Convexité dans le plan discret. Application à la tomographie
Texte intégral
(2)
(3)
(4) "!# $&%')(+*,.-/!0
(5) 123" 465. 7988:,;=<. >9?@A:,BDCE< F GJILK&M H N CO?BEPRQSCOT3UV;,8WUYXZCO8\[?]^X N P`_O@A;[; acbEdfehgji9kml^gonqpsrct^uwv)g6kxu/ehy{z|}ku~x K& yjdu~|)n~u/e3y n~b}k3uweh^ui^g N BV:qV;,8WUV:,;;qUVCO?hUY;,8 ?; N ?T3P`X` ?;q@;q8WU N B f \E= PZ;¡ O ¢[:,£q;,@¤TBD;¤¥§¦O¦O¦ ¨ X©UYBD;ª< « ¬ ¯®±°³²µ´f¶· ¸º¹»º¼ 6½ ° ¾¿½j¹» ¸º´j¼\ÀÂÁ°Ã¶ Ä ¾º¾¿½j´jÀù¶´6¬ Å ½j¹ ¶ ¬»Æ ¬»Ç»Á¹È¾¿Éº´6° ]^X`BD;,£qUV;,?BÊ[;UVËÌ,D;ª< Í ?BVXZ£,;¤ÎÏ`ÐÑÒ Ó ?BÔ^< Õ XZP`P`;q9Ö9×0ØhÒ.ØjÑÚÙÚÛ)ÜhB ÝNN COBDUV;,?B 7^P`Tf;,BUYCßÞà×0áHâªãÙÚä}å 7&UVUYXZP æ ãçÑ)Ü#B ÝNN COBUY;,?3B Ó £, ?;qèéÑÚêåëª×0Ø Í ?BDX`£,;¤Î¢Ï`ÐhÑ3Ò Ó ; 8hìwí}X`;,BDBV;¤î×#Ð=×0Ïáháï0ðOÜ#B ÝNN COBUY;,?3B ]^;,8XZ¢îÏñjò.Ñ=Ø.Û. ó T COB UYCÝX`BV;9[; ó COôOX` ?;§Ü[6QS7^PZôOCOBVX©UYË@AX` ?;§Ü3;õUÊ[6Q÷öw8høsCOBV@ UYXZ ?;[;ùP`;qBV@ACO8WU¢ ú }û 7È¥h üOý ó TfCOB UYCOXZBV;[6QSöw83øsCOBD@ UYX` ?;79P`ôOCÝBVXZUVË@AX` ?;Ý<þ#CO8[3;,@A;,8WUYÊ;qU7 NN P`XZ£ UVX`CO83ú ÿ Í ¦.
(6) ¥.
(7)
(8) . Ó ;UYXZ;,8BV;q@;qBV£,XZ;,B6< £q;qUDUY;UYË!Ìq#"V$B; A Ü % '&$(!)*&
(9) +,-+.0/21 ;qU43 &50/7698 9&
(10) +,+:&!;<&
(11) =0&
(12) '>$( [6Q @? COX`B £,£q; N UV:¤[;¡B §NN CÝBDUY;qB D@CE 0F+UV
(13) Ë9Ì,G$D&I;ÝHÜ N=JCO0D ;,BàN COPZ;,?3¤B9Tf@COQ8K 83? ;,CO¤X`B ôO??;qXZD[UY: XZCON 8;q¤8 [ ? X 8WU@£qQSCO;,8WU ? N ;,UYBDBV@; XZ 8[3M;ßLöw@P ;A øD ?6@AÜf[X`P`Ì,X BVPZXZ;V;q[3B :,KT? ?3;,U^£\[PZ;; [3CO@ X`8B; A Z "
(14) B &
(15) +,[6ONQQ K PR? CO&
(16) XZTBESU£,F/BVF?VF[N Ìq;q9 U PZW;0[+.:,T3)?3U^+,X/;qY8&
(17) P [ N COìw£,?3CÝB8 UV? CO
(18) ; ?3\hUYXZ;qUV=0: A 8CÝÚøsBV?3£qUY?3;,?D;,}[XZV£,?3VVXZCO8qܧ;õUª@A;,BV£qX ;õ%QUc T3X`;,8m;õ5 U]BAUVCOUY?CO?3£,;qPZX? ;,\éA@?3;qe; @¤dqTQ BV;qX}£,_§[UY? CÔOó.:qó 79 ?éöù T?;qUBV; ;,8?ÂNù &BDf<UVX`W£,?3P`X`;qB P^&
(19) /7'(_; Ü 9i G$`#0+.1 ;õbU a 0/ +.0+.1 Ü &
(20) +-VF&^vÜ)3 F+:G$gh&
(21) jJ(lkmJ`hÜ`#;q0UV/70£ 6L c Ü Y o W c Ü c )n-90+ ko`F9&
(22) +,+
(23) prq70 & "$J(
(24) 9&
(25) /sP^&9V0Fp q +:NJ/7tu7&*Cl0#10F J(
(26) wx/Y&*)F+-V0m&
(27) .
(28) ü.
(29) ¬»Æ. Æ. ¹È´6Á°. *q "!j5Ý"s!f yjb . . yek3ugol^gjxÃdb)tv)gEgjx. . !f
(30) .!#s. L© ^] :#8XZUVX`CO83<L!L!L4L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!Lº L ¥ ù CO883;
(31) \hXZUV:;õUÊ£,CO8 ? ;,\hXZUY: L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!Lº ü L ¥XL© ]^:#83XZUYXZCO8 L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!Lº ü L ¥XL ¥ ó XZ;,8Ê£qCO88;,\3X©UY:§Üf¥"ìw£qCO8 ? ;
(32) \hX©UY:ÝÜ0£qCO8 ? ;
(33) \hX©UY:¢UYCÝU P`;[ 8 Z L4LIL!L4L!Lº L X¥ L û}U[ 8 Z L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!Lº ¤ O!ff
(34) js . X¥ L© ]^XZBV;,£õUYXZCO8E[ 83 Z L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¥h X¥ L© L© Z CO?~ìwBV:qV; X? \P`XZ:,n ¡[;qX? \[X`BD;,£õUYX`CÝ8 LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¥h X¥ L ¥ ùCO8 ? ,; \hXZUY:VXZ@ N P`; L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¥ X¥ L [ ìw£qCO8 ?
