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f-pqFprKn"AAye6po A.O.Morris Ka04y4ti1 tott Topta {on @ecop4nrdt' M a04yaarc6v UniaersituCollegeof WaL...
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Mr6eioqyary4or4
f-pqFprKn"AAye6po A.O.Morris Ka04y4ti1 tott Topta {on @ecop4nrdt' M a04yaarc6v UniaersituCollegeof WaLes
Mer69poo4: A.l. Aepr(rrbr4 Kaqnynqnn na!&ut wiav Kpitt4s
'EK66oerq F.A. flveuporrxoU E 829.2.823 AOHNA
flgolrey6pevcyr,6.rfv'E?')'qvlrrl'Ex0oor1 :ri OuorxoUaoltrarl|(4>Iolrl rot nov€nrorqF[o! KQid]E lx€r Ko].].!€eyrt0€i tl 1664,i 6noio 6Lo xat neeloo6reQo6txd!(iwsId! p6 16 ncQaoForot Xe6vou, r4E pcrdgQsorlg anroqFqvrK6v ouwQoFFdr(l)v, noi €lvar 6oxrFoop5vo l(dt dvayvoQloFcva dr6 nol,trd riavcrrronlpra roi aEor€Qlroi xa( riQooq6Qovrd! yrd riE oro!66E notr etl,orJFevd roQAolF€ ord ndv€rron4F06 Fag. Il|ol€rJoulr€ 6ir y0d Trt qr(l)xi Elr,4vu(latgvlytlI \&=t
\1-t
.
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nc= (iBnyrr\ \
a=l
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^Gq= ( i., / i p*,rr)) \l=t llnx'! \ *=l = I > > @,*9vlwt\ \ r,=tt=t /./,
=
(AB)c
'O
dvayvd)oqE 6poE €d rQ6n€r anlnl6ov vd 660€[ ,rQooqo4,6oov dqoQA ri< ,rQdE.[Enlvdxov, ord dx6rlou8a noQa6€ay'laro
Ol1
\z
r\
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6d,a6i Aa = 0, Al]'d oils 6 A otre 6 B €ivar 0. (ii)AB=AC'}B=C. 6rt1,a&4 A(B - C) = 0 FroQ€[r,6lo{6€rF6A + 0 | NINAKAX
"E!a4,n x n Etua.wst1c lapeic,
lt r o c tr t \1a t a c Kai oopSoli<Etdrpt diae(Ir, . . ., \).
)tytmt 6taytb-
EIAIKOI TYNOI NINA]TNN
nap.1.5
25
3. ANTDTPOOO> IIINAKA: 'A^t A = (d0 efudtZ1/d1nx n r.1,td.t q ror. t'ta1n x n riltdkdg B Mpe 6a erltal A d.rl t\tq, &.!riotpoeoe @t trtvd.tcd, AB=BA-ta. 'O A,,/tttEpoeoe t,/6e d'rana el'lsl poeaiurhe. AHMMA 1.11 AnOAEISH
Toto C laoEd).l,oEdw(otQoqogrot A, x6r€
AC=cA=1" (cI t c = ctn= c(AB\ = (cA)B - tnB- B 'O (pov.r6rK66)Avr[orQoqoErotr A ouF6ol[Eetar A -r . F,6 TadE ntud.KasA roi b(& dwiotQoqo l6'yEtorF1i-l6rdlrov{ dwrorg6qqog rtvcKcE, rr dv 66v E1er.dwiolQogo, l61srat a6r6l0tr t Fi-avtrot86qrpo6 ,tivcKcg. AHMMA 1.12 '4,,/ AftaiB cledt.d..'n@qlvlpol n x n nt|aftq, rbre6 AB et /dr t,nwptwqloc rai (AB)-I = B-rA-r. 16r€AA-r = li = A-'A Ka( AnOAEISH "Av A rot I slvaravlrotQ€1i)[!,or, 8a-'=In=8-'A.T6eo (AB)(r-'A-r)= r, = (8-'A-|XAB) Ka[deo 6 AB €lvardvirorQ&rrpos Ka(aq6oo1,djr6t6 ArlFpq1.11,6dwiorQoq6s ro! €hor l,r.o1a6uBr*di1.1 lqxrl
\A=l
= { : 4Br, l \(=r
/ a._
: (/1B)r, 6n(olv€rrirE, ol oiorxeld)6€lEnivcxeg elvcl ol rpeig anq,tEvo0t6noL ol 6roiol dwrpstadxqporropodE'yQapp6\,4+d4, ri+ rj, orollorlv orot! oror.leldloer.g rt + rl + drJ Avtlororxc:
(i=t,...,n) ", (''", ,)
( r . j= 1 , . . . , n )
TPAMMIKEE E1IZA'EIE KAI IIINAXE'
30
Au@)=
Ktp.I.
