 6 r l v P A = L U ' i l L - t P A = U . O t t e X e o r n i e E n t - rlpcrppig tou L-1P nptaer, vo cror€lotv 6doq 1o! dolol€Qou pl6sv6xoQou-6L6rr roLlcrl,crodlowdl [€ rov,4 x.rr 6lvouv 116 pl6Evrr6E lpappdE ror.r [/.ETicvdt'd$6drolg€: O dQlor€Q6g [nbev6xoQo! n€Q!6la ro]JE ouw€],€fi6E r[o!ouv6ltilow rq 1pa[FdE roo ,4, oor€ tc ridQdyow to Fl6Ev,r(6 6!dv!oFo.

:ro 3 sni, 4 noQddE!/pd pcE, q prldevLrrl lpappri lrqv (1qc$pt 3)-2(1qo$i2) + 5 (yod!4rn 1). Ar.rt6E qtcv o i,6ro6 oovb]JqoudE pe rov \ -2b2+501 ttqg6€E!(k aLEUedE. nou o6nyouo€ rqv rd,,evrcitr eliooorl orq goaqri 0 = 0. Kaidouvdnerq o! owrfid)seg xolJ g 0q €i.vo! 5, -2,1. To 6ldvuopa o!t6 €il(r| 6sonrou oprorqorl pl6sv6xo)Qou, r iou 6XE! 6rdoroorl m-r =3-2 = 1. Aw6€ivcrr rl i€l€]Jrcticr 1pc44rq rou.L-rP nor, ncgaler q pl6erLttl '/oqpp{ ior] {/-oo1vd prloQoljp€ va 1o borlpe logiE va lrroloyiool4le qr , r. A! grdooljF€os droyvuJoq. UnoQouFr nLivror! vu l.rroouF. 10 ArJ, = 0.

A1,c1yv(r)Qil0) 6fl ro 6[6],io, pd1pr. rcirpc, 6ev qavdg@oe rov )'oyo y,c tovonoio paE e\,6LaqdQE! o 1r(,47 ). Av yptirlo ge n}.d1c ypa4rr.a o etqroreg6gW6*67a-ooE tivat enioqE o4panLxi;0a ai"tat a.oor6 oLlo xa06lott trcloflra6.H xotdorcol 6c1,r!{i)verc! oro E i6p€!o xeqti}'oto, 6rou 6qioxouF€ trLd quorxrlep|Jnv€icr xoo g,

Tri'iJr rtrror yvurQitowe rL5 6rcordosrE rov rroodEDv xole,('}v rar unoaoqrrvo xiE ouvoqriooup€ o€ €vcv nivcrc, pciLrotc rpctrercr, oorm6 ro vcr t< 6r,oqrpi-

OEpE,"tb6Es ecbp\tto, aE Ieayy.xi; AAyE6ao4. MipoE IR(,4) o xdaoE or4L6v rou,4. qsr 6rdmqot r.Ar(A) = o u6ev6xtDpog iorl A, 6xer 6doraor n r

R(,4") = o t6aos Yeat+rin, rou ,4, 61er 6rriorcoq r.A/(,4") = e splslraos t],ltaev6ltlgog tou A, €1e', 6ldoraol m r

2QL

4.

t10 r - A N ) : M A r O r \ . , 1 ( ) r ( q l . r ' a \ r \ r l K l r ' l ) . : r I t

l i , l[ qPAAEfn4A A= - l l ' , ' ' t - ) . t - l

t r o l

1. o xi)Qo1 oa4^6v nqr:xEt ol.a ra rollanliroLa -. fll. s o',i-"n't

oritU ri.var orry i6ra l{orruorlvot rar 5€a suv€roqtpeL rirore v6o.

2. o pqb*ttTt poq x,te!6xr( oLc rc rro).1.o'r),.do* -, | ?]. t" a'a*"r,"L , l

crmo. rcr06g rcrL rcl ;roLl,orl"tiotr tor rravororolr to .4;r = 0.

3. o xialq yaapyor auprr4tr o).u rrr nol) . ' ln l{o, ,( -" l j ]

** *

yeirq(u ocr' 6[dv!ou(-orrf,q. crg,o!. U(],dvrog (omled. n€Qrix€r(r oroaxd)0o ox{ld}v ro! ,'lT.

| _?l4, O ootortoia pnblixaoos rr'-DrsxEr otr( ru noil,irn).rc'u m!

| j

l.To brrlvwpu curi Lx<rvoro!:i ro ,47y = 0 -o olvatrcopirg rnrv lpctrpdvror ,4 pc ouv-rrLcordg 3 {c1! I 6i!e! 10 1lqbe!.

:ro fioQabelylra aur6 ,6r ot tQ€t< rrdxaQor tirar {ue€iral Aw6 ei,v(tr, crriFpcrno! og€ilEr(rr orr< r = l xo! n - ? = 1 xc(t'}r - r = 1. O! oorylo€[E 0(r axolvp€lcI]Jr€QI iro!x!l,icr.

:q[Elcool€ 61r ov (].ldEEr€ ro rEl€uxcfo orolx€io rou;{ ar6 6 oe ?, r6reol.rq o! blcorcioe[q utrldlo!!. O x6QoE oxll.dv xur o xrleo< yaqurdlv 6xo!vbLdoraol r = 2, o qbr:v6Xopo; rcrL o cprmcp6g fI6tv6XoJQoE rrtpl6touv Bolo 1(6Lcvrdpcr( .rr = 0 xc! , =0 xut o niturxtt< tivor ut,umQh!,qo<.

'YfioQEq a!'t[orQdp(',!

f\r(,rQiqouB€ or! .11, o ,4 iler opLorego nvriorpoqo (l]A = 1) xdr 6€ELo avri-otpoq,o (,4C = 1), r6rc oL 6no alrinaoqoL Eivar toot B = B(AC) = (B A\C = C.Me q 6o110lcr rrp rdllq ev6g iriv.rxcr, eiv.( ruliolo vc arorpcoiooqrt: roror 61orwcfJro!< torJE c\rr[fiQd(foljs. ML]drvrq nQolerQu, urdQXEr traa uwiarpoEos p6-!o! orcv I rti tildt n ltFydnrQn suvtrri.

H rirll Lxnvo:ror.ci :rchvrote r1v r 1 rrl Kr0tirg xcr' rlv | < n. E\Lt< rtE:ri rr nivcrxcc 6rv propei va 6xer :rreroo6reQeq crn6 nr. crvrltiprrye,< lpcrpp6_< t1, ov€E(iarlr€S orrltrr<. Arv u:rdpXer ltirpoE yrc rr€Qrooorupolg (mo ;t) oblyotsi rr0rooor,Qou! ( .1o , , . O€Lor ' t , f vo 6r iEouF€ orr orov / = n rorr uruCXFl60EroE .r\,riorQoqroE x(lr or(r! r = 't rorr lJfidol€r oQrorrQof .1vri.fiQoqoq- :qv,'iQ6rn ri€Qirwool xo ;{.r = b aI€( 7i(ivrorE Lioq. :rI 6Erix€QI reQi.rrro}q I ),!ol(r , r ' ono! t ' ruQXfr) r iv 'ar Uovn6rrt . Moro ;v, ic rFrQrl ' ! ,vrvol r iv ' ivr tc pnoOti vrttXf l t = ut aot . =, , . d0o tro!. ;v ' , , r rrALiT-\ ' ro( r ivuxl( FnoQts: va nrrux, lruuroxpo\a rlr.rqEl z"r povoo4p.rvro. Movo iloc rfrauyoJ!,zoc nrruzu: ixcrbi:rirrpov avrimporpo.

2R YIAPEH: To otorlBcl,4r' = b iler picr rou),irlLorov ],uort x lLa zdo€l. av zar [6\'01 ov or" mril"q rcrpirlou rov n-, onore r = m. : trrrtrtv

tv r€Qi.rr(,rot rJn.iQX€r Avoq fr erii rr 6rEr6< crvriorpoqog C ./tc rov onoioAC' = 1", eivcu. o rcnrrorrz6E ;rivcrcc rdlr:trr nr. Arro €iror 6rNar6v p6vovorcrv m < n.

MONOTHMANTO: To o!otqllu ,4r' = , aX€( ro no),ri pru hrol lLcr xcioe6. trv x.1r. F6aov av o( lpqr!6q €i\,(r yacftlrxri crveldprrlrrg. on6re r = n.: au-rrtv rrl! r€Qi.nrarot urdrQz{ i:rug rr rni zr uprottp6; crvriotporlog B.lfn 1o1 orroio RA = 1,, ltut o touroxrl{os rivoxuE -rdEetDc rr. Aur6 riv.rLbwcrov troaoa a'r(v rr > ".

:rttv np(;)rll ,r!Qinr{')oit l{(r brrlcrri inort eiv(r I.f = ('r. c{on rorF ,4.r =.l( h = l). Ot. un(iolor,rl q(')q zrrr ri),ir: ),tcn:rq rriv r,ncipTotv z(r iri.),or b.Eroi

:rrt beirFAt,rr0inro)ort. r(iv L,n(i07r'1 tr(r inol ror ;1.r = I (L,rit n0lnrr \'(ri|Lt r = lltl.r = Blt. M:roQri opr';; r<r 1rr1.r LrrciplrL ),ror1.+

Ytr(iQloll orrl bt(i0r_ori lL(q anioi rlixoL y(r rouc ceLorrporic rttrr c\r:roi((l.i'lrorQo(tor,q. flJi (ioo! cr-roi t|n.iQlor!i

: . t o I l F : :F t ' r : oENtEAl l lCEl : ynOX!2pOl l t l

l ] = ( , l t , 1 ) t / 1 1 y . u t ( ' = . t t ( A A t ) I .

Bdixir( lJ-.I = I tt ( :\ = /. AllT6 6tLoc :ror brr tirar rooo oil,orpo ri.vcr. tootr oL '1/'{ xriL l-'11 i,irur dxr,,q {rvrrorQiit'r1Lor. :ro Kr.rl.{i),(ro 3 &ixvoutrr-orr o Jr -1 ipL:rpri.ipcrrl olriorAo(o or.1\, r1 rri!1 r.ivalr rrrr rjrr o :1:1r rilcri o i J r L j r ' , 1 , , u c u r u ! r t r l r : r t ! , , r , . , F r . r i 7 , , ' \ ! , , ' . , ( , , r , r v(r,trii,;q rt rdEI riv(' rt tri"/[orl bL,rcrtrl xcrL rorr naoobroaitolr(r oL trov6xi{.!t'oL

Yrdoxsr fr io ' l ! f r r t nL,) oror/ fr( ;btq rr-xr l i l t rort . Mropot ' r- .v( c l(qlrr t -oo|trI flal ri\'c1x( a' r!0ririolra< orrtit naoq or)i],rt rrtl

.- lc=1 rj -1l.rL .r ': r,,,1 =L€ ., . €,,,1.

Kd0E oTliit ror C. noi),Giri(lorGi6[wr] [t-ro! ,4 6ivrr pLc otliq ror rcrurorr.zoijri!crz(. ll( v( i.nooqr€ ro -1.r', =., X0fd!!r.r' rri 6Lr^uoprrtrr dLvttru.lpirruv c;lct nrar67orrar oro\, yd)po oritldr!. El{! (rroq rEpLa}Er alr1.i r.r 5K'vrioF(r(. ror€o Tropo: orlid).v:rpra:L vc eircr 6io ro n"'l H blhor(ori rori (t ratE'l) roi'iEL!c €i.v|lr | =rlr. Ei.\'a I rEpirrrrot lq n:r.1aElq". or.rv or oIliiF< n(1phi,our ro1

r 'Io nt.ritro: tu^ tiac.r'oll\ a.lrirtulrl lr.lonriiltr.i. . ra i) rt . .\(:) dflf nra rr.nrl

l l l ATANY'IVIATIKOI XQPOI KAI I-P/\MMIXE> E:I'O>EI:

IAPAAEIrMA O€ll)arlore rov an],6 2 €rri 3 ri.\,o]iq rdEs(o( 2:

E:rer6r1 r = n = 2, ro 0etip4rc e!co9ol,i!r! dvav 6€Erd a!, orpoao Ci

,=[; !3]

,'=[fi ! lt[l i] 1;

,[;l]t*ir[t;]

ti i il:tI i] |;

[l ilirt lll

rln0dyFafl.un,iQlouv noD.oi b$Eroi avrioToo,toL. H lr)*urci( yp(y1rri ror., Crtvct.1r^fl(')< 4LiH(flQEr4. Awtl rivcu uru nFpinlrDorl rlnupEl; 61r op,ra rcrfovu&xorrFo6. O:rivaxq ,4 bcv iler uprrnrpor avriorpopo Artitt 4 ri).fi;tctr.onl^t rol r./l rrvuL oryou0a Blijfv r. 6€v Fno$i vrr ouprgurvti tlr rov 3 Eni I

fr' Ctwo ro n.1Qd6ElyFCr, o tirog C = Ar (AAt )-r bivrr r1v er.6Lxr1 nlrl,oyr1

O dros € Ll,€yer rLE (ruoc.iQrrrE orso€p.< (rt Lcr. cil va €iv(r pr;biv. ArnrlrrlLrl Fr( nrtnrrurol _ro1J Wfiour oreignr,. _ivqg reonog €rl,oyrig rv6(r-rorl{ou (lwlor€roq ou. or(rv b€v undQll t xdnoto< xrrvovlxo< rQonoE r:fl },oyi(. nouuvrurooooqre oro nprirro llaQd$ryo.

, O cvdmQo{loE,rou A 6iver 6vcr ncpir6ellpo mrlv otri0€q xar€i8wol, F€cfi erpollq aplorf(rori( dvri,orpoeouq:

?lTd)Q( reLr:ialq otrOcrlp€rl siv(rl 11 rplrrl otrf.q rou B. Afi6 t:ircL ru:rrx6 vr.(!rlv. 'nFei'rli,'(nl

Fov(&r.oqr(( . dr(v or n ('ryl).€S rot, ,4 rivqr ypcpgrxrirgov€Edeltr€E. H rdErl €ivor, r = n xg| 6€v ndQxouv €),€!0epeg Uero6l,ti66 cqoj,r - r -{). dQC uv undpt(€r pirr trrior1 0c rivcr xur povubrxf. Mrroptiri nr, a.irrofl rr) nltQddE!.fFa (luro dleL gra ).Ioq:

Luv€aoL 6ruv h=0.

'Oxcrv ro ,3 €ivqf II66v, I L{ot (Fov(l6Lxnt) ei\r u" q = (Il\by 4 = (115\b2.

].,] OI TE:'EPI> OEMEAIA ET: YNOXOPOI 3

flc ivsv a(rQaLtlq),6yQaFFo nivsxq, 6ev yiverm va €Ioup€ iqor6lpolotnnplrl zar povoorltrrclto. Eriv ro nr brogd:qrl crn6 to n, bev pnogei va 61oqrexar r =nr ltct t = rl. te dvav xsrQslov!&6 fiiloxa ouF6oiv€I 10 o\rtL0€to. Edl,rr = ,r drv tr:roQri vrr iXoupr q pic r6rorlrrr Xru6rr'g rcuroXpovc va iXoqre rcL tqldl.rl. Evs< rFrp,r'fulvLaoc rivcxuc dpt iv(! aQrotreo trvlioreo,to av xrrr tlolovcrv ilrr 6e!r6 rnrtoreoqo. Yrioe1a p6vov ivng trwiotporgoE. o B =C = A-t,Or(r! o nil,(rc< eivuL rerpcyarvLxoE, t')jteQEq otj,tltdf€rer ttol,ada.jrqrq x.rlyovu6uinlra ourtiniy€rar [ar(,€t. H ouv€rtxl ./i owriv qv &wlmQ€1'tForrtrseia(l, I rdcn vs €l1,oL l Uilt('Trl 6uvsrn: r = m = n. Auro FnoeouFrj !c robL(rundooulr€ zor !€ dD,o rQ6r[o: Avoyxdu xcr Lxtrwl ower]21 yr(I v(r €ivcrlivrr< rrrprrrrrrzog rivtrxus / FI r6LotroQqoE. riv('r va rx.rvonoLrir(r| tritt (;ro nErnol€v€g o!1'0ix€(:

(l) or orritr€( aapdryo!\' ro! A" rc! !ro! ro ,4r = b 6xer $tn rorLrixrororhrorl '/!ct xdoc /).

(2) Or oril.e( eivdr yQdpplxd( trvtEdpllt€E xcrl ttol ro ,4r' = 0 d]€[ rItrov(6rxi Luol ;c = 0.

O r(rdLoyo: (1ur6E lrnoQri 1,u li1,€r fiol,u l(leurrQos, s(6rx(i orcv xQ6EoW€m(l r:r6peva xerpc .ocr. K(ioe ouv0rlr{q rou xcrra},6you €il(rr roo6i,v(tul tl€ xc0€r?,L1 xar rlcrogul.ilrL 6r! o A €ivar trq r6r6[oQ*0os.

(3) OL yasBIa< rou .4 nuQdryow lov .F1".(4) Or yaquag -ror ,4 eivcr| yQutllrxriE ov€Ederlll€S.(5) H (rnd,oLqri [noQ€i vtr olox].qgr0ei: P-;l = LDU. ye 61't ro. d1 I o.(6) Yrrdpt€r trvq< rivcrriaE ,,l I ylo to! onoio .4A t = A-t A = L(7) H opiloroa ror ,4 6e! IrnbeliE€ror(tl) To ttl6fv 6ev €i,vtu !6rotq.rrj tou .4.(9) o AT A silo! 0Er!r{a oQ[oF€voq.

lbou llru rurLzri eq(ppoyi. @€coQrtor€ 6),( r( nolu6vuFa P(l) 6c0Fou n 1.To p6!o drolo xoludvu[o no! Ft6€vi!,€ra! o€ n 6oe6r'ra ory€k. lr. . . . . t,, €iv(lro .l'(i ) = 0. Knv€vc dd.Io rol,urirv4to 6a0foi r I 6€! pnoeei vc 6t€t , ai(€r.Auro rivur tvuc rl}j(r'0rotE Fovubrx6rtlrE rur or'vrn.lyrror fv(v Lolu0rotrounaQE$: Ao0ivrr,rv onoLovbr]:rorr rLt(i)v 6r.....6,,, unrie1€! 61tr not$d)vutro6a0porl n 1 :rou :rcpep6hiJ. t-ro o'( l rn6s rK rqr€E: P(/ i ) =6; ^/Lcr i =1,. . . , rr .To (lLorp6ouro ei.rnr 6 6loup€.ra rdA,ol]p€ FE a\,{rv xExQ(./ovrx6 ,ri!cx(r. To:rl.li0oE t{l)v oxr€l€ordrv rou P(i) (fiou €ivdr n) rolQLa(E[ F€ ro r],ll0os x()vFElod)o€ov. nedytcrr, o! €Elo{ioElE P(ti) = li ti.T'<rL r,ooStvcpaq pe to oriorrp<r

li ; : .i' ;'l [] t;l

I l.l r\IANY)MA1'IKOI XOPOI KAI fPAM]\']KE: E!I:A:EI:

O nivaxqc ouweleordv ,4 rivcr n e;ri n zcr rivcr y\,oror6< oc1, nivaxo< rotvandemonde. Era\aLoip6civorBe -ro ouLlolopo: Agou I 1,r = 0 6XU! xit trovobll{r]l.not r = 0 ([€ ci].],a ].6yK P(ti) = 0 €ivar 6wcto! polor 6rcrr P = 0). 6n0ror. orro ,4 eivcrr yr1 r6LopoprloE. Aao ro ,4i =, e1€r plrl ).iol €rc nol.!6v![o trropsiv(1 riEeva ofi6 ono!€o6rinote trp6q 61 or n SLcrrpoperrxci olpeio fr. Aay6r€Aa 0o6ao1,tr€ rltr oQiloro( ror ,4 r(lr 0a 6ouF€ 6rr r0ciyFarl bsv $rD€viq€raL

liva,.€E idE€(')g €l'a

Tiiog f0(ivon[€ orlv €rrzotr6repr1 rrpimooq. oiror I tdEi] ullcr I !!xQ6r!:atblrlcr i (€rr6E rou trlS€vLxor rircrlcr rou ileL tdrEq ttl6f!). E''( urr'r( oly(ln-x6r€po lilfil[or(l r(|)! M(erlflclrlxclrv cl,''(rr. 6o06vroq r!o( o!v06ro! nQ(i1ruaroq,1,( 6Qoi[E n(ig Ufioe€i. \,n brco]i(o0ri or' lrr].oiorr0(r rol4rdrr(. :rl faolqrLxlAL./€6Q(, x( an),olorep(l i.rolldrL(l €i.v(r ]rirax€E ?.i€srrg 6ta. r = 1. To €fi6ttE1'ored6{rq,Ia €ivar X(l0axq0lorrxol

2 | r ' l4228442 - t t l

2 I 1 l I r l t24 2 2 t | 2 lh 4 o l= l o l2 7 1 l L 1 l

,lKcio€ yQqrtri €ivo! nol,trc(Jl),iloLo rrtc rpd)rrt< yoclf)ic r(r o X(i)oo; yQct+(nveivcr povobrriora-rog. ne(iy[dfl, pfio0or][e v(r yQa(lroulte oL6x),rlQo.ov niv..xalL. 1()! c,_T6[e!o e!6!,{6 1Q6fio, aa\ lo\itEl'o 81,6< [n1,[olarc<-ot]i24E lt il'at\KtuMtte-),a44oi.

1 1 l

To yrv6p€vo ev6; nivaxc 4 erri 1 xar cv6q:rivcracr I tlri 3 €ivtrr 6\,cE nivcra,<4 !-ri 3, no! (xer rh!1 (vu. XrpuLrirote 6r t(rJr6XQo!.1 ot oqLrq cf\,(rt 6).8<noL),(nL(io!o rolJ Lbio! 6!c1r,roF(ro6 ot)ir,il6. O Xcireoq orll6v rxrr ErioIE6L.iolcoJl r = 1 r(rL (w.r?€x(L ar Fr.r €u0tia.

To i6ro :rpciypcr Ocr oqr6oiver zot tt€ xd0e d),}.o 7iiva,{c rdq€(Dg ivc:

Kd|E nildxa1 dEEoq 6,a nF r4ll anAi yopqn A = utr.

Or lpcpp6q eivcr 6L€g io),Lcn)"doLcr rou r6io! 6lclloFaiog 1)? xol oL orlltEt:irur 6l"uE rol,),crltiorcr rou !6io! blo\nloparog u.

OI TE':Efi' OEME IOAEE YNOXQPOI

ill2.4.3 Be€ir8 q brdotoort xar lro 6dot icJr iroodg,,v e+€Lro6drr lroldeov rov rLvdxcrv

AIKHEEIT

2.4.1 A),!0tE rt !rr'!66r: Ed! 'n = D, 16r€ o X(lQoq yaolrl6.v ror A rcouar Fr ro! rdQo

2.42 Ba€iff rI brdoaor xar ao&Geuiore FLq 6dsl r(oa cood$rv unoxdcxdv no!.D(ediol,i(| F€ 'rae€ 6vav ar6 rol5 rivsxrs

,=[3 ]t 3l ..,,=[3;

f r 2 0 r l l l 2 0 1 l,1=10 r r r ) l xG t /= 01101 .

L r 2 u r l L0 0 0 o l

f0 r0 l,4=10 0 r l .L 0 0 0 l

2.4.4 tlrqryqdlFu ro1ls rdoo€eq orp€].rd)l'€r, rno)((i)aoK ro! rq6rdorcro! Xdeor, ,ro!trtri' (ol,r(l' F€ ro!

2.4.s E.iv ro yr!6!!vo 6uo nLvdrov €ivoL o rl16r!,rG varq, -4]] = 0, 6EiEr. 6r ot(i)aos mt).6v rou B nsqti1!"ror ('Iov !rl6.vdX{'ao iou ll. (Eniorc o t6eo('yAq4rd!tor A nEeLirlaot orov .lA10r€46 Fitbevdroeo ro! E, aqon rd0r yeoltri ror Anol).(rlaordkr rov A rar 6iv€r lla $r60vrri yeq$i.)

2.4.6 EEryiotE yLori ro ,1r = , eilur r]IrMutlo, rdr€ xot !6rov i6ie dlav rdEn(,4/) =rdh(r'l), drou o,4'orFori(erdL neaodrovroE ro bodv sri nLdo! oril']. YndbrGrlH idEr eilar r 6r'ioid6't rolJ td,aou oflt).{i,t'. n6ft r trqootrirt FiaE sri rldo! onihsaq)ivrx nt brdoraol a!o),Ioi(0nti

2.4.7 Ynoe6mr 6rr o .{ €tv(l nivql.eE /|l €nf n rdEtus r. Yro roies otlvoixrE yf avronqtot'q aeL€Forl<u) o n i(tr 6irLeuao at'rioto(x0o:,44 | = A I A = I.6) ro ,4r - 6 dx€L .in€ra€s Lid€q lrc.r xd0e ,:

2.4.8 fLa 6sv ufliqsL nivdxoE rou oroio! o xd'qq yqap&v a( o F$rvot(l)pot neera-Xow sot or duo ro b 'dwqrc [ l I I l I .

2'4,9 Yroolmt 6 I F61rt ).nol 1l)! .4j' = 0 (tn €lLo6oE$ FE n ayv6mot5) eivor I r = 0.noLa rilar r rdh ).or yrodi

11.10 BeHirr 6!ov r|lilro I €ri 3 ro! onoio! o [46ev611oqos aroft].€irdr or' dra r.rbLovt|oparc ro! Rr yro ro onoia rr + 2r2 + 4ca = 0. B€sir€ i!o! ri\,dro 3 F rt 3 perov i6ro tldfroxoao.

2-..[rl Edv to /r - , el€L rdrot€ fia ro dxrmo! ]'iol, 6€iEr€ orr 't trd\.rt ]'iol ro!r1'g =0 €ivoL I I =0. Yr66€Gry IIoLo €irqr I rdhi

AIANY'MATIKOI XOPOI KAI |?AMMIKE' E!DO>EI'

2.4-12 Ea\ ao Ax: = o ix€( Lrlcr pI Fr6surri liorl, 6€iEr€ ort ro ,47 y = / 6€! ;1& ltrol 1r0oawp€\,sq 6€E*s n).€!qd6 J. Kdrao.lEr{iore dla naado€ryra #rcor n rot J

2.4.11 Bqoir€ rI1, rdEI ro! '4 ror yqdw€ lov ni'ara oc.v A = r.7:

f l 0 o 3 lA=19 9 9 91 ,"" A=l; - ;)

L2o06 l

2.4.14 Edv 6000rr! rd (l,b r{sr c, pe a 10, rdrq ]Iadn€t va €n €lei ro d, (;ot€ o

o=1"" i')vo dx€r rdErt tvdi M' auq q\, €nr.orn tou d, raqayo\5orooift€ m! ,4 oq poqqi

2.4.15 Ba€ift ro! aa|o$a6 ovrioreoqo ratrt rov b€E!6 dvlioleogo (6rov rlndraxeo YI.l1or-{

. I I I r J t , . - l o l,=L; i ; .., ,,_Li il *, .=l; ;l

2.4.16 Ed! or ortL€g ro! .4 eildl taat4rr,t(i av€Edqrlrs,c (o A FI,I.II m eni n) t6ft I rdErlrou rilar - )|ar o pl6€v6xluqog eivdr - xci! o x6eo! YqupId! €ivqr

,tu unoQx€r €vsg _ .r1]fl(nooaoE.

2.4,17 t Eft ndsd6o6o' Ynooia'F ot| .t!_dhrout(e rov 6€Ero avriotoo.fo 1o! A Tors. 1 8 = / o b r y r : o n t v A t A B = A t \ B = t A t A ) A t o E o u , , r s L x c v o ' o r € iqv 13.:l = I xr ei\](lr aerm€e6s o'riorpo(pos. noro 6I!d srov o1JD,oyrotro Fi'vot

2.4.18 E;v v €iv,n o unor-qos o roedy6,rFvoc o,ro rd

f r l l1 l [1111 , 12 , l s l

lo l Lol Lol6e€irr d!o'v niv(r1.a '4 ftou 6X€' roa y do! Id)ao yqapptnv xar 6vcv rivcx<r N :rordx.L rov V odv Frp€voroeo.

2.4.19 Beeir€ pn 6i.ot yu xdo€ 6!d on6 rots #ooEeq lrox.bqoq rol

f 0 12 3 4J f l o 0J l 0 12 3 4 lA=10 12 4 6 l= l l 1 0 0 0 0 1 21 .

l o o o 12 l Lo 11 .1 Lo o o o o l

2.4.20 fQdl'r€ a!o! nivaaa p€ q LTonp€vr flo {dr(D tb!6tttd xaL €Eryrtm€ yrari !'idaxo!!latoroL tri!dzsS.

24 OI TE::EPD OEME.A]AAED YIIOXOTOI

I I l fo lo' o xdeos Grru,v ,rra(ix.L ra

L3l Lll.ox.eos ryI,i ' nreLdcr ra [;l, l;]

6) o r6eos or4r.i," a$ 6r, "" lil,.

f1l

L1l , p1|6s\,4o,sos dx6! 6dor ro

L2ly) O xiloos oql.6v =Ra, o ;(draog 'feapp6! =R3.

2.4.2r Aibs:rar &r o ,4 el€r io15 t6rol]g tdoo€aq 0e1r3ld6srs 1'lrr6Ioao1ls [€ iov B. Ioxi€I

2.S T IPAOHMATA ILII AIKTYA

A€v si.Bcl o).6r8),c rzovoror+r6voq [€ iov 3 €ni.4:rivoxu t4 npoqlorg€v1grcrpalpilqor. Ano o€ulQ,qiwli oxonrir slll6g lrcrv aol.lj r.rcrvonoqrLxo<: oLri:oo€arg u-rotrDaor tr(rv uroLol,i,ortor ztlr FI r€rQL[[6aor. O].€! or 6rt1ordor:r:IorJq lir I.f.rr T rFav [I $16€vr1.€q. O rupcrcivco trilara_( 6$(r)g r1r01,r€7flrd €nrloltrivoq zor 5€v r0ocex6r(rv 0716 11(inorc llriora Eqap$oyrl, "/i anr6' l rv,r , tvrp,, 'v l zr l nooo npuTtrur.ru,,qrf t rrr : ror l r rur uuroi ur unoxuaor.

Aurt I ra0dyoc(fos €roarlrx !(( xcrl./oQia nrea),LlloyQdFF(,)r nn (ix(Dvnor txoll! bno rAor€rrl$ora; ri!c( cTrLol zcr, tom6lpo.vc avbLcrpiQo,|'1€q. Al)1o|Fi!C[ 1f'(|loroi, ouy fiiraxtE ftQoofitrboEtoE zor ri(e€ orotlri,o rorq (ooulrl( [€l. -l t 0. Aur6 nou ri.vor (lE|(xnpri(,Jro €i,,'(rr 6ir ro tbrc ow6ci.v$ xa( [rrou( L ) .r . r a. zuourc z,u lu r ,( b{(oDta brtrvvutruru ruv rr oo,,e, , 'v u:ro/, ;p, , 'v.Or €v ),o1(,) ur6l(.raor naiiouv 6rcrv zrrrpLx6 a(ii.o or1 0ECJpic bLxtnov z(lLoll e€0)0ic ./A(1qrrudrol. O n|roz(q npooirrd)or{,)q ireoi0lr:r(r zcr' euesicr.,'(fio 6,'ct 1Qate41(r r(lL tsrN(ilr U avo €rblxo n(Oatberyl,t(l, ((fxni roatoolfrr:6Tl |l ],i:I yacigrlfrcr" bev orprlil,r:r lpriq4pa prcr< o!1'(iQllorlc (6xo),< LZ. rlrclQ(6otr{ , = J:). Yn(iQu€L pr<r 6eireql orllc.oi( -fiolr crrcl.o €Ely€irar-r€).ei(0< bftfoQtrrxrl, x(r :rLrlor iortprl TrQoS rry fiLlQoq o0Lxrl ;rllpti :rpoq rov(lrr€rQoorlxd ioylofro. A! x(r 11 rtrpci.Tpuqroq olj1)i ei!(r nQo.le€rrxrt. bi!€!,ororooo. xlv s!}lc(Qkr vft 6oli|r rapd.Ulo./ea$o1lg fiiv(r)ieg e! 60ho!-r, x00(i)Ex{1r ro;rrirq e!q,C1ri.!etal oxo ldl,oq o r€lo(ryoJvLl{clq ouBBerqLroq r[hc(rcrg..11.4.

'*u ypaq1pa dx€r bro o!drcr!x.i: ira orivol.o xopulirv q x/@a1r' x<tl6vcl on!o),o 606ptov rl axlrrir, no! rLq ouvbdouv. To yQ(i(t4lt( oro:X- 2.4 dX€L,1 z6tr6orlq ti(! 5 crldg, r{cr 6ea ouvb6er r{he€ qe!./o< ri6pdi,)! yt prcr r41. KcirLriroLo blv efvcu. a;rcpciqro (xur 6popor nor lexLvoir on6 z(lr xc-rcl.1yow orovi6Lo rop6o c:rcryoqerovt(L). MorrilrL tri.orlq B'iv.l obLl6 Xclptl. pr: trg :roLer.q oiqitou6ollq x.lf rorq 6a6[or( o)g (atiq. To 6rz6 trtlq ri.tcn i..tl. :rpoourrud.rcah,oygi./ 4tl11. 6rorr xci0e bpopoq 67eL ivc 66l.oE:ro! xoooAiEr.r rI! xcrEu01]\'oit ror,.

O ttivaxag rqoonrtbata6 axptht-xitpqov ei.v{{ 5 rni 4 r(r ortl6oli.E€rclfs ]| O riva'1.is .lrr6q ixer prcr laatrtui yra r.l0r arprl ror 6rixveL touE bnoz6p6ouc nou evcover q an$\. Ecr 4 axpit ttnyaiw an6 t" x6p6o .i mov xop6o

1 1 3 IANY:IUATIKOI XAPOI KAI TPAMIlII(l)J I.:3IIO:EI:

- l I 0 00 I l 0

0 l0 l00

!t{. 2.4, 'Eu ipooavdto}.rotdlo lAl{fqlo x(L o xivaaaq nAoorr6o€ox dzp(;rr,xotr60r!

0l

I

- l

01

].5 fP,\OII\IAI A KAI IIKT\,\

l,', rtirt q ypapyil ipr I m4v 61i1n j xat +7 orqv ori q t O ;rivcz(c;rpoo;rr6oeoq ittx ruirr'rO€i bi:rLcr oro lpirrplpc. H yacp1tr] I 6eitvet tuc trzfnlcr6 rov r6p6o I orov x6$6o 2. H yo(4rtrli 5 rpo(plctuL (l:ro t4v:ri1rar4 rrx1t1:rorr :rirtL u:rri rov zop6o 1 orov zolttio 4.

)4rt:rdrote au6 nou orF6(i,'r-( ('rlE t]1rlLr(. H laiq (]ril.l biv€! 7r),IQoqroQi.sqyrcr rov xoptio 3 lr(g ),i€r noror 606fo! €waQlowo! 1r1r noror €Ffpxovx(lr. OLbQ6tror 2 r(L 3 tto€QXolrrlr. o iQoBoE 4 EEaQXrrcr. O A ]'€y€ror xaF[rd qroail z.rLri\ete. ct't,txttrdrqra( i ni\(rLcq ntd,oi'iu;. ou\rl0rlq br- izr( nL-oroo6r€e€q11x4rp6< rcr' o.rL orit.t-:. Or(r ro yed{nr(t €Z€r ,n tl1.}lt< z(r n ,.otcrouq.ror€ o ,l €i!(r ,r fni fr. O avirorpogq rou €i\'( o nivtlza< neutnrri)o€{'):

3*w(4u'tr€ ro! !q6€vot!,ao ror -'1. Yn(ial!'r ouv6n.roFoq r(l)! ortLi;)v nort8i1r( ro lrtbar: Korovrxd Io:rdvr4orl bibrrar" pmir qr cm(),orqJl. rbr;) 6po;<rpxri pLr prrtrri. Ot orl).t< t'yn,t AAQony tn lqiE xli ont)'r. z\t'rrn(t otrtbrvox(l)aoc rirt'rirxfl ro arirvuoFu r0rr to!(ib()\,r r = (f, l, l. I ). 1I(r ro otroioLoxirFr ,{.r = 0. II },iot (cv udQzs x(itro!() trtc eEio(xtl( ,4.r =, bev €iv(Ltrovcb(xi . K. l {) f ort lerQo 6r( i r l r r t r( i . f =(. . , ' . . . . ) 3:ropr i v( rpoort0ei ot l rai|otl nl( Ar = , x(r rcr btiroer :rcil.L ),rrorl.

' lorrro r /rr : r , i roro !onl ,u. , l \ l l r " ' (" ioor ' l ' ! r rc or ' \ ' ror, j ,orc. . . . , . . . . ' . . . r o 'r t,tvaprxd mory xiq6oug. Tort ro 6L(ivl'oF(l -.1.f bir€r rrq bl(qooiq buvquxon.To -{r a/fl nivrt or,rLoldJor': (n rQ(;rrt riv(r xr -.fr (1no r(1 I I rI< xadrl< ya(rtt'

; r l : r4c . l t nol brvolv r : drdq'r4is b" lqrrro' x,rrr i gr izo; r , ' rv:rr :vrr rr4r, ; r:riafn(;)( rt rqio('rort L. = , {)arrL ro €a(irr4rc: Adjtvro)! r!)! brcq o(x;'! ,i. . . . . rrijr)firr r( brvutrrl('i .rir, . . . ..r'r. Toi\o (lu,,q sivar ctbnv(tov! Mnooou[c r'ct ttriEri-ooupe rl\,( t(frl;)ooutrs ro bwcpLxci xcrri r1v i.bLcr orc0epr'r r. Xrp|; vu crl,l.dlouvor bL((tor)i. rouq. rpiryirft nor |nl{jetj(1[(bvr'' orL ro j = (c..,..f) r:iv( rtrov [I-6rv67rrpo ror, -,1. IlpcinrcrL. (ur(i civc! ru 1ovrr 6Lc\rropcl( olo! lrt6rv6x(DQo.6r6fl l.f - 0 orycriver ior 6rwr4 xci xurd tojito< x.i8e a4utc. O FtbFloxtDao(ourorl rou rilczcr npooiTr(;o€o[ eivtt lovobrriotuto-q- nQoobrcuiloqr€ r6pcror! uno).orrouc rar-r,( zr;Qor\.

X(iAo; rntg(;)v: frtr norq brqor)dg br,.. . . r' tr;roQoittc ra ),iooryr to -.1.r' = b:fr.r vc ttaoulr (v( (4rEoo lprrrieLo ;rrirpt :rioor o{ov xiatlx(. To doaoLoFarlra yQqrB,i)v I 't(r 2 rivcr I yoctrtoi 3. :r1v brlrd rl,trpri xpetcr!6pcrort rrpbL + rr = |lr, brrrtioA€rrxri brv u:rrlp1rL ltlo1. OgoioE. io dr0Qolopa t()v ./Q(3p(i)! 3zar 4 rival I ypatrli 5. Apu rl 6e!rh rLrr.rpd npinrl !(/ r,{avonorri rt\r ,3 + rr = b\,6orr r1 (:r(l.orgrl vtl rcrrohi,E€L orlv 0 = 0. fru va owol'foo1l[r. orav ro 6 ftlltxrLrrtov Ttirgro otlitirv. 16r:

d r +b r b :=0 ( r )

:r'lrti,iovrc< (t'rtiv rry t!drno1.6piozo4rr eaior6 6rr ro d0porog( tov lpcpptirv1. 2 zcr,l €ia(1r I yao$$r15. Tor,ro 6pu,r: ber rivor rinorr rio. (qoi ;(Qoo0€rovrlr;rr< tlrorirong (1) raipvor$€ (pdooE b + i,: + rr = bi. OL ndwe ouvLor(;)o€q

AIANY:MATIKOI XOPOI KAI TPAMMIKE> ESl>A:El:

rzavonoronv 6no owe{reg, br.6.n o xrirpog o1|l},6v e1er" brciorool 3 = 5 2. OrowOrlzeg cluraq 0a [nopouoov vo 6pegoriv rr"o ormlparrzd pe cnc].orq{, c}.},cif6u irpa,rr r vu iIUUv zor xu:roro o'yooio yru ru yAdq qILr.

O v6[o! €ivc! 6rL ot itactopiE 6uvaprloi o' ivav .eLoui x{x2o 4itrctl'e ixow lqQotolto l14dE'l. Or 6roqoQ6! xard $ylzoE rou cv(,)r€Qo! xlirLo!eivcL 6r,b: x.r|" b3 (ro [€i.o! crarr€iroL ].6y(,) qE xrneuorivoerug rol 66],oq).fra va npottl)ArioouF€ orov xuxlo xo'. a& qrdooup€ rioor mo i6r"o 6uvaF|,1.6lpeLitlerct vc ropeL b1 +br 6r =0. Ioo6r;vopa, or 8rrrrpoAdE 6wnprxor nqt:rer.vo rxcrvo'rolorv rry (xr rr)+(rr i]) = (r, r3). Kor'rwoloyiaa,I arairnol6: + 61 6: = 0 npoepxercrL cr6 rov xdro xuxl.o. )IFerdoxe 6ft or oryll,eE rou /r.zcrvoror.orv crneE tr6 6no ancrtrloeLg, 6:to4 rcrL np6ner", crpol ro ,4.r = 6 luver<u.ordv dxQ!66q to 6 nept6letcL oro! XdOo orl),cb!. Y:ttiqouv tpe|E crveldrglregoflitreg xcr! n ldEl €ilor ? = 3.

Aprmep6E p46n6prqog: nor,ol orlv6uaoBoi tow lpcpprilv 6ivouv q p$evrxrllQe[plll Kc[ o' our6 €ntoqE tii,1,our oni(wlo4 o! xilrl.orl Td blcadoFcfia fioulxsvo'IoLouv Tnv Ur ,4 = 0 Ei,vs!

y( =11 r - i o o) xa t y ! = [o o r 1 -1 ] .

Kd0e auxLoE :raqtil,er 6vo br,civuopo y ,ro! cQloi€Qot p16€v6xopo!. H ouvlordoc+l rl -t 6ei.Xv€! xqxc fi6oo xo 66l,oq r1g anprjg 61eL qv ibLa rar€iou,ol tlErlv n((rrrr0uvorl rDrr xuxl.or,. Or ouvbriqotloi ru! ,t nqr ,: rilqr €:riolg orovoQlor€Q6 pl6Ev6l{l)Qo nQdy[drr yr + y, = (1,].0,1.-1) 6iver" rov clorcE1{611rixl,o ro1] YQsgltlcros.

Bl.cners 6tr o l6QoE fiI),6v xqL o oQror€Q6g trr16ev61ropoE €iva! or€laor!6€6q16vor. 'Orqv o qgLorepd< pq6ev6lorpo< n€Qlix€r. ro gr = (1,1, -1.0.0), rq6Lcvtiopcrrn rou xbpor or1)'riv Lrcvonororv r1v ,r + r, - Dr = 0. Aur6 qrrveqdv€rrov xovovo ?/?-b = 0, fio! 0o yi1]€! orl1liopo to bflri€Oo pdpoE rou "@qr€Ld6oqOe{DQ,iporos rns fQCFU,arig Al.1e6QaE". frpiloule riioo oll y6\'rx{ ]r€airrorotxcrL xcrrcrl(Dqi.louFe o]lTi rr|, €r6rx{ TreQinr(ljot o(\r d\r(r vopo rqg o€oQi.cgxuxkDlrirft]v. ]vrooro oa! ro! Ka\'6va Ax1'a$L]i6r' rou Kirchloff-

25 Tc( 6tavuoFord rou ooror.QolJ Frl6r!6XUQou a!.rororxow oTouE xi-d.orE ror lqnq{parog. To xQmielo ylo vo cA,rix€! 10 6 o-rov ,(6Qo orrl].dJvEiv(JL o Kav6lug Aurdltrxolt rcu Kirchhoff,

To tiqpotolta tav ttdqoq(br turapu,oi tcdtri pixot xit lott,tQirwt va eava. p46w.

Xrirpo5 ypcpsriiv: Au6g eivcrr o rel.flrrrioq ur6lopoq mov onoio bibsrar volpaFe ryt 6<l{0€rtl tou yQaqripolog. O ltirpog ypcqrl6v rrprdpL 6rcvtropcrc rorr€rAabLdororou XdQolJ. oX! 6poE 61(r ic 6lo1,ioFoxa oeori I 6rdorcorl rou Eivol

2-5 IIIAOHMATA KAI AIKTYA t2l

r" = 3. I.rl vo 6poripe rpelg avelripnrpeE yao+g€s F toQotp€ 1 va xorrdEoqrcoqv cr<rlroqri rl vo xor,rilor4r oro lpdqrpa. Or tQelg ttQ<,r't€S yQaFpdE €ivateEaQrq[ives (6!6rL yqcpprl 1 + 'rpdppti 2 = 'yeqpp{ 3) ev{b or yaauFdq 1, 2,nor" 4 eivar" cr,eldpr4teg. Ieolrp6E dvfl,oyolxoiv oe ozp6E xdr ygcgp6E eivcrcveltip,rrpeg eq' 6oov 6€v 'reQ!€touv {dd!or,S.

or yaoll|rds l, 2, 4 dnoftloiv 6rio4, clld p€ rr. poldlouv o! oo!6!doloi,roug; >€ xd€E yQc44rri re orory€iq q(ollv deQorop(l pq66!. :!v€n6E xdoEow6r;ooF6E 0a 6te! rnv i.6r.c L6tdr4ta. ntr:r I = Ot,Jz,ft,ft) eivor 6vcE1Qo$p!xo! ouv6uaop6s rov lQal1[6v, 1616

h+12+h+ la=0 (2 )

Aux6 €lvdl 6!0 1{QftrlQlo c6o-rs 10 f va a\aircl fiov I6Qo yQq4ldw. KolidqovtaErd nQonyotgslo! 0c npiner vo ufqQX€! dnorc oXdoI ps ro 6Ldwopcr c =(1,1,1,1) ro! Fq6€v6toQou. AoraE or rdooeE"g $o!d6e6 oflp €Eioootl (2) 6€v

|'rJroQei vcr Eivcr! oupriorpqrrxagi

Eiv tu I ft€Qtlrcial orot' ylipo ypapptiv xar to c tfiQldlctal atu1t pnd€uj)(iQordu JTx =0

Ko! ndll qvr6 6ivel pr,o erxdvc xo! Oe1lsl[66o!s O€toQrrlpqro! rIE fqqr$lxilgff'y€6esg (M6QoE 2). Taur6xQovo iouto trQodQxeis! crn6 6vc €elreLuldq v61o116 oeorgicrg &!&XLoudr(l)v, riou rdQd €lvc! o KavdvaE Petpdxog \ou Kir€hhoff.H d,rxti qoi oe xdqE *tip6o ebu prt6lr. Or aero[ol /r, /r, /3, la ohot "trrw€Ee€ilcrto!" oro1,S ,{op6ou6. H fillyrl ft rQdner, vo €E[ooQQofisl ro -91 - ]13 - V5rou eivcL q porj rou gelry€! qn6 1o1 '|6[60 1 ,(atd Ufr(oE lov qxpdv 1, 3 ,ts! 5.Awri ei.vqr n re6q eEioacl rcu ATy = J. aQ6Fols loxriow xd! olor6 dl.t'o!gxdp6olg: ororrlqrloq rou eoQrlo! crficrrEi "€!o€Q'x6pEvo qoQrio = sEspx6F€voqooxio". To cDQ(rLo sivot" 611 o drtiotpoEoE t'on A efuat akql6th6 o xdrd).LtlloEtivttxdE ytd. to\t v61to tou eetpstog.

To ouorqpc ,4"g = I l'tv€rq! 6rqv io f ovlxe! otov Xd)Qo ot1lcDr, tou ,47 ,riou Eivcrr, o XCDQoS yQaFF6v ro! A:

2S' Or, t6ooepr,g eElod)oelE -ATv = J, yls .rol5 16oosQK rEr6oug ,[o1J

yQaq{paroE, ExeQdqow ro\ Ka$6ro lott Kirchhon

To ouyohua,6 gaipa aE x&oa x6y6o sire. p46it.

o v6pog our6g g.nogei vo r,xcrvonolei, aL F6vov 6rav T o ol,r,x6 qetlpo tor-trrqQdxer0|, eEoftQr,rd oroiJg ,iEr6oq €ivar fi + 12 + li + Ja = 0.

Ed! / = 0, Torc o KcvovcE PeriFo-rog rou Kirchhoff €ivc!! Itg = 0- Kot,Lacrvonor.eircr cn6 xriOe peupo nou 6r,opp6er, wcr, ltux)'o. 'Eror ot xtx?"or 6ivowrc Srcvuopcrc g rou pq6ev6loqor tor.r ,4?.

121 dIANY)MA IIKOI XAPO] (AI 'PAMMIKE: EEDO:EII

Euv66owa 6i1,604 ,ar al€EdQtrFeE YQaFFcg

Eivo! crtloolptitrrro ort r.dr0e otol1€io rtl;v pl6evo6rttvuolLdro)v r xa! g

eivcr 1 r1 -1 rt 0. To i6to al'rte€rJ€r 1(c( 1|tl tor,5 rcprilowe; otn't PA = LDU.rro! nQo6Qxovr(r (tfi6 a'I(ll'olqi. Aur6 FfioQ€i vo Fn Qoiv€IaL tr€Qi.€Qryo' cAouctri0rur yLu rov nivcxc ireoofitcbo€c,rs F€ lol' oTroio Estawtoal8. Om6oo ic +1bov rpir€I rc 0€(,)0o!r'rar cuIovoqrc ?io[ FfioQ€ira $l! 1{}.IQovolouvral orlo rL<

i .,t(r L. A! Erxr\noorlr uno tov,4 = | i i l. rott 4 caoloLq ri 0u rup<i1trl ' , l

6v(. 2 r,4 treirepo o6ryo (zar oE oQiqol'o(r €nio4,(). Al'Ios o tlircxcl! ,4 6€v €[!ot^rvczq nQoonr(')o€(oE.

fro ,(ivr.r€E nQoo,'lt(Do€(DE. ,{(i0r 6i[a rlg ctn(l]oqrl! o4ldv€r ,{(h filt'ttaro ./pciqnrcr' €xrFLoritrt (rur(i r(r 6if(1tc o' €vo raQd6er1F(:

-r rol0 - l , ol0 0 -r , ll 0 0 - l l

To ntxi)lo 6rttr(. nooo06rer rr1 lpcrpprl 1 oq lpapprl 4 y.( vcr {vet Blbtv I xdt(,JaQror€Q(i y{,)vic. Aut6:rcpriytt lrt vi( rd1oQtl YeaFFYl 0.1.0.-1, rou €nionsn€Qral€L rl xcrl o riaazoE eivcr taicn6 6vag nivcrxcrq nQoonlci)o€ox. H v6s

lpr4rpf avrrototXei orlv;.rnTrl (4rrt rou YAcq4rdroq. cn6 ro 4 oto 2 H trsi'(rtduzuri uno ro 4 oro I ibtoot rn oiol rrF o sr'Ttiv rI viu (.zltrl

To r:rr!rrv,r 64yfi r1r u-ralorqrlc. Xplolro:r,rr,;vrr( rl latqrn 2 o.lv Ypupurio6rtyou 0a civcA ncQ6BoLo. nQoo0arolr(E rrl YaaF$l 2 orn l'ao YQol4ri 4:rcphye-r(L to (0,0,1, l) flol, €il'al Plo ldq nxFrl uro ro 4 oro 3 H ).€trr46roy{ivrc cxtri rrQdreL fidfu va avtlzcr(roio0€i ori6 q vdcl }.sfitli axFYl (oro

;rdrvor ptpog ro! olrildroq). Aurli oup6(liv€l 1'(1 mlYd.a€I crfiio€re tl]ro ttll'caxLr{ rou 3 nQo,< 4 bt6Tt rc 6€)'1 4 - 2 xul 2 - 3 ouv6ljd!,ovtct l((tt blvow 1()4 - 3 .

To re),sut(.io 6rtF(1 (no).oLqri6 e[cqcviler rc (r,rtiv 14 vdo crxp{ z-cr atprlvtt

tr6€\,ra(i mqv r€lcrQlll laaFFrl. AQs o U €ivar o i6toq B€ rov A €t{Iog rIErdrut(li<re yor4r[nE rcov g$evrzdrv. Or tpe6 rpdneq YacFF6E rou ,4 rlrov

YOoltrLxrbE cr!€EdlQrr]x€q.Auro oblY€i tra).r orl y vrlol FQ,it4on: Ilorrc ]aolFac tvos:rivuxo ,'l0o_

o:rrDoruf rivut avrEciArtrc(: H ,n41't4oll Fivtr:

Or ypa1ul< ixtr artEipr4rtqtt1' tioot,or aniororTt< az-1lq h ox,yttritoLt xixlo.

Yncaz€r tvcl olotrc oq o€cdaic yaaqrudrD! lta on!o),c ultrdv nor 5svnrpriTolv mrziolg. Al(cr Ltlovror ddyitpo. Or rfoo€p's a,'p€g oro rrrpclor'vuio!r(iq yQ(iq ItLu 6uv o7,qr(tilorv c\irbpo zcr oL r(ciotpLq 1p(|1rig ror,4 bevfirar (a€:('t0rrlrr'(. Qorooo or rQr-r: ,re(;rrtq Gfttraq (xur" otlv rpcrllorrxdrrlruororfobl'ror! ra!L( riz;ric) ro rrerzoiJ 11xrp{grro< crott-}.oriv ivc 6tvlao. Toi6ro ort6trirrr rioL ll€ 6uo cxfiq ll xc( lrr_ lri( lrolo.'Ev( 6dr16(,o pnopri vcr circ(lttxQo, rl1r(,)q t:ivcr q'r.orx6 rcr avalltrlooltrE ro 3€yrr,Lrrtpo ballAo.

'E!.1 56vb0o nolr cylil0r ziror x6tr6o rorJ yp(fiFaro( sivor iv( ourdioyrt f i , l0o. Or ,r7t Ic ro ' n,(L! i7l1, \ ' rn ]rxr ' t ' t , r ' ( " ' , r

or , r 'qrtrr j r , , r ' i r ,$i ' / ( ' r 'v r t ,Z(irpo 1e(|tr(ira. :rlv nQo1lroruiorrFcr or lpolLfr€i crurtq (lrorrionv $o 6arol rorrXrirpot lprqrlri;v tor,,,l xc( rOooolxl Ur"(l. (i),).l]q yA(f$tc 0rr.( d).7,1: oztLitE){l(1 oz'ltr('lrlf iv(\, xux}.o. El,rr ouvc\iol birbpo riwr i1(1 rioo to c\riv(tortrrl(linrrQo bi!60o. Ed\, 6vo or,!€xfl1r6 ./A(i(tltrc 6Xfl , 1t(jtrdolq. rdrr' dofmirnior 6iy60o i|a n 1 axpts. Arr6g riv(L o (1a ltL6: r0)1 (wgcarrir0,!'1, t , , t I r ' ,v rou . t v ' , r r , ( ' rdlcov ( r t r i ! ' t rou :r iL, ,r , (

nQarr! rniorF !c lJri(iQlo!! z(1! , - I a\r.E.iQlli€q olrl),r:. )uvo),Lz(ir,:rripTorr ii - I oTli),!'i. xo)\, orol!')/ op0)q 1() cioQololc 0i!cr ! [116r\,rzll oilLl.O frlbevotopo< ror.! ! {.iv(! $L(1 r!00i.r bLrQloll{,lil cfio ro plb.,'obrl1\,ljop.1. f = ( 1 . 1 . . . . , 1 ) . O ' b t ( o r ( i o [ r : r a o o r i O p v r c r b i a o v r ( < ( r - l ) + l = , . o r ( l , <trnrrLtr:i ro Ory.irrirr\r'q Orrirprpa r1r fg4uLrxrlc ALlriilxr<.

To 0fd)ailtr( (116 btvrr friort< ro1 (pr0tro rov (vsEcariir('rv znd!r!, notlr.iraL 1 6trirrrcorl ror uprorrpou l46tv6Xtut'or. Aurr'1 rirur ,r / r'l ri r + l.'Ed! ro./Q.iqrlIC ntQ(d1trcr o' r!vu uri:rt6o, y:ro1u\ru vu uvu(ryrjoorps opiotu:rorq xirziorq tip6lou -oto :X. 2.,1 lJn)ipx(\, bro riroro! pLxqrri rreLor. obr trr'"/i(ioc lEorr€prxo( rnzLo< bfv iir.r./ .1!€Eharitro<. Axoirl x.rL oro! roladq,Itra ri6pX€r(1r. rou lrur6bor raaal6vovr.r< or\rrrrr]o- fq(xo]"orJ0ai '\r.r(y.tr rt r + | (vfgdaqror. xur),oq. Kcio€ zotrSo. €v6( otll€zrlrou yp.rqiiFaroqltnopfi \(r or\'5()ri trr- x.i0f ('tr'o xclsr6o r:rrii,7rr trLtr bL(bpotni cxtrdrv t€rrEiro\ r.r(r or rbrorrFe! rou ri\,.r,(a npoo'rrrloftoE ou\'oliito\'rcr o,( !E'l(:

t l4 6t Ldyjopo q: 6Lioruort 1, rFprix€r ro r = (1.. . . .1)X(Ltloq otttub!:6t']rmnort i, - l. onoLf.r6nnorr r I o lLc< eivuL uveEcip FtqXriteoq y puttlttiy: 6,"imcol ,r r. .rv€EdprrF€q ypcpliq cir6 zdOe owb€ov SdvbpoAr.lurttgri< 1t4t\tvri7tt1zoqr 6L.irrrcol ,r n + l, nepr6Xtr y c;ro -rotrg xuz),org.

f r u x r i 0 r 6 L r i w o j c r l o r o v T r i , p o . l p c r ; L g t i ; v e r f = f t + . + . f " = 0 , r c r e l r o r e p L x t igriycrrcr eTouv cr0porolc trlb&. fra r/re€ brdvrioga b mo'v lcopo or4l.tir grb = tl.

] 5 TPA ' I 'HN A ' IA KA I A IKTYA

n .\Nii .i!aL o rnio< !o! Eul0r. iof uoburiEr(r iiio\ rrE raqrtrLzij dil,€iipur ,r r + 1z iz i .oL + (n i f (n r . zo t ra t ' ! ) la t r i0 . i , la t r ( l ) ! ) + in i i l l ) , ) : a r l r tu ) |

l:l AIANY:IlIATIl(OI XOPOI KAI TPAMI\IIKF) F3I)A!EI:

or brcrqoa6s 5wc4[r.ori ,i z(rd. urizo! €v6E xrixl,ou €Iouv d.eaorott(r trl66y. Ar,'rir6iro1,rst uiro ro\ z(l\'o!t< lou Kirchhoff z(rt ro N6tro rou Ohm. nou 0u 6oipeqraooJg. O !oto! clno! ow6d€r ro.) llE ro, zcu szqedE€r rbroqr(r rou ui,r,{orixrr oXr r ,ru r ivrrz,r ,rgooxrJJoro6. AE:rcrQol€ivor, trr neog or|} | r lv mov nivrr ,{ . r, l y,r prrr rqrrppol l : rou porr i l r r uoo6cq4 Xc'rpi : vcr eivcr.

H xardroE{ fioiooqarQwd)v oF06ov

Xr6 rdl.og rldr0e :rapL6bor, oL €xl.iliroq€S r(qroxClooonv rrE zo},€yLalidE rio-6oogxu"qLx6E opdbeq. H zpiol aivcL lrroii€L[€vLhr], 6<ror!6ger'q oe pra Bdq tt1rttrr6qrrov. nor liverar noi'u crprpi6oLl perti ng:rgdne< 6dr6exc opd6eg. Erei606l.oupr vc xcrrccci[oule 6Leg tr,g opci6tq oc 1rttr nearoo6re0o Fae+rclrllii 6dot.

To irQcino 6{gtr eivtr vt 6poryt ro .f0(i(t)rtfttr. Eciv 11 opciba j drcrLEr lrrlv l. r6rc ur[aQx€! [ia (xpri prrcE rouE. Or olr.id€( tL\ctL oL x6!6ot xaLr.<t r ryiA etvru or ./4r1(. YndQloll1' l.orn6v tr€Qrzis cL(rovr(i6€6 z6F6or,ra! [€Q!r€: Ztld(b€S crzFa< 1e zurrri0rvoq u:rri tqv rrLox6irrpr.cr ogirbtl ;rpo;1I (pr.oE€louoc oldlbo. To oxritr( brin€r f€QoE i()v r(oQL9ci(o! opdbov xo!z(i^o$v (iLi.{r! oo6apdrv opci6ov. xrrOd4 ;rut rvcr xo}.iyro :rou 6ev rp4pilercrL yrrrro no66oqc!Q6 rou. Eurul6< lt crvro ro zoLiyro (oro onoio yQd(r,{D curd. r(.)'o./ft) to lpdg4rc b€r'eivat ouvsxrrr6. Mdr6vrrc pa04yrcurci. 6er paopotyt v(rno6EiEoup€ or! ro MIT bE! ri!(r 1() nedlro (€r{16; (v n(it€! n(rl!i6! €v&trfo!

usc

1./l , /l , /t /

!X.25. To yq.iq'lua yLo ro ,robosq(rqo.

Eirv ro nob6oq(eo {rav ewe},ti4 }.oyrx6. 0.1 F roQoionFE va c(lrrorolli-oor,}t€ €1'o "bulctttzo" rr oe aso€ oF(i6a. T6re. srlv I otrd6a r'6narq€ F€ rnv/r. arlr{:rou eixe ro pelc}.urego 6uvcpta6 0c z€a6!!€. :qv davD{i 'r€einosl.r1 6rcgopci 6 tuv nrorel'royriror Ocr ircv io{ pr rqv brcqopdr r/, rr rrlrv6uvaptx6v tou;. Mt.rmc 6r:v 0a XprLcr!6rav zcv vc:rcilow ro rcrpi6L! :'crurriv r1v aepinrool 0a ei1<4re rlrlpr; ouptpr,rvicr 6rr r1 opri6a ge ro trvdrrcrrobrvqrzo €iv(lr I xd,ut€at.

H g60o6o< armi 6xet 6uo (rou),Axroro!) 6uozo),isq. Ilaoona0oi1t€ va 6aoupt6acr coropd r ./LC xir0€ o[c6o. 610r doft .r/, - r

" = 6i yr,a xcioe ;rurx1,i6r. Aur6

orlFaiv€r F€Qrr6q XrLrdbE( €Elooo€rg xcr lrovo! Brerzi.! €r(.rot'rdrb€E ayv6mor\.

Mreh,Edn No l re Dnne Ceor8 ia Tech

rP;\OIJI\JT\TA KAI I]KTY,\

Or €ELo(;o€Lc .rr, :r', = I, o11pcrri.lou dvc .Tpcrtprx6 orlorlpc AJ = l. oto onoioo -;1 €ir(L dr(i! rivard( nQoonitoEat;. Ki(u€ rcltzvi& 6Z€r tl|rl ypc4rpl !c +l otrlrlr)i1,il } .r(r 1 mf orrt],I r, :ror briTvorv rorr; old6e< ncilorv o'cu6 to

lla(,)rl brozoLia: Eirv ro b lrv nrpr"iTt:tu, orol X(irN orrti(i)v.isa rrirol€tii,ol. It{ cror€7"toucrc rpi:rtr vc ruLQr(iEouv ano}.rrr,)q. dJ.rcrc 5rv rivatbrvarov rcr 6pr0orv uzpr6tl 6u,ctrr(i. A€irrolt 5ro;aoi.i(r: Ed! o ,{ aZEr ttlp1c\rrrxd 6rlvuoprru otor ptl6rrtip,po tt1c. rors rc buvcrtrtzir r bEv opi.lovrcrLtroloor}G1r(. :rllv r(Q(;)rit r€oirr(.rol b.! rarig1€L-ro:r'. ort b€lr€al urlIo 6€\rivor uovuc\rxir. IIr0trror(rc 0( rr(Qoror(looorv z(r or bto broxolieq.

O p1ltno7,r';,ooE ti',(r r-rixo),o< xcr (t(r!cgi)vrL rirrL .rqrooltL€i(Dro. n€ariz€trirrrort- ro &thr,olc r(r1 trol(i6(,r4. {(tori o A ol4roril€t ltolo rt: dr.l(opl(/r, r,. fr( 1'( r(L,oo6rLrpi0oL4La rfr 6wrqLz.ri lroporile vtr:rrlporpr trr0criprtcrro bua(tflr{d ror HaNa v(l filru trtbi!. Atlro eircrt ftro),ljrr;c brz(ltoiol![(vo($'vo(; tr{1el]lrrlnri(;:). Ecir opr,\ to 1ptlt14cr brv !i'r(r ou€rr(26. roirro 6!rrn(lonri. K(igF lrr_lLor(]rlltlo xoulldrrl 1()| Y0(l(t]ltrcroc ouero<liOrL av(t 6rirvuottr<rtor pli\rrir7,r'reo. : (llr(il nru6zr,r(( xtlr ro (ivrjo$(l r'Nll - I z(lr ol(l 1(ri],i.o./r = 0. I]p6:rrr Lor:rov vtr lrrriroolpe 6Zt po1,o 1() H.rrvad (tLd r(r tri(ol(ib(l or },|('r{h (olNfxrrxd) xo|lir rolr y0ft(irllt(1roc. (Ar-! (lllaootr€ v(t!a\(!(vrroroLTtirvrag |lb|arro b!v(|r1ti). Etiv ra riD,0 br,llaltx(i r.ivtut lcilo) (r]rdro !rilbi\,. rors I lfro4rivlt otuibo zcr(lr(ioocrftt nQ(;)!].) H bL.ior(art ro!|lrlc\efiJXlr0ou rrrorir(lr lrr ror (el{)|(t r!)! xotLtL.lrLr;! ro!i yQ.l(tiitrc-tos -rooi,rclr

!r ro xi)i(b. r')! ij((\!irv {.Ll.1r0.Q!irE ro1, .r. H 0i!r10!pio ontrl uvurpriruLdr(0rtx)nouivr(1E i\c b!!(!rrz6 oF r1.i0r zotrtrdrr zur bt-v tivaL c\lvurdv vrrz(rr{r.i:ortrs bno iItoarord zotrtrdr|(1.

O xri-rpo: orl).drr ;rtqrtpr(lr.rur nro 6iozoic. nola cror€i6otr(r.1 r(Lpdlortc;rri).urrr o ivr oriv<no 6wr4Lr:rriw: xiyorpa to .1.1 = l, b€'\, i.nl€rcr (i' ro Hrrvidvrzrior.r ro Y.rle, ro Yale lLr{riocr ro PriDccti'r zaL ro Princeron 1rz)ior-r ro Hrrvard.Mriirorrr (r{o1 nrQLodorrQo. noinFL or braqoaiq r(')r cnorr),r'oBdrrr\, ,rdt.it'4zos tti. xtlxt.ot rti Frot t aqAorcpa p$tt:

bnr + brp + l ,prr =0.

AlrT6< rivc{r o N6tro( Allla$rxon ror Kirchho l or 6l.r(poQi< 3u,.1tLrton xtlrfrpizo; tvog ,{nxior :r-o6irL v( i7oL,r (i0Qo,o[.r tttbi!. AltoE tiv(r Eriol< xctvoloc r1q la(quziq rii1r61xr<: to oloqga -'h = D i.nr,€rar 16r€ (xplb(irE. oravr u f u , , r ' o t . | 1 I r y ' r r o , r o r r i r r c r h t ' r ' l C ' t t r t . t z r : , : u Q r r t o , r c t . r f l : 7 p u t , , u : r ' t :(a,dr.pri. rl.|lrQd!. T6ra I furc).ouf)i obry€i mlv 0 = 0 za' rrdalow 7.uor-r.s.

:rt! np.rytr.rrrx6rlr( ro D. olcbov:rdv.rorr.6rv reprfxrrur mor Trirpcrot11,ri)\r. T(l .ljroxriiou(1rll ror :toboorpcipor bev rivcft 16oo :ro),n ioYtzh. Ooontro< 1Q6no. 1L(r v(1 tprrloor;rr or pLa apclparrzrl ncrrhrail ei.r,ar r1 t!0o6oqIti)\ .q(otut| rttQtryl|vr)r 1l1 q,60ortrE ro li ooo ro bucrov jrilortorra(l orob. Aur6 yi'v€r(L oro K€q/r).c'o 3. trll}ro!€rortr€ 6tr)q,.c, t,r(lv rh).),1 66p0r,ot1.O vLzlrlq rcripver 6!ra€d! 50 ri cx6frr z(lr 100 6oe[oi! ni1!0) ffro rI Sl.qroaat

l t6 AIANY:M,\TIKOI XOPOI K I TPNMMI(E' E3I'O:EI!

+ O Di Lcrlc (Noh Dame) itodd. !( rirlan odiNrt mo rcArodrxo Mnnasenenr sciencc in

.rnor€1.€ofirroJ!. Atcttpoperrxci ro vc xep6iq€I g€ 1 'rtiol, civor fiolu xovtd oto volciver ge 6r(l ^1xo),. Toito o6ryei ta l]no),oyr!6tl€!( onorE),ao$ato no),ri ro\,I(i

Ileoo0itxrt x.lrd. q 6td,p0t"oq. Aqou alQoq,o omylv rry nqQdlQ(rqo 6arp.a rotnnprvn dpoQo or l ! tqqFt0i6a N/! y,al nnr"\ :

"lra rctrtxd npoyvomrzch yru to 1985 o uol.oyrnlE xcnetcle qvMianri (10-2) rcivo aro r1v Tennesse (9-1-2). ortv 666o[n e€ol. Aiy€Gprprs ptrci. rigXrocrv vc xat(q,0dlouv oro cr0l.1rLx6 qripo rtDv l),r1r6tBclr(t F€ noeroxdl,la 'iar yet4rlcrro gljtl('travrov on(lbd)! q< T€nnesse.H oayri nQotax€r(rt (rno ro yFyov6g ort q T€nn€sse ol^'6renl,€ oiouqlr-),rzoi'< rlv Mirmi (35'7). T€),rzd rl Tennesse xcroritlolz€ 161ae t xcLi Mia iolBclvrlri. nro r(i(,).X0eg to npoi 6rptctoctr oql crnoOtixrl ewdc laproxr6rirrtc pe ;roproxir-Lr(. IIQooQlotloE ro!( liicv ro vooo)io[€io Bellevue xcl! olrvob€tovr{{v Feqr rao€rdonoilon 6rr I roL6rrrcr x(r ro rsQL€lo[€vo r(,v r[oero]io].rd!€i.\'ctL .4rqi6oi,u".

Aexrr(i Yt'au i rn! rqoatro./rl qs fQalrtrlr(i( AI1€6Qq.

Airrua x(lr AraxQrfli Eqcqpoopivc Mco4pctx'i

'Evs yQdq)qr{r pft(iQdnfic! or dixruo 6xov (wrloroL1ioo![€ qQlopouE, . , . . . orq,rxtr ic. O aurqt i6 " , r :Topri va oru{, ivfr rr l ' to6 r4c azprlE iri lwcrgrx6 ri rriol (av:reprdXeL €),.rriarc) ri cltlylp6tr1tc (crv r€pLdt€! rl\rti.-ortrot). O! cpL0troi alroi n€ev(ivt o€ svcrv ,r €ni m 6trty6!!o nivcxcr C. AutoEa\iavar,a "Lbrorqr€E rou ufuzou" or ulli0rol F€ 1()! nlvcz(r nQoonl6o€0\,:1,tror 6i1,€L JrLlQoq,oai rg ./r( lrg orrybdor|(. O ow6u(rolLo! svr(i! roJv 6rio n!vc!?ov,C ?|ft ,{, e[qjuvil€ror orL( €€F€],[6brL( €ELo6orlq rlq 0€oQi(l6 6Lkr{ov, rtq oroir(0a eilyrjoorpc.

H nrolya(irt traq 0a 6(oLo0ri orov r1l.urprop6. lqv 0xprl i 1 cyoyrpoultctt:ivnr c; rcrr 1 crwioraol l/c,. O \r6$oE iou Ohnl ld€! 6r! ro et:!p(l nor r[EQ\,ddtio rlv ovlior(lol ri!(L

Ji = cre,, ri (a€wtr) = ((yoryLF6ryF(Xrrd)ot 6lJvcrtrLroi).

Tolrro yQL(p€r(lr €fiiorlE xcr. q .O = 11?. 1 nr6oa 6w.rttr1ao{ roo!.(or |r€ ro Q€ulcr€ril qv {1lrforoon. O r6poE roti Ohm o€ 6rovuop(lrrxt FoQq{ xcr yro 6L€E 1rE(l1pdg 1.rtrolQololi lQLiqr:r(rr , = C..

] . J T?AOHIT IA I 'A KA I \ IK IYA

frc 1c oqrnLlA(,roortL€ ro nicior.o rQlcoiG( XQ€LGEotrtlor€ 11clt rorE Kcv6vssr," KircIhnl l l L i ro luru!,r?o }ur yr,r ro u,1f | :

KAKr Or rrriloErq blt1ctrrxon zar(i $ltzoq ziir),or, ixouv doaorotro(1louv,

KPK: To d0Aorotrc r!r! A€Wirr(D! o dv(v zit$6rl Loorr(r trt Ulbi!.

O xclol.lg bw.4(Lxoi rltrptr(r'r vt crvrr"ororTioorg 6wr4(r1t (i .rr. . . . , r,, oro1,qxotr6ortq. Kcr6rrN or 6r(rpopi< x(td |rlzo; eroq zrrx),or t\i.vow (opoiqtrrrrr rim,;gro (rr .f ) + (.r, .f') + (r'r .f;) = 0. oi( oroi.cr r(r ;rirvrcr (1l.h1io(1\,(lrQo!v1(lr.O r{clolqq ro| Q€ri|(roq f((q r:rft1opflrrr" v( irAoog6ooutrr r(l Qr.ipcrrc jrororrpq€oN orov z6p6o 7. Lr o:roiog rrupLorirrrr(r oro.v ,4 [€ ri] orlj].lj. Auo< ix€l+1 yrcr ({ztr6( nor.ro.0Xovr(L oror rop6o zclt I ./r( (rxtrag rori rqr:Alo}icr. Ono),).trrrLcorcoloq A7 .v ;rpoo0irr:r r(1 a€lrtrcrc tr! ra ooorti ;rp6orpc (orpq urutLf r{1 d(il t('Jv (xtr(bv). To 6rrirropu uur6 ,,1ry rr,;v ouotLnd;.v pruprirorr, , r o . c ) i ( ' . ' r , u ' r f r f , r . t r . o i r . a r , , v n r v r ' : n r r l L \ , v r E { , , r r f r a r c r , . r ' r ' - : , r ' r r t r r t vi r e . r r " i r r l " K , r . t t , r : P t t y u t . : r o r K i , , h t , , , n . i t a . 4 t I - r .

fr_lruorcQ(l. r.nrrptirou!r !Ecrx€Qtx6q nqyiq. Ecir i.oLa6v oroLl( xo!6o!E:rap!7orr<rL r lorrr .prrr i qrLrparu /1, . . . . / , . t6te o ropo: yivr.r(L , .1r, = / .

H rii),1 r.!iooo4 fia(! o Ndloq ro1r Ohnr. c),}(i xer.L(iErrcr. v(l ijQotrtL€ ro ..norj ti!(1r I nr(i)olt buvr4ttJaoi zard ytjzoq yurq ttrhnuo4q. O rol,LLrr]^uattrctltgl.r ibLbe ri1, b(4o0(i 6u\qrrror per(Eu zog6(1)v. A\'r(o-rQd(irovrcq-rr1 rQoolfr(.:roiaroritr! ro --4.r. ;ror,6ivrr rrN zr{,jd)/ bri1ratrrxori. MaQo; fi1,c nr(i)olq fnoor.i!( (ntri).!r(r or prc pzarapia 4 o:roi.<r :raatlltr rnlv unpl tdol b,. To rr6),o!norI< lrd)olq o(l,citrr'rtrr orill alri.orcol r(r bi.6rrcL c:ro r1v bLaq,oph r. = b - ,4.r'.l u - ' o \ o r o t r o r ' ( l ' r ' t , f - , ' r ' ! n i r r r o / l r r

tl = ('(b |l.r) ri

Ar! npooncloontrE nliol w ).iroorrrr ro /l./ = , (:ror r'ltcrr bnoxol.o 6LorL !;'1pyuvneprrloorrp€q chor;orLq cro nlrirotor4). YrhAX€r r6Aa ivcrq v6oq oOoq a rr.nadytr(tr. r srbLr{ ;repi:rra;o4 onor ro onorltr(t ;1.r = i, iyer, oL,p:rrorpctrzrh.),ri(rr] ri.r,Qr. r1 nrAirl1ol]'l orll! o:roic 6rv r|rdpxer po{ pan[.rro:. : (1Urrj r1v;rrpirrool or.6rcrgopiq arorciroudr(|]y ror no6ooq&iao! { or trr r.eiar iX,n,v.'r(r.i [rizo,< nixitlv, d0poLopcr glbi'v xriL 5rr r:triplur crdlxl peirprtoq.

Ai'vorlu. e[(fcol all< 0€tlELrr65EE e€rolitotrg rooppofiittg or o:roie< owbrdr-louv rov'\,oFo ro! Ohrr $r: ro\ btlo za!o\rE< ro! Kirchhoffl

I]Qox€nar yro 6vcr o4+r€rQo.o oumruc ar-r6 to onoio to c E)(€! €Eoqqvloo€i. Otdo/voloror €ivc! rcr per,1rclra , ror xq 6wc$rxa o xar ro 1Qa$ptz6 aurd qiorruo

s:ropet vu lsrrqei o" poqqrt Fnlox ,"c clig

12lt AIANY:MATI(OI XQPOI KAI TPAMMIKEI E=DA:tsI>

=tilMnopoi[€ $oLrora lc xrivor.rge cnci onpri o' s]Jriv q [oQpri Unlot. O obly6Eeivat C r. o ;roJrl,o:rl<roLcroqg eivat ,{? C znr 11 crguiptoq pl6eli(eL ro -ztT xriro.rc'l rov obl].|76, To crrior€},€opo €ivcr

[i, f ]l:l (s)

Al CA.t = At ( ' t ' - 1. (6)

Kcr6rr!\'lre ald6poln arllx(fid(oroo! oro nQ{i)ro qr]oqpo nciQvoupt t{c to l/.Asv undQxsr ti,/ror€ ro tllorpLdbeq otr[, sEiorool (6) jro! nQoliritrt€r cvr!-

zs0!orr,)\,rq ro u = a'( h Ar) o:'nt AT !) = t. T(1 Q€ull(rc g e!<rtreirpowtfl xcl!bivow FLo eEao(,ol yrcr ro ,r.

llltttrt'j/.i naaaNieqo4 lldlror€ 1!' cwds ttE rFLorootLs €i.\ro! ()1jL'r(r0rlla6 v(rmcro€eono!€fr(|| u( r(,)v rporipov 6v(r buvuprx6: x,, = 0. O n'oor6E xop6og €iv(!./sotl6!o; xcrr n n-ootri orri),rl rou cpxLxoi fiivartr :tQ6onlo)olg ileL nqcrqe0ei.Tov flivoxq:rou rypoxurrrr o46olllour rtil,r tlr,4. €llar ilaE triv(t,{oq rt, fni, - I k(l or ontl.rc rou riv(r rlvsEderfiE<. O rerpay"rvrzoc xi'*l:"'a< At C A;rou eivcL ro .er6i yLcr vc trrioougr ro oudrltrc( (6) o\ neos r, €i.v(r! €vog6rlorQ€q'+roq filvc2[tq rdEs{,x n - 1:

, r . r i D € r i | 1

To oior4rn rol npfire! vo trtle€i urs t[QoE ri. 6ibEra! otro rnv rsi.€urcrio yQr4r[11.

IIAPAAEIIMA Yro86or€ 6rr oro xtxLt.)ttcr nou ou!,ll1{oopE trQolyoxFav()Eirpoort0€ral [L(r Fncrrapia. p!c. mry4 perlpntog (l..lt tr€w€ onrmdoeK):

fc r ,a l fu l I b lI o -,4rcAl l ; l

= l t - Atct, l

=1"*]ftlA ' ILI

A

TPAoHMATT\ KAI AIK I YA

t:ly r y r s r=o . . . I I 0 - t 0 t ll t - t ] : = ldxnAr= l I | 0 0 0 l.q : - r r - l / r :0 L 0 I I -1 0 l

To ,rQdto nQoypa ,rou rQ6ns| vc €E€r(ioou[€ €ivo! o Kcvova( rou P€iry(rxogorou( ?6l6ouE l, 2, 3:

fLCr xov x6F6o 4 6€v yQdqolp€ eEfo(j]oyl. :' qur6! rov r(6p6o o K(r!6voE rouPeitrrcrog 0c {rcv yr + ys = 0. Towo 6:rersr sfi6 i5 d),l,sE tQ€q €ElodoeE, tlovonoiov ro doQoropc €ivclt arerBoq -j,rr - ys = 0.

H til'I1 elior,ror1 ti,vat C-ty+At =6. To 6wcttLx6:r ow66€r@ p€ toQ€rlps g Faotr-r rou N6Uou rou Ohm. O bLayrirvLog nivc*crg C:reEiler 16 r6r"tecrvrrordoerE c, = 1/Il,. H 6e€L<i t).etlQd €xqpd!€! fllv titol q< [n(rlcQicrE 6 = y

n < r L q p o p q r l p r ) , o x i l e t , r o C t g + A t : = b n i * @ . 1 f l 6 r c A I ! t = J l

Ua

- l I 00 -1 I

- 1 0 I0 0 - 1100

00v

0UI0

Rt&

[i; r] [x] = &

To oriorlFa eivar 8 eni 8 F€ n6w€ Qeii[ara ro! rQi.c 6woF[,{d. H ano},orgrio6ry€i. fio 3 €ni 3 oiorrlpu ATC Ax = Ar Cb- /. O fiivcoios o' aw6 ro oufirua;rcpL61eL rct nvriotpogu c; = l/Ri (br6u. mtv crnl,oqrl eoeig brcqeim ge roqo6tlor6). O rivorsq rruog eivaL o At CA xo.r. a\ilet wr x6no va rov borlr€,olpn€eLl.qr6dvowo6 xal r1v rercpq lqqt1lrl xaL orrf,1 rou lerop6vou rog6ou:

-.5 (ridF6oE 1)0 (1i6[60( 2)

-r! (rcF6oE 3)q + cj (roF6og 4)

n51 0 -1 0 - 1110000 1 I - l 0

l l0 AIANY:IVIATIKOI XOPOI KAI NAMMIKE' EEIXO>EI!

;Ib --------+ e = h- .,ft __------------+

C

* Auil or.!Lo.iro.t( flv.i7''a "r.

oL,rlridrorlra tn4oer,€< rqroliro.r< IEkrd{(r dro.?.r{i6r,

!(( Fri(irollima Eqdpfoolivc M((hlKirx( (W.llesle! CrDbridce Prcs. Box 157. wrllcslet M (J ] lN t . )

MrroQ€iT€ v( 6€i,i€ avtov rov Jtivcrlis ao),trsrd,aorirlovrrrE rov AT p€ Tov ,4 -qv1|aror (rar('xgadr yo)vio Fc t4r,adoar belLir y,rvin rou fl ).6y{D 8 €rri 8 r vclxa. Tone6ro olorx€io €ia(.r I +l + I ri q +ca +c5 6rav orF:reprl.dr6ouge rov C. or oeF6< 1.3. 5 qyi_tow roa x6F6o l. To er6pwo bLaydrvLo moLreio eivcrL 1+ 1 ri cr +r::. u:r6rtg uzliq ro L' u).yi!o w rov zop6o 2. nup,olotu. ro ' . tr, '11 ;rpoipr;rtuL rr;16 rovxr{'do :1. Eriro. rrF o'uyr'rviou ru r qrq uvi(oncr Ir cpvrprzo :rpoo4Fo. r;7r ,;lrrr<xu ot axftE :rov ayyiEotv rot, ytuopiw x(46o 4. Aur€E crr'rlxoul oqv tar(QrrtyQ(trtll lral or4L4. no! noQcri.€inovlo 61av q ri.raoq'ypafFi (lqce€e€l an'rov/. f€(;vovrq rov rsL€vr(io z61160 .t1'(y6F€u.r o 6vc orlmru(l rdErog ll - 1rfr. nAriyl'u nro orllrc(vruio. or a!.rv niv'(x.r AI C A nou €ivur cwrfiQi\rruoq.O),€g or yQol[6g ro! o! oriL€E rou 4 erii 4 ri.vaxc u( 6.61v(1r, d0Qo!o[a llldav xc!ro (1. l, l. l) 0o (1vrixs o,rov llqb€v6x()Qo.

:IFrr{in€ 6xt o /"C1}l €ivcr( o![trErQrx6q. o avdorao(pd( rou (.,1rC,4)" =,41 Ct At l ti\ctL ncttrr o ATCA rrrr t1€r o€rrxorrq o6l1oug. aw6 6parq rorqrlvoupe yrc to KegriLtuo 6. IlpoxunrtL cr6 ro 6uo!x6 n),aio!o €Oy(oi(1s 10)!Eq,uapoof(al(l)! MoenBclrrr{(i)a. riou (n€ixovi,!€r(r oro oXrlICr:

>:x. 2.6. To r)dioLo eqfodk( yL( rI! rroeaonic: nryis , ror /, nilozo< .l7 a'-,1.

fr,r 1).rar-or'ai xlDrtr(Ulr(rr,r ro .r n€tnFrl€ buvuli,zu :tcu ro 9 :rrgrerXr qrqrcrrt.: l tLrlxsll)tll ro r zaL t/ Iilolrnr pcrcroTrloeLE r(1r r.io€!S. :ra qad Eivor fii.€olx(L rrlxurylrc eois.* :rrt marLmlxt ro (i €L\'aL ro oq&l+o xaL or EErodroalgbi\row q\r xal.rir€ol 61]aani naoocp$oyrl etrclioro)! I€rQoy(i)v(Dv orc br6or6rc.To rQ!a),6 ./L!6[€\,o rorv;1?.C xcL ,1 owbuirt€r ra xQia 6itpora ror ri].(roi(nJr(ry(oi(lg oe aaov F€trovoF6vo nivaxc nou €Idyxe! qv Looaaorria.

T€),€r{b!or}r€ -ro xtqd),cto o (l!.16 ro i4criplo olltl€ior r{ drdf,iir(roq rvo<0€[e),r66ors rlo6lliu(ro! mo rq,oQtroof€vcr ]ra6|]lrclrLzA. :uXv/r rouxo (or(rr€inraLooorspl 6rdo0ryt ffi o,It I l.iort rou nQ'o6lrtFaroq. :(tv na6ro 6rttrcorrt laq{rrzrt dr}'ys6a&. }'rio.4r€ oro K€qcilqLo 1 'faqrFrz6E €Ero6o€r<. otr('\ il6t(ru;roo1 tr:rcrirrpe q 6a0vrsat yv{i)ort rou Krrpc}.niou 2. H ouvrLoqopri n'rvlc0nrcrur6v 'Aar r('v crvoedrnov 6€! €iva! or urol.oyrouoi d.ldr I €lvuicr.

:5 IPr\@t li\'l]\T;\ liAl alxltA F l

Mro Foxrd nQq ro €tr?rQoE

Oq(oc4rt ror Z(i)Qo orll.(lv (,r< ro onloio r(uv btclnoudr('rv.'lr'r't(L loacpLort:1xi ;t16tro7opo r,4 ro olvoio ror ),roelrv -ior., lTtl = 0. 6r6n alJr6< or

lr(0ltrcrrxdq rrrororrioerq )(prrirlovruL otrg tr1,(atLoi,i< rt(r 6izrr(l. ro Ar bi!€lrrq 5r(1(1opdq 6rr.rpLxor;ror rr.Gloiroroi,! rol aotro 61J1cttLxon. To ]/ rzcrtoirot€iror v61o plrlLclroq. Mr lo!.rbL(irq trlrLor(iof( (al = 1) oI rq'o6o€q toopaoiri.c<

At r l( 7 )

H ta(qrtrLxrt ( i i " / . i io{ l (r t x( l rcr 'nor icv r l lT(x(r( i ( j r (ort) o6ryt i or1v,, l i (6.,1.r) = 0 z({' o rro}.o1Lon1c irrrrL ro ,1'.4r = -,1rr. Yr(ia1rt 61u,\ ptcr co.opl:rr 1.ArborE nori ap€nrr vc )'r1q0ri u irtl,t'.

H r{,L,ur(i( ;rr11l rivar 1pr'4rrtpiu rou outr6abiltr tt{, rrtv (iilr6po.tivrrL iilrx bLrrloprrLzrl aro tllrfl]v. H ,{(r(ao[4 r(,)a 6r(vuotL(ir0)! oro z(,rao' r , z c i o r t . r t . ' r z , 1 . r t a r t ' r , . \ , r r : r , n . , ? , 1 r n i ] v o ! r i , r . , r 1 . c t r r t , , r \ ' r t , i v ' -o1r1/1r1(o/p!',ra Tori<. :ro nuodilll(1. arlT6 rou oru6(i,!et ri,v(! tnrun!Do|(zo: rol.f Fi!.lr xdotro oro i/. Or br((loQaq bLvcpLaori eivc{ r.i.io€rr-g oto L)roll(rcl To(i{laoLqr(l Io1rf rivrr r, dlxr z(r rouro ro bril.vriopc, bt(ffxr(irilt ot buo xdo|TczotrtL('rr(r -ril! naotxtrl ro! .,1.f n{ivri) oro! Xri)rlo ariti,ri)v

"(rt rit', lao6oi4 ro!

t/ rov(D orov (ar'rr[rJo tr]of!o7(')0r,.A1rTrl 0( 0i!(o r(r rt o!v€roq,oadr roI Kr'(I(aaiotJ 3. nooouarrr 1lt ./For[traic

orrtv ii),lri'Q(r r,',v rj(iori)! xcL trw uo16pox. zcu qthvtr orrq oo0oztrrorrniqini(rr'rq z(1 rlrori,c o!*)oy6\,ror: unoxopori<. Kdarr r:rio1< (Lrri :rou 6rv propci't(zrhrr r1 rhi./|itilra Frpis EEor€errrt 5o)i0r'rc: 6ivtL [r(r c;rrl\,llol ^/L.t ro ,{r = ,ritut,ro l: 5* trtpriyttttt otot,rbqo arfl,r|]'. I{ fqi.o('orl, oro! bir[t(I, bs! 6i(€'inot. Il( ra ril\r Lioorpe rpi:rtL vc (r1.(rpdottL'tLt ro UdQoq ro! b no! 6aiola€rcrl|z16 rou Xdrpor oritid;r xur rrirtr rr1lirll rrbu!.irl. Aur6 nou cro[6v€I silat i]f.iotDo)t (7). )i :1r-,1.r =.1"1.:r,rr o6rlyri triorlr rI< yE(,)trsrpicq oro K1ilrrao

ATKH!EI!

la( i l rs ro! . l .n i I n i ldrd raoorr6oEo. . l tou roLlo{zor ' /ac{1, r t f rdr .< t , .3aijtrrjor,. ro! {tliualo< rcqaz.hro. B$iu f{o iuol ro! Lr = 0 ztl n.aqe.hF. oia1( bdlrotrar( tor tLl6evo/oaor ror .1. B$irf fnio't. tud ifttrt 10! ,l? y = 0 zdtr€erya.ilirE oid rd bralnorrora ror (oorfao! fr1lb.!6'ln,Lror ror --1.

frc ro! nqoruolB€vo i eri I iivcr(. lfiErf aor'.lre.i(! dtt tL: m'i]..F ror ortrar0s 6rirvlofr( ror Z(;aof o0lL(i! ror e( LzdlonoL.i rl! br + 6r h = 0. Atilrt rcti6ro xa(i]tr(/ t.flr,, r.r rar(i)! yad[$dr! ior rov E:rodou!]v rou otodlr(lroq 1.. = b.Ii o'}rci!.r (uo ?LC rL. bK|oei! 6N(frzo! rard lolror eloi riizLo!:

:.5.1

1.5.1

r32 A]ANITMATIKO] XO?OI KAI IPAMMIKI]' E:I'O:EI)

Kdrpoq l

Kdlpos 3

2.s.3 AeiEis l.ai EU9Ei.1! oro nsTa.lt'lis orr ads8 bdvuoFd J ior r(baol yaop$(i'v 0at,tovonottiiry h +/, +i = 0. AEo6€iEr€ ro ibrc nadYFa 1{c{Tl.1 tlq rasK sErorbo.LsAru - I. IL ultru.\rl ulru oT(! ru / t.!(. ro alur.oro nod roeFxolrar oroJc

2,5.4 Yno).oliore io! oltrp€rQrro 3 €ri 3 ril{lra 4",{ zdr bsi&.611 sildr orptreraLrcsul)d r6rcpoe$o< r1oro 6nurotrord rsAr61o\ror o'io! pnb€!6lo)06 to1\1 Aqoe6vrcsq! ff],zuroia srrth ro! A ('{or rr! rc},ruroia yeu|lllri ro1r,,1') neoxljnrer 6la( 2€ni 2 ni\rcxaq o l! (i!o ooror€Oci yo)!id. deiEre orL cvros Eivar t|I r6lopoqq,o(.

2.5.5 Bci],rs ror b|(ltdvro:rivoxo C fc ra morI0io (rjcricrr oro fdoo! x(1r rno).oliorFror 11C,4. AriErr ncih ljn o 2 cri 2 rrilcutuE mI! dvo se|groed ro)lid circl(1\r10r0a!nIo5.

2.5.6 re(ilJrE io! 6 sni 4 nilaro ,4 yLa ro b€niFQo yQd(ru(l ior crtrtBaro<. Io bdluoro(1,1,1,1) rfrar moa plbeldxoQo ro! n. d,Id rqoa 0o uldalorv ri - n + I = 3clFE(iA Fa bLaruoFura no! rzovorrorolv qv;l/ 9 =0. Becirj reia irsvlloparc c){ar du\n iarr rn t r rnrc

" r r " lor< rot ye( , ' t i r ( , roc.

2.s.? Edv ro rqdqru.r c1r6 rcArmdr€r dEr n.[Xvtbr( p€raE! lroodAor! opdbor\, .rdr or6Loqoqiq to! anot€).eoldror €ivor 11....,16. rdrs Eilar5lvaro\' vo c!,rlorortioorp€6!vd[La.i orLq opd6.<, ardL rbor. or 8rdtpopa< 6wapr].ou !a oqrrirrol,v oxar6drs p€ra rr M€ dl,Ia loyrd (Bd Kirchhoff i $ otralo'qi) derift tLq oulolkcq lra to b,60r€ro ,4r = 6 vn ),wer(r.

2.s.8 reLltr€ rLs 6Lodtrid.rs lrrv r.oodqov e.f.),!or66! wrotdg!! yr( orndv rov 6 ufli ,1rircxo rqDontdrotq rcL pR 6aot yrc xa0E ur6rorao.

2.s.9 Yftoioyidr. ror\ -,{?A xor ,4"CA, 6rori o brald!'os 6 dri 6 f,ivaaas C it{r'ororx€id .rj.....6. noto €ilar ro ororunopd ntr }llaiaE b{oy(')\,iorJ i"C.'l r6qpf,oeeir€ aa dnoqdveeir€, 016 io yq.iqntFd, ftoLo . ed eprpalilovr(lr ontv j yaat4[i;

2.s.10 orr.iErs ivo yqLqlFa F€ oeL0FryalEs adr nqood!fto).rdFiv€s aaBdE (l{dr da!ep$i-vr ' s

"ot ,60.s) r ' 'v ororu! o r ivorog reoorr ;o,d ' vd

K6lpoq 2

tl" EI ?i o t 0Loo-r

Eivsr utlro ro yadqrua 6!v6eoi (Eivar or yeofllts rou ,4 ov€E{iqrlftd A€iErsrir( uq,orecl1{a( q! Gl€Moia aari nqornnrtu dao ow66ov 3av6eo aoL t6rg or, rri).orrF: yCaFFiS rivar Fu 6dol tou _:

2,5.11 'Eto!ro! crqdrado{ q' ft}.ntroio ori}'I oro!,reoryou$o .,1.,{or l|€ roq a0!epot!l. 2. 2. 1 oq bra.rclvro rol, C, yqAwr ro 7 cni 7 oooqto

I'I'A'I'HMATA KAI AIKTYA t33

ATc

Arcr),r'iqlor"roE Kt 1Jt,tblt.uJ F6vow rqs6 rELsdoeLq ATCiz = / TLo rq. . r , r , , rh. Alore f ls eEror i roc lq ' t tor / = (1.1,6) . M'avrd ta e€npara v( ra.aixovtar ororE x6F6oq l. 2. 3 to! xul.dporoq. riord eivo! ia 61jvofr1{d otolgzdpdoliE ncr ro A€nBsra ot4 axtrt(i

2.s.12 trriv o ,4 eilar ni!{rKs naoort(iro€o( 12 sni ? evo< or\,ettLrou yOctti|r(rto<, noLcri!(r I r(iEt rour n6o€s €ildt oL rlru0lqr9 !ft(6),r6q ail MoI tou nr: = ,in,id.c Filar or €?LruoLeLs Frru6hr:; rq Lml ror',,17 9 = / no,IE oxti( nadn.r !'q,uLqdlouv yra v crqrEivfl dvc ouv6dov 6iv6qor

2.5.1J Barirr xcr ra 16 ouvdiows dav6ac o av(t yEiqrrs !€ 4 '.6I6orE xs! 6 a4ris.

2.s .14 EnvLxu'Hr ivur rns.yovrxoi r iv(xrc. roro r ivo to yrvqrFvo rov r tvdxdv orIrr^o7 toe,trl

' l r yeolud€ f.4 Bl [ / t Fl Dr yeop#:rrr lqdF|laE la ,j [G H] nr yeapF6s

x(., ro,d ud flvor rd oxrifdro r'1v ttrloa do trldF.vo:

2.5.15 Ed', io MIT vrxlio€r ro Haryard !e 35-0 ror ro Yale rooQaQio€r ro Harvnrd xar ioP nccron lrrlioFt io Ylle 7-6. iror0s 6ro$oAir onoi€).Foldr(Dv di.r l]no),orna 3 nar-Xvlhrc (H'P. MIT'P, MIT'Y) 0s 6doora br(eoads anor€),roldr0)v rorl s]Irraurouvoutrnr)ort Fr 6Loqoqds 6uloFrro11i A! or bro(toadi tuv dorrl.€ordrorv sivcr yrooriq'rrc rc rc(yvibLa N6s ov66oros bavaoo!, €lva' yv.)otds yu 6),0 ra r(Lyvib(.

u,r6 o) noror otvor or lafrE !6por ro! a€14raro( A"y = 0 |'[orE tul y€(Dpivols x6p6o!s'

6) [(;s o xoroldr- ro! qE+Laros mov ye|{l)$6ro 161160 ouvd?doL aro rLs reurs fr].6y0) €ELo6oEr<i

t34 AIANY'MA'I'IKOI XOPOI KAI TP MMIKE! EEI:Q'EI:

"r) noro €i.ldr r rdh ro! A";6) nselypd$e rq MoELs ro! ,4? y = 0 ttdsr,r rov xrratov iox xlxl,6patos,

2.5.17 Mtiirq oq f€0o66 Fos naorturoflxd,v Jro6ooaoiqou 0o €-'rqene va Ld6o!p€ !n' 6{rvro l€y€oos rqE ol.riordols -ri to ilowe i6I oqseq .d6eq

2.5.18 Edv unds€! frc o'rpri llfidEn '.de€ E€rryo'Js xqr&ov (6va 'rl,1h€s yQdqllla) lt60€ra4rds unde')(ouv owoln'idi To redqrllra iI$ fl xo$6o'r, aa! a4.E qn6 x6t6o orovtour6 ror 6N eaLtq€rouoc

2,5.19 l'rd 6vdv r€rgifolo 6a6lo rou 6IEr 6€ffi xi$6o!E se dos n?,rwd lou, €,tsboci'oftrov iuro ro! Eul€r rrr oe]'i6as l23r x6p6or-oxtrsE+zwtor=1.

:,6 TPAAITIIKOI META>XHMATEMOI

2.6 T IPAMMIKOI METAIXHMATITMOI

t cur6 ro oqr€io p()Qi(oUl€ rd)E tl€rsx€rQi.!€rcrr €vcr,< jrivc(x(E oerotrivouEur6x(o0oxq. O Ft6€v6IoAoE rrd€r oro fqb€vui6 brdv]JoBc. 61sv roLLcnL(torir,ooult€ ps rov A. 'Ol.n-ra 6rav oparq ndve orov X<irpo orl1,6a, 6rctr, ro rli€ivtll o€ rA0€ nepi:ttoorl tvn< oUvbrrcoFq rov orrltrdv. >rl1.[o[a 0o 6€ix€ riolxdtr no),u cupcrio otr o A otil.vrr rov lirgo yq<4rprirv rorr orov Xr|po orrl).drvrori itcr( o owo{q roljE ldrQo!(, 6L(im(olq r, €hclr 1000/. crvl|"oraili+oq. Aurrlrl 'IQuyl((rrlxt 6pdoq rvoE;rivoru rpudsld *d?r(,)E (r]i6 tol,s l.lrDev6ltDqouE xfrrroq oQlmeQolj( [Ibr!6loJpo!q, no! €i,vu| oQooyolr.or. ndr crolougoiv rov 6r"1,|6rorJE 6Q6po (npoc 10 pn6(v) 6rcv 6pa4 o 11 €LrcI t€rQc1()lrx6E rcr nvnorq6-rl,+oq. x6r€ or. XdQo! clroi.6ev ilow orpcoi(r. Au6 rou 61eL olpcoic eivar trouB6aiveL [6oa orov Xd)Qo 6n],c6{ [doc oro! n-6lciorcrro l6po, 6rdv o A €iva!ll €ni r. A1116 ro oqr€io XQ€dl€r(L nLo ,rQoo€xrlx{ [d,6fi].

Yro06me 6tr ro r €tvar, n-blAoraro brdvuops. 'Orsv o ,4 aol,Iofi),ootaEel10 r'. [,'IoQoult€ !u 0€orQlioollF€ 6r! ro ,rrrao'JlFarils, o av(( v6o otdwog(l ,4ri.Aurd ow6qivur o€ zsO€ olltr€io r'rou /-6dfi(rou Xd)aou ,fi|'. Oi.6xLqaoE oXdJpoE ll€Tc(olllsril€Ta! ll6oo rou nivcx( .A rl (r€lxovil€r(a! orov €ou!6 rou.Aivowr r(oorQu n(Qu&iTluru !rrruoxnt(flotl'i,! r(ou r(QodQt(orft (116 niv(rxrE:

1. Ilol)'unL<ioto tou ranroil1|oi rIi,ts:,t"&, A = cI,,rcoriXl€r xdtoe 6r(ivuopc tr€ rov i6ro oulr€L€ofli c.Ol6xl.taq o X6aoE 6ram6lJ,€rcu ri ol'oI€l,l€rc(! (i zcrdxa;toro rQ6no:rrpvir ytoto 16 cqXrjg oqv ori0t:rrlnl€lQc( 6rqv ro .r €i!a[ aQvnrrx6).

, = [ ; : ]

2. niv{rx'rE tttQtolpoqrii nou oTQdrg€r d}'o rov XCi,Qo/ |

',' -.i I wgu rr:r'rrlv rrp 1. To rcoddnyuu ouro orQie€r o).q ra

| | 'rl 6r(rvuotrurcr x(rd 90 FrruoxrtFariqo}"ra6 ro (1.0) orovdEovc r: oto (0. l) '{'a! or€IvovraE ro (0, 1) oxov dEovqV. oro (-1,0).

,=ll.1 3. fiL\tbutE 1v xLaory nou [Exaox+roriEer rde€ 6rd-r l \ ' r ioucl ot lv €Lxovc tou. u;ro r4v r i l l l grprd rv6c xu-ul OQdtrr4. ! urLto to rceciburysu o zo0ainrtE 'ivur I

ruor'iu trt x).iol 45" y = .r. x(r| iru o4reio or'uE ro(2. 2) 6ev perc6dl,l,et<[. 'Eva o+r€io 6n(')E xo (2, -2)cvrLorp6q,eror (fto (-2,2). :' 6vav ouv6uaoF6 onax o(2.2) + {2. 2) = {4.0), o r iv(} toc cqlvsr ro dvd FiqoEi6to xar avftotQdq€r ro dl0.o. To anoril€otr.r €ivm vrr€l,ctJ.(yoiv rcr y ,crr .r r.a! va naorr$€L ro (0, 4):

ttl= , fl,1. 1_11) =(lil. [ ,'])tt;l [l] o

A'\NY:]IIATIKOI XAPOI KAI T?AMMIKE: E!DO:EI>

,/z_._ _,

IX. 2.?. Mtrdollfrurlottol tor.'triri6ou ttiolr rlosdeow nLv(xur''r.

AuroE o ni\,ox( ?"cronrQrolou €ivat €niolE xat rivax( p€Iso€olEl Mdlt-or(l. otai,vol'ru< ro ({. y) olo (y..r). itu! c}'Yt6Qrzt 16tJ() @r}.o. nou arQu6trI yc{rtr€roLxl tri(ov(. To xdrqQro ,[(Q(i6€tYtr(r €i1(! cj .6 x((r cTro n< 6io

4, nt\ex<tE nQo6oLig nou ordl.vFt 0)'61{}.180 ro Xdpo o'dv.rv u61n)Qo trlrQ6tEQIE bLdorcoqs (lf clt1o btv [no-pri va rir(r (wrrnp6rirt[oE). To r(adb€r1to F€I.rq(ryo-riEF' zdor h'ivuoF(l {r'.j/) rou rnLnd6ou oto rYYirrQoortF.io (/.0) rou oQlfovrlou dEov(l. O dEoa.l< .ruros€i1(! o tdpo: otqLclw rou ,4. Elo o xcio€ros ciEo\,(q(;to! npotJci).Lrrur mlv Lrpxl) r'rvul u UIbtvL;XruQoc

z.orpoai rotd 90' auitrlqon

^=[,11]l

Tu ,l(reub€tlpuls u!'rd 0(l FnoQo!ocv v(t lollxtlj0oiJv orig IQ€LE br(roxdoeLgY[(i!,Xouv nialrn€E trou 6Lurx6]J'orn'r1 frl li qa tr€QLorQ6qol'v ri xrp (rv(tz)'orr!

i'E nOoE 1o €nin€6o rou tr4ptprvou (petcrcl4rnri(ovtcrE tov 6oQ€to o€ \'olton6i.o). Yn(iat€r 61'qq nivcr1.(q nou reo6dl'L€t r(( ncit'rtl nslo o'(tnr610 Etrin€bo(xcr rolE 6uo ,[61r}!-< oto xdvrpo). Ei1'cr! a'taYxaic xar' etpurci zuL ci),i,ctrz6frl rraQt{b€iyirolo. Qm6oo nptnel rc cvapt'-rpiootrpt 6rr or riiwx€E b€v

trr(opouv vc xirvorw rcr:rcirwn rcr l(itrdor perrtopprrrLopoi 6tv tirtrL 6Lrvcr6rv( n(loco1.t€ouv Fs ni!(r2€5:

i) Etvm t(bnvarov v( petaro:rLooei 1 ttp i. cqotl ,'10 = 0 tolutr ltcr zd(oc

ii) Ehv io 6rirvuolcr.r :rcieL oro r'. r6re ro 2r xgi::ttL va m]Yclilt( oro 2/.IcvLx6repc ro ar rgdr€L vtr niYuiv€r fio cll! cqrou ,'l(cr, = (/,f)

iii) Elrv rc 6mvto[(r( :t v.an g ni','t olt:t' 1{a! ?/, x61. xcrl 1() doQoLoFc loljE.. . , npi :r f l vu nrryuiv| t oro .r" + y ' . ut t o! . l (J - , ) = ,4 r ' Ay

O no).l.cnlaorao}r6: 'rL,\,rizr,I' srr6dtiJ'et atroi; roui zctvov€g olot,s tr€lclolnra-tr,opoug rou lcbpou. Or bllo npd;toL tivcrr eiao)'or zo! o 6rur€8oq rcQ!6XcL ro!npriro (ntipre c = 0). Ei6aI€ rov zav6va (iii) €v 6pdlo€! 61ov ro 6ri.vuola(4,0) rcrofixQioolxr o{ nQo! rlv €uo€to x}io€(Dq 45" Ei.Iop€ oLdoraotl or.r

2.6 rPAMrlllKOI META>XIlMAl ltMOI 13?

(2,2\ + (2, 2) xlJ:r Ie 6rio p6p4 tcro:rrpiooryrcr ltoprord. To i6to 0c $noaoio€vc yivu no !e rq npo6ol.6E: 6r.doaao1, le1copr.ot6g nQo6ol'€E ncr nQoo0€ol rtl;vnpo6oi.rbv. or xnv6reg ouroi Loxriorv yla ,?.i0€ pENa6l4p&1topb lrou IrQotQxeTc.Lsno aasv niwx.t. A6'ro qE onoubar6qrdg rou(, -rouE odoqFe 6vo olo[(l:M€raollFarrotroi :ror.r rncrro:rorouv torLg xcrv6veE (i) (iii) L6Yomar yAdFliroi

Or ow0llrrq .lurig FnoQouv v(r ouvbuaoeouv a ivuv xct uovo zcrvovu:

II fra 6),ouE tol,s aQl,opoug c t{oL d xq! 6r'0 to 8[ovuoFsis o r{d! 9, oflollarXdoloo$6E tiwdxoJv lxdlofiolai, iov 1is!6!d ?Solr|'|lxoxlTasi

A(cc + dY) = c(Ax) + d(AY)' (1)

Kde€ p€rqolllporlou6E fiorJ lr4cvoJrorsl dtti6v lov xqv61(, €i,!o! €voE },8ol'urlro E ler cdm pal topo E,

KoOr ni!(r)ra( oo4y€i oFdouc o evd yQaufrTo ttPr.roxlFattotro nro e\'btaqi'

Qouos eivcrr rl crvtiolQoqq €Q6qol: Obll€i rde€ YQal$lxos [€rooxqFffitotlogo€ lvov Triv(14crl A!r!)iei[€vo crl]to! ro! )ieqdtroiou €[!s! l'q (rrc{ltto€i (xq_rcrqclrLil. o1[E ?r 6(c!ordo€!E) o' aorl]! rlv epdtr1o1. Arnl I geoQio qaor€]si

q o€pEliooll rq6 foawutilq Ai.w6psE -EE'({ldvrcrq qr6 rrlv 6[6rnrcr (1) xcrLcvcrrr:ooornt4 trg owenellE r4 nou eival nolu rlo oql8np6l'll qro filv fiuQlafieoo6yllorl crurori ro! 6r6l.iou. nQorwrloa[o \r(t E€llluloo]JF€ ?iot' E]Jo€icv F€niv(orr.g aor. ropc 0c 6ou*€ fi{iE trqQroroi! YQcl[ttlLxois [€roofuffilotlolJE

nQdner !a 6(i)oo!IE i[qool oro 6t! dvcq [€xqo1rpctllopog 6ev eivd! ovcrlrolva mdLver rov 1dao -E" orov i6to rov X6Qo &" Efvdl erotrfio et[!xQ€-llt6 va [€rcronprrriq€r brovrropoia rorl,61']r o€ blctvuopara er6g 6L<rrpoperLrori

x6aou R'". Aur6 capr6dE oup6civaL rar pe evav rivcra r,, €7ii, nl To 08'

rr.z6 6rdvropa .r eltl n o!\'r€TcyFdv€q x<rL perno lpoti(ercr o,ro drcivuoFo Aj:ror €pL rrr olvterrrypcveg. O xrwovnq lpapprx6rlraE rt al'onoL€iro! €q iootlxol.d otr6 fiapoLl.Ii.6y0uppor:E nivcreg xcr 6ror traQdYouv xr cr'tltoi lQapprllouc[€raorltlarLoFouE.

'Elovrng rprtioer 16oo pr.rxpLti, 6ev urdpler, Lolog vc orqrcniooupe- or:rpcifieq rrcrr epqavilovtcr orq orlv0tlil yQop[rxdxnxdq (1) €ivcr! n na6o0€ol6Lowolrdrrtov xnr. o noll.crrLoor.coF6g pe cpropoUE- Qot6oo -rc I n<rr I Bev eivarodrl}iq v<r eivcr b|0viopdto oxil.€E rou .R". O 1d{og cvtog ei.var lprloqogaLlci 6Xt xot pova6tr6q. E! oeqro!, rr(i0€ \Kfluaparw6g xtbpoe uttQitrtrouv\uaouotis t4€ lopqis clt +.llt 'tc "6r<wuogcrc" lr kory InoQottv orlv:rpcrp<rtrr6rryc vc ei,vft ro).rtivr4rc, ni'raxeg v1 rcL owcpIJloe r.g :r(t) r0[ g(t)Eq'6oov 6vaq F€roo"tnpcrtoF6q FsroEu 1610r(,)! x6E v uao\,oxlor€i qv (1), €ivaly0q4rx6q.

AIANY>I!1A'I'IKOI XAPOI K I fPAMIIIIKE> L=I!OIEt:

ncriQvo![€ oav ropd6€ryFc rotx ldQoli P" rov onoicrv rc 61,trvuo[urot i . ! { l l no}.odwpcl 6oe$on (ro no},n) f l . Aorc! !1ow qv poprpr lp =or+rrt t +. . .+o,,1" r{or I b[dorcorl rolj Xdpolj cilror rl + 1 (i.oyor lor oro0raou oAo!. u7ldQxorr!n + 1 orrvrcLt:orec),

ITAPAAEIrMA I H nprilr1 qq ,rapcyr6yrolg. / = d/dt €i!(r yactrl.xri:

.tA r - i k r + a n + . + o , t ' t ) = a t + . + n a n t " t .

e )

O 14bevo1r4og t4g eivcL o yovo6rdrorcrq X6QoE r(ov mo0rpriv :rol.urrviporv:.h /.1/ = 0. O X6Ao: ort),{i}v €il(rr o 'r-brcioioro( I6Aoi P,, I xo q brlL<i d,eupirxl; (2) r€QLax€rar ntrl.Ior€ o (vtoa rov l6Qo.

'lo dreAoroF(r nF lrlb€vtx6qrcs(=l) 1{(r rts xdh; (=n) €iloL I 6liloluol xou sQzlxou t6aou P,,.

IAPAAEIIMA 2 H oloxlipao\ ui ro 0 itDE ro i €ivq! €riolG yQ([$Lxt (iiorrxiLrtr ro P,, oro P,,*r).

r 'A r =

1 , , k , ' - a t t - . + ' t , , t " t t = , , t + . . . + - " "

t t " t . ' ( 3 )

Autn rrl qoed b€\, unAp1€r U6rvo1oAo< (€"ro< lou Frl6€vlxou brc(vuopsro<,on(')q ndl.rtt!). ottoE tl oLoxll(xrorl 6Ev nedry€L o),o rcr ao)rurirvqru otov P,, ,1. Hb€g(i nlruad q< (3) b€v:teprdleL mo0r:p6 6po. nroclv6E rcr orcr0€Q{i roLudvqrclu(l cnorrLolv rov (lQLor€Qo tul6€v610)Qo.

nAPAADIIMA 3 tlorlarlcro.aot//.iiVt i,!.t a1)yxexQLpdlo /roLu6vuFo, 67rorq ro2 + 3t €iv.{. yQ(1ppll{6s:

A p = ( 2 + 3 t ) ( a a + . . + a " t " )

= 2 o a + . . . + 3 a , t t . I .

Kclr n(il,r tlr,.r6g gercoTrpati.leL ro P,, oro P,,.r.1opiE trufev6gopo, ext6c rou

I' ntrri ru n(tQa6ei.yFctrcr, 6:rr,rE xrL or 6Ln oXe66v t< :r(lp(r6ri1pura.rt lQcpprxorqro 6€v €iv(r 6ioxoLo 1,c 6rrrnLoro)0€i. MA,rmc. 6€v gciv€r(rLv(r oLiJ €vbraqiAorrocr. 'Orqa undtQlrr. aivaL npnrtLxc (16ivorov v(l rrlvn(10(61'( outr€. Ilcrp' 61,( cwri eiv(|| r1 mo oqpcwtni r6r6trp<r nou pnop€iv(l CXer tvo< t€raoflBarLo[og.** Ororrrl. or nepr.oodrepo! [erdoxl]Forrotroib€! e|lar ./Qollprxoi yra fi.rQir6e!"f[d, o rrtprrytovroF6g no],uovupn)v (,41 = Zr)r l r(SooHFoI TL'u | { ,4p . p- I } I ro vu xpnrdur ' rouc 0rrrnoq ouvr€trroI i !{A(+ - rj) = r). Or ypt4rprxoi usr.roIrFftT|ouoi r(rr lovov drroi siv(r nou U( FLIEobllioollr niouJ oror,S rilor€E.

r !.r.E.r 'E!q( rtoicmrxo< llc (!.iirsl: 0( !$oioE djJ.o?ri rt: tls jtlnjs taoc doo,.ozirlE,tK croi ro r.u'rari!rd, oor doo do")isr'ti

" r Io , ( ru n l ra 'o ' t r 'o u ' r 'd \1 'z ' t rN durner , t ! rn ' r t r ,

:.6 |P MMIXOI MFIA>XHMATI'MOI l19

ME'rao rorloFoa nor, fiaeot(ivowal F€ nivorasE

- H yQclptx6xnrs iXeL prc xpior,pq ouvdnen: Edv .yvrrrpiloqre ro _42 yrc x<i0e6ldvuoFq Flog 6dory, r6ft ./1'o)ei,EouF€ ro -4s .f!a xd0€ dLdvuolrc o' oi6xLrporo1/ X6Qo. Ynoo€or€ 6rr ra n 6Lcrvrioporcr r1, . . . , r,t oJ-ror€;,oriv uLo 6don. T3r€xd0F dL),o bldvuo|rc.r r.ivar owbuaoFoE aurdv ttrv ctbrxc,lv brnvuouritrv (:rouncpcilour tov Xtipo) za. rdr€ I yeo$$r6qro '[Qoo6toQl!€! xo ]4nl

€ d r a r = c r r l + . . . + c , , I , , $ t e A x = q ( A r 1 ) + . . + c " ( , 4 e , ) . ( 4 )

Aqou (r'loqooioolxr 1t 0(L oqr6sl ora 6(o!r(d dlqvrrouotq. olov ,4 6€v (orouev€rruppic rLtuotgicr. To u:ro).orfio rou lreraoxnrorroFolr xaeoeilFraL lrao0r rrl<yQaFuLr{or4rac. O x.rvovoE (l) yLd blo orovlro[qro.r xcr 17. o6ryet otov 1d1Ir(1 r 6|(1v oFaro.rr,.....r,. O gsrqq(1|ForroF6g 6I€! d.sut)€Qia ndvo otcrorc1,liolar(. qS 6d.o( (nou €ivgl av€Edpnlra). 'Ora! 6FoS uurd xaooerooolrv,t6re dleL nceoqr,ooel xcr o^6xlnqo5 o [sloolrFqrrop6<.

nAPAAEIIMA 4 EQ6rrpI: [oto5 lgry|rxog treroqlrpcrroyog ori].vcr ro

'', = [l] *" ,",Aurdq ne6r[€r v(r €i,!(l! nol]ar],cror,oo1rdE pe rov nivcxc

!ez!v6\,roE cn6 6r.cqoperuri 6dol n.X. (1, 1) xdr (2, -1), cr.n6g ei.vnr enlolg otrovcrblzoE yeeggrz( percroppcropd6 pe

til=[i]'."'-',=lll

12 4lA=13 61.

[4 8.1

'ttl=[l] ..,,,1-?l=li]. :rI ow{,Xe!a ooxoi.olpseq ps dvo lao nQ66l,ryo -$ ro vo 6QoWE 61,qv

rivaxa nou nuqr.orriver rnv rae|oldyrorl xur 6vnv nivqrc nou ndp|mdvrL rlvoLor{}.ylao,oq. Aur6 yiv€rar bwdr6v (rnd rtv orrly{ rou 0n br<r}.d!ouur gra 6dor1.ILa fio).rdw[o 6a0po 3 (o XdQog* P3 rou orolou r1 6rdorcorl ei.vnr.4) u:trip1er,Frtt quololoytxrl €nd.oyrj T{rrv teoodqxov 6aordw 6mvuoudrov:

P t = \ P 2 = t , p t = t 2 , p e = t r

* !.r.E : YroloEi rolui)ltllo 6cqron ..lld7er xqr 3 .. T( tMpiix torro6doF( nol,uiratu (6n0j<roL a.il)r dD.o! l.tiorrdou (i(oBon /0) b!! dnorc).ofi rooQd!6( 6@nqo(x6 x6ao.

IANY lN lA ' I ' IKO I XOPOI KA I f PAMN, I IK I i : F - I :O 'F I :

H 6do4 aurtl 6ev ei.vor, govobrzl (rore 6ev eivor.). eiva 6rtr;g crvi,yxl vo enrL€'lorg giu zur au{ rivor 4 nro rcrci),l.dq. EEctdlorpr tr. xrivr:r q ncpayi.ryrorlo' oriTc T( Tdoo€o( 6oolxd 6[oruo$oTo, O[ trcpdl,uj./oi lolJf Eil,a( 0, 1,21.31r rl

O ,4 6pcr ouv (vag nivozclq, GILd noLoE nivcrxos: Yro0lore 6.[1 6pLoz6praortrvorov ouvit0loFtvo r€Teabtcor.rro It|)Ao l€ rI owleto$6q 6dol ra brsvuoFaraour1tsr(yug\ror1' 2r = (1.0.0.0), z ' = (0,1,0,0). p1 = (0.0.1.0). zr = (0,0,0,1).Ti,tr u :rir.r'au! rou uvrrorotXr i or4' (5.) 0u 4ruv

A!ro; €ivo! o "nl!(x.rq fioQ(1yd)ylorlg '. 'l o ApL Ei!c! I ne6q ofiltrl ro!. no!fi'vc1L lllba!. To ,4!: €i\rcr I brrJreQq orliLn rou rou €i!(! ro ! . To Apr ei,vcrr 2plzur to ,421 ei.rcu. 3pr. O [Ib€!6XoQo6 rrroLiI€r ro p1 (1 rcrpdyo.To< pLaE orcr€eqci<Et"tcL plbi!). O Xtirpoq rrrt).ri)v nrp!s.1r! rc pr, p1. /)r (r1 :rcpti.lo1og t:vriq zr6Lxoriro),u,rvri1ou efv(t|, 6!a be1]1eQo6h0[Lo). H noQaryoyoq e\,oq onololb4fioIe i(),Lo!onvblcro[ou 610\ L.X. xo p = 2+ t - t2 - l] 6Qi.orrrar [ioo t4; 1q($La6n]rcgrcL (trro 6or civcL ri.rorr vio -ulroq eivcrL o povo6Lx6q t-ooroE :rcqc16lortE.OG tr(v noadl,oyo vc crro['vl[ov€noou$r- rtr nuafiyrDyo 'rir0€ ]rolr|tl)vnFo!.

O :riv zcE p:ropei vn n.lQ.ry(.lio€r (11]T6 ro xrol,udr\ruFo:

, \ t = o . A p t = p 1 . A p t = z p r , A I l = 3 p r . ( 5 )

=Ap_ - 1 2 t 3 t l

f0 t 0 0 l. t0 0 2 0'=

lo o o : ] 'L0 0 0 0 l

ln;:l til l:l,lt

:uvoqiLowaq, o nLvaxag peruqiper .eq q oL)ot@rlxi< n^4QoEopie<. Ed\, i tur 1v" 'orr1 r1 6dor1 ac0d: xLrr , , !o, tvumo. "d,

o ]Qft t , t r ' ro(gercroxrpa.nop6q.

H loSLxorioilol ut< ,rllporpopicq oivor cnLrt. ftrr lr-rcox1]rorLo[o{s- €!6ExcbpoU otov aaur6 ro!. cQii€i $ro 6doq. I-|c [ExcomlorLoFoiJE €.,'6E XdiQor o'I vut u; . Io. xpf lar, ra, pr, . 6unl . / rr zoH' Xt; ,c. .

2.6 fP MMIKOI META:XI IMA|I:MOI

2U Yfioe6or€ d'ru ra blavlroFcrc c1, , . . on qroElouv 6dsl rou Xd}QouV xat aa yt,. . . ,ym qnorELow 6dol ror, W. Tdr€ tidoe 'yQq+LnG pera-ol4,ror,oy( ]4 o,16 ro V orov W rs$ordlsr(rl. Fe 6!a\, r vdxo. H j-orrlorfil,q 6pioxercr. eqoqrdEoviaf ro\, .4 oro j-or6 8rdvuotrc! rrE 6dorE. Too:rot6)'ropc ,4o; etvcr, *vcg ow6r,uopfo ro)1, 3ri ,r(rr, or, ol*rslsoi66 ourodrou ol,v6u(ropor, rieQ'vd\,€ oq qr{},n j:

(6)

:rov nlvdxo fiaQol6"ylolE, I oflil'I I nQodxuqr€ on6 ro nQ6io 6!dvuol.lc! rllE6rior16 p1 = 1. H nopdlcoldE rou r]rav pq66v xcr 6ror, q oni?r4 1 rircv pfidv, Hr€tlewaio ooiln ,rQolrule an6 qv (d/dt)r3 = 3t2, E rEr6i 3t2 =Opr +0p2 +3pr +0&, q r€l,eurdia o i).1 n€QrEiXs rs 0,0,3,0. O rio!6vcrs (6) rar(lox€liao€ ro!

Kdvow€ ro i.6!o xo! 1l,(1 qv ol,oxL{gool. Aurrl o#Ivfl rrGrxo nohxlvqrdo€ rfiaQro6d€Ula, !€roo)(| rarilow<rE rov V = P, atov H/ -- P{, dQo XQem(6-[(.m€ F[d 6doq rcu I;l/. H guoloLoytxli €ttt oTri ELraL yt = 7, !12 = t,!3 = t2, lja =13, ys = lj, aou rcQriyow ro ro)Lu6vqro 6o0Foi 4. O rivaxqs Oq €ivol tn €ntn ris_eri 4 xdl rQoxdnr& xdvowaE oloxtrriQ{Dol* oE r.do€ 6ldvuopa qg 6dorE

t t t t - | |

I td t = t i AJ . t=u : . . . . . I t rd t = 4 f i A r ! -aus .

JN JN'E o! o riivqxcE rou ,rcQrordv€L rlv olorltQrlto4 €lval o

Arj = atjgt + dzjg2 + ' + anjArn

l -0 0 0 0 ll l 0 0 o llo + o ol .lo o * l lL0 0 0 i . l

IlaAadpqa4 :x€q,r6Faor€ t1v rcpalt|ylor1 xa! qv oLoxl{Q('ol oclv a/r/-aqoE4 rpriltrg. Toulaxrnov ol,o 'lE. o1 u.xo),ou0oipu4 rn6 xg.payhytol,obryei rioo orqv apllxrj owdptno+ I.ld vn 6oripe cv our6 oqr6oi!€t orotl!rivcxeE, 1per.c(6pcore iov nivcx(r rop'd./6yrorE an6 ra r*aQro6d0p!c! oloxu6lxd (rol,u6vwa), nou €lvdr o 4 €nL 5 nivoro!:

f0 l0 0 0 ' l, l 0 0 2 0 0 l^ ' "s=

lo o o 3 o l1000041 1",,1

+ t r E ^rirr qv nontlsii{q qo nqEibhnru 2,

112 AIANY>MATIKOI XAPOI ICAI |?AMMIKE: EE>A>EI'

H noocyri)tr.ol eivar spror€Qi qfiiolQoEq rlq oloxliatDollg. OForE raaol-itloypa[[or rivax€E 6€v Froa8i va 6touv 6ir],€1jao1,s aarloraoqoul :rlvavriorpoql SrrftrlEt 6sv pnopei v' clq0rurL 1 ,4..,4""n = 1. Am! cnouyldveLortr nporrrl nr 6l,cg orrthl 6;rou ro yrv6yao 5 e?ii 5 €IEr Flb€vlzd. H ?ioailyrDyoEprcrg orc0epdg eivot pq6&. )trg dr,l,Fg ot{},tg tor -ll,,i./",,!, n!:erdxers( o rcuto-r!x6E xoed)s 10 oLox),{Qcul(r (:.i.E.: cfi6 0 6oE l) rnE fi(Qoy(i)yor.r rrlc f" €ivc! q

traoqdg Q, [ao6o],6g P, xai Avoxrdo€[E ;I

Alri I ncptiypnrpoq lexirloe pe orpoqdg zcri 90'. npo6ol.6E orov dlovc r1d! crvcrrl.i(o€K olg fiOo( qr eljg€io zLiorlE 45". O! nivdrcE rol,s trdv r6loirepo

Olorrri. ol cwtiororlol '/Qcu[[1t oi [€roox4[arropoi rou €rr,1166<:! ;r:,9 eivcl e!isotr or),oi. Mo! goiv€rs( b€ 6r[ Eiva[ oxs66v ro f6!o Eirolo vcr avx[],Iqi0o!peorqog!6 xarci ci),l,eE prvi€g. nQo6oLiE oe dl,l€c eu0eieE licl cvcxl,<ioerE cr.r5 rpogril.).cr rriro:rrgcr. Kqr quroi ei.vor uriolg lpcpprxoi pt:rcoplpcrr.opoi, t:<1' 6oovI ,eX4 n,(Q,$ivFr or,rofQdr r10 = 0 xar noanft v({ n(lerolrivovtclr Fr rivaxcE.Oa).o!uE \" (iv(ir(),urtiouFr c,rrror,E rou! niv(rEE. xQlouonord\.rcr( rl quorxrl- . f l l f o l' " " ' r

l i l l "" ,Ll l1. traoqif :xo rpjpc 2.8 nzovilmcL [la otqoqyl wtra yavia d. To ofipcr6eilvcL snior6 ro a:rotdleopc rcivo otc 6tio 6aorxdi 6nviopara. To nptixonqlcivcr mo (cosd, sind) rou oroion ro pryog rivcL :rdh 1 xcr nepr€xercr onlv"d-€!0rio'. To 6€u1€Qo 6do!x6 6d\,!op(l (0,1) orQd.p€I(r o1o ( sind,cosA).

o=l? il(otao(fl])

tl

[ ' ]Lo

,x. 2,8, traoqi xord ,: r lEoFEaia xc! o vardr.

tl

"=l;31, "=lt;l(nao6olrt) (dvdxtoo)t)

f r - s l r=cos0? '=1"

" r=s inetl

2.6 TPAMMIKOI META'XHMATI:MOI

:ulqovq Fe xov xdv6vo (6), o[ aatepoi ouroi fi(iv€ 016 onl],€E -lou riivara raarXQloqrorototp€ ol lplioerg c ?ror, s yra ro o$.Ipirovo xar ro qpiiovo. H orxoy6-v€to ourrl rov mQo96v Qa 6iver, pr,o elorperxr] euxcr,qic v<r 8rcnlordroorpe .rqvotr,oroLlta perc$i perqqlqpailopdrv rio! riladxcDvl

Eira| o apriaeoEoE'Eou Qs iooE pt Q 6 (otqogrj lqog tcr nto(l) xct<i B);Nd[,

143

Eivat. rc rcqdyavo tuo Qe loo pe xov Qzo (orpog{ xord q 6rr}.dorcyo\r l ,d) ;Ndi .

a,a-, = l: -:l

l_:

aa-l: -:l[: -:]Eiuu ro ytv4pevo tou Qp xar rcr Q6 ioo p rou Qp,6 (otpoqri xnrd lrov{,o dxor ror6m1, xfid d); Nat.

f cosdcoso s inds ino Ivdqo =

L sin o cos @ * cos g sin o -l

f cos (0 -@) - s i n (o -@) l ^= l" i " te * ' i * , ia*,1] =v'- '

H t€l,sulqiq neqinrcuo4 oupneqdrnp6dver, rr4 lrgdxeg 6uo. O avrlorpogog ep"qwltfio! 6to! I d yiv€! -d ro! ro r€rQd^tovo eppcvllerol 6rcv r1 { yi,vel +d.Ks[ oL iQ€LE ou6g egorloe6 cncvrrlorplov 1QrloLuorioldlios tgr,.ytovoperEz6grslr6rlF€E (ro! 6ivolv dvcv v6o rp6no 1r,c vc €up6ocote cutr6g ttg rcutdtr;.reE).<Duorxti 6ev rjrav nloio ro 6rr, dl,eg ol snavfiio€q rirclv v(r!. To yly6tt€1'o tu-vdxav oQ[E€xar xdt' ao iv rov rQ6To 60r€ to,ltrdpeto tt|d,xav v'amrotoryeioto,laritlt1to * tur ltclaoylpataotrttbv.

2V Yno06oft 6fl r4 rol B €l,vsl yQq{.rlxoi Fs[aoxlpctr,opol er6 ror, I/oro't W xo;r 0r.6 xo.t U otov V. To $v6urv6 rorg lenuvri p' ivo 6ldvropoa tou LI, rqyciveu oro .Bu tou V lraL ra&i.s*ifFL W xo ABu ror W. H"owoeot" cr'ri ,{B ei,1, ryd}'r, 6-vcg yqcppw6g petoolqpcropog (ord tov Uotor W) nol o nivcxcE rlou to1/ ropLordva, etvcr, ro ylr,6pwo rtov nrvdmvnoo ncrprrdvow ror.rg -4 xcl B.

* t.r.E.: OwL,ri io ,tLvitpao id pdqdxrya(oldiry" lfl €ilaL roQ{i I oMioltq o'ocLn

s l 11 0 lcl =

lo r l

I i - sz -2cs I fcos29 - sin2d I= l 2 " " C-sz l= ls in2e cos2o l

AIANY:M,\TIKOI XAPOI l( I IPAMMIKE! E:DA:II:

Aur6 6Lanrordr0ry{s nQolyoopir'(,)E yrtr xor' ,4ruo,4.io 6nou o oriveetoE Ferco1n-pcrrqtdq frcrv o rcvrorp6!. (Ka06E €niorlg rior. yc tov ,{.;o,,1.". nou prp6vr,lro),€( rLs mc0€a€(.) fta orpoqig oup6uiver 11 oerptr rou :ro).1,n;r).oorcrqrou va pqveTrt oqpcoia oL lrirpoL t/,V trar I!- ouvt?iL?r1.rv FE 6Lo ro ertne6o r:,9 xcl oQpQ!, €[!Ct o (bro( [€ tov QdQr. l-ro !{(r {'tQoqi x(! trlo (!dxl,aoi, 4 bldr(Eqn(il€r 6vc ao),o.

TrVufi :rpmfip4a4: fru ra xcruozeudooqt€ niwx€q ;(Q€ra!6goo1€ 6dotlEr(',1 l/ x(r| ll'. i.c06( r.cr i(,)v U rcrr I/. Kpurriwrcr< qv ibrcl 6.i.D'fr(. rovl/ ro ltv6pwo fi!\r(ixov ri(ier oorolcr cn6 rrl 6dol rou I/ oq 6(i{rl rou Il'.Edr zdllor4lr bldxeroq lr*ogu rou l€rc(oxlF(flolrori A ,(clr iou nivard io! (ro!oroio c{< o14|6oLfoorlLc pr [Al). t6rr: o zav6vcr6lou ylvopavo! 2v ylv€rc.! noliorivtopogr [,48] = [A]['B]. Enc{r,c},c[6dvoupc 6rr o rcr6rtrE notrLontrcroioo[o!nLv(xu)arou K€{.c}.(iou 1 xc0oQio0l}i€ jr},rl{xD< ({7ro C(urrl qv arcrirnol -6l},116ll6rt nptmr vc rcrpLciltr pr ro lvtiprvo lpcppLzo)v F€raoXrtpcxroFdv.

E/itotQirlouls rdJQ(r ftir),r oe ol,yrrrpt[&{r nc(Qcr6riparcr pc vdouE nivcrxr<,

:x,2.9. naodoli nilvro utt l-tloti(: I luopftqia xa' o niv(a((.

2. Ilpo6ol,ti :ro :11pcr 2.9 erxovi(etar I rioo6olt tou (1.0) :thvo-r mq d-er0ei(r.To tuizoq rl< raodoLrlg €ivor c = cosd. :qll€ldor€ or! ro o?/rlo fiQo6oLrtGbr! €iycL ro (.,r),6r0)s cy(i ozaq,{)|Ira },(l1'soqriv(. To 6L.ivuolt(.luro at€LFrixoq I (€i.v(lr r oiaoql rou (1, 1)). H nao6ou lou (1, 0) €iv(a .: qrcOdE"auro ro trovo6(i.o brdwotr( ri (.r.(.1). noe6Fo!(. I rA()60),{ 1()! (0. l) 6XrLpri;ro; s zcr ;r6q'rer oto (cu.sr). Ar,rri 6ivtt r1v 6rw€et or{},1 rou n[1,(1xorrAo6oLrtq. O rivaa(lg cw66 6sv €X€r otrioxQoqo, 6rd,i o [€rsolqFdrrotrog6ev 67er uvrimpoq,o. >tp€i.r 6rcrlg ro (-s.c), rdvD mrlv zriOcq erl0ric,nQo6t Jo\'talrrlv (exi. H su0€itr clrnyi€iv(L o F16ev61arpog rou P. Tcu6lqovarcr otpeicr t1; d-er0eiag rpo6drl.7'ovrcL mov €cr,'r6 1tl116! M€ d),}.c }.dlLa, ro vc:rQo6itl,oupe buo rpopdq eivcr ro i6Lo B€ ro v( rao6cil,oupe gic rlopdr zuL P'z = .l':

[l]r,i:"- ["' *l

[ . " , t ]

d ( r j + s r ) r s { ; + s r ) l -c s t ( - + s _ | s l . - + s ) l

]6 IPANII1IKOI tr'ETA:XIJI\{ATI'IVIOI

OroLxti, ci + s: = cosr 0 + sin'd = l. 'EvaE nivazaE ngo6o),it noitdr ttt 10rxrpdyav6 tou. Eni,ang €i!c! ouFB€rerro_(.

3. Avdxloot To: tFo 2.10 6€ixv€r ritv (ldrL(lol rou (1.0) oqr d-erotic. To

lr1xog -rou crc1{].uarr:vou roorlial p€ io oQxr]io, onc,r! ouv€6crv€ xor lr€ q olQo{l]

roorooo or buo peroollu(xLo$oi. tivar no).u braqoptrLzoi. E6or 1 d't0rio peverorl e;ol qc. H xd0€rl o' (rljrriv roeia avrroraiqr-r rrll ]icI€lrouvori rltE xar o].(r-rcr olpt:icr nrpvrive 6ui p6oou rolj raga€nrl, c ' r1v n1'),q gtpri. H lpr4przortltctraxrorioL€i T( l|n6LoInrl-

tt. 2.10. A!.iai(ot a{ reos qa d'.!0r,ir!: rl 1B4r!:rQi.1 x(r o nil(ad<.

O :rivcrxcrq cnr6< rler r1v cErulpeirorrl Lbt6tlrcr Ilr = I Aio avaxlda€5yaE qiqvout ao mo apyx6 orpcio. E lo$6voq $Ia ovaxtraol roolrlot P€n1v urr imqorr4 rqc. H H . :r1t iy1t, ' ro ' f rvur npoq',v ic,1:To rr l yr 'utrrr , r ' , .hlorupo 6pt-r< jreoq(ldq cjro rov ri\rdzci rlq. Mio roooa !{tt I(]lJ 06Fdro<

livrrcr pdotu rIE otdot{r; (!.ri{}.lto€t'rv zcL rpo6o},tiv: H = 2P L Xxloo1;r(ivtr orL .l1r + r = 2Pr I €r.zova otjv ro cpxrxri LiroucL ge to 6Lnldoto rttErpo6oiilq. Touro €nr6€6c(drer" uriollq 6r!

H ) = ( 2 P - I l = 4 P , 1 P + I = 1 ,

ogro{ 61.€q ot nQo6o}d! tx(1\,otroroi! tlv Pr = P.

Or rpflg cRroi $r-rcotlu(nofloi ri .r(pit\rouv ra Fri]{l a!a),l.oicDrc1 (oraoqtszcL a!oLl.do€q) i n€QroQilou! 1() $rtroE (rpo6ol6_(). All.or l€-raoxryurtottoip:ropci vcr ar[in'ouv ro [rito,<: 6!c1oro].6< xnr icrpcpoprptboetg 6i6ovrdL otquo"tof,c. Kci0r rup. i6r ' rnru ixrr a, , t ;v ' t t n ivuxLr nou ro nap.oruv,, . r ' ,1o onoiog eivcr ro nipro oepo o ouril! rlv rapdlAtlqo. Yndet€L otroE xcr rolrlrrpa qE €r'Ioyi< rrE 6riq5 xdr. roviqorl€ tr€ 6y9rcy 6rL o taxa< r[aptdrota:t6 r4t ni.oyil r?E 6rio4(. fra naai{6€Ly[a:

trlIjl'''"[| [i]

, , 2c' | 2.$

L 1(.' 2r- l l

12.1' l l

IIANYIMANIOI XQIIOI |\AI I PAMMIX!: I-:I!OIII,:

(i) l!(l nQo6oL€E, uro66ore 6rL to :rpcirro 6cor,z6 6rrivuopc repLelmcL mqvd-€uO€ia r(t ro 6€ur€eo €iv([ ',idoEro o' or.ml. T&e o jriraxas ?rpo6olrK livercr

:rai, r - fl !1. o "i".,*ur

.,vr& zcrra.rxrudlrr( onux ovwtor'<: I nadrtl u l r

oql,q tou :rpodplcror ca6 ro :rqcbro 6qorx6 8rdvuoFa (ror fi0o6{i},1€ro! orovrc(w6 rou) xc.r q 6€ur€et oniLyl :rpolplercL cr:r6 ro arurepo 60orx6 bLirvuoFcnou nt'o6dL).rrc' i'ro Flbiv.

{ i r r f r ( r r v c r ^ a o r < . I r b r u b d o l l i O r r r f = l , l ! . [ o O e t n r p o t , u o r r oL ' r r l

brdvuoycr cvcaLcirrrl oro crqvrprz6 rou xcL napd"fe! q b€ultpl otrlLl. O fiivctrtaEff tlcroLou0ei vct eivcr ioo6 ge 2P 1. 6rcv lp4otponororlpc rlv fbrcr 6(i.ttl./r.(

(iii) fr( oxaoq6E, 0a Fnopouoc4rs vs 6tol.€EouFe rriLt povc6Lcic bravuop(toxcrd prixog rq< lr-€rl0€Lcr< r<rr r1g xc0irou trg. Qn6oo o niloxr4 6€v e' oltra(€.

or tuur irc uut is rrgdreirovrqr rr i l . t rcrtd t / *" A=]: -*1.

Or, '4 xo.

nAolyor4dlor<.

To 6l,o 0dpc xtg €rrt'oyrtg rrtE rcl.r,reg$ 6do$ €i!o! lterte!,{t( otlltcoiog xo!ft. €mfiQ6vo1,l|s o'uurd mo K€qirl.rlo 5. O ot6xoE eivar vcr liver o nilcrxcg6L(ryd,vLoS, 6nx4 ouv66r1 pr ror,5 P zm 11- ftcr vc liver o Q 6raldrvro< Xpettilor.rcrFLy(brrd 6r(rvuoFaxo. 6rorr ol.o rc nQoyFarlxci orQigowcr.

Mv1po\,€roup€ €d{; qv €ribQ<rort ro! 6X€L orov ni,,ttnLcr p.n d).cy/1 6do4E,6rcrv o gercroppurLop6s rcpcrp6vet o i6rog. O r ivctag A ( i Q tPiH)t,frarpirsrar orov S-rA5. Apo. ivug r,u Fovo trrr(ottFuftq'dg raom(i!Frorcri6 fiolJ.o{q bLoqoQ€rurol,s rivax€E (F6orD b|C1{oA€rrx6v 6.io€ov rou .r}'tvreo-oorrerowur <l]ro rov S). H o€(0Qiq r()v rbrc6*rvro$dtrov 0( o\y1oer o' cur6vtov tn:ro S L,4S xar orrlv rch,'repr1 buvcnl 6tio1.

AEKHTEI'

2.6.1 IIoLos nivoxos 6e( Fg cnord].EolLo v( ma&psr xdo. nLdvuoFc aar.i 90'. a( act6trlvva to rQotjdl€L mo! (iEovd r;

2.6.2 noLoi nilcrds noerordv.r iitr reodoLi trdvar oror dEova r:, arolo!0olrr.los aroqv neo6oll riivo mov dEoro vr

2.6.3 To .rrvo|€vo 5 o!.rx),riorov zor 8 otpoqtir mo €nin€bo r, y naqdy ptd mqoqli ri

Br(r dvax&oli

2.6.4 o n,v,rx(l(,4 lo i]

'*"i" uts btcorotillt:lr! qv xorslJuuvor r. :Ir'6ruorl

rov rn ,o "c, + s, = I xar .yrl€s an' du6v ra oitl€ia (2,, !,/) ftou rqorimoNno).ldr),aord,towdg re ro! -4- n oxiFd 6x{ aunt I aqrr .r;

26 l tAM\llKOl rlttA:xIlTIATEMOI

2.6.5 Kd0r E!0Eio tadfli rcqcpirEr €xeEio p€rd oro avcv ]qapBrao f€toompafloBo.Eciv io : Eircr ro Fioov tl€ia:n tuv ar y, l€iErE orr ro 11: €iv( io Bioov BftoEn

2.6.6 O uru iu( I i i l ropuyr r \a \ r ln4ol r10 r r !6 dqid, lodlE rou uq.r" , , ro '

Gor( v alc).Ioioro. Adror8 f{L( {xovo x$ 6airoqq rou qov.iEovc r. bri.rvovroq1 nf te( i ! . r 1( ' (1 ,0) . ro (2.0) , i ( ro ( 1,0) , ro06q z(r t rdr ! r l t rcoxl f rcr | i r rqro).oxirlaos o (i:ordq.

2.6.7 t loror r ivuz 'c l r r , . { r ' r0 .n?ar 'o 'vro c t rn. rdt r i ( rot rorrc rni) iqo6(i].Lo!v u.ie€ 6(i!udLd oro EnirESo r, vr

ii) zolorlaiiolv rires Sldvlorro.r( rqoq ro srin€io r.l/iiii) orpiqo!! ro lrinrlo r, V zold 90'. cqrtlovroq rol d!ov( . orce€Ao:iv) orar:{oL'r ro rxfnrbo ri y. x(loxr! ro rji; ,1or nororw ro y... o}.a ncr{i 90'i!) ezrfloi! oLf( r< naortloulrar,< olaoq]i. d,id r(110i y(,)r(r 160''i

2.6.E [oLoS nivaxq raaFrdrsr rtv d.:ldt: oror Tieo Pi tu)v artjrr(i)! noiw)wt()vi.K(1roo,t|ldoir rov rv ioyo 4 0ri 4 rirorrl Xqlollorororr(lq rrt orlrt0rotLtq d(o!l . / . , : . , ' I l u r u c ' , \ , ! u t , . l o r \ u \ i u e u s . u i ' r l o : r r r t 4 , ; , \ k , . r o ' , . , r l c r T u r u ^ L u r u t uior (vlxolv o (rroriq:

2.6.9 noro<ni ! ( r i ( (naQrord! . lFrro! io . f f i I . loL.ropor lFro(2+3i) roror i i lFrrcaut j rzr iroirriwr4c ror P1 oro aoiulvu& rilcero! 6(0Fo! ro! Pr: O' miirs duon rof 5fni l ri!(Ll1 dx)t!(ri!o'T cr eq,rrOrLrjior"r(q tov FerrlolrltLclLoFo o. x(iee i!( ca(i r({jcoLai brda!of(ra rL = l, ..i = i. .fr = ir. .fr = lr.

2.6.10 Or )Jd.(q rtF rQqLttraris bL(,{oe'htc fiioootc r/rr/r//r = r oula (orry iluv bu'uoralLao td)ao (drc! orlb!(oltoi ),!o.ov.i.a(| Ir(iiL Lnoe(). Baeil. hrio (!rE.ior$eii u | ; rur ,dJr t .u ,v 6ur t ( ' ru ' rdr /u ' !JU.

2.6.11 loro( ori!bv1o$d! ri)v 6(ldLr(iN nrrvldt(ir.r! qc Aox. 2.6.10 lr,ler rl! €iid('),rl. peeTfzA<r4Le<r=rxntd| / =J, . / ta I =0r o f l ] .dr r l . r . ldx l l larLoloq,( t : r rL!e)(rri( r+riq. oq ].!o11, si!( yadFtraoi. norcq eiloL o aivaaas ro! ro1 rdEdtliv.ri(X(npLporo ' l to t t r =1, u =0 roL r =0,9 =I oor 6dof ro! t ' zat ot 6dol dc( to !I I ' ) .

2.6.12 Erd).r10.!or€ ,or' €!0€idv dro nl' -c:+.r = l, 611or nivcaE. ovdriodrt zovonorc!!

2.6.11 Yio0imr orL o,l !i!dt dldq yeaBgrzoq Fftod)(IFarLoBo< ror €rLri6or r.y olo!eulro rou. Ari lr orr o ,1 | rilar srioT yeaFudo( frEraq(ruarrotro; (sq, ooovM.iat.L). E(i! o .,{ rcprmtlvrr(r aro ro! nilc,i( ,\/. €Eryrtorr-r&ri o .1 |

r(atdt.iv€Idr da610! nilnro r'11 I.

2.6.14 To tLloBrlo (,1B)4.' yqq[La.b! rt€ldotrFdtroFda, EEaL!.i F €aa 6Li1rldFd r, ida.itdr av( br.illorro u = alx, rcr6rL! 5. o,olou0€i rov kdv6lc 2v .qdqF6Eovtd( rovJB oro u rc, tptihvc, oo (-,lB)C'r.i) EivaL ro anoriirc']Ld ro i6ro fs duo tro! naoznnll €qdet'olouoq ro! t' zdrotrrrror ,B rar rf).oq xo! .4i

t I r F \ r r r r ' r uo ' l p t i , ! o t l t r L l r u f l u ! f t r t ! ! J .

AI NYIMATiKOI XQPOI KAl PAMMIKE: EfI:O:EI:

2.6.15

ii) Eivor ro arordlEotto ro i6ro p€ our6 rol'reoxljrrr€r fqaarolo!'roE 10! BCorol,o!0olp€vov qfld ror ,4i Edv !u| 161. o( raetlgaofts .i.!cr rrprTras rar oreoo€lcLqLm!rcq ldlos (/B)C = A(rC) qd€L yLo raall+lLaouq F€ruoxil|lafloFo1,9.ttvbr!.riorrsvt rr. ro! tavova yrvol6rcr 2v. out'i Fiva! r xol.nrFQr uob€LEit to!i6rou x(l!6ra yrd nilaffi

Aeiltr 6u ritov o t eav(lr yeoFFrro< tr€'(oxtFcrroldq (Lx. ror R] oio! Rr). rdrrio ibro €ilat rar o A:.'Ib

onvo).o o)ov rov nL!d.ao! 2 ui 2 7i€A#rE! ro rioo€ao 6(oLxd ' dvnoFor("2.6.16

t; 3t, t3 rl [3] t3 llo€oq1inlt ro! yecpFud Israorryorrqrd dd.rQoq,isrdos 2.ni 2 nivoxo l{ar 6arirEtov rivofi rot '{. o,c rpoq dmiv qv 6dq. rlcri LoxllFr Ar = 1:

2.6.17 BArirr rov rivoxd 4 sfti 4 aoD naqLordvfl rM xuxl,Lxli Frrdftq: xd0e 6rciauors(r'r..rr,rl,rr) Irraq(rlp.riii€1ar oro (r2.cj,:rr,..r). nora eilat it bqdot ro!,.11:AsiEr€ o1| i i = .4 ' .

2.6.18 Barirs ro! 4 r,ri 3 niv(x.l ,4 nor rio0Loldver q 6rqrd p€iatonror lrcrd try onoiaxd0r brdvuopa (rr.r'r.l\) f€1(ox4rr(ri(€r(| oro (0,rrr.12,rl). B€rift eriofi rovniv(xu rtr .'Arorrqis fftaroittoK g oro to Rr fliuto rno R] nou srrrrqlFarrlcL to(,'ir,.r'. ri. cr) oro (J', 'r, cr). nol(l rivot io yLv6pe!.r AB xcr B.,li

2,6,19 :ro! bro'!$otdflr6 tdJAo y dlo\, ltr.', x1J6rx6! Jrol.uo,v!!o!t P = o0 + l?r? + a'rr +arrr. ido S io 1Jloonlolo rw ro),mv{rov pr

"[] dr)dr = o End].)t0sums oIL oS rta(lt r'n6xoeo< xot 6a.tt€ lto 6don.

2.6.20 'Evu!

llrt yQ.4rltLtq strcqlpcrtoroE owrorpiq€lql ora! roxrr€L nrdeEr xaL Fova6F

"ontro: J(r) =, txdr fia o1|Qr6d,s ),nq yro xd0r 1,. To naQd6€r'yra /(r) = i: 6r,v

li!ftr dvrrorqdlqro. 6dr! I rr = b dl€r trio LlJosrs .lla e.fl.,td , ror n(lttpia ),ioyl t|Coaqrki 6. no|o| ard ro\ srr6lrvous F€rdoxl$qr|opo!< (qrro roris rradrparrrc!(aA$FouS R' orotE reorudtrxols aerefous Rr) .iv@ aELdrQd!'+roLi Kavdvus bev€iv(rr yrxqryr.l&. orne xa! o (c).( ! ) / ( r ) = r r ( b ) / ( r ) = € , ( c ) l ( r ) = ? + l t ( d ) / ( . r ) = c o s r .

2.6.21 noro; .i!dr o .iEov,lc rPqmeoqry, xar r pvia or0ol ic rou urrrtollyarropoi ro"ord) . ! r r to ( r i r , r r ,J1) oo ( : r : , , r r , i r ) :

' :.r.E: Adft raoryolNq lrDo4rEi0orl.

AIKHttrD rIA EIIANAAII{,H: K€ga].oro 2

2.1 Barilf t(o 6dort to! €Eric uordEov tor Ri:o) Ic itoatiot'dtd ln ia onoia rr = ?.rltj) Td 6lolnorLora lla ro oroio x r +..r +..r = 0 tor f,r +zr =0' r ) rov l roxopo ror nqa401c' cno id (1, l . l , 1) , (1.2,3,4) l {c , (2.3,1,5) .

2.2 tsqioxowc pro 6aq ror, ffqryqdue 6rcr xnox(oao rou Rr ro! 6ev r€Edi(€r zavivo(no lc br( r ld [cra o lvrsra] [ i {ov (1.0,0) , (0,1,0) , (0.0.1) .

2.1 ALr0iq ri 1,.!66q (p. q'rltrdadb.!rua y|C ro !J€!56():() Edv to 8r(vropora xr, .., dr,r fi oadyotr tov ln6loqo S. r6re dim S =,,r.6) H tolLrt 6no unox(ba!)! 6rd!1,op(taon )(draou 6sv Eivot nori xtrtl.

l) Ed! -4.. = AV. rdr€ r'= t/.6) O rdaos laatlF(i)! ro! I 6rst pro po!o6u)t 6(iot nou pnoaEi va !no).otto0€idldlowdq ro! ;l oE &l4rozoflt [oa9)].s) Edv 6rcq rElpatorltxq rilax(ls ilfr aasEiianF€c orrilsq. io ib|o oltrdoi!| tat

2.4 not(l €flor )t xl4rsiorl lroeq,rt t,'ror rivaxa

\:KH:EI: IIA EI]/\N/\AII'IIH Ii€t] ]

norsq ria( oL b((rdfttr< ro,! rfoodAo! 0+eiro66v uro)(dia()!:

2,5 Be.it€ tv r.ig1t rar rov prtd.loroao ri,!

2.6 BQoire ai(1ofrE ro! r€od.iqov e+Elto6.bv lroldreor! nol, rednroaiiollor oro ious

._ f r r l "_10

0 l , . l r | 0 n l' L r o l " l r 2 l -Lo I o r l

2.7 norc s i ldr I t r .vrar t ] . iu t ro! ,+u+b = l , u t ' =2:

2.8 d) Koidoreuior. ivaa rivorc ro! onoiol, o plb.loxorQos n.qL.l€L 10 bLdtvloF(r = ( 1 , 1 . 2 ) .6) KdrooxEldmr ivo! rivaxu rou oroior o etdtEeos FI6sv6XoeoE ntqdxu ro

v = ( i . 5 ) .

]) K{rraor€ljdor€ i!a! nilorio ro! oftoio! o xibqoF orrl'ii)! rtlAdyer(lL ono ro6L.ivuo[a (1. r. 2) e(rr xo! onoio! o X(iao6 yqaFp6! noed,rrar ano ro (1, 5).6) Edv 600()r! raia onorobi^or. t'Lo'lopald io! R" ud xaia onorobrtnore b!d'llioFdra ror R5. lriiqt€r av(s rivdroE 6 sni 5 ro! oroiori o l6qos mt)..br rar(eoy, -d, oro r ( rqulo rq.o ?olu ' , . ,o i , r ' . l , ,Coc yeduruv vo roedyFrar ororc 6sursqa rAioi

I r 2 o 2 rJn= l r 2 1 r 0 ,

l l 2 - 3 - 7 - 2 )

l r i o r l f00 r ]Jl - l o o I B lo o r l

11 1 1 l L r I r 0 l

150 AI NY:M]\TIKOI XA?OI K I IP MIVIIKE> EEDA>EI>

2.9 trov ldreo tuv flvdro! 2 €ri 2.o) dva' ro onvolo tuv nrv{irov iiiEeos 1 ln6xoraos;6) roro,c u6roqo; noadysr( aro roUs rivax€c Fndedo€oqrr) noLo( lnoxoqos roadl€rar oro rors onr"ouE trivc"EE (olld ta arj > 0)i6) roro-c rn6Xoqo! naedr!:rar a{o rous awrmail,rrroq nivaxss:

2.10 Karaorcldm€ aad! bravuoBftrr6 td,ao ftor !d n€a*}ft 6).o1lE ro1jE raoturrol]gtrErootruarrotrois dno ro oaio€rc rov raroaa raoooool]E.Iloto €i.!ar ! Srdolooi roli

2,11 o) Bafilr' otr rdLI ror ,4 r'i(| odJotc trd 6dot ro! rotisvor(iraou ro!:

f r I f r 2 0 | 2 r l^_ , r ,_12 r luo22uol' ' - ' " -12 t t lo o o o o r l

1 .1 2 I r_ l L0 0 0 0 0 0 l

6) A w (All]0ic tpe'r6as) ot nadrreq- ta€|r yacpp6s ror U dror€Iow 6{ic! rourdqo! reat416' ro! ,4.A 'l,OL qrrlle< l, 3, 6 ro! U arorrlonv 6dol ro! XdreolJ otl),d'v io!,,1.A tI, OL ftos€qq yeapldr ro! ,4 ororsLonv 6dol ro! Xd)ao! rQopp(;)v ro!.y) Bacir€ 6('() ro 5!vn16! neoroo6rF0.r o'€Edoqto hn,\'tjopaift i) 1lft ra oroid rl,'{r: = 0 61r, t,to tri*.b, :ntv urulorqt To! 4. ruo rol).or).os(o rnc reLrnc yesFtlK a,fore€ir(.r ,'drs !dBrl6€!|o0Fi 1 Iitaaq yqappli

2.r2 Ecir o A sivoL rrir(,ts6 n €ri n - I xst I rdE)t ro! eiv.lr n - 2. no|cl eir{u I bri()rcq1o1r llo€loloQorr ro!;

2.13 xqouororim€ ardLoLqll tls va 6eEire rollE reryovrroljs ncAdyovr€s ,4 = tral,

f ' r 9 ! lL t L h h l

A = l , l .l a b . . ll a b c d )

ftdror. o!!ei!€( tt€ ro\ core[otJq d, b....1..i.\,(t( or orlt].eq yaatlt1rrdE clsEdeqffE;

2. r4 Anor. ) .o! ! ta bra lnopara (1,1,3) , (2,1,6) idr (1,4,3) Bro 6dor lo ! Rr ;

2.15 Adrotr napdbriyBcrd trn.iaov.'l lLo rols oroious o dEeF6i t.)! r"!oedr! rolr -,tr = lJ

i) 0 )t r. a!d).oya $ ro ,.ii) €, (!€EaarirlE ro! 1,.iii) 01 {. al(i)'oyo Ft ro l].iv) 1, avcEaqntrl5 ro! 1,.

2.16 >tI! naoryonp€r1 dor1]ot, rdrq o!rb6r-_rar ro I pr x( m uar n @ x.ioE f,Eeinrooli

2.17 E.iv r €ivar brdvuofa ro! R" adr r?y = 0'yn a.ie. y, ij€iEr.6rr r =0.

2.r8 E(i! '1 €ilor ilaq rivoxcq n fli n zcr ,{2 =,,1 xar ri.Er A = n. 6si&s drr .t = 1.

A:KIDED Il?\ lillANA lt'IH: Xeq. 2 l5 l

2.19 I]orcs uaoXoqos ro! vduov 3 frri 3 traedYtx(( ar6lo\ mortrr(irl'.tq ri'aa€s- Eij.

F€ 6va oqr 6(cytblLo Kr ro nor! lvc Frl ttqb€aL"o moL)(Eio ano r&o;

2.20 11600r trilot€S |€ra0do€o5 5 r:Ii 5 t'trdexow: Eildr YqcFfra(iE o!.E.iaDFoLr na_

oa1otry oio -ov /6ao tu! 5 fni 5 f,rltxt'vi A€v xq€rdislor !d tot5 xatclqd+€Ic

2.21 norc fiacl it idlll xor rivcac r sri no! adec moLl€io toll .iu lr Ilorc lilctr rlrdEl ror "riwx.r dxaarou ps oi j = 0 6t(v i + j €ildr dano aur oij = l Iia' i + J

2.22 a) Y,To noir,c ori!0ri1|.! l|a ro b 61Fr 1() I r = b Fra lnq. otov

6) Bacirt t('o 6.iot ro! |Ibfvoxoreou rorl A.

]) Baeitr q yrv'xn ),!ol ro! ,1r, = r. ot,r! ntrdaxFL hlonl) Beritr BLo 6dol ro1, xdraou mrt),6' xor .'1.€) no(r Eiv(L it r(i:rt ro! ;11 i

2.21 tltis rrroerir. v( z((oxflJddfrr ivav rivct<c:ror praol+radirt r(1 rsic a'(vi-oF(r( drvr . rnyfa l i ) ! . r . . ' . . i o . ra i ( r 6o0a! ' ru 61uvio l lo1( . r ' r r . i r i norE. iv( t6!a( raroroq riv("ci awtoraaV!+tosi

2.24 E( i ! . r . . : . . rF ivaLoroX6aooot l tbvwo(r i lcad3tr i5 , t lXrr r t r . ( l t ros bFEro ideLm.qo (vriolao(roi

2,25 Yf ,ourr r urr u / rv( , i ! , ' : /L , , t l , l , , roc ur t ( ,d / ( rynr 'ouoc tou R'norJ orr ] .ur ru

u l t r r o ( r . r " d ' n r o l x - , r " r ' r " x r . n r e t { e o \ n r r r x n v t L o 7 - _ m o q r u i u

2.26 Ai.noar ri \t!€!d6<: () K(ier !'I61oao( lou Ri rilaL r$tvoxoeo< ?(tnotou triv(x(6) Edv o I ixfl rov idto p$rldt.lao 1lr tov -'1? . torE o rivozo< fi!(lt rtraa'{ovt7or.y) O f.r(dxiltr(1otr(i9 xo! n.i.L ro x oto ')D + b €iv( YeafFtxd: (cno rov Rr mo'R r ) .

2,27 Bq| rr (Jio,.c nrv n oodo 'v 04.r1+,,^,iv ' roX. ,0"'! r, ,v

2,28 d) Edv o|rQ.4q1is ro! 1 si.vc' yqo$1|xdr altEdarlr€q (o .4 silur m €ni n) 1610 ilf.iEr €ilcr - x(' o t6eoc o tldia sildt ?o( o oat{n.qoq frlb 6loraos

6) Eirv o .1 eivoL rivozas 8 .ni 10 !s btb(im6o Frl6€art)(o)ao, deih. 6fl ro .1:x = 6

Fxoari !a ir'0ti Yrc zA0€ b.

I r 2 r r 3 l f / , ' l. {=10 o ' , ' ,1 x ( b= l / , . 11

L2 r n r l L l , l

f r 2 o 1 l f r . l' l i;;;l "" r' l i lrr 'r

Lo o o 4 l

152 AI.\NY'M TIKOI XAPOI X]\I TPAMMTKE' E=T'OIEII

2.29 I].ALyadln€ tolE yA.+Frxoi,,c Fsrdl}xr]ForLotorr rou €,!aa6o! z, s no! aderoftivo_rar os reo( il o\,l]croFaq 6dot (1,0) iq|, (0. 1). fE tols nivaa€s

A,= l l 0 l . / ,= l t 0 l r t 0 r r' l0 r l ' ' ' -12 rJ ^ ' - l - r o l

() Eov o.l sivar rrrqatovLxos rivdxo<. briEr€ orL o u4brvrixraqo< rou A/ rrqrrXcrrov |rprvoxn,oo ror' ,4.6) AEiErr dr o Xd,aos oqLd,v rou A? n€er€I€rcr mov r{i,ao otltrolv tou A_florF o rivdxd: rdEF,.,c I. J = du7 . dXfl /? - 0.

a) BQ€ire f|( 6dol orov un6xDeo 6).0,v tu,! drovuorri1(,rv, rou R6, rou LrcvoroLonvrLs r:t +rl = -rr +rt = rs + J6,6) Be€n€ rvd nrrem nou il€r io! rqo,tyoqrsvo urdloqo u4 gtbFvoxoeo.Y' nQfrrr rvuv nrvux( nou axfr ur,{.j! ro} urrtloeo 05 x60o oq}dv rou.Yno06ord d1 or riva,rsr 1',{ = rU .tvsi. dl,rimoix(

2.t0

2.31

2.32

fo l00 l [0 0 r - ]2 ll r000 l l : 14 2 lJlo00 l i l 4 2e 141L0 0 l 0J [ : I 5 - l 5 l

f r 000 l f2 - r4 2 t1_10 | 0 0 l l0 0 I _3 2 l

l r I r 0 l l 0 00 021L2 r 0 I . l Lo 0 o 0 oJ

a) noLa sivar I rdh rou Ar6) Arbors pra 6dol to! Xd,ao! tea$rd' ro! ,4.y) AhoiE 'i riEU6*: or yE4{ds t. 2. 3 rou ,4 eivcr yaalllllxG avsEdqEF€j.b) d6ore ll(r 6dol rou x.0ao1, onl),6v rou /.€r noa ewfl I btdmdglt tou dQtotpeou F lbrvoxoQou,ou J,l:!' rrord Hvor rt ?rvrxrl,'uort tou ,4r =0i

OP@O|ANIOTHTA

3J ! (AEETA AIAI{YEMATA KAI OPEOTQNIOI YNOXOPOI

AjI6 ro nQolloi[€vo x€qd].o!o rrr.,QiEow€ fl €ivo0 6dorl. At!.f€6Q!xd, sival6vcr otvolo ov€EcQmt(0! olovuoFdr(l)v nou rlqQdlow rov l6po. f€o[€rQlxd,eil/ar 6va o6\,oLo oE6v(')! ot^rera1|r61,ov. 'ElcS o[cwouarrros l6Qog oQilerdrov€EdQrrFa sn' curork rolE dEovr6, (l)01600, xde€ qoQd nou q1{6qroFc[ ro€rin€6o r.g ri rov rQrbrdorcfio X6Qo i rov R", o! dEov€E €iv(!| naQ6w€iFdLwxo €ivoL ormi0tos xd0€ro[! Or ilovtg otrtnaypivtot nou xataoxtttd.ltt4 qctraoia eiva4 rpaxrtxi<, trdvrou op9oytivror. Arql6lottdE FLd 6doi,lrivor.4rr va nQotrFiooqrE Fr(r oQooydvLa 6dol.

Edv 11 r6da rqg 6rioqE eivcr an6 rE €€F€l,Lo6elE qE ypopprzrig <il,1e6pcg,r6re q eler6ineuol oe plc og0oy<lvlc 6rioq 6ev nder, no),u nro:rioo oe onou6sr-6rpo. Xper,c!6poore Ulq 6doq y,a vc petarq6rlor,pe y€cop€rQlris rqrqox€udgor cl.yedqr"rotg unoLolLolroilE, Ncr oQ0oy6vLe5 6doer4 yr,c vc arlorLote{ooqredlnoiE iouE urioLo.floFoiE. YndQXsr [!d cl6pl eledixerol nou xdve! q 6dorln€Qirou !6av!xl: Ta 6lqv{ouarq 6c npdleL v<r €Xouv larir{oE €!a, Kcr! our6F:ropei eriory vc enter,10ei, olhti 0a rqd:rer. vc yft{ilor+e

ro prixog evoE 6rcvuopcrosro lalrrlQlo ra0er&waE 6lo1,uoldr(Dvr(iE vc! r(crcrora€,udoow€ rado€ra 6la1,Uolaro, E€ral(irwog oro yQotr!!-x6s ov€E{ierrFa 6ravrjoudrq.

Eni aL6ov, oqv rLx6vc :rpeneL vc ool4rer6Xow xar ur6ppol Kr duroi €nisrEpnoqei rc eivcr xiOeror. @' qvaxatdpouF€, 16oo 6pop9c xcr sn)d roueivdr an6louol vc ro 6l.enovpe, 6rt ot rtooepLg 0rye1ki)6EtE rniyaqot eivatoegoy(bvrcL p.raEi touq. Ei\laL xdee,ro! old 66o, 6uo orov R- roi 6{'0 olovR". Aur6 0o olo,{Llp6oe! ro Oep€r!!d}6€E O€6Q'rUa qE fQoFl.l!){lig A}'y€6Qds.

(1 )(2)(3)

OP@O|ONIOTITIA

(0, 0. 1r)

( r1, o)

(0, a2, 0)

( r r , 0 , 0 )

(a)

| : r I = r l +11+ . . . +z l , = 17 , r . (r)

(tl)

tx,3.1. lb piros cros br-xdr 0!E lAr-6rdorato! 6ravnoparoc.

To nQdro 6{p(r eivd! 1a 6Qotl$€ r.o ptilxog *6E 6,dwop@?(rt, ror., oup6o-Lilet<rL pe r l. x(rl ('116 6uo broora(o€rE npoxurir€r c;t' rqr uoteivolocr ev6gopooloviou tpr.Iirvou (:X. 3.1s). To rEiQdyovo iou |rli)io]jg 6Q60nrs, €66 xarno),! 1ro!Q6, crrr6 rov nu0a16pcr: .rl 2 = r] +;1].

!r[E rQ€rg bloordoelg io 6rdvuoud e = (r1,r:r,r:r) eivo! I6lqydvlog ev6Elaolrrou (tX. 3.16) 1tcrt to pirog riQoxinrEr €qxeF6(ovr<rq 6io goQaq rov niro ro!nu0qy6Qa. H 6lbLdorcrq 7r€pifir(l)oq rdrloriorei rI 6r.ay<irvr.o O,4 = (rr,r:,0)

nou 6rcrrpdler rr1 6cio1 raL bi,veL O,4-' = cl + e1. Artrl olqpcti,leL og0r1 prvic pc11.,' 1ti(0€q nl.€uQd (0,0,.rr) xqL exor $rroQotilr€ vq €fi[rcrl€oeouF€ rcrL nd),! 1olfluocryoQa (oro Enifir6o (,)\r OA xar /8). H urror€ivouoc xou rplyarot OABr:ivcr to pr]xoE rl trou bfb€rrrr, crfio 1o1,1urio

t r )=o4' -AB =, i - .1- .1

H t€vir€rol o' atlo 6L.i\aJoFo r, 6norooeoJv ;r = (xir, . . . , r") rivcL l4reor1.To ltixog rl Ey6E ara|lopani rou R" Ei|d. q qdrxit tttQayauxil ptla nn

Aur6 o!d^/€r(lr ̂ /€(,)p€rQrxti or egcppoYll tou rurrou rox I!0a^/6Qo r I qoQ€E,rpooesrovr(rE fuo 6ldolaol oe xci0e 6{gc. To doeoroFa rrrn, rerpny<irvo;vorp:riarrr. pe ro y,v6pevo e":r; zcr" ro prlxog ror r = (1.2. -3) eivtrL r47:

lilr ' : = [ ] 2 - l l

].I KAOETA dIAN]AM TA K I OPOOTONIOI YNOXAPOI

:X. 3.2. To €ttLtiFbo rALy{"vo pe rle1]ad< rj y n(tr r - 9.

YriooaolE rd)Q@ 6rr" 6i6ow<[ 6uo 6lavuoFnrcr E xaL y (2X. 3,2), n@EFrloQo{F€ va crnoqdreo{'pe o! sh,o! xdo€ia; Me dDla lollo, noto shut roxpr.r{pro ogeoyorvr,6qrqE; H €Q6rrlol owrj prope[ v' onafiIoel o' dva 6t6rd-otuto snir€oo F€ rQryovoFErQio. E66 XQ0ral6goor8 rq l,svirsuoi orov R'r, ol,ldpnoQouF€ ndlr vo roQcgeivou$r oto erinebo ,to! rlcQolouv tc r Ld! y, Mtoao' ctw6 ro eniae6o, ro c €lvcr oQooldrlro oro y, €q' 6oov oxr roritor , &aopoolirvro rpilovo. Qg xQlfilQro, )Gnlo4rofiorodg€ Eovd rov nJro rou llooaldQc|:td r xat ! €ieat ltdeeTa 6ra!

l r l ' ?+ ) l 1 t l ' ?= l ! : - 1J1 ' z . Q)EgaQt6Eowas rov no (1), yr"a ro prixoE. rcrlpvor4re

( r l + . . . +z l , ) + ( s f+ +y l , 1=1 r . t - t ) 2 + . . . + ( s , , - Un )z .

H crpLmepf r),erpri avcr.get pdqog qE 6e[rti5, lo! ovofirioo€tc! oql

(r i + . + r '?") - 2(oryr + .. + ;r, ,y,,) + (sl + .. + yl).

Apa ot|v (2) nYuer 4 Lotjrnxa, 6tat' o dqpotopa ruv "peurdtv 6puv" tirttrtt46ivt Ie. ): xat ! Eit. op$oytbvu 6mv

:rlferdots 6tr crnrl 1 roo6trpa eivcL orpr,6drg q i6Lc pe ro ;r"g, 6n1'c6n ro'/!v6Bevo €v6E rivaxd 1 eni n (ro 6r,rivuopcr-1pop[ti xi) d! €v6F r{vsxq rl €rli 1(ro brdvuoud-oxrtU g):

r r g t + . + t h u n = u . (3)

l;lr ' I a =Lq . . h l = E t U t + . . + r n A n . (4)

156

Aur6, pe rov oup6ol,Lop6 iolt n0poiopcrog eivor Dz;yi. (Eitiolq Eival xatgr':n.) O ouv6ucop6E cut6E eUtlcvi.lsrcr ot 'idOe brepeuvrlorlr4g leropetgtag rorn'bdorcrou xdpo!. M€Exdq qopdE ovopdlerar, 6c0proto yr,vo$vo yl ylvotr€vo-rtsL€io ric! or4r6ol'iE€ror pe (r,tr) ri r I. E[€i( ndaroE roor4tdF€ vo roovopirEoupe €o(o?€pr,.ri ywhFxvo xa', \c\ lr,rQc:ltivor4re o-rov oup6ol.rop6 r"y:

3A H noooqtc xrg etto:t to so0ti€Qr,ri6 yrv6p€vo lo! 6t,cluopdlc0v("oql'tlv) o xcr g tou R". Aur6 €lvqr pn66v, €dv riat, F6!o! ed! ts c |.ot 3r€i,vor, oQ€oydvla,

OL r66€E ror tulxolE x(lt tou eocr)reprlor-r lvopdvor 0r]!660\5(r! p6oo iou rltrou,r?

" = ei+ +ri, = | r lr. To F6vo 6!dv!opc pe pr16evrx6 pfnoE -pe cill,c l,6yrc

io tlovo 6(dvuoF(r, oQooy(buo oro! €ct!56 ioo- €iva! ro pl66a. To otdluogori = 0 €i\,dl oQ0oyd\rlo o€ xdee 6!d\,!opq rou R'r.

NAPAAEITMA6uo axolrv $nxog

eivql oQooyolro olo ! / = (-1.2.2). Ka! rs

:Tllv €fi6p€vyl rrdQd"yQcreo €E€rdq,owa! 6lavrJopsrq fio! 6€! 8L!oI oSoo"ycbvtcrq! yro rd onolo ro r"9 nQoo6[oQiq€( q lerclrr roug "yov[.o. A1J16 6i!E!€!d! (puo!ol,oy[]io oQro[6 Tou 0111'l[[16!0! olov n-6lciotdlo XdQo xq! &o0€Tqblqlriopclxq 6xow cos l/ = 0. :' cr!fl11, trp traQdyQaqo treQroQl|6paor€ orLg oQ06EyrDvi6g. :r6IoE po< €ival vc rctcvorloope rouE r6oo€8K 0ep€t'!d)6€t6 ltfl6x(oQouE

e!6 rl !6r6qra fiou lsg €vdLaqdQ€r €ivaL yl oQ0oy(0!!6qrc.Kcl' qQId6. urdQx€r, F|a crhl oleoq F€ rl yQapFDdl qv€EaQxloia: trir

t r t p4 y46tuxa \Lar lopan u1,. . . , r :u t ivat oQeoybrn p€xa, i xous (KteE6!ci\,!opa €ivor x.i0cro o' 6l.a r' ciLlc), rdre eirat xu yQaqu'.6€ awtdQrlrc.

Ard6E44 Yno0€or€ 6rr" crur + . + c^,u^ = 0. fro vo 6€iE€r€ orl ro cr €iv(rlFl6av, ndrJre ro €o0ir€Qr# yrvop€vo xrov 6no ?rl€uQ6v pE ro or:

uf l (c1q+ + c^ r , ,1 = t , /o=0. (s )

H oQooyuvrot4to 1u\ ' r 'oqIvFr ivov F6vov ooo or l ! (5): rr ,7, =0. FnF'br lra orovl ,otror. ' u:r f laolodv try l F$fvrxa., f , r , 0 r<ar owFni, 's ' = 0. Io rdro(rl,qoeuer xa! 'Io. "a&gE ct. 'Eror o $6voq ow61j(ropo6 r(l)! u; trox rop.il€r" ro

tlrlbdv ei\'crr o r€TQLFF€voq. pe 61'c ro q = 0, {o! €fiop€lr,)6 rd aLcvtopcrc eivor(,.,'€Ed$Wq.

To filo e\,6[a{piQor' rrdQriaerypc op0o1r'rvi,tov petalu torg 6rcvttopcirrov eivcLro onlolo r|0! bravuo$rinov owr€To"flrdvov er. c2.. . . . e" ro]j Rn. Arrd €ivor oLorrl),eq rou tcurorrxou nivdrd, ofiol€Lo{r'rrlv or}.ouor€al 6i(ol xou R", €i','orpovcbLcic bLcvrio[crc xciOe 6vn toxE 61er. prlxoE er = 1. ?ior &ixlow nQo!

To ;t = (2.2, -1)4+4+1=3 .

rtr ridrsroNot rrD! cEo!(ov or','x€ra.rud'vtDv. Edv orAdUo!$E (rrro ro otror+ra,1() arord),€o[a €ivar Bio !6c 'aa0o1ru,o1\v.i dtioa", ntl(6rt €14 v6o ouortBoop0o"/rDvlcov ricr trovtr6lui(u! 6rovuotritrtDt. :ro €nin€6o. cr.lrrt I orpoqrt r(adY€l

,1.1 KAOET,\ .\IANY:IIA I A KAI OIOOTANIOI YNOXOPOI

, , , = Gosd .s ina ) . r =( s ir / r . cosd).

* llair.r v( nrcdbqoir 6l o rrraoorwds aat o ni(i166 roi/.o( lvoa iol,tllo! f.Lilioft lea.i0.ra enirEb( rof Rr K(rd ro! oprqrn [a( o!{,< roiro &y (irtoEi.t] Yr.iQl.!! .!0fl8< . n(L

u ro! bio roixor io! nfl rirrtoliir rari oa0)1 ioid. Iladl|{(r'. rl . r.i.r r(r.i Frixo< nF pYi(q

nrarf7tr(r zdr d.ota art roiiio!< u( oit'olpc bE! €ilcL olrFolr;ard mor Ea!r6 rrlE

Aur6 rivar ivu <rx6pc ncpd6trypc povcr6rui.ou grtlroLiq G,lr = 1 zar r'fr'3 = l)'.cr oaOoy(uvrorilrq( t'l t = 0).

oa0oT(i)vror lJrr6ltoQol

Eax6[am€ r(irac o l! oQ0o1()t6rlro bto rfioz(i)p(,rv. Aur]l ararrFt tl;roxriec drrivropa io| tld! !nol{;reou. va €ilor oQ0ol{;)vro oc '(de€ 6ldflJolorou al,trou. :xo! rer6raor&ro 1tbpo. ot rii(6x0)Qor na0torcivovrat ltr- €u0ti8l r{cr€fiintbc nor 6riexovror an rry oQxrl x(rr orrg (txQ.ri€E reQln16oer,q oro rlv oQ lptivl q< rjo),6xLqpo rol XdQo. OI rno7"()aor Frop€i'vcr i1o!lr$ 6rtrordo€q 0. I2 ri 3. O lln6xo)0og {0} ti,rc{ op0o.^irvro< or 6)'0!( rolJE utrox(bqor(. Mtcr €uettd

Irr(oQei vo L_ivor oQ0oy6vtc oe [La t!0eia rl or 6!c €fii.;rtbo. o)'].d i/lr lnl-ardo[tr yroqti ru tirrtL op|o1,tin'ro o ii'u rntft|oi Ot'6xtr1pog o Xtlqo< R3 ei.var.op0o"/rirvroc [6vo orov {0}.:rrq r br(lordotE o oQ!o[o! €iv(r o rErt<:

3B A60 un6xoQor l/ zcL I,l/ ror Lliou tdJQou R" eivo oQ0olrinror6rd! x{ioE 6rovtto|la u roo y eivc! oQooydvlo o€ rdo€ oLcivuopd 1ir louI l : r , ' r r , = I i l , r"d xddr u xcl 1rr ,

fIAPAAEIIMA Y:rooiorr 6rr. V eivtrL ro e:ri:te6o:ror n(adYerdr (Jro r(1fr = (1.0.0,0) ' rcr f r = (1. l . (1.( l ) xtrL I I ' e i lcL r1 €r0€i( no! j raphY€rar (1j I6 ror, = (0.0.,1.5). rorr. Errrb4 ro u eivc{ op0o1<,',vLo rcL orc bro r,. 1 eu0ei.n II-0c ri\,ar opoo"/ovra o o)'6r).Ipo ro 0nifiebo l'.

: oljIt rt! r€eirrr,rol rolv tlnotd)ao,\' 6'.lmdocos 2 rcll I ror Ra, rrrriqltrx(rr. roqoq y.n 6r'cv rpiro ltnol0rQo. H Erro€ia , ro! r!-Qvo ofio ro : = (0,0,5 1)ai.vcrr zd0€rl raL oro I,- zrrr oxq! ll . Eriol: or 6Lcrorrioe6 o0Qotlopeveg bivouv2 + 1+ 1 = '1. xtlr polo xo p46e1rx6 6Li|vro|cr tivo! 1d0€ro laor ororiq rQfl(u6Xoporg l',II' nclt. L.

158 oPooroNIo tHlA

To €v6nq6Qo! Ua6 1ta oQ0oycrrl/ior6 uno1.beor5 brxaroloy€iraL tog e!{g:O! ,no sv6raq6pow€E on' sulor.E 6€v eFqaviEowcL rrr(ciq za| ndv€ ndwcxorlr E€d11. :qv naoyForlx6rnrs rivo nvcn6gar,cror.: Eiaar or 0;peJu'66E.grxtfuugotl Ei,rcn r'toaep4 urr{(opot xqr ndve xard lEufq.

To tprirto laryog eivct o FqtuvO(toQoE xo! o l{ipog ypogg6v. Auroi eivqrufi6ttoaor rou R", o! yQc446E q(ow n ouvrfi(l)o€g xor ro i6!o oqr6alr€[ nct perq li,oeE rou Ac = 0. IIQ€'& !a 6€iEoWE, IQrplpono!6wqg F6vo r4v rELooorlA!: =0,6xL o,IQa tq fjvar op0o.l.brre6 oro 6.dwopd:r.

3C O 1daoE yeqpp6v €ivar oeeo1dvro6 orol/ lr46svdlopo (otov R") xcr,o IdQoS oql6v Ei,vq! opeoy6vlog diov dQr,ot€Q6 pqdsvdxoeo (orov R-).

IIA.ht4 dfi66e.E4yfio0dore 6tL o elvcr 6trivuogc ror pl6evolcirpov, Tdre ,4o = 0xcr ro oi'orr r(r 1ov rr sEro(i)os('v F ioe€i vo yQog€t cwo).6!n6r6Qd orq Fo0prl

To olr,oo4rei<oro oup6criver. rl6q onlv rpticl e\torloa4 4 ypcpyri 1 tivnrog9oytitvn mo c. H rliooo4 Lder &r ro eootplxd rovE y,v6pevo eivor. pq66v.H ber'l€Ql eEio(l)o{ }'d€r ro i6ro nadwo yr(r rl lpopp{ 2. Aolo rov pqbevr,xrirvrrlE 6eErd5 fiLeuQdE, ro ,j €i.var oQ0oyd)vro oE rid0e lgcprprj, dpcr eivor, op0oltirvlottL oe x60E olrv6uqop6 ft)v ypcrpprirv. Kd0e o rou pq6evolrirpou eivcr, Lor,n6voQ0oy(l![o o€ xdo€ 6 ivropc rou 16qou 1qc4r[(i)v, 6tlq6n ,^/("4) I R(,47 ).

Or, dl,loL opooldrvroL un6xopor 'lQol.(intow dn6 \q''! AT ! = 0 i uT A = lll

ar A =l l l t . . . . 01 .

(6)HI|l ' Yoqqni I

I YaaFfr z

L. . YQol.r[I rn

fo o lt1 1l

v,,,r l l l l=rot " ^ l14 1 lL I n l

(7)

To dldvuoFa g €lva0 oQ0oy6vlo o€ rdo€ orill. Au!6 poivetar on6 lqv EEio(|)orl),oy(l) x(ov Frp€'vrx6v orn 6€E[d r]'eued. AQa ro u €ivor oQ0oyd)vlo ,ra[ o€ xd€€ow6ucopo rrv oq),riv, xcr, xqrd ouv6rac ro runrx6 6uiwqr<r tou cEmeporiprlbevoldrpou rivcr oQooydvro mo tdQo otr/.drv: "M1,4r) | R(/4). Awo uordqerge xo rQdrxo Froo xou 0€oQrilroiog, 6nou ro ,4 dwrxcrOtoror<r pe to A?,

AEnrEQq dn66E Ett To i6!o nQdyFCr o€lorrF€ 1,' onob€iEoqlE ,io! p' 6vcrp6no aveldgnpo cn6 owr€xcyp6v€E, H cwiOeo4 perolu rr,:v 6iio arodeiE€ov

biv€r. oroa nv.rfd)orl ircr XprloLpo ncrpri8eLnrc p|.(1< "(rfl1prN6v1g" €ra\'ir" $Laq''oryz€xp$6q<" [E065or ori].rryLopon. @o r10t:],o vc, riEepa roLcr eivcr ocq€oreal

rcr erzol.6repc zcrcrolrrl.Y n o e , o r o r r r o / , r r r ' r J / , r a r , ' r , ' ! J r t 6 r ! o / , r f t ' d " , r u

" r u X ' i , p o 1 p r 4 q r , i , vTott .4rr = 0 xcLL L, = ,4' : yrc rciroro Srcivuqrc :. (To 6d!!orrc u rivoLorl6wotr6q {rDv yecrtrgrirv. crqou neprprar orov X(;po yAqp(i,v.) Ocr 6eilorprroe ptu 1pt44r{ otr eivuL op0oyrlvrcr:

I I KAOIT AIr\Nl:NIATA KAI OIOC)rON]OI YIIOXOIOl

t t . t = ( A I . ) t . t = - t A r = z r 0 = o

IIAPAAEIIMA YroeiorE or. o A 6Z€L r(iEI l, (ip(l oL X(;)Qor orllo-rv xurlea[$o)! xou €ivrl| €ug€r€<:

(rJ)

tiilOr y0(trlLiiq {,iv(r noi},cn}.(inc ro! (1.3) rcrL o glbev6ilupog rrplixcr ro ( 3,l)not| €i.!({r. oo0oy(iraro orr<./pa[[6q. nQ(irylurr, o $i]bl!ol!')Qoq rivur, crxpL6ti( Irirt)rq eu{)ri( ror Rr. :ro! rx(aoiorsi

I l f " l ' x,u 12 r, t l ' I , " , r .r 0l f ' l=(,

:s (vrLbrootoiri. o!.lti),or 6rio r:rdlrrgrr :repriXovrur tnov Rr. O XrirQo: orn).dvr iv(( r l fu0!- ic b|( i rou (1.2,3) lcr o 0QloxeQ6g pt6e!ox!)g'E t iv0r iva ! : r i : rc6o.AttoE npd;rer vtr r i r rrr . ra xa9no € r |do gj+21/.+3yr =0. To{ro ( ,{Qr6( i)E r . iaorto neqLrTriFr:r,o trF yT.,1 = 0.

:I[Er.d)o.e orL ro ra(lro q€r,l,oc (oL 6uo ru6eieq) st]E brrrordocr< I + I =2 oror Xrbpo Rr. To SrilrQo lerilog (:u0eic ,aL rrin€6o) rile brcrorltoeLEI + 2 = 3 oro\r Xdrpo Rr. E! ydler o 1<irpoq yQapld.v xor o $r]60v6ttoeoq 6lorlv6ruorrioerc rtov oroic,^, ro duaorop.i rir,(r r' + (l, ,) = ,. To hl),o lsuyoq6i!al aoao(]F,r r +(ii. n = rr. Ebd ou[6ci.v€' xdrL nsQrod6rE$' (lnd rn!rtrU oA0o1rtlr6qia r.rl nQin€t vc urrmrLeo0d qr uopovrl ocq ./f .r!ro rorpoo0rro Xcrpc0rrrlpLorLx6.

:iyougr eivcr c),r1Oertl 6rL o p16e'v6Koqoq eivcr xriorro5 trtor yri,qo 1pc44[irv6er aivrtL olttD_q oL6z),Ipr1 q cr],{6erc. O "\ (,4) 5€v r€ardtsr .rn},(i)E 1{ilxora 6LC

vroFrlra oaool,drlr.r orov lrirpo lpcrpprirv, trtl,it trtpGxu tttitu iroLo drivupu.O Blb€vo/copoq .rrorrl,€frc( (1r(6 6r,€E rr( l,jorK ror ,4i = 0.

OPIIMO' Ao06r'roE ]jnotrhQou y xolJ R", o XtirpoE 6l,t-rv rov 6nwopcircorror tivcrL op0oldrvrc mov V ovoFdlercr oQ0oy6uo 'rpttliaapa rcr VraL ow6o).iE€10r p€ Ya.

160

M (m1 r1v opoi.oyi<r. o p46tvoTopo< rivcrr ro oqoo.ltirvro orlr),rta(o$(ror, } i ,por lqqprbv: -v(/1)= (R(, l r)) t . Tt l r , rolpova roX|er zcl l r1 ( l1 ' t i .otporprqo7dorl: o 7ri{o: .1pr4qr(i)r :rtprrTer ir).c rc brarr,qrcrc nor €i.vcrL oe0otrix(r orovtrl6€\'oli)Ao. Atno &r eivcn z(r looo neoqcld<. Aqoi iuvovro; ro l.r =0.Eehvrlorpt'<ra6 ror Xri'Qo./A($[(ix rltlL 6alx(1Lr o),( tft I ror r.ivcL op0oytirvrcro crur6v. rtirpcr n(4r€ cvriorQoq(. Yro06orF iorir6! orr rri;roLo brrivtlolu :ri\'(( oe{k)y6!ro orol' Irlbr'\'ol(')Ao (r),i(i z(lr r?ro! ror, t(;0o1, yaqtr(i)\'. Torr-r;rrorrvd:novrc( ro : o(v rri ni6ov yactrlrt ror] --1. 0< tr€y(1,(;rctrt- ror l6ao1pcr14rd,r' rou -,1. 7,r'rpiq v( ci),crEo|tr€ ro! tot6€\,6Xtl)po. frrdpiio|pr 6!r'rq ltrrr,:rcipTtr ircrq oiTol'qo; rr,ro: | + (?r - f) = n:

dnn(/(;)aot'./a(qrll(;1') + dirr(tr)tbr'rd7(Daor,) = r),rleo( ont),6v.

A(ot'. fnror,!(inro1'r(c ro :. b$'(l.i(iior'! or bio rlitr,r(ior cAr{)t'oi. €i!(rr c6n,!(rov !r1 (),).dit-r z(r o ;ro{i)ro;. :qr:T€A.livortrr 6rr zd0r &(ir,r.l]ttcl oA(n)T(iyrorxor 1r16rvo 'rpo :rtpr"iltrt( orov Xtirpo 1p(pp(irv: n(-{i )= (,v(-,1))r.

O i oc or, 'oyro!oc. rq (ptrototrr-voi oro\' -{. x(e(./!r ro bLiilt o oUrrilxrotr(lo uau,tu;. r<\tri/_!,t!o. .\'\.-lt ) xat o fueo< ot4)titv R(.ll ti nt opth4titrunry' .,lait&re. Or ({ordor( rotr bilol,v d00on4(r (/r , ) + | = tlr. Tourootyn),ttgirv!t 1rl bflilr.(x) tr(oo ror Oettr ri;bouc Orr,-rprjpuro< 11< fpapgrxrl<Ai.yrtiA(r.. To ;ro(iiro tuo6 d&Dor- n< b((trrioe(< ro^ r€ooir(xLl! (l€I€l.ro6(i)v|nox(i)oorv z((k;J. r(r ro yryov(t. orr r(h:rl lecNr(;'l = rdlt ot1l.t;Jv. T{np(./\,tLrQiiouB! oIL r-ivrrr x(i{)r-ror. n(r pci},rmtr orr rivrrr opUo"/irvL(l o4ur}.1po'r[rlrcr.

3D SqeilubDxE Sedg4pa t\g lpayptxilE A1,yt6ga6, Mipog 2O Flb€voXDQoE €ivor ro oA6oyd)\'ro orprr).rlpo$o ror! X6Qou lQctl.rFcbv orov

O t1(nor€e6g pq6er6Xorpog eflctr ro oQeoyrivlo oqnllgopc rou r6gorol4Ldrv oio! Rrrr.

E:rt 'ai(Idcivorlt orr cruraq oL rreor(io€rc si1'(r (artoradvLBe(. O;((rQo( ypopp(;vn0QtdtFr o.rrbrinor€ Eiv(lL oQooydvro orov tr)l6r'16X('lQo. O ,diQo< ()rll.6! n€ardtrLo.r6inor€ €iv(r oAeoldvro otor crpLorep6 gr16tv61arqo. Aurrjrfvur gLtr ap6too1.zqr.ypdvrl mq peorl ror 6r6l.iov a otolu ooriao xoAopiitt :ro6< t[ntla q €i\arrr,lr;rrrrrr;! Bi6aovrir< rrlv d{$Eott. tt ,1r =, crrarrri ro ;reEilercL ro I orovXdrpo ord,(i)v. BLirowaE €ppeor, curl crrttrei.TLcr ro 6 vc eivcr. zirOtro oro!(larorrpo FtbEloX{DQo.

I l x.\oETA \tA\\:nt.\I,\ x.\r ol,ooraNtot Yflo\opol

3E H €qi.otrlot A.r: = b ),wsrcr €dr! xcrL F6vov €.iv 6Ty = 0 zdrO€ qoed

H (4'tr€ol npoo6lryLoil ritcrr "ro /: rp€rcr vc.ivor owbr(ot{oq r(uv orl),6v . H6$tcot1 :rgoof ro4 ei.vur "ro ir rg€:rrr v( €iar( oaoot(i)v(o or ra(i{)s brdvtlotrujTor' rtv(.r r'('(1oy6\ro orr! orri),F<. Ar'16 b€v (rzoiF/rr(r o(v iioi,rioorl (11(.vc ro fi{qrQ.idorfL€ inl{l). Or(rv citrrl( urirplo!r' ro},}.iq orll,ts }tt . trovovir.l i buo 6r(\'r,otr(r( ooooy(lxt(r o au6<. r6rc €iv(1( noli f|roior€Qo 1'(€:rci.rt0trtoor'trr crtriq rrc pi(r ri bio or!0rtz€! 14< popqrf bry = (]. O Ndtro<Ar'vrlBlzo! rou Kirchhoff. rfq n(raclya(iq ou 2.5. t-ivcr d1,( xctro n(0(i6€rn(. Tovcr br(iTrorr;oor,ll€ }llbr_vroBo zcr(i llllxo: ztz)'(lrv ci.vcrL:rol,rr rutoiritepo o16 rov( (!cyvr')eiootl(F or'vbrr(lotroi: r!)1 orrl).(rv.

nAPAAEtfMA Or rrpLotep(q nlrupig rol':rrrgracira; tlroriroe('.v biborw (i0aoF

lI ili;'.lli'l'ii

Ot litotirorr< ).rntx'r(L .nr xu gtho' ctir ro i9gouttu rnr otorT_riror rot,t \ tE ( t t ' : t ) . { 'at i r Li t \ut l rd6,. Eia(u €r*o}.( i leoo vc L- i iyFo+rE o! /J +br+tr=0

nor, z(i!r.L ro , op0oli;\,Lo oro , = ( I. 1. 1) rolr cpLorepoi |l6evtiynrqor- rrrpriv( tcrLorroortr€ ur ro 6 eilta our'6r,cotr6: rov orri.rirv. Kcr(i ro 6rpri,r;6€<of(i)Otf(r r(r |r,o our(i rir(' roo6ia(tc.

O iir(rraE z(rr or un6l(|,por

'fot'ilotyt :rrii,r 6rt. ornr or brcor(iori ror'( ri.\'ut ao).u pLzp6<. rorr or I' ztrLII'1r:ropri v( r'ivcr ooHotdrvroL. 1r,piq v(l riv(L o|p:ri1pr,4<rtLzoi. H €uOri(l l'rou jrcoir]€r(r lt'ro 1() (1.0.0) Ffvar oQ0oyrir1,ru orlv tLr{)r,i.( II ror:rtlix'rlercr(mo to (0.0.1) .liiri o q rQerq 6Lcrordotrc o li bcv orl;ri;rrer pe ror II . TooAooydJvro oupr),rlAqrc ror Il €ivcr 6r6Ldorcro. Auro civcrL 6vc rni;rebo zcrr4 er0ti.c I' rivcrL 6vc ydpoq tot govo. 'Orcv ot 61oorl(o€L( rivur orDor6c, rcirroAooy(;uor r,rTo/olaor tirftL a\'alzaorrrh opoo.ld^'rcr ory:rirprirp<rrcr. Artrl rna.rrl x€(rnr(Dort ror 7oQor' lQ.rtrloJv zar ror' $llbr-vi,Z,(|)aotr. zar I cn6SFLirt luoAFi1( €( i ( latroooci y€vLxhl

rcl t r I I - l = I |

OPOOI QNIOTHTA

Ix. l.l. Oq0orclar( orfn).nad4to( oro! R'r rni,r€6o ,ttr EUorio.

Mr lrl),c l,o./t(l, f rr = Vl Or. 6Lqord.oe q r.ut V nctt, II' €iv(r[ or or,rmtq xuL oXdQoq (va),i€rcr or 6io oaoo./drs paQq (tX. 3.3).

'Orcv o X6po< (vd!ri€r(! o€ 6io oQ€oydNLcr y6pq, ro i6ro oul6(iv€r r(!trL- ,.d.fi€ 6rAv!oF((: .r = r + u. To 6l(ivuo[a I eivcr 1 :rpo6oi'r] mov urrdx(rQol/. H op0olirvrc ouvLorriroq u €iv(r! I nQo6o),i rolr ,r orov I.l'. Or ur6gevegrc(xiyQctqo! 6€i1vo1jr fid)E 6piozovrcr aurdq o! nQodoliq. E6cb unLd4 tr,EXprloLltorororpe r(r otrl1ow. nLo(\,(iJS. oto nr,o evdrorldpov olylpo rou 6rtiliou( :x .3 .4 ) .

To :XrtF(t 3.4 owovf!€r 1() 0rIrLr66€E o€dJp4r(r rr]E yQoFULxris dl,t€6auE,not (n€tzoviE€r rrlv nQoyF(rlxrl 6prio4 erft nivaxc -crw6 rou outrd(iv€rL(to (rn6 rov €nre(lvct(xo rolJ,(jrluorooF6 ,4i. E'\,o [6QoE tou o€oeriF(rroqtlQoo6!6Q!l€ r.tc, 6tao ouE rov unol<igr.rv. To xL€!b[ nrcl! 10 6rr o Xr60oSyQa![dv x(t o XdQo( ottl6v €touv rtv i6La brdoraoq | (rlv rdErl). T05OoyvorQ|qolll€ rrlf rov rpooet'etolnld l,l!D\ ftoaAptov lcbgrrv. Arin ur61apor, eivcrLoQooyd1,la ollpnl,lQdlpcrra mov R'/ )ic[ o! d],].or 6io orov R"'. O [d€v61{Dao5rul€rxoviE€r(r fio lrrlbc1,lxo 6ttrvr'opa. Ti:rore 6€v ord],v€roL mo! aetor€Q6trl6€voxrDpo. H nQ(l.fpsrrzri 6Qdol oqr6cliv€r lernln rou 16qou yQq[rdv x(!tou l(i)Qou oll).ri)v x(lr FrroQ€fl€ va qv rataLd6Er€ '|ow(iEovrrq 6vc xunrxobtcillroftc r. Auro El€L tra owtm6ou "otov liQo.fpollfr6v" xcL |rLa ouvlor0ioa''orov pq8€votoJQo" ,) =:rr + rr, xc! 6rctv rrolLcril.dolso0ei pe rov .4. 6ivcL

I ow!m6o(r cttov fl6Ev6lcrrpo oneLxoviletc( mov !I6av: /zr = 0

I ouvlor6o(( oro )(6ao yeclf+r6v c.r€lxoviE€-rar orov Id)Qo ortld)v:

3.I XAoEIA AIANY'MATA l(AI OPeOTCINIOI YNOXO?OT 163

44t

t\

* fiQayrralx.i. 6sv.tuoLEolll€ n6q !d ot &ddo1lll! 6no oeooydJv@ls udroQol5 ouimaqE rraL a - r- Edv xqGr.q6qila€ due! r€ 6@dd@E xdr q! oaodr@dtitrd, F tv dqilds w oor

!I.3.4. H 6gdol ro! ntvdxs,4.

To ndvro, qwlrd, cln€txovigowcr drov Xtilpo or4Lrix -o rl,vsxcE 6ev FJroQslvo xriver, rinme dlrlo- xcr, ro op]pa 6eilver n<i6 out6 oqdclveui

3x' H qJrerrdvlql ar6 ro1| XdQo .yQdgp6a oiov I6Qo otlL6r slrol,oqv npqpctw6opc, awr,org6ryqq. Kdos 6ldvuopc D tou 16poo or4l,6vnQo6elero.! dn6 dvd ria| p6vo odvuopq o roo ,6qou "ygcpp(it.

Ar.<66erF t Ea,r fi D reQdxe[ar orov ,(6Qo oql,6v, r6re eiror owSucoFog :{cr(t)v oql6v. ErIv rQcr.ypdnr.6rrFc crr6 elvo! to ,4r, F€ r, orol,IdrQo Teq{r6!,qqof l ot'vlordoa o@r prl6ev6xopo 6l.vEr.4rr =0. Edv rdroro dl).o 6dvuo}a.r'. orov 1<irqo yqopp6v 6i''eL Ae| = b, riNe A(8, - el) = 6- D =0. AudlI q(6orltoto0sr€i ro rlr - sf tdoo orov pq6w6lorqo 6oo ra! orov l6Qo yQagp6v rar 10ridv€r oQooyci)vio orov edur6 rou. AQc cffi6 el,eq! io prl6C!, 6!La&l c, = cl xo!oto 6 afi€lxovig€ral 6vo rar, p6vo 6rdwouc rou rirpou 1gqrp6v.

Kd0e t ir.txq.t pewoyrynti&t an lieo ygalupdl tou otov Tbqo ongfuin au

>' owol6 torlg r-6rdotorous rnoldrpouE o ,4 eivcr cwr,orparpqog, evcir orovp466v6xopo ro! Flor\,tEaraL Aur6 sivot rrQoqovEg 6rov o nivcxcg -A eivcr,6la'y6\,toE. O tlnofihdrads rrou rr€QL6X€! rd pq pq8evrzri r Eivar av,rr,orQd1lpo<.

9(A)

fv<opiloqr€ nl,€ov ot! cruro crqe€ €r nirr.ror€. E:riolq o ,4r naer nQoS rylvovrio€q

"(tl€u0uvol. cr^6 ro! R"' ^io(l, .ftov R' zar tln6 rov ,?(,4) ri.o{D mov

R(,{1). o!ou.(i. o crldorQoqoq 6€v €iv(.! o ovrifieoqoE! O ,41 a:rer.xovi(erotoorir rouq rirpou;. crlJd 6X! r(t l€t(ovroFEv(r 6rcviiou(rcr. g rqrri atllri avrix*oror' ,4 r. rq ooov rorrroc unLrQXEr -zirl x-'ru0lr I Uovo\' orcv r - m - rr.ArarloprrLxri, eivcr o<rv vo !rycitr€ vcr nirQoutr€ riorrr ol.oal,qpo tov pqbev6T,oporr:16 ro plbevLa6 6LitvuopLu xo p6vo. npirlpcr nor 6ev gropei rc xtivrr, ncrvevaq

'Oxc\' 6ev uTrdQI€r o,4 l. trrop!:irr rcr dprite €vn 9ror"ol.o"116 woxurci-or(rro. Aut6 ovo${iqol(lr t!e!6o4t,ioreoqo1. oW 6ol.ilErqL [8,4r xar uvxlorQiq€rrov ,4 €x€i. fiou crrr6 €L\rcr 6wc1r6rr ,4_,4r' = r yLcr r otov xirqo .lqcrpgtirv. Xtoveror€Qo UIb€\r6IoQo 6p<,r; 6er pnopt:i vu live! rinors: ,4'? = 0. Apa o At crvrFo-reiq€L rov A sz€i fiou (ur6( €t!(lr c1rrlfieaqJrloq raL o uno].oyLop65 ro! 1iv{raLoro n(lQdQrnp(. O unol,o./[o[og oln6s 0q(Q1(i1rrL c;r6 pta er t(]}r o:rou6cLorilrcovficQalovroror4o€ov €1og :rivoxc -r1v raQayorroftoaqo4 r6top6Sqoty ltp6rnQr! trn q\r onoi(r terrhlruL va Erpoupr rc :repi Lbror4rri,v.

AXKIIEEII

3.1.1 Besir€.cr l l | iz l i ( l r ro toorerao trvofrvo r . ,v J =(1.1.0.2)xsLy=(2.-2.1.3).

3.1.2 Adot€ (no! Rr 6!a rcedbrl.4ra avcEaaniro)v b(vriotrdrov nou 6€! €ivqr oe0oyi;!(rrersEi ror(. A{;oi€ .,riots a!( n(adb{yFcl oa0o?oviirt' b'cvro}rdtu)v ro! 6fl da(

3.1.3 Kcr(i nt! avd,utrxrt yroF.tai(, nlo .ne.i.. ro! {nlnd6o! €ildr xaB.ffs. (jr(! rolrldprlo r.)! r.ioodr! ror!.iv( l. E{c0!6oi€ io ora b(cvno}r6a r =(rr.rr)'t(t/ = (sr. sr). r.)v onoi(i! or r{Liuf(< rilor r:r/rL x(r y:/!r 1l(l !' arobriEur rdLL rlow0rtal oq0oytrv(i,rtrG .'ir t/ = 0.

1.1.4 fldrq ./!oei(o4r. 6r I rs4rlr] i r!i\ (vrrirrettuo! riv(x( .a eivu' oe0oyovLo oritn , t L r j r o r B r , 6 t d v i l i l

3.LS oora (ufl uivoL oqooydvra, F.rcEi ro! bruvL,spdrov

3.1.6 Be€ir€ oro! Rr d i ( ra6rc ro! ( r tunot '€ i ! (LoAeoydrr t (m((r ,1.1)7aL(1. 1.0) .Ar'alr.i ra 6LclLlotar( qrrdErr 6!u oioqts oQeoYlnvio! Frraqn ror! zaL Fovobtaio! 6Lavuol'dtov (iv.r oasoxarovuo ouonud) rox Rr.

1.r.7 Ba€ift ird brd!to|( r oa0oydrvro (no! t!(i,Ao yACFfdv. zar 6!a 6ran'oua v. oaeo-ydivLo mov t(lao ('nli6! ro1,

'lil",lll"lll

f r 2 1 lr=12 4 31.

L 3 6 4 l

1.1.8 Edv I/ l{dr ll €ilcr oaeoy(;vLoL uotoQo(. d€iEt€ dt| ro troro aorv6 ro\ 6'dvuola€ia(1l ro u6rllzo d!NFC: I'; n It = {0}.

1.1.9 Berir€ ro oasot6vro oFn),iaDF( io! €nrndbou nor rcqdyoxv ta 6rovnoFlt(r(1.l.2) z( (1.2.3)- 0loadr1'rq t.r dcv fqallllds roD -4 aat iuvow(E ro Ar' = 0.Orlnt0€itr ol ro oaFtri.risilc €ivor FLd oloxlrur sl'osic.

l.l.l0 K(rraozrrim€ rLC orrot€d €Eiooor Ft rofls (yvdrotor!. q! onoi.rc oL ino€K tivoor yo-qqouoi oxv6laqrci rov 6(!'rop(iril'v (1.1.2) aa' (1.2.3). Awi tiv( I(Eidreor4 rI( iporvoltrrvt! doar|ot.. neoq(ads. 6F.8. t.r 5no rao6lritr(tto ri!.rl10 ib(o roa-Itr(.

l.l.ll To e€!L€),(b6€( 0rdJantrc rlF rad[FLalis Aiys6quE brarlrrrbrE@ oulrd oq lqdt(rulnrntukrdi rcaq;j tu,i Frtuth.tht: I'o at0E A t(r x(i0€ 6. dlo rur tt6loa iv( dnor0 snolrlu ouqriFar( it(s( ].nor:

3.I KAOE'I'A AIANY:IIIATA K,\I OPOOT!)NIOI YNOXOPOI

( l ) r l , . = , 1 2 ) A 7 r = l ) . U r b l O

Mr rillu IoyK. rt ro17 i€aLaxui(r oto xri'oo drl).drv "(,4)

1rn/ralFr 6v( y otov,(..1). 6rd, (i,or€ y7 {,10. afiErf dr, ro !c rxolv ru (l) )tC| (2) tcrut6;(pow l!ofr<.t'v(' llvflit(lll{o.

-r.l.l2 llofirr lll(l 6(i.'I ro'r p$cv'jx!,pol' ror

r=tl I l l .' - t l | . l i '

"d( .idiito.um. &r cmi .ia(' oq9oydM( mo! X,i)Qo yecpl6a. Ao{)fwq ror

.. = (3,1.1). bftordole to os rtkr owrm,;dc tor rdpou yqcufri)\,.r, rct triaoulLorooc roli fln0voxopo! !,.

-1,1,13 A(;or. pro .L!o!d rl< 6pcio6 rou A7 pt (rtl r( (!d).oro ro!:X. 3.1, miLrow(lsro! R(/t) rioo mov tdrAo yQ(4Lp(;)v xor rov (arorfad ltl]6€ldtoqo oro fDfa

-r.l.l,l driErr orr ro I y riv(r oaOoyli,vro uro .r + y. .rv ao uo!o! d!, r. I = j/ .

-1.1,15 ua.ir. €vc! rivaac ror oroior o ri}aos 1qc!!t6v nf$ix€L to (1.2,1) zaL ro! onoio!o Fr6.!6xoao( xraralet ro ( l. -2. l). t (lnoh.iFrE orL &v urdexfl tiroLos ri!.rxar.

-1.1.16 Ba€irE oio ra 6ravnowt( xori riv(r xd0tr( rr( (1..1.,1.1) ad' (2.9.8.2).

-1.1,1? Ed! l filar ro oaeoydvLo dqm),isD|( ron II mov R". lrdaxfl xi!ci.5 !e td,qoruqqr(i,r l' K{ rr$rvoiloqo ll i :*rvr;wa. ffi6 Fra 6d(nt rot' l'. 6ri&c rGxcr(oz.ui!€rar ivc< rdroros nivaz(s.

I . l . l8 E ' i ! S . {ur r iv( , u 'noj ( ,dAos iot ' Rr i ru ! xr0, ix ' , fdvo qv dent . f ,u , , ' r iv , t roS-: Edr o S trcady€rat and ro (0.0,0,l). roro eivd( to Sa:

-1.1.19 Ai)t0iq ri rt'!!6!E: d) E{i! o v Filar oe0dy{;Mos orov ll', rorE ror o V'! efvoroa0ol(llroE mov Il'4.6) Ed! o v.ivoL oqeoy6vroE oro! l.l'xn( o ll: rfl(lL oaool(lvlo< mo! Z. ro1s raaro l' €ivaL oA€oy6lLoE mo! Z.

166 OPoO]fAMOTIIAA

Tor0r P ro lrrtr€6o (6Ir l'r6xlreos) too rel6{-doftro! X6Aoo , tE ?Bn{Nq x +A - z =6. Be€lle qv ttldoort evq .n.ni6ou P' raeo],],ifou rou P, o i 6uaxqd1,ou ari6ttn' oef,tl. BQelre rJdo$ 6tu otdfldpa ,.doEro s' aud 'a !Jdne6a. nolog rtvdr{o66x€! ro P' 0x Frpsv4o,e6 ro'r;'Eoto ,S o llrdloQos rou Ra nov risQdler ft 6|,a1/DoFord Ie ,r + ,2 + 13 + 14 = 0.Boslr€ p|a 6dol rou Sr, o oroto( 1rsQldl€r 6ld rd 6rduJopdfi rd oeooy{j)1,la orov

3.Lm 'Eoio 3 undlu:pog ro! Rn. EEryisis r! orpotver, q (5r)r =,9 xa!"rnrl €tl/dr

3.121

3,122

3.2 E:ATE?XA TINOMENA KAI IIPOBO E> )E EAOEIE'

3.2 T EEATEPIKA TINOMENA KAI IIPOBOAE 'E EYODTEE

fv!)Qiloltrs 6rt ro €o(0r€Q!,.6 p6FEvo 6uo 6rawopdra)v r ,(c.! y €ivor o aQr0p6!.ri j/. Mixa' r60s Fqg €vbrirprpr lr6vo ro n6rr ro €o(lrrqLxo 1v6F€vo €ivdrFr16sv $s dlJ,a,!o'fLq, n6r€ ra 6io d.a}llolrqro €ivar oQ0oy6vrq. 'Hp0€ q 6eavq rEfldooutrr rry nFainrrool pr1 p{tvurrirv soorrptxdrv yLvogivtrrv rcL yovrtirvriou b€v €ivor oA0dS. OiLo4re vc xc(cvor]ooqr€ q ot6o4 eooi€Qr,r{od ./!vop6vou

rclr ^/orvi,K xdod)S x(rr rI oIaol €o(|)lcQrxoi "yrlJop€1/ou )ior ov.rorQ6go!. :roK€qd)dlo 1. rorcoxel]do(rpe io! o'\dorooqo, yuQilotrdG dvav fiivox(r, oav vo{tcrv oge}.rtu. Kriu xcl,urepo 0o rp6ner va xdvoupe.

Ed(v o€lrloot]fr vo owot ioow€ ro lln6rloL-Io rou x€qolqiou, 6€v UJIoQoiF€vc( snoqnyowr qv ropcrtqpqo4 otr 4 opqoyrivn npintaal th'u rc).i notl6taEipotou. Yfloeior€ 61 6i6Dror €va onU€io r, orov n-6rLor(rro I6Qo. rouonoiou 06l.ouF€ vq 6eo{p€ fll! (rn6mool ono doe€ioa €ljo€icr, l.X. 'qv €ljo€iaotrll }{crterieljvol xou 6l(rv{oudros o. AvqEIrdFe, xdrd Frixos rIS €uo€laE, roolp€io p 7ro! €i1,or nlnot6orsQo oto Ll. To Nl€t6[ €i!&! orl yeol'lploJ H Etq4hrfrov otvaier No b pe o p (I tjlorcrc$dvl €uoEia oro :X. 3.5\ Elrd'. ltdgenloro 'tdjntopa a. Auti q d,ie€n 0a pcg enzpttper xt 6po1ic o ntrqoiaftpooqptlo p. xatrltu unotroyioorF€ rlv afi6oroori rolj qfi' to ,. nqQ' 610 nou ro6o€6ws 6lovuofaro 6sv .i.vo! op6oy6v[u, 4 IuoI rolj nQo6l,i]ruros n€pvd Faocan' q1, oQ0oYov[6qr(.

H it qrsoraorl eiv(! [6!0 lE er€l\n xord Trlv oroi(I, ovri llo xc,l€i,euvoy] d,bi6rrcr 6vc erbre6o, f 1€vrx6r€Qo onotoobrinox€ 1J[6XoQoE.9 rou R'r. Sovd ro/rQ66llltlo €irdr vo 6poiF€ dro olll€io I ro! !riox6Qo!, ,!o1r rivd ?rtlqordorEQoor.o b. To a4pdo p tivet 4 ttQo6o^i rou b otov uniTapo,'Ornv qlpoq.re pLtr

q. 3.5. MLa Fovo6rdora.rl 'tao6ol'i dro! n-6rdorato r6eo.

':r "u'n,\^, 000 00r

trrz iFo'!!':3 D'ddnlnz.r (; 3'l B0n/ tu rro! rxlr0t'd'3r'h noDnlro,io orD,lra oiuplJi orNoiYtn Dtr)qrlo VUH^r,r jtbtdDnbr,rrx r)trrdtrh I rromauJ lu:( btatnr tet]iu\,,,a)

^Lr!. Drrr{iilriln ^(0n.d,r,r,l$! ^, ,torl,1tr ̂ .or Jo$rlJirl Lr 'hzrorDr! ltr. Drz rxrloôzro rrt *

u 5tu oôxrrlL^Roor lDx oôlrriu o.r. r)1, rdiol(r 'ODO noô).t|(-)r nor Dono^r3rorn Dr. rD^r3 olorlool loDrrio+^Drg oo1 Soztoi or. rD^r llD rro jDuo:pro J 199 X3) xDô:r,)ôr 3 D rdon^rrg ot rrllrDdtr)b nou Drôi I D^rr D uo 3roiooul 1lo(')]rj|drutr]Dro)rlara hrc t gnltlq nld]3 oô)/,rdr ̂ tuo iliilo/".r31j)D1Dx oriorrr)bfto ôlrinodg D^ Dt 1or1, 5nlâ1 'lL, oôtFlh^nn ot ?11D

'o^3r19^r/, o"rl,rr.oo3 oI! 5Dr3gm .!D 1D13!rq^ll() ro]l irl^(')^ [ irtprdxD rD^]3 ̂ rg r-r9 arroo]rorllD1g ))gDdollldJ ^qrlô),I, rD, ^.o&rlo^11 ̂c,oxd oor ldrl! lroulllno lr] lt^ri{3g

zr?,r{qJs ftor rrhr,oor^D L 11L n^srio^l,L trrrdrr(t,o!

