View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector 69, 173-187 (1992) LANRUOJ FO NOITAMIXORPPA YROEHT Strictly dna Uniformly enotonoM zcilrO-kaleisuM secapS dna snoitacilppA ot tseB Appr~ximatiofl W. CRUK etutitsnI fo ,scitamehtaM ytisrevinit fo ,irazizoP Pommi, dnaloP detacitrumnoC yb naiilr sukniP devieceR enuJ 21, 1990; desiver lirpA 30, 1991 In siht repap lareves yticinotonom seitreporp fo eht grubmexuL mron ni zcilrO-kaleisuM secaps )&L dna )p(,E revo cimotanon erusaem secaps era deziretcarahc ni smret fo eht noitcnuf 4. roF )g(,L ti si devorp taht lla eseht seitreporp edicnioc htiw eht etulosba ytiunitnoc fo eht mron dna ( > 0. emoS snoitacilppa ot tseb noitamixorppa era ,nevig neve rof lareneg hcanaB .secittal a‘ 1992 cimedacA ,sserP .cnI .1 SEIRANIMILERP teL X eb a hcanaB ecittal htiw a ecittal mron .jI.Ij gniwolloF ,2[ .pahC ,]VX eht mron .11 /I si dias ot eb ylmrofinu enotonom MU( j, fi ref yreve E > 0 ereht stsixe )E(U > 0 hcus taht -fII jlg > 1 + )s(q revenehw fr g E +X evitisop( enoc ni ,)X II l/f = ,I dna /I jlg 3 .E We lliw llac hcus a ecaps a MU ecaps ”BMU“( ecaps ni .)]2[ We yas taht eht mron si yltcirts enotonom )MTS( fi +flI llg > fI/ I/ rof lla evitagennon f; g XE htiw jl >llg .0 nI siht esac ew llac X na MTS .ecaps We lliw osla yas taht X sah eht MU ro MTS ,ytreporp .ylevitcepser nA tnelaviuqe noitidnoc rof X ot eb na MTS ecaps si taht --f/l )lg < /lflI afrevenehw O>g dna .O#g NOITISOPORP .1.1 ehT gniwollof stnemetats era .tnelaviuqe )a( X si a MU .ecaps )b( roF yreve O>E ereht stsixe a EM ,0( )1 hcus taht -fiI //g < 1 - )c(b revenehw f > g 2 ,0 II f II = ,1 dna llglj > .E )c( roF lla evitagennon secneuqes ,),,f( ),g( ni X gnijfsitas j/ ilxf, = f dna j/ ,f + ll,g + 1 ereht sdloh 11 l/,g +- .0 )d( roF lla secneuqes ,),,f( ),g( ni X gniyfsitas I/ lil,f = ,2 ,f 2 l,g 2 ,0 dira II ,f - /l,g +- 1 ereht sdloh lj ,g jI +- .0 ehT snoitidnoc )a( dna )b( nac eb desserpxe ni smret of a suludom of mrofinu .yticinotonom ylemaN X si a MU ecaps ylesicerp nehw )8(~ = 371 29/5409-lao $3.00 thgirypoC ,% 1992 gk cimedacA .sserP .cnI 114, sthgir of noitcudarper ni yna mtof ixvreser 471 CRUK.W +fll(fni llg - :1 J ,O>g llflI = ,1 /lgl( O> ,ro ,yltnelaviuqe =)s(6 )~2 ,Oag>f:llg--fII-l(fni ,l=/If\l ,O>}s>llgll ebnactI.)l,O(Eserehw deifirev taht rof hcus eht gniwollof seitilauqeni dloh :eurt E lE@ ~~4(E)b6(~)/(1--6(~)). )&(6-l ,ylraelC hcae MU ecaps si na MTS .ecaps nI MU secaps eht mron si redro suounitnoc EX( ;)A( 0 Q ,x OJ seilpmi liaxll ( +- )0 dna yllacinotonom