J, ;)
6:rou rrioe d),),opf-6oy6\,ro oror)(€io€lval pl66v. AHMMA L16 (i)'Ae ttna 'nx n Ehla*acB pttoptl rA flpoxt'Ver&n6blav , x n Ei,tatd A Leapp6lowactrd otorytto64 p$aoxnparlopb lpdpp'br, roE 6 . B cbat laoC, p.t fr ytrhpdo roAd&lDtoLUou n x m ororvevb\n fl|,td,,.o,lnl dt A, 64)aiil 6a e chal tval o.ot7zub6e4yaaopyntwpg ypappAt,/t6tE B=e(^)-c(t^)A (ii\ K60t otoryei)649 ntuard1 d|dr tvnoryigqlol xai 6 d\,.torpow ,cAee onxet66n fua,.atllar arcutl66nq AnOAEIEH Td (i) r'aiu4 Tliv ttoQtiYQogo,66t0'oupE6lr tndpx€|, Fr6 rftom dn€r.r6vr.o4xcf 6rr arlt{ | dlerrdvrcr; eivcr por'o6rrrl. 'Ay A = ( o ' u ) + ( r r , r : . . . . . r , ) e M , ( t O , 6 t t o o4 = ( a n , OEOPHMA2.1 p6 Anea@' dr...., din) (i = I,... , n\ slvar ot lpaqte @it A, riE SftApxeL
nap.2.2.
n x n oPIZOYZE8
47
ewLDt M\(K)) KEro.AnE (1)'HD tl,nllpapnrfidtt p6sdlrpo.t ptq'ooA,64f,ziirfi i = 1,2, . . . ,n D \ t | . . , . , d \ + r i , , , . , r r ) = d D ( t l , ,, . , ' r . . . . , r n )+ D ( r j ,. . . , r i . . , . , r , ) , (u)'Ae 6u66w)afteypapptcrot A €l1/ lasc,64f,a6lt,rk = ry+| lrA kdno.ok 'ewr(nowk = 1,2,..., n - 1, tbtED(A\ = 0, (III, D(1") = 1,6&00ln ebd 6 tdo&nAie n x n Eivaras,'E D etua.pora6*i 6prop6,tndftt d.e l6*t4a (i). (ri) roi (lli) Kai li"ye&r 6pr df {,/ -Ero., aq(rAF6lo1,ras anaver).rlFF6va riiv (o) 1116 trdyo, naiQvoul.r€ ^(f,.r:.....rn) /\
= ^ l I o r * r * . .. . ) ( = r " , , r " /rI \^=r \.*:
( = I
6nou 16 deQololro6rarp61er. roi< n'r6porg norl ].o]r6dvoworanrrafJlowoErri tr, *:, . . . ,1,, vd rcipvouvrlpaE B€raEi ron t kai roi n. 'ADd, tu6 tt4v (F),,4v Kde€tvd dn6 rd d*,€ivar ioa, A(c(,, dt.. . . . , = 0. -EroL "e") ^(rr,rr,...,r,) ( * , t, : , . . . , r , , 1
nap.22
n x n oPIZOY,EZ
(tr, t:,..., 6rot 16 d0eoroFa 6rarQ4€! 6l'€Eifg pera06o€r.E &,) xdv (1,2,...,tr).'A16n4v(y)xci(d)trnqcEnercr6r!,dv(11,1:,...,1,,)€avorp!