1230274

Convexité dans le plan discret. Application à la
tomographie
Alain Daurat
To cite this version:
Alain Daurat. Convexité dans le plan discret. Application à la tomographie. Autre [cs.OH]. Université
Paris-Diderot - Paris VII, 2000. Français. �tel-00012136�
HAL Id: tel-00012136
https://tel.archives-ouvertes.fr/tel-00012136
Submitted on 13 Apr 2006
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
"!#
$&%')(+*,.-/!0 123" 465.
7988:,;=<
>[email protected]:,BDCE<
F GJILK&M
H
N CO?BEPRQSCOT3UV;,8WUYXZCO8\[?]^X N P`[email protected];[;
acbEdfehgji9kml^gonqpsrct^uwv)g6kxu/ehy{z|}ku~x€
K& yjd‚u~|)n~u/e3y„ƒ†€n~‡ˆb}k3uweh‰^Š‹uŒŽi^g
N BV:qV;,8WUV:,;;qU‘VCO?hUY;,8 ?; N ?T3P`X`’ ?;[email protected]“;q8WU
N” B
•–—f˜ ™›š\œžEœ=Ÿ
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£,;¤ÎÏ`ÐÑÒ
Ó ?BÔ^<
Õ XZP`P`;q9Ö9×0ØhÒ.ØjÑÚÙÚÛ)ÜhB ”ÝNN COBDUV;,?B
7^P`Tf;,BUYCßÞà×0áHâªãÙÚä}å
7&UVUYXZP ”“æ ãçÑ)Ü#B ”ÝNN COBUY;,?3B
Ó ” £,’ ?;qèéÑÚêå‚ëª×0Ø
Í ” ?BDX`£,;¤Î¢Ï`ÐhÑ3Ò
Ó ; ” 8hìwí}X`;,BDBV;¤î†×#Ð=×0Ïáháï0ðOÜ#B ”ÝNN COBUY;,?3B
]^;,8XZ¢î†Ïñjò.Ñ=Ø.Û
ó ” T COB ” UYCÝX`BV;9[; ó COôOX`’ ?;§Ü[6QS7^PZôOCOBVX©UYË@AX`’ ?;§Ü3;õUÊ[6Q÷öw8hø[email protected] ” UYXZ’ ?;†[;ùŽP`;[email protected]¢ú }û 7È¥h üOý
ó ” TfCOB ” UYCOXZBV;[6QSöw83ø[email protected] ” UYX`’ ?;†79P`ôOCÝBVXZUVË@AX`’ ?;Ý<þ#CO8[3;,@A;,8WUYÊ;qU‘7 NN P`XZ£ ” UVX`CO83ˆú ÿ Í ¦
¥
Ó ;†UYXZ;,8BV;[email protected]“;qBV£,XZ;,B6<
£q;qUDUY;†UYË!Ìq#"V$B; A Ü % '&$(!)*& +,-+.0/21 ;qU43 &50/7698 9& +,+:&!;<& =0& '>$( [6Q ”@? COX`B ” £,£q; N UV:¤[;¡B ”§NN CÝBDUY;qB
[email protected]” 0F+UVË9Ì,G$D&I;ÝHÜ N=JCO0D ;,BàN COPZ;,?3¤[email protected]” 83? ;,CO¤X`B‘’ ôO??;qXZD[UY: XZCON 8;q¤8’ [ ?” X [email protected]£qQSCO;,8WU ’ ? N ” ;,[email protected]“; XZ” 8[3M;ßLö[email protected]“P ;A” øD” [email protected]Üf[X`P`Ì,X ” BVPZXZ;V;q[3B :,”KT? ?3;,U^£\[PZ;;
[[email protected] ” X`8B; A
Z” "B ”& +,[6ONQQ K” PR? CO& XZTBESU£,F/BVF?VF[N Ìq;q9U PZW;0[+.:,T3)?3U^+,X/;qY8& P ” [ N COìw£,?3CÝB8 UV? CO; ?3\hUYXZ;qUV=0: A 8CݏÚøsBV?3£qUY?3;,?D;,}[XZV£,?3VVXZCO8qܧ;õUª@A;,BV£qX
;õ%QUc T3X`;,8m;õ5 U]BAUVCOUY?CO?3“£,;qPZX? ;,\é[email protected]“’?3;qe; @¤dqTQ ”BV;qX}£,_§[UY? CÔOó.:qó 79” ?éöù T?;qUBV; ”;,8?ÂNù ” &BDf<UVX`W£,?3P`X`;qB P^& /7'(_;Ü 9i G$`#0+.1 ;õbU a 0/
+.0+.1 Ü
& +-VF&^vÜ)3 F+:G$gh& jJ(lkmJ`hÜ`#;q0UV/70£ 6L
c
Ü
Y
o
W
c
Ü
c
)n-90+ ko`F9& +,+ prq70 & "$J( 9& /sP^&9V0Fp q +:NJ/7tu7&*Cl0#10F J( wx/Y&*)F+-V0m& ü
¬»Æ
Æ
¹È´6Á°
*q"!j5Ý"s!f
€yjb
Š‹y‚ek3ugol^gjxÃd‚b)tŽv)gEgjx
!f .!#s
]^: #8XZUVX`CO83< L!L!L4L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!Lº
L ¥ ùŽCO883; \hXZUV:;õUÊ£,CO8 ? ;,\hXZUY: L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!Lº ü
L X¥ L© ]^: #83XZUYXZCO8 L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!Lº ü
L X¥ L ¥ ó XZ;,8Ê£qCO88,; \3X©UY:§Üf¥"ìw£qCO8 ? ; \hX©UY:ÝÜ0£qCO8 ? ; \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 js
¥XL© ]^XZBV;,£õUYXZCO8E[ ” 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:qV; ” X? \P`XZ:,n ¡[;qX? \[X`BD;,£õUYX`CÝ8 LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¥h
¥XL ¥ ùŽCO8 ? ,; \hXZUY:[email protected] N P`; L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¥
¥XL [ ìw£qCO8 ? ; \hX©UY: L!L4L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¥
X¥ LSü íÚCÝX`8WUYÊ ” X`P`P ” 8WUY L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¥ X¥ LSü L© ]^: #83XZUYXZCO8Ê;qU N BV;,@AX`ÌqBV;q N BVC N BDX`:õUY:, LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¥ X¥ LSü L ¥ ùŽCO8V: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`83V:,£ ” TXZP`XZUV: L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¦
X¥ L XL ¥ ]^X`BD;,£qUVX`CO83‘Ë3COBV"X !,CO8WU ” P`;,Ž;qU ? ;qBDUYXZ£ ” PZ;, LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ¦
X¥ L#FL X¥ L© 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;,8DX`CO8 [XZBV;q£qUYXZCO8ECO? PZ?R L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L
X¥ L÷ý ó X`;q8 ”@? ;,£P ” £qCO8 ? ; \hX©UY:¢UYCÝU ” P`; L!L4L!N L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L X¥ L ó ;, [ ìw£,CÝ8 ? ; \h;,ÊøsCOBUY 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 ? ; \hXZUV: L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L X¥ L L X¥ L© 7 ? ;,£PRQSX`8D:,£ ” TX`PZXZUY: L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L üW¦
X¥ L L X¥ L ¥ 7 ? ;,£P`;†[:õUY;,[email protected]“XZ8 ” 8WU L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!L ü
X¥ L L íÚCOXZ8 UVÊ ” XZP`P ” 8 UV&[;q [ ì/£,CO8 ? ,; \h;,9øsCÝBDUY L!L4L!L!L4L!L4L!L!L4LIL!L4L!L ü ¥
L©
2
3
2
ý
$j!fÝ"s!fÚ O!ff jAÚ OW
L© ]^: #8X©UYX`CÝ86Ü N BD;,@AX`Ì,BD;, N BDC N BDX`:qUV:, L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L
L© L© ùŽCO8 ? ;,\3X©UY: L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L
L© L ¥ ùŽCO8 ;,\3X©UY:DUVBVX`£õUY; L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L
L ¥ ù ” Ê[;¢P ” [3?X`@A;,8DX`CO8H L!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L
L Í XZ8X`@ ” [6QS?8;¢øsCO8£õUYXZCO8\DUVBVX`£õUY;[email protected]“;q8 U‘£qCO8 ? ; \h!; L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L
h}!ff"12Ús#
ü L© ]^: #8X©UYX`CÝ8;õU N BV;[email protected]“XZÌ,BV;q N BVC N BVXZ:qUY:qÊ[;, N COX`8WUY[email protected]:,[3X ” 8 L!L!L!L!L!L4L!L4L!L
ü L© L© ]^: 8XZUVX`CO8 L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L
ü L© L ¥ í}BVJC dD;q£qUYXZCO8 L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L
ü L© L ù ” B ” £qUY:qBVXZ ” UVX`CO8\PZCh£ ” P`;[3;, N COXZ8WUY[email protected]:,[X ” 8 L4L!L4L!L!L!L!L!L4L!L4L!L
ü L© LSü ó ;[email protected]¤TBD;†[; N COXZ8WUYÊ@“:q[X ” 8 L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L
ü L ¥ 7 NN#” BUY;,8 ” 8£,;[;q N COXZ8 UV[email protected]“:q[X ” 8 L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L
ü L ó ;,Ê@A:,[X ” 8ED?BÊ£ Ë ” ’ ?;BV:,D; ” Y? L L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L
mgjd‚b)t^xWeki^d0ehub)t
6
ü
ü
ݦ
ݦ
h
Ýý
Ýý
ýO¦
Oý ¦
ý
Ž
*q"!j5 Ý" !f
¨ OC @ACOôOB ”§N ËX`;^£,CO8WUYXZ8 ?; G$NJw2F & +,J&51 ONJw N5V0+.7`Fp L!L4L!L!L!L!L!L4L!L4L!L ý
¥ ¨ OC @ACOôOB ”§N ËX`;^ôO:,[email protected]:qUVBVXZ’?3; L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L ý
¨ [email protected]ôOB ”§N ËX`;^[X`D£,BVÌõUY;_LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L ¦
%ß O!fÚõ 5 Ý"s!fm. à O!ff j
XL© N :,B ” UYX`CÝ8E[;¢£,[email protected] 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;qVDX`CO8[;¢PRQS; \hX`UY;,83£,;¢[6QS?8;VCOPZ?3UYXZCO8 N#” BE?83;à¥ì/ø[email protected]¤?P`; L!L!L4L!L4L!L L!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L O¦
XL L© þ#[email protected]¤?3P`;,E,; \ N [email protected] ” 8WU
XL L ¥ þ#[email protected]¤?3P`;,E,; \ N [email protected] ” 8WUÊX
P ” [ E(j)
ì/£,CO8 =? ; q\hXZUV
: L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L 3
XL L ùŽ[email protected] N P`,; \3X©UY:†[;¢PQ ” PZôOCOBDXZUYË[email protected]“; L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L 3
XLSü û \UY;q8VXZCO8 N PZ?E[;¢[;,F? \[X`BD;,£qUVX`CO83 L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L Ýü
XL ùŽCO838; \hXZUV: L4L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L XL÷ý
;,£qCO8DUVBV?£õUYXZCO8 K” ? ;,£P`C£ ” PZX` ” UYX`CÝ8 ”N BVXZCOBVX L!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L XL ö[email protected] N P`:[email protected]“;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!fq 5 Ý" !f
# #ý L© ó ;,% $' &ÊXZUV£ ËX`83ô§ìw£[email protected] N CO8;q8WUY) ( L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L #ý L© L© ]^: 8XZUVX`CO8E;õU N BV;[email protected]“XZÌ,BV; N BVC N BDX`:õUY:, L!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L #ý L© L ¥ *}[email protected]“:qB ” UVX`CO8[;¢' &ÊX©UY£ ËXZ8ô§ì/£,[email protected] N CÝ8;,8WUY L!L4L!L4L!L!L!L!L!L4L!L4L!L O¦
#ý L ¥ ó ;†UYË:qCOBVÌ[email protected]“;†[‚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 ? ; \h;, L!L4LIL!L4L!L!L4L!L4L!L!L!L!L!L4L!L4L!L 3
#ý L X¥ L ¥ û \hUV;,8DX`CO8 ” X? \ [ ì/£,CO8 ? ; \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/£,[email protected] 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!L4Lʦ
q
j
ý#L ¥XL ùŽCOBVCOPZP ” XZBV;qL4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊOO
ý#L ÿ983X`£,X©UY:†[6QS?8;¢DCOP`?3UVX`CO8\[3CO88:q; L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊOO
%ßO!fÚq §5 Ý"s!fm Ú"! L ¤ O!ff j-/!0"
L© 7 ? ;,£[;,?F\[X`BD;,£õUYX`CÝ8L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊO
L© L© ó ;q ?§” BDX ” T3P`;,ŽTCCOP`:q;,883;, L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊO ü
L© L ¥ û \ N BD;,DVX`CÝ8„[;¢P ” [ ìw£qCO8 ? ,; \3X©UY: L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊO
L© L ó ” TfCOBV8;†XZ83øs:,BDX`;q?BV; L!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊOý
L© LSü ó ” TfCOBV8;†D? N :,BVXZ;,?BD; L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊO L© L ó Q ” PZôOCOBVX©UYË@A;^[3;BD;,£qCO8DUVBV?£õUYXZCO8 L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊO L© L÷ý ;õUYCO?<B ¡P ” BD;,£qCO8DUVBV?£õUYXZCO8,; \ ” £qUV0; L L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ"¥Ý¦
L© L ùŽCO88,; \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£,CO8UYBD?£qUVX`CO8 ”ÝNN BDCh£ Ë3:,; L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ"¥h
L X¥ L ¥ ;q£,CO8UYBD?£qUVX`CO8; \ ” £õUY; L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ"¥
L ;,£,CÝ8DUVBV?£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 ö[email protected] N P`:[email protected]“;q8 U ” UYXZCOD
#
! 5Ú s!fmW Ú"!! 12¡!f5.f "
# L© ùŽCO8£qP`?DX`CO8 L!L4L!L4L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ"¥ # L© L© ùŽ[email protected] N P`; \hX©UY:†[;¤î†×fñjå=ÙÚðÒ.؂ãñ6Ò.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? ? ;,BUY L!L!L4LIL!L4L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ
# L X¥ L© ;q£,CO8UYBD?£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`;[email protected]“;q8WUÊ£,CO8 ,; \h;, L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ
# L X¥ L ¥ ;q£,CO8UYBD?£qUVX`CO” 8 ”ÝNN BDCh£ Ë3:,?; L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ
# L F¥ L X¥ L© ;q£,CO8UYBD?£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`DV;,@A;,8WUE[3;P ” £,CO8 ? ,; \hXZUY: L!L!L4L!L4L!L!L4LIL!L4L!LÊ ¥
# L F¥ L L ¥ ]^X`@A;,8DX`CO8 L!L4L!L!L4L!L4LIL!L4L!L!L4L!L4L!L!L4LIL!L4L!LÊ ¥
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 .s h } 12 sL.hH 0!0§ 12„. "O!fÚq §5 Ý"s!fÈ.
¤O!ff 6h
L© ;,£,CÝ8DUVBV?£õUYX`CÝ8[;q [ ì/£,CO8 ? ,; \h;, @” ? ;,£PRQ ” P`ôÝCOBVX©UYË@A;9[3?£ Ë ”§N XZUVBV; LIL!L4L!LÊ ü
L ¥ ;,£,CÝ8DUVBV?£õUYX`CÝ8 ”ÝNN BDCh£ Ë:q;H[;, [ ì/£,CO8 ? ,; \h;,ßøsCOBUY 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! 0" s
6
*,
¡"!s! f" 6
­±¶Á¬ ¸ºÀ鶴6¬»­
ó Q÷[email protected] ” ôO;,BDX`;¢8 [email protected]“:qBVXZ’?3;à;,U†?8H[CÝ@ ” XZ8; ” ?JdDCO?3BV[6QSË ?X=;q8 N P`;qX`8„;qVDCOBuL †8 N ;,?hU[XZDUYXZ8ôO?;qB
N BDX`8£qX N#” P`;,@A;,8WUŽ[;,?X\ ”§NN BVC£ Ë;q‘UVË:,COBDX`’ ?;q‘[3;£q;qUDUY;¢[X`D£,X N P`XZ8;ª<
P,; Q \h”§;,[email protected] N BVN CP`;†£ Ë?;A83£,;[email protected] UVX`” 8 ôO?;;ª;,<Ú8CO8³8X ? V?; ” N?N CÝ[V;¢;\ôO’ BD?X`E;ß;qPDQ÷[email protected] ? ”?ôÝ;¢; £,[email protected] @“? 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;à[email protected] ” ôÝ;;qDU†[CÝ8£ N PZ?3UY_ÝU9£,CÝ@“@A;?38;àøsCO8£õUYXZCO8„[; [ ” 8 ” 8EPRQSCOVBVXÚ[3COX`88 ” ” UV;,¥?§B‚ý <
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`XZV;,BE?38;ôÝB ” 8[3; N” BDU9[;,‘@ ” UYË:[email protected] ” ì
UVX`’ ?;,E[3: d5¡; \hX`U ” 8WUV;,u Lfÿ‘8,; \h;,@ N P`;†[;¢D?£,£qÌ,‘[;¢£,;qUDUY; ”ÝNN BVC£ Ë;†;,U9P ” £,[email protected] N BD;,VDX`CO8 Ó í)û Õ
[;qŽ[email protected] ” ôÝ;,}’ ?Xf;,DUŽT ” D:,;^;q8 N#” BUYXZ£,?PZX`;,B}D?BˆP ” UYB ” 8Dø[email protected]“:q;9[3;^þ#CO?BDX`;quB L ó ;^ôOBDCO N BVCOT3P`Ì,@A;Ê[;
£q;qUVUV; ”ÝNN BDCh£ Ë3;A;qDU’ ?;PRQSCO8H8;¡£qCO8WUYBV_ÝP`; N#” †PZ;,†;qBVBV;q?BV¢[?3;, ” X? \ N BVC£,:q[:,[;¡£ ” P`£,?39P L6í ” B
,; \3;[email protected] N P`;¢P`;q‘BD:,;qP`‘VCO8WU9VCO? ? ;,8WU ”§NN BVC£ Ë:q N#” B9[;q CÝUVU ” 8WUY‘[CO8WU‘PRQ ” BVX©UYË@A:qUVX`’ ?;;qDU9;,8£qCOBV;
@A:,£qCO88 ?0; L
ó ” V;q£,CO8[3; ”§NN BVC£ Ë;E[CÝ88;E[;q ” P`ôOCOBDXZUVË@“;qÚ’ ?X3VCO8WU)[XZBV;,£õUY;[email protected]“;q8 UªX`@ N P`:[email protected]“;q8 U ” TPZ;,=[ ” 8
P ”àN B ” UYX`’ ?B; L#í ” B&£,CO8WUVBV;¢P`;q&@ ” UYË:[email protected] ” UVX`’ ?;q&[?[X`D£,BD;qUECO8:õUY:†Tf; ” ?£qCO? N @ACOX`8 [: ? ;,PZC NN :,;,
£ ” B£,;qUDUY; N BVCÝTP`:[email protected] ” UVX`’ ?; ” @ACOX`8¡[‚Q÷?8 VXZÌ,£qP`0; L Í ” X`£,;q“[3;,BV83X`Ì,BD;, ” 88:q;,A[;[email protected]¤TBD;,?D;,
BD;,£ Ë;qBV£ Ë;qªCO8WU:qUY:)ø ” X©UY;,.[ ” 8.£,;ˆ8CO? ? ; ” ?¤[[email protected] ” X`8;ª’ ?;ˆPRQSCO8 ”ÝNN ;,PZP`;}@ ” X`8WUY;q8 ” 8WU $¤ôO:,CÝ@“:õUYBVXZ;
[XZV£qBVÌqUV; (YL
ó Q÷CÝFT dD;,£õUYXZøf[;ÊP ”^N BV;[email protected]“XZÌ,BD; N#” BDUVX`;&[3;E£,;qUDUY;&UVËÌ,D;9;qDU)PRQS:qUV?[;E[;q):,’ ?X ?§” PZ;,8WUY}[3X`V£qBV;õUY}[6QS?8;
8C§UYX`CÝ8é£qP ” DVX`’ ?;A[;AP ” ôO:,CÝ@“:õUYBVXZ;ª<jP ” G$NJ/7=0& c L.û)82; ;qU[ ” 8P`;£,CO8WUVX`8 ?2P ” £qCO8 ? ; \hX©UY:;qDU
£qCO88 ?;[; N ?3X`ÊPRQ ” 8WUYXZ’?3XZUY:^;qU ” [3;[email protected]¤TBD;,?D;, ”ÝNN PZX`£ ” UYX`CÝ8&;,8ôO:[email protected]“:õUYBDX`;ÝÜ3;,8 ” 8 ” P©ÔhD;¢;qU‘;,8
C N UYX`@AX` ” UYXZCO8oL
û}P`PZ; ”” ?3VVX):õUY:A?3UYXZP`X`D:,;àUVBVÌqUV_ÝU;,8 X`@ ” ôO;,BDX`;§Ü N#” B¢,; \3;[email protected] N P`;P ” øsCO8£õUYXZCO8 ’?3XˆBD;,@ N P`X©U†?8
CN OPZÔôOCÝ8;VCÝ? 9ì cXZ8[C &ʏ Ž[;,@ ” 8[;†DXjP`; N COP©ÔhôÝCO8I; ¡BV;[email protected] N PZX`B ;,DU‘£qCO8 ? ; \h; ? COX`B #ôÝ?BV;¡ L
ùŽX©UYCO8 ” ?DVXÚP`;£ ” PZ£,?P.[;PRQS;,8 ? ;qP`C NN ;à£qCO8 ? ,; \3;¡’ ?XÚ;,U?8 N BVCOTPZÌ,@A;¢£,P ” VVXZ’ ?;à[;ôO:,CÝ@“:õUYBVXZ;
” PZôOCOBDXZUYË[email protected]“XZ’?3; ” CO¦ L
XZ;,8 ’ ?;“£q;VCOX©U¢?8;8CÝUVX`CO8LUVBVÌ,¢£qP ” DVX`’ ?; ” [: #8X©UYX`CÝ8H[ ” 8¢P`;¡[3X`V£qBV;õU8‚Q÷;qDU N#” †UYBDÌ,
: ? XZ[;,8WUV;9£ ” B)X`PÔ ”¢N P`?DX`;q?BV}ø ” CO8ˆ[;ÊôO:,8:qB ” PZX`D;,B)P ” £,CÝ8 ? ; \hXZUV:^[?“£,CO8WUVX`8 o? L ] ” 83 PZ; N BV;,@AX`;qB
£ Ë ”§N XZUVBV;)CO8 ?ݔ‘N BV:qV;q8 UV;,BÚP ” £,CO8 ? ; \hXZUV:ŽP ”EN PZ? N BVC£ Ë;ˆ[;ˆP ” £,CO8 ? ; \hXZUV:Ž[?£,CO8WUYXZ8 ?†< P ” G$NJ/7=0& c
PO” NJ 0 '& L ] ” 8ŽPZ;Êc L‚£ ùŽË ”Ý;õNUVUVX©UY;¤BV;^8CÝ¥†UVX`COCO88LXZ8ô UV” BVBVC[[;[email protected] X©U}” ?P`ôO8BD;9:¢8UVCÝCOUY?3XZCOU†8 [3N;, P`?3N ˆBDø C ” N XZTBDX`P`:q;EUV:,;õUŽPZ[3X`:q;¤; P ”£,£,;qCOBD8 U ? ” ,; X`\h8XZ;qUY :¤[3UVX`CÝBVU;q”£qP`UV;ªX`CO<08CO‚8 <
65G$NJ/7=0& N ;,?3U ;,8 N#” BDUVX`£q?P`XZ;,B}[: #8X`Bˆ?8“:,’ ?X ?§” PZ;,8WU [;, [email protected]@“;õUYM L ó ;ÊUYBDCOX`DX`Ì,@A;Ê£ Ë ”ÝN X©UYBD;9;qDU&[3:,[XZ: ” F? \
øsCO83£qUYXZCO8}£qCO8 ? ; \h;q&[; Z [ ” 83 R L ó ” ?VDXfCÝ8A;qDU £,CO8høsBVCO8WUY<: ¢[;, N BDCOTP`Ì[email protected]“;q}[;‘[: #83XZUYXZCO8M L
ó ;†’ ? ” UVBVX`Ì[email protected]“;†£ Ë ”ÝN XZUVBV;¢:qUV?[XZ;†[;, N CÝX`8WUY N#” BDUYXZ£,?PZX`;qBV&[;qÊ£,CO8 ? ,; \h;,,ÜP`;q N COXZ8WUYÊ@“:q[X ” 8u L
2
2
2
"!$#%&#%'()+*,)%#.-/!0)1#%)+"2&3'-546#7 89:* ;8<(=>(4 8<35;5)[email protected]>A/$'=40A/)%B8<!C-40$C'-6CD/E!0FGH&A/>!EDI3JLKMB5)ON
¦
"# !
"#$&%(' S &$( - +:NJ'(lp 7&$(r1F&7NJjpKVFNJ/Y&$(ltum&E+$& w2j KNJ/YG -9NJ/
(@NJh( I6
/21FN (
] ” 8 P ” [3;,?X\hX`Ì[email protected]“; N#” BDUYXZ;ÊCO8 ”ÝN3N P`XZ’?3;‘£q;qUDUY;‘:qUY?3[;^[;q £qCO8 ? ;,\3;qP ” [email protected]ôOB ”ÝN ËXZ;0L ùŽ;õUVUY;
UY;,£ Ë38X`’ ?;£,CO83VX`UY;I¡ BV;q£,CO83DUYX©UY?;qBEP ” UYBV?3£qUY?3BV; ]›[6Q÷?38CÝTFDd ;qUQ N#” BUYX`B [;¢V;q‘[email protected] ” ôO;,&B ” [XZC§ì
P`COôOXZ’ ?;,9£,P ” VVXZ’ ?;, ”ÝNN ;,PZ:,;, N BVCJDd ;q£qUVX`CO8 uL ] ” 8PZ;à£,CÝ8 UVX`8 ?LDX=CO8H£qCO88 ” U^PZ;, N BVCJdD;q£qUYXZCO8
V;,PZCO8 UYCO?3UV;,¢P`;q†[X`BD;,£õUYX`CÝ8†[6QS?8 N P ” 86܂CO8 N ;q?3U ” P`COBDBV;q£,CO83DUYBD?X`BD;PRQSCOTFDd ;qU¢ôÝB Ý£q;A?38;¡UYB ” 8ì
ø[email protected] ” UYXZCO8;,\ N P`XZ£,XZUV; sP ” UVB ” 83Dø[email protected]“:q;[; ” [CO8 #L ù^QS;,UÊP ” UY;,£ Ë38X`’ ?;¢?3UVX`P`XZV:q;9[ ” 8&P`;^V£ ” 88;,B
@“:q[X`£ ” P9L
ùŽ; N BDCOTP`Ì[email protected]“; ”” ?DVX0?8A:,’ ?X Ý? ” P`;,8WUˆ[ ” 83ˆPZ;E[X`D£,BD;qUj< X`P3,Q ” ôOXZU}UVCO?3U)DX`@ N P`;[email protected]“;q8 U)[3;ÊUVBVCO? ? ;,B
?8HCOTFdD;õU[;¤PQ÷;q N#” £,;¤[XZV£qBV;qU* N#” BUYXZB9[?„[email protected]¤TBD;à[; N COXZ8WUY^[ ” 8^£ Ë ” ’ ?;¡[BVCÝXZUY; N#” B ” P`PZÌ,P`;I
£,;,BU ” XZ8;,E[3X`BV;q£qUVX`CO8ML ùŽCÝ8 UVB ” XZBV;[email protected]“;q8 U¤ £,;†’ ?XV; N#” DV; N CO?B P ” [email protected]“CÝôOB ”ÝN ËX`;‘£,CO8WUVX`8 ?;ÝÜhCO8\V;
P`X`@AXZUV;e ’?3;,P`’ ?;q^[XZBV;q£qUYXZCO8Ê[; N BVCJDd ;q£qUVX`CO8oL ó QS:qUV?[;à[3;£q; N BDCOTPZÌ,@A; ” :qUV:àUYX`@à?P`:q;†[ ” 89P`;
[:,T?hUà[;q ” 88:q;, O¦ N#” B†?8;¡UV;,£ Ë8XZ’ ?;“[3;[email protected]`£qBVCOD£,C N X`;¤:qP`;,£õUYBDCO8X`’ ?; [ ÿ‘7> ¨ ö ¨ û Í C
PRQSCO8£,[email protected] N UY;†[3X`BV;q£qUV;,@A;,8WU‘P`;†[email protected]àTBV;¢[6Q ” UYCÝ@“;qE[ ” 8Ê?8£,BDX`DU ” P9L
ó ;,9£ Ë ”ÝN X©UYBD; “;õUý“CO8WU N CO?B‘T?3U^[;à@ACO8WUYBD;,B‘’?3;àPRQSCO8 N ;q?3UBD;,£,CÝ8DUVBV?XZBV;àPZ;,‘;,8D;,@¤T3P`;,
£,CO8 ? ;,h\ ;,=[3; Z ”&N#” BUYXZB6[;}P`;q?BV N BVCJDd ;q£qUYXZCO8 ”K? ;q£)V;,?3P`;,@A;,8WU.üÊ[XZBV;,£õUYXZCO86[; N BVCJDd ;q£qUYXZCO8ML 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©Ô[email protected]`83CO
ì/£,CO8 ? ; h\ ;, ] ó >^í Oý ” ?X\ [ ì/£,CO8 ? ; h\ ;,ML ó ;Ž£ Ë ”ÝN X©UYBV;Žý^:qUV?[XZ;,B ” PQ÷?38X`£qXZUY:)[;ŽP ” DCOP`?hUYX`CÝ8oL
û)8 #86Üf[ ” 8ÊP`; £ Ë ”ÝN XZUYBD ;ÝÜ#CO8,QSX`8WUY:qBV;qVV;qB ”“” ?£ ” ‘C PZ;, N BDCJDd ;,£õUYX`CÝ89DCO8WU^;qBVBDCO8:,;q N#” B
; h\ ;,@ N P`;!¡ £ ” ?V;[;, ”ÝN3N#” BV;qX`PZE[;¢@“;qV?BD; L
)
?
Ïä+*
-,
/.1032&414658714 91:;7 <
>=
[email protected]
&A
CB
ED
GF
2
HJI
GF
CKELNM
C4 8/<4C B A/B5- 35"'#%[email protected])!%B=3!08=BIC&4 8<)%G )%# [email protected]! ' @A/)% [email protected]!6540A ' E-5 B5* &!0)135'=B#%-M86(=- 408"BBA/G3H8<8<4C!C!C;5)%&2G A/&8!C# )+ [email protected]' -5E,A5! - !C4CGH *8<4 -5B 3')%[email protected]! A/ # )%GH8(
O
QP
SR
VP
UT
#W R
Á°€Æ
·m¬‹Æ
´ Á° ¾¿¹{Á\¶´6°
·Ã¶Á„´6° ¸ °€¼ Àì»­±®±°2²µ°€¼
O
« ɺ¹{¾¿´f¶Á°
« ¬‹­±®¯°³² ´f¶· ¶¬ ¶¹È½6°
_ R ] ” 83 R P ” £,CÝ8 ? ; h\ XZUV:“;qDU†UVBVÌ,†DX`@ N P`;R\[:#83X`B6<?8 ;,8D;,@¤TPZ; F ⊂ R ;,U¢[XZU†£qCO8 ? ;,\3;
DXUVCO?3UD;,ô[email protected];,8WU dDCOX`ôO8 8WU9[3;,?X\ CÝX`8WUY9[; ;,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©UYXZCO8V?B Z <
\ ª" !f / 10tum& E &$( 2 65G$NJ/7=0&X&.G$NJ/7=0&X&#0+^(@&9V0w & /7( 7NJX+ONJX 7NJ/7
A,B 1F& E '&$( 7NJ/7(^& /79& +$(R10(@&9V0w & /7 [AB] (@NJ/7 1h0/#( E 8 N ;q?3U ” ?DVX.£,CO83VX`[3:,BV;qBÊP`;, N” BDUVX`;,E[3;
’ ?XjDCO8WU9UVB ” £q;,Ê[;¢£,CO8 ? ; \h;,9[; R <
Z
\ ª" !f " /Y& #0+,9& E 1F& Z &$( R10O& ONJ90'& w & /7G$NJ/7=0&X&( R& e(@& F'& w & /7I( n
n
n
n
& '( O&IF/rG$NJ/7=0&X& F 1F&
R
n
O& tum&
n
n
E = F ∩ Zn .
Í ” PZË;,?3BV;,?3V;,@A;,8WU^£,;qUDUY;8CÝUVX`CO8;,U^UYBDX`£qUV;,@A;,8WU N P`?3‘øsCOBUY;¢’ ?;P ” ¥"ìw£qCO8 ? ;,\3X©UY:£,[email protected]@A;PZ;
@ACO8WUYBD;†PRQS; \h;,@ N P`;†[3;P ” #ôO?BD; LZ0L
S
Ïä+*. L©
65G$NJ/7=0&X&*/YNJ/ NJ90'& w & /7nG$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 ? ; \h;à[3;
£,CÝ8 UV;,8 ” 8 U F L0ù^Q÷;qDU9PRQS;,8D;,@¤T3P`;†[;q9T ” BDÔ£,;q8 UVBV;q £,C;A£,XZ;,8WUY N COVX©UYX©øs&[; F L
] ” 83 P ” [:#83XZUYXZCO8 L© LZÊCO8 N ;,?3Uˆ£,CÝ8VXZ[:,BD;,B)[;,ˆ;,8 ? ;,P`C NN ;qŽ£qCO8 ? ; \h;qŽ[; k N CÝX`8WUY ” ?P`XZ;,?
BD;,8[3BV;¢[;,‘V;qô[email protected];,8WUY6<
)
Y/ & # 0+,-9&41F&
Z2
n
R
n
[; N
\ ª" !f "
/ 10etum& E &$( k 65G$NJ/7=0&X&( 7NJF+ ONJF 7 NJ/7 A ,A . . . A 1F& E '&$(
7NJ/7(^& /79& +$(!1F& conv(A , . . . A ) (@NJ/71h0/#( E 1
1
k
2
k
ü
( (
ÿ‘8; BDUYXZ; [; ;,DU“UYCÝU P`;[email protected]“;q8WU“£qCO8 ? ;,\3;„VX‘;qUߏD;,?PZ;,@A;,8WUߏDX E = conv(E) ∩ Z L
;,B O¦3ÜO L© L #L ý ” DV?BD;¤’ ?;UVCO?3U N COXZ8WU^[;PRQS;,8 ? ;qP`C N3N ;
£,CO8 ? ,; \h;[6QS?8; N#” BDUYXZ; F ⊂ R ;qDUA;,8³ø ” X©U¤?8ÃT ” BDÔ£,;q8 UVBV;sL£,C;A£qX`;,8WUV N CODXZUYX©øsà[; n + 1
N COXZ8WUYÊ[; F CO8;,8[:q[?XZU‘?83;:q’ ?X ?§” PZ;,8£q;;q8WUYBV;¢P ” kìw£qCO8 ? ; \hX©UY:¢;qU‘P ” £qCO8 ? ;,\3X©UY:¢UYCÝU ” P`;§<
"!}!f s !f / & /#(@& w "u'& E ⊂ Z &$( ONJ 0'& w & /7G$NJ/7=0&X&<( & (@& X'& w & /7 ( UU&$( n + 1 65G$NJ/7=0&X& û)8 N” BDUVX`£,?3P`X`;qB.P`;,UYCÝU ” P`;[email protected]“;q8WU=£qCO8 ? ; \h;qˆ[3; Z VCO8WU}PZ;, ìw£qCO8 ? ; \h;q P`;q=£qCO8 ? ; \h;q N” BÚUVBVX ” 8ôOP`; ó ;, N CÝX`8WUY9; \UYBD:,@ ” X? \[6QS?8H£,CO8 ? ; \h; [; DCO8WUP`;q N COX`8WUV UV;,P`‘’ ?; F \ {M } ;,U
;,8£qCOBV;\£qCO8 ? ; \hB; L Z X E ;qDU?8; N” BDUVX`;[3; FZ ” RP`CÝBVàPZ;, N COXZ8 UV¤; \UVBVM:,@ ” X? \2[; conv(E)
DCO8WU
”ÝNN ;,P`:qÊ[email protected]@“;õUYE[; E L
ó ;†UYË:qCOBVÌ[email protected]“;¢[; †BV;qX`8hì Í X`PZP`@ ” 8 ;,B ݦ3ÜjO L÷#ý L J ” “[email protected]“;¢’ ?;¢UYCO?hU9£qCO8 ? ,; \3;£,[email protected] N#” £õU K
;,DU9PRQS;,8 ? ;,P`C NN ;¢£qCO8 ? ; \h;[;¢V;, N COX`8WUV‘,; \UYBV:[email protected] ” X? \ L † 8;q8[:q[?X©U9PZ;ø ” XZU&V?X ?ݔ 8 U
"!}!f s !f k NJF & #0+,9&ONJ 0'& w & /7*G$NJ/7=0&X& 1F& Z #8XZ; &$( < & /7=0& 'N 7& G$NJ/7=0&X&
1F&e(@&$(*(@NJwxw & (
ó ;Ž£ ” PZ£,?Ph[;,}[email protected]“ @A;qUV=[6QS?8¡;q8V;[email protected]¤TP`; 8X3[; N COXZ8WUYª;qDU}?8 N BVCOTPZÌ,@A;Ž£qP ” DVX`’ ?;Ž[;&ôO:[email protected]“:õUYBDX`;
” P`ôOCÝBVXZUVË@AX`’ ?; ? COXZB N#” BE,; \3;[email protected] N P`; ” C O¦hÜ LO N#”
”
2
†B9P`;†UYË3:,COBDÌ,@A;àE[;¤ù ” ZB ” UVË:,C[COBÔ 2
n
n
2
n
n
n
R _, 4R, a yt9u/e3u~b)t^x
c
I COXZ£,X6[3;,Ê[:8XZUVX`CO8&UVBVÌq‘£qP ” DVXZ’?3;,Ê[;¢P ” ôO:[email protected]“:õUYBDX`;[XZV£qBVÌqUV;ª<
=Ú"s!f " PR& %7NJ/7( A,B ∈ Z (@NJ/710("!76 h1$#MFG$& /7(*( ' &I=0&$G O& F+ −AB→ &$( 1h0/#( V = {(±1,0),(0, ± 1)} P^& J 7NJ/7( A,B ∈ Z (@NJ/7 10 (&%J6 h1$#MFG$& /7(n( v'&Q=0&$G & F+ −AB→ &$( f1h0/#( V = {(±1,0),
(0, ± 1),(1,1),(−1, − 1)} P^& J 7NJ/7( A,B ∈ Z (@NJ/7 10 ((' 6 h1$#MFG$& /7(n( v'&Q=0&$G & F+ −AB→ &$( f1h0/#( V = {(±1,0),
(0, ± 1),(±1, ± 1)} /Y& #0+,9& E 1F& Z &$( 10 & k 65G$NJ/7/Y&X&7NJF+ k = 4,6,8 ( 7NJX+ONJT( 7NJ/7( A & B
1F& E & '( O& X/ k 65G$`7& wx/ +:& j0/7 A & B ) G &$( +*E10+:&F/Y&x( F & 1F& 7NJ/7( 1F& E ,
(@NJ/7 6 h$1 #MFG$& /7(
A = M ,M , . . . ,M = B O& (Rtu7& M & M
ìw£,CÝ8 ? ; \h; &$( IF/Y& #0+,9& 1F& Z 1FNJ/7Ik /7O& +:(@&$G -9 NJ/ 0=0&$G_ONJXO& 10+:NJO&
/Y& #0+ 9&
`7NJ+,JNJ/7 0'&^NJs=0& +,-9G50'&R&$( G$NJ/#( - c &R1F& 7NJ/7(xG$NJ/#( c G X :(
/Y& #0+,-9&"!765G$NJ/7/Y&X&- /79&R1F& Z &$( 0h($( 7& c & N COPZÔÝ[email protected]“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
$ ( ;
$ ¤u~gjt^xÃdbˆt9t^gÊu/ehy "d‚b)tŽv)gÊu/ehy †d‚b)tŽv)gÊu/ehy e3bªeh|)n~g l^|)t^x Z2
] ” 83 R P`;q¢;,8D;,@¤TPZ;,¢£,CÝ8 ? ; \h;,àVCO8WU¢UYCO? dDCO?BV¢£qCO88;,\3;q,Ü.£ ” BPZ;¡V;qô[email protected]“;q8WU [AB] ;,Uà?8
£ Ë;[email protected]“XZ8¡BV;qP`X ” 8 U A ;qU B LWí ” B}£,CO8WUYBD;E[ ” 8 Z P ” £qCO8 ? ; \hX©UY:Ê86QS;,8WUYB ”CB 8; ” ?£,?83;E[;, kìw£,CÝ88; \hX©UY:,
s£,CO8DX`[:qBV;qB
] ” 83£q;qUDEUY;=V;q{(0,0),(2,1)}
£qUYXZCO8 CO8 ?§” @“CO8WUVBV;,B†’ ?;¡DX}GC F CO8 V? N3N COD;P ” ì/£,CO883; \hXZUV: ” P`CÝBVPZ;,†[;q?X\
£qCO8 ? ; \hX©UY:,‘[: #8X`;q N BV:q£,:,[3;,@[email protected]“;q8 U9VCÝ8 U‘:q’?3X ?§” P`;q8WUY;,M L
!0h1é " "
1 W NJ E F/Y& #0+,9& 1F& Z 0'NJ+:('&$(1F& E +,w 09NJ/#(R( F=J0/7 &$(
n
2
(@NJ/7 c tuF=J0'& /7 &$( ,
S & /#(@& w "u'& E &$( NJ90'& w & /7G$NJ/7=0&X&^& ' 65G$NJ/7/Y&X& S & /#(@& w "u'& &$( 65G$NJ/7=0&X&R& ' 65G$NJ/7/Y&X&
ùŽ;}UYË3:,COBDÌ,@A; ” :õUY:)E[:[email protected]“CO8W2UVBV: N CO?BjP ”&N BV;,@AX`ÌqBV;ªøsCOXZ 6[ ” 83 †X`@ 3<9Lÿ‘8;)[:,@ACO8UYB ” UYXZCO8†[X`BD;,£õUY;
;qDUÊø ” XZUV;[ ” 83 CO8 -L 8 ?ݔ ;q8[CÝ88;,BÊ?83;8CÝ? ? ;qP`P`; N BV;,? ? ;0L
\ ª" !f " " W NJ9& /7 v & v 1F& =0&$G O& X+:( 1F& Z /Y&x( XO& 1F& 7NJ/7( (M ) &$( <X/
G$`7& wx/ 10+,V c #0+ (v ,v ) ( 7NJF+RONJF i '&R=0&$G & F+ M−−−M−−→ &$( n X/r1F&$(^+:NJ'(x=0&$G O& X+:( v )
v NJ v + v Z X v ∈ {(1,0),(−1,0)} ;õU v ∈ {(0,1),(0, − 1)} CÝ8 N” BVPZ;†[; ìw£ Ë;[email protected]“XZ8 ì/@“CÝ8CÝUYCÝ8;0L
8 §? ” [6Q ” TfCOBV[@ACO8WUYBV;qBE’ ?;,PZ’?3;,‘P`;,@[email protected];,6<
1³12 " W &$( F/Y& #0+ 9& qv65G$NJ/7=0& X
& ' 65G$NJ/7/Y& X&R1F& Z 0'NJ+$( 7NJF+4ONJh( A,B 1F&
E
E U& '( &!F/ ' 65G$`7& wx/ _qv6Ow NJ/YNJ NJ/Y&I+:& j0/7 A * B c
ùŽCO8DX`[:qBVCO83Ž?8 ìw£ Ë;[email protected]“XZ8“BD;,PZX ” 8WU £[email protected] N COBU ” 8WUˆ?8ß[email protected]¤TBD;[email protected]`[email protected]¤[email protected]
[P ; wN CÝNJX`/#8W(UYM+.L 0Z -9QSX`NJP0/ 86, QS;,U N#”  ì[email protected]ÝUVCO8;‘X`P#; \hXZDUY;9UYBVCÝAX` N BCOXZ8WUYˆ[X`UYX`83£qUY
[3;^£,;9£ Ë;,@AX`8
2
1
1
2
1
2
2
2
i
i
1
i+1
2
HJI
1
2
2
s;qUÊ[
HJI
” 83‘£q;qU9COBV[3BV; UV;,PZP`;,E’ ?;¢PQ÷CO8 ” X©U
M,N,P
yM = yP 6= yN
8
CO?
xM = xP 6= xN
N ;,?3U ” PZCOBV=£,CO8DX`[:qBV;qB}P`;&£ Ë;[email protected]“XZ8¡CGFCO8¡BV;[email protected] N P ” £q;&P ”^N COBUYX`CÝ8¤;,8WUYBD; M ;õU P N” B}P`;&D;,ô[email protected];,8WU
’?3X;qDU N PZ?E£,CO?3BDU‘’ ?;¢P`;†£ Ë;[email protected]“XZ8XZ8XZUVX ” PO< £qCO8WUYB ” [X`£õUYX`CÝ8oL
[M,P ]
2
c
c
1³12 " W NJ
A(0,0) & B(a,b) 1F& 7NJ/7 (1F& Z = + - 0/7 0 < b < a & n+$& j ( #0+
F/ ' 65G$`7& wx/ _cq 6Ow NJ/YNJONJ/Y& C 0l1F&$($( h(1F&4s10+$NJO& (AB) & q = b c 'NJ+:(4f& '( &RX/
G$`7& wx/ 10+,V #0+ t R+$& j0/7 & /YG jh(!1h0/#(* & /7=0& 'N 7& 65G$NJ/7=0&X&e1F& C & ( c & /7C+:&I_10+:NJ & (AB)) ) & ) '&^G$`7& wx/ AC B )
COXZB #ôO?3BV; L L
P c w NJ/#( +.0-9NJ/ , ó ; ì/£ Ë;,@AX`8 C ;,U ì/@“CÝ8CÝUYCÝ8;ÝÜhX`P6;qDU‘[CO8£¢[;¢P ” øsCÝ[email protected];
2
a
b
I
0
HJI
”K? ;q£
C=
ri ≤ s i
;õU
ri+1 ∈ {si ,si + 1}
C =
8[:8XZU‘P`;qEV?X©UY; 0
(ri )
8 N BD;,8[ 0
;qU
{(x,i) : ri ≤ x ≤ si }
L 8 ?§”
i=0
0
b
[
b
[
£qCO8DUVBV?XZBV;†PZ;†£ Ë;,@AX`8
{(x,i) : ri0 ≤ x ≤ s0i }.
BÊBV:q£,?BDBV;q8£,;ª<
(s ) N#”
0
i
r0 = r0 ,s00 = s0
L
i=0
C0
VCO?39P ” ø[email protected];
ý
( Z ? NN CODCO8 @” ? COX`B£qCO8DUVBV?X©U
L Z CÝXZU
K” ? ;,£
CO8 N VB ;q8[
;qU N CO?B¡UYCO?hU
Ü N OC ?3B‘UVCO?3U
(
” PZCOBV
rj0
s0j
j ≤ i
s0i = si
i0 =
0
0
0
min{j : q(j − i) + si ≤ sj }
i<j<i
rj = sj = q(j − i) + si
0
0
0
ri0 = q(i − i) + si ,si0 = si0
i0 > i + 1
ri00 = q(i0 − 1 − i) + si > si0 −1 ≥ ri0
0
i0 = i + 1
ri+1
= q + si ≥ si + 1 ≥ ri+1
ri0 ≤ ri00 ≤ s0i0 = si0
C0
C
(1,0),(q,1)
2
;qU
L
CO8 ”
L
OC 8 ”
L
] ” 8ÊUYCÝ?ÊP`;,E£ ” ÊCO8 ”
L
ùŽ; N BVC£,;qVD?¤[: #8XZU[CO83£P`;£ Ë;[email protected]“XZ8 ’ ?X N#” àB £qCO8UYBV?3£qUYXZCO82;qDUàX`8£qP`?[ ” 8PRQS;,8 ? ;qP`C N3N ;
ìw£qCO8 ? ;,\3;[; ;qUÊ;,U
ìw£,CÝ8 ? ; \h;0L
2
ZZ XX
B
C
y=i
ri
si
C0
A
)
8;,8[:,[3?XZU‘P`;ø ”
1³12 Ïä+* L XZUED?X ?§” 8WUj<
NJ/#( +,mG 9NJ/l10EG$`7& wx/
C0
W NJ A(0,0) & B(a,b) 1F& 7NJ/7 (I1F& Z +:& j c ( #0+*X/ ' 65G$`7& wx/ ' 65G$NJ/7/Y&X&
qv6Ow NJ/YNJ NJ/Y& C 0l1F&$($( h(1Fc &^ 10+:NJO& (AB) 'NJ+:(R & /#(@& w "u'&x1F&$( 7NJ/7( 1F&R & /7=0& 'N 7&
NJ90'& w & /7G$NJ/7=0&X&!1F& C ( (x& /7-+:& (AB) & C &$( 1h0/#(e & /7=0& 'N 7& 2 − convexe 1F& C †8 N ;,?hUʏV? NN COD;,B
P c w NJ/#( -+.L 0Z9CÝNJXZ/ U , ” †PZCO8 BV §? ” @ACO8WUYBV;q;qB&U‘P`£,;^CÝBV8:,VDXZ?[PZ:,U BD” COU 8N#Ê” P`BŽ;†BV£ :,Ë£q;[email protected]“BVBDXZ;,8 8£q;[D?CO8B 8max(a,b)
: N#” BEP`;†P`;[email protected]“@A; N BD:,£q:,[;q8 UuL
0<b<a
ùŽCO83VX`[3:,BVCÝ8ÊP ” UYB ” q8=ø[email protected] c” UYXZCO8 ” A8;¢V?3X §? ” 8WUY;ª< C
2
a
b
0
f : (x,y) 7→ (x − qy,y)
;õU
VCO8WU)BV;qP`XZ:, N#” B=P`;E£ Ë;[email protected]“XZ8
L ùŽ;E£ Ë;,@AX`8
ó ;q N COXZ8WUY
;,DU H I ì/£,CO8 f? (A)
;,\h;9ü§=ì/£,(0,0)
CO838; \h;ÝÜÝf[6(B)
Q ”§N BV=Ìq}(aP`;, N mod
BVC N BVb,b)
X`:õUY:q}[; C CÝ8 N ;q?3U)[CO8£ ”ÝNN PZX`f’ (C?;,B=) PRQSË Ô N CÝUYË3Ì,V;
[;BV:,£q?BVBD;,8£q;\£,;£ Ë;[email protected]“XZ8oL.]^CO83£APZ;, N COX`8WUV†;,8WUYBD; C ;õU (AB) VCO8WUTXZ;,8 [ ” 8¢PRQS;,8 ? ;,P`C NN ;
¥"ìw£qCO8 ? ; \h;A[; C L ó ;, N CÝX`8WUY;q8WUYBV; C ;õU C VCO8WU ” ?3VVX}[ ” 8†PRQS;,8 ? ;,P`C NN ;A¥ìw£,CÝ8 ? ; \h;A[; C N#” B
; \3X©UYB: L ? COXZB #ôO?BD; L÷ü H ìw£qCO8 ? ,
2
û)8€ø ” XZU“CO8€; ;,£õUY?:HPQ ” P`ôOCOBDXZUVË@“;\[‚Q û}?£qP`X`[3;„;,8WUVBV;LPZ;,[3;,?X\€£,CCOBV[CÝ88:,;qß[? ? ;,£qUV;,?B −AB
→L
] ” 8†P ” #ôO?BD; 1.4 VCÝ8 U¢[3CO88:q;, N#” B9; \h;,@ N P`;¤UVCO?3UV;,PZ;,^UVB ” 8ø[email protected] ” UVX`CO8‘8:q£,;,D ” XZBV;q N CO?B
P`; ? ;q£qUV;,?B (55,34) L
0
0
0
0
$ ( ;
$ H0
H
f (H 0 )
B
f
A
H0
H1
H1
H3
H7
H5
H2
H2
H4
H6
H2
H3
H3
H4
H4
H5
H5
)
H6
Ïä+* LSü
H7
H6 H7
s8<NJF+.tumNJf'&$( 7NJ/7 (R0 1F&$($( T(!1F& A & B (@NJ/76O (R1h0/#( E
( (
ùŽCO83VX`[3:,BVCÝ8@ ” XZ8 UV;,8 ” 8 U¢?38; N#” BDUVX`; ¥"ì/£,CO8 ? ;,\h;ÝÜ ìw£qCO88;,\3;
[; L Z COX©U PQ÷;q8hì
V;,@àTP`;“[;,[email protected]@“;õUY[;PQ÷;q8 ? ;,PZC NN ;“£qCO8 E? ;,\3;\[3; E L 8 N ;,?3U N#” B- U ” /7ôO9;q& B S Z;q8 S ;qU SS ;,8hì
V;,@àTP`;q†[;,[email protected]@“;õUY^’ ?X=VCO8WU†BD;, N ;,£õUYX ? ;,@A;,8WU¢D?BP`;TCÝBV[Hô ” ?3£ Ë;¤;qUV?BPZ;¤TfCOBV[L[3BVCOX©U^[;
Ü BD:,£,XZV:[email protected]“;q8 U6<
convE N
2
g
Z COXZU ) sBV;, N
£,BVCÝX`V(S
 ” 8WUY;BL
1³12 i
d
Sd = {(x,y) ∈ S : (t,y) ∈ convE ⇒ t ≤ x}
Sg = {(x,y) ∈ S : (t,y) ∈ convE ⇒ t ≥ x}
(Ti ) P ” D?XZUV;Ê[;, N OC X`8WUY}[3;
Sd
BD;,
N
Sg COBV[3CO88:q; N#” B}[;,?X\hXZÌ,@A;E£,CCOBV[CÝ88:,;
W NJ E F/Y& #0+,-9& - /79& NJ90'& w & /7sG$NJ/7=0&X& ' 65G$NJ/7/Y&X& 1F& Z & (@NJ9& /7
c
c
(S ),(T ) 1 - /79&$(*G$NJwxw & 2jh(n`#0X NJ/#( 1 +:NJ/#(Q1F&$( ' 65G$`7& wx/#( _q 6Ow NJ/YNJONJ/Y&$( C & /7+$& S
& S & C & /7+:& T & T W NJ M X/ 7NJ/7Q1F& & /7=0& 'N 7&R NJ90'& w & /7eG$NJ/7=0&X& 1F& E
0'NJ+$(R0sw NJ/#(I F/Y&41F&R1F&$(I+:NJ'(R0j & +,/20=0&$(^&$( [email protected]+.09&
S & 7NJ/7 M &$( f( cc & /7+:&!X/rG$`7& wx/ C & _10+:NJ & (S S )
S & 7NJ/7 &$( f( & /7+:&!X/rG$`7& wx/ & _10+:NJ &
& '( O& iM& j & 1F& J 7NJ/7 ( U ∈ C & C V ∈ C `7NJ+, J(TNJ/7 T0'& w )& /7 0jV0/ c (!O& (tu7& M
&$( ( F+I'&e(@&9V0w & /7 [U,V ]
=0NJ+ -UV0F+:& i
2
i
0
i+1
i
i
i
i+1
i
0
i
i i+1
i i+1
0
j
i
Tj+1
Si+1
Ci
U
M
V
Si
Cj0
Tj
T2
T1
)
S1
Ïä+* L T0
S0
8NJF+.tumNJfONJT(!'&$( 7NJ/7 (R1F& conv{S ,T } (@NJ/76O-6O (!1h0/#(
i
i
E
i
P” PZCOc wBV NJ/#;õ( U +.0-UY9;qNJP`/ Ê, ’ Z ?? ;ªN< N CODCO8ß[CÝ8£„’ ?;LPZ;,“[;,?X\ N BD;,@AX`Ì,BD;, ” PZUV;,BV8 ” UYX ? ;,AVCÝ8 Uø ” ?VD;,ML Z COX©U
i j
;
$ ( $ Z CÝXZU U ?8 N COX`8WUA[; C [6QSCOBV[CÝ88:,; y ;qU V ?8 N COXZ8WU“[3; C [6QSCOBV[3CO88:q; y £q;, N COXZ8 UV
;,\3XZDUV;,8WU£ ” B¡PZ;,¤£ Ë3;,@AX`8¡DCO8WU ìw£qCO88;,\3;q LˆùŽ[email protected]“@A;PZ;,¡[;q?X\ N BV;,@AX`ÌqBV;q ” PZUV;,BV8 ” UYX ? ;,àVCO8WU
ySi ≤ yM ≤ ySi+1
ø ” ?3VV;q9CÝ8 ” sø COBD£,:[email protected]“;q8 U
i
yTj ≤ yM ≤ yTj+1
0
i
M
[6QSCGFP`;†BD:,D?PZU ” UKL
M
xU ≤ x M ≤ x V
2
P c w Z X NJ/#( ;q+.D0U -9#NJ8/ Xj10[6Q ”ÝN ` BDc Ì,NJÊ+:> P`;qw ‘& PZ;,@A @“ ;q, Z COL XZ¥XU L E;õU?38L ;q¥F8L ý¤V;[email protected]¤CÝ?3TU P`; N 2CÝìwX`£,8WCÝU 8 ? ; \h[;; ;qU ìw£qCO88;,\h;0L ;,DU9;,8ø ” X©U
M
convE ∩ Z
[ ” 8 EE ;qU‘[CO8£ E ;qDU‘TX`;q8\UYCÝU ” P`;[email protected]“;q8 UÊ£qCO8 ? ;,3\ ;BL
ùŽCO8DX`[:qBVCO83‘@ ” XZ8 UV;,8 ” 8 UÊPZ;†£ ” ÊC E ;,U / - /7 L
L ó ; N COX`8WU M ;,UÊ[CO8£†?38\T ” BDÔ£q;,8WUYBD;¤£,C;A£,XZ;,8WU N COVX©UYXZøs [6QS?8
Z COX©U
∩Z
;q8V;[email protected]¤TP`;M- /7∈ convE
[
;
O
C
`
X
W
8
V
U
}

