A KAM theorem for generalized Hamiltonian systems without action-angle variables

Transcription

1 ho fo gald aloa sss whou ao-agl vaabls Yo u Jo u wa Jog : aual L UG Uvs Pogag oa Popl s publ of oa : Faul of ahas L UG Uvs Pogag oa Popl s publ of oa bsa povd a ho o s of vaa o gald aloa sss whou ao-agl vaabls s a galao of h sul of d la Llav al Llav 5] ha dals wh aoal aloa ss wods: ho vaa o gald aloa sss ao-agl vaabls al gabl ss oduo hs pap w povd a ho o s of vaa o wh a opha vo fo a gald aloa ss sudd gald aloa sss whh a o b pubaos of gabl sss o a o b w ao-agl vaabls lassal ho pss of vaa o s povd fo pubaos of gabl aloa sss : al gabl aloa ss wh ao-agl vaabls ologoov 95]old 963] os 96] 98]Pöshl 9]Fo ]Fo ] o-agl vaabls alwas s a ghbouhood of a vaa ous owv a hag of oodas bgg h ss o ao-agl vaabls gal ao b pll opud Llav 5] Llav 5] obad a ho fo aal aloa sss whh a h pubd gabl sss o w ao-agl vaabls h obad h ho o solv h paal dffal quao whh vaa o ap sasfs b quas-wo hod ao-llav] appld hs sul o h ual opuao of vaa o Luqu-llauva ] povd s of low-dsoal vaa o aloa sss whh a h pubd gabl sss o w ao-agl vaabls Jog-Pak ] povdd a ho o s of vaa o fo dffabl aloa vo flds whh a h pubd gabl sss o w ao-agl vaabls L-Y ] povd s of vaa o fo al gabl gald aloa ss wh ao-agl vaabls h appld h suls o sah lad o h pubao of h dsoal sad Eul flud pal pah flows Lu-Yh-uag 5] povd s of hpbol vaa o fo al gabl gald aloa ss wh ao-agl vaabls Lu-Zhu-a 6] povd s of low-dsoal vaa o fo al gabl gald aloa ss wh ao-agl vaabls L-Y 6] osdd khooshv sabl fo al gabl gald aloa ss wh ao-agl vaabls usg ho a paal applaos o has o osd h sss ha a o los o gabl ss bu vhlss o has so appoal vaa o wh sall ough o obad fo apl b usg a o-goous ual hod so aspo paso Llav 5] O h oh had gald aloa sss of g s ad gg fo

2 apl ologal quao of pdao ad p Loka ola odl hfo wll hav a sgfa o gal h suls of Llav 5] fo aloa ss as o gald aloa ss as aodgl sah a ho o s of vaa o fo a gald aloa ss whh a h pubd gabl sss o w ao-agl vaabls h poof fo h ho follows h poof poss Llav 5] fo aloa as owv h a so opls h baus ha osa suual a aloa quao vas o vaabl suual a hs pap s ogad as follows: so w osd vaa afolds gv b a paal dffal quaos lad s ouga ap h pobl o fd h vaa o fo gald aloa sss s fd o fd h soluo of a paal dffal quao vaa o quao aodg o h sul so so 3 w osd Lagaga pop of h vaa o ad appoa Lagaga pop of h appoa vaa o so w osu a asfoao o appoa dubl p fo vaa o quao ad osd solvabl of h dud quao so 5 w of o-dga pop of appoa soluos of vaa o quao Lasl so 6 w pov s of vaa o of osdg gald aloa quaos basd o ovg of quas-wo hod fo h vaa o quao Paal dffal Equaos gv b s-ouga ap ad vaa afolds Fs w df so oos ad odu so oaos L Z do h ss of aual ubs gs al ubs ad opl ubs spvl do s of all ogav gs b Z do aah spas as E F G E F G oao U X as ha U s a op s of opologal spa X f a subs Y of opologal spa X sasfs Y Y h Y s alld as sadad s of X oao Y X as ha Y s a sadad s of opologal spa X L X b a subs of spa X d all da sup{ d ; } as da of h s L b a fo fo L / Z b a -dsoal ous L U b a op s of o U wh a op s U of L : U b a al aal fuo assu ha b U s a al aal a-valud fuo suh ha: b b b b k b k blk bl bl l l l l Oda dffal quao d d