{i $€rdo€ol rfi, (1, 2, . . . , tr),r6rE a ( ? r , e h , . . . , e i , , )= t ^ ( ? r , e : , . . . , e , . ) KATEToL A(rL.r:,..., ri) = 0,6ttc0E dficli€ilc! f gva ,rydao4 1 M6 6poro afir1€{Qrltld 6rl(j)E16 r[16ndvco,|rjroQeitd 6€r10€i o,rt D ( r l , r : , . . . , r i ) = > a g o 2 r : .. . d n kD t ( e k ,e k r . . . , e k , ) = ) s e n ( i ( r , & : , . . . , & , t a r r ,a : * . . . . a n r , , 6rrou16,iOQolousdratQ6xsr-riEnlp€xa06o€!6(kr,k:,...,Ii)r(nv(1,2,...,n) r(oi lgn (kr, i(?,., ., t(,,)= (- 1),' 6nou p elvcL6 AQleFdElvqll,aTiE yQappdv ttorl drsLT€iTc[yld vd rdQoup€ r6v rcuroxLK6r[ivqKa Ii dfid x6v fiavcKo (dt,, e*., . . . , er,,).Atr6E6 nlrios ouxvd6iv€rq! odv 6Qro[6Enls6QiEouoqsEv6E fiivqKa 6 ava"yv6orns 0d turoQoto€vd 6E6at6o€! cdr6v r6v fliao ofl1v nsQirroot 'r = 2, 3. 'An6r6v6Qtop6 qE EWE ao42 D, Efi€rcrltr! dv EvcErlivqKcgAEX€rptd or{'E 1,1dn6 $q6EvuIP(6OZ
XQrlo+orordrwoE r6v dwforQoqo roi rsQardvtl) nsQd6Erypo hoD!€
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65
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\z
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l. XQnor[onoirpEq F€0o6oainls tis roQdledqou ./rd vd 6QsiSrots dwlorQoeouE rd,v6Ei0 ixdQX€rt.voqaraparo-.x ralo()< dJorsa,,, - ar < € lrdl,i, n > -\) 3. noL.i a1r6rA inritr€va i,,1ooirvo1,( roi l/,,(R) (x = l) €i!(tL rtrovlotr(nKoi I(irQori
AIAI'JYZMATIKOI XAPOI
Keq.3
( 0 1 ( d , , a , , . . . . d , ' ) l a ,+ a : + a , : 0 ) ( i i )l ( a , . a r . . . . @ J l a ,+ a : + d r = l l (iii) {(d,.d:....,dx)lar
= drdri
( i o )l ( o r .a : . . . . . a , , ) ll a r l > o ) (r) {(d,.ar.....a,,)la,
a: = a, - d:')
4, nor(r dr6 td ttr6p€14odvolq rQqY$arrKdrvouvoQltio€ovf, R - R €'vdl R-x6poLi (t T6 ouvol'o tt)'(l)vldv nQaYFartKdvnoluovtFolv 6a0po1]AkQL66qn' (ii) T6 oivo)'o iil (l)v16v rQqYF(rrrxdvriolucl)vlipov6q0po1l> x -/(r) (iii) T6 ouvolo 61(t)vr6v n€Qlrrdv ouv.rQnio€o)v(6li'alti /(-r) = ytd ii).q rd r € R (ir) Td ouvoi'o ijlov 16v oragoQtolFuv "uvaptio€t'v' C[0' l] €lva! R-Xrneot; roi inooivol'o €n6l'l€1,a 5, nord da6 Id
(t)l.rfe cto,tll/(t) = o) (:10. (ii)Y€ lll/{l) = I I (iii){/€ ct0,ril/t0)=/(l)} (ir) {/€ c[0.l]1.[l/(r.lt= 0] = l} (u){/€ clo.lllJ,lt(t),rt
3,3