[; E L ùŽCO83VX`[3:,BVCÝ8 ” P`COBDª?8ABV;q£qU ” 8ôOP`; R = [a,b] × [c,d] £,CO8WUY;q8 ” 8WU
N
F
L ó QS;,8D;,@¤TPZ; E = E ∩ R ;qDU?8Ã;q8V;[email protected]¤TP`; #8XRܪ¥"ìw£qCO8 ? ;,3\ ;£ ” B R ;,DU“¥ìw£,CÝ8 ? ; \h;0L}öwP ;,U
F
ì/£,CO838; h\ ;\ £ ” ?3V;[? P`;[email protected]“@A;ß L ¥XSL ü 6L ]àQ ”ÝN BDÌ,¢£,;’ ?X N BV:,£qÌ,[; E ;qDU¢UYC§U ” PZ;,@A;,8WU†£,CÝ8 ? ; h\ ;A;qU
[CÝ8£¢P`; N COX`8WU M ;,DU‘TXZ;,8[ ” 83 E ⊂ E L
2
2
GF
2
0
0
0
M eml^|)t^x Z3
] ” 83 Z [;q?X\ N COX`8WUY A Ü B DCO8WU“[3XZUY 6ì ” [ d ” £,;q8 UV +:&$( 18ì ” [Jd ” £q;,8WUYqÜ 26ì ” [Jd ” £,;,8WUV VX
−→ ;qDUˆ[ ” 83
AB
V = {(±1,0,0),(0, ± 1,0),(0,0, ± 1)} +:&$( V = V ∪ {(±1, ± 1,0),(0, ± 1, ± 1),
Ü
(±1,0, ± 1)} V = V ∪ {(±1, ± 1, ± 1} ùŽ;, ” J[ d ” £,;,83£,;, N ;,[email protected];qUDUY;q8 U‘[;¢[: #8XZBEP ” kì/£,CO883; \hXZUV: N CO?3B k ∈ {6,18,26} L
ó ;†UYË:qCOBVÌ[email protected]“; N BD:,£q:,[;q8 U^8;¢V;¢ôO:q8:,B ” P`XZV; N#” Eø ” £qX`PZ;,@A;,8WU Z <
"! }!f "s!f " # U& '( O&IX/Y& #0+,9& E 1F& Z %J65G$NJ/7/Y&X& ) 65G$NJ/7=0&X&*w 0'(4tuXf/ &$( #J(
3
6
26
18
6
18
3
<
65G$NJ/7=0&X&
3
í CO?B ”K? COX`BA?8€£qCO8WUYBV; ìw;,\3;[email protected] N P`;„XZPʏV? Uß;q8 ø ” X©U“[3;LV? N ;,B N CݏV;,B“?8 UVCÝU ” PZ;,@A;,8WU“£,CO8 ? ;,\h;Hü§ì
£qCO88;,\3;[; Z ”K? ;,£?8L¥ìw£,CÝ8 ? ; \h;8CO8UYCÝU ” P`;[email protected]“;q8 U&£,CO8 ? ;,\h;0L ? CÝX`B #ôO?3BV; L÷ý )
2
Ïä+*.0L ý
Y/ & # 0+,9&R1F& Z %J65G$NJ/7/Y&X& ) 65G$NJ/7=0&X&Iw 0'(!tuX / &$( #J( < 56 G$NJ/7=0&X&
3
¥Ý¦
( "!}!f s !f " Y& '( O&<X/Y&
#0+,9& E 1F& Z ' 56 G$NJ/7/Y&X& ) < 65G$NJ/7=0&X&w 0'(QtuFv/ &$( #J(
ó ;¢£qCO8WUYBV;; h\ ;[email protected] N PZ;†;,U9[3CO88:!P ” #ôO?3BV; L L
765G$NJ/7=0&X& !
(
Ïä+*j L 3
Y/ & # 0+ 9&R1F& Z ' 65G$NJ/7/Y& X& ) < 65G$NJ/7=0&X&*w 0'(4tuFf/ &$( #J("!765G$NJ/7=0&X&
í ” BÊ£,CO8WUYBD;¢P ” ’ ?;,UYXZCO8D?X §? ” 8WUV;BD;,UY;CO? ? ;qBDUV;ª<
i '( O& 6O-6OvF/Y& #0+,-9& E 1F& Z %J65G$NJ/7/Y&X&^& < 65G$NJ/7=0&X&ew 0'(I/YNJ/ !765G$NJ/7=0&X&
A5q !f
û \ N :,BDX`@A;,8WU ” PZ;,@A;,8WU P ” BD: N CO8D; ” PQ ” X`B)8CO86Ü ;q8ß;;qUŽV?BÊ¥ üW¦O¦O¦O¦ 3ì/£,CO8 ? ; \h;, 6ìw£,CÝ88; \h;qŽUYXZBV:,
” ?Ë ”  ” BD[[ ” 8ÊP`;†£q?T; {0, . . . ,3} ” ?£q?886Q÷;qDU9ü§ìw£qCO8 ? ;,3\ ;BL
)
3
3
3
« ɺ¹{¾¿´f¶Á°
¹
À¬‹­±®¯°³² ´f¶·
] ” 839PZ;£ Ë ”ÝN XZUVBV; N BV:q£,:q[;,8WU†CO8,QS;,UXZ8 UV:,BD;,VD:^P ” £,CÝ8 ? ; \hXZUV:à[XZV£qBVÌqUV;àP ”N P`? N BDCh£ Ë;[;
£q;,P`PZ;)[:#83X`; N” BÚPZ;,ôO:,CÝ@“ÌõUYBV;quL Í ” XZÚXZPW; \hX`UY;Ž?38; ” ?hUYBV; £,P ” VD; UYBDÌ,ÚX`@ N COBU ” 8WUV;}[;Ž£qCO8 ? ; \hX©UY:ª<
£q;,P`PZ;[3;, N COPZÔÝ[email protected]`8CO H I ìw£qCO8 ? ;,\3;quL.í ” B; \h;[email protected] N PZ;à;q8H£,[email protected]¤T3X`8 ” UVCOX`BD;¢CO8L8; ” X©U N#” 9£,[email protected] N UV;,B
; “£ ” £,;[email protected]“;q8WUßP`;q N COP©Ô[email protected]`83CO ” Ô ” 8WU“?8 8CÝ@¤TBV;[; N COX`8WUYA[CO838:, £§Q÷;qDUß?38;[3;,ôÝB ” 8[3;,
’ ?;qDUYXZCO8Ê[;¢£,;¢[CÝ@ ” XZ8; @ ” XZ&CO8 ” X©UÊ£,[email protected] N UV;,BEP`;q N CÝPZÔOCÝ@“XZ8CO H I ìw£qCO8 ? ;,\3;q EÍ þ L
û}8¡[email protected]ôOB ”ÝN ËXZ; [XZV£qBVÌqUV;EX`PV; N BDCh[3?XZU N ;q? N BVÌ,}P ” @ ,@A;ŽVX©UY? ” UYX`CÝ8†<§XZP386Q Ô ”N” }[6Q ” P`ôOC ì
BDXZUYË[email protected]“; N COP©Ô[email protected]“X ” P s ” ?høªVX.í †>^í N CÝ?BÊBV;,£qCO8UYBV?3X`BV;¢[;q N COP©Ô[email protected]`8Cݏ ”AN” BDUVX`BE[[email protected]¤T3BV;
[; N CÝX`8WUYÚ[ ” 8ª£ Ë ” ’ ?;&ËCOBDX !qCO8WU ” PZ; ;õUª[ ” 83ª£ Ë ” ’ ?; ? ;,BUYX`£ ” P`; 2C; Ýý Ü N” B=£qCO8WUYBV; £ÝQS;,U N CO~ì
DX`TPZ;DX=CO8HBD;,£,CÝ8DUVBV?X©U[;q N COP©Ô[email protected]`83CO HJI ì/£,CO8 ? ; \h;, ] ó >9í Ýý L ùŽ;qUVUV;¡BV;q£,CO8UYBD?£qUVX`CO8
D;,B ” [:õU ” XZP`P`:q;^[ ” 8ÊP ” [;,F? \3XZÌ,@A; N#” BDUVX`B; L Í ” PZË;,?3BV;,?3V;,@A;,8WUÚP ” £,P ” VD;)[;, N CÝPZÔOCÝ@“XZ8CO H I ì/£,CO8 ? ,; \h;,ª8;)qQ÷:õUY;,83[ N#” jø ” £qX`PZ;,@A;,8WU E[;,
[XZBV;q£qUYXZCO8.’ ?;,PZ£,CO83’?3;,u L ó ; T?3U=£,;Ž£ Ë ”ÝN XZUVBV;ˆ;,U}[;Ž[3CO88;qB=?8; 8CÝUVX`CO8 N BVC£ Ë;&[3;, N COP©Ô[email protected]`8Cݏ
; \3;q^@ ” X`Ž[: #83X`; N CÝ?BE[;,F? \[X`BD;,£qUVX`CO83E’?3;,P`£qCO8’ ?;qu L
H I ìw£qCO8 ? ,
, 4s ! Z2
9ÿ 83;^[XZBV;,£õUYXZCO8 N ;,?hU qUVBV;9[CO88:q;^DCOXZU N#” B ?8 ? ;,£õUY;,?3B p~ ܏VCOX©U N#” BŽ?38;9ø[email protected]“;ÊPZX`8: ” X`BD; p L ó ”
ø[email protected]“;P`XZ8: ” X`BV; p = ax + by ;qU^P`; ? ;,£õUY;q?B p~ = (−b,a) BD; N BD:,V;q8WUY;,8WU¢P ” @,@A;[XZBV;q£qUYXZCO8oL Z X a
;õU b VCÝ8 UÊB ” UYXZCO88;qP`&CÝ8[3XZUE’ ?;†P ” [X`BD;,£qUVX`CO8\;qDUÊB ” UYXZCO88;qP`P`;BL3] ” 8E£,;¢£ ” ECO8 N ;q?3U N BD;,8[BD; a
;õU b ;,8WUYXZ;,BD9;õU N BV;[email protected]“XZ;,BVE;q8WUYBV;;,F? \YL
K )b i9x "k3yjxhgj|)i n~u~yjx ¯l9gji l^uwkgjd0e3u~b)t^x
8àD; [CÝ88;Ž[3CO8£ˆ[;,?X\[XZBV;q£qUYXZCO8.[XZDUYXZ8£qUV;,
;qU q = cx+dy ”@? ;,£ pgcd(a,b) =
p
=
ax+by
Ü
L
1 pgcd(c,d) = 1
ó ;†[:qUV;,[email protected]“XZ8 ” 8WUÊ[;,Ê[;qX? \[XZBV;q£qUYXZCO8 det(p,q) = ad − bc ;,DUÊ83CÝUY: δ L ó ;,Ê[;,?F\[X`BD;,£õUYX`CÝ8
DCO8WUª[X ‚:qBV;,8WUV;,}[CO83£ δ 6= 0 L [ ?X©UVUV; ^£ Ë ” 8ôO;,B (a,b) ;,8 (−a,−b) CO8 N ;,?3UªV? NN COD;,B δ > 0 L 8
8C§UY; p~ = (−b,a) ;õU q~ = (d, − c) PZ;, ? ;q£qUV;,?BD ” DVC£,X`:q ” X? \\[3X`BV;q£qUVX`CO8qÜCO8 ” q(~p) = p(~q) = δ L
8Ã8CÝUY;
;qU¤
@ [email protected]“; hi,ji ;q8ÃPRQ ” TV;q8£,;[6Q ” @¤T3X`ôO? UY: P`; N COXZ8WU M ∈ Q UV;,P&’ ?;
hi,ji
;qU q(M ) = j L Z X δ 6= 1 ” PZCOBV^£,; N COX`8WU†86QS;,U N#” øsCOBD£,:[email protected]“;q8 U¢[ ” 8 Z ? COXZB #ôO?3BV;
p(M ) = i
X¥ L© ¥h
2
p,q
2
¥O¥
( U0
$ ( U1
L0
V1
L1
L2
V0
L3
5
4
53
=×
33
×0
4=
3=3×1
2=3×4
1=3×2
0=0×0
3
2
Ïä+*¥XL©
S &I+ c (@&50
1
L4
0
0=0&$GI'&$(410+:NJ &$(R1F&!10+$&$G 9NJ/ p = 2x + y & q = x − 2y í}P`?3 N BV:q£,X`D:,@A;,8WU9CO8 ” P ”¡N BVC N CODXZUYXZCO8ߏD?X §? ” 8WUY;=<
"!}!f s !f S&7NJ/7 hi,ji &$( Q1h0/#( Z ( Q& (@& F'& w & /7 ( j ≡ κi [δ] N κ &$( <F/
& /7-9& + 2+:& wx9& +!0=0&$G δ /Y&R1 c 7& /21h0/7 tum&R1F& p & q P c w NJ/#( -+.09NJ/ , ó ; N COX`8WU hi,ji ;qDU‘[ ” 8 Z DXj;qU‘V;q?P`;[email protected]“;q8 U‘DXjP`;†DԏUYÌ,@A;ª<
R¥X©L i
x
a b
=
j
y
c d
?” 83;DCOP`?hUYX`CÝ8\[ ” 8 Z L
\ [CO8£àX`P=;,h\ X`DUV; u ;qU v UY;qP`^’ ?; ua + vb = 1 L
ó ;,^;,8WUYXZ;,BV a ;õU b DCO8WU N BV;[email protected]“XZ;,BD;,8WUYBD;¡;,?XL
)
Z2
2
2
2
†8
CGF
”” P`CÝBVEPRQS:,ô ” P`X©UY:^[3;@ ” UVBVX`£q;,
κ = cu + dv
L
a b
u −b
1 0
=
c d
v a
κ δ
ó ” @ ” UVBVX`£q; M = u −b ;,[email protected]“C[?P ” X`BD;ß[CO8£CO8 N ;,?hUø ” XZBV;P`;£ Ë ” 8ôÝ;,@A;,8WUß[;
v a
V
B
X
T
`
P
;
[ ” 8 %¥XLZ ’?3Xj[; ? X`;q8WU
x
X
?ݔ ”
=M
y
Y
X = i,
κX + δY = j
’?3X ” ?83;DCOP`?hUYX`CÝ8\VXj;qU‘D;,?PZ;,@A;,8WU‘VX
∈ Z.
2
†8 N ;q?3U ” X`V:[email protected]“;q8WU9ôÝ:,8:qB ” PZX`V;qBÊ£,;àBD:,D?PZU ” UQ“P ” [[email protected]“;q8VXZCO8
Z
X
6
P
ø
3
?
U
P
`
O
C
D
B
Ê

“
@
õ
;
V
U
Y
U
D
B
;
P
@
V
U
V
B
Z
X
,
£
” ”
” ” ;
n
[; N BV:qV;,8WU ” UYXZCO8[3;, n [3X`BV;q£qUVX`CO8ŽVCO?Êø[email protected]“;[3; H ;,[email protected];0L
ó ”&N BDC N CݏVXZUVX`CO8 N BV:q£,:,[3;,8WUY;ˆ@“CÝ8 UVBV;ª[3CO8£}’ ?; Z ;qDU.P ” BD:,?83X`CO8†[; δ V CO?~ìwBD:,V; ” ?X\ L ,L , . . . L
[: #83X` N#” B
j−κi
ad−bc
2
Ll = {M ∈ Z2 | q(M ) ≡ κp(M ) ≡ l [δ]}.
0
1
ùŽË ” ’ ?;[BVCÝXZUY; p = i CO? q = j ;,U[ ” 8†?8HD;,?P}BD:,V; ” ? ;qU¢£q;,¢[BVCÝXZUY;q8;¡D;¡£,CO? N ;,8WU¢[ ” 8
’ ?;VXÚ;,P`PZ;,‘VCO8WU^[ ” 89P`;@[email protected]“;BV:qV; ” ?oL6]^CÝ8£à;q8„ø ” XZU‘£ Ë ” ’?3;¤VCO?3ìwBD:,D; ” ? $¡BV;qVV;[email protected]¤TPZ; (
Z
Z @” ? ;,£P`;q‘[3X`BV;q£qUVX`CO8ŽËCOBVX"!,CO8WU ” P`;qE;qU ? ;qBDUYXZ£ ” PZ;,ML
í ” BÊ; \h;,@ N P`;†[ ” 8ÊP`;†£ ” Ê[;¢P ” #ôO?BD;¥XLZÝÜ Z ;qDU‘P ” BV:q?8XZCO8[; 5 VCO?~ìwBD:,V; ” ?X\ L
2
2
2
δ−1
¥
$ "
4 ^ vu \ ª" !f " /Y& #0+,9& E 1F& Z &$( G$NJ/7=0&X&( w2'&(@& 'NJ/_10+:&$G -9NJ/ p ( 7NJF+QONJh(
7NJ/7( A & B & (*tu7& (AB) &$( #0+.0'> '&(* p '&$( 7NJ/7(e& /7-9& +:(*10 (@&9V0w & /7 [AB] (@NJ/7f1h0/#(
E
û}8 N#” BDUYXZ£,?PZX`;qB†P`;, ìw£qCO8 ? ; h\ ;q¤VCO8WUàP`;,¢£qCO8 ? ; \h;q¡VX`@ N PZ;,¢V;qP`CO8éP`;,¢[XZBV;q£qUYXZCO8 x ;qU y L
ó ;,A¥ìw£,CÝ8 ? ; \h;,AVCÝ8 UPZ;,¤;q8V;[email protected]¤TPZ;,’ ?X&VCÝ8 U£qCO8 ? ;,\3;qDX`@ N P`;qàV;qP`CO8³UVCO?3UV;,¡P`;q¡[X`BD;,£õUYX`CÝ8
B ” UYXZCO88;qP`P`;quL
\ £,CÝ8 ? ; h\ ;,¢DX`@ N P`;q9:õU ” 8WU†£qCO8 ? ; h\ ;¡DX`@ N P`;§ÜfCÝ8 N ;q?3U†[:#8XZB^PRQS;,8 ? ;õì
ó Q÷XZ8WUY;,BDV;q£qUYXZCO8L[;à[;,?X„
PZC NN ;£,CO8 ? ; h\ ;àDX`@ N P`;BL
Z XCO8 N BV;,83[L?8L;q8V;[email protected]¤TP`;[;à’ ?;qP`’ ?;q N CÝX`8WUYÊUYXZBV:, ” ?LË ”  ” BD[6Ü0;qU^?8;à[3X`BV;q£qUVX`CO8 p XZP.Ô ”
[;¡øsCOBUY;q£ Ë ” 8£,;q N CO?B¢’ ?;AP`;q†[BVCOX©UY;q[3;A[3X`BV;q£qUVX`CO8 p N#” DV;q8 U N#” B¢?8 V;q?P N CÝX`8WU Ü6;õU[CO8£
’ ?;“PRQS;,8D;,@¤TPZ;“£qCO8DX`[:qBV:DCOXZU£,CÝ8 ? ; h\ ;\[email protected] N P`;V;qP`CO8 p L ó ” £,CO8 ? ; h\ XZUV:ߏ[email protected] N PZ;¡;,U¡[CO8£A?8;
N BDC N BDX`:qUV: ” VD; !¢ø ” X`T3P`;0L
ÿ98 ” ?3UVBV;ŽXZ8£,CÝ8 ? :,8XZ;,8WUª[3;ŽP ” £,CÝ8 ? ; h\ XZUV:ŽDX`@ N P`;);,Uª’ ?; P ” £,CO8 ? ;,h\ XZUY:ŽVX`@ N PZ;}V;,PZCO8 N P`?DX`;,?3BV
[XZBV;q£qUYXZCO8\8; ? :qBVX #; N#” Ê?8 ” 8 ” PZCOôO?;†[3;†P ”¡N BVC N CODXZUVX`CO8L0LZ L XL0ùŽCÝ8VXZ[:,BDCO8ÊPRQS;,8D;,@¤T3P`; E [;
P ” #ôO?BD;ߥXL ¥X6L öwP);qDU ìw£,CÝ8 ? ; h\ ; 8X9L ó ;, N COX`8WUV;q8WUYCO?BD:,VCO8WUP`;q N COXZ8 UV†’ ?X)8;VCO8WU N#” 
[ ” 8PRQS;,8 ? ;,P`C NN ; ì/£,CO8 ? ; h\ ;“[3;, ” ?3UVBV;q N CÝX`8WUYq܂£,;àVCO8WU¢P`;q ” 8 ” P`COôO?3;,^[3;,†VCÝ@“@A;qUV N CO?B
PZ;, ì/£,CO8 ? ;,h\ ;,uL Í ” X`ÚPRQS;,8D;,@¤TPZ; E 86Q÷;qDU N” =:qô ” P 9 PRQS;,8 ? ;qP`C NN ; ì/£,CO8 ? ;,h\ ;E[;&£,;q N COX`8WUYqÜ
N ?3X`V’ ?;¢£q;, N COXZ8WUY&ø[email protected]“;q8 UÊ;q?X\ ì[email protected],@A;,Ê?8;,83V;,@àTP`; ìw£,CÝ8 ? ; h\ ;0L
2
HJI
HJI
HJI
H
I
H
I
HJI
)
Ïä+*¥XL ¥
/ & /#(@& w "u'& _q 65G$NJ/7=0&X&RtuF / &$( #J(! & /7=0& 'N 7& _qv65G$NJ/7=0&X&R1 X/ & /#(@& w "u'&
wx/7wxFw 1F& 7NJ/7(
4 ^ ] ” 38 ^£,;õUVUV;D;,£qUVX`CO8;qU9P ” V;q£qUYXZCO8„D?X ?§” 8WUV;CO8V; #\h;à[;q?X\[X`BD;,£õUYX`CÝ8ÊB ” UVX`CO883;,P`PZ;, p ;qU q
[;¢P ” ø[email protected]; p = ax + by Ü q = cx + dy ”K? ;q£ a,b,c,d ∈ Z Ü pgcd(a,b) = 1 Ü pgcd(c,d) = 1 L
íÚCÝ?BEUYCO?3U M ∈ Q CO8[:#8XZUÊPZ;,Ê’ ? ” UVBV;¢’ ? ” [B ” 8WUY ” ?3UVCO?BÊ[; M N” B6<
;qU
R (M ) = {N ∈ Z / p(N ) ≤ p(M ) q(N ) ≤ q(M )}
;qU
R (M ) = {N ∈ Z / p(N ) ≥ p(M ) q(N ) ≤ q(M )}
;qU
R (M ) = {N ∈ Z / p(N ) ≥ p(M ) q(N ) ≥ q(M )}
;qU
R (M ) = {N ∈ Z / p(N ) ≤ p(M ) q(N ) ≥ q(M )}
ó QSX`8[XZ£,; i [; R (M ) ;,U&;q8“ø ” X©U [ ” 8 Z/4Z [;9VCOBUY;‘’?3; R (M ) = R (M ) L ó COBVD’?3;^’ ?6QSX`P
86Q Ô ”LN#”  ” @àTX`ôO? RUY:§Ü.£,;,¡’ ? ” [B ” 8WUV¡VCO8WU8CÝUV:,¤DX`@ N P`;[email protected]“;q8WU R L 8 N ;q?3U ” PZCOBV[: 8X`BP ”
2
pq
0
pq
1
pq
2
pq
3
pq
i
2
2
2
2
pq
4
pq
0
i
¥§ü
( $ ( R3 (M )
R2 (M )
M
q = q(M )
R0 (M )
)
/ì £,CO8 ? ; \hXZUV:ª<
=Ú"s!f " [
(
Ïä+*f¥XL
R1 (M )
p = p(M )
S &$("! tu h10+.0/7(
/Y& #0+,-9& E 1F& Z &$( 65G$NJ/7=0&X& . G$NJ/7=0&X& #0+!tu h10+.0/7R(@& 'NJ/ p & q
2
û)8 N#” BUYX`£q?P`XZ;,BßP ” £,P ” VV; [;q [ ìw£qCO8 ? ;,\3;q„D;,P`CÝ8 ;qU ;,U„?38;éDCO?~ìw£,P ” VD;é[3;,£,CO8 ? ;,\h;,
VX`@ N PZ;,àD;,PZCO8 p ;õU¡V;qP`CO8 q L ó QSX`8WUV;,BVD;,£õUYX`CÝ8³[;ß[;qp?X\ [ ì/q£,CO8 ? ; \h;,;,U¡;,83£,COBD; [ ì/£,CO8 ? ; \h;ÝܪCÝ8
N ;q?3Uˆ[CO8£E[:#8XZB}PRQS;,8 ? ;qP`C NN ; [ ìw£,CÝ8 ? ; \h;‘[6Q÷?38; N#” BDUVX`; E [3; Z L 8[:qVX`ôÝ8;Ê£,;qUDUY;Ê;,8 ? ;,P`C NN ;
N#” B QCON V (E) L ó QS;,8 ? ;qP`C NN ; [ ì/£,CO8 ? ; \h;D;¢£ ” B ” £qUV:,BVXZV;¢;,8ø ” X©U N P`?3‘DX`@ N P`;[email protected]“;q8WUj<
"!}!f s !f " " W E &$( X/Y& #0+,9&R1F& Z 0'NJ+:( QCON V (E) &$( & /#(@& w "u'&
(∀i Ri (M ) ∩ E 6= ∅) =⇒ M ∈ E.
2
pq
2
pq
QCON Vpq (E) = {M ∈ Z2 | ∀i Ripq (M ) ∩ E 6= ∅}
L=]àQ ”ÝN BVÌq¤P ” [:8XZUVX`CO8éCO8 ”
P c w JN /#( -+.09NJ/ , Z COX©LU íÚFCO?3=B ”K{M
∈
Z
|
∀i
R
(M
)
∩
E
=
6
∅}
COX`B)PRQSX`83£,P`?3VX`CÝ8¡X`8 ? ;qBVV;9X`PD? UŽ[3; ? :,BVX #;,Bˆ’ ?; F ;,DU [ ìw£qCO8 ? ; \h;BL
?
F ⊂ QCON V (E)
Z COXZU M ∈ Z UY;,Pj’ ?; ∀i R (M ) ∩ F 6= ∅ L Z COXZU A ∈ R (M ) ∩ F L#]àQ ”ÝN BVÌqÊP ” [:#8XZUVX`CO8\[3;
XZPf;,\hX`DUV; B UY;,Pf’ ?; B ∈ R (A ) ∩ E LùŽCÝ@“@A; R (A ) ⊂ R (M ) CO8ߏ ” XZU ’ ?; M ∈ E ;õU
F
[CO8£ E ;,U [ ì/£,CO8 ? ;,h\ ;0L ? CÝX`B #ôO?3BV;¥XLSü 2
pq
i
2
pq
pq
i
2
i
i
pq
i
i
i
pq
i
B3
pq
i
i
pq
i
B2
A2
A3
M
A1
A0
)
Ïä+*¥XLSü
B1
B0
8NJF+.tumNJ {M ∈ Z
2
: ∀i Ri (M ) ∩ E 6= ∅}
Žù ;,£,Xj[3CO88;†?8 ” P`ôOCOBDXZUVË@“;EUVBVÌ,EDX`@ N `P ; N CÝ?B&£ ” P`£,?3P`;,B PRQS;,8 ? ;qP`C N3N ;
#8Xj[; N COX`8WUY‚<
&$( 65G$NJ/7=0&X&
[ ìw£qCO8 ? ;,3\ ;¢[6QS?8\;q8V;[email protected]¤TP`;
¥
C
8£qCO8DX`[ÌqBV;¢[6Q ”
TCÝBV[P`; $ #N ” B ” P`PZ:,PZCOôOB ” @“@A;)(
8
∆ = {hi,ji ∈ Q2 : i,j ∈ Z, min p(M ) ≤ i ≤ max p(M ), min q(M ) ≤ j ≤ max q(M )}
N ;,?3U ” `P CÝBV‘£ ” PZ£,?PZ;,BÊP`;U ” TPZ; ” ?[;àTfCCOP`:q;,8
ôOB DÝ£,;!¡P ” BV:q£,?BDBV;q8£,;ª<
E 6= ∅
M ∈E
M ∈E
VXZôO8X" ” W8 U
M ∈E
(V0 (M )M ∈∆
M ∈E
V0 (M ) = “R0 (M ) ∩
00
]^;à@[email protected]“;CO8 N ;q?3U†£ ” P`£q?P`;qB^P`;q^U ” TP`; ” ?X\
;qDU ” P`CÝBV6<
L ó QS;,8 ? ;qP`C N3N ;¡£qCO8 ? ; \h;A[;
V0 (hi,ji) = V0 (hi − 1,ji) ∨ V0 (hi,j − 1i) ∨ hi,ji ∈ E E
QCON Vpq (E) = {M ∈ ∆ ∩ Z2 : V0 (M ) ∧ V1 (M ) ∧ V2 (M ) ∧ V3 (M )}.
” [CÝ8£Ê?8 ” PZôOCOBDXZUYË[email protected]“; `P X`83: ” XZBV;&[6QS;,8 ? ;,P`C NN ; [ ìw£,CÝ8 ? ; \h;^;q8AøsCÝ8£qUVX`CO8[?“[email protected]¤TBD;Ê[; N COXZ8 UV
8
[;
V1 (M ),V2 (M ),V3 (M )
∆
L
!M I 4x
a yt9uwehu~b)t^x g‚e  k3g.Š‹u 6kgjx  k3b  ku~y‚ehyjx
c
ó ;, N COX`8WUYE ” X`PZP ” 8WUV&VCO8WU‘P`;qÊ:,’ ?X ?§” PZ;,8WUYE[3;,‘[email protected]@“;õUY N CO?BEPZ;, [ ìw£,CÝ8 ? ; \h;,ML
\ ª" !f / 7NJ/7 M 1 F/Y&)#0+,9& E 1F& Z &$( nX/ 7NJ/7 (p,q) 69(u00/7 )
( M 6∈
i
c
QCON V (E \ {M }) / h" (@& /YG$&x1 0w "uV0 ( F+R'&$(x10+:&$G -9NJ/#(NJ/ #0+,'& +5 ( w2'& w & /7
1F& 7NJ/7f(u00/7 1³12 " / 7NJ/7
M &$( F/7NJ/7 (u00/7f1F& E ( & U(@& F'& w & /7( v& '( O& i & Ytu7&
R (M ) ∩ E = {M } PD;,?c w PZ;,@ANJ/#;,( 8WU‘+.0V-Xj9NJX`P/ ,; \h]àX`Q ”ÝUY;N BVÌqUY;qPPj”A’N ?3BV; C N CODXZUVX`CO8„¥XL L ¥CO8L ” XZU^’?3; L M 6∈ QCON V (E \ {M }) DX=;õU
i
(E \ {M }) ∩ R (M ) = ∅
2
8 ”” X`8DXj?8:,8CÝ8£,:[email protected]“XZP ” XZBV;Q
P ”¤N BVC N CODXZUVX`CO8H ©L L Ž<
"! }!f "s!f k2NJX 65G$NJ/7=0& X& #83X &$( & /7=0& 'N 7& 65G$NJ/7=0& X&1F&(@&$( 7NJ/7 ( (u00/7 ( c
WL öwP3ø ” ?3U)@ACO8WUYBD;,Bª’ ?;‘£ Ë ” ’ ?;
Z COXZU ?38; N#” BDUVX`; [ ìw£qCO8 ? ;,3\ ;‘[; ÜO;õU
’P ? ” w [3BNJ” /#8W( U+.”0?3-9UVNJCO/ ?, BE[; ME £,CO8WUYXZ;,8WU‘?8 N COXZ8WUʏ ” XZP`P ” 8 ZUŽ[; EML ∈ E
ùŽCO8DX`[:qBVCO83 N#” BŽ;,h\ ;,@ N P`;PZ;^’ ? ” [3B ” 8WU
L Z COX©U&[CO83£ P`; CÝX`8WU&[; E ∩ R (M ) ’ ?X
)XZ’?3;0L RùŽ; CONX`8WUà; N h
@AX`[email protected]“XZV; (p(N ),q(N )) [ ” 8¤PRQSCOBD[BV;ßPZ; h\ X`£qCORôOB (M
Ë
\ X`UY;\£ ” B E ;,U #8X ó ;
”ÝN
N
N COX`8WU M ? :qBVX"; R (M ) ∩ E = {M } ;qU‘[CO83£ M ;qDU‘?8 N COXZ8WUʏ ” XZP`P ” 8 UuL
2
ùŽ;õUVUY; N BVC N CODXZUYXZCO8;,DU)[email protected] N COBU ” 8WUY;)£ ” Bª;,PZP`;ˆ@“CÝ8 UVBV; ’?‚Q÷XZPÔ ” ?8;ŽTX Dd ;q£qUYXZCO8¤;q8 UVBV;&PZ;,Ú;,8D;,@¤TPZ;,
#83X`E[; N COX`8WUYE ” X`PZP ” 8WUV&;qU‘P`;q [ ìw£,CÝ8 ? ; h\ ;, #8XZuL
7 ? ;,£P ”¡N BDC N COVX©UYX`CÝ8¥XL L ¥¤CO8;,8[:q[?XZU6<
!0"! s# " W S &$( < & /#(@& w "u'& 1F&$( 7NJ/7 (^(u00/7( 1 F/Y& #0+,-9& #8XZ; 1F& Z 0'NJ+:(
7NJF+I NJF i ∈ {0 . . . 3} ) & M ∈ Z NJ/ ,
2
pq
i
pq
i
2
0
0
0
2
2
S ∩ Ri (M ) = ∅ ⇐⇒ E ∩ Ri (M ) = ∅
¥Ýý
( $ ( P c w NJ/#( -+.09NJN / CÝ, ?ó Bà;A?V8 ;q8 [3⇐CO88;,D:ßU;õ£,U P ” X`BML ?3Í 8 CON 8WCOUVXZBV8WCOU¤8[; P`;D;,8 ⇒LªùŽ<jCO[email protected]©U @“M; ?8 N COXZ8 U†UV;,P)’ ?; COS8 ∩”
R (M ) = ∅
i
N
R (M )
R (N ) ⊂ R (M )
L
‚
]
Q
V
B
,
Ì
^

P
V
B
C
O
C
V

©
X
Y
U
Z
X
O
C
8
X
¥
L
L
¥
6
8
S
Q
,
;


†
U
[
O
C
8
£
9

[
8
9PRQS;,8 ? ;qP`C NN ;à£qCO8 ? ;,\3;
Ý
”
N
“
”
N
N
#
N
”
”
R
[; (NS Ü0)’∩?3XÚS[6=Q ”ÝN ∅BDÌ,P ”AN BVC N COVX©UYXZCO8L¥F÷L ü L ;qDU^:,ô ” PZN;R E L]^CO83£ N 6∈ E N CO?BÊUYCO?hU N ∈ R (M )
9& R (M ) ∩ E = ∅ L
2
"!}!f s !f / & /#(@& w "u'& S &$( <X/ & /#(@& w "u'& 1F& 7NJ/7(x(u00/7( 1 X/ 65G$NJ/7=0&X&
( & (@& F'& w & /7 ( 7NJX+IONJX M 1F& S & '( O& i O& tum& S ∩ R (M ) = {M } P c w NJù^/#( Q÷;q-+.D0U9?9NJ8/ ;¢, £,CÝ8V:q’?3;,8£q;¤[?P`;,@[email protected];¢¥XSL ü L ¥FL
⇒
ùŽCO8DX`[:qBVCO8à?8Â;,8D;,@¤T3P`; £[email protected]“@A;[ ” 8¤P ”N BDC N CݏVXZUVX`CO8YL Z COXZU
L
⇐
C N CݏVXZUVX`COV8à¥X(S)
L L¥
Í CO8WUYBDCO8}’ ?; S ;,U}PRQS;,83V;,@àSTP`; [;, N COX`8WUV= ” X`P`P ” 8WUY[; E OL ]àQ ”ÝN BDÌ,ªEP ”^=N BDQCON
CO8 ”
¥XL ¥ E = {M ∈ Z | ∀iR (M ) ∩ S 6= ∅}
Z X M ∈ S ” P`COBD“XZP9;,h\ X`DUV; i UV;,P‘’ ?; R (M ) ∩ S = {M } Ü ;qU[CO8£ N CO?BAUYCO?hU N ∈
CÝ8 ”
;qU‘[CÝ8£¢[6Q ”§N BVÌq ¥XL ¥ CO8 ”
ÜCO8;,8[:,[?3XZU
R (M ) \ {M }
’ ?; M ;,U^?8 N COX`8WUÊR ” (N
X`P`P ” )8W∩UES[; =E∅L :q£,X N BVC’?3;,@A;,8WU‘VX M ;qDU‘?8N N 6∈COXZE8 Uʏ ” X`PZP ” 8WUŽ[; E
X`P6;,h\ X`DUV; i UV;,Pj’ ?; R (M ) ∩ E = {M } COBE[6Q ”ÝN BVÌq ¥XL ¥ R (M ) ∩ S 6= ∅ ;õU‘[CO8£
i
i
i
i
i
i
i
i
pq
pq
i
2
i
i
i
i
;qU‘[CO83£
M ∈S
i
{M } = Ri (M ) ∩ E ⊃ Ri (M ) ∩ S 6= ∅
L
2
)b t^xyjd‚iÊehu~v&uwehy›l9gjx x|)u~nn~|)t ehx
] ” 8UYCO?hUY;¤£q; N#” B ” ôOB ”§N Ë; E [3:,VXZôO8;¤?38; N#” B UYX`; #8XZ;¤[; Z [ ìw£,CÝ8 ? ; \h;V;qP`CO8 (p,q) ܂;qU
PRQS;,8D;,@¤T3P`;†[;¢V;q N COXZ8 UVE ” XZP`P ” 8WUYu L
S
=Ú"s!f /E(u00/7 ’ ? ” [BD: &$( F/ G$NJ2'& (M,i) N M &$( F/ 7NJ/7 (u00/7 1F&
W
E & i ∈ {0, . . . 3} &$( eO& tum& R (M ) ∩ E = {M } M &$( IF/ 7NJ/7(u00/7!0'NJ+:(_'&
c
c
à
@
?
Z
P
V
U
X
Z
P
`
X
,
£
©
X
N UY: 10 7NJ/7 (u00/7 /óYNJw ["u: +$#&!8XZ1FUV& X`CO78NJ[/7;q  (4tuCO#XZh8 10UV+Ê( (M,i)
$
&
(
7& &
”
N
” XZP`P ” 8 UV&;,8WUVB C” B 8;¢’?3;UVCO?3U N CÝX`8WU‘ ” XZP`P ” 8 U M N ;,?3U qUVBV; ? ?„£[email protected]“@A;
?8 N COXZ8WU’ ? ” [BD: (M,i) @ ” XZ^£q; N COXZ8WU†’ ? ” [BD:¡86Q÷;qDU N#” øsCOBV£q:,@A;,8WU†?8XZ’ ?;ÝÜ N” B; \h;[email protected] N PZ;¤VX
P`COBD ” ? N COXZ8 U (0,0) CO8 N ;,?3U ” VVC£,XZ;,BÚP`;,ÚUYBVCÝX` N COXZ8 UVª’ ? ” [BD:,
p = x,q = y,E = {(0,0),(1,0)} ”
C
G
F
L
9
>
Ý
C
V
U
O
C
8

R
P
S
Q
,
;
83V;,@àTP`;¢[;q N COXZ8 UVE ” XZP`P ” 8WUY&’ ? ” [3BV:,‘[; E L
((0,0),i)
i 6= 2
S
=Ú"s!f P^& J (u00/7 (Rtu#h10+ c ( (A,i) & (B,j) (@NJ/7QG$NJ/#( c G F- ,:(e( nNJ/ X/Y&x1F&$(
2
i
0
( G$NJ/210-9NJ/#(
i=j
i=j
i=j
i=j
{i,j}
&
&
&
&
NJ
1F& ,+:NJ/7-9> +:&
,
N
N
= 0 R0 (N ) ∩ E = ∅
= 1 R1 (N ) ∩ E = ∅
= 2 R2 (N ) ∩ E = ∅
= 3 R3 (N ) ∩ E = ∅
= {0,1}
{i,j} = {2,3}
q = q(A) N
N
N
N
N
N
&
= hmax(p(A),p(B)) − 1, max(q(A),q(B)) − 1i = hmin(p(A),p(B)) + 1, max(q(A),q(B)) − 1i = hmin(p(A),p(B)) + 1, min(q(A),q(B)) + 1i = hmax(p(A),p(B)) − 1, min(q(A),q(B)) + 1i q(A) = q(B)
E
& &$( </YG jh( 1h0/#(^X/ 1F& wx6 20/
¥
C
NJ
& p(A) = p(B) & E &$( /YG jh( 1h0/#(RX/ 1F& wx6 20/
F1 & ,+$NJ/79> +:&
ZP4X‚0CO/#8 (4” G$&xPQ÷G5?J83(^;NJ[3/E;,Ê/Y[NJO;,& ?F\[;,BD8X`ÌqBV;,Ê£qCO8[X©UYX`CÝ8ECO8[XZUʒ ?; [AB] ;,DU‘?8 29&51 [3; E L
{i,j} = {1,2}
{i,j} = {3,0}
,
p = p(A) ACONSB B
A
8 8$6 5 8;678 /8 / 5$8 ! ! 8 1 6 57/
6
8 1$6 4/ /8"#$ /1% 8 & 8$6 5 8;678 /! 8 / 5 $8 )( 8 1 6
$/ 1 8 8 1 6 / 5 8' 5( E
(A,i)
(B,j)
E
B
A
sS &$(R1F& sG5J(R1F&RG$NJ/#( c G [email protected] c * +$NJ909NJ/2+$>$(
8 ” [CO8£Ž@¤?8X
[6Q÷?38AUYBD?£qUV?BV;E[;ŽôOB ”ÝN Ë;0WL í ” Bª;,\3;[email protected] N P`;&DX3CO8¡£qCO8VXZ[Ì,BD;EP`;qª[X`BD;,£õUYX`CÝ8
S
;qU q = y ;qU^PRQS;,8D;,@¤TPZ; E [;¤P ” ôO?BV;¤¥X÷L ý ” PZCOBV‘P`;q^V;q?P`;q^BD;,P ” UYX`CÝ89[3;à£,CÝ8V:q£,?3UVX ? XZUV:
p=x
)
Ïä+*‚¥FL#
0
DCO8WUj<
(A,0)CONS(A,1)CONS(C,1)CONS(D,1)CONS(E,2)CONS
(E,3)CONS(C,3)CONS(B,3)CONS(A,0)
ó ” BD;,P ” UYXZCO8
CONS
[:8XZU‘[CO83£?38£õÔ£,P`;V?3B S L û)8ø ” X©UE£ÝQS;,U ? B ” Xj;,8ôO:,8:qB ” P-L
0
(E,3)
E
(C,3)
(E,2)
(D,1)
D
(C,1)
(B,3)
C
B
)
Ïä+*¥XL÷ý (A,0)
A
(A,1)
/ & /#(@& w "u'& E 65G$NJ/7=0& X&(@& 'NJ/ p = x & q = y & (@NJ/ G pTG '&1F&$( 7NJ/7 (n(u00/7(
c
tu#h10+ ( -P40/#(^G$& &X& w2'&IONJT(!'&$( 7NJ/7(41F& E (@NJ/7 (u00/7( "! } !f "s!f #
S &V0+.7`7& (S ,CONS) &$( F/rG pTG '& 0
¥
$ ( ( P c w NJ8 /#N( -;,+.?30U)[email protected]/ ;q, UDUYùŽBVCO; 8VXZ[D:,COBD?CO}8P‘P`øs;qCݏEBVü[email protected];q; 8V;[email protected]¤TPZ;,
L
L
”
”K? ;q£
†8 „
Z
P
O
C
V
B

s
ø
Ý
C
V
B
,
£
q
:
“
@
q
;
W
8
U
L
)
X
`
X
ª
P
Ô
©
X
U
3
?
8
O
C
`
X
W
8

U
[
8

Z
” ”
”K?§”
N
”
P” ZCOBVA£,; N COXZ8 U“V;qB ” X©U[ ” 8 £,;’ ?X^;qDUß[email protected] N CODVX`T3P`;[CO8£
L
]^; N ZP ?
;,DUX`@ N COVDX`TPZ;¤[Ì,’ ?;
Üj£ ” B C
;,Uà[ ” 8
L#]^CO8£CO8 ”¡N BVCÝ? ? :’ ?;ª<
DVX |j − i| = 1.
(M ,0)CONS(M ,0)
ó ; ôOB ”ÝN Ë; ,CONS) ;,U[CO8£H?83; £ Ë ”CB 8B; LÊ]^; @ [email protected]“;ø ” CO8cCO8 N ;,[email protected];,B\’ ?; P`;q
ôOB ”ÝN Ë;, (S (S
Ü
;õU
VCÝ8 U‘[;q9£ Ë C” B 8;qu L
,CONS) (S ,CONS) (S ,CONS)
ùŽË;qBV£ ËCÝ8@ ” XZ8WUY;,8 ” 8WU¡P`;q N CÝX`8WUY¤£qCO8D:,£,?hUYXZøs ” F? \ N COX`8WUYà[; S ’ ?XE8;\VCÝ8 U N” ¤;,?F\ì
@[email protected]“;qE[ ” 8 S L Z X 1 < i < n ” PZCOBV M 8‚Q ”” ?£q?8 N COXZ8WUE£,CO83V:,£q?3UYX©ø=8‚Q ”ÝN3N#” BDUV;,8 ” 8 U N#”  S
£ ” B E 8; N ;q?3U N#”  õUYBV;¢£qCO8WUY;,8 ?[ ” 8‘?8[;,@AX©ì N P ” 8\[;†øsBVCO8WUYXZÌ,BD; p = p(M ) CO? q = q(M ) L
íCO?B i = 1 CO8A ” X©U ’ ?; M ;,DU P`; N COXZ8 Uˆ’ ?X#@AX`[email protected]“XZV; (p,q) N CO?B)PQ÷COBD[BV;EPZ; \hX`£qCOôOB ”ÝN ËXZ’?30; L
Z COXZU N PZ; N COXZ8 Uڒ [email protected]`83X`@AX`V; (p,−q) N CÝ?BÚPQSCOBV[BD; PZ; \hX`£qCOôOB ”ÝN ËXZ’?3;ÝÜ"P`; N COXZ8WU N ;,DU=øsCOBV£q:,@A;,8WU
[ ” 8 S L ó ;, N CÝX`8WUY)[; S D?B P ” [BVCÝXZUY; p = p(M ) VCÝ8 U ; \ ” £õUY;[email protected]“;q8 U M ;qU N L]^CÝ8£9PZ;,}D;,?PZ
N COXZ8WUYÊ’ ? ” [BD:,Ê£,CO83V:,£q?3UYX©øs (M ,0) VCO8WU (M ,0) ;õU (N,3) L
þ=XZ8 ” PZ;,@A;,8WUŽCÝ8 N ;q?3UÊ@“CÝ8 UVBV;qB&’ ?6Q÷XZP‚86Q Ô ” ’ ?6QS?8;¢D;,?PZ;†£,CO8D:,£q?3UYX ? X©UY:†;q8 UVBV; S ;qU S Ü3?8;
V;,?3P`;\;,8WUVBV; S ;qU S ;õU¡?8;\D;,?PZ;ß;,8WUVBV; S ;qU S ;õU¡’ ?;\£,;q¡£,CO83V:,£q?3UYX ? XZUY:q¡V;ßøsCO8WU¡;q8WUYBV;
N COXZ8WUY,; \hUVBV:[email protected] ” F? \H[;q†£ Ë C” B 83;, S L †8H;q8 [:,[?3XZU†[CÝ8£¤’ ?;P`;ôOB ”ÝN Ë; (S ,CON S) ;,U¢TX`;q8
?8£õÔ£,P`B; L ? CÝX`B #ôÝ?BV;X¥ L L 2
Si = {M : (M,i) ∈ S 0 }
S0
S0 = {M1 , . . . ,Mn }
p(M1 ) < p(M2 ) < . . . p(Mn )
q(M1 ) > . . . > q(Mn )
R0 (hp(Mi+1 ) −
1,q(Mi ) − 1i)
S0
Mi CONSMi+1
GF
Mi CONSMj
|i − j| > 2
Mk
i<k <j
R0 (hp(Mj ) − 1,q(Mi ) − 1i)
i
j
0
1
2
3
0
0
0
i
i
i
1
3
1
1
1
2
0
1
2
2
3
0
i
S3
S2
N
M1
p
Mn
S0
)
Ïä+*¥FL S1
8<NJ/7(*(u00/7(41 F/ 65G$NJ/7=0&X&
q
1
¥
C
p=x q=y
p = x + y,q = x − y
)
Ïä+*‚¥FL sP^& Js&X& w2'&$(41F& 65G$NJ/7=0&X&$(
¦
$ ( R _, ùŽ[email protected]@A; N CO?BP ” £qCO8 ? ; \hX©UY:éUVCÝU ” P`;Ý܎P ” [ ì/£,CO8 ? ; \hXZUV:é86QS;,8WUVB ” 8; ” ?£,?83;2[;q„£,CÝ88; \hX©UY:,
[X`D£,BVÌõUY;quL ? COX`B #ôO?BD;¥XL ] ” 8£q;qUVUV;\V;q£qUYXZCO8ÃCO8 ?§” £ Ë;,BD£ Ë;,BAP`;q¤£,CO83[XZUVX`CO8 N CO?3B¤’ ?6Q÷?38
[ ì/£,CO8 ? ; \h; ” X©U ” ?DVX.[;q N BVC N BDX`:õUY:,Ê[;¢£qCO88;,3\ X©UY:BL
R ( CB
\psu~t^xyjd‚|u~n~u/ehy
] ” 8ªP`; N BD;,@AX`;,B; \h;[email protected] N PZ;Ž[; P ” #ôO?BD;Ê¥XL 3Ü"CO8 N ;,?3U=BV;,@ ” BD’?3;,B=[3;,ª[BDCOXZUV;,ÚËCOBVX"!,CO8WU ” P`;qÚCO?
? ;,BÊUYXZ£,£ ;õ” UVPZUY;,; ª’ BD?C Xh8BV;&XZ:q£,UVCÝ0: 8 L UVX`;q88;,8WU ” ?3£,?8 N COX`8WUÚ[;&PQ÷;q8V;[email protected]¤TP`;BL ó ;q};,8D;,@¤TPZ;,ªXZ8V:q£ ” T3P`;,=86Q ” ?BVCO8WU
N#”
N N
=Ú"s!f W NJ E F/Y& #0+,-9&I1F& Z ) p F/Y&!10+:&$G 9NJ/ ) pmin = inf{p(M ) / M ∈ E}
& pmax = sup{p(M ) / M ∈ E} /r10<tum& E &$( /#( c G5h"u'&1h0/#(R 10+:&$G 9NJ/ p ( 7NJX+
NJF i & tu7& pmin ≤ i ≤ pmax 10+:NJO& p = i G$NJ/7-9& /7 0 w NJ/#(*X/7NJ/7 1F& E ó ;,Ê;q8V;[email protected]¤TP`;q9£qCO88;,h\ ;,‘VCO8WUÊUYCO? dDCO?BVÊXZ8V:q£ ” TPZ;,Ê[ ” 83‘£q;,BU ” XZ8;,Ê[XZBV;,£õUYXZCO86<
W E &$( ' 65G$NJ/7/Y& X&Q0'NJ+:(<v&$( /#( c G5h"u'& 7NJF+Q'&$(10+:&$G 9NJ/#( x) y "!}!f s !f "
W E &$( %J65G$NJ/7/Y& X&!0'NJ+:(I &$( /#( c G5h"u'& 7NJF+I'&$(410+:&$G -9NJ/#( x,y,x − y
W E &$( !765G$NJ/7/Y& X&!0'NJ+:(I &$( /#( c G5h"u'& 7NJF+I'&$(410+:&$G -9NJ/#( x,y,x + y,x − y P c w NJ/#( -+.09NJ/ , ]^:[email protected]“COÜ 8WUVBVCO8 P ”N BV;[email protected]“XZÌ,BV; ” [email protected] ” UVX`CO8 ”K? ;,£^;õUP ” ?[8X`;\BD;,[£qBDUVCOX`COXZUV8 ; p = [6xQSL:,’ Z ? CO”XZU UVX`xmin
=
O
C
8
inf{p(M ) / M ∈ E} xmax = sup{p(M ) / M ∈ E}
D
x =
,
;
£
L
S
Q
,
;
3
8
V

,
;
à
@
T
`
P
;
[
q
;
X
?
\
€
q
£
O
C
@
O
C

W
8
Y
U
q
;

/
ì
,
£
O
C
8
3
8
;
\
h
,
;
ß

’ ?X
ó
@
”
?
”
N
”
x
£,CO8WUYXZ;,883;,xmin
8WU¢£ Ë ” ≤£q?8x;¡[≤;q xmax
CN OXZ8WUY^[3; E Ü[CO8£ ZE £,\CÝ8DUVX`;q8 U^øsCOBV£q:,@A;,8WU ” [email protected]^?8 N COXZ8 U9V?3B
£,;qUDUY;[BVCÝXZUY;BL
ó ;, ” ?hUYBV;q£ ” V; @ACO8WUYBV;q8WU[;éP ” @[email protected]“;Lø ”J CO8c£ ” BPZ;,\;q8V;[email protected]¤TP`;q Z \ {y = cste} Ü
Ü
8;ʏVCO8WU N#” qÜBD;, N ;q£qUVX ? ;[email protected]“;q8WU ì/£,CO883; h\ ;ÝÜý ì/£,CO883; h\ ;ÝÜ
Z \ {x − y = cste} Z \ {x + y = cste}
ü§ìw£qCO88;,\3;BL
2
0
2
0
2
2
2
2
a uwkgjd0e3u~b)t^x ‰^b}k3ub)t eh|ˆn~gjx g‚eÃv)g6khehu~d‚|)n~gjx
c
Í ” X`8WUY;q8 ” 8WU‘CO8 ?ݔN BD;,8[BD;PZ;,E[XZBV;q£qUYXZCO8 ? ;,BDUVX`£ ” P`;q p = x ;qU‘ËCOBDX !,CÝ8 U ” P`;q q = y L ó ”
HJI ì/X`8D:,£ ” TX`PZXZUV:ÝÜ3P ” [ ì HJI ìw£qCO8 ? ; \hX©UY:ÝÜ#PZ;, H I ìw ” X`PZP ” 8WUV&[:,DX`ôO83;,8WU‘BV;, N ;q£qUVX ? ;[email protected]“;q8WU^PQ÷XZ8V:q£ ” TX+ì
P`XZUV:ÝÜhP ” [ ìw£,CÝ8 ? ; \hXZUV:ÝÜ#PZ;,Ê ” XZP`P ” 8WUYŽD;,PZCO8PZ;,E[XZBV;q£qUYXZCO8EËCÝBVX !qCO8WU ” PZ;,E;õU ? ;,BUYXZ£ ” PZ;,uL
.
ó ;ªUYË3:,COBDÌ,@A;}V?X ?ݔ 8 U6[CO88;}?38;)£ ” B ” £qUV:,BVXZ ” UVX`CO8^[;,.;,83V;,@àTP`;q HJI ì/£,CO8 ? ; \h;,Ú;qU ì/£,CO883; \h;,ML
!012 W NJ E X/Y& #0+,-9&x1F& Z 0'NJ+:(4'&$(1F& J 2+:N 2+ c c (I( X=J0/7O&$(!(@NJ/7 c tuF6
" #3O!fÚ.
=J0'& /7 &$( ,
S & /#(@& w u" '& E $& ( _q 65G$NJ/7=0&X&x& ' 65G$NJ/7/Y&X& c
S & /#(@& w "u' & E $& ( 6 _q 65G$NJ/7=0&X&^& _q 6O/#( G5h"u'& 2
V (
;
P c w NJZ /#? ( +.0CO-9VNJCO/ 83, ‘’ ?; ;,U ì/£,CO8 ; \h;à;õU "ìw£,CÝ88; \h;BL
]Q NN BVÌ,àP BVC ECODXZUYXZCO82¥XL#FL ¥CÝ? 82 XZU¤’ ?; ;qDU¡XZ8V:q£ TPZ;D;,P`CÝ82P`;qà[XZBV;q£qUYXZCO8 L
”ÝN ”N N
”
”
öwP.BD;,DUV;x N BVCO? ? ;,B9’?3; E ;qDU [ ì ìw£,CÝ8 ? ; \hE;0L Z COX©U M ?8 N COX`8WU‘[; Z UV;,P’?3;¤£ {x,y}
Ë ” ’ ?;
,
£
O
C
W
8
Y
U
Z
X
,
;
8
3
8

;
3
?
8
O
C
`
X
W
8
U
[
;
L
N
R (M )
>‘CÝUYCÝ8 ∆ = R ∩ R 3L û}8\A£qCO8VXZ[E:,B ” 8WUE?8 "ìw£ Ë;[email protected]“XZ8;,8WUVBV; A ;qU A CO8 ” X©U&’ ?6QSX`P0Ô
” ?8 N COX`8WU B ∈ (∆ ∪ ∆ ) ∩ E L ? COX`B ôO?BV;à¥XL Z ? N3N CODCO8‘’ ?6Q÷XZPj; \hX`UY; k ;qU l UV;,P`
HJI
HJI
i
2
i
i
i
i+1
i
i
i−1
i+1
i
R3 (M )
R2 (M )
∆2
A3
∆3
q = q(M )
∆1
∆0
A0
Ïä+*‚¥XL A2
M
A1
R1 (M )
R0 (M )
p = p(M )
8NJF+.tumNJ M &$( 6O 1h0/#( E
’ ?; B ∈ ∆ ;qU B ∈ ∆ Ü ” P`COBD&8CO?3‘ ”K? CO8 N#” B ìw£,CÝ8 ? ; \hXZUV:’ ?; M ∈ E L
Z ? NN COVCO83&’ ?; N CO?B)UYCO? k CO8 ” XZU B 6∈ ∆ L 8 ”” P`COBD B ∈ ∆ ;õU B ∈ ∆ ;qUE[3CO8£
N” B ì/£,CO8 ? ;,\hXZUY: M ∈ E L
]9;P ” @[email protected]“;ø ” CO8VX N CÝ?B&UYCO?3 k B 6∈ ∆ ” PZCOBVECO8 ” XZU‘’ ?; B ∈ ∆ ;qU B ∈ ∆ ;qU
[3CO8£ N#” B ìw£,CÝ8 ? ; h\ XZUV: M ∈ E L
] ” 8EUYCO?3‘PZ;,E£ ” ÊCO8 ” M ∈ E ;qU‘[CÝ8£ E ;,DU [ ì ìw£qCO8 ? ;,3\ ;BL
Z Z ? X NN CO;qVUCO83‘@ VCO” 8WXZU†8 UV[3;,;,8 ?X” \ 8 U ECOXZ8W[ UY^ì [3; ì/£,COXZ8 PÚ? ;,; \3h\ XZ;0DLUV;¤?8 8 §? ” ì/@A£ ËCO;,[email protected]`8L;qBÊBDP;,”¤PZX N 8WBDCU N BVXZ:q;õUVU :†V?X`X 8Ý? ”£q8P`?UV9;ª<[ 8
N
” A B
”
E
LE A B
8 §? ” ø ” X`BV;“?8;ßBD:,£q?BVBD;,8£q;D?B
L Z ? NN COVCO83¡P ”N BVC N BDX`:õUY:
max(|x − x |,|y − y |)
? B ” X`; N CO?3B^UYCO?3 A Ü B UY;qP=’ ?; max(|xL 8Â− Vx? |,|yCݏV; − yBà|); \h≤;[email protected] l L PZ;ßZ CO’ XZ;,?8W; U A ;õU B UY;q;õP`U¤9’’ ??3;;
NN N#”
N
max(|x L −CÝxXZU |,|y − y |) ≤Ü l + 1
Ü
L Z QSX`Pfx; h\ XZ<DUY;†x?38 A UY;qP
Z
y <y
A (x ,y + 1) A (x + 1,y ) A (x + 1,y + 1)
’ ?; A ∈ E ” P`COBD&[6Q ”ÝN BDÌ,‘P ” BV:q£,?BDBV;q8£,;CO8 ” T3X`;,8?8£ Ë;[email protected]“XZ8;q8 UVBV; A ;qU B L
Z ? NN COVCO83‘[3CO8£¢’ ?; A 6∈ E Ü A 6∈ E Ü A 6∈ E L
ùŽCÝ@“@A; E ;,U ìwXZ8V:q£ ” T3P`;¤XZP=; h\ XZDUY;?8 N COX`8WU M UY;qPª’ ?; x = x + 1 L Z X y ≤ y
” P`COBD A ∈ E N#” B [ ìw£,CÝ8 ? ; h\ XZUV:¡£,CO8WUVB ” [XZ£qUVX`CO8ofL ]^CÝ8£ y > y ‚L ]^;¤@,@A;XZPÚ;,h\ X`DUV;¤?8
N COXZ8 U N UY;qP ’ ?; y = y + 1 ;qU x > x L Í ” X` ” PZCOBV A ∈ R (A ) Ü M ∈ R (A ) Ü
Ü
;qU‘[CÝ8£ A ∈ E £,CO8WUVB ” [XZ£qUVX`CO8oL ? COX`B #ôÝ?BV;¥X©L ¦ B ∈ R (A ) N ∈ R (A )
)
I
0
k
2
l
0
k
H
1
2
k
H
1
0
3
H
3
3
2
1
I
HJI
A
A
A
A
B
A
1
B
B
B
A
A
B
B
B
A
A
A
2
1
2
A
A
3
A
B
A
i
i
I
3
M
2
M
N
2
3
A
1
3
N
3
A
A
M
A
A
0
3
3
3
2
ó ; N BVCÝTP`Ì[email protected]“;[3;†£,;qUDUY;¢£ ” B ” £õUY:,BDX` ” UYXZCO8ß;,U‘’?3;PQ÷CÝ8ß8;¢£,CO83VX`[3Ì,BV; N” EV;q?P`;[email protected]“;q8 U‘[;qE’ ? ” ì
[B ” 8WUY[email protected] ” X` ” ?3VVX [;q.[BVCOX©UY;quL 883; £qCO88 ”CB U.[CO8£ N#”  ”ÊN BDX`COBDX ” ’?3COX BV;,DV;[email protected]¤TP`;)?8;,8D;,@¤T3P`;
[; N COXZ8WUYŽ ” X`PZP ” 8WUVˆ[‚Q÷?8 H I ì/£,CO8 ? ;,\h; "ìw£,CÝ88; \h;BLû}8ßø ” XZU P ”N BVC N CODXZUVX`CO8“V?X ?ݔ 8 UV; N ;,[email protected];qUŽ[;
BD:,VCÝ?[BV;¢£q; N BDCOTPZÌ,@A0; L
¥
$ ( ( M
A1
B
N
A3
A
A2
Ïä+*¥FLZ¦
s S &R1 c "uX 10EG$`7& wx/ & /7-+:& A & B
"!}!f s !f /b& /#(@& w "u'& E &$( qv65G$NJ/7=0&X& ' 65G$NJ/7/Y&X&x( I& Q(@& F'& w & /7Q( Qn&$( qv6 65G$NJ/7=0&X&^& f( 7NJX+IONJX (x,y) ∈ Z NJ/ '&$(!tu 0+:& c /YNJ/YG c (*( X=J0/7( ,
E ∩ (R (x,y) ∪ R (x + 1,y)) = ∅ =⇒ (E ∩ R (x,y) = ∅ NJ E ∩ R (x,y) = ∅)
E ∩ (R (x,y) ∪ R (x,y + 1)) = ∅ =⇒ (E ∩ R (x,y) = ∅ NJ E ∩ R (x,y) = ∅) R¥XL E ∩ (R (x,y) ∪ R (x + 1,y)) = ∅ =⇒ (E ∩ R (x,y) = ∅ NJ E ∩ R (x,y) = ∅)
E ∩ (R (x,y) ∪ R (x,y + 1)) = ∅ =⇒ (E ∩ R (x,y) = ∅ NJ E ∩ R (x,y) = ∅)
P c w NJZ /#CO( X©-U +.09NJ?/ 8³, ;q8V;[email protected]¤TPZ; ì/£,CO883; h\ ;0L Z ? COVCÝ8à’ ?;
L
NN
⇒
E
E ∩ (R (x,y) ∪ R (x + 1,y)) = ∅
ó Q÷;q8V;[email protected]¤TP`;
” [3;,?X\ £,CÝ@ N CO ” 8 UV;, ì/£,CO883; h\ ;,
;qU R (x,y + Z1) L \(R
D;,@¤T3P`; E(x+1,y))
;qDU&XZ8£,PZ?)[ ” 8ˆ?8;9V;q?P`;‘[;^£q;, £,[email protected] N CO ” R8WUV;,(x+1,y−1)
ˆ£,CO883; h\ ;,
ó QS;,8(x,y)∪R
;qU[CÝ8£ (E ∩ R (x,y) = ∅ CO? E ∩ R (x,y) = ∅) L ó ;qßUYBDCOX` ” ?3UYBD;, ” DV;qBDUYXZCO8ßD;
[:,@ACO8WUYBD;,8WU‘[;¢P ” @,@A;ø ” CO8YL
Z COX©U ?8; N#” BDUVX`; ? :qBVX" ” 8WU&P`;qˆøsCÝ[email protected]¤?PZ;,R¥XL 3L ]Q ”ÝN BVÌ,ŽP`;ÊUYË:qCOBVÌ[email protected]“; N BV:q£,:,[3;,8WUEX`P0D? U
⇐ [;[email protected]
CO8WUYBV;qBª’ ?; E ;qDU)X`83V:,£ ” TPZ;0L Z COXZU}[3CO8£ x = i PQ÷:q’? ” UYXZCO8¤[6QS?8;Ê[BDCOXZUV; ? ;qBDUVX`£ ” P`;0L 8
V? N3N COD;’ ?;^£q;qUDUY;^[3BVCOX©UY;Ê8;^£qCO8WUYXZ;,8WU ” ?£,?8 N COXZ8WUŽ[; E ;qU&’ ?;9PRQSCO8 ” ?8 N COXZ8 U A ∈ E
?ùŽ:[email protected]" @“” 8W; U x ≤ i ;õU‘?8CÝ8 N COXZ8 U B ? :,BVX ” 8WU x ≥L Zi LCÝXZZ U ? NN COVCO83PZ; 9’ P`??;¢ PRQSCO;q8 UVXZU¢” ;,y8WUVX`≤;,B†yUV;,P}L ’ ?;
” R (i,y ) ∩ E = ∅
N N
(i,y ) 6∈ E ‚
jmin
L
w
ö
ª
P
,
;
\
3
Z
X
D