(35) ; \hX©UY: L!L4L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¥ X¥ LSü íÚCÝX`8WUYÊ X`P`P 8WUY L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¥ X¥ LSü L© ]^: #83XZUYXZCO8Ê;qU N BV;,@AX`ÌqBV;q N BVC N BDX`:õUY:, LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¥ X¥ LSü L ¥ ùCO8V:q£,?3UVX XZUV:¢[;, XZP`P 8WUYr L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¥Ýý X¥ L ó X`;q8 @? ;,£P £qCO88? ,; \3X©UY: L4LI L!L4 L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¦ X¥ L XL© ó QSX`83V:,£ TXZP`XZUV: L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¦ X¥ L XL ¥ ]^X`BD;,£qUVX`CO83Ë3COBV"X !,CO8WU P`;,;qU ? ;qBDUYXZ£ PZ;, LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¦ X¥ L#FL X¥ L©
(36) ìw£qCO88,; \3X©UY: L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¦ X¥ L#FL X¥ L ¥ ü§ìw£qCO88,; \3X©UY: L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L X¥ L XL û \UV;,8DX`CO8 [XZBV;q£qUYXZCO8ECO? PZ?R L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L X¥ L÷ý ó X`;q8 @? ;,£P £qCO8 ?
(37) ; \hX©UY:¢UYCÝU P`; L!L4L!N L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L X¥ L ó ;, [ ìw£,CÝ8 ?
(38) ; \h;,ÊøsCOBUY L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ý X¥ L L© ]^: #83XZUYXZCO8 L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ý X¥ L L ¥ ó XZ;,8 K ? ;q£P [ ìw£,CÝ8 ?
(39) ; \hXZUV: L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L X¥ L L X¥ L© 7 ? ;,£PRQSX`8D:,£ TX`PZXZUY: L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L üW¦ X¥ L L X¥ L ¥ 7 ? ;,£P`;[:õUY;,BD@XZ8 8WU L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ü X¥ L L íÚCOXZ8 UVÊ XZP`P 8 UV&[;q [ ì/£,CO8 ? ,; \h;,9øsCÝBDUY L!L4L!L!L4L!L4L!L!L4LIL!L4L!L ü ¥. 2. . 3. 2. .
(40) ý.
(41). . . . $j!fÝ"s!fÚ O!ff
(42) j AÚ O W L© ]^: #8X©UYX`CÝ86Ü N BD;,@AX`Ì,BD;, N BDC N BDX`:qUV:, L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L L© L© ùCO8 ? ;,\3X©UY: L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L L© L ¥ ùCO8 ;,\3X©UY:DUVBVX`£õUY; L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L L ¥ ù Ê[;¢P [3?X`@A;,8DX`CO8H L!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L L Í XZ8X`@ [6QS?8;¢øsCO8£õUYXZCO8\DUVBVX`£õUY;q@;q8 U£qCO8 ?
(43) ; \h!; L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L }!ff" 12Ús# h ü L© ]^: #8X©UYX`CÝ8;õU N BV;q@XZÌ,BV;q N BVC N BVXZ:qUY:qÊ[;, N COX`8WUYE@A:,[3X 8 L!L!L!L!L!L4L!L4L!L ü L© L© ]^: 8XZUVX`CO8 L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L ü L© L ¥ í}BVJC dD;q£qUYXZCO8 L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L ü L© L ù B £qUY:qBVXZ UVX`CO8\PZCh£ P`;[3;, N COXZ8WUYE@A:,[X 8 L4L!L4L!L!L!L!L!L4L!L4L!L ü L© LSü ó ;83CO@¤TBD;[; N COXZ8WUYÊ@:q[X 8 L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L ü L ¥ 7 NN# BUY;,8 8£,;[;q N COXZ8 UVE@:q[X 8 L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L ü L ó ;,Ê@A:,[X 8ED?BÊ£ Ë ?;BV:,D; Y? L L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L mgjdb)t^xWeki^d0ehub)t. 6. ü ü ݦ ݦ h . . Ýý Ýý. ýO¦ Oý ¦ ý. . *q "!j5 Ý" !f ¨ OC @ACOôOB §N ËX`;^£,CO8WUYXZ8 ?; G$NJw2F &
(44) +,J&51 ONJw N5V0+.7`Fp L!L4L!L!L!L!L!L4L!L4L!L ý ¥ ¨ OC @ACOôOB §N ËX`;^ôO:,CO@A:qUVBVXZ?3; L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L ý ¨ CO@ACOôOB §N ËX`;^[X`D£,BVÌõUY;_LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L ¦ %ß O!fÚ õ
(45) 5 Ý"s!fm. à O!ff
(46) j XL© N :,B UYX`CÝ8E[;¢£,CO@ N PZ:qUYXZCO8 L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L XL ¥ ó ;,EUwÔ N ;,[;,[BVCÝXZUY;q LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L XL û \ N BV;qVDX`CO8[;¢PRQS
(47) ; \hX`UY;,83£,;¢[6QS?8;VCOPZ?3UYXZCO8 N# BE?83;à¥ì/øsCOBD@¤?P`; L!L!L4L!L4L!L L!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L O¦ XL L© þ#COBV@¤?3P`;,E,; \ N BVXZ@ 8WU XL L ¥ þ#COBV@¤?3P`;,E,; \ N BVXZ@ 8WUÊX P [ E(j) ì/£,CO8 =?
(48) ; q\hXZUV : L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L 3 XL L ùCO@ N P`,; \3X©UY:[;¢PQ PZôOCOBDXZUYË3@; L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L 3 XLSü û \UY;q8VXZCO8 N PZ?E[;¢[;,F? \[X`BD;,£qUVX`CO83 L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L Ýü XL ùCO838
(49) ; \hXZUV: L4L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L XL÷ý ;,£qCO8DUVBV?£õUYXZCO8 K ? ;,£P`C£ PZX` UYX`CÝ8 N BVXZCOBVX L!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L XL öw@ N P`:q@;q8WU UVX`COD 8 L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L Ú W# s Ú "! ! "12L. sH " O!f q
(50) 5 Ý" !f # #ý L© ó ;,% $' &ÊXZUV£ ËX`83ô§ìw£qCO@ N CO8;q8WUY) ( L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L #ý L© L© ]^: 8XZUVX`CO8E;õU N BV;q@XZÌ,BV; N BVC N BDX`:õUY:, L!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L #ý L© L ¥ *}8?3@:qB UVX`CO8[;¢' &ÊX©UY£ ËXZ8ô§ì/£,CO@ N CÝ8;,8WUY L!L4L!L4L!L!L!L!L!L4L!L4L!L O¦ #ý L ¥ ó ;UYË:qCOBVÌq@;[Q÷?83X`£,X©UY: L!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L 3 #ý L X¥ L© ù Ê[;,ÊUYC§U PZ;,@A;,8WUE£,CÝ8 ?
(51) ; \h;, L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L 3 #ý L X¥ L ¥ û \hUV;,8DX`CO8 X? \ [ ì/£,CO8 ?
(52) ; \h;, L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L #ý L X¥ L F¥ LZ íÚCOP©ÔôOCO8 PZX` UYXZCO8[6QS?+8 $, &ÊXZUY£ Ë3X`8ô§ì/£,CO@ N CO8;,8WU ( L!L4L!L #ý L X¥ L F¥ L ¥ ó ;, Dì N COP©ÔhôÝCO8;, [ ì/£,CO8 ? ,; \h;,9øsCÝBDUYn L!L!L4L!LIL4L!L4L!L!L4LÊ ¦ q. j.