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our6vaap6g rob S. O inoxanpocl'nob ) t i - y et r K - y p a p p r k 6 g Eoi) K a r a o # u A o q 4fkt e a p a at . ! al i y E m t t n o l 6 p o q apAyttar( h pi itflo:d)po an6 i)nootw).o S. Oa oupSotri .'rl= 0) ) '= -A) { Ae . l r t ( Rl A
(r,r)
3.4 fptrpprr4
'Avtltrgrloicr, B6oq rca Ardoraorl
'Eoto v &os K-I(nQo
K.e.3
\
l=
dul:=o,
t6 6noio 6iv€r
L
Bj Uj = 0 Kqi 610r16ouvo).o{a1.. . . , y,,,} elvq! yQqpFll(daEoQq[Evo'r{iv{o
016 (. Air6 6p(l)EElva!drono. T.0Qaino06rouF€6rr t6 oivol,o {vr,..., v,,) Kai 16 ofvolo \U,.....U.1 €lvol l(-6do€tEroi y. O€(l)Q6vroE 16 {yr. . . . , U,,,}odv EvalQaF[lrd av€€defl]ro 6noo{vol,o or(i nori nsQd.t€rat dn6 rd {u1,. . . , u"l. 6 ]116n6vo aq(uQLo$6S X{irQo b€i fl 6ir m = r. Oporo. 0€roQ6vrog 16 otvo).o {u , . . . , u,} oriv tvc .ypcgprKdaveEdottlro iitoolivolo.n6 Xrirporoi nsQriy€rqrdtr6 16 lyr..... y_), taeror iirr n = t|r.:ljverds Elor,Fsrn = n, itno)garaw€iiar. f T6 esd)Qlpo oir6 LrdgantrQ6n€!vd d6ooup€ r6v aK6l,ou€o ol[ovrlx6 6qrop6. OPI:MO> 3.f 4 'O 6plepl)g t6,t' otolxeiay l!t61 K-Bao4Cyrd tlay nu.tpaotttrla nopo,y6p€1o K-XaDpo V lty.&, 6rAorao4 .oi, V nara 06 K,ra.i o u p S o T t | ep t ut rd l t n v i ( V : K \ . ' O V ) . l t e r a t6 t a y u o p a t r t e 6 c X A p o s E e n t p a o p 6 \ tn 9 6 L d t o t d o 4 c . TIAPAAEITMATA 'EE€rdlovroE nQooeKflKdrd ropad€tyForo no6 660nKavnqiv dfl6 t6 naQqJrdvo 0€drelFo,6l,6noup€dr! (0 (Vr(R) R ) = :
(i0 (v,,(() (iit (%,(c) R ) = 2 n (i,) (M,(K)
(,) (M,(c) R) = 2nr. T6 afl6pevo8€d,pqpddeifv€r 6fl irdQx€r lid K-6doI Kai rdur6xQovabiv€r [.ri nea|Clrri p60o6o KarqoK€l1 drrr > BJrj+ ) j=l
r=l
rlvl=o
*= I
T6rr Exoup€
) a r u r +X Q t l = - 2
i=l
J=t
*=l
7 1 9 t € wnr w r
'Eg6oov16 ouvoLo {ur,..., u') efu(r!F!6 K-6dortroi wr n wr, tttdQlouv 6r€K(i= l,..., r)r€rord ror€
X v*ur- ) 6,,,
A= I
l= I
'Ee6oov
r6Qa 16 orivoXo{ur,..., u', yr..... yrl €lvar pld K-6doI roi w: (kdt oo!€rrirEyQapplKd dv€EdQrqrondlo ot6 K), trp6'r€t7r =... = 7r, = 6r =
"Opo!o, prloQ€ivd!dno6€[X8€i6iI
B r= p . = . . . = B n = 0 "AQaExoup€ !