3 s alld as gald aloa ss wh aloa ad suu a Zhao 995] Fo so s good ough fo ad b o b fuo bu fo 3-6 sos al aal s ssa do o-pologabl soluo sasfg al odo U of dffal quao b s h au sg val of soluo fo h al valu pobl s a op val volvg L do h Eulda sadad sala podu o L U b a dffal -fo o f h ss a a-valud fuo U suh ha U U h s alld as psao a of dffal -fo assu ha a s vbl fo a U o d suh ha d d d w assu ha fo dffal -fo U h s a dffal -fo La s a spl fo o U ad holds fo a Poof s o-dga skw-s * J s also o-dga skws hfo s dffal -fo o U h assupo d s losd dffal fo so s spl fo o U s Posso ap a 6] holds akg h vs a of boh sds h las lao w hav ] d ulplg o hs lao fo h lf b hfo hs as * J J ad fo gh b w hav fo Gv k k Z l k do h absolu su o k k of k L f h s ad suh ha k k k k Z \{} h w all ha s -opha s alld as opha odo do h s of all - opha vos b fo 3 L us s a gv posv ub L U { ; } ad Fo U w l 3

4 U { : U ; s ouous o U o-pod all s vaabls o U ad al aal o h U } h U bos h aah spa wh o sup udoé 96] do U U b spl L { / Z ; } h h s s dfd o h s U { : ; s ouous o U ad al aalss o L U b gv f sasfg da U ad l { ; f } } h s a ghbohood of s U o hs so w laf ha wo daal sss a souga f ad ol f h h ag of h s-ouga wh so s-ouga sasfs a paal dffal quao hs as s-ouga bos h vaa afold of h sod daal ss hs sul fs h s of vaa afold fo daal ss o h s of soluo fo h paal dffal quao fo L o uppos ha s afold X : s fld o ad do h o-pologabl soluo of h al pobl b : wh X s a op val volvg X vo X 3 3 L X { } ; X } h X bos a op s of volvg { } h ap s alld as asd 978] gal of vo fld X o gal : X ; : X bos pop of soluo of auooous ss 3 fo a X gal of dffal quao 3 baha- ap dfo of gal ad 5 s s s s 6

5 hold f X holds fo a h h vo fld X s alld as opl baha-asd 978] f X s opl ad X gal : of X sasfs h ap s s s : s alld fo a h : bos fo 5 L f fo a flow of X hs w df s h s alld as vaa s wh dffal quao 3 o wh s s dffoophs ad sasfs gal vo fld X f a vaa s wh dffal quao 3 has h suu of h s alld as h vaa afold wh dffal quao 3 f vaa afold wh dffal quao 3 s hooophs o : b : X of afold ous / Z h s alld as vaa ous wh dffal quao 3 fo 6 L b afolds X : Y : b vo flds : X b f lao holds w all ha gal of X : Y b gals hs u s alld as gal of Y ad u : b ap u u u 7 : X ad : Y a s-ouga bw ad X Y s-ouga wh ap u fo 7 L b afolds ad X : Y : b opl vo flds L : : b b hooophs f h lao holds w all ha flows flows of X Y spvl L u : u u 8 : ad : hs u s alld as opologal ouga bw ad a opologal ouga wh u ho L b afolds L X : Y : b gal : X of X ad gal : Y of Y a vo flds s-ouga wh ouga u : f ad ol f u sasfs paal dffal quao 5