'DrDion^r)rQ Dr^.rlogdo,xo (,Jol! JD1ôi(rdD^D od}ooord3u Lrngxr 5norp 1-rot rinoohrotrowD r^r3^!!do!D ̂r!ort)) € t o$rd/,DdD! ^trl() rDrr^nl rDx 5noor3liDdDri JrdrDoDtd3!,rd ̂ nr^(l)IDd13l ômlx'Jyr oeoe?i [m ]rhoDu^?r

'hooxlri]iru lxrr^Drl[o orr lr'LunV nor Uor.)9 D|l rrJoorolodrctrdx lDr a(I)roDloDre ̂(od, oo(niru od(oIorft^t^t .o 3 no1lrrgod]l ^D],9 59v.ir) ,rrrrdrDlr rr^3ilr)oDr .I or nrl :o!n! o'jo)i(nrJlg 53r^cnl5r1 rDit o^3ri9^11 orrd3ro,oJ or ri U.{ogodr! Û1hrnor irlnoo3g^noD0 rD] o(1p 30 o^D)r don^Drg ,î :ix)lrgir Snobolbr)rg ,ri r no{?godri r){)hîio!] ^trlo rDz o,bDdAr)dD! ̂trr trl'nr .3 Dl30m Dri D)^lr jodo)(oun o^Dro o(oxo3 rD^rr '{I o! Dl/, fto! r itrordft3 :tu 'D lupir odir}rg ol

*'trt4rgordniD^D rD^lr itrlorldDJ! 5trr JLrnD DroDriUo H'!r/,oyr!3 }l)r^na Udr4yl)11 ̂tll ^Do / olirltro ()1 l3d(')ao ^.)^(rlDdrr ôur rID13 ̂0)I

Soqooirrl H [o+1 r]Idr)!r ^3Q lDx tror o!l)r?li 5aml)grglinon lr)^t,l Jr3ocoollr ro'ô1D^9gD D^lr ol'nor rrg rrlnorilg nodqrx nol ftornD ̂ rDlDrloo^Dtg ^(ortoDp ̂{')lgioDoq^no ̂ Do I ol iino nd/, t)^ 3inoot€Dlood! ^V odaJ)(o!,r ')r^r0og ôroD)rrl(3rd3! D^ Jo(, LtJDl D.( {o! DdD! rrLrri3! rD? jrio!&ozoorlliq rDx ))D)iDdr3r9)rD ̂ noxj+zodr nor 'r)îdoa3Q nr r3,rrr1o10D! q t)rion^rrrg oI no,ro)rrdBt^fitotIrrF ât ogo|rn llt rd So,'nnlltoao ôqlotdotqoodttd?ta tgg 'ltol.(J|rl t) t.tggdr or ri irDgrdxD orqr ot tD^13 'Dxrd)ii(0i/, o^gtj 3inr[Dd/,]d3r.jtd(J)r rdl?d nou Jnd Ditrygodr oI '1r)^ Ddnollo D^)J :t3o[rl^n!r rO

:d Ul09od!^Lr Dr/, :o!nr Jororpr ri)Gprn nod(DIolln not hopg Dld 3rinolg ̂ Da (Z)

:J9/,orldDd3 i;xrrxDdr 30 Drlhtggdl! or grnD rlDn/.DdnDrri1^Dina (I)

:t6hr^rrnr rDtôiD|3dT ool,513o[t!,x]i onq irDrig,lnoldDr^ S'ôrdqolom rDH q ̂ rDl3rltro olD!3i ôroDroox,Dr1l.t199dlr olLronb olo [o+1 tr'{rrD Drrl rD^]3 lunl) pl{ldl3nr|)3/. :rr^(J)IN odori(9ljnô1 13^ti3x lr),30D? [r no!9 o]3 ho or rD^1r d or !S' oro Q or 9rjl) tlr3epx

3.2 E ATEPIKA |INOMENA (Al IIIOBO E> >E EYOEIE> 169

b =(br b, )

:x. 3.6. To su,l]pitoro ntE ylr1,ios d = l, - a.

A , A 1SmO = -----i. msO = -:,

To i6to cl'.r1Oeuer xo y,o ro b ror tr1v cvtlotoln ytrrllc ro! p: 10 lpli,rovo si,l,o!L'z/ lD xcr to ouvrpltovo eivcr b1/llb . Erier6i r6Qs I d eiro axpL6drg B - a,ro 4pirovo 6i,6etcr cn6 fllv rQlT ovop€rQr,rrri -rollt6trpc not xcveig 6ev propei vcrl.ryUoviio€r:

cosd = cos rcoso + sin d,h. = fiff.

O cqr0prpri6 o' cu6v rov triao eivrn crE6ri4 ro eooregr,r6 yrv6pevo rt,rv b xcla xol 6iv€! rq ox6on fiou llFdpe:

To ouvqpltovo tqE ltoviaE perolt 6to 8lcvuoydrov o xol 6 rlvo!

a I D t l Q)

:nFsLdorE 6rr o rufioE €irar, ooot6g cug ngog rr4 6r.corrioeL5. Eri.v ourl.cor,tiooqrero pr]xog T or 6, r6re 6r,aLcror,nletor, ror" o cgr"opryrig xm o acpovopcor4g evcir tooxl.qFirovo rraQstr&€! avol.l.oioio. An' qv dLlq F€Qr{i, d.ldlovraE xo nQ6olporor b, clidler rcL ro rp6o4po tou cosd, evrir q lovic cl,}'ri(er rcrti 180'.

E1$iaoq YropXa rar dvas dr.LoE xcvovaq rrF rQryovopFrQiag. o vopoE rououvrpr,r6vou, nou o61ry€i ,ioi' euo€iov o-io i6ro oqrn6qcopc. Arr6E 6€v €ivcr, 16oo euxo),oprng6vernog doo o xunor (1), otlox€riE€r 6FoE F{xn n}.€ua6v

170

onorou6{nor€ rei ydrvo!l

l l b - d l l : = l lb l l? + l ( i r 2 l l t i l / ( l l cosd . (3 )

Orav l Tovic d eivcL oQori 6aior.ouF€ rrdh xo Ilu0cy6a€Lo O€60Ipc. AV€EAA-rlra 6pcD< crn'ntv d, rt noQdorcol lD-.r 2 FnoQ€i va ovol.u0Ei o.rv (b-a)? (6 a).x aL 16rr I (3) ./[vsr(l

b r b - 2 a r b + a r a = b r b + a r a 2 l l b l l d l c o s d .

An),onon;vrcE ta btb nu aTa orLq buo rl€rjA€S r1g cliooor6. arayvcopiloprrov Tirio (2) rox oulllprrovou, Xrrlv otoicr. arir6 (I7ro6€rz\.usr rov tJro Touonlrl[lIovolj otr,q n bL(otdo&g, crpo! pnopor.rpr lra rrr:QloQ[olo{[c fio srifi€iorpiyovo Oo0.

Zlrd4rs r6ec vc {iporige ro o4reio apo6oi,rjq p. To o4l€io (lw6 nQdnerv(l Ei!(L zirroLo no),lsn)doio p = r-l? rou 6o0irrog bLovrlollcros tl -xd0€oqF€i.o rlq €rleric.l( €iv(ll ,rdfioto ao)rLorltioto rou d- !(.r ro nQ66L4lo €ivcfl.,'u unoLoyioolJp€ ror ouvx€I€orrl t. To $61,() nor 1pnr,0!6pcrme yi c1or6v ro\rutroLoyrotro ri.vqr rl yr(|)$crQrzrl (lLi00!(. a,r! t stredt afi6 @ b olo ,tLqortorEQooqltio p = io €ivat xdqE'aq oro ariwoFe a:

l t , - ' . r t t - , t . r t 1 r ( i , - r , r ) = 0 . t , = , L | J , ( 4 )

K(r( afi' (11116 6xoupr io! rrlro yt.r ro t rc! lo pl

3H H 'rao6oLitou l, rdlt|) oqv €l,o€io norJ oQi.l€icr! Clt6 tc O

t ) , . DP = E a - - = i - a ,

(hrbxara) - @7 0,

(5)

Artr6 luE rnrrQ€n€r vcr Ecvaole6r.ciooqrr ro lpjpa 3.5 pr. rov oolor6 dno .y!c 10p.

Afi' (ulo rov riro t:Edy€ro! dv(l oEroo+rei.@ro /r6Q!o[a, rio!' n!e(,!d]E €ivar Irro tv6|oqdQouoc olro6rrp<r r<rry Mcr0qpcrr.x6v xdt oqtrsAd.otr6dv€! oov €r6rxirroinro'oq ro yFyovos oT' o aprHFtrrxos Fioo< (r + V)/2 rivo' FEyatrurrAoE androv yco)F€reLr{o v,5t. (Arnrt €i1]sr €rfolE looal'vcpl -b€ift qv Aoxryq 3.2.1p-[€ q! rQLyo)arxt ovlo6rqr& yLd 6rcA,uoFara.) To nopLopa gniveraL va lproxrinru,oXE66\r ox$rrt,rFot!1ld, dn6 ro.TeyovoE 6-n l mrQsyovtoF€vl or6orqol lllt ?l rmo :I. 3.7 i€! F roQ€[ va efvnr. cpq'Lrl:

("r "\

>0 .

:]2 EIOI'EPIKA |INOIVIENA (AI TIPOIIO E: IH BYOHIET

b = ( b t , . . . , b , \

b -p

u r t > x 1

3I Auo oaol(rblitroi€S.hrvan

6Luvriopctu Lzsvonotorlv qv trvto6ltltrt rou

lo16t < l la l l l l r l l . (6)

rx.3,7. H :r0o6oii tou l, sni ru! u ltr "o', = 3f = d+h'

En€rbrt o r€l€r,rcrioS cQ$[ryrk 6$'€ivat trori ogvrprn6g, 'Xonp€ (l'"ll)kTa) >(orr)r -xco rclielo\'r(E rr.rQaY(ol'ords Ptq€E.

!ryri{'rorl : uFqtDvu fF ro! lnno (2}. o }.o1o! Frruqu r(l)v buo ntrfuct;! Iff avr-

oointu< rou Sclwarz rivur uxpr6d-r< lcosdl Aeorr 6L' r(r 'ruql'irovu tt€Qrrlovr..l

" ' , ' , t ' , i ' " .nu" - | r (o 'd < I d,"ro nivrr urc ci) 'Lr l dlr6berEq r l ; (11): \ t t t ' to61\ro

rtr Sthwar tivar 11 i6n pe r4r' cosdl < l. Kdtd tdtroro lQotro, s1n4 €tvat

rro euzo).ov6r1o1 ""o1.,E 1 tratl rct ow4ritovcl €ivrrt mo otrreia Kar o! 6rlll

nnobciler.q Lopow o1o1' Rn, olllerdlot€ tDo-r6oo 6tr. I 6!x4 Fdq uvdlolov o€

d'cA, curi,6 uao)'olLopo tou I D /l r rou ritrw gl ((pl,lFuto t'(! 0a rcQau€iv€t Frlupvqrr6 6tcrv cgy6reqa erocyipupt vleg 6rvcl6rrF€E Yta $!lxq xar €o(|)I€Qrxcr

yrl6trev{I. 'Etor, rl arLoorllo xou Schwarz qfio6€t7'v6€tcl Xoptg !c( €rl'rol€oiou|r€qv rprloroperpicr.*

* M (d,i .|! c!rd6q( a7 b < lLdl l)1. drq6irlu{ EnioE ro 6!o!d ror clucht 'u( o'

P.noor F.iirm( .lr olq,iaota d(v drdot|]fu rrN Cauchv Schrvln Buniaro*rkll ()| 'moauoi rd

M(O$r(rt"nrv q{i!r1(( va dl(qovorv rtE I (!c(,o$i otu! Runi owsiv drdnorQi!'r( ml!

OPOO|ONIOTITIA171

MLa I€l,rrt floaoxrtalorllLa qv lalbl < lal ,b H rcirtlrd lq|ttt Edv xal

uohtv t iv 1, t b t ivat t td)ort) ' , ioro t t ' t o H luv;u t ;vLrr d - r l ' I d - l ) iu xar ro

ouvnrurovo rn: I r l - l . : r r1r ' iv rnv ncarnru,oq 1{) b roul i l t rur Fr r l l ! nQooo^q

,n]J p, n" ' t1"no-"o'1t ' r rcEu rou L xur tnc ru0! ic( r ivor !rDcv

nAPAAEIIMA rlpo6tiltcrre to 1, = (1,2,3) oql' sxoEic 6rit rot] a = (1' 1, 1):

tLT l , b -' , 'n I= '

H rqo6ol't eir.rr p = 2a = (2 2 2) To ouqllxovo €ivaL

., P "\t

H uuoolftru Schwarz r ivot , ,r ' ,1 3.n' bt y6 S,nt/E Edv loc'+'ouurro 6 acrv ,..46. d!'ro rivat roooilc.Fo tr€ tlv y'Iz ' y'14: ro oLtnurrolo ervdt

[r1(Q6tsQo ro! 1. IoX!€t n (rv|o6qra 6161[ ro b 6ev €lvcl TraQt(]J'n]'o oro ('

fiQo6o1'66 triEeorE 6vs

H npo6oLrl ror I nchto otlv cuoei,a 6Lcl tol lr' ei'\c]r p = a(a'rbfo't a\' Ar'16a

,,n,., o iinn. lrug P = 7d. ypL4tper'o: orc.rE )iYo orQu6{i: To brflluotlo a 'Ztlro:roeetndui nttv ro! ottdto 7 = a / , d' o YndQXEl iv(s IoYoE firoo Lrno curn'.4,1n,uo1,.u,na totp,uytvl. cltr.ryrl H :19o6o).rl nrrv'u oe Frn €!0tiu trl€)Fitctl,r:ro evuv 'zivaxa zon6olit" P nou qu!.Q!,rvtr(r ora! r4 YeqlioljFr F outl rYl

6tctrc4:1. Ebat o ttivixug nox nd')nn)aouiEet ro h xat wrpayer n 7::

(7)

Aur6E €i.\,ar FLd oriLl €ri gic ypt4rprl 6rnq terp(ltdvrxo! nivcla dE- bralQep€voE

ge tov cpr"0po a7 a.

IAPA EIIMA O trivaxag x(o! rao6d)'L€r n'ivto orll' eu0€io 6Id rotr 'r = (1' l' 1)

Ei.v(1rt \ r 1

P 'u, ' - : l r l l r r r l l i l ln 4 r l t | : : l

O :tivarcq art6E 61er bno rbL6trFeg, tror Oa 6oup€ 6rL eivcr xr4arrlpr'orL#gT tl)! nQo6old)v:

(i) O P €ivoL ouFB[,[arz6r ri.v0x(lq(ii) To rflptryo\'6 ror rooirar F€ rov €arro rou: Pr = P.

Airor€).€i. sri.oitq oro!6cio ri(Ad6€rygc 1cr ilv norav6lol rrDv r€oociaov 0E$rLt(r&i)! unoldp(Dv:

H tit(q tor t:ivcrL r = IO Xcbpoq ord.rirv to! c'iorel,eircr c;t6 rlv tueeia SLCi rorl .r = (1,1.1)O $t6cvo ')Qoq xor c(noreLeiroL ccr6 ro uinebo ro rdti€ro oro o.

Kiroe or!11 eivor noll,GnL(iorc ror a. oQC io Pi, repr,ele-r(1r ortv €uocic 6r(irou d. Tu bL(lnolara ror nlxr6dl,lovx(r oro 1, = 0 iZonv tbLc -repo evbrc{6Oov.flprjztttcrL yLcl rcr bLcrwoparc nor Llicrorororla rrl! (T, = 0 -€i\rdr zii0€ro oro( 1tc[ r] olrvtor6o.1 rou; xrrh pryoE r1g ru0eitrE r:ir([ l1l16iv. flriQralo\acu Lo!no\'trro rd0ero eri,7rr6o. irol ei.vuL zcrt o pl6er6rrpoq rou ni v<rr{a ;rqo6otrrl< P.

:rIv rQoltrftrr6qrc ro naQir6sryFa crr6 ra0(l!:iv(r ril,rLo. O !qbev6XoQoi€ivcr xri0lroq orov X6Qo orli.6r,. rafl cEoxpo€v!1l6. O $I6er61!rpoq 0c inqrrevo ei.v(r rdorro< orov X6Qo l,QcruB(i)v. Efirrb{ 6$!)E o P €ir(1! stutlcrQrxoq. o1r'rq'sc r.rr4trr'rv rotrlln(tt Fc rov X, rgo yt-,,{11! 'v

naQffi;letlorl ttTlv allsyi xlipaxo€ Ect\ xo 4 6rd,croloot)€i. o riil,({&( nQotio}.rlqaor / 1a f i (eopdvcr o t6tocl

]2 E>AIEPIXN TNOIVIENT\ KAI II?OBOAE':E EY6EIE: 173

fiiitt i i i lL ] ] ] J

1 l?lnl;lll

H €ni)ei.1 6rll-rou.r sivar 11 i.bLc xcr totro eivcL ro g6r,o fiorr €\r&o(f€peL rovri!.r,{a nao6o),Jtq. E('rv ro a iloL [rpoq i:.vc, r6te o ncrpovopnorrlg ei.vcL aTo = I'{.ft o nilaraE r.iEtr,rE ir,o r:ivar axaL6drq P = oar.

IAPAAEIIMA 2 IIQo6o),rt ,{(r(i r1v xoreuOrvol d" mo €ninc6o r.y. H| ' r0f u nrpva uno ro, - 1.o'd. rrd; rru o r ivox,rE r ivur

t.t.' I t " " l - '

l s . < - cl : +

' l . , l l c s s lr , . | l

l - l

EAdr .r €i .vor.ro cosd, s eivcr ro sind xar i+sr = i otov ncpovouconj.To! Ifvoxo J] tov crvnxcr).urlrape :rpo41or+r6vtoq, oq! IlcQdlQoqo 2.6 r€AiyAq4rLr6\' F€roonB(lrrprirv. Mnopol[e rdpa vc npolcop{oo4re ripo aro tlv

1 7 1 otooraNlo lHTA

:rpo6oLi rou €ruri6ox r. U o0 trLcl €uosid zar 1a r)noLoyioou[€ rlv riQo6o].ri PyLcl tw\' oroLobn]_ror€ aer0[6 SLoordo€ov. To\riloup€ ridi.! 6r[ aurol na8o1€!rlv Jrao6ol,rt 1,:

In tqv fteo6o1i toi b aro n, ftoJJatlaoiaa€ pE P: p = Pb,

O avdorQoqog svdg ivoro

Tr).errirvovtcrg, snmpiqroqre ororrg crvdorgotpoug riltlx.q. MdIo rdpooptltrpc rov ,4T crvcrxl.riwa< rov A. og rpo; tqv zripta bLayrirvt6 ror. dtoL tirorroL ypuplti< rou,l v,r Tivow nrrtlr< rou /7 zur tr,'tiotpxrqu. ME ,La tr.r/ta. roororlrio ror r1T o1l lpcrFFll i ){dl orl or|iLl j riv(r ro oroLl€io (j. t) rou A:

( A t ) i j = ( A ) j i .

O ,rluorQortoc ilr'r Fr'r 6{r0r,.rrQl oltruoi( nou neoifxrrc{r ftno rl orfvrl ox€o4rorr gr rc eooreqrxci yrv6pcva. IIBttpcrL. l o160l ol1t [noQti va IplorpoTrollo8iy(r r( oQLoo€i 6vc(( 1'6(.)( tl(r noLi nlo "q,qa4r€vos ' oolollo! rou avcorpogou:

3J O sr,dorQoqo6,4T progei v<r oEo0el an6 rlv en6gev4 r,6L6trpo: Toeorouprxo ytv6pevo rou Ar ps ro !/ loo1rro! [€ to sooreQrx6 ?r1,6[evo rou cF€ ro AT.u. Tufi)rd. qwd (lJ1L6S o4raiv€r orr

o oprotr6q cu16< tIfl 6uo ot6loug:

(i) Ma,< 16€r :rtiq v( xplorFotrotorip€ rov c(vdrorQoqo. 6rgv aoipvoupe toEoux8arzo 1Llopsvo p' ila 6[(pQ€x[r!6 rQ6no. Arno djroarri L6r.afreg rnlpnoicor lv rrpi : rruor l rrrv l r lycbrx' i ,v aer0pi 'v.7ra rouc onoioL! ro v, :o ro,urrprxnyrv6Frvo opi!.er(lr orrt! n(lp4aaqo 5.5.

(ii) M(! biv€r rv(l! (|),i!o rQ6no vs €nsllo€uoolrtr€ rov rurro yLa rov avd-otQoqro €vo< lrvoF€lo :

(AB)'t = Br A't . (9)

Awo €fl686cd6v€t(r €qrsQ[6(ovrdE 6io qopiE rov (8). ne(;)ra y|a rov A i.(ttzardrlLv Yra rov li:

(A:x)'t u =:L'r'At a = t:1 (A'ra). (8)

(ABx\t u = @x)r(Ar !) = x:r (B1 Ar u\.

],2 E:OI'ETIKA TINOMENA KAI iIPOBO E> )I' EYOEIE'

Or (v(iorpoqo! lpdqovtar rar' ql' orQoqq irdttotyt or4 6€Ed rd.€uQd, 6nursortQt66g or (t'tiotQo.por rivcxeg orov ovd).oyo rino (r4B)-r = B ',4 t. Oqri-EotF€ trd?u 6fl o[ 6{cl ornoi nlaor owfuc!6pevor bivorw rov c[Loo4reirrrroT riro( A t \ r = ( A r l - t .

AIKHEETT

J.2.r () Aool\{(t)l 6!(] esnr(jjl oqt0rdv r r(L y,0faei6axb = (\/C, @\lata - (fi, rtr.Eq,oepdsr€ q! ovLo6rnro rcr Schw{z xaL oq1{eivar€ ro1,.rAr0Fryrx6 Faoo (r+y)/2l'. ror y.r4rerALrc l|€oo v?t.6) Y:ooimr or ErxrvdFr Fr ivd brdvuotru ar6 qv..Oxri iurs ro o4rFio.f xorzarorrv rreoo0€rolrr. 6va 6rdvuolcr lrixorE yll owleovrq rc.f F€ ro r+y. Hrpitq nl€rled rou iqLy6vou rcift an6 qv cqri oro .r + g. II tpqauni avo6t4taLotltpietnt .jtt tt aa.jnaot (tt'i 6tr Foe.i ra.irot tltyai{rce| oa; ro igeootatue 6no qti)rutl

r + Y i < r l + l Y

Avol0€ii€ otltr (1 o6ritrd schwnp. trrqqovLloras t'E blJo rl€leds kor orarftio-uor.ro< ro (J + r)"(r + J).

1.2.2 XqrtoLponoriwq rov dflo (5). €rd,r0rtor€ 6 to rrlixos its raof,('lrE €irclrl p = l t l l c o r r l .

-1.2.3 flom ro .sr].dorc tou o = (1, l l) eiv(| n],ryrtmsao oro oru.io 6 - (2,4,4): Beeift€niofi io olltrio tI( Eu0€ics brti ror b, aou eivaL rlryrimreo oro a.

3,2,4 Efurnor. yrori 1 avtodtlru ro! SchwtE yav.rcr lsorrlT( mlv JrrQinrlrort flou ro .rkoL, n€p#row.lr otrtv i6ra €ljo€ic 6Ld flts darls rat p6vo! o'ouril, lr ftrei,rroq.T; our,6diuL otav r.$i1ovrd' .rdras,efv iIS dqlis:

J.2.5 flord rrvio (',(r]toriger m 6rdwo[( n br.rordo.o,a (1,1,. . . . l) F. rolE dEov€s ro!rrrM€iqlG{nvi llorq siv(r o 7iilaxus rAo6olrk P ndlo o' (u6 ro lr(ivnqLc.i

1.2.6 Ot(! ra ( adr l, xavovLxoroq€oDv fx nov nqordEDv. I (uo6ntr(r ro! s.hwazanodflzlurlcr oc !r(1 Yao$ut:

arb=La.pi .D,, i b:r <D&58 =;- := " t

nora aft flc naorlyoup€!.( do'{io€rq 6rxoro}.oy€i io p€ooio 6'irtd;

.1.2.7 A'dlalorra! ro 6€Ero brdvuora b oqr @ro6qro lox schwM. aflo6ciErr ftr

( a l + . . . + o , , ) r < ? r ( a i + . . . + a ; ) .

I]oc Lotne I @oqrar

-i,2.8 To F6aro to! leeolforJ cH{ sivd 6(el8eqpivo oav vd rtro! ro dropo ro! dleaokamt FaoI lor ra idsoea.r drorro ro! !6Qo"rovo! arq roquqis. E{i! oL rcewi(roroe€qeolv ma qre ia (0.0.0) . (1, 1,0) , (1.0.1) rat (0,1,1) -cr tFeLd,m€ 6n zaL

OPOOI'ONIOTHTA

or iEr qxpds irow rri,os r4, dpa r" rered€6ao €i1,or ravovrr6- nom sivsr roo!\ruimvo nE yo ds f€IaEn io! axrtm! nou flivc uft6 ro,r€vrao (i,;, j) nqosrq xoal-(pas: (H torlia 6€qori, 6vqs ndlLos yv6q$os @v xlFn{nv. €ita! ftairo!109.5' . )

3.2"9 T€rqayovior€ rov nL\o:xs I' = aa'r /aTa no r€o6dli$ nd\,o oe ILo €uodq '|(6cift€ oll Pr = l,. (flaoo€Eft rov sorgu6 oro oro pioov ror nivczc aa' ocT!)

3.2.10 Eilor o rt1,o1{as reo6o).rt5 awrmetvuos; rrori li 'rrori 6xrl

3.2.1I ft) BarilF rov ,ri'.,'tc nao6o).,8 4 oq! ruoria brd -" . = lll xqod5 'or rov

niro'|a ./)2 no! nao6d].).!L rdvor ort\, €u00kr qr xde€q o"ro o.6) Ynoloyior€ iol! q + P, '.or R P, zor FEmiolr.

3.2.12 Baeit€ ro! ntvo'|a no! iiao6 J!| ndoE ortpEio rox Errintbou rdvo oq! suOcior + 2 ! = 0

3.2.13 Ano6riErF 6rr ro 'lrvoE'10! P =&'t /eTa -Jro!.ivdr ro dgeoropn ro]v bralovi()volor:(€i(d! rori rLrar ndworr ioo !e 1.

3:.14 noros niaftos P neo6du,€r xdtu on}|€io ro1j Rr n.iw dqv e$osiq roFts tr,r,rnrrr66rDv o+v+t =0rcr r I =0i

3rl5 A€iEr€ orr rc tifxos rou /' roovrdt F€ io litxot 1or A't E.6r.r\ AT A = AAr.

3.2.16 YJTo0dm€ 6rr P.Laar o nivs'rsg reo6olrk r&o) oiit! ct]e€Ld 6Ld io! d.o) rr.n{ ro €oorFerx6 y$6Ftvo iolJ r !€ ro Py Loovtar g! ro €s(l)tsalx6 tr!6p€vorou P.r !f 10 s:6) Eivar or 6no 1oviE6 or i6Esi BQelrs ro orryFirovd drar a = (1,1, -l).

" = (2,0,1)

x a t u = ( 2 , 1 , 2 ) .y) flod ro €o(oreoLl.lj ylvdFevo rou Pr l|f ro I'y etv(rl ndk ro i6ror nor( Et\,aL ly.)vio ll.raEi alll(i! rlur bioi

3.1 ITPOBOAEI (AI NPO'ETTI:EI: EAAXI'TON TETPATONAN L11

33 I IIPOBOAEE KAI NPOIEITIIEII EAA)(ETON TETPATQNAN

M6pL rcbqa, 6vq ouoiqga At = b ei.re iler prc l"rio1 ei,r€ 6€v 6X€!. Edv io b6ev :repdlercr orov yirQo oq).6v R(,4), ro onoqFq €ivql Ft o46!6aor6 r.or, Ic,rci.oqrl Gauss a:ronry11riv€!. Aur6 oqr6qiv€r, ox€d61 oi.youpc, o' 6vq oiorrycp€atx6v €Elodoe{Dv F€ tris U€to6}.rFi. frs rroQd6erlpc, ro otorqrc

2tt = bt

3a= In4r =b\

eo L[v8tql, F6rov 6rdv or 6eEraE n],€!Q6E q(o!v l,6yots 2i3:4. H ]"rioq c eivcroiyor.rQo Uovcrbrxt, eq' 6oov uridQls!, x(rl 0o uridQ)(e! F6vov 61cr! ro |t cl1]ol orn!i6r.o eu0elq ue rc 6rdvuoucr

noQd rlv s6uvogld Luq€dE rou5, firlv r[QdEI oovawdvrs! pI oq.l6[6qo[6g€Erodo€ls rou €iva[ ovdyxl vo luooriv. Mla 6uvcrdrrpcr €i!a0 va rQoo6[oQi-oouF€ ro r l(Qrptllonold,was [tQoE rou ouorruraiog ,{q0 oYvo6vr(5 ro l,n6lorro,nadyFd 6ioxoLo 6lxoLololorlrevo 6rqv xor or ?rr €E!o6o€6 nQoiQt(owor ar6r1v i6tn n1yi. Ami. v.l €nl6t(bxow€ driQi6aa oe p€Qlxas €€1060€6 ro! p€'ydlaoqi{l+ata otK un6}.o!t!EE, €tvc[ fto],{, 4q},rix€Qo vc 6td,t[oupt ro r nou €La-Xtotonotsi to pioo oqd-trpu orr( m EEn(boELt. YfidQXow nol,tlot rQ6rro! ./ooQlooeL dvds r6xolos p6oo5 dQoE, qlLd o flo r.a@^lq^os Eixar tu dqQoropa ftor

E! = (2r -b)2 +(3r -k)2 + (4r -U)2.

Ed1, undQXE! Fro drQ!6rig Lrloq tou oc = b, 16r€ io etrdxroxo oqdlpo ehs! -E = 0,:qv r o ,rload ?i€Qintoo4 nou ro b 6sv €ivo! a1,dl,oyo rou o, rl ouvdQrrlol -82rivdr tlio naQ,a6oi.t Fs €ldxroro oro orueio or(ou

l2la= l3 l .L4l

Ai\,owoE (l)g rQoE ,, 6loF€ 'q l{ol €Iaxior('r rerQa'f6vo.}v rlg oc = b nou €ivd!

- z b t + 3 h t 4 1 ' 1-

2 2 + 3 1 + 4 '

:rov qQl0prlui ovalvoqileL x<rveig to a"b evti; fiov ,rcQolo[ooni ro a?tr.To i,6ro oqr6ai.vEr xa! pe q y€vlrri trEQifirool. " ivolFs" ro a:r - b

€l..txr0ronod)vrog ro

-a;=21(2t - br)2 + (3.t: - h)3 +(4.t - D3)41 =0.

E2 = lar - bllz = (a$ - bt\2 + . . + (a^c - b,,,)2.

Ol)OOIQNIOTIITA

H noQd.f('ryo; rrc t? sivar FI6€a oro o4r€io ll(t ro onoio

( o r . r , r l a r + . + ( d , , , . r b , , , ) o * , u .

E),alLororloLouF€ rlv (!7i6or(ol c1lr6 to 6 orqv n0ei.c 6Ld rou a xclr o cr[€!Qo,ott*6s ),oyroF6q bivsr ryl'i6[e afidtrrlol tt'arriv nor 66are nqoryor$evrr4 r11€0rF€rQi(ll

3K H 1'60{ iorv €l,cq(foro! r€rQaTdwov o' Eva rQ66},rud ax - b F|Idelerc6lrFlF €ivo!

- aTl)

B),arrer8 6rr orw€16( enlorQaqo!$€ orl l€orp€rQnrj eppqvei(r ror rqo6Llpa-rog .Loxior(l)v r€reayd)vov -ro !(r El,(lr rofioflooupe pxrv un6o-rao1. Il<rpa-1,.r1i,!orr(g t1v Er xcr elrotirvowqg rlv r[(rQ6/@1o tl€ lt46av. tAryrponorol]F€@r€r8oottx6 l,o.1Lop6 1Lo vs €flr6r6(Ldooqrr fll ys(,U€rQic n6 lpo4yoqr6r'4Etr(rQ(Yeiq,ou. Eniolg rl €uo€is ,rou ow6i€r ro .l [€ ro p nQ6rirr 1,c( siv.r! xcioerrl

aIP, aal , t 6 : , ;+,r , ="

:o\r be!.r€Qctouoq nqQqrrlQlol, oll€rdor€ rI1,r€raL$i:vrl rr€Qi.?rr(Dol .1 = 0.'Ol.a i( nol.Lc{n}.i(orcr rou a oivcu. plbdv x(L 11 eu0€iq €xell),iqrr(r o' dlo orlp€lo.'EroL p = {) rivo! ro B6!() 1]noltrlqro wr tqr rqo6o}.rj. Qoxooo o flifioE lLcr io/, avc Xo)oi( vorllla 0/0, carcvo*Ldt oorotil 10 y€yov6s orL xo rol,L(r'rLcioro Itiv(l (.brlv((Iov v(l rQoobroQroo€i. 'Ol,€S or rlpt rou ;u bivor,v ro ibro ),a0oEE = l0.r 6 i xo 1 Er eivcL [rcr oQr!6'v"r(. €oO€io (vri 1c ;rcpu6o],r1. MLoo',.ontlorqta lIE l,ruboalrroxaoQor, oro Ilc.Qie1l[a, rivuL vo crrtLororyrioerrAnor(r rrFi mo X. r(! o' (uriv rlv repinroo{ 6a arrrolorxouos rqv qul t = 0.Truri nou. roul,(ixrmo!, qcirerur, v'orote).ei pL(:rro ouFprptx{" enr}.oyt aJy'orYoLov6inorE dL),o! careF6.

flQo6]"iltlora Elalioi{rv Txpclrirvorv pe mpr.oo6r6Q€E F€ra6trF6E

Elpoote :vl.eov €ro4ror yt.t ro oqpatuxi 64Fa tqE nqodor,ilg rou b o' ivavutritol(,o. utti qS nao6ol,{6 oe eu0eia. To nQo6},rya crur6 npoxumer toq eErlE.Ytroe€ote 6t! E€"rvi4r€ rr:ro qv Ar = 6, aLLr'r crurl q tpopri o ,4 eivcr ivc6rivaza< nt eni rr. Awi va e:rr.rp6rpqre iva Fdl'ov dyvooto. F'rivcl Fovdrr6btitvuolcr-oolLtl d. o rivczcg eler rcirqa n mr1Leq. Oeopoige erio$ 6rr o cpr0$6gm r(D! tl(tQfiIQlio€(IJv €ivcrL pelcdrriregog ar6 rov aar0po fl r(l)v cr"yvd)firDv zal

] ] I IPO ] ]OAF : KA I I IPO) I f I ' I ' E I : E , \AX ] :TAN TETPA |ANAN

6ror ;rrp4ri:voqre 6rL to ,4J = b 0c tilcrL Fit ouF6L6N1o. nleaviq dlv ea L)naaXEIe 'oyrj rox r nar la raeliELr u42r6Li< pt Ta bE'ottiva 6. Mc dD,c L61Lc.:rr.0crvri;: to bLirrroprcr I 6E! 0c 0i.vcr oN6rcro$69 rorl orlldrv ior A, oLtrd 0cFivar !zr6c rolr XdAor nrlldrv.

Kar n(il.r ro :tp66irpn eivcL vu e:rL),ilorpe io 7, 61L.'[ d)ors vo E],cXtoro-i<,rrjooltrF ro trqirlpn zcL I e),cTroroioilot crunt 0G 1i.rt:I tlr: rrt $€0o5o

.r(l)v

rlnxiorrov rerpcrTtbvo;r. To orli pcr €i\r(r E = IA'r" - b , ql.o6{ 4 an6otaa4ror b .tn6 ro a4ptio At tou T6aou o7416v. (O!$10€i.r€ ort ro Ax €ivotor lv lucrotr6q r0;\ 'or l ldrv tr€ or ' ' , t€Luor6!.rr . . . - , r" . ) Etorrolc (1rdpe 11 Luot lrnrv rlcrTiornlv r€racldrruv 7, fiou EicXroror[or,€i ro ]i. [iaar ro i.bLo;re ro vcr

hrrilLe ro otpeio 2 =,47 nor .i\r(t nLlordortQo oro b. ar' oL(r r' duo oltr€iarou Xr,)0ou orl)"rirv.

ttlr(ipos

f,; l

t.,l

ori),r 2 =

rx.3.8. nqo5olri otov toqo mrl{nr doq rivora 3 €ri 2.

frc !(1 tlno),oyiooul€ io 7 lr]IopolJFt !( lQno||lororitoouu€ y€tDtt€ral( IcreLpoorrz6 lo1Lop6. lrlg ?r 6Lcotilo€L< nQox+rougr ro 06).ylrpo rIE l€ort(€-raiaq:ro 2:rpi:rrL vcr tivar "1 rqn6oltj tor b:rcivar orov Xtipo mr1l.tirv". To 6rdvuoparou otpri),patoE h Ai ,tQtittt ts t&at xdqno o' autdv tov ybgo (2a.3.8).

o rlnoLoyLou6( ro! t xol q; nao6o)'rtq p =,'1.r €ivcr 1600 0€Kld)61q. 6<11€rov xd.!o!p€ [E fu]o rQonouq:

1. Tu 6rcvrolrcrr(. r(l zdro€ra oro! Xdeo orl].(dv. n€ar6lo\'rct fio! aQ,orfi(xju46rd'yapa. Er.at r.o 6Ld!!oBa rou oqciL$aroE , ,,17 np(ner vu repdzerntoro! [16€,'610)0l) rorJ ,41 :

A' (6 ,4r) = 0 ,i , rTAr=ATb

180 OtOO|Q^"IOTHTA

2. To brdwolo ror oqcillrcrog rp6ner vc eiv.rL xa9Exo at. xd9€ od^rt rcr A:

l,,-,.]'ol(, A-) =0

'.

oj (6 ,4e) = 0' [1]

.[i:],1:l

H te),erxaicr €ivc! 'rol! I €Ei,oool /"(6 A]) = 0 i A7 Ax = Arb. 'E tc!--rpitoE rp6nog sivtrL vq ncpoqrt prpLxlg rnpolcblolg qg E, = (,4rr - l)r (,42 - 6).Atn6 6Ltet 2Al At: - 2,,1Ti, = 0 evd o ou'ro46repoE rp6:ro5 1!o 6o0av o{orrlpc,,llr = 6, ro! 6ev l,river(r\ tL\sL K( fiol\ctfilaoraootpt cr.xptatitg pt tov Ar. Kato[ tQslE trurdq roo6uvcrpes pi0o6o[ rcQdlow tov retprryowrlt6 fiiverc AT A,lo! €ivct o![[€rQrx6E (o avdorQoq)66 rou 6cv €i\rdr o A,47 !) xar o e€[€],t66q(ni v(rnc< crljro! rotl ri€o(,t'dior,

Or eEro6oe|,g AT Ar = ATb €ivcll y!o]or6q oril orcrrlmlzri aa\ ,,xsrorrl#E

eErc.!0oEr<"1

3L H ldorl €ldxloiov xerqalcirvorv ev6g pq oup&6coror1 ouo-rrporog ,4u = 6l€ rn EELodloe[g ro! n o1!6o[otE uiovorot€i rl]v EEio(l)oI

Ar ,4i = Arl,.

Edv o! orrll,€E rou ,4 slr,ol lpqrpr,x6g alelciqtrpeg, r6re oovTro[Qeliltlos xar

i = (Ar A\ | Arb.'Ers!

I nQo6olri rou , n<ivco orov 1<irgo orqL6v ei,vor,

p = ,4i = A(Ar A)-tArb.

(1 )

A" A efiat,

(2)

(3)

Erd.dyo4re 6vc rroQd6€!4to 1|"c ro onoio I 6raio0lort poE €irar ro i6Lox0).rl lr€ rolE rurroljE:

En€r5{ xdr or 6no o-[i}.€E rr-I€Ldvouv F€ FI6€vD.rd. €ivar €ririol.o va qavraoooitr€6I|" o xipoq orl)d)v rou '4 eivar to erineao z, g oror rp!6uiorcro Xdpo. Hri0o6olrt ro! 6 = (4,5,6) 0o €ivo! ? = (4,5,0). or r xaL g ol'vror{i)oeE rrcrQop6vow

3I IIPOBOAEJ' KAI TIPO>E|IDEI' EAAXE'TON TETPA|ONAN

o[ i6t€s €v6 ro z = 6 eEaqov{Eetol Au[6 srrl6Ecorrirv&ol xcr an6 roLE njriotE:

ara-lt t ot[ l 3l tz sr^ ̂ -tz 3 oJ li i l=fs r:J

i=(A't AtrArb-t f - l l f l f Sl l i l = trtL _5 ' l L2 3 0l L;.1 -

Ll l._ fl ?l rzr l l lp=A== l r 3 l l ; l= ls l .

[0 0J L' , r Lol>' auri rrlv -€lbuoi rleQirt<oo4, to rc?r.ureqo nou FtoQotge vq xdvoqlt atva! vcAuoowe rK iito nQ6i€S elto<iroeq tou ,4.r = l:

! 1+z t x=4a t +3c2 = S

Oet+0or=6 .

AQq .l = 2 r{q!.2 = 1. To ogdlpc oqv t9[t1 ellcool shdL avol|cou,r(d 6.

nae@il/itdl4 l Yfrogtote 6tt ro , rleEdxera[ otov X6eo oq],6v tou ,4, 6x],q6tt€rvar owouaolrds D = Ai. r(l)v dtll,6v, Tdre 1 ,go6oLrl rou 6 el.vcr to t6ro io 6:

p = A(Ar A)-t Ar Aa = Aa = b.

To rl,r;o#oteqo olpelo p rivdr ro 1,6!0 io ,, JrQdlpc ovqrev6gevo.

n.reetltpnoll 2 >d,r dLl,l qr.Qal,a reQlrltool, 1)noe6ot€ 6rr ro 6 stvql rdoerooro.I(oQo oqLdn,. Enedri eivoL xdoeto oe 1.d6e oflllq dxowt 4"6 = 0. doo o,aurl fllv nEQintoo4 ,rQo&il,lEqr oro F46€vrr{6 odvr,oUol

p = A.(A1' Af t Ar b = A(,{",{)-ro = o.

naedqewq 3 Oru\r o A etvqr tetQcyr,rvLx6E xcr cr0orQ6uqroE, o x6Qos oqld)vrou €uvdr o^or{lt.il€rog o l@Qos. orow rad0E 6dwopa rQo6d.tr}.etqr orov €out6 rouxo ro p roorhcl pe to 6:

.p = A(A7' A)-1Ar b = AA-t (A\-t Arb = b.

ADnl vat 4 lrjvq ,ttainao4 m4v otroia pnopoipt va aya)iooDtt€ ro (At A)-1xat va fo yQtitpot)ttl oa" A-t(41\-1. Otav o A vat Mpal44tuiypaltpogTrvTxaE, auro ttvqr aoovdfoy.

182 oPooroNroTttTA

Ilaeadpqo| 4 (nAo6ol"1 oe €o0€lo) Yro06or€ 6r! o A €X€r lia F6vo or{Ll,fiou n€QriX€! io bldwola o. T6r€ o ril,sxa( ,47,4 rivsl o cpl8p6E aTo xnr roi et\u" arbluT o.2' &w4 qv nepimtooq proporp€, swi vc orcFqrioorlr€ orov(,4? -4) r, !(r 6rqe6ooqre pe rov -4?.4 xm 1,c 6eoriF ror nQorlloiFsvo fltno.

O IlivcraE eootrgrxoi pvopevou -4T'4

To p6vo relvLx6 l{r4pc notr cro[dv€r eiver va €Esrdooupe rrE L6r6i4-ftS lor At A. Ariro< €ivor oilorlprr oopp€1pui6E: o dvdorQogft rou slvql(AT A)T = AI ATT no! €[vd! nd],r o ATA. To ororlelo lou (i,j) .iror ro€oorrQrx6 lrv6u€vo rrE oniLr|E i rou .4 Fe q orliq j rou .A xal or+rrin$r F€ro morx€(o (J. i). nou rivc( q oril4 j €ni q odLq i. 'Evq 04ra.xl€rbi €ivo! rluvrrmprrl4r6qrc rou .4'l ,4 x<'rt rqrcil €uruXi ouprntoorl

o Ar A iyet nv i6rc p4681'6Yttpo lts tor A.

:iyouQa 61ov ,4r = 0, t6re xu. AT Ar =0. T(r 6l(ruropord E rou pq6€\r6x0r-Qo! rou ,4 r€EtlovroL EnioIE r{q! orov FIb€v6XoQo rol ,,1?A. fLcr vn ndpralt(otQova. unoOdol€ 6rL /4r A.r = 0 xar ttdptc ro roc,-rrtQtx6 yLvoFevo F€ ro .r':

.r1 ,4" rl.r = 0, ri ll,A,i l'1 = 0,

AQc ro r n€Qdx€rft mov gl6€v6l(oQo rou A ro! o! 6io FIb€v6XoQor iarJtiEo-ws!. Er6!xn, av o r4 dl€L avEEd$rp€E fiileE (zcl p6vo ro r: = 0 :reprdletcr otovFl6t1'6XUAo) r6re to i6ro o).ylorxFr raL yL.t rov ,4r,41

3M E6,t o A ta* lgqrpr*cirg ovddqnrpeg ouil,s6, dtr o AT A efuar.tnpaywvLx66, aupyctptxriE xat at umphptpoE.

AQl&€Qo 0d 6etEolp€ 6rL o ,47.4 €ilal sniolE oerrnri opLogrivoE (6Lor, or odryolxar or rbrcrudE eivq! o€rrx€E).

!'nurrlv t1v nepinrooq, rro! ehcrl q nro roLui xcr nm evdlaq6pouoc or'6l€6, or xavovlxag €EloooerE €ivdl 6wcl6v !c ,lljooiv (l)6 nQog -. 'O]rox xor otodQr0Frlxle6 noQdd€r a, o ,4 €l& "F€'yroq iaErl oqtr6v": or n orrll€E rou Eivoltrve[riprqcE, rp<ipc 6Xr ordvro orov m-dldoTnro 16p'o, 6rov m > n, <r].],ri 611xat crnogrir,5 crl,qoig. II(iwtS orcr erdpevcr uno06roup€ 6r! roIUEr.

Ilivarcg rQo6or,6v

Or urol,oyropoi Fcg cni6eLlct 6u ro n\o#omqo o4reio oro 6 eivor, ro'p = A(Ar A)-t A1 b. o daoE auds uqpclu pe nivaxcs, qv xaraoxtul1 pLag

, : n P o B o \ L : l \ \ l n ? u : L l t l : L t : F \ A \ l : t ! l \ l t l P , \ l a N o N

tr$EkK z.riqtrI. oro lriAo 6r,l)-.;r,(ll'Ton ror' rn:rou €iv&r 6v(. nil,(zc.

ror |l tu1' .\ti!!tttu a:t6 o b. O ,ti\@.nrnao6o).i: Jrou 0c our6ol,ilosr 3r P:

P=-q ( . . 1 r - { ) - . l r

O :ri v(]{q< cmoq nao6di},€L xho€ bLirvrop( l :rrit,nr mo xoqo oql6v ror A."Mr ril.).rr ).6yrc, ro p = Pb eivar 1 owlorrixrc rou /: oro 1o;qo orll"drv xcr roorlriipu b ,1)D eivcL r1 olvrotdoa oto oqOoyr;)arc oxtmlta{,4ro. ( H, 6moq eivarq,torxt5 ru:roupc, o 1 P sivar €rioitq ivtl< :ri vuxc< :rpo6oLrlq, no! nqo6irl),erro l) oro opooldvLo olpnl.1g.rpcr rct q npo{iol"r1 ror b rivm (1 P)b = b Pb.):rryonrLx(i. i/oltr€ rllnor,: trr nivcxrq trr v( br{rojrc4(F €v(l bti(urolcr o€ buox(i{)rrFc orllror(boe(. To P, nrari7r-rrrr otov Trirpo m1}.d;r ?(,4) xsr I d),ho|r'rorriroo (1 - P)0 xr-Qrtllr(l oro! (ordrrQo llbrlo7arQo "\i(,17). rToli r,iv(llop0o./(;)uoc oro /(;ao tJrtl.r;\'.

A|roi or iri!(lx€c rao6o).ric i/ou\' (Lyr6$l{ri x(r y€!)tr€tQlzrt onlr(.oi(r. Ano'rr'7,ofi'tr(l orzoytv€rc nrlirzoJ\'It flbr)aa: rbrornr8; no! H(.IQr]otlrotrotn0ouvtrplorrpu o(r'bolrzoi i.iHor lr( o).or,: rol'. o|trlrtet,{or'( jri!c)t€:. fr' atrov ro).o1o 7(iaor,xr iv( trLzAo br(i).€rt{'( 1L(1 \(. r-:rrdoor'l|r rr; r6rotlrt; rou P. aprtr;rorea\rcrtr€ or( €i,(i/ror( r€re(i}')v(t.

3N O nil'(xtl( nQo6o),rtq P = A(Ar A\ r,4? 6XeL buo 6do!'adE ibL6qr€(;

(i) Ioorrrtrl p€ io rerp/tfor!6 rour Pr - P.(ii) Iooliar [€ ro! cvdorQo(p6 rou: PT = l".

A1,limQoe(r, tiro€ oui4r€rQr)i6( ni.vdxtrq !s Pr = P ri(rQrorov€r l.r!o r[Qo6o),]i.

Atu r€'g'l Enzo),c iiLrlnoqrt y{1ri LoTuer P: = P. Eirv Ernrvrloorrllr cnc! xrl:roLobrrlvuogr l. r6tt ro Ptr:rtpuleraL otor l;r61opo ;rrivor orov oroio rpodri .oupezr(. z(r(ir ou\,if1rtl.l. 6rar E@vafipo6d],],ot'pt 6t1' d..ltiEtt

"lroz€. To 6rhvuog(

P, Ei1(r ib't oror r,;r6xropo y.u rc PlPl,) rir(L ndlL ro Pb. pr hl).cr ),61cP: = P. Ato. rprL; ri zct :relrilru rao6oid. 6i1oul ro ibro orlpeto p gr rry:rodrrt npo6oiri:

p - l { l r l r t / . { ( . { / - { r " . { r t { . l r l ) I 17 - p

" loo: rf,iq/€r zdf,oro: zi rbrro: d,rt /rdrF I n ror: f,ir(zr: !r.r(0eo&Dr f,or orriotr:Dtr r 11i,t:

n€ ro ,airqr( l'. o ziYDBo! ur(nn rir,! nrz|})i z(L nlroor(ooiu r( rr!.lrqoiio\a(r rori zd,

oL i io a laoLr : . r , [ i? ] 'a d . t ido

oPooroNro t IA

flcl vc 6€i.Eoriy€ 6rr o P €iv(r €rioylg ouyp€rarx6E traiQvotJF€ ro! avootQoq6ron, Ek 6ti :rol.Lonl.aorrilor4e 1o!g ovaotQoqoljg pe cvriorporpq oeq<i xcrxprlo4ronorot'pc tqr turr6rlra (r-1)r' = (-B") I pr B =,4'l,4 yrr vc rdpe ai,or,r

p r = ( A 7 ) r ( ( A t A ) t \ r A r - A ( ( A r A r 1 ) t , 1 - A l A r A ) ' A r - p .

In ro ovriorqorpo, :tpo;rer vc olp:rrprivouge ctn6 tLg Pl = P xnt P] = P6rL o P eivcr :rqo6ol.ti. omog xcL zri0a ciLLog nivtrlaclg, o P 0o orllv€r z(i0€6rcivropcr b mov ldJao orll.6v ro! rur ro Pb ecr €iloL olvbudopo( crlr(i)vrc,w orll.tirv. lra va crnobeiloupe 6rt urrf xLvat q qo6dli rdvrll o ot,t6v rovu;r67.topo, c[n6 ;rou:rptn€L aa 6eiEoutl€ €iv(r orr 10 paQo( rou , ]roo snoF6v€r -robttiruqc ror otit\rnrog b - Pb -rhu opqoyri|ro orot, D:ftj/.oao. IlQlu'!$,rt,] t rr zd0r brciruoprr P,. o uw6v rov un6loi0o. ro ro0lrFpLx6 IvdFEvo r, ivFrlofv:

\ l ' . P L ) t P . = t . r l l - P ) t P , = L I \ P r : r = 0 . ( 4 )

Agc ro i., - PD eivcr" og0oyd)ro o'crvr6v tov ur6lorpo xoL to Pl eivnr q npo6oLrlrr.iv(') oxov X6po orl)d)v.

IIAPAAEIIMA Y oOao,r€ 6rL o A €iv(l! ;irl! (r1'Irmaal'rtro(. Edv Ei't(tl 4 tni4. rtire or riooepq< orr1l,e; rou rilcrL crvriripqreg xrrl o 1ti-rpoq otll,(i)\, tou €ivulol,6xilQoq o Rr. no!(l €ilar n nQotiol,l nrivo o'otr6xrlrlQo rov ldrQot Eivcr orulrrorlxds ni\ruxcq,

P = A ( A ? A ) t A r - A A - r ( A 1 ) t A r = I ( s )

O r(vrorrx6! niv(x(q €ivco oul4rerpLxog, 1r = 1 rur ro orpirLptr D 16 eiv&rlllo€a.

To aErorrQ6oozro ors ci ,o nrracr6€i1rora €ivrrr 6r! crut6 nou ow66t or1!(5) 6 rnnalftult. Enava).aF6rivou$€r 6t(r\r ouroi or ri,\,otrEE_€i1,(! TioglLLl-).61puppoL.6* ynopoipe.,'o olrloreivo]Jtr€ [slovL+dva roriq ,{r xcl|",4. Aur6<nou clrroTQ6(pexcL €ilcr o retp(lcwLn6g nivana< ,,lTA.

[aQ€F6ol,il o€ 6E6oIEva |rt q F60o6o E],aliorrDv TEreoTdv(,|v

Y:ro6eore 6rr, xcivo4re pr.c oeLpri rerp<4tirov rar n€Olgdvou$€ 6rr ro sEcr"/o-prvo n:rord).eopc b ()cr rivcL lpcpprx{ owriprlol ror rlooy6pero! 6s6opa!o! f.Avclqrrige pic er0cia ll = C + Dt. frc TiaQdbtirlpo:

(l) MflairF€ oe 6Lirrpopeg lpovLriE olrytr&, qv tl,-rdorcon .vos 6oQW6Qo!,zco(ii: zcI€uot'v€rc.! o1o\, Ae+ :'otrrl rrN n€Qinr(l)oq t Eivar o lQ6voq

"o!b Ei1,(r q or6orclol. Err6< rrlc n€Qrnroo€rDq rou 6ouLeu€r o xrvl]Tr]p(tq yl yl

6aprqrn eircu clro0$ri, o 6oprq,oQo< 0( rrvEiraL F€ oxEbov or(0tQrt raliTltr.rr ' : b = b o + i , 1 .

(2) M€rc6hlloqr€ ro qoQtio n.ia(D o ivcr rtrtlro z(r llt-rQG[E rry rarol irotrbntoral€irtl'. : ouro ro np.ia.4( ro t €ivcr ro rpopri o zcL ro I Fiaar il 6vb€tEilror fr{ilarFri r(iofirrs. E}iroq iIE n€pLnro)o€0)q riou ro {loprio rivcL ftelirlo zar torl,rzo {u uroorri p6rqrrl ;rapq6qr1ool. [l( yac$$I'rlt o1ior1 b = C'+ Dt tivar"or1I0n4r€11 oq t]0(,raia €)..rorrxorlrc(.

(3) To x6oroq rcaaltDyric i 6r6),i('rv. on(,r! (rJro €6(;. €irct oz€bov yat4rurzo.h = C + Dt'.1Lr C 1ru tr] ororz€Lo0uoia xor a:rt1L6).etcr zar D yLu tqv txdrcxrrlz({ ro 6iortro (ro x6oroq yrfl xdeE Eni rii€ov 6l6)'io).

To eprir Ng rivcr. :rr,rq uro),otiltr z(vr:iq ru a zct D c;r6 ra crnottl'tqcturur0r4r.;r(')v. Ehv I ox6ol €iltlr ne(iltr(xr lpalLplrrtj rctt lrr rrciqxtr ot{ d)'p(oro :rriprqc. 16rr brv r:rirq1€r rediiirltr(. Aro pttprlotLq torl /l exolv lLtt v(:rpoobropi.oouv ;rirl-o('r< rtr r'r0fiu , = a'+ ,t. Ol€: o( tni n),4o1 trtroitdfr< {lc:reqLilovr({r o (l(ril n1r €rr0€irr. E(i! 6$,r; r.urirpTtL oq(r}.[(l xrlr rtt l:ri :rl6ovoltrri( (6l.11(rroia vo r0oo"/rL(,0or! orltl €!0ri(, r(;rf 0u tr06jrrt v(1 rroqr(oroltr€1'(r :rripoupc ftioc( r4Ldg ra)1 fir(Q(yrirr,r'v x{1r rtr 6oorrftc 1111 r(),!reQl bri1'(rrlf r ,0{. ic. H Fr9f i . ( l ( ! f l ibr1 0(rQinr !( orryX6rr( ! tL€ l ! ! : !oei( n( ia0J orrtv onoiaapotiriiupe ro ir o-r1v;rqoqlolpevr1 ruQ(i./Qo{lol n0i11u(1t!. cq,ou uficiQXolJa brorhlvororor ('xra /) no! x0an€r !c nQoobroQro0oriv. (kr bouL6lJoult ftt nQo6oiiqar 6b otuto r,:roy,r'rpo. To rQo6),Ittu t),(Xiorr)v rtrpruyrltrorr, nQoztllrft cnfu0r.i{r: (ln r( nt!(x4(rl2.i rort}.[o[(rc

].J NPOBO,\E: Ii I l]PO:L,I I IIFI: EAAXI:TAN TETPAT!]NAN 185

C + D t rC+D l1

(1 )

C + D l ' " = b " '

Arrir circr ivu r:rrqrpoo6roprotra ,'o oiorl!|1 !ri ,r €qrot)ocr! r.1[ bxo a./'r'6ororizdr €d! lrc'rpzouv o.ihip(rcr bcv 0c ixtL ),nott. ToviqorttE oi' o nilczcc .1 axtrblo orrliec zcl to iryw'xrro 6rrivuoga .r' iyaL 6ro olvtortircte<, r1v ( raL t4v f:

rl[ ; ;l;l

H x(inrsal ),iort C. D eirc{ rzeirl ror t),ctXLoro:rorei ro

Er= l , , l i ' = (b ' C D t , ) : * + 1b , , , C D t , , , ) ] .

It: oqoioliu nLldrzrDl. 6r.11€lortLE €x€ivo io :r1Lc ro onoio ro p = -{t €ivalro niloLdnteqo 5u,crr6v oro 6. An' 6Le< rs €lr0EiEs i,paFtriq b = C + Dt.

6ro)'eyoutl€ €x€iq nou rop€F6dl),fi4r 1.ord rov za),ur€Qo tQono oto 6e6op6\,c{(I. 3.9). >rrf 1a{rqwri naQdoraol, ra oqdUara ei!.rr or xata*68uEq.ti.oottioEt' b - C Dt (rn6 1rlv €uo€ia yao$Fli (6[! o! ,.40€rEE o' oudl'oiromdo€!(!). Aur& or zcror6Qw€S qnoordo€K €ilcr! 'tou 1€rQctYoviqowo!,o0eoi!o1rror ra! €lax!fi oriorouvrqr.

IIAPAAEIIMA Yno0€ols 6t! 6[6o!iar rQErS Fe'rQrio€ts, dtios olFEldvovlot oloqQLor€Qo oxqFql

b = l y r r r a - - 1 , 6 = 1 y r a t = 1 . b = 3 ^ I L < L | = 2 .

Iqgerrirort 6rr or ttF€g tou a nQarl€! vcr Loardlow. o nee($ot(6F€vos FttoQ€iva €nrl,6Eer r(t'dl,Ln),s! qr6E (axdtro xar evrFtx6E) toQi( xq4ti.(. €nitrr(Dort orq

Fc.e+rcrrfxil. To fiQ(i)ro 6ipa irctt vt xaraypaEoil, ot t€nd)otc not 0a iofr)u'tit t1 tu9t pr,'1ottat n n4ldu xur ui6 Ia reia onuFio:

Eirv rircv 6wnr6r, !a luoou! (lna( or €t!o6o€l5 A.r = l, 6ev ocr unrlpxcrvoqduartt. oI(l)E tc olu€kl 6€! n€Q!6xowar! oe Fic elo€iq x(r| o! €Fto6o€lq 6€vl.r,vovrtr gtr or ibLeE, crllit ,luvowcrr aur€g tov €),(lioIo)v rciQ(y(i)v{o1':

At Ai =Att , , , "" , f : ?- e =l i '' ' * L2 6l LDI -

L6lH l,to1eL.ruLe =9/7,D = 4/? xar I xal'ritrQ! er0€io €ivft I (9/7) + (4/?)1.

c - D =rC+ D= l nC +2D =3

^ l \=3

t7

&. 3.9. npooiyyrol ll. €lo€id yadFFri r.lr oviioiorxtl nAo6oi.i.

li iltgt lil

, l;ri

' 7Io06 'rofli6l.:

uL7

:] IIPOBOAF: KAI NPO:EfTDEI: E AXDT9N TET?AIANAN

napcrrrlprlor: rrg rDpcieq oxdoeLq prrr(u rorv 5uo otlFdrrlrv. To r0o6l.il|laeivcr ro ibLo clJd ro o7rllc rprriverar brcrpoprrrzci. :ro b€!r6 o11a ro b 6€1Eil,Gr or','6r(op6q r(|Jv orl)"dlv. ro (1.1.3) brv eivcrr o|rbrcop6q rtdv (1. 1. l) zaL(- 1. 1.2). :ro ca,orFao, rc rai( oryr:irl br-! rFaLtxovror o€ €re€ia. Ta €),irlrororsrAd"/ov(r cvr.z(l0roroil ro l; pe to 7r xru tcr oqprrio ;rol 6er repr(zor'-rcrr oqr€r0€ic trc oitFoirl ro! nuQritorxur o (rrrl r. MI ynoqr;Jvrug vn Luooutr ro-q.r '= D. / . tvotr fr t ro -q.r = /r .

Azot{r zal i r€0a: H r$ei( (e/7) + (a/7) l 6trL iul 5/7. 13/7.1717 o{ouq/Aovor( trtrartolq 1.1.2. Atl1ir ra oftr€io n€0r6lovror orrll €L,gr:i(. :xa.ndr<ro brirrrotcr p = 617.1317.1711) ncpLilrr(iL oro X{i)Qo orq)'(i)v. Eni n),6ov. it€!e€i(r cr,rll €iv(rr I no),irserl x(r ro 6r(ivrrr!(l eriol)( ro xd.urr:Qo, 6t6r[ ri\'Llr rlnpo6o1,{ (rou 6). To 6cEr6 olr}a ai.ror. orLq rper.q 6(rordo€K (it r,, 6r,(ror(ho€rq.c^ uitqlouv nr oltr€kr) xar ro crpLorep6 o7ri11( ci.vrlr orrg 6rio 6Lcorrirrtr< (r1 ,c\Luotriottq. uv rrnirplorl' n ncrpdperpor).

A( l( i r0(;) ! r(c ro.1./ ( i lo 6 iXoupe ro oqi po e = (217, 6l7,al1). Axrivor rc z(rcz60r,(l1r )dOrl roli (rqor{Qoi, olilll(ro( zcL or ouvrolo)o€q roub(rzfzot.ttdlo! 6a),ouq oro btrEr6 o7{trc. To 6rirvrotrc c!r6 Fivrlr opOol(;vro orrt!nadri t od),r t ( l . l . l ) bt61t2l7 617 +11'7 -0 ^o! o0ooy(i)aro or l 6r i r€0rt orr t) , I(-1.1,2) bt6t! 2/7 611 +8/'7 = tl. Ei vrrL Lomo! op0oy(i)aro orov Xrirpo orrll.drv.(iQ( neQt6xer(u oroa (lororf0o llbr.vox(l)Ao.

EQtirttory Ea\ o! cQzrxd! frerQ{o€r; d6wcv, = (2/7, 6/7.a/7). roft 0o trcril r{d.rlTeAil rr0si.r .l.lr. ro z.d,ureQo t: AfilllrJloil: H pqbevLxrl crr0r:i( q o;roi(o4ui.nrtr pt rov oprlrivrLo dEolc. r(r ro 7 = 0.

MroQor[c vc orJlolrioo![r ya{yoac flE r.ELo(irogrc rcp4rSo}.riq yL(l trL(lft0ri.c. H nedrrit or)il.it ror ! nsQ!€Zr! $o!.ibrg z(! I 6 lrr0l :rtpL6Zt:L rouqXp6vou /i. tipu o ,,1",4 :rtqL61eL ro h0poLopcr rLLr! ,r $ovd6oi!. ro .iflQolrr[o rrrvi i ,{ar ro d0QoLofc r(Uv li:

3() Av 6o0odv or $orprloelqe€l,e C + Dt nou €lqx!fionorei

,4 r A l l = Ar t , it u l

11 . . . . , 1 , , , o ra o ru r i a 1 r , . . . , r " , , n Ex -ro -er npoxurift{ ;.rt: -r1 p60o6o elaliorov

| ,n r r , l fe lrr , l t i Dl= lx l , , I

LI t r l , r l '

n.,,e(riiQqottf (l(rtul.oltrclrrrfi r(j)v €i.aXiorn! r€rp.t./rl.vorv brv nfliLrr L6lr1ix€Qoao)"o ro 6rI ;rciltp6ti),Lorpe 6u0ei.eg ypaff€g o€ b€botriva. :E ro d n€rA.iparo6E! urdAlrr io1,o< r,(l n€Q(pa\row€ gru ypcppLxrl oXiorl ncL 0a rlrcrv rcpc .olova Uir/loutr€ i,ra [r.1 r€ror..1. Yro06or€ 1,.r. otr t:!erd!oqr: rdnoro pcrbLevepl6

!).rxo. To €Eoy6tr€vo l] 0a €ivar n a\'6erEl €l,oq [erQlrrl ceiger o€ 6roqoQe,<Xeovr"6( 0111!16( t. Mnog€i vq Tv{oQ|loqrr 6Tt io ul,Lz6 Faq €ivor" Fi a 6iopcbL€vEoydv nllxdrv ev6o€c,Jv nDv oroiov 1'v()ailouF! eniorc r()16 Xe&o]JEuno6ril.(1or(rouou ({ Arretro{E 6rdoraots), rr}.ld Fropri !c BI yvo)aiEoupn ][600an qa xdo€ €voorl rl€Qr6lsror oro triyF(t. Edv (rurds oL 6r'0 dyvoJm€E noo6rr]r€S€i\i(! C xar D, 16r€ or e\,6€iE€|E rou [€rarFri ceiger ou[nEQlq]iQofiar oav rori0potoyu 6uo ex0erra6v (zcr olr oo! lua €uosia):

l ) = Ce-^t +Dc-t ' t . (8)

frlv rQdEll, o flinog crl)Ios bev uv(:rcrqci.ler(L szQr6dE cn6 rov perprpj.Avr i yf t ( ! r6v, nci ,Q!ol ' !e €16sLEe!q 6r. . . . .1, , , oe te6lous tr , . . . ,1, , , x(rr I (8)rz(!oroL€irsr x(rd nQoo€yyron:

( 'e ^t '+De t ' t x ht .

:

C,{^t , , , +D(,_rt ̂ _ b,, t

Ecirv 67on[F n€Qroo6r€Qr'( (n6 6uo €vb€iE€L<, nr > 2, r6re x(rrri :tdroc :rr.onvotrFc6tv propor,lr v( ),uooutrr 14 apog C raL D. H ceXrt 6ttoE tov e).nXiorr,rvterprlirvov 0c 6rioer, 6i.1,tr,ottc tU6E C rcr D.

H zcr(lorclol 0c rlr.l! rrL€i.(,)< hcrrpoptrrxri av pr,rpilagr: tq:roo6TrlrrgC xaL D xcu fieooncr0oio(1F€ vcr cv.rrallnpoltlr rorjg pu0pous bLdon(olE )]a(rr /r. To rE).eurqio €l't<tt p4 ypupprx6 ql6),r1pu t),ayiarut rcqaydt'av xudQC 6r'o1aoi.6r€Oo. K(i nli),r 0( 0Eoeouo4r€ ro ,l-r'r. 6ql,c6ri 1() (ioQototro r(l)vrfiQcyov(l)! l{,)v oqa}+droiv r(r 0q ro E}.oXLo1ofioroilo.4re. Ehoriwovtq 6poqrri n(OC.y(n./o!S rou Fs ro trl8iv b€v 0o ,rQoaruntov ypc44[zi.-< elLotloeq lLcrrc 66).110r( ,\ rcL yr. IttE (roxrlotr( nclQqtdvoutr€ o€ lpcl4lrx.i rxQo6lriFctr(lr).(7ioxu)! xfi Qaydrvo!.

AEKH!EIX

3.3. 1 Bq€ir0 llt!- .,td,t|rFQrt Inot r)drimov T .rqcldrv(,)v - yra r(< lr = 1 0. 4r = 5. noloo{d}+a Er u},cxrmoJToruirati Erc},ileEUmr drL ro drdvmpc oqd}.!.noE (10 3r,5+r) rivaL z.iorlo oq onii,t (3.4).

1.3.2 Ynoolor!:6rL oL r'tdE br =l rG 15, =?, orouE re6vo\ lr =l xar t2 =2 obnyoUv(}qv raqEF6o).'i Et{lri(! b = Dt 66exotliv,rE @ad olr ae ti. Li)arc t< D = 1 x-Il2, = 7 r'e .).(ilrca 1rreiirovo udL oAbrir.x€ q 6aitorr .la€id.

lJ.J Anot€ to Ar' = b Fr r).d7(mq r€Iq.iyola xft 6a€iff to p = -:lt 6r(rv

,liil,lil

3.3 II?OBO E> XAI II?O:E|II>EI: EAAXI'|ON 'IETPATONON

Eral,I0snm€ ofl rc S inoFo oqdl.laroq b - p €ilar ndo€ro orq ont),E ro! ,,1-

3.3.4 Avcrn&€ rc t'? = l,4r - a, r ror oErodrm€ p€ plldav rLs rdqo?6yols ro!, os rqoE

[1 0 lA= 011 ,

L l 1 l

:u/nai'ar€ n4 ,raorurrorofE €ELo(;ro€16 Lrx rry AT Ae = ,4rb, €Ea),I0€iovia< dttroso o aneraomrrds lolro}6E 600 1|aI n leop€iaio 6i!€! tE ravoflt{tr €Erotiro€rg.Ba€irs q ).iot t aar qv nQo6oli p = AE. fldti. IorXriet p = tri

3.3.s To sndp€vo oiortFs 6s! 4s,l.nol:

ndaeB6d),ar€ pio €1jo€lc no! oqy€i ss s)dx'sroroinol ror rerpa'yovr.xou ro)'r](rrr-!o! (C - /) . 4)'1 + (C s)'? + (C + ,D - 9)'?. nom €L1or rt ?iao6o),ii to! b rdvo oro.'xdreo oql6! rori Al

3.3.6 Be€ir€ rI' reo6oLi ro! 6 rdvo oro\, tdrao oql.(irv to! Al

Avd,nore ro l' oe p + q, ono! p trw]rFr oto! Id)ao oflt]'ilv 1{ar g eivsr xdo€ro oudov tov xuAo. noros aro tous rdooEArg X,oeor{ nre,dxcr ro q:

1.3.7 Ba€ire rov rivoxo npo6ohjs P nriv<o sror :(6po no! ndqdy61ar ard ro a r = ( I , 0, I )r ( l r r o a 2 = ( 1 , l , l ) .

3.3.8 E{i' ]' sivaL ftivdxdE reo6o},ris o' aaov l-brdotoio !ri6xo0o S rou x{iQo! R", noroi€i!o( o xdraoE o1nkl' ro! P rat noro ei.ldr rt tiiEt rou;

3.3.9 a) Edv -a' = 1'".I', 6tiEtt ofl o P ti'v<rr triroxo< rAo6olt(.6) xs roov urdroqo reo6du& o rivaras P = 0i

1.1.10 Ed! ra bn\,lofaro dl, a2 1r(1r b ei.!o! oqeoy6vn, ron €iror ro AI A xat ATbinord €ivar I rao6oLi io! b riivo mo ef,ift6o tua ar xar o2i

33.11 Yroodm€ 6fl P €ivar o rivdkog f,qo6olrts dov ur6roAo S tar Q tilar o riiloro!f,qo6o).rtr- mo oe0o/rbvio oryr)'lEqrd Sa. nolor oilot or P +Q ,tar POi A€iErs 6iLo P Q tauilera pe tov orioiaoq6 tou.

3.3.u Edv v Eivor o !tr6roaoE ror' ridadr€rar cno ro (l,1,0,1) aor (0,0,1,0), 6q€ii€d) Bn 6dot ro! oeoololiou ouFft),rq.bparos v'6) rov nildza ftqo6oht6 P rdvo mo Yy) ro 6rdlLiofo ro! y, ro rilrlo!*or.ao mo b = (0, r , 0, 1) ro! v ' .

[i i]r;r [i] ,

,=ll-i],,=lllL 2 4 ) L 7 l

19{J OPOOTONIOTHIA

3.3.13 Basift q 6d),rrorrl €l,eEis rrda€p6ol.rig (Eldxrmo r€iadplo) yra tl( FElAiopE

t = - 2 . b = 3

K(r6rLv GeEirr qv nao6ol.i rol l, = (4.3.1.0) ra!(D orov xdrAo aql6v to!

3 . 3 . t 4 T o 6 | a u r o l ( n o a r = ( 1 , 1 , 0 ) r o t o : = ( l , l . l ) r c q c i y o l r a l d € r i r e b o r o r R r . B e e t r rrov nilor( flao6o).rtq l' ndvo oto fnin€6o. rd0tlE ecL 6rc p1 p4btrLxd 6Ldvrropc irrou naotjdl).rror mo Frtbdv.

3.3.15 Eiv P €iv(L o trivdxas nqo6olt ndv.u s! rue.ic rou sntnf6o! r.V. orrid(nF arodnipo no! v(l ri€aryadrvrr r! badot ro1| Tihoxo cv{i .doits tl = J 2P. EEqyriurt'tr(')p€rarxd )1or fl).1€6aLxd, yrlli IIr = /.

3.3.16 AtsiErs yurti. yrq Fov(h(io oravlord U. o nrvi,xdE rdEr',,r< i!i! P - rrr €ivornilcr.s neo6o),,iE, 6l)di aafl 16 L6LoulrEE (i) xdl (ii). At(,,ayowos { = a/ la . oP rtlcrot rQotiol.rl ndlur mI! €r0rlc 6d to$ d aa' ro Pb eLvar ro uru.io p = -o.I]qo6otr{E r.iEro( a!(r avxrdror,(oul (or(r6dE oc neo6litpar(l da](iqro! ftreardra!ry

3.J.U noro6 nivur$ 2 eni 2 neo6(i),I$ to €rif,rbo r:, y oq! -.15"^ slerid r + t/ = 0r

3.3.1E Or)'o!!c vu noQe!6'iloultt trirFdo y = C + Dt + .a: ma 1doosqa oruskr

| ' 2 l^liil

: = 1 r , = 6 y t a I = 0 . : = l

: = l l / = 0 Y t a l = 0 j : : 0 .

3.3.19

t L20

(l) Berift 4 EEro6orq fr taeq oy!d,mo\ yro vo r.odoete dvo .rir€bo aro Iariooeoo otF€ia (r{' 6oov Drder(€r raroto €rir.bo).(2) Baeift 3 rELo6o€E Fe 3 aladJoro5T Ld q 66l.flmt l.uol rl(t(imov rsraoy(i)mv.

Ea\ P(, = A(AT A) rA? eivor 11 rpo6ol.rj rdvo mo' rdqo oirl).6v ro! A, ro|afi!tu rt rpo6uh Pn ndw, orov Xr,qo tsorq,'i,!: {AFv rilor a Palr

Edv Pft €ivoL n rao6olri orov Xd,Ao yAoltll6v 1ou i, rota EivcL I reo6o),i PN movrl]b€loxoqoi (o( buo unoxoaor €i'or oa0o16!rcr p€'a€n roq.)

Yro0eme 6tr rr rivar ! EUsEio 6rd ts ae,Ois rurd q 6L€n0!!gllrou dr ,{ar /,2 €irarI nre€f.r 6rd ror, 6 xatd ttt 6r€u0!vcr4 rox 02- rA(iln€ rLs €Erod)o€'5 aE nqos rr aorr, ftou rl(rxroronobnv ro I rrdr - cror b I, yta va 6acift ft nlrloLdor€Aa o+€iar1o1 xq b+r]o, o6 6rro €$€irE. Aum€.x neog r 6r(lr ar = (1, 1. 0), a2 = (0, 1, 0).b = ( 2 . 1 . 4 \ .

3.3.21

3J IIPOBOA.E: KAI II'O'ET'T-EED EAAXETAN TETPATANON 191

3323

3322

t3.u

1326

Be€it€ rt 6€lv.{tnl s!0€ia naa+6olri6 C + Dt lto ra , = 4,2, -r,0,0 o€ rc6vorEt = - 2 t - l ' O t l ' 2 .

A€lEr€ dlr rl 6€lt|oq ropep6ol,rl €Xdltoror istQoy61aov Lrldg oQrqdwtaE €l]o€tdg-p€ dl,La ldyra $rss oraoEgis olJvdqqqs y - C-,rro cvo orvol,o F€raioe(Dvt/l , . . . , 3m 8hd! tl paol rqri tol)E

C = 9 t " " + U n '

:e ord ooxrl oaoloyicr, q sdoft to! y norr sl,olroronorsi ro oqdlra P =fur - vl2 +,.,+ @^ - gf ei1ql o ldot 6{os io! det]|raroE rur rc nQoxlrniov t2€h(rl rl 6.aaroed (r1 Slo nFdrol) d'z.

Bqette q 6dlacq euoeic rcq4r.6o),{g rov :nqrgvov FsrQ'tosov xd! q(s6rdoft Il1lton o(rtl

U = 2 | t a t = - 7 , g - 0 , y r a r = 0 ,

y = - 3 y | a , = 1 , g = - 5 ^ t l . r t = 2 .

Yrio0d6e dtl ovti flo sljg€to ,roa€p6d),ole, ola 6€5oF6vo rns *eornofuevrE doq-qS. Fta ,rq0a6ol'i: y = C + Dt + Et2, norc6 €hul o ,rtwxo! ouvtr&or6t A, todYlllrqro &dvuqro c ,rcr to 6rdvlolro 6s6oFft)v 6, oro Fq oql66oc6 otorud.4, = 6 nor, riQoxl Jrtst ord rdoosa(6 $steto€6; Asv xe€tdlstar ro oror\o{oE€ ro 5.'Brds dvoe(Dnos ro! p€ooi{rvo erqrql{rhory€ o' 6rov raox6 6aooufi{ell,)v oe FrlxrlL = 5t6 xo,l 7 ro66v, rurdrw s4aciroyriE 6!vd+€orv F = 1,2 xar 4 r6v}{Dv.Yroortovtdg 6r! roJ0ie! o v6trcE rou Hook€ , = d + ,F', 6ee{ft ro xdvolw6 iou tVoEa f€ 1erion €l,axtorov E!aoyd,vo1,.

l')7 OPOOTONIOTIlTA

3.4 I OPOOTQNIE! BArtrlI, OPOOTANIOI IIINAKEI KAI HOPSOKANONTKOIIOIH'H TQN GRAM " SCHMIDT

:€ BLo oa0oydJvrq 6aol. xdr0€ 6(i1,uog( eivur zdOero oe xdOe titrLo tudrvuopct.Or riloveg ouwetqF€l(])v Eivor oQ0oydvror percr$ ror4. A!'16 €iv(It ox'i66vrbcrtz6 r(rl q p6vn 6wcmt 6e),ri(ron ./ivrroL €il1{o),a: ArsrcofF€ xdg€ brdrvuola

F€ ro tllzoE lou r.rcrt 1o zavolF€ Fovatjruio 6rd.vuopc. To 6lpcr nr,t6 perctpi;reL

ltro oe0o,!6vtd 6dorl at Vn oQqoxarovrni 6dol. O(l oupdol,iloulE rc 6oorxdbrd.,tiol(rc p€ q. lQnowoTlor"ri)\'raE s!r6 ro yai4r$( yrc 1(r bltrd]voutl€ rlvooeozavovrn<ltntc,

Tc 6ravuoucrcr gl, . . , q[ €ivcrl oQooxcl/ovlrid 6Tc1

q! qt I o 6-ovi lj, rio! 6i€r fllv oQ0oyDvr6qrai =j, nou bl,v€[ rqv xcrvovrxdrnrq

'Orqv oQeoravol,ua(i 6r(liol(rrs livo\,x(L onl).eE rvdE nivcrxcr, o ;rivcrzcrg aur6<0u oup6oLiletnt pr Q.

To nro €lbrcqieov naQir6ellpa t:iv(L 11 "out'i0&4ti)rl 6do4".zfi enint:boJr.l/. o! nlo l!{roloi cEov€E ber eiv(L p6vo ldo€tol alLd xor opt!615Lol 11a!xcrrnx6qw,or aar I 6hor1 eivcL e1 = (1.0), er = (),1). O Q civc! o ralJror.r6<:rivrxcg. xtq rr bLamcio€rE rl ol]qoloB€vl 6c.ol c(rorckir(lr nhfu t1'16 rL< o,Iil,e(r o u Q = 1 :

lrlAurrj 6* riv(L 6e6crLcr q lr6vl oQoox(rvovlxr] 6dot. MnoQox[e va orpdli'ouFr roljsdriovrg, xopig \r(r t€rtl6d^or#€ nE oQ0€s y(,)vi€q r.rra rq orroi.€g ouv((vr(i)vtc!.Auroi or ntvsxeq mQoqrl< 0a €roq(oouv iroa(rxirro oto rcrpc6ei$crn :rLvctxovQ. Arr' rnv d,trr1 pepLri. ar ev6Lcrgrp6gcme 6xt yL<r ror R" d!ld. 1!o xcrnotou;r6Xorpo tot, rore rc ornl0Lqrdv<r 6Laviopcrc c1 prioQ€l vcr FIv n€e!6xovtcrL orolrrr6lorpo. I' trurrl r1v:repinrooq 6ev etro 16ot> riQoecrv6( 6t! pfioQ€l !a 6Qe0€ixtinon opOoncrvolrrr] 6irol. Oa 6eiEor4re 6pog 6rt lrcipleL nirvrott prn r6toLc6(irnl, rou xsrool€udleral ge cnl,6 tp6no, EcriLldrvta6 rrn6 pto ofiola6lifior€6dot. Ami I xcrco:reu{ ror ptrarpin€! ivo r},ay!o'fdr!o ouorqpo aEovov o'€va oAooyrbr'lo. eiv(l! y\lJJori aS xdvovutoxoirlol rav Gram - Schmidt-

:woxr''rd. r'r raiu 6tr0rzri Odguta uw6 rq; ruQayQdgou rivur:

(1) O oqroFr6q xcr or r66rqxe6 xov oQ0o1Dviov rlvdx@v 8.(2) H luoq rou Qz = b, suvqoroF&n rl p€ €tdxlorc xerQdy(l)vc.(3) H F60o6oE

T (l)v Grdln - Schmidt rar rt €Qpqv€ia TlE (r)6 trqs v€osnoaoyowonooio€orE ,4 = Qr?.

Og0oyivror :tivuxeg

'Evrs oQqoylbvroE ftivaxdg ti.!(^ (rnLd,s 6va5 '€rQl,.yaux6s t ivd.x.rt ItE

oe0o,1o,ror,x€g otfi,,Eg.* XprpqofiolodF€ io'fQdppa Q ll,d rov riroxo ra[ io{r,. . . r {tr 1!o xlE oq}.€E to!. Ol r6!611F€E lov oqfui^t it a'v qT qi = 0 xat ql q = t&! qn[6 0s dnQ€rr€ vo |r8rcrqQdqeicr! o€ !6[6rnrEE rou nivdxc 8. Ms dlJtri qp€rCrqQoon, flou Eivsr fio].{ orl{, 0a qav€i ndh, I crEicr iou oup6o}.ropo{, p€fiilroxe6,

3Q Edv ol or{}.eE rou Q eival oQ6oy6lleE r6re

Aaa QtQ = I xo)" QT = Q-t, fta oQ0oy6vrct5 rri"to:xq,, o ayiotQoEogEirrd,. o awiorgogos,

O-rav q ygcpprl i rou Q" notrl,cnLcoriler rq orrlll j rorL Q, ro nrordleopa eivoLqlq; = 0. Eror. neoxdnrow ro pq6evr,xd enr6g qE 6ralorviou. :rl 6loyci)vrc,61ro1J i = j, 6xo!e ql qi = i. 6qlo6rl qv ridvovu+onoitot o€ blcvrlopotc p.lrougl .

:np€t6otE otr QrQ = I rox{EL ox6p1 xor" 1Lc noaal,I4.6yedFpo Q. T6r€6pog o Ql eivar, p6vov (rQ!fi€Q6E o.lriorQoeoE.

IIAPAAEITMA 1

3.4 O?OOKANONIKOIIOIH>HTAN GRAM-SCHMIDT 193

* Fla drreao 6rcfc oo rituv opooxdovLxG nivaaa<. a].).d EivoL roi.rj aE/d ./[ v qUnEs

ro olopa. A.t lrdax.r edqq aro6sar6E 6po< yro dvoa napaMl).o./paF+o rivdo tE oaooxalovLxicm,i).e<. rLa dd rno@ .iarcisloo!!. !d ladqoulu Q, cX?{i bur ro! oloFdtoFE oeooydaroiivax( dv 6d rivoL tmadlorxd(

'" [f ]l+ + 1tl n: :t

e, =e .=l_::;f :*9]^ l c o s d - s i n , I' / = L sind co,d '

194

O Q otp6qcr zdoe 6rdl,iropo ,{ard q }ovfc! d. ro! o qr ro orQ6e€! cwio€rc!.zsra l(ol,ta 0. fIQoq(A/6E, o! orll,€E €ivor oQ0orovovir66 xcr. rorjro 6rortcosr0+sinrd=l.OrivceuEQT eivar oq0olt i rvroE, 6nt4 cxpr6r i4 xcr o Q.

IIAPAAEIfMA 2 Kdo€ nivcxcrg F€rs€to€o.E P rivar oQ0oy{i1,ros. OL oT|i},€Erou eivct oiloup<r povcr6L(i( 6r.lvlioFors r(rr 6a6oLo iicl oQ€oy6vt(r, 6L6rL yl

trord6o gFqovil€roL o€ trcl drrrqoQ€uri oaon o€ rdr0€ mytLl:

:lfo 1.0, o= ll 3

fo o 1 l16T€ p- t=pr = l r o o l .

L0 1 0 l

Ei,vcu, :tpogc\,dq otr, PTP = 1. fuLo6i o cvdorQoqoE Eival o ovriolQortog. EvcEirlJ,o< P, ue Prr = P" = P3r =1. or6trvrr ror.rE tiEoveE r. g, r srouq.?,3/,l,,6ql,(rbiiva 'beErootQoq,o" o m4rc o' dvc "aQLorEQofiQoqo". @a rjrcv l,ourdv ),riooEva urovotlooqre 6u zde€ oaeo"/dJvLoE Q r(pror/uel p.ta orpotpil. MLa arri .co4

tiut tE ioor t iutet:tr i . o,( i1'uxuq P = [ ' ] l l a* ""nrr,

orr< orpoqt'< rouL , " l

nrccb€iylraro( 1.6!6r! 6ev ulriplet rLpf qg d lou va r[(lQdy€t rov P. Avri0€x(.o P.rvsr{}.d rdo€ o4l€io {.r.J,) mo (r..l"), nou eivot 11 zcronrqrz4 etx6vtr rot'nr :rpo< rrp euoeic pc xLiol 45'. f€{DF€rQrLd. €vCtE opoolcirvLog Q eivuL lLvdpevoFra,< mQoqi! 'toL flog ovddqorE.

M6vtr adprl gic eoLvrl Ldr6rlra rov mqoqriv, rtov arcxllioeorv xaL 6l,ovroJv oQ0oy()\ri(r1, /rtlduo)v. 611 6tuDq 7c& t(l)v riQo6ol(i)1,, noD 6€\, €ivcr! opgoydvttq.pdl.rmo orire ovtLorqdrprp€S. OL rQo6o).€E FDiQ(rivow xo Frl'aoE €v6E brcvusuqro:,rvtb oL oq0oyirvror niroxe< 61orv pLc L6r6rr1rc nou eivcL lcrpcnrprmrn$ rcL :nosvbrcl(ptQouoo ar' ol,€Sl

3R O noLlsntrsolqop6E pr oQooyd){o nh'qrq orsfllQ€i r(t piftqi

l l Q a ) = l a 1 y r ( r x q o e 6 r a v u o l r a s .

A[a?]Q€i €niorE ,r(r ooorsQrl.(i y!v6Fs!c xc[ ilg yurvi€s, 616rr (Qx)r (Qa\ =rTQrQy = rrg.

H brtrrrlpqot ruv F4xdv npozunl€r <rn' €uO€icrg ur6 rnr Q't Q = I:

l lQ" '=llr I 6L6rr (Or)r(qr) =,rqrqt=tr:t.

Orav o l6Aoq rrsaroxaaq€i yl roxorrQroe€i, 6rarqeot}"toL 6Lo ra €onr€QLzctyvdprvc zcr rc pr1x4. To prpoE:rpoxumeL puorxc ord ro roorrrprro yrv6pevotou broviopcroE pr tov rcruto tou xct ot ycovirg cno ro cot'-rrrprz6 yrvoper,o

r a , l . r r - \ \ ' r \ t t , r n i r l l t _ - , l a N L R \ I l \ L H N l t r 195

rolr.r It ro g. br6-rL-ro cosd 6i.6srcL.r:ro ro xidrot{c t:rltl r' 11 xcrL o Q 6evc).)di€L uur6 to mt),i"or.

EpXdlLcore rripc oror r:roiolo;rd nor XQlorponorr:i r1v erbrzl rbr6rlrcr?r = Q r. O€('ralrrzir. o rnoioyLog6-. p:roqr:i lri lirtr ot: o:rorcrbrjrore 6cior1.:rry nqdEt. trrog rivct elcrLperrzci caioq y.n prc oqOolrirvru 6rio1 cpy6repcr5fi)(rorir!: dl curo< ob1.1ei o.qv rbic-x).erbi :rior,r rm'rt< ouaa< Fourier. Edr bi.-Srrcr $t(l 6.l(]rl. rdrE zd0t brdwolcr rirc[ oubr.roloq r(,)v 6coL1r(;)r 6r(rluogirr!]vrct ro npodltrtr r'iacr ta 6poi'yt tou< ourtc1u#< ttttoi' tor ott'6tnatot:

fpdtytz to b oav &w ou,t6tiaop6 h - !Iq + r2tf + +r,qn.

fr(1 !( iialirf ro J . r;rrial€r 6!(1.rxio r6zv.rotro. Ilo).Lcr).corirore ztl| rLg b!o;rlar,p6< 14< elioarory pr ql. :.rI' (prorcprt r),€!p(i n.liaaorgc .tJr. :.q b€ELirfl(l(t.lviiovr(tr iiior or irpoL (itrorr q1' .1, = 0) rrroq ar'ror rOr;)ro tipo. Arro :ror

, t ! ' =, ILEnrLbq q/q = l , i loutrr t iQt i ro.r ' . . ror: roor: t<t pr 17f6. ogoiorc, o 6r irrpoqorvr.).corrj< tivuL .r'1 = i1J l, o U;roE tnt!(i)r 6Qoc, 6r.1! noil.ftri(ior(ioorjl(s l1rlq.'. ()r |n(i),olnor oQC'! i!0(i!ou\, ),6./(,) o00oyrrrvrorrp(c. fru zti0e xol[cirr rori/) i/orrlLr_ 6vt1a (1JT).o luno ..r(! (v(1oulbr(ilovrcq r( r{otrtr(i-rL(l nci.evorrflr ro il:

Kri6t ,uiwopu L rroit0, ' t t t tu{])Jt/, + (q! Dq +... + kl l ,Dq'. (t)

I1( v( ro xdvoutr (1J16 74r'r((rlraorr llru 6dor1. tip( o Q rir(L terQ.1yrr^,Lndq.A f ! I - o o , ' ! , r v r r o r , r O J o r u r n f L , u d t r o . v , r 7 0 r r i r , i ( , r . v

ni\,ctlr€c. A!(i4touo(tttt rolq ouvre),toris crrtt bLclrlrotrcrrrrt EEior,rort .r'rqr + +r',(, = 1). nori €iv.rl ra1rT6ot[n Ue r]t! r (o(l)ol :rLyu,ttr Qt = 1,. (OL oriti.c<mr rJ no).),c;r).crolir!orw rrq olrlordroaq ror .r.) H )'tort rol rivar I = Q rb.

ErFrbii otroE Q ' = r?r r6(i, civ(r :ror r:rtrot-oxtro rl oeoou (volrxorrlrc r).ror eilclr eriolq z = Q7 b:

t : = Q r b = [;;]l,ll;l:l (2)

Ebr,; tggcvitovtcrL 6l.rq or oulror(io€q rou.r nou, oftog zclt ;rpollo4rerrrr;. eirarrcr tocu.r:pLxir y.vopa.vc qf 6.

H €Eio(lroit riL\,.ixov briTrtr tlli.o$ rr livrrcrL 6tcr oL orri).e< 6w t:ivuroaoozcrvovrz€q. To v(1 E}r(irQiroor[€ io ir ouv orvbrcro;16 rra L +. .. + r.,.r, €ir(lro ibro t€ ro \'.r itoouua ro ouorlltr.r ,{r = l. Tcr 6corxir bLcvrolcrcr livo1,rar

oPooraNloTHTl

or{}.€q rou rivaxa, onor€ ua€rt{!6Foor€ ro\r .1 r. ro1J (\rc}. F6hvst q 6o!)'€Lci.t-rlv oa6on avovrzil n€ain-ro;oq 6p04, Xgexr!6pnore ;r6vo T ov Qr.

nasantp4rr4 1o ),,610.. (tThl.rTq r:pqcvi.otrlxt :,lrt rpo11o![drrDq, 6rcv apo-6irLale ro ll nitvr,; o€ €!0€io. Tore €iXC4r€ rlv €u0atcr 6Lci ro! d xor I nao6oli

lrav (or6/aro)o. Trirec 61o+e rtv ruo€ia 5Ld rolj q], o :tcporopcrotrlE ei.var"I xa! I nQo6oLrt €trar (q|b)qr. 'Elorge LoLn6v pLcr vea epplveic tor trnorl' =LQT t),!, non 66Ex€ oro nlaioro Kriat bnruoyc b titrn ro aaponya rotpot,o-dukrturtrt,ttsoddiv to! :Riva a c tu9eleq 6ft ru, rt.

Kcr x(rr @.6$I. EneL6rl cru€g or:rpo6otrtg eivnr opooyci-rvtrg. o lu0ui,oQo(nL,r.r f l n,() .r v ' , Fn,d.tAr,rrr ,n. To rrr0uy,, 'vo r ' tc u:roT, ivououc nAinf l v ' , r iv ' , .:oo F! ro,(0. lo.nU,r r , ! rrrC,ryu,v,"v ru r ouvror i iou'v

t' lr = kJi,)r + ... + ((Jr):.

Arno nQirlcL 1( r'iv(r ro i6Lo pt ro i-, I = Qbl : nor @ob€iEc4r€ npoqyorltravor;.

ncqatip\o4 2 Ere!6rl Q'l =Q L 6to![€ €nior]q QQ1 = I. Orav o Q eivcr

U : r r , " o r , r , r r T " r o v Q / . r n r r n n n L ) . ( n ) . n o t . ( o t r o g ' l T r u l v t r r o t o u r t v r x u T r v o d r v orr.,n yqapluit, ro! 0. (:rov QrQ tlrct! oL otli)'cg.) Er€rb{ ro ufioral€oBu6ivur n(i).r o r(ljrorrx6: rivr?icE. dtow€ 6!(1 (!ftrcvrexo oripniQcro[ur 'Ordv olmrlLeE evig tetpayovrxoi ttir,axa tfuctt oeDoxslrot,txis, rtjat @ i6to axpo&irtlxtt Pt lra fQd.ltpiE rou. Or YQ4rllaq. orco( t{rqiv!Tc1r 1{(t olov srr6[r!o ]_ri!(rx(r.zcrtr0llo'vr(rr nooq r€ieit,r 6rurpopt:rrxdg rcrnrrhlaotLE (n6 rlE otli),eE x(I b€rtJ),i:rtu yet'rgrrprrri yori rlalayti(ilowor vc sivrrt oa0ox.tvovLt{dq. oroq rpEFtrrt

IIAPAAEIIMA:

0

[oAo]tlld^/AaFFor rilcxeg pe oq0ozcrovrxi; mir],eg

'Io x€qhLcro cru6 croxoleirc[ pe r1v .4r: = ]. xtrrpiE o A ,,.l ri\,.r1 a!.1-

lxcorrndr-rtrpc1ovllr6q. : ulltri rrlv n(edyQ(({o 6t2o$.1ors ErioIE ril! ISLcrbuur6rlra lLcr rqr Q.r' = I UroQei !c rJrr(ioXoul' :repLooorepeg ypnpliq ar'o.tr mlLr<. Elope ,) 6r(l\rio[(r.r qi. nori silaL oL orrf.e; ror Q. cLi.],cL -J.|[.a

ru 6ruvtoFcrrc dxolll /D > n ou'\rrox(i)o€E. M€ riil..r 1.6./La. o Q eivar nt eri ttri.vcrxag zcrr 6rv npf:rrL rc nt:p4r6ro1rp€ rrzar6it l.uol rI-( Q.r: = b. Aq ex rotnorr q ) t : r c " p 1 t t t 7 p i 0 , t { , ' t ) ' , y ' n ' u L t t ' p o y i v u v .

A! uri.ialFr x.irior(1 8'xaroouvit. or oa0oxdvovLx6_< orril.€E 0c €raEnr: vcdir ; .onorouv ro n9o6).qru. Auro a;. ' toJ ur y,( r f rA( lu 'vrxl ' , r : : r iv, ,yf : vu. r ,"Q,

i!fi1l|fi

. I ' ' f ' a ' ' 1 . \NoNl l \onu l l [H l1 \ \ ' iR \ \ l \ , l l \ l l t ' 1

0c bortr€ oTl (iII0FUEL xtlr yLu rouq tr(rQ(iJ,tL6yQqrtror!. To ]lLfrli tiv(t Irca(rri0'tdrt (jrL ! Qt Q = I EEexolar9.i rt ctiqlrt L:

t;; l]r::l']=L';1(3)

O (!)r brv r,i\,{( rl,ior (rarior0o{o: ror q,. x('-r(rlra!Fr ol.c{ aQlor€QdE oYiorQo-qo; zrrr (itrr6 eilrr o.rr /prLdriercr yrc rrr riritXLotct trtptl.prvt. Ot rctvovtzi<|Ero(;ofL< n(loizl,\(11 ror.ft;r),corriiovt(( io .,1.r = i/ tr! ro! cvl:ior0oq,o ni!(x(.: rpr i lpcr:roL, ibrr l qr - , { / , l r =.1rr. Eb(i) .4 r i1(r o Q ncL oL xcrvovtu!;rErot i roerc uivar Qt Qi - Qt l , . otr ! ' ( o ( .) /Q r, ivcL o rctrrorrxos r ivtrxrrq! AQcr1 ),rnl|] rirru r,t/1. rire o Q rirrrL rr,rp(1yr,r'vrx6c n(L rl iuol €iv(r (lxal{iitq. cirf o(2 r.i1(1r x(0(r),l.itiot p(ltrloc krlr ixoritrr rrl ),r(tr1 r,),<rXioro! rrrprri,ri;vur'.

3s Edr o Q ilsr ooeozcr\'ovrzi; orrt€s. rorf ro :rp66Lryn d,crxio&rvr€rpcydJv(l)! uir),onot€i1crt:

{/.r' = tr (:rcpu1J,1l61Q(l}(t1o ouorqpc x(,)QiE }.ust yr.c r& nF0roo6rra( |))

t|t qt =qrt (x(l.volrzrl €lio(,)ort./L( ro 6dLtroro.r orrt! onoicr QTQ = 1)

, = Qrb (ro rr, LoontuL pr r/,11,)

r=qr (4 rpo6o).{ ior | / : orr"s oql. tE r iv<l l (q!b)q,+.. +Ql,b)(h, l

| = ()()t b (da3 o ni,,'t(LxcE rOo6oLnc Eiv(! P = QQr).

or r f i . f ' , r ( ior rn;ror ! 'orhlorv re rou; t = . . lx tu P =. l ( .1 ' .1) r . l / . :Tor,(\ilorv r'lv:raooo),ri z(tr ro! rivaz( rooibirti 1c rtTou( -1. Or(v or ortl).r;r i \1rL oa0oz(rovzi ,< z. l r o,- t roonr(( pe t l . rort o niv(2(<,- l / . - l l i l l r ( r o' , r , r r r r r r o c a i t ' r z ' t Q ' Q l . $ o f , , r o ; o t r i a u c l t f r ; u T r o r u t ! r f r L ! r 7 ' ; ! , ' vf:.r(t(\,i(rr(( ri(r r.r brcvio$cr( ria(r oooozclol(1a(,r. OI naodoi.a. orou< (iiov€bt1, (),i'l),o08(0r0)vr.A rcr ro tJ ei"l/(t (i0aorott.r .rucin, rt'lv frovobr(or(rtr'vnaoitol , r ; )r i 1r = (qLl6)ql + (qTL)q.+ +lql , I ) .1, .

Tr-rviior,tre 6rL or :rpo6o)d< artd< btl cvcrrqir.lorv to L Au6 ouv6txrtveorrlv :rcpinrrn4 6F! ollrtiui'\rr( 6pcr; orty :rctpa).),t1)'irygtpptl ;re-ainrnnrl 6:1rrl ir > ?r. or noo{loidc (urac xupcilow ro p zttL oyr ro ttaxL,{o^r,^

"nt. , ( , AtTo f iv,r ro zrr) .rrrFpo :rou l r : r , ,L,o, i , r lLr rr Ar lr i l lpt . orn! f r , .p

7rn\ irsQrooorrerq FErodorLq tlno (11'(;)oror;. oaore rcr q 6el n:ror€).ou! ri.6o!

198 OPOOTANIOTHTA

6{iol. O rlivazog rQo6o}.rlE, ?ro! owi0oE €ivqr A(AT A\ 1A1 , (o[loro€ltqt

P = Q ( Q r Q \ - 1 Q t i P = Q Q T . ( 4 )

:rusrd)or€ orl o Q't Q eiwt o n €ni rr icwottr6g ivazr4. erlrii o QQT eivcrrgrcr nr e:ri rr fieo6o1,i P, nou ou[nitrlet [e rov rqvrortx6 otL! ori].eE rou Q (oP flE nqrlver cperd6l,rFeg) a}'Ld tlqb€viq€raL oro opeoycbvlo oqrtrl'{Qo)[q (rov

Fr6€v6IoQo rou Qr ).

IAPAAEIIMA 1 H €r6gcv4 aepian,roq eivcr crir].ri cl].d r!trtx{. Yt ooEm€ 6ttnao6dLl'ow€ 6vcr oqpeio 6 = (l..3/,.u ) fio €trim6o r:, J. H nQo6olrl rou eivs!p = (o, t,{)) xar sr,'[6 €iva[ io deQololra 6uo pqrovcuptrarv lqo6o),cirv :rdvo; otovdEova r ,{or t/:

xuL (q[ h1q1 v.sl lf Uq, =

O aivcunE npo6oltjg oivcL

l;ll:lttl'[i]

tlllilli:P = qyf + qztrT = lll . ,.* nol

nAPAAEIIMA 2 YndQ,xer. |rict ;rtetmt.rol xcrir rry onoia rl ]roQ€[tioi'ri €u8€i(rEo6l1€i o€ oQooxavovrx€g otrll'E<. E.iv livow p*prjoeq yr,titiyr o€ XAovou(rou 61ow p6oq flFl pqb€v. ),.1. tr = -3.t' =0J11 =3, rox€ n irQoordo€td !urrspsp6dLoljle rlv y = C + Dt obny€t o. lQ€E eErod)oe[E cos trQos 6io oyl(i)o1olE:

dqovoq q2

,l'tl" b

d4ovoq 4l

p = Pb = qg,t h +q,q,tb = QQtb

Ir.3.r0. nAo6o).i o0 rJtin€6o = ii9aorola rA060].61oe oe0oralovLad qr xar s2.

3.1 O?OOKANONRONOIFDTTTAN CRAM.SCHMID'I'

C+D6=g tC+Dt2=y , fC+D6=9 ,

ot 6{o mfi)r:g efuu oqqoydvrEE, o.Qo [fioeobl\E vo r[Qo6dloup€ ro i/ oE xdo€ Fiadrr6 rlE oiniu€E t(q! !o 6Qorue rouE 6dl,rr,ororx ouweTreordg C xo D [e1o:pLmd:

li ilrgt [ri]

:np€r6or€ r4v r6lcrireQ4 o:rl'6r4rc rou e = (y1 + 1t2 + !3rf3, \o.J ai\tctt o ltiooadpog row 6e6opivtov zar 6iver rq 6i).rtor4 noQsp6ol'i p€ pla oQl[6vtlcr suO€ia,evrir 1 Dl e{vcr, n 6!},t!orn ndQr[6o].ri p€ eue€lo nou 6r6Ql€rar afl6 qv oQt{.Etrttdi oL nfi)fg civttt ogqoyti|Ltg, to deQo\opa avt6r tu)v 6io €Ex.t)Qordvyep6r tiraL 4 60,trn4 napep6d,fi pttctli ,au ruv 6uvaui.w €ue€kbv. EneLdi6pog ol orri)'eE bev eivor, povc6nlc liravtouct(r -tcr dmvrlogctc 6ev eivcrloggoxttuot'txlt- o tiwog qg nQo6o).is neQdr(el orov nqQovopafiri ro reiQdyovotou 1.r{xov5.

O[ oQ0oy6vleE mf],eg ei,v<re rdoo nolwrpeg nou rrEite[ to! *6no vcr pe-r(6oqrr o' au*!. Edv I Ftoq r$ri nov lQ6xov ncQofllallorE 6ev €irdt tt{b€\,-I pdol rrFrl eivo! i = (tr + . + /n)/m- r6ft rt aQr(ri t(l)v xQ6ruv FnoQ€i vapercrorroeel xcrd i, A.r'i arE 1J = C + D!. rpryol6pcore ge tlv y = c + d(t - t).H 66k!oq e1Josio ,il,o! q i6ra! Ofi(l)E rar sro napti6erygc, 6qtonor.rpe

!(rr - i)vr (s)

I(tr -i)':( f , - i ) . ( r - - i ) l l r r . . r - jT

To 6dl,iro1o a €ilor rl F6ot r4ri ror q(oqr€ uriory 6vc rcrritrLqXo nino yrcm d. :rou npoqyoripevo A't A rs exr6i 6rcpviou ototXeicr iral tri, ,ro!percronilovtag rov 1p6vo xnrd 7 retulope rov trr16evlop6 cwriv rov otot-

lsio)v. H [€tdr6firoq oufll €[va[ ,iqQd6€ryl.ro rIE 6rs6lrooiss Cram-Schmidt, rouopqoyawrctroBi* r4v xar&ataaq ex rwv xportpav.

:q! oQloFnrrxi fpollrlxl A)'y€6pa or oQooydvrol nivd*es €hd! oqFovr!-xoi 6rofl 6ev fipoEel,oiv <rmrioetq. KootiE ro g.r]rl raQap6vow a!(rl,?'oi(ora, rooedlpa orQoyyurlonolqo4E ei.vcl un6 6]'qr(o. ff aur6 q oQ€oKrvoltronoilotl6tcvuopritov q(€! yivtr p0a crycrwlni reprx{, rrOdvds 6s1fteQ! oe onou6amr4ro, |lsrd. rlv oJralorqi zaL o64ye[ oe pr.a ncp,cloworoiqol ,4 = QII el ioouqrpropivrl pe rp .4 = IIl.

C= [1 I r] [yr y, yi]?'1 l + 1 2 + l :

(, - l.) ' + ... + (t,,, - l) '

o [-3 0 3][v' vu Yr]", u = ----------;---;1 .

( - r r ' + u ' + J -

* ).!,E,: :qr ra($(flzmiFc o olnaeaeiqt EdoEi oeooaavolrxonod

H 6ra6rzooio t(Dv cram-Schmidt

Yro06orr 611 6i6o"!rctl .raicl ov€Ei{arrFc 6rnv6oyara a.6. c. Edr trwd €iru!oaoottoaovrrri. rorc rt lo{ ci/(L enxo}.I. Irs vc nao6irlEr€ 6vc 6Ldv|op( r'ororaoxo. uno).oyit€r€ ro (orr')o. rrc vcr ;reo6dl"l,etr ro i.6ro brcivuopc oro rni:re5otorv 6uo ;rpdrrov upxei vc npooodoere (arrr)a + (l"r'.)t. frr va npo6cilec orov!no)iu)Qo. orov oroio or ir[orrg 6i6ovr(r.m6 rrr n,6. c. cpxt:i nci),1vcr npooO€oerrrq rperq npo6otrirg. To p6vo rou Xpldtrrar o 6),olq ror4 rifiol,tryrotrrolq eiv(Lr(1 €olrI€Qli,id llvo[€vc aTl',67u zcrr c? r. Qor6oo, .1Lc vcr ei,vol Llvro ('!]016,rQanlr 1,( Er'zr\rfooqr€ ).€"1ovrcq 'r{iv ei.v.rL op0ozcr.vo.vrrri . Ilpo0col pcrg td;pu€iv(l( vo liQortlF t!(lr ra6no va ro xirvolj$€ oeooxcvovLxd.

H t(60o6o( tiacr an),{. Aibovrrlr ro. a,b,c aa'" !Ird[€ rc .]r..i:..1r. r((1ro q| 6ev rnirpXer rAo6l.rua rc! lfioQ€i !c b€it!€[ ;rpoq rlv rars!0uvol roua. Arurporlpe pe ro prlzoq x(r ifro( .lr = (l/ (l Ei,!ci. Uoldbrclio brdvrloBo. ToIQ(Ypcx[x6 na66],I[a rcporor"riler(L oto r13 ;ro! rQ(rrr! vG rii,vrl! oQeoycb\rrc oroqr. H r6(c riv(! ,'c E€){rvlooqe pe ro 6 x(r cv ro 6r(ivuo[( a!r6 lXc[ oulroIo)oclr'f(r(i rrlv x(rrlrl0wol .lt (7ro! €i.'\ro[ xal ! zc(rslJetjvol rox ). 0u lrQilwt raaqaQiootFt atJliy tI our,ror(boq,l

b '=b (q lhyx .

"I

(6)

To l' €irar rl.iov oe0oydllo oro 4r rftr €t\'aL ro p60oq ro]J /, rror! rcrrs!0ri\r€xcr!xL)o; Fr(l !a(l r(rftrouvorl. brcqoQe.rxyl €x€i.vrls rou.r. :1o:1. 3.11. ro r/€irc{.la.ioExo oro ( 1ro! x(l0oAi!,rr r1v xcrrm0worl rotj bravlopoxog r/:. Enr.r611 ro4r raar€r v(l si\raL x(r tLovab(rio 6Lrivrop(. drarpoirpo ro l' pe ro grizog rou:

:' cllro ro otfri.o 61outrr 6eEi at qj xa q\. fL(i ?l! roirl oe0oyd.vrrlr(r€n0uvol Eez!\r.4t€ (n'ro r.. ro o:roio brv 0& rEpLilsraL oro cfiix(e6o rorv 4rzaL qr nor r(1uli.!ft.a p€ 1() rrinr:6o rorv a ri(r r. To. LLror6oo FnoQei \r.l aI€Lowl.artirocr o' aur6 ro uri:rE6o :ror ;rpintL vu {1lt,(es0gi.. (Eri! o!$6€i .' = 0.rorto oqpcriver 6rL rt u,b.c bEl, 4ra\r .lvrqhprlx.l, .1n' q! oa1lt.) AtjT6 norl(rot{6vEL €iv.rl I olJlrdrtirc( r/ nou ppeloUBe, bl}.cbri ro traeoq ror . rio! 6rixlrLoe pr( vit1 6|,!:ueuvol r.{dlo€rl oro exiijr€bo:

!r. 3.11. A.larao+E rt or{mri,oc i?, ror l,icr xovovraororonfrs ro o zdr b/.

3..1 OP@OKANONIKOUOIHTH TON (lRAl\'l SCHMIDT

ArTll €i.var o),q r! oll I !6€(r oL6xlqQqg rrlE 6!(l'6rrcoicE r{Dv cram-Schmidt: v'aEarAiooxpx a xdqE rio 6.d,wopd 4E arry@r'boE< xsrd rE 6.eu0ivouE rayinwopdzat ftou i4ouv i16tt 6pEd€i. H r66a ar,tri lQrlolporoleiror qald rcrtuvd." Eriv undpxrr xcr rdropro brrivuoFo. cqcrgoupr rrc ouvrorLiors ro! xordrlv zarEoervol r(,v qr,qr,qj. OuoDid ro povo8rai.o bLd!'ljopa q3 oxrtpoxio0nx€(rfi6 to c', 6r{ e6lrdE Lle ro F{xog tou: q3 = c'/ ]c' .

IAPAAEIIMA Y,ro06o1€ 6rr ro oveEdpqrd d|dviop(lr(l €ir(rl

flo !a 6Qou[€ 10 q | , p€-rorQdriolJll€ 10 fiQdro 6[rir,uopn oe povc6r,cio: q1 = a/ V2.flo v(l oQorlp€ To q:, s9olQou[e aa6 10 6E1Ji€Qo brilloola 1l owroldroc 1o1)x crrd rtv nQd)rYl rqr€rlolvoli

c: =c (qfc)q (ri! c)qz (1)

To xovounoftorl[6lo r.l: si.vqL xo ,'bneelivo lre ro pfxos iou b',:rol| eivq!| /,./2:

I r l \51-=l- 'r i-1.'-

l t ta)fLc va 6qoupe ro q crrpcrqoupe cno ro c rLg o!\,!o[6oEE x(rrd fll 6rcu0u\ro1 -[(jJv

qt xcrt q:(t = (' - (q! 4qt - (q! .)q,

t2 l l t r r t ) | t / r t1 fo l= i l - ' /2 o | .21 o I l r l .

[ , ] 11 rt) l t rt) Lo]AUT6 etvnr r]6q Fovc8Lcio. dQo €[!or ro q3. Exsvcr pEydLl nQoondo€r(r dJor€ v(rr€Aroaio(lj T o nl,nooq x{Dv r€reayovr,r6v e!Ed! (fiou crToxs?'oiv ro xouQomrr6[€AoE Tl]E Gram-Schmidt). To anor6].soFa €ivar 6vr{ ouvo}.o oQooxcrvovlx6\r6!(\,!opdrov qr, qr, qr nor ndve orr.g mrtl.€S ev6g opeoyovio! nivcxa Q:

.lil,lll"[;]

b'=b-(qtb,q =lil ; llif] =lL]

t I l1t, /2 1trt ole= lq t q : ,? r l= i 0 0 l l

L l l l r t 1trto* A! ra6loE q ddferja. o CDn, rL @a&.I!s tLo ro! S.hnidti

201

Eivc( rpave-oo otr or orit),€! .rnr6r €iv&r oQoo]tcrvov|x ag.

3T H p6eoSoE cram'Schmidt lextvd cn6 oelcipttlto 6lov6opoicat j . . . ,an r ,sr xolabi l€! oe oAeoxovolrxci bra\r{opotd ( . I1, . . . ,qn. : roj-or6 6rlpc aqalQ€i (116 ro a; rLg owr"mdreg rou xclo ttg 6!€u0uvoeLq rou€Iouv i6rl 6Q€0eil

a',, = aj kT ai)at , - (,f,-s;),ti. r. (8)

To qj shor to povd6rdio 6rlivuopc oi/ aj .

Ilapatilg4otl oTori unoJ,oytoltoi,t, Eircrr €wolor€Qo !c oQgo./c0'vr,ofio!4ooripsbru1,uo1raro. lorpiq vc cr(rLrrlooljFe rc [lixr] iotlq vu t:i.vur ivrt. T6rt or-r€roa-y0)rrxd( aiL€E El.rvoviio\ac(t o1o r!),oq, oroa bt(rrpotlge 1 cll-rrd tcr pti)in. Tabr(rvuoFcro ai €i\rcr Is i.brul

aj = a,-(rpo6ol. ! rou a, oo o\) - -( Ipo6oIYl rorr aj oro l l j r ) .

To p6vo:ror 6rft.fJ6Q€L ci.\,(! o flrno( rilE rrpo6ollE, ro! zQqot[orrotci lC) prlxuvovrxonor4r6lo 6L.ivuo1u a' crvti tou povc6(rio! 6nvt:opctos ql

npouorr r,), ,, -.,,; = !l+" = rri dr,/ =nuuuo^r ri,u,,r oro r .' l l t l ' t t l

:ro nQonyo pcl'o nnpcibeLlpc 6a tilqre rc i"6n h' = o.\ v.o.' = n',, ltopiqrL,< r.rtxryfirlLrdE QiCe<i

H rcQoyovlonoilol A =QR

Etxr$iorrps an6 rov :rivcrcr ,4, rol onoiorr or orri),eq rtrcrv 4.6.c xclz.rraL{qagt orov nivczc Q, ror olroion or orn},€! eiv.rl ql.qr,fi. noro sivat

I nl i .n t l ! r ' l : " KLf l no. ' f iv, , r I o/ in4 df r fprooorfA' :SLcrordos|c, 6r.lv E€xLvi'B€ cno xdL '{arorrtyo![€ ora qr.. ... q";OL:rircrrg A zcr Q rivor n €ni t. oxrrv rcr SLcvropcra neprdlorrct otovm-brciorcro 1d;po zur 0o nQdriEr vo xtidpt€I d'vog rpirog ;rivczcg ;ror ra ow6der.

' [i] ;[i] .-'-,- . [:] lil,l il

: j . r ( ) f { ioKAN()NrKor lo lH:H rQN cRANr scH\ ' r l rJ I

II r6ec tivcr vn lprilrortrr ru a otrv owbtcopoiq r0r! q. frc riuQd6€rypc.ro 6r<ivu4ru I oro 11. 3.11 €ilcr orrbutll6( rtrr! oa0o,.arortr6\, 6r(rLro$.11o)1tlr tet 1) v,<a lLriirma lvr,rQiiou$e noLo! €iv(L a!r6E o ow6!(1o[o!:

b = (ql qq1 + hl Ur.

Tonro cldtErrlr. rio(l) orl! €Ei.o0)ol oro nlcioro (1). Kcis€ Srcvrotra ror€nm€lor €i.v(l' ro doaolotrcr rorv qL zc{ s: onlroiu)odrv ror. IclaofroLa. ro c €iva'to ci0poLorrrr rr'lv qr . q:. zcrr qr o!vror0)o(i,r i ':: = ful4q + (|l | ),1 + lqTc)q:. Edrlrzrlpcioonpe rorno pr nir(arq. rcripvorye t4 via napayovtonoi4oy A = QR:

[,.]1," "l f".. r'ii :'i:)ll(QcrrtQrtor€ r(1 lrtb€lr.i( orov r€],€urcio ni.|'co,'Cl. no! €t!c(r rary0)!!xog. ),oyo)tor tq6aor 1rr rov o:roio rztd,rircrL r1 brcrbrn(1oi.( GranrSchNidt. T( TrQdr(brrrviolrtrro a z(r ilr rreoliXorr(r orlv i6I( r!0{ic. KcrorrLl, ra |,1) x<A tti.(t.;rrprilovruL oro i6ro frire6o. Tcr rpiru bruurop<rr<r r rur r1r brr owtroiq,rpavrirors tr[20! ro rpiro 6{llc.

ll nuQ(./ovrorioilort troLtiler. pr np;1 = /,Lt. exro< ror 6rr rrlp<r o:rprinoq:rapdlov Q 6rer op0ox(rorrziq orllu<. O 6trrtpo< ru-ori.1,r'v ovopti(r:rar R.br<tL rc p1 yrlbr.vrzri ororlcic t:ircr orc 6r.!rit r1g xiir)ic! 6ra1)1iol,+ (rcL ro./az+qr.1 t,r dxer rtbrl Zato+tonolrl0ei). To Err6; bL(./rD\rior orolzaic ro! li ri!(lloL trprolroi q/ b = tlrt xol qi

" = ,t! , = rD :ror i;plurlxav npolyorlirtDq. Tc

d l , ' y r ; v l , , . J l , ! / r . , r r r r , r r - , r l r r l r r l r t . t , / 1 . t y ' ' c o : r o i r o u u i o c . r r H o ) . ' rir(0.1i,o,'1ofi orlorl ti\r.rt

:rit brc{ydvro rotl ft 6}d:rere rn s1x1 rov a'.6/. c'. Err6< qr brcrytovior ei.vnr. ra:roD.u]r),iroLcr 1(r\ qt x\r e:ror aqrrpi0qx(v pt: rrt 6La8rz{1oi( Cra r Schmidr.Tc i6rc rc oooorc1,ovrL(i 6r(r\rirolL(To. fioLJ ei'\,eL Ie q\jqi,qr x.ir cfior€J,or! io.lvrLrr4rrlo rnE oQoo1ac\,o''lxofioinorl!, fieQr6Xolr(1r. orov ra6ro r(Qd./o!r(l Q.'lor'r<

r iea1,olronoirol Q-4 Elv etvcr rooo opcricr ooo r1 ZU (L6yo rt'l

Ir( q o€tDpia q; raaFtr.z'ir A),./€6acq x.l| €EclLa€rr]{a x€vrarr6E yrtl rorEr:rol.olLoFonq. EU(! I ZU 0ErDprtoEi I AEH, 16r€ I QR €iraL o OTE.

H yEvLxrt n€airft'rol €ivrlr (lipL6cdq 1 ibLc. O :ri.v.lzcq ,4 etvar n e:ri nzcL .ro oroLleio (i.j) altor el,tcLt qi a;. Au6 civcr prlbir ora! ro i €i.\,cl

f l l r l f l ,2 r ,2 ol l r t t r t . /11. l lo o r I o o r l l | r t , t -eR

lr ' r ' r I l t , : r ' / : ' ' l l r l

: :r.E.: Ri-lhr - Ae:Ld:.

tr€ycl),nrsAo rou j (ro q; xcraozarcior4xe xdOr:ro .'to .rj) xcr dro! o n €ivtlr dvo)r0rytDluaoE. Tc orort€ic rou €Fqavi,to\'rql orov ruro ({3). rbtoir€Ao 6rav ro.rj(.vrLraraorcori gr ro llr';llqj:

o. i= k [a ; )q + . . .+ tu j ,n r )q ; r+ la ] l v i . (e )

H b€Erdr n),€!pir €iv(r exar6(bg Q €rri Il. .feauu6q rlvcLlJlrr{d1.

3U Kao€ ?n €rL n nfvoxcg A ge 19o91.r[1{65 avEEdQrqr{,g orrlleE paopeivo r[sQcyowonolqo€i ot A = QR. Or oq]"eg rou Q eivor, op€oxovovr,xlqtuL o l? eiv<o <ivar tpLlrovrxoE xc! qwrorA6rlr4rog. 'Orav nr = n ecr 67r.or ornivcrxeE eivcr r€rpcrlovD{oi. o Q yiverft oQ0oy6floE rdvqxqs.

Aev radn€r v.r E€Xdoco ro oq[c!i(rorsQo ol!€|o t1E op0oncwovtao:roir1o1g.fio! (nLonot€i ro rpriELrp(l el.crltottr;v rrrgcrldxr'rv ,4;r' = /). Or r(vovodgeElocoo€lE !:i!cr fi(i).r o(,)o16E. a).Id o,4",4 cvrr"mpdtpat.rL s!xol6r€Q(. z(r

A 1 . A _ R 1 Q t Q n = R r R'EroL

I 0€frLr(;)6fi €Eioororl Ar',,1r = A/ tr uri.o:roLeirct orry

( l 0 )

Avri vo Li(nJF. q! Qnr = l, ,rou 6s! yillercr trrivoupr r1v Illr = Q7|. roulirttnL:roLl yprryopa. 66rr o ,R civcL rpL1r,rvr,z6q. To,rQu1,lru1ui6 x6o-rog ei.1,o!otE nrnr npciltrq qq Gra r-Sch idr. nou .1r(rror,.vrr( x(I (ex(i( 1(r 1,( 6Qoull€roq l? zct Q.

H ibto tbia op0oltuvr6rlrcg ecpcrpg6lrcL xcL yro ouv(errlotg. Tcr +rirovorcr otvlpirovcr ravar oQooy(iraft, 81,6 oL bwc4€r< l,.r'..r'r. . . . 6er eivaL- Or(tvrt / yqo'trr'n f,( or'vnrlrolro( nFrrov!,rv z,( ouqurlovi,v. nporri:rrer prtr orr4rriFour,ler. Kl(o€ 6po< oivcL rpo6ohj :rcivo oe gLc E1l6€lo, rlv !:u0Ei(l oro Xd)Qoollvrlprrlor{j)v aou rlr:Q(62€! 1(l fiol),(nlLior( roJv cosr.r li sinnr. Aur6 €iv(rlr_vr€I.;f n(ooporo [r rI! nseitrrosq rov 6L(r\ruolz.l(r)'\, xo! 1600 €vbLaq6Qov.iror (EiEet !(l F€),€rlo€i. Td.rr{ri dXoqre r.1r gLc 6ou},€!ci .fto rov Schmid| vaoQ0o".ivovrr(ono!{o€r rrq bwd|€r< ncQciy€! r(r nol,u(i)\,!itcl ro!Legendre.

XdlQor ouvoQtilo€ov xar o€[Q66 Fourier

Aurrl €i\,o! [ru olrroll xar rpocLperr.rrj ncpdrlqcrl,o< gt xriprnooeg xo].6Erpo0ao€LE:

(1) !.1 €!osl,i{f€! rouq nro {rl|ro$6vouq tl]rEreobtaorclrox 6rovuo}rcrlrou</6pouq.

R1 I?r _ Rr eT b i Itx =ert).

. ' r , r t . o x \ \ . \ n i o n o l H : H t . \ , , F 4 \ r { N u D l

(2) rc Efi€xr€t!€! rq r66€q rou Fllxouq xur rou €o(r)TcolxorJ y[vou6vo!. ur6brrrrrloput( r, oE ouv(ado€r< /(r').

(3) !c (lr(ll\'olaio€r q o€rad Fourier tllE /(i) rDq dOaorotr( lro\'obrclordru!:rpooor.rirv Or ('r,(lo7,"vrr: mr'/.r: :rou :rrrpdTou! rov 7riL,o filrrr ru |llrirotuzrrl ltr ourlllilo!(,

(,1) vtl €{(10tr6o0r oaooxcvovLl{oroilot Gram-schmidl orc ;ro}.ucbvupcr1 . . r . r r . . . .

(5) v.r 6a€r rit 6al,rrorn rpoodprol rq< / (r) crr6 pr.a eUoei.cr 1ptrppr1.

On :rpooncr0loouge vr. (zoloroloourr F€ ouorltr(lrr1to reojTo (ruro ro;rpo1pcp3cr. :rol rvoip 6r'cr ol.6x),4po :rr6io veov rqugroyri,v t4; fpc4ptzrl<A).y€6aq.

1. Aqou trsLFrilooul€ 6l!}!( rolJq X0)0or( R" r€r€e(otr€{]< brrioruolg. eivnLqlorn6 vcr ncicr r1 uxilrrl pr4 orov R-. ror:rcpr61rt 6Lc ra 6ravuopc.rc r,=(i .r':.r':,. .) fiou ilor! [rc( iorErQI (xotrouoto o]r!!or0)o(i)r,. O X(l)QoE (rlJioq.oruv 6fl Lrn(i!'XL'r. il.e11o< ra^ ouvLotoodr r,,. r:iv(r ro).ri l(€yciLoq y!(1 v(lei\'(r( XprjorFo<. ML(r noli xd.urrprt rbi(l ri.v(r vc r{a.rrrioortrf rov y\'o)orooQrollo rolr |rizol'a. o(rv rlv r€rQoy('rvr).ri Qi!( ro! nu0(y6erLo! ooQoiogsroa1r')\' r€rQuy(;)\(')1,. tu \.t qtueiooultE privov ExEiva ra 6u,r.ltoltsw. rou ii ,vrrcrryaoptto pilxo-<. H .ln€rQrl osro(i

. := r ' i+ ,1+, ' i+ ( l l )

nein€r ac oryxiilr,r o iva r€nsQ(o[6vo (0aoLoFcl. M (rJr6v rov r€eroaot(toropfrtL lxr5prltvtr o:reqo6Lcioraro otrvol.o brcrvroprirrorv rot o4:rrpL),crp6rivr.rro brd\ ' r ,ol l r r ( l . l / l . l /3. . . . ) cr) , l . r i t r : roxl .r t f l ro (1.1.1.. . . ) . Tc br(vuol(r{r1t rr:rtgrrlrilo grlzoq rorort),orlv brcruopcrrrxo Xrirpo. propouv rc rpooreOorvtr [rcEi rot , . ( ' +t l ! l f l+ la l )zat vu :ro)J.ar] ,(ot(ro0oiv trr ' (pr0troi<. ( iQ(.oiorEi.oul, bL(rvlolc(zo /(i)Qo. A!'ro. ritur o nrQi(plfro: /(lpoE HilDert

O I6Qoq Hilberr r-iv(L o (lxo!o),o./rx6q rei)no< !.r .r(prlooltrr riq b|(lor(iroeq vai,ivor! irrerQr.( xrI rl1r,r67eo\ra 1's xe(rrl('orJtr€ rrl ./€r,4rr.roi( rou onqoLotrtrlorEr.'xitL6riou 2rirpot. Or e),ir-i1rtr< lLvovrct r:re rpo6rriorcrru €),r€rq,orLbri. ol:rug(16oLdg yt'ro'rtcr ;ruQC6oLo€!fui. z(lr o!,.ii0r1E( €1J0€iEl (\,(y!(DQiEo!ra! [e xo!i6() rp6no. orf)c z(r jrAollotlilt)gr T( br($loB(I( | z.rL ( ri\'ur o00oy6!1.1.r i r , .y ro,o.,rrgzo ?r lolrrvo ror. : r i r 'ur 1rr16n.

lru = r ' r r i l + r ' : r r ' : + r .

Aro6Ewvulr(r orr ro iloporoF( c.w6 olr/xLiv€r, zcrL 6tr 1ro 6uo onoLc6ti:ror€Src'r'noBctrcl rE(roLoro€i !.r LoXrEr I cv(odTlF.r rou Sch$'rrz r'7 r' < L u, .Ar6$t 1(r orov /6ao Hilben to orwrlpi.rovo 6er rivcr :rorr geFltrr:po r4r

O ldrqog au6g 6xer zcL picr dl,l.I oELooqlEi{D1l rb!6rrlra: Tov owovrqt€u6 ror ).cg petcprpLdo€rg. Ta "6ro!uo[(lrr1' xor lnoQ€i v.r Ei.var ow(rQr4o€L(xL aur6 gcE rpdpreL mo 6eiitepo o4reio.

2. Yno06or€ 6fl g€()Qoi,p€ rt owdQntoyt /(r) = sinr oro briroqFcr 0 < .r <Z1t. H I ewi lorirq€r Fe 6v( bLcivuqrc rou e1n oi'6xLrpo ol,v€/jq (|'t6ouvLordo€q. rLE rqrig qg sinx vcrrir trixoq rou 5r(otllltrrro!. Ei\'c! (biacro:r)dov va eqcrpp6oo4re rov ouvl0tqtdvo xov6vc( fiQ6o0€o1g ro'\, rclQo'/6!ovrov ouvLorarorirv yrcr vcr 6por.rpe ro grlog ev6g t6roro! 6!c(vuopcro(. To o0Qoro[quL,ro .r!'Trxaolorurur. xcrri rvcr rluoroLoytrti traL dv(tnd.fsu]rro rpo;(o. dtrd trlv

( r2)

o Hilb€n xsiQog pcs ayrve rdoo6 ouvde4aflov. To oravuqrara €i\'(rt ouv(p'rrlorq. 61oqe ivc tp6no v<r F€rQdF€ ro Fryros rous xoi o Xd)poi TrEQ#I€! 6i.egrq ouv(prrlor( nou dlouv ;rarep<roprvo lulxoE, 6fi(l)q (xq{ir;! )id[ nQol]loljF€-v(DG orov (11). ALJIi)q 6s\, nEQtiXE! 1,.X. q ouvdqrry4 a(xr) = l/r.6L6rt tool,ox),ip{0trc rllE l/.r: elvcrr ci;rerpo.

H ibrcl r6ao q< (vrrzcrrdrorcrol< ro c0eoiol(rros F€ olor().i$DF(l rcQdryrrv.6 ro to.r',ttptxti ltrtltEao 6io ourtrQrrlo€tor". Edv J(.t ) = sin J zct ,(r) = cos r.i6rr ro so()1€Ox6 rou( yr!68€!o €iv(rr

tla t2t

\ J . ' J t = | l t ! \ ' l . r t ! ] . r / { i n , c o s . r ' i . r = u, ln Jn

( 13 )

Aut6 porcilrr pe ro 6rc1!!o!(tu{6 r:o(,)rsQlx6 llvoFevo /7, zar ouv6ier(rr. :rh],ttrr ro lrixos lio(|) rq< (/.J) = l/llr. H (vrodrnr.r schwafl rxlvonorirclL r.ornciL l ( / .a) l : / l ly I . ouorzt i .6no ouv(0r iogrc. J:ru,( I \ in. t z,1r r lcos.r . r( 'JvoroiLoa ro €on)r€eurc llv6Bsvo €ivcl FI6av, 0a ovoBirLowcrL oQ0oy(i)\'!EE. Aur€Ssavr( vaL oQ0ox(vovrxlE gerci crn6 brripeorl pr ro pryoE ror.rg y6.

3. H oEeti Fourier !L(4 olJvoQrIoIE €i!(rt [!c ovd].rorl oe rpitovc xcL otvrpl-

v(r) = ao + dr cosr ' + rr s inr + arcos2r + 6r sin2.r + - .

ft(l \,u uno),oti.o€rr 6vav owr€I€ori oroq o 6r, rioLl.(nLsslcior€ 1r(r xq 6ioir).crpeg pe qr crvliorortl ouvoprlol sin r xor ol,oxlqQdor€ atro ro 0 6oq ro2;r. (H owdpqo4, oQi!€ror o'arro ro 6!{iorqto.) M€ d}.l.a },6yLa, rrdQr€ roto, ' r r |Qrzd yrvottrvo rr f l ruv 6uo n).rrpd,v 1te ro . in.r :

J , , u t , t ' ; " . ^ t . , = , , . ,

J , , , nJ , t J + i l '

. 1 " co \ r s in rdJ + i , '

/ , ( sh . r rd r '

l.l OPOOKA\ONIKOnOIH:HTOtr\ (;Rr\ltl s(rHN{IDT 201

:rI b€qLfi nl€rpar, xir0[ il(r (lfio xc o],oztrIp(i$(ro €l!(r [r]6ty, rd,ry €v6E. orooroio ro sin r nolLcrl.coLirirr ror €orrr6 ror. Te tt!iroyu xet ou,4ylrota dt arl.ro.Eli ro* oA9o),6yui, oroq orov (13). Aoc ro 6r siaar I cpror€Qri ftl€lQ(ibr"rrr"pqrtlq ge crmo ro 6!a trl Bl6€vrx6 oiozT,Iao4rtr:

. / , ; " ! (r) s in x.h

G1',., . str-r)

O ouvre),Eorrt! Fourier 01 0( XAryqronororor ltl cost orl etol ror sinr. zrrr oar 0r1 Xorlo+roroLolor ro cos2r.

To rl\,6rgq,aao'r o uurong rolrq uro),olopoug rivur v.r borpe rlv arnlolictr€ rq r!o6oLi<. H o!vror(;ro( rorl bruvr4crtog 6 rcrrir lizog r1g €!0€ia: r'iolrcady€r ro ./ rno).olioUIr€ nohl v(opirr0(i

:f lr(r orfQ(i Forliet neo6d)A,otpE tnv ! nd.vnt rto sin.r. H owror{i)oc rou /)o (urri rrlv x(re!0lrvorl rivcr czod(i)g 6rsinr'. (frc1 bL(rvl'rolLft1( Jlr.rv 7n.) OouvrfLtollg 1)t €i\r(t I Luoll €).(lliol('t' trrp(Iirr0w 11; p1 r:rrLriorprlq lliorDDq;1,1 rn .r' = r7, p r, (litr( i,oyr(. Gvro( rir6o.r€ | lc) 6 sin .r 6(10 ro brryar6! n).IoricrrfocorL' t/.

'Io ib|o (ilo!!rr 1!( 6l,o!E loq 6Qo!: -r1g or:lpriq. blLrrbrl xri{)!- 6v(1crivllr rpotlo],1 rou t/ rd\ro o! d!{r lliro'r'o ri ollvlui.ro\,o. Ene!6rl 1(1 ltLilovczrrr rrr orvlpirovrr tivtft op0oy6!(, ? d€reri Forrb bit,rt t< otN'rtrq'tiyt. ror''tturLloruroq

tl LDc ireoq iru otl1rr.o (tLnxial|,y tu 40o.) /rfiiron aE61,o1,.

4. Yn(irptlr ririeo; Zolo4r('lv ouvrerltoao,\,. bl.(iroQcrrx6\r .rno IfLiro\,c Lcrdr,!Itriro!(. or onoirq bfl ri\cr nrivrorE oe0oi,drvr€:. Or .ri),ouorcQe< Fi.|'irr r(1noi,lri)!r+(r xur 6troluyri;q bev lndpXtL brcioriya. oro onoio. oropl r.1r. oL lQcrExlsirtol .iEovrq oL nwupr{oer: 1..r zct .rr rcr eivar xriueroL. (To EorDrEprr6lv6pero tov 1r(1r.r ' : t i . r ( lL nr ' rrrort 0urr l ' 'o,6LorL t ivuL ro oi .o ' . ' ) , r iaor[c r lq.r'r.1 rr' crrt6. rl ri),toL€(rreQl ncrxl6o),rt orrt ('uy.iQ.rror ?/(,,) 6tv tivcr ro(ioaoLor( ft)! iTao6oir)v rI( rirr('r oro 1. .rr. 6a UrcipTeLaraq olrbErrz6E rpi.xoi,6n(oE (Lar6ri)q o (.'lTi) | u-r1v :rtpi:rr(l]orl :rrvdrzcov xcrLIrir),rm( I br.lori16€(]1 bi5Erllr.1ft6 ro.,'l(lkriE r(rcm(iouoJ< nilcra ror Hilbert.: r o b r d o r n r r l 0 a r < 1 ,

O ni\,.I'rc( arr6-. 6XEr .lvriotporpo nor nr:pL67tr pre./(iLolc apr0polg. bL6rL rc 1. rzrlr /' o;rf1orv roin aro ro va Eiv.r. opooyrirvL.1. ,4}i6[rl xer yrr tvcv r;ro),o,

t {1 .1 , {1 . . , ,1 . . ' l l j l l , f , l l l l l''' 1,1;l ',',,';il,',,'.",1-11: i; i:l L . :]

I ! . iL l r o toT i r lo , i r | , r r ! f \ r , t r r '1 r (L r r t t r lu0x( L .

OPOOIONIOTIITA

yrorl, I l.roxdox(torl 'tivErc.t cA,un6qopl, 61av trQoooioorF€ LiYoq (rx6Fi dEov€9.ii"ai rpr.xnxri

""uFqoeo ncr ),rjooufts to A'I Ai = ATb yr<r vc 6poripe ro

nLrpdoT€eo nohnbwBo 6(r0Foi 6dxcr.A4r66or€a(. dror oor"gtpopo vcr l'rlooupe auro 10 o{oqFo pe o'tcr}.olqi

c ss: xdr0€ oqd)$o otQoyyul'onoinorg 0c qrqe0{vero F€ orrl'r€}.€od F€Ytl}.i-r€ao on6 l0'r. An'rylv iri't l F€Qrci, 6€v na6n€t xa0 va sv66ooult€. H tlQooil"Ytolpe :ro),udrv4rc :rp6:ret v(r ei,\,ct bwarf. H otootl t66.r €ivo! v(l ir€Qdooulr€ o€ oQ_ooydrvr,ous <iEoveE xa! roulo onlqir'€r op0or(vovr.xojroilo{ lord Gram-Schmidt:AvtllNdF€ ouvou(loporrE rcrrv 1, r rcr, rrl, or onoior vc eivor, op0olcirvrot percrlurollE.

Eivcl ox6r[Fo vq eQryoo0oipe pe 6r.irorqpcr oqrprEr0H6, 6fi(0E 1o -1 S ;r < I,bdrr rotro xa0Lord 6).ss t[E n€Qtrxls 6wdp!:t6 tdosreE o' 6].€s rl( dQrrcsi

, . . . , , ) = / , " ,a* = o.( r . . r ) = / r ' l . f = 0 . ( '

t ,'Eror.

FrroQoul€ \,cr E€xrvrloouF€ q 6lcdL1.ooi(. Gram'Schmidt ono6sl6B€vot t'r = Ixar r' = r' oE rolE rrQ(i)toq buo xdo€rous <iEovsg, xcr to [o1o aou alouF€ 1((broa06oouF€ €i.acrl rl yo)!i( tr€rcqu rou I n(.| tou r: Kc.td rov rufio (7). to tQilooegoy6\,ro no)'0oauFo 6f r.ll

- ( l . J . r ) . ( J . . . r i ) f ' , " i i , r If ' = ' -

( l . D l

T( rrol.ui)v4c ,rou xorcoxtrdlovrtl !r' cw6v rol tq6ro ).d1ovtcr no;xdvlgarou Lcgendre rcr eivar oa0oY';)vt(l B€r(1Eu totiE. oro ltr(iorqlc -l S x < r.

Enalitut04

To nLnor€or€Qo roludrvuyo 6<r0pou 6ixa €iva! tr)iov ufiol'oYioL[o, l(DQiC afiE'pcrvro)'oyieq. naoddUolt(s ot x<i0e 6r(l o16 ra rpdna r0 (Yl 11) nol'utwtrcLegendr€.

5. Yro06or€ 611 06i,ox[€ vc apooel"Tiooupe rlv ]/ = 15 F€ Fr(I €u0tio Ya.4+ui('+,r, olo briloqFa 01 .r I L YrripXow tpelE roulirxrmov rQotroL vo 6aorrtr€(.|'rn IIv €uo€io. nou €drv rol\ ouyt{eivstE [€lcrli touq. F]roptitt v( xctcr'd6eftol,otI0o ro'r€qAl&ro!

I a ' 1t l ) An. lrr ro l l r l l ' , . | =. .s se eLriyrol( rrrQdYtlvu. H r ' l ioroorl Ar,4.t =

L ' ' )A?1, yivetcu

L,r , . t I

(1. / l )=/ r . " ' - i l , l , = l ; ; l = , 'J I J L J J ] I

[8,i] 8,;| [5] =[8::]l '' li il t;l =ltl

1.1 (.)POO(ANONIKOnOIH:I|TAN GRAtvl.SCltI Dl

^ ̂ r 2 , r

_E) .u / l o rono (no r r ro E '= /u { . r ( C - D r t , t . t l ) - ;C - iD -C , ,C D - t, D . Ot r(p(iliuyor uE r0o< C rzat D. ;rrrri uro bruiqriort u, ro 2. obrwoi!aioro orLg ztnovrxiq clto{lo€q rI: $€uobou (1)l

- *+c+ l r=o zc r +++r,++r=0.(3) Eqetr6or€ br(brxcoi( cram,schmidr yt(. vc clvrrzcruonio!.rs ro.r tr€ro . . (1,r) /(1.1) =.r .- l /2, nor r ivar o0oo^/cirvLo oto l . To;pc nl ,6ov oL

yololLdrorcrrr; :rpo6olts 6ivo!! rI 6i).rrort €uo€i(:

c.D. = f+i.#++r- l, = j.ir"- jr

ATKHEEIT

3.4.r d) KarcTedlft 1. riddr(,r( .Sro,i'ofl. ie€rr6oiis nE I = a'+ D/ ot( f,.6oFd!(

l t - - . 1 f ( t = 2 . U = 3 . t \ t t = - 1 .

! = - l f t e | = 1 . ! = l ) . t t c t | - 2 .

a.aiu rtt' or onjir< r.i!(r oq0o?dlr<.ti).llq'.irf nl tti],ldnt .re.k1 tocf|1i, s.{r6r.iqrf iva rA.iq,,ltc lic, tqdlrr io oq{i}.tr(

(./) EF yrl,m r, lelovrt o ro ort(ii{,c €iv(r rrrbiv t.. ot 6o,je€( ror (elraonou'rllt,('rd; rRnis!! s:Lodoro)! adr bno a./vd,d(D!. nor daioxsroL rt 6EE,.i riiled.d. o7.o'l f. ro! Zdeo drrt6v 't(r ro(r .iv( rt rao6oLi nta.pl

3.4.2 neodd),(rr io , = {0,3,0) dF x.ie. dla a,o r( oq0ori(volLa.i arc! ,pdr( .rr =(2/3.2/3. t /3) 'aa r j = | t /3 .213.1/3 j zar z( rof lv 6a. i rF ntv Jroo6dj i to ! .poro €r.irrdo r()v.rl z{rl .t.

3.4.1 Bo.ir. srid'l< rry raodoli to! t, = (0,3.0) mo ar = (2/3._1/3.2/3). nqoo0tor.1: re'r. rorod(odtr: i(,ul;{'t.: ?i{ r!17rlotr ro uroriir,xu. fr. r o p4 r ' ' - ' . ' - 4 , . ' i , , r r r r , r r r f r o ! n r F o n r c l

3..1..1 E(iaorQr z(LQr €ivdr oa0o.r6vror niv(z.r zat €rouilo{ tzalo,rorony tl! ere =1.b€i;t€ o1 o QrQr siwr uiorF oafio.r(;vto<. E.iv o er rivcL ,t *qnq,,i *,;,,i ar(r o Qr.i!r( it orAo(tai !(r(i d. ioroi sia(r o erer: Mroa.ix. rc 6qrit{,r;ra'.pvot'{q'zrs r(r,ronlr.. yrc ro sin(r+o) zot to cos(d+d) oroa ,ror.d).(dl.rolloxrvn?o, ! QrQ.r

1.1.5 Edv u eivct trovq6raio 6ra!Nfc.6rit$ 01 e = / 2ru? ciraL op0olirvrog rivuraq(Arro. 0ir(r ru( (!dri(olt. ylo)oloa.:riots o..v rr€rao1l||ralorrdq ror Househotder.)Ytro io l iors roa e. ora!

" r = l t12 t12 r /2 L/2.1: .

210

3.4.6 BaEir€ }lro tqirl odh. alor (ior o ftiraaag

l t / !5 t/ . /u Ia = tt,/1 2/r9 |

l t / , /3 3/\/r1 I

!d eildr oaeoy6lLos. Aurt nqlalr ac €ilot alo ttorabroio dduJola. oqeo-r6vtomrE dl).€s 5no milsq-. n6on !].€1]0taia arop6v€( Eirallo€lqrE 6t! rdu6leovo or

leoFrric ti!ovrqr cdloFdru( oauoxuvovrric.

3.4.7 >npdriqolras ar' €le€ics ro b? b, 6siEr€ 6rr o xalolas ror n!0d./6qd loxnsr

?ro ororov6rinoft @\,6moFo b =r|q| +. +rtnq,r oe0oaalovtkda bKrloparo\,:t 1=4+ . +t1,.:€ o!B6ol!op6 pt nt!{r.t( b = Qri. dad a!r6 6si.xas' nd}.,61,

brcnlaourar ld Frlzlr Qr '1 = x 2.

3,4,8 nao6dlcre ro brdvr|qa b = ( I , 2) rdlo o€ 6to b|alnopoto no! 6t! €ilor oeooydlta,ro or = (1.0) xar ro a2 = {l) l). A€i&€ d( ovli0srd !e rlv oe0ollbvtc reQiftiool.ro deoo|opo tu! 6!0 llorob|aoidrov rTeo6o]dry brv |oou.lt !e 6.

3.4.9 Edv ra bl.1\,1jotato gt, q2,93 eivor oqoofcirvru, no|os ouvblaopds ro! qr xrl|g: eiv.lrrhtor€ot€poE sro q3i

3.4.10 Ed! rc 6nuro[(lr(r qr xcrr g" €ilor oa0onu'ovrxd, noros swbloot16g .i!cr nuot6-mseoE qro b'dvuoFo bi Ena).leenot€ 61t ro 6rdv!sp( sqdllaros eivd| oa0oyrb!'ooro qr rar gr.

3,4.I AsiErs orr or aras oeooyrluos tri!.lxas- ei!.rt rar dvo reryovutoE rors eivoL bLa/!b-

: r .4 .U noro nol lar ldoro ro! a l =[ l I ] oan€L va oqaec0si oro ro o, =[4 0] l r ( l !a

l lv l l ro onord) . rdud oeuoy,rv.o oro , ' , : naqayororo ' rorF tov I i ; ]

oc On ur

oQeordlov'"d 6Ldvnspard mo! Q.

3.4.13 Eqaap6(}16 6l(r6rxaoic crlm-Schmidr dro

,aL ye.ivtt ro oro#hopa fi! pooArl I = Qn.

3.4.14 Yrooiotr orr ra 6o06vrd 6romo!o1.r €ildl

"tilBa€ir€ ro oAoorololtrd bro'udpdrd qri qr, qr.

3.4.15 Ba€iG ira oqeoaavoarro o!vo).o qr,q2,ih an'to oftoio id q1,9, noe4ouv lov

r6ao oql6v ro! rivoro

"til,[i]"lil

"t:1,[i]I I 1 l

A= 2 11 .2 4 l

],.1 OPOOKANONIKONOIH:H I'ON CRAM SCHMIDT zIT

noro! 0strsl.rdr6lq uoxoqoq roatdX€r ro q3: Ilotc.itoL rll.i.n €Iaxioro! rEraayii)lo!ioLi ,4J = I, orcr io lJ = [] 2 7lr:

3.4.16 Eaqe{iorE rt! oqooaavovraoroiryI Grm-Schmidt ror

f |Uq-1: ' "= l lt 2 t r )

ox A = QH. do0avro! n 6tovlopdro! oi. ndo€ 6!(l an6 ra onoio 61.1 'noi,v,ot6o€q, aota civd, r( o ittoto tura A.Q rar Rr

3.4,1? rt(l ro! ibro trilana A ftoL tre i) = [1 I l]T, ]aryuo'ontors qv .4 = Qfi yLo aailo€re p. .idxrora r.rad, orra io rao6).rua A,r = b.

3.4.18 Eda .,1 = Q,e. 6aatrr dldv anl6iljno .l,|c ro! rivoxa rao6o],rts P ndlo mor X6aooqldv ror n.

1.4.19 AsiEre 6r' ra rAoronorru6w cram-schmidr 6riloro

<" = t: - kft:Jtl t =c't oT c't)q,

trfta(iror,! ro ibLo nr(ilrotr(t c/ 6i(',s o1r! (7). Yno).oroo'td. cirsr rotr! rTro rloraoagro \( n, l (ro^uJr rk rco(o) i : urc{ nooc Lr io

3.,r.20 Bqeirf ro p,tros rou 6lcrv1|opalos u=(/\/2.11/1,11J8,...) li(lr fl]s- orv(iorr(nt(/(J,) =.r (iro 6ftrrrpa 0 < .ri < l). noro eilor ro €oorEerrc tw6revo rol eir rar. ' s ciljro ro brdsrllc.i

3.4,21 nor(l eilllr I n).loriote0t or,lda tml a oos r + b sin r orl olJriiqrlort I = sin 2r:. oro6rzldotpa -tr < r < trr no|( etvaL I tr].ryrdot€et cu0sio ri + Ari

:1.4.22 EE'odaowas qr rcei./aryo f!: ptbl!, 6erift ql rLFn ro1, bl no! .),cXldtotioLef to

b | . ' ' r e : - / , L r i n J - y t J r t ' J . ,J ' r

:ut"Qilart rouo tf rov ,r'J!rF).conl Founer bi. E.i! !(r) = cosr, roo €ivoL ro brr

1.4.23 BqsirE tou,( oulr].EotiE Fourier a0,ar,r1 x$ x).1!(,iorxriq ou'daqofi r(r). ]ror'oouxar p€ 1 oro b'dl'qr( 0 < i < tr tft p€ 0 oro dnopdro! StdoqFd tr < r < 2ii

( e , l ) ( a . c o \ J ) , ( e . ' i n . , J" ( l , l ) k n i . , c 6 / ) ( \ i n r , . i n / )

1.4.24 BqEit€ ro sr6uElo ro).!.bv!Fo Lesendft oro brdoqfra 1 < J < 1 tv( z!6rr()ao].!6wtto, oa0ol6vto mg l,:t zat :t1 I/3.

3.4.2s IIor( €ildr I r).ryLam€fl €le€ic yqaFpri, ftr! rdqa6oht y = o']. mo braoqfor : ! f , < r ;

2t2 OPOOfONIOTI{TA

:rov nEo (7) to,v Grd-scbmidt €nd)'Ioerror€ 6r! ro ct ei,vo! oe0oy6vro oro q1 aar

92

Bqstft tla oe€orol,ovrri 6don io1j rrryol6eo! no! noady€rar drd ra ar =(1, -1,0,0) , a , = (0,1, -1,0) , a3 = (0,0,1, -1) .

Eeoap6oft it pd0o6o Gram-scbmidt ord (1, -1,0), (0,1, -1) xa! (1,0, -1) yra 106a€tr€ p|a oa0or.aro vr,rii 6dorl ro! enrrddou ol + 12 + oi = 0. Ilolo eh(rt 4 oldoidorla!'roi rou rJro16ao! xar ndoo prl p16s!ud oraviopold rQorttiol]v an' irp C'ram-Schmidt:

O TA \Y I \ I I IA ) \HNJA I ] : \ IO : FOLRIER

3.5 T O TAXYT MElA'XHMATI'MOT FOURItrR

:ro r6).oi 1rl( r!i€ur(i( 'r(1acyairqorj xi{v(llE lv€il:r orq o€L0d! Fourier jrorijr(r laa!$Lxri ar,lctilxr o5 dr"r€rQt< 6rtroriro€r<. Tcr 'brcrvtotrara'

lrov oLntrprrlo , r : / { . . | : r o , j u . , 6 , r } . . r . ' , , , ' . , , t r r ! ' \ i , / , u u v , t r r o v , r : T A ' . / r , u : r o u n ( L ' , l t ( / ,rorrq outtieortq Fouricr. An trurl rry rirrrpl (l]4oioroi( lllrLrol(|)v x.rr or-rr {rovor, no}.icr7"sot.rolL€v(,)1 [s rorl< orjvrf].coli! Ior]c Fourier. trnoAortr€ v.{(1r(in(r,(l1or,trf qr .f(.r ). Art1rt €iv(l' I ri(o'7.r] r€ai.rroo'1. 6r(r. rrl o|ri),c6r oFoufief. :rot( ,'recrrLzou rtnoio]roloiq .)orooo. crro nou rfioioyi.iorrtrr- Eilcro 6KrzAtf6, ttEiaol\ltdnopq Fourier. O Fourief €nrEfi, dirl or|( nFirrQ(otrf!€g

ll t).,"0i( ui\'(iu,r0oal yac4rtu{rt irit€6p(. 6(o'i6$[1,rt orit! (,oft)r,u(;qr(.A!r| rIE oulir0rrlort< ./, r( 6r'bot,irc rtlc,r rrr( (xo)"o!okr o0rstL6! ,1 . . . . . r,,l o : r t r o ' 1 r , \ ' ^ r . r r r t r . , r , d . i . r , . ] . , ' ) o ' o r , r , b r o ' t r f y o ' : , \ ' T r t u , , c , , n r , - , r l c , . h r -Lor,0iac. irtr orvo),o rr owrt ),r,rxr;rv Fourier . ),. . . ,.,, L. H oXaol tr€rft€! ro)! tti(L . fi\,(( laq(trlxri. rip(l :rpirrL v(l bi6cr(r {l:rti iv(v rircr<. Ar.rtoq rivaL oltit, xq ltoutier. (:r<! to.v o;roio e!(prrir(r ,acL ol"6x),rla! 4 rr,Tvoioyia ent!rp-./(1oi(lc rl,rlqL(r(;)v orltLrir('v. Aburirpfr(D< tol c\r naoaoXrl(lr cro olr].io. {rx6a!q,(rorlrrr,r(ir r{irt(rr( rj ril}.fr1!ko!r,( )!ieq (1 crx6pq z(L (viTveLrorl :rr.rpc},aior) r0olpa r l rr1lLo:rort i rcu. Awo [r ,1cor] [(r i ( rrcr ur ro! ni lcz( F xor xcrronLrlrnoQfr va (v(xrrl0L'i. 6(Jr! r'( n($1700i 1l (QXr) l flro.,'(. Ei.v(' (noq.(o'o Xilqor ((ri.(( ro (ifl z(1r or 6iro tr€r(o7,r}tcrrotroi trroQori! \'l1 rxlri{.oiJoL,v r(1zir(r(:

( I ) o irlrior('oq,o. rii!(o1.lq(2) or rioi),(tnl.(o(r('t()i trr

I noantL !( rL\,(r cn),o(xur F ' :rei:rer vo rivctL 1q)iyoqror.

:ritrftxr ;riiol drt0lnouv rar r buo. O ni,\,.lri( F ' tivcr Ftooroq uroTprivLc rct portiirL ;ro),r pr rov a. npriyl.rrr o ]' rilaL drtrtrcrArzoE x(loqrooi,rirvLoq (rzrrlq (ari, iv(v ol!1,r€LsoTli v];) r(r ita' iv.r p6!o! r-i(lrr04ro:r( ororlri.r rcr tLttLL prlabrxoi aQ'orroi. At|T6 €i!(L 6v(1 trlrQo ("yrirLLro roonoio nairEr r( rirta!'roo1+r! zur to :rir}prirvorlrr naacz.ir('J. Or buoxoiirsfl,(X|or{'roL,,i,vr(L ca6 ro leyovoq ort liu ru aroLTth rot' F xuL tor I r

tiro.r hLn'lytL; rwi< portbLxol apL9ltor\ u,. Alri ttcq r),)iaou! €Lrr(yo)tli< ororE. r ' / ' a r . r o ' c u o , r i . ' o r r - . r L ' i , 7 ' i i , t r ' ! 1 7 , ! r , , u u ' t . , , i o ' v u / 0 r t o . t r o I o . o , j , , r r t L(lirooljriorrl eiioorol Lr " = l. nox ncQril.r r11itovu. owryirovc ncl €k0Erlraaqi)r,v(iu€rc zcr 60io'{€r(r orrll r(10bd rrl. 6r(1k0Lrrt,c (v(i),ro]]E Fourier.

H rrzo).i.( (\,r(oraoqriq ror F rivctL (EloorttrEi(,)rl z.rL (xopc r{r trv (116rlrnv ro poro:rpotdplpti ror (Lcr t(6/ar ro 1965 rtruv 6vrrrE ro $6vo). o 6r(rplroqp.r.1ozltLl1rLoro. xdir 0( z|l-rriX€ tl|.r €.vbra.{,6poroa odol. :iiF€Q( dtL(u: urdATrlzdrL jr.1ar1n.i!r'). Or no ,(ricrorcrqoi ge ror I ztrL-rov -F I eir(lL drNcrov rcrlilorv irE i\,ll.,' F.Efu'aEflrr? 1prl.7opo xm iEuwo ra6ro. Alri rorv nr lrTorpLorri,r: r o ; . ; . . 1 : r ) . 1 { o r c o t r o ! . r n , : r r , . f 0 } . o ! r | 1 r , , : r o r ( , u r o r / J r u r o , n r r d n u . | \ , . y ! ^ t r f r o;rrvdztj)v oc|v ro I re a;raLrri ;toro (rr logr)/2 6{rrcrcr. o ;ro),icr}.aoLaotro!

€i.v(|! o ibro( al,Ld 1iv€rft F€ Fto xal.yt (tva6idraql. H ava6r&(lErl (lrq ).ry€rcrTafiE Mewct\paiaFdg Foxrier (TMF).

H :rcpcrypcgo< ou{ larrvh B€ ro u' xrrr rLg rbr6rlr€g rou, ?i.Qvil olov -I'xcl| xcrcrLrller. orov TMF, rov rnlu petaoppotrop6 lic! rrq Eq(1Q[to16E torJ. Ho:rou6cicr rycppoyj ovopir(era orvd,l'rf4 zat ro xtreL6i qg ta!ruXfo( ri!(t oxard tq rN ou'i).Gn<.

MryabrxiE eig€g rIE Fov'i6oE

M rQL?6( ,rQoylrorLz66 €ELoriraq paogri vct ilowptlcrbLxtq ),uotrq. H elioorot.rr + I = 0 obityrto€ ornl' errvolol rns t (rcftt xcr Ilti il). Alri o€(lJQtstx€6rL eivaL pL<r ),uot 116 elioarol; xur ro 0ip( elqle. Eirv ztinoLoq Q(r)rol.ro€ yltrqv .r'r - i = 0, unrlple crnirwlol: Or trlQayuiltxaq eiqrE pryc6rx(i)v crQreldveivur nril.r ptLwbtxoi aprefroi. Ill)inr1 !a orlodElo€irt: rnvbr(opong .r'+ iy. penQrypcr(xd [6QoE r xor 9(wcorLz6 p6qoq y xor 6ev lperdLovrar rcQrooor€patr,priopatcr. Kcr0r :rotrurirvupo 6cr0pou r 6p,L rr crxpL6ri< pi!r< (€v6€Xo[6v(oE

[lpbLxrq xuL tv6eXoF6vr';E roi.tra:ri.t<). Aut6 €ivat to O€trt-].668q Oe(;)QIla qEA).y€6e(K',rc! I LaEI "trrydrridq o4r:rrpr).c4t6d'tet q bu1'(16try( flb€\,rottourou q,cvr(orLxou pipolg J = 0, on6rr o er0tog €ivct :rpclntattx6q.

Mr4 [v6L.rqaaow €ELo{i)o€( 6no; l.rr = l, rou 0c dnQtnt- vc alorr! taoo€QtE),uorLg. oL o:roir< 0c iapr:re va €L'!a "ot tiootQr| piEtg tt 5 pord6ot". Or bi,oterpcrytovrxi< AIE€E qg [ovdboq 8i1,al I I x(L I -1. O! rir(Ql€< QiE€S €il'ctr oLrfrQ(./!,!rr6E aiE€6 rco! x€rQ(yr)vrxd)v pLqdv. 1 x(ll 1. i v.&t i. O uqrfu6,< i0c1 rl{cvonorFi rt\' ir = l. b!61! !x(rono!r:i l! i: = -1. fio lG (iy6oeE QiE€Erqi f(ovirbog XA€rcE6p(1ore r|q rerQ(y{Dvrr{dE QI(EE xlg I xar roliio pcE oblyei oE$lyoi)rxorlq ((n0pouE 6r(r'rg o ui = (l + t)/y'2. T€redyovilovr(E ro r/, ,rQo){rJrrEl10(1+2i +i))12 nou €i !c!r io ibro ge to i . bL6rL ro 1 + i r €iv( l ! Ftbdr. Aqo!10 rrre(irrDvo eilaL ur = i. n oy6ol 6riva11 u,s = ir = l. @c:rpdret vcr u:rtrplrLariror<r oum4rcrLzrl pd0o6o< t6rir :rdpo.

Or FrynbLzoi rrpr0loi rou nilcrztr Fourier ri!<rt rel,€itog €rbrxoi. Ttr nQcytt.trrzch rox yr:p1 r ' iv,r owit tr i lovc' xs' iu 1, 'vrronTd rou6 t '6A1t ivrr qt i rovu:

I l l

( r )

Y:ro06ore 6tr to rpa4atrx6 pdpo< orpeuirvercu. mov tilovc r zaL to rpcvrcrorrz6piQo: orov (iEova u (:X. Lll). Torr I unoorao4 ro1l / Clio rlv upx4 r'ilulcosr i9 + sinr 0 = 1. o u rcad)a€rcr cnov pova\nio xJxln z"cr oxqpcrtileL pe r1vopt!6rrLo q 1tDvicr d. )' cu6 ro oxtlpcr zdr0e pryc6Lz6,< npL0p6g : =.r + ig t1uprc 86o11 oro eaiirc6o aou ovo[dr!€rclr ,rryaarxd E ir€ro. O],ozrqQo io €nir(€6o€Eercl!€rclr oro K€qd.clo 5. onou €FqsvlqoficL ur./clbLxoi ceroloi ('( L6rorr[6g

]5 O TAXY' ME IA'XHMA TDMO' FOTJRI ER 2t5

tI.3.r2. To !ryd6|1t6 eriltr€do r(rl o pol,obraios rnxloq.

(o{6!q r{(tr nQ(r.rufir'|d)l rinr{ir{ov). E6d), lla va }.uoor4r€ qv rr" = 1, xe€rcrg6-froor€ [olov sr6rrad orueicr ur, rou aeqrdl(rvrcrr orov govdbloio l{tx].o.

n6g urol,oyiEol4l€ qv n-oorrl 6wop1 too u; :ilouQ,a -ro €rQd'/ov6 ro1Jl.lJroQe[ l/a 1Jno],o1[oeoi op6oos:

L': = (cosd + i sin d)l = cosl , - sin2 d + 2i sin d cos ,.

Kur cwo rpoivercr dqryo;ri1qr vo rqrlveueei Fe tov nitto qs 6utl'dolsg'torviag trE rerTtovo$fieioE. To :rgcpcttx6 gdpog cos: d - sinz d toolttot tr€ cos 2dxclr ro qcvrcorlx6 p6pog 2cosdsind Etvar ro sin2p, (!4rer.rirore 6rr. ro i 6evouF r€QlrlqF6dv€iqr: ro gavraqrlxd F*QoE eLv(rl riv(rg trQdwdtlx6E aQ!egG.) 'EIo[

rr! = cos20 + i sin20 xo'r ro r€TQdyolo rou rl, ,reQ!6xera! sriiorE orov Fova6ratoxrjx},o, ol,ld dler 6fi1don yavid 20. Avr6 Fos l.dv€| vo urorlilaoeoulre 6rr ror/)" ollpdrile! rI yoviq rd, ripd./pa fiou (ltqo€u€I.

Y?rd€r(€! 6vdE xol,trrgo< rpdnog vo oxnrorioor,trl€ 6wape6 rou u, Ooo!6!clo[6E 101) ouv4lrt6vou pe ro rpirovo eivar, pr.c pr,1o6r,x{ exoerLxrl drivaprleupor.rg dvo zar. lovfoE qdoeo4 d:

c o s d + i s i n d = e i a . (2)

w2 = e '2e, u" = c ine , ! - " t " (3)

OL Krv6v€g no),lari;,ootoolor, rior 6loiQ€orE 8wrilr€tov, 6mx (€)(e) = e5 ridlct = c t. €Eorol,ou0oril lo Lopiow, drcw or. ex€6reg eivm qcwcorrnoi.:w€r6g or 6uvd{r€E €ivcr 6ruog rr5 neqqr6r,orye:

116 oPOOfQh_IoTHTA

H n.oonl buvopl snlariqEr 1 Yolid nd. 'Ord! tr = I o awiolQoEo1llu omltati$t tn ttoria 0. Edv no)'Lar).ootdlcrF€ rov cosd+Isind F€ ro1'

".(-al I i ri"( A). Os tna€ne va n(iQow€ l. naaYp(l1l, €n€rbri cos(-d) rootnot

gr cos /J (ro otnlgitovo eivar ciptK ouvriqrlol) 1((t sin( d) loouro! tr€ s'of/(io qrirovo eivsr. n€Qltrt oul'darlol). o rollonLootcloF6s cul6i 6i\'€r:

c idc i3 = (cosd + i s ind)(cosd - i s ind) = cos: 0 + sinr a = t .

tryr€i.ro{ @l'pdgct nou r1p0e grr gep<r 6vtr lpdrtllro oro MIT c(no avav qulazF

otrivo orrt N6c Y6orrt. ;rou Qorouoe cv o runog (2) itsv al l0rl( Eivm. nadYlt(xr

raqtibolo. tlrov ro 1t(1),oox€qrrsir€. titr tpeq owcrqrrloe6 x)'€r6rci rov pa0q{(r-

rrx{iw olviud(ovrct $fiati lolE B€ 6vav 160() sQtrovtxo 1Q6ro. H rd'vl€prlrlr(ivlrlorl |((< rtcv !( tlr,Deriooull€ 1I bwutroo€l(,(

. h - 1 a l i 9 J : - l i o l ' - . ."

= r _ , , - _ t _ 3 , .

_

TonQaypcl l t {or$Fapo\1-Al lZ+ civcr ro ouvqptrovo To 9(vx.ror l t6 qEp6pcr 0 l9r /6 + . . etv(t to lpirovo o oro5 €ilct 00)016: ]|CrL 0o er,TEror)v v(l

eilc4rr orri),tt ptcr opcu6teQrl (a66ttEl

M' c!'rov to! ruro FnoQottl€ vt( ),loorill€ rr]v EEioool I," = I Al,rrl1i!.1o!r ''' = l. onore ro frd nQatr€! v(r tr(l( (l6Q€!, 6(loxtqovraE to'v l(]jxlo. tlioo oqv&ati. II ),ioyt€irc! v( €TLt't\oupe 0 =2tfn: H qurael.Hi * n-oolri ailanlg

(4)

H ,}-o('Iti 6nv.art tI: €i!(r eril rou loor'rt(t F€ l :rnv rr€Qinr()orl ,r ='l x(rn = 8, crurt rl :rpd;q pilu eivcrL

H raroQq aiqa ofl[qri!€r d = 90d r&! I oY6o4 , = 45". Aurdg eivcrr, 360"/4

xcr 360"/8 cl,Lir 6tv eivcr ot Bovcitzd! r6rcQn€q zd! o'/6o€q pi!€E tqg trovirbos!

11 1ru ] = c o s - + l s m = /

7 n l + l, q = c o s - + i s n ; = i i .

| :.r,E.: H xa6rt'

OI un6l,oln€( r6rcr€,rsE Qiq€E €iva! o[ br.rvdFetq i2 = -1, ir = -i xor. ir = 1. Orri:t6LoureE o16org pileE eiror or 6uvriperg a'!, ro!,. . . , url. Or pileE loc#xowlr€roEu roug. navo) orov govaoloio r.ur,o, l(oxd 16lo 2zr/r. :4rettnme lcvd 6rrro r€rdl(,)vo tig urr eivar xo 1l)r, nQoyFa rrou 0a ano6€tl0€i ouol(rortx6 Yrcro! rotrt F€raolyu(.rlolrd Founer :nr€16oft enLorF 6it 10 ti0eorap6 61ar Norvg r l l b r e i va r pq6€r , f Lo qv ru1 = ;cn r6€ i vd lo1 {a t66E1+ i -1 - i=0xa ry t s

1 + trs + rrri + . . + u,l = 0. (5)

Mr( (rr66e!€n rival va no],?,(tfil,aordoow€ rtlv dQror€pt nl€uQd [r€ to t , 'IolJrqv crqrl\,€[ o\,o].].oio,rn. GlQoxufixet ro u'8 + u'i +.. + u.r!, c]J'ti ro rl! Looitcl pel.) To otr6 onr€icr olQ€qowo! xcrd yoviq 45o dlld ncp,qr6vouv ouvol'll(d taibrcr o4pein. EneL6{ ro pn6iv eivft o U6voS spr0Fos nou ntrQo!6v€0 av<rrltloiotoE,6r(lv aol.l"(rrl.cor(ro0ei F€ ro trb, ro d0eolo[o trQ6nfl \,o tiv(! Url6€v.*

O fiivcr,icE Fourler xor o a1'tiareoq6g rou

:rl ou!€ti nrairro)(''l I o€lQd Fourier avanoQdyer rrlv /(r) o' 6tro ro 6td-orrl[s oorouou 11( xQlo$onor6[oE dnrQ( r ri.lovq r(0! ouvr|tdrovo (i Ex0€r0x66buvdFrL6). :q btcxglTrl :repirroorl. 6nou 6rotr€youtrE !6vo r ouw€).€m6E. 6€vpnopoulrr va reptpivnrpr xrirr :rapoporo. Zrptige l6vo I[v ur.irtlru ot lt a\Fit.npdrypc rou 6ireL rr rltorirerg. I' dvcr orLx6 rp66lqpo p€ n = 4, or €ELod)o€[Eno! naQqlow lrg r+16g 2, 4. 6, 8 eivor

a r ' + c l - . ' _ " = 2

4 , + ; . , - , \ . i ' , = 4 ( 6 . )

a l - C l + c : ( 1 = O

q - i c r - c : + i c r = 8 .

H oxoLolJofo r(l)v 6€6opfvov €ivgl y =2,4,6,8. H or.to).ouelct t(I)v ErFou|ravoveivcr r:6,11.r:2.r:r. Or sErdorxE lrFo{v va riQooaloQioolv prc o€LQq Fourierr€oooQ(l)v 6Q{Dv nou loerat€! ora 6€6op61'a ̂ d r6ooeQ(r rfirn6xo1'xo olF€ia orobrdoTltFa ano 0 6(l)E 2?r:

2 Y t a t = 0

, . - 4 Yc . r = 11/2a o + c r e _ + C l P _ - + f r e - =

tt Yl,t( ;t = 7'

I \ t 'a ! = 3n /2 '

J.5 O TAXY> META:XIIMAAEMO: FOIJRIER 217

* t aw,i utv &eirtuotl r, = 1rr. oror ro tr .i!olrioi,o.. oL qi!< rtil(tiio,ry. fl{i 6no, ldry,t @nl)iror. om6oo o' xnlxa( ailEq rou I 6i6otarrimF ao€otora F|l6ir.7.olaic !o dti.rtiouvcrElM( dld bi!.

Au€_< rivcL oL tioorprg elro6oeLE (6). flc .r = 2r q o€rQd €firorQ6q€l orqv cQllzlirlq l4rrl y0 = 2 t{cl orr€Xi(€r n€Qto6lzit.

:TrErd)ore 61| I o€rQn Fourier yQirq€rar orxaorrQo ort tLLy(l6'.1{ri rrts Foeqrt.oJ; orrlbuoolLo( r(,)v €n0firz6r,c'N" Fdd).ov. a(rpri o4 owbuaqr6E 1FLr6w)v x(lLouvrN!r6vo)'\,. Orav oxFneoo.cF6dvovrar 61.€q oL (rppovrxi<. or 6uo rp6;ror eilur|oobuvqror. ),oyro rlq rcortrcs

( L e i ^ ' + c * e ' ! ' = , a c o s A r + b i . s i n l . r . ( 7 )

Mroporlpe va:rugcpeivou$s nrq nQcrurlrr1t iG o€reds. (LLir 6t(! (urr-E oidlrotor]!orov r6tcrpro 6po. ancrLrrirrrl etbrrri :rpooopl xar ot flinot 6€v €iv(r io!1'oi. M(rza),trepq pd0o6o< civaL aa lQqorponoriooqt€ tq ttyclbtxri poa{ti zct !o i,r'oou!r*rry €;io()ol (6) r)( naoE rot'E oult€lrtxd; r o.. r.cr. c1.

Aur6 b€! €ilcr 61j('z0l'o !c yi!c!. E(iv fipoo06ooul€ rK r€oo€er! EErodo€lq.tlrdQXE! r€QdorL( (),l,ll,oaacliQEol orl! aerl'I€Qi Jrl€lQri. To cnor€).eo[o ri\'clr,1.1) = 20 x(r lror ro cf aiv< 5. 1 pr:q rqrrl rorv olp(irr'! 2, 4. 6. 8.

Yrrirplt-r rniolE ro6noq crQrol< rou .r. flolJ,cntr(or(imr rlE eEro6o€q F[l.-i.-l.i x(r reoo0ior€. T(t ri.j!!x( (Qtor€Qd d.Lll.oav(1Leouvrcr rxroq ror4cr nou roourcr tL€ 2-4i - 6+i l t . Iweacir i ct = -1+i (or lF€(i)or€ 6rL sircrtrycbrr6;). neinrr !( uncQl€l zcl n((xilorq !16€obo< yr(. ro .] z(L ro (r.

H 6!(16rx(oi( q,ui!cr(! roo(Qor€Q( yQdqo'\ra< rr< rELor;)o€rq Is nivc(z€<. Orr ioo€QLq r:ro, ;ofrc Hiv(t Fc = u v. tu o niv,rx(tc t ivul

Ta 16 .lr,r.i ororXsic Eiv(tt r( i6L(1 trr rous ouvre).rm6E rrF (6). A}.ri rou i'v'yQilVaF€ rQotyouFdvorE i, ro i" ouFnirr€r Fc -1 Lcr ro i' riv(t| i. M aL.ntq popqri o l' ' crvclvarpilerar ruxoL6repc. Elroq (n6 1o1, ou\aeL€orit l/4. oa\,rior(xxloq nilrcrxcg dXE! rI! i6r( oxQr6cirg FoQqi!

O ,(cr' €uO€icr\, rol,l(l,r,(lorool6t 6ib€! FF I = /. fru rsQcldelyFcl, I 6€ur€QnyQ(lutri rou a €ni q 6€lr€arl ori),I ro! F I ei\ror (1 + 1 + 1 + 1)/4. floQ6por( xolr(r ur6Lomc bLcrydrvr( otor"leiu rooiw(l! pe l. Evc xufirxo yrvo[ o no! 6i6EloxoLt€io €zr6q qE 6Ky(,)!iou rivo! ro 1+;+ir+ir = 0. (Oullo€irE orr ro iroQo(opcrow rttirprror qrldrr aivcr Bq66r.) An6 rnv rilvco yqc$rt n6 c = I' 'tl 6).€nouF€;rhi.L 6rl ro cl eiruL I triol rLgrl r<'rv reoociprurv oruirrov, r"D = (y0 + vr + gr + 9r)/4.

t r r r 1 lr ._ I i t : / i l

l t t r t , i , l

f l I I I ' l

I r l l r { i ) ( / ) r ( t r l |- - l l r { i ) : ( / ) r ( t f l '

L1 ( - rF ( - ; f ( i f l

]5 O l AXY: IVIFTA:XHIVJ TDMO: FOIIRII-]R

Fr ' = ur t l

219

z\FEiao\ 2t fiiv(4€E Fourier Ei\'crt qrrlotol,oYu{o vu aatglouu€ rL< lQou[6q larrrq o i),rE (lJro ro 0 6ar! ro n 1, cvri ci(o ro 1 6(,)g ro n.

H br(lbwcctic ror 604xolt€ :rpoqyorpivog 6N r€aroaiLr:rtlL o€ ,r = 4 1-t()aLie' , . o ,r ! , (z,rc :ro" ouv$11 ro J trr ro . t r)rop v ' , / r i , |Tt ' r ! rrr z,(r Iaavrrorpuq,ri erixol.a. O nivcrza< orno< ;rdptorirver rr tEtotiroeL(,}itioe lrcr c;ro ttEonoir:q {ln(rrri .1][6 r1v nEn€a(otri,'rt (Jrrpti cr + cLc'' + (' dQot) v( olj[nirr€L

tlt ro J (os n o{F€ia). H na(i'rll o]Jlxr(,]ot ouu6rdvuL Yl(r 'r = 0:

q ) + f + + ' r r r = r / l )

To burtrpo orpt:io rivcL ro .r' = 2r/rr. rorr t:Lori'1at tov zpior"po crpr0po rr' = rr"'/".

a o + c L n , + + . , , r r r : ' i L = l / r '

To rpiro oiltrcio .r = 47./,r reQrtrqLti(ivor ro ern'/". ;ror.r tircl ro rr,r:

Att6< tivat o ttiraxdi Fourier F'

I.La r = 4 o crpLo0rig rr, )ir(r ro i. 1 rtropq aii(t rttE $ovdbo<. E! Yal,cl o rt'roourur pr: trp nq6rrl n-oorrl pi!( rr1,< povltboq .'rr'./'. To rtorlti.o tou -F ot4r

.i ./a4$ii zar A orrtl.I €ivur pLc 6wr4r1 tot u,:

a o + c L r , l + , . . + . , , - I r ' : 1 ' - l r = r r ,

Tc r'noLolnc olllrri(r eroriloLrv (vrixepeE buvdf€L( rou r| zat i€)'[xa xo ouorlB(l

f l r ,-- ' l f , , II r ' l l J I. ' ,t ' t-t,, lro't l

l r , ' , - , " I t , , , l LJ, ' l

I-I na(int yQ(1$ri'l i1€rj =0 rcr r n$irt'l ortlLl 61er I =0,6r(Jl o)'a r oroll€i{r

frc 'r.r 6QouU€ rcr c :rpinrr r(l avttotpirlo4re rov F. :qv n€pirrtDol.l Eiri 4 o olrlorpoqoq :rivuzuq neo|€iue btn,irlerq-ror ;. u€ r!] Ia ].6YLa oFr ,..rr(orsua,t6rav ar6 ro l/t. Arr6q €ivaL xtlr o lEar11o! zalotaqi oF I :rpox!:ner crn6 tor prwbLxo apL0f6 u, L. nox oxrtfr.lri.!€L ltDvia (r€ rlvopL!6vrLo) 2;r/rr.ltir o ur oxrul'rtq€ ytlrvic 2'r/r:

220 OPOOTANIOTIITA

3v O cr'riorporpog ?iiva,is( r, e'rt n llcraoxerdlerar on6 rq 6!vd[€!E

1

)tLg nrpL:rrri;ottg n =2 rcr rl =3 auro oTraiv€t 01[ o

. . f r r lt=L t t ] t r€ l

, - - ' l8X€r , r ' = -

x),fLbi €ilaL vcr :rapcrrrlprlooqrt 61r o! 6Qo[.rlrroi eivclL or 6wchpe6 torr

1 + I t ; + t j i r + . . . + l I ' ? , L = 0 . ( l t )

O (0'0utr ll Eil,ar nld.L lra pi(cr qg povd6oq: ]i-' = r''iu ''r :rou eivcL iooFr 1rI /' = 1. En€L6ri xo j €i var braqoQsxuio (tJro xo l. to II' eivcL bLcrpoperLz6rou l . f rvf tr ouvr ruq ;r iL l crro rr ,c dl . t r c 4, i1-. tn| r ivu roozlrrutf t ,q,rvcc nnvuorov zxllLo. 'O[cDq 6].Eq audg or pi(tg txaronooUv t4v 1 + Ilt + .. + I,I"' | = 0.To r€Xv.rogrr Eivcr ro i6Lo tr€ €xEilo rr< (5): Io).Icr:r).cor.irore pe ro il' rou 6tvrr.lirLrr" qv crproreprj .eupd (e:ruL6i to Ir'" oTo d]lao b€ELcir roonrtlr ll€ ro ]tou aprotepou irxpor). To oOQoloUG, rollcrd,aora!6p€vo $€ ro Il'. noQcF6v€Iavcriioitoro. dpn rp(ner vo [q6€viE€rar.

111. r , l u , ( " ' , 1 .

,,,1u-,, . ,,, ,,, ,,'l.;li

, =;[i _i]f l l r

' l

l l ( 1 " , / 3 e ! " i / !

L I c { " i /3 p $ ' i l r . l

f l r r Io F = l l i ' r n r l r P r n ' l i I

LI rr ' ' lr .r"i/r. l

(e)

flqi:rtl vu tl<rl,q0er)ooup€ 6rr o.fl?-L [oouro! [c roa rctl1ottx6 rii!ci1,(cl.firr rlv liiQlo b[uyci)vlo roiro €i\ral oc(pdq. H leoulli j rou F-r ctri r]l

oflt),yt j rou a riv(lL (l/?rxl +I + +l) no! €ilst L To 6uoxol.6rsQo peQogeivar ru oroL6ic rztrig rrlE 6royo!io!.6lou neir!'r v( 6!:iEor+Lt 6t! I YQ(i4rn iror F eni q ^1pcggr1 I ror.r F-r roor-rrtrL pe pq6dvr

l . l + u ! r r r ' + r r r r r : ' + + u , ( n r ) r u , ( r r ) a - 0 r h \ ' . i l l i ( 1 0 )

ToI I '

I i {) lA\Y: :,[r,\:XHl\lr\T]:\fo: JiOt Rll-R

E;rsL6l I I -" = 1.I caror€pri nirrpd tul6t :1i ! rr( l l . To i I 'ouoq b€v ci .v. l f . i rpo orfi.cirrcioc n(adyo)! xp6r€r !c1 $ilSrriLfrcr. Ot dilr,€t lori [ €h,ar oq9oytirvtE;.tlcp.16i.6novr(i< (ifl ;rrlipvo|pr orr!!1ciq p11(16111(, < (,{ ri - a,f) 11cr brora(irvr(trE ro n. ltnoooitrf 1( noipe 6rr o (vdorpogoq rou F t:ir(r" o lt r.

O TaldE Mri(roltFcrrol6f Fourier

H (lvrll.rlol Fourier eirct pLc lripoTq 0co)pic, cD.d crrt6 :ror-qv xirverz(1r Fvb(1({ieoLro( r'i1,(( orl €XFr jTA.rxrLx6q €(t(0troy6(. O x(LurtQo< raojToc lr(l\ , , u t u ; ' r i n o r 1 . f . r r , . . r , r r r r i . , r e r o . t r r t i , ' r U . r , , r n t . o u i r r lrta (trorriol1r. H .l!r|oroo(f1t 6r(1brx(oi(l tr(l( ollyri :rionr orrp x rqrtrrop o0rl rl.fL(r {t,r\Jr'rorl< z(lr tr(0ltrcrLzoljc i6you. (n fr{)!r$1i< rnwaprlorLg dlouv lLrtrrt\trri.rr.p1 orlpraric z(l l(fioQoul(r- rc brcrxpivorpa prc zevrpLzrl (lritr yr' (r!ro:edr raqayr'tyiont a1t c'N' t r4t, ,oxi\Uiotrs i plr(tq,lptrl ro.t oro r + lt,to anor .eopu titat tih it'tt tt ),n .dorc qi i'a'. OL !zf]t'I!xt< orrycortoflafivcL noiir x(r(iiiit).{.< yL.r b(1{totnnic fgLo(i!Jr.G. oior},lQo)rriri< clL(rrirrrr.L: xr1l!.!Lor;d{.Lc b(qop(;l. Kd0r oul,rtrr(;o( ouTvrlrlFaq rTpoT,orpd tr0 ro! brx6 rIClQorro 2c! 6irq tLoii or!1sr(rq,dAou! ('rt ),iort. Eror rl c1d),rq x.rl or\,u.orlorlrdrtrv o uroiorutpLtiq roL, r c:r6 ro y xrrr ror I crro ro i- rirru irc rrlriroq

" r r t , 1 y , . r r , t { i } . , r u r u , ' : r a r r l r o v z , ' . i . , r t r , , . r r ' , < , , '

O( 6riEoLrrrt orr I brablr(oi.i ltnopr.i rc rnrr),r ci tarj"/op(. lo rturbidpiflrEi(L orrl o7ilo1 ror fl trr rov 11. ri rd,rirrQir Ir 6tro avrilp(q,u ror /r.nor opiiou! dl'irv,1i!tr'r( i.i:

n o a o i a \ o n M . ' t t i . i . \ ' r , . , 1 , , i 1 r r ' L ' y . r r , , L i t , J o o : L { n r t \ r , , , r . r r r , , ,

I I I - ' = ( l I l - ) ( l + l l + l l : + + I I - " r ) . ( 1 2 )

"'= li i

(u'0,)' = u',,. I

| | l r ll l

' ' ' ro ' d)rn Lr r r I

r' i'.1 | r 1

O a{ r€pLaXrr 1|! 6tNrht( Tltg rir(pt4q pi.lcq ror 1. r,r =;. O F nrAr6Xrr rqbuvar$€r; rori r,r = l. rI: r[rA(ly(U\,r'trtc pilclq rou 1. :r[rELarorr r6r(irrtxr orr rrrIroti oroLXeicr rou i'.' ri,cr p16t!. O trr'rcl'/ltltrrrotroq 2 rri 2. zoFrl r{cL o{|1!fxrf),€irttr bno qop6q. cr.lLrri r)l t ort trovo 6()r],€rci cn 60I Ta€Ll{!Fr.1L i!(tc .l'r'rrotir< gctcroTlpcrtr.opoq,l Eni 4. IIcQotLor(1. r'rir irc( grrcroXlpcrT|.ot6q 6.1 eni6.1UnoAoro. lcr alrlr.ltctlxcr0ei (lno 6lo uercoTrycrrolLoug 32 e:ri 32. 1 6()uierir0c perov6rcA zard to rltrror (a v ro L()lJ10: ovconlo€otq rt,)v anorsLEoB(iro,v).:i1t\, na.ih. curo alloorrL xnr eivcrr eq,Lzro. ),610r.r1< cr:r),iq o/tq< Usr.Eu r(lrr

O}OOTQNIOTHT

H 32'n ailo €i!(r o€ 6rnLirorcr tlir6orool orov z{x}.o cr]i'qv 64-t. Eirv ud = I,rorf (tr)])3r = l. Ev 16vE!. ornv to tn eiror ro pro6 ror n. r6te r1 rn-oorl qi.ln€fr(r! ro r€ro(iy(D\,o rqE n-ooqE QIE(E:

II ralurlra rou TMF otytr rljnoriotnFtq [o(ql 'lou noQouordqou|rs €bri.eEaprdrcr cr6 r1v epycroiu pe pelirtrrg 6rvdper< 6nl4 11 2r: = 4096. O nivaxrrsFourier ecr €I€r ar morltiu. dtpa ppi< tov rqu peraollprttof6 lprrirlor'rur(4096)r = 2I noL),(l't),clor(ofroi lcr !( nQoxrhlrr ro .F €ni ,r. Or noL).(nLoorcropoir(r' rnctv(rl,$fi F€ ro\r F./[vo!i(l loln6v n(!d1riQ[6or. :c (rvri,0r:orl p'c{!r6,tvcrE tcrXu-q FttaolrlparroF6q lrnopci va zriver rciot jro)'Lafi)'uoluoF6 o€ 6 .2rr6tlyuru. rou eivat ffgtoofteQo ar6 6N qogiE rcrrirseog. 6161 awLxs0Lmcirv( n(Qdryor,r( 4096 Ir€ 6. E\,y€asr. 6r(l'\, ro /r roorn(lr pE 2*. o l€rc.ox{r(troFoscvrLxo0rord ril fioL),arLcor(lo[oui F€ (r[)/2. :uolsriqovxcq rov fiilcxc 4i pr]buo (!riyQc1q( rou .F,i/,. 1.((ronr1, !r rioo€po (fiilQcrqo rou 4,/t 1.((r r€],!xd

lrr fr (vrilQsq( rou Fr (nou Ei\,(L 1€rQrltr€vos) rs rrr ouvloroll6l'( 6rilsro(a(i./oars! sri (, log, n)/2.

flp6:rcr vc 6oripe rrirq ro y = -F,,ir (ivc 6rrivropc pt: n ouvrorrirocq) proqelva avaxrl0tf cr:r6 brio 6rcrvlopcrcr nou 6Iouv ro Frc6 Frtxo;. To :rptbro 6tlgn€ir(( v{l 6rc(QGooupe ro ibro ro c. To 6rdauolra (q,..r.. . . ..r,,-r) 6r(ron(.r(( osbrlo ttrpor€a(. Xo)oiEowcE r.E dQrLr< Gr' rrs n{QLtTfq ouvLordo€q;

Or ouvr€}'Eora< xcr.lvalovr(l €vd,LdrE ma / ?.rL ./'. K(rrorrrv, cr' .(!rtira ftr"nvuopcrru oppnrilorpr Ia U'= ]",t' xuL tltt = Fnctt. rou €i.var bionol),arr).cortlotroi tr€ iov prxp6rapo nl\'cracr F,,, (0$t0!:ir€ 6rr ?,' =,/2). 'O'r!)E

rtt| ort! (!r!i(liror(.ol tou Fr I€ rov I,-. I bou),€rd nEQloQiofilx€ olo Fro6.To

".1'rprr6 rao6).Wa €iv(r va av(n(lQciloupr ro t! c!f,6 x( t[ooi ]r€./60ors

6rtr opcna y' zcrL s", na(iyFcr nou d6sLEr.v r(l< yivsrcrr or Cooley x(r Trkey:

1 , ' = ( q . c 2 . . . . . c " 1 ) x 1 L r , , / = ( c 1 . 1 1 . . . . . c , , 1 ) .

3W Or :rptireq m ridr or releurai€E rn ouordroeg roo 6Lovr-'opmog

yj = a' j + uti ,e'; , j =0,... ,nl 1

Yj-n=a: u)J^Yi, i =0... 'm 1'.'EtoI ra rpia 6r]gcln eivcL: 6rrioncoe ro c mc C v.ct" c", 1rtrqo1t rdrro€a'j)14 owt !l' xctr U't xo:L zdroox€u(roE ro g dn6 t4v €Eioool (14).

(14)

, lt(Ly irr =

2n.(13)

] 5 O TAXY: META'XHMA TE[,IO' FOURIER 223

o(. €n(l,tloeioouFE al#o('s 6rr o!r1l I 6!o6!xooio 6i6€! ro 0(0016 9. (MrioQ€ivo npor|Fsr€ 1() ),oylxo 6dyQdluo Qo{E cvri lllg d),}€6AoE.) H orlpcoia qg€iv(r| 6r[ o !fio],oyrop6E iou 4, xQ€ldgErcrr F6!o 6!fi]dora 6{p(1rq (rfio rov .l',,.ow xooq 7r €rri nl,dov nolJ,crlclol(ro[oiE Fe rol,! ?n dAroFoug 1u]1, oqv (14).Eni l)dov 1 r66n propei vn encrvalryp0ei. Ilporopi4re cn6 ror Ilori orov {srlotov F156. Ar l*r.vrjooulr€ F€ rq 6rlloFl n = 2l rcr npoxo)plooupe 610 io 6Qolo[6lQr ro D = I 6nou bEv XQ€trlercli ,ioll,aJrlaolc(oFos- rors o orraoLrroEcrer0lr6s 6nrir1l')v €iv(r (r'l)/2. O oAro[os (rlrlos n1/2 unaxoi€r orov xov6vc roub6oqr€ nQolyoul€vo\: ro 6Et^do@ ralt Gq{irtDrt fn ro tt. our m ti nltot':tdJ.a:riururoyoi, 6i6u ov aprqg) 64ptinn,yn rc n:

2(1m( l l ) )+ r r r = la / .

To r6oloE €.il(lL xdxr ),i1o n€Qloo6rEQo ar6 yQupp[x6 xal o ,loydpLopo( Irrvrcvtral,ir totq noL),crLnoLcopouE uti,ui orry Oa). Ev rodro€ ro ,rll2 Ftrar16trtr roli rro plrp6 rm6 ro r2, (i)oc o TMF cr),l.dr!eL rrLeior q Foaqi rIE6tctptrl< cvdLuo4 Fouricr.

End.i0tuo4 roD riitou (14, ytu ro li Lvraflctaft zdr0€ ol'vrordo( rouy = Fr or ivu ltiQog nou rQofQl€rqr (rnd to d (iIE d.Qtt€E owtoldJo€< c:t) t(cllo€ 6!u lteos nou nQosQx€rtlL (ln6 ro /r (tt( n€Qft16E ouvtotd)o€( .r(, | ). Tolno6rao:rti rov tt;ro lLcr to yj o€ 6t'o pipt: ro

v;=!udtr:1 rouiltror pe ro ! r. ' i f. ' ,r3p + ! u,j ir.ru.' 'u, ' .A.nr {J [ ,1r

Kd0e d9qor,optr 6eEri 61er rn = n/2 6qolg, Eredrl ro uri roournr pe 16, rtr 6uocrOgoiopctc eivcL

yj = t url;i ci+,,,r, t u,flcil.

Trr aot'oiol(lr( (lwi. €iv.lt axQ!6dx tu y', xut lli nou rpoxrimow (rrd rougpercoxrycrr4ror.< prooi peldOou< I,,,c' xu F,,t!'. To 'radxo F€aos r()v (14)riv(lt ro ibro tt€ r|tv (15). Irn ro 6ewepo p6poE lQ€lol6Foor€ ro j + m avri rouj ra! rorko ,rQoEE€i d,L<ry1 rpooqpou:

$6oc ortl qoeoiolara rn ,,llr"") ncg<rpd'tL .,'l;/ 6L6rL ul;]" = 1{ = 1

c!c'.r cra' crwri. ro rr.,/,*"' = ul, bl6lr u'tt = e2rinlr - cri - -L

H aLr.cryri nQoorlpoo 6iwr ro b€ur€ao F€AoE fllg (14) xar o 1linoE lrd rol,TMFix€r t-rclrlo€ue€i. H r66n rpononorettar €uxot!c!, {bm€ v& roxu€r t co ?r(r dl}.€E6uv.iF€LE nQdr(l)v clQLotr{i)l/ ( or oXr F6vo./!a 6wdF€LE roo 2). Edv ro i.bro ro n€ivoL na61oE. XQ€LaLer( sv&( I€).€i(oS 6rs{poQarrrrcE d.t6ar€FoS-

(1s)

IIAPA-AEIIMA To 6riu(ro 0716 n = 4 os '7r = 2 €ivar

trl-lll t;:l

[]I;;]l'',,l[',',]""tlii

- [ , ]

O rcrh).).Il.oE ow6rcrop6g tov tpttlv 6q[dru;l nol,],ur],oordEel ro c pe tov -F1ror 6iver ro li. Eirer6rl xrioE a{Fa si!o! yQcuprxo. rQ6.rrs[ va fi0otel[r(lL crrodvav :rircxc xuL to lLv6FEro drnd\, tu)"' fiw(irr'rv :rqiner vc eivcL o 1l:

ilAv(y1'(Dei!srL' aorjrald)E r<r 6uo avriyQc(r( ro! lt: oro xiwqo. !rr1 belLd prprdrrivor o :rivcxcg petc0eo€(DE riou El(ofict xo ( oxe r/ v.sr t],', :ra cpror!:o(i oi!(lto trivazog trou rol.Ltr:rtrcrordler pe ro url,. Eri! t€xlvoloopE Fe rov nivcncr F*, opeoaiog rivtrnrrg 0c reQt€ilE 61Jo alrfyQclqa rou Ft xut xdqe lva arl'auii ec6ldofi6Nar 6ftotE xdt flQotlloulrh'a;. 'Eror o TMF avcly€r(lr o€ [t(r ytyolrtcioI , ,C, (yovi .r oi ' tot rou;Tivt l r(r Fou're' l O ol lyztsrQrFi loc :Tiv.x, tc F"uner pe n: 1r11pqbevLxri oror,2efc eivcr dvn 1L'v6puvo c:r6. xurd nQooiyyLon, I = log: n rivcleg1r ' rvLr olvolo uno n/ t ry l Fl6rvrx,{ ororxr(1.

O rrltQIE TMF ror I ri.ral"oi6a

To nQd)xo 6n[o ro! TMF .lvrrra0rord roa rol].cr].corcrop6 uri -F,, pt 6tioro).Ianl(lor..roFonq Eni -F", = ?',,/,. Or ouvLordo€E ll€ ciarLoq 6€i,ax€E ({,.], . . .)

F€r.lolllarilolror 6r.crpopetrxci ai6 rK (rir.c:,...). Ai\rolll€ 6vc 6rd^/aaF[da o t q y L a n = 8 :

],5 O TAXY: META:XHMATD}TO> FOURIF:R

.]r'r

| | , . .

ueraoxr[arro[6q

2

- l

H zevrpr"rrj Lbda tiL,l(tt rd drflxdraarilooultE xdqe xouli F! pt ltrc ltd.Q61totttnxivt nou trtltiytr tuo xourrd f:. O vioc n.rptiyu,v r = r sivar ro r€rQtiYdvoro! n(rfcrori Lr = u,$ = .rtrils = 0 + i)lrt2. To d\ro p!o6 ror.r lgcrprltrcrrog(rl,)dler (1n6 /,1 o€

_1,6

ri

!i

v\

Koronr.v, rdgr ivcr nr' trurti rc rourLci y,a ro -e = [1 11 .t'"' u'",,.u.".

lreraottucrrop6q4,,'vro l orue{c

2 '

226 olooroNlot-H1A

001

0 t 0

0

t 0 l

l l 0

U l

100

I t 0

l u l

l l l

)uvbudlovrog rd yQdgtForo. pnop€ir€ vc 6€ir€ o),olilllQn fllv €Lft6vq, noubeilrer Il1 6rirrclq F€ i{v onoicl ro r (ro fil,ioo6) r: er,oiplovrcr orov TMFra06G x{:r! ro logn io n),r1€og 61parn lor advouv o'aurlv -6€lx€r €riorK riq!qv cn),6qrd fil( Lolr k:

Khoe 6{pa lperci(ercr n/2 no}.i'criLoor(otolE. otr6rE o x€fuxG aQL0p6( €[v(L(rl k4n)/2. Y:rripxrr tvos €znl,ixrrx6s xrrv6vaE./Lo rI F€ra€€ot nDv c rralv rrtv€[0060 orov TMF: l-Qd.W€ ro1'! 6€ixr€E 0,I,....7 o€ duodrxrl 6irol 1|0! arrt-otptptt 14 ottpl rat' lrnqialr ro1tE. OL 6eirreg eprpnvilowor '!€ alriotaoqqbLdraEl rlqqiov" m1v cpLoreprl nLe4*'r tou lQ&q{patos. AQrlo! aprOpoi npo4-yoiwrrr rcov rrearrr6a (.ra }poi nou r.ls(i)vorru pt 0 epqnvilowur nprv rougcrpL0gorig :rou teLerdrvow tr€ 1) x .rut6 €?ravaLap6dvftar o€ zde€ 6it.tr.

Arrd qv urqa€oio ql.ezrqovr.zou rolu6popeior n"r/ib, 6rcri,e€rar &raEdvlvcrg xd8txaq (obryieq mo nopdQrtFcr C).

:] 5 O 'I AXY: MIT :XHIVIATDMO: FOURIER 227

A'KIIEEIT

3.5.1 A! F uilar o 4 sri 4 rivaK< Fouder, rioro( !:i.vci or -Fr x& -a4i

3.5.2 noro ri.var ro neoyparL"d rar q,orraairx6 piqos rov rq6! ru6rr(ll arld! rrc

3.53 noro-q livcL o uoniv(ads 3 eni 3 nor' r{ctdo.lEudlsrar ai6 ror 6 €trt 6 nt1flo f6.laororaroq t6ro ro corr€ia qs nadq<, rai.rtc xdr rilnils yaduris n(l| orril.Iqr

3,5,4 t4r€rdJot€ mo Frydb|ro €niftr6o 6l,rq rE d1 t€s qi!€s ro! 1 . no|(l €i.\'or rt raordql|1ilqila u(, (6q€i.re ro rqaruo rd ror vdvrdoiu{6 pdaor qs)i no|d 6i.vo[rl fl]s rt,roorndt tE l/ur6i Me rr uo1]Ior 10 1 +.1'+ t,l + rr3 + ul + r)5i

3.5.5 Bq€ift o).€S 1ts L!u€r! rlc eEioooits c,r = -1 aar <j),€S rrs l,!o0< ntq cio = i.

3.s.6 lloro fi!(( ro rrtpdyDvo xa' I ftrad.ro!|lt1i eiqcl lS u'r:H, Ea(raoxuis 128,onisAi(cs rou li

3.5.7 Anft. ro 4 rni a snsrnra (6). drcv or bsELds r),srufE rivclr y0 = 2, At = 0. r' = 2,tr = 0 Mr ol].., Ioy,r.. Ilutr ro F+ = 9.

3.s,8 Aioi. ro ib'o o,joTltrcr !d ! = (2,0, -2,0). y\,(,)oi!o!ruE rov r! | ncrr u:tololiiolrcsro c = { ;ry. Ers}. Iocuole <i tL to q1+c1r: ,r '+r: '2: [+c1e3ir roiprr t tq tLpdg2,0, -2,0 ma qrysio x.= o, i t /2,1t,31t /2.

3.5.9 (a) E{ ir y = (1,1.1,1) 6eiEtr 6i ro. = (1,0,0,0) rrs lo 'Torei q! i ic = t / .(6) Yno06or€ rileo dT'y = (1,0,0,0) .rc1r 6oFfiF ro .i.

3.5.10 faaln€ io ,0 cno rtv rrq6q yadlBri tn.- (14) aar lo /r uno q 6eursel. yr(1 ll = 2.rru r' =,1. xotdtl|onoLiore try nqdqTqcpprt r|(1 va 60eii€ ro i/0 xdL ro Jr ra! r!btlrFel yru vd 6e€fft ro,2 roL rol/i,6l.u ouldq tort 1.,:J! .s' xat y" .

3.s.11 Yro),oyidro ro s = ar. fE ra reic 6rtpar.1 ro! TMF, or(v . = ( | , 0, I . 0).

3.s.u Yno),or ime 10 , = t i . !d ia reis f t tor{r ror TMF, 6taa . = (1,0,1,0.1,0,1,0).Eravd).d6dt€ ro! 1lnol .o-r 'op6 $ . = (0,1,0. 1,0,1,0,1).

l.s.ll rqi.qr€ tous rirro\ yK ta qr,.l.cr!.r 1Lc rov:riro.,rc 4 rni 4 ror Mdl'loend1€drr dtav to J €ivar fterfl6, rdr€ ro. €ivar nrqEr6. To 6Ldvrdtd / €ivc' rrerrr6olav 1,, r= f. rrr n = 4 cu6 oryairsr 6n J0 = 0. "[ = -.fr. /! = 0. routoa{r, eciqsraL (ro m . r(r .rro xo (sino, sin r/2. sinr. sin 3rl2) r.! obrysi o' ivartali f rraomForrotto r}rLidvo!.

OPOOTANIOTHTA

3.6 T ANAIKOIIH'H KAI IIPOONTIKH

To ";rprbro pro6" rlq yacg$'rolE d)'1e6pcq 61er nl6ov oLoxL4$uO€i. Amo E€-zivrpe cr6 roq tt-rqcr1r'rvLxotr-< ni.rnxeg. npoxr,retor mott< noQ(trl.lL6ypaFfouqr.ict €ft(orQrllru orov rctp(yt,rvr# nivsxo ;1T,,1, rirQroxQ€q6[evo lipo (,ln6 roororlpo ,4:r = b xcr cncLrtbvtog n€e(oooruQo <1116 ro pr.o6 rot €Eo[rtlou (rou-)dlri'Iov oro Fi(t\[a ror orylaoa€o):

IKEqiX,ato 1. AloI nt A':t' = b yrc arrnrpil4rr: ,4 (re nrnlorprl).T(ie<p&),olo 2. ,11a4 rcu Ar - lt yn oroun''finor ,4 ([€ rot,\ raoo€ptg uroxd)-

aol5).KE<taLaLo 3. Arion ayiarotr *:rpuytbrar, (re rpo6ol,r1 n<,r! t1v ,47 ,LT = '41 bl.

Noto0to rci;ro4 buotiprorc rrou ouv6lnoc o),6xl,lQo ro eitla o iv(v rooo [r1i0o,i(rx(i],olo! Aoq,o),r;rs 0(r (v(yvopilere 6tt o cttx(< rlc rQfl6./a(4$6E irs0lxL€isr(r!n).10(, : l r 'aonu{r.x, ,v Jvoocov. ] lct l uTr 0( FnL'Fn! v( l nouL,r ncQtoootLQ( lLu.,{ , r lh l l . , r rou. Korru,vtc.S Jrgo( r( ntnr, . , r | rn:rou el tvrv. dlouL.r l t lv t t rz(rgi(v(r zirvoufle vle( ouvblou< rolr livovral ooqdoftp€E ex ro! uoraQcDv. Aurdqr,v,r . nr ' t rr) . lCr 'Frrtreg z. l r notzr) ,ouv u:c:ruoE rI ol . r(oi( . rrolooo ot utrolrr0o.rivar teLFt0E eelE),rdbolJs oqraoi(ls xor or rb6€E 7io! ,tQ!6or,1c{t ,rioo rolq 0(' t rrvrt- , ' , r )ouv or.(v HE(lelootr t t Io Llogu.ourr rou,( r( l tnv loL. i rou:.

H crcrozo:rr1o1 EtxLvri. (5n(1)< x(t ro 6t6),io. pe wol.o./lopouc. To xt\'rQrzoo1[tio eivuL 6rr ari0r.)"uon rQodrl.lue Gn6 [K noQ(yo wonoalol 1()! rivotis A:

Kr, l r i ) , ,1,r 1 aar, 2: 4 =ZII Ko<1,ti),tlo 3: ,,1 = Q-l?

Eirr trrptryulvrzci ti-rt :rcixd),11'61poppo. to ororlpa .4:r = /l r*cLttrcL ct 6tto

r g u l r l , / ' z , ( r o n , v / r - : ' U L t l t

na, ' ru Q, t r l r ron.v B.r = t = t t QtL

Te€'q or' ano{q rollg ;rirczt:c -ot L ti nat ?? tivcL rpr"ycuvLxoi. O privog:raphlevo< rivcrL o Q. nou dTtr rr 1pt44r6q xnt p6r'o rr orrllt< xut qv eloLpettx{rbr6rlra or orrllrg rorr r.1 ci.v.rf opeox(vovLxi:q nu. n6a o1lvirftr(t.. o Q1 Q \4r-iv(r o rurrortxdg nir.rraE. H luorl elaxioror' l€rpoy(ixov tq; Qc = 6 cilcLa = QTr. IEo[rxQrxd. ro a rQorrrr€r .1][6 rrq lolo6t('ifiot€E 7rQo6ot6E ]idvco o€rci0e uia un trq orltl,tE. To rs),r,z6 r €ivaL ro i6Lo pt t1 l"trrrl tor A? Al =;{rb.

'Ero' ro oimltra.4r = l, truc o6ty€i (oz€6ov X(,)piq vo ro 06l.ow€) ororELU zo.L QR. or :rcpcl,owororlotrE riooxirrornr an'r.r 6rtFart1 rtE brobrzuoio_( Iuoeog. fl).Ioraoa[e (]1oF.r zdr. or ra€L_( dl.].€g rapoyovrorroLrlo€I,<, rroucrroor(l):r1oc4rr T<rla zolrrnoouFs 1(r x€ift€ro rrro L_utlv(r1\'(uoro. H fiQr;q. rou

pvqFo'\,et0T.€ nQotyoqrfwS xor, ryel o4paoic Y,o rq eqcrpgoytE, eivcr q el6mipoAql4qa A = LU i, xal,,tftpa, dE A = LDU, 6rov o / el,vc! oqrp€iQrxq.

Keedl,aro I H fiaQd.f ofioftoi\oq Nou Clrores*/.tr,o oq4.r€rptr{6 ,4:

A = LDLT = (L D1 /')(LD1/2)'f

K8edrlqro 2: H ar4.yttvq rapalotzottoiqoq -$v6gao zl efii r xo:t r efii n,yto. xaOe xlvttxa A: A=!!Ifuedl,cro 3: Ardj,no1 rdtoltoeqwdtv tpd'v 1o onolovirinote ,4;

t=arDaTT6 €Eerdq,oqr€ pi.a nQo< [ia,l.'Orov o,4 eivft or4rlsrQotos, o U rautilcroL pe ror,.f,?. 'Eror rl noQayovro-noLl4i'l LDU ^tL\tldr A = LD L't y.cL tl 6eELd nl,€uQd stvd! r€l,slos oqrlriQtxt.O Chol€sky nqoldprlor 6va 6tpcr ricrQori6Qo no brdonoor rov bLcycirvLo D,rlqiQvovrcg xl5 r€rQcy@vlleE pileg rorv o\.1drv (uloodrowcg drr, eivcr, oernol.6qLc6ri np61i€uror, pc tr1 "o€llxd oeloUdvrl" ,r€Qlrrr(ool ro! K€qd],aiou 6.) T6r€o /4 o'\,o},i€xqt oE 6i,o tquy<ovrxoug acqrilor"teg (LDt /2)(LDt /r\T .

E66 ow6qiv€[(dt! r€gl€Qryo: Edv o Cholesky E€xn oioe cr6 rov.4r,4 (ro!6X€| 0€ruroq o6ryouE), ol 6io lcptiyowis iou Oa iiav axp6d4 or .8" xcr _Rto! ylvoFdvou cram-Schmidi M€ dl,ld l6y!a

A7'A=nTR (6.rj1 A|A=n!e!'eR xcLt e,r'e=I),

].6 ANA'KOIII{'H KAI NPOOITiIKH

3X Or nolnla:tLoorootfu fu rou qrol,otqtk causs tou Ar A:.o:1|itlo:'.iar"Fe roug ro),lsnl,ao|,aotd6 nou opgoyovr,ooolorw q dn4lE rIE .4.

IIAPAAEIIMA (o livc,tc5 rou nporrprirge yr.c rI 6lo6txooio Cram-schmidt)

- l

2- l

KondEr€ rrQ(irra rov /1,4. H aJrc).orqq nqooo€r€r 1/2 qS riadrtE rcqrl,(rE orq6€ueeq xol 2/3 qE Datr€qr61poplk otqv eirl. OL oi.qtot et !sr2,3/2,4h,'t;r.dv€! rl 6m6lr.aoi.a cmm-Schmidr mov ,4:

Edv o naorwoiFslog loxrEop6E ei,var oromoE, 1/2 qS nednE orrtlls ,ro1.J ,4ngood€eror o-r1 6enrepn ort).q. To ffiorilsouct Ei\ ai =(/2,112,-1,0) jrol)

;ll?L_;

,'=li:ll'i.

Eivat oAeoyri)vro ottv nebt\ 6r4r,t. Kordnrr riQoorio€l1cL 2/3 rI< SFI|TEQI(orli),qi ollr raiil or{lq. To crioiii,€o$o €ircL ai = (r/3. l/3.1/3. 1), :rox si!.rra?]o),tirn ou< 6i'io na(i)r.q nri)€q. A\triL 0ia.1r ra 6rtFcra r$ Granlschmidtror obli/ouv oE 6vov ril'.lr.i.r !r€ oAooyd,vrrq or4],sq:

: ila $qa),rrrao lraairbrrltrc. I EroF€vI orli),I 0o dazr!€ t(€rl1,r l4.r l4,t14. L Am6 rivsr 6vcr oxrri .cropc opsoyruvit;r oqltbv ron qai-vovt(L r'-lpcrieq z(L bEv F|ZC Erlltlb€i [(XAr ol0r'A(. Afioui]\,r[ vc b[([Qtoo!lr! IGort l ; . f : l l l ru tu ix4 lou<. [o rE uqtr rzo t | . i ' s rvru ot r to t t tc ' iy- ra k" ; u t tx ' i ,uu,r i ( rut t g t roq o64yoiE ror AT A:

1r + ( -1) : = 2, ( j ) '+ ( l ) '+ ( - r ) ' = j , (+) '+ (+) :* ( l ) r+ ( -1) ' = 1.

Avro btl pnoqet r,cl ei\ror ,rux{ri.ol Or orli),cE rou '4/ = A(r7)-r €iluroQoo./d)!!€E x(1! 1(r i€reilyor'(r 1c1)r, plx6r'ro!( Jr€Q[6lo!1cr! olor' Dl

&N ^r A = LDLT r6re (/1')rA' = L tAr A(t,r ) I = D.

:ro rr'),€]J1aio 6rtBC1. I'c vc x avov'liono'rioolpf fl; ortlr.<. &(llpon$r'$c fl;tsrOrrrovLxd,( eiluc awrirv rlru o6rryrirv:

Q = A Q 1 ) t D t / ) \ A = QDt t )L ] =Qt l

tiiil

Atr16 Fcrq obtysi rriL orov Drlrlr ror ctrotesky,:ror rcuilrro pe rov -R rr1;Gram-Schnridl.

H tpr1o,vLx6.r1rc ror ,B cvrovux),ti u1 6trittr!1: no),l,cnLcicu rov rpr:11o1F€l(.]v oq)drv aqarponvrar tr:ro flr6[eveq orr?.eq. O I xcrqphqeL rcr 6{pntca:rcLoLrll< ror A7,.1 rcrL o ft xarclpirrjer rc d1[crrc op0olDvronoilory ror ..1.

2, A = LII rivct q 'clvllprvq :rcpcrlorro:roir1or1 '. I{ rcl.Eucri.cr, 6:noE ncrl r1or-rtrrpetpLzl ;(rpcloxonoirlol /,rLT. €i1,or nlo EEtooQQonn$tl'q (1ro tqv;rcl,Lri.{ = la,I, cl"}..i FE lLd 6ro.toQ€xLri 6!vo((1.'Or(1r, o A 6XEr rlrqn r. lp€rol6p(or€oTlv na(y$.rr'r6r4ra Uolo r orltl.ss ror t raL r yqagg65 rou U. Or rE).€mair!rr r ^/Qcf4l6E xo! a/ €ivo( 6uvox6! vo noQc[€Qrooori! (oqo! 6LEE ro!( ]ira!FISE\rrri!). EE i.oor €ilar 61J\r(r16! vc ridQa)"erq0ori\, z.r or n r orn).€! -ro1J

l, (aqou $' aur€q ro).),c:rL.lordEclE Er'r€ivEE rq Fi]bEvrr€E ypr4rBiE ror U, y,(l)ai.<za!6v.l .lrord),€oFa). Mcq crop6volv or" I xcu" U, tcov o:roiov ro p6sero eivcrnciL o i6ro_( rivcrrl_( ,4:

3.6 ANAIKOIIIDII KAI nPOO[TlKll 211

I

'EvaE ntvaxag tdltug r pttopxi va napayovtoxorq9eiot yrv6ptvo ndE n rri r zivaxa xar n6s r tzi n zivaxa,

)qv mpinrc,roq nou o ,4 lpeLdletor evcL),oy€g lpappd;v, or6te P,4 = IIr,yir,etol pLcr prxprl oL).cryj: o ! o11pcri,letor ar6 llE apd;reg l onjLeg tou P L-L

clri xou ,. :€ xdO€ n€eiTrroroit o( r ornl,Eq rou ! clrtor€lotll,6do1 rou XdAoxorll,(;r rolr ,4 xnl ol or r lpnppiq tou q cfioxeLo!'v 6aoq rox X(i)Qo! yQ(tltld)!

3. H atjitl'Ja\ brcloeqottbv rLp6r (AIT) eiv(r| fiol! EvolcqdQouod ]r( !(ouvouro0ri o€ pic,r noQdyQdqo xor €ivcl ro (rvrlxri[rvo ro1.l rloQaQrtFfioE A. :'(rur6 Tor[o0er€iTo( 6voi 6rol6vloq rivuxcq : [€rcE! 6uo oQ0ol('rvlov ,r[vox(r)v,ro! oivouv oQ0oyd)vL€! 6do€16 Tov r€ooiroov liJroxdQ(|)v xcr obqyolv orov1,su6oavdor0o(fo A' xar orov "careF6 xcrdoraotE". H AIT 6t€L cro6€rxo€ie€[€l!66o!g ollllaoicrE yrcr rouE e?rrorqFovrxol)g liroLoyropotjE.

Aicluolcrlxoi ld)Qor xcl !Jr6x0]Qol

A:16 ror.rE unol.olLoporrg orQrq6ptro'r€-r(ilQo orlv riL1e6pc. Au6 leri.vloe1.1€ rlr, f66a rou 6rcvuolral.xoi Nti)Qo', 6nolJ ait't.{ 6u\raldE 6do rQdEeEl

ir.rnraFdfixit nQ66eEo4, ): + !J, ,.ar trdJ.t l\doladp6s pe oQfipt, t:r:.

O! rQdE€!6 cMiE oQiEowa[ €lxol,s orov 'l-brdfisxo lcoQo R". fiolr €iva! roeli:Xov rcrgdber.lp<r bravuoparnou Xdeou. OL rradE€6 arirds oQilovrqr efiiongltrL tvr6q Frrpordpcuv ouv6fu-rv, 6ror< or eu0eitq rct tc enint6c nou repl6Xow rlvaQtil lo! Rr. Yn6X()QoL trivar u?ioolvold rio! .rn6 *6vc1 1o!g.iv(r|" 6rdvrouarrxoiIdQor, rcL or -reooeo!< 0€F€Lt(i)6eiE u7i6x('J0o! arior€),ou! io xle!6i rrF Ft:ldqs

'YncaEt rou r

Mo!G6[r6TrF( To! r

to b nqdreL v( resLilerol orov xdpo oql.6v ?r(,4)

ro I nptnrr vc eiro xd0ero otor cprorrp6

pl6evolupo rV(,4")

o pq6€v6l(,)Qof .^,r(/) TiQdnE! v{:r 7r€Q!6I€!

F6!o ro pl6€vr,{6 6rd.,'ropc

o lrirpog lpnpprbv R(.'1") fiQdn€! vo €i\ro[ 6].r)< o A"

'Orrr! yro z{iB€ I rncipler gLc luo4, r6re or ypalqrtg etvar avelciprlreg xcL Irdh €ivar l' = rr. orov r ).nor eivrtr povc6Lxrl, oL or{}.eE eivcr cveficip.qresxrrt, lt xdE{ €i.!a! r = r. Ito rt{0€ Tiivar{a. I rdEt eivnr r1 6rdoroorl 16oo rorXdtaou orn).drr, 6('c' rcrl rox x6aou yQa$Fd)\'. Edv r = m = rr, x6r€ o ,4 €iva!mrpc.1o.vr.z6g xnr. ovrLorgcrl4rog.

OPOOfANIOTHT

Ilp6ner" vc nolFe 6rL rc xegirlcr,cr curo ar.ol,orjoyloav 6vnv oryzexpL-ptvo o167o: qr zcncr6ryr1 rou ,4; = b. Kdo€ via 16€rr xcl ogrop6q

oul:teqD,([6ovop€vcuv r(Dv blcvljopcxr,rdv Xdp(D! x(rr rlq yaau "riE aa€Eopryl-oi(q, q< 6dorE r{o( oldorooils. rrl_( rdElg 1{(rr rolj ll6€vox(ljpor. rou Eoror€arxory(r,opirou LcL rrtg oQ0o.1<Drr.6qraq erori10l, fll mrpj roo lpero!6tav. 1'mr-r6v rov trovo6rli6 oxori6. Tdaa Ealariorrril€ irg i6teq arlttq tb6q yto va 6,oorrprtr€0lxonE ouot8rlo$o{q rou naoon€pdootr€.

