ADAPTIVE MULTISCALE HOMOGENIZATION OF THE LATTICE DISCRETE PARTICLE MODEL FOR THE ANALYSIS OF DAMAGE AND FRACTURE IN CONCRETE

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1 Ssl gg Gologl f ls (SG Dpm l ml gg om Shool gg ppld S s llo 6 US DT ULTSL HOOGZTO O TH LTT DSRT RTL ODL OR TH LYSS O DG D RTUR ORT Roozh Rzh w Zho Gl s SG TRL RORT o 7-/57 Smd l ol Solds Ss 7

2 dp ll Homogz L D l odl ls Dmg Roozh Rzh w Zho Gl s Dpm l ml gg ohws Us s L 6 US S 55 Ws S S Dgo US s Th pp pss w dp mll homogz shm sml dmg f ss wo-sl homogz mhod oplg mso-sl d pl modls mosl f modls mld dp fmwo m mll fl ld f-sl smls ss dp mll fmwo h ppoh smls s who sgg ml Rps olm lm ( mo-sl f f m dp ms d mll homogz fmwo dd fl Th lds ssl d ompl os sp lodg ds ldg dmg lolz whh l sm po msh hd wh homogzd Sl ml smls d o sg pl dlopd dp homogz mhod dd dld sd ompl os pmd Kwods: dp ll Homogz; dp ; Dmg od ll l m-md gg mls hogos lgh sl osop mhl pops mls sh Yog s modls sl ompss sghs hgoml hs sgl dpd fs s hogs o s mhl pops mos gg ml h dl dpd mhl pops ggg m dms ggg sz d ggg olm f mplog dld f-sl modls whh ml hogs o l od o m ml How ml sml lg sl ssms sh dms dg ps l pow pls f-sl modls ld omos m dgs fdom poh ompl os Th h mod shs dlop dff gos mll ompl fmwos ospdg ho -ml ddss: g-s@ohwsd p smd ls 7

3 whh f fom lzg mhl modl ls lg gg polm wh l ompl os ll ls fmwos f wo pm gops [] fs gop lds hhl mll modls whh f-sl polm smld os o d mhl modl opld mo-sl polm g smld Hh ompl fmwo slhd hogh solg mo- f-sl polms dpdl whl m flows w sls dg ls sd go lds mll modls whh polm dom domposd wo sdom sdom d homogos m modld o whh dms sdd l g whl o sdom modld dld f-sl modl s wo sdom d hogh ppop ss wo sls sold smlosl ll homogz hhl mhod whh h sl sgd o p dds ll homogz mhods w fs dlopd d mml qs ff ml pops hogos solds d sol f-sl od l polms [ 5] How s ppohs lmd l mls wh smpl pod l ss sdd d sm dm flds ompl homogz mhods w dlopd q s lms s mll fmwos Rps olm lm ( f-sl ml sd sgd h ompl po homogzd sps sdd ff ml ho o spf s qs dfd mo-sl Dg h ompl sp s Gss po ppld ld ml od d sol ll homogzd whh sfd ompl po ls ompl homogz h wdl ls ompld gg polms [6 7 ] Smh l [6] dlopd mlll f mhod whh oh dom mshd wh ( mplod ls pd ol mls l [7] slhd mll homogz fmwo glzd whh s o Kozso [] mld hgh od homogz shm whh fs sd od gds dm fld ppld od ds show gd hd homogz mhod -m dm flds podd wh od l polm h l [] ompl homogz ls pols mls d lg pl dms dd ompl homogz mhod h opl d modls (D Go l [] sdd l dsp sps gl md d m l lodg d opld /D ppoh Wg [] l dlopd hhl mll hgo-mhl modl sg fld flow gl md smp ps Homogz (H mo mm hoogh ppoh mplos mp ps fld ls d gog qs ospdg od

4 ds mo- -sl od l polms [] h l [] lzd h ppoh sd ho m ssms sg sl sp oh sp m H o mplm ompl sws h sl shs sd mhl sps mls Hs H [ 5 6] plhd h-pp w whh psd dls H o s f mplm l mls wh pod mos psd o w ompos ml wll sl polog opmz h l mld H fmwo sg l-pl ho omposs [] gl h [7] poposd dd od H ppoh sml l ho f ompos mls wh dd ompl os Ghosh l omd mp ps ppoh wh oo ll lm hod ( dlopd mll ol fmwo ppom l pops [] sg l-pl sps [] omposs wh om mso-s Wh l mll homogz mhods ompl dlopd fmwo h mo sgd pd ompd ompl ff mo shoomgs lsl homogz mhods whh ml s sgd mo-sl s fom gg ls mdos ompl pss ss sol od l polms h g po sp sls dg ls gg polms dlg wh f fl ss g dmg s lolzs g s ml dom m l sgg ml s g pos spflos h gd Ghosh l [ ] poposd dp ml-ll modl mll sml f hogos omposs whh h modlg sol lls modl ss m dmg mhs modl d mp homogz mosl smls omposs pmd fmwo mld g gog s w d sls Rgdg ls ss S L [] dlopd dp mll whh g modl ms spf pld f-sl dz og hog h mll modl ho sdd l p p dp fmwo lmd smpl D polms ps sd ds d m homogz mhod l dlopd hos [] dp fmwo splzd [] Rzh s psd homogz fmwo opl L D l odl (LD [ 6] whh ml s smld LD sgd s hd ls Two-sl ml mpls w sold ompl dlopd mll fmwo w fd ompg wh fll f-sl smls pp s lzd wh oop l l s q Dg sml dp shm m ds s m d sgd ml Th sg lds sdl sg ompl os

5 whl sll dmsd sf Rw L D l odl (LD L D l odl (LD ss goml ps mso-s hogh followg sps: ( os ggg ps whos shps smd sphl odd olm -- om pod ggg dms dmd smplg smd ggg sz d f Sphl gggs d pl dog spm dpd g ( Zo-ds ggg ps (ods oml dd o l sfs fl ppl od ds ( Dl hdlz gd ggg s sod h-dmsl dom ssll d o o wo gl fs sd h hdl show g g llss po hdl sod wh s fo ods ospdg fs omg sh pos fom hdl d sm od o ospdg polhdl pl whh loss sphl ggg Two d polhdl pls g hogh shd gl fs dpd g d olhdl pls sdd gd gl fs whh s qs dfd l m smd pol ml fl los g pss polhdl pl ps pl dog spm shold sd sphl gggs gd ld d modl whh smls l mso-s whl o dl ml sol pod od sphl gggs d od h pp sod polhdl pls goml s mplod ml ls Th ss qs ss ompl d modl fmwo: df s h f s q whh ls f f s pl qs s df Rgd od ms mplod ds dm l/pl ssm dplm mp od h f df ms s ɛ α α ( α L (ɛ f oml s omp; ɛ ɛ L f gl s omps; lgh l s ods shg f lso sod hd dg (s g ; α s dfg f lol s ssm f sh ohogl f L f gl s (s g l s qs l s lw gog ho ml

6 ( K ( L L od (d L LD lls olhdl pls ( g : ( Sphl ggg d pl dog spm pl LD hd g fo d gggs s sod ssll ( Thd po sod wh ggg (d LD polhdl pl losg sphl ggg ps ( olhdl pl ps pl dog spm mposd od h f l gm oml sh s popol ospdg s: ɛ ; T ɛ ; L T ɛ L ; T α ; ff oml modls; α sh-oml oplg pm o s s d l lm msosl l phom hzd h mhms ospdg f ll l s qs fl dsd low ohs d s s-sh o sl lodg (ɛ > fg ho mld hogh ff s ɛ ɛ α(ɛ ɛl ( L /α whh df f oml sh s ɛ (/ɛ; αɛ (/ɛ; L αɛ L (/ɛ ff m l (ṫ ɛ ms sf ql (ɛ ω (ω p [H (ω ɛ m ɛ (ω / (ω] ( whh m{ }; ɛ (ω (ω/ ; (ω ɛ / αɛ T α/t whh ɛ T 5 ɛ ɛ L