6 Poof ss: uppos ha 7 holds F u X Y u X Y u 9 abal Fo 7 fo all u u holds dffag boh sds of wh sp o w hav u h akg h las psso w hav 9 u u X Y u uff: L s suppos ha 9 holds F abal L s osd h ap : u ; u X Y d d u u u X Y u Y d d : X Y u s a soluo of h dffal quao Y sasfg al odo u O h oh had fo h dfo of u X Y u s also h soluo of dffal quao sasfg al odo u Fo uquss of h soluo of al pobl of oda dffal quao u u u holds fo a u X u Y oolla L s do o-pologabl soluo of aloa ss sasfg al odo U b Fo ap : U w df opao b h ap 6 : U sasfs f ad ol f sasfs h fs od paal dffal quao ss ak h as of 7 s gv as follows: u f g X ad Y f g f g u u ho L E F b aah spas s E afold s F afold ad 3

7 X : Y : b paal dffal quao 9 h u has a suu of E u u vo flds suppos ha afold o u : sasfs 3 f X opl h u s h vaa afold fo dffal quao f u s losd s of h u s h vaa afold fo dffal quao 5 f s opa h vo fld X : ad Y : a opl u s h vaa afold fo dffal quao ad u : s hooophs oov flow : of X : ad flow : of Y : a opologal ouga wh u Poof : uppos u : s o ow w odu h suu of fo s u do h vs ap of bo u : u b afold u : u L { U J} s h alas of pu u U fo a J ad df ap : E as u h w pov { J} s alas of s u Fs of all w hav obvousl Fo a J J J u u U u U u J u U u U U U s op s of aah spa E ow w pov ha ooda asfoaos a u u u u : U U U U : s hfo { J} s alas of s u s u has h suu of E : w pov ha afold u fo u u Fs w pov u L w h h ss op val : ad ap : u wh suh ha w ] ] s h quval lass of uvs o whh a ag o a baha- asd 978] ow l s a l sasfs u ad df ap ~ : as 7

8 ~ u h w hav ~ u u u w pu v ~ ] hs w hav u v u ~ ] u ~ ] ] w u hfo w u holds w hav povd u w pov u L ak v abal h h ss op val : ad ap ~ : wh ~ suh ha v ] ~ L s osd ap u ~ : u h u ~ u pu w h s u v u ~ ] u ~ ] ] w u w hav u hfo s povd 3: L s ak abal h h s a uqu suh ha u X s opl w hav u u u h s X Y Y u u u fo a u fo ho s h vaa afold of X Y : L s suppos ha 9 holds fo Y o 8 Y Y u : u s o subafold of gall bu has h suu of afold slf as w s ad h ag spa of a s gv b L s suppos ha u fo so w hav Y Y u u X u X u fo 9 Y ~ : ; Y ~ Y dfs a vo fld o afold do h gal of dffal quao Y ~ 5 wh vo fld Y ~ : b ~ : ~ L s ak ad ~ abal h s Y d ~ ~ ~ ~ Y Y ~ d ~ s h soluo of sasfs hfo w hav ~ Y Y Y ad ~ ~ Y ow w pov ~ Y f w d hs fa h h ss s Y Y suh ha s ~ o s suh ha s ~ osd h fs as h Y Y agu fo h sod as s as sa as h fs as L s f{ sy ; s ~ } Y h s sup ~ h ss l l ~ l h s s losd s Y s s Y Y