etelpmoc EX( ;)B( 0 .J,x< dna ,pus Jx]l < oc+ ylpmi taht ,pus ,)XE,x ,Z[ .pahC ,VX meroehT .122 nI rehto sdrow X si a BK ecaps [ ,21 .pahC ,X .tceS .14.4 teL su tniop tuo taht X si a BK ecaps fi dna ylno fi ti si redro suounitnoc dna yllacinotonom etelpmoc rof secneuqes .ylno roF ,elpmaxe secaps-,L htiw 1 < p < cc+ era MU ,secaps tub eht ecaps ,L si ton neve na MTS .ecaps ehT melborp si woh ot hsiugnitsid MTS ro MU zcilrO-kaleisuM .secaps ehT remrof stluser gninrecnoc eht noitaziretcarahc of eht MTS dna eht MU ytreporp ni zcilrO secaps nac eb dnuof ni [ ,1 meroehT ;4.4 ,4 meroehT ;14 ees osla ,S[ meroehT .143 nI [ ]4,1 emos rehto yticinotonom seitreporp era .deziretcarahc nI ,4[ 15 eht esac of eht zcilrO mron si osla .deredisnoc ehT tsrif mia of siht repap si ot wohs taht a zcilrO-kaleisuM ecaps revo cimotanon erusaem ecaps dewodne htiw eht grubmexuL mron si rehtie a MU dna ecneh na MTS ,ecaps ro ti si ton neve na MTS .ecaps nI eht esac of ylerup cimota erusaem secaps ereht tsixe MTS zcilrO-kaleisuM secaps hcihw era ton MU secaps .]91[ ruO hcaorppa si aiv snoitaziretcarahc of MU dna MTS zcilrO-kaleisuM secaps )p(,L ni smret of eht raludom (+I ). dna neht ni smret of eht noitcnuf 4 .ylno ecneH ti ,swollof ni eht cimotanon ,esac taht zcilrO-kaleisuM secaps era MU secaps :yltnelaviuqe( MTS )secaps ylesicerp nehw 0~4 dna yeht evah redro suounitnoc )ecittal( .mron ruO dnoces mia si ot ylppa eseht stluser ot emos tseb noitamixorppa .smelborp ehT MU dna MTS seitreporp nac eb deweiv sa eht mrofinu ytidnutor )RU( dna eht ytidnutor )R( detcirtser ot eht evitisop enoc +X ni ,X -cepser .ylevit nI ygolana ot etaidemretni ytidnutor seitreporp rof RU dna R (WUR, CWUR, ,RUL ;RH cf. ,6[ ,32 )141 emos etaidemretni seitreporp rof MU dna MTS era denifed .woleb ehT noitaler neewteb eht MU dna RU ytreporp sa llew sa neewteb eht MTS dna R ytreporp swollof yletaidemmi from snoitisoporP ,2.1 ,3.1 dna eht snoitinifed .evoba NOITISOPORP .2.1 ehT gniwollof stnemetats era .tnelaviuqe )a( X si a RU ,ecaps ,.e.i rof hcae E > 0 ereht stsixe a )c(8 E ,0( )1 hcus taht f(lI + 112/)g < 1 )E(?C- revenehw II f )II llgll < 1 dna II f - l/g > .E ZCILR4KALEISUM SECAPS 17.5 :fqorP )a( * .)b( fI )b( si ton deifsitas neht ereht stsixe na ls > 0 hcus taht rof yreve ey E ,0( )1 eno nac dnif f, g E X gniyfsitas j/f j/ = ,I !lgil > la il)~+1(/1s2~ htiW siht ~E dna O>)rs&=~ from )a( ti swollof taht -f(l/ /12/)+f+ = ,),s(d-l>))~~(d+l(/l a .noitcidartnoc )b( 3 .)a( fI ,ton tuohtiw ssol of ,ytilareneg ereht tsixe 1a. > 