7 T L ω pm dfs dg w sh oml lodg ɛ m ɛm αɛ Tm h dpd l ɛ m ɛ s lodg pos p sg modls dfd H (ω H (ω/π sg p H sg modls p s (ω π/ pssd H / (l / ; l G / ; G msosl f g LD pods smooh s w p s p sh (ω wh pol sgh g (sold g (ω s ( s(ω s (ω α os (ω/ s / [α os (ω] ( s s / sh sl sgh o dhd l g pss sgh dom ospdg ɛ m / T ss ɛ ɛ T show g p s (ω p sh (ω π/ ω π/ sg α G s 5 l g pss lodg-lodg l dopd h wo ms ff ss ff s lshp whh ɛ (ɛ m / ml pm dfs sz hss l omp po ollps ompss oml s ompss lodg (ɛ < ompd hogh ql (ɛ D ɛ s-dpd od dpds olm s ɛ f d s ɛ D ɛ ɛ olm s ompd olm hdl ɛ smd s fs lgg g hd d l lm modls po ollps (ɛ ɛ ɛ l ol g olm s wh sffss H omp hdg d po ollps lm (ɛ ɛ ɛ / omp s whh po ollps ss ɛ κ ɛ omp s gg hdg κ ml pm msosl ompss ld w (ɛ < (ɛ D ɛ ɛ ɛ H ( D ( ɛ ɛ ( ( D p [(ɛ ɛ H ( D / ( D ] ow H (H H /( κ D κ H ; D ɛ D /ɛ (ɛ > D ɛ D /(ɛ ɛ (ɛ whh ɛ κ ɛ H H κ κ κ ml pms s ods show g d ompss oml ss s D D lodg-lodg ph lso show D whh d dsfd oml modls ml pop LD pms sdd plog s s 6 G H / 6 H / 6

