AN EБPLICIT TOWER OF FUNCTION FIELDS OVER CUBIC
Transcrição
AN EБPLICIT TOWER OF FUNCTION FIELDS OVER CUBIC
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L?bd ? L?bd R I stI Yt{ Qwh( Mw?( ) (DD ) 8h( w wrDD ) 8h( w?bw? L ? b dDD ) R Ak N88s dd $$$D WBb b {BtR$I s }Bj h BP ? $t B Vs$t PB8 eJwddD b BAs$t Ys 8h w?D ) bR stI 8h wrD ) R b d RB Y PZt{$Bt 0 U) w?rDd $R s }$8 j8t PB h yR $IZ{$Aj }BjktB8$sj B% w?D $R d ?R L ? b d , c x Rd b ? ?RLd w+ I w0DD ) 8h w?d c 0R1 D ) R L wR b 1D ) 1R b 1 + wx D ) x R b stI Yt{ Qwh Mw? ) DD ) 8h VR st sRk {BtRJZt{ b BAs$t PB8 _B}dn Ys wPB R k 1D ABY -tR$BtR Bbw?D stI BbwrD s tBtTpsjB$R qY$R PBjjBbR PB8 Y s8$h{s$Bt AYs%$BZ BP Y }js{R w? ) D $t Bbw?D R} Y }js{ wr ) (D $t BbwrD 1 is9'P'{s'Ct stI tZR 3$R b $tBIZ{ RB8 sII$$Btsj tBs$Bt w%sj$I YBZYBZ Y$R R{$BtD R $R st sjAs${ {jBRZ BP $R stI p ) I R L ) dJ3 )$ ) ) wBd , B1 , Bn , 333D $R Y RJZt{ BP PZt{$Bt hjIR B% bY Bd ) w?d D $R Y s$Btsj PZt{$Bt hjI stI PB sjj 7 w d B7Ld ) B7 w?7Ld D b$Y d b ?7Ld ?R7 L ?7 b d ) 3 ?R7Ld ?7 3B s }js{ < YwB D b R <i U) <MBi PB i ) d, 3 3 3 , w13dD M F=7]eNyW! eeiiV ViWVN*! pVi]yV <eWWyWp 7qy]<qeW!q< stI b b$ PB > IJ ?i ) s < $P <Mw?i ) D3 N88s 1d 3B sjj w d Y PBjjBb$t YBjIRU $D >B U Bd H ) R d $$D qY }js{ w?d ) (D BP Bd b YsR s Zt$JZ -tR$Bt < YwB D qY$R }js{ < $R sjRB s 9B BP Y PZt{$Bt ? stI b Ys% w<Mw?d ) (DD ) R d stI w<Mw? ) (DD ) d _BBP 7$t{ eJw1dD $R st JZs$Bt BP I R $t Y %s$sAj ?7Ld $ $R {js Ys >BiLd U Bi H R PB sjj i w d stI YPB >B U Bd H R d WBb j < YwB D A s 9B BP ?d $t B y $R RZ^{$t B RYBb Ys < $R s 9B BP ? stI Ys w<Mw?d ) (DD ) R d stI w<Mw? ) (DD ) d qY {sR ) d $R $%$sj stI b sRRZ8 Ys BZ sRR$Bt $R Z PB RB8 G {YBBR s }js{ <I YwBLd D b$Y <I M< stI R <d U) <I Mw? ,?Ld D k _B}dn b {Bt{jZI Ys <d $R s 9B BP ?Ld b$Y w<d Mw? ) (DD ) R stI w<d Mw?Ld ) (DD ) d y PBjjBbR Ys w<I Mw?Ld ) (DD ) d stI w<I M<D ) w<d Mw? ) (DD ) R Ak VAYkst,s:R j88s <t{ w<I Mw?d ) (DD ) R d c R ) R G b$jj RYBb js wR qY81D Ys /wBn D k d stI YPB b Ys%U ]BBjjsk 11 qY RJZt{ ) ) wB7 D7d bY${Y $R IhtI {ZR$%jk Ak eJw1dD $R s Bb BP PZt{$Bt hjIR B% stI PB sjj i w d b Ys% >BiLd U Bi H ) R qY 8s$t Bsj $t Y$R R{$Bt $R B I8$t Y tZR /wB D PB sjj w d Yt{ b 8ZR RZIk Y s8$h{s$Bt AYs%$BZ BP }js{R $t Y Bb ) !Z t- j88s PBjjBbR $88I$sjk PB8 _B}dn %$$D N88s 1n N w 1 stI {BtR$I s }js{ < YwB D $D 3B d i c Y PBjjBb$t YBjIR s Y }js{ <U T $P ?i ) ( B ?i ) Yt ?iLd ) ( T $P ?i ) PB RB8 p Yt $Y ?iLd ) B ?iLd ) d T $P ?i ) d Yt ?iLd ) PB RB8 p $$D 3B 1 i b Ys% s Y }js{ <U T $P ?i ) ( Yt $Y ?id ) ( B ?id ) T $P ?i ) d B ?i ) Yt ?id ) PB RB8 p T $P ?i ) b$Y p Yt ?id ) d WBb b stsjk9 Y }js{R < YwB D BP Y PZt{$Bt hjI B bY${Y s s8$hI $t Y -tR$Bt B bBd V q!Gei !3 3=W]qy!W 3yeN*7 !zei ]= y] 3yeN*7 _B}BR$$Bt 1; N w 1 stI j < YwB D A s }js{ bY${Y $R s8$hI $t Y -tR$Bt B bBd qYt < $R BP Bt BP Y PBjjBb$t k}RU qk} dU ?d ) ( s < qk} 1U ?d ) s < qk} nU ?X ) s < PB RB8 X b$Y d c X _BBP i{sjj Ys <i ItBR Y R${$Bt BP Y }js{ < B Y hjI Bi t B PB d i 7$t{ <M<d $R s8$hI Y $R RB8 i bd RZ{Y Ys <iLd M<i $R s8$hI <t{ Y }js{ < s8$hR $t Y -tR$Bt w?i , ?iLd Dbw?i D Ak VAYkst,s:R j88s wR${jk R}s,$t Y R${$Bt BP < B w?i , ?iLd D s8$hR B% w?i DD y PBjjBbR PB8 _B}dn Ys ?iLd ) ( B ?iLd ) s < WBb b s}}jk N88s 1n $$D B BAs$t BZ sRR$Bt qY s8$h{s$Bt AYs%$BZ BP s{Y BP YR n k}R BP }js{R BP B $R JZ$ I$&t PB8 s{Y BY b b$jj I$R{ZRR $ $t Y RZARJZt }B}BR$$BtR _B}BR$$Bt 1x wqk} dD qY $R -s{jk Bt }js{ < YwB D bY${Y $R s 9B BP ?d stI Y PBjjBb$t YBjIRU $D w<i M<id D ) R w<i Mw?i ) (DD ) d stI Qw<i M<id D ) R PB 1 i $$D Qw<M<d D ) RwR d b dDbwR b dD _BBP G Ys% sjsIk RYBbt $t N88s 1d Ys ?d YsR s Zt$JZ 9B < $t B stI Ys < $R s R$8}j 9B BP ? y 8s$tR B I8$t Y I$&t -}BttR BP <i M<id stI BP <M<d G R hi U) <Mw?id ,?i D stI Yt Qwhi Mw?id ) (DD ) R Ak _B}dn $$D 7$t{ <id $R s R$8}j 9B BP ?id Ak $8 $D $ PBjjBbR PB8 N88s d1 Ys Qw<i M<id D ) Qwhi Mw?id ) (DD ) R y8 $$D PBjjBbR Ak $tIZ{$Bt PB8 $8 $D ZR$t Y stR$$%$k BP I$&t -}BttR wR >dEH ]Yyyy;D _B}BR$$Bt 1E wqk} 1D qY $R -s{jk Bt }js{ < YwB D bY${Y $R s }Bj BP ?d qY$R }js{ $R s {B88Bt 9B BP ?1 , 3 3 3 , ? stI b Ys% $D w<i M<id D ) R w<i Mw?i ) (DD ) R b d stI Qw<i M<id D ) 1R b 1 PB s{Y $tI- i b$Y 1 i $$D Qw<M<d D ) 1wR d b dD _BBP $D N < YwB D A s }Bj BP ?d y PBjjBbR PB8 N88s 1n $D Ys < $R s 9B BP ?1 , 3 3 3 , ? stI b }B% Y 8s$t$t sRR$BtR Ak $tIZ{$Bt B% i qY {sR i ) 1 PBjjBbR PB8 _B}dn WBb j 1 i bd k $tIZ{$Bt Yk}BYR$R Y }js{ <i $R s 9B BP ?i b$Y w<i Mw?i ) (DD ) R b d N hiLd ) <Mw?i ,?iLd D Vs$t Ak _B}dn b Ys% Ys whiLd Mw?i ) (DD ) R whiLd Mw?iLd ) (DD ) d stI QwhiLd Mw?i ) (DD ) R k VAYkst,s:R j88s b {Bt{jZI Ys w<iLd M<i D ) R stI w<iLd Mw?iLd ) (DD ) R b d stI Ak N88s d1 d( F=7]eNyW! eeiiV ViWVN*! pVi]yV <eWWyWp 7qy]<qeW!q< b BAs$t Qw<iLd M<i D ) wR b dD c R b wR b 1DwR b dD ) 1R b 13 $$D 3BjjBbR sR$jk PB8 $8 $D Ak $tIZ{$Bt B% qY s8$h{s$Bt AYs%$BZ BP }js{R < BP qk} n wR _B}1;D $R 8Z{Y 8B {B8}j${sI qYR }js{R s {Yss{$9I Ak Y {BtI$$Bt ?X ) s < PB RB8 X b$Y d c X G 8ZR I$R$tZ$RY Y {sRR X s d 8BI 1 stI X s ( 8BI 1 _B}BR$$Bt 1 wqk} n stI X BIID N d c X stI X s d 8BI 1 stI j Y }js{ < YwB D A s }Bj BP Y PZt{$Bt ?X qYt b Ys%U $D qY }js{ <X $R s8$hI $t BX bw?X D b$Y s8$h{s$Bt $tIw<X Mw?X ) DD ) R wXdDb1 3 $$D qY R${$Bt <i BP Y }js{ < Rs$RhR T d $P 1 i X b d, z R b d $P i ) X, w<i M<id D ) R $P X c i c 1X stI i s ( 8BI 1, d $P X c i c 1X stI i s d 8BI 1, Z R $P i w 1X3 yt sjj {sRR bY w<i M<id D ) R Y I$&t -}Btt $R Qw<i M<id D ) 1R b 13 $$$D qY I$&t -}Btt BP <M<d $R T R w1nXLnDb1 b 1 z Qw<M<d D ) R wXLnDb1 b 1 Z R wXL1Db1 b 1 $%D N \,X $P X $P Ld , 1 Ld c X stI s ( 8BI 1, 1 Ld c X stI s d 8BI 13 1 U) 'I< YwB D < $R s }Bj BP ?X J qYt T Ld Xd $P X , R 1 z Ld \,X ) R w1Db1 $P c X stI s ( 8BI 1, 1 Ld Z R wdDb1 $P c X stI s d 8BI 13 1 $P V q!Gei !3 3=W]qy!W 3yeN*7 !zei ]= y] 3yeN*7 dd _BBP qY }BBP BP $8R $D stI $$D $R YsI stI $ $R Y {Btt BP 7{$Bt ; $$$D 3$R b {BtR$I Y {sR X Ld $ Y {sR w 1X b d k $8 $$D stI Y 1 stR$$%$k BP I$&t -}BttR Y I$&t -}Btt BP <1Xd M<d $R wXnDb1 Qw<1Xd M<d D ) R wXdDb1 3 c wR b 1D L w R 7 D c w1R b 1D ) R wXLdDb1 b 13 7)( Vs$t PB8 $$D stI Y stR$$%$k b BAs$t Qw< M<d D ) R w1XdD 1X 3 c Qw<1Xd M<d D L w R 7 D c w1R b 1D 7)( ) R w1XdDLwXLdDb1 b 1 ) R w1nXLnDb1 b 13 yt {sR Ld 1 c X stI s ( 8BI 1 b BAs$t $t s R$8$js 8stt Ys wXdDb1 Qw< M<d D ) R wXLdDb1 c wR b 1D L w 3 R 7 D c w1R b 1D ) R wXLnDb1 b 13 7)( yt {sR Ld 1 c X stI s d 8BI 1 b Ys% wX1Db1 Qw< M<d D ) R wXDb1 c wR b 1D L w 3 R 7 D c w1R b 1D ) R wXL1Db1 b 13 7)( $%D k $8R $D stI $$D sjj }BjR < YwB D BP Y PZt{$Bt ?X Ys% Y Rs8 }Bj BI ts8jk T Ld X R $P X , 1 z Ld w<Mw?X ) DD ) R b1 $P c X stI s ( 8BI 1, 1 Ld Z R wdDb1 $P c X stI s d 8BI 13 1 7$t{ R d ) >B U w?X DH ) \,X c w<Mw?X ) DD b BAs$t Y IR$I RZj _B}BR$$Bt 1M wqk} n stI X %tD N d c X stI X s ( 8BI 1 stI j Y }js{ < YwB D A s }Bj BP Y PZt{$Bt ?X qYt b Ys%U $D qY }js{ <X YsR s8$h{s$Bt $tI- w<X Mw?X ) DD ) R wX1Db1 $t Y -tR$Bt BX bw?X D d1 F=7]eNyW! eeiiV ViWVN*! pVi]yV <eWWyWp 7qy]<qeW!q< $$D qY R${$Bt <i BP Y }js{ < T d z Rbd w<i M<id D ) R d Z R Rs$RhR $P $P $P $P $P 1 i X b d, i ) X, X c i 1X b 1 stI i s d 8BI 1, X c i 1X b 1 stI i s ( 8BI 1, i w 1X b d3 yt sjj {sRR bY w<i M<id D ) R Y I$&t -}Btt $R Qw<i M<id D ) 1R b 13 $$$D qY I$&t -}Btt BP <M<d $R T R w1nXL;Db1 b 1 z Qw<M<d D ) R wXL1Db1 b 1 Z R wXLnDb1 b 1 $P X L1 , 1 $P L1 c X stI s ( 8BI 1, 1 $P L1 c X stI s d 8BI 13 1 $%D N \,X ) 'I< YwB D < $R s }Bj BP ?X J qYt T L1 R Xd $P X , 1 z L1 \,X ) R b1 c X stI s ( 8BI 1, $P 1 L1 Z R wdDb1 $P c X stI s d 8BI 13 1 _BBP 3$R b sRRZ8 Ys X w ; qYt b ,tBb Y s8$h{s$Bt AYs%$BZ BP Y }js{ < $t Y hjI w?1 , 3 3 3 , ?X , 3 3 3 , ? D PB8 _B}1 BB% Y }js{ <1 $R Zts8$hI $t Y -tR$BtR w?d , ?1 Dbw?d D stI w?d , ?1 Dbw?1 D Ak _B}dn Vjj sRR$BtR BP Y $8R $D $$D stI $$$D $t _B}1M PBjjBb tBb Ak VAYkst,s:R j88s yt Y {sR X ) 1 b s}}jk _B}1E $tRsI BP _B}1 B Y hjI w?1 , 3 3 3 , ? D y8 $%D $R }B%I R$8$jsjk sR $t _B}1 R IhtI Ak eJw1dD qYt qYB8 1 N ) ) wBd , B1 , Bn , 333D A Y Bb B% ) $ Y tZR /wB D Rs$RhR /wB D RL1 j$8 ) 3 R 1wR b dD B }{$Rjk b Ys%U V q!Gei !3 3=W]qy!W 3yeN*7 !zei ]= y] 3yeN*7 dn $D 3B s ( 8BI ; /wB D ) d wR Ld L 1R b 1R wL1Db1 b 1R b1 L RD b c R w1Db1 c wR L dD3 1wR b dD ; $$D 3B s 1 8BI ; /wB D ) d b 1 w1Db1 wR Ld L 1R b ;R wL1Db1 L RD b cR c wR L dD3 1wR b dD ; $$$D 3B s d 8BI 1 /wB D ) d b d wdDb1 wR Ld L 1R b R wLnDb1 b nR wLdDb1 L RD b cR 3 1wR b dD 1 _BBP G b$jj Btjk {BtR$I Y {sR s ( 8BI ; Y BY {sRR s R$8$js stI b js% Y8 B Y sI 3B 1 X b {YBBR s }js{ <wXD YwB D bY${Y $R s }Bj BP ?X stI b ItB Ak wXD QX U) Qw<wXD M<d D wXD Y I$&t -}Btt BP <wXD B% <d U) <wXD MBd VR $t _B}1 stI 1M b j \,X ItB Y tZ8A BP }BjR BP ?X $t B V}}jk$t _B}1; 1x stI 1E b R Ys Y I BP Y I$&t BP Y -tR$Bt B bBd $R 3 R b R d I *$& wB bBd D ) L 1wR b dD L QX c \,X 3 Rbd X)1 V{{BI$t B _B}1 stI 1M b R}j$ Y RZ8 ;wiD U) 3 QX c \,X QX c \,X $tB PBZ }${R sR PBjjBbRU PB i ) d, 1, n stI ;, X;-i bY Y R -i $R IhtI sR -d U) IX L X s d 8BI 1 stI n X w L dDb1J, -1 U) IX L X s d 8BI 1 stI w L dDb1 c X J, -n U) IX L X s ( 8BI 1 stI 1 X w L 1Db1J, -; U) IX L X s ( 8BI 1 stI w L 1Db1 c X J3 w131D d; F=7]eNyW! eeiiV ViWVN*! pVi]yV <eWWyWp 7qy]<qeW!q< 7$t{ s ( 8BI ; b Ys% Ys -d ) IX L X ) d L 17 PB RB8 7 b$Y d 7 c b;J, -1 ) IX L X ) w L ;7 b 1Db1 PB RB8 7 b$Y d 7 b;J, -n ) IX L X ) 17 PB RB8 7 b$Y d 7 b;J, -; ) IX L X ) w L ;7Db1 PB RB8 7 b$Y d 7 b;J3 3B8 _B}1 stI 1M b Ys% wdD w;Db; w1D b; wb;DL17 wnD b; L1n7 w;D b; wb;DLd7 ; ) ; ) ; ) ; ) 7)d 7)d wR 7)d wR 7)d wR wR n7 b 1D c R 17 ) d 1R 1 wR b R wnL;Db; D b 1 wR w;Db1 b dD, Rbd R bd b 1D c R w1Db1 ) d wR wnL;Db; b R wL1Db1 D b c R w1Db1 , Rbd 1 d 1R wR Ld b R wnL;Db; D b 1 wR b1 b dD, Rbd R bd b 1D c R 17d ) b 1D c R b1 ) d wR wnL;Db; b R wL1Db1 D b c R b1 3 Rbd 1 WBb $ PBjjBbR PB8 eJw11D Ys I *$& wB bBd D ) 1wR d b dD L ) 1wR d b dD L b ) 1wR d b dD L R b R wdD w1D wnD w;D L ; L ; L ; L ; Rbd d wR Ld L 1R b 1R wL1Db1 b RD Rbd 1R wR L dDwR w1Db1 b dD b c R w1Db1 c wR L dD bd 1 R1 d wR Ld L 1R b 1R wL1Db1 b 1R b1 L RD b c R w1Db1 c wR L dD3 Rbd 1 qY <Zb$9 tZR PB8Zjs $%R 1/wB D b 1 ) b1R d L I *$& wB bBd D, stI Yt{ 1/wB D ) d wR Ld L 1R b 1R wL1Db1 b 1R b1 L RD b c R w1Db1 c wR L dD3 Rbd 1 V q!Gei !3 3=W]qy!W 3yeN*7 !zei ]= y] 3yeN*7 dx n is'Ctsj }js{R stI Y j'9' CP Y Cc yt Y$R R{$Bt b RZIk Y Bb ) sR s Bb BP PZt{$Bt hjIR B% Y ht$ hjI $f BP {sI$tsj$k f ) R n 7B b Ys% Y PBjjBb$t R$Zs$BtU ) ) wBd , B1 , Bn , 333D $R Y Bb BP PZt{$Bt hjIR B% $f b$Y f ) R n bY Bd ) $f w?d D $R Y s$Btsj PZt{$Bt hjI stI B7Ld ) B7 w?7Ld D b$Y ?R7 L ?7 b d d b ?7Ld ) 3 ?R7Ld ?7 wn3dD WB Ys >B7Ld U B7 H ) R stI Ys $f $R Y PZjj {BtRst hjI BP B7 PB sjj 7 w d wsR PBjjBbR PB8 ]B11D BB% R$t{ Y tZR BP s PZt{$Bt hjI B% $f IBR tB {Yst $t Y f Y PB8ZjsR PB Y tZR /wB D $%t $t qY81 sjRB {BtRst hjI -tR$Bt b$Y ) $ YBjI PB Y Bb ) {BtR$II B% $f !Z 8s$t Bsj Y $R B RYBb Ys {s$t s$Btsj }js{R BP Bd ) $f w?d D R}j$ {B8}jjk $t Y Bb ) $ Yk R}j$ {B8}jjk $t B bBd PB sjj w d qY$R b$jj }B%$I 8stk s$Btsj }js{R BP B b$f stI $t {Bt`Zt{$Bt b$Y qY81 Y$R b$jj }B% BZ 8s$t RZj bY${Y $R qY8nn 3B sAA%$s$Bt b $tBIZ{ Y s$Btsj PZt{$BtR ywx D U) xR L x b d dbx stI Kwx D U) 3 xR x G {BtR$I Y RR f RLd ) b dJ stI U) I $ f ywD CJ3 C U) I $ wn31D _B}BR$$Bt nd $D G Ys% t $f stI ' ) RwR L dD f stI sjj YR $$D N qYt Y JZs$Bt ywD ) KwD YsR -s{jk R BBR $t $ BBR AjBt B Y R G }BR}Bt Y }BBP BP _B}nd B Y tI BP Y$R R{$Bt stI b Isb hR RB8 {BtRJZt{R BP $ qYB8 n1 N ) ) wBd , B1 , Bn , 333D A Y Bb BP PZt{$Bt hjIR B% $f b$Y f ) R n bY${Y $R IhtI {ZR$%jk Ak eJwndD stI j qYt Y }js{ w?d ) D R}j$R {B8}jjk $t sjj -tR$BtR B bBd qYPB Y tZ8A BP s$Btsj }js{R BP B b$f Rs$RhR ] wB D w >B U Bd H c ' ) R wR L dD3 _BBP G {js$8 YsU ]js$8 $DU qY }js{ w?d ) D YsR -s{jk R d I$R$t{ -tR$BtR < YwB D dE F=7]eNyW! eeiiV ViWVN*! pVi]yV <eWWyWp 7qy]<qeW!q< ]js$8 $$DU 3B stk RZ{Y }js{ < Y $R RB8 I RZ{Y Ys < $R s 9B BP Y PZt{$Bt ? b I y $R {js Ys Y sRR$BtR BP qY8n1 PBjjBb PB8 ]js$8 $D G }B% ABY {js$8R