1. H rolti 6ito o.awopalwri| yhQar H nevrpLzr] LSeo cvi(y€tol o-rov oQtotl6$,6q 6lc1'1ropcr,xou Xd)Qou ]icL €v6E urold;pou. oeorpcbrag 6I! p6vov 6vo!tr€[ov(D[6!o or6XCDQo yl ivov pe$ov(4r6vo :tivczc ,4, al.].ri rous ouozs-rropoljE$sr(lEi 61io urox(i)Qov rl [er(r€u 6uo rr[ldr4o}v, 90dNo41€ o€ v€o e€(l]llpcta. Toaq6ro (lrrya eivar ro mo ortpcrlrlx6l

3Y Ed! y r(c I'l/ ehdl ufi6xoQor dlovuoForrr(oli X6Qou, ro i6!o sop-6niver, xoL ge trlv topri l/ n W. Ts drqvriopqra rlou ov1)liouv ttor orolE 6{0un61coQot5 aror€trolv ivov vio ur61cuqo.

H on66€rEl ei,lor d$orl. Yfio06ors 6rr rcr / zo( :t a\alrio]Jr, orrl! y n ll/, [sd ,a L6l!c eivcrf 6Lovuogana *ioo tou V 6oo xaL ro! II'. T6r€, fier6i or y r<rLII_ eivor cno Fovor totg drqvuoFqrrzoi X. i ,q"r , rc.r-y x,(r , . r , rv4xouv x'(r oro!V rqL oro.v llr, Ta anot6apsro rnt ftQ6o0EotlE xat tou lrol dfrld,ordo1toipa wfipoiE xaqcptaour evN6E rqE roFig. f€opererxa. I rop{ buo enlr6dov6!(i llg oQI{; (rl "ufisQsnurd6ov" rou Rr') eiva! '[dlL avos !fi6xoQo<. To i6rof,( ro7!fl '!or yr.r rops6 reproooriqcuv uroxriqtov I rlropl xLrr yru ropic rrft ipoun).rlOorq urollpcov.

IAPAAEIIMA 1 H roFrl 6uo oqoo./ovi0rv lro16pnrv V rcrL I,I/ rivcn o ua61o-pog ev6g xrrl p6vov oqpeiou {0}. M6vo ro p16€!!x6 6!.r!rr<19rr €i','ol oQ0oydr,ro

IIAPAAEIIMA 2 Edv or urdx|oaoL V xcr If'tcrrtlowcr pe rcr ouvoLa rt.rv

eiva ro ouvolo rov 6rcloviov rwrixov.+ A!16 €ivar oiyouQa avaE uri6I0)QoE.naooodrovroE bio blay6vro1jc nivax€q ri noL?.aril.(1otlil,owaE p' 6vov aQ!0po.:rnipvotrpr ncft 6vcv 6rclti.rvro rivcrxc.

IIAPAAEIIMA 3 Yno06oxe 6rr o V eivar p45fl61copoq rou n rar o W- €i\'aLFrl6ev6 Dpoq rou 3. T6te o V n II1 eivar o Fr,xp6repoq lll]6ev6ltt)pog rou

e I r . E : E s o E r r d n ^ n .

3.6 ANA'KOIlH>H KAI TlPOONTIKII

IrelsAUT€pOU irrvcL a

c=f4l' -LBl '

II a'r = 0 onctrr€i rour6xaovtr Ar - 0 xnl B;r = 0. oir6te to ;n fiQ6nc! v(.aept6lerct xar orou< 611o 5r16evo1tirpox.

2, To tigQololta 6io dnwopatxbv rrrordporv >vwlo0x, aqorl ouqrlrylooqr€xar ,rcicrvo{oor!€ fllv ro[ll 6iJo ow6l(,)v, €ivar guolr6 vo 6oi$ ?a! rylv dvoorirors. fr(r brcurolorriori5 1tirpor,5 ev roltror,g. cnr6 6ev €ivar quoroi'oylx6. itfi\,tor V ull' d{o unoy.bgar 6Ev dru t:1' y lr€ | urdttr.,ooe. @er,rQrioc rov irFovcr,r x(r 1(J! dEolo y oro €ninr6o. KdU€ dEovsE an6 povo( ro! eivar uno1rDQog,x(rL (Dor6oo or, tjrio [tr(i 6€v ei.vo!. To creQo[qlq r(|)v (1,0) xqt (0, l) 6€1 €iva!oe xav6v(v ciEola. Touro ollF6(rLvr[ rldwor€, €xrq T16 fi€er"fircilo€o\ no! clc(crr' touq uoXcbpoug nepLdlercr orov ci),l.o. Tor€ F6vov I aloori loriE €i!c1[ rdtr!uno1opo! (rTou oupri?rrer. ge ro peyalrritcp6 toug).

n(O 6t'a clutd, e6tloqr€ va ou1,6!/loouF€ 6io rrnoxdQow lior yr' auto avrtyra r4v dvrrori rot< orpeeoF.rorF ('To d0poroFd ro1l(.

OPDMO! Edv V xaL W efiar 6io !r6)(@Qor. 6o€6woE I6Qou, 16r€ rod€Qo0olrd rol,E y + ty eil,ar eri.olE !fi6[oQos. AuroE dror6]€iral ar6 6lousrorg ow6laoyouE c = r, + ro, ofiou o €ivdr, r1r(6v ordvuolra rou y r(dt 1r,sil(rl n (61 6rduisuq rou I{/.

Auro bev eivcu, rinorc dL),o naqd o 16qog rou nnptipral on6 ro V u ll,.xcL eivnr o prrp6repog uTr6XLoQoS riou nEOdXE! 1600 to! y 600 x(rr tov ll'. TodOpor.opcr tou cilovcr r xar rou dEolrl 1i Eirdr 10 ts?iirrs6o lrl, r. To iblo o1416ai!slxcrL pe ro ri0porolrtr 6rio 6Lcrpopertxcbv $gtd)\, xu9itov /1 p4. Ea't V t&al ociEova< r r(.|" Il" rivar r1 €ooeia x),io€o( 45. ri = 1/, r6re &d0€ 6Ldvoo[q, 6noE ro(5. 3), (t!.r,uerar o€ L, + u = (2,0) + (3,3). Aqc ro I/ + I;l; eivm o).6d'lpog o R:.

nAPAAEIrMA 4 Ynoodo1€ 6tr or l/ xcrr l;l: tivur op€olcirvrc oqd,rpdpcrtr,o il(rq rou dl,l,ou mov R". To doQorotd rous €ivnr V + II' = R". Kdoe r eivcud0ao!o[a qr rrQo6o].rt6 rou t olov y r.(r| rtg npo6o).i1q rou u'orov II-.

nAPAAEIIMA 5 Edv V €ivqr o ldrpog rarv riwo rpLyrvrzdx mvclx,rv xst II'ria(rt o XriQo! ro)v rdft) rpLltrvLxtiw nrvrir.rov. t6r€ o y + U- €i1/at o X6€oE o)'ovrarv nv(iL{'rv. Kri& vczc6 p:roqei va yQ.rg€i octv aOaoropa svG du'),.crr €v6E?dro lery(',vr ori fiivnzc( pe rotr),oug r06roq, bt6n ro 6tcry6\,[o olotl€ic 6€voQiqolrcr [ovo.tFfr]"ra.

nAPAAEIIMA 6 Eirv V eivut o X6QoE oq)d! iolJ Tiivara '{ xnr I'y eivqlo xrbpoq orrll,rirv rou nivnncr B. r6:e o V + l4t eivnr. o ylpog oq).dl' tou

OPOO|ONIOTHTA

F€./ailrrr€Qou ni.\ar.o. D = [,4 ,]. H 6rcorrol tou y+If F roQei v(r siv(ttprxp6repl u:rd rov ow6uao1r6 r<ov 6lcror<ioer,rv 1(o! y xar tl- (br6fl o! 6uoX6Qor tl'Iop€i va 6xow r.o[vd. onF€iq) xal q..lo6€rxlrlErcr €uxol(t 6r!:

dim(V + I l ' ) = ttdq rou D. (1)

Kcrd ,rapd6oqo rQ6Jto, o 1,nd.oyrc!69 rcu V nllr Eivar noLtj nn fumdtirnla. yrro06or9 61t blbovrdl buo 6do€r.S ut,..., ur )io! rrrr... r rrt xft ovo(l-torlFe 6tiol r1g roprlg rtov bio ufioX6EDv. :l'fouQa 6€v qrdvE! !(r 6lonror6oouFsq\, rdrloro cfi' Ts u roor'rfl [€ iJrolo ar6 ro r), O! 6uo X6Qo! eo F ropoioo!.rri6Fn r(c! va iouii,Eo\ro!, y = l4/, rcrL naQ'6t!d durd or 6do€q ror,S lcr sivo!r€).€i.c|]6 6Lc(f oQcu.x€(,

O rilo onolel,€o$Clflr6E tpd,tos civdr vd olqpdtioo4rs rov i6ro nivqxslJ, rou o:roi,ou o[ of l l lEg €hor ,] , , . . ,?rr.r l ,r j . . . j ta,t 1{c(! \r(| ] jnol,oYLooupe iovpgev6rpo,A,r(D). @cr beiloqre 6r pLn 6rio4 curori ror.r Frtb€v6xorQoo o6qf€i o.Fia 6drol rou y n tt' xtrl 6\r oL bio X.iQot lxouv r4v i6m 6uimaa1. H )wotoolrou tllbrvoloQou i!6yerdl Url6FvLxotlr(' r{oL iror

dim(y n ll'') = }l'|6€v|n6qro rou D. (2)

Avrd obll€i o' €vc1v rrino nou €ivu! (116 [6vo6 rou €v60ae€Qo\,. Ileoooito-w.rg xq (l) ru! (2), q(ouIE

dim(y + I'Ir) + di'n(y n lt') = triEn tou D + g4b€vrx6twa torl D.

.AJI6 rouE ]Jno),oyroForrE lrog [s r()rJ6 rtoo€QlE 0€[r€),rd,6€!E rjnox(irQous, .fvoQllows6tr q rdEq ouv q prlbevLrdrrFo loowo! pe ro rl{0o! rov ornL(i)v. :' duni qvncQiirtoorl o D 6XE[ k+l onj].rg xnr, ureL6rj ft =dimV xar. I =dirnl,l', grirvor.rproto (rrol,ou0o oulr'rioaoLrd:

din(y + tf) + dim(y n W) =dimy + dinw.

Aev eivm xct 16oo rioplpog nlroE.

IAPAAEIIMA 7 O! Idrpor. V zat ll/ tov civo xcr zdro rpryovrxtirv :rrvcixrov*€)(ouv x(.l o0 660 rq 6doroo! n(n + 1)/2. O XdiAog y + I;lr 6Lov xnv nlvdl(ovelet 6rcot<ro1 n2 xcr o XCi)eoE y n ll/ tov 8td.ftovitov mvrixrov 61er" 6r"rim(ro4 n.O,rllE neo66l,€VE o (3) 2 + n = n(n + l) 1 2 + n(n + 1\ 1 2.

(3)

+ :,r.E.: Ynoloci n x n.

3.6 ANA:KOTTH'H KAI NPOONTIKH 215

EQX6Foor€ r6Qa oxnv cnoieLEl rn6 (3). fla nQ6ryl qoad o' awo xo 6r6Lioxo €vbrcrq6pov ^/La rov unol.oyropo €ildr Fr.rA6rEAo arr6 ro €r'6rarp6eov rounaao or"dE€r r r€tr'rxt rnE ar66€iErE. Eivdr rl Foq qopd 'rolj 0o xaro+ronor{-oo![€ ro rdlvaoFa rnq '{arav6lorc €v6q tdraou rorpLdqol'rdq rov F€ €va! i(l.Io.)r1per6me rct' apldg 6rL o p$ev6lopoE tor D elvor 6vag rr67,opoE ror .R**',evd; o V n / e i,vor 6vas rr6lltDpoq .ror B-. Ip6re t vo 6e i,loqe 6rr or 6rlo cwoiXd)po! 6lo!v rqv i6ro 6rdoroo4. To 1€Ivootrcr €i,vor vtr 6ei[oqrr drl or, 6uo oltoiX,npur rutqLdtouv cao).uro. piou r4g tnogrv4g ,1111s16r1iug.

Ao06vroE olcvljoparos rx rou prlo€vo1orQou ro! D, lQdlE€ rqv €Ei,s(r)orl,r = 0 F€ rll 6on0sro rrov orrlLrirv. u5 elrjgr

s t t l l + , + : , t t r l ; + C t r t l r j + + c t , t l l l = 0 ! ( 4 )

Tf , l l i l + , . + : l i t ? r t = r ? i / i +L l l r l , . f i t + t l l r i ,

+ Ynuat{ ivaE irXlo! root{oaqLorr6q Feraq! ro! tdrao! oqLdr a( ro! xdrao! 7aq4rd!. ro!irow ?o, oL 6io 6ldoiaot r.

H crQror€Orl nl.€lQd xrK-r€l,€]JrdiaE €Efo(roIE, og ow6udop6E to)! r,r., dqx€!oro\r y rd! I 6eEd fiLeuQd d1/lix€! orov lt'. Efierbi o1jrd ro 6uo €iv(ll iod,/roEoldlolJ! dvcr oL<i\,oopn ll $1J V aW. A!'r{ I 6!e6!xaoia oQi,qel qvcliloiolxio p€raEi 1oo 6lsvuoporoE r rou ^r(D) xtu roo 6rovtoFc1oE lq rouy n l . Euzol,o €ndl.rloEustdr 6r! n (rrrrololxi(r tjtaqpEi fllv fiQ6o6€orl x(rl rovnoLldfiLootcop6 [e oQ!ep6: Ei(v ro ,r ovrlororl€i olo I xo! ro r' c1\sroror1Eioro y'. 16r€ 10 J + c? olrrorolIEi oro l/ + J/' xcr ro cr o,vrlo1orxEi o1o cll. EfiLnLlov. &<i0c ! orol' y n 14/ TrQoiQl€rqL qfi6 iv<r r<n p6vov dv<r ;r, oror ,V(D)(Aoriqon 3.6.18).

Aur6 €ilor 6\rd i6LElo /icrQ,:t6tr'l!rl toopoQEtopo, [EroE! 6uo bL(rluoldrlx6vxdQc0v. o[ xri)Qol, €ivql 6GqoQ€r[xoi, a)'lq ),ra dt€E xLE aAy€6QLxiE e(paQpoyieeivat axet6d)E /dror. TcqLdlow sn6lurqr Tq yQqF$rx6S qv€EdQrrtrq oivo).aLrwrmoryotiv o€ yQo[trxd]S qveEd$rFa orivoJrtr ]{(r[ xd0e 6tiol rou er'6g trl'tL-oior"I€i, o€ kdnorc 6doq ror dLl.orJ. Eror o[ 6rooidos6 rouE €tlor, i6[sE ,{or(1!16 oupri),qQ(i)v€r rtv crr66t:rlq rt-rv (2) ra! (3). flQ6ffina! 1ft 10 €i.6og rorrstrorEL6o[cro! fio! enL6[6xe[ 6\raE aLye6Qrfiri!: vq o!'ol€rio€r 6uo 8rsqoQ€rr i[aoryarrrd al'r,]i€iB€va, nou xar'ouoiav €ivor" ro i6ro naciyFa.* Eivar y€lovog6iL 6!() IdrQo! F€ i6!o ouvoLo cpr,0ptirv (6o0ptorcir1) xo! lq! i6!a (n€ri€Qooldq)6 ioraorl Eiva! fi{ivrorE loopoQqo!, cD,n aur6 eivai fio}.x Fvrx6 y|a !q €XerLtir"tltrrqo ev6rcqsqov. To EvbLdqdpov otpeil.ercr oro rcigrcropc tirio x€leia4cv6[oLcuv ltirprDv, 6mog o ,^/(r) ral o y n /.

IAPAAEIIMA 8 y €ival io €riin€6o r, y ii(t|, 14/ €ivc! ro €nifie6o r,::

or 2 nQdi€E ollll€E : 6do! rou Y

or 2 teLeurcieg orlt'eg : 6cio4 rou W

[1 o 1 o ' l

llr ; 3 ?l

H tcilr1 ror D rivor, 3 ncr o y + lI' si.vrlr, 6),oq o -&r. O pl6ev6Xr,rpos napdXer 1or = (1,0, 1,0) x({r 6xEr 6Ldoiaon 1. To avriororxo brirvuouc !/ €ivaL l(oT'illl 1)+ 0(orriLr 2), xcrl 6X€( ,r1v zcrenewol ror tiEova :r. nor eivcr 11 toprl V n I1'. Orrno< (3) yrrl rr< 6rcordoeq -rtor V + II; zor I/.t ll' yi.a€raL 3 + 1 = 2 + 2.

Or 0€Fd,r66€rg u?t6ltDaor lra yrv6p€t.r A,

,4.,t6 r(l lurrllt rnold)rror, :reprdpe r6pc otu lLrop*tr a|vrinnrv. 'O;rtog nttvrcr,brr eivar" t{r fl€po\'(Dpiva ororloia ror /B fiou (xo!a [btcirspo svoraqrieov.Arrd €lb€/otr€\rorE bE! izol,v 1.i.r[$ic opoL6qta Fr rct orolX€io r.tn'r A ,r.n" ]1.Alrio€rc(. or rbr6qrr( ltov .4 z(L l-l nou pnopou! vlr xtrqQovopl0orv atr6 ro A,rzgpirlovrcu" trcil,l,or orl llrirooc tov blawoFolo! -rLS ./ao$6g rl rq o-rlllEgrlrv nlvdklol. rr(Qd [€ ra ororx€ia 1o{. k(rr ]idL 6$(Dq, 6ev €ivor -rooo o!

tr€tlov04rtles 1pc14rdg xar o! oriL€E ttolr rsilouv A6l.(). 600 ol tJn6loQot fiox:r(pri1ow. Or un67,<rpoL c{rJroi r(rQryQd1ouv oL6x},lQo rov rilaxo blo p!as.

To 6uolrio eQ@ry)p(i tr{l( aiaar ro €Ellq: flo[€( €ivu! o! oxioaq lterdElr ro!rrooirQov 0rgt:)n0)6d)v rrrol(i)Qo)v fio!1 c1.l[olor,Xo!v orov A, 1o)! r€oodQo! tro!ovrloTorxoul oTov , z(lr i(o! tsooi(pcov ,rolJ uvrlorolxouv oro Yl'r'6p€'r'o lBlC)lor crljloi or nivcx€S prloQ€i !c eilclr ttciQoLtrq),o1Qa$Fo( 1{a! u]iclQxow l6ootQlE6ooLzlE o16or16r

\i) O t14 d€!6ppo< ror AB iu:Qti'/f t ro1' p46tt 6pao nr B(ti) O )(i1oa orr).i",rot AB ittpiyerut orot, ybpo n4tuLt tox A

(iii) O aa$rrQ6q lnbn'rjZoAoE nt AB ttQirct rov aQlortQti l46t1,6|uao rorA

(iv) O /.i)ao< y0tt46! ror AB mpiygrat orot ybs ypappiv rot B.

or 0fio6Eiqer6 €ivar antrouorarE!.

(i) E(iv ,r = 0, 16r€ l{ar ,48.r - 0. Kri0e c orov plbev61t.rpo rou B (\,rixE!rniolc orov,\r(AB).

(ii) K.ioE (]rrtU rou,4A eivcL lpcrpprx6< ouv6rooF6q rc,rv orll6v ro!,'1.(iii) Eriv y7 / = 0 x6r€ xr'.," sr AB =0.(iv) KhO€ lp(ftni ro! ,48 tilnr.lqcB[rrog olvbucrop6g ro, ^1p<44rdrv tor B.

A;r' nrxdg t6 lvtboer.q 1Lc rouq ur6xcupolq nor crltLotoLlouv otov ,4,B rporri'nro|v q0' iyp' i ro 7rr f lc nraoraof lc rprd'r r l : l , turor i<:

IIOPDMA H rd€I r.(dr I |1nbsvrxortrtl roo AB ri{cvorioroi! ,g

T(AB) <r(A\ r(AB) <r(B) ain^/( ,48) 2 dim., t / (B).

231

A€v yi,v€rol nQoorid0eto vo aro6Ert0€i 6rt o AB €pL peld'urepo prg6ev6logoctn' rov ,4. [ad],!qt! 6er pnoporipe vo egrq0odp€ xdft *ro!o. TT.X. o A xcLL Beo lnoeouoav v(. Ei.vo! awiororxu prl6€llxol ni\orL:4 2 Eni 3 ,tzuL 3 EnL 2 xol o,48 vc qeL rov prxp6tepo pq6elolrrrpo d. evd, o -4 q(€! rov !€yolutEao .6P.

@q1€eirc ril,oE or i:'rcg onotth'cxag C oxryoliq€rar naQo),Elnowcg xo-no€E (rl

"otrFis) qr' rrq yQolqr66 tou A x(rt xdfior€E ({ xol[icr) sn6 rtE orrll€E

tou. Aev eivcL bioxoLo vc p<r'vrirpoqre 6vcv neqloEoF6 .f|,a rqv rtilq tou d.

],6 AXA:KOIIH'H KAI NPOONTTKH

Yno0dore 6tr, o ,4 dvcr n €nl n nhtrxaE rd[€oE r. T6r€:Kri€e unorivcxog C eival tdleog ;; r.'EvcE ro!ldl!o!o1/ r €.-ri r !fiorivqx(rE iou ,4 €X€r rdE! qxQr6(iE r.

Atr66EtEq @a ulsl(ryoqr€ ro! A orov O oc 6110 6ripdrn. To :rgrirro otprlveL ro),rlooE r(ov orlL{iw afi€iQa1io x(r! c(p(lLQ€i, Fovo ils lQop[d< ,iou 6s! e6].o![E

orov C. O ltirpoE yqcFptirv olrourorr ev6Lripeoolr fiivors t r€Qldl€rar nQoqovd)Smo1, X6Qo"yQqrBdJv ro! ,4 x(rr dror tdEq(B) <rdEI(A) = r. Xro deutego 6rjp(,o B ordyelcrr otov C rlced,eirovt.rE rL{ c.v€n0lpqies orril,eE. 'Erol o XdrqoqmqLtiv rou C neEixer(.t oTov )(d)Qo mltr6v rou B ror rrih(C) 3 rd!q(B) S r.Aur6 cno6e!t(1,!€r ro (i).

frs filv crn66nEn rou (ii) r;ro06ors 6r! o B orotel,eitcrL a:r6 r ove[ciqrrrptEyQcq4r66 rou A. T610 o td)Qog yQa[[drv rolJ , aI€! brdoraol r, rd!1(A) = r,zcl|" o ldqos ornTrriw rol D ne€r€L vo €lrr rlp i6Lq br,iroraoq r. Ynoo€o1€ otlouvex€to 6rI o C arrorel.€irsr" un6 r dvrlriqtrpeE oTlLr:g ro! r. T6re o Xd4ogofltl.dv ro! C 6I€! 6rdoiaorl r ,(dr rdEI(C) = r. ,41]16 o).o1|}.lQ6!s! rlv on66€!EIrou (ii): Kdo€ ,rivoxoq rirEEoE r€Q!€X€r tv(v pl !6r6poQqo |Jnonivaxa , €ni r.

nAPAAEIrMA O€oarior€ €v(.v (.116!I rfraxtr 3 €ni 4

32(D

(ii)

. ^ f l i lFr.unonLvaxo . =

lz .t ] .[ r 332f

2 6 9 5 l

[ - r 3 3 o ]

KirO€ 3 f.;ri 3 uroniva].ag rou A el.vcr Lbt6popgog, svd o C 6ev efvcr. Apc 11rAqt (rou A) €fv(r| 2.

A€v .rEt!,E! vcr b6ooupr r8rnireprl epq(ort o' trur6 ro 0€6aqra. ErLqq-velozdr. porl!!€! F€ avc.6l.rt|)E €!6!o96Qov ee6p'4rc, rou rfirerco rou tX. 3.4. :'€zrfvo a7ro6€[Eap 6tL xirOe ,4 sivcrL 6rcrg avrLorpirpr4roq petcol4rcnop6g rour.brcorcrou ldrQou"yaappdra rou oro! r-6rdororo Xrirgo or1?"dw rou. EmivoLor Xd)Qor r(l! o [€rcollForrol6E fieQ!€Xou! 6],rl qv 7i].qQogoQtd ./ra rov A. O1rL't&xai e4 pnoqouo€ \,(r rwcnnpcrXoei. oL6zi"qpog rrJr6 q yrdoq clltou rorperno'x4rcrrr.ouoi. E6<ir ro Eilrlfro €ivo! \,s 6Qoritr€ €1'av swlorQ6vrIo unonivax(

C xnr. Srv n(ilrr xo.vtuG p6Lo;ror"ov crn'6),oE 0c 6r.rLdEo4L€. 'Orioq or[6ci\'€lznr oio naQdb8ryil.r, Fnoeei vo rl'rdaloll\' Trol),oi twrLorpdqFor. riv<raq rciieog,. Io 116\'0 v6o rolj 6Xoup€ civc evag r"oobrlvcpo; oAroFoq rlq Kt_EEora Ar)rtit i "ut 4 ru!4 tot ! ty t7,1' \ : t ' t t I ,6t iyopso,t r tontraxu.

EtooFrotr€vo Erdlr.no T€rQdyo)ra

Aq €inoredrfo![E ]iQoE onytliv oro anlouoraro si6o: nrro6],4[(roi !:].cl,/iorri)! r€r0(y(i)v{')v, rrry !'xriplotl 7 rou ddAouE €\'oq .rooevorJ<. 6(io€t 6rr()n(Air tolnl ' ̂ t = L br . t = r : . Ezros on rtv :TfCriTr, ,aI nou or t l r lQlurrsrcuriioviclr, avqfr(rnilorLpr i:vo aoqr6i6amo ouoqtra 6uo €€Lo(i)oeo\r pe lvcv

f r t f / , Il r l r ,r= Lrr l

MtXL)r rd)Q( oeopoioc4re ttg bro parprioe( e! ioou (Eronlors< xar hroljoaBs rylrr+L]i 'T ro! r),c1!mono[ei, to Er = (,r: - 0 )r + (:r b])r:

-::: = 0 oto .T =b t+b :

2

To 60,11010 7 ri\,o! I fLdol rL[rl rov BrrQr]o€ov 11q! ro i6!() o]jlnaocopo naiQvoqrda:rti qr eliooroq,,{1,4:r =,,{"1. tlpriWarr. o ,'l?'1 ei.vcu :ri.r,aaa< 1 eiri I x(,L It;to{x,rl }iacr(r 2.rl = rj + r:.

:Ilt!(;(j-ir, ort o{wo$rt rtr(!Zr},qq{ pc(s, x(r irl! npo6o}.r1 ror L, :rcirrol r ' l

o r i t v s x 0 € i . ( b r c l r o u a = ] 1 . H : r p o 6 o L 1 r i v c u . p = r a . $ v = n r b f u r tL ' l

z(r o loyo: (rlToq Ei!(r nd),! rt [aq \$ (Lt+b)/2. Au t lom6\' Ei!(a I(tu(]rol.oyu{li crdYrlol. exroq (.,' fl€(1)Qnoo!$e qv e!49 rllpc}.Inlil. nor dlcLngl1.rLno fvbLcl.t60ov.

v'roqiln( Liu or\ 6no lEratioEtq 6t1' a:ro6i6orpt rry ibn aEntuoxia. Hrr$ri.f = 6L ['roorf'ra ]rpztnrer crn6 pLc crxpL6ior€al zl,iFarc -ri. o€ or(rc'rui6nrnjijl4rc, cno trq,c).nrapo 6af?[o irn 6,rL I (tul ], = rr. n(10' 6Ia a1]Td o[ll)s,bsv oiLo![€ !G oQxeorot,pe m1v.r' = 61. br6tr.1 S[Urrpn p€tplol reqriXer xL currtndno'c 'rLleo(toQia. O u:r),ouorrpoq o!p6r6(op6q Eivar 1c 6r{r}.€Eowr: €x€ivo rot. ro oroio E),atroronoLei. xo Aqeorot& rerpay;tot pt 6dp4

Er = L, i ( . r - b1)r+ r i , i ( r b1)r.

fuv " x ' . rorr u:r116.6oqrr ' 1rr yulurrpl o4Lrrnrr ort ! nr,ul t nuvurte,tortxLr l

I SLo6rxooic €).a1roronoilolg rrpoorcOei ncQroo6rtqo vc gr,xpuver. ro (z 61)r.Yrolol,flowu sL,xoLo 6rr

]6 ,\N :KOI1H:H KAI TIPOONTIKH 219

zar €Ero(;r'ovrdq ro B€ ro pI6€! noiavo4rs rt !6c hroitf,,.:

Alri-nt: lt€olE rLt{tq r(Dv bL r{ar i)r nou r({iAvo$0 or(1 1r:r = rir = t, ro 7", €ivalivt< ola0prcpivoE ptoog -rtll! 6€6o$i1,u)v. o onoio< eir<l ;r),IoL6orapoq ngog to

To owt0rd$.1,o np66l.Itrc !:Loliororv retptrldrvorv nou oatyri oror i;,,,bt(rurc,rarr(rr rlzo),c xcr :rpor{nra c}.l.ir(o.vrcrq ro Ae = l: nro vio ouorlp<rIr Ar = II'0. Att6 dJdler t4 li:oq axd t o€ t,1,. :ro no0d6cr./$ir [us

, = lll ,1u,OL zcvovr.xtq eiLodoer.g p ro vi:o ;rp66Lrprc. dlow rov II_J orl 0aon rou,4 ncrro ll', otl 0dol ro! r. Kcrr(i r(l {i},1(l br\, ol"}.(iicr rirore xc(r o rir,ox(( ll'7ll'euqrc'vi!€t(rL xcr orrq b]Jo nlruQiq rov oroelrold.lov xuvo\rr,{(i)r eElocijoeov:

H 1io4 a{iortttv r|lQey(itxo| rou ll'A,i: = Il:b nqooitoQi etu dtr6

(At rl"t w A)i,,, = A7ll't rr,b.

TL ouv66l (]rq lecupetpr.rrl crzclvcr, onoU ro 6 npod(L).6r(\, (]ro /1.T: Kl(urn {lo rpdncr ririol<.vcr lleL rr},l.i;er. H r0o6o),ii ti.vcr :rd}.r 6vo olpri.o-rorZtbQoti ord.d!, norl €iltlr nLrtordorrQo jrpo< ro i,. 'Opr,r;

I ),i:EI r),to|doftao '6X€r 6\,( lao v64pc. 6t0v ro glxoq ownral),(lll6hvsr. rov II'. Eplci6Foore F'!:!( or(l0trlotrd\,o prjro,< ror.r. nou roonrcrL pe ro nuvrlOr.ogdvo Frtroq rou II r.K.10trorlra bev orpcivrL:rl.dor yr.r'=0. :ro v€o ouorltrc, ro rprrrlpLo tivcl(ll'y) / (l I'.f) = 0 1taL oto [iaov €l{a.viqEr(lr o nivax(E Il"/ . H rao6;],ri Ar,,,ztlr ro o.tiri$.l, Ar". si.l'{rL n.iLL xri6Ero. r!Jr'orn{v rI v6a 6r,vo(u.

:rI! r€ir-tl1ai.l :raprhygrrpo reqL.lqdqortt ,a ra notprxa ywoptttt:Alrd rporrrirolv cr;r6 cnrLorperypouq ;rivureg II'. nor orlv ;rpulpn-n#t4r<roqL6ir).louv [6vo o'6v(v olpF€rau.t6 orvbtrrropo C = II.7ll'. To roculeqr#yrv, i . , , ro rou rrr ,(r Jrr- .r . :4Ln' ;orr or Trrt i l , rv opOoyovror r r u r u I I ' - ( ) . r u o ; : o o u v b u u o t , o c , r \ , . r a q , ' T Q - / . , u f o u r p , z oy"rqevo ba,\, rircr ri.rore vro r1 6rcqoperLx6. O x(0opLoF6< nrl, 6crpcov n;r66!c! opooydllo nivcrnc. Srr d1r:r zcgpicr €ri6Qcot, aqo| r1 :rrprmporprl roUXdQor. c'qrtv€r ro €o(ur€prx6 yLlo$r:vo avcl),i.oi.tDro. Kir0r cilloq II rit),ri(tr" xcu.1(r tritzo( xcr ro €oo)r€QLzo yrvo[€vo.

- ui6 + ulbtrlJi + r';

" =[t ,1.1

fru xtitu uvrrnpip4to :tivav.a ll', ot .tt6lFvor xuttit tg opllow iva vioEoorEaH6 Yu'dtu vo ,a !tixo<:

( r . s ) , , , = ( I I ' V ) T ( I I ' i ' | ) l r l " , = I I I - r . ( 7 )

Errrbrl o II' eivar crrroreau+oE, 6ev und€,X€r brrivuopa $ Fl6rvrr6 !r1xo_< (ur.r6qrou plbevrxou 6(ovuoporoE). Ol,(t rcr €ooieQlrid 1r"v6Feva nou eloprd;vtar"tactr tu{drs q,"T( i r(r : r 'xar y xoL r i !s! 0€r lr(o 610\ ' . . =] / I0- oQt! ,o!rc| , ,acr 'nuor tov rporo. ge r4 6ori0slc r{rinorou niv(xc ll'.

:t4v:rpirl1. ro evbrrtpipov ;rgo6)qgcr eivur rl srrD.oyi rou II'({ tou C).H zuLritrp4 urcivr4orl irpodpXera cur'q otcrtmrzri ].cr cQllxci cln rol causs.Mnopei vu lvtopiloupc 611 4 peorl rqL{ rou ogdl,paror €iv(L pn66v. Aur4 €i\,crr1 "uruprv6ytv4 rtpril" rou oqd.l.trarog rou i), rrce'd),o nou b€v n€Qqrtaootr€tlQcltt(lrrx.i frl6€!u{6 oqirLFal M:ropei r:rioqE vn pr'railoxF€ rl Faon rLpri ro!{!:rLnr/(;wor, lou oqil}.lloroE, ;ror rircr 1 6nonogd. E.d'r ra oqd uro r(,)vparprjororv b1 tivu! crv€EdQrlr( tr€r(qi rouE xcr oL bLc1){14l(ivotr"q rouq eivcr o].r6tr rcr oootti 6rlall riraL rri = 1/ot. 'Ooo xd,ur€Qo lv(DQi(oL,pr tu(r p6rpryI.trQ(.4rc nou (!t(l1,(x).drcL oe pLap6tegl dLurlFtrvo1, 16oo nro gelcl),o eivc0 tooaQos rI;.

fleprrv t1c rrvrrlro:rrmirrg lov uviorDv xr(e(rlerioeov rivru zcr ro orL. c!r,?icytopei ru y4t thu uwEiArtrt< tttt.aEri rorrg. Eciv ra orpri\rcra si,L1l,oeE(Q-Id)vr( -r( r(x)l\,{Dft!xci ytc ro'r' :rporbgo 6w eiv(L cvelcipr4rc an surd lL(llov ls()or'dL(rmi ,ic! 9!orxi( ourr- c;r' cuti ./t(l rov crrtrnp6rbpo- 16rr o IldXtr 6porrE exr6q bL(yovio!. O nlo rllooQQon4lrdroq :tivuxaq C = II'TII_ si!.rlo (vtiorpoqo( r.or rh,ozc bl*rprivotuq. ro! onoiou ro oroqrio (t.j) €ivo!t (v(4rrlvdf$,Jt flFn-rou (o(p(il+1oroq ro!l)i) t::ri ro (oqrcil4( ror b,). H xlQrs6rulrhuoq ns0rszsr. xlg b!(rxullcivor!( aj. rou rivcL r1 $orl lrlt rou (oq.i].Fcrro<

IIAPAAEIrMA Yro06ms or 6uo ouvircLpor xcdxre< rou bridge l(\r€uouv(Bt1d tt< &yop€r) rov olnz6 (Ar0F6 on(0Lri! nor zcrrdXour. fr( ){d0r r'?riprtolrtt oqciLt(ttr 1.0.1 dlur 6ulcr6v vtr 6xow r1v i6ro n$ca6rryrr l/3. To(lvotrEvotr0ao o{(i},F.l €ivcL 161€ ttnbav x(r I br(xutrcvorl €ilcrL 2/3:

t ( . ) = i ( -1 ) + i (0 ) + i (1 ) =0rG:) = +(-1) , + i (o)r + l ( r ) , = i .

Ot 6 o rzrLptioar"q eircL elcrpqptrrq.6rorr. orpilovraL orlv ibLC neooqoe(i.ciprD; 6t,! rcrurtlovrcfl. crqou rpood).inorv oe 6r(qo0rrLltd XaQt(. Yroofore 6rrI trt0(.v6qta \(r €iv(L i.ft or 61l(.) roli ul'Il€f t }ft or bio no),i tqrd,6g ri'rcry46dr'. (iJri 4 arunv6r4rtl cr\.rLoarov ),(06v €ivcr 1/3. Tdr€ E(( rer) = (1/3)( l)2,fl o (vrlorooq o: rou -riv(x(l br(lzllt'(voq: riv,rl

I I r _ t u I | _ / " _ t t - t t t .t :

3.6 AN 'KOIIH:II KAI NPOONTIKH

O zir,azcrg auros p-rcdver oro p6oov tan, mn€plof6fttv xovovtx6v €E!o6o€ov.

AIKTTEEE

3.6.1 Yno0lor€ 6iL S,ar T €i'ar ln6Xo€or ro! Rrr l|t dim.9 =7 aar difr? =8.u) noro €ivar rl !qa),u.et 6wdri 6Ldsrsot rou 5 n 7i6) IroLa €iror t] Frze6ftq 6uvdri 6rciorsot roti .9. Tiy) nola €ildt r Btxe{ft€qll 6waii drdoroor ror S +7:6) nora €ivar I p€yolrrr0At 6t|vorn bdoro(nl to! S + ?i

3.6.2 no|ts rivar or roleq ror enot6'0rv gFUydv rrToxdE!'iu) To! €rr1JI66()! i:,.v r{ar ror €rTrfldbolr 1/,; orov Rr.6)

.l'lls ruoeias brd ro! (1,1j ]) rur rou ult,,Ia6o! bd rrJv (1,0,0) 1iol (0,1, 1).

'l) Tor pl6elrxo! 6Lovuofatos )rdr ol6x)'tao1J rol) xdao! R3.b) Tox enlId6o! ro!

"cr06rou oio (1,1,0) rot rou €,rurd6o! ro! ra06ro! oro (0,1,1)

no,i, eilcr ti, urt€,iolrdro aurdv rov !n'y,ir! uol,;g !:

3.6.3 Maos orov x{'ao ro)v rLvizo)v 4 €rt 4, dor(0 l/ o xtroxoaos n\'etduryari@rnrvdxov xar II' o ur6x()Aq an'\ d\o ety()rt,i'v rrvdxo'\,. n€QlyAdln€ tov1trr.jx()eo v + ll'. ror onoior,l.r oroqrio riv( or dl(o Hcss€nb.rs nivarcE, t oo(kxur ro! un6xolo v n ll'. Er.rlte€uorr rov lu:ro (3).

3.6.4 E(i!ol/nII fl€ert,($ p6!o ro rrD rxo bl(ivuolc. ror€ I (3) yilt-rar dim( t/ + ll') =dim li + dim ll'. End{0riors ror,ro 6rdv o V :ivcr o ;aoE yq{4lpdN rou A, o n'ria(r o rrdraO({'oos aoL o,,l .iv(l nl .ni tr. r'iEru5 r. nor€s €ilar or 6roo|lio€|<i

3.6.s Adm€ nagi6rnrc cov Rr y(r ro oroio o ll'n v rrqra7fr p6vo ro lrl]bEvrt6brdvuopa. d'ld o V 6€! ei\,or oQ0oyd,vros orov II/.

3 .6.6 Ei iv vr l l l '= {0} , r6r f l o y+[ / € t rdr ao E1t0 i . jqeoolu !o, t V ' lo . rw;anrolF6o),iqrrdr Ft v O l,f . El o v roady€ia! otro ta (1, 1, 1) ro, (1,0,I). b!fl]'dEt€lnoxopo ll/, dror rlotr Ir + I,l/ = -R3.

3.6.7 EErlrrtorE yrcri ,id!0€ 6dvt|spa j oro s$n deaorFo v 0 W lroari !a ya(rv€l !pivc xc, povov iv<r rp6ro o(1r r = 1) + rr' (!s u oio! v rar r, oro! lv).

J.6.E Beritr pra 6dor rou aoAoiop ro< y + H/ io! Ii,qo! y ror' noa4€rar oro ra or =(1, 1.0,0) . 1 , r = (1,0,1,0) xsr tou : (dqou W no! naadr€ra! on6 ra ur = (0,1,0,1) ,rlr = (0.0,1.1). Bqsir, €nislg rI 6rdor(ol rou y n II,' rdL FK 6dor ro!.

3.6.9 AriEtt r€ raadberyp( or! o !n6€v6,(oeos ro! ,4D b mqrdxa zft' dv(i.|v{ rovF'lb€vox()qo rou ,{ ,.ar or! o t6eoE orl}'d'! rou Aa b.' r€aFtftar aar' ovdyxt oroxi,qo aqLn! lou B.

1.6.10 Bq€ift rov rr€.rLmo awLorQ6{,qro rrnoriv.u.d xdr 1rl! niErlr{|f

tlfiltil.-

l 0l 00 l0 1

f l 0A t = 1 2 O

L: o

lll oPooroNloIl ITA

3.6.11 Yiooim€ dr o;l .ivcr n rni n xur o I €ivd r ui ri F€ n < m. A€iEr€ ftr lo1v6F€vo -{B €iv( rbrorrogtos ri!s"q.

3.6.12 aFtEr€ un6 rca no (3) 6r' rdqr(rt + B) < r.G(11)+id!r(a).

3.6.11 Edv o ,.1 Eilar rftacl()vrxo( (vrrotAdlrtro!-. briErr 6( o ,{lt dxd tov i5Lo !N€vo-)(opo rrt rov , (zd0(bs x(l rov iho lii,Ao yaqr!6! 7or rt i6ro rdEn). Yr66!Gn:Xqlortrorotrlmt q oxdol (i) x(l| oro yrvdFrlo tou '1 I pe ror '4B.

3.6.14 loeolovronoLrln8 ro! .4 o. Tr!6fF!o tlos 'n sni . rilaxo I zat €!oq- r eni 'i

,.=t3 i I 3l xc' | ' . r ' r^=[ i :

3.6,rs nol,ierlao(iiowqE ud0r urt;\r tou , Fr rrv cEioroLTrt yQ.rr4ri iou g ,(( trqo-o0ilolroq. noo1trrnt€r ro yrv(i|lFlo .,1 = r(1 o(a n0aororo r nvdrru,v rdF.r{ 6!l|K(r0o"c!(iore iotF ! roL q x(o(i)s acr rorF bio niraar: rdEr(E ara rTou 6xow(ieQotoFa rov

. 11 I 0l'= Ll i ,ijJ.6.16 AriErr orr r roFi rqLr;v 6'trL.iorcti)v lrot,;arDv ror Pr btv tivur 6v( fova6rao

ottp€io {0}. Yro6{Ert: n6oo Fraart FropFi lcr oilst rt torri fu'' bio rtni'r'!

. } . 6 . l 7 B e f l t f l 1 ] ! ] i l e , , Y i l v t o I o i I t ' l A - I . I ) L l N a 'ndpdyor r ! ro ! ( lD 1 / / l r r1 r . y r r rov

, t l r2 l- = l 12 as l

3.6.18 Erclle.ldt€ ro! roxuoLotro, oI| "rd0r , sro f .ll' nqodex€tdL dno ira xar p6!ovild .r io!.V(r) , r.qtyadqoras mj4 yu booia .l,/ ftryaivoq'e rioo oqr eEiooorl15) l.G 6aio"olFr ro ..

:r.6.re ri o1l6di!€L mt daeF,dFaq Gor r'fi ro = (ui4 + ulbrt(L'i + ul) ,tr(v mf,qdrro 6riqos L'r reoo€r/ilsr ro |tbivr H ri:a'lor l,r rivc{ r€}.€ia5 alc,EronLoq.

1.6.20 I r ( Dr . rv€E( iaqr t r Fseio&S br , . . . . ,nr rov oqtrypon oa<, $ 6dal r . r . . . . . r i " .doL'x Erw' o mdgrrotravo! lliooS no! (rwrz(l0rm.i rov (6): H zdLi'rra1 r"rirllorlyivdaL olav oL m(tTLonxi< b'ntldvosri €ivqr di = r/ui.

3 .6.21 Fi i ! I I ' - l ; ; l

6a. t r . - l l -€oorrp,zo lvoFFlo ro, ' r - r ' . t r ,u , rou s ( l . l r

zor ro ll'-}lrlto! rox r. Ilota lr(vlotfftri e!e€i( ei!.r ll'-z(i€ftn mo u:

il

3.6 ANA:KO[I{>H KAI IIPOONTIKH

3.6.22 Baeir. q ),uol mafutopa!0)! e),axioI(Dv r€raay6v.)v '!, lou .4r = 6:

12 0 0 ll0 I 01 .Lo o 1 l

,[i;],[i] w-

3.6.U

arortor6otc 6r, r Eqo6ol.i ,,14, €Lvo, rdll ud0Ein mo oedMa 6 - ,4-u (oE trqogro tl/-roorrs0r6 rr6p€\'o!).

d)'Eotu orr pan€nst€ qv rllxid ror ao0rynti ods, rdvolidsrd l.'igrle = -2, -1,5px nto(l!6a\a':t 1/2,1/4,1/4. End).I€enor€ 6rr ro a!dp€!{ir€!o oqd}.pa t(e) €i.',arpt66! ror 6a€tft nt psro6ol{ t(e'z).6) E{ir o xooryryir Bcvre!€r enLoft (i ,raoo,ra0ci va 0![|]0ei), ivoyids roo(ri).pcra 1,0,1 ps rroavoqra l/8,6/8,1/8, no|a 6aet urr rar urz 6ilo1Jv rlva&onrcria qs brxis oos €l{Tfpto$ rd! rts €xraplots rolr '|aeryl]I4i

Yno0dqrp orL rJ yqrypds raar q mrileE, l,ot6ovdperss paEi, reodrow 6Ia ra molxeioro! r1. AsiEie 6fl I rdh 6w €Lvor p€yal.lji€al dn{t p + g. ndoo p€yd}.o TiAdn€r !(tei.lclr tva rrreorovrxd pn),ox pn6€!rx6v o0 prd yovia evdE rrivaxd 9 tni 9.'yro vo€Eoop(lligeror tu o rivuraE rilar r6ropoeqoEi

AIKIDEII rIA EIIANA IIpII: K€qdr'oro 3

3.1 tsauirr ro Frixoq rou o = (2, 2,1) zor zoroyqd\i,ors bno alEEdqrrrc oanoparc'ro! oiva' l{(io€r(t mo (1.

3.2 Ba€iG 6),c ra brcauopara ror ft!(l| rdoora oro (1,3,1) xar (2,7,2), noiplorraq roftleUr(i.o ox T aapp6.( ror A rd! ).!vomas io Ar = 0.

3.3 Iloro €ivft I yovic rr€rdEn rou o - (2, 2, ]) rcL ror b = (1,2, 2)i

3.4 [oLo €ivar I rao6o].i I rou b = (1. 2) 2) rdlo .no a - \2, 2, l)i

3.5 Ba.ire ro mrl]lllirovo rI!T(Dvi.or tr€raEn nur braauofrorol (3,4) nor (4i3).

3.6 no! n6({,rFr I nao6ofrl rou|) = (1,1,1) ndvuJ oro rrinr6o rro! ropdy$dr an6 ro( 1 , 0 , 0 ) x a ' ( l ) 1 , 0 ) r

3.7 To o!ort[.l .1ir = l, 6XEr ).!ol, t6G a( polo rore, oraa ro 1J eilor oAooy(iraro ssoerotL|jrorE dro rous 0eBa).6b€4 lrnold,Aouq. noror 0i!ar ar"roir

l.E Ilo|C e!0eio yeu!!r)t 6irsr qr lt(rLursql ruqe!6o),ri srs tje6olivd: D = 0 ltd t = 0.b = 0 t n t = l , b = l 2 I n t = 3 r

3.9 Kuruurruuurf To! nr luro nso6uhi : .1 'nuv@,rruv I ' ,ao rou nf tcotr rnt n:Tn i f t( 1 , 1 . 1 ) z o r ( 0 j L l ) .

3.10 nord oraored swcierrtol ,ivar n),Iqdoreq! rli y = r'r, mo 6ldorltc 0 S,i S l.rro ritr iwo(r Io)v rlcriori)! ftradrdar)!,

3.1r Edv o Q Eilar oe0oy(i)!toc. olptjoi!€r ro i6ro rat fF rov Qri

3.12 BeEift olors rous oq0oy6!Lo!s rTildasE 3 uri 3 rov oroiov ra ororx€ic rilarFtbFlLt(i tar fovdbEE.

3.r3 lloto nol.).(lr].doLo ior or nQir€| va aqd(eE0ri atro ro or, dms to dno l,sopo !a.i!dr ooeoy(i)vlo oro art Kd\rr dla orila.

3.r4 loaqovto]ioLnorero!

fcosl i s i ! r l

sin d 0

or ?ra6tl3!o QR. raeaqadwd( 6rr I na(iq orriLl e|wr rtbl plobraio b(d Jotlo.

3,15 Edv rd0r morr€io €v6q oe0ololio! trild"d si.vdr l/a I l/a, ,tooo lt€y ,o< etlar

3 . 1 6 Y n o o e o r e o r r r | ] l 6 r a v u o B a r d q r i . . . i q n r i v a r o q 0 o r a v o l L z . i . E i r v b = c r q r + . . - +.,cr, bdrm. e!o! xnro yra rov ra&o olvrE)"€onl (ir, ouvceflioEr ro! b zar rov q.

3.r7 Tr reaqadqo!! or.Ero.irorLs iT,{r = A76. ro 6rdwoua p =,4u = Pb roL o trild"d(P = l ( ,1"1) 11" ;

l . f8 l nv,n oq.Jo/ovuq/q i t rov"oforo . /_ \2/ 3. 2/ 3. t / 3) xat L - \ t / J , 2 / J , 2/ \ )I r v o r o r r l | : r o u Q . r o r o r , i v n d I v * ' , Q r Q / r Q Q ' T . a . r n . u U u Q Q r r v o l6vas rivoao( rao6orrtc- (rirvo mo €rift€bo rov qr aor q1).

A>KH:ED IIA EIANA HVH: K.q. 3

3.19 Edv !r,... j u' €ivar oaoorovovrri 6dor rou R", 8€iEu 6n ?rtf + +t,,uI = 1.

1.20 A1,roiE rt g€tjbeqr Edv ro S|awofara d! s Eivdr oe0oy6(d 1|ar P €i'at nao6o)'1.rol

"qr tu PJ Xar Pq I ivar rriorc oqgol;uo.

3.21 naoonoEore 'a roe€F6dl.e€ pla €!00i4 b = C + ,, ord otF€Ld 6 = o't = 2 ^cLb = 6.t = 2 ra! ieiqrE ofl o! xaroru{6! eEro6o€|,5 aoraeq6o!!. tt€6!do1€ oI€E rq66].rross Eloei.s rol s].axroronoroiv rc rsradyoao tuv 5no oqd'tuftov.

3.22 noro orM€io io ! €nmdbou o+l / z=0sLvarr iT.nodor€qomob=(2,1 '0) i

3.23 Ba€irr fra oqool{cvovrlalt 6{iqt ro! Rr !€xtvdvldr ar'ro dtduJopd (1i l, -1).

3.24 o|r6€E prxavds arri!(o! x (oaao#s cT) eE€tdlo1Jv rov aos€qi oftti 6Lcaopertxd56r€!0!as€|s ror raadyouv dl(r1 nt'vo)ra Jto! 6ivet Tts mxvdqrq mw om6r, rct mrrmdry os adg€ ol]!r€io. MdEparrxd ro rreo6lrlla eivft !a 6A€0ei avog nlvaras faootu)! roo6old)! io!. :fltv ,r€eirrrorol 2 rri 2, Froa€fre'vo 6aeite to't nivaxd A. orav

1loeiqsr! ro doeoropo r(n oroqeiror td€e lpcpptjg xdr ro d0aotopo rlDv srotl€i(,l

3.25 Mroa€tl€ !a 6q€irc 6vo ni'ol{a 3 ent 3 6rn! 1',oaLe€te 10 doeotspard rov or!).d]'ro!, ro (roeoiuparu 1!' yqcrFpdJ! rolJ, xa0d)d eflLofi xar ro uoqot'ryoro rcrd piaogrrg rueius 6royluviou ,.or r0)\, reooda0)v dll'or raeull.rilorv 6tu'yo!i0rv:

3.26 BQeirc t r l ,oe0or!ovrr r t 6dol io ! Ffn i6ou r -g+2 =0xet rqosbroeiorE to1rTilora .a', o onoios roo6dl).€I enL to! enlni6ou avroi. notos silor o tr'l6€!6loaoE

3.27 'Eai(D A = [3 I - ll xar v o pl6sv6ro)Ao9 to! A.d) Bq€ilE pLd 6cjrot ro! v aai pt(l 6dol tou vi.6) Bqeire Fra oqoorcvo rn 6do1 to! Va xut ro! r vo,d rao6olls 4 flo'rreo6dllFr ra brourott.rro io! R3 €ni rou v L.

y) BaeitF rov Eiv.rx., nQo6otrir P2 rrou reo6ci '€r ro dLduJopcri.r lo! R3 €fii ro! V.

3.24 XAlortonouior€ q 6n5u{aoiq Gnrn-Schnidt"r|0 va llaroolt€rjaoEr€ avo oe0oravo"vf lo !E!ro< qr , q2 sn6 rc dr = (4,5,2,2) xor a, = (1,2,0,0) . EaqedsrE ra dr aor a2oq o!6laoporE rov ./r xcr q2 xc|rot{tfodlJre rov tetyovrx6 ,I? qs A = 8R.

:1.29 Ar iE* nrL yk, xd0. .4 .b, r xdr s _( , ) E. i ! nr = b roL vr ' { = 0. rorr ,1 / , = n,( i i ) r o v , 4 J = 0 x ( r A r , t - l ' . r a ) t t r T b o .Iloro ordqllfa 1ra rorJq o€p€h 6€q lrox.bqoq cro6 xwErar pe atroi

3.10 YrdOXer nildroE tou otroiou o X6eoE 'yeaBFdla 'ita#Xe! ro (l,1,0) xdL io! oftoio!o [$€v6roeoE r€qr*€, to (0,1,1);

3.31 H atr6mool ro! !:rLri6or a"r = c on6 rlv aEll1, oe tuov m-6r<iooro 1tlqo, eivorc /l o . [6oo pcxqLd tivor to €driobo rr +r, rr .r = 8 ono qv aqlrt tot 'iotooqr€io tou civar f,l,rordor€ao o' o1l11lrl

3.32 tro raqol).n)dyarypo !. xoalqds oro 0, o, u id! u + t 6€iEr€ ort ro doqoropo to'terpatdvov r<ov ptr6v tuv rEooriEov rl€r'ad! 10r' tool]IoL F€ 1o 'ieqoqn tuvr.rady6!o! rov prrdrv to! 6no drayolio! to!.

245

246 oPooraNrolHlA

133 o) &€ir€ Fn oq0oxavovdi 6dq rou r6ao! orr)'dlv iou

I r -6 ]13 6 l

.4= l4 81 .ls 0 lL 7 8 J

6) rpdl'r€ roa ,4 o!'t toe{rj QIr, dror Q d1ct oeooravorlxaE oni},e{ ,|Cr o R €ivoL

y) Besift rr ],!orl e).urioiov tslQq.rdr,lu! rou,4r = r,6rav b = (-3,7,1,0,4).

BAeiff flt! ro[t y n I,I/ xd! ro d€aorofct y + W, 610!o) y= p$€vdxoeog nL\]oxd ,,1, xdr W= t(dreos mll.dv ro! A.6) v= or o1Jppsrsu.oi ni\,or€E 3 rni 3. xor lt = d1o rorylorrroi ril/arcg 3 snL 3.

noto rilar ro ll/-rotorepLx6 yLv6pelo rou (1,0) xsr (0,1).6rav o Jrtvdros 6dq.iJ!. . , , t 2 I l

E t ! ( | i ' = [ l 0 ] :

3.36 l-r( !c luoor+F ro rce.d.lrlldryagLrFo ouoqFn A-r = /' aELxdo'oroqr. rov A ' (rorbFv urdsxr') Fr ro! (/4r /4)-r,4 r ('rou

"'(i('xFr ord! o A ixfl avrEdar{rss cnilrE).

A€iEt€ 6r! dtEos eivor 6vaE aEoffQG ovrforAoqos tor, / dnd oxr xa! b.ErGovrioraopos. noLldn)'dordgovros ar6 r' oAroftQd rou ,4 nAoxnrft.r o ra1noltxos,fln6 icl beEd I n0o6oLtt P.

Boeil€ irp €r'oFia lOa$ri (] + ,l nou 'loeq.l6dl€ift 6ilrroi.. oirs pg'tAtio.Ls b =0,1,2,5 rord rotE XEi ,o l { t = 0, l ,3 ,4.

Ba€ir€ rlv LolljruLl y = C + D2t ro! 6tvct lltv xol,vr€elrl noe€t6o),i €laltororvt€rAay'nrov orE terqllse4: g = 6 yto | - 0, I = 4 yn I = 1 j s = 0.rrq I = 2. Iqdqrsq ra€q €Ero6oFt5 no! ),r,aowar dr(l! I xapnil.I ddQxFror d?id ra rQia oru€id, xor6aoi1€ ia 6d).iror0 C 1{.t! ,,

Ed! or oril.€s ro! A pi\,ar oeooydv|€E peftE! ro!s, fl pnoAeaff vo reirp y|a fllForxfi ror .4' A: Ea! ot mirpc ?lvar oaeoxavovrxi(. rorr fl FnoAFirp vn rprrp:

Yn6 ftoid @v0ixt yra q oniL€6 to! ,4 (ftou FftoqEi !d €ivdr naeal,).rl6yeappos) oAr A €ivor arror€dvrpos:

3,39

OPZOY>E:

4.I T EDAT(TTH

Eivc(! bioxol,o \,cr c/roeooiooup€ Tr rrQlri€r, vc roi$€ llq r[( oQi,!o!o€E, rrour[Qlv on6 €6bo[r]!rc XQovlo qo!\'6vro!oc! vo 6xow r€Qloo6reQo €v6rsq6Qo!crfio roug nivcrx€g. clfi6 roljg onoiolE itor nQoliLoav. H loroela tar oQrEovodvrou Muir €y6pr!,€ r6oo€pE r6Forg. To Mdenpsrlria. €v rouro[q. a],].dqo!vowEICbE ror€ljewol xcr! o! oQ[(ouo€E €ilqr nl6o! psliQ]Jq sr6 To xdvrqo rouev6lqqjaQolroq mq fQolrp!,ali Alye6Q( :ro ,4dx(')-xdroJ, 6!aE ordros (rol0p6g 6€vturoQ€i rlcred vo rl€Qdx€r 7r€QropLop6\q ?rlqQoeoQicl "|]!o 6!c, fii,voxcr. ncrQ' 6L'ewcr eL.var y.e1urrl.Ily.1r{6 1(J 7i6oo B/ioQ€i 1(r xav€l auTos o aQ!0Fo6.

Mla qnoq'q Eivcr! I €ErlE: H oQiEouoq bi6€L evov ovol,wrri6 "rr.rno". prcror,"perpLpivrl troQqoxqol o€ xl,erorrl $oqqrj, 1c rov unoLolop6 pele0tirv oavtov,4-1. O ri:rog aur6E 6ev qlJdleL rov rp6no p€ rov orroio u?roLoli(oup€ tov,4 '

{ ro n '0, pci}.Lora xcr 11 i6rc 11 ogilorocr unolo"yi!€tcrL p€ a7rotro!$rl. >qv:rpcrnrcrrrz6qrc. q orr(llorq{ prioo€i v(l 0e0)Qq0€f (,)E o nlo cfiorel€oporlr6ErQ6fioE clvll,ioriloroong 1(r)! orol1€io! evog nilcxa n eni n oror r]jfio, Aoro:ror t:rrruylirvcL o rr!:roq eivar vtt bei.ler tov tp6no pe rov onoio o A ' elaprdrcrcn6 rc rr morxeicr ror vaxo, xo0dE rrrt io nd)E (u16! [etrr6cr]J,€xor. orc!F€x(6dLl"o\,rft .r1}Id r0 ororlef(.

AvdqiQou[€ tr€Qrxi6 oro rrq ruercr€e€q Xerto€Ls qq opiqouodE:

(1) naa€X€r 6vr1 r0mlpr.o cvnorqsl+lqdqraq. Edv \ oqilouoa tou A xivarp$nv dn o A zivat ddpoAEoE. Eriv det A l0 rritx o A zbar wrtotpirprpo; H:no ev6nq6powc eqapgoll, rcL rcm6lpovc o L6yog yrc rov oroio ro arrpciLarocu6 eivar orpalrLx6 1Lc

.ro 6r6f io, aqopci mqv orxoydverc nrvcizcuv ,4 )1. H:rapciperpog ) aqcLpeirar trr' 6).c -ra braldrvn oroLpic rcr ro tq66Lrpc eivctlc 6pe0oiv exeivq or rweg rou ) (or rdrorrylq), rra tg oroisg o .4 )-r €i.\,c!r6(6poQqoE. To rQ[iliQro riQo]irifii€l €EroovowoE rqv oQiEoroa oljrou rou fiiv&xa

2.1lt oPtzoYtE:

t : " '

/, /

/ - - .

' , r ' ' , . )a-" - /

i

\a \ , ar , a31)

"I ,', :>,:.-',.

; t v' . /

. . ' l a , q ! a ! )

Ix. 4,1, To ruqc).i,I),enifl€6o rou arol.',€irar urrt T|s yqafilig rou ,4.

to Fr|biv. Oa 6oupe on I de(,4 - ^1) eivoL adr.rxilvr4ro tog:tpoE ,\, 6a0pori D,xc& ow€rd)s, perptirwag xaL lolr.lnnl,orptE, 61er l oxpr6ti4 pileg. O aivuxogl,om6v aX€r ?r L6ro UtE rol uur6 siv.r! pL( (r],riO€!c nou r€roxrjnftl cnd rov rrirro14< opilouorq xur o qno 6vav utroloyLorrl.

(2) H opilouoo roo A tooiJlcr Fr iov ritro rou fiaQd),l,nlerLnd6o! p roo,-brdororou Id)Qo!, rou oroiou or crkp65 oQilolra! an6 16 yQal4laE rou ,4 (:X.4.1,).4 'IooJE qqv€i nsQdbolo ro vc 0dLoupr vc ufiorloyi,oowo 6vcv rdroro\r 6yxo.:qv nAdEl ro P €ivo! oul\rd 6v(r onslQoor6 oror,lei.o 6^7nou, o' Avo noLlonl.6oLoxl.risrpc. To corlouorEQo moLleio ti,r,cr, tvcg pr,ap6g x.ir6os dV = (lxxl.gdz.on(,E rFqavitfror oro ol.o1{l,i€r"Fcr [ [ ]lU.!J..)dV. yr(ot iorr 6fl. yLa vatr),onorioor,tll ro o).ozhiQlulr(r, uxoqaoilor4r| vc pnru6or4re 06 r{uhvberxiEouvreroyF6v€< r.d. : . Or oldoEq eiv<rL.r = rcosd,y =,"sind,: =: . T6r€,6;n g (xq!6d\ np6rer v(. 0up6Fsdr€ 611 6vc( trlxao orort€io d} 6L(rm6lJ,Et(L oTo(ctt I tluldu -otw ro u arrflserord( ro r, o' 6vc tr:rl.6 ol,ox),iQ{Dtto dro! xar ioorotleio 6yxou d/ = dtdgrlz brcor6Ll,etar orq rpeLg 8Lcorrioerq. To r.l,€ur(io'ti\ErlJt Jdrdodz, 6no1r q lax@&avi*a opilouoc atvnr ro rpL6rdmoro qvdrlotorou ouwrh:orri 6raoro)'riE dri//ri

l(htat ax 00 axta.l f.".rr /.,n, 0lJ = l ar /a" 0u l0b 0u l0z l = l . inp rco\p 01 .

lDt , r az /ao 0 :102) L0 0 l l* o, (4,& o( ,td buvar6v t! oaaEor( a( @6 1( mi]< tou ,{. rcdn( ro! 0( 6ertE Ev(

r.t.en( b'tutqtrr'ii .ruea,].drrrrrbo pr r,r io'n q'n( iryzu.': :,r,E.: ,\ro ro! na6oo tu0+rd!x6 K,c,J. hcobi (l$01 t&51).

I I El:]\f!2r11

H r+ui cL,rric rlq oailoL,oq !iv(r ./ = | xot.ntro €ivat ro r rol zLillabQtrororoLlriorl oyzor td 0t:. rt'ru €irct ro tuzao 1c: ;rupcr),?'1itri:re6o. ('Orar:rpoo:ra0rlooryr rc ro oTrbLrnrorpa, ntro qttivttctr zqrrL,idyLx4+Lo. (il'.i tr€r(-6ir,iFr(u cr- oxr661 €r(hi./0q+ro zcu(;q or (ztric yirori(lr uaeLpootir pLxpeq.)

(3) H oaitoro( 6irtL rrnoLc y,(t roLrE obl./oltq. O€!)alnz(i, 0( t,nooor!oqr. vt1 rrt xalortronorloorlrr 1u vr1 nao6),il,orift€. :r6tt oe p(c 06trl o6ryonH(r Elq(!ro0fi Ir]c\ir. LrrLtrr 0( ZQfr.ro0€i €rft)'icryrt yaq{ll(irv. nrc L'16r((i60.)Irrz61rrl rirur ro 6r( o rino! opilotoc = +(yrftpto i(rv ot',y.tr). obt]Ei oroo$nisrrotrcr rn urtEtitrttttt uitj t iultuEu 14. und,ot.Iti., ro )1n\)!Ltt'o rtDrob4ytiL rupuuit+t to ir\rt. tnrj< utni r,r\txtiyeLa u .nytj r?o,/it,'r,. npL! ffio

IrqLxti lgouc. rortro obry)los orl\' ;rr;roielorl. i)rr rlr(1 {1ozojro f (tiTofrllzertircrc roii przpriq (,6i)j,dr trr (r(ryt tac4rtr(irr. br(;r fvir,/ottia.)g o lurQogollyoq 0(i tr(l< :rpoiri6or r(pcrxri-r|r. A!ri) i4(r( nor'orttfttilrr orivrtg(Dc orlrrpriEq. rirrn' l|v (:rogeryrrar iva: crorlvtt0rrtrt trz0(jq oblloE. ri!.r orr (r,ro<(1zoioriorircr r(rti r6ba: (l,.Ii) iv(a (o',v't0torc ltty(iio ob,tyd. LroL !n(1(1-rlir.rcrr{ ro iLvo[t-vo or (t,roro),o.,/rz(i trryi{ht. zaraotpirl,ttriL ()tr(r< i) (l0l{)!rrtrrt|lj

(.1) H o0iiouoll terit(i rrlv lE(;arrlort rori --l I rtnri to tla(iorort' drorlrio, . I1(A, d f \ (r nf i lnqr( (dJnlE!I Unr n(0(4r ' r0o: r l bro0{)r .{ l t - i t r (( n{14(11)tQrt( ' i t . o

' ' rnrrrr i r rrr t : r ' : r ibsrrrr tq (Jro.f =. l , s i \ (rL fvc nf i ixol oL)r(orxn; l .

YndoTer x( l r ivo { i i io na( j i ih lgr ur.rL( opi io!o€<. Af! f i \1r t r( i !o birouoi.o\,( (xut(1\'{)oitr. lLir rrl rt}(roi(t roLrq z(I Tlt Oioil roL( orf (lr,)r)i( rrF fQ(trtrl',|ticA), i r t i r -rr : . rr i ) , r i r i raL broxoi.o nt oqrq l rvrprr lu. 2(tr d i i ' (v oarotro. I I !o(t(rv;)c'l dcr -l bfl |ivct zur rooo u:r),rl olrriptrpl rlDr /r tLeroiritlrd)r. braq o4t.Tlrr't o--t 0( L,noioll s6r(1! ;ro),,i f,,x,)i.irr.a.i .lJI o,r' ortri;(i\,fr orlt! nt'itytrrrrlt6irtt(.O (l\,rli|rrxo: ri,noc orltv fl(1x1./0.(fo 1.3 Uu on(rrrlofr (l'Jx{a: fxrEityrlo!I! )1tu. t r / , , r t r "

" r , , i ' ^ r r i r r . - \ t o , r r r l , r . n . r ' u n o t o r , r ' L ' r r r i r i . r r ! r l - .

Tu dt .oiorEaa rgdypatc yiplt fti' lnv oQi.louoa 6(v ti|ut ti'.,o olavalvazoi rnror 6ao ot duitttll1' rrot' try' Tapaxt4pilovt. Arird rrro6trzd€t!r( qlo,oiolLzo olprio erzi!r]o1;. H opiio,rfu t,roari !o (z(t o!rt'r. 0()olro0ri ffio rr< rpFr.orlrin(,rbaorrorg r6lot1tr6! ntq. Eror ro xaooi.fl(r 0( eivalvc bei!oLyr, :rrir:. Tplortronor(i)!r(! ororr}rcrrzir ttrri: rr: L6r<ittrlg, p:roporgt rnr;roi.oriioqrr nrr ooitorna. npirllLa:ror 0c trt< obtlt'rto.L ;tioo oltl cl:r(iott{riCliusi r.(1r oro yL\,6t1€vo r(1)r obr drv. To nro 6Lro,1oio oFr,ralrrzo npobi.i}r(l 0|1r r , , , 1 , , \ ' ; j . , i r o f l u t , : . r p T , l r , r L i : r , ' r ' u r i ' r t , r - , 1 ^ r o r L r ' a t o r . , o : 1 u , u , j , r :

L6r6r)tr.q. ro c:rorlieqc eircfl :I(irtore ro i6ro oL tcpozrlpronztlc tbrdrrlr€!ril'(r ortLbrtiaorr; trr-rc:n rorlq.

Ir4r, r:ro1Lru1 :rtrptilplq,o x(1r.ltacqoqrt rt: xuQa,(rlQrorLXiC Lbtoritl€q rI;opiloL,ocrc zcr rr< r() €r6l(rttiporor( ollin€raq ror,:. K(roxlv. orll IIopa./A(rgo.1.3 biror-pr 6Lrnl,opor; tnror< lra rlv opiqorou irar ardurt)1o lrr rl oaor,:.i!a! di,io ll.1tr,l1zo rn;ro zcr (vav reiro nor readqr6civ€L obl,or,c (rrro1 o'roio rroiol,ii€rc' ort! nadht rt orliioroo sloq trFltiior niv(l,'c). :rll

oPzottE:

Iloqdygcqo 4.4 XQ'Io+onomqre rlv oQilouoa y[a vo 6eo1jrc ao! A 1 ,ul.va ,lliooup€ cDE rrQoE c = ,4 16, nou eivcr o fljros .ro! Cramer. T61,oE, F€Frs t[QoatQstlx{l rlqQffirlQnol oxetlad p€ xt"E percoioeq, ono6euruloupe 6tr, orr66xllrEE 6iva[ o14r66dor6g psrqEd roos, dQa 6Ev !r[dQ{€! oodqstd orov oQlopd.

.J.2 IA]OTI{TE> TH> OP]ZOY'A'

4.2 I IAIOTIITE! TIIT OPIZOYIA!

E6d; 0o npoznpr 6vcg trclipig zcrdl.oyoE. EuruX6E 6pr4, or rcv6veg eivcr€lrolovorlrol x(li €firEqyouvrar o)i6[q €1)}iol6reQ(r o' 6vd noQd6Elyla 2 E ii 2.'Ero! 0a €rcLl0€{ooupe 61[ o oolqorop6vog oQloI6E orrp fi€Qifir(l)ol 2 efii, 2

6x€L rK Lbrofllr€( fiorr ovcqeQowo!. (>npelcbore rouE 6io oq16o),!ol1oug riou(rtioosloFdole l,!cl

,rnv oQilolod: li det r4 I /.) Afio rlv i&{lQiq l6!6qrorel p€Io 6Ei1voi4l€, FE 7io!o rQ6fio dr.,rli ripoxtfi-r€r an'1q fiponyoupel,cE, ofi6rExi(e€ r6r61lla €ivdr o!v6ri€!o 10rv iQ!6v 7rQ6rov. ToviEoulE x(rr, ridh 6rr o[rcvoves €qaQlroEovro! o€ x€rQo1cov!&oriE fiL\axt( onorot'iroft l1€figot,g. OnpdroE xcv6vcE pordler pe rtlvcopc, eivar 6po6 xqiortlog,

L H oai€o on tlaqr&ar ygappLxti atui r1v nQdxn yqappi rr1g. Yxo0tar.e6ir or nilcrx€E A, li, C tcrwilovrcu. .1116 q bslrsol 'fadppri ro1JE xcrr [erd, s!d)11 rptirtrl lpappr] ro! ,4 Eival ollvbuaop6E rt,lv nQ61o! ypcppcbv ttirv B ror, C.T6€ o ,ov6vc{E u?r(,'/oper.rer,: 1 det ,4 eivcL o i,brog o1JvbuaoF6E i(l)v det B xdi det

foopll!,{oi ou\r6ldouoi nQoru?rrow Fs 6rjo nQdE€r<: riQ6oeeorl 6lovuoFdlorvxcl 7ro].lffrLoorcrop6 pe cQrepouE. AQcr o ,iov6vog our6E Ffiop€i v(r ava],!6€i oE6uo lton:

d", [. ,.1 =t c a l

'.ll=,t:

" t ' .1=oa a..c a l

i li,l.': xld

!1pr:ubmr 6rr ro rpoxo p6pog 6evtcurilercrr. pt q tro!0ao[61q !6!6rrFd det(r+C) = detl +detd. Asv [rioQ€ir€ v(r nQooodo€r€ 6],es IIE yQoFtl€E: tlovo fi,ayQ(frfrli ErirrQ6xrexft \,' d.).d(Er. Kar or 6!C' rrl,€uQ6E 8fvoxv ro i6ro dnordl.€opaatl + a'd tu b't:.

IldQ6pol(r, io 6€1lr€Qo p€Qog 6€\r rorjrie€ro! pe rn L('1]0aop6vn !6!61niade(lA) = rder,{. O 'ril,a1{ag ,,4 6I€L rov rdedyovra t os ,rioE tadtrtr{ (,.(or 1.id6€6r'aE rouq rolllanLaordler re).rxi rqv opilouoc F€ ,). Auro FoLdE€| |-l3 rov 6yloev6s rourLo6, nou 6leq tou or" a4r6g 6nor€),l,ovrft xard 4. :rq n 6Lcmiroerso 6yzog rcL 11 opi(ouoo 0a 8raotdD'ovrcv x.Lla 4n. Eal 6tnmcll.6tcv q picr[6vov (r,{pri. o 6yxoE ,{ar I oQiEouo(r 0o 6!oord},}.owo! xffid 4, 6noE L€e! ,{o[ o

O €7r6p€vo! ,{avovoE 6€iI1& 6r! q 7rQ6q yQotlt( 6€\r ei,vc! iiclr ro !6roileQo.

2. Edv €vd).d\9oiv 6io ypappiE, q oaiEoxaa alldEEr ra6aato.

c da b

OIIZOY:E:

Ar' orJro ifi€ror orl. rl oQilouoo €rEoQrdx(lr lQo[lLX.i an6 xdqE yQcFpit r\EEtxaaund. EiLv o ruQdl()v 4 rioLl.Gn).oor.(io€L rl 2t lpa[[t, FnoQoiF€ va rlvrval)dEouu€ [s rrv ]t tp.rtrtlrt Gt ]dEovraE ro rp6olltlo rrE oQiqorioaE). va€Edyo![€ rov ncedlol'Io 4 (zorci iov xavoaa 1) ]ror vr1 Evoudqo4r€ ncil,r rrqyp(BF€( (€nrorpiqovraq niorD oro rpoottro no! E€ !rior4r8). EroL o ravovos 1€q(la!61€rc1t oF L de€ ypuBFrt.

O }'a!o!aq 2 6z€r ano Fovoq rou €abLaq€aov xaL obly€i. orlv o0iEouoa€voE nivarc [€rco€o€t'r:. M€ $La o€Lad €val].nlt,)r trroaontr€ !a avciyou[€ €!a!nivtlt c tL€roeao€oq mo! rorrorrx6 rivozs. Kde€ €vaLl,oyri y0orrf{: a ,irqEl1o raool[o rtq opilonoc( ]'ar oro ldlo: r0€|r,!6[aoi€ ri]v oai.lonoft roulcuro-ftL ou rrivclro. Arloq €ivor o onl.orm€po( ar' 6),orE roug lrcv6vsq.

O! )acl!6!e( I zar 2 oqri1,or! (mQoobr6Qrorl llm fioD,anl(o!(rorlxll oroeeQd.nor xaoooile! o xa!6va; 3. YnriOZr d\]aE "polo6r(ior(rrog I(i)OoE 6wor(i!oQ!!o!o(o! n(1! o r4c,'o!ag (11J16g xofloQi.!€r lic ofi (u16E.0€o]rtio,'rcq 61r ! dcl,a oi.v.rr l.

H opilowa tytr .r'tt, xct)o12ut9ti ,i1peq,.rLLl( roL\o 6e! ci\raL x(061,o!nuuq , rvic. Q: f x rouro]J ud xQlor u onor YloulrJ r rorc x(vuvtsc urro r'c ormYlu afl xLr.Trrr vr 6r,oupl r tv o 'rr louo,r lv j : or"."u6{r"rr :r ivrz, , .

3. H oai.tol\'t tur rurrortzoti :ti1\!t tnat

4. Ear bio yoayplq tor A Ei1'ar iot<, r.jr. de(,4 = 0.

a b, = a b b a = 0 .

Auro 6r8rar.!ro ro\, ".1\,6vc

2.61611 6r.r! evdD.dooo\rr(r 61j() lQq+€s u'iorisrrar orL r oaiqolloa a),l.dqEL naoolll'o. Arr4 o[rD: rapctr6v€r a$€rd6hrt. s|orrztrL o:riv(zcrq rcpo[6v€' a[€ri16).ToE. O [6vog cpr0[6q p'aur]v r1v L6r6.qrn€i\,(lL to [t6dv r'r(r fioL detA =0. (To r:rrz€i.alfa ?{uluaQinr€rcr 6ruv 1= l.riQ.i./Fo nou roXlltr orqv tiLlt6pc Boole. T6r€ o t{ov6!q 2, oc\, pprrrl-qprmraf[6L6rlro. 0(l ifiQ€xr€ vo c!i(,4(r(rmco€i o7i6 xo! xo\ro\,o 4.)

5, H moLxErLl'tt< nptiEn rI< aEaia€a4< n la ,aaiox \La< ypappij< a1nv A14 a(rtji,er t41, oaiEona (41Era61rrr.

l 0 t )010001

l 00 l

, lr 0 ,, !r

,i,

42 TA|O tHTE THt OPIZOIT :

o ra\'6\'aq 1 Ocr 6l.ep 6tr undpqa ivnq cll6pl 6poq /c t I. d i o oTrolos opoE,

6rlparo cnttrLorrptlg 6evxcrir rov xcv6va 4, eivol pq66v. Eto! rrr olq0lo$€vosrqa€dqouv rlv oailotloo.

6. Edv o A iT1 p46wrxi yqaypi, rdrs detA = 0.

L::

a0ub0d

"=l.,'.,",,1

Mr& crrr66slEn €Ioupe irpoo0arolras xdnor.cr d),I,q lpnpprl orl pl6evr)ii. Kqtdlov zd\r6!(t 5, q oQiEorJoa 6€v qlldler xol €n6l&i o nivotio( 0(1 aX€! xdaq 6uoio€g yQalt|l€(, )io,ro rov xqv6vo 4, ocr loll]e! del,4 = 0.

7. Erbo A eh'cu tptlalrutti€, rdr€ 4 det A €hn*ro yn6tteto a ...a,,ntavmorxeiav t4€ xuQiae 6@yaf io . I'rcfteq 6iav o A ixE! 1 xaxd pfrog14E btctftt)riov, xdx€ del A = L

Ard6€r€4 Ynoo€oie 6r! xc 6Lqyo)arcr orolleio Eiva! brseoQq ioo [q6€v68.T6i€ ra 6rl[qr(r Tl]6 crncri.orvig |r1roooiv vs €Eal€ivolv ol(r xq [I 6ro"f(i)vtc,ororx€iq. XoJQiE !' dl.LdEo!1, q! oQi,Eouoa (rsrd rot l{dv6vo 5). Edv o A €i,va[rilro rQ!yov!r6E, ra 6ri[qrcl yivovror" tr€ I o!1q0roF\al o€!Qd, E€rlvd]vrqs str6tov nprlto obq16. Edv o ,4 €i!&r dvo) rQr1,orvr,z6q, r6re zooagilo!$€ Trpd)lo tnvrel,eorci.c mr1Lr1 lprlo4roriod)lrq Trol,lattLcroLa ro! a',r rq5 xcirto /owlog. Kc!orLg 6io negLrrrboeL< xorol.rl"youp€ oro [email protected] nivqlio

lLa vc dpolpe tv oQilouod -ror erpnqp6lor4re llnopoverr,rci iov xcv6la 1EEdyolp€ ro\, fiaQdyo\,rd ar1. xar6nrv rov o22 xor dr'og 1ov li",', otr6ts dfioFs\'€r"o rourorlx6E nivqxoE. :ro rdl.o6 eqcpp6loupe tov tiav6vq 3!

d e r D = a n a i 2 . . . a ^ n d e t I = a 1 1 a 2 2 . . . a n n .

Avriocra, al iva \taytbuo otorxeio eivar p46lv r6re a am)'orEt:j ga tapdytLpia pabevui yqappit. Ka1.61o!,iovold 5 cud rc 6jpcrc crlrcLorgtig 6evcrllcilow qv oQiEouoo ro! xcrx{i rov }iav6va 6, pq6evrxtl yaoFp{ onFai.v€L

lrrl6€r,r,a{ oQiEollo(. 'Exo! arobsi€aF€ T ov xav6vc 7.

OPIZOITE>

'orcv €\rog rQry(L,,,'lxoE ni.vcxo! efvor 6r6FoQqoE (l.oy(D FYl6€M"rou oilv xuar(t6!ayd),vrc) I oQiqouod ro! €ivar Fyl36v. O €r6$svoE xav6vaE 6€i1v€r 6rL ro i6roo1j[6cir€r p€ o],ouq rorE r"SmFoaqouE ri.a(lx€E.

8, EdvoAdvarbujyopEoq, rdzedet,4 =0. Eavo A etvar avrncpit l tyoq,rc det A 10.

. l ' 1 l , ,va, ro,opopvoc. ov x. , t lovov av ad b, 0.t ( t r )

Etiv o ,4 eivcrL rbL6poqgog, t6te 1 arcloLqtl o61y€i oE 6!tt! flL\l'a.t U IElritb€vrx{ yaq+rl. Kcrrir torg xov6veg 5 xdr 6, det,4 = det ar = 0. Edv o ,4 eiv(rrtll rblo[oQqoq, n ono].orqn obylysi oc 6vuv rivco rprycovr,n6 Lr pe pq plbevrx<ioro(Xsi( rlcrd tolzo( rn( 6!(ryo!io1J. A!'((i r:iv(rr or ooryoi /r, . . . , r/,, x<rr xrtdrov xnr,6vcr 7, det,4 = det U = +d.i.1.. . d.,.. E6tit nvartdpx rov xp6to tlnoyrt t4t opilouoa. To o]j\' 1i rrl.Iv np6o4ro €EoQrdrdr on6 ro cw o! Eva),Iol,aEyQapucbv €i\rdr d$[sE rl fi€Qtrrtg 10 n],{eo6.

H rflourvrl rdrorrlro rivur yl nro qnQoodorrrrl.

9, f@ 61jo onorctlobitore th,axtE n. uri n., q opi(ouou rot yu'opit'ou AB€ivcu ro y0,611E\,o ret' o?t€oloti)l,: detAB = (der A\(det B)

a e + b o a f + b h- ce+t lg cf +dlr

ML(r er6lxll fieQinr(,)ol ouror] rou x(r!6vo, biver, qv opi(ouo( rou ,4 I norsi.v(l! l/ det?1:

i i'l

(de r , {Xde l A r ) = de r AA r=de r1=1 . .Qo der,4 I

,tl

:ril! n€airrcoot 2 €ni 2 o xar'6vaq rou yfvo$Evou €ivaL o ibroq ll€ qa

(od - bc)(ch. Ja) =(oc +bs)(c. f +dh)-(nf + bh)(ce + dq).

Xtqr nrpinttr;o1 rr rni r €n€r6li a1J16g €i\roL o riro cTrQootoxrFoE ri(l!6!cq-nQol€ivorp€ 6uo oro6€i,Es[E, Kor or 6uo ufioodToll\,6T! or A xo! , Ei,v.rlFn riL6$oQqror. blo.loQErlxd x{1! o r{, 0o €f\ror r5rcFoQqo6 }{ar I EEfo|l]otdet /B = (det.l)(det B) €na),q0E{Eror €llro),c. oqorj, kcxd ro! xov6vo 8, ocr

(i) o€rDaoulls ro 16"1o d(,4) = det(AB)ldetB rcr 6eipoqre 6.r €1€r. tqr6Lciqres 1 3. 'E.roq ener6rl ctrrrg or !6r6rlr€q opilorv trv oai.Eouocr, o .J(r{)

t t IA IOI H IE : f tE OPIZOI>AI

nQ6rfl vc oulurinrfl FE rlv detl. flq ncrQdd€ryFa, €frv o,4 €ivq! o r(urorrx6g.x6rE oi'youQ(r d(1, =derBlderB = 1, dQcr o'rav6vq 3 txavono!€irar att6 rovd(,4). Edv €vu),Iqouv 66o ypq4r6E rou ,4. ro i6ro oug6cfveL )tft p€ tq i6rcE6io 1pc44r6E \ou AB lraL eaogdvroE nL)'tiler ro ;rp6o4po rorl d. 6n(DE cricrli€io x.d,t6tug 2. T*l.oE, tvcrg ypcFprr6g ouvauootlos 7ro! €pq(rvi,l€rc{! ornv rrQcbxrlyact4ri ro! / 6i!€! ro! i6ro yedF[Di6 olv6uaop6 olr]v TrQdq yA(rFpi rou ,48.'E-ro! o xc(1,6vcrE I ^/!( inv oAi,lo]joa ro! ,4r. 6rcrqepdv4 pe rq orc€eqrj noo6rllTcd€t B. obly€i orol' xcv6lc 1 yr(r iov ),6yo d(A). O d(A) Lotn6v ouynlnreL per4t' opilouoa. Aurd o+rci!€r 6rr det AB I der B = dct A, nou rivcr xcL o 16:rogtlc.( Yra ro 1v6F€vo.

(ii) H 6€lJcal ffi66€lEl strar l'ry6r€Qo r(olrlrl. E€r4!v(i uro86row(< 6r! o,rl rtvcrt 6rclrirvr.oq nirurK D. Tor€ o rtnoE det DB = (det DXd€l B) rao,irinr€run6 rov x(!61( l, .Edyovr{:l( sro xd0E yQaFFrl rov (rwimoLlo fiop{ilovra d.fltr 6r,a yevrro ri.voic '4, rdvoqr€ crvqltl;yrl orol D pdoo nrrv 6qpcirov 16crrraloL{tiris "Cruss - Jordm" -Fs rc owylolot1ilo 6riBc(r(l, naq€ cnd tov A orovU kcrL lorloqronoldlrdE ,l(g€ 06l/6lL(r vc qrlaEol]$€ Blb€vrxa afl6 n.rvo rou.ni(ll€ crfio ro! I/ orov ,. H oQiqooo( 6€v ().ldq€I. rrsQci q nqog ro nq6orlFo.ydl0€ qoQd nou rl(1l,i'irooovror bi,o 1Qq4r€(. Tu i6ra 6iForo 6N)( cviryow rovAB arcv DB xft 6xouv rrE i6re< crpr6ti4 Enurrdo€rg oiqv oQilorxr(r n(l roaDB opra yvopiloupe orr o xovdvc( q tivul oormo(.

lO. O ut(ioeoq o< rot' A i7t r4r ihn opiloLtoct tit rov A . det Ar = det A.

2"r=bd

Ksr rdh I rr€Qfnrruol lbloldQqou ̂ i!o1iq eivnr rn].{. O,4 rivaL Lbr6pogpog.rorc xaL p6vor drsv o ,{a eivtrt rbL6poprpog zcr dxoqr€ 0 = 0. Edrv o ,4 6€veivnL Lbr6poprpoE. rdr€ eivft 6uvari FLo ncrQ(ryovronoitot PA = LDU xelc{crpF6(owa( tov rpolyoqrrvo xnv6vn 9 1La ol1, oQi-qouoc lno[€\,ou. 61oqte:

detPdet,4 = det rder Dder U. 0)

A'vctolQ€.eovttlE qv P/ = LDLI txolJw Ar P't = fl t D1- LI Kl eqxrgpolovtcgfid}.r, rol xc!6!a I,

der,4rderPr = der U/ r lcrDI der LT

A\rri eivar rrnloumeQl cn' 6,lL qcriversr, 6L6ll. ot L,U.LI xol {Jr tL\aLraryr,rvri.oi Fe trovdb€S orlt 6rcydrvr"o. Karir rov xtrv6vc 7, 6kg ot oqilouo6E rougriv(L €va. E:rto1E xci0e 6rqrirvrog nfvd14c1g r(rurilsro! F€ iov (r1,dorQog6 rou:-D = ,' . Ano[6!ou! Lorn6v or nivar€E [er(e€o8og.

(2t

OPIZOY:E:

H oQi.Eouoa rou P €ivar oiloueq I yi -1, aqon reoiQlet(rl on6 rov icutorl'|6iriv(rra. iorEQa ur6 €vo),latdE lecB[(;v. naaaqarlois enioff 6rr PPI = j.(no),1.r!rd,oor"d!ovrtr5 rov P [€ rov P1 . io 1 qE nQd)t{E lOoBpfE rou P x(tQ!d!€rpe rr1 povribcr 11< nprirt-c orril.Ig rou P/ ear ei,\,(! oe 6tcrqoQe-rttti 06.n1 an6 rtrI t(,v driJ.(|)! o1ll.d)v.) 'Ero! det P d€r P / =det1 = 1 :rc(! or P zut PT ngdret vo6tol'v rrg i6L€S oQiloljor!. or onoi€,< fJo €i\,( I { -1.

:ir[ri€Q(lvoulr€ or| ru }vopevc orL( (l) z(tr (2) €ivqr ra i6rq iraa detA =detrlT. To lrlovog ovro 6url(ord!€L rov x(rldo16 Fcr< Fe tL< rfu6qrrg,616rrxoo€ r(lv6v(rE rou ioxot: yLa ypc4ruiE rri)Q(. 0a Lolirr" xnt yLcL arh\E<t On1'F\ AAaoortet hto otil),tq 4 oailoLtou u r'tEEL 46o4p, 6io iot< orfikE ft$!( ornl,I grDF1a irv) iraqAyou, y4bevaij optEoma, 4 opilotoa tEaqr(ircLyQutltttzu Rn; x'iet oq)4. H ffiobHEn npo)iinrfr uvuorpiq owcr6 rov riv(xrrzcn 6ouLeiowag pe rrg lpcrpprg.

Mou g(iver(r 6rr Eivor o)Qo vo orqlofiloco licL vtl (rvqxqQriE(.) rov rtlldl.olonLlleil. AroF€vrL vct 6QoiBe 6v.rv owxex0L[6ao runo yLo xrlv oOitouoo r{(! 1(ltov 0iooupe or 1grlu1.

AEKH!EII

4.2.1 nd{ olvbdo\'I(l| oL de(2.'l ), dc( -.1) acLd.(nr) fF qv der-4.6ravo A €ivar rivuzus

4.2.2 Aeiftr {ftrrl(i,vxa( ).d0e drt!( oe ia( nc.qci6fl./F( 2 Etrt 2 oI' pr( €!c},I(yituv yeqqr(;rv i xaL i FroA.i t'(1 rceorori rAooodtolt(r< q 104$i i otl tqotrli.j.zaront! q,(Ladrvr(q q lao yqa[!t'i / (l'rd rI yqatLtt i, x(rdnL! nqoo0s:rovrai n] vdoyslptli i mI lacl'lui j x(rt. rd].o!. noi],on].corciiowc( rt yeqllli i f€ L norol(zovovrs 0( fnoQonoof€ vc Xqrlorlorouiooqr€ yrc v( ortrrrq.iloqrr ro! T.rvova 2:

4.2.1 Eq(qF6;oE.1i flc f,€irlas fft(lln rqaF|dv. ro! ohyon! cor ir!0r rQLy(ovL"d Lr.uro).oyr crre lq

f r 2 2 {J l l z - t 0 , r l-liiill*^lxiiilElaiLiErF rLs yaaFFis I z( .l rou 6si{s€ou triv(lic xoL Edvantroloyin€ lor5obryons xtu flt! oeiloloa.

tryi./adrl MsArrof alorrdore< rroai(o1ry rlSrt ;vsr rllno rro rrq oqilouueE 3 oni:. no! ixe! 6Er oaou. (eEioool (2) rlls.noF6\ats raq(yqdqnt, lriiqolv reFrE roxnd!. r.rpdD,ti.u nao( rltv r.upra 6rot(irvro xdL reEK d),).or no! r.iv€ avcinoba. $nq6otrro Frjio!. Eia( q,!ou6 vd lLntloq€ o ivav reoforo rnro yro riv(a€s,{ rni4. Yraexfl iv(rs runos. Trei/:t 4k . 11. = 24 6aovE \xu .jxt ott6 rivor. Kt i'tot.-qovrq,i'vorv ot tro|r vrc o('zlidrLc. b€v tuoorir' !d €rors otTofaor di ro ,rE oltrot€io! o!u6s6t!€t rr{ rt' diit 6rotdaro.

.r2 TAToTHTE: TH: OIIZO\2A' 251

4.2.4 EEI !,!oxe 'yrori

[6o€s €vo].],aydE raerdqovrar yro ra ftdFE qr (n-yqdpp1i, (n - l)-yqopp1i, ..., 1-yaaFprt ofltv aavolrzri 6rdlaEl (1-TaaFFrt, ... , (n- l)-yaaFpi. n-TeaFpd; noft€i!o! derP = 1 aar noft detP = -1, 1ld ro! n snt n nil{rrd l€ft0ioetos P, pe 1otry d?,).I 5lcyd!rci :q! npolyoulsMt doqo! €ixo$ n = 4.

Bprii€ iK oaf(o1jo€g

(a) e!6c Eivara rdEe0E dlo

(6) ro! d\'(|) re'y(Dlr,tou r luxo

= 1 .

121

[0 I o o lt 0 0 I 0 lo" ' lo o o r lL1 0 0 0 l

l 0 0 0 r ' l'"'13 ? ;31

L1 0 0 0 l

l r lA=l4 l t2L2 )

14 4 8 8 ll [ | 2 2 llo 0 2 6 l 'L0 0 0 2.1

[ 0 0 0 2o D 2 60 t 2 2

1 4 4 8 8

(y) 1()! rd1(r ratr,'!|roi nilal|.l Ut.

(6) ro! dlriorAoqo! ri'vatcl U-r,

(r) rou "(!ctno6a reryorlrxo! rivara, no! reo,{nnrsr p€ E atr},ayds ypsfrrd'r

,11 =

{.2.7 Be€ir€ ft6< rao,tlrre! o ftold'c< 6 (der = 0 6rq' pld yAapIi ei!a! }rl6d') on' €lre€ioqaro rous aa!6!€S 1 roL 2.

1.2.8 Yro€dm€ 6rr xiil|J€ r(tf iolQor(l 6!() nadEeE mlE 104{Ld< F€ro6dD.olra< ro!

t o b o a . b n dc d c l a d l b

Ba€ift. 11€ rov xa!{tva 1 i aar tl8 cfl' rl|0rics lrio}.o?roFo (aor arl,ofloitor), qvoQilolda ro! v6o! rivoaa.

4.2.9 Edv Q €ivdr oe0oy6uo5 nivcras. or6re Q"Q = 7, 6€iEt€ 6fl derQ LoouaL p€ +li 1. Iloro ri6oE raqol].n].€mr65o! oflFatiLnar aro rLE yeduFds (li oIt].€s) €voqoaooy(iruo! nivdro Q:

oPtzoYrE>

4.2.10 Xsp4ro,roLric€ nadEsq F€ YaaHrtE Yta 1(r €noM0€lo€r€ 6ir n 3 €ni 3 'oailol,os

vmdermonde toofiot I€

f l o o ' : ld e t l l t , , ' ? 1 = ( 6 a ) ( c - a x c - b )

L l c d l

H ftaiJrruot 4 sni 4 n.qLixEcr ofls ao io€tg rts €tra!d)'rri'al'

4,2.11 o) E (lE uvrtouppsrou{dE 'Ii!(lacts rxalorol€{ qv KT = -1{, 6nos o

. . I o i , lIL= l I 0 .1 .

L - b - c 0 l

rrati oritv rtgtnlorol 3 ctn J roxnrt dc( l( ' = ( |)'dcr Kr EE aIIo! dct '\ r =

der./i (r(\.rolr). :qrrl('avcrf 6rL dct ( = dcl /f x(r 6tt I oALtouoQ flA'r{ vo

6) rqd{E€ rvov avlrqq{rerat.,{o 4 sni 4 nivox( loU oroiotr { dct /r d€v plnFvil'tat'

4.2.12 Abeds rt !,srbds, FF brxa|ol6flol rlc . ii0Fra x( al'rltlaa{iberYla Ytd {s'rdoql

a) Edv or A rot , t(rvdqow(ri naworJ. €ttt6g q< fldv0) caLorEQliE Y )via<. ono!

, l | = 20rr . r r i r€ dct1] = 2dctA'

r H o0i!ota'. rivnt ro Yrlorevo rnrv o6ryd,v

y)Edvo,'l stvat ovt'o1au!,tpos acr o /l tbdpoAqJo<, rdreo,4+D 'i\'ft al'rloratvLlos'

6) Eda o A ritar owrorAill,rpos xd| o B rbtdroegod, lot€ o ,4.8 €ilot t6Loloegos

42.13 Edv xd& yqauri 1o! A !X$ d0qotol( moLXFio! ttlbd! beift' 6tt dctA = 0' EdN

ioE laupprl 6rtL doao|gpc ororlei('v 1. 6€iEtt drL dc(A - l) = 0 a€iF€ F€ rdtroto

raadbFrYpa dfl to 1.l'Plraio 60v ollaLv€! det.4 = I

a.2.la Be€ir rLs oetLoM€E 4 .nt.l I€ ardlotgri Gauss:

l 1 2 1 3 1 4 1 | I t t r t 3 l\ t z z l l z t l l i I i / : l

de , , ; ; ; r r r a l l t d ' r . r l r : r r i lL;i ; ; ; ; ; l f r ' i r r l

4.2.15 BqEirt tts oeilouoo! r()v

. l 4 2 1 . r f 3 2 l i \ , _ 1 4 - , \ 2 Ia=Lr : l . r ,_ro l_r rJ . "_" ,_, , 1^1.

fro ftor€q t4l6E roD ) o rl )1€ivoL r6dFoaqo$

4.2.16 Yio).oliole fltv der:1 avrilomos ror / oe retYorvtxrl poeqn (t{ov6vsE 5 xor 7)

f r r 3 l [ r | ] l l r I 1 lA-10 4 61. B= lo 4 61. ( ' -10 4 6 l

L l s 8 l l r r o I l L l 5 e l

4.2 IAIOT}IIE> TID OPIZOY:A>

nor.es eilcr ot oql.louoss t oa B , C, AB , Ar A ^,tu" dr i

4.2.17 Y^os..ole 6 CD = ,C rcr 6eetre ro ).d0os orov &rdtrevo o!L1oyqr6: noQvo-vras q oeilolos5, €xoq€ (detC)(detr) = -(detrxdetd), dqa rvcE ar6 rors Cl{dr , 4€r prdevuol oallollod. >lll,s,iriE, I c, = -rd €tvdr 61]vdT'i p6!ov dro! oCioD€lvar|6dpoeqor.

otIzoY:E

4.3 I TYNOI TIA THN OPIZOITA

O rQdro< rtrog 6XFL {tu] a[qa'vroo€i:

4A Eirv o,4 eivor t6r6popqog.'r,6t A = P |LDU w.u,

det ,4 = det P-r det t det, det U= + (y!v6pe!o ior! o6ryy6!). (1)

To na6o4ro 11 €Lvol n opiEoljoo iou P-r (f rou P) xul elaprrircl! ono roav io riLleog rov eloLlay6v yQc4416v eiv(1! dqrlo i n€Qrrrd. Ol rQLl(l)llxoir{(rQdyolrsE lxow det, deru =l xar det D =dt . . .d, .

:rqr, ,isQijrrcnol 2 €rri 2 I ouvtoroBdvrt rcpoyovro:roLrpl IDU eivcl

f , i l _fr u l f , , , ) l f r / , l? lL. , / l

- L. I l lo { , , , / - r , L, l Lu I . l

To lLv6pevo rt,rv o611riw civcr ari bc. E(i! ro nod)ro 6t[(r ci\,cr pro E!G),].(ryll

", l " dl l r ul l , u Jfr ,r ,J' " -L , l l -L , , , r l l0 { . , , / , , ) ' I l r ) I l

To 1,r.votr€vo r(Dv obnyd! €i.var rdQa - de(,4.

IAPAAEIIMA O ni\rnzag :renepcroirdvov bLcrpoqdiv qg ncprrypiqou 1.7, ei10rrlv :rcrpnlomo:roir1o1 ,4 = -LDIl

l? I ) t2.,^ l112 7 I | 3 /2 l| 1 2 l= .1 4 /3 1L lt l t lL - r 2 ) | O+1) /n ]

H .0r i^uo.r rox f rvr i r ro Trvoprvo rov ool , r i rv :

,,",, ,(j)(1) ("" ) ,, 'Ezr6g on6 noL{ rr6rxoq nivcttg. crtt6g tir<tL o tQ6troE B€ to! onoio uno}.oyi-lo1,ror oL oQiEolo€<. O xtl6rx(E oro r(redaquo f GpioxEr rI! der,,{ on6 rorq

4.3 TYnOI rlA t-HN OPTZOY:A 261

o5ryoL!. :rlv rQ(ryl(rr[1{6trFo, o! oblyoi €ivcr ro aror€).€ou(t rIE ou[rninv(l)orlErlE n;.lQoqoeiqE nou n€p€X6r(.v aQr(r.rd. ora n 0rorx€ict rou rivcrxs. An6 qvo€oelrrxt oxonrd. €v rourols. rl owri'1{v(r,ol 6}.4E qE n},ruopoQisE orol,f 06I-1orig, 6let €vc perovizr4Fn: €ivoL rr),€ov cr66vurov va 6ouF€, F€ rlorov rQ6no rlcllL(l}ri €\,oq olorx€i.ou €nru€dg€l rylv oQilooo(r. Iaor€ivouF€ Lom6v v(l 6poiu€6v(! cval,ulrxo lurio /rc tqv opilouocr, ouvcpnrioer, 1ov rrr orolleicov t1g,

I-la n = 2, 0a 6sLEolJU€ 6rr. to ad - 6c ei,vc{ scDor6. 1-!o rr = 3, o o\rri,ororxoqrtroE sivaL r.il,t oQxErd ywror6E:

_ (r dlt{l]] + .tll(lr.ll + al3arlnl: e)-c l td2r( | t l - a l l3ar l la l33 - a l to l t . t l l .

-l: 3-13

: 3l='

:t6IoE FaS €ivr( va rlqQdlor4r€ flinow, :ro! fiOoxtrwolJv xqr' €ue€iov ofi6tlE !bt6t$€E 1-3 rns rlQoqyo]Jp€qE ncQoyQdqo!. Eriv enrfegyao€o{re 1tEr€Qltrrdo€lq , = 2 xur rr = 3 Fe ouor]FarLx6 rQ6fio, 0o 6orip€ q 6[s6!xqoio ro!yrct 6),oug rol)( rlvoli€q.

3€xr!d!€ braordw(l\- rdo€ lpolrUri or 6ttrvuoldrd, r.idrh rl bLslfJwol rov0Eov()1, ouvrEr.ryF6vov:

I a 6 l = l d 0 l + [ 0 l ] x o t [ r : / l = [ c 0 l + [ 0 d l .

EgoQudqowe acronLr r1v r6L6qtn'nLeL6t 161Qa[Ftx6fl]roq, q€X(oe!fiir o€ rd.O€lpappri -:rprirro orqv ln .fQat4.ni xcr! leldr or1 2q.

4 0 l 0 ,c t i - t : t la 0 i l ( l 0c o l -10 d

,) (3)

Kir0r: ypqprl ev6q rivo.o n €fii n Otr crvcrtlue€i o€ n aEov€S ourrrrqr€w)v xcrlI trva},loq crunl bi.vsL n" 6porE: :qv r€Qinrorori p<rg ritnv 2: = 4. Ernuld4 6pa4,oL ;repLoo6rrpoL tu' ouroug (6:rog o :tprirro,< xcr r€l€lnqio€ oeoE Trnaonavo)pq6€\riEo[(! (luroFhr(lrg. 'Ornv 6rio ypc44rlE *1ollv qv i6r"u r<rrtr.r0woq. t6te 11[io €ilcu rol,],anl.dro|o rIE d].].tc xs!

060d

YnAaX€L FL.r oflll,l ono pl6cvrxri xcL pLo plbevrxri opilouocr. EtoL elerciloqrep6vo nq ypapptg irou beiyvouv ttpos 'rcgogtrtxie zanxqiyott< n p4 p46cwzrimoqda rpirct ra ryqsvieot"t. oe 'rcEoeE-.rxts oril€$ Ynoe€or€ 6T| rlnadq yAaUFrl 61er 6vc pq trr46evrr6 orolxsio orlv a oTlill, q 6ewep1 lpqprj6I€! pq pl6€vrk6 olor"X€io orlv I orllq rcu,-relog I fl-oorli yeq.q.ri €Xe! prl

OPIzOY:E>

FI6€![x6 oiolx€io

:qv reQi,TrT()on 3

orl! y orrtl,I. Or a0!0$oi,onor€),ouv avaarciraEr 4€n i3u rdAxow6opor :

I(t)r ont),6v o.3... . I/ €ivaL 6),oL

1t.'tti9Eoq 1.1ut]r' aare$6v 1 2.... . n.

. r l , a : t

f l t :

I

11

(4)

Enava),a$6(ivo!$r: I d\rid,ro{ 0o 6rQ€n€ vc r.€Q!i.X€r, 3r = 27 6porE, cn' tou-<o;roioug 69a5, 6l,or rzt6! arnclv rtl)v 3! = 6 €ivor" pl6iv 6Lorr, lXou nol.Lonl.doreEottiLeE. Ev fver" c:ropivow ll 6por. (YrdQIow n 6lvcrt6rryeq cnd,olrlq yLc fl1vfiQ6q orlLl o. n I 6uvor6r4req efir2lo./llE yrc( q m11"1 /J rcL ril,o: pia xa!frovo bwcr61tlra./td r1v teLeurci.tr orlLl /, oqorl 6Ev ea 61eL crnoFeiver :rupciFta Fovo ofllLrl. xcocbE €L[oo6[€eo cvilBsocr orL< m{},eE rou n|voxcr). M€ itl.LcLoltcL, ufltig'|ouv nl Wifiot pErdqeo4€ tnn dpfiFioy 7,2,. . . ,Jt. fLc !c 6oouBsrr: oariorollEq [fia06o€[8. €E€rdloljle rrlv cllio],ouoicr rco! aOL0Ud)\, fiD\r oql,6v.Or iEL 6Qo! orov (4) nQodQlorrcr ajr6 rq €r6tuve( orlLEEl

( o , r J . / ) = ( 1 , 2 . 3 ) . ( 2 . 3 , I ) . ( 3 , 1 , 2 ) . ( 1 . 3 , 2 ) , ( 2 . 1 , 3 ) . ( 3 , 2 , 1 ) .

no! €i .vor or 3| = 6 Fercr0ioe6Tol ( I ,2.3), I nQcirr ' l roJ! onotov €tvc4 r l rqlJror lxn.H oQi.Bouo( ror A cvtrlcrar rrl6ov oe dEL aol.ri Gnl.orlorogeE oqilouoeg.

EEd./ovraq xorJq iroQclo\,r€( oij rrQoliunr€r ivcrq 6pog 1r,n ziroe picr cn'rr.g elrIerc06oer.q:

. r,rr-a.l ' o

(s)

Kdo€ 6Qo< afvor 1N6p€vo n. = 3 oror.1ei.t']v a,;. 6noua(i0€ yaag$rt xcr zdee oniLrt€[qali.Lovrar Fic riar l6vo qoeir. M€ ril]..r ],6y!c. undpxfl 6vaE 6pog y.(r xdor6rcbpoprj :rou 6rrtp6t€r" rov rivczc nqoq ra xdro rcu" 1p1or"po:ror.ei dr6e orrll,qFio xaL F6vo qoQd. Edv oL m{I€q 1aloLtronoLq0o{.,' [€ rr 6rdx&Er @. !],. . . .u)ror€ ro yLv6$€r,o €i\,ar (r.,, o2r, . . . j d,r/ €ni rlv oQilorlocr rou ,livara F€rdosort<P". H oAiloroo ol6x),IQou xo1-r ,-rivoLa €ivar ro ci0poroptr ormiv rcov 6pov xnr.to dgQoropa at'r6 Eirar o arar\utmdE ttfioE nou yupxlape

4,3 TYNOI fIA THN OPIZOY:A 263

aet,4 = I(a.r., az,.r, . . ,a"")detP".

*=1"dX€r (troLou0ic orll.d)! d = (3, 1,2),

.", =., "" ", ll ?] r l . .o ]

= " , ' , . , " ' . " , '

\ 1 ) Q Q I ' ( )

(6)

l.o €vav aivazc 'r r7(i '. ro doaoroFa srrsiv€rar o' 6lEE irE frl ptxc06oelEo =@,.. . ,u) r0)! cQlopd)v (1.. . . ,n). H p€rde€ol 6iv€r 1o!s cQ!0$o!6 r()vorl),(i)r, zc(ki; xnre6crivor4re r"q lpctr[68 1or nivoxc xo! xo0oQiE€L enioylE 'rov

ni,l,oxc p€ro0doeoq P": Or povi(6€E ror P" €[gcvilovror. mlc 0do€6 rou,iciellcrv Id (],iJ OlO! A.

Anoplver vo 6QouF€ xrtv oqiloron tou P". EroLldycE yQoppd)a rov ovdyouvolov rorlxorlx6 xor 'rdOe €va).),ayi dr.rlotpte€! to xp6onpo rqE oQiqolocE:

der& = +1 ri -1.trtihoyd ItE ro ctr to uliqos trr Ev.tl aytb\, sivar dpno I xqtn6,

t1 lP" = 1 6xer cxol,ol)eic( ornLclv (o, p, v) = (1,3,2),

L1l' ]

O rQdrog XQeLtiEeto ptq sval.).cryrl (6€iisQ,rl B€ iQi,rrl tQdtrplt). deq det P" = -1.

O 6eurepog lpetd(erru 6r.ro 6\roLl.ay6E y(s vq ader srov tcrurorrz6 (nptlrtq peNurrpyl . uzoLou0orrtrr ln ( f io bcurrQrl Ut rpir4 ygauprj t . dOo delP- = { l r r=1.Atnd eiv(r 6uo cro tn 6EL nQ6onFq nox €trqqliloYrru mry (2).

H €Eioo)on (6) 6i.veL pr,cr ctrcT'.urxrj :rr,oryadg{ tqq oQilouocE rsr. €ivdLstjxoLo .''c e.fi(lqo€uooups rrl! n€Qtrrrr')ol 2 EnL 2, ot 2! = 2 pexcrodoEq €ilaLo = (r ,2) 1lctL o = (2,1) xcr Erop6v0q

A€.,'trnopei. zaveiq !o roxUprooei 6T[ o ovoLutLr6g runog (6) eivcr Lbttti.tepc(7il'.6g. Mropo]j[E oFtDq vcr 6oipe }cri ucvoro!€i xorE xo\,6v€E 1 - 3. H lbroqr&3, ro y€yov6! 6rr detl = 1, eivar gruornrl q cnl,ouor€Q4- Ta yLvop€va r(l)v aij €ivaLnrilxore FrD6v, exr6q qg:repinrooqg tqg atolouoiaE oql6v r. = (1,2.. .., n), [€lll.l,a l6y'a qq raurorrlig lerd0rotg. H !6!6qxa 2 ea €E€rao0€i o tv €n6$€\'rlncrphlpnqo, 6L6rL ebd lroq €v6LaqdQfl neQrooox€Qo I L6r6T1trt1 1: H opilouoo0c:rqdner vc r lcprdrf t ypalprxci ar6 r4v:rqthrn 1adFt t ar,ar: , . . . ,or ' , . f rctvc brcnrmci-rooqre 0rt6 ro €i6oE eEdQ,rnoqE, €E€r.iqortr€ ro1,5 oporg rqs (6) Tiounapr"clouv ro all. Atlloi oxqp(rriEolror 6rcv 1 enr).oy{ pcg lttr tn\r nQd)xrl oni}.rt

oPlzoYtE:

€ivor o = 1 .,io! 16r€ aroFEv€r trid perd0€oq o' =(0,...,v\ rov uroloifi,)vorlLdv (2, . . . , n). :uyzcwQdvoup€ 6l,ouE aurous -rol'5 6QorE ool [oQrpri arr-4rr,6fio! o ouwel€o iE iou arr efvcl

IIaQ6Folo, 1().ftolx€lo dr] no].tlatrtrqodE€rqr F€ trLo roQdorqon Atr. Mot€uo-l'ruE 6IouE ror8 oQo1E noD EExrvouv F€ ro i6Lo ar;, o nJjro5 (6) {vercL

d e t A = a , 4 l | + a r r . 4 l r + . . . + t t t , , A t , , ( 8 )

:wsfi(i)E I det/ €Ecpl&c|! IQoFU[rid cn6 ra motleicr d|l.... .ot,, rr]S nedqs'reaFFiE.

IIAPAAEIrMA fls dvcv fllvorc 3 €rt 3, o xQdro( s!x6q oFqdorol4olE TtDv6pov 6i,veL:

delA = a (a:ra3r - o1ro32)+(t tr(o2rar - a2ro1i)+ar3(o3rdrx - t t : ror) . (9)

Or "ouprcQdyowcS" An, Ap, Att ,.€Qttlov..r. <tttE t4gfl; n.tpfv0to'I'.

Al,dirruEl trlg def A m ooFnaQdlol,r€g

xQ€lcil6[(tors 6!.r! ax6pq flrro yr(I rnr oQigouocr. Oo riicv rollj 6uoqQ€oioo! 6ne€rir vo E€ro.rioootl€ ori' ro fq66v, tiltt\ tot' ttltro ror ixovle i64c1'u,r'.u1i1lt!: Eircu o (8\, xar ro p(ivo ttou neilt€t vd rEQl.yQtill)o l1p €ivat oratgttoprityovtg AIi.

fv(l)QigouF€ 6'l o oQrqr6E ourdE,4rj eEaardrqr n:r6 rLg ypdutriE 2,...,'l,ev6 q 1q ypaprpil dler l1q0ei un' 61rrv pe tov:rc.E.towo .irj x! €tr€!6il to arj€i!Cl! I oul€rorpoQdr q5 j'oofilE olrll,qE, o olJluraQdyovrG rou ,4rj 0a rQ6n€rvc eEaQrdi(! croxl,€rorrrid an6 i6 d),},eE oxrlls(. Or i6re6 lpr4rFdE Ercg, rtoiriIEE, 6Ev FnoAtri vc lprlo4ronorqeow lavd ltr rov i6ro 680 1c1! oulo tro!1t ill'ouF€ orry fiQoyp(rrlx6qla €i!o! vo blcrojrdoolF€ 1rl1, oQiEouoo oro €Jropevod0Qolotlcr:

/ r = t { r r r r . . . a , " )der P" , . (7)

L o ' :

l';; ";,I|a gra opilouon rdrler4 n, I 6rdonool osr{ 6iva n Frxe6r€Q€6 oQiqouo€g (nouLt'lofi.crL ciltfr.ooovq) rdEa4 n - l. O! ufionivc,.!€E 2 €J[i 2 €Fqavigowor olrl6€Efd nlEUQd. O unonivoriog ,44j o.tll|tta1tle1o.l ittlQd.ti'tov&g t41' l\ ypappfi

].] TYNOI fIA THN OIIZOY'A

xaL r4! j oari o'ir,4. Kd03 6AoE oln 6eErd 'Il.€uQd siv.r lLvo[Evo rox arj raLrrt< oQi.loro(ls xou -ll-irj. pe ro oom6 ra6oltro olrv ri r},ry. Ttl rA60+ro auri{rval.).flonovra' xoo,;'( jTQoX"0.rtrf x,rrrr trtzoc r4c yeolrtl'tc xor or ouU:ToQalo!'fl craurilovraL r€ILXd [€ ra

'41, =( l ) r . r ' dcr I1r; '

l-ro rc0d60L.rua, o 6€ur€Qo( or+lJl(lo{ilovrag Arr eivar orrar arar, rrou ro!"ri.--Ler.rr [€ q! det,l.1r2 €rrt (-] ). H i6la r€rvrrrt eqop$6lsrar xar o€ r€raoyovlrorlqri.v(rli€q ororou64ror€ pe160o1|q. Mra rlpoo€xrrxrl raadrllalol rIE €ELorl]oIE (7)ri?rr6€6(rtrixsr 6rr o ,411 eivcr" l oqilouoc r$ xdrtD belrrig y,rvicE.441.

Mra cvdLol.rl avaTrroErl !ol!e! &ar yra rdoe riLl.q 1pc4prj, 1..x. try i-oottj,rQd./trc ro! aroa€Lxvier(lL walJ.dooovraE rlv i-oorrt lootrlrl FE rlv 1I:

48 H ooi.Eouo(r rou .4 elvdr 6!(rE olrv6uoo[r6E qE i-oortg yQotltlllE ra!rcrv oqrndQayofiro! r4g r-omri6 1qc14ulE:

der A =atA;t + aizAi2+ +ei,Ain, (10)

O atpngdyav A1i sh(,i I oQi,louoa 10! ,41i.i l.l€ ro oocrr6 t[Q6ol[o:

(11 )

O ,l4t oxnpoxiq€ro! raeal€iro!5s6 rnv iooryllqcppri xcr rrp j-ooryl owtl,rl

Or rnror arnoi rxgpdloN-r1v det,,l oq ouv6roo[6 oQlEorJo6v r(iE€oE n - 1.Orr ptropotiotryt ).otmir t,u oplnn4u tttv opilotoo. pt LtutftD./i tD< nQo< n. tLrrrolq riivafq I Eni I 0l1 06roFE det, = a rar 0a XaloLtro'roLouoaFe qv (1) yrava oAiooul€ 6La6oXLXri rL< oAfloxo€c nradxorv 2 €rri 2. 3 €ni. 3 x.o.r. €ri' irn€LQov.Iloor4lnocgE 6[(,rE !o oQi.oor+re q! oQiEoxoa F6oco rr,.v Lbrorrlrnv qq, ro]j eivcL:rr.o euzo).ov6treq r(r rc oqrrepcivorpe, cn'arrdg, rov rnro (6) rcr rov rnro pero!< sxFnap.ilovr€E (10).

Yncigler prcr cx6pr1 ow6ner"c rou orpeiLercrL oro leyovoq 6rr det,4 =det-4T.H r6r6qrc au{ pc< enrredn€r vo avarirljooorp€ o€ orl1traQoyolt€g plaE oailt?(nvri ypc14rig. aE rpog r1v o.n1L1 j €1oqe

Aij - G1)i'i det II ij.

dcrA = at jA\ . i + a2jA. t j + . +a ' jAnj . (12)

H ar66EGyl 'rpolitrr€L dvanrtooovraq rlv opiqouoa rou A' ors naoE rousoq,nopoyovlc rr1c j -orr lc yqcplr tc rou. rrou r ivor I J-ort or i lq rou A

oPIZOY>E>

IIAPAAEIIMA O€urQnorg rov riivaricr 4 €fii 4 rieneQoop6vo! 6!aeoQ(i)v

H p60o6og-rov ouprcpdy6r'ftov €ivc! rfl.o Xprlo!|ll'/!o laop[6E nou 6xou! FEldl.onl.leoE pq6€!!xdv. Ebd n 1l yadFFi 6iv€r 6uo g6!0! 6poq. o oup?r(rQdTDl10! ajl fiQoz{nrE! fidQdlsinow(rE rl! 1l yA(lFpl ,a(rr rt! 1q odLl, nQdrud no!c(qli!€r 6vdv rrilox(r 3 €fii 3 [6iou ox€6[{toForoE:

f|(r ro./tz JroQqt!€f7r€rqL I2I oT{},I x(rr crur6 roo rrQoxirlr€r efv(rl

- 1 1, l

detA" =2(derA,, r) - detA" : .

| 2 - 1 0 0 ll - 1 2 - 1 0

l 0 121L 0 0 1 2 )

l2-1 o l,41 = l ' 1 2 - r l .

L 0 -1 2 )

f - r - r o l , .( - l ) r "der l 0 2 - l l=+der l i

L o -1 2 l L - '

:rlp€lci)orE 10 or, o! o]JF rapd'/owEg Eaval$loLuonowlolxa! oro rEl,€u1oi,o 6rlpalArtrl q rpopci r1 oc'rorri enrl'oyi rirav q 1q oryltq (66rr crrnrj eile iva gdvo g1

[l6e1,rri6 oroq€io, 10 1, rio! roorlydye io 0efl,a6 ]io6olgo), ori6 q1 oroionporurii€! Lllc oQitouoo 2 €ni 2 rrc i6!ag popellE. Aufll €iv(l! o ou[naQd"/ov roucrpltxor.r morleiou o12 = -1. :woLLlict. (tfi6 qv 1I yQdt4lrl trQox{rw€t I

det/a = 2(det Al) detA,.

H i6[( !6do eq]cQFol€roi ory)v n€pi?ircDol 5 €rii 5, pe-ro filv 6 erii 6 xo! orqv n

(13 )

E lef6ri poQi.lor4l€ 6xt det A\ = 2 14(rL detAz = 3. (urG o (vay(Dylx6g xufiog6i.6€r xlv oQi.touoq r(l)v peltr),ordptov nLvdxrov cruniE rry poprpr16. :E 1,(do€ 6nFoI oAiEouod rou ,{,, €i.r'ol n + 1, 6rcx! auro r.r,|avoror€i xlv (13)l

t r + 1 = 2 ( n ) ( n 1 ) .

H rrFl n + 1 oljuninrEr. ror. Fe T o .yr16F€vo rov o5ny6v, ortv cplrl a!4q qqncpalpiqou.

4.3 TYIOI fLA THN O?IZO.?:A 267

A'K}I'trII

4.3.1 l-ra rov rriv0r0

6e€ift rov pold6|to prl pD€vt'{o 6ao nolJ €pqo Ltxa! oro! rlno (6) -ro.', f6vore{tno €r .oyrts r€oodao)v oroq€i(0v Trou nratdlovrar s€ 6!a{oa$rdg aii}.!s rrlrSloqoqnrl,iis yqoFpes. to)qiq va snrldEoup€ xdroro }1116011lt6. Yroloyiot€ rtvoAllol]od der,4, eE€rdlovros o! a1)ni rl p€rdosq €ivqr derrs t rEqrIIi.

4,3,2 AlontnEie ill] oailoljod oe oqrraAdyoli€s r$ fta(brls.yeqtrris rou reolyoip€vorrnllara A rar avayd'yei€ n]v det,4 o€ plo oAiEolJoo 3 €ra 3. Kdlre ro! [610 rcr yior.irri t! oqilo1Jou (noeuqed\res ndh lov 'rdedtovld (-1)r+J) rd! €i.$.1ld6ft€1ld q! neo'irJnrot)ss 2 €ri 2 oqilousa. TdIoE, tlnoloyiore qv derA.

4,3,3 AU0IE ri $e!6tsr (l) H oailo\Joa io! S ','lS lool]rdr rr€ tt! oaiqouoa ro! ,4.

(2) Ed! dcr,4 = 0. r<jre dvoE roljldlxrmov ar' ro1js sulnaqdtow€s- nein€r !du6eviLe.lar.

(3) 'E.',as rivoads rou oroio! ra srorx€is r.ivar 0 rar 1 txer oqiqorlsa 1. 0 n -1.

4.3.4 (a) Baeift fltv noadyo}'ronoLrlol ,U. iol]s obryot rdl rI! ooilolJoo ro! 'riva,{a 4oni 4, ro! onoio! rcl orortefo eir(lt a,r = ro pr46ffao r(')v t,.i. (facil're ne{ira rov

(6) Beriff rnv oeiqotoo tou niloro ts n , = ro FrxAor.ao ruv ', xdr !,. froxr =2. ! =6. , , =d. , r r = lu. Mno€ci r r v i 6qcrrc drnv yrvrxo xnrovo yra" r rh6cn r s n i < n l < n ' 1 i

4,3,5 Eoro t, t oailoljoo lo1] 1. 1. -1 rAdlayri,uol| nilota (r rni n)

I

1 - 1

t l

Avamlooow.K oe or+uraao.rom€q 1i0rd pi)ioq rllq hq yadFFrts 6EiEre 6tr ,n =D" )+D., ! An6 ndeiiy.r yra rK oeiEolloes rlv arc^Dlgid tolr FiboMcci1,2,3,5 , 8 ! 1 3 , . . . .

Y?ioofn€ orr o A" €ivor dvaE rer6ny6vLoE n Efti n flivaads pE I n.rvrori mi{ ra€E

, I r l , - l

| 0 l4, t r t . 4r - t i i l . A. - I I l l

L 0 1 1 l

zroip€ xlt! oai(oroa r" rou 4".

[ 0 10 0 l

,= ; ?; ll100101

".*l'-i

264 oPtzoYlE>

a) AlajtruEr€ o€ olllrdodyovreE TlE 'redrK yedl4r4 ror A" "a'

6€iS€ on D" =D " t D n z .6) E€arvdEds dnd Tlp rr = 1 xoL r4v D: =0,6aeii€ rLs ,r, r{, . . . , Di. nqqo-rrtedrqus nd5 auoi oL oer0poi ovo'.rd,6vowar (xot Fr noLa n€qio6o), 6eeir€ rnvDrono.

4J.7 dl Yno)oyiorr rolr ouFn(adyovrFs rls redqr yaoHric rls

7 7 7 71 2 0 12 0 1 21 1 0 2

6) Eno).'t0enoi€, dpded,'wqs rnv h orrl),r c'rd s r,'rdlome! ror Eolotfloi,olito.

4.3.8 EEryliot€ y@ri 6voq rilsxsE s €ni s ro! neqdxs! tva! s$ewd uonircxc 3eni 3 0ivor oi'yo']a(l $L'fuoqqos (av.Ederrro an6 rc 16 u pn6€vrxd ororxFLc roumHerovol,rar $ 3):

eivql u6rv.

4.3.9 AeaErr 6r|yrq niraxe! 4 eri 4. rv ytv€L Lot$E!

d c ' [ 0 ; ] = d c ' / d c ' D d ! " 8 d c ' l a ; l + d . r . 4 d c ' D - d e ' B d e ' c .

ot A, B,C,D eilsr s6dr ril,(rr( 2 rJti 2. 60r€ r.ladd€ryps no! E'ro).ro€!fl flt!alro6qrd qro 6€ir€Qo 16A()S.

4,t,10 YrololioiE rr! oQilouoa rou

l 0 0 0 r : . r j ll o 0 0 r ' r lL o 0 O j r - l

f0 1 I 1 ll r 0 l 1 l11 1 011L l r 1 0 .1

xQtorrorornwd€ nqdE€lg trftEn leo$rlv yr.r ra roedy.iyE€ Fd€vLxd i o!o-nriooowo,. oe orsft(E'r'rovrf; rm nElrlg yeqptS. BO€ir€ €trioE rK oeilowssruv przAodaov rLvd"ov ]1r xar At, nou *ow ro i6ro.rx€biaola Fr6Evrxdv oq6raydvro 'ror Fovd6€s ftaEou alloi. Mnoqrire vq neo6],61'srr q! derAnr

4J.ll nooor roD.ari.oorooFot XqrtdlowoL yra vc unoi.oyroori r oailowd n €ni r sn6c) rcv avaruu6 nino (6)i6) rov nro F€ rorE oupnoad.yow€s (10):y) to! flrno F€ rolE ob{rois (1)i:!!bdm. mo (6) ro nl.iooc y|C n, pe ro rfiOo5 yu n I braorAo€q. :io (y)0uproEic fltv onaloqri.

d ] IYNOIT IA IHN OPIZOY:A

4.3.U :s t.',(r 'iilcro 5 !'ii 5, ro trr6pero r!)r ororroi(l)r rls dlJ,IE 6rarolio1ra1\n4r11dtto5j 6lfr nAoo4ro + rt I Me dD.d l6yn, €ivor rt o = (5,4,3,2,1)dafla 4 reqrrriizrrtlooq:'ta aEaea'ta\d trF- oxaxLieaE lE ra rq6olBo + ro! ol1lrdeay6\'rov, d€"Sivow ro Eq6o+(o oE al K&r drorc 0a 65n r oro ovrrbraydrlro 'yrv6p0vo dr3oro3rro ldooq tra6o4ro +. np6'ier !o a'ioqalo€i,re av l] p€rdo€or €ilar darr{r ri n€arlni.

4.3.13 Ed! o,4 €ivdr m sri n ror o B €irar a €ni m, 6€i&r o

der : '. - der AB. I vndd.rErr nouanlasrnme or' Tu bfE(r ur : : . I

L r t t ) \ " L U t ) /

OrrdEi€ 6vdnaad8€ryuap€ m < n xo! 6vo |rt rr > fr. frdri oro 6dftao nopd6€rypealol'rll! der,,lB = 0:

4.3.14 Yno06oie &! o nLvaxor ,4 ndaap6ver op€id6].)tro!. error ro! otl, r!o! [€ra6d].].€rararTo - s€ +e. Adors rroed6rirpars yru rs onoio I dcr,,{ si'ar 7r{i'rorr [16!! il6.v eildr nord prbrv. ^.i&. xdt{tftIl ((llo iry avdmlJEn o€ olprdadyo\a€{ (8)) 6tloirs 1rn6lofteE ri€anido€E I dct,,t = 0 s1jti6oivsr yrd pio axar66g iqii io! o .

44 I EOAPMOTE' Tg)N OPIZOYTQN

:'t.l(ti rtv r(lai(yaaqo €Esrirloxtr€ I€ 1l o€tQd xr-c sqaeFoyaE ,rou n€QrlQdlpotr€orqv €ro(y{Dyrl: atr@teoglE, },to4 nv* A:t = b, 67xot xat o64yoi At,td5 sivcr

F€rolu r(,)r, r4oQror60o\' unol'oyrotrd)v iq( ya(lF$rl{lG dlF6acE (rotL livovrcL peofiotrolqri) xa( rdQ(l 6lou$€ rufioq nolr divolJa rLE Luotq cl'a),uI[xq

L o t)itd,oy@Ftg roo ,4 ). Au(iE lezrvti pe r1v oeiloljoa avqnruyF€vrlz(.Tir rouq oulfioQdlor'I€q qc i'oorlli yQ(llltnis. OuUl0€ile 6rt ioc morx€i.onoiJ,(crl(oldi€r(tt I€ rov ouFncQ(lovrc. tou:

d e t A = t l , t A i t + . . . + a , " A n . ( 1 )

Karcroxrrdloqe evcr llvdpevo nrvirr(,)v olo onoio q r€/oruori arti 6i1'tt to

furcngn, xnn y4xo| t tc xueia< 6(tyal iot l

)r1per<irote 6tr o 6€!'rEQo6 livoxo( -o "nh4&Q€ rll, ltrttltaQaydnov" - 61eravaoreaqei. EnQErls 1'a 6dl,oul.rs ra ,4tr.....,41, oxlv fi86t1 fiiLn xat oltoqv nQ6fil 1pagr1, droL ()or€ vo no),}(rt ,aotdoouv 1( d .....{rr, tictr lG6{i)oou! ro b(ryrivLo ororx€io dclA. OL ua6}.oL:r4 lpt4tpdE rou A xft or oII}'€<r.,)v outrrr(xryovlolv ror5 6ilouv ro i6ro ((nord.Eoucr, delA, ml 6tq6vro. Hzpio4r4 ep6trprl cir<rrlrui ript'otpr 1t46a'ui navroi, txti< ttt< dnyaviatEd.\, ou!b!iroo1'!r rcl oroLtcio rn< llq lQi4rFrig F€ rouE ouFnsQdYovr€E xlE 2qE.

d i l ,4 l l + {111,4 :1+ + .11 , ,? {1 , =01 (3 )

H crdvrqol ei.vcr": Aur6 fiotl lltroLolilouF€ $€ ro trQoqYori$€vo croQoLo[c, €i\,arorlv rO(ygorLz6rlro I opi(otocr cv6E vdou ritclro B, lou rivaL o ibrog p€ lov,4, rxr6; ccro rI 2I yA4+ti. H nptitrq lpuyptti rou ,4 6X€r avancQax0€i otrlbeurprl lpcrpprj rou B. 'Erot, o , €t€! 6uo i6!86 laag[iE xcrt deiB =0. HsEiotDort (3) €i!(rr ro aldtrruyF( ro1'' det-B zorti trrt)aos rrts 2ry lAq$rrtE rrlE, 6rouo B ix€L rotE ibrouq ouprcQdlowts ps rov r1 (6torL, 11.1 vc uttololLo€ouv ot

210 otlzol:El

* !.r.E.: H tLitrt /.arlol r.)v Aeoeo! .ror.. zdr '.ntc . .:aar.idr (rrt rc (v 0Eoaoirr orr

iiroLt,f oio t$( 'l tEid!'dr". awidto,r(.

l;:, "', :::] |;: ;.(2)

f dc tA 0 . . 0 . l| 0 de rA . 0 |=l;

i ..i,.1

1..1 EO]\PMOTE: taN arr.rzo\:eN 21r

ouFrraedyol,r€9, ncrAcllsirr€rol I 6€ir€QI yQo[[ri l.icr povo ount I yaaFF{ €ivaLa'aqoa€rrrt aro rov .4). 'E1or. ro aqloo4F€irDro yr\r6p€vo rrvdxorv (2) €ivdr

To lLv6gevo auro 6iv€! cll€orDE.ror,4 r. :tqv oaror€ai nl"€uair I]]: (2)61()4r€ €!o rouanldoro rl( x(uroi!x{q. detr4 €iri 1l

(A)(A-, , , ) = (detn)1. (4)

Ebdi o ,4,,,,r, tL\st, o ,tivaxaE ny o ptapayiyrat i "o ouluyrlg nivaaaq" noue$(poviErra! orlv e[iocuorl (2). OWIo€irs 6r[ o ouprioet(yoiv ?ro1r 7lQoxurxcl [t7rcQdtr€t1,tl 1lS i ypcrl4rrjg ricL j oril,fi rorl,4, rdeL o-r1 .l yaolqryl 1i(! i orlLnroli tril(rc o1J$napc{lovr(l)\'. ArcrQ{lvruq pE rov cQrep6 dctA (€q' 6oov 6evr4brvrq€rar) Ict iUvuuLrr ro.r rdno ]r{r rov .4

4C Ta orollela ro1l ,4-r eivo! o[ oqfi(rQdyol,rsE rou ,4, qv€orQoul.livo!,6no4 ot4 (2) rar 6!d!Q€pd!or pe xnv oQiEoloct dme vo nqoxlrpo0v rcAl1il det A:

l ,

det A

Edv deie = 0, 16r€ o .1 6ev ovrr,orpiqercr.

IIAPAAEIrMA I OL ouBfioQa(yot5€E 1{)!

( , A ! = b , A r = a l

{ .4L4" , r= 1, ' , , . ] l d - t , l _ , t , t L, o I' ' " " t , ,1 l t . , l - | 0 , , , / b . . l

AraLpdlx.r< p€ ro od b., rox eivar I oQilouon det,1, 6Xoupe ro l.roFevo A e;riA r

1,, , : l =__!_l d -b (h,l . o l o d _ t , ( l t a )

:IF!-16or€ nci),r rI! oloo-rAo(t1i tt€otl orov ,4.u orl 6sErd nl,e!Qd.

IIAPAAEITMA 2 O (vtiorpoqos ror

il

(s)

[: li] ',.',,,

f l l r ' l , f l I11 = l0 1 1 ELIaL - j i l l = l0 r

lo o r l oet 'e lo o

To nQool[o tr€iov +qdvi.,lercu. 6r"61 or" orpncp.iyovr€< r€erEtou\r ro ( 1)'.r

o?lzoYtE

2. H Litor1rcu A:t =b. H 6€rn€Qq clwi €qaQ+ro1i npoxri:ner nolJ'c,-r),cortilo'wn< ro b pe rov nivoxa,4 ' tIE (5):

r=A tb = --LAn,b.

Aur( ci.vcu,, ar),,oriorcra, ro l!a6[€vo Ev6E tri\,oxa F' 6vq bldv]Jopa, 6rorp€F61ope rov cpr0g6 det,4. To a?ror€leo[a oF(,)E pnoQei 1'o yaaqrl1|o[ [e 6vd gll.llo[avora6no:

4D l(sriqtt ton crsmerllj-oqoi owrordtoa ro! o =.4 rb elvcu,

L = ------+, Otro! tt =

:Iov Bj ro 6ldvuope , oltrra8totd q! j-ottl).l tou .4.

Anri6rrl4 Artrndlre fiy det Bj o€ oUEqQd'yorteE qE i-omft ot{}'$ (roueivco 1 b). E:rrr6r] oL oup:ropdloweg dyvoouv dntrl q onill, ro onor6l€o[q

d e t B j = b t A t . + b . 4 , + . + b r \ L j .

A!i6 or4urtfir€r pe r1v j-oor1 oulloto)oo iorl llvopdvo! fii,vqxa pe 6ttivuogn.4",,,0. ALcrLprbwcg pc qv d€r.4, noi.Qvoup€ rnv j-oorr1 orvr.mdoc rou t.

'Eror xri0e ouvrm6oa tou jr €i!(! firlliltov 6rio o8!!o!o6v, €v6G ?iorluDvtiFou6a0pori n 6ruLpqr6vou pe rilLo :rol,utiwr4ro 6aqloi n. To y€yov6E outo 0(rInoQouo€ vcl 6rtrn!ol(')0€i xar onN (r'ratroqi Gauss, rot6 €v rofiol{ 6€v 610v€

nAPAAEIrMA H Lrroq rou

l r t +3Ji2 =0

z t t+4x ,2=6

I;;;;:]

r 0 l2 61 6 ^

i : - - - - ; - i= - - -= -J

2 4 l

(7)

l0 3 l16 4 l t 8 ^.r'r = i:--=-i = ---;- =,r.t r J l2 4 l

or nnpovopcotdg eivor or i6!o[ xs! adEor€ ioor $€ det,4, e!6 o! 6€Er6E ri),euQdE0 x(,| 6 qrg(triEolra! mqv rQ6iq oxyll'rl yLa ro rr x a! oq d€lhEQl YLo 10 t2. fLq

II EO,^PMOr!! lON OPIZOY'OI_

1000 €Ero6o€tE, g(r tQ€r.oq6Faordv 1001 oQltouo€g. ME iQopo ovc.xdrluvo 6vorial!6 6[6],io riou owrorodo€ x{i, r raroro (i€t6r'tos or' dtQqora qv ariatlotqrl):

"ficr vo ),riooqre ivc :rQ66l.rlFo rio! nsQl6ler r6oo€erE F€ia6),qr6E u, u, rr, z,npdner:rqci.rrc v<r otcl,elvoljF€ Fia cE o!r6v, 6ore vc ,iQoxtq'ol'v rQEr,s €Erod)o€Ls(l)g raos rQcLS Llard6l,n16E, 1ioi6r v vo EJrsvcls6oup€ rn 6La6lxqoiq, dol€ !drraor.ii!'ouv 6io sElodo€K. o avalvoorqE fiou 0a err"leqr]oer xcirr, riroro, ocrdoxlol. 0a 6!('rlordoe[ n6oo eoceQq rouQqmrxri livercr q 9€0o6o9 ord.otg{E,6ra\, dloqrE r€Qroo6r€Qeg aa6 rper4 gerc6),rp6E. H draniortool aufli o6{1ei mqvnvc!{qnq prcrg rclreeqe F066ou" (nou €iv@t o xov6vos rou Cramer!!).*

nc €!6flo18 aivcneg, olou o xov6vog tou Cramer €qaog6qstor eixol'c,6IUUtr (xQl6ior(lrl ,rtrqeoqoarlq yLa rI Luqn.

3. o tiyxoE etiE tapall4le"l'drou, H ox6on tr€roqrl oQiEouoag ear oyeou 6€!€ivar xa06t!o! riQoqovnE. Ynoo€orE 6FoE 6r| 6?Leg ol lo.rvieg e/r,cr opoi<, fu1,c6r1o! c(xU(! !:ivclr xdoEi€E pe1sEu rollE xal, txorJlr€ 6vc oQeoydr![o xouti. T6rs o61xog etvc& cxgL6ri:g ro yw6F€lo rov [l]td)\r to)! ottl6v: o'/rio6 = lr h . , .l',.

6c 6porge rov i6Lo turo 1.(o! ?!u rlv oQl,lo!'oc. OWlo€Lr€ 6rt or qxp€s ro!xor:rtou rjrcrv or. ypr4+6E lou nivaxa /. :filv fl€Qinrool roir oQeoy{oliou pos or..ypr4rpec uurdc tivar gctcElr roug oe0ol6vlrs x(rl 610r

I roouui t Ioo, =l I ILlSql4ln n.l

* A!r{t !o GiNn@ln Ei!(r d6 ft Muevt ntz.i f n earor&up.a (t.r.M,: qloeti,rd tred

ro0. o! rjncllor Hogb€n (1937)- Edr qirdr@ r( zartorono,ridd rov "a!6va

ro! Cnfti €o ro

ovirtui! Mue, nred ya tula El.atosreu\1pt,

f '; o I=t",1L0 t ' " )

A Q

F TP Brl rt1 1 l

Ts li eilar rcl urlil (ov yQoplrdrv (crx[tirv) x<rr r<r ploevrui exrog qg 6r.gyoviou:rpoepxor"to a:r6 qv oe0oy(Dvr6rw(l r@\,lQaF+6v. flaiavovrs6 oot!,ouo8E rarxQrp|llonor6wog ror! ,{(.vov€E 9 xs0 10, 6xouF€

ti ta . . . C, = det(AA I ) = (dcr.,lxdet,4?) = (dei,4)?.

H r€rQ(rto1ari ailo qg €EiooronE biver ro trForiFevo: H oqitoroa rcoita.pt tot 6yxo. 'fo nptiorlpo rqs der,4 6elper cv or uxp6g o1lpatiEoov €v("6EE[6orQorlo" oriorrlpn orwcrolrdvrov, 6n0iq ro owlololdvo oriarli|lxl. lN - tJ - z11 ercr cEmeq6orgorpo oimrpc, 6na4 to 37 - r z.

Edv ro rrdQal),I1.€nir€6o 6€v eivo! oQ€o"y6uo, iox€ o o'r.(oE 6Ev €iva! rl'601'ro 1rv61revo rov axgdrv. :xo srin€io (>I. 4.2), o 6yroE totl raQotlllloyQdF[ou

oPlzoYtF>

ldua 4.2 'O1ao!- raQa),]'l]loyadt$ou = der,'l

roolir( $E rt 6tiol 11 erl ro ul,oE /r. To 6Ltivuolrc 2, prjxoq ir tivut ioo 1tt.q 6eUreprl 1pc4rpr1 6 = (a:r. aI) p€io! rrlv nQo6o),1i rrls orryre6q ya(ltprt. Hr(patrjprloq-z).a6i €iv(! I rErl(: Kcr(i rov rsv6w 5. I detA bcv tr€rc6hlJ,srt(t6rcv ca6 rq 6€ur€ot roqrFi (lq(lrQ€o€i dv(l nol).qni'drolo tI! nQ6r$ lQcpptE.T(r,rolQorc o d1xo5 eivat o ibLos lr ezrivov €v6S oeooY{oliou 6dot-o)< lt z(tittpotg h. Mtopoitpt lor:dv va uluyril,ou\t to ;tp66),4ptt ar4v oa0oytit\lnpi:rtoa4.6tot: iatL ilbn sro6€rx0€i ort oyxo( = oQlqouoct.

fr(l vu Ftt(r1Q6uo![e iau :rtrp(l),h/c:rint6o o 6ro opooyirvLo xowi, o16 rrbKol(ioeLq. xp€LdEeror n€Qloo6teeo( xoroE. I rbio E-La-rq riv(lL 1 fb!(t. Ork. ot!.1xoq oUre q opi(ouocr (l),Ldr!ow, 6r<rv ororlporrrd 1lo tq yQa[[6q 2,3.... . n.cr<pcrLpoqre (.Jt6 rdo€ 1ea41t r1l!,rQo6o)"rl rrls oro! It)Qo,[ou tr(Q{il€rcL arorL6 :rpol1or-4ruvdg tqg lpaBuiq. o Uari(ol-[t(< fro! dva 6!c(\,riopq urlro1j6". ortDqro pl, nor rilnr xcioero orrl 6(iol. To (1rior€IEopc aurlg xqs 6Lcr6lxooi(lGcram-sch'nidt eivc! dvo ouvol,o oQooyorviov p€rolU roq lpc4rpdrv pe t1v i6r"nopitolm( xctL rol ibrc 6ylro ior.r oQluori fica(0.Ul€rm66ol xL eirEt6{ 6yxoq =oaiiouotl yrLl rlv oQooyrivrc n€Qinr(ool. rl ibrt( Loorlra nQitr€! !( loxu€t )|(t lt(ltrc (.Qxrx€g Yo.4tlt€<.

T(r :rupn:rdrvro ouFr),rptirvou! to orrolertop6 dyxoU x(rt oQilouo(r6. AEi(€t6pr'6 va ernorptrporpe. 1!c lLlc cx6Fl <popir. m4v (orl,ouorEQl n€Qirr()orl.

*'[l l] = '. d", f l 9l = 1.t c r l

Or oQiEoL,o€E aur6< 6ivowloq 61xor:g -r1, znl,rlx€Qa, xo €F6a6d, o.po! EipumtorrE 6!() 6roordoe|g rcDv 'napoLt'nL€trla66o)v" rox :X)iFaroS 4.3. To nQd)tosiva! drv( poloSraio rerpdyuwo. rol ofioiou ro €F6a6ov €ival oiloljQd L Tobeilreeo eivar €va naaotJ,qL61pcr[po pe govc6Lcic 6ciot xor polo6loio ul],oc,rou o:roior ro EB6a66v €ivc! eai,ong 1, aveEdtprrFa 0JI6 q 'me€6l(,)oq" tro!n(Qiry€L o ouw€Lem4s c.

r . ! L 'nAl 'NrorE. ! r ( :N or lzo\ . !oN

2 = (0. t )

. / p o p u i l = ( 1 , 0 )

. /pdt ! r i 2=( . . l )7

,/ Ypalqtrr I = (1, 0)

!r,4.1 EF6cbciftradrdJrorzar naao)"lll.oyad$ror.

4. EvaE iittoE yra toug o64yoiE. H retrertoirl erpappoy{ oqoptlt oro npo-6l1pu ttuv plbevLztbv oblyd\'. Mnoporrp€ €fii 1€troui vc 6[ctfi!old]oo![8. n6te€i'\ro! bur(q n oficdorq{ Gauss. loptg ro XQ€[dEowc[ evcri.L(./aq ./QCl4l6!. H7r(Q(lT)ierlol-zl€rbi €ivsl ro orl or A fiQd)ro! o6lyoi oQiEovra| r].lA!)E cfio roauro;rivcarr ,4a oqv :rcirlr cpr"orepci yowicr tor ,,1a. Or. t:n ,octt< yOctttti< tior4Le< ou A 6tt' ixo r, e 69ao4 a attlj t4 yat la rcx :tpo6)sjpato<.

NAPAAEITMA

:i./oriQ(l o nQdroS oblydg €EcerdxqL $ovo ano rrtv rad)rl yeoF$rt l{sr orllql' l r ( ! , / , , . o 6rrrrrgoc o64yoc:roL' nueouo(ior4nr au€or 's tur( . ro n0u'ro6ipc tu_rd,oqrlE, ipar (rrl - bc)la r(! €EcrQrirroL trrovo cra6 ra oror7ei.G a.b,cx.1! .1. To !fi6),o!rio rou / 6e! n.rQouoldlertrr noQd oxo\r rgiro obryo. :Tll1,fiQayF({rolorlxc, 6e! €ivc! trolo oL o6rlloi, a),l,h ol.6x).r1peq or rrivrD lovieq rarv1,. D zcL I/ ror rpoobropi.lorcr o16 qv rltvco cpLorepri lcovi.a tou '4:

f r l t , l l1 hla-)A=LDU= c /a 1 | @a-b /a I I

L* * l l t - lL r lA1JTo fiou 6ldno1JFE olrg 6uo ,_rQ(i)r€E ^/Qqr&d! r(r m{r,EE €i.'ror oxQ!66E rl

:rapcyovto:roir1or1 ror r:ro:rivcrxn A. = C ,) Atr6g eivcL 6va,c 1ev,z6q

zurovuc. I q ooov o, ! undpTouv rvc). Ia7rc ypuppuv.

4E Ed\r o ,4 fioQoyo\rronoL€iio! orn loQqt ,Lral, x6re o! fidvo) oQlm€Qit"tojvl€E llid\rofiolouv P

Ax = L*DrU*.

frc zdrOe A, o urorivoxcg A1 uno6d .flar xar cut6E oq 6rr{ -rou o?rololpri.

^=1,, i il-1.'i, ,n!*,r. r,,t t*trcf

276 ouzoY:F!

. dctA, detA, d(|4,.d i d , . . . t 7 , , = - - - - - - - - :'

d c t A u d e t l r d e r A , , I det ]1o

H ar6berl4 livncr eitr 6idrovrcrq 6tL n y()!icr auni propei vcr tcrxroroL{eirQ6rt. rAr! va ;rpolup4otr 11 a:rr),otr1{ rcpaxtiro. rirr T.prpqrorortirrug tougtc!\6\!t< ftolkritlao&,oltot Lrrrix@v xard trftAox. Or znv6veg awoi €iv(! o!i6Lor l€ Tov raovovo no),},(.' ,aorcolou ororl€io! tr$\ fiorx€io xcr q aru = 4

[ r * 0 | l r , 0 | [u* F l _ lLkDku, , L : ,D, ,F IIB a ' ]L{) r l l r } Cl -

IBDLU^ BDtF+ct( ; )

Eqi ooov o! riaax€E 6louv 6roonaoe€i xqrqiJ,I),o -o! x€rQa./1')vr)ioi i n.lQaL-L1l61gt4rpor uro:rivnxeE dlouv tr.E aarirl,l,1Leg OLcornorq. rirore vc propotvvo :rol.Lqnl.aorco0oril Ei.val 6uvqr6v a(r fiolJ,onl,qoiqsooiv xcrri pnl,ox, p'crt6v rov re6no.* :lyxpLvovrac rov x€Lrurufo riaax( Fe xo\' ,4, 6ldnorpe 6rLI./(,){cr Z(.,^.LL ou$ninrE! Ll€ ror',41. xu 114E eircrL oa;or1.

O[ 1uno|fl(t ioui obllou! rrQoxirwow noiQvol,r(rq oQ|!o!o€E:

der A* = det r^. det rr dett l t t = det Dr = 4d. . . . dt t . (8)

To yu,ilttto rat zp<brut, k o64ydtv th,at I oAiknau w Att. O r.ov6vc.( (uios./r( rov ,4/: €i1,(.! o i6roE tls rov r((1!6v(. nor' yv(opiEouF€ n('rg roli€! ylcr oi,6zLruo1ov Jriv(r?o A = 4.. EnErbri I oailouo(r rou Ar.-r 0c 6t6€tcu. xci' ovol,oyi(.!.un6 rc tl ltl . . . tl 1, L FnoQori[€ vo @roFov6ooqr€ rov o6ryo ri1 o,u\ ,.n),axov6io oq$ouotbu.

d ( t A x d 1 t l . . . n l, r " "a^ '=7 '4 . . . ,1 , , " ' u t , r

lro nqot./oqrvo rupir6eLntir pcrE, o beurepog o6ry6< rlrdv axa!6ril5 xo ntlixov(o.1. bt)lo, no! 6ev €i!o! dLLo cr:r6 ro nqLixov qq opilououg rou A, 6Ld 116oeiEo1]oaE 1or Ar. (:utup(,wodlle 6r! der,40 = l. €1or dor€ o nQ61og o6ly6g v.rFi\'(r d/l = d.) floLl(JlLoori(lovr(1E olouq rouc o6ryoig p(Li EarancliorouBsrtv

An6 rrtv (9) troiQloqte tii.og t4v orirvt4orl mqv crpl.x{ gq epdrt4orl: 'Otav

,5j,ot or aQfipoi det At" Eirat pl 1tr16*txoi, t6tE trr otorxeid o'r$ etoat 'rnro64ydv tiv.r t pn yqitvtxti.

4F H cra).oLtprl causs ytc to! rtivarct ,4, €xr€lsirar XoQiE EvsLlqyasypcpprbv, r6re zcl p6vov r6x€, 6rdv 610! or.

";ueloL uronivcrcE -4r,,4:,. .,.4"

eivar pq r"6r{ropqor.

* II(a 6).0 ror dcv ror ollartdatt t(sd & Ks ii(N l. o ,a!d!ns alro< .iaar aoin rcrio4ro.

,]..I E<DAP[4OTE: TON OPIZOY:ON

M€ eqdiaeorl rI! naoarA€rrzil olp€itool, nor llroo1€0yp{ope mlv oQXi xolJx€qal,aiori, 6xoup€ oupnllp6o -ro o€ya tov oplEoro6v. H o+r€i(,)orl ex€i,vnaqoAouo€ orl o![6!6oot6qts ro]v loqoxfllpronxdrv Lbio-ntrtllv 1 - 3. To d.€6irlrcv 1 r6r6r1rc 2, qS ol,la.y{q Tiaoortporr nord tlv evcrfLcll lpapudlv, nouobityloE orylv oeiEouoq EyiE nieaLa pddqio^aE. To povo alqLo6yFriortrooqr€io orov ovalurLxo dro (6) €ivoL ro eE{sr Ei.r,cr altO€rc 6rL, aveltipqraon6 xtr €r6r,a{ axololjoio €\'al}.oyri)v yeopFd}v rou o6n.youv 1ov P. orovr'rlnorza. ro r).'t0og roJv rlalloy;v rivar navrorF rlQro 4 aartotr :rrprtro:Eriv o1jp6oiv€r, fiQdyp(rr xdrr -retoro, r6te bLxar.oripraore vq oaoldqolp€ rn\rcvriorolxl [€rdecol "riprLc" rj "nep!irr]", xcrr. 11 oqilouod rqq €fvo!, xdrd ro\rxal61(r 2. xol,(iq oprop6vrl xcL LoorraL pe +1 11 pe -1.

E€} ldvroq c(n6 t1 Fetdoeol (3,2,1), Fe Fiq xar $ovo6r i evall(rlli i(l)v3 xor 1 0(r $rdvcp€ oll gl)olxri bdrdEl (1,2.3). To i.6rc 0o r€rr1ai\rcq€€va).)dosovxqq xcr 3 y,st 2. To 3 xor 1 rc(!, 16troE. rcl 2 x(!| 1, Kq! orr!6!o cxo),ou0ieq, ro r(trioo( rov €va),Ioydv €f,'ar riEQ!ft6. lotljol!6uaore 6(Etxu,tinru< atr6 u (3.2,1) 6tt, pnopotipt nod t'a ftri4ou!€ tn rpuor.xt \uiraEn,t ra dat@ ,tLieoe etdllaybr

16or ptu <r:robeL[r1. @ecoprlore ro !,€!yl crQ!0[dv r$ Ll€rdosofi x(rr 6oruN ro rl,10oE rcuv E€uy6v, oro ofloicr o npd)roE oQle$6E €ilcr o Fyq).rir€Qog.Xilouqcr N = 01!o rl qruorxl brdraEl (1,2,3), 1r1v r<nrrorLxrl tl€rqoEon xarN = 3 ylcl Tlt brcir(lEt (3,2,1). oqorj 6Lo ra q€r,.fl (3.2),(3,1) rs! (2,1) eiv(lr(lvLirobcl (6rqrexc(./tdva). To llFltlcr €i1,crr !o 6siEoup€ 6rr I F€rdoeon €i!a!t'rprLc t1 replrq qvdl.olo pe xo c\r ro N €i,vsr i(Qrro ri nsplrr6. Me dll.oL6y!a. texrld)\rraE C1?r6 onor"a6rjrore p€xdo€oq, va b€iEouu€ 6i! }1i(0e elcl,lclyriBerd6dlJ.€r ro N xcra( ivov ri€Qwrd o0!0p6. EroL. yrc vo qrqooupe oro N = 0(q tpuorxrl 6rdrcEl) XQ€Lq!6poor€ ircr rl,leog Evcl.Lo/ci)v rou a!'roi runo!(itptro 11 :rtprr6) pe to cgXr,r6 lf.

Eriv ro letl1oE nor eval.Lciooor4re onorel,€i.rc! arr6 ycwovlroljE oe!01ror(,.ox€ jrQocp(,'(i)E ro N [era6dll.ucr. xcrci +1 rl -1, fiou €ivcl! xo! o! 6r]0 rrsQt"rroi.H atr6b€IEn o),oxLlptirvtrcu, U6)'6 :rapcqqyloorFe 6tr. rri6e ev(),),nyr1 propriv& yivsL [E rn 6o40ela el,oE neQlrtou r)'i0or,c r:vc].lcrdv yerrovu{drv qeroudv:ro o0QoroBa r€prrxou nl.n0o!q rreErxdrv oQ!6p{iv €ivot n€Qrri6E. H noparri-pnorl atlrn enl6e6ord)lr-rcr *xolc o' 6vcr :rcpci6ernrc: yrd vd €\ra),1,{iEoqr€ rob€nftao [€ ro r6ropro orollEio, rrcrQo,{o10r. fiou oqr6cive va eivar 2 r.iar 3.lprlo4roroLoupe ;rewr el'o ,a./sE (nEQLrrdg oQ!0p6g) yrLrovr"ativ oroLleiarv:

( 2 . 1 , 4 , 3 ) - ( 1 , 2 . 4 , 3 ) - 0 . 4 . 2 , 3 ) - ( 1 , 4 , 3 , 2 ) - ( 1 , 3 , 4 , 2 ) - ( 3 . 1 . 4 , 2 ) .

Ev ./6v€I, XQEKq6Fcore i k evcL).c1l6q ler.rovrxrbr apr0prirv ./LC la n(ip€ rooroLleio rqg 06orc I o[I e€or1 l. Kcr6nrv lper.cl6gaore I - k - 1 ev<rl"1,c{y6gllrr vo nd$€ ro ororleio rol cpXrxci rpcv mq e6o1 I (ztrL r6Qa 6Qioxcrar or106on I - 1) r i ioo' orq 060I A. Ener6{ o ( l I )+( l k- l ) €i !c! n€Q!ft6qI cr65€LEn Eivor nl,nanE. H oaieouoa 6xL p6r'ov €xer 61.€g r|E rSL6rnrEE no!6A{xaf€ npotyouF6v(l)E }.}.d rc! rJnirQl€r.

oPIZOY:E

A'KHTElr

4.4.1 BqriTf q! oailouoa ttar toug .wio ouroedyorrs rou

11 231A=10 4 01 .

1 0 0 5 1

tfi!ftLms rard'rrv rov nirora ro! onoio! ro oiorx€io (i,j) ei!d! o qwroedyov,4ri. EridLl9€1ron€ drr A erri Adr! €i\,dr o ro1jroflx6E nilaxas €ri q1 ooilousc.I lotoq ervftr o A-'];

4.4.2 MF rov rlJro ,,lry/detA, 1ro iov avriorAoeo, €Eitl.rlore y(ari o A-' sivd| dv(])ratYolvlros, 6rov o ,4 €i,!sr dlor rerylrvor6g (1.ar awroiadvLlos).

4.4.3 Xalouonotlior€ ro\, nilsno sr{uraeo r6!'rov 'yro 1]a awtotA6p€ie rous

f l 1 1 l11 2 21 .L l 2 3 l

4.4,4

4.Ls

Me ro! flrno,{oq,/ del A, yLs ror avriorqo9o, €EnliorE yrari o A I €Lvdr olppsro(-r.165, 6ra! o A eivor oulpstArrds (rdr ar,lror0dvuos).

Bauir0 rs a,g rar z p€ rov lro!6va ro! cramql

r+4a - 2= r1 1 . 1 r E + ! + 2 = 0

2 r + 3 2 = 0 .

4,4,6 o) Ba€iro fltv oeiLouoa dtar ro srdworrs r a!'&roornd q! j-ood mi).r ro!

6) Eda nr = b, &i€ft 61r .4n1 eivar o ni,lo'ios ,j nE rqioooff (7) adi r 6eEdrl€lad aJ EB.{aliLnoL mtv j-omi mi),r.y) :lpfteeiivar€ rov ,akjva to! ctunet, ndie\onag oqiloloes mlv ,4,i1{ = Bj.

4.4.7 B$itE rI! Idro6Lo'1] opf(om<r J rn rlv dJayi 016 fls oqeoy6vL€s oD'r€rayfdr€Er,y ,z mq oqaerais r , r ,4 i : r =rcosdcosd,g =rshPcosd,z =rs inqr .

"'^'l'i,,]

4.,1 EOAPMOTE> TON O?]ZOYION

a.a.8 a) :xebrdme dro raiyolo rol onoio| or roowi{ rtvcr A = (2,2),8 = (-1,3) %o!C = (0,0). o€0,ad,waE ro oa! pLo6 'raaallqlqaappo, €Enrioft trad ro q6o6or

,L,6doov {ABC, ]"' t i ll6) Yrio06oft 6tr r reiq rosrqn €tvdl d = (1, 4) a! y|a (0,0). Eral.r0sime mv

I l , r v . t l I | 2 2 r ' l, p 6 a b o v { / I , ( - ) ; d e ' l r , e , l l ; d e ' l - r I I

' L . t r g r I . l -

L l 4 l l

vz.id!r6r: Aqae6lrcs rlv releuraio rpo#ll cr6 ns 1rrdlotneE rroiqrolpc

| 2 2 r l I r 6 0 lde ' l - l 3 l l=de ' l -2 I0 l=de ' l ; ;

L r -4 1 l L r -4 1 l

:r€6!dore 16 aoouris // = (1,6)jtr = (-2,7),Ct = (0,0) aor BeHirr rrs q(dssr.ErolJg !s rrs A,B,C.

4.4.9 EEil}to..e xeryUorot(i)waE 6yltor,s, yr{rri dct3,4 = 3'dct,4, y(a rd0c ir sri n nivdno

4.4.10 Ediv ro pftl,or.oiirryG ,4 rirar srrrorqd{rpo. rdre ara}.o|(9ri rcrd pnlox 6ir€r

I I 0 t f A A t f , { t s 1l -cA- t r ) lc , r l = lo D-cA- tB l '

O riraaas.D C A-t B Lqe:ol oLrtrlije@rc ',oLr Schw. L.i*s 6n r oeilorrsd rouErit q! oatEo1joo ro! ,4, toovrdL p€ ntv oai.loroa ror cslwo! ni.vcxc or pr).or, orcseror€Qd. Edv AC = C,4 b€tlr€ dfl a]Jd €iv(,r ior pE det(AD - CB).

4.4.fl I1oror xdr6v€s ro! rrropavo! AE reodet(ovrar ano rous xovolE( rol).or).oo!.1ofoiautd pf,l.ox trr tdxerd l'rrfli pftLor no! rivar onil€( b xcr r xcr yQa$6( ri

4.4.12 nao6l6lire €}{ t(Dv:rqotdqov rouE ohrois xd! €rdh0EUor€ FE oral.oqi :

12 121 12 121,4=14 5 0 ra r a= la 5 3

L 2 7 0 ) L 2 7 0 )

4.4.13 Ba€ie o),q rq r€a!ft4< $ra06o€rq ftD! aAroldlv {1,2,3,4}. Aljx8< €i.',ar o! F€ra0do€Ero! reodqxoroL dft6 n€arr!6 ftl,ieos €vaLi,ay6! xar obTroljv o€ derP, = 1.

4.4.14 Ynootor€ 6rL r F€rd0Eot o mdlvEr ro (1,2,3,4,5) mo (5,4,1,2,3) .

^" l.l, ,,]* 4B-1" - ]l;]

OPIZOYIEJ'

o) n xdv€' , I I oto (1,2,314,5);6) Tr xdv€! n d r oro (1,2,3,4,5);

4.4.15 Edv I d elvft n€altni t€id€Eol, €En4ot€ yLad q d2 etvd! dalld dl,ld r1a-r etrorr€a|titl. A60r€ sva naed6€r'fFo dra! n = 3,

(416 Acttse &r €dr ow€Ito€ft 1u floDcnl,qsLdlste rov A Fe m! t6to rdvara FstaedosD!P, xdnor€ I nE q leat$i 0a €moreaqer ,rino cnp oelflt tm 060+

4.417 Edr o ,4 €i1lor nivor(os 5 €nt 5 !€ 61a ta lairl < r, i&e dct,4 < i (Asv yvoeilo 1onqlvleeo qeriylro, an'i or 6y,(o! tl o t$nos'(6), i or oSqyot 0a lJte€tn 1|a 6[vow,va dv(o qed$a Ta qv oeilowq.)

:KH>F:l> rlA EflANA HII'H : K.q.'l 2El

AEKHTED IIA EnANAAHTPH : X€odlaro 4

4.1 no! ne{ner ad 6d}6€ Fr6€vLxd of arcrv fliraxd 4 €ni 4. tarloqonor{i'vrsg roh?d$aa blJvoid, al,Id aqrtrd yto vo €ivdr I oeilol)sa plbdv;

4.2 no! nQ6rFr !d 6'il€rs frbEvrrd xft pold6€g o€ d!a! Jri.voxa 4 €ni 4, xqnoqonoLlb-i'r(s r( hy6$aa 6uvc(i, d,]'d o0t .rd .trs vs civsr I oeiloxoa dvar

4.3 Bqrir€ s oailouo€s rov

l r | | r ' l | - 2 | 0 r lf l I l 2 l t - l 2 | 0 ll r r : r l ' " '

I u t 2 - t lL r4 r r l L - l 0 -1 2 )

4,4 EC\ D = AI r,4.U. y|dti raxrrer dd, = dcl,,tr AuiEre sniqs 6fl der A \ B = t.

4.5 Ai,nF c1{fiaad6Erypa qE de('4 + ,) = det,'l + dei B. rt(r roto ldy€0os nL,',dl{o,!LorurL dwcr< cuni I reoiddrti

4.6 E.xr!(i!rss qrd roa niraxo ,4, rol.ldtr(rodor. qv na(lflt yedHri tot' I€ 3. (;,ot€v(l nQoruri,€r o niag'z&s B, xor6nrt, dqaLquort c!r6 i4v ra(;jq yaol4ri rot' B Irlb€l'{.ort. ri,ne vs reo'rw€! o d. [(;)s ('wbacrql I dd C lt€ qv det,'tr

4.7 Auotc io ooorqro 3u + 2u = 7,4/ + 3r = I I ll|i rov aat61,d rc$ cnmer.

4.t Edv r(l ,xorxlio ro! ,4 €ilar udeoLoL aoL I der A Eal(lL I 'i - I, n(i)s oqlrEadil/ttr otrr.r morx€ia ro!.,1-r €ivor dl{dAorcri Adroft ila naAdb€rlpo 2 €nf 2 (ijXr 6ury6vm).

4.9 Ea! to ororrda rou A )icr tou ,,t | 0lvcr u*q(ror, flri,g ono6axdt"ror drr. rcr. or6no oailoLiorg €ilor 1 i| -1 Y tr66ul4: Me \" wourqr I det,4 tni rtv dct / r i

4.r0 BA€trf rous @Frae(yovtFs xcr tou( oltrnqd,toL( ru!

t 3 5 l l d t t 1 I d b l lln o1 "n ' l r , , l ' " ' l " a l

4.ll llorq .ivdr o 4aoE roD raednt].€nu66ou F€ r6ooeAq xoeueag oto olf€io(0,0,0), (-1,2,2), (2, -l,2) 'rot (2,2, -1): non ri'aL ot ti4lrs xoqup€Er

412 n6oor oAor @u+84(ow oro crldnrwld !ro( oqiioljods 5 €ri 5 '!ar rr6oor €i'aLoi'youeo !rl6a!, or0! a,l = 0l

4.13 norc €i\,dt ro Ar, otdv 1{do€ ya.44r1i io! A dx€r plb€vrr6 z{oeorsfo oroilet(D! roLio r eivor brdvr'opo-mtll |l3 tLold6eEi n6E onobewvlndr 6rr derA = 0r

4.14 f|ort ro n?'4eoq 6),0! rov Frrdodo€(')! to! (1,2,...,9) €ivar darm l{ar yrsri or lrodsdrQl66s aft' du6s €iva! r€ernEst

4.15 Edv o Pr €ivaL d$os rivaras p€rdeioE(os rcL o P2 €tvar fterIlor, own.qdvdr€cn6 tnv r]r + Pr = Pr (Pfl + P; )4 tu det(1'r + P,) = 0.

4.t6

4.77

BOeirE tr! oqilofoa rou A, oi.t\* aij =i + j.

Edv o -4 !,(er e€rrri oeiEoxoa. bsiftr orLFroa€i vao!!be0ri rrs io! taLrotrx6 niaoacF€ ria {Dsx'l d.Doiba f,Llidov. rou 6xow 6IoL eftrrri oailowc. O A F€r(6(i)J,EoLr6ft mov rouorr.o. xopiE vc liv€r rtn4roEf'oq oto 6(xito. (To €lorr/qqpo Folondttono ro! I moa I, itol nearypair€r(r! ono t! ailtiirf6o rLv.iro! i(/) =i+t(I .4).o6ry€i nadytoT| aro to' .4(0) -,4 orov ,1(1) = /, ('Iov 6q6Fo 6poE o /(f) Froaeivu yivEr rf'fipoaqiog. To re66)'l[u 6Ev rilat 7dr roqo diro].o l1r xci0s ].!oI €Llcrrrneoo6€zq cr6 rov o!'yyqoqia.)

Eq! o J riluL [I 'tu.!o{-)rloc. briEtr 6fl:einfl vo urd0x' r xdroro< iiv(aq;F€raofo.r,! P y(u rov oroio o Pi ar€L trt FitbFvr"d moLr€ia oqv xiqLa 6Laydrvtliro!. Auros 6.! .ivu( o P tl]s cnd.o(q4iq.

Ehnrtorr trdri ro olt'oio (ii,y) n€adx.tor ollv nosLd hL(i rdv (2r8) xd| (a,7), dro'

4lE

4.19

4.20 :r ovaloyi( rLr ritv rQotlourlrut .iorrtort, roL( Ei\,or I €!io(')ort notj rxaloriorFito ( r . .y . . ) { t ror r raLdx€tnr qro l r i r rbo 6rc i to ! (2,0,0) , (0,2.0) xaL (0,0.4) : To!'irl,a (wLllrrlxiiEra, F€ 6vav nivoxa 4 .ni 4.

4.21 [ lo , ( oei io lgd ! ' ] r ' f ! i ierar . 6r( lv rc d , . iu ( . r . v . : ) , (2.1,0)xar ( l . l . l )n€qr txo\ . Ioro€ €ni rsbo 6 leqxo$ro d6 qv sqx4i Etv( rc 6LsvnoForc (1,0, l ) . (2 .1,0) , (1,1,1)!Euetlt!6!(l 1 avlEdeqrci

4.22 Eiiv a(io. yedufri ror I rFQrir(rL i 6va +l I irc 1 tj dro cn'to xa0lvo m!! (xc(1( uno),ornd .'torxff( ro! eildr I'lr'iv), 6ciEre 6rL dcr.1 = t t -l i 0.

a . 2 J E r i v ( - [ " ; l r u ' , = t ; . - ] . ' o r r ' l a D - D . m n ! A o r r o t l . { . 1 4 y i v r u ,

CD + D(' =o i l

l " e I ld c t l 2 8 l l = 0 .

L,r 7 |

() Be.iff trtv oailoru( dutou ror 4 .ni 4 flva"o -4.6) AFiETF orr der ,4 =0 oe 6no f tq" i r ( i )oFrsrd +.1=0 ! l l id tc =0.

> 6).!s rLE Zr).].€s nsarnr6osrs, C, = - /)C EiloL 6uv(tov F6vov 6tor D = 0.

a.2a EEt}.'im€ yrdri t 1 cni ,l oetiol'o(r vandemonde

i; "i'":, ;llll lil

l l 1 , b r , rl 1 . d d l 'L l " " r

, i l

* :.r.E.: E:!r@o'ldur or i7.o\r. iia(?( tr x n.

:ror pDno:FoL rD^13 Dlou (u ot 9!,.D tr^31i9lDl3) rDx d :o3o?gDt3i srxD^l,l 5ol1orcF D om^F boll (t "e 'Z) oro (u " rZrI) or r319oDt3ri lror:!g,,p$li !rxq{o1r H

'(r - cxq - rXD - txq - x)(o - )')(E - q) ='A

tr |?urqxodll '0I Z t Lrol.o.ov ̂tr.lgultr so oor Er ol?dDrdno ̂ot 5Dt q)]orodoL'dx, = r nx q = r'o = t n o t$?wllr^F odn goyo], 01grnD FD 'mx r sodr tu onn^qnlor,91 gnti rr^F D^ r3r!?0,r

|dit : HinH\ cNvlra vLI <IgStDI<v

IAIOTIME> KAIIAIOAIANY:MATA

s.r r ElrArg)rH

ALiro ro xrrfl(),cro €yxc{v(iqsr ro dcurrpo proo" rrlE 0€{,)Qicq nLvdxov. To:rprirro pdpog 0c),r,ro!o€ ol66ov (lnod.eroltxci 1(r lQcr[4u,r(i ouorlpcrru A.r' = |z(r I 0!tr!Ld)blE r€nLxi rlr(v I (rncioL(ti{. Ar[6 rdQo xC(! oro t'Ei( or rt'cileq 0(r:rct(ow dv(v rlooovcr q6),o. Tn !d( rrAo6l.rlfcrc Ocr ),rrvovrar, :rld.L pe aal.o;roi.r1o1rof ri.l,(n( -li(ivovrarq rov br(./livro rl cilo rQry(Nrxi> ul.J.iLto 9tp (b&q 6i!t1ntr |u ri.ior to 1" Qq cu2oriTt h,n ttdint .Tou; yurq yaevtiE un(j !rcfl a ,tt.Ar:'t !trq evbL(qdper r),io! \r(r 6rcrltQoutrr rov Xcirao 1pa$$6\, 11,6S niv(1i(. c],Ill!( brrtaoupr rrq 1610r+16< ror. oL onoirq gr'r(6drll.oar.rL zcrL rn! (nal.orqrl.

To z€q.il..lro yL( trE ooi!oro{ (lJror€isi. or1)v ;rp(yp(lrLrorrpcr. pcrh6aorl,(no ro rlr.Lo rpo6),I[.r ,,1.r] = ir, oro v6o rip65].ryo rrrJ.,' r6Lorrldv. K&r otr.Ebr')o nta|nrdro€rE, 1 oqilouc o611t:i or: B|(1 .too[d.|,orrrt ].1ror": ro! Lc.v6.vc1rori cLrnler yLir ro j = '4 rb zar ro ro).u6wtlo det(.4 )1), rorl oroio! o' QiEr<rivcr oL 'bror+dq. (Tovi(oqre 6rL oLor oL rirarEE ri\,aL rdaa rr:roct(Dvrkoi. OrtbLorrpiq n.lpcdtrtL6yqcpiror ;rir(n(. 6atDg ncrL 4 opi.iouJ( E!6< rirorou, 6evalorl! !on$.r.) 'Onorq nfivra. I opi.qouo.r prioQri rphy$.rrL !a Xplo+onoLl0€iortl Luo4 :rqo6),1ytrroq , 6my n =2 'i n = 3. fL( stlhicr rr, o !:ro),oyroF6Er(Dv rbror+(iir ei.vtrr nro 7povo66po zcL b!oro),o EylEiQl[rcr (1:r'6,11 11 ),ljor] rou-,1.r' = D. trdtrLora x(lr o i,6ro; o causs b€v 601611nol. . Al,),(i nsai ouroi &py6r€a(1.

To :rqrixo 61pcr sivdL v& x.rrdl..!6o1J[r -rL €ivor oL rbrorrFsE x.rr ./(tri €ivcLXprio+r8E. Mi(l cno rq €r1relo1i< rolj(. x.rt lrdl,Loru exeivl ge rqr onoto f]c rr<€Loa./d./on$€. €i1,ar I ).uol o1j$10o)v SLoq,oQr"xd)v rerodo€(D!. A€! 0.r 1liro06ooqrEx.r6oio! 6rL o crvclvtiorlq €i..var rl!0€!ri(t orK 6rcqoprxd_( elLotiroeLqt Edvfrnoa€ir€ va naQoyuJyior-r€ rL_( otlvlero$ilaE ouvnprloer"g, 6nc,rE or r,,. sinixor .i', yvopiq€r€ cpr!-rir. lcrv eLbrzo anpdtur..4rcr oer,.rprjore ro !en1o_( roj,v

5.I EEAIATH 285

orJl€lylrdvo! €Ero6orov

: 4 f - 5 i . ' . r : 8 y t a l = 0 .

= 2 r - 3 t ) , r { r = 5 y r a l = 0 .

du= j r l . x = l l o y t u / = 0 . (21

du= tlrr Yt(r I = u.

dx

1it

( l )

A116 rircrL rvcr rp66lqgo c.Qxrx6v r!tl6v. O! cqvooxor xa€oQflovr(rt rl xQovlxrlmryull f = 0 xcr 6l! orc 6io d,ecr xaroLou lpovrror! brcorrlporoE, Eror0!6roq€Q6paor€ '/!o ILC p€rcr6orlrri xaL 6x! Flrl orrior.pq xnrrimnoq. To orioqpcrEsxLvo un6 rlg aplLxdg T

4rdE 8 xcr 5 rcr! ro :rp66l,4rc eiv(.r, rc rr€QryqqlioripEqv €E€l.!€ri rou.

To ouorqpc 6lururt|rtrol €r.rxol,s FE rlvoxEg, Eorol 6fl rr €ivd! ro 6ld!!opcT(r)1' .rYv{i}Oft)v, u0 I qQXLXn 10! T4Ln &0r / o nftdxct rcDv orjvrel€otdll

-r 'rr) l -r l . o=11 : lu(r ' =

ld i / ) ] ^ = is t L - - r l

M' (ur6v rov oqr6ol!op6, ro oriorlpcr ;rcrfpveL popxpri 6rcwopcrrLnr]E efiooorg:

Auoi Ei,1(1! I 6€fE7'rc66lg or,cfir.rnaroq rou rpo6lnForoE, ln|,l€L6ore 6tr nQ6zerrcLlLu cEioorol fiQdrrnE rdEEc)E -6w orppertlow r<rQc'rcuyor, cvc,rrdQaE rdEecu6-xtr p<ilr,orcr yQaptai ,oE:rpog roug nlvriroroq. Eni nldov, 1 rlioooq rptoru1tpoit< otw ,carlE: o ntvcucg ,4 6ev elnptdxdr crri'iov tQ6vo.

flri6 6pioxoupe r1)"ioq; Eciv un{QIE ivdE p6vov ilyv(,)otoE, c!'ii ya 6uo. rl(rjtdvqol ofllv eQotqon 0c nro! urxoln. O(r €ilo!€ ror€ Frd epr€FrFlxri, ovrLyLc! brclvloudrLr{i A'.aqoQLl{yl eEiorDoyl. Ed.! eitar olroy€vrlE xrt pe mnoepokowrcl,cmiE. 11 p<iv1 pop*9ri :rou p:ropri la 6Xtr eivot

(3)

H )'uol tn! €i\,(!! ro F6\,o nou XQ€ldlfiar vs EtQer€:

11(t) =culr tu (4)

Tlv cpXtrri lpovrxri mrpri a = 0, I u roolrlc! [E zo 6rorL e0 = l. H ]t(Qdyrryogrqc tot eler rov cncLrouprevo ficlQd'/ovtc o, r:,or r-lor.e d.u/dt = au. Eaolircvoror€iraL r(1r rl aplrxt our0ixl ricr! I €Eioooq.

l I lO l ln tF : K / \ l lA lOAlANYt l t lATA

llgerd)ote n1 o4raeptqopir qg u l|cl tr€Y.llouq la6vouE. H €EiotDol €ivot

cor(roylg 6t(v o > 0, o!66rsQc. €uorcrO{g or(rv a =0 ncrr euotclollE ora! d < 0'

611l<161 11 ),uo1 teirct oro dnerpo, :ropcp6ver qpanrdvrl rl reivel oro 9166vEir' .; ; tlr"" Flya6tll6g daL6p6s, a = a + i!1. 1610 trr i6ra 'QL1{0!o ga

el crppolovruv oro rpoyFortt4o UtQoE rr ' To rrYobrxo lrFpoE JroQCiYtr r(Lt!'rdr-ocrE, , .' = (o,.1r L sin .t/. (l.Lu I ruoraUrra rogoQit€r..r 0:16 to! truQdlowo e ' .

Ar"rcr, 6oov (rpopd oe pla prpovop6q €Eio(.)orl. oc( ttQol(,Qrlooutr€ roacr

zcr' €u0cio\, mn oumriFatcr xal 0(. (rva!+{oou[€ )'uo€tg fiou 6xow rlv /dra

ueYdti idarqot ai6 to I tt'(.l ria nou l6).K 6Qlzotre'fts trlv sQtottltlxrl

r( t ) = rr tY rs\rr ' ( t ) = r^t : '

i or dL(lllJol((lrrxYl btatlJltooYl

Auro €ia(r o),'j1l),ruo 1() ,!€rbi lt(r rLE bluqoQtr& €Ero(;o€Ls du/dt = Au:

Avc,lqrtiarrt ttt n@$q(ft ex'''txii lio€rs. At'rtxc0tm6!'r(l! orYlv €Eioool tK

t = r^ty uan u =(^r: .6Qioxoqr€

)er ' l / = 4t^r i l 5e^t:

^(,^t - = zf^t u,3e^t z.

O ficQ.ilorv trt t:ivcr xorv6E o'<llouE toug 6poug xrrl pnopaL vd rcrQrrlettp0€i

Awri r1 a]rcrLoqrl eircrr, nct o ) 61oq ̂1Lc tor o;roio uno0aocF€ 6rL o! 6uo dYloolor

€Xouv rov i6rc !v 06rq .\. A'roFavE! lom6v to

4 ! - 5 . = ^ ! t 0 \2 Y - 3 : = \ z '

:rou rivtll 1 6aorzrl eliooorl xat 6trrrlnol€vl pe ri.vczeE 61et fll ltooql 49 =

^J. TIv rEioooyl curi E(1'(t6aiozrrt orav /elorp onorrlor 1r o'{lvuotroilTYl }'uorl. r - , \ ' . r -er(v upr0yo , \ ' : rou auEdvFr n qHivfr I ni dvo or{rHfQo btdluol i ( r '

H a v r t x a r d a r l o r t q 5 u = , r ' o t n v l / ! l t = A u b r v t ) e ^ " r = A t ' r i ' x a t 4utalorEi fiaQdyEt 14!

(8)

TdQs €XouFe ltl 04r€)t66n €Eioooil arroi rolj ,a€qotroiou. nou ]I€Qrlil€i€t

61lo alvriroror5. to ) xnt ro r, xor €zqQi(q€t ivd c),Y€6arx6 ,rQ66)'lpd. MtroQoqr€

(6)

i L F l lAro [ l l :87

1,(. Erxdool,l(E rr< 6((fio0rz€( €Ero(iro€rg! O (Qr0[6q ) iiyercr r6rorrtrrilou riv(r).c/ z(r ro brcivuoF0.r €iva! ro (aiiororlo ,Jto6uiLr!6pa. &oXog lruE eilcl| rcr60oitr€ z(lr acl XploLForroLrloou!€ rrE (blorlBig rcfl. rcr rbLo6L(r'vrrolc1x(1.

Or Ario€rg qg Alx =.\c

:It'€6mr orr I Ar = L.r eivcrL prl .lpuypurl rliooorl to ) :ro),1.r:r),coLh!erro.r. E(ia trioaoi,octr€ vcr arczn),rn1n11e ro ,\. r6te 4 rliooo4 0a rlrrrr 7pulltzrlt,.r; :rpo: .r. Ilpciypurt. 16r€ 0c ya(iq utrr t/.r'. uui TLcr )J .' 2(r 0( qial(ltr€,rtroy roy opo ont! (lerorrai ir).rltgd:

To X).!''bi y|tl rI l.nol ro! rao6l{f(roc €iv('r ro €EtCl

To ltd.tltotttr'r' n€Qt€xetr orov tut'E,fliy.aao @! A - ^I

o qtEt6€ ^ Eivat irotos d'aff o A ^I ru lIEt pqtlvdx.l.l?o,

Bid(Lo. zci0€ xivcz(( 61fl Fllb€aorDpo. Eiv(r (or€io !( jroi,t[ zdrr b(r{oarrrzo.7(1rci(6ci!rrr otr{')q n !-!!oontrr'. ElvoonF€ r/r t,/rr'r,rzui rbrobrciluotr( r. To6r(ivrotr( .f = 0 rzclorTorti :rrilrore rrl ,lr = A.r zcr:rtprtlTtrrrL ;rcivrort orop16r'v<iprpo. 1rr rr1 ),ro1 opog 6rutloprrrirv rlro60€(D\, €h'cL tllQt(]ro. :r61oq [Clqrilcr v(l orlwl(4rolot4ooutr€ ril ),!oil r(/) Xolo+rorordw(< €?io€rrxiq.lr.r. rcrpc< evbrcrrlipouv 1rr-ilrrl utit'e g ot rr[ul< 4ti< ror A, y n tq oroitq unip'tt tur74 p46ruxti u\ux\uiruaprr.ri. fra lu ixrr lortota letlolporllr( o tqbrloloQoEtor :l )1. irear& vo fleQrdlcr 6r"urropuru b!(trfoQExrzll rolr llrl6r!i)(. Mt liy(IoTr'r. ,' I \/ netttt ru tivat t6ti1toet1li,|.

H oui: , r .uu oiv ' lvd (rrorf ; jd lrr f lzu,rprrt lso ?r dr"ro.

5A O (a!o[o! l eiv[(r r6lorqrll rou A 6rav rctr p6vov orov rollJ€!

(())

Auq Eiv(! I XaQarqQrm[rli €Eio(Dor] xaL o€ r{d6s }.rjol rlg ) crwrororXei6\'.l rblo8rdvuqLa ,c:

d e ( , 4 - . \ 1 ) = 0 .

(,4 - )1)r =0 (11 )

* \!qri:{(' d rduorrzo. rit("(. t,'( r,r iilrr(r d(/.oarol6i xtr(zd. l(Al\[(iror(oL l11 ' ( ; \ . H { : i , \dor t l , l , \ )x = 0 e i } (L .Nro t r ( i i fo l t . r r i . i . i !n f ld r i r l< i ! \o r r i .

( '1 - )I)r = 0. (e)

IAIO] IME: KAI IAIOAIANY:I\IAI'A

:1o n(lQd6er$d FoE, tr€1fior qow(1E rov ,4 xclrd )I rrQoxifi€L q

To 6€ut€oo Laro6rirvuopc €fvco oJroro6inor€ nollftrldolo rou

,-^r=[o;^ _il^])ilperrbore 6rr" ro,\ crrpcrlpeircr p6vov ar6 qv xtpro 6Lcyrirvro (6r"6n roLLcnlct-ourlr t rov ./.1. H opilouoc rou ,4 .\1 rivu.

(a ) ) ( -3 -A)+10 r t ) ' : - ) 2 .

Avr6 eivaL ro "llapaxt4prmvi zd.uituyo". OL pileg rou.'f!o tK onof€( Iopi(ouou yrlltrvilrrur. rivcL or LbtorrgiE xur rpoximow urri rov yrvrxo riro1rc rLE pileE te-rpuyuvLxrlE r[iooo1< { crn6 fl1l ]id(xryovronoLryI X - ) - 2 =(,\ + lxl - 2). nou Fl6€!i €rar 6rcv ) = -l i ) = 2, 6mr6 ear6e6nLriver aC[ olcviltoq TUrOq:

, -b +,'E anc | + '/9"= 2 "

= Z

= t , l t .

Y:r<ipxouv bto l6ronp6g,6r6rL xcr0e rerqclnrvr,xtj eltoclrol dler 6uo ('lleg. Kdee2 tat 2 xi,rctxug '{ - )I {X€r orlv oei(ouori rou to )r (c}.},ir 6Xr pelci,ureqrl6uvcpri ro!). Kd0r piq ar' cw6g rr.E er6Lxdg rqrdE, ) = -l xar ) = 2, o6rry.i o€pr.c Luol ror ,4.r: = )r 11 (,4 - l1)o = 0.

'Evo< ni.v(rc(< I€ frlbrl,lri oeilolto(l ri1,c! [6roloQqoE r{o! enoF6vo! opq6€v6Xc')QG rou n€Q!€X€r 6v( Frl pqaevLzd 6rirvuoFa+ r. :qv rQ(yFc11!'a dnF.lo prlbev6Xopog repdler pLcr o1L6rl,rp1 eu0efc qn6 lbro6rovuogot.u trirrr! dlaEunox(oQoil

\= r : ( .a-^ /) .r= ls -sl l r / l - r ' r lL2 -2 l l : l - lo l '

H i,uol (ro nad)ro rbro6(ivuota) Eilar onolobrinor€ nollo/rldoro ro!

f1 l" '= Lrl '

O ljno),oyropoE lr(l ,ro ,\2 liv€xc! E€l(,)Emd'

(.4 - ),1)r = 13 :] l:] = [l]]

r ' = l : 1 .

x Etd. ).ilovr(q rc (A ,\./)f, =0, ohwllr.ift oro r = 0, r&F I ) bLv.i!(L Ldorfri.

Z4p&oary not woloytolld Ldlo'ntuopatuv >qv n€airir(|}orl 2 efiL 2 xu or6uo lQ(ppar rou ,4 )1 0G €il'al no).),anl.iroro 1or r6iou alclvrropcroE (.i,6) xatxo L6r"obr.ivuo1tc eivar r6re onorolrlnote nol]"sr doro rou (-b,a). OL 1pr{rB€qror ,4 ):1 nrcv (2, 5) rar ro rbrobldvlopo rtrov (5, 2). :qv r€airroot 3 €ni3, 6rrcu orlvci srcr olvrmd;oc tor z iorl pt: 6vn nrt l6vo ro (,4 )1)z = 0 tl;q npogrLg ir),l,eg ouvr.or6oeq. aD!o!xd, orc! ro r eivcu" r"6ro6rdrvuopc, ro i6Lo oup6civrLrcrr [€ ro 7r r.i({e6E 1tc[ ps 1() -;r. 'O),o ra 6ravuopcrcr rou pl6ev61opo! totr.4 ,\I (ror ro! ovopdq,oul€ I'ktIDAo) 0u lxavonotouv qv Ar = ll'. :xorcrpdber.Ard [crq or drox(Dpor eiv.A or €uo€i€g 6Id to]v

"r = (1, 1) l,iar n2 = (5) 2).

fler.,' €floxQiuo!$0 orrtv eqaggoyl gcE (q 6Laqoprtirl EEioqrol) urioyQoppi.-!,oulre iG 6llpolo ylc rllv eritrriol xou n0o6uForo( 16!01416',,:

l. Ynol,6ytaE rre oQitouaa Nor A - ^I. Or.d't (,qdeee€i 10 ) an6 ro 6!oy6![aoxoL1t:ic. I oQiqo]Joo (rur1 eivar dvc noluivu$o 6d6[ou n.

2, BaEs NE ai+g awon Nou floltoteit o1r. Ot n piEsq €iv(rr o! r6loxrpdE,3. rn xdqt ttis t6to1.t 4, t,iot to oiaqpa (A - .\I)r = 0. E Isl6ri I oailoljoc

€i.\ro! [lbiv, lridexo!! ],uo€!q 6ragogetudg tlg .r = 0. Arxig etvnL tcr!brcb[Clv!opcx(1.

:rrE nrr{ toorxE( rErooro€ls. cr! . rd xoQcyrr rrc $6rxi< } .unf lE r - e\ ' . r . nou eLrcuot l.eecOd €xe*uit L1io€q

5I EDATO|H

, ="'' ',t, =" 'll] l1ctL . =,,r, ' r , = ",t [ ! ]

.

u = cte l ' l , r r + r , : r ,At t l : (12 )

f r s l f . , l f8 lIr zl l . . l

= lsl

(13)

Or. orne epes ei.v.rl. cL = 3 xu q = 1 xn'" \ ),iorl rq< aexdti€ EEioaott€ tirdr

Mril.ror( or 6uo audg eL6rxdg Lljo€rg. noLtrarl,soloto[€v€q pe 6uo orq0ee6E.ilr{(r .l x.r! nQoorro4rE!€g, o6lyori\,firI yevLzrl )'.io1. Orcv 6tio owcrpttloeq u]rcr! r.r2 lxovofioLouv tq yeappu.ni €liooorl du/dl =,4u. to i6Lo o![r6oi\rs! ,tc! p€rn! Lr + Lt. :w€fidrq, riiroe olvbrcol6g

eiro! fiaLL Luoq. To qalvoB€vo au6 ].ayerar u#aqEol xur €9(rQFo(€tor xoooo€ 6nqoQrmq €ELorilo€rg (opoyeveig ypnlprzdg) 600 xor o€ oLye6Qrx6q EELo6o€rE,,1;| = 0. O lll]6ev6xDaog !:ivor nlil,ror€ iv(E 1jtr6t0reo6 lar or o]JlduaoFoi }'ljo€(,)!€ivo! xioL! LuoerE,

'ElouBs €r(JL 6io s).Eio€e€E rnpcpnpolg (\ nar c2 keL €i','oL ex),o./o va:rrprprvoqrr orr p:rogouUr vo r ' : r r l . r :Eou[r ' r rg rrpiE rour i ror -orF l rr rxdvonorf i ral

I aaXLXrt ou\'otrl r = zo yra i = 0:

,,(,) =." ' [i] . "" [;] (14)

IAIOTII\'IE: KAI IAIO IANY:I\IATA

Ipirrlorrcrg rL_q 6uo orvr"ortboeE lc7,opr"mci. or,to olpcillx 6rl

, , ( 1 )=3 . ' +sc r r a(t ) =3e | +2e11

' Evc n(a(b€rytrd nor aore bE! 0( nior.!(. Otrtui. rpdl|ldl ro 1831 z(epaEloE trd liqra(

Eux ola €raUoEuovror zcrL or apllnic ouveizes r'0 = 8 r(lr 'ui0 = 5.'o)'. orr(i r{,ci1,€xcr[ v(r $crQftpolv fi(,)( ro z].€16i. rl< sEio()orl< fir,(1r oL L6ro-

rrBiE z(lr r(l L6ro6r(vrwF.lr(i rIE. Aur6 otr{Dc rror 6€v qafvrrar oro :rcrpdbeL.4ra0ivcrr I qruor;ari olltraoia crr6a rrDv nQaylfircDv, no! Eivor oltravrrdr rao €al,'[crzar olr ffil.o'r-( oav [600! €vos r€trdolaroc yro rlv €na€ol rlc u. EvbE1otr6atl)cro nLo orr{€io rrciQd6€r1[(i* eiv(rL (1!ro rorv or0crnr-rdv rolj ]r€ovouv llr"c yiqua(r.:lJvloi,kror va oi.rlflronv ro 6l[crLotro ]iar !a.rlv n€aaonv n€ariardvr(1,(. O).oyoc,,vt l , or,r : roqr i v ' r o '1ror i vrr utport !ouv,rc o,ryvorlru rot naoc tr .LrttDv rbLor4rorv qg 16qupcrg. I orroicr r6re 0u crpxileL !o rc),c11clv€rclL. (Oncuqazpr6rb; pru ncrbrztl zotvtc. fptlloptr ltverrt cr!r),Int1i I qltorzri r$ o1Jf,6qr((rcr (ro(lw(l( l! ou\,rolrotra ,'(. oral,lrrrlrll! koille lnll,or€Q(.)'Eve< pIX(vLXoqnQoor(Or,i v( 1tt)(lritorL rrc q,!oLriq cr1jtv6rrtrr-( -rr1;

liqnrpir-c rou ri roU arp(nilorJror tL(ra!(i cri6 aljTdg tor cvdpor 11 toug z0o6c1otroiE tlE luUovrlq. An6 q!(i)").1 ntrFro(i, 6vcr( Ioll.1(lrrorli: Eobslrsr q !0)i ,ro! nQoonce(i)vr(q vcl o!vro!r-oeF.t lr€ rrq (fuorzdc o!Z!6-rIr€E r1l-( (lyoodE. O! lbror+raE €iror 1C) rlo o{fl(!flro7, rqrrrzr4grnrrzo nTcltov rr i r t i r 'vr{Frrou ou.t l i lnto;

n€air.$It ror noAo6€iyFoio

)rr4r(rout{€ r6Qcr €bdJ wr','a o1l!o!,ioorr[e auril rror ltrv(v ha|va bontrlrr (rotrlvrL \ra yiver. Am)i it t'o(./(j)"/ri qcvi0(UoF rov q,tlor]{o )i(' (lriro!(lrorp6no pe rov o:roio epq(vi(owar or Lbrorqrig xcr rcr rbLo6Lcviopcrtc rou A,circv ),i,votpt rr1'r rlulnt' = ,,lrr. Tirorcq €grir{i)orq ator,! xa1eaA u1nrxiEtrineq r = r^t.r xa oL r6LorLp6< 6ivow rov p!0p6 ({Eqqq ri c,-roa6eotq. €\ri) ro6l.ivu(r$.r x EqEiioorl(r tL'(ur6r, ro! QU0[6. Or rro],oLrE< ),no€Lq Ei.\,ar onloEoilrdroor xc6apd;v Luoeov xcrL r1 onvoeol rc0opilercr iror 6ote vcr rzcvo:roreirtE trpxrxtg ou04nr;.

H eEioool-xkL6i eivcu. 11 n.r' =,\e. Tc reproo6repcr r\rcvropcrcr 6* r.xcr-vonoloiv llru rd-ror"tl u[i.oroor1. 'Evcr rr;rLzo r], orov fioll.an].oor"dl€rur 0€ ro!-{. c),l"illrr 6€!0u!ol }1cr o!v€no); ro AJ] 6€\, €i,'(L fiol"l,.fil"doLo -ror J]. :u!cr/,o1.orol]. p6wv oAnpivot tt,uoi uQrqpoi ̂ xiru t'torttis xdr pti|o! oAnthutdua 'mrirapKa:t: tiler t6to6tl,JiopdN(. ouorltd. Edv o A {r./! no}.),anidororou r.ltrorrron. rorE r(116r,(1 bLd\rrota 6€v 6 irl).atE rt1r€!6uvol 1.'(' o).o raSr.crvroF.lrcr 0(t rlrav [6ro6LcrlrioF(xcr. :!v{otl)q oFrDq r.r r5Lo6|C!!oFCr( €ilaLli./a x.lL cnoFol,(DFsvr1. Aurd 'raarorilvorv rr.q "7.(vovrr€q xaraoriroEr.E" rou

i I Et :ArorH 291

oumiFsro< xs! €rr|6eoriv crve[riprrpcr ro 6vcr an6 r' d].].o. Mr[oQor,lrE v(r rlqQc(-'iol.ou0iooulr€ irl oup r€gqoqti xri0c r,6ro6rcrviopcrrog zcL vcr ow6uriooqr cr,r6qrrc xavovLxdE zqtcordorLg yrc vc 6potige pru Luot. To i6io nad/la e,iqgd!€rolzcL pe 6vcrv dl,tro rp6no: o avriaroLytg triwxaE ptropei t'a dnyut'untoq1ei.

Ile6Ldloqre v' cawpdroorlp€ ilv nqQdfaoqo 5.2 orl eEcDQt.r 6royolto?roiq-oIS 1lcrt llg err6F€lrg ncpaypdrpor,5 mq eqcp4roy€g tlE. nQ(blc. otK tErodlo€lsbr(qoQ6v. oiot\ cAr0llol\ rou Fibonncci xct mtE av€LtE€lE tou Markov, l.tot xc-r6nr! orra irtupoQLrriE €Ero6o€r(. :€ zdg€ n(rQdrbErylo. E€xrvciB€ uro),oyilowas1rg rbror4l6s r(|L rc [6ro6r(l!i,oForo. Asv l'ndQx€! povondfl lKr vs n(rQ((l{duv€rzovrig crvrA r( r[QaTlaro. Kfiojrlv 6p('8, ra rlaQ(r8€illrfia obrwoiv o€ 1600no),I€q r.(r€u{\i!o€K. no|J Eival o6uv(.rov va 66ooqrE ou!,rotr€E ,r€Q0}'{U€r5.MnoQooll€ ollo,E v(. 6(i)ooup€ €[qso4 oro 6rr o! ou$r€rQrzoi rlvcr€E €i,v(! eEcr-Q€Tlxq €urolo!. ev6 oEop€vor, tii.),or, "rtrcTT()UcTLzoi rivqri€(" €ivqL €EaLQ€rud6uozol.o[, O! r€treuaior oreqowrar ev6g n],r1qouE ou6lou r6ro6rcvloprirurv, 6ev€lv(lr 6rqforvLono iorl.lor xal o6lyo v o€ r{ordpeeuol rlF telllxrlE xo! x(!rour.r6v)rcrroorilor0J!, :iyolQa fiQdn€r culoi 1,( ouEnfllooi!. AEv rouE €.flrQdjror4r€6po4 vo zcrol,ri6our ro f€y<rl,irEQo lriQoE xou 6r6),io1J.

E€xwillr€ tt€ Jr(Qob€iypcrrc rbr([r€Q( x(Io)v firvcx(,)v,

nAPAAEIIMA I To ndl'ts €ivcu rQotp(vrl. 6r<rv o rt cfrcu 6rayrivrog

:s rd00 r6lo6rdvlops o A €n!6Qa /roQotlolo Fe rov rorJror!)i6: -4r1 = 31' ,ot,4r: = 2r'.. AIX( 6!(1vioF(ra, orrrdE ro li = (1,5), €i.vo! olJvbxqolol. J]l + 5jrr rovEro rbLobr(vuol.rirxov xaL 6rcv o ,,1 e:rL6pd oro :r:. 6iveL

Arnd rircrv 6vc rurtxo 6rcivuopn:r xa 6Xt 6ro6rcivvoycr omdoo r1 6pcioq rou,4 :rpoo6Loprl6tcr :rril.r c:16 rc drobrcrvtiopcttc ra! nE r6rorLFis rou.

IIAPAAETTMA 2 Kal.rj eniolq eivcr. xct itn€QiTir(|lon $LaE rpotox'ii

!] ur" ^, =, u. ,, = 1,1] ^,='*,",=[l]

l-11

,=l;

a" = ,1,r, * s.l'r' = [r3]

.

L L 1 1 I 1 lP- i i l i x f , \ , r r r f r , = l i l . ^ . OFf - r ,

L t l l t ' t

or rdror4ric putE 4o6otris dvat 1 xar 0! Elortl€ l = 1, 6tav to bLrivlropntrQ,od(.ll€t(rt oro\, €(urd tou zcr ,\ = 0, 6rcrv aQo6dtrLErar mo Fq6€vrx6 6rdvuoFc.O X6Qoq fiIL6v rou P y€FiE€! [€ !6ro6krviopcrrct, xo 1610 €niol! ow6qtvE! zsr

I.\IOTIIUE: KAI IAIOCI ANY:I\IATA

F€ rov Ftb€vo7,crao. Eirv ouioi ot !fioxopor.6loriv br"cimaol r xt{r n r. 1o1€ 1l,\ = 1 rncrc),c4r6rivercrL r-rpopiq zst rt ) =0 €ravc ctr6irv€rat n r rpopig;

€ X € r , \ = L 1 . 0 . 0 .

t - . t 4 5o J-) 60 0 +-)

f l 0 0 0 ll0 0 0 0 ll0 0 0 0 lL0 0 0 l t

olu'\, o .iD €i,vo! 4 €fii 4. l]]Idpxol,.v ncilr r6oo€arE [6tot4r6E. o! 1t ct 6€v €i.v(rr 6)'€qSLarlopsttzig pttrr!u rorlq.

:q!€[(ioi€ 6r[ I ) = 0 6tv tfuar notE ro EEaerrrrd. 'o]rorE zct xo0t o].tro(oareB6E, ro i(I6av trnoQrli v(l €ilor rarorLlrt rl $ro0Ei xtll $l! siv(tt. Edl Fiv(ll,roTrl 1(r lbLobRlvroF(rra rlq r]trlvonororrv try ,4.r = 0r:, ETo[ 1() ,' rrolix€r(ll oro

$I6w6X0)Qo ro! ,,1. Mtbtrrunt rSro-r4rt olltroi,v!:r (irt o i 6XFr €Ecaqttiv€( or{}.sqxar 1pr41L6; rcr r1 opiloroit ro! €i"'rct l(16(v. OI (vrrorairlJLttot rivax€q axouv6tc: rrg ) 10. €\,d o[ lbr6poQq,or rivarie,( dxot'v ro Fl6€v (lvc(p€oo ottg tb!or[[a(

IIAPA^EIrMA 3 Or r6rorqdg etrcrt ea|orlE (,ovrQdE 6r(v o -4 €iv(r rAryavu69.

dc(,4 - )1) = =o ))(1 ̂)(j-^)I-I opilourr<r rtv(L ozptdriE ro lv6pevo rov 6rulovio)a ororxeio! 1L(! [nb8!iq€rcl!,ora! ) = 1,) = i ,i ) = j. Or r6LorrpdE trorror 6piorovtcrv 11611 orrlv xnpLu

To:rapd6ernru cu6. oro onoto oL LbLor4tdq ('vcrlvopi(ovi(r pt- trLtl p(fld1.!:ro6erzvueL dvcr xuEo 06pa (mor xo! r{Eqa},aio!: Ntl $sraoxlllrcri,oorue rov,t ot dvrrv 6LclivLo ri tqr.7r,rv.zo 'riv(rc y,ropic v'trlt'riEorpe rls L6!or.F€E tou.fr( [La .rzo[l qoad rovi.qoqlt 6( I roQcyovro*oiqoq Gauss A = IU 6ev eivc{n.rrriLLrt),I "/f crt,'rl q 6or),t:th. IInq 6l,o nor oL L6r.ot4rd< tot U tivcr. q(v6p6Eoq\' 6ft./d)!m. (ur€g 6rv rivar oL rbLorrlig ro! ,1.

fr.i rouE r€aroo6r€aoug nivcxeg, ro :rpo6),tpct L6!ox+d,' ei\raL, lt,rQi.q rtlqr-6o),ia. lro).o1rmu{d broxol6rcpo cr6 to ,4.r = b. H cltpL6rtq }.uol yaatrtrtrdrvouorl[cirov nAoixuuE t€ dlo netiEQ.ro[6vo d.rtooq 6ruitrtDv 1t clt o€ n€nta(lolr€vo1p6vo. ('H. roo6rirc4ra Stnrr;rt,4r6r'o. o z(v6vaq ro! Crmer €dt,)o€ avav orQr6lrxno yru xrtv $' Lolll) ),ljot.) :qv r€airrrDol rtov r6Lorrpdv 6t:v gnopti valrlrdpzou\, orli€ xdroK, 6ilara orre r6roroq runoq, 6rcrpoperLxti 0o tQitclle rox6zoLo rou calois orov rtirpo rou. To lapcrztrlprmrno ;roLutirvulo evoq jrivozc 5€ri 5 €i,'cu 6\,(r r€tuixo6ri0$Lo ro),ljdvwo raL au6s cr66erEe 611 6r:r proQei vorn('tQz€L (1ly€6Qrzog riroE y.cr flq pileq €169 r€rorou roLulvtipor. To F6vo rotl

I E €nrra€r€r (o calois) eivcL $eprz6,< ci],fq enu].l0€!o€r!, [srd ro! rro],oytoBottDv L6ror+tbv. Avaqiaoq: blo tr:r' ou6g:

5.1 t [ArQf l l 293

5B To .iapoopd rc,)'\, n r8lorq.r6v |oouror tr€ ro d0porogo rov n 6royar-

(15 )

To doQoro$s q!'r6 €i.vqr pcoor6 o4 iyrog rou ,4. Eri :t1,6ov to yrvtipevo torvz l6ror4rdv r.oorncr ge q\ oeitouod r.or A.

o ni\ruxcr rrQo6oLi( P eile brnTtlvr.c oroLt€ic +,j acr L6rorqreq 1,0 xol ro

I + I o![niarcL $E ro 1 + 0. 6rc'6 (rvr4rdverar. To ittro oul6ai€I ]tar [€ -ri]v

oQiLouoa. nou €ivo[ 1 0 = 0. Bl,irorLpe rri].1 6tr 6vcrg t6r6[op'qog nivctxog 1tepl6r:lolri oQiloloo d1n $ic 11 :reqroo6repeg i6rot+r6g tou ioeg Be BIb€v.

A€v rrQdn€r va lirsrar ol,1,Irol tl€r(rEr.r 6rdyrDlicov oro(X€iov xo! [6[orqrdv.xr dvcv rpryovLx6 nivcac cwir oqr;rtn-rouu, ol,ld rouro €ivo( €EaiQ€on. :uvrl0oEo! oblyoi, rc 6!c./6.,'[c olorxefo rior o! r6[oxqrlE €ivor r€],€toS 6rarlopetrxd xcr1,[c( avav rii,r,cxo Z erf 2 ro .1vogiloupe 6lct

l o 6 lo [ "

)] txel ix"oc o + '1, xcl ooiLollocr ad - l'c

. f , r - . \ , I . 'de r | , , l = r - - ( i xuos) ) - "u i ! c , r ,o , rn ^ l

, ilvos r: [(ixvog)1 - 4oPi!.]1/r

Tc 6uo .\ 61ouv ri0poropa ro ixvoE. H AoxryI 5.1.9 6€f1!€t 6rr l,\1= i1vog.1laol,ouc rou: ri!(rreg,

A'KH'EIr

s.r.r Bq€ilE rL-( Lb,orLuis acr rd r6ro6{alnordlo ro1, ,\,ara 1 = l] I]. e'"rre."-.

o1 io i1vo6 Loouar F€ ro iioqoLord to! r6ror+16a xar r oailoroa rooljrdr p€ roltroprlo ro!E.

5.1.2 \ t ro d.o r i ld"o 4. ) .uo- ts r t droqua,x i ,€rn ' 'd l d , t , , , , . , , " l : ]

5.1.3 Yrooaoi€ 6 Feraloni(owe rov rqotlonfevo I oqdradwdE 71:

"-' "''l! llnols_c .i!or or tbro4riq adr td LbLoindlnouara io! -D ,oL n6-q @!6ao!rar $ Er€ira

IAIOTIME: KAI IAIOAIA,\ Y:MATA

5.L4 Anoxe q! drldr = Pr. ono! P Ei!a! Fro nAo6o).t:

*=l:lH owrordoa ror r mor t(drao mlld,r cLiEda exoorftd. r!d, I oruordod ro! orovFISE!6Xoqo naadFir€r orde€art.

5.1.5 Be€irr r{ rbrolrtrfr yar 1o r6(o6lor.lrqrora r(ur

:ilEnoh0€nme drr ro )t + 12 + -lr wonrd| !s ro iXlo!

"dr orr ro lt)rl3 |ool]ra! !€

s.1.6 A{iroiF r.(d6€rypa non beiper drr o! Lbror4lis |lnoqei vo alldEouv, dra! ro}.},u'n).doro prc( rAa$rtq trOooreeel o€ !r{r'v dl}.rl.

5,1.7 Yrio66ole 6 ) €ilar 6Lo1qr110! A litrL r dlriororlo rbrcbld\,lofd rou: ,4i = )i.a) Aei.lre dfl ro i drJro €ivc.r 1{d! L6robrdflopct lor B = A - '7

1 xar 6q€ift Itrb,orLrrt ro!. Auro eo nein€' vo.nr6F6ud!er rtv Aoxrort 5.1.3.

6) Yno0imlias orr ) 10. 6eiEt€ 6u ro r €i',ar FEi.c4E rb,ob,<ivropc ror il ' rcr6eoir€ rlv ovrtororn r6rolrtrt.

s.1,8 A€iEr€ olt I oaiEot1oc rooirft pe ro y!!6p€lo row !6to1$ri,!, or€noFFvo, 6rL rox',edzrle,drxo rol !!'po rdqcYo\.Tono'si'd, mI uoeql

de(r l - ) I ) = ( ) r - . \ ) ( ) r - ) ) . . . ( ) I - ) ) , ( t 6 )

1i.{ rdlolras pft! ifunr! €fll,oyri ro1, ).

5.1.9 A0iir6, ti6 6no 6i[ma, orr ro irvoE rooltoL fE to dl0eolorra tuv r6LoT|p6!. BeEir€na(;r.l ro! dri!'r€I0ot'l 1or (-))" I ('''l brErri rl|led qs (16). KarorLv, 6eEirro].oug roug 6eorq tljq

det(,1 ^l) = der

a" f l 1 l

" = t i \)" t ' '

13421 l0A= O 12 B=10

1 0 0 0 1 L 2

ll,^ro! rEeL).dF6.i!ou! ro (-r)z-1. EEryyior€ ./!dri ol"c cuti rqoiq,(o{ar cr6 roYrv6FElo aor.i Fliaos qq heidE 6roFri.o! xor 6a.i.re ro! orJlx0lrm't rolJ ( .\)n L

mtv dqLm€qrt r).€!e{i rIE (16).:!rrairare.

s.Lr0 o) Kdrdor.ldm€ 6!0 trildreE 2 $if 2, dror rirm€ or t6rorqdE ro! Aa vo !ry €ivalrd yrv6F€va tuv r6Lofltr.irv tuv A ldL B adr or Lbrcr4dE ro! A + -B !o ft! eil(tt raaoaoiolldra rov aErmoirov r8roflF.b!.

5.] EI:AIOTH

6) Enolll0€noi€ €v ionrcs 6 ro doeorolo 6lolv rov rdrcru6v ro! ,4 + B lool}rdrtr€ ro doaoqrd 6kDv tuv rSrdqr6v tu\, A ror B, opoi(os 6e yla ra "rw6p€!d.rrori alrlo€lJe! aur6;

5.r.r1 A€iEr€ drr oi ,4 ftd! .{7 6xou, q t6Es dror|46s, onyaqirowas ra Xaaomrplorrrd1or5 fto^!o\1)pa.

s.l.u Be€iG rE L6,otLFrs xd rd ,b,ob,awqrors r* , = l i _or] *,-"r=l; : ]

Edv o B dXfl 14 r6ffwq l .2,3 xd! o C &Fr rE rbronpCs 4,5.6 xo! o D 6XEr q

L6Lotrpde 7, 8, 9. :rot4 eivar ot tbLorqi< rou 6 e'ri 6 rhdrd 4 = | ; ;lr

BaEare qv rdEt )tor rrs r*offeq drofltr!{ }a rors 6to n{!axp{, tov [o!d0o1 1.dr

5.1.13

5.L14

nors lbro6rdviopola d'!,!roro!lori! o16 p1 p16wrx{ r6rorqdg;

5.r.8 noLo elvar I rdEl xar or r6rorudE, 6rov or ,4 xut C t4g :tqoqyolFenl doa4ofi et'cLr €nl a; o$ll€€aie drr ! r6lmrpi -\ = 0 enal,alap6dwrar r - ' qoeis.

5.1.16 BA€it€ 6).€s 1E 16!0rr!4 ro! A - I, 6rsv o ,4 sLvar o 4 €nL 4 ni'vdxos po1/d6o!,(sr6/xeLvarE p€ rry Aoqot 4,3,10).

5.1.f AraL6Ere qv rei,"I yQappi iolJ "o!vo6oi niraxa"

dror rbo"re ro xaecxqerarrrc iou noi.!6w!ro 1,4 - )I voeivq|'-,\3+4.\2+5,\+6.

5.1.18 Yfto06or€ &r o nivarag .A 6t€r q dro4rds 0. 1, 2 p€ l6lo6lol1rorara !o,!r,1!.n€efrQdqn€ tov f1l6Ev6r.ueo xq|' rov r6eo oni,6v. Ariorc rry eEioool ,1o = or + r:.AEiEre 6r! rl -4' =,0 6Ev 6t€r ll]ort.

l 0 1 0 1 ll1 01 0 ll0 1011L l 0 1 0 l

| . 1 1 1 1 l. t l I 1 1 1A=lr r r r l "o '

L 1 I 1 1 l

f0 1 0 l'= L9 : il

IAIOTIME: I<AI IAIOAIANY:M,\TA

52 I H AIAIAMA MOPOH ENO! IIINAXA

3€xL!d[E opdoo]S F€ ro! [o!a6rr{6 e€[E].[66I unoloyr.o[6, 7io! €lval Eqoe€ru{darL6q ra! ocr lelo+rorrorle€i o€ 6t!eE 16 fioQolQdqous ourox rou x€qal,oio!.

5C Yroodore 6rL o n enL n nL.vaxcq,4 61er n lpq4u,xrirg ovelriprqtc16roblauioFata. Edrv rq 6rcvuopcrc autd enrl,elouv oov miLFs rvo6 rivqxa5, rdre o S-',4S elvcr, dvo4 6Lay6vr,og nhcrxqE A, p€ rtE !6rol.paq .rou ,4.xarri prixog t4E 6lnytovlou:

OlopcitouF€. rov S "nivc,i( !6robKvuoll(ir(r)v" xct! to! ..j vdxcl r6lorrFriv,'XetoLFono!6\'ras xegcLcio ldrpbcr. ldlo rcov prxprirv 1,cip6(r 1<r tLg rbLorLpd<,

AndOElg4 Tonoo€rliore rcr r"6Lo6Ltrvriop<rta r1 orlq oxril,€E rou .9 xqL unoLolior€ro yrv6Ftvo 4'9 o iLl nQog ftrii.yl:

^ ' '

^ " ]

0)

[ " ' " , ' , , ]

^tr"l

O€orpdwqq ro oav d.orllorl lou riot'l.dnl'aoroopori rLvdrxov. ei.vaL orpcvtr,x6 vatoroOerorltrre <tttoriE roug fii.vcrieE orl ooorrl 6Ldtn!1. Eriv o A egqcvr(6rtrvrQr! ono rov .9. lrvri y|a F€rd, 16r€ ro ^r ec rlol),onl,aoicr(€ ra morFin qg

llrl 'llTo r€Xvqolra xar6filv €iv<rr vn nvcrhlooqre rov re],erncio livcxo oe evc 6r,cao,

I[ ^ ' r r

^ r r , . . .

J ! H AIATONIA ITIOT'I,H ENO> IIN KT\

TrAd)qE lAopllig, evd €F€iE 06i!o1,1t€ va €trqavroo€i orrtv nAdrl orytlq. OmDSri.v.r! yQc4uivo, 6xo1jpr ro ooor6 yrv6p€vo .9 .. :uver6g

O niva'aoE S eivo ovnmp6qLpo_<, 6rfu oL mi).eg rou (tu r6to6rcluopnrn)uror€ory(1, ov€EdQrnxs.

Ilagcroiroqe r6ooepq ;ropcnp4ortg norv bdooop€ noQqb€lypara { €qsQ-Fol€s

nsQ.tlriQq t I Eov o nivszo( / 6€v 6xE[ €nql(llsF6qv6pevc6 r6ror+r6E -oLcqr0pol ,\ L . , , . ),, €ivq! 6rqpoO€rtlol FExs(i rou5 - t6c to n tb!o6!o!to[uroei.vcr" auopcircrrE crveEdprrFcr (6€fts to 5D fiopoxdro). AQCI. rdO€ fiivoxrrg UebloqopErlxiG [€taEi rols lbrorrpa( |t,roQ€i va brqyovlonolqo€i.

naQat4'Qqorlz O brcyovronordv ntvaxaE .9 dEl, eh'ar lovuiutjE. Kcr' aqldqavo r6ro6L('.\,uopc r [noQ€i vo notr],(rtrco!(oe€i pe ptc orcroeqd, rnqcqrtvortcgrbro6rdvuotr(l. :w€nols pnoQoup€ vc ro),,),anltoorriooqre ltg orril,sg rou SF€ [q FI6€vLx66 orc.o€Oig ror va nse.iyouF€ ivov vdo ortrytovtorotot'r.ra S.Ennva),np6av6peveg r6LorLFd6 oqilolw or6Fl nspLoo6teQrl €v;dQ€rcr, rol 'f!c ioreiQllrtli\,o nr.acibEry[a A = 1, oJtoloobinoi€ aw!oiQ6r].,Uog ,9 xav€L rll! boutr€Ld:o .9-11.9 eivaL rhvrore drqydvrog (xrrt o 6rqdvroq r vaxdg A .iro[ axQ!668 oI), Amo avrnvcxki ro leyo!6E 6r! 6Lcl rcl brollroporcl Eivo! !6ro6!ov!oporo

IlaQatiQ\ott 3H EEio'oq ,4S = SA r,olurL privo orn! rreQirw(ool fiou or. ornlesrou.9 ealc! ra r6lo6rovuolorc rou / r{at oe rugpicv rid},r1. AL}.ot wxeq S6lr' fu :trtpriyout' tol bnytiuo A. H arrid yr' aw6 6Qio,ctot otouq xc1v6v€sno).lar ,ooruotrou. Ynooaor€ 6r! rl rQ6rrt firil1rou S eivnL r1 y. Tdrc r1 rq6n1or{},I rou 5A €ivcr n )1y. Eriv cw6 o4rdmer pe qv rptirrrl or4},1tot ,4S rouor,pgovc pe rov rolJ.cn),coLaop6 :rLrdrxor rfvco .49. r6ft to J, ttQ6tfl vcr 6i1tltL6ro6[irvuola: -49 = )rg. nad.lpc.rr, rl 6rdrqEl firv !61o6!c.vuoFdrov orov I r{cr.r(,)! !brorr!6! orov A €ivdr ouro[drog I ibl(r.