0 dna secneuqes ,),f( ),g( of eno-mron snoitcnuf hcus taht -,,fI/ /l,g ,> ;E dna ,a = jl ,f + 2/ll,,g > 1 - .n/l teL ,h = ,,f( i .),,a2(/),,g nehT li,hlI = 1 dna ,,YL + .1 roF ,z = ,f( - ),cc2(/),g ew evah I/!,z/I .2/*&a gniylppA )b( htiw 2/i&=& ereht stsixe )s(8 O> hcus taht rof ,k dna ,z nevig ,evoba ew evah Fxam ,,h/l T jI,z > 1 + .)s(8 nO eht rehto dnah ,Z/II //,z- = ,!o/l +- 1 dna siht noitcidartnoc sehsinif eht .foorp Remark. gnitoneD eht suludom of mrofinu yticinotonom yb dna )E(+II eht suludom of mrofinu ytidnutor gnitluser from )b( yb )e(q ti swollof taht .E<)E(+~<)E(~ // f I! revenehw lihl~ > .0 ehT elpmis foorp si .dettimo gnitcelloC lla eseht stcaf ti swollof taht =RU MU .fc( ,2[ elpmaxE ,4 .pahC ,VX ,)]41 MU + ,MTS >=RU R * .MTS We llac a hcanaB ecittal X yllacol ylmrofinu enotonom )MUL( fi htiw)NEz~(O~,g>f IIfII=l, ylpmil+j/,g-fII otdiassiX.O+ll,grC eb ylkaew ylmrofinu enotonom ,)MUW( fi rof hcae evitisop lanoitcnuf lj,f I E X* eht( hcanaB laud ot )X dna lla secneuqes ,f > ,g > 0 htiw // = 1 eht noitidnoc jl ,f - llirg +- 1 seilpmi ),g(1 +- .0 We llac X ylkaew ylmrofinu enotonom ni eht dnoces esnes )MUWC( .fc( eht CWUR ytreporp ni [ :)141 fi rof f,, B ,g 2 0 htiw 11 II,,f = 1 dna evitisop slanoitcnuf I E )*X(S eht( tinu erehps ni )*X eht noitidnoc ,,f(Z - ),g +- 1 seilpmi taht I/ /j,,g +- .0 -azilacoL snoit of eseht owt seitreporp ew( redisnoc tnatsnoc secneuqes ,,f )f= dael ot eht stpecnoc of WLUM dna CWLUM ,secaps ylevitcepser ees( ,6[ :32 ]41 rof eht evitcepser ytidnutor seitreporp dna rehtruf .)secnerefer ,yllaniF ew yas taht a hcanaB ecittal X sessessop eht +H ytreporp na( ygolana ot eht zseiR-nodaR ytreporp ;H ,.g.e ,32[ ,)]41 if.f b ,f > ,0 j/f j/ = ,I dna eht 671 W. CRUK kaew ecnegrevnoc f, +-- f ylpmi eht fo’ecnegrevnoc 11 f - f,ll ot .orez ecniS X si a hcanaB ,ecittal ni eseht snoitinifed eno nac laed yltnelaviuqe htiw eht slanoitcmif 1 from eht elohw ecaps X* ro ,)*X(S .ylevitcepser ehT gniwollof snoitacilpmi era :tnedive MU 3 MUL * WLUM, MUL *CWLUM + +H dna MTS ,.e.i( ,)MTS+H MU * WUM * WLUM + ,MTS MU -= CWUM * .MULWC We evael eht yduts of eseht seitreporp rof hcanaB secittal ot rehtona .noisacco teL su tniop tuo ylno taht eht +H ytreporp of hcanaB secittal syawla sdleiy eht redro ytiunitnoc ni .X ,revoeroM a hcanaB ecittal X si a CWLUM ecaps ylesicerp nehw ti si na MTS ecaps htiw na redro suounitnoc .mron teL ,T( ,C )p eb a ,etinif-o etelpmoc ,)laivirt-non( evitisop erusaem ecaps dna d(r, :)t R + x T + ,1 eb a noitcnuf hcus taht rof .e.a-LF