9 of b h assupo holds h ss soluo of wh f w df ~ ~ Y s s s h ~ s s s h soluo of wh hs oads ha Y ~ s o-pologabl soluo of wh hfo w hav Y ~ Y Y ~ fo a Y ~ s vaa afold of Y 5 uppos ha s opa h u s also opa ad vo fld X : ad Y : a opl Fo 3 u s h vaa afold of bv ouous ap fo opa opologal spa o ausdoff opologal spa s hooophs u : s hooophs h fo ho u u u u u u holds fo a ow osd h gald aloa ss d : d Fo l us df u u oolla uppos ha h paal dffal quao h u s vaa ous of u : s hooophs s a opha vo ad o u : U sasfs u u u 6 3 h flow of sd o s dfd o ad s opologall ouga fo hooophs u : wh h ok s flow hfo h sd flow of o s quaspod Poof L df vo fld X o b h b X : ; X 7 flow of X ods wh h ok s flow df ha vo fld Y o U Y : U ; Y 8 do gal of Y b U h 9 s w b 6 hfo fo u s vaa afold of ad u : s hooophs s vaa ous of Fo ho sd o s dfd o Y ad 9

10 s opologall ouga o h ok s flow wh u : hus u u Fo oolla h pobl o oba vaa ous fo gald aloa ss fs o h pobl o oba soluo of ola paal dffal quao 6 fo 8 asd o oolla w all 6 as vaa ous quao L s osd ap U U h w all fuo U as o of ap fo quao 6 al ub sup{ ; U} as o-s of ap fo quao 6 d w all as appoa soluo of quao 6 wh o all appoa soluo of 6 wh o o as soluo of 6 f U U s soluo of 6 h fo a U 3 Lagaga vaa ous hs so w show ha h ag of h soluo of h fs od ola paal dffal quao 6 dv b h gald aloa daal ss bos Lagaga vaa ous wh spl suu wh ad h ag of a appoa soluo bos h appoa Lagaga vaa ous hs fa plas a ssal ol o g h s of h soluo applg wo hod o h fs od ola paal dffal quao 6 L b aual ubs L us do o b L us odu au o fo a l of f dsoal Eulda pa as a akg a as h l of h au o s gv as a{ a ; } O h oh had w df h absolu su o of holds df h opao o of Poposo 3 L us s a sup \{} as h a as sup a s l a ad h h followgs hold:

11 Poof : a b a b k k k k : hs s a spal as of 3: Fo k k k sup sup 5: Fo opao o of a s dl obad 6: L a a a a a a a a a f w appl fo h a Fo a w hav a a oov a a a a a 7: Fo 3 ad h dfo of au o 8: Fo 7 ad 6 Poposo 3 L us l l a as h a l ad a l l as df Poof L l l h sup sup a a sup a

12 Poposo 33 auh s sa L ad opl fuo f s aal suppos ha fo a f a a{ f ; } fo a h w hav pu a 5 Poof hlfos 966] oolla 3 L ad U { ; } L f sup f fo boudd aal fuo holds Poof F s f : U h fo a f f U U abal akg abal fo losd ball U wh adus holds Fo auh s sa f a{ f } sup{ f U } f oov f f oolla 3 L b aual ubs ad w l Fo boudd aal fuo Fo U w l U { ; a } f : U w l f sup f sup a f U U f f f f f L f b a opao o of a f ad f sup f h fo aba holds Poof Fo a U Fo auh s sa f f U U holds F abal f a{ f } f

13 ad fo 3 of poposo 3 ad oolla 3 w hav f f f a ; f w hav f f La 3 L d d d b dffal -fo b dffal -fo gv b d o U ad : U b a L a a h h followgs hold; 3 ap L a ad h psao a of dffal -fo s gv b Pull-bak * * of dffal -fo b s gv b a d 3 h psao a of h pull-bak * of dffal -fo b s gv L a a fo 3 o 99] L b a -dsoal spl afold ad a - dsoal subafold of s alld Lagaga f ω ξ η ξ η fo a p f -dsoal ous U s vaa wh h gald aloa ss ad U h s alld as Lagaga vaa ous of h gald aloa ss O h oh had l L b h psao a of f fo a ad a d h s so suh ha f h L holds s alld b appoa Lagaga vaa ous ho 3 f bddg : U s h soluo of 6 wh opha vo s Lagaga vaa ous fo h gald aloa ss Poof Fo oolla s h vaa ous of h gald aloa ss Fs l us show h psao a L of s osa fuo Fo * d p v l us df ap v : b v v Fo oolla w hav 3 h fo la ad 3 w oba * * * * * 3 L us l : ; ad osd asu psvg daal ss of ok flow s aoal dpd asu psvg daal ss s god old- v 968] Fo a w l * f h s w oba * * f f f