o1r1- 6
t6 odvo)'o 166noio6iv€!dr = 0 (i = l,.... r). aqdoov {ur..... u"l €lvaLpui X-6doI roi Wr n w:. Aqlq6rl, ar=0(i= 1,....r),
Bj=00= I....,n),
Y*=0(l=1...,,n) dv€EaQqro ndvo 016K.'AQa16 166rioiodno6€rKvri€r 6r! 16s €lva!yQspFLKd T S slvorpld K-6doI roi wr + w:. ff otr6 16e€d,QqFa 6ivoop€x6 6EnsnaQdd€r'tpo: NAPAAEITMA -Eoro wr l(ai wr oi 6EnE i'roxdeor ron vr(R). lvr = {(a. P. r. a)la = F). 6:2p\. wr = 1(a,p.7.d)l+ dB:.y (( l, 1.0,0,),(0.0.1.0).(0.0.0.I )l Eivot 6rr16ouvo,lo Etr(olodrrodELKvriErar R-6dot ron wr, [rd 16ouvolo1(l, 0, l. 0). (0, I, I, 2)l €ivaLsrd R-6doI roi lvr. 'y€."vti€t r6v T6 ouvo)'o { ( 1.1,0.0),(0.0, 1.0).(0.0.0.l ). ( 1.0.1.0).(0.I. I . 2)|
AIANYZMAIIKOI XOPOI
Keg.3
wr + Wr ndvo ot6 R, d),l.dFxouu€
t . 0 J+ - 2 ( 0 . o 0 . rt )( t . r . 0 . 0-) { r .0 .r. 0 )k a ri q d o o1v6 l o - ' l ' , 1 1 t-.'0( 1 . 00) :. ( 0 . 0 . (0.0.0.
t.0). l ) . ( 1 . 0 . t . 0 ) l s l v a ,y s a F u r x d 1':nottt. dvfEdQrlrofidvo ord R qrrt6dnorrl.ri urciR_bdolroi Wr W-. -Av(d.8. y.6) € lyr o ry,,16r€ B:A
29=a a 9 aw , l - { ( d .a . 2 a . 2 a } ' a€ R l . n p o e q v ( nt 6s o i v o } . o{ ( r , I . 2 . 2 ) r flvcr Frcf wR-baort rot Wr n ry:. :nFcrovouF€6rr (wr : R) = l, (W,, nt = Z. (lvr + w,: R) = 4 |(ai(wr n w:: R) = t. rd 6noio 6€ixv€r tir! ( r y r: R ) + ( W : :R ) = ( w r + w : : R ) + ( w r n r y r : R ) . 'Ao|(1ioslg 3.4 l. Nd 6€lxr€itjrL rd ouvo),o {( l. (,..]).(5.2, t). ,(i) ov | €rvorrvq6 ('raBrQ6S {0. I, 6)I rlvqrurdR_6dqron v,(R), (tr, neqytarr,(dgd0r€86;xdi /.tr(x)= t. !,{:r) = = x + r.(,{r) - { r - r)..16rr. t6 oJvolola,,{-, I{,) €lvartrd R-bdolroni:{R). 2. Nd anobuxrr.i 6rr 16 orvol.ottr. :. ot.'(n.S. ,r. i _,. L l,i ."f,",'*,li *l -.", 6e€eonvot ouvrrroyra1,r:roi {r). r-r. r 7) roi ron .r:o 1,( :{lT-,,r, (,): ."d ru. fiQoE ourrtrrt6don. 3. 0) Nd.-b€rxrsiijir 16 oivoLo {(3 + \D, I + \/2.). (j, | + 2\fa} €tvar -./aaUFL(6 aEqQqp6vo nd\,(o016R d).}d./Qqp[r*aa".gapir1."na",.iO q.
lii) Nd6axr€i6rr16on,nu {r r _ i. r. iz, _ r .