Ilsodrioqott 4 A€v dxoov o),or or nivaxeE n lgqrprxri cvelciqryFc r6Lo6!dvrr-opclrc lic! olv€ridE 6E\, €ilaL 610l or nlvcr€E 6LcycovLofioL4oLtlo[. To tl]rn'{oncQd6Erytlc rv'ic lLall(!ll,rflroi aivoxs Livcr o

,4S=S , i 5 ' ,4S=A. i /=S^S r . (2 )

,=13 ,1]

298 I.\I(JIITIE: KAI IAIO \IAN\:IIATA

Or drorLptE ror eivcL )r = -\r = 0, 6L6rr €t.,'or leryrDvrx6q pe p1ae'vr.rd mr1

' ) l l . .d e ( . 4 . \ I ) = d e t . . , , ^ .u - ^ |

tuv ro J tr . r \ ' r6r06rrv.ro;u.0o irpr:rr vu rrcronorr i r1v:

" l;lIcQ'6l , t lnofn)=0€i.vcrr"6mlr1l6rorLp{-1a) 'yedprxt i rc),2a i r4r i qq i . tut ,2 tlcr cla povobfdoraro Xd)oo L6ro6r"awogirrtt\.lI fEotEerxli tu).t.unlirnnir4E eiruL I -udpXeL 6r'cr p6vov cr.vrldprlro r6Lo6!dr,!o[o- ncu, 6ev g;ropotlpc'vc ncrr(oreucloolur tor ,9,

Iboi xuL pLc ;no ripeol qr66e!!q rorJ 6rr o ,4 buv etlct 6LcryorvLo:roLrjoLpoq.Ererbll ^t = l: = 0. o A ecr 6rpEjre v<r eivnr o plb€![x6q /ri!cr(lg. Ed,a 6I{,}:S rAS =0, 16rr: nolJ,anLc{ori(qo\ac(q c,_r6 oprmepri $r rov,9 xut ono b€Ela pErol S | 6pioxoqre 6rL / = 0. En€rb)i o ,4 b€v ei\ror 0. I crvrirpcol cno6erzvueLorr x(\rdrft< TdToroq 5'b€v urroQet \rcl ,4crroQ0(i)oeL qv 5 1,,15 =A,

E),nt!or ro nced6s!.Ao vo Uylv €i,vc[ ficr0o7r],a!{-nr6. H trnorrryia rqgbto./(,)vtonoirlorF ,ry o(irri],ercu, oto ort tllbElilovroL or 16rorr[6q. Or rivazr6

lliil"lsli

f , - 1 1r.(clr ,4 =

l; o' ]'=i;ll]

Eniol< 6€\, ei1,or 6r.cr"lovLoroL{oLpoL, cLl,a o! !6torud,( rou! €ivtlr 3. 3 }rsL t. l. Torao6)q[( el!(lr q dl.LErl'I (Dv rbrobr.lvuo$llr(l)r' rror (n.lrroivrcr Y|cr ror S, xcL(rl1o ripirEr va ro\aoo€t:

H ,uty(rryrofio|,ory'tqrlt od,ieKu pE n r6n6wiotaie,H uvtneetlt4tdnFa orw;itlu tt, tq t'toitltia,

AE! uniralEr oXdol pucEu 6r&"1arvronor.ryqr6qtn6 Qr ovelriprqrn Lbr.obrcrvU-or&ra) xar a\rftorQerllpo.qrc< (l1l1 flb€.vrzic r6Lor4r6E). H p6v1 6v6er"!r1 :rou:reotplrrnr" ftro rLq LbLor+rEE riilcr I eErtE: A 6rttyauoto14a4 a:tonypttL p6rcorlv nealnftDol 2xa1d.a!6urc!6!ar 6ro4t(iv. Ay.6|nl taL x6rs 6poE. 6€v.rnol!yzi|v€r rcivror€. O ni!.rr.1(,4 = 1 dXer rrg ei.r..+6(v6$evoE |,8rorrpiq 1,1, ..., 1 aL),ci r:i vcrt {bI br.rydlro<l l curl tqv repi;rroo1 bev udAXeL il,Lsrllf]L6to6rcvuogimov. To xqr.rrlpLo r:i.vnL vcr efrrdoorpe, y.n pLc Lbr.or4rl nou ercr,vcrl,up6civeraL 2 rpopd;. edv r:nriplorr 2 cvr:Ecipntrc Lbr.obr.nvtropcrrn -pr dLlnl.o./ltt. Ed.\r o ,1 A1 6xeL rcill n p.

frtl !a ouun).rlQri)ooqr€ um6v rov xrzl.o r5€cbv. rQ6n€! r,a 6eiloulr 6rL ordraqopszzl( LSror+sr 6* nnporor"rtlow ;rp66).rpc.

5.2 H AI^IANI MOPOH ENO: TIINAKA 299

5D Edv 1{1 !6!o6!dviopota rr,..., ixt al'tlororxorrv o€ 6IaqoQ€'rL1i€E 1610-r!p6E ,\1, -.. , ̂ e, 161€ ro |,6lo6lfl{opd-rd dmir €i,vd[ fQq4[rdE dv€Edaqra.

Ynoe€or€ zar' aA1dq 6rr k =2 xu 6t rdrorcS ouv6ocop66 x(Dv rr ,.rcrL :r2:tapitlet to p16€v: crxr + crcr = 0. noLl,cnl,coL<ilowas pe rov ,4, 6pioxoqreq)1rr1 + c3.\7r3 = 0. Aqorat,;lroE )u rpop6g r1v npotrouFevl eliooroq, e[cqcvi-Eo!p!: xo 6(dwopa ix: I

c , ( ) L - ) r ) r , = 0 .

EncL6ri )r l,\: xdt ;,r 10. nQ€fisl cr = 0. AvtiLolo ct = 0 xo[ owefici)E rd 6uo6r(lv1jop(tra riv(ll cv€EdQirFc. oqo! [6!ov o rEiQltu6voq oljvbu(lopoE TraQdy€rro FI66v.

O i6Log oul,loy,op6g €r€zreiv€rq! xtrl o€ ]rd0€ rlrl0oE rblo6lovuotlsitov:Ynoe€to![€ 6-rL xdnoLog orw6utrop6g :rcpdyer ro p166v, :ro].].cr:rl.<rorri!oupe perov A. uqarQolJl€ ln qoQd( q! oQxuol eEio{DoI, €Ectqalilofi.rq ro 6!d opc. ctzaL a{privovTqE 6v(I! ouv6r]ooF6 ro\r ;r | , , . . , 3/, I no! ncrQ(il€r 10 pI6€!. Efia-!a).a$6d!oarcls rcr i6rcr 6rjptrrcr (rj triyovrcr6 nE 1,6EtrS ta9npaxfti ncty@yi)xcrclql,outl€ s'6r'q roD,ofitldo[o rou sl no! noQdy€! ro [I66v. Auro €Eovovad-!€r ro (rl = 0 rq! T€l[xq rdo€ ci = 0. >w€rr6g, !6!o6[ovuopolc no! fipoaQxovrdlo?i6 bIqqoQ€rLXdE rdloruiE eivql arJrol.ldr(0E a!€EsQilro. 'Evcq nivcrrcE aor.r61er rr 6roqopenrdE r6ror4rd6 pnopet vc 6rrryovrororlOei xqr qw6 €ivo! I r!filxrlriEQinl(r)oil,

IlaQoi€iTFars 6]ayo)l[onoa4olg

Efi[orQ6qo]Jtr€ oxo riQro 06pc o!r{g qE ricrQayQdgou, 7ro! rlr(1r l S r,4S =A. O rivqxq6 rbrobLsvuoFaxo! ,9 petctpeneL Tov aQllr6 / orov ni,vcrxorbLorLF6v A, trolr Eivqr 6!ay6vLog. A!16 turoQ€i vo 1,ivE! ovrrLnfiro y!c1 fiQo6oL€Exor orQo.pa<.

f r r l r r n ln c P { A E l r v A I H n ( ' o 6 o } , ' i , t | , I i x r r n i v u 7 u , o r o f l r ' i , v A l , l ; l

L ' = ] L ' , " 1Trr r6Lo5nlrioF(x.r fiol' ufioloyi.oorlrov Trpoqyol)td1,{,E rr!:Qvdau oirg orl},€q rouS:

H r€I€rrraio EEi.oooyl snoUosu€r.r. F€ Fr.r Fdr.i(. Aao S r,4S =,a..

nAPA EIIMA 2 ftu VLU .ftQoqi or i6!€E o! (6!or!p6E 6er eivor xct 16ooqa\'€Aaq:

'= li "]

r=li J] "" der(1t U) = \2 + 1.

IAIOTIME: KAI IAIOAIANY:M TA

Arpon o ni.vnxcr< orp€q€r ro eriin€6o Nqrd 90', t G Eir 6waN6y irs \uiLyttapsltt ofeirtErdr xa. rar16NQoys rs 'nqAti qt i6rc xffidqrvor: IleoqcrvdE 6€v€iv(t 6uwrov -eet6g rr av €ivo! to Bq6€vrx6 6L<i!rola. rou rtvu rilpqmo.IIQ6n€t 6Foi ao ondexouv r6rorLg6< xar ne€nrL vo eipcorE o€ 060I vc Luoortr€rlv rlr/df = ^r1. To X(rg0rtqpLmr.x6 ao),ro;vrtto l: + I rQin.r va 6I€L nciLr buoaiqrc -c).Ld or pl!€: ariri< b.v r:vLll nQc(yfaflxas.

Auttf eivcr q 6te!o6oc. OL L6ror4rd< tou 1l eivcr qavr@rtxoi e?tqttoi. ̂ r = iz(lr lr = -i. T(. r6roaravuoF(lra bsv €ivor €ntot]< ne&yFGrrzd. Ktrrdt rdaotor0o;ro, mQt{o!,r(S rarir 90", fio},1.(n},ootci!ovr(L [€ rq i x.t[ -i:

(11 -.\r1).rr =

(1f - ): Ih: =

,' ll [:] = l3l '{cr, " = ll,lr ,'l lrl = [s] x.', ",= lll

Or Lbrorrfi!. (v ztrr (1(vr(orrziE. eivot brarpoprrLxdE xcL r( r6ro6r.rviolrututi1,at cv€Eciprrlro. TonooertirwcrE rn orrq oriL€q rou ^9 rai.pvoupe:

'=lt,,Iog(|'ri0{o,, Al,rLtt€ro)ni[oupe pL(l avcrn6rjeurtl )acrhor(ot, tu]Laoi ro Tryordgott XQtt.tEdltaort t'|'rtr6t tois aQopoig at 61nt x. Ta flQawatxotg flivarE7Mnop€i vc( un(ietow noL{ Liy€,< nA(yF(rLr.€E lbror4rdg. uadpXow dpc,4 nriytolstt pLya6Lxdq r6roru6q. (Or [1./o6r{€c orlur€Qd,c4rdrivow tr"g rpu.lpatLxeg, .71( rgoaoiFq ro qrcrvrcorrr{o [[Qo! .i.v(1! pl6tv.) Edv uridaxorv ro].ri Li.1n L6ro6r,qvu.o!(r( orov rqrrrypctrx6 x6opo -Rr rl rov -R", qxipoupr cnor Cr { mov C,'.O Xl)Qo< C" n€QLiX€r 6r!€E rt( mtil,€( tr |r1./s6r1{€< ornmriroeg ;r(L ileL drl].or_qooroForlE 1(r !o BrlroE, ro €orDr€Qrxo yrv6tsvo, x(r xiv oot oyrDvL6qra, Xorei<rct ei.var 6uo1{o),6repoq ofid rov .6|,'. :rlv [aQdyAorpo 5.5 zci\'ouFr- pra tiroLlrAoo(Iatoyrl orl ltlycbrzi rroirr(1)on.

Auvdp€r6 rar Iiv6FEva' Ah xol AB

Yn.irQX€! tric ctyopl n€Qin(ool. mI! onoic or u;roloyropoi €ivr(r €ixorloL.YnooioTr orr iXoLIU ylbl 6pfr rr: rbrorruic rut tu LdodrcvioFuru rv6: rivuxo, '1. T6re oL r6Lorr.pdq rou,,1r eivar, anE6ri4 )1,. . . , ) f , xaL zr iOe rdro6Lirvr-ofto rotl ,4 ei.vc! xcrL Lliro6rdwopc ror -4r. EexLvcige al6 tnv ,4:r = )a zcr"roi,i..rrLrloL.lioqr€ ntr).r p€ rov ,{:

l] ,."' s'11''=l; :,]

A: r=A l r= ) r l u= ) r r i . (3)

il ll \lAlrlNlA l\ lt'liH ENo: niN \KA

'EroL 1 ,\r eirur (6lotqr{ ror ,,{r pe to i6Lo L8ro6'civtiofta r. Eci! o nQ(;lo<

noilcn).aorcropo,< ;re ,4 cq'rtvet rI! Lc1€i0rvol ror l cg€1d6),lril. ro iStooqrdaivrr zlr t,€ tov bcllr€po.

To i6ro <r;rordLeopc nqoziirrtL aro tI Stcrlorvto;roiqo1. Ed! S r,'lS = A.luI rrrLnryi"vr:olruc 7ur rrc 6' ! , ,1 l . fLrf fc :TL{.avoulrr

(s r r ls ) (s r .4s)=^: r i s l r l rs=A: .

O :rivozuq ,,1r bL(yorLonorpirct tr€ ro! ibto r(ilrrxc "9 x(t iror-rc rbrobr(\'uo$ttr(r;rcrpc4rirorv ru i6ru. uvd) or rirorqr€q r€lQaY(,)liloarcr.

To ibro EEcro),o1,0r:i ru Lol er xnr Ytcr rir0t 6r, vcp1 ror,4:

5E Or r6rotry6g rou Ar €ivaL )f,.. . , )i, or [-tl16( 6wd[€q rorv tdlolFptiv tou,4. Kiroe [6[obrdv!o$o ro!,4 sivcr| fla},! !6Lo6rdvloPcr 10! A{ tioleirv o ,9 dnyrvronorei rov A, bro'tcouofior€i €?rior6 rcrL ror ,4r.

At =(s-r /sxs lAs). , , (s-L/s)=s r ,4 's . (4)

KdO€,S I cllcrro€i, lvav,9. crr6E rorrtrpxLxori 9-r r(lr ro! rel.r{orj S.

vEtir o,4 r.ivcL ovrLotpiltpoq. o x(!6!(< auoq tq(pt6!.tr(l l(t olo\r

ovrii'rgnto 0r€ flt bnlctrl ,( = 1). Ot 6rcflp19 nv A\ ti!41 tl^. Ar1loq(ilcral}iar XtDai< rI 6(ry.)lloroirtol:

s d v , { . , = ) . r r 6 r e . f = ) i ' , r , , L l . = - a ' " .

nAPAAEIIMA Eir! 1L €ircft I orQoqrl x.rrd 90'', tote /ir eivcr l otporfll zctcil8{)' n(1r l( r rilaL l orlxrprl r(th 90':

. . 0 | | 0 l , . , f 0 l lA , , " , , u - ln . ] v , , r , ' -1r , ,

or rbrorrpdE €ilcr i xar i. rc rrrpdyorir tot; eiv(r 1 xtlr I xor or .rvriorpoqigr o ! : E i / ( l l / i = i v . c . l l ( i ) = i . M n o Q o r l t t e € r i q E t c , l c g e x r l r , m l v 1 i r , : r o r€ivdr tLi(t nl,rtQyts orpoq)i 'aord 360'l

' ' n l l L0 { t l 'o l l

H buvau ir eivcr 1 z(1! o :riraxcs oraoqrir '.'(rri 360' Eivar o raurorrzog

301

I IOTIME: KAI IAIOAIANY'MIITA

IldF€ r6Qo alo y.*iqeto 6io ,t.v,i,qav 1.or ovatrFqrE 16 [6[oxqr6s rou,4,B. EivqL rol,{ 6r},ecorr.x6 va lpqoqrororfoor4re rov i6ro ou},}.opog6, oc pLn;rpoondoer,a v' ato6eilouF€ rdfl rou [" y&€r 6€v a.lqrsirr. I6oi Fra ,lalo(lopewlan66E!EI 10! &r o AB 61er, r1v roLotqrl prl, 6t(r! o A 6x€! ilv !6rorr[i l r0|, oB lqv l6!or!!ui p:

A B r = A p r = p A u . = t . ^ x : .

To ldoos ayx€lraL oro 6i! uno06oo[€ fi(,)E o! ,4 ro! .a 6xow ro 16rc r6lo6ldvuopor. Ev yiv€! rouro 6e\, loxier. MnoQ€l vc! 4(oul.l€ 6rjo nivori€( F€ Fl6€vlxtr6roltF6s. r(l)v o olov ro IW6!€vo va €r(E[ fllv rblorqni ) = l:

Td !6ro6lqviiopqra our6v xov ,4 xql B eivqr €w€r!d6 6&qoQ€r[xd r(s! ow6sLrar ro o vIo€E. l'!a 1ov t6ro t!6yo, or !6ror{rds tou ,4 + B bEl, ixolv ilnots \rdxqvouv F€ rnv .\ + l.

H trav0aou6vn dufll otr66eqrl uno6elrivri€! rlv a],rio€[a. Edv oL A NoL B6[ouv ro i.6!o r6ro6rdwoFa, r6rs o[ 6roqr{s ro}J.onrlaoldlovt(rr xq[ o .4.B 6X.rrqv rbrorLtni pl. Ynd€t(r! 6Fos xo! xdi! 't|o ev6ne6Qo\,. Yrdexe! avqs et*olosrQ6riog 1' ovqyvtopilol+e r6t€ o[ ,4 xcr B fl1oov 6vo rl,riQ€E orivol,o ,{o[v6rrbro6roruopdr(l)v, xor dur6 €ivaL avo 0qra-xl6r6i orrp n6ovrourD(ovrriri:

5F Edv or, ,4 xar, I eivol 6r,cytovlorlolrlo4rol i6r€ $(ouv io! 16!0 nhoxq6ro6Lc!.loudtov .9, €d.v 1td\ l\6\0\ eA\ AB = BA.

A:l.66e$q Ed.v o i6rcg .5 6raloto:rorei xcr rorg 6tJo. A = SA'S ' xft B =SA:S-r, rore pnoQori[€ \,cr ro lcri],aoldoowE xa0 ]r€ ttf &jo ErardE€rE:

,48 =SArS-'sAzS I =SArA:S-r xor 8,4 = SA:S-rSArS-r =sA2ArS I

EfisL6ri ArAz = 2Ar (or 6!aydw!o! nhdrsG ol'1F€rdrioslroL ndvtorr) 61oqreAB = BA,

flo fil./ a,\rrioieoql xol€dowol, ufio06o!€ 6It /lB = ,8.4. Eet{lvdwaE ar6qv Ao = ,\e, 6XoW€

A B r = B A x = B ) x = ) , 8 r .

Aea ro ;r r.dl Bt eLvo;" xcn, xq 6110 r6rcblcrrrolaxa rou ,4t oE npoE rrlv[6rrr L6ror4ri ) (6rcqopunad Bii = 0). Edtv rpog offfpiv rlrroe€oouF€ 6 orr6!or!!r6E iolj ,4 €lvar orogoQ€rlx6E FfiaEd rolls o[ r6r6loQo! eil(rl 6r!0r ror-5Fovooldorcro!- rdx€ xo ru nQirl€! vq €i,vo[ riollofildolo iou o. ME dl],c ].6?[a

,,=13 tlll 3l=lt 3l

ro r €ivcL lorc6ldvropo nar rou B xdr rou,4. H ar6EerEI F€ €Trovololr6av6u€v€Er6rorL[6< €i\r(L ],i./o po,aa{r€At.

noealriAqot Z1T|\ x6&\ro[ryav|,z4, €r€il,or. oL oTioio! rnoq(pow cr6 qv aprir4S doEfauiqNat aox Heisenberg, €irol or nivor€g rou 6€! pexo-ri,e$'ra!, 6nox rl060I P xor I oQ|lll Q. H 06o11 eivar o4$etpLx6g nivnrng, 11 opprj crwrorl4rerpr,x6grcL or 6no pali r.xcvonoLor,.v qv QP PQ = 1. H nqlyl rIE c6€6or6q1cEnaorijrr€f an' rrt! cvrs6qlo rou Schwaz (Qr\'(Px) < llQr ' lPc qqIloaayadvou 3.2:

; t l 2 = r r a = 7 r 1 q P P Q ) r < 2 Q r . l P x : l .

To lLv6pevo rutt lQx I rll )idr lPc / l;rl rou ncpLordvorv rtr oqdlltrltro€otl; fl! oo$ylE, orqv q )iularood!(iQrlorl €i,lrcr! 4 l- etvcrL rouLdlLorov 1/2. Ei-vcl cbilcrov rd rdvoup€ xa! ro 6!() oqdlpcrr(r Ilit Qd. 66r! 6rc(\r nQoonqoourcv(l tl€rQlloo]JF€ q edol ev6g o(Dgsrr6iou, a).ld!o!tl€ rllv oQpti roir.

EnrorQd$or.Jtl€ raLoE orl! ,4 = SAS I, rlv rroQayovxonoilorl noo nQo1tu-nI€t cn rLE r6rorr$e<. A!fli Eivo! r6loi,r€po xffi{tL},nl.I yro !o tJno},oyioolBeltE 6ri!dp€!S roo ,4, rlQd1ro flou qdr1rllz€ orlr .rrl.oior€Qyl reQiTrxoon iou,41.H :rcqolowonoirprl ,LLI eilcrL nlarorel.eopdirxrl, 6rqv r€rQq1tovw6€i, Evcb ISAS L eivaL xatr€la. To rerQd./orvo €ivo! SAr,S I, rc r6ro6r,cvuoparo lopc-ptdvow crvcl.l.oitrrro ro! qxol,ou0d).,'tcrg (uld id t6lo6rcrvtoFara 0o l,]Joorip€€EKdo€rg 6rcreoQ6! xqr 6lsqoQiriq €Elociro€lE.

ATKHEEI!

s.2i ndq(yollonoti{'1e ro1( rropdors raxeE otl! Boepri SAS r:

5 ] H A IA ION]A MOI I I 'H E I \O : N1NAKA

Batixr ro! riv(aa A, ro! onoio! or L5oi4#< .i!dr I l.(r 4 ra( 1(I drodrcvnouora

*, l i l* | ; I dw,coix-c. , y,od.,r4: .4 s^s )

Baoitu olsr rlq r6LoiLp6s ,.1r '!.r !6robraviopora ro!

l;;l

5,2,3

.[iii]tor aaic-yadrF8 5no 6nqroa.flaoi. 6ldyolronoro!\rtq rivczrs S-

5.2.4 Ed! 6ra! i.vo tar'fou"6s 3 .ni 3 rrivo,roE dx€r 6royrir.'aa morj(sid 1, 2, 7, ndisf.)ai!ft€ ofl uroaEi vo 6rd, ovrcroLl€Eii IIorcs eivdL o A:

IAIOTITIL: KAI IAIOAIANY:M TA

s.2,5 noror ftr' rorE €ir6! ous niror€g 6€v sivar awordv ao 6tsyo)vrcrorleonv;

1' = l i - i l ^.-11 :,1 .=[: !]s-L6 t) Ed\ A2 = I. noLrs riv.rL or 6ri!(ri! r6Lorrlic ro! ,4:

6) Edv auos o -4 €iv( 2 Etri 2 rot 6rqoa€rzo< rov l xor -1,6e€ir. ro ilao{1t(lr qv oailo'rod rou.

y) Edv I na(int raaNri rivft (3, -l), roLa €ivoL I beEretr

s .2.7 Euv.4=Li 2 l

. 6ar i r r l (e . ' " . r ' ro . /ovrono( iEf t rov .4.

5.2.8 Yno0afl€ ort -,1 = lul rivor gu onii.l eri sta yaqti (ava-. nivaxcc rdq€ox 6!a).u) nol,),ari(oLdloro< rov A rri ro r, briErE orL ro r rivar r6tobtawopa. noro

Eilor to .\:6) norrs €ivor or . r! rbLoflId( (xo|rt({):y) Y oi.oyim. ro ixrq (,4) = ,7 u fe 6io re6rors. sn6 ro {i0qoLoFo qs 6ta ro)vio!

rar ard ro d0Qorota ro! .1.

5.2.9 d€i&r lrr on {r'e.io< uno}.oyro}o &r or l, a( a,r 6xoDv ro i6ro izvos, dr(v

. ' ' = l , ' t ' s -1, ' .1." - t . d l t r r l

:Wn€edvat€ i'1t AB - RA = 1 €irfl obuvaro!. {Mnoa.i 'o oqr6€i p6vo ortsdneer< t'roordss<.)

s.2.10 Yroedoft 6fl o /l iZ€L rL< r6roqris 1.2.4. floro €i!6 ro iXlos ro! /rr rlorc eivgr 11ooi louoc 1() ! ( / ' ) ' r

5.2.11 Edv or t6ortp€E ro!.4 ri!a! 1. 1, 2. ro|( (l,l ro.ndlovo ei.lar oiyoueo o).'t0€ro:A(i'ote 6rxfto)'oykr 610! ( rt0eiow 'rar orrrnsqdd€rlrs oqv n€einrooit ldeots:(l) o A Eil(lr cvrroredll'4rog(2) O .4 ri!d| bLaT olmrouiorFor(l) O -,r 6€v FivdL 6rayovroroiN+rog

5.2.U Yro0tur€ 61! r(l B6vd rliro6nniopdto ror A €ivor. ro trouarlddL.r ro! r = (1,0,0):

A q' O -4 6ov (wrorad(p.t0rA w o ,1 irsr tlld roltdrli t6LorFiA \Y O ,t nev 6L(10\,ro'ioteit0r

lA =.d,I0GE. 'Y = V€r,bdsl

s.2.8 A,oybvroro,ioTs ro! rTiv(.xd .l = l+ i] *' ep.l" ,'" '- rs rrroayDv,xdc eitrc

rou -i!d! nilazc reroro' dor€ Rr = 1. n6or! rgtady.'vu.i\ aile( 0a undgouv;

s.2.14 Ed! o A sivsL bra'yovronouior[oE. 6eiEre &r I oQalouoa ro! A = .SAS I ei.vdr royLv6F€lo to! L6rorF6!.

s.2.15 A€iEr€ o zdoe flivaxor' €ivaL d0eoropa 6110 srl r6rcF6Qqov rt'd ,,v.

' :rE E!,r.qoirr(rrsrlauqai!.

r l u l L : l : !2 :L I A lA t u lL lN l \A l U l A \ \A^ lL l ) 4 r

5J T OI EEIXA'E|' AIAOOPAN KAI OI AYNAMEE A[

Or EEro6oErE Sroqoadtv 5€v siv.r" rooo 1vrDoriE ooo or 6raqoarx6E eEro6oE|r,a! xaL 6a 6nasn€. Or na(Dr€-< Iwovror F€ 'r€r€aooF€vo nl.rteo: r€rEpaoFtvrDv6qpcirtuv, rvt; prc 6Lcrpopr.z{ eEiotool hlveror pe 6vcl dneeo n),iOos cneeomd}v6rlpittor or 6lo 0er,-rpirg rtrot6oo crvcfirliooo{o! €tt€},(i)g riapit}.}.q}.cr. Eivor r1i6((1 crvolol,io ur:xaEu 6lcxQ(rolJ ltc! ouv€loiE.7ro! €p(f(r!i!ero( Eo,,'d xal Ecldoro [ot)r]$orrxd.

'Ioo)g il r{(l),lri€Qil €7i€qlyrlon eiv(rr I erEr€vrl ror 6€r oxrxi!€xor$€ n -6rtior(rrrt ./QcFtrLr'iyl dl.i,€6Q(l. ctpo{ oL xarce6o€6 orlv rQirr€la eivql povo6vc( e!0$6(.

Yro0aorr or( xori!€rr $ 1000 yL(l Trdvr€ XQovta pc 16r() 6%. Edv o roxo(11cTc6al,),€Tor dfiaE rolJ dxolE. Tor€ To reqd,o[o ro),],a7r],qorqE€Tor p€ 1,06 ha[P1,- =1.o6Pi. Alttti tivn pm tfknoq \uultoptitv pt:7polnl 6/1pu t:,lg ixo <,ow6€€r ro r€qdl,aro perci cn6 A + I 6q pe ro ]{€qdLoro rou lloolyorrlslori6toq acL Luvetct €!ro),a: M€rd 5 6q. -ro (,QIr)16 x€94(Icro ,& = 1000 6I€!rro ,ofiI(loroo0€i 5 qoQaE zcr

R = (1.06)sl i = (1.06)s1000 = $1331J.

Yroe6or€ -rd)Qcl 6fl ro lQo\r[]i6 6iBd fi€QloQi!€Ic& o€ ivo Luivcl. H licl €Eioool6t(lri,oQri.)v et.vct tr.* = (l + {),06/12))t. M€Id 5 6q rl 60 Bri!€E 6xo!t1€

p6,, =(r+(0,06/r2))orrp,) = (1.00sI{'.1000 = $1349.

To rn6pevo 6rpc eivcr o roaop6g crvd 1p6p(:

/ / r i r r \ 5 t n 5{ l - = ) 1 0 0 0 = $ 1 3 4 s , 8 3 .\ Jo) /

T6l.05, y|0 va rpnrorv roUg xrc),Lt).ouE roUE oe oraoX6lr1o1, or. rq{ineleE:rqooqipow ouvepl toxrop6. Or r6zoL nqoorieevtcL avd :rcioc mrlp{ xcr r1rEioooq 6rcrpoptirv xcrcrpp6er. Ilpirlpctr, propei vcr vopilue 6rl o rcrpicE 6wq€Q€L aTrEeofirro l,o^/roF6 {(r 6e\, [fioQEf va ].oyapLdoer rr o.1< XQ0rord.

'Eler

6t[DE bto braqooEr.r6E 6wcx6qr€E: Eir€ vo roxiq€r ro lisqcilaro olo€r'a 'irooulvri, zct vc 6pioztr" 6rr ro 6pro eivcr

/ n n K \ ' "I | ' "J " ) rouo . ,n 'n rooo ,H l4q .87 .\ r r /

€ir€ !a orp.rq€i os F|a araqopL'iyl EEforooyl ro 6ero Fr.lq eEiordolq 6|(1qopdiv?/,*r = (1+0,06 41)p/,. Mercrq6gowas ro p1 oqv aqrorea{ nl,erpd xcr" bnLpdr.raqtr€ ro xpovLxo 6rtF.l At r(tiplorFE

?u.k;po =o.ooau 'ro! r€ir,€r orrr.,' ff =o.oon.

IAIOTI}IE> KAI lAIOAIANY!MATA

H Luorl €ircL p(l) = eotuar zar B€rd 5 6rt cur6 rivor,-rdl.r $ 1349.87. Torrrt)dl.rrro i.eag6v€r ir€n€aaolilo, ncrad ro 6rL nvnroxiltrcr cvit :tiroo mrlyti

xor I br.rqoaci €i\'aL Fovo 4 o€lrc.To rc0h6€L.4ro au"'6 r€arc)df6c!€ 16rro clrotioeLg bLcrpoptbv 6oo xoL 6tct

rpoprldq ellociroerg. Or ng{;)rtq 6rerv(v orq S tepeE, xo00)E xo xaolrz6 6YlF.1y"vorcrv o),otrcr ;rxp6reeo. YnriqXcL 6[arg rsr r]"rt0oE €Ero(ijo€('rv 6!cqop6v rotlor(xovrtrr" trr6 p6!ug rolr< xor ro Setnepo ncpdbtLnrit !r4 'iQo€Q1€iar atro rlvnFAiql]tln (1xolo!0i(i ro! Fibonrcci:

0 . 1 . 1 , 2 , 3 , 5 , 8 , 1 3 , . . .

Ev6€Xo$6v{Dc xs-ra)"a6crtvere q 6!(6(r(okr: Kcio€ oarottoE Fibonicci Eivcr rod0QoLo[c rorv luo rQor]youptvu)r rou

Avri €i,'(r lLrcl eEloool 6lqqioQ{irv. rFqrcliletct oE [!c{ 9crlruorlxt rorrrtri(r€(t-taQltoycb! xa! aqiq€L \rq q< av$a(o0ei. 6\ra o)'6x),lQo 6r6)'io. AYla(i0I(r x{1!rfruL),( 6),(orci!o!,' €),[,aorL6dr:. xor ot d!(r,{IG6i. 6(ropo!Q1.crs 1p1}.Lti< t] 6tl(l-vLbrciq 0u 6petrt 5 ex6l.uorrlot( yro zc10€ 2 r€QrmQo(l< .1uptr rr:rti ro 1tl.c6i.H (xLcbd i:x€r 8 €r6Laoqo€rg y!c( xdor 3 r€aruiao(!6: |r(u q tt!(i. (rxolrl trto:roLri:rLoxc. dxE! 13 er6),(torito€K ./r( n(iOc 5 ntQlnQo(pd<. O 'Ia(']x(t0Lrtrri<(p.rivErar vo tivcrL ro r1),rorptinro xo! D nicl T. O'Connell (scientific ADrefican,Noi!6aro( l95l) ro! orroior oL ofioQor b|a),iyorlv 6vcrv o1e66v (mioltuo loloF':lFt. = taalT3.*

Mr' rroLo 6r(qoQefl,a6 .Qoro F roporF€ va tieoul€ iov Xr,toord aQLoBoFibonacci, (vri ro1, 6r.1EoXrrct rrol.olLo[oi ft:r6 ror,< Ii = 0. -Fr = 1 rnL otrouqlorE €!6rd[eaoxE. +eXpL to,FLr)00; O ot61oE pcg Ei.\,(t vo ].uoorFE rtv €EiotoolbL(qoAdrv -at : = rr-r + Fr rc! a.rv fiQdrro 6i[a tmopo!$€ v(l rlv avoydyo![Eor prc rliuro1 E,'6s 6ri[.rroq ut,-t = A1]r.. Avq Fo(iq€L FE rn! €EiooJoI.lv.nolrroBor. Pr , I = I , 06P1 . ;r6vo nou r6pc o riYlororoq a ivcr" 6va brirvuopo xoro rol.).arl..loLaorrtE A nainr va oi,'cl i\,.r! rivcraq: sdv

, r= [ . , : ; ' ]

* f( rq +uarotli (ri( o l Bor lLzil. r.ift ro ijL6i.io ror D Arq Tbonrpson. O, G,,rlr,

,r/ /'r,r (cxnrbridee Lrnir. IEss. 19.12) rt ro nDatolfto btbiio ro! Pcrcr SFven Pot.n6 nt Nunrr

(Lnrl.. BroNn. 197J). Ez(ou,il.: riti.( r6ror!r.. rol 4,, iToN bnrrootr!r!i uro r!aL6L7n /:'r,trd,ri

Qrdd l O Fihonrccl .tala.raL ao.tra.arilo< ro! tlflty(i! roIS (aciiL"oii aq(r[.iq otr]! Lra.;rrrt.

( r )

5I Ol l:l:a>ED ^IATI,OP!)N K,\l OI d\NAMEI: '41

t r = t t t t t f l t l?-^.., = F*-, Yrv€Iar ur.L =

Lr o lu^

Alt6 €ilcr 6vc rrl;ru{6 ii1.vcoyo yra $ra €Eio()or1 rirlrog s. H 6o0rioc clioarrlotlvbudE!:xcr" Be s 1 rerpLf4rtlvrg e!rod;oeq<, 6aroE rl F*-r = Il*r, yLc vo 60)('€r6rc orot4rn er6c 6rlpcroE. 1-!G iolrq Fibonacci lrav s = 2.

H €qio{')oq lr/,,r =,4ui. }.r-rveral r!zo}.a, EaxLv6vr(( oflo ro r0. :ro 7rQ(iro6rlpcr ;rpoxumeL to rr =,41 . :1o b€ur€Ao 6rlua, xo 112 LootrrcL pe .{U1 nour"oournr pr ,t1:rrn. 2t xtiqt 6i1ya yiwtat ii,(tE nolltrlaa&ot16. ue A, xat yllaA 64$aia 6Xo!! yiv€!,( notr],cnLoolc1o[oi:

H Tioq turr u*.t = Aut dl|ttr ux = Aktta.

To nQcl./lrfiLx6 n0661,Igo rircu vc 6poqrr: xino[or yQrlyoQo re6ro !fiol,olropoljr{ov alndlea}r A" rclr 6ror vc rqoobLopi,ooq.re tov Xrl,lofio oQrfJB6 Fibonacci. ToxLelbi rrsQr,€Nero! oilq !610r$6( x(t rs r6ro6m1,uol.loro:

5c Edv o ,4 BnoQ€t !d 6!o1ov!oJro!le€i,. A =.9.4..9-1. t6re

u^. =,4ruo = (Sns-rXs s-')... (s^s-r)110 = (s,\{s r)u0. e)

Or ot{I.€( rou.9 €ir,ar la 6[o6rqv opqra rou,4 rq! O6rovrs6 ,9-ru0 = c,1 ).tor1 livetdL

H ),rioq eivct oubuoolr6E rov "cd.cbv Lioeruv" ,\fz;.

OL rlnoL all1oi bivouv 61Jo 6LnpopaLx6g npoor^/.yio€6 rIE ibr.rq LuonE,/ <n 'q , . , . O rpr iro< rnroc ((vrryvJ,pr i : r orr o ,1t r , twi!rrur l r r n,v.s\*5 ' rur 6rr t r rocouuuUr va m,{ l rr toorrur fr f i H n€ir ' (q nQoocy}r l r46[urq 6riTvar ontp6orepn qv cva].o.1in pe nt ],iol br&qopL]itr €EiotoolE: Avriruw zc0npir exe*rrtirv Luoecov ertiz; 61orpe nl.aov rLE dnL6E 6lrciperg )lr;.OL rcrvovr.riq lar(rordo€rq €ivaL rri],r ro (6rc6ravlopara ,ii xo( o€ xirB€ 6tgrrcvLoTllovraL tr€ rrq LbroT|$6g Ii. :lvbudqo\rr.rE durrlg rr"g erdraag lioeLg. eror" d;orevtl r(lr.QL.ilow lr€ ro u0 -r(t €6dr r:i\'aL rrou urE|,odQlor.iar ra c, 6Qi.o,ior4r€ rIooorrj Lroq r = SnrS 'un.

ur =S^'|, - f" I f^' I f', 1 =. ̂,.- -.. ^,,,,t l l r i , lL", ,J (3)

IAIOTIIIIF:: KAI IAIOAIANY'[IATA

>e rire€ Et6t1{o ri(Qd6rtyFo. 6norg c1!"t6 lou Fibonacci, io ro{i)to 6rlf( eivarvo dQoiF€ rrE rbrorlp6E:

H 6€rir€Ql ./Qotrlrrl rou,4 )I eivar (1.-)). cipo ro tbtobtdvuol( €ivclr (A.l).or nadror rloretroi Fibonacci Ii = 0 xqt tl = I t(siQvorw rlt Otot rou( mo tto zu!

,=l i , \ ] ' dc(,4-)1)=\ ' ] . \ r ,

t + t / 52

Av[iE siv.l! or orce€o6q otnv ua = .1)jl lr.1 +.",\1n2. En€t6i q 6€ureQl] o!!lor{i}oaxcrL tt,-rv bro r6to6Lcuropdrlrv rivct l. :reonlnter, -F/, = r'],\f + cllf oq bsth€Qlouvlord)o(l iou ut:

- 2

c = s u, = li' i'l lll = lll)i; ̂ ;l,l

,- rf )1 *[(+).(=4).]Arnrt eivar I ).io4 not' !ryouoqr€ rlaL atr6 Frtr drrolin1 Eiatll Fdr] ]'ov araoobottntl.616r! o r(lvov(lg 101J Fibonacci Il,: = Fl*t + F1 r<rgrlet nirvtotr trxdpcLorSoQro[or,\, €vd e0d; rcrc],{Ec4re ot xlnopcrc 'iq! r€tQolol|xd:E aiE€E. Ksrdra:roro rpo:ro ru npnylrata uur,i xt\dnrr vu all.rLo(1,cttQorrvl(r (;orr v onotivrrfvrrq cxr:puroE. nedypalL. rnsrhyt o olurpos 6QoE [( l - y '5)/2)]* ' r '5 r i1 ' (rndwolc lrr,(xlrsQo6 xou ].0c petcrtonilt:r tor:tqrirro 6qo rpoE tov nl'loLdorepoa:ttpurt-r H rtqciqrorl .tpi1v.r Fd1'o to (xiQ,no gipoc zur

F,u,, = r)'ryrimeoos ,txip.,o. ,^' I lf '- *\ "' l-c L\---l IAr,t6g, rpuotxti, tir,rrr 61'aE x€QirorroE cpt0p6< rnL o /?rou 0c eivcrL ardtl peyaLulEQoi. Eivo! ol.otprivrpo 6tL oI ),oyor 0a livol{tl| ol.6teLc croip<^T or,ouyxQlvotlrlol pr rotlq i6rcu( tols axaQo(orE. O 1.610< Fr00r/I.10ft nQ€fi€L !af.i!(r no),i xovro orov aare[6 (l + !6)/2 ! 1,618, rov onoio ot 'E].].r1veq ov6Fc-g(i\, Xprrol roUI .t Mr dl.L,r LoyLc or )l livovrur corlrcvror. oulxpL'oplvor pero)f zut uLoloc F^. /F/. rc ivfr t rpoc to ) i ' / , \ f ) r .

A!"16 ri'\,.lr 6vc ru:ru6 naQdb€rypc ror o61ri otls bwdtr€E rouf r r ' l

A = | i

" | *" :rrprrl(r&rvr ro y'5 bttitt cw6 -rrselor(v ong t6rortrtc. Ear

* lii o xorrt'.. oaooli)rt.r ./.ol\ t.6to rl.ro{iD 1.618 raoa 1

5-3 {JI E3l:A:El> AIAOOPIIN KAI OI AYN MEE A}

6!sl,6Eoups avov 6[aqoQErtx6 riivc1,tq, ,rolJ onoiou o[ r6|,ort|.l6E €ivs! ct6Qcres,tlnooouFe !a €oftdooqrr rlv nQoooxrl FdE oq1] aJrldr4ra rorv !'rot!o.trou(5v.eEoi drcy ar,rcroti oo'rpE tov lti11c*ai

, f -4 -s l f r ' l [ - r l^=f ro r r l rxct ̂ r - r ' ' ,=L-r l ' " , - " ' ,= lz l

t r 1 r r i r r n l r ' t l f 2 6 r t _61 I=t" "- i l ; ; i l ;

L . - r L - - . , , , r l = [ - z+2 .6 r - l +2 .6 t1

or 6w<ipeq 6a xcr. lF, rou eFqcvilowqr orov relEuroio rd.vqxc ,4x ow6udlo-wdr u! ic L6[o6tqMrouorc,

Tovl(oupe ro zrlplo lcgcxrpr.orr.xd r4g ell,ooo4E dr.cqogrirv 21"1 = 110.Edv ro r ai!c! 6!o6!dwopq rdre

ltts Swrtt i t iot l Eivo/ t1rr6 = .r, u1 =).r, u2 =)2r,..

'Ortlv ro aQrylxd uo oqr6ei va eivcr, r,6ro6r.rivuopc, r6te I )rriol eivct: ot = )lc.Ev y6ver, ro un 6ev eivar, 6ro6uiwopa, 6tcv 6gog or4r6ei, va efuat aut6vaop6€r6r,o6Lavuopdrurv, 4 Lrlo1 tr1 6i6etcr. o:rd tov i6ro ouv6!c!oF6 ou6v to1, €6!N61,hio€ov.

5H Edv un = s1e1.!..... c,ar t6te u1, = q),fq a...+c.)fo", O C6l,oEftrv ci .l,vq! vo wcQgolrtgow q lto! nQog q qQr(!r{4 owoi&rs:

."= l', ."] l;]="' , ro'L c=S-'ut. (4)

:1e€q6[ame r6Qo o€ €v6|,aq6QooosE sqaQpo.y6E r(ljv €Elodro€(l)v 6!c!qoe6v -riowdFeov nNdriov.

Av€liEeG too Markov

YmjgXe pro doxqol mo npdrro xegdtrcro.lc rr4 percnrujoe6:tqog xcL c:toqv Ka),lq6pvr,c, nou crliler. pr.c Oerireprq pmrd. Or, zcldveE rftcv or, elr]E:

Kah yptiw fi rcu rlt19uopoi, €xda RaTrytietlng, tyxa9iotataL o' awtjvxat rt ov .49uopoi r4E r4r eyxata).eirouv.

To Fyov6q cw6 uno6eLavtier ULq rEioororl 8rnqopdrv. 3exr.vorige ge y11n),40uopo rz16: xru :" rwog xar oro riLoc rou :rptirou ypovou ixoupr roug:tl,rl0uqroriE ezt6s ztrl evt6s

.I0 IAIOTITE: KAI IAIOAI/\N T:Mr\TA

Yr =0 .9 r0+0 .2 :o

:r = 0,1U0 + l). l i ;o I 3:Slltrl

l'r,']l '^l .F€Ta6sivol+€ aJro ro'\, A

t r I f r , l1 2 ) l4 l

f l t | . ! t=(rn+aD l i l+ ( ro-2 :0) (0 ,7) ' 3 , l .

L : t l L - t l

Awrl civcL q ncptiotoo4 q)f ;r1 + c,:)f r1. O :tcrpclov )f = 1 civar xpw€voqorov npdro 6Qo zar e{zolo 6iano14r€ rr yiv€r(rr ifirQ(. nr6 geldrl,o xpovtx6

* EnioF 6n, ,..4rd{ivF( ,(odtot rr'd\ra rt l6oQia, Kll(h $d zcrlrot(ol u( I rEcpl(ircrun \o u r6 qv

F\,6t7otri\u:7,( or i6rE< llaa or iDar v( siy( (rFii:s. Mrkor. uiiri E -ri!o ao4 o/t.

" :r.E.: H trqbi1.

frcr vo 6eotr€ xo\' ,41 rnL rrlv n<rrcvoprj perd i dr1,

To ?rao6l,+ro qrlJorrd ,rQo6rlve XoQiS nQaruc(xu{d ororlei.n, 61rr. 0[(l)6 rlE 6uoIapaxrl0lorrx6E rordrtrr( pL(lq cvdtr4rrg Markor| O ouvo).LxoE nl'IolJoFo< nctQ(r-pever orn0ep6g xcu ol :r),r10r.ro1roi ext6g xcrL evr6E 6ev livovrcr, nor6 npv4rlxoi.'NH rrQdq L6r6irtrc cvr(lvuli),cir(l! oro yryovog 6t xn or.ot'/.Ei.a xdqE oril,nt tNotrvs0aonps l. Or nci\t€( sTioyQcigoaror xor bEv ldr'€rar x.rv6vclE. H o€ur€pylf6rcrqrc clwcrlcrzLdrrcA oto .1ryor6g 6t o ,tircx(tg 6Ev [Xer sgrrtrud a@qEi/.E{li ooo\,Ict o€,X[rid( to 1t(r i| rivcL 0r:r!x(i,** 1() [610 0( o14l6siv€r rcr! fr€ rcr r]x.r:r, r(r Ut r(r[:] Lo.z. Ot 6wdpe6,4l €ivo! o],€E ltl oQ,rlruiaE.

'Eloqr€ rlv agoOeoq. :196ro vo Inoow€ .rvnt qv €Eio().n 6mqoe6v(xprlortlonoro)vtcrE io! rrlno,9A^.9 'urr), x(t6rtv vc( boi$€ av o:rlr10uop6gr€iv8r nQo< tric( "ordorlrl xcrlirorcorl' xcr ralo! v(r o!!,lflloo![€ ./rcl 1rE aleliE€KMarkov €l y6!€r. frtr rc Etxtrrlooqrr rou! l'otoprcloFouE rpdaeL aqdrtcr v(r6rs1o\ronorrloou$€ iov,4:

lr I = [3:?

'= [3:? 3:3] , dc( '4-)/)=^r-r '7)+o ?'

[:l] = ,^ [l:] = li i{ I | '- ,, ,^l

i j*l

r r " l l : _ I : C : F l : \ t \ i t o t ' A \ { \ I o \ \ \ \ \ f l r : _ 4 r

H;H:,?jl;T ""rci1cur, (0.7)r. .iiv€tcll :rot.t tt 7.e6< x^ q tior t,jive, o. tu(,

l::] ='' ."'iil

;$ti;{iffi p;ffi li5ffi #$'itt''l-'"'x**l';i:;:l j.ri:*rr,ffi [:r1#:'ll1$:i::1T:J.:i

! ot:o:a::ywl ttvgt ro toto'tstt,ott( rot, .t aot frtototrci arnt. d'tor u

;,; l.iil'il;',11,',i,ll:'L'll',i,11,,'.: 1,?1, "'' "'1'u" ' ".'o "' "" 7,"n,"" r,'r,,,INo, l , i ior i t rF T' l Of l , , r , i ( r iv ( lu. i iqr,^,r( , r Marko!:

3ir',.',::l;lillXli,'Jr".Yil:iffi:;;i: Fr .qv4nx6q niac]ia< ]d.tr<r) )t = I eivqt rbtorqri

'..,iir;l..i;.ffi:',|:ii]ifl:f ."'| €iv'1' Fr (rprqrLko xcr !:iv'r'| Iy) or un6),orr€6 L6LorrFic rxcvofioLow rlv )rl < I

rsti':i:i .ii:it',:- ti ;:,:,f,,:li: :# r",ii:r;,;,lT llt;i,.;.J:Jl:ro/.rorl.iroro rou.rr. nou rivur I oro0r.Ad x(rrior(orl u-.

frtt 1c .iaoitro ro ooord no),l.cn)doro rc

mfuil*l:ruff il ".,,,:ff ;i#.iHl':',1"!jil':,'.il"r,?,i.r',,:'#:',\ ,b,. z,,r,..o,,,r H. ,{,,, f,,r7r ,{,,, i,,,. , ll "'t't''t' '', ' "*,lX'.ii ,".]l;i]].

, ' ;,,;i:':J';::';''"il i;t;:t,:fi:l: :: T,,l:t ;l) H,,;::,,' ""' i;;, "". * :.r4priroo4 O rp6:roq p6 rov o;roio jrrAryudT

;:l;:li:l lt.:;l ;il:f ;tilli"r;i,|l;i'tJ I;iJ,"i,r':lT:lr :,, ,od, , !,, .,, u,7r,u,, o,\ ,,, n,u(vorrrrc ,,^'i" ;,.j Jlilii.;[J',t;1ru,;:J

ii?iilfil=[i]"".=.-

zrv€ilar raog ic €\rr6q p€ TrL0av6qra +. Eriv €ivar evroq, xrveircr *Qoq ttrr^t6E pu rit0o!6qxc -1. H xivnori rol yivetor pto ruTaia 6La6rmait xrn onivtrzcrq,4;ror rrlv ncroopi!€r l€y€rar rivcxc< pENs$aoEn9 AE! Edpoltl€ rl,€ovrou ' rx0rbu: I r lar. o] .) .d 1.o zr ior i roc ot Uwrur ' ;or; rul r , - ,1/ , ,n z 'r i ro0i louvr4a nrgav6-rlr.r va Eivar €ri6E rqE notrrr€iaq rc! rlv rl0nvorlrrr va €ival €r'rog.Or nL0av6q-r€E crt€g €1orv d0eolopo I -o{po{ rairer rdrou !a Gai.or€rarxcl| bev livor"r@ Trole oQrritrr}iiE. Aur6 Uaq o6rysi oo orr4 6uo 0e$elrrir6e(rbr6filrrg ro! nivorio pero6{io€(,)E: Kcio€ orrlU eler tiopologo ototlciu)! I xo!ztrvtlc orollrio 601 rjiv(L oQvrlrrxo.

To 6uorxo Eil+rc olq 0eu)Qi.c eivcrva rctctvotloouFe ylori rl l = 1 €t\ror;rtiworu t6r,ot4r1 xo! 1[dri ro L6ro6rdvuo[d rls €irar I mae€ea( xctdorcol.To fiQdrro olp€i,o sEll€iraL €!kol.a: Kdee otrllq -ro! A - 1 axet a(eQotopaoror1ri.Uv l - 1 = 0, daa or 1pc4.rpeg tor .4 - 1 i:Xo!"' doQoropa q pqbuvlxrtyAolpyl, €ivor yQcl4r|1id 6Edor416v€8, o A - 1 Eival rbloBoQqoq ro! I ^r = ]eirar L6r.or4rr1. Err6g €rbrrd)! n€erfirdo€or.i' ro r/r ocr reanar va rs.iver oroc\rr i ,orol lo lbrobldvuo$cr. Aur6 qoia€raL oro! rr i ; to u4 = 4'11,1, +.+c,, l i r t ' , i .orov oroio xqrlia Lhoqri 6e! FrioQ€i. lcr €ivc([ p€yo],1jr8QI ro! l. (Arcr9oQ€r!]riloL ft0(v6rnr€E ?la 0o cljEdvolrcrv €xQrlr(rr11ci oa\r rolE oQlo$oiE rolr Fibonacci.)Etir )'.orr6v or riLl,eE r6r,oxqris eivcu ylloi.(Dg ftlroot€Qt( t$ ^L = 1, ror€ o fiQdros6qog rou nr:ror.r xuqrcrglei cr:r6l.uc, xc0t14 oL ctL),€! )l rEi!o!l'oro plba! xtrr I

To rrLcuraio oivcr nc0(i6etypo €voE (rfi6 rq x€vrQLxq Erppot<r curorl ror1t€q( ato!: Av 6o0o!v TytrqQoqoQiE6 yro rov ,4, vcr 6pe0ouv :rLlpoqopieg 1xt tt;rbLorL[aE ro!. Ebd 6artr<4re 6rr ),,,,, = 1.

YrdQIer lra qa\r€Qn 6r(qoQil p€raEo rov ce!6pd! ro! Fibonacci i{ft 1,)!G\,€LtE€n)\r rolr Markov. O! aQLopoi rr, tirowar o1,o6vc pr1cl.!r4or, errir, t!oQlopou. xdo€ "nL0ov6xrFc" reqrlI€rlr|" gercE! rou 0 xur rou l. H €Eio(,)oIFibonacci civor aor.rdri( xnr. ro ibLo oup6ai\'rr" $E rqr €Eioc'rol 1o,{!oFoi Pa-r =1,06Pt, 616rr ro x€q{ilcLo pelc}.tirrrr o},odva r€Qlooot€Qo. Ed! ot rrroc\,orlt€SMdkov dr€n'av oro $Ibev. r6te rl rlioool0c tixo! erJoroo4g. Otr{tE 6€v rEivotiv,616r or: xd0r: (tF( ro rioQoro[d' rolls na€'r€r" va sivaL 1. Kcrir ouv€]r€rq Ul{la!6LLqt Mdkov €i.var

"xblrEaa El,oxaeis.

Y:roOroc rd;pc 611 6i6€r(L p!( r:Eioroq brrttpoQdr! r^,r = Au1 xdr OiLo![€vo lr€L€ ioo![E rr] oupfi€QlqoQo rq: ,{aodrqTo A - -. YroeirovraE 6rr o A6Loyo)l,Lofior€i1(1. 1''(,)QiEotBe 6rr I l.ilol ua 0c eivcr o!vd!co[6g cnl.tbv ).t]otarv.

1 2 IAIOIL\II! KAI IAIOAIANY:MATA

ue =S.a.aS ruo =r:rAfr; + +c,,)1,r,, .

+ Env oior oL ezro< rHamrolu(r rvi! ,cr 610r oL dr6( !.r(zLvon!t( cxr6q, rdrE ot

niloNtroi (urm!{orruL @,i iroE au' 5.! u.icrL or(0.ail "(amaol.

o riaca(< !er(6droeo<' , ' , ' ,

I . o l r | n n o d y ' u . ' o ' o 4 n u g ' U , r ' u u n . J ' u r ' r " , ' ,

5I Or L - r r3 r ) ! r : s t ] \q roPa i ! KAr (J r , \YN NIEI : A^ l l l

H lrrrc6o),1 rclv rrr. xcreopilercr unr! ro4 napdlowe; Ai\ rar I Fvold9tnlE .Eaadw and tt t,torqttE.

5J H €E(ooJorl 6toqop6v u&+r = /1rr €i\,ot

€r,oladiE 6rov 6L€E or rW6E rxqvorroLouv rrlv ^il < I

ot)6t1€pa aralaqi€ 6\@ xdroreE )il = 1 xcr or, urolomes

l ; ( 1

corc0rl6 dr<rv pic rouldporol dtortgtl rler l,\; >1.

ft4v euorn0r] :rqimororl oL bwdqrsK Ar r€irolJ1l nQog ro loD6v, ro! ro l6rcoulr6cliv€r l€ q tlion r/i = Aauo.

nAPAAEIIMA O niv(rxoi

.4 = l1t 1lI ' r r l

ciror oiyoue( {uor(orl< bLorr Eiv(L rLnf)vrr6<. or r6LorrF6E rou nrQLiXovr(r orrlvxuer.r bLay6lLo x(r riv(rr {) zaL l. :€xrv0)l'rcq c:r6 oroLobti:rorc upxrx6 6rinuopaii x(l| oxo).ou(kilvrcq ror xrw6v( ap.; = -4ur, :rtprl6uoqLe ntog 11 Lri<q Ocr tei.vr:r

,ti=lil.l.,lM:ropeic v<r beitt rov rpo:ro [€ rov ono(o I p€y(]rirrarl r6rorlFi ) = Izc0opiirL tll op(rpuvorl. MFrh lo nQdxo i;lltl(.. z/r0€ drdlluol(( lI1. eir(L ro pLoo1or nQolr/ortrdlo! 1or. H trQa.Ar(lLxri ('uvdr€rc ror :rpdxou 6r11tcro; eivc{ rrrbroondoEr ro ll,, orc 6no rbrobravllo$rlirl rou A.

lo l f 'J l | 8 lIr l = lr l . I o l '

z(lt vcr p$evioeL ro 6eirepo r6LobrrivuoFc (rou cA.nmoLXt-i orl\' ^ = 0). Ton06ro rbLo6Lrivlo!( tro ,arrLcrordEcr.l o€ '{iroe 6ripcr yt to ) = 1.

O€flroi IIiror€E ,rsr EqaQFo'yiE

Arn:rruooovrcrl rLg L66eq ror Markov lroaoutr€ vc 6potpe dvcr przp6 Tpuoo-prysLo (tvtd.tbq ryouerrrzd) erlc4polirv trov nLvrixrDv ortv OLrrcvogic.

fo l f1 l"'=I' l ' ' = +

IAIOTIMEI KAI T IOAI A'\2MATA

IIAPAAEIf-MA I O niwx(rg ttotp'oltiv.Jv-tEteropivl],v ro| LeontiefAur{ €ivo( lis sao rq rQ(6x€S F€y(i),€g €nrruxi€_c nDv F(loqFqrL'a dv OrxovoFrrda.Try €Eryoulr€ ?cllaox€rdEow(lg a\a'! ,tlvakt ,aruri),tuotls, orov onoio ro .rij6iver r1v ;roodrlrcr rou naoi6l'ros j nou Xa€rdlercL 1Lc tcr :rupa1Oei ptc povci6cr

10,4 0 o.l l (ato<i),r)A=10 0 .1 0 .81 ( r0oqr )

10 .5 0 .7 0 . I l ( roy . ,o r ( r r

To rQdro EQdxl[a eircL: Mnogoupe vc rraQd.yor4r€ y1 povir6eq crocrl.toi. y1pov<i6e< rpotpriE r.lL ,1 povd6€q €Qyooi.aE: fLo va yiv€L rdrrr riroLo :rp!ner" v'coXioor4Le ll€ I€lclur€Q€E noo6qr€E pr.pr.pr, 616rr r(inoro [iAoE ror\ x(rrcv(-),ioz€r(r orlv ibL( q 6rc6Lzcoiu :rcrptqoyrig. H noodtqtc nou x(ruvtrrliox€rc!tivcrr ;12 xcn rnopivrr. 6vc u:r6).orno xa0nptiE :toac"f{,)'fng / AZ.

nAltqtrya Na 6atod \uit'rapu p iror tbou p- Ap = yti p -Q A) ty.

EnLrp<rvcLarri. ro p6vo:rou lytr(iB€ va 6aoups sivcrr cv o I-,4 eivcrL ar"rLmprirprpog.Y:rdqler 6po5 xcrL prn crrairrpq 0erLx6rtltcrg oro nq66llp(. H lntlol xcr InsQoycoyriy',(rrr /, €ivcr prl opqrLadE. E:raLbrlro /, €ivqL (1 ,,1) ry.1nq<ryp<nrrf

fU,iirIoI l'o rov niv'r'r( nou no).;.ontr(ordqrL ro, rivcr:

I tot tod-A) | r i rdt t I aevqnx6€, t tvu.t t i

Ki(\,o\rx(< pr,( ,rodXErQI €rritqon, o .4 6r:! trnoQ€i va eivar noLu pcydlos. Edv rlbrclbrzaoicr :rcqa1(rlig qvcrLiox€r rdp( nol,},ti. 6er 0' nnopdveL ri:rorlyr"c qvrapcycull. To xLcLtti 6i6er q peycrT'riteprl LbLotL1trl ,\r rou,4 no! np€n€L rc €ilar[rxQor€Ql rou ll

E(iv lr > 1. 16r€ o (-I - ,.1) | 6rv eivnr g1 trpv4uxd<Edv lr = l. idr€ o (1 ,4) | 6rv u:rtippLEd! \ < l . ror l o ( / Al s iv( l r ddOorou(r ut uevt l f lxt ;v r , ! . r , { i , 'v:

To 3 €]ri 3 :r(pci6uwcr ix€r )r = {).9 rcr ra tlepz6Frvcr Er:reQvotiv rc eroeq6-lr€\'c. H n&aay(|)yrl Froe€i vc ]eoxrDQnoer.

OLc crurd <r:ro8rrrvuowar ruxo},c. 6rcv yvropilorpe ro xuEo lcqcrrrpL-orrz6 yr"c p1 uptlttxorlg nivnxq, 6norg o ,4: OXL p6vo q peyrd'uteprl rbrorqrrltou€ivar O€rrxrl. oiJ,i( ro i6ro ou'lr6aiv€r r{ [e ro L6lo6rdvuold rr. T61€ o (I A)-'6I€r xo i6Lo rbrobrcivrotr.r U€ Lirorui l/(r -.\r).

( 1 A ) r=1+A+ .4 r+ r t r+ . (s)

5.] OI ESDO>ED AIAOO|AN KAI OI AYNAI1EE AI 315

Edv q )1 lenepvir ro 1, r6rs o relernaioq arlt6q crqrep6q eivcL crgvrpLn6g.o nivcraE (1 A) ' 0a oril,vsr ro 0srlx6 ar.ivuoua 11 o' 6vn npr.41166dvropc r1l(1 ),) xal o' alt{v r1v nepintooq o (1 ,a) ' oiloupc 6evOcr eivc p1 opvrlrLx6g. Ei(v )r = 1, .r6re o 1 ,4 ec €ivar L6r6popqo!. Hr<r1xr1<ryLztl nepi.nt<uo1 ulerori5- oLzovogic5- eivor )1 < 1,6rou or 6uvtiyer"q ror,,1 r€i.volJv oxo [n66v (€wrdOeru) rcr I dfierQl orlpci 1+,4 +,4r + ouyrli,ver,.Ilo].),an].oor.d!ovro! avoi rrl orxQo pc xolr / - r4 cno[tu€r !6!0 o -r(rrrortx6g

nivcxcrq 6).eq or dl.l.eq brJvdF€LE qLl,lLoovorQoriwcr- xnl 6rol o (I .4) I eivor,dt€porouc pq crpvrprxrix rLvirxrov. Aivoqre iro naooaeiylaral

,=l? 3l6x€r,lt =2 xcL rl oLr{oao[iq na€l orl! xoicolQoqri

. fo .5 2 l II = | i- ,r11 rxrr \ -

t zut rrnoaodlt va rapc,Youpr orr6rnorr'

o, :rvu,.rc t.I- Ar-r rivcr o dur.r rrq br,o ,T"q,n..i,o*. l-i _i] ^,

l3 !]H oEi q( 4urv€uolE lol Lconlief 6Qior€1crr olo orl €60)06 61,(1 Bo\,dLo rou

IQ{orf(onor€i "y1]Jlo!(l ororl€kr cno rla rioo$arrrJl orrovo[l(r. To oLxovolrx6nQoyQ(ll4rcr r(,)v H.n.A. ro! l95li ?r6or€ixe li3 6ropnla\,ieg, xd0e [ic. o?i' rrE o]roi€S6orEl..,'E 6\,cr\, fiLiQ{ "fii,l(1xc €Qworlrv" 1{(rc\r<iloollE xcL n<rprryr,ry1E. H Or:ropi(!tou rqoXcoqd rdpc ur6 T ov (1- l)-r xo! gii(\,€[ oro\r xc0oQrop6 quorolo"/lxorvrqr(;v xar o! o4raro 6€l.rLoronoiylorlq. :lrvtoos to eglcrLxo bwcrpl]ro €i,'aLfi€Qroplopdvo t{o! nQdri€t alr €lcxloronor€ircr, xcr <pror,r<i 1 or"zovopiu 6ev rivctrravrois Ypqtrur.xrt.

IAPAAEITMA 2 Or rtpis o' iva xl.f'ori por#to trctaxopivov- xEtryoptvlrDt,To por,T€),o ),eyercrt "xl.sror6" 6rcv xaraval.ioxctcL 6,Tr oerBdq rcpciyercr,ItdQi.s rfrior€ \,.r 6./aiv€! d_&r or' ro o oqpc. :' currl qv ft:pinrrcrr1 o ni.vrrx.lg ,4rilaL 6va! rrivaraq MrkDy ,r,&L or oliXEa iXow ttqQotopd otoqtier l. Mlropo'ttt|rutr vrr srl.cpc /rd rlv ntrl rou uroa).,on. rt1c rqoq'tq n.{, r1c rpluoic-. uwi ./uro :rfiooE ror.v povti6rrx. n(L ro 6L(i.,'1Jop(r p nc0Lolover, rrpdq ovri 1Lc e:rim6o

Y:ro0torr 6rr ro po eivnr €vn bLcirlopa r4rtlv. T6r€ ro ,{p0 ro}.}.ar}.ao'dEErxUdE p€ noodntl€g Tra vn 6rboer" r[E -[+6g riro€ raoi6vrog. ,4.!16 eivr{. dva vsoori\rolo r[(i)v, ro oroio xprprpo:rorri ro otronpc yra rov rr0opLop6 rtov v6covrp,nv 4r7,, . To rU'rr lulr f iv,r rrv ur rurc or '1rrc xcrucirco4tooQQotrio,<. YndQlouv rrBdg yro rrE orroi€E p = Ap x&L prq o6l),ri &rlTrj Ifu.rbrrcoio rioQay{Dyliq o' aulaq;

A1'ot\,(r)Qi(€r€ 6tL to p €ilor ro ([q trpvlrrz6) r6ro6rirvuopn ror nivcxaMdko! A, rro! cvrLoror.X€i. orlv l = 1. At|Tn €i\r.rr I oroo€Q{ xordol(ol1,- xor:rpooe11i(e-rcr cn6 oaorc6rlnore cprx{ zarrioraot p0. H oL}iovo[io €EcQrdrol

116 l^to lblE: t\At latoalAt\] Y:IIATA

an6 to 6r! €novolo[dtivovrcg r1 6r"o6rrnoic Eovci xor Eavti, or rqr€q rrivouv

Iro nLnQ6xlra 6i6o!trE lrc yAryoQt €E lynoq torv xproiprov rbtotftcov r:v6ggtruot niraLa, fio! 6€v fiQdfie! !c or,yxdexol lr€ 6var, 0€xirid oQlolr6vo riivaxo,o oroi.oq €ivar oufrF€raiz6E rar 6t€r 6L€E 116 r.6lc)r+6! ro]j 0€rili6(. E6(i),or:rrxd€ivaL oI0 ra oror"X€i.a a,j.

5|( H pe1<rl"tirepn1 r,6lor4ul )l ev66 0€rrxoli,livdxd €i.vcL rpcyrcrlxr] zor,Oeuxr1 xcr ro [6lo onp6oi!€r *dr Ue rrg ouvlo[6o€E rou !6rcblquioporoE irl.

A106€4r, Ynoo€or€ 6fl ,4 > 0. H x€\rrQlxn L66c Eivo! !o 0€coQ{oo1]F€ 6Loqrous uQropotE I yLo ro]JE onoiol,s ,,lxi > fu y!c( ridlroio pI cQ!4i!xo blirvuopor (6rffpope-nr6 rou :ri = 0). fTllv ,4r: 2 l;r €rirrQ6rout-rc rrp ovLodrrlro lrcr!c 6loupe noLlok rTrolriqroljg 6€r!xor,! L frr qv $6yc1l.rrreQl xr"l.rl i.,," (fiorJrnrrwxdv€r(lo 0s bei,Eorlr' 61r to|iEt tt todt\rd: As = fNrr.

Al(rgoQ€rlito, ov Ao > i,,,."? 6ev {iov toorrlr(r, flolLanrloold(o]Jlr€ !e ro!A. En€lbll o ,4 €i\rcrr 0€r!x68, nQoxd/ir€r" [!a l1,iorcr cr1,ro6rnrd A2ri > t,,,,,/4r.AQs ro brclvuopcr1/ = Ac txovotrolEi Iry A,V > t,,,,1:9 vat o t L,i ea Fnooouo€!c p€1,cLcio€! a€Qloo6r€Qo. H crwiqaorl o]j1li eri[6sLl,E! rI\, rodqrc Ajj = 1,,,,,"or(lr 6Xo!$ p|"C !6LorL[1i. To idloblavuopo n €i!s! o€r!)io b[6r! or' aQlo!€Qa qELoofllr(lg. to A:r: €il.rr oiloueo o€rrlio.

fL.l vcr 6€i.i€ 6rr xcrpBicl larotqri 6 lrnooEi ra €iv(l! trEycll.rrrsarl TlE i.,,r,u:ruoiorr orr ,4: = \:. En€rdri rcr \ zcr : p:roqei vc:rrpLdXouv sQvlrrxo]Js IFr"ya6r"xotq cpr"o$o!q, rciqvoupe an6).r.lteg rr,[6q: l)] .? = ,{: S,412, ouuqo\'apt rrlv 'rprylvLxrl cvLo6npa". To zl olro eivar pr1 cqvltrx6 6rdvr:opo, dQq1rl l)l €ilor dvcg nn6 ror,5 8rNcroug rroplqrorq L AAa I ) 6ev p:ropei vcEEn€Q.,'(i q,' f,,,,.. rl oroic nptnar vc eivcrr r1 p61Loq r6rorqrrl.

Auro eivaf ro O banp ftDy P,rro*Frcbeni ! .yro o€xrl.ioig rrf1,orEE, rooroio dle! pi<l (t{6pq €goppoy{ oq pn04rcrlui Orxovopio.

IAPAAEIIMA 3 MoviiXo alanit)outipeq€ Obto|opid€ rot Von NeumannEnrorpcqou[$io(, orol' 3 Eri 3 nivd]{a A rrou 65Lb€ ry)v 7.ara!d].oort o€ arod].r,IQotprt trcrr €Ol|otcr. Eciv tc e[eqx6pevc eivcL sr,i,lr, ro.rr rc oranongeva€ro€Qxop€v(l €ivdr

o,1 l f3r I0 .8 l i Io ,1 l L l r l

^ l:{0 ,10 ,7

= Aur .

:xtv Orzovolic I rliocoot brcqoqdir nder nqoq rc riorDl A.!"[i .!L& 1r1 = Aut€Uouue ,0 =,4ur. Orov o A eivcL 1,uxp6q (6nog xcr eivcr) r6re q ncpcy,r"ll

5.] OI E=IIO:EE AIAOOPON KAI OI YNAMEI: AA 317

6ev rcrql'ctlioxE! xc ndvro xal I orxovopia qvqfliriooercl. Or !6LorL$€6 rouA r 0cr xa0opi,lou crr.nriv rqv cvri:rrufq. Yrriqlel 6ptoE rrdfu Fla qroirrlol0€r|x6xrlrsE, ra06q ro srodl!, I rQoqrl xsr. I €eyctoiq 6ev FroQ€i, vd yi.vor.rv(rp,vlTlxiE ;roo6qreE. O Von Neumarn qvqltqoe rov ptloIo puoFo crllrp1E Ilr€ rov onoio FnoA€i r' slqnrnj(o€i I olltotoFi(l cEaxo).ouqtbwaq w tutpapivtt!4 aavtt tu1, rrox o\vui ler orr r>hro>0.

:r!€?i(i)E ro ria66ln[d oftorr€i ui > tAul xat €ivo! oo! ro riQ66lqld (DvPerron'Frobeniu!. p€ ro ,4 cl/16 qv d).ll [€Q!d. on(l)E x(rr nQolyor]tl€!0)6, I ro6rliooxi€! or(av ro I yiv€ror 1,,,,^ Jrou €ivcr I lbrorrpll I onoto crvrlfiorXEi oro oerrxorbro6rdvuol(r rou ,4 r. !' (|rn6 -ro rcpdrberlpo o ouvr€l€orrE 6lofioLr1g eivcL f;.

ME crodrl,rrQoqri-€(ryooi(r oE avaloyia l-5-5, I oLxovoFio ovonrrioo€rol rorolurego bwcr6v nor o piyn@s Qtgp66 svti$rtuE4g ztuar 1/)1.

A'KHIEE

s.3.1 o) B0nrr rs 6wdFr,6 e:. Ar. .rr r.r rrvo'{c tou F bnn (i A- l i ;] * .".*,"(xflorpo ordlrsE ro xoitrslo) vp,1r0r).

6) Bprirr ror a-r"r 6'" ' . d = [ 7, r2.1.

5.3.2 Ynooam. &L o Fibon0cci E€arroiop tltv dxolo$eia rox Fr F0 = I xo( Ir = 3,(ao),of06rag rdL rov i6Lo rd\,61.l at,l = Fl*l + Ft. BA€il€ to 1].o (Otrr66Ldvlota r0, iol]g \,60r[ o!'!rFl€or6s 4 = S rao xd! rouE !601,5 der0Bol! Fibonacci.AriEtr. ull nr ).otor F^. r// i rfilou! flu).' nooc rn Xeuui rorrt.

5.3.3 Edv xde. aeroFdE strat o ttiooq .jeo| rntv nno neortybrtlvar tuu, Cr-! - (Gk,1+C^)/2, 6eEic roa rivca( A xaL b(ryovLorooidrE to!. !€ara{nEus ono rd c0 = 0r.rr Cl = + 6A€iie 3vd! flrro yta l(t Ct r.o. !trotroyiore ro oqro. ro0d,6 ro * - oo.

s.3.4 O B€midelli pe)'6qo€ avu ol|a€dar "no! l0r rai.l lovo Xpolto xcr dvonoadr€rarorov reiro xaovo". Ed! r opdbd e!6E drorq €ir!6r6!€r !e 'rred!6qra l. rr oio wavgr rr0cv6Tlpa I xcL 1 rpiq raqdysL 6E. erl.uxd xar r€eoiv€r, ror€ o vcxas €i!a,

f r l lo.4 o o.rJ f r l fo.el or= l5 l r ( l r 1 r=10 l r .1 0 ,8 | l s | = la .5 | = hx .

L5 l 10 .5 { r .7 0 . l l L5 . l 14 .5 .1

f9 0 6 ll+ 0 01 .Lo i o. l

Aei*E 6rL A3 = / 'rft Jraecaoi.oxoitmE qv xaravopli iov odooa(lv yta 6 xaolto!.!r!ri,wc rr. 1000 o?oodaru oudo€ oF{ibo t},uids

l l l r IAIOTIME: (AI I AIOAIAA'Y'MAT,\

5.3.5 Ynog€or€ orr !ftdqxa lia Eftr6'uio. oqv oioio ado. Filvd or prooi aft6 ror5 vrl.tsapa(Dcoivow xdL 6vo rrra$o d6 rolE aeeddots tr€Ooivow. Be€ir€ rI ors0Eqd'rcriiordon

rra qv ovrioiorm oviLEI Mdkov

5,3.6 K(lttrrQdwe io.',3 €rit 3 ?iivd,|a peroddosrrE p16s Fa0ipdtos Xry€Lds fto1i yiaEa!o€ 6rio q"iFoE. &.,! xdo€ €66oFdda I Tor qufuctos A rsL I rou r!1ipffos at |rtnalriaow to prie4Fa xcr I xtiet nrrlFaro5 :nrycivo[v oro d.Io lurlFq.

5.3,7 Baeit€ tl< oQrd?dc rtrt6s to! I/t xar :r.(l - oo) &(v

,r+r = 0,8r1. + 0,3.r i/l] = 0

:r . . r =0.2Yr. +0,7:r :u =5.

Bpeire rnlo6 rurorg 1tc tc y1 ror :1. cmd rry :1r = SA[S-1.

s3"E o) E€ r(lvrag on6 io 'yeyoro6 d'Ir lrt o!'tht + 2't oflih = 2(3't orrl},rl). dao or ofll},r<.ivqL yqdFfrxd cEsqqpavr€. 6$ir€ pic dno rq L6Lorqt6s rou I 'ior ro awiolorlo

6) Berirt lts (m$ L6lor4rds rou ?r..y) Eri! i0 = (0,10,0) 6aEitF ro 6qto iov i^!0 yLd I - oo.

5.3.9 Ytio06mr 6tt ur<iq,(our tptc xlpLc }1drtpcr tvorxn(opdtor qoeiry(lv. Kdo€ pilard prod ofl ourd ?lou 6e(o1t(^'roL ont Boordrvil yar ro Aos Awl€L€s r{i1,€ (no:rx4o. r( d},),o prod favoljv ont eaurl ro1\ roL atitd rou :txd.ro! porodlolror e€ioor FftoEn Bom6wE l{(r Ao! Awlr],rE. OndEr. rov 3 €ni 3 nivoro p€ra6doroE,.1 x(lr 6qrir€ q (r(aored r(lrdoraor an no! alatotoLlFi- oq! r6rcr+trl ) = r.

5.3.10 a) IK rorrE rqr,q t.)v a xor b €ivar I ffiqr€q €Eio{ool prd ola],rEll Ma.kovi

, | 4 6 I l l lLt - " t t l ' * ' " [ t l

6) vro).o1iot io ,1 = gArs 1,0 yLa xdge o xaL ,.y) Yn6 itoi€s (ry(hix€s yra rd a xar 6, ro !* ftQoo.{rild ava nen€qaoEvo 6ero,

'<40d,! ro I - l)o ff.I roro €ivaL drtr6 ro 6aloi Mirq o n nq6n€L vc €ivaLnivor06 Markov;

53,11 no}"llrovrxiE €iataEiss otE H.I].A., ldJtu*ia u& E!A{i}q 6rouv eto]redE€rs 4 raroe-rarotFrqi(l)r $. :rI! opt0i 2 latoe'tflotrpnaro $ €lvor on< H.n.A. lt(r 2 rALoEro-rolltrnan S sqv Exqd,nrl. Kdo€ raoro + ior xarBdrov mL< H.n.A. per€! orov i6ro

f r / . , ,1 f r 1o l fa* t

L;,rl =li i il L;rl

f 0 ,2 0 ,4 0 ,3 ' l1= 10 ,4 0 ,2 0 ,31 .

Lo .4 0 ,4 0 ,41

5.] OI [:l:A:El> AIA@OPs)N K,\I OI AYNAMEI: Ai.

1610, j nda oqv Ianovia J(or np EttadJrr. Aflo qv Ionovia ,lor rrv ErJq{i,nrl

F;'€L cov i6ro ioro "a,

+ n'is. orLs H.n.A.a) BedrE rov riva1!0 no! 6i6€r

6) Bq€i€ rL( 6lotqrdc xot ic L6|ol|avuot(lla rou r1.y) tspri'r rrl! op|axli rcravo$1i rov 4 rero€xoroFpuqio! $ rard nt outtdlel( iolJ

xoqrou.b) Barift: rr! xdtalopri to 6roE A.

5.3.12 Edv / €i'cr rivcras Murkov, d8Lk! dfl io doAoropa ru,' o!1rmos6! ro! ,{rtoontot rr€ to doAorop.l t(')! mfloroo(l! ior' '. :l,lln€ad'aT3 orr edv .lr = h $F.r / 1, or ouvroi6o€S tor r6ro6raviotLolos di(orv deaoloFo FI6i'.

s.J.fJ H lidl ror Jt dt - An - [; 0 lu ((btotrrris i r( -i) 6tsrqdg€r ivov turl.o:

r = (cosr.sinr). Yro0tors orL nqoo€l.yLgolrr !€ eEtodo€E 6tdqoe6r on6 elnqos.aao rio.) iar aro q Fdot:

O r , , , r - n r = A u " i u " i = \ I + ̂ ) - "6 ) r , , r - ! " = , 4 u , , r r i u " , r = ( i / l ) - r ' "y ) r . , r - ' " = / ( r , , , r + ! , , ) / 2 i ! n , I = l I - ^ l 2 r - t ( I + A / 2 ) ! , , .

Ba€ir. ns 6,otq,6s rc! / + A,lt - A)-' xa (t - /t/2)-'(t + A/2). rtu notc€ltoorol 6roqoedv n(raalivcr rt truot ndvD o€ xntloi

5J.14 f( tro€s qrig rou d rq)xlrrrL ((n(e (,rro oi'nn'( r,,,r = d0", + tr', ), d'', r =

5.3.15 Baeir€ rts piyloles r4ds ro,! a,,5,.yl(l rs onoiei or sn6pevoL trivdxrE riaoL Euma0ris

i o!6i1eQa EUdr(eEk:

I a 0 ,81 [ , 0 ,81 fc 0 , r ] l10 ,8 0 .2 I ' t 0 0 ,2 t ' 10 ,2 . I

s3.r6 nolr,anl.cord(oro( oqo trqo< oqo, rnd),rpelgtr rry (1 - /l)(1 + A + Ar + . .) = r.H dL,r€Lsn awi o€ed noarordv.L ro' (1 - ,1) I xnt eivar Fr uqqflxoE. 6tcv o ?tsivar f'l deqtrzos, x( r$ 6oov dr€r n€rEeootdvo deeotdld- Md Laaui oNeiix'l'|'tAno rivnt 1)r < 1 KAE| qv reoo0rol nE orrEis raL maiq0€ncs &r rod0qorqrd ldoi'tdr F. (1 - ,,1) r, yro ro! riraa( xor(\,(t}'6o€os

l 0 t1 l //=10 011 , , ,

L0 0 0 l

5.3.17 BqEir€ rLs 6u!i4r€q ,1t (tou ,,lD oJpffaL).qr6.l!o!!!o!) tft 6€iEre aluluutd drr ro

risao,opd xoq roorno, r,€ rr -,rrr..* ' a = [! l].

fH l fH ll r l =Alr lLE I * . r - . . LE I * i - .

l^loTlntE: Kr\r r^rodlANY:MAT,\

5,3,18 EEtl.im€ pF taOnlcrrzd rl orrolopi.r, y(cti I'sy{rldrvovras r{inoLo Gorreio rorl''riivora adroval6o€oq' /, aljEdlFL ,rc1i cldyxry xor io t,,N = )r (aar F(rblFr(lt rl(!'inr!El tqorltolol|faq).

53.19 norc .ia( ra 6aro. t(( * - x, ( orc0splq a(rando€L(), to!

t 0 . J r ) . 2 1 * I l t I 0 . . 1 0 . 2 1 ( t 0 t t 0 . . t 0 . 2 r *lo . r ' u. i l lo l " " ' lo . r ' r . r . rJ I r ] *" ' lo .o r , ,u l '

5.1,20 AIEIF "i x.iljr rqiruc uerFF6c Fih.nrco rjrlor uQnoc

5..1 AIATDOPIKF) hsl)o)Ll)j KAr r(J EKc'trIIKO Cil

5.4 I AIAOOPIKEX EEI'ATED XAI TO EKOETIKO eA'

'Onore ourclwdre dvn oim4ra rErodroet:v, crr'rt yta pia €Eio{Don. I0€0)Qianrvclx(|)v ix€r. vq roiE€r zriroLo p6),o. Aut6 iollJE yla €Eto6o€K 6raqoQdv, 6trourt ).uot r/,, = /luo elaprdco c:to tLE 6uv<ipeq rort .A. To i6to lolier',.ct! ytcrbrutlopudE rlrorirorr4. onou 4InoI u(t) =(rriu tEoQrdr( an6 \\ ntoFrtxilrttrtitQr4a'4- rolr ,,1. f[(r va oQioouF€ crw6v tov €z€erl1io ni1,axtr xat vtr rovxcr(rvoriool4lr. €E€rdtoupe 14.16orog evo ncpci6rLnrcr:

l 2 l

t tu I -2 1 l. = Au =

11 _2 ]u , ( l )

lqragtiro 6rlpc civ<r vc 6qoipe rLS L6ror+6E xar ta t6rcblaauoFcrrr:

f r l f r l | | I I r I,{[i]=(-')[;] rLj,l=r-:rl_rlKar6n!v. ur[dQlolv r[o],]d< olvnr6rllTeg ror o6q1o!!, 6],€8, oq! t6!c1 ,'uotl. 'Ioorg

o xcr).r-rreqog rp6:rog eivm vc|yprirlor,ge q 1wft{ }.rrol xcr! !a qr, ,rQoooQFdoouF€oro up x6 6uivuogcr 1r0 yra t = 0. H yevlrin l,uol €ivq! ow6uaoF6E xntsoQ{iexOsrxtbv l,uoearv, oL ofioieg ei.vtrr, qq [oQgrig ce^tr.6nou,\ eivar r6orqrti rov,4 xclt .r crwiororlo r6ro6r(i!!oFd rIE. Or l,uo€K (rul6E lxsvonoloiv q 6loqoQr.r.riaEiooorl nrgori dldt(ce^Lt) = Alcc^t rl. ('Ht(v rl €LosloTi paE otLE L6oqrdE,orrlv cpp] tou zry(ircior.) :' auro ro 2 rrii 2 n(|Qit6€rua urdQ,Xow 6io xa€.rQd(rx0rrLxaE ol'vcQqo€lE ,rou flQtn€L va ow6uao0oiv:

u ( l ) = . i r r ^ r J r + c 2 e ^ r r r " t i ,,=tlTI XpovLxrl olryp{ pl66v, tn er.toetlxti ei,v(|l, e0 = 1 xqr I u0 xc0oeiqeL rcr cr ror

11, =. r ' / , + r r } l n . = [1 ',] l::l =*E6rir avrqvopilere rov dvcrxcr l6lo6tavuogdr0rv .9. Or. otcr0ep6g c = .9-rtiri,nr or i.6r.rE p' uur€g no! xQlorloflowlooge (}rq €Erd)os'4 draqopdw. Avr!x(r-olor(l)vro< x|{ orry (2) 4(outr€ fil luol rou nalo6LiBaro<, I onoic, ft€ trivclxsg.y0(,!qsrft

',]1"-' " "]l:l] o,

.,,,- ll "J

[' ' " ,,] l:l] ='[' ' " .,]s ,"

I6oi o SrlrLiirit,lqt dnog autii q€ naQsledqot)t & = Se^rs-rzo, o oroios l,weLq 6Lcrrpoprarj rfiotoorl,6nog crxpr6ri6 o SA*S-rug iLvv€ rqv EEiooron blogoQ6v.O! fii1(rkEE-rl.€L6rd €iva! ol

n=l t .] *o, ,'=1"' " "]

I IOTIME' KAI IAIOAI NY:MATA

Aro npcipn-ra cx6pq rp€ner vo zavo4rt p' orx6 ro laQd6€Lype. To rp6roeivcr vcr oupnl.rlp6oo4e rc pa0rycrnnd, 6ivol'ra6 6vav an' €ue€iaq oaropo'ror E \Etrxol ltirsxs. To dIIo ri,var, r<r 6(boor$€ quor"r.ri eqpqveic-16oo ovlveliotDol, ror 61er j(pfor,pq rqxrppoldg, 6oo xcr. oq l.6oq vlE.

Aa1i,lolF€ pe xo er0€r[ra6. f|(l o|oychvlo nivarc A, sut6 eivot suxoi.o. Tor:^i e1er, tor,5 n apr0porg e^l or1 6ray6vro (6nog otqv rpoqloupevq rupi.nrororl).fla 6!ct! I€v[r6 riv@ra,4, 11 rpuoLo],oylzt] !66a €ivct! vc plpqoow€ rov oQr"oF6pe bwcpooerpd

,r2 11€ " = 1 + r + ; + ; + .

Ei{v a\'flldroorlooull€ ro E [s At ,is! xo 1 [€ 1, ro doQoroua €iaar 6!ag ?rhd](dg

eAt = I + At + (At)z

+ (4t)1

+... .

Aur6( €ivql o Er0€rrr6E rtvaxaE rou Ar. H oelQd oqx].iv€!dOporopti r1g dleL tLg L6t6qreg:

k^Nrk^ t ) = e^ { ! , o(e r rXe - r r ) = I

ft.ko') = t"o'

(3)

(4)

(6)

An6 ql 1E),€woia 6lano1JpE 6r! q r = cllrro l,rjver. q 6loqoQlr1l eEio0)ol. HLuorl crrnrj :rp6nrL vo rivcL 1 L6ta p€ €z€ivl 1l tloQqli roo xQloqroroulsq$oro! nQoilyoljlelo 1]Jrotroylopd zaL no! irov I z = S('^1S-1?4]. fra vq beiEo4r€6r! o! 61J() aur€E ?'uoer,g oupnimoov, oqqoeite 6tr zti0e 6uvcrpl (SAS ')r

ol]lrrrxrJoo€xor. oqv .41 = SAfS-l (6r.6tr" o S I 6Eor6€t€80)16t to\r S), dQc,o).oiUeo; o Fzof fl rio6 niva1roc brayovrorror.iror rcutoxQov.r:

s ^ q - r , r S A , S / ', , ,_1rSnS ,_" , ,1 r , 3 : ,

-s i r - , r r - r11t ' * t t i r ' - . . . \s ' -s, ' ,s .\ 2 t t /

I IAPAAEITMA o er.oerrxoq :Tivcxcq rou e [-] l l . ,*, .| | - t l

, ',=s,',s =fl Il l" ,l l ILr r l l " l L r r l

= t l " , . r t , t+ ' r i l

5t 1tA4,(JPI(E: [!I:A:EI: KAI TO F.XOt-] ftKO rf'r' 323

f!c1 I = 0 ficiQvollil€ ro,, ror,"[orlx6 lri\@ra: ctr = L H dfi€lai] o€lair crr 6i,b€[q l!|o{ yr..1 r.{d0r /, oDd rno),oyilErc( 6ioxo}.c. H Foag{ sc'\rs bi6€! qvibra ),uorl. orav o I €i!G! 6rcy0),vrono[1o4ro6, o;r6re urdpXolJr .l ove[trpqrctrb(obravuo[crro oro1 S. H poprprl crwf fuq eivcL roLi airlouorrprl rtrL obryeiot ouv6ulop6 n xc0cpci er0errxrirr owcrprriorov c-\l.r eircrL ).oLa6r q xcrLureprl(ln oir'. rr. tropq€q ),uo|lq:

5L Orav o -4 bra.fovrofior€ir(rr rcr ,4 = SAS I, rore rl 6r(rqoQ!1{ri€Eio(l)ol du/dt = Au 6X6r q ).iorl

u(t) = en'ro = Sr:^tS ruo.

Ol orrjLeE ror.r 5 eival rc Lblodrtrvtopcrrc iou .4 xci 610[

(7)

,,n, = lr, .,,1[''I ^_,

= cte) ' l l r + . . + c, ,ar"tr , . (8)

H l€vrxi Lrlorl €i1,sr ou\,6ucrop6E xaooQo Ex0€r!x6v ows€mioeorv x(.! ororrr0€QdE .ri nou rrQooclpg6(ovrcc oqv <41wri ow0irq rr0 etvcl c = 5 rup.

ArlTo (lv ro ouyxQivcrr F€ ro 5c bivrL trLa nLrlQn (.!(1l.oyia F€ iq €:Lodo€L<6[q]oQ0)!. K(t ottg 6uo ntprnrcloeq uo0ioulle ot! o ,4 propri va 6rcr^ltuvro-norl0t:i. brol! bl(qop€lLxri fkr ti.1u Lrl6rrpc (rn6 D ldlobm\ruoporu rur 6s\r 0(1tlpion(pt: <rprerdg erdLr{dE ),xoeLE. :xlv fiQ(^/pcirxorllxa. oL }.l]o€rE nolj Lrifio!!',r(ialolry, d.L(i €ivar nLo ouru€rEE (n6 xG it(0ce.i Exqeav.ia y.ctL oxeriqow(Lpr "lcvrreupdvu r6ro6Lcvnopcrrcr" raL :rcprilovrr< 6roq o tcrl. naa' 6L' ourir.o rnno( r(l) =.'rrui, rc4c4rivrr oL6rrla olrot6s.*

O aiv<rrcr6 err €irtlt nrivrorr ojttoteivtpos. Mra an66€rEl x(ipvorl(€roLrtirvtc< rL< rbrorqr( rol. Ed.v ) €ivcr t6Lorarrl ror ,4. r6re e\' eivcL 4cvriorotTl r6Lor4rri rou efi zcr dvcg cQrutr6! lr1; Fogtrlq z:l 6tv riva nor€utbia. Mrs ciLil r.poo€ ryror1 dvar va u:ro).o/ooq(c rlv oQi.lorlotl. nou €i\'(r ro.r/llollcvo r(Uv LororrF0i!:

det e r r = " \ ,

r " . t , t , , . "1 , , t

_ r i7 \o : l l t ) .

Kul rcil!. cr,ro 6tv pnopei vc pqbrvilrraL. T6),oE, o xcLrirepoE rp6nog rircL r,'uaytapioorpt rot e '\t oav tn mpoqo ),6y0) tlE (5).

r li( r( fnololiuor$r d!.1qr r).crot.rn"it flernlul fnoaorlE ve ratorlorooloorjrr rl

troa,fri J,trLi ror n(a(r,r,ir(ro< B zdr yd iipoiM ro! . ".

IA]OTIMET KAI IAIOAIANY:MA A

H ovrrorQ€+qr6qra ourrt €ivaL 6€[€Lr66qE trc rE 6r..rqoarx6E sErotbo€K6i6rr 61er qv €qfu ouv€]I€[a: Eciv rl. l,tior:tE ei.l,cr yQa$Frrdq avEEdprylr€E yrot = 0, 6\E rdaatitouv ypatqrxtbE avEEdeqrcE yla tulnc. Ecr\ r'cr ,;.pxLlto.

6rcrvuopcrc rivcr 11, ..., o., tuloQorip€ vc roro0errlooqrt rq Luoeq crio oe 6r,cv

[ . : . 4 1 u ] . . . e 4 ' u n l = e j i l r r . o , , 1 .

H opi.loroc oqv opr,mepf nlerrad 16./€rcr "lovoxLctnj" opilouoo+ lic! 6t!pqSevileror, not6, 6L6tL eivc{To lv6gevo 6uo prl p16evln6r opllo!o64. K(lr or6uo rivaxeg beErd eivcr! cvrlorQilJ!0or.

DrtpEir"aq LE\ sivcl 6),€E or, 6lqqo0lxdE eErocbo€lq riou o!!c!rd$!: suorllrqrqnadrrl< 1dE€c0( xrt( $oQqis du/dt = Au. nQdyptrrr, !7roQei, vrr EsxLlrtoow€ ffI6

ULu Ft[ovcuuivt tEro,uot av.rrAdg rdEturs. dtru< I

l l " ' -3y" +29' =0.

f[o vu $€rarpavoup€ Frc( r6tolcr o' dvo 3 €ri 3 o!orn$o, €rodyouFs El(loE rqg

, )rq[ rlE 6o10qrr.ldE potcr6trIt6E ?] = pt xat w = y", Ol 6tro ou6g €E!o6o€!Eowblti(ovrcrL r6re pe rrTv apxlxi 1{a! bivolrv to

r l 1 l ' =

'Exoutr6 ricrl.! oiorltl(l fiQ6rrlE rdE€os. To na66lq[a Lo!ti6\, ltroQ€i vcr LrJo€i tr€6iJo rQ6fioug. f' 6vo Friolpc tiLcqoqrrdrv sEro(i)o€(,)\r. 0' o"4!x(0!orouoqx€ orylv€Eio(,)on xQiqE rdEcll)E ro g = €r^':

(lr - 3.\2 + 2))e^' = o ,i ) ( ) l X ) - 2 ) c ^ f = 0 .

li I il trl

O! io€lg ,fo0apd e:r0erLxdg hroeLg tivcL y = e01, U = et ro:L u = €2r, Xoroiq vautrErodaxovrar LbrobravuopoT&. :' €1/( p(iortprr raapFu{tlE A}.1€6aaE, raolt|}Qiltr€.6rtLrq otl\noi(€-ro! yr( ol]ornF(rta rprirrlq r.iEFo<, dpioxotcrg ttg !6LorL[i< rou

de'(,4 \1) = = .\r + 3)'] 2.\ = 0. (9)

x :rE.: An6 t. 6ao[d ror fdioroiovon q!orxouu01|l(1&r JH w'onslii (naoqaon(t

f ) I o ll 0 ^ lL0 -2 3 -^ l

Kcl! riolr, +qctr'i.Eo\rar or i6LoL rgeLq ex€6-rtg: .\ = 0, .\ = 1 ncr .l = 2. Au6q€ivo! 6!oE y€vr.r66 rar'6vdq, ro! xciver ttg 6uo F€066o!E ou[6r.6aor6E. OLQuo$oi p€xo6o).{E r0rv Luoecov ei\'cr dpplx-rc oxr,6€6€[i!ot ft€ ro rQo6]'rud, raL

5.r A|Ai|,(JPIKL! lal:AtElI K I TO EKOET!(O €i,

rcrpcFdvow or i6ror 6rav or €ElodJoELS dJdqo!.r Foaqi. H ),uot 11; eli.ooo1grair$ rdE€n)E [c.( fqiverc[ ()fiooo roxir€Ql].

:rQrq6Fcore r(irQc orrt qloLrlli rpplveic q< rftrl,lt = Au,!€ A = l-? ] i| | - 2 1

nor €ElyEir(! elixo).a xcL towdlpovc ncrqouoLirleL npclpcrrxo ev6r,cqepor,. ilbr"aqopLxri elioorol repL.lg<iqer, pru 6r"cbr.rcroia drrirl,orlE, ,rou €Ety€ircr! 6!c(QdJ-rra< ivcrv orrrLlvc crei.Qou lrizoui oF raoor.eo rgrilrattr. Tcr brio geoaic 4ulpcrtcrrirc{ :rrreg.1op6vcr xcrr ra 6uo (rpLuvcl rivut ryrrunr:pco}tva (:X. 5.1). TrIAolrzti olygri t = 0 rcr ar;rapcrop6vc 4tycrcr r€QL6/puv zdnolo trltt.,{6 bLd},qrcrovyz$'rguoqE r.1 zcrr 16. Trlv i6L<r Xpovrx4 orrlp{ u},}dr zut or 6},eg rLg lpovLzeEortwt_q, 4 or.1z6vr6nrq oLqliFsroq or( 6uo cr;rerpo qrnFara €ivoL {). A6yo)tou anripor 67rou,1eLx61,c (urri q( tiortc ol,yxivr(rorot; rou brcd.itrtlrog, or((inuLec rtLtttcr(, €ircr ooorl xcr" perri rou 0c upXioeL q Xqpr,rq ouoiu va bru16-0r(r. H brdXUoI Eezrvci o€ Xe6vo I = 0 xat xooopi.lerqL qrr6 rov rjri6lr:vo v6[o:Et xdgt XAovuri otqti t o eu0tt6g \ta|rioEtt a rcfttEi 6io lErto1ltr(i)y Nln uitft.yftoilar rc q tuqoed flot ouft tyre(;)otor tort 6ts1.{ttdlog Oovr(q6Faorr 61toF ncior rtritro rt orryx61,rQ,(Dol €iv(r oporci|og1,1. To:rqo6l,npo rivnr olll'!1dE rog:reoq ro! Ieolo. d.).d br(xQw6 o): nao< rov Xrirpo. Or [6vo! dyv(l)olor ril'(lr o!orh/zrvle6orrq r(l) e(.! u(/) fic 6 o €oor€Qlr(il 4r1iFc(q .5 r ,((|!.S:.

H ou.fztvrEDoll r. tr€r(6(i).lsr(! 1rc 6lo io.yorg: ),oyr btaXrox,4 oro(QLorrQ6 trr-yd),o rt il( 5r z.rL l6y{,) 6L(Xrlo€(l)i oro ri s 6 ro 5:. O o}.rr:o<eu0l'(x f r-t(6o).itE €noF€loE, €ivcr

7r

in = ( 'r, - ")

+ (0 ir,

= (0 '1,) + (r ' ' / , ) .

€r€rbt I or,yrdvrQ0)oJl oro ,50 rivo! rolrot!*Li Frt6fl. nopduoKr.

'Er('r ro oidrrllr( rrlrard.q ffiol,ur( trs ro nced6€ryfr.i ttea dn/lt = Au aW€qio(l)ot (l):

| 2 r lf , , l du | - : r ' + u , l'=t,l a=l' -z.l

:X. 5.1. Mowo,o br(i)tuort<

q ,t,

IAIOTIIVIEt KAI IAIOAIANl':Mr\T

oL rbror+rtq 1 r'iar 3 zoeoQiEor! rrl oupfiEQrqoQo xI! LriolE. 6ivovrG roprr0p6 crpoi0r6 ttr;v oq,zt:wprirotrur. H ,\r nai.lel ortFavrlror€ao aolo Srorrlolo 6!c ftol,{ E!6!ro oljvolo cQzw(bv o!v0lz6v F roA€i !a obrlyio€r o€ Fra"ureQcQoi.(')oli Be ro1 0u0F6 e 3r. uriLrorcr. or ouv0lr€: crrr6; npdner vanpodplovrcrL ca6 rvcr rbLobLcivrola (1, 1) xar rirv to neipopcr arr6ileror s6vo41 rerrlrLxdq ol,ylirvrprboer"q, I unepcpsiorol ri.vor cbrvcq ricr o oarox6E are1roq0u eivct e '. H ffioI rou r:ro6d),Lercr or trpcrit;o1 pr or,'r6v rov pr0p6,crr.rLororxei mo r,btobrcivro[o (1. 1) xol olJv€fidr: o[ 6uo o1,?x€:vrQ(io€[( 0o yllouvou€bov io€g xoodrq 10 f - Dc.

Kcr 6va crx6prl ox6Lro o' aur6 -ro nopitbeLlptcr: Aulo cjrot€Lri [[c 6rsxQtrrlrpootyyrol. pe b1Jo p6vo! crYlarorouqj rilq oulsxo{E 6roxt|o€!}! no! ngerlQag€Ic(l[€ Iq !€a!1tq brcqoarxri €Eiooon

0u6l

= ; ; , r ( 0 )= , / ( 1 )=0 .

H eEloool o]Jrll nQoo€yyiqficr xQcrdlrcri ru 6uo onelQ(l r[ll[orcr o€ plo8vrloiorh/xa\rrQ(,)ol xc! 6rcpeQilovtaE rcr peocri,a tpr]psro rou ocolllls o€ otrodlcr Ur,rQ6-r€Q( ,'lullrcro. llixo]Jq l/N. To brcrxQlxo olm1pcr gr "lf c.lvtirotouq raoopllerar

. f , , 1 l -2 | I f l ' ], t | | 2 l l7l r l l

1 , , . ' . l L l 2 l La r l

( 10 )

Arltoq eivar o rtvcxa; re:repcopdvolv br.aqopd;v ;re to potido 1, 2, r. H 6rErilr ) . ru.o'r ,1r. 1rr 11 6u10rru fvu: o.up0u'f lxou:r.rp, iyovr,r A- nou nuurQxrr ' r lcn6 ro :rpo6Lrpa eolc, r€ir€r riooE fil 6€rri€Ql rcprilo.To r/ru/d,rr. opLcxir6rov Ir' - {. .tr.i\,onF t. oll4\ tEi.,rD.''t StQtltiiitaE )ult)t =D2ulDt2.'n1gotroitLgor ).uo€q €ivcr rirl.L ox!6!ao[oi x.lo.lad Er6€rrz6v ].uo€r'rv. rr'rv oroiorv r6earo n),{0og rivcr irr€Lao. Avri yrc 6vcr rbrobrttvuopu rqE ,4;,, = ).r dloqr prcrrbloou\r.iarrtot qg tPtLllxr = )1l, ;ror Ei\,ar rt L = sin(n?rr). EroL. n l.{or rr:EqtorLrorE o€auoqraq aivar

' ' = ' r i n n r . r .u \ r t = L t : n €

OL ora0Ep6< c,, zaooai.lovr.lr, 6ftds rdvra. ar6 rLq opxrz6q orvortr€q. Orp!0poi cr6o6eo1q eivcu. oL r.6LorLBd(: )" = n')n). or crnliq lioeLq eivcr or.r610rluvuorrto, r : xur o l rovoc vcorrpropoc ' ivar or r vo ouvuprrto, rc ror o/r6rcrvtiopcrc, 6rorr, ro np66ir1pc rivcr ouvc16< zru 61 6rorpn6.

:J AIAd,OPIKI: [:D!]:El: K,\i TO EKOETIKO €.-' ll7

E {'td0€r(t brcqoQlzd)! rElo6or0lv

'O;ro)q ozpL6(b< i,L.r sElo(loftq 6L(rq,oa{i)v. (lur6 nor (lJloqaoiE€r rit ortur€ar-q,oe( F r(t). ,ta00)!-ro I . !. cil(r or rirorLld<. Eqi 6oov o 1 6Lcr1(-rvronorei-r(r. 0l:l t)loruF n zaocac( €r0€rr;a6q l.ioFr! rl! iL((toQLr11t< r:!io0rorl: zcu" ziroe115.ai ; ' , ' t . ,0 ) 0,r , . ru, o, vJ dotro: T, lc t ,of , t 4(

r / ( i ) = S ( \ 1 , 5 L r u = c r e r ' / . t r + ' . + r ' " e r ' / . r , , .

H cLiordr0fLrl z(0oAiltrar crro rolrq;recrlovreq cr'r Ehv 6).or rriroLrv rrro gq6(v16r,' rt r/(i) r€i*r oro plb(v. eirv rcrpalL€voll r1s(ylivor r6rc rt r(l) napqrirlu((tec./trill. t dl ra(it(o(o< <;r' ulroLrq clE(vEr ai.lcrobti4 r6re. rzr6q cv ri.wl :rol.r€rbrx€: or r1011r€q oraor}e<. to i6ro 0u zdrvrr z(lr I ),r,ot. Eri ridor. ro pdleiJoqroLr cri rlaprirrct lLovo c;r6 ro :rpc.lpcrLro gipoq rI< -\. Mdro ta ie&yparudFiaq ,o1oaiEouv r\r Et'oritqEKt: Eiu ) =, + lr. rors

To rr.ifL,r(io (1,[1ir|r yr( d < 0. rtrQ([fa!'r or(0!:Qo 1rG .r = 0 z(1r r.r.t0ri1!urcL yr(d > {). To {l(vr(orLlo piqx\ all$i1er, r(ri.r\'rr;)o0L<. ()J,(i ro r'liAoc n0oiAXer(l(xo ro nLxlltL(rLzo trf-0oq.

5M H bl{ .r t , rurzn fEro(Uort ' / \ f4t = 4 d\el

Er6(,J45 xet l:rLr r 0 6rcv oLo ro Re ), < 0

oL'6irEQ,l twiaqrit oauv olc ra Re .lr < 0 xft zdroro Rc lr = 0

dota9|S x(tr o t:'tt eivcu. p4 tpqclpdvoE, 6rav rdnon 6rorL[i €I€rR e ) i > 0 .

Kri:roror oqlparleiq o'votr.i!ou.|' Ilt ouv0ri,.t Re ) < 0 (lo4nrorrLxrl rroriroorcr.bLor eqooq,clilrr crr6odeol oa tLrlhior< taolou! L To €nrleiQir(i ucq orlQi-70lr't€ oro ofl axortrr- r xoB.lad €xo€rrzaq ino€L<. dJd ro roArq(l roziH x(L6rar axopl o ,1 6r'v 6Lclovro:rorei.rcr (o:r6re r:rirp1ou, oqor rlq loptprfi n:]r):O)"tS ot ),notg tEivot'! oto tntdiy, t6rE xar p6rov r6tt, ttlw d,ts or t6tot4tiiixolr aArqnzi 4ayptnxi yiAot.

Ibr(i.t6Q( 0rxoi(l 6ra:rLordrvercr r1 rrorhoerc o'ivcr otrotrlp( 2 Fni 2 (nou€trqaviqrr(r. noiir orxv<i orrq rq,oasoyts). H €Eiocool eircL

. \ t t t t t i l ' t = . ' , , ( c o s r f + i l i r r t , t ) z c t . \ , - . d i

[ : , ]

IAIO IME' KAI IAIOAI NY:MA'I'A

r(lr tQ€!d!€ra[ va E€poulr€ n6rs x(1! o! 6uo !ti!or!p6g alrroi iou aivoxc 6louvaQvnrlxd npoyfr(lllxd [aQq. (:nF€LdorE ad]i 6r[ o! !6Lox+69 pr(oQ€i. vq eivs![lyabrrci oQloFoi.) To xQrfllQlo €uorao€icr( €[v(r[ no),u d$€oo:

fo ipo6 a - d npimr vq tivat aqv4ttx6.H oaaEoxaa ad bc t eit et va €irst acr.xi.

Oruv or rbrorrFiE rivcr :rpqpturrzdg. ru xgLniprc crnd ryyudrwur orr uurt'g tivurcprnrLneE. To lv6pevd ror4 eivcL q oqilouotr xat edw eivcr. 0enx6. or LbrotLlti<€ivqL xaL or 6io yi O€rLzig rl cq\TurdE. To d0aoqr{i rouE siaar ro ilvo5 xaL sv€ivsL qQvrFlx6, 16r€ xqr or 660 rQirl€! v(l €ivqL qQvylrlx6E.

Tc aErrlpr.n r.opow xcL 6rcv or Ltiror4rdg eivcL dve !€q,der Frya6ut drv r+irlto il\roq €t!c,| 2, (rou €i!o! < 0) xar I oQttouo(r €ilor (c + iyxx, - ir) = .rr? + !2(nou sivor > 0). To olipd 6eiprL to retcrprrp6pLo €uorqosic(, ,(a0(ig €rLolEx(! io aqeo6ohxo olloQo p€xoEu nQa'/garlrdv xa! [ryo6rr6v r6rorq6v. Hra0c6oln oqeil,€xqL orlv €Ei.o(l)oq yLa rK LbroxqrdE:

. l a - ) , I . , . ,a* l - " '

,1 -11= l ( i r 'os )^ + (oa iqouoa) =0

Arni €ivcu [Lc! ii€v[€Qo6ti0Frq eEfo{'rol, dpo or rblorrFaE €iva!

,r = ]lix*s t /t,x*tl, -l.oit"*.1J. ( l t )

fld|ro cn6 rrp ncqc6or'ii, o (ror0p6E oio u'l6QQrEo €i.vqr dpvnr!x68, dQcl io ,\ber eivcr nqcparr"a6. Erri qE n.lQd6oliE, n rEraayolvrkrl Qilo Fl6€!i!sr(r!, dpaI ) €nc!a),qr6(r\r€r(l!. Kdro aJr6 qv rdQo6o),{ q ) €ilrcrl

.t::t t- ( ,1 ' : , ' -

). <0 *!, Ir>0 trmde,t

!1. 5.2. n€qLox6s €lmdo€ias 6s riv(la(l 2 eri 2.

:.r llA4,ottK[: [!l:!2:[]: li^l To Elio[TIKC] .ir r t9

rrp(1ytr(1rLrrl. Kdo€ ou[lErplr.o< fiiv(oi(! €iv(( €ni fil! fi(Q(,lio].ll( 11 rcltr,-r crno(nirr i . i rof l (11 b =. ror€

( i7vo!)r a(oai! . ) = 0r + df 4(od t) =la t) t +4t > 0.

Mrt(brzac 'brorrtlac €irar blrcrrt< 36vov orcrv ra D zru c dTow avriorra apooltcz(r €ir{r (akt.T(i trfl(;},(l.

ENA IIAPAAEITMA ANO KAOE TETAPTHMOPIO

:r(1 oi,vo!( (L br,rn6prn, rrrrrprlpo0iou 1l f:io,'r1l fivur oubr:rru,r elrorrr0ric.A(rozilovr(: uuir t(l orvopcr irrav(4rr or (or(i0rr(. :ro! op'ioll'o h:o!c rtt r i ( r lLo,4r i t t i l ( l t t r lb i , ' (brorr I oQi ior. ,oc €ir( l r )r l : =0) )ro1 xdu€ro i r :or(zar :rrilo u:ro rrl1, (QXll, o! btlo !bLor!!i( €iv(! z(ouQd qruvr(orlza< (b!6( roi.7!o( 1,i.r(rt |lba\'). Aljrai oL br.(iarorrE olr\roQo)r rtlcr or (irur,}.t(i)brL( 10ono! ILrro!( onoio!< UroQ!i v( X(lUfi I {rior.iof(r.

H nrQirwol('ll ri c;rt rr eir<rL nlo biozol,rl. I'r.r r( 6Qor.!tL0 d1,( (1!(./r.GtoraL rx(v6 xqLtrtpLo !ror.l0Ei.r< -a!( nirie8( ,.er1liQro 1,[(r ro riorr Rc ), < (l-tln(i0){oll1 bno bt]lcrorrtre<. Mi( r'i1'(r v( (1(rp6goqrr oror( Routh }iur Hur$itz,nrnr ' rLnp(! lLrc ofr txr caroorlr( lv lLrr rc rnor lr ' i r r d ' j . Ar: ! lo lLiL(D ( j toc ( ; lrl naooayyrort .1r,rri sil.il %(rzr),Lrt).'l y(i 6\,.n trry.il.o ri\,(,..r. O 'lnoLo./roriiEp:ropei ioro< vc dpri rq rbrorqraE pe [orr),!req1 oo/olprd. cr'o.1. !c c:Er.ioercrNfq rrq ci1rr.oorilreq. H d).Lit br1(r6rfr.r t1vt1'{.rii,.i,0l}r€ crn6 ror Liaprnorr(, 189'7 zaL oryiaruu oto ru tjlt]ri h'a. thaxu< &pr|t I1-, iru aot ruotltqrctib ttijxoq 11'u(t) t'rt c|lirct nivrort. Iltiv r:rripltL i!(< rfroLoci l . ro1! to tr ' = - , l i r rQineL vc €i l i r t rutr(r0i ! . To I I r Otr qOi lr l 6({}r i rqrpo< ro trtb6! rcrl tr8rci Llro ii./a oz.r$r.trrdl{o$nr.1 zar ro ibro ro r nadral 1'.1rtivcr oro plbi:v. II ir$ryfrGnxrt (Eio r)1. gr'066clr, ror t.iapunov cvcyvoai.!r-rc'oril! reairr(Jrot ut ya(!+r',{6nF(rF. r.rr.i r1l'v orioi(i 1t Eqiooron $noo.i !(r €i!.rlabrlrarov va i.r0€i, urtratrL otLo! iv( il 'r(l) nor qoirEr br(pzrbs 6rtJr trropri'v cnob€rto€i €rcr/ro€ra lorpi; vcr yvopilor;re tvcr1, rnro "/rcr 1itr u(t).

li :l l,' "l

lJ .,,1 l,' !l

IAPAAETTMA r drl,- = [l ,,t

E:rtLbi iTvog = 0 zar oaiEolloa = 1.

1/. xilvr-r ro ! vc oratqr:rc' o 6vrl xnzio.

6/0lttl€ za0cpd qavr.rorrz6; LbrorLFaq:

| \ r l -, . l - \ ' l - r ' u a l , \ ; r . u l

I^]OTIMFI KAI IAIOdIANY:MATA

Tc t6to&cvuoFcra eivaL (1. i) ,|c.! (1,i) xat 4l'rion €ivat' ' t . . " " f1 ' l .u( t )=cre" L_/ ] * L ' r

Aw6 rivaL oroor6 aiJd 6I! toQaio. Avtlxc0lor(iwag ro c't tr€ cost + i sinl (xc!ro c 'i pe cos a i sin t) Eovcl€trecrl,ilovrc! rlQqpar0r{oi aQr€Fo[:

, , , , , _ | ( " r + ' r ) cos r+ t ( c r c r ) vn t I' " " - l - i t . , - . r ) cos i+ ( . r + r ' r ) sm t l

(12)

E6d) axorjtl€ ,rd{fl €EolQelrrdr e!6!((t€eo!l O r€},€uraioE riivco.lsE no}.lafiLoodq€Lro r.'. ripu npi:rtr vc €ivcL ro €x0€rrr{d p'll. (Ouulorir€ dfl yl }.uoq €ivdr rrFpo5x1r1E tr - r''rrr). O aivorcg crnoc trv ornlprr6vurv xcr rprrovc'rv tivcL roxnaLo raEi6€1./trd tro6 €v6E oQ8oyoviou nivrrxc. O! or4Leg fXow prlxog tvcacL ro eooxrprzd lv6gevd rouE €iv<rr pl66v drpc 61oqre Frcr €n!6€6citooq rouriroll€vou oswdolou qc[voI6vou:

Ef,e o A Eir@ av'rot tttttrQtxos 'lirF o p't' Eiv oaaoTnlros nivaxai

Auroi cxE6d4 eivar or:rivcxe6 tov owrtptqtur6v ouotrpdtor. 6rou 6ev ldverar€laQry€lc ),6yo rQ!6rtg { 6ruItlor-('Js:

fra f = 0. 0r[6r€ cos, = 1, ro nLo :rdvor 0a urge". ""

orlr^i"-* 1,. "o "o

= ff l.oe( or oele[oi o xar 6 ftO6nsr ra Trol"troiLoordeouv ro cosl 1ao! rl tl(l) xarcl],{1,€l

.,= l;:::l;i:;il = l:rl ;ll'l l;l (13 )

AT =-A xcLL (eri)? =€ '{r rru lcntrrul = rrol.

H tel.eurcia elioool exrpghler. prn orpcvttxrj t6dqrc.rov op0oyoviov nrvrixo:v:'Orc! rio).1(Eldoldlouv 6va bldll,Uopo. agtvouv ro I{r{o! rou aFt1d6i'rFo. Tobrclvuopct 1.t0 anLdq xqioraror rl€Qloxeogrl xqL rouro :repLlqdqer rr] hlon roodtLldl = Au: rnnf neptrg€peraL ;rdwor o'drn xtxl.o.

:'(uri filv noi.li oorwriototl :tepinnoorl o elr proQei \r'al,oyv(l)eroed r(ar'

ruoridv uno rnv d;rnQ4 orred. :IrErdor€ drr o -4 = [l o' ]

O., A: - I. xcL

xQlorlono!{fi€ rouro zar' €ns!dl,$l,l:

. 1 , I * a t * \ A t ) : t l A l l r t . .

f(';. )=l('*. ) !-'.,i. ll=

l '-2. )i

5 AIAOOPIKE' E=I'A'ED K.\I TO EKOETIKO E'IT

I IAPAAEIIMA 2 H rerouol broXroEco< d l r / r = | , ' ' , ' l

u l rcr! ruorqoic.

g € ^ = - 1 z q r l = 3 .

IIAPAAEIfMA 3 Edv z),€ioouu€ ra dn€!Q( rpi[ara. 16r€ 6€v 6toVEW€L rinor€

du I l 1 l. t r =

| t - r l ". dr /dt =u' t '1 6'1,17 =, - n

d :u | 2 11=Au= l t 2 )1 r .

3t l

Ar,rfj riv(t| tr|,( olt,t1Iq ari)r[4 Markor. A\nt rn percxLvoiwcL .rvri 6rog. tcrooJtl(ltibtc F€rarrvo{wqr &d0€ orL^/Frl. O owo}.u{o< cpr0p6g roq r, + tL, eivaLot(r0rpriq, on{,E rrpoetlnreL npoo06rownE rq 6rio EEro6o€lE dEErir: d/.lt(u+t,) = 0,'Er,(E /iir,(xclg Markov 6X€r doQoLo$s orrlldv ioo tl€,\,,,", = t.'Eva( orv€Xil<rLv('iGE Mffkov. ̂ /rd 6roqoQ!){6E eEtodosL< axe! doQolo$t orli.dt! ioo p€ ),,,,,, = 0.O A €i.vr( nivc0iog Mdrkov, r6te xcr g6vov 16rr, 6ra! o l] =,4 ,I eiv(lr. orwelrlE,ii!ur{(q Mffkov, H or(ro€od )i(rr(llor(ol zc! yr( toq 6uo eivcr to Lbr,obrtivuo[ctI< .1,,,,.. ro oroio orlv eliourl ir(reoQdv nol,l,clnlsoldEerar lre ro 1* = I xcLort( 6(q,oQ!1iq eElodro€{ F€ ro err' = l. riQ( ber }retoetvtirtrL.

:to t-noI€vo nc.Qil6€L1ll(t I or(o€Qir x(Tarorcloq €I€! | = L.

nAPAAEIIMA 4 :q\, 'ruo4vtxri FrllavLxyi. ivuq crwLbqoorrjpc.( L6ftur xg1or,tro< or(Ia €iv(r ou6€x€Qo Euorc{oi!. on6rr: 4 brironcot rooQQonei lr. rlq (lndlc|€g.MtxAorrAl trdon(ol rov rda!:I rur'T(0rl i1 rttoxpiayo ,ict! rcrld.riysr os ar6-otjrol. Aorc0llE 6Lltoncrol orl[ailre! (rrotrLxrl 66F6o.

EEr(i)o€r6 6Eul6potl 6aelot

Ot \'6tror brriluortE o61.yqocv o' evc orimrp<r rpti:tqg trilea4 d ldt = Au.To ibto oqr6uirrr xcL p'€\'c nli0oE hl,kDv €qrrAgoyriv, ort lilp€tc.. q 6rol,oyia',i.r! d,).oi, aUdr 6Xr (nor rLo er6tctpdpovrc v6po rlg quoLzl;. 21o\ 1,6!0 rouNerton [ = ma, 1 emrcilwoq a civct pLu 8eurrpl ncpriyuryq. H c6privrLarqQCty€L rEro6o€r; 6€ur6pcr; rcilerr; (rp6:rn vn l,noo4re qv Xuldtl = Au rlr'ri16 du /ll = At\ taL o m61q pq €irar vcr xarovoioor+re. :16g arrtrlrl Btrdr6nolott; 6eurepeq ;rnpnlchlo!6 €nL6Qi! oq Luoq.* To e6to €ivar rrQoaLQtr[rd lr( qfe(Furx4 Al.yr60.l. oyr 6tru: xttr./rcr rI OuoLxri.

H olyrarol 0a €ivoL r6l.eLc, av rp<rqooqtr tov i6Lo .1:

(14)

* lii@t€< i(pdtolor sirdr €riol:6w(ri:, drlt 7t|W 6ozur. airni 'tq io't,l (i\rr(r a( frlroit..r rq iOo/osrio€r xdaand\u dno ro riodteq.

Auo €ivar oL aQx[rdE olJa0{x€E nolj fiQitl€r. !a roooQlogoliv ̂ /|a , = 0, 60I€ vo

lenrvrlocL to onot4rc: 1 "FsroioTrLo,l" tr = tro xcrt n "ro1trrFcr" duldt = u:,'. fnvct rpooaqr6oor4re oLr€E ns ow€ri,{€E, cvri 1n n 0c xQElacr[ori[€ 2?r xaoapd€xo€arx& Luo$s.

Yroe6oi€ 6r[ lg4oLForoLoqre ro u awi rorr ) ror yadqoqr€ q1n6E rEEr6rxiE ),110€16 oq popqt r = e'dtirr. Awltic0lordvrcg cur6 ro €x0stw6 otl6raqoQrxll €Eiooon, ra6n€! vo 6IouF€

IA1OTIMI]I K,\I IAIOAIANY:MATA

*u '' ' u = on''' ,, ri -.'rr = Ar'

u(l) =(rr cosurr+i)r sinurrt)rr +.. +(a,,cos.r"t+r,,sin,r,, t)J, ' . (17)

t h = a l r l l + . + a r i x n 1 l u o = D c , I n = r r r ,

To ,uiewlls t: fiOet e. vr Eiv(r. 6.o6td?atopa @u A, dtnx stal&;tg r.ar nQort-yor'rriv(r|E. H ovrioxorxl rbLotr.pr] eivur rdrpo -urr ror r; oqlv6qtcr t., ow66et<rrpe ro qu0po cn6o6eo16 ) p6oo tou v6pou -ur = ). KdO€ eLbtnr] )'rlo4 e^t.rqE eEio<uotg nadrqg rdEr{DE obqlr:i oE &io erdLxig },rioeL5 c'dto ir5 €tioooqEbrur"aq rdEro4 x.lr or buo €r{odrFs rivoL ",,,= ry':\. Awo t{staoQi€r uovovortrv ,\ = 0, :rou 4(eL Fin rerpayrwr*ti pilct. T6t€, 61sv xo riLobldvuoFa €ilor r,or 6uo er,6rx6E ),uoerE eilcl! r xdt tl. Evog yvrlorog rivcxag 6iaXuoEor< 61€[ 6],€51!( r6Lorqdg rou aQl'ntlxds r(q! efioFdv(Dg 6t!e6 ol owvo,rntEs r,, €i,vo! tipcrf[qxF',,,6l Keeaei 6ui7}tor7 p*arpirctat ot xeeaeti rolawao\ o! naQdYorr€g etdtraQ(ryow ou6dr€Qyl €uord8€rs. H l,rioI ow€ auEdv€r otnt goivet 1{or I ol'tx4€va(rycrc rrcQ(Fdv€! drdeegr], petagipowaE ow€Xo)E xo ouolqpd €66 1(t €r€i. Hpvrxrl luoq p:rl?u/ritl = Ar\ 6r,uv o A dteL oevtrLxeE r,bLotqriE )r, . ., ), a<rr.orcv lrj = v-^j, €rrr'ct:

u(t ) = (qei ' | + Ae- t ' t ) t : t + . . + (c , ,? i - " t + t l , , ! . ' ' " t ) r " . (16)

0s)

'On(l)E rdwc!, ot oraoepdg xaooQilowcr c.r6 rL6 qQlrxtg ouv0tneG. AuloliveruL nm euxoL.o (ge cvrirrgo drltr etli nlaov rutro) p€ta6oivofioE a:t6 tEral,qm€uEr€ve( €xe€ar'|i! or(. tiLo 'fv6Qqta r rirova xar oljvlFitovc:

H opXrn{ getat6nrorl fttoFovriv€ro! eixo}.s; I = 0 oll[toi'\,€! linul = 0 xotcos,rt = 1. 0'16r€ unoF6v€r

H perar6rrol neooaQfl6!€ta! F6 rc a ,{at oS1y€i. oro ,9 I u0, 6noE xat neolYou-

F6vq. Kar6nrv, ncpclt ryilowcE qv u(t) xdL 0eror'ros t = 0 nQoo6toeiEovratrcr D q6 crpryLx{E toltitltag:

u6=bru r rc r+ +6" t ,u : t , -

5.] AIAOOPIKE: E:DA:EI! KAI TO EIiOETIKO €I'

( , , ) r , =t, . t ' = l ; ]

- | r - l(6)ur = r'3, rr = ,

L , ]

ti(. 5.3. H oa.l,ilxqr I ]aiyoat rc).d'roo1t.

H rEioororl Inv€rcr (lruaorordrvrns ra o rcu i' orov-r!no ltc Ilv 1l(l).Oc a(|]r1o$ooo1j[0 c!ro!( rouE rUrouE oro ficrpdb€ryllc. O! lbtor+r€i 10!

etrct )1 = lxcr)r= 3. crQct oL oux\r6qtsg Ei, l ,a[ r l =l xc!rr =y'3. Eavlooriolrltlc Esxlvd cno ezr\,lotcl (n aQzlr.nl rcX]Jxrlro lli r:ilcr Bqbiv). ror€ oL o0or,sinr/ 0o fEc{l]C'vro0o!v. K<rr cv o:rprirrog rol.crvrantlg 4trcfio;rLooci ncrtti pticrpor<i6tr. 1 116 = olj|L + al:rl Yilerc!

f r l f l l f r ll; l =" li l-",1 rl r

H iro)t s'rodvr'r( !i!aL

1

I f t l I. , ( / - : (^ . , r , r -^ .*v i i , ' , It l t z L L l

M:roqorpe vc efupoorpe rr1 tproLal olpcoia crurl< qr Luoq< t'r< r!1c:'Elorpr 6ro ptirlrq ouv6e6tpivt:q Ft:rolr toug xcr. o€ 6io otq0€Qoig roitotgle rQi( i.6!( rLarnQLo (:X. 5.3). H nadrrrt [dEa [Exl1rorfqErar aelr}iir oxl66(J1 2,0 = 1. E'vd il bsur€al tl€r€r oq e6ot qq xa' .q raolu'rt oflytoi r = 0cgrlvorw ro owlllpc v( rrvrl0ti. H t{flnorl rotlc u(l) yi\rErar o F€ooE 6aoq6io r{cooQrir\, r(l1,o,'i(iror(,)\,, fiou .rvrLororlouv ora 6no L8robr"cvuolar.l. :rlvnedrn ar),)i luoil. or 6no [dLEq zrvonl,r.lr. olvrovrogdva rcr ro r],crrrlpro or1p6orl bev revrtirverar :roxi (:r. 5.3l:1). H oulvorlrcr -1 = 1 eivc[ I i6rc [' cur]vev6q eLrrlpiou xcu. pLcrq $6vo [ci!cg. :.rlv rclnrepq ct:r].r1 ],no1 12 = (1, 1)pe ouvrorrlo4 crvrr0frou :rpooruou lraL [E ollv6xnro v6, or. FriteE xLvoivtqL:rpoq avri0creq xareroivorLc crl.Lc! pr ioeg roxurnr€,< (:I. 5.36). H 1€rrz{ hioleivcr ouv6lcop6g cud;r' rcuv 6no o^)drv Luoe{Dv, xor I 6LXrt poE €!6!xn }.uorlrr:arlrar6d\'E' 1r'orl air6 xdO€ ric.

Knedq culdver o 7p6vog. I xivloq eivcr. 6110{ l(Bt:, "oxe66v nr:$o6irrl'.Edv o ),o1oq,r1/,r2 rpcv r)"iropc nx€Qoio,,. rore oL 6to $riq0E 0o €xi6orQ€qa,'

IAIOTIME> ICAI IAIOAIANY:MA1'A

x(iror€ orlv aaxrzyi 06ot r, = 1 r(! u = 0 x(r[ I 6tr1 6taiu{aoirr 0rr €rava},rlF-6av6rcv cz6 r4v crpX{.'Ev(rs ouvtiuoqr6s r{ov sin2a zar sin3f €o €its r€eiotjo2;r. EncL6ri Erarq o V3 eivo irpplroq. ro ncd,rnepo ;rou p:roporpe vcr rorgr ei.var.6tt oL Fdrleg 0c eacv6t Oo:w atrcu6une tot'tri orqv apltxrl 0€o4. Mirl'Lorcr 0nnlllotcioow (rn€eroerfio xor'ld i((a! orlv sw(0fin rordorcrol . = 0 xor xr = I,€9 6oov n€etll;voup€ yt(l (.QL€1d 6rsornlc(. 'OnoE

F€ Fl(( !ndr,l(l lou ln !!drQ6ounoll crvcrmlod o 6vc r€),€i{lJs },€io rQ(tj#tr, I olLrlri Ev6gyEr( n(rAoF€a€r ('Too.Qi.Aeld n laryyoea or FirE€E rllorzrlow onoLqorinoi€ xaxsorclol al€r auri rrlv

A€v tlrroQoi$€ rdl,r v'arprlooupe ro np66l,r1pcr XarQiq vq 6€iEoW€ rqv r(Qd!-Lrll,iu pc rq oue1r1 :reqim(rort: Avri yLct brio pdleS ri trcr N pd(eS, uTrdOXer dvoo!vcIe<, KclodrE oi FdE€E raL ra ElarrleLs olJylole]Jovr(lr o€ pr"o or€QEd Qd66o,or "6€itsesE 6Lcrgopeg" nor b|bovral crri6 rolc olliel.eordq 1, 2.1 ytvovroL6rutrptg:rupt iyolot. H optuzr l au4 xurunrrrcq:rnpLypdgrrur Fs r tv xrtarrxr igiorloq 01uI0t? = 02u/dr:!.

AEKITXEIE

5.4.1 Axo).ou0{,wa_( ro radro rcad6€L r( c!rt( Ilts naeayedqot', 6e€ir. rK rt}roiL#s,r( rbrodr(violala

"(|L ro! €ao€rtxo riv(xa.'{' ro!

, r= [ I l l" t | - | .

r(r ro! naollonpelo Jrivcr&. oAlitc I .t.!L"ri ),not tou drlli = r1r xcr ritv €6r i),iot pe (axrftd< olv0ixer !0 = (3,1). nota eiv(r \ dtuqeA'1 tutAotuol xa0.i4ro I - 6: (Avri sias' pro swsxris .rvaltfu M.*ov. ad| ) = 0 qe lro bruq]oqrrirEiorDot alrLoo')(tsi oro ) = I Frcq €Eiooofi 6Loqostbv, 6Lo r"' = 1.)

Ytro0ior6 ofl alrrorAd({,etor I {iioQil ioll Xqolo! roL 6i60( tol rilaro -/tl

l u f l - 1 1 f 3 l7=l- r r 1 ' re "n=l rJ

Ba€ir€ rt! L(r)xd! 6riEI€ ('I| auEdvr' ovtEi).erxr( clri !cilei.let, t(06( ro t - x,.(H 6rapor €ivaL Fr avdmedl'rrrl x(lr 't fEiooq e€aF6oFas 5€v reoxoed rqos ru

5.4.4 Eit\ P.fv.n 6vd! niv.rxd! nqo6oirl<. 6EiE1!. Iarpqrorord,wa! nt! (inerAl oerA(i.

. p _ t + 1 . . 7 1 8 p .

r r 0 ls.4.s fv,( b,u},;u.( rivrr.ac. o:rr5 o ^=l; j]. Borcrcrfi rov o.q€rqrrvo ruvov(

e'\(1r?) e^re^? j 6Ldrt o xca6vog avros rsxn€r yto rdOr btoyd,qo ororteio.

5,4,2

5,4,1

' ! . r F l * . , .o oJor , r . r . !u "n

( P . . t - r r - 'P

5..1 AIAd,OPIKE> En>A>ED KAI TO EKoETIKO .7r' 315

o) EEIyim€.yr(ri cr(t*") = "r1"/T.

xptlo,t ooor,i,uo( rov Il]no crt' = se^ts I.

6) AeiEr€ drr .^'B eAeB 6ev ol.Iesnsr ylo nivax€!. Xaryuoroltilras ionoedb€rypo

, I 0 0 l - f u l t, - l r n l u - 0 0 rbe l i r cu l xe ry r o r ra . i , v rou . , a "o , "a t

s.4.6 H sEioool c!0xdao! rdlso5 y" + v = 0 Frossi va yaa9ei ool orloqpa rqdq;-rdEslls, telorFono(iMas q! rolinFo s' oav rq6oeEro ri-yvomo:

d l l l l ! ' 1 l v ' 1atLv ' l= lv " l= l -v ' )

noros civdr o 2 tnL 2 nt\$a% ,1, or I neeandlo oxdon yQaq€i orl poeqridtrl.lt = Au: BA€irc rq r6rorqd< rsL ra r6ro6rav1jspalo xa! ffiol.olLo rt LrJorl nouEe{'ad pe yo = 2. !i) = 0.

5.4.7 MetdrA6!'re qa y" = 0 o0 oiornro nA6flts rdEetrs dtuld.t = Au:

1f ' l= f ' , '= 'o r l [ ' 'd t - ! ' l - 0 )

- L 0 0 J l c ' l

Aui6s o 2 €ni 2 niraftas,1 Ei\,(r €tratlof.nrtos (6xEL dvo poro rSro6rdvljsla* rorbe! pnoaea ld 6rayorrorout0ei). Y,roloyiore qv c,rr (ln6 rt! oeLed I +,{l + . . . rc}ryadF€ try Morl .'!rilt|, E!l{|rdwa,( ano ro !0 = 3. /6 = 4. Era}.lo€l)sre or! rt },nsl

".s , =

L;,1 ,zavono'€r qv ," = u.

5.4.8 Yno0dolE orr o! nUoxoBoi I rov nou!€),r6v "dr

u rov )"1Jrow 6drowcr sro rLs

dr_ = 4 r , Z u i .

t L u = , , " ' .

() Eilor ro omtFd alrd .!moedE. orr66rFaa ero-.a0ds i cma0dqi6) Edv dax|td r = 300 xoL u = 200, roror €ivdr o! nl]lfJlJotroi q taouxi mrypi tiy) noro €ircr. ro nllizo! r()! aoNd.rrbv nqos tolq ).!ror5, perd nol! Xpovo;

5.d9 EE.rdoru ros f,qos tt! €lmilee|,(l ro u' = ,.1u, yra rols 6r6p€!o!s rivdrcql

" ' i l l (6'| A ll '?,]

] ' 4 f l r , l (d) A-l | - l l

s.4.10 EE€r.iore !E Tiaoq qv sudo8c i doldeeta @ dl/dt = to. dloltLL = r. YadetElai.rou liot fto! qofr€':

* :.r.E.i AlqiOFaL qc taalgLzdq avoqdartra.

IAIOI'IME KAI IAIOAIAXY>MAT

s.4. Me l{orriero ro ixvo( rols xor rt\, ooilorio'i rous, 6esift o€ troto )(0&o t or €r6tr€vorftivqx€s al,ldlouv l€rdEi €rora0dJv lre rqa?pdirxtg rdrorrll€s, rljord€& Fs Flyo6rx6Eldroq4 aal aars06v:

, t r - r l , t0 4 , l . t r - t l^,=1, r l . " := l r _zJ ^ '= l r r l

5,4,12 BA€ile t4 !6LoqdE ror re 6tobrd.viol.l1o rolJ

nl4 EfAftE xoeLc -va rov uro?'o1iorre. ort o "4' uq stlor oe0oyduos rlvalag xordfl ro li(t) l' = ui + ui + ui €a rilor oto€Ee6:

5.4.f r( ntv (E,o+Fsretxi €qiooorr

o) 6e€ir€ ( ri,4, ul rg|, €'ro.l,r0piot€ rrtv 11,r + !!r, + air3 = 06) sll,ffqd'oft drt'o Filro! !t + 4 +,3 eivql orao€po'y) 6Q€ire ns |t)orqrdg ro!,4.

H l'nol 0a noarcreiqErar ffat iov dEora o = (d,6,.), brdTt ro ,4! €i\,or ro"EEoftaixo y|dlltl,o" a x u -ro! sivar x6!0rro ('ta ! xaL u.

5.411 noEs €ivar or L6rorrtras l r.dr ot orrxvorlr€s d ror.r

4-t-s .1. . ., h : _ 1 4 . 5 1 , , ,

Br)rirr q yertxli ).rlo1|' FE rov raono rm EEfo(x'4s (1?).

s.4I5 Ai'or€ qv €Eiooq brurdsds rdE€oE

E " f s 1 t t l t , f O lac = I - r -s)" P" * = lo l

5.4.16 :r€ fteood€e€s reoapoy€{. I elfotool 6udeoE rdE€ciE e;el qv Foegi M!,,+ru = 0, orou or 6.urse€r ndQdl(')yor roD.an]'doto.\ovat F iNsr trivaxa FiEd1.Anrxaraorrloar€ trt'', idoaeli er{0t1 i 6q.ift ro 1o!.r€!pe!one66).rua r6!or+(lv" ?io! rqin€r vs l.!o€i {,E nAog flt o!)ry6qrq @ xor ro 6rivrJoFa

s.4u Awt|.oloGioaIe lto *o€oed €xoettxi tr = e^'a xa! 60.iI€ ro Grqoyovrx6:rqo-6l,qFa r6ron+d)v os ftAos .\, yrd r4a €Eiooort r" + Fu' ar = 0, !s ,rilol.a ra6tk

+=^"=[-" i ll,o ' Lo -a oJ

+=,,,-ll. ; ;'lflll, , , L t , _ , o l1, , ,1

5.4 ATAOOPII'E E€T'A'ED KAI IO E(OB'IIKO C,'

5.418 BQeit€ qv qol irE SeGeerE pdlo6, dtcv I nO<!q oE ledvo t = 0' o)olttar

le * = [3] ,", 4 =

fll' *"r".e'*cs qv *toooq (14) to! lt:qqrdtou F€ovlvdnTtq I rcr, /5.

5,4,19 'EvdE ritvoxas F€ l,tooE pq6av lnoact 1d'fQdqel oqv poQqtl

,_ r d 6+c l .^= la -c - . l '

Aeqft d or !6loqrds ro! ,lr,cr tioo'flrotr,x€, &dv {o p6vov dtcr a2 + f > d.*

5.4.4 M. at(r60q|wli awut(lidotas4 { trs itrolo1t-otl6 to\, ttto6tavloldto\,, }'t'ot8 to

, t r2 r1 111{=lo : ol" } . , 'o=lol .4r L0 0 4.1 L1J

. t.!n.: Arodec!@ c rQ.ry1lalqi p5@ orclr€i@

i 38 I,\II)IITII: KAI IAIO \IANY:MATA

5.5 r MITAAIKOI IIINAKEI: IYMMETPIKOI KATA EPMITL{NAN

KAI OPOOTANIOI X,{TA OP€OMONAAIAIAN

A€v €ivcr' n),€ov buvtlior ra eprcrEogu<ne p6vo [€ rpay$a.rt]ti1 brovno$cr( xcrt;rparua-nro{q rivsxeq. :io TiQdxo [!06 rou 6r6liou, 6nou ro 6crou6 ;tp66],qpu{rcrv 4 ,,1;r = D, ripcrmcrlr oi./orQor orr. €r|' 60(]r o A zcr ro 6 eivaL rpolltuttir ti,zol ro :r 0c t:irct titoLo. Aev unrlple lom6v crvciprl 1c ptya6r,zotlE apr0trtotr< x<tr€cv roljq Enl1odnoue, 6€v 0(l irA6o0€rcv riror€ vto. TdrAC 6pu)q 6cr ptrooouF€ t(ltou (noqirlolJtlr. 'Eacc rQqYFalrx6q rilcrftcrg 61€r nQoYllortxolrE olJweL€o168oro luQ(lxxrlQloru{o ro! rrol,ui)vqro. c lir or L6ror4r6q rou (ofioE orrg olQoqdq)trr(oa€i r( fia(lr [ry(lbrxiE.

Oc erocliryor-rpe ),oui6! tol ZdQo C" r<ov arcvuopdrax F€ n trlYc(brxe<ouvte-rcrlpirrE. H 7iQ6o0€ol xor o rro).1.<rnlcorcoLL6g nrvi(xc,rv oro),oueoriv Io!!iblolJg p€ rotf nQonloupevo]Jq z(lv6v€-<. OF{r}: to ttixog Ev6E ,KrtiofaNo€ ltpilttrv ( l t ) , f t6rr . t , iv rolrr } , f l fd ' '

- - ." i - - . , i . tort ro b| ' twot i , . f r

. ' u \ 1 o r , ' o , q ( 1 . ; l r r ( r t T c F r t z o g U 4 b t v : 1 : - i : = 0 n e F n f r n F , ' , ; v , ( r T r r t . t z o :v4 bl6tl ro €IQo./!r!o 10! Frlxow €i'vsr lr + I r. H cdJ,rr./rl clrltli to! 1Q6fio!rno),oyro$o1j rou luixorrs, ollysi or'[(,a orrqri ri].low (d,l.a./6\r. To eooieQlxoyrvopcvo 6uo orrtvrrol.cil' '\'. n dvsmqo{uc rvo: n:v(x,r. or oproFoi .Jtrtr.tcrzoi.(vrLo!$r€rQlr4ou ,.r(r oQooyovro! nilall(l. 61,'uurh, :rp6rer !'cl),l(iEo!v [E rI!€p(ir(ivron r()v [ryoblxcb! eQ[0p6\r. :€ xcio€ rrAimt'ro1. o v6oE oqtop6< or.rirainreLUr yo\ rd).ro. ordv Ttl druvuo/uru xLfi o. :rivrrxcc tivru :rQ(lu.tTr2oi.

Tr,( al),(yig cur6s dlolJue rcrcrloQi.o€r oe zordtroyo. oro 16),()g (1rnlls fllirrcQa./QdqoL!. aor p:ropei. 'vc O€rDet0ei oft! 6ac ),e!!xo. yrtr q Ftxdqpaol arortv aqan(rl,'|rt or1 pL1u6rxt1 rieQinlljjon. fl€Qt6X€I tniolE rI! trLlptoreolbuv(rr1 rhlpoqopic y!( q oiorl rorv Lbr.or4rdrv xci6€ [rac o,"[6 flq 6t(irpoQeExcrt"To0it:,< nlvrixorr. rp(iytL(, er,riEow€, Xetorpo 1,|c rov cvr{vtiroq. I6xri.teprtpa( e!6r(fidQou! or dtlpEiQtxoi niv&xEE noi 6qioxol'tuL or 6nt4d< roq

"att L6nftepo iynL ru u\utbtttrtioptnri rory; Qq nQo: rL,( riQo,{tltrd! tqraQ[oy6E,cruriq eircL or. rro evbLc.rdpoloeq esDrrtotrq oq 0eolairr lijrotr$d),'rar €qloro![€rr]! :rpooo7rl orc 6no xlQrc orr[n€Qiro$oxc:

1, 'h,a9 outtttctpbnis fltuaxas ixEr nQayFEitxiE t'ro|tpiE,2. Ta ,,rc6|olnosua tor t&at 6utcr6v ra Enrltyoir xdq.rs pF',aEi roui.

I].rQ(t66q(n;, y.t1 rd SEiEoup€ 6tr. or. LbrorL;rdE eivcr rpclpnrLzts, (QXi.loWE FErn! c\{i0rq 3uvnr6r1ra, :rqdlpa :ror [aq o6ty€i oro!( pr,yc6!xou! rrQrelou6,brcliJopcrrc rrlL fii'\,.rxE<. Or dllaydq eivar etrxoLcq zor rr(oQx€t ila €troEnQoi,Qd[F(xo! ]isQ6o<r Bpioxorl$€ opLopfvoug Bllcbrxorq nilcrr€E nou 6xor!0fiiol< rQcyF.rrrr d< rbror+rdq rcL opoo,acvovr.zd r6ro6rdliopct(. Aurof eivaL oLI qr f la lor ni ! (rx,q . , {dr 6u borru' noa(zd,u, r ,"c rp rFL,, ; touv rou- rpa]truf lxotcor$perprxorg rivcrzrg xor 6xow ol€g 1(E ouoLcorL?i€E tous Lbr"6rlr€q.

5.5 NJIIAAIKO1 NINAKE: :YMMITPIKOJ (,\TA EIMITIANAN

Mrla6rzoi agrOpoi xcr or oulupig rot'E

O cvcTvtirm4 eilcL, ev6qoyrvo4. €Eot*rrlrivo< tr€ ror< luyo6Lxoug npL0-Fon-q. Encr6l6!loE Ze€ralovrnr Fovo rc! nlo cn),a L(lpaz lerorrxc.rouE. FTroQoiF€€6ro).c va 6(boow€ trcl ouvroFl n€QiLIV{.* O aa06vnq pt':pi.!fl noq,6,rr. xr rnrivcrL ro i, Lxavororri. :rcirrl4 rlv ir = - L EivcrL dvaq r&0(1pd Vuvr(oru{6E aar9-[6q xclL ro i6ro oqddrrL pe rcr :rol.].anl,cioLir rou i6,6aor l: civol rpcrnrcnrx6gcar0poq. To h0poroycr t-v6g rpr4pclLmu zcl tv6g qcvrcorLxoti upr0poi eivarivug ptlubLr6g (Qr0t6q o + lD. zcl ol€Orit!ftar [€ q!on6 tp6no oro pr"1c6Lx6riin€bo (:1. s.4).

Or natlytr(flxoi (aroFoi (yr& roL( onoiorK lJ = 0) v(lr ot {(l1.Ic.orrxoi ttaroFoi(r = 0) otrrrrQri.(ltr6(ivowcr. ocrv eL6ra6g :rrprnTriroerg. cvcipeo( mor'< ptlcrbLzouE($0Foi! -,&( 6Qioxovrr( n(iv(o orov €!( I fio! dl,l,o ciEola oul,rerqf €1,o)\,

t r + i r) + (c+ id) = (r + f)+ i (r+ d),

rc! noltrcfiL(ror(i(oli(r( IQlolllonoK)v1(rE ro! x(rvovo i: = -l:

( o + i r x r + i d ) = a c + i 6 c + i a d + i ' b d = ( o c b d ) + i l k + a t l l .

O prya6u6a ottryis rou (r+ia' €i!(r o (lQr0F6q .t - ib.llE at'riqtro ro vQ6or tott't g'tttuortxoi lt;Lr't c. fru4rrrotzd.(ruroc fiv..r I .icroxrpr"ri frr{(iv(, uno rI!(ii.it ll€arir rou 'rpqtrarrxori rilovcr xcr xiroe :rpcr1Fctrz6511g0p6s ouprimer pe1o\r ouluyrl ror'. O olrlufllE oull6ol.i.lEr(r llr llls fiauLo o +ilJ = d - ib xo! 6letrpfrS r'!brcl(l6Qor(]eE rbLoxnre(l

qaw0oflK6q {i{ovds

o=!

npd'ypdrtK6q dEovoq

d - ih=;+ h=re a

!r. 5.4. To t{ra 'to sJrinEdo p. iluv trLy(6rro aerolo,rcr rov ouEllyri ro!.

1 O' o+rd\tzi. rbir. rir( o t[1d6r"G o,!1yili x( 'l doruoq 1+t.

IAMME: KAI IAIOAIANI:I!IATA

(1) o oulq,{s rou l|voldvor eil(tr ro.rwot€vo rov oulwd)vr

[ +;61c + rdl = 1nr - b,]t - i \ t . + od) = ta - Ai i ' i4. ( l)

(2) o ouEqllE rou o0Qoiouoro! rooiior [€ ro dopoloUo ruil ouluyriv:

G;;t: lG:A = (,r . .) - i\b + d) = \a- i6 + tr -;4.

(3) O rolLanlcrorqoF6E o:rorou6ti:tore d + ill 1te -rov ou!u11 rou a i6

ruprityer f vuv:rpuptrrtxri cpr0p6. :rou eivcr ro rrrqdlo,vo rqc unortivououE oro: I . 5 . 4 1

(2)

H cn6or(tol r ovoFd!€r(t a iTr.og rt,ri rou clQxlxou a + ib (i ltiea rot)a6pi) r,<\ aul6o],iL\€roL |r€ i.{iO€r€g roril€q I e + ibl = r = yrr\E.

Tdl,oq. q rQr./o,vop€reicr ow66e! iq rl€uQ6E ll r(lr b B€ filv urior€lvolrocl r

. r = r c o s d , b = r s i n d .

Iuvbuirlowcg nrrr6g 116 €qlodosL5 xar Xa,rlo4tonot6\rq noLo{6q owrer<qpdvegnC1[evoqr€:

(a + ilxll tr) = o'? + ,r = rr.

a + i 6 = r ( c o s d + i s i n 0 ) = r r " . (3)

Yritpler prc rv6rcrgdpotoc t:r6r,xtl naqinrcoorl, 6rav ro pdrpo r oLvar 1. T6reo fryc6rr6E c(Qrgllos €i!o! (rer66q .'d = cosd + isind, ro! n€Qdx€ro! oro!towinio *lLlo l.o1J l\y(6!xou en!r*6ou. Kcotig to // F€rct6cr!}'€ror (('[o ro 0dr,r; ro 2r, o agL0p65 ei{' reprorq6gercL yipo curd qv fle1ri o€ fitl0€Q1 (xrlvlxll

ctn6olgol Pid = fi*tal "i"T

= r.

nAPAAEIrMA .r = .r - 4r e:ri rov oulrlyi ro! 7 roourar pe ro rLre(.yi,'!o rl<cnd.!.ut: rr[rts:

rJ = (3 + aix3 - 4i) = 2s = r? dea f =lr l =s.

fLo ro blorQdooqe p€ rov 3 +4i. nol.l,cn).corriloqe cE6prprl xcL ncgovopconiFe 10! OUqt,Yrl r()lr:

2 + i 2 + i 3 4 i 1 0 - 5 i3 . 4 t 3 + 4 i - 3 4 i 2 5 '

O noLLa:rl.corcop6g xnr q 6rcipeol yivovrcr eririolo o€ roILXdE oul'r€ra1dvsE:

rfis xnL \ei0. dler ar6Lrrrl tLgrl rlry rar y(or,icr dr +drrrei', 6rd r"eid: €Xer ar61'url rlpi rr/,", ,.at \a\Ls 01 02.

ji Ntll AAlXOl flINAXE:: :l-\InlETPlKOl XAIA EPNIITIANON

Mrlxq rcr ovriorqorpor. crr; prlcbrxrl neqintoo4

EnrorQlqoutrl€ orrl lQ(rpprrri A)!yE6Qs rc([ n(l[r c]i6 rol( np(.4rcrr,nougoroq prlcrbrzo!;. To fip6ro 6iF.r rival !o (tjro6€X0ouFe FL./(6utd brtlvuotrorclrcu cnrr6 brv :r<rporord!€r xav€\'c nao6).I[Cl: El opropot o xripo; C" nsQrdlsro).u ru burvuolr(r ,r . . l r t n l rrT.rbrzic ouvrrrupivsc:

3.ll

var l ly I =25

' t ' l l = r t l l t + + x r l J n (5 )

l':l'f( b|(vro$rtr(.f xrlr , roooriu€vraL. 6nt|)c x(r nporlyoultivrog. ouvrotd-roa ;rpogorvrordJou, c),l(i o noLlcftL(lorooll6s yivr:r(lr r(i)Ocl pe ptloblxoug upr0porig.'Onrrq xrir nQoqyol4l6voE. tu brcrvrlopct( r,1. - . . , iJ1 ei,r,el lgupprxri4 e!(qr4dv(t.6ttrv ircE p1 rerQlBpivo< ow6n(olog .rr,r + .. + .{ r'{. nopdyrr ro trr16cvrlt6btirvro3c. Trirpu rc r, 3ropei vc eivcrL puycbrxoi. T(r 1ro\,(6!oia b!(v6olaro rivolac?,1 orov C". r-xiolE ei.v(r irci),! svtEirQrrlr( z(t (rote),ou! 6aol. Erolr€1,{DE oC" ix lr . r 'x ior1:. bt t iot .rot bruvuotrurrxou / , ;auu r.

Tovioc4r r1b4 6rr o oprop6s ror prlouE bruvuopcro,< ,rpdTret vtr d,i.irltr.To rtrpiryorvo tv6< yrnr6tzoi cpr0pou 6tv rivar rcr' crvdTz4v 0rrtz6 xcr r1

.r r =.r] + *."], 6tv p;ropei :rLtov vc xprorltonolilO€l. O a6o< oarolo< riltrtrti,t_ir'x { r'oro}.oyLxri:r to rl cwrr(0io1(1r( !r r.r laxQo.ori .r, r ztrL ro grlnog

nAPAAEIfMA.r =

Iic xQ(ylt(rLri(i 6r(rvuol(rI& unrlQlr [r( mrr1] oXfol I€rcGu rou [rlxow z(lL rol,roarrprxou lvogavot: rlr =.r''.r. Oiloupe vcr bLcrlprlootpr crrrl 11 oX6o1.fi clto. to cororeprx6 lv6pevo :rpd:reL vcr tpononorrlOei drot tirote rc tarprdrlrtt rr rov vdo oQrotro rou fr lxouE z,n o x(. t lLrAo[ ivoE rAonoE r i .*o oto ron' trQrxoyr',,6[E!o 1,c noipvoup€ ro orluyd; rou;rprirtor 6rcvriopcrrog. A1]16 olpcri!€r 6rrro.r c(v1r7u0ior(r(r lt ro.r rjr 10 $@tForxtj yr"6paeo rov ! xar lt yiuFiar

l r+ i IL2 4 i l1l

(1)

'Eva runr.x6 eotrxeqrno ylv6B€vo oroa C2 eivcrr. r = (1+ i.3i) xcr" 9 = (4,2 i):

7"e = (1 t)a+(-3tX2 .i)=1 rl i.'OiG! fidAoW€ ro yrv6pevo xo! r. p€ iov €c{rr6 ror, enrorq6rpo4re oro -rngd.lcuvo

e " r = ( 1 + r ) ( 1 + i ) + Q r ) ( 3 i ) = 2 + 9 = r l r .}perdlme orr to g-?:r rivcr.6r<rrpogerLz6 ofio 10 77y. An6 rtirqc rrl oto eErlg,0o rrQ€rer !c riQoocxorF€ q bLdraEI rov ola\rooldrorv oro sooreQr]i6 .yrvdp€vo.YnoQXFr zur ivus utr) .oc vrorroroudc: L.dv uwu{uruorqoouuf ro./ Fr ro ' . r . roeooreQrx6 1t1,o[Evo ro)v r xary roi,lcfi].crolciE€ror 6X! F€ ro . a],].d pc ro a.

A1J16 paE o6q./Ei oro vc xrivoupe Ui(r axo[I oLlcyi. Mro (r].].(11,ri no!6Ev oqopd rqpc! tl6vo ro\r oup6ol.rold xcr! ollrrruivdv€t 61]0 oi$6o].q oe 6!ciA,'ti. rlg nail.cs yl(r rI sl]guyid xcr" rou ? "^(r rlv d\raorQoeri, ouv6rdlowsqortE flq fiQdE€lE 6xq oulwi d|soqoq)i nor ouu6ol.iEorip€ p€ lov €re6q Il.'Ero!

"' = x' xcr ro! i6ro or4r6ol,rotlo EgoQtl6lo1Jtt€ oror-rE nivox€(: O o!(!.ytE

d\ra(oieo(troE rou / el!o!

7 ' = , { t . u s o t o r x e i u r A I r t = i l r . { 6 r

Eciv o :rivcrzcrg ,,1 eilcr. nl exL n,1618 o Att sivdt r €ri fru. fls aaQd8€rypo.

12- i 4+ i o l=L-3r s ol '

To on[6o],o Arr ei,rcr eriorlp4 crvcl,vcirprcnl rou y€"/ov6roE 6rr. yr.cr pr^/o6Lrid oror-X€fo, nol.! on.i\r!( XOELc!6paore p6vo rov cvtrorQoqio rou,,l. Aur6 rrou ole66!or ol.€q x6 rr€Qrrrx6o€K ono6er.xvrernr. xcrti),LqLo eivaL o orrlqrjE nvhorporpoE rjEppmav6E at'riarpoEo(.* MroAorU€ €tro).a I'a ouvorpioolJF€ I|"E rpononoDioEqnou .n,rrouvr(r . / ra roxr Fry,,brz, ,u6,rur0Iouc:

5N (i) To €oor€Qrr6 .fn/6F€vo r{Dv r rio.! 9 €i,vot ,Dsy, xcr. curti ei..r,nlncroeto 6tcv aFy = 0.

(ii) To ptroE rou r €irlar lrll = (:Id,)]/'z.(lll) (AB\T = B'r AT , &q! rrqiQ ,ovrdg ro orEry6E rdo€ oxory€iou.

0:U16 l\E1,41 (AB)]I = Brl AH .

IAIOT'IME: KAI IAIOAIA\)2MATA

* O ni!(2i( ,1tr ar(l{la-ci ol7]a oaa _,4 Ealll(vo! . tlainr bE u.imr nao.larLzoi drmr!d brdaaiwE (116 ro 6!oua un6 q qpdol o A 0i!aL EA|trTLdIo<'. no! olltLai{r 6r o 1 exni.ar

Iz+ i t i l sl4-r s IL 0 0 l

EAFnrovoi nivoz€E

:rc naollorg€vr{ 1i€.tdLuru p rtocirt 1.(1 oul[€rarroug ri]at{€,c: ,4 = Ar.T(,rpcr. nou ixor!r ni!clr.i!q [r tri,(br1td ororleic. r1 Lbdcr r1s ourtrrpiaq np€nrrv , r D x r u 0 r i . H o , n r t y r r i z f u o t 5 r v r . v o . u . : r . v ' , / r c , 1 r , , r o o ! ! r u r r ! r o vGa(ior0oq o rolrE, rr|Ln oL lirsxEE flolJ rcoi,vrar F lov oa-tvyti @ioiQoEri rot'.Alnoi €iv.rr oL EpFLrraroi nivnzeg xaL tutr turlxo naaditr4.r €i.v.rt ro

i.5 ITIIAIIKOI flIN,\KE:: :YNlIlE I PIKO I KAIA lP\4ITIANAN

, = [ . * ' . ,

t , t ' ] =r ' (7)

Kl i i i f oro /F jo ro! ,1 o lv lo, l r 'p , r mo. , a. l . '

,., r" 1; i ' l l , ] , ' . . ' l f l l' t J | r / ) l L , l

= 2 u u + 5 . r , + ( 3 3 i ) l / , ' + ( 3 + 3 ; ) r ' , .

O r ' o r ( l , , , v r o r u q , r r i v u r r 4 , , y p , , r ' r o r . d o n 2 d r , ) r ' , r l r 5 r r 5 , ' .

T uezr6g 6Lc1c'rvior otorTeic €ilor oulu/i ro 6va ror ci),Ior, zct 6toL ou'6rcilovtcuzcr 6f!o1,'\, ro bLnl"doLo rorr r0aw.irrron [6aorl: rou (3 3?)zf. Eno]rtvu)Er-r) .o.<) 1,1t 4.ruv'ror,(a ' t . r / ' Lr " v ' r rQalFarxrt .

frc-r1 ltvrzrl (or66FrEn. UrioQo F€ ''.r r:ro).o1i.oorpe ro 1zF;1:l1F. npftttrcr ;rripoupt: ro\r o1rlLryrt rou I €ri. t ri\ralo r',4r, ][c rov o'roio, :rpdpurr.'EclclioioxolJlt€ ro! iSLo aarolo: (xH,l:.-)H = IHAHTHH = /F-4.r. AQo o.rar0[6E (luro< €i.',aL nao"/larrxoq.

:Itr€(i)or€ 6rr ra 'ruytbrret ororyeia Nir ra dtatL :r(!tLi'4.r.r2.i, 6LorL ira€ntlvc :rapcrptvouv crperci6),Ircr L ott! I o!!!y|c. K(i0€ moLttrio arro( rI: btay{lviorrtrprirleL 1te ro xcrton1ptz6 1(}! !)q trQo( rrl! z!0tc btcydrvto. orrti)c ro 3 3i ror€i\rcr 10 ou.frh/(< io! 3 + 3i. 2t ,AAt tu,Oinn')ol rolri€r o,j = At. To /roQd6e(111c(1,r(i {)(l !::T/rto|r rLg rpr:4 6r1orz6q rbtotlr€! tor! Ee!!r(!dv ,rLvdxo!.

KipLoq or6los pcg rivcr vc cnobeiEouls curlE r(q xe€q r6Loqte<. lttr ttgo;roi r.< ;rpi:rr L a(l),1 v(l rovt ooull€ 01! roluo!\r xcr 1ru orjgg mpuoug :riv(lxc5, ?r,agtraq'!urtxl< a44ureLxdq u&cE tilru oiyorpa Eptrirard<. ficr rQulB(lrrrousriaclir'( b€r uri(ipxfr brcrl.opri percll tor; ,47 rct AIl. Anoq,uoLorlxo eircrr ro cv1t olaorpo(l,I {rq.ivr:! rol ntlcrc ctr€td6),ffo. E<iv L.olutL uur6. t6rt or r6rorrpi;r , , , 6n,, , : r \ ( lc i9() l r r rqtn",c- t iv ' r nsdlFuf lx i : .

t6r&rFa 1 E.iv A = AH. rirc y!(1 6l"a rcr p!ya6!xq d[svuo$arc x. oupL6[6< rF,4rr €ivar raay[(rrlx6E.

3l'1 IAIOTIME: K,\l IAIOAL\NY:MATA

l6r6qia 2 KdO€ l6mr+q€v6E EQprrlovoli Tiivarcr Eivc! ripo.fpo-nliri.

AtdtE.t{ Yno0€or€ 611 1 ) eivcL rdLorLpri xcu" z ei.var 6v<r <rl.iiotor1o !610,SLdvuo[Gl ,4r) = Ai.. To rixvaops r6rE xfuar yo ,tolTsttlsouiooutt pErrII :.r/l,4lr =.\.r1r;r. H trproreprj n),eupd eivcrr, xcrci rlv r6r6qrc 1, nQalBarrxrtxf l I6€Erdrl .€upd:rH:r:=:r l rei .vor,rpcryBcrlx ixar0€rrrr t ,616rrr l0.:w€rrbg ro ^ rrQ6r[€! va €ivcl fiQc(yI(rlx6. :ro ]ioai(tt€ly$d [aE, ) = lt yl ) = -1:

= \ : 7 )+ lo t - l ; l :

.tltrsi{ro' oqilexa[ 6i! rl anod€lEn ed llrdv euiol.6tepnl ru npcrylrurLror ,4:

,4r: =,\;r biver it:t A:t =^r'1 r, apn ,i = Ip ei.vcrL nqa.1Forrr6q.

Aw6 ),oLr6r qxrivcrar oo.1016,lla xd0e nQoruollr6v riil,artc ,4, €no[ivo.)< ][Qi?[r!\,o rJndQIsI x<inor<r ra.7i6c{ To t6tob rvala x: llroQti tia *rtr. pLyu6urj,Movov 61rrr,4 = Ar luroeoip€'vq €ilqor€ oiyouQol 6Tr xcrl ro ) rct-ro r eivcLrp(rytroll1d. Mcrl,toro, (rx6[I fi€Qloo61€po. tu bro'tuttiolara Eiyar /AAEK(.rrH, = 0 orl'\' npoTraxlxr] oltltl€rQlrin fi€Qi/rr(,)ol x,]r itT tt = O oq [!yqb!xl]EQFLxlovrl fi eQinl0)ol.

l6r&rFa J Tu t6ro6rcvuoycrc €vog nQqypalr{ou orrpptrprxorr nilJa)id q€\'68 EQ[nr(rvou fiivarc! riorj ovrloroixouv s€ oLaqoQelrx€E !6toll[6g, si1]dropooychuo percrlri toug.

H cnobrLEl EElirvd d'ro rrt\r nl,tQoqroaio nou 6i6€ror, At = \p. Au = \* ttcrL

(\p)H u = (Ax)H a = tH Au = r:tl (^2a\. (9)

Erer6rl rc ,\1 eivnr. :rpnlpnrrxh, oL oxQoi.oL cQl0poi t:ivcr )r.rHll = ,\rrttlXpqoLpo:ror"o!;re r6Aa qv un60Eot ̂ \ I ^, nou o64yd mo ory qaopa 6ttt11, = 0. :ro nap.ib€LyFd Fr{E,

(A s1)r = 1..u,,

' ."] [;l] = [l] , = [,

(a+fr=[.,r,,,;.'] lil]=[3],,=l'

o. u = tr.|, 'r--'il= ) ? - 7 ) - 8 = ( , \ - 8)() + 1). (8)

l,l.nl

5.i NlIr .\lKOl nI\,\kE:r :Y\ll\,IETPIKOI KAT,\ EPIUITI]\N!lN

Tc 5no (urrl LironrcanoF.lr(i €ilcL o00oy61ro:

. L H l J = l l I 'rl ' ' '] =o

olo'rr1. rE ioor r.'crid rb'obralrolcrc 0c rirc! xar ororcbrnorE no),}l:l-r),ciorc r/o xtlL e7i. Y:roerorr orr :rtlipror|llr ,J = .r' . ororrru.i/o zcl u/,J rivur lLovabLcriu br"cvuogurrr xtrr. rc r6Lo6LuvL,olLcrtcr ixorvzavouzorontu€i rirorr va 61ou trtrioc a!c. Aqoi, rltuv rlll op0o1d;vrr. eircrrr.tprr aaeox.ttottx.i. Ear €n €lonv oav orit),€q ror niv(zrr S. rore (o:ro;qory6aivrr :rriworr. orav ru rbrobrcrvrolarcr rivcl or otq),rc) r1orye 5 r,4,5 =A.O 6tqoroaor6! airaxoE iXFr oAqoxttrovui, ortil€i.

:rt! nr0inr(')ol nor o -4 ri!.lr jra(r1r(rr,{oq z(1r orutr€rarxoc. or rbroi4ricror., €rv(u nQcyll(lnr(t<. ortrf!)lc tr€ rlv rlrortr( 2, z(r rc, rbro crli,olcrrir rotlrir(L oQ0o1r;)rr(i crLrp(t(,r1(1 pr 1l\, (brorlr( L T( Lbrobrcaiq(rr( c|r(i fiwrr, : r io1c:rprr l l r rrrrxr i (ralor l i r ivorv r( , ( , , l ) / ) . f = 0. in,"c orov pr l \ r .vr i7"rporol,Kr(t(),{1iou 2l r(L r( p1xrl rouc cilcr bN(1r6v !(1 ,'clolrrononl(loLJ! or 1.l r \ o I f t o / . , { n n \ ' , d r , ( n r t / . , c r v n : n u l r n 7 ' ' r . o ' r r v ' , r ' ( :

I l d v A = A 1 ,

o nuryowrnouir rirwts S poA?i vt filEr oQqoyl;rvu'< nirwK Q.

il o0{k)x{1vo!Lx(irrlr( rrrrv orlLrirr r.ivrrL ou!{htxl roobiv(lLrl pf Tllv (,)rf.) = -1 ri()t = Q t. H or,1,rto'lr br(11t,,.,'|oroi'loil S rl,t =A sEFrblrsn€rar xor tivrrarQ r AQ =^ 'i .1 = Q^Q i = Q.^.Q/ . ordoatlr o ir( (:ro rc o:rorbc,6rrpr0rurpritl(r( rrt( yp.rtrtrrl4iq i1),./r-tipcq:

sO 'Ev&q na&ytnrrz6q or[getpuoq irivcrzcq p]roqti rcr :rcrpcyowo:rorr1-Oei oq popgrj A = QLQT pr rcr op0oxcvovrxti rbro6rcvr-ropcrrc mov QzlL rrE rdror+r8E mor' l\.

:rt fs!+LcrQi.1 4 o1lt Mry(lrz)i cruro rilcL to Ottitprlpa tar KrylLotAAnaN v.&t bibEL ril ooro t €nLl.olii ro\, .lEov'r! lLr(1< 6L).€r1.|li]q. OL .iqov€<cuoi encr xrioeroL percln roltq ztlr xareuotrvovrcrL xcrrir rcr rbLobLcrvropcrrcrror crrrioror7or 'rivazo. (H Iloaciyac{qoq 6.2 ou\'66€r ortrtrr.IaLxoll< nivox€q trrn-bldorar€; €i.],€i!,€G.) :rI Mt7clrrrri. rcl Lbrobrovno$.rr.1 bivorv rLq ;alar€_<zcr€ror'vo€q, ,,,'crrir lrixor r(j)! oroior.v urapxEL zaotlari orttni€o't ri €q€izrotro!

z(rir rr: iri),E: z(rErorro€r: li\,€rcr zctr ' ).o:cuoit" 1'

i Lr.ir.i ,\nitrrFr2rt n(Dotr.a.t, nl

lJOTTME: KAl lAIOAIANYlMATA

:ro Mootpax(,'1d, o xtnos A = QLQI €ivar 1v(,or6q a.. O.tot&rlx6 @E<ip4pa. EcL'r nol,L(rrrl.Gorciooul€ orrlleg pe ypcHr€9, o nivoncg ,4 'riv€ltr! o.','6u<rop6E povobLco.rdroa rpo6oldrv nou eivar er6rxoi *ivozEg rdEe<,x 6r,<r, tqgpoprp{q rr? :

Lr

1 = Q ^ Q I =

L,,

= ) r . r . r r - \ / . . " r | . . ' ^ . . . t , . . ; ! . .

To 2 efii,2 r(oQd6efypd pc( aze! r6lorqr6s 3 xar 1:

nAPAAEI IMA ,a - - , . i = l : r r - + 1 1 l

To rbrobrc!iotrc.rc. rciov'ro:rc,i41,tu" f,iprlxoiur,i;irni

(10)

r t l l l f r l, ,=71_rl L(, ' l :=/2Lrl .

'Exo! oL ni.vcrxe6 ofll 6€!r.ci:rl,euqd eivo:r1.rf xal rirrrl -orjl6E rri yaappa4-ncr rapr.mdvorv :rqo6oLd6 mqv eu0ricr 6Ld ror.,rr xd fil\' €tlO€io 6!i1tou jrt.

To.,'i!o!ps 6r! nrJrt sivar. rriwote 11 ibLc 6n.1ovroroir1o1 pe qr A = SA^9 I

ncr ro g6vo :ron .Tiverar eivcr vn e[rdr.rou0ei yra S = Q xar S I = Qr ,ao.L \Qbraonao6ci oE xopgcirLc, ivc yra xriOe r6ro6rivrop<r. To (nor6leop( ei\rcL 6rl6lor or olppe-rqrroi :rivcrxeg cvcrhiowtrL <re po1,o6r{iotaxc< neo6o}.69 nou €ilaL1r| (lrrdE o!lqe1Q!,{oi. nilnxeE rrilerog erc.

ndAaliA1o1F.t\ &,l.enivcxcrq eiv.{" npclfclrrx6E xar or L6rorrFtq-ror o4t6ei. vcrvar zr Lrurrc npuypLrrrrig. rorr ru r6ro6ruvt'oyor.r rou rivor ftri.t+ t(prrylrar'zu.6r.6rr. rilcr oL ),uoeq tou (A )I)i =0 x(ll eiror. Sulcr6v vtr u:ro).oy.o0orirv geurr(rlorq{. Auyd 6$oE 6€v 6tl €i\rot oQooyd\,|,.r, ntrqd g6r,ov 6rcv o ,4 sivcLorlu€Ialr{6g: .4 = QAQ? o641ci orqv ,4? =,4-

Eciv o :rivnrcrE €ilar fipaltraxlno( cLld rdfior.E6 rSLor+6s €ivaL FryoErz6q,r6re or" r"6rorqr6g audg eprpcvi(otct d'rit !€in. E.i, 4 a + ih tivru 6rorryi1qayptnrxoi vaxa, rt to i6ro oup6oivtL tdr ts rt! a - tb. H oai.loxoa rou,4 )I eivor nol,r-rrirvupo p€ nQoy$oxlrioiE owrt),eot6q ,icr 1Lc r6toLc nol"uti)wpcor srlcr6r.xig pi.leE:rtire dvd qEiW. :qv r€airrool 2 eai 2 o tnaoE rtov prldrvnegrdleL qv *(6r +ac)1/2.

.ttrri(Do4 fro v( F4r.lorr crardrii.. ro.tcrotrcrrrro o€d)Altra 1= QAQ" ro c:ro-6t-iEujt- 16vo or4v rapinroorl riou or 16rotqrig ror..l rivcL br.cqopttr.zdq lercrElro!<. Torr riraQlorv oi1o4(1 n <rveirl(lrltu rbrobrcrrlqarc ;(rL o,4 6rrq{]Jvr"oroL€ir{1, (?rFrG. E! rorro'q €llrrL (i)iotr(t (6t:irr rrv Ilcairyacrqo 5.6) 6rt axlaraxtL f& t:toetrerdevdurt< tbuttryl<, lt'u; otl4ttrpLx6< tit,ctxtl. €E&dnr9t iKl 1^/11 r\\t rltiaEE arlar4ya opeoxctrorrztitt tbut\o.,ott.ir.rr. H uf;p"ti( rEai-nr(,rort €ivctt (uttl 1or r(rrorLzor rivczcr.;ror ixor ) = I srclcl),cu6clorr€lr n{oat(. I(lraiq vo [61(;'lfrcr ro nLi]ooE rrD! rbrobL(uJotr(ir!)a.

fitr ru r€),rrorvoUrs p€ rrl $1./o6txrl nrpi;rrclro1 lpercl6lLcror€ ro (1(iloyof!o: xa(ytr(rrLTorl o0Uot(;)vror ri\'(lxrl xcll Inoprire va ;L(vrirlrrrr.rr orrlrtiu(reLLrr qr crrcriqol Qt O = L O (ldoreoqro( 0( tllru.rcrcoroo€i ll€ rov or'!r,iior?rrpocyo ntt l ollletzl {kr tiv|l a,'s f,' - /. To yAirtq{cr t,' (rvr(v(zlri to rtir6()tt(1:r' 'Ev(<

rrryad'dg ni'a1tuE pr o08ot6,ortriE ortil.Eg,.iy.raa oqqopo,t u6ttior

OQ00Fov('6r(Ii{)r nilax€;

N { r ;rL)orr iror]ir 8 tr i ( c\'. L},oyrtL Er trg EpyrrtutriE 1t ir xttE IEtoQt i r( otrWAnE itt'tvuv nluy!(rui qtEri x(t ir(< oQqo{orultdoE f.iyuzuE tuto|ti rd ot'yxQtqEilt iwv qrqsi tolt loradniolt xri,)"or, -ivnv lLryrr6rxri ({n0tr6 ffio).ririt( rrtl]t<L fr( rq rbrorrtri! rt olyxeLdrt (urlt fi\(r rdrr rr-proo6rpQo und rn(7.o1ia: Tu) rircr ;rqrc.Tpcrr.zti or(v lil = --t x(! nr$axolrcr orov tro\,(brai.o zlrio or(vLi t t Lt = L T 4 L br obrrrvtrotr(r( L^i.vcr. oQboyrllr.1 ri rL r i vi rL <\rruarcir vc t-:r L),r1oiliror dror! r.r d1o!1, po!(bruio 1njlloq zcL ra ltvou oQ0orirlouz(i.,!',

Or LC\drlrrq auds r(coq,(l"irrlzcv 1'( EptrLrrrl\,oljq rivarr6 (or+tnreLiqL-h(1orLi!0r! rr'rv o!!r[€r0Lrri)\r). (],1".i 6Dt ilouv (lrotrl (nob€rX0Fi yr.r oQootro!(bk1io,,; niv(ze. (oltrn€ptir46nvopira,r ro* o40oyL,r!i(]).v).'Eror rpoyorA.4La zllr't-!0€itt! dtr! rQfL< L6ri)rrFrq rol L'ror tlr,rLoror.Toril o-rLE rgollortrveq Lbr6rlreqI ir,r.3 ro! L ori[i]urirr' 6fl o L' r'iZE oa0ozalo1rrza< orrll.fr:

\ r r r \Atxol nI \ \Kf : : . \ t \ r ror | ' \ \ L t l l - l l ^^vr \

I rHLl = I , L i L t t t = I . i t U H = U 1

AllTo obitlai ?.1r'€uor:i(lv orlv lbLorlrc 1/. 6rr o rio).),anico(op6; ge Ii beviTer e:ribpllr4 ot(1 ro0tesrzh lrropercr ri rLE prvi.eq rt r(1 Frtr.rt. H cr6SErh €iytlrLricr 1pcrpp4, o:ror; azpr6ci;q itrav zor fLc rov Q:

' : r F : :r( cl'i.Lz.i opbtro!.iltrroi . r,r unJll (aoqnuio:)** A.i(irlo( uL:,z!i\ortr. ?!r!!a[rrtr dr: f,i1(r': rf &O(ori q({(olroi: (s(\oii adL

''z(vorLzori" ni!a2.: li.rfyr(6rzoiq (arotroi.-d ib. E1d: anranqTr'oi( o{H,,y,rw Lbrobtrrviotr(o.df1 ,^rp ot zq{trLd (n o!r( r:

"ar' dlnr.7,d ri\r( $rf,..i. i:!r dn (Nrt otr (ut.i,ir

IAIOTIME) KAI IAIOAI NY:MA'I'A

r6.i'aryo. t (u r)H (Ue)prllrt 6larqQorrvrqu

= rHUHua = cFy xar (olaMyowog y = r) rs

l lt/oll '? = llslP- OD

H €noFEq r6r6t$cr €wofiiE€r q 0tol rcov r6rctqr6v ror U xct 6eipeL 6rL xrioe) fi€eltxer(r|, oro\, [ov(r6roio rxxrlo.

l6r,ftrF(r 2' Kdo€ rborqri rou [/ 61er. cn6],vq r$t lll = 1.

Ar,'t6 6r€rcr apaoo( trl[6 qv Ul: = );r:, orryxpfvowaE tcr lrirl rtl! 6uo ]rL€!Q61:I Ur: = | x xord qv l6r6rllTd 1'. tsv(i) ncrltote A;r = l)l l.r . AO(l l)l = L

Ioldrqio 3' I6lo6loviolroxd ,torJ dvrlororloriv oa btoqoQsrx6E r6!orrF66eivct oqOollvr,o.

H or66eLf4 uro0dter 6tr" Ul = ,lrr. UU = ̂ 2!J v.a ftti?"tEt too)t€Qrrd Yrv6tt€v(rgoQ[6(o!ra< qv l6dqrct 1':

t:tt lt = (U a)H (U.!) = (^r .r,) d (): y) = fr ): r' /r u.

:i,yxQilowuq rqv cQlorEQrt pqq 6€Eui ir).eUpd, .xoqre )r^r = 1 i rl'y =0.OFo{. 1.Crrri rtv l66rlra 2', )r)r = I xclt rtor 6cv propei v(l rolri€! )r,\r = 1.AQn ?Fg = O xcr rc rdLobLcvioycrtc tivaL opOo'ytivrcr.

nAPAAETTMA l {/ = [ ' :"1 --i". ' ' l .

Ls rnr cosr lOL r6rorqrd< artd;v rorv orpoqri;v ei!c(! r" xnr c 'r fiou exolv ori6lrlfi1 trttlivo. To L6lo6lcrvuoForo €ivo! n = (lJ-l) |,(a! g = (1,i) rou eivcL opooyrivLtr.(ouFrl0Eir€ 1,( ,rdQ€rc roug or(uyeiE oto r:/ry = 1 + i'] = 0!) AraLodvxo< rrr pe valivor'I.tr oQoox(lovrxd, zcr 4 6tcycovLo,T oilol rou U eivur

fcos t - sinrl I-= t--E ll r'llt, tl ["" " ,,]I4rer.dlom 6rr, ro rq6olpc tr,rv i ro -i dl.l.alcv xctci tqv conorpoqrl. IIpdypcrtL.I 6€Er.i nl€ued eivru lv6pevo rpLriw govc6rcitov nLldrorr trou ncptrler 6vcgIolc6niog rivcxcg mcr cprorrpri. Avd[€oa o' 6],ous rol,E U!"yabu.o\ ni.l,arcE.ro en6gevo ncrpd6errucr 6iv€! rov nlo €vbrcq€povrd oQooFova6rafo nivtrra.

: ' I l l l \ \ l l \ u lO -NANL \ \ ' \ l L f t IK i r I K . \T \ fPV lT l \ \ ( r \

IIAPAAEIIMA 2 O nircracrc Fourier Lr =

yarq{3{ I roL, a,'1r €ri o t}.I 2 ror t' €i.rol

1pr41L{ i ror tIF rri orilit .l rorJ ti €iva'

O rcpi{10r1 v4i or,Inrelt:r fl! ori}.q o€ lolcbtciu br(vuog(rc. (Kri0r oloLr(€io61rL cr:r6),!t1 trgrl 1. <ir,( x(iop. orri),I rori cazrxoi F ixu p'i'to< !fi.) To "/€Yo!oE6 . I F I , I r \ , , r I H f t r ) . r u r 4 c . l r i r 4 ' , r r . r . . r r , r , i r r n l r r v o r r ; r , r u o X q r u r r u . r u uFourier r1cr linp.!0rf(iqorlLr ro rrvrprz6 orpr:io t!! I]ca.r./pdqou 3.5i

l i t + r , o r l r + . +u , " r )=0 .

! 1 t + t t - + t t - r + . . . + i t " 11 = o .

_F1

"n "';t

[ 0 I 0 0 llo o 1 0 ll0 0 0 l lL1 0 0 0 t

)tlr ;rq('rrt :rrgi:rrroill o BrY(rbLir6g ce!0[oq x, tivor llt(l ?r-oorrt QiL(l roul . nlpf i ) igr( f orov pov0brtr io xrzl 'o | r l tor i ( e =21t ln,v.( t rooxrcf l t rr r : r ' i ' r

roL, o;roior or bw(4(€!! eiv(1[ ol.oro|opq( n(r(veprlptveq oto,'xu]i],o. A!r)i rlz(r(rvolL! t_!(lo(|d.ilfr orr ro (i{)0orolL(l r(rv ?r lu!(4te0)r rou r{ nol) ti.!cr o}.8( or., 'or i : ( oi . ! r-< ro! I - s i ,v(! t r l66v. Alys60rr( i . ro coQoro[cr f ivuL ({ ," l ) / ( l l ' l )

I t 4 h t v t i t t , , t .:rl bfltT!'al rrpirlIn)ol ro II_ tlvctt llt(l btraoll ror rr'. A! l !oo!x(! pr ll,i-'

rcr tiv(r n(i),r pi!(-qg povribog. blaqroQfrrzri rou l. 6L6r! cq€rdiolit8 ororxti(rzrt i< rr lq 6(ryoviorJ tor ( i t tU vu ouvt:rr i r< I I . l . OI bur. lp€6 tor I I '6 i l tot tvn( i i , ! duaoro[a ( l l ' " l ) / ( I I ' I ) = ( ] .

O t,I Lonro! Fi!(r oaootro!(lbr(ioq xiluaq. :tcr nporllo!|eva ntrr(lprirpr4rezcL ror nmi.otpoq6 rou, nou dXe! rll ibrcr crzpt6tlq poaq)rl. t:xr6< uno ro ort ro u'(vrLzaoiorarcr cr:r6 ro u ' = r: ', = u. Avcr.At'lafLolttrc r{i)QG (1!ro iio! or!a6rt.En€rbi o Lr ir(! oe0ot(,!.rbl(io:. o m'tiorpoq,i,< ror tiai0rtza o\r(oraa.toarhEror (rori bcr d.I(igtL rinort) xrrr riciQvovraq orirltl otoLTein (rol (lLir!€r ror, ori r). o cvrioraoqoE tor I/ ei'vcr o L'.

Kcrci r1v ISrorlt( l'ft,r! opooFovobtui.tDv nL,'ihxo)!, ro lttizoq ro! 6rt1vn-rrtrcroE.r !i'\,. ro ibro [E ro tlrtzoc ror Ui. H a\,601€t.r orov ttireo rcrcor(iot(o'rLoonrar pr r1v erfpY€ru lr€r.i ro'\r [Er.ro1ltlcrLo$o. H t-!t!eyt( €t\,.rlro doQoLotr.l

rour6lQo!( ro hoporola r(j)v tt€]rov(1)[tvrD\r "aABo!r.1cbv". 'Evtr

6rrivr.ogu t!:ros to r = (1,0, ....0) n€a'a7€L io€E ouvrLorpogeq aro zdor orvtotriro(o u x l o r i r ' , ( z ' , , r o i r ' ( 1 . l . . . . t t l f i ; a n I r i o r y L 4 z o : | ! r 1 .

Oqrilorpt 6tL ro tir g:ropei v.l rinoioyro0ti yQityoaa tt€ ro! Tczn Msr(-oTrlp(rrLoFo Fourier.

TIAPAAEITMA 3

I

a=

350 ]AIOTIME> KAI IAIOA]AN1'MATA

Amoq €ivar 6vaq op0oy6v|"o! nL\\tat xctL xdrd qr ISL6qra 3' 0a 'rQ6r€r va€xer opoolcbrrc r6ro6rovtlot.roro. Arrci eivaL orrll,eq rou rivc'ia Fourier! OlrbLorqr€s ror rp6ner vc 61orv cn6l,ur1 r4r{ 1 znr eivcr or apLopoi 1,tu,...,tu'' r(rj 1, i, ir, ir yro rq oqxexprperq 4 eni 4 nepinrool). O ;tivcxcq a!r6q eivarnElYpoxrlioE. oll,(t o! lotottliE lotj,acl ia t6lo6tavtiopctrit tor eivcr pr,lodLxri.

Kar lro r€],uri o4rei.o:o1. Or, qvrFEo$rrrsvoi. rivcr€E lrcvoTro[olv fl]vI{H = K,6na,; crpr6drg or cvrro!$$erpLxoi raavonor"onv qv.l(r = -1i. OLLbL6qt6g tol,\ rpoltrnrow apdorog crn6 r1 orevrj o1eor1 roq pe ror,g Epprrr,avoriE

Biv o A Eivst EQtu1fir6, t6te o I{ - iA xfudt drrt"EQtnrny6g.

nol,Io?rl,aordlou$€ p€ t, dQq or !6ror+d6 ro! Ii €iva! xs0sQci qovrqottr6qavti yLcr xcOcpir npcypcrLx6g. Tcr r6ro6Lcvrlopatc 6ev cl.Ldlow. To EQ$rrrov6nc(pitbel./ps ro! nQol"you[61,0r.,, oe].i6cw eo odll\/orio€ oro

, " I z l J + 3 1

L - r + r r ) l l

Tc 6ro1(iwro orollsid eil,crr ]io].l.orlcror({ ror i (pl6errd eriliQirroviq!). Or!6rcr!p6E €b6 €i\ro[ lli xo! -i. To r6lo6lo1uo[cfio €ilcr! nd),! oQeo./d\r!& xc!iXoupe nci2,L Ii = UAUI/, ire dvav porc6r,cri,o tI a! ,1[c! nQay[arui6v oQ0oy6\rrcQ. x, f r r , r 8; rur - i orr t brqyJ,uo rou 4.

H ncpdlpcrqoE curti owoqilercr o€ dvd rivoxd cvtrl.oyr"tirv peroEu :rgc.1-pcTrx6\, xcr! p!"/ob!xd)v,

nadyFatxoi baflt Mryaiun'r

R'i 16eo< b|dlNpllftrv tte n + C"= x(i)ao< btovrropdxo! !€ nred'ItrorLasE olumdG,( FryadrzdE owrotdo€S

r[ iro-c: 1,,]2 =,,?+.. +,.; , * Fircs: l l :d '? = 1", r+. .+ "" lr

dvdmqoqos: Al; = Aji + EapLrldlri'< dliioraqros: .4/j =,a1

( lB)r = Br AT * (AB)H = BH AH

tootee. lLrdFevo: "7 g =,1y l +. +r ,v , + €oor .a.1! !6p€vo:r l ly = i tyr+- + iny i

(At:)r u = i:T (AT.!r) * (.A:t)H y = :tH laa u)

og0oTuvronlro: , ra 0 - ogOoy' 'v'orrya. rrrg n

5.5 lvllr AIKoI IIINAKE:: :YMMETPIKOI K,\TA ElMl1 IANON

o!pl€Iarrci K"€s: ,41 = ,4 * Eofrrtdloi rivaa€s: Al = A

n =QAQ '=Q QT (rqqyporL'{os A) e A =UAU I=t/AuH (traayForL"l's A)

auLotrFeqLaoi iivo,r€!: fT = -I( * (wFEQpnroroi rivcxtE:r// = -X

oq0oy6vnr niv.: Q"Q = 1 ri Q? = Q I - oOooFon"tsiot tit.: uI! U = I i U E = U-\

er ) r l e ! )= r t !Na t lQ . l = l l " - l u t ) t t ( uy )= l :Eyx l ux l = l l r

Or ofll).€s, o! yaoHrdq riq! rq ldrodLcvioF(rrct i('tt Q xctt U €i,vo! oQeox(vovLxd1(cr tia0r ^ = 1,

351

AIKHEEI'

5.5.1 ru, rous tryobr"ouc nArqForK 3 +.li xat I iq) 6e€irE r$ e6oas ro$ mo FLyd6rx6 rnir€bo6) 6O.i$ ro doeorols xqL ro yLvoFrld rot!y) lto.irF touc o1|tvrfl: zdr flc nxol.r,rc( qr< ro!'.

fl€qraxowor €vro( 'i €rro< rou trolo6ruiou xix].o!i

s.s.2 TL rnoqsire vs n€ire.r|(c) ro d0qorouo €!6s $ry(tbrxon oeLolorr xc! tou orJlqous ro!:d) rov o!!u/i €v6s (arsBon orov rrovd6rdio xrrxLoi1) to ltv6p€\,o 6!0 aeL0lt6r oro! FovnbLdio rux),o:6) 1() rioAo'opa 6no aqLoFdv orov polobLaio rixtroi

5 . 5 . 3 E c i v r = 2 + i x c r . s = l + 3 i . d e u i l r r o l i . r - . j ! , | / t " , 1 . a t r l r . E r a L i t e . n m € d r r Iofl6lfft rLrrli rv ,oolidL t. I' pni lrl "or

ofl r d?r6).!fl1r4q 1/r roourar F€ I6'd, l' .

5,5,4 Bq€it€ t(r o rar 6 tuv Fryo6rxdv aproF{;t d + i6 orov Folq6tdio a!"llo. t. no}.r)d('rovi!5 0 = 30',60',90'. Ef,a),I0€noi€ !E on e!€Fias utroloyLoFo &r ro r.rody(,,!orou rqdrou Filar o 6, Ln' 0<r.

"ar o '.r'6os rou rq6rou rivar o rqirog.

5.5.5 c) Etiv r = reid, rotc rivcr tc "?. "

I . xnL r ot roLLriE mnqtrdv€<i noi 6ai-ozovrur or FLya8r"oi oerfuoi ron i,(ow i-r =r;

6) O rLya6rzos deOF6E e( L')t Loor,rat !e I yra r = 0. :Xe6rdorc q 6ra6eoFi doFryc.6Lzo €nir€6o, xaed)s ro r alEdvnsr s,t6 0 oe 2r.

55.6 Bp€ir€ rc pnxtt lloL rc soorEqLr6 yLvopevo rov

l 2 - 4 i l 1 2 + 4 1 1'= l + ' I "= l + l

t52 1AIOlIIIL: KAI IAIOAIANY:MATA

5.5.7 fadw0 ro! rTira,ic As rar lioioyims rov C = AFA, orcv

r= l l | 0 l" i 0 1 l

Ilora €ivar I ot6ol prrcEn tot,C var ror CHi Mriao\ dlnl to)(n€' 6ra! o Crftroox{r1.iifrd, {ir6 a,iro'o! /".,1:

5.5.8 a) XaloaronotriorE crdlor(trt yra vo ino€x! xo ,li = 0, rt{ ro! lraoryontr€vo .4.tj) A€iEt€ orr o plSrroroqoq 'ior tro),r! ffioT.oyioor€, .ivar oaeoy(irvtos orov

'R\Art) xar 6y orov nvit\n ztitpo ypatufu R(AT ). or rtoo€arr efpsL(;bELsrnot raor orr frydrart rsainoof riror -\I(1) rlor R(,4). ono,s ftrrr rrqr!. d,I(ior dll,or si.!.A r(irao y',/(A H

) r.at R(AH ).

s.s.9 o) ndrq o1|ri(rror rt opilouoc rou /E p. durrl ior .,ri0 ' A,rEr , orr l . rar lo 'a , , ' \oc Fep,r ,avoJ r i 'uz. ! ivor :aoauarr iq .

5.5.10 i , r t l , ioouc { ( ,n, ro ic tLu ' r ,q .oc rXf l rva( nvcyt .urr ruc out ' t , f r ( "xus f l ! , , r ,$ .aaft npof!(lluor b|clydllos nivoxaq ad( ivas ra(ytLa(xog oaeo-r(i)!6s nlva-

(H radq otrd\'rlon €ilat ro dr0eoLspcl ro! bljo {ii}.o!, b'6rL ,'1 = QAQ".)6) A€ilic 6tr or EaNrr{lloi ri'orrs 3 rni 3 axo!' 9 6o0po!q €leuoeoi(ls x(lr ot oa-

ootolobrcrior alolv 6. (Toao oL milrs ro! U pnoqolv !d nol,tronlaotiilolt{lt

| f oaLoto iJs, : " ' . )

5.5.11 ra(ilJre rorl; EnlipcvorK rirdnes gTt poaq,rt .t1..1r{/ + )1.4rj/ ior q(lopairzorr

. . f 0 l l' q = 1 1 o l ' " - 4 - 3

5.s.r2 Aods rra rlriytot or(! d!a( or)oro x(l (\'r|Jr(adbErrp( 6r.1v.ivcr liioo(:() Orc! o I 0i!(L EAfrfl(v6E, 16r. o ,1+ ?I sivoL dvrLmail'+oq.6) Or.r! o Q eivcr oa0oyrirvroq, ror€ o Q + (1/2)l rlaaL (vrLmailu1oE.y) Otdv o .1 .i!oL rqcyBdflros, rorf o / + li Eiac'r a!rLoiAa!'tt1o<.

5.5.13 Yroe!ft€ 6tr o A .ivar oltttErerzo( rild"dq 3 rri 3 fe r6roqrdE 0, 1,2.o) |lolt5 ibrorlt.s Lo{nolv oiyoli(x! y(r rl1 rbldblalnd[atd z.!, ui6) o.$yAdpre ro! lt1lb€!6toao, aerct.q6 Ft6E!o1oqo, )(dreo'yeo$dv roL X.bqo

mti6v tou A. tL. ot 6ort0Y) Bq€irE dld 5'i.v!opo r yr ro oroio .4x = | + r). Ef!(r ro r poldbrrot6) Yr6 noiq o!v01t"EqTn ro b dxer I ,,lri = D Lroti, ) b u v r u , . i u , L U l u t m l / , - r o ' o r s r r u r s I . l 5 :

5.5.14 IIoL€< adttloei€E trrriraov, oo! En6F€vo rarcilolo, mpr!1ou xo! ,1 ror no[€(ft€Ad1ou! roa B:

r 1 ! lt i i l

lii;:l'iliilil

r MllAA-^OI nl^Al(L: :\MN{L^Pll!O- L-{lA LPMIIIANQN

oqooydrvror. dvrLmq6il'qror, reo6ori, psrdssor. EqBrrrov6E, rdEsox l, broyovrotrontouoq, Mdlo!. Bp€i1€ I(- nroqrdq rou A zor 1o! -D.

5.5.15 nora €iaar I brdmoon ror r(baor S oI(D! ro! racy$orrxdrr oupp€laridrv rirvdror ncni nr To qaofdflx6 e€ aruo ).ier 6rL r{ie€ o4rF€ieraoq ftiaoaos €ivoL olvdldouoEu flvdxov nqo6oi6a. II6s eETr€irar ro y8-roa6q 61 ourl I brdmoor Een€elii io ni

s,s,16 K(rcyad!'ir qtro lic orltloEL*rt Lit<jn]ra yra fl! r6rorrpr(a) Eloq riaqwartxon olpprarzou niloza6) ErG c!or(oo!\ rilaro, bllabi .!o! rtloya /, yLo rov orloio! ol,r_c or ],lroet-c

qs drldr = /z r€ilo1]r o1o pfba!y) E!o! o00o rdra|ori nilc'to6) Elq nia(lr(l Malkovf) Evo; r),(11ropolr'{oi nila'to (p! 6ldyoaronor)torpo!)i) Elos r6ro$oa9ou rtv0'{a

s.s.r7 A€iEtE 6rL dv or i,r raL y siwr.oa9opoldbrdioL, r6ft ro ibLo ror!.' ,d!T lcr rov t,'y.Xatoqronornore to

"atniaro a/" aI = L

s.s.lE A€ift€ orL r oai(olqa evds oaoopovcd|Cio! nilaaa 'raaorolsi itv dcr U = l, s).].d! oAiioroa be! roour.A an(lQaitrFd FE 1. n€AtladlEe ololq rori( oQeoForabraiolqti.\1xts 2 €ni.2.

5.5.19 Baeir€ ntv iafq oli)"I. 610r (i,ote o

r( et!a! oe€opov(16L(!ios. n6('lt d,flr0rai( lndeX€r drrtv €n!).olrll

5.5.20 Ard/u!,oro 'ior, rov dvr,.Fp! r'd!6 2 rrr 2 riv(rd r(, ro! oroior o).d _d mo,.xr .u r ' !u ' ; . Yro) .oy id, ' ; A ' - s \ 's I xc. uu) . r1orr ior orr ' , K ' r ru. roQeopovcbnio!. noro uildr r noer4oy$ ro! 1K t = 0:

5.5.21 |I€ef[q.ivi. o].o!S tous rirarei 3 rri 3 ror €ta(L loltolqoad Eepftdvoi, oeooBo'vcbLaiu uaL bKydrvrot. o6oor ro rhioos iElo'or rivdrr-( lrdQj(olvi

5.5.22 Kd0e rivdroE Z SKotrhrdr oe i!d! Eqpr dv6 ,aL dlav dvrr EeBrtraa6 Z = A + K,oroE oue|66g 6!0; Frya6raoE aer0[or : 6raordrar or a + rb. To rqqForao pieoEror , .i!d, ro G +:)/2,iiMo naoyFdrL,6 p{aos" tor z ei\ot (z + zL)12.Ba.i\toldioyo no yLo ro qcwcmra6 Fdqoq -4 adr 6LooriiorE ro15 En6F€vo\ niror8s

- l 1 + i 4 + z i l"= lo 5 l z = l i i l

I rl,'5 ilto lt J = l 1 l l 3 0

I itu4 tt'4 )

s.s.23 A€ik€ ofl or mrll,cE to! 4 Etri 4 nivcac Folner r civar r6ro6rdvnoparc ro! rildxo!$(06of.o( P rorJ Ilaa(6aiyparo! 3.

5.5.24 In -ro! i6Lo nivaad F€ro0do€os ye{iw€ ror z\N:Lnb ̂ i\axa Ct = cal + qP + c2P +.rP'. (o rivd,oE '3ror+r6v rou civoL r.il.L ivas riloaas Fourier.) Iqci1l[€ €riots fl<

I IOTIME> KAI IAIOAIANY:MATA

rdoo€aq owloidro€s io! ,ynopsvo! 'ltearc sr 6ldvl)strs. Cr, rcr eL.eo.L rt ow4rEttT ou c = (co, c1 , c2, .3) p€ io c = (rn, 11, 12, q).

5.5.X frdrt ,yla svdv r{rrxLL^,,"6 C = FAI.-I enal o yerlloeog o ,roD,ar]'oo!aqi6s tr"€-io.yF-', psrd !€ ior A,ror psrd p€ ro! F (o ,{ov6vds ou/Cl4rc) riaqd o an's1lo€losriollonldordoll6E p€ rov C;

5,6 ]!IETA:XHMATI:MOI OIV1OIOl'HTA:

5.6 I METATXHMATI:MOI OMOIOTIIIA'

OuorcorLzd! Lcie€ 6rt[o amou ro! ]{eqal(riori sfrrr€Qr.€i1€ ro ouviucopo S ',4S.

T( r6ro6!avdo$o1c ror A €roroe€ronvro olq orrtl€! ror S xcr" o ouv8lcroiroqS ',4S rlrnv iras 6rcryrirvrog rivczog (nou o!F6oli!o[€ 1le A). Otcv o,a rjrrnotfB€rplxoq. l$i(fope Q crlri yLo I y,c vc uev0rpiloupe otr rc r6ro6Lm'nopcrcri.vcr 6war6v'va urrlslolJa oQ0oxovo'\rlxd. :q pryo6liit] ri€aijrltoo1,6;ror o,4ircrv Epprrrar6q. lpdrqrcpe U xcL nri).r.1o rov irirlorc (61o6!c1,!opdrov. TrbQo, o'currl rrlv rrLeucric ncprilpcrgo, elerrilor.rpr citrlolq owblcopolj! rl1 1,4,i11. nouoxrFariqovr(lL Br ror i6!o rQ6no, ul,l,a W orrcLofiilor€ avuorp(t1t4to lI madtEtd xu nt' rurioqogd to! ot' oeLoftea. O fii\ratraq l6[o6ttrwopdlo]v $noe€ivcr $Iv udax€L (I slarra)Forrrrt n€oinroot) { prroQ€i !o tlrt! €ilcr yvoror6g i$noAfl l4ct vo [I ito; Xa8raE€ror.

Kdr' ao1ciE r6ou 6va: v6oE o0os yr(r va fieQrlQqvoup€ q o1aol ItEicE!r(!r1, rt xo! ,ill r,,1,ri1. OL niraxeE cwoi l6lovrnr dpomr. H [erd6cq (r.6 xo!i:v(v oro! cl),},o tl,\ut f\cL< tltrsovlpatop6a oltot6qre6 O petcroll[ctr"op6gatl16E trtraL 6!q q,loroLoyLito 6{pa. riou yi!€r(rr 6rov eQryog6pcor€ [€ 6rdqoerdEelrotboeLg,6uvripe6 ev6q rivqrc ri L6r,orqrdq -6no6 urQ!66E qr:orol,o.flxd rlrovr{qL ra 6{Bqrtr snql,olqliE 6iav €Qya(6pcorav F€ qv ,4c = b. (H cr,_rol.olqrlro),1.(r),coiu!r rov A uro rcr crEorcqcl pe ror'r-r. olld 6X[ xa! o7i6 ra 6sErdpe rov Z.) Kcrvovrxri, rEQL[dvool.r€ v(r ]JndQl€[ tus o),6x],qQn o!|oy6!€!a ofi6rir(*eg tr1g poprprlg.\/ 1,4,\/.6l,oL 6poLor. ro! A, xtrr undQxolv 6ilo O€p€tr!6bn

(1) Tr'ro!\'i. ororX€to dto!\' 6Lo! (luroi o! .\I 'A,\I;

(2) nota e!6txn poprprl pnopor4ro vc nrrulorge 1d rov r'l-f r,4,41. €d\iEnrL€toupe zo-rdl"Lt),o rl.a;

H rtLLxrl u:rcivrqol 6i6eraL o16 at poeqi Jordan F€ rn! onoio rel€!{;v€! 1()

AEi(tr .rov a6:ro vc 0rp6pcrore rd;q b1[rolplourtcL cruoi or. ouv6urogoi,11 r;1,\1. Av 6o0ei prcr 8rcqoQLxn rqi.ooon i €Eiotooq bLnqropdrlLcr rov cilvroorou, u:ro0dore orL ;[c al.I.lyt tr€r.l6].Irrtq" L = r'Ift, ELodyEr [r.i r,€.r [exo6l,nd rr.Tor€ rl

, t , , a r- = ,4r v ivrrcr - \ / - =,4n1r, . n

u. . , ,4 u.. y vr rcr , l / , . . ,4, \ / r , . 4

I =,rr ' r , l ru

r,,.r = r'11 'A.ii/u".

O v6og :rircota< oqv elioool ei.vnr o ,41 ',4'l-1. :qv e'6rr{ 'r€Qiriloort nou.\1 = 5. ro olroqpn cr:roowb6ercr, 616r' o A=S r,4S eivar. 6rcr16vr.o;. Orcrl,6E xorooxiro€rE €E€Liooovror oveldpqrc. :rI "/Idooa ro.v lprqpr.rrivFEraolitFarroBdrr,. nor exr|oevrcrr rcrpcrzd.ro, ra Ldro&cvuo[crcr eaLl€yovrcr

I IOTIME' KAI IAIOAIANY:M/\TA

oov 6dol. AlJfli 666oLc eivcr 1 p61m1 6uvcrrl otrloroilol, cl-],ci unciplorvxor ol.).f< l.ryorlQo 6paorrxcE cnlo:rorrlorL<. or oroi€! rivor fnrolc XAiorFFc.EnldLdJxortrc Fr.o FoAq{ yLa rov ,!1 r,4.L1 nou €ivar. nro erilpqotq o:16-ror i,6ro

H:tpr6-r1 epdlr1oq aqoQoxo€ rn! olroy6velo rcDa nwo:x,oy luI tAlyl, noln€Qll,op6dv€r xor ror i6ro Tov A, €nll,iyovrqE ocv ,41 tov rorrorlxo d,aoxo.Orioroo&irio1€ cr' cwotg rorg rri.vclxEg proQ€i vc epqovLooei o-rq 6rcqopr,xiq€Eror;o€q n eEroci)oEq 6!cqop6\', F6oo qS al,lq{E pEro6}.rFrtE 11, =,V1r. 'Exo!

or ni.vcxeg ouoi 0a rptmL vo 61ow '{irrr riora6, ror, r[Qci^/poi!: 'Oporot flharEs6xow N$ i6tEa t'tottltit.

5P E6.r B = M-1AM, r6ie oL A y,aL B 6xow ilE i6!EE !6rorqa!. 'E!a

!61o6!dv!opq c ioo ,4 slJirorolxei o' 6v(r l6loidvu(tlrat M-to :,oD B.

H dn66efEq €ivo! dpEor',, 6t6r,t A = tuI B tuI -t :

A.x = ̂ t .+ I,IBM ro = )c + B(,4,1-rc) = )(,e1 'r:).

H ,\ Ei,1,o! ){dr r6[oqrrl roo B. To !6ro6!d'\,!opd ro],l,qfll,qoldoirlx € U€ ror ,4.I l.XQlo+o?ro(blrqE rov xa!6!a ywop€rlou./rd oQilouoeE, propoipe enioqg r,<r

6rqrlorci]oo![E 61r o! opilouoeq rrov ,4 -.\1 xdL , - )1 iolxiEovra[:

de(E - ) i ) = de(I ' I ' ,4,1 '1- ) I ) =de(r11-r(A- Ar)M\

= det,[.t-r det(A - )1)der,l:I = de(,4 - )I).

Of 6io oQilouo€E io lseqrqQroruic( rol.ldw4ro turv A wtt, B- eLvat, ia4.Kcrd orvdrer.a or QileE ror,s or r.6ror4rdg rov A ^Aur B- EL\u or. i6Leg. Xro€notrEvo nqQd6€lTlo 6Qi,oroupe oEop6voq rivar€E no! €i,vdr 6poror, rou 14.

t r n lnAPAAEITMA a -

16 6 ] . O'W;r.c l r rb,orua< I xor 0.

e", V = f l l l .u" u n, ,q,rr - f l " '-u , l , , 6 ] ' r o t ru ' " ' *6<p ' ' 6 ro r ru rc l x r l ' 0

r " " r r f l , I | - *a=u ,4n1 l l : :npo6o). r u .ororr r iE Ixa, 0r- , ' l t ; ) -

rd" ,\/ = i: 9l .-. e r/ A1/ = ono,oob4norr nivoxds us rbrorrl i. It ' : a l

xct 0.

:rn! r€).erjl(l{r ri€einroor] F roQotF€ !a arpr,oupry{ooqe onor.ov6{rox€ rii,vcrrr.eler crn6g rLE r6rorpeg. H neginrool currl civtrr €rixol"I 6!6rr or L8roxqr€g 1 1|o! 0

5,6 I\fETA:XHMATI'MOI OI\{OIOIH'IA: 357

€ivaL SLaqoQerLtiaE. H $oagi{ Jodan eo iazrorior{o€r rt! rr€Ainrorol €novolo$-6.rvo[6vo! L6rorrFd]v xor. €vOEIog6\q. €Iarl(ooitE l(ljv t6ro6rovroF(ir(,v r6pa.ouro nou rolUarlo[oor€, €fr(1|, or! o ]i1 ',4,11 6I€r ro i6ro n].r1gog aae[cp tto]rrbrcijrcvlo[(inDv trE rov ,4 (610r. r(l r8LoSravnoFora ]io].].Gn],ao!dlovro! FE lov,11 ,) .

To nadro $6Qog ei1'ur prlrorroptr,o xol lrLo 0otopryrx6 elotooq trov ypc14r"-zdv F€rooxruarroBdv nou nQx6olxc! Trlou) cfi' rolE riiv(lr€E. Au6 pog obryeinio(n orry nopdlpaqo 2.6, orolJ €pFnv€i(4r pro orQoqrt { plc avtix),cor1 ripLc npo6o),j pe leoperpLx6q 6wor"eq ocv $ro €viQy€lo fiou o!p6uiv€! fiovr'brciororo Xrbpo. O $€r(loxlFqrlo[oe 6€v dX€L avd.ftI rl] IQctltur(rl ALfs6Aa. HfAcl+rzt AUe6Aa oF(D_( 6r€riro1,{r€r, 6roxr rov l-r€:rorQar€r oe ro}.Icj ,oo!ooF6

Ar,).oyit 6doqq = M€rso1p11atrop6g opor6tr;tag

Eftv ,:rnrp;\'ouL€ oro nl.uioro ruv yQutrUrlrdrv utroo14ucrlroFuv. rl oxro4o[of6rlra: r(|]v rflvdlic,ry A x$ B = ll-tAXI livetur e[<rLperLr.tci oI€\,]i. OWn-{,rirF rl! r6iu-r}.tld:: KaeE yQappu6| trttrdo{rlltdrtoqog nsQnravftat k t!a!flit(xt. Ev(t EltL ! \to\ rroQropa r$ IaQdyQdqor 2.6 ilia\ y.dt 1o't1ot O nhaHaEtlugubu rn6 r4r utioyy'l 14< 6do4<. Ed\r qLl,ciEo!$€ r! 6dol. oLla(E€r xc! ont,'cl1{a< xclL r(i)Qa €ilraor€ oe 060l vq eE€idoori$€ rL €nibQcoq dXE[ oro! fiilor{oI a dyit 6donE.

'Oporct niraxq flqrcdtouv tot i6ro pEnompdrnpd,|tt nAoa ,nqoAeflxiE [ti6Eta.

To a),ys6Qrxo [6Qo! €ivar,1ol.! dr].o. Yno06orr 6rr. tXoupe pernollpcrLopo ?(6rorE n orao{{) t{.u 6dol ,rr,. ..,1,,,. H j oltlt rotr A :rpoxunrrL erp(pg6lovrcrq

Tu, = ouvbucop6q nDv t . i =atrni+ +at , ln.

Edl, n. iQourr€ pru v(cr 6cioq l4, . . . ,y, , , r6re o veoE:r ivcrcq (dotro B) n<rrtrotreudEeror ps rov i6r"o rp6ao: TIl; = owbuco[6E rcov 14 = Ij lbrrll. Kd0sV 6ptDq npitiu. ./o €|vc! raL ouv6rcop6g rov :rcLcrrlv 6corxd;v bLcvropdrcov:V, = | m,, r',. 'Orcv ro povo 7rQd.#o rou oul6dilrr. e ivcr" I al.].crl 6rio$, r6te orivcrcrq ,41 n.erord$L mlrr rrQcypcrr!)i6qra rov raurorLxd pnaop pau.oprj (l).AxoLo!66vr.rE ro!< xov6v€! rnc flcQolQ(lqo! 2.6 €qc1p[6ooFs rov rct'rotrx6Ir€rooxltr.lrLoFo (nor nqrlver xo y op€1d6llrd) ncr ypcirlcpe ro cnorc),eopcotla ouviuooFo rr']v r,. O d\'ri.orQorpog nlvoxoE ,i11 L rcrpLorriver" eniolg rovraulorr]{o lr€raollFor.o[o, orov oI].dEEr I 6doil, (rno r(r 1r orc y. O rdv6vdErou lvopdvor 6i.ver ro o4rnigcopn ror 0dloqc:

IAIOTIME: KAI IAIOAIANI:MATA

5Q OL ]I,fuaxEi A xor B fiou ncrQtordvow iov 16r,o yqauplxd perco l-ponop6 ?, a4 npoE 6rio 6ropogerlaeE 6doeq o xar"V eL,tat 6porot:

lT)v re.4 v =lll" 'c,x v [?]" ,o.s "t4y ,!,x ,B = tr'1-t A LI

H un66eL!1 (aou orrlgilwtr mov xcrvdvcr rol yrvopevou) eirnr erintog plorrjprcranL o rd,wrpoq rp6log e!r1 olg €ivc( 10 naQdb€$lo. Eoro 6fl ? Ei.vo! nnpoid.fi orryt atqeia L xitia..]o4 0. Aur6 ei.r,or, lgopprxdg pt:roolrlptrtropog xcrireQryQdgeiql rl,iQrlg lopig u1 6ori0erc 6doIE. l-!o vq rov rlaQcotlooutrre opoqF€ avav nivo:{o, XprLtrl6p(ore rQdyF(lI! Fla 6doq. rrr. ro )1. 5.5 gnvepdrvtr6uo e;rLl,oy6q. Mic €iv(l| I orwtiolE 6dol or = (1.0), r,, = (0.1) rar rl .illneivnt pLa 6iro4 €rbrr.d €nrl,E./F€vrt yrs rov ?. nQiryFsrr TVl = V, 1o*o1, ,o y,r€Qld1sluf oxrtv €!0€i(1 L) xut TVz =0 (oqo{ ro % civur zdorro ulnv €!rosio).QE fiQoq qurrl Tll 6do{. o r voxaq ehql 6[sy6!!os, 6!6r! ro 14 1ia[ % eiv(rl!6!o6!oviqroro:

B = [?]r',r-* r,

i0ll ' ' -L' l

. [l1 t* z*'""q'6*''.B]

"fj,l,* "^",'-.w*e*'V]

;l&. s.s. AI).od 6.idrE nor rdre' rov ri\,.!ro-nao6o).i 6lay.;!ro.

To d).),o oqra €iva! o nivcrcE c),Lcry4s 6dtoearE ,11. fr.c vn rov 6po{re, exqpir,louge ro V1 oav owduoot6 r'r cos 0 + 12 sin d zcr roaoorrorue rorr ouvre).eor6cm4v lr1 mr1l4. Kcrt uvu).oriuv. xcL ro y. {'i ro J%. onou ogcrcox4gunopoq i

I r o lL0 0 l

_kt-t,l

5.6 NIETr\:XHIVIATEMOI ON{OIOTIIIA: 359

€ilar o rrlrrorLx6E) €ilaL ?! sh, + ?rr cosd roL naady€r qv 2I oriLit:

i1 = Lr r \ ,q{ , = [ : ' . : ]t s ( l

O ovri,orporpog nivcrzcq ,il-a r (nou e6d; eircrL o cvriorpoqo_c) nnr:L cro ro uoto l/. nc ow6lc!6p€vog F€ ro\ B rcr,41 6i.vor rov aivcza r0o6oLitq, nooxaroloallrc[r: naoqyorpilo]q. oqv Iloaciyacqo 2.6:

:!vol,ilo!tr0 ro xuQrc o![ni]Q(lo$o. 'Eort-r orr 6i5r:r(1t:ri.voxog 6ir0)E o ,4. Orp6nog cmLonoilorl,< ror orrlv ag<rlpcrrrr6rlr( 6[a1t,r!r,onoi,IonE- €t"''c(r vcl6poryr rtr lUobrcrroScrrd rou. Aurrf livovrcr orlLts ror ,\/ (i1 5) xoL o11 r/,'11 fil,(r 6ru1!bvro:. O (),1e6pLorr1g ),eeL-rr i6ru npdyp(rc rx1 y),tiroo<rr"\ lL!qrtrni"v Lrr( lu}r l . , ' r r .LrLr(rv: f ru \ 'u ;rHrrJZout l t v,r :nrC,,or loofy 'r rovpetuoTlpcrLop6 l pc 6rcyirvrc riivcx(l, €r.rl.i./otpr pr,o 6dor1 nol urrort),eirorcln(i rbtobt(1\,io[cr(l. H orJari0$ 6cior] r,o6{./r1oe oro!,4. no! 6cv rlr(r\, (rnl.oE.H oo,orrl 6(irrr1 V ob{11oe otov B. nor rlro'\, 6lc(ydrlroq.