Ak $tIZ{$Bt Bt qY {sR ) d $R $%$sj stI b sRRZ8 Ys ]js$8 $D stI ]js$8 $$D s ABY Z PB RB8 w d N < YwB D A Bt BP Y }js{R jk$t sAB% w?d ) D Yt Ak $tIZ{$Bt Y }js{ < $R s 9B BP ? b I b$Y I qY PZt{$Bt ?Ld Rs$RhR Y JZs$Bt yw?Ld D ) Kw? D qB stk RBjZ$Bt BP Y JZs$Bt ywD ) Kw I D {BR}BtIR s }js{ <I YwBLd D b$Y <I M< RZ{Y Ys <I $R s 9B BP ?Ld b k _B}nd Y JZs$Bt ywD ) Kw I D YsR -s{jk R I$R$t{ BBR stI Yt{ Y {js$8R $D stI $$D sjRB YBjI PB L d 3B stk Bb 7 ) wxd , x1 , xn , 333D BP PZt{$Bt hjIR B% s ht$ hjI Y j$8$ w7 D U) j$8 ] wx D /wx D -$RR R >H Vt $88I$s {BtRJZt{ BP qY81 stI qY8n1 $R tBbU qYB8 nn N ) ) wBd , B1 , Bn , 333D A Y Bb BP PZt{$Bt hjIR B% $f wb$Y f ) R n D bY${Y $R IhtI {ZR$%jk Ak eJwndD qYt $R j$8$ Rs$RhR w)D ) j$8 1wR 1 b dD ] wB D w 3 /wB D RL1 3B Y sj tZ8A \wR n D wR Iht$$Bt $t Y $tBIZ{$BtD b BAs$tU ]BBjjsk n; wptsj$9s$Bt BP $t,:R jBb ABZtID 3B stk }$8 }Bb R b Ys% \wR n D w 1wR 1 b dD 3 RL1 WBb b }B% _B}nd 3B Y$R b tI Y PBjjBb$t bB j88sRU N88s nx N Y RR C stI A sR IhtI $t wn1D qYtU $D 'C ) R L d stI ' ) RwR L dD $$D yP C Yt R 1 LRLd ) bd yt }s${Zjs Y R C $R {Bts$tI $t $f $$$D qY R $R {Bts$tI $t $f f BP Y _BBP $D k Iht$$Bt Y R C {BtR$RR BP Y 9BR $t Y sjAs${ {jBRZ $ }BjktB8$sj W wx D ) x RLd b x L d 7$t{ W wx D $R {jsjk R}ssAj b Ys 'C ) R L d 7$8$jsjk Y R {BtR$RR BP Y 9BR BP _ wx D U) x R L d x b d PB sjj j8tR C qYR }BjktB8$sjR s {jsjk R}ssAj Yt{ ' ) RwR L dD $$D yP RLd ) b d Yt R 1 LR ) R b d Zj$}jk$t Y jsR JZsj$k Ak b Yt Ys V q!Gei !3 3=W]qy!W 3yeN*7 !zei ]= y] 3yeN*7 d 1 R LRLd ) RLd b ) bd WB Ys R 1 L R L d $R Y tB8 -}Btt $t $f b$R stI Yt{ Y R C $R {Bts$tI $t $f $$$D N qYt ywD ) PB RB8 C $ b Ys% d b ) R =R$t $8 $$D $ Yt PBjjBbR Ys R n 1 n 1 1 ) b R LRLd c R ) b RLd w R DR ) b RLd wd b DR ) b RLd L RLd c R ) b RLd L c w R DR ) b RLd L wd b DR ) b RLd L b R ) b RLd L b wd b D ) wb RLd L b dD L ) , 1 stI Yt{ Y R $R {Bts$tI $t $f N88s nE yP Yt KwD C _BBP N qYt Y $R RB8 C b$Y R ) d b 3 <t{ KwD ) R L b d R b R ) ) Rd wd b D )U #, stI b Ys% B RYBb Ys # C G Ys% 1 w# RLd b # L dD ) wwd b DRLd c R d b wd b D Rd L dD 1 ) wd b b R L RLd D R b wd b D R L 1 ) b RLd c R b wd b D R L ) bw R DR b wd b Dwd b D L ) bwd b DR b d L L ) R b d L ) (3 <t{ # RLd ) # b d stI # C WBb b Zt B Y }BBP BP _B}nd _BBP BP _B}BR$$Bt nd $D qY$R $8 bsR sjsIk RYBbt $t N88s nx f KwD CJ 7$t{ IwKD ) R Y {sI$tsj$k BP Y R Rs$RhR $$D N U) I $ ' R c 'C ) RwR L dD k N88s nE b ,tBb Ys t stI R$t{ ' ) RwR L dD Ak N88s nx b {Bt{jZI Ys ) qY$R {B8}jR Y }BBP BP _B}nd qY PBjjBb$t 8s, bsR }B$tI BZ B ZR Ak ] _ $tU i8s, n w{P >1(HD y $R bjjT,tBbt Ys BBI jBb ABZtIR PB Y JZst$k \wfD }B%$I jBb ABZtIR PB Y PZt{$Bt f w&D bY${Y }jskR s }B8$tt Bj $t {BI$t YBk 3B Y Iht$$Bt BP f w&D b P B >dnH 7{E1 B >dEH ]Yzyy1 qY p$jATzsRYs8B% ABZtI RsR Ys f w&D w d b If w&D PB sjj & >d, wf b dDbfH, wn3nD dM F=7]eNyW! eeiiV ViWVN*! pVi]yV <eWWyWp 7qy]<qeW!q< bY If w&D ItBR Y fTsk tB}k PZt{$Bt qY PBjjBb$t ABZtI bY${Y $R AsRI Bt sjAs${TB8${ {BIR bsR hR }B%I Ak qRPsR8st zjsIZ stI $t, wR >dMHD f w&D w d b & b \d PB stk \ b b$Y ( c \ \wfD3 wn3;D 3B f ) R n b {st {YBBR \ ) 1wR 1 b dDbwR L 1D Ak ]Bn; stI %sjZs$t Y ABZtIR wnnD stI wn;D sAB% s &( U) wf b dDbw1f b dD b htI sR$jk Ys d b &( b RL1 k d b IRn w&( D PB sjj R w 3 1wR 1 b dD qYPB Y qRPsR8stTzjsIZT$t, ABZtI $8}B%R Bt Y p$jATzsRYs8B% ABZtI $t s tBtT8}k B}t $t%sj {Bts$t$t &( PB sjj f ) R n b$Y R w ; is9'P'I }js{R CP k} n yt Y$R R{$Bt b b$jj }B% Y $8R $D stI $$D BP _B}BR$$Bt 1 stI YZR {B8}j Y tZR {sj{Zjs$BtR wR qY81D PB Y Bb ) ) wB7 D7d $%t Ak eJw(D i{sjj Ys Y -tR$BtR B7Ld bB7 s $%t Ak B7Ld ) B7 w?7Ld D bY d b ?7Ld ?R7 L ?7 b d ) 3 ?R7Ld ?7 qY$R 8stR Ys ?7Ld $R s BB BP Y }BjktB8$sj ?R7 L ?7 b d c x R L x b d B7 >x H, ?7 stI Yt{ b s jI B RZIk }BjktB8$sjR BP Y PB8 yx R L Kx L b$Y {B^{$tR y, K, $t s PZt{$Bt hjI B N88s ;d N obI A s R}ssAj -tR$Bt BP PZt{$Bt hjIR B% s hjI BP {Yss{T $R${ k ( VRRZ8 Ys >o U IH ) E s ( 8BI stI j o A RZ{Y Ys o ) Iw D 7Z}}BR Ys Rs$RhR st JZs$Bt y E LK