8 dp ll Homogz dp ll Homogz dp ll Homogz dp ll Homogz [] T [] [] T [] w ' whh ql pls Usg s lss wh q w O( f sgl f sd gog w gog qs sgl pl sd sd O( -sl qs h qs pl sd gog O( -sl gog R O( -sl qs gog pol ; ms ds lg s ms O( ss ' po s fs os-sl qs gog O( f -sl whh olm f O( os-sl qs polm qs ldd polm ls olm ; d o olm ; ms ds ol qs o gg gg ol qs qs q polm fs m q 6 whh O( f sdg hh op f s w os-sl qs gog polm sgl w polm ml gog f O( -sl qs gog polm os-sl q O( qs w f lsl pl dl q gog O( mpl q O( q ld ms f O( pos f od shd pls lol low-sl ood ssm q 5 qs p ml oml qs df O( lsl pl dl q go omps sdg q po ldd fs m q 6 lsl pl dl q go O( f whh w f whh w O( whh f w olm ; ds m q gl s w ms low-sl dplm how g op s s fs s s fs po f whh f w olm olm ml mpl q O( q ld lsl p s gog polm olm ; lsl p ml mpl q h sll sll d qo mplo pl df sll q o whh ldd m q 6 O m ; q ml ml q q O( s lsl omps whh fs f ldd w f w ( ml mpl q ml mpl q lsl pl dl gog mpl O( q m O( O s po ms ds whh wh olm ; ms ds dplm sll lsl pl dl gog ql ml lsl pl dl q gog olm 6 s sdg q fs m q sll q ospdg homogzd q ospdg homogzd m dfd olm ml mpl q ml mpl q O( whh osop polm q gos polm follows oml gl s w ms low-sl q gos polm follows q gos polm whh follows wh sld ; ol qs gg qs o R ml q ld sol sl O( gog follows fs (6 q ls sd s ml q O( ld mpl sll ms ol hogh p m q O( q w show g op po s fs q homogzg f whh hogh s ds hogh homogzg ms sd fld ls l dpd ssm ms ood m sld dl fld ol l dpd m pl dl q q oml lsl m m O( sdg f oml gl w ms low-sl q m o gl s omps sdg ldd m 6 6 mpl mpl (6 sl fs olm m q f (6 q ldd m q ld po sd m ml q O( q ld sl q gos polm wh homogzg lsl pl m q gos polm whh wh lsl pl d dplm gg qs o pls dplm ml ql s o q lsl pl m h q m w ms follows ; hogh m mpl low-sl ol qs whh q po 6 m sld o olm ; po m dfd ol qs q ol qs gg ol gg ql ol qs gg qo ol (6 o (6 O( os-sl qs gog polm ol q lsl pl dl q gog o m m hogh homogzg sol p m q O( q ld sll lsl pl dl q gog m m hogh homogzg sol hogh homogzg sol R m m ol qs gg ql d ls f oml gl s w ms low-sl dplm ol qs gg ql o 6 qs q po po m dfd po m dfd mos olm ; ms ds m q lsl pl dl q go polm sll q ospdg homogzd l ol gg qs o pls mo sd m q (6 l O( -sl qs gog polm ol qs gg qs o pls mo-s m sll q m m ol q osp m m gg qs o pls mo-sl ol q ospd oml gl s omps sdg ldd fs m q 6 o m m m m ml fl mpls qs whl s o s l dom smoohl ospdg mom s s oml po qs q df sd low-sl ol ms dplm (6 gl hogh homogzg sol polm dd m s m gg olm ; mo-sl w smoohl f w low-sl m ol m q ; s ms dplm mo q 6 mo-sl whh m slo pl dl gog q p ol qs o pls mo-s qs ds m q 7 ld lsl mpls (6 h sll ospdg homogzd pssd ds qs s m (6 sll sgl f-sl h o 6 sll ol polm m m hogh homo ol ql q sll q ol m po q gs polm polm m m m m osop s qs O( -sl gog ms f m olm sol sll wods olm ; polm ms d sll q ospdg ospdg hom p sll q ospdg po m dfd ql s sll homogzd lsl dl q gog ol homogzg q q sll q l ls m po pl gg qs o pls mo-sl hogh sol hogh homogzg sol po ol ospdg l ol ospdg mo q ospdg mom q ospdg hom d ls q ospds ls sll q ospdg homog po m dfd s hogh homogzg ml mpl q O( q ld sll oss m o fd s dfd sll q ospdg homogzd pssd po m mo sd m q 6 po sll q ospdg homogzd pssd l d ls q o lsl pl dl q gog q po s s spl s spl s s spl f-sl df f oml gl s w ms low-sl dplm ospdg homogzd pssd qs o pls mo-sl s h f o wods sol polm l m m s s spl dom H ospd dplm o flds spl l olm ; ms ds m ol q ospdg dd polm dom H ospd dpl osop s mo sd m q 6 ospdg lsl homogzd m po ( follows sol q R olm l fl po o hogh flds q do m q 6 s polm gog whh pl dl polm sld lsl pl dl q pods ol q pssd q gos wh sld wods sol polm m m ol ol sll q s h f o wods homogzg sol polm dd m ol polm osop spl m q q ppld fs g gs f-sl s s spl gog sll s h m o lsl pl dl q gog s oss o fd h po m dfd d po m df d ls q hogh homogzg sol s s spl ospdg homogzd pssd s pod l ol q ospdg mom ol q ospdg s s s spl s s spl m ows wh sld polm s o shd s spl d qs qs gg o whh follows wh sld s so q oss m d dls ospds o qo mo-sl hd o l ssm f pls po m dfd pls q fd ol ospd q gos f ss d ls ospds ol f o s spl d pssd ol h m l f od shd w pls l lol low-sl lsl po pos od w pls dd lol low-sl ood ssm q pos pos od w lol ood ssm flds f-sl mo sd m q 6 ld mo-sl polm s f o wods sol polm sdg f-sl m hogh homogzg sol dd fl lsl pl dl d ls q g f pss gd gdg df ' oss m o fd d ppld fs g gs sol mosl s h o wods sol d low-sl polm pos f od shd w pls low-sl ood f pss gdg pl s g s s spl gog polm ood s s spl osop s o o f wods sol polm hogh homogzg sol polm m gs sol lsl dl d lol gd dd shd od shd pls lol low-sl ood ssm hogh homogzg od w pls lol low-sl ood ssm s pod fs ol q ospdg sol os shold ppld fs g gs f-sl sol s h pl ms lld sfg hogh homogzg sol polm dd s pod oss m po m dfd d ls s so sll q ospdg homogzd pssd ( d ls q ospds pos f od shd w pls lol low-sl ood q 5 q pos f od shd w pls lol low-sl ood ss q 5 qs oss m o fd d d G s l / m & omposs ( pos f od shd w pls lol low-sl ood ssm q 5 qs s pos f od shd w pls lol low-sl ood ol q ospdg m g f m wods show d ls q ospds lsl ol gs polm s fs oss o fd d dls o Rf [] po pos od shd w (7 lol ood ssm low-sl show g pls op s fs dd show g po s s qs w q op po s s fs sgl op dls mosl s s h f o sol polm hogh homogzg sol ssm f (5 s shold fs g f-sl sol ' ql Usg lss lg wh q w fs ood (5 shd q ospdg ' 'fs ql Usg lss sfg os shold ppld fs g gs f-sl sol o f od shd w pls lol low-sl ssm 5 qs sgl pl sd s ol pls ppld s h pl ms lld s od pls lol low-sl ood oss m o fd d q 5 qs pl sd sd op fs ol q ospdg dd show g op po s s l fs g gs f-sl sol ms lld sfg s hogh l po s s fs s pod op po s s ol q ospdg sol (7 sod os h pl ms sfg s qs ol q ospdg O( f lld gog f os-sl polm (7 ds show O( os-sl qs ls pos f ldd (5 d q m olm R o oss m o f ; f od shd pls lo m d O( show g sdg po s s R whh olm f show g ol po s s s O( f fs whh w g op oss po s s fs O( op f whh olm f R q w o fd d dls o 6 oml o op oss m fd dls o qs Rf [] op show g (5 po s m o gl sdg q fs m q 6 d ms oml o gl s omps q ldd fs q op fs w O( oml po s s fs omps q O( g o gl s omps ldd m q 6 sdg fs 6whh fs s Gss po s s shold ppld g gs f-sl osop ms dplms O( os h pl ms lld sfg s sdg olm ; ms ds os s olm ; ds m d df O( O( s sol w op po s s O( f f d op po s s fs ( f whh f w ( smps sol od ds sol h ms lld sfg s h pl ms lld sfg s pl omps sdg q ldd fs m q 6 oml o gl s omps q ldd fs m q O( sdg q ldd fs m q 6 ol q os-sl gog polm O( os-sl gog polm qs (7 os-sl qs gog H(ω omp ds O( sol Lpolm pl ( dp ll Homo qs fs (ω h opl pl s lsl shl po pod od d lsl d m o lsl dl Gss op f-sl ( polm s ldd q g po s fd low-sl pl q os gl s omps sdg q ld low-sl ldd fs m s oml o gl s sdg fs qo dplm q qs gog polm m s gog g oml o gl omps sdg q ldd m oss low-sl f show 6 omps po pod oml f-sl df f oml gl s w ms dplm sgl df f oml gl s ms m /os sd oml oml f-sl df o gl s w ms dplm oml o gl s omps sdg q ldd fs m q 6 w ml mpl q O( q sl dp lsl ppld pl ml mpl ql gl omps sdg q ldd fs m q qlm qs whl od ds d sol ms dplms h pl ms lld sfg s sgl pl sd pl dl q gog s gog polm lsl pl dl q gog q 5 qs sd sd L ll Homogz α gl s omps sdg q ldd fs m q 6 l s omps sdg q ldd fs m q 6 L T T p opl s dp ll s ds d s sol po oml olm ; ds mos h hl ( f oml gl s w ms low-sl dplm low-sl opl s oml gl w ms dplm pod od ds d sol w ms ms Homo ( gog omp f po df gl w low-sl Gss 5 low-sl f pl f-sl ml os-sl qs polm ppld osop polm O( s ml mpl q O( q ld sll os-s ml mpl q O( q ld sll qs sgl sd sd (ω gog O( olm ; hogh homog mpl q O( q ld sll fs q 5 qs sgl pl sd sd hog ppld oml o gl s omps sdg q ldd s gog polm f-sl df f oml gl s w ms dpl omp opl f-sl df f oml gl s w ms low-sl dplm f-sl df f oml gl s w ms low-sl dplm po m dfd O( os-sl qs polm po m dfd olm ; m pod ds 6 o flds ld mo q m sol low-sl po O( qs gog olm ; ms ds m olm ; ms ds m o flds whl mo sd m q 6 opl dplm L po po o oml mo sd m q gl dp olm R polm ( o moso ol qs gg qs pls mo-s o m m m sw f-sl df f oml gl s low-sl 6 m o ol qs h -sl m m qs f omp w os-sl m flds qs od d sol polm m hogh homogzg h q O( q o f q whh f d w hogh homogzg qlm q O( s olm ( O( qs gog ( ld ms qs gog polm f-sl df f oml w ms pol m m O( df f gl s w ms low-sl dplm -sl fs s f f oml gl w low-sl dplm Homogz ; q-l mls lsl pl dl q gog wh olm ; ms ds qo olm ms ds m q omp ω sol mo sd m q 6 opl ms sll lsl m d dpl mo sd m q 6 m sll po s o m o o m dp ll po s mo m m fs ppld O( os-sl qs gog polm po po q olm ; m po m olm ; ms m q O( q ld sll omp opl ms ds m q 7 po dfd qs qs m q m o flds sd m 6 po H ol gg qs o pls mo-sl m ol qs gg o pls mo-sl ol l qs gg qs o pls mo-sl ol q ol m m ol q o w o q-l mls s lolz os fl l O( f whh f w f-sl df f oml gl s w ms m m o flds mo sd m q 6 po o flds mo sd m q 6 poo d ls q lsl pl dl q gog h d ls qo po m dfd o flds mo sd m q 6 po lsl pl dl q gog q po m dfd mosl s s h f o wods sol polm lsl pl dl q gog m m olm ; ms ds m qo mosl s s h f o wods sol polm m m ld sll mosl s s h f o wods sol polm lsl pl dl q gog q l m m olm ; l ω o q-l mls q 7 m po m dfd sll q ospdg homogzd pssd osop s R lsl pl dl osop s ml s qs mo-sl qs ds mo sd homogzg l dfd fs q sll q o gg q osop s qs o mo-sl polm flds gg pls ld d ml mpl q mo dp lsl ld l ls l o pl ospdg po l ol q s pl l T ol ospdg f opl s q m m m omp O( sll m ll Ho m q O( q ld sll ml q O( mo sd m q 6 po po ms m m q 7 d mo sd q 6 6 po q mplo lsl dl q gog q pod hogh sol polm ml mpl O( lsl pl dl q gog q pods d q o flds pls mo sd m q 6 l po po s h o wods sol sm polm m ( s h f o wods polm m m sol l mpl f lsl g ml do polm q m m osop polm m dp ll Homogz osop s q ospdg lolz oss o po m d s po m df d o lsl pl dl gog q gos polm whh follows wh sld sll q homogzd pssd gg qs o pls mo-sl d s pl dl q gog ql sll q ospdg homogzd pssd s gog q pods q gos polm whh follows wh sld sll q ospdg homogzd pssd α d ls q ospds h dfd 6 oss homogzg mosl s h m (ω lolz h g whl m o q-l mls s lolzo m mosl s f o wods hogh sol polm m ml ds ml q O( q ld sll wods s m ls sol s mpl ; q 7 po o flds o mo sd q polm hogh dom h m o fd s s f o sol m hogh homogzg sol ospdg oss m o fd mosl f o wods sol mos sol po m dfd (6 m mosl (6 o pls mo-sl po dfd f-sl polm m h f wods polm s shold ppld fs g gs hogh homogzg sol p d ls q osp d mqs q pods s shold fs g gs f-sl s f s s m sol d q ospds ppld ms ppld hogh s spl s shold fs g gs homogzd pssd lsl dl q gog pod olm m l s s spl do f d homogzd pssd l d ls q ospds lsl q homogzg sol dd pl dl q m m m d s ospdg s ms q-l mls s lolz fl Th h osl s h o sol polm s sol sol gg qs l lolz f-sl g ml s homogzg o polm ol q l ol gg qs o p s wods ol pl os moso s s pod s pod dp ll Homog dld ol qs qs o pls mo-sl pod gg qs o pls mo-sl hold s h f o wods sol polm hogh homogzg sol polm dd fl mool q ospdg s ol qs gg qo d s h f o wods sol polm gog q pods hogh sol polm dd fl mo-s q ppld fs g gs f-sl sol ol q ppld fs gs f-sl sol d ls ospds h g o q-l mls s lolz mosl s s h f o wods sol polm m sgg ml f l d d lolz oss d shd ls pos f od w p homogzd pssd 5 pos f od shd w hogh homogzg sol o O s lsl (6 sol hogh homogzg sol pos shd w pls f-sl sol zg polm dd fl mo-sl s m pod ppld ppld sfg (6 o f o oss Bos dospdg qs qs o pls mo-sl polm m m lolz h pl dl q gog q pods sgg f dz ml od m m mls gg pod oss m o fd d dls o gs d ls q dol pssd ( fs q-l s lolz os fs dd fl mo-sl q-l m ol q ospdg m 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sgd h ospdg dd ls s m o fd d dls o qs Rf [] oss m o d dls o qs Rf [] ls lolz os l fl Th dmg oss o fd d dls o qs Rf [] ms g dd fd oss m sg ff dp homogz fmwo whh f ol gm lodd ll f s ds s ss s ompl os mdosl ss s ompl os mdosl m ml ls ls h hp ppop fs dlopd ls h sg h ls h hp p sg ff fs f ms sgd f lolz g ml dom whl dom m l g sg ff h lolz ml dom whl ml dom m l gm ls ss sdg po m dfd dd o q-l mls s lolz sgg ml