t E ,T ,O& )t = ,0 (d ,. )t si laivirt-non suounitnoc( ta orez htiw oreznon ,)seulav ,xevnoc dna .csl revoeroM tel ,r(Sq ). eb ,elbarusaem rof lla r > .0 zcilrO-kaleisuM secaps )p(+Z ,12[ ,51 ,61 17 tsisnoc of lla elbarusaem-p snoitcnuf j t-T ?i hcus taht a&.J=)fro(,Z t(fI <&)t cc+ rof emos 0>10 gnidneped( no f .) nehW dewodne htiw eht grubmexuL mron .11 dII ti semoceb a hcanaB ecittal rednu eht larutan gniredro ,.g.e( ,)]12,42[ erehw 11 f dII = >l{fni :0 )A/fJZ < 1 .} hT e f noitcnu 4 si dias ot yfsitas a ,A noitidnoc 4( E )>A fi ereht tsixe a tes oT of orez ,erusaem a tnatsnoc >K ,0 dna na elbargetni -non( )evitagen noitcnuf ,h hcus taht rof lla t E ,,T\T dna r > 0 ereht sdloh ,r@ )t d ,r(dK )t + .)t(h nI eht gniwollof ew lliw ,etirw rof ,trohs 4 O> ro 4 < ,oc+ fi rof .e.a-p r E T eht noitcnuf (d ,. )t si yltcirts evitisop tpecxe( )orez ro semussa etinif seulav ,ylno .ylevitcepser teL )p(;L = {f e :)p(,L fI I 2 ,f OJ * /I ll,f d }OJ eb a ecapsbus of snoitcnuf htiw redro suounitnoc mron dna =)p(,E fi E :)L&q- )fu(,Z < oc rof lla xc > .}O nehT )p(,E c )p(;L c )&L sa desolc slaedi ees( ,42[ .p ;71 ;21 .)152 fI 4 < a+ neht )p(,E si repus redro esned ni )p(,L dna )p(:L )&E= ,42[ .p .191 )u&L sah na redro suounitnoc mron ylesicerp nehw )p(,L .)@L= ylraelC eht mron ni )p(,E si redro .suounitnoc ecniS 4 si suounitnoc ta ,orez )L&L si elbasopmoced .fc( ]7[ rof .)yhpargoilbib ,oslA fi 4 < oo+ neht )&E si elbasopmoced sa .llew nI ralucitrap siht snaem taht ereht stsixe na gnisaercni ylimaf },,T{ c C of stes of etinif erusaem htiw ,Jl ,,T P= T hcus taht ,1 )F(,LE ro ,1 E ,)&E ,ylevitcepser rof lla n E N ees( [ ,21 .pahC ,VI .tceS .)13 We llac eht raludom &,Z ). ylmrofinu& enotonom ylmrofinu( )enotonom fi rof hcae E > 0 ereht stsixe na )E(]Y > 0 hcus taht f 2 g > 0 ni )u&L htiw )fJZ = 1 dna )g(,Z > E .pser( htiw II f cII = 1 dna @llg/l 2 )E ylpmi taht fJZ - )g Q 1 .)E(]Y- oslA (,Z ). si dias ot eb yltcirts& enotonom yltcirts( ,)enotonom fi f 2 g 2 0 ni )p(,L htiw )f(,Z = 1 dna )g(bZ > 0 .pser( 11 f I) o = 1 dna II ,/lg > )0 ylpmi fJZ - )g < .1 We lliw yltrohs etirw taht JZ )= si ,MU& ,MU ,MTS& dna MTS ,raludom .ylevitcepser ylraelC hcae ZCILRO-KALEISUM SECAPS MU raludom si MU& dna erofereht .MTS& oslA hcae MTS raiudom (41 ’ ) si .MTS-$ ,revoeroM ew evah eht gniwollof .noitisoporp NOITISOPORP .4.1 ehT raludom JI ). si s?)awfa .MU%q y!ineuqeszroC eht snoiton fo MUbq dna MTSbq sraludom edicnioc .)yrartibra-L/( .foorP teL E > 0 eb .yrartibra oT evorp eht tsrif trap of eht noitisoporp ew llahs evorp taht )E(S = ~)2/~( si a doog .eciohc teL f 3 g 2 0 eb hcus taht =)f(,I 1 dna .