14 f f 33 s god -vaa fuo f s osa alos vwh old- v 968] oov s f s ouous f s osa fuo hfo L s osa fuo Fo la 3 h psao a L of s w b L d fo d O w hav L d O L s osa fuo w oba L O hus holds * ow l us ak as spl suu o ad s * dffoophs fo o akg a ad a h s a uqu suh ha ad h a uqu suh ha h s * -dsoal ous s Lagaga vaa ous of La 3 Gv h U w assu ha h h d s o h h s a uqu al aal soluo v : U of h fs od paal dffal quao v h 3 suh ha o-pod all s vaabls ad s avag s oov h followg holds; v wh s osa dpdg ol o ad h 35 Poof üssa 975] üssa 975] h as of sg -ous as / / / s gv b 3 fo 3 L us assu L fuo of s dfd b / Z : U U b a al aal ap h o θ U 36 ho 3 L d d d h h followgs hold: h ah opos of vo hav o avag: f hus f s soluo of h h s b : ad * a suh ha db a d 37 ~ 3 h ss b : g : ad a suh ha oov * ~ db a d g d 38 g 39

15 5 h ss a osa } a{ dpdg ol o whh s dod b poloal wh posv offs wh suh ha h psao a L of * sasfs L 3 Espall f h L Poof : Fo of la 3 h pull-bak of d d ud s gv a d d * a d Fo of la 3 h psao a of d d d s gv b J ow l us alula a ] ] } ] { ] ] ] } ] { ] } { ]} { a 3

16 6 ] ] ] w hav ] ] ] 3 h sa wa w oba ] 33 Fo 3 3 ad 33 w hav } ] { } ] { } ] { ] ] } ] { a hus w oba } ] { a

17 h avag of lf-had sd of 3 ad h avag of h fs qual o o w hav O h oh had fo la 3 fo a avag suh ha holds ad : h fs of h gh-had sd of 3 w l 3 35 h ss al aal fuo wh o g 36 h ] h h b la 3 h s al aal fuos b b : wh h o avag suh ha b h h b h h f s h soluo of 6 ad h g ad b hfo h b a { b h } hus w oba a b osdg fo la 3 w hav hfo akg w hav w hav * * b h h a a a b b a d a d d a d a a a b b db a d hus s povd 3: a { h h} b b { g } g { b h b h ]} g 7

18 b b a g ] b Fo b g w hav b b a g a a b b a g subsug abov psso o * a d w hav * a d b b d a d g d ~ hus akg b b b a a a * ~ db a d g d : Fo 3 * * d d g d holds ad fo la 3 w g L g g hfo fo 8 of poposo 3 w hav L g g g g g Fo abov psso auh s sa oolla 3 36 ad la 3 h ss osa a{ } dpdg ol o whh s dod b poloal wh posv offs wh L suh ha aabl asfoao o appoa dubl p hs so w du h vaa ous quao 6: o a appoal solvabl fo b a vaabl asfoao L : ad L s df F : U U U b 8 F h pobl o solv 6 s quval o h pobl o solv ola quao wh ukow : F od o solv b quas-wo hod l s dv h lad quao F F 3 df F : U F : U b