(53) ý#L ¥XL ùCOBVCOPZP XZBV;qL4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ O O ý#L ÿ983X`£,X©UY:[6QS?8;¢DCOP`?3UVX`CO8\[3CO88:q; L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ O O %ßO!fÚ q §5 Ý"s!fm Ú "! L ¤ O!ff
(54) j -/!0 ". L© 7 ? ;,£[;,?F\[X`BD;,£õUYX`CÝ8L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ O L© L© ó ;q ?§ BDX T3P`;,TCCOP`:q;,883;, L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ O ü L© L ¥ û \ N BD;,DVX`CÝ8[;¢P [ ìw£qCO8 ? ,; \3X©UY: L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ O L© L ó TfCOBV8;XZ83øs:,BDX`;q?BV; L!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ O ý L© LSü ó TfCOBV8;D? N :,BVXZ;,?BD; L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ O L© L ó Q PZôOCOBVX©UYË@A;^[3;BD;,£qCO8DUVBV?£õUYXZCO8 L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ O L© L÷ý ;õUYCO?<B ¡P BD;,£qCO8DUVBV?£õUYXZCO8,; \ £qUV0; L L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ "¥Ý¦ L© L ùCO88,; \3X©UY: L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ "¥Ý¦ L ¥ í}P`?Ê[;¢[;qX? \[XZBV;,£õUYXZCO8< L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ "¥h L X¥ L© ;q£,CO8UYBD?£qUVX`CO8 ÝNN BDCh£ Ë3:,; L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ "¥h L X¥ L ¥ ;q£,CO8UYBD?£qUVX`CO8
(55) ; \ £õUY; L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ "¥ L ;,£,CÝ8DUVBV?£õUYX`CÝ8 ÝN3N BVC£ Ë:,; K? ;q£PZCh£ P`XZ UVX`CO8 ¡N BVXZCOBVX L4L!L4L!L!L4LIL!L4L!LÊ "¥Ýý 8 L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ "¥Ýý LSü öw@ N P`:q@;q8 U UYXZCOD # ! 5Ú s!fmW Ú "! ! 12 ¡!f5.f ". # L© ùCO8£qP`?DX`CO8 L!L4L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ "¥ # L© L© ùCO@ N P`
(56) ; \hX©UY:[;¤î×fñjå=ÙÚðÒ.Øãñ6Ò.ÏåªÙ LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ "¥ # L© L ¥ ÿ98XZ£,XZUV: L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ "¥ # L ¥ í}BVCOT3P`Ì,@A;,ÊCO? ? ;,BUY L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ # L X¥ L© ;q£,CO8UYBD?£qUVX`CO8 L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ # L F¥ LZ L© í)BDCOTPZÌ,@A;£,BVCÝX`VQ: L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ # L F¥ LZ L ¥ ¨ CÝU P`;q@;q8WUÊ£,CO8 ,; \h;, L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ # L X¥ L ¥ ;q£,CO8UYBD?£qUVX`CO 8 ÝNN BDCh£ Ë3:,?; L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ # L F¥ L X¥ L© ;q£,CO8UYBD?£qUVX`CO8£,CO8WUVX`8 ?; L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ ¥ # L X¥ L ÿ98XZ£,XZUV: L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ ¥ # L F¥ L L© 7 X`TPZX`DV;,@A;,8WUE[3;P £,CO8 ? ,; \hXZUY: L!L!L4L!L4L!L!L4LIL!L4L!LÊ ¥ # L F¥ L L ¥ ]^X`@A;,8DX`CO8 L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ ¥
(57) j. q . $ O!1 }!." (. R7 L© º]^XZBV;,£õUYXZCO8 {x,y} L!L4L!L!L!L!L!L4L!L4L!L!L4L!LIL4L!L4L!L!L4L!L4L!L!L!L!L!L4LÊ R7 L ¥']^XZBV;,£õUYXZCO8 {x,y,x + y,x − y} L!L!L4L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4L!LIL4L!L!L ,üW¦ R7 L ]^XZBV;,£õUYXZCO8 {x,y,2x + y,x − 2y} L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ ü ¥ !1 s
(58) .s h
(59) }
(60) 12 s L.h H 0!0 § 12 . " O!fÚ q §5 Ý"s!fÈ. ¤O!ff
(61) 6h. L© ;,£,CÝ8DUVBV?£õUYX`CÝ8[;q [ ì/£,CO8 ? ,; \h;, @ ? ;,£PRQ P`ôÝCOBVX©UYË@A;9[3?£ Ë §N XZUVBV; LIL!L4L!LÊ ü L ¥ ;,£,CÝ8DUVBV?£õUYX`CÝ8 ÝNN BDCh£ Ë:q;H[;, [ ì/£,CO8 ? ,; \h;,ßøsCOBUY K ? ;q£HPRQ P`ôÝCOBVX©UYË@A;\[? £ Ë ÝN XZUYBD; L!L!L4L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ üOü ¡"! s
(62) ! 0 " s. 6.
(63) .
(64). ¡"!s! f " . *,
(65). 6. .