Torilorpe xur :rtiL orr or ouvbutropoi 6nc0q o ,i\.f -r,411,i b€v €p(p(r'i.1o1,1(rlo1.r\, ),i1,olur ro A.. = l). :' aurl rrlv efi.orool, q 6ooLnrl rQarEI {r(! \,cnoL).arl.eorl1loulc 1o1,.,1 (p6vo tr:t6 rc cgnrra(il) $r avav rivcxo no! a{crQrirrol.fcrl.doro laatrprtE (no $L.r\r .llLI. E\,(r; rdloLog prraollptrrropog brarrpolosrov lrlbrv6}(D0o noL ro'v XdAo lec$d! ror A nar b€'\, efxe r(p[t( ox6ort$e rblort[6!. Atri.0erc. or $erao]nroiroBoi o[oL6qraq cq)ivori! T|,S rbrorq6g.r[€r.i6].nrt; xc! oq\' 'reri[r1 or L6rorlpig (rrig ri:ro],oyilovrnr pe gLcr aro).on€icran)dv o$orori'trrLr!. M'(ruriE. o nircx(q {rrriyurcL 6cr0prcricr oe rpr.To.rvrzrj popgrjzaL oL rbLorqraE E[q.r!i.!ovrc! 6a0prcio orr]v xnpLc 6rcrltirvro. (MLa rarorcoxo).oroic nrpr"/aciq€rar oro KE.{fr},a|o 7 r(1r i\,n 6rlFn qg e!rytitur r''qv i66oplrioxqol r(quxhr0r.) Au6 eivcr :roLn za).nr€ao. ar6 ro \'.r npoon.reoi[r !(l1jnol.o./ioouB€ ro 'ro),nrirrrrpo det(.4 )/) ror o'roio! oL piLe< €ia(r or rbrorL[i:.fr( ivov ur-ydl.o rilaro. Eivar aer0[nrrxir .r6uvdro! rc orp;ruzvriroorire o].r1(uri p r)"ruoqroQio oro ro).!dr4ro xdr. ri:1rorL! !a rill' alaxritoouu€ .1n'

T0ryrDrlxfg FoAqig F€ oaooFovo6raio M

H iradrl FaE rivlq. n€pa ono rit oxvrtq rrQi'rr(r)oyl ,il1 = S, €i\,or l"iyont:pirpp: Avri 'vc eecoprjooqe 6rnr levLx6rcpo rl1. nd$€ clriO€rtl zor ro!ttpnpilorltt rc tirut op0opot'a6nio(. To rp66).I$.l l.omol' Ei.vcL !c 6Qoi[€pLc cpxrtd un),1 popqrl yro rov rl1 LA,ilI, lra' atr6v rov :repLopLop6. OL orrllesror r11 = I,- aircrr€ixor lcl rivoL opooyrirr"cg (or1v rpcrpcrtr.rri repinro]oll 0a

| ": ""A=A IB ]T = ' ,t . r s '

IAIOTIME' KAI IAIOAIANT''MATA

yQdqa$€ ,1,1 = q) {c! €rirog fiE n€Q!firoo€o)E fio! ro r6rc6tctvuoporo €ivaloQ0oloJvra, €ivar c61i!c!rov !(! tr€xrlouFe o|qycbvlo A. OprDg ro ").tF+ra rouSchu." raQdy€r Fra rioLu Xaylort-rt [oaqyl, rouldtlorov oq o€oeia.*

5R flcr 1.id0€ iQr.fovoi6 riilcllia A,.u[dQx€r oQooFovaoralog rlrqrdgM = U , tror" ritor,e o U - 1,4U = ? vq eir,sl svo rgr,1orvur6E. Or, tdr,orr,p6g rou,4 eivcr or. l6reg p' cr,reE '[o! ? 2iar, epqqdloarol crrrlr, dQ[d OlaTdvro rou

Anods.gt, Kde€ nLvcxcg, r.1. 4 eni 4, 61er |,tia toljLdxto-ror !6toruri ^1, rio!o-rl X€IQ6I€QI rieQi?rrool 0d €nololop6c!6rc! r6oo€Qrg qoQ6g. 'EXE! ow€n6go rivox(rs A 6vo lbro6ldvriopo €t, Jroo [r[oQoupe vq xqvov!)iofiolliooup€ o' 6!cpovo6r.cio bLdvlopa r rcr vc ro roro0errloolrpe orlv nqdrq mril,1 rou nivcrcU. I' at'r6 to mdbr.o, or dLleg tpelg otil.eg eira! adr,""arov aa riQoo6loQloeoil,xcl oupnLlp6vo![E rov nl\r(rxc [rE rdrioLo rp6no, 6io! 6ole va fiQo1,(uv€r 6voEopoolovcdLciog :tivoxoE nou o!$6oli(otpe p€ Ut. (H 6lobrroois cram-SchmidtEEaoeoLiq€! clnfii q 6r!cr16qio.) To va €X€[ ro lw6pevo Lrr-r,4LIr, rou].dxtorov,qv ?rpdq oniLl ro! orl ooori poQgri: Acr = )rcr oltlcri1,€l 6rr

:To d€rix€Qo ors6lo €Qyqq6lrqor€ TorQo p€ Tov :rivars 3 eni 3 fiou rqrc-).ry6riv€L fltv xdr@ 6eEr.ri l<rwiq. O ni.vanc6 aor6g a1€r Frq L6loqrn )2 rsr 6vqr6rc6r{ivuopo rr ror eivar dwat6v vc roro0srn0ouv orrlv :rg6rt mrlLt ev6g 3

T6l,oE, oro xel,niroio 6{po, Torioo€reiTor, a\,a !61o6!dv!opa xou ,(L'!uxo 2 Efii2, aqe x.Ix.) 6eE!d 10)vfaE, oe daav oQooFovc6raio ,41r. rror.r F€ xr) o€r.Qd ror.,roro6meiral oq pvi.c ror Il l.ar.

u1t(ut1u11Autu,)h = =T.

* To !16].0110 ror xeqaloio! EildL o{rEa.D[No nepLoo6reao aq oeoaia. ndad mrE eqaafoyd<.

H roaqni Jordan oo 5U eivaL oaeqdarlr! rlq xpr-to{aia Uoaqiq mo 5R.

'" "|i iii] i u'\AU ll iii]

f1 0 0 0 l l ) r * * * l

fl * ] "'ut1(u\1Au1\4=li tiil

1 . \1 + * +110 . \ " * * l

lo o ,t , - ll 0 0 0 * l

5.6 NIE] A:XltNlATDMOl ONlolOlH l A:

To 1Lv61revo U = Utqq rivor" e;riotg op0opovcbtuioE riivclicg cttt6 6lcrnrorue' ]r ' orqv Aoz' lol 5.5.17 vl l r rror r /oupr ro! on,1rroi l r ,vo rory ' ̂1zot i I Atr =7.

ErrL6rj crrr6 xo l,ltplc ts{GaFoleror os 6}.o!s rotls rivl:lr,i€q, ol1vci [oq€nrroin.L !(1 crfio(filoll[€ rlv !1160101 6tr o ,4 brc1tDvro:rorcirar. 'Eror, eaprioQoroqrr alr ro tQlorpo:roLrloorpe 1trr vc beiEoup€ 6rr or dlvdr€rE Ar r€ivolvon pn66r, 6ra,i|g or Ai < 7 ,tl ot ExqErrxiE poAgit e"' ftiyour on p\6iy,6tu! 61s rcR. ), < 0 xar trd),ro-ra T,orpig vo lperoooo{pe-ro rl"rlpeg otlortporbrobromo$rir(l)v. nou uroeaoqr€ o l o€aleic €ljoro0tslcE, tor I]cAoyAdqorv 5.3raL 5.4.

i r r lI IAPAAEITMA ,4 - 1;

" l ,xr . r t ! ro,orru' i \ I rouo qoeac).

L , " l

Mi( €uoElo rbLodravroBcirox (or1v rpcrlpcrr.x6qrcr q ptovobLxrl) repvtt cro to(1. 1). Ar(edrvro: rrlv p€ ro y'2 alox$e q! 7rQ0)q ortl,rt xou at xor rt di,tr)t oTltlt€i\rc{ oQ0oy!i)"r'[c{ o ourllli

t l r t1 l r 2l1l \al =

lo t l. ,_, , , , f r r t t r t l l2 -1 l l rvtu r t=Lr\) - I \2 l l t r r l l t ' [

Auroq ei'vqr o rprl,arvLx6< ? [r rL< L6ro-rl$6q m1 6tcyiwur. fltr l€yatrtLt€Qori'vcru, ro :rptixo 6rjpcr 0c r.6r6e rov t/r xar (Jc oxoLor0ouo(r,' or Lr2, Ur, . . ., rC[-ro y.vr\ll,vo riLil)r aurdrv 6iarr lov [I. O r:ao],o1Lop6q tou ? rlev eivdL xaL 16ooe!Xdproroq.

A'ayovrono'drlrag ouFprrprxouE xcr rg;Irrcvoug rivaxrg

Erpupgo(orr(E r4r nro,_r(r!() xQrya)vLrin [oarpi], e.r b€iEou[E or. niroE o4r-p€xQrr6E li EQFLrro\,6< 7ri!or(6 -eirE 6l€1 6raqoQ€rrx6E r6roruiq rir€ 6xr at€r6vrl rineEE on!o),o oe0oxrlvovL}{6v rbrobLalrloBcirorv. XprLc(6p0ore €vcv opOo-povc6rcrio :rivc0ta. rov onoio f6llc pac r€uoq/),roe -ro ),Itttt(t xotl Schur sR. itolrirole o U ',{fi \ro ei\roL 6r.rydvro!. frc !a ndle cno rpL.Trrrr.zoUq oe 8r.rydrrolq,Xa€rdrqo.|'rar 6no 6rtFara:

1. Et|v o,4 r ivcL EppLrrnv6q, ro i6ro orp6criver x.r l [€ to ' , 'U rAU:

( u t A U ) H = U H A H ( U ' ) t t = U \ A U .

2. tuv orrutrrrprzo: r t FAtrr f luvo:€i.var 6raycbvLoq.

En,1oq o / / r ,4U f lv l l r 7u, FQp[ruvo! no, rQryDvrxo(. | |vur duro.r t i r ruc6roy(i\,ro<. Tollio o1Jpfi),JlQ(i!€r xl\r o7i63EGl €voq 0€[€Ild6ous o€o]arllaroq rlqfAt:lBIrrrtE A].y€6atl!:

362 IAIOTIMI) KAI IAIOAI NI'MATA

SS (AaopffiLx6 OedQ\pa) Kct9e neq!"aflk6g oul4retpLxog rilcx<rE pro-Qsi vq oLay(l)vroriolne€i pe q 6orj0elc ev6g oQeoycDviorl nivqric x(rl xdo€EAgrriavog rri.vorisg UnoAEi vd 6loyrovlonort0€i F€ q 6of0€Lo floS oQ0opo-vtr6roiou nlvcxa:

(r{Qowdtrxri neQi?rrorot) Q 'AQ =L (pr1o6r,xrj neqinrcooq) a/-1,4U =

Or oriL€E rou Q (r1 rou U) neqrcloov 6vc nl{QEE orflol.o oQoozsvovuid)vL6r"06!qvuopdrorv.

no,oalial,on I :rtv nQ(rypartl1ti orggerpr.xrl rcpi.nrororl, or: rcie€ 6rlF& rolj 5R,or Lb[orlpi:q zcr rc raroorsv{opcr(l si!d! neayporrrd. Aoro naodye! dvavr0cYgcrrlxov opeo[olcr6roio LI, [€ d].lo 16./rd rov Q.naautianq 2 f[d s]]p!€rQr{olg ni.vaxeq, uzr5p1 nc! |t€ e]rava}.qF6(wop€ve<[brolt[69. eivor oi./o!Qc1 1.Grco,orF6 ro 6rr urriglrr, 6va nLrlpeg ouvo],o o00of)-\ri,ov r6ro6!o!!o$ril0)v. MnoporL[e vct e€rDQioo![€ rov .4 oo\r 6Q!o or,l4erQrlcbvtt!.,'dy{Dv [€ oLsqoQ€rrxiq dror+6< rcr. xq0dq rQoo€lyi,!€rcu to 6QLo, rq t6!o6[o-luolarc rroqqrd:vo!! xd6€1( ft€rcrEi ro!q. :€ q.vri.Oeor1trr' or:t6, or. p1 orpporpr,xoi,,!ivc1n€q

dlow r"6r.obr.ou.ro$cncr (1,0) ror (cosr,sind). Kaod)s d - 0. ro b€uT€Qo LbrobLL,,tlroprl n0oo!./yi.!rL ro rrQdxo, fiorl 6ivtr! ro p6vo r.bLodLitvropc rou p1 6rcrycDvLo-

f o L lnoLrorriol l nrvox( lr 0]

IIAPAAEIrMA To qoo[oit"r6 oedprucr rope! rrirp<r ra orppetqr.xo niv0zc,

,,r,= li :i;r]

rou ilEr €n(lvala[6avoFEv€6rbro6r.l!uo$.iio\' €ivor. I

= 1, \ r = L MLa rnr).oy' i

lo 1 0 lx=11 0 0 l

L0 0 1 lr6LorrpdE: ,\r = )l

. t r ' l lo l'

ol [0 o o l0 l+. \ , 0 0 0 l0 l L0 0 1 l

A!rCE €ivar orrt),er €\r6q opooyrivrou nivcxc Q xnr, ,4

|''lL0lQ r = QAQ" yi\'€r(lr

f l I0 tl+ iolo 0 ol

,4 =t f i +^,", ' l=), l l +

Lo o

5,6 I\lETA:XHMATI'I\'fOI OI\lOIOTHTA:

En€64 )r = l. or 6uo naoxr-: nao6oL(E (zit0! uicr rdEru)E tw| ouvdrdiovro',(rr 6i!o1r'', Fr(! ri0o6orlil J,r xirE€(')q 610. onor€ o A yQirq€ro(

[u ru l f . o l f 1 lu lI 0 0 ) ,4 . \ r , t r l ; + , r l { t ) l - l I 'o l

Lo o t l l r , r r l 1 , , r r u l

YnrirQler (vtr o1,62),qpo uirr:bo 'bro6rtrvuouirft)! ror cvrrororTotv otry,\ =I ricL ra ,a1 ri(( :r: no! 6rci,(!ope rlrcrr xafiorg (r0ciQt.1(r, 'Elor xor ruprpovr'4r(ra.rI.r I zcrr .r 3:r'f eircrL E:rio6 xdaru; cr0criputa xcu. yoro.ro irOporopirroq nor rivcr 1 npo6o),1 Pr o oloxllpo ro c;riae6o eivar povoorlpuvtooQLop€\o. (ride EpynLavLiq niraxaq Ue k |La(roperLxis ftnr4ti< t7.tt t46e-i ror "q@!.trtxli @t .uot" o( ti1porcya A = \tPL + + \tP*. ircu Piti1\ 4 :raoiolti oror dtdXoao 14( )i. Er[F6rl urdplr[ dac fi]niQr< owo].o[b'obr(!rtu[(ir(,,v, or n0otiotraE ixorJv (ioaolo$r rov r(worrlt(t. K(lr r:rfl6{ ortbL6TtupoL eircrL op0oluvtor. q LLi.cr :rpo6o),tl crxo),or0ouyrv1 aro qv rilhl np6:teL1'( ir(odle! rrl trlbr![z{; fjl, = [1.

E(Lcore noiti xovr(i orlv c,_r(ivrlorl 1rlr( $lotoLoyrrll< tQ(brlolc licrL fno-porilt vu xcl.rirpolpr rov 6p6fo rou cnolivr: frc:roLoq:riv<rxrg orllrnirrer orpL.1,r'vrr6q ? pr ivcv 6Laldlvro A: Epprrxrvoi, avrL-EppLrrtrvoi zcL porcrbraior:rivuxfq uvlrolv o clri rtv rurqlopic xcL cvrrmorlniv or: cai0!o!: rolfiQ(i,[a1!xo1r .i!o!o, ro! (]uvr(1orr11or! hEol'G xcr lou iro\,(r6L.1io! ,+it .or. Zr1-rl4r€ rdrac oL6"Ual Tlt! xarlyoaia. ror (1'vT'oror.Xei oro olvoLo ror prrrbL,rdvl(n{)p,;!. OL :rivcxc< clroi owyirlovrcr "xavorrxoi '.

5T O nivoxqq ? civcr xavovrxri5 orcr pet<rri0erur p€ rov Na : NNs =Ir'"N. l't' crt'rotrg xcL p6vo 1L' oulorls rorE nircxeg, o tpr"yc,wrx6g 7 =U ri\ral €i\roL o 6!oyd)\r!oq A. Kcvol,rltoi rivoxfE €i,v{l( er{€i!o! crpr6d)g, oronoLor ibou\ &a fi7iQE€ oivoTo oQqoralorrxti| r6to6nrwltila!.

:rlucrd)orr 6rr or EQp(rL(l,'oi (rl o4rfr€xQLxof) ni\r.ryr; rir,(lr (]i.lorQc 1ic!o\,(xoi: Eh.v ,{ = AH. t6rr ,4,4// raL ,4//,4 rcmilovrcL xcu. oL 6no gr rov ,4r. Orop0otrov(brdioL :rirc:req ei.v.u. r:rio4; xavovLxoi: UI,lH r.cLt LIHLI raxriqorrarx.ft o' bno B€ rov rctlrorLxo. :'au6! rrq bno €rb'#q n€or'rrcbo€rr arob€iqal€,oc bno 6rl;rcrcr,6rr ? =A. Tn ibrcr cuir 6lpctc zcivouv rrl borlerci xnr yrc 6vnvo:rorov6lrore rcvovuo nivcxa:

(i) Eciv o N Eivcr rcrovr.x6g, ro i6ro ow6ci.ver xtu" ur rov ? = L- r.vlt:

T T H = ( i \ N U L I H N H L I = ' / I N . \ ' H ' I = T / I . \ ' H N U

=L'H:{HLII I t ̂ 'L ' =THT.

'Eror. orav o N ei.vnr. zorovLxoq. o rgLlrr;vr.x6q U 'NI/ lqcnel vc Eivcr 6r(ryd-vroc. Err-rbrt c1rrr6; eXer tq i6Lrq rbrorq[q pc rov lr', :rpdnt:L !( €ivaL o A. T(rrbrc6rcauoltlra rorl N eirqL or orrtl€q ro! U. ron rivcr oo0oxoaovrxtg. Avntf iv, t l r l zLdI nrpinruort x ' f l nroFrtotr . .orf rd,aLr or4 l rvr7' t rrr) inr i ' r rr t , ,no rorrz(l1t'tr_1x'r,q 6r'vurori; rirt.zr< ororq lrrQot€eor5 bwcrtoug.

H FoAqrl .lordan

K(iv{lllr_ llaxa! rorQo. o'c11rli) fll\r rcQd1,Qo.lo. 1C) rcrtrir{,(x) bwarci aouturoaonoaFa tr€ oQoo[ovobroi€q otror.orrlrfs. Arcrr(i)vr(c bi]Icbrt o I-l v'r rilcrop0opovcrbtcrtoE ni.vcxra tt.,pipapr rov I1 r,,lI.I oe rpry'rvrx1 popqrl 7'. Trirpnaiponpt awov ror ;reqroprop6 1c rov 111. E:rrrp6:rorar o:rororbrtrorr iivcrzr;zrrr rrroloq pcq eirct rtr zirrorpt rov ,\I r.4lll ioo n 6r'rut6v :rtpno6rpo

To tcrori).rqcr crwf; rry yiyrurlc neoonafl€(.E 6rrryovro:roiryrlq civcr IFoflti Jo'daD./. Eci\'o /1 eXtt r),{Qr< oi!o},o alrEceoiftw L6robt(l\,uolrcrro!.r(iploutrf 11 = S r.& qr.ilorlF ('rltv ,/ = S-r,4S =,^.. onor rt tro(n{,n Jordanoqrn|]r1sr trF rov bL(ry(;)!rc A. Auro ri\'ar c6uv(rov yrc 6.va.v rlarroF(rrL'iofifvur( xcr, y!( xi.0c !61o6!(i!xotr( ror Lri.nr! I [oAAn Jordan. 0a i:X€r I ozpLd{irgrdlrr (r(i xc oroLl€ia tll; xupi(q 6r(Y0rviou. Or rbroi+6q r:F(fo!i!o1,ia! orJl\ribLc bLcyr;mo, h6rL o J eircrL rpL.Torvrrtiq xar or br.ryop€trrd5 L6ror$iq pnoQoul'rirvrorr v( b|alr'qeouv. Movov or rriavcLaF6crvoB€v€q l Ei!(L €xrivsg nolI'iop€i v(l (xcrrrtoolv (ri ncr ac Frtv orcLrrtoouvl) 6va €vroE bL(lyrriou ororxcio

5U Etiv o ,4 ixrr s uvrldprqru r6tobtovuopcrc. r<irr rivat 6poros:rpocrov ntvqxo [€ s pr[lox:

i6J lAror rntE: KAI lAIoAI,\N\':MATA

(ii) E!a; rqr'/onz6E T not eilcr tcrvovrzo< irp[ner. va eivrtr 6tuyrirvLog( A o x t o € L g 5 . 6 . 1 9 ' 2 0 ) .

Kdree p:rl,ox "4 ei.rlcr evcq rqLlovtzos ni.vcxcE pe grc povc6e{ r6Lor4rrl );xar. 6r,cr u6vo t6r.o6r.rirrouc:

Jl

J"

56 MEIA>XHMAI]:MOI OMOIOTIIT : 365

't]

!ll;ilt; [;1]=,' K.i0c oufiqdqid( reoordo.i !d ad!.r 6(iiq 1r< .! lEn' lprroFaa.'ls aur nLorruo orL I

cn66aq! ro! Filippov Ei!c' 1l "d,ir.oll.

Etrdt i6oo anhi no! oi.6o! lLnoa.i !a o!dAi$' rt!

or{tl4oit r(q acL !c encraqtaEr qv zdrcoz.ui ror J d6 rc Il(ellpr4r( mo x!eio]. a€+s!o.

l'''Orov ro F rl,ox sxei rdEI m > 1, I !6!ox!l- i )r €fisvsl,q!6dvexo[ m qoQeg ,(a[u,-rtip'Xouv m - 1 povd6€S ndv@ o:r' q brnyivro. H i6lc r.6rorqrj )1 p*opei,rc epqcvr.o0ei o€ n€Oloo6ftptr Fnt'or, 6rdv rrvrloiol1sl o€ negroodrepacv€Ed$Ila !6ro6lduiopota. Auo nivamg nou €Xouv rr]! t6[d [oQpn Jordan

IloLLoi owypogeiq -roroeetouv cu6 to oetiprpc orrlv roputpl ror prcrorl-[cx6g xou! 1!S fpopltLxrlg A],1e6pog. E .txpni( nlor€i<l; 6tL aw6 eilct ],ci0oE.:i./o!pc Ei'va! c',nOercr 6T[ 6€r, 6[o./ovr,onoro!\,xo( o],o[ o! ni\rcx€q xar 61r r][oA(fI Jordan ri.vor" 1 nLo 1e'vrzr] rrcQi.rircDol. Awrl 6$(l)g azQ!6dq €ivc! rt orli(rno! I xc(tc(or€rrll iorE eivcrt xnr nepireTrl acL elcrlprtLxri cotce{g. (Mro FLxpytulJ,a./ll orov A. lnoQ€i vcr €ficld(pdQ€r 6],c, ro r6robrcruropdrd no! l€f7row,(o! 1'( cEoqrcl,i.oe! 61.€g rLg ear6g 6roltoriou po\,d6sE.) Kcrd otlvdrr€ta ro scuor6[dQo! yLa !c 6o0ouv 16,_rlolrdp€r€E €ilor 6vcr rcQdpqla* ]ior o rqfuirr€eoE rQ6rroq1[o \,Cl E€z!\rllo€r rcl\,€iE rl poo$ll Jordan €i!c1r vc eE€rcro€r prpLad eL6rrd rtl|,dxplora rcgo6ef .4rora.

- l I l - f l ( r l

nAPAAEITMA t r - l ; i ] , " ' / - l i , ' ] " " B=Li i ] obryouv zo ', f 1 i ]o r r a r : K o r o ' v J = l 0 l ] .

OL rdooEQL< .lrroi rrivcrriE< ilow rLE LaloxrFdE I xc/l \ pt itur privo ftnbnwopuzcL dror. o J aroreLeirot atr.6 lut privo pttTox. To eTroLl0€ljou[€r Or oQiqouo€E

€ivar o).€E i.o8q Fr 1 rar r.l iI\,I (rd doeoi.oFtlrrl r!r\i ororxEitir! fits nJokrEb'oyoJvior) rivar 2. Or rdror{.rtq Lxavonorotlv rrq 1 1 = 1 rar 1+ I = 2. :rorisA, Ll aaL .l no1) eivcu, rpr,ytuvraoi, or r6Lorr+dg epqavi(ovrm oq drtrrilvro. Oc6ei!ou[e 6tr or. rfvme< cmoi ei.vcr 6poLoL -aqxo!./ 6),<lr oqv i.6lc orno.1d1,€Ld

x.rr o ./ , ircr { FoAqrl lordm aurlc rlc orxoyivr r.rc.A16 xo\r ?' no\, ,-I. ro Enror[€r,o €i.!o! !(l tlEr(61.r]0€i ro 2 o€ 1 L cL roiro

yiv€ror. [€ dvdv SLayd]vLo nilaxc:

!l=.V 'rrr = li

IAIOTIME: KAI IAIOAIANY>MATA

Aro rov ,4 r'Tov ./ ro Elroufr€vo €iv.ft va yiv€r o nivaraE raryovrroE x.r v ar6'rrvv' allciE€r ro 2 o€ 1:

x.'L xar6nLv .v rr1= [; 1]

= ru .{r =l; il=1O tI r,4tl rjtcv to rcqdb€ry[o rou Ml4orog tou schur (oto 5R naolyor4r€vo]g.)xor o U uii ro\, ,il/ obqyri, oro! J.

Aro rov B orov J ro llroopr vo rivor avcotpoq4 zor ror.no r:rrrq,xuvrrort{€ €vov n|voxa [€roe6o€(1)E:

?llt;lltll-'lo r 2 l To o r ' l

nAPAAETTMA2 ,4= l; I i ]""

r= l l ; l ]

fo l u l fu r u l fu u u l' ln ;l '-lt : :l '=lt : :l

.,"=l? ;]li

En€lbri ro tu1b6! €ivcll prd rQUr].{ l6r,otr,p1 zct orouE 6!0 nivoxE(, dur6 eorlqqvi.E€ral o€ xi.o€ [n],ox Jordan.'Ero[ Jl !fioQx€r 6\,s lo\rdblx6 [r].ox 3 Er(i 3,4 wciQX€r ila pnLox 2 tlri 2 kaL t\ct 1 eri 1, ll rJnciQlolJ! iQi,a [fiIor 1 €ni 1.Xuvcrtirg or 6ucr6q poQqds Jordan €ilql

:rlv n€oi?irorol ror ,4, ro p6vo Ldro6rdluoFd eivcr to (1, 0, 0). Aaa Ipopqtl Jordan dltL dvc p6vo p;rlor xm xcrrd ro xupro 0€6pryo 5U, o A 0o,"ra€rstvrr Eir,af 6[oroE [E ror, ,/r. O 1rl.\er.ea D 6XcL xo Efii n].do\r !6ro6!i1v!opc (0, l,0). olve:rd;q yl poaqi ror Jordln €i\'aL J, },'ar 6.l na6r€r va x,ldeto ' 6no F rLoxadrti FrtxoE rq; 6xrytoviou. Ooo llc ro! ./3, crur6q eirrlr nr6 p6voq rorl pLco),orLnen xarnyogio. a(po! o p6voE ni\raKrE 6goroE ro! [r]6€\,rrori rii1,o{c €i.ror,o ,41 '0.r i1 =0.

frc va npoobroei.oorFe rov .-I o' lrurl! rq 7rrrQa6€i.]$axo. cQxo[o€ FL(rl€ralol Lbrott aauouirr0iv ]1|, aur6 €f1,.l|,'rdlror€ 61jvar6, 6rav 6e! undAl€rzdrL nro :roLunl.oxo a:r6 lra rErU r6LorLpr]. :cv 1evLz6< rcv6vcq 6poq. 1rrxvrzri cuni qg Firpto$ obrleirft oe eaq1xr.lili ricrdoraot Fe qv rc).eucic

EEuQpoyig onE elrcdocq 6raEoar;'y xar crtt dtlqoauiE EEro(;'oxr1 @uvtry€rq xalEt-qntlri; poaEtq). Ee\ o rt 6!cyo\r!ono!€[ro[, x6r€ or 6urdp€rg rou A = SAS '

6pioliovraL Enxola: ,4^ = SA^S t. Edv 6€\, 8raytllr,roxiorci.ral 6lo![s ]id].l

5.6 iIEIA:XUNtATDITiOI OMOtO[HTA: 167

A = I'IJII I p€ rylpoagt Jordan oto Feoov xcl ttirpc Xqer,o!6Baor€ rK 6wdp€K

At '= (M. IM- t r$ tJAI 1 ) . . . (M. rM ' ) = , t / . r r ,V '

O J €ivor brqd)vros crn6 gn).or xaL or bwripeLq rorv pal,oz u:ro).oyilortcrEcXopLorh:

Auru; 0u un€rod0)(Frql url Luol uru< rEiou,orF broq opdv. otov u:rripxtr rprnr,rjra'ror'pn r(r rlq Folo rhrobrqvuola. Oblyri r:rio4g orn l.!ol rlq .rvrlcnorlnqornqn0lxl: rErou.)C'4q:

Avrd nQozunrrr (rno ro a0poroyo t16 oetpd; / + J,t + I,J,t ): /2: -. . . flou nanciyrrro | -.lt { = ' a' ort bta.rci,vro z,rl ro ii \, q7Qr6d{ (rnd ndv{, rnq.

IAPAAEIIMA H rQirq orr]11 ror r,/,r ouFB€r€Z€r oq ], q rrt.. duldt =,li1ll

To ollotrpo l.uvrror F€ dvabQo[rxr1 (tvrr,zordorcol (bL6rr o niva)icq €tva!1a'yL,JvDtoE). H rr) .cur,r i t r rEioarort oivsL x, =, j ' . H r l ioooq yrcr r ,1n , l . i , " ,d,u / .d,1 -

a :a u, xur 4Inoq t4E civcrt r , \ ' . H r l iouorl mqv ropulr l r ivurn . /4t = \ t t +, , xd' I tuol E l iv( ' ( i r , \ ' l l2. : . iv, . fnLoy l , ry iool< ,n

rrno Irtl rbrorLlri no).Lu:r),tit4rcc ,, Ir ivrr trovo rbrobLdvuolt.,_ o rrt :r).tovnapar/0r! I €pqaviq€ra! D I eoadg.

. Or 6uvdUrrc. uurrc rcr oL FXor l rx ic Uopq 11: rou J ctvru Uipuq rou ruox rrF

/,uorlc. ro f1I o ulQoc flvdr o .11 :rou ouvba€l rov dQll{o,4 fr tov nro E1JXAIoro

€av u$l = A&k 16r€ u^ =,4ru0 = r \ IJ*I1 1?10

tct\ duldt = Au i^tE rL = eAL a = AJt Jt l I 1u|.

Orav or 11 i.cr J eiyel S xcr A (t 6Lalovr.onoL{oLg4 n€Ainrool) auroiolfrlnrol'v tr€ ror5 nrror\ q; 5.3 zor 5.4. To laadQr![a B smmQ6q€r oq FrtbrqovroroLrto4ryt r€QinroJoq zar 6€itv€r ?rdE err,rulxrivi.rct 1 goprpi loraan.

'

l . \ I i l l " J \ , , r " ' , ( ,_ l ) \ ' , r ' l. / ;=10 ̂ I I -10 ) ' / , ) " I

Lo o ) I [o o )" ]

fer, /crr ltrc^r Ie/ ' '=10 €A' ic) . l .

L0 o e^ ' I

l l r , t l , t ' l l \ I u l l r i , l rOtl , t , t : / r t t l= l r , \ t l l , r l Erx ,1 ,d ,wds c l td r " , , ,= l0 l .L , t r , / , t t ) Lo 0 ) l Lr i . . l L , l

IAIOTIME: (AI IAIOAIA}]Y'MATA

El.riE(D o €n6F€voE nivotio( vo 6iv€r tro suxplolt or]voli,"l].

n ir dk aE M e N ao mp az r Win O p o tit4ffi E

1. O ,,{ €ivo! 6rcy.De@,rotiotpo6. o[ otri],€E tolJ S ei1,ar. r.0r.o6Lcvtlopcroxctt o S-tAS =A eL\at \taydvng.

2. O A eLlaL otfraieetot. oL onii,Es xou 1U €ivd! [6!o6rc!uoFoto 1is!

levrneup€va dloolavuoFfia tolj A liq! I U.oeqri Jordan M-1AM = Jti.\crL 'Mytielos p€ ln\ot .

3. O A Ei\sr cu0siaaio6 rol o y oQ€oFovs6rsioE; o [/ ptioQsl !o stilrle?€iiror r.iorp o U-|AU =T re ti,rctr rotywuxtig.

4, O A stvcr! ,afirbtdg, AAH = AH A : o I/ p:rope[ vo €ti!].€y€i. 6xoltlor.e U-1AU =,^.. Er6[xdE rleQlr[ici)oe[E 1o1r 4, 6t!€E F€ oQooriovovlridr6[06rcvuoForo:

d) Ed1, o ,4 €ivo! Eo[!r!c!6E, to-rE o 6ha[ tQolpatlxdE.o') Eriv o ;1 eivsr ?r0(r1Ffi[r6E outlpElQrri6E, xdt€ o A €i,vcr, nQ(r'tpd,rF

xdg *aL o U = Q oq6oltirvroE.6) Edv €ivol aliGEQtllrtolog, 16r€ o A €ivql qo\,tdoilxG..y) Eriv o ,4 eivor, oqOo1d\aos { oQ6oFovq6laioE, rdi€ 6treE o! l)ii = 1.

AEKHTEIE

5.6.1 Ed' o , ri'or oFoLoE f,poE to! ,1 .lft o C eivdr oforos trqos ro! ,'r, 6€iE1€ ort o Csi!ft 6lotoE reos ro! .4. ('Eo1(D 6rt B = M I A\,1 Nat C = N '8N.) nolot ni!a.,te(silur ottoror pr ro! x(vrorrx6i

s .6.2 r ,e,yedr-r rF, ) .oyru d,o4 ro '5 r rvarrc rou ' i !d ' opo,o, - , l j

0 ,1* ' e . . . .

6uo otr' durons.

5.6.3 Efu.l,lioiE 1lo o A 5€v €ivar rot€ ottoroE p€ ro! ,1+ L

s.6.4 Baeir€ dv(v bray6lro M, ftou dnoftl.sit& ono +1 ror -l ,tdr aeift€ otr o

s.6.s A€ift€ 6 (oror o B €ivar aw'mq6![poE) o BA €ivar o]'oroE to! At.

5.6.6 o) EZl! C, = ,(] (,dr o D si.lar alxrorAd{rpoE) 6€iEte ott o C €ivdr 6FotoE ro!C.

f . r I [2 - r ], 'i j,l ' '*' -"*,o" " l-' ', ,' I

L r2 l L l 2 . 1

5 Ii I\fETA'XHMATDMOI OMOIOTHTA> 169

6) tlllJl.adlcft &t oL Lbtorqras lox C ,rqaiEt !o .Fqavilo\{oL d!{i le"Yrl awte'-

turv oqLoFdlv.

1) A€i&. cn €!0siag otr av c' =.rr tor€ c(Dr) = ^(Dr)

5.6.7 Oroeim€ onotov6rtror€ ,4 xot Flc l1rluio €ninFbl oraoq'i' [I:

s i n 0 0 l'T'llA|CIa!ff rn I.)vic 0. €ror dJott !a Fl6€viofts rc otorx€io ot'l edol (l l) 'o! triv(x'r

I4t.ioq Aw6! o Fll6slrolos 6€! ow€xi!€Iar €nzo).o. 6Lo1 or orqoq6s nou

troEiow F$aa oq eaorl rou I zor tou i' 0o aqraoretwuv ro "o

|$€v!zo crrl

r.uia. npt,rern ",p'ioow€

pio 6raY6aLd Bd'io dr6 'I! xiqLq xat !o o)'oBl'la6so1|pt

iov rrcL,),yLoyo n"r tbrorqtdr! pe 6nq{r0.r'xo tadrio Anqoa'flki o! pnoaonoap'

vtr 6LclovLo:ro(oorpe ro' ?r sdL !o 6onF€ iq L6LortpiE 1o!. 0d 6aioxol'€ q qil€s

ro! rolurvttro! de(.4-)1) )Qlouoaodvra! (roxi.t.|mrxd t'roaYoalais oil€r' aoo

oei(orv ro cosr, x{i1| a6nvaro!

s.6.E no'os nivqras 'r/ crLLdltL fll 6dq vl = (1,1) v1 = (1'4) oril 6doll 11 = (2,s)'

"r = al.a)i Or onirrE rou rr naoxnnro .xqqdlot{(rs rd vr r{n ri od!oNbuoolotl!

s.6.9 fro tK i6€s 6.io€'q, etqedoft to 6d{o8d (3.9) o(! g1lv6lcouo crri +'2v? rc069

erio xat ocv rlrr r + l:rr' Endtrl0s!@t oerOpryrtd 6( o rrl o366tr 1o ' F€ to d:

5.6.10 Efler6aruorf rlv lrlruro,( douryrl oi.lP6erxri E{iv vr = "'r" r _'r:.u:' vr =

ni l . / t r r , . ) ( r = l r x( r 4r7 l ' l i r t2 i ' = 112. TOrts td o lov loF(r4

.ril +.:la aut,ilfr +.11t2 rowilowoL Auro!.ivat o runog of_)'(fk 6do€oE'

s.6.ll Eiiv o petsorill1oflopo! ?- €t!g| prd ovdxlcol oE ireos €loeic llioEoE 45' iotr

€rua6o!,6a.iu loa riaoadr 10! oE neos t'l o!"vllq 6doi ul = (1'0), !2 = (0'1)

zo€(;E .ricnl! ad ox naq rd r1 = (1,l). vr = (1 1). a'iFs &! ot riv(reE auoi

5,6.t2 O rtrtotx'. l.tuo1q4afu4uiq me)'veL xtiee tLriwopc cor Eam6 1o!: ?r = r'

Bqsir€ 10! otiGotxo nirdx(l,6ra' 'l re6q 6dol eivdL a = (1,2)' u = (3,4) ?d' 4beueon 6dq ri"at

"'r = (1,0), u, = (0, l) (^€v €ivor o tanottxq varo6l)

5.6.13 H traadfo'yog rotl a + br + cr2 tiaor t' + 2.r + 0r3.o) Karayadl'r€ roa 3 €'It 3 ri\,ato D Yta to' onoio rox!€t

f n 6 . l f cos / ,A= l , t P J l , r r = l np

Ls n i . l L 0

"ltl l*l6) Yrolo-yior€ tov Dr aar reFt!€lJor€ to onodl€oPd [€ q 6o'ieec rdeoY6Yov'

370 IAIOTIME> KAI IAIO IANY>MATA

y) not€s €i1,ar or rbrciqlls 1|ar ro ioLob|d\,roFaro ro! ,i

5.6.14 A€iEr€ dtl o p.roorrtparror'6r "J(.)

= "[0' l(t)dt 6sv 6x€r r6'or4rds, €v6 o

"/(r) =

.f/dr iXEr xde€ oeropo sor r6rcrurt. Or orJvoario€E osoqorrlror 6rr oQi(ovra' tta

s.6.15 :ror r6eo tuv rrvrix@r 2 €ni 2. dot0)'.l o Fe1aoruaflofo< rou cvqoredq€r xdo€vdra. Beeire fls rbLo f6q ,{qr rorE 'lbLonila.,{rr" ro! r (tor! trivdiEs rox

rxavo'oLonv A7 = -\A).

s.6.16 d) Ba€irr 6!a! oa0oydrvto Q dloL dor€ Q rAQ =,^, do!

l l l 1 l/ = l l I I r o L A =

Lr r l l

Ba.iir xot6Ew 6!a t€ljrEeo lrlyos oeooxdvorla6v r6to6tawopri'rr,n r1, r2q q ' \ - 0

6) EnaLte€].jor€ orr r dxqeoor P = 'rri +rri,j €ilar n l6rd rat y'a r( 6no !nir.

5.6.17 ^.tft., o€ 6!0 6fipalt!. 611 1tdg. oa1ory,rabtaioE nivardE ,4 eivqt 6royovLoroL'ioLFos:o) Edv oL t rcr Lr €ivor oQeopolablaior, ro i6to rori€t x lraaoaT =U tAI/.

6) E 09 dr(l) raryorrxos "

rcr €iroL oAooFovd6raios €ivar 6ro16vLos.'Emrq

orr o rerlDvL'rG 7' rilar A 1t(lt 6rr rde€ oAoopovo6roio< rtvdx.ls{Fr 6iiqoemzdc i rI r6ronFi() iIrL ivq r},i$c orJvoi\o rdto6rdwoFdrov:{.f r,lU =,^. O}\ls or r6'or+is Lxuvororouv qv l.\l = L

s.6.1E aas|r. iva lcvovrxd Jtildxa rol, br! fi.!o! Eegrtralos. alatsEetLrLoaos, opeoFova.6l(fos tt 6l.1y(;!ros. AriEt€ 6tL 6IoL ot nLa(aes rerq06oros ei!d! xclolrxoi.

s.6.19 Eoro o1? si.var alas .i!o rary.)ltxdg nivaras 3 €fli 3, p€ ororxEtc ti'. t]Jytail.ltero otorlei{r r!)v ??t xar rrlr ncL teiEre ou edv €ivar ioa, i6rr o

" trA6ner.',a

ri.',ar 6lal(ix,rcq.

5.6.20 Eav o. \ 's i !d ' rovou"o<. berEf t o l t Nr l = | . rFr , . f lu xdo. i ' rLdvuoFo r .:4d.Odvdr€ ofl rl, leorpi roll N axrx io i6ro frixoE pe 14! i orri),r. -t?t€@ol: Ed!o N 0iraL eflio4E (dru) reryovLxo<, ntr6 o6rr€i n.iIL mo oqr'td0aotc dit JredrEr !o€ivaL 6Ls./6rrc{-

s.6.2r d.iEr. drr, dna{ raxuaitEat xa! ro 5T, avas rivaxq F€ oeeox(volrxd r6rc,6LavnoFd&. rqar€L va .tvar xololr js: Edv U rNU =A i N = UA{/,r, TlltrrN/ l - . r r rN

5.6.22 Besirp ildv oqeoFolaoroiu U x(, avo! rQttuuxo ? . rror norf U l.4U = f. yrs

; t , i q i A =

s.6.23 Edv o / dra i6ror+i< 0, 1. 2 nor€E eilor or rbto Frr ro! A('1 IX.a - 2.r)i

5.6.24 a) A€[Et€ !€ rar' euoeiav nollan].aoLao]o, orL 6vo5 rqryortxo! nivaroE, ]..x. ivoq

l 0 0 0 ll0 0 01 .L 0 0 3 l

[ 0 I 0 ll0 o o l .L l 0 0 l

,=l:

5.6 META'XHMATI:MOI OMOIOTIITA:

3 rEi 3, txcvonor€i q raedrqe,mLai mu €Eiooor: (" -,\11)(T )21)(")31) = 0

6) Arrrdermdvla! Kt T t U )AU, @prEqdvac ao 0.6AqFd tu! Catte!-Hamitton: Kd$e nilaroS tr@onoEi tt x@Aadr\Ataqri No! EEioao\: 6 --rjIX,4 )21)(A )11) = 0.

s.6.2s rn 1rcaxrraror txo nordu'uuo - , , - l : : ]

r ,va, . \ : - ro , ,1 t^ . ta4 t , . ) .

Eno),lodors p€ or' suesios oflxardmosrlro o€{ir(rltro ro'r Cayley-Hamilton: A'?(o + dtA + (ad bc)I =0.

s.6.26 tro laed6€ryld 2, mo r6l,oE ro! x€qo).ato!. 6Acire io! .14 ro xovoro€i rt q(6orlM ' B M = J .

5.6.27 Etr a1i = 1 rdar) oro ntv nnqro 6reyd,lro xar oij = 0 ra\"ro! d).),ou, 6q€ir€ Trlloeqnl Jordan (1.x. 4 sri 4) rqoobroetlowas ro |6ro6rdviopaio.

5.6.28 dei.E1€, 6orqrdloroq daav M la| orror,"yxdrovrag, dfi or poAqdg Jordd ro! 3r:ri 3 noeo6€LlBatos 6Er eilat 6FoEsr .Jt / M-tJzM xcr'. 1/ M tJtM xo:t. J 2 l L r - ' J 1 M .

s.6.29 Almp rI! 7re6rl €liooor) p€ o!dbAo[!r{ (r1rr.1{d1dmoor])rq| q 6eureq! lre avaloyliorl lroPqrl r4 = n1..IM ':

d u - 1 5 l l t u l7= ' ' '= lo 5 l [ , , , ] ' ' '= ;lltil,-=[;]

5.6.30 YroLoyisre rol\ ,lto. Ari eat.Ar,6\sr A = MJM-t

, f 1 4 9 t l 3 - 2 1 1 2 l t 1 3 2 1^= l rc 1o l =14 3 l lo z l l . r : l

s,6,31 rqdltrr o].es iLq 6!vord< fo$pd< Jordan €!6s ftiaoao 4 €ri 4 ro! A€r ro pl6d!dd! rerQarrrt r6roflF1i. (Aft6 oLorla,j ndqdSori. ro pnloa yivowa' ffie6fteoadoc rqox.nadp€ traoq io adro dov rivorc J) Ea! rftiiqto!! 6no ole6dQrFardro8(avnolaro, 6siEre 6iL lniiqto!! 61io 6nqoQ.rrrdE 6lvar&tt€E ?rd ro! J.

f l t l ! t 31 i x a r - - = A D = l ,

IAIOTIME! KAI IAIOAIANY'MATA

AIKIITEIT IIA EIIANAIHYH: K€qdl,Glo 5

5.1 Batiff rl5 t6Lorqr€! 'io! ro !6to6tawolora xaodE 'ro. rov dLayNrcrololt!,ia ldxa

. t r 0 . " | 1 2 l4

[ r 3 ] x . L d -

r s - 4 j

s.2 Beeir€ n5 oailoloeg ior,{ rar ,{-r, did!

, r -s f | ,o ls , .

sJ Edv o ,{ ax€L r6Loqr6s 0 aaL I pc alriorortd L6rc6ravqlarq

i l l | 2 1lz l " ' l - r l '

ynti proA€ir€ va ftao6)1,!,sr. 6rr o ,4 €ivor ot4rere!,{dsi noro €i\,ar ro irvos rar IoAiloloci Iloro< Eivar o /4t

5,4 nortc rLvdL or r6rorqr€E r.aL ro Lbrobrovnsuara tou 4:, t$ rqoryoujrcqc acDltot9:nord EivoL rt oldcn rou ,4r F€ rov n:

5.5 YJrdgx€L irds nivoxos A. aror 6m€ oMxtru,rl I orroyav€La / +.1 !o Etl/ar drr,orqap4rl. yrd ado€ fr.ya6t1t6 aQr0tro c; Ba€i1€ dlor nAcyFqrLao flLno'xo A + trswrorei{tFo .rc xdO€ rao$a xo /.

5,6 Anor€ xcr ytd rr€ 6rio oa)(rxds flpds rqt xai6nr! 6AEaie to' oerlotor,(o ent:

d r t 3 1 t f | f o lar

= [ t : i ' * * ' u =

l o ] xo r o rov !o = l l l

S.7 Od nqoq.oioaE va avdroxil€rs ord rqiFtvo FE 40% ro! Iaovo ri dvd Eros p€ 50yol

s.8 AIrpdE i UeuidE (F€ awlnaq(i6sr.rpo yla ro !r€!66s):a) Edv o a onpaliqEror dn6 rov A .vdAdooovrds dlo yeqrpt<, 16rE o B €iaoL

'i,po!og tou A.6) Ed! Cva( iaryolrr6E ftiro,ias eirar dporos eaos bra.r{i,vro1,. r6ft €i.vo'. 116l

6ral6!ros.y) Kdo€ E€1ryo9 reordo€ov owerd"pr@ qv rqiq: o ,4 sivft Eetlrrdvos, o ,{ .i!al

po!a6|afo(. /' = /.d) Edv oL ,4 xa! B 6royovlonoLonwaL, to i6ro oqr6diver aaL p€ rov 44.

s.9 Ti oqr6oiv.r F€ otv oaoLou0io Fibonacci, drdr lQovtrri rri+r€ d!.rioraoqo, raL r{i5(m&€iar o F A F€ ro! It; O xav6vas Ia|, = I*. r + r'[ €Eoxo),oo0€i vd LoXr]n r.ard q a a 1 = L

s.l0 Barir? q fEltni ).not qs dv/dt = Au &o.

f0 1 0 l.4=l l 9 -r I

L O I O J

At(HtEl' rrA EnAN HVH: K€{ j l7l

Mroa€i.re vd 6Q€ilE 6vo! ra6vo "

orov orroio 't ,\nor z(") .Edoro).ilno, 6n 0d€nroraall,e! oq! aexrxri 'it|ln z0r

5.rl Eda P sivar o riaaaq no! f,qo6rirl"l€r rov Il" o'tva! unoxoeo S. tEnim€ y|Cri1.dos 6rdnoF( ro! S eilar 6@&dwopo '{ar ro t6ro ow6oivrL

"or F€ ro' s1i IIor!ei!o! or rtuor|l|iii (>ruerdm€ q! oxaot t€ rtv P2 = P, ro! ortpoivsr ^2 = ).)

5.12 A€iE € &r xdtu rivc'as raE€arE > I ei!.rr io doaorolo bno LdroFoqq(')' nrvdxov.

5.r:] o) A€ik€ orL I 6roqoqmi €Eioool ft,!dxo,! dxldt = Ax + xa 6x€, q l1]orlx(t) = e-Aix(0).Br. 6) Arob€iEt€ l't| oi Iuo€E rta dx/ttt = AX xA 6row ofrdO€ rao'rrrt orlrpi rL6 i6q lbrofltrts.

5.14 Eciv oL LbotLrd9 rou n €lvar I xar 3 $ r6todtdvnoporc (5, 2) xor (2, 1). 6qeite rKIuqsq rou dul,lt = /r rd! ro! rt,r = ,4!(. EExt!6no( ono ro r = (9,4).

5.15 Be€irr rq rbrotrfa< as( ra rdro6raanotrdrq rou

. [n . ' 9l" - |

' :L u - ' r ' l

fiota dtdqra (tlqdlFr€ tto ro lbrobr(lvlloparoi AIIo€!€L redyFsr(:

5.16 fiqoof,de(iJwos !a ].uo€ft nl!

l n { , l [ { i , l _ [0 r l_ ,t c , l l t c d l - t 0 0 l "

hei€re 6lt o .4 bfr ix.L l.reor{r)rrxrt aiid. A .dF€ rd 6rdt. vLa motrda ro! 4 or 4a(r 6e€tft p t8radyolLa1i eiBa ro!.

s,17 () Be0ire n{ rdnrqriE raL rc L6ro6rodopct" ".,

, = l! ;l

6) Aums q!./,/./r =,12 E€rrvdwcE a,io io r0 = (1m,lan).y) Edv r(,) = 6oob( ro! xenL(i1|mdv adL u(r) = 6006(1 tov r€Iorti,!, xa, o. 6no

a)J'rto6oreonw(L F€ rLs drl.tt = 4u xat dur/ttt = !/a, ron ftiv.t to tnlltrc!.'/r' xao(i)< ro t - oor

5,1E Al'tois i rP!6is, t'€ d'(oloyia lLo to dloiE xat dwrraQdb€tyFd yLa ro t€r'baq:o) Iro xd6e ri\,oxo A, !f,dAr€( oI a\s irltit = Au no! Es ld a d ro !o =

( 1 , . . . , r ) .d) KdoF autctai qrwos rivaxu; btayolrororr ir (.y) Kdof brdyouorolioqro: nivqroc owrorarq I ror.6) Evd.lddsovro! r\ yqo!+s5 €vos nivoxa 2 €fti 2 dwLorAaqoroL ra rqoolllla

lov Lorort(ov rou.s) Eda r( rl,ro6rdvnofarc r raL y a,tLoroLlow oe 6raeoa€it'r*E r6rm|lla6, t6r€

s,19 Ed! o r .frsr o{'o1,t'$raL!& rivunaq, b.iEle orr Q = (1 - ,IL)(I+ ,() ' Eildt

oq0oyii)\,Los nflaxds.+ Bq.t". -r A 6.* r = |

0, ;]

' ).r.E-: EEU(xo,1.K on o I( €iv(, "(

nod/faraG niraz.< (no! €[email protected]., zd' r,l!(u ,maq ' l l l 6qrc rorJ+r ( ) .

314 I^IOI'IME KAIIAIOAIANY>MATA

3,2n Ew KH = r (awFEafrrtavds), i6I' or r6,orrpls €ivo, qowomrxts ,{ar rardro6raruopara oaooydJvrd.

d) I16s ,IvoaiE€rr &r o Il - I €ilqr awtmqh!+ro!;6) nd4 yvoeiEer€ (iT| r{ = LrAUs 1La ivcv osoopova6roto Jrildl{o y:'r) l'lari o e^r eivor oq0otorobroioqi6) l"rcti o eK' eivar op0oporo6rcriost

5.21 Ed!orr l€ i roro i roy6uosr ivorss!€ooL1ciod,r r '? , / .noLoseiv{ l lo , l , / rArr /xat roLts €iraL o! r6rorqd6. 610!

ilti

sin 3r'l' i n6 r l F r d= ;$in 9rl

5.22 Ed\ 42 = -/, ro|€s eilor or l6Lo fi( rou ,4i A0iEre 61t to n ne6rsr vs oivor detrc.dic! o A slvdr ftea.rpaI|xd'( ntlar(ls i Etrl n aor 66m€ noadb€r,ylq.

5.21 Eda ,4x =.\rr rar ,4? v = ):y (6lc rpoyFc rd) 6ei€re 6rr r"9 = 0.

s.z Mc traeoD,oTrt to! rilqxo Fourier .iaar o rilo.ras olvttFrrouo!'

j I s i rd

V2 L. in 3,

sin20

sir 6t

5.26

s.21

Enrd,rl0slor€ ort Sr = S-r. (Or onlldg €ivqr rd |6Lo6|anJolala ro! tAficlldvrc!n ivaa( -1,2, -1. )

o) Basirr trl p1l5€vrx6 rivdxo N ps Nr = 0.6) Edv Nf = ,\r, b€iltr 6rr ro .\ roaner ro €ivdr Lq66!.'y) A€lli€ 6rr o N (ovopo(6pr!o< "lt)6.vo6wapoE 'rt!axas") 6s! proasi vu €ivar

a) Bpeirr ror r{vora P = ra? /a'a nor npo6dner rdo€ 6L'iwopa orlv r!0€iq 6tdro! o = (2,1,2) .

6) noLa Erlar I Fordbrri u r$rvL"ri rbtonFri rou P xdr roro ro ovriororto

'y) Auote to r^.,r = PLr Erl.|r6vrus cr6 ro,0 = (9,9,0).

Yroeaor€ 6t. I rEi'q 1aa|r|rti toD ,a !ivs! (?, 6) rar o! !6Lofllds ro! rivor i, -t.Bq€ire rov /.

(l) rLd nor.6 r+rq iov c xor d, ixer o ,,1 ned.yportu6s !6Loflt'ds ror oe0oy6!L{tD|o6ldvlofarci

11 2 0 lA=12 d c l

L 0 5 3 l

6) rtd notd . aar d Fftoaoi$ aa 6eofu€ roia oqoozdrouri bl(rvtiol(lra ftou vd€iva' (mduooloi iov oql6v (plv ro.rd!ftrl):

5.2t

AfK I I ! I I ) T I , \ EN ] \NA , \ I I ( ! I t t i q i

5.29 Ed! r( cviotrola ir rar rt firuL otiti.( ro! 5. noL.; €iacL ot t6Lorttric l(' r(rorodL(.!un01a turv

..,=.,1; il, /'=sl: lls,,5.10 IIoo .i\,( ro opro. zae(;. ro r - a (rtdlllke.i a(r.iorcot Nlarkor) ror

tilll],;t-t;t,

OETIKA OPDMENOIIIINAKE>

6.I T EAAXTTTA, MEII'TA KAI EATMATII(A 'HMETA

MdIQL r6Qs, 6€v €iXoI€ l6yo \,' aqov)(oriF€ lLa rs nQ6olpq rov r6[orqr6v.I]Ei afl, 0(r r]rav r[Q,ooQo 1(| Qx,)rlloou|r€ cv io ,\ rfvcr 0€r!x6 { (e1nxrr6,rQr! Udoow€ crv €iv(1r ,[Qc1Fcrrr6. >ro Ksqdlolo 5 (,)mdoo, ono6€iEa[€ 610or nto rv6rqqaQowes nivclr€c -ouFfrrqnoi qqv necyF'rT|xri n€Qinroon xalEaFrrlclvoi mt piyrrblxr1- elouv npqFcrrxeg r6ror4rig. 'E1er ),our6v v6qpc, vce(|)qooup€ n6ir uuroi €ivor e€rlxoL' (vo! on6 rorE or6xous [qE €ivo! vc 6Qoi[e6!a xQLxliQro nou pfiopri !c €q.rQpoor€i xqr' elo€i(rv oxo! oltrtr€rQrx6 rLvcr.r,, XDaiE va ljrrol.oytoow€ 16 rdLor+6E, ro onoio eo tEdaqaliEet 6fl 61et ol|jtorqtit eiru orrrr.dt To xallrlalo oriax€dE& rQ€K an6 tE 6corx6tepeE rb6egrou 6t6i'iou o6qyoug. opilouorg xut tbLotryfg

flpr.v cvclrF iooqre 6vc droro xp{Eo, 0c repLypril.'oupe pLc vea xnrd-oroorl ̂ /ro rl\r onoio or l6!ot!F6E Fivar o+lol1Lxag. Aur{ 6!oq€8€r r€LEiors on6 roE4xlltrs rIE €umdo€r(rE rov 6rc{poQr,xdj! eELorboe0i!, 6no! XQ€[dlExa! vo alorF€cpr.rFudg cvti yra O€rr#c rbrorLgdE. (Aev 0c snpene va 6rc!6pcora o' au6ro o+r€fo,6FoX ro rd!o!F€: Edv o -A rrcvono!€i 10 xQlrlQro riou ridxvoups,x61t i duldt = ,,1u 6XeL rp0ivouoeg hioeLg e^rr, [€ xclo€ !6rorqrl l < O rc|!1d'zuldt2=Ar 6X€! naoaQd xalawuudF€v€q fio€q ei"rr, [€ oi = y'l-.) Hvec xarirmnoq epgcvilerar oe 16oo ro1,l.6q enrmlpovrxd< eqcqroldg, 6nog q

[4xa1|r{1i ror, l{ire€ ]rQ66L4ra 6e}.rroronoiloq<, nou elniloqe 6rL o cvolvrlorlEbe! ga 0€)"fo€r aa d[Vlo6lrrloEr ro xlqlQa ro! 0a neox(o€nlo€! xol' €ue€i(,r oro[selB(rrrr6 na66Llpd.

To ra66l.rtFa rfv(tt our6 qE nvcrx<ilv1.qq evoE €laXiorou, rou €ErWorjF€ F€6tio naa06€i'f[ara:

F(a, s) = ? + 2(:: + y)r ssint/ o3

6 I E AXD'I'A, METDTA KAI >AIMATIKA :HMEIA

J @ , a ) = 2 t : ) + 4 r u + s t .'ExEr xanora dttLt nq F xat ! dLdTrmo orc oapEio r = A = 0.,

24p.iao4 1 Or 6por. pabet,Lxl1q tu[soq -F(0.0) = 7 ]io! l(0,0) = 0 6sv €nlp€d!o!v qv andvritoq. Ar),ors ov!1,6vo!.v rl Xo[r]],<irvow tcr ypoqlpcrc rruv F xnr

tqpEiaol 2 Ot" ytlupptxoi 6pot 6i"tow pro avoyxaio ouvorixnr frcr vo uridex€!zdnoro rpoonrl'{t €),uliororr. ne67r€! vc 61or[e orrioLlo ortp€io. Or nAdrr€g:rcpti1o1or" npdner. vo pq6evt(ovtoL 6tav :r = r7 = 0, 6roE rrQdlpar! o14r6oi,a€ri

aIA.'

=4 ( r+y ) -3z r =g xa t 6q

=4 ( r+y ) - ycosy -s i ny=0

A f a l; t .

- 4 ! + 4 J = D x q : ! = 4 ' - 2 u = u

'Eror" 1 uqXt €i\rar oldo4o olp€to 160tl rr15 F ooo ncr! 14 /. feorperqLzir,

I €nLqrdr,€rc ; = I(r.r) eivar €qqm6[€vl xorr oQt!,o.,'dou rnr.n66ot z = 7lqrI crrrqcivElc : = /(i|, y) €|var eqcrm6peq ro! efi!fi6bo! ? = 0. To €Qrilrn$oeivar. av or, It xal / 6otoxovrdr rdr) 0].6 olJxq rc Er[ifie6o, xcr06E xrvotlF€Oonropcrxprv6pevor. cn6 ro onpEto €ncqllE ri = y = u.

D\pekoort 3 Ot i€NQoyaytxoi tiQot, itou nAoaAXowat dfl6 Ntt 6EitNtQE< rwq(6-yo1)S tivar aroqaontui€ o\p$oiuE:

,): F

a, t =+ O ' =+

F AlF0t 0u )yDr

* = 1 + i / " i n u - 2 c o . y = 2 ,

I a'!AiA! }UA:t

d.J 1all)

''

AuraG or rapciy(oyor. rreQrdlou! rtv onirvrlorl, zcr €n€16{ ei\r(r o! i6L€q yrar1v I xcr l, 0n :rpdrtr r1 on(i.,'ilol vo civcL 1 i6Lc. Or 6rltl orlr,cQuio€L(ouprieQuppovrar xorrd mlv aQ1ri rioi('r ro! i6ro cxpr6cbq rp6rro, hpc q F 6prt].itxntu dtx xar p.jyol rdtx 6wr, \ I ixer E\&Xtoro.

ndaariAqol 4 Ot iraadyoyol alartpaq rLi[eaq 116 F bev rr4ped(ouv roE{qpa undaEto,. cr6g ronrrou €}.nxiorolJ, d,l"d gropei vc ro €[robiqorw vatL.vEr oiud Eld1Llxo. Xto :rcpirberlpci Fr{E. o 6eoq ,rr, 0o o:r-o6!er cpyir {yartyoaa rlv I'nQoq ro x. nv{cip-qrn rou rL orF6aiv€L xo.,'Id oro r = ! = 0.

OETIKA OPEMENOI IIINAKE:

'E\rc rdrolo €v6€X6F€1'o eivai o6uvqro yro rI1' I, i./La onorqdrlaor€ dl.l,rl"rcre(1ytDy uti poaE4 " , rro! 6E\, ri€Qldx€r 6Qoq nvortipng rdlEtog.

Kdge reqayo*rxfi FoaCi J = azz + zbra + ctJ! it or4v up7r1 mdo4tooapdo, 6nou Af /Ax = 0l /0y =0. E(l.v orc x = y = 0 6t€r ro'rL'{d €},4(roro, r6rero cqf€io our6 €ivqL xar ol,ry6 €tldlLoro. H mrrpdveLcr z = f(:r, g) 0c{ ofipr.rilerUrLr xo!:rc nou axouLt:rci F ivc or11.L|'io qg oqv aqXi.

>uvovilor4le: To €Odrl$a ronrroi e^(rl[oio! .flo rnv -r eivot too6r]voFo F€ro i6r.o rpd;r1gc 1c qv /. Edr ro orci(orgo olueio jg r' rircv oro c = a, g = Bavri tou r = y = 0, rt Udvn cll,afrl €o {rov v<r lpqot4o:roLrioor4re t6 6eurepeEn(Q(YdJYorE mo a, /r:

/(".v) = .-f ig@,0)+rufifuA2 t'- ,,2221;: ; { o , d ) + - ( ( l , d ) . (r)

A1rlrl I ftraay(ovlr{rl poprp{ o4r:rrprtpdprro zortd oro (0,0) pe rov tbro tporo:ror 1 f oup;reptqeqercr xol'rd oro (rr,lr.

Yridpxc! Fla r€Qtrwcool /ro! rp{ner va cluLpdoor.ge. Awrmor,lei, otrl 6u-v(rrdrnrc a'l = 0 x.rL ei.vclt rQo[€QdS riolozdqcLoE, ar6gl x(rl ylq oovdQrlprlFr(r( p€ro6LyFriE. Mncivow r6ft oio rlQo6r$lps xa! o! rpl.rrg naQdyoyor, 6L6r!o! 6€ur€Q€g adt'varouv vcr 66oow nLrjg1 clriwqo4. fid v' anoqnyoltl€ ouoi rl6uoxol'ic, onalror,F€ or,v'liorog crn6 to rerpcluvtxo F6Qog vc siv.rL Fl r6r6FoQqo.frd va dlol]tl€ /rQa./For"ri6 €LoXLoro, I / 0d rrqdn€L va lrl6sli.q€rdr F6vo oroc = g - 0. Otov rl / tivor ./vrlotog 0€rrxt o€ 6t!(, rd ril,l,a oqpei,cr, t6te )r.61ercr

'l'o rAo6lruc rtipa ovdyercL oro €EllE: Aoo€ioIG owoQfllo€oE 6io gsra-6lLqrdrv .r: xcr l/, noro ri'\,o! io ooor6 unor(ctdororo 116 otrOrixrE F,, > 0ifLc pr<r p6ro perc6lwpr], ro rrQdol[o rfi d€ur€Qlq acpu./6you anoqaoiE€l avd-F€o.r oro €Ldxroro L(1! ro F6y[oro. Tripo 61oupe iQrq 6€1Jr€QsG naqc.yriyolE,\r. F.,!) = Fu" xctr FvtJ. Ot rQ€lE oljroi aeLeFoi oql(ouv rr1v I xoL 0a npdn€!1,o nQooblopilouv crv rl a (xo0rilE xar n /) €X€r I 6ev st€r €)d1roro. [or€5 €ivs.ot o$totixq pEnEn r.,r r,h xat c ttotJ tEdoEoJ.iEow ti,t rt f = e!1 + 2hru + cuztivdr oEtt tit oqrotriyrt,

Euxo).d 6QioxouFr Fr.r avdyxoia ouvoylxl:

(i) Edv a J rtuat 9etxd opnpiry, ftjtt nelrrer a > 0.

Elerdloqrt ro orpei.o.r = 1, 9 = 0, 6aor I az'? + 26rjy + (V, loolnar tr€ a. Aqo{q / et!c( 0€r!xd oQrop6vl, cUT6 npatrL ra eivcL 0€r[1t6. Ilqyai,lo1acE rio@ ofllvF. rodro nruoiv€! 6fl a,F/de, > 0. )tc0epororoiF€ ro I = 0 xqt cqrlvowe vcrFero6dl.€r(l|" F6vo ro c, tirme 1c vc 6Xoupe eLdxldro rrQ6nfl F,, > 0. Avd).oyc.€dv otae€eorolrloouF€ io r = 0 xsr F€ro6dloqle ro y, rdiQ /oqr€ [La oworir.I1o rov or,wre).eorri c:

6 . 1 f \ A \ l r . A V t t t : t \ I i , \ t ) A t V q r . ^ A : H V t , , 379

(i,i) Edr a J dnL q€rt'/"ri ogLoplt't1, uftt npitra c > 0.

Elooqc)"ilow ciqolr urt(g or oua0{xeg, .r > 0 }rar c > 0. 6rL it / €ivor 0€rrrli;H cnriltlot eircL 611 o 6lxr6q 6pog 2br9 propei. vc onpc5leL qv I xcirco cn,ro $166v. ov Eilar apz€.rd $€yd),I.

n A P A A E I I M A f = t r 1 0 , , y + t . X q v n e p i m c ) o q t l r . , r r t a = 1 x c r c = 1€ivar 11a' ru 6uo e€rrrci. Eorur er rouroq 6rL rEerdlorrrc ro r = y = 1. Er€rbri/(1,1) = 8, crurl r1 / 6er ri,var,0rrrxir opropeul. OL owerlxeE o > 0 zcrrc > 0 rlooqa),ilou 6rr I / al)Ed!€r zari( rLE 6!€u0ilo€rE x xar. y. al,Ld tlJloa€ivc qO|vot itorri lixoq pr(rq riL),1g €rU€io!. H orli(Arlorl eivcL cqv$rx1 rrrivororlv 0r0!:ic.f = !. tjrorr o l, = l0 rr-€01o7'sr r(ov o ricrr c. Ei.vG! u6uvorovvc etcapr"6tiroo4re ro orrrrd ooroptro €E€rciqovrqE rq ou'rdprlol 1i(rd [r]14oconotoulryrorF nfTifQ(oIdvou rl{fkni( rr0eLd;r' xr.rr" rourrl 11 I rxteivetcr. :rrircrrur rtito; or' to |q6iv.

flpoqreld;q to 1) enrlpt:dlrr to ;rq66)"rpc x<rl oflt! aQxrxri lla( / rtre! eEr!Lo.Mt1:to4 cur6 xt(!€t llv I orrLxrl r{or €E(oq,{1}.iqrlr. ivo el,ciloro; KuL ircil,L 11tu_Tciarlol ei.\,c! 61t to ;rpoo11.to tol b 6ev:r<ri!eL p6tol flapu ro ytyolii< irri),ot .)r otil'ttl€arl< ror lrar 9truoi, to tqXLzd tapciittyltrt 2.t:2 + 4:t:u + 3t1 dtyir@ |tn"d oau4iin. Ont I F oitE q I irot1l tXax@1o. n(ivc,) oilv €lJoelor ' = r / . I J € i v o r o Q r t r L ' o t r / ( 1 . - 1 ) = 2 - 4 + 1 = - 1

Eav gi).orttl q J w titttr qErtrd oAtopiytt, twd ttor trQind w ,ifEot)pttirar n liytqoE torr b oE oYioq pE ft n ,at.. 6a tiNitls rtlqcr dla cepLtidgzt)Lr4oro. nor bi1'€r tuo (l.valt{.lia 1r'.1! lz(ll,li ouvo{xit dsxrzriE oororBoTltraE. H.'fi),oiolEpl rrlvr.ril ci\raL !a "ouBd.IQ{l)oor[E ta rrrqhycora"t

! = ar'1+2h.ry + t:!t1 (2)

O nA6roq 6aoq 6eErl1 bEv ei.var:rori cQ\rl r{6q,6LorL ro rrrQ(iyorvo roD.lrrr}..rorlrlEraL [€ rov 0€rrz6 oul,rt roIli a H &qxtLa or,lotzl (i) ro1[er ux6pr1.To r€rAdytlllo 6ptr;q Uoqei ra ei.rtrL pq6iv zaL r6x€ o 6€rrr€AoE 6poE rQ€n€r voeirrrL 0trLx6s. O opog yr :ro).l.cr:r),noLrilercrr pe tov owrr),eorri Qc bl)lnyaLt0Erororrtrci rolr Ei.r,ar rl rsi.r-rir.1ta ffioirltot yr(1 \'c 61o4re 0etLzrl l:

(tii) otur | .l ti!&L a{L1ti, uht: qiteL at > b2.

:l[€r6orF 6rL or ot|ve{xEE (i) zcr (iii) owe;riryovroL cruolldroq rI owonxr (ii).Edv a > 0lror dc> 6 >0, x6rE oiyorpt{ c>0. EEco.tal . i .E€rcr loLtrov I0etrzorrFc rlq 6rErdc nl€uadq rIE (2) xcrL p' at,"[6 arcnniocrpe, :ni rdlouq. rqvaQ6rIoI:

= . ( . . : , ) ' . ( , , : ) , ,

OETI(A OPEIlENOI nIN?\Kf:

6A H r*prryalvlxrl Voltr:t | - urz + 2bxy + r:y2 eivat oetltli opr,oyevrlr6re ?.ar p6!o! 16r€, 6rav o > 0 xc! ac - ,2 > 0. Awiororxu zcr 1 -F 6xerivu (FI '6'oFoepo) ridxroio oro r = y = 0 6rdv xor Fovov or(rv or. .rpdrtcncady{oyoi rtE plb€viEovrar ',.o1

Sro,o>u, [#r.,'r] l$fr',,r] ' [*rar r]'

OL oul0iizri yt.l t(6yroro dQiozolr(1L !1,2o),s. 6rorr rt / a7r'r lEroro,6rcrrt -/ rXL'r.L(iXLoro. no! oltr(ivFr trvrrorpo{trt irpootparr rov a., z(1 .. Aurrl((r,rtv€r (v(),LottDrt rl 6rir€at uu!0rli'l r. - rr > 0: H rerpcrY{|rvLxrj poprlq rivar"qqrua oarottivrt :"ort xcl p6vov rort, ritcr o < 0 xur" o. /;r > 0. TI! ibttrrdJ,tryt1 Xpt:rri(t:tot rrtr q ,1r

H rcrpal'rvrzrl tropqirt livarcr rbrogopqrl. 6rcrr ac |r = 0 xcr curi rivcr"r1 atpi:rrr'rorl:rot a:rozitior4r. Tore eEcrqcriiercr o bti,rr-po( 6OoE ortv (2).trq:rlvovrrrq tr6vo ro rmpciytovo xar zirlorrcr: q! / €ir€ 0rrrxri ttprcaroqilq.or(1' a > 0. .ir€ doyrrrrrri tl//,oA,oftrtt. or(v , < 0. 'lo nA60rtrc rytr' (r{rt}rLorrv / 11 bt'var6r'trcr vrr rrrowlt p tr'tbiv. 6n{'E oulLd.rilrr oro.'rtpeio r'= a),l= o. ftllurOLxti, rt trqillF(r : = /(.r. r) (16 yvrlorc xori;r( r-x(pu),iq€rc! ortrru ZrnpLerl ox{ir(t'rl. (:fil0Ei1€ 1rl! rnr(t).i!rL( : =.r'r orov trrL6Lrirotaro X6po. Hoz&rl rzrfiv{,rcr ]{(ril [4roc ror) (iEovc y. nda!) r(r r(ira). ]{tlr xri0r: ryh{iaor(lro11 tivrrL r1 ibLc :rcprdo),{ r = .f'.) Mlrr trrlo[n ]r!o tbr6troQq)11 lelQcyolu(ilFoAqrt €i\or at,til nor! Ft6evilt1ar :r(vrorl. a = 6 =. = 0. nori ri\,(r rrrmoleour0crLrd z.r evinz.i rt[LooroB6\'I xcr tt zon]r( 6/[t yiler r!),ri(')q rnirrbt.

frc tu( ol'v(iel'torl lf(.t). or lli( brdorcrorl. or 6u\'rrrorrlrrq 0n ei;qav fbrirr:(aritori: EirF 0( r:r'tpx€ t-i,('rt!oro tite 3dyrmo rirr 0( trcv F" = 0. !rt<bio 6riror(idrrq. er ror,rorE. c;ropdvt-t trr"tr roliJ rv6l(qdQouo( 6u\,trr6rrtrrrr Oou'6Dao!6q a( ll pto(lti t'tt ti,tu rr{lnlrr2.i(. A1JTo o!va{j(l(lvr oro rapirbrL.4l(igcq. or(v ro tttyrooq ror ir r,:rtoioltre Tote I / rir(v 0crratl:rpoc op$pireq rcrrer0r,vcterg zoL cplqrr"rtj rpoE .il.i,a!. :tlp6(i!rr rntonE x(r yLuonor.r6itnor0 l. orur t(l a zft. a7orv (!ri0!rtl raooi}r1l. T(;r! or riEoa€q r0r1r zcr y bi\'ouv (vri0|ra cr;rorel.ropura oro! A'cr hlowr (ulrivrL q / zcr orovri).io.{eivfl. Eiv(r 7o'jo+ro rt. 0ro)orlooutr€ 6no erbrzr; :rr-0r:rrd;orLq

h =2t:tt

:rtr npdq :rrpi:rtt'ro1. :.rrpLcrpTri ro b rcL a = c = 0. !r1 6rurp4 rcr a xcr rrTow crvri0erl ;rp6o11cr. Kcl orr; 6r!o ;rrpL:rrriror! i1orltt nc 6r = 1.

6I EAAXI:IA. MffETA X-{I :A|MATIXA :IIM[, .

!,(.6.1. To odyBd I = 2zy aar or roocdeptadE aapr!).Es ro!.

OL retpoycovLx6E cr.ndg poQgiE Eivs! do(xor€g 6r6rL prropouv ra ndpor.rv xcrrrc b]Jo nQ6orll(t, .f > 0 lroL I < 0, ovdLo^/d p€ ro c ,{c[ y. Eloup€ 610!6r,cr ottior,po oq[Eio ]io! 6€v €irc1! oui€ F6yloro olJrE E],cilrolo rcl ro onoioo\op.a\eljjr oaflarrrti otpElo. ([!oovdE 6(6x[ n €rrlqdvEra z = l@,1/), L.X.; =:r? - 92 ixer ro oxriircr otiypcrog (oopcpr.or.r) (tx. 6.1). Kqrci rrl! xqr€uowonrou dlow y :rrier :rpos rc rtito, 6:tog lpridlerctr 1cr tc n66tcr, xor xnrri qvxax€uolvoll ro! dEovo rc nq€l neoE xa Jrdvo,) MroQ€i va nqox[tlcll€ Tl]v dnolry)6rL ,rpox€rrar ltd 6eoBo trou neQvli avd[€oa o€ 6{0 6orvd. To avdrrdro or]tr€iorou 6p6pou r:ivar el.rilo.ro toq 'rsog rc 6ouv.i ror utdvovrar ora rLciyltl, €ivcl6ptx gdyroro rcrci grprog rou 6p6gor.

Tct acLypact 2ry xrLt, r:') y'? eivcr npca-nati4 rc i6Lc. Etiv orpdlro4r ro drcx(1rc 45' noiQvol)l€ lo d],lo. Eiva! €Trioqg o1e66r o6u\rolo\r ro lo ot€6[doou[€.

o Afi€rQoonr6,( AolLop6g 0o aQzoioe 'rm !' orio6eiE€L rr,E ow€r1xeg pcrEElatiorou: I]" > 0 xd' I]".F!u > -F;v. H raaF|-l|"ri Al.y€6ed ErDq l'roa€tvc reruxer. neploooreqa, p6IrE avrrlTp0oi$€ ro ncbg or oh'releorig tlg /6rnr<rooovtcL oe evcv orppe-rpra6 :tivcrc A. Or 6poL orz xcr, r:y'? rpgovi(ovrcloq 6!a./6\r!o xor, q p€rxln noQoyoyog 2bn, 6loondxo! o'6!(r ororl€io rio!0r xc!E\ra la(irr). Eror n / yi./srar r.ruroxu.rd i.on p' il,c .7LvoF*o :nvtilov

181

a ; +2 tus+ rs | = [ t y ] f ' al t , llo . l ls l (3)

H iow6iq1a durn €iva! 10 xtrerti oLou rou r€rpslaiou. IQdqelo[ / = jcr Ar,lEr,rr.i€u€rrrr ap€o{Dg or.E n 6raotdoe6 ]{Cr|, 6qprcre.y€f pra onvro|llt p60o6o p€l,6xrE'ror reo6l.rit.r(1roE prelimov ]{ar s),axi.or(,v. Ordv cvd rr,rv r xcr s 61oupe rL{

OETIKA OPI:MENOI NINAKE'

l r€ra6UrdE r1r. . . rrn, rrE olp6ol i lowe !€ 10oulrpEquo nivoka A, n yrvipeto J = xrA:tpoaEi:

6tciwopc-mr1l,1 x. lrc xdgxetuar pra Tttots ftracyayui

x1 Ar = [r .1", ',f;l : l::l

"", o=1tr tr)l?il

rl,i,= a | j x 1 + ( r ' 1 2 1 1 I , + 4 7 1 x :

- \ - \ - , , . " "

Aea urdQxo!'v our€ p€yatrudQaE oul€ prxQoraQdE rd!eo-< 6poL, rcrgd p6vob€ur6QoE rdE€{DE. H orvdAqorl pl6€!i.!€rcr oro i = 0 ,1d[ 16 i6Lo oqr6aiverxcu 0€ r[( nQCi)l€E iroQcrydyouE rylg. To €earrro[€vo €rirebo €i!c! oQ!86\r!o, rori = 0 sivor, or'oltlo olpEio no! 7ip€nsr vd eE€idoorFe (tv eivu p61Loro, eLdr,oro

(4)

IAPAAEIIMA 1 / =2t2 + 4x11 +y2

IIAPAAEIIMA 2l =2cjl tcLt A=

- odyLratrxo otlpelo

+ ouytlarLxo 04tt€tt:t

nAPAAEITMA 3 I = 2rl - za,q + 2r\ - 24xr + 2r{ ovrr"mor.lei orov

- i 0 I2j -.1 - El,r1XL'xo

F(r) = F(0) + r"(grad F) + lz?,4r

+ oqor 3$ td[e.os. (5)

+ :r.E.: An6 ro 6!.$d io! r€atalor 7.{D[!rrq L.O. Hesc (1811,187a) no! ro! noitq. mc18,11.

,liAVr€g €f1,q! y\'{oreE rsrpcltovrx6< oU\'cqrrto€rE. Qordoo orro|a&inol€ o!.ydQ,

qol I.(.rr, . . . , r,,) nQooctyi|€rot p€ ro1, i.6ro i(ro;io. Aralrltonpe tc orrio4rcrolFeia. 6nou pn6€vilovrcr 6).e< or npdrreg :rcpdloryor. :'aur{i ro oiiF€io o,4€ivar o "rill]zoq {ov 6rurtpov rcpcytiltov" rt "Eoornvag* :rivcrng" pr mor.-atLa aii = 0'Fl0x'0r1. Arlros ei\roL alJtopdto4 ou14€rprx6g, 1Lc rov i6ro l.6yonor Lapt )lFl1z1y = &FlAyAx. Tixt, \ F ixtt tltixnn 6fti t wialaftWayanxi yoppi f = t:r At civar 8E txi. zat\ / rlouv crnopovro0et or" 6poLSEUriQo! rdE€oq, or oroior" el{1oUv rlv -F oqv rrEQrolyl ror fi(iorlou o r€io!.onrrlE rpor,€Q6\rtr q o€refi Taylor romi{ oro r = 0:

b.I EAA-XI>fA TftfIfIA KAI >AIMANI\A:HMEIA

le xcioe ordoqo o4r€io, io srad F - (0F/0q,... ,1F/ac') €i!a! 6ldvuoFctprl6evrzriv xcr 1 retpoyt'rvrarj popql et,4r o6q^/€i rry a aQoE ia rdvo ri tidro.Edv ro oraoLuo onpslo €ivcrr olo r0 (rlri ro!' 0. xore q o€[Qd E€ "1'd Fe xo F(clJ)nclL ro ,0 orq 6€Eri( nl€upd tqg (5) ol,)'rilel oe :r -:16.

H Fnoutvl noQdyQaq o(, :rrprrXrr ro xgrrrlpLa yrc ro :r6tr 4 / -r7 A-r sivoreerrxl. Ioo&iva[o. ou[a drroqaoilow yrq to roTe o :rivcncg A eivar oertxtioQrcpdvoE, rrQd'rus nou eivcr o ;ripro5 m61og rou x€qddiou.

ATXHEEIE

6.1.1 AfiEre 6fl rlt€raorolttri loEpl / = rr +a.v+2er drcL oclpcllxri oryrio aqv aqxi.naed ro oI| or ovw.Ircas qs €ivaL o!flaoi. Eovdlad1]E€ qv / q dLdqoAd 6no

6.1.2 AroQd!8€iie nv.ril(lt ri o1t 0€lrxd oototavor or naeaxdro nivca.s xarrqcilr, ul!avr imotxl t / = r ' 4 ' :

. I t l{d i l j ; l {6) l_ ' r r ' l ( r ) l : ! l 'b) [ : _r ]

H oaiqouoq ont (6) €ivaL F tb6v. Kqrd F ixos roras e!€Fias !i!a, 4 / p$i!:

6.1.3 Edv 6vc.s oullF€re.x& ril(lx.rg 2 €ri 2 r€Avd ra xarfltAlo a > 0. ac > b'z. L!.nrtv tetQ(t(ovrai r idoot det (r{ - )J) = 0 xor 6eifte o or ail.< qs eivor o€ILaas.

6.1.4 Aro{po!0€ire o! {xo!$ s}dlLsro, tliyrolo 11 o.!'frronao oll|lEio "t,|a r\- o!!.1Qtr)otLql) l . = -1+ ak ' r ) 5 . s in ! + 6! ' o io cnBEio , = , = 0.

6) F = (r' 2rrcose. p€ otdo$o ('tlrio oro r = 1, !/ = r.

6. r .s ( r r - tc ro,orx det(hor5 bo r ivaxoc / = [ j l ] , , * , * , * , , * , " , , "* ,

6) Baeitr nt! troasyowottoitll(n1 A = I.D l,T. o$! o 6 6(taixsr ro n€6io o€rLxliqoprdrporry(l:.

y) BaEir€ rtv s).'itrair qri q9 (r'1+2br!+9!'))/2 /. yr(l, €wos d!"ro! ror

b) noLo €irar to.).drrco 6ra! i, = 3i

6.l.6 Eoro orL oL 0dLroi o!\r€Iem€< a zcr. lneerotl'ow ror b. f€.ip awon a +. > 2b-EEadpdlilfl ovr6 6rr uc > 6u xoL 6rr o nivdxas rivaL o€flad oQrolivos; A6c€(no6E,et { artnaad6'jtyla.

6.1.7 o) [oLoL rildx€s 3 .ni 3 drLmoqonv orLg

t ' =,1. " i .

' ! 2!t.2 - 2tt4 + 21:2,1 xul

J, = "l +2,i 4,1 - 2aF: - 2rr er 4r:rri

6) ^€iEr€ ofl rl /r €ivdr fvd 1.L€ro rsredyovo, dL-)d oxL eetLx.i oerouiq. nonFr6€vf!flar rl l r

383

384 OEITXA O?I'MENOI IIINAXEI

y) Exqedo!€ dtv /2 os ri0eorola 10!6\, rsladydrwv ua! raeayoworodn€ io,', ,4oq poeqi ,rrr .

6.1.8 Eriv .4 = l; "

I riv( o€I,'.d oaLqr.los. b€iEr€ ort o A r rirot osftid oEordvos.

6.r.9 H ftiaoyrv|'{i poagit J = 3lq +2.2\2 + 4r1 .i\ar 0s!r{i. B€€il€ 1ov niaa'id r4s .4,naqarMorodc. t@ orr Foe(Pri ,Drr xa! ouv66oE€ ra oftrr€ia iov , ).ot , !€q! dqxu.i t.

6.1.10 Edv R = ii i ] reaur ro" n' rc €nqh0€nore or rivor osrLx'i oqrol&vos. sxrq

rIE rr€purr6oeoE 6no! o a .ivar L6L6poepo!.

6.t.ll d, Ed! / = l;

-l dvq EepLflavds kir /,lad,xl b) 6e€ir€ rotr obryous rolj xaL

qv oqilowd rou.6) tlJp'il.IadorE ro tsied"yolo y,o it p,'yd6D('i I = .t,4. = ol.r 12 + 2 n"

clxr 'z=a4+g/a)xr l 'z+t1) no|a €ivor ia l.qrniara yla qv t > 0 nou €Eaoqol'i(o1jv 6r! o A stvoLeEru.d

, I I l . i l 1 3 4 + i l ^o, Ervor or ftvax€5 [r _ i z ] *o, lq _, o I u8ru.a oqrq.fvoLr

cl.l2 Aroqololir€ ar n F = ;!r2 - 2, - 2! txer ekixFro oro qpeio ' = y = I (aqon6€iE€ft ofl o! nEireE naedltl'ror Fn6flilovror q' ovr6 ro oryreio).

6.1.13 Yr6 noi.E owoixcE y|a ro d, b,. Loxu.r I q'2 +2boy +csz > 12 +f 'tta 6Ia td c, e;

XPI IH I ' IA OET IXH: OP I : IN IO IHTA :

6.2 T XPTTHPIA OETIKET OPIIIMOTHTAX

Ilotot otpprrprzoi ;rivlz€a aXor,! rrtv r6rorrtrc x/ -{.r' > 0. 1La zritu pr1 prlbo'tz6.r : Yr iqTorv r ioofprc 6rrrqoprrrroi rponot, t ; rc i \ .Trtort< u , . r ' r t qv fpt i rrr tot zurr7.n.!ourrr v,r r , ' r ' - i ,Ln4r o).ou: I r4r : rCotyoulrcvt : r 'qrr lgrqo. uoxrouur przirnorEq urobeiErLq ]ra tc neooiltru tlrv rblorrtrr;rv. ul"Lrl l oulllrqol 6mr6q otrlItaol L(lL ro Eirll.r r0)1' Lb(orqrdv i6t)oe rI oaol rol d rv(r !€u/oq ouvol>.(i).v

. l n 6 l. l = 1 , l . n 0 F : r t t , r ' 0 , a , / ' : U ,I D C J

:roxo< trt< fiacr rrr }'rrrrnoougr uuic rL< or;verjze< ot ivtv:rivarc rriEcr,tq lzcr vu tiporye q oydo4 Br r(r neoo4r( rr)! rblorrld!. frrl! n€airr(oort 2 Fni 2.rorkizrorov. ot ouvoriTt( ol$oi\'oria orr z(r or 6rio rbrorrrrfE €i|at f.ttxi'. 'fo

./r\,6trslrj rort eivct 4 opitouoa al b: > 0. <'rqtr or 16ror+16< €ivtrr 4 zcL or frio011126. il r(r o( luo q)vttrxeq. nQan€r 6tror< vc €iv.rL 0!_rrra< brol! ro (i{lQorolrd.rorE Ei"v(r r() itvo< r + . :' {).

Ei\'(lr (lqLooIFci()ro nooo nror.! alr(!(lxtrouv (|ri< or 61(' jrQoo€lliorrg ror)eil(roi rl $ic c4rForl. ruro).o1LorLx{, il .il},l nro xolr(i oflq ryyE\,1-iq Lbrdrry€qrou ;riruz( (rrc rbrorr|r<)- r( btto la(rt ror' Sr6Lior!. Ila.iyl(rr. zorrdvrqjraoorzrrzorrA( ro unoi.oyror(zo zQ(rriQLo, [noQoilLr- r'u 6rczpfvor!c zur r1vttr(i(ivrolt rr')v obw(iJv. Auroi oltrtrrllitcr oTI\, (!(ii.rl('rl lrtE / o€ (i0porot(c

, t . , : - : t , . , t - . , ' - , , | . , t L ' ' ' L '

\ , , ! r ) , , u ' ( r )

Ot ourr).rrrrt< u z (a( b )l!' riv(n (7pL6d'< oL ob'tloi ror 2 eri 2 ;rivczr.Ed1 (r(ri rt (]Zi(]t r-ic1rioi,ou6t-i vc roznrr l|cr [s./(l),rir€Qou< rivs.lr<. 0u lLu<rnrraa'l,rr ilrl noln trr),o xprr{pro {)€r'r)i: o0rortr(irrtr(ls: Tylv €iar(rdt i(|)vobrt./(irv. T(ur6IQor,( 0( (1or1[ pL<r ro),n (n].rt r:riyrlort: 1t r? .4/ si1,(( 0frLralopLopevrl. cv zcl p6vov u\' ./pi(q,Ercr oo\, (i0eorogc n crvetcrprrjrtuv rergryrirtov.

K(r Ii( (zolL,l nooz(rtlpTrrrri n(0(rrriarlol. To bi() Fiar (urol ror, 6r6iiouofvbri)\'r(1, lrioo) rrt< or-(,)aiq tov optiot'otirr zcr r-noFi\'q €iv(lr rn),oyo !u(xNrlrrour€ rL p6),o n{rilorry oL oQiqouor-<. Iiyot'Q.t 6Er darEi r' alraftiootptu1r6 ,p oAieovot to|, A va tivu qert*ti. H cnafrrtort crrl rz(voroltirar 6i&va = r = - I rcr b = 0. bivovrac ,.1 = I, ro! €ivaL &o.vlrLx(i oaroF!\'I [ooqt. Toahoolltiroto €ivot ort ro xprrrjpro 11< opilolocrq tq,(rpptilrrar.6TL povo orovibro ror -'1. 6i!o!r< rrc l/ > 0. aii,ir ;{aL orov 1 rni I r;ro,.rivcrz( (l. ofitvtrii\r) cQ'orrprt yrDvic. tl quoLzrt yerizrUorl Otl outrrror)..qr6Av€L 6Lors ror! ndw uptrrtrpotl; no:rh,uxt;

:l.::1]_:11 = [d ] . r1r =' t ' "' "" l. r, = l,.i

I arr

OETIKA O}I'MENOI NINAI(f-:

l6ou ro zuE"o e€6elFa raL FLa },€7rrop€OtlE 01166€r!I:

68 Kde€ 6vd arr6 1c sri6pevo xerfliare €rqQdq€r pi,o ovoyxaia rorrlicvrl ouvotxl, yra va €ivor o ,ip(rl[d-r|z6s oul.rlrEr.eLxb' ltL\axal A qxuaoQLopifot.(l) r".4s > 0 ylo r{d0s FI pq6€v!x6 Ordwopd o.

(II) 0),€6 o[ [6!orq6f tou -,1 rxqvofioLouv qv )i > 0.(IIl) Olor ot dvo oQr,oi€Qd unonil,oreE,4t 61ow 0erxrl oqiloro<r.(Iv) Ol'ot o! ofu^/oi (Xopi,g 6vall,oy;E yQaLUdv) lrovonoro{v r$, di > 0.

Atd6€d4 H ouvOllrl I optleL 6rro oetr.xri oqro06vo ni.voxd nar ro ngi)ro 6{[(lpaq 0a eiv(rr .v.r 6€iEo1Jpr rlp roobuvaBia qS $€ rn olJverynl ll. Yro0iroltl€l.orro! orl rz(rvoro€lro! I I x(rr oupri€Qdi\rouEe 6rL xd0e rbrotLprj ) npdner va€i.v(l| 0r:rrxll. O ouLtro^1Lop6g ri.r,ar, arL6g. Emo.) 6tL .r:; €i!(rl ro cvltolollo[o\ra6rcLo 6rri1,!ope, ror€

,4r i = ) : . r , c iQa :c! A.t1 = t l )1,x, = ) , , ,

cqot r '1-. , = L Enrrbl r l oul f ixq I rnXUfl 7r( raor x. ec( rolrr f l 6rra r(rr yraro rbrobrc( lunL^u.r, . xur r noodlrF( r j ,4. / = \ , rQiner vc civ(( 0rrxl . Eror.ivsE etruti oQtoptyos ttilsras l|ot 0t1tx6, dnupit.

T6Qa TidpE nQoE rrlv dL),I xar€liou,ol, liro06rovroE 6rr ol,€E o! )i > 0 r{oroqurcpaivovraq orr cr Ar > 0. (Axr6 nQirel.," ano6erl0€i yra xdo€ 6lcivuo[ctu. 6Xr p6\,o yro rbLodLa\r1lop(1ol) En€r6rt oL outrB€xQrrci ni.vaxeg ilouv rhipego!vo).o op0or(rvo.vr"xd'v rbr.obr,c6,uopcirtov k)oo[oll,r6 6sderyo), [noQotFe voypcilio]J[E orlor"o6rlfior€ l oav ouviuaopo c1r] + + c,,;r,,. T6re

A ) : = ( t A : t j + + c " A i i = . r ^ r : r r + . . + c n ) , I ? , .

Aoyo iI; oA0o./ovr6iliaE x(r rrlE xovovrxonoitolq r,/ zi = l,

t:r A,, = Gll i l + ... + .,,r, lx.r.\,,,, +...+ c,,)".r,,)- r: l)1 + ... + cl,),,. (2)

'Orav ,tdo€ ); > 0. r6re I (2) 6€ix!€r. orL ,,?Ar > 0. 'EroL I oxlolr{I II

ouvr-,-riryEraL q ou\,ori1Ll L:rasqoFaor€ rdQa orLq III xar lV, r(l)v onoi,olv I rooblll'aFio F0 1rl, I 0rl

aro6eq0ei oe tpic 6rlpctcr:

EAy nxnEr I l, roJdu y"u 4 III: Kar' .raxdr, q ooiqouoa €voq ni.,'orc €ilar roylvol€vo T t')! rbLorL[(bv rou. Orav rotu€r ro I, ./\roraiLoqE 6rr .l!rd! oL roLorrp€E€ir(rL 0€rLx€g:

d e t . 4 - ^ r ^ r . . . ) " > ( 1 .

KPITIIPIA OETlKlE OPDINIO'I'H'I A:

fr( v o:rob€iloul€ 6rL ro i6ro rolucr zcL ylo o].orq roq rl]roni]ax€q ,4a.Scilvoqe 6rr. orav o A rivur Osnzd oero!61o<. ro iSro tozr€r ,{aL ./La I dOE ,'lr.To r€xlco1c €i.vct vu rlrtcioouur: tu 6Ltnropctct tcov o:roior or ttLeuruieg r Aor.l!ror(bor-c !ivur trlbf!l

,,,.= j"r fr l l; l . ';r,,' ' t l L U l

'o-rcrv.rr . ,1r > 0 yr( x. i ( le l r i t $ i l50vrr6 i . r6rE €rbrzd rJ.4/.?r. > 0lrc x( iO€ t t lpl8rrrzir.r4. Eror 1l duroriril I roTrrr yra ror ,{r, acfl eqrrrp|loErrct o rrLtttir oibrog oul,loloy6g no! r'q(euooritzs xar yrc rov A. or r6Lotq(g 1!r! (nou 6€!eircrL ra iSrcr , \ ; l ) nginr l t i t ia(r 0l , r txdq, I opi iorou t ivcrL-ro' / rvol t tvo torg zrtLorl€ri(ir< o( (i!(') (pLor€oi< ooiqor.o0! €ilcL 0€rr^6q.

'Orw n7ft.1 4 lll. rt utyittt rar r/ Iv: Arl1o Gro6r(xruur(r1 [ixo].c. 6Lortu:rrippr pLc ripcol o16o1 petcl.r rr';v cpr0pr;rr det /r. z(t ro)a obtYd)v. Ko-ril rrt!tl(a.i./Q.qo ,].4. o A-oroc obtloq (lr. €iv(lt (1i0r6{;rE o i6yoq rlE dcl ,4r iTeoq nt!der ,'l/. r. A1 Lolriw oif( or orrilouorq riv(r 0rrrz6q. ro ibro 0c rox!r-( t{(r y(ro!< obti,oi\ -zrlr 6 xpcrirlolrcrL €!crD,(li< lQctrtrdv./r( 0m;ad oaw[iaou

'Ortt ' royl i t I I \ / , k j rL ut ' ) t i " t r .1r, I : Eb6 lvoQlqor: tLe 6r! or o6lyoi r i lcr

oerrroi rc! rQirifr rc bsiEo$s 6rr J r r1.)r > 0. AlJIo xcrlcptl t{(t oqr nfpinroolI lri 2, (Jqrnl.ta(irvovrcc ro rFro(iyorv(. O! obnloi rtr('v ot (A(\'oi i:(l) (n(ir i ( r i r , -n(7 'v i r . rr ' ( v, , nor. t . f r . i , : onlr"crrv" i ruro of nuL.rrLrrxou; nr!(r 'uconoLolbrlnorr' Fr../i0ou;. cnr(]rpi(oUrr m1v (lrrirr r0rv obryd,l -t1v .t:ra).oLtptlC . r u . . l l u r r I \ . l - / . n I - . t ^ , , , r . , , t n , r r , r , t n r z i r o r | l . o r , - y , 7 u v u : r l u r , J r :trttr axalor.t)i Guuss ttri; otypt:rpLxoi' tit,tLxtt. o rira qryt,tLLxt)< U tiwu out,1un7oc1o; t t r t"r i ru rpryarLxoi L. Apa o A = LDLI yi t t rut .4 = LDLI .

IIAI'AAEITMA

noiitlr).crorirqovrcq a:ro rc cq,rortprl Ltt;r7 xct c:r6 rcr beltci 1rr .r.:rci-ovor.pt6rc ri0soLopu rrrpcyiro,!, ttf olr,rtl€maq rou: oblyorc 2. I rcL l:

t r f i I [ r 0 "1 f ' I f r u lr - l r i r r , ' | | o , . LDLr

Lu r r l u , r l l . lL" " r l

|l ;' ,,] [i][1 o o l [2

. / :1 . r= [u r u , ] j r o llo i 1 lL

=,(" i , ' ,( , j ,) .i,.'Al}Ioi oL o€rLrol o6ryoi.. noD,ar coririovrcE ral"€L( ftrQ.r./or!(. rtrrorir rrt

?-4r tlerrzrl. EroL I o]rvorixl IV ourr(iy€ro! rit o]Jvotxl I. ortrrLlprlvol1cq

try (rnd6€!Ell. T_o 0€6p4rc 0d ir@v srQL6d\ ro i6ro oqv n€airr(l)ot EQ[mav6v

Eivar rro),n o)Aaio ro orr 6uo 6aorxis liciaox€ld5 -oTiaLorgri rtsr ouprlit,Qc,)oI r€rpaytD\'(Dv- €i"1,er, orrlv ouoL[l, ro i6ro nQilyfro. To nptlto 6rjpc [email protected] aqqLQEi ro 11 an' 6l,Eq rlE un6Lorn€E €E!o(i)os4. Oloi(|)E. to np6roterptiyrvo oulzevrpcirv€r 6).ouq ror,s 6poug rou:r./,4c trou nserb(o!1, ro rr.A1,y€6Q!zdr. ro noporogo T€rpcydrvo.rv (aou 6iveL r4v o:ro6eLlr1 6rL x? , jc > 0)

OF:TIKA OPI>MENOI I I INAKE:

tr At = 1:t:r L11D11Lr t:) = tLlQr.L:)l + dler:r:)l +.. . + d,,(Lt r.)i,.

Or o6qyot dr €iv(4 €E{D. Or nollffil.aora('l6E lii €ivur u6oclt B}.!mrc oto,roecibEryB(r ror5 .lQlglol'< - t/2 r'tt -2l3(x(rr oIoxUOFs flg ori).r's rou t);rouqqclvlio!.rar F6os oru lQtc( r€redyn)vcl.

IIAPATHPHEH Oc {rcr }.tiooE vc orprlooqre qv cvflrnool 6r!, olrl ouvoiknIII. o! ir\r(') aoLor€Q(i ufioriv(rx€E ,4r. €i.\,(t| rirx! €EolQerLxd. El ioou xd.ci 0oFnoQorroqr€ vrr XQlortrojrouiooqlr rouE xdro 6€EL<i unoniv(zrg. fr.\,!xdr€Qs. OcrFnoQoioq€ r(r XQlorlonorrioouF€ [L(1 onolcbinor€ c],1Joi6cl ,t!Q((l)v unonv(i-x(')a. 0€op6vrac o(rv ,rQtirro :riv(r( 6vrr onoLo6rj:rorr rnor.Xeio a;, rrF 6r(yorvioljxt.| nQoooalovla( 6va v€o lE]]yos yQ(FpiE ra! oq),q< oe xir0s 6{Fa, Er6!x6r6ocralo! axorFf x(r prcl d?dyxaia oxyerixn yra 0€tuit oerot!6t4te: xtigt brayl,nuroryt i ' u. , nut l t t v r ihu d.r tx, ;-Tu :Nlr ! . ruio urJro r ivar axQr6cJ< o ouwr).€-ottjq rou r;. 'Olrrrlq 6Fo< 1l,(DQiqouge and Ia jropc6ciyttctcr, ro vq xolrdr!€ g6vorc br((y6vlc! ororx€i(, anix€r nol.u .rtr6 1() vcr siv(r x(lL rxrv6 zprrrip|o.

Or o6lyoi (li b€v nQ€rr€l \,a oyyXdovrat pe rla rbLorLla!. fic dv(v ru;rrx60€rlxd oeroBdvo 7ri!axa, efvqt buo trl,ri14 6LcrpoQ€rrxd otv0La 6€flx6v a0L0p6\r.:ro 3 €ni. 3 noQdbr:l4rd pcg, to xgLtripr"o r4 oqi.!o!oct< €i.vtrr. iotllE ro .!xo].6r€Qo:

det A1 = 2. 6., ,1, = 3. der;4r = der / = 4.

ftc rvcrv peyr ,o aivcrxcr. rivcrL a;d,,orioreqo vo :tcrpcrlgr]ooupe tor,5 o6qTorg rqgrttrcl.or.trtE, rou €66 siv.r ra nnLLxu dt = 2, d, = 1, d1 = {. Katci ravov<r roxQtrrtQto rbLorLpcbv E[voL ro rLo 1povo66po nl.].lt €b0! ./vorQiEooF€ 611 61.€g Ei.l,c!0crLir€E,

^t =2 rt. ^1 =2. ^j =2 + v5.

Towo, raA' 6),o :tou yLc 6e6o1rdvo :rivcra ,4 eiraL ro 6roro?r6repc eq4po!6,tr€vot o€ O€orerFlxd l4t{pcrc eivor, to nr.o Xprioqro. K&k trQr:riQrc xiyar afli

getrzd opropGvor tircx€g zar €ldlrora rstQdl0rva

Eini-i(D v( pou FtrrrAdyere 6rcr crzopl zpLrrlqo 0€rrzri; oQLo4(6q1c(. Ar.16€iv(r rt6rt iroi,u ro1,ld zm biver lLrc rr),rntp1 rcrrcrvolol rr.rv nd.croraAorvxpLTltaiov L(tr r(Dv oXao€drv ror< pt ro u;r6Lomo rox 6r6l.iou. :w66oc[€ 0€ruiir oeroBivollq niv(xrs pr o6ryoriq (KFqrii,(to 1), oaiqouo€E (Keqxil(o 4) rcLrlLor4(tq (Kr:q,ci)'(1ro 5), TriQ( r|< oq€r(ilonp€ orc ;rpodLrp(rc t:l.crliotarv -rerpc-

y(iv('rv ror Kt_q,c),ciorr 3. t(o! reol{ nrorrv ruTo napal,l,Il.olpu|polq ni.v(lxeE rouK€q (li,tliou 2.

O l? tirur o :rcrpuii4l,olecFpo< ntv(.z6 zor ro:tpo6l4ru tluyiotrrr rerptr-1r';v,rv tivrL ro n.r = r. xorr €t€r ,r €Ero(i)osri. pe rn ) a (or1:rtprl.t4r6riror.rcrLzc[ rrtgcYo^rzri ororrN(lrc). To y0irFF.r -.1 ro lQr((lolcort y[( to! ortrtr€rALzonivrzLL Rr R;rrl tlqrrviierat orq riaaoarra< {!Lo(io€rc: ? tl.rtu1,a &qioruvrtra\yir\Dr r t^\a 't ltio| rct' RtIIt =,R? tr. 'Ororq 0o 6!i!oule otr6o(u(. oxi,ruxtL< RI R brv.iv(r troro oulltlt'Iarxirc (I.).ri xcr 0rtrxc! opr"op6voq. r$, ooovoL n oqle< ror n eiv(r ypapFrxr! rr.vtlrienlrfq.

O rlivclxq( A €tvo! oullfr€rQr)rds xar o€flrci oQrop6vos 16rr xo! p6r'o

(V) Ynd(alsr 6vo6 rrivozqs R pe av€EdQ/rrF€S oflll.€q. rdloros 60r€A = R r R .

6C

6] K}ITHPIA OETI(H: oPI:II\IOTHI :

Bi,6nouttr o1t o Ill,4 eivcrL 0r-rLxti opro[dvoq..lvcprDqi.lovrlrq 6tL ro."rF?R."EilcrL ro rsrQ(i./(')!o ro| prjzou J?.r r. rou btv g:ropei vcr |iwr llQlll1r.zd. A€!Fnoari €xiort< v( lil'(lr ttl6dt'(xopig va LoX|er.r = 0). br6l o t? 6/fl (vrE.iprnrs(orrl).t;: Eth'ro.r ri!,ll pr1 sq6rvLzo torr z(r ro RJ riv(r p't l'rt6prrro. EroL orr.1.r = 1lR."ll: fiv(r 0111.16< zm o .{ = n7 n tiv(r 0€rrz(i o0rol6vo<.

AnolivrL v(r b€i:ou[€. orl or(v o .{ rx(roror€i rrg ouvorlzr. I - lV. 16rr'tza1oroL€i ?L{ lrtr v. Ilprlnrr rdre vc tjAortr! 6vcv n tr( ror oroio rl = Fr-Rz,ar ulro ol fh, ,v ro r /ortr f ) .uvf l bno qoqic:

(i) :ro lt),rt:rcio d)i3o ror ,t !pi(n, 0co)0rt[aror, 11 nrcloxl,r] Gaurr ncpoloro:rot4oe tor' -:l otrl sopq,4 ZDlr. Or obryoi eivcL 0erLxoi nu ty o,,/D iy.xLrrq r€rpcyovr"Ig eiir'< rou nurd BizoE trt( brcryr)liouJorr il = LAyrDLl ."Mr(l €rrr.oyfi ror, R €il'(r o d!(rr rQL.rr'ivrr{o< tiwtai '/DLt .

(n) Mrtl (i),t t nQoo€yyiot, ,1o! n(rQiry€r ivw 6tcrqoprrrzo fl. ri\(r v.lIpttor$o;rooioor,tr€ rtt II swl yl( lttv lV 0€rrz€g r6rotqri; uvti tr( 0E1rzoi!-<

" ,\r{'t.i!tr, rt ,tr'ionaal o! Cholcskr. (\n.f o obtyoi dr(oi.;nr(r rdorrf0Li): I'rr(:'i r)v.ir(! 2oL z.irb iarinrL?(r)! rrE$,r noi'i iii,) drllrrafl (No dno qv n(a(lovroi irtdrt ro' crus\-,1 = Irr".

"(r dllrluor r!,onr() ! ,ia.(ri./rd 10! rnoiorrtd audN ror rfrt(10nrx,tv ,tr!ri!

OETIKA OPIIMENOI NINAKE:

o6lfotq. To !6to6taviqr(ru llfi(rilouv oe 6vov oQ0oy(ivlo ni.v({xa Q:

A =e^e1 = (evq)eqe') = Rr R.

Mrn rpirq 6uvor6qrc e&at R = Qt[XQr, \ ovtrpstQuti Ottt ,i$tacyevtxi aias r.o1r A.

(4)

oQtota|q

O! IQ€LE avr€s urtLoldE rou I? irrrv 6leq rerecyorvrxoi nivarcg. asvXpetcroojz(pe qr d,.tu0€Qio no! €iXoF€. vcr rdrpoupc rov F ;rcpci.).11'6ypr4rpo.YncipXow r,nrdoo:roLl,ig r ,i,eE eaLl.olrg, rerpc.4ovLxig rj acrpa).Lti6ypappeq x.tlnoQoi[e vo boulre ro yrori. Ecrv no1,tr(rnl.rordroec Ulq €inloyrl Fr orolovaijror€,'riv(rx( fiou 6x€! oQ0oy6lr€E orri),e6 (6mt,r Q), ro lrvopevo rzcvo:rorci rrlv(QR)t (QR) = RTQTQR = RI IR = A. EEor o QR €i\ 'q! Fia ax6Ft €nlrort .To aEroottrsio)ro €ivcr 6rr rc rpo6lrtFora €T,cxiorolv rEroayd)v(|)v obtyloa!or 0rrLxd opLop6vouE :tivoxe5 xur um6E eivctt o L61og tro! €$q)ol,ilovttlt otlqaq(rQl oy€q. *

ncAariAqon :ro Fdoqld Fou, olavd F€ OCDrdl€ yla Bq oupp€ie!1{oui eEflra(opropi:vor,g :rivaxeg. nod 6t1, xelorFonord oulo ro! 60(). 'Eva( r[0c{!o6oqtopos rivcr 6rr ro ouppwqre6 Faaor (,4 + A1 )/2 nQ6n€t vo silul OEtlxitoeLo[dvo. (:rr] prycr6Lzrl aeqimoorl :rp6:reL v' curnrtrloovpe vc drat eetziroqopivo ro Equ.rrav6 p6pq (A + AH)/z.) Aut6 crQl€i yt( l,cr rEsogul,ioorlp€6rr rcr :tpcrnratrxir IrdQ4 t('Jv !6rorrll6a €iv([ g€rfts. 681, ei.!c! 6tl0rq avs'y]ioao,(irox 6€ixv€r n E/i6pE1,rl (]r66€rEI x(( ro n(Qdb€ty[(:

L{o1,n or]\roixl1(1 vo sxo![€ Re ) > 0: Ar ,{Jj = Ar, t6tr l|s, rr = )c//r:xttL LLH A,t =\attr. nQoo0aiolroq, ro fiQcrylrqrLr6 pepoE ei.v<l (Re .\)otc =(1/z)t :H (A + AH)x > r .

l r a l - f r , lo1 avulxcr ic: / = i j I i l rL Re.\ > 0 ulJ, t i (A, A' ) /2 . 1 ; ; l . ivu,

L " . l L . . l

Eli€r!,ofl6i orrE n t]oot|io€rE

fl(ll'rou o' nl]xo io 6[6Lio. niolJ) sn6 rI! d].^/€6Qo rllvar{cov xQu6ovravl€(|)perQrrlq xcrrqoxerriE. :ro K€qsl(rro 1, Fra yqcFpr.rl e(ior,)ol1 lcgr,orrivovrcvpe dvtr rrine6o. To oumqpc,4.r: = b o6rlylo€ oe prc roprj enmd6ov xrrl oe prcxir0eq ngodol.rt (€).dxrora r€rady(ovd) 6rav rcr erineaa 6€v olvc1!rdr'[c.v. Hogi(ouocr rtcv o 6poq cv6E rcpnl'l.r1L*une6ou. T6aa, Tlo dvo 0srLzd oerop6voni!.rla A zdL yrtr r4v rnrpuytuvrz{ owripqorl / .rr A.r. :rcipvoqrr rrtrrxri rva

* Eq(afofa ror 0m',.i oa,otrito,! lr(ixor a€arero\rdL mo (f(r,ir{'o I ro! ddiior tor.ht'.drttn r tut Appli&l Muth.,,rr'.r (\vclle\l!. - C.mbridSe Prs).

' r t . r .F . . : Yr i r l r f \Toru ro r ! r I f ( ' .n " i , r r t r r , 'p r r t "24 opr f l r vor r r r rx ( .

6: KPI IHPIA OFTIKH: O?I : IyOTHT, \ : l9L

zr l lu iuloolr tuc.nou€i1u\ l l ,R,futVIorLq6t l06r(orcoEr:r ' ic( t^rcr i l tu lotLdiq

H olJagitxl ror 0sDaoutr€ €ivar r",'l.r = t. Ed1, o ,{ ei.vcr" o tctroruroqnilcxc!. (1r j crr),orioreircr oqr .rl +.Ll + +.r;, = r. 'Iou €ivct I

'ltovcrtcriu

o(prripo . :ry€rt,lor€ orr ur o A nofiunLrror(o0€i [E ro 4, 6rot cbort A =,11.

1 orpcipcr livercr trzaorcar. H t:liocrorl gmc6dtr}'ttcrL otlv 4:r'i +.I:fi + +ar; = 1 zr l | avr i lcr olpt : i .cr i )n0'E ro (1.0,. . . .0) iT€evir t l ro ol trs i( o '( ! rq ro(1/2.0. . . . ,0). To r(vrQo rrl< 6l.LprlnE rllcL orqv up11. cqon. (l\, ro 6r(ivLro[(r.r'Lzavo;ror"ti rqv.'",4.r = l. ro i.6ro orltr6tliy€r xLlr l€ 10 (ari0sro 6r(lv1ro[c

.r'. Eplcrlogrrotr trr yl1iot€< r€.rp(ry(,rvtrdq popq€g xut ro tn6tr!'ro 6ritr( -ro

orl;Luvrrxo 61pcr tilcr !c n(41€ c:ro rov ruuortz6 nilar,a a€ irlrr Aq,(it'K)

En€!b4 r(1 ororzsic fil(iL (x(lr gcrlndl) tltr(6ci1o1jtrf (lno l]q(1i0(, or

MroQriirf v( bfirr rLq 6uorx6( truo€t( rlg cEioool(. Ml( eivcr it t = (i.0.0),unit" l\i1to< ro! nQ6ro! <i!ovc. Mi(l (il),t ot''clt it r = (0.1.0). mo! clEovtl .ri..:nt', rQirl 1t(r[ir{Jr,1o11 t:ivrrL ro .r = (0.{).3), rou civuL 1 pqaLr'rt:p1 6w(1r1c:rootuol oqv onoir (nki1{.1 ro cD'. t1fo0r666. T( ol ts io ( i .0.0), () . 1.0),(0.0. 3) fiv,u r( ril)..l rirpu .ror pLnp6rrpor dEovc. ror pto(iorJ (iEo/r, x.r! ror

litroroLr .?!o\,.r. Aux6 trordqer ;rt pru p:rri),<r ro! Q(i./xftir. o)(r 6troq oxpl6drcro orr}rc rr . r t ) . rrruia; r . ivt l l n i l loLtdrcQo *". . i . . .1 .( ] ) . ,1 =1. or 6rro

ioor owrci tor ic ro zr ivorJl l t t l r i l r6, t , lE nQoc ro l r intbo.rr . . r : , r i r o tLalLoroqdlovaq q,rriver trixQL ro (0,0. vf). H [rh].o orixr:r(r 6Q0r(l xar ti!.1l, I'iyo-ro].x.irorprl i,1( ),rixrrogo, .lv aEoreEooutr€ orr ro trro6 q; r:ivcr Xr,r[6,'o mo ib (toq.

or.Llortrr rdiQ.r zaL mo rt).Eucrio 6rilc, onot sitlrpinorls UI trllbt]tzcioror.I0io srr6< ritq bl.ryc0lior rou .4.

TAPAAEIT \ | { I f l i l " " . r r ' r . r 5 , , i - ^ , , - s , L Aur { , , v ( , u . , l4 r l

6Ll€rq,rt oro rrinr6o r, r. To raarQo rrl! eivcr orlv crpxti. aiih or irEov!:c rlq 6elei.r,cr.-16oo quveQoi.. T( r-xr6E 6royrrJ,'i.ou rEooatpLa aqivon! rov rivczc !( €i'r(Iotrrzd oproutvo;. d,)..L orQi(torv rlv diLerq, 6roL ti-rote r:r rilorf; r1g ,\,o gtlrrr0urpcrrrpiEorttu 'd.do/ ltF tolr< l(:ov€-( ouwr:royF6!0)\ (:7.6.2). A,'If '/L' .rtt[o,0(1 6€iEor$E 6rr o' d6or,!s rqs i rrvnlr xwEvquvolit ,a1i ttt 'rcnsrrott r.ivdo\tavwlterar nr A. EnErbri o ri\'czc( €ivar o![$Frar]ioq. ra rbrobr.'vuottcrrdror fiv(1r oQ0olrirll(. Eror o 4irTroro< riEovcr< -q< 6),1,tLtp1g. ;ror trvtrororXei m4przgdre6r4 L6rotql)j tor,{. eircL zci0eroE mor t),(i7roto ilEova. ror avrLorotz€iorrt $r7.riir€art rbror+I4.

rro !a €vrori.oorl€ rlv iLtrerrln uo1"o.1i!orpe rq LbtorLtt6q ^L = 1 zar .l: =i.Tcr rb'obLarro[cra. xftvorrronorqfd\,o ot [orcbLaio Ui)to: €ivar (1//2. l/V2)

.i,.='1,i l[ ' '

OETIK4 NI I :MFNOI NINAKE:

l>l ' '

13

s,,2 + au,, +,,1 = (# - #)' . ' (# i) ' =,

o

- l

:x. 6.2. H dl,l,crqrl1 5rr + 8r! + 5!, = I roL or rr:aror dCold5 uts.

2,ut OIvAJ l\4), offii(.riqoul, yo)vt( 45" Fe tolJg tiloveq r. t,, cl,l,ri or. 6reu0w-o€K tor{ ouF,1i.firorw }re ror4 ciloveg qg 6},LeruE. fi(t lc 6orjF€ qv tl,},€!lrl,nQirrEr va €ataypcVortrt r4v giooal:

I-F

(s)

OL tbLotqrdg 1 ro! 9 €ivar €rrtos (ov r0rQoy6ltllv. Tu r6r.o6rcvrjopctc eivar€woS.* To npdxo trrgilrovo tooLnfl pE I olo o1;l'tto (ll\/r,-I/\4), qordlos_rou Bdyrorou dEovc. To b€urrpo rrrQdyovo Loornur Fr I fio ottrrio( l t ' /2.1/\ /2) xf t xpeLaEEroor€ dvq! rcpt iyovtc 1{; l ycr vc cvcrqdoor4rc to 9.O FrxQds dEovo< €i.!ar to 6vc rpi,ro tou prixorE rou pe./dto!.

Edv r{'rruvo'ioourt rr ouvi64 ;.rt tqv r'liouroq (5). Inopon!€ v dn}.onolll-oor.l|E trr rov ido rQono xcr xrior < r.Lo el),nrponb€E .rr,,1.r l. Onoq fidw(r, iotnoqaoorrlr.j 64!a itav w \nyanonotfloortE rov ,4. I-€l1)gErQrxd, croxut!6-pap€ qv €Lr.ovo mQaqowcE tox dEowc. ALys6pLxd, rq r6m8rsvliolaia rou ,4parlxcv or ivar oQuoy( 'vto nivaxo Q xaL A QnQr . AuTli rlrq! ura ano fl<xiQrrE rbr'rg rou K€qol,oiou 5 x(rl p6LiJ I f€rd6(ort oro y- eil rapriylr ivaouQoroFo r€rQ0Yov(r)'\,:

rr AE = (r't e)^(er E) = sr ny = \,gl + . . + ),,,y],,. (6)

H EEioo]ol €iv.rL i6QCl ),yl +... + \,,y1, =1yoj at = (Qr dt rivcL q owLor6oarou s r.drd. FrixoE ro! nQ6xoo !6ro6ro}rjo[oroE. Ildoo peytr].q F roQ€i !d €ivdr

* Ado sivo' I'uqoadflaO 016 q o+drie@ot rov ftrcaydvov o.5(z +ar/5), + gr:/j, ll€ ro!<,d?i/,,ri €a(t nv raC€lolororv.

62 KPI IHPIA OETI(tr: O?I:IMOTHT/\>

dltrl; Eivor 10 nolLi llJ)1 6ad\ 6).@ rs dl,lq x€tQdyovq €iva! FIE&. AQo opiy totoc tilovac,. xaft $rixog rou Lbrobravuopcro< :rou cvrLmorXri oq prr<grirep4Loctr+h. qftvn ot an6o:&a4 l l1/) ,1 an;rcxiwpo.

Or d).l,or. rriproL dlove6 ei.vor. xqr{i prixoE rov dllov r6lo6lclvuopdrrrll.Tc prinl rooE ei^rat,1l,/4r,...,"1/1/),. :rue!6ore 6tr, ro .l rg6ner ra el,varggJ.m(I -o niuaxaE ngheL va eiyat. eel\xti oQtoplro(- olcrqop€tlxd 0o Exoolrsrg66l.rpa pre rL! r€rQay@v!1ii6 Qi!€g. H EEiotoo4 A? - 9AZ = 1 riEQ!"r/Qdqsr grdu:req6olri xar oXr Frc dllerqt.

H dll.qyi dri6 io t alo g = QTs fiQoE6vqo€ [ra orQoqri itov a[6vc01 rouICbQou, cboft duroi vq fiQoocQpofiow oroug dloveE rou EXlelqroa,6o{E. :rlEFetct6llt6E g n eEtooo! yh€ta! 'ioLu rlo qr[l,i xar 6lano1Jp€ o[dooE dil 0o16€llor €vo €).1,s!Vo€[66E1

6D 'Eor(l) 61! o ,4 et1,dr oErrxd ooropivos: ,4 = Q,{QT pe }; > 0, T6t€ 4orqoql 9 = QTc c:ri,oloul, qv r:"Ac = 1 oq poqqri

arQLQrc =1 ti yt.,rc =t i1 )1yl+ ..+)^f^=t.

ew\ etlor q €€ioool svos €tri.€r1loe!6odE. Or, rilov6g rou q(ow f,rinq1/\/\,,..,1I\/8,$erQorrFevol on6 ro ridr.rQo, r,io,! oro! dQt{lri6 x6Qo rtovo!\'rfiqTr6v(Dv s, xclleuedvowor cin 6rci0won tuw [6!o6ro!uopdiov.

6.2.2 Aftoqdv0Eir€ dv sivd! i 6xr 0erurd oerspd\,or o!

A'KIITEIE

6.2.1 noro €ivar ro rr€6io rur aet0f6v o rar 6, yld iolr onotol! or Jrivdx€g ,4 ror A €{!dl0e.!ltd oAroudvoq

6.2.3 Kftaox€ld@€ i\ar a6at6o+ ^iaaxa no\ Lx ad p€rdlm€ed oroq(€io ro! orll!rnQta 6!a'y6\,rc:

f , r r l 12241A=l r a | | a= l2 r 8 lL ] 1 d l L 4 8 ? l

12 r r l 12 r r l l r r r2 l ' :A- l t 2 - r l . r - l - l 2 r l . c - l r0 r l

L | | 2 ) L - r I 2 l L2 r0 l

f1 b -b l, {= lb 1 b I

L 6 6 1 l

. >.x.E.: Aeift lno{r1Uri(iot o.390.

ps , < l proe€i va aIEL detA < 0.

OETIKA OPI :MENOI NINAKE'

6.2.4 A€ilt€ rrr rl! tSrciuas 6rt 6ra1 o ,4 €i!(1t e.l*d oetou6vos. ro i6ro civo( xoL o /2

6,2.5 A€iEre 6rL ov oL -,r rat , stvot 0srLxd oetoFarot, ro i6ro €iaaL aaL o .4 + B noLdar6 fls (N0rtze< I - lV tirot xe1io'Il (lvni q eoed;

6.2.6 xery$orotdnoc rou( odryois. r'c ,b,ot'/ic r- * ,b-b,""ior-rd lou .4 = [: l] .

yadFf ro! ,l oav nTn f€ rotls test( te6nor5 rou raoftilolTar oq! sn66rri{ nE

6c: ey6)k6Lr\(e\/i)ll'.Q') "", (au6a' r@lia' \.

6.2.7 Ea\ o A = QAQ? €tvor owp€retr6i liat 0cltxd oqw!6vos, 16r€ o R = Q\.LQI'xirar q orlryetpLx4 xdl |Erlxd. oetollq retecytovui aild ror' rlotL ctwog o R6x€r raaypaitxdr r5lor4rds: Ynoloyiote ro! ,q koL €roho€nol€ o1 ,q' =,4 ytq rors

. I t o 6 l"= [ -o ru ]

6.2.E E(iv o A Eilat ouurlerarlt6E. 0.tLXd oetotiloE xa( o C ri'or lll lbL't!o(xl,o6, b?ift.6tr o t = cTAc eivoL eri'rts olrqrlarzG xaL e.rLxri ogLoltirog.

6.2.9 Ed'r ..{ = nTn briEr€ q y t'rsqrdvl (lvto6nF.. lou s.hwar, I'Tltvl1 <(.",t"Xy",rg).

6.2.10 H iu.uqq r: + +.r = t cutorortci -- , = l; l]

rEiwr tts ,b,ot,t "c

aut tu

ltto6tsliolora,{ar q(E6rdmr n]l i}'}'€t1t,rt.

6.2.1t AvordycrE r1v rlioorr{ 3ar - zyart +2r1 = I o 6!c d0cowpa tHAdYd,vt)!.dQ{orclr(s itg L6loflIii rou ovtiototlo!,4, xs! o)(€bL{iorF qv 6Ufl1'I.

6.2.12 'orsv

6tra ra )i > 0, trfte 4 )1r]+ )rrl + )1Jl = 1, :rcrpror<iv* orLq tetLs 8tooldolrsc'a rI],sLqJoFr6aE. no0tyadlft 6I( i( 6naoe€1rftd eifu MLqovtltiJ"' nou 3pq,a!i'govrar dq 6€rtxd rumatdIiv)l n€Oi1l(D(n), 6no! Iia ri rcqoo&tets lb|ot4ds siaat

6.2.t3 rqi,!,rE fls trir€ oNoi)r.s nou eEdoQoLilouv 6tt dvcs rivcros 3 !fli 3 eilot.Lsv'ttxd oeotituq ( A riv( 0€rLrd oeLop6vos) bivovraE rbrdi1€el reooorl orlorryo'i"it III: ndE oxtf,iL.rtu rl dct ( .,{) Fr rI! det A;

6.2.14 Eirv ava bLardvro ooLxeio tivot !I6dv. deiEre o( o ,{ b€! lJroe€i vd .ivc' dAqr|xd

6.2.15 A,ro$(rveeire noLo! oro rox enoptvous ntvo)tE! eivot €erxd oarolevos. cO\aFLxdoaLopdvoE. {rLoerouavoi i a6eLoros:*

2 06 - 2

- 2 5 LIi l

,[j;i]"li| :.r.E.: AEirc unodltrioot o. 390.

, C = B , l ) = A l

6.2 I(TITH?IA OETIKIT' OPDIMOTHTA'

Yftdqi(€r nesypaTr{i }.uon r1E z2 syz gz2 4ca - 62, - 8!z =1i

6.2-16 'EgIo &r o A €i.var oq4rsrela6s, es xd oqqrdros rar o I oeoo,tdv!%:

A q, o e",4Q €Lvar 6ny6uos nL\,drdg

A rp o 0",{Q €Lrar ouppslah{6s, o€flad oerqletos

A v o Q",4Q Exer fl! i6',Es !6rortpq pE roa -a

A V o e-^ eivar oqrp€iArx(, 8elrrd oaroldvoE

6,2.17 Edl, o,4 et'',(rl €slud oerqrdros r{ar auEl0€t ro o1r, 6€lg€ an6 mr! o1lpnaqd.yo}ae{(itl aljEd!€r ror I oQLEolJod. AeiEre p' dvd rrsed6€r,tpo 6fl rodto 6e! Io)0rer &av o/ €ilar aoetoros.l

6.L18 An6 r\\ A = RT R 6etEre lrd eerrrd oero}dvolls nivax€g 6fl der A < d nd22...a^n.(Y't.t'e4at To txixos alo i€iqd,yovo tl)E oir1l.1)5 j !o! a, €ivar @jj. xeryrpororiare6fl oellollo0 = 6/4os.)

6.2.19 (KArfl1Aro rolJ Liapunov, yrd rt! €ljoido€ra ror M)'Eotu iilr AM + MH A = -I ttE0€ rd ootoldro A. Edv Md = )c 6€tEr€ 6fl R. .r < 0, (yrdd6r6tr nouanlaor{iorsqr,:rq6q elioolq pe ca rcl c.)

+ ! r E Ahre unuqllliuol o. 190

OETIIi,\ OPE\IINOI IIINAKF!

6.3 I IL'WOPIIMEN-OI XAI AOPIITOI IIINAKET, ,4u =.\-ryIu

T(;a(.. nor a/ou$E (nob€igfl n).rltx'). r( ztn.ia(r yr(( avclv e€rrzdr ootoli\'oni!(lzc. o oroToq clllT'j( rrF noact(xiq o! fiv(L r' (.ni(roort,€ ru iiztr( $(! ii?o:rr f .nool oo. TorLi :Torrol .4r,r , , r i / . ' r1I z(. l r i \ 'o, \ :

( l ) rar ipr( yt(r 6vc! o€f lxr i ' } (o0rotraro nra(x( l .r . r) r t n7, ot t r f ru:u ro.ul .r , i+ r ,{ , 'nrtY' ;r fvoc or ' t r turarTou r iv,r i , .(3) ro y€v' l€r l€vo n066],r}rc 'broTltr i ) \ , , '1.r ' =, \11.r ' .

'l o ;rporro ocr tivar ;ro1,ri orilrolr(). bL(irr rt bo11l.fr(i aX!r i6t yiv€r. T(l xpLr)iar(ryLopLoLg6rlra: er Xcie{ir(tolw rr< ̂ /\,ilorr'< r1\,Lodrrlr!(.,7.,1.r > 0 z.lt ) > 0zftr d > 0 a(lr del -',0 !flruinovr(. rrtr 0t{l(ia(I tritbfaLz(i)v. Or (60roroa:ril(zr; urLtpinou' lniort< (ol'rtrrziq rbrcr(trlq r{cr obtyo!: z(1r $oq nir!€ oron11aro ounaa(otr( (1,rric rrl: nca.rlari(l or,: lh e6ot r( 7{'rv t6torttr(,t' r..rar{itol'vlt te rA6oryKt 7l'ov o6\/irv. A|roc fivr{r o rt)1ut< url,2uLtitt<" . Tiio< :r<illr orll1'.1.. - .\.11r. ro| rtrq (!ittrar oL,r'r.yri,< or :root,),ritr(lr( tUU(loioyrzriq clhilolq.E{l oool o .1/ ri\'(( o1,t4lt.r0r26. ztlr 0fr(x(i oarula!o:. ro ]r.\17.€q.F!o rao6inr(r{i{,r ,.(orii.i.Ii.( trr ro yvrrr6 -1.r' = ,\.r' rrr t!t(ptuorlt orr r( r6robKvlo}r{rrc.iv(tr oQ{lo)(;)\'r( r,:16 tri( vi( rryvor(r. :. xoii.ai rqtrp|o16: o -1,1 rir(r ir(crnazq fueq. : (r'ol ll irra(hlrn(to !"irrr4'f i!( rr:rz(; :r(lxilfl1tr( zcrorrtl ll(odTaaqo 6.5 :r(aottjr(liortrr rrl trr:frodo tur, :tt:tttltrntilun,nror/_(itor.

| , , ( r l tuu,)ru(/ ! , , , : . r , ! , , ,ar : . ro or l txorrzi ' r , t , , ' r , r , t r l r rr , , 'oorrr t rr : , rr ,o.o7ir :pc rq 0trrzri oqLrrpirq :rrpi:rrr xrrl.

6E Kcio€ 6vo rlro r(t rfi6p€r( xQLrri0r( anorsL€i. crvcryaaic xcr rx(lrt]dulU4zt l ro vd r iv ' { o A.qErtxt i t l / .xoQtott i rog.

( l ' ) rTAr > 0lrc xhOe 6rduro[a r (nr6< r ivcrL o oprop6g).(U') O1,€< or r6rc Fiq ro! ,4 r,{ovonorow qv .\i : 0.

(Iu') ol,oL or x{pLoL uronilax€q axow [I cevrFL}{6q opiEoro€<.(lv') A€! rlnripxorv (lpvrlrrroi o611oi.(V') Yr(iAxer na1,(,{(( /1. €\rbsXolavo< pr uQtrQTl}rdvq orril.€q. droL dror€

Or.!! o --1 t/n rcql r'. rorc rt ./ =.ri .{.r rircr d0Qorotr(t r r6),€()a r€re.rydrr,r\'.

I { oTioq l l t rulr ! tov.r / l . f : ( l zr l t ) , :0;ror t i lcL r t nLo or rnrrrzr l . r i \ 'arI i6rct trt try ;roorllorir€rrl: H b(yov(rtroilorl -l = QAQI o6rtet orttv

. ,r A.t = .t l ()tQt ., = tt7.,r 'r = ,\ ,yi + + t, ,r i . ( l)

' :.rE.: \lr! rtroortut(n,' ...rLJ{l

6.1 llMloPl:MENoI KAI AOPITTOI lllNAK!!. Ar = ).11r

A!flj €ivor trl cQvrFrrii, 61{1v ro ) Eival [I dQv1rrxd. 'Orav o / axe! id[qr. unaQXolJv i p1 p16*ln6g lorcru€E raaL r rd].€Lo r€rQ{il@aq. 'Ooo

1c rqvoQilor.ros. qrrri eivnr,To y,r6pevo rorv ,\, lcL eivcr miory p.q coqrlffi. Ks0dg€riio$ oL zuQror unoriil,<rxt:E tivcr rpLoqropivor. ot rbr"or$eg rorq xcr or opi(ouotgror,5 eivaL er crxeg > 0. 'Eror ouFn€odlaqi{, qv III'. ( EvaE zJpro( rrfionircxcEonUurilerar ;rrlpol'ei:rovrc_( milcg xar laq$i( ratn6lp'ovn -),.X. m{),tE xrulpappiE I zat 4. Avr6 blaqQei q ouFB€1Qio 1r(tt rnv 4{opro4r6rrF( Or(lv."r Ar' 2 0 yro zrioe :r. r6te q !fu6rll1(r clrln €EorolouoEi 1](r roli€! x(! or.1vI nQ6rq x(lt r61(lerl ouw€roll€vrl rou:r trr|6€viqor.rar.) H ).crlvorolriq €iva!6r! n III' €qaQl6eero! os oLous rou! ?.uQrors ufiofiivqz€E xol olr p6vo o'(r,roiE lrlE d\ro aQrorEQdE yor1,iag. A1' 6€v ouv66orv€ xqrr rdroro, b€v 6a

[noQouoaF€ !a 6!c1,ipivor4r€ 6!J0 ,[ivo,i€E T(Dv oroicDv o],eE ol cvo cEorcprioQi.qorooE [I6€viEo.vrar:

l r ) u l fo o l| 0 I |

.tutt orttrct tlurootolfl'o< xul lu t ]

,,uo' o€Nnlttr(r lltoct.rF.vog

T( n(lQ(bEiylcficr aurd .p(v€Qri)vorw orL FrloQ€[ vo xQ€raoooriv €vs],l.ry6<yQ([tl6v. :rl]v idluI€pl n€Qlnrooil, ot,rQdr€( ] dvo sQrsr€Qd oQi[ouo€Etiv(t yq pibrvrzdg -o uzorirux(g r rni r orl l(,)vicr €iv([ 0€rrxir oQrdFfvq-xut txoqre r e€rlxorE o6rryouq, I(oQiq tvulLq€E y(|<rtrl6v. Or r€L0urdsq,, I yao$€E Ei.\,o! [16€ra1(ds. onu]f ouvl0(l)E ou[6siv€! F€rd rlv .rn(.].or(Pl.:rrtr 6Xr zur rooo xcdlt n€Qinl(l)ot, )(a€(r(qKor€ €vo).}'q'fas yQ.rtrtrdrv xcLcvri.ororx€E €vdJ,qy6q fiI}.dv. dorE v& 6Lclrlorlooott€ rl ou$r€rei(l. O rivoxasTi.v€rcr r6re PAPT. yc r 0€r!).ois o6lyo E. !q leq6repq repbworol 1 nupr(!6r.<rlirvLoq rivu! 1epdq pe prlbevrxri xrrt rc pl6evLari curd eprgavilovtco acr m1vP,4,Pr' -r6re 6ev u:rtiqlow oblyot. 'Evag rdrorog:rivaxcg 69o4 bev pnoqei vaehcL qrLoqLopi:vog Xcupig vc eiraL tcur6lqord $t66v!r{6E xcr iror, o' ouni rqvn€ef rft 0)orl tel,€Ido(l[€.

!rov R o6rlyouprcorr Ur nq ibrrq xrrrrroxrldg. d:rug xcL nporlyoupdvog.o:rotu6rlrorr roro tLg ' rLloyr'g A . (vEr\vELt ) xuL A = \Qtti)tt^Qt tcxol4 xar A = 1Qv{Qr)(Qv6Q7 t nuqdyrr r4v A = /?t B. 1:r4v rpi14 r:rLloy4o .1? eilrrL oupperptdE ',raL ),iyercL 4yutprt4ulry vqayon rxti pila rcr A.) TiLogsJrrorQdqor[€ cno trN V' orry l'. 616t| xdre€ riv(|xoE fllg poprprig /?TR ti,,<rroErt i qBroQlopdvos (rorLdxrorov). H r€rQ(l.y(,)vflri troQqri tTf?,R} LoouraL pr

ll-R.rllr x(l! 6€v Fnoa€i va eiv(L cpvltrxrl, rgciwc nou ol.oxlrptirver. rov xuxl.oTC)v ClrooeiE€ov.

NAPAAElrMAf , - r - r ' l-

_iA- l - t 2L I | 2 )

rivnL +rLopropivog. ).6ro rov r:roSivov xpttqpiov:

tr8 OETIKA OPDMENOI IIJNA(F:

(l') xr.4r = (x r rr)r + (r1 r:): + (lr,r rr): > 0.(ll/) OL rbror+r€_c €ivo' ,\r = 0, ), = l: = 3.

(lll/) OL xupror rnonivcrz€q lxow opiloroc 2, 6r(v ri.vct 1 €ni. 1, oaiqoroa 3orcrv €ivor 2 €nt 2 r'icr det ,4 = 0.

(lv')

f2 o o l- lo I o l .

Lo 0 o l

.L l [2 o o1l- o I l

z ) L0 . r ;ljrI

21

,4 - atT,4(: .7rc rcirorov pq r.6L6popqo C (.2)

nuoaflioqott Or oltloix€g ilroeroqro ti(g 0o UnoQotioqv e:rio1q vc cnober-Xoouv a,_r6 rrg crexLx6< olvoizeq I ' v, [s ro €rol€vo r6lvaoF(r: noooo€or€ivrl urxao nol,l,orl,(ioro ro! i(1,rorrxori, or6rE ua rda€r€ rov e€rlra oQtotltloriilcr( / + €/. A{{,rior€ rcroftv ro € vc-reiveL oto pqblr. Efi€lbli o! oQltouoegx(L or r6rorr!6,c eivaL ouvetsi: otlvclorrio€rE tou €. 0c t[!c! oe,txeE [6XQI rrlvrr) . fur,o, ,orrynt orvrnu,c yr l / . = 0. o ni \crr ' , : nefnfr v, t r iv,u +I , r t tvryt ,ro:

M€rqololFarloFo( rco6lJvsltiog ?,tqr o vriFoE rrlg a6Qdl8taE

Nopitepc o tru6 ro 6L6l.io. otlv ancr),orqrj rtrL rL6 Lbr.or4r!5, boocpcipq,(ol cnrq oror,1er.<i6rL< nQilEELs fiou '|ri\,ow 6!c! nilcr( onl.o1loirao. :€?Li0s ncQirroort. ro olpam116 rlrcv rcr lvorpiloupe :roLtg !6L6qte! 10! nivolc:rcrqc[6row ovc].loiorsq. 'ora! 6va no]J,(;i],i1oro trLq yQqrtutE aqarQ€i.ro crr6prc dL),1, o rcrriloloE rorv lrc),loLtltov ' trcv rexetti tr$/tll.o(: O Xd)Ao<lpt4rpdrv, o xrlpo,< mlLd!. q xarEJt zcL I oQilouoc rcq4rrvuv 6l.c-ra i6rc. lrlv:rrpi;rrorr'rl ror r.6LotLprirv, r1 ftrolxri iedEt itrcrv o prrcol4rarr"op6g olror"6rqtcrg'{ - S r/S (ii ,{ - ,11 r,.1,41). Aur6 nou ,roat4r€vrr c!c},l"oi.{Dro Ei!c! o!'brorr l rr r fzdr r t fop,r I Ior, l ,n). Tr l ! r6ru I vJ'r tot 7 ' lv.ryr r ' i '0, , T, ' r . tupfrpr l :ou:r.ivut<ts: IToLtq ti),aL or aroLy.tulfitrq npriitr< ttur ot at'a .oiuis tol1; yta lt,.t:T A:'::

H GaoLxi naciEl o€ tr.t rEraa]t|lvLxl Foa(|,li t:iv(L q cl.)..ryri U€rrl6Lnrdr,. 'Evtl

vi:o bLcivlolc s oxeri.lrrcr pe ro.r pdoo, evoE p4 Lbto$6arpor rivaxa. r = C!, xar

1 rupcrlotrl pogpl livercL UrCr ACy. :uvErdq, I raaEl perclu :rtvirxov,no! ri.\,clt 0€p€Lrdr6nE 1|(r rL: r€rQa^/o!r.z6q [oaqfq, €i'vclt o l€r&onttarLolro(

H ou|llEraic ro| ,4 6r.ct4peircrr, br6rL o C"iC fioQa[dv€r ortltlErQrxo!. TorcuTtrLrT."o , aur4Fa-rir6),r1re_( xc-rd ror pcrcroTlpcrtrop6 roo6!vc$i.r!: H andvqot 8i8€raL ano rovNdIo (!60avria6 ror Sylvester.

6.1 HI I IOPI : \ t t \ ( ) t t i : t AoPETOI n INAI iE : . . - l . f = )11 . f

6F o ,ri\.d1ad; CT,4C 6x€! ro iblo r[),{ooE 0er!a6! 6rorr$d!. ro i5ronini0oi ((Qrtrrxrirv rbrorrFdv zcu 10 iblo n).ioo! FI6€vD.(i)v r6roqr6\' F€ ro!A .

Me ti),i( it!L(. u:rp6o4ycr rf/v rbtorqrdrr (rt1r o/r or i.6Lf< or rbrorLtr6()6r(rrluoulr(r zcrrdr rou< gercollpcr(ot(or< roobuvqriq. :rtv un6brrErt 0cu,-ro0aoortrr. 1c ruxoiicr. orr o ri riv( p1 rbL6gopqo;. Tor€ o ('7..1C r-iv(r€:riolF lul rbrotro(xjo! z.rL 6€\'r:r(irATouv tuDrlrzi! rbrorrl€! v(l trc: (lrtoryorrv.( Ar ((t oorrrzch Fnoool'lrr v( bor').€rl'ortr r trr rorS tol r ololxl or': .,{ + r / ztfl .{ e1z(n !(1 0lr,arioot,lE on ro t' - o.)

O{1 b(1not}outr€ drft r67r(opa r1c ro;ro),o1iag zclr crapr66rrrrlxr r1q oyo'ro;ri(q. Yno(liorf orL o C' olvbiercr yrt ivrrv og0o1;Jvro ri'ttrit<t (,) gioo lrcr<oulelolc {1)"Lroil'uq rrvdz(rv 4'(/). nou r(vi./rrq tor;q b€! Ei.\,(! lbtopopqoqr fL(| = t ) r a t ' = l . ( ' ( 0 ) = C z ( ( a ' ( l ) = ( / . ' f o r r o L r 6 ( ) f l t r a q r o r a ' ( l ) ' - { a ' ( i ) f t rt(lr(l6iiilovr(lr 6(UtrL(ift, x(10r;rq ro / trfro6(i),1€rcr (rno 0 or. l, (;16 rLc rbrorqLi:<ror, ('1 .1(' orr< LbrorLltc rorl Qi .,l(,). E;rflbrt o a'(t) b€! €iv(L noti rbrrlropq,o<.zrrulr i r t r t : rr i r r< t \ rorrr i ; atr i< i tL u:rogi Lrt rryyi i r r ro tr4dr 'v (xoi i , i r fuLooi)-rrpo bt. vtt ro 6LuoTioer!). :r'vt-tr(i).. ro ri,rjOor to;v rbrot4trirr br.!rri tor' llbrro;z(r ro:T).r j0o: r tprorcpri ror, . t i rur ro i t \ ro yLl rov C/- l ( ' zr l 7r(( ror Q/. .1Q.O il|]{\rd: (rro: rivar uriorl: o i6ro: zur tr( ro! -{.:ror, dlfl rri i6rri (zeLo(i)<rbrotrlri: yr. ror i o(, ;rnrat t) | ,t{l =Qr,!Q. Ar,rri Eiv(r rt (:y66nE t.*

nAPAAtrf TMA I Erir .1 = /. 16rr a'7 -la' = ('t (' v,tr.t o ni |'(x(. (lrlro< r.iv(l!ijrrrx.i orr(rlrivo< n(iQr€ a' = il (lrrl (rr\,0rlrrl V. Toc'o o r(urorrz6r. (!(ro xr( o(" C'6fom , (l|Tr).ri< rbrorqraq, L'nL6rti(L(;)vovrrl( rov vofo (bpovri.1q.

n A P A A T l l M A : n , , ' . r f , r , " , 1 ' o ' r ' , ' l ' l r , / ' . , r r r t r z ' l t ' L , r : o , d , ,I r l

d ! . r a ' l - l C = ( d . r a ' r ) ( d c r - 1 ) ( d . r a ) = ( d c r a ' ) ' < 0 .

Eir€r6rj (rrrj rl oaiiol,ocr €ivcL ro trvo|fv(, ro\, r6torLLlt,)r'. o C/-lC :r1x!:rer v(6trr tucr oErrrrt zoL tll( (lpu]rrzrt rbronfot.6;r.)( r(r o l. r,ncroi,ovr({: ;r(iiL oro!ropo ubqrrvei t( .

nAPAAEIIMA I lo nrptl5er.1;tft a|tri rircL ro flo €vbLa.t6aol:

" E , tiyr ror /,-r.L.i,f'tzr 't oa0olrn1(jrqt,( i,r ? utrr z(i)t uri.oi,rt (if Q ri!,n !n

Eq(arooor r t l t , \ ! , ' \ ' r ,n r i , ,GrJnr Schnr id r o r l : , J r i i r :1 , r 'a l r j rFa Q/ lz ( r , td . rdL& 1 ,v . i2 i r !

iw ! ( i l ) lQ- r l r )41r l H o 'zo : , i r . r ( ( l i r t r r r , r tn i , \n aor . i . t rnd l r : Cr {n Sc [nndr . a : . r . r

Q/ ln r i rQ Arn t , , ! . , nnd i r r i ' r r | , t . n 'on oQ r i !u , ( ! r rd ro . \ r ro .7n , . r ! . , r ' r ' r , ! . r I - t l l ) ,4

.100 EE'I'IKA OPI:MENOI IIINAKE:

6C na rdes oq4rfiQrx6 ttiraxs, ts EQ6aIra lar obtffriN otluimowItE ia ,t!6o4pa tne l,.lorrrlttiu. O nllaxaE !6|otqr6! 1tc[ o r ,1/qrcErolv o64yov ,, 6xow ro i6lo rlti€oE etrlrtd)v, oQvqtt6v ra! prD6vrx6vorollsf(,)v.

O(l uno06oouFr 6rL o ,4 €rL66x€asr q ou'rlE nspulowoaoilo4 ,4 = IDU (x4i6€vdlcydq yaoFFdv). Emr6rj o,4 €ivor oopt€rQrr66, o U €ivqr o ovdorQovoExor L v,sL A = LDL|. 2'ormi! rqroQt{6lowe ro v6[o qg cOqcveiagr o A€XEl ro i.6ro rl.{0oE 0€xrzrirv L6rorqrdw pe tov D. 'OFI,E or r,6lotlpis rou D eiv<rrazpr6<16 rc 6rcyrixrc oror.leicr (or o6q1oi).'Ero!, ro n)'Ieoq t(,)v 0erLz6v odny6!rcrrprriler ge ro ri,r10og tr,rv 0€rurtov r6!or(F6v rou /.

Arn6 rivru r(.L r,rpcrio xcu np(xrLx6. Apcio eivar bL6I! ovv6€fl (ytc outt-trrrQrror.Js ni\,cx€E) 6io [aQl ouroi rou 6r6],iou :rou rporryoq6v,)E xe(louoqr€Xr,tEora: oitlyotiE ltol tborryl<. flquxrLz6 etrct brrtrl ot o641oi 6Qi.ot.olrureuxoi.c -xrrr td;po pnopoiv vc €wofiioolv rt! !6IorL!6E. O€oQnors tou( nivaft€E

f3 3 oll 3 l0 7 l r ( !L0 7 8 l

f1 3oll3 8 i l .10761

O / €I€! €€irl(oris o6r1yor,5. o ,4 2I |XEL 6"to.t (1a1rl1[1{6 obqy6. :wetr6E.o ,4 dleL 0errrdE rbrotrpdE xcr 4 tia an' ar#g th'ar" lrxQrjttQn ro 2 -ELoxlcrqcrprivrcg 2 rq! r(rF6doc!€ x(tro (n6 ro [y i6v. To €nfus\'o 6t1gc eEerirlerrov A - 1, 1Lo vu 6€! av n ldrorr[ri eivcrL prap6tep4 tou 1. (Ko! ei.vtrL. 6L6rL o,'l / 6xEr aQvnrlxo o6lf6.) EErrilowcs rc rp6o4rn. €lc11(,)vouttt xcrdr ro'it'rou ro dliorrttr'r ('br6croqrug or xd0' 6{F(.

Aud irav I :rqdrl rqaxrLxrl F60o6oE ufio),oylopou riLorlFrov xor eupLaq-loroe 1ipo.r ora 1960, F€id cn6 [r(r tsv6laqdQo]Joo 6ci.xioor] -vq yfv€L nedrao,,1 raririay{i,lro(. Or oblyor rorr rJno}.oyiqovrur oe 2/' 6ylF(ra uvrL yLc rr'/6.H rr:ra).olqri yivercr 1p{1opr1 nru 1 ala(riqol !6tottpdrv ontroufiEtlEro!. AurteivcrL pomrj ot\ lleoboq Cilens. H tQ€Xouoq nQoritrlot tivnr 11 p60o6oE QR;rou 0u reElprirpor.4rt mo Krrgir),cl|o 7.

To np661,qla t(')v yslrx€rJF6vov l6rorr.Fov

An rigu t6oo oiyoupoE 1!( rlv Orxol,oFia. .d!trd I Ouorxrt, I Mnxalo).oyiaxru q :r(rrmLxn ilow ouvtt0ug tqv Euyil'rto vo ncQdlouv oWUrr0|rcuS nivuzrqora :rpo6).ipatcr r6torqr6v rorE. Mroqji d|llDq va xpoxinaow 6io trivaze< avriyn ivav r.at 1 ,4r = ,\n v' nvltlrqoioraroL ar6 qv Ar = ,\?11:r. Eletcilot4re,oav ncQdSEr./pa, qv xiwpq 6rio cvioov pclriv.

63 IIIIIOPDMENOI KAI AOPDTOI nINAKE>.,4i? =li/J]

K(lr or 61() r6rorrFsE €i,' r aa\rlrrza!, €vdi or btlo o1,tv6rlr€sTo !6!o6!c!1ropoxc ..i !fio),oyi.Eo1.r.1r. [E rov ouvrlel ra6no:

l r t 1lr , 1 ^ _111 . . ) ' i . r , { r .t 1 \,/31

lII. 6.3. Ev( rolovtElot{€vo olo tFo Ftr cittotq pd(rq.

O vopoq ro! Newton aiv(t Fr = mr or ./ro rt',' riAd)q [dlo xat F: = tn:.4..,rGq berrepl. Mnoporlpe vcr reptlpirlror.rpe ro rpror,x6 nQ66),Ip( pt pic rp6r<ro1:or Fdteg fle-raronilovtoL xcrrcr r, rior ur. onco; oro 2x. 6.3 k<rr r(l €),arrloL(rtlq (lJeou, rQoq Ic fiio(J) $c brrvaprl F| = 2r; + u; x./J. lz - r' 2L'. To v6oorqpuvrLn6 ororlrio :rpoirplt:rur a:t6 r1v cprorrpti ;tl,erpri rorv Eqloci)oEco!, 6no!cpfclilovrar oL 6r;o yit&q:

rr i , - - i = -2t + rd l - ; 111

d 2 t , L 0 D r 2

Ar or pci!$ rlr(lv ioeE. nrj = lrr. = l. .rlrro 0a {xa! ro 'Io}.ro oloryl[o 1r" = ,4r]..TrfQe ei\rar ll tL" = Au. pt i\u\ ":tilard laEuE' ll.f. To nQ66),![c t6rol[d',':rqoxnnter t6],q crvchtriooqr€ or0r-ruia; liJo€rg €''rll

, i11u" = , lL yivEr(rr i I ( )o) '1t ' ' t r = A(i"r t : . (3)

A:d.o:roLcbvraE to c''l.r'xar. yphrpovrcg ) cvri 1a (iu)2, alro yir,€lot

' l . .rf ' : ul, ,4).4 r - \ .1 / r . I l r ' : u ,1 lY:rcipleL l.uorl g6vov 6-rcrv o ow6ucrop6< ,4 Arl-a ri.vor !6ropoQ(poE ni.voxaE. .iQaro ,\ eiva pila qg rro),trt,rltlprznE eEiooronE det (,4 ),11) = 0. H Erblrrt €nrloyri.{/ = / [q o6ryri. rio(, ort ou./rtol €EiorDoI det (,4 )1) = 0. E:reErpyc(6pcoredvc :rnpd6er.1pc pe ?]r1 =7 v rry =2:

d'11 | -2 1. l t ] | - t

€ivor ,r, = v/ \ .

lt,']'

OETIKA OPDI\f ENOI NINAKE:

| r . , / i l

r .4 \ . . r / ) r=l-- I " . , , . "

' Y 'I t t * , / : l L I l

Tc rbrobLaviotrara atl[d Sivouv tq zcvovudq xcrrtrorcioer"q qq rc]dvroo1q.:rI zatrtior€al oljlvorl.ro uL or 6no trarq€E ralcllT€uovrnr I'aqi €rroq or'ro6rr I nqo;fl1 ptrlu xrrri.rcr p6vo g€1pr".r1 p€1oq rrn6orcrorl r,6 t = O.r:.:rI! rtl/Ur€ait zorirorooit or ouvrord)o€E iou .f txoll! crliie€rcr npooi}rc r.rorarrouvrcrL oe (vrie€r€q lLFrexvocrE. ALml q qopir 1 gLzporepq girln qOdvrr

H cwioroLll 0ecupicr nioor crn6 cur6, r!ryeitcr erro),6tepo cv brconiroorpetov ,'11 or l?r,Q. (O )I woriAerut d,rL elyu eerLxd oproiriloq. oiros moficpdbFrytru.) Torr rl ovlu{or(1or(oq l/ = .Rr U€r(tial,it|, ro

,1.r = ),l.Lr' = )?r'?.r' oro AR 1x = \Rt x.

:!fr(jo qoar(< lrt C rov R_ xttl fio),].crJ,eoldqowcq $r' (RT) I = (-'7. or.rro7iv.r , . rv,r , 'u\ ' t r r . ' t ,cvr ' : r r- ' r , i ) . r Iur rrrorr tr , ,v ]r( r ! , ( ! x".C'| AC:

C t A C y = \ y . (s)lll; (QXlxIE .{.rOL Lbnr4rd< ei.vor o! ii5!c< l.s rI; (QXlxrlE .4.r = 117., x(! xc lbrobr(\rlopcr(1

rrvoru\rdr l r r r lv r / , = nr, or ror i r t l r r 'c Tou or 'Lr.LTg'2ou:rrrcau c7 lcLDmr4ra<

" Flcq r!/ir.!,). raii!! a( x(ariToLlrf irc! rur{,'lorrivr nir(n( 0( rt(r y( nlaorrrE np1l -1u = 1.. dli.i o,1/ L,1 l.! Fila oq,t'.i,tL2d:. FidL f,l].i'r'1i.a,'. !( a,irarrr. r. t(o,! rof

.11 i (ro 2.i0. 1i.ro.i r.! ,1. (;dr. rn i/oqr r.v aqllnorzd a-714

obryonv tr:r !!0€|(c or.S (vri.oror1e< rbr6irlruq rl< /.r =,\,\/.f:

l. OL lbLol+rds sil,Gr nQcypcrlrdg.2. K.rrh rov v6[o .rbenreicrq. 61ow ra ibLa re6or]trd tr€ rrq ou,'rto€r< rbrorr[6(

3. Tn rbLo6Lcvropcrrcr !J, ror CTAC eiv(L 6lvcrrov vc t:rr"lelouv opOozcrvo-vrzri, dpc rcr r6r.o6rcrvropcrcr .r; ri.vur" ",\/ op0ozuvovrrti ':

t ! l l t , = t ! R t R . t . , = u l ! ) = I 6 1 . u \ i = 1 . 0 6 x c ! i l j . ( 6 )

4. Kor' .r\rd.oyi.r\,, rl Ax i = ̂ j.tT Xlx iri.1,(r. rt )r 't 0. OL rir(t1L€.E A rcr r1/bttr.,,tu vt o:ro ro it tttr rttt ypot,n. Ehv o S 6xer tc rr orrq orrt),€q rori. ror€t- . l .s n l lur 5- 1/ c / . : lLf .u 'orr or uurL' : f rvr t l rLloxqr{rror.dcL&6tn<qkq, pr: tov 5" m cQror€Q(i ticr oxr p€rcoxlporro1ro( olLoLor1r(

fEo4LErarzd. omo 6trr 6r,.1 .,'oitF.1 nor 6sv ro rar.1).o6lli!or,[e no),i r.1)d.Irrl Oetrzh ogropdvrl :rogi:rtroo4, oL a:rrpirvrLeq rr',,12 = I xcL.r'7r\-l.r = l sivcLEiiELl,oEr54. Ei\'.rL rp(leQo orL .I = S: oQi!€L Fr.(1 v6o €nLl,oyil orNr0rcrl[dv(D!

6.3 HMIOPDMENOI KAI AOPDIOI nb-AKE>. Af =,\il/r

6€v €iaar xa0aQri oxQorlrl, 6L6rr o 5 6er rivcL oQeoydvrog nivoraE €ror 60r€ro 6lJ0 sl,l.€11'o€r3yl ro e!0qecppilovroL ooord. Auci eivcL

..r Ac = z'1 5't AS z = ̂ 1zl +... + )"zl, =t

. , r t l . , z t s t i t s z . i - - . i = l . " '

To b€ln€po €II€iVo€16€E €ivol !!o ogaieo! Euxo].o l,otr6v ouvorlii!€ro! xo ,iuQloowireAsoFcl rIE o€rDQicst Ee' ooov o .{1 sivo! oerr,zti opto}6vog, ro yevr.nelpdvorp66),qpc Lbr"orqrdrv ,4r =,\,;l-fo oqrruprq6petor 6na;g co4r6ri4-ro :{r = )e.

AXKTI'EIT

63.r redlrs ro r?,4r odv dgQoLo|d nno reroaydlorv raL ro r:"At oor dvn retprilarvo,1l.1 ror< l[roerolttvotE nivoz€<

-1 ' l r r r r l- l | { r uE ' l s2 ) xu , ,B= l l I l l , r dE$ l r2 ) L r 111

6.3.2 E(i,aq[oore rs reio .lam]aro oe 1.(i8F i.',a! ard rous:rivsrrs

t(11 ffio{|a!0r:i.rE cv €i.!d! 0rfixii oerdptlor e€nxd lF'oA|opavor I d6ermoi.*

6..1.r r,n ,o,< . = l ;

"r ] *, , =

l i ; ] r 'd,rernotr ol o, ,b,oluis rov .r 4(x(1r ,4 6to!! rc i6n treooruo. Koraot€l,dme plo o).rioi6a Fr '6roFoeqlv ftrviirovC(r). I onofo !a 01]!668r rov C ps 6!dv oe0oy6vLo O. l'r.rxi fivaL a6nwov vdxcraoxdurioolrE rri.d Bt r56poa.n olroiba ow6iouoa ro! C !E rov idrso a6

6.3.4 Ed! or o6ryoi si.vd' 6).0r F€yalii.eo! ro! 1, eivor o! L6rorqtls Fryal.uEeEs lo! liAoa+tddr. pr xouc xqbrd.fdrvrouc 1,2, I ni.!da€q.

6.3.s xqryqorooiot€ iolr obryol,\ ro! ,4 11 yrc v'cnoqav0eite cr o A dl€r (bLor$rl

pupdrepn tor, l:

. t - . ^ 'L - r - i

f r r r l 12 r 2 l/= l l l l ro r a= l r r r ,

L r r0 lL2 t2 )

r l 2 ,s 3 0 lA 1 = 1 3 9 . 5 7t Lo i t .s )

* :.r.E.i BUn. uodtu.ioq o.390

OETIKA OPI:NIENOI nIr"?\Kf >

6.3.6 MLd d.yc1aLxi atu'€tEi ro11 rtitlor d'a@€iaq Euztld p€ oqooxorouad 6'a!nofcro /r.....J,p ror A, aou dvflmorxouv oE rirotrtdE.\l > 0 aar oqooaclovrxd

r6rodraauo[crd vr,. . . , rq rou C",lC ]Io1, amrmoLlolv os r3rotry6q pi < 0.a) f rd vo tu iEtr rE or t to . t r , . . . , i tp , C!1, . . . ,CtJq x ivor cvtgdpt l ro, r i t rooaol f . o IL

xdnoto! owbricopoq dia to pl66v:

a t J \ + + a p x p = b t c t ! \ + + b q c 1 t . t c : 1 i . / " )

AEiETE orr .r,4: = ,lroi + + .r,,,i > 0 aaodq €trioiq x':LL '['rL ,r 't, =

p lb l+ -+y , , t ' l , 3o .6) >tur.edvns oir rc a rcr ta i, rilcr plbdv (nadlF( ro! dEobetxrrjlr rtl

'racp[rxri arEgdpqold). :1jpreadvm€ oti cu6 otr p + q < ".

t ) o ib(os o l l l lo1to!6E t$Tin - | aavr l r rTr i ) tdr to r -q 0el |7d / r , b i ' fu| - p + | - tj S r. (YnoearolFE rdlt pn u6dvrre( r6rorqti,<. no! prroaorr! !dbor).rronv x(da'ord). a€iEre 6ir /, + q = r, daa ro nl'ieoi- t, to)! 6€rrxd! )roorr( pe ror nltt0og r - 4 ro! etrLati)! /r, ro! eilcr o !olo( ooesleios.

6.3.7 E.iv o a,' Eirat Ft 6tdBoAqor. betEre 6rr o 'a tat o CI AC 6low 1I! i6to rclEt.(Mnoaeite vs xe)touonooii'rrr rr! n(adyaago 3.6.) 'Etor, txo]Ja ro ibrc rllteospt6e!'xd)! 15'olrpdr!.

6,3.8 Begire nseup(rLa.i ro rhieos low {lettxd!, cqutT|,td! aaL tlbFadir'', Lbrcrq(i)! tou

^ _ l r^ B l"= l r l o l

orcv o urorircrtrg a (tdi€0'< a/2) ei!u! pI rbtoFoeq,oE.

6.3.9 Ix(lorTolo!1v nciuotr o' ,,1 xot C1 A(: ..c! !6!0 obaaleias, ordv o (r bw Filatrftea?oarao,ci

6,3,10 >ro ).!pivo rTdqir6cryFd ro! 6r6).io! r1€ 'rr = I not n2 = 2. enail0tnmc orroL KaovrkiE rctaotriocq eivai ,U-oAeol6vr.s: ri,\I:d, = 0 Edv xat ot bno

FdLrs oropaaauloolv li.1rd q pold6d cr6mco$ ac' [dtd arp€ootl! e].ene.a.(, t€,'0 = -1 roL r,o = l, 6grire rous ouw.trr'drlr ai ml! riMtot no! 0a neoa).totitr = drcosdL/rr + a2cos,2txl. IIoL( .i!dr I lsIa).urat r.&xrl u6('iaart nFra(irxfi [d(oE 0116 q eSot rdoeqorioE (orov ro 6t'o ol,u]ftlold €i'oL +1 zdt l):

6.3.11 Batir€ xrE r orpi! adL ia LSrcSnvuott(td ls

l-u, ti it,6.3.12 Eda or olrrp€xarroi 2 enf 2 nivcre5 .1 aar .11,1 €ivar aoqroror,* 1ox0 I ,b = ),\1r

pnoqci vc p1v 61u npclp.rrrr6E r8rci+#q. KdroozEldorE 6!d 'iaadb€rytd.

3 t )h I 1 8

* :.r.E.: deift rnooll|rrioOt o. 390.

6.4 T APXET EAAXIITAN KAI TO NHAIKO TOY RAYLEIGH

! ult1 rry :rcrprilpcqo. xcrerirg n),rordEcL It: rdioq rot' 6t6iior. Ocr Eeqiloupt1La:rprinl rpopir cra6 rr: voatLtrxi< t-ELoriroeLs. O irlvootoq:r b€v Otr 6i6troL otrrinort ror Ax = b 1

-roLr ,a.r = ,|r. al.id ec npoobroaii€-r(1l. (lJro x(inoL( cotri

I r ! / , r , tn) jy. t .y.n ro nono no;.r .oi r{r . rr r1: , luotr4t rx,rC.t :^!rar , 'ccrpT(: t-),u7iofurr,. To "/syov6; ),.x. orr ru 6crqrd ryprl zat(ziroovr.lr orov 6reo;rQo,(inr€L (17r' rrlv sL(XLoro:roi1o1 qg bw(prxllf rol]q sa€p]rr(rq. Orav rde€ottoe urcr xcrpdniu rl E<rrlr;mra o' 6!c ra€66rilr ic ticrriQrcr rcirrotrororvror arolrborr lu t),cr;lorororsittL 4 ev6pltrci tou. tiloua(( !rdl)(oLrv xar rro no)'inLorarcrpclti1yutu: 'Er( zriaro ttoL(iE€r tt€ ltLcr :ro)'l;rtror1 xcpdxitr:rot oryrtbvet toic\Lo ro iiripoc rq< x(L rl g€trEi(rlllq (laXit rI( 6o$Ixilc Mlx(vtxtq €i.vor I €),(ltt'oronoitort rrlq olLriq .vaay€Kr(. :q Oro!i.{ !]Idozor,'v AoT.Qclltrc(vi:< z(ll' , ' i . , , r r . r1- ' iprrr ,r oC',o" c E\ ' . , .2|1,, '

" rv ' , norrtcr vrao ju.rr t . rr i . r?14t iv, ' .

, \ r 0 1 r 1 ' , ' t i i c ' r z o ) . u r 0 r , r , r o , ! , h u I r i ' , , o - r r . r q r ' i v r r o r o r . ' ( n , " ( , ( ! . Y i v , r , , l

Ilpi:rrr va:rori1*: ap6ot,r(, ott truraE ot tvipy'Lt(' 6Fv Fivor rinoro (iLlo coro0t1\t(i o?tottit+< toa(tl|(. Eniol( I fi(Qoyo./o< ttLqq r8rQ(1l{,)!tt{tlc lloo(|llE cilut

1(x4rtrrzri. Eror I !I{r}r('roxoirlot 0(1 ft(E obtyto€r nicrr) dTlg }v(i)oriE yQctl$rttrlqrELodrosq. orav prfrvioorpc rt( no(;)r{r; rr(rQoydlouq. O no{i)roq orozoE (1rrrl!16 :rtrptrlprirrlor r:i!(l va fQoirE Nlv .twi eiaxioto! nov ,'otit|v(ttti tto rttrAt = b xdritE,u txtiltt ltor too,rNatti ttt r4v .4.t = )r. Oct xcivoqLt m(nlnrQ((Jtrd,'r! br(ori1ofl< tllro untrLdrirg rou o LoyLotro; trsro6oid)v r(ir!!L o'6vcr owtli< :r1xiiiL4rrr, ti:ror o trlbElrotLoE ttov nprirrox n(e(y(ir./o)r, 5i.1,el tl|.lbr(rl.oprzrj rEiooxrrl(tr1 |rqio(l)ot E lcr). :t x(io t: xt:t i rro)ort Eitt.lorc a).rnoreot vtrarulrprlrrtqLr. sLrr LUoq r$ lqcqpLxriq eEi.ot0otq I dv(l ttrri2Loto rtlq terq(./(,)vrltttc

zuL or. roilir :rpo6),1pcrro, 6r(0( 0c q,(vt:i rrqv tn6pn tl :rcqri.Tprtqo. 4 c\tnrt:p16ucr6r1F( brv 0.r rp6reL vc crYloeirc1t.

ro iatoro 6)itLcr rilclr noil r-i,rolo: oil"o+ra la 6aoitr€ rlt rcp(d)o),ri' P,rlq onoi4 ro r'),(ixro-ro €nrruyld\,crat oro orycio o:rot, i.f = 1,. Eri! o I 0fvarlv(lg (r)r0BiE. (1rJr6 yi,'€r.1r €n?oLa:

6] A!X]J I].\AIIJI'ON KAI TO NHAIKO TOY RAYLIIGH

P(f) = +-irr 6x. dP' |QL , = t l . t h .

L&;r 0( iXoLrFr ciciTLoro, 1L6ror 6rcr o ,1 r:irrtL oerLro<. H noacl6ohi tlvoi/€Lnpo! r( ndvr,; z(L o trlbrrrop6; rlq :rprirtr1< nrrprqd;1or bi.rer ,1.r = b (:1. 6.1).

" Eitr(L nfxf drril.i Ljl ruil2dr .nda{Dnot (r(nriouo\r,r ditr(t1'rn !. oq7< .i.(troronoqqq.

Nbt/ltLi1r,! 61 7,d rl iara rl dlonn,Ef ro! noitrUtoi ( 'l(oi:*,n dr ildr !Lj$r ei.ittolE ro.iot(.

H !\(z,rrUl riro.n !o|[)r.iw' ro ortrt.lt;nr( nr[u yL( !( n.pa..!!. (f,o a(adrtafo(a ut

r !trt$ r: .tatr.!r.v.n. a(L 0( naintr v( ur(a/o8 a .iii.r (2atrl rotLot ei.(Turonoilorti. no1

i.a+t\.u \' onoz(irqllor!. nq dLlrlriiq.lorri!.1 2d mf BroioTi(.

OETIKA OPI:MENOI N'NAKE'

i l

l

' 2

P , , , - - r b t / 1 r h

{9) tuKdet 6!0 &ddrdcov

!x.6.4. To €).dj(rmo tuoE rcrAolovral< owdaqofi p(').

:e rr€Qloo6reQsg 6Ltror<ioeLq I ncras6ol,ri p€ra6d),1€rcr o€ rrqQo6oLoe!66E qlld'ytc rrlv P eloxoLor0ei v<r toXrier, o t6LoE rljjroE. Ev6./rc vct €E(logql.i.ooul1€r l(Xr(rro XQr'rdErrdr xul prc o!\'erl1{n U€T|xorrlrsc.

6H Ed! o 14 Ei,vor, oul4rerQlxdE ro! esrtx(i oQlop6vog nivorcs, rdr€ Iouvdaqol P(r) = (1/2)Er Ac - $Tb 'XEL sl'o:$loro cro orp€io oriou l, =,xar o' aur6 ro oflFEio Prt,t. = -(1/4b" A-tb.

Aftt6€4t Ynoo€or€ 6xL J] €i\rdr I Luo4 rol .4i = r. f|q xdoe 6LdvuoBo u,

P(a\ P(r:\=(112\utAu -yrt ' 1tlz1r'| At:+:,rb= (112).u't Ay - lr Ar + (rl2)rr Ar:= Ul2)(u - t)r A(.!t t:). (t)

Enrr6{ o ,4 eiv<t OrrLrti oeropdvoE, orlfl1 I noo6xrFa 6€v FnoQ€i nore vo rivctcpvrprnrj rcL eivtrr p166v p6vov 6rnv y r = 0. :' 6l.c r' dl.l.a ol$Eio n P(r)€iv(r $yal.tlr€Ql afi6 rlv P(z) zoL dxo! xo €L{ixLoro srrlrurltdv€ror. oro ,r.

IIAPAAEITMA E),nyr,oronoLriorr rn\r P(r) = ri 4x2 + .t:l bp1 - t4:r2. Ho!\,riolE npoo6yyrol ror 06paroE Be aTrEreoorrr6 lo1o116 livercr €Ero(i)vovlagrLE ricoa.ldyouq F€ Fn6€v:

AP/AI:1 =2t: t q U=00 P / 0 x 1 = a r 1 2 a , 1 . . = O

n, . \P \ - : L \ \ \ t : I ON l \ \ t tO n l l \ l t , r IUr RA\ l L ILH

H .lpcrpprzf ch),1tdpc 6rcrrr:rd;vrL ro ibLo orp;rrpaogtr !:16 q [Lrpqrl ,.l:r = b.

r , ( , - n : o \ r L r c n q J , r , ( r r l P r , , , ' 1 , r t : r . , / | 2 ,

, ' , 1 , . . ' , l , r o f ; r ' l r , , r o

r l rr" /- / ( ivtrr ! ont 06ot.r ' = -{ '1. c i : rol

f . , . . = t l : r i . { r r l . 1 { . 1 r / , r t . f l , r t l , - 1 ! , 1

. . 1 r b r : r

Irrq rqcrqgoyd< ro (l/2)rfr.4.f €i1,cr rl co0)rtQu{tl .,u6p1.,u *n, ro .r'TL, rivarro 0q(i)rsArzo iAyo. To q,uorx6 ororrpn :rrirr (luro$.iro)! riQoq ro ,r'. ro oqlrt:ioLooppo;riaq. o:ror, r1 oiLxrl €v60./€L( P [iv(l| t],(ixrml.

A€i'r€ao: oro/oa tr(a rir(\'v( 6A(]r,111 6v( npo6i|l1r( €).(Ioro:toilot( roo6rivr(llo Ir rqv -{.r' = .\/ Artr6 6€! Fiv(r x{ rooo rizoi.o. H ouvdernol ;rou ̂ pin|L\,(r €i,.r/roronool0fi br-! tr:roa€i vcl ri\'(L tu( (L.Iirl r€raq()vrzrl. bror' 6L((roas-rtrri 1 ntrpa}irlo1 irF tkr ollyorlo€ or ]$4rtrLxrt rlio(j)o1 zur ro npotiLrltr(rbrcflfr(;)1 !i1(11 r(r(i i;{ior! trl yActrtu,rlj. To rdTrcropcr rivur vr rEFr(iooutr€ ro!2.i,., bio rrr0.1./0)!rzd)! trooq;w x(iL (rroq nou tr(< rErnrtofrri {,i.,'crr./!(Jor6qtre.t r'l.ixov ro1' Rrlteish.

, . , r - . 1 . r 'l ( l r \

. r t . rl l fqv( l l l1 2{rr ' r fO€i( ! oro zuero or{; 'er[xr:

6l H qYi 'orr Rafleigtr: To nil),i.xo ti(r) sl,sIroronorrircl crio ro nQ(btor 6to6!dv!op( ,r = :r'r acr 11 eltilLotl rLprl tot eivar" I Fr.aQ6raprt Lbror4rn ,\ I :

n(..,,=':iil' = *rlr =.\,

fFoJl|Ta!1i(i. irliflt0ri.r€ or. ra(lriit'! r(,\, er0lr]rri ioo tr! I r(11 xri.,'olfrE ro.vri.raolo$r1orrt ooo trr-yuLlilrQo yi!rI(r|. O (laLoprFli; .r'Ar = l opi!tr ivrrrD.E(Uo€rbiq, rori(ilroro! or(v o -4 fiv(r 0rrrzd oero[!vo<. O :r(povop(or1Eeirar .r'r.r = .r r. ripa r1,dr7r,oupE .1p11 ro oi}r€io ror El.i.Lnl,olbori( ror' €i\,or onttu-,1orr l , r : r .n,r 'nDl ,r : ro nty rcyrt r" ^r i , \ ' r ,o d.. : rol .7f l ro l r i l 'oro rrr l roc.A;ro qr :r,-ror11oilrv1 nFpryacqri ro| r;.i.€r!'o€rboi\ yviDeiio||r orr o lrayrmoir!!ort! rot' dTcL rqr zcreiOrrol rou :rgirror Lbrobrcvuoparo:.

A;"T€iilnz.i. (1u6 livrrcrL tnxo),1 lrtr),1mo (7t.rpiq rlr a:rni.rryrt e€r'rl].oQro(Udtrf(g). Atcrytxro:rororpc rov .,1 pe 6rcr op0oltilvro nL'ra,r.a: Q' AQ =t.[(ialo\,r.,r .r = Qv.

lQ ! ) t , 1 (Qu)lQy ) t (Qy )

, rA , )q i+ + l , , r i. ( r l

! l ' u l l t + + ! l i ,

To ei.drlr,oro rqg -R unruylcivercL oiyoupn oto oltr€io 6irou ,l = \ *!r !!1 = . . =y, = 0. :' aur6 to otF€io ro rrll,ixo !oo!.tc! F€ At. xar o€ xdo€ d),lo orllreio xo;rlLiro eivar yelcLurepo:

)1 (e f + 7 j + . + s i l 5 1 )1y f + ) ,y ! + . .+ ) "s i ) .

To rnt).ixo rou Raylcigh 6€v 6I€L rLpri nori ttua nn6 rc \ xca. ttoti ndva o''n6ro ),,. To FI.iX'ord rou rivcr mo t&o6Ltivuqru.rr xur ro ti-ytor6 rou tivru rno.r,,. Tcl €vtudqr€o( rblob!(rvriogato eivcL oaypauxd oI$ia.

Mru rvorqdQouo( ouvi:rrLcr: Kti0r 6tuycirvto otorlrio. orors to nr'. reqLiletcL percEu r0r1' )r 'io! ,\,,. Ilptiyptctl, (1 6oxtftdooup€ 1o 6lirvtlopcl. , = { 1 . i t . . . . . 0 1 . r o m i ) i o r o u R n y l e i 8 h L o o r r r a L F r d .

A0t6E minimrx yra trE r6rotry6g

H booxo),icl !r ru od.r[aflr(i olp€i( €iv([ orr b€v 6XouF€ r6dc! ylcr ro.rv I./l(.]]) dQioirErcrl lrivo 4 r<irc'r ufi' aurir. Aur6 xqv€r b{oxo},yl rnv r:uesol to}vevbLr4rdoc,rv lbrorqrdv ,\r,..., ),, r. >rl,E sq.rQ[o/dE. ou!6 nou nQcr"/l.lcrttxa tivs!Xprloq.ro rirot xd:roLc nQlli peliolo! I r).dxi.o1ou. Avchroi[e ],our6v F[(l rdrolaap,Xri, anbLcirzovrcg vn dXoqre el<ilLoro t1 pfutoto oro j-oot6 t6Lo6rdvuofa xj.*H L6d(.-z}!€rbi orruilercrL oq 6aorxti Lbr6qtc nrx oqrp*purtirv ar.rrixtor,: to e;eivar xrieero orcr v:r6i'otno !6ro6ravriolrata.

6J To ?ldtroro rIE -F(x),6rcrv un6x€rrcr o€ r<futotov leEoqropd rrz =0, 6ev urep,6clvel rore ro 12.'Orqv to z €ivcl. to rQdro !6to6ldwoFd, t6t€orn6 ro ur[6 ol,v0|ixr] €lrtlroro tqg ri toorhor pe .11:

)" > min -R(c) r{dL }: = marlrin J?(.r)]. (4)

A!fli €ivar FLcr "(lQxrt n,rximin" 1L(1 ro )1 ncr 6iva dvc tpo:ro exriFrlot; ttgbeurrq6 16rorqnic. X(')(,ig vu yvorpiloupt r1v :rp,irr1.

nAPAAEITMA ArcryQcrq'r€ tqv ld,svrc(i,q lQaFFrl x(r|, mriln evog outltl€rQtxoi

OETIK OPI'MF,NOI NINAKE>

- l2

-1O A=

l . r l' y r v E r a L u = t ! 1 lL - t 2 l

* AN6 to eau( rirur o.:.L6La{l@o, rr(ir rl F.oo&x r.^ rcnsecotrov mor7..iov d.tai:€(L

drlf li(/rmonoitdrl rol P(r) "oL

,q(r) Mroesift /.oai< troToli( vr n.ir€ 7ar' r ).i(! .rry

;,1l '.L;'

Tort 4 turiq4 oe yiyt9o< Litor4nj rolt A nte6 l1,tt r4 pmpdrp4 brongtjro,, -/]. (: (116 to rcrpirbrrylLc or aar0loi amoi €i!a' 2 z(L ].) H prxs6tFpilL6ror+r4 rou '4. nor r-ivaL 2 r,4 ry 0.6 ri.vcr /rzprizrp4 r$ $Lxpoteprlq riLorqlrt<Tnu l t . r : . r .F.. H lrr1ur ' l r rpr| broTrtr i rou . l ' t 'u. t ) rr r f t r |nr0h.r\ f . r ftr€yc),nr€al rbrorLtui ror -D. nor civ(r 3.)

Atd6€r6{ Orc! r = (.rr,.rrr,0), ro rrll.izo Rayleigh xolr B oulninrEr ttE ronq),ixo Rayleigh rou ,4. Audr eiv(r rc SLcruiopar( r(l zai0€rI( oxo ; = (0.0. 1) L cryr' (uro roxx€r rr /l = r? B.r'. To a)dlLoro 1c 6), cu-i.l ro .f €i\,rrr I przp6r€arl6Lorqr{ ) j(B) n(r|" brv trropei rcr ei.var" pr.xp6rrpo an6 q pr.nqoreprl r6rorqr! tou-{.6t1;rr uurl ti.var ro rl"irlLoro, oE :tqo_( 6Ia r(1 :r. Aor6oo ro }1(B) go rivcrL[u{QoIoQo orio q bcLrtepl rbr"or4r1 iotl ,4. H crri(i: Kri:roLoq ouvbucopoq r(Dv6io lar;rl)l rb(ob((vuopi{irD! rou ,,1 ri.vcL xdr0e-rog oro :. E('11 6(i}.oqr c!r6rrov ouv6ucopo oro q),izo Rayleigh jT(iavoutr€ 6!a! (prutr6 $rx06r€ao rou ):(.1)r ' i . r r Nyd.ureQo rou ̂ r(B). E;ropir ! ;q , \1(A) ) , \1(3).

H r),1pq; eLxtlva tivrrr ltit ttlhtloittptttfiol.ri 6tort1t6r.

l (n) < l r (a) < ) r ( .4) < l : (B) _<. . s ) , , - . r (B) <. \ , , (A) . (s)fec4rrrpLxri, I (5) al€r luo (luor)al €E{ oI. Yro{Jdor€ 6r! ir(l r'i},Eulnsrbaqrltr'r'er(! (id eninrbo f'rrQt6[cvo rr:r' r1r upXr]. T6rt q ro|i1 ril'(lr ncr2'r a\,o€i],€rl'o!rbiq, lrLxQdrrQo rct(i lic 6dorcol. O pdyroroq (iEolus (llrrlg rnErofotc bev |noeri \,.1 EilcL leyaL1]rcooq (a6 rov p6loro (iEoa( o],or1].itQo! routLirLvotrboi\: )L(B) > ,\ (,,1). Qol6oo o p€yloroq cifov(q rI< roFrtq cilcLruuliTutrot r&n pt.1r0.o<, 6ao o 6€lreo< AEold4 ror crpXLrol el,),rLtpo!L6orlgr\ { F ' ) . r , l r . K u r ' , , v , , ) , ' , y r r v . , ' r ' ) , i r r , , n ' c , , : , N , ( : n F r , l | l t c f l l u r t r . " a o r p o (

6.1 APXF: F AXI) I QN KAI TO NI'\IKO TOY RAYTEIGH

al).dooov d(ov €rx{iporaq ropriq

rrciq(ov d6('^

Fiqov d{(Dv 'lkdpol'4 roriq

l i ' t

. ' . .1

Ir. 6.5. H deldg naxinin laL minimax: .\r < rr < ,\, < /rr < -\r

OEl IKA OPDMENOI NINAKE:

on6 rov crarrx 6 b€ur€ao irlovc ncr pelcLtrrepoE tr:16 rov cp1.x6 rLciloro dlolc:I r (A )< ) : (B) ! ^ r (A) .

:r pia erbLzrl lepimorq. 6rcv.ro r(gvov rnirebo r:i,vcr" zd0ero orov appr6$il(oio dEovc. 6xows lr(,8) = )r('4.). Ev 161,€1. 6xur r6$\roulr Fr: ro oa(16'rt!ocni.ri€6o rir = 0, rirQL[drolfrr (!(o611FC. H rd0ero< oro rdllov Eriri€bo €i,vol: = ({). {). l) z(lr ffiopQi.rirouue rllv r€}.€uraf(l yQcpp{ xcr mr1}.1 rou ,4.

To iSro roiry$a F'roao{$€ aa Soutr€ xaL mI MtX(lvLx4. Av 6o0€i. €raru).awolprvo oloqpc elcqpiorv zcL paldv, r:rooeorr 6rr picr pcilc e[avcpri-!€r(1r v(l Looooo'r€i oxi.vlrl. T6rc I Fu{Qor€0l ouXvorrya rlrEdvciftr a},Id 6cv!rr06ai!sr rt .lr. H tr€Ta).nr€Ql ou/lorlra €),arrdrv€rar aLl(i 6Xr xdr(D ffra)

Eivcrr zcrp6q vc cvtLorpdqoqr curri tc eeorprlpr.rrn rcr vc 6r"crund;oorpeB!r1 c.ori minim"x. IIAoe€orl pcrE €ivc! vo EntI€LAIoottr€ nadrr( va /1r),roro,To,i-ooUt€ ro nq),ixo R yleigh zu! lL€l(i lc( 7i8oo6[opioorlrr rqv e].ir1Lotr1 buvu-rr1rL$1(1uro! Torl llrYlmorJ,

Yir(i0louv 6r(i(t,o0or ra6ror va ro €rrrulotlF€. €Ea0r(;)$€vor t'r(j ro ror(l!61()1rlr{ i,1l1l(!1€. fra !a 6Qoutl€ rl lldyroq r6rolr[ll )]. [eyrororrororrle ro!.1?(r)'|(! q 6otoxolrp€. Yfio0dorc 6f((,)E 6rr '/!o€uo!fr€ rI! ):. Enclbrl to nl),ixoRryleigh civat

.l ' iyij ):.r1l + .+ l,vi .

x i+ t t : + + Y) ,

rl /iuoboc 7ru ru rcienu.rr rn .\- 'Dc Uryroro. rivcrr vu u:rurrioou.rr .,r. = =

,, = 0. Or , 2 atnoi :repropropot uqnivor! p6vo iv(! &6r('rorc.ro ur6lopo. :rorrr(1Q.Lyri.4 (1'16 r( nQ(i)i( 6rio !61o6(i!riolcr(, ! (ur6 ror €16rr{6 6r6rilol(rounolr,)Qo, ro [i./Loro xor /?(.]r) ei\r.rL ,\: -6[nE r(r rbLobn\aro$.1rc }r 1{c!.]rt ef\r(rltr6eoE rox rrpo6),{taro5 xcL ouven66 dwoioiq.

Orcrv eilcge rn\r .lvrioror./,I zcrtdorcrol Y|c rrl, ae7,rt DraxnDin. I 'bdazl.rLbi rirc! vc €l.axrororontoouB€ $' 6vav aueci.e€ro r€eroaro[6 r'.: = 0.Awrlpdrporpe lorr6'v rtl.r Lbta xu ptyLotororolye ral R(:t) o' itar at9aiptto'Lbuioruro uniyapo. Ae\r dXorJ[E 666(r.r rQ6no \ro yvrr,Qiqoqlr c,' criro< otlnolopo. rear6x€r rbrobratUolarc ro r€}.€urai.o 6€v €ivar rr0lr,'6. Yrdpt€r €vrorjror,( trn (uoortrc yLcl ro Uiyroro. Iporyow6v{DE. ro €ldlrmo uro roa:rrpnpLoFo .r:1 : = 0. ira! [!xQ6x€8o ro! .\r. r(i)Qo ro pdlroro orov urd1t|)eo 0o€ivcL frE./.r.ur€po rorJ 11. Eir $ iu rrxeioq 66ui.ft6oq utdxoaoq ror R".

mrx I?(r) > ),.

And6r44 Kriroo r' orov ufiri7ureo ei!(r riiro€roxar yf drro ErbLZdr ro ,r lv(DpiqouB€ 6fl ,q(") >

(t

oxo nQ(i)ro !6robrdvuopa .1r,\r. H oaxrl minimax an€I(rl

6.4 APXE EAAXI'TON KAT TO IIH MO TOY RAYLEIGTI

6K Eriv pey,otonor,orlpe to .R(o) o' 6l,onE rorg 6r,6rdototoq undloqorgSr, 4 eld ,oqTr4r] dltov quldw iov peyloro! €l,vot q )2:

)z = minlmar fi(e)]. (7)- sr '.€9,

>lj$qr)ve Fs tI! (6), I roo6trp.o evtdg trov cpul,rirv 6e1 elvor fiql6 Fne6-teQnl T ou )2 xar orov er&,x6 rrt6loQo, rol ncQdletol at6 tc 611o aqdrcr,6r,o6uaulolrccc, uoorircu pe )2.

KlElvoq.r€ ont{ try noQdyQqqo Fe &io rclQqqerioer{. EtrriEo 6rr, sxdpqxar, 1ogl,g lemopepeiE ono6stEs!8, I 6lalo0qorl o<rE oupqorvei dn eivot ooordg.

naQdt,lQqort 7 H crqxi minimax €rerrelv€icr. crid 6!6rdorqro1E uri6xoQol5 o€j-brdorc|torg un6lorporS xar nqgdl€! inv )jl

): = ninlrqar J?(c)].

naea'I4eqoq 2 Or i6rcg aut€g caliE lolrrow t{ct y|,(r to lsvtxeqrdvo ttQ66)qFar6rofl$6v ,4r = ),\1:r, drov ol,lrrifoqr tov rcpovoFcoq rou ,qtrbcou RayleiShr:rn6 c't c ae al lllc. Awrxc0lmdrvtcg pe c =,92, drou o! ofiil€s rou ,S €rvft tc!6ro6lcuio[ara rou ,11-r,4, cnlrorolouFe to 8(o) oq Foqqi:

^ . t r Ax ^ t z? * - - ^ " z lJI\x) =-'",'i--;';- =-Z;::=4- (Y,

Ar'!f elvar rl oroic T|6 raur6lpowls 6loyovlorloirlorE rtov -A xal .4y' orr|vndQdyQaqo 6.3. Ko! oL 6rio rstQ(ll0rvr.ri€E poqe€g livowar doQo[+(r tE),et(t)vrrpoldrvorv xar )r = ninR(r) roeri5 edolg *ar )n = maj( R(o). AndFl xary"a dtaoeg $r\4, o' dra rolow€!6Be'\,o oioqra, I orryaftIol Frsg Fdgag o€rooQponio 0o cruE{o& rI ldt. l1,6r€Qlll onjfl6qra, xar 0a €larni,o€r tI FeyoLfteQrlovlvoo4ra,

ATKHEED

6.4.1 Oelueiors ro ouoqtro 4r - b trou 6lSFtaL dnd q!

4t1

(8)

| 2 - l 0 l f - r l la ll - l 2 - l l l . z l=101.L o - l 2 l L r j t a l

,112 OET]KA OPI:MENOI IIINAKE:

Ko1oo,{€,.,do1e qr ovrfmorn reracy(,rar i poa(lll l'('r,'r,ri), xno},oyio!. rLsp€arxiq ri(aqlir,or! ar'lrri xor €no].to€rJor 6fl lrrtj.'iiolrdr o'ar66s ]rd ntvefle!|trd Inq.

6.4.2 >!pn^ta6or. ro f f radyoJao or t ! P = ( I /2) ( .? . . t " ) " r 'D=0/2)( '

lL6)? l ( "-,t L6)+oto0€q.i. H oroe€qd aunt roour( F€ P,nn hori o oao< ftrqoqd qs b8 rivorrori aqvllIr"o!. (InriJ

6.4.3 Bq€irE io €ldr)(roro, €q 6oov urdq)(€r, qg 4 = (1/2).: +.v + er :e ,o, .15P ( l 2) , J0. I rotc r : \aro: A ovrroro.Xf i mq\ P ' :

6.4.4 Mrd a).).I r.raay.rrLari poa(|i nor oiyonac aXsL €l.drLmo oraa A,j - D sivdr I

a(x) = \r12) . iLr ' , r=(1/2h"/r.4r-xrArb+(I l2)brb.

AT!o(l1.rq rI! or(gfad (1/2)1,"i, rar ouytqtlolroq rov Q ps roa ],, roLo oi{mlFd€!ro(i)os.,! raialolrlF oto €IdTLmo ro! Q: norEc €i.!( c!16q or sEro o€(- EouLtr to\ r :owot xc l r oqv oto,A:u r ) ,qz ior , "v r r reolu\ { ̂ .

6,4,s rL(l rlrolra olr.p€rQ'xo riroao,l. !troloyior0 io t?,fro R(r)1'd rry er6,xri otrl).oy'li = ( 1 , 1 , . . . 1 ) . n o r o r i l i r t s l e o r t r o u u o q o i i r t r c l r o i o ) . o v r ( q r o q r - i o ! d i j t r s r l s

6,4,6 BaFir€ nt! Fnr],oyrt .f ro1, lilFr i!(t F(r) $r.l()o-i.Oo clro io (tQdlf(1 )j S 2,

r ru, lxuu, \o uru T( b,u. ; \ ,u . ' ru . r ,1u ru. . ' = |

' , l ' ]

ou, ' , , , ! , ! r r r ru; ( ,nrr

rq'ri ro! F(r)r

6.4.7 ELiv o a !i!ur 0srl"d oqrqrdvo<, 6rtE!r' pc 1lt 6ort0fro rou rdi"ou Riyleish arr ItrLaaon0r '6roitri ror ,1 + a uiv(r |EyolnftOf aro q [LxAolFAn 'brolpri ior ,.1.

6.4.8 Eda,\r silor rlrr"qorrar tbLoltrLlt rou / ao r1r .ivu' it fraaor8ar rbLorrrlri rou ,,1,hiErc on rlfLxA6ft()l] 16ror(fl ,r rorr r{ + A .i!(r rolil(itnrro! rddo }01d.)t doo ro)r + !r. (Ao,t4(im€ ro (mimorxo 6robtdwopa r ord rrl).i"d Rayleish.)

ttltltiaol ot 5!o dudq odzryetq dldr io(Ds ro rLo tlrLzdr aar er&dqaqorodrfn.Qdofr@o, ro! raor!f,rolv .nroio ar6 q! err Raylci8lr. lllld p€ r1€ydh6uoroi,ia dio tr6 ibr.( fl6 .qrodro€L< rbrot+r.ir!.

6.4.9 Ed! o B rivar eE a.i oeLopavos. 6.iErc Fs tr! cqxrt nininax (7). 6 r 6sJz€p,/tlljap.,rcar l6roqrrl u! i.lurar $ qv naoo0Dor xo! B : lr(,1 + E) > ^r(i).

6 .4.10 Eiv ur"LVi , r ' r rF 6 io or t . rE rou A. ru. , ; uv,uul i t r , ; rFUUr ! r , . , .Fru i i r I9 r , r7a6r -

Q)lq r orrlriq r, xo! arotr6lo!'roq rilar{r rar x('r! cQi(r,i6! .\:

6.4.11 Barirs rLg rl,a7Lmsi trFic rorv

i.(., = + raLri + r:t

0 ,4,12 Ar-r r uro qv ( r . or ro 8(n 6.v r : !u

"? " , " . + ,1

[cyo).!ftpo otro rI uE dlit€al

6.4 AD<E> EAA]<I>TON KAI TO TltjA]l(O TOY RAYLEICH

6,{.13 H oe)d minims Tld rlv j-oort r5rcil|lt ri€Ad,ap6dver j-6idororol,s llnoloeols Sj:

\j - minl ma R(r)]." S, res,

d) :qr€Advfte 6i! &a! ia .\j €ivft o€.Irr{i. rdre rde€ 5j rreqElst 6ldMJopo ' peR(d) > 0.

6) :wn€edvar,e 6rr xdo€ Sj 'reoqet dduJol.td y = C-tx pE F(y) -

srcT AC!/\T! > o.y) :qln€edl,aw d'r6 qv oQxri minimaJ( 1rc to! CTAC,61L t j-ooril l6rciq14

ro! €ll!ft €rrlom oerr l, ono6eut'lio\"rds grol'dl.L 1.ov rdpo a'aatetat. ('ftt€qaa|orLs{ pou oro Ilovsnronlpro qs Minncsola.)

6.4.14 A€t&€ 6ir r Ll|rcdrefl drcrrFri ]r rrq AE = ^M,68'v €hd! t-tsyd]'0!€at o'i6 tovXcryo orrlm1 tov yovuxdrv oroqrirov.

6.4.15 norcs €r6rx6s urdxoeos, oqv aaxri minimax (7), €i1/d, d1nds fto1J 6h€r qv €t'dxtoir)rrui .\r; Ms dD,o l6/yra, yrd 'ioLov s2 d\d! to Flyroto qg .R(c) too t|! )2;

6.4.16 Anoqav0rift, ndoF{ rbrctqrq ei!a! 8e!I#E, ri6o€s oavlflt 6g xot t6oe5 pq6!v, Xoeis\,c rrs l)riololaoere, Ylo 10\,

413

l 0 0 1 l,=l;;3'l

L \ 2 n )

OE1 I KA OPDNIINOI Nh'AKE>

6.5 T H MEOOAOT TON NEIIEPA'MENAN 'TODGIAN

arro 4ru\ or auprc r6rfc l tc rpotyorrtr ivt : napuypurlorr :

(i) H ),rio1.qg A.r = 6 roobuvap€i F€ €Iaxloronoiryr\\< PA) = \112)x:r A:l

(ii) H Luol rlq Ar = )r., Loobwapri tru d(lroro;roiqol rou n(r) =

O(t rQoofi(Orioouuc td)Ao vu €EIyrloou$s. rtDq audg oL r66€q €ia(r 6!vcr6v vacqcppoorouv orqr, rptiEl.

H rorop.u r .v,{ . :T,{L(. }r f t r i or roo6uvul. i t : (urdc l ] tuv yluor ic nprv anoilc o!{l)\rd ror fi),€ov. tE o0lo$61a x),(ou{d rrQo6}.i[cr( rnq TEl\,rxtlq r(1r l:O].!otztlq. on0:E eilcL ro Luyr.opc FLcg nldxog I oL 0€[€].r(i)6€q r(lrumdoergcvoq (t6por (r6!orq[q xc([ rbroolvcQr{o€rq) TprloLporoLouwcv cud5 or. r}.cr .-oronoL{oe6 1Lc vc 6Qr:0orir, lor6Qrxiq sxrprioslq ilq nQrrypalrxng }.uoqg. An6lr!(r (tjroln. o! nQooelYiosrg ri-rcv u'taYxrcrtxa Xov6Qo€r6siq. $rc not| 1(r p6!(teQyoLtk! {tcv 7apri. r.tcr fo},i6r 11 prc prxprl prycvrj. Or po0npolrr6E oQXdErJnllQXOv bt(rrro)Fdv[< (l],l.ri 6€v $roQorJoul vcr €qctQ[oo0o!!.

nQo{cldq ot lrlqrLcxoi. unol"oyror6E eivar rou 0( 6qrrevaa rllv fiqldorcloll.citrd ro rQdro 6iIC lir(,, \rc roofionorl0orrl, or cpx6< el.uliorc'rv rou rjrav:roLr.rjrol.L6< xcu" qpya<. Er€rbn {x(v E!ro),o vo 6s! n(1,€tE ,_r(i)E "6roxpLrororcirqf'[I.r br(r(f]o$xll rliorror1. 11 p60o6oc r(l)v rr€rr€e(oBavcDr, 6!(rpoQo.'' eile q! fiQ(,)-ronoQia. Try €ibapo otl! nopilyocq,o 1.7. !a avrrli(0roid xaoe nced./rDyo pe6v(1 rll}'ixo braqoadr,. 'Io qloLri6 fi€6i.o oQropo! raLinlcrct tur6 ivc :rl61pcror lpei .rrrv xtr or ad0r orpr io tot :r l . ipcrto6 d1o4re -rr1*1 +2l; ui 1=1t1111.To nQo6Ln[o 6Ldrrirr6vtra oq poptpj Au = J xat" l AQL0FJ]lLxn A1,i1l.!ol rIEbEr(€ri.1< iolJ lq50 1ircr! rcl.eitl;q cr:roqqoqllivit cr6 rr1 trvrimr!1 yp!1ogrvtra666{t! €nrl,uoec}S ouoqpir tov. ror etvcr ro}.r [€yh]..l zaL no].il oQcLd (:.r.M.lni'vc,aq tr€ roDd Fn6c1,r,r.(i).

Aurorouhrv yulr avrr l . lq 0r ' i : r lqpL'r : r ivur of l d; ioFI ?ar or r6l f4rropivrgbr.qop6< yi'vovrar arLoxrlJro nol.inLolieE or na(lylrorrrd nao6lr}rara T€xvLHnE.tinoq oL rhoe6 o' 6va ceqo:rl.clvo 11 oL rpuor.xi,< oq\ror1reg rou cv0por:rivout'tv.tqAl\on. H xeeyttarrxlj duox ia bet'iyxttxar ornr miboy rav eErctbouy.t .i artt ilrrttnoon zorq. I.cr pLc aouplerql r€pLod te€rrll6poorr ooli$[€r(xrni6ytrrr, anor€ior[El,o cno tpilt-rva 1 reTpdalerpn r1 rcrpheSqtl. y.o"t rr,ax6xrL\X0€rtlqoF.ror€ 6v.A, oroiillclrz6 rQoro rAootwrolE rrDv rnoze4revov rploLxrirvv6ftl)!. M€ ciL],a ).61r(1. o lroloyrornq rp6ner rcr 6or10rlorL 611 p6vo or4 hrorlruu bra}A,roi :T0onl i l r , t roc. u).) c i zor ml o.ort 'nuo' l rou.

Mroa€ir€ va [am6v€rs r. orr,€6n. Er(l\riloo\r or ncrl"r"6q ll(0o6or, Be sLcvio o$oq 1660 zar v6o 6vopa. To ldo 6vo$d €L\o.L p/0o6oa ny n€ttEegopiyayorotxxiov. ̂ u\ q v6c r,6rcr gn6peolvc rz[€rrr],Ieu0Ei rit 6u,o[r1 rol uol"oy"mrloro v.l i4drcal€ldro€r pru braxprni npoo6ryror1, vn 14 lUoer. x.r| !c rrcQo06oE!

itrror€iiotrcrc. rrcQrooor€oo un onorcbiror€ d ,I r€Xvu{rt orrtl €nrorlfolrxriri;roioyrorrxri.* ro ri8rSi !'il(t, !G b({qariooltr€ rry ar},6rlli.1 rlt< 6uoLr.)iqrbtcc. zr c( €ivor or rqcQtroli< trol"!riior.(€(. f cruiq rr: €rpo0tro./6q. I iFqrooillrrrrrto:riorlxe rrno ro! oTEbr(lolro (isQo:().(ircrv orry crorlcr),rrcr rcor, ;rrp4rLxdltrrrL6puorll'r'rv x(L rtbpcr:ror "lpdrpo cruo ro iir6iio untipTrr prcr oi.c{1 brcrpciTrl./I(t tv €naxrcort rrtc ort! rbAoStlvctrz)i. fro n0o6l.iltr(r( (llTitc rrlg zi(L(z.tq.6r'c ttx0 r;rrlppc, olyt-i.o rircrL ro rooroq rotlq tpo6ong(L 6rr trcr brorxarolLpr--qro bo7,i.(iQro ria(r pL( o!vr1p1rLx1 ezrilrlo4 yrn rn lLdXg rrirpu iloc\a. E),nilroorr tr€przoi .1aan(;ortg 0a iTorrr crperr6 evbr(tpdpov 1i(L rnL[(,\,i1. r;]or! vr{tivouv rrloLor rrt< tr$o6o! rtx ;re;rrpuo;Ldvr'x ororTcirrr.v r(r 1( 6€lr(;roorv rrtx0rtort rlt<.

fr( y.1 rEryrjooqr! rrl tri0obo. Etxrrirpr cno.qv t:rw ij (Wti tor Rartcish-Ritz r{(rr xcronr! rLod.putr! nt 1ttl (bi:( r0)r rEn€0co[tv(r).v rxorXrit,Ja. Mnopoilrr rtl rp.laoroi.llr trr rrla i6L( 6L(qroa(zrj FEio('roit ll" =.f(.r') xar rr< i6Lt<oLrvo0L(rt< orlvgrir'irc r(0) = r/(l) = 0, no|l tr!),rrt0rtx(v nporyo|p6wr; po rr5xf;r0r)oirnaatq bL(q'oQi:. O|oLo./orlL61(')q. ro n06ti),ltr( t11rr6 t-irctl "anri.porvlr(rcri1or(,)! . ro , i7!r (r'rrr1(r(to1(0ri (116 rq ouirorlolt ./ zcL o :rir<rxcq .4(ro lov rr_),r.orrl r/r/d.r:. Mnoporpt o1t.rq rl<r nQox!)Qrloorrlr crril.op zar vrrr{{r1(tQ(i1l,or}r{.rrP rrrgr1(,^,Lnr1 popr1,{ r1q orroi(rq (qtrir{1r r1 r.).(lroro:roirlorl.(\'rLx(r{)r(Jr(;vr(rq Io0)rrpr.xti lLrolurrcr 1rr. oiox),1girp(lr(:

6 i L l U IoOAO: TA i * nEnE I ,A !N IFN !2N : lO lXF la \

i , ( . f ) ( i , " ( . f ) ) / i . l r ( , ) l ( / ) , 1 / .

( , r ) ' / r l f f ' l l - i r l aa P ( r ) (,, ' )r/ / . ,

n :i7.).i.r. rror n(p(act.!r dlro ror cll).rotrdBa. l!i)nl0 drL,t rLioobo: lnoafl r( Brtr

H nil({1('rirort orTli nalr€L \,.1 rl'(1lroronorl0ri. r)q neoc r)lrq rLc orNcerrjoeLq t,rolj L)r.rlonouiv rL( ouloaroxi( orNo)it{Eq zar t owdQnta4 nv 6irtt n .layoro0d titut q 1i'or r. II 6rcq,opLzr1 r:io0Jo't trErnrp.lntr. oa coxil rr,(ziorou r(rcnotr6'i'rr !c oLori,rla(i)i'oupt xurri Iisr1:

I'/ , ( i , ) = ( r /2 ) , ,? l i , - t t L , = 11 l ! 1 , ,

t'= n t2) 1,,1,,'t "t 't"=l,O ol]or | r / r . ivt l ( BIbi ! , {crr or.r bno drA(.5rorL to iSLo oqrt ioivrr I r rr ] ! r , .Eriotq o r€ra(y(D!rr(t! 6ao< /(,1)r tivcrL otrplerprzoq, onoq r1 .r.?:1;r. xc0rirrrioitc z.tt oErrxoi. rip.r o!(oq <r),ilor$r 6vcr r-),ri7Loro.

Ilri)c i;ai.ci)loqr€ crjro ro E).frzrorot o ().QI61; u;roiolop6: ri\'cr roobr-\,crpoqlLf rrp (rQr6rt ilol rlq bLcr.ioprzrlq r'Eio(,ro1q xar ro rp6ti),qyc uut6 ei.vcr. cr;reipovSr.lorhoaD!. II cAXi rrDv Rryleigh- Rirz :rcrqxl.fL iT( ,, 5r(iorrlro np6billl[r.br&),6i ,o!Toc , , t ,o!o 6ozu(or[afq or l laprrto€Lq i = 14.. . . , r ] =1,; , . rr ; robeT6per '4oiorr< ro!! orw6!(otror: I/ =.qrl,i(r)+ +t)"lt.(.1:)xu uno),oyilovrc( ro! €rbLx6ori!3!(otL6 (o!ou{iorr- roy U) rror €).azroTo'ror€i rrl! P(1,'). En(vciarr6irvo4r€:

OETIK OPDIIENOI IIINAKE:

H 16(0 €ivo! ro eLolLororortoouf€ rr€adl,o) €r,6q unoXdpor, cvti unepdvto 6Lorvro;v 6rvorriw I xcr 1 ouvdrpqol nou 6i!€! ro eLciFoto ei.vcr" I a/, ovli rn< u.Etni!o!$r:6T|, artdq or btlo eo €ilor zo!r{i, rl lra oqv cil].I.

Avf l ruUroru'rroq rr lv r t rr r lv y.4 rrrpuyuvrxr l l luqq n 7i \ ' f rul

P(v\ = (112\ j, (y, I{ ' ,+ @tV + . + l t rUt)Idl : . (2)

Oupq0rtte 6tr ot !' efi€l.dyqoav €x rcuv fieoriQo! ticL oL dYaoorot €ivol rc(1/1.. . . .1/ , , . Edl 6d).orge crurr i ro 6{ iQrl o '6rG 6rdwo[n j / . 16r€ I P(y) =(ll2l)!r Ay - yl b $lctyvropi!€tct (xQr6d)( ooa prc cr:r6 nq retpcrlorLzdg popq,(:<yr rq o:roiog ixoupe eEor)rcloesi. To orory€io ,4'j rolr nivdrtcr €i,vo! J l4lyldl =

ouvr€l.r:('irlG ro! t/,r/r r{cr o! owtordoeq bi e&at .l Vt.fdx =_owtcleoilE xcov yi.M:roporpe 6e6oLo tibpn vc 6Qo $!: ro a).rixLoto Ir1E |l2)llt A! - yrb, npaytt'nlooburatlo lrs fil }'rrol riF ,4i/ = b. Ta 6tlpolo l'o[n6v rlc $€0o6ou r(ov Rslicigh-Rirz €i.vo! (i) drd'lf6e dotq&aru,tis o1)1.(rQtio€ta (tl,) I)tu)'iyLor roxE aDrt eod<Aii zat bi (iil\ |lae qt, Ay =1) xo! (iv) lphle rq! rQoo€fyrcrwti ).i\rlU = u t v j + + t J | V r .

Ta acivr( eEaQrrirvrar tln6 ro 6rlpa (i). Exx6E rI (!v ot olv(la io€tg yj Ei!c!€Earpsxr,zd qrTl.6q, r(l hl),c 6npct( ec €ivql. orlv r0(1[(r!t{6tltc1. clltc(toQgcoro.E{iv FnioIE 6rv lm(iQxE! o!!b!qo[68 rror V xor"tr'r o1{! nQaypcxl1rri truol 11.Ia 6n[c.c( .lur.! 6cr eivcr cixplm<r. To /iQ66],rucr €ilaL vc ouv6rLlooolv '((!rn br-ro. bnvcr6qrc uoloyrolrorj x(r[ cxQi6€!.l, xctt 4 dia- ,Er6i ltov Xttr]Qtot'it! Fntruxia aflt! ltgobo rrov ntrEQcottF*or aroqriotv, tir/lv q xainq q,|ttrffitxaltd"rlorntttxt'r ovvdQliata\i ors 6ox4taoNul6v awq aeuv V.

To or),ouor€Qo xat n€Qtoooreqo lqlo4ronoLoriytvo arct1Eio ti,"rttt tytl ptn altypr4qLxo. Etxtt'fj:ytt 6crEovtcq a6[6ouq or( or]$ei( 11 = A, rr = 2I. . . .u,, = rrr,or(']< crpL6(.]E har mlq rurt:p(optvtg 6tctpopdq. Ira ol$r:i.r /D = 0 xar l,, + 1 = I,oL orvoprtrxdg otworjxeE tr:rano& cr6 rd0t V vrr rilot ttt66\r. Tor€ I yj €ivclq "owirpqorl-oporpr1" :ror LooucrL pe I orov,16[6(r fj zcL prlbev oe 6lou< tottE

.),orE ,i6u6oq (:X. 6.6(). dac ri.r,ft oqxevrEDpfvl o' tvn przpo 6Lrimlpnyupor ar6 ror r6tt6o ttg xrrl [I6€vil€rar ncvron cdJ.on. O x.i0€ ow6uao[6q'!t LV +.. + u"V, 0o nQinrr !a 6z€r rlv rr[t yj olo] x6[6o j, 6L6rL otr€q or ir].tr€qV rivuL p$tv. EroL ro ^7qiup4[d rllq oltbLcilerar €lJxolo (:1.6.66).

Autn ouBn)qprirvouv ro 6ri[a (r). XrI ollvil8lo lJrro].o1i.(oxF€ rorg ouvr€-Lt:oreg;{1; =.lvividr rou "riraxa ar{,pl,ic(" 4. H r}.io4 Vl Loolrcrt [et/h oro gLxp6 6rirorlpcr cprortpri ror rr rar 1/n oro 6riroqptr 6t!r.d torr.To i.bro rozi€r zar yro tv {' tomci orov xoF6o rlg r; zar €.t' ooo\' orrd rtt"nn)rlL 6Laorilport" 6t riptovra4 n yn6po ViVl dvar rnrotLzl pa6&.Tcl brcornFarc tdulo\.rcr p6vov 6tav

65 H IIEOOAO! TON NENFPA'MENAN :TOIXEION 111

v.,

'1Y = ytvt+.. .+y5Y,

2=F,

u te=4n

(o)

tr. 6.6. xllaeniorLs-oaorpiE xfi lpojFwoi tot6

(D)

i,s! 1,,,;,,= I (i)'. | ( i)'

xft 1,,,;*=lG)( ;)=+:w€nr)E. o nivrrx.r( sxopryicE eivnL. oqv agalpsrrrdt4io. rQt6loy6llo!:

, l- i z' -re=; l - t 2

t, ir]Aw6 [oldt€r pe rq :rerepcropeveE 6r.cqopdg (!) zal, o6fyrloe oe r.i,uibeg ou-ErFioer!'yLa q o16o4 gerclri r<ov 6uo F€066(,)v. nlo :ro1r.tft)roxa nenepcoy6vaororlekr -no)'urirvupcr gryaLurdQot 6ae[o6, nou oeil,or'tcrr o€ rQflorvo ri tErQd-nl€uQa. yra ffarxds 6raqo0)dE rfrotirotrE -roqxilow rriols u0ltrous nivax.<,1. MnoQrirF va 0Ftuprtorrr ra nfnFe((oFdvo morxrio ouv fv(rv oumnrarrx6ro6rio rcrcrox€vrlE €Ero6o€o)! 6r.rqoQ6! oxpr6doE yra ooi,FB€rQo r[],6"ypa1a,o7r6rs rcr fisn€Q(rolr6vo oioq€lc 6Qio1{owo! fiqv "xoFri" r(,)v pE06&Dv Rayleigh-Ritz r{dL r€n€Qcop6luJ! 6LoeoQ6v, To orisloorrxo eival n .rrtroxl]ra quld]v rc,lvrgrpcnrui loLuovql.rdv owcgnjoecr.rv, l,61ar rlg onoicg or xl,ioeq toug oe xiOeurotjrdmr r(r r:ivor exxol.o l'a 6Q€0ouv no! !a ol,orl,rurDeouv.

Or owror<iroeg b; m4 6elrd rleupd eivcr. 6LaqogerLx6g. Alri "yra qp4rovop6rrl r4ui t4E I mo rj,6rlco5 orL{ rurepcopeveg 6ragopdg, et!o! tdQ'aprtt p6oq rqr{ qS f Vngo ar'aur6 ro orpeLo: b1 = !V;Jdt. Kcrt6ntv, oro6i!a (iii) Iwour€ ro rar6t(ry6vro oioqF(( Ag = 6, nou 6iv€0 rouE ouwEl€ot€grrc 6oxtgaotrrrig, elcruoro:roLoiroag owrierlirorq U =ytU +...+ V"%,. T6los,

J18 oETIKA OPDMENOI TIN KE'

otlvb6o'v.rag 6I(l tlljtd ro uqJll sj $€ lra iselooF€rq laouui, €1oqe prc er.z6v(lr1g apooellLorLxrlq Lrolg aI.

IIAPAAEIIMA u' = 2 $E u(0) = 0, ((1) = 0 xor Luol u = r-.rr. H naoo6yy'ol0o XQlolponodo€r rpi.o Sraorrt$aro, [€ l = 1/3 ror 6!0 owaATrio€L! oQoqdq. Ojriv(rxc! ,4 xcr(oz€rcimlx€ rraeardv(,) rcL ro 6rdvuopa 6 arraLr€i o).orl,1la{,)olttls ouvdprlo4q'opoqfq €ni ritv .f. E r€rtui -f = 2, ouo jroady€r 6io qoa6! xoq6c66r xriro ru6 t4v opoq{:

l . 1 10 =

l ? / 1 1| - 'l 2 /3 )

H ),io1 tou ,4y = b ei',tut, g = (219.2/9\. lwe:rd4, q 6t|aax\ U EL\&L (219)U +(2/9w. 2' currl qv reqi.ntcuorl, ox! ffier mouq zdp6orl€ tou .iytt.ttog ttrt l t1. exardi ̂ [ar u =.t - x: ! = (1 I 3) - (r I e) ( i t (2 I 3) - (4 I e)) .

:'dvor firo nol.!tr),or'ro ntrpd6eL.lpcr. I fiQoo6.l,yrol 6cv 0a €ivc! er{q[6rlEoroq x6B6ouq. al,l,d eivcrL clroorpeiono ]16oo 'iovro riaea[6r€r. H 0€(l)pi.(l roli6piorerar;rtoor r!16 arix6 crvdL!€ral oro 6!6],io rou orilyQcrpta Mrn ct,i7w1f4< rce6dor tDr it€TttQao!61'(Dr otorlfia!, \Qc44r6q (116 xolvot [E ro! CeorgeFix (Prentice Hall, 1973). A)rLc 616),icr 6i,vorv trlvr,xiq rgappoleg kcL e66Lh€qlou l)nol.oyroirl. To oapc. rlrv fi€7i€Ocol6l(l)r ororleic0v ix€r,.yiv€r 6vo o4rr11,rLx6ll6Qoq tlg €,incia€uolq ro1r illx(r\rlxcb1, ){ar exri0!:ror flo q.vq).!r!zd olo vdo [o!6t61,,i,o Enoy ay I aru €quQpoapiva M ae4 ltt rxd (wellesley-Cambridge Prcss,1986). Ex€i fiQ(ry[cr1€u6[cor€ €riorF [€Qrxis- b(sqoQlxdE €E!o(i)o€K, orr< ojroiE<I U€Hr,6i ,E 6Uio.,{rrur nQaypdf lzrr oro. ' ror l ! i i r rq: .

IIqo6l,rlpcto [6rotrprirr

H lbao rtDv Rayleigh-Ritz -d!(Xrmonoiqol rlrrpil1,tl) Br.trq orxoyil,€!(E ou-laprrlo€rD! y, rrerr€Qcopavo! bLcmiroecov, ovri ^/L( o),€E -rK 6uvardq zr- eivolIQio|"lrt 16lJ0 orrE €ErodoerE or(lo€pd< xd.raordo€{Dq 6.'() xor ora rao6l.rt[ararSrcrqtdrv. Axr{ rI .topd €}.cr1Loronoreir<1. ro nr1}.izo Rayleig[ To naoypa-irro rou €Idrzroro €il,(lr I o€F€l.rdbllq oqv6r1r.1 )1 xm to zatci :rqoo€Wr.r4€Loxroro A1 6.r lruqno€i 6rdv nepropioorpe rlv or.xoyt!€la tuv bol{L|laoxrzdvouv{ranio€orr aro rLE ri orrs y. Kc! nri}.r ro 6tcrpn6 nQ66IIpd pnoa€i va6r€u0Erl0€[ I oaxri ]rpir.Araft eqaap6(etc{, p6vo:ror 0c:rpdlfl or ouvop-rr1oer,qyj vc uiioLo./i(ovrar €rzo),o. 'Eror. ro 6rlpcr nou elve ra mLrtltoic 30 1a6\'rrlrFcrv reLei.oq quoro),o}n6]{cr. cvcrn6rpeurro: vn rrpcppooooriv or. verE r,6reg nDvrr€rrEoooptvo! ororx€i(l)! o' trurrl r1v cr6 pcrpol xcor.rqrreeion trretcr6ol,r,xrlBoeqrt ro! nAo6)"rparoq LdrorLgdv.