V
U
¡
;
[
O
C
8
¡
£
?
8
O
C
`
X
W
8
U
Y
U
q
;
=
P
’
?
;
;õU x < i L
N
R (i,jmin) ∩ E = ∅
−1
ùŽ[email protected]@“; (i,jmin − 1) 6∈ E CO8 ” øsCÝBV£,:[email protected]“;q8WU RC(i,jmin −y 1)=∩ jmin
L
9
]
CO8£„CÝ8 ”
E
=
∅
Ü
Ü
Ü
A
£,CO8W∈UVB R” [(i,jmin
XZ£qUVX`CO8 ”K?−;q£1) B ∈ R (i,jmin) R (i,jmin − 1) ∩ E = ∅ R (i,jmin) ∩ E = ∅
CO? E ∩ R (x,y) = ∅)
E ∩ (R (x,y) ∪ R (x,y + 1)) = ∅ =⇒ (E ∩ R (x,y) = ∅
”K? ;,£ x = i ;qU y = jmin L
2
þ=XZ8 ” PZ;,@A;,8WU Ü"ôOB Ý£,; ” ?¡P`;[email protected]“@A;&¥F÷L ü SL üCO8¡CÝT3UYXZ;,8WUªP ” £ ” B ” £qUV:,BDX` ” UYX`CÝ8à[;q};,8D;,@¤TPZ;,ª[3; N CÝX`8WUY
 ” XZP`P ” 8 UV&[;q ìw£qCO8 ? ;,3\ ;q ì/£,CO838; h\ ;,‚<
!0"! s#s" / & /#(@& w "u'& S #8X &$( F/ & /#(@& w "u'&1F& 7NJ/7o(u00/7 (Q1 F/ _q 65G$NJ/7=0& X&
' 65G$NJ/7/Y& X& #8X ( & (@& F'& w & /7( 7NJX+Q NJF M 1h0/#( S o& '( & i O& otu7& R (M ) ∩ S = {M }
& ( S = c +, -&I'&$( KNJ+,wxF'&$( R¥FL )
2
0
2
1
3
0
2
1
3
3
1
0
2
1
3
0
2
0
2
0
2
2
1
3
1
3
A
B
3
B
A
B
3
C
C
1
0
2
1
3
1
3
0
2
D
H
I
,
B
i
V (
;
.s 7 ? ;,£AP ” ü ìw£,CÝ88; \hX©UY:¡CO8 ” [;q£ ” B ” £qUY:qBVXZ ” UVX`CO8^VX`@AX`P ” X`BD;,‘@ ” X`CO8 [CÝXZU†@ACh[3X"#;qB^PRQ ” [Jd ” ì
£q;,8£q;[;qÊ’? ” [B ” 8 UV6<
"! }!f "s!f / & /#(@& w "u'& E &$( _qv65G$NJ/7=0&X& !765G$NJ/7/Y&X&( e& Q(@& X'& w & /7<( Qn&$( " O!fÚ.
_q 6 65G$NJ/7=0&X&4& ( 7NJF+I NJF (x,y) ∈ Z NJ/l '&$(41F& J c /YNJ/YG c (( F=J0/7( ,
E ∩ (R (x,y) ∪ R (x + 1,y + 1)) = ∅ =⇒ (E ∩ R (x,y) = ∅ NJ E ∩ R (x,y) = ∅)
E ∩ (R (x + 1,y) ∪ R (x,y + 1)) = ∅ =⇒ (E ∩ R (x,y) = ∅ NJ E ∩ R (x,y) = ∅)¥XLSü
P c w NJó /#;†( D+.;,08-9 NJ/ , D;[3:,@ACO8WUYBV;¢£[email protected]“@A; CO?BEP BVC COVX©UYXZCO8¥XL XLSü L
Z ? ⇒COVCÝ8’ ?; :qBVX";àP`;q9øsN [email protected]¤?PZ”;, N R¥XNSL ü ‚L û}8 BDUYXZ£,?PZX`;qB :qBVX";àP`;q9ø[email protected]¤?PZ;,
⇐¥XL ;qNU‘N ;qDU‘[CO8£ ì/E£,CO83? 8; h\ ;0L Z ? NN COVCO839’ ?; 86QS;,U N#”N#” Eü§ìw£qCO88E;,3\ ? ;§Ü0X`P‚;,h\ X`DUV;¢[CO8£
;õU B [ ” 8 E ’ ?X8;àDCO8WU N#” 9BV;qP`X`:q N” B9?38Lü§Eìw£ Ë3;,@AX`8„XZ8£,PZ?‘[ ” 8 E Lû)8„£,CO8DX`[:qB ” 8WAU
?38 "ìw£ Ë;[email protected]“XZ8L;,8WUYBD; A ;qU B CO8 N ;,?3U¢D;¡B ” @A;,83;,B ” ?H£ ” †C A ;qU B DCO8WU ì ” [ d ” £,;q8 UVuL
Z ? NN COVCO83 N” B†; \h;,@ N PZ;A’ ?; A = (x,y) ;qU B = (x + 1,y + 1) L ó ;, N COX`8WUV (x,y + 1)
;õU (x + 1,y) 83;ADCO8WU N#” ¢[ ” 8 E L 8 ;,8é[:,[3?XZU¢’ ?; R (x + 1,y) ;õU R (x,y + 1) 8;
£qCO8WUYXZ;,88;q8WU ” ?£,?8 N COX`8WUÊ[; E fL ùŽCÝ8 UVB ” [3X`£qUVX`CO8 ”K? ;,£P`;,&øsCÝ[email protected]¤?PZ;, ¥X÷L ü L
2
ùŽ;qUDUY; N BVC N CODXZUVX`CO8ß[3CO88; ” ?DVX.?8;¢£ ” B ” £õUY:qBVX` ” UYXZCO8\[;q N COX`8WUVE ” XZP`P ” 8 UV&[;q N COP©Ô[email protected]`8Cݏ
#83X`
ìw£qCO8 ? ; h\ ;q6<Ž?8 ;,83V;,@àTP`; ;qDUA?8 ;q8V;[email protected]¤TPZ;[3; N COXZ8 UV¡ ” XZP`P ” 8WUY¤[6QS?8 N COP©Ô[email protected]`83C
ìw£qCO8 ? ;,\3;àVX.;qU9V;q?P`;[email protected]“;q8 U9VX N CO?SBEUYCO?hU M [ ” 8 S X`Pj;,\hX`DUV; i UY;qP’ ?; R (M ) ∩ S = {M }
;õU&VX S ? :,BDX"#;^P`;, ø[email protected]¤?3P`;,R¥F÷L ü L ùŽ;õUVUV;£ ” B ” £qUV:,BDX` ” UYX`CÝ8 ”K§? ” XZUŽ[6Q ” X`P`PZ;,?BD):qUY:9UYBVCÝ? ? :q; ”@§? ” 8WUŽP ”
[:#8X©UYX`CÝ8\[;, [ ìw£,CÝ8 ? ; \h;,ML
ó ;,¢ø[email protected]à?P`;qR¥XSL ü N ;,? ? ;,8WU ” ?VDXõUYBD;A:q8CO8£q:,;qà;,8 UY;[email protected];[; N COXZ8WU¢ ” XZP`P ” 8 U’? ” [BV:=<DX
;õU
VCO8WU†[3;, COX`8WUY9 XZP`P 8WUY‘’? [BV:q ” P`CÝBV‘PRQS?8H[;,^[;,?XL
\ £,CO? N P`;q (A,i − 1)
(A,i)
+U 2) ?VDX?38 CON X`8WUʏ X`P`P ” 8WUE” ’ ? [BD:0” Lf] 839
CO? (A,i +(A,i
,
;
Z
P

;
Ý
ô
B
3
Ë

;
3
[
;
q
£
O
C
8
V:q£,?3UVX ? XZUV:?38 N COX`8WU
”
N
” ”
”
”
”ÝN
1)
 ” X`PZP ” 8WU ”§NN#” B ” UßUVCO?JDd CO?3BV[ ” 8[;q N COXZ8 UV\’ ? ” [BD:,£,CO8D:,£q?3UYX©øsuL ó ; £ ” [? N CÝX`8WU C [;
PQ÷;,3\ ;[email protected] N P`;[;†P ” #ôO?BD;¥X÷L ý83; N ;q?3U‘[CO8£ N#” EV; N BVC[?XZBV;BL ]^CO83£[ ” 8EP`;†£ ” E[;q N COP©Ô[email protected]`8Cݏ
[’ ? ìw£qCO[38BV:,? ‘;,3\ £ ;q¢BEVCO;,8 PZCO8L;,?hPZ;,Uʏ[[:XZ#BV;q83£qX`UYBÊXZCO[8X`BD9;,£õËUYCO;,[email protected]"!,;,CO8W8WUÊU ” ?P`8;q9£q;õÔU £qP`? ;!;qBDUYXZ£ ” BDPZUV;,X`B&XZ[PÚ;q;q DU†COXZ8 XZ8?3UVUYEXZP`; P`XZ;,P`P N8WUYCOuXZ8WL UY^ ” X`PZP ” 8WUV
”
” N
N”
N
” ”
2
0
2
1
2
1
3
0
2
GF
1
3
HJI
H
I
i
CB
M &ehgjt^xu~bˆt »l^uwk3gjdfehu~bˆt9xmb)i  n~i^x
Z X q; DUˆ?8;,83V;,@àTP`; #83X[;Ê[X`BD;,£õUYX`CÝ8,ܧP ”e[ ìw£qCO8 ? ;,\3X©UY: N ;,?3U õUYBV;EôÝ:,8:qB ” PZX`V:q;E[;EP ” ø ” CO8
D?X ?§” 8WDUY;=<
\ ª" !f E ⊂ Z &$( 65G$NJ/7=0&X& (@& 'NJ/ D ( 7NJX+ONJh( p,q ∈ D O& (_tum& p 6= q
& /#7 (@& w ;,£"u'£,& ;õEUVUV;&$([ : # 865X©G$UYNJX`CÝ/78 =0&XCÝ8&(@& ;,'NJ?h/ UpôO:q& 8 :,qB P`XZV;qB¡P`;ßUYË3:,COBDÌ,@A;„¥FL#XL H[6Q ?3UYBD;,A;,8D;,@¤TPZ;,A[;
?
N
”
”
[XZBV;q£qUYXZCO86<
"! }!f "s!f # W NJ D = {x,y,x − y,x + y} /Y&%#0+,-9& - /79& 1F& Z &$( "!765G$NJ/7/Y&X&
2
& ( w2'& w & /7nG$NJ/7=0&X&(@& 'NJ/ D ( & (@& F'& w & /7 ( & '&x&$( /#( c G5h"u'&*(@& 'NJ/ D & 56 G$NJ/7=0&X&
(@& 'NJ/ D
2
ü
( $ ( W E &$( - /7 !765G$NJ/7/Y&X&& ( w2'& w & /7<G$NJ/7=0&X&I(@& 'NJ/ {x − y,x + y} 0'NJ+:(
E &$( 65G$NJ/7=0&X&(@& 'NJ/ {x − y,x + y} ó ” [:[email protected]“CO83DUYB ” UYXZCO8\;,U‘VX`@AX`P ” X`BD;¡ £q;,P`PZ;†[?V;,839[XZBV;q£qUÊ[?UYË:qCOBVÌ[email protected]“;¥XL XL L
P¥XL#Fc wL NJ;q/#U‘( P-”+.0~9OÜNJ©/ ì1F&nOÜì 2ì +$[ N 7ìw£[email protected]( 8 ? 9NJ;,/\3X©UY :9 [ '?, PZ⇒;,@A< @“P ” ;†OÜ¥X÷L#¦ F L ì#L ¦hÜZ ì [ ìw£qCO8 ? ;,h\ XZUY: N BDC ? XZ;,8WUª[3?àUYË:qCOBVÌ[email protected]“;
Z CÝXZU φ PQ÷XZ[email protected] COB N ËX`[email protected]“;[; Z ’ ?XªUYB ” 8ø[email protected]; OÜ÷¦ Ü ~OÜZ ;,8 OÜ÷¦ Ü ¦hÜZ φ(x,y) = (x −
L ó QS;,8D;,@¤T3P`; E ;qDU„ü§ìw£qCO88;,3\ ;é[CO8£ φ(E) ;,U ì/£,CO838; h\ ; £ ” BPRQSX`@ ” ôO; N#” B φ [? ü ì
y,y)
?;,DCOU^XZV[X`8 CO” 8ôO£ ; [6ì/£,QS?CO8 883N ; COh\ X`;à8WUŽ;õU;,D~UÊOÜXZ8¦ £,ìPZ? ¦3VÜ©;9[ ” X`88DE:,PZ£ ; T P`ì ;? COsXZ£ VXZ8 B ” ôO;9;q[DU ;PRQSOÜX`@÷¦ ” ìôO;ÊÝÜZ[? XZN 8COV:qX`8W£ UuL Tó PZ; QS;,8;qDU9;,@¤[COT83PZ;£φ(E)
XZPj;,DU
”
”
” OÜ÷¦ ì ¦3Ü© ì [ ìw£qCO8 ? ; h\ ; N#” BP`;)UYË:qCOBVÌ[email protected]“;Ž¥XL XL L 8¤;qE8¤[:q[?XZUڒ ?;
,
;


U
ÝÜ ¦ ì~OÜZ ì [ ì/£,CO8 ? ;,h\ ;0L
†8 N ;,?3UÊø ” XZBV;[;¢@,@A; N CÝ?BEP`;qÊUVBVCOXZ ” ?3UVBV;q9£qCO? N PZ;,E[;¢[XZBV;,£õUYEXZCO8uL
< Z CÝXZU A ;õU B [3;,?X\ N COXZ8 UVE[; E L †8\ø ” X©UE?8;†BD:,£q?BVBD;,8£q;D?B d (A,B) = |x − x | +
⇐
L †8V? N3N COD; x < x ;qU y < y L Z COXZU A = (x + 1,y ) ;qU A = (x ,y + 1) L
|y
−
y
|
í ” BOÜ ì ìwXZ8V:q£ ” T3X`P`X©UY:CO8é ” XZU’ ?6QSX`PªÔ ” ?8 N COXZ8 U¢D?BP ” [BVCOX©UY; (A A ) L í ” B~OÜ ¦ Ü OÜ© ;qU
¦3Ü© ì OÜ© ì [ ìw£qCO8 ? ; h\ X©UY:}CO8;,8[:,[3?XZU’ ?;
CO?
[6Q÷C PZ;ˆBD:,D?PZU U B.BV:q£,?BDBV;q8£,;BL
Z X x = x CO? y = y CO8 ” PZ;BV:qV?P©U A” U ∈N#” EB ~OÜ÷¦ A CO?∈ E¦3Ü© £qCO8 ? ; h\ X©UY:”VX`@ N#N ” PZ;0L
2
"!}!f s !f W NJ D = {x,y,x−y} /Y& #0+,9& - /79& 1F& Z &$( %J65G$NJ/7/Y& X& & G$NJ/7=0& X&
(@& 'NJ/ D ( & f(@& X'& w & /7 ( & '&x&$( 65G$NJ/7=0&X&(@& 'NJ/ D & /#( c G5h"u'&*(@& 'NJ/ D ì
P c w ì NJ[ /#(ìw£q-+.CO08 ? 9NJ;,\3/ ;B, L ⇒ó QSX`< DCOZ @AX ECOB N ;qË3DU“X`[email protected]ý ìw; £qCO88;,h\ ;XZPŽ;qDU ” øsCOBDUVX`COBD[X ; ì/£,CO83UVB8” ; 8h\ ;øsCO;[email protected][; CÝ8£ Eì ;,U (1,0)
ì (0,1)
(0,1)
− y,y)
;,8 (1,0)ì (0,1)ì (−1,1) L 8ß;,8\[:q[?X©φU&:’ ?(x,y)
; E ;,7→U (x(1,0)
ì (1,1)ì [ Zìw£qCO8 ? ; h\ ;B3L ]^;^@(1,0)
[email protected]“;Êø(1,1)
Ý
C
ß
8
” CO8 ”
P ” (0,1)ì (1,1)ì [ ìw£,CÝ8 ? ; h\ XZUV:0L
< Z COXZ;,8WU A ;õU B [;,?F\ N COXZ8WUY‘[; E UV;,P.’ ?;àPQ ” 83ôOP`;†;q8 UVBV;OÜ÷¦ ;qU −AB
→ ;,DU^DUVBVX`£õUY;[email protected]“;q8 U
⇐
£,[email protected] N BVXZ;,8WUVBV;A¦ß;qU¢ü[3;,ôOBD:,uL 8 §? ” £,CO83DUYBD?X`BD;¡?8 OÜ÷¦ ìÝÜZ £ Ë;,@AX`8 N” B†BV:q£,?BDBV;,83£,;V?B
L Z CÝXZU A = (x + 1,y ) ;õU A = (x + 1,y + 1) L Z COXZU M P`; N COXZ8 U&[;
|x DXZ−
x
|
=
x
−
x
UV?:¢V?BÊP ” [3BVCOX©UY; x = x + 1 L
E
ì ì [ ì/£,CO8 ? ;,h\ XZUY:ECÝ8 ” A ∈ E L Z X y > y Ü N#” B (1,0)ì (1,1)ì
ZX
” PZCOBV N#” B
ì[ /£,CO8 ?y; h\ XZ≤UV:y£§Q÷;qDU A ’?3Xj(1,0)
[ ” 8 (0,1)
L COXZB #ôO?BD;¥X©L O 8 ” [CÝ8£ A ∈ E CO? A ∈ E L
E ?
1³12 2
1
A
A
B
B
A
1
B
A
A
B
A
2
A
1
F
1
A
A
2
2
B
2
2
B
A
B
1
A
A
2
A
A
A
A
M
1
A
M
2
1
M
B
A2
)
A
Ïä+*¥FLZO
A1
N
sS &R1 c "uX 10EG$`7& wx/ & /7-+:& A & B A
2
B
A
V (
(
L ó Q ” 83ôOP`;&;q8WUYBV; ;õU A−−→B ;,U)[ ” 8}PQ÷XZ8 UV;,B ?ݔ P`P`; ]0,45 ] £ ” B
Z ? N3N CODCO8}[CÝ8£E’ ?;
qQ÷XZPf:õU ” X©U N P`?ˆôOB ” 8[’ ?; A45 ∈Ü ” EP`COBD)P`;, [BVCÝXZUY;q y (1,0)
;qU x − y = x − y 8;^D;^£,CÝ? N ;qB ” XZ;,8WU
=y
N#” ;q8é?38 N COX`8WU¢[; Z L Z X}£,;õU ” 8ôOP`;¡;qDU; \ ” £õUY;,@A;,8WU:,ô ” P 45 ” P`COBD†P ” [BVCÝXZUY; (A B) ;,DU
;qP`P`; ì[email protected] ,@A;9?38ý ìw£ Ë;[email protected]“XZ8\;,8WUYBD; A ;qU B ;õUÊ[CO8£^CO8 ” TX`;q8\?8ý ìw£ Ë;[email protected]“XZ8ß;,8WUVBV; A ;õU B L Z XZ8CO8
[6Q ”ÝN BDÌ,P ” BV:,£q?BVBD;,8£q;\X`P},; \3XZDUV;?8Âý"ìw£ Ë;[email protected]“XZ82;,8WUVBV; A ;qU B ’ ?X N ;q?3U õUYBD; N BDCOP`CO83ôO:“;q8³?8
£ Ë;[email protected]“XZ8;,8WUYBD; A ;õU B L
Z X A ∈ E [email protected];¢[;P ” @ [email protected]“;ø ” CO8’?‚Q÷XZP;,\hX`DUV;¢?8Lý ì/£ Ë;,@AX`8;q8 UVBV; A ;õU B ’ ?X.V;
N BDCOP`CÝ8ôO;[CÝ8£¢;,8?8„ý"ìw£ Ë;[email protected]“XZ8;q8WUYBV; A ;qU B L
ó ;, ” ?3UVBV;¢£ ” ‘qQ÷;q8„[:q[?XZV;,8WU N#” BʏÔ[email protected]:qUVBVXZ0; L
2
◦
1
1
◦
A
B
2
B
◦
1
1
1
2
2
R 4 ^ ! ùŽCO8WUYB ” X`BD;,@A;,8WUeß £q;¤’ ?XªD; N#” DV; ”K? ;q£¡P ” ¥"ì/£,CO8 ? ; \hXZUV:¤P`;P`X`;q8L;,8WUVBV;¤P ”_[ ìw£,CÝ8 ? ; \hXZUV:¤;qUP ”
£qCO8 ? ; \hX©UY:¢UYCÝU ” P`;;qDUÊUYBDÌ,Ê[email protected] N P`;ª<
"! }!f "s!f W NJ E F/Y& #0+,-9&!1F& Z S&$(R1F& J 2+:N [email protected]( 9NJ/#(*(@NJ/7 c tuF=J0'& /7O&$( ,
S & /#(@& w "u'& E &$( NJ90'& w & /7G$NJ/7=0& X& 8NJF+I NJF &$(R10+:&$G 9NJ/#(I+.0-9NJ/7/Y& '&$( p & q ) & /#(@& w "u'& E &$( 65G$NJ/7=0&X&(@& 'NJ/ (p,q) P c w íÚNJCÝ/#?(B.PR+.Q 0” -?39UYNJBD/ ;}, Vó ;q8QSX`@Ú£qN COP`8XZ£ D” X`[UVX`:qCOBV8 CO8⇒.?;q8DU‘;,83£,VP ”;,@àX`BD;0TL P`; [ ìw£qCO8 ? ; \h; V;,PZCO8†UYCO?hU.£,CO? N P`; L ]àQ ”§N BVÌq
P ”N BDC N COVX©UYX`CÝ8m L© LSü„XZPˆD? U¤[;“@“CÝ8 UVBV;qBà’ ?;“EUVCO?3U N COXZ8WU M XZ8WUY:,BDX`;q?BR?82UV(p,q)
BVX ” 8ôOP`;
C A,B,C ∈ E ;,DU ” ?DVX)[ ” 83 E L Z X M ;,DUV?3BPQ÷?38é[;q¢£,CÝUV:,[? UVBVX ” 8ôOPZ;ÝÜ6£[email protected]“@A; EABC
;,U
[ ìw£qZCOXZ8 8? CO;,8³3\ ;COV8Â;qP`£qCOCO8\8UVDCOX`[?ÌqEBV;ßP`;qÊPZ;£,CO£,CÝ? ? N P`;,P`E;[3[;;ß[3[X`XZBVBV;q;,£q£õUVUYX`XZCOCO88q Ü3XZP6;,U ”C øsCOBUYXZ;qCODBVUX‚P ¥"ìw£q[COXZ8 BV?;,£õ;,UY3\ XZ;CO8³;qU‘[[; CÝ8£ M ∈;qU E L
”
N
) CÝ8 q
;qDUP ” [3X`BV;q£qUVX`CO8[; (BM ) ;qU9UY;,P’ ?; q(A) ≤ q(M(p,q)
;) qU p(B)p ≤ p(M ) ‚L íCO?B‘UYCO?3U (AM
”
i 6= 2
L †8 ” øsCOBV£q:,@A;,8WU
£
Ž
B
D

`
X
8
O
C
“
8
`
P
‘
;
Y
U
V
B
X
8
O
ô
Z
P
;
q
;
D

&
U
[
8
&

Z
P
”
”
” ;
R
[;[email protected]“(MX+ì N) P∩” 8\{A,B}
[;øsBVCO6=8WUY∅XZÌ,BD; p = p(M ) ;qU9[COp(C)
8£ M≥;,p(M
UD?)B9?38L[;,9£,_ÝUV:,9[; ABCABC
£,CO8WUVB ” [XZ£qUVX`CO8oL
]^;¡@,@A; q(C) ≥ q(M ) jL ]9CO8£ C ∈ R (M ) ;qU M ∈ E N#” B [ ìw£qCO8 ? ; h\ X©UY:¡V;qP`CO8 (p,q) L ? COX`B
#ôÝ?BV;¥X©L "¥ 2
8 N ;,?hUÊ@ ” XZ8WUY;,8 ” 8WUÊ:qUY?3[X`;qB‘PZ;, N COX`8WUVE ” XZP`P ” 8 UV&[;q 7NJjphNJwx/[email protected]( UYC§U ” PZ;,@A;,8WUE£,CÝ8 ? ; h\ ;,‚<
"! }!f "s!f W E &$( 4X/ N COP©Ô[email protected]`83C ONJ 0'& w & /7G$NJ/7=0&X& 0'NJ+:(E1F& J 7NJ/7 ( _q 6
(u00/7(^G$NJ/#( c G F- :( A & B 1F& E = c +, -& /7 ONJ #KNJF+:( |x − x | ≤ 1 NJ |y − y | ≤ 1 c
Z X ;,U&?8 N X`;q[CO8 x = x CO? y = y L 8 N ;,?3U [CO83£^V? NN CݏV;,B ’ ?6Q÷XZP
;,P \3XZDw UV; NJ/#i( UY+.;q0PŽ-9’ NJ?/ ; , R [AB]
;õU R (B) ∩ E = ∅ L 8V? NN COD; N#” Bà;,\h;,@ N P`; i = 0 ;õU
(A)
∩
E
=
∅
L Z COXZU
;õU
Ü#£[email protected]@“; A ;õU B DCO8WU‘£,CO8D:,£q?3UYX©øs
p(A)
−
1)BV;qP`XZ:,
CO8  ” <XZUp(B)
’ ?; R (N N) ∩=E(x=,y∅)&L í}?NX`D’ ?=; (xA ;q−U 1,y
V

O
C
W
8
U
N” Bß?8 ü§ì/£ Ë;,@AX`8 CO8 ” ’ ?;
B
L ]^CO8£ VX |x −x | > 1 ;qU |y −y | > 1 L ” P`COBD (x −2,y ),(x ,y −2) ∈ E Ü
[AN
]∪[BN
]
⊂
E
[CÝ8£ N#” BE£,CÝ8 ? ; h\ XZUV: (x − 1,y − 1) ∈ E £qCO8WUYB ” [X`£õUYXZCO8 ”K? ;,£ R (N ) ∩ E = ∅ L
2
û}8„ø ” X©U9[ ” 8^PZ;ôÝB ”ÝN Ë3;[;£,CO83V:,£q?3UYX ? XZUY:[;, N CÝX`8WUY ìw ” X`P`P ” 8WUYÊ?8 N COXZ8WU^;qDU^UVCO?JDd CO?3BV
£[email protected] N BVX` ;,8WUYBD;†[;,?F\ [email protected]@A;qUY ? CÝX`B&D;,£qUVX`CO8H0LZOÜhP`;qE[email protected]@“;õUY VCO8WUÊ[;q‘ ” X`PZP ” 8WUV N#” BDUYXZ£,?PZX`;qB L
í}P`?3Ž; \ ” £õUY;[email protected]“;q8 U&VX A ;qU B DCO8WUE[;,?F\UV;,P` [email protected]@A;qUY ? :qBVX ” 8WU R (A) ∩ E = {A} ;qU R (B) ∩
;õU
P`COBD}P`;q N COXZ8 UV} ” XZP`P ” 8WUYª;q8 UVBV; A ;qU B VCO8WUˆ; \ ” £õUY;[email protected]“;q8 U
R = {B} 0 ≤y − y ≤ x l− x ” m
PZ;, N COX`8WUY S = x + i,y +
N CO?B 0 ≤ i ≤ x − x L ? COX`B #ôÝ?BV;¥X©L 2
GF
GF
i
2
,
A
A
i
A
A
B
B
i
B
0
B
B
0
A
B
A
0
A
A
B
A
B
A
0
B
B
A
B
0
HJI
0
B
i
A
A
B
A
A
(yB −yA )i
xB −xA
B
A
0
ý
( $ ( C
A
M
B
Ïä+*h¥FLZ"¥
)
NJ90'&
8NJF+.tumNJYSo 65G$NJ/7=0& c (@& 'NJ/ NJF &$(<'&$(<10+$&$G 9NJ/#(*& /7+./Y& 6O-65& '&G$NJ/7=0& c
4 ^ IU
] ” 8Ê£,; N#” B ” ôOB ”§N Ë; D [:,DX`ôO8;¢?8;,83V;,@àTP`;¢[;¢[XZBV;,£õUYXZCO8&B ” UYXZCO88;qP`P`;quL
ó QS;,8 ? ;qP`C NN ; [ ìw£qCO8 ? ; \h;V;qP`CO8 N P`?[3;¡[;,?X\L[X`BD;,£õUYX`CÝ8^;qDU ” DV; ![X A£,X`PZ;Rߣ ” P`£q?P`;qBuLû}8
;;qU†DX E ;,DU^?8; N#” BDUVX`; #8XZ;à[; Z ;q8L;;qU QCON V (QCON V (QCON V (E))) 86QS;,U
N#” †øsCOBV£q:,@A;,8WU [ ìw£qCO8 ? ; \h;V;qP`CO8 PZ;,†£,CÝ? N P`;q (p,q) CO? (q,r) L ? COXZB #ôO?BD;“¥XLZ,ü L öwP=ø ” ?3U†[3CO8£
; ;,£qUV?;,B^PRQSC N :qB ” UVX`CO8 E 7→ QCON V (QCON V (QCON V (E))) dD?D’ ?6Q “£,;à’ ?6QSX`P86Q Ô ” XZU
N P`?Ê[;¢@ACh[3"X #£ ” UYXZCO8u L3ùŽ;’?3X N ;q?3U qUYBD; ” VD; !P`CO8ô#L
öwP0ø ” ?3UŽ;,V ” ÔO;,BE[3;[: 8X`B P ”^[ ì/£,CO8 ? ;,\hXZUY:^[;^ø ” CO8 N PZ?ŽôOPZCOT ” PZ;ʏV?BŽUVCO?3UV;,&PZ;,Ž[3X`BV;q£qUVX`CO8ML
íÚCO?BŽ£,;,P ” X`P#ø ” ?3UŽ[6Q ” TCÝBV[\;,D ” ÔO;qBÊ[;BV;q[: 8X`B P`;,Ž’? ” [B ” 8 UV N CÝ?BŽ[;q&;,83V;,@àTP`;qE[; N PZ?Ž[;
[;,X? \[3X`BV;q£qUVX`CO8M L
2
qr
qr
pr
pr
pq
pq
/ a yt9u/e3u~b)t^x
c
=Ú"s!f /l
W 8 2+:&$(utu7& 6 1F& wx6 20/ ) ” P`@ACODUìwV;[email protected]“X+ì N P ” 8; x(@& 'NJ/ D &$( F/ Π =
R (M ) O& tum&7NJX+ G$`#htu7& 10+:&$G 9NJ/ d 1F& D & '( &^X/Y& 1F& wx6 10+$NJO& 1F& 10+$&$G 9NJ/ d & 1 NJ+ V0/Y& M & G$NJ/7O& /7m&!1h0/#( Π
û)8Hø ” X©U?8 7 Z ío;qDU¢?8H:qP`:,@A;,8WU¢@ ” h\ X`@¤[email protected] [ ” 8†PQ÷;q8V;[email protected]¤TP`; {R (M ) | p,q ∈ D} CÝBV[CO838:
N#” BªPQ÷XZ8£,PZ?VXZCO8oL ÿ987 Z í N ;,?3U ” ?VDX qUVBV; ? ?¡£,[email protected]@A;Ž?8[;[email protected]“X+ì N P ” 8 ” ?3’?3;,PCO8 ” ;q8P`; ? :E?8 R
@“XZ8X`@ ” P9L ó ;E8CÝ@¤TBV;E[‚Q÷7 Z í ” ?3UVCO?B}[6QS?8 N COX`8WU M ;,U ;q8¡ø ” XZU}:qô ” P ” [email protected]¤TBD;&[;Ê[;,@AX©ì N P ” 8
[;àøsBVCÝ8 UVX`ÌqBV;à[ ” 8 D ;qDU N” V ” 8WU N#” B M ‚A X`PjÔ ” [CO83£¥ |D| 7 Z í ” ?3UVCO?B9[6Q÷?38 N COXZ8WU M L ó ”
N BVC N BVX`:õUY: N CO?B R (M ) [6Q qUYBD;¢?87 Z í 8;¢[: N ;q8[ N” Ê[; M L
88CÝUV;
[email protected],@A; PRQS;,8D;,@¤T3P`;†[;q
UV;,PZ‘’ ?; R ;,DU‘?87 Z íUL
A
A
(i,p,q)
ùŽ;õUVUY;¢8CÝ? ? ;qP`P`;¢8C§UYX`CÝ8 N ;[email protected];qUÊ[;¢[:#8X`BÊP ”[ ì/£,CO8 ? ; h\ XZUV:¢øsCÝBDUY;=<
=Ú"s!f " / 65G$NJ/7=0& X& KNJ+ .NJ& /#(@& w "u'& KNJ+ O& w & /7 65G$NJ/7=0& X& !&$( F/s& /#(@& w "u'&
W
E O& tum& 7NJF+I NJF 7NJ/7 M 6∈ E U& '( O&!F/s 8 Π & tu7& E ∩ Π = ∅ ó QSX`8WUY;qBVV;q£qUVX`CO8[;†[;q?X\ [ ìw£qCO8 ? ;,\3;q‘øsCÝBDUYED;,PZCO8 D ;,DUÊ;q8£,CÝBV;†øsCOBDUV;,@A;,8WU [ ìw£qCO8 ? ; \h;¢[CO83£
CO8 N ;,?hU9[3:#8XZBEPRQS;,8 ? ;qP`C N3N ; [ ìw£qCO8 ? ; h\ ;¢øsCOBDUV;¢[6Q÷?38; N#” BDUVX`;†[; Z L
pq
i
pq
i
pq
i
pq
i
D
,
,
2
pq
i
)
Ïä+*¥XL©
8NJ/7( qv69(u00/7(41 X/7NJjphNJwx/YNxONJ90'& w & /7G$NJ/7=0&X& ùŽ[email protected]@“; N BD: ? ? £,;õUVUV;„;,8 ? ;,P`C NN ;„V;„£ ” P`£q?P`;Tf; ” ?3£,CO? N N P`?Aø ” £,XZP`;[email protected]“;q8 U’ ?;„PRQS;,8 ? ;qP`C NN ;
ì[ £qCO8 ? ;,\3;V;qP`CO8 N PZ?VXZ;,?BDÊ[X`BD;,£qUVX`CO836<
w
"! }!f "s!f S & /7=0& 'N 7& 65G$NJ/7=0&X& ,KNJ+,O& (@& 'NJ/ D 1 X/Y&#0+,-9& E 1F& Z /YNJ c &
c
QCON V F ORT (E)) &$( 1FNJ/7/ & #0+ ,
2
D
QCON V F ORTD (E) = {M |∀(i,p,q) ∈ A Ripq (M ) ∩ E 6= ∅}
ùŽ;õUVUY;9;,8 ? ;qP`C N3N ; [ ì/£,CO8 ? ;,\h;øsCOBDUV; N ;,?3U&[CO8£9V;^£ ” P`£q?P`;qB ;q8UY;[email protected] N ŽP`XZ8: ” X`BV;Ê;q8øsCO8£õUYX`CÝ8[;9P ”
U ” X`PZP`;‘[? N COP©ÔhôÝCO8; ∆ = {M ∈ Z : ∀p ∈ D min p(N ) ≤ p(M ) ≤ max p(N )} L
2
/
N ∈E
N ∈E
¤u~gjt^xÃ|.v)gjdJn~| db)tŽv)gÊu/ehy
? 8[ 83;9£qCO8WUYX`;q8WUŽ’ ?;9[;,?X\“[XZBV;q£qUYXZCO8qÜWP ì/£,CO8 ; \hXZUV:‘øsCOBUY;‘V;qP`CO8 ;,UŽ:q’ ?X PZ;,8WUY;
LP ”l[ [ ”ì/£,CO8 ?D; \hXZUV0: LˆùŽ;qUDUY;:,’ ?X ?§” P`;q8£,;8;\D;ôÝ:,8”4:qB [ ” PZX`V; N#? ”  N CÝ?BàUYBDCOX`¡[3X`BVD;q£qUVX`CO8MLªû}?§8 ” ;;qU
PQ÷;q8V;[email protected]¤TP`;^[;†P ” #ôÝ?BV;X¥ L©ý¤;qDU [ ìw£,CÝ8 ? ; \h;V;,PZCO8 D = {x + 3y,x,2x + 3y} @ ” X` X`P‚8‚Q÷;qDU N#” 
[ ìw£qóCO8;¢? P`,; X`;q\38 ;¢øsCOB;,UÊ£VP`;,;qPZ CO8ì/P`;q£,‘CO8 @ ;,\[email protected];,;,9E;,[3DU‘X`BV;q?£q8 UVX`CO;,8? M L P`?E£[email protected] P`XZ’?30: L#íCO?BÊ£,;qP CO8 ;q8£,COBD;¢[:#83X`B
”K?
[ ?
N N
N
” ?§”
[6Q ” ?3UVBV;qEBV:,ôÝX`CO8E[3; Z <DX p,q,r VCO8WUÊUYBDCOX`&[3X`BV;q£qUVX`CO8Ž[; D ;qUʏDX r = αp + βq K” ? ;q£ α > 0 ;qU
Ü PZCOBVEPZ;,‘ý¡BV:,ôÝX`CO8Ê[:qP`[email protected]“X©UY:,;q N#” BÊPZ;, [BVCOX©UY;q‘[3;[3X`BV;q£qUVX`CO8 p,q,r ;õU N” V ” 8WU N#” B‘?38
β>0 ”
2
( E1 = QCON Vpq (E)
E
E2 = QCON Vpr (E1 )
E3 = QCON Vqr (E2 )
$ ( E4 = QCON Vpq (E3 )
p
q
)
Ïä+*Ê¥XLZ,ü
0'G F1F& & /7=0& 'N 7& 65G$NJ/7=0&X&_(@& 'NJ/
r
3x − 2y ) q = x ) r = y {p,q,r}
1 X/ & /#(@& w "u'& E 0=0&$G
p =
M
M
M
Rqr
2 (M )
Rpr
2 (M )
Rpq
1 (M )
U
M
M
M
)
N COXZ8WU
M
Ïä+*¥XL©
sS &$( % - p 7&$(R1 W 8 7 NJF+ D = {p,q,r}) p = x) q = y ) r = x + y Rqr
0 (M )
DCO8WU^8C§UY:,;q6<
V0pqr (M ) = {N
V1pqr (M ) = {N
V2pqr (M ) = {N
V3pqr (M ) = {N
V4pqr (M ) = {N
V5pqr (M ) = {N
/ p(N ) ≤ p(M )
/ p(N ) ≥ p(M )
/ q(N ) ≤ p(M )
/ p(N ) ≥ p(M )
/ p(N ) ≤ p(M )
/ q(N ) ≥ q(M )
Rpr
0 (M )
;õU
õ; U
õ; U
õ; U
õ; U
;qU
Rpq
3 (M )
q(N ) ≤ q(M )} = R0pq (M )
r(N ) ≤ r(M )} = R1pr (M )
r(N ) ≥ r(M )} = R3qr (M )
q(N ) ≥ q(M )} = R2pq (M )
r(N ) ≥ r(M )} = R3pr (M )
r(N ) ≤ r(M )} = R1qr (M )
Z XjCÝ88‚Q ”N#”  α > 0 ;õU β > 0 Ü#CO8 ” PZ;,Ê[:#8XZUVX`CO83EV?X ?ݔ 8 UV;,
ZZ XX α > 0 ;õ;õUU β < 0 Ü3Ü3COCO88 ” V = V
Z X α < 0 ;õU β > 0 Ü3CO8 ” V = V
û}8 ø αXZU<P`;q0 ìwβ£,CÝ<8 0; h\ ;,øs” COVBUYßDCO=8WUV P`;, ì/£,CO8 ; \h;,’ ?X :qBVX #;,8WU?8;
”
[ ?
[ ?
?
P`:,@A;,8WU X`BD;ª<
V? NN
”
pqr
i
pqr
i
pqr
i
p,−q,r
i
−p,q,r
i
p,q,−r
i
N ;qUVXZUY;£,CO83[XZUVX`CO8
D
M
)
Ïä+*ˆ¥XL©ý
65G$NJ/7=0&X& (@& 'NJ/
/
&
#
/
@
(
&
w
u
"
'
&
65G$NJ/7=0&X& KNJ+ f(@& 'NJ/ D D = {x + 3y,x,2x + 3y}
,
w 0'( tuX/ &$( # J(
r = r(M )
V4 (M )
V3 (M )
V5 (M )
M
q = q(M )
V2 (M )
V0 (M )
V1 (M )
Ïä+*¥XLZ )
sS &$( %+ c 0V 9NJ/#( V 7 NJF+
p = p(M )
pqr
øsCOBDUV;,@A;,8WU [ ìw£qCO8 ? ;,\3;
( & R(@& F'& w & /7R( 4I&$( ¥XL (∀i V
(M ) ∩ E 6= ∅) =⇒ M ∈ E
c
8LV? NN CݏV;à’ ?;
;qDU^øsCÝBDUY;[email protected]“;q8WU [ ìw£qCO8 ? ;,\3;BLjíÚCÝ?B9UVCO?3U
XZP6Ô ”
?P 8Hw 7 NJZ /#í ( +.;q0U-9[NJ/CO8, £¤⇒?38H’? ” [B ” 8 U ’ ?Xª8E;à£,CO8WUYXZ;,8WU ” ?3£,?8 N COX`8WU^[; E L 8L8; N M;,?3U 6∈NE”  ”K? COXZB
£ BÊ?87 Z 퀣qCO8WUYX`;q8WUEUYCO?JdDCÝ?BVÊ[;q?X\ V £,CÝ8V:q£,?3UVXZøsuL
∀iV
∩ E 6= ∅ ”
`
X
W
8
V
U
,
;
8
W
8
UAV? CݏVCO8’ ?; E ;,U [ ì/£,CO8 ? ; \h;;õU ? :,BDX"#;P ” V;,£qCO8[; N BDC N BVXZ:qUV:0L Z CÝXZU
Í
⇒ ;q”U‘DCOXZU ” ?8 NN
[X`dDCOX`8WU ”K? ;q£ E ;õU‘@ ” \[email protected]¤[email protected] N CO?3B‘PQ÷CÝBV[BD; N#” BDUVX`;qP‚V?3X §? ” 8WUj<
M 6∈ E
Π
R (M )
"! } !f "s!f / & /#(@& w "u'& E &$( p = x) q = y ) r = x + y 65G$NJ/7=0&X&4& 7NJF+I NJh( p,q,r ∈ D ) k ∈ {0,1}) M ∈ Z NJ/l
2
pqr
2i+k
2i+k
i
pq
i
DVX R (M ) ⊂ R (M ) CO? R (M ) ⊂ R (M )
Z ? N3N CODCO8E’ ?; Π = R (M ) 8‚Q÷;qDU N#” Ž?87 Z íL3öwP6;,\3XZDUV;[CO8£†?38;†[X`BD;,£õUYX`CÝ8 r UV;,PZP`;^’ ?;
L ó ;,ˆBV:qôOX`CÝ8 ;qU £qCO8WUYXZ;,88;q8WU&?8 N CÝX`8WU)[; £ ” B)VXZ8CO8 86Q÷;qDU N#” 
Π
@ ” =\hX`[email protected]¤[email protected] =L ZVX V = R 8;9V£,CO8WUVX`;,8WVU N#” Ž?38 N CÝX`8WUŽ[3; E ” P`COBD,ÜF£ E” ?V;^[;^P ” [:Π#8XZUVX`CO8“[;
P`;,†;q8V;[email protected]¤TPZ;,
;qU
£,CÝ8 UVX`;q88;,8WU ” ?DVX}?8 N COX`8WU†[; E £q;’?3Xª;,[email protected] N CODVX`T3P`; N#” B
≺ ì ìw£qCO8 ; \hX©UY:0VL0]9CO8£ V
? B [3X`£qUVX`COY8 L V (M ) Ü V (M ) Ü V (M ) £qCO8WUYX`;q88;q8 U9?8 N CÝX`8WUÊ[; E ;õU‘[CO8£
(p,r) [
,
£
Ý
C
8
V
U
”
M ∈E
2
í ” B ,; \h;,@ N P`;9[ ” 8ˆP ” #ôO?BD;X¥ L©ýP`; N COXZ8WU M ? :,BD"X #;9TX`;q8P ” ø[email protected]¤?PZ; X¥ L Ü N CO?BDU ” 8WU M 86QS;,DU
N#” ‘[ ” 8ÊPRQS;,8D;,@¤T3P`;ÝÜ3[CO8£¢£,;õU‘;,8D;,@¤TPZ;¢86Q÷;qDU N#”  [ ìw£qCO8 ? ; \h;¢øsCOBDuU L
8 ?§” ? [email protected] ” X`8WUY;q8 ” 8WU’ ?;¡P ” [ ìw£qCO8 ? ,; \3X©UY:¤øsCOBUY;;,DU¢?38;¤8CÝUVX`CO8„P`:,ôÝÌ,BV;[email protected]“;q8WU N P`?39øsCOBUY;
’ ?;P x” [ ìw£qCO8 ? ; \hX©UY:ÝÜ3;,8\; ;qU&VX6CO8 ” dDCO?3UY;^’?3;,P`’ ?;qEËWÔ N CÝUVËÌ,D;, ” P`CÝBVŽCÝ8 ” :,’ ?X ?ݔ P`;,83£,;;q8 UVBV;
P ” [ ì/£,CO8 ? ; \hXZUV:¢øsCÝBDUY;¢;õUÊP ” [ ìw£,CÝ8 ? ; \hXZUV0: L
Ripq (M ) ≺ Rjpq (M )
pq
i
pq
i
pqr
j
pqr
j+3
pqr
j+2
pq
i+2
pqr
j−2
pqr
j+1
pqr
j+1
pq
i
pq
j
pqr
j−1
pqr
j−1
pqr
j+3
pq
i+2
pq
j
üW¦
" $ ( ( f ~(©,O Ú! Žù [email protected]@A; N CO?3BP ” £qCO88;,\hXZUY: D;,£õUYX`CÝ8 ¥XL PRQSX`83V:,£ ” TXZP`XZUV: §? ” 83CO? N ;[email protected];qUVUVBV; [; BV;q8[BV;
N P`?DX`;q?BVÊ8CÝUVX`CO83E:,’ ?X ?§” PZ;,8WUY;q6<
"!}!f s !f W D &$( !X/ & /#(@& w "u'& 1F& 10+$&$G 9NJ/#( +.09NJ/7/Y& '&$( & E X/ & /#(@& w "u'&
/#( c G5h"u'& (@& 'NJ/F/Y&<10+:&$G -9NJ/ p ∈ D 0'NJ+:('&$( -+:NJ'( 2+$N [email protected]( 9NJ/#( ( X=J0/7O&$( (@NJ/7 c tuX=J0'& /7O&$( ,
E &$( 65G$NJ/7=0& X&(@& 'NJ/ '&$(xX`; G$NJ2'&$( (p,q) 7NJF+!ONJF q ∈ D E &$( 65G$NJ/7=0& X&(@& 'NJ/ D E &$( 65G$NJ/7=0&X&(@& 'NJ/ ONJT(!'&$(xG$NJ2'&$( (q,r) ∈ D < E &$( 65G$NJ/7=0& X& KNJ+,f(@& 'NJ/ D c w NJ/#( -+.09NJ/ ,
P
 ¥ [?3;* Z B
BV?38;qUDUYXM ]¦O¦J L Z COX©Uˆ[CO8£ q ;qU r [;,?F\[3X`BV;q£qUVX`CO8)[X ‚:qBV;q8 UV;, [; p ;qU M
⇒ ?8 COX`8WUˆ” ’ ”?X086QS;,DU
N
N#”  [ ” 8 E L 8 §? ” @ACO8WUYBD;,Bˆ’ ?6Q÷XZP0; h\ X`UY;‘?8ߒ ? ” [B ” 8WUŽ[3;^[XZBV;,£õUYXZCO8
” ?3UVCO?B [; M ’ ?Xf8;9£,CO8WUYXZ;,8WU ” ?£,?8 N COXZ8WU [3; M L Z X E ;qDU&£[email protected] N BVXZ [ ” 8Ž?8[;,@AX©ì
q,r
N P ” 8„[;øsBVCO8WUYXZÌ,BD; p = p(M ) £§Q÷;qDU£qP ” XZBuL Z XZ8CO8£,[email protected]@“; E ;,DU^X`8D:,£ ” TP`;CO8L ” XZU9’ ?;àP ”
[BVCÝXZUY; p = p(M ) £qCO8WUYXZ;,8WU†?8 N COX`8WU N [; E L [ ?3XZUVUV;^ß BD;,@ N P ” £q;,B q N#” B −q ;qU r N#” B
CO8 N ;q?3U‘V? NN COD;,B‘’?3; p = αq + βr ”K? ;,£ α > 0,β > 0 L
−r
L ùŽCÝ@“@A; N ∈ R (M )∩R (N ) CÝ8 ” R (M )∩E = ∅ CO?
Z ? NN CODCO8 q(N ) >L q(M
)
8 ” p = αq + βr ”K? ;q£ α > 0,β > 0 ;õU p(M ) = p(N ),q(N ) >
R (M[)CO∩8£E = ∅
ÚL ùŽ[email protected]@“; E ;,DU [ ìw£,CÝ8 ? ; h\ ;ߏD;,P`CÝ8 (p,r) CO82;q8³[:q[?XZU
q(M )
r(N ) < r(M )
’ ?; R (M ) ∩ E = ∅ CO? R (M ) ∩ E = ∅ L ùŽ[email protected]“@A; R (M ) ⊂ R (M ) ;qU
;õU
CO8 N ;,?hU‘UVCO?JDd CO?3BVEUYBDCO? ? ;qB i
R (M ) ⊂ R (M ) R (M ) = R (M ) ∪ R (M )
UY;qPj’?3; R (M ) ∩ E = ∅ L ? COX`B ôO?BV;¥X©L 2
,
2
pq
1
qr
2
pr
2
qr
i
pr
2
pr
3
qr
3
pq
1
pq
0
3
qr
0
pr
3
r = cste
r = cste
N
Rpq
3
Rpr
1
Rpq
2
Rpr
0
q = cste
Rpq
0
)
Ïä+*ª¥XL© R3pr
R¥ ⇒
Rqr
1
Rpr
2
Rpq
1
Rqr
0
M
Rqr
3
Rpr
3
p = cste
Rqr
2
q = cste
M
M
pq
0
p = cste
/ E"u9& / R (M ) ⊂ R (M )) R (M ) ⊂ R (M )) & R (M ) = R (M ) ∪
(M ) 'NJ+:(utum& p = αq + βr 0=0&$G α > 0,β > 0 ó ;£ ”  q(N ) < q(M ) ;,U‘VX`@AX`P ” X`BD;0L
X`Pˆø ” ?3U¡@ACO8WUYBD;,B¡’ ?; E ? :,BVX #;\P ” £ ” B ” £qUY:qBVXZ ” UVX`CO8³[3;\P ”LN BDC N CݏVXZUVX`CO8Â¥XL L÷ü L Z COX©U¤[CO83£
UYBDCOX`E[XZBV;q£qUYXZCO8E[; D Ü k ∈ {0,1} ;qU‘?8 N CÝX`8WU M ’ ?X ? :,BDX"#;
q,r,s
qr
0
pq
0
qr
2
pr
2
qr
3
pq
1
qrs
(∀i V2i+k
(M ) ∩ E 6= ∅)
?L3§ùŽ” CO[email protected]“? [email protected] ; COD;,B;,’DUŽ?3; X`8MD:,£ 6∈TEP`;‘[[ CO8383£  NP ” B [ XZDBVX`;[email protected] £qUYN XZP`CO; 8 [ Ü ìw£,£,;õCÝUV8 UV? ;^; [\hXZ;qUVBV: 8CÝXZÌ,8 BV;‘” 8(∀i; V;q?3U  (MõUYBD)∩E
=
9
;
[
8
”
”
”
N
N
”
”
∅)
E
p
L
O
C
Z
X
U
`
P
;
O
C
Z
X
8
ˆ
U
[
;
V