(66) ±¶Á¬ ¸ºÀ鶴6¬» ó Q÷XZ@ ôO;,BDX`;¢8 ?@:qBVXZ?3;à;,U?8H[CÝ@ XZ8; ?JdDCO?3BV[6QSË ?X=;q8 N P`;qX`8;qVDCOBuL 8 N ;,?hU[XZDUYXZ8ôO?;qB N BDX`8£qX N# P`;,@A;,8WU[;,?X\ §NN BVC£ Ë;qUVË:,COBDX` ?;q[3;£q;qUDUY;¢[X`D£,X N P`XZ8;ª< P,; Q \h§;,N@ N BVN CP`;£ Ë?;A83£,;COXZ8W@ UVX` 8 ôO?;;ª;,<Ú8CO8³8X ? V?; N?N CÝ[V;¢;\ôO BD?X`E;ß;qPDQ÷UXZ@ ? ?ôÝ;¢; £,NCO@ABVC @? X`;;q8 ?3U¤8[6;Q÷?3øsCO88;ߣqUV[X`COCO8838:,;ߣqCO8WUYXZ8?3;ÝÜ N#L B f : R → [0,1] P Q V B C £ Ë ; 3 [ ` X V q £ V B õ Ì Y U ª ; # < O C 8 q £ O C 8 V Z X [ , Ì D B ; R P S Q ` X @ O ô Ê ; Y U q ; ` P ` P ^ ; ? ÷ Q q ; ` P ` P ^ ; , ; D Ê U Y U C £ Ý q : ; [ § N N ?38;àXZ@ ôÝ;;qDU[CÝ8£ N PZ?3UY_ÝU9£,CÝ@@A;?38;àøsCO8£õUYXZCO8[; [ 8 8EPRQSCOVBVXÚ[3COX`88 UV;,¥?§Bý < Z {0, . . . 255} 83X ? ; X? \[3;ôÝBVX`M L ó AN BV;,@AX`ÌqBV; ÝNN BVC£ Ë; PRQ K?§ 8WU ôO;¢[; N CO? ? CÝX`BÊ?3UYXZP`XZV;,BE?38;ôÝB 8[3; N BDU9[;,@ UYË:q@ ì UVX` ?;,E[3: d5¡
(67) ; \hX`U 8WUV;,u Lfÿ8,; \h;,@ N P`;[;¢D?£,£qÌ,[;¢£,;qUDUY; ÝNN BVC£ Ë;;,U9P £,CO@ N BD;,VDX`CO8 Ó í)û Õ [;qXZ@ ôÝ;,} ?Xf;,DUT D:,;^;q8 N# BUYXZ£,?PZX`;,B}D?BP UYB 8DøsCOBD@:q;9[3;^þ#CO?BDX`;quB L ó ;^ôOBDCO N BVCOT3P`Ì,@A;Ê[; £q;qUVUV; ÝNN BDCh£ Ë3;A;qDU ?;PRQSCO8H8;¡£qCO8WUYBV_ÝP`; N# PZ;,;qBVBV;q?BV¢[?3;, X? \ N BVC£,:q[:,[;¡£ P`£,?39P L6í B ,; \3;q@ N P`;¢P`;qBD:,;qP`VCO8WU9VCO? ? ;,8WU §NN BVC£ Ë:q N# B9[;q CÝUVU 8WUY[CO8WUPRQ BVX©UYË@A:qUVX` ?;;qDU9;,8£qCOBV; @A:,£qCO88 ?0; L ó V;q£,CO8[3; §NN BVC£ Ë;E[CÝ88;E[;q P`ôOCOBDXZUVË@;qÚ ?X3VCO8WU)[XZBV;,£õUY;q@;q8 UªX`@ N P`:q@;q8 U TPZ;,=[ 8 P àN B UYX` ?B; L#í B&£,CO8WUVBV;¢P`;q&@ UYË:q@ UVX` ?;q&[?[X`D£,BD;qUECO8:õUY:Tf; ?£qCO? N @ACOX`8 [: ? ;,PZC NN :,;, £ B£,;qUDUY; N BVCÝTP`:q@ UVX` ?; @ACOX`8¡[Q÷?8 VXZÌ,£qP`0; L Í X`£,;q[3;,BV83X`Ì,BD;, 88:q;,A[;8CO@¤TBD;,?D;, BD;,£ Ë;qBV£ Ë;qªCO8WU:qUY:)ø X©UY;,.[ 8.£,;8CO? ? ; ?¤[3CO@ X`8;ª ?;PRQSCO8 ÝNN ;,PZP`;}@ X`8WUY;q8 8WU $¤ôO:,CÝ@:õUYBVXZ; [XZV£qBVÌqUV; (YL ó Q÷CÝFT dD;,£õUYXZøf[;ÊP ^N BV;q@XZÌ,BD; N# BDUVX`;&[3;E£,;qUDUY;&UVËÌ,D;9;qDU)PRQS:qUV?[;E[;q):, ?X ?§ PZ;,8WUY}[3X`V£qBV;õUY}[6QS?8; 8C§UYX`CÝ8é£qP DVX` ?;A[;AP ôO:,CÝ@:õUYBVXZ;ª<jP G$NJ/7=0& c
(68) L.û)82; ;qU[ 8P`;£,CO8WUVX`8 ?2P £qCO8 ?
(69) ; \hX©UY:;qDU £qCO88 ?;[; N ?3X`ÊPRQ 8WUYXZ?3XZUY:^;qU [3;83CO@¤TBD;,?D;, ÝNN PZX`£ UYX`CÝ8&;,8ôO:qCO@:õUYBDX`;ÝÜ3;,8 8 P©ÔhD;¢;qU;,8 C N UYX`@AX` UYXZCO8oL û}P`PZ; ?3VVX):õUY:A?3UYXZP`X`D:,;àUVBVÌqUV_ÝU;,8 X`@ ôO;,BDX`;§Ü N# B¢,; \3;q@ N P`;P øsCO8£õUYXZCO8 ?3XBD;,@ N P`X©U?8 CN OPZÔôOCÝ8;VCÝ? 9
(70) ì cXZ8[C &Ê [;,@ 8[;DXjP`; N COP©ÔhôÝCO8I; ¡BV;q@ N PZX`B ;,DU£qCO8 ?
(71) ; \h; ? COX`B #ôÝ?BV;¡ L ùX©UYCO8 ?DVXÚP`;£ PZ£,?P.[;PRQS;,8 ? ;qP`C NN ;à£qCO8 ? ,; \3;¡ ?XÚ;,U?8 N BVCOTPZÌ,@A;¢£,P VVXZ ?;à[;ôO:,CÝ@:õUYBVXZ; PZôOCOBDXZUYË3@XZ?3; CO¦ L XZ;,8 ?;£q;VCOX©U¢?8;8CÝUVX`CO8LUVBVÌ,¢£qP DVX` ?; [: #8X©UYX`CÝ8H[ 8¢P`;¡[3X`V£qBV;õU8Q÷;qDU N# UYBDÌ, : ? XZ[;,8WUV;9£ B)X`PÔ ¢N P`?DX`;q?BV}ø CO8[;ÊôO:,8:qB PZX`D;,B)P £,CÝ8 ?
(72) ; \hXZUV:^[?£,CO8WUVX`8 o? L ] 83 PZ; N BV;,@AX`;qB £ Ë §N XZUVBV;)CO8 ?ÝN BV:qV;q8 UV;,BÚP £,CO8 ?
(73) ; \hXZUV:P EN PZ? N BVC£ Ë;[;P £,CO8 ?
(74) ; \hXZUV:[?£,CO8WUYXZ8 ?<
(75) P G$NJ/7=0& c PO NJ 0 '& L ] 8PZ;Êc
(76) L£ ùË Ý;õNUVUVX©UY;¤BV;^8CÝ¥UVX`COCO88LXZ8 ô UV BVBVC[[;?@ X©U} ?P`ôO8BD;9:¢8UVCÝCOUY?3XZCOU8 [3N;, P`?3N BDø C N XZTBDX`P`:q;EUV:,;õUPZ[3X`:q;¤; P £,£,;qCOBD8 U ? ,; X`\h8XZ;qUY :¤[3UVX`CÝBVU;q£qP`UV;ªX`CO<08CO8 < 65G$NJ/7=0& N ;,?3U ;,8 N# BDUVX`£q?P`XZ;,B}[: #8X`B?8:, ?X ?§ PZ;,8WU [;, VCO@A@;õUYM L ó ;ÊUYBDCOX`DX`Ì,@A;Ê£ Ë ÝN X©UYBD;9;qDU&[3:,[XZ: F? \ øsCO83£qUYXZCO8}£qCO8 ?