L)( b$Y y, K, I stI j h YwID A s }js{ BP I RZ{Y Ys 8h wyD ) 8h wKD ) ( qYt Y PBjjBb$t YBjIRU $D yP 8h wD w ( Yt h $R Zts8$hI $t obI $$D yP 8h wD ) bd Yt h $R Bsjjk s8$hI $t obI stI Y PZt{$Bt d $R s }$8 j8t PB Y }js{ < YwoD bY${Y j$R B% h BB% Y I$&t -}Btt BP <Mh $R JZsj B Qw<Mh D ) 1E b 1 V q!Gei !3 3=W]qy!W 3yeN*7 !zei ]= y] 3yeN*7 _BBP d k sRRZ8}$Bt Y }BjktB8$sj y c x L I>x H K y B% Y hjI I wx D ) x E L $R Y 8$t$8sj }BjktB8$sj BP $D yt Y$R {sR 8h wKbyD ) ( stI 8h wbyD w ( Yt{ $R $tsj B% Y %sjZs$Bt $t BP h stI PB sjj < YwoD b$Y <Mh b Ys% 8< w I w DD ) 8< wKbyD ) ( y PBjjBbR Ak N88s dd $D Ys Y }js{ h $R Zts8$hI $t obI $$D N < A s }js{ BP o jk$t B% h qYt 8< wyD ) 8< wKD ) ( stI 8< wD ) b b$Y ) w<Mh D 3B8 Y JZs$Bt y E L K L ) ( b {Bt{jZI Ys 8< w D c ( Yt{ 8< wy E D ) E c 8< w D c 8< w D ) 8< wK D qYt E c 8< w D ) 8< wD ) b 7$t{ >o U IH ) E Y$R $8}j$R Ys ) E stI 8< w D ) bd stI RB Y PZt{$Bt d $R s }$8 j8t PB < qY 8$t$8sj }BjktB8$sj BP d B% I $R 'wx D ) x E L K y c x Ed L stI Ak N88s dd $$D b BAs$t Qw<Mh D ) 8< w'I w d DD ) 8< wwKbD c w d DE1 D ) 8< wKD b 8< wD L wE b 1D ) 1E b 13 R $R st sjAs${ 3B8 Y Bt b sRRZ8 sjj tBs$BtR PB8 7{$Bt 1 yt }s${Zjs ) $ {jBRZ BP $R stI Y Bb ) ) wBd , B1 , Bn , 333D B% $R {ZR$%jk IhtI Ak Bd ) w?d D stI PB 7 w d B7Ld ) B7 w?7Ld D b$Y ?R7 L ?7 b d d b ?7Ld ) 3 ?R7Ld ?7 w(3D G tI B $t%R$s Y s8$h{s$Bt AYs%$BZ BP }js{R BP Xqk} n PB X BII" qY$R 8stR wR _B}1;D Ys b {BtR$I s }js{ < YwB D bY${Y $R s }Bj BP Y PZt{$Bt ?X PB RB8 BII X In, 3 3 3 , J 7$t{ b b$jj IR{$A Y s8$h{s$Bt AYs%$BZ BP RZ{Y }js{R $t sjj -tR$BtR Bz bBd b {st sRRZ8 Ys w 1X RB b Ys% X ) 10 L d b$Y 0 w d, stI w ;0 L 13 7$t{ ?X ) s < $ PBjjBbR PB8 T ( z ?i ) d Z #7 N88s 1n Ys Y PBjjBb$t YBjIR s Y }js{ <U PB PB PB PB i i i i w 10 L 1, ) 10 L d, ) 10 L d b 17, d 7 0, ) 10 L 1 b 17, d 7 0, w;3dD 1( F=7]eNyW! eeiiV ViWVN*! pVi]yV <eWWyWp 7qy]<qeW!q< b$Y {s$t j8tR #7 p ) I R L ) dJ, PB 7 ) d, 3 3 3 , 0 N obB A st -tR$Bt BP PZt{$Bt hjIR RZ{Y Ys B t o t B G Rsk Ys Y }js{ < YsR s8$h{s$Bt $tI- $t obB $P Y }js{ <Mo YwoD YsR s8$h{s$Bt $tI- B% Y }js{ <MB 3B8 _B}BR$$Bt dn b {st sI B& Y s8$h{s$Bt $tI- BP < $t RB8 RZA-tR$BtR BP B U T d R z Rbd ) d R Rbd Z d $t $t $t $t $t $t $t w?7 , ?7Ld Dbw?7Ld D w?7 , ?7Ld Dbw?7 D w?7 , ?7Ld Dbw?7Ld D w?7 , ?7Ld Dbw?7Ld D w?7 , ?7Ld Dbw?7Ld D w?7 , ?7Ld Dbw?7 D w?7 , ?7Ld Dbw?7 D PB PB PB PB PB PB PB 7 w 10 L 1, 7 w 10 L d, 7 ) 10 L d, 7 ) 1, ;, 3 3 3 , 10, 7 ) d, n, 3 3 3 , 10 b d, 7 ) 10, d 7 10 b d3 w;31D 3B 0 ) 1 Y$R }st $R RYBbt $t 3$Z dU x1 = 1 1 x2 = 2 1 q 1 q- q x4 = 1 x3 = 1 1 q x5 = q-1 1 K(x 5 ,x 6 ) 1 K(x 3 ,x 4 ) K(x 7 ,x 8 ) q x6 = 0 1 q x7 = 0 1 q x8 = 0 1 q x9 = 0 1 x10 = 0 Figure 1 (for t = 2) k VAYkst,s:R j88s wR N88s d1D b {Bt{jZI PB8 w;1D Ys Y s8$h{s$Bt $tI- BP Y }js{ < Rs$RhR sjRB T z d $t Bi bBid PB i ) 1, 3 3 3 , 10, ) R b d $t Bi bBid PB i ) 10 L d, Z R0 $t B10Ld bw?10Ld D3 qY$R }B%R Y $8 $D stI }s BP Y $8 $$D $t _B}1 =tPBZtsjk VAYkst,s:R j88s IBR tB s}}jk PB i w X L d ) 10 L 1 R 3$ 1 V q!Gei !3 3=W]qy!W 3yeN*7 !zei ]= y] 3yeN*7 1d ? K(xs-1 ,xs ,x s+1) K(xs-2,xs-1,xs ) q-1 q 1 q K(xs-1 ,xs ) q 1 K(xs-2 ) xs-2 = 1 q-1 q 1 K(xs-1) K(xs) xs-1 = xs = q-1 q K(xs+1) xs = 0 +1 Figure 2 y b$jj A {Bt%t$t B {Yst Y tsBR ?7 BP Y hjIR w?7 D Rj$Yjk s{{BI$t B Y }js{ < G R l ?10Ld7 b d PB d 7 10, 7 s ( 8BI 1, 7 U) ?10Ld7 b #w7LdDb1 PB d 7 10, 7 s d 8BI 1, V U) ?d 10Ld .7 U) ?10LdL7 PB sjj 7 w d3 y $R Yt BA%$BZR Ak w;dD Ys Y }js{ < $R s {B88Bt 9B BP Y PZt{$BtR d , 3 3 3 , 10 V, .d , .1 , 333 G {st b$ eJw(D $t 8R BP Y PZt{$BtR 7 , V, .7 sR PBjjBbRU R b 7Ld 7R L 7 ) R 7R L #w7LdDb1 7Ld L d PB 7 s d 8BI 1, R 7Ld L 7Ld b 7 ) PB 7 s ( 8BI 1, R 7 L d 7Ld L #w7L1Db1 w;nD w;;D dR b #d bd) , d L #d w;xD .d b d V R b V Rd b d ) , .dR V Rd w;ED R V bV Rd 11 F=7]eNyW! eeiiV ViWVN*! pVi]yV <eWWyWp 7qy]<qeW!q< d b .7Ld .7R L .7 b d ) PB 7 w d3 R .7Ld .7 w;D qY PBjjBb$t RZAhjIR BP B b$jj }jsk s {Z{$sj Bj PB Y I8$ts$Bt BP Y s8$h{s$Bt AYs%$BZ BP Y }js{ < G R PB