f dz m o q-l mls s lolz sg l ss pssd s lsl ol q olm dd fl mo-sl fd d dls o qs Rf [] sll q ospdg homogzd pssd fd d dls o qs Rf [] l hogh homogzg sol polm dd fl mo-sl fs s f ms h f l B fs dmg dlopd sgd fmw oss m fd d dls o qs Rf [] ls ss sdg l oop s s g lshold os ml dd ls h hp ppop fs oss m fd ls hp ppop dlopd o q-l mls s lolz os fl Th d ff dp homogz sg ff dp homogz ddz ppld Th gs f-sl sol lolz o g ml dom whl l l o q-l m po g dd ml s s shold ppld fs g lolz o g lz B ml dom ml dom m l gm po m dfd ms dom m fmwo homog sg ls s lolz fl ms dmg ds whh f whh m l whh f ol g dfd l sgg ml f h sgg ml f dom sgg ml f d l ls h hp ppop fs sg ff dp lolz sgd f lolz fs whl ms f h f lodd ll f s l lolz followg g H ls dom whl ml dom m l gm om dff l d s lld sol ls h hp ppop fs dlopd ml mpls sold whh f ol gm l dp whh f gm l followg dp d q ospds lsl ol q sg ff sgg ml f dz ml dom sg ff sgg ml f dz ml dom sg ff dp ls ss sdg l oop s dd sg ff dp homogz fmwo ss s ompl os mdosl mplm fs f ( lodd ll f s s q lolz g ml dom whl ss o dd Rf [] g dom whl po m lolz dfd ldp homogz l ð Þ¼ ðþ ( f ff ospdg dd po m sfg D ml sg 5 ol qs q ll H ml l ospds g sg D d/ mdosl d ospds lsl ol q fmwo ff dp fmwo s sgg f dz dom s lolz ml dom whl ml dom m l g q dplms os q h pl ms lld s ml g olm oml s: lodd l d whl ls lsl q sgg f lhomogz dz m lolz ms g m fs f pl dplms os h ms l d ls q ospds h ( D sgg ml ls ss s ( ls ss sd dom ml dom m gm mplmg om dm ls ss sdg l o ss s ompl os ml s ol ss ompl os md ss h s ompl os ml ms ls þ h hp ppop fs dlopd so s sg sg ff dp homogz fmwo sgg ml R lodd ll f s s q dz dom (7 sg ff dp homogz fmwo oss m o fd d dls o qs Rf [] (7 ls ss sdg l oop s ho l ls ss sdg l oop s ho f ss h s ompl os mdosl mplmg whh d df d ompll ss s s ompl os mdosl mplmg dm olm whh f ol gm l followg dp sh hp d q ospds lsl ol q Homogzd f l lodd ll ml f sgg ml f s d ls q ospds lsl ol H sgg smd fs ml pms fs dm f ms ls ppop fs dlopd som ml mpls d m polm sold s ompl os mdosl mplmg whh ls h hp ppop dlopd som m Rf polm sold dss d dls o qs Rf [] oss o fd d dls o qs [] m qs whl pod od ds d sol sgg ml f dz ml dom wg dp shm poposd h sd oss m o fd d dls o ss s ompl os mdosl mplmg qs whl pod od ds d h sgg ml f fs f lm fs f ss s fs f ms dz ml dom ql sm qs dfd h mopl whh f ol gm f ol gm l T whh f ol gm l followg lodd ll f s s sg ff dp fmwo d ompss-sh ml sh s ps som ompd poposd o lodg (ompss homogz oml ompd ss ss sol < ss h ls ppop fs dlopd mpls l osdp mdosl mplmg h dm ( dp ll Homogz fs f ms ml sgd ss ll fs f ms s ls sgd h ( LD ll Homogz whh f gm followg dp shm h ss LD sol l l whh f ol ompss followg dp shm poposd sd Homogz hp ol d dp [] ls sdg l oop s ho f ls dhomogz dls o qs l Rf mopl modl [6] followg hp ss os md ss ompl os mdo ff gm oss m o fd d o qs h Rf [] Homogz ff oss m o ml sg dp fmwo ol homogz h f dp shm poposd h sd mposg (Ddls 6 ompl Wh ods lodd ll f s ( ql p ( 6 ( pgm sg dp homogz fmwo o f o p p s f omp opl s ss s ompl os mdosl mplmg dm lodd ll lodd ll h qs f omp mos whh f ol gm l followg dp shm poposd d pod B: ss s omp lodd ll f lm ls ss sdg l oop s ( whh f f ls ss sdg l oop s d pod B: l os mplmg dm o s ls ss sdg l oop s ho ( mdosl d--olm s o / dp ll Homogz ( ( λ ϕ/ λ ϕ/ λ pl mlpl wh dp ll Homo D D sd ( ( L poposd dp ls h hp ppop followg T L Tff L whh f homogz whh f s ql ho smd m whh f h oop L sdg L s gm l shm h lodd ll f s fs lodd ll f s l sg dp fmwo ls ss l s ho f o ls ss sdg l oop ho f o s fs f ms sgd f q od sg ff sgd ls ss sdg l oop s ho f o l whh f ol gm l T ompss smd h l whh f ol gm l f sd o q-l mls s lolz os fl Th ms dmg ds o o q-l s lolz os fl Th dmg ds poposd l: Th ¼ ls fs od modl d sffss q-l mls lolz os fl ms dmg ds s mls L h hp f ls h ls h hp whh ol gm oop followg shm ss h sr h hp ppop ms dg ls ss l s ho f o fs ms f ms ppop R whh ol f ms L sg gm l followg dp shm h sd ls sd fs l poposd pol lodg-lodg ds ϕ λ sdg dfd ϕff ppop dp f ( fs ldg dfd ollps ls soop mls s lolz os f fl ms ol (modlg po followd mls s lolz os fl Th ms dmg λ pl l po m ls lodg l lodg-lodg dd ls s L dd po m Th dfd ds dd ls ss po m dfd dmg ls hp fs dlopd sg dp homogz s f o h hp ppop dlopd so l lld fs l fs ms sgd f s sffss s dfd fs f ms ds sgd f s l lld fs f s lm f h lodd l ll dp ll lodd ho ms q Homogz l ( fs f sgd f sd ( m 7 o q-l mls s lolz os fl Th ms dmg ds ls ss sdg l oop s dfd olm l ; dd sd s q ls ss dom sdg l oop s g ml ms ds m q 7 fd ol l dd o q-l mls lolz dom whl ml ml m l gm lolz g dom ml dom whl dom m l gm (modlg omp hdg lolz ml whl ms ml dom m l gm l g pl l o sg ff sg ff h lodd ll f s sg ff dp h lodd ll f s ls ss sdg l oop s ho f o lodd ll f s fs f sgd f ls ss sdg l sg ff dp homogz soop s ho f fs f m < l fl lw sh sgh smd ml dom whl ml dom m l gm s fl Th ms ds ol l dom whl dmg ml dom m l gm dp ll Homogz d ls q ospds lsl lsl ol q d ls q ospds dp ll Homogz d ls q ospds lsl ol mogz fs W q sg ff homogz fmwo w sg ff dp homogz fmwo q fs fs f lm f lodd ll f s pods s q fo sgd sd sold lodd ll f s dp q s ¼h ls hp ppop fs dlopd som l lodd ll f s s q q pods opd som ml mpls ls q ospds lsl ol q lsl pl dl q gog q s q ospds lsl ol q sgg ml f dz ml dom lolz g f ml dom whl ml dom m ðþ l gm sgg ml dz ml dom sgg ml f dz ml dom ms lolz ms g ml d f R fs f ow dfs o q-l mls s lolz os fl Th ms dmg ds ls h hp ppop ls h hp ppop f mogz ls h hp ppop fs dlopd fs f ms sgd f s lodd ll f s s q dd f dz ml dom fs f ms po sgd f sd ml dom m l gm f dz ml dom lodd ll m dfd dd 6 oss m o fd d dls o qs Rf [] dp ll Homogz > D oss m o fd d dls o qs Rf [] D l oss m o fd d dls o qs Rf [] dp l D <hp h hp l lo h ppop mplmg fs dlopd som mpls s lodd ls hp ppop fs dlopd som ml mpls sol m s o s q lodd ll olm ls h ppop fs dlopd ml mpls sold sg som ff dp homogz fmwo dsfd oml modls o lodg sg dd fl fd qs lolz Rf [] hogh homogzg sol polm dd fl mo-sl fd d dls mo-sl o qs ls Rf [] d ss s ompl os mdosl mplmg dm ss s ompl os mdosl dm posss ml ss dls s ompl os mdosl mplmg q-l dm sgg f dz ml dom d ðd ; g sgg ml f þ h mls dmg 6 6 ds o q-l s os fl Th ms dmg ds lodd ll f s Þ¼ H ð D Þml D lodd ll f s o mls s lolz os lz ml dom whl ml dom m l gm D ls s lolz os fl Th ms sg ff dp homogz sg ff dp homogz > sg ff dp homogz fmwo lsl ol q ompl os mdosl mplmg dm dz ml dom mpl os mdosl mplmg dm d ls q ospds lsl ol q lodd ll f s s q fo s w h ( ( /( : ls h hp ppop fs dlopd ml m s s q som h mp f f D/ D Dlodd D ll D ls hp dp ð ml ( [ p( ] (5 homogz Þ ðsdff Þ' mpls ow sg ff fmwo sg dp homogz fmwo s D Þp½ð D ÞH ð D sg ff dp homogz fmwo op fs dlopd som sold ls h dd ol q ospdg dd whh f ol gm l followg dp shm poposd h sd whh f ol gm followg l followg dp shm h poposd h sd whh f ol gm l dp s poposd ( sd ls s lolz os fl Th ms dmg ds h od qh pssd f ss s ompl os mdosl mplmg dm ss s ompo lz gm g ml dom whl ml dom m l gm o q-l mls lolz os fl Th ms lolz hm g ml dom whl ls h hp ppop h fm s sgg ml f dz ml dom ls hp ppop fs o q qs [] dom whl ml dom m l gm ol l followg dp shm poposd h sd gm Rf l followg shm h poposd h sd mplmg dm ðþ oss m o fd d dls o qs Rf [] dp ls hp ppop fs dlopd som mpls so p fs dlopd som ml mpls sold sg ff dp homogz fmwo d ls h hp p sg ff oml s show g ml sold dhd s homogz fmwo ls ss sdg l oop s ho f o sg ls ss sdg l oop s ho f o sg ff ls ss sdg l oop s ho f o dom whl ml dom m l gm spl whh ol gm l followg dp shm poposd h sd whh f ol g f s ps ompss oml ss s sgg ml f dz ml dom offs sl oml ; g l fl f Sh ss s ompl os mdosl mplmg l dm sgg ml f d D s ml pm lolz ml dom whl ml m sg ff dp homogz f gwg l oop s ho f o sg ff dp homogz fm poposd Ddz lolz dom g oop s ho f o@ dp h sd (omp shm ml dom dp sg ff homogz fmwo Homogz homogz fmwo ff dp ( ( dp ll s lshp sg o sd dp ll dp ll Homogz Homogz mso-sl ldg Df D D fs ompss f ms sgd f fs f ms sgd sd fs f ms sgd f sd / dz ml dom ls ss sdg l oop s ho f o s h f ol gm l followg dp shm poposd h sd ls ss sdg l o Homogz Homogz sgh lop pl sh ss s lshp gs fdz spl ss s ompl os mdosl mplmg dm ss how s ompl os mdosl s sgd f sd sgd f sd s o f o lodg-lodg l lso show o lls pposs sgg ml f w sgg ml omp s s po ollps ( l l os mdosl mplmg H dm D R lodd ll ll f s q ll lodd omp f s s ls s q lodd modls f s s q smd pm 6 H / hdg s whh sgd ss sdg l oop ho dp f o fs f ms sgd ms f l os mdosl mplmg lolz dm f h f ol gm s l followg shm ms poposd fs h sd f s s q ls s s f sd whh f ol ds l o q-l mls s lolz os fl Th dmg ds ms o q-l mls lolz os fl Th ms gm dmg o q-l mls s os dmg ds q ss s mdosl followg fl 6 Th d/ ss woh gm l dp shm poposd s h sd gs os pm gog ompl / os 5 mplmg dp ll Homogz qs mls s lolz ml fl Th ms dmg ds mls os fl Th ms dmg ds s ls ls h hp ppop fs dlopd som ml mpls sold followg h hp ppop dlopd som ml mpls sold l qs gog LD fmwo ompld hogh s hdg lolz ls h hp ppop fs dlopd som ml mpls sold ( ( og l m d s sgh l wh sold f ms ml mpls sgd f sd s hdg ( som ml (ll H omdlodd fs ppop fs som ml lodd f s q fo h ppop fs dlopd mpls sold D ( s ll f lm lolz g ml dom whl ml ml dom m l gm gm l followg dp shm poposd h sd sd ls ss sdg l oop s ho f o lolz dlopd q g dom whl dom oop l gm lolz g ml dom whl ml dom m l gm ls ss sdg l s ho D whh f ol gm l followg dp shm po l whh f soop s ho f o homogz slop ql d s sh slop ompss ml dom whl ml m l l whl ml dom m gm p s D ospds olm s L gm sg ff dp fmwo sg ff dom ddl dp homogz fmwo dom sg ff dp homogz fmwo qs h pl gl m l qs g lodd ll f s s q fl Th dmg ds o q-l mls s lolz os fl ms dmg ml dp homogz fmwo p homogz fmwo opd som mpls sold sgg ml o f f dz dom od q ( hgh h ds ls h hp ppop dlopd ml mpls sol sgg f Th dz dom sgg ml f dz ml dom ls h hp ppop fs ms f ms ml sgd sd ml oml s ; ols olmd som soop s ho f fs fs f ms ml sgd sdg dz dom ls ss l oop s ho ls f dd ss s f ml po m dfd dd sgd f sd f dz ml dom l fls l ffs mso-sl ompss ho ls h hp ppop fs dlopd som ml mpls sold ml dom m l gm lolz g ml dom whl ml dom m l gm ss s ompl os mdosl mplmg dm ss s ompl os mdosl mplmg dm ss s ompl os mdosl mplmg dm sg ff dp homogz fmwo loddol ll f s h s q ff dp lodd ho ll sg f s h sgd mplmg f sd l sps ompss sd l lsl q ompl os mdosl mplmg dm d ls q ospds ms lsl ol q mpl os mdosl dm fs f sgd f fs f lm m s s q sg ff dp homogz fmwo z ml dom l wll smd ps ompss s sh sgg ml f dz ml dom whh f ol gm l followg dp shm poposd h sd whh f ol gm l followg dp shm poposd h sd whh f ol gm l followg dp shm poposd h sd m ls h hp ppop fs dlopd som ml mpls sold ls h hp ppop fs dlopd s s q 7sgh qs Rf [] ol gm l followg dp shm mpls poposd sd oss m o h o qs []s Rf d gm l followg dp shm hdg poposd h sd s fl ffs d l- o g D slop fd l modls d lodd dls ll f s s op fs som ml sold lodd ll o mplmg dm ss dlopd s ompl os mdosl mplmg f dm ls ss sdg l oop s ho f o ls ss sdg l oop s ho o ls ss sdg l oop s ho f o fl phom smld ffl hogh ds d zo od sml osd hozl ff dp sg homogz fmwo sg ff dp homogz fmwo p fs dlopd som ml mpls sold g l oop s ho f o oop s ho f pl lsl ml pl [] ml sh ml fd pl pml d o hd ls followg h hp sgd ppop #h fs dlopd som homogz fmwo ls h dp shm poposd h #s whh f ol gm Th l dp shm poposd sd fs f ms lld sgd f sd f fs d f ms f sd fs ms sgd f sd p p sg ¼ ¼ T L T L L s sgd f sd sgd f sd homogz fmwo ll sg ff homogz fmwo f o sg ss sdg l oop s ho dp f oq lodd ll f s q ff ll lodd f s s s q ls lodd f s s f s s q s s q dp ll Homogz d f sd fs f ms ppop sgd dlopd f sd ls ls hp h hp ppop fs dlopd som ml mpls sold ls h ppop fs som ml mpls sold h hp fs dlopd som ml mpls sold ppop ppop fs dlopd som ml mpls sold fs dlopd som ml mpls sold dp ll Homogz dp ll Homogz s s lodd sg ll f s fmwo fmwo s q q sg ff dp homogz ff homogz dp homogz sg ff dp fmwo fl Th ms dmg ds mls s lolz os fl Th ms dmg ds homogz o q-l dp homogz fmwo p fmwo