(77) ; \h;q&[; Z [ 83 R L ó ?VDXfCÝ8A;qDU £,CO8høsBVCO8WUY<: ¢[;, N BDCOTP`Ìq@;q}[;[: #83XZUYXZCO8M L ó ; ? UVBVX`Ìq@;£ Ë ÝN XZUVBV;¢:qUV?[XZ;[;, N CÝX`8WUY N# BDUYXZ£,?PZX`;qBV&[;qÊ£,CO8 ? ,; \h;,,ÜP`;q N COXZ8WUYÊ@:q[X 8u L 2. 2. 2. "!$#%&#%'()+*,)%#.-/!0)1#%)+"2&3'-546#7 89:*
(78) ;8<(=>(4
(79) 8<35;5)+?@-5>A/$'=40A/)%B8<!C-40$C'-6CD/E!0FGH&A/>!EDI3JLKMB5)ON .
(80) ¦. .
(81)
(82) .
(83)
(84) .
(85)
(86) .
(87)
(88) .
(89)
(90) .
(91)
(92) .
(93)
(94) .
(95)
(96) .
(97)
(98).
(99)
(100)
(101) .
(102).
(103)
(104).
(105)
(106)
(107)
(108)
(109)
(110)
(111)
(112)
(113)
(114) .
(115) .
(116) .
(117) .
(118) .
(119) .
(120) .
(121) .
(122) .
(123) .
(124) .
(125)
(126).
(127)
(128)
(129)
(130)
(131)
(132)
(133)
(134)
(135) .
(136) .
(137).
(138).
(139).
(140).
(141).
(142).
(143).
(144).
(145)
(146) .
(147)
(148).
(149).
(150).
(151).
(152).
(153).
(154).
(155).
(156).
(157)
(158) .
(159) .
(160) .
(161) .
(162) .
(163) .
(164) .
(165) .
(166) .
(167)
(168) .
(169) .
(170) .
(171) .
(172) .
(173) .
(174) .
(175) .
(176) .
(177) .
(178)
(179).
(180) .
(181).
(182).
(183).
(184).
(185).
(186).
(187).
(188).
(189)
(190) .
(191) .
(192).
(193).
(194).
(195).
(196).
(197).
(198).
(199).
(200).
(201) .
(202) .
(203) .
(204) .
(205) .
(206) .
(207) .
(208) .
(209) .
(210)
(211)
(212).
(213) .
(214)
(215)
(216)
(217)
(218)
(219)
(220)
(221)
(222)
(223) .
(224)
(225).
(226).
(227).
(228).
(229).
(230).
(231).
(232).
(233).
(234)
(235) .
(236).
(237).
(238).
(239).
(240).
(241).
(242).
(243).
(244).
(245) .
(246) .
(247) .
(248) .
(249) .
(250) .
(251) .
(252) .
(253) .
(254)
(255).
(256)
(257) .
(258)
(259)
(260)
(261)
(262)
(263)
(264)
(265)
(266)
(267) .
(268)
(269).
(270).
(271).
(272).
(273).
(274).
(275).
(276).
(277).
(278)
(279) .
(280) .
(281) .
(282) .
(283) .
(284) .
(285) .
(286) .
(287) .
(288)
(289) .
(290) .
(291) .
(292) .
(293) .
(294) .
(295) .
(296) .
(297) .
(298) .
(299)
(300).
(301) .
(302).
(303).
(304).
(305).
(306).
(307).
(308).
(309).
(310)
(311) .
(312) .
(313).
(314).
(315).
(316).
(317).
(318).
(319).
(320).
(321).
(322) .
(323) .
(324) .
(325) .
(326) .
(327) .
(328) .
(329) .
(330) .
(331)
(332)
(333).
(334) .
(335)
(336)
(337)
(338)
(339)
(340)
(341)
(342)
(343)
(344) .
(345)
(346).
(347).
(348).
(349).
(350).
(351).
(352).
(353).
(354).
(355)
(356) .
(357).
(358).
(359).
(360).
(361).
(362).
(363).
(364).
(365).
(366) .
(367) .
(368) .
(369) .
(370) .
(371) .
(372) .
(373) .
(374) .
(375)
(376).
(377)
(378) .
(379)
(380)
(381)
(382)
(383)
(384)
(385)
(386)
(387)
(388) .
(389) .
(390).
(391).
(392).
(393).
(394).
(395).
(396).
(397).
(398).
(399) .
(400) .
(401) .
(402) .
(403) .
(404) .
(405) .
(406) .
(407)
(408) .
(409)
(410).
(411)
(412)
(413)
(414)
(415)
(416)
(417)
(418)
(419)
(420)
(421) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "#$&%(' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "# . . !. S &$( - +:NJ'(lp 7&$(r1F&7NJjpKVFNJ/Y&$(ltum&E+$&
(422) w2j KNJ/YG
(423) -9NJ/ (@NJh( I6 /21FN ( ] 8 P [3;,?X\hX`Ìq@; N# BDUYXZ;ÊCO8 ÝN3N P`XZ?3;£q;qUDUY;:qUY?3[;^[;q £qCO8 ? ;,\3;qP UYCO@ACOôOB ÝN ËXZ;0L ù;õUVUY; UY;,£ Ë38X` ?;£,CO83VX`UY;I¡ BV;q£,CO83DUYX©UY?;qBEP UYBV?3£qUY?3BV; ][6Q÷?38CÝTFDd ;qUQ N# BUYX`B [;¢V;qXZ@ ôO;,&B [XZC§ì P`COôOXZ ?;,9£,P VVXZ ?;, ÝNN ;,PZ:,;, N BVCJDd ;q£qUVX`CO8 uL ] 8PZ;à£,CÝ8 UVX`8 ?LDX=CO8H£qCO88 U^PZ;, N BVCJdD;q£qUYXZCO8 V;,PZCO8 UYCO?3UV;,¢P`;q[X`BD;,£õUYX`CÝ8[6QS?8 N P 86ÜCO8 N ;q?3U P`COBDBV;q£,CO83DUYBD?X`BD;PRQSCOTFDd ;qU¢ôÝB Ý£q;A?38;¡UYB 8ì øsCOBV@ UYXZCO8;,\ N P`XZ£,XZUV; sP UVB 83DøsCOBD@:q;[; [CO8 #L ù^QS;,UÊP UY;,£ Ë38X` ?;¢?3UVX`P`XZV:q;9[ 8&P`;^V£ 88;,B @:q[X`£ P9L ù; N BDCOTP`Ìq@; ?DVX0?8A:, ?X Ý? P`;,8WU[ 83PZ;E[X`D£,BD;qUj< X`P3,Q ôOXZU}UVCO?3U)DX`@ N P`;q@;q8 U)[3;ÊUVBVCO? ? ;,B ?8HCOTFdD;õU[;¤PQ÷;q N# £,;¤[XZV£qBV;qU* N# BUYXZB9[?8CO@¤TBD;à[; N COXZ8WUY^[ 8^£ Ë ?;¡[BVCÝXZUY; N# B P`PZÌ,P`;I £,;,BU XZ8;,E[3X`BV;q£qUVX`CO8ML ùCÝ8 UVB XZBV;q@;q8 U¤ £,; ?XV; N# DV; N CO?B P UVCO@CÝôOB ÝN ËX`;£,CO8WUVX`8 ?;ÝÜhCO8\V; P`X`@AXZUV;e ?3;,P` ?;q^[XZBV;q£qUYXZCO8Ê[; N BVCJDd ;q£qUVX`CO8oL ó QS:qUV?