d i 7 0 wR 3$ nDU I7,i U) w d , 1 , 3 3 3 , 17 , V, .d , 3 3 3 , .1i1 D, o7,i U) I7,i w.1id D Ui U) oi,i w.1i D3 BB% b R h7,i U) <MI7,i stI <7,i U) <Mo7,i 3 qY t- YB8 $R Y ,k RZj jsI$t B Y tZR PB8ZjsR $t qY81 qY sRRT $BtR BP qY8;1 {BR}BtI B Y t{${jI s8$h{s$Bt $tI${R $t 3$n L 3 = F 13 1 E 3,3 q q H3,3 E 3,2 q H3,2 L2 1 E 3,1 E2,2 q q H3,1 q q H 2,2 q q E 2,1 q H 2,1 q L1 1 q q E1,1 q q H1,1 q q q 1 1 1 q 1 1 1 q q q-1 1 x6 = 1 x7 = q-1 q 1 q 1 q 1 x8 = 0 x9 = 0 x10 = 0 x11 = 0 q 1 q 1 F1 x1 = 1 x2 = 3 x3 = 1 x4 = 2 x5 = 1 Figure 3 (for t=3) x12 = 0 x13 = 0 V q!Gei !3 3=W]qy!W 3yeN*7 !zei ]= y] 3yeN*7 1n qYB8 ;1 N d i 7 0 qYt Y }js{ < $R Bsjjk s8$hI $t Y -tR$Bt o7,i bI7,i stI $ $R Zts8$hI $t Y -tR$Bt Ui boi,i BB% Y I$&t -}Btt BP <7,i Mh7,i $R Qw<7,i Mh7,i D ) 1R b 13 VR s {BtRJZt{ BP qY8;1 b Ys% Yt ]BBjjsk ;n qY R${$Bt <i ) <MBi BP Y }js{ < Rs$RhR $D w<i M<id D ) d $P 10 L d c i c ;0 L 1 stI i s d 8BI 1 $$D w<i M<id D ) R stI Qw<i M<id D ) 1R b 1 $P 10 L d c i c ;0 L 1 stI i s ( 8BI 1 $$$D w<i M<id D ) R stI Qw<i M<id D ) 1R b 1 $P i w ;0 L 1 WB Ys ]B;n $t{jZIR sjj Y 8$RR$t sRR$BtR BP _B}1 stI Yt{ {B8}jR Y }BBP BP Y s$t qYB8 _BBP BP ]B;n $D 3$R b {BtR$I Y {sR i ) ;0 L d qYt Bi ) U0 stI Bid ) o0,0 stI < $R Zts8$hI $t U0 bo0,0 Ak qY8;1 W- j 10 L d c i c ;0 L d stI i s d 8BI 1 qYt Bi ) I0,>Ld stI Bid ) o0,> PB RB8 > b$Y d > 0 b d wR 3$nD !AR% Ys I0,>Ld u o0,> u o>,> stI I0,>Ld u U> u o>,> stI Ys I0,>Ld $R Y {B8}BR$ hjI BP o0,> stI U> k qY8;1 Y }js{ < $R Zts8$T hI $t Y -tR$Bt U> bo>,> stI YPB < $R Zts8$hI $t Y -tR$Bt I0,>Ld bo0,> Ak N88s d1 $$D WBb j 10 L d c i c ;0 L 1 stI i s ( 8BI 1 qYt Y $R RB8 > b$Y d > 0 RZ{Y Ys Bi ) o0,> stI Bid ) I0,> stI sjj sRR$BtR BP $8 $$D PBjjBb $88I$sjk PB8 qY8;1 $$$D 3B ibd w ;0Ld Y s8$h{s$Bt $tI- BP < $t Y -tR$Bt Bid bw?id D $R ) Rbd Ak qY8;1 wR 3$nD qY }js{ < $R Bsjjk s8$hI $t Y -tR$Bt w?id , ?i Dbw?id D b$Y I$&t -}Btt Q ) R wR _B}dn $$DD =R$t N88s d1 b {Bt{jZI Ys w<i M<id D ) R stI 8BB% Qw<i M<id D ) wR b dD c R b wR b 1DwR b dD ) 1R b 1 y 8s$tR B }B% qY8;1 qY$R }BBP b$jj A I$%$II $tB bB }B}BR$$BtR bY${Y s Y {sRR i ) d stI i k d R}{$%jk yt ABY {sRR Y 8s$t $Is $R B s}}jk N88s ;d =tPBZtsjk Y XBA%$BZR" tsB BP Y {BR}BtI$t hjI -tR$Bt o7,i bI7,i wR} Ui boi,i D IBR tB 8 Y sRRZ8}$BtR BP N88s ;d qYPB b 8ZR hR }T PB8 X}Bj BI IZ{$BtR" stsjBBZR B <sRR:R }Bj BI IZ{$BtR PB V$tT7{Y$ 1; F=7]eNyW! eeiiV ViWVN*! pVi]yV <eWWyWp 7qy]<qeW!q< JZs$BtR wR >dEH ]YyyyD qY PBjjBb$t tBs$Bt b$jj A %k ZRPZj 3B s PZt{$Bt hjI Bb s }js{ h YwB D j8tR Fd , F1 B stI st $t z b b$ Fd ) F1 L 2wzD s h, $P 8h wFd b F1 D w z yt }s${Zjs b b$ F ) 2w(D s h 8st$t Ys F $R YBjB8B}Y${ s Y }js{ h i{sjj Ys h7,i wR} <7,i D ItBR Y R${$Bt BP Y }js{ < YwB D B Y hjI I7,i wR} o7,i D PB d i 7 0 3$Z ; AjBb Yj}R B ZtIRstI Y Rs8tR $t _B};; stI ;x f3 1 3,3 Q3,3 q q P3,3 Q3,2 3,2 q f2 P3,2 1 3,1 = 3 Q 3,1 q 2,2 q P3,1 3 Q2,2 q q P2,2 q 2 q 2 f1 1 P 2,1 q q Q2,1 q q 1 Q1,1 q q 1 P1,1 q q q 1 1 1 q 1 1 1 q q q-1 1 x6 = 1 x7 = q-1 q 1 q 1 q 1 x8 = 0 x9 = 0 x10 = 0 x11 = 0 q 1 q 1 F1 x1 = 1 x2 = 3 x3 = 1 x4 = 2 x5 = 1 Figure 4 (for t=3) x12 = 0 x13 = 0 V q!Gei !3 3=W]qy!W 3yeN*7 !zei ]= y] 3yeN*7 1x _B}BR$$Bt ;; wqY {sR i ) dD 3B d 7 0 b {BtR$I Y PZt{$BtR +7 U) V d c 7>)d 1> I7,d stI 7 U) V d .d L 7 > >)d wbdD \> #> +> o7,d b$Y \> U) > E)d R #E 3 qYt Y PBjjBb$t YBjIRU $D 8h7,d wV D ) R 7 l 8h7,d w > D ) R w17>LdDb1 c wR b dD PB d > 17, > s d 8BI 1, R w17>Db1 c wR b dD PB d > 17, > s ( 8BI 1, 8h7,d w+7 D ) bd $ Y PZt{$Bt +7d $R s jB{sj }ss8 s Y }js{ h7,d $$D o7,d ) I7,d w 7 D stI Y 8$t$8sj }BjktB8$sj BP wx D ) x R b 7 B% I7,d YsR Y PB8 d L #d c x L wbdD7Ld \7 +7 b _, dR b #d PB RB8 PZt{$Bt _ I7,d bY${Y $R YBjB8B}Y${ s Y }js{ h7,d $$$D qY }js{ h7,d $R Bsjjk s8$hI $t Y -tR$Bt o7,d bI7,d stI Y PZt{$Bt jB{sj }ss8 s <7,d qY I$&t -}Btt BP <7,d $R Qw<7,d Mh7,d D ) 1R b 13 $%D 3B 1 7 0 Bt YsR +7d ) \Rd c +7R b +7 L 2w(D s h7,d 3 7 %D 3B 1 7 0 Bt YsR 7d %$D N Wd U) .1 c R d L d c d ) #7R c 7 b #7 c R 7 L 2w(D s <7,d 3 Ud b$Y d ) #dRd qYt Ud ) od,d wWd D stI w.dR L .d b dD c WdR L w.d c RwRdD D d c Wd L _d ) (, PB RB8 _d od,d bY${Y $R YBjB8B}Y${ s Y }js{ <d,d %$$D qY }js{ < $R Zts8$hI $t Y -tR$Bt Ud bod,d _BBP qY $8 $D PBjjBbR sR$jk PB8 w;1D stI VAYkst,s:R N88s {P 3$n G b$jj hR }B% $$D stI $$$D $t Y {sR 7 ) d eJw;ED {st A b$t sR w WR .d .d b d ) R V V wV b V Rd b dD d 7 $R s 1E F=7]eNyW! stI Ak eJw;xD Y$R $%R w eeiiV ViWVN*! pVi]yV <eWWyWp 7qy]<qeW!q< .d V WR b d L #d .d d L #d d c L R c ) (3 dR b #d V d b #d V w;3MD !AR% Ys od,d ) Id,d wjD b$Y j ) .d bV stI Ys eJw;MD $R st JZs$Bt PB j B% Id,d BP Y PB8 yjR L Kj L ) ( sR $t N88s ;d b$Y 8hd,d wyD ) 8hd,d wKD ) ( <Bb% N88s ;d {sttB A s}}j$I I${jk R$t{ 8hd,d wD ) bR G Yt }js{ Y ts$t j8t .d bV BP Y hjI -tR$Bt od,d bId,d Ak Y j8t .d b \d #d +d 3 V $R s 9B BP Y }BjktB8$sj d y PBjjBbR PB8 eJw;MD Ys d ) wx D ) x R b b$Y /d ) \Rd #dR +dR b \d #d +d c d L #d c x L /d dR b #d w;3D d L #d d L #d d L R c Id,d 3 R d b #d d b #d V WB Ys #d $R1 R$t{ #dR L #d ) d qYPB b Ys% \d ) #dR $R1 stI \Rd #dR ) #dRd =R$t +d ) 1 bV stI eJw;nD b BAs$t /d ) #dRd c +dR b d ) V d L #d +d d L #d d c Rd L R c R d b #d #d d b #d V w W 1R d L #d 1 d L #d Rd #d c Rd b R c L 3 V d b #d #dRd dR b #d 3B8 eJw;nD stI w;xD b Ys% 1R wd b V D wd b V Dw 1 L dD 1R ) ) Rd Rd R V V bV w d L #d DRd stI Yt{ /d d ) V d ) V w #dRd wd b V D d L #d 1 d L #d c w 1 L dD b R c Rd L R Rd w d L #d D d b #d #d d b #d W w W 1 wd L 2wRDD c w 1 L dD L wd L 2wRDD c Rd b d L 2wRD #d w W d 1 ) 1 L Rd L 2wRD V #d ) w;3d(D 1 d c L 2w(D ) \d +d L 2w(D s Y }js{ hd,d 3 V #dR w;3ddD V q!Gei !3 3=W]qy!W 3yeN*7 !zei ]= y] 3yeN*7 1 G Ys% ZRI Y Ys 8hd,d wV D ) R stI 8hd,d w d D w R qYZR b Ys% }B%I $$D $t Y {sR 7 ) d qY sRR$BtR BP $$$D wPB 7 ) dD PBjjBb tBb $88I$sjk PB8 N88s ;d wBAR% Ys Y }js{ hd,d $R s 9B BP d stI 8hd,d w+d D ) bd Ak $D RB N88s ;d s}}j$RD WBb b RYBb Y $8 %$D k eJw;D b Ys% w.dR L .d b dD c .1R L .d c .1 b .d ) (, Yt{ RwRdD D d w.dR L .d b dD c WdR L w.d R1 d c Wd b w R 1 RLd D d L d c .d L Rd R d K d ) (, L_ K d od,d bY${Y $R YBjB8B}Y${ s Y }js{ <d,d 7B b Ys% B }B% b$Y RB8 PZt{$Bt _ Ys Y PZt{$Bt w R1 d R 1 RLd D d L d c .d b Rd R d w;3d1D $R YBjB8B}Y${ s <d,d y PBjjBbR PB8 w;D stI w;ddD Ys +d ) b#dR w stI Yt{ .d bV R d L dD L 2w(D s <d,d , ) d L \d #d +d ) ) d b #d w ) b#d Rd R d ) b#d R d d L dD L 2w(D w;3dnD L wd b #d D L #dR eJw;xD stI w;ED $% V .dR c ) w d L #d D c R V .d b d w bd #d L #d #dR +d d d L 2w(D L 2w(D s <d,d 3 W L 2wR 1 wR b dDD s <d,d , stI YPB .dR .d d )b L L 2wR 1 wR b 1DD s <d,d 3 R V V V G {Bt{jZI PB8 w;dnD stI w;d;D Ys d ) b#dR c V Yt{ V ) b#dR c stI .d ) R1 d R 1 d .d c V ) w#dRd DR c V R 1 Ld d c wd b c wd L R 1 LR d b L 2wR 1 DD, R 1 Ld d L 2wR 1 DD R 1 Ld d L 2wR 1 D s <d,d 3 w;3d;D 1M F=7]eNyW! eeiiV ViWVN*! pVi]yV <eWWyWp 7qy]<qeW!q< 7$t{ d ) #dRd b BAs$t w R1 d L d R 1 RLd D d c .d b Rd R d ) 2w(D s <d,d 3 qY$R {B8}jR Y }BBP BP Y $8 %$D {P w;d1D qY sRR$Bt $t %$$D PBjjBbR tBb PB8 %$D stI N88s ;d !Z t- s$8 $R B }B% Y $8R $%D stI %D BP _B};; $t Y {sR 7 ) 1 BP Y PZt{$Bt +i b Ys% +1 ) ; c +d stI Yt{ +d ) +1R c k Iht$$Bt d 3 c ; +1Rd 3B8 eJw;nD w;;D stI w;xD b BAs$t V Rd 1 w n L #1 DRd c w d L #d DRd c wd L 1 DRd d ) c ) R R 1 ; wd b V Dwd L ; D +1Rd c ; ) w#d #1 DRd c wd b ; D L 2wRD ) \Rd c wd b ; D L 2wRD s h1,d 3 1 G Ys% ZRI sAB% Ys h1,d $R s 9B BP d , 1 , n stI V BP BI w R stI $ $R s 9B BP ; BP BI wR b dD 7$t{ Y PZt{$Bt +1d $R s }$8 j8t s h1,d $ PBjjBbR Ys +d ) +1R c \Rd c wd b ; D L 2w(D 1 ) \Rd c +1R b +1 \Rd c +1Rd c ; L 2w(D 1 1 c +1R b +1 L 2w(D s h1,d 3 ) \Rd 1 qY$R $R Y sRR$Bt $t $%D PB 7 ) 1 k w;D stI w;ddD R d RB R d b b d L #d c dR b #d d L #d c dR b #d d L d L #d cw dR b #d d d L \d +d ) 2w(D s hd,d , d R c w\Rd 1 +1 b +1 D ) 2w(D s h1,d 3 #dR qYPB w d L \1 #1 +1 DR b L \1 #1 +1 D L d L #d d c \1 #1 +1 b R c +1 ) 2w(D R d b #d #d V q!Gei !3 3=W]qy!W 3yeN*7 !zei ]= y] 3yeN*7 stI Ak Y Iht$$Bt BP Y PZt{$Bt R 1 b 1 