19 9 F F spvl d w df ouous bla ap L : b h s w hav F 5 ow w df L U f U g b g f 6 spvl Fo L U U w df ~ U b ~ h : ~ U U L U s ouous bla ap ad hus ~ ~ ~ d d d d U d L U d 7 ] ~ ~ g f g f g f F w hav ~ g f F 8 Fo h dffao foula fo oposo ap ~ g f g f F 9 holds Usg 7 w hav ~ ~ ~ g f g f g f g f g f g f F Fo aps : L U ad U u : w df ap U u : b u u f w hav f fo oposo ho w 98] h sa wa g holds f w d ad fo ] g f g f F

20 hus O h oh had s F 3 F s ouous la ap wh w oba F hfo fo 3 ad h lad quao 3 s gv b h s b k k k akg a-valud fuo whos k -l s gv b w hav hfo f w df k k k b k b U k 5 6 h h h 7 U 8 9 U h h lad quao 3 fo quas-wo hod fo s w as L s ak ha fo uppos ha θh h h 9' sasfs h followgs: Y U s o-dga a do h vs a of Y b h avag of a-valud fuo θ: { s o-dga a h w all ha do h s of o-dga La Fo ]} s o-dga b 3

21 : f w df f a 5 h a a 6 holds Poof do h o-pologabl soluo of : d d sasfg b Fo f d d 7 7 w hav f d d 8 Fo h dffabl ho of h soluo of oda dffal quao wh sp o al odo w hav a f d d 9 wh s dsoal u a 9 s Posso ap a 6] 3 holds ffag h boh sds of 3 wh sp o w hav ] ] ] d d d d d d Fo 7 ad 9 w g ] ] a a akg h las lao fo 7 ad 9 w hav a a ak h h h h a h h h h h a 3 3 hold fa f w df h h L ; : h w hav f pplg h dffao foula fo ouous bla ap w hav

22 a h f h ] h h h h h h h ubsug h U o h boh sds of h las lao w g a h f h ] h h h h h h h ubsug o h boh sds of 6: a a ad akg a ld L us f h followg dsussos La f s h soluo of 6 h U Poof h psao a of * fos h bass of s gv b L ad h ss a osa a{ } dpdg o suh ha h psao a L of * sasfs 3: L hus f s a soluo of 6 h Fo a vos L s ohogoal o E s h la subspa of ha spa h E Fo h odo d dpd ad s : s la soophs { } { } a lal a also lal dpd oov fo d E w hav d E { } fos a bass of E hfo bass of L : s h la soophs fos h bass of s also h L do -dsoal u a b ad -dsoal u a b θ :

23 L θ h h followg la holds La 3 assu / θ : 3 L b h osa ho 3 h h ss a osa dpdg ol o suh ha: f vs s gv b wh whh s dod b a poloal wh posv offs wh h s vbl ad h 33 ] h ss a osa 3 dpdg ol o 3 dod b poloal wh posv offs wh suh ha Espall f h Poof fo b b w hav h h s o-dga ad whh s L L

24 Fo dfo of ad 3 L 39 ad h ss a osa dpdg ol o whh s dod b poloal wh posv offs wh suh ha Fo h assupo of h ho ad fo ua s ss ho s o-dga ad ulplg 37 fo h lf b ] w hav ] osdg w g ] ] ] hfo fo h lao ] s vbl ad h h ss a osa 3 dpdg ol o

25 whh s dod b poloal wh posv offs wh suh ha 3 La f s h appoa soluo of 6 h Espall f s h soluo of 6 h Poof L : L b h ouous bla ap gv b h dffag h gh-had sd of : fo h dffao foula fo opos ap hfo b h dffao foula fo ouous bla ap: w hav d d d d dl d h h h h h h h h h fo a h hfo 789 gv us ] ffag h boh sds of lds La 5 Fo h soluo of 6 dfg b 3 w hav 3 Poof f w ak vaabl asfoao basd o la 3 o h lad quao : h w g ] ow l us alula s h a soluo of 6 hols b la O h oh had 5