[;à[3;£q; N BDCOTPZÌ,@A; :qUV:àUYX`@à?P`:q;[ 89P`; [:,T?hUà[;q 88:q;, O¦ N# B?8;¡UV;,£ Ë8XZ ?;[3;A@AX`£qBVCOD£,C N X`;¤:qP`;,£õUYBDCO8X` ?; [ ÿ7> ¨ ö ¨ û Í C PRQSCO8£,CO@ N UY;[3X`BV;q£qUV;,@A;,8WUP`;8CO@àTBV;¢[6Q UYCÝ@;qE[ 8Ê?8£,BDX`DU P9L ó ;,9£ Ë ÝN X©UYBD; ;õUýCO8WU N CO?BT?3U^[;à@ACO8WUYBD;,B?3;àPRQSCO8 N ;q?3UBD;,£,CÝ8DUVBV?XZBV;àPZ;,;,8D;,@¤T3P`;, £,CO8 ? ;,h\ ;,=[3; Z &N# BUYXZB6[;}P`;q?BV N BVCJDd ;q£qUYXZCO8 K? ;q£)V;,?3P`;,@A;,8WU.üÊ[XZBV;,£õUYXZCO86[; N BVCJDd ;q£qUYXZCO8ML í}P`? N BV:q£,X`D:,@A;,8WU Ü3[ 8&PZ;^£ Ë §N XZUVBV; Ü3CO8 [ ÝN UV;?8 PZôOCOBVX©UYË@A;E[:Od5¤£,CO88 ? N CO?B&PZ;, N COP©ÔOCO@AX`83CO ì/£,CO8 ? ;
(424) h\ ;, ] ó >^í Oý ?X\ [ ì/£,CO8 ? ;
(425) h\ ;,ML ó ;£ Ë ÝN X©UYBV;ý^:qUV?[XZ;,B PQ÷?38X`£qXZUY:)[;P DCOP`?hUYX`CÝ8oL û)8 #86Üf[ 8ÊP`; £ Ë ÝN XZUYBD ;ÝÜ#CO8,QSX`8WUY:qBV;qVV;qB ?£ C PZ;, N BDCJDd ;,£õUYX`CÝ89DCO8WU^;qBVBDCO8:,;q N# B ;
(426) h\ ;,@ N P`;!¡ £ ?V;[;, ÝN3N# BV;qX`PZE[;¢@;qV?BD; L ). ?. Ïä+* . -,. /.1032&414658714 91:;7 <. >=. 6@. &A. CB. ED. GF. 2. HJI. GF. CKELNM. C4
(427) 8/<4C B A/B5- 35"'#%=-B@6)!%B=3!08=BIC&4
(428) 8<)%G )%# 35B@40! ' @A/)% M3B@!6540A ' E-5 B5* &!0)135'=B#%-M86(=- 408"BBA/G3H8<8<4C!C!C;5)%&2G A/&8!C# )+ ?@' -5E,A5! - !C4CGH *8<4 -5B 3')%B@! A/ # )%GH8( O. QP. SR. VP. UT. #W R.
(429) . Á°Æ ·m¬Æ. ´ Á° ¾¿¹{Á\¶´6°. ·Ã¶Á´6° ¸ °¼ Àì»±®±°2²µ°¼. O .
(430)
(431) « ɺ¹{¾¿´f¶Á° « ¬±®¯°³² ´f¶· ¶¬ ¶¹È½6°. _
(432) . R ] 83 R P £,CÝ8 ? ;
(433) h\ XZUV:;qDUUVBVÌ,DX`@ N P`;R\[:#83X`B6<?8 ;,8D;,@¤TPZ; F ⊂ R ;,U¢[XZU£qCO8 ? ;,\3; DXUVCO?3UD;,ôO@A;,8WU dDCOX`ôO8 8WU9[3;,?X\ CÝX`8WUY9[; ;,U£,CÝ8 UV;,8 ?H[ 83 L 8 ;,?3U^BV; BV;q8[BV;§ÜUV;,PZP`; . . F N N N F ?;qP`P`;§Ü£,;õUVUV;[3:#8X©UYXZCO8V?B Z < \
(434) ª" !f / 10tum& E &$(
(435) 2 65G$NJ/7=0&X&.G$NJ/7=0&X�+^(@&9V0w &
(436) /7(
(437) 7NJX+ONJX 7NJ/7 A,B 1F& E '&$( 7NJ/7(^&
(438) /79&
(439) +$(R10(@&9V0w &
(440) /7 [AB] (@NJ/7 1h0/#( E 8 N ;q?3U ?DVX.£,CO83VX`[3:,BV;qBÊP`;, N BDUVX`;,E[3; ?XjDCO8WU9UVB £q;,Ê[;¢£,CO8 ? ;
(441) \h;,9[; R < Z \
(442) ª" !f " /Y& #0+,9& E 1F& Z &$(
(443) R10O& ONJ90'&
(444) w &
(445) /7G$NJ/7=0&X&(
(446) R&
(447) e(@&
(448) F'&
(449) w &
(450) /7I( n. n. n. n. & '(
(451) O&IF/rG$NJ/7=0&X& F 1F&. R. n. O&
(452) tum&. n. n. E = F ∩ Zn .. Í PZË;,?3BV;,?3V;,@A;,8WU^£,;qUDUY;8CÝUVX`CO8;,U^UYBDX`£qUV;,@A;,8WU N P`?3øsCOBUY;¢ ?;P ¥"ìw£qCO8 ? ;,\3X©UY:£,CO@A@A;PZ; @ACO8WUYBD;PRQS;
(453) \h;,@ N P`;[3;P #ôO?BD; LZ 0L S. Ïä+*. L© . 65G$NJ/7=0&X&*/YNJ/ NJ90'&
(454) w &
(455) /7nG$NJ/7=0&X& Z X F ⊂ R CO8\[3:,VXZôO8; N# B conv(F ) ;q8 ? ;,PZC NN ;£qCO8 ? ;,\3;[; E P`; N PZ? N ;qUYX©UE£,CÝ8 ? ;
(456) \h;à[3; £,CÝ8 UV;,8 8 U F L0ù^Q÷;qDU9PRQS;,8D;,@¤T3P`;[;q9T BDÔ£,;q8 UVBV;q £,C;A£,XZ;,8WUY N COVX©UYX©øs&[; F L ] 83 P [:#83XZUYXZCO8 L© LZ ÊCO8 N ;,?3U£,CÝ8VXZ[:,BD;,B)[;,;,8 ? ;,P`C NN ;q£qCO8 ? ;
(457) \h;q[; k N CÝX`8WUY ?P`XZ;,? BD;,8[3BV;¢[;,V;qôO@A;,8WUY6< ). Y/ & # 0+,-9&41F&. Z2. . n. R. n. [; N \
(458) ª" !f ". / 10etum& E &$(
(459) k 65G$NJ/7=0&X&(
(460) 7NJF+ ONJF 7 NJ/7 A ,A . . . A 1F& E '&$( 7NJ/7(^&
(461) /79&
(462) +$(!1F& conv(A , . . . A ) (@NJ/71h0/#( E 1. 1. k. . 2. k.