To ko) Lrr pu :ruQci6r rylrd, ivar ro r(n).orrorf Uo:

$€ 11(0) = z(1) =0.

h5 H I\JFCJOAO: TAN TEIlEPA:MINQN:TOIXEION . ] l9

To riQ(i)ro r&obrrivroFir rou €ivcr rl = sintir. F€ r6rorrll ^L = rr. H olvdArloitr, = sin trr bi.v€r ro €)dl.oio oro rlvrioror.ro nlit'io Rayleish

n(.,) = Ll '?!41 =1,",',t,,

li!!1,,,'a"

Afio .{'uorzl (inolJrl. (r.16 ei.var ro :rrliixo t$ bwcFrxrlq rpoq try zr.vllTrrriErip.ler"o. ;ror LooQporouv yr(l ro rSro6dvuoF(. Kcrvovr.xci (rrro to Lbrobr"tivuopo0c llIclv dyv{'xrro x(r, yru va ro trpoo€./yioo|tr ocr ncripvr4re [6\,0 unoqJlqr€qouvrrqrrlorr< V = ltlrt + + u,,V,,:

Q)tVi +. + t, ,V,!,):d: ' :u' Avn(v) =lUtV) + + th,V,,) !d. . l

Avr4rfroni(oulr€1{;e(rorpo6), I [crqEF].cxrorLttroir lo lgrouytAyfyT\ ly.Eri! o nivcx(l( r1-1 4rav o rcurorrroq. oljro 0a ()6)l./0!0r.oro o!r,{(}roldvonpo6),1ptr r6LorLpd;v -4y = )ii. Olr(,\ o :ri.varitq pcg .11 0c rir(ll rqL6Lcrl<irvLogr(L (frt .rLQ!6(;)q I x(r1.lor(lot eivcL :ror obryei orc y*rxarylrc ry6624yctitor4kL1,. H tl,it.ltrtt r\p{ Ar 0( eivc! rt [!]ra6repit rbroT|.Fl rJtq l1, = ̂ .\11l x.rr0o civcr zovrh oto;r:. To criororlo rbro6rd.,'1Jopc , 0c 6i\'€r trro n$loayyK'rtr lq tbLooNriontolq: I i = y.V1+ + tJ, ,V,, .

'Ono),( rtlr mo or.1x[zo rAotj],I[a. .rl B€0()6()S lropr-i .va o1l1,ol]{o0ri orraooEa.r tl4[(lr(lr (i) bLci].€EF rc yj, (ii) !ro].6yLoE rotrE '1 xor ,\1, (iii) Luos ro.{t/ =,\,\1y. (i!) drtl)o€ ro ,4, ,a(r tv ti. A€v Edo() lLuti t1r16 oroLltleL d!(lbroezcrropg[qro 6o].l.ciprcr.

ATKHIEI'

6.5.1 xorto+rorotridt. rer:L! olvcqoio.L( oeorti! tr. n = t/.1. yrc !d ).no.l€ l! !" = 2 [E!(J) = !(1) - 0 roL.iairtetnoxr o1 1t nqooil./rrl rllodftr mo!. zoBtiolE {f rr!

6.s.2 At!m. rt! ," = r F€ r(0) =,(1) = 0 aaL xal6rL! ).iorf nt! nqoo€yyLmrx.i B. bioo'ilaQritotL( oaoqiq uor ,5 = r/3. non or[6dilEt ro Bry{r}.!r.po oqL].rLo:

6.s.3 Em(o orr u" = z t€ oL,roQrazls oueriz.r d.idlfd!€q os "(0)

= u/(t) = 0 HolvoaLorrl o!v0'i"r yr( tt! L' .i!(r "qrloroio.rlrri a(r b€! ta€L(i!ftaL !d i.e€i drr.doz+'amLEc oi,lcprttoa( frd n = I/3 !tr.iaxtr llic €ai nr';o\ tuoti-oaaEA I:), aorr.iu aro to 0 oio r, lsra:u ro! r = 2/3

"aL .. = I yao].oyim. ro .1l = J(r.l)r.iil

J, "r ,"v,

Lt,

l ( l + u ) / l= v 3n'>idodo-l3oLrnD^flo un]''/" xp[AlA J = fr'rr -qDilrn tD,tD^rr o 1D^]3 5olou 699

'I I'€ 9 LoUmV ^hD rruY = rY ^tDnrrclgr

Dnu1990r or 3bty 1Dx rplAtt.l = tlN 5n\n11nxD t! z rB z ôr 3Dt,tolorro 'rlz

= t!Z= r lDx t/l= t1= ! D1o 53^?nords l!a h rD, h :?drodo-r3oFdo^M 5rt DrJ 899

lz! = y frn1o19r lr^rotlltD ̂ h 9l!n Ld,irgdxin !. udd+XDI3n rD^ta lrlltt : Yt ! \ r ^ l ! i^ lq ̂ mnlrora l p ' ih , rgoo! I r l r I n I +:A I = t { tDf ; . r ! ' - t t t t - V DLrlo,Io(om Z | = r oro U^rn",0urt,!r ,1 tdodo.[ohdo^ô [^rl, ô']J'l Drd !U t S9

ld, ̂ i's fnr udr4dt b 1r^19 (l rl 'l) : /nor o1D91rJD11o! oro! '(0101t) = j nx r = v Ma (n ûm r,] 5tl1 5trcr0t/4. sL'r

tar3lx o1 1D^13 oinv) lrt?^t/, 90r1d or! orJ9 9113a o09 orqod! ̂ o1 ti^pl{ grn)r1\9\g'q |-V oto tt odlo?lrrp or ro^l! ^@o!$D^no ^qoruoDnltog ̂ oo9Xorrn 59^3o^po?!h otollf13 oJ 9r y = 4 o1o lD13^tll.nt r? d J|'ll oErXPIs orntglD oJ

' i Fv r,tLzl t) - (q I v - n)v,t\t,y - qQ/I)= arn - nv r.nQ/i= (nJ,J

u rlJ^F (5o^-9noo/,I$i! 3 011o!onroL0x) i?6dod :? ôlDdr3l r.rr/, Dtlrymrt 5tl(lrllx3nrg nrN 9S9

'qlyatnlalDs oQo]?n I 3n 91rx Y trgl D1 13^19!rr.,r/,, I 119 srltaaorn ll)li { = ,t' ^Lro 3l3oprd !^ 'r,t tdirl tlDr :1o.,D0tr$o1o

, p t , 1 1 | - , p ' 1 r 1 ' 1 i " n - - i \ : 6 h r t - , /' | - J

'1 '6'*30 Drô,l.oedo m^F r^ / :h rr)r ,/,- tll,I ltlnBn pdodm4 L iloo lo1? rt) ol?U!3h DL 11Jz xALn + . . . + r\ ln = D ilrr ro^r3 t'o+.i LzrroDnrrcg H d 5Lar^.Dlno$]etl1 t$D (/ = ,/n- r'1) t'oootl3 f!,dod,D,g tr out 1)^1f;3\ ulyrtbD )oeo1?n II s's'e

3n ̂!r13rmm ^(0mdo,d3,.r)i rmrr^.ood! ûr 3rD^rdzrno , j"i;,:= ii)T"t'Xfill]j?1{Drdo^no tr n]J ( r/ ll.oLudDôo Lu trmD 3m?Q3x3) Z/I = . 'ro! t1^.

'tll = ' orc5t|t o9rl9r ôt 3ryD9 DyyD 'ldodo tlotllo?^r1o o^9rl D! Dti 3n z = ?/D ^ln 3to+V t S 9

' tA$+ztzk+ tArnôt3Xloro ̂ o^9ioDd3!,! L'ilt ht 5od! !D'! - ny ox 3roft lDr rPttz J = t9 nk

3:lxvNlll loN3Nildo v)llllo 1zt

YTIOAOfDMOIME IIINAKE:

7.I T EIIAIAIH

Izo:16< auor-, ror 6r6Lior rlr(v va rlllyrlooulLe [€Q!xi! cno r(1 [dQI xl]E 00(DQi(rq:rrrrizrw nou 6lorv 1p1oru'g t<1<rppo16g.:t dui,tiQLorl lr€ 1( o!,tl0!o[ar,a 6r6].ic((fiq0it$d!q; fQqqrr,{rt! A),y!:6Q.rE. I o1rr.z{ 0€opic b€v il€r prqrzaq cd.},("/iq.E1|1 (no r(1 r{oiursalr Xtacrzrlarmrz(i ro! oilroroE r.ivcr (jrL I 0rr)t)k1 ci!c!;r1xi.4LurL cvcpciu lLu nE Fq,(epoydg. Auro nou Eilar br.rqoeetrzo €tvol rlytrt :tur4 tt< 4t(l orl< 'Ior o(t8itr€r.r. o! IL(1 vilr o}ioftil. H antl].oqt rorCruss turozr.i trEy(),rirsal oi}rcoi(l .rro ro vo 5poi$r' (rldrc trcl 6dol rou t(i,eorryactr$dl. r(r I br(brzaoia r|,Jv crnm-Schnridr bev ti.vur povo pLa (;ro6uL!1tor ritt,xh0€ rnot(oaoq lt€r opOozcrvovrzrj 6cio1. Avri0e-ro. trlrog lorr olr or (dy6Qr0Fo(cruroi tiluL:rpri.4llrr cr:raQoirlror, xpr'(!61rorJrr: xirL pLt1 ,.'irr.hil't].rt ntplyQ.ttt)i. l / . r ' 1 I r 1 R r r p r 1 y , r n r r , ; r o l r r r l o u v .

:to zaqrdLaro auro 0a 1,i!ouv tt€prrd azotrit drifclrcl rpoc rtv i.b'( xor€r].orvort. Mo! q,Giv€-ror orr, r(1 6rtucr(l u!r(i Inr(r(Quouvr.rl .!no q! Mo),oylorlr'riicv(lzuLoqru l(iD,ov rcQil cno flt! zo$rportrrr, ti&r 6€v qiao nr :rp6reL vcrlrt too, orhnvdr$rt i,i (ur6 $ordlolr' zti:rtDq e!rht4[6va, nlt6 6ftl_( eiv.'r ).h-ooq. : o]]Tfr EE€rrliolrar tc :rcd.crL6repa xcr :rLo 6e$el.rd;h :rpo6Lrl;rcru rouxldbor, rot eircu or rlrodoeq ,h = b v.cLL A:t = )/ orrtv oloic 6grc, ri.iOt6vc cm audr rc 5rynra crro-rriei i:1uvcw4 trlq noQolol( yu!ri1< [o0q[(rtx(i)!.:rrlv Aprotrlrrnrt Ard),rol udaz€r 6vc Ei6o< e;rL6irLrol< ror tolrrQoraQo! lrcLOai.or1€ \'c r€prypirqorpe pe-orz6,( r6€F( nor €Iouv tnr6r(i)o€t UaIQL r(i)Qo. AlttiExtl)atLowcft oE ra€Lq o$irb€E:

L TExvrxit Eitilr'otg r4g,{r = b. O cr),y6pr0[oE ror Gauss €i\,cll ilcE til.€LoEc),l6pr0tLoq. rzr6q (x'rlv irsQinr(')ort rou ro r8l(irtpo nao6Litpa €ZEr xti-

Yfl OAOf t:lvlol ME mN.\(E!

not.; €!6txdE r6roqr€E nQdrruc nou ouF6(iv€r oX€b6! Fr xrio€ nA66l+ra.H otl!,nfltori Fra ea €nx€vtEr0€i mtr r6r6rrycr ql allukjrnraE. drrult 1tn€Qrooor€Qo fiorl€i( ro!,,1 €iv(! Fn6av. zs06s

"or orl! av(inruEl rnavaXrr.

nt|x6r, ar api6av, pt066ov luorls rfi Ar = r. Mlo €n(vd,n^rrlrl !ra0o6o<c1,'robloe0dv€r(r|". €rovdlup6civol'ra< It( 6rc006o€!E E(vd r{or Eavir. AEv(fodv€r ror€ oq\' arar6rt (ndrvrlot. oroloq 11< oFo.rE sivar va rylv nLlorci.o€t roturr:ea aro rlv (rJr(ltroLq)ri. :€ FEpLX{i rAo6l.iF(rc1 cruro trro0€i. vcryilsr. t:vdr or nollir dll,c 1 cnal.orrprl, erp 6oov l"r4rtidver ur' 6rpr"v r1< ralllb€!!x(i, €ivur nro oi./ouQl n(l[ filo lQtfopl. O ow(ryoJ{ofos bEv qcri\rer(l!vcr ftI€(i!€t 1{(r ne61o-i orotoE [tlG €iv(r rcr av(xa].trpupe q owoyPiqrou €E(oqclii€r or,1zl,ro1 oqv:rpaypattzrl Lt'oq ,4 rl zm xcreopilet qvtctutlto rll(. Ksr6jrrv. otry n(lAdryA(lqo 7.4. €q(atroloul8 aurti rI ow(ht,arlmqr, tr:rrpycldrpooq tctt or r?,l,or5 r:rurul,rlrtLzox zarrivrE.

2, T€Xrtxi; titilrrotl€ r{E Ar = .1... To nQodLntrcl [bro!!tr(iw €ivclr [K( cn6rL< olpcrvtu6; otrLrlxi€< xrts Aar0FyFrxrtE Afir).1jorts. Ti0€rur Eezdorcc1,q rrror6cr6trpir tor eivar npo<1crulg. 6${,)G BiXpL n06orfctcr zcrvel( 6cv{E€Qc fi(i,q vc 10 }.rto€r. noor(i0ixc! berd(b€s oL1,oQi0t&Jv, no! qt;olrdl €i\rcr(bri!(Iov !( blcrlaXeoiv c1.it)r66q. cno rov x(),ir.eo fiQo< rorr X€!O6i€Qo.bron tu ;(d!lc( €E(Or(i)ficr (Jro ro l6y€eo!. rr! r6r6fllreq rou ,4 ,(rr ron),ri0oi 1o)\' rbrorrllor Trou (W(ilr€. ME dri,),( )'oy|tl. r-i\,cr €nrztv6r'\,o \,(!$(4r€ Fru !,'toQowi\,( (,'ro €v( lJnol,oltoflxo xdvrQo. /,oQi( 1,( E€QolllerfnorF yr"(( ro n€Qrexolevo rrl:. (ELnilo vcr Flv XoFr(orri. lcr d.6yE€rFxiror evroLrl rIE FORTRAN.) Arcl.6lr4re 61lo-rprLg LbdE nou le'r€p(ocrvol€6d\i oLtq rrq nQonyorip€!(< rouq: tur a)\ytjJQtqtn QR,In1, o!xo"/dver( r(|}v"pt0ti6uv r/ov 'rrr(rpEav"x(L rrlv rQorn€E€Qryooic av6q o!$r0iQ!xou nilrox(rrou rov r(i!€r ?pr6raydvro.Or buo:rprirrrE pt0oboL rivaL r:rcvcl'ryrrLxdq nri r1 relernuin eircl ripeorlrcr o),ozi'rlpriwercr o€ nrn€e(op6'\,o er0F6 6ruirrtrlv. Aev zaraLrlyer mLq[6rtg tg tbrottltd; aLi.ir tr6vo o€ eva roLu crrlouotepo :rivazrq o oroio;x04orlronorsalc! or.1 en(rld.Inrlxdl 6{Fc1ra.

3. O 'Eixiqt xwaoraotoa evit nivaxa. H llaadyaoqoE 7.2 Eflt€Lp€i vaFclQnor( fllv 8rJc&oonoio n "!:uni(o€Ld' $!aE ),uollg: Edv oLlilouv overai'ootr(r r(r / xaL b, rooq ei!(L rt crribQaol orrp r = ''l-rl: flQLv (rQlioou!€

F' crnri tv €pdrtol, o€ioutr€ vcr qovsedrooure 6vc rtnobro (ro q€n€Q-vLirm enxoLc). Ilp6:rer vc u:rirpltr rp6:ros va parplooupo rtv c ,rryrt 6,4zcrL vtt ertrFrloouu€ ro ibro ro F6ye0oi rou A. To tolLo< 6r&vuo}(.To< 67€Lt64 oqLo0ef ;rcrL tdpn xperclopcrorE q oraettrla cvti; :rivaza. Eror o6€ifilg rarcorao€co: xcrr I €r'cLo0loia rou ,4 0a :rpoxtitlrouv cl.r' €u0€iaEon6 tlq oti10!€! rov A zcL n t. Or riwvtq o atft) ro x{tdJ,ato eha

H:T]\OI\IH K,\I O IEIKTH: KATA>T,\:EA: ENO: N]NAKA

7.2 I H 'TAOMH KAI O AEIKTH' KATA'TAIEg' ENO' fIINAKA

'Er( dqditL{i x(tl s1(r Zolrpd idr0o! €il{rr roi S(1(t,o0srrrri r!xii,!r(r(. To. ' t { t ] , , . , \ 1 f t r . ' r t r . } C t r ' , n i ( r ^ n r n l . r H , ( ! " : ' ( \ 1 , n o l , ' ? r l ' , x n t . ' t z ' ( . ' ( : T oircr r67,:Lo r.1eltr.1.ftxo rt i!(! ri),Fro 1]:roioYtorj. E!.l /ovra6 i.iooi Eivflrroir oo6(a6r€ao xuL le1cr),urpo. rcrrir lrcr rir!1 fr€lfeorq iorlirxroro!. Orovo rroiolLorrF orqolj,ri€n€r 6vftv clarol{o oro olboo brzcrbrx6 r]n19io. auo eivarirc og,thiju. circr or';< tvc nso6),t5cr rivaL rcioo rpolcarrzd nrcrio0lro. tirorr 11orgolluitL,ort 1cl ui),cilrr ruiticrg 14 ),ro1tor. rorr o7e6or oilorpcr rci:roro< iTrLxdlrL iltr /ovrao )d0oq. :ro1o; pcrc o crLrtrl try :rcrsdystrq,o €ivur r'crc),riooL,ptrrr r;ri6Quo! r(';v oq,a),lt?ror. (ror ti;ort- v c;roq,€r,1ovr(' xolrpd ),ugl.

Ir4r oroiu orrrliior,pr sLcr otli1r1ort;ror riplLor oro Krqri),c,ro L pe ro!<

Iolxqrorrtzctrr orr o .l' rircrt or zairl xorirorcrol z(r ozt rblcirrAo rlcriogtro< or.or!,oy11,],o'roirlort frr(i( f(il rt (tn(ior(lr] causs ut(0trq'rri I a!(r ilii0ro ra(ino.oir(irr o ni\rri.(q livr.r(lr rr.ir,irr< r,Linc(hj.. Eiv(L 7o!r0(;,rq(iitr( !( br./{)(nlllr.ro 0.(Xl0l ! r< r0(;)ro odl i ,d. laru. npi :r | r vtr ! rr f t t i \ 'out l ( ,ot ! (r t r t l ( ( ] , t rr tQ)l r( l [:no ,riyoripi) 0nLioi,'1. L,1,(ii(io(Jo1rr(( n< i,aqrtraq ror, 1l'. Or(r 'l ltrorla1lo6rjllorl tvoo4rtrrrirvrror drol' oL./opLutro olroioqr'F. onort o r:roirryLorfcxol:r,iif|rL,rotriTl,). ?(l rov $r1(},rirtpo obryri. rrirr br! br{rzrvbrvFi,fr(r )l(t r,tnxrl irlri.trir(i'l .tro (rl riitrrl orrnr/yri),rr'orK.

I ldr! tLrra.itrf rrt (irliouoj ( rrrj .1\,rior(oil r(r JI6( (:ro(ir(oiqorfLo 11! o ni.1az.<fivl' of 1t r7,rl i zcz'] r(r,iirrci'lr l-lii\, trfrroirl0ri 7,i1, ro l, rj o /1. n(n'o trry,iirlfilcr rt lLr.r(16o),rj nol nQoxrinrr 1r( lt iljq.fi

:urvri1u. pr prtL d.1a'1ti ttt< itiuiq aLtryli.. (1x6 b o. 1t + lr. T,, o(rll"U(1ariro tLiopci \,.r nQoiQzcr.( an6 :rrrQ.lu(1rrrtr &bot'a\,.r ii (1no orpo1./uirro'1.lvlroaontrf v( rro06ao1,to orr ro 16 fi!(r trrrao. .ri],.i rt r.1rrn0!!o)i ro1,.i.,'llrt-xr6; rol eLilTor paq. H ).nol !L[1(6c(),isr.1L a:ro .r oe .r' + l::

l(.r' + 6.r') = l, + ll. hpcr rr{rcrprirvrclq 1(b.r) = bl.

Arnj eivuL prc r6Luirrprr rr;).r'1 nrpirtotrl. Oeorpougrt ti),rq rL( 6r1r(Airli<1rb z(L ezt4rdpc Ilt\r n0oknnrorio.r 5r.rr(p(1Xri 6r = I 6b. Eril o I eirnLtr€j,dioq. ror€ 0( npo:Ev€ir(L tLEy.iirl .rii.lyri orl ).nq rorE o -{ €ilar oX€bo1rbr6;roprp: xcr lLciirmc ctrl Oc eirct rbrcrirepcr selcr),1 6rcn'ro l| eivcr mqvzcrrer,0rvoq :ror trroxr€rc{ :rr-prooor€ao cno ror ;l

A: ltzrrloorSr Lr:ro0irovtc:6tr o I tiv(r otrtrtrt-tpur6q xrrr, or r6Lotrut< rolr i rur Ottrzt : : 0 < )r a , \ r < . . . < ) , , . KciOE brciwotcr 6l c iv(r ou6ruoudq tr , Icvtrotoi lovlLolc6LcinvL6ro6rcvrol i rrror toxtr-oorrpooq,di tr t rorgliuivtr orlv zcrrer0rrol ror, :rpdrror rbro6ruvrotruroq :rL. O ia-odrlrv ruobri{;,!rL orL ru ll zcr lr eivcrr lLrrpr?:

,=ll 1,,,,,,,]

eh t = at1, rc;rr: r|r = 11,,/,\1. o)

YflO OTI:}jO] IUE NINAKE:

To liTEqoa ro! aq $arcg 6b' teraxnEft. xa r ll^b nol tivu q tttyd.ir|arldro?a'i ror,{ '. H erioporl rivcrL siY|orq, 6rrrv to )r EilaL zovrd mo [I6€v.&A nl.ot oL rtu tlJuio9ltor tit,ru or ay5Liv LbL6popqot rlue.t<.

A . r t t r t . , i r r ' l o r r u u n A 4 n r , a : , / r r f ! u t r o f o t r r l o f , / , r t . i , . . . r , ' , . i J , : , i ! ' ooo6(ao. Eol.) orL rroli.r'riaordqortr€ -rcr oroLx€iG rot I t€ ro 1000. Torr zcr ro)r rk1 roi).an),aoriri€rtlr tr€ ro 1000 x(lr o nil(lr1(! 0(1 q,(il[rar ro).n ),ryor€aor5r6!roa(foq. TollTo lLcg npoo6irD,r'r ritl (io0loit io! rilror norxlrbtot. Ar!, r . r u , : r r L i , t r ' L i : r ) . f . i ) ) , " f / ) . , ' , r ' , ( ! , ( t r " r , ( r * r r T , l l r l t r t

" , r r , ( o r ' , o ' t r f o c

n | q l , . 1 o | " u / ' l l ' E r r L r r . r , ' r 1 0 r r , t t r - ' r u " ' r 0 ' ( | | ! . t | | . | x n n T | p o " ' l T u10(Xl, (1],1(i ro ibro e(l oul6ci!€L i.clr trt: rrl ),ro1 r'=1 rb. To o1€rth6 oqhi$tll.f / .r' 0c €ilcr ro ibLo. O nupcilalv ;r' oroa raeovottlorri xavorLzo:torti

ro r0o6ilf(c zcl| ro brcrotpcr),iltr cfto rr,rQ(trtriv.q c}.icyaq x},i.ltcr'i((. H |b(rrcrrorr"xonoitor propri. ,'tr lirr-L x(Lrl(j/porc 'rct ylcl ro Ib. 0;16r! ro ra6dlTrtitr(< r'ir(r 'f'( ori,i{pivor\gF ry axtwti ptnfld.ti 61, / , gF ro orfttki aqi4t(

t . r / / . f .H XrlrQorrat :rr,pi:rro)(rq tiv(L titcr ro A.r' rirar lrtl(i)'o -o1 bli1r(0(lxirq

ri1orN ril! x(1rr.llftry l ror rbrobr(11,rrotLtlro<.fL- x(! or(r'v o ri(eo1o|lcorrl( .rrivtl pLnociq. : rrurf np jr€0inr(')ol rt ttit b(rr(a(lfavrl )'iort.f (kr in0exE !(18i1(( oooa ro b1jaor6r |11}106rr.0l. ot: olryrlQlorl [f ro lLl b|(ro0(]!avo r. Aur(iolptrirtr orr ro qxtxi ru16)qyrt -4.. = b aa iqtrlt tt thut oro tffitt riz,po.l l , . r , ( r r t ! . , r , ( r i r r , v , , r t r n l r r i . r . r , , r o . r . ' l , r r r , r r l r r o : . r :

Auros o oulbrlaofLoq, tt = .L:,, lu IL = t .r 1. 6i1cr ro Uayd.lrrpo buv.rro olt.I'zo' n t u ) l r d . A ! r , : | | \ { r d r ' o t u ' ( . r c : r r q r : T r " o r r c r i ̂ r ' o t [ \ ' u \ u v ' o o r r I , ̂ :

7A l-!o dvdv o€rLrd oqLopdvo niva1rcr. r1 ),uoq r' =.4 rl rC! to orpdLBnJr =,'1 '6b rzcrvo:rontv nciviore lrE c!(oorrF€<

e c r r l , = . r ' , , , r 6 v t = t \ ' t t = b l s , , . ( 2 )

l " l > It, r c , 6 ,1 1<61,11.\r

l'vracbg ro oFrLx6 oqal.rc gQ(ioo€roL cfi6 ro

6r l , , \ " 66 l lil,f : \ fr,f

(4)

Au6 ro :nliiro c = ),,/) r = I,,,""/),,i, ovopirleraL dEixtqa zaftaldoxut r.o\lA.

7,2 H:T,\OMH K I O AEI(TH: KAI'A'|A'EQ! ENO: NN I<A

IAPAAf,IIMA 1 Or rbrotry€E rou

f 1 I lA=li ' .obo'laivar. xcrti nqooiyyr,oq,\t =10 a/2xat )z=2.Zu\rs?rdq o 6€i,l{q5 xcrroords€6Erou ei.vcu. liqo oro c = 4. 104 ]ior 0o fiQ6ri€t vo nepr,p6voupe 6inr,1 c).l,ay1 qgI!o1E cx6p1 zaL 1a nol,u orwrtolo}€v€g ai'),qyiE 6€6oUiv0)v. :ro K€gdl,aro Ior,7zpi,vc1rc rL< €E!od)o€$ Ax = b xu. Att = bt:

t l+1.U1011,+2.

u + t ' = 2

u+1.00011l + 2,0m1.

Ot 6€E!€E nl,flQis [{,ra6d},l,ovr0t p6vo rcrd ll6b | = 10-{. Tar,"r6tpovq, oL },toetEB€ra6d),Lovror o'n6 u =2, t; = 0 o€ ?1 = 1] = 1. 4!16 sivs! 6vc oxrr${6 oedl,lrdqq iarqrt)q

' r i l - {-l l) l - f -^' ' 'oo'1"" ur 2 l{r ' l !!!: l 'l l . |

= l (10) l l t t )

Ail(os v( 6XouF€ rdw€L r.i(noL(r €r6rxli €fltloyri qg 6rcrcrpap]E (ro :r gcrE rcrL ro6l o4yatilow y,rvieE 45' p€ rrE X€rQdr€e€s nsQr/rr(iocrE. ,retrylcr nou dytldv€rsrpe rqr iiJ,errpl ror ncgriyoYrc( 2 p€rcEiJ iolJ 2 l0r xcrL qg nxqoicrq bwc.r6qrcEa=4 l0r) o!v(6t Fro rpopeprj crl,Lnlioq l,uoq.

tn$sL(i)ore 6rL o 6€ixrlE x(rloordort|)q r 6E\, €rnlQear(€i&! dlreoc cn6 ro$6y€0oE 1o! ntvcxc. Etiv ,4 =.f t oxdprl A = 1110, o beixt4g xctcorrioeoSei!(L . = ),mr/),i,, = 1. :€ o],yxQlql p.' ,urr.6\. tl oQitoroa arot ,ei atraiontxrip4o4 rou oo xani dvar 4 xrtttiontt4. EEaQrcircrr o1r povo ar6 qvxL4rc!xc( dJd xut u:r6 rtv rdrEl r. Eir! ,4 = I/10, r6re q ogilouoa rou ,4 eir,<rL10 ". :qv nQqlfir)rdr4ra, arno! o "q(e66v rb!6uoQpos" ,liv(rr(.s 6X€r xnv r[!ozri.{ 6wtmi zardmao4

IAPAAEI|MA 2 O€alQrrlole tov ?r efii. n riavoxn nenepcoydvcrrv 6Lnrpoprirv

A=

2- l

- r2. - 1

- 1 2

H gtyclwtpl t6tot$ri rou €ilat li = 4 xar 1 pr,zp6repr; eivcl yip<r mo)1 = rlln!. :r,,.lstlog, o 6€it{1qg zaraotd.o€6E rou eivar xarir rrQoo6Tyrorlc = rf 12 xa," qvni q qopd n €Erieqol atr6 rlv rdrErl n €i.'\dr tvriola. Oooxtrl,rrepc npooepilorpc rlv u" = J, q!'Edlo\,log xo ii),ri0o5 lDv <rl.v6ottDv.

YIIOAO|]:MOI ME IIINAKE:

rooo 6uoxol.6r€Qog li1,€r(rr o uroloyloIos rylE rreoo€Wrolg 'OX[ Fovo 6roQ-

x€i nrAroo&€Alo XA6vo, qDd r.uL €nqQ€dtsrar n€eroo6r€eo afi6 ro oqduc!oralo'fyuL€uotg. Ano 6v(l rieioruo oqF€io r.oL n€Aa, Fra (tnEryI rou r, orlvna.rlFtrlrz6rrya. 0(I o6ryrio€! oc iv(l n€vlra&€ao ffior€},€oF(r.

Eurux6f lLc ro\' trq/g!!x6, ctw6 to trpfo$o orltr€io 6eior.€tct sx€f rou11 <rzpi6cLc eivar {bI aQx€ril raLi. EraS tunx6g urololoqq, 6oul6uor.rosl€ cnl,rl uzAi6€rcr. FnoQ€i va xi!!€! oqa).tloro fiQoyyir,€uols fili rirE€(oe rou10 v. E(iv I rQoo€Wrol XQrlorlonolet 100 a./v(ifioug. ori6l€ c = 5000, i6r€ 6vorsroro ).(i0oE €vroxu€rsr drorE lcr €ivctL, ro r[o].u. TlE rdE€o( xot 10-s -noox(rr jrril,L eivo! jrlo oxQr6lE aa6 onolcorir[or€ owl0wl6rq f6rQno4. @cr urcrpler6F(DS ra66lqFo p€ 10.000 q!6oro!S I p€ rlQoo€lfio€lE ,lrn€Q(rolavor! blaqoQ(ivelLotioeov pelc),udpug t<i[rtog, 6:rr,6 q drl/dr{ = /(r), 1lo rtv orioio o bei,xqExcrcrorhoeo6 crl<ivel dl[o( ro fl{.*

H cvtil.uorl pag egcrgpol6r<rv pdlqr rdrgn cnoxl'er.mr.rri o€ oupp€rQLxorlgnivox€E [E g€1rx€E r6rotrpaE. El]koLc 0r. pnoeouodp€ rc naqcr].eirpoupe qv1)16010l rlg 6€1[x6qrdg nft \,( XQq('r[onor{oo![€ crio],ur€E rr[aE, o7i61u obefxqE rcrtcrmrioetoq 0C1 yrv6rav . = max lli / min l), . fLa v' c!'ropuyow€ Er{D<rar tlv urr60€ot rrts or4Lpergi.rg. rd.rl nou eirnr oiloupc uu0qrry6, rqi:rer vcxdrvoupe prn pe1<rLurrqq d.Lqr1. Arn6 qcrirerar edr.o).o orol6 nivtlx€S

, = l,i 'f ] "", , '= [i -'J']

O! !brortt16< €i!(r xclt o! 6uo l. d,ldr oiyouQ(1 b€! Eivo! a),io€!o 6Il ro oxnrtr6oq)dl+( ro! l qQiloo€rct an6 q oXerr.xrl pcrc6o),i rou b. O xardl,].ql,oE 6€ixqqxaraordo€or( 6€\' €ivcr ro ),,,"-/),,,,,, = 1. :qr{Qivc.c rLE lus€lE

fo l f l {n ) l f roo l , , f roo lr=Lr l or4! , ,=Lr l . . '=Lol o.n ' , ,=/u lM€ra6o),li l% 1('! , neoEi:vlo€ enctovtcr:rLdolcr crLlcyrl rou :r. pr ouvteLeotlFrliowo$ 100:. fnflbyl ro . ,rvflnQoo(J)flru€r tva {ivd qQdyFa uwrlE qcpe1eowcqg,0n np€neL vcr eivc4 touldrlLorov 10.000. H 6loxolia F'aurouE ror,srr[!.rz€E €fvsr 6T1 6'\,( p€yd.o ororx€io €rro< 6rs./ovio! mov A. oqBoirEr tvd €Eioou peyr .o moLleio orov,4 | -o€ avrio€oq !€ rI brsloorlrlz{ aw(},i1nl,6rr o,,1 I 0n 6:rpere vc sLzpoivsr xc0(i)s (uE.ivu o A.

l-ur vc 6porige eva oroor6 opLopo tou 6eizrrl xartrordroetoE, redrl€r v<rE(.\,(6o{g€ rrtv Eliorool (3). Ex€f nQoond0oriooU€ vtr xirvoqre ro r gupo xcrto b = .4r' peydLo. (H or|Qclia n€pint(oorl ow66ct!€ yra ro |6robrdrvuoua .r,,,6jrou o )dyoE rou ,4r np'oq to jr itco axQL66E ),,.) Aur6 :rou clJ.rileL 6rcvo A 6ev ei.vcr, o44rerpLx{, €tvct 6tr !., ltiyrmo rou lAzl/ lxl pnopti t'u

(5)

* o drsrorq quErplxo< aovowq. ron rrahplulrdr x( xHaqdrLa.i. Eiv(, on o Molsrhk

Zrilc! log. 6Exu6rzti oEo€rq an6 rd ltiLott orQot )..r$ts .np urd.oL{iat Caus.

?2 H >T OMH KAI O EIK1H! (AIA>]'A:EQ: IJAO: ]T1NAK?\

|) l < lra I lr i ) isL .1$l S I A-r 6bl.

6 t l - ^ l 6b, t - ' - ib

Edv brdxaQdEoupe iov nivoac ,4 ovri rnE 6elr"dq 'r),EUQdg 6, 6xoupE

15r l - -116AiFT;ti -']7ir

121

).a!6.11'rrur o' ivu 'dyxota r rcu 6ey fuar L6n6 1tuopa. To p€lloro ovro€i vaL rdlJ, 6r(r EqcrQ€xrd gtrpo rou pey€oor,5 rou A,l,i\ar,.,r ordqpn xor itt\.\z.ctraL ou[6o],i!€ra' F€ 1,4 .

'lB H ardgpq lol.J A EL\dL o qQrqrds nou oQlqero! an6 qv

, , 4 t = . * l 1 " l ." ' ' ; ; o ' l " r , l

Ms dl.l,a 161n, o ,4ll qpdooer t4r, "evLolurrx{ 6dvapq" rou rdvcrq:

| ,4r I < l,4l | :n I .f!o rd6€ oldvuopo ie.

H ro6irFo loxie! 1!d 6vd rorl.dxroiov l1ll prl6El,Lx6 o.

O[ nilclr€q ,4 rcr! ,4 I orrl! €Ei.oool (5) 0c 61o!v ori(eB€E ri(no! p€rcrElr1()! 10(Jzc1r rorJ 101.:€ li./o o(l rrE riroLo./ioo![r crQ!6clg, cr,L(i no(i)ra 0dlowevo o![n].lp(i)oouf€ rI o1lor1 perulu orcOptlv ncr 6s!zrd)v xcrlaordo€o5. Efi€L6rl| = Ar ncrL 6.r =,,1 L6t. o oQLo[6< (7) fl]E oide0rli r|\,oxo bt.,'s! c416oo)g

(6)

a)

(8)

Au6 eivcrL to unorcrldororo r$ (3) 6rav o ,4 dev ei.vor o!FF€rQLx6<. :1!oq4rcrpLxrl reginr@ort I 1,4 ei.vcu, 11 i6r,cr pe qv ), rct 1 lA-r si.var I ibrq $e1r1',' 1/^r. To oaod tnoxtqaar@o @t ^,./^t titwro yudpn'o I A A-1 ,:rot titu o 6t:ixr4E xrnamTorc<.

7C O 6Eix|'.t6 x(rNsaNdlaEa6 ro1, ,4 Eivs! c = r{llllA lll xcr, ro o1enr,6oqirl+rs lx(tloriolsi. fil!

(e)

(10)

H Glrooqra (9) €ivar ro "/LvotrElo rrov 6uo ovLoonix(!,! orqv (8) xcL LolneryLa xd08 b xor rdo€ 6r. Eivar clLoolpeitoro 6rL o i6rc< beixrqg )iolaordo€rDE

YNOAOII'MOI ME TIINAKE>

€lqavi!€ror / o[ oq (10). 6ra1' or(rrap{ioos[ar o i6roE o riivcrlio(: Etiv ,4r = 670, ( 4 - | A)(.r + r.f., = t,. rorr aq arp-vrcg iXoqrr

o noll.cnlooroog6g F€ ro 61 €vrotlle! ro aidvuoFa 6 mproo6repo cr6 tqoriroll 6.41 x(! perd no ,ortlaordlovtcg Fe rov r{ L dur6 svwI1i€ic! 6I!ri€Qloo6ieQo 0/16 l'4 | . Xwerrbg

A6r+6,4(r+5r)=0 rl 6t=-'4 r(r5,a)(e + 6z).

l b : l : S / - r l 6 , 41 r+6e , ,

rt

r r t 6 A____:]- < n-r l ._4 =._r_-. 'j x - d . r l - A

O! crlro6 y€g crr,lriE olpoivoov 6tl to oqd),pc orqoyyul,€uoqg fiQo6p1€1o!<rn6 6uo :r1ydg. H FiC1 €i!a! i qirooti €uatoeqoia roo rQo6tpqros, rou

[€rQoir(lr U€ ro .'. H d],Ll 6i!dt ro ,iQcr'l,il(rrrr6 oqd],$cr ,, yl 6,4, Aurl tir<rv r16hor1 rlg crvd).uorl; ocpillloloE ro! wilkinson. Efi€(61] o q],y6Qrqros arsl,oqnE,o W ri0ol[orLx6rlic(. naQolE[ nQoo€yyioelE ,/ ]t(lr U', sur6 ?ro! rdls! €i\rd[ !oLuvE! rl! €Eioooll tl€ rov €oqolldvo r.L\axu A + 6A = L'U', e\ni ,rorr oooloini'runtt A = tlr. O wilkinson lorn6v snd6erEe 6r 11 pepr.rrl obrinol eivar){cr(iLtr{Ll yLo l,cr xQofilo€l ro 6.4 un6 t),e11o -6€irs ro dQooo to! "RoundingEnln ir Algebraic Processes"- iror o 6eixr4g xurumtiota< . Eletr 610 ro6dQoq tur oqdltero< oxQoyy EUoqE.

'Eraq droE 1,rs irtv oidoFrl

H fi/roFl xou ,4 F€rQil rI fr€yr.oxq nooorlro F€ rlv onoic €vtolu€xo!onorobirrore 6rdvuo[d (L6r.obr.irvuopc r1 6xr) ;ro]"l.anldorol6p€lo $€ to\, nfvoxd:1,4 = max( Ax I xl. n orciopl tou rduloflr{ou ni.vcxc ei.vo( 1. fLCr !a

urololiootpr au6v rov 'ouwel.rmrl eviopolE", lrtlJ(i)\,oup€ r.irrt rt5 6uo n).€uQdE

l A r , t r A r A . ,/ r ' - max -; l - - .*

. . , . . f l r i

Aur6 gaE oblT€i orov oup$€lQl'io .4?A l|at ro ,{crrd Rayleigh rl}.irc ro!.

7 . ] I I : IA ' - ] \ IH XA I O AE IKTH: K I \T ' \ : T ] \ :E ! ] : ! \O ) I I INAKA .t:9

7D H ordo$I ro! ,4 €i\,or I rflatryor\rst oila qq $clrrlnr€alq tbro.i'uitsrorl 41 ,.{: llA = 1,,,.,(,47-4l). :q! n€Qiri(ool oqrperpLzor A.,1r A = A)11a' I ord0trt Eilclt I rrEyd,{rFar (:..r.E.: anoiutDc) r.btottrti: ]lA =Inax );.:s ]ttro€ n€airr(|)ol. {o drdvloFa riou €vroltrErt{r ir€erooort-Qo €tl(ll rocvrimorlo rbrobrrir, r-ropa ror ,47,4:

t T A T A i . r - t \ n L , r l \

Iqtttkoott 1 :. ngrzrrzri xao6irttr(r(. rl ol(ir{)!(l z.rt o 6eixqg z(raoriot-r'-'<brr u:roioli.iovtar;re(yr,({Lz(i. rrLiri lLin'o xatti apootyltol. brirrL bul r,'r(ia/!lzaoloc r( ) , ! i l€ i al( neoi, i r l t r (1 Lbrof l t t ( ;v ( , )c neo< r l t ! . \ " , . , . ( - ' l /? l) .

vlttrilto\ 2 O bfizrrtc 11(r(or(o!(,)c r-(t(ffr(jitrrlr ort( xcrovrza< t:ro(iotl(.-l / .l.r = .-l/ ll tor nporiir}r(it ,\, fI(17ior"v rfrr)uyri)lo)r. O l!izr)t< ,i{1r(or(i(r'(Dq,111/,- l ) f i lcL to trplyato rotr ,1. ,1). O ulr}r t rrrotr t jq tor, '1 l ;1. t r : ropt i l r r

Irrcrpilrr.r nrr r,lLi( ira(it;),r}Lrr o ivr ttd)f!i: r'r(L trnoa{.i \(1 xr-n'L((trfi v(l

ta)t( l r t rononl(n), ,1, f i n l rLr l r Iz{1oi( G,!n,-Sch,r l i (h ' i r r t b ' .1or(r{ ' l l k\r(4rd0(t( .) \ 'rqrrr ' :1 = f , l1: f .11.

tt ttioo,t 3 'lil rJrorxFirl o, ro! bft/dvtoti nia{lx(l: tL!i{ {)L rjujltoet!rs rrtiEror, ,1 zor ra t ry i1,ort i nn,< t i ru or tbtorLl iq tui ' At .1. 2tr l ' [xd(. a\ o( di) ,oqrt,ror yr( ,I)t Lrrritltrf fiv(r ,t =o-.,,. oL oQooi,dv'ot Qr rtt r?r tqilorry rtt

fLrtxrt t1!rfr(i6),itr(1 orlv I | = QL:QI .| rat 610r o trtydiirft'oq LlJxurL,a6(:rapir lorrc< eivtr ro pi loro o.

l,tlttioo,t I :q.ri)+Llrrrt mlroy./ril.euo'l( rn€roiaxovr(r 6lL lLd\ro afll! lr = ,rri),ri zut or4v lr' = t.f. Arlro fyr-it)r-r iv,r v,io tQdrr)}L(.: Ilotoq Ei1'cr o'5€izrl< rlcrcrorirort,)q ror, ;rqrc6)"rtp{1toc rSlor+Ld\t i H naoqc!rt< o;rdvrrlorlr-iv(L i(v0(1rltrrrl. Arn6! 5!1 eilc(L o ibro( o c\t-izrrti it irrrior('tr.!,E ro! -{. A'vriyr' (rr6v. rfr tlLo9ttoi& tu,)r tbrorLrtbr pntJri o r\tixtttq ratuoraa o< rorr\ tn ' , , rovtotonlvto< (eu S. En! l , . i ral FLrr Lbtotqut ror,1 +-8, t6rt 1, . : r u n r L i o r t r o , ( : r ' i n ' , ' u , , 5 r , ' r . , ' t r , , , L i \ u l

1 , , \ < S . 9 r - 4 = r { . 5 ) E . ( 1 2 )

:rltr;r0Qirrr(,)ort:ror o S eircr oq0ottb\'roq r(i!c11c! Q. ro rrp66irlto tbtortt(irl€i\( o€ dQrorrl vardor(o)t: f(Q) = I z(, r fLti.16o),)i 4 ) orL( tbLortfLt;5 0i1'.rl lr0"/oirr€prt an6 rI l€rcito),)j E lo1t n|\'avl ,'1. Arro orftociltil6tcrr rcr rbrodrcrrrqcra :rol eircr or oriiit! ror S Et\'.r op0ozcloltzd. IIzci.ueprl :rrpimtoo4 iomor o4tdrilrrL 6ruv o ,'l €i\,.t oulfr€-ralz6q lt y€vLr6teaa

YIIO OIDITOI ME TIINAKE:

AT A = AAr .T6r.. o A €i.var dvcg rcl,or,t,aoq nivorcrg. o 6LcltovLoaorcirv S ei,vnropooro1,o1,r,roq Q (ncpirlpa<poq 5.6) xor o! r6rci|"tl€q rou ei.vnr" oe elarqerr,xdrcLrt xardorsol. TIv rnporoia roU S p:roqeirr ru 6eirr otov dro 6r"crapcX{qyra rdo€ U€troroU6'\at r6rorqrli: Edv 11 etva|" I A-orri otrll,q rou 5 xcr yp eivar 1i:'mrllpcpFrl ror 5 ', -rore

7r1 ,\1. = 918:11 + 6por tcilx'l< E lr. (13)

)tqv:rpit!1 ro v7.,4e1 etvnr, pr,cr qruLtotrxrl ezriplolltq pet(6ol,rlg qq t6ror,Fig.)e rti0e zaL6 ol,16pr.0po, I r6a(l siv(l! 1,o ,iQc l0Ei o niraxaq t 6oov ro 6!vcr6\r[lxQotrjQoq €rqr€vovrcq ouvrlOcoq va 6Xo![€ opeoycb]ror,\ ,ri.vrrxEg o€ rdO€ 6i[o.o.n 'c 0o xovrr 4 :rr4crlovroroi4un Q,4 ortv rroFfv't n.lofty0ft'ro

A'KHEET'

7.2.r Eit\ . t\ r.ilor oa0oydltoq- niv(a(lF Q. b.iErs orr 0 i = I .lae(i,c ac' (Q) = 1.Or oQ9otdrln rtvort5 (r.ar ra no},Iffiidotd! rols nQ) 0ivar or trovor nilaa€qeEureorrxd rari< afl rdordortE.

7.2.2 noru rieQilil]lut s'rsorrrcr bilcr rrv l(1+B)rl < /r + Bi , r.or trori. onoq v ( 6 ) . n F r a , 6 r L r + a l S i l + l , l r

7.23 Eqryimr Trori lAtu | 3 A al r ullr oupncedllor€ aflo tn! (6) 6rL €In€r.48 < ,11 It . AeiEre o c1!ro o!!.n(iyFr(' ritv .,(,,1r) S ((/)e(a).

7.2.4 yr.r) .uy,urr y ( rov osun oe.durlo i = |

' , l ' ] , . A =r \ . . .4 - \-

"dt ro! .(,'l) = ):/.\r Ba€ire prd b!:L{i n}.!ued I' vor pr.1 b|(lraacri 6i,, 6toL .i)ors ro

d(ri)+t( vc riaaL ro xceoreeo 5rv(ro, 16r ll .,: =.1 6bl I lbl .?.2.5 A.iEr. a)r' yLa Ki0s L8rorll'rl ) rox ,1,1J. = .\r'. rorn{ l.\l < ,1 .?.2.6 RQ.ir8 rLs o4t6.i( oriiel.q ftw r(!da!ra drr (5).7.2.? Afiqxd ono qv Aorlml 5.6.5, orlzAi.volr(5 tE riro4ri( tov A,t A xat I/17, 611

A = ^ 1

7.2.E H 5(io^oq Cholcsky tLo dvov 0E .i oqtdldlo,,1. .i'aL ,1 = LDLt - Rt R,6norR = !6Lt . ^tiErx to.r' ruofiu! uno ro 7D ort o b.izilE roroordoool< ro! t? ri.var't t€xAqouui qild ro! 6€iant rorom(io.or ro! / Ar' clto in€r(r 6rr, lla oft'rdoqLoudvo rfraric. o u).toqr0troq tou c ss 6€! Xa.rdirrar rvd).Lq,ic leoBFr;lr. Huarirmaot 6€! x€Qor€a€!€r, oqon .(,,!) = 4nr ).(R).

7.2.9 aqioro{as d!1nc0ab€iytr(r( 2 sni 2 tur .\,,,"^(:1+ B) : )DA(A) + )D}(B) uoL)nri,x(18) < .\,,,".(.4))-!i(,e). bFiExs 6t | .\Dd, ri or6Fr rcL 1 max l,l/. 5uv tivcLr?clotrorrrL,i( mci0FE(.

7.2.10 Yno06o1€ 6r didEer ro r , cno ro E!"i€i6do p,t"o! ('i + . +J; )L /r m1 "orrieplurl ioror" l i "r- ordorrr" : l r (ndadb.Ll fcr (1, 2,1) n = 2.)Ytrol,o./iots rtt! dlIimorio mciofl flv.irov

. l r ' " 1 - f r 2 lr r , l f \ L r 4 l

H ) lA( ) I t l i , \ t ( ) I K I IE K, \T / \ : lA : f ! ) : t rNO: n IN, \KA

7.2.11 r . i i r t o lo, r f 'k ) f l , , i i . . , . r = f ! r r l

t . ' , ) I ' ' " " -o ' i . oL 'nrorror ' r ' r t : r l r ' l : lo i I

l dErrr: .\ozrlr(iorr roY Er.

7.2.12 ( ) EZoNo,.1 r ( ( -1 | ro! inro &rzt1 l7er , ( ' r , ( , tsr , : :ii) Ar(i7.o],. tr ro d\() qe.il( (9) t((r ro oq.ii[(. (aoiti:rf ro z.ii, ) ,t!(itu(:

. l I i ,, t r . ) r , , , f r t , r r , r o I , , d I 1 . l " r l

YNOAO|I:MOI ME NINAKE:

7.3 I O YIIOAOIITMO' TAN IAIOTIMAN

A€v urira/,€! 6vqS 6€).Itmo6 raono( uno).oyrofoi rrDa !6rorqr6v Evog fiiv(!,it..Y:rirplow 6pc,4 oilouqo yeprzoi unnioLoL tp,oror. irou 6r'v 0a 6:rpeae va 6o-xtlao0oiv no16, zc0d;q xcr! $eprx6g L6deE rou crlilow plc p6vlp1 cvapriplorl.Es,{wo[€. ,ieQ[t,p{iqorroE prd rioh] on),ri xoL Xor6QoE!6ll fiQoodwro{. rrl p6eo6or(0! 6riwiF[(')1,. rlE onoi(.i or !6rofiFE( orixhol( €i,vo! €u,{o}.orolr€E. Kcl1o-nLv 'rrvor4r€8o ow€I6E rQo( €v(v noli fi€QLooor€Qo r€Qir€Xvo olyoQreFo. fiouE€xlvd [fi.r6d1,],ovraE €v(rv ouFF€rQLr6 rdvr(iiq o€ rQi6K./6v!o xcr x(rral{l€rxc{vovrqE ro!, ofil! ouoiq. 6!oydw[o. To r€],e]J1qlo 6tFc fllE 1iv01a[ [€ rqb!((brraoia cram-schmidr.lf cur6 n tr6eo6o( €i.vc! ylrrort oE qR.

H ouvrltlolr6q F60obos rlJ)v 6uvd[€ov leLrouQysi F€ qv ibra (QXrl 6noEr(rL lLo €Eio(,)on 6r(qoQ6v. !r'x!vir ono $L(r o(ryrxrl €'rritnlol lio xor o1r roxit€ibrcr6oltzir rLl ur = -4u0. u. = Aur aar le1'rx6teQcr u1 1 = ,,1u1.. Kcioe 6ripctiw! €lcG nol,LarrLtroLtrop6g nivalic €ni blciluopcr rcrL perri cn6 I 6tipctcr,'rQoeur1sr ro l/t = /'(ro, ov r{u! o jrivazcrq ;1r 6ev elrgar,itelclr xcr06},ou. :qv:rpc1potLx6r1ra, ro o oLdb€! €ivcL I onlduF( roo jrotrl@rtrcoKoFori !s rov,4 -et'n o:ri.rcxcg eivcr pqtil,og, 0o rlrav xcLur€acr v.r rivct <rpcrog- 6L6rr. r1orryxl.ro1 nqo6 ro rlircbldduotro [iv(r o!X!A notri ag1r1, Yno0drorrrq 6r o ,4rXrt evo d.rjpeg owol,o rbLobrovuoFt(r(ov .rt . . . . .:r,,. ro fudvuops ur ocl div€rsru:ro rov.rrrvlHrq'ivo ru:ro yr< elior,ro4: 6rrtqogrir':

1r r =4r ) f r r + + ' r " ) f ' r "

Yirooior€ 6rr o! !6rorrlaq tilar sQr0!q161,€E xor' (ltouoa rdEI rclr 6r! I [EyrorlrbLorrFl ),, riv( Foa(Xrlqq. 1op( vcr enavo),r4r6.1v€r(.l ,((r XoQiq vc urlirQxouvit) , I ,< L6LorLtr ts tou r6iou pryd0or,s. 'Sro!

l ) r l < . . ( l l " ' < ^, , . T6r€. €rp'6oov q cp1Lr1 exriprlol ut) lrt1ttLl,e xarout or\tlordoo rou r6Lo6r(luio$(1roE ?,),aror ofi€ c,, 10. I o!!!or6oa cur{ 0( .fi.!Er o!"fir'oryd 6€or6louoal

u L t \ \ * ' I

ii (i1) (t ) " .". .)Ta ,Jw{o!ffia rrk xarruqlt'ovrut tt( o^otvu ttryalirtQrl laQiqEra rQo. rlt, v.(r-rtt0uroy rou:r,, uutooutry ,emi< ovyxtriotae Eh,er ro rqlixo t =l^,, t)l ^il.Au16 FoLdgrr Fs rl olryl{},rorl orll ora0eQ{ zardmool, rou lr€l€rrlocrtrr fir! c.1,€-),tF€Lq Markov. F€ eEqiQ€oq ro 6rr q t6rotqrrl ),, gnogei rripn vo 6[(l(')aQ€r an6to l. [a(iryForr, 6e.v Wroqilolge rov ncpciyovtc )i; orqv (t), €l'd ea 6'rpc,"r€1,c1 €Loalir'/otF€ zci:roLo roi.].a,Ii,aorcorrx6 ouvrd.rorti, 6t6rr 6ngoper|xd tatl. otl {16,bwnr6v vcr culdvortctt ndrpu nohi ri aa FE!6vowu[ acipn;ro).it,m|( fi€plnrdosrE 6rou ),, > 1 { ),, < 1. :rMio(,E, nQw neot{r)QrioorF€ fioEn6[€vo 6r1[cr. turoQoiF€ !a 6rd!Q6oo!F€ r{dos r{ p€ rn! /rQd}rl orrvloid}oa ror]oA.. M€ (uni v1v rutl,ri c).Layi ri.4rcxog. I [r€eo6oE r(l,v 8l'vdlr€ov.fiv€rarur-r = -Aul/ot itcu ou'yr!i\,€! o' 6vcr noLL<rll<ioro rou .ru.*

* Or olrcledaE ulroi a,, d.lalilow uiorc ngx r1p ,lr.

' .' D \ no \orl:nto: T!)\ I \ oTI\llN

IIAPAAEIIMA (e1tt rtr Kd(tderru) o;rou t( rr,, trirorw rpq.ro rbro6rr!fo .667 lvror'( 10.333

:

. fo .9 o .2 lI = l;. i ;. i] lrc! o ni!cx(< lt€I(xirlors nLloror(;v.

,lt,l.=lstlltiil I r ) ,7 rJ l I I o .74 i l" '= l rr , : rol " '= lo,zsr l

M uuo ro ;rupci6crlp( qr('v!0(irvfr(r o oo6(lQorrQoi rreQro(no$o<: Ed! ro i €iv(rxorrci o1o l. 161r: I o!yxl,rol [i!(r 7ro].u c0yrl. :r 7ro].],iq r{{.ap$oy6_( r > 0.9 nouorlpaivet otr lprrrilorrcr nrproo6tcqtE cn6 20 rncvcrlrllrtr< Ic va nepropioorptro ( , \ : / )r)r x( lx( i t r ic buvctr t ror 10. ( : ro n(pr i6rrpa ai tqr€ r '= 0.7 zt l r Iorlzl.rort rtrcv;rril.L (layi.) oroLxri. r' = I olpuirr.r ), r = )" zt{r beu troutrtx(rr , ' ; . ( ' , nu/.a)ror l Y:rr ipTorr : roL).oi rU'r ,r yr ' , r , ' , , : ro, l uyoqrH aLJruv r , ' rrr t ncl . ' ,o z, , t U,, atqt lQrt \ 'ur ' l l r rurr : ' ,Jr ' , r ' ror ' :

(l) H rliAo6dg r.rr 6r,vr4rftDv ,uit ttldox bor,iflif-r Ir- ;rcproo6cpcr bra-ri,o!(r( r(r'rolQow. (vri l(l ivft |r|ov!4rr1'o uf. E(i! Es){vlioouttE (.no /,o(ruoz(l\olrzd bL(viotrcrc. r'r ;ro).).(nl(ot(ioorllr- 6ic F€ rov --l ;r(r rgrrato-ool'|r! br(lbrz(oi( cftnr-Sch idt yr.1 vc r( oatlololuororr]ooryr ;rci).r roitori1(o i1(1 trrtrol,olaro ijrl(r rrtr ll€0obor rorr ro cxorair-otr( ocr €ilct !(r:rrqropiooqr ro :rliizo or1zLtor6 or / = ^,, | / ^,,. E:ri nl.6o\'. 0& 'rciQollrr(lr,r(;xrrrlfg nQooclyioer< /, b((toerlrxr;'v Lblor+rdiv 7cI rt,l tbloblaaM!(ir(')v

11) H tti0o6ot r..)v @rtorooqov dr,vdff(dv 6orir-iJ€! ge ro! A r. cui prrov .,{. Eva ptporrr4rivo tjtf( rrF €gio{Dolq 6r(vo(r;! .i\,(t lr,r = -4 rlr,.

;ror orllLcrivu orr iil,oufr ro yQ(fqux6 orolrl0c '1r\-r = 11. (zaL rpcrtqr rorE]rtrlrrilovrrE L i..et Ul). H dEoQi( i(rJQu r:qcoq)(r.iLcr afit'.l.Lo1cr1 gLxgdrrp4irtttyl. aq,' ooor o ouvrrltorrjq orllx),roqE r'" = .\r / Ir €ivcr rtLnp6ft:rx)q rlsIrordboc. )rxvci. curo nou arurtcirct orLc t{crptroycq €iv(L I gLXQ6reQI r6Lor+{ri.o:r<irr rnrii./srrrL furtogr.lr(l)q rl .r\,liorQo{{.1 rn(,v'd.tnrLnii 6rc6rzc1oia.

(3) H tri0o6os tl,r trc'arortopivtor uvrrcrAdqov 'r'rtitr|rDr Ei!G! q za),urrpl .l]r'oie;. Eoro on crTlr({)rcrorJllr rov,l $e ro! .,1 oL Tox€ 6),tr olrbLorrtriq I' tr€r(ro:riiovrur zcrri ql ibr.cr ;rooortru o z(r o rcA.i./r1)r, orll:lirotc

lLcr rlr avriorqog.4 u€0o6o l€rctj(i ,.1c! or r"' = lr o / l: n . :lrv€ndr(. €.ilro o t:niqei !u r-iva! p(r ).ci,rt nQoo6.71tor1 tou )1. to ; "' 0( €ivft roi,n lLr.pozcL rl oiyzirol0c €r(r(r7ri\'€Icr lQoB(zrrzd. Kir0r 6ril( or"r|li rI< [E0o6oL' ).r'\'rLro onor4yc (-l o1)r'1.-' = r'1 211s (t14 r1 t!iot'r4 6Lruloqdrr Lz(vo;torEirdL oiro

( \ r , , f ( \ r , ' ) " ( \ , o ) "

Eo' ooov ro o eivc[ xowil oto )r, o ttQoTo! qn' aurouE tou! trcQo\,ouaortg €ivcll

16oo zovrci oto g166v, tiote evc ri 6no 6rnrors apxorjv 1cr !c xcivow ;ttrqpt'rg

rov rqoro opo zupicexo. Etbr1rd. sdv ro )r aXrL Yl6l unoloYroe€i [r xanorov

riij.or ui,ropiOto (o:to. o QIi). r61F n.iiQvouF€ orrv ̂ qv uroloYrolivrl rLltrl

r"r, \. u ",'ndrout'n

Jrcbrzcroia Livor v{r napuYowonorrlouupe rov ''l - "/ oe

l U " x , v u l u o o u F t r o U r j = I I . 1 . . I l r l r u v o b Q o F r x r i ( v r L x ( r d m o o 4

Eirv ro )r 6ev dlet 1161 rqooellotlei trlro xrinoror uvtEdpqro ol16qL0po,

r6re n pi0o6oq ltuv FErcllortqrdarov 6wd4r€ov TrQan€L vc fuglorpltioeL r1v

brarl r1i rrrlo'n1 tor o 4. ,qori FnopafiF \' d')'dlot'Fl rI ytrurortorl ot

r{uor dnuu rliv ro strlHul,rouFr' Ha JrQtn€l vo Etrl^€Yrl ro rrr ttou ry'9(rvt{rfrol

ornv (,,i - ul/)ul.r = --1. H cr'l).orimrQl bwqtoqrs €i!(t vs bool€riow€

pe r(prilorrt< o\,(oro0lltonE ;rou rpipvow xci0€ rrrl o '\'( LoYlx6 [4reoE'Yniror_nw 6l uE ]l(tL d)'i.€( r{dLtreQ€E €trt},oY€g :q ou|r|t€rQtrli tl€Qitrtoorl

A =-,,1?. tt nto onolt).€oltcttxli e;rtLoyl guiveraL ao €ivft to 't 'ixo lot.'

Yl lOAL, r l :MOt r t l f n lNAKE:

Ra!Ieish

* A16 frotiri ( {{(!si on rt!a( o€ tE(!a.r"ri x(x1i z(riot@rl, (qd! o i - dI €i!( 'ioo

r.orooor.a. L6L6trdslq illaaL Eutr.ir< olr.r< 10 d(.i)+( 'rL'rua{;!rr( 6la x(rtnollol ro!

rbro6tr.!\iolrio. Fia6'l 6r ziOE noii.dind(o Lbtod(r'nqudor .irar r6roid!$l'u. ro rritro nof

raoor(0o!!r !( MoLoliror|lc ci1(! d!r'l I br6!0!1d'l** rpaysrail oi/zlo,l uruoird on ?d.od 6iFe io)ioarao(i!'t ro o(lrrtq( | tlv(r oft'oEao

t " . , , " ; , ' . l . l mcu l " r z t r d t /a , rq d rurdr$r o ! ro oqr i i+ ' r t ' ro ( / " ' i : rn t r d ' / ' r rF h r l l ' u

" .u ' -uu- " . " ' - ' l | , ' , ,n .n r *u" ' . , . / r ' r , ' . f ' r r , t Y 'u q ] !o l t r : J l ' r I ' K r6 ' r

n . j , i " " l "a , " ' " .

o " 'o ! , rd tud , , r r r ' ! - .n r n t " 2 " n | ' tO i ' h ' r ' l u r ' r r rdc u r ! u

fvcuQiqoutl€ llt! 61r ro trlLirrc clu6 €1€! 6vcl €Ltrxloro mo anQl6ig lb!obldvt'o['1-oL n(Qcry(')yo! lou ll €ilclt Fl66! x(lL ro'fQdqIFo llordl€l [€ 1() xirl(') F'Qog

r,,oq ".rp,ienitlq. )rrvurri4 to ofir\rtr .\ - 1rr' r1E r,btotLprlg €i\'rrt Xov6Qllii1

t,t i..p.'iyr". iou orp4trgcrq loll L6ro6tc1'uoFctoq Or o!!€l€oti:q olYYlIonE

r"' = lir - orl/l)' oa l tt)')'irgow ot tdt0€ 6flpcl x(.! oIry nQoYltrrt',r6r$rr to

i.bro ro r"' ttiveL oto pl66v. To rs),|x6 crfiotd).€oFa awdlv tcDv [etcronioE(')v Ft:ql.ix(r Rayleigh, €ilcl I ,rt6txi o1')yr^tor1** ll(r,\ or' nQoq rq! )1'

Tor6ra1dvra FoQqi ,or FoQqt Hessenberg

H lrioobo< ru)v 6uvtilrer'rv tivrtr ftnorE)'rouarrxll u6!o Yo F€Yd]'ous rcl

nnLri (earou< r ivcrc<. Orc!:ro)Id {L-r ' r {r olor lFir t I i l t r r Fr l l4orvrxu tor!

n *,f'.ft "ri.W q! p€06bou €ivor ld0oE. AvaarDtt6pttole loltr6v tl! un'iQx€!

;d"*u; ;"i6(ia6;;E 1,rr lcrooyorrge F1b€vrrci, rio! our6 6llEou[€ 6d! o16xo

ron, rrol,t6vo! rcpqpdqrov.

Oa nQ6ti€r 1,o ro{p. ono -rtv cprl 6rL, B€rd rov llno}.oyLotd rou 6Foro!nivcrx(r U tAU $€ ri0Qtoo6r€Qo yrlbevr"rci on'6.rr o,4, 6er oxon€xou[s ro€nrorA€!,orF€ orl Fi0()6o raJl, 6rvtipcruv. Ynriglow nol"u rro ftpirrTaeq rcgc),-),ay€E xoL I xo),nr8Al an aur€( (pciv€roL va €ivar o olyoArepo( QR. (H $€0o6oErcuv pcrcrorlol€vtov cvnorp6tpr,r'v bnvdpetrw lQfl.dE€ror oro x6log, yLo rlv €!-puorl ruw Lbro6tcrvropcirc,rv.) To nadrro 6{[a €i\'al va raQaxoo{\r 60|, ro 6lvcrovtr€QLoo6rtQu plb€vrxd xcr 6oo ro Suvnr6v rcrprr€Qa. O F6voE neQroQ(op6E pa<onlv raXUqro 0a ril,cr 11 1pr]or] Fovctir"ciov ({ oaeoyovir'}v) [€raoxryorropd,r,rou brnrlaouv rI olrpperQic rLrr 10 FtlI. Eti1, o,4 eirsr oqrpa-rqtr6g, ro i6roow6(riv0t xer $u xo,' fr' ALr zcr. rcur6lgovcr, xcvdrcr oroLleio 6ev p:roqr:i vcpeyc)droer rnLxiv6wc.

fl(r vo fidpr: cno iolr A orov tI rALt undpTow, rlpirrE,6io 6rrv(rr6r4tcq:Ei.r€ vo r(rAd"/o$e avo pt6r1,rxo o€ rde€ 6rlpc (6ntDg xdver xar" r1 c:rcloLq,rjc.russ) €i.rc ao bor]l.srlJoule pc pio otroriLqQn orll),1 ic!r6xpo!o. fLcr evcr pe$ovo-pilo lllbcvrzo. aor€i vo XerloL[onorrioo!gr: Ua enin€6l orQoqri, 6ntt)! rtov€AdJv€!I tEioool (7) noQcxdror, notr lplorpo:ror.ei ra cos ll r(tr. sin Fl oe iva pnLox 2 eri.2.'EroL 0o [no0ouoope vu rrrrEeeyoorolip€ o1l oeLpd 6l.cr rc oror"Teic rdro: <rn6tq brryd;vto. bLcl.dlovrcq ot: xtioc 6r'luq pLo orqo(|,l d. ,!o! ncQdy€r av{r [16€!r&6.A1)tt ti\ral I xeaTQLXI 166Cl rlq I€0ddo! ro! Jscobi. I onoi.cl Er(oE 6€r reltl.laiveL!a bL(ly{i)vronorrloer rov,4 or fi€fisQoolavo ]rl{eog 64rd1(o\,.6rorr rc pr$evr.rrinou 6lprorpyouvrcrL oxo ,rQdTo 6{B(trcr. xcrclorQ6gowcrr olo! 61pLolplouvtcL

fLo 1,c1 6tarloiloouFr, c|rri tc prlbevLxci yoL vo ora[artoolr[e F6Lr< naQa,70oiv. rrQdfiE! r,c €n[bdxorpe xdrr LLyr5rrpo un6 rpLl(Dvrxrl poqrprl. Err,rgeaoupej.ol|I6\ lut rI ltt[enxi] dtayiy& xiur oni rttr xitpur t\utyrttn. Aurrl eivcr 11IloQtti Hessenberg. Etiv o nivoxcrq eivcL or-rpprEzoE, ror€ xo rrdvo rQry(ovtz6[6Qoq t)rt.rvtlypdtplir ro xdro rElorvLx6 piqoE zcL o tbro< o rrir,oxcE u( ril,a!

Kar or bllo arJrdE loEpdg llnopEi vc l.1grOouv ge prc oeqri orQo{fbv oraorooici rrir€6a. Aur6 Ei\r(L nol.i cnoi€l.€opalLx 6. O Housholder 6aix[ raL6vc v6o raono va xdvEL ro i6|o crQ!6d)E nQdypo. H 6€o rou r{dvrL rrtv'r0o€roL[.rof.r yro rr fi0o6o QR+

'EvaE Fraampa cp6€ Househotdet il

mor7, r 'udr: u, dx, lddrtp,ra r rvrrr ivrrq rrvuxr,c r l ; Ior_jq ic

, , r l l' ' - l r '

Iqvir ro u xcvovLxo:rorEiror rcr yivric! [ovo6Lcio drrivuopn u = r'/l a.o:r6reo tl yivercu.l 2rur. KaL orrE 31Jo rreQurr6o€lg o 11 ei.vor xor orgprrprxog rcroaooydvroE:

7.3 () YnO,\Orl>\tot TAN I IOTIMO\

Ht l t = \ t ) . ' , ' r ) l l 2u r r1 I 4uu r q r , r , r , ' , ' 1 L

' Nlnoalire i( reoz.4LrFre (rflt t'tv rpo.ro+Ladi.r 7ar !n n.itF x(r' .lofiar mor aiy6ar0Fo

Q/i Nlnlo ?ra uo/.oyrorro!< mfr xa.i:t /pfl.iid(r v( ido.it,.r. ioini( rn |!ti.! iii.

YNO OTI>MOI ME NNA(E:

'Etor E = HT = H l. Iq tru1o6rxri :trpi:moo4. o nvriotorT"o< irivcxe4 I -2uuHrivftr raL E(pLxmv(K xaL oQ0oBovc16taio-q. :?ono< rou Householder tiro\, aa:r.{O.llFr ll5Fvrx.r Xpqotponorti,vrag alnoic rol,c nivoi.r( xar q r:rrruTic rou[EoorciraL aro Tl]v efi6p€vl rdnroxllia:

7E Eor(l) 6r! z eivcrr. ro 6uiv0opc-onil,I (1,0,...,0), d = ll.cll {aru = x + oz. " f6t t . E x = -oz = (-o,0,. . . ,O).

A1t66ErEr

ll raworllru aurri p:rogct vo lelorFonotlosi of€ooic. 3€xrvdf€ l€ rlv nedrrlonilrl rou,,l nct 0u16uc(mr 6rr o rri,rxo< U-rrlU apdru vcr eivoL r$6ruy6vro<orrl o FF€rQ[l(n ]r€Ainroorl (li €v ldvrr H€ssenberg). :w€r(is 0a olpll€ dlouvtnnt nt n - | q ,uXt i t xr i ror t t t r i r4 , \ r , ryutLt r , :

t r t F ( x + r : ) t ( x + a : )

(.2)

(3)til

: ]

t|i:lM6XE" r6Qa o ni.1]crxog Householdcr 11 rivcL rrile(lx ll I ror. yL' cr]Jro €1,riroo€rdlotrlv zirror beELd 1ori.c ev6q. n).{pox brcor.ioeox. ,ri.'ror{o ULI

=u . t . xu u t lAU t=

A6ytr; rou 1 oilv dvtrr aALor€Qd ytrrvitr, o nivalia< Ur cqlivrr io mor1€ioall teLeicrS cvi;rugo xcr, ro or11tovrrx6r€Ao, 6tv lritpci(er ra gt6tvrzd nou€$q,oviLo\,.raL mlv (3). 'EroL ro ne6ro 6fFa ileL oLoir),q6fei xo:t o U, |AIltiTtr nlv u:rurroqrtv4 nadJnt orrtht.

To 6enrego 6r1rn rivcL :rcp6por,o: to z c:rord,eirot crfi6 ra r€}.€v[oitr r 2orol1tsio fl1,< ticr{€Ql< ori},$. ; eivtl ro pova6Luio 6rdvlopn ouvr*<ryr€vtr;v

000

tl

0 *

I

7t o Yllo^ofl:Mot ToN ]aIoTIMaN

!6iov 6roordo€oll )ror o H2 eL.vrrt n 2 enL n 2.'Orov rotro0€r1e€i oE 6vav Uz,ncprileL roug

Ttog, o Lr3 r(rlftono€i filv rpi'q orllll xcr yr,o 6vov trlvalia 5 etr[.5 ixoupesnmJX€L Trl $oQqfl Hessenb€rg, Ev 1€v€I, o Lr 6hal 1o 't,lvopevo r(nv ?llloxovU1UJ Un !xc.Lo oQteB6E ,rt,r! trQdE€@!, fiorl trticllouvrq! ll<r Tov ufiol,o"yl,ol6rou, €h(rr t4g rdEECoE rou fl3,

NAPAAEITMA

,.=l?l''=lll'"=1" -Jl

, ,7u=z+ l la l 2 ,4= I -2pp .

Tono6€rd\5oE rov I{ u6oo otov U 61oupe ro roL6to't/dvlo qfiola)'€opq:

O U-l-4U eivor 6rng rivaxsg 6rolpos a' ono'|oLu!-r€l tlE l6torqras xou -o dL^t6-

QroFoS 8R 6iv(l! €rorpoE va E€rwrio€r- rdQexrillvouF€ 6pols, fiaos ott"y[nl, yrsva enLorpdroqre 6uo riLl€E €qqQFoydE d!10)v r(l)v p€rqqrlporroudv Householder.

L H Eaeayovtonoirlon A = QR, Aurrl ritov ouvroF.oyQsqio rlq 6la6tr{aoictEcranlschmidt rou K€qaLciou 3 Tdrpa p:rogei va exretreooei rurloumepc xct"ge peycl,utep4 euotd0er.a. @r4l10eire 6tr o rR ng€fi€Lva €ivdr dvcu rQrTtDvlxog.Td4a mn, 6ev lpeuilercr, va 61oupe eni nldov FI Fnb€vrri 6tdY6!!o xdxo)ur6 rqv nripn, atpou 6ev 0a r.Dtrip'1ow t/ { Il vcr nol"lor}.nouilottv(rn6 ra 6eELd (onog mov U rAU) xa| vc 1o}'ow tn p16€v!,.rd rou nbrtixolv 6ylpl.ox0yq0€i. 'Eror to rpdro 6ripc oqv roicrox€lf rou Q €ivdL vobooldtout.r€ F€ o).6'i1,rcI qv na6rrl m{}.q rou ,4:

ilft il 331 f::U = lo o | =Ur t . U . t tu t rAUt \Ur= lo

133" I t3 3

11 o 1 lA= 01 1J ,

L i 1 0_ l

l r o o l 11 -1 o lU= u u - tJ . U LAU =l -1 0 r l .

L0 - r 0 l L0 l l . l

|;'l |,]

H npdxl onlll rou 1I1A eivor 6n()! c,{Qr6drg q 0il"o4r€: rcolrct tr€:r ". 61er glbevrzir zciro tr:16-qv zlpL<r 6ro1ri;vr,o ncl tivcr 4 ;rpcbrrl

otlLq {ou R. To bEur€ao 6rt$a 6or},€n€r pe rr1 6er:req1 oniLn rcr .qrA,on6 ror o6ty6 zar xirrcLr, rcpirlovrcg ro'r EfitA:ror rircL pr166r, xitru.1r' arlT6v ro1, o6t1o. (O 6Lor ul"loproyoq por.dler aol.lj tro rlv (rro),oLqr{Gauss rflr orlv TrQoruoi(zoll{cr rivar prcr ),i1o nro apyi raa(Ilqt iIE.)To c1jror6lso$o r(Dl rl 1 6lpdrov uivnL tvcrq dvc,r tprltDlrz6q B. lnrpiqotr , 'c o )r ivuz,{ : nou xurnTpnqrr ru 6rporu vu | i r ( t iv(E rdro rOry0,rrroc-L. Avri 1i cur6, o raleurnioE ei.v<r dvcr yvopt..to Q = H)8. .. . H,, , norrroAri v,{ x{rr .rY0a,r. i o ulJnt r I ! ncooTowunornl iv| t roAq i . XupiE norlv(l unoloyrsu€i cvd.lJTLl{d ro llvoBrvo. Aud ol,ozlqptbveL tq brcrtiucroicrCranr-Schmidl.

Il. H |L(ioiuon dLopigqov qrity Q( AQz -t. :ro ffoAdprl$o 0e boqrctl€ noro rAoro (lvr)i I bldon(orl bi\r€r q 6i),r!oil Lnol r onoro!6iroi!:nQo6),{ftcrtos e},crTiorrDv retpcyriwarv. O : eivo! 6[a./d.,'rcq ni\rozdq, r6i.ovUr ro\r A 6!(oriroeo)1,, xm rc oroLleic ro]Jg (6r6poeqreg r[U6S) €i.,'ar o!rerQc(l,ovrlrlE pi!c,( -r(rv rbtor+ri)v ror .4r,4, Agou o[ pcrsollUcrlo[oiHousholdcr nQork,4rd(ou, [ovo q ],!oI rou rQo6)niparoE !6!or!tl6v. X(,)OIEvcr ro ^u\row. b€v tr,looolrF€ v(l fi€Qr$6louBe qr'(r!ro!q !c JrcQslow ro./:. Avri tr,' c1J16, n(o(iyo!! dv(v l'rdr(.r),djvn :tit a<r, $ plb€!!za. rcl\irorj,extrlE cr:r6 rrtr )ruQi..r x(! fl1,'cr[do(1)E nd!{,) a:r6 atnr] 6LultirvLo. Ererbri or;rtvartq 1l ti.vor" op0oydvror. y)rrQo€Qyaofa clrrt €lr'crr oQl0[qlx(i ruora0i<.

To rod)ro 64$( €ivft a1Qr66E xo i.bro p'cur6 ror QF rupu:rdvtr-r: ro:r.eivcr I npdrl orlil.l ror ,4 zar. q I11:r: eivnL gr16d\, ridr(r) crio rov o6qy6. To0n6pevo 6ripc eivur rc roL),cr:r).corrioor4re cr6 tc be€rd F€ d\,cl, d(') nou0a rcQlrye! pJt6€!u{d xold Uixo< rlq nodqE ypcpprjq, |05 eqligr

YNO OT I : . .MOI ME N INAKE:

t- - .A- l . r ,1 =

l0 + *tu * *

- H.AITL\) = (4)

ElHIAEI 'J*0

OLoxhlpdwoqre [e btlo re],eunioq percollpcrrlo[olE Flouseholder:

[it::]- l- l

il

ilT-;

lolrAu t Ei.var r 6rcltivra popq{ rou !ryouocqe, *d 4 ge0o6oE rpcrvr:qriver"nd),r. qv rcrltrrpo ge qv oroiu or" percoxrpcfiroFoi Ho seholder nao(iyollvfI6€vrlici.

O ALl6ar0FoE Q,R

O a),y6Ar0[oq €iv(r ols6o1 0r_( 6rci t{alri.o< oni.6E. 3rxLvti nno ror nivuzc;1n. rov o;roi.o ncrpcrlovrororei oe QoRo xar crrcrSrntrioorr. rou! rr(pEovrs(i,lr = Rr80. Ar.rr6E eivct ivcrq vioq iivcrcq, oporoq rotl cpruxo!, Qi'AnQn =

Q;'Q' R'')Q,, = .,1r. z(r r brcrbrxooia orvrxiLrrar xtupig rrrro6olit rov L6ror$drv:

Ar. = Qt. .Rr rcrL zar6rLv Ar.r=l?r.Q*. (5)

H bf(b'Koir{ crlrit nEprlaal(p€r w li ttr nonu4dro alyiatqlo QR. ito\)ollziiver xhrnr cr:r6 cpxmri "/svu6q ouvOrtx€q: o ,4r. r€iv€r oe ircr, rQr./(r!ulonivaz( x(rL olv€nc,rq ra 6K"/6vrc ororXsic rorJ r€ivouv rlDq n( L6(or(pirq ror.not olrtrntnrour licrL tr€ rrq LlLor4rdg ror oplLxon nircxc.{, *

: (lt t lit [oagrl o criyoptOlrog ei.vuL xol6; o),].ir o7L roiu xu].6;. fra rc rorzdvotlt(r 8q11er:nx(j /Qrx(iqo{0! b{o r€lrlorlor,rtofr.(r (l) I]ain€r !a orrtpdrlouprlLeroronioc|q ril-q (exlq x(ir ii) nA€r€L v(l !E(1oq(d.ioolpc orr rl n(Qcryo.,'roroilolQ/l 0o eivcr. oe rir0r 6rltr(. no)"u yartyoal.

e) O lt€rcttorfuottil,o< d.y6pL0yoq QR. Eit\r o crQro[oq n1 si.vcr rovrri oe[(1 tbto-rt[rl. 1o1. lc} drltra (5) (kr noarrr (lpior'rq v(l per(6],10ei oro

73 O YNOAO|DIVIO: TAN ]AIOTII\IAN

4 1 . < t 1 . 1 = Q 1 R 1 . z q L i { c r 6 r r a . 4 t t = R t , Q r + u r , I

ALJr(i brrarcIo"/!:ir(L c:ro ro orL o l/,, r ei.v(r o|oroq rou -4l.:

(6)

* o,1L o,[i]oi.i:., roa nhl,xa ttr ro! onoio lEaLri o (iy6aLorLoq QR. Ed] ura:e zirnor(npoero+(oi.r !r. rirdr.< Ilotr\eh.ldcr i.f oSrlyildn! E iA 6Lai,di!L( rrolqri. rirs o 1L oNdirrcr !eror ruri{i,. (o7Lro rirdao .1ttioo, r., dror Li '_{Li =.11

Q t ' A l Q t = Q t , ' ( Q t R r . + n i I ) Q t = / i ,

Aur6 nor o+r6ci.1,cr orlv r4ril1 tivur 6-rL ro (r,r) ororxfio tor A7, -crur6ritE )rirr(l) beELdE lnr!k1( ri\rcr 1() rrQ(i)ro nor :rpoot11iler lro rbrorL[t. ,4.!16Lornoa ro oror1Eio afv.1|, I.rn),ororeQl x.lL nro 6r1por|LLrlq urLLoTl Tc 11Ir€ruronLoil .r/,. :r./riu(l)6. il orJ.,'in[r.1 (r,rnE xIS €nr,oy]iE ril,ct !o npoxurir€lr[ra(1,ov'xri ouyzlLoi] zar [hiLorn. oT't or[tlErprxrt nrQinTl,ron. r$Lzt aliyttl(}l:tpo5 t1 srLxp6rtp4 r6rorqr1. Merd cro rpicr-rioo€acr 6rif(rc ror Frrarorrofrilouc),yoar0tror. o rivar.ri ,{r UoLd(rL $r Iov

YI]OAO| EIVIOI I\IE NINAKH>

To ^1 tror lnol,oyioag€, -ro 6€16[aor€ oav Fra rrolrl rc],i nQoo€yyrol iouirAayFarrllolj .11. I-!( vu 6pei rqv a6gev1 Ldr"o-r4rr]. o cll6pr0FoE Ql? o!!€XiLrjrpr ror pr"rpdrrpo nivaxo (3 sni 3, oro flaatrb€ryfa). oqv trvto apr.meqri lrrrvio.Tcr r'16 qr'brcltirvLo ororx0io xou 0o 6tod [r,ap{ver" nd;ro{, 161or rov npdrrr'rv6tptitt|)v tor Q1i xcr buo-roicr or6$n 6rl[crcl €riclQrour yt"(l va 6As0€i.I .\]. Aur66irtt yrcr ororlycrrr"z{ peoo5o €npEoIE 6}.(,)v r()v r6!oxrpdrv. IIAd./p(lTr. I p!0o6ogqR i|tt i6tt iEAryesgEi ttli?aS. A.|{.6 rolj qno[6v€L eircrr vo erovcxrrioorpero r6Lo6rc\rrlo[(ro aou eirsr €v.r rcL povcbLx6 6r1p<r cvtiorporlqg buvaplgxcr vc lplor,poirorrloorge to prlSevLxd nou tilproiQylo€ o Householder.

6) O oxor6S x()v fiQofi(Qoor€uoo-rri{cirv $€raoxlllrflrrol(i)v Householder, no!q€avolj\' rov ,40 os FoQqrt rQr6!o./d)!!o ri Hessenberg, frcv vc xrivow xrioe 6rjpnrou Qn no),rr yQrtyooo. Kcvovr.xri,4 6r,cr5rx(oi,cr cram-Schmidl (nou €ivor o Q,4)0c ondwouo€ O(rt) npdlaq. €vd yrc( 6vcv ni!(xc,r Hessenberg tru6 yiveraO(rrr) x(1!./rc a\r(\r rQllLcyoxro O(n). Xarqig currj rt 6eLrioorl. o oLldpLeFog eorFcrv ('n(lpd6exrcr crploq ria! €q 600\r r{des v6o; ni.vcrxr4 .41. 6€! €ival nciLl o€trooqt Hcssenberg i rQL6(1ci)vr(. I6Elliooq eo €qdAtrol6t(v privo rxo rprirro

Er"tUI6E crul6 bc1 o!$6cli.v€!. fla va 6siE€r€ 6rr, o ,41 eleL tqv i6r.c poqgr1 pexol, Ar beir€ rlv o1dol

Q, =.{ol?;r =

A, = nl iQo =

. t t - - - - l*1 0 + * ** l 0 0 * +* l t0 0 0 * l

t l

ts

-t f. - - -t* t . . - * l- l0 . - - l '* .1 L0 0 * * lli

MrioQtilo ml{ol.(l vc branrordroer€ ox! cmoE o roLl"cnl.coLcrop6g crrplver. tov QxFE ro fbto xo{.c pl6e'\'r1irh, 6rojs rcL orov ,,10. O i6rcS o Q0 dxer loEti Hessenberg.Kuro:r ! , ' 1r 2,11,(oxcu(i t l rL{r i rvad.urLiooolrd. rous nr(Cdynwrs. onorr

0 +0 0

Tc i6r( reic trlblr'rxir E[(t.r.,'i!ovx(r xar o' cuo ro yrv6pevo. Otar o Anriult niy(#as Hesse"hen, tuixt @ i6o mpduivtr. xar. pe ov Ar. H oljtrtrErorrrinepirtool rivcrL cx6pr1 ra).irepl, 616( o,4r = Q;r,{0Qx rcpcrprver. olpperpLx6q:

al =q{ a;11a2 tY =q,'.4,Qu =.'t'.

Arro rov oul.),oyr.oF6 rrou tl6l"[q otroL),Ipdroalr€ npoxunrer 6r o,41 ei.v(r( €riiolqHessenberg- E:r€LAyl 6por; eivtrr rcrrr6Xqovc or$perpLx6q r.lr Hcssenberg. o ,4rtivc"t" rptdtaytbvnq. O iSLoq ouLLoylo[6q utpupp6lsar. rcrr .7Lc zdoe rvcv crrotouE :rivuxeq ,41,,41. . . . , on6rc xnqt 6/1pu rou QR lexu'd ye it at, rqdruyLbtao

73 O YIIOAO|DMO: TAN IAIOTINION

To r€),suraio rn1[€io. 'rou F6v€r va sEry'looup€, €iaar I iSro I naaoyovro-roi.r1or1, ror ;rcpciyer rorq Qo xcrr. ,40 cn6 rov cprrr6 ,40 (xueoE xor rols Qr, xorR.a ax6 xa\e A1. r1, mlv npcypcrrr6qrcr, o116 rov Ar, crr,I). Mropotlls valprlorgolroL{oo4re nti}.1 rov Househoider. aL}d Ei,l,u( on}.o{or€Qo vo $t6€r,ioou[€tvu-6\ro ro fiorl€io,adror cfi6 I 6[oy6vlo p6o0) trLcE "€rri7[€6q< orooqylE". H

(ii) sv6E 6iFdroE ro! pstaromoFdvo! dltoqioFo! rr = (A or) ru0. lls o =

"T .l'....,t,[q.

To oaLorio nLdrl]o|ra,- ril([ rdad ald nollar).iidro io! ii .ou tbLoitLdvnopo-ios (1.1) .

PrALt =

0*

'Io orolx€io (2, 1) o' aur6 10 1!6[ero eivoL o11 sin d + o21cos d xcr brd],(1or4rtrI yo\rid d !1or cxp[6d]g. tirote <rurdE o ow6lruolr6q v(l pq6€viqsrq!. Kqr6rrw 1lPlr err!),ay€rsL [€ fiqQ6$olo xQ6ro, oor€ vq Fl6€ailerdr ro oror"X€io (3,2) roljPr:P:r.Ao. Merci an6 n - 1 16ro!€9 morlerci6eq orQoq€E, ro -r€I!x6 ofiordL€op(r€i!c{ o i{1,c1) tQrycl).r,lit 6q :tcrpdlovtaE &r:

R,' = P,,,, t . PtPztAo (li)

Alrh eivcr 6sc 0a progouocpe vo nor;p€ rllrdpxouv ax6pq nol.u nopwoo'tepc or 6r6lic ttg cpL0Fltrxng fQ(l[plltig AL"y€6psE- ̂/r(l avo! ori6 roug TrrooEloorl[€io,ro1jg crly6Qropous ro! EnroqFovrxo! loylotlo!.

Avo{piQor4re [icr q]{otrn Fdoobo -ri] tr!0060 rolr Lanczos- ll(r pryd]'orKoQaLouE nivox6E. Auril oQ0oyorlro?ror€i flW cro),ouoic, Krylov. r,4r,.4'zc,.. .,r(r n€QlyQilqrel(r! orrl 6!6tr!oyQ(1qi(.

AEKIDEIE

7.3.1 Eqcqfomr raug qoaiE rit Fi0o6o rov 6uv(tl6o! rr.+j =.4rr.! oxlt! calLxi exriprto)lt 1 ] 1 2 |' , l ; ]

v ( r c v r : w u r l l - . 2 l . u " t a ' o r u r c \ r - r a , ' . ) : - r n . ' . r i v o '

xo oanao bL(ino[d !-:

7.3.2 rro tor i6Lo .{ a(, trv dq4rri &ri1rro'r n0 = []]

r*^Lr*' * -*al**"(i) reL.bv 6+tdrov tI6 F€eij6o! c!flmqoqo! blldFeor

jl".

l"' ""t , ,] ["i'il (7)

, 1 l 2r ^ .+1 =A- r r= i 11

YNO O|DMOI ME ITNAKE:

7.33 Ari*€ ort ro 6!0 6taaoaerlx'i 6rdvtorara rou r6tou rrixolE. thll = llvll, r.nrroyri! ={-, o6rryFt o dvar FFooxItdrLoF6 Hoeholdcr, 10 ro' onoio," =v xa,

7.3.a Ytroloytoi€ rd d = llall,o = "

+ a a&aot ni'ara Ho$hotd.. r, lta ft

r 3 t 1 1 l

Erid.}.l|o€ior€ &! Il, = -dz.

735 Xaqoqroiord'vro6 rqv Ao4oI 7.3.4, 6a€tff io1/ rAdiolbvro fli],tol7!o, IJ-1ALI ^ot)nQoxirrfiet urr' ro?

7.3.6 a!'(t(imos a'r'ror a = [_],,

-rl], o"re'" -r o Lq Fmaronqr&vos dlydsopos

QR rogdy€r rr rLxer povo 6sldoqr Ar - (ri s) []1

-j].

737 E r€l,aote cva povo&x6 6fuo rou Qn orcv

, f c 6 , s i n , l"=[""a o l '

trp q pPtar6mo[ a - a22 -nou o. qut|i rlv ,t€otnroo4 o4rotver Io(& prar6,rrorl,aeo1r.a22 = O. AetEre &r rd o.re 6(qy(|,riou croryrto psra6d*ovtc! drd sin, os- sin' ,. nou €ivaL ltd n€eirft,!,at xv6txis aqxjons,

7.3.8 a€l.Ers du o lor6loy,t vtos,4 = [l l] r,O'* .r""aerrr*s *,do€ 6ipd rcrl a].yoet0-

Fou QR xd! ouvcrds dnot€).rt dva an6 ra (onduo) avrtnoola6etyrdra ovv,.tuok.Aud 6(oA0d,verar stoaydyovrsg Bto ar€dieftq lsladnlorl

ru.9 Aeihr Is Frayuryrt, 6rL xo,(iE pgtadrtdrl. (eoer . . . e*XR* . . . Rr lto) elvur a,ier6,iEI ndAdyovrorolryrl QR roD a"*r. Awi I rauTrol ou\,6!a q! e.R Bs q nd0o6o rurv6wdtE0r aar o6rryet oe p|,a sEfyryI qs olh'{rro'iE rtE. Edv l) r, > l))l > ..> l)nl,rore or 'oLoqr€E avtEE 0a qrqavigovlaL slyri_oryd, oe poivoljoa 6uiraE$ rcrd lrirosflE rljeia4 aoyoviou rolr A!.

[1 3 4 l, { -13 1 01 .

L4 0 0.1

7J [N]\NA,\IINTIKE: \IEOOIOI fIA TO AX = O

7.4 t EnANAAHIITIXET MEOOAOI IIA TO Az = b

:€ (llTio€ol !r€ rr! rbrorrtr6E o:ror 6r'v Fi./.r[E arii! €nrioyrt. lrc !u ].iroow!ro ,1tr = ,. bFv za€r(!6laor€ snaaftrrtr(Dq 'uc Erralc).I;rrrrri [i0o6o. M€I.i(no n€r€Q((4rdlo uar0uo 6iNdr0rr. z(r €q 6oov o r{are$6q .ruoq siv.lr orioi,rr.; o0Lri. I uro),orq,i Ciuuss e(l orcrul:l-rrto€r ont irot r. Ano rry ci).i'ttrcard. edv ro r,r/3 si.v(, rfa(?orLo, pnopr:i vcr Tprrcrorri vcr axrlrlnloolle 4ar p o o i , - 7 r o r r o r ' . r . n ^ , z ' r r ' r l t a r " . i r l r r u / r l r r u ' , r \ " ' r o \ u ) l L r r o u t , f r \ utriaoq trovo 1ilq ffi(iorq)i< z(lr rcxdirLr vG or(trcrritoor.tre. 6er lplorpLeler" oerijrorr. :ro1o( tr(1< ]$rrov tilur r( n€arlAiuortl! lt€066ou nor lezLvonr crot r . , , i r r 7 , t r t , x r i . ' r t n ' . " . 2 , , r r r g " ; : 7 u ' r r . r , , u ' i . r r " 1 r r v 4 r p o o i T T r o q . f J . { r : T o r r t vnpolloryrvrl n9ooi.11rofl .fr. $Ltl 6r(161z((ti( rou ac xfptrtrriL€r(( x(r(i 6orriloit.

Eirc{ euoio !.r srrvolooUrf. !ll(1 riroLo Fi0o6o. (lJIl.(iE draor(,ivra< rovnnoxc .t. Er?r ,1 = 5 'l . 16r. rt r:ior,)o'l -,{.r' = ir oirtu Loo6iru4rit t,r' rrtlS.f = L +,. :r en(ir< Iroporpt va rnLTeLploorgrc ro rncrrcr),1:rrrro (jljp(l

ot jorx( i , b! ! !Zo!tLr.x.rplLi i r r ' ty i rr l (J ' l drL {1r,rr i r l p i{)oboc {)(1 ! iv( l r z(} , ' j . ndvr! ,rqIr( r'rrrriTri< bdor(rorl 0.i fidi'nfL v( Lxcvonorfi bio 6L((l,()Ar.rLziq npoiiro0ior.(:

l. To lfo bLiivrlrtro .fr naa'rfr 1( rroloft(r?.rr fljxd2.r. )u!sn6< o ,5x-Din!r r( si.,'(r rir),tiq (zur rrrrLorq,!\J!1()cl) ni!,1'r{1:. MroQfi i.}. 1( 8iv(lrc\L(lydvro. ij rpr101rt{o<.

2. H (zo),or,0iir .r/ 0(1 noixfr !( or,t/r.iafl or!\ na(ytr(rrz'j ),noil r'. E.1!\,ft(t.ilQioor+$ ro |n(v(hl;rflxo i;rjt'o (1) oro rrtv rp(ytrl:lrrxri lEiorro't Sr =1.r'+6. ro (roril"totrc rivuL nuq rirroq aol rrQr)qrodarr tr6'vo ra o{d),Urrr(

ALmi ri.\,or (xp'6drq tro 0!ioox}t 5L.lrpodJ!, iolJ E€rrv.i I'r ro cazrrct5Lt 'n,r lu r ,r rcr ucrh tr : ro i 6r l ; rcrrc:rcpdler ro v6o or ic i) .ga e7, = (S rT)r. , r .

To npodi.'ltr.i tr1; ouyrl.LorF rivcr ro i6ro (rqrdd)g ut ro rp66ii}ra rrtq€rlorfro€1.tq: r/, - x rorr orQrdd)E, 6rc\' .ra ' 0.

, 5 r r ) = I ' c r . i 2 )

ror€ xar Foao rol€, oxuv1. O p!0y6q orlzLrortq rllqau\ qaopstrx4 axrrva xot'

7I' H er(1la).47rxrxrt ld0o6oE (l) ody"lir€trdOe r6LorLp{ ) rou S r?'r.zororoni. tqv ) <rEcprLirnr , rro 14 pryrort \ . nou r.vor yvtuor4S ' ? :

p(S rT) = max . , . r . (3)

Ouplotir€ 6xr Frc rtrrr.rrl l,io4 Trjq e1'1 = S '7c7, eivcft

e ; . = r : 1 ) f r 1 + . + c " ) 1 2 , , . ( 4 )

Ipoqcvd\. I p6yroq rc,-rv lli Id,rtd 0G utL}{parrlo€r raL 0a xaeoaio€t ro!pr-r0p6 1e rov oroio ro r:^. orryra)'iltr oro FI66v-

ot 6uo oratrlorq yro rI! ercldlll'I trnooEi !a oqaeouovrar. :ro eludxeo 0c lr(oQouoopt: vo :rerulor4re dptorl orlx]'roq pt 5 = A xoL T = 0, 6nouro ,rQd)ro xcl povcbr,n6 6r1trra rlg urrodr].rl11g ec frcv Arl = r. :' duri flNnrQifirorol o fiir,chrrE oq,dl[(ro< S '?' eivcrL $Ibdv, or r6rctqraE tou no! ilqicopcr,rlli urlrivo €ivoL $n66v xcr" o 0u9$6( ouYxtrlonE (nou o1jvrloorq oQllrro!o(rv -logp) €ivcr dr€rQo(. ALId o S =,4 [noQ€i 1(1 cvltolSatft-ral buoxol'Q,:rprlpa :rou rlrcv ro cpXLx6 eiqtpo rqE b(ron(or6. Ei.va! cEoolpt:i(Dro ro:16oo cnL-r! 1g lr'roQ€i lcr €ivdr plo oflLrl €/i oln rori 5 xar e!€xdEow€ re€IEbrvar6trpes:

S = 6r"olirvr.o $dpo-< -rou -4 ($6eo6os tou Jacobi)

S = tQryorvL*6 [epoE ro! A (lleeobo( ro! Gauss-Seidel)S = ow6!oop6( ro! I rclr 2 (brci6olri{ll unsQxc)dec0orl rl sOR),

O nia(lxcrg 5l6y€rol €nior'lg :tponaqaoxtuaotuds zqL I €flloyrl rou civcr xoiorttlYrc 1I! c8[0!qM{ cr!d}.!on.

NAPAAEITMA 1 (JACOBI)

YI IO OTI:N4O] I\{E TIINAKE>

, = l-', ,']. "=['r] li;lEriv oL ouvr.orrior; ror r eivnL r, xct ri,. ro 6n[Q hcobi Sr],*L = ?ra +z' €tvat

l .2.

'=l? il "'"=2 r - . , r .+1 , r | . l f 0 , l f , I f6 2 l: , .1 ='1-r. a l , l ' l . ' ' l 1,1.. lr , zl

o cndl)(rororLxnE oq$oaio! nivoroE S r?' exer. Lbrorr.|rq +1/2, nou ol[oiv€r6rr oE xrioE 6{tra ro orpd),Ua gerbvetcL oto ['06 (ivtr cx6[I 6u6!x6 lrn(iio.liverar ooo16). :' aur6 ro :rtrpci6€ry$cr, nor eivar r(ol"u t1txQ6 yL<r vo Ei\,.11nrrr.:rqooo'relrrr6, 1 ouyxhol rivcr 1prj1opr1.

Eirv :rpoonc0rjooqre vc rp(woo0oripe dvov [E^/{r.[rEQo ri.vaxrr ,'1. .lvrrF€-ro:rilolge tr.r roln npaxrrx{ 6uoxo)'ia p€ xrt! €ti.r','d}.ll1pt rou Jacobi. Ari{aflat$i ra r.pcnipx 614 rta ouvnriraEa iou .:xk pi|el€ 6io1J oTothteogEi outoAoyrcp6g ror rk 1. MLc :rol.u rro rpuorxtl !66(r, 7ro! Giror.rti ro Fr.oo rqE trpol-Toup v' t ; Uv' tp ' t ; . ' ivor vu uexrooupf vu t04ortrororol ,pf xt iO, oulroruoa rouv6or drcrvrro$trto; ;ca*r . op6o(DE lt8xd rov rro),o"/roF6 rI!. 'EroL ro rr.r raiav€t, t Aiq ro,, . / , . d lvrnr ' ;dl l rco- ouvrorLioa. rar ro r{ U:rooci vu zaraora(vf i

7.1 j i l lANAAHI l l lK ! ) M lOOAOI l_ lA 1O,4r = b

rar'r61aova [€ rI 8lfrorpyi,c rou ra*1. H fi-odtl] oruoxooo fiaQoU6vE! n i6ra F€

o f i ( r 1 ) / , + r = ( . o L , , r , , h + t r .

lo, i roFrvo 6l t , . l Soulfun aF,o,uc Ur ol ,r ' t r t ! 'u ntr t ro,r .rr .

orr(r ' : )a*r = f l l ( r , r)r . . r +(-ar111 - . . r : , r , )^+b:.

Kcr I r€I€rrai(, €EiorDoI oro €nava),Intrx6 6itrc( 0o Tpqotporotri crozletortzirrLg v€€q lqr€q.

n , , ( : i i , I I r , + I = ( . . n m L l l i r , r ) ^ . + l + o , .

Aurri ovo$(ilr:ror pt0o6oE ror Cru"-s-seidcl. nv xu xa-./J tcr qar.v6pev( j.rcrciyvororl orov Oauss evrb o Scidcl 6e! 9at.,'€rd! \rd l orrvroroljo€. Ei,vc! prufisQi€Qll loroQto. b!61r I B6€obo; b€\' €i\rc(r xuxli. :nrr!6ore orr orur ol,o! or,6por oro r1,-1 [ercrx!\4leo1r\r ofi]\r oQlor€or] fi],€uQd. o r/rdrd( S yit,trur to xri,rtoqLyutx"i yi11oE ntl ,4. :1I 6eEr(i fll.euQir. o ir,LoE ni\rcrxcrE T rls britonoolE.€l\r0[ l\,rloloE dvo rQr^/(,).,'!x6q.

NAPAAEIfMA 2 (GAUSS.SEIDEL)

, -2 - l - . . [2 u- ,_ fo r l . . r1 _ f t r .1' - r 2 , L r 2 , -10 0 l " -10

: l

To blpd rI : pfoobou caus-sr idel f rrdddl.L r , / . . , ( r{ 4 / orc

2t :p , t = a ;^ 11 , t1l

t2 o l fo l lI r u l " ' r= lo 0 l r /+o '

Or L6ror+dg ro! .9 '?, Tior ri.\,ar j rcr 0, 6xo!v nd).r .lroq.lormr}irl olFcoia. Tooqill+rrl 6lae€irar os '.id0s 6'i[a ilE ro 4, .iaa rc ii,a 6it1a Guuss-S.i.tet aEiE€l6ao 6no qittaxa.ktahi.* E'r€rbi ror oL bno pr0o6or c:rcrrouv rov i6ro upr0p6:rpirlecr'v lptoqroror.oige crxpr.6r;q r1 vdc rLp{ cvri q< :rcrl.crir< rcrr oqvrpapcrlr6qrcr elor.xovopoupr pr'{pq I F(0o6oE causs'Seidel uirct xaLutupl.

* O xcliv(lq ur6s rdxiu o. xoiiiq {r.At'.],i:- a! 2d.i!ar bNnro! 1( aat6ac!(o0o!v

r(en6.i$ald 1trr r( .roi,r 't tie.Soq rr.ohi olt,aii!fl. r1(; rl crs-seidEl 6er olr,).tffr (ri "('(rridoo((). H or!+'ftarai r.QinrrJort liv.rar da c!0ci(q: Av o/.c rd a., rivdr > 0. ror. I treoobo<

' 'u . . s . r ,1 . . 1n / . v . I r \ , r ' ,n "o

oo ,o t . f \o . .

YNOAO|I:NIOI ME NNAXI:

Y:tripler pdhorcr rp6ro_< vc .ritv xirroupr ax6yr1 zc),uepr1. Xrc 1p6vr"c rovlnoloyro0drv 3e ro 1epr. crvcrrcr),rqoryte (pd),Lov ruXcit1) 6rr I otryxfuorl eivcLxclti€Qit, td\r npol!-rqrtooultr n(Ao ono rI 6L6A0oroit Gauss-Seidel :r( 1 r/,.MLl.drvrciq Fl arJorilQo. n ouvl]0[op6r,n p€0o6oq olJy,rtriv€l govoidv(|)E ltor orrpoo€"/yio€L_( i r naAaF6vouv .l'16 rn! i6rcr peQr.i rrlr 7rQo.ruorft)i< I"rolq r. Eivcrr)"orr6v 9ror"16 vc er.ocrlciyorpe t!8 otltrEltori utE%atdAaoq€ u,, (ior€ r,cnE{€ rcmrr€po mI l{ol. rra.r = 1 Eavaroio.vorFs rlv causs'Seidel, €!(i y!((!, > 1 rl p€eoboE €ivor y\'crod $€ ro 6vo$a d.d6oxt'ti r,flEwar.daaan I.SOR). H6(2'r!otq €rif.oyrl tor ', €Ecardtor oro ro nao6l.1pc. clJ"dr nord 6ev lenapvtr ro2. erd; oqvd 6piorerar" o€ $ra n€arofrl ror f,9.

ftcr va mpLypdrl'oupe q p60o5o nLo avc),rtrzci, €ortrl 6rr D., rar U eivalru btuycirvto. ro yv{otc xuru rgryu,vrzo )i(t ro yvqorrr dvr, ri)ry,rvrxo 1i0oE rorl,,1. (H 6ldorcoil c(r'ttrl 6€v 6XE! rtnor€ vo xi(l€! FE rlv n = rru rL( snai.orgriEri(I, oolv 1(Qoy[oTli(o1q1(. cln6 no! !ol!€r r6Qc( e|vdr ,4 = .L + D + U,) Htd0o6o: Jacobi iXfl ,9 = ,D oritr (QLoTEQI n),er.rpt! :rc ? = -/, - [,/ oll bEEri.. €r!]I €fltroyri qi 6(ionaolE CAuss-Seidel etvar S = D + ,L zar ? = -LI. Tdou, lL(v(r €rr,loxu,r,olrpE fll o!?z),rorl, lQilor[onololp€ 1l!

I D + a L l t ; t = [ ( 1 u ) D - U j l i r . + . , b . ( 5 )

ncro(rrlQylm€ orl ./Lu o = 1. be\r ljndQxer €rrlToxu\,or] z(r €filorQ6qow€ ntoo)orlv Gauss-Serdel. Av€EdQ[lTo otlo( ano ro rr. o n|vcrzclE orc, (lQtoreQar €tlrcrrxriro rprlDvrroq r(r (or6F or( b€ELd €ir,(! cil'('i rQr./(D1ur6q. lrJverdq rcu rrll,rrl v6( owtoldo( loulJ,i.*L [rooei lc oallx(rloorto€I rlv fiatrld ro! rir., o$aoorqg€rd rov !|tro).oyrot16 fllq. Er,.r rurrr6 fi.1QribEr^/$(r elv(r! ro

. 1 i i ( . r , ) t + r = t 1 , , ( ; t , ) a + u [ ( - a i r . r r - - a ; i - r t ; - r ) r , r + ( - a i , r i - - 4 i , , r , , ) r + b i ] .

Eirv il fiaL|(i E,,(ri$rlorl rrr. o!\rd6c!!€ \r(r ('lJ$fid('e! [e qa noultrurutll Luorl r,r6re 11 vlc extigrlol .r1,r 0a TraQiBevE I i6r(1 xar I rrcQltox(lol fi6 (ryr!),€E 0clllldr-vrqorcv.

IIAPAAEIIMA 3 (soR) rLa iov i6,o rivcl]{cr a = | ',

.t I, xhoe 6r1pc rrqt - . . 1

SQR iIeL nt loQq]i

| 2 o l| ,.' z lr-t =

ul f : , r -Jr , I I r - - 2 I2 l I o 2 { l - r l -11 - , t l ' | ,+ - '14

l '(';' ',,':,,,]"*."0'Ecir' bLaLpdoorF€ $E ro "r, o! 6{0 ol)roi ni.!ox€,< yi!o!ro[ or S xcrr, ? r1cbr.iro:r(o$ ,,1 = S 7', rl'd) I e7r.r.,'hLll1l{ rTiLorQiqr€r orrt poQg{ Srr t =T:tt +b.'Eror. o zqioLpoq :rivnx(l< S r7. rol onolor or lbrorlFdE xo0oQilow rov 0u0lr6orplLotq. eir,ar o

"" =l:,

H 6dlrLorq trrri,oyr] xou u, €ivcrt €x€ivq rou Ldv€L rl F€yqlvr€qrl lbloxqrri rou,- (6t),s6i q qc.olaxlvri ar.xivq rou) 600 ro 6warov Fn{Qor€Qr} To,ttrpm)o4ptio tt1s rtrqXatuipao4; tuu va at'axalnytt aur6 ro 6 .roto u. Toylv6F€vo to)v !6rorLgdv woirol F€ rrN opilouoa ;rcr. crv 6oupe rouE brioielY(Drlriois nflQcrlolr€E. r('v ono[(o! to yrv6F€vo 6ivoL tov.L", o:tptirrog €1eroaiqo]Joo 1/4 (perri qv ovrrotpoqrl) xc! o b€rir€eos 6Xs! oQlqouoa 4(1 ur)'z.:uv€n6E

)r): = det.L, = (1 - u- ' ) ' ] .

ArnG stvar 6rcg levrz6E xcv6vag: To 6rr o rptirrog nivorrs< (r+.r-L)rou!€lo9€pet qv d€tD I oqei)reroL oro 6rr. o tr 6pioxeror xitc'r cr6 q brcryrirvLo.xo! ro 6fl o oeurseos nivcx(rE rnwcrorpiper rqv det(1 -)D oe€ii!€r(l! olo 6r! oa/ 6Qiotfl( t((iv(o ur6 rl bLuydJvro. Etor. orlv rr rni n rtpinttroq. ro ytvqrrvorouq eo €iv(.r detru = (l -

"r)". (Auro rl6ll rEry€i .t!clri. 6€! ndp!: nor6 n6Qa

(l,[6 ro |, = 2. To yLv6[s1,o rcw dlorqdv 0o {r(v ro},ri B€ydtlo. FI €nrrQ6r[o1,raqyro 6lq rs i xqv l), < l, on6i€ q €fialdlrl I 6€! 0q ouvdrlLv€.) naiQvollpE€niorg [kr oio04ol yld tlv oqrncgrrpoqdr rr,;! Id!oi4r6!, rorr rirttl 11 elr]g: frc, = 1 of rdlofl+liE Gauss-Seid€l 6ivcr! 0 xci l/4. xe,r" xo.g(irs orEd!€r ro ur. orr6Loi$6E co,5ig ,r),rtordEow [EraEu rol8. f& tu iiltroro a, ot bto dtoaplEth'at iott xut ftift n aplt zar rav 6{o ttptlet fii th' u I, iror 6fte nytt'riptt'ri tovg |a tarpt<i(a p t4t, opi(ouo<r.a H rrprt (Alr{ rou., uno}.olilEtar€uzolc!, brorr ro (i0Qorofa r(l)v r6loqrd)v olpnirr€r ni(!'toft [€ ro o0eo!o[(r r(l)v6rsyovi()v ororl€i.ov (ro tlvoE rou ,.). 'Ero!

I 66)'rrofll cruri ,r(eolerQoE !rrpoobropilerct rur6 qv rlioororl

) + , \ ; = 1 ; . , . , , - l ) - ( r , , , n l t = 2 2 ) . , 1 . . - ' i t j \ , 4 . ( 6 r

H r€rQuydur{t ,rrrrl tEioc,oq oiv€r .ir.,i, = a(2 - "/31

= t.01. >uvrndJ< or brjoiorg rdror4td6 €tv(, xsrd rrQoodyyrot. , - I = 0j07, rou nnor€},€i €EqLQ€rr i6€),tioot. ov or-rperesi F€ qv rrFi causs-Seidel ^ = 114. 'fn , = l. H o(l)ollie^L),oyri rou !,, o anr6 -ro ntlea6€ryFcr. burl,ooico€ :rdl'L rov puep6 or'yxho4g.6Lorr ( l /a)r ! 0.0?.

H (rv(lralul'q 6rr tu(t 1610|(1 6€;.xt('ol FrroQ€i va /v€r x6oo €rjxol,q ox€b6vpclx<i, urlpEe 1 oqetypio Fras repdmlcE 6pcrorqpdrqrcrq eni Fkr eLzoocericoTlv AQl0prylrill A!ld,!ol. To riAdro nQ66),Ila rlrcv v( avtrrlrijlg€i xc! vcE r€rrtlo€i.I F60o6oE rlq urr€Qxa)dQooq( xor ro 6!6qxroQrx6 ro! Young, ro 1950mpr,eile trt l.1rol 6vsv dr .6 rurro tl<r ro r.l. H Lb6o-x),€r6i ircv vo ouvde0oljvo! [6roqr€E ,\ tou nivaxc Z- pe t6 L6Lor4rd< p tou cpXrxo! nlvclri(l JacobiD )( L - U). H ouv6eo4 cuq exgpri!,€ra! p€ riv €qiooorl

7.4 E[ANA Hrll lKE: litEgodol rIA To A:t = t/

() +. ' - l) : = ),r: t : .

+ Edl ro d d!!I(ki i' iilio. oL ,b,ofluiq /i\ovtu ir( L.rroi oir/d! $td), x'i)r l.a, qd,! nor

i.ow iu, o, 6!0 l,\l = o l- onir ro y,v6uE!6 rotr ei!( r.,1, (, l): roi io $roo ror\maodu;dva rnq'ixallu t'! ro !.

(7)

YIIO^ON:)\'OI ME NIN KE:

Auri ro1usr llc FLa €ua€ic i. crlyoeia nwdrlcrv n€jr€eaop€voa 6!(qoQ6v xa! clvrrdeoufr€ *' = 1 (causs-Seid€l) 6ivrr. ): = ,\l?. :uv€noE ,\ = 0 lor ,\ = pr. Avt6€r!6s6ar6vErar (rr6 ra [ae(6€iypot( I rc( 2. 6not ! = rl/21$ ^ =0, ^ =1/4,tuirtz6 lapcrr4prmLz6 TtE l'x€o€o8 !€t(rEu tol, !€Oo&Dv Jacobi }iol Gauss-Seidel:o)r.oL or irivuxe< tr6 rL(orl! ro! Young 6xow L6ronF6E / nou €Fqavilo1{ar ,rord!€lryl crQMtru{(iv 0€rLn6v }trlL q (?) briprr or or <rwimotleE ) eivnr 0 xcr 13.Xploltlonord)waE rL< rsl€urai€q rQoo€yyio€rE rou .r, 5ur]'crordrlouyc rov pn0g6olYxLLoIE.

To €v6icq6Qov :rq66).4rn rivur !c 1{&lur€Q€uoouF€ t1 p60o6o. OiLoqre!a 6rcliEo!$€ ro l, 6xoL rilox€ rl rblox!$i l !c €l,.rxloronor€frc(!. E!trl6< tonQ66ln$rr ooro 61eL rl6q ),r0ei.. H !:Eiorool to! Young (7) 6€v €itlat ricpd IloQ(rrqQlmrxi €Eioo)or1 1Lc ro 2 rni 2 n<rpri6€rruc{ ,d zdr io xd},ljrrQo u rllovrx€iao nou 6r{ars xa! rlE buo pilsg ) v(l eiva! i,oEE [s o - 1.'O7ros (rq6d)E oqv(6), 6rio! /rr = 1/4, or;ro o64yci otqv

( t - r ) + ( , - t ) = z - z e \ + L t ' { | ' . l r =2( - J14)

Il povu6Lxrj brctpog'r eyx€lrc! oro 6rr. yr( t\,(\, tl€y(itlo nivcrx(l. uuro ro Fod6o0a urovcr?tc4r6drvercr yrc €vcr n)'ri0os 6r(t)oQ€rrrd)v !€vydv i/;, €vd nQan€! vcrnLl.6Eoqr€ 6vc r.(! lrowbrx6 kr. To I€ycrlu1€Qo or'r' crurd ro !euy1 6iveL 14IEy(l,r't€Ql rblorrpli Tura Jacobi, 1h,,., x(0dg €nioqg 'tcL 1rl F€fd'U1€al r$rllou J 1{o! iou ,\ =,..' - l. Iwrrrir(. crpou o ot6Xoq Fo< rirov vcr xd.!o]+E ro ^ooo to bwat6v prnp6repo, crn6 eivaL to arpoio quio,< rori x(0oQilet rot tqlxcLli€Ql €nttro?rl rd!r,.:

,0.,. =?lL-r4:le;) r.,, ),mx=,,,j.)- -r.

At}Iog ei.\,ar o rutroq rou Young yr(l ro! 6d).r|oro orvr€).€orrl lj?r€Oxatrdpools.M:roporpr vu urol.oliooqre q 6cLrluorl nou rrrLt iL,ur n J. ollv n€Qitrrool

tou rivcxa 6raqoQrbv 1. !E orolx€id -1.2. 1 1',.r1.d trtr{oq r(,)v rQr6r'xuat(oviuryDlioiv. :r(t n(pabetpcr<r o :riv(rx(rq auog tirav 2 €ri 2. Yrooeotsr,irp.t rirr rivur n r:ri rr dror, drmr vu uvrLotor;1ri or'&cydprol f€ un6ctt(orl6!a6ol!x6v x6u6(ov h = l lQt + 1). Kard. rlv AoxIoI 7.4.3, rt F€'ylorl !6rorr$1irou Jacobi 0a r:ivcr tr.,,, =cosrh. AQo q tri-ylorl !6ror+rrt ryt( Gauss-Seidel 0scivar g:.., = cosr a/r zoL { lr€. onl 161()140l fiE soR 6Qlox€raL or.rrzoormdwo(otry (8):

. 2(1 sin n/r) (1 - sinrl): r sin rh

Arn6 ynopei vn entqqori F6vo tl€ or.ry1.i€xQ[p6!o nspdSeL.Aa. 'Eor(,) oil o ,,leivcL tdlerlg 21. F€ye0oE cQr€rq pirQro. T6r,e ]r =1/22. cosn,h = 0,99 xar rl

(IJ)

_.r hn, \ t i ,LAHm txF! Mtoo^ot \ToA. l =L '

p!0o6oi Jacobi €[vo! C(Ql|i. H cosr 7rlr =0,98 or|tl(llver 6rr (rioFn ra! I causs-Seidcl U,r .Lr.rlrFi :rdQq nn).}.d( fnrrvu).Iv€r6. Aq ou otru< rm,:, = tnm =0.P.

I 6iLrrorl F60o6o( ursexr{l.da{rofi ocr ixsr. orriva orl1,X},(oqs

, 0 . 8 6 - _ -, \" , . . = I i4

0.7s. z, f l r . , , ! I ' 1, , ," . = 1.7s.

Ir rri0r 6r1gc. to oq,ril.pu repr.opi.(er(rr. xcrtri 25"/" xo.r trc pepoyapiyo fiipa rniSOR toobuvapu pE 30 lfitarc qs p|06bou Jacobi: (r,99Jr" = 0.75

Am6 €ivcL xo [o].x aTrQoo6oxnro ofior6t!€opc !!a( 16clrl crLrls- !6aa(. Ol,IaayFarrx6E Tllq rq(rQ[oy66 6€v rjiv( fi.r [ovoblaorera nQo6],ipc1a (oD{o€r!6La9oprz6-c elrori)orL5). FLa xft LX. lvs rpLbrcy6ro ouorrtptr /r =, ).w€rrrrruxol'u. H urrplai'tipt'roq 1iverru evbLcrqipouoc orK n€0Loo6r€e€S broordo€q,ot &(lq)oetzdq cito(i)ou.< tr€ tr€Qlt ig ir(a(cf6youE. Eriv cvtr.xcrrnorrlooupr toFoa(16r{rfo 6rdorru( 0 < r S 1 p€ ro [ovcbroio r€rQd]('^,o 0 S r.y S I, x(r|,ul.LirEow€ xrlv rEio(l)ol (116 -(,.,. = / or -t.r,.- luu = /. rorr: ro q1Jorx6cvuLoyo ror ,riloxa nrnepcopdvorv dLcrrpoptiv ei.var" to "opjpc nelre olpeiorv",T( morxrl(r -1,2, I x(rd q 6r€!0uvol I ol)vduitEo!ftlt [€ ra 1,2, -l xcrrllq 6rr!0uvoL y[( v( bdootv pta xiqrcr brcrlrirvro u:rorr],ouFev1 a'16 ro +4z(n r(oo€Oc €:[165 brayoviou otorXcicr l. Qo:oao o niruta< A 6ty iy.tt eipoq,lra4rdo6 5. Aer u:rcipyrr rQonog rrai0Ftot< tur Nr oqptitrrr tou br.xrrlou o dvcftrL',.lo'vo. alor ,;,trr xri0u o4rtio v,r rivur xo14c x(r olo rrruore(l yrtrovrxri rou.Al'ri F.i\ur n "xatiea r@1, ,to21d)1' ^uKttdotei, x(r or rqxil,Ll).or rJnoloyrolr:c0(( rI {)eocnsioouv tt€01l{6<.

Eciv l 6rdrcEq yi.vsL 1p<4rpr1 npoq 1pc$Fli, 16r€ xd0c o+reio rrQirisr" vc:rtp4r6ver pur oLox).4pr1 ygrrgyrl lu vu qtdon oto yrrrorrx<i uno xrirru rou xuLo nivsz(< r(l)v nivrr o4frfurr" dxrt Rrpoq tcL1rfng N:

O rivcxcg arr6g rOir6IE€ n€AroooreQo r1v :rpooopl xcr rotlLopyri0ryi€ [EnoLu TreQroo6r€eor\ 6tuqopr-rLrorig rp6rol5 c:r' ororn6tiriore dL].I lAcgFrrirlEEioool .4r' = 6. Nolilo 6rr I rciorl oipsQ(. €ivcr! I €rrorQogri orK &1' su0€irr5f€oobouq, ror mqQfEovrar oE p!(r 66c. r(l)v Golub x(t! Hockney. AraI€aiqo-vr(: of pritQ6rFpor'c ni!0:{rc uf rov ourolo rpono. rrttu}.r11oupr of oQro}rivous

YllOAOl IIIIIOI lvlE IIINAKE:

"anoor '6s5€[6vo\" F€ra:{ roug r]16rxo!-< nivar€q. (Auro 0o lrnoQorio€ !c ou-^/rQr0ei Fs ro! raXU $ErootitF(rfl.ol6 Founer.) naotY{elL(1! or ffiiraLtltrltll'Spigo\ot tvd,r.qit xaftleiloea' k6A t olroi€E I riao(1lcq€Q0€ioo 6uiorool6rcxtoprlr rov rprbrcrldvro ri.vaxcr zctci t1v zcreuOwol ,r al6 €x€t'vov xard-rr1v xcrerorvol !/. Kar fiQLa aa' cr,.rr] xuptopXoioe I uraaloli(ac'rol. 6ro-rt ilr6rorr[i Jacobi l^,,,. = cosz/r eircr" 1 idr"u, 6truq a(r <x1 pi.a 6rhotco1, 6rcug i6rog€ivol rcr o oljr5€l€orng u]r€Qtol.trQcDoit! ,ri€ii . :e xh00 r€ainlorol. I 6uo|{o)'icl]x Elror orlt trFrd6aol an6 ru rpo6Lrlpotc-povrilc oe apclpctrzir rpo6trqputn,)tor ,4c0€ $L(l ar' all1€: rrq $ueo6ot1( iX€! rrE 611{6g q-< du'crt6qttg npooogpoyiSo€ e€ro(;)o€Ls l€aLx6r€Qeg n:r6 tlv -u.,., - r,,', = J x.rr. yrr4tetqieq 1evlloteqeg(1fio GljTrl! €voq r€rQaYdrvolr,

At! pfioQoriFe vc rl.sioouF€ X(l)QtE !o xavolJl€ lv€ic oq F6go5o rc|Jv or,gt yorlNiiotlttt. norr rpcLv6t0v neocrp&q oL)d E.!(f,vtrd qorvrdv€1,€ no),! x(t uiv(rlpril.Lov dpcorl, noQa( €novol,rywlrri, |r'Iopet 6ptc|rg. ovtieelo p€ rll! (lJrol.oqrl,v( orc4rur4oer oln trrol. Af! xprr,i;rrat guoL:tcr !d ro\iooruf or. r:rnpr. vcrE!1qd.|'loe€i ,or t-ro r€].El(DE veo !664 zo[ 1(r x€Q6iorr ro ]rarYvibt.* OoilEtulotco: 6ix<rLo va :ror-r|re otL 1l]! nQcYpci(Ilxri €raa(ioraol orl Luol 10! Aju = l)dqepe 1 perri6crol an6 to 0.99 oro 0,75.

A'KHTEII

?.4.1 Iu lov ri!(a(

. [2 . ^ ' o . lr l = - l 2 - l

Lo 12 )

!€ r6totrrric 2 v4.z.:*v4. apetre rou "ia(ad

J.cobi /) r( /, l/) li(lt rq r6too[i.tou, toa ni.vaxa causnseidel (r+I) r( {i) z(4 rLs Lbrortpdq ro! roo.irs zdt tou,caar0polq ,'j.).,. xcr ),,,,, nt6 soR. Arv tettdt.tdr !a rnoLo1id€r. ro! rivoac I..

7,4,2 rLa to! n eri n nirdkd ,=l, t' , ,,1r.aqQd!,r€ ro! rildro lacobi J = D '( t, f,/). A€ik€ 01 10 6L(illofc rr =(s innn,s in2nl , . . . ,s inrnh) Eivor rbro6r( i ! !d [ ( 1o! J fE ' l to t+r1 i , \ r = cosnn =f " ' n / r , l )

7 .4.3 f rd toa i6n, l . 6s iErs 6rr ro 6td lNtrc ra = (s inrnr . s in2l in / t , . . . . s in , l t rh) . ivoLr8ro5'irwofa. nol).ar).aoraor. to ra tL€ tov -,11l4 va detire rry (lr'ri.mot7l LbLoflFt

' i Mru rpooqaq.nr ) .oy i l r i rd r r t 5 = IoLL.6nor .6 r , . ! naaat ,onoro t . i r ( t o ,1 . r ( tu "ador,trlri.,rk)! naq|Larrz,;r I x(r U €!,o{lroF(r r,r |,tni! Ami olotLi,lr(t ,r..,lti.ILr",i a..;,j.

7 ] ENANAT\IINTIKE: MEOOAOT TlA TO AJ = 6

o/i. Enoite€l)or€ orr orlv ftqirtuof 1or 3 €nL 3 or rbLor4ris crraq €iaor or2 .,,5.2.2 + ,rA.trylEtuo,j: Ot r6rorLp6s ro! ni.vaad Jacobi J = ( L ti)12 = I i/2 lilcr^r. = I a,i/2 = cos ltr/r. Alrd ra ^ €Fqorilovtdr ){ard !e!.fl aqlrrra6! €.fur(b!xtlo! xar | .\Dr\ €ivdr cosn/,.

Or €noFw€E aorrio€r"E Xa€rdlovrar xo "Oed)Aq[o r(Dv zuxkDv ' ron cershgorin:KdeE rdrorqt4 ror A rEaizEraL o' iru1, .to1rldxrcnr an.j ntq xlj ,ouEC1,. . . ,C,,, 6rou o Ci iXeL xivrpo .rot n |nytitvro anLlfto aii zru. 4 axritaror e h'ar r1 =1,r, u 11) io4 yt rc agpu4ut ray crdljrur z4dy tur ororTeittvttt € rtr6^otnl< yaaltli€.

A'rcit€r6t H,4ii = )ir obry€i. ofl]1,

( ) -a ; ; ) r ;= la ; r : r ' ; , ,1 l . \ - " r , <) ] l , r j l9 .t t , l / i

1

E(iv I tl€1,(l,weQl ou\r6r(i)os ro! ? €ivc1[ I i,. 16r€ ro re].errTtlic c.uto nllixn€ivcr < 1 r(rr To , l r r€Q!6x€1o! orov r i tz),ot ̂ -ai i <1. i .

?.4.4 O niloroc

f l I l lr= l i 1 l lL z z ) )

).tytrt\ [&f'uri@< inni."or. 6Lo rd0e la;; > ri. A€iErs o ro II6dv 6er pnoaFi!(t r€etarrr(r de ,td oto! ffio rolE x!a).o1E rsr s1ltLJr€Od!(lrts orL o ,,1 eivoL ttl

7.4.5 fr'(urov ro! brorovios be.r6!,ovrc l.yedw€ rov rilcxa Jlcobi J aar dp€ir€ ror!iQn( zltlols to! cershgonn yr( ror ./. AEiElc 01 ot o)rrf!€s ror! Lioaororcnv rt!r, < r lar oxt I €ra!.i).fu Jrcobi dqrltr'eL.

7,4,6 H rqayrdrrht iu.rl ror, .1r = 0 riraL ).t./o 6r(.parrLxi ano q tnol i1t< ord.o(t4isLUx\i = t). O 1 ,Lr tir! ff!(L pllbtv, ).oyo ro! o(fid)+(ro! mqo].r!1r!o1ls. MLdb8(rdqrll ri.vur vo yi.!of! r( r.il1ll *. b'ibi czei6{a, dJd ivds

"a}.ureoE aar

xarnrreos reoftos.i!ar r rn@ahlardlj tx1latu,o,t: M.6Lr).ri drqi.6ELo !troloyiorltrovo 10 ijldluopa , =, ,{x0, }.!dir ro ti.rr = | aar reooodolr rry oatimoLH),uoll mo r0. nad6),Ifur flol].otrldo'.lm€ ro rr = ,0 + rr llt rov ,1,', yadlm roaroriLr.oua oE bLdordol .s'r =

"r0 + D roL €ETflor. l,raxi ro T rivdr eEdLa€It,{d

Fdeo. To tlr*Ao auro 6rtFd lt.iq o6ry€i, o1r6ov d,E6oq mo r.

7,4,7 fro 6vdv ?€!r?d rilaac 2 rri 2

!= f " l lL d

6peitr ror nirazc erortilTlDE J.cobi s 1T= D r(L+U) xa'rs L6'orLrds ro|4i. BAdrr miotq ro! ftivaro Cau$-seidel (D + I) ra,I. rr! r6ror4dq ro! l, roLanoqdoim€ fti! ).,,. = ri-.

fPAMMIKOUfIPO|PAMMATI>MO>

KAI @EOPIA IIAIfNIAN

8.I T TPAMMIKE' ANIEAEEIT

xw10r,r. 11 6rcqopri prraEu A),yc6Q(E xcrr Avr0,wr6 rivur iilo tiq:ro).i t<rur6-,1pr1 pr.rr1 6rcrpopir 1*(E! €Elodo€rLr! zcr clvrodroeov. n(l(6isa&. I dlcX(D0Fo-rrxri ./Qop[4 [Ex(Eri Io]v dUo qrutv6rcv va civuL oaqrrlc. Tt)'rrri ti|ur< xtrrd)'(6<t6rr o lpcrpprz6< :rpoypt4pcrrLop6< anorelti 6\,( (\'imcrQdbe!./tror A{poe.i .rvlo6-qrr<. u),Lri tivur <rvrlpq,ro6ltqtc $€Qos rrtq racgtrlxitq Al.y€6ac( rtrll !d),rorcr (rQrxrll(i lorioluo -oL cno(doEL< otq €flXerprloeq gui.vercr 6T| lArr(ilowcLrrQroooroQo ro,, lQqrprxo rpoypctrtrarro[o, rcph rLq opi(ouoe< tl rlc 6Lor!dg.M€ rQ€rS rQororq $roQonFr v( rpoo€ylioorttf ro Muo+turLral eni r(ol/ onoi(D,'or lQiqfrar o ?put,Frxo: npoTp{(/ l rLrroUoc: r no:rr7'1. ui f l ' ' r r ' : 1r, , 'urr0 c<. r 'xo-i.oyorrrh. yioo r1c ar0ri6or sinrplex,i.r.rly!6pl)iri, [io(lj tI< oEupicq buio$or.A\rrof o! rQonoL raooil./roli alarruoool,rcr o€ ronrl tv mraciyaa?o, rc06qzdL orL( 8.2 '{cr 8.3. Kc1r6:rrv, orlv Ilcrpdypaq,o 8.4 coxo}.oi[€o.r [t nQo6Lrtu.lr(,6'[0)q c!.rc! r(ov yr4rDv. r(o! onoi(D\r tl }.ior) Eilar d]16aaLa. :rijv It.ladi,acqo 8.5o!E1lrl41€ ro fiox€Q. xc06: l'|.r. d),].a rar./!i.brc :rrvhzcov zcr oro 16].o< rEryorgero 0adral[( nrininrax, ro o:roi.o xpu6etnr:rioo cra'crurir, xn0rirg rcr q oliol rort l t ro UfJpr ru burol .oi rou yon,rt | |xou nao]aqrl , rnL'L,orJ.

: crur1 rrlv rpiq €z5oot, I rlaadyacgoE 8.2 ntQLdxer z('1r! !.o. H U60o5o;sinplex avray(l]vi.!€rcL noiir ivrovcr fvo r€),€ioE 6rctq,oqmrxo rQ6fio EZr6l.eoI<rolv rnololrqutilv, ovopcr!6pcvo pd0o6o to! Karmarkar. H !6rr crlri !6611 7inQ.pe1,ciLr1 t)rpooL6qrc. ev gtper 616rr or upxLze< uvoqoQ6c tozrQi!o'\4(,' oxr. €iv(L50.f,opig raxrlrpl cn6 q 160cl6cl snnplex. O rol!p!o$6! lotia. ttroFit.r).id oXt1( onoLo6rirot€ np66trrpo. H ,'6(r [ieo6o< Forti!€L va sivaL anorE].Eot'arrzri lLatr€i/(ilo cQoril nQo6l,rtBc/r(r -ora\r SIIa6rt o ni.vczaq I 6t€L-ilv trvir],oyl bo$t

3.I TPAMMIKE: ANI>O:ET: ,153

I (t, 2)

!x. 8.1, EErodo€q a(rl dvLorirrr€r.

€v(i) q [dgo6oE simplex EEoro).ooeei va 6€or6E€L or no],Ld drJ.d rQo6],ri[(rro.Tlror€ 6€1' €ivcr rrQxer<i oarpd6, 6L6rL o nrir6Lxag 6qLoxetnL oc otd6lo cvCrtrulqgor(. €Q,loortpro AT & T Bell ro! €i\,cr! afi6QQ[FoS. (O lesFlrx6E rlQoyQaBrot!-opoE €iv(! 1600 €v6rae€€(ov xar 6rab€6oFvo6. 60r€ or 4rnoQ!r.i66 €qqQ|toldq touvd 61ouv aroxrriour l€yciLq orpooitr -crur1 ei,vcu, prtr uowri0Loq xordor(or1 .yr,cremo-q$olrxo uno),oy!oF6.) Euqd4, oL 6corx6E r6dr6 aXoljv 64lootororlesl xql€tot fnoAolrt s vd €Elyrlool,p€ its oQ'XriS rrl< [E066ou Klrmarkar fiI! ncrQdyQo(0o8.2.

Auro ro ,€qdtrarro pnoQei va rorio0rrleei o{i oftoroorirror€ orlF€io, percr roKeqdloro 2. @c utdqlou,' naQa),Lnl6yQtrppo! riilc.tsE, IoQIE 6ptrE rc aovrcrvo €ivoL yQol4u1(6( €E!or6oe6. IIBrlForr, eva xl,fl6i crutor.r tou mgdrciou eivcrL!o bowe nl l€olerQrnil €eFlv€i(r r,i,\t yeapputiv ava.itoe1Dv Mra dviooon6r(lrAri rov ,-d'doruro X'i)Qo os 6to t/ilr,r(rou5. iv.rv orol' onoior, rxovonoLFirqlI a\,iocuol kqr 6vnv otov onoi,ov 6ev rxcrvonoreircL. Turrr,{6 ,lqQcib€lylrd €Lldrrt r + 2U > 4. To orivoQo FsraEu nDv 6uo r|tuxdla{l)l eilar I euO€io r + 2y = 4,onou I clvlo6rrlxo toxlieL oEaxir. Iltiwo n:r' cruri qv €u6€ic €lvar o or!(oF6voEutloxalpo5 tou:1. 8.1, l(dlto €ivs! o ovri0eroE un6lcugog, dlou n avlo6qrarqQo6lo!€tat. To oxtiFd €tv(l|, 01€6()r ro i6!o xd! orLS rQ€LE 6LcorrioeLE, tool|opo yivErgl sv(. €nire6o. 6r[|Jg ro r + 29 + z = 4, €r(i on6 ridll]) rou €i.vct! or rixoaos :r + 2y + z > 4. :rK n 6r(ol.io€q €Eaxolou0oq.re vcr ovouttlor,1re ro(r 1)-6Ldm&ro oivoQo "entre6o".

Erir6g crrr6 a{o6qrs6 crJrou xou €ibotE. uridQxe! xar ri}'loE leEoEop6g.aor €ivdr 6ep€l!66|1< yra rov ./Qqqr!x6 fiQo./QqrFollop6: Td :l| l.oL g cnolrEftclvo eLraL p4 upttlttxd., H afiqlqol ourl €nqQdEsrut p' 6vo E€tryoS av0o6o€ov,x > O xaL U >0. )uverdrE 61or4re buo <rx6pq 4pilarqoug. rarv onoiov ra orlvoparl€Eixolrql oxous dEorE ouvrerc.4r6v0w: q r > 0 cno66letnL 6],c rc ol eic6€Ed rou ;r = 0 xor I v > 0 silar o ryrix(roog r.ivo qr6 xov v = 0.

rPAltNlI(o: nforPA,\tnl TEIto> Kr\t ()LoPtA nAfl b{IaN

To €Arxr6 ouvolo xat q orlvdprqoq z6orouE

To €!6!o(l6Qov 6ri!G Eiv(lr yl ouv(1l,{oE!ol xc! t(']v ra(irv c\'rs(;o€ov r + 2y2 a, r > 0 x(r y 2 I). Aw6<. ow6uatolL€v€<. blvow to ozrcrolivo xopioro!:X. 8.2. of iou t l f iopr i rr vu uvu./vupiorr8 rr lv rar, , ; ruv rQl;v l t XL;AUv.Auro b€v t'i\'ft r),6()! Itri;(oAoq ali.dr ruflx6 rcpirdeLyp( auou iror o ytrx44u.z6capolpappcrtropd< ovopctltL tqwd ottrol,o. Mi)dJvr(i F€ r€eroo6reQ4 (xQi6sltr,€'rcr €qrrro oulol,o anor€Leitc1l aa6 lK trro€LE [raq orroyivE!(< yQal$rxd)v

lo or,ortu( A.r = b. pr ' , r r l rodor,E x(r , r cr7\, 'drou.. rrgrypd, lrr rr lvroFI ,r €nlTrdiiD! ivc lLcr xri0e €:io()ot. (Orrrv t = n )tar 1e eri:rt6c eivct(v€Ed($rFa, rdlvorror o ivc orlprio. rou rivcLro.f =A-rr.) Kar' crru).oyicrv.drct ouorrpc ?r cv|odoe0)v ,4r. > | nepr,lpirger" qr toprl rrr l]lr(Xdtx ,. Ell.,'(r,tcl|lrioorFF rnrnldov x(i0r ouvrordrocr tor.r vcr r-ivcrL p4 cAv|l1r.ri (ro onoioo![6o].iE€1cr! Fe pLc burwqLcrrLxrl cviooot .r > 0). a!.r(i 0(r rAoo0do€r hIIorE/r ro nli0os UriXo)LmrE.

'O'to nEeroo6rEQo!( nepLopLopoUq €fir6.ilJ,outlr. 160()turQ6r€eo yiv€rur ro [qLxro or]vo].o.

Eurol,a paopef !o o![t6€i ro Fq]r,lrr6 orvol,o vcr eivor eQuy[r6vo i qz6$I).rlr xtvo. Eov d,Ldloug€ ro rcrpribtLlpri FaS r(r OEoQri<loWF rlr i + 29 < 4.xQ((I6vrcs rr< .r > 0 zc! y 2 0, :rcriqvorpe ro rQi1,lDvo O,48. lln'bucilow(tqltaf iq buo cvio(iro€lq .r + 2lt > 41rur .t +2ll < 4. ro olvol,o n€QrcQiEel(l os $itltu0si( or bio (wi0trot tr€QtoQtolroi o6lTot ! (vcryxcolr,rd fiIv i + 21l = 4. E(ivnQoooaoo u[E i:vav cvrrpcrLno aepLr4)lopo. 6noa l] .r + 2! < -2. r6re ro e9 Lzroouvol,o Fi\,c(! zE\,6.

'Ev( F€Qo< rt1; p€),r'rrlE pcr; rivcr l drl.yr6pc rrov ./Qcttltux6! (vrodJoeov (i (,)v

Eqrxrd)v ow6kDv). Qmooo. oror 1pt44[a6 :rgo.1pr4r$crtrrrp6 untr"odpletcL xcre vc cil.lo owrrir6e< ouol(rur6: M(lG €v6rqdp€t 6y .o ro oit'd,o rav x]rzrtitta4peiuv, dlti lta tdbtd alttio rov irol duXuno:rou:i i1 pcyrmottod pruoAKrylyI "o1jyaQal i ,o ' l , xr iororJE'. >1o r i (rQd6€ryl tc . t +2tt >4, }>0.J>0,nQoo0dror4L€ rI ouvdarlot y-6aroua (it at'turrytvtxfi ouftpraca) 2:r + 3y. Tonpa roTtrc nQo6l.rya 1(,! lpnpptxou 'I0oyp.lFF<rrrouou tivcL vn dpoupe rou4eio ,r. y :rou rre4rdleru atu dttxri o'roho xat Eha|nnlrotEi ro xdaro1.

To;rp66l,qra rpt'xiietcr( tln6 q yrrrrFelpirlro!:1. 8.2. H oLr.oyev€ra zomotq2.rr + 3y oAi.!€r pr,cr orrol6ver,<r fiaQC 'nl,ov €u6€!cir! rc! e€lows lo 60ou[€ roel.dttoro x6mo<. p€ dl,),c l.6yrc. lv rpri)rrl t:u0€is nox r€lrv€r ro €qu{r6 ouvo),o.H roFyf ornrl. oup6cfv€r nQoqcvdE mo o4reio B, oirou r' = 0 xat u' = 2,raL al,1rXrmo r6droq €iva! 2x- + 3'lJ' = 6, To brdvuop(r (0,2) L{yE:.et €(rzftibL6rr r€er€lrrsr oro €Vrzr6 ouroio :"'ot" eLyu 6 ,runo 6L6rL el,c orororei rrlouvciQrrtq r6oro!E. To e),ir .oro x6orq 6 eivm l r4rl tou :rpoyptq.4tarog. Tcr6itrflor{. 6ruvuoFdr( eu rli outr6oLiEo$r' Fr cortUro)to.

MnoQsir€ vs 6€it€ 6I| ro qtkoro bkituopa 6eiaxE't.rt oE ptn yavia(xoeuqi) io'.J €qw@l otldlou. Aur6 sErtofo}.i!€ror ar6 nt y€(')[Ereic, 6r6ri oLerOeitg :rou 6ivow q ouvderloq 16010r,< (i ra eni:rebc. 6rcv :rirlrr oe aeproo6-

I.I f?AIU]V'IKE: ANI!O:EI:

l r - l r = 0 : r + l r = 6

\ - l r = 1

tX. E.2. To Blraro orvo).o zer I orracieqml boto\ 2r, + 3V.

r€arq F€rq6),Irr() l(rr(l(lrouvr(tl bKQx(;)a rQo( r( n(ilor lilQr a( 4rloouv 10r{{(zro o!vo},o. H :rQ{irrl €nc(fri ),orno! :reitrrr !(l dll(tiri z(rd lulzoi roUol'vopor'! H Fd0o6o< snnpler tr.lQ(t!a!r-r o c|(6 to oli\'oeo. trfl(6crivovrtr; r;rorrt tu( "o0l,{frt

ror' f(rzror, o1,.ro).o! ont! i).11. Ba7e6 611)r, tiori rI! zoQtqn I€ro lrrtx;rEAo x6oroq. Avri0ercr rt F60obo( ro| Klrmarkrr n0oort./iir-r iI 6ai1!orl),ro1 cr;r6 ro roorr4rrzri ror rqrxror orrv6l.ou.

.E lflti.Dot fr( trr(t br(r(rcocruart orvi{arlort xoororK, I rotri tlnoo€i v(l t(Iv livcr!1( trstrovo+d\,o orl!.io: Ea! ouva6cr1,r-io xooro< !( €i.1,(1r r+2y. 16r€ oL6l{),lorl|l (r4oi. tr€rcEri r0)v il xcL l]. 0( irrllr lcL"rolQoac o),o ro eqrzt6 orivol.o.z 'n 0u r ' : r '1p7, yrr Irnrrpiu dr i . r iorrrv tur^ uqrurov z,rr , i ; r4zo: (r(r i . r i t : , (zt" j :(:1. 8.3). H rLtut riv(' :rcir.ror€ po!(6rzr1 (ro .r' + 2r' Looril( pt ,1. lLrr oL( ru(;€),!!or( 6rc\'Uotrcrr.r) x(r orre:rriq ro ;rptir'ri1p(1 €).cXrororoiltorl< ilrL nctu FL(lI'oloo'ilttltl (n(i\,lto1. An' rrp til.I1 pcprlt. to rp66lrycr pr.lLoro:roir1o1; bevixr:r l.uoll :ro r{trxro ll(< orj\'o}.o, io r.(6{troq |roqei rtr lirrr ooobinorD F!:y(i},oxot to giyroro x6orog rivcr dnrLpo. 'Erlq 6Ltrq)o0€irr6q rp6rog €€liootq arrrtEr|l< bular6rlr( r-i!c(r !a a\rr!fiQal,ourF ro r{6oroE oo .r 237. ;r(pcrp6vowcrg oeneo6),riF(tr(1 r).cTroronoirloff. Torr ro r].d/rolo eilaL :!:. 6nor< o1o:t. 8.3. Lt1rnri).r 6rr r.;rnpTrr ).riorl. KuUr r0ob;.rlur Tpuy1rrzou:rooTpt4rlurroFou €urioorrurot li(l cn6 rL< r0€r< br!\'cri< y(r lyoeir'<;

(l) To rq,rxro o11vol,o Ei!c! r€v.i(2) H oudArlol I omoq drtLL y4 qpayylt,4 oro €qrxro or!o),o(3) To roolo< 6tef d!(l.r2dyLoro (iy yiltoro) or,o rrprxro orirotro.

Or n1r,r . ; 6io 9cr i rgtr l vu r ivur :ro),u o:rr ivrr : 7ru i ru ynloto orzovotuzo Ir€XvLzo rrQo6inr(1.

2r +3r = 12

456 IPAMMIKO: TIPOTPAMMATI'MO> KAI OEQPIA TIAIfNION

tr. 8.3. ErdfidE r€orm(i)o€K: ola ru oryeid rlno 1o A d04 ro t el'.txroronotona qvr + 2il. To Eld1rolo irtg -ir 2i, eivd! -6.

xar,aQig FE-to6l,rFiE

YficiQler evss qr).os tQ6itog peiqrQofitE tou ovtoolrltou rl€QloQlo[ou ot€Efo{DoI, €roci./ovrcrs 'ffi

7d,ea4 rctu6)4xi ut = E +2y - 4 nQdYpclt!, ct!51isivcA I €EiocDorl $qG! O riQorffoll€vos 7r€olooloF6E c + 2:y > 4 FerarQ6?rEtc! o,rov(, > 0. ,[ou rarQra!,e! c1n6l1j1c [€ rouE uft6lolttouE tt€QtoalouolrE ii > 0, I > 0. M€our6! oxQ!66g rov rQ6r[o 0o E€xll,rioou[€ xq $60()6() si'npler. XQI]o[o]ror(i)wcrEprcr 6on0qr[xli p€rq6],yFrl yla xdee qvioooq, 610! 60r€ o!'volLrdl vo dXo![E povo€Erocooe|E ]{or antlolE fi€QroQlstlo!( Fq dQ14ltilt6qrq6. To 6aotx6 /iQ66?'llFq ro!yQqp[r]{or riQolQ('uBarropoi li.verot:

E|'aytmotoi4aq t1g r::r:' utit rov€ tt€QtoQroltoiE At: = b xat t 2 0.

To 6r"dvuo1ra-ypoppli c rugLdlcr ro xo0e nootog, 6lroE o-ro raad6elYp{i [dq6nou c = [2 3]. O riyvoorog r r€QdXE! rLE crglLx6g pno6].tpdg xaodq xaL rGonoleo&irior€ Xal.ap€E B€-r(.6LntdE. H orvortrq ii > 0 p€xaq6Q€I ro fiao6Upd ortlpq cgvrprxrl yo-rvin tor n br.ltorcror xtipor. Ol crvLodroer"q aL'dE oLl.qLerxt6aorrvpe rlv cncl,oLgrl causs ncL lper.itlrtcrL pr.c er"re)'.riq vdc 1664.

Mra €pllvri(t lou nQo6li,raroE rar tou butl{ou to$

Aq ErLorpiuouu€ oio cQxr.r6 Faq napci6erlpn, lt lrooxo\ 2x +31J, trdr (4

ro brorurrdoouFs [€ ].6yrcr. Aur6 eivcL ncpciber,ypn tou "npo6ltipctog 6tniqE"orov yQaFF|,1{6 npoypcl4rcrtr.og6. ge 6io n1y6< riQu)xrivnq, 1..t. qLl6lo xaltprorLxo6ourupo. KdO€ xLl,6* qtonro6or,tupou 6ivsL [r.rr Fold6o na{DrEilnq,

* :.r.E.: o olyyaqidE cu(q,fa.! L4mQtq a{i aLLirv (1 ).iAna( = 0,4s'r xriir)

r r fPA l l I l tE : aNl !Q!E l :

xdoe 1i|},o q ,6101 6f'rcL bro Sovci6e<, sv(i) yrcr q 6icrrc ansrror,w(r lioo€QlErouirilLorov govri6rg. lrJ1,€nd)E ltL(l 6i(!rc fiou nce[tX€L .r] n|,id qloru{o6oiruQorx.rr, xd, qr!trdror r.fQroQiltsrcL a7r6 rlv i|+2, > 4. zclodr< x(L rr!.r > 0 rclLj/ > 0. (A.r propri va d1o4t oprrlrL:io q().6ro { qrorLko6oiruao.) Auro €iv(lrro rrt,rr.ro oi1'o),o 7t1r ro n0o6iryt! ri\'(lr vr. tr€rdooutft ro ;rooro:. Ed! tv(zL),o q rmrzo6oUrlpor morZi!€L 2 $ z(1r ivrr zr),6 qL),rroU otorxil€r 3 $. rort roouvo),rxo ;r6oroE 116 tlicLrtrg cirar 2.f +3r. EurrU(iJ<. I daitroq bi(rrtr tiv(Lpolo fri€ro: r- = 0 rcr ]/' = 2.

Kder 1pc4rlrrx6 nQolQotrruc, 6nrDE ro ,rQonyouB.vo, €x€r (vo 6ui,.d. Ef(vxo (071'46 €ivcr nQ6dlrlp( €),cxr,oronoi Ioilq, rorr ro brriro roli riv(rL rQ66l.illlat(r./L{nonoiilol:. H ),iiD tolj rvoq o6rl1ci. zcr' Er0€i(lv orrl },ioq ro! (i}.},orz( nQit./l(srr. 1() ei(ixlo1o tou uoXLrol nQodii[croE nodnr! v( roorircr I€ro tr iTruru rur ' hrr)aou ruu. Ar"ro f lvur orr t l ro,r11r.r f lzorqr,r ro zrrrpLxr-rorl||ntQ(utr1l r|E 0r(Dai(( ro!./0(qqrrr{ou raoTa(4rtr(rrotrou L(r {kr €:rvloriort! ncadyacqo 8.3. E6(;) :r€AroQrlotr((lrr oro nAo6irll(r tli bitrlrtc x(ln(xntfic0outr€ vc r:Iyloolue to luizo rou.

Avri lLc ror ayoa(orri, rou 6'cltyFr tr[r(Flj q tjorrxot oi,r Lirx, u x(lr. qtl.iror.ti;orc va:rriprL n5xurtivrl !r €),illloro rooro(.1{l brizo nQoliln[( (wlu€r(Jjni(r'rrllrcr6 rol ip;ropo. o o:roio<:rorl.ci ouv0trratl :rgoteivl zur olrv(y{,)!i.!€1u( ror1r),iro rat ro (luorrzo6ourl'po. :r'\'(1'rcilt ctr€o(,)S r( br'o rtuotcrrzr. r'!69runrzon laq4rLzo| npolprilltr(.To<: At"rq 0i),€L !c frlroronoLrio€r q! lrtrt /,r1 onoicr opor< u:rozrrrcrL oE tpctrtritou: nr$otnoloi<. H orrlnnxti :rpotrit4btv ;rpi:nrL vc xooriia nctn('ool8Qo un6 lv fiQlx€ivl roLr qruorLro6ourL,eoL)(nou 'ir(! 2 $ (v(i frov(ib() z(lL rou (t'rlirou (rou trcv 3 $ clvd bio po!(ib€<).'lirlrtozaovtr.

1rarr1 :tpi:rtr vc r:i,rct IrI (pvrp,ai. bnq,oatlrnri o dgroQor brl0( ror,),r'r. Aqo! I i)i(lrr( (n.rtIri rao{'l'!r; Boacibe( ;rgorr.ir4;. 1 eiorguE41()| itr^oo.r 0(l tir(r .l/r z.lL ro b().i) noodi.nra ril(r rrxpttidq rouro: N.!ptyi'f[o'otq0ti ro 11t, r'iri rot:'S trtptoptotoiti p < 2, 2t < 3 zar /, > 0. Aurotiv(L 6!.r raQdbfrltr(r oro o:roio ro 6r,lz6 6t€r ivtl fri,vov (i.7vr'roro zcrL irivrt(Lrr 'o fuzor.Lr,r : ru ro rru ro. O r,L)! , ) :6rrror?u: nrr ' , ' , r ' rotroc f iv l l r n. ' , r ,1 ' , ,1r , ':p j 3 rcr I liyldll rrtlj rrF nrilUF1lxiiq r$Drrill]q ai!.rL l,5o $. Erot I !6yLorqf i o r u r h r i r a r 1 1 , = 6 $ .

'EEr 6o),rig<r rjrftv rniolq r(L io n6oroc ror .rp/rxor! nlool.itF(rroi. onorto (yoa(olrii zcrci{pr ru 'rirpdrvrL ro i6ro yrtl r't q,r('',ri z(r rI ouvor xrirarDrtiatt- Aur6 cil(r z(r ro v6nro ror' o€oariLr&ro< t'riotrou: ro si.{iz(totoor'i.rt In fo I'Eyt6ao.

TrJnrx66 €qoQFoT6E

H r;rofr€\'t nceita((o! 0( €nrr€\'rp(,)0ri mrt iiot r(l)v r0o6ir}irro! yaqL'trLzoi naoyaa$1(notron. :uv€nd,: r,irgr rirct 1 zurriiJ.{.1 mr1p4 ra rtgt-yQ(il,oul€ oaroliav€E ;rorrxrrxtE 'tcrcfillo€LE. nio(o cfi rK onoir( 6Qio,(Elcrl ro

TPAMMIKO: 11?OfPAMMA'I1:MO: K IEEOPIA NAlrNlaN

Fooruarlrd €Qdrq!.o: l,a d,ulloxonoqeE[ 4 ra pEftotofloLtlq'i pr(i yAalllHiouvda'Ena4 xtiaro E, urd oQotlilooE yQaptmoiq rtQtoQtottoi,e.

L ZlE6rcop6g ,.seoyotlriE, Ynoe6or€ ortI ceneral Motors x€Adil€r 100 $ ytqridoE Chevrolet, 200 $ yro )rdo€ Buick xcu 40O $ 1!o xdo€ Cadillac. Ta ornoxivrFcarlrd 6ralioov svrioTorxa 20, l7 Nclr 14 Fil,lcr cvd l,irQo* 6€vEivrc, €vd) roKovyxpdoo erLFdvrr. 6rL 11 pdo4 cu6booq npdner, vn ei,vcrL l{3. To €€ryooidotoF rop€l vo owoqro].olEi Frs Chevrolet ovd ].€nx6, pLq Buick .rvd 6uo l€nrd xcrrpro CadiUN rrvd rAis l.€nrd. lloro eivar. ro $d1oro liiQdog o€ [!c r][dQa 8

nQ66141ts Elu1nronofolt to 1(nr' +2NU + 4tJ)z lotd rouq n€eroetottotg

2 0 x + 1 7 ! t + 1 4 2 > _ 1 8 ( r + V + : ) , x + 2 u + 3 2 < 4 8 0 , r . , y , 2 2 0 .

2. Kstdrolti tftriioattrt 'Eoro 6rL bror{eolrs! [erox6( rQd)v rrino!: xQsrtl&S.F€ ioxo 5olo x(ltllloQlog A, hporLxtg. pe t6xo 6010 xdrytyoQi&S B, xor F[ag€rolQ€iog ouocviou p€ 16'(0 9olo. x(rl4yopidE f. Mnop,oqre v'olopriooqrr noodrin.g 1(!! : trorJ b€t Ee]reQ|loiv ro olrvol,o x(ov 100.000 $ . To ,rQo6lrllls €i1/cr rltlr_yrorofloilol rorv r6xov. uno rolj( nsgoQrotrorJc

(i) yrs oupdl'|,o 6€v €rlrQ6n€rcr !c En€vbuooiv n€QLoo6reQ(r 0r[6 20.000 $(ii) I ldo4 raoqyoaia ryF xor&voptE rrpdrr€! vq €ivaL rouLotLorov B.

Ilpit6),4pa D,ayototrodax. rtty 5x: +6!l +92 vtrti rot4 neproptopot'g

ATKHTEI'

E.l.r xx€drdot€ io equn6 oBo)'o,6tor or reeLoeroroi sivd(,:+2V > 6.2r+ a > 6, r > 0,v > 0. Iloro oIIEfo 6Alo"owaL orLs rQ€rs aoewdE' ouorr roD ow6].or:

t.t.2 t:wuritftr) no'o riv(r ro rkixroro nE duvdoqo|lc xi;orol5 r + r. mo reoryoi.ptvo €qtno oivolo: :r(€6Ldd€ qv ned'rl €uo€id f, + y = oroosQd. ro! €qdtrrErdLmo €frxd onvo).o. nora oryEio €),crrmoro.onv (s owaanissrs xdorolr 3r+y xoL

8.L3 A€iEft 6n ro €qrxr6 o!vo).o ro! n€orooiha, cro rts r > 0, y > o. 2! + s.! < 3,3r+8/S 5 eivdL x€a6.

r+v+r< lm.mo, s S 20 .000 . :S r . : r . . v . :>0 .

* >.r.E.: O or/yaqqid< aroq6a.r torJ.6vru qqi lirE ! (1 yd,i6r (U,S.) = 3,78 iirco)-

8I TPATIIIIKEI N NI:O:ET: .l5t

E.1.4 A€iErE orr ro €n6t$'o nQ66trWo ax.r sq)rar6 ouvolo pr qeoyp6lo. sropdlog 6€vdX€r 6i),Irmn iuol: M.yLoronorlmE ntv r'+ y. lro io\ r€proarotorrs r 2 0, , > 0.- i r + 2 y < - l . r v < 2 .

8.15 nAoo0aor| drq r > 0. s > 0 Fn dviooort, €rq- re6f,ov (ir@E to €erliirt @ro).o nourTqoxrJr€' vs nPeLaxet ivu tr6\,o orltttio.

E. l6 noro r i lor roa lFa rou.{Lzro i ouv6lou x >0.y > 0. ; >0.r +1. /+: =1a( t noroEiv(rt ro Ftrrmo rI! r +2y + 3!:

E.1.7 Aror€ ro nA66l,rua qE xc'rdaop'i< rov en.'6no0ov tro! 8tftln(i)elx€ rAw it(

E.r.E Or oweix€< trr (1agtflx&rr(l: r. y. : > 0 .rsaiaow rL(r ro r'qLzt6 oivolo qs cencrulMoro6 dva $/boo rot' rarbrdarcrotr r(6eou (ro e€itao oy6o+r6Eo)- fldx rilllrtcL'flo q:Tu r" duo rn.rrf^(r ror'lei!o"v ro'5 rFeroerqrris zar roro nvaL ro Lrrntr.rro! rqutron olNoloui nds pnoepi !d 6Lonrotooti (ln(5 q roaut)fs tou. &r I oll1onrror,: blo rsqloqrqrols a(r !6vo. oq 6thoq ),uolt 0( ttq.avilorcr l|6vo bno tlror

aJ.9 nei6lt$ ttrsqoad,. Ylro9tale otL os ai.eE ptc dn'flc nokftiFE rou TlEcq.nF Kd,rqr6alr{r<

"crr rls A},dor{sE naadyovial cn6 tl( ${.l1o!trJgo 6aa!h(1 flFtQ€-

).(io1'. >ro:rx(iyo, not, tjaiot8rar o€ (n6oruq't 1000.2000 xc! 3000 p!).iov, olnoloi-

xn\. ono ror\ ra€t< rono'E naqa./(,yri5, xe'ldlolacr 8m.000 d(qall(l }|qt oq NdaAwiia. nou o{il$ 1500, 3m0 xct 3700 rii.r(l..!\notoiXo{. Xa.r.iiowot 2.2m.000d.etLra. Eav ro ,6oroE r€ra{i,oQdq rilG lkr loldh{r old 6(ai},L4r}.Lopsrao. roroyeapprx6 Trq6yqcppcr fu ndvro €eroeLdFou-lsdrFsq trqdrfl !( )"!0ri yrrl v( d.uxFrtonor(rri ro x6orq t€rcqoe.t:

I P NiMIKO> IIPIJI PAMNIA I [.\IU' KAI Ol]?PI,\ NAIIAIS2N

d.] I H MEOOAOI SMPLEX KAI H MEOOAOE (ARMARI(AR

H rctp(iyat{qoq cuoi aolo),€irar $s rov ^/Qcl[rzo rrQoyeclLtlarroBo l[(r n I€ra-6)'Ir6E zor ft n€QroQloloui. :qv rQorllo{F€vl traairyQuqo €ilofe n = 2 z(lrn = I, untiax(lv bl)'ab 6uo Frt qpwF(r{6q $€rc6l.rlrsg xat €vaE Fo\'q6Lxos reQro-prott6< r:+2y > 4. H y€Mr6r€al rsoinr(Dol 6€! Eil(lr no),r 6iorolo vc tEllleai,ber rivur 6por,< rut ;rol.u eixo),o vc ),r0ti.

O xcd.ur€eoE ra6nq €ivrrt v( bLrlruirdloor.tr€ ro :rg66l.qpa F€ riv(z€q.

(l) rvcE nilclx(lE A ,, rni n{2r iw brirluolu-oni).1 6 pr rr ouvrordrorE{.1lr ! . l i r iwotncyQqrtr l . t r r , uuvroru,orc.

fLu vc rivclr ro 6L.iauolcl r "€(f rxro". rreinlr v(t rxcrvo:roLci ttE r 2 0 rar ,4r ) LH '6i).tLmr1" iirorl eivru ro rtpLxr6 6rdvuqc el,o1iorou roorout. 6nou ro z6oroqct\r( [ . ; r =clr ] +. +c,,1r, , ,

nQ66lqpa. Elayotottoiq.r4i E],aYon:roDior| rnv.r,v:ki totr< nrQloek4ott( r > 0, ,,1/ > 6.

H lr(oB€,rQrxri ovsnoQ(iorc(oil rf!(l €{xo},q. H nQd)rr] ow8rlrr1, .ri Z 0.;reptopiteL ro btdrvuoyc mo tlFr!26 !6aro rou rr-brdrot(rou ldrpou nou €i!c! Ixolarl rEQlolri (i),o\r 1o)\, It[XlirQo'r, .r , 2 0. Au6. orri 6io 6r(orcrosrE. r'i!c(r ilr(rircQro ro!| er!fid6ori x(tr rtxrE rQ8lg dvo olboo ro1' XdQor. E!( r!7cLo 6],Cv!o[((rou r-6L.ior(rou X6Qou) 6XfL [io orLq 2' n0.rv6qr!g v(1 €ivaL [t (QvnrLr{o.Or un6),ornoL nr :rtptopLopoi opflouv nr :rp6o0eroug ltrLl{;)aor( zor r( sqrzrabr(vuoF(ra rx.rvonoLouv o).ou< ro\ r,r +/, r€pLo0roFoiq, avou 6aio){or'r(1r orop6pr"o .r' ) 0 rct rcr:rriXpovrl tx(votroroi! nt! A:r' ) b. Me ciL),c l.6yLc. ro upLxr6ou!oLo rivar rl roFn nt +, rllrldQ('\, kc{! FroQri !a €f\,lrr itr(xryfivo (LE €nir[E6rqni,€uaql) I rr't .t0(tyll6vo ri ('.ot't z(r z€ao.

H owirprrtot '{onouq .r €roilyrr oro r0o61,ryc Frc ol(ix},tal orroy6\'€rarcpc ),tjLor' €nur66rDv. Evrr F€troq r4g orzoyivercrg, ro p6ro nou 6ra07.rct a,"r6rnv (QXli. €ilur ro €rin€6o (,r = 0. Or(! ro :r u.ovono!€l (lrlrl1 q! €qiooorl. ri\r(rbL.lauoBa 1rl6(vrzorl x6fiou!". T( un6Lourc €nin€6.r r.i = m(0rpri. bivow6).8< rr; hlL€E 6uv66E r4r€( x6oTo!f. Ko06g F€ro6dl,).Ercl ro x6mo;, rs eni'tsbcnrtd ocrprivorw ol,ox),IQo rov n-6Ldo!(ro X(i)Qo 1t(r ro 6i1r@ro 6Kilva{t )ltrtxrftAvExlt oro ottlrb ditolr ),re rQdrtt qoad ayfiEou,ro EErxr6 ottud.o.To 6dauoFo i' €ivc1t eqrzr6 zor ro z6mo< rou c.f' riv.lL rlcrlLoro.

O oro1o_< tro! o'ar,.nl rtv mrpdlpaqo €i\,or 1,c( ll]roLoyiooup€ 10 r'. O(proqouoc4rr vo ro xd{vorJF€ (xax'(exrl\') 6Qioiovrdg 6),€g rr.E xopupdE ror€.tLxroi ouv6l.ou xar urol.o1ilowcg ro n6orog rtq xhe€ !t(r<, ordr€ €z€i\q iro!

II2 H NIE6OAO' SII\'IPLEX KAi H I\'IEOOAO: KARIIARKAR

!I. E.4. Or Toqr,itds xrll or aqris ror!ft,uron ouvo^or

dlrr ro prxp6rrpo x6oroq ea ilcr1, r1 6(:i.rroq. Irlv rp(i o[o)s trn r6rotclrQoorol eivar crbulctrov \'(, r),ono[I0Ei. Yrciexo!v oroexalolluola t{oQl)qaE xc.t6el paoporpr !c rrg xnolol,ioorlF€ 6l,s(. Alri li uur6 )'or;r6v or0e(6[clot€orrl pt0o6o simplex. nou 6X€t rclt(orei plo sr6 rlg :rro qlpropdvec r66eg -rr,x

u:roloyrorrzrirr pft0qpt(1rxri.,'. AlJrrl c.v(t'Irlxolx€ $r_ ouorrl[(lrrx6 rQono (oro to''D nlzig, y!c1 q ),!on yQatrur"r6! rpoypaypriror xct rir€ otl€t)€,r(! ot$, ruxrl.eite oe pelcr),oqu1 ip:weuo4, eirar xarurhlxttx(i onrtuji(. @c. fiQoor(0riso![r':rptirr(l va ohlYloowr rc 6rl[(rc Ils tr€066or simplex. ro! cpyiJrf-Q( 0c or'-'vollJioofrc. Kcl6n!\r 0c fieQr^/Qd1,or|€ $La vdc [i0o6o rou nQol(io{xe etr6 ro\r

I€oF€tQi(r rivrlol xorri pirog trlrv crpdrr

flLorrL('r orr aur6 :ror qorriler rI pioo6o €ivct q yeo4te.Qrx{ eQ[I!€ia. Torraoro 6i[a evto:riler cr:rLi{ prcr xoEnxll 1or €qrr{ron oll!o},ou. Aurri rivaL 't

"oiol I' nor uno0dlo4re 6rr dtcr o),orL1pr,l0ei. Kar6rrv, r1 1rr1i 11< pe066ott:flrlurxlr',1 drrl i), urL0t 'piot r1j. trou r4yaivrr Mo xopuqn ot xopt.ti. xafltptlxoE nr axprtu nr cquroi auft)"ou. A;tit at'r0r: rrnLrtl tloeuqrt 6Iox[E nnnLl.o.76g tupcbv. rar ex rov o:roi.r,rv opopdveg nnopcxqwor! ctno T t 6€Lrrorla).ki tiyvcoorl xopupl, rvrir hl),tg obryotir 6.r0FLairl reoq .ltlrrta. O DanrzigbrhleEr qv :ropei.a ror xcrrd piro< pl<t< cxp)i: rou $E 6€6aLorlra F€itov€ rox6oroq. Arr{ 1 cr4oj o6ryei o€ 1rro vio r.(oQl(pit ltLxp6rcpou x6ororq. 6i1oqSwat6qtc €lrrorporj{g or o.rr6rpore 6c;rornp6teao. T€}.Lxi' qrdvoqr€ oF rrcerbLxrj zoprgri, crn6 qv o:roic o)"tg ot cttp6! o6ryon'v o€ urfilor€ao xooroq: Tox6orog {1eL e },cr7rmonoq0ci. :qv xoarlq'i nrnj eivcr ro 6t)'rLoro 6rdvuogu;:'zur I 6rc6rxcroi( oxc$or(i.

To fiQulllafto nQo6),IU( Eival vcr U€roqpcoosi .rljrri I r56a o€ laaggLxlrille6prr. Ilplrro '(airr!( !( €QpIa€ ooll{E rI-( iiE€Lq xoptqti ,to.t axpl orLq n6Lomiro€rE. KoQU(pii €i1'ar ro oltr€i.o ro$r]q n tu.qope-ftlcirv urLrtbcuv. zcroivo(no r.1 oroi( ra0rt0i(frr(' cro $La €Ei.oc,rot 6ncog axpr6rirg rpic erireda(),.2. ro rcircofra. o ri€rpLx6q rcr o ylrpoorrr6g roilo5J 64rorplorir pru zopuq,lof l : rAf lc b.L{oraor.c. O+,ldr rrc orr. oror TAur.r ,rxu _r-DUyr, ' rULUrrot,o. ro r ' t rzroorrvoLo opi.lercr un6 nr crrrodror( Al ) 4, xtlL r (l'rc(;o!r< r > 0. Kir0t zoQl,q,lt. rurou rol ouvolou naov'nnlr '1o pr l r rpo.t t l roc cQloriorrg. )iu[ neoobloQlo[o Iq( rourii r()v rl (t!rd)\, rrura6(,)v. MLa €tbLzrlt n . o 6 t t ' r v n e l l o o r o e o ; v € i l , u ! , f t = 0 . . . . . . . , , = 0 1 o n o i c p c C t p d p v e r o q vapXrl rr,-rv cEorr'lr. Ontu< opo< xuL trrt o),t-E rtq (i),},[E 6uv(riE €7rL].oy6!, dr?.i r(,, t4ytto rol t jc 0a , i rut 7r4o" ' zoat, t t j . t ,o, \ , t ' t r t a,( : ; . t , " ' , tzuL rorq d),iouq tn np(,0roto1i<. A!(1qro0[1rl{d iln eivclt rcleitoq cncrlinl. (Vouorirr x(lv 0( (vrtzrL oro Fqrrro o vo).o. To:rcrpdbeLlpa orlv Aoxtol 8.1.1 €iz€n = 2 tr€ra6)"tr6g rrlL nr = 2 ;reoropLolorg ̂o! uni(exow !!r ,5walaE i5oU(E nou.turvorr, l r oro:Xftr(( ̂ 4 Terrc ono L(ul ic rr \c.r n! ' (yLr(rr aovu\rc ruu r , l rxr.rr-rouvol.o! . ol t rsr( ;)vovr(r yt P,Q.R , trLL ooi lona rc 6|(rvuotrcr( , (0.6), (2,2) rcr(6,0). To 66kroro 6r(iaro$o ro6r€r vc civc( 6r,cr.rr' ol)rd (exIoi (n fll!nrQirrrool no! ro sidxlmo xooroq sia(r i'r). OL hl),FE rO€rE roouqraE, fr€rc1E!rc0v onoi.ov zal I sQX4. ri\r(rL (Ir(rrl],iE.

fs1,rxci. urdQXo!\, (rr +?n)1/lr!r,rl bu,claE ro[aE. dLorL uur6; eiv(lr o aqr0p6Erov 6u\,ordr\r EnrLolcov ,l €n|nabcD! crfi6 avo oivoio 7r + /r !no!,Iq,iov." Eda roeqrzr6 orro),o rivcL zev6, r6re xuyyia trr'curic rrg ro[6E 6r,v 0c rivcr ptloLtrroQu(l,rl. Avrlxei.[cvo r1l,( O(i(nlE I civ(r! oxQL6d]: ll \,( 6QeL [La lvrlo[a roQlJq{ li'v.r .lfiobei.Es! 6r! xo e(prxxo o[rol.o €ilcL )tevo- :]J!€X|Eorlltr olrDa lro0irovraqorr ixrr 60s0ri t'ro yvrtor( )ioarq,rt.

l'dra.r. ooov (l(oari orl\, ox$ii: Yro0ao,rr orr r(e(],sirou[€ ila (1lro r(1 /rrepv6pova eri:rebo, o:r&t r:rolivouv rr I tirodrouL( rcL owt:rrirq t]{r< 6c0[oqe],€u0spi.(. T.r o4petc rol rxcrvoroLouv cltdq rL< r 1 rlLo6o€Lg tr:iol€Ioivprc u:tplj, :ror :rt:pvd tr:'lci r1v cr I<!r, zoiluptl. fe(4,€rQla/l. rl .rzgn .1tltin€QryQd(fci(! ('.M()$4 , 1 etrLn66(nv. KoL rdir o lpctrlrzoq rpoi,patrtrarroiroqtraq ljnolasljl€r'va r(papeiro|pe oro eqr:.rr6 o!voio. bitrd: buv.rr6rlr( rnL),oyrtcnurrrOrvot-rlq e:ri r1c cryrl<, uq,on pic pdro zcter0udrl €|!(r oiro8€rtti ((urnrcx(i 1rlv onol( $0Lri^,0r.rr ro,{6oroE). EXor+€ oor6oo rI 5uv(r6qrc flrL},o"/]]qcvcipeo( oe n br(q,oa€rradE oz[iE. neiryptl ior ro ftro.{(oi!€r I Ohoi] IL

fr.r \r(1 ncQltQdrtJo![r (rtr] rq rpciorl. npdreL vc lavayphqorge rqv,4r 2 bzcrd rp6no rr),e it'rq (rdri.o1,o sxsivou rrov n crd,cbv arpLopropriv :r'1 > 0, . . . , r'" >0. ArJ16 ro p6),o neit ow ot w).aelt FEra .4tlE u = A;t b. Or :repLopLopoi.{.r 2 6 percqqfrEolr&r ororE ur > 0.. . .. u,,,, > 0 [€ trrc 6or0trrxri ir€rc16LrrrtyLrr rir0s ypctr4t rou A. T6re 1 eliooo4 nr =,4.r b. 11 ,ar tu = 6, lpirqrrcrL 1rr

rPA\'IMIKO' IlPOTP,\JV'I\'I TI:IVIO: KAI OEAPL\ NAlf NIQN

+ To rr!r0o! (uroi ror erllr{oi.lu rrtyrii! r) ?(r ,. a.i!.r ro! rrotot'rqrn n}]d! rdrv zo+(i(i)v

8r.t,i( (!iilriro .r,t! na.i:r

r : l l \ l t , ru \o \ I \ rPrF\ l i , \ l l r rLoo \o: ( \R[ !AR| \R

nLv(x€< oril loQqrl

rA r f ' l =r.' t u lTo r-{[zro onloio ;r(0ot]ilrr(r ano rrc ,r cur€q €Fro6or-rq rc(l(ir: zcr( rLC /, +,rrrr i . i : t r rroorr l rr : . r ' r i . r ' 0 z,( l r , ; r ' , ' i /of l rr nqroptopoig pt rc6tqrq zt t[t (elryrz6rtrs.

fLc( vrt oiozirt$;oortre rr< rd,i.c16;. Siioljt(E l(r rE.ri.ei{o1jtr€ r(! onor0s5Lr<popi< perolu rou (plLnoi ri ncL ror vso(qlz0drog u. H pi0o6og simplcxbsv {i7'..fir[ brcqoQir rc! fJ(r rFCv (ioxofio'\,c or.wrlloo|pc pc rouq olp6o],ropo!<

I; I) l;:l'Erot ortt1oiitot,ltt ror utyd.rlrqo :rinat xih pr 'l *u n yaxsftrso.\kirt'ott& ,tdit 4r .r', ororr o! nrpLoerotroi .lptiqovror o1l1 t'oaiti lr' = b z(ror tl +,r {1n),6: (lvroorlrrq yoil(|ovrcr .r > 0." To aQlrro brdv|(]ltc xoororK rTpttilttct e:rizrcrol 1r rrr trrrl.dov ouvrorrirrrr.c. iTo! t(! ir(iQlo1rf€ 6L€; llqba!.T6rt to x6oroE c.r tivuL.lrc r(l vdu.r' iL( .. ro ablo llr io n(l,Lo. To !l6v(( a}llrou (q 'ivr 't 6ot0ryL'{rl 1eru6),1r{ u rilo or &cmciol: rrr t:ri r, + n tou liol:rivtrzu ..l zuL ot n + rrr ouvrtrqp6vr; ror'laou r. A:ro rov q)7rzti orp6oholrirzpcrrilrt trrnci rc otoLltil p6ro zrt arplvor,pr 1o. Dt :.rlr n (pt1(i6Lrycr. 1(l!( !(4 oulaiorJa (lura) nol, or,'va6ll. 'Ero! ro ;reo6i{r( lttrcrQ6.'r€x(L oro:E).ayoroflorior€ tu t .t, t'rti lout tsQtoQtoForl5 .r' > 0, ,,1.r' = lt. XrptLriroro 6tt o,,1 tiv(r :rapc ,1L61ptrp1oq.

IIAPAAEIIMA :ro npo6irltr( trt ror( nrproproyoti; .t + 2r '': 6,2.r + | > 6 r.ulro Looto< .r'+ r. :rou rilyrir(r Ie ro :2. 8.,1. ro v6o otor4lr( iTrr

. f r I | 0 l [o '. r = ; ; , , , , ,= l ; x , , ' .= l r I o o lL . ' . ' ' ] | . ' ]

E/o!r ' ( yoLivf l t rur r , ,u: rr0roQ, ot, , ' fc ' no tros, l i tv , , \ ,nlr{ 'v. t , i roAo'1rr vrrlrx(ujoou$E rI Ia0obo snnplcx. O!}lt0rire 6rL q Ociorl I iTrr 4b4:rpoo6ropiorrprc zoprrl4 rou q rzton owoT.or'. 6ror, rdlvovra r t:riatbc: rr rl;ro rr; c1t7rz6<(rrorir4rr< .r' > 0 zcr ..1.r > , (ri .r.l,rd]< r' : 0) IrErcrpa;(olrcL oe €-:ro6ocr<.Mt tiriic l.61Lcr. xoptttgil aitar iva oqydo, ato onoio tt orlvtoriot€ rou viou

rcriopanE x (ro r(lrc r' Ba(i. Fr ro u) drar p46tu. lrry ,,1.r = 6 or rr(rlriq trFr(16).iF6E rsiEouv xo pol.o E,l.rrr:e(,r, tftru'trIrdi' E\6 or (rotrirololqrr trFr(6l.tr6E €ilcr ot 6uouiq lErad)tl ri .<.

'EroL. €Erodvoar(< rrc n Ei€!0rpfcf!.Icijl.lr[< Fr $tba\,. raoo6rooiloxtr€ rL< r(tric r(D! n? 6rcLvdx F€rc6ilr6\'(no

* ll Orzo\lrri( zo rl'1./\rzrl {f,oalr r( r'( nriorr f,Earoar.r,.i! r( kn;rrr.a Eroi! f: ,rarri:..iLi!. rr rrit]oiJc \inrrle\ :ri'\(i zG E$Eio! do rl! -,1,r' = b. /(,ai: a\.i?x't7at.(a,;! trnaiiirFr,r

I-IANIMIKO! NPOT?ANIMA TI:!IO: KAI OEAPIA IIAITNIAN

tr< 7r €Ero6o€!_c ,4r - r. H ),ioq cwq r li'ldq 6aov-i. yrc v(l rovLoosi yl€,E(iQrtoi rIE o/ro q( oldJvwEq !tEt(6]'nr6!. Auti ouFnirr€L $€ rnv tdtzi l.riotqE flrqcrypritpou 2.2 )anL anowl,tL yvioK xoquqri. 6rcrv or m pt pq6flrx€goril'r.or(i)o€E qq €iv(r o€rLxd-<, oTror[ avrlli€r olo crprar6 olvolo.

6A ot xoerr@s Nou tquNori ruvdlou €iva[ ariQr6dx ot aaarrcg ),itxtsrct Ar = b: Mn I'ia4 eLva! 6aoui 6rav n on6 rq n + m owrotoo€g tlsetvcr p166v zo egLxrrl drov rxcvorolel o1v t 2 0. H Odon I qS Fs066ousimplex 6eiox€r Fla 6oo'.riri €qLrrri ltoq r|o! rl odon II naoxoA€l 6ipq-6tip(1trQo6 rn 66)',[toq lu{ol.

nAPAAEIIMA To o4|eio P flt! xoarvrtE rou :X. ll.4 6eioxclur mlv roFrirlq :r' = 0 xq! nt( 2r'+ t/ 6 = 0: buo ouvrmriot-< ron r eivcrL p$rv aur b]JoouvLotrloeg rou ti.vcr" io€E F€ iE!. Eror ro x €i!ct[ 1.(Cr 6oorzo xo! €e!xr6.

. l l 2 l o l" '=l : r o -r ] = l:l =,H

t3lM6veL vu ).ryloei I xQioLtll rrioqidol: !e:tor( zoprJg,q vcr 4olopriooupe;

BQilx([€ pLC xo11rqq. 6rou 'rr = 2 {ruvrmdoEc rou .r €ivor orrLxt( tt(r! orun6),oLrrc !Io':v.i(,It t,.l.oulr vLr nAolrdcl.touttr xutri Fixr5 Flltg (xpr]< oe pt(ytLtortxl zopurl ri. Io lryovo< r'rr or OlJo zot'r{tic tivur Ysrrovrra( InFuiv€l6rr. (-(no rE ,,r 6(|(J126< !Er([6l,nriq, ot nt -l irletlttt,orv 6aoqi< ! vtit ltiay6r,o yivtttrt .lr'rirrQ,t (FI661'). Ttrur6lpova, pi( (l:16 rL5 F€rc6lrya: nourtr(r €),€u0€Qrt ./ivtr(r 6uoLzrj: I tqr{ rrF uvetuiwL ndvor ur'ro prlbdv evtir oruno),oLnt_c rn 1 6coLzdc ouvrmdoeE $rratjoLtro\'rt11, ?r.rQaFdvovias 6FtoE 0Et!1i!.H ouorc.rurli cn6rpcoq cgopir ro fiorc FElo6),rlli nQin€r v a(ncLQro€l on'rrlli(iort xaL tror( \,c aQoor€oei. Motr( Fdoo+€ no$q F€rc6l,nr6q €ivaL GaoLxdE orlvdc xopuprj. r:rolo{loare r4v rq{ roq ).wor.rq r'ta A.r = i, rrtt trrb€vi!o1.ra,<ttE tra6lorneg ouwetaypdvrg rou l.

naQdtaypa pn€ aoqYdlt.qq xat FwE a:'fQ'X6tttrtt; ItEraqlntilE

DluX'oh' . t t r iorE 14y 1.t , r- - 3. ' , uno rcuc nUt 'Utottouc

r r + r 1 + 6 r r + 2 r 5 = 8

xr + r ' i +3.r ; = 9.

lexrvlqt cr:r6 r1v xopu.lrt 6rou rr = 8 zai rr = 9 (omaq €ivar oI 6oo!,rdElt€ra6l,fi6s) rd! irl = rr{ = :rr = 0. Or n€eroQloFoi !,{oloflorouvrol. dlld

S] H N'IFC)OAO: SINIPI FX KAI H IIEOOAO: KARIVIARKAR .165

ro zooro( trnoQ€i va trlv €),c7rororror€irar. lladllarr, I orrydaqol xooron<naoo6roail.L no'o ffio rrs- r-i€re€a€E uEra6),rFaE natr€r vcr Fror{Xo€i orl 6ciol.O( {rcl ri(Qir,oyo vc xd}oqrr: {o x\ ectrxo.6rur o or!-i€}.€orrlc ror €ivor +7,t(t naooric0o U! rr t_l.olLoronortlooupr: ro aooro<. Aolx(< rpcivovr(r or rr rirr xcr 3rd.6yo!F€ rlv ir. 6rorr o olJrr€i€o i< rlc 3 EivG! o rlo tlAqrrx6E.Erct t tuitg^/i!1:i,tt tttr.tdvri ti u r ):r.

frtl lcr €roi),eEr it rr oq 6dol, na€r€r vcr quy€r t.rr ri I.rr. :rl! ra(i)rlrEior')or. cruEirvoqre ro.r'. zar elcrrcbvorgre ro r bLcrrtptirvr.lq .rr +2.r5 = 8'Or(v ro.f( qlciorL ro 4. io rr e( 160€' oro frlb€v. :q beurepl eliotool xpari4ooru0rQ6 ro r': +3.t5 = 9 zrr ro .rr $ro0€i. €bto vu auEleei g6vo [dxpL ro 3. on6rr.f: = 0. A! naox(uartoorlr rcrpsn€pcr ro l: yivrraL cpvlrrzo. Aur6 ev.ro:riier ql,iur zoAul,I oto oitll€Lo .rs = 3. :t: = 0 zct ono rI! nAd)rl €Eiooroil nciavo4r0.!j =2. H a!fta/ittrr\ rcru61tFli Eh'u 4 .rr z(( ro xooro: trrL(irvrrar" oro 9.

tlprirro{ (iAr! mouq nr:QroQrol(o!< r(1 7rl},f}i( r(,)! 6r!t(ir, fi),e!Q0)! riQo;1o1r!orvreirord; np rroepxi4cvl-c lrr.ltUilTrt(. 312 y.a 913. To ;1lxp6rpo nll.izobript :rot<r trFrc6trInl ftxdAXtlru. Oer,rpoupr p6ro e€rrh(i ntiiztr, 616rr cv oolJvrr),eor{( To! ,a5 rlr(v -3 (\rIi, y(l +3. rors (!!(i'\,o'\,1c1q 1() ,ri ocr ariqc.r,.1o!x61Qo,,cr ,{(1L ro .rr (ytcr .r'i = 10. I bsirFQI FFio(1rol 0o 6bNF .rr = 39),uno[a\,(l,< 1C) /: b€\r lkr d(|0ft\,e nord ro lll6i!. E(iv 6l.0r or owre),rordg ror.f5 rllc! rrQvnrLxoi, fkl !:i7qrr nl tr4 qQ1fttit,\ neQl.111ttto\1. o(r [roooio.4rc \,(rztirlolltrr ro .r's ooobinorr trrl,hlo ]1(r \,(l QiEoltrr ro rooro( [aXQr ro -.o. :'o1116 ofor< ro nr10(ibr'ryllo, tu b€{r€aa rylixo 9/3 Eal,tQtil'u 6rt 1l dttlt€Qtlynuni4r i l :4 ir t r rr t qt lytL wu rcur ixtovrr divcr .r : = 3.

To s! ),o./o 6rt[.r x.'rc],)itrL oril \,6rl roprqn r = 2, r, = r, = rr = 0. .rr = 3.To enotrevo 6ritr.r, o[(t)<.0( €i!.rl €nxol.o, trolo\r (1! or elLorioerq ouvrXi.loll!vl iTouv 6oirxr] popqal gr rLq 6aoLxlq [s1(6llriq qci(oarmd, o:rtrg orrd6crvetr!- n(.rr..rr orrq (1p7rr6< €ErodrocL;. fi clro ro ),oyo obtyoil€" Inro'vrc< rIbttirrQl t:!iorxrl r'rq rrpo<.r' xtr lr,rLxa0rord;v.ra< ro n:ror6),ro1c m1 owhprlorlx6orolq z(L rlt! nQ6rrt pEfo(,)ort; 3.fr = Lr: .r'i. To.,'do nQd6),tpc. Erzrlrirvru;cno rll.va( noa!.{,ri. €i'vcrL:

Etruy"utrotroQa4 t t ; '7. t : ; . r : (9 :r : rr)=J,r+s.fr : r1 9, uir6nr;

.rr (2/3)Jr + (1/3)Ir +6.fr =2

(1/3)r, + (1/3), fr +, ,5 =3.

To tnop€l,o 6rpo eivrrr" t:6xoio. O govubLx6q opvlrLx6c ou'rrleorrjq oro zoolo_(ot[(i'\r€( orl" ro rir ;rp6:rrL rc eivcu 11 ero4X6I€vI t€ra6].Ti. Ta n!i.i,y.u2/6 tr&L3/0. t,r,6F.EI(irv riitra(i,r. 6rurpcplvrw pe tr orrtLr rrq .?ir, 6Ei7f,,oLN 6rr" I :r'r r-i.varI arEalotLEvit [Ero61,tr4. H vacr roQu.tr] 6Qior€xar oro ollpeio .r'r = Jr. = i. = 0.rr = 1/3. rr = 3 'r(L ro v6o ,t6(]1()! Eilar -e+. :ror Eiv.lL 6s).rroro.

TPANIMIKO: NPOTPAMMATI'MO' KAI EEQPIA NAfNION

:' 6\t( Frydd,o nao6l,qFo. unoQsi Flo cr€QxdF€q g€ro6llri va €nrota6v€!ogy6rrQu oq 6don. Kuedrg 6purg ro zomog gertiruerur owqrig rxrdg r4E€vquLrol€vrl! rleQlrlr6o€@E- nor6 6Ev 0noQ€i vct o4r6ei acr or rr 6oondEF€rd6l.nr€E vo etva! fbr€( I€ rtE rleolyo!FEv€q. flor6 6€v Eovo€t roxefir6$sorEgLa zopup{ xcr" 6ror. l6rcr6rxooicr zsrct),'i'r€! oryt 6€IrLoq zopqrj (r1 oro -x6rcv to x6mog 6ev rppdoo€iq|. Aur6 nou €ilaL clr.oorpeiarro eivnL r1 rcl,.rrpoye t4v o:toic EpLoxetct ro 66l,trmo.

Ei)voli| Or sQro$oi 7,-1,-3 ofltv rQ6q roQuqt ]{ar l,iij-1 oq oeir€Qlr{opuql qrroqdsrsqv yL(l xq €LoeQX6F€v€S F€ra6},ylrdq. (OL sAloFoi orLroi. pn0i-votrv oro r. xo xeisluo 6!dv!o|rq fio! opiE€ror. ro0axdr(D. 'Orqv €iv(t! oEril'iol.oroFar{iF€.) Tq fil}.izs crrloqqoroqv yra rrE ar[px6$sv€( $€ra6},qr6E,

2wilro Er.a rou €xqultot ori MLct ,roQwri l.€yewtt txrptitopir'l 6rov Bqb€vilo-vtcr n€Qooor€e€E qrr6 xK ouvrlo€l< ?r F€ra6IT6E rou r. An sur4 xr|v xoauqrln€Q1d^,€ 16t€ nsQloo6r!:ea ar6 rr r:ri:rtbs xcr ltcptx6g a;r6 rq 6crorxig per(-6l'l]16g o!l6cd\,€[ vc( Bl6€1'(lovr(r. AUIo yiv€rcl owtrlrn6, f€rtr qv obtylorl,ano rrp +qdvrol €vog [Ib€vrno{ orr] 6€qL(i fi},€uQd rov €ELo6o€ov. To nl},ixc!nou xcrooQflow rlE crr€Q,X6Iev€E tr€rc6Lrlrdq 0c n€QLilor,.r plb€vrxci 1.(rt I 6donpnoQet vcr ll€rs&,l0€i X,)QIE v(. (l.,i'drqoqlc xoQwi. O€(,)QrF[za. 0(r trnoQoloc4r€vc EovaluQiEouF€ €n' dl^€lQo mqv i6(. 6.iol.

EUIulG, otttv rQdEI6e! ou[6ctv€! xdrrx6roro. Mdtrrora €ivc!! 1600 ordvto,(;)m€ or ?6brz€S nou 'ru',i).oqoQ()uv mo qrroQlo r1v crlvooriv. Auoruxdg otro\o cx,puhopo; rivur lEurplrLxd ouXvoq ortS equq$oydg -ecir rurri,vrrr ro xdoro<tr€xd sr[6 zdee 6l[c iIE simplex, 0a 6cir€ 6rr E lsvd.c{r6dv€r(! cQzEr6E (poQ€g.naLv I Fe0o6o6 si plex 6p€r tr.r ,{(1}.i (lxFri, onor€ ro x6mos e}.arro!€rr(|" rd).r.

To Tsutt?'6

Kdo€ 61i[o lIE [€066ou simplex nEprLo[6i(v€L afioqao€lE ?ror 1€rQroporislpcp[rirv :rq€ncr va sfiL],€t,ouv or €!o€QX6[€v€E xqr qr€QX6p€vEE F€1a6Iqi6Sxc06E rcr! o TAonos F€ rov onoLo 0d [nouv ]rar 0& 6youv. 'EvaE rp6noqoQrivoorE ro! 6iBarog Eivclr va brardEoqr€ r( 6€60[€!(r o€ 6vav ]r€yd.Lont"raxc'fi wpn),6. frc vrl EEiy{oor4r€ rE rQd(E€LE. xLg Ex1€}.o1lF€ 7iQ{i)lc orrtyl.6ooc rov rLvhxrDv raralriyolraq q' lvsv riro 1Lc ro x6oroq (r:c0ft ricr^1tn ro xprntpLo rrpgcrLopori, nou Fovo n 661,rloq ).uo1 rr.avorror€i). Aib€rcr6vcg rircxclE ,4, 11 6elLci :r).eupir d, ro brdvuoptr lr6oroq c, o:r6re ro rt4r:rl6tzziv4o4; rivur. u,r).drq. ivu< peyuLweBq nivuzuE:

T-

3I H N|EOOIO! StNIILEX (/\l ll UEOOI\O: (ARl\I RKAR

To xcr.rr lo €ircr r?+1Eni Dr+n+l | l t r l t la Ejr l rrLiot lpqrtLt l xar ott l l 'q. Iqvnp7ri. p:ropt-i or 6corzig peta6),qriq rc tiro crvapepLlptrt: y€ rLc d.ii.t( TclL ro;rpciro 6fua 3u< eivcL va eptpavioorpt ptcr polcbLz4 6cotzri gttr6).4tri ot zti0e

lpaggrj. Mttri <rr' uut6 or anoqrioeLE )'q6drorr(t €i1tol.1. A).iil(or'rc<. svcvhyzt. oai0Filol, cq ljnoocooutr€ 6tt .rr. . . . ..r,,, €ilc.r oL 6t1oL'{€g $8ta6LNdryro rI! acQouoft xoprqrl x(1t 61l rutl un6trorna r'€ilaL €L€i0€acr 0(46fv).T6rr or :rqrireq m o-rrlLeg tol, -4 o11p(ri!ow 6rcr rtrQ(YorvLx6 :rircxcr .B (ro6couo;riv(lzc arqrt< lrl< zoauqti!) z{lt ot r€),€rrrci€< /r firii'Es q{nf..Ii(or'\'r l rA , , rni , , : r ivuzu . \ ' . Krrr rrvrr l .oTirr t . ro i r . ivtolrrr Tooror! , bLlro- ldruLoro [rs 11] 7rrt o i(yvt)oro,(.r orov l.r3 .r'11r. Irlr i6rc ql zopurgrl LolnerJ N = 0. Aml.).cyF6 t otr6 rL( rv L6yo ts).€r0€Q€q [€r(6].Iriq. I €lioo)ol A r = 6

tror(reirTfr(f o:lrl Btu = b nor rooo6roQiilt lrg 6(rou{aq trtrnd}ry(E .fr' ftavc rirvorgt nedErq olo roFn),6. ro ox€(F6F(or€ 6(4ltQloltvo. f€ nQdrrrq rL<ii(orza: l'rr .r-<: ,l:or 6.rorzi< tr€r(6i.rtris 0a u:roptivotw p6vtg torg. crv p:ropiooupe va apqrrvioouptrov 1cl'ro1lz6 rir(tzc oTI 0iotl 1()! B. AUr6 cl:t)'(;),< orltr(rivfl ou!l0rotl6\'u6rigar( (rrriorqrlg (:rptiEtrq pctcqu rov yactrttdt\') dorE !(l q{)irooutrt orov

b

t)

l"l , ,N u 'b

flcrp 6),o ;rou rori ber 4rgrtlilerar ;rlxrTptrrtzti. o :rircrxcr< rou yerctoltlpcttilerrov B orov l ;rp6:ret va tivtl o B r. Iuve;rri\ 6),.€c ot (qrclaaoErq Frtlr Yaa$lEu:r6 ILc ir)J,r1 rpd:rtL'vcr arr'r1ovt(r. ot: ro)'Lrur)'uort(r1rd ge tov B r' (!t1vnoay$urrxorllr(l elre),.oi3r drcr 6rl$rl tqq $r066ot GrussJo.dan. (lgou br(tQo![€pe rot< o6ryorl; rirore rtr ncQou[€ [ovirb€< otr] 6Ln1bvLo. rct pq6 Lzri trnto ttnox,hr, , ooo zor uiro zdru.) f l r ru rr l r r , ;oolpr ' . . rqurr,or i r t , I qopi: ro :rcr,"uioo: (x ro z(irrD:

To rcqn),o eivaL 6ro4ro v(1 FCE 6(i)ot1 itt o(Dmi €aflv€irl. To :reo6l+rti trag rll.r!vc d.olLoroaoLrjoorpe ro c.r !716 rL,< ouv{)rlxEc t > 0 xar ;ll = b. H eEiotrol.4/ =l] no),l,onlaorho01xe pe rov lJ I i{(r| atx,)o€

!a\ - aBB

. r ' s+B ' l ' . r ' . - =B ' 6 . ( 1 )

rPr \ I l l \ l t l i r ) : t ' o ,A \ i t \1 , \T l : \ to : I i , \ t oEaPtA A l l \ l l lN

z.u ro 70(Jro: ( . r =.B.r 's +. \ . r \ | r r(bi . r i0 lz€ of

. . r = ( f \ . / J l l L \ 1 . r 1 + c 3 3 r b . ( : )

To olLrcrrrLzo rirar i,n r',i.rr r<, rvbrrrltpotrrt 3rii01 flq(.viiot'r(r oro rctri;.o.IrrN crzpcr brlrri !'rar(i r$q(\'iio!r(r oL i,(or2i. tr€r(i;;.rtr€r rr = B rl) (orei,fl,orqF< tr€r(6i.Iri< !i\1( {&prtxD: /\ = o) To rA62()! zooro; Fiv(r ... =trB lt. ;ro.U ,r!Qri7€r(r ol'l! z(irnr yrrvi( tr! (alrFrzo rsoor}ro. To 'ror f r ) " ' t r u r , \ | l f r u " r t , r ! ! o ' r , , \ , . , t n r \ 1 , ! ' . r r " . u 1 r ' ( r n , ( f I r u u u ' t r ] r ! 'o i) , r lort r-E€1(i !o\ ,r(< ro r '=r ' . r ' r '1 l l r ' \ oro;L6oov t4q zdrrt-r lpcrSgrlq. I rr i lr r . n o r r ( 0 1 u ' 7 r ' u r u ' r f l \ , . r r . r ^ , t r . r i 4 . r r ' ) t ' ! , r r r t , r n f f . r , , n r g r o v r . r r ' . , 1 2 . , . ' r 1nlo noili. :rrll tlioorrrl (l) trnoponfrf vo x,i )rtr ro /.f,\ < 0. Ax rlv (i),hlpL,pLri. trir r' > {). 6xorjtLr tjofr rrt\' z(}.irfl.'rl zopryrl. Auo rir(lr ro xprtriltro,tuxortq \ nv$ix't id"rtnonoirors

EB E(iv ro 6(iv!otr(. r =tt tnB 'N €i\'(rt Fn uevrtt!26. i6tr: 6r!p:ropei !( yi.ver 6€Irioot rot. xixrtor5. H zoQ!.ll] s(vc( r16tt dd),Tloq x(lLro €).dlroro 16010< rivu c y B r b. Eror 1() xoloiQro 6r(zonll( si!(( r > 0.Orcl ro zQ(rriero curo cnoru/y(ivsr. rors xiro€ Clolrltlzri ou\'rot(io( lot,,'cA.rrmorlci oc pio aarr1. zarci yqzo< rq< o;roitrg Ic) 2oolo! ltd)!tr{(. Huw104< mgtrryrz{ eivru rrr urr} . i7rru r t : ro r .0vrtr tz i or 1ur, ;uir r , ' r ' . .

H "ri0r

or!1rori;ro( ro| r fl1(o ro utrypivo ".rimoE

btilu.L fl drorTiirt) l tQ' tor l ! rr< f i r -u0fprF trrr(( j i , ' lnt i t t t ior t rr6:r) t ' t toLxort4rt i . I - ( i r ro cn'f fuFiLr. x6oroq (oro . , ! ) r i r (L prxpt i r fqo (rro ro r ' :oLl io1otLoi[ t !o (no! o(tv i) , f r((oe ir.7orcQ1 7e)imlrlrv tnrrrLx(irv trr.rutLrtrri)!), 16r! ulileL la bozqr(iootitr urriirl! rl!-r0rpt tLtr(tiirtfll.*

'Eoror 6rL to ;rro rrprlrrrrl trvqyltiro xooro< rivru ror,. Tirrt 11 ; ofl1 rruvLorrirort ror .f,\ l)(1 livr turLxii. Arr)t ril'c 4 rrorpxrip$'4pF ir6r.,,z,i. nori .lliE.iveL (n(; ro t lbi! or tr(( orflzr]r'r'il .'.

Ilorcr €i\(L rt Tl||)i Tll:,,: A|rrl z(llloaii[r(r riro to ci],io ci){ao rrt( (r|ijr.Krr{)r ;r ro, l r iuL l rLorpTo|ru1 o|vrrrr i rorr . f , . or r i i iE: ofuordrorq ror, .r tuTop| i\'( trrLo\'o\'r.lL (aror dorr \(l br(lrrtarir(r rl -{.. =/,). H :rgi,Tl1 o|r'rorri,orr .r1nor [€rurlcrcrL zcr qOd\'fr oro Lr166r lirerlr 1 artpT6pevl pttu .ttrri zttrl r rrarpi ;rerrrraroocorzr ldfr i . r .11(1€!nl .Ort tvuruorLorr;ntrot.r 's, 'htru4r\ iL.iToL'1tt (t9aa ot trt ,!rLrt,,Lz,t zoar(rti.

I , , r r , l , o o r r r f u , r 1 7 ' , , ' , , ' t r r , r o r , , r ' r r ' , r r i , ' . r , o , , r r ' f l z ' f l r g r z r u z , rir(orLo: Eq(rro 6rorr dzoqre {rzolrl .f : 0 z(' 6((Jr:a(r brorr 67oL!r rriir rrpr4bt\'tzi.- o--vrotriroug. H i o1|i or,vror(;Jou ro! r\. nor c;ro ro lulbi! aq0(o!oro d. (vrLza€Lorri trI,L onl or'!ror(;)o( ror, .rs. nor IL€rdrortnE dr< ro tot6ar. Orfriir! U€r(6iilri! ror- .r'/r. trer(6iil0)ttr(r tr€ zirroLo raoro. iB€rra'\ otr(])c 0ErLriq.

t O uot.olkrtroi ror / iri,11 df,,rqr|,,/ r n t{tr(oirtlii!

II.2 II MIJOOAO: Sh'PLEX KAI H IVIEOOAO: KARMARKAR

Aurl rou iyrre p16ir eircr exeivl nou 6irEr to €ldXloro 'qlixo o.qv (3) Oouro0i:ooupr otl o npr.gg6< rrrrtoq eivsr k.

8C 'Eoro 61, z ei,vcr 1 i-otr1 mr]1.1 tol -l[. o,-rol :r; €ivql I €Lo€pt6p€vrl

F€ra6llr{ -no! enrlifo4r€ ra F€roredl,orp€ ono El,€uoEaI oe 6oou i. To1€rl ouvlordoa rrl! s€ Fre vdc xopurprl eivcr to el,riporo nrllixo

. \B 'b ) , tB 'L t *" = ** tEni = iD:r;t

I

O€oQoiFe avr6 ro €ldIrro xa06E 6lsrQdloups F6!o tE Ofilx€E ouv!fi(i)o€Erou , rr1. Ed! bev urtip,lolv 0€xtr6E ouvlov6osE. q ur6pevrl roquqileivct crmiprog pdrQlo, ro x6oroE prnoge{ rn pertlve, crr, €t[' dt€LQov xsl, roEldxrolo i{6fioE €iloL -oo. trrg dl.}.€E n€plttrtlourE 1 l-oq na}.d!d ot{}'Irou .B ofidQxero! (rr' rlv 6dol 1{(t! €Io€QXero! I v6q otrlLq u.

:roreLrx6rcqrLdn€QLixovrororotr l ) , tcB-r lzurB-Lrt( l lJrrreircrLpLootr l t r lrol Zl LN rou 6qiozet< rdvo C1r6-ro riro c(o!rlr[26 orolxsio l( r0].6u1oklEyoollLriE r). To 6rltrcr uuro rtq simplex ou'ovi!€rcr! Uer{'r aro 6vtt n{{irbe!y[oz,( l x.r tox.! I o,od.tuord E,(v,rptrC,,1 f l r L\ t . , \ tu zur, lq l .

IAPAAEIIMA Mr rit or.lrciprlo1 n6oror5 r + t/ .,tcr. rous ri!'QroQLololiE rr +2, -

rLi = 6 var 2.t + lt - ri = 6. nou €ilrrl€ nQollollL6vorc. ro nQrirlo 1.4l7rlo Ei.l,ar

6

6

Ecir qurr',lioorlgt ono t4v i.8Lc zopurprl P ror, :2. u.4. 6trot tt r = 0 -r€pv€r rI\2' +t/ = 6 (xior ol$criv€r t = 0). r6c oL 6ooLxiq [etc6ltr6,( eiv.r or dl.Ieg 6r!o. yx.rr Lr. fLa vrr Eilcror€ ouorll|tal'(oi.. €,'d,LdooouF€ rrq o-rrllt-< I zo( 3, 61('1 6orEor 6aoLriq [Era6hr6q va ronoOsln0ou\r rQr.,' rrq e].euoepe< putr6lqtdg:

TPAMMIKO: NPOTP N'MAT]:MO' KAI OEOTIA IIAI| NIQN

Kcrronv q orcl.oLqj noL),a;rlcor.irleL qr,:rp6r4 lpcpprl p€ 1, 1cr vcr 6droElpovn6rcrio o6ry6 ncrL 1p1o4roror"ei-q 6€rlrsAl lAoFFn./ro vc nuqdler plbeM.ni{oq b€xr€Qrl oql,I:

ui3

EEEttiEoupE tuQtbNtt ro r = [ 1 1] orlv xdro ypnprprj, .4!16 6ts! ivo dQvnrw6ororlsio, orrore I r{oQouoc ropuqrl u' = I - 6 xar ro ra06v k6oroE +6 6rv r:ilrcr6t).tr"orcr. H cqvrpr,xrl orlrl[mcloo €i,\,o! o l orlll.q 3, on6rs I rQi.rrl Fero6],Irll 0aslo6Le€r oq 6dol. H orrl),r1 ncivo crroTo apvqtlr{6 ororl€i.o el\,cll B lr, = [3 2]"xa! t(, rtlLtrfd roug [€ qr te].euroi<r orri],4 eivtr 613 xclr 612. EleL64 to :rptbroalLi.rov ei.var, ro tllxporsQo, I psro6lryrl ?ro! ecr andl.o€r qir' rrl 6aon €i\rd[ nnQdrq (xaL radrrl ori),q rolr 1apnl,6).

-. To !6() ro[n].6 elc(l.Ldoo€L rlE o-rrl].€g 1 rdr 3 xcL 1o61yr1or1 $e aacr),oLq1

0 l

I

0 r /3 1 /3

2

2

4

To r = [1/3 ] /31 €iaor 0errr6 xcl ro xpLnjpro 8rcxonjq r,ncrvonoriror". H aopurprlEi\ a = u =211eL xo 1t 6moE qE +4 €ivar 6€lfloro.

H oaTdv('ort €.y66 GiFaioE qq Sinpl€x

KaroQedoa[E rI p€xo6aol o16 q F4retpic qq F€e66o! sinrplex ori!ciLy€6pa cfi6 rn lL(iooa itl)v "aopurp6v" o' currl raw "6corx<irv rqrrnrtirvl,io€(D!". 3€Qor+€ r(i)Qu 6rr croqclororrrl or1[coic dler ro oLdroop(l r xor ol6yoq a 1tar 06Lo!$E .,'o Qi,EolJF€ cll.Lq prc pcrntr orov urol.olLoFd ro!E. Auq€ilar I1ru i rrtE $E0o6o! simplex ror FnoQei vc o€ryavoo€i [€ rp€L! SLoqoQ€rrxoli_<rQ6ro!g:

3.] H iIEOOAO: SINI?LEX KAI II MIOOAO: KARMARKAR

(1) M'€va ra[r lo.(2) YnololiqolroE rov B ' xar avav€cbvovrciE rov. 6rcv 11 i-orrl mrtl,r u rol

,V avrrxaerorh rlv A-o{ orr1r1 rou B.{ l t Yrn)trr i lovrrrc rct B L(. xo, uvuqJvovr( Lrurouc rou;:rcpr i lovrrq

(vTi rou B l .

O xnrril.o.lo; (r)'r6q ri1,(r riQ(y$(lrrriir [(a o1,,',xop1] (oroQio rtE l1r066ousinrplex. Kcrd rdnolo rQoTro, To filo €trxlrorlr6 ora6ro rlrov ro ,rQd)ro rorclnld Tiou xuQlcQlilor: ororr xld6o ylc fioL].d XQ6vrcr. f[C rorg fitQroo6luQotjqcra6 epdq :repr 66(i),e rov yptrpprr6 rrQoyQqrB(flo$6 [€ U(r (tQ(l [rJfiilQiou,l{uai.{'r:6rorL rcroQ0.x€ la croq€i./€I. ox€6ov r€tr€i0)E, rov o!$6o},[opo Trrrdr{o)v(1p1oLFo:roLd;w(q tr€ €n'60E'or1tr( rqv cvcLurLxrl x(rcypfffl] 6l.rrv row :rrvri-xor!!). Qor6oo, yrc unolo}orrxou; oxonor;. oL p6peg ror rc4.rn)"6 pe elcipeolIc( t1t28d nQo6hltlcrc or bLbcrrrLzit 6L6lia- re),eioocv.

f((l vo 6ei,Tr ro yr(rrl, o]Jtrl0€ift 61r c(qrou ur{o}.o1,[o0ei 10 . rcl o filo(pvrylzog ouwe)"r:urr'tq uro6cilr:L 11 olillt irou 0( Ltrrn orll 6dot, rlqr tt:r'ttE cil.Leg oq),e; rirvor c,_r6 rlv r 6Fv rQ6rsrr(r v( Xoloqrororlo€i. EroL ourololrcfr6E rou< sivcr onsr(i),I XQ6ao! r(lL o ev<r ge"lcrl.urtpo aq66l.1pc. 0uuro),oyiLovrov €xcrrovrci6€: oq),tbv. Ecvri x(lr" lcrvri. :repLpilovrcrs [€ rI osleqrolE rr1 lnorv o-q 6(iol. H €ridLr_ol rrl( crrc),ory,ris !s r6ror( rtrIoorilr( trrrop€i','( (|oli.ior q 0E(Doto, orry no(iEl 6[oJ,( 6Fv 6!rc!o],oyrilcr.

Ei[aoxe €rortlo! \ro Eyzol(r!€tqio1jls lC) rc0fi),6. Eilar nro loiyoQo nalrol.rld rro arLo v.r xor.r.iEo!$E ril ll€0obo shplex zaL lcr bouBe no!o! l]nol,oyLoFoit:ircr :rl,rrnrurLnti uvuyxcioL. Kti0t: 6rirttl r!c),),(ioorL pLc or{11 ror N pt pLcrox)iLrl rou .B xo! n06n€r \, (rfio(p(roio€L (ono io r ra! ro o) nol€,( orliir: vaefir),6qEr. !rzLr,o1\4aE (mo ro! 1Q67o!ic 6(oLro ni.\,(r'{.l , xor q\r rQdxotloo l.!ol.r'a = B 'i,, ro 61pcr curr6 .lrarrEi ro,' .l}16).(r!00 rljrd,o rrphEErovl

IlirdxnE 8.1 En 6ipd trtE E0d6o1t sinptex

(1) Yno).oyro€ ro bdrrJopa-yaappi ) =.sll ' 1tar ro irdlrrop(l to! a\rtypalo!x6mou r =rr", )trI.

(2) Edv r > 0 orduritrpe: I rdeonod ).nol €ivar 6al.flmt. Ardpoq€fla.i, dr r;€ivdr tl nro dAqrrti 6rt'rotrboo. 6dl.eE€ tv i ori orlil.r rcu -N os erdEer6fEvloq 6dot. :q66),ros fllv r€l€lidia us u.

( 3 ) Y n o ) . d y ' o € i o , = B r r .(4) Yro),o'yro€ ro q].ira rol' B rb q apoe B r1!, EEftdloraE p6vo rrs 0rnxds

or;vLotdro€s ror B rt. E<iv 6rv Mdfiou 0rtrzdg olvrorboeE, ro rkiltotorooioE €ivor @. Ed! ro rueorEeo rt).iao ).ajr6'iv€rar er! A-mi ouvLm6od,i6r€ I t'mi mri).l ro! raalowo< B eo ora).efr.

(5) EMtFiEidE ro! B (1 i B 1)rarqt ruol r r=Brr .Eniorae{€oro1.

|PANIMIKO: NPOIPANIVIATI:\1O: Kr\I OEAPIT\ NAITNION

Koryrd qoQd, yrcl vr{ yiv€L I brdzArq ar6 rrq naciE€q oc roprr),o, xoljrtovot/rE€xo( c1do€rrQttriy4 pieo,oE si,',plex. )r1v npalFarLx6qrn si!.r I i5Lftrt $60o5oq sinrplex. orpri rJxvu)Ulvl.

H orErtqort ftcE Otr ol,okl,tQoJ0€i !oiq onoqooioorge ntbg vcr unolofioor+ero 6lt'ftr( 1, 3 raL 5:

(1)

O nro blloqllrtq raa,roq ri.vo! ro €eytlo€o!$r: or' croricrg ye rov B r, uol.o.1i-Lovrci; rov cvtrl.mzri o l, ne(;rit nopuqrl. Kor6rtv. orrg rn6peveg xoprxp6<, 4obrtytol eiwL crl,rt. Otnr q or!11 A tou raurorlaou ;rivcrzcr cvrrxcrcorcOei ;rrr1v rr. toto 1 orr1i.r1 ,('ror E I G,'{(x cr0iororcr ur6 qr r'=B rr. H c:rclouprl.yrcr rc rnr"orpdrleL orov rquro(ro nir,trr(.0( fiol,l,onl'cordo€t -rov rctrcd, I

A = r : r 8 1 . L , = 8 1 u . to = lJ rh

floL),oi c{ntroi ^{i)6rx€q Xplorttorolttrv a"4.v "rrolJ.anLaorco7uli1 poptpil touar,floaQ69ou". no! cnoF\]tl[oreurL roq c:rl.or]< cwoq riaa}|Eq n-r. cr\,ri !auvuvetlvrr. cn tr6ei.(1g ro! /l r. Or(v zoc!{ro0ri., (uroi trlupp6lorcr xcu. orc IxcL r6. Io zuvovr.xh bLaori}r&ra (e\r6€to[d!n)< x/100 40 6J]Barc-rtq sirrplex) olJ-r i(vcil;ro),o1i(nur rur oL E-r 6rrlpcirlo'vr(r. H eEiorDo4 (5) end.trlrrl€raLortv Aorrort 8.2.9.

MrCl rro rAooqarrl rrQoo6yl[oyl ri.r,(! \,c lQnoltlo]roLloouv or owl0Lopd.ve<[i0oboL T]t. cAr0[Frzrtq fpcFtrz)i< Al.1€6a((. 0Er,rQd^,r.r< Tlq (4) ocv rpFr<tELo(iloErq nor dxolv xor.v6 ro.v nivczc owrd,eotrb! Bt

(5 )

A B = c B . B t : = . B r B = 1 . (6)

. ' .

r r , , / r , a I

K.tronr!, n ouvt0ro[ir,1 n0pcyorroroilol B = LLt (i P B = Lti, V.E ala].Iayogypclrpdw 1rc :reproo6req4 slordo€La) o8tyei orL; rQe( ),xo€L(. Or L ya tl(rlcv€(;vovraL crvri .va uroloyiEovraL Ea.vd

Krir rdrL ux6FI: niaa 6iwra si'Vter Xaadkrat ya xlyorpt: Al[6 ttLtcnd6u\rarov r,( crnc\,lr]0€i ex rov :rporipnrv. H :r€ipc b€itv€r 6rr n [i0o6oE cyyi!01n€pinou n rt- 3ml2 bLctpopt.rLntlg xopry(g. ror olpcrivrL dvcr:rhi0oE :rpdrlrow

"/nao oro nrn. Aur6 €1.,'(r ory).rQiorllo pr rrl o|rr1or.opivq cr.l].orqrt 1ro xoAa = b v. Eir'oL n arria t6 s'Irruxicrg rl]E [e066or simplex. Ta Fa6lF(rrri(ioor6oo 6Ei.Xvo!! or, ro ttnxoc qE 6rc6ao[ig 6€r p:rop:i airvrore vn rppricoercrrrro lLitroLo orc0eQ6 riol),trnLrioro rj 6{rcrprl rou /fr. Ta X€ra6r€Acr €qrzrir ou\roi.o(o Klee xar o Minty €q,€uQa\r 6!a! riroro lio[Fivo x{6o) p;ropouv vc clclxcioouvrn tr€oobo simplex v.r SoxLlcioar o].r< rrq zoQuqroq. MdXQr nQ6oqara b€v rira.v

r . - I I \ t f nn \4 : \ I \ tPr f \ r , -A ' rJ v f c . v ) ) t \ aR\ r aRr Aa

yv{,ro16 adv o ypqr[r,4o! i[QolQqlpcrrrolrog Lrlr]rlr orlv ClJQaia rarllopifl PrrDv €rrr).riorFrDtl o€ noim)vrpr,{o taoao { orq qo6t-p{ xcrrtlopic NP. 6itDq

-ro np667.rya ror rl.av6Sror jrn)Llrrt. 1-(r r(r NP ;rpo6l,rlpcrrcr ;ftorctcrcr. 10;pigvc i:Z€r crrob€LZo€i. orr oioL oL rQoo6roarorrzoi .r1../o0L0por €Xou, rx0€rLz6 /A6voicLtoupliag ulxpr vc rrplLcrrioorv (zcL :rri),1 oq Xsra6r€al 'r€pinrt,ot).