;;",
r;.;I,;;E;; ptvo ndr,o) ot6 C aLld €lvar yQaF[u(d av€Eaor4to D} novo ot6 R.
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rsi'r) kai16o{vo}o t orvt}'crva, {e"r1p
5. Nd drjrtr€i6tr 16 inoouvoto {(t,0, l. I,), (1, O, 2. 4)} ron vr(R) eivqr l,Qo[lrrd dv€E{iQfl]rondvo 016 R rqt vd an€jX. 1). Td 6io rQri,ia dr6 id 6r6Fsra noQa6etlFdra ostxvolJv 6tt 11iwor,o roi yeaF$rxoi peraomlarlolro0, dn6 6 o 6norc6rlrolE 6LawoparlK6 td,Qo o' lvcv dr,tro 6rovrio$drrx6 llnQo, elv@ Fr-6y€v(x€lrol 6e@p€vov noti lvoordrv
96
f PAM MIKOI MET A'XH MAT I 2M OI
Ktp.4
"FEroolnFarrqp(iv" 016 6r6rdoraro X6eo V{R), fu}.a6r1rlE ,r€erorQogtg yiQo dr6 &crol!.io xa( ri6 ouFpsrQtdg 65 np6Eptd €toe(qyQoFpt. NAPAAEITMATA l. Eoror L trrldr erleeio{6noia6r6Qx€ra[ dir6tivaQXi xai Col(]Tt drErK6vlql tl6noiaor6l,v€r rdo€drdurola 016oullrsterx66t|i!!oFq 6Ene6sniv€rioE(s L.
:xiPd 2
IP AM MI KOI ME TA' XTTMAT TZMOI
nop-4-l
2. "Eoro R i r€eroreoq{ yuQodri6 r{v AgjOttlv dl6vw€]trig, inaQx€. Faa I - 1 dwr0lorxad F€raErl roi ouv6Lou r6v lQapprrdv
IIap.4.2
III NA] a u x J (yr....,y.) = )
(i=1,...,'n)
j(ay,a1,...,a,,y)
:uv€n6! 16 (0r. . . . , yD) elvcr Evogypcgprr6Eou!6udoF6Etd,v orql,6v roi' A Koi dQaoi mrli,€Erot A yE1'\dv€fliv irrtT ndvo oi6 K. Aii6 orlFdiv€r6116de}r6E ro0 T = 6agp6sro[ A drEfiQ6EraEoritr€E,{inoE dnaw€ira!. :d! n6QloFo dr)ton ron 0€(t)Qrtlrdrog F oQonF€r6qc v{i dro6eiloupr q(€dgo!'ror 6QloF€v€E neordo€rEnof Fa 16 6o0F6 6v66ravaKa d,SnQ6EriE oril,€E. I
TTOPI>MA (i) ?vda nivaKaeA e Mn(K\el,,ta.r dwoetvrpos AyKatpbyoAy6 (be 6a0y69toa A np6edc 6nlae e|val ioocpt n. (ii)'At A e M,t(K\ eiwr Elac a1'rwtptvrpoc nitn6dc' tAtt ot 6a0poi ni,t AB, BA Kai B 69 pq,ia oq)2e, dval llxr\ 6nov B xtuar t,ta4,rufrr E rdka4 pi n vpapp4. 'H d,r66erEtron (ii) yiv€raL6do€rroa O€oe{Faro!4,12. 'Enmliov F roQotF€a' anod€iEoutr€ @EQPHMA4.l9 Oi ypappoladinapar ntuarEcEpvlt rolt i6a Bagp6dtqftpos ric or1l4. AnOAEIAH 'Av 6 B €ivor yQagFoloo6trvapoE ne6g r6v A, 16r€,dn'16
TPAMMIKOI METAZXH MAT I'MO I
Kcq.4
nivak€g tr, Er, . . . , tt rEroroL(ior€ O€6Qnpo1.17,rlndQ)(ouvoror.X€ld)6€rE B=Er4r...EtA 'A116 oi oror)(€.66€rsn(vcxq elvcLdvrrorqfqrpor nivaK€sKqi iror, dn6 rrl! (ii) nr6 ndrro, EXou[€ 6deF6Eron B drEnQ6ErigoriL€q = 6aep60&u+0.