?
ˆ
B
P
[
D
B
O
C
Z
X
V
U
;
L8 N ;q?3UˆV? NN COD;,B N” B); \h;,@ N P`;
Z
N
”
{q,r,s}
N
E
p = p(M )
8
qrs
2i+k+1
’ ?;
ü
Ü[ 8&£q;^£ ” &P CO8 ”
#ôÝ?BV;X¥ L© ”
L
G$NJ/7+.h109G -9NJ/ ? COX`B
N ∈ V0qrs k=0
;qUŽ[CO8£
M ∈E
N#” B [ wì £,CÝ8 ? ; \hXZUV:V;qP`CO8
(p,s)
2
s = s(M )
M
r = r(M )
N
p = p(M )
)
"
Ïä+*‚¥XL© sS /#( c G5h"uj c
(@& 'NJ/ p & /7+5/Y&e 65G$NJ/7=0& c KNJ+ O&
q = q(M )
,
f s„.W §12.#
W D = {p,q,r} &$( X/& /#(@& w "u'& 1F& +:NJ'(10+$&$G 9NJ/#( O& #tu7& | det(p,q)| =
1 & E X/Y&)#0+,-9&x1F& Z 0'NJ+:( E &$( 65G$NJ/7=0&X& KNJ+, (@& 'NJ/ D ( n& (@& X'& w & /7 ( E &$( 6
G$NJ/7=0&X&Q(@& 'NJ/ D UY;qPª’ ?; | det(p,q)| = 1 ;qU E ?8 [ ìw£qCO8 ? ; h\ ;AV;qP`CO8 D Ü6CÝ8
PD? c Nw N CONJV/#;\( £[email protected]“[email protected]/ ;, Z N COBVXZ:qU £,:,D[3;,[email protected]@“{p,q,r}
;q8 U’ ?;\PQ÷CÝ8 ” ?8 N COX`8WU M UY;,P ’ ?; (∀i V (M ) ∩ E 6= ∅)
;õU (∀i V (M ) ∩ E = ∅) L 8 V? N3N COD; N CO?B¡DX`@ N P`X #;,B k = 0 ;qU r = αp + βq ”K? ;q£
L Z COXZU A ∈ V (M ) ∩ E Ü B ∈ V (M ) ∩ E Ü C ∈ V (M ) ∩ E L
α > 0,β > 0
Z COX©U N UV;,P.’ ?; p(N ) = max(p(A),p(C)) ;õU q(M ) = q(N ) L ó ; N COXZ8 U N ;,U^[ ” 8 Z £ ” B
L
| det(p,q)| = 1
ìw£,CÝ8 ? ; h\ XZUV: N ∈ E L
ZZ XX p(A) ≥ p(C) ” P`COB BD N#” B ì/(p,r)
p(A) ≤ p(C) N” (p,q) £,CO8 ? ;,\hXZUY: N ∈ E L
] ” 8EUYCO?ÊPZ;,E£ ” ‘XZP‚Ô ” £,CÝ8 UVB ” [3X`£qUVX`CO8 ”K? ;q£ V (M ) ∩ E = ∅ ? COX`B #ôÝ?BV;¥XL ¥Ý¦ 2
Z X)CO8 ”N P`?3[;UYBDCOX`[XZBV;,£õUYXZCO8,ÜjP ”N BVC N COVX©UYXZCO8 N BV:,£q:,[;q8WUY;ß;õU¤P ” £ ” B ” £õUY:,BDX` ” UYXZCO8³¥FL SL ü
@ACO8WUYBD;,8WU†’ ?; N CO?B ”K? COX`BPQ÷:q’ ?X §? ” PZ;,8£q; [ ìw£qCO8 ? ; h\ X©UY:¤øsCOBUY; [ ì/£,CO8 ? ;,h\ XZUY:¡XZPڏD? U¢’ ?; N CO?B
UVCO?3UŽUVBVX N P`;õUŽ[;9[X`BD;,£õUYX`CÝ8ŽCO8;,8 ” [;q?X\ [CO8WUEP`;9[:qUV;,[email protected]“XZ8 ” 8WU&;qDU† hL í ” BŽ; h\ ;[email protected] N PZ;P`;qŽ8C§UYX`CÝ8
DCO8WU}:,’ ?X Ý? ” P`;,8WUV;, N CO?Bª?8¡;q8V;[email protected]¤TPZ;E[;E[X`BD;,£õUYX`CÝ8=[;EP ” ø[email protected]“; D = {(1,0),(0,1),(a,1),(1,b)} L
í ” Bà£,CÝ8 UVBV;ßP ”N BDC N COVX©UYX`CÝ8 N BD:,£q:,[;q8 UV;\8;ߏqQ÷:õUY;,83[ N#” ^
üL[X`BD;,£õUYX`CÝ86<.X`Pˆ; h\ X`UY;“[;,¤;q8hì
D;,@¤TPZ;, D ü†[XZBV;q£qUYXZCO8ª’ ?X3£,CO8WUYXZ;,883;,8WUˆ[;,?X¡\ [X`BD;,£qUVX`CO83=[CO8WU)PZ;&[:qUV;,[email protected]“XZ8 ” 8WU};,U 1 ;qU N CO?B
PZ;,V’ ?;qP`ÊP ”[ ì/£,CO8 ? ;,h\ XZUY:†øsCOBUY;¢86QS;,U N” Ê:,’ ?X §? ” PZ;,8WUY;¢P ”[ ì/£,CO8 ? ; h\ XZUV:0L ? COX`B #ôÝ?BV;¥XL ¥h "! } !f "s!f ,
2
pqr
2i+k+1
pqr
2i+k
pqr
0
pqr
2
pqr
5
pqr
4
2
ü¥
$ ( ( r = r(M )
C
q = q(M )
r = r(M )
N
q = q(M )
M
C
N
M
B
B
A
A
p = p(M )
p(A) ≥ p(C)
)
Ïä+*f¥XL ¥§¦
8NJX+.tu7NJ 06O-65NJ/ NJ #KNJX+:(
/
p = p(M )
p(A) ≤ p(C)
N ∈E
z b)u~t ehxmx|)un~n~|)t ehxml9gjx dbˆt v)gÊgjx b}ke3x
ó ” [:#83XZUYXZCO8¢[;, N COX`8WUY. ” X`PZP ” 8WUY‚[ ” 8P`;}£ ” Ú[3; N PZ?.[;}UYBDCOX`.[XZBV;,£õUYXZCO8j;,U N B ” UVX`’ ?;,@A;,8WU
P ” @[email protected]“;†’ ?;¢[ ” 8ÊP`;†£ ” ‘[3;[3;,?X\[XZBV;q£qUYXZCO86<
=Ú"s!f /7NJ/7 M &$( F/7NJ/7 (u00/7 1F& E (@& 'NJ/ '&$(x10+:&$G 9NJ/#(41F& D ( M
/ &$( #J(41h0/#(e & /7=0& 'N 7& 65G$NJ/7=0&X& ,KNJ+,O&*(@& 'NJ/ 1F& XZP`P }8WUY^[‚Q÷?8 ì/£,CO8 ,; \h;øsCÝBDU
Õ B DÝ£,x; ßP ”N BDC N CݏVXZUVX`CO8HX¥ L L CÝ8 N ;q?3U¢[X`BD;à’ ?;P`;q DN COX`8WEUY9\ ” {M
”
[ ?
VCO8WU9P`;, N COXZ8 UV M UV;,PZ‘’ ?6QSX`P‚; \hX`UY;¢?87 Z í Π ” ?3UYCO?3BE[; M UY;qP`E’ ?; Π ∩ E = {M } L E
8 ” ?38 ” 8 ” P`CÝôO?;†[;¢P ”¤N BVC N COVX©UYXZCO8X¥ LSü L N CO?BEPZ;, [ ìw£qCO8 ? ; \h;q9øsCOBUY6<
"!}!f s !f # k NJF 65G$NJ/7=0&X& ,KNJ+, #8X &$( & /7=0& 'N 7& 65G$NJ/7=0&X& ,KNJ+, &Q1F& (@&$( 7NJ/7(
(u00/7 (
ó ” [3:,@ACO 8DUVB ” UVX`CO8„;,DUP ” @ ,@A;à’ ?;¤£q;,PZP`;à[3;¡P ”“N BVC N COVX©UYXZCO8HX¥ LSü L <UVCO?PZ;,97 Z í ” ?3UYCÝ?B^[;
£ Ë ” ’ ?; N COXZ8 U M [;PRQS;,83V;,@àTP`;†£qCO8WUYX`;q88;q8 U‘?8 N COX`8WUʏ ” X`P`P ” 8WKU L
%ß12 465. & & 2+:N 2+ c c &$( ;,M0T($(@& 7NJX+Q'&$( 65G$NJ/7=0&X&$( 8CO8 ,KNJ+ ( , F/l& /#(@& w "u'&
65G$NJ/7=0&X& / &$( N#”  ,KNJ+:G c w & /7^ & /7=0& 'N 7& 65G$NJ/7=0&X&l1F&$(7NJ/7 (ltuX4/Y& (@NJ/7 #J( 1h0/#(
& /7=0& 'N 7& 65G$NJ/7=0&X&41F&$(40F+$&$( 7 NJ/7( q2NJ+ -UV0F+$&
8 N ;q?3U ” ?DVX.[:8X`BEP ” £,CO8D:,£q?3UYX ? X©UY:BLíCO?BÊ£,;qP ”
CO8£,CÝ8VXZ[Ì,BD;
q; U R (M ) ∩ E} 6= {M }
]^;q?X\’ ? ” [BD? N PZ;qUY (A,i,p,q) ;qU (B,j,p ,q ) DCO8WU^£qCO8D:,£,?hUYXZøsÊDXCÝ8;qDU‘[ ” 8‘PRQS?8[;q9[;q?X\
£ ” 6<
;õU^PZ;¢’? ” [B ” 8 U [3CO8WUP`;qÊTCOBD[‘£,CO8WUYXZ;,883;,8WU ;qU 83;à£,CÝ8 UVX`;q8 U
(i,p,q)
[;, N CO=X`8WUV(j,p
ß[; ,qE) D;,?PZ;,@A;,8WU\V?3B\R£,;,TfCOBV[3 £[email protected]“@A;„[ ” 8\P`;q\ü AN BV;[email protected]“XZB;,BD£ ” \[;HP ”
[:8XZUVX`CO8¥XLSü L ;qU {p ,q } CO8WU,; \ ” £qUY;[email protected]“;q8WU†?8H:,PZ:,@A;,8WU9£,[email protected]@¤?8 r Ü0CO8 ” r(A) =
ó ;,;qÜ 8V;[email protected]¤;qTDU“P`;qX` 8{p,q}
£qP`?¡;q8WUYX`ÌqBV;[email protected]“;q8 U“[ ” 83“?8 [;q[;[email protected]“X+ì N P ” 8¤[;øsBVCÝ8 UVX`ÌqBV; r = r(A) ;qU
r(B) E
S 0 = {(M,i,p,q) : (i,p,q) ∈ A
0
0
pq
i
0
0
0
0
pq
i
ü
Ïä+*¥XL ¥h
S & /#(@& w u" '& E = {(0,0),(2,0),(1,4)}
$
&
(
65G$NJ/7=0&X&!(@& 'NJ/ D = {2x − 3y,4x +
w
0
'
(
/
$
&
(
#
J
(
5
6
$
G
J
N
7
/
0
=
&
X
&
K
J
N
,
+
2
@
(
&
'
J
N
/
3y,3x−2y,−x+2y}
D ) 7NJF+, 0/7 det(2x−3y,−x+2y) =
)
3,
1
q; U R (B) VCÝ8 U¢XZ8£,PZ?[ ” 8¢PRQ ” ?3UYBD;¤[;[email protected]“X+ì N P ” 8L[;¤øsBDCO8WUYX`ÌqBV; r = r(A) L6] ” 8
£q;¢£ ” ÊCO8[X©Uʒ?3; [AB] ;,U^?8 N XZ;,[[; E L
? COXZB #ôO?BD;à¥FL ¥ 8 ” @à?8X
?8; DUVBV?£õUY?BD; [; ôOB ”ÝN Ë;ÝÜ&£[email protected]@“;H[ ” 8\P`;H£ ” [; [;,?X\ [X`BD;,£qUVX`CO83ß£,;
S
ôOB ”ÝN Ë;9;,UE?8\£qÔ£qP`; ? COXZB #ôO?3BV;¢¥XL ¥§ü L ó ” [3:,@ACO8DUVB ” UVX`CO8“;,UE[email protected]“XZP ” XZBV;ÝܧCO8\[COX©UŽ[3:#8XZBE¥ |D|
;q8V;[email protected]¤TP`;q S = {M : (M,i,p,q) ∈ S } L
p0 q 0
i
Ripq (A)
0
)
pq
i
Ïä+*)¥XL ¥O¥
0
/ & /#(@& w "u'& 65G$NJ/7=0&X& tuF/ &$( #J( & /7=0& 'N 7& 65G$NJ/7=0&X&_1 F/ & /#(@& w "u'&
wx/7wxFw 1F& 7NJ/7(
üOü
$ ( ( q = q(B)
A
B
p = p(A)
88 6 5 8;678$/28 /$5 8 8 $1 6
57/
8& 8 6 5 8;678$/2! 8 /$5 8 )( 8 $1 6
$/ 1 8 8 1$6 / 5 8' 5( E
(A,i,p,q),(B,j,p0 ,q 0 ) ∈ S 0
E
(A,i,p,q)CONS(B,i,p,q)
p = p(A) = p(B)
q = q(A)
B
(A,i,p,q)CONS(B,j,p,r)
A
q 0 = q 0 (B)
)
Ïä+*‚¥XL ¥
sS&$(R1F& J p 7&$(R1F&xG$NJ/#( c G [email protected] c 1F&$((u00/7 ( KNJ+,( ,
S1qr
S2qs
S0pq
S2ps
S2pq
S0ps
)
p=x
s =x+y
r =x−y
Ïä+*‚¥XL ¥§ü
S0qs
S3qr
s8NJ/7 (e(u00/7(41 F/ 65G$NJ/7=0&X& KNJ+,
,
q=y
)
ü
Ïä+*f¥XL ¥
/ 65G$NJ/7=0&X& KNJ+,f(@& 'NJ/
,
{x,y,x − y} üWý
( $ ( « ɺ¹{¾¿´f¶Á°
¶°€¼
¬‹­ºÀ鶴6¬‹­º¼ À¬‹­±®¯°³²µ° ¼ ¸º´j¼\ÀÂÁ Â
] ” 83E£,;†£ Ë ”ÝN XZUYBD;CO8 ?§” :qUV?[XZ;,B&PZ;,ŽøsCO83£qUYXZCO8 £,CO8 ? ;,\h;,‘[; [ ” 83 L ó ;q?BEË[email protected]ôO?;q
[ ?£qCO8WUYXZ8?‚Ü=P`;qøsCO83£qUYXZCO8£,CO8 ? ;,\h;,A[; R [ ” 8 R dDCÝ?;,8WU¡?8ÂZBD_OP`;AUYBDÌ,R¤[email protected] N COBDU ” 8WUà;q8Â@ ” ì
UVË:,@ ” UYXZ’ ?;,qܪ;,8 N#” BUYXZ£,?PZX`;,B;,8 C N UVX`@AX` ” UYX`CÝ8oLÚùŽ[email protected]“@A; N CO?B¤PZ;,¡;q8V;[email protected]¤TP`;qCO8 ” N P`?DX`;,?3BV
N COVDX`TXZP`X©UY:, N CO?BAP ” [: #8XZUVX`CO8m[;„P ” £qCO8 ? ;,\3X©UY: $¤[XZV£qBVÌqUV;%( L 8 ?ݔ £ ËCOX`DX`B£,;qP`PZ;’ ?X‘BV;qì
D;,@¤TPZ;†P`; N PZ? ” P ” £,CO8 ? ; \hXZUV:£qCO8WUYX`8 ?;BLfùŽ;,BU ” XZ8;, N BD;,? ? ;,^[3;£q;¢£ Ë ”ÝN X©UYBV; N ;,? ? ;,8WU N#” B ”6B UYBD;
£qP ” DVX`’ ?;qÚ@ ” XZj;,8ø ” XZU.;,PZP`;qDCO8WUª8:q£,;qV ” X`BV;q=DX ? ;,?3U K” ? COXZB?38; 83CÝUYXZCO8XZ8WUYBVXZ8VÌq’ ?; [3;ˆøsCO83£qUYXZCO8
£qCO8 ? ; \h;[X`D£,BVÌõUYB; L
ó ;,BV:qV?P©U ” UV¡[;£,;£ Ë ”ÝN XZUYBD;\8CO?D;,B ? XZBVCO8WUA[ ” 8PZ; N BVC£ Ë ” XZ8£ Ë ”ÝN X©UYBV;D?B¡P`;q N COXZ8 UV
@A:,[X ” 8M L
n
n
_ R _ U b)tŽv)gÊu/ehy
" Y/ & KNJ/YG -9NJ/ f 1F& D ⊂ Z 1h0/#( R &$( xc F/Y& KNJ/YG 9NJ/ G$NJ/7=0&X&s( & (@& F'& w & /7 ( 7NJF+e NJh( A , . . . ,A ∈ D & ONJT( λ , . . . ,λ + & ( [email protected]( - :(I1F&(@NJwxw & 1 NJ/ \ ª" !f ,
1
n+1
X
1
n+1
λi A i ∈ D
í ” BÊ; \h;,@ N P`;†P ” VB ;qDUYBDX`£õUYX`CÝ8
"! }!f "s!f W f
=⇒
f
1
k
n+1
X
λi A i
,
n
!
≤
n+1
X
λi f (Ai ).
Z [6QS?8;¢øsCO8£õUYX`CÝ8\£,CO8 ? ;,\h;à[; R ? ;,BD R ;,DU‘£qCO8 ? ; \h;BL
&$( F/Y& KNJ/YG -9NJ/ G$NJ/7=0&X&R1 c - /79&*( X+IF/ & /#(@& w "u'& D ⊂ Z 1h0/#(
1 &$( 7NJ/7(R1F& Z 0'NJ+:(
∈D F
i=1
R & A ,...A
,
n
k
X
i=1
i=1
n
λi A i ∈ D
n
,
n
n
=⇒
f
k
X
λi A i
!
≤
k
X
λi f (Ai ).
Žù ;õUVUY; N BVC N CODXZUYXZCO8^dD?DUVX"#;P ” [3:#8X©UYXZCO8„[;q^øsCO8£õUYXZCO8^£qCO8 ? ;,\3;quL.í ” B£,CÝ8 UVBV;£,;qUDUY; N BVC N BDX`:õUY:
86QS;,U N”  ? B ” X`; N CÝ?BPZ;,†øsCO8£õUYXZCO8’?3X ? :,BDX"#;q8 Uà[email protected] N PZ;,@A;,8WU f (λ x + λ x ) ≤ λ f (x ) +
< Bj; \h;,@ N PZ;}VXOCO8†£qCO8DX`[ÌqBV;)P ” øsCÝ8£qUVX`CO8†[:#8XZ;}V?B E = {(−1,−1),(1,0),(0,1),(0,0)}
λ f (x ) N”
ü
i=1
i=1
i=1
1 1
2
2
2 2
1
1
ü
( ;qU
DX
N#” B
øsCO8£qfUVX`(0,0)
CO838; =N#” 1E[ ” 83f9(x,y)
PZ;†[X`=D£,BV0;õU
(x,y) 6= (0,0)
Lfû}8£,COBD;?38;øsCOXZEP ”
#
[:8XZUVX`CO8[?„£qCO8WUYXZ8?8;
c
ó ” [:[email protected]“CÝ8DUVB ” UVX`CO8“[;^£q;qUVUV; N BDC N CݏVXZUVX`CO8A;,U&T ” V:,;^V?B £,;,PZP`;‘[?ßUYË3:,COBDÌ,@A;9[;
ùP ” Bw ” UVNJË/#:,( C-+.[0COB9NJÔ / , ? CÝX`B ;,B ݦ3Ü.O L© L XL ýJ 8 ?§” [:[email protected]“CÝ8 UVBV;qB^P ”¡N BDC N COVX©UYX`CÝ8 N#” B‘BD:,£q?BVBD;,8£q;¤V?B
L0íÚCÝ?B k ≤ n + 1 £§Q÷;qDU‘P ” [:#83XZUYXZCO8oL Z ? NN COVCÝ8Ê’?3;P ”¤N BVC N BVXZ:qUY:†DCOXZU ? B ” X`;9UYCO?3U k < k ;qU
k’?3;
Z kCÝXZ>U An +,A1, . . . ,A ∈ D [;q N CÝX`8WUY)[; D ;qU λ , . . . ,λ [;,&BD:,;qP`)UYCO?ˆ8CO88?3P`ˆ[;^[email protected]“@A;
;qU = P λ x ÜfCÝ8 ?§” @ACO8WUYBV;qB9’ ?; M ;,U?8„T ” BDÔ£q;,8WUYBD;¤[; k − 1 N COXZ8WUY N#” [email protected]`;q A
1[;¢[;,M
X? \ø J” CO8Ê[X ‚:qBV;q8 UV;,ML sP`;[email protected]“@A;^[?£,;qBDø ì ? COP ” 8WU Ü ? CÝX`B #ôO?3BV; L© L
0
1
2
i
1
k
k
i i
B
i
C
M
D
)
A
Ïä+* ©L
S& 7NJ/7 &$( 41h0/#( c
w 0'(Q&$( ^0h($( I1h0/#( conv(A,B,C)
& *1h0/#( conv(A,B,D)) M& QONJF &[email protected] conv(A,B,C,D)
+.1FNJ/7/ & "$0+,phG$& /7) -+,tu7& 1F& M 1h0/#( (A,B,C,D)
&$( F/Y&
G$NJw "u/20'(@NJ/ [email protected]( =0&<1F&$(<[email protected]+.1FNJ/7/ &$(1F& M 1h0/#( (A,B,C) & 1F&G$& '&$(n1F& M 1h0/#( (A,B,D)
8 ”
[CO83£P ” ø ” @AX`P`PZ;
[ ; N CÝX`8WUYE[;
;,DU ” A8;[email protected]“;q8WU9[3: N ;,8hì
[ ” W8 UY;BL8\k ;,≥8\n[:q+[?2XZU&PRQS; \hX`UY;,83£,;[;^B (A” UVX`)CO883;,P` α , . . . ,α UY;qP`Z ’?3⊂; QP α = 0 ;õU P α A = 0 L
Z COXZU i ;õU i UY;,PZE’ ?;
n
i 1≤i≤k
1
1
†8
”
k
n
i
i
i
2
λi
λi 1
= min −
τ1 = −
,
αi <0
α i1
αi
τ2 < 0 < τ 1
L0íÚCO?3BÊUVCO?3U τ CO8
λi 2
λi
= max −
τ2 = −
.
αi >0
α i2
αi
”
M=
k
X
(λi + τ αi )Ai
i=1
@ ” ZX †;,8 N VB ;q8 ” 8WU τ ∈ [τ ,τ ] CO8 ” λ + τ α ≥ 0 N CO?3B¢UVCO?3U i ;qU[; N P`?
[CO8£¢[6Q ”ÝN BDÌ,‘PRQSË Ô N ÝC UYË3Ì,V;¢[;¢BD:,£,?3BVBV;q8£,;CO8 ” <
λ + τα = 0
1
i2
2
i
i
λ i1 + τ α i1 = 0
i2
f (M ) ≤
X
f (M ) ≤
i6=i2
L© (λi + τ1 αi )f (Ai )
i6=i1
X
(λi + τ2 αi )f (Ai )
;qU
L¥ ; ( û}8ø ” X` ” 8WU τ1
ü
A −τ
2
τ1 −τ2
CO8COT3UVX`;,8WU
X
f (M ) ≤
λi f (Ai ) +
i6=i1 ,i2
X
=
τ1 (λi1 + τ2 αi1 )
τ2 (λi2 + τ1 αi2 )
f (Ai1 ) −
f (Ai2 )
τ1 − τ 2
τ1 − τ 2
λi f (Ai ) + λi1 f (Ai1 ) + λi2 f (Ai2 )
ùŽ;¢’ ?Xj;,U‘PRQSX`8:qô ” P`X©UY: ? CO?PZ?;0L
i6=i1 ,i2
2
W f &$( X/Y& KNJ/YG -9NJ/ 1F& D ⊂ Z 0'NJ+:(* c 2V0+57`7&e1F& f &$( #0+,9& E
c
1F& Z × R 1 - /79& #0+ ,
E = {(M,y) : M ∈ D & y ≥ f (M )}.
"! }!f "s!f W NJ f X/Y& KNJ/YG 9NJ/ 1F& D ⊂ Z =0& +:( R 0'NJ+:(x'&$(< 2+:N [email protected]( -9NJ/#(!( X6
=J0/7O&$(*(@NJ/7 c tuX=J0'& /7O&$(
c KNJ/YG -9NJ/ f &$( F/Y& KNJ/YG 9NJ/ G$NJ/7=0& X& 2V0+.7`7& E &$( nG$NJ/7=0&X&RG &$( *10+:& E = conv (E) ∩ (D × R) < 7& '( O&nF/Y& KNJ/YG 9NJ/sG$NJ/7=0& X& g 1F& conv(D) ⊂ R =0& +:( R & '&tum& f &$( e+:&$( -+,9G 9NJ/
1F& g * D P c w NJ/#( +.0-9NJ/ ,
L 8;,U&[ ” 8
Ü [CO8£9CO8 ” XZUˆ’ ?;
(1) ⇒ (2);qDU9L T Z ” COBDX©ÔU £q(M,y)
∈
conv
(E)
∩
(D
×
R)
R
;,8WUYBD;à[; n + 2 N COXZ8WUYÊ[; E Ü0X`Pj;,h\ X`DUV;[3CO8£[;, N CÝX`8WUY A ∈ D ;qU9[;,
(M,y)
\ ª" !f "
,
n
n
,
,
n
,
Rn+1
n
,
n+1
Rn+1
£qCh; A£,X`;q8WUY N ÝC VXZUVXZøs
λi
[;¢[email protected]@A; 1 UY;,PZE’ ?;ª<
M=
n+2
X
λi A i ,
i=1
i
y≥
n+2
X
λi f (Ai )
i=1
@ ” ZX &[6Q ”ÝN BVÌqÊP ”¤N DB C N CݏVXZUVX`CO8 N BV:,£q:,[;q8WUY; ”§NN P`XZ’ ?:,; ”K? ;q£
f (M ) ≤
;õUÊ[CO8£
L 8
L
N BV;q8[
(M,y) ∈ E
(2) ⇒ (3)
(3) ⇒ (1)
ùŽP ” XZBuL
n+2
X
k =n+2
CO8 ” X©UE’ ?;
λi f (Ai ) ≤ y
i=1
g(x) = min{y : (x,y) ∈ convE}
L
2
Žù ;õUVUY; N BVC N CODXZUVX`[email protected]“CÝ8 UVBV;[CO8£’?3;¤PRQS:qUV?[;à[‚Q÷?83;àøsCO8£õUYX`CÝ89£qCO8 ? ;,\3;[XZV£qBVÌqUV;àD;¤B ” @AÌ,83;R
PQ÷:õUY?[;†[‚Q÷?83;øsCO83£qUYXZCO8\[:#8X`;¢D?B R @ ” X`Ž£,;¢86QS;,DU N#” &øsCÝBV£,:[email protected]“;q8WU9?38;¢TCO838;¢X`[:q;0L
n
ݦ
( #
b)tŽv)gÊu/ehy›xWehk3u~dfehg
Y/ & KNJ/YG -9NJ/ 1F& D ⊂ Z 1h0/#( R &$( F/Y& KNJ/YG -9NJ/ UYBVXZ£qUVc;,@A;,8WU G$NJ/7=0&X&
( !& (@& F'& w & /7Q( 7NJF+ONJX A , . . . ,A ∈ D 1F& Z & eONJh( λ , . . . ,λ + & (%[email protected]( :(_1F&
(@NJwxw & 1 NJ/l
=Ú"s!f ,
1
M=
,
n
n+1
X
2
n+1
1
λi Ai 6∈ {A1 . . . An+1 } =⇒ f (M ) <
n+1
X
,
n
λi f (Ai ).
Z X f ;,U&?8;‘øsCO8£õUYX`CÝ8“UYBVXZ£qUV;,@A;,8WU £,CO8 ? ;,\h; ” P`COBD)CO8 N ;,?3UŽBV;õø ” X`BD;‘P ”N BV;,? ? ;[;9P ”¢N BVC N CODXZUVX`CO8
ZL 0L ¥Ê;,8¤BD;,@ N P ”,” 8WU=PZ;,X`8:qô ” PZXZUY:q L© ;qU L ¥ N#” B[;,ÚX`8:qô ” P`X©UY:qUYBDX`£qUV;,ML 8àCÝT3UYXZ;,8WU=[3CO8£ª<
"!}!f s !f W f &$( <X/Y& KNJ/YG 9NJ/ ( +,9G & w & /7eG$NJ/7=0&X& 1F& D ⊂ Z 1h0/#( R & n( cc
c
A , . . . A ∈ D (@NJ/7I1F&$( 7NJ/7 (_1F& Z 7NJ/21 + ( #0+ λ , . . . ,λ + & ( [email protected]( - :(_1F& (@NJwxw & 1
0'NJ+$(
i=1
i=1
,
1
n
n
k
M=
k
X
1
,
n
λi Ai 6∈ {A1 . . . Ak } =⇒ f (M ) <
k
X
λi f (Ai ).
†8 N CO?BDB ” X©U ” ?VDX‚;,D ” ÔO;qBÊ[;^UYBDCO? ? ;,B&P ”!? ;qBVDX`CO8 $¤DUVBVXZ£qUY; ([3;P ”N VB C N CODXZUYXZCO8 ZL 0L÷üÜ@ ” `X ˆP ”
øsCO8£qUVX`CO8 g [; ? B ” qUVBV; ”¡N BVXZCOBVX‚?8;¢X`8WUY;qB N COP ” UYX`CÝ8ß[;¢[;,ôÝBV:¥¤[; f L
i=1
4 i=1
M v4
Z X f ;qDU?8;¡øsCO83£qUYXZCO8H[:#8XZ;¡V?B¢?8
” P`COBD
[:qVX`ôÝ8;¢P ” øsCO8£õUYX`CÝ8\[:#83X`;¢V?3B
D f
2
X`8WUV;,B ?§” PZP`;
[;
N#” B
Z a
[a,b]
[a + 1,b − 1]
;qU b N ;,? ? ;,8WU qUVBV;X`8 8X` So KNJ/YG -9NJ/ f : [a,b] → R &$( vG$NJ/7=0&X& ( Y& 7(@& F'& w & /77( D f (i) ≥
0 7NJF+I NJF i ∈ [a + 1,b − 1] So KNJ/YG -9NJ/ f : [a,b] → R &$( v( + 9G O& w & /7fG$NJ/7=0&X&( & (@& F'& w & /7o( D f (i) > 0 7NJX+
NJF i ∈ [a + 1,b − 1] ;õU9[3CO8£ f 86Q÷;qDU N#” Ê£,CÝ8 ? ; h\ ;0L
P c w Z ?NJ/#N(N -CÝ+.V0CO98NJ9/ ’, ?Z ; X D f (x) < 0 CON 8 CO?” BÊfUY(x)
<
CO?hU x L Z XCO8 a ≤ i < j ≤ b CO8 N ;,?3U‘ø ” X`BD;¢P`;,‘£ ” P`£,?3P`
D
f
(x)
≥
0
V?X Ý? ” 8 UV6<
D 2 f (i) = f (i + 1) − 2f (i) + f (i − 1)
"!}!f s !f " ,
2
,
2
f (x−1)+f (x+1)
2
2
2
f (i + 2) − f (i + 1) = f (i + 1) − f (i) + D 2 f (i + 1)
0
0
0
f (i + 1) − f (i ) = f (i + 1) − f (i) +
i
X
D 2 f (k)
k=i+1
j−1
f (j) − f (i) =
X
(f (i0 + 1) − f (i0 ))
i0 =i
= (j − i)(f (i + 1) − f (i)) +
j−1
X
(j − k)D 2 f (k)
k=i+1
f (j) − f (i)
= f (i + 1) − f (i) +
j−i
j−1
X
k=i+1
j −k 2
D f (k)
j−i
]^;¢@,@A;†VX
a≤j<i≤b
V h
CO8 ”
i
X
k−j 2
f (i) − f (j)
= f (i + 1) − f (i) −
D f (k)
i−j
i−j
k=j+1
Z CÝXZU x ∈ [x ,x ] Ü#£,[email protected]@“;
1
2
D 2 f (i) ≥ 0
;qUÊ[6Q ”ÝN BDÌ,‘P`;q&ø[email protected]¤?3P`;, N VB :,£q:,[;q8WUY;,9CO8 ” <
f (x) − f (x1 )
f (x2 ) − f (x)
≥ f (x + 1) − f (x) ≥
x2 − x
x − x1
[CÝ8£
(x − x1 )(f (x2 ) − f (x)) + (x2 − x)(f (x1 ) − f (x)) ≥ 0
[6QSCGF†<
x − x1
x2 − x
f (x1 ) +
f (x2 )
x2 − x 1
x2 − x 1
;õUÊ[CO8£ f ;,DU‘£qCO8 ? ; \h;BL
íÚCÝ?BŽP ” UYBVXZ£qUV;^£,CÝ8 ? ; \hXZUV:X`P#D? UE[;9BV;,@ N P ” £,;qBˆPZ;, X`8:qô ” P`X©UY:q N BV:q£,:,[3;,8WUY;q N” B [3;,&XZ8:,ô ” ì
PZXZUY:q&DUVBVXZ£qUY;quL
2
f (x) ≤
4 R ovUu R ^ ÿ983; øsCO8£õUYXZCO8UYBDX`£qUV;,@A;,8WUª£qCO8 ? ;,3\ ;ŽV?B R ” UVUV;,XZ8 UDCO8¤@AX`[email protected]¤[email protected];q8¤; \ ” £õUY;[email protected]“;q8 U=?8 N COX`8WU,Ü
CO8 N ;q?3U;qV ” ÔÝ;,B†[;£ Ë;,BD£ Ë;,B†?38; ? ;,BVDX`CO8„[X`D£,BVÌõUY;¤[3;à£,;õUVUV; N BDC N BVXZ:qUV:0L Z ? NN CODCO8[CÝ8£à’ ?;
VCÝXZUÊ?8;†øsCO83£qUYXZCO8\DUVBVXZ£qUY;[email protected];,8WU‘£,CO8 ? ;,h\ ;à[6QS?8UYCÝU ” P`;[email protected]“;q8 U&£,CO8 ? ;,\h; D ⊂ Z ? ;,BV R L
f ] 839PZ;†£ 
”
” n = 1 Ü3£,[email protected]@“; f (i + 1) − f (i) ;qDU‘?8;¢V?3XZUY;¢UYBVXZ£qUV;,@A;,8WU‘£,BVCÝX`V ” 8WUY;¢CO8 ? COX©U
’ ?; f ” UVUV;,XZ8 Uʏ[email protected]“XZ8X`@¤[email protected] ;q8 ” ? N P`?E[;q?X\ N COX`8WUYE’ ?XjDCO8WU ” P`COBDŽøsCOBV£q:,@A;,8WU‘£,CO83V:,£q?3UYX©øsuL
û}8é[3X`@A;,8DX`CO8 2 £§Q÷;qDUT; ” ?£qCO? N N P`?†£[email protected] N P`XZ’?3:¤£ ” B¢?8 N COX`8WU¢[; Z 8‚Q ”\N#”  ? B ” X`@A;,8WU
[; N COX`8WUY $ ? COXZVXZ8 (’ ?XV;qB ” XZ;,8WU^£qCO8D;,B ? :, N#” B9PZ;,ÊUYB ” 8Dø[email protected] ” UYX`CÝ8EP`XZ8: ” XZBV;qE[; Z COBÊPZ;,
UVB ” 8ø[email protected] ” UVX`CO8jPZX`8: ” X`BD;,.£,CÝ8V;qB ? ;q8WU)P ” £qCO8 ? ;,\3X©UY:&[3;,=øsCO83£qUYXZCO8ML 8 N ;q?3Uª8: ” [email protected]`83ÚTfCOBV83;,B
PZ;†[email protected]¤TBD;†[;¢@“XZ8X`@ ” [‚Q÷?83;øsCO83£qUYXZCO8\[; Z [ ” 8 R L
"! }!f "s!f " W f &$( F/Y& KNJ/YG 9NJ/ ( -+,9G O& w & /7QG$NJ/7=0& X&^1 c - /79&I( X+4X/ NJ90'& w & /7
G$NJ/7=0& X& D ⊂ Z 0'NJ+$( f 0-O& /7n(@NJ/ wx/7wxFw & / 0 2jT( ! 7c NJ/7 ( PR& 2jT(x( f 0-O& /7
(@NJ/ û}wx8/7ø XZwxUEFw£q;qUVUV& ; / !BVC 7NJCOV/7X©UY(4XZCO08ß'NJ;,+:(^U‘G$?&$8( ;!£,CO78NJD/7:,’( ?K;qNJ8+,£,w ;& [/7X`BD ;,X/£qUV;¢#[0?+.0PZ;, @A'[email protected]“V0+.;90Vwx?wX & 8W Uj<
”
N N
§? ”
c
1³12 " " W NJ X/Y& #0+,9&41F&
E
Z tuX = + -&
M_u
s_
n
n
2
2
2
,
2
,
2
∀A,B,C ∈ E
conv(A,B,C) ∩ Z2 = {A,B,C}.
0'NJ+:( G$NJ/7-9& /7 0 2jT(! 7NJ/7( P^& 2jh(<( &TFG & w & /7 tu 0+:& 7NJ/7(I0'NJ+:( &$( F/
#û}08 +.0; ‚ Ec ;õ'UN5V0PZ+.;,0 wxw CÝX`& 8W UYC m?8;øsCO8£qUV X`CO8 DUVBVXZ£qUY;[email protected]“E;q8WU£qCO8 ; \h; UVUY;qX`8WU“[email protected]“XZ8X`@¤[email protected] E:,BVX #;,8WU
N
? ”
?
£qP ” XZBV;,@A;,8WUEPZ;,E£qCO8[X©UYX`CÝ8E[?P`;,@[email protected];0L
’ ? ” UVBV; N COX`8WUYH[3X`DUVX`8£õUYH[; E L Z ? NN COVCÝ8 [6Q ” TCOBD[ ’?3;
P c w NJ−/#→( +.−0→-9NJ/ , Z CÝL XZU ó A,B,C,M
ø @AX`PZP`; (−AB,
L3öwP‚; \hXZDUY;†[3CO8£†?8
→ −→ 86Q÷;qDUÊ[CÝ8£ N#” E?83;†T ” V;†[3;
| det(AB,AC)| 6= 1 ” ”
AC)
Z
GF
2
O¥
( #
−→
−→ −→ Ü
N OC XZ8WU N 6∈ I = {αA+βB +γC | α,β,γ ∈ Z, α +β +γ = 1}. 8 ”†” P`COBD −AN
= uAB +v AC
−−→
−→
−→ ÜWCGF
@” ? ;,£ u ;q;õU U v 83CO8AUVCO?ˆP`;qˆL[3;,?X\A;q8 UVX`;qBVuL Z CÝXZU N ∈ Z UV;,P0’ ?; AN
u =u
= u AB + v AC
mod
BV N 1∈ conv(A,B,C) L 8;q8„[:q[?X©U^’ ?; N ∈ {A,B,C} ;qU9[CO83£ N ∈ I
Z X u +1 v v≤ =1 Ü v” P`CÝmod
£,CO8WUYB ” [XZ£qUYXZCOo8 L Z X u + v ≥ 1 ” PZCOBVˆP`;9DÔ@A:qUYBDX`’ ?;[3; N N#” B B ”ÝN3N COBnU (BC) B + −N−→C ;,DU
[ ” 8 conv(A,B,C) Ü#[‚Q÷GC F N ∈ I £q;¢’?3Xj;,DU9;,83£,COBD;?38;¢£,CO8WUYB ” [XZ£qUYXZCOo8 L
]^CÝ8£CO8 ” = αA + βB + γC ”K? ;q£ α,β,γ ∈ Z, α + β + γ = 1 L †8Ã[: #8X©U ” P`COBD
UYBVCÝX` N COXZ8WUY A =M −A
Ü
Ü
L>‘CO? ” P`PZCO8[email protected];qB
+B+C B = A−B+C C = A+B−C
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
B 0 ∈ conv{A0 ,C 0 ,M }
A ∈ conv{B,C,M }
B0
A
C0
C 0 ∈ conv{A0 ,B 0 ,M }
C
C ∈ conv{A,B,M }
B
A0
B ∈ conv{A,C,M }
A0 ∈ conv{B 0 ,C 0 ,M }
)
Ïä+* L ¥
s w & -+:&!'& 7 NJ/7 M
’?3; M ∈ {A ,B ,C } L Z ? NN CݏVCO8[CO83£A’ ?;£q;“86QS;,U N#” ¢P`;¡£ ” ,ÜjCO8 ”” P`CÝBV†P`;q N ÝC VVXZTX`PZXZUV:,
V?X ?ݔ 8 UV;,6< ? COXZB #ôO?BD; L ¥ L
ZZ XX α ≥ 1, β ≤ 0, γ ≤ 0 ÜÜ ” P`P`COCOBDBD A ∈ conv(B,C,M ) ;qX`@ U‘[CÝCOV83V£ XZTAP`; ∈ {B,C,M } Ü#[email protected] N OC VDX`TPZ;
Z X α ≤ 0, β ≥ 1, γ ≤ 0 Ü ” P`COBD B ∈ conv(A,C,M ) X`@ N CODVXZTP`;BL
Z X α ≤ 0, β ≤ 0, γ ≥ 1 ” Ü PZCOBVC ∈ conv(A,B,M ) X`N @ CODVXZTP`;BL
Z X α ≤ −1, β ≥ 1, γ ≥ 1 Ü ” PZCOBV A ∈ conv(B,C,M ) X`@ N CODVXZTP`;BL
Z X α ≥ 1, β ≤ −1, γ ≥ 1 Ü ” PZCOBV B ∈ conv(A,C,M ) X`@ N CODVXZTP`B; L
”
N
]^CO8£ M α∈≥{A1,,Bβ ≥,C1,}γÜ ” ≤?hUY−1
BV;[email protected]“;q8WUÊ[XZCU ∈ conv(A,B,M )
0
0
0
0
0
0
0
0
0
{A,B,C} ⊂ E ⊂ {A,B,C,A0 ,B 0 ,C 0 }
8 ; N ;,?3U N#”  ”K? COX`B N P`?¢[6QS?8 N CÝX`8WU¢[ ” 8 E ∩ {A ,B ,C } £ ” B†P`;¡@AX`PZX`;q?„[;[;q?X\ N COX`8WUY
N#” [email protected] {A ,B ,C } ;,DU[ ” 8 {A,B,C} £q;\’ ?X&;,UA;,8£qCOBV;\[email protected] N COVDX`TPZ;0LÚ]^CO8£ E ;qDUPQ÷?8Â[3;,
†8
0
0
0
0
0
0
V L S & /#(@& w "u'& D = {(1,1,1),(2,1,1),(0,1,0),(0,0,1),(0,0,0)} / &$( #J( /YG jT( 1h0/#(
'&$(e(@NJwxw & (^1 X/ #0+.0 c c 2 7>51F& 8NJX+,90/7ONJX c +5F>510+:& * (@NJwxw & (R1h0/#( D /Y&G$NJ/79& /7
07G X/7NJ/7 1F& Z N#” B ” PZP`:,PZCOôOB ” @“@A;, {A,B,C,A } Ü {A,B,C,B } Ü {A,B,C,C } L 8 ” [CO83£¡TX`;q8 @ACO8WUYBD:¡’ ?;VX
PZCOBV |E| = 4 ;õU E ø[email protected];†?8 N#” B ” P`PZ:,P`CÝôOB ” @[email protected];0L
|E| ≥ 4 ”
2
%ß123§4‚5 " " P40/#( '& 1 c "uX!1F& 2+:& F=0& ) 7NJF+s1 c w NJ/7-+:& +_tu7& | det(−AB,
→ −→
c
%
W
NJ/0F+50 2 Xj'(@& +s'& ` NJ+:> w &E1F& 8 9G$g =0NJ+ #0+lc & X& w 2'& < tuXR10xAC)|
tu7&c =0+:1&
.G$NJ/7/7m& 1 X/7NJjpuVFNJ/Y& P *(@NJwxw & (41h0/#( Z &$( Vh0 * I + − 1 N I & B 1 ( V0/Y& /7
'&I/YNJw "u+:&41F& 7NJ/7(^& /79& +:(I+:&$( 7&$G -=0& w & /7 * /7 c + 9& F+!1F& P & ( F+I'&R" NJ+.11F& P %ß123§4‚5 " So 2+:N [email protected]( -9NJ/ < < s/Y&4( c & /21 #J( MFG '& w & /7 *s 10w & /#( 9NJ/ 3 80+
&X& w2'&!( D = {(1,1,1),(2,1,1),(0,1,0),(0,0,1),(0,0,0)} 0'NJ+$(^ NJFO& KNJ/YG -9NJ/ G$NJ/#( c 90c /7 &1 c 6
- /79&I( F+ D &$( ( +,9G & w & /7G$NJ/7=0& X& ) 7NJF+, 0/7 D / &$( #J(R/YG jT(^1h0/#(4X/ #0+.0 2 7>51F& =0NJ+ -UV0F+:
& < <
)
Ïä+*
3
0
0
0
B
2
2
,
+,
§ü
( #
« ɺ¹{¾¿´f¶Á°
°€¼ ¾ ¬‹´j­±¶¼ Æ
·€¸º´j¹È­º¼
] ” 839£q;£ Ë ”§N XZUVBV;CO8 ?§” :qUV?[X`;qBÊP`;, N COX`8WUV‘@A:,[3X ” 8Ê[;q N” BDUVX`;, #83X`;,‘[; LfùŽCO8WUYB ” X`BV; ì
@A;,8WU ” ?X\ N COXZ8 UV† ” XZP`P ” 8WUY:õUY?[XZ:, ” ? £ Ë ”ÝN X©UYBD;¥ß£q;, N COXZ8WUY¢VCO8WUVX©UY?:q ? ;,BDZPQ÷XZ8WUY:,BDX`;q?B†[;
PQ÷;q8V;[email protected]¤TP`;§Ü[email protected] ” X` N#” B ” [CK\ ” P`;[email protected]“;q8 U&X`P` 86Q ”ÝNN#” BUYXZ;,88;q8WU N#” &UYCÝ?JdDCO?BD¤PRQS;,8D;,@¤TPZ;£,CÝ8VXZ[:,BD:0L
] ” 8 S] ó >^í Z 9P`;q ” ?3UY;q?BVßCÝ8 [email protected]“CÝ8 UVBV:’ ?;LP`;q N COP©Ô[email protected]`8Cݏ H I ì/£,CO8 ? ;,\h;,£,CO8WUV;,8 ” XZ;,8WU
UVCOJ? dDCO?BD9PZ;,?BD N COXZ8 UVÊ@“:q[X ” 83‘D;,PZCO8„PZ;,‘[X`BD;,£qUVX`CO83‘Ë3COBVX"!,CO8WU ” P`;,Ê;õU ? ;,BDUVX`£ ” P`;quL ó ;T?3U9[;£,;
£ Ë ”§N XZUVBV;†;,U‘[;ôÝ:,8:qB ” PZX`V;qB&£,;¢BD:,V?3PZU ” U¡[;qÊ[X`BD;,£qUVX`CO83E’?3;,P`£qCO8’ ?;quL
] ” 83‘?38; N BV;[email protected]“XZÌ,BD; N#” BUYX`;^CO8 ?ݔ [‚Q ” TfCOBD[:õUY?[3X`;,BEPZ;, N BVC N BDX`:õUY:,E[3;, N CÝX`8WUY&@A:,[3X ” 8Ž[;,
;q8V;[email protected]¤TP`;qÊ’?3;,P`£qCO8’ ?;q^;,83V?X©UY;¢CO8 ? ;qBVB ” P`;†£ ”  N#” BUYX`£q?P`XZ;,B&[3;,Ê£,CO8 ? ; \h;,ML
2
_ R _ U U U _ !ux _ ! a yt9uwehu~b)t
c
*ªU ” 8WUÊ[CÝ88:,;†?38; N#” BDUVX`; #8X`;
[; Z ;õUÊ?8;†[XZDU ” 8£,; d CO8 N ;,?3U&[:#83X`BE?8;†ôÝB ” 8[3;,?B
EBEDÝ£q;!
3
[
;
O
ô
P ” ø[email protected]¤?3P`;
O
C
?
&
B
Y
U
Ý
C
3
?
U
Ý
C
`
X
W
8
U
N M Z
D(M ) N
2
2
D(M ) =
X
d(M,N ).
N ∈E
}í P`?3 D(M ) ;qDU N ;qUVXZUÚ;qU N P`? M ;,U N BVC£ Ë; [;&PRQSX`8WUV:,BVXZ;,?B.[; E Ü £,;q£,X ” @“Ìq8;Ž[CÝ8£ŽP ” [:#8XZUVX`CO8
D?X ?§” 8WUY;=<
\ ª" !f / 7NJ/7 M 1F& Z &$( F/7NJ/7Uw c 100/s1 X/Y& #0+,9& - /79& E 1F& Z 7NJF+
10'( 90/YG$& d ( & f(@& F'& w & /7 ( 2
2
D(M ) = min2 D(N ).
] ” 83†£,;£ Ë ”ÝN X©UYBD;¤P`;q N COXZ8WUY@A:,[X ” 8DCO8WU†UYCO? dDCO?BV^P`X`:qeß?8;¡[3X`DU ” 8£q;¡P`X`83: ” XZBV;ࣧQ÷;qDU!
[XZBV;†?83;[3X`DU ” 8£q;[3;P ” ø[email protected];ª<
N ∈Z
−−→
−−→
d(M,N ) = α|p(M N )| + β|q(M N )|
Ýý
( C
C F α ;qU ⠏DCO8WU)[;,?F\;,8WUYXZ;,BDˆUYBDX`£qUV;,@A;,8WU N COVX©UYXZøsÚ;qU p ;õU q VCO8WUˆ[;qªø[email protected];,}PZX`8: ” X`BD;, ” VVC£,XZ:,;q
G
„[;,¡[3X`BV;q£qUVX`CO8B ” UYX`CÝ88;,PZP`;quLÚùŽ[email protected]@“;A[ ” 8àP ” D;,£õUYX`CÝ8Ã¥XL©ßCO8 N ;,?3Uà@“;õUVUVBV; p ;qU q VCÝ?¤P ”
ø[email protected];ª<
”@? ;,£ a,b,c,d ∈ Z,pgcd(a,b) = 1,pgcd(c,d) = 1 ÊÜ CO8 8CÝUV; ” ?3VVX£,CÝ@“@A;H[ ” 8P ” V;q£qUVX`CO8
X¥ LZ δ = det(p,q) ’ ?;ßPRQSCO8éV? NN COD;ߏDUVBVX`£õUY;[email protected]“;q8 U N ÝC VXZUVXZøŽ;qU κ [:qVXZôO8;PQ÷;q8WUYX`;qB[3CO88: N#” BàP ”
N BVC N CODXZUYXZCO8¥XL© L© L
p = ax + by,q = cx + dy
z k3b gjdfehu~b)t^x
L
8 N ;,?hU ? CÝX`BA’?‚Q÷?8 N COXZ8 U“@A:,[X ” 8m83;„[: N ;q8[ ’ ?;L[?€[email protected]¤TBD;[; N CÝX`8WUY“[;
[ ” 8
£ Ë ” ’ ?;¡P`XZôO8;¢[;[X`BD;,£qUVX`CO8 p CO?„[;à[XZBV;q£qUYXZCO8 q LùŽ;,[email protected]¤TBD;,D;,BVCÝ8 U^:qUV?[X`:q;,8„[:qU ” EX`P.[ ” 83
P ” [;,?F\3XZÌ,@A; N#” BDUVX`;[3;£q;@A:,@ACOXZBV;0L
=Ú"s!f " So 2+:N #K&$G -9NJ/ 1F&e & /#(@& w "u'& E (@& 'NJ/ _10+:&$G 9NJ/ p /YNJ c & X E NJ X &$( KNJ/YG 9NJ/l1F& Z 1h0/#( N 1 c - /79& #0+ ,
p
,
p
Xp E(i) = |{N ∈ E | p(N ) = i}|.
}| k|)d0e3y6k3u~xh|ªehu~bˆt n~bEd‚|)n~g l^gjx  bˆu~t ehx Š‹yjl^u~|)t^x
ó ;, N COX`8WUY†@A:,[3X ” 8¢DCO8WUPZ;, N COX`8WUY¢’ ?X)@AX`[email protected]“XZV;,8WU^P ” øsCO8£õUYX`CÝ8 D L.íÚCO?B ? :,BVX #;,BVX)?8
N COXZ8WU M ;,UÊ?8 N COXZ8WU&@A:,[X ” 8ßX`P0ø ” ?3U&[CO8££[email protected] N#” BV;qB D(M ) ¤P ”4?§” P`;q?BŽ[; D N CO?B UYCO?ŽP`;,
N COXZ8WUY[; Z ÜCO8 ?ݔ @ACO8WUYBD;,B[ ” 8£q; N#” B ” ôOB ”ÝN Ë;¤’ ?6QSX`PªD? U¢;,8Hø ” XZU9’ ?; M VCOX©U?8 N COX`8WU
@“XZ8X`@¤[email protected] [; D V?3B‘?38 N ;qUVXZUÊ;,83V;,@àTP`; ” ?3UVCO?BE[; M L
=Ú"s!f " S &I=0NJ'( /[email protected]&41F&) c +:N_&$( & /#(@& w "u'& /YNJ c N ,
2
N = {~u | ~u 6= ~0, − δ < p(~u) < δ, − δ < q(~u) < δ} ∪ {~u | (p(~u),q(~u)) ∈ {(±δ,0),(0, ± δ)}}
q = 0
0
N
)
p = 0
Ïä+*0ü#LZ
sS&!=0NJ'( /[email protected]&I1F&) c +:N 7 NJX+I'&$(410+:&$G 9NJ/#( p = 2x + y & q = x − 2y ; ( ': C
]àQ ”ÝN BDÌ,ÊP ”¡N BDC N COVX©UYX`CÝ8¥XL© LZ¢CO8 ” XZUʒ ?;
N = {hl,mi | − δ < l,m < δ, (l,m) 6= (0,0), m ≡ κl [δ]} ∪ {h±δ,0i,h0, ± δi},
;õUÊ[CO8£ª<
|N | = 4|{(l,m) ∈ ]0,δ[2 | m ≡ κl [δ]}| + 4
= 4(δ − 1) + 4
= 4δ
í ” &B ; \h;[email protected] N PZ; ”K? ;q£ p = 2x + y ;õU q = x − 2y ÜhCÝ8 ” δ = 5 ;õUE[CO83£ |N | = 20 ? COX`B # ôO?BD;†ü L© L
ó ” £ ” B ” £qUV:,BVXZ ” UVX`CO8\PZCh£ ” P`;^[;,Ê@A:,[X ” 8E;,U ” P`COBD&P ” D?X ?§” 8WUY;=<
!0h1é /7NJ/7 M &$( F/Ew c 100/l1F& E ( & (@& F'& w & /7 ( sü LZ ∀~u ∈ N D(M + ~u) ≥ D(M ).
P c w NJ/#( +.0-9NJ/ , 8 ” <
D(M ) = α
X
Q∈E
û}8V: N” B ” 8WU‘P`;q&UY;[email protected];,Ê;,8 p ;qU q OC 8 ”
X −−→
−−→
|p(M Q)| + β
|q(M Q)|.
Q∈E
D(M ) = DP (p(M )) + DQ(q(M ))
CGF
DP (i) = α
X
|i − i0 |Xp (i0 ),
i0 ∈Z
DQ(j) = β
8[3:#8X©U
X
|j − j 0 |Xq (j 0 ).
j 0 ∈Z
” P`CÝBV
S0 (hi,ji) = S0 (i) =
X
Xp (i0 ) = |{M ∈ E | p(M ) ≤ i}|,
i0 ≤i
S1 (hi,ji) = S1 (j) =
X
Xq (j 0 ) = |{M ∈ E | q(M ) ≤ j}|,
j 0 ≤j
S2 (hi,ji) = S2 (i) =
X
Xp (i0 ) = |{M ∈ E | p(M ) ≥ i}|,
i0 ≥i
I
S3 (hi,ji) = S3 (j) =
COXZB #ôO?3BV;¢ü L X¥ L
8 ”
X
Xq (j 0 ) = |{M ∈ E | q(M ) ≥ j}|,
j 0 ≥j
S0 (i) + S2 (i + 1) = S1 (j) + S3 (j + 1) = |E|
sü
L¥ ( C
S3 (j) = S3 (M )
S0 (i) = S0 (M )
M
S2 (i) = S2 (M )
M
q=j
S1 (j) = S1 (M )
)
p
Ïä+*ü L ¥
NJ
q
S &$(!/YNJw "u+$&$( S +$& 2+ c (@& /7O& /7 '&4/YNJw "u+:&R1F& 7 NJ/7(R1F& E 1F& ,+$NJ/79> +:& # 0+.0'> '&
p=i
*
,
i
;qUj<
DP (i + 1) − DP (i) = α
X
(|i + 1 − i0 | − |i − i0 |)Xp (i0 )
i0 ∈Z
=α
X
((i + 1 − i0 ) − (i − i0 ))Xp (i0 ) +
X
Xp (i0 ) +
i0 ≤i
=α
X
((i0 − i − 1) − (i0 − i))Xp (i0 )
i0 ≥i+1
i0 ≤i
X
i0 ≥i+1
−Xp (i0 )
!
= α(S0 (i) − S2 (i + 1))
]^CO8£
DP (i + 1) − DP (i) = α(S0 (i) − S2 (i + 1)) = α(2S0 (i) − |E|)
sü
L
DQ(j + 1) − DQ(j) = β(S1 (j) − S3 (j + 1)) = β(2S1 (j) − |E|).
sü
L÷ü ]^;P ” @[email protected]“;ø ”J CO8CO8 ” <
†8;,8[:q[?XZU6<
DP (i + 1) − 2DP (i) + DP (i − 1) = 2αXp (i) ≥ 0,
DQ(j + 1) − 2DQ(j) + DQ(j − 1) = 2βXq (j) ≥ 0,
L
sü # OC 8Â;,8Ã[:q[?XZU¤’?3; DP,DQ ;qU¤[CO8£ D DCO8WU[;,¤øsCÝ8£qUVX`CO8£,CO8 ? ; \h;,ML ? COXZB #ôO?3BV;\ü L LÚû}8
N#” BDUVX`£q?P`XZ;,B DP ;qU DQ DCO8WU‘[;,EøsCO8£õUYXZCO8E[:q£,BVCÝX`V ” 8WUY;q N ?3X`Ê£,BDCOX`D ” 8WUY;quL
í}BVCÝ? ? CO83[email protected] ” XZ8WUY;,8 ” 8WU&P`;‘UYË:qCOBVÌ[email protected]“;BL Z X M ;,DUÊ?38 N COXZ8WU&@A:,[X ” 8PZ;,&£qCO8[X©UYX`CÝ8 sü LZ VCÝ8 U
£,P ” X`BV;[email protected]“;q8WU ? :,BD"X #:q;,ML
:,£qX N BDCh’ ?;[email protected]“;q8 U,Ü3D? NN COVCO83‘’ ?; = hi,ji ? :,BDX"#; sü LZ LöwPø ” [email protected];,B&’ ?; M ;,U‘?8
@“XZ8X`@¤[email protected] ôOPZCOT ” P[; D L Z CÝXZU i ;õU j M
P`;,Ê[;q?X\;,8WUVX`;,BD ? :,BDX" ” 8 U
00
00
signe(i00 − i) = signe(i0 − i) signe(j 00 − j) = signe(j 0 − j),
i00 ≡ i0 [δ] j 00 ≡ j 0 [δ],
−δ < i00 − i < δ − δ < j 00 − j < δ,
!
; ( ': C
? COXZB)þ=X`ô LÝü L LW]àQ ”ÝN BDÌ, P ”N BDC N CݏVXZUVX`CO8“¥XL© L©ÊX`P ? X`;q8WU j ≡ κi [δ] [CO83£9CO8
0
0
”¢” ?DVX
j 00 ≡ κi00 [δ]
DP
p(M )
i−δ
i00
i
i0
i+δ
DQ
q(M )
)
;õU[3CO8£
Ïä+*fü#L
j
0
j−δ
j
00
j+δ
j
sS &$( KNJ/YG -9NJ/#( DP & DQ ,
ó ; N COXZ8WU
;qDU£,CO83DUYBD?XZU N OC ?B’ ?;
L Z ? NN CݏVCO8Ê’ ?;
Ü#CO8 ”
L
ùŽ[email protected]@“;P ” øsCO8£õUYX`CÝ8
;,U\[:q£,BVCÝX`V ” 8WUY; N ?X`“£,BVCÝX`V ” 8WUY;L;õU
øsCO83£qUYXZCO8
;qDU‘£,BDCOX`D ” 8WUV;D?B
L#]^CO83£
8 ”
[CÝ8£
[3CO8£ACO8 ”
Ü3£[email protected]@“;
P”
N 0 = hi00 ,j 00 i ∈ Z2
N0
N0 ∈ M + N
DP (i00 ) + DQ(j 00 ) = D(P ) ≥ D(M ) = DP (i) + DQ(j)
i0 ≥ i + δ
(+δ,0) ∈ N 0
DP (i) ≤ DP (i + δ)
DP
DP (i) ≤ DP (i + δ)
DP
[i + δ, + ∞[
DP (i) ≤ DP (i + δ) ≤ DP (i0 )
00
i≤i ≤i+δ
^] ;¢@,@A;VX i ≤ i − δ CO8 ”” ?DVX DP (i ) ≤ DP (i ) L Z X i ∈]i − δ,i + δ[ CÝ8 ”
CO8 ” ;,83£,COBD; DP (i ) ≤ DP (i ) L
]^;¢P ” @ ,@A;^ø ” CO8CÝ8 N ;,?hU N BVCO? ? ;,B‘’ ?; DQ(j ) ≥ DQ(j ) ;õU9[3CO8£
L
DP (i00 ) ≤ max(DP (i),DP (i0 )) = DP (i0 ).
0
00
00
0
0
;õU‘[CO8£
0
0
;õU
i00 = i
00
q; DU9TX`;[email protected]`83X`@¤[email protected]±ôOPZCOT ” P[;]RL
2
í}BV;q8CO8‘[;,?F\; \h;,@ N PZ;,ÊX`@ N CÝBDU ” 8 UV&[;¢[XZBV;,£õUYXZCO86<
[3Z XªX`DCOU ” 8 8N£q;BV;q[8;[éPZ;,†[[: XZ#BV;q8£qXZ;UYXZCON#8”  B p = x ;qU q = y ;õU α = β = 1 ܂P ” [XZDLÚU ” ] 8” £,83; ¤d£,;,;\DU £ ” ”  PZCOBV^P ”
d
N =
;qU‘[CO8£¢PZ; d N (M,N
COXZ8WU M) == |x(i,j)−;,xU‘@“| :q+[X |y” 8DXj−X`P y? :,|BDX"#;ª<
{(±1,0),(0, ± 1)}
D(N ) = DP (i0 ) + DQ(j 0 ) ≥ DP (i00 ) + DQ(j 00 ) ≥ DP (i) + DQ(j) = D(M )
D(M )
1
1
DP (i) ≤ DP (i−1), DP (i) ≤ DP (i+1),
M
N
M
N
DQ(j) ≤ DQ(j−1),
DQ(j) ≤ DQ(j+1)
ýO¦
( C
£,;’ ?XjV;¢BV:q£,BDXZU
S0 (i − 1) ≤
#L ý ü
|E|
|E|
≤ S0 (i), S1 (j − 1) ≤
≤ S1 (j).
2
2
” ?3UVBV;,@A;,8WU¢[X©U†P`;q†@“:q[X ” 8[; M VCÝ8 UPZ;, N COX`8WUV†[CO8WU¢P`;q£,CCOBV[3CO88:q;,¢VCO8WU¢PZ;,†@A:õì
[X ” 8;,¤[3;,¤£qChCOBD[CO883:,;,à[;, N COX`8WUV[; E L '$¤@“:q[X ” 83; ( ” X`£qX DCO8V;q8¤U ” UVX`DUVX`’ ?; L
8BD;qUYBDCO? ? ;PZ;, N COX`8WUY[email protected]:,[3X ” 8E[3:#8XZÊ[ ” 8 S] ó >^í Z -L
Z X^CÝ8 N BV;q8[ P`;qß[XZBV;,£õUYXZCO8 [email protected]/20'&$( p = x + y ;qU q = x − y ” P`COBD“P ” [X`U ” 83£,;
;qDUA:,ô ” P`; 2d GC F d (M,N ) = max(|x − x |,|y − y |) L}] ” 8A£,;£ ” “CÝ8 ”
d
;õU^[CÝ8£ M = hi,ji ;,DU^@“:q[X ” 8DX.P`;, AX`8:q’? ” UYXZCO8
N
=
{h±2,0i,h0,
±
2i,h±1,
±
1i}
V?X ?§” 8WUY;,‘VCÝ8 U ? :,BVX #:,;q6<
sü L S (i − 2) + S (i − 1) ≤ |E| ≤ S (i) + S (i + 1)
∞
∞
M
0
N
0
0
M
N
0
S1 (j − 2) + S1 (j − 1) ≤ |E| ≤ S1 (j) + S1 (j + 1)
S0 (i) + S1 (j) ≥ |E| S0 (i) + S3 (j) ≥ |E|
S2 (i) + S1 (j) ≥ |E| S2 (i) + S3 (j) ≥ |E|
¤gJt^bˆŠ ^kgol^g  b)u~t ehxmŠ‹yjl9u~|ˆt9x
8 ” [3: d5 ? ?¤’ ?;
;,DUª?8;ŽøsCO83£qUYXZCO8à£qCO8 ? ;,\3;BL Z Xh[; N PZ?=CÝ8¤V? N3N COD;&’ ?; E ;,U}X`8D:,£ ” TP`;
D
” P`COBDÊP`;,‘:,ô ” P`X©UY:,% sü L# ;õU^P ”AN BVC N COVX©UYXZCO8 L ¥FLZ@“CÝ8 UVBV;q8 U9’ ?;àP ” øsCÝ8£qUVX`CO8 D ;qDU^DUYBDX`£õUY;,@A;,8WU
£,CO8 ? ;,\h;Ý܂[3CO8£¢[6Q ”ÝN BVÌqÊ P ”¤N BDC N CݏVXZUVX`CO8 L L©¢CO8 ”
"!}!f s !f W E &$( n/#( c G5h"u'&4(@& 'NJ/ '&$( 10+:&$G 9NJ/#( p & q 0'NJ+:( E h10w & <02jh(
7NJ/7c (w c 100/#( P^& 2jT( ( E &TFG O& w & /7! w c 100/#( 0'NJ+:(sG$&$(w c 100/#( KNJ+,w & /7*F/
#0+50 'N5V0+.0wxw & Õ B Ý£,;!¡P ”¤N BDC N COVX©UYX`CÝ8¥XL#FL ¥¤CO8;,8[:,[3?XZUj<
!0"! s#s" W NJ E X/Y& #0+,9& - /79&R1F& Z
W E &$( 8 65G$NJ/7/Y& X&I0'NJ+$( E h10w & 02jh( !Ew c 100/#( 7NJF+I_10'( 0/YG$& d W E &$( 4 65G$NJ/7/Y& X&I0'NJ+$( E h10w & 02jh( !Ew c 100/#( 7NJF+I_10'( 0/YG$& d eU ! _ !M _ 4 ] ” 8 S] ó >^í Z fPZ;, ” ?3UY;q?BV CO8WU&@ACO8WUYBV:‘’ ?;PZ;, N COP©Ô[email protected]`83CO ì/£,CO8 ? ; h\ ;,E£qCO8WUYX`;q88;q8 U
UYCO?JDd CÝ?BVEPZ;,?B N COXZ8 UV&@A:,[X ” 8 N CO?B&P ” [XZDU ” 8£,; d L †8 §? ” ;,D ” ÔO;qB9[3;†ôO:,8:qB ” PZX`D;,BŽ£q;†BV:qV?P©U ” U
A [;qÊ[X`BD;,£qUVX`CO83E’?3;,P`£qCO8’ ?;quL
! ,
D
2
1
∞
HJI
1
Žù ; N BV;,@AX`;qB9PZ;,@[email protected]“;¢[CÝ88;¤?83;¤£,CO83[XZUVX`CO8V? A ” 8 UV; N CÝ?B^’ ?6QS?8H’ ? ” [B ” 8WU†£,CÝ8 UVX`;q88;¤?8
N COXZ8WUj<
W
1³12 W S (M ) + S (M ) > |E| 0'NJ+:( R (M ) ∩ E &$( e/YNJ/ [email protected]& S (M ) +
& ( R (M ) ∩ E &$( [email protected]&40'NJ+:( E &$( /YG jh(41h0/#( R (M ) ∪ R (M ) P c w NJù^/#( Q÷;q-+.D0U9X`[email protected] NJ/ , CÝZ VXVXZRTP`;9(MVX ) ∩ E = ∅ ” P`COBD S (M ) +L S (M ) = |(R (M ) ∪ R (M )) ∩ E| L
ùŽ;,P ;q8WUYBN 8; S (M ) + S (M ) > |E| DX
” ” E = E ∩ (R (M ) ∪ R (M )) S (M ) + S (M ) = |E| L
2
i
Si+1 (M ) ≥ |E|
i+1
i
i
i
i−1
i
6B
i
i
i+1
i−1
i+1
i−1
i+1
i
i+1
i+1
i+1
VC
ý3
C
ùŽ;¢P`;[email protected]“@A;^UVBVÌ,‘[email protected] N P`;e¡[:[email protected]“CO8WUVBV;,B ?§” qUYBD; N [email protected]“COBD[X ” P[ ” 8Ê£,;£ Ë ”ÝN X©UYBV;†;õU‘V?BUYCO?3U‘[ ” 89PZ;
£ Ë ”§N XZUVBV;†V?3X ?§” 8WU‘V?BÊP ” BD;,£qCO8DUVBV?£õUYXZCO8oL
!0h1é " " W E & '&$(R10+:&$G 9NJ/#( p & q = c +, -& /7 '&$(I+:NJ'(xG$NJ/210-9NJ/#(Q( X=J0/7O&$( ,
S & /#(@& w "u'& E &$( 65G$NJ/7=0&X&(@& 'NJ/ p & q 8NJF+ONJX 7NJ/7 M 1F& Z & i 1h0/#( {0,1}) ( E &$( /YG jh(*1h0/#( (R (M ) ∪ R (M )) \
{M } 0'NJ+:( E &$( /YG jh(41h0/#( R (M ) NJ 1h0/#( R (M ) S &$(410+:&$G 9NJ/#(I= c +, -& /7 δ = 1c NJ (δ = 2,α = β) 0'NJ+:( E G$NJ/7-9& /7 ONJT(I(@&$( 7NJ/7(Iw 100/#(*(@& 'NJ/ p & q P[ ìwc £qw CO8NJ?/#;,( 3\ +.;§0Ü0-X`9P6NJ/;,3\ , XZZ DUVCO; X©U MUV;,Pj?3’ 8 ?; N COX`[email protected]“:q8[;¢X ” £,8 CO8W[UV; X`;,E8WU ’” ??Xˆ£q?88 ;N D;,COB XZ8W” XZUEU [N#; ” L[ ” 8 E Ü.£,[email protected]@“; E ;,U
R (M )
V