(463) ü. (
(464)
(465)
(466) (
(467). ÿ8; BDUYXZ; [; ;,DUUYCÝU P`;q@;q8WU£qCO8 ? ;,\3;VX;qUßD;,?PZ;,@A;,8WUßDX E = conv(E) ∩ Z L ;,B O¦3Ü O L© L #L ý DV?BD;¤ ?;UVCO?3U N COXZ8WU^[;PRQS;,8 ? ;qP`C N3N ; £,CO8 ? ,; \h;[6QS?8; N# BDUYXZ; F ⊂ R ;qDUA;,8³ø X©U¤?8ÃT BDÔ£,;q8 UVBV;sL£,C;A£qX`;,8WUV N CODXZUYX©øsà[; n + 1 N COXZ8WUYÊ[; F CO8;,8[:q[?XZU?83;:q ?X ?§ PZ;,8£q;;q8WUYBV;¢P kìw£qCO8 ? ;
(468) \hX©UY:¢;qUP £qCO8 ? ;,\3X©UY:¢UYCÝU P`;§< "!}!f s !f / &
(469) /#(@&
(470) w "u'& E ⊂ Z &$(
(471) ONJ 0'&
(472) w &
(473) /7G$NJ/7=0&X&<(
(474) &
(475) (@&
(476) X'&
(477) w &
(478) /7 (
(479) UU&$(
(480) n + 1 65G$NJ/7=0&X& û)8 N BDUVX`£,?3P`X`;qB.P`;,UYCÝU P`;q@;q8WU=£qCO8 ?
(481) ; \h;q[3; Z VCO8WU}PZ;, ìw£qCO8 ? ;
(482) \h;q P`;q=£qCO8 ? ;
(483) \h;q N BÚUVBVX 8ôOP`; ó ;, N CÝX`8WUY9
(484) ; \UYBD:,@ X? \[6QS?8H£,CO8 ?
(485) ; \h; [; DCO8WUP`;q N COX`8WUV UV;,P` ?; F \ {M } ;,U ;,8£qCOBV;\£qCO8 ?
(486) ; \hB; L Z X E ;qDU?8; N BDUVX`;[3; FZ RP`CÝBVàPZ;, N COXZ8 UV¤
(487) ; \UVBVM:,@ X? \2[; conv(E) DCO8WU ÝNN ;,P`:qÊVCO@A@;õUYE[; E L ó ;UYË:qCOBVÌq@;¢[; BV;qX`8hì Í X`PZP`@ 8 ;,B ݦ3Üj O L÷#ý L J BD@;¢ ?;¢UYCO?hU9£qCO8 ? ,; \3;£,CO@ N# £õU K ;,DU9PRQS;,8 ? ;,P`C NN ;¢£qCO8 ?
(488) ; \h;[;¢V;, N COX`8WUV,; \UYBV:q@ X? \ L 8;q8[:q[?X©U9PZ;ø XZU&V?X ?Ý 8 U "! }!f s !f k NJF & #0+,9&ONJ 0'&
(489) w &
(490) /7*G$NJ/7=0&X& 1F& Z #8XZ; &$(
(491) < &
(492) /7=0&
(493) 'N 7& G$NJ/7=0&X& 1F&e(@&$(*(@NJwxw &
(494) ( ó ;£ PZ£,?Ph[;,}DCO@ @A;qUV=[6QS?8¡;q8V;q@¤TP`; 8X3[; N COXZ8WUYª;qDU}?8 N BVCOTPZÌ,@A;£qP DVX` ?;[;&ôO:qCO@:õUYBDX`; P`ôOCÝBVXZUVË@AX` ?; ? COXZB N# BE,; \3;q@ N P`; C O¦hÜ LO N# 2 B9P`;UYË3:,COBDÌ,@A;àE[;¤ù ZB UVË:,C[COBÔ . 2. n. n. 2. n. n. n. . R. _ , 4 R , . a yt9u/e3u~b)t^x c I COXZ£,X6[3;,Ê[:8XZUVX`CO8&UVBVÌq£qP DVXZ?3;,Ê[;¢P ôO:qCO@:õUYBDX`;[XZV£qBVÌqUV;ª<
(495) =Ú"s!f " . PR&
(496) %7NJ/7( A,B ∈ Z (@NJ/710("!76 h1$#MFG$&
(497) /7(*(
(498) . ' &I=0&$G
(499) O&
(500) F+ −AB→ &$(
(501) 1h0/#( V = {(±1,0),(0, ± 1)} P^&
(502) J 7NJ/7( A,B ∈ Z (@NJ/7 10 (&%J6 h1$#MFG$&
(503) /7(n(
(504) v'&Q=0&$G
(505) &
(506) F+ −AB→ &$(
(507) f1h0/#( V = {(±1,0), (0, ± 1),(1,1),(−1, − 1)} P^&
(508) J 7NJ/7( A,B ∈ Z (@NJ/7 10 ((' 6 h1$#MFG$&
(509) /7(n(
(510) v'&Q=0&$G
(511) &
(512) F+ −AB→ &$(
(513) f1h0/#( V = {(±1,0), (0, ± 1),(±1, ± 1)} /Y& #0+,9& E 1F& Z &$(
(514) 10 & k 65G$NJ/7/Y&X&7NJF+ k = 4,6,8 (
(515) 7NJX+ONJT( 7NJ/7( A &
(516) B 1F& E & '(
(517) O& X/ k 65G$`7&
(518) wx/ +:&
(519) j0/7 A &
(520) B ) G &$(
(521) + *E10+:&F/Y&x(
(522) F & 1F& 7NJ/7( 1F& E , (@NJ/7 6 h$1 #MFG$&
(523) /7( A = M ,M , . . . ,M = B O&
(524) (Rtu7& M &
(525) M ìw£,CÝ8 ? ;
(526) \h; &$(
(527) IF/Y& #0+,9& 1F& Z 1FNJ/7Ik /7O&
(528) +:(@&$G
(529) -9 NJ/ 0=0&$G_ONJXO& 10+:NJO& /Y& #0+
(530) 9& `7NJ+,JNJ/7 0'&^NJs=0&
(531) +,-9G50'&R&$(
(532) G$NJ/#(
(533) - c &R1F& 7NJ/7(xG$NJ/#( c G
(534) X :( /Y& #0+,-9&"!765G$NJ/7/Y&X&- /79&R1F& Z &$(
(535) 0h($(
(536) 7&
(537) c & N COPZÔÝCO@XZ8C 2. 4. 2. 6. 2. 8. 2. 0. 1. HJI. n. i. i+1 2. ,. 2. A. 4. ). .0/21234/257678$576. Ïä+*j L ¥. A. .9/1:3;/257678$576. A. .9/1:3;/257678$576. sS &$(< p 7&$(R1F&xG$NJ/7/Y& c 6. 8.