26 6 ] ] h fs olus of a all o L us do h las olus of b hus ] 5 O h oh had ] ] 6 akg o aou 6: 7 akg o h boh sds of ad usg w hav ] ] ] ] ulplg h boh sds of h abov lao b fo h gh w hav ] hus w g ] 8 ubsug o 5 gvs us ] ] ] ] ] ad h θ ]

27 7 ] 9 ulplg lao 9 fo h lf b ad usg la ad ak w hav ] ] hus w g 5 s h bass of b la h a uqu as suh ha θ θ θ θ θ θ θ θ θ θ ω 5 ulplg h boh sds of 5 b ad usg 5 d s b Lagaga pop of w oba 5 θ θ θ 53 O h oh had ulplg h boh sds of 5 fo h lf b ad usg 5 9 w hav ]} { b 53 w oba h asso of h la

28 La 6 assu ha s h appoa soluo of 6 ad L Eθ 5 h h s a osa 3 dpdg ol o whh s dod b poloal wh posv offs wh suh ha E 55 Poof od o oba sa 55 l us alula E s h appoao soluo of 6 ffag h boh sds of h abov lao wh sp o fo la hus wh ] ] akg o h boh sds of w hav ] ] ] ad ulplg h abov lao fo h gh b ] ] ] ] ] ] ] w hav 56 8

29 9 ] ] hus w oba ] ] 57 Usg 57 ad ] w g ] ] ] ] ] 58 ulplg h boh sds of 58 b ad usg 3 of ak w oba

30 3 ] ] ] hus 59 s h bass of wh s h soluo of 6 f h o s sall ough h also fos h bass of hfo h ss as ad uqul suh ha 6 ulplg 6 fo h lf b L 6 f w d 59 fo 6 h L L 6 ulplg lao 6 fo h lf b

31 3 L 63 ulplg lao 58 fo h lf b w hav ]} { ]} { 6 f w opa 63 wh 6 wh w hav ]} { O h oh had s Eθ 65 fo 58 w hav

32 3 ] ] 66 ow l us sa F abal suh ha h fo w hav ρ 67 l us sa ]} { : w oba 5 6 } 5 {6 } 6 { } { 3 hfo w g Fo 33: w g

33 33 E E E E E E E 69 O h oh had E 7 holds w g h sa L d h fo 63 ad w hav L E ad ] w hav ] E h las lao ] ] ] holds d w hav

34 3 ] ] ] ] ] ] osdg h abov laos 69 w oba 55 ho L b a appoa soluo of 6 suh ha h h followgs hold; h vaabl asfoao h lad quao s asfod o h followg fo w p 7 wh θ E pθ wθ oov w 7 hold wh s h osa la 6 f w ak w 7 h 7 s dud o p 73- p 73- oov fo a h ss a soluo U of 73 suh ha h s a osa 5 dpdg ol o sasfg 5 suh ha Poof f w ak h vaabl asfoao fo h lad quao

35 35 : h w oba ] hfo w hav ] 75 Fo 5 w g E E 76 w l E f w l w p 77 h w hav w p 78 osdg w oba 7 oba h sa fo fo la 6 ad h sa fo w fo la 3 : f w ak w 7 h 7 s dud o p 79 f w l 79 p p p h w hav p p ad h 79 s dod as ss of quaos wh ad suh as 73 Fo 3 37 w hav ] p hfo w oba ] p p

36 b ho 3 ad sasfs h opha odo ~ usg la 3 73-: p has uqu aal soluo U wh o avag ad ~ p 8 ~ holds L b a fuo ha sasfs l p fo la 3 p ~ p 8 73-: p has uqu aal soluo U wh o avag Fo h doao of p h ss a osa 5 dpdg ol o suh ha ow l us sa Fs w sa Fo 8 p p 5 8 p d 5 Fo h dfo 3 of 8 ad p h s a osa 5 dpdg ol o ha 83 suh ~ ~ d 8 5 h fo 883 ad 8 w hav hfo wh 5 ~ 5 5 ] ] ] 5 ]

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