O
C
W
8
‘
U
[
3
8
‘

Z
P
;
O
C
Z
X
V

Z
X
8
O
ô
;
L E
8;q8[3:,[?X©Uʒ ?;
Z X δ = 2 ” P`COBDEP`;q iN COXZ8WUY h±1,
”
?
”
± 1i
N
sü L DP (p(M ) ± 1) + DQ(q(N ) ± 1) ≥ D(M ).
2
i
i
i+2
i+2
i
Z X δ = 1 £[email protected]“@A; DP (p(M ) ± 1) ≥ DP (p(M )) ;õU DQ(q(M ) ± 1) ≥ DQ(q(M )) CO8 ” ;q8£,COBD;
P ” ø[email protected]¤?3P`;sü L L
û}8€BV:q?8XZV ” 8 U ü L Ü sü ÷L ü ;õU ü#L CO8 ? COXZU“’ ?; S (M ) + S (M ) ≥ |E| N CO?BUYCO?3U
L&û}8 N#” BDUYXZ£,?PZX`;qB S (M ) + S (M ) ≥ |E| C i ;qDU\UY;qP^’ ?; R (M ) ∩ E = ∅ LE]àQ ”ÝN BVÌq
jPZ;ßP`;[email protected]“@A
;ßü L ¥X©L \CO8Ï ” X©U’?3; ;qDUAX`83£,P`?3¤[ ” 8
)L ]Q BVÌ,P
R (M )) \ {M }
[;q?Xh\ X`Ì[email protected]“;£,CO83[XZUVX`CO8CO8 ” X©U^’ E?; E ⊂ R (M ) CO(R? E ⊂(MR) ∪ (M
L) Z ? NN CODCO8 N” B9;,”Ý\hN ;,@ N P`”;
L 8 ”” P`COBD S (M ) = 0 ;õU‘£,[email protected]@“; S (M ) + S (M ) ≥ |E| CO8 ” S (M ) =
E ⊂ R (M )
[CO8£ E ⊂ R (M ) ∪ R (M ) L B E ⊂ R (M ) [CO8£
|E ;q∩DU¡
(RXZ8£,PZ?(M
)
∪
R
(M
))|
=
|E|
¤[ ” 8¤P ” [;,@AX©ì/[BVCOX©UY; R (M ) ∩ R (M ) Ü E ;,U¡[CO8£\?38V;,ôÝ@“;q8 U[3; Z XZP
E
£qCO8WUYX`;q8WU9[3CO8£UYCÝ?ÊV;,‘@A:,[X ” 8,Ü3£,CO8WUYB ” [XZ£qUYXZCO8 ”K? ;q£ M 6∈ E L
2
Z XCÝ8¡CO?TPZX`;Ž?38;E[;,ªUVBVCOXZª£,CÝ8[XZUVX`CO83 ” PZCOBVªPZ;ŽUYË3:,COBDÌ,@A;&[; ? XZ;,8WU}ø ” ?X¤\ £,[email protected]@“; P`;&@ACO8WUYBD;,8WU
PZ;,E; h\ ;[email protected] N PZ;,Ê[;¢P ” #ôÝ?BV;†ü SL ü L
j
GF
i
j+1
i+1
i
i−1
i+1
i−1
i+1
i−1
i+1
i
i−1
i+2
i
i−1
i+1
i−1
i+2
i+1
2
i+2
!0"! s# " " W E &$( !765G$NJ/7/Y&X&R& qv65G$NJ/7=0&X&40'NJ+:( E G$NJ/79& /7 (@&$( 7NJ/7(!w c 100/#((@& 'NJ/ d W E &$( +' 65G$NJ/7/Y&X& & 65G$NJ/7=0&X&^(@& 'NJ/ x + y & x − y 0'NJ+:( E G$NJ/79& /7(@&$( 7NJ/7(
w c 100/#(Q(@& 'NJ/ d P c w NJó /#( +.0BV-;,[email protected]/ X`Ìq, BV;£,CO8[3XZUYXZCO8L[?LUVË:,CÝBVÌ,@A; BD:,£q:,[;q8 U;qDU :,BVX #:,;¡ôÝB Ý£q; ?LUYË3:,COBDÌ,@A;¥XL XL L
?
”
ó ”” [N ;,?F3\ XZÌ,@A;£qCO8[X©UYXZCO8 ? XZ;,8WU9[?ø ” XZU‘N ’ ?;
q
;
U
83;†VCO8WU N#” Êü§ì ” [Jd ” £q;,8WUYML †8 ” δ = 1 [3CO8£¢ECO8 ∩N R;q?3(M
U ”Ý)N\N PZ{M
X`’ ?;q}B&P`;EUVË:,∩CORBDÌ,@A; (MN BV:q)£,\:q[{M
;,8WUK}L
"ì 8” [J?3d ”UYXZ£,P`;,XZV8W;¤UVu£qL ;qUVUV;“øsCOXZ†£,X)’ ?; E ∩ R (M ) \ {M } ;õU E ∩ R (M ) \ {M } 83;“DCO8WU N#” 
2
Õ B Ý£q; ” ?¢P`;,@[email protected];}¥XL XL 3Ü CO8 N ;,?3U ”ÝN3N P`XZ’?3;,B6P ” [;,?Xh\ XZÌ,@A; N#” BDUVX`;ª[3?£qCOBVCOPZP ” XZBV; ” ?X\ ;,8D;,@¤TPZ;,
ü§ì/£,CO838; h\ ;,‘;qUÊ£qCO8 ? ;,3\ ;q[email protected] N P`;,ŽV;,PZCO8 x + y,x − y L
1
∞
ED
i
i
ED
i+2
i+2
ýW¥
( C
DQ
DQ
E
DQ
3
8 6 / 9. 5 8 ) 6 6
E
6 8 5 ! '1 /8% 8
DP
8" 1$6
671 /8% 1 6
Ri ∪ Ri+2
DP
DP
DP
8 %;678 1
)
α = 1,β = 3
Ïä+*fü SL ü
DQ
6 1 ; 675 8'
p = x + y,q = −2x + y
) & 678
NJ/7-+:& 65&X& w2'&$( * #0+,O& /20/YG$&!1F&$(Iw c 100/#( δ=3
E
E
U 6
;
!s _ ý
ó ;UVË:,CÝBVÌ,@A; N BV:q£,:,[3;,8WU\;,U ” DV; !„[:q£,; ?§” 8WU\£ ” BAP`;,A£,CÝ8[XZUVX`CO83AD?BAP`;,A[XZBV;,£õUYXZCO8AVCO8WU
Tf; ” ?3£,CO? N UVBVC N øsCOBDUV;,uL‚ÿ98;ø ” CO8H[3;¤£,CO8WUVCO?BV83;,B£q;qUVUV;¤£,CÝ8[XZUVX`CO8;,U¢[6Q÷:õUY?[3X`;,B9P`;q@“:q[X ” 8
D?BÊ£ Ë ” £,?38„[;q δ BD:,D; ” ?X\ L [:#83X`Ê[ ” 8ÊP ” V;,£õUYXZCO8L¥FLZ L© L
û}8; ;qUED?B L UVCO?3UY;^[BVCOX©UY; N#” B ” P`P`ÌqP`;Q p £,CÝ? N ;UVCO?3UV;[BVCÝXZUY; N#” B ” P`PZÌ,P`;Q q L 8\;,UÊ[CO8£
£[email protected]“@A;[ ” 8ÊP`;†£ ”  δ = 1 L
\ ª" !f " W NJ L F/ 1F&$( δ p 6 q (@NJT( 6O+ c (@&50J / 7NJ/7 M ∈ L &$( X/ p 6 q w c 100/
i
i
( F+−−'&→+ c (@&50 L ( M &$( F/ 7 NJ/7Uw c 100/_1F& E ∩ L 7NJX+10'( 90/YG$& d(M,N ) = |p(M−−→N )| +
|q(M N )| ùŽ[email protected]@A;UVCO?3UY;[BVCÝXZUY;[;¤[3X`BV;q£qUVX`CO8 [; V;¤£qCO? N ; ”K? ;q£¤UYCO?3UV;à[BDCOXZUV;[3;¤[X`BD;,£õUYX`CÝ8„[; CO8
;q8H[:,[?3XZU¢’ ?; p(M ) ;qU q(M ) @AX`[email protected]“XZV;q8LU9BV;q N ;,£õUYX ? ;,@A;,8WU DP ;qU DQ £ ” PZ£,?PZ:,D?B E ∩q L L
]^:q[email protected] ” X` DP ;qU DQ DCO8WU9£ ” P`£q?P`:qEV?BÊPZ;†BV:,D; ” ? L L
8 ”” PZCOBVŽP`;UYË3:,COBDÌ,@A;[‚Q ”ÝN3N#” BDUV;,8 ” 8£,;V?X §? ” 8WUj<
!0h1é " " W E &$( ^X/ 65G$NJ/7=0&X&_(@& 'NJ/ p & q & L &$( xX/ p 6 q + c (@&50bO& *tu7&
E ∩ L 6= ∅ 0'NJ+:( ,
W DP NJ DQ h10w & X/ (@& F7wx/7wxXw 0'NJ+$( E ∩ L G$NJ/79& /7 NJh(n(@&$( 7NJ/7 (<w c 100/#(
W /YNJ/ DP 0 & /7v(@NJ/ wx/7wxFw ( F+ [p ,p ] & DQ ( X+ [q ,q ] 0=0&$G p < p & q < q W NJ9& /7 A = hp ,q i,B = hp ,q i,C = hp ,q i,D = hp ,q i S &$( w c 100/#(1F& E ( F+'&
+ c (@&50 L (@NJ/7n'&$(7NJ/7( 1F& c L G$NJ/7 & /7h( 1h0/#(Rc '&#0+.0 c 'N5V0+.0wxw & ABCD W E ∩ L
/Y&RG$NJ/7-9& /7 #J(! NJh(e(@&$(!w 100/#( ) 0'NJ+:(!'&$(Iw 100/#(!1F& E ∩ L tuF (@NJ/71h0/#( E (@NJ/7
i / #0&+,T9FG G XOj& 9w& + & E/7 ∩A,CL G$NJNJ/7-B,D
9& /7 0sw NJ/#(IX/ 1F&(@&$( 7NJ/7 (!w c 100/#( P8;9c w£,CONJ8W/#UV(X`;,8W+.0U -9” NJ?/ £,, ?Z 8 CONX©U COMXZ8 U 6∈[; E ?3L 8 Z N ? CÝNX`N 8WCOU=[email protected]:,[3X N#” ” 8BŽ[;,; \hE;,@ ∩LN P`; L †8à ” L3X©U=ùŽ’ [email protected]`@“P; ; h\ X`UY; i [email protected];qPhX`8’ [email protected]“; XZRV; (M )
)
DP
;õU q(M ) @AX`83X`@AX`V; DQ CO8 ” EDP (p(M ) ± 1) ≥ DP (p(Mi))=;qU0 DQ(q(M )p(M
±
1)
≥
DQ(q(M
))
[6QSC CO8[:,[3?XZUÊ£[email protected]“@A; N BD:,£,:q[;,@[email protected];,8WU‘’?3; E ⊂ R (M ) ∪ R (M ) L
[3Z COX 8DP£ ” [@A;qU^?8HV;q?Pª@AX`[email protected]¤[email protected] ” CÝPZCOB¡BV£,[email protected] @“” ; ;,8Lø ” XZU DP (p(M ) ± 1) >CO8DP (p(M )) ;qU
” S (M ) =
S (M ) >
,S
(M ) >
E ⊂ R
(M
) £ ∪ R (M )
q
;
U
q
;
U
[
O
C
8
|E ∩ L ∩ R (M )|
S (M ) = |E ∩ L ∩ R (M )|
S (M ) + S (M ) = |E|
£qCO8WUYB ” [X`£õUYXZCO8 ”K? ;,£ S (M
L
;õZ U ? NN COVCO83ª@ ” X`8WUY;q8OL ùŽ” 8WCOUª@A’ @A?);&; >PQ÷CÝ8¤,S;,U}(M[ V” CO8)8W=>U=P`£,;ŽCO[38;,?XU h\” 8WX`Ì[email protected]“;,};ŽV£ ?3” B ML 8 ” ;õ[U CO83£ p ≤CO8¤p(M
)≤p
,
;
à
8
[
:,[?3XZU
] [q ,q ] ’ ?6qQ ” ?≤£q?q(M
8 N CO)X`8W≤Uq[3; E ∩ L 8;DP,DQ
:? ,BVX #; p < p(N ) < p ;õU [pq ,p< q(N
L
^
]
O
C
3
8
¡
£
V;,?3P`
)
<
q
PZ;,[email protected]:,[X ” 8 A,B,C,D N ;,? ? ;,8WU qUYBD;£qCO8WUY;q8?39[ ” 8 E L
ùŽCÝ@“@A; DP (p − 1) > DP (p ) = DP (p + 1) XZP.Ô ” ?38 N CÝX`8WU G ∈ E V?B9P ” [3BVCOX©UY;
? COXZB=ø[email protected]¤?P`; ü L , L ùŽ[email protected]@A;
CO8 ” q(G ) ≥ q WL ]^;Ê@,@A;
pXZP}=
p
E
⊂
R
(M
)
∪
R
(M
)
; h\ X`UY; G ,G ,G UV;,Pˆ’?3; q(G ) = q ,p(G ) ≤ p ,p(G ) = p ,q(G ) ≤ q ,q(G ) =
L 8;,8L[:q[?X©U N#” B [ ìw£qCO8 ? ; h\ X©UY:à’ ?; ;qU DCO8WU[ ” 8 E L ó ; N COXZ8WU A
q ,p(G ) ≥ p
8‚Q÷;qDU N#” Ê[ ” 83 E £ ” B A ∈ R (M ) L 8 ” p(M ) < p B£ ” BʏDDX`8CO8 B ∈ R (M
Ü#[3;@,@A;
)
£ BDX`8CÝ8 D ∈ R (M ) ÚL ]^CO8£ C 6∈ R (M ) ∪ R (M ) ⊃ E L ó ;qà@A:,[X ” 8
q(M ) > q ”
”§NN#” BUY;,8 ” 8WU< E DCO8WU9[3CO8£¢; \ ” £õUY;,@A;,8WU B ;qU D L COX`B ôO?BV;¢ü L 1
1
1
1
2
2
2
1
2
2
2
1
2
1
2
1
i
GF
3
|E|
2
0
|E|
2
2
3
1
3
2
|E|
2
0
1
1
|E|
2
2
0
0
2
1
1
2
1
1
1
1
2
3
4
2
2
1
2
2
1
1
1
3
2
2
3
1
2
0
2
2
0
2
2
1
3
4
1
1
1
1
2
2
3
I
0
1
4
ýÝü
( C
G1
D
G2
C
q = q2
M
q = q1
B
A
G4
G3
E∩L
)
Ïä+*fü#L#
p = p1
p = p2
s8NJX+.tu7NJ (@& X ( B & D (@NJ/71h0/#( E
2
öwP‚86Q Ô ”àN#”  [ ;P`XZ;,8A[X`BD;,£õU&;,8WUVBV;PZ;,Ž@A:,[3X ” 8ˆV?3BŽ£ Ë ” £q?8\[;q&BV:qV; ” ?X\\;õU&P`;qŽ@A:,[X ” 8ˆ[;9UVCO?3U
PRQS;,8D;,@¤TPZ;0L ? OC XZBE; \h;,@ N P`;qÊ[;¢P ” #ôO?3BV;¢ü L÷ý !
i #/ (@& w "u'&$(*1FNJ/7f'&$(*w c 100/#(V0'N0"$0JR(@NJ/7 10 c +$& /7(*1F&$(*w c 100/#(( F+*F/ + c (@&50 S&$(R10+:&$G 9NJ/#(*(@NJ/7 p = x + y & q = x − y û ø
5
ó L Z Z Ü
?
?
¤ü ì
\
? ,\
[ ì ? ,\
W
W
" " / & /#(@& w u'&
E &$( 65G$NJ/7=0& X& .G$NJ/7=0&X&I(@& 'NJ/ NJ+ +, (@& 'NJ/ s10+:&$G -9NJ/
' 6 h1$#MFG$& /7( ) C & D (@NJ/7
p ( 7NJF+xONJT( A,B,C,D ∈ E & M ∈ Z O& ( u7& A & B (@NJ/7"
' 6 h$1 #MFG$& /7 ( ) [AB] ∩ ∆ 6= ∅ & [CD] ∩ ∆ 6= ∅ N ∆ & ∆ (@NJ/7I'&$( 1F& J 1F& wx6 10+:NJ &$(
1 NJ+ V0/Y& M & 1F&410+:&$G 9NJ/ p 0'NJ+:( M ∈ E =0NJ+ -UV0X+:& ! û
ì ,\
Eø
[ì ?\
W
" W E &$( 65G$NJ/7=0& X&(@& 'NJ/ 1F& 10+:&$G -9NJ/#( p & q & ( E &$( ' 65G$NJ/7/Y& X&
0'NJ+$( E &$( 65G$NJ/7=0& X&Q(@& 'NJ/ p & q )
Ïä+*3ü ÷L ý
!#"$!#%" M &
$' )(+*-,/. 10 ,, 2
0 (+33(+45,0 '(6'7. 08( (+*-,:97;< =;>,;?, [email protected] ;>083(+4
@+44*
@ 4
CB(+*D0 E-E0
[email protected](+*-,GEH( . +
@ '7'70 I*J, ;K0>,GA7'0L.>@' 0 *J, ;M37EA7DN.>@BO37E*-P A; 0KPQA70REH(
.>@' 0 *J, ;TS
UWV YX[Z]\^Z`_FX
Oa
b
2
0
0
c
$'[email protected](+'#,eE(
gMh^_ $_FijZ`\Z)_FX
.>@'7'70 *J, ;K.>0>,,0M'@+, *[email protected]'f0 D,e3EA7D `@4, 0KPQA70LEH(
.>@'
0 I*J, ;TS
U 6
;
ý
∆
B
M
A
C
∆0
)
Ïä+*0ü L P c w Z NJ/#( +.0-9NJ/ , M
Z
ì Ë
SA
"
Ü
L A
B ?
B ∈R B ∈
R ?
W 65G$NJ/7=0& c
D
VË DÌ
ì
L XL L
K'fAI, *E-*D50eE-0 DCB BO0 DC*-97; 0>DNPQA70K3 @+A74 E( 97; [email protected]'7D, 4(6, *[email protected]':97A:, 7; @4 BO0 @*J,
A7' 3 @*-'#,:970
,0 E Q
P A70 , @A7D/E0 D
.>@'#, *0>'7'70>'Q,WA' 3 @+*'#,
'
2
R (M )
A ∈ E
10 ,
@'?D (6i*-,eP A *-E[0 *-D5,0 970>A [email protected]*-'Q,D (+9 i <(+.>0 '#, D
.>@'7D*-97; 4(+'#, A7'
. 70>B*-' 0>'Q,40
Bi
i
i+1
; 45* (6'Q,
01,
@*4 A740
0
0
i
i
i
i
j6=i
,\
ô ¢ü L j
X\
A2
q=
∆1
∆2
A3
R2
--
B10
B2
B1
B20
M
R3
LŽû
A1
R1
∆3
∆0
B3
R0
B30
)
L∆
∆ 6= ∅
Z
Zì ? \
Z
@6, @'7D
\
p=
A0
Ïä+*0ü L B0
J
+ì
s8NJF+ u7NJ M &$( 6O1h0/#( E
L
R (M ) ∩ R (M )
[B B ] ∩ ∆ 6= ∅
E( 970>B* [email protected]*-,0
i
B00
b
K'fD(+*-, PQA70
i
i+1
i
0
i
i
@A
[Bi Bi0 ] ∩
A733 @[email protected]'7D PQA *E 0 I*-D5, 0
0>,
,0 E-D PQA70
01,
([email protected](+4
0
0
k
l
[B
[B
k Bk ] ∩ ∆0 6= ∅
l Bl ] ∩ ∆2 6= ∅
. @' 0 I*-,;LD50 [email protected]'
@+' (
p
M ∈E
* [email protected]$, @AI,
([email protected][email protected]'D (+*J, PQA70
0>,
0
0
]
k
[B
B
[B
[B0 B00 ] ∩ ∆3 6= ∅
k k ∩ ∆0 = ∅
1 B1 ] ∩ ∆1 6= ∅
01, [email protected]'7.
3(+4 . @' 0 I*-, ;LD0>[email protected]'
q
([email protected]+4D%@' D(+*-,OPQA70
0>,
0 B M>B∈0WED* [email protected]%, @AI,
0
0
k
[B
B
]
∩
∆
[B
k k
2 = ∅
3 B3 ] ∩ ∆3 6= ∅
0>, [email protected]'.
[B2 B20(+]'∩D ∆
M ∈E
1 6= ∅
,@A7D E0>D .j(+D
i−1
L
Z ì ? ,\
L
L
L
c G 2+$N u7&&$( M0T($(@& , & /#(@& w u'&x1F& -UV0F+$& ! &$( !765G$NJ/7/Y&X& & o
S
+
65G$NJ/7=0&X&Q(@& 'NJ/ x − 2y & x − 3y w 0'(Iv/ &$( #J( W 65G$NJ/7=0&X&(@& 'NJ/ 10+:&$G -9NJ/ x − 2y h " M ∈E
b
2
,
Oa
ýOý
( C
M
)
Ïä+*ü L (@& 'NJ/
/ & /#(@& w u'& !765G$NJ/7/Y&X&Q& 65G$NJ/7=0&X& (@& 'NJ/
x − 2y Oa
{x − 2y,x − 3y}
w 0'(/YNJ/ W 65G$NJ/7=0&X&
"!
!#"%$'& (*),+.-0/21435-%6 +.$'& $87/:90(<;>=<?
@ACBEDFHGJILKJMNBPORQEFTSVUXWYDK Z\[^]`_aDQbK GJSVQHc4DFdK efKJgJDNhDGYQEgJMfD_iA4g0QEFHDNjkSVFElfKJImSVF BED BEA4FEG
LR Pn A4gYjkSVILG0opAqjkS4FElfKJImSVF f FHD_.DfQHK _iA4GreKsgJDNKJgJSVQbcVMDtBEILgJDflfKsDfh'DfFdK hqA4ILGua_iA4grlfSVfFdKsgJDvuPRSVFT_aDQHK
hDGYQEgJDfgwoxOyA\KJKsMfFzQiAvKsILSVF BPORQEFgsA|{VSVF}zQEIPKsgJA|cVDgYGJD5o~ORSVUXWYDfK L
€DfcVDfFESVFEG'_.S4QEgtGJImh_EoLILiDfgA4Q‚lA4Gqƒ„[^] L† IwoLDgsA|{VSVF‡DGˆKqomATBHgJSVI‰KsDBEDcVDlKsDfQEgqBEILgJDflfKsDfQEg
QEFHILKŠAvImgJD ~u DfK8_EA4GJGJA4FdK€_iA4g M A4oLSVgJGfuHBiA4FHG%omDfG%l|A4G8oLDG%_HomQEG%GYImh_EomDfGuXopA}zQiAvFzKJILKJM0h'DfGJQEgYMDr_aSVQEg
lfD gJA|{VSVFDGˆK2DŒ‹HA4lfKJDhDFdK#
Z
3
∞
f (M + t~u)dt.
Xu~ (M ) =
@A'jkSVFHlfKsILSVF DGˆK A4_E_aDoLMDN_EgYS\WYDlKsILSVFŽGJDfomSVFomAqBEILgJDlKsILSVFBED5cVDlKsDfQEg L @ACgJDflSVFHGYKsgYQElfKJImSVF
lfSVFEGJILGYKJDBESVFEXlt‘CBEMKsDfgJhImFEDfg f ‘C_iA4gˆKsILg’BEDfGr}XQEA4FdKsILKJMG X (M ) L“ Fƒ„[^]`~uoLDG0_EgYS\WYDlKsILSVFEGrlS4gˆ[
gYDGJ_aSVFEBHDFdKNDŒ‹EAvlfKsDfh'DfFdKt‘>omAKJgsA4FEGˆjkSVgJhMDBEDq2A4BHSVF<ILFzKJgJSXBEQEI‰KsDA4Q<BHMUEQHKBEQ”ƒ4•v–s— ˜aGYIm™flomD L
š DKJKsD2KsgsAvFEGYjkSVgYh'MfDrANMfKJMt›œDK8DGˆK€DFElfSVgJD|†KsgY™G8MKsQEBEILMD L @ANUiA4GYD0BEDG8hMfKJžESXBEDG%BHD0gJDflSVFEGˆKsgYQEl[
KJImSVFClfSVFEGYImGYKJDrDFBEDG%jkS4gJhNQEoLDG%BŸO ILFzc4DgYGJImS4F}zQEIŸBESVFEFEDfFdKwjkS4gJhDomoLDhDFdK f DF jkSVFElKsImS4F BEDG X L
›¡cVSVILg8_iA4g8D‹bDfh'_EoLD¢R£¤S4jœ¥ „¦  L
§ ¨ ©85 ©%#ªf«' @#D'¬4gsA4FEB,BEMjœA4QHK5BED'omACKsSVhSV¬VgJA4_EžEILDlSVFdKsILFzQED'DfGYK5}zQEDqoxO SVFTBES4ILK GJSVQbcVDFdKlSVFEGYImBEMfgJDfg5oLDG
_EgYS\WYDlKsILSVFEGGJDfomSVF­KJSVQHKJDGtomDfGNBEImgYDlKsImS4FEG Lª® F”h'S¯{4DF°BED BEImhImFzQEDfglfDFHSVhNUEgYD BED BEImgYDlKsImS4FEG
DfGYKtBPORImh_.S4GJDgBEDGlfSVFEBEI‰KsImS4FEGNGJQHg f L:± F”_.DfQHKtGJQH_E_.S4GJDgt_iAvgND‹bDh_EoLD }XQHDoxO S4UXWYDfKlŠžHDgJlŠžHM
DfGYKžESVhSV¬V™fFED L ]0AvFEGtlDl|A4Gtop‘omAjkS4FElfKJImSVF f DGˆK'‘TcvA4oLDQEgYGBEA4FEG {0,1} ²l4ORDGˆKopAŽjkSVFElKsImS4F
lA4gsA4lKsMfgJImGˆKsIL}XQHDrBPO QHF>DfFEGJDfhNUEomD E L
@A<_EgYS\WYDlKsImS4F³BED E _.DfQHK A4oLSVgJG'efKJgJDBHMfiFEILDŽ_EoLQEGqGYImh_EomDfh'DfFzKP†omA_EgYS\WYDlKsILSVF³GJDfomSVF‚opA
BEILgJDflfKsILSVF p = cste DfGYK#
SV´ µ DGˆK’omANh'DfGJQEgYD BED @DfU.DfGJX¬VQH(t)D5BED=BEµ({M
ILh'DfFEGJIL∈SVFEµ L : p(M ) = t})
± F<_aDQHKNA4QEGYGJI†Imh_.SVGYDg BPO A4QHKJgJDGlSVFdKsgJA4ImFdKJDG GJQEg E lS4h'hDt_iA4g D‹bDh_EomDopAlSVFdc4D‹bILKJM L
“ F¶µ¯¥ ZHu:·’Avh'hDg'A_aSVGJM>omD_EgJS4UEom™fh'DGJQEI‰cvA4FdK#:lSVhNUHImDF‡BEDBHImgJDflfKJImSVFEGGJS4FzKqFEMlfDGYGsA4ILgJDG
_aSVQEgBEMKsDfgJhImFEDfglfSVh_Eom™KsDhDFdKtQEF°DFHGJDhUEomDlSVFdc4D‹bDŽ‘T_iA4gˆKsImgBEDG_EgJS„WYDlfKJImSVFHGGYDoLSVF‡lfDG
BEILgJDflfKsILSVFEG4¸'›ˆ¢ ·2A4h Z ¦ 
š D _EgYSVUEom™fh'DqAŽMfKJM gJMGYSVomQ*lfSVh_Eom™KsDhDFdKDF”BEImhDFHGJImS4F”ƒ¹ªQEF”DfFEGJDfhNUEomDqBED BEImgYDlKsImS4FEG
BEMKsDfgJhImFED KsS4QEG'oLDGlSVFdc4D‹bDG GYIwDKqGYDQEoLDhDFdKCGYI€lDFHD>GYSVFdKq_iAvGqomDfGtBEImgYDlfKJImSVFHG'BEDfGqAvgJefKJDG
¥
−∞
u
~
u
~
u
~
p
!•
BPO QHFŽ_.S4oL{X¬VSVFED gJMf¬VQEoLImDgw‘QEFED5KJgsA4FHGYjkSVgYhqAvKJImSVFA FED5_EgJ™fG›ˆ¢ V• ¦  LŸn A4g2lfSVFdKsgYDlD5gJMfGJQEo‰KŠAvK
FEDBESVFEFHD_iA4G0QEFThS¯{VDFTBEDl|A4oLlQEoLDg0oxO DfFEGJDfhNUEomD‘C_iA4gˆKsImg2BEDGr_HgJS\WYDflfKsILSVFEG L ]’DG Avom¬VSVgYILKJžEh'DfG
D‹bImGˆKsDFdKh'A4om¬4gJMKsSVQHK›Y¢ V¥ ¦ u0¢ A4gs¥ zuKJžEMSVgY™hD<µ L ƒ L ƒ ¦ h'A4ImG'ImoLG'FEDGJSVFdK _iAvGCDFElfSVgJD
GsAvKJImGYjœAvImGsAvFzKJG8BEQ_.SVILFdK€BEDrcbQHD KJžEMS4gJIm}zQED'›ˆ¢ A4gs¥ XuiFES4KJDµ L ƒ ¦  L
5 5 f %
š DKJKsD8jkSVILGˆ[ lIESVFtGYDwBHSVFEFED€QEFDFEGYDhNUHomD E BiA4FEG Z L @Ar_EgJS\WYDflfKJImSVFtBED E GJDfomSVFtopA0BHImgJDflfKJImSVF
DfGYK2A4oLSVgJG%KJSVQHK€GJILh'_HomDhDFdK8oLDrFESVhNUHgJDrBED _aSVImFdKsG€BiAvFEG€lŠžiA4}zQED BEgJSVI‰KsD0BHD5BEILgJDflfKsILSVF p 
p
"
$#
&%
!
('
2
Xp E(i) = |{N ∈ E : p(N ) = i}|.
I¤SVFA'QEFEDNlfDgYKsA4ImFHDlomA4GJGYD BED_EA4gYKJImDG2BED A4oLSVgJGwomD5_EgJS4UEom™fh'D BEDNomA'gJDflSVFHGYKsgYQElfKJImSVF
GJDoLSVFQEF>DfFEGJDfhNUEoLD D BED BEILgJDflfKsILFSVFEG8GfO MfFESVFElfD jœA4lfImZoLDhDFdK#
]’SVFEFEMfD' |D| (F ,D)((p ) ,p ∈ D)
N
SVgYKJImD
E∈F
∀p ∈ D∀i ∈
± FA'AvQEGJZGYI#XomDr(i)_EgYSV=UEom™fph'DrBHMlILGJILSVFEFEDfo#A4GJGYSblfImM
]’SVFEFEMfD' (F|D|,D) ((p ) ,p ∈ D)
N
QHDGYKJImSVF,
E ∈F
∀p ∈ D∀i ∈
® F>Avom¬VSVgYILKJžEZh'XD2(i)}zQEI#=gJMfpGJSVQHK
L
gJMfGJSVQHKwAvQEGJGYI
€MlfIm_EgYSb}zQEDfh'DfFzK0QEFTA4om¬4SVgJI‰KsžEhD }XQHIgJMfGJSVQHK (F ,p,q) _.DfQHK0KJgJ™GrGYSVQHc4DFdK efKJgJDt(FA4Bi,p,q)
Av_HKsM
L
_.SVQHgwBHSVFEFEDfgwQHFED GJSVoLQHKsILSVF DfKwBESVFHl gYMGYSVQEBEgYD
± F lS4FEFiA LK8_EgJILFElIL_iA4oLDhDFdK¹BHDQb‹lA4G8SV´
Ž_aDQHK€eKsgYDrgJMGYSVomQ DfF KsDh_EG
_.SVo‰{XFESVhIpA4oV
@A gJDflSVFHGYKsgYQElfKJImSVF BPORDFEGYDhNUEoLDG }XQHDomlfSVFE}zQEDfG A|cVDl BEDQX‹ _HgJS\WYDflfKsILSVFEG
›
A¯c4Dl |D| = 2¹gYMGJS4omQED _iA4g€8{XGJDfgN›ˆ¢ 8{XG Z ¦  L
(P(Z ),D)
@AgJDflSVFEGˆKsgYQElfKJImSVFCBŸO DfFEGJDfhNUEomDfG8· w[ lfSVFdcVD‹bDfGwDK v[ lSVFHFED‹bDG%A¯c4DlroLDG†_EgJS\WYDflfKJImSVFEGžESVgYI‰[
SVFdKsA4omDfGwDK€cVDgˆKsILl|A4oLDG8gJMfGJSVoLQED _iA4g %AvgJlQHllIDK’Avo L ›ˆ¢ ¹][email protected] n ¥ 4¦  L
® FtBEDG:UEQbKsG²BED%KJSVQHKsD8omA’BEDfQb‹bIm™hD8_iA4gˆKsILD%BED8lDKJKJD%KsžE™fGJD€DGˆK:BED%KsgYSVQHcVDfg²BPO A4QHKsgYDG:lomA4GJGYDG²_aSVQEg
omDGY}zQEDoLomDG€omANgJDlfSVFEGˆKsgJQHlfKsILSVF_.DfQHKwGJDrjœA4ImgYDrDFKsDfh'_EG8_aSVo‰{bFHSVh'ImA4o L
± F0KsgYSVQHc4DgsA€BiA4FHG#opA%GJDflfKJImSVF L µ L µ²BEI .MgYDFdKsDfGlSVh_EomDŒ‹bILKsMfGPlS4FEFzQEDG¤BEQ _EgJS4UEom™fh'D
2
)*,+.-0/214365872+936:;-0/
Z
<
=> ?;@BAC=
p
kRlm:;1436*,/(+.*
q
DFEHG IJA>LKM=ONG PQ=
i i∈Z
P[APQ=4A\S]^IJA
[email protected]=> [email protected],XYPV?<
_Y=?`?;[email protected]_cN [email protected][email protected]?hUHPQ=OE iK? [email protected]
i
<
sr
=> ?;@BAC=
DnEHG IJA> KM=oNG PQ=
i i∈Z
[email protected]@t?;Iu>LP8AvAPQ=4A\S]^IJA
p
i
[email protected]=>[email protected]?<
N [email protected][email protected]?hUHPQ=SE iKW? [email protected]
)*,+.-0/(14365x7(+936:;-0/
ukylm:;1436*/2+.*
OkRlm:;1436*,/(+.*
z)*,+.-0/21 365x7(+936:;-m/
{
|
)*,+.-0/(14365x7(+936:;-0/
)*,+.-0/(14365x7(+936:;-0/
2
d
|
c
~
‚
j}

6€
Y)*,+.-0/214365872+936:;-0/
L
v !µ
M4
f : R2 → R
f : R2 → {0,1}
f : Z2 → {0,1}
~u
!#" $&%'()" (+*#$
¤µ
3y:'465
!#" $&,',-" .+*!$
|87 AC=:9
@;CTVAC=oN [email protected]\FU=< KbGT> ?hA^<
3
4
3
1
!#" $&/#" 01%'2-$
2
!ƒ
]’MfGJIm¬4FESVFEG_iA4g Q(p,q) opA*lomA4GJGYD,BHDGDFEGYDhNUEoLDGiFEILG [ lSVFdcVDŒ‹bDGGYDoLSVF oLDGBEILgJDlKsILSVFEG
DK q L @DUHQHKBEDlDTlŠžiAv_EILKJgJDDGˆKBEDTBEMlfgJILgJDŽQEF A4oL¬VSVgJI‰KsžEhD_.S4oL{XFESVhIpA4ow_aSVQEgCgYMGJS4QEBEgJD
p
L š ORDGˆK’oxO ILGJGJQHDrBPO QHFKJgsA|cvA4Imo.lSVhhNQEFA|cVDfl AvgsA ¹gJQHFEDfKYKsI L
(Q(p,q),{p,q})
]0A4FHG¹KsSVQbK8lD0lŠžEA4_EILKJgJDvuHSVF GYD’H‹bDrBESVFEl’BEDQb‹qBEILgJDlKsILSVFEG p DfK q DfK8BHDQb‹CGJQEI‰KsDG¹BPO DfFdKsImDfgJG8‘
GYQE_E_aSVgYK€iFEI (p ) DK (q ) La± F_aDQHK€BES4FEl BEMfiFHImg
q
p
)*,+.-0/(14365x7(+936:;-0/
i
j
pmin = min{i : pi > 0}, pmax = max{i : pi > 0}
qmin = min{j : qj > 0}, qmax = max{j : qj > 0}
lp = pmax − pmin + 1, lq = qmax − qmin + 1;
£¤SVQHKJD*GJSVoLQHKsILSVF E DGˆK,ImFElfomQEGYD­BiA4FEGToLD­_iA4gJA4omoLMomS4¬VgsA4hhD ∆ = {hi,ji
L
DfK
pmax qmin ≤ j ≤ qmax}
]O A4QHKJgJD _iA4gˆK2S4FGJA4ILK€}zQED oLDrl|A4gYBEImFiAvoPBED E c4AeKsgJD Mf¬dA4o#‘%
: pmin ≤ i ≤
› L µ„
BES4FEl GJI › L µ„5FPORDGˆK'_EA4GNcVMfgJILEMDSVF”GsA4I‰K}zQPORImo%FPO {*AvQEgsAŽ_iAvGBHD GJSVoLQHKsILSVF L ]0A4FEGtopA>GJQEI‰KsD VS F
GYQE_E_aSVGJD BESVFHl }zQEDq› L µ„¹DfGYK€cVMfgJI‰iMD L
©%rf5 ¹5 ©%
n S4QEggYDlS4FEGYKJgJQEILgJDqoLDG [ lSVFdc4D‹bDG }XQHISVFdKomDG UaSVFEFEDfGN_EgJS„WYDlfKJImSVFHGSVF<cvAQHKsILomImGYDg QEF
A4oL¬VSVgYILKsžHh'D>¬VomSVQbKsSVFr8SVF lSVFHGJImBH™gJDTKsS4Q\EWYSVQEgYG>QEFHDUaSVgJFHDILFHjkMgYImDQHgJDTBEDG>GJSVoLQHKsILSVFEG α ⊂ E
A4_H_.DfomMD NFES¯{dA4Q qDfK¹QEFED0UaSVgJFHD’GJQH_.MfgJImDfQEgJD β ⊃ E A4_E_aDomMfDNlSX}zQEImoLomD L £¤SVQHK%oxOyAvom¬VSVgYILKJžEh'D
lfSVFEGJILGYKJDwBHSVFElw‘ A4QE¬VhDFdKsDfg α DfK†‘ BEImhImFzQEDfg β DK¹‘rDfGJ_aMgJDfg}zQPORImoLG²BEDfcXILDFEFEDfFdK¹oLD8_EomQEG²_HgJSXlŠžEDG
_aSVGJGYImUEoLD L
± F lSVFHGJImBH™gJD’BESVFElrBiAvFEG%lDKJKJDrGJDlKsILSVF BEDQX‹C_iAvgYKsILDG α,β BED Z DK8o~ORDFEGYDhNUHomD E(α,β) BEDG
_iAvgYKsILDG E ∈ Q(p,q) A|{dAvFzK2_aSVQEg8_EgYS\WYDlKsImS4F (p ) DfK (q ) DK€KsDoLG8}zQED
A=
X
pi =
i∈Z
X
w
qj
j∈Z
w
h
v%
#
q
2
i
α ⊂ E ⊂ β.
!Z
j
!
8 }
&^
w^
& @#DGwSV_aMgJAvKsILSVFEG8BED lS4h'_EoLMfKJImSVFCc4SVFdK’_aDgYh'DKJKsgYD BED KJgJSVQbcVDg α ,β KJDomG€}zQED _.S4QEg8KsSVQHK E ∈
SVFA
E(α,β)
A4QHKsgYDhDFdKwBEILK E(α,β) = E(α ,β ) L α ⊂ α ⊂ E ⊂ β ⊂ β.
± FBEMfEFEILK8BŸOyA4UaSVgYBoLDG8SV_aMgJAvKsILSVFEG%}zQEIŸGORD .DlfKJQEDFdK’GJM_EA4gJMfh'DfFzK€GYQEg€lŠžiA4}zQED BEgYSVILKJD L
n SVQEg¹lDomASVFCFHS4KsD α oxO ILFdKsDgYGJDflfKsILSVFCBED α A|cVDflropA5BHgJSVI‰KsD p = i ubDfK α oxO ILFzKJDgYGJDlKsILSVFCBED αp A|=cVDfil
opANBEgYSVILKJD q = j u β o~ORImFdKsDfgJGJDflfKJImSVFCBHD β A|cVDflropABHgJSVI‰KsD p = i uEDfK β o~ORImFdKsDfgJGJDflfKJImSVFCBHD β A|c4Dl opA
BEgJSVI‰KsD q = j L
± FBEMfEFEILK2A4QEGYGJIV
0
0
0
0
0
0
&
j
i
j
i
g(αi ) = min q(M ), d(αi ) = max q(M ), g(βi ) = min q(M ), d(βi ) = max q(M ).
:SVILlIEoLDG†SV_aMgsA\KsImS4FEG²lomA4GJGYIm}zQEDG ⊕, ⊗ , , BEMlfgJILKJDG†BiA4FEG [%][email protected] n ¥ ] h'A4ImG
A4Qb‹
BEImgYDlfKJImSVFHG8}XQHDomlfSVFE}zQEDfG› δ BEMGYIm¬VFED det(p,q)
I α 6= ∅ A4omS4gJG ⊕α = {hi,ji : g(α ) ≤ j ≤ d(α )} L
•
• ⊗α = {hi,ji : d(β ) − δp < j < g(β ) + δp }
I α 6= ∅ u hi,j i 6∈ β A|c4Dl j ≤ g(α ) AvomSVgYG β = {hi,ji ∈ β : j > j } L
•
I α 6= ∅ u hi,j i 6∈ β DfGYKwKsDfoP}zQED j ≥ d(α ) A4oLSVgJG β = {hi,ji ∈ β : j < j } L
I α 6= ∅ A4omS4gJG β = {hi,ji ∈ β : d(α ) − δp < j < g(α ) + δp } L
•
± F'cvA5A4QEGJGYI.BHMfiFEILgBHDQb‹'FESVQbcVDoLomDfGSV_aMgJAvKsILSVFEG DfK }zQEIaMomILh'ILFEDFdK:BEDfG¹_aSVImFdKJG†GYQEg¹oLDG
BEgJSVI‰KsDfG ⠍
I p = 0 A4oLSVgJG β = ∅ L
•
I hi,j i,hi,j i 6∈ β A|cVDfl 0 < j −j ≤ δp A4omS4gJG β = {hi,ji ∈ β : j < j SVQ j > j } L
•
›[email protected]'BEDfgJFEIL™gYD5SV_aMgJAvKsILSVFGJQE_H_EgJILh'D BESVFHlomDfG€GJM}zQEDfFElDfGrBED h'SVILFEG€BED p _aSVILFzKJG€lSVFEGYMlfQHKsI‰jkG2BED
 ± F>BEMiFEI‰K€BED h'efh'DroLDG8SV_aMgJAvKsILSVFEG8GJQHg α DfK β L
β ±
F”c4ADfFHiF‚gJA„WYSVQHKJDgtBEDfQb‹°SV_aMgsA\KsImS4FEG¬VomS4UiA4omDfG ⊕ DK L†± F°gsA4_E_aDoLomD }zQEDomDfGGJQEI‰KsDG
BEMfEFEImDfG0_iA4g0oLDGwjkSVgJhNQHomDGN› L ƒV€FEDBEM_aDFHBEDFdK }zQEDBEDfGr_EgJS\WYDflfKJImSVFEG’DfKr_aDQHc4DFdK BESVFHl
S (M )
efKsgYD l|A4oLlQEoLMDGwA|cVDfl5oLDG8GYQEILKJD (p ) DK (q ) L
S4ILK M = hi ,j i KJDo#}zQED _.S4QEg8KsSVQHK k SVF R (M ) ∩ α 6= ∅ SVQ S (M ) + S (M ) > A
I M ∈ Z A4oLSVgJG ⊕ α = α ∪ {M } L
•
I M 6∈ Z A4oLSVgJG
•
I p 6= 0 A4omS4gJG β = {hi ,ji ∈ β : j − δp < j < j + δp }
I q 6= 0 A4oLSVgJG β = {hi,j i ∈ β : i − δq < i < i + δq }
›kcVS4Img%i¬VQHgJD L ƒV
oPjœA4QbK€hqA4ILFdKsDFEA4FdK8h'S4FzKJgJDfgw}zQED lfDGwSV_.MfgsAvKJImSVFHG%GJSVFdKwcvA4omILBEDGP
M ∈αi
M ∈αi
M ∈βi
M ∈βi

i
i
i
i
i
0
i
i
i
i
i
0
i
i
[email protected]=
i
i
0
i
0
i
i
6€
i
i
0
i
i
i
i
i
0
0
i
i
00
i
0
i
0
i
00
00
00
i
i
0
i
00
i
j
i
j
0
0
}
k
i
M
2
j
M
k
k
k+1
0
2
|
0
iM
|
O
iM
0 jM
jM
UH>LK
@BUH> @
1
M
iM
jM
M
∈ {⊕, ⊗ ,⊕0 } _
2
∈{ ,
M
iM
M
iM
M
jM
M
jM
,
0
,
00
0
,
}
UHP
G
E(
1 α,
2
β) =
š SVh'hD α ⊃ α DfK β ⊂ β SVFAKsSVQ„WYSVQEgJG E( α, β) ⊂ E(α,β) L
n SVQEg%oLDG¹S4_.MfgsAvKJImSVFEG†BEI ŸMfgJDFdKJDG%BED ⊕ D K opA lSVFdcVDŒ‹bILKsM’GJImh_EoLD2GYQ tK8_.S4QEg¹hSVFdKsgYDg¹}zQED
L
E(α,β) ⊂ E( α,
β)
E(α,β) <
i\[email protected] @?hUHP

1
2
1
2
0
1
0
Q
2
M
!
&:4w4
Q ?
? ? ?
⊕
?
?
⊗
? ? ? ? ?
α
?
? ?
β\α
?
? ?
pi
Z2 \ β
pi
?
?
? ?
?
? ? ? ?
?
?
pi
00
?
? ? ? ?
pi
3y:'465x
Lµ
|
7
? ? ? ?
AC=oUCTViK G @?hUHPQ=ON AzgCUH\
[email protected]?hUHP
=> K
g Q
> G ^>,AI ?< P8A
’] Mfh'SVFdKJgJSVFEGtoLD omDhhDq_aSVQEgomDfGSV_aMgsA\KsImS4FEG ⊕ DK SVF”GJQE_H_.SVGYD}XQHD E ∈ E(α,β) DfK
DGˆKQEF_.S4ImFdK BED Q c4MgJI‰ A4FdK R (M ) ∩ α 6= ∅ SVQ S (M ) + S (M ) > A
M = hi ,j i
_aSVQEg:KsSVQbK k L rg 4lfD0A4QCoLDhh'D 4.2.1 S4FCGsA4I‰K¹}zQED’_.SVQHgKsS4QHK k SVF A R (M ) ∩ E 6= ∅ L ]’S4FEl0GJI
uE_iA4g [ lfSVFdcVDŒ‹HI‰KsM BED E SVFA M ∈ E EomDroLDhh'D’DGYKwBEMfh'S4FzKJgJMr_aSVQEg8oxO SV_aMgJAvKsILSVF ⊕ L
M ∈Z
QE_H_.SVGYSVFEGr}zQED M 6∈ Z L# SVILK A,B ∈ Z oLDGr_aSVImFdKJG’BEDomAqBEgJS4ILKsD p = i DfFdKsSVQEgJA4FdK M u
lvO DfGYK'‘,BHImgJDCKJDoLG}XQHD q(B) − q(A) = δ DK q(A) < j < q(B) L:± F‡GYQE_E_aSVGJD p > 0 L o
DŒ‹HILGYKJD BHSVFEl QEF_.S4ImFdK N ∈ E KsDfo#}XQHD p(N ) = i La I q(N ) ≤ q(M ) A4omSVgYG%DFjœA4I‰K q(N ) ≤ q(A)
BES4FEl N ∈ R (A) DK N ∈ R (A) L ± F<A R (A) ∩ E 6= ∅ l|A4g R (M ) ⊂ R (A) L ]’D'hehD
L ]’SVFEl KsSVQEG2omDfG R (A) lfSVFdKsILDFEFEDfFdK0QEFŽ_aSVILFzK2BED E BES4FEl A ∈ E L ]’D5h'efh'D
R (A) ∩ E 6= ∅
GYI q(N ) ≥ q(M ) AvomSVgYG B ∈ E L ›¡cVSVILg%i¬VQEgYD L Zd
]’SVFHl {A,B} ∩ E 6= ∅ BESVFElwKsSVQbK¹_aSVImFdK N ∈ E GYQEg¹omA5BEgYSVILKJD p = i c4MgJI‰iD q(B) − δp ≤
uiDfKwBES4FEl SVFAtUHImDF E ⊂ β L
q(N ) ≤ q(A) + δp
2
“ ŸDflfKJQEDg KsSVQbKsDG SV_aMgJAvKsILSVFEGrBEDlSVh_EoLMfKsILSVF_aDQHK GJDtjœA4ILgJDDF
omDfG S4_.MŒ[
gJAvKsILSVFEG GJQEg5omDG5omIL¬VFEDG _aDQHc4DFdKtGJD'jœA4ImgYDDfF O(lplq) L¤n SVQEg5omDfGSV_aMO(lplq(lp
gJAvKsILSVFEG ⊕+Dlq))
K u S4F<cVSVI‰K
}zQED0oxO S4Fq_aDQHK¹l|A4oLlQEoLDg¹oLDGU.SXSVomMfDFEG V (M ) = “R (M ) ∩ E 6= ∅ SVQ S (M ) + S (M ) > A
ILFElgYMhDFdKŠA4oLDhDFdK5lSVhh'DSVF­oxOyAjœA4ILK5_aSVQEgoxO DfFdcVDoLSV_E_aD [ lfSVFdcVD‹bD L š DCl|AvomlQHo_EgYDFEB*BHSVFEl
S4_.MfgsAvKJImSVFEG L¹® FEDjkSVILG'lA4omlfQEomMfDG'lDGCcvA4gYIpA4UHomDGtoxO SV_aMgJAvKsILSVF ⊕ _EgJDfFEB A4QEGYGJI O(lplq)
O(lplq)
SV_aMgJAvKsILSVFEG%DKwo~ORSV_.MfgsAvKJImSVF O(lplq(lp + lq)) SV_aMgJAvKsILSVFEG L
n S4QEg’A|c4SVImg α,β ImFdcvA4gYIpA4FdKsG€_iAvgwoLDGwSV_aMgsA\KsImS4FEG€ImoPjœA4QHKwgYM_aMfKsDfg’oLDG€SV_aMgsA\KsImS4FEGwA4Q_EoLQEG |∆|
jkSVILG l|A4g ‘>lŠžEA4}zQED'jkSVILG }XQHDqo~ORSVF,D .DlKsQED'QEFEDSV_.MfgsAvKJImSVF,lŠžiA4FH¬VD|A4FdK α S4Q β uSVFjœAvILK UiA4ILGJGYDg
GˆKsgJILlfKJDhDFdK |β \ α| L ± FDFBEMfBEQEILK
0
M
M
2
0
k
k
q
2
k+1
k
0
2
2
M
M
0
1
iM
2
3
i
2
2
o
iM
0
iM