'H dn66€ql ANOAEI3H €fusr €trKolq, naQo6€ayparoE XdQ{, dv t'=(a,p,f'),u=(a ,B,'t ) 16rE (Iu, '.,) = Iaa' + IBB' i Ayy-
+ BB'+ w', = r(v, u) 16 6noaoAro6€rKvri€rrd (,i).'E raolElq(nsr d? + B'?+ 7? > 0 p6 a, B, 7 € R dv xoi |.6vo dv 6vo rolldxroro! &ri rri a, I i I €lvor brdgoQo ron ptbEv6!, I hl,a6{ (a, B, ?) + 0, rd 6roio dro6errv6€r t6 (iD). "Otrog ot6 K€qdldlo lll,6aqroniKol€ 06 W(l)orEEEworcEyLd vd ddroorpe xar6rlv r6v 6Qrop6r(iv 6[cvoo[crrK6v X6Q(tl!,610r '(r C6d, oriv En6p€vyl rraQdyQago,0'i oqQr)(rotue 016 nL6ndr(l) 0e6Qnpayrd vd 6eloorF€ aorDx€QF rd yrvdpevcod 6noLov6rlnor€6rowoFarrK6 XdrQo. @d Fno€loriooF€vd o€(oelioooF€6rawopdrrr(oriE X6QouEaqobtaou€vooE 116 6va oupl,tEtQ[k6eortlrsQlK6yw6FEvo, 6l^o6i Evo ao(l)r€prx6 yrv6u€vo noi! vd iKavonot€irig l6lt5rlftE (i), (ii) kaa(iii) ron O€(oeipaxor5.1, dil.'t i l66qra (io) { 6nofc eivcr mevd ouv6€6+Evrl p6 16 prl(oE 6rawoBarog, niv drr6orool p€raEf olp€iov Ka[ trl yovia prrclr3 66o 6rqvuo[61(!)v d6v FnoQ€i vd y6!tkb!0ei rdvo 06 6ron6{nor€ o{iFdrc. ff crlr6 16 }d1o, on4v 6n6u€vrl tctQdyQago,06n€QroQidouttE fl4vrlQooolli pogoa blavuoFarlKois X6QouEnod 6Q[Eowa0trdvo 016 o6Fatc r6v neoyForu(6v raa pryadtK(i)v dQrou(nv.
t36
XAPOI ME EEOTEPIKO TINOMENO
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'AaICio€q5.1 q€6l'rl6nvuopdtov€lvctrrioeroperoli roug; l. IIod tur6-rd6n6F€va (i) (2, -1, l) Kol(1,2, 1) ( , i ) ( 2 ,I , - 3 ) K o i( 1 ,1 , 1 ) (iit) (?,s,3) xa{ (1,-2, 1) rd 6tav6oFaro(2, -1,2) Kot(1, - 1, 2). 2. Nd 6Q€e€i drdwopa Kdo€rorrQ6E 3. Nd 6es0oivrd Firn rrirvi!r|E 6rcvrogdtov ( 0 ( r , 2 ,1 ) , ( i 0 ( 3 , - 2 , 5 ) , ( i i i ) ( 1 , 0 , - l ) . 4. Nd 6Q€0€[ r(i)v6EiEE€uy6v6[ovooFd-r(0v i l(Dv(aL.€laEtr ( t ( 3 ,- 2 , l ) K o i ( 1 ,- 1 , 1 ) ( i t ( 2 ,l , - l ) xai (1,0,2).