(538) . ;
(539) $
(540) (
(541). . .
(542) $
(543) . . ¤u~gjt^xÃdbt9t^gÊu/ehy "db)tv)gÊu/ehy db)tv)gÊu/ehy e3bªeh|)n~g l^|)t^x Z2 ] 83 R P`;q¢;,8D;,@¤TPZ;,¢£,CÝ8 ? ;
(544) \h;,àVCO8WU¢UYCO? dDCO?BV¢£qCO88;,\3;q,Ü.£ BPZ;¡V;qôO@;q8WU [AB] ;,Uà?8 £ Ë;q@XZ8¡BV;qP`X 8 U A ;qU B LWí B}£,CO8WUYBD;E[ 8 Z P £qCO8 ? ;
(545) \hX©UY:Ê86QS;,8WUYB CB 8; ?£,?83;E[;, kìw£,CÝ88;
(546) \hX©UY:, s£,CO8DX`[:qBV;qB ] 83£q;qUDEUY;=V;q{(0,0),(2,1)} £qUYXZCO8 CO8 ?§ @CO8WUVBV;,B ?;¡DX}GC F CO8 V? N3N COD;P
(547) ì/£,CO883;
(548) \hXZUV: P`CÝBVPZ;,[;q?X\ £qCO8 ?
(549) ; \hX©UY:,[: #8X`;q N BV:q£,:,[3;,@A@;q8 U9VCÝ8 U:q?3X ?§ P`;q8WUY;,M L !0 h1é " " 1 W NJ E F/Y& #0+,9& 1F& Z 0'NJ+:('&$(1F&
(550) E +,w 09NJ/#(R(
(551) F=J0/7 &$( . . . . . n. 2. (@NJ/7 c tuF=J0'&
(552) /7 &$( , S &
(553) /#(@&
(554) w "u'& E &$(
(555) NJ90'&
(556) w &
(557) /7G$NJ/7=0&X&^&
(558) ' 65G$NJ/7/Y&X& S &
(559) /#(@&
(560) w "u'& &$(
(561) 65G$NJ/7=0&X&R&
(562) ' 65G$NJ/7/Y&X& ù;}UYË3:,COBDÌ,@A; :õUY:)E[:q@CO8W2UVBV: N CO?BjP &N BV;,@AX`ÌqBV;ªøsCOXZ 6[ 83 X`@ 3 <9Lÿ8;)[:,@ACO8UYB UYXZCO8[X`BD;,£õUY; ;qDUÊø XZUV;[ 83 CO8 -L 8 ?Ý ;q8[CÝ88;,BÊ?83;8CÝ? ? ;qP`P`; N BV;,? ? ;0L \
(563) ª" !f " " W NJ9&
(564) /7 v &
(565) v 1F&
(566) =0&$G
(567) O&
(568) X+:( 1F& Z /Y&x(
(569) XO& 1F& 7NJ/7( (M ) &$(
(570) <X/ G$`7&
(571) wx/ 10+,V c #0+ (v ,v ) (
(572) 7NJF+RONJF i '&R=0&$G
(573) &
(574) F+ M−−−M−−→ &$(
(575) n X/r1F&$(^+:NJ'(x=0&$G
(576) O&
(577) X+:( v ) v NJ v + v Z X v ∈ {(1,0),(−1,0)} ;õU v ∈ {(0,1),(0, − 1)} CÝ8 N BVPZ;[;
(578) ìw£ Ë;q@XZ8 ì/@CÝ8CÝUYCÝ8;0L 8 §? [6Q TfCOBV[@ACO8WUYBV;qBE ?;,PZ?3;,P`;,@A@A;,6< 1³12 " W &$(
(579) F/Y& #0+
(580) 9& qv65G$NJ/7=0& X & ' 65G$NJ/7/Y& X&R1F& Z 0'NJ+$( 7NJF+4ONJh( A,B 1F& E E U& '(
(581) &!F/ ' 65G$`7&
(582) wx/ _qv6Ow NJ/YNJ NJ/Y&I+:&
(583) j0/7 A * B c ùCO8DX`[:qBVCO83?8
(584) ìw£ Ë;q@XZ8BD;,PZX 8WU £qCO@ N COBU 8WU?8ß83CO@¤TBD;9@AX`8XZ@¤?@ [P ; wN CÝNJX`/#8W(
(585) UYM+.L 0Z -9QSX`NJP0/ 86, QS;,U N# ìw@ACO8CÝUVCO8;X`P#;
(586) \hXZDUY;9UYBVCÝAX` N BCOXZ8WUY[X`UYX`83£qUY [3;^£,;9£ Ë;,@AX`8 .
Documents relatifs
To test whether the vesicular pool of Atat1 promotes the acetyl- ation of -tubulin in MTs, we isolated subcellular fractions from newborn mouse cortices and then assessed
Néanmoins, la dualité des acides (Lewis et Bronsted) est un système dispendieux, dont le recyclage est une opération complexe et par conséquent difficilement applicable à
Cette mutation familiale du gène MME est une substitution d’une base guanine par une base adenine sur le chromosome 3q25.2, ce qui induit un remplacement d’un acide aminé cystéine
En ouvrant cette page avec Netscape composer, vous verrez que le cadre prévu pour accueillir le panoramique a une taille déterminée, choisie par les concepteurs des hyperpaysages
Chaque séance durera deux heures, mais dans la seconde, seule la première heure sera consacrée à l'expérimentation décrite ici ; durant la seconde, les élèves travailleront sur
A time-varying respiratory elastance model is developed with a negative elastic component (E demand ), to describe the driving pressure generated during a patient initiated
The aim of this study was to assess, in three experimental fields representative of the various topoclimatological zones of Luxembourg, the impact of timing of fungicide
Attention to a relation ontology [...] refocuses security discourses to better reflect and appreciate three forms of interconnection that are not sufficiently attended to