17 pi0o6oi rou Khachiar rira! €"€i.vl nor 66€rEr 6T| o tocFtrrzor ftaola(tr$artopoc gnoiJri vcr irOri or ro),uuvrlLuro zaolo. noaitrtvc pioc oro r{ rzniorvol.o 7(l oui),(4r6(rlr q i,ro1 p(oo; trrcE oroio!0i(q orpArkvor$0v(D! ei),€r!Jo€tb6v. K(ro?rLa t1 pt0o6og tot Karmarkar, rol ;reprlpirqrrcr :rcrpcrxrin'r,cnoSri.Xrllz€ aw(yrrarotLa! o7r g6vo oq Orcupi,u cdl.ir xcu. oqr :rphir1. 0o16oo.6l.<r (r,ro xo 6t(ioirltrc. rt tri0o6o: sinplex 6xcv€ rr1 Sorier.ri o dvc 4rizr Xirovo.nou olobriZ0rp{u r(i)A( (1(rr p!(r ncAdJ,(1i fltq ol;vlercU&qi F€06bou) 6rr ailarro),u,xr$rxoq. flc xrifioro ),o1o. ,{Qlrlr[dao orr] tr'rlr€reic llrv no].u&i(or(r(,rvnolufbQorv. rc kexo. FI,t1lJn orlvo}.( riv(r or((ilL( zct tl [(ooboE simplex €ivcrrruxIQi).

H FaootioE ioo KarmarkAr

EQlolL(orc !i)QC1 oro nro r\rrl]n{Dolozo yryov6: rlE reooq(rl; rorooluq ro!nQolQ(fqorLopo!. O K nn ftar nporr:rvc pra pd{)r)6o 6(loro[ivq ('e bip cr;ri.i<r6[:r_c, 1l o'[oi.|1. orcl nELQdlr(rzr ro!. !nr:Oiox!o!: qE [et)66o1) sinrplex. K.irtjT{1Qo[uo o[(0( brv orira6l oc AJ.(l n€rQrhF(rx.r. [r ciL]..r rpo(i].rp(rrc. H r.:rLLo./ll),oln6v ron nsnj),it(rr(\ x(r or turTrofrip!rtq ror zrirc\rzcr uiruL e! ioor xqi.oLpcrt1r it br([dy,I {]Qio?Er(r cv cqcliqrL. Qordljo or rbdrc riwL 16|j() q,uoro},oy.1ta(r(r rcri,rti(ou 16oo ro),n oro r),(io'o 14; €(t.rauoopdlnE rgr|l|rrzrtc Al.yfaiailc,ri,ort vrr riryoi|vrar oe ).i.7rq rcpc./qilqorq. Ocr icrL\,rior'r ttt 1.'\ eb,ljaL(4{)avao il\aotlq kL xljlorrv 0.r €rrorldlrD rrq €rbrxdq bL.rbrxaoiEE ror fnil$lr(vmol Kamrrkar ! rrllo6riEa q unpirol oo :roho'vrf Llr6 Xao!o.+

H nQorrt L6i( cilcL 1tr qcrrvrioolpc cr6 ivu oryti,o piaa mo ryM6 olv .o0c lroodoo4rc orr .rur6 rilaL to .r:0 = (l . l . . . . 1 ) xcr rc :rpoxtuprlooqre oe

lr\ti. x.tr€tAr1wtl tlrkDoIq turi x6orDrq. ErErbii ro rolJtoE ri.,'(r .rr', I rc),llrral'{rtrcu0ulort tflcr x(r(i ro .i. :!!ri0rDc clt6 lcc 6yi1!€L ar'ro €qLkro orlvoLo.crrpo! 4 xivrlol rpo! (rT'i fl|, x.rrruoulorl 6tr 6ruqpt:i r1v :1.r. = 1,. Edrv,1.r'o =l) rcL n.r'r = b. r6te A.r' = rr r rQ6fis. !.r rx(vofio(Ei lv 7lA/ = 0. To6r1pa N nphta w ntptlnar ony pISEy6Xapo nu A. Eror. lr(t vo 6Qoitrfrt! (no6€1{rn x0ttr0uvoq ror eivc{ r}.1o#mrp1 m1v ra),!req1 x.trEiJo1]!on,qoir}loryt to r ;rrivo orov g16ev67,opo. Auo eivar ro qrorzo .d.Id 4rrr.

' To ri'looa ro! ra.i:rDr (rDiloo.ro 60 rrvritrLi r()! Dr x,l ti. rji,,: ontr G(ifrlit zur.t dr(nrr(or( Crxni.S.hnriJr ltrr r.v (a.!xro n$riQq rilqio rnL t|! c\.i/.!ort or iptiro!<n(ailour!. oior or I'rooroi oitoorlJtLor ro.trii.vrnL.z0.nror'i tolj!.f: trrt/n,tllrf0t x.!irrtut

ro r.!i(rlo x b).ftr( lr I -\.]r drdrr(r oro r'dobri:ollc ol t'k rirotrr r!.tj)rttrar,i 6.!

|a.!t.i !n i r.le/.' not1t,lrtrLzi. d.ydarotro<.

r!AMMIKO! IIPO|PAMMATI>MO> KAI OEAPL\ IIAIINIAN

boraqQo (:-x.8.: arr6 d('[olil r.ooror,E 7€rrouQ'lios H,^r) 6rlFd rrlE BE0o6ouKarnarkar.

H rAo6olrt ndvo oIov $16€vdloQ,o xou ,4 B€).sdgtx€ otry Aoy"qol 3.3.20.O :rivrrzc_( 'ao6oliq eivcr

P = I - A I ( A A T ) - I A . \7)

O tei'erncio_< arnrj6 an€rxovi!€r xa0r 6rd!uo[a otov pqbev6l4o, 6rtu ,4P =A @AI )@AT) r,4 - 0. Eciv ro :r €ivft ri6{ orov [16€!6x{Deo, 16r€ Pi = ir.6!6it /.r = 0. Avoyvorqi(ere 6tr" bev lqerdletdL vcr ndpeL xcveig npaypcrLxri rov(,4,41) I o or"rimpogo( b€v uftol,oligcts!, Alti 1f cw6, Ltwoqre 6vc lqnprpr,xooliorn[o xal aqalOouFs on6 ro .: (fiou oc(|)QoiFe rripcr ocrv 6rdvrogtc-orrl].q) TTpolrvLotdroc lou nou nrQtiXttut uto 1ri-rpr-r lpcrltltdrr:

AAT y = Ac xct xsi6tr!1' Pt: = c - ATU. (i3)

To 6li[s Ar €iv( no],].addorc qS nQo6o),iE -Pc. 'Ooo peyalwepo utvcr ro6ri0cr. rooo neoroo6reQo n€eroeil€rcr ro r.6oros -rr4iE dptog vc 6ycivoupe cl6ro €rprxr6 oi'\'olo. To nol.L(.n).doro rou -Pr €ndiyer6 iror 6ow ro rl v(. siv(rzowri. <r)-L<i tr'yo z4rog ra /t:u.r, mo ouvoQo rni rou o:roiou gtcr owroraroc rour llb€viE€rsr.

n={0 , 0 , 125 )

i ' i ' : r , +:r. +r. = s

2, 0)

r ' = ( 1 , I , l )

'R=1.

,i't.olx, +l,r:x. +o,olx, = s

X,

It. 8.5. BiFa ro!

xl

G)

alyoAiopo! or.l(nd8lroitql rE reo6oli, rr olo €

I! ] H MEOOAO: SIIT'PLEX KA] H NIFOOAO: KAR}I]\RK,\R

.4 = [2 2 1] ( i0o A.41 11 = Ac r ivrr(L e, = 27

IIAPAAEIrMA 'Eoro orr 10 L()0-10g c7r = llxr + 2:?rr + 7rr xor o n€pLoproFo!,1J =2rj +2rr+.rr =s =/, . : ' r { l lTr i i lv n€pi.nmor1 to nlrzro onvo),o eivar rotpilr'r'vo fQ,R ror:1. 13.5. Arro riv(ll. p(pog etrLr66ou ro! rprbrirmcrou X(Dpor.jrolr (oror(lv€ror cn6 rorq r€eroo(otror< .r I > 0, rr > 0, rr > 0. H 6(trLoq )"uortnaan[r r( !:ilr(r IL( cxo nq xoAr(f6<:

P : ' | = (5/2.0.0) se z6oroE cr;r ; = 20

Q: . , =(0.5/2.0) yr looroc cr l = 5R I r = (0.0,5) tr€ x6oroq rrr = 35

'Ero' ro r = (0. 5 / 2. 0 ) 0i vuL 6d,rroro xuL tL;r i.lorge 6rr. ro 6rlpu A:r ecrrrrUrvcr(L:rpoq ro olprio Q.

O uxoioyrotro: nQoftiL),€L ro r = (8.2.7) ncilor oro urirrbo 2J +2rr+.rr = 5:

To brh\r1JolLft (11|T6, nor| beix\rr! fll! x(1-il.t{)rrvorl rou 6rlllllro! Ar'. 0(l fnarnsvo [ia(r oro! plb8dxo)rn ro! A = 12 2 1]. rA(i./r(r rou 61,T0\ o1+16circL.Kttolirrlor rlro to o4reio exnivloqc .r = (r,1, l) nQog alnri! rlv xtlrsuot|1orl

lx l f r l [21f u= .A t r l =2 - - r2=1

t7) ! l l4 l

, ; '

LI as l

[ { , s2 l f0 i l, = 1 . ,16 1 . r v r i . ru , B= l l .u

L 0 ,041 10 .01

fr{r toiro< 6r}rcroE s = l/1 rt rpirrt olrLotrloa rivaL p16dv rcL [o rr]! rrrl,oyii(ur4 qo(ivorrr f mo oi}raio r lou oy] i lcroq, t rr ourrar. lytr i !€E ( l /2.2.0). Arro€i.vct ri.Iolddr€ao oro 5d),rroro Q. d)d rEclrL{;lrcq ror beo$o tr!7$ ro ori.voao,[ivr:r no].n ].iloq Zdrpo! 1rr ro ca6pevo 5qpcr. MroQri. \,d orpori (n(a' 6),() ro!rbr'r 6€\' or+rti(lilr-r) I trlbnr.zrl orrrorriroc oro B v.l trry €i.rar tLI6d! mI iji).rrorlr{oprqij. fr' .luro oraB(rdtr€ ).i.yo nro rar! imo ro B, e),crrrdr.vortag ro s (rir 0.25ot 0.2.1. O ro).1,(xi,(oruorrx6q arroq o!\,rE),€(]tli< n = 0,24/0,l5 = 0. 96 €i\,(llX(rAonrlpLorL'{ oE 'r(1! rt llrzart rrtl]t .r = 0,25 nor, 6t0oor€r0lr€ ?ra nQ&t (togl!c;r'ro! Krmrrkar, cnrii:70ryt y.tl v .rrobEi_Eal t4 oiri,zir"ol o€;rol.u qrrzo1p6vo zcr.6XL yL.r !.r 6ol0rto€r, rov avayrcborrl rcr;rrtipr Tl1 ouyxl.Loit ornQqtrsrr){6 Xao1o. OrG! .5 = (0. 96)/4 = 0,2a. ro 6rlp( xcrtcr}.rl1rL oro

Ar!16 o),ozhtQ(;l€r rtv radq r56a. qg '(Qo6oi)i<, nou bi.v€t rlv o anororq€Erxri rt€kDl. Aqo! I ouvfzrort ritg noo8i(E rAoq rlv ibro rc{€n0uvorl siwLdoxorirl. ro 0no[Eao 6rlFa ararlL_i ptu \,6c L66a-

H ngoruot rou Kh,tnar" at it,\ {L 1tFrcoxttltuttoovpr ro t) vu ra n :rd1ttaior't mry xzvrprxil 0dorl .r = (1.1.1). H c),Lup uw1 ror [sra6]IrtD! otr(D,dq€r zar ro ;rp661'rpcr. zcl t6rs ro bentepo 6tyu ;(ao6dtrL0r ltl 'vio r mov

1L4brv67topo ror r(o1 ,4. H clLrryri F€rc{6Litrd)! riou 6zu1€ o Karmrrkat (nol ectbciroourc apy6rtpu) nttrr prl 1pr4ptxrl. Aot6oo o ct().ouor€aog [€rcloxlpurtouoq,,vur. u,r) . , ' ,c. t rru avamriOgro4 ru,v uluv' ' r .

f ru yc tr€raq,60our€ 1e .rr = (0.52, 1,96.0,04) oro xevtpLz6 o4reio r : ,crpo!1rr rLg (!riororXrs ouvr€roltriv€E 1id0t 6(Cauotro-roq F€ rou: rQ€rE crlrrorrq

(1argtrow. Mr ri).),a L61Lcr. 6riLo4re aorou< rouE .1Qr0ttor( o€ ivo! brcY(iJrtoril,(r{( ,D z(r avanteyiEoupE 6\ct rc 'rcvitopatt pt tov D t:

|P NII{IKO) I IPO|P ANIIUATDNIO: KAI OEOPIA IIAITNION

fo. s2 l1 , ) - 1 . 9 6 v . u . L D t ' =

L 0 ,041

Enroreiqor[€ ),ofi6'' niour oto rlvrpo, o:rou 61o4re Xd)Qo vcr l!{l0or1u. Qoroooro €qrLxro ouro).o (ro l0iy(,)vo PQ,?). xoodlg xclr 1() brilvljo[( x6oro14 c 6Xo!tu ) . ] , r i g € r ( : x . t t . 5 6 ) . H ( 1 , c o r d O t r t o n c r 6 r o . r r n o X = D r . r ' i / t L b u o o u v i : r e r r : g :

o nepropLofroE ,,{r' = b li.vutuL ,4DX = l.ro xooloq .r ./ ./ivrr(r .rD.I.

Erjt rtl 01tol ror:ti1'a)ttr Attnipt,tt o ufttq.4D )au tI 0io4 roLr dtwttotuno<tr truiszvtL ro dLrlt'tnpa (1 /). :ro nc$ibelypd [cr6 crtr6 bivur

AD=[2 2 l ]D= l .o4 3 .e2 o .0 , r lr i r= [8 2 7 l I ]=14 .16 3 .e2 0 .281 .

To !6tl neo6Lip( €ir,trr rtirpc 1 c),u7rmonoi.4orl ro! cr'r,Y tl:r6 ro!,( n€QrcprotrorqADX = b xrt" X > 0 (riou €i\r(' 'oo6wr4rt g€ rrtv .?i > 0). :' dva tr€ycd.ir€aonaad6r:1./tra. o A 0o ri.te m.Tpnppd< .'vri }u pic. To 6a1jr€ao 6rtFa rao6dl)'€r 1o\,do 6rd.,'rotL(1 z6orors rrl'vt') oro! Ft6e!6to)Qo rou,4D. H €Eio(Uol (8) yiv0l.rl

A D I A r u = A D 2 c z c L 1 6 r r P D c = D c - D A | u . ( 9 )

'EXor+€ Lotn6v ivcv o),oaLqpo oLlopL0Fo, szrog an'rlv r:nxilttoq la.lL roorr4Lrirrpir ror. Oa p;roporoL-va E€rl!d rr:16 dva o;rorobrj:rore tot-rtegxo olpei.o.rr ror rrprrron otv6lor. :€ r'tci0r 6{$tr,I ra€tomc exrigrlorl :ur uvool(0[i.!er.rtoro orlpeio e = (.1,.1.... ,.L) xo' i6r€ I rpo6olt (9) biv€' rlv x(tr€rlouvol rolj6r1[croE tDq npo( rlv a\icoro0FtoF6vl F€ra5],llTri. To eEQl6[€vo rou 6rt[aro:

A 1r 6 A $ poq Awn tto 1ttot19

(l) KdtoozEnoo€ ro broydr!rc nivaxo D an6 q o1rylmdro.s io! rk, dtot .bors

, 1 r a = ( 1 , 1 , . . . , 1 ) = . .

(2) M' a!r61 ro' ,, lroLo'yrof. rtv ftao6olrl Prc qE sEiooorg (9).(3) naoob|oqro€ aar0f6 s, 6ror rbore ro c sPrc vo 6x€I p'd pn6t1r,{i olrvroti,oo.(4) nlar6erog a1,'r6 ro .r r' 6vr'' ouw.],€mri .i (1..1. o = 0,96).(5) To rdo 6rd!uo!a €i!(r r/i+L = -l rrPrc.

:qr clcor(roFroB€q [€rc6)'Irl. ro 6rltrcr AX rr]yctirr:l oTiri to ( oro r - .rP,D..:rLq (e1rx6q $rt(6),qriE. rl"/ai!€r a;r6 to.r-a oto .t^ - sDPDr. A-( xirvorJpror-rro!; toq r;ro),o1to1ro1JE olo ncrQdbrlYy(, Yr<r v(l;rripe (lati ro .rl oro.r':r

S.] II IUIOOAO> SI}IPL!X KAI H MFOOAO: KARM RKAR

AD:,41 l t ADzt yi !€rct 16,45, = 19.7 i t , = 1,2

X '= t (0 .33 )PDt =

* :.r E: nao.trv{nq o o!,1poqid< iTEr mQot"/ri€urL ro @orii..trr .1o 2. 6.2drro ]lll(nr

t'r( r. nsirr.i npilEELq. zd' mo lo I'ro l: br!rra!!.

Etiv uqrurporoapc ro sPD. cno 16 e = (1.1.1), ro tr ixo< 6rtp(roc s = l /2.91,0c dxa\rc q! nQLbrJt ouvloxdotr [$6!.

'E!(l rro ouvrlQrF!){6 ti{[(, ci.1,(! to-{ = 0.96/2,91 = 0.33 Torr ro vi :o Xr xur ro

"eo .r ' r = ,)( : Ei !( I t (

14.16 ' l I r .01 l | 2 ,er Il r .e2 -1,2 . r .e2 l= l 0 ,7r1Lo.28l 10.041 L o,2r l

[0. o4o l I o. o2.l| .25'7 xs1 .r:r = 12,,16 ..

L0, e24l Lo, o4.l

Akrn|orriJ1,o1+r! 6tr. ro .rr :repLdler.lr oro tqLxr6 orvolo: 2.rr + 2.rr + J]i =0.04+4,92+0.04 = 5 acL i , t r t : ivuL no) ' [ ]o, ' rd ort lv xoprl({ t1 Q. t4; onoiccor olrvr,or(i)oE( €ivoL (0.2.5.0). :i^/oxpd €flam€ apxmci rovrci olo ltl€u(rioti4[( tll,< F!.0o6or ror Kamarkar vd 'oi1tE nout xoQt.lit dvat xo'litEQt xuta n4Slloryx o' aurn1l,- 'O),oL

Eipollv 6rr ro 6d),rtoro onF€io €i.v.rr Fi(1 cro rL;zopuq€q P. Q. n xcL 6ro 6rlpatc ror o),yoai.0$ol Ur6Qro.r\r ','.r z (0oaioo1j'v rIoromi xopr.lri Q.'O),I

I boul€rir 6yrv€ oro 6r1[ft 2 .ror cr],1ogi0pou, ror ti.var FLa or.loFLoF€vlrao6oii. :q "/Lri)ooa rlc tl.laal,adqou 3.6, o ni\,(t".rs 6ftAd)\r Et\,(r Il' = D ralro 1Lr,6pevo II'rlI. LoorrcL se Dr. H ore1ltottirl xarot'tz4 t[ioan4 wpaytt

TPAMMIKOI IIPOI'PATIIVIATI>MO: Kr\I OEA?IA fiAITNION

ro ]/ xar yl Liol rI< Eil,cr a1,c 0€p€),r66€g np66lTra qq apLou4rr]1]i< .yadFprxiE

dlls6aaq:

00)O zavovrr6,c rp6rioq rlrro).oy.olo ror y eivor" 1 rutclorprl, :rpd"1pa rou 61trt:nrnrlio o tvc prap6 :rp66l1pn r1 xor oe pelci),o. 6rsv 6),oL or nivari€q ropaB6vow GAoloi. H €va ,ozrLzt Ltrol r!)v cram-Schn dt Eivor I oA0oyrDvronoilorlrnrr otq),crv tou DAr, nou crv rcL a;r),onoLei. rov uo).orro uroLoltop6, p:ropei.vc xorni.leL nol,u. To qo6oai yrc pelcrlurpcr rpo6Ltporc ei,vor I Fd0o6oErot ouluytfu xliotor. iro,J rivcrL oF66v d$EoI x(lr 6ivfl rlv Llza!6it crtr-11lol oQn/6r€QC un qv anal"orrprl. c),).ri p:ropei. vc orcpcrrtlo€I, oq $60l rISirofuzttoic,c. (A€v Urioa€i.i€ 6g14 r'cr orcrpcrrjocre oro llioov rilc oroLotqllE.)H r56c r()v oullr/(fi rl,io€{'}'\r. xur o "neonaQaoxrraor{E r[o! rq! xd.,'E! €nt-ttr1t]. :rcpLlprirperr( oro 6[6),io po! Erod.yali ota Eqtepoopiya Mag4ptrNrxa(wellesley'Ca rbridge-Press).

O cQllzoq c(),y6Q!0ttoE ro! Karnnrkar nQozr0Qolo€ n6Qo on6 Tll! avaor(i0-prol D rr:. rro! xQc(ouo€ ro x6otog .1pr4rprzir. A\'ri yr'(1ur6, y!\,olav 6!crg"xpo6o),!r6E per(oXrlparLopog" tou .r oro , Lr/e"D Lrr. Arrro ZQslcloict!otlv cljrobrrEl qq roLuovt|ur,xti< orj.y)itrlolg (l)E npoE rov 1g6vo, ,rqy @dtttt4toE f!'c;ru1, rction rt< ai,uori0$rtg. Arrrl q orr1pr1. o' 6lov ro\r x6o[o oa),y6orqro< 6oxLldrEexc([, rQo/iofioreircl! ?{cr e;renreiveror" oe UI ./QoBplxll 6€LrFororoilcq. No[i.!o 6rr, I pdooboE 1()! Karnrarkar riv(L g,roLrrj, e].ruotrxrl xaL ocr€trtlrloe!. OroJ< oup6<river nt4 U€ cll,tr€E tbdE; rou €nto lroltrou lo1ropoil, oe[€QLza trQo6liporc 0(l rnrruxrL l{c!. o€ olJ,a 0a c.rorl]Ier. H F€[b6oE sinrplexrc0(i)S rcr ro ol.o 06[.r rou yQ(tttuxo! :tpo1pt4rlrrrro|oU -nox cArcrcLljeonrc({d),'c< peril iLg yQalrlrrrdS eELorirocrE .4.r = b. orlQi.E€r(lr rDor6oo orL< o€F€].rdrb€LsIb6€E rnE fQ(4rpHlE Al.yL-6aaq- €EoxoLorl0!i r,( Ei.r,rrr sE(rlp€lrx(i aEd),o"/o. An'6),rq tq r6deE, (unt $€ rI f€yaLrir€ol 0[6i].ELa. aivrrr o 6!iop6E no! €gerdq€-r(r

A!KHTEI!

8.2.r El(trorororimr qv .r +.r xr Mo rtE d!!erl'if<

2rr ,1:.r + rr +:..r =.1

3 r L + 5 r r + r r + ' r = 2

nor( otro rLs e1,r,,r1 nqin.t !d rLoiiodry oq 6.iot a@ noL€E dfto 1tq rr,rrrafntL vo ardl0ou!: Yrolo,im€ to lio i.lroq tuv 6aoldw frera6)'ll16v Kr 6eeir.ro rooro; ot1 vio zoqupi.

8,2,2 M01d ro nqoryolFEvo 6)tFa sinplcx. fro+rdor€ xo vio 6ito ror Ezr.}.;ms ro.

8.2.3 :ro roedSrrytro tr€iii ro 8C. !froeaor€ orr xo ?6010! uivar 3r' + g. Ald6rdrridoovld!1t( $rc6).IlT6q ro irdvuorld a6mo1E liv€idr. = (0,1.3.0). AEiETE ot r > 0 zdrow€r6E |

"oar,qit P sivor 6d).rrorr.

\AD: A r ) ! = AD) . .

3 ] H VEOOAO: SIIIIPLEX K,\I H I\IEOOAO: X.\RIVIARK R

8.2.4 Eiia I aarrzli oriv.iartorl zdororc irdr .. ?/. oi6rE [. rry dwnrdra$t olrtv roauliiP €c €i1(ll. . = (0. 1.1.0). fro/"oyior. io r xcr croqrcoior. nora nit].I , e(iipur u Fn€r oq 6ciol. Kcrofll rroLoyiotE ro E r u xar 6€iEt€ $€ ro rloontoror or rori 5i.v 0( oNcvniourr dl.r xoor{{ri. Auo nol o}rdcivrl riral o1oaua(t o].(ir!o ft, I oro! dEovo y rof t{. 8..1 ror ro .. y r(ilfr oro !.

8.2.5 :ro rbro rcadbfLflu. (iiiErt: ro xo@oq of i +ly. Er(iriJriorf orr r rtr0o6oqsimplex obtt.i blano)(rz(i (tnt\ n P do Q, $o R tu o1r 1l barn{.il R .ia{{ tirirLdll.

E.2.6 H (foort I i)(fl oroto N .ra.ol graq 6dorz1lq dion.z i! ii,ort< ror .lJ = 1).Ai)rilovr(i r( r!oor}r(. (;or. !c ti!.r b > 0. e€.rarior. io oortcrtrLzd nad{t}.I[(. i ( t rmonoir toK r lE L r= r ,nd n( ou!0r i lFq.r > 0, r > 0. 1. .+r i , =LOr(! rt -,1.r - b i/.( fra |t daulrLrit iriol. ro .l,.iXroio aooro. o (rr(j ro r!!jliir}r(0c oivaL r{rDEv. trE u'' = 0.(a) AriEn orL rr' quo ro rio ne66LrN(. r xoqlqri r = 0. ( = t, livqr xaL 6(olxri

rdr f!Lrrl. 'EroL

r 6''i ro. q(io! I rlXfL oioriru()0fi z(r rt tri!0oboq simplexfr roA. i v f t iLNIr )u lo. r dr l l ! . !aeq ror t j i l lnof i . i lo fc r ' ' , r ' E( j ( ! r ' = l ) ,rors ro ..' silcr rl irtronp€U rooufri ior' (arLaoi iootiirit(tro<.

(6) K(l(1?aGlrf ro iiort0rFro ioodiffr(, ro 6rdvrotr( r = 0, ( = l, rrt! rf(iorF rol,l .z( r ro l i rLoro bL( i r fo t ( r . i raa I =[ t t j r ( r l ,=L3] . uor i rs f lp zoaul , iior F{rLzror i orNoior | - . r ' = 3. . / > . . . > 0 x(r d lFnr( idrF | r ( sLzdv( ( r ro i l

8.2.7 noro tr( iir(r to aaLrlq'o lLuronll( y'( ro / a(r rororE zdvoar. 0d dzoiouooldlfr( orliirt ro! ,\ tro v( lird doorzrj a(r I'ro orlj)jllor /, ./tri vc lirfl tii.norgll. (roaidl. v., t.rtldroro|lirJoltrf (vri vc fi(7(dronolidorr(f ro a(trrd. (tLr ..1., =/, ,(1' | : o l i

E.2.E E7d)(rororor imr tv 2. r r+. f , uro rouc r raLoeroto lc . f +. r , > 1, . ' r+3rr I 12,.rL )rt > l). .'i > 0.

8.2.9 Enci r le f i ,dr r r r l l (5) x , r rs i :x f ( i rL o Ul i i7a Br = t d t tv t -o l jdr r i l r t ro ! 'Err

D BI ; . i !a t o ooJot t i ! t jddtr6! r i ! (za! ro! InoF.vor 6ntd1o. , . I r , | . i ! ( r odlrioleo(oq to! acr rt ooori cldvAomt lo! 5ad8on rilau.i tiret.{ tr. to! -' I

8.2.10 'roro 6tL 0i).o!tr€ ao .i(7Lo1oaotri{rorB. ro rL rr rno 1rc duvo1la.r

2 . ' r . l x r + J ' r

j . r r + r jJ , * , . , = t2 ( ' t t t o i " t " ' ' . ' u)

Td.rL ri ro.fr e( iia.nr !' ulErioortr.. cno qv rQazolo( trlb.rrzri r+oi iori. (v:EziloldqG ontt ro r = (0.0,6, 12): n6oo nor'i Inoaoia ! (rErl6oN. lro, {;,d. olrlroo@lq ! a!(Y/roorv ro rr 1t io r.r !d ti!€r lLrbivi Iloro €iaar to x o au6 10

8.2.11 fLaro! n i \ taoP=I - f / (1- f / ) r -1,6€iEt€ ot r dr( ! r € ivc i oro Fr tb. !6x. )at )ror -,1. rot€ 1'r = r. Arti(6ri o utSElotoqo( rdqativ8r dvailoiorros os nqos itv

E.z.12 H ai.iayri ro! zoorou.-. oro nq6ro 61itlo trts F€eo6o! ior Kmtuktu Eirat c? ae =stTPr. ltiqrt orr (uro roonrc{ ru r .1'c r. zcr o!!snd,( | lsra6oiri si!(L

oal1llrali zdL ro ao.ro! e).crr(b!€rcr.

'UOd no^9/d1nol' sll,)rdor 5|llor1u9 5h Drioo^prg or arl zo or 3rD tdrdnolt1! $rtl8(@) no990fll s|lt Dnl€ 0&439 or Dn,r,lagpdDll or 9ml) p 3ro?Ftxg SIZ.8

'rErEuuE) r{fg ol'ipdl! or ptarlis= xcy o93:!11r3 o:'D 1D13r.old9 nolioôl,ldr gr|{rtu o! 3ro?r9ix3 d Drn I, ohfDlDe ôr 5l) odpdlD!fl.odo ito ol,^ldll Drlf4 or 5Dt pmdDro 86'0 = i, or 3d 5lrol,D.op 5Uâilolhod! X]r s 01 3ropr3w tI.Z.8

'gxdun! Lrlrds - a 01 ]rdlllDe oo! s ndFg our&n ol.31ladg 1Dx h r rl = y ool onol9mglrd ̂ oro or\pr, (s'r.s) = r or alDxpgodu (9)

'0 < ,r ^!odo4do4d3!ô)1 o^_o1 rDr.]lrgriouD oo1l oo^lDl1dr oot Urb.d' 5?6onox n,' 5luolpnrxoq .€= tr + eo + ro og3u1!? or.o ,tg + zr1, + try = r ra 5o!D9x or 3rotuo:lor-orlo€ (o) €I.rS

NitlNJM VIdUgo tV) 30t l3LLrr'tII MlOdII 3o)IIy\lIrMJ

8J I OEAPIA AYI:MOY

Iro Keq,rii,(Lo 2 lrzrvrlocp€ i,alo\,l(iq orr. acp otro nor rl r€IvL?|rl riE an(),orqrls6ivEL tlrr nQoodyyrol qq Arj = ,. [L( bra?oQr]rrxil x(L 6(0rnrpr1 rcrrtrr6rprl{or 0dF.lro< [i!c1[ rnlol< 6wcrrl. To iblo (hQr6(i)( rozr€r xar" "1Lcr to lErpprxorpolpr4qt(rLou6. O Blxoarotroq rlq trr0obo! simplcx 0& LtrorL ro lpapprx6nooyA($rc. oBo( o 6riioloq €ilcL nA(trsru{(i olo r€lrpo rtG 0r!)0i(< nouund0t€r niot') cno (uro\'. naoz€Lrtlr yrtl Irc rottll'ri z(l! r(uro/aov( 0r-l(rhdbl.7r,r f l i rqLeroTrc. r rr z ir HLr r t ! f : r t7r lu,r ' lu oon zui, i f lau I 'no! 'o, I ' r .

:!7L\(i (Jr6 ro rrinLx6 ;rpo6),rlF(I:

ti oEoPl,\ a'r'l:l\t0\

ng nn'9ixE< r >0 x(u A! > h.

* Yiao/.r iircll.olqrlrai( trEr(:,ra,.r/r?oi ?ur n{zoi :Fznrldr4r do .i.grdraroi 't.'t. qr':

'l !il)oho. \irnplcr !l(e!(ol.r( f: iror zd..i zor oi'|l,rtdrororlmt er r(dl ftqrrrnnnl. z(r r(

.io rL-$oirirtrtr( ii\ur(r rarro7ooro: !r .N idrr rrirrD6o.

APXIKO D.c/laro$ori orE rav u:,

Torptl otrlr, uvri vc 613r.orpylotr avtl Loo6ri\,q1o irAod),4t( Itt rEro(irrlrL< o11leioq rr';v (rrorirrru,rv. o bljiouoc bllll|or]elr'i i!( rri.sico< 6r((i]oe0rL}|6 r0otj).I[(.To 6t'ix.i ne.;O,'t!& :€xrvti az6 rt k\u A.b xat t zur avrntli+ tr tu :ttirra.Zrcclelrzo. 1() . rir(v Jl or\'(iorllol aoorou< x(r ro r llrcl' o n€QLoQloF6a. :ro bui76 orAotor (r'r(;r! ro^'6'(\r'op(ir()\ r\'si),.iixrln'fto. Errion<. ro nQoolllo rrt< lllio()ol!d.id(fl. Jt !6( t(rr(ii),toi ! riv(r fi(h,rlolr(-yeqr[t] ror ro r(tLxr6 oivo].orzirvoir ,ntsl rr t \ ' rA' . r \ I t rn: . l / / , I i ; .o: . 1i ! ! rdr lu l |orr ' ; roiry1 uvr i 1|ur').ci(rrrronoirtorl. To [ovo irp(iypc rorr napc4rtrtr" cptrr'r6L1ro eivat q cmcrirrloll ) . r .xorrtr(c. 1-o,( ]vrntro - /€Irr / , , uuvlrrr(yr ivrc rrU :rpixf l vu rr ' ,vr ' t r ,nfr r lv, > 0. Me Li1,(l l.6yL.r, ro iiriiz6 !1(i( n0o6Lritraro( rl(ltroionoirtortE €ivcr ivcrpoii),1p( pry"lJTonoi.rlolq:

L\IKO M ryrnoroLtiaft t4r gb, r'nd rt€ ouvqilxrg ! > 0 x@ UA < c

To 6rtz6 .rlrorj ror' ;rQo6).ruqrog eivcr nri),r ro rrplLzo :rp66).t1pc t),g7roro;roi1-orlq '

n0o(tcvr;q, oqri),rD rcl oaE b{i},i(| ]i(inor.l €:i./Iot yr( (iir; (1ur6s riq .lvrr-orpoqrdg. Y:toxpr:now rov crvrcyr)lrop6 prrtrlu cicrTrmoaoi.tlotlE )rur t{€1oro-:roi1or1,c xar I rlrl11orl rrQo€eXercL afio 10 nQ66Lnra rl< 8L(irlq, fiori tLriiirD \,(1o n(Qo{oLorgrlo€rr yru [r(l (x6[l (loQd. To fi0o6],I[c €]..11Lororoilo1l( dIEL, cyvdororrq nor n(Qrorh\rorn,}r rQoqa;. nQo; x(rcvd).rDol, o€ (rrl (Qvrlrr1(tE)Troodrr l rsq,.r . . . . . . r , , . Or. rrr : repLopLo|oi r(Qror. i r louv rr cvcryx( is< t j r raBi.v€(

.]3: | ' , r \ \ l \ l l l i r ) ! n t r ' l l \N l \ lA l I l lO : ^ . \ l L ,L ! ]P l . \ n \ l l N la l \

nof (vrrz(ororona 6\'(rl raolloqrrro :rraroarol6 &Az€rri.,-Taurdvrti. To oror,7 , r ' , ' , r i l L n r t i r a d o r r r i r r t : , . o d l : o r r ( q r i r 4 : o r r t \ ' / - o n T r l r p o , l I " , . r I ; . o l l1p(1$nt ror! -{.r } b (vrrlxciirr fit bicrr( v( nrpLictl6clrer rrurri q tjrr.rtrivit or:rooorqr,r ruu). , i7,mor I . f r) .u: . , tv r lc. / -Lrort : r0o,r I( .. 1,t: + + t ,,.t:tr = c.r eiv(L ro xooroG rrlE 5i.r1G. To xooroq (1uo rTpinrr v.rrlLcxrorono Lrl0!- i . Arro €ircL ro rppr6 np66l1pa.

:ro brliro. o (firatrczonoro( noutr(i! ],?nr(r orrctriv$ ovri 1ru rpoqri. OL. tqrigror, ,, flv{lr neoo.rplooLlrr:. n(ili!)< trl ((rutrrzas. O ;r€0rcpLotro(-zi€Lbi !xJroool l ! (( l T, , L 'r , . rr t ! ] , iH' routr t . l r t t r rorrr r , r r4 Tpl ' i ,or r : r rrruor lpo,rro o.rr .rI . 1 ' , 2 ( ; ' t : A ' r o r " t i - o o n l r v o , t | l r r o r i T r ' o l l ( ] r i r f : o r r o , , o r r t f l : , i . / . r t l r l | t T o Iqq)l(zoxoror, .1Lcr too6i'r,(Iq noddntr( or- orrqriy€q 6€! I:ropti vtr Er;reprioerrrp rr;o1 ror, 3:r(zrii.1 rJ' A!ro. !,ir(lr o j ooco: ;reproprq6: orl! r,1 < ..Ao11),riovr(< fr' crlroug rolq n€e|o$olotlq. p:ropri r(;r0riryrrr :rooor4rcr 6, (l:rrizd(h t j ( r(4r ivI zur" ra 6Xrr o).rxr i ioob( ,r l r + + y, , ,1, , , , = r /r . ; ror o() ,rL v(

O(t xQirll v( (1a(1tv0)piqrrr orr r( {,(trxr(! owoic 1r! bi,o n0odUtr(ir0).v!i.a(t rr1'li{,)q 6r((toA€rrz(i. To ng(i)ro ffvrrr urooivoio ror 8". nor x(0oai!€r(r(:rri ror :ri!ftxrr .{ xuL ro :rrgLournLzr'r brtirrqcr 6. To bturtpo riv(L r,:roourol,<rroL, n"'. nor,7(0oQi!r1(r (lno ro! c\(ior0o(o ror,-.1 z(L ro rii,;,o brdvlotrc.. Oorooo. {.v oqrn€Ori,(itroqrr rr: or.rrrprrirrrr: Toorolr. r( 6io nAod).rtu((r(:TrA(i .qr{xir( ' rv r t r i6r( belol i lu.{ .1zar c. H o).r ] or( ,)Oi( rol is4r[rzor inpoypctrtr(rtotroi ontpiicr(L onl br(ofi'6r-ort ror,: z(! ra7otr(ol z(1r' rL,0ri.rloro 0f trriri;brq (rrf rniA(c'tr(:

8D @tdtQqpa Auioltoi Edv ro cp r6 ra66l.ru( 4 ro buax6 6XEr €!a6dl.rLoro blrivuqtcr, ro i6ro oup6aireL rar pr ro d),}.o xrlL or lpaq rolrg si.\,rrL'LbL4. To tMymo o1g c.t rcottu ye o piyrmo r4g t/6. M€ dl,),u l6y!a. a!6ev u:rrip,Xotv 6iLrLma 6LaviJollcr(. rot€ 6{0 €iatrr or bw(r6rlr€6: 'H zaLta bio €qrxtir oi'vol.o €ivar r.rlri rt ro €1,.1 Efa(l zw6 ra to i ,o ;rp66l4rcr€i\"1 t,t {eou6vo (ro Ityrmo eivat +E riro €irilroro €i\'(lr -:.).

E(i! rcr r( brio nao5iri$.rr( sX(n,i, r(t,ritrri br(\'noF(r(. 16r€ rnrirr1oL,! oiirrorc6t(1r ixrt t ( lu.r ' xc! j /_, l icL €rLr).rov..r ' = r ' r .

An6 tr((hltr(rlz)i rltroqq. (1116 r(zro;rorci ror, (ar.1./!)aLcrtLo perclU torg:rrrxri)"1 xrrL ror qaptraxonorori To utorr:1tayu €i]'(u tu|rt'r' kro:ra1ia. E\rLn({Aotroro {)ci;)r-n}r( minin x ,(! pL( :rcrqo$oL( rooagonic tkr uLrcr.wrloo4Leoq @eo;gift ncr1ri(Dv. Ta o€('rQrttr(rc (1,1h 6e! o4luilorv o o zaravci.olrrt!b€! ni.rto(;vfl (1zAL6ir $(r Itii.rlorl bi(lLt(l rl orL ro:r(rTvibr $t rov;rivazctiycr (qrFQo;.rtrro zcL }c rof. 6io ;rcriztr;. Auro nor' ortl(liloN rilcL orL oz,rrrrrr ; . ' ' t r l : l r r r7r ' r r .o7o tr rg, ' r r1r,r urrurrrrr I , rrr i r r , t roo1rl ' rz,rr l l z 'n rn o9 ' ,Lrrr tzorord- r ' ;1,r i rur or. ,or{ry, , \1:rrLn o, oi . r- r , : lL '1, ,1ic lo, rn\ . , : r , rzLi i r l

II.] O!QPIA AYDMOY

xar ltrl ot{QI66< 1Qo969.6fi(o( xo qrum(2()6()1,5!A(), noo}.d p€ przA6r€A€E rUaE.O(r &iqowr 6rr oL oliQL66E rQoqdE pftoQei !a xQ(rrt0o{\r e&,r oro r1 6e}.rroIr1biarrrl, oror€ ro a'ror6l.€o[(l U]ioa€i. !a €fvt1r (zaL 6wo6 ri.r<r) rooncr],i.cr.

At'r6 snopei vc porhLeL pe xl,ripsE a6r6Eo5o xcr. eL:ri.!(D vo [Iv 1160€r€ o€crroyv0rol. TIv xpioqrl r).ruoqopic r€ar.61ow ra 66).rLora 6r(vuo[crc. :roup1z6 n-o66),qBo. ro r' ),&r orov rurcvcrl.rorrl rr ra6n€r va xciv€r. :ro 6rlir6, roy' xu0opilcr rq qroroloyrztq r$€s 6 "oxrrt6:r9 trt ig') $E rrE onoiEq 9a d'rQ€r€,ttt l,t:Ixorqr/ri q orzorogiu. Erl 6oov to 1pc4rprr6 povttLo tclg trvravaz),ir rtvrQ(rlucTuall olxoaopicl, xc 6r(\rrouclr1 (1rld fi(QroTdvoljv rG Lrroqdo€|q fio! eoinQ€r€ ltt ),lq,0our. To 0rr;-rpl1ltcr 6rio4ou 4irg .tuvsprilrr" qv nLo olpovrr"xl rougL6(6-rlra, XQ€dE€r(1[ oflorq \,(1 !fio].oy!o0o!v [€ rI paeobo simplex.

A< _Er:)rvrloorpe q.v ur6bt-r!1. Modler iqoq(vt; ro 6r! o qdQllozono!6:$rrop€t vc ovsGdo€r rr( rqri.( rou (bort 'r'c grdorr rov 6$rroQo, ul'trd 6!! €ivo!.Mit),Ior civcr p6ro xcrri to:rprirto pdpog: Aq,oi xd00 rQoqJrt pnoQr:i \rc (.,'Ir,xc-loorcrO€i tr€ ro roobuvcp6 qg oe 6rrr4iveq. 7"{upig aljho1 tou x6oro4. ti}.eg or.buvariq 6ialreq fiQdnn 1,c( €ilcr rouLdlroro\r 1600 clxpr6d! 600 zar lrd(ee r+r! trsrlv onoia tu Xp6ov€ o (l(Q!@rono!og. AlJlll tilor !!cr UolonLelJel a.,'[o6111o,4nj tur q,cpTuxottoutilranj tuu ryrd(lo1t. nor eivcr o[orq 0tpel,ul6qq. l,!1erars.oqeviE 6xiop6q t(tr (fio6€lrvu€r(! €uy ol,c yrs rao€ lQcppl)io nQ6yQcq4ro ro!

8E E(l\ lct E lrctr g €ivcr! oriord6inols €qrrrd 6lcA,ioFoid rcD! ftOo6l,I-$drov €l,sliororioi,Iolq zcd lr€lloiofioirlolg svrloroix(oE, To1€ VD < cr,

Ato6.r5t Ennf, t r ' , I r ,vt 'op,,rrr t iv, l r ' r rxr, , . r7 'rv.nnrn, lv r E

A : t > b l t A < t .

Enml6ov. €r€r6ii I .rroSoXn ori$r€arE).d$6(./€ r.lt rL< r > 0 rdr. g > 0, $noQoigrvc rhao!$t- €oor€Arrd yrvo|E'vc1 X(.rai< vcr Xcd.ciooul€ rrE a!|,o6rnr€(:

yAt 2 sb g,- l .r ' : rr. ( 1 )

ErErbii or apror€ai! rl.ELrQ6! ri.r,cr fELrE. nci.Lr\'or'tLr ro! coo€vri 6ri:ou6 Jl < r.,i.H Fovod.€xpl ar'ro6rnrrr 6X€L oQr.o[dv€! (rfi],6! orll,ifierE<. IIQ(lro cn' 6l(r

ffioxl€i€r rI buvarortrd x.lr. rc 6uo nao6lrtFara !a Ei!.lr tot rtQ./l[ivo. Ed!ro t/l eircr cuoaiper.r [eyir),o, rore 6ev U:rcipler €qLXr6 r. aqou 6r(rloQsxrrd 0(rEiXa[€ rlv avri.q&ot 96 < cf,. Ktlr'avd.oti.av, €ilv I a].(r]Lorotroinolt sil,or tulqpc),livl eclv b4,cb{ ro ce p:roqei vn xare6er d\ to --- ,16r€ tr.o ro 5!ix6b€v lrllop€i va urciax€r €qrrro g.

Mur 6erftepr1zcu" eE ioor evbrarp6poroc owinerc eirc.6rr,6ocr brtrvuoF.rrcL2(vor!ololr\, ltlv roorlrcr, yD = ar, npd;rer vcr eircr 6tLrrorc. f' cr,.r6 ro olpeio

r?AMMIKO> nPOTPAMMAII:Mo> KAI oEQPIA nAlr!_IaN

I rqri rou lrtclliddl toolkor B€ anv TLFi ror qdQ$cl|otroLou x(tl, I 6€l'rlortl

disriq, ""eG ""t "i

oit'"t*.q T $€{ &tolrr,vcirv' cv<ryr'opilovtar cn6 to 6rr' o

xorsvotroqq 6€v 61el linor€ va 6td!6E€u

tF Edv ta 6lov{|oFata o ,((rL 3l €il'cl eqlrard ,cL ct = gb, t6t€ td ,D t'olgr sh(rL 66k!dro.

At{i6stg4 :uF,luv(r Ft ro 8E. ti(1'tvq €qrrro, 6tv FroQEi va xcrl'Et ro rt

l,lFya].r,.t€Qo qno,.r. E:rttbrl to nuQotrovo tL6t*6 3r tnttu}Xdv€L awtl tYlv rLFn'

iircrr 6d)'rLoto. Ilcrpogola, xa1/;v(. t$!xr6 :t 6tv FnoQ€i \,a QiQ€! ro ct xdr0) ano

tov crEeg6 gb x<rr owenoE, onorc6inor€ c €rtxvl(dv€! owo ro g,ldl!fio JlQetl€!

ra eivcl 6€Lt!oto.

Aivoure n(Qh6erYF( Fe 6rjo rQcx{r6g xaL 6io 6noPivt5. :r rEtd)m€ lov lQ6no

ur rov otroio +racrfi(,€rut to ,47, otav lqtiyor4rc 10 butzo. cqou rA S ' Ytabrcvnopctcr.ygr4rpdg oqpnivrt A/y'l 5r1 lrn mritreg

APXIKO E).axrmoro.lime qv rr+4:rr AYIKO Ml'Yrmonoriott qv 6vr+7yl

rnil rouE nrarcarspolrs xir > O, r? >0, xtid roq rta$eLgpolE 3/r > 0, 9r >0'

l \+ xz >6sri +3x:z >1.

2h + 5at 37g1+3h <4 .

H €fl!).o'f{ ,1 =3 xar,.r: =0.ival €qrrrli, F€ xdoloq i|l + 4'r: =3 lto-buixo

apo6l'4rc ra u = 1/2 xo.t" y2 = 0 6i!ow qv i6t<r rpri 6!Jt +7U2 = 3 'lcl

d|clviopoia c(t'td trQ€n€! !o €il,(.t 66lttota.A!16 qaiwrqr fieQ6olrrd (rttl'6 AEiqeI 6poE la Qi€ouFs [!a fiQoo€xtrxt

Farrd lla ra 6QotF€ t! nQoYFotlxa ouF6oiv€! rl onYgri rou Yl ctltoorrFaslt S ce;

livetcrL ro6qtcr To npcilP(. ttot.iqet g€ rov cl'rerQofitx6 loYtog6, 6nou 610! lvtD-pilouv tI ouv0rlxl lrc lreltoro 11 e)dxrmo: Ot nQdrEe rtQAyavot EivcL p46iv

ln' T1v ri),1,1 [€Q!d 610l E€Ivot! 6t! n owonq orj1ri o]']'dq€r 1€I€i(,q F€ xrlv

roOouoin r€QtoQtold)v. To xohtr€Qo n(rQh8eryPa €ivat 6va €u0i?Qaplro 4l.l]lrq F€xl,ion naoE ra trdrt,). H ttoEilr,ryG tou 6€v Fq6€viq€ro! trot6, o an€QomuiE 6€v

6o40&, ,"i to p6yt*o erltr,Yxdv€to! oilortQa fio dxQo tou 6lrrm{Paro€. Amrl

uxpr6d4 rivcr ror I xataoTt{ol otov lpap;^16 naoYQ(rtruortolol YndQXow

rrQlooorrQrg Is(a6lYFdg xcr ro brrio-r1pc cwtxaeiotcrar F fvd €qll{To ouvolo

7(rQrooorieoJv blooTdo€ov. :rdh dF0X ro F€Yloro rtnttY)(avrroL trdvrorF oe Floxoouqil rou tprxroJ ouv6lou. :q Y).dJooa rnc ureobou simPler. unaolet eva

6i),rtoro r rou rivct 6aorxd: E)(Et. bl].abll. trovo m tll Flbcvtx€E olJvro-ro:)o€q'

To naoY[cr'rx6 rrQo6]'lFc tor.r 1p'qqlxou :rpolQouraxDFo{ eiva[ v' aroQa

oioouue rora eivcL cutrl I zoaugl l]o€tr€! va oFol,oYfioor,!€ 6fl o Afi€rQool!'lG

s.l cJlj!2flA .\\ irNro\

Ao1lr4ro:6r\' eir.1r re),ei!;q cbrv(poq lrqoorir o (r,ro ro n066Ltl$.1. :rft6ailEtro 81s),(;< .lari0rro. 616rr o ;rllcrrro;r6g r!-rv noLicni(o1.ff161 rot Lrgrange'

tra! €n.1\'.1rtip€L orov glberro;ro r0rv n(puldryorl ot( orlpelc xi.ty.i"oro,'. natpeliorol zar orlr :rqcrlpurr"zotqu 6t\xt< ltte6lttriq ! ruI axa6ti< otnd.^aii,aarcErla Ltskl/r.(c ror, ri0oii).4tcro; €i.rxroionoirtoitc rrF (:t. Arl1o tircfi xcfl ro x),rri\i tor,1rt r,r.](trlrzori nQo]Q(4rtL.rrrotron. Or orivortzlE yr( Ficlzrcrozclt piyLcrtLr uno ir€eLoQro$orl! 0c 6r.rrrljrrDooriv tr l(0illr(rrl{o{,q opoL,( oltlvFEio(rort (2). Oii,') qr( ,! n0,;)ra vc rtq exqptioo ge orzovolriu,q ciport : // diaizat x t , t , | .

t t , : t . 1 t , L ) t t a t , - o . t ,

1rr O 1r:rorr i .4c nru).r t u4o"vrxr1 nonor4r. l rao{tc oruv urrr t I l t r . (rc( l;rFQroootrQo Gn6 1() roo6tivapo fl1-c ot 6rtc$iveg.

(ii) O qaatffonor6q zparixeL pe plberrzrl t4tn onoK6{rior0 6rr(4rivrt uJI€a-Toplleircr otrl 6icLrcr.

:ro nrlrt,ilfr./tr(1. r: = 0 bLtirL r1 r\rtlrrgtl tpory,il tivlt no],i (1tt0Ltnt. II ntnirr l< r inlpdcivr rr t ! ntn' l roLr ( t (0tr(zonoLori , bLolr r t ! +lr : 5 a t iwr ]artorc(lrroorilro: l/2 + l) < 1. n(tni|orc. y, = 0 6r(i1, il t{xtrij ijtrqriul fi!(lor u:rr!)7oprt1qorl. brtLlbrt 'xqr1'lrrcr darqrtrit, , noL,Lirltroi\,fr on fiv{1r Xr)L)iqorlprroi(. To n(t$i&ryu(r ((n(l|ronor rnrd portibr: r1< 6rrlr.Qrl( ll(r(iri!rt<.rv(; , ' l b i(rr( l : r t - joo!( l t0f 5. /r +3,r = l5 z(L i . ror { la i i t rorB! t / r = 0. Mrropsir f!( c"r-i.rf nd)< oLonirlg;)0rlzf o bLriofLoE: Bil.rroro !fn"/oq ilot,tlf tL(jvov (irir!rzir\1)noloiviin /.uL o L t\ ti .11,\\o\1at..

:rrt y),r;do.1 rorv ft1.lxo)!. auriq or 'ra otrrs .irr,rioto! Fi1,(1( 0itrol.o vazor(1vorl0orl,'.:ryr't0ivor[r! ro brdvrlotr(,'l/ tr€ ro b(i!!qL(, (fhruthirr orr ro\ , , 1 ! ' d " ,

r z r o d r u r _ . 1 1 , ' ' / i o \ " r 7 ! , , ' i . , , ' r r r t o r o , , : r lclLoorrpcr cr:rorqTrlver. A1116 (!TloroLT{.i ,}t LJr.p}oalyontrcvf 6rrattivrt. tiac rtr+oi rtc 0(1 filcryi = 0. ToNdtQo!( dLhr,rptvoupe ro y-1 pt ro r xil :rtpLlrirorpr'drr 6),rE or lyrto|€q (rloorrlrt-i (crar66q raoq,i<) ftr .i\,Trl]loLlo!! (rt r, = 0(cqclip€ol (1r' rrt bil1rrtl). Alllrr< t.irLu <tL ary .4sayarLxlq ftpt9oanxi.olldrjzr: ror, 1er4{Llzoi npolprippctro;tol xuL at olit'hixrq KrhtTu(l. t rotif rI lQGtr$rxoi, nl]ol0.rulL(1rolorJ:

8G Ocrbpqpa nopqoniag Eot(D or t.r etpLxrc? bLctrlopctcr e xr{ gr.xcvonoLouv rLg ur6pr:vtg or+tnLnQ(!4rff 1166 ouverlaq 1ol(porlrcrg:

F t l ! ( i r ) i > l i 1 6 r r r i = 0 , x ( I d v ( i r , 4 ) r < c j r 6 i € r j = 0 . ( 2 )

rP.\M\! tKo: ntnrr \ \ r \ rATrrNlo: KAI otaPlA nAlr\ro\

T6re ro o xcl y €ivs! 6iLrrorq, Awiorqoqc, 66lrroto dtcltolsrs ltqvo-noroiv rrE (2).

Ard6€r6{ OL €ELo6ocLE-1|leLbrcl €ivcr or

Kavovur.ci. povo 1Lc rr1 grocicr rlioooq e(rcrme oiloupor.. fr"a rrlv rgrirq rliooorlriFaore oiloupoL 6rr y ) 0 xcl! ,4.r: I b {ipcr xr,l, 6rL t/6 < t/(A,r'). Eriotg, unirax€livug p6vo rp6ro-( .1Lc vrr rolu:r I rc6rlro: ,a(rid€ qopA tu iynyr yrilo&6nqop(i b; 3 (Ar)i, o nca(i)\Dr li nor floL),unlao\ittt dftlq rK ou'utr(i)a(<:reln fi eit'at 1t4dir'. Torc t 6!.(loQri (urll 6rv ouv€roqlQ€! or( €ol,irEQLX(i.lLvopna rtrL q ro6rlru btttodErrar.

To t6to roxueL r(lL y|( ilv irli,t €Eioool: Etprxr6 olpcivrr .r' 2 0 xcrty. l _ , . c ipu y.a 4Ai _, . . lo ' , r i1u :ruipvoupr p<irov otrrv rzrrvorrrr i t l

I 6frkral or'verlTtl n€Qro('r0lou: E(iv undQl€r un€priuol (.41/)j < .r, ((!'nl;rp6:rrL vc eltooppo:rr10ri ro),Lrcr),uorcilolrag F€ .r, = {). To r€),r'ur(io trtl( 6iv€rrrtv vlr = c: ot1v eiiorrxrrl-zLtt6i (3). zsr I ro6qls cruri €E(o(td,iLtr ("urr:(.oq.(ri,ii€r(|| ono) io 6tt tu .r xur y €i!c! 6ai,rror(.

H rur66alq t4E 6uix6trpoE

f!(1 Trl tlovontrelQl .rvLooxlra ,, < l|.f, (E sQreorotrpe o' crrrri rou ci:r<r[r, lIc:rri6rrh1 qg rlrc'v elno).I. ib(,)o€ dvc lQtyoQo ltpr.rrlqro 1r.cr 6sltroxa bL(rrrtrparc(trt p€r(radnouv or roorrycr) xcr" p6Lr.; d6tooe rtr dvc oivo).o crvclxairov zcrLxruvrirv ow0ry.t v lalirpcoo$. To [6vo ]rp(irypc:rou 6tv ryrve rlrcrv l orobet!1rou 61r { lo6filrc yD = r:r rtvuL npd{yFc{fl 6w(1r1. To 0€dralItl buio[ou bev0(t aX€L oloz).IQrD0€i ,rQLv cno6€fiou[r orL d€].rroTo 6r(vto[or(r €ivft 6w(rov:rpirTpctr vc :rcpaxOoiv. rar aut6 6ev p:ropei vc Tirer 1e erirotor,5 c:r).or,5xrtQrolou,<.

fr(l va n&p.iryoqre dlor(. brcrliolcrcr, €mmQ6qoqrE ('rt Fi0o6o simplcx,trr rt! onoia unol,oyioole ri6t ro 6d].rrmo r. To np66tnrd F(l( €ivsr v(lnaoo6roaioouF€ rcr,r6lpov( ro 6di.rroro g TrlL va b€iEouF€ rirr 4 pi0o6o;orqrcr(i Er€i rou ra€nrr yr ro 6rix6 :tp66l.rpc (,.rcp' 6i,o :rou znrcozeuciorryrc"/!( !( ),!vsr ro cpxrx6). Y?r€\'suFilou$e :rptir.rt1 n6g !*rv{or4e. A),Lirlape).rct' (ela: rrE rn crlro6rryeE ,,{.r' ) 1l oe tErotiosLg. eLotlovrcq 16 6or101tLx6gp€r.r6t!$d< u, = Ar b xa| Ecrl,.ryQircpo\,raq q oworl'{q rou Evrxtot urlo iI!ooqn

l t b = y ( A r \ = l l l A \ x = c x . (3)

r, - ,r [;:,] =,. l;] - " (4)

s,l oEa?t]\ dYDIltoY .187

K(r!in[\r. I 6Lc16lx(loio ffi€],qE ,r or{1.€( rorr [slol"ot ni\r(lLs [,4 1] ocv6ttrru6g :,raL tL< gtri1,r1F (roul.riXLotov 0turp1rL,{ri. a1, oXr x(! {nroLx.i) ornv aQfi.Aurrl r(rprl./C./€ 1o1, riv(rra [B N] z(r I (1'rimorll $€r.lronlol oro F€y€oljtrivobLcivuolt.t zo('TorK I. ()1. ((v(tbr€raooE tq orwtmcires o€ [clJ c\ l. H ouvu'irtbtszojnk. nor 6qeQ\,€ oe iepaq rrt p€oobo simplex. rirnv r' = c1 - caB rN 2 0.

Err€rbri ro n).rtUo< torv xopultiv eivcL renepaopivo.lvoqiloupe orL aurri 4our94x4 r ' b. nlrxd uurctor6rur. ' to zomog rorr i r rrv

f p r , r. . f - I . n . \ l " ^ " ) - . s D - L . r o r L d X r o r o x o o r o e . ( S ,

L " l

Eriv yto to 6r'ir(, €tfltp'rErol rn AtEo.tttE t! = ccB t, ;ort oiyoup.r 0dfuou1tt Ab = cx. To €lci/roro zcrL ro p6lroro Ocr cirar ion. nQdn€r i,ourov vcbriEoUrE or! ro 9 u(vo:rotei rorlq bl{xow n€QroQrotroii yA 3 c xflt y 20.tuvol.Lrd. nQanr-r v(l brtFo4t€ rrl1r

r lA - / l < [ .01. (6)'Ot(v

I $d0o6o( si pl€x ('\'(6!((rdoos! rov llcl,iQ! nivcz( x(t 10 6rirvuofc.ddlovr(r< r( 6uoud brcrvuqraro otrp :rptirq 0ior1. u1,cbr((rtioorr xcll rouqnr-eLoproFoui (6) x;i!o!rt. row

y[B N] < [ . / ] f r l . (7)

Mc xrl\, €filloy4 $r1( t/ = cnB ',ro rprirro pLoo livercrL Lodrlrc xcr ro 6€rJrrQotroo €i.vn! ll crvo6rlrcr ca8-rN S ('N. AttIri €i.v&r I ou,Orllal bL(lronrlE r > 0rou yvoQilou[r 6r! rxcvonoL€ir(!! Eror ro y tivur r.qlxro x(r ro 0€dJQI|(buiotrou ile( caobt-Ly€ri. Erro,'rilovruq ro\' r{pio4ro rrr sai l,? rivuzc B. :rouFiv(r Frt lbr6troQ(to( r-rt, ooov iXotle rlxayoprrlor.r ro! €r,.q^,In+o. ncahyouFr trs

lF€{)obo simplex ro 6a),rroro r' r(ur6xQov( !e ro /.

!xrdi6€lE flF6E

Ili4 prrcr6idJ,trur" ro el.dlroro zooto; cv cr),Lirloqrr q 6tlrri r),rrpd Irl ro bf(ivuolr( ',i6orori( .l Aur6 rivcr 6v<r egrirt4tn tr1; drt .uorlt tuo,toqrloio,gron [c< €n|reanrL a( (lnoonaroougr'(no q F60o6o sinrplcx n],ri0o; rrQooo!()v+r€ar€XoF€\ov n),rc(xto0Ld)v. frc 6!c! oLzovotol,dyo ri dlcrv 6t€U0r 'rit, n],.rftoL €a|(nrrloelg yiQrD cmo ro :ttQrqoeutztj z.joro< EivaL oL rno evbtcrqipouorq.

Edv €rr!r06!rouF€ tr€lir),€f olLu./dE oro 6 rl oro .. r6re 11 ),rio1 o4r:rrpLtpdperor,[c ro],i (tor!€lll to6ro. Kood)E crElt\lct I r+lll r(,)v .r!yd]!. H(l uri{AEcL 6v.rotF8io orou 0( eqftqr.rlroooi! (16 q bl(!r"rcr. :q yl,.droocl ror lpcFprloUtraoyAo$$arLoFou. rl tr€r(6llrrt r,,./,i ec1 tr€rarrtbrlo€r a:ro tq 6cor"zeq mtq€),€ue€a€: F€rtr6),fl4q. fLcl \'.r Fd,efliooulr€ ro lrjrlpa ooorci. 0c 6;rp€]rr vn

I 'PAMMIKO: NPOTPAM\ I \T I :MO: KAI OFOPIA I IA I f \ I ! ] \

Ero(yiryouF€ 01)16 fiou o\rollci!€rdr ncrQaF€rQLX6! nQolQa$Fono|ro9. Eciv oFrDEo[ q].],ay66 Eiv(r. U!xa6q, rrQdypo nou €ivaL mo qlroko, r6ft I xoeuqi ,tou

irav Eiknrq *rQapirEr 6i),trool xaL rl ntJ^.oyri rov 6aoirdrv Ferc6LrF6v 6€vcd.).dE€I. M€ tr),Icl 16./r(. or B zo N :rcpr4r€vow ot i6ror. fe<rperprzri. crur6 :rorzdvape rirnv vc gercro;riooupe Liyo (alldLowas ro 6) ro €qu.t6 oivo)'o rt vtrrlcrytdrooqre el'crr1pti4 rry orroydvrrc rorv ururi6{rv 'rou ro fttrlouv (dJ,(i!ovr(to r). Etiv. oo16oo. or ar!I,c../€! €i!(r !uxQ6<. ror€ q €n(rqi or4r6(i!€r yt( rrQlirrl,toa,{ orI! i6ru (f}.(qa6i trflnro,trotl€vll zol,w rl.

:ro di.oq qq [e066ou sinrpler. 6r(v rivft lv(oorl I olrsri €nr].oyri ft)v6(orx(ijv tl€ru6),Ircir!. or (1.ri,ororxrE ,) mrii.e< ro! ,4 (nor€i.ou, Tov 6.1oLaoni\ruz( B. : ourrl\' 1I\, i4oQljrfri loliE! orr

r ] .cyr.no xdl t l r r : "sB tL = t 'L.

Mlc perat6nroq 'icrd noodxrtrcr db al.Ihler ro el.cil"oro r6oroq xurh y"Ab. Hbuil'ri Lnol y' hi.ret ut, Qv0!6 ulluyi < rcu etraTiotox xiotn < (1I.,' ri(adloJydro!) oJi nQoE ,r[E p€ra6otr6i ro1' ,. OL dwrtx(1|6vsl ror.r y' €ilCL ot oxtiriEt|npig n(lr 6lo!v olvcrq6q [€ ro ovotrir rouq v6+r(. Edrv rl (vdy](t yLCl drrctrivr]Br (lu9tori xard A. l r4rrlrou tpcgpcxonorou €ivaL l/i. r.qr qr'r65 riat.| nl,iEnE(.w(r./a)vrm[r 66 rQog lov Fr('t.i],Il. r6m ro x6mo; 6Loirq5 (rou y;r('rh].1 r1 ror(t'(rQF().onoLoi) 0s auEllori 2(1r.i ,iA. :rlv n€ein(oorl roo ro l/i eiv(L FI6dv, rldLrqrivrt Br €ivcr "€),ei8epo npoi6v ' 'r(L prxp{ tr€r(r6oi,rir$ ,t'irrto'iq r$ b€v 6X€!r;rr ; rrr i r rrrrc -1 6iarrrr :rrprr iyr r ib l : rrptooorrpr l ruro rry uvrryruiu:roodt4ru.

EpX6pcore rripc o't'r'cr bLcqnperr epti:tlpcr. Eotl;6tt trryivoult- l biuttu!( rrrQrdlsr [!( €ldlLorn nooorrtr( (rry(;v. O n€QLoQroB66 p1 agrlrLx6rlrr<r,,,r,, > 0. tu raororoL46ri mov.r',,,",,1 > <1. fkirE erL6Oir cuo oro x6oroEr

Eirv tcr cqd nepr,eXovtcv orl 6dLtloq bituIo. 6ev rrrippr cr),Lrryt1 -q v6o(7r(irlol ibl rxcvonolr:irur x(lr 6r! lroorlq€! linor€ En[nl6o\r. E r otroq dev,[eQ[rI6Tcv orrl6fc!Tcr. io !G o![ncQlld6oupe irlv rroooxrF( 6 0c rioo1iEfl 1tu1r,ll (!|EIol bEv paopei vrr rir(lr olox),Iel rl r.oo6rilrc c(r,ltia,6((5xl [noQoul€v( reproQioouF€ dll.eE raoqrdE z(r. 1,( €FLooeQorrrlooxfr€ trEoLxdq ro r(ooroi,H rru!1or1 oqv :rpqsnrrr6r4c xa0oailtrTcr uft rc "av\yFtvo xtiotoq" T'tr(u/dv -lrlv rrF4 rorK tr€r'.rv rtv riln nou n),nQd)rol,]r€ yrn roo6uvqrl qr0ryor€Qrlrao(t,ri. fr(r rov unol,olLoFo ror r;rrmpdrlouyr orlv €EiotDo't (2) rrt< ne(ya(i(tou8.2:

zooro< = t ' . r { 'BB rN)r. \ +crB 16=r 'rr '+rsB r i , .

Eda rcr auyd ei,\,ar. 11 rprirt4 r:).ruOrpr1 percr6L4nj. r6rE I ( Eryt rn( nQdrngollvrorr6o(g iou rN r.rrh .5. 0(( cuErlse! ro n6moq xnrd r1li. lwrntlg ronp(lt[lI|]io romo( sil'cr rr. n(Qotro(, ro I biv€r ro rQ(./fr.1xr1o 16('r()< ).cL6),t'.rv torv hLtrrov pq 6doLxtirv rQoq,d)v -rrE a).).ay€q rou ofu"nou k6oro!( ftcoc'iEr<r pl6evrxri xciro qqdluar( rolJ ri (or rr€QroQLoFoi FI ap!IT|.r6qr(() 1,i|vouvrsL:tpog ra ndl'or. rvopilorpe 6r r 2 0,6LorL cu6 {rcv.ro zprrrtaro 6roxonl<. Horxovogru gci< ) . icr ro ioro:rpururr.6rtrr6t l orr ro uvrt lucvo rooro; rr , ,v,rr ,Ti 'vb€\, tuioQ€f vc €ivcr. r.QlTrry o. bdfl 6L(poQelLrdr 0c ri.pv p:rer oq bi.ctra.

si aJEoPtA \) I:t\!o\-

H 0roQia t(Dv ovrodrorol

A€\, r:r(ipt€r trovo ivc: rpirnoc 1u !rr prLerrlootlrE rol blrotro. H ,Toooay1rorlnor czo].o1$Iocrtrr- -v (nobriiorrtrr orL g, < {'.r rcr xcrlixrv !.1 XQIo+rorollj-tJottr€ rI tra0olo sinr ex y( I'.l :rhpoqt rr]\ roorrlr( rtr(\, 6oirzri brolr €i7qrrribl rrrarrnltr q lri{)06(). O|orz(i. €ilqft !|( 2ertlottt, oro.rl arddrri4. ((toiru .r' .lrrr ll uroi.liorlz(v ortlv nedlrl. Pizvor[r r6aft tr(l oivrot[] u(Illi oftll( ol(1q oor'flzl nQo'tryl|ol, ro! (q]t!rL rriorr rlg rov trIX(1,roU6 rou (ii./orri0-t'o! snnplex z(1r 0:fr(ilrl :rLo ci|r.o( rr1 1ttr4rerpi.1- Notrilo) 6 o( r6ar(-z),€(br.i u(yi!o|y t-: i,'(n' oa(tri< {rlb€loll6v(,)< z(lr rr({t6orrQc<) €(iv x(lQ(i,€irl,or1r€ [€QLzi:

II rc),rirrAl e!r1yr1o1 crrr{< qc trr.0(i6ou n(Qolorirooitj1 r oro Ortr€).(;rbr(OrioArl$( Tll< fA(4rt(zri< Ai,ytr,p(l(. To ,1006).r}r( oro Kfl{ii).{(o : ,trc! y(iri,x)utr€ rrtv -'l.r = t. pe d),iu i,61(l 1( (taoutrr ro /r 3t1rlr r'ro Ir;,Ao orl)dl,1r!'l Mrrd rtv (n(),o'rti r.i(r rorE rdoo€Q( Llxo1(;'1,(,rc, (1rjro ro Fad)rr}rcfnri.rjortrorlr(. :ntL,r' (x(ivrrlol orry Aoxrtol 3.1.ll tr' i!c! r|t€i(,)c br((toer.nx(;

H ,4a = 6 i i/€r llot ;t urripXcr y.6toL riote 3y,4 =O x.u lh lO.

Alro rirrrr to eetiptlptt ttlg ztciJ"axtrx{g 6rncnir4rag trcirr ro vrr r:rripTo|r,z(r ri( bio (.. z(rL u) €[v(L (briv(rol,: E.i.' .{.r =,. rorr u.l.r = uh l0:ro,,t(vrlqaoz€r orr[ ]/.-l.f = 0.. = 0 :1rl t)"{;ooa rr)v llno/i;)tri'.\, ilro D x€Ari7sr(rorov l6Qo tJrl).6! rorJ -,1ri brftrlopot0rti iTrr pl prlbr,\,Litil oljlrordo( oro! 1i(i0rr(,|no/,oao. nor! r-r!(r o llQLorrQog p4b$'o7.r'rQoc ro1,,{. H or,l,rmdro( urtTi r.ilcr

O6ioutr. v.1 iiAoutr€ av( :rupo|or"o 0rrirp4p( yxr crvLorirorr<. To orroro orlurior,z'ril''lollq cf\,ttr. ro |bro olorllrrr A.r = l. pcii. 6!crq trr rov ip(jot){.ro nrproarotro/ : 0. notr |ircrplr(. !i/r Ir( liol rou .{.r = r. d,t.{i tr{r /t dAy4nr.t ),io\. Mt(i).i,c io}(.;r6rF ro F(tLrro l]rvoi,o ror jraorjirurrrog 1tr:rrprogLopoi; roorrlTi')1,

f(r r'cro\.rrioor'tft- or- (ll'rri ro rOrirr+rc rirrr'rlot,ut :rrt).r ror,! orj\,6ur.ouoi:t(Jv orrti.dr! ror.1. :ro Kr{tcir,(Lo:. orcA,r:rrrpiroprr o:roL(6ryrorr rrtol oror. ro b [iroQouor vr1 yi\rer ororobrtrorr bLriruoprr ror ](i)por o1it],drv. T(ir!(lftrrQ€nollirt tto\o ltl uav4rtzor< orvdn(ouor'! x.lt ir(]L rr1 b b€\, r-E(vrioirx),r:o! oi.6zi,rlQo ro! uio/rDQo. Alri yr' (t'ro. yrpitotv gLa ztDrrzrj :rrqroTrt onrr;(trrl rou:1. 8.6. f(l 6vcv:rircrxc rrr rni /r 0(l Ei1a$r n orili,rE orov ,, 6ldor(io

8H

i Rtilrf .irtt ir rl r(r{it€r:I ($rl td.iwur.r(.,nr(,lzit To rr(ilo ror -,,rUrri:.rn. ri!r(dl ro b i7.L drqm.du tlrr !rD€\rzrt) dror n.L.lr $) told.rd7, r!$. n(j.,_rf ir 2.i ro l]0,i (vri2. dror

,190 I'PANINIIKO> IIPOTPAIIIMATI'MO> K/\I OEAPIA NAI'IIIQN

Ir. 8.6 O tr;loq r.)v trit (eettixdv ouvbuuorrdrv to! ortidr!: 6 =,,1r, p€ r > 0.

il,)Qo xor o x6vos eo }lvorov pt(l nuQqri6o X(j)Qi( 6doq. >ro olllFa, o 4 €fvcr 2sni 4 rr(r! undQt(ow r6oo€Qq orrltr€( orov 6r6!(ictr(ro X6Qo. Edv 10 , n€Q#X€r((ro cur6v rov x6vo, 16r€ undg(€r FI aavtrrri l,6o11 rou ,4:r = l, 6rcrqoperrxd bevursQx€r.

To ra66l'41d FoE eivor v' ov{rrdlivoqre fl or4r6oia€r orlv €latlLo)iflxrlrlsQifircoon: ?r du!6ahEL irul' to h qpioxtrur €xxdg rcu xdvoq H brlvar6rllxc(ruTrl rcQloTqv€rc! oTo :1, i1.7, dno!, [€ pl(r pox{q, 616rr€T€ Tl] y[{DF€TQrxnel1y1or1. Yr<ig1er dvc "brclogLorr.xd c:rine6o" rol :reqvci ano r1v cqpl nar 6lerro 6 (rn6 q pro rL€uQir rox xaL ol.6rl4Qo ro! r{6vo cln6 q! dLlt. (To ?ip60qr.r!niQ- ro 6d!o!t1€t p6vo yro va loriooqr€ 6fl o crQL0p6E ro\r 6ldordio€or1, fnoQsi!o €ivqL p€ldLo!- To €ri.7ir6o, 6nog fid\,ro, c.aor€},€iraL dr[d o],a xc 6!(r ou(lraror €ivai x.ioero o'6va o[oo€ao bLdvljoFa y.) To €oofterxo yrvoFevo F€roEutorl b t,rct rou y ei.vnt c4v1ttz6, br6t! o patilouv yrwitr peynl.weq rrrlv 90".€!6 ro €orDx€Qrzo ^/L1o[€1,o U.loEu rou .l/ xc{ orolc(o6{rox€ fir]lqE ioo ,4 €iv(rro€rL1to. :q y).ri)oo.. r(,D flv(ir,rDv, a!\6 on[cia€L oxt y6 < 0 z(rt y,4 > 0, nQaryfcrnou naarmdrv€L rlv €1,a1').orrLxri rr€Qinruort nou avoqrFonoqre.

tI H ,4l' = 6 i Aa, p1 cpwpLxrl ),rior1 ri urdQter v croro d,or€ v,4 > 0xor, ?, < 0.

| :.r.E.: orftr mr ].tsidrd ror:1. 8.?.

3.3 OEOPIA AYDMOY

rI.8.7. To b €rrq ro! x6vo!, Xoeil€rar on'dur6\, pE llneAerin€bo tdeero oro t.'fo'ko 10 9E(i)Q1pa rov 'tctx@Qtort,toi (oft€d€tunl\ou shor 0€[e d6eE yroq ldoq[dtlri Ottlovogia ncryro grc dr[66e€ll ro! r[dQari6Frowe oro (l)Qqio6161,{0 rorJ Gale ndvo otl o€oQla ro! yQa[$rr6v ol'iovofu,&cb! po!i6r!o!,

nAPAAEIfMA Edv o ,4 eLvcr o rouro, Ln6E nivcrcg, r6re o d)vog rou Eivo!ro 0errx6 reraqopogLo. Kti0e 6 o' ar.'r6 ro rerqgrlyoqro €ivqr fn oQrr1tLr{oEow6r]dopo! -lorv oq),6!:

fld x<iO€ 6 dEco qn6 to terqQrnFoQlo, ftQdnfl vd ro1d€! I €vql.l,sxrr,xri r€Qirltoon:

"',,=[3], "0"",=,fi].,1?]

.u, r=l_"],t6r€ .ro V = [0 1] 6h€! y,4 > 0, 6!1(Dg 3/D = -3.

E66 o rilovcq z (xd6eroE oro y) 6rqxraQiEel ro 6 cn6 T o 0err.x6 rercpqp6qr.o.To 0e6qqgc cur6 o6ryei oe pn ol.6rd"r1q1 oepri cn6 acpErorcg eval.l'cxn'i6E

tepwtttire6 (6eite mor cale). nedypdrr, 'iorolfyer lo nlm6rper" xcveig 6r", cn6&io olJ.nlodriox).€rErveg negmtcloeq, pic roul,ril,mor ;tq6ner vo d"1Oetler,.Ila nqQd6E!"fpo, 6vas ui6xoQog ,S ro! io oQ0o'y6!|,o o!pn)'{€[Dpd -iou S' 6€veivcr,6uvo-r6v nol ot 6io vo n€Q#xoljv oeirrd 6[0 ioud-'d: To eo(,r€Qr,],i6 rougyr,v6pevo 0o tltcv 0e.rr,x6, evtn oqeo.rdv',o 8rovuopato elouv p$evr,x6 eoorrpr,x6y!v6p€vo. An6 trp <i)"1,r1 pepui, 6E! €ivdr, ro! i6oo ol'youQo 6r! io ,9 ri ro ,9rr[Q6rler !o r[€Qri1ouv o€rlxa 6!o!uopoxo. To .9 pxfoQ€i !o €ivo! o c[or,<rg -rtov l'lcr. ro .9! o dfovcg rr::v y, orldr€ rleQ#Xow U6vo rs "IFro€rrxd" oLcvrioparc

|PAMMIKO' I1?Of?AMM TDMO: (AI OEAPIA NAjINIQN

[t o] zcL [o 1]. Au16 nor €tlor oErooq&ioro €iv(rr ro ofl t ooe€v€oi€at cruq€\'d,Iaxrrxi nepi;ttt;oq 6r"ro;E bolLeuer. 'H o S :rtpL€1er 0€ ri6 Srdvuopo rrl o 51 rieq€t€! Ipr0€rrx6 brdvuola g. 'Oraa or S xa S' dvoL opoolrixreg€uotsi,Eg xol e7(!7.d6ou, xore €iv.rr €rirol.o vo 6ouF 6rL I pio t] I it),Ll fip6jler vcelodplowar oro npr6ro rercqqp6pLo. Aut6 6poE 6€v io 6tr6rio Trolu xa0oQe o€reqroo6repeg 6r"crordoer"g.

Ev6totptpouoeg evo)'loxrr.nig ncpurrriroe6 eptpcvi.lowcr orov lpappLx6 npo-yeappclilol6, 6icv o[ neQ[oQlqloi. €ivsr avroci]oer.g ovri yLo eElorioeLS:

8J H,4c 2 bri dler l,toq c >0{ uridpx€r, 6rdvuol.rd y r6torc 6olE'UA>0, lb<0 ,U:0 ,

H 8J dn€ral €uxo],(l afi6 qv 8I, Xoloqrorord)lrdg r[E 6olerylx6E tt€to6lli6grur =,4;r - ll rou BerarQdrorv qa alLooloq oe eEioool:

I.4 -4l:: l =r,,. . 1 , ]

Ed! durrl 6€v €Isl triorl p€ l| > 0 rio! r, 2 0. r6re. rord ro 8I, 0o udQlsry

gIA - I l>10 01, er<0.AlJIt oxQ[6cbg I6!vo16rwa €ivqL n €vs].lqrm]oi rI< 8J. Auro ro ouFnepoopo€ivc! fior., obly€i o€ FLc

'[r] xqrco]i€uqorDdl cri66E!El" ro! 00(,)QlltloioE 6uio!o!.Yroole0rpcpe 6proE vc :reqrogr"ororipr oq leorperpki, ri(Od'€ittornaE cr].ye6Qltig).enropdpewg xar xpcrrdp€ oufip fil! rln6o1eol.

E.3.1

AIKI{TEIT

I]oro rilllL ro 6r'iro rorJ fnotLdvo! rao6].tttroioq: Eloxrmotrontore ro zr + rzr, lroro! ! n€alooropol l i r j >0.r r>0.2rr > 4, r t +3! t > 11! Besi rdq } ' !or l a t i ro lJro! reo6).rtparo5 a( ro! blikon ro! aar €ral,l0enote ot|io eldxroro roonrd! pe

IIoLo .i!oL ro 5!iko ro! Eftofdro! nqo6liFalos: Mqrorotrontort ro U2 !16 rolJ!ft€arca'opolq sr > 0, sz > 0, yr+v, < 3; AlmE ro xood,q xfiro inJiro ro! n0(i6).rud.

8.3.3 'Eoo 6n A tiror o tnuot,lci5 ?iilaroE (on6r€ m = n) xat ao stdvnopola 6r(rr c oilar p! oa\rtT|)|d. EEry1im€ lLoii r' = b €i'oL 66),flmo yLa ro trqo6l'ryo.).dtroroftoitots. 6eeit€ io s' ro! fteo6l.iFdroE uqrmoroillotE rar ttrd,I0€nme6rr oL bno rLpas €ivar ilLEs. Ioro Eivd ra ,:' tor 1/'. 6rov I na(irrl olvLm(ldo io! bsrvai aqqrrxl:

8.3 oraPlA AYi:l'tOY .193

t.3.4

E.3.5

Karcor€uir{'r! a!( noqd6{./tr( I .xi I yr( ro onoio ro ..1.r > /,. .. > 0 it€L zEvo.rfralo oivoio xar ro 6rii,.o rAo6ht[( ea!( trl ,fl]oltraao.

- l l 0

7a' u bno €(t'zr(i otvoi(r ..t.. > b, r > 0 z( rl < ..l/ > 0 v( ria( zEld.

E.3.6 Edr oia rd orotTrio rlra .,t./) z((. eivcr 0!rr,{d, 6€i r€ orr rooo ro crpi(tro. ooo r(rro blizo axolr trl

"tad .(frLard onvo),d.

8 . 3 . 7 A E i E T E o f l t o r ' = ( 1 , I l . 0 ) a ( r / = ( 1 , l . 0 , l ) s i l o r . q r t d m o c 0 j ( L n o n o ( o 1 o b r ( , 1 6rq66inra f€

f0 0 | 0 l f r l f | lr_10 | o 0 l , l r l l r l' '=lrrr, l"=l ' l ' - l ' l

L l 0 0 l l L l l L : r l

Kotonra, aqron rno).olioeI€ tc cr ruL Xl. r'Erytiote }dri riv(L 6L!Ir!'r(.

8.1.8 End.I0rndr. otl r.r brclnoporcr rI( nQolllonp€ll< ds't)tot( rx(vonoro!! i(- o1r|ln),l-qiralx6c olloixrc rd.aeoqraq (2) tar 6erirt $a ((!'oonzrt ollori"rt tclLirqoqrd(1(r ro eX'zo zc' ro dli"o xqo6i,{r(.

E.r .e rd 'or , ' - l l ] ,= [ ] ,1* " . | l u* ' " r , r 'nm(r ' "u , , ra ,rnoirtl)riorf n< oq'n).10{D!(rrr i< or,a{hir1r< Xd.aAootrsi 1"(0dx x(r nt! y6 =.r).

E.3.10 Edv or rearoALoFoi ro! (elLkoi rAoti).ryaro< rirar sEL(ri'd.rs crri l|( alro@ofrsaiaxorctuoiort rtt' r): 1,ti ro1'< tuqtoet4aiq.4r = 1, xor r > 0- rdtr.mo b!i,to, rl (rT{riqort ii > 0 noao}.rin€xcLr M€ytorolto lott ra }jb riri rotl<atatueL'tuti. !t,1<.. Arilrr ol rl trolixl.Eler (vroorlrft t/b.-.r €Eaaolo!,0ftv( ro.4n.'. f(lri r !,2 0 ir(! (wyx(ic otr! (1) cl,i..i 6rr .bri): o ao0&iq cl,ro<6r,iotrG p:roeri v( oioz).r[xDe.i oe niiar briiotro.

8.3.1I (d) Ei.rTtgtonoLrtm! ro r6oIq 5r | +3r, +.l.fi !tri, rd5 r€OLoQro|oi! :r I +r'r +.h > I,xl > 0. ,r, : 0, .,'1 > 0. Zoiors fll lt0obo simplcx.

(6) Iloro si!(( ro drir& ro!.{prrrou ul]volor:(r) Ilo(' fi!( ro 6riia6 reo6].)lu( a(r nor.i EivaL r ].!drt Tlru r:

t.3.12 Edv ro (atrz6 iX!( rrio trolabrri 6Fi1(otr llor.ri' rcL l1trc{iir}.olBs €iaqqd io...:rt?'lm, 7'arr ro.r' r,ei{'a!.' orilrdr,t ii,rr

8J.rl Ed'v ro {.d,,ro '.oo.ii.r .r = 3$ xar ro quonro6onn'ao .r = 2S. 6ilor(cr c\aiororz(6uo zcL Fia roa(it'o ra(orEiut< (€vd (n(rrorvrcr laoo€eK). dqcirt 1! o"rd'br|< r+t<q,( tr(xDrliwF aar ro aqyF€vo romoE ro! q.fmtro6orrreou.

8.3.14 Eav .4 = l; i ].

traLladtm rcr rd,ro r()! rrr aa\nnxdv d!r6!ospd,v ro,r olf,rJ!.

[oLo Eivd' ro rql,tri) or]!o].o.6rd! ro ir nrarrxurG do! r(;!o, i.t. b = (3,2)r Ed!ro I rivut exr6,c. i.x- b = (0. I), troro rivxoFa s eo (z(vonorsi flt! .v.r.).oatr"rt

T"AMMIKO' IPOTPAMMATEMO: KAI AEO?IA ITATINIAN

83,1s Mnoa€trs !a 6qeire on{ ra€E 6roordo€6 orvol,o iE! srovuotdror r{!1, o,ro{ov or6vos t{)v pI devtlrxdw ow6uaold'v yqrllE! or6'l1,mo ro\, I6eo: Tr l,sle,yroraos€eo 6ld1topoidi

E.3.16 Xe'Iolporio!{loi€ io 8H 1r(r vo 6eiE€r€ 611 6€v lJndafi€r Ldoi (6rdr! r('l(1Jer rt evdaa-nrxrl n.elrwool) ro! riao6l{poros

12 21-_111t4 4 l ' -

t1 l '

83.17 Xeryuoro',rioft ro 8I yla vd 6€tEsrs dfl 6sv lJ,lderE! pll aeqflrri l,1,on (6r6fl |,ags!q svaual{Iuol ,ieetntoot) ro! nQo6tripdros

f1 3 -51 1211 1 - 4 - 7 1 * -

t 3 l

8.3.18 AslBt€ dn o! evatAaxru{4 n€eu!.do€K oilv 8J (,4o > b, o > 0, y.4 > 0, gD < 0,y < 0) 6€v efvd! &rvoi&/ ld ralto.Uy ft.0:'6:/:eo\o.. yt66EtEttt UAs.

IJ.1 }IONTEAA AIKTYON

8.4 T MONTEAA AIKTYAN

MEprzd tpq$rx(i nAolAd;rpc-rcr 61ouv boprl :ror e:rLrq6:rtr ro),r,1p11op1 e;ri.),ror1.Ki{-ft l:rvl:r,o./o i.oXU€ mrq ypGIUrx€g eElocboeq "/r.r rorvLoFrb€i! r(t.,'tlr.i€q. Orc! rc[q plbcrr"zir ororTei.cr r]rcrv or,lxuvrpr'4rivcr rovrir orl 6r.rt(irvLo. rore I .4r =,l,uvor(v Fnltoic. Xro i,p(plLrzti:rpolpuppcrrro|o. orov onoio o:riv(zuc;1 rir(rnreu),l,r ol,crtrtrog. €lSrof0aoFcors lrcr pLu rrbLzrl za-qlopkr :rpo6i4prir{]Jr.'flnali au.\ ftaoqltipstd 6'xrn(,n. : alll(i o -4 rirur i.rtLs ttiraza; zpotmtitoxos,r(l ororX€i( ro! rilcr 1 I +1 I (ra n€0rooor€a() p16ir xrrr" rcr 61por.1 obrlnorFn€QriG$6(ir','ouv lovo naoo96orLq x.rL ((tlrrQ6our<. frrl rirorori< ;rirczn< p:roporirrrt ).r0oiv no),rr gelcrlrrrpc :rpo6irjpttrr cn 6.rr or.rviOoq.

Kcxd €lJIrXri ouFrr(|)ol. 6izr!( urirl(odexovl(l o o),cr rcr ei.b1 row trl,rrp-t r o t i ; ' v H r t . , r , i o v n ' v | , , v u f , , r , , ! | , d r , , z r v ' t , . " v r z , r v o r o r r i - o r r o p oa€rrlLcrrog rou KirchholT: oro1]r z6tL6ouq b€v 64[orplorlrur outt zctcrorpdqovrcLGLJroxivtrft. O neolQ(4rtr(x(olro< brxtnruv :rcprllcrl€ ol]orriprrrrr (lr,r1rirr aeqi.oLJ.rurQ€i(i.ori l.cr v€oou ro! r,i.vor r(rd €)iorolLlltlQru boi.7,cor(l qfhllorfp( (1rorttria( ro! 0(r ozebti((ovrcv 6|r1lo(hlrr{.i. X(')ei( Tll d€ltlororoilol. Ai.vr.L rni-a\l r.o nQ66L\Fa ror ytipntr -n6t v(l lFlLoronorl0fi o (Q!0fro< tdtr('!. 6r(vor vrit-tF( 6zow brr(iol( (iQvlols. M;ropsi au6 v( trl! €i\ror ro nQcrlflftrrr6ne6ti),illl(. eiv(! ollrJ)< fiOod).Ttu xor ),ri1,!r o npoyp(B|(rLot(o< brrrlro)!.

Iro 11. 8.8 q(i\,!ror a\,(1 povriio brr-nior. To rpo6).I[o lizei ri.!(rl v(ltrryroionorl0L'i I Ao4. (1'ro rn! rlyri (z6ltjoc l) orlv (roAQol (xop(tog r,).OI neQloQrotroi ti'rur or ygp4rndr4rtq rot uxytir. OL poiq b€v trrroQsi !.(!r3p6onv rI< X.r(xtr!11orrtr€( xcL or zarll0rlou.E rtov SeLdrr 6cr r.:rrrp!:rerar. ru(wortnq,onl. lo !(11161,() (urd p;ropti al1 !xLl.!0ri /opiq rl Xaidrl ot:{oai.(1(.nor.l L'i.v(! | lr'yrdq poi (1'ro r aaLmcall naoc -r( 5tlLdr

) nuo ro rpcid),1pcr or hy'vorotoL eivoL oL poig .r:;; c;r6 rov x6[do i orol'aott6o .i. Or ;rtpLopLopoi Ttopll1rzorrFcrq eivcL .r';, I rit. Or t'or1q rilcL trtu \ ) v 4 r l z l : r o q r n o o r " . v r , { . r , r L r u , } . r t r r d r 4 r t I t ' c t r t n t t * , ; , ' r , , . : r r r u r 1 , , 4Qori ror| trropon$r r(1 ro[Br 6( fnrorpiq]Er xcrci B)i'ro< rI! 8r.lrrxoltavrF yaqr-

2 " . ; q ,

I

5

3 .

6

6

1 ' .5 . .

IX. E.6. Mo!ri].o rL. jrt/i, aroaqori. xoeryrzonF€s roL 31dtoF1i, n0o6i r( (6tLoq( qori.

fPAMMII(O' I1?Of PAMMATEMO: KAI EEOPIA IIAIf NION

p{<. Melorororti-wto< tq porl utlotqogfg 161, FelLmonor.oqre acr. rlv o}.Lxr1 potioqv onoQQori.

Ylrip,Xer xar rill,oE irrproelolos ,rou ,re6ner vc ld6ouF€ un61,rl. Aur6s €ivo!o "v6FoE 6tofltQrpn6", ro]d ),6tr o1r q traEex6rcv4 poi oe xdp6o, noirat rc qv.lEqolEtn. ITe6ttttret lrcr xov Kovoao P€{$oroE roo Kirchhofi

I . , , - I r ,u=0 y lc j = 1 .2 , . . . ,6 . ( t )

Or €to€QD(6F€\,€E Qo6S mor x6p6o j €irct r,j xrrt rpo6plowar cu'6lor,5 rougr6p6ou( L Or elegldpeveg podg civcL :r;1. xft xqr€u0ivol'rar nqoE 61014 roq,i6[6ouE l:. H €ElooaQ6nqol r+ €Eto{DolE (1) FnoQei vc Taaq€i cnyl Foeqri'4ri = 0, 6ftoll A dtuu lvaE rtuax E npoamrintuq xdpjav.axytitv p,e gLcyQr4qrrl yl(l xdo€ xdF6o ror liu oniLn yta rdor ff4rn:

I 1-1- l I

- t 1- t I

- l

J t3 i'2.t 'It5 ;r1l it15 .|:.J6 ,l]i6

l t- l I I

I23456

- l

I

O v6po( xo! Kirchhoff Ar = 0 €ivcL €va ousrqp(t 6 frii 9 no! 6X€[ oilouQq luo€lq.Zlrdut url npvltrxac Luolrc nou 6n' Ernrpvor:v rq loplrtxorrlrr<.

EqrrriEolr(|! 16 XoQIFlx6qreE rolj €to6Qxow(r[ rnqv crnopporl. $.6:roupe6r! I Qorl 6rv Fnoad vrr E€n€Qd.o€r xo 6 + I = 7. Eivar rou1o €qrl(loi Mra Qoi2 Bovd6olv F roQei !' .rxolouorto€r rt 6LobeoFi 1-2-4-6-1. Mro ao{ 3 povdbov[roQEi 1'oxololjeiot! rylv ]-3-4-6-1. M|,0 riQ6o0€ryt porl 1 povd6oq proqet v'ctrololj0rlo€r rlv rordrsAq 6!(6Aolui 1-3-5-6-l. To ouvolo sl"tdr 6 xol 6Ey tiycrhlrl'eri !Ey41{fte4 Qoi. fl(i)E FnoQou!€ v'ano6EiEouF€ 61! eedocqr€ ro pdyrotoi

H F60o6oq T)tE 6() tuiE EivoL nrrmrri. d,ld rc F(€yucrrlzdr 6iv(rt c6KFqr-o6iqtcl: To d.eL6f eivar vrl xaraolrsuiroouFc pLc 6raropri ror 6rr:tuou, r.ord$yPioq xlE onoiaE 6L0q or 6u\raprx6rqi€E axou! eE(rl'ri.t0ei. H 6rnroFrj xorpileLrotK x6F6orq 4 N<\r 6 o1r' roq UnoLomoq. Or. rpt:15 rupdg fiou &croxiEow fll6r.<rtopti npoE rcr prpo6 €1ow o),rrt 6u1'oFLr6qx(. 2 + 3 + 7 = 6 xaL xi,lrortl€Qtoo6ftpo 6tv 1trrop€i. vo n€Q.roe!. O tro0tvis 6uio1r6E unaloprrier" 6r zri0e6LrlroFi 6[v€! E"vo gQdyt(I r4g oilrrri( aoiE xar o nl,iallE 8uioFog unayoa€ier 6Itrl FrxQ6r€Qrt X(DQlrrz6rita (4 .i7rm4 6ntopri\ gnopxL va npuypororrort0si p€Fra Eqrlrrn QoI.

EK Su;lpryld 4e7long eo1fu-dn*oqt 6.e.oqiE. H 1ti:ywtl gf a' ino.birruo r,oornal ps rl x(}aqir,).6qrd irE €fuitroqE 6[aro]rrE.

3,,] NIONTEAA AIKTY!]N

, , , : - =s i ^ . I

, ' '

'

. ,

, . 'T ' - ' . ,

t " t

I

I

. . 1 , ! -

i l\ \ l - : i ' . r

. l

t - - - - _ l

a . a -I

U.8.9. Aiarro tr( io nQ661,ry4 rov ltipun.

Mr),{irvr( (lrorrloir. tkr axQ€n€ ''( ),atr€ Jrrcr r}.<ixroq 6rtrropl' aq,ou briqoQ€srolLdr Fnopolv

'\,c d7ow ry ibL<r 7,opqnz6npc. Ar(r-rofLn ei.vrr! 6vct< 6lcXoQropoqfl)v,{6p(io),'oo bl]o opdbe< 5 zcrL ?. pr n1v mlyrl oll'' 5 x(1! rry afioQQoil oril!?'. H Xrrprtrlzorlrd q< rirar ro ri{)poropa riior tov T,top4ttzottlttov tt'rv axpt;tvnor rtivt u:rci ro S rno T.

Atr66tl4 rou ritt piyrmt Qoi = tlti4torn trcropri. Iilorpa 1 pori 6ev p;ropetvu tivuL pci,cri,rrrtprl cn6 rn 7,nrQrltlzornra ororoo6lrort: ropti;. r4g t)"it1Lott1E6|(lrotLri: outLneQrl.4r6(vo[ir,t:. To broxolorrpo ne66l"tpc. ebri) o.f i,it v.a a'oi,611LlQo ro,'6r'i.'[6. ci,'lrr','a bEiEorill€ orr I roorlr(r 0fiop€t !(r cfilrtrrloci.

Y:ro0i:Lrru 611 1 porl eivor. B6yLorI- KotrdEtt 6lor( ro!( xtiu6oltg rno\oiroi.ou; UnoQai vcr qriroer acrvrig cr:r6 rTv rntyi [e tiQ6o0arl aoii. y.tDpi< v.rie:ropvir q 1op1rLx6r1rcr zcpLt'rE (xp1g. or rotr6or (lrroi., x.lgtiE 1tcL I Jntyli.:reprixou(r olrt,'oFd6.1 S. H &ropport :rp6:rtr vtl:reprdxer<rL crn! lli.Lt o[i.b.lI (6r.(ioarrlrd 0.r lnopotloE vc be7oei nrproo6repl Qo]il). Khos ar[{ rou5rcroXilEr ril bratoprl rpdrn rc rEcvr),r:iroL, 6LcqoqerLxc! 6cr p:roporoe vcr:rcrpcroTe6ei:repLoodreprl porl:rpoq 6',c1, zog6o mo T. 'Eror

I port €EcrvrL€i rrty,rprlrrr6rrlrcr rtq roprtE xor I ro6rlrc €Zer F,rtr€q0ci.

AIJTo ro rrtreuraio unobeLxvtrer ivcrr rpo:ro ,{orcror€rni< tI( Fd.lLorrE porjq:EEEralol€ (! llndQxsr SLtr6qo[rt p: crlploL[oroii]rl XoQlrn6rnr.r r.r raoou€m€

Qon ,i(r(i [itzoq nnr{q t4q 'erauldvouoag 6Lo6ao}ri('. K(ios 6rlF.1 uro}.oyil€trrE (no[ivorJo€! Xr'lalrrzortre_< x{r croqcroiler or n cnoppoti drozojrr€raL ffior1v nry1 rl av e:rLtp€rerar fla6o0€rr aort. H o).4 SLtrbrr(oic opl,cvtivrtcr ge rovtiy6pt9!o etttypaq,iE. o oroioE €n!./Qdqr€L rri0€ r6p6o tor 5 ge tov n6p6o an6

TPAMMIKO: lltOr?AMIlrlTDMOt KAI {:THOPIA IAlri-IAN

rov onoio nAofl.o€ Aoi r.crL €irLrean€r qv c(aabaoFLzl qleua€ot rIE 6'.cbAoFri<qq €rLnl€ov aloris.

To :1. 8.9o 6ei71er ivc 6€ut€Qo 6izrro z(! BLcl €qtrq ao{. O),€E o!X0)plrrx6rlr€E (an6 tleror€ed npo:6€EL(i) €i\'( io€g Ir€ 1. Maopei va gair*trL()1[ llolo or 6uo fi(i!(r z6U60! xor ol 6rr() xdro x6p6o! prloQou! v( rlQoofi€lco0ouv(tjro fieQtooor€Ql Qorl c!rr6 opo( 6ev d.Io€uerl H erurl,aov Sori F roQ€i va fid!:[xQo-< ra i[oo n$4 lixog pmC axp{E 1(r €ror va arGLQ*or! Fra ]JndQlolroc Qorl-€{d 6oov (!!rrt qidv€r r€Iud oxltv (noaaot (:1. 13.96). Oruv naoox€o€i auni qpotj. q owo),rzri yivetcr 3.:rou eivcr pt1Lor1. ) cn'ro ro oplpcr dleL o1e6r(meizru I rl,ixLor4 bturop{ :rou ripvtr 3 :r),lprrc urpi:.

To ne66lqFo t{ v ydF(',v

'Eoiro o( 6lou$e r6oo€a( ilvba€E x(I| 160('10r< yrl\'cir€E. M8arr(i orolcl 6r,x({:qr lsryilpro €i!o! ou!6r6(oro. r:vciJ a(l.Lo 6l)oftX{i}S dxr. n6r€ €i\rarbuvsr6v r( 6aou!€ €!a nLnQ€E r(iQL(ro[c, dmc 6].0r v.r n(vxQsrrouv: Aqolr llloutrtrxrt irLyr6a( lmoeEi vo 6oul.iq'€r o€ 20-6rdor(ro!q XdlQorq, oiyoueo turoQ€ivlr x(rrr?fl.rorFi rar F3 ro rrr0rlrlrfvo:rQobtrnru rlrv Y(rtrc,!.

Yn(iQ,Xo\,l' b1'o rQonor v( n(Qouo(lor€i 10 nQo6),qF( -Ir ii'\,uz€6 ri F€le(qtiF(r.r. O ,"riv(z< n€Qral€r F{6evrxri acL povci6e<: .i, = 0 6rsv rl 'ywai'/.(

i r(1r o hvr0q j €ivcr souF6i6(rolo! ){(! ai; = I eirv oi}.ouv v( ;rpoo:rc0rioouv.'Eror

11pcpp1 i biver 16 r:rd,oyig tq; yuvcrixc; I ztr 1 orrl),r1.1 crwtororxei orovd1,6Q(.j. E\,( noQi(bE[.4lo ei.vut o

l l 0 0 0 ll r r r olo 0 0 lL 'L 0 0 0 1 l

To (l\,Tiolorlo yQaq+rt( neQralt-l(r oro:X. 8.9- Ayvo.;\,rs( rrl\, tnlyri z(t In,oroppo4. ur{qcv o'cruro doorQrs x6F6or aarol€ad }.ft r6oo€QrS 6€Erd. Oraxftq ytrciEri ror< evrtmotxolt' ort. povri\ts rov nitaxa. Ld ur<ipxer czp{FEroqn Tll( rredfllg yuvrrix(tE xc! rori rarc(Qrou d!6QC }i(ll oro! niv(lxc au = 0.

'Er'(l r),yla€q raieLaola (€q]'6dov filaL brvar6!) €fvar olc olll,otro reoodQ(,)!povcitiov otov :rivcfia. Aqoi 11 6ry(lpi(1 urcrlopeuercr, 0(l nQoaalovrar o16ldoo€elq b!(leoQ€ttxd! orriLeg xcrL tdooepLg 6LcaoperLx€q yp.rFFd5. Au6 pordrlrLtr€ rlv clcr!,ritrlol ev6E nivarn prd0roqg rrldrproc ortr FI plbEvtrl(i cnol1€ia rouA. :ro ./Qd(gtltr(, 0o €ixall€ rot€ r6oo€Qr,< or.p€g xolaiE zorvorrf z6p6oug. EirvEalulonooEtorioq€ qv nlyi zar tqv (.troQQori ztrt 6drlnpe povc6rcic pori zcrriprizoE zdoe c.r[i,( r(lraldoloroE. I oi.lrn Aoi €a ircv 4. H tl€ytoq ao{ €ivcrFrrQ6rsQll sn6 4, r6re aL Qr6dlq 6rc.v ro fil,rlQ€E rciercoucr €ilaL (6!'\rarov.

H BqloII Qort o,ro riaQd6r:lyp( rli(l! 3. Or ydtroL 1'1.2-2.4-4 €ilar buvoxoirxauuc zur exrrd d).}.d ouvo).n ydFcovr. drv urriqXrr ogoE tqonoc vc tpraoo4rr

S,,] IIONTEAA \I( TYON

rouE raoo€er\-. A< €EErdooupr rlv tlr,ricr y[' (ur6 F €lcvra6no no! 6trL rq cAlroyi]o onororbrlnole trila}|( $E 0. 1l

:ro yadqrr.r. to nao6i+ra 'ir(a 'l rL(i/rorr hcrotrit. Arrrt xoai!€r rLq 6noyr\'(izr-< zdro) (Qrol.Qri u;r6 tor5 tgrr; (i!60€< jr(ivr') brltci. Itt;6tio pvr;izrg&:roFiv€r a( 6rtd,6Eour, 6rnv govor (irbQ( rou i€\' €n(lQz€f. OL tpei,( dry6gsgilor\, 61J.) [6r!) y11\,(1ixE( (rq 6ft) ,"rQ(ir€(). H €Lxov( r'ia(r nro }ico(rQr] orov

(l ) O! yaotrBdq 3 n(tr 4 €tvcrt pt 1r16ti!rzdq p6vo otl miLl ,1(2) Or oni) ,€< 1.2. 3 ! i11( Ft t ( I ( t€vLtdf t r6vo mG./Q(fq(!< I z(L:.

'Ottrv uttrig,Ttt ita utroatvoAo k yttvatxti'v, cttE or.oitg ortyd.txd apioouv).tytirtQot oJt6 k tiv6QtE, tirE tirat a ivaro i|a fiAieti raiQtnopa. Aq bri[ouprorL (rJro fia(r ro (noq,(or(Jrrzo xgrriQro 1,m nil(xcq orroLoubiinorF tr0y€Oou<.

To or4ln6p(oll( [noQr:i vcr €zfa(o0ri pt :ro),Lol< bLctpoptrtxoriq rponoug:

( l )

( l )

(3 )

ltro oxtixt Eir( trbrrcrtov rrr tiriLtrr rioorpt<;rrpyouq rrr ttrp(iynrv( pr:porirbtg. tror rirorr zuvdvcr< :r!g1oq v(l tolv zttril8t ,{(!ivuv d),io nioyo.'(ttot, bt'iotui) OL lo1(ibsS oroa :ria(z(r i(piicivovrar ffr6 LydreQr( (norfoo€ar< yQ4rtris. O r).(irzloros cQL0F6i kr).uxro1'odJ1' laqrFi;rl (oarlolri(Dlri zo0irr0v) r'i1(fl roFr( z(r CluroE roour(l| trr rov FayLoro (0(\ro r('^,ldB(Dv.llrl fQqttuxi (0.),€6Qal KA$ ni!c?{r< trr ro ibru plbrruri. on('Ji rou ;1

rLc $ovtibtq rrnopoil v cvrrzclrcorrioorv oroLcl6rirmr oroqric- eivctr6rotroQq,o(. H oAiqolo(i ror f-iv(r trtbiv.

Outrl{)€ii€ 6rr I ooitol,oo tirr( ircr ri0potr4ur 4l = 24 6E,rv. K(io€ 6ao(Zeno4rojror€i raoorar< yo(lFtra< zor orrti,€s. Aoy!) to)! IrlbrvLz(;! rou.1. z(1r or2.1 ripoL eilrn grfrr zat ot!rn(;( I oAiiollo( tiwL ltbi\'.

To :r},rlprq r(ipL(rop(. o ullrd ro:rtrpri6fln(. ro t-ur6brl0 fr( lnl.oH ffi6lr lbe\ ' ! r( i . O unoniv(r){cE rov lpc4rgt i -rv 3,,1 x(r l roJv oqld)r 1.2.3 alccrrrori\rco1.rE 2 rfii 3- (iror€l.Eir.rr B6vo (no Frl6Evr.rd- O yr'l(zoE '1a!o!qlrrr dv<r ;rivcucr rr o:ri rr eiva ro 6rr dva 2 Eiti tt 1u ,ox ltyttvuht eptto|i(crro rsiQrcopd, uf i)oor p+tt > /r. H Aoxloil 4.3.U €izr alc 3 e:ri 3 p:r),oa}lrtbevrzd)1 [E ,l = 5. ]rou cvay"(-lE rt! oaiqouo.r v( Ei!.11 Ft66!. :io nQo6].+r(rrD! ydrtltDr. or rafl!./u!dzEE otl F rolDiosv va navrafUroriv tr61o rol! 5no'r:rotrivorrr: tirnor:. ll|N p 7r\uizr: F.r,'aou! \'( nLrlrpfr"r,,ri! ;ro\'o n qrir'ftptq zcL p > n ./ (nor €ivoL ro iSro tr€ ir(l F,'riox tlttb€a|zrirv $e p+q > n)rore 6v(l:r),ipeq r.lipr(qrc rivnr cb!v(ro.

To [.r0ryulrz6 aQo6lqpo €i.,'LrL v' uno6EiEouu€ ro avriolpo$o. Oc 6€fEo!'IrorL ora! xo\,6!c Ni).ox prl6c\,Lx(D! 6€v €i,vor noLr Uey.i).o, 16rr n oei.Eoro.l [roari

r :.r.E.: E\ao.ir(r orL n ri\(zua (\armaL7ri d. dz(zLia( !E rc mor/ri( u lr (\ruor/oN

rPAI'lMI(O' NIO|PAMMATIIMOT KAI OEOPI I1 IrNT!)N

vtt €ivcr FrMielwi. Ftir G ,,ti0E oivoX.o p Wvawbv aqioouv torMpotovp dvieeE, r6t Eiru 6aral6v ias. xTlgeg aignopt. Awi ei,rrut" l ouveix1rcD Hall * eqcpF6lerar oe otivo)'a onorouairoft F6-y60otE: oe xde€ Twaixa7rQ6n€L v' cQ6o€! 6vaE roorldlloiov dv6Qoq, o€ r.ido€ 613o ywoircq rpareL v'aQ€oouv 6{0 iolldxrolov i1v6Q€S xc oiro xoe' €E{E, tl€lQ! nou p = n.

tL 'Evcr nLriQ€E ioiQr,oopa slvdr 6uvdi6, i6ie r.c! F6vo1/ i6r€, 6icvLxavoroteiror, 1 owo{xl iou Hall.

:iyouQ.l n olJvorixl iou Hall €iv(.[ cvclxc{io: 'Evc rci.Qr.(.o}rs €ivcr c6wsxo,orcv oe p lwcixtg trqdoouv hydr€Qo! qr6 p dv6qeg. To 6uoeo).6rego irq66l.4rneivcr vc 6ge0ei 6vc roipr.oopc 11 v' aro6eLl0ei 6-rL utdgler, 6vo- 6r<rv Loltiet 11olvOllr(rl lou Hall. Mra oQlouiq riaood.tyr"cnl lplolpo7io!€i [o0tllroflrili €noyo"ylioE nQo< n, loQiq vo x(rlooli€udE& r[Qd'/[(rrr 1o l(liQruolc. Ep€lE fiQorrpdp€ vc{rlrrdoolpt ro avrimorlo npo6L4rc bLxruou xor q pilmq gorl ror.

H or66€rEq €iva! onloior€Qrl 6rcv orq or4r6E, nou 6Qio1o1,r( oro F6oo!.or X(l)Qrl r.6fiFss noo crl'troroqo6v oro I rou nil/(lxc1 €[!u| r awi yro 1. Ort(j)Qlrrx6flF.( ro\, o4r6v nolj €E6elovtor (r't6 qv rq,Yrl qQrorrQd €ivc1r 1 x(rroUIiE nou €LotQlovrcu orrp onoQQ{xi €i.v(.! nd.},r 1 (:X. 8.10). Edv I FEyrolIQoyi €ivcr n, or oxpig oL clrprydpeveE an' rylv rllTn rar €ro€QfidpEv€E oqv(lnoQQoli €i!r{r" n},rlpeq -raL 11 potl nepvdr an6 to dvct drkQo oro drl,}'o, rpciparou x(r0oQilr:r ?r lapou(. OTov ri'\,&r cbrllcro €!a nLllQ€q ioi,Qr(ropa, r6rE Ip6yrorrl po{ elvcr pLap6rEQq snd r xcL ur€xowl./f qord nQdreL l'(r €ivqr xdrolq

AurI I oraroFri 6q €xEr )(oQnr0xdxnr( FrlrQor€QI ro! ,r. Yfio{J6fie olt orz 6 p 6 o L I I ' 1 . . . . . l l r o r n c r Q ! f i € Q , c r x < r l l \ - l r . - . . r i l ' o r s 6 e E L d , e t v c r o l o o w o L o SqE nqyriE.* H XoqlrLl{oriFtr zcrri Frlxos rfi 6rnotriE eivar n p nr6 qv :tr1linQog r|E anop€vouo€E llv(rir{sE x(1r r tlr' lor.E dv6Q€E crvtouE npos qv anoQQori,En€Lbi I t{,)QrFr}i6rrtrd qS bLorotuia €i1/sr tttrQor€Qn roo ,r, rol4r|a szF{ 6€vtlitoQ€i rrr dlao1io€r trt 6rarop{, nr6 16 lll1,... ,Itr,, ororg onoFtvoweg d.v6qeE.'tG p yuaixFE ae6oovt, ptivo r debQEq.'Op0X I XoQlru.o,r$(r r p+r €ivo!grxQor€QYl ro1J rr orov p > f.

'Orov I ldyrorl Qori eival [wQ&€al rou n. r(l rlcrQc.ndvo 6el;lvouv otr. q

ouvo{xrl rou Hall 6€1, roxi€!. Orav rl owO{rq rou Hall roxu€r. i6r€ o ar!y6Q!0FoEE rryQsgriE rou 6Qifi€r rI F€:yloq Qoi, 0a 6Qiorc! rc1ur6)(Qovo o' cl,Id roeLdrxo iiixruo- xar 6vo r[rlrlQ€E r(1iQlaola. >€ xir€€ r€Qinroorl 0o fuiox€! ,ro

Fiyloro 6uv(rr6 ioiQr(roFo.

'r r.r E M. ft yd,r+,elc lI,' xar ll.I rrorooirat. avticorxo, luv(ixqr xq, d!5aEq

8.] MONTEAA AIKT\AN i01

Euv66o1ta 66v6Ao ,'(,r o olT6AroFoE onl,IariaE

'Ev( 0qrr).K;r6€E t,ovriio 6rxrnou €ivuL ctrt6 rou nao6)niBoros €I/rxrorrgbLabpoFriE. oro onoi.o or Gz[6< clIi 1(r /(lr0rlrrzorlr€<.6Xour pilxr]. zu( hrdl€rI orNrotrorrprt 6|,abao$il cn6 rlv nl./rt nAo! rI1, cnoQQort. E(il or ox$dq€ivar rl).EqcrvLx6q ypctrt{6< xar rc trtzrt €ilaL Xpovor za0]JordpryJl]<. 1lJd7,\,o!tr!l-q orNrotror€Al yacl$Ft yLa ivc qLcqdvrya. Eciv or r{otr6or ci.vcr. 1jno}.o./ror6_c.i:1or[r ircr i\i.xrro rrrolo.flordx, 6n(,]q ro ARPANET. norr ],ivrr" brcrpxdq ro;rp66ltpu rlirxror4g brcbpoprjq zur tri{)"Lora o7eb6v azcrpraic.

Yaciplcr ivtr nol. o!././€a6( fipo6l,r1pr(, lorpiq rL( tvaoLeq rilq rrlytlq rc(rI: anoaoorlq. Anro ct(rilrd Io tTtixtolo otv,by 6ir64o -Eil'cl 6!c or','oLorr 1 cr4tbv ror ;repLdT.xL 6.or< tot:< xtilt6or< rou 6rzruor. A!Ii. v( iral,iorrtr€lplyopct ro lLtrorq4cr pcrcrEr bro z6p6orv. 0aloljlLc rdoa va c),.1-rr(i)oorit{r rori)oroi o!\,btsollg ol,ruv ro;v x61t6ruv. Atl uitpTorv rt;xlor. br6( ro x6oroqxLFl('4t(rro< oroq xur),ou brv €i\r([ (fi(Qcrirrlro -,ro g6vo iro! (qra(tlr_ €i!c(rk(inoro; roono: va $cr(&rivollur (1'ro orolov6rTror!: xof6o o8 onorovbiliroredi,Lot. 'Eva ow'Jiot btt,<\po at tutuuqx4t6or< xaakw o'/,1ttntErt tiJ).or;,xc(! Elrc4r€ ro rLcrlrolo bwcrr6. lbo! 6r(E iTro(r,oE cll.y6Qr0froEl

(l) EExti\iott e1t6 orotol,'rinort xdt()o s reL t tw).i6ftt ro rl/1q 6ite:naooAlorr rrl oltrrottdr€Q)t a4oi :tou ort'6ittro rQiT.or dh,6po y h,at,io xtiy1o.

:ro :I. 8.1I ra [r]r.rl rora c,tfd)v 0o €pqr(lvilovrLlv BE rl orrad 1. 2. ?, 1. 3, 6. TorE).€tt(io 6riBC rln0pnJ16ri trry arptl FrlzoL\ 5. ror zLt:irtL ror zrrzl.o. To ol.rx6Irit{oq €i1&r 23 Di!c! op(,( sL.ixroroi A€Zorl,a(4r€ ro qlnpo pn)aou; 7 ;roL6.vt,rpiq.€vd o b€urEpoi rr.l,oAluuor ro .1{)i\,fr oro r€},or.

l i r . . ] :

t , r r

I . l L - ' . , ' r i , , , , - -' S t

r' a : : _

l " ! t l 1

- I| j i \ !,1i.

!I. E.10. Xoettrzoqra &(otoi. [Laeor€el lo! n. To rirtq€( tdiqtdotto .irdt (birloto.

i02 TPAMMIKO: TIPOTPAMI\IATDI\'O: KAI OEOPIA NAIfNION

2

:x.8.11. Aixruo rar otlllaol 6iv6qo tril{olE 23.

,:,,r.j.

Q) ndaft azplE pr ar[atdyta !ix1, aitoopittronaq axpk itou tdvout'

TdQd or dx$ag €pqc{vilowoL pe il o€!Qa( 1. 2, 3, 4. 6 (crnooQirrrolrcrs ndllro 5) v.tn 1. Aur(:q €i\raL or. ibleq pe tq npolloupevrg -risp' 6Lo no! a1]lo6€v oup6ci.v€L ndwore. To ouvol.rx6 pr]xo< eivcr ro t6r.o -xut tttn6 otp6ctivtrricrltot€. To nOo6),IFcr lou ouv66ovrog 6€\,6pou, nou LuvercL pe plc xivqol, riv<u.eEoiQeol.

:rI y),6oo( tou yQclFpuiolj nQoyQappcrilolou, 6pioxor4re nprixtr q 6d].rroTl1l(oQuqrll To povr6Lo trlE B61torIE Ootg {1ov o/rlouor€po cfi6 rc rrsQroo6i€O(rlqt4rpLrci rqolq<ipporcr. Kc[ o's!r6 opc,rg lerr,vr1oc4re ano [io Qoll. fllv o?roi(l6c0prcic 6e),rrdroape. H pd0o6og snnpiex treLrorp.lo]Joe. u).tr<i bev ritcr cvciprl\rcr rl\r €fi!orl[d\ro![6 o A rlla! o ]rt\r(rr(E npoofiTd)o€r}; xar, 6ou].€!a$€ cr]r'noefcq g' c1trI6v. TrbQ.r nill,L ro r[e66].l]lro rou o!\,66ovroE 66v6Qor ),i1,eIftlpu xdrL :rou pordlrL pe cvc6qo4xrt alrLaardordort, XtoQiS bozL1laorrxd 6ipara.>X€66r xdOe Ba0o6o<, rou rivcL 6rlrr.oq oe xrioe 6r1pa. pa5 oblyei o' drcr(uiot6trtopu. H ejr6p€vl luvur{ 6rc6rraoia LryrtuL a},ydAfipoE ditl1oaiat

(3') Karur,-ExAorr [t'baa not tn:ttpr.tTou' xet rou< t1 y.d!6orq n.flutu!.6ttuonaq ro elijs 6ipa: zn^tExE o,toto,i'roft dl6Qo *at naoaqlnt rttyuxpri J.uyiorot pijxouq ml t[iprrlL an' au ro 6h,6po.

Tc 6tipcrc eEaprd;vrar c:16 q oeed urd.oy{q tcDr biv6pov. To \ra naeopdloutr€oro i6rc 66!60<l.6iv€r rov ol.y6ar0po (1). To vc:rnipvorpr culovrc grpl eivaro 0l./6010g0; (2). To vc elt:rrtloqre 6l.a ra 6(:v6-oo oq oeecl eivcr. ivcq vioqoly6ar0$o5 (a). Arorry€x(r €!,{oLo, (llLd oe €!(r l€ydl.o rp66].rpn 11 6opr1 rovb€6o[6vo.v yiv€rar xQioqrl. M€ lf].ror{ rloF6o!,<, gropei vcr urdp1€r o1e66v6va €,rarou[teLo al'|F6v rcL 6e'\r 0( 0€],or€ vo brorQdl€rE dvcr r6roro zornLologepLx6g 1Ltrr.hbeg qop6E.

Alrir outnl.Ia6voriv rlv €roaytDyri [aq o€ Fovr6].o 6|,]1r!rDv. Yriraxox!tE ioor etlzotrcr ncr, ev6rcrp€povrc rpo6).{pcrc nor olerilovrcL [€ rpo6hiparorarpLdoUaro!:

S] :IIONTEAA IKTIO\

L 'Io itetiqLqpa t49 6 ,t,'lr.tE anroroi|rolg. Eor(.] 6-[r a,r U€rQd rt! t{noboorlrou rrrol,ryrro! ? orl! €AYooio .7. Avt(fiorxior€ rAyaoi€-< 6roL do-r€ v.ru€Ioronorlo€i€ rtr o7"'nri anoSool. tutic6t ro ireporo[( ro,v ,;; o' 6].esrL< rptctoi.€q. (Ed1, oic ro oir €i.var 0 rl l. crro orprinr€r tr€ ro nA66l"Tra

2. 'lo t A6 qpa ov pttaqopbv Ai.bo.l.ror ro rri0c z6orog C,; trrLq (xpfq. ortruaclyr')y€q o€ n oru€i( xrlL or lryioel< oo rr cyop(<. Arcr),6!rc crnootoi.tg.r,r (lJro xoaclrDyon< ot: rcrorcr),cureg :ror e),(l1l oro;ror"olv -ro o),r.zo zooro:lC;;:r;;;. (Edr 6).eq or n(ecy(Dytc lrcL or. !1r1oer,q riv(lr I, aur6 orJurim€r,F€ ro nQo6ll[.r 6ilflmq( (litoroitrol<. ora]..vorr(g 6vu hropo or zti0r

3. Pott d.al ionu xdorour T,: 'Q( or b0otror i louv X,,a ' t rrxortr ,{ n,r ajoro;.olrvbr,(i(owo: ro r0661.ryc rtr tritlo-rl:porlq tr€ ro na66Lnr( ro).v tr€raq,o-C , ! . t l o , , r r L v u r . ' t H r t ! o r , U I p o r l . u ; r o : r r g r o 0 r n r r o r ' < X , " q r 4 r . r o r l r , a c :

Mttl yorlr€rrrflrl rLeuQi( curoi ror! 0i$(1o< ei.vuL q (vr!nrr,E4 dyoQt0t(orv.Avri./tc e€o)Orlrrlai ffrobtr!1 bri opor, 1p1or p on oro{trv aqtxl l\Llqto at ,riroqt qyJx6 4rlELtkt at 6dAoa .tru \,o 6porrtra q 6i),noq orrroroitrort rt xrlg,0ryottql porl. Arro por<ileL [€ q $60060 sirnplex. xc0(;: lerrrri[r trn<i prcr8q,!)4rrl Qorl (!l{1 zoQuq,ri) r(r noooodro![r'prc vlu po1 (jr(4Lc or, $|cl./€rro!!zlizoQu{rl). O! c},./6$Ofror slrroi et\r(r c!6Lzoi 6LorL r.lbrzcl !:i!c! x({r-r{r neo6}.rlpcr(6Lxruo.w.

Y;rdpXrL eiri.oqE 11 rqvLxi r.o1r Suyapuott nAoyQolrtrdzrolor1 nor orlAiE€r(r.o0 tL!(l oxlni rb6(: EZr\' trL( 6rrr6pr4llj cr;r6 rr1:r4I1otr1v rr:ropporl eiraL deitLorrl.rorL^ r(r rdo! qrjtrc rI. blobQo[4,c 0( riwL 56],11010. A€\' proatri a( ll]rllQxeLr(irr r(),!reQo. roro ivor,zopSo Trrivr,r ('1l 6|tlbAoFii rqo6 t+ rcroppo{. Hiiol xlrraor€rditt.{ uvcrbpoprzir. pr gLG frL(6rxcoiu brcboXLxdl ll1roq)doet'lv.:€ x.i0!_ orllblo, I.rTroorcorl u:ro rrlv oropporl rivcr ro el.dlLoto trrLuq viuqctrorrrcola or! lra tr.r,or{i cn6oraoil:

(ndotoort r. l = c),(troro u.q.iao 1n,v, Ilq (uaordolt( x.,V + amnr(ollq y.t).

Aurl rivcr r1 nro cn),ri rop.{ri rlq Elior,)onE 6wa;rrxo! :rpo.7pr4L;rcrrolor roUBcl lmao.

Mttzt?pl r,c r,m1p2e 7ri;poq yrcr rieproo6reqcr ngr?lpcrrcr ppo c:r6 rcr 6in-ruc.nou r iv ' , , roou u:T).Lr dtr i . , i a, , r ,16. ra, , ' .

AIXIDED

8.4.1 IlAodein.3 o0 acioE /oArtr,zorlra ro!:I. 8.8. Be€ir€, f. t.u trorL(i. trt fiyLo l LrolixaL q! €l.i'trnt 6r0totr1t.

8.4.2 BqEilE tl Fiyrrq poi x(r rt! €ld/rorl 6(rotrj yrc ro Er6F€lo biaT1,o:

503

501 |PANIMIKO) I I POTPAMM]tTI'MO: XAI OEAPIA NAITNION

t.4.3

E 4.4

noL.I dJoyi 0o nqoErvouoe q pryd.rcal onhol r4c- Fi'yr(rq9 qois. dv FroqouoorEvo l|Cra6dlLlt rr roelFra6DFc xdo€ oyoryoi. oro naoryoiFFlo bixruo;

:Xr6rdor€ 6va Sixtl)o 5 1(6p6ov !€ XoqrlllxriTlF(r , j F€ra€n io!.rop6o! , x( roDx6f6ox j. Ba.iu r't lsyroirt 6war! qori rut6 ror xoF6o I orov x6r!6o L

: ilo yAdSlFo. o pdrrcros daretds ddoqoFu! ;(oQr9 so,lic r(xFEst. rsonrar $ iov cldxroio aaretd oxfdv. rov onoi('v rt oq(lA€ol anoo!"',6!rL ro sand ro l. :tll'trliorE ro tr€ ro e.(;q r.r fdyrarnE eoriq €)dXrotIE b|Cto!1is.

Bafir€ ro t!"yLoro orryolo.rdFov (fl f,!v.1r6v, dva n).'ietg ratarao}la) yrcl ror\E,4.6

T l I 0 0 0 llo r o r 0 ll0 0 r 0 1l r I l 0 0 lL l 0000 .1

T00 l00 ll l I 01 l l

^=10 1 l 011l0 0 I 1 0 l100010 t

:xr6rdorE ro 6i'irm tou l' fr rro )(owa€s yeq+i<.rra rt! cxfds toD r(OL(iofdrq.

E.4.7 IIol€E raopp6< rdea6Li.low n] owelix'l rou Hall, orov neoryoiFe'vo nirdac rl,I6yo ror orr nre(aXou li-y€s $or(i6ss; Ilorcs p €ni ./ unonivalt(ls |tl]bevrr6v ir€L

E.4.E llod€s lealltl€s (oqLl6wEF rd| r{io.r.() xe€Ldqoll(| yro va aa}.u.!0ouv oler orfovdS€s ro! traotyofu€lo! ,{r Ehyio€ yra o c60enis buiofd! d,ro€trar ?raxde€ fti!(xcr Eciv riv(L 6xv(oi I ydFor. r&€ yt( !a '|0].w0ow o].€s or rrovdd.i.,@€Ldqowdr ro!'ldtrmov * leqqds.

8.(9 ., Emo o rd0! 'yaol4ui 1{ar 1idee (miII ftqrdr€L qtqr6(k 6rJo povdb€s. A€ik€(i1t €irar bulard! 6r( ri.lae< rflfardolo. (asiErs o oL poadb€E €i1,(r a6walovva xa).u@onv F€ IL./6r.aE( crTo ?r yqo$dE).

6) Baoitr iredb€LyFa u€ 6ito i Ietoo.;ueq ro.tlt6Fs oE xdo€ yaal+ri rar mril.r.yro lc onoirs rivor Sword dva nL'ia.( raiaroolo.

&4,10 Edv evas nivaxns 7 €ri 7 ex€r ls Fovd6ES. 6EiEr€ 6r €mrQin€t 3 roulAxuov ldtor!.

8.4.1r rLa dzcrgr ouvola, Froq€i ro Frlv ria(r bwdr6v dlo n).riq€E rdiEaors, o,ioFrtxdr orav r?crvoflor€irar r @!0iq ror Hall. Eir! rt ne6flt yqcpprl rtqtiltu p&o

. { MONTEAA ATKTY(]N 505

Fovd6sq ft(r xcrcnrv eii I = 1. bEiErE ort oftoEobrlroft | ladFB;s 6louv poadb€s

o€ ror).dxtolov p oriirq €! lontoLq 6o r,n{iala n),iqDq xcianola

s.4.U Basit€ rr o1j1rop6rsel 6'aEpoplt an6 1o s @o t, acoc x(t xo €liixroro @!bio!6ivbao. €dt ro XI. 8.E ncaril.r |[lr! omr .lld IoqltrToqr.q

s,4,13 Eqcqpoqr rors d'yoqi0Foq (1) aor (2) lto aa 6q€irE ro oltoFo€eo owdaoa66vbao tor brzrio! qq Ao,ttofi 8.42

8.4.14 a) rra 6orL.n.L o o).16at0poq an].Imios 1(1r YLa to trq66lruo to! o!6io1Tos6a!6Aou

6) A0tEr€ f 6\,c ropdbfr/trd 6fl. E0xl'(irraq and q orJllopoxsfl axfri' 0a

BroaolodBs aa anoirrlol'pE mI! oEnra€oil rIE olllopdrfflc SLaSqotrl cro to

8.4.15 Ed! o niac,io( .rl Eivdt 5 |:ni 5, F€ pordlbllE oaa'6(;5 nitv(u r(t xdro dro 1rll aia(

8tqo ' |o ,6a3i rEa) ila ollolo yaotr$drv rou 61tx poldb€c o0 ?'rloreaEc otrl/'o(6) avo snvolo nli(i,! xou [1.' tro!d6€< se ].tTotEaeq lqetqras1) tvoa lnoEilsxo 2 fni I B€ [l6evrxd zor 2 + g > 5b) #oo€qL< lqa$ds nor xd,!mo!a 6/'€s rrg lovous

E.4.16 To neoolq',, F.y.oll]g qots ;xF, rion0'llrxac rFr4J;rric a r = . - J,, Yrrr rnI'r.foq; ffn,;f I' ,0llrxorlr',;

"ot oo,lc A,,,ru,ddTr To rqo6)'t[.. ln'r :X d i orrv

reotjtrft (l YAo4rLTo! neoYAqlralrol'rl

8.4.17 Efu or! nd5 oe xdr0€ to|6o a(ir(D (rTo rI bffoprl ro! >1 89 llnooti !u q'r.ioFL

nrerosorFQl oori -(aopl xot olo! aofrdo oro! oroio drtr ibl qrcirott an'erooloi

rP \'tNIlio: nPorP,\lv\IA I lINlo: K]\I OEQPTA OAITNION

8.5 I OEAPIA NAITNIAN KAI TO OEAPHMA MINIMAX

O zri.urtpog rqono< FEiIpI: rro< ;rrrrybLol ;rirrrzu tilur rcr brirooryt irutr(Qd6fri,tr(. Arto ixr 6rio;rcrizrec xuL or xurorrc ri.rrtL, or xrl0r 1tpo, oL i6rou

O ryairrlE X ov(;a€r ivc rt zcr r( 6io rol' I€elcl z(tL (r\'€_EdQnFd crfl (nnov.ro i6Lo z(iv€! x(rL o Y. Edrv ono{l(loioow 10 ibro. o Y xep6i!!-r l0 $. E.1!(noqr(oioo!\' svrio€r.l i.F06i!€l o X. 10 $ Gr olztirorL ro e1,( UaQ!',.ar 2{) $(\, olx(ilo€! xul r.l buo. To ruo(rQo x6p6oq Tou X neQLyQ(iq€ror €rJzola p€

, I l o 2o I-*=lro - lo]

€1( XFrn buo ldprrr

xou X iorJ X

iv(l XraL rou Y6no xapra rou Y

A0zri ).iy,t ozi\,'l1'( !( ii'-)rirf nt! zd.r,rfQrt orpcrrytzrl. Eirt(l ;rpor1(16<ri t r o X <\r t 'p;ropt i ! ( z( i l r r ro ibro ra( iy| la z. i {k qoa((. bron br(qoarrzr i o Y(l( ro\ ' ( . \ ' r t l rx i( t f l r (1r {nl zaQbi(fr n( i l roi8. I l ( iA(; tun( o Y b{ ' \ ' t r i ro0r- i \ rr rr€proproTI.i rre pLo orqrrrrl?rri. br6rr o X 0( zdl,L'r ro (!ri{h,ro. K r or 6i\, 'l(ixreq;rlinlL vu TQlortronorrioouv lrc lrrraj arertr4yrz-ti. tu.L ttri niiov rl FnLl.oll oEz ' i t t r 7 ' 4 o r , . , r r r r ! , , r i v , , r r ; . f i , ' : , , u : , r U r r t r ' t ' , r , , r ! : n | . ' f i o r ' t | N u : 7 r ' L n , : .Ar((toerrr2d. r(i! t, (.vri;rui.o! (!rl).rltOfi iv.1v lr(zeoTa6vro 07f (qri). turoQ.i\.( ro! r4rrrd-t.F"{), i. Azolrt z(lr tr'( ora(rrl?(7'j ojr',< z(ivr or,r'i/r,,r rrl! ibt.le:rr).o.|j. r(t 6{10! r{.0bi(fli zrr( ri}.i(:f troir< r.!ofr( fil(r raoq(1ri). troro(io.Mtrri rl:rri cQxtrd icrlvi6rtl, o crvti;rcirjq oac Ocr pr'rpiLr.r rL crardd)q noingL va

A , r u d \ ' ' . / z , i ' : r r r u , - - 6 r ' o r ' , . r n : \ , r z n ! . r \ r . ' \ r r o t r r o , a , ' . r ' 7 r o , , o : .X trnoaei \r (ioif(oiifl !a olzd)oFr ro i!( Z€ar gr or'/vorrtrc .l.r 7(r r( bnoTigrrr pc or7v6nlra I r,. :r z(i0r yL,po 'l rrr).o7rl (r,l tilta rr,7ci(. Krrr'(\'(ioyi(\'. o Y tr;roafi !c rzflt(i r(( brz€! ro| nr0(^or]trr! trf- ,r z(r Lr(vr i , ' roryc. KoBtr i ( l (x crrac b!\ ' r : rrrqi ieraL vc r i lur t l r j 1. 6Luq oprrL,rci oavri:r(),o< 6rop0tirvor rr-r :rrr.y'vi.br rou xar xepbileL. TarlTotOorc.6cr !i.vaL oarldsor! {Jc n0anfl !c io€q tLr 1,/2. b'orL brc{toarrLxir o Y {)(1 }(ivr-r 20 $ ro}.ir outvfr.(O(i drult ?0 $ ro dvc rircrqro rou 7p6vor. 10 $ ivc (iii,o r6rcrpro zar Ocrzi('ltir l0 S oro uroo ror'Tqoror iTo\r(; lltc $io't flr'j (cr6i,Er.r;2.5{) $. io|tirtrL :rtprooorrpo tr;r' t;.fl Za€rrii€rcr.) Ooo nEqroo6rreo oll(l)< o Y rrivfl or rLcrztr&r,Dtt otqrr4lzrt trur 6io Xeqr(br. 16oo reproo6rrpo o X Ocr zLvrircr ;rqro: ro

To 0€tLELfri)6r< nQo6irtF,r ti\a n 6QE0 !u! t('o?Qozl'a. Yt((ie/r-r [t( trcrrii

85 OEA? IA NA I |N IAN KA I TO OEOPHNIA ' I I \ I ] ! IAX 507

oracrrllLril ,r x(rr y: nor. rlv zalortroroLl0€i F€ or!€rr€ra ano rov Y, va Ftr,'rap6X€.1 €r6rzd( nQol,o$trl orov Xr MroA€i o X ro 6rc),i!er rrg nL0ov6rl-riq rorl.rr xc' 1: 6roL (i)orr o Y !( pry tpL xovtucr t6yo v' uvarrnE€r rlv brnri rolorAcqlLxil::'6vc r6roLo orltrFio rooQoorri.GC. Gv u<ip7r:1. to g(oo ztrpbog ror Xecl txr:r .triro€r a Ew oayFclbti a tEio: O(i ci.aar F€troxo 6oo.v agoed rov XxcL rr,c(j(roro ooov cqoAci ro\' Y. To v(r 6ar0€i €vc r6roLo orlFrio r:i.rcr rooiilvrqotr€ ro a( i.ro€i' ro n(ryvibr.

'Evnq tgtinoq uv{?}.uols- rr')v unoioyroiro! rotl X Filcl o €Etq: Alt6q ouv'6!dq€t rq 6io orril.Ec tr€ rtt 6irQI :rL rclr. I rr zor nsairy€r Urcr !€r1 orri),I.'Eor{,)ott yqloqonoui tcr 6ciait 3/5 xcr 2/5. oror€ nupir.Trr rrl orrl),rt

l

5f l0 l 2 f :u l l2 lI ro l- i I rol =

l : l'O,it xu ya xrivEr o y Erarriov .Iultis ltt; nAdlnywtiE, 0a )rto 2 $. 2i xitgttr!tror,!4ri1,o yriao.lo z6aboq €i.v(! 10 $ i 20 $. Eitr 6prrs o Y ollzri)ver frtiv4rc ro6v& /60r. r(jr! 3/5 ro1| /Qoro! xsQbiler l0 S xar 2/5 ro! ZQo\'ou t(ia€r 20 $, tLetr int ,( : r ; 'hr{a 2 $. t , rv t ' Y reort , , i ru bro X;e.( r t (v(.1,r7!rJf , 1r! orcarqyr/F:-or onolo6]iror8 uvd.oyio, To orroTi),col( fic(ooui\rr! orce€Qo or(1 2 $,

AllT6 6fv orlfr(ivfl orr otre! or orQ(rrl1,!xi< ri!(L 6ai,r[o1€q yL(l rov Yl E(ivtfr(r rcp:ri),r6 xcr" or6xtruL. pe ro ivc Xlpr url-r(i)lll!o. o X 0(r c),}.dE€r zcr 0((lpliorr vrr atpbiirr 20 $. T6r€ 0c d.),(i,Err o Y x(lr r(rorr! 0' Ldi.dter x(! o X.T€iLtt.!. al MoUiooup€ orL xaL ot 6tio tivru ilu:rvor. rtiuo o Y ooo xu| o X 0ar{crc}.4EoN (r8 trr( 6lirLorrt (!d[rGn. A1l1o on[oiver 6rL o Y ftr ouv6lri(r.r rrg7r, ' , rd.r : r , ' r . f l , , i r ' r t . /

"{ f l | . r : ruu,,nu0i; ,vr,K !u ' r lqr,(r iofr t | | . r v iu yad/t [ ]

no! [i,var ooo! xo 61J\'(Tov pt,{Q(i1eoq:

y r I l 0 2 0 ] + ( . 1 y r ) [ r 0 1 0 ] = [ l r ] 2 0 u , l 0 + 3 0 r , 1 .

To orDmo l€h/tra zdv€r. rrq 6rc or!,t,rorli]oEs iD€q: 11 z(llt = l0 + 30rr, rouorlu(1iv!' yr = 2/5. M ottTlj rlv FnLl.o?]] zct ot bno owLordroE< Loon\,rar F€ 2rcr I \r6d ./aa[[it lf\,ero.t L2 21. Aeu p' awit t4 mpct4yrxrl o Y 6E| pnoAEi yd

xrtoEr r€arcootAa an6 2 $. O Y Elatrororoi'tor 11 p6.1Lorrl tor anrirl"r:Lrr zar .ro

miDimax ror orpri;rrrr pr€ ro nuximin nor 6prtze cvelhprlrc an aur6r, o X. Hr4oi tu" :tuq,yr\tui thz! mininrax = naxinrin = 2 S.

'Evc tdroro onllLcrrLxo o4rrio Fi.r,.ir.rELonJtpeil,)ro 6LotL nqrcriver orL oX rci(er q bentepl orpcrrrlyLxrl rox 2/5 rou XQovor Uo!o, rGQ'61() nor Iorpcrrl./'al crrrrl ror biveL ruxcu.qic yrtl 20 $. Taur6Xaov.i o Y rv.r./,.(ioorlt !:vu orobrTe€i $ra orA&rrlyrxrl 'rou XdrvrL av zaL 0rr rlot],€ 1'(t EivG! roonol,oir€ rov X. zaloL[oroL€i. rr< c.''tieere; rluclorlr€! 215 xd 3ls. M;ropriru vcelorprd{|)oerr 6rr o X z€abi.!€L 10 $ trs oux,vorrra (3/s)(3/s) = e/2s. x€a6i.E€r 20$ pe onxv6qrcr (2ls)(2ls) = al2s n(L r.irv€r 10 $ F€ rrv anoFd,vouoa oux.,'6rnr.r12125. Orros clqrilrrur, cu6 crnocleper o crrr6v rvc ueoo ripbo_< 2 $.

508 rP,\Ml\iIKO: n?OrP,\NIII TDIUO: K I OEAPt]\ n^ll NIAN

lp6n€r v(:l oltr€rrboouB€ 6ao oqdl+rrl nor Tiv€rcr' €rizo),a. AEv ol.leEnErrcivror€ 6rr ro 66111010 fr€i.4rcr €ivar $La laot4rrl ge ioc oroLleic. Yroodor€ 6rrr :rrrpinrror mor X lrrc rpir l orpurlTrx4 m1v oroic rc o'1zLiv, r rprf t 7, a,u rulvu zepbiio" 60 $ 6rcv o Y oryovn tvcr acr 80 $ 6rcv o Y oTrdrv€r dro. oiii vuxcc xtp6ou< livtrol

, l 10 20 60 l" - l ro 10 80]

O X tu 6(rl,ir't€r zir0t <1ro{tti trt 1,!( otQurty(x{. qroalilowor rt; or1}.eg pe ru6(iQt iriL = 0. J: = 0 zu! /; - I (xc(06Lou rruoirl) zur retuluivowcrg e)"ir1Lorot f f n , , : r ' 0 $ . f r . u r i l a o r u . o Y , , r i l r r . y , r p r r l t . r t r " ' v 7 p u p | r , ; ' v r o u r i r u r o o o r o5uv(r6v pLxptirepo. A|(r.6./rx ndwor!: rry rprirtrl .1p(pgr1 ncl! 1pt1[ml (]rcrlrldrou civar 60 $. Aa(i ilouflt, ndtrL nraxiDrin = mininrax. (].].ri ro ocrlpcrrrro olptlor-rvct nuv0r oq Y('rvr(l.

O o(')oro( li(lv6v(l( q(rivrrcr vo ri'v(Lr orl 6i:Lr(orrl firrtrol{ ro! Y, qr+Lrt ror ,-TctrylrbLou atrqrlli.lsrcL (o;r(1)s rc 2 $ xtll ra 60 $) tr6ao ,nq orrilfEfiou ZQrloqlororci TrQcry[ofl o X. Kctr' c!a],oyi.c1r,. oro 6a),rroro frei./|rcr oqLdvr, ,u X I rhr,r I |dr l , t t , r r . t rr ,o o ,rr .vrc f lE lC,$tr ic :roI nr l r t .Fr iXo|v nrt66lltoll orQrlrrlyrl{ri -rou Y -or rill.ec ypqli< 6i1ou! zirrr :reproo6tego r<l oY llE ftnoq,tiyfl. O xcvovc: (vrof cvrroroLXEi n),r]0(i)E orlE ori[r].IAt'4r(tu iqncp,b,,Qr, ,aiE owhrt?rE r, ' r , yu,+,.r tzo,: nt , ' l t r tUp,rr ,ol . , ' i .

To 0€6QrlF(! Minimax

To :rLo 1evr.z6 noryvibr fiiver{.r" $or.!ier (xQr6rlg [s xo fi(Q.i6rr./tr(i p(ic.

fr€ [t( evbtcrq6aouoc bLcqopar: O X €X€r n s'r.i.oi,iE Har o Y €Ifl. ,r. O;riv(zcrq xipSou;,'1. ror:rcrqcldvrL o ibLoq o€ zdo€ yuqo tou ncrl\,Lblori, 61rL,, yaouttg r(lr ll mi}'tc. To ororx€io a,r rcaLorarv€r ro r{60boq rou X. orc'vauroq ErirLi./El rrl1, .l o(n1i rrra(rrlyrxrt ron rirL o Y t:nLtl:r r4v t-ooli 6rz)irou. Ev(l aavlnxo ororxEio oit[aiv€L cr]djq apvlrrro r6a6oq, q),abi x606oq1r.r ro! Y. To cnordl.soptr rivcL:rcil.r 6vcr zaryvi p4btrrxoi a0qoiopatoE 6iorarrrav. O.trbri:rorr Xd\,!rdr .rno ro,\r i1,.1 n.rirrl xEQ6i!€r(lr un6 rov .iiLo.Aot6oo, 1 lrcrpEl €\r6q ooyFarLrou ol[riou r.ooAaonkl< 6r! ei\r(I pE ,{o!tvc

'Or{Dg oro rr(Q.i6€r./pc. o X €t,,cr €lerlO€po,( \ra €rrL€E€r orora6rinoft [€rxrrioraarl./Lr.irt r = (rr, . . . , i,, ). To $€iypc crr6 ri!(L brrivuop( rr0cv6r1rcrg, 6rorr.l jl], Ei.v.lr. [I .rQ!1lruri1 r(lr 6zouv d0QoLo$a ]. OL olr,r"mtiroeq crurtq 6ivorv rrgorXrorlte! 1t,rt rrE n bLtrl1loptrr"niq mp(tqlLzeE zu oe xir0c yipo rou:rcryvr6rono X Oct aroq&oiq€r 1Io ptt1 crni ollrdq Ft ivcv Fllcrvr,opo ruXdcrq errLoyrjg,xaruo"ruuotrrvo I ror uorf I r-oort .rrr . r ITrat v rvr wr 'r iyrr(r f r ouxvorlr l /rr. O Y a vr+r€rconiq€r avh).o1e_( crogciorLq: Ar(1],ii,€r 6r.ivuo+o y = (rr . . . . . y,,, )

S,t OLOPIA Nr\ITNIQ\ KAI IO G)FOI'HI\IA IlIININIAX

€niollg Ft ri > 0 rol tV' = l. lo otroio biv€t rrg oul1'oIrye: rlq bLzilE rorllflnIlc or0crrl1,|zrl<.

To n.lrlai& siv(r rll(rio ?cL 6rv FxoQo!v 1'u ln.iaEouv :teoy\'t)orzd. yr(r ro{(nor€i.€ol( rl'dq tr€|ovo4rivou yuQou. O[o<. zcrir $iooa 6A(). o ot!r6wo[6! r|lii-ooni! orp(ntyrx'i< rou X r.or rlq i-oori< ma.rrtyLr.riq ro! Y 0( r'i(1,rtut.rvi!r-1(lr tr€ nrocloritrc .fr t/, ro lvotr€vo rr'-'v 6uo tr€trovr'rtrivoJv fto(rorrirara. 'Orav

olfr6aiv€r cxro, ro xdplo( otl €ivcr dij. :uvE,-rdlg. -ro (v(trrv(5tr€vo riaboq rouX un orno! Tov cLbrxo owbucropo 0(l €ivcr! o';xrlji xut 10 o)\nti @clE1,6!trcxie6oE xti9t :rt(!ri6a< rcu diov trtr;,t'u\roit 9a €i,ar ttdij.r'rJ,. loliloqrrnd.r ofl o:ro((6)inor. (rno ru ororl€i( rij [nopei vu uivcu upr'ryorir. OL zav6ve<€i!s( or i6ror yr(r ror'< X ztrr Y zar ru'ro trot, tlnoq,(loil€r iroto. z€ebiq€r ron(|1vi6r c iv(r r( ororz€ic ( l , , .

To cr(4r€aoll€vo raQbo< ye{i(lflcr ft|zol6r€Qu o€ o+l6o),LoFo nradir.(')r: To6rr),6 (io0orotra LoolrcL azpL6tirq pr r--l.r'. i6lrD ror roi),(n).(o(rotrori! nr1('ly.,v

y A t = l u 1 . t 1 , , , 1 = o| . t i / r + + / r , , ,J n l ln , .t;:"i:,,:,I l;

,l [;'] (;)'. .(;)' ;

Avto €lvat 10 zie6o< t/-4.| no! o X oiLrr ru peloto;roulotr z(lr o Y {li}.€r1(

IIAPAAEIrMA I E ror ox! o .ll ei!r{ o rrnrorrndq rr rlTri I niv(rq. r1 = 1.T6rr ro (vr4rrv6yrvo xipbo< l ivtrur ul . t = .r tUL +.. +.r , , t r , , xcr b€v ei .v(rbuoxo).o vcl .Enyriooutrt q o4pcoic rou;raLyvr6ron: O X rLrii€r vc n€dl€r lvi6r,r trLLoTri lt aurir ror Y. on6r. rrabilr-r a,, = l$. Tcur6rpolcr o Y ilrriv (roqu/$ ro! X. on6r€ 6e! 0( 7er'(i(rr(r r( :r),tQ6ofl. 'Or(v o X rmLiy8rrrt (Jrri).tt i. o Y flt|ialrL pL( tu(qopttrfii ypuptLri j z(r ro ziQbo< tiv(r .ri., = 0.Eci! o X b(Ii:./rL trr( orp.rrtyrzrt rou olzv6r0act oiro rq diJ,r'g, ror€ o Y B'roe€iv( b(qr€!h/c! ol t lor€Qcl. i rpcr ro r i iLtroro lei lpc €ivcr / ' = (r /n, 1/r , . . . , l / / r ) .K(r' (rc).oli((! o Y brv FnoQEi \,( ,roviqr:r rb({riree( rrifiorc olQ(rtyr}.|1l t!)Qi< vfurov cvcl1{(1),iJrr€r o X. rlpc I 66Lrrorrt or0(rrllLxri ro! {X!{ rriollq rq rLoavorrtr€q!t ' =(11tt . . . . .11n). H nrecr6ntrc v( l rxL)dlouv z(r oL 6uo rr l otpcf i lyrzr j irivrrL (1/n)r | (t to d0AoLqra riturv currirl rt,rv ouv6urogrilv ti'r cl ro alcpevoptvozip6oq rou X. H o),rzri mFri rou 'rarylrbror ti!(r n qoQi< ro (l/,): rj 1/rr. o:tr;€nL6€6ft6r€r(r z(L (c16 rry

o' A,, ' = I, 1,,..., l,, l l

K(od)! (uEir!E! ro n. o Y 6x€r n€Qloo6r€Q€g ruzc(pie; 6L(rptry{<.

rPr\VMIKO: nlOrI,ANl lvlA I l) Nl O: KAI OEAPL\ n]\IfNIAN

:It€rrDor€ ro orr I oulFerQi( xolJ rii.,'tlzc .4 = 1 6ev eyyrhrtrr. ro trpep6).ryrrotol, ncrylrbLou. OvrcDq. o tv rpcy$(rru{6rrtru o4rduiver to €vr€},cbq ol-lio€ro:O artuntpyeryxdq t,u.da A'1 = ,1 ti.tru urroq ftou tyytarur iya d,ft)titqaprp614:tto taryr16r.. Ercrg-r€Torog aivarcq crr4rmarni!rr rol< 6uo ;ruixrrg pcrcrpoBoL€q &no{tdo€L(. uqro{ q enr),oyrt qq orpcrl"/|zri( .l cn6 ro! X xor rtc iano rov Y ffiofiQ€L ?iiQ6og oij orov X. xaL I €naoyi -rI< j (ljro ro! Y xar rlgt (1ro rov X anoq60€r ro i.6r"o lriAboq oro! Y (ra0(irq o/, = -d,j). Or 66),rro1r:qoxQurlllz{E x' ror ]/' nedrr€r. vo €ivo( o[ i6[€E lrcr ro cvr4rev6pevo x6q6o< rpd*crva r: | !cr u 'Ai '=0. H r+nt ror n(tL. /1rL6rou. o:(r Ar = A. €ivar tr l6 iv. HorQ(rrlytxt of(o: na6r€r rirl.L vc 6Q€08i.

IIAPAAETMA 2fo -r -r l

A=l l 0 - rL l 1 0 l

Mt lo./r(. ot X rar Y bld"€youv ror or 6rio ivrrv tlg!0[6 tr€rdu ro! 1 rrrL ror3 rcr cxei.voq rou (Xrr ro\r [rxp6r€po nsQ6iL€r 1 $. (Edv o X brcr,aE€! ro 2 rcro Y ro 3. ro z6Qbo( Eilct nr: = 1$. Eda br<r)'iEouv rov i6ro oQ!e[6. €i$(rorsorlv xuQro 6rcr./6!ro xcr brv z€pbi!€r r<rveiq.) Ilpoqcvdg xclei.q naixrlq 6€vF,_rop8i lq rjr[].aEer olQclrl./Lxrl rrou vc o!Fr8Qr],qr6d\re! to 2 r] to 3. lurpig vcrTiroeL rm rov ril.io ro! 0( x(166€r nlo x(i(,). :uvrrd<. or c[!./Ei< olQollylldq.r' = r/' = (1,0,0) eivcL 6d),rLoteq -zor or 61Jo n(lixrr,( 6Lc).d1ou.v zrioe qopir tol- xa! I r41n €ir ' ( l r t / 'A.r ' =r l l =0.

AEi.q€r a( ol[erd)oollpc 6fl o rivtlxag roLr cq{v€L ol,EE xrE Gro({](iorrEr1vcl.).oiorrrg bsv €iv.{ o rcruorLxo< :ri.y(llrc< al.Ih o ri1,tl}iaq l' nor ils! /ir,.l ruororz€i..l lou .ir io(l lu 1. nQoo0rixil ElloE noLl(1]_r).coiou -rori E orov ni.r,nxcxipboll< .{,r4 - r'1 +0E. orlpciver. dn)drs orL o X xep6i!fl oe xh0e pqo dl,nnQoooEro xrooo a. H rLtlli rou 'raL./\rr6Lou arqdlel xctir o. cL),ir brv uirpXeL),oyoq a),1.a./4q ior .r" ,act rou y_.

Enrorp6qo$€ rdiQa orn yE\'rnli 0lrrQio, ronoo€rrirlTa_< totl( r(utoriE Faq orle6ort ro! x. 'EoloJ 6rL alroq €nrl .d. /€L r I $Errxr i otpurlyLxr j . r '= (2r. . . . . r . , ) .Tor€ o Y a{d.aF6irv€rc( rel.rd crrt{ q orpdrlyrxrt r(L trLLrY€r ro y:roU€lcxroronor€i ro xEQ6oE 9,,1r; itoL o X Ocr 67er. xdpboq min,, r.,,4.r. E.r(q (!u ogn.rirq; X 0(l r:rrl.6Ea dvn ar.dl,uogo.r- (p;ropri vcr p1v eivcL govn6rzo.) nouFtyrnoitotti att6 ro ELdxro"ro. M' aurn 1ll, .nr),on o X eivcu. oi.yorqoq 6rL ocrx€p6io€r ro!)dlLorov ro noo6!

min?1i" = mnxn nr lr ' (1)

|I€arooor€ao. 5E! ['roaei vcr r:pqri.ver.O nai.xrlg Y rlivEr xo ovriO€ro. fra ndo€ Fr.r a;r6 T|,q ptr,xt6g orpcrllxdq

ror ,. nQar€r'va n€pL$€!€L 1]1o X 0o ovolia),rv€r. ro SLciluo[o rrolr p€yl(xonor,€i

\ { HFAPIA nAITNIA\ I ( \ r r t ) i JLAPl t^ tAMl \ l l lA \ 5 l l

ro yAr.t AQ.l o Y 0c 6rc},6E€L ro l€fy$rl y" 1ro1r ElayoronorEi auri o ptyrctox,n r'lYri(iiar orr nr v 0u X(or I rr prooolFco ,(no

nrax y'ljf = min nrttx jlAr. \2)

O Y b€r [roQci vc ]rteqiv€r zirrr z(l,rirt:Ao.E},:ri!or va 6i,i:tete roLo €c €ia(L ro unor6l€otr(-zi.€r6i. rda €iv(r (r,t0rro.

Ot),o!tr€ I nooorlru (l) nou €iv(r elcrorl,u),ropivo 6rL 0G z€Qbl0€l on6 ro! X.a( otll|nitrrtr yt r1v roo6rqr(. (2);rou ocr tivat rxclonornt6vo( a( Xdro€! o Y.Torr t ( l lcrylruru.f ' x( l r l / ' q,r n(oi . /o! ! ro o,rytr ' r f lxo oyl t [ io rooeQoni,rc z(r1() nc!./vi.6r 0(l tXtr ),ueei: rciv o X llr-r(rxrvrloEi cjr6 ro .r". rors tl61,o InoQri.1( Xdoe!. xco(i)< xo! o Y. ecv lr€rcxrqoti crr6 to y'. p6vo vc X<iorr tlrroQoi. Hutr(Q:l curo! rorJ oc1ruatllou olllleiou turo6tipqze cm6-rov von NcunrItnD xcttrt!(l yvoJorrl ocl! eEbQryta minlmsr]

8M fra trdte€ m eni, tt fii\ct1,,a,4, 10 minimax or€QdlcD 6)orv torv orpc-fil/r1r6v toor'rdr [€ ro maximin:

marminr,A,] =n nmax yAt. (3)

H roo6qra cnr{ ei.var, n rqrl rou ,ralyv!6roli. Edv ro tl€yloro oro oprorspdlaF6dv€ro! oro o'rcu, ro e)rcir,oro ora 6e!ri l"cp6tiveror, oro.q'. r6ff o!orQdrqy{lidE avr€E €ivql 66t!r!or0g xar rdQdlow to oayrorr.nd oqpeio, andto onolo xaveiE 6ev 06t!€! vcr BEx(xLlrlo€i:

: (.vr6 to ocyycrr',r6 orprio. ro .r'' €iv(( rou),rizrolo €€ iooU xrd,o IeoroLobrttroft cll,i,o.r (rtporl y'Ai 1g',-h'). flrrpoFoicoq. o 6ri,rE()os x(izqE.€I{r1rt1l€inor'r(1g ro y'. fro\'o n€eloodrreo €c( pnoQoio€ !(r Xtro€r.'On(oi clxQ!6oc )acr! o t o€oaicr brioForl, (laliEouF€ tr€ ttt Fo1,on),LxAt ulroo-rrFa: DraxitniflSminimax. Arr{ 6€\' rilaL nr4d dvc< ouv6uaoF6q rou opLoFoi (1)lou x' licL xou oQLoFou (2) ro! y':

U'A?. < !t^ Ax' 3 yAx' ynx&9exxat y. (4)

nrAxmiry^r =lnint / ,4.r '< y 'Air i " < mnxg'Ar =minmaxyAr.

T Arro nN trroed !a 0ivar ro r' F(i! o Y uo0.!ond. !r1 ra.lr racnl/lai. r6t. o X 0(zrpdr:! iroro.orrsr (r'oo( et, fitu I { l). H oroei( furllni! nlrinr' ra rnooio.r 6n or i(i7r(

5 l : rP;\MNll(O> Il?OfP NllvlA'l llNlo> K I OEOPIA IIAIrI\ION

To [6vo tror ].de! I (5) €tvar 6( c(v o X iX€L Eyyurlr€vo ]:6Q6o< roui,crxrorol'a.x(r o Y €Tyl'+rA'l €ld.xrml (n6),€rq ro roLi J. r6re o 5 3. To erirer,ypa rouvon Neumann rlrcv 11 crno6€'.Ell rou 6rr n = J. Aur6 €ivft ro 0€6Qrus minimru'ort$criv€L orL I rooxrtxa re€n€r" \'a rotu€r" olrtv (5) xcr I rbr6rllT( o(lyFarrxouorttttiou (4) ouvirysr(r (n' au4v orn AonIoI 8.5.1().

To nro cmpoo66xlro rprilyu o cutl tlv ur66ttlrleLror,r,o 6rL xp4aryoiroLtit.! i\tu uxAtbi. rah rtrtxo tr t,t Hr@Aia tut ytJt4quzot:ia.'yAullarrclou.A(uoellrrzd. (rur6 €ivsr q(€66! nQo(l(!€(. Or X x( Y nc(iEolv '6uizous'

ad),olsz(t 6rcr,6yow xcL or 6uo mQCrqyL1..( ojrd ro "@rob€1tr6 oivolo' brnvuoprirr'ntit0(lvorlixrDlr: .fi > {). tr' = l, U, > 0. Iy, = L Eiucrl €2r[),Ix1[x6 xo 6rLcr(oltll lcr o von Ne mnnn 6ev (vcl,vci.rproe qdorDq 6r[ o! 6io oE{,)Qi,eE €ircLro ibro nQdyFC. (Auros cn€68Gr ro e€dpnpa nrininrax ro 19213. o yQupF(roq:rpolqcrpgcrrLoF6; ieaivrpe ro 1947 z(r or Galc. Kuhn x(.t Tucker b4rosi€xo.rvrlv ned)rn crob€rEl buiqron ro 1951 -6coi!,6trsvor €v rourorE or o+r€r6o€rqrou Von Ncum.rnnl) H rur6ba!{ rolE 6ryoor€1'0ry€ !di'rot( oroa i6to t6foon, 'u o DJntzr! t r : r i6rr :r tnu roo6w,4riq tov lcr4t;rrrcov:rpoycr4r lrr i tuv ptt t rrr(lryvidLs firv{il((,,!. 'Ero!, orFraQ(ivovx(lg ro o€dpnrc nrinimrx rur6 rov 6!ioF6,cvlrnrQ6gorFE rlv roroQi(1.

To 0e6Qru( nrininrirx [nopri. ev ouvroFi(.. v crno6er"x0ei rDg e(i;: Emr,r ]ro brtrluqLu-orrll.1 a;rd rl porribeE xcL c ro brhvuqrcr-lpcrpp{ cro rr povci6e5.OFa)Qriox€ rs 6uix(i yaaFFrr.d naoyadNoro

(P) E),otroronorYtor€ xrtv cr)llno rlE otlv0rtr€E ,4.f > l). r > 0

(D) MryroronorYlor€ rrtv y,]Jno r( ouvolx€< yA <.. y > 0.

frrr vo €qcaF6ooutr€ 6uiopo. rp6rer vc r(rcorr oiyouaoL 6rr*! rc buo 'rao6l,i'yarn eivar eq,Lxrir, droL rciv eivcr cvirynrl ;rpoofldtoqr€ rov i6!o F€ydd,o crpr6p6o or 6Lo rtr ololt€ic ror A. H aL),cryj crn{ 6ev €nler(q€r ltg 6i}.tror€< orQa-rl'"/!)44( ro! rc!y!!bro!, brolL riro€ raQboq auE(ive! xurt a. xc! 1() iblo ou[6ci!€lxco olrr nrininrax xcr( maxinrin. frct rov fiQo,aujt,rovro ni\r(rzo. Jro! ou!6oliqolF€ir(h)'| Fr I, ro , = 0 €i.!aL fftD.rd oro Suiro ,r(( o:roLobrl:rort fpy(il.o x €ivol€{r}.ro oTo (.Qtrxo.

To HruQlt'cr ou-rq",r' rou./Qct't'rzou;rQoTQat't'.l1rdtrou ryyudru' fti,0u oflun(iaryouv r' x(lL U' tr€ rytv Lbi6rrtrc cr- = J'r. AoyrD rrDv Fovd6{rv mc /)x(L c, cu6 orlp(iver 6tr. t/i = tvi. Eirv tG cropoiopc.ru aurir Looil.[(l!tr€ P, r6rs 6!c!06vraE pe ro d. to ir0poLoptlyivetur I xctt or ttpoxtmouat<rcuris arpurnl'ui< .t'/0, xu y'/0 elrat 64tutrt<. fro ono!€obritrorc dl).eEorQ(rnylza( r x(t y 0c rozi€!

. 4 . r ' - ) 6 o w u c i l e r u r y A r ' > u b = \ t u L g " A : t ! c o r n e r c i l e r o r g ' - r t r ( a r = 1 .

To evbLcrpdpov 6pioxercL mo 6rL y',42 < | < UAt . Arcrprbvrc< pe 9, tpavrpti-r-vEtul 6rr o n(i.zrlE X b€v F,-ropea vcr z€p6iorL jreQloo6r€Qo cur6 l/d evawr qgorpcrtriyr,zq; y'/d zot o noi,ztt,< Y b€! trnoQei !c Xcio€r ),!.r6reQo ('16 l/d 6va!1!

R. Of . t ' ta I q I N to^ t \A l -o o l ]ap l lM^ MINIM, \ \ 513

rIE,rr'/d. Aut€E Lom6v €ivar. oL orAaq./LH€E nori r(Q6n€r vc €rrt €yorv rar 16r€m a x i m i n = m i n n n a x = 1 / d .

Axr6 ol,oliLrtQ(i\r€r q e€Cl]aio afldr oq{v€L avand.!"rrtrit xnv ftro quorol.oyrrrisAdrrryI on' 6LEE: Ilor(r ofi'rcr ouvlerol€va naryviSra €ivaL rp.i1$oxr looo{vopcBE

r'naryvi.6ra nL\ny.l]Jy't Er.ttioo€7at io oxtixt, to bridge xa. @ ir6xq atu,tLaiorc lnE qEatAIaE von Neuman4

Mou qoiv€rar orr ro oxdxL 6€! €.yrdoo€rc( noLu laa)d. yla 5uo L6yo1,!.Kcr' ue1ry I otqarllLxl yLa ra lau{d 6ev oriorr),€iro( ofi6 ro irvorytra xol$ovo. Alj1ll fiQari€! vcr r€ad.(IF6dv€r or6qaol yrc ro r6i 0o ofio.'rr4o€i I irecirfllxivtoq ro! $o!aou. rrcrr6nrv :r6q 0a rrncrvirl0ti I 6srirrQrl. rtclr rlcp6[oLa $61arro 16),()( 10! n(!y!!brou. :€ rdo€ 6rl[0 lrldplouv 160() fiolL€! eva],l.cxrLxegoao€r( l,rc rov X. dror€ oljroE va ax€r oryl! 6rd0e(ni rou 6lorxorol4ruala a[!yo)lor0arl"/r16v. )iar ro i6!o ro1!€! xar yra tov awiacl.6 xoo. :uvtfidq tc r,rxcr n 0ivaL arior€rir(l $€t,d),c(. Ear6g owou. 6€v 6L€rr,) lidroLo 06),() yld q.,'ruXl. Ea(\' o l.€uxoE ttroQo o€ ve 6Qe[ luc orpcrqltzrl nou xep6i.leL f eriv oB(uQoq gJIoQo]Jo€ vc 60et gtcr orporlyrxrl roo:rali.aq -nQaype nou b€l 6X€r yivsrlr6lAf orlp€A(r- (tuio eo 660!€ 6rcr rd).og oro axar.L. d'wty,ir ro nc([y!ib! edouv€Z!!6rav v(, nci.!ct(1. o:rtog I tQiLLlo, oL),/( o €leoljorcoF6q 0q lrrrve vu

Avrio€ru B€ 10 axsy.r, 10 bridgc neo#l€r xir:roLc eEcrrdrrlol -y[c ncQd'b€ryla or crroqdoe( oE !!c1 ouv(1Llcr'/ri. Mroo6i vrl 0r_o0l0€i, oa! rary\ri6!fii.\,oxa, olLo)g €i1'or criocrvo ll€yad,(.

'loo)g lilno!(l B€uo!(uBivcrtraQl ro! no[y\,[6!o! [no0orioa.v ao ovs),!0o{! oS nQol rr] tjd}.rloIq orQarly|"1iri.To ibLo or[6(i.vsr xar tr€ ro nd!a].nJ.6nor o r:xrtl.rorrjq x(r o tep[croqu].oragrlQoon(loor]li !o [ovr61'ouv ri]v €trd.o q; [EQraE o d\rcq iou ir].].ou.,N 'Eva

lrrpo [6QoE rorr ]iaryvt6roti Ocr proqoroe :rh].1 !a crro$o\,(,)0ci. xo! va (1vt1],!0si.A16 qv lrD,I tttQtlt, ro bla&Jdrk 6t1, rinL :raryvld rh,.rxo (xo\l,iN.to-to\

oro zaeivo) brorL ro x.rli.,'o o.roLor0si oro0sQoq rcv6ve<. Orav o qriLo,< lolrEd Thorp 6pri]{s rtlL bn$oof€lJ(]E [r(1 orpailyuirl nor xep6iler €q.1vrr./1ahEo!r(1,<o€ pra ol.Lcyli rov rnv6rov rolr rr.1ry!(6rou oro Las Vegas n oxF6oxhi rorEE(pritro n),I(x,,E tarir 1v xcrralpcrqrt ro! xtler6.,' :io! ri.tcv 16I +{pavro0€i.Arv umtpX€ ororl€io rn716, irpa onre mu" ptu-flt orQcrrlyrzri r'.

YTi.iQXsl r:'riolE ro dir,'i,tlpc xotr at'lctuo1titor. 6r(v o!),Luy6tivovrcr. 6iooriv€Ayoi.- :rov xtioE tvc leXorpr.mci rrivercu. dvaE outr6r6aoBoEr OFo),oyIo€ r{arsioaL €),€!g€poi. un6 rov oqo o-ft o oUv€vo16g oo! b€! 0(r oFol.olno€r xf axr6q(roft o ori!6voloE x.lra8Lzdqer(1| oE l01p6.vrc rpr),cz{). Ehv o[o].o1ltoor\r x.rroL 6uo. 16rr xci6e ivcq zar.rSrxd!€rc o€ 6 Xp6vra. Ed! b€v oFo).oyrioEr ro!ri<,rort ro Fovo ro! crrobErrlntrcL €iv(Ir 6vG rl"u{taxrrao 6"1x1.1[c (2 1q6vrc 1L(rro,'t(10€\'c). TL va zcilotlv: O fi€LQootld< €t.,'(lr no),! p€./i,.o! ol.Lci, av ea$noQoroov vcr 6croLo0oN o 6rd! oro! dl,i,o, 0o rov dwelcv. Auo eivnL irn

r : r.E.: To d!0!urz6 naoildcrt'tla r.r d{,ya(l !o rir(r no o!r0{o. ci6 ro Tdao ro! bNebrll,

[. ro oroio o i].irt!a( (!(ivriiorr< bE! Eildr. (o(fdrli. .Eora.rDBi!o..

fPANIIVIIKO: NPOIPATIN'-{TD]TO> X-{I OEAPIA IlAlI'NIAN

'rcr.i,vi6L trl Fla€vrxon ae0oi.oFa.roq. Kor" or buo $rropEi vu Xrioouv.To r(1"€ro nrlQdiEryFo ol,vrl0ro[6rou r.rryvLaron rou €i.var €ri.otE raryviSL

rivozc €il,or ro Ji(ix€Q. H Frl.6qa €ivar ouoLao-ftrri xcL lcl va €ivoL cnror€IE-oFarrrri 'ra8r€L va FIv €ival nao6),i1,qrl. Eriv 6€t€or€ rI e€FELrdrbrl ltlT6e€or tcOropins ncrL./viov.6rr o crvrinaL6g ocrg rivor €lunvq. r un6rpuol t4 pn).6qu<nAir€L va LIqO€i ewelcbg rryaio. (Ed\r o ofiifioLoq crvtriqqrOei xcinoln encrvc),Irrrxri obrzooia eo rupbioer.) Or nL0cr,6rltlrq L'nip i r1(xd rL< gnLdq)c<ocr eEup-lrir'\,tar (lno r(r Z(lerr(l fiou €iv(! (rvolxrd rc! r(! oror ipara. :-rlvnA!ryp!r{!x6fl11o. 1(} n},f0o\' rt)v €1c)'.},crir!x(iv buvcrorllrov, xa0lora (lv6(frr{Tlrql e1jQsol I([aE cno),!I(']: 6[),1101l( orQarlyrxric r'. noQ'61a (r!'[d iva( xd'6<fiaix ls'ro! noreQ nQaTrt:['\ro qiJ(iorl fio].i xovr(i oro ra'. ]t(r! ToiTo pTroQoups v(rro ),oycQrdoorJpe c,{e!60)s, rq' 6ooa 6!?&nlgr iltv rn(ipsvrt r€adorrc cirloroirlort

o noixrlg X brorQc.y[oleiwdr lvcv 6al,! il iv(r prl]s. trr io€E fir0a!6rF€q. evd)u Y nciQvt ' n(!r(rrL vr! i t r ( . A,r , ' ,1 ,) X zorr, iE.r r(r X(,or{ i , r l t Xror{r ror ' . Firr:rirar :riroo xcr lirreL ro rorripropd rou rrou tilcrl I $ eir€ enolEdv€r BE 2 $.Edv o X enoljEio€!,16r€ o Y eitr Jra(€r Jrdoo zur Xdver ro brno ro! fiovrdQ[oluI $, ci-r6 c1noq,cloiler r(1 Briei ora 2 $ ,rc1! va bri crr o X pfiLo9i(Qer. :' c]Jrllv rlvnrQi.nrr'rorl ro xcLurcoo Zaori rlrQ6i(cr-rc 3 $ nou fidvrcrQe o c!.,'li.fic!loq.

Arolll xcl! o' ctlro 10 ctntr6 a!]aidr. o! 'ha0aedi orp(rlywds" 6r! rira!rc06),or nQoqruvsi( ,}icr! €ivs! 6r6oxrrx6 r(l rLC xoi(r1Q(hNu[t. O Y €ZcL 6lr()6uvcr6-rrlrrg, c<1rou trur6( ro p(i\ro nou ,ri1v0 ri.vcl! v (1fi(r\4(i olov Xl

( l ) Edv o X novr( iQ€r. oYn(icr ndoo.(2) Ech.v o X novrciasL, o Y ru 6li:rer ruL n(ier tra rc 3 $.

O X 6Icr 6Zcr rio{rcQ!< orQ(lrrtyLx€5, peQlzd; lolrz6s zaL [Epr]r6< rrdQd].oyE6:

(l) EncrEcivrL xcrd 2 $ yLo dvo! Qny( rclr nd0L ndoo ?L( €vcv 6cr,i.(2) MnLoqclpcr xftr orlc 6i,o :rcpr:lrtixrurg.(3) ndr€r rcioo ltaL urL,< b6o n€Qffildro€rE ,{or ld,'€r I $.(4) nci€r rcloo mov anlcr rcr enuuErtrer otov 64tr6.

Xa€'irq€raL Ii./I uroFov4 yra rov rroloyroBo ror ril,rlxd x6pboug:

.rrr = U 6LorL o X ldlEr 1 $ rov llnt6 ZQo'\,o $r 6.r].i xar x0abi_kL pa prty( (o YJrirE! Jrooo).

arr = 1. 6r.6rr. o X xriveL I $ rov $Lo6 ua6vo roL reqbiler 3 $ rov ciLl.o pro6 (o Y:rpoxcopei, :rcp' 6),ov :ror o X ixer p{ya).

ar, = 1.6rorr o x iraoT.ora€i r(r o Y nd€r ndoo (I $nloqt{:r€ raiv€o.ar, = 0. 6rcrr o x zraailEL 3 $ uE rov arila ;aor xirv€r 3 $ F€ rov 6a)d (rt Fr),6qa

anorrlxdvEt.

S.J OEOPI IIAI|NIAN KAI TO OEO?HMA MINIMAX

H -rai-q or0arlyrrrt td!€r rirrror€ I $ rcr I r6roQrl ei.,'(lL r(cil,L urrrrtr,Ti,<:

,=l? ;"lH 66ir(mI orporll'1111i yru tov X o arrt6-ro rurtvibr tivcr" vcr p;rLorjirper. rov pLo6

Xoolo. . r ' = ( l /2, I /2.0,0), roL o crr)({< Y raart-r . ro rrL), !1rr . yt = (112,I l2). Hr+Lri rou nar"/vr61oi rivcr 50 o6vr!.

IrAirAyoq o roonoq ctl16q $€ rov onoio r€).€li;)v€r-ro 6r6Lio, trc0(ivoard<oac xrlc v.l nrlilrr0 frLa ncQqno(llrdvil rr(pri6c rdl€Q. fltor!'i(!) olr(u< o{r !!xo[rlz(rL ro :r6rep €tfL rt {)6('l rou or4 fpapprzil Ai./!6Qo r(r c tq,cQtro./i-q rlq.Eiri.l(D ro 616).io vc oa( rlJr,?o./{;,ylor.

ATKH)]Elr

E.5.1 ltri)( 0( r'nrtaf(!'()()ir or 6i),1urfs ora(nlttx!: rc'r exrro(, x(rl!|bLorr. r(i! xa 20$ (fq,l0oi,! ds 70 S z(r nc)l( !ir(L ,t r'trit (ro trioo aiaboc ror X) rot, aiou cljroi,

8,s,2 Elrfiitors rov rtro].otro|o ror Drxi$h (ino ro! X a(L rot, nririnu (ino ro! Y. yLrl

r n \ r , v . ) < a x ' . , \ o , : . 1 - i ' . n o , ' c o " r . " , 1 1 . r ' ' r r : . ' , L r . r ' f i , { r : . . ? . d , ' .-

L - i l l

8.s.3 F{ia ro ail ria(r ro rri?toro olorl.io drlt la(trftri to! zat ro trL,a6tc0o d l doiL,lror, tr(ri d X 0( bLal,irr i(inon.rrl oritirt j a(r o Y r)i! blulf/fr advrorr nltaqqoi ; (ollqiiQilrc (no ror,lrroioLno :ri!(xc): Afi:r0 6rL rl rQortyoi,rrr.!rt dozrpruirr ild riroLo drotT€io. rfu ndt.Qa r(I(dzaudrr da(r A iToriaa pI! iler.

E.5.4 Ilqrirr trl di)mmrl ore(otyrzli Y.6ci(ou(< ii.iAll U zdL t y dTti ?aqrlrda rolr Aa( a.ilovtos ro yqLiqt[( zoL roa terd'r orvrdroo.b!. O X (k1 o!??€mA.)efi dn]lel(Ii,rfrnl duu]x{lou (fraloltaqrtri) r(' o Y neinsr r( sro1tdronoDio( clro rotta?toto. [oLo Ei!( ro I' a(r ioro.i!(r ro minimax tirloq (6ro".x,'t,trr:!rl /,]qrt|j)i

I , + 2 ( l r )

r+ l ( l - r J

I

, = [ ; : I

f P MMIXO: NPOI-PAIVTMATI!MO: KAI OEOPlA NAITN]ON

E55

&5.6

Besire q 66l,rrorn oiaaTtrrfli rou X. yLo rov fbLo A, fta! 6efE e 6rL xqrptroroGiF6vo rq 6io oni}.c {qv rdrrlxst reirl} nou ouvaw,;wdr oro orltrrio mimar rou

Bqsii€ ns br,o 661.flm€s cedqltxdE xsr Tltv r{r'1 rox rsLyvt6roi, orsr

8s.7 Yrued(]tr ".,

/ = [; ;] nora 6deq ..r xor (r rrr 0q 6doou! oq milr r'l

powi tu ulT ior noLa 6dql| yr xor I gr ons br,o.raqppas. Oa 6d,oow 14 #ayac$,i l,i rl: a€iEft 6rL u = u.

E.5.8 BA€irf r{r r' rar y' ror rlv ittr4 U tLu io.v

f r 0

"= L3 3il

E.5.tl

E.5.9 Ynoloyim€ io

E.5.10

J''9, Int$ (rrer +.,rvr).

EErl}il'tt xder |!a an6 rLS av$dqlrs orry (5). Kar6rrv, ceon ro 0€6erya minim0xrLs t.rcraa!,€r o. rd6rlrcE. ownradvcir (Jr(ih UE ).6yra) tl! €Erodroas (l) to!o(lYFaflxou olF€Lox.

Ytrolotiloros r( y-4r' acr y',.{.r a(r .ndl't0€no''!oE t's ow0irrs (4) €!G o.1tF.r-r r "ou or t r for , 60iE e & ot r ' = Q, /2, r /2.0,0) zat ! t ' = ( l /2 .1/2) . ivdL 6drrror?EorQatrlYrre< mo iorq.'

'EXer oro6trr0li &( 6r! fnoAFi va ur.iqr€L mqarlyrxrt ro! [ouao)! no! )tea6iE€rnswort oto otdxr; flrmfrr(D otL .lur6 Eilcr1vom6 povov 6tav or rciars dxo!! 6uoxrvlseE o€ xd'o. riao. Edv o paieoE riXr xdnoK mA(fl'tytxli rou z€e5lE€r, o ),Eu,irig0( fnooono€ v( rLMio€r ivo! inno !,rao< zar rioo ,rcr zor6nLv v( axolouo'ioFt atcwq (Ei nr cqdnryLz'i, ned$a nou e.r o6qyoior oro anopd6€zro oq!r6A(ot.!6t| €(l

"6a6Liav ,ar oL 6no.

E.iv o X brd.dy€r ivdv fte&o aer{)|lo ,{ar ro1'r6xaova o Y !ow€n& a! our6q Firorr00tri6q i 'ianos (xdlowos rj ffobilo!'rd( I $), ftoLor dxer'r),covtxqpai

8.5.12

E.5.I:|

' :.r.E. Ala(iordt mo ndaab€|Yl(( fiK or(rsidq.

3,5 OEAPIA I1AIINIQN KAI TO OEAPHMA MIMMAX 5 1 7

E5.r4 Ai.Snar &! o x €ildr onlo0o$i)'oros no! dx€r Eft 'oYlt l/a iqiq€r i 10 n€adoot,*xar o Y €i'vor o dAnyoE rIE d[wo! rro! qpllveal €vovriov to! ta€Eiparo! li rouri€aiiopdros. 'Eoro &r o nilsxqs adeSorr (oE'yrde6t5) €irdr

A=12 8.1 dFwdivdut ratELFdrcc

th ol d[w( Fvav( nraaqroroE

redE$o Jraeasto

noq et\]di or 6d)'rloieE otaorl"rfllg xdr ro pdoo xde6os ad6€ raIT !16rcU

+ t.r.E.: Aadqdanar mo dllsarxdt{o nobdoaqrao.

IIAPAPTHMA A

AIA>NA>HIAIOMOPOAN TIMON

KAI WEYAOANTDTPOOO>

Tn! rairl trt_Td).I noQcyolronoirlorl. nou olllrd,rlQ.iv€r rr]l ta,' rqq oncloL.priix.lr rn! QP rrtq oQoozcvolrxonoi$rlt. (6r(6l,{qoi€q G.iuss x(lL Gron-Schrnido.r lv oqlo,r trr yr ' i . ! ro ro rcrpLiArl t ru. Arr, t rr)n r(r dvoprr auvt log zru r iv,rryvoort ocrv 'AI'f '*

n 6uiotrto4 droytiprptot, rqlon. Oo ;i(nxtnc0looup€ vo xrtvn€QryQ(hfoulLrl, \,( q! arlobEiqouue xo( vcl ouqrlr4oo4r€ xr6 r(l(loFoye< rlq -noxeivaL ;rol,)dq rcL trulclvolrar olveldrE.

H AIT riac! ouv(i ouv6ebepivrl pc qv :rupclovto:roirpl tbror4rdrv - L6ro6tcr'vuopirrov rvog oupyerprron :rivazn: A = QAQ1 . Exei or rbtottpd; mE6Xorrcrorov bray6rro niv(za A zcrL o nilrrza; lo)v t6LoiLcvuoldnol Q tL\er oa9oyti)-w,'<: Qtq = L bror' r,( rbrobruvu'rtrurc. Fvd< ollrtrFrorr{ori:riv(.z( trropFi rrl€n).elouv opooycivrcl. f|c( rou< rrserooor€eouq rircrx€E (L'ro 6Fv tlU0€[€r Lt(|y"rl rcrpcrll,tL6ypr4rpouq nivr.?icE rtvat lc),oio. Edv 6trorq EjTttQ6qoupe o Q drtr(rgmEeo rio! o Q/ oro 6€E!d vo €i.,'ur d!d.liQtror oQqo)\b|'nt raxe| v.u oXLavdorpogoE o ivcg ror dl).or r6rt: 11 ncpcloworoirlol lfvetcL rdl.L dwcr{.Eri nftov o 6Lcyti-rvroE (a),Iir n.ec1l.Ll),6yad[[oE) riva],"aE oto paoo! trnoasivc yiv€r frl cQqflroE,0o o4r6oLilera| B€: t{(| rc e€Il1til ololx€kl rou. rouovopirlovtcr duipoaEq rltis aor ,,1, uc oug6oLilowcrt tt€ dr , . . . . rl,.. Ar.rdrzutci,apddnou! t4 r nQdr€g 0€o€r; orry z{Qra bralt|lvro rou:6nou r €il(.r Irdria rou .1.

To zler6f ot1 8oulerd g€ rtlaui.LlloyQoppouq nlvaxrg elrcL. o1e66v :rciwa.n eEtraon tcor ,4.4T ztrr /?A.

' t,r.M,: Ard ro Adomml l6r!1riaq{)! Trt(drv.

NAPAPIHMA A AIAlNA:H IAIONIOPOAN 'I'IMON KAI VEYAOANTI:1'I'OOO!

L.a6ndlt t t6tolt6Pqr,r t$i* Kdo€ nl €fii n nivc'roE -4 noQqyoworror€iraroq Foeq{

/4 = Qr:81 = (oe0oy&!oE)(6rsTdroE)(oeooT6vros).

OL onii,eE rou Or (m €fti m) 6ivai r6!o6tau!opc-rc' ror .A47, nar, ol odl,eEror Q2 (n rni, n) €lva! L6ro6ra iopqrcr' :,ou ATA. Or r r6r6goqqeg ql6Eoq 6iayov.o rou > (m eni n) eivcr, or, rerpc"ycovrz€E Qi,leE rov t|!l p{6ev[r{d!rbr,orqrtiv rdoo rou ,4,47 6ao xt rort AI A.

,I..porfprro{ I fro O€trxdr oAroF6vols nivsx€e I irce(ryoworoiryI ouq rsurtE€,rcu pt tr1v QAQT. fLcr croptmor4 ;rivcrzq. zci0e crpl,tFrxi L6ror4rylrou A./iv€raoxov : 0€rrrri 1i(L o Qr €ivar blcqoQl:rrxoE .r/16 rov Q2. ncr prlc6uotg :ri.vcrxq,o X ntrpapdver npay[(rrlxo(, a],]d oL Qr xsr Q2 /,vovr< oafu4tolailaiot (1oBll,abtxo a\ri(),oyo roJv oq0rryoxirov) xcL ro .rrr,4 = UlfU,//.

nses.dAnoq 2 Ot ari]:,eE ttov Q1 nur, Q: 6i.vouv oQooydvlc( 6(iodt xtt yKr tou<r t oo€ Q | < 0t lt € luibt t € un i fopout

oL ;rpdrreE r orr?.eq rou Q1or rtrl.urcti€E m -, ofill,€< xou Qr

or npdre< r orrlt'.r< tou Q1o! T€),€urcis( ?r , orll),€S rou Qr

XdrQog oq)dv rou -,1

.rQrol€QG Fn6€vo1(|)Qog 1ol] .4

tdrQos 1Q0p[(0v ro A

ttlb€aot(oooE ro! A

no,pnliend|t 3 H Ll'l Enr),tyfl curFE r!6 6doe|; F€ €vc(v eE(|esrl].(i €!6r',10 rQ6no,drms vo €[vc(r r.dLrr n€eroo6r€Qo afl6 oQ€ox(lvoytxts. O 1] o A no .airlaorAE€lgta ar4).4 tou Q2, t6rr neozi:trtt ptu orti,lZ rou Qr. Arn6:tpoxrinrrL 1416o14crr6 r1v nQ, = Q1:, c1' rlv €€€rdooopr: {mi},t nAog otl},r1.

naanqavI 4 E r ' ooov o rnro: Q|IQI rivnr oruoroc. ba :rQinn v(r roXuorr!

xrr or ouo)(cfloFoi trrruEi r,," AAI x,rL ,4 fA. Alno q aivrr,.r Fl1xo).'1:

AA' = (Q,zQ!,(Q.2,qI) = Q,>:rQl *( oFoi(,E ,4r,{ = er:':el. 0)

:uFqova Fs rlv nQ6flt, o Qt ite!,ttt la Eivat o itivuxea dtotptir, rcu AAT.O nivczcq r6LorLp<irv, oro p6oov, eivcrr 2!I 6omi1o€{Dv r}r €ni rrr trc oi. . . . . r;oryl braydvro. :{pqov( [€ xl] 6€ur€QI. o Q1 ftElntr ra thtar o nivuxuEdLobtat'uopriron, rcu A",{. O 6r.a^7<irvlog livorcrE Ir I iler tn i,bra o],. . . , a],€iv(rr otr(l)E n €ri ?,,

* :.r.E.: F!rr@oi.(,. 7(\ovtzoroqdru

Or 6mn!or6o€[E or'tsE eivar euxor!€E,6€v o''o6Er1.vuol,v 6Fq t4v AIT. Han66€!EI 6€v €iv(lr 6uor{o}.rl rat turoQ€i vo oxo}.ouoio€! F€rd ro nc1Qab€iYFdra,(o! rE eqsQFole(,

IAPAAETMA 1 (o ,4 6ra1d!tos)

nAPAAEIFMA 2 (o ,4 cxrL Fia Fovo orytU)

520 NAfAPTIIMA A AIA:NA:H IAICJIVIOPOON 'I IMON KAI q'EYAOANTETPOOO>

Ebd o AlA ohqr I €ni 1, t\6 o AAI sLv(rr 3 cni 3. Kot or 6rio dtouv rrtvlbLol$ri 9 (qs orioi,ClE I terpcryrovrxrj QtEcr etvqr to 3 tou >). Or 6uo p4bevtx6grdLorLFaS rou -4,47 crgiivow rd!'rora €tr€oO€Qiq oru tbroblcn'uoporo ttov orrl)'6v2 xcr 3 rou Qr. H €n .oyfl [as l(ov& cwov rov nivoxa oQoo'fd,vto.

nAPAAEIrMA 3 (o .4 eivo! {bn ootsoydrv|oE)DLrl A =QII etr.e A = IIQ f o,x6pn A = (QQr)IQJ, olyouao 6Folq: =,r.

nAPAAEfTMA 4 (o A civar nivoxcE n€'oonrdor(oc AA' = | : I I' I * ^ =

3 . t )

o=fi,' '' il=1,'lt !,#llt i',1

EOAPMOTE' THT AIT

@c xcrn:ucororigr pe peQLmE €v6lsgiQolJo€E eqcp4to16g, crpou ogc4 toviooul.tt6vo 0fu(I-zl.€r6i. H AIT €ivar 0E Q€rrxri lta aer€trryrxd suorc0€is unoi'o1rolor-5.O rla6rog l.otoE et!tu 616r! or. Qr tr<n Qr ei.v.dr oQeoydvlo! nivon€s l.at b€vcl,},ir!ouv ro16 ro t{xos bnvuopsxos. Ener6rl lQcl: =.rrQr.r = l;rll:, o

ti +l li ; il [i i]tl rr| ' l I l t t z l3 2 i3 I f3 l

,1 =12l=12/3 - t l3 2/3 l lo l l r l .L 2 I L 213 2/r - r l r j Lol

| 1/vG -zl\G I l\Gj| - t lv2 ol . tq r l !21 .| 11"5 rl'/a tl\4 )

* :.l.E.i O, ,6drrft ro! A?A eir(i 3. l.ll

ro],1.(rfiicor(ro[6! U€ xor q 6€1r ricrrcrmQ6f€L fiord rillr rdEi] [Ey60oug. AE1,$rropon$€ vo rolUAro0oiF€ ro i6Lo 1,|,(r rov ci ,o rr(rQdlo\lTa :. MrioQ€f lano),I.lriaorcioo![s $ 6r(1 F€td),o r rt ('rro ox d) va 5LaL06oon$F F'€!c trrza6o x.l|" va E8rr€acioortrs ra 6pLa ror riro),o1rmrt. Kcr ncitu oprDc o 2 ti1\u &J!)xd.irepos yh,trat. Arora),lr:tr€r trs axpi6€ra .n eivar peldl.o zar rr eilcL prrp6zaL 1 euzotrq np6o6cor1 o nlrtrl .r1v :r),Ipo.topic eivcl o brurepog ),61o_( qqbruorrzorlrcE -rI: AIT. E;rrorp6qolpe o cur6 ro 0tgu mrl 6elrcpl ryopyoyj.

1. EflrEEafdoiu tbaivt1 F.atu) 6I| iv(E 6oqrrpirpo_( rtr'rro./Qn{tlrL r(lL 0i},eLva m€i).€r rI (p(Drot$(fi(l (xt ft. H €rxova trnoQei..v( n€$6Isr 1000 rni 1000''rrzoloororx€i(it -ILXQd rr:rQ(10)1(iz!a [€ xoooQroBivo XQd)0(. MnoQo![r rcxo6rzo:roL{oorgt: i( XQl;tr(r(. o 6vc :rr6io pt:rolri pcign x(r dorrporr. z(rvo or€i)"o4tE o.q ry 1.0{X).000 aar0[ou:. KdurEoo otr(1)E €iv(L vo 6aouf€ rr]vouorcorLzil r).toofooi(r ror 1{)(X) eiri 1{)(X) ria(,{o }rcr. !c or€iloutt€ ctlrrt xcL

Yro0imr: or( yvoeiEo!$ rrlv AIT. To rl,rLbi €iva! o! 16(oloeq)€q qriq (orov:). tuvrl0c);. LLepLx6g aivtrr rirl[cvrrzis xcr ir],Ie-( cir(rr" efcrperrxri! prxpdq. ErivrQcrlioowe 60 ̂ o! ,r€rc(EouBr 94(). 16re ordL.''ouf(€ g6vo rlq cr\rriololleq 6() orrl),eqr(,)v Qr ,(ar Qt. Or drLtrcg 940 oniLeg floD,onlaorirlortc{ olo! Q :Q.{ l€ $LnQilo. 1c orroi(r ayroorrlr€. nOdy$arL. trtrogotlp wL x"rit,otryt ror nlLu:ttunnr4drn<ixu, or4 p11t1l otij).a ni y(!qqi:

Q E Q ! = u p L L ' l + u : o r ! ' + . + , ' , . " , . t ' ! ; . ( 2 )

Ebdr rc u si\,(r or oril.sE ro! Qr zcL ta r, rivcrL or" o'rrjl.eq rorj Qr (ro |r' si!a!1 :rptirq lpcqprl ror QI). Kci6r :rivaxo< ei'vct d0poropc l rrvrixtov tdlrt\ iva.Erir zpurrloouFt ptivo 60 opox. 16rr oriLvoUpt d) (|ooa( 2000 (10l1)$oi( ovri 1l,(l

K.rodrE fiQofii,0e r'ic! oLo xc! firQl(]oolrQrg LfuopoQ(lr< x+6G, o! rrnov€s Eir(!np.il$crr ExrLnxxrr€E. :rn.,' cQlli 6€! 6r,6n€L( rifioxc x(rr E.rrp\rrzri (1\,oy\r(,)QiL€6

2. H utoitkolltt&i |t14:ro KEqd),arc 2 bE! Srordoo$E \rc opiooutr€ rn\r rdEtt-v6c rivcrlicr. 'ro! {rcv o crpL0p6< rcuv cveEapnjn'rv ypclgd;v !, Loobrvr4ra, oapL0priE ttrr crtlug!ruv oq)"drv. Arrro propei vu tivcr 6rronol,o vcr crorpoL'ooEi. F€ xnol.o./roFonrl H (lQtr.zri F.q Ud0o6o! lrov v.r FErodF€ rorJ! o6nyoi; xolatl[o sircr ro o(.oro orit\ xrEpirroiol r$ .rxpL6ouq .10r0$ilrLxrtE. :rnv rrQ.rltL.rrrxrtce'OlrFrrrt otr(l)q trroa€i v.l raat1r].a!I0on$€ ro vtl raa(6laroritrs otror< rorEprxpoiq o6ryouq bev eivcL 11 oroonj cmdw1orl. oeoip{ore roug

IlAPAPTH\4A A AIA:NA:H IAIOI\'IOI'{,ON TIMON KAI'I'EYAOANT]:TI'OOO:

l;;lII(,Q' 610 fiou ro oqc),pc mQoyyuL€roq! 0o rioQily€r xoxa nooo rreovornro€\ra\r SEUrEQo otutyo. o riQdrxo,< rf\,oxoE 6X€r rdEn 1. K(r or 3 o obryoi 0c ei.v(r

'"' l: ,i.ll e ze lI t z1 ""

IIAPA}THMA A AIA'IIA'H ]AIOMOPOAN TII\'ION KAI IUEYAOANTIITPOOO'

Frl{Qoi., rr6oong }.olfio! vo oll,orioolJF€; O 6€xreQoE 6ler 6vo $LXQo (:)6rl^/0, 6evfrnopon$s 6trt'i! vo ro1xQroooljl€ 6rr I yaoFFyl ro! €ivar" aotF(tvrl. O rQiroE€X€r 6uo o6lyouq kor q rdErl roo €ivar 2. I "a'ror€l.soFarrx4 rdEI" ror 6[(l_( 0a€ra€n€ v0 €ivol, 1.

Ilepvdye topc o' evo ;tio euotc06g p6rpo tylE rcifieoa. To npdrro 6rpcr eivar1 1prlo1 rou Ar A i AAr, rou eivcu, or.pperprxoi c),Ld 61ouv qv i6!(l rrihp€ rov 14. O! !6[o1r[aE iouE ro iE-recryrDra (D\r r6!o[6eq0r\r 1udv 6E! Ei,1]o!rraQc!7itrG1.rlrr#q. Mr 6dol filv nxai6Elo ro)v 6r6oFa\'{Dv, cu[oqcoilor.rpr Vr,c evarr€Qrgd)Qrc 6rcDS ro 10 6 xc{ nQoolnotroyi(oqrs [6vo lts xW6( rrolr ro EEjr€Qlouv-6rol uroi.olitoulre rrp (1]rorel€oFarlx{ rdEq. :xa fiQollyoiF€va ficQoa€iylr(rd<rlr1 eivcL rtivrore 1. 6rov ro r eivtt prrp6. B66artr alri €ivul oouvefig, rirov1 r6r6poprprl rqrrl EereQdoel ro 10 6, eivql 6po6 trlyor€Qo jrQo6lnpar[xri 0r{6.ror ouvq0topdvo cl.1e6pLz6 oqrop6 -:rou eircrL crowefg cno a = 0 XruQiS vrrsrr!186n€r 1r({16!cr! 61.€ /Io.

3. Ilolui 6urofla6rl K'Joe Uy(r6rxq crQ(0U6S €i.vor 1rv6[€ao EvoE Ul qQu]rlnou(rprepou r x.r! e\r6E crQre[o! c'' rou Fova6rdior x r],o!: z = rcru. Aur6 eivorI noQdorool rou : oE "nollxig ori!r€ro1[16v€E!', Edv 0EoQouBe curorjg Ior)EaQropouq oa! 1 fii 1 Jrivox€E, 16r€ ro I crvlLolor:lei, aE a\d\ eetfta 4proQto!!1ioth)@ra x(Jr 10 e'tt nwtaioLtji oE ercr'r oQeoy6!to r/l,dzd, ArQl66oraQcl, €nsr6ri oe i3 eilcrr Ff ycb!ra6E_,{rrr, ! xcr\oflorlL 711r e-i0 ei0 = 1 , crflo:.c\EL i\o\ oQeopora\krio7 t:fii | fih'axa: UttU = I. na @r ay:\fiarlop6 rou yrl e€oQoul€ rov lryclbrr6oullyYl rori sv(iorQopolJ.

H AIT €ft€l{reiv€i arni rrlv nqpqyovrorroiloq o€ rivqx€S ororcudyFror€

Kci0e npcypctlx6g r€rQsfovlx6g ntvcrxcg ncrpcyowonor,eltcu, orq popqri/ = QS, 6rou Q et\aL oqqo,ltirr.og xcl ,9 €ivs! ouppa.E 6E xu qeaxdqproqrapivq, Ecrv o A ar.nqrQ6qEraL 16r€ o ,9 €ivcr! eerlrcd oQlop6voE.

fra rtv anobflEt. on),r.,E n(Q$6dl),ouFr rov QaQr ,1 oro lioo! r'K AIT:

A = Q,zQl = QIQT)QIQ!\ (3)o raodrory 5 = Q:)Qf eivcr nivcrag outqr€rpLx6g rcr ee(rd oqrop6vog (br6rrxtrL o : Ei!a!). O rcrpttltllv a = araT eivcr, op0o1d;r'rog :rivczcg (616rr

QrQ =QIQTQ:QT =t) !tr1 $rltr6tzrj neqin.rt.;oq o 5livtrn EplrrLcv6s cl.riYLa o4+€reLli6g xol o Q l,i.\,er(rr. oQ0oFovobl(rioE clri ,yL(i oQooYovloE. :tq!crl"rLorqeqqrq nepinroroq o : eivc! oQlol6voE 1,ic! To [6!0 ouF6civ€r ]i(r|" ]r€ lov ,5.

TTAPA ETTMA n0^rKrrt atArllarHr A = 8s

f r 2l lo r l l r r l13 - r l r 0 l l | 7 l

NAPAPTHMA A AIA:N :H IAIOMOPOON 'I INIO^" KAI \YEYAOAN II:1POOO: -itl

NAPAAEITMA ANTIETPOOHI I]OAIKHT AIATNATIIT ,4 = 5'Q

l l 2 l f2 r l lo 1 ll : t l = l t 3 l l t o l '

Or aorioag qrcvrpdrvow nds orlv crvriorgorprl 6Lcirc!1 o S (l)'Llllsr, ol')d oQ nctAdtr6ler o i6!oE. Or ,9 acu 5' t'ivcr" xcr" or" brlo or4perprxof xcr 0€rLX(ioQLopilor. br6r! o -4 t:ivcrL ovtr.orpdrlrL[oq.

ttlttirloq Oe p;roporiocrpr vcr E€rl\'rloolF€ cno qv A = QS rnr :rq1ni.vo-!rtl! neo< t( :rioor va 6p,oiFr rqv AIT. Erer6i o "9 €ivoL ouFtr€rer"o<root'tat [e Qr:Qj cwa eircL r1 ou'10rqiv4 acpclor.roroiloq tbLor4rrir-16ro6!q1,!oFciro)v qAQr . I6vo :rou 61Lvt rri.),cyri outr6oLrotrou. r(ll

^ =QS =QQ.2Q! =QEQ!

Fi!a! I AIT. knoprxdr. 1no),r,zri 6(hor(ot eera;pefto 6rr uivcrL nLo oqpcvrr.xrl-ryd 6[c0E ttroxci(l} ott q 6laofi(or] r6totLoQ{{,()! r4(dv €i.\,(t flo Ortd.rdrbtc.

EEaepoyi qt A = QS MLcr o4-Lcvrtxil rqnppoli t4; ;rotuxriq bLcicr.rofi {verqtortl BrtXcvu.li r(l)r ouv€I6! tr€o()r (rur nro nQooqcr(. orrt aoFnorlxl). :F z(io€nclQcrtl6$tnlol tir(r oqpcrr.trx6 ru lqt-rpiooultr tov €{n€Muotro (rJr6 rr1 cnp6qrla(1! sltto (xQr{i(i)( nt1uloiv€t rl n(1(xr1,o!lonoi.rlon Q5. O oQ0oldJ\,roq fia'\,c1,((r!Q eivc{ prrl otpotprl. ev6eloFerog cxo}.ou0oqr!:vl u:r6 crvch:+}.cro1. (1}.}.(y!< qrrorlq ofiofe< ro uLrxo be! upi.orar(r nirol. O o1jptr6rQu{oq nirurcrq 5 dler rL<rbroqrrq o1,...,a, fiori €ivrr! oL oLJlrlrLrlorl( erp€l1,lroFoiJ (ll oqrfiiroqE). HbL(lyolvLoroirtorl rou {I,(v€Qdrv€L r{ur6,< rLE r6Lorrtraq biv€r rI rpuoro},oyrxn €nr}.ollirt-rv cE6vc,-rv nou, orro< roL orLE €).l.rirprrE xqg noe&./edqou 6.2. ),i1o,nc!" ,igtottitEovlE. O S rivcrL crn6; nou x.r0oai!€r rlv r+aagoyrj rpyou rxo vILz6 rcL(lnoollztu€r €].Clotrzri s\'dQF/€r(.

>+r€d)r'outr€ ott o S: tooirc( yr ,,1/,,1. rot' tivcr oqurerprz6g ra(r 0rrrxd.oproF€voq, orcv o ,,1 !:iaaL cvrrorp6l'rtro(. O 5 €ivcn rt oxgF€r0rxrt O8rrxd opro[6\,rtrelQo'rt,rvrxri pild tou ,4r,4 xcL o Q eivcrr o AS 1. Iqv npclpnrra6qrcr. rcr ibrc6c i.o1uc1, xo! orlv rlEQinr.)ol norr o A rltctv rupu "4).6ypa1tyo<, ttt'iaov oA' A nao,ly + It:rnd opnpivoq. (Ei.vo! ll rEQinrt,lor) jro! ouvovrdp€ 6!(Q1.d)!,6ql.c6ri 6rnv o .'l dler av€EderyFcE orylLr6.) >rlv cvriotporprl 6rrirtrEq ;1 = S'Q.o:r ivuxu< 5' r iv(rL I ouNurprzt r f rprry,rvLrr i piLc rou AAl.

4. D.dvora rt1Qriy."r.r:ro K€qirlrrro 3 60r}.qr€ q L{ol €i,qio(')v r€rQ(y6vrDvrv65 rcpcrll.q)L6ypr4rgou oumilparoc ,4r = ir. ;rou rpo€zuyr .116 (g zavovrzdq€Erorlo€|E A7 '4:i: = ,lrb, g' €vav ro1up6 nearcp|,ou6 yrtl rov ,,1. Or onli.eq rordrqur va eivcr (rveEdprrF€E 11 rirfi tou 6:rpur va €ivaL n. Alcqoaerrir, oAt A 0o il,r't F1 avrlorpdrlJt{roq lrcrr 10 o- 6€v 0c propo,joe !d nooo6Loeioe€i

,.ido€ 6rd\'1Jol1u orov FIE€\r6XorQo 0c [fioeorioe !a rQool€O€i oro t. MrioQoliF€

rdlAd v(1 oriFr),Ia6ooqr€ ro K€qcii.aLo 3, &(I6yovrcs- 10 "zrr.ur€Ao" l1l yro 1.d0rlpcr[fur6 ouorlla ,4f, = b.

Y;rdpTorN 6uo :tr0crri:g 6roxolicq ont ),ion rori Al = 6:

(l) Or ypaBld< ror,4 pnopei vl:l rivcr elcrpqF€vrq.(2) Or orri),€E rou A Nropei rcr eivar eEuqntU6',t,<.

Irrlv rprirtrl :rr:qi:rtt'roq, or t:€Lirr;ooLq lnooai v(l $t! 6ZoN ],uot. Atl[6 o!$6ai!€r.€rir(6rl ro t r:i!c[ exrdq rou X(;]Qou orltr6v ro! A ror ro K€qr(i),cro 3 6i.6rr I0epurricr: Ilpo6oLrl ror ir oto 1i;po orq),6v. Aari y<r to ,4:r' = D, ),rlrorpe ro,4- =1. fio! Eircl 61Jvcro1r f](rl!6rl ro l) €ivar orol Zdeo orl].(i)v. H n€pirrool(2) n(eouo!(lqrL ivc bLurpopcrLx6 s$ir66ro. E(iv o A eler elrerlBdveg orrt),e;.r6rc I Liol x 6€a 0o riv(L povcbLxl. nptnrL rc uc)dforpr 1ro €tbtxlt lrol rouA : 1 , x , o I , : r . ] . o y , i ] r v , r ' o o u t r ! , " v , , L , f r o r r n o t u v o x u v o v o :

H $ilt@fl] lttott tot' A.t = | €ir.It ErlEir\ flor aXtr ro x).dYnn pi"og.

Awl rq 66),rLoq Lrol Otr orp6olilor4re Fe .ir'. ea €t",'rlt il sn .o1tl no!jrQorlpor.itlr. cul lq),!reol ),rro1 rol ,4rr: = b (:rorr 6ev sixc iiol), 1too{lE eniol;r{tlr rou At =.1) (ro! r'i1€ no '6(). O or{onoE Fiv(r vc nQoobLo0ioo![e ro..r ),|ftlEerrvorpe p dvc :rcpir6erypc.

IIAPAAEIIMA 1 O ,4 Fi\rar bra./dvro< tr€ cEaprrt[6!€< oril€< x(lr €:aAriltri:w<yut4+ric:

n \ ? \ P l l l M ^ A \ - ) n , , \ - l l l l a l u M J v l ' , : N t i \ - \ r ' ! \ v r A \ r r ' r l . r 4 i o .

,'=l

lil

'OLec oL oirtLr-< reLertirvow pe pr16dv. lrov Z(i)Qo dtql"(irv, ro ri),IoldoftQo6LdlroUG olo b = (rr, ,:. lJr) ti\ro' to p = (br. br. (l). Aurn Eilar I npo6o),rt 1{nI roo{tri}+rc (0.0.6r) €tvar rcioeto o'6}.e_< rLq m)ileq. To r(alurepo ror [ropoi$e vclrirvouyt pr ro.1;r'= ir eivcrL vc liooulu 16 6tio rQ6t€E eErodr<rerg. uqroiJ q tQiqEqi.o(,)o't ri.r,(r 0 = rr. To d(t.il+l.r o .ru t fit! EEio(l)oI 6€v Uroa€i !a rsa'oaroeEir€proo6rsao, ra oqd)+nrc 6gco_< otq bro npdteq elrodoerq Ocr ei'vc1r llbdv:

. r r 0 0 0 l

sr33l

tl : t :1Ello o o oJ l_l ' l

Epi(oFaorE rdeo oq SEUr€prt 8ljozoi.ia. Or or{},€( sivaL sEaaqF€\rE< '{cr ro .r'bev ei.vcr ;rovnbLx6. or. 6no ira6r€q orvrorcboe,( eivcr bLlot v.u blor cti.Ld ottrr,€E ouvrordro€g 7r zar zr €ivar r€),€i.rDs cueaiA€rFq. fta vcl xdvor[€-ro t ooo

ro 6uvcr6v Frza6r€ao. braLiyoqre 1f nur6q ,16 ouvrordroe-( rlv trul yq6€'/. Il^uon t).a|iotoD pnxovt rov Ar' t tirut .t

IIA?APTHM A AI :NA:Ii IAIOIUOPOON TIMON KAI U'EYAO N I'D IfOq'O) 5]5

(4)lil';rl ['[ 'f i] [li]H eEiooorl (w1 €ivcr €vbLaqdooloa. Iaoo6roQiE€r xo iJ zal 6€itv€l

xnLanq rot, :th,czc tox ttupdyrt ro :t. anj rc r. O triv(lr(lS (rwo,< ],6y€rolllt€t\oawioreoEoE 1ou A, ou[6o],illtaL $e A* xot ttot (Xolpt r' =.1 b.

Boo!!68€vor o tr!16 ro rupri6eL.7pa, pnopoipe vc 6qori$t q 66)'rrorll trrlot.r'* xs! rov Ur:rbocrlriorqogo ,,l orororbrl;rore btcyci)uou trl.v(!x(r.

I / o '

E(iv ''l =

O rn e:ri r :rivcaa< A tler r p4 [16€!rxh oror1sia o;. O yer6otrtiotpotpoEtou,4. eivcft r eni r), [e r';r] gnbevLr.id ororyei"u llol. olet or r€v6( 06osrqn{rQ!ft (i!orl\r $lbE!rzd.

:rlsr6or€ orr o (,4')- €ivtlt rdl( o A. Axr6 $ori'q€t ll€ rlv (,4 r) 1 = A,cr).ki rdpcr o -4 6ev avrrorpeqmcL. Ed! to b 6t€I ourrordJoe,( (br. . . . . b,,,), 16r€11 rpo6o),rt I rTer ou,Lorr,'rorg (br,. . . ,6,,0.. ..,0). H l,uorl Fr.ze6rEpo! [rirotjqrou ,4t = p rir,(lr ro .r* nor 8ivetcL rcparttvo-r. Aayorcac 0.1 ouFdol.ilow€ rot61o1d)r'r.o nirrrra F€ : xnr. rov rledocrvrimpotpo tou [€ :..

f([l ''( xiQol(')ortooutr€ n6pc ajr6 rollq 6rcy(i!Lou< nilar€q, as E€xrvloor,lt:(lJrd ql €llrol.il reQi'rrroorl. 6aor o ,4 eivcr nvrLorpErpqros. H €giotuol ,1.,i =/)61er pin trrovo l,ucr1r .r = A tb. t anfi t4v repittroxrq u r nolrar pn r* xuLoA ItiunA. Orur o A tirur at'tLorah!4u,E, r.jr| a lptr'oatriorQoEoq Eivalo i6nq lE rot, a1'rioeoEo. Eiral.rte€{orl€ rL! suo orrorlzeq: to ,4r €ivar ooo toSrvcr6v elprepo oro 6 (roourar 1rr: ro b) vaL ro .r eivcL q Lrio4 tLcrlimor p{xoug

1Lc ro ,47 = 6 (eivcr 1 pLovo6Lzl l.uoq, orqv :repiartDol :ror o A rtrtlorq6qttal).

526 I1APAPTHMA A AIA:IIA:H IAIOMOPOON TIMON KA] '!EYAOANII:TPO@O:

0

".t:S"

&. A.1. H badot ro! !,r'l.r6ouvrtorpo$o!.4*.

Bpioxouyr: rrirq<r ro .r'* xal oq levrxrl :tepimoorl. EivnL q ),rio1 e),c1iotorprliaoug lm io AT = p xsr roluQrqd0(lor€ orr ro rrrt €ilca nawote atot ybpoyqqptbv w.4. @upIo€ir€ or! r{ooe 6dwoBc( "| bloondrcr o€ $o oljvrol{ioqlar(i rov X6po lQcrB[d! x(& p[u xord rov prl6€!o7oQo: 7 = t,. + t,,. TQeq €iaclol ev6togdBoroeg l6r6rqrrg crLnjg rqg bLriorcoqg:

1. H oljvrordrscr ro! X(r)Qou yQc4lldr! L!.,'E! Eniol( to ,4x, = p. 6r.6rr"

2. Or orvloxdoeE ei\roL oQooyd)\rleE xcr rxor,o;rororv ro ffu0u.16pero oetir-pryal

r ' _ = . 1 , . r l J ! I f t - D ( r r o r I r v L f i f r ' l u x r m o o r d v ' f ' . = u ,

3. OLeE ot ),tiorK ro1l Ar = p 6Xox! xlv i6ra ouvror{io.r XdQor yQq4r6!I | . To ' tdluopo au tEiettrrot :+.

To or]lrc ror nadnEr vo 01lg6paor€ €ivcir ro :I.3.4. rou :rapiotcve ro O4r-LLrir6tg Oed;prpn q; IeaFFrr{; A}.yE6aa6, Sri1vovr.rg n6E o n or6}.rEr. rir0r roro X{i)Qo mlL(l! ro!. To o ila 66€LXr'€ Eriolq trd]( ]ido€ p orov I(i)Qo oln],(i)vnpo6QX0r(L c7r6 6\,4 liar p6vov €vc brdvropa rox tdjeou yaaFF6v. (Ed! AJ, =1)xcr e:rio6 ,42i. = p, 16r€ /(2,. 7|) = 0. H 6roqoadr n€ar"61ernr" orov pl6ev61opox.l6dq zaL rov l6ao yA44rd)v 610r ro t. 7i eivar op0oYtirvro orov enu6 rouxar ra6'r€r va Eivar prl66v). To p6vo rou xdtoupe tivar w 6d.lEo1vE andn bLdruopa r+ =7, oar t4t xd,irp] ).6ory rou Ar = 6. Aurl civtrL q l.uor1€l.ali.orou F4rouq TrE 'il1o#orepqg (mqv opyLzrl) e[ioootg ,42 =p.

* :.r.E.: Yrcvo.i ro F .

NAIAPT}II\{A A AIA:NA;H IAIONIOPOJIN TIMQN KAI '!EY OANT]:]'POOO: 527

O 1i€u6ocv{orQ,oqo< or.rlfiQtqEr fllv r{ar€iotlvol ror' l'xliForoq. 3€rwd sn6ro l, r.ar €nLoTQ6{pfl oro r' . At'rtoreiE ror A .x€i nou o A Eivat antmaitl\goe-p€laEu rou X{i)Qo! yAqrFdv rqr rou XoQou oql6v. O V€uboa!.riorQ(x{)o<@toFo\,6v€r rov (QrorsA6 FI6€v6XoQo ori),vo.\,rdq rov olo tl466a z(t! 0,-10].66€!rov Fna€\'oxoAo 6|,c1;€yowuq ro 7,. ocrv :c+.

A€v 6€iEaF€ (?(6Ft 6I| u?rdQl€r €v(g niv(xas ,4' rou 6i\,€r ,-rdr.ror€ ro.r*.o[o; !ndAX€!. Auro! 0o ef\,o! D €ni rr. b!61 or€].v€r ro 6 rou R," nlorD oro .r*ro! R'r. A< bouF€ 6v(( (xolll naed6€ryltc. nQrv 6AouF€ ro ys\'rz6 ,4-.

nAPAAEIIMA 2 ,4rr = i, €ivo! rt ,rir + 2rr + 2lljr = 18.

