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¡ ¡ ¡N(‰ )‰f‰2S;‰ =‰ =‰ =‰ g1 2 1 2
¡q(rb)¡q(r )¡rb>r c =
¡rb¡r
¡q(rb)¡q(r )•0
orforwases,kcaoth;Rrighet,ebproblem..Insuchright).negativ5,lineandofeeFiguraeendingLeft-halfthat(se3.3withwne.ointh3.2.1ontainspinthestateswithagetherseento3.1,,agativesp.vRemarka3.4aWhenofwehwantnotrtodenoteconsidereshockssetoftozerifoifspesespetodareinonthealeft-half.prvoblemthe(thatPropartheeesctitioushaasositivtheysoaronlyealoe:ccaterarefactionde,onthetheeaxe,spinee),negativweeed.haveFixtomodifyahalittleshothespsetdThenandpisthe:,Ifnegativinonlytheprc,asewaves:withtheeseeed,gmentorderWknoandwhattakethewithattainableextrtheointsspwemevextrWwithhagmenteseinthepro:ofemeositionpthatoints2-wandv(sealweysxvaplefteIfeed,left);w5,canehaFigurestate1-wandvandeithereshoalkwaewlov.depPronosignof.rTherok.whicaremeanswparticularcases:isequivalentoftoesearcsphingLemmaanrarticialandrighrtstateIfLeft,5:Figure,forethevrighatcrofstateseedattainableby.intheaofproblemointslikthatesolution(3.10)(3.11).theWwhiceisareehereandonlyifiniemannterestedobleminifthecwonlyalvstudyThereofttheoleft-half9problem¡ ¡ ¡† r ‚r q(rb)•q(r ),rb‚rc
¡ ¡ ¡† r <r q(rb)•q(r ),rb‚?(r )c
¡rb•r x=t=
0 ¡ 0q (r ) x=t = q (rb) rb ‚ rc
0 ¡ ¡q r ‚rb‚r r ‚rc c
¡† r ‚r r •rb•1c c
¡ ¡† r •r ?(r )•rb•1c
¡‰
¡‰
¡‰ =0 N(0)=f0g[¢ ¢ =f‰2S;‰ +‰ =1g ⁄1 1 1 2
+ + + ++ + +‰ = (‰ ;‰ ) 2 S r = ‰ +‰ P(‰ )1 3 1 3
‰•=(‰• ;‰• )2S1 3
8
@ ‰ +@ (‰ v(‰ +‰ ))=0> t 1 x 1 1 3<
@ ‰ +@ (‰ v(‰ +‰ ))=0t 3 x 3 1 3‰
> (‰• ;‰• ) x < 01 3: (‰ ;‰ )(0;x)=1 3 + +(‰ ;‰ ) x > 01 3
+r •r T ‰•2S ‰• +‰• •rc r 1 3 cc
+ +r >r T + ‰•2S ‰• +‰• • ?(r )c ?(r ) 1 3
+¢ + =f‰2S;‰• +‰• =r g1 2r
+ +(‰ ;‰ )2S min ‰ v(‰ +‰ )=01 1 31 3
+P(‰ )
+ 0 + + +P r >r P (‰ )=¢ +[f‰2S;?(r )=r >c r
r‚0g
+ +¢ + ‰ +‰ =r P(‰ )1 3r
+‰• ‰
+r•¡r
+r•‚r fx‚0;t‚0g
+r••r r •r••rc c
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