0
i
i
q
0
i
0
00
i+1
0
0
&
UH>LK
@UH>[email protected]
α0 ,β0
[email protected]]^IJAC=jTQG KIJAC=zUCTViKbG @w?hUHPQ=N A
2
O((lplq) ) <
gCUH\
UHP
TVA>[email protected]@wKMUH> E AK
[email protected]?hUHP
PSTVA> @RU ]^@BAPV?;K
α
[email protected]
[email protected]@AI=^>,A
β
O((lplq)2 (lp + lq))
E(α,β) = E(α0 ,β0 ) <
AP
?;PVEHG K?G [email protected];=RTQG K
UCTViKbG @w?hUHPQ=
@BUH> @BAC=IJAC=oUCTViKbG @?hUHPQ=
=^G > ,
α,β
0
AP
!
8 R3 (M ) ∩ α 6= ∅
S3 (M ) + S0 (M ) > A
&^
w^
& R2 (M ) ∩ α 6= ∅
S2 (M ) + S3 (M ) > A
M ∈ Z2
⊕0 α = α ∪ {M }
R1 (M ) ∩ α 6= ∅
S1 (M ) + S2 (M ) > A
R0 (M ) ∩ α 6= ∅
S0 (M ) + S1 (M ) > A
p=i
∆3
R3 (M ) ∩ α 6= ∅
S3 (M ) + S0 (M ) > A
R2 (M ) ∩ α 6= ∅
S2 (M ) + S3 (M ) > A
δqj
M 6∈ Z2
∆0
δpi
K?G [email protected];=zTQG K
[email protected]@A
G I
<
IJAC=
G PQ=
UCTViKbG @?hUHPQ=
A^]
⊕,
Lƒ
∆1
|87 AC=zUCTViKbG @w?hUHPQ=
⊕0
0
[email protected]
pI G >[email protected]>LKFN UHPVP8Ap> P8AS\[email protected] V
> U^N A
,⊗,
UHK?;@'> \FAP8A= [email protected]==?;\
β = β \ (∆0 ∪ ∆1 ∪ ∆2 ∪ ∆3 )
R1 (M ) ∩ α 6= ∅
S1 (M ) + S2 (M ) > A
R0 (M ) ∩ α 6= ∅
S0 (M ) + S1 (M ) > A
3y:'465x
0
q=j
∆2
AP
O(lplq log(lplq))
<
TVUH>LK
U ]^@APV?;K
UCTViKbG @w?hUHPQ=^<
00
TIJA\[email protected]> KI UCTViK G @?hUHP
α
[email protected]
β
?;PVEHG t
G IV
> A> KMA> =4A\[email protected]
<
± FT_aSVQEgYgsA4I‰K A4_E_EoLIm}zQEDfg0lDoLDhh'DA¯c4Dl α = ∅ DfK β = ∆ L A4oLžEDQHgJDQHGJDhDFdK GJIªSVFT_EgJDfFEB
_iA4grD‹bDfh'_EoLD (p ) = (q ) = (1,1,1,1) uPA4oLSVgJG’A4QElfQEFEDSV_aMgJAvKsILSVFŽBEDlfSVh'_HomMfKJImSVF>FEDhSbBEI‰iDFHI
uiFEI β L
α
± FDGˆK2BHSVFEl SVUEoLIm¬VM0BHD gJA„WYSVQHKJDg8BEDfG€_.SVILFdKsG8BiA4FHG α Li“ FjœAvILK8SVF cvAtGJDQHomDhDFdK€lŠžES4ImGJILg8BEDG
_.SVILFdKsG }XQHIDŒ‹bKJgJMfh'ILGJDFdK p L ± F­_EgJDfFEB­BES4FEl α = {U ,U } A¯c4Dl p(U ) = pmin DfK q(U ) =
L o¤{>AqAvQ_EoLQEG lq _.SVGYGJILUEImoLILKsMfG8_.S4QEg2BHD5KsDfomG α LŸ SVI‰K2A4omSVgYG α,β BHSVFEFEMfG’_iA4g2omD omDfh'hD
qmax
L µ L ƒA|cVDfl α = {U ,U },β = ∆ L† I%SVF‡A α 6⊂ β AvomSVgYGSVF”GsA4I‰Kt}XQHD E(α ,β ) = ∅ L†± F
GJQE_E_aSVGYD5BEMfGJSVgYhqAvImG
0
i
0
j
0
2
1
2
2
1
2
0
0
1
0
0
0
{U1 ,U2 } = α0 ⊂ α ⊂ β ⊂ β0 = ∆.
]0AvFEG’omA'GJQHILKsD5SVF>cvA'KJgsA4I‰KsDfgwoLD5lA4G
‘lfD lA4G€DFgJDfh'_EomA |AvFzK q _iA4g −q L 
q(U1 ) ≤ q(U2 )
L ›x]’DKJSVQHKJD5jœA fSVFSVF_.DfQHK0GYDgsAvh'DfFEDg
!!
h
p = iM
p = iM
R3 (M ) ∩ α 6= ∅
S3 (M ) + S0 (M ) > A
N ∈E
R2 (M ) ∩ α 6= ∅
S2 (M ) + S3 (M ) > A
B ∈ Z2
B ∈ Z2
M 6∈ Z2
A∈
M 6∈ Z2
q = jM
Z2
A∈
R0 (M ) ∩ α 6= ∅
S0 (M ) + S1 (M ) > A
q = jM
Z2
R1 (M ) ∩ α 6= ∅
S1 (M ) + S2 (M ) > A
š A4G q(N ) ≤ q(M )  A ∈ E
š A4G q(N ) ≥ q(M )  B ∈ E
3 '465 L Z
E
A
B
s§ ' Cª ± F­cvAŽh'SVFdKJgJDg5}zQED α DfK β _EgYDFEFHDFdKtBEDGNjkS4gJhDGKJgJ™G_iA4gYKJImlfQEomIL™gJDfGGYQEgNlŠžiA4}zQEDBEgYSVILKJD
SVQ q = j L
p=i
@ABEgJSVI‰KsD p = i GYDgsAtBEI‰KsDrBEDrK {X_.Dª
uEGJI β = ∅
t0
uEGJI α 6= ∅ uiSVFAAvomSVgYGP
t1
N ∈E
y:
c%
|
c%
#
|
c%
UH> K ^>VUH?
#
[email protected][email protected]@wt?;I
UH>
c%
i
|
i
αi = {hi,ji : g(αi ) ≤ j ≤ d(αi )}
uEGJI α = ∅ DfK β DGˆKwlSVFEGˆKsI‰KsQEM BED
_aSVImFdKJG8lSVFEGYMlfQHKsI‰jkG La± FAtA4oLSVgJGŸ
βi = {hi,ji : g(αi ) − δ(pi − |αi |) ≤ j ≤ d(αi ) + δ(pi − |αi |)}
|
t2
i
i
2pi
βi = {hi,ji : g(βi ) ≤ j < g(βi ) + 2δpi }
|
uŸGYI
DfK DGˆK lSVFHGYKsI‰KsQEMBHDBEDQb‹GJQEI‰KsDGrBHImGYKJImFElKsDfGrBED p _aSVImFdKsG’lSVFEGYMlfQHKsI‰jkG L
± FA% α = ∅ β
SVQ d(β )−δp < j ≤ d(β )} A|c4Dl d(β )−g(β ) ≥ 2p
β = {hi,ji : g(β ) ≤ j < g(β )+δp
t3
i
i
i
i
i
i
i
i
i
i
t0
pi =0
α
β−α
t1
pi =2
β
t2
pi =2
L 87
± F>_aDQbK€BEMfiFHImg€BED opANhehD0jœA S4FoLDG8K {X_.DfG
}
|
S2 (hi,ji) > A
[email protected]
AC=oN ? [email protected];[email protected];WTVAC=oN AoN KMUH?;@A
t0,t1,t2,t3
i ∈ {pmin, . . . ,pmax} [email protected]
S3 (hi,ji) + S0 (hi,ji) > A <
UH>LK`@BUH> @
i
pi =2
t3
3y:'4658
i
^>VA
GYQEg2oLDG8BEgYSVILKJDG q = j L
pi 6= 0
?;[email protected]
j
@BAI ^>VA
S1 (hi,ji)+
i
!
8 w^
SVILK jmin = min{j : S (j) + S (i) > A} L ]’SVFHl
i\[email protected] @?hUHP
š SVh'hD
&^
1
pi > 0
i\[email protected] @?hUHP
& 2
S1 (jmin − 1) + S2 (i) ≤ A
A − S3 (jmin) + A − S0 (i − 1) ≤ A
S3 (jmin) + S0 (i − 1) ≥ A
SVFA
7
S0 (i) = S0 (i − 1) + pi > S0 (i − 1)
[email protected] [email protected];CTVA
uiDfK€BES4FEl
L
S3 (jmin) + S0 (i) > A 2
I p = 0 A4omS4gJGw¬Vg 4lfDN‘qoxO S4_.MfgsAvKJImSVF S4FŽGsA4I‰K’}zQED α = ∅ DfGYK0BHSVFEl p = i
DGYK2BEDrK {X_.D t0 L
I p 6= 0 uwS4F _aDQHKA4_E_EoLIm}zQEDfgoLDoLDhh'D_HgJMlfMBEDfFdK L ± F GJA4ILK}zQPORImo D‹bILGYKsD j KsDfo }zQED
DK S (hi,j i) + S (hi,j i) > A L†± F°BEMfEFEILKA4oLSVgJG j BEDopAŽjœA\[
S (hi,j i) + S (hi,j i) > A
SVFGJQEI‰cvA4FdKsD
I q(U ) ≤ j ≤ q(U ) A4oLSVgJG j = j L
I j < q(U ) A4omSVgYG j = q(U ) L
I j > q(U ) A4omSVgYG j = q(U ) L
@D_aSVImFdK M = hi,j i cVMgYILiDBESVFHl ∀k R (M ) ∩ {U ,U } 6= ∅ SVQ S (M ) + S (M ) > A L
›kcVS4Img%i¬VQHgJD L V
p=i
AC=oN KMUH?;@BAC=
t0 _ t1
t2 <
UH>
0
i
i
0
i
1
0
2
|
0
1
|
|
0
3
0
2
1
0
1
1
1
0
2
1
2
O
1
0
0
1
0
1
k
p =i
2
k
p = i
S3 (M ) + S0 (M ) > A
p = i
S3 (M ) + S0 (M ) > A
S3 (M ) + S0 (M ) > A
U2
j1 = j 0
k+1
U2
q = j0
U1
j1 = q(U1 )
q = j0
q = j0
j1 = q(U2 )
U1
3y:'465
L
S1 (M ) + S2 (M ) > A
q(U1 ) ≤ j0 ≤ q(U2 )
|
U1
S1 (M ) + S2 (M ) > A
j0 < q(U1 )
U2
S1 (M ) + S2 (M ) > A
j0 > q(U2 )
⊕0
PnTVA> @[email protected]>efUH> KM=oGTTI ?^>,AKI UCTViKbG @?hUHP
0
UH>
=> Kzg Q
> G ^>,AON KMUH?;@A
p = i<
I M ∈ Z A4omSVgYG¬4g 4lfD0‘5oxO S4_.MfgsAvKJImSVF ⊕ SVFqGJA4ILK}zQED M ∈ α DK%BES4FEl0opA omIL¬VFED p = i DfGYK¹BED
K {b_aD t1 L
I M 6∈ Z A4omSVgYG%SVF_.DfQHK€D .DlKsQEDfg2oxO S4_.MfgsAvKJImSVF DfKwBESVFHl omANomIL¬VFED p = i DfGYKwBEDrK {X_.D t2 L
2
o#gJDGˆKsD5‘h'SVFdKJgJDg8}zQED omDfG8BEgJS4ILKsD q = j GYSVFdK2A4QEGJGYIBEDrK {b_aD ti L
7
q=j
t0,t1,t2
t3
;
g 4lD A4QomDfh'hD0_EgYMlfMBEDfFzK’SVFGsA4I‰K€}zQED GJQEgwlŠžiA4}zQED omIL¬VFED p = i SVFA
2
0
&
2
i\[email protected] @?hUHP
AC=oN KMUH?;@BAC=
0
[email protected] [email protected] CTVA
UH>
<
|βi | = 2pi − |αi |
“ FGJS4h'h'A4FdK8KsSVQbKsDG€lfDGwM¬dA4oLILKJMG%SVFAtBESVFElª
|β| = 2A − |α|,
› L ƒ4
w
x
+Cx Q
=
!¥
QHg8omDG€BEgYSVILKJDG q = j ui¬Vg 4lfD A4Qb‹S4_.MfgsAvKJImSVFEG ⊗ DfK SVFGsA4I‰K8}zQED
00
|β j | ≥ 2qj − |αj |
DK€BESVFEl
X
|β| =
|β j | ≥
X
(2qj − |αj |) = 2A − |α|.
š SVhhD0SVF>AtoxO Mf¬dA4oLILKsMN› L ƒV¹SVFDfF>BEMfBEQEI‰K2}zQED _aSVQEg8KsS4QHK j SVF>A
j
h
j
|β j | = 2qj − |αj |
L rg 4lfDq‘
I
S4F­GsA4I‰KBHM^WJ‘}zQED
DfGYKBEDK {X_.D Lª QE_E_aSVGJS4FEGN}zQED
oxO SV_aMgJAv|αKsILSV|F > 0 SVFGJA4ILKr}zQEDtoLDGrGJQHqILKsDf=GrBEjD_aSVImFdKsG’lSVFEGYMt1lfQHKsI‰jkG BHD β SVFdK A4QαhSV=ImFE∅G p _aSVImFdKsGfu
lfSVh'hD SVFA |β | = 2q Imo#{ACA4QŽ_EoLQEG0BEDfQb‹>KsDoLomDfG’GJQHILKsDfG LŸ ORImo{ŽAqQHFEDNGJDfQEomDGJQEI‰KsDomA'BEgYSVILKJD
DGYK0BEDK {X_.D t2 L# O ILo{>DFTA'BEDQb‹A4omS4gJGwDomoLDG2SVFdK0KsSVQbKsDG’omDG2BEDQX‹ p _aSVImFdKJG’DfK’BESVFElopA
q=j
oLIm¬VFED’DGYKwBEDrK {X_aD t3 L
2
j
j
00
j
j
i
j
i
7
AC=N A>anIJA\z\FAC=`TKMiCgCi N [email protected] [email protected] ?J=d=?`UHP
?;PVEHG W
K ?G V
P @ =uTQG c
K IJAC=UCTViKbG @?hUHPQ=UCTViKbG V
P @6=> K JI AC=dI ?< P8AC= ⊕,
@UH>[email protected] , H
U ?J=N G E UH;? K0TKCi G I G]^IJA\[email protected] [email protected]>,i > P8A , UH?J= JI AC=UCTViK G @?hUHPQ=
α
β
[email protected]
< <-9
UHP
TVA> @ON UHP8gvG KWKW?;E AKvD
2
O((lplq) )
A [email protected]=?
UHP
<
UHKW?;@
>
@BUH> @BAC=
IJAC=
DSgCUHPN ?;@w?hUHP
GTK C=dI G
N KMUH?;@[email protected] A
KMA\SG K ^>,A
@;WTVA
ti
?
UHP
p=x
TKMAPN
q=y
=^G > ,
G IJUHKM=OI G
gCUHPfX
00
I UCTViKbG @?hUHP
?
?
?
<
>LKbG @?hUHP
G IJUHKC=SIJASIJA\z\FA
N UHPVP8iCA TQG KOI GX
?
?
?
AP
q ª #
r
#
aLA\
?
?
>LKMA
?
?
[email protected] , G >a <
TKMUHt
r
β\α
?
pi
TIJA\[email protected] [email protected] GFP8iCgCAC=C=?;@BiON AI UCTViK G @?hUHP
00
<
q P€¹ # C f5 r ¨§
^%
P
< [email protected][email protected]]^IJA
α
?
<
5.2.3
<
?
|
c%6%
00
A [email protected]>VATQGH=pI UCTViKbG @?hUHP
L
P
[email protected]
UCTViKbG @w?hUHPQ=
3`:'4658
g G KSI G
qj
b"
U
00
,
\FAFN AvgCAvg >QGT?;@wKMAn?;I [email protected] ?;PV> @w?;IJA N A , G ?;[email protected]@BA [email protected]?;\z?J=^G @w?hUHP
2
N [email protected]> @A , G UH
P >LP8A gCUH\ TIJAba!?;@i AP O((lplq) (lp + lq)) <
TQG [email protected]>[email protected]=OIJAC=
> KbG @w?hUHP
0
UCTViKbG @w?hUHPQ=^<
UH>LKnI G I
g Q
> G ?;[email protected] GTVAoG > KbG
<
> P8AgCUHPfX
,⊗, ,
⊕0 , 0 <
=> TTVU4=4Ad=4A> IJA\[email protected]
#
%
ª ± FŽGsAvILK0}zQED5KsSVQHKJDG0oLDG2GJSVoLQHKsILSVFEG E lS4FzKJDFiAvFzK {U ,U } cVMfgJI‰iDFdK α ⊂ E ⊂ β LŸ I α = β
A4oLSVgJG α DGˆK UEImDfF,QHFEDGJSVoLQHKsILSVFŽBiAvFEG E(α,β)
lA4gro~ORSV_aMgsA\KsImS4F ⊕ Imh_EoLIm}zQED opA [ lfSVFdcVD‹bI‰KsMtBED
uiomDGwSV_aMgJAvKsILSVFEG , }zQED X α(i) ≤ p DfK2o~ORSV_aMgsA\KsImS4F ⊗ }zQED X α(i) ≥ p L.n SVQEg€KsDfgJhImFEDfg
α
oxOyA4oL¬VSVgYILKsžHh'D€ImoŸGJQ tK2BED gJDfKJSVQEgYFEDg α L
1
2
q
0
0
Q
p
i
p
i
V•
8 &^
w^
& I SVFNFED¹_.DfQHK_iAvGGsA|cVS4ImgªBHImgJDflfKJDhDFdK²GJIbImod{A’QEFED%GYSVomQbKsImS4F Lv± F5c4A2D‹b_EgJILh'DfgªoxO DŒ‹HILGˆ[
KsDFHlD0αBPORQE(FED2β GJSVoLQHKsILSVF_iA4g¹QHFEDrƒ„[~jkSVgJhQEomD L š SVhh'DwopA GsAvKJImGˆjœA4lfKJImSVFBPORQEFED ƒ„[~jkSVgJhQEomD ›k_EgJSVUHom™hD
ƒ„[ £w_aDQHK†GYD€jœAvImgJD8DfFKJDh_EG²omILFEM|AvImgJDr›Y¢ 0n £ ! ¥ ¦ udSVFA4QHgsA QEFqAvom¬VSVgYILKJžEh'D†_aSVoL{XFESVhIpAvob_aSVQEg
L
gJMGYSVQEBEgYD
lŠžiA4}zQED>_.S4ImFdKtBED M ∈ β \ α SVF‡A4GJGYSblfImDopAŽcvA4gYIpA4UEoLD V }zQEI€GJIm¬4FEILiD M ∈ E S4´
L lŠžiA4}zQEDA4GYGJIL¬VFiAvKJImSVFŽBHDGrcvA4gJImA4UEomDfG ILoªlfSVgJgYDGJ_aSVFEB,QEFDfFEGJDfhNUEomD ) L o
E ∈ E(α,β)
omS4F p DK
GJQ tK8BPORD‹b_EgYImhDg†}zQED E(V ) A5oLDG†U.SVFHFEDG_EgJS\WYDflfKJImSVFEVGDfK¹}XQHD E(V ) DGˆK [ lS4Fzc4D‹bD0GYDE(V
L
q
p)*,+.-0/21 365x7(+936:;-m/
M
M
Q
!#"$&%('()
q
=q
± FGYD b‹HD5QEFED5BEgJSVI‰KsD q = j L.± FGJA4XILKw}zE(j)
QED5lDfKYKsD5omIm¬4FEDrDGˆK0BED K {X_.D t A|cVDl i ∈ {0, . . . ,3} L
I#DoLomDrDGˆKwBED K {X_aD DfomomDrFHD lfSVFdKsILDFdK2A4QElQHF_aSVImFdK8ILFEBEMfKJDgYh'ILFEMrGJILFESVFSVFc4AlfSVFEGYKJgJQEILgJD QEFED
jkSVgJhNQHomD F Q DŒ‹b_EgJILthqA4FdK X E(j) = q L
I q = j DGˆK2BHD K {X_aD t1 S4Q t2 A4oLSVgJGŸ
q
j
i
0
j
q
j
|
F Qj =
|
^
Vhi,ji ⇐⇒ ¬Vhi0 ,ji
^
Vhi,ji ⇐⇒ ¬Vhi0 ,ji
hi,ji,hi,j 0 i∈β\α
I q = j DGˆK2BHD K {X_aD t3 
i0 −i=δqj
F Qj =
hi,ji,hi,j 0 i∈β\α
I DfGYK [ lfSVFdcVD‹bDDK0cVMfgJILED
AvomSVgYG’omDfG q _aSVImFdKsG2BED E GYQEg0omAqomIL¬VFED
GJSVFdKwKsSVEQ\WYS4QEgJG8lfSVFEGJMflQHKJILjkG Lٱ FDFBEXMfBEE(j)
QEILKP = q
i0 −i>δqj
q
gCUHPVE AbaLA^<
7
q
UH?;@
APQ=4A\S]^IJA
j
j
V >LP8AvGH=C=?< PG @w?hUHP N AC= EHG KW?G]^IJAC= ][email protected];IJAC=
E(V ) E iKW? XjA Xq (E(V ))(j) = qj [email protected]=4A> IJA\[email protected]=? V
^>,A
E(V )
q=j
[email protected]+*jt
E iKW? XjARI G-,fUHKW\z>LIJA
F Qj <
t1
A
B
? ? ?
? ? ?
α
VA ⇐⇒ ¬VB
?
qj
β\α
Z2 \ β
B
A
t2
? ? ? ? ? ? ? ?
VA ⇐⇒ ¬VB
qj
B
A
? ? ?
? ? ?
L ! 7 -,
FQ
]’D5hehDrSVF_aDQHKwBEMiFEImg8BHDG€jkSVgJhNQHomDG F P DŒ‹b_EgJILhqA4FdK X E(V )(i) = p
t3
3`:'4658
|
VA ⇐⇒ ¬VB
AC= fUHKW\z> IJAC=
i
j
p
i
x
+Cx Q
=
Hµ
- !#"&%-' ) % -' - " )
± F‚cvAh'A4ImFdKJDFiA4FdK'D‹b_EgYImhDgopA [ lSVFdc4D‹bILKJMBED E(V ) L o€jœA4QHK'Imh_.SVGYDg}zQPOyAvQHKsSVQHgqBED
lŠžiAv}XQHD>_aSVImFdK
Imo{”A,QHF‡}zQiA4BHgsA4FdK'FEDlSVFdKsDfFiA4FdKqA4QElQHF‚_aSVImFdKtBED E(V ) ›kc4SVImg
i¬4QEgJD L d M 6∈ E(V )
š AvG M ∈ β \ α L @#DG'BEgJS4ILKsDfG p = p(M ) DK q = q(M ) GJSVFdKCBHDK {X_aD t1 u t2 S4Q t3 L
š S4FEGJILBEMgYSVFEG€omDfG 'BEDfh'I [ BEgYSVILKJDG ∆ = R (M ) ∩ R (M ) L
I α 6= ∅ A4omS4gJG ∆ ∩ α 6= ∅ S4Q ∆ ∩ α 6= ∅ L
QE_E_aSVGJS4FEG¹BES4FEl α = ∅ Lb SVILK k ∈ {0,2} KsDfo }XQHD |∆ ∩ β | = max(|∆ ∩ β |,|∆ ∩
± FAA4oLSVgJG |β \ ∆ | < = p DK€BESVFEl ∆ lfSVFdKsImDfFdKwjkS4gJlMfh'DfFdK€QEF_.SVILFdK8BED
β |)
_aSVQEg%KJSVQHK E ∈ E(α,β) L
E
]’A4FEG€KsSVQEG€oLDG€l|A4GwSVF_.DfQHKwl|A4oLlQEoLDg€QEF k ∈ {0,2} KJDo#}zQED ∆ (M ) ∩ E 6= ∅ _aSVQEg8KsSVQbK
L ]’Dh'efh'DqSVF‡_aDQbKKsgJS4QHcVDfgQHF
KJDo8}zQED
E ∈ E(α,β)
_aSVQEg5KsSVQHK E ∈ E(α,β) Lª± F”DfF°BHMBEQEI‰K}zQPORImo¹DŒk‹bImGY∈KJD {1,3}
Kl JDo%}zQED_aSVQEg∆KJSV(M
QHK )k ∩6=El 6=SVF”∅A
_aSVQEg€KsS4QHK E ∈ E(α,β) La± F_aDQHK’A4omS4gJG€h'SXBEMfomImGYDg8omA [ lSVFdcVDŒ‹bILKsM5BED
R (M ) ∩ E 6= ∅
_EA4g8opANjkSVgYhNQEomDª
E
› L Zd
V →V
_aSVQEg%KJSVQHK N ∈ (β \ α) ∩ R (M ) L
š AvG
DKwomDG€BEgYSVILKJDG
DfK
GYSVFdK’BHD K {X_aD SVQ t3 A|cVDl5BiA4FEG
oLD%BEDfMgJFEILD6∈g²βl|AvGKsSVQEG²oLDG²_aSVImFdpKJG=BEDp(M
Gβ Y)QEg:opA’qBH=gJSVI‰q(M
KsD%BEQ) hehD%lS4KJM8_iA4t1,t2
g²gJA4_E_aSVgYK:‘ M L ]0A4FEG
lfDtl|A4Gfu¤lfSVhh'Dt_EgYMlMfBEDhhDFdKS4F_aDQHK KsgYSVQHc4Dg l KsDfo†}zQED_aSVQEgrKsSVQbK k BEI ŸMfgJDFdK BED l
DfGYKNFHSVF,cXILBED'_aSVQEg KsS4QHK E ∈ E(α,β) L¤± F<DŒ‹H_HgJImhDA4omS4gJG opA [ lfSVFdcVD‹bI‰KsM
R (M ) ∩ E
_EA4gP
› L z
¬V
_aSVQEg%KJSVQHK N ∈ (β \ α) ∩ R (M ) L
š AvG M 6∈ β A|cVDfl QHFEDrBEDG€BEDfQb‹BEgYSVILKJDG p = p(M ) S4Q q = q(M ) BHD0K {b_aD t0 SVQ t3 A|cVDl
DFdKsgYD%omDfG BEDfQb‹GJM}zQEDfFElDfG:BED¹_aSVImFdKJGªImFHlSVFEFzQEG L š S4h'hDS4FAwH‹bM%oLDG BEDQb‹_EILDBEG U DfK
M
SVFqGsA4I‰K%}zQPORImoaD‹bImGˆKsDrA4Qq_EomQEGBEDQX‹ }zQiA4BEgJA4FdKsG
DK
}zQEI.FED0lfSVFdKsImDfFEFEDfFzK
U
AvQElQEF_aSVImFdK8BED α La± FhSbBHMomILGJDrAvomSVgYG8opA [ lfSVFdcVRD‹bI‰(MKsM _i) A4g€opRAjkS4(NgJhN)QEoLD
› L V
¬V ∨ ¬V
_aSVQEg%KJSVQHK€_aSVImFdK N ∈ (β \ α) ∩ R (M ),N ∈ (β \ α) ∩ R (M ) L
2SvKsSVFEGwA4oLSVgJG F opAlS4F\WYSVFElKsImS4FBHDG8jkSVgJhQEomDfG F P u F Q u› L ZVu› L du› L 4 L
q
O
|
}
i
|
0
i
|
i
i+1
2
i
i
i
k
i
i
|βi |
2
k
i
0
i
2
i
k
k
0
k0
q
k
w
N
M
l
|

q
k
w
}
N
l
|
1
2
l1
q
l2
w
N1
1
2
l1
l2
€
w
qj
7
i
APQ=4A\S]^IJA
[email protected]=4A> IJA\[email protected]=?
V
(
! " )
[email protected] j
* t gCUHPVE AbaLA(=4AIJUHP
E iK? XjAdI G f
, UHKW\z> IJA
E4?N A=?`[email protected]=4A> IJA\[email protected]=?mI G
E(V )
,fUHK\z>
IJA
F
F<
w
}
w
j
p
[email protected]
q
UHP8gcI APQ=4A\S]^IJA
[email protected]=^G @?J= &
, G ?J=^G]^IJA^<
%-(#" ) &
N2
∀i,jXp (i) = pi ,Xq (j) =
=4UHI > @?hUHPQ= E(α,β) [email protected](P8UHP
[email protected] iKW? XjA
N AC=
š SVhh'D F DfGYKrQHFEDtƒ„[~jkSVgJhNQHomD SVFŽ_aDQHKrQbKsImoLImGYDg€QEFTA4oL¬VSVgJI‰KsžEhD omILFEM|A4ILgJD _aSVQEg’gYMGYSVQEBEgYDƒ„[
£”A|c4Dl%omAwjkSVgYhNQEomD
›Y¢ 0n £ ! ¥ ¦  L4 I F DfGYK²GJAvKsILGYjœA4ILGsA4UHomD4u¯oxOyAvom¬VSVgYILKJžEh'Dª_.SVQHg gYMGJS4QEBEgJD8ƒ„[ £
F
dƒ
8 q€MGYSVQHK)*+.-m/(14365872+936:;-0/
&^
w^
L
u
D .DlfKJQEDgwSV_aMgJAvKsILSVFEG8BED lS4h'_EoLMfKJImSVF
KŠA4UEILomI‰KsM’S4Q α 6⊂ β
α 6⊂ β gJDfKJSVQEgYFEDg f_iA4G8BHD GYSVomQbKsImS4FEG "! u
α ←α β ←β
# $ U ,U ∈ β KJDo#}zQED p(U ) = pmin,p(U ) = pmax % & L
u
α ← α ∪ {U ,U } β ← β
' D .DlKsQEDfg2S4_.MfgsAvKJImSVFEG¹BED5lfSVh_EomMKsImS4F
( )* KŠAvUEImoLILKsM2SVQ α 6⊂ β
α 6⊂ β gJDKsSVQHgJFEDfg2‘omAtUaSVQElfom+D N_.S4QEg8lŠžESVILGJImg8QHFA4QbKsgJD (U ,U )
,! l|A4oLlQEoLDg8omDfG8jkSVgJhNQHomDG¹U.SXSVomMfDFEFHDG F Q ,F P , › L ZdŠu¤› L z¹DKN› L V
lŠžEDgYlŠžEDg2ImFEGˆKŠA4FElfIpAvKJImSVF V BEDfG8c4AvgJIpAvUEomDfG¹cVMfgJILiA4FdK2lfDG€jkSVgJhQEomDfG
ªGsA\KsImGˆjœA4ImGJA4UEoLD gJDKsSVQHgJFEDfg E(V )
(Q(p,q),{p,q})
α←∅ β←∆
&
d
d
1
2
1
d
1
2
2
d
&
1
h
j
,! "! gJDKsSVQHgJFEDfg_iA4G€BED GJSVoLQHKsILSVFEG
i
h
}
2
w
& x
Q
+Cx =
VZ
∆2
?N
M
∆3 = ∆ k 0
?
VN ⇒ V M
∆1
∆0 = ∆ k
?N
M
¬VN
? N1
M
¬VN1 ∨ ¬VN2
?N2
3y:'4658
L
|87 AC=-,fUHKW\z> IJAC=
Aba&TKW?;\SG [email protected] G
*jt
gCUHPVE Aba!?;@Bi
BES4FEFEDtQEFEDtImFHGYKŠAvFElImAvKsILSVFBEDG cvA4gJImA4UEoLDG0gYDFEBiAvFzK F cXgsAvImD L#± F,DF,BEMfBEQEILK BES4FEltQEFEDGJSVoLQHKsILSVF
A4Q,_EgYSVUEom™fh'DBHD'omAgYDlfSVFEGYKJgJQElKsILSVF L ImFHSVF ,U },∆) = ∅ u#ImojœA4QHK BES4FEljœA4ImgYDQEF­A4QbKsgJD
lŠžES4I‰‹ L IPKJSVQEG€omDfG8lŠžESVI‰‹ SVFdK2MKsMrjœA4I‰KsG%ILoPFPO {E({U
A_EA4G€BED GJSVoLQHKsILSVF L ›¡cVSVILg€A4om¬4SVgJI‰KsžEhD5µ¯
KsQEBHImSVFEGh'A4ImFdKJDFiA4FdK#omA8lSVh_EomDŒ‹bILKsMªBED:o~O A4om¬VS4gJILKJžEhDV›kSVF lSVh_HKJD²omD²FESVhUEgJD:BPORSV_aMgsA\KsImS4FEG
MfomMhDFdKsA4ImgYDGPiAvGJGJIL¬VFiAvKJImSVFHG%DfKwSV_aMgJAvKsILSVFEG€A4gYILKsžHh'MKsIm}zQEDfGŠ L
o.{A'AvQ_HomQEG (lq) lŠžESVI ‹_aSVQEg8omDfG€_.SVILFdKsG U DK U L
@#DfG†S4_.MfgsAvKJImSVFEG:BED€lfSVh'_HomMfKJImSVFt_EgYDFEFHDFdK O((lplq) (lp + lq)) S4_.MfgsAvKJImSVFEG L ›œoLDhh'D L µ L ƒV
o€gYDGYKJD‘,c4SVImglS4hNUEImDfF‚BPORSV_aMgJAvKsILSVFEGtGJSVFdKqFEMlfDGJGJA4ImgYDGC_aSVQEg'l|A4oLlQEoLDgtopAjkSVgYhNQEomD F L
@#DfG€jkS4gJhNQEoLDG F Q DfK F P _.DfQHcVDfFdK’GYD lA4omlfQEomDfg8DF O(lplq(lp + lq)) L n SVQEg€omDfG%jkSVgJhQEomDfG
› L Zd€DK› L z€SVFA WYQHGYKsDU.DfGJSVILFŽBEDH‹bDfg M DfK N LP“ oLomDGw_aDQHc4DFdK BESVFElGJDNlA4omlfQEomDfg’DF
Lzn SVQEg:omDfGjkSVgJhNQHomDGw› L ZdªSVFAr_EgJILSVgJIEUaDGYSVImFtBED8H‹bDfg M,N ,N L A4ILG†DfFjœAvILK|u
O((lplq) )
AvQqomILDQtBEDwH‹bDg M SVFq_aDQHKGJDwH‹bDgopA BEgJSVI‰KsDwBED€K {b_aD t3 SVQ t0 GJQHgomA4}zQEDoLomD€DGˆK%omD€_aSVImFdK
La± F_aDQHKwBES4FElrKsgJS4QHcVDfg2lfDG8jkSVgJhQEomDfG8DF O((lp + lq)(lplq) ) L
M
1
|
2
2
1
|
2
d
2
|
w
h
2
}
j
i
w
1
2
2
8 }
± FDFBEMfBEQEILKP
$ [email protected]>
KCiC=4UHI EHG [email protected]
)*+.-m/(14365872+936:;-0/
< UHK?;@'> \FAP8iCgCAC=C=?;@hG
(Q(p,q),{p,q}) <
P
G I
w^
& O(lq 2 (lplq)2 (lp + lq))
[email protected]
UCTViKbG @[email protected]
€ Œª€†f5
± FtBEMGYIm¬VFHD8_iA4g omA0lomA4GJGYDwBHDG [ lS4Fzc4D‹bDG†GJDfomSVF D L ›kcVS4Img²BEMiFEILKJImSVFƒ L L !  Ld± F'_aDQbK
L\± F
MfKsDfFEBEgYDjœA4lfImomDfh'DfFdK#o~Q(D)
O A4om¬4SVgJI‰KsžEhDª_EgYMlfMBEDfFzKª_.S4QEggJMGYSVQEBEgYD
(Q(D),D)
GJD'b‹HD'QEFED'BEImgYDlfKJImSVF p ∈ D L ± FGJDH‹bD'KsS4Q\WYSVQEgYG5BEDfQb‹_EImDfBEG U ,U ∈ α KsDo:}zQED p(U ) =
Lª± F,_aDQHK jœA4ImgYDomDfG5SV_aMgJAvKsILSVFEGrBEDlS4h'_EoLMfKJImSVFTDFD‹HA4hImFiAvFzKrKJSVQEG omDfG
pmin,p(U ) = pmax
lSVQE_HomDG (p,q) ∈ D L
± Fq_aDQbK¹KsgYSVQHc4Dg%oLDG:jkSVgJhNQHomDG F P A4GJGYSblfImMDfG8‘5lŠžEA4lQEFHD0BEDGBEImgYDlKsImS4FEGBHD D AvImFEGYI.}zQED2omDG
jkSVgJhNQHomDG › L Zd¹DfK› L z L
@AGJDfQEomDBEI 'lfQEoLKJM DGˆK'BEA4FEGopAjkSVgYhNQEomD› L VuªDF°D .DfK'_.SVQHgNKsgJS4QHcVDfglfDfKYKsDjkSVgJhNQHomDCSVF
QHKsILomImGYDqoLDqjœA4ILK}zQED U DK U GYSVFdKtBiA4FEGBHDQb‹­}XQEA4BEgsAvFzKJGtBEI .MgYDFdKsGfuSVgNlfD FPO DfGYKt_EoLQEGomD'l|A4G
omSVgYGJ}zQED o~ORSVFlSVFHGJImBH™gJD QEFlS4QE_EomDrBHD BHImgJDflfKJImSVFEG (q,r) A|cVDl p 6∈ {q,r} L
QH_E_.S4GJSVFEG2BESVFEl }zQED U DK U GYSVFdK0BiAvFEG’oLD }zQiAvBEgsA4FdK R (M ) L.± FŽlSVFHGJImBH™gJD omDfG2BHgJSVI‰KsDG
A|cVDfl i ≤ q(M ) SVQ q = j A¯c4Dl j ≥ p(M ) L† I¹o~ORQEFED BEDlDfGtBEgJSVI‰KsDfGDGYKtBHDK {X_aD t1
p = i
A4omSVgYG0SVFT_aDQHKrKJgJSVQHc4Dg QEFT}zQiA4BEgJA4FdK R (M ) u k 6= 0 }zQEIlS4FzKJImDfFzK QEFT_aSVImFdKrBHD E _.S4QEg’KsS4QHK
L
E ∈ E(α,β)
I o~ORQEFEDBEDfGrBEgJS4ILKsDfG p = i A|c4Dl i ≥ p(M ) DfGYK BEDNK {X_aD t2 SVQ t3 DfKrQHFEDtBEDGrBEDfQb‹TBEDfh'I [
BEgJSVI‰KsDfG p = i,q ≥ q(M ) SVQ p = i,q ≤ q(M ) lSVFdKsILDFdK5_EomQEG BED p _aSVImFdKJG5BED β uA4omS4gJG lDKJKsD
BEDhI‰[ BEgJSVI‰KsD0lfSVFdKsILDFdK2jkS4gJlMfh'DfFdK2QHF_aSVImFdK€BED E _.SVQHg€KJSVQHK E ∈ E(α,β) uHSVF>_aDQbK2BHSVFEl5A4QEGJGYI
KsgJS4QHcVDfg2QEF}zQiA4BEgJA4FdK R (M ) u k 6= 0 }zQEI#lfSVFdKsImDfFdK2QHF>_aSVImFdK8BHD E _aSVQEg%KJSVQHK E ∈ E(α,β) L
± FŽ_aDQHK0gsA4ILGJSVFEFHDg’BHDNopAh'efh'D jœA S4FŽGJI o~ORQEFEDBHDGrBEgYSVILKJDG q = j A|cVDl j ≥ q(M ) DGˆKrBED
K {b_aD t2 S4Q t3 DfKrQHFEDNBEDfG0BEDQX‹BEDfh'I [ BEgYSVILKJDG q = j,p ≥ p(M ) SVQ q = j,p ≤ p(M ) lfSVFdKsILDFdK
_EomQEG€BED q _.S4ImFdKsG8BED β L
]’S4FEl<BiAvFEG>lfDGŽlA4Gop‘”lSVhh'DTSVF lSVFHFiA mK>BEDfQb‹ }XQEA4BEgsAvFzKJGSVF _.DfQHKŽDŒ‹b_EgJILh'Dfg>omA [
lSVFdcVDŒ‹bILKsM DF>_EgJDfFiA4FdKwomDfG8h'efh'DfG%jkSVgJhNQHomDG › L V L
QH_E_.S4GJSVFEGrBHSVFElN}zQEDNKJSVQHKsDfG0omDfG0BEgYSVILKJDG A¯c4Dl
SVQ q = j A|c4Dl j ≥ q(M )
)
GJSVFdKtBEDCK {X_aD t0 u t2 SVQ t3 uªKsS4QHKsDfGomDGBEDhI‰p[ BE=gJSVI‰iKsDfG p =i ≥i,qp(M≥ q(M
SVQ p = i,q ≤ q(M )
)
lSVFdKsILDFEFHDFdK’DŒ‹HA4lfKJDhDFdK p _aSVILFzKJG€BED β uHDfKw}zQED KJSVQHKJDGwomDG8BHDhI‰[ BHgJSVI‰KsDG q = j,p ≥ p(M ) SVQ
lfSVFdKsILDFEFEDfFdK’DŒ‹EAvlfKsDfh'DfFdK q _aSVImFdKsG8BHD β L
q = j,p ≤ p(M )
]O A4_EgY™G omDfG jkS4gJhNQEoLDG F P SVQ F Q S4FGJA4ILK }zQEDGJI M DfGYK5BiA4FEG (β \ α) ∩ R (M ) AvomSVgYGrImo
D‹bImGˆKsDBEDG_aSVILFzKJG
KsDoLGN}zQED V ⇔ ¬V ⇔ ¬V DGˆKtBiA4FEGoLDG
jkSVgJhNQHomDG F P DK F PQ L ∈ R (M ),P ∈ R (M )
š SVhhD5ILo¤{ACQEF}zQiA4BEgJA4FdKr}zQEIªFHDNlSVFdKJImDFdK AvQElQEF_aSVILFzKu.QHFBEDGr}zQiA4BHgsA4FdKsG0AqKsSVQEG’GJDG
omILKYKsMfgsA4Qb‹'‘ MfgJS LE± F'DF BEMfBEQEI‰K¹}zQED0BEA4FEG¹lŠžEA4lQEFBEDfGKJgJSVILG}zQiA4BHgsA4FdKsGKsSVQEGomDfG¹omI‰KJKJMgsAvQb‹tGJS4FzK
M}zQEILcvA4oLDFdKsG L n SVQHgwDŒ‹b_EgJILh'Dfg8opA [ lS4Fzc4D‹bILKJM ImoPGYQ tK2BHSVFEl BED gsA„WYS4QHKsDfgwomAjkSVgJhQEomD
›L
V ⇔V
_.SVQHg€KJSVQEG N ,N ∈ R (M ) L
q€ #f5 &^
v%
v%
#
$#
#
v%
q
c)*,+.-0/21 365x7(+936:;-m/
1
2
1
2
2
w
h
}
h
&
‚
1
2
1
2
0
k
i
k
j
q
{
w
i
j
2
1
1
2
3
M
P1
P2
~
q
Q
h
N1
$ 1
AP
2
7
O(n )
UCTViKbG @?hUHPQ=^<
N2
?
2
n = maxp∈D lp
G IJUHKM=
)*,+.-0/(14365x7(+936:;-0/
(Q(D),D)
TVA> @
@KMApKMiC=4UHI >
` L S4Q t2 SVQ
t3
P1
?
t0
? N1
M
?N2
VN1 ⇔ V N2
?P2
t3
L¥
3y:'4658
|87 G
,fUHKW\z>
SVQ t2 SVQ
t0
IJA=>4TTIJi\[email protected] G ?;KMAONG PQ=IJAzg GH=N AjTI > =oN AoN A>a
N ?;[email protected]?hUHPQ=
“ F<jœA4I‰K5S4FKsgYSVQHc4MqQEF<A4om¬VS4gJILKJžEhD_.SVo‰{XFESVhIpA4o h'efh'DfG5GYISVFh'DKoLDG BEImgYDlKsImS4FEG5BEA4FEG opA
BES4FEFEMD L n omQEG€_EgYMlILGJMfh'DfFzKwoLDr_EgJSVUHom™hD A4oL¬VSVgJI‰KsžEhIm}zQED
]2SVFEFEMfD'
D
|D|
((p ) ,p ∈ D)
SVgYKJImD N
E ∈ Q(D)
∀p ∈
D∀i ∈ Z X (i) = p
_aDQHKwefKJgJD gJMfGJSVoLQDF O(n d ) SV_aMgsA\KsImS4FEG%SV´ n = max lp DKwDfK d = |D| L
)*,+.-0/(14365x7(+936:;-0/ o+.-0/
Z
P
<
APQ=4A\S]^IJAmXYPV?
N AoN ?;[email protected]?hUHPQ=C_
Z [email protected]=>[email protected]?<
P APQ=4A\S]^IJA
p
=> ?;@BAC=
_=??;[email protected]_oN [email protected]
DFEHG IJA>LKM=oNG PQ=
i i∈Z
[email protected]?hUHPQ=
E iKW? [email protected]
i
7 2
p∈D
5 CΩ
@#DqKsžHMSVgY™hDƒ L L ZŽhSVFdKsgYDq}zQED oxOyA4oL¬VSVgYILKsžHh'D_EgYMlfMBEDfFzK'_aDgJhDfKBED gJDflSVFHGYKsgYQEImgYDComDfG [
lfSVFdcVD‹bDfG \[ lfSVFEFEDŒ‹bDGqBH™Gq}zQED D lSVFdKsILDFdK {x,y} L:“ F‚D ŸDKCImo¹GJQ tK AvomSVgYGBEDc4MgYILiDfg'_EgYM|A\[
omA4UEomDfh'DfFdK0}zQEDomDGr_HgJS\WYDflfKsILSVFEGrGYDomS4FlfDG BEImgYDlKsImS4FEGrGJSVFdK lfSVFEGYKJILKJQEMDfG5BPORDFdKJImDgYG FESVFTFzQEomG L¤ I
lvO DfGYK omDl|A4G0KJSVQHKsDGJSVoLQHKsILSVF [ lfSVFdcVD‹bDtDGˆK A4QEGYGJI \[ lSVFEFHD‹bD L ImFESVFILo FŸOR{ŽA A4QElQHFEDtGJSVoLQHKsILSVF
\[ lSVFHFED‹bD L
& \[ lfSVFEFEDŒ‹HDfG
D
{x,y}
F
*
cVDfl D = {x,y} u¯lD†FPO DfGYK _i(FA4G¤,D)opA€h'DfImoLomDQH gJDlfSVh'_HomD‹bI‰KsM²O(n
lS4FEFz)QED\DFD ŸDKªBEA4FEGªlfDlA4G¤op‘8BPO A4_EgJ™fG
oLD¹KsžEMfSVgJ™fh'D2ƒ L L ZromDG [ lSVFdcVDŒ‹bDG \[ lfSVFEFEDŒ‹HDfGGYSVFdK†D‹HA4lfKJDhDFdKomDG „[ lS4FEFED‹bDfG· 2[ lSVFdcVDŒ‹bDGu
SVg8BEA4FEG ¢ ¹][email protected]]0 ¥4¥ ¦ omDfG€A4QHKJDQEgYG2BES4FEFEDFdKwQEF>A4om¬4SVgJI‰KsžEhD’DfF O(n ) L
n S4QEg’omA v[ lSVFEFHD‹bILKJM5lvO DfGYKrQEF_.DfQŽ_EomQHG2lfSVh'_HomIm}zQEM L. QE_E_aSVGJS4FEG2KJSVQ\WYSVQEgYG’}zQED D lSVFdKJImDFdK
L @#D5oLDhh'D L ƒ L µ5hSVFdKsgYD }XQHD α DGˆK0ImFHGJMlA4UEoLD GYDoLSVFomAtBHImgJDflfKJImSVF p = x L ]0A4FHG’opAGYQEILKJD
{x,y}
SVFcvA A|cVSVILgrU.DfGJSVILFTBEDo~ORImFEGYMlA4UEImoLILKJM5BED α GYDoLSVFTo~O A4QHKsgYDNBEILgJDlKsILSVF q = y LŸn SVQEg lfDopACILoªGYQ tK
BED H‹bDfgoLDGBEDfQb‹°_EILDBEGGYDomS4F°omABEImgYDlfKJImSVF q DfKBPORD .DlfKJQEDgomDfGSV_aMgsA\KsImS4FEGNBEDlS4h'_EoLMfKJImSVF L
_EgYImSVgYIªlfDopAqcvACjœA4ILgJDNhSVFdKsDfgropAClS4h'_EoLD‹bILKJMA4oL¬VSVgJI‰KsžEhIm}zQED BPORQEFjœA4lfKJDQEg L A4ILGrSVFTGsA4I‰K
}zQPO A|c4AvFzK†BED8H‹bDfglfDG:_EImDfBEG:omDfG†BHgJSVI‰KsDG q = qmin DfK q = qmax GJSVFdKBED8K {X_.D lqti udS4F'_aDQHK†BES4FEl
GYDQEoLDhDFdK€lSVFHGJQEo‰KsDg¹BEDQb‹ lŠžESVI‰‹qBEDr_EILDBEG8_aSVQEg¹lŠžiA4lQHF>BEDfG8BEDQX‹lŠžESVI ‹ L @DfG8BEDQX‹ _EILDBEG8}zQED
oxO SVFAH‹bM hNQEo‰KsIm_HomImDfFdK%BESVFHl omANlSVh_EoLD‹bILKJM’KsžEMfSVgJIL}zQED _iA4g 4 }zQEI#DGˆK2lfSVFEGˆKŠA4FdK L
q

q
Q
Y
G IJUHKM=
?
[email protected][email protected]
)*,+.-0/(14365x7(+936:;-0/
TVA>[email protected]
q

[email protected]
=?
@KCAKMiC=4UHI >
[email protected] G
gI GH=C=4A
AP
7
y
N AC=
UCTViK G @ ?hUHPQ=^<

Y
+
jt gCUHPVE AbaLAC=
c
5
n}
Q
&
2
8 &^
w^
& _EgJ™fGqA|cVS4ImgD .DlKsQEM>lDfG'SV_aMgJAvKsILSVFEG DGˆK [ lSVFdcVDŒ‹bDGJDfomSVF
DK'ILFEGJMfl|A4UHomD4u²ILo%DGˆK
L¤ S4ILK
BESVFEl \[ lfSVFEFEDŒ‹bD L š SVFEGJILBEMgYSVFEG BESVFEltGYDGlfαSVh_.SVGJA4FdKsDfG v[ lSVFHFED‹bDG {x,y}
α = A ∪ ... ∪ A
DfK y = max L QEILKYKsD2‘5gYMSVgYBESVFEFEDfg¹oLDG A S4FC_.DfQHKGJQE_E_aSVGYDg%}zQED (x )
x = max
x
L
š
DGYKlgYSVImGYGsA4FdKJD SVhh'D α DGˆK [ lfSVFdcVDŒ‹HD2SVFDFqBHMBEQEI‰K†}XQHD y DfGYKlgJS4ImGJGJA4FdKsDwSVQBEMlfgJSVILGJGsAvFzKJD L
QE_E_aSVGJS4FEG2_EA4g8D‹bDh_EoLD }zQED y DfGYKwlgYSVImGYGsA4FdKJD L š SVhh'D A DGYK \[ lSVFHFED‹bD SVF>A%
q
&
n
O}
1
q
M ∈Ai
i
M
i
M ∈Ai
q
i
i
xi = max xM =
i
i
min xM
+1
yi = max yM =
min yM
± F_aDQHK2A4oLSVgJG%DŒ‹b_EgJILh'Dfg8opA v[ lfSVFEFEDŒ‹bILKsMrBHDGwGJSVoLQHKsILSVFEG%_iAvgwoLDG%jkSVgYhNQEoLDGP
M ∈Ai
n
i
M ∈Ai+1
M ∈Ai
M ∈Ai+1
+1
z}
Vxi ,yi ∨ Vxi +1,yi +1 .
›kcVS4Img%i¬VQHgJD L µ¯•d
O
A3
?
y = y2
?
A2
?
y = y1
?
A1
?
y = y0
?
A0
3y:'465.
L µ¯•
x = x0
r
|
a&TKCAC=C=?hUHP
x = x1
Vt gCUH PVP8Aba!?;@i
N AOI G
Vx1 +1,y1 +1 ) ∧ (Vx2 ,y2 ∨ Vx2 +1,y2 +1 ) <
A>LI=
x = x2
t ,fUHKW\z>LIJA
TQG KOI G
IJAC=`TVUH?;[email protected] =N A
β \α
(Vx0 ,y0 ∨ Vx0 +1,y0 +1 ) ∧ (Vx1 ,y1 ∨
?;[email protected] APG [email protected] PQ=
[email protected][email protected]=^<
± FSVUHKJImDFdKwBESVFHl
& G IJUHKC=
Vt gCUHPVP8Aba!?;@i
IG
? D [email protected][email protected] {x,y} [email protected] =? F C
A [email protected] G gI GH=C=4A N C
A = *jt gCUHPVE AbaLAC=
)*,+.-0/21 365x7(+936:;-m/
(F ,D) TVA> @ @KMAKCiC=4UHI > AP O(n7 ) UCTViKbG [email protected] ?hUHPQ=^<
[email protected]
7
G >[email protected]=4UHI >[email protected]?hUHP
,&G
?;@AONG PQ=
\[email protected]@wKMARN AC=`?;P8i<G I ?;@iC=YI G K< AC=RNG PQ=YIJAC=
IJAIJA\z\FA
<
Rk+1 (M )
gCA
^> [email protected]?;\
.t gCUHPVE AbaLAC= Vt gCUHPVP8AbaLAC=
0 r
gCUHPN ?;@?hUHPQ=yN A`I UCTViKbG @w?hUHP ⊕ <
PFA [email protected] GTK C=
jTVUH> KIJAC=
Sk (M ) + Sk+1 (M ) ≥ A [email protected] Rk (M ) ∩ E = ∅ G IJUHKM= E ⊂ Rk−1 (M ) ∪
E [email protected] Vt gCUHPVP8AbaLA^< E [email protected]@BAnUCTViKbG @w?hUHP IJAoP8U ;!G > α
[email protected]> @BAC=IJAC= =4UHI >[email protected]?hUHPQ= E ∈ E(α,β) [email protected] gCUHPVP8AbaLAC=^<
< nIJUHKM= ^>,A
[email protected] ?;[email protected]\[email protected] Vt
7 v[ lfSVFEFEDŒ‹bDG
}
TVU4=C=?]^IJAIJUHKC= ^>VA
&^
w^
&
L
J
Q x 4 !
€ ¹5 P#ªf5 € % rf5 f ]0A4FHGqomD_EgYSVUEom™fh'DBED>opATgJDflSVFEGˆKsgYQElfKJImSVF A,_EgJILSVgJI8S4F³Imh_.S4GJDA4Qb‹‡GJS4omQHKJImSVFEGtBEDcVMfgJILEDg
L n gYMlILGJMfh'DfFzKP
α ⊂E⊂β
]2SVFEFEMfD' |D|(F ,D) ((r ) ,p ∈ D)
N
α ,β
Z
SVgYKJImD
E∈F
∀p ∈ D∀i ∈
Z X (i) = r
α ⊂E⊂β
@²OyAvom¬VSVgYILKJžEh'D€_EgYMGYDFdKsM _EgYMlMfBEDhhDFdK8gJMfGJSVQHB>AvQEGJGYIŸlDr_HgJSVUEoL™hD L oŸGYQ tK8BED0_EgYDFEBHgJD A4Q
BEMf_iA4gˆK α = α DfK β = ∆ ∩ β L
0
)*,+.-0/(10-0+
%
0
<
=> ?;@BAC=
p
i i∈Z
2
XYPV?hAC=
0 0 N A
Z
P APQ=4A\S]^IJA
! 0
D
7
EHG IJA> KC=SNG PQ=
[email protected]
D
=> [email protected]?J_dN A>aSTQG [email protected]?hAC=
_`=?`?;[email protected]_cN [email protected][email protected]?hUHPQ=
p
i [email protected]@AI ^>VA
p
^%
0
A`TKMU ]^I \FA
0
0
)*,+.-0/21 0
-0+
(Q(D),D)
@?hUHPQ=^<
E iKW? [email protected]
Q
TVA>[email protected] @wKMA
KCiC=4UHI >
AP
O(n7 )
UCTViKbG t
. ©€ €t¤r
@²OyAvom¬VSVgYILKJžEh'D<µ<BEMflgJI‰KBiA4FEG>lD,lŠžiA4_EI‰KsgJDA”MfKsMImh_EomMfh'DfFdKsMGJQHg>h'A4lŠžEILFED Lw® FHDoLM¬V™fgJD
SV_bKsImhImGJAvKsILSVFrA€MfKJM†jœA4I‰KsD²GYQEgªoLD²lŠžESVI‰‹ BHDGª_HImDBHGP\A4Q5oLImDfQ BHD†H‹bDfgªomDfGBEDQb‹5_EILDBEG¤GJILhNQEo‰KŠA4FEMfh'DfFdK
SVFqD .DlfKJQEDrBEDfG%SV_aMgsA\KsImS4FEGBHD0lSVh_EoLMfKsILSVFqDfFdKsgJD’omDGBEDQX‹ lŠžESVI ‹'BED0GJS4gYKsD’}XQHD0_iA4gˆjkSVImG†QEF GJDfQEo
_EILDBGJQ Kw_aSVQEg€_aSVQHcVS4Img%gYMBEQEILgJD omDr_HgJSVUEoL™hD ‘ƒ„[ £ L
@#D,KsA4UEoLD|A4Q BESVFHFEM<‘”o~O A4FEFHD‹bD L µ,h'S4FzKJgJDT}zQPORDF jœAvILK KsgY™GGJSVQbcVDFdKŽA4QElQHF _EILDB FPO DfGYK
‘b‹HDfguDfKCoLDGCS4_.MfgsAvKJImSVFEGBEDŽlfSVh_EomMKsImS4F‡GYQ GJDFdK‘<gYDfKJgJSVQHc4Dg omAGJS4omQHKJImSVF ›œ}zQEI2DGYK A4omSVgYG
QEFHIm}zQED¯ L š DflIBESVFEFED'BESVFElqQEFEDqlfSVh_EomDŒ‹HI‰KsMh'S¯{4DFEFEDqD‹b_.MfgJILh'DfFzKsA4omDImFbjkMgJILDQEgYDq‘ O(n ) DfK
lfDlI#ILFEBEMf_.DfFEBiA4hh'DfFdK€BEQ>FHSVhNUEgYD BHD BHImgJDflfKJImSVFEG L
£¤SVQHKsDjkSVImG'lDfG gJMGYQEoLKsAvKsG'D‹b_aMgJILh'DfFdKŠA4Qb‹‚BESVI‰cVDfFzKCefKsgYDŽlS4FEGJILBEMgYMGA|cVDfl_EgJQEBHDFElfDl|A4gCoLD
¬VMfFEMgJAvKsDfQEgrA4oLM|AvKJSVImgYD5QbKsImoLImGYM5_aSVQEg’jœAvUEgJIL}XQHDg0BEDfG [ lS4Fzc4D‹bDG DfGYK GsAvFEG0BESVQbKsDUaD|A4QElfSVQE_ŽKJgJSV_
gYQEBEImhDFdKsA4ImgYD L
&
L