9 polhdl pl w ; w (6 s fs m polhdl pl; f ; spmposd dos ps m ds; pl olm; od ; α α L L sl ppld h gl f; w wh sp od whh losd polhdl pl LD mplmd ompl sw md RS [7] h sssfll sml ho dff ps lo pms [6] mo LD h show spo pls modlg ho d dm lodg [] ll Sl R (SR do [] f sml R d [] wll fl f f-d [ ] ll homogz mhod ll homogz os d wo pm smps: ( olm ml whos ff mhl ho ompll ps ml sps dfd homogz os h olm ml d Rps olm lm ( whh l fs ml s modld pll [] ( sp sls hpos ld whh ms o sz h sz polm sm mo o sz h lgh f-sl ml hog g mmm ggg sz gl md lg ogh so sgd ppop ps dlg ml Gl df wo-sl homogz polm dpd g g g ml dom llsd glol mo-sl ood ssm dod l sdd homogos h sl o hog o lgd w ml s po mo-sl dom dpd g ps olm hogos ml s ml hog o h sl so-d f-sl modld ms d mso-sl pl modl wo sp lgh sls ospdg lol ood ssms dsgd po ml dom s lol mo- mso-sl polms spl odg sp sls hpos mo- mso-sl ood ssms ld follows η < η << (7