&>)g(,I teL =A et( :T .}.Z/)t(“@d)r(g nehT =)~JIQE )A?.Tlg&I+).lgJI ylpmi .@>I.,-,lg(& ,txeN 3)5-f&~-)f&~ 4df 1 A.’T )g-f(($t-, ).,..I 3 )A.7ff(~J)2'!&-1(-),~.lf(,r 2 x!Q%J( lg(,I T )A 3 2)w = ,)E& jt(f)2/,&-l(<)t(g-)t(fecnis no ,A\‘T dna eht tsrif noitressa si .devorp ehT dnoces eno swollof yletaidemmi ecnis hcae MU& raludom si .MTS& NOITISOPORP 1.5. fI )L&L si na MTS ecaps neht 5q > .0 ehT emas si eurt rof )p(,E reitezrehw 5q < .cc+ foorP We deecorp yb a .noitcidartnoc teL CA { Et :T )O>r(3 ,r& )t = 0 j, )A(p > ,0 dna =B .A\T ecniS T si ton na mota ew nac emussa taht )B(u, > .0 erehT stsixe a elbarusaem rotceles )r(g E )t(I = Y{ > :0 ,r(d )t = ,iO erehw I E .A ecniS j&L si elbasopmoced tuohtiw ssol of ytilareneg ew nac emussa taht ,lg E .)p(,L teL h E )p(,L eb hcus taht l=,ll,lhlI dna .O>h enifeD .Alg+Blh=f ,OZ.lgg>fylraelC ,O#,Lg dna =1 Ilhl,ll,< fll ybll <),lh(,I=),lg-f(,I=)f(,I .1 yltneuqesnoC 1 = fI/ =$Il II f - ,lj,lg =( ,)~llelhlI a .noitcidartnoc roF eht ecaps )u&E ew deecorp .ylsuogolana nI siht esac ew ylppa eht ytilibasopmoced ’ro .)P(,E .krameR tI nac eb devorp taht eht MTS ytreporp of eht )p(,L ecaps seilpmi taht 4 < .a+ ,sihT ,revewoh lliw eb ylticilpmi deniatnoc ni 4 E .IA 2. YLTCIRTS DNA YLMROFINU ENOTONOM ZCILRO-KALEISUM SECAPS llaceR emos cisab stcaf gninrecnoc snoitaler neewteb eht raludom ).(dI dna eht grubmexuL mron II /I. 6 ni eht zchrO-kaleisuM ecaps )c&L dna ni eht ecaps .)p(,E erehwyrevE woleb p si demussa ot eb cimotanon eht( ylerup cimota esac si deredisnoc ni .)]91[ NOITISOPORP 1.2 kizduH( .)]ol[ ehT gniwollof stlfeznetats era -aviuqe .tnel )a( Q seifsitas eht ,A .noitidnoc jb( fI/ =dlj 1 seilpmi =)fJZ .1 871 W. CRUK (cl IIfnI14 i+ 1 implies &(fJ + .1 )d( )L&L seod ton niatnoc na cirtemosi ypoc fo .,I )e( )u&L seod ton niatnoc a ecittal cirtemosi ypoc fo .,I )f( )p(bL = ,)p(;L ,.e.i eht mron 11.II it si redro suounitnoc no .)&L .skrameR ehT sfoorp of eht secnelaviuqe )a( o )b( o )d( nac eb dnuof ni ]9[ eht( noitpmussa no eht dednetxe ytiunitnoc of eht noitcnuf (4 ,. t) ni ]9[ nac eb ;)deppord ees osla ]7[ erehw 1~ si demussa .elbarapes morF ,21[ .pahC ,X .tceS 14 ti swollof taht )f( + .)e( oslA )a( + )f( ecnis )&E c )p(;L c )L&L dna )a( seilpmi taht =)U&E )c&L .]51[ ehT noitacilpmi )b(*)c( si laivirt dna )c(*)b( swollof from ,9[ ammeL .15.1 ,txeN )d( 3 )e( si laivirt dna from eht foorp of meroehT 1.1 ni ]ol[ noitacilpmi( )d( * ,)a( .p ,)16 ti swollof