1 d L #d c dR b #d 1 b \1 +1 ) 2w(D s h1,d 3 w;3dxD k N88s ;d Y }js{ h1,d $R Bsjjk s8$hI $t Y -tR$Bt o1,d bI1,d stI Y PZt{$Bt $R s }$8 j8t BP Y }js{ <1,d ) <Mo1,d BB% b {Bt{jZI PB8 w;dxD Ys d 1 R 1 7$t{ 1 ) d L 1 b \1 +1 ) 2w(D s <1,d 3 L \1 #1 +1 Y$R $8}j$R ) d 1 b #1 \1 +1 ) ) b#1 R 1 L #1R 1 1 b #1 w R 1 L 1D L 2w(D L 2w(D s Y }js{ <1,d 3 <Ak b Ys% }B%I Y $8 %D $t Y {sR 7 ) 1 WB Ys b Ys% sjRB }B%I Y $8R $$D stI $$$D BP _B};; $t Y {sR 7 ) 1 Ak w;dxD WBb b }B{I Ak $tIZ{$Bt N 1 7 c 0 stI RZ}}BR Ys Y sRR$BtR $$D T %D BP _B};; YBjI PB 1 > 7 G bst B RYBb Ys Yk YBjI sjRB PB > ) 7 Ld k $tIZ{$Bt Yk}BYR$R b Ys% +7 ) 17 c +7d ) 17 c \Rd c +7R L 2w(D s h7,d 7 stI Yt{ c +7 ) \Rd 7 17 R c +7Ld L 2w(D s h7Ld,d 3 R 1w7LdD k eJw;nD stI w;;D Y PBjjBb$t $It$k YBjIRU R 17 w 17 L dDw 17Ld L #7Ld DR ) 3 R 1w7LdD w 17Ld L #7Ld Dw 1w7LdD L dD 7B w;3dED 17 Rd ) #7Ld c wd b 1w7LdD D L 2wRD s h7Ld,d R 1w7LdD stI {BtRJZtjk Rd R +7 ) \Rd c #7Ld c wd b 1w7LdD D+7Ld L 2w(D 7 ) \Rd 7Ld c R +7Ld b \Rd 7Ld c Rd w 1w7LdD +7Ld D w;3dD c +7Ld L 2w(D s h7Ld,d 3 WBb b BAR% Ys Rd 1w7LdD c +7Ld ) \dR 7Ld L 2wdD s h7Ld,d , bY${Y PBjjBbR sR$jk Ak $tIZ{$Bt wZR$t w;d(D stI w;dEDD stI Yt{ w;dD k$jIR R +7 ) \Rd 7Ld c +7Ld b +7Ld L 2w(D s h7Ld,d 3 w;3dMD n( F=7]eNyW! 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Figure 5 iPt{R >dH >1H >nH VtjR stI ] s$ V tB Bt s8jk s8$hI BbR BP jBAsj PZt{$Bt hjIR 3$t$ 3$jIR V}}j M w1((1D 1(u1dx F 9s stI V ps{$s V Bb b$Y tBtTpsjB$R R}R bY${Y ss$tR Y *$tPjITzjsIZ ABZtI B s}}s $t F WZ8A qYBk zp *$tPjI stI 7p zjsIZ qY tZ8A BP }B$tR BP st sjAs${ {Z% 3Zt,R$Btsj Vtsj $ _$jB9Yt d wdMnD EMuE >3Zt{ Vtsj V}}j d wdMnD xnux;H ;1 >;H >xH >EH >H >MH >H >d(H >ddH >d1H >dnH >d;H >dxH >dEH >dH >dMH >dH >1(H F=7]eNyW! eeiiV ViWVN*! pVi]yV <eWWyWp 7qy]<qeW!q< W* ej,$R e-}j${$ 8BIZjs BbR $tU _B{I$tR BP Y qY$kT3$PY >dH VttZsj VjjBt ]BtPt{ Bt ]B88Zt${s$Bt ]BtBj stI ]B8}Z$t wq sRs stI V zsIk IRD =t$% BP yjj$tB$R s =AstsT]Ys8}s$t wdMD 1nun1 W* ej,$R e-}j${$ BbR BP *$tPjI 8BIZjs {Z%R $tU eZB}st ]BtRR BP sY8sT ${R w s{jBts 1(((D zBj yy w] ]sRs{ZAs sjj IRD $,YsZR sRj w1((dD dMudM s$%U8sYWq\(((xd;( w1(((D V ps{$s stI < 7${YtBY V Bb BP V$tT7{Y$ -tR$BtR BP PZt{$Bt hjIR ss$t$t Y *$tPjITzjsIZ ABZtI yt%t sY d1d wdxD 1ddu111 V ps{$s stI < 7${YtBY !t Y sRk8}B${ AYs%$B BP RB8 BbR BP PZt{$Bt hjIR B% ht$ hjIR F WZ8A qYBk Ed wdED 1;Mu1n V ps{$s stI < 7${YtBY !t s8 BbR B% ht$ hjIR F i$t Vtb sY xx w1((nD xnuM( V ps{$s < 7${YtBY stI qYB8sR !t BbR stI {B8}BR$s BP BbR BP PZt{$Bt hjIR B% ht$ hjIR 3$t$ 3$jIR V}}j n wdD 1xu1; p %st I p stI %st I zjZ Vt sRk8}B${sjjk BBI Bb BP {Z%R B% Y hjI b$Y $Y j8tR Zjj NBtIBt sY 7B{ n; w1((1D 1dun(( O yYss 7B8 8s,R Bt Y tZ8A BP s$Btsj }B$tR BP sjAs${ {Z%R B% ht$ hjIR F3s{ 7{$ =t$% qB,kB 1M wdMdD 1du1; GT]G N$ < sYss` stI < 7${YtBY Wb B}$8sj BbR B% ht$ hjIR $tU VjB$Y8${ WZ8A qYBk >7kItk 1((1H w] 3$, stI * BYj IRD n1unM < W$I$ stI ] $t is$Btsj }B$tR Bt {Z%R B% ht$ hjIRU YBk stI s}}j${s$BtR ]s8A$I =t$% _RR ]s8A$I 1((d FT_ 7 7Z j tB8A IR }B$tR s$BttjR I:Zt {BZA sj?A$JZ RZ Zt {B}R ht$ ]i V{sI 7{$ _s$R 1E wdMnD nu;(1 ) eZ%R >d1MH FT_ 7 is$Btsj }B$tR Bt {Z%R B% ht$ hjIR Zt}ZAj$RYI j{Z tBR Ak 3+ pBZ%|s <s%sI =t$%R$k wdMxD < 7${YtBY VjAs${ 3Zt{$Bt 3$jIR stI ]BIR 7}$t zjs j$t dn V q8,$t <$jA {jsRR hjI BbR BP PZt{$Bt hjIR B% ht$ hjIR stI jBb ABZtIR PB \wRD F WZ8A qYBk M w1((dD dMu1d( V qRPsR8st 7p zjsIZ stI q $t, BIZjs ]Z%R 7Y$8Zs ]Z%R stI pB}}s ]BIR A Yst Y zsRYs8B%Tp$jA ABZtI sY Ws{Y d( wdM1D 1du1M F GZjPst sY8 q[ Z8 sjAs$R{Y 3Zt,$Btt,[ B} _Y* YR$R weRRt 1((nD qY $t, *ts$Bt BP 7Y$8Zs RZPs{R stI s }BAj8 $t {BI$t YBk $tU 3ZtIs8tsjR BP ]B8}Zs$Bt qYBk >]BAZRH wN ZIs{Y ID 7}$tTzjs Wb OB, dMx x(nuxdd
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Table4-Conserved in All Ecoli
1761493_s_at
1761494_s_at
1761499_s_at
1761502_s_at
1761505_s_at
1761513_s_at
1761516_s_at
1761522_s_at
1761524_s_at
1761526_s_at
1761528_s_at
1761540_s_at
1761541_s_at
1761543_s_at
1761544_s_at
1...