Q
q
3
!
8 &^
w^
& ± F A­cbQ BiAvFEG omD>lŠžiA4_EI‰KsgJD_HgJMlfMBEDfFdK>}zQEDToxO SVF‚_.SVQbc4AvILKCgYDlfSVFEGYKJgJQEILgJDQEFHDGJSVoLQHKsILSVF [
lfSVFdcVD‹bD€‘0_iAvgYKsILg BHD¹GJDfG²_EgJS„WYDlfKJImSVFHGDFNQHFNKsDh_EG_aSVo‰{bFHSVh'ImA4o L A4ILGªSVFtA’GYDQEoLDhDFdKlfSVFEGˆKsgJQHILK
QEFHD GYSVomQbKsImS4FPuHILoPGJD _aDQHKw}zQPORImoŸ{ DfFAvILK€UaD|A4QElfSVQE_BPO A4QHKsgYDG L
I F DfGYK€QEFED’lopAvGJGJDrBHD0_iA4gˆKsImDfG¹BED Z DK8GJI D DGYK%QEFDFEGYDhNUEoLD’iFEI.BED0BEILgJDflfKsILSVFEG¹A4omSVgYG¹SVF
BEILgsAN}zQED o~ORSVFA GYI#_.S4QEg8KsSVQEG E ,E ∈ F SVFA%
(F ,D)
q
2
/2:;+.: 3
1
2
k› opA lopAvGJGJD’BED2KJSVQHKsDfG%omDfG¹_iAvgYKsILDGBED Z †DK%GYI D DfGYK¹EFEIPA4oLSVgJG†SVFCFŸOyA€WJAvhqA4ILG
n ¦  L. I C BEMGYIm¬VFHD5omAtlfopA4GYGJD BEDGwKsS4KsA4omDfh'DfFdK2lfSVFdcVDŒ‹HDfGrA4oLSVgJG
L
 L A4gJBHFEDg8DK nUL rgJI‰K h'A4FEFqSVFdK€h'SVFdKJgJM4ubDFdKsgYD A4QHKsgYD4uE}zQED GYI |D| ≥ 7 SVFA (C,D)
± F*c4ADfGJGsA|{4DgqBHDKJgJSVQbcVDgtQEF*KJžEMSVgY™hDA4FiA4oLSV¬VQEDC_aSVQEgoLDG [ lfSVFdcVDŒ‹HDfG L š SVhh'DClvO DfGYK
A_EgYImSVgYI%QEF‚GYQ\WYDfKBEI lImoLD4uomA_EgJDfh'IL™gJD MKŠA4_aDGJDgJA,BHDjœA4ILgJD QEFHDMKsQEBHD>DŒ‹H_aMgYImhDFdKŠAvomD L:“ F
_iAvgYKsILlQEoLImDfg%SVFc4AtBESVFHFEDgwQEFED h'MKsžESXBED _.DfgJhDfKYKŠA4FdKwBPORMFzQEhMgJDfgwKJSVQEGwomDfGwlfSVFdKsgYD[ DŒ‹bDh_EomDfG’‘
oxO QEFHImlI‰KsM0BHD KsA4ImoLomDwGJQ GJA4h'hDFdKw_aDfKsI‰KsD L
Œ¤² i¹5 5€ I
(∀p ∈ D Xp E1 = Xp E2 ) =⇒ E1 = E2
P(Z )
/(:;+.: F3 =(P(Z
›¡cVSVILg5¢%][email protected]€
),D)
2
2
2
/2:;+.: 3
~
q
Q
c%
$($
%
2%
' " )"-' ) !# &" ! !#" ) #@ SVgJGY}zQEDo~ORSVF”ABEDQb‹*DFEGYDhNUEoLDG F DK F A|{dAvFzK'omDfGh'efh'DfGN_EgYS\WYDlKsImS4FEGtGJDoLSVF D uoLDG
_aSVImFdKJGwILFdKsMgYDGYGsA4FdKsGwGJS4FzK2lDfQb‹>}zQEIFED5GYSVFdK’_EA4G€BiA4FEGwo~ORImFdKsDfgJGYDlfKJImSVFBED F DK F L.± F>MKsQEBHImD
BES4FEl8GJDfQEomDfh'DfFzK†oLDG²DFHGJDhUEomDfG E = F \ F DfK E = F \ F BPORSV´topA0BEMiFEI‰KsImS4FGYQEILcvA4FdKJD
,! ! 7
G€I‰KslŠžEILFE¬v[ lSVh_.S4FEDFdK
(E ,E )
+
−
+
−
ASgCUH>4TIJAzN
I GvgI GH=C=4A
7
’] S4FEl DK D L
AC=
F
−
APQ=4A\S]^IJAC=
I([email protected]
/2:;+.: 3
E
[email protected]
E
+
+
+
APQ=4A\S]^IJAC=
[email protected] APQ=4A\S]^IJAC=N AoN ?;[email protected]?hUHPQ=
−
+
D
=?RUHP
[email protected] ?J= efUH?;[email protected];=
[email protected]
−
+
−
−
[email protected]> P
TVUH> K
GFIJAC=oN A> a
[email protected]\
gCUHPN ?;@?hUHPQ==> ?;EHG [email protected]=
\[email protected]?hUHPQ=
=4AIJUHP
D<
DfGYKrMf}XQHILcvA4EomDfFdKr∪‘'EoxO ILFED‹bILGYKsDfFElfEDNBED ∪ GE€I‰KslŠEžEILFE¬v∪[ lESVh_.S4FEDFdKsG _F.SVQHg
E0 N ?J= efUH?;[email protected]
(F ,D)
E ACg
−
+
@AI=^>VA
V¥
+
0 [email protected]
−
0 [email protected] PQ=
<
F
¥V•
8hw
M 8W
' &^
fw^
± FGYQE_E_aSVGJDBEMGYSVgJh'A4ImGw}zQED DfGYK0GˆKŠA4UHomD _iA4g’ILFdKsDgYGJDflfKsILSVF L.± FŽ_aDQHK’A4omS4gJGw_EgJDfFEBEgJD5BiA4FEG
opABEMfEFEILKJImSVF L µ L µ o~ORDFEGYDhNUEoLD EFGJQEI‰cvA4FdK#
\
DfK E ∪ E ∈ F }
E = {E ⊂ Z : E ∪ E ∈ F
“ FD .DfKwlfSVh'hD DfGYKwGYKsA4UEoLDr_iA4g8ImFdKJDgJGYDlKsImS4FS4F>A
DfK
DKwGJI
FPO DfGYK:_iAvGBEImG¡WYSVImFdKªFA|cVDl E ∪E A4oLSVgJG A4QElfQEFDFHGJDhUEomED E ∪KsDEoX}zQE∈D FE ∪EE ∈∪FEDK ∈EF ∪E ∈EF
FPO DfGYK2BEImG¡WYSVImFdK€A|c4Dl E ∪ E L
T
“ FjœA4I‰K’GYI
A4omSVgYG E _aDQbK0GJDl|A4oLlQEoLDgwBEDNopAhqAvFEIm™fgJD
= {F ∈ F : F ⊃ E}
GJQEI‰c4AvFzKJD E =ENVP(E)
V
S
´
lim
F
0
2
0
+
&
−
+
−
−
−
0
+
+
0
0
−
+
0
0
n→+∞
n
F0 = ∅,
Fi+1 = (ENVP(Fi ∪ E + ) \ E + ) ∪ ENVP(Fi ∪ E − ) \ E − )
]’SVFElSVFT_aDQHK’KsDGˆKsDfgrGJIQEFTlfSVQE_EoLD (E ,E ) DfGYK QEF NG€I‰KslŠžEILFE¬v[ lfSVh_.SVFHDFdK ¬Vg vlD‘CoxOyA4o [
¬VSVgJI‰KsžEhD ƒ¹
’MgYILiD GJI (E ,E ) DfGYKwQEF tG €ILKJlŠžEImFE¬\[ lS4h'_aSVFEDfFzK .uiDKwl|A4oLlQEoLD E
−
v
#$
X E
& p∈D %
6= X E
gJDKsSVQHgJFED €®
+
p
,! "! E ∩ E
gJDfKJSVQEgYFED €®
+
−
6= ∅
−
0
− p
+
+
"! F ←∅
F ← (ENVP(F ∪ E + ) \ E + ) ∪ ENVP(F ∪ E − ) \ E − )
F ∩ (E − ∪ E + ) 6= ∅
gJDKsSVQHgJFED €®
,! gJDKsSVQHgJFED r
F
FŸO DfGYKw_EomQHG8h'SXBEILEM L
E0 ← F
$-
' % )"$-'
" )
"'
(
! -''()#
AvImFdKsDfFiA4FdKSVF0cvA8GORImFEGY_EImgYDgPlfD†lfDfK¤A4oL¬VSVgYILKsžHh'D¤_.S4QEgPl|AvomlQHomDg.KsSVQEG#oLDG#G €ILKslŠžHImFE¬v[ lSVh_aSVFEDFdKJG
ImFElfomQEG BiA4FEG²QEFDFEGYDhNUEoLD€A4gJUHILKsgJA4ImgYD:lSVhh'DQEFgJDflfKsA4FE¬VoLD ∆ BED¹KŠAvImomoLD l ×h Ld± FQHKJImomILGJDvu\QHF_.DfQ
H¥ µ
lfSVh'hDwBiA4FEG%oLD’lŠžiAv_EILKJgJD0_EgYMlfMBEDfFzKuiQEFA4oL¬VSVgJI‰KsžEhD€¬VomS4QHKsSVFŸuXS4FClŠžEDgYlŠžED BESVFEl’omDG¹G€ILKJlŠžEImFH¬v[
lfSVh'_aSVFEDfFdKsG (E ,E ) KsDfomG€}zQED
+
−
^4
M α+ ⊂ E + ⊂ β + ,
α− ⊂ E − ⊂ β − .
± F _.DfQHK2A4omS4gJG€A4QE¬4h'DfFzKJDg α ,α DfKwBEILh'ILFXQHDg β ,β ¬Vg 4lfD ‘o~O A4om¬VS4gJILKJžEhD’Z L
± F¶_aDQHKTAvomSVgYG>jœA4ILgJD,QEFED­_EgYSblfMBEQHgJD­gYMlQHgJGJI‰cVD ›œA4om¬VS4gJILKJžEhD d_aSVQEgŽA 'lŠžHDgŽKsS4QEGoLDG
NG€I‰KslŠžEILFE¬v[ lSVh_.S4FEDFdKsG ImFHlomQHG%BiA4FEGwomDrgYDlKŠA4FE¬4omD ∆ L
“ FA4FEFHD‹bD GYSVFdK0_EgYMGYDFdKsMfDG BEI .MgYDFdKsDfG’oLImGYKJDGwBED G€I‰KslŠžEILFE¬v[ lSVh_.S4FEDFdKsG SVUHKsDfFzQEDG
¬Vg 4lD ‘tlDK2A4om¬VS4gJILKJžEhD L
® F'D‹bDh_EoLD’Imh_aSVgYKsA4FdK:DGYK¹omD€l|AvG
L ]0AvFEG¹lfD’l|AvG
u
›[email protected]#Drl|A4oLlQEo._EgJDfFEB
SVFCKJgJSVQbcVD A4QElfQEF NG €ILKslŠžHImFE¬v[ lSVh_aSVFEFDFd=K CIQ(x,y)
mFHlomQHG%DBiA4=FEG8{x,y,2x
QHFlA4gJgYM+13y, −× x13+L 2y}
¥4žGYQEg€QEF š DfomDfgJSVF dZVZ L  š DflIImFdcXILKJD ‘tMfFESVFElfDgwopANlfSVF\WYDlKsQEgYD5GYQEILcvA4FdKJD
! Ln A4g'lSVFdKsgYDBiAvFEGqomA_EgYSv[
AvomžEDfQEgJDfQEGJDfh'DfFzKCSVF‚FED>gJMfQEGJGYImgsA_i(Q(x,y),{x,y,2x
A4GC‘BEMhSVFdKsgYDgq+lfDy,gY−MGJQHxoL+KŠA\K 2y})
lŠžiAvImFED GJDflfKJImSVF>SVFcvAh'SVFdKJgJDg€QEFKsžEMfSVgJ™fh'D BES4FzK2o~ORQEFED BEDfG2lfSVFEGJMf}zQEDFElfDGrGYDgsAopA_EgYDQHc4DNBED
L
(Q(x,y,2x + y, − x + 2y),{x,y,2x + y, − x + 2y})
s§ ¤ ©8 € f¹Œ©
+
−
+
−
}
H
‚
}
P
/2:;+.: 3
G
<
/(:;+.: 3
('
$
%- ,! ! 7
$#
) )#%- &' )
(' - _aDFdKsD
P1 (R) = R ∪ {∞} < 7 APQ=4A\S]^IJA
N AC=pN ?;[email protected]?hUHPQ=zN A R [email protected] AP ]^? [email protected]?hUHP G E ACg P1 (R) < K gCApD I GTTI ?hg G @?hUHP
: ax + by 7→
Z
b
− a < P gCUHP ,fUHPN KbG N UHP8gn=4UH>LE [email protected]> P8A N ?;[email protected]?hUHP d G E ACgS=^G
(d) < P8A @wKbG PQ= ,fUHKW\SG @?hUHP
a+by
[email protected]?;E [email protected]> P8A ,[email protected]?hUHP N A P1 (R) NG PQ=I > ?;t\ \FAoN AOI G ,fUHKW\FA y 7→ c+dy < [email protected] G @K G gCA
=> KIJAC=ON ?;[email protected]?hUHPQ=ON AI GTTI ?hg G @?hUHPI ?;P8i G ?;KMAON AI G ,fUHK\FA (x,y) 7→ (ax + by,cx + dy) <
GzN KCUH?;@[email protected]?;E AcKMiCAI;[email protected](I APQ=4A\S]^IJA
2
,! ! R ∪ {∞}
7
An]^?;[email protected]
^>QG @wKMASP8UH\S]^KMAC=nN ApI G
N KMUH?;@[email protected]?;E AzKCiCAI;IJA
P1 (R) =
[email protected] i XYPV?xTQG K
x2
x3 x4
G E ACgIJAC=
N A
_aDFdKsD
x1
=
x3 − x 1 x4 − x 1
(x3 − x1 )(x4 − x2 )
:
=
x3 − x 2 x4 − x 2
(x3 − x2 )(x4 − x1 )
gCUHPVE [email protected]?hUHPQ=OiE4?N [email protected]=IJUHKM= ^>,A
∞
[email protected]
APOefA> <
7
Ao]^?;[email protected]`N A
^>QG @wKMAoN ?;[email protected][email protected]
#@ D8UEImgJA4_E_aSVgYKBED BHImgJDflfKJImSVFEG²BHImGYKJImFElKsDfG²lSVFEGYMlfQHKsI‰cVDG†DGˆK:KsSVQ\WYS4QEgJG²GYQE_.MfgJILDQEg†‘ 1 Ld“ FD .DfK
lfSVh'hD = uvS4F_aDQHK GJQE_H_.SVGYDg}zQEDomDfG _aDFdKsDfG x ,x ,x ,x BEDfG ’BHImgJDflfKJImSVFEG#c4MgJI‰iDFdK
La± F>A'A4oLSVgJG = SV´ f (x) = La± gwGYI x < x < x < x f
x <x <x <x
DfGYKwBEMlfgJSVILGJGJA4FdKsD GJQEg ]x ,∞] DKwBESVFEl f (x ) < f (x ) L
n AvgDŒ‹bDh_EomDGJI€SVF”_EgYDFEB‡omDfG}zQiA\KsgJDBEILgJDflfKsILSVFEG
omDfG_aDFdKJDGqGYSVFdK
gYDGJ_aDlKsI‰cVDhDFdK ∞,0, − 2, udSVFtcVSVI‰K:}XQHD2oLD8UEImgJA4_E_aSVgYK²BHx,y,2x
DG†}XQEAv+KsgYDwy,BH−ImgJDfxlfKJ+ImSVFE2yG²gYMSVgYBESVFEFHMDG†DGˆK#
IJAO]^?;[email protected] AIJA>LKM=
[email protected][email protected]?;E AC=^<
c}
2
0}
d c
b a
a b
c d
1
&
3
1
f (x3 )
f (x4 )
x1 x2
x3 x4
4
2
3
2 3
x−x1
x−x2
4
4
1
2
−2
1
2
0 ∞
=
( 21 − (−2))(∞ − 0)
=
( 21 − 0)(∞ − (−2))
1
2
− (−2)
=5
1
−0
2
1
2
3
4
¥dƒ
8hw
-
V36:;-0/
M (α+ ,α− ,β+ ,β− )
A¯c4Dl
fD K i ∈ {min p(M ) . . . max
]O A4U.S4gJBopAN_EgYDhIm™gYDrlSVFHBEILKJImSVF BED G €ILKslŠžHImFE¬
X β (i) < X α (i) S4Q X β (i) < X α (i)
gJDKsSVQHgJFEDˆAvQElQEFG €I‰KslŠžEILFE¬ #$
p∈D
p +
M ∈∆
p −
,! X β
p −
p +
= Xp α− (i)
α+ ← α+ ∪ {M ∈ β+ : p(M ) = i}
β− ← β− \ {M 6∈ α− : p(M ) = i}
,! X β
p − (i)
M ∈∆
p(M )}
± F A KsSVQ\WYS4QEgJG
α + ∩ β− = ∅
lA4g β
DK
oLDrGJDQHo#_.SVILFdK
KsDfo#}zQED
∪ β− ) \ (α+ ∪ α− ))(i) = 1 Xp α+ (i) = Xp α− (i)
M←
M ∈ (β+ ∪ β− ) \ (α+ ∪ α− )
p(M ) = i
β+ ← β+ \ {M }
β− ← β− \ {M }
p
,! "! +
]’DfQb‹bIm™fh'D lS4FEBEILKJImSVF BHD G €ILKJlŠžEImFE¬
F ←∅
F ← (ENVP(F ∪ α+ ) \ β+ ) ∪ ENVP(F ∪ α− ) \ β− )
F ∩ (α− ∪ α+ ) 6= ∅
gJDKsSVQHgJFEDˆAvQElQEFG€I‰KslŠžEILFE¬ ,! F FŸO DfGYKw_EomQHG8h'SXBEILEM L
% & = Xp α+ (i)
α− ← α− ∪ {M ∈ β− : p(M ) = i}
β+ ← β+ \ {M 6∈ α+ : p(M ) = i}
,! X ((β
fw^
p + (i)
' &^
β− ∩ α+ = ∅,β+ ∩ α− = ∅,α+ ⊂
β+ ,α− ⊂ β−
8W
β− ← β− \ (F ∪ ENVP(α+ ))
β+ ← β+ \ (F ∪ ENVP(α− ))
+
⊂ Z2 \ α−
^4
M j/(7 ,56*
' -
,36:;-m/
α = β
¥VZ
(α+ ,α− ,β+ ,β− )
GYSVFdK2ILFdc4AvgJIpAvFzKJG
(α+ ,α− ,β+ ,β− )
α+ ,α− ,β+ ,β−
α− = β −
+
(α+ ,α− )
DK
A lŠžED
! !
lŠžES4ImGJILg M ∈ (β ∪ β ) \ (α ∪ α )
M ∈ β › α ∪ {M },α ,β ,β \ {M })
+
j/(7
+
−
+
+
,52*
"! M ∈ β +
"! j/27
,! −
› α ,α
j/(7
−
+
−
−
,52*
+
−
› α ,α ,β
∪ {M },β+ \ {M },β− )
,56*
+
−
+
\ {M },β− \ {M })
n S4QEg€_EomQEG8BHD BHMfKŠAvImomG8GYQEg€omDrUEILgsA4_H_.SVgˆK8SVF_.S4QEgJgJANgJD¬VA4gJBEDfg ¢ ¹Dfgs¥V•bu L µ ¦ L

,! ! 7 APQ=4A\S]^IJA E = {A1 ,A2 , . . . ,An } ⊂ R2 [email protected]> P TVUHI ; < UHP8AFgCUHPVE AbaLA
=? ∀A ∈ E UHP G A 6∈ conv(E \ {A}) <
7 AjTVUHI ; < UHP8A E ⊂ R2 [email protected]>LP D tJTVUHI ;< UHP8A [email protected]> @AoN KCUH?;@A d N AON ?;[email protected]?hUHP NG PQ= D E iKW? XjA
|d ∩ E| ∈ {0,2} <
7 AYTVUHI ; < UHP8A E ⊂ R2 [email protected] ?;@YG P8A\[email protected]< >LI ?hAK =?2?;[email protected] >LP8A @wKbG PQ= ,fUHKW\SG @?hUHPG P8A
2kπ
2kπ
]^? [email protected]?;E A φ [email protected] n ∈ N @AI ^>,A φ(E) = {(cos n , sin n ) : k ∈ {0, . . . ,n − 1}}
EHG [email protected][email protected]
<
9
<
<
<
<
<
|
D
= u?;@g
=4A\S]^IJAN AoN ?;[email protected]?hUHP
D tJTVUHI ;<
D tJTVUHI ;<
D tJTVUHI ;<
D tJTVUHI ;<
I([email protected]>LP
I([email protected]>LP
I([email protected]>LP
I([email protected]>LP
7
> P
APQ=4A\S]^IJAN AON ?;[email protected]?hUHPQ=
[email protected]> P8A
@wKbG
[email protected]@UH>[email protected]=oNG
> ?;P<
D<
t gCUH\
[email protected]
TVUH> KI G
BG >[email protected]\[email protected] ?;@ UHP
P
UHP8A
gCUHPVE AbaLA?;P8gI > =oNG PQ=
UHP8A
gCUHPVE AbaLA?;P8gI > =oNG PQ=
UHP8A
gCUHPVE AbaLA?;P8gI > =oNG PQ=
UHP8AOG
gsAvKJImSVFHFEDoLomDG 7 AC=
=> ?;t
Z2 <
Q2 <
R2 <
P8A\[email protected]< > I ?hAK^<
N ?;[email protected]?hUHPQ=OgCUHPQ=4iCg>[email protected]?;E AC=ON A
l4ORDGYK2omD0KJžEMS4gJ™hD L BHD'¢ ¥ !|¦ L
lomA4Img L
(1) ⇔ (2)
(2) ⇔ (3) ⇒ (4)
dTKMUCTVU4=?;@?hUHPQ=
gI GH=C=4AN [email protected]@ G IJA\[email protected]`gCUHPVE [email protected] APVt
/(:;+.: 3
(C,D) <
GTQGH=
D [email protected]
2
PQ= ,fUHKW\SG @?hUHP I ?;P8i G ?;KCAdKbG @?hUHPVP8AI;IJA φ N A R
1
PQ= {−2, − 1, − 2 ,0,1,∞} <
AC=O]^?;[email protected];=ON A
i\[email protected] @w?hUHP
|
UH?;@
i ^>L?;EHG [email protected]=
[email protected]> P
<
PQ=
{ 43 , 32 ,2,3,4} <
@BAI;IJA
^>VA
[email protected]=N A
φ(D)
¥
}
|
O
8W
}
}
|
M ' &^
fw^
c4SVImg2¢ 4• ¦ SVQŽ¢ Avgs¥ zub_iZ v[^Z \¦ L ›œ@D2omDfh'hD L ƒ L ƒ GYDgsA5QEFED’¬4MFEMfgsA4oLImGsA\KsImS4F
BED lDKJKsD Imh_EoLIml|A\KsImS4F L 
l4ORDGˆK’oLD0KsžEMfSVgJ™fh'D L NBED'¢ ¥ !¯¦ L
(5) ⇒ (6)
SVI‰K d ,d ,d KsgJS4ImGBEILgJDlKsILSVFEGtlSVFHGJMlfQHKsI‰cVDfGCBED D }zQED>o~ORSVF‡lfSVFHjkSVFHB‚A|cVDfl
(6) ⇒ (7)
omDfQEgJG8_aDFdKsDfG L oPDŒ‹HILGYKJD QHFEDrKsgsAvFEGYjkSVgYhqA\KsImS4FC_EgJS„WYDlfKJILc4D φ BED P (R) KsDfomomD’}XQHD φ (λ ) =
L š SVh'hD λ ,λ ,λ ∈ Q lfDfKYKsD KsgJA4FEGYjkS4gJh'AvKsILSVF<DfGYK‘ŽlSXDfjp[
0,φ (λ ) = 1,φ (λL ) = ∞
ilILDFdKsGgJAvKsILSVFEFEDfomG š S4h'hD d ,d ,d GYSVFdKlSVFHGJMlfQHKsI‰cVDfGS4F*DF*BEMfBEQEILK}zQED omDfGN_.DfFdKsDG
BED D = φ (D) \ {0,1,∞} GYSVFdKrFEMf¬dAvKsI‰cVDfG LŸ SVILK x o~ORQEFED BEDNlfDG0_aDFdKsDfG LP± FŽA
L ± FDFBHMBEQEI‰Kw}XQHD x ∈ D = {−3, − 2, − 1, − , − } L ]’SVFHl {0,1,∞} ⊂∈
{ , ,2,3,4}
L š SVh'hDrSVFA%
φ (D) ⊂ {−3, − 2, − 1, − , − ,0,1,∞}
BES4FEl {−3, − 2} 6⊂ φ (D) L
=
BES4FEl {−3, − } 6⊂ φ (D) L
=
BES4FEl {−2, − } 6⊂ φ (D) L
=
BES4FEl {− , − } 6⊂ φ (D) L
=
BES4FEl {−3, − } 6⊂ φ (D) L
=
± F>DfF>BEMfBEQEI‰K€BESVFEl }zQED φ (D) DGˆK2ILFEloLQEG8BiA4FHGwoxO QHFBHDG€KsgJS4ImG%DfFEGJDfhNUEomDfG2GYQEILcvA4FdKJGP
L
F = {0,1,∞, − 2, − 1, − }
L
F = {0,1,∞, − 3, − 1}
L
F = {0,1,∞, − 1, − }
I φ (D) ⊂ F SVF_EgJDfFEB φ opANKsgJA4FEGYjkS4gJh'AvKsILSVFComILFEM|AvImgJD’A4GJGYSblfImMfD5‘ φ
I φ (D) ⊂ F uXlSVhh'D φ (F ) = {0,1, − 2, − 1, − } ⊂ F A|c4Dl φ (y) = ubSVF
_.DfQHKw_EgJDfFEBEgYD φ = φ ◦ φ L
I φ (D) ⊂ F SVF_EgJDfFEB φ = φ ◦ φ ◦ φ S4´ φ (y) = L
o¤GYQ tK0BED _EgYDFEBEgYDoxO ILhqA4¬4DrgJMlfIm_EgYSX}XQHD5_iAvg φ BEQ>_.SVo‰{X¬VSVFED gJDf_EgJMfGJDfFzKJM‘'omA
(7) ⇒ (3)
i¬VQEgYD L µ²
(4) ⇒ (5)
|
8hw
1
2
3
1
1
2
1
3
1
1
0
|
−3 −2
0 1
|
− 12
−3
0
−2
0
− 12
1
−3
0
|
|
1
− 13
1
− 31
∞
− 13
∞
1
3
1
2
0
1
3
1
2
1
1
3
1
4 3
3 2
|
2
2
1
9
8
9
4
1
2
1
8
3
1
3
1
x 0
1 ∞
1
3
1
1
3
1
2
9
8
1
1
3
9
8
1
1
|
1
2
1
|
2
|
1
3
3
|
1
|
1
1
1
2
2
2
|
|
1
3
1
2
2
1
2
2y
3−y
1
2
3
1
UHP8ADp=4UH\z\[email protected] =NG PQ=
Z2
3
1
y
Q
3y:'465
Lµ
|
Z
P
D tJTVUHI ;<
G E ACg
D = {−2, − 1, − 21 ,0,1,∞} <
2
^4
M )' "('
¥
%-
-' - ± F _.DfQHK%BHMfiFEILg%opA [ lS4Fzc4D‹bILKJMrDfK8omA [ lfSVFdcVDŒ‹HI‰KsM’jkS4gYKsD’GJQEg R DF gYD_EgYDFiAvFzK¹KsDoLomD2}XQHDomoLD
omANBEMfiFHILKsILSVF GJQHg Z L š DflI¤A4h™FED A4QX‹BEMfEFEILKJImSVFEG8GYQEILcvA4FdKJDGP
,! ! iFEI E ⊂ R
; <
*
,
D
∀A ∈ E
A 6∈ QCON V F ORT (E \ {A})
gJAvKsILSVFEFEDfomomDfG 7 D
q
q
2
2
Z
UHP
<
<
UH?;@
> P
[email protected]
> P
TVUHI
L
I
D
D [email protected]?hUHPQ=
> ?;P<
= u?;@Bg
<
UHP
P
D tJTVUHI ;<
jt gCUHPVE AbaLA
APQ=4A\S]^IJASN AFN ?;[email protected]?hUHPQ=
AC=
[email protected] ^>L?;EHG [email protected]=OG >avN A> aOTKMUCTVU4=?;@?hUHPQ=
I([email protected]>LP
UHP8A
TV/(UH:;P8+.A: PV3 @
t gCUH\
GTQGH=
*jt
[email protected]=4AIJUHP
=?
<
D
@'>ViCUHK \FA
2
APQ=4A\S]^IJA
G
<
P
OTKMUCTVU4=?;@?hUHPQ=SN >
=> ?;EHG [email protected]=
N A*jt gCUHPVE AbaLAC==4AIJUHP
D
TVUH>LKI APQ=4A\S]^IJAON AoN ?;t
(Q(D),D) <
,[email protected]=4AIJUHP D <
± F”c4ATBEMfh'SVFdKJgJDg ⇒ (8) ⇒ (9) ⇒ (5) L @²O ILh'_EoLImlAvKsILSVF (1) ⇒ (8) DfGYK'lopAvImgJDq_EQEImGY}zQED
}zQPORQEF5KJS4KŠAvomDhDFdKlf(1)
SVFdcVD‹bD%DfGYK [ lS4Fzc4D‹bD¹_aSVQEg#KJSVQEG¤omDGDfFEGJDfhNUEomDfGªBED†BEILgJDlKsILSVFEG†›œ_HgJSV_aSVGJI‰KsILSVF
ƒ L L µ„ L @¤ABEMfh'S4FEGYKJgsAvKJImSVF BED (8) ⇒ (9) ⇒ (5) c4ASXllfQE_.Dfg8KsSVQHKJD opAiFBED lD lŠžiA4_HILKsgYD L
<
I([email protected]>LP
UHP8A
gCUHPVE AbaLA
q
! ! !
( ! ! !
lSVFHGJImBH™gJD’QEF tG € ILKJlŠžEImFE¬\[ lS4h'_aSVFEDfFzK
]0A4FHG%KsSVQHKJD0lDKJKJDr_iA4gYKJImD2SVF
qBED [ lS4Fzc4D‹bDG€FESvKsM
L
(E ,E )
± F‡cvATlSVFEGˆKsgYQEImgYD>QEF D[ _aSVo‰{b¬4SVFED [ lfSVFdcVD‹bDjkSVgYK E l4ORDGYKC‘,BHImgJDQEF‡DFEGYDhNUEoLD E }zQEI
c4MgJI‰iD
› L µ„
∀M ∃(i,p,q) ∈ A R (M ) ∩ E = {M }
]0A4FHG omDG _HgJDhImDfgJGCDŒ‹bDh_EomDfG }XQHD,oxO S4F‚jœA4ILK ‘­opA<h'A4ImF‚o~ORDFEGYDhNUEoLD E = E ∪ E DfGYK
BEILgJDflfKsDfh'DfFdKrQEF D[ _aSVo‰{b¬4SVFED [ lSVFdc4D‹bDjkSVgYKrh'A4ILG’lDFŸO DfGYK _iA4G2KsSVQ\WYS4QEgJGroLDl|AvG›kcVS4Img’i¬4QEgJD
L ƒV L
+
q
−
q
D
pq
i
−
+
q
E+
D
3y:'465
Lƒ
!
Z
|
P= u?;@g
[email protected]`N A*jt gCUHPVE AbaLAC=
^>L?uP
I M ∈ E ,p ∈ D A4omSVgYG M DfGYKwQEF_.S4ImFdK€BED
+
E− = ∅ <
> ?;P<vgCUH\
E−
UH>LK
p
@UH> @uTVUH?;[email protected]
M
N A
E+
?;[email protected]> P
ASP (M ) <
[email protected]=> PFTVUHI
E−
KsDfoP}zQED
Π
;<
UHP8A
*jt
gCUHPVE AbaLA^<
p(Mp ) = p(M )
G > @BUH> KpN A
M_
L
@AI^>VA
Π∩
± F,GJDtH‹bD
L¤ SVI‰K KsDfo²}XQHD R (M ) DGˆKh'A\‹bImhNQHh BiA4FHG oxO DfFb[
GYDhNUEoLD {R (M )|R (M ) ∩ EM ∈= E∅} _aSVQEg8i,p,q
oxO ILFEloLQEGJILSVF L
± F_aDQHK2GJQH_E_.S4GJDg2_iA4gwD‹bDh_EoLD5}zQED i = 0 La I R (M ) FŸO DfGYK0_EA4G2QHF rbn AvomSVgYGwILo#D‹bImGˆKsD
QEFHD'BEILgJDflfKsILSVFEG r KsDfomomD}zQED'opA BEgYSVILKJDBHDqBEILgJDlKsILSVF r AQEFHDqImFdKJDgJGYDlKsImS4F<A|cVDfl R (N ) M¬dAvomD
P
[email protected]
pq
i
pq
i
+
i\[email protected] @w?hUHP
pq
i
−
pq
0
p,q
0
¥
8hw
M 8W
' &^
fw^
L ]2SVFEl
L QEI‰KJKsD‘CgJDfh'_EomA4lDfg _iA4g SVF_.DfQHKrGYQE_E_aSVGJDfg
A|cVDfl
L
L ± F,cvAh'SVFdKJgJDgr}zQED
QH_E_.S4GJSVFEG }zQED
FEDtlSVFdKsILDFdK5A4QElfQEF<_aSVImFdK
L0± F A R (M ) =
BED lfD}zQEI GJDfgsA°DF lSVFdKsgJA4BEILlfKsILSVF A¯c4Dl,opA*h'A\‹bImh'A4omI‰KsMBHD
L AvImG%AvQElQEF _aSVILFzKBED FED0_aDQHK%efKsgYD0BiA4FHG
lA4g¹GYIŸImo {'A|c4AvILK%QHF
L ›kc4SVImg%i¬4QEgJD L Zd
KsDo#_aSVILFzK AvomSVgYG8SVFA4QEgsAvILK
‘
q
{M }
r = αp + βq
αβ > 0
r
−r
α > 0,β > 0
p(Mr ) < p(M )
R0qr (M )
−
E
Ripq (M )
R0pq (M ) ∪ R1pr (M )
E−
R1pr (M )
N
M ∈ QCON Vrp (Mr ,Mp ,N )
qr
0
Mr
Mp
M
q = cste
Mq
N
LZ
3`:'465
|
UH> K ^>,UH? H
; t G [email protected]?;I.> P
r = cste
p = cste
G > @UH>LKON A
M
E− <
N ?J= efUH?;[email protected] E ACg
I p(M ) > p(M ) AvomSVgYGwS4FA q(M ) < q(M ) u DF>ImFdcVDfgJGsAvFzK p DK q SVF>DGˆK’gJA4h'DfFEM5A4Q>lA4G
_EgJMflMBHDFdK L
n SVQEgNoxO QEFHImlI‰KsM'Imo:jœA4QHK2WYQEGˆKsDCgYDh'A4gJ}zQEDfg}zQED GJIImo:{*A|cvA4ILKBEDQb‹ rHn Π DK Π KsDfomGN}zQED
AvomSVgYGwILoP{A4QEgJA4ILK€QEFHDBEILgJDlKsILSVF p KsDfomoLD }zQED opAtBEgYSVILKJD p = p(M ) GJDgJA4ILK
Π ∩ E = ∅,j =L 1,2
BEImG¡WYSVImFdKsD A|c4Dl E
2
r
r
1
2
−
j
−
"! ! ?;[email protected]
A∼
A,B N A E + [email protected]\[email protected] G ?;KMA\[email protected] ^>L?;EHG [email protected];=
i _ p _ q _ N ∈ Q @AI=^>,A A,B ∈ Ripq (N ) [email protected] Ripq (N ) ∩ E − ⊂ {N }
A>aTVUH?;[email protected];=
2
^>,AzUHP
A ∼e B
G ?;@`IJAC=ON A>a
[email protected]=4A>LIJA\[email protected]
gCUHPN ?;@w?hUHPQ=
=?c?;[email protected]
(i,p,q) ∈ AD
=>L?;EHG [email protected]=
e
B
?;PN ?hgCAC=FN AC=
=?
@AI=
ASP (A) = Ripq (A) [email protected] ASP (B) = Ripq (B)
ASP (A) ⊂ ASP (B) UH> ASP (B) ⊂ ASP (A) U IJA TVUH?;[email protected] N ∈ {hp(A),q(B)ipq ,
hp(B),q(A)ipq } @AI ^>VA A,B ∈ Ripq (N ) _yE iKW? XjA Ripq (N ) ∩ E − ⊂ {N } <
omD¹}zQiAvBEgsA4FdK²}zQEIXcVMfgJILED%opA’BHMfiFEI‰KsILSVF L ƒ L ’DfK}XQHIEGJSVI‰Kªh'A\‹bImhQEh
GYGJI R (O) ⊂ R (O)
› L ƒV
R (N ) ≺ R (N )
SV´ O = (0,0) u›klDK’SVgYBEgJDrFHD BHM_aDFEB>FEI#BED N FEI#BED N L 
± F_.DfQHKwGJQE_H_.SVGYDg i = 0 L
QH_E_.S4GJSVFEG†}zQED R (N ) FPO DfGYK¹_iA4G:QEF 0Hn uzBiAvFEG:lDwl|A4G†ImoEDŒ‹bImGYKJD2QHFED2BHImgJDflfKJImSVF r KsDfomoLD8}zQED
A|cVDfl αβ > 0 Ld± Ft_aDQHK:GYQE_E_aSVGJDfg†lfSVhh'D%BEA4FEG²omD¹omDhhD¹_EgYMlMfBEDFdK α > 0,β > 0 L
r = αp + βq
I R (B) ⊂ R (A) A4oLSVgJG:SVFqA R (B) ⊂ R (A) DfK R (B) ⊂ R (A) hqA4ILG ASP (A)
lSVFdKJImDFdK R (A) SVQ R (A) ›¡cVSVILg opA*_EgYDQHc4DBEQ BEDfgJFEILDgomDfh'hD¯ L ]’SVFHl R (A) S4Q
c4MgJI‰iDFdK’omDG€lfSVFEBEI‰KsImS4FEG8BED opABEMfEFEILKJImSVF L ƒ L DK2BHSVFEl SVFA R (A) R (N )
R (A)
BiA4FEG¹omDw_EgJDfh'ILDg:l|A4G¹DfK R (A) R (N ) BEA4FEG¹oLD’GYDlS4FEBPuHlfSVFdKsgJA4BEImlKsILSVFqA|c4Dl0omA5h'A\‹bI‰[
hqAvomILKJM’BED R (N ) L
VS I‰K R
_.SVQHgwoxO S4gJBEgYDrGJQEI‰c4AvFzKP
i\[email protected] @?hUHP
pq
i (N )
p0 q 0
j
pq
i
0
p0 q 0
j
pq
i
0
pq
0
pq
0
|
qr
0
pr
0
pq
0
pq
0
pr
0
qr
0
qr
0
pr
0
qr
0
pq
0
qr
0
pr
0
pr
0
pq
0
^4
M ¥!
I R (A) ⊂ R (B) SVF_.DfQHKwGJD gsA4hDFEDfg€A4Ql|A4G€_EgYMlfMBEDfFzK’DFMlŠžiAvFE¬VD|AvFzK A DfK B L
QE_E_aSVGJSVFHG}zQEDo~ORSVFFHDqGJS4ILK _iA4G BiA4FEG5omDfG5BEDfQb‹BEDgYFEImDfgJG l|A4G L± F_.DfQHKGYQE_E_aSVGJDfg}zQED
SVQ p(N ) = p(B),q(N ) = q(A) L n opA fSVFEGˆ[ FESVQEG8BEA4FEG€omDrlA4G
p(N ) = p(A),q(N ) = q(B)
DK

p(N ) = p(A) q(N ) = q(B)
± FAA4omSVgYG › p(A ) < p(A) DfK p(B ) < p(B)%SVQ<› p(A ) > p(A) DK p(B ) > p(B)%l|A4g
GYImFES4Fr
L
I r(B) ≤ r(A) A4omS4gJG A ∈ QCON V (A ,A ,B ) ›kc4SVImg%i¬4QEgJD L zu
pq
0
|
|
pq
0
r
r
r
r
}
|
pr
r
p
?;\
r
TVU4=C=?]^IJA
r = cste
q = cste
Ap
Br
B
A
Ar
L
3y:'465
p = cste
}
|
r(B) ≤ r(A)
YGH=
L
I r(B) ≥ r(A) A4omS4gJG B ∈ QCON V r(B ,B ,A ) lD }XQHI#DGYK2A4QHGJGJI
± FDfFBEMBEQHILK²}zQED (R (A)∪R (B))∩E = ∅ SVQ (R (A)∪R (B))∩E = ∅ Ld QE_b[
_aSVGYSVFEG#}zQED²o~ORSVF A4I‰K#BiA4FHG#omD_EgYDhImDfgPl|A4G L„ SVI‰K N = hmax(p(A),p(B)), max(r(A),r(B))i L
± FA R (N ) ⊂ R (A) ∪ R (N ) ∪ R (B) DK0BES4FEl R (N ) ∩ E ⊂ {N } lSVFdKJgsA\[
BHImlfKJImSVFA|c4Dl opANh'A\‹bImh'A4omI‰KsMwBED R (N ) L
|
q
pr
0
pr
0
pr
0
pr
0
0
pq
0
r
q
qr
0
−
0
pr
0
pq
0
?;\
r
qr
0
TVU4=C=?]^IJA
−
p,r
pr
0
0
−
0
2
± F”BEMfiFHILKtopAŽgYDomAvKsILSVF ∼ lfSVh'hDComAŽlo vKsQEgYDCKsgsAvFEGJI‰KsILc4DCBED ∼  A ∼ B GYI8ImoD‹bImGˆKsDBEDG
_aSVImFdKJG A = A ,A , . . . ,A ,A = B KsDfomG8}zQED A ∼ A L
± F>BHMfiFEI‰K ∼ u ∼ BED opANhehD0jœA fSVFGJQEg E L
e
0
1
n−1
e
UH?;@
r ∈ D<
n
i
?
e
i+1
−
A ∼e B
Ar ∼ Br <
G IJUHKM=
VS ImDfFzK A,B,r cVMgYIL AvFzKoLDG lS4FEBEILKJImSVFHGCBEQ omDfh'hD L%± F FESvKsD A = A DfK
L¤± FTGYQE_E_aSVGJD}zQED A,B ∈ E L¤ SVImDfFzK p DK q oLDGrBEILgJDflfKsILSVFEG0KJDomoLDGr}zQED ASP (A) =
B =B
DfK ASP (B) = R (B) S4F³_.DfQHKCGYQE_E_aSVGJDfg i = 0 š SVhh'D R DGˆK QEF 0Hn SVF³A
R (A)
A|c4Dl αβ ≤ 0 L.± F_.DfQHKrA4QHGJGJI GJQE_H_.SVGYDg0}zQED α ≤ 0 DfK β ≥ 0 L š S4h'hD A ∼ B
r = αp + βq
ILoD‹bImGˆKsD'QEF<_aSVImFdK N ∈ Q KsDfo†}zQED R (N ) ∩ E ⊂ {N } L ]’D_EomQHG5SVF,_aDQHK5GJQE_H_.SVGYDg5}XQHD
LŸ“ FMflŠžiA4FE¬VDA4FdK A DK B GYIªFEMflDGYGsA4ILgJDvuPSVFŽ_aDQHK
u
p(N ) ∈ {p(A),p(B)} q(N ) ∈ {q(A),q(B)}
GYQE_E_aSVGJDfg r(A ) = r(A) ≤ r(N ) ≤ r(B) = r(B ) L
A4ILG’SVFGsA4I‰K’}zQPORImo¤D‹bImGˆKsD i DfK j KsDfomG’}zQED R (A ) ⊂ ASP (A ) DfK R (B ) ⊂ ASP (B ) u
BED _EoLQEG i ,j FED _.DfQHcVDfFdK’_iAvGweKsgYD FzQEoLG€l|A4g A ∈ R (A ) DK B ∈ R (B ) L
0
i\[email protected] @w?hUHP
0
r
pq
i
r
+
pq
i
pq
0
0
0
−
0
0
0
e
pq
0
2
0
pq
0
i0
pq
0
0
0
pq
0
0
pq
j0
0
0
¥
!
|
8hw
M 8W
' &^
fw^
o#gYDGYKJD ¥AvQHKsgYDG€l|A4GŸ
L ]’SVFHl*SVF A R (A ) ⊂ ASP (A ) L0 I M ∈ R (A ) A4oLSVgJG A ∈
u
i = 1 j = 1
DKBESVFEl
lDT}zQEI0DFdKJgsA mFED R (A ) ⊂ ASP (A ) L ]’D
QCON V (A,A ,M )
opAhehD’jœA S4F R (B ) ⊂ ASPM(B 6∈) L E
I p(B ) ≤ p(A ) A4oLSVgJG R (A ) ⊂ R (B ) DfKwBES4FEl A ∼ B L
I p(B ) ≥ p(A ) L
0
pq
1
0
0
p
pr
|
|
pr
1
0
0
0
0
0
0
0
{
+
pr
1
0
pr
1
pr
1
0
0
0
0
pr
1
0
0
0
0
e
B0
N0
M2
N 00
M1
A
q = cste
0
B
A
L
3`:'465
p = cste
|
i0 = 1 _ j 0 = 1
YGH=
p(B 0 ) ≥ p(A0 ) <
[email protected]
SVI‰K N = hp(A ),r(B )i L ± F>A A ,B ∈ R (N ) L
± F cvA€BPO A4U.S4gJB h'S4FzKJgJDfg#}XQHD†_aSVQEg#KJSVQEG M BEA4FEG R (N )∩E SVF A R (M )∩E =
aGYI M DGˆK’BEA4FEG ASP (A ) SVQ ASP (B ) l4ORDGYK’lomA4Img L ImFHSVF>S4FA p(M ) < p(M )
∅
l|A4g†GJILFESVF M ∈ QCON V (A,B,M ) SVQ M ∈ QCON V (A,B,M ) L ]’SVFHl’SVF'FPOyA
FEI R (M ) ⊂ ASP (M ) FHI R (M ) ⊂ ASP (M ) LV I R (M ) ⊂ ASP (M ) u\lSVhhD
SVFDfF<BEMfBEQEI‰K M ∼ B DfK5BESVFHl R (B ) ⊂ ASP (B ) lD}zQEI:DGYK
B ∈ R (M )
L ]’S4FEl R (M ) ∩ E = ∅ _aSVQEg8KsS4QHK M ∈ QCON V (A,B,M ) L
SVI‰K M ,M ∈ R (N ) KsDoLG}XQHD p(M ) ≤ p(M ) Lw I r(M ) ≥ r(M ) SVF A
DfK²BHSVFEl M ∼ M Ld QH_E_.S4GJSVFEG²_EA4gD‹bDfh'_EoLD%}zQED r(M ) ≤ r(M ) L
M ∈ R (M )
SVI‰K N = hp(M ),r(M )i L ± F>_aDQHK2A4oLSVgJGjœA4ImgYDropANgJMflQEgYgJDfFElD GJQEI‰cvA4FdKsD'› H 
I M ,M ∈ R (N ) ∩ E KsDfomG8}zQED r(N ) − r(A ) + p(B ) − p(N ) ≤ n DK
A4oLSVgJG M ∼ M . › H 
p(M ) ≤ p(M )
DGˆKwcXgJA4ImDrlA4g€SVFA R (N ) ⊂ ASP (A ) ∪ ASP (B )
H
QE_E_aSVGJSVFHG H L2 SVILDFdK M ,M BiA4FEG R (N ) ∩ E cVMfgJILiA4FdK p(M ) ≤
Ld SVI‰K M ∈ R (N ) \ {N } ∩ E › O ILoHFPORD‹bImGˆKsD8_iAvG²BED%KsDfomG_aSVImFdKJG²A4omS4gJG
p(M )
 L± FTA p(M ) ≥ p(M ) l|AvgrGJImFHSVF M ∈ QCON V (M,A,B) SVQ
M ∼ M
0
0
0
0
pr
0
0
?;\
pr
2
TVU4=C=?]^IJA
2
1
pr
1
00
2
1
2
pr
3
1
0
e
n−1
2
1
e
2
pr
1
r
0
0
r
1
2
2
1
2
n
pr
1
|
+
pr
00
−
0
qr
2
0
0
1
|
pr
1
−
pr
1
2
pr
2
pr
2
0
0
1
0
r
+
1
pr
1
pr
1
r
e
pr
1
pr
1
0
0
pr
pr
0
pr
1
0
00
0
1
00
r
2
2
00
+
00
1
2
n
0
pr
1
0
−
r
1
pr
^4
M L ±F A
Li“ K2BHSVFEl
Mr ∈ QCON Vqr (M,A,B)
M ∈ QCON Vpr (M1 ,Mp ,Mr )
≤
r(Mp )
r(M )
V¥ ¥
l|Avg GYImFESVF
p(M1 ) ≤ p(Mp ) = p(M ) ≤ p(Mr ) ≤ p(M2 )
r(M1 ) ≤ r(Mp ) ≤ r(M ) = r(Mr ) ≤ r(M2 )
] O A4_EgY™G’oxO žd{X_aS4KsžE™fGJD5BEDgYMlQHgJgJDfFElD M ∼ M u M ∼ M DK M ∼ M BHSVFEl
L
M ∼M
± F>ABESVFEl UEILDF A ∼ B L
u L ]0A4FEG€lD lA4G€SVFADfFElSVgYD R (A ) ⊂ ASP (A ) L
i =1 j =2
I p(B ) ≤ p(A ) A4oLSVgJG¤SVFA p(A ) ≥ p(A ) l|A4g GJImFHSVF A ∈ QCON V (A,B,A )
SVQ A ∈ QCON V (A,B,A ) LP“ KrBHSVFEl A ∈ QCON V (A ,B ,A ) lD}zQEIªDfGYK
L
I p(B ) ≥ p(A ) uXBED0opA hehD2jœA fSVFCSVF A B ∈ QCON V (A ,B ,B ) lfD0}zQEI.DfGYK
L
A4QEGJGYI
u La± FA R (A ) ⊂ ASP (A ) DK R (B ) ⊂ ASP (B ) La± F>BEMiFEI‰K’AvomSVgYG8opA
i =1 j =3
GYQEILKJD (M ) _iA4gP
I‰j p(A ) ≥ p(B )
M =A
ILj p(A ) ≤ p(B )
M =B
M = (M )
›¡cVSVILg%i¬VQEgYD L 
1
p
1
0
pr
1
0
0
0
pr
?;\
|
0
0
p
qr
TVU4=C=?]^IJA
|
2
r
0
0
|
r
2
0
|
p
0
0
?;\
TVU4=C=?]^IJA
0
pr
0
p
0
0
0
pr
0
p
pr
1
0
0
pr
0
p
qr
2
0
pr
pr
0
0
0
0
pr
0
0
0
pr
0
p
0
i
0
p
0
p
0
0
i+1
0
0
0
0
i rp
M2
M3
B
M0
0
M1
A0
q = cste
B
r = cste
A
3y:'465
L
p = cste
|
YGH=
i0 = 1 _ j 0 = 3
[email protected]
p(B 0 ) ≥ p(A0 ) <
± F A p(M ) = p((M ) ) > p(M ) lA4gGYImFESVF (M ) ∈ QCON V (M ,A,B) SVQ
L8“ KBESVFHlŽSVF A­QEFEDGYQEILKJDŽImFHEFEImDŽBHD_.SVILFdKsGCBEILGYKJImFElKsG
(M ) ∈ QCON V (M ,A,B)
L
ILFEloLQEG%BiAvFEG2QHFDfFEGJDfhNUEoLD EFEI#lD }zQEI#DGˆK
i+1
i r
i r
qr
i
i r
i
?;\
TVU4=C=?]^IJA
pr
i
µ¯•V•
u
8hw
L ± FA
I
M L
8W
' &^
A4oLSVgJG
A4oLSVgJG
I
u L
I
u†lSVhh'D
S4F A
L
I
DK
A4omSVgYG
La SVI‰K
o#gJDfGYKsD oLDrl|A4G
DK
I
DfK
SVFFES4KsD
L
jœA4ImgYD0QEFED gJMflQEgYgJDfFElD lSVhhD0BiA4FEGwomDrlA4G
fw^
i0 = 2 j 0 = 1
R1pr (B 0 ) ⊂ ASP (B 0 )
|
p(B 0 ) ≥ p(A0 )
B 0 ∈ R2pq (A0 )
|
p(B 0 ) ≤ p(A0 )
A0 ∈ R1pr (B 0 )
i0 = 2 j 0 = 2
|
p(A0 ) ≤ p(B 0 )
r(A0 ) ≤ r(B 0 )
q(A0 ) ≤ q(B 0 )
B0 ∈
pq
R2 (A0 )
|
p(A0 ) ≥ p(B 0 ) q(A0 ) ≥ q(B 0 )
A0 ∈ R2pq (B 0 ).
|
p(A0 ) ≥ p(B 0 ) q(A0 ) ≤ q(B 0 )
N 0 = hp(B 0 ),q(A0 )ipq
pq
M1 ,M2 ∈ R2 (N 0 ) p(M1 ) ≥ p(M2 )
N 00 = hp(M2 ),q(M1 )ipq
0
0
i = 1,j = 1
|
|
I M ,M ∈ R
DKCBES4FEl
L
aL ± F c4A
KsDfomG8}zQED
fD K
A4omSVgYG
› 
SVILDFdK M ,M cVMfgJILiA4FdKtomDGlSVFEBHILKsILSVFEGNBHD,› H  L: S4ILK M ∈ R (N ) \ {N } ∩
@#D_aSVILFzK M c4MgJI‰iD p(M ) ≥ p(M ) L8± F A­A4QHGJGJI p(M ) ≥ p(M ) l|A4gqGJI‰[
E
L ]’Dh'efh'D q(M ) ≥ p(M ) l|A4gtGYImFESVF M ∈
FESVF M ∈ QCON
V r(M ,M ,M )
L ]’SVFEl SVF>A%
QCON V (M ,M ,M )
∩ E−
p(M1 ) ≥ p(M2 ),R2pq (M1 ) ∩ E + = ∅,
R2pq (M2 ) ∩ E + = ∅, p(A0 ) − p(N 00 ) + q(B 0 ) − q(N 00 ) ≤ n
M 1 ∼ M2 .
Hn0
1
2
1
pq
0
2 (N )
+
r
r
q
pr
pq
2
0
n
2
r
p
r
00
q
1
q
00
p
2
p(M1 ) ≥ p(Mq ) ≥ p(M ) = p(Mp ) ≥ p(M2 )
q(M1 ) ≤ q(Mq ) = q(M ) ≤ q(Mp ) ≤ q(M2 )
]’SVFEl BPO A4_EgY™Gwo~ORžd{b_aS4KJžE™GYD BHD gYMlfQEgJgYDFElfDSVFA M ∼ M ∼ M ∼ M L
u L @ABHMhSVFEGYKJgsAvKJImSVF DfGYKwopANhehDr}zQED BiA4FEG€omDrlA4G i = 1,j = 2 L
i =2 j =3
u L ± FA R (A ) ⊂ ASP (A ) DfK R (B ) ⊂ ASP (B ) L
i =3 j =1
I p(A ) ≤ p(B ) A4oLSVgJG B ∈ R (A )
I p(A ) ≥ p(B ) A4oLSVgJG A ∈ R (B )
u L ILh'ILopA4ILgJD2A4Ql|A4G i = 2,j = 1 L
i =3 j =2
u L ILh'ILopA4ILgJD2A4Ql|A4G i = 1,j = 1 L
i =3 j =3
2
@AgYDopA\KsImS4F ∼ DGˆKQEFEDCgYDomAvKsILSVF<BPORM}zQEI‰cvA4omDfFElDCGYQEg E = E ∪ E L² S4ILK D , . . . ,D oLDG
lopAvGJGJDfGtBPO Mf}zQEILcvA4oLDFElfD>AvGJGJSXlILMDfGDfK G o~ORImGJS4UiA4gY{XlfDFdKsgYD BED D DK P oxO DfFEGJDfhNUEomD {G : i ∈
L
{1 . . . n}
@#DNoLDhh'D5_EgJMflMBHDFdK h'SVFdKJgJDN}zQED_aSVQEg’KJSVQHK i ∈ {0 . . . 3},p ∈ D Imo D‹bImGˆKsDNQEF j KsDfomG0}zQED
_.SVQHg€KJSVQHK M ∈ D omDG€_aSVImFdKJG M GYSVFdKwGJQEg D L “ K€BES4FEl SVFA p(G ) = p(G ) L
]’S4FEl SVFA
7
“ F_iA4gˆKsILlQEoLImDg P lfSVFdKsImDfFdK’AvQPhSVILFEG¹KsgYSVDImG¹_.SVILFd; <KsG ›kl|A4g8ILoŸ{ AtA4QhSVImFEG¹BEDQb‹ BEImgYDlfKJImSVFHG%BiA4FEG
L
D
1
|
0
0
|
0
0
|
|
q
2
p
0
qr
2
0
pr
1
0
|
0
0
0
|
0
0
0
qr
2
pr
1
0
0
0
0
0
0
0
0
0
0
0
0
+
i
i
[email protected]> P
−
1
i
p
APQ=4A\S]^IJA
0
j
tJTVUHI
i
UHP8A^<
n
i
j
^4
M µ¯•Hµ
p,q ∈ D < PSN i XYPV?;@,IJAC=9TVUH?;[email protected];= N1 = hp(M1 ),q(M2 )ipq _
< IJUHKM= TVUH> K
N2 = hp(M2 ),q(M1 )ipq _ N A TI >=zUHP=>4TTVU4=4A N1 ∈ Ripq (N2 ) U Ripq [email protected]> P @UH> = r,s ∈ D IJAdTQG KbG I;IJiIJU=< KbG \z\FA {M : r(M ) ∈ [r(M1 ),r(M2 )] [email protected] s(M ) ∈ [s(M1 ),s(M2 )]}
[email protected][email protected] TQG K G I;IJiIJU=< KbG \z\FA {M : p(M ) ∈ [p(M1 ),p(M2 )] [email protected] q(M ) ∈ [q(M1 ),q(M2 )]} <
[email protected]
M1 ,M2 ∈ Z2
SVILDFdK
›
[email protected]
lfSVhh'DBiA4FEGomDqoLDhh'D L¤± F­_.DfQHKGYQE_E_aSVGJDfg
DK
A¯c4Dl
DfK
u
L ›kc4SVImg%E¬VQEgJD L ! 
L ]’DChehD
BES4FEl DfK
u#BESVFHlqomDt_iAvgsA4oLomMoLSV¬VgJA4h'hD
DfGYK
L
DGˆK’ILFEloLQEG%BiAvFEG
DK
M1 ,M2 ,N1 ,N2 ,p,q,r,s,i
i = 2 ?hA p(M1 ) ≥ p(M2 )
q(M1 ) ≤ q(M2 )
r = αp + βq,s = α0 r + β 0 q
α ≤ 0,α0 ≤ 0,β ≥ 0,β 0 ≥ 0
N10 = hq(M2 ),r(M1 )iq,r
N20 = hq(M1 ),r(M2 )iq,r
00
00
N1 = hr(M1 ),s(M2 )ir,s N2 = hr(M2 ),s(M1 )ir,s
q(N10 ) = q(M2 )
q(N10 ) ≥ q(M1 )
q(N20 ) ≤ q(M2 )
N1
0
0
N2
M1 N 1 M2 N 2
M1 N 1 M2 N 2
M1 N10 M2 N20
M1 N10 M2 N20
M1 N100 M2 N200
2
i\[email protected] @w?hUHP
L% SVI‰K
fD K
š SVhh'D
S4F*A
GJS4FzK5BiA4FHGomDt_iAvgsA4oLomMoLSV¬VgJA4h'hD
L
ILFEloLQEG8BiA4FEG
]’D hehDrSVF>_aDQbK€h'SVFdKJgJDg€}zQED
r = r(M2 )
r = r(M1 )
s = s(M2 )
N100
q = q(M2 )
M2
N1
N10
s = s(M1 )
N20
q = q(M1 )
3y:'465
i\[email protected] @w?hUHP
7
L!
N200
|87
N2
M1
p = p(M2 )
?;P8gI > =?hUHP
;<
p = p(M1 )
N AC=
TQG KbG I;IJiIJU=< KbG \z\FAC=
*jt
gCUHPVE AbaLA f
, [email protected]=4AIJUHP
4S ILK G QHF_aSVILFzK€BHD P DfK D opANlomA4GJGYD BESVFdK G DGYKwoLDrUiA4gY{XlfDFdKsgYD L
± FGJQE_E_aSVGYD _EA4g8D‹bDh_EoLDr}XQHD D ⊂ E DK€}zQEDrKsSVQHK M ∈ D c4MgJI‰iD ASP (M ) = R
A|cVDfl p,q ∈ D La SVILDFdK pmin,pmax,qmin,qmax oLDG8D‹XKJgJDh'AtBED p DK q GJQEg D L SVImDfFzK
APQ=4A\S]^IJA
P
[email protected]`> PFTVUHI
UHP8A
D<
+
pq
0 (M )
B1 = {M |pmin ≤ p(M ) ≤ pmax}
B = {M |qmin ≤ q(M ) ≤ qmax}
L²± F­cvAŽh'S4FzKJgJDfg}zQED D = B ∩ E GYSVILK M QHF*_.SVILFdKBED
m
o