taht eht evitcepser yrtemosi detcurtsnoc ni ]ol[ si osla a ecittal msihpromomoh os taht )e( * .)a( AMMEL .2.2 ehT gniwollof snoitidnoc era deifsitas rof .)L&E )a( t,ll,hlI 1 seilpmi ),h(,Z + ,1 revenehw 60 l,hI ~fdnaf< .)p(,E )b( )&E seod ton niatnoc na cirtemosi ypoc fo .,I nI ralucitrap ni ,)u&E fi II f dII = 1 neht )fJI = .1 JoorP teL su tsrif evorp eht tsal .noitressa ylraelC JZ ). si etinif dna xevnoc no .)p&E ecniS )zZJZ d ,jlklI d 1 ti tsum eb suounitnoc no .)p(,E ecneH eht derised ytreporp .swollof gniylppA eht emas hcaorppa sa ni ,ol[ ammeL ,15.1 ew niatbo htiw ~,=l/llkIld )11~~( taht ~),h(~Z*),a-2(+),h2(~Z)l-,a(~),h,a(~Z=l ,a( - )1 )f2(,Z + 2( - ),xt .),h(,Z ecniS f E ,)p(,E )f2(,Z < .IK+ gnitteL n + + oc ti si nees taht ti tonnac eb taht ),,,k(,Z < lt < 1 rof yna ecneuqesbus .)kn( ecneH )a( .swollof ylraelC )u&E sah redro suounitnoc mron os yb eutriv of a lareneg tluser ees( ,21[ .pahC ,X .tceS )11.4 eht noitidnoc )b( si .deifsitas AMMEL .3.2 teL $ > .0 emussA taht f 3 ,g > ,0 II f 4II G ,1 erehw rehtie f, ,g E ,))L&E ro f, E,,g )L&L dna $I E .,A nehT ),g(,Z +- 0 seilpmi taht lIg,II,+O. :foorP emussA ot eht ,yrartnoc taht ,10 = (I ,/l,g 3 E rof emos E > ,0 erehw tuohtiw ssol of ytilareneg n E IW. gniylppA ammeL 2.2 ew niatbo taht 1 = JM&Z < kl( - )1 Jg&Z + 2( - Q/l J&Z htiw Jg(,Z +- 0 dna ,,a .dednuob yB eht noitpmussa ,,,g + 0 .e.a no T htiw 0 < ,,g J< IffE )p(,E neht )f2JZ < .oc+ ,oslA fi fE )L&L from eht noitpmussa d E ,A ew evah )y(,Z < .oc+ ,ecneH gniylppa eht eugsebeL meroeht ti swollof taht ZCILRO-KALEISUM SECAPS 1~3 ),,g2($1 + .0 ,woN yb eutriv of eht ytilauqeni ,evoba ew teg a -cidartnoc ,noit os eht foorp si detelpmoc .fc( .)]31[ ecniS 4 E ,A seilpmi 4 < ,GC+ from snoitisoporP 1.2 dna 5.1 ti swollof YRALLOROC .4.2 If )L&L si a MU ro na MTS ecaps :lrezlt )a( <%q 0 ereht stsixe 6 > 0 hcus taht -f&Z )g < 6-l frevenehw O>g> dna .&3,ljgI/ ,woN ew nac evorp ruo niam stluser of eht .repap MEROEHT .7.2 roF a, cimotanon eht gn$wollof stnemetats era .tnefa~tiuqe )a( )p(,L si na MTS .ecaps )0( )p(,k si a MU .ecaps )c( )i( .0>5q )ii( 4 E d z ,,>ltnelaviuqe( I/ \I. d si redro suounitnoc j. 081 W.KURC :JoorP noitacilpmI )b( .s= )a( si raelc dna )a( + )c( yb yralloroC 4.2 dna noitisoporP .5.2 oT teg a)c( )b( ew lliw evorp taht eht evitcepser snoitidnoc )ii(-)i()b( from noitisoporP 5.2 era .deifsitas ,tsriF ni weiv of noitisoporP 1.2 eht -noc noitid )ii()b( .swollof ,txeN ni weiv of noitisoporP ,4.1 eht raludom &Z ). si syawla ;MU-$ ,.e.i rof hcae is. O> ereht stsixe ),E(,S O> hcus taht ia-ld)g--f(6Z revenehw ,,&,>)g(,Z ,O>gaf .l=)f(,Z oT wohs eht MU of JZ ). ti seciffus ot evorp taht )g(+Z 2 is dna )f(,Z = 1 nac eb decalper yb 11 llg 4( B 1s dna 11 f 11 b = ,1 .ylevitcepser nI eht tsal esac noitisoporP 1.2 noitidnoc( ))b( nac eb .deilppa nI eht tsrif eno noitisoporP 2.2 ,seilppa os taht rof yreve E > 0 ereht stsixe )E(Y > 0 hcus taht ,l/g/ > E seilpmi )g(,Z Z .)E(]Y ,ecneH htiw ,E = ,)s(q ew teg j&(6 = ))E(]Y(,~ hcus taht )s(86-l<)g-f&Z ,O>g>frevenehw ,l=61lflI dna ,/lgll .EB ,woN from noitisoporP 5.2 ti swollof taht )L&L si a MU ecaps dna eht foorp si .dehsinif MEROEHT .8.2 roF cimotanon-u, dna 4 < cc+ eht gniwollof era .tnelaviuqe )a( )u&E si na MTS .ecaps )b( )p(,E si na MUL .ecaps lc( .0>4 :foorP noitacilpmI )b( 3 )a( si raelc dna )c(*)a( swollof from noitisoporP .5.1 oT evorp )b(*)c( ew tsrif evorp taht 4 O> seilpmi taht (,Z ). si MUL .raludom nI weiv of noitisoporP ,4.1 (6Z ). si syawla MU& dna ecneh .MUL-IQ morF ammeL 2.2 dna ammeL 3.2 ti swollof taht JZ )- si yllautca MUL .raludom gniylppA niaga ammeL )a(2.2 ew teg taht )p(,E si na MUL ecaps sa .derised .krameR nI lareneg ew tonnac tcepxe taht )u(,E si a MU ecaps fi 4 > 0 .ylno nI tcaf from ,S[ .p 1357 ti si nwonk taht fi 4 $ ,d neht )u&E sniatnoc na cihpromosi ypoc of .Oc ,ylraelC ni siht esac )u&E si a reporp laedi ni .)&L morF ,21[ .pahC ,X .tceS 34.4 ti swollof won taht )U&E si ton a BK ecaps ni( tcaf .)ecaps-B nO eht rehto ,dnah sa saw ydaerla ,denoitnem hcae MU hcanaB ecittal si a BK .ecaps eroferehT )L&E tonnac eb a MU .ecaps .3 SNOITACILPPA OT TSEB NOITAMIXORPPA teL X eb a demron ,ecittal f~ ,X dna cK X a xevnoc .tesbus ,etoneD =)f(KP EU( :K llu-flI ,Ehfni= (I f - h II .> llaceR taht JZ si .cimotanon )A( ehT stluser woleb etacidni emos ygolana neewteb eht elor of tcirts ytixevnoc ni eht yticinu melborp ni tseb noitamixorppa rof demron ,secaps ZCILRO-KALEISUM SECAPS 181 dna eht MTS ytreporp ni hcus smelborp rof demron .secittal We lliw laed htiw detanimod tseb ,noitamixorppa ,.e.i tseb noitamixorppa htiw tcepser ot K rednu eht noitpmussa taht j-3 K .)S>K( NOITISOPORP .1.3 ehT gniwollof stzremetats era .tnelaviuqe )a( X si na MTS .ecaps )b( roF lla f E X dna redro slavretni ,a[ b] c X gniyfsitas f 3 ,a[ ]ii f( < ,a[ )]b ereht sdloh ))f(,6,,cP(draC d .1 jc( roF Na f E X dna lla secittalbus cK X hcus taht f b K O>g hcus taht O#g dna 11 f - llg = fI/ //. enifeD =K ,OC .]g nehT :0 og .)f(KP aI weiv of )b( ew teg a .noitcidartnoc )c( * .)b( sihT si ,raelc ecnis redro