Il eltoruol EleL ivo ol.6rl,tQo snin€bo Luo€rDv. :upqo)va p€ q O€orpicr FoE, q l,uon€}.(li,otou l.rlrious rrQ6/rel v(l nepLilercL oro }irpo lpcppriv ror ,4 = I- | 2 2). -fo

roLLcttl,iroLo currlg qg lQcl1rpis ,-rolr Lzavororei qv €Ei.oalol €ivct f = ( 2.4.4),Yr irQxow xcr d), l .sE l ,us8ri 6noJ,< or (-2.5,3) { ( ,2.7,1) r i ( -6.3,3). rouri.ra! 6Fr'Lq lrlsLrirepeE* rfi r''. ('OLr:< ilolv p1 pq6t:vt*6g owrordrorg oro[1br!ottllao.) O rivaxoE;rol rughyer ro r'' cn6 q 6aErir r],eupir 6, rorr rlrcrvro noLu [rkQ6 6rriwols lllil. riv(r o V€u6o(v1iolpo{og,4'. Ev6 o A nr(v I€nf 3. o niva1.rl< ,4' €fvft 3 €ri l:

A-=L-1 2 2 )= 1.(r A' [18] = (s)lrl| - t /e ll2 /e l| 2/e )

O Xcirpog mr1).rlv rou A' rivar o yrirpoq lpqr[rirv tou,4. Kcl rrirqa biroqrr rvcrv

'Em(,) 6t! I odoraorl r6top6pq(l)! tu(iw lou A ei",\'aL A = QFQT. '161'! o

!,euooovriorQoeog rou 24 €{.var

A = QiQ:. (6)

Ot L6L6poppeg rq6g a1, . . . , or 'l€Ql4owfl or4 6ray6vlo rolj : (m €ni n)t{o! ot al,"ridrQoqdE roq 1/or,. . . ,1/c, ae1fuot'tcu, or4 6rc1<irvlo rou X* (n€rti tn). O rlreu6oavrimpoqoE rol r4r eivor ,4** =,4.

To llsed6€tyFc1 I n€Ql€it€ q 6lcry6vr( neQinroorl, ot(v o .{ irar, :, o qr = 1('n €ni. n) xar o q = I @ Efli, n). To nuQdb€ry[( 2 €ix€ €riofi Qr =[1] rarI |6roFopg,rt tlFi riror' 3, blLcbi I r€ro(rrovrai pilcr qg r6ror4r'iq rou .4,4?.

* : . r .E : Mr n ' l v lu 'a n t ! r r .n r t ' r ' do t ! u r t d 5 : r5

II PAP'I'IIMA A IA:N :H IAIOMOP@AN TIMON KAI IUEYAOANTI:TPOII'O:

:nF€i6m€. naQ(xqld. roq : 'rdL t' ors €ir6Fsv€g AIT:

f - t t 1 ) t \ ) t z 1

I | 221- l1 l l300 l l 2 /3 1 /3 2 /3 || 2/3 2/3 1/3 )

I t /3 2/3 2/3 1 l r / r ' l I r / , t lI t22 l=12/3 t /3 2/3 1| 0 l l r l =12/e l .

l 2 /3 2 / : r - l l 3 j L 0 l l 2 /o )

MroaotF€, r{ar'[uo€i(rv, va brcrnrord)oolF€ orL o fiiv(rrqg,4+ obly€i orl6etrtm1 l,uol ror ,4:r' = 6.

H i{o4 ,alinot pfixous rou eodrilane tlu2inav teqaydwtvy& tu Ar = tt Fit4r t' = A'b = qt'Ql b.

Ait66ttEq O nol,L(,il,clorso[o! $€ ro1 oQoolci)l,ro xi,ttrxtr. Q'! crgrjreL ta prjxrl

a.r llt = ltQr>QJ.r r| = >aJ.' Ql rr t.

Otorpriote oco vdo riyvororo ro J = Q] r = Q;'r. ;rou rXrt ro i6ro pqloE pe ro r-Tott, q €trcr1loioroiloq io! I Al-, rivcrt tcr,t6oqp4 pr qv el.<qroro;roiqorl roulzy - Q{b .!' (vnlr o ;riv(xcrg €iv(lr 6(ryd)vros xg|yvr,4iloqre 6rL 66l,rLoto

tri\'aL ro J. =:'Qi b, doq rl 6!trxl(xl luol r- €i!(L

L' = Qrs. = Q22'Q( b.

O(l ffioQorio4rE v(l brcmord)oouF€ x(r-r ruO€iav 6fl aur6 ro / :reE61€TaL movX(;)()o yQ44r6v xul 6Il rravono!€i lry Ar' =]p.

KstnoxEui 'qt AIT'EXoaraE r8}'tldlo€r trE rrc €qaQtroydq, €rrorQdq)oufr€ rriorooq 6Lironcort !6rop6aqnD! rLIdl,. otry oroi(! otlplxorlxcv rn ndw(r. Ilpurervtr briloqe dtr. xir0e rivazcq ;ruprryowo:ror.etroL ot lLvdpevo,{ = Q1IQT, ge6L(y{i)vro : x.4 oQooycivro\ niv(rsE Q.'E'r'.l< out4r€rQlx6q ni\,axq.6noE o /7A ixeL 6r,cr nl.ripeg ouvolo oa0or{cFvovrxdv |6LobLqr'1Jol1(iloJ! rj. nou Fnsi,r,our filE ofiiL€E rou Qz:

A t A t l = \ , : t : t p t r l r , = t x o : t r : ! . r 1 = o . 7 1 " i 7 i . Q )

(Irrl grp6Lz{ :repimr'ror1 n},}.irloqre rov ,4? oe ,44 xcrL ro e? oe :rfl.) flcipvowcrq ro €o(|)r€Qrl{d yrv6pavo pe ro :r:;. nvnrulunroup€ 6rL 61,a r(r )r > 0:

t lAl At; = \1: t : l t . i I At:1 1= ) i . (8)

rL\PAPl HN{A A dl^:nA)H IAIOMOPOAN |.|"n* *,r, ,r,U" n,rO*t,ra",raaD 529

'Emor on or rSLorLp6g )1..... ). eivm 0errz6_c zcr ra un6Lourcr n r ro nl{0os.4r, zur ) . , r ivur rr tbiv. rLtr r ' j orr txr 'c. opi ioupr o, . J\ xut q, Art /o)(Or oaL0poi rruroi, rr, . . . , r. 0c oroi€laoow rrE r6!6FoQqeE trF6s or4 6(r"tdrllorou :.) :Ip€r.rbo-r€ 6rr, r.ictri-r1v (8), rc q; Eirar Fovo6rdtct droluoFoto iou t?'i?ioL x(lir llv (7) €i!c! lcrdEll rouE oQ0oy6vrc:

(tT urtT^ rAr , ^ , r1 . , , ^

o io j o ,o j,!ts i l.i. (9)

:u[(f(r)vcr [E q 6[c6[x(roic Cram - Schnlidt. ra I oQooxavovLxcr (lora q FJroeo!1,vo €rrcxr(eo{! or F(r fi)liQrl oQooxavovLnrl 6cioq qr,...,9-. AurdE €hcl o!orrl).eq iotr Qr. To oror1si,o (l.j) rou yrvopivo! Q|,4Q. rinu' qf,4l'; (1q.qrp'je:ri rivnzu uri m{Iq) zcr rcr cnorleic trurci rcr !€poupe:

q! Ax1=0 6t:ru j > r (6rot , ' l :c; =o)

ql A,:1 = q! o iq1 6rc! j < r (br6rr At:.1 = o.1r1.1).(10 )

Qor6ao ql q1 = 0. exr6E orov i = j. AQ(r xrl [q Fn6eltrici ororXric oro yr.vdprvo2 = QI AQz €iv( r(( nQri)r( r brsydvrc ororl€to, nou dvcr a1. . . . . o.. Emrb{Qr = (rl r. CXorIr ,4 = (/,fQ:/ . o:rc,X ,r;rarrrirur.

Y:rrv0qiloqre :rirfu 6rr nr.rh ra l zcr .t tclptdrqow orouE i€oo€QrE 0tttr-l!6bEr( u?i6X('JQo!E rot A. Ta r nptirta;r cnor€louv 6tol rou xoQou yQ(ptl{i'!rcrr r( r - r re?'.erJtuftr (uior€Iouv oQ0or(vovr,r{i 6dsl rou Flb€!6xoaou (6no1j,4r', = 0). Tn l nprirrc g crnote).or;l opOoxcrvovrxrl 6cior1 tou ltipou mql,rirv xcrLrc nr - r rd.rurcricr crore).ouv opooxovovLxri 6(iorl, rolj op!fi€aou plb€vol(l)aou.To orr or 6riocLg cutdE u;rirplow 6ev eivct. rpolzs, rcrQqE€vo. To yvoQiloqr€!To eE(retl[16 ytcr tcr r 1{(rt 4 .t1)cl| olr A.ti = o.tq.'. O tti\,(x(r! 016l\,€[ 6(ll'tx/rblcruiop(rl(r oE 6aorxd 6ra!{olrar(l, or(v oL 6c!o€( €ivc! o(,)orc €rrtrelpdveg.lr4rqorvc pe rorq xuvdveE rcrcoxeulE ev6g :rivcxc. xcrri ror,g onoi.org o ,4upcpp6lercr oe pra 6dq rorl ,Fl'r xat lcl anort).ropc exgpdlercr leqol[onold]-w(q lra 6{iot ro!1 A"'. o:rivcxng rrrl:rqoq cutds rE 6irocE 0a €i!cr! 6!al6!!o!.Awq (.zar66g €iv(lr o :.

ATKH!ETr

Berire q buionaq r6roloaqxD! rLrtdv xsr ro! 1'€!6oawimaoqo 0' tor tt'tbtet/.u1i

BaFit€ rl 6Li'oroot rbLotr6slov flr[i'v aoL rcv U€x6oawiorQnpo rovA.2

A={r111t*,,=li I 3] * ' , '=l; ;]

,{.3

nAPAPTIllilA A AIA:fl :H IAIOMOP{,QN TtMeN KAt ruF:YAO?$aTDTPO4,OI

Edv o n €ni n 'rilanoE Q dr€r oAeor{rvolrr6s milss. noros rilar o V€!6oa1"imeo-9os- O*i

a) Yro),oyicrr roa A",{. flr rbrorqdr xft ra Lbobrovuolord rou. aar q o€arad o-Eo}ar,'] ftrao.rouzi eila iou .S. 6raa

J r0 6 lt0 8 l

L) YnuluyrsrF ntv ro].r)t'l broonoo)t,4 _-es rvd0t(olras rov e. 6efil ta!,rvl|('leogd b'c(trr(yrriut 6rodndon / - S'q,

A-5 nora sivcr I €i.(lrimo! FiaoE IuoI e).(rtio(,v rnaayd.r,ov.r = A+b ro!

Mnoa€tre va t|no),oyiofte rov .,1. rt vq 6o€Lie rt.r€vrrt l,n.nt ntE ,4T,1J-.= r{11, r(lrc 6rd).dE.r€ q lljon no! fteQdlrrar mov Xojao yQapptiJ! tou ,{. To rpo6llju au<jneq16di.rfl xald 66lrroto rQono to rnine6o C + Dr + /': oro, = 0.i( r = : = 0.6 =2 oto i6ro ol l l€ io. xsr, = 2yrr I = : = l .

A,6 a) Ed! o ,4 dx€| oleEdAqrsF oril€q. r6rE o oerm€Et dvrimQq,os (,,lr,l)-r,l?'eivru o V€ubosrriotQof os.

6) Ed! o 7r aX[ oveEdeq'€q 1g4qris, rdre o b€Etq a\,Iioreoqos ,1t(,1,1r') Irf r(r o 1'.!6o(vridre0qros.Kar ol|s 6!0 regun(i,o€'( snolrl0cnm€ o ,- =,1+, ea\rd| oro ldeo.fsat4d;!xnr A ' A ! ' = A t h

l.z E<iv o ,,{ = Qr)Ou I 6do aoe€i or es, (awioraoqrt no}.,xi 6rdonoor) rotor civa!or Q xst s'i

A.8 loxuer ndworr yta l' 6oqvriore{x|,ol( 6tr (Aa)* = B'n-r A€a ro ,itor€ur.A.9 KdeE trivd'iai raecyovroro(iror o..rLv6f.vo / = Itr. 6rou or r mir€< ro! !raqdlorv tov xdqo oq),6! tou ,,1 x( o, r .re44ds ro! q ftaqdyot, t;, xde;'req4r(& (ndqdyqdrpoE 3.6). Eraho€nor. ro! aaolulrxo ro

A' = !!r lallr r t (L, L), LT.

y;rdd.rA?: frori l6yo rorr !l ror rivot Frqoord, ro rr., €ivourdvroft movxdeo yed$6v rat yrdli Lo)(rr€L ,{r/tl*6 = -4? b. nar ripo ro e? 6 rxcvororeiDtv rsvol'ali €giooorll

A.r0 Ehyrioi€ ytnri or rirorq AA* xtj't A*A eivdr nivaxe< fleo6o].dr, (1.or oulsnri)<otJpfrtqflol). n{ivo oe rona 0€p€},1d6!xs lr6xoeous rpo6ti}.}.orv;

, l̂=f, i

," = li 3 il l;l = f:],Lr r r . l L r ] L2J

IAPAPTHMA B

H MOP@H JORDAN

fio 6e6op6vo nllcxc ,4., 0dXor4re ro 6potpe nlvaxo M 6rot 6o:le o M-|AM!o ehdl 600 nsQLdod/rsQo olo.ytivrog .ytveror. >"qv adlot@€Qrl JrsQtrftootl, oA d1er, 6vc al,rlqeg otvolno !6ro6[d,v,!opdrov or, oroleg ,yl,vowol otri]LE tou M-lvotltot rar oov S, H gofi Jordan J = M-|AM = , xaroqrisodtsidl 4ol,ortrtQou dnd 1 sfil, 1 UJrl'o& Jt = \t xat o drdxos rou 6r,cy6v!ou ril,vdriorJtrt1}'yxdlerd! nl,te(os. >q ysvr,r6rlQrl lto! nto 6d(l'lolq r[€Qlfiroo$ Islfiout,oQr,opdva &,c\aroporo rol 4 6lo11ovr,c poqrg{ etvcr, c6dvctq. H ,ieQlftool aurrlel,vcr, trirgo to xrlguo 06pc gloE.

Bnaval,att6dvoq.le ro 0e6Qr[rd riou 06louFE va cao6el!or4r-e:

NAI 'A THMA B H MO?OI I JOR 'JAN

Ilapti6ttlprt er6E rdrorou rivaxtr Jordan €ivaL o

H bLTrLrl l6lorllui ) = 8 t1er ivtr povo L6Lobrtivuopc rcr(il rrl! r{crslif}uvorl rrlgnQ6ttg ouwer(ytrivnE er = (1.0.0.0) i.o! 610r ro ) = 8 rFqcrvilerar F6vo se6'\,c! !n),ox ,,/r. H tpLr),I LbLotrlrl l = 0 6t€r 6uo rbrobr(rrrrol(lrs er ycr €i. rroucwLmor1oriv orcr bio lulox Jordan ./: xsr ./3. Edv o ,4 ti;1r 5 rbtobtcvuopcra.6),c rcr or,ltrlo'ac 0o frov | €ni I r(r o J 0s ircv 6(ryri)vlq.

H €Qd)rrto1 xL€rbi Eiv(! I rEtcr Y :tott< auv9ilztq U toaqi Jokku €uj<illor 5 arl 5 rh'c!*t A 0t th,ru o ibrcg nfuaxrq ,J: lllc uittiQy! tya< II,i toL tbor l l l - A.\1 =., / i Kor 'ag1i< a<i0t:6$orog r ivcrcr<,4 0o dXcr 16 i6Lng!6!orudq 8. U, {),0. 0. Auro op{Dq undlel r1jI6 to va rivcrL cpxer6 o ar(yrinoqrilsftas I crixiq rlg lbrorrli! 6sv €iv(r 6loroE ror.r ./- x(r q eo(ftlo1i $cr( aqoacizuQirDE ru r6!o6(1\,uop(r(.

ftcr va r1v u:rcwrjooupe, lcrvaypcigoqre rov ,\,t r-4,\1 =,/ orrtv snloim€Qqloeg{ Arl/ = n/./:

EzreldntcrE tov noLlcr:rLcrorcrop6. mril'1 npog oril),y1, e1outr€

. ^ ]

,4rr = 80r x r { . l J . = 8 J : + . r r ( l )

13,11lsil

I0l

f 8 l 0 0 0 l

lo r o o olJ= lo 0 0 l 0 l=

l0 0 0 0 0 lL 0 0 0 0 0 l

l",1. ". ̂. * 1",,,,,4il)1 = Orj ,4.r)r = 0rr + cr r.((t! /rii = 0.1-. (2')

Avcptrrpi.(oq.rs rriipcr rrg ouv0{xeq y.rr rov ,4: IIp€reL vo iI€! rQi.( yl,Iora!6ro6tcr\'iopota, &ra4 czpL6ti< o .,1. To 6va F€ ,\ = 8 0<r pnciver fiIv nQ6rqmri),4 tou ,1.1, 6rq crxpt66g 0a dproLvr zcL m4v 're6rt rrrrtU rou S: ,4xr = 8r.r.Tt( dLlrr buo. nou err o4t6oi'iloupe F€ rr xar ri. Ocl Fnouv orry rpirrl 'r.rL niFnflloni).r1 rou 11: ,4rr = Ar; = 0. T6LoE, 0c rpdrct vc u:rirqXow 6xo crx6Fl €L6rrd6Lclrlopc(tc. ic{ "ywrxerpivu drobnt,toparu" ;r2 xcL 11. @eroqo{pc 6r ro l,nvrlrer, o' ivcrv oett&06 |Lwuopdtay nor Eavouv pe ro r1 rcr. neqllqrirgovrcu,cr6 qv €Eiscooq (1). :xt! nQrrf[crttx6rtro, ro povtr6r.z6 d],].o bLcilropcr aorouror op[rcoou etvo ro r:r r..qL ro ovrafiolZo trrd,ox ,,I1 eivar tdleo< 611o. Hrliooorl (2) :repLlpdq& 6io 'ruq.oQ.ttzoliE oetu.rddri<, o' €vcv on' roug onoiorE

rIAI'AI'THMA B f] MOPiDH IORDAN

ro rir ifl€ror ro! ur €vrh orov ir].Lo reqlixer(l! [6!0 ro c5. To cvliolorxo l]7rloxJ: zqL Jr rivar 2 eni 2 zcr 1 eni 1.

H a1'a€irna4 xnE poaqiE ,h,.txt ro! A ct'iyrrut ot4v avaQitaoa utrtitv rot' og1ra06t' |Lovuotltiral', xa9lvag uni rouq onoioDg EExLt'd and i14rdrchtavDottu: nu xdqE i.

Trl bL.rvriotloro.r, prcrivow mq mr1Leg rou.4I xql rido€ oe$c(06E ;rcrpripr rvcFovabLx6 Bnlox mor J. Oljotaorrxd rqd:reL ro 6oupe rtirE rcraoxerhlowcL.lrrof oL oqpc0oi.lLcr xqoe rivaxn ,4. Kcr67iLv. €d! or oQFdooi ar.roi raLalaqouvot|c rrbrxi< eEroriroel'( (1) '{aL (2),I trop{riJordan rou A 0a €ivdr o J.

Mo! ecri!€ro! noq ll [66a rou Filippov, fiolr bl][oo!€tolx€ orov 16$() 26 ro!Moskow Univershy Vesmik, rciv€r ilv xar.rozFUrl Is ro! nlo o&qi xo! anl6 ra6rTorou .yiverur. naoxo)aei [€ Bagl[atrxrl €noy(|J.frl, EE1.wdw(rS an6 1() yryovd! 6rrlvaq rtvax[g I ETri 1 €ilar rlbl or $otxt{ Jordan. Ynog€roupe orr n xor(loxcu]6Xa! ibt (1'ro6€!X0€i. lta oloug ror,g: vnxeq ri.E€@S Flr{a6r€afi rou n aurrt rivtrl11 eaayrrlLzrl uro0eoq"- xcu €ttyoigt x(r6ruv ru 6riparcr:rou Xpetdrlowcl lLcr6vcv fii\,(i(( rciE€os r. XQ€tdqovr(L rQf( 6{Fora. fiou Fura ono Ft(r ysvlxrlnr(nyQ(rq,{. r(r €qoQl6ior'lrr o€ 6!( €L6rxo ,r(rQad€L1l'rc.

Bt4tpc 1 EA\ uno06oouFr 6rL o A eivcrL tbr6lop9o;. r6te o xo]Qoi otlL6!ro! tlrr 6r('loruol r < n. nsQroQr!6prvor o (urov rov [[1.!QoirQo N(i,Qo,rE(oqr(l,iEorrtr€ xlv avayo./n ont FoQ(pt Jord.ln aro rlv €rol(l)yurl uroo€ot

np6:teL vcr unhqlow r uvelcipqtc brcvuopcrrn uJ, oro tdlQo otl),cb1,, 6to! 6ois

eire ,4r, = );.r, €ir€ ,4..i = ̂ ,:rr +ri r.

r i r r , 1 r ' = , \ , r r i r e , 4 x . ) , , , x .

(3)

(4)

Brfura 2 'EfirD 6r. o frI6€!6I(0Qos xclr o I6QoE fiIL6! ro! ,{ dlou, lucr Iotrlib(aor(orl! p. Ouorxh, zA0€ 6lrlvuol( rou !I6€vol(o0ou €iv(r Lclt |6robdvuol(1nou (vrroroLlri orlv ^ = 0. :u!rn{iE, 0(. nafn€r va undrAlow /, o(}t(roof oro6I!(l l. nou cQxiEow r1'r't.lrtrl rrlv rbrorrlui. LoL €v6L.rqEp6Faore yr(1 r(( xirou riv(r l]xo r6Lo< dur6v r(Dv oQfrsodv. K.iB€ 6v(r an crnir rc p brcrvriopcrr(uwper oto Xrirpo or4Lrirv, cipc civcr ow6unopog ruv or4Lti-rv tou A: u; = Ay;.

E{pa 3 O p$ev61opoE 6x€L filifiore bld{or(to4 r r. AQa. (v€taQqra afi6t4r p.dLt iotnt4 ruyr] rou Ir ro\ ' X,"po ot l ) , !Jv. npinl t vu aeQ'dle' t t - t - p f i ; ,fi).Eo! daorr( bluvuol'rr|1 ?.. cr.ierfi.oFsvft fxrd( uLr)i; r4S roui;,

Turotr, 1r<r v<r croS€i.EouF€ xo o€dQr ro rou Jordan. ou1'6!.i!ou[€ i( r0io

N P PTHMA R H MOI.TH JORDAN

Ta r olovriopoicl rdi, ra p 6lo1'doFaxa Ui xaL 1cr rt r p 6lqvuopsro zioxnpqriqorJv oQpoooriE Jordan rou nivcLq ,4. To 6rovdopara owd eivo!yQol.rl.lt1|dr! crv€EdQxrFc ra! Fnailolrv 0116 mrl]"€g rou ,U F6o0r

-rou onotouriAoxljJ-rr€r I [oQpri Jordan J = ],,1 1AlrI.

Edv 0dl,o4re vc encrnprg$tooupe allrd ra 6ravuopcrn oq lr:1,. ... r,, acll va tcraerciEowE trr rq elrotioeLq (3), r6re rd0e !/i nQ6n€! va rorooernoei, oBaoorEpe-rir ro rrrl ru6 to oroio npo6zu1e. 'iqr 0o o4uiLlecirv€r 6vav oQp(tgo.y!o rovoroio ^, = 0. To . prnivouv .rro r!l,ot. xiro€ ivo oroa oQF(l06 ro!. xor rl!6rorL[rl €ivul fidL [Ib aqolj r(r z ,[€Q[!Zo\4(r o-rov FI6€16X0Qo. Ta |r,rr,.oztl€ FI FqAevLXf !610rr[11 6tour {6I r€}.€L(i)o€! oro 6{BC 1, ra Fnl,or Be pt6sr,(raEtbrorlF6E F€ynl.ri)vow 1{(rd pic1 yad$[]i xar" pkr orrjl,rl oto 6r o 2, sv(i) ro 6rl0q3 ow€toq6oe! rn I €ni I prLon "4 = [01.

Aoxr[dlolju€ rdA(l .ve fiClpobErllro rcr Tlcr ao $lv ano[uzQureor4l€ crr'rLErrpolyolJBsv€( o€1.i.6€q. n('iQvotF€ rr( [6[or!tl€c va ei.1'c1t 8, 8, 0, 0. 0:

Btipc I O 16pog mqk;v aX€! bldoraorlr = 3 xcr ncrpdlercL cro tn 6xrvtopcrrcrouvr€rarui.r,(D\, €r j c2. arr, fla va ]reproQLoTolJpc o' GUr6v rov /dQo, c..yvooupc 11,iai,il x.r rdrqQxrl ypr4qrl xcr orrllq rou ,4. O ni.,'craE rrou crro[dv€L dXEr xlglbLorr[a< 8,8.0 zaL poQqni Jordan nolj trAo]{u]rr€r en6 xc &ov6o]rard

f8 0 0 8 l i ll 0 0 0 8 8 l0 0 0 0 01 .

t3 SUlt il

lillillllTa ?lJr nsQrilol,rar oro Xrbpo ord.dv, or+lJllnQd)lotiv iov opFa06 y.o ) = 8 xaLE€xrvolv xo1' oQFCloo yra ) = 0i

A1t ' r =81,r, Aur, =8L-r+tur. , rLrr l =0uir . (5)

Bitld 2 O Fl6€1,6toraoq rou A nep$1er rc e2 r(1r" r:;. ripc 11 roprj rou pe tor 16poorl)da r.lQdyero! cln6 ro cr. :uv€ndjq p = 1 ?iar, 6roE nvapd1,ernL, r:rdpx€rtvuq to!o! oaFa06! mq! rliocLrol (5) a!.Iioxorxo< xnE l = 0. To 6LdvroFo 1/rl

N I PTHMA B H ]\IOIOH IORDAN

+rqaviqeror oro ri],oq (xoodrE €riioitE lrar orlv oAXt) auror rou oQl(loou xcLtt \ = A(€r €r). AQo r = c{ ( , r .

BipdS To noQd6€(ypo 6i6€r n r ?=5 3 1=1ncr.ro ar"r ivuopn.z =e.r€ivaL €xr6q ror 16pou orq),ri:v. Art6 Ocr ei,vor to z ror ;ropirleL ro 1 Eri. i |lnLoz

Eir! ou/r1€vra6ooqr€ xor ro Trdvr€ 6!cr,!o[rorc. or fi],llaElE oA$ceoi 0a €ivor:

A r l r = l 3 i | r . A r r r r = 8 u : + u r l . A r r r = o r f i ! A A = 0 . u + : U ] | ) A z = 0 2 .

:r.\,xQ|r,o'\,roq pr ttg elLocboeLg (1) ri(r (2), 61oqe 6vcr reLero r(riQloopo -il

$oaq,rt Jofdan rotl nopcb€iructr6E Faq 0a €iaflr crxpr"6ri4 o ,/ norr yQalJopeireolyowiro!. Tofio0ficlrlr(q 10 ri61,rE 6!o!!o$oTc{ orLg o'r1}.eg ror ,r11, rpinrrrc dlorpe nl1 = M,l ' f Xl- tAII = J:

'EZou!€ aQx€rl €priroroolvl orq pqoll(lrlrs (1 eipcote cpxerd r€pfiaLn6cg)cborE r,(r [rl, exlsL6oor$€ rov rol],crlaoroo[6 ,l/-r,,{,\-f.

To povo reXvLl6 olpeio orl\r xoroox€urirou Filippov eircr 16Lcniotcr.rorl qE(r.,,eEoQrloiaE ro! ouvolor tcov tui. yi rar .?i. Aq !fio06oo1jp€ loldo\r 6r! rdtroto<otlvbraoF6E (11J1d),' 6|ve! ro fq6€v:

l , : p , ; + l , t 1y i + l l t i z i = o . (6)

(7)

f8 0 0 1 0 ll0 I 8 0 0l0 0 0 0 1l0 0 0 I 0L0 I 0 0 0.1

nol,LcirLoo!(ilolruq [€ rov,4 xcrL lprloLporoLrivrdq r( EErori]oEr! (4) yL(l ra uri.vagtt< 1,.&L x!\ Azi = 0. riaiQaow€

r",f . ''0"

l.r,',u, oL ^ r " , + x , r r l

Tcr,4y; eivnr. ra €r6rxlt xr, nou €$qovtLowar oro 16).og trov op$.lorl.v rorcl\'rrororxouv or4v lr = 0. dQa 6€! FnoQei.l,( spqavilovraL oro nAdro iroporo[a.(Ilo),Lanl.coLcilovtcL gr p13ev orc )iui.) EnErbrt r1 (7) rivnL ow6rcog6q rcovt i, ro! ouFrprDla F€ rn\r rr.ry(,)^/Lr.iti rfi60eon €i!o[ nveldpr1rn, o4rrepnivorpr6rL xdet d, qlttL n gtdEl/(!ro,. Enror0iqo\rt<( orrv (6). cur6 6iv€r I c;u; =

19121 nct, 1(Aror€Ari rLclad rrrQLiX€raL orov I(i)Qo orq)d)v. Ener6{rc-zEi!.r c!€EdArtra ar'cm6v ro'v ltirpo, zde€ g, nesr€r va [n6€\,i(€xo!. T€),rri1,I c;u; = 0 1s1 q ou.goprrloicr rcuv u; acrpdyel c; = 0.

ho(ooll3 ̂ u19!D 13xr01od! /t D on^tirQ or rDx I = (l 'Z DtUg oD :!oE^n3 I,n 01grJD rDD/,'/dDr rrr ^91[ro odlDx ̂om nodf.(xoyo :3 rDl31?d3u sodco)(9^3qhn O

:/ti DXD^PI ̂ oro ll^lDt, ,f '.,n r,n 59en do o'rJLr.{3J

[J] = o -^" *0.,, u,r"",u,

' r , n + i i n 0 = i i n F 0 = ' 0 r /

fn n ol f r r r o lf = l l ' r 0 l = / lY . / t 1D t l ( ' tO

t0 r 0 l L0 z r l

t:l

fll = [l] " ,,, ,,= f8l "[ r ] [z ] L l . l

fo o ol' 0 .00=y rd l r 0 0 l=r .LZ I O ]

:531!uo blDm3l ola Jr], 3ir?r]Jr:lrho D^ r3^lDgrino nou rin '',n

9oDndo 59x19Dôli rDlt 5D^? r3ldpur ^9'ttrl,oodoll ôro ,o?l l Drlon^Dreorer oôri D^3 rDx Z = r [qD! 1313 5otoD 5DXD^lr O

YITIJITvvdVU

'loy?r oro 3ho^prd 53l3^9,l,ox1o 30 toh g^rlDr^lr1 lunn .t{ 1o1^frl,D€ ̂gg.rxo rD^lr nou ixD^lu ro ^noo3dD sno ôo9 ,rh3-Joddo g oru o D^F f o nriî/,oxro 3el)x rZ '1o1otl? 1D,t13 tttQ talr^r^()x1o,tpxitido'1,1e otrq 531tott1y tott3

'(( to1 'on! 1Dx) snot l.r,]jtl 1o1on9 nV)Dttl) 1Dx uDp.tof lirdotl D1q ab MoX? 5Dt),9loxto 5D1Q! 5L1sex ,tlu 10 blA:Dru.rgrgr u^3rigrr3 ̂lrt ari '^(F3^3loxro Joolqll D^9 ,o rDr?rrorrq aoxD^1)! ̂ (n!^(')y9 olo^+o or rclE, t, (lor uEprot Fbdorl FxraDôrl Dr r3ldrit,' I gxtgoôtln^. Sodt! Sotono Dltlr Splnn xoylJli ̂ or LlDrDrqD^D Dronpx 9!D 59txg 1.uEprof DxD^lu ̂ D^? 3n 5ororlo 1D^1a I/ 30?x uo h:r3ggrrD ̂trx 13^lDdtrlurioo c/mv

.r = J) + 1I = J[rr I \ ! + ] [ 'V \ I I=I \ IVI IV

:/V orql ̂ or, lDx )ooeDddoSnolal 'ror proronroUdl IV oor uppot !68od tr ̂ rl!orD) rz 1r)? rn l.n Dr orrD rx^(I)r *.ro€D Ao tror 5rrô.{DdD!',f = ,[,yr I[ lolnD uEprof trddod trlo ,]. ôr13^dgd 50 €ldg.ttD O ( y oo1 l,oirlolqr 3rollfgDlorjo rrr^13 D^ ^9r4o.( l3dorrl

'5od)-dorigtq1 1D^13 D^ ,y o 3roqo ro1? rDBl?vra r Dd3ODD h no O,) IJ V = tV ^6a^Drôl9ddDd3 D0 DrDi[g Ddr Dr '5oddongrql ̂Dr[ me y iglrxbD o ^ta

NVCllor Hd,JOt{ H A VllH.ldVdVU

I IAPA ' / IHMA I J H MOPOH IORDAN 5_t7

Eqaepori oqv n /dt =.4u 'On('E r(ivr( onlonorcriF€ ro rao6Lrytl 6tol{rQi'EovlqS roug o}'v(DmorE. O 6L.lx(l)ploloE civt6s €ivol nlrlQrK !6ao orqv n€Qin1o.n]rlou ufidQX€L rlrlA€6 orwolo r6roiL[tvuoFdr(Dv r.cu u = Sr'. H zoi,ur€el dJ,qrlmrwnq$6v{Dv oqa n&pouo(r rr€Qiorcloq €ivcu v(r fidQooF€ ri = ,Ur. Auro r':r-adyfl q vda €Eiooot .l1drldt = AII r i dt/dt = ./r,. :ror eivcr. 11 n:rlotmrprl6wnnl un' crxeq rE ouv0rizeE. YrdgXer ow6eo1 F6vo l,o1o rory FovriSory err6gqi br(r./(,lioo o€ xdo€ FriLor Jordan. >1o fiOolfoupevo ndQdr6€ryFa. nou €ix€€vcl Inl.or, I .lr/.tl =,4o ./iuji(lr"

To o{oqpa l]Jv€rsr ofi6 xcro fiQos r(l novo 1.(C[ oE rde€ 6{[(r €$qoliEEro![!s \'6s 6il(lln roo t. ('Evo pn].ox,l, cni. /, i1er, bwcitr€lE [€XQ[ rior la-r.) O!rx0rrLxoi :rivczeg tou ./, o cur{ rqv mpintrool ur0tirg rtrr ro rrQort1oip€vo 5eni 5 ncQi16Ery[a. €ilal

MrroQ€tr€ vc( 6€ir€ n66 €Iquv(lo!'r(|t or o1r\,rr),rmds n.6 n(r c orov agtirto€r0rrrx6 nivcoiq, roL oxo 6srl[€Qo n(re(ifurytrc Fnop€ir€ vo 6€ir€ roL rL( alt€"tr6Lxi< ),tiortg" Irc du/dt = At. Tpr4 nt6 cr,'r6s aivcL xc0cp6g ee0etrxdg,r =p8rtr,L1 = eOr rl ltc! u\ = er)r ltr tro!' oxqBdtilowat, or! ouvrlOo{, &?i6 ta tQicr!6!o6!c1,uo[cio ro! A. O! 6io d),LEq nEQr6xow id "/€lLri€up6lcr !6ro6raviopcrrc

t r : = c 3 r 1 1 . ' * , ' , l i c r ' / l = e 0 t ( t r l + r r ) . ( 8 )

H nLo Te\,|"d ).rion qS dufdt = Au eL',tfjir t\q a!v6uco[6q .ir,r + .+.sui r(1! oouv6 noF6g nou rcer.ilE! [€ ro ul yla I = 0 rivaL nd].I

? r 0 = . i r f r + . . . + . 5 r i j t i u o = . 4 1 c r i c = n I ' r o .

Awo o4raiv€L 6rf rr = ,\ Iert trI-t un xut ott ot S lttr! A orov naLr6 rino pe ro'\,Sc^tS 'u0 avrryarcmri0leov crlo rouq,\1 xur./.

AIKHTED

8.1 Ba€ift rr! lloE t< Jordln (oe rai( 6tiIor(!) r('v

, f u I 0 l t l ' t t l t = l ' L t = L t ' . - L t t - , . ' t : 2' , i = lu , , t l ' i l t ' . , i t= , r h= t , , -cn t-

L0 0 h ) d , td t =o

", li ;"i'] x(,, ' ti' 1 ;ii]

f0 1 2 ll0 0 01 .L 0 0 0 J

,=lillB=

NAPAPTHMA B H MOPOH JORDAN

8.4

8.5

AeaEr€ tu n €16'144 Morl ,a ftry eEiooor (8) rxdloror€i qv drldi = /r!. axa,6drsIolo ro! oapcoor /rr = 8rr, ,44 = 8' , +, t .

rra ro! nsoryoilrlo;rt!o"o a. xqloqrorouiorr rrl Foaqi 1r(r'.1/ | yrd vou:rololiomr -rov rx9nuo niv.ur( p!' x(r oyxAivdrd rov uF q bwalooflAdI + B t + ( B r f / 2 t + . . . .

ApiEre orr 1|(i0r prr),ox Jordm Ji uivor 6!0|0 fteos ro aldcfiaoqo ro! JI = p r J,p.IQitorlonorovros ro! nilax(l ltro0{oeos P pr foldbss '{ard fixoE rts €.yndqoLosh(yi)viou ((nd zriror cprmeq(i JrQos ra nrivo b€Er{i). :rtJr€odvcl€ oll r(i€Fniv.1"ds'€ivdr 4rotos rqo5 rov cvrimpopri ror.

Berirc, Bs Fr( F(rrd. rrs lroqqdc Jordan ro1,

B.6 BesirE q !oa{4 Jordln .rar ro! rtvaxa rr11 , 1 . r , - 1 ) :

l ' l ' l

A ltot B lo B 6I€r r6ror4d\-

11 231'=13 l;l B=

t0 0 I 0 0 ll0 0 0 | 0 l I

, r=10 0 0 0 l l 8= ll0 0 0 0 0 l IL 0 0 0 0 0 l 2

- l 0 - l l2 o r lr - r l I- 1 2 0 l

nora .i!st I ).fo't nts duldt =,{u xaL noro< eivat o err:

8 . 7 E m , D . f l , { : = , { . A f i E r ! d r r r t F o ( i l i J o r d a n , , r r o r t . . , = t / I . t t / . u < ( v o n o r . i r l vJ - J. E:rfl6rlru br-a?;ud un).ox roeduivou! xuerol'ivc. ouro ouniwr ofl, yLaki0r tuLoa, rol!$ Ji = Jr. AsiEi! !E oft s!0sid( !fto).oyroIo o ro Ji 6Ev }lroe€Lflaa{ !d Eilst €!(r I rri I tln}.ox Ji = [0] 1Ji = lll. Enodvq o / €ivor oloro<rAo( alav 6Rydvro niv.1xa rov{idov x(t pld€\,rxri,v.

t{rt.i.Dl Awo .ivdr pta runL?'j r.Airrr.q ro! e.rprircros F. ro oroio rFIe,6-!o1,tr! ro 6r6),io: O aivata. A bnyawototeirar, d! dt !t;rot' Edr to ll,.jltlro( i - ) l IX,4 - )21) . . . ( / ^ r tJ \ar 6ev nei ,ut t ' i t ,d mavel i l lu< tu! ) " e iyu ls?dir. Mra oliAotc n€ai.rrr0)ort ei!a! €x€iut io! ri.vaxo p€ 6&qroa0rfl6q r6Lorr[€s.To 0rdeltrd ro! Cayley Hamilron oqv Aoz|lol] 5.6.23 Ld€r ort p€ n 6LaqoQertrcrrqropdyoEr,. rr - .\t raiqvo4Lc Jrdvrore ro Fr66r. To dlto d,.eo €tvor o &urorLx4nivora<, ro! €ivar ndLt 6Layovronorrio+roE (,/, = 1,(oL A 1=0). O pl6Lc.prvro-

t 0 l lnoLioLj 'os r ivoxac / -

l ; r ; l ud ldof l dxL rnv.4 0. o; \ ) ,d Fovo qv 4: =0p|a € tooroit |E errovolaF6ovoprvl ai(o.

IIAPAPTHMA T

KQAIKE> YIIOAO|DTONrIA TH |PAMMIKH AA|EBPA

O( z(ivoulrr brJo (zoFrl 6rlI(lrs fiQog ro moxo Fo<. rou rirav va 66oooF€ orov(v(p(iorll 6a( 6r6l,io XQrlorlo. To 6vc €il'(r au uobeiEoqrP irry6!

"ol.riv rao-

./$4qr(irov rfi .rQrotrryrxrE feo!{r!zrt,< Ai.y€6aq. To €ulAproro tr' cluro ro nriF(r(t)v Ma0ryoT|'l6v livcr 6fl €ivor xrlr np.r){rr6 xcr (l)A(io rar $dt(riv€rcr ,i(lrirr(virLoyo 106no. MiqoE 11; rpaxrrxlg nLFriQClE Eia(! !' rrrorfiloE! xcrvDtq r:tlrrFaf( ntq fQa$[r ig A],y€6QcrE XQloLtrorrou;vr( rrt!. H oslDekr ixe! [€xoqocootrLoa x(i)6rx€< €nLorrttrovLlioi ).oyroNxou xa! (k uob€iEoulL€ ivc qdoFcl buv('rorlrr)v -an6 pqd),r; &6),ro04etg :rpo.lpagpittorv itx zrbbtxsg 6laflOatrrvorfri.liOepc p6oar d,rzreolrzoi rcXu6poprior' oe elbrzrutrtlcr ttozEr(. t( onoistirur bwar6r rcr TprprtronolloolJv or- ivo lcroryc fQ4rprxrii A),y€6Qq.

To 6etirepo 6ryc eircL vc lrobriioupc rptoto:ropLurci trtlQir|rarrr rou 0atnoQolJo(! !(l rixr€l,r:otio1]r $'atlrouq lorrg rd)61)iFE. A\rli vcr b1pr,o1]glrloorFt'rllnrxd,rcea6€i./F(rr{r IQ([FLzrii! €Erori]orov I rrao6],tFdrorv cloxior(ov raroc-lirvt,rv f :rrvcirtuv Mrrkov. nAooro0ouF€ !( nQor€ivoutr€ lda nQo6Lrit(rc -1oonoi( o qorrrrrlE trrroQ€i vcr Eiv(rr o,a(xrio; nou oct 6d)o€r (('rAwrloyl. EEuno-]aousr(.r orr 6€v 6i6ow(r ).uo€Lq. { 6ev propei vcr bo0ouv. Oa ocE riFouv ro),nruyu;Fuv fdv I 'ou yo' i ' t , r rr l io y ' t r rvb(( l i0owu rrrpd ulr .

'Evr 6cor.z6 otvo),o rorbfxov €ilcr ro LINPACK, :rou <rvrr.xadorloe lLl.Lri-6rE uropourLvcov, cn6 rr,g oroieq dl),eE rlr(v znl.fq, cilkq 61. 61$orp),4pdveE ot:tolllil.c uo),o1orr,rci :repr6cil.lovra.

To LINPACK olrbrltoerlx€ iroL drorc vct eivaL culriprlro p1rrvrl6, (ljro-rrtr€oFr(It1|6 zor on),o. H €rsq€Q./ooi(( 6yLve ojro rorg Dongana. Bunch. Moler,t((| Srewan n(rr o! obrlli€; lt(l rov /,artml Eivar €E((la€rrxa<. AqoaFrt yL(. rtvcvirrru!1 cmdrv rov xr,.r6iaov ur{ple 1 rlrrruxic rou EISPACK.rra ro ra66}.ryol6rorrp(i)v. 6y(l) rolj EISPACK yv(,aiqoup€ nl,aov qv oEi(l r(,)! rdl,oyQatrll6!0rv

NATAPTHNlA I ' KOAIKE: YNOAOI I ! ION TIATHIPAI ! 'MIKI I AATEBIA

$su4r.t(thrv nQoYQoFFor0rl.:ro LTNPACK o).oL or niv(xeE anoo4x€rrovrcr zar' €uo€!(l! olI pvlFl rou

1j][o),offfiri. :E noL;.€E $lzcrdc cur6 .filrQ€n€r IQlioI o).oxl,nEDv nLlditov ro!onol(D! rl tirtl (.v€e;{Erclr o€ Fr'Qrx€E €xcltomc(be! lacr rorvro€r6Ei! riv(x€q ro!onot(D! I rdE! €tlclt tr€alxdE N!1,!ri6e!. :t€b61' ol,c rc ,rpo6).iFc1rc1 XelorFonorou',dio unoQourilE<. Lris 'f|(I va n(rQcryovronorioolv rov riva;{cl out'r€).€oldv L qr

Fi(r lrcr vc. 6Qdoow o€ zirnor.r srbrxri b€ELd rl€uaA. OnoE rovioolx€ olrtvn(aa?0.r'to 1.5. (luro rEoLxovot'ri undlo.frorrx6 7e6vo mr: nralnrd'oflS ;rouunda/€L uro),ou0icr 6€Er6v nr!€ua6v ouo1r oFivo;v pe rov i6ro rivczr ourtrLe-mcbv. O o6q16g yrn rov Tprlorrl oqpudrv* 6r "cr!.rl I r.ftdoraol €Fq,cviletcL16oo oqAci xtrr rt oLxovotric 1g6vor eivar 16oo orlpcvrlr"{, drctr bev r).tlrt,Otpiq4rvu yra q liol ev6; pryovropivou orortllnro( fs lia r{clr !6vo rnoaouriva .

Aivollrl 6!<rv cnoo:raopcrru6 xctd].oyo rov r:toporxrld;v. O 1p1or1;ynoqrri vo 6raL6!ar avcipeoc oe tlrihi t{tll 6l'rll crrQi6€ro xr& (vi4l€o(r oEra(ywllxll ll tuyoblxi cQr0lrqflxi. Allltg oL 8!&oli€E ricrvoli! ro noxdxo oxrrLl{riFoy.irLo (roob]Jaqro p€ 80.0001aa9F6<). I boFri xou 6For eivcL Cllrhl. Or xribrxr<ncpqovronorouv rov,1, rl l,riow ro ,1:r'= l. r1 unol,oyi.louv r1v opilouoc.4rov (vrioreoq.o. flo rlE e:r6pevsq rvliu x(rtfoQi€s ntl,i(r((J)! ,4:

GE yc\'!,|oi<cB yr\,Irokrcrvro€[6riqPO 0r1[zd( oQrolrtlor!PB ocrlzir oQtopdlow r(llvroft6ei<SI or'B!€rQU{ouSaoQroroqHI Eat(rLavouEcroproror(TR rAryovLxoiqGT yeulaouqr0drc!yri)!!o!qPT 0Frrxd oQroF6\'oug rQrb((y(iwrot\

Yn(iplouv ra€LE €ri '.6oa x(;)bue( rou ]jroloyilotlv rt briron(ol Cholcsky-4 = 1l'ft. tlv op0ordrvLtr-tpr.yrwtxtj briron(.ot cram ' Schmidl A = Qn xur rtd(ion(ort |6roFoerqi1)v nfrar A = Qr:QJ. f|.1 rtv Cholesky o niv(z(; €iv.rlrrreqt)'\nx6< zar O€rLxti oqLopivo; xo o lil aivaL azpr6r;q o IvE. f!(r ra r),h-lora rtrpcilovc e4ilo4re 6tt u:rdppt I r:rtl,o'yr1 crvi4reon orov Q.fl :rotr eivntcpr|tuFrrdr €uora0yi:, qv AIT not' silcrL zurdrLlll.I yrtr 1or5 no),ri oto),cotrror'Ex(L q 1prloq rou ,,1r A zcr r{r)! ,crvovlrdrv €gro0lo€(|)v nor €[v(r n lcllu1eQl loJvrQrci)\,. H ou\,i0qE €JltLoy{ €flcL o Ar A. f[(l ouB!€xQrrois ri Ep[LxrctvolE |r[1'czt:qr( nQ.jtnutl roJv !6[orlp{ir', orJp(p(ovo [E ror NoFo A6pl1v€r.cE (ncted./a(1g]oq 6.3)r(rQrdlow oro TrQ6onF(1 (Dv oblydv xul u!ro ri.r,cu. prpo_< rax elclopercr.rv torrl]aoQorrlrvd)! 1[o rr< x(rrnyoQieE rov nrrdx{rv SI rc! HI. @uBiEouF€ or[.1610 rr]cou$rrrQi(lE, xo rdoE rootoE rEQloQiqEr(r ol€66v mo FLoo- YndQXE! eri,ollq npo-6l€ln yL(l avav€Loot zrrL €navar0oo6roarqr6 TtD! raaay6vr(l)\' Cholesky, 6r(.v oA Ffrahl.IH.r. :TAooofrovruc 11 uqurlr,i,vrrr: piu ypr4rp4.

. . A I ' A I ' ' I ] \ ' \ T K ] \ I K r T I N N \ N T I : T E \ T I A T H T P \ \ I I ' I I K I I A \ T F B P A

MLa 6coLz1 unoQol-,riro €ivct il SGEFA, :ror ncpcrlovtonoLei ivov levr"zonivcrxa oe cr;rl,r1 GrQiduft. xQry+ronori peprxrj odrjgol rcr o.ro A 6r1Fa rll<u r ' , ) .ur1r1- r .rr l*1rr f l , ( ! , ,btyo ro 1r, yur.uf l po ororTrio fnr I i ( ," ' cao rr t6(./d)vLo rt< ,+ ofli],1q. O ar6aaroq IPVT(K) anoollr{curr.q yau0p{ o6rtyitol<.Ml6sv[1{6 o€ 06ort oblyou 6* olpcivfl 6rr I ncQoyolxonoitolt itv fivsr(L(yvrDpilorll€ an6 ro K€qdicro 2 6r', PA = ,a/ uiv(! n(ivror€ 6ttvcr6v rctL t1LINPACK ouvrnoloyiler rov P pi:ocr orov t). Alro rnlprtiver 6rL o U tivcflLbr6poprlog xar otqv crvcr6poptzrl dwrxcrdorat;rl 0o Eittlp€ 6roip€ol ti€ Ftb€v.)irtv n-odEit. brcie€ot U€ pI6€1' oq8il,€r(L rr( n0Qrooor€a€g qro06g o€ ki0og 10rlrpoypcippcrros', ertb ivcg ct666v L6r6poq9o< l,t rtcv€a(ba€t apt0$rtrLxt euord0rto.

O obrtyo< ./Lc xov XQrtoq 6ivcr tlta ouvoql rou alloaieFor ncaoyo"rionoil(nls:

D O k - 1 , . . . . n \6Qf< I raroto ri)(tr!r .r/(. = ^xa. .r r.^acrc ro I olo itr,f(A)llr, 04. = () nl6o oro r6),oq rolr €ncr1'cr)'.Inttrou xi[)'o! yrc ro l_r\,dl,l,otr rc (llt xcl[ (/r'D O i = k + 1 . . . . , t l

t t t t r = -a' t fur ' r ' ( i idLt ro or l 060l . r , r ) 'D O j = A + 1 , . . . , t tr1,(i),LaEc rc atr x(r[ a,,D O i = A + 1 , . . . . |

rI' i - rltr + tt t i r + l]l,.i

T t ) i . r t t t 11 t , tu^ , \ ' t t x , " t z ' i zh" ' y 'u to j

l , ) to . tou , : t . t td4nrk , ' t t x t t a y t ' t r t ' L

O ,(try, iL,rr)J, , : 'v yrvrxrc ysrppiE. "rvur orrf ioc. oLr{ui Ur, , : ru :r0oorxr n1t , r , , r , icnoxdlurw€r €rril{EcLi. O c),y6ar0[oq bfl cqaeei xcr' e!0ticrv :ro],Lur],<iorarrtE yp.rppri< o6ri o$ cr6 rL_< cr6 xirro 1pqlprc. Mr- ril"),c ),6yLu, 60! rd''t!:rpciierq ir€ ypclp6(l Avri 1,r'cur6, o xdr6rxrrc rtnoloyilEr olor'rlQrt ril oriiln rorv:rol),cr;rlcor0ord^' ,rii z(1r zcrdnL\r naoz,{Da€i oTrt}.rt rpoq m)i}.t. xarto4roroLrb-\,r.rr .lrtrouq ron< nol.Iarl.aoLaor€q. O ).6yoc ri\'cr 6r q Fortran MoqrxEiE!fliraxet orilrt rcos arilq v.u iror. ;rpoxtoprivttrE (]1|! (JT)il.E< nQoq r.r ndro,blprorploiw 'oflprcrxrl" :rp6o6coq U,nFtE. fLfl [8./dl.ox( ri.vat{Eq rou .l'Io-0rytrovrcfl or oxilpu oe),i6os. orro $roQei \r' anoq6a€r ottpalrtlr)i otxovopiatQ6vou. Eror o €o(DrEQrxoE 6QoUoE (t = k + 1... . , ?r) roraore|ciLercr dror tirmE1,{r €rr€}.Ei. nAdE!.l( $EroEn ort}.dv '{.lr 6Xr ypa[ptiv (orryzpivarc pr qv Aorqol1.4.9 yr.1 ro\r no),l..rrl.aoLaoF6 ,4r [e yqaNtq { [€ oflil€E). Kct' ov{r"ot1lov ol,((i6rxe( Eni).rq< SGESL. SGBSL. SPOSL. ... licivow avabeoFrrrl Lrwrx(rlootconorrt),11 raoE otriLit. O rr).eltcioq ciWrDoT o_c oro rpLlovrx6 orimlpcr Ul = r: eivath, = c,,lu"" lirrr zarorrv aro xlv c otpcrpei-rcl| 1 rdeuciu ol4trq iou U €rii ror,,. Arn6 rro! oropdv€r eivcr onorqpc.rirEcoE r l. irol[o yln to €ri6paro 6t[aqE avabpoBr.xriq a !.ILxarcioraonc.

IIAPAPIHM-{ I KQAIKE> YNO OIDI\)N TIA TII |?AMM]KH ATEBPA

FI c),Lu1j arnl .rorv:rpdleorr [e o ]),tq oqri].erclr orov Molc. zar r.r

npoyQriurrcrtn 6titrrbOryrcrv 6a0grcicr 6olr vcr rdvou rzriprlol aat r.o,r t\tiztrlxrtruottiota< (rou opioOrlzt otrlv lcrsdlpuqo 7.2). nari1rrorL, I olrriery€flioyrt oro LINPACK 6|la tiacr I SGEFA (l,Ld q SGECO. nor orFrEAd,atL6rivErL l r ' r q r t r , z r t r r { i t i r ' ! r r | 1 n t r f , , q ! r X C r t . r v , r , ( r , ' u , , o ; L r l 1 u v f l l t l r \ U a v r } r uvo naozrDaio€r orlv r1v(160oFl i cv(rcrrdolcorl lt[ roa x(ir6rx(1 SGESL. TGxpolpc4r$crra ollTd €E(ar(i)vr.1' ano sva orlorlfr( r:ro;rqolqaFlxir0rv B(ou41qfA(lLtuxitg Aiy€6pag (BLAS) ror rirvouv:rpdlaLq,6:10r l npoqlorprvl orrlouro;rrrxl n€ArlAoqri ror a),yoAie$or. rAooe6rovrag ;ro),ianl.rioro fll< tr((l-<otri,I< o-qv dLil. To ra)"1x6 cr:rorilaopcr aivnr pLa ori),o11 r()6ixoa nori|loroLei eaqrdoLo rit 0e(lrAia rlE foallrxriq ,\].yr60a<. O otu 6< ro! )(Qliorrlbrcrio€rcr ort br€nervort:

Socicly for l rdnsr ial and Appl ied Malhenrat ics (SIAM)117 Sol lh 17rh StreetI 'h i ludclphi l 'A l9l03r (215) 564-2929

'lol|LoljtLc cjr|orF rrt ol$ooio rou EISPACK yra ro n0()6Lr[ra rbrorrt61,.Ot x(bb!,a€E cruroi eivcl bry6oLor. IIepri1ovrcr oe no),ii( tjdLo0lr!< uno-

. ' , ' " rrr , : ,v zrr l fxrv(41.f , , , 0 ro uqlr{r \rrzLc rr ;1e: rpo7gu,, , , i rLr! 1.u fnr\ , rr l l , 'vrxo

l , ln lcmatron.r l MalhcmLrt ic.r l rnd Sl.r l ist ic l l L ibr. ' r ics ( IMSL)750 Belhi.c BoulevrrdI-k uston TX 77036: (800)222-4675

2. N!,merical Alsor i thms CRnrr (NAC)l l t l l 3 lst SlreelDownef s Grove lL 605151 (3 t2) 91 1 -2331.

:rI o!\,alcrc c(frorortrc rrN nQoooxri of tllc vlo ltor Ei€!0€pIE rpoo6aoll<nlyri ror )"dyftcr "Dellib'l Arril ri1,ar t C !f4( Lr:rrlptoicr :ror 6rt10t'rsL rL6rrlgrlroAourivL'E. oXr 6r6),ro0iizs<. 1r.r l.crrol'Qy0i [E olox),liQou 1e r1),t:rrpovrrorc/ LibQolei.o trio(|l rrDv brrru(ov ARI'ANE l . CSNE I , l clcDcr )j Phonc Nct. K.rU(i)E6Ev lroo],ati€i .iv0Qo,no:. €ivor 6wcrov (rcr ;rL0crvtiiq rTro ]aiyopo) va ).d6srncrveiq.'c1 naotpriFtraro. .ix6trl ror rI lulIo. To p)lvrt|'C t{cs yAo$llic \endindex r-vralonorri olmllc eErirlQirrlolq :ror, or6],vrr !\r(.,' xor.i].o./o pa rot l . r id0.\ r , ' r '17,6qrtrLr"o. Afv , n11u7,1 ,nrtp,oru rvrt . , , ouorl : z.rr "pr14nr1. ora\rrd),i.ry[o 6F,D;11 ('r6raroq eivcr cipeol,.ro z6oroq elcixrmo. zdL 1rr.ov€ct izboort rdlror€ 6ra0toqrrl. OL 6rsu0uvorL< 6ev avc.lvcopi(owar c:r6 ro

3. trtlib(!antrncs (ARPANET/CSNE f) )i resedr.lrrel/,b (UNIX nenvork)EISPACK. ITPACK. LINPACK. MINPACK. QUADPACK,TOEPLITZ, TOMS. FFTPACK. BLAS, FISHPACK. ...

H e:rdpevl :r1yr] rircr ior'4 r1 zrd,Litrpl. Ilp6xer.tur. Y|,cr 6vcr 6rcrlolLz6na6TaaFtr( no! ovo$dE€ror MATLAB lprqpero rroLxci Y|,cr r:roLolLogoiq pr

NAPAPTHNIA f KAAIKE: YIIOAOTI:TON 11A TII TPANIMIXH AATEBPA

ni"voz€q. Arari"0€raL yLo IBM PC. Macinrosh. SUN. APOLLO x(r! VAX. O r66FzoE uuroE xar€).IE€ vc €iv.{ zcroL€EDBaro E1|'raL6€u1r1{6 €Qyd,€io. r(e6q €niqgt!(rt ro 6corz6 nadyAoFF& ./Lu rrtv €e€wcr or oQroFrlrLrn fAalrprzi A).ys60tr.ToliEoritl€ rllv rcq qrt(. r( fAclq,Lya. rov rclxi Ferco/xl(rrroll6 Fouri€r. rov'',ir.!,ItruloyQ(iqo" Io)v iradE€{D! ttu rivar€q. rard rI arciQx€r.o rlE €zi€}.Foig ro!q,

4. Math Works2{) No(h Mrin StreerSherbom MA (11770: (617)651-1415

>rI or'!tl€(r .plol.rcr rEr €xn(16€vrrLoi r66Lx.< ytc PC. nQ6z€ri(r yr(l6\'(v Clnoonco!(rlzo xutdLoytr.Ir.a zcr. xdrir raoy0irFFar.r 0( €E.1'roLoletloo!!va epqxrviLovrcu 6pnrg 1r(r r(1 cn6!€!a iloul€ axouorr" xc],ci ol6lLcr:

5. Mattix Akehrc Cdl(xn,?or (Herman and J€pson). broxdru xaL 6L6l.L<ipLo prolarli€: r'qeFoy6s: Brd'kecole. 555 Abrego Street. Monrerey CA 93940.

6. MATRIXPAD (Orzech). orF6(rLxrt €i.oobo< nrv(ixcov xcL aiFi G{O!OFI,rrxli. €xbr6dllr-r,o tu_T6 ro! D.C. Herrh, 125 Spring Srrcct. Lexingron MA02I?3. n),rlaogoQi.€< o116 io Malhcmatics Dcpanrnent. Quenn s University.Kinsston K7L3N6 CmMr.

7. Ltrrt (Fr.tleiSh). p€ cwioroLTo xeipevo: 0-201-15456-0. Addison-Wesley.Reading MA 01867.

8. arrprr-,(n (Anton) $E (vriorol1o xrff€lor 0-471"830U6-0. John Wiley. 605Third Avenuc. New Yo'k NY l0l5r.

9, lllatrit Cdkulutor (Hogben and Henlzel). 6co!r{ig egrcrppoldg oe p1tr1(1OI0flrLr{rt .rxer6€i.r6. Condui(, Universily of Iow.r, City IA 52242.

10. MINITAB. orcrrroru{i xcr 6ooLriil lQc4qrl.ll ciLy€6Q(l. 3081 EnrerpriseDrive. Srate Collese PA 16801.

O[ xdblxe! ]rc Apple rcL Macintosh c!(tjrrux0lxov brorpopetLnir zru :rr.oca./h. Mra(rir rQoyQqF$clto Eoroye(1qo1,t.r, nl).ci dvc n),rjpr< rphopc rgo-YQctrttirrov fAaFFrxrtE AL1,€6Q( $&i!81(r !c €fvar reptooorrpo pr( pe),iovrLrrl6E6.(16rry( n(Qd tll(. n(lporioa 6uvcr6rrlt(..

Mvryov€ioxge mo r6Lo< pLa rpooc,rnrxl :rporigrlol lLcr 14 6!6tlozalic rrtqleoobor si plex oxo! lpaFulxo nQoyQ(1[i1crrrop6. Aut6c eivar dv(< a].y6Q0-log $€ otro.ld.o€r< o€ L{i6€ 6ilt(: pLn etoepl6pern yFt(l6),ryr1 crwreaormir pL(l€EEAXoI€vtl. Or :rpd!er; mr< yQCtFFri( tlou ({zo],ouooi! rnv onoqool 6€v €iv(tt€v6Krq,6Qouo€E (rp6;reLtcr 1L( ev(l F€y(i),o rQodLngo orlv uri6e{r1 qg rpc4r-[rx{( A}../E60cS). nl6rlwaq mq vdc :ropurpq -nor [loQci vo €iv(t rr(].nreelrl X€LAorseI { 6tl.rroll o rci6[raE "MacJirrplex" e;rrtpd:rer otor Xg]oq vcrdv€L ldort zo! !a €nLotpfper :rioo. 'EX€L ye&qr€i ono lov Dr. Andrew Philpofi(Engineering School. University of Auckland. New Zealand) yLa v( l€rrouQry€i oeMacinlosh 512K rar Macintosh Plus.

nr\PAPIH\{A r KQAIKE: \TIOiOrlIlON flr\ TII fl'AMl\llKH,\"\TEBPA

IIPOTf,INOMENA IIf,IPAMATA ME TON YIIOAONETH

:r6loi llou o curdt flE ritlQoyQ(iqoug €ivcr v( nporr[!(') [!c( xcrl€{orJlol a€!-e.4raror !l€ utrol,oyro (. ,iou (,roqsrryoul' rL< nro oljq0rolav€q zol oo6crQa<6r.o,{o).ir: roLK. AYrd nou orryrloors ooF6uivfl rivllr ofl o rporrlt: floiy'r dvuvnivor(. rcxd 6vc rowni riaL brcr6irl€L rtv oravrtol. nollyt Tvri)ol 6$' €rr-rr,y1d\'€ror. np6lirLxqL r€proo6r€po yrc tra cnl,rl riorlol ncpd yrtr eva cl10rv6:rcig '4rrr . To braoygd..orzo l rr rrr l rcrHqlrcrrxt i s i1 '(L I r iprnt f i iuv nA(l ] t , (rdvxur :rrorsuu ritt o r':roLoTLorrt xrrlrrord buv,rro xrirr riroto.

Mtcr t6ia'z),trbi €ivcr l(( xQqorlloirolrl0ei y€vuirQro rlrl.riov oQLoF{bv yrsr4v rrtpcrlulpl tuxri'r,v :rtvizt,,v. H 0rcugi(r o ,twov tov rolri( sivaL or vl:rr(2iIl.rlikl )icr uuE(ivFrcL lQrlyoQo (7r(Qc(zlvou$E\'l o' quro €v B€e€r qro ro oculiro|(rr0o6Lr}ror(1 ro! nQo,{]Jnrouv oTtq q)(orx6s €nrorrtttsE). Or lbLorLtris. rt r(r [€y60Inov obryriv 'iaL tu\' noi.i.cr:ri.rrouorrirv. i or opflor;oq. ri txfu1 zaL q yrl-crwrrnQrrlrrp6rry(l rrwr'rr r4q (vrrorQslp4r6qtug otrp 0 - | n€Qim(l)ijl or,(.(ur(i obryo!'v o€ (iz$Dq €v6mrl6Qoro€q €{[Dlrlo€Lq. Y'loolxxvrlollr Icerri6q ou0(1 lroQo]Jo(v v( rroouv raL orL< oiyolro(r 0(t or€q0€irtr rr dl.Ieg. TrrdoraaLutroi ouqll1otvrar orrl oe),ib(l 553.

Mr( 6€rilEQl !6€( €ia(a! 1,( x(r(roe€loEouF€ ni!(xrq €!6( oryzexQrtril,orilinou. not' vc( r-nrrpan€r try crliEryI lou f€./d{)orE rox evrir 4 xcrroozetr{ rou<ncQqrd!.r €irol.I. 'Evo! 6sorz6< ruro( aQoiQxex.|! (rn' rr( er.eQcro[aat:E 6L.r-q,oeiE, 61(\r rQooglyi(o!! bLrrptrqr:r6g eEr,o(irorLg. MrzQ(tvovrcq ro 6iB.r i.crEcivrrct 11 i.ih ro nivcxo, evcb o rivcrcrg nupr4rdveL rpcrL6c xcL er1pr1-oro:. no)L(i rQoiJ),tjt',rlr rnrottirovrxou ).oyrutuxoi tr€Qr olerlq . )rrat lipo rt;rordroLou< nivc(',€q: rr< r6L6rw!q roq, q oupruq:Lq,ogci rou; 1( peLoipevo /r xcltrb6€E ./L.r anox€L€oF(rLxou( unoloyrolor)q.

Ouoildlc( n(xaru .rfi orJrd norr cvcqaQ(pr: fiQoilyoup(v0)g neQL€Xo!! er6L-xouq ni!nx€( no! lrroQci !c itlqooiv an' El|0siaE. Aur6 €E(ll,€lqr€r. ro:rp66l.r1pcrrIE rrlJ(]6(r11. piqog riBt4 rarv F((h)F.lrrxrbv nepriXercu. ac(! orrl bLcrnrncoorl rorraod).itrcrog. Ttr n€lad+rnra rcpcrxdro oup:rLrutivow urndE rrq aoxrlor6.

Twaiot 0 | nit'axtg 0t tni tr)

f.f noro qriFc rorq €ilur r6L6poprpol IIoLtE rivaL oL oul!6rrF€5 ru)! 6!cq6-Qolv oQ!!o!odlv: flo(r eivcr 11 peFlutepl 6uvcrn oQflouo(.i llotog €ivo!o tl€yd,ulsQos 6uvarG obty6g (ps l€Qrxrl ooytyipl) x(lr ^6E quEd(v€r ip€orl 14ol lou tr€ ro n:

1.2 Ba€irr rr5 r6Lor4r€q (reproaiEolroE ro! A va €irc! oxt{r€rQ!*68. €irvro cncrLrei. o acb6Lra< zcL o7$tdlovtri<.r< €q' 6oov ro €nrlpdr€I ozri)6ri.c(). flolcr eivcr r1 geyc).urtpq 6tor$rj ro! 6eiypcrroq xcL roraeivcr I pdol r4rq tt'-rv ptya),ur€Qtrrv tbrotrldly:

lJ n6E d,Iirlow oi GrcrvqoeLE 6rcv r1 :rL0nv6rlra ft-'v I cro$(lxaua€raLcl'Io To ;;

I IAPAPIH ] \ ' IA I ' (O IK l l ) ] Y I IOAOI I ) ' I ( 2N I ' IA fH I 1 'A \ ' IN l IKHAATEB}A

Tuxaior nivsxt€ xu ircynopals (n Eni. n. n €ni r xoi n €ni 1)Ta oror.Xria r()v A ]{or b em),6yovrar [€ Frrr yEw{rQra 'nryai.ov aE9[dx a?ro6vcr opLopivo ne6io rrB6a iift 6I' $6ao ro 0 1. Taonororrlm€ rc, Ev avd./zl,€ror orme q peol rLprj (q avcpev6pevl r+rrr) va eivar pg€v. Yncipxow strobuarorlr€E yra oLa ra n€racqara:

(i) v(l 6oul,6!Joup€ licr' El0ricv fl€ xc ,4 rc! l)(ii) vo E€rruiootlrro an6 orcoopci Ao rcL 6o xsr vG o€(,aou[s rov A xar ro

b o(lv SLcrrcrpcldq €),iyz€r€ rI orati€ad orirotroq: c zor boul€r€rs F€rov ,4n + c.{ zcoti5 zcL ro fu + cb.

Y;rirplorv owcrpramLxi rporrlpcrc nou btv dlow ox6p1 ancvrrlOei. pproon6 xl olulrorlr{t qS ufid,olgnlq Gauss (orio! o A xcl io , nQoaQlolrc! dro

lloro Eilc(r l rcralro$ri rov det, i Ka(\,o\,lc9 dficl.orqr)i z,orai.q oblylol.prroQ€irr' €iriorlq vcl 6Qeirr rLE oQiqolo€q rcDv (il,(D (Qror€ed ur.orlvtrov.1i(io(;S ?i(lr rou< o6ryouq li(lL ro!F rrol.Larrl.ooL(lo-riE. OL<r curri ilouvpeyritro r.v6rrr1dgov.I lorcr eiv<l 11 p6o1 trp{rour=,4 Lbr:rh/vQivovrcg [r ro, ] 1=,4,;r l ) , , ,noLo civcL ro doo $ryoq ro! .f = (r{0 +.,.1) r(ro +.1,) rc{ki)c xc' rorJoq(A[(rloq ,. ,lr|iAraLrriore o!4rtrQilr ro! r{, orore llrL (rr: + r)/2 crvuldqrrpcr tquiuoroLrrio, licr tiptiru rr; rdror4rr'< ror. Bpeirr uriols r1v F(l)=nlyloocrbLor+[ir! llL11QorEQr1^, xo! l. no|C ai.vor I zr[o!o[r] r('\r 1,,,.,i ndgcr),IdlrL crr,'r6 pt ro /r; To t pciqruLr x{( F 0c g:roporot vcr quvri prrdcr:r6:rol).d rrrqciprtcr. (Y:rhqlrL iva< 'xcv6vt(

Ip'rilxlio!" notr ).i"/orixorw rn,vcvnjrrrr. H F pordlrL pt ro o).ozLlpopcr rlc (l /r)'/r.[€rl1 (mo a\rcfid0[rol r((r I nuxv6rlla r(l)! r6torrp(i)v g,r(ixlsL daorl[rxu,a),'o.) Ilotl1 €iv(r r ;ric1 tLprl rr1< rct0vopj; ,\",i"(?10 +.].'l)rKarcrol{€uiror€ ru}.ri.ou: rivax€q r? €ri ?r rar €g€rdo.r€ rorq obilyon:., , r ' , 1 ' , , 1 r 8 , , o I n ' , r , p i c r , ) ' l r I : r , , , r v , , r r l y r ' 1 1 r : v , , , , ' p p r r p r r , ' cx&r Oetrzdr ryroprogdvoq.

TtYaia tirrr't,(t ttilexta nQoon ;ota< ()n tfii ti)Aooivrorv n 16g6cor:ror e'vcirvovrcr to:rolu pc (rr' n)/2 cr:{piq per(iu rorq.rono0€lrtors ador corsrl se :rrocvoqrcr I (Er.rrvrlorc B€ p = 1, iva nhip€q1 . r , r 1 q r " . x , , r o r r v 6 r ; r r 1 , l / 2 , . A r L L l r u r i v u r L r c r p o o : r u , o r u c , r r l r r cyQc$Uli ylcr xd0r lra crn6 rrq nr crxgt< (Ilcrprilpurp< 2.5). Il€a#x€r 1 x&r +1orr"g ottjLt:q to!. rcr oroic cn'rrororlorv oroug buo x6g6ouc:ror orvbtrl 11 cr41.B ti'trLt o tn1ypivry rtuwlq jrAoonrioroE, ;ror r:ivcL o i6roq F€ rov .4, Xorai<oufr< r I oTriLt n. Cei.var.or u n. f t lvux q a1i!6$4( [s. i j =1€ri lo!x6F6o!; zu, ; oLrv6, ovrar

"ar . r - I I ra} 6rv ou\SiovrL|.

2.3

2,4

I]APATTHNIA T (OAI(E: YNO OTDTON TIA TH IPAMMIKH AA|EBPA

3,1 flrlold stvar I r(0s!6qra E\65 ourtltxutoli dl}|ilou; :' dufli rI'\, ?reQt-?ftcuol o ,4 dx€r rdqq n - 1. fkis eEsQrdrqL aofli I rll0rrvdrrlrrr ori6 roDi

3.2 florc 6rivqrq C[ etvcr t1 ekilLoq pe 6].a rq moll€io qE o€rlxd, X(l)QiEFl6€vrr.dr. ,{c.! roro sivqL ro !64ra lou * oe o16o1 ye rrE 6!s6Qo}r€EF€raErj 'r6[6ov: tldE [€ycd,6v€! ro t oulacmioer rou n;

33 H oeitouoq rou -Br B Veec- ro ihiqoe tu)v ouv'eivrav 61,6gay. tl6g€EaQrdrsl rowo ono rs p xq0 n; TlndrorE yra [wQ6reOc tQqqriFctc(n = 3.4. s):a) p|,@ 6dorl rou Fl6€v6xoQou rou A6) Frq 6dol rou FlbcvoxurQou rou Ary) xq m,r zar rrlv rdEI .6) rov Br B xo! rI! oQi,Eorlod rou (io nllOog lov ow6e6vr<rv 66v6qov),TcrQldor€ rq qfior€leopsro ;re ro yQaqllq. €r6!xd ro IaQoE (6) !e touE6QoNouE.

3.4 BQ€i,,r€ rrE r6rorrpag ro! 81 B xcrl 1ot) C -er6lnd qv zctcrvolti rov^-". zqf fllv licfiovo[ll r@lt 6€Ht(6y xaraoraoErog c =,\."-/)",6.

3,5 noLog €ivs! o .Br B r(|! or r6roruaE rou, orry nepintrorl tou:rl,rigorgyQqeiForog, 6aou urriplow o|ta ot (n'1 - nr/2 qrqds: Tundox€ xol,snivcrz€g 3 €nt 3 xor4 €n[ 4, pe n =4ya! n =5 f i l 6tcr ldwro zaL-1 aLLori. Bpeire rry nsQqyowonoilcrl LU xat tat6xw t1'r LDLT

6l,6rcre ro pori6o nov lo)J,qrLaoloor(l)v orov ,;

IreneQd.op.'ei 6rcqoea6: -utt + cut = I trs 0 < r < l.

4.1 fla c = 0, r{qrqox€udor€ io! rQ!6ra"yci)vLo r/vazc 'eurtpaq 'tcqopAE.4n p€ -1,2,-1 orLg 6Lc.ycuvtoq, Efiqllloe]Jor€ 6rL detA, = n + 1 xctl!fio),oytore ro! /;r. Ts ororx€iq rou (i,j) 0a nQ€n€r v(r dXow q

Boeqrl ar(1-oj) nd{D cln6 -r1 6LclcbvLo. Bgeire rov qQ!0I6 rr xrrr Vdlreyla lori6o xdro dri6 rrl 6rdy6vro.

4,2 Enol,rloedfi€ ofl or r6loru6E rou An €ivdL )j = 4sin':j7rh/2 xaL 6rr rat6ro6rcvrloparc porh(ow F€ rpirovo -dnox mi! Ao*qol 7.4.3.

4.3 Korcrox€ldfor€ rQ€rE 6wdioug niar€6,"pdrar dnEoprbt: ro,t F- trot:aXsL I oqv ziQLo 6roldvro xsr +1 on6 rdvo, rov F rou 4(& +1orrlv zua!( 6ray6vrc x(rt -l qrd xdr(l) zc( rov .F0 - (I] + F-)/2 nouiX€r +l/2 na\a a:r6 q 6royrirvro xa -7f2 tctl,lJo on6 auoi!. Elooo6qeo ne66LrFo rc|]v Flxdvlx6v efvar rorcv F vo XQloqrorouioow€.Xlperr iore or 1 11" + cl , r / npooeyyi!€ror ox6 $! AI =h'A+h-)cF,onox h €i.var ro 6lltto.

xore rl! -u" + ca' = 1 [s u = 0 otd ouvoQr.sxd onp€io "

= 0 xa!r = 1. (IQooridoriot€ tre rry €cr xdL rlv .lr + 6-) orr,dlre, el 6uvar6v,ro lpriq4rc q; <rxpL6o{g LuoEoE "/!a c = 0.1.10 roL d},},€E rrpaE. Ator€xcr6mv T|"q I1,u = e i tcu l \ I -u = e xar r l1ou = €, 6nou e = (1, 1, . . . , 1)ecrL or. :11 rporqlovrcL an6 10!6 rQ€16 -F. fLcr peld).c c, 4 nQdJrrt

NAPAPTHMA f KO IKE: YNO OTDTON NA TH |?AMMIKH A TEBPA

ndpdloryoE car €ivar olFdvrrxt: florc enr,Loyi rot F owlolol€; Ilordetvct q roLdrepl enLl.oljlLc pLxpri q lqxqivcrE [€ qv r]6rt ]JndQXorod

4.4 YrrdQX€r, xqv6vo tlori6o orry der-il1 i oro! ,il1 l, yro xrio€ dvsv qri6rolg rivazeg ,1d,,il/ .,110;

EEiaaan fit Laplace: -ur. - Lr r! = 0 o' dvs polaoraio i€rpd^/orro.

O-tcv to 6r]pc eivcl h = 7f (n + l) xwa tg tr,a:ruOuvoerg r zol y, 16r€ ufidQxorvnr dylorolor" [6od olo r€rQciyovo. AQLo[rlor€ ro1,5 aeliEorsog c(n6 q! xdrLD1Qa!4r1, xcrl6rirv r!\, 6€6r€pl n0os rs rcv(,) xc([, r6i.og, rlv nd\ro ./QoB|'lri. Ord'/\'(,)oior ofllv xorco aQlor€Qd, zdto 6elrd rcr. :rrivor 6elLd lorrfa aQloBouwar$€ Tq Lj n zor nr cr'lLoloiloJE.

KqrqoxerJciore rorJE nivoxEg 6Eu6Qov 6roqoQd)v 11 r{dr y (oarg6vnq xqrzde€Io) [€ -1,2,-1, rdE€cus n:. O I{ ;roi ,pveL 16 6raqoQ€E 14crd f l l ! c)isr€rlorJvol, zdrd [lroE tcD1, 1,Qd[tFcbv, zqL 0q €ivs! rQ!6[qydvloE -6pcoE ra -1ndvcD , cr xdror qro fll 6roydrvro 6xow rs\rd €x€i rou orolqroljv o! yQqFF6E. OV ;roi,qveL rrE 6logoQdg xord qv,{oI€uo!!ol y rcl rd -1 €EqqaliEo!'tot orrEn-0166 Eraycovioljs (Ento{; $ r€1,i1).

BQ€i,r€ r$, opiEouoq xqr ro! alri.orQoqo rcr yaxo 6raEoedy riyfto'tpEiav L = E +V. Mropeite vc 6ei.re xcinoLa [ori.6a, xa06E auqci\,€r

Bpeire rrg r6LorL[d5 rou ,L xctr ouyriQ|v(rr€ pe ro croQoio[arcl )j + )rtory l6roiqrdlv tou ouvrloorq nivcra n uri n 6€!r6Q(,)! 6taqoQd)a ,4.rAuro: il.r ,4 r&or.trrc noLr uvaq iqo4xov rqo4./o4dro;. tipcr u:rdpXouvnr (1oQoiopatd.) M:ropeire vc o|vbdoere rLq r6rcrrF€E rou ;l Fe t(rr6ro6rav o$.lra xou l,;

Auote ro -Lr = e. 6:ror e = (1,1,.. . . 1) dleL prjxoq n:. XQno||lo?ro lo-r€:rp6rtr ornl,ottptl y.cr rc 6pei.re prc nxpL6{ }.no1. (ff aur6l rol-rarvLoerarl :rivcxo ro r {oog rov xrQdleov olEdv€L cvciloyc :rqoq ronr. ftlE rpo6).dn€rar rouro dtro q! naod.yQa(po 1.?) Xpwr[oro|ltor€,toxont! lLE €novallnrLx6q f€e66o!q ro! K€qa).aio! 7 -Jacobi, causs-Seidel xo! SOR zar ol,yraivars rlv .lnoo6sol rou oqilLpoxo,< e ri l[etd aro ,L eriovaL]tqJ€rE. IloLc .,.., efvnr :rrLpcpatr.zd r1 xo].urepr1 otlvSOR;

5.1

5.3

Oo {0e},c vc ouv6lcioo roq rrlaioq :ri.vnxeg xol 1o!,< ri[!crL€E 6raqopd]rF€o{D rtlg naQ(roraolE ,4 +c-F, 6:ror c rivcL aLcyrivr.oq [e ruloicl orort€ia. TorrrooLro: prvur re). t r , 'c vto nfo.o rar bfv lvupr;u ourr xcv rr : oomrE rporr lort : .

NAPA}THMA I' KQAIKE: YIIO OTI'TON TIA l'H fPAMMIKS TESIIA

KAAIKI'}: TIA THN A =,U KAT THN ANAAPOMIKH ANTIKATAITA'H

fra eurohi( rorr 1aioq Sivoup€ ),iota €wo)drv FORTRAN, y,t)Qi! riolJ6g ),€-:rroLo.7ieg. oLtir znd .I)rq yrtr vcr 6t<rozr6dr(et zcrv€ig n€tQCF.rrt(6F€voS tr€ roAx = 6. H tr€r(i{p(xrorl o€ BAstc i PUI €tvar €{"oll. H €iooboq r€pd.qu6irv€Lrlv rciEl N rou ni.vclr(l. zue6( r(! q 6L.i('Tcol NDIM rou 6rc1vioF(rrog. jrour ivur tvo ritvo q{-)rilFrr Tlu ro N. O unoloyro i: xurrrviprr rirpo ouFqula pt t4vNDTM. €vd ro N FJIoQ€i v(r !€r(6dl).€iaL ovdl.oycl U€ rc 6togoQa ouoti[crt(tr(,)v oroio! yivsr.rr I smE€Qrycoicr. O ri6brzas (lvllxof)rotd 6ifcl fiaoq 6ll[(rrov rivqno,4 Fs rov U rxo ioar rgLpvrxo FieoE xcrL !t. 10! 1 -L olo lvllomxirro) xQrlovrxo FdQoE rou. 'EroL (vri vcr cnoolr(€{o![F r(l |lb€vrzd ]rou €[9(l-vilovroL zsrd( xtv c:rci,orrprj, u:roOqzrriorfr€ ro4 no),1rl;r).cor(otfr (tt ttti),lovtrt ttpr,4rrxci tat' nd.lanleat arit,. i.6'yo-r rou 1 - Z). Eiv( cQr€rd onrtrlrllt6ro 6rr I ard.oLt'll brv lpedrirtcr e:ri nLror Xri:po l,(r roug Xr:tQrotrou!. Ecivril(rr anclooiqlr-q 6rd{op€s €vc},),(ydE yp(ppriv. rore or oiilE( ro]J 1- t 0culii,orcrlicl perc06oc{ rou lrovo o nro unoFoverLx6g uvcrlvdrmlE ec 10el* v(l€0[rr:a661€r. Aurd:rou dler olpaoicr rivr& vo €Q[lvctlthiry ouJotd cflo rov zti)-6rxo dro{ieotl|rriG crrlxdrdor(oli. (o /, reQl./ecrqercL ocv rlJUxololLr(cr" rdrr'}rpryDvLxog. IrIv rrpirl(')o{ jror ! 6lq(i!ro( rou ,4 €i'r'cL rr(Ql(i)< 6€or6Eouo(l,b€v ()cr yivrL obriltol ect ecr Loxutt ,4 = LU.) ^fo r.L€woio tr6Qo( rt( IPVTn€pr6lrr tl. crlidoy(l trr ro uv ro :ri,l0og tov e!clL(y(;v ri\'( oQrto I fiEQLrro.z(r 1 opilouou 6pioxet(r (lJr6 tov tuno IPVT(N)*A(1. l)*...*A(N. N). Edv curriri!(r [n6av. 6r(xonrorJll€ rov unoloyrotro.

Aivolltrr- rdA(l tov xrir6r,x(l roJv uroQolJrL!6! DECOMP ,!(! SOLVE, roufl(Q(1lovlorioroL'v to!.4 tiar 6Qioxonv ro 1. Aivor-rFtr €niorlg dr(l 6ei.1ycr zrlpiorl,IQoyQ.iBtr.rroE nolr bt(6dq€r bebogdvcr. xcLei flE lnoQorrilt< xaL 11Jn6!tr lrr

cC AEI|I{A ISPIOY IPOTPAI,IIIATOE fIA N E:I'O'EI' ME N ATNOETOY!

cINTECER IPVT (10)

n E A L A ( 1 0 , 1 0 ) , B ( 1 0 )NDIl,l= 10

ccc

NDII,I EINAI TO

N=3

A ( 1 , 1 ) = 2 . 0

A ( 2 , 1 ) - 4 . 0

A ( 3 , 1 ) = - 2 . o

ANO {PA|I4I TII' TA=II' N

IIAPA?THMA I KOAIKE> ]IIO OTDTAN fIA TlI NAMMIKII A |EBPA 99

A ( 1 , 2 ) = 1 . 0A ( 2 , 2 > = 1 . 0A ( 3 , 2 > = 2 . a

A ( 1 , 3 ) = 1 . 0A ( 2 , 3 ) = 0 . 0

A ( 3 , 3 ) - 1 . 0

I,{RITE (6,6) SDIM, N5 FoMAT (X, 'NDIM=', 12, ' ! {= ' , 12, ' - -- A: ' )

D0 10 I=1,10 mITE (6, 20) (A(I , J), J=1, N)20 F0tUlT (x, 10F7.3)

mrTE (6, 30)30 F0RI'1AT (X)

CALL DECOMP (NDII'l, N, A, IPVT, DET)

tBrTE (6, 35)35 FOR}IAT (X, ,I,IETA THN AIAE: NAPATONTEE THE A:,)

D0 40 I-1, N40 I IRITE (6, 20) (A(I , J), J-1, N)

I ;RITE (6,45) DET46 FoRl, lAT (X, 'oPIZoYEA =' , E11.5)

U R I T E ( 6 , 3 0 )

TF (DET, Eq. 0.0) SToP

B ( 1 ) - 1 . 0B ( 2 ) = - 2 . 0B ( 3 ) = 7 . 0D0 50 I=1, N

s0 mITE (6, 60) r , B(r)' a l , r , / \ _ , F7 1 l

CAIL SOLIE (IID]}I, N, A, IPW, B)D0 70 I=1, N

70 r i rRrTE (6, 80) r , B(r)

NAPAPTHMA f KOdIKE' YNOAO|ETON fIA TH IPAMMIKH AA|EBPA

80 F0RMAT (x, 'x ( ' , 12, ' ) - ' , F7.3)

STOP

END

SUBROUIINE DECO}{P (NDIII{, N, A, ]PVT, DET)

INTEGER NDIM, N, IPVT (N)

REAL A (NDII I { , N) , DET

REAL P, T

INTEGER NM1, 1, J , ] { , I (P1, }1

cC EIE0AoE A = nINA(AE TYNTEAE ToN IlPo! TPIfoNonoIHEllC EEOAOX A NEPIEXEI TON ANO TPI|ONI(O U XAI I'{IA METAOETHC TOY |NHTIOY (ATO TPI|ONIXOY I_Lc A=LU rE nEPInTotH M YnAPlllt ENAilAroN rPAlll,l0NCc IPVT (l() . AEI(TH' THE K rPAIt{l{HE 0AH|H!HE, EKT0' THEC IPVT (N) : (-1)**( IAH00E ENAAAAToN rPAl1l loN)C D E T = I P V T ( N ) * A ( 1 , 1 ) * , . . * A ( N , N )c

D E T = 1 . 0IPVT(N) -1

IF (N. EQ. 1) G0 T0 70NMl=N-1

cD0 60 (=1, Nul(P1-l(+1

cC BPEE TON OAH|O P

cU-K

D0 10 I=KP1, N1 0 T F ( A B S ( A ( r , X ) ) . c T . A B s ( A ( H , K ) ) ) U - I

IPVT (K)=il

IF (1.1. NE. K) IPVT(N)=-IPVT(N)

NAPAPTSMA T KOAI(E' Y[O,\ON'TON fIA IH IPAMMIKH A IEBPA

P-A(fi, |()A(!r ' X)=A(K' x)A(X, | ()=PDET.DET*PI F ( P , E Q . 0 . 0 ) c 0 T 0 6 0

cC YIIOITO|IEETOY'NOA'IANAAIIA'TE'c

20 D0 30 I .(Pl , I3 0 A ( I , X ) . _ A ( I , K ) / P

cC ENAAAATH KAI ANAAOIOH (ATA 'TIIAEXc

D0 50 J.KP1, IT.A(1.1, J)l ( , J ) - A ( X , J )A ( ( , J ) - TIF (T. EQ. 0,0) G0 T0 50D0 40 I .KP1, NA ( I , J ) . A ( I , J ) + a ( I , x ) * r

40 CONTINUE50 CONTI}IIIE

60 CONTIIIUEc

70 DET.DET*A(I, t{) *FLoAT(IPVT(ll) )c

RETURN

END

SI'BROI'TINE SOLVE (NDIM, }.I, A, IPVT, B)INTEGER IDII{, I{, IPVT(II)REAL A o{Dr}r, n), B(n)

cC NDIM EINAI TO ANO TPAT}O TH' TAEEOE N

552 TIAPAfTHMA f KAAIKE> YIIOAON>TON |IA'I9 IPAMMIKH AAfEBPA

C A IIUPIEXEI IIAPA|ONTEX IIOY YTOAOTI'E 11 DECOMP

C B EINAI TO AIANYII'I4 TH' AE3IAX NAEYPA!

C IPVT EINAI TO AIANY'I'IA OAII|I{XHE IIOY BPI{KE H DECOII,IP

C ITHN E:OAO O B IIEPIEXEI TH AYEH X

cINTEGE& Nt l1, K, KB, KP1, (M1, M, I

REAL S

cC AIAAOXIru ANAAOIOHc

rF (N. EQ. 1) C0 T0 30NM1-N-1D0 10 K=1, NM1KP1-X+1U=IPVT(]()

s-B(M)B(l ' t )=B(K)B(x) -sD0 10 I=(P1, N

1 0 B ( I ) = B ( I ) + a ( I , K ) * scC ANAAPOMIKH ANTIKATA'TAXH

cD0 20 f iB=1, NU1(}11=N-KB

I(=KM1r1

B(K)=B(K)/A(K' K)

s=-B(K)D0 20 1=1, I (Ml

2 0 B ( I ) = B ( I ) + { ( I , K ) * S

c3 0 B ( 1 ) = B ( 1 ) / A ( 1 , 1 )

cRETURN

EIi'D

I€witQro Tulaiov oQroldv

Ensr6{ ra n€lQd$orc rlou npor€ivafs rrQolloqr€\r(ug Gr(crrorv fltaioxqap!Opo]]E, trQooo€ioriFs l.i./o ),oyr( yrs rov rp6:ro anp(rYlrlitq totg. Apxeroi!nol.o./Iorr!g rot\ dxolv ri5r1 droqro{ npog 7prlo1 (rou}.rilotov oqv ;rrpinnoorlnotl 06).oqr oltodpoptptl xrrcroprl oro 6Kiorit1ra ar6 0 atr;E 1). :n]! nAorua-rr"r6qrn rrrrroi tl"rc"t" 4ttudorrptu (lpr0poi. rou npoxun.rouv ar6 6\10\r nl.{a(|Jsr"retepprrLorLxo (),1opropo, mlv npd!1 6truoc olpn€Arqcaovlar o(rv rrxoior aaro,poi. lia va z(trayQoq€t evcg Lxnvo:rorTr.rdq ul,lcipL0foq anbtq orA!6cl(l<unoloyrotr6v. aQxri tuo povcbrxrj lpqrp{:

l ] r0 =0. ,r = (25'733n1) j + 138a9)( od 65s36), :c i =tn1/65536.

O aA!0p6( 65536 civor ro 2rr' rc1! 6Lc ra 6rltlara yivovr(r| $€ apro[llIlrli a1tar6ei.cq(ro! 0a i:rQefir .,'a €ivclL rnn6pcrq or plcr pqlorl 32 bit). H dxqeaon 'mod 65536 '

olttcr|rsL orr ro ,l.i Ei.v(l! ro un6lorno [r:rd fl] brdiQeol rou (25733/,j + 13849)$' ourol'rov a0r0[6. 'Orq\r ro ,rr btalQ€iror $€ ro 2 6, r6re ro .r; eivcL ivtrrl.do$o [srcE! 0 xcrr ] -io 6uob!x6 olllt€i.o prloxr1,€iraL r(lrd 16 e€oeK.

YtrdQX€t eniolE raL ivaE orL6g rQ6no< yra vtr:rcipr cn6 evcr ler,yrigrrtrotov, o[oloFoe(pcl rqxaveplltl€v(ov. r]airDv crQ!0p6! (),.I. ? ,{ar y) o' ava\E"r\oQt ttavoruti y.tr.(rvep4rcvcov rqcri.orv cpr0[d)v 11 r.io! r, (nou b!€novr{l <rn6ilv )iopnul,! Oau$$. $ o/rl$a )i6bovoi). To no(irro 6r]pcr eivcrr va percrro:rioorpefltv otroroBoQqn xolo!o[4 oro drdo1lpa cii6 -1 d{lrg 1: 6d],re ul = 2r - I xoL:=2t l l . : r lowdX€rrro lpar iorr : ro r = r1r: + , : r . T6r€ x<r u xtrr r i laoo r ld!

NAPAPTIIMA T KOAIKF)] YNOAON:TAN fIT\ TH IPAIVJMIKH AATFRPA

/ - 2 l o c s \ r

dlorv ravovrxrt xnr<rroprl, pr prfie.vlz6 gdoo 6po xur povcdLcia 8rcxtpavoll.Mr(r ltall naAano[]r{ eiv(lr o 6€Ur6poq t6pog ritE orlQdE ior Donald KDUIh r/??Art of CDtltpLtct ptoya,ntins.

frc tv EnLLoyi FsruEu 0 rar I F€ rrL0cr,6 tr€S /J rior 1 p, ruroLoyiore qvopot6Fopgrc xaraveplpdvn pera6),Irt ir, 6fur: rportyorpdv(oE, xor rdax€ 0 6rd1,ro r €ivar lrlreor€Qo iou p.

BIBAIOIPAIDTA

A0qeqpi\1 r eapytx i Alf e Sga

F.R GdtDachcr. Theory ol Matrices". Chelser, 1959.P.R. Halrnos. Finitc'Dimensiotal veclor sprc€s'. vr Nosrrdd.Reinhold. 1958K. Hoffman and R. Kunze, Lincd Alscbrn . Prentice.Halt. l97lT. M!ir, Detemineis". Dover. 1960.

Eq aauootttrq r eawu i Alr EqOa

D. Arganbright, Mtrrhematical Applications of Electlonic Sprcadshccts'. Meccraw'Hill. 1958.R. B.llnan. lnlroduction to Matrix Analysis . Mccra*_Hill. 1960A. B.n-Israel and T.N.E. Grcville. GcncraliEd Invc6csrTh.ory and Applications wilcy.

1974.A. Bcman and R.J. Plcmmons, 'Nonne8ative Matices in the Marhemalical ScienceJ Academic

Press. 1979.V. Chv6tal, 'Linerr Programming'. Flecmsn, 1983D. calc. TIe Theorv of Linee Economic Models '. Maccrow-Hill, 1960D.G. Luenberser, IntroduclioD to Lincar and Nonlin.d ProsnnminS Addison'wesley. 1973.B. Noble dd J. Dmiel. Applied Linear Als.bn'. Prenticc'Hall, l9??.C. sreg, 'lDrrcduction to Applied Mathemltics". welledey.Cmbridgc PEss. 1986.

AAnpry ui f eappei Alytt0a

C, FoBylhe md c. Moler, 'Conpuler solution of Linear Algebraic sysremJ'. Prcntic€-Hall,1967.

A. Ceorge md J. Lill, "Conpurer Solution of Lege Spa e Positivc Dennire SysremJ .PEnticc-Hall. 1981.

G. Colub and C. Var L@. 'Matrix Computaions". Johrs HoPkins Press, 1983C.L. Laqson md R.J. Hmson. 'Solling Least squms ProblemC . Prenice.Hal, 1974B.N. Pdleit, "Tle SymmeEic Eisenvalue Problcm". Prcnlice-Hall, 1981.C.W. Slewait, "lntroduction 10 Madx Computations". Acad€rdic Pr.ss, 19?3R.s vega, "Marix Iterative Analysis". Prentic€'Hall, 1962.J.M. wilkinson, "Rounding Eron in Algebnic Processes '. Prenlicc.Hall. 1963.LM. wilkinson, "The Algebraic Eigenvalue Problem'. Oxfo.d University Press. 1965.J.M. Wilkinson dd C. Reinsch. Eds.. Handbook for Automalic Compuradon II, Lin.d

Albebra". Spnnger, 1971.D-M. Youn8, "Itcr.tive Solution of Ldg. Lin.ar Systens". Acadcmic Prcss, 1971.

AY>EI:EfIIAEfMENAN

A>KH>EAN

KEOAAAIO 1

1,2,2 L = b\ - b7, t = b2 - b3, tu = b3.

1.2.4 OXr 3 €l ,gsieg 6rd r ls aexr is 3,0,1. L2,6 (-2,1,3).

1.2.8 1xEE.(1) 1 x eE.(2) + 1 x €8.(3) = 1 + 0 = 1 -1 (3, -1,0) €ildr ttfa ano rrE trrrssq.

1,2,1I yt - y2 =1,,y\ - w =2. 1.2.U (1,1,1).

( u + ! + u = 21 . 3 . 2 { 2 u - 2 d = - 2 , u = 3 , r - - 2 , u - 1 .

I z .= z1.3.4 O ouyi€I€otls +1 6d !€rara€n€ io orrootpa oe r6ropoqpo.

1.3.6 H a =0 dndrrei €vd.rdy1i'yeoBp6r, alld 6€', €i\,dr l8ropoApl. H a =2 €i\,dr.Lb,6Foaq (ivds ob4yos, dff,aia lioEo.',). H d = -2 €i\'d| rSdpoaar (d!aE o6qy6s,'roppid ).iod.

1.3.E n3l3 = 72 €?aropprjera reciE.Ls, 9000 6ellee'ii\€fta oro Pc, 900 6EuEA6).€ftd oroVAX ,rsr 6 6E!r.q6l,€nra s' 6'a CRAY.

1.3.10 O 6eifteoE 6e% b. + dd Eild (a + 6X. + d) ac bd (!ovo 6!dq rra6o0eroqno),).oJrldo(dop6q).

L3.12 O,gu + O,h = 200.000-000 0, 1! + 0, 8, = 30.000.000-

T2ED Eru Ef MENAN A:KH>EAN

f7 l f2 l [3 ]1 .4 .2 l8 i , l 5 l . l 5

Lo l L8 l L7 l

1.4.20 \AB)i:) = ir, (A

t .+4 , , "xn nrp ro{^a '^ao,douo, . , . . . . , - [ i i l , - f l - i ]

L4 5 l L I r l1.4.9 To ra6ro re6yedll|ro 6qidsr rq o 10'1600r ror ,4r E€roarmd. taloqronordr'lo(

rls yqdt4riE ror ,,1 pio neo! [io. To 6€ir0ao 6aiok.L ro ,4x oo' owbrdou6 ro!oql61 ror ,4. rno).oltloficE mqh nqos oril1.

r .4.r4 4- l i ; l 8-L; ; l c- l i ; l . r - i r . r - r -L i _ i l .1.4. rt e!'ro ro r, ro t. sivar r nq6q miU ro! B z(lr ro (/a)i Eil(lr I nadrq otlil.I

,.,., ll ll [i 1]. _ - | r 0 t I r 01 | 1 0 l' "o l r : r l lqs r l l -e r l1.s,8 (o) Ar6q ot mry$ri '!o! xers$onoriror rua o6rydE yaa$t, our! oxlp.l Eetor c1fl6

ro' u xor 6xr (no ro' ,4. (6) KolovoE (iii).

l.s.l0 :rrp r. = b, o 4v(ooros cr 6Qioxeror !€rd sno pia rqiiEl, o,? crrctTri 6no radEfLg,t.o.x. :!vo).r,td6 rqciEors: I + 2 +. + n = n (n + 1) 12 - n! /2.

1.s.12 AraloLqi ro! causs ro! E€rwd an6 qv ie),Euioia yes!$ri (! ev6e1op6vr4 r1vradflt on17.d.'ox'.

f r r ru l f0 r r l l r 0o l i r o o l f 'o 1 lr . s . rs r0 0 r0 r l - 0 ro l l0 r 0 01 1 l

L0 0 LL2 141 L2 l r l L0 0 - r lL0 0 l l| 00J [ r 2 r l l 00 l l r o oJ l l 2 l J0 0 r 2 4 2 l - | r 0 l l 0 l { r 0 r0J .

l r r I r i l r I 12 0 l jLo o o lLo 0 0 l

f r 0 0J1.5.17 I I 0

L 2 0 1 l1.s.19 Ha =4o6ry€ioe E a ad yedBrrbv r36+10a =40o6ry€ioe tbrcpoaQo nivoa.r

r. =0 oblysi. ou da).}.a-yi yeappdJ! I c =3 odly€i oe rolopoaqo d.vdxd.

1.6.1 ,,, ' = [! i] , /t, =

{ o l , 1 co "o "no1i + l ' ^ . = [ s i "e .o 'o ]

BC)it = 2ft2.

ti il]t: i i] ti iilli iil

, f 1 0 0 l1 ,6 .6 ,1 r ' ,= | r 1 r l

| 0 0 1 l

t . ' i i t 0 tA t = l+ 1 + ,A i = 1 l

l r 1 r 1 0L r l l l

il

AYIEI,! EN1AETMENAN A>K}DEON

16.7

1,6.9

lil

l,/ttt t1z I | rt12 ttz 1I ttz rt/2) | tlz rttzlYao0lore 6t q rsiq yeqF i ro! A-r €i4d +8d =I + b8undar(€r lior.

Iu o o + l r l o.4,,=lo 9 i ; l ^_ =li I

l0 + 0 0 l - l i i

L r 0 0 0J L j iI d _b 0 0 l

I l _ . a o o l^ ' = "a t , l n o d t , lL 0 o . a )

I0 l Iu 0 i '

vor (d,6. c . d) , A- i ,4 = 1 + 20 = 0, a+3, = 0,

1.6.10

r.6.12 (r). (2), (s).

t . 6 . t 4 t A A T f - l t l l t A ! - A A t . t A r A ) T = , q r l , q T f = , , t T , , t . r , i p t , = l l : ]

A/. = [ i ,o] , / , , = [ i , i ]r.6.16 (a) a(r + l)/2. (6) (n lhl2.

l.6.lt {() O cwimqogo( evo< zdr0, (drot raryr)vrxou nil(lllo €ivqL nci},r 6vaE ridr0, (dvo)rQtyo,lrlios ftivorcE. nol,Icurloordorr6< bno x(irr (avot eL.yovLzdv mrdxov 6i6ctrdl xdro (dvo) rQryrrvrao rdvaro. (6) Or

"nerq 6roytiruor rorv LltL2o2 xttr

o ln l ta I , t i rcr or i6rq p o! rd( rov t r iar , r dwi" . t rotX. . . L l t t ,zD = Dt I t l I . t .dea Dl = ,'. >u./.rQi.\,ovtfiE ta pxros b(yrrviol GotXEio rN Lt I L2D2 = DtIJttJ;loup?i0Aaivo4rr 6rr rar or 6uo € ivcr nny( i )vror r r i rcx€E. Lt ' t ,1D1 = D2, Dtu lu; t =tl, o rr ri!(r (tlrLfiedq,rtog + LlLr: = I. LtQ, | =I =, Lt = L2,Ut =t/2.

0 l f , 0_ I f l t ' / u lr l l 0 d - (b r l 4 ) l l 0 1 l

1.7.4 (,r. ur. rr) =("r/8.0.-,r/8) (lvri ft,v nac.$allxd;! rqrdv (1.0. t).

XEOAAAIO 2

2.r,2 ((). (6). (s). (E).

2.1.4 O x6eos 6)oa ro)v trrvdzov 3 cnt 3, 6r6tr xd0E riaoaas pnoaei !o yqav€i oovdoaotofc €v6s orrF€rqLxoi rar Flds l{ciro rerlorrxoi. O x6qoq 6kiv rov tlra.r0)vi(D!

f r 0 0J f ] 0 r rJ f l 3 s l r If . 6 . 2 1 I l o 0 . 1 0 l l u I l l

Li r r l lo o 2 l Ln o r l t "

I L r ro o i t , , t [ j lr .? .2 | rn r3 - ro l l , : l= l l l .

L 0 to : : l L , , l L j l

Y'El' ENhEIMENON A:KI{:EON

LL6 . +2! + . =0 o Po €irar !n6i(('eo! ro! R3, €!6 o P 6€v €ivoL.

2.2,1 Eald nou, oqrir€ALlorr6avodvou rou nivsxq !r ld€vrxd ororl€id.

I 0 0 l f l 2 0 r l2.2.J J0 I 0l l0 I I 0lo, t r xd( u s ivar 6uor"r< Frro6l$dc or u zqr y s).e i0sqss.

Lr o r j Lo o o o ll2 - - s1 [ ' ] [ -11

, | _ u | _ l l 0 t -n y r ! , x r ^ !on . , va , r _ | . l= "1 r l - r l 61 "=2 .L v I L o l L r l

r o I r l ) 1L2,4 t -

10 o o Ol oL, . " . , , i * , 6aor i ic !€ru6i '4rdszdr 4 d r l .uo?et Hyruxi

I lo i t rou / . = 0 € i lar . = ( r . -4t r ,o,y) . To Ar = b. ivsr ow6r6doro €d1/.2 - 2br =0 H y€lt,ari llorlror,4' =b €(vot

| , I f r l | -o l fo l'=lu';o' | =.131.,1 il.,131L u I Lol L o l Lr l trl

l - 2 r : - 2 r J l | 2 l | ' l

2.2.10 (o)r= | ',: =., I ll-"" l_9 1 ",",,, ,. .,.,.drnor€ a*suo:.1-;l l L;J L;l|-"-3ul

' -?' -11' ' ' '-l

I I "L i]--l ?] *"''^"r'('bin"'Fdq''''

f , l f -2" - 3 l f -2 ' l f -3 ' l"n Ltl=L ; l='L ;l.L9l

z.m ['o ? _r] [;] =[]

2,3,2 (a) avrldaotrd (6) eEaeqpiva (y) eEaa tFdva.2.1.4 N..r .r(0r +!2)+.r(rr +r j3)+.r0Jt+13) =0 r ( . r +.2)!r +(ar +.1hr +(a +.r)rr =

0 + c r + c t = 0 , . l + f i = 0 , . ' + . i = 0 + . t = . 2 = . r = 0 + u l . u l l u r € i ! a !

23,6 (o) O dEovas , rox Rr. (6) To €in.6o v: ro! Fjr. (y) To €ninE6o v. rox R3. (6)o R 3

23.7 (d) H.,lI = d 6Ea l.iryftaL. Asd ro, tuv n€ArsxErcL orov xr6xoqo. (6) To L.(!F tiTopig ro t'r) noqdlow rov Rr.

I n - - ^ I t f 0 0 t f 0 0 t ^2.r.r0 \4'd 6{icnr onoftroiv o, l; ; l l ; 0l l; ; l l ; i l o' r l |Fdz-ro'flva)!€6 noadyoJv rot( .i!u rp,l,uu"orrg nrla"Fs

YtEIt EnlAEllrlENoN A>xHtEON

23J:2 Eortr) ur = (1,0,0,0),...,!4 = (0,0,0,1) ro 6ldvuoloro oL'\r€ld rivov. Edv II'€ivar rt eloEid 6Ld rou (1,2, 3,4). l.d!6\,s ord rd | 6Ev n€qd:(€rar oro.', /.

2.3.15 (i) Edr o1Jrri 6E! irav 6don, i6t€ 0q lroeonocF€ !o nQoo0dso1jp€ €flr).do! dv€-Edaqra 6rou,oFard, aor 0a urea6ahop€ rr 6o6€Loo 6Ldmaq t. (ii) Edv 6€v lio!6don 0a proaorooFE v' aqaedootjtr€ [€Ar,.ri 6rovio[ora, oqrlvo us ].ry6r€Aa dfldrI 6o0sfou 6rdorool l.

2.3.17 Ed! rs ul,r2.!3 cnot€l.oriv 6don to! r/ .lar rd ul!ul.r3 6{iol ro! I,t', i6re ro6Er 6Lqwotqra 6€v FoA€i va €ivor ov.Ede/qra

"d| xinoLos ow61'sotit rolls €q

nq€n€r vq r't6€vtlsroL: t.iri + t diu, =0ri Dciri = -tdii!'i €ivor 6rdluqraxo(vo rov 6rio urolda(Dxf l o r r n r r

,rr0 l ; i l adt l i ; l 23.22 (r) A), i lo is {2) ryFubd;.

2A.2 RIA) t . = 1. (1,2) ,^, f ( ,4) | n - / = 3. (1,0,0,0), (0, -4,1,0), (0,0,0,1) r i (Ar) :r = 1 , ( 0 , 1 , 4 , 0 ) N ( A r ) , - - , = 1 , ( - 2 , 1 ) R ( u ) r 0 , 0 ) , \ , r ( L ' ) : ( 1 , 0 , 0 , 0 ) .(0, -4, 1, 0). (0, 0, 0, 1) 7e(1,") : (0, 1, 4, 0) ff(.r") : (0, 1).

2 . 4 , 3 A B = o + A ( b r , . . . , b n ) = 0 + r t b l = 0 , . . . , , { b i = 0 + b r € . 1 / ( , 4 ) , . . . , 6 n €,\r(,,r).+ ,?{a) rspraxErar oror nr(A).

2.46 }1. = 6 €m).t'ortro <+ b e 7?(.4) (+ R(1) = R(rl') <+rdqlA=rdh ,l'.

2.+7 \a)n=n=, (6)n >n=r . ' .4 . r0 t l '4 t f l i :1 .L 3 6 t 2 l

2.4.1:2 Ar = 0 ryer pI [r]6euxri l!ol+ r < tr + 7?(r1r) €i'dr pr.,taorsao and 10!R'r + ,,1r y = / 6E\' ).!!€tdr y(d a.iroro /.

I i - i l2.4.1s o A.r€r dlav 60Eto ovrioraoeo

I i i I. o,rr lxet ivov dEG€a6 svrimeoeo

t i l - l t - . .^ - ' . i

lJ . r r ,o -a,o: r|_j i I I Edvo /0. o r ixf l orqinr.rueo dw,oraoqo

l i i ; , ; )L _ ] 1 J J

2,4,1? O A? A 6ev €ivor rar' aldyxrv oE!@ed\trFos.

f1 r l2.{.20 (s) | 0 0 | (6) a€v urdqx€r rdroroq flivaxoc: r+ (n r) = 1+ 1 /3. (y) Aev uldaxer

L l 2 lrtro@s rivsxosr o 1drAos oq).6v ,aqr o Xdeog yQ(l!!d! €rolv qr i6ra 6rdor(rol.

2.51 :€ xdo€ odU. ro doporouo r<ov 6!o nptirn.rv ororrEio! Loovrar re ro raito. Aadairos o!v61JostG 0d rrarorot€i br +h - D3 - 0. / r = b + xr,? = dl . r i - 11 = bl ,rr -11 = 61 + br +h 6r = 0. AUro o|tpailer oIl ro 'ioaoLola rov 6'aeoAi,!6waFL"on xord pi"o< €!G zi"riou €iror rr$av.

2.s.3 To doQoLoto rov morx€iov zd€€.rqdpp'isrilqr pll6i!. Aqa zd€e @v6uoolG 0al'/$ tftv ibra r6r6qra i + Jz + h = 0. A! y = f .) lt + !3 = h, u + uz = k,

AY'ED EII,{E|MENON A:KH'EON

lz lB = 11+ h + h+ lr =0. Aut6 oruaiv€t rh rc orL# eEr+c, rc €torqr6p€voffi *Eo ei'aL rll6dv.

lcr +c] .r .3 I2J.s I cr .r+., -.:

IL -cr -a .t + crlczo) / ( . r +q) .

f - l I 0 0 ll - r 0 I 0 l

z.s.e ,= | I : I 9l| . ' ' ' " ' |I - r 0 0 1 lL 0 0 - 1 l t

I 0 0 0 - l I 0o + o o lo lo o + o o l oo o 0 l o o - l| - r 0 0 0 0 0I 0 t 0 0 0 00 I 0 - l 0 0 0

2 5 . 7 b r + 6 j - r s = 0 . h - b r + D 6 = 0 , r r - 1 5 + i 6 = 0 .

2.5.10 q(r. M.rd qv aaoiqtol rqs Istr€ursiqE ya.44ttig. ot tqsq no! coFdvo1jv €LvaL(!€Edaqrc{ zor cnorehnv 6tior rou rriroot teo$drv.

25.11

2.5.12 6 1.6.6.

2.5.16 (d)-sr-ua-11 =0,y1- i ! ! =0, lh+!u- ls =0 (6) I ]aoo0alo! ' idg r ts3 cEro{ i rd3 '6.(y) 3 (6) Avrtotorxo!! stouE 6io avEEdq trouq xnxlorS er srer 'ta! $t/5i4.

( r . 0 . 0 . l . - 1 , 0 ) , ( n . 0 . l . - 1 . 0 . 1 ) . ( 0 . 1 . 0 . 0 . . L I ) .

2.6.4 El],erlT. 2,6,1O et ,c-t 2 . 6 . 1 1 : r ' + j . . l i i l .

Ynoe6m€ 6rL r ='1-r' t.rt o = .4-'r. Enerdi o A €iva! teorl}lLrG. r1(.! + du) =. A u + d A t = . ! + d ] 1 . A 0 o . 4 - r ( a r + d y ) = c u + t h x a t o , 4 t € i v o ! l e q 4 u x o s .

Eft€L6i or rfto.rx|}|(litoloi rftalororoiv rtv A r,1 = /, o rcv6vds toD yrvoldvo!(Dfndyrr( or o' dEiorori(or riwx€c 0a rruvoiroro['v ntv ll 1/ = l.

:l; | . b'rli dlaotaoq{ ro'r nivu"u bi6r tov dex,x6 rivuaa

i l

0000

f -1 I l - . i l"= l- i i | ,= | , ' ] , I .

t - f | | T ILoJ l+ l

[ 1 0 0

,.6.16 l9 9 I[ 0 o 0

!(r),q(r) € s + .[,itpt'l * aq(')),r* = "f nt"la" * afi ct"la. = 0 + .p(r) +

dq(r)€s+sri ldL!n, i ,poE - i + , , - { * "r ,

- J + "3

eivor 6cio4 tor s.

Y:EI: ENlA!TNJENON A:KH:EON

KEOAAAIO 3

3.1.2 (1, r) aor O,2) (1.1) raL (0,0).

3.1.3 lx:- lxr)Qzl =-I +xLyt + rz:r , , : =0+z?y=0.

3.r .6 ol .c ra ro). l .onrdo'a r .ra (1,1, 2I \1/ , /4.1/h,11\/3), O/\4.-11,,4,q,Ol,/G.1,/yG. -21\b.

3 , 1 . 8 r € r / , r € l l : + . r ' l r = 0 - , r = 0 . 3 . 1 . 1 0 r r + r r ' 3 = 0 .

3.l.ll AT ! = 0 = u't b = y",,t" = 1yTl), =O, "ou "v.rrpdoner

nqo*- trlr yl r lO.3 . 1 . 1 4 ( i r ) ? l r + r ) = 0 < > . t I , + , 1 . t - , t r

" - t t T y = ( J . + , 1 t = y T 9 a ) l x = y ) .

3.1.18 Rr. To oq0oldrvro ol4rtr].rtErfo napd.l,uidr oro rd (1,0) 0,0), (0,1,0,0), (0,0,1,0).

3,1,20 >rl|ldiltt o xdr0€ 6rdrr0B0 op0o1<i o rqo< 61c ra bL(lvrlspara ror eildL oqooyd-ltd rou s, nfel,Xdror oro S.

3.1.22 Mta t idor l r ia( ro (1,1,1,1).

3 . 2 . 1 ( t ( i + , ) / 2 > \ i i l T . ( 6 ) r + , , 1 : S ( . l l + l s l ) : + ( r + { ) " G + y ) < i l , +2 1 1 . . e l + l l i / r + r 7 y (

l . r 3 1 .

3.2.3 (r0/3,10/3.10/3) (5/e. r0/e.ro/e). 3.2.s ilcco(r/v6) fl l , 'r, ,r,,Lr l

3 . 2 . 7 r = ( 1 . . . . r ) . 1 = . . = a , . ! . 2 . e p t - ! ! : ! ! : - o l o l " ) o r - " " r -= ;7;7;

= "'\",'r,

= -,7 = r'- l - - l t .1 '

3.2.11 {dr i0 .r _ . ' r , " ' ]o , ] l i i . r . " , reol+n

, , . 'p"e"t- 'Ltr t i l I L- ] i l l i l l

rdlor m 6uo a.ieerrq F$Fi09 6i6!r ro saxlr6 btdvlopd. H reo6ol"ri or pra E!0cio'tat rarofl! o€ rrLov d .r (xd0nr reoq qv nad)Tlt) 6ib0r n{iator io {0}.

| ! I l l323 ! ! )_y . . . * " : ! t=1 . 3 .2 .14 | ! i _ i

t * -1 *lr.2.16 (d) pr = p.+ (pr)r ! / =..rpre =,"?ips). (6) oxt 1,1\4a,t /3\/a. (t)

p2 = t' .+ lp'\T pc = al pT p, = .r ptu = r'1 e1).0" ri tso. (ftd'o oqv ibra

3.3.r 2 (10 3.) ,+(5 ,r ' )2.(4, 3X3,4)7=0.

f l r I i rr.r.r '=l i ,=l i , ,=l j ]

L j r L - : t r

3.3.s 6+15/2)I (7/2,6,17/2). 3.3.6 t(iraor mtl"6v s, ri.qr a.

561

A]>ET' EII] ETMENON A:(H:EQN

! . 3 . 1 0 A T A = I , A T b = o o .

f i i o .11t t I ^ l

33.u ro) ( r . r .0 .0) . ( - r .0 .0 . r ) (6) | | | i ; r r r ro .o .o.ur .Li i ; , l

3.3.13 61/3s - (36/35X (r33/3s,9s/35,6r/3s, 1r/35).

3.3.15 I l 'z =(1 2p)z = r -4p +4p2 = I -4p +4p = L

3,3,18 (1) ar +, + ,' = 3, C + 3 E = 6, C + 2D + E = 5, C = 0 (A€v ljndql€r +ioro €rinE6o).

f4 r s l lc l f r4 l(2)13 5 3 l l r l= l1r l .L 5 J I I ] L E J 1 2 6 ]

t.t.t:, A1 (AA't )-| A. 3.3.21 | "1" , - "1" ,1

; . ,| - o i ' ) a i a z l t t 2

x = ( 2 , 1 ) .| .lblt-"; t, )

121| 9lt - r lL -51

C +2D (6)

t r - r r t r r - r3.3.2{-3 10-(12sx 3.3.rsl l I l l l i

L I L 1 J

2 + t ' ' 0 ( y ) T o 6( n ) 4 = C - 2 D , - 3 = ( - D . - 1 = ( + r . u =Jt€QraXFr( oro\, r(dreo ml),rbr. To i6ro ro D.

3.4,7

(-2/3,t13, -2/3\.'to o1aorop(l €Lvdr ro Lbro ro b. :ryevn xe 6t', a1u'1 , a2o! , a3al!'Flvor or reo6ol.4 ndlu o€ rqeq oq0oy,luEs 6rFueivoFrs. To dgqor;un roiJs rivdrneo6o),i s o).or].nao to xdroo xcr roourar tlt qv ravrorrxi ansrrovror

| + -; +1t - zuu't"ll 1t-2,,r1=, - 0,,' * 0,,',"' =r'O=l-i + +

riii( r rs r+ . .+ ' "q" )? ( . rq r+ .+ , , , ,1 , , ) = '?+ + ,? , + i lD l , = r? b = '? + .

- , 6 <_- ^ 2 , /21oqt_ntz. 3.4. t t l ; " i _ , " i ) lY ,ur l

it-il-'l

f0 0 1J f0 0 r l l r r r l0 r 1 =10 10 01 r l .

L r r r l L ] 0 0 l L0 0 1 l

t1 1 t f r l r 0 . l . " - -l2 3 l= l2B r / tn l i , . 'nL2 | j L2/3 r / r t ) ' " " '

3.4,n nr = eTb.+ i = R 1e"6 = (s/9,0).

3.d18 o 8 dr(€, iov i66 r6eo oq).6v p€ ro!,4, dal P =Q(QTQ, 1Qr =QQr

\:EE ENhETMENAN A>KH:EAN 563

3.4.2n L, \n,z - 1V2 !.4.22

3.4.24 H lJ sivar i6t oagoyd,!'a r0os irs {ialles o1jvoadosE I xo, "'?- {. rla vc

agoLAdooqr€ rt qxvrm6oa TlE ,aord q 6€u8ulo4 r1E ftqLFis oxvdpqsrE e,uoroyiEowr (rr,r) = J] rrdr = i *" t","1 = 1., "'a"

-:. r" €'6p€,onoludvr,$o rou lrsc'drc €ivaL J - ((i,.)/(r,r))r =.r - ir.

tLn ( / \/1, -1tvz.o,r),lr t,/B. | /\/e,2/\tG. ot,l-t /2\/1, -r /2Jj,1 /2J1. -1 /rt1.f4 0 0 0 l f 16 0 o o l

r . r l 0 0 0 4 l l 0 16 0 o l- - ' l 0 0 4 0 l 0 n 16 0 lL0 4 0 0 l L0 0 01o l

!.5.2 r, -r/2+ e5/4i,-r/2 - e4/2)L3.5.5 r = (21+ 1)', d = 2ttr + a/2,1fi.!(l| adsrrcs.

3.5.7 (1.0, l ,0). 3.5J (0,1,0,1).

3.s.r0 yx = y '6+ y!, j . y =vto- l t lh = tr i+ tr i i ,u1 - ! \+i l i t . ! ! -u3-ui ; , t t1=l i - i t i ' .

l r l r r l f r l f r lt o t t t I t o t t n l35.i l l r l_l ; l_l ; l - l r lLol Lol Lol Lol

3.s.13 .n = (.10 + .fl + h + lr)/4, q = Oa - ilt - !2 + ih)/4. cz = Uo - h + 12 - h)/a.a t = a ) + i h J ! i h ' ) / 4 l a = 0 , ! 2 = 0 . 1 r = , f r = r . 0 = 0 . c , = 0 , ( 1 = - f r + . .er!ft erilorts rrEarirrl.

3.6.1 (d 7 (6) 2 (r) 8 (b) 13.

f. ...,3.6.3 Ov+l i nror iy.r rol 'c 4 rni 4,r ivarrc

l u 4 r : n ' l[ 0 o a j r

f , " r : o o1uvrrr r r rpLq( f l ro | ! r rvdxr ! | ^ ; -

- L

t u u d r r a r r IL0 0 o a { l

dim(r + Il:) + dim(v n w) = 20 = diny +di'nw.

" l . a i . rv * r l r = r : ., - ' ]

d i m ( Y n I l " ) = 7 ,

3.6.6 H Huoeic 6rd rori (1, 0, 0) (6e! €i\,ar l{ar'ardyx't! kiont mor y).

3 , 6 , 7 x = 1 1 + t u t € V . u t , u 2 e W , + r r ! 2 = u 2 t t e V n W ={0i + !r = !r, r'r = u,

4 6 . e , 4 = [ r ; J . 8 - [ ; ; 1 . o f ' l E . l o x o q o s r o ' J n B - [ ; ; ] * " * E f x " - [ i ]

O Xdreoq oql,6v rox 4A 6€v r€Or*€adt oro\r X{i{o oq}.6v ro! A.

AY:EE EM^ETMENAN A'KH:EAN

3.6.rr rdEr(,48) < rdEr(A),rdEr(,{) < r, n < m + niEl(, B) < n +,4, dvai 6tdfoa-

3.6.t rdh(A + B) = dimllL(A) +R(B)) S din7?lA) + dimTt(B) = rdEl(a) + dEn(B).

3.6.r-] .^/(a) - N(A I AB) ) N(AB) 2 N(Bt =+ NIAB, = N(B). o rd,ao( yqoFr(i)veivcr ro oq0oyd,vm ownLrtg0lt(r ,at I rdEI r:ivot I 6tdmaorl tor. {'iqd awd Filstrc i6n ytd tous 'B tcr Aa.

f l 0 t - _ ^ . f l t l 0 lr .6.rs , r= 10 r lL; - l

Yl=l0l t r r01+lr l t0r - r i .L I I , I ' - " L I J L I I

3.6.16 dim(U n tt) = dim(tr) + dim(v) - dim(U U v) + dim(t/ n v) 2 6 + 6 - 8 = 4 > 0'dim(U n r n lr) = dim(a/ n /) + dirn(l!/) - dim((L/ n v) U tl') + dim(U n v n ll') >4 + 6 - 8 = 2 > 0 .

3.( l t I € v n t l a+, = rr fr + . . + rr . ! ( . 1!c.L €nloff i I = -r t+rt , r - - rr+rur(*)<+rr f r +. . + i r+rq = 0.c} r = (rr , . . . , r**r) €, l / (D). A6Y(') l rE lovcblaorr l rqs ruvrueuorrio€ov (*). ro v roadr€r av.l xdr Folo r oto! "A/(r).

J.6.2u r,, = -r3j--I-;1-. J.6.21 rl 5 r eut)?ia ( rof (t. -4r

rri + + u'n,

3.6-22 i.. = (+, +t ,{i,,, = (+, +, i). b - /i!, = (-+, +, -i ). (.Arri. )ri"r'$'(b -.4 i , , . ) =0.

3.6 .23 ( . tE{ f ) = +( -2 )+ j t - r ) * j t s ) =0 , r1 . '? ) = j t -z l ' * l t - l l ' n lo ' l =*

(6 ) Dr =Vr i , r !=2 .

3.6,a A = B + c, 6nou o a n€qrdx[ uurd( rK p yqo+{rds t (t lrD€vtrd nowori duon.c = ,4 - B. rdEr(/) < rdEr(a) + tdE (c) S p + .? s eri s.

KEOAAAIO 4

4.2.r 2"de(nl (-1)"dcr(,lI (de(,l)):. 4.23 20 5.

a.2.6 (d) 0 (6) 16 (y) 16 (6) l /16G)16.

4.2.7 nol.?.er).aordorr q p1]6*Lxri ypoltprt eri 2. Au6 ro)'1(l{I(lotdltt qv der,l !e 2€!0i rffioxeova o 7l raAdFivet dvd\t'oiuos + del,4 = 0.

a2.E (1 - ,nixad - 6.).

l a o 2 l 1 o4 . 2 . 1 0 | b b 2 = u h - a b 2 - a z = 0 b - a b 7 u z =o-.)k . ) ( " - r ) .

0 . - " " : o : l 0 0 ( . - . ) ( . , )

a2.u (a) ̂d€os.d€, [1 i] +z ", [f l] tar "ao.s a"' [f l] = :. - "r'r"r -,

€ivar l. I aLl{i yivftqL wa}.}.au YqE!+t{i'v. (1) A}'leis de(,4,1t) = de(A) de(-B) = 0.

Y'FI: EIII ETMENAN A>KH'EON

4.2.8 nqoo0droros 6Leq 1|r onil€q rc!,4 oqv nA(irrl nao)r1]rixs! rrto Fr5€vL*i oni).t, dqudct ,'l = 0- Edr to i'eaorolo moqriov xdo€ yQalti< roor,'iQ+€ 1. tor€ xd0e tQ({+i

rou A - / 6x.r d0aoroF( cotr€iow 0 + dcr(A - /) = 0 A = | i i | . dd(,1 - /) = 0.

o p o { d e l n = 0 l l . L : l J

4.2.14 0 (l - r?)r. 4.2.15 dc(,4) = 10. de(i-r) = r0!, ) = 5. ) =2.

4,2,17 naievowq n< oaiiolo€a iloqrE (dcraXdct/r) =( l)'r(derr)(derc').

431 dr! , Idr jdrr '€ i lar der&, de(,a) = l .

f 1 o o l4.3.2 t re( , , r , - ( r r r :d",

l i 0 i l=,-r ,a" l l , l l - , ' r r r r ra"4r1 =r.L 0 1 0 l

4.3.3 (l) Aholq (2) rv€!6d5 (3) \,uud!<.

4.3.6 (6) 6 Dr )0 = D6xroo,r = Dr = -1.

4.3.E >ror rnro (6) stvar 6t6oro dir ro nr,, -dr,, rilor !Ib6!. yra 6Ia ta blvorA(o, . .. r). H. aard ritv 3,6:4, rdE t,4 S 2 + 2 = a.

4.3.9 Mr trrr(ror olrotoir yAa!!!ii! oror< riluffc ,l Ftri 4 FFldro6noulr€ ro\ n x.lr Do. dvo reqovwoi<:

. I . 4 B lde' l ; ; l

-4 r ' : rd d:: =der(.r)de(D).

{ 3 1 0 d c t 7 t ' 1 = - 3 . d e t , 4 1 = 2 d e r i r = - l d e t A , = ( - l ) " i ( r - l ) -

, t . 3 . 1 1 ( ( ) ( r - l ) , r r ( 6 ) ( 1 + 1 / 2 ! + + r / t , - t ) l ) ' l ( y ) 0 r r + 2 r - 3 ) / 3 .0 . l l f / 0 t f 4 B A t f / 0 1 | 0 , 1 1

t l [u t ] - l0 r l " . ' le 1 l= ,=d. r l_8 / l -

" , 1 ; ; l -dc( .44f t r .x . r1 [ r 21. ,e- t i ] d . , l_ ; ' i

I - . -a" , . ra '

. . r = l i ] . r = t r r l d " , l ; ; l - 0= r re i . 8 ' . E rHb , i o . r a , i v . . r n r v , , x r l < r r rrdl(.{a) < rdqt(,.1) <, <,,.

4.4.2 To otorXeio (,.j) rou rq rol tivcr o orluue{irov 4j, €iv(r p$G! 6ra! i > j.de( o n eil(r dlo rAryorLxdq.

, f 3 2 r l | 2 - t n l4.4 . . r . r - r = l l2 a z l .s r - l I z - t l

. L t 2 J l L o _ t t l4.4.4 O ,1 o!llIerqLrcs + o €Idooor! (i,J) rivor o avootQo.lo; ro! El.ioooros 0,i)

:+ Ai j = l j i .

4.4.6 (u) der rr,f = 't (6) A€iro rI qrttr j ro! ,41.I nolr €ildr .,1r = r. 'o).es oL dLl€s mitrsEtou rlrl/ rildr oL i6L€q r' cr'(iE ror ,4, dea ,4r'11 = Bj. (y) det,4 det,l/ = deraj +r j =dctBj lderA

AY:ED EIIIAE|MENON A:KH:EON

4.4.8 ( o r ro R,bcdov ro ( roaqurl),oyeri,rlof fi!d, *, l-I

j l. ,o' r. rit,* o'a'c'

drer ro ibLo €F6o6o!. ArId iXFL fFto$eao€i m'l! uq1il.

4 .4.10 -4( '= CA.+ A(A r =( '+ der .4dc( , C,1. ra) = d"( .4(D - c .4 'a) =dc(..ID - Cn).

4.4.11 tr) :ni),es rou .,18 = ll 8ni @iirs ror! B. (6) ila €iaar ro deaoLdta ro! aql6v HnirK reatqds (trivdkrs rdElq 6vo).

4.4.15 MLd rsrdo€or ro! rol).ar].ootdlFtar !f n€aLml lsrdo.q a).l.dt!fl Looqti(." Aa.l

I /, = a? €ivar. dc((l. Edrv n d ' tirav daiLa. I od ' ea 't(! r.otFi. OI0E tloo = r sLvdLC ! t r ( ,

4.4.16 OL b dtrerq ror /' Fivor d).or ni!.1xrs tero06osoJs, dAd is),rtd ){(iroror nadrr€r vorr(!.r,ry0ei. Ed! P' = Ps rorr Pr ' - L

4.4.17 K(r.i ro! I1!no (6). det /l < 5! = 120. H ).6yo qs dct / +yxo<(,{) S (\6)5. Hl6yorovo6T/{;v,der, l < I x 2 x 4 x u x 16.

s.r.2 ,, = ., I ll "r . .,

5 , 1 . 4 r = , l + 1 ; 1 .

KIOAAAIO 5

-ll "' ' =o [-] I "" -e [-j] ""

s.1.7 - . l f , =.u + ( , t ?1). . =() -7)r ' . .4r =)r + r =)4- ' r + ,{ ' r =( l / ))r .

s.1.8 ALcll.aErE I =0.

5.1.9 O urivr€l.oorri< silal )r + .. +^,. :rrv dc(,4 - )1), !!o< oQoE fiou orpt!€e0,at6dN€r6vu srr6< 6LaT ryio! moLIEio n,, anor,ei€r x0rr ro aii - ) tr.l| ro ajj -.\. Aqaivog tiroroq 6eos 6rv n€erdxEL ro (-.1)" I. o oulaE).rcris to! (-I)" ' netntt acneolar€rdr dno n]v xnqLa 6rqr;vro zor rnoErus ti! =;\r + '+;\n.

s. r . r2 . \ .=5. \ . i s r : l i l , ,= l_ i l ) r =n-b . ) . - , -^ , ' - l i l

I t l" :

= l_ t ls.1.14 idErt(-a) = 1, - \ = 0. 0. 0, 4, . . r - (1.r .1,1), tdE{B) = 2, ) = 0, 0. 2. -2,

! r = ( 1 , 1 , r , 1 ) , r { = { 1 , 1 , 1 , - r ) .

s , l . r s r d E r ( A ) = 1 , , \ = 0 , . . . . 0 , u r d h ( c ) = 2 , r = 0 , . . . , n 1 2 . - n / 2 .

s.r. O Fl6Ev6xoao! Jtsqa-rnor cr6 ro ro r = eto + 11 + lrz rx. r1. 12 tivaL (veldpqt(6roqoq.(ziE r8rorqGs) + f0 6fl rsersrEar fto xdao oqlri)v tror noe.iysrar ano

* >.r,E,: H Ld&Ira nFdllo 1 (='pM'r").

12ED EIII E| MENON :KH>EAN

s.2.1

5.2.4

5.2,8

s.2.9

5.2.t2

5,2,14

f r r l [2 0 l l r r l ' l r r l l2 0 | | rL r r j t n 0 t ! - | L0 2 j t 0 0 l L0 2 l'Eter Snqoe€flais '6roqrdE 1,2,7 A=dia-q (1.2, 7).

OL /r aar Ar 6sv ;rd'Iovrcrololvral

(a) .\ - 1 i -l (6) ixlos={, oeLEo1]oo= I y) (E, 3).

(4 /r = Lrru - ('rz),1 + ) = rrr. (6) OIes oL d7.1.€g riroflrriE €ilar !'l6i! 6161td i m , \ / ( , I ) = n - r .

ipos( ,4r) = ipog(8. , l ) = oq+6s+c/ +dl + ix 'os(AB' BA) =D -+ AB BA=I

_ t ' r r 9 0 -_ l IV .A ,A . 5 .2 .13 ,4= . ; l l ; ; .. - . l t u r - I - l' l z r l ,

11 2 l -

dc1,4 =dc(s^S ' ) = detsdeddet"g r=ded=)1. ) , .

(,) "'= [? l] ,E =l) tr1,/ = l: il,, - = lilll ill]tur"=f i t . l

dror /J ror = IJ

,.r., f9^,1 = l i i lr r J i r r t L r u l

i; ?l t_? _;l-,'-',,, =,,(-1)-1' =-1,

[1i_1 ^ =' "--j l?;]='",.. 13'l=pr r l f r { n l f i i l l - l. i - i . In r-- '*1 l i i - [ol u'=t '- ' - ; ' . ] r-;

5.3.5 :fl] drs0€ert rg1dmool oLot 4(ouv nu0.ivtu d- = 1, s- = 0, a,- = 0.

l '+ i, ' ls.3.6 i i n] s.3.7 ,- = 3. :- = 2. ?r. = 30 (0,5)i),:, i =2+3(0,5)4.

s3 . r0 k to<d ,D< l, - , _ l r r l , r l l

- r f o I ; t ' t - , r r l l r I(o ' , * - l r _rJ o r "_rr , ]L | | l r l

| 2 b l - a - b , . 1

4 n , i l , . . 1 |- 2 ( l - a \ l - o b , . , I

L D a - l I t a - l I

l zh l(y) !/. | ' , ,, 1 -l | .a' d ,l r. oxl

L / , - - i l

AY>EE ENI EIMENON A'(H>EON

r l I l rsr .n (a)A=l j j i l CIr ) r =r , ) ,= j , . r , =o,

"1 = (2,1,1) ,4 = (0, - j ,1) ,

l i o l l*3 = (2, -1, - t ) ( . r) (2,1, r) (6) (2,1 _ ( l )^, I * ( l )a).

53.12 01 lDLG6orE rou /a 4ow do€or(Vo .r + ,2 +,3 (d ororx€i.o xdoe onil4s dlouvd0eoLora l, dald.'.ov€is 6€v x,ivsroo. or mrytoi6oss,* r a.- aopoii"y"l(cr + ', +.r). Edv ) /1 ro .r + a: +.3 na3'end r7l6Fri!!'ror.

5J.13 Or 6Lor$dE io! .t + ,4 €irar 1 + r, apa )\ = rt,Inot aoro8rls. Or r6ior4d.- ro!r /- .4j rr t ldL(t+i l r .dqait l ) l =I .v5 frrol r umrl0r6. Or rqn!< rdrcrwd<;tvor(l+ ') '(l -r +0. dpa l)l = t. oubfrse(l s]Joraot9. r rn.nt flual noqdudlrL rdvn oF

5 . 3 . 1 5 o = t . 8 , 6 = 1 , . = 0 . 6

s.s . r7 Ar=r .a ,= f9 ̂ l l .e ,_ [9 -0_5, ' f ^ ^ . r ' - r ' l' .0 0.5-r " r0 o.s.l A" = li

";1r^ 1 ,u > ','

T . L t t 1 t f l l ls.3. re i l , l i l . l i i lL r - r L i r t I I l

s.4.r ) , =-2.),=0r, =1r.-r1.. ,=1r.r l ."n/=j l_: i i l - : f l : i ]

s.ar , ( r ) = | _f : , ] j ] *oasr-- . " : ,_. - - .

5 . 4 . 4 P ! = P + e t ' = I + p + p 2 l 2 t + . . . = I + ( t + 1 / 2 1 + . . . \ p - I + G - 1 ) p - / + 1 . ? 1 8 p .

5.4.5 (a) . .1(,+? ) = .g.^(t+?)S-l - Sc^re^1 .9 r - .q.^.S-rS(^ 'r 'S r _ E1r€n?.

s47 e", = l ; i l d,* = 1".- ' li .4E (o){=[ i - i ]

, r ,=: ," , = l f ] , . r ,=2, , ,=[ ] ] . - .0m

rer " = [;] = roo/ 11] .100".,ll (y) o r,jyos neoo€nil€r to 2/r.

5.411 o ,1r rivor ou6!ftqo €Mdois yr(l r S l. aora0is .trs ! > l_ O ,,1: €ivq! ootaeisyu r < 4. oudsrre(r maoft yrc. r =a. flm(rtlrE Fe:rqowdlrxd _\.rdl a < I < j,tMdois !€ lryobrxd l yrd / > 5. O r{1 eivar oid4 $ neoyFa(,.d ) yLd I < -1,ou66r€qd euoraeft yro I = 1 rsr aoie0is yLs r > -1.

5 . 4 . 1 3 ( ( l ) r 1 = c u r - h 1 t 1 . 4 = . d l + a r l , " \ - b " t - " . , 2 +

t t | u t + L i L r + t i L 1 = 0 .(6) Errl6li o e'' eiv(r opooyri,vros ' 1,d)ia5, ll(t)l, = lc,r'q,|, = ll!011, etvaloa€€qd. (1) .\ = 0 'i +(\.4, + 6tl7)r.

5.4.14 lr = -1, -\' = -9, dr = 1. uJr = 3. !(!) = ('r c6, +ar sin l),r + (d? cos3f +h sin3r)rr.sdl7 --1r =.rI'r + )2' ri (A .\a -.\r.r)r =0.

5.4.19 O' Ibrcflp3-c Io!_€i!(r trqc./Fafla6s r+ (ixros)r 4det > 0 <f ,11 "2

- ur + "r1 2O . : n ' + h ' > . '

I ir - 6"11 + f1ll5.4.20 != l 61t + 6, .11 |

L ""15.5.2 {i) Eivor treayBdflrli. (ii) nfqdxrrot orov trovo6rdio unzlo. (iii) nrqdr€rar Enioff

mov po!(6roio are].o. (iv) Il€aLti](.r(r mo! rnx],o c"rivos 6Uo rt mo €oolrqtx6 'o!.

5 . 5 . 3 - = 2 i , . . i = 5 , . r ! = - r + 7 i , 1 / r = 2 / s - ( 1 1 5 ) i . r / t l = l l 2 ( 1 1 2 : ) i x ! ) l - \ 6 0 =x u , l t l t = r / yB = r / r .

s.s.4 \/a l2 + (112) i. \ l2 + (.rt P)i. L

5,5,5 (u)r ' r=rre ' ru, r 1=( t / r ) . - ig . . I=f t i0 , repLi le tcrmovpour6rc i .o f i lz l .o .

I i ls . s .E , ' ) l j ; ; l l ; i ; l = r ' . r J -usd ! ,o . r : v ( ro^ run ' , (o ,o rou l - ' l

L l l'lo

brdr1jolcl cuto eivar xd0rro sIrs orils!- 6Xr ror /7 (yaof4rcE ro! ,1) d,Id orrsmi),uq ror,4rl.

s,s,10 (a) "(D

+ l)/2. ,. '(r - l)/2 (6) Tc roi(t ororxeia oq 6ro.r6vro €ilor rollyrlorrxd.T( iQia oiort€io, ndlvo cr6 ot br(ly(irvro. 6lorv 6 neqla1rxol]< 6o0ponq- .lErt)eeics.Tu rpi.s ar6 x(iro' riw' ro i6|(. ded or oljvo].rroi rQayparrroi 6s0!oi e).f$FQiaqrov EqprrLavdv flv(ixov Fiaar 9. fra poas6raio!< ni.!(xFq auroi filor ndl,r 9 -ReuLr. Imr , Rcur1. l rur . t . ReLrt . Rcrr : , I$r [ , Rcu]1, Rcar1. Oaiw(L orro a/AiI n iXfl 9 + 3 = I 2 dc0Fors s).€!0€eio<. d Ds oL 3 ralorrlotroro1rllsr. tt(r !ctrol).anrdo'co0on! ot o )].Fs (bLoordrnqpord) p, (1t|e(iafto c/d, adt !-Eaq,aailo'rormo 10rtr16 lLv6frvo.

s.s. | | p:^ .n.) . - r . r _r_ / : ] . , - l ; i i ) u,^, _ ' .^ ._ r

f t v t l I r ' r l ^ l ' , s - | r . s l' -ti,, l ',-l i "5 8 ) 5 ):- '" -l; kl ' L ;i. l

s.s.r k Eivar oeOotrbvn p.r(irr rollq. (6) o Fr3€!6r.)aoi ncpdrerar ono to u.O oeror.aoi llbEloloeo( Ei.ldr o ibLoq f. rov tI6E oxoeo. O l(i]ao< yQqr[iixtraadyddr dro r( ?, rar !,. O roqos @tid)v €i!.rL o ibLoq Ft ro! r6eo ?qdFp.;!. (1)." = , + l. r;1q ptopotpt va nqoo0aoollE mo r onoro6riroxs tro .or)..idro ro! z.( i ) D ? , = 0 ( s ) s i - s ? , 9 r . l s = l t i a B ( 0 . 1 , 2 ) .

s.s.14 O / firtlt oa0oy.bvLos, a!rrorad1,+oq, SnlovronoLioLttos zdr Maario6rdv6E. o,.ivft rivaxcs nqo6oii(, EptnLcrG. tciEtox dro. 6Lo.ro)vrorTouiorfoE. Or r6LoflFiq10! 7r ei.!u' jl roL +;. or 'btolLpi! ror, a rivar 0,0.0.4.

s.s.ls H bt.ioroort rou s sivaL n(n + l)/2 rdL 6rur. Kd!0€ ou!ftqtft< nila)icq A civaLow6ldorr6s n nao6o).dJv,6[os or reo6o).iq tr.rd6d].lovrar aoo.bq- d,].21(€! o n. Arvurdlg/€r 6ciot otr6 r oiooraon-q aivarcs reo6o),1q.

AY>El: ENIAE|MENON A:KH:LQN

A\':EI: ENIAEIMENON A:KH:EON

5,5.18 H oqiqo1roo rivaL to }v6pevo )r ....\n p€ on6l1lro rqr4 .rr ... ,\,, = I (idetLt 2 = det IJHI l =der l - j ) . O 1sf t i 1U =[r ] a I€r detLr = i . E! lda€;

. , f d c o \ d h s i n 0 lf / - f . . , n p d , o , r ] o ' o u a ' D t - , - ' . t _ t x d t a b . , , 1 o .

s.s.19 H iqiq miu Ftroq€i va €irar tr!6usro its (.,2,i)/yG vE onoro6i,ioft aqLep6(naoyForrro n Urlabrro) orolurF F'ts i!a.

^ r ^ | I ' ' - " ' " - t ' " ' 's.s.20 ^_l z, l '_l_i i l .^-="" ' , ' ' ,L i_;,, , i , , ,1 r,",

i "o, _ p, ,1 _, . ., t t L , i l - '

f " o " r " : , , : 1 l q r i , o+c r r r + . r ' r+ . r r3 l

S .S .U c :=1 "3 c0 ' r cz l a j r= l ca io+ .0 , r1 +c r .2+c r .3 ]l / : . 1 4 ' . r l l ! : m r . , r . r c n : - d r J 1 lt . ' ' L 1 , l L . t r , - ! : r ' - . , r : - r u r 3 l

5.6.1 d =N-'aN = N-1,1r-r/r1.1N = (r}.rN)-r,l(ruN) o rauro1rrcs flivox(ls.

5.6.2 Aljioi. or nivcxeE dlow rb|or4ds 1 x(lr -t iX'oE 0 xor oQilouou -1. fl.I.f r 0 l f0 1 lt2 -11 ' L1 0 l '

l-r Is .6 .4 , r r= | ) , , t=Ma t .

L -'

'ls.6,6 (a) C, = -D( ,> c = D(-c)D | (6) Ed' )r , . . . , ) , , r i ,ar r t j rol t tds rou c.

rors - . \ r , . . . , - ) , , l ivcr r6tmrpiE ro1| -c. Ano ro (a), oL C ) iar c dxorj !rts tbrrE r6!or4ds. dQo o( ̂ r,...,)n 0o Ftqtl.viqolrar xoid ot|\,.r),ry q€vfi. (y)Cz = ̂ x .+ DCx = ̂ Dx .+ -C(rr) = )(rr) =+ C(rr) = -)(Dr).

s.6.7 To ororxeio (3, 1) eivsr cosd+lsinp, rou plb€vitFrar 610! ran, = r/t

s.m .,r=f 1 9ls.6.r0 .r I / r +.ry, = (m .r +,,rrc?)/r + 0nlcr + ' : Ic)b =dr\+rbx2.

s.6.r oE reo. r f t . . r o ' ,vaxd( tou f ivo, 4 - l l , l * r0o< ra y, .y, f ; !o,

I f l 91. r -" , , r - | l l , r - , r ra,r i

l 0 1 .0 ] l 0 0 0 ls . 6 . r i ( o 'D

l 9 9 : l , u , r ' - 19 9 9 l o ' r i vu r o r { . ) i r o - a r voxcc rnOuruy ,u r s .Lo 0 0 .1 L0 0 0 l

Or iqir€E naadroyor ro! I, r, r, tivoL g1i!dr. dao rr = 0. (y) -\ =0 (rEtr).i) d,arr6vov iSrobLdvl,opa, ro (r,0,0).

Y:ED ENIAE|MENQN A:KH:EON 571

5.6.14 H I = c^. rivar r6ro6rdvuofc ro! d/di oc nao( rr! rbroxqrn.\. Edl' f l(r)dt =^/(r), r6rs ndqayo,or rov 5no r).s!qdv @vsrdr{or l(r) = )/'(.) dac /(.) =c€'l^. opo4 olotru6vovraq aro ro ro 0 ioq ro ' ixoqre cl(."/r 1) l.rJ(r).

s.6.1s Or rb(orqraq €i'ar I (ra|n),rt) xar 1. Or 6Loni'ca.( €ivar| 0 0 | 0 0 0 l lu o I o l o t l ' " ' L t i , r

5,6,17 (i)'lTH = U IAUUH AH(U r)tr = /. (ii) Eriv o r eivdt rary{,'tt6s ad! oaeopolcbroioq, rorf'ro 6rar;rra ororx€tc ror (LbrorLpas) ra6nsr vd iloN ar6tltl flBri!!o. Tor€ o),o Ia fxr6E brar)'iou moLxeic raaftr !d pl6s!i!o{aL, oqro! or oflt}'€sraaner va eiloL polobraic brcvnoFdrd.

5.6.20 Nr r = rrNrA.. -.//.v^'//r = lxHr lr.$ . = ei a]Jrd o1jlrndleror I orn),rli = yaoB!11 il .

,.u." , =1,r,# ,,/,fr1 ,=|tr rr1 jls.6.23 or 6ror+raq ro! ,4(,4 - I)(t - 2I) €i!a' 0.0.0.

s.6.24 Ario ia 6rcq(irlro rLd€vrxd io! ?-)r1, ?-)2I,I-)1J. ro rr'6p€!o P(I)o1jldv rovnfvdr(r! eiva'Bt6d!. T&€ A- Î = U\T - Î)U- t, tlea (,4-)rIX,4-.\r1)(,4-)3I) =Ir(r - trrXr - !rX? - kl)u-\ =u\u-t =0.

s.6.27 O rilar(ls iI€( 6va polabrrc rbrobrdluoBa, da(! I poAei ro! Joldhn al€r ald polo

5,6,28 rl.I r./:r,\/ = 0, dQa or buo lrlo1ir(1ieE alrodrlreE rir.rr €!,{o}.€s. .\1./r =.rr.t/.EaldylllE& flt! redrq mrt).I to! ,\1 oE Fl8slLotro. dled iouoq 6sv Ff,oesi v

s.c . ro . , f2û l l ; , i ' l q ' " . : " t l l 451 " , r , . : '

r1 o 'r r 2 " 1

- I b r r 5 a l L l b r l l

5 .6.31 J l -pnl .oa4Er i4. Jr =3. f t t3 roL I r r i l . Jr =26i"2x 2 xai , 2 , J . ! = 2 Eni ,2, 1 sri I

"cr I €ri 1, J5= rado€eo prtroa I ef,i 1=0. Or J2 xat Jr ilow o16 6uo

'6r()6ralnolara.

KEOAAAIO 6

6- t . l ac b2 =2 4 = -2 ,10. y = 1" +zs)2 zsz.

6.1,2 (o)'orr (6) orr (.y) Ndr (6) orr. >xo (6) 1l p$€!it€ro, rord pi)ioq qr €1,0eio!

"=[3; ll "=f3;L l 0 0 l L 0 0

,_fi,l"t ',11

572 AY:EI' !TIL\E| MENON A:(H:EON

6 .1 . r de (A . \ r )=0 .+ ) , (a+ . ) )+a . -6 r =0 .+^ r =1 "+ .+yQ i f iF l l z ,t. = 1"*" uQ,if iaP)/z. ^r > 0 €i!o! Faoqov6s,,\z > 0 616ft (d +.), >(a - c)r + 4bz rooiuvcpei pe ac > br.

6.1.4 rrvoxrc b€ueo(,< n""",,"., I 1 -51

6 . 1 . 5 ( o ) l < b < l l b ) l : l l l : - . ,t r r l t u e D ' l

6.r.6 Orr , i

f 1 r - l6.1.7 (o) I -1 1 I

L - 1 I 1

I r l l lr(n l-r 2 -21 \ t t = (.r . : , i1)? = o 6ia1

r r - . r r - r ' 3=0 . (1 ) l r= f " r - " :

- . r ) : - 7 . "1 .

I r 0 0 lI = l - l I n l .D=J r r l r l . l . - 7 r .

L - l - 3 r l

6,1,8 O ,,1 {ivar 0eflxd oAroFdlos ++ )r > 0, )? > 0, l/):! > 0 <+ ,{-r €ilar OFrttcioa!4d!o<. Alsoo< a) . .yr (osr nr I=c/ (4c-r2) > 0rardcr ,1-r =1/der ,4 > 0.

, . . , f 3 6 t t l 0 t t J 0 t t ] 2 tr,.r.y .r= lo r. l = l : i l L_, ql [o i ] rn r ' !r ' ] .rur{c rdv t.rony.iv,,v si lur or

oqroi rou ,. fvdr or qlwelsord( !6dc ora r€radyorcr 0i!st or orril.s to! /,.

. t - 1 , . 1 . r , , , r r r6 .1 .10 n - -

| lU ,_ , , - i _ , : ' l ' r * r * ' , " o oq 'q , i vos o tp r , , : r r , : - t : ' . " : r p . r : o

(3r fi)r > 0 <+ !1 - rr /0 <9 fi €i.!dr pt r6ropoa(fio<.

6. r . l l ( (1) or o5ryoi . ro! f i \a a. tut . b ? /a tdt . ter ,4 = a. t , r . (6) /= l r1+0, /o) . r r r+

k b ' z / d ) " . 2 .

r l l6 . 1 . 1 2 O n i l d " d c 6 , r . q ( ( r l l o , , " ; ] o u v c r a 2 . n e , , t t d ! r X f , . ^ o x r o o m u t t . I l .

6 . 1 . 1 3 | > 1 a ( ( a - l ) ( . l ) > 1 , r .

6.2.1 , > 1. A€! lriiQX€r D xo oroio vd kdv€r rov B eerrad oeroBdvo.

6,2.2 OL B yat C xildt g€xr,.i! oq'oFavor, .v(ir o '4 o1r

6 . 2 . 3 . r e r , l = - 2 b r 3 i , r + t < 0 1 r r b = ] .

6.2.4 Edlv o -4 at{ eeruii( L6roflBtc )i. i6t. or rb(or+rig ro! Ar €ilar .rir rar or r6roqristou A I rir(! 1/), eri.oru 6kq eerrriq. Kord ro kgrrteLo II or Ar ror A I eivaro{raa oEotrivoL.

6.2.5 Edv rl Ar > 0 adt ]" B.' > 0 In tlr0€ r 10. rdr..r(A + B), > 0 o!'Oixr (r).

(Fldl,mo). | ; ;

(FFrroru)

f l b t 1lo r l (Y)

Ae 6:l (nr ar! unoaxEt

AY:EI: EIII ETMENON A:KH:EAN

6 . r .6 P=* [ : 1l "=;11 -ll,^=l? 1l

| | r , .nC 2 , 7 E r a 6 i ^ > 0 . n = : | :

' ;. z L - ' + v r

6.aE Ed! rT4r > 0 yLo xri0r i /0.(\'nora€qnfoE. deo c. /0).

" = l-i -llt tc, = (c,lT,q(c,l > o (o c €tvar

6,2,s rtA!11 = l"T nl r iylr = l(p")1 Fy I < In"l ' ] l lRvl ' ]=6rn7n,1grrtrttyl=("T,1"Xv"/ty).

6.2.10 )1 = 1,,\' = 4, or ltodEovrE rlyoivo!! sro (+1, 0) xdt (0, t+).

I r - .61 / t "4 r ' rJz r V6 , 2 , 1 1 A - / 1 , "

- | t , t s . \ = I x ' ( a . l - ; r - . . : r l r 4 { - r - - " 1 = 1 .

l - { t z I \ / : r , / t . / \ \ , / 3 , / 3 /

6,2,12 M(a Flb€vtai r6totrFi E€I1]ktv€r lC) €)')'er!,osLt!d< o' dvcv rirnqo rul'rrb{ro ) r ,i +)rrl = r *od lircg ro! rairo! dEovo. ano Fit6€lLris ,bro ds (q,ivouv $j!o10 6io €nin€bd yr = +l//t. Ta.'< l'tarvLxas rbror$6s bibow 0=l (bFv oQil€t(r

6.2.L3 (l) rTAr < 0 yto xAO€ !I [tb.vrr6 6L.iv$otd r. (lI)'OI€( or Lbrofld! rou,4rkvonorow .\i < 0. (lII) dernr < O. d€tAt > 0. del,,lr < 0. (Iv)'Olor oLodryoi (toAiE €vol.I(lyes yAqqrdv) urcvonotonv di < 0. (v) YndqxtL niv(lxas Il $(t\,Eldafltr.q- onii.€s iror (;,n? A = -nt B.

6.2.14 E(ir io i-ooro tnaydvro l'roLxFio plhfvtl.roL, aoro . =.i, 16r€ 'rlr = {}. AQa[i!4 o6ilarov o ,4 !d.iv,r o0\a]rxd oAtorrdvoE.

6.2.15 O B €L\,or op:rlrd oaropdvos, o a- Firor aA\ryr'd oQ|opd\,oE, or /t tlar./l r,ivdt uoarotor.Yrciexer rro naorparrxri )\tl.'ll, br6rr I reiaay0)vrrri foaqri raiav€r aa\.l]nxdE r$dqx{rr io r pnoeri va crv(draefroe.| (nore !d a(i)oer -1.

6.2.16 't. A. A, A.

6.2.17 det,4=ar.4r + ..- Ed' o,4 €ivd 6efl{i oe$pil,os. td1. All >0. Koed{ dDldv.rro au, arqdv€t rar ro duAll €vd, to t"r6lotrq 6€v p€rd6dlromaL + I derA 6€v

6.2.18 drj = (reo ri i iou Frxorrh i ro! R)=€rpriyovo ror prixor5 n5 cil4s i rc!R. der / = (dern)r{6yxo! rou raAd.}.rti,E rLnd6ot! R)r 5; lvopevo fuv rcrpcydwvrov pry6v tuv oq)6v ro! -R = a I d:1 .anr.

6.2.19 .H AlIr+ l I I i f t t . , { r = el l r + }al ,4r + ( Ir)q,4r = rH.+( l+t)rF,4r=-r'lr + .\ +.\ = 2Re-l < 0.

6 . 3 . 1 2 G r - ] . r l q ) : + i ( . : ' r ) : t . r + , : + , r ) r .

6,3,2 O ,1 fivdr doeroc. o a pildt eeflx(t lrloctotri!ft.

- l + V 3' * " ,5 ]

r6re .Tc

6 . 3 . 3 o L A 1 . ( r c r e t r l o w . \ , 0 . ̂ ) = 0 . ( ( r ) = r Q , { l , } O n . 0 = [ ; - l ]

" =

l; ;] o c rxFt Fid oavnrrxi xo, Fia ernxi (b,orq'ri. F!i; o l,vrortxoc

nilards dX€I 6!0 e€tr#s tiirortFis.

6J.4 OtL.

6.3.s Oro6ryoito!A- !,f €ilqt 2.5, 5 9, -0 81. dad Fia t6rclrprl .ivdr ooqtlarl + o ,4

6xa ttrorLp4 pLxq6reer to! I.

C3.6 (6) r+a < x 6rcr €ivar (6i'otov va !ftdQlouv n€Eoo&€qo an6 n ov€Edeqts

6.3.7 rdEt(c"-Ac) S rdEit,'I, rdEt(c? ,4c) > rciEl((cr)-'c? ,'lac r) = rdhA +iEr(c"/c) = rdEf.

t.l.E O,4 ixft ln e€rlxdE xor lr deeFsrc rbrorrFtg.

6.3.9 OIt.

6.r .10 rv6-r r t [ l 9 l l ' -Y ' l=0" ,=f - l ] . = i " ,=- l r f 'xqor .erFdra. _ t , , , ) l _ ) l

r6tiv$ pdxpr ro y/3 e\,dr q lurprireqrt nod bfl Etn$pvd qv aalt)4 qs l|Torortq I

c.3Jr ,r1 = s.l..r3 = 5.r/s. 11 = [-l

r l l, " : = l r l .

6.3.D ,, = [i -l] ,v=[f l ] , Lr-.r. ,rr =- . \ : - t=o+)=+ t .

6 . 4 r P = r i - r r 1 2 + . r ] - " 3 , 3

+ " 1

- 4 r r - 4 r 3 d P / a q = 2 q - 4 - 4 , 0 P / 0 4 =-! \ + ?tt ! - r ] ,0P/0x! = - ,4 +zxt - 4

6,L2 :ra0€Ei=-,1b"A-rr. drorL o,4 Fivor o'4r}l€aqLi(j! xor osrrxd oetoEvol

6 , 4 3 a P t ? r = x + ! = O , a P t l \ l t = r + 2 v - 3 = 0 + r = - 3 , e = 3 H D 6 € ! 6 x s t

€Idxno (ftdeie y - oo). Aw(motr€{ orov ryroarofd'. "t*,. l; 3]

6.4.4 ELoxLororoi|]olt rtE Q o6ry€i cE zavovrxt €ELori,ott< ,{Tir = A1 b.

6.4.s frdptc c = (1,. . .,1) mo rillitov ro! Rayleish (o naeovoFo('trls -yivtlcr isos Fe ,)rdr€ rAr < (d0Qo$pa 6ko! tov o,, S nln

AY:ED ENIAETMENON A>XTDEON

H €)'dlLoq lqri €fvft rl(r) = r, s€ r = (1,l).

Ener51 '7a" > 0 yn i()e pt-pl6e!rr6 6tdwot o ",

ro r"('l + a)o ec eivaL

trtd'wepo ror r"1'.

64.6

6.4.7

T T A ' ; I ' B '6.4 .8 Ldv( ,1 . a r i= r ' r ' . ro rF) r ' r , r S - .a . f - f i e t

| , / re . a r " l I . ra " lc4 .e \ . ( . {+E) -minLna - i r

l rT :L , t , f , , r , I = ) r { .4 ) .

Y:El: ENI ETMENQN A:KH:EQN 5?5

6.4 .10 , \1 !a ! . \ r . C+17\ Q- , f r /4 .

6.d12 )1si + . + ),,! i , <.\"(ri +.. + vi) + n(r) < ),,.

6.d13 (d) . \ j = min,1[mu"s.e, , i?(a) ] > 0 +r toxdo€ 5 j , md.€s, r l ( r ) > 0+ Si

n.qL4€r 6r{i1'udlo ' $ R(') > 0. (6) ?./ = C-'. =} r = C! .+ R(!) =

! ,TCa ALt , ' t A, ^ . ^- - - - - - =-_r = r ( ( r , > uu- 11

- T t "6,4,14 x =FL, ̂ r =m'n += < j r r

6,415 Aurdv no'J r(A{i.touv ft r6ro6rdwolor.r rr xo! 12.

6.4.16 Mra {r$Litli. Fra sev/Frst it( r 2 FrpslLxis.

| 2 - t u t l i l l - ; l6 i . r A-41-r 2 l l r= l i l .A!=h+!= l * l .u=f iu-{v : . iv ,=L 0 I 2 l [ ! l t *J

oro,rg zafeolg "

= j, 11, J,tr = i,. -|,=[i]

6.55 Olord{Qd,orr rord Fiqr I

6.5,6 1/3 ro! v 6irer qv €ldrroil trpri irE P.

6.s.7 n = 4, ru = J. ) = 12 €ivar lt€yatrr'r€lq dn6 qv oLno,Mi 'dLoflrni.

t I _ L l6 . s . 8 l l - r { I J . b r i n q ! 6 3 l l y r u t o l r o l ' o r n o .

L 6 i l

6.5.9 l/6 goetE rov l, 4. 1 rar6rdyd,!$ r.ivdrd.

6s .2 - t " = ' , r (o ) = r ( r ) =o= ' r = - i " r . i " , r= : [_ ]

f ' l / < (n ' o t ' l -A t r=L+ != lT .U= ;Yr - ;Y := ( :+ r - : t i S rS iLdr t i - i " is,<'

,/1l / 3

-, 0l f i l f : l-i-il'=Lil''='''=Lil,=r8rr,u=iy, ={i" *, i:::i-viv1a, = Jlvivla" wlvjt:tr = Ilv!vjd" =166

," =,. u, = +, , =, [-1

,.r = 1{1. o = 1z1.,rr = r =r

KDOAAAIO 7

7.2.1 Ed' o Q €L1,Cr oaeoyiv,oE. t6t€ llQ =md(llAd / i|ll) = 1, 6l6ft o I 6toi4a0i rop4rr: I A. | = rl ytd adoe r. Entsff o Q I ttror oq0oldJvLoE xot !1st urdopl !vo.dQa (Q) = 1 .

?.2.2 H ra, rovrx'i av,o6qro lt(r bna$or.tta t'i6€t ll-ai + B.ll S llA"ll+ ll8rll l.sL 6rovbnafoouFr r,f to llrll aoL Fqrnonoqoo+E i0€ oeo. to orori).tolq riv( Irqryovr-di ovtootTc 1Lo tE otd0lrs rLlazo!

7.2.r Kotd rov oqLsto r$ ot i0uls r .)u A.IAHr|< '1, ' l ' l Bi Ix( x(onrv Iar /

ls I Jr. ArurpJrc Ff r , -r l .c x" ' rcytmol lou r . lc ryo,rF l ra < 1,4' l Iro.brc o).r10€i€r xdt Yto tor tr!'rioleoeo lB ',4 'll < lB tl l.a IlL nolirtllla-ord(ovrq rLs 6no arto6qrcs naielotlrt€ .( / r, ) < .(/t).(8)

?.2,4 lA ' | l l =r . l , r ' l =r . , ( .4)=3 rnqt , r - " : - [ l l t - , - l l l

7.2.s >rov ooLol6 / = mu( l/.l|/lltl ) bflld!$ oav r to tv ).oYo €rbtx6 6durolurrite ll,,tr =l.rl

" o:r6t o ).6roE tivar l + o tLiYrotoE ).oloE eivaL rol)'dxrorov

) .

7.2.6 Ar 1l= l,* ,,r,ri I ^t - luo02) - I = 0. ̂ -." = 5001 -{5001: l)r/: H mrigtrir

€ilar I ftrprryrlvrx{ oi(a trou eivcL i<r rc | .a-rll

7.2.7 O\ /lrA a(.{nr ixouv rts i6tts LbtorudE (cx6pr1 xcr. av o ,4 titav rnroFoAq'os',ro! riaoL oer()di nteinftrq t$ (idnilo$). Aro dlryxExQllld!(. A'Ar = ^t =+

AA?IA!) = A(Ar A!) = )(/r). kdL rcdtnto T(uv tqLottov r6ror'ldv 6i6$ l/ | =

lA" l.7.2.8 h.rbi A = R'I R x.L ,{-l = n-r(n?)-r, i/.oq,t l l / l l = l l l l l lr ' i(t In ' l =

| (nr)-'l I = ll/?-r lr. (Kcrd rI! nao{yor4r€vt iroqcq, o clricnpoqo5 rytL rlv i6tcmd0s+) Aea.(n) = ((lr)):.

? . 2 .e r ' ; ; l , = t i ; ] )m ' . ( ' / 1+ t ' > \ - ' ( ' 1 r ' A . , . ( r i r ' ( ( ! r o1J l 0 -0 ) r ( '

tniols .\.*(-'lr) > ),D(A).\"',,-(l1)

7 .2 rn E ,J r -L - ; l . * ' r , - | i l * 1 , . a " , . / l l " =7 A l ' n i r i vd i r c r xod io

r€ainool ,.cL llrrl - = 7 liyLrno (ioeoLoFc aftoi'tn(,)v r+6v motl€i!)! lLlag'raaHrft (ro a4oio r ax€.! (D|or6o€E al)

- l n T r , ' l7 .2. t t H - ^

o^ / ; r l ix t r ,b tonuds d ' / rxr r r In - n urr tvrxret Aqd or r6ror4ds tou

-B Eilar +di ,rcr r, - ft F|lb€lrtag.

lD i , l - l , . r l l l d r l - I r l r i ' l7.2.D {a) No,. (6) .,r-'r -, , bli s .-t ,ll '

it" i . ; ttrttr r . r 5 1 f l 4 t f t l7.3.r ,u - l ; l , - | i l , , : [_; ] . ur -

l i , l , , = l_: l

AI'EI: ENhETMEXON A:KH'EAN

AYIEI: ENIAETMENAN A'KTI'EAN

7.3.5

7,3.6

7J7

"" = [1],, = j t lll, - = ; ltrLl, * = * l!!l ****"0'' " = f, ,",25 I24 251 t3t 11721' ' =ae lzs zqi [+] - Lr r r l '

n, -, - 1" - 6-!!- j fL. -r - (r. ! ,=y. rorr f l( l i r) f lv$va, r =t r - ! r _ ( r - v ,

II!.

-11,,,=aq,=i[li -3]cosrsinrl

- f 8 t . . I f - t - 41d= ) ,

"= [ 4 ] ' . , = i [ _4 : l

f r 0 0 t f Iu=lu - i - i l=y-r .u-r , ru= l -s

Lo -+ i l L oI 2 - l t ^ ^ I f 2 l l I t 5[_ r 2 )=L tn .q=- L - t 2JF to

l c o s d { n d l _ ^ D _ l c o s p - s i n d l l lI s in, 0 I -- -" -_

t s ind cosr l l0

Ra = l'(t +i) --1' ]

- 5 0 - l, t 2 l

6 t l r

.t o ,4 sfvar oe0oy6vtog, dsq Q = /, R = / s( RQ - A na)].

7.3.9 Yroe,oi. rfti (Q0... Qt-1Xn[-r ... fto) Eirot I naaaloworotrpl QR tou Ae, no!€i1,dL aoqa)dE d)"I0!s 6tov t - 1. E,{ tarafl€uk -A!,r = R/'Qr, i Rr, =,1,,*rQl -

aT. .a!' Aa, ..a*AT. noD'u')'aotdlowqs dn6 tuEui Fe (&-r...rb) '.atxerpuonoL6wos iry rifli0€ort. €rolrt€ Rr . . . P{ = QT . ,i{ o*' Msraq,qot'idgia Q mrv (eimuari rls]Jqd n.rlevoljF€ to hro{$Evo anota},cspo yrd rov A^*'.

fo i o l [ r i o li .A.r D-t t -L- r / )= l i 0 l l . r -0.+t / r t@-L) ' r -urlo I ; l , r . .

Lolol_ - Loi i lqfaO,O,I/2 d6,\ , =4-2\, /2,W ^ns =3 - 2\ /2 -0.2.

?.42 o J 4(.l orory€icr ioa Fe 1/2 xali !rir.o6 rov 6royoviov ruv l€[email protected](;v treos frpaueid 6!d'y{i}aro,}{dr trl6evr i ravrou dl.},ot. Jrr = l(sinztrr,sin3nh+sintrn,...) =

7A3 hL = (2 -?.osht 'h,i. J,r = lGin2l"^,.in3r'l + sin^'h,...) = (cos li'a)'r.

?.44 O xnxb! $po ond to aii 6€v rice#Xer ro trrl6tv 6rav I axd.vq ro1J "

ei\,arp*porep4 ror la;; l. :werG ro rrdEv 6fl .ivar '.6rorqri K! avos xard rry &ay.irvLo!ft€Eoxll(ov ril,dr.ds 6sv lr'toqEi vo €lvsr r6l'tFoeqos.

fo I i l z | 4z.ts t=-19 0 j

I o! driv€6 €tv* n = ;, ",

= ;, "r

= ;, or au o! dxolv a.\roo

l ! z n lro urbi":dqA 6i; o, l^,1 < r.

57tl 'I':E]: ENIAE|MENAN A:KH'EON

zq t o t r t + t t = l 0 . .t -./{I

) = 0, ,c/dd..\.". = ir;,.

(D+r, ru = [3

KEOAAAIO 8

E.l.l oL yovi€< €aa(rt mo or}r€io (0.6). (2,2). (6.0) 6Eits to:r. 8.4.

8,1,2 H i+v €IcrLmoroL€ircr sro (2. 2) p€ x6oroq 4. H 3r+v €l(xroronocirsL oro (0.6) |c "6otoq

6. To €).dl|gro nts !' - i/ eivqr .o p0 r = 0. v + c..

E.rJ Or mA|optoltoi. 6i6ou! 3(2r + 5r) + 2(-3x +8!) < 9 - l013lv S -1 no!.rvlrqdo}{rrotl! ! > 0.

8.1.4 ndQre ta r rar y va civur iss x.Ir rolu p€y(i).a.

E.r.5 r > 0,, >0,. .+i / < 0 bal€ror } ldvo 10 ot! ! (o (0.0).

E.1.6 To rqruro o1]lolo €l!(1r 6v( 'oorlruqo rqiyovo, r.OtEl6Frvo oro €nirs6o r+!./+. = |F. roautaE oru (r , ! / , : ) = (1.0.1)),(0,1,0),(0.0,1). H r€),€u( l iu.rovio 6iv€L rIFa-rLoq rqi 3.

&1.7 J = : = 20.000. u = 60 mo

E.2.1 :rI! nqeonoc a(rdm(ot ir = 4 rcL ri = 2 au]xow uq 6dorl xft ro x6o|ios rivuLf$a!. H eLo€e16p€v[ t.t(6].Id nqiJr.t vo €i!.!t rt ir, rou rrL6vfl ro x6mo<. Hffi€AxoF€q Ferc6lllTt rpdnfl vc fiv(ll rt J5.616t|ro 2/l €iv(L FLre6r€eo ro! 4/1.Mr ro rl x.rt :1 or1 6tio1. o. fteoerofoi 6i6o!v 11 = 2. r{ = 2, 1r(r ro r6coEyivPr(r r6qa rr +.r 11 = 2.

E.2.2 Anofov(i)lo,|'ras ro rr orov 6.usao neproEopo 4oupe ri = 2 3xr 5..r x5.Arrrxuordr(i)rrdq allt6 oq oulaeflpit riootou r(trrov ne6ro ft€e|oolq1o, n(i!eol|!rro €Erls ne66).rtpc dnlv6(l xoarq{: E}.atroroiroilot qe 2 + 4:dr +6xr + 15. Er.tnrior orfirhrmig 4. 6 ?ar I ei!o( 0errxoi. io e),i1l|oro Fr nryldvEa( €rL 6r(v!r =.r =:r5 = 0, aar €iFoot€ ibl dq dll,rrort roewri. M. (i)Ja 1.6yro, ro aeLderobrs'aomt( r = (4,6, l) > 0 d,r0EnEr ,ot ro eL(ilt()ro romoE eivoL -2.

E.2.3 | = 0 ll. aAa r aoew,rt€ivaL 66l(oril.

. . , . . . | | 2 I 0 o l f l 0 J 2 6 l8 . 2 . 4 l ' : l - l o | 2 - l 6 l , l o r 2 | 6 lr ' , / v u r

L o I I o o l Lo o I | " l| = [3 - l]. aa( r 6€hEqt oni),r roD ̂ €rodqxnar ont 6do[ H oni],r (wi Eivar

t 0 t - 2 1, - l_ r1 ,. "

, I r I

F'!'( ((r'rT|xo. dqc ,t (rp't '-rfl onneo ruFos xuL torl..i/,rmo a6moq €ta(t oc.

8 . 2 . s : r o P , r = I 5 3 1 . > r e 0 , r = t l - l l . g t o n , . > 0 .

8.2.6 (a) To E€r"yoq i =0. r, =, si,\,ar l|t aQvrlnr.o, rxovoroEi qv Az+r, = r rar €iror6doLx6. 56rL x = 0 o1r o9a0u I p)16€vrxdE orllLmdiorE. (6) To 6olerl lto ned6Lrya

-'l"1 ,=*(*)'"

AY:EI' EIIIAEIMENON A:KH:LON

rldxtorono,€i 10 or !'ro n-c oworixEs or ) 0, t: ) 0, ur > 0. xr rr + or = 3. Tobrduioto q! odolE I ria(l| .r) = 12 = 0. or = 3. To 6llrtmo 6rdlaqto dlct .i = 3.r; =Li = 0. H aoelrFi 6aiafttct oto rr =3,': =0 '{(t 10 €qtxd oivoto.ivot frq€!0€io aEexrtFsvr ar' d$r6 ro olp€io t€ '{}'ial L

8.2.? To ,$niaro braxojnic yivrru( ' S 0. E(! cuo uorL:(rL zut I i-ooni ouvLn6oo€ivaL I F€yal.nEql, r6ft I (vrlororr! oni).I rou N noaaxercr aq 6dol. O 'rs'orasEC yLa ro an€qx6FEvo 6l(ivuolra (jt6 q 6(iol €i!C| o ibroi pe rov neoryou$vo.

8.2.8 :r r((l roqugrt. 6uo arT6 rolJs n€ercqqrons yivovra! toorrfts E!6 lo1jt6XAoadr.lovonoronr.ler or dl).or rQotE. Ilvrnci-r< or xoewiq tiv(r .r = 3. 12 = 3 xot oi = 12,..2 = 0. To dwisrorxo x6sroE 0i!sr 9 '{ar 24, dqa r trediq toquai eildr 66},1toil.

8 , 2 . e B E = B I . 1 , . I = [ . . d . ] , ( q , o ' r / , 1 J = r .

E.2.10 To 12 0d 6nQsn€ rd al]he€t trdlAr ro rr = 2, mo onoio .r = 0 tdt ' = (0,2,14,0).Ertrr./xdlorLpe ro €),dxrrno ].1 r l"r

- z = 2 oe dvd 6iFo.

E.2.u -4r =o+ Pi ='- 4r i1. ,r?; re" -"

t .2.12 p2 =p,pT =p+ -",T p" = -""r pT p. = dp.)rpc = -sl 'pc 1,.

s.2.13 (c):ro P =(3,0,0), cIr = 15. I ro Q = (0,3.0). c"r = 12. l ro l l =(0,0,3)..ri = 24. AQd c"r elqrotoL:toreirur oto (0,3,0). (6) Pc = \-2/3.-sl3.t/3).

5?9

E.2.14 orav, = J{o.et) = o.,tz,I r .28 I

D = l l , ? 0 |L 0 , 0 2 1

8.2.15 ADzAt! =,tDr. y i ler( .1.53y! = 0.98/0.82 = 1.20 + ,Yl =

ro lirit',r r.ffl;v.r co {1.28. 1.70. 0.02..

1 0 . 7 6 i , = 4 3 6 . P D . = ( n P 2 . 0 6 1 . 0 0 ? ) . -, - r t a D t = ( 0 n 2 , 1 ? 3 , 0 . 9 2 ) - r ' = D X L -

(0.03, 2.9,+, 0.02).

8 , 3 . 1 M . y r c r o n o i n o n q E 4 u + 1 1 1 2 . t r ' l J t > 0 , ! j 2 > 0 , 2 t ^ + ! 1 2 < 1 , 3 t r , < r ' . i = 2 .

rj = 3. ei = j. !/j = i, r6otoq = s.

8.3.2 EIsXIoto oilol q( 3r1 rnd ro\ n.atoorofous rr > 0, xr > 1 vi = 0, !i = 3.ri = I, xomoE = l.

8.3.3 To l}lrir.o p€yroroiror€i rI! ,b ff ,1 > .. >uverd5 r = lt tdt J =. €ivor €qrtrd taL6ivow q! i6La flFi .6 yLd ro i6dros oro .lAlr.16 ror io 6!a'r6 nao6l'rua. An6 qv EFtreirn vo rivoL 6ih@(. Av ,r < 0. r&e to 6d]'lrdro r' dJril€L oE (0, h... ..6")r c t J ' = ( 0 . c r . . . . , . n ) .

83.4 A =[ l ] , b = !1, . = [0] 6rv. ivd ' sqrxd. To buizo Fryrootror€ i ro! iyro, > 0aoL -lv < 0, rdL EivdL prt qaql'avo.

8J5 r= t011" ,c= t ro l .83.6 Edvror €i lor t rol i p€y( i) ,o, rore ,1r > bzoLi > 0. Er iv y =0,r6trvA I rzot

e > 0. Aed aaL id 6no €ilaL €qLxi.i.

580

E3.7 E,ttL6rt ci = 3 = yr, rc ! aQ y €ilar 6;IrLc.1. dF{oaa l€ q' EF

s . 3 . 8 , 4 . = [ ] l 3 l l r > b = [ 1 ] 1 1 l T F € l v t l o r a a ' | o o f l l r o o < r e o E q a r a i n l

o!\m6oo. Aqa I tQirl oxvrotriroo tou v naii€L !c Plt'e'ilFr(tt naaopoioE

r//r = [l 1 I 1] <. = Il I I 31. xoL I rvtloro avtoonFs lnoxqEdrlEL 10'r = 0

83.9 r' = tr 01". v' = ll 01. lt y'b = I = d'. Or 6EusssE (vto6qr€s x(l lLa ra 6no

,1I' 2 b rar v',a ( < tivoL-y\aiors. dea or bs!{tqtE o]Jvrsrdo€E rol r" xoL v' Eilat

8j.10 ni =,6ibtt e,'tr =vl, eirt t > 0 Eift olf i/A < c 6i6rL v-'r. < 'r agon i > 0

:vyxailowu9. l/1, < ....

8.-r.I ior J; - u, j; - l, J i = rr. ' ', =: tC) ttutrL to rquto tcrncouoqto pr ro trrcrGh('oqrn iooro i ier t r droxor t i '1yo ' iu {11 Mr} tqororr io t ' ro ! t ' ro rL: o! 'vsr l " 'cy l > 0 , y r < 5 . y r < 3 . r r < a r ; = : l

E.3.12 Tc €rirE6( .i =mdotsld fi!a! El((tadrs €nLa()'ll,1rotlt!(!, dld I rledJrl xoeufll tou

.ittaron (11116).0! ont! onoi.r sq,(inrordt, FilrrL I i6t( l$ nP raoryoi$vl'

83.13 oros ,.or ct! n.1a(iYacqo 8.1, io 6lixd F./rnororri ro 1l rnd ns onvorlxfi

tt 32.2p 33, p > 0. H iuol tivot p = 1,5$ I 0lt|66$ r+ln q; naoteil'ls (curl

,oourc( pu r1v t4lri rou prxprurton. 'Io! tLv(lL I 6!1'rtotrl 6iorrc) H ovryldv'l

r+ttl ro! q!o1t7o6ovr!eo! ciaot ?$ - 1,50$ = 50 oe!'rE rot' €ilcr e'tLlari' (iad ro

q uorLao6owleo 6€v nEqrtls"rqL mrl 6a)'troq 6ictta

E.3.14 Or orilsg r(edlo!v ro! tlilvo Ftt4! 10! 0ttrxou tilova r zot t151pEle€ic's r : r')'qr rptirq neqirroolt r'= (1,2)r , = (1, -1) tttovotrorti nlv eao)')'oxtr)lrl

m.ls Or..rll.. rrr. l;

I

83.16 n.iAft , = [2 1].16rs !/,'l = 0. s, /0

8,3,t? [(iAr€ y = ll - 1]. r6tt e,r > 0, vb < 0

8.3.18 ,,4 > 0 bi6tL r,4. > 0 ,4ir > i,6L60r !'1r <,1, < 0

&41 H FF/ronl qort tivcr 13 pt tklrq roprl ('{i1 ro! 6(txo,eiia tov "oF6o

6 on6roLE lroLo'JtotE.

8.4.2 H Bdyrcr eoi eivoL 8 !e .I.ixroll lopi (univ ,Io! 6LdX0)Qi!€L lorE ro|l6o1K I - '1

f f i 6 r o 1 ! 5 - 6 .

8.43 AlEdlo\rq q xoalttrriqta tuv aYoTdr! otro rov zoF6o 'r mov aol6o 6 rl aro rov

z6!60,1 orov 7df6o 5 nAozur.r rl p6Ytoq atlEryI qq tttYrm'ls eorir' H F€"rrotl qorl

(llldvEr dn6 8 de 9.

S.aa H piytont 6wani aoi (lro ro!'r|)160 1 oroa xo!6o 4 tivul 3

8.dS Ytroodmu 6n or xoaflrxoQltE! Iov czFdJv eivar 6i€q I To ldYrmo nli0os

a.llqo0,lrrur! bQo/nt uro 10 ' l'ro I 'ivar I FrYrcI eol O {}dXrooq @LULrogo4r;v ruv otoiuv I nqaiqrot (ro(mbiFr ro q aro to / f'vnr I r^ar'rrl roFI

(616( 6isg or aaFas 6xow T,oqlFl-l6rn1a r) T&€ I trsYtoq eotl = sLdxraq los'l

A\:El' ENhEN\IENQN A:KH!EQN

0 - r 0 rJ l l r 0 0 - l lu 0 - r 0 l r l u I 0 - l l .r 0 0 - l l L 0 0 I 1 l

AY:El: ENIAETMENAN A:KTDEI)N 581

8.4.6 Evo ldrroro o! !o) .o ydFoa yLa10,4 s ivot 1-3,2 r ,3 2,4-4 E ar ] ' r la8sra io l ,dotru ror B Eivd! I 2 ,2-4.3-5,4-3,5 1, .

E.4.7 OL yeot{dq 1 . 4. sor 5. O urorivdko< ro! nQoxuJwtr on6 fls Yqappig I , 4 1|(r 5 xalrq oql€q 1, 2, 5 aX€r 3+3 > 5.

8.4J Xq€idEowar 4 yqcFFiq. Or I r(i|or swroroqo!! os /i ttovd6€E no! redlowsr o€6Lrqoq.rads raoAds x( anl)'€s. Xq.rdlottat i|yao$t€! Ym v( a(Lw0ouv at'tt!or l'ovc6rs. Enol6ao! XqerdloEdL toulAlnov l taall|r4 Y@ vc xa).r{,eon! oI!;

E.49 (, O nivsxqs Fr€r 2n Fovd6ls nou b.v €ivat dwalov va zclwoo& I. ILYorEplE(1lt6 tr yaorqGE. 6rorL x'io€ yecprit xoilunrtt 2 $at6rix sovri6r.

8,4,10 XQfu(ilovruL ro!).dxroro! 3lacft#( rra 'a aclweorr! xot or 15 Io!db.(.

E.4.ll Eild nQoeola9

8.a.12 O orryrqrdr€aos 6Eipos ono ro r oto I ri!(lt o I -2-5 6ri I - 3- 5 -6 Evqlldxlmo orJ!6aov 6d'6qo rivar lo l - 2 - 4 - 3 - 5 - 6.

t l . 4 . l 3 I - 3 . 3 - 2 . 2 - 5 , 2 4 , 4 - 6 N a t 2 5 . . l - 6 , 2 - 4 , 3 - 2 , I - 3 .

E.4.14 (a) o cly6Ar0[o6 anlr|mids EEat'd F€ , [rlovtuavd 6iv6EL :e xdoe Et'd bLolareti!( 6ilhao

" adr neooosr€t p(a! o4rli . .).olimo! a6oo$. Yno0aor! orr d. xiitroLo

6riFa ra lrriqro\'i.1 bdlbes (ror T flJFrsqJdF6o1,otdvo!) 0ivoL fdarx tou ELalimoro!!6uo!1o< balbeo! S, aII(i I lia o4i p (tdttvoloo ro

") d€! ei.l,(rr liaos ror s

>irouo(1 ro s nredt&.i;\h (11!1i (' ttol, o!v6d& iaa",rop6o 8\'ros rot, ? p. d'ovx,irrtio f xroi rou T. br.trt.erlttu ro .i .)l v ed ltdv owbiov :rYoueo snd$ I d4rrlcurri Lr6tv rivur olwoporeer q6 ., rou ira! d4ut eldximo! r6orotls t€ltvoroa 1()

". Mtroaoupr ),o!ro! !a cq<rLqloowe r1v r' cro ro s aaL va qv (wxcll(lotliooup€

lr nt! p. xoeis !a auEioolre to oi'o.6 p{aoa. To s 0a .EarcIo!0ci va tilor.ldxrco ary6aov 66!bQo. (6) O .rdx.dos 6e4ros 6.! n.erax€L 'tur' avdYxltv q!€).dltoq atFi! a€it€ r.X. qv &4.2.

&4.15 (d rqq$ds 1, 3, 5. (6) :rIL.E l. 3, 5. (y) o !trotrild"d€ nou nqoxinr€' an6 rtEyedFFt! 1.3,5 roL rLs mi l tE 1,3, 5 (6) faotpt< 2,4 zd miLt ! 2 ,4.

8 .4 .16 Meyro tono , ' r , r , r o r - , u ronso , r ye1*€s '1 i ' ' - 191 . [ . " ] : u .

t / , l t r l t t l t r l -

8 . s . 1 - l o i r + 7 0 ( 1 - ' r ) = 1 0 . r - l 0 ( l - l r ) , r i 1 1 = 1 , . , = j l 0 e 1 + 1 0 ( 1 - y r ) =

70!r - 10(l u),,1vr = l. rz = I prlol rql]l reqsols 6.

8.5,2 A|al,tryoros rdmor ol! beljteet oflill, o X x€e6iLs! oiYotlaa 2. Aratr6rovraq qvradltt tqdFrrli o v rd!€r diyoueo, ro ro),!, 2. Aq( ," = [1 0] ad! r' = [0 r ]' .

&5.3 Atalrrovta. qv o'th j o x teQbilEt oiyouaa oir,6ton auo ef!ft to przqor€qoororxcio ahk qs otrtlrE. Arol,€1oltaE rlv r€q$i i o v xdvet to,to)'11dij bL&t

f l I 1 I 1 ll r o 0 0 r l

(6 ) l l0 0 0 r l .l l 0 0 01L t 1 I I l l

A1':EI' ENIAETMENON A>KH:EAN

a!.r6 .ivql ro lsytoro ororr€io ouris q( ypdI|ri!. :qa naolYonf.vrt dortlot dl] = 2fuaa orysio Looeqoniss ahon ror FibolE. Ed! 6g{ €vd.IdEoqr€ to 2 xar ro 4 4ft6adto. xaldlo col1.io 6ev €r€t tLg n0oTyottrt'sE r6r6nFei aor ararronlra! F€i"u5

E.s.4 H rorri r,/' @F6oifti &(v r+3(l - y) = 3v+2(1- s), dao y' = j. rn,:,pos 1mini.*rrrrn) circr l.

8.5.5 H 661,1|ot|t otadntro.ri y'q rov,Y oDvbud!€r rLs 6io yaq4'is ?rc vd r(od?lr Ftooarld1,tla yaotlln i!,oq I ro! eyluircr cr.no ro noodv. O ouvslr.lopos sivdl

:(3y+ 2(l - r)) + +(y+3(l - v)) = I. daa o x nuite u5 m1).es pt otp6qrn 1,0, l.&s .6 v '= l i i l . , '= t i0 + l? . !$ , i F r6 !v .

s.5.E Edv d = (fi, i, i), roff o x 0u xeq6i!e, fi dvdw, onoLdobinor!: oraarryr ir ro!v. Edv i/ = (*, i, i). r6t€ o v rdv& fi dv.nfl oJro,ao&lnorf i'rqdrryrxris ro! x.Arr1i r toopaonicl Inltt io ndrxvfbL.

To ro(orExo ItE/Lmo €i!a! F€ydl,rnrpo ro! i/t trat v2. :!'yx$radrvot$€ olo ro 'o cl'ro. Koi6nr!, rtraxlmono rwas ro $eyoi.ueQo uno r't olv9rtrrt yr +,1 = 1,:rai.proup|r1v cr:rrirlot ].

:ro (5). mine.4r" S y",4f,'. 6L&r ro eldxruro ur€Eilrr olov t(l)v y b.v rivolps'ydLilfao qn6 rlv flFi yi cuo to tL6tz6 y'. nus4otd /'nr' S naxe',4x. Ecivoqv {s) L.rruEL rt ro6rrFa, orotE minyA.' = r".1r", r6t€ yla ioe ! dwo eivclpdq6tuqo n ioo tor i/,l,u'. Avr6 €lvar ro tjsireao )irr(o! otE (4) ,.ar ro rqdJro liplo1Jatr€t(rt cnri ntv maxy'.4r = r',1r".

*' = ll ll? zcr y,rr' = lrr * lv:. rou ro"*", 1,e I rLo riks < rnocqrrxle tot'v . v ' e = t l I - r - r l " a rv ' . { r=+ r r+ l r : - r r -4 : roub rypaoe€ iv (1m€e6€ iio j. tlo ev6ldp6oo rivar y']la" = +.

Ei!.a bi,{oro naLyribL.

" ' = ( ; , ; ) , v ' = ( : ,1 t - pdco rdqbos r i rcL !P.

&5.9

E.5.13

8.5.14

t 4 l^,^ lil , r;r; r srls;:] " ii i] " ttt l l

NAPAPTHMA A

A.r [0],, , , ," =/m,,!t01,,,, , , / , , ,", I0l; , , , , = t01,,,, , , .

o +slbo(lwimqopos eivar Q* = Q? .

r"r,r',r=lrou ,fl=;tl -ilt;

,:ll-l llA.3

:l

4.5

'I:EI> ENI E| MENON A:KIDEON

. l f l - l | 2 0 | r | ] t 3 l t - . ^ , ^ r I | 3 l l' - t [ ' r l l o q l l t r l=L l ] l ro ' v - ' r u l s [ - t : J '

^ , , ^ L 1 1 1 8 4 lJ = " " / = i [

4 1 2 l

r€arxit).notlt=(1.1. i'. ,t'). Auor €loriorou pri"o'r "'

= (1. l. l).

(o) rra oa€qdaDF€s cihs, o xdroos ygoppr'v eivft o)'o ro n". iT'lr' -.{7l,

daar nao{frdsis. (6) To '17(,4AT) rt, eivcr orov xiqo lpoFFdJ!. 5tol| .4?- qoe6<onorodrinor€ brdwolo auinrr o cuov lov 1drqo. AI Ar+ - AT AAT IAA't ) lb =ATb.

A=e'e; t =ee; le i :e tL =es ' .e =ere;1. s '=ez>ei l

,4 * li piaoL UT qoA6s ){dro(o b(irl'dp(. (iea qvrlxtt oro! XrirAo YAdppdr tou U =/irqo(

yqafr r6v rou.{ . Enions.4 ' AAT' = ut LTLUur( lAt) 1(LrL)- |L)Th- A '1 L.

A =QEQT ..> A' =QE-QT .+ AA' =Qr>:.Ql + (,{,{ ')r =Qrrt*>>'Qrl =

Qr22'Q{ = AA-. ro ittLo oxnfl B(r 'rr( tov ,4t4. Or ,4,4' uaL A*{ r€o6duo1'clwisr.J|Id oov;(.;eo otrt),'i,v s", ,nov tt;eo yqqq'dv.

A.l0

NAPAPTHMA B

B.'J=ti3t'=[lil]

t 2 t fI 0

I = .I + 8i br6rL ,' = 0.

r . z f = sJ ,1 ; " , * , 1y+ rs , " r , au . = "s t ( iA . . , +^ r r ) =es r (8 re r+8 : . :+ . r r ) .

_ f l

"''""'=Ll8.4 Krior n,!(,.dc 4 civgL riuo,og roo: iwv riv(x,, lo,dd J - 1/-r.1.1t x(t Totd

rn /iqo: (a) J - PJl/' rEdd o P ,ryrydfltru urlor noos Fn?'or tlnd rq

F&a0ioa< etror drcfovio! nou nacyFdroro.orJltat oe ado€ |lrloa Jr). :u!En6so ,4 €iv(L 6toLoi iotr nT: ,\/ 1n,\, = J = P.IT P I = PII"/T( 1/r) ll,-r.

iI = l t t t ' t tr l tT 1tt ctrtr l 1.

f l {J 0l8 . s . i =10 4 0 l , i = l ; ; l

L0 o 6 l

EYPETHPIO

AeeoLoI (0416010) 24, 31, 156.AoaoLola bnalopdrov 6.

!rordreo\,) 233.Aoeorotd ierQdy6\ro! 177, 380.

383. 387.

Axltit Il8. 462.A).1.60d Bdt. 252.Att60rgtroE anl,Iqris! 502.A),ydqL€$or €m reoqrt 49?.AD'an 6dolx 146, 357. 369.A 'an F€ra6)'rd'! 355. 399.A {lorru$t6o)'i a09-AlDoiaa fladaor 282. :103.Avd6aoll ardaldc(ol 1.1, 17, 548.Ava6eotrxi ((1!di!.''l) 168.Aw0sulerlLdq (!deobo<) siDpler 472.ArdxLoort 135. 145, 190, 194,209,435.

Atatx.]0lwrt 69. 476, 471 .Aldoreotpo< 54, l 7 4. 255, 34.

AleEoQqoia 92, 94, 299.Avlyldvit raQdyo!.ronoiryt 229.

230,242.A\atyFavt xLLFaaori Foeqli 89.A\aMrE\,o xoorog 468, 488.Ave$oros ror F€oaiova 191.AvLodtrttd ro! Schwu 168, 171,

175, 206, 303, 394.Avioool 452, 453, 449.Ar'rc$n.ovrtg 347, 350, 374.A'!'rrx€qr€vw) ovdqrlol 454.Avn0rQ6!n+o! 48, 53.A\sioiQnpo yrloFavo! 49.AvrioiaoqoE 36, ,18.

Avrioteoqos ovomq6qo! 55.Awrc!$eteta6s 58, 258, 330, 336, 510.

AEo.',!s 153. 345, 391.Aoqrorn (popQrt) 381.

Adaroros (nivoroE) 390.Anol,orqri l. 15, 36. 50, 8.{, 89. 275.Ardlo'q,i G.!ss t, t3,44.275.And,o'ql xard oq'nLo'.o 279.

Anflpo6rdmaror 74. 205.Aft(llol€vrl trsr(r6l,llnl 466. 468.Anl.tE 6wd$€rg 307.

Atroorool 245.ArorsL€otdflr'i rdEr 521.

AaordE 60. 414, 420. 544.Aar0plnr6s pdooE 170, 175.Ae$pog xarAoraol|t 42, 424. 427 ,

429,54t,546.AA(r€aoE arricqoqos 48, 103, ll1.

115. 141, 197. 530.AermreG prp€v'ioqos 103, 108,

r20. r58.AOtrFLrld oQrolEvos 380, 394.Aqnq perdofq 277. 548.Aflri 0&6.rr6ntraE 303.Aqdi €IatmronoiwrtE 4O5.AqIr,6 (reo6l a) 48r.Aoo€vrlE 6dq(i5 483.Ao(eis 67. 313. 32?.Al.EtiE LU._A = LDLI 56.229.3a7.A = LDU 41,2'75.A = LPU M.47.

E1?ETIJPIO

A = LU 37.38. 16. W, 242, 54a.A = L4n0 .530 .-4 = i . /Jrt. l r 36.1,531..4 = QAQ7 345.362. 3$.519.-'l = Q.9 s22.A = Qr:QI 't:c. :tsI - .9AS I 296.A = xrT 1t4./-4" 58,5l l t .iT,{ 58, tno. 2.16. 518, 52i.AT a A t2g.

BaeF6( e).€r,0sai.(r< 86. 98, 152.Bo0loro (yrvoprlo) 156-B.io4 97. 1.15.357. 369. 519.BqotTri |.r(6l.ttd 84, 88, 464.Beirrororoirprt 468, 484, 485, 524.BoF6d 3: l l .Bridse 24(1. 290.Buninko*s*y 171.

r€Lo)lalos 128.r€lr|aerJpivo (6rodrdvtlstrd 323, 532.frvr'arlJlalo nq66l,rud Lbbt4r6! 336.

400.403. 4 ,419.rtvrzi Indt 85, 88, 123, 537.

fLvdlevo l4l, 236, 254. 279. 102.Irlofelo r.)! oij'|].fdr 260.reoFFri Jteog yqalFi 23. 2u.fqdr'rrxi av€Eaqqofa 92. 94, 299.req'prr{js F€uofl partqrG 137,

14t, 357.fQappL,{oE neoyedFpsrlotroq :152,

457, 543.fAo$rLr(6; ouv6laotd'E 7.fqi.q|]lLa l18. l2:1, 545.

c" ioo, ::c. :so.Califomia 20, 309. 433.Cuchy l7l.Cayley-Hdmilbn 371, 538_cD = -DC 32.25r .282,368.Cholesky 229, 389, 540.Cooley 222.CRAY 19, 32.

Crour 41.

AEH 203,A€iznts x(rroordo€o( 422. 424. 42?,

429.54t.546.A,v6qo r22.AeE|o ori.orqoqo 48, 103, 111,530.A€Erdmaoqo {ouorlt[c) 1e4, 273.^mbox'r'i (.nal,oqri 14. 17. 35.

Ardnop{r o(ri\rcro( 1E0, 183.AmlouoroLlot 296. 298.drayolLoaoriotro\- 297, 302, 361. 538.Arc.ydrvtor ni!(1xaq 41. 296. 518-ArayolioE 6ron6!or 451.

Ata4rr6q FEr{loxitFarr4rds 2l 3.ArovljotrdtLx4 X6eos 73, iio. 231.A|airiqot 325. 334.Atdon.lq rf'Kioro6v rLtdv 164.

229.: l3l l ,519.528.Ardsnoo|l ftiraxa lg3.A(onoa{i 191,2:10.dldo ol86, 98,99, 109, 234.

A'oroFi ,19b. 5(x).dra.noaft t,iar rbrortFds 297.Ata.poQ|l{rt .Eio(i,on 60, 285, 323,

366. 53',7.aLaxoaLonxd (lnre).nirrno 490.

Ai,inlr | 26, .195.A!i!o.- 456. .181.A!io}16< 482, .186, 499, 512.allvqt riv(lr( 300, 366.

Aworlr"G treoye{44r(6rof oS 503.

O-rr ig *t , SlZ.

Er6Lxi ),uor 68.EL*6va ruv yQor+rdv 8. 23.Ert6vu iov orll6v 8, 23.Elo€axdFFao.eE€qXoFr!s 314.ELo€exo$€'w] fera6i,ryrt 464. 468.Eaarou+ruqortos 273.ExodLai 321. 330, 366, 537.

E?qlriropoc 466.E).dooo! 264.E].aftr)pcl|"ri 291, 297. 323.E).dlrmo ifted,{rra 125, 178- 180.

239, 412. 524, 5Za, 540.Eklltoro 376, 380.'ED,ot!q 345.El.i€r1l,o€rba_c 205, 390. 409.El,.neearl |fta6lryi 84, 88, 463.Etr6ebar 274,279.EldD.oT rt laof4Ld,r 27, 42, 252, 263.Elol).axrrx1i 165, 489. 491.Ev;AyEra 405.EEaairu6!1] 92, 95.EEioooI 6errreaor 6a0Fo! 331.Eliooorl braqoedv 61. 305, 309. 319.EEiooor eeaF6qras 326.EEo}ieeu{o yrlotrelo 336.EnsluLoir6sv6Be!€E r6rcr+dE 298.

362, 364. 532.Enav0),!nlrx6q- 421. 443, 151.ErcErq/soicl eflovflq 521.En nsuos 76. l15. 161.489.EnLFeQLmdrt 29.ErirTeb! orao(ii 369, 441.Erarsbo 2. 5.EApmav6E (nivd"ds) 343, 350. 361.Eod|.1rc (ni!aa$) 382.Eoorearxo lw6foro 22, 11, 156,

206, 239. 341.Eldtdqoi( 124, 487.Euosio yqaFpri 186, 190. 19,1,209.Elou .ieAo|oFa 241.Enqos tdrvios 64.Et}tuorig 285, 313, 327, 361, 395. 520.

EISPACK 539.

Fibonacci 267, 306. 317, 372.Filippov 365, 533.FORTRAN 32, 541, 548.Fredholm 165, 489.

Gale 491,512.Galerkin 420.Calois 292.Gau$-Jordan 50, 53, 89.

Caus-Seidel 444, 547.Ccshgorin 451.Gilens 400.Golub 449.Gram-Schmidr 192, 200. 204, 210,

229,399, 438.

zllaLxrt l'7 .

HLrora6nro 306.HploqLstr6vo! 380, 396, 522.Hpii(i)Aoc 451.Harvard 125.Heisenbclg 303.H€ssenbels 241, .136. 440.Hoebcn 273.Household$ 209, 435, ,l42.Hurwitz 329.

Oepe),x;)6€q 8e(irprua 109, 120. 160.Oe!€).ri)b'ls lr6l(reoq 103, 158. 352, 519.oerlrd oarordvos (ri!(lras) 62, 229,

378, 386, 406. 522.orra6E (rivsrqs) 316.O€6Qttp(t nrnlimax 482.

I0t0)6ft!rt oqi(or1sa 248, 278.Ibd(Dv. brE lbLosoeqo<.IbLobrd Jop{r 287, 289, 341, 368.Ibroper4 r41ri 429, .131.Ibr6Foe{po( 8. 15, 43, 254.

I6rorrpi 287, 290, 301, 386. 432.

loopoqqopog 235.IooAAonia 130, 407.'Ipos 176. 293, 294. 304, 328.tBM 17.

Ko0aed €ao€rLai ().ior) 286, 289Kd0oroE 155, u9.Kaonndtg 243.K0ti xardaido4 65, 423.K(1!6t0c 6woFruo! 120.Kdvorrc Iptft !x).i.o! 545.Kdvoaas q.llforoq 121.Kaaovag ro! Crmmer 272, 278.Kavovrxdg sEr6o€(- 180. 524.

EY?ETHPIO

Kdvorralt rardoraol 291, 326.Kdvo{r6s nir(ads 347, 363. 368, 370.

Kar.!e!r6p€!o radTruc 116.Kdtu rALyovraos 30, 37.K6oEoFIXalruI 303.K€pboE 508.K).amoq z!z).oq- 120. 123,501.KriFdxo 83.Klqtaxoroi trirczds 89. 9,1Kop6o( 118.Koe1jqrri 454, 461.Komos 16.Katdp'o bnaoDis 468.KurirxoE ,rfvdxaE 353.Kup(rLa1i sEisoot 334.Kna'os dEolos 3,15, 392, 523.Kne|os unonivaru( 396.Kdlos 490.Kamarkllr 452. 455, 473.

Kirchholf 120. 12'1. 161. 495.Krylov 4,11.Kuh!-Tucko 485, 512.

,^'tutrd ro! Schur 359.

Legeddre 208.

LINPACK 539.Lyapunov 329, 395.

Mdyroro 100.Md'{nq aori - f}.dxror! 8rtlloFrl 496.Mdeobo! Jacobi 435, ,t.|4, 547.Mi0oSos QIl 400. 439.Me0odoq rinplex ,155, ,160, 471, 543.MiOoSoS 10! NewLon 434.Mi0oSos ro! dvlLme6E!! 6!rdp.!r!

,+lt.MieoboE i(')! 61Jldfsov ,132, 442.Mtoobos r.)! rf.Tif aoopalo!

MErxxI maqn rzrl 506.M€ar'iri obinor 67. 69. 514.Maq 4rl .V3, s45.MEooF oeos 191, 199, 5,14.M€rdoEot 151, 262. 277.Mtrc0unaoE 26. 29.M€rao1ruarropoc 135.Merario€l1ar 30. 33. 302.M€raronLolt 433, 439.M€rarl1ris cci-qd 188.M€rao 340.Ml dallrrads 311, 314, 4u9.MI brd, ov'oronlo+toq 362, 538.MI rbrcloaqos 15, 16, 44, 53, 82.MI opolorni 86.Mt c'qr6r6aoras (eErirdseE) 9, 82, 17 7.MI retaLt4ldros 92.Mtbevlrn L6ror+lri 292.Mrtbr'rrxo ff 0io! o6rtlon 15, 43, 276.Mrtbe'orcg uroni!(1.l.1c 243,268Mrl6carx6 Fcl 106, 233. 236.Mrtbuvob!!0fo< 374.M&taororeoc 79. 85, 105. 158, 287.Mrtxo; 1s4, 194, 205, 239, 338. 341.Mrr06rr6 €rirFlo 214, 339.MffobLaos cqL0!o( 213. 300, 339.MryabLaog dlL!"yrtg 339, 343, 350.Molob|(fo btdlrJopa 192.Mo!.rbftio< x!x).os 214, 340.

,l/ -oeoozaro!La.i 402.MoAral y(vortivo! .172.Moqqr i Jordtn 355, 364.371,531.Mtr).or 413.Mtr].ok |11lb.! i[,! 243, 268, 499.MrqoorLv6s roiloq 157.Mac Simplex 543.MATLAB 542.Maximin 408, 507. 511.Minim.x .110, 507, 511, 515. 516.MIT 124.

Nopo< rltE obAdr€ro( 396, 198. 404, 413.NopoE tou Newron 331, ,101.Nopos ro! Ohn 126.Ne ib 226. 5,12.

589

'Olaot 248.273.ofu'r6s ll. 14, 41, 83. 249. 275, 388, 521.O8llyonoc brdmcorl 54-1.Otrcvolia 313.Oiod,rta(lol 138.Opaba 71.O[el6ic 167.OBo-ysats 79. 86.'oporoq 355. 358. 361, 368. 532, 536.

oq€oydrlros 155, 206. 350.Oagolo!|o'ioitol 199. 208, 229.Op0o?oilro r6ro6rcwspora 344, 368.Oqgo16lro oqrnl'rtgrpa 159.OaeotdlLoq rilcxos 193, 330. 345. 519.Oa0o16!tos lrn61oeo! 157, 160. 352.Oq0orcllovrzoE 157, 192, 196, 202.363.OaooFovcbKios riv(zoq 347, 351. 360.Ooilouoa (innoq) 51, 247, 251, 260.

262. 264, 293, 499.OAf aeoc'brob|.ll1Jopair(.! 532.O!6ft.0(l.ror(0ric 313, 327.

naifuo 506.tlaao, o{otroiitq 2. 37. 224. 548.ncQ(rroaro,roildl rrt_4r, 275.naed{orotroilol,1ltr 56, 387.fllloedyorotroiitort LU 37. 46. 81.tlaaqo{otroirtol !U 230, 248.n(raa-rovrotroflol QIl 202. 204, 439.

Ilaadrdt,tot 138. 1,10, 370.Iloed]Jdsriff 60 2,18, 273.noad;0.IloyedBlo :1, 2?4.Iloed].],I].or- 4, 18, 88.IloaEf6ohi or be6oF6!a 184.ndaeleid€r6 26, 29.[€bio t+r.ir! 106.[€ied[d 544.Il€fteoofiq drdqopd 61, 327, ,114.ltarcpLopos .156.Il€ercEop6s ro6qxar,163.[rArmAoqi 135, 194. 299.52t.Il€an1ri [€I.ie.dl 277. 279, 548.Il€ic?.orbd 225.llLiro Rayleigh ,107. 412, 418. 43,1.Ili.vaxo! 21.

nivcxaq (liord) 350, 168.Ifvoxas 0 - 1 70, 198, 514.Ilivaaog a{aFfiag 417.Iivara( ald,ti,aofi 145, 190, 435.Ilivaxoq ovdmqoqoc 54. 1?4, 255.ntvoxa( awr Eaprfldv6s 3:U, 350. 374.IivaxoE dEro_rairluoq 53.nivaxoE orimeoqoE 36. 48.nivaroq ovrwq4rerAr,{oq 31, 510.Ilivoxa( aoqLoros 381, 393.nivaxoE SnyoltorotllotpoE 296,

297, 361, 368. 538.ni\,ardc 6ny6!tog 40, 518.nilon0E drcanFoaofi 240.niloror 6raqoa6v 425.I]i\,arac €re€rtxo!- 322. 537.nildkoq .I(rrortr6rog 291, 297.

323.3()2.niluno< Eaprrrs!4 143, 350, 36Lnivozrtq EEor€A|aon ywotdvo! I82. 197.nilazo< 0erlrd oqtsiravog 62. 229,

378. 386, 405.nilara< oerD{oE 316.nivdrds rrr(oQtotr6los 396.nivdroE r6rc6uwo}1riturr 296, 519.nils.ld6 r6|opoeqoE 8, 15, 43, 247.ni!ali.!< rb|or4d! 296.IlilaxdE aalov'roE 347, 363. 367.ni.ldad5 adtdvri).oots 314.IrildrdE xdro rary(wDios 37.ni.!dad6 adebolE 509.nildad! a).+d"ord6 83, 90, 94.tlilara! x!x7'!r6< 353.flfluraq pd(aq 336. 401, a19.IlildldE F€ri.6oo$ ]12.Ili!d7d( tt€idedo€o! 42, ?2, 252.Ilildzoq F! aaqrLroq 311. 31,1.Ili!d"d5 Fd€vo6lvasos 374.Ilirario! p)] bloyolronoutorpoq 290.

291 . 323, 362, 531.I]lvdxd! Ft L6L6Foqrto! 14, 15,

.14, 53, 82.Iliadad! oeeoy.nlLo6 193, 330, 345, 519.flivoro! oQoopolabnfo! 347, 351, 360.[ivc"d! oBootrFdq ]55. 361.Ili.!o,io! n€neadolilor 6Ldqoq6v 61,

260. 425. 450, 546.

ElPETHPIO

nil(rxaE nqo6ol.ic 145, 172, 182.197.291,174.

nivax.r( naoorndroroE 118. 496. 545.nivaxas o€ xa,tt xaldoraol 65. 423.fiivaxas ddTton 151, 295.nilcaos morxsL66x 25. 35. 53-niv(lxu! maoqrk 33, 142, 299.nil(xoq out4r€raolG 56. 338. 361.UivaroE ouvoaoE 295.flivoraq ouvolis 545.nivaxaq ratlroor6iE 40, 60, 5.{0.nivoKs rdEro! dra 11.1, 304.nLvoacs raltorads 25.nlvoxdq rqt/{lrLads 30, 253. 360.ni!cxc\- iAt6roy6ltos 62, 440, 540.[ilaros ftr! ntvr€ o4r€ior 449, 547.niv.rxd< tu'! olJpnaqayovrov 270.nivcras rov oM.l.emdv 21.

niv(rds Hessnberg 241, 435, 440-nivaxas Hilben 69, 207-nfroxu< Iordm 355. 364, 371, 53r.nivaxos Morkov 310. 315. 318. 331.nod6oqarqo 124. 517.nottEa 514.noD'ontraotsop6s 21. 22. 27. 32, 143.nolldl(oLoopos on6 pnqomii 29.no).l...'IltoLft apoE acrd Ftrt'o* 34, 2'76.nouaal(or(1oto9 mrdxo! 22, 25,

26. 27 , 33, 143, 522.no ,or]'ssraolds LasruDsc 485.[o],],arlsoraori< 13. 37, 229. 388, 548.nol).(lrl6qlo 2,18, 29?.nolrxdg s!a'r€ ypa!.E 248, 340.nol,Lxi bLdondot 522-noll(jvllrtG t(q6vos 4?5.nol!6n$o I 1 3.Iloeroaci],I 126.nEiEI 1?, 46, 5r, 63-nE 6),410 6rdiDE 4s6.481.nE 6).nra to! yrirrov 498.II466].tFa tov Fdaqoq6v 458, 503.nao6oli 136, 144, 167, 190. 291.

358, 474.[qo]raeaoxe!0mud< 4'44.nqofftaLercrLali lblorrya 26, 143.nQoo.,rorArotrr6E 29, 33.nQoolpa ior.', r6rofltdrv 396. 400.

nq6nt5 rd,Esoi 285, 324.Il!0dy6eag 154, 196,210. 526.nN{va( 106.

PA = LU M, U.541.PC 1?. 19.PeEon 316-

Pi(sq olq !ovd6cq- 214, 216, 349.Povortav,t (oeLloma) 32a.lR" 73. 99.Rlylcish-Ritz 415, 420.Roulh 329.

S lrts 146, 296.soR 446. 547.xdrFaf lao ql€io 381.507, 511.:Eedr Fourier 195, 2M.:Era.i T.ylor 382.>xdxL 513,516.:zdrL6A(r l5l ,295.:)<td64q lri 4{t3.487.:ra0red xordotclort 20. 60, 311.zrrist\ 422. A7 , 429. 430.:rsqropdlos 238. 477.>roflmrxli 180, 191. 240, 545.>rI),I rqos mil,rl 23. 50, 541.xq),Pq PrrL lAo$dg 33.242,521.:torxP!0orts nLloxug rr, $. rr.:r0d6k')oit 147, 2?4.:uE!"yic 339, 351.>ulryis arciorsoqos 342.:!!vyris xl.ior 4?8, 480.:!F!!'re$.oS iraaas 56, 338.362. lE2.:qrnoeliyov 264.:WtraA{iyov nivdzaq 27r.:qrJrlieru0 I 60.:1lllnlria6l|lo rou Schur 279.:qJruE4rorHe! n€ar€oaraxf! 485.:qnlriE o'l ro! ffraay6lo! 3?9.

384. 412.>wceutq aomou 454.:wdatryr-oqoq'i 417.:wduaopog yeo$rdlv 25, 28.>u.',6u(rop6s oqldrv 23, 28, 76.:u!dl,El 213, 3s4.:ri!6do! 66v5eo 123, 501. 546.

EYPE'flIPIO 591

:l,qtirovo 168, 171, 205.:teterc}lavn 6dor 192.>we€ol 143.:uveizl rou Hall 5m. 5{}{.:!vo6o6 rrildr(( 295.:uvoar0 i m'v9ix't 60. 68.:uvr€ruylGq 6. l9:.:r6X€ o$ao. iTi!(Iz( 271.:rFlaooti( 313.

:qilua oreo.ryu),eriol1r ]3. 65. 426, 522.:xEb6v r€Arob(xrt 331.)trruroflFo 5.

T(|'|o€rbrk nia(a(l< 40. 60, 64, 540.ToieL(ol(l 498. 5m.To).dvrioq 286. 332. 376. 401.Tap).o 466. 169.Tavumu{6c or}rdohopo< 24.TdEI 89. 92. l16. 150. 236, 231, 521.TdEt y0r{rl'd,v = rdEI oq}'6v 107.'IdEr

alo ll.l. 172. 3O4.T(uorLao< ni!('r(S 25.Tdrirox!'ovos 402. 41 1.TaXi( tr€r(oX1tF6rs}6s Fourier 213.

2?1. 3.19, 353, :l49. 543.T,odroq 0!F!lL66Ers ln61oAor 103, 519.Tataey( 'vrx6q 288, 377, 406.Torecy.,vtx,i afto (ri!axs) 304,

390. t9.tTsrAdrbqo I75.T€rarFtilo( 92. 306.T4oi 507.TFIF(IIrt ro).No!!rud< olJv(ledoe!<

{16.T6z0( 305.ToFi ( r"roTdJao!) 212-TeL./rrvrzri dvro6qt( 175, 430.Tar/ovru !i f,(aayo{oioitot 37,41, 84.Terfrlrt6( iilora! 30- 38, 253, 360.TQLb(rt(irlto< 62, .ll0. 5,10Trrros (vrimqoqou (ni!a,{a) 270.Tnroi f|C rou.- oblyoLr 271.

'Ilacaoi sQt0F6s 545, 552.

Tho.p 513.'ftkcy

222.

Vandcmondc 114. 258. 282.von Neunann 65. 316,511.

wilkinson 65. 428.

'Ynaoh l1l.YrrdaeEorl 289.Yrr€Aft Oootiroq|l'!6ro (olrorlFd) 168.Ynte,(uideoorl 444.Yno),orronlE 2. 16, 31. 544.Ynoflir,(ros 237. 275, 385. 396.Yn6Xireos 75. 81.Yr6x(,eo( 0€!€).ni6$ 103. 109. 157, 352.

O(l'rufitx& qpr0po! 213, 3m.odot I x.rr II 4&, 479-oaot(rllni (irivq 444.ooopcrrxo 0€lbqlud 346. 362.Or).axtofalos 216, 513.oloto].oyLx'i owoaroai swExr 419.oum|to6on qo 456.

Xo).deis F.ts6).Wdq 456, 462.Xoa(lxqAlolrxli €Eiooorl 287. 371.XaaortrlqrmLxd noluilvwo 288.XruLx6i 176.X6OoS 73, 80.X(;Aq y0otrtr6v 103, 104. 158, 163,

352,526.X6aoS Hilbcn 205.X6eos oql6v 77. 106, 158, 163.X6Aos owaanioror ?4, 206-

tY€!6o&riceoqos 112, 1t1, u8,

WTprcd!'rroraoqat 226.

ΓΡΑΜΜΙΚΗ ΑΛΓΕΒΡΑ

Documents

Transcript of ΓΡΑΜΜΙΚΗ ΑΛΓΕΒΡΑ