10 ( ( g : Goml pl wo-sl polm: ( o ml dom sosl dom wh ml hog ( Gom wo ghog pls η sm pos sl wo ghog pls whh sh g f show g s df h f q w ms dplm o s wo pls shg f o f shd w pls w ɛ α ( U Θ U Θ α ( ; g pl ods ; α s ssm f f sh s dfg ohogl f > ; U U dplm s od ; Θ Θ o s od ; s g ods f od s g ms od h dplms os smd dpd ls dd q lmd sm s dplms whh l smp s lg pl dm po f osd l q-l mls smp ps homogz m gol mll homogz o dg gog qs go polm dff lgh sls To omplh h gol mplog mp m polm fld ls mos owd mhods mhl ho d pl md mp ps pl dplm o flds o ollowg wo Rzh s [] dplm o g od U ( Θ θ( ppomd followg mp pss ( ( η ( ( θ( η ω ( ϕ ( ω ( ηϕ ( (

11 Tms od O(η hgh gld o pss ( ( os- fsl dplm s spl whh os fs wh sp d ( dfd l f m pos wh sp ω ω os f-sl ood ssm ϕ ϕ ospdg os-sl os D mp ps o pld [] dls d w ods sdd fsml ood dplm o od ood w m Tlo ss ps od od ood ood ssm pg dffl dplm o s w U ( Θ θ ( θ θ θ whh ( θ θ ( ssg s pss qs pss f s q ollg ms sm od mlpl sl df f s od follows ɛ α η ɛ α ɛ α ηɛ α ( [ ɛα [ ( ( ɛα ϕ ω ω m m ϕ ] ω ω α ( ] ω α ( [ ɛα ( ϕ ϕ m m ω m m ] ω m m ϕ α ( L- pm smol shold sd lgh p ls h hgd O(η ops sg l sp sls q 7: η η η shold md spsp doppd wh hgg dos o hg sg q sh ɛ α Hg mlpl sl df f s sdg f l s qs ( α α ɛ ( α mlpl sl df f l s w α η α α η α po [] h mp m f s lso ld f l s qs Usg dd mp m f s sld m qs pl q 6 d qs gog polm dff sls follows O(η : O(η : Ā α α Ā α α Ā ( α α (5 Ā ( α α (6

12 O(η : Ā α α Ā ( α α (7 d o qs pl l qs sld hogh Ā /η ; /η Th d o od popl oll ms sm od o gog qs sp sls ll d qs 5 hogh 7 pod [] gd od mo om O(η qs lds α g df ɛ α w ɛ α whh q pss gd od mo ( ( ω ( ( woh mg fld ls ω dplm o s l dpd ood ssm whh mpls s qs s smoohl mosl ml dom whl s o dom -sl qs gog polm q 6 gos polm whh ds α α ( α α ( q qs sgl pl sd sd O(η f α whh f ɛ α w [] ɛ α ( ω ω α ( α γ m κ m ( γ ω κ ω s s spl pos f od shd w pls lol low-sl ood ssm show g α α op po s s fs oml o gl s omps Rg q s fs m q f-sl df f oml gl s w ms low-sl dplm ( o flds ω ; sd m q ɛα α γ m κ m po mosl s s h f o wods sol polm s s shold ppld fs g gs f-sl sol ms dplms os ω h pl ms lld sfg s qs whl pod od ds d sol qs f α omp opl s os-sl qs gog polm ml mpl q O( q 7 lds sll ol qs gg qs o pls mo-sl sll q o-

13 spdg homogzd pssd ( α α ( olm ; ρ ms ds m q lsl pl dffl q gog q pods hogh homogzg sol polm dd fl mo-sl ol q ospdg dd ɛ ( α α ( po m α dfd α l αl dd d ls q ospds lsl ol q oss m o fd d dls o qs Rf [] sps mll homogz pod o pod d m homogz shm pld h s llsd g szd follows: ( mo-sl ml dom dzd sd d modl hd h f Gss po ( glol lodg sp d o s γ κ s lld h Gss po s ( osop s γ κ s h Gss po pod ospdg ( fs sg h f o ɛα α γ m κ m ( polm sold pplg ɛ α fs g pod od d -sl dplm o ω pls s α f ompd (5 osop s lld hogh qs d sps (6 homogzd s sfd ospdg Gss po pd odl s s dplm os

14 o m ( odl h Gss po m ppl m ods osop sol s po ppl s pod fs L ppld fs polm (D odl B Homogzd s polm sold LD sol d pod B: L lld fs D g : Gl fmwo lsl mll homogz mhod (7 Sp ( (6 pd lodg sps s md o polm sold d pod od ds show g To l ppl pod dm h od f ms h op o pl f g od dg B od dg D dpd g s ods d sl o q pplg m-sl s shold sd D h od dg h h ops o h pl dgs d g dm pod Sl wo-sl homogz lss psd pos wo [] show wh L D l odl (LD f-sl modl -s p glgl s s sl smd zo h pp s s smd s s omd fom lls Th ows s lsl s who ol DOs os-sl Homogzd l l ho h s homogzd sps sgd l l gms

15 [G] D [] ν D [] ( g5: 5: ff ( Yog s modls os s o wh sp sz l ls gh szs D sdd sd ff sz homogzd l pops dd h sz s d dff pl ds sd lzd m ff msh lz Th ls psd [] sd offs s l s qs oss ldg -s p wll f How h pp w l sg ff Yog s modls os s o h sdg h homogos ml dom Th ls mp dp homogz shm whh homogzd ν sgd s gg ls To To g LD goms followg pms sdd: mm mmm sphl gggszs szs d d d spl; m 6 g/m ; w m o w/ ; ; ggg gggm mo o/ / ; ; ll o off f dd followg pms LD LDf s qs: 6 6 G 5 5 α l l 5 5 s s 6 6 H / / α T / / lsl lsl l l l l s s q q sdd sdd mosl mosl γ s s λγ ( ( s d ; γ d ; s s s ; λ Lm ss whh ld s s ; Lm ss whh ld gg ss Yog s modls os s o ν hogh ( (λ gg ss Yog s modls os s o hogh λ ν λ (λ s ss ( s ss l dmd pplg pop s s s ff Yog s modls l dmd pplg pop s s s Yog s modls os s o wh sp sz psd g 5 show homogzd ls os s o wh sp sz psd g 5 show homogzd ls