slavretni era .secittalbus morF smeroehT 7.2 dna 8.2 ti swollof won NOITISOPORP .2.3 ehT detanimod tseb noitamixorppa ni )&L )p(,E( htiw 4 < -I- a ) htiw tcepser ot secittalbus si euqinu fi dna ylno ft # > 0 dna ,AESq .pser( .)O>$f gninrecnoC eht ecnetsixe of tseb noitamixorppa tel su tniop tuo taht ni )A&L htiw 4 E ,,A ro )p(,E htiw 4 < ,cz‘+ ew evah )f(lb,aCP # .0 .deednI rednu eht snoitpmussa )p(,L .pser( ))p(+E sah redro suounitnoc .mron ,suhT yltnelaviuqe ,21[ .pahC ,X .tceS ,14 lla redro slavretm era ylkaew .tcapmoc woN ti seciffus ot eton taht eht mron si ylkaew .csl nI tcaf ew evah a elttil .erom NOITISOPORP .3.3 roF eht zcilrO-kaleisuM ecaps )D&L eht gniwollof stnemetats era .tnelaviuqe )a( .,f-d )b( ehT detanimod tseb noitamixorppa htiw tcepser ot desolc secittalbus syawla sah a .noitulos ,revoeroM eht noitidnoc )b( llits sdloh eurt rof )p(,E revenehw 4 < CZ,+ 281 CRUK.W foorP )a( 3 .)b( teL f > K dna ~,h K eb a gniziminim :ecneuqes =d ,.,fni +,,mil=,llh-flI m+ 11 f - jl,zI .d ecniS K si a ecittalbus ,,u = =;V I ,h E .K revoeroM 0 .0 gnicalpeR fi yrassecen f,, yb 1( + )~/l f, ew niatbo II f,ll d > /I ,A ,+ .dII ,nehT rof eht ecittalbus =K ,},,f{ ~0 ,K =)0(KP .@ nO eht rehto dnah K si )mron( .desolc ,deednI tel rof a tnemom II f,,- gll, g+,,fnehT.K$gdnaK~,,frofO+ ni erusaem dna yltneuqesnoc g = .0 suhT II f,J ( + 0 hcihw si .elbissopmi ,eroferehT ni weiv of eht ruo ,snoitpmussa )O(kP # .@ sihT noitcidartnoc sehsinif eht .foorp gnitcelloC snoitisoporP ,2.3 ,3.3 dna smeroehT ,7.2 8.2 ew niatbo MEROEHT .4.3 ehT gniwollof era tnelaviuqe rof zcilrO-kaleisuM secaps htiw eht grubmexuL .mron )a( ehT detanimod tseb noitamixorppa ni )u(,L ni( )u(,E htiw 4 C- )CC-I- htiw tcepser ot desolc secittalbus stsixe dna si .euqinu )b( )u(,L .pser( )L&E htiw 4 < )j(c+ si na 44TS .ecaps )c( 4 > 0 dna 4 E ,A .pser( 4 > .)0 nI tcaf a regnorts tluser naht noitisoporP 3.3 nac eb detats sa a yralloroc from meroehT 6 ni .]3[ MEROEHT .5.3 roF zcilrO-kaleisuM secaps eht gniwollof stnemetats era .tnelaviuqe )b( roF lla desolc raenil secittalbus cK ,)L&L #)f(KP 0 rof lla f E I&L foorP )b( * .)a( sihT swollof ni weiv of noitisoporP .3.3 ehT -acilpmi noit )a( * )b( swollof from meroehT 6 ni ,]3[ hcihw setats taht rof ylkaew yllaitneuqes etelpmoc hcanaB secittal hcae desolc raenil ecittalbus si .lanimixorp llaceR [ ,21 .pahC ,X .tceS 14.4 taht ylkaew yllaitneuqes etelpmoc hcanaB secittal edicnioc htiw BK secittal noitceS( .)1 nI weiv of -isoporP

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