5.2 Eir?,ei0rorrca UnitaryXdrQol :6 61,1atTi fllv raQdyQoqo,F6K 0'i ou$6o1,i!ooF€ €lt€ i6 o(ipo rdlv tiQa'fFortpt'/o6!kdv dQrolui)lc Koi pEv Bd oop6olG Kdv dQro[dv R €ir€ 16 od)Fctdv Eo!'F€hav K-XdJQo. OPI:MO:5.2 "Eta toattprr6 ! r v 6 p 3 1 t o L n i t u aVv E l l t apt w d . E t r xbvro4 i 6notu Anmoryti ot KAee6rawwtltivo leiryoe &avuopha'r u, v e V pi:yeeos(r,r) e Kto 6rro:to"e$ads$4c t&6qres: Ew Aa9p@roapfipnn. ( i ) ( r + ' u ,u ) = ( l 1 . r )+ ( @ , r ' ) , r w 6 t a t u r . u , u e v (it\ (au,r') = a(u. u\.ym6tatu u, u e v, a e K ( i i i ) ( u , a )= ( d D , r d 6 ) a f t . t . r e v (io) (u,u)>06ru+0. i)nov i na a i66 bno6errvtet 6n gEapoupx 16 prya5rx6 oulvyie,. "Evo.s, 'wvvopaanc6s xbpoe V teotnoptvos p' Eva ioateptr6 ywbpevo 6vop6.1aar. y | 1 t 6 p e e o . I 6 u i e p a , E l a sR - I r ' p o sp t pt naarEptn6 xdpos yr"r6pevo My!'ar Ebx)Lei6ros to@rtprk6 Xapo s ka.i twe C-XA)poc t l i i o u t t p t $ ' y r v 6 q v o) i y a a t u ^ i t a r g I d p o c . nefa 0€oeloorp€ naeab€iypdrd, oi[€r6voup€ 6r! oi (i), (ii) Kqa(iii) 6ivouv (au + Bu, t) = d(u. ut + B@, v) - a!.r\ + F@, u) (u. o.tr+ BlLo) yld 6).a16 a, P e K, !, u, $ e V, aq6oov
nop.52
EYKJXIAIOI KN UNITAAYXAPOI
(au + Bb, u\ : (du, t) + (Btt. u\ : d(u, u) + BQ,u)
117
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Kai
(u, o.1, + pu) = (,a1;+FF6l;1
= ,) + B-('LD "1,, = aQ,u)+ EQ.a)
a'r6 Tl4v(ii,) an6 rrlv nqQarrdvo) dn6 qv (tii)
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d , ) , u = ( B r 8 , , . . . , B h )e v , ( R ) , 6 Q i o o u r 1 r r6
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r6x€ €tKorla driod€iKlt€rc! 6rt ddrd €hat Evo loorr€QiKdy[v6p$o arri ro1] VIR) xd{ deo 6 yi(R) €lva! b/aE EiK},€i6roEX6QoE.Air6 €lvar F!6 gooLKri ylvop6vouaJI[rot V3(R),16 6roio dvde6Q0nK€ T€viK€uoIrorl ao@r€QlKoir orli! traQdyqaqo1 Kaa0d 6voF6!€xoroufll0roF6vo (sfa4d4r.l)ao{Di€QrK6 1lv6[Evo
antron vn(R).
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(rrt l(,,,)l< il,llll,ll
( o ) l l u+ D l l = l l , ,+l l l l t , l l . AnoaEI3H (i) llaull= (au,cu)'/'!
= ("a(,,,))"' =l"lll'll (/l) oiri rQoKlnr€rapcooEdn6 fllv (tu)ron 'OQraporl 5,2. (lil) dv r = 0, r6rE xd{ id 6fo p6},1€lv(lt pI6C! Kai ouv€r(ig loIriEL. ' A v u + o , o f r o u u € e= ' - j l j - , , . 5 . . . llull'
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nap.52
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