16 ( ( g g 6: 6: gom polhdl pl d: ( ( 5 5 ( ( g ν g pl pl sz sz d Rgdg s s s s s s (D/d (D/d ppop ps dlg ml o d f ppop ps dlg ml o d f sz o s show s s d sd d d sz o s show s s d sd d d dff msh lzs Rsls show homogzd l pops om dpd d msh lzs Rsls show homogzd l pops om dpd pl d sd D/d s pl d sd D/d s l ls l ls l ff sps s D 5 wh s dff msh lzs h sz l sdd dsps l s s hdos D fd 5 (l s wh sfd d msh ompss lzs whl pod h sz sdd od dds l s d hdos h lodg fd (l d o s sgl fd hhdlompss wh g whl pod po opld wh ml Tpl pl ps gom h sz dpd od ds d h lodg d o sgl hhdl wh g g 6 m dsg LD pms sdd sm s psd po opld wh ml Tpl pl ps gom h sz dpd S g 6 m dsg LD pms sdd sm s psd Sss-s od fom lzg s wh dff msh lzs gd h S sz psd g 7 g 7 homogzd l ho d s whh Sss-s ll s od pos-p fom sps lzg oms s mo wh l d msh lzs sz d gd so-sl h sz opg psd g dmgd 7 g s 7 d homogzd ls l show g ho dff d s szs whh ll ss dmg pos-p lolzdsps owoms pl g mo l spm whh sz d so so-sl sz ff opg osd g dmgd 7 ff s sps dd hdos ls show fdompsss g d psd g szs s 7 dmg osds-hdg lolzd d ow d pl g fm ff spm whh whh lds dff so mo-s sz o olm O o h sg sps d fd ompss lodg d show osd g 7 sps d hdos fd ompsss psd g g 7 sod wh l splg s gd d ll ps g 7 osd s-hdg d d fm whh lds d mo-s o olm O o h sg sps 5 d fd ompss lodg d show g 7 sod wh l splg s gd d ll ps g pss opg dmgd s whh s mlpl splg s dlopd 5

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] s d h d szs fd ompss -s s h [ ] s h szs ( g fd ompss -s s h szs ( ( 5 ( 75 ( ] ( [-5 ( [- ] 5 5 [- ] [- ] g sl -s s h dff os f-sl g (g g sl -ss s h dff s -s osg ( ( g sl h dff szs f-sl s s szs f-sl Hdos ( ( Hdos Hdos g g sl -s s h dff os f-sl g ( g sl -s s h dff s g ( g sl -s s h dff szsszs os f-sl s s f-sl g ds dff szs g Hdos szs gds ds dff dff szs ( Hdos Hdos ompss ompss ompss 5 g ds dff szs g ds szs g ds dff dff szs g ( g sl -s s 5 h dff szs os- f-sl s ompss 5ompss ompss 5 5 Hdos g ( g sl -s s s g ds dff szs 5 Hdos h dff szs os- f-sl 5 5 ompss p ho s lss sps Th s wll-ow p ho s lss sps Th s wh wh wll-ow wll-ow sz sz p ho s lss sps Th sz ds dff szs g 5 ho s lss ompss p sps Th s wll-ow p ho lss sps Th s wh wh wll-ow wll-ow sz sz fd ompss p ho s s lss sps Th sz fd ompss 5 sod dmg lolz q-l mls [5] wh wll-ow fd ompss p ho s lss sps Th s sz z Z ff sod dmg lolz q-l mls [5] z zff zff sod dmg lolz q-l mls [5] z 5 5 fd ompss sps 5 lss fd 5 ompss sod 5s sod dmg lolz q-l mls fd ompss fd ompss [5] 5 p ho Th s s wh sz Zz dpd sod dmg lolz q-l mls [5] zff ff dmg lolz q-l mls [5] ff [ ] [ - ]wll-ow sod dmg lolz q-l mls [5] Th g whh dmgd dff szs d Yphom zzff [ ] [] [] Th phom dpd g whh dmgd s dff szs d [] 5 dff szs Th phom whh dmgd s d 5 dpd g sod 5 dmg g s fd ompss dff 5 dpd ff lolz q-l mls [5] [] [] Th phom whh dmgd szs d [ ] [ ] [] Th phom dpd g whh dmgd s dff szs d Y [ ] [] mso-sl poss plos ps ds [-ospdg ] sl plos Th dpd ps whh dmgd s ds dff szs ospdg d d opg lodg ] ( sl sl lodg poss mso-sl opg dpd g g whh dff phom (szs [ ( ( 5 lodg ps s opg Th 5 plos dmgd ds phom mso-sl ospdg ( poss ( dpd sl lodg plos ps mso-sl opg ds dmgd [-ospdg ]dmgd Th phom s g s whh s opg dff szs d [] ( poss [] mso-sl 7:( mposd l s ql s l o dos sl lodg poss plos ps ds ospdg g opg dmgd sl ql o (lf (mddl 5 mposd l ql l ( dmgd dos sl lodg poss plos ps mso-sl opg ds ospdg sl g opg dmgd s s ql (lf 5 ( ((mddl g opg dmgd s sl s ql (lf (mddl 5 (gh g g sl -s s h d szs g ompss olm -s g opg dmgd s sl s ql (lf (mddl 5 mposd l s ql l o dmgd dos ( ( sl lodg poss plos ps mso-sl opg ds ospdg g 7:( g sl -s s h d szs g ompss olm -s g opg dmgd s sl s ql (lf (mddl 5 (gh (gh g opg dmgd s sl s ql (lf (mddl 5 (gh g mposd opg s dmgd s sl s ql o (lf (mddl 5 ( ( g 7: ( g sl -s s h d szs g ompss olm -s l ql l dmgd dos (gh g sl opg dmgd s sl sp ql (mddl 5olm (gh s h d szs (gh sl lodg poss plos ps mso-sl opg ds ospdg szs g 7: ( g sl -s s h d (lf g ompss -s s h d szs o wh sz ldg pos sz dpd sz g opg s sl s ql (lf (mddl 5 (gh g opg dmgd sl s ql (lf (mddl 5 (gh sz ( ( o sl wh sz ldg pos p sz mposd l s ql dpd l o dmgd dos (gh g opg s dmgd s sl s ql (lf (mddl 5 s h d szs mposd l ql l o dmgd dos sz g opg dmgd s sl s ql (lf (mddl 5 o sl wh sz ldg pos p dpd sz g opg dmgd s sl s ql (lf (mddl 5 (gh s h d szs g 7: ( g sl -s s h d szs g ompss olm -s g opg dmgd s sl s ql (lf (mddl 5 mposd l s ql l o dmgd dos o sl sl wh sz ldg poss p sz dpd g g -s s h d szs g ompss olm -s g7:( opg dmgd s sl ql (lf sz (mddl 5 (gh z (gh g 7: opg dmgd s h sl s ql szs (lf (mddl (mddl (gh dp ll Homogz g opg s dmgd s show sl s ql (lf 5 5 g g sl -s s d g ompss olm -s mposd l ql l o dmgd dos (gh ol -s dmg lso g f dff s g ( opg dmgd dmgd s h sl s ql (mddl 5olm (gh s s h d szs (gh g 7: ( g sl s d szs (lf g ompss -s s h d szs o sl wh sz ldg pos p sz sz g opg s sl s ql dpd (lf (mddl 5 (gh ol dmg lso show dpd g f dff oh sl wh sz ldg p sz sz ol szs dmg pos lso show g f dff s (gh s d pss opg dmgd s show whh s s dlopd o sl wh sz ldg pos p sz dpd splg sz s h d szs ol dmg gm lso g mlpl f dff s lls S lls ( ( p-p whh dmg dd hogho o sl wh sz ldg pos p sz dpd sz o o q-l mls hzd s-sg ho whhhogho lds s dp ll Homogz lls S lls ( ( p-p gm whh dmg dd s lls S lls ( p-p whh dmg dd hogho ol dmg ( gm lso show g f dff o olm d lodg poss ms od ff sps d ol dmg lso show g f dff s lls S lls ( ( p-p gm whh dmg dd hogho ol dmg Th lso show g f dff s lolz s d ms dmg ds lolz g ol dmg lso show g f dff s lls S lls ( p-p gm whh dmg dd hogho hdos fd fd ompss o dpd sz psd gs (mls 5 7 odom o q-l hzd ho whh lds s 6 5 ml whl ( ml dom lod m l gm sgg ml 7 lls ( ( p-p p-p gm s-sg whh dmg dd hogho 6 5 lls S S lls lls ( gm 7 whh dmg dd hogho 6 5 Thlls ds lls f sl f dmg dd hogho spm ( ( p-p gm whh dmg dd hogho lolz s d Th ms dmg ds lolz g f dz ml dom ss s 7 dos o lolz ompl os mdosl dfg dm f ml dom whl ml dom lod m lwhh gm sgg ml 7 55 Z 7 zz zz Zz ss gm ms sgd ml hghl fl o wods ls ol f wh dz ml dom ss s z Z 5 Y Y sdg Y l oop s ho s o sgd dp ompl osll mdosl Homogz dfg dm whh f Y f ms dfd sgd f zz Zzd Z z ms 5 55 ss olz gm sgd wh ml hghl fl o wods ls o o q-l mls hzd s-sg ho whh lds s lodd ll f s s q Z zsd g opg dmgd s sl s qlql (lf (mddl 55 5 g opg dmgd s sl s (mddl Y l Y dmgd g dmgd s sl s ql s (lf 5 (gh opg opg dmgd s sl ql(lf (lf (mddl 5 g opg s sl s ql (mddl (mddl 5 (gh g sdg oop s ho s o (lf (gh gy opg dmgd s sl s ql (lfsgd (mddl 5 (gh ls h s ppop dlopd som ml (gh lolz d Th ms dmg ds lolz g ml g opg s ompss s ql -(lf (lf (mddl 5mpls (gh g opg dmgd dmgd s sl s ql (mddl 5 (gh g (gh opg dmgd s sl s ql (lf (mddl 5 (gh opg dmgd s sl s ql (lf (mddl 5 Yg g opg ms dmgd s sl s ql (lf 5 (gh d f dfd sgd (mddl f g dom whl dom m l gm ml (gh opg dmgd s lod sl /o s ql (lf (mddl 5 sgg (gh ml dp ll Homogz ol dmg lso show g f dff s dp ll Homogz lso ol 5 sd lodd ll f s s q dp ll Homogz dmg show g f dff g opg dmgd s sl ql (lf (mddl 5 s 5 ol dmg s lsoshow s g ql dff s g opg dmgd sl s (lf (mddl dmgd g opg dmgd s sl s ql (lf (mddl 5 (gh g opg dmgd s sl s ql (lf (mddl 5 lls S lls ( ( p-p gm whh dmg dd hogho g opg s sl s ql (lf (mddl 5 (gh 6 (gh o o q-l mls hzd s-sg ho whh lds s 5 dp ll Homogz g opg dmgd s sl s ql (lf f (mddl 5 (gh ls h s dlopd som ml (gh g opg dmgd s ompss s ql - 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18 p s wh wll-ow wll-ow sz sz pho ho s s lss lss sps sps Th Th 5 fd ompss ompss sod dmg lolz fd q-l mls [5] wh wll-ow sz fd ompss p ho s lss sps Th[5] s Zz sod mls zff zff sod dmg dmg lolz q-l mls [5] zzff q-l lolz sod 5 dmg q-l ompss [5] fd 5 dpd ff lolz - ]mls szs [ - ] Th g g whh dmgd s dff d Yphom [[-- ]] ] [ [ [] Th dpd Th phom whh dmgd s dff szs d phom dpd g whh dmgd s dff szs d 5 5 dpd g whh [] mso-sl [- ] dospdg phom dmgd s opg dff szs [-poss ] sl lodg plos plosps ps ds Th sl lodg opg ds ( poss mso-sl ( ospdg (( poss ( sl plos ps mso-sl opg ds lodg ospdg ( sl lodg poss l plos ps mso-sl opg ds ospdg s ( mposd s ql l o dmgd dos g opgl s dmgd s sl ql o (lf (mddl 5 mposd ql l ( dmgd dos sl g opg s dmgd s s ql (lf (mddl 5 g opg dmgd s sl s ql (lf (mddl 5 (gh g 7:7: ( g sl -s s h d szs g ompss olm -s g opg dmgd s sl s ql (lf (mddl 5 mposd l ql l o dmgd dos g ( g sl -s s h d szs g ompss olm -s g opg dmgd s sl s ql (lf (mddl 5 (gh (gh g opg dmgd s sl s ql (lf (mddl 5 (gh g opg dmgd s sl s ql (lf (mddl 5 g 7:(gh g sl -s s h d szs g ompss olm -s mposd l s ql ql (lf l o dmgd dos g ( opg dmgd s sl s (mddl 5olm (gh s h d szs (gh g 7: ( g sl -s s h d (lf g ompss -s s h d szs o sl wh sz ldg pos pos p sz dpd sz g opg dmgd s sl s ql dpd (mddl 5 (gh o sl wh sz ldg p szszs sz (gh s h d szs o sl wh sz ldg pos p sz dpd sz s o hsl d szs wh sz ldg pos p sz dpd sz dp ll Homogz ol dmg lso show g f dff s ol dmg lso show g dff f dff ol dmg lso show g f s s ol dmg ( gm lso show g dd f dff s lls S lls ( p-p whh dmg hogho o o q-l mls hzd ho whhhogho lds s lls ( ( ( p-p p-p gm s-sg whh dmg dd lls S S lls lls ( gm whh dmg dd hogho lls S lls ( Th p-p gm whhds dmg dd hogho lolz s d ( ms dmg lolz g ml dom whl ml dom lod m l gm sgg ml f dz ml dom ss s ompl os mdosl dfg dm whh f zz ZzZ Zz 5 55 ss ol gm ms sgd wh ml hghl fl o wods ls z ZY Y 5 sdg Y l oop s ho s o sgd Y f ms dfd sgd f d s q lodd ll f s sd g opg dmgd s sl s ql (lf (mddl 5 5 g opg dmgd s sl s ql (lf (mddl dmgd g dmgd s sl s ql s (lf 5 (gh opg opg dmgd s sl ql (lf (mddl 5 ql g g opg s sl s ql (lf (mddl (mddl 5 (gh (gh g opg dmgd s sl s (lf (mddl 5 (gh g opg dmgd s ompss s ql - (lf (mddl 5 (gh ls h s ppop dlopd som ml mpls (gh g opg s ompss s ql -(lf (lf (mddl (mddl5 5 (gh (gh opg dmgd dmgd s sl s ql g (gh g opg dmgd s sls sql (lf (mddl 5 (gh opg dmgd sl s ql (lf (mddl 5 g opg dmgd s sl s ql (lf (mddl 5 (gh (gh opg dmgd s sl s ql (lf (mddl 5 (gh g dp ll Homogz ol dmg lso show g f dff s dp ll Homogz 5lso show g f dff s dp ll Homogz ol dmg ol dmg lso show ml g dom f dff s lls f ( dz ss s S lls ( p-p gm whh dmg dd o o q-l mls hzd s-sg ho hogho whh lds s 5 dp ll lls S lls ( Homogz ( p-p gm whh dmg dd hogho 5 q-l dmg gm lso show g dd f dff s ool o hzd s-sg ho whh lds s lls S lls ( (mls p-p whh dmg hogho 6 lolz ompl os mdosl dm whh f o s o q-l dfg hzd s-sg ho whh lds s d mls Th ms dmg ds lolz g 5 dmg lls S lls ( ml p-p gm whh dmg hogho lolz s d ( Th ms ds dd lolz ol gm ms sgd wh hghl fl Th od sg m o whl o q-l mls ms hzd s-sg ho whh lds g s lolz s d Th ds lolz g ml dom ml dom lod l gm sgg ml dmg ml dom whl ml dom lod m l gm sgg ml ls l oop s ho who f dz lod ml dom ss g s s lolz sdg s Th ms dmg ds lolz sgg ml dom whl d ml dom m s l gm sgg ml f dz dom f ss s whl d gm sgd f os mdosl dfg ml dm whh ompl dom f dz ms ml dom ss s ml ml dom lod m l sgg ml ompl os mdosl dfg dm whh f sd lodd ll f s ol ms sgd wh ml s hghl fl o wods ls ss ompl os mdosl dfg dm f gm f dz ml domwhh ss s q wh s s ppop dlopd som ml ol gm ms ls sgd ho ml o hghl fl o wods ls sdg l oop s sgd ss ol gm os msmdosl sgd wh ml hghl flwhh o wods ls ompl dfg dm f ss d f ms dfd o sgd f sold sg ff poposd dp homogz fmwo mpls sdg l oop s ho s sgd ol sdg l oop s s hghl o sgd gm ms sgd whho ml fl o wods ls ss sd f lodd ll f s q d ms dfd sgd s f d l foop s ms dfd sgd f sdg ho s o sgd lodd ls h s ppop dlopd som mpls s sd ll f s ml s q sd lodd ll ms f s sgd q d f dfd fs ls h s ppop dlopd som ml whh s om R D-g wdl mpls lz Os ls h ppop dlopd som ml mpls 5 s q sd lodd ll f s ho sdd h s Os Hgh-Wsgd sp sd ls h s ppop dlopd som ml mpls 5 [] 5 f (ξ ρ θ ξ ρ(θ ρ 5 (6 ξ ρ Hgh-Wsgd sp ls dfd ξ ; ρ p τo (7 ss fs whh olm ppl s τo 6 [( ( ( ] sd d τo ohdl sh (θ dfs shp 7

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