A
l
m
o
4
A
J
G
Y
G
D
D
{M |M ∈ D}
L ]5OyA4_HgJ™G oLDG0BEMiFEI‰KsImS4FEG’BED pmin DK pmax ImoªDŒ‹bImGYKJD A ,B ∈ D KsDfomG0}zQED pmin =
E ∩B
LP± F>A A ∼ B u SVFDfFBEMfBEQEI‰K’BES4FEl5}zQPORImoDŒ‹HILGYKJD A,B ∈ D
p(A ) ≤ p(M ) ≤ p(B ) = pmax
KJDomG8}zQED A ∼ B DK p(A) ≤ p(M ) ≤ p(B)
I q(M ) ≤ max(q(A),q(B)) SVFGsA4I‰K€BEM^WJ‘}zQED M ∈ D L
I q(M ) ≥ max(q(A),q(B)) A4omSVgYG q(M ) > q(M ) lA4gGJILFESVF M ∈ QCON V (M,A,B)
DKBHSVFEl R (M ) ⊂ ASP (M ) SVQ R (M ) ⊂ ASP (M ) L “ KBESVFEl A SVQ B GYSVFdKNBiAvFEG
DK2BHSVFEl M ∼ A L
ASP (M )
DK
2
1
+
+
1
p
1
0
0
0
0
0
0
e
|
|
pq
0
pq
1
e
p
p
pq
µ¯•dƒ
' ± FDfFŽBEMfBEQEILK0A4QEGYGJI²A|cVDflNomD5omDfh'hD L ƒ L µ¯•'}XQHD D = B ∩ E L ]’Dh'efh'D D = B ∩ E DfK
S4´ D = {M |M ∈ D} La± FBEMfEFEILK2A4oLSVgJG%oLDG€ZtgJMf¬VImSVFHG8GJQEI‰c4AvFzKJDG
D =B ∩E
DfK
V = {M |p(M ) < pmin q(M ) > qmax}
DfK
V = {M |p(M ) > pmax q(M ) < qmin}
DfK
V = {M |p(M ) > pmax q(M ) > qmax}
›kcVS4Img%i¬VQHgJD L d
8hw
M 1
2
−
2
2
8W
&^ fw^
−
1
+
2
q
1
2
3
p = pmax
p = pmin
V1
B1
V3
q = qmax
G
B2
q = qmin
V2
3y:'465
L
|87 AC=oN ? [email protected]=KCi< ?hUHPQ=
L @#D _aSVILFzK DGˆK’oxO ILGJSVUiAvgY{d[
KsDfo#}zQED
QH_E_.S4GJSVFEG€}zQPORImoP{AvILK8QEF_aSVImFdK
lDFdKJgJDNBPORQEFEDlopAvGJGJD D L.± FA D ⊂ GV ∈∪ PV ∪ V L ]GO A4_E∈gJ™fRG0o~ORD(G)
h_EomA4lDfh'DfFzKwBEGD G o~ORDFHGJDhUEomD
FED _.DfQHKw_iA4G€efKJgJD BEImG¡WYSVImFdK8FHIA¯c4Dl V FEIA|cVDfl V L ]2DQb‹lA4G2_aDQbcVDFdK2GJD _EgYSbBEQHImgJDª
D
L: SVI‰K
o8D‹bImGˆKsD
u
KJDoLG}XQHD
opAŽgJMf¬VImS4F*BESVFEFEMfD_iA4gopA
BEMfEFEILKJImSVF A L ∈ƒ L VL ± FBA ∈RV (N ) ∩ D =A∅∼l|A4g8BGYI A,B ∈R E(NA4)omS4gJG R (N ) ∩ E = ∅ DfK
GJI A,B ∈ E A4oLSVgJG%KJSVQHK€_aSVImFdK€BED E ∩ R (N ) DfGYKwM}zQEI‰c4AvomDFdKw‘ A DfK2‘ B L
± F>BEMfBEQEI‰K2BHQ>oLDhh'D L ƒ L µ„ƒN}zQED B ∩ B ⊂ R (N ) lfD }zQEIDfGYKwDFlSVFdKJgsA4BEILlfKJImSVFA|cVDfl
L
R (N ) ∩ D = ∅
o#D‹bImGˆKsD A ∈ V u B ∈ V u C,C ∈ V KsDoLG€}XQHD
A ∼ C ∼ C ∼ B.
u
u
BESVFEl_aSVQEgNKsS4QHK M ∈ {A,B,C,C } SVF°A
D ⊂ R (A) D ⊂ R (C) D ⊂ R (B)
La SVI‰K M QEF_aSVImFdK€BED D L
ASP (M ) = R (M )
QE_E_aSVGYSVFEGr}zQED A,B,C,C ∈ E LŸ± FTA q(M ) ≤ min(q(A),q(C)) lA4g R (N ) ∩
L¯“ KBES4FEl M ∈ QCON V (M,A,C)
SV´
E ⊂ {N } N = hp(A),
min(q(A),q(C))i
L
lD }zQEI#DGˆK
QE_E_aSVGYSVFEG A,B,C,C ∈ E LX± FqA q(M ) ≤ min(q(A),q(C)) l|Avg†GYImFES4F M ∼ A L
± F*A A ∈ R (M ) DK A 6∼ M BESVFEl R (M ) 6⊂ ASP (M ) L ]’DChehD C ∈
DK C 6∼ M BESVFHl R (M ) 6⊂ ASP (M ) L¤± FA q(M ) < q(M ) BESVFEl
R (M )
DfK R (M ) ∩ E 6= ∅ L“ KCBES4FEl>KJSVQEGqoLDG}zQiA4BEgJA4FdKsGqBHD
R (M ) ∩ E 6= ∅
L
BEImgYDlKsImS4F p,q A4QHKJSVQEg€BED M lfSVFdKsILDFEFEDfFdK2BHDGw_.SVILFdKsG8BED E lD }zQEI#DfGYK
@²ORDFEGYDhNUEoLD P DGˆK BESVFElQEF D[ _.SVo‰{X¬VSVFED [ lS4Fzc4D‹bDŽjkSVgˆKCGJDfomSVF D L%± F A<BHSVFElŽ_EgYSVQHc42M
o~ORImh_EomILl|AvKJImSVF (8) ⇒ (9) BEQKsžEMfSVgJ™fh'D L ƒ L L
0
0
0
1
0
2
1
rs
i
e
+
0
2
3
0
e
pq
1
pq
0
pq
2
|
0
+
pq
2
p
−
pq
0
pq
3
p
p
p
pq
TVU4=C=?]^IJA
|
pq
2
pq
0
e
pq
3
0
?;\
+
rs
i
rs
i
2
rs
i
1
−
rs
i
1
|
0
2
2
rs
i
+
0
3
1
|
pq
0
0
p
e
p
e
+
−
p
pq
1
pq
2
p
pq
3
p
p
p
e
p
p
p
+
p
q
+
?;\
TVU4=C=?]^IJA
}
^4
M µ¯•VZ
! ! % 0] A4FHGwlfDfKYKsD GJDlKsILSVFSVFcvAhSVFdKsgJDfg8o~ORImh_EomILl|AvKJImSVF (9) ⇒ (5) L
± F'cvA A¯c4SVImg:UaDGYSVImFBPORQEFED’FHS4KsILSVF'_HomQEG²jœA4ILUEomD8}zQEDwopA D[ _aSVo‰{b¬4SVFiA4oLILKsMBEM^WJ‘ ImFdKJgJSXBEQEILKJD€BiA4FEG
¢ Avgs¥ dui_iZ ¦ L
D
,! ! P N ?;@ ^> > P8AdN KMUH?;@Aj=> [email protected]
M
E
=> [email protected]
=>[email protected] E <
Z
P APQ=4A\S]^IJA
D
NG PQ=
UHP
D
=4AIJUHP
E
E
=?R?;[email protected]
D tJTVUHI ; <
[email protected]> P
=?2AI;IJAdN iI ?;\z?;@A
Z
E < POTVUH?;[email protected]
KMUH?;@BA p = p(M )
> PvN A\z?;tJTI G [email protected] [email protected]
>LP8ASN ?;[email protected]?hUHP
p∈D
UHP8A,&G ?]^IJAn=?yTVUH> K
@BUH> @BAC=
@BAI ^>VAzI G
IJAC=
N
N KMUH?;@BAC=
d
N A
N ?;[email protected]?hUHP
G
|d ∩ E| ∈ {0,2}
UH>
(|d ∩ E| = 1
[email protected]
d
=> [email protected]
E).
D1
D
D2
" ()0 /!$
E
D4
L¥
3y:'465
|
D3
Z
PAbaLA\
TIJAdN A
D tJTVUHI ;<
UHP8A &
, G ?]^IJA^<
7
AC=
^> G @KMA
N KMUH?;@BAC=
D 1 _ D2 _ D3 _ D4
=> [email protected]@
E<
]0A4FHG omACGJQEI‰KsDtSVFTGJQH_E_.S4GJD}XQHD DfGYK QEF D[ _aSVo‰{b¬4SVFEDjœA4ILUEomDDfK [ lfSVFdcVD‹bDjkSVgYKrGYDoLSVF D
fl SVFdKsDFEA4FdK’AvQ>hSVImFHG8Z_.SVILFdKsG L E
n S4QEgwKJSVQHK2_.S4ImFdK M ∈ E ILo#D‹bImGˆKsD QEF 0Hn Π KsDo}zQED Π ∩ E = {M } L. I M FED5GJQE_E_aSVgˆKsD
_iAvG E A4omS4gJG lDK 0Hn DGYK5QEFEIL}zQEDqSVFomDtFES4KJD'DfFElS4gJD ASP (M ) Lª ImFESVF E DGˆKNlS4h'_EgYImG BiA4FHG
QEF<BEDhI‰[ _EopA4F,BEDjkgJSVFdKsIL™gYD p = p(M ) A|cVDfl p ∈ D L o†{<AA4oLSVgJG5DF<jœAvILK5BEDQb‹ rbn }zQEIFED
lfSVFdKsImDfFEFEDfFzK¹A4QElfQEF_.SVILFdKBHD E GJA4QHj M uXILomGGYSVFdK¹BHD2omA’jkSVgJhD Π = R (M ) DK Π = R (M ) L
I
lfSVFdKsILDFdKwBEDQb‹ _.SVILFdKsG8BED E A4oLSVgJG¹SVFFES4KsD M oxO QHFEIm}zQEDr_aSVImFdK8BHD E BHImGYKJImFElK
)
L² I p = p(M ) lS4FzKJImDfFzKNQEF”GJDQHo8_.SVILFdKBED E A4oLSVgJGSVF*FES4KJD
BED M pKJD=o%}zp(M
QED p(M
)
=
p(M
)
L
M =M
q
1
pq
i
pr
j
2
p
p
p
,! ! p,q ∈ D _ i _ N ∈ Q
2
P
N ?;@
@AI=^>VA
A,B ∈ E [email protected] gCUHPQ=4iCg>[email protected]? ,M= [email protected] ACONSB
A,B ∈ Ripq (N ) [email protected] Ripq (N ) ∩ E = {A,B} <
^>,A
=??;I
[email protected]
ACONSB [email protected]=4A> IJA\[email protected]=?RUHP GFI > P8AON AC=zgCUHPN ?;@w?hUHPQ= =>L?;EHG [email protected]=
Π = Rpq (N ) U N = hp(A),q(B)i [email protected] Π∩E = {A,B} _ Rpq (A)∩E =
[email protected]`> P pq
i
i
pq
{A} _ Ri (B) ∩ E = {B} <
µ¯•
}
8hw
7
cVSVILg%i¬VQEgYD L µ¯• (AB)
G
N KCUH?;@A
[email protected] AON ?;[email protected]?hUHP
q = cste
M D
NG PQ=
8W
' &^
fw^
E<
[email protected]=>[email protected]
B
B
A
pq
Ri
A
p = cste
L µ|• 87
ACONSB
± FgJDKsgJS4QHcVD BESVFEl omANlSVFEGYMlfQHKsI‰cbI‰KsM BEDfG2GJA4ImoLopA4FdKsGBEMfEFEImD DFƒ L ! L Z L
š SVFEGJILBEMgYSVFEG€BEDfQb‹_.S4ImFdKsG%lfSVFEGYMlQbKsILjkG A DfK B La SVI‰K i,p,q,N lS4h'hD0BiA4FHGwomA
BEMfiFHILKsILSVFBED CONS L.± F>GJQE_H_.SVGYDNBED _EomQHG2}zQED R DGˆK0h'A\‹bImh'A4oP_aSVQEg€opAtgJDfopAvKJImSVF ≺ BEMiFEImD
¬Vg 4lfD ‘tomANjkSVgJhNQHomDt› L ƒ4 La± F_.DfQHK€GJQE_H_.SVGYDg p(N ) ∈ {p(A),p(B)} DfK q(N ) ∈ {q(A),q(B)} L
I N 6= A DfK N 6= B A4omS4gJG¹DfFCjœA4ILK¹SVF A R (A) ∩ E = {A} DfK R (B) ∩ E = {B} Li± F
GJQE_H_.SVGYD'}zQED R FPORDGˆK _iA4GrQEF rHnL š SVhh'DBPO žEA4UEILKJQEBEDtSVFGYQE_E_aSVGJD i = 0 DfK }zQED r
DGˆK’QEFHD BHImgJDflfKJImSVF BED D KsDfomoLDr}XQHD r = αp + βq A|cVDl α > 0,β > 0 L
]’Dr_EoLQEG¹SVF _aDQbK%GJQH_E_.S4GJDg p(A) = p(N ) u q(B) = q(N ) u r(A) ≥ r(B) › š D0}zQEI.DGˆK8cbgJA4I
‘QHFED ImFdcVDfgJGJILSVF_EgJ™fG2BHD p,q SVQ­›kDfKŠ A,B L 
I A 6= A A4oLSVgJG¤_aSVQEgKsSVQHK M ∈ R (A) S4FA A ∈ QCON V (M,A ,B) BESVFEl R (A)∩
lfD}zQEI¤DFdKsgJA pFED
A¯c4Dl
lD5}zQEIDGYK
E = {A}
L :SVImg¹E¬VQEgJND =L µVµ hq(B),r(A)i
L
DFlSVFdKJgsA4BEILlfKJImSVFA|cVDfl5omANhqRA\‹bIL(NhqA4)oLIL∩KJM2EBED =R{A,B}
p = cste
3y:'465
|
AC=oN A>a
g GH=
U
<
i\[email protected] @?hUHP
pq
i
|
pq
i
pq
i
p
{
qr
0
0
pq
i
pr
1
pr
pr
1
p
0
qr
pq 0
r = cste
q = cste
B
N0
Ap
N
A
M
p = cste
L µVµ
A 6= A
I A = A A4omSVgYG E DGYK2ImFHlomQHG8BiA4FEGwBEDhI‰[ _EopAvF p ≤ p(A) DKwBESVFEl B = B uiBHSVFEl E DGYK
ImFElfomQEG8BEA4FEG q ≤ q(B) uElDflIImh_EoLIm}zQED E = {A,B} lD }zQEI#DGˆK’ILh'_aSVGYGJImUHomD L
I N = A S4FA R (B) ⊂ R (A) La± F_.DfQHKwGJQE_E_aSVGYDg i = 0 L
š A4G p(A) = p(B) L„ I q = q(A) FED:GJQE_E_aSVgˆKsD:_iA4G E A4oLSVgJGPILo4DŒ‹HILGYKJD†QHF5_aSVImFdK A 6= A L
I
A4omS4gJG A ∈ QCON V (M,A ,B) DfKwBESVFEl R (A) ∩ E = ∅ L š O DfGYK
DFElfMSVgJDŽ∈ cXRgJA4I’(A)GYI q = q(A)
GYQE_E_aSVgYKJD E ›kl|A4g R DfGYKBiA4FHGComDŽBHDhI‰[ _HopA4F°}zQEI’FED
lSVFdKJImDFdK_iA4G B  L ]’SVFHl2oLDG†GJDQHomG†_.SVILFdKsG†BED E BiAvFEG¹oLDwBHDhI‰[ _HopA4F R (A) ∪ R (A)
›œBED UaSVgJB (AB)%GJSVFdK A DfK B lD }zQEI#lS4gJgJDfGJ_aSVFEBA4QBEDQb‹bIL™hD lA4G€BEQomDhhD L
3y:'465
|
YGH=
p
|
p
q
pq
i
pq
i
|
pq
3
pq
q
pq
3
q
pq
3
pq
0
pq
3
^4
M µ¯•
@D lA4G q(A) = q(B) DGˆK’GYImhImomA4ImgYD L
n SVQEg€oLDG€BEDQX‹BEDgYFEImDfgJG€l|AvGwS4F>GYQE_E_aSVGJD p(A) 6= p(B) DK q(A) 6= q(B) L
I
FED%GYQE_E_aSVgYKJD8_iA4G E A4omSVgYG q(A ) > q(A) DK q(B) < q(A) BESVFEl A 6= A
DfK pA=∈p(A)
lD }zQEI#DGˆK’ILh'_aSVGYGJImUHomD L
QCON V (A ,A ,B)
I p = p(A) GJQE_H_.SVgˆKsD E A4oLSVgJG E \ {A} DfGYK>ImFElfomQEG BiAvFEG p < p(A) DfKBESVFHl
DfK
GYQE_E_aSVgYKJD E BESVFEl E \ {A} DGˆK%ILFEloLQEGBEA4FEG q < q(A) DK%BESVFHl
A = A q = q(A)
lD }zQEI#DfGYK2Imh_.SVGYGJILUEomD L
E = {A,B}
2
± FTACBEM^WJ‘qcXQTBiA4FHG0opAqGYDlKsImS4F<ƒ L ! L Zq}zQEDoLDG0_aSVImFdKJG0GsA4ILomomA4FdKsGw}XQEA4BEgJMfG0jkSVgJhDFdKrQHFlf{XloLD L
± Fq_aDQbK¹MKsDFHBEgJDwjœA4lILomDhDFdK†omD€gYMGJQHoLKŠA\K¹BiAvFEGoLD2lA4G†BED’lfSbSVgYBESVFEFHMDGFESVFjkSVgJlfMhDFdK¹DfFdKsIm™fgJDfG L
]’Dr_HomQEG8ILlIŸoLDG%_aSVILFzKJG%GsAvImomomA4FdKsGGJSVFdK€ImlfIŸKsS4QEG%BED hNQEo‰KsIL_EomILlILKJMrµ GsA4QHj¤oLDG8_aSVImFdKsG¹GJQE_E_aSVgˆKŠA4FdK E
}zQEIGJS4FzK’BED hNQEoLKJIm_EoLImlfILKsM 2 DfK2BESVFdK0oLDG€BEDQX‹_aSVILFzKJG2}zQiAvBEgJMfG’GJS4FzK2lS4FEGJMflQHKJILjkG L @A'FiA\KsQEgYD5BEQ
¬VgJA4_EžED (E,CONS) DfGYKwBESVFEl omANh'efh'Dr}zQED lDfomQEI#BHDG€_.S4ImFdKsG€}zQiA4BEgYMGP
|
|
p
pq
p
q
q
|
q
7
<
A
KbGT>VA
(E,CONS)
[email protected]> P
g;gIJA^<
A,B ∈ E [email protected]=4iCg> @w? ,M= [email protected] r ∈ D
r(M ) ≥ max(r(A),r(B)) UH> r(M ) ≤ min(r(A),r(B)) <
?
G IJUHKC=oTVUH>[email protected]>[email protected]
M ∈ E
UHP
G
EQ _E_aSVGJS4FEG }XQHDo~ORSVFACQEFT_aSVImFdK M KJDoª}zQED r(M ) ∈]r(A),r(B)[ L @ACBEgYSVILKJD
FED _aDQHKw_iAvGwGYQE_E_aSVgYKJDg E lA4g A DfK B GYSVFdKwBED lŠžiA4}zQED l 4KJMrBED lDfKYKsD BEgJS4ILKsD L ]’S4FEl
r = r(M )L.±
FcvAGJQH_E_.S4GJDg p(M ) < p(M ) L
M 6= M
QE_E_aSVGJSVFHG'}zQEDo~ORSVF°DGYK'BiA4FHGomA,_HgJDhIm™fgJD lfSVFEBEI‰KsImS4F°BHQ‚omDfh'hD L ƒ L µ L† S4ImDFdK
lfSVhh'D BiA4FEG#lfD†oLDhh'D L|± F GJQH_E_.S4GJD i = 0 L¯ I p(M ) ≥ p(A) SVF A M ∈ QCON V (Mp,q,i,A,B)
›¡cVSVILg0i¬VQEgYD L µ„ƒVuŸGJImFHSVFS4FA q(M ) ≥ q(B) DK BESVFEl M ∈ QCON V (M ,A,B) Imh[
_aSVGYGJImUHomD L
i\[email protected] @w?hUHP
r
r
|
pr
qr
r
r
Mr
r = cste
q = cste
B
M
A
3y:'465
|
,! ! E<
|
YGH=
p(M ) ≥ p(A)
QE_E_aSVGJSVFHG’h'A4ImFdKsDfFiA4FdKw}zQED (AB) GJQH_E_.S4gYKsD E DfK0DfGYK’BED5BHImgJDflfKJImSVF
L h _aSVGJGYImUEoLD L
M ∈ QCON V (M ,A,B)
pr
N A
L µ„ƒ
p = cste
7
r
APQ=4A\S]^IJA
ME
[email protected](I APQ=4A\S]^IJAdN AC=d\z?;I ?hA>azN A
N A>a
p∈D
L.± FAqA4oLSVgJG
2
TVUH?;[email protected] =gCUHPQ=4iCg> @w? ,M=
µ¯•
TVUHI
;<
UHP8A
*jt
E
?
D tJTVUHI ; <
D<
[email protected]`>LP
gCUHPVE AbaLA f
, [email protected]=4AIJUHP
8hw
M 8W
' &^
UHP8A &
, G ?]^IJA*jt gCUHPVE AbaLA f, [email protected]=4AIJUHP
D
fw^
ME
G IJUHKC=
[email protected]> P
VS ILK M QEF_.SVILFdK€BED ME u M DGYKwBES4FEl omD h'ILomILDQCBPORQEFGJDf¬Vh'DfFdK [AB] BED E
A¯c4Dl ACONSB L
I (AB) GJQE_H_.SVgˆKsD E u E DGYKBiA4FEG5QEF<BHDhI‰[ _HopA4FTBEDjkgJSVFdKsIL™gYD (AB) DKNBESVFHlKJSVQHKsDfGoLDG
_.SVILFdKsG€BED ME GJS4FzK2A4QHGJGJI#BEA4FEGwlD BEDhI‰[ _EopAvF L
ILFESVFŽ¬4g 4lfDA4QTomDfh'hD L ƒ L µ S4FGsA4I‰Kr}zQPO ILoªDŒ‹HILGYKJDNQEF rHn }zQPO S4F_.DfQHK GJQE_H_.SVGYDgrMf¬dA4o‘
DfKw}zQEIPcVMfgJILED R (N ) ∩ E = {A,B} L
R (N )
± F*cvA,hSVFdKsgYDg}zQED R (M ) ∩ ME = {M } L QH_E_.S4GJSVFEGt}zQPORImo8DŒ‹HILGYKJDQHF‡_aSVImFdK M ∈
L @#D _.S4ImFdK DGˆKwomDrhImomILDQCBHD
A|cVDfl
R (M ) ∩ ME \ {M }
L
rg 4lfD A4QomDhhD L ƒ L µ SVF>_aDMQbK2GYQE_E_aSVGJDfg A ∈ R [A(A)B ]DfK B ∈A RCONSB
(B)
@ABEILgJDflfKsILSVF­BED omA>BEgYSVILKJD (A B ) FED _aDQHKN_iA4GNeKsgJD BEA4FEG D lA4g R DGˆKtQEF rbnUL: S4ILK
A4omSVgYG
omATgYM¬VILSVF°BESVFEFEMfD>_EA4g'oLDomDhhD L ƒ L µ L†± F‚GsAvILK}zQED DK GJSVFdK'BiA4FEG'omD
_iA4gsAvomomRMfomSV¬4(NgsA4h) h'D p ∈ [p(A ),p(B )],q ∈ [q(A ),q(B )] Li± F DfFBHMBEQEI‰K|uE¬4Ag 4lfDrA4BQ omDfh'hD L ƒ L µ„ƒ
}XQHD A DfK B GYSVFdK2BEA4FEG R (N ) ILh'_aSVGYGJImUHomD L
i\[email protected] @?hUHP
pq
0
pq
0
pq
0 pq
0
0
0
0
4
0
rs
i
0
0
pq
1
0
0
0
0
0
0
0
pq
3
pq
0
0
0
0
rs
i
B0
p = cste
B
q = cste
M
A
A0
3y:'465
[email protected]
r
L µ|Z
|
UH> K ^>VUH?
R0pq (M ) ∩ ME = {M } ?
2
E
[email protected]> P8AoN ?;[email protected]?hUHP
[email protected]> P
UHP8A,&G ?]^IJAC_
N A
[email protected]
D tJTVUHI ; <
D G IJUHKM= Ar
Br
A
B
[email protected]
[email protected] A>a
TVUH?;[email protected] =gCUHPQ=4iCg> @? ,M=
[email protected] A>aTVUH?;[email protected];=zgCUHPQ=4iCg>[email protected]? ,M=ON A
N A
E_
E<
L o2jœA4QbK BESVFElh'SVFdKJgJDgq}XQHD DK GJS4FzK
± F FES4KsD
u
lSVFEGYMlfQHKsI‰jkG L @DfG_aSVILFzKJG A DAK B=GYSVFdAK¹lfSVBFEGJMf=lQHKJBILjkG†BESVFElwSVF_.DfQHKgJD¬VA4gJBEDfg:omDG†BEDfQbA‹_.S4GJBGJILUEImoLILKJMG
BESVFEFEMfDGw_iA4g€omDroLDhh'D L ƒ L µ 
± F<GJQH_E_.S4GJDq}zQED A,B ∈ R (N ) A|cVDfl R 0Hn DfK R (N ) ∩ E = {A,B} L ± F_.DfQHK
GJQE_H_.SVGYDg p(A) > p(B) u q(A) < q(B) DfK r = αp + βq A¯c4Dl α ≤ 0 DK β ≥ 0 l|A4g R DGYK
QEF rHnL oPFPO { AtA4QElfQEF_.S4ImFdK8BED E BiAvFEG8opANUiAvFEBED r(A) < r < r(B) Li I A = A A4omS4gJG
SVFA E ∩ {r ≤ r(B)} = {A,B,B } DfKwBES4FElrImo.D‹bImGˆKsD N KsDfo#}XQHD R (N ) ∩ E = {A,B }
DfKwBESVFHl ACONSB L BEDh GYI B = B L
L
ImFHSVFqSVFqGsA4I‰K¹}zQPORImoaD‹bImGˆKsD
KJDoa}zQED
DfK
E = {B }
± FFHDr_.DfQHK8_iA4G€A|c4SVImg i 6= i,jj l6=A4g80ImoaFPO { AtA4RQElfQE(AF_.)SV∩ILFdEK%BE=D E{ABi}AvFEG8opRAUE(BA4FEBE)D ∩r(A)
<r<
L
š
f
D
5
K
E
B
V
S
H
F
'
l
V
S
F
E
F
D
.
_
f
D
H
Q
K
E
_
4
A
G
E
_
p
o
4
A
f
l
D
g
4
S
'
h
h
D
f
D
5
K
m
I
²
o
Ÿ
F
R
O
,
{
A
A4QElfQEF
r(B)
A ,A ,B ,B
i=j
_.S4ImFdK€BiA4FEG€omANUiA4FEBED r(A) < r < r(B) SVFGsAvILK€}zQED A CONSB ›kc4SVImg%E¬VQEgJD L µ z
0
i\[email protected] @?hUHP
|
0
r
0
r
pq
0
pq
0
0
pq
0
0
pr
1
0
0
pq
0
0
pq
i
0
p
0
q
0
0
p
pq
j
0
0
q
0
0
0
0
0
f}
^4
M µ¯• !
B0
A0
q = cste
B
A
Lµ
3y:'465
f}
|
p = cste
UH>LK ^>VUH?
A0 CONSB 0
QE_E_aSVGJSVFHG8}XQŸO ILoŸD‹bILGYKsD p ∈ D KJDoP}zQED p(A) = p(B) DK E DGˆK€ImFElfomQEG¹BiA4FEG8oLD0BEDhI‰[ _EopAvF
L oD‹bILGYKsD
KsDfo}zQED
DfK R (B ) ∩ E = {B } L
p ≥ (A)
}
L
š S4h'hD0_EgYMlMfBEDhhi,jDFd∈K’S4{1,2}
F_aDQbK2hSVFdKsgYRDg8}z(AQED )i ∩= Ej D=K A{ACONSB
]0AvFEG8KsSVQEG€oLDG8l|AvG A DfK B GJSVFdKwlS4FEGJMflQHKJILjkG L
2
|
pr
i
0
E
?
pr
j
0
0
0
0
0
0
D tJTVUHI ; <
[email protected]> P
UHP8A &
, G ?]^IJA+*jt gCUHPVE AbaLA f, [email protected]=4AIJUHP
D tJTVUHI ; <
0
UHP8A,&G ?]^IJA
*jt
gCUHPVE AbaLA f
, [email protected]=4AIJUHP
D
G IJUHKM=
ME
[email protected]> P
D<
± FCA¹WYQEGˆKsD2‘ hSVFdKsgYDg:}zQED ME DfGYKQEF D[ _.S4oL{X¬VSVFED%jœAvImUEoLD Lz S4ILK M QEFq_aSVImFdK
BED ME L š ORDGYKoLDhImoLImDQ”BED [AB] SV´ A DK B GYSVFdKClS4FEGJMflQHKJILjkG L8 SVILK r QEFEDŽBEILgJDflfKsILSVF‚BHD
Lª SVI‰K A = A ,B = B DK M oLDqhImomILDQTBED [A B ] L ]O A4_EgY™GNoLDqoLDhh' D_EgJMflMfBEDFdK A DfK
D
−−→ BHSVFEl
GJSVFdKrlfSVFEGJMflQHKJILjkGrBES4FElNomD5_.S4ImFdK M DfGYKrBEA4FEG ME L#± FTA −M−−M→ = −AA−→ + BB
B
L oEGJQ tK†BESVFHl8BED€h'SVFdKJgJDg²}zQED€GYI M = M A4omSVgYGopA0BHgJSVI‰KsD r = r(M ) GJQH_E_.S4gYKsD
r(M ) = r(M )
L QE_E_aSVGYSVFEGwBESVFEl M = M L
E
± FTFEDt_.DfQHK _iA4G A|cVSVILg A = A l|AvgrGJImFHSVFTSVF,AAvQEGJGYI B = B lfD}zQEI²DGˆK5ILh'_aSVGYGJImUHomDlA4g
A4oLSVgJG r = r(A) DfK r = r(B) GYQE_E_aSVgYKJDFdK E DKwBESVFEl E = {A,B} L
QE_H_.SVGYSVFEG A 6= B uESVFAA4oLSVgJG B 6= A L ]’A4FEG8lD lA4G (AB) FEDrGYQE_E_aSVgYKJDr_iA4G E lA4g A DfK
GJS4FzK’BEDlŠžEA4}zQEDNl vKsMBHD L ]2SVFElS4F_aDQHK’GJQE_E_aSVGYDg R (N ) ∩ E = {A,B} A|cVDfl R
B
0HnUL š SVh'hD R (N ) DGYK QH(AB)
F rHn DfK }zQEDomDfG0BEDfh'I [ BEgYSVILKJDG [AA ) DK [BB ) GYSVFdK _iA4gsAvomom™fomDfG
SV_H_.SVGYMDfG S4F,GJA4ILK0}XQHDoxO QHFEDNBPORDFdKsgYD'DfomoLDDGˆK BiA4FEG R (N ) lfSVFdKsgJA4BEImlKsILSVF L ]2SVFEltGJI M = M
A4oLSVgJG A = B DfK B = A hqA4ILGBEA4FEG%lfD0l|A4G¹opABEgYSVILKJD (AB) GJQE_H_.SVgˆKsD E DfK8BHSVFEl ME L š D’}XQHI
hSVFdKsgYDrUEImDfF}zQED ME DfGYKwQEF D[ _aSVoL{X¬VSVFHD’jœA4ImUHomD L
2
i\[email protected] @w?hUHP
0
r
0
0
r
0
0
0
0
0
1
2
0
0
Q
0
0
0
0
0
0
0
0
0
pq
0
0
pq
0
0
PvN UH?;@.KMA\SG K ^>VAK
E < G KOAbaLA\ TIJAd=?`UHP gCUHPQ=?N
D tJTVUHI ; < UHP8AG E ACg D = {x,y}
0
pq
0
0
0
pq
0
KMA
\SG
^>[email protected]@[email protected]
0
[email protected] ?hAg G KdUHPvG
]^?hAP
P = ((0,0),(1,1),(1,0),(0,1)) UHKbN UHPVP8iG ?;PQ=?0G
?J= MP P [email protected]= D tJTVUHI ; < UHP8A,&G ?]^IJA^< E UH?;KuX <
IJUHKM=
> KMA
UHKbN UHPVP8i
P
<
[email protected]> P
µ¯•
!
8hw
M D
8W
' &^
fw^
E
ME
Lµ
3y:'465
Z
|
P
D tJTVUHI ;<
*jt
UHP8A
gCUHPVE AbaLA\SG I(UHKbN UHPVP8i
@#D _EgJSXlŠžiA4ILFomDhhD’hSVFdKsgJD }zQED opAKJgsA4FHGYjkSVgYhqAvKJImSVF
ME
E<
A>aSTVUH?;[email protected] =SN A
N AoN A> an=4A< \[email protected];=ogCUHPQ=4iCg> @w? ,M=N A
M
lS4FEGJDfgYcVD opAlfSVFEGJMflQHKJILcXILKJM L
[email protected]=4iCg> @w? ,M=o=?
[email protected]=4A> IJA\[email protected]=?j?;[email protected]
VS ILDFdK A,B,C KsDfomG5}XQHD ACONSBCONSC Lª SVILDFdK M ,M oLDGhImoLImDfQb‹BED
DfK La± FcvAc4SVImg%KJSVQHKJDG€omDfGw_aSVGYGJImUHImomI‰KsMfG¹BESVFHFEMDfG2_iAvg8omDromDfh'hD L ƒ L µ L
[AB] [BC]
µ|Dg l|A4GŸ oHDŒ‹bImGYKJD i,p,q,N ,N KsDoLGª}zQED R (N )∩E = {A,B} DfK R (N )∩E = {B,C} L
@#DGN_aSVImFdKJGBED E }zQEI¹GYSVFdKBiA4FHGopAUiA4FEBHD p ∈]p(A),p(C)[ GYSVFdK B,B L @#DGN_aSVImFdKJG5BED
}zQEI:GYSVFdKBiAvFEGomAUEA4FEBED ]q(A),q(C)[ GYSVFdK B,B L SVI‰K N KJDo:}zQED M ,M GJSVFdK5GJQEg
E
L QE_H_.SVGYSVFEG}zQED>FHSVQEG'A¯c4SVFEG'BEDQb‹°_aSVILFzKJGtlSVFEGYMlfQHKsI‰jkG DfK
omDUaSVgJB°BED
KsDfomG8}zQED omDrhRImomILD(NQ'BE)D [P P ] DfGYKwBiA4FHG R (N ) L oLSVgJG¹o~ORQEFBEDGw_.S4ImFdKsG A,B,C PDGYKwBiAvPFEG
omDT_iA4gJA4omoLMoLSV¬VgsAvh'hD p ∈ [p(P ),p(P )],q ∈ [q(P ),q(P )] DfKBESVFHl¬Vg 4lD,A4Q omDfh'hD
L ƒ L µ¯ƒ P DfK P FED_aDQbcVDFdKq_iA4GqeKsgYD>lfSVFEGJMflQHKJILjkG'lSVFdKsgJA4BEILlfKsILSVF L:± F‡DF‚BHMBEQEI‰Kq}zQED
DK2BHSVFEl M DK M GYSVFdKwlSVFEGYMlfQHKsI‰jkG L ›kc4SVImg%i¬4QEgJD L µ 
R (N ) ∩ ME = {M ,M }
i\[email protected] @?hUHP
|
1
pq
i
2
1
1
2
pq
i
2
p
pq
i
1
pq
i
2
1
pq
i
1
3
q
3
1
2
1
2
3
2
1
2
2
3
1
2
1
C
2
N2
N3
M2
q = cste
B
N1
M
M1
A
3y:'465
Lµ
|
p = cste
M1
UH> K ^>,UH?
[email protected]
M2
[email protected]=4iCg> @w? ,M=
vƒ Dl|A4GŸ B = B L†± F°ATA4omSVgYG p(A) = p(C) DfKtBHSVFEl (M M ) DGˆK'BHDBHImgJDflfKJImSVF p DfK
GJQE_H_.SVgˆKsD ME DfK€BHSVFEl M CONSM L
Z4Drl|AvGP (AB) GJQH_E_.S4gYKsD E L ± F_.DfQHK€GJQE_H_.SVGYDg€}zQED E DGˆK2ILFEloLQEG%BiAvFEG {p ≥ p(A)} L ± F
AtA4omSVgYG
DfK€BHSVFEl ME ∩ {p ≤ p(C)} = {M ,M ,M }
{A,B,C,C }
L]
SV´ M EDfGY∩Kw{pomDrh≤ImoLp(C)}
ImDfQCBED =[A,C
2
AvImFdKsDfFiA4FdKwSVF c4AgYDfcVDfFEImgwA4QX‹ D[ _aSVo‰{b¬4SVFEDG jœA4ImUEoLDG L
|
1
p
1
|
2
2
1
p
−1
*jt
gCUHPVE
−1
p
2
P8UHP
E [email protected] D tJTVUHI ; <
AbaLA,[email protected]=4AIJUHP D <
?
UHP8A+*jt gCUHPVE AbaLA f
, [email protected]=4AIJUHP
D
G IJUHKM=
M2 E
[email protected]
D tJTVUHI ; <
UHP8A
¯µ •V¥
SVI‰K QEFC_aSVImFdK†BED E L š D0_aSVImFdK†DGˆK%omDwhImomILDQBED2BEDQX‹CGYSVh'hDfKJGlfSVFEGYM[
hh'D5_EgJMflMBHDFdK lDG BEDfQb‹_aSVILFzKJG0_EgYS„cXILDFEFEDfFdK5BHDNKsgJS4ImG
lfQHKsI‰jkG M ,M BED MEML ]O A4_EgJ™fG omDNoLDM
_aSVImFdKJG8lSVFEGYMlfQHKsI‰jkG A,B,C BHD E L
SVI‰K r QEFEDBEILgJDlKsILSVFBED D DK A = A ,B = B ,C = C DK M ,M ,M omDfGrh'ILomILDQb‹BED
L ]O A4_EgJ™fG’oLD5oLDhh'D L ƒ L ƒbµoLD _.SVILFdK M DGˆK0BiA4FHG M E u._aSVQEgwhSVFdKsgJDfg
[A B ],[B C ],[M M ]
}zQED E DGˆK2QHF D _aSVoL{X¬VSVFHDrImoŸGYQ tK’BHSVFEl BED _EgJS4QHcVDfgw}zQED M 6= M L
QE_H_.SVGYSVFEGBESVFEl M = M L š SVhh'DBiA4FHGoLDomDhhD L ƒ L ƒVƒŽSVF°GsAvILK}zQED M = M DfK
hqA4ILG†A4omSVgYG AB BC DGYK¹QEFC_iAvgsA4oLomMoLSV¬VgJA4h'hD†DfK (AC ) DGYK¹_iA4gJA4omoL™omD€‘ (BB ) BHSVFEl
M =M
‘ (AA ) DfK2_iA4g’lfSVFEGYM}zQEDFdK C ∈ {A,A } DK C ∈ {A,A } L ]’SVFEl A = C S4Q A = C,C = A L
AvImG’oLDlA4G
DGˆKrImh_aSVGJGYImUEoLD _EA4g’žd{X_.SvKsžE™fGJD L ]2SVFElNSVFŽA
u.h'A4ImG −−→ −−→ L
]’S4FEl B = BAuE=lfD5C}zQEI#DfGYKwImh_.S4GJGJILUEomD2l|A4g E DfGYK2QEF D[ _aSVoL{X¬VS4FED L A = C B B = AC2
A4ILFdKsDFEA4FdK’SVFcvAI‰KsMgYDg€omD _EgYSblfDGYGJQEG L#± FSVgYBESVFEFHD E _iA4gwopAgYDopA\KsImS4FBHD5lfSVFEGJMflQHKJILcXILKJM
SV´ A = A DfK A CONSA Lz± FBEMfEFEILK†A4oLSVgJG E = (A ) _iAvg²opA’gYMlQHgJgJDfFElD
E = (A )
^4
M 2
i\[email protected] @w?hUHP
1
2
0
0
0
0
0
0
1
0
2
0
r
0
r
0
1
r
0
Q
0
2
0
2
0
0
2
0
0
1
0
0
2
1
0
0
0
0
0
0
0
0
0
0
0
0
i 0≤i≤m
0
m
i
A0i = Ai , An+1
=
i
i+1
n
An
i−1 +Ai
2
+
2
n
n
An
i +Ai+1
2
n
i
1
1
1
= Ani−1 + Ani + Ani+1
4
2
4
SV´oLDG€ImFEBHImlDfG8GJSVFdKwl|AvomlQHomMG8hSXBEQEomS m L
rg 4lfDA4Q”omDhhD L ƒ L ƒXµ SVF‡_aDQbKth'SVFdKJgJDg_EA4ggJMflQEgYgJDFHlDGJQEg n }zQED A DK A GJSVFdK
lfSVFEGJMflQHKJILjkGCDK BESVFHl}zQED E = M E L @#DŽomDfh'hD>_HgJMlfMBEDfFdKh'SVFdKJgJD}zQED E DGYKCQEF D[
_aSVoL{X¬VS4FED Li“ FjœA4ILK%SVFAtQHF>oLDhh'D0_HomQEG€_EgYMlILGP
2n
n
Ai ,Aj ∈ E
?
n
i
n
i+1
n
E iKW? [email protected]
i 6= j _ p(Ai ) = p(Aj )
G IJUHKM=
n∈N
TVUH>[email protected]> @
UHP
G
SVFdKsgJS4FEG#lDKJKsD:_EgYSV_EgYImMfKJM_iA4g#gYMlfQEgJgYDFElfD†GYQEg n L„ QE_E_aSVGJSVFHG#BESVFEl:}zQED p(A ) =
L ]OyAv_EgJ™fGoLDGCoLDhh'DfG L ƒ L µ ! DK L ƒ L ƒbµSVF GsA4I‰KC}zQEDTomDfGCGJDfQEomG _aSVImFdKJGqlS4FEGJMflQHKJILjkGBED
p(A )
GYSVFdK (A ) DK (A ) L ]OyAvQHKsgYD_iA4gYK0BPOyAv_EgJ™fGromDG’omDhhDG L ƒ L ƒ DK L ƒ L µ ! SVF
A = (A )
GJA4ILK8}zQED omDfG8_.S4ImFdKsG8lfSVFEGJMflQHKJILjkGwBED A GYSVFdK A DfK A L ]2SVFEl
p(Ani ) = p(Anj ) <
i\[email protected] @w?hUHP
n
j
n
j
n
i
n
i−1 p
n
i p
}
n
i+1 p
n
j
n
j−1
n
j+1
{Ani−1 )p ,(Ani+1 )p } = {Anj−1 ,Anj+1 }.
D K€BESVFEl p(A ) = p(A ) L
2
A4ILFdKsDFEA4FdKrSVFŽcvA'cVS4Img0}zQEDoLSVgJGY}XQHDNo~ORSVFjœA4I‰K2KJDFEBHgJD c4DgJGroxO ILFHiFEI A4oLSVgJG lSVFdcVDfgJ¬VD
c4DgJGwQEF_.S4oL{X¬VSVFED0A FEDhDFdKwgJMf¬VQEomILDg L š D gJMGYQEoLKsAvK€DGˆKwlSVnFEFzQ>BED_EQHImGw]0A4gYU.SVQX‹(E›Y¢R]0) A4g ! ¦ .
n+1
i
n+1
j
gCUHPVE AK< AE AKM=> P
n
|D| ≥ 3 G
D tJTVUHI ; < UHP8AG
?
IJUHKM=
IG
=>L?;@A
En −G
π
cos2n ( m
)
G
U
[email protected]
[email protected] A
<
KbG E4?;@Bi
N A
E
P8A\[email protected]< > I ?hAK&<
š S4FEGJILBEMgYSVFEG#o~O A4_E_HomImlAvKsILSVF’oLImFEMA4ImgYD φ : C → C BEMfiFHImD²_iA4g φ(z , . . . ,z ) =
SV´5omDfG¤ILFEBEILlDG¤GJS4FzK l|AvomlQHomMGhSbBHQEomS m Lv± FgJDfhqAvgJ}zQED E = φ (E) L
( z + z+ z )
@²O A4_E_EoLIml|A\KsImS4FComILFEM|A4ILgJD φ At_aSVQEg€h'AvKsgYImlfD0BED gJDf_EgJMfGJDFdKsAvKsILSVFr
m
i\[email protected] @w?hUHP
1
4 i−1
1
2 i
1
4 i+1 0≤i<m
m
0
n
1

0 . . . 0 14
1
. . . 0 0

4


1
1
0 0 ... 4 2
1
4
1
2
LL LL LL L L L LL LL
2
1

K = 4

1
4
m−1
n
µVµ¯•
' ± gromAqh'AvKsgYImlD K DfGYK BEImA4¬VSVFEA4omILGsA4UEoLD Lan omQEG’_EgJMflImGYMhDFdK omDfG’cVDflfKsDfQEgJG _EgYSV_EgJDfGr_EgJS4_EgJDfG BHD
GJSVFdK Y = (e ) A4GYGJSXlILMG€A4Qb‹Cc4AvomDQHgJG€_EgJS4_EgJDfG v = cos ( ) L
@ANjœA4hImomoLD (Y ) DfGYK2QEFED UiA4GYD BHD C uESVF_.DfQHKwBESVFEl MlfgJILgJD E = P λ Y La± F>A
8hw
i 2πjk
m
j
M 0≤k<m
8W
2 jπ
m
j
m
j
&^ fw^
m−1
j=0
K
j j
m
−1
2
E = λ 0 Y0 +
SV´
X
(λj Yj + λm−j Y¯j ) + λ m2 Y m2
BHMGJIL¬VFED opANlfSVF\WYQE¬VA4ImGYSVFBED C uiBES4FEl
z 7→ z̄
k=1
m
−1
2
E n = λ 0 Y0 +
X
cos2n (
S4ILK r = max{j ≥ 1 : λ 6= 0 SVQ
j=1
jπ
)(λj Yj + λm−j Y¯j ) + (−1)n λ m2 Y m2
m
λm−j 6= 0}
j
DK
E n − λ 0 Y0
n→∞ cos2n ( rπ )
m
F = (Bi )0≤i<m = lim
lSVhh'D ) > cos( ) _aSVQEg 0 ≤ j < j ≤ SVF*A F = λ Y + λ Y¯ u F DGYKBESVFEl
o~ORImh'A4¬VD’cos(
_EA4g€o~O A4_E_EoLIml|A\KsImS4F R[ omILFEM|AvImgJD φ : z 7→ λ z + λ z̄ BEQ_aSVoL{X¬VSVFHD Y L
š SVhhD8omDG:_aSVoL{X¬VS4FEDG E GJSVFdK†KsSVQEG:BED€¬VDfFEgJD µVudDK}zQED Y DGYKBED€¬VDfFEgJD r u4SVFqArjkSVgYlMhDFdK
L n oLQEG8MoLMhDFdKŠA4ILgJDfh'DfFzK€S4F_aDQbK2GYQE_E_aSVGJDfg2}zQED oxO SVgYBEgJDrBHD E c4MgYILiD
r=1
p(A ) = p(A ) ≤ p(A ) ≤ . . . ≤ p(A
SV´ p DGˆKtQEFED BEILgJDflfKsILSVF<BHD D uªlDfGImFEMf¬dA4oL)ILKs=MfG5p(A
GYSVFdKtA4oLSVgJG ) cV≥MgYp(A
ILiMfDGt_iA4g5)Ks≥SVQEG. .om.D≥G Ep(ABHSVFElC) Df≥F p(A )
_iA4GJGJA4FdK0‘omAtoLImhILKJD2SVFSVUbKsImDfFdK#
p(B ) = p(B ) ≤ p(B ) ≤ . . . ≤ p(B
SVg€GJDQHomD o~ORImh'A4¬VD’A FED BEQ_.SVo‰{X¬VSVFED Y _aD)Qb=Kwc4p(B
MgYILiDfgwlfDG€) Im≥FHMp(B
¬dA4oLILKJMG L ) ≥ . . . ≥ p(B ) ≥ p(B )
]’S4FEl r = 1 u lSVhhD Y = (1,1 . . . ,1) DfK λ DGˆK0oxOyA ‹bD BEQlfDFdKsgYDBED5¬VgsA|cXILKJM G BED E u F
L
DGYK2opANoLImhILKJD2BED
]O A4_EgY™G2omD omDfh'hD L ƒ L ƒ GYI p(A ) = p(A ) A4oLSVgJG p(B ) = p(B ) L ]2SVFElGYI φ DfGYK2UEI WYDflfKsI‰cVD
A4omSVgYG F DGˆKwQEF D[ _aSVoL{X¬VSVFHDrA FEDhDFdKwgJMf¬VQEoLImDg L
QH_E_.S4GJSVFEGtBES4FEl }zQED φ FPORDGYKt_EA4GtUEI WYDflfKsI‰cVD L @²O ILhqAv¬VDqgYMlIL_EgJSX}zQED _iA4g φ BPO QHF°_aSVILFzKNDGYK
BESVFEl KsSVQ\WYS4QEgJGwlSVFdKJDFzQEDBEA4FEG’QHFED5BHgJSVI‰KsD l|A4g φ FPO DfGYK2_iA4G2lSVFEGˆKŠA4FdKJD › λ 6= 0 SVQ λ 6= 0) L
± gwlS4h'hD Y DfGYK’QEFŽ_aSVoL{X¬VSVFHD lfSVFdcVDŒ‹HDILo#FED_aDQbK’{A|cVSVILg’ZGJSVhhDfKsGwBED Y A4omIL¬VFEMG L ]’SVFEl
_.SVQHg2KsS4QHK M ∈ F = φ(Y ) S4FA |{N ∈ Y : φ(N ) = M }| ≤ 2 L ]O A4QHKsgYDN_iA4gˆK’oLD_aSVoL{X¬VS4FED
DfGYK%ILFEloLQEG:BiA4FEGQEFED2BEgJSVI‰KsD LH± FqANA4QEGYGJIaGJQE_E_aSVGYM0}zQED D lSVFdKsILDFdK8AvQqhSVImFEG²KJgJSVILGBEILgJDflfKsILSVFEG
F
L SVILK B QEF³_aSVImFdKBED F u:Imo8DŒ‹bImGYKJD i ,i ,i BEILGYKsILFElfKJG'DKqBHI .MgJDfFdKsGqBHD i KsDoLG'}zQED
{p ,p ,p }
L š SVhhD5SVFFEDN_aDQHK A|c4SVImg2QEFGJDfQEo KsDfo }zQED
uaomDfG’BEILgJDflfKsILSVFEG
p (B ) = p (B )
A¯c4Dl l 6= k GJSVFdK0KJSVQHKsDfGromDfG’BEDfQb‹Mf¬dA4omDfGr‘qomAqBEImgYDlKsImS4FiBED F lD}zBQEI =DGYK0BImh_.SVGYGJILUEomD l|A4g’DomoLDpG
GJSVFdK2BEImGˆKsImFHlfKsDfG L
I |D| = 2 A4omS4gJGrKsSVQbK_iAvgsA4oLomMoLSV¬VgJA4h'hD5‘lfS4KsMfG_iA4gJA4omoL™oLDGrA4QX‹<BHImgJDflfKJImSVFEG BED D DfGYKQE2F
[ _aSVoL{X¬VS4FEDrA FEDfh'DfFzKwgYM¬VQEoLImDfg L
D ±
FABHSVFEl BEMhSVFdKsgYM
j0π
m
jπ
m
m
2
0
r
r r
m−r
r
0
1
m−r r
r
r
2
jmax
jmax+1
jmax+2
m−1
0
m−1
0
n
0
1
2
jmax
jmax+1
1
jmax+2
0
0
En −G
π
cos2n ( m
)
i
j
i
j
1
m−1
1
1
1
1
‚
1
k
2
i
3
1 2 3
i
k
ik
k
i
ik
l
?j?;I`[email protected]> P
š D5}zQEI#DfGYKwD‹HA4< lKsDhDFdK
G
P8A\[email protected]
2
D tJTVUHI ; <
R <
(9) ⇒ (5)
> I ?hAKoNG PQ=
UHP8A
*jt
gCUHPVE AbaLA f
, [email protected]
BiA4FEG€oLD0KsžEMfSVgJ™fh'D L ƒ L L
G IJUHKC=
?;I`[email protected]> P
D tJTVUHI ; <
UHP8A
Ch 4 µVµVµ
-%(" #
& |D| ≥ 7 _yG
?
IJUHKM=zUHP
G
/(:;+.: 3
(Q(D),D) <
“ FD .DfKwo~O A4GJGYDgˆKsImS4F,› ! ¹BEQKsžHMSVgY™hD L ƒ L Imh_EomIL}zQED |D ≤ 6 L
2
]0A4FHGoLD*lŠžiA4_EI‰KsgJD”_EgYMlfMBEDfFzKSVF A KJgJSVQbcVM”QEF`Avom¬VSVgYILKJžEh'D,_aSVoL{XFESVhIpAvo_aSVQEggJMfGJSVQHBEgJD
L
(Q(D),D)
]O A4_EgY™G%omA _EgYSV_aSVGJI‰KsImS4Fqƒ L L µ’QHF'KsSvKŠA4oLDhDFdK†lSVFdcVDŒ‹bDrDGYKKsSVQ„WYSVQEgJG [ lfSVFdcVD‹bDvuEBESVFElwGJIaSVF
GJA4ILK’gJDlfSVFEGˆKsgJQHImgJDQHFEDGJSVoLQHKsILSVF [ lfSVFdcVDŒ‹HDDfK GYIlDKJKJDGJSVoLQHKsILSVFŽDGˆK QEFEIL}XQHD4uP_aSVQEg0GJA|cVSVILgrGJIILo
{ŽAQHFEDtGJSVoLQHKsILSVFŽKsS4KsA4omDfh'DfFdKrlSVFdc4D‹bD'ILoGJQ K BEDtgYD¬dA4gYBEDgrGYI²lDKJKsDtGYSVomQHKJImSVF [ lSVFdc4D‹bD'DfGYK
KJS4KŠA4oLDhDFdK8lfSVFdcVD‹bD L
&
i\[email protected] @w?hUHP
)*,+.-0/(14365x7(+936:;-0/
q
q
q
L
& ? D P8An=^G @ ?J= ,&G ;? @yTQGH= G > a C
g UHPN ?;@w?hUHPQ=vi ^> ?;EHG [email protected]=
)*,+.-0/(14365x7(+936:;-0/
(C,D) TVA>[email protected] @wKMAKMiC=4UHI > A P O(n7 ) UCTViKbG @?hUHPQ=^<
G IJUHKM=
N >
@'>ViCUHK \FA
± FAtA4QHGJGJI#QHF>lfSVgJSVoLopA4ILgJD2_.SVQHg8opANgJDflSVFHGYKsgYQElfKJImSVFBiA4FHG Z 
<
<
fl SV_EomA4FiA4ILgJDG
“ FŽD ŸDK0SVFŽ_aDQbKrA4_E_HomIm}zQEDfg2oxOyAvom¬VSVgYILKJžEh'D’BEDgYDlS4FEGYKJgJQElKsImS4FTƒ4] BiAvDFEG’lŠžiAv}XQHD_EopA4F>; _iA4gJA4omoL™omD
A4QX‹BEImgYDlKsImS4FEG%BED D L
© # fCf5 ©8
4QEGY}XQHD0op‘rSVF'GORDGYK¹ImFdKsMfgJDfGJGJM0‘ oxO QEFHImlI‰KsM8BEDwKsSVQbKsDG†omDfGILFEGYKsA4FElfDG¹_aSVGYGJImUHomDG:BEQq_EgJSVUHom™hD€BED
omA€gYDlfSVFEGYKJgJQElKsILSVF L A4ILG¤S4F_aDQHKªA4QEGYGJIzGJD_.SVGYDg¤omD:_EgYSVUEom™fh'D²GYQEILcvA4FdK#„MKŠA4FdK BESVFEFEMfD¹QEFHD_iAvgYKsILD
BED D‹bILGYKsDŒ[ Kˆ[ ImoaQEFED AvQHKsgYD0_iA4gˆKsImD’BED A|{VA4FdK€omDG8hehDG¹_EgJS\WYDflfKJImSVFEG4¸ n oLQEG8_EgYMlILGJMfh'DfFzK€oLD
_EgYSVZUEoL™hD EST U N IQU E(F ,D) DGˆK’oLZDrGJQEI‰cvA4FdK#
EST U N IQU E(F ,D)
]2SVFEFEMfD'
E∈F
QEDfGYKJImSVF,
F ∈F
F 6= E
∀p ∈ D∀i,j ∈ Z X E(i) =
n Avg5DŒ‹HDfh'_HXomDtFGJI:(i)SVF,_EgJDfFEB F = P(Z ) uDfK D = {x,y} u#SVF,GsA4I‰K }XQŸO ILo²FPOR{T_EA4G QEFEImlfILKJMDfF
¬VMfFEMgJA4o8l|AvgomDGtBEDfQb‹”DfFEGJDfhNUEomDfG E = {(0,0),(1,1)} DfK E = {(1,0),(0,1)} SVFdKtomDGthehD
_EgYS\WYDlKsILSVFEG›œopAgYM_aSVFEGJD5_.S4QEg EST U N IQU E A|cVDl E Mf¬dA4o¤‘ E SVQ E DfGYK0BHSVFEl  L.n A4g
lfSVFdKsgJDroxO DfFEGJDfhNUEoLD E = {(0,0),(1,0),(0,1)} DGˆKwomD0GYDQEo#‘tA|cVS4Img%_aSVQEg%_HgJS\WYDflfKsILSVFEG8oLDG%cVDflfKJDQEgYG
L
[2,1],[2,1]
@#DG€lS4h'_EoLD‹bILKJMG8_aSVQEg8BEI ŸMfgJDfFzKJG€lSVQE_HomDG (D,F ) GYDgJS4FzKwBES4FEFEMDfG2BEA4FEG€opAGJDflfKJImSVF L µ L ƒ L
“ FNjœA4ILK lD_EgJS4UEom™fh'DDGYKªjkSVgYKJDhDFdKoLImM†A4Q_EgYSVUEom™fh'D†BED%omA2gYDlS4FEGYKJgJQElKsImS4FA|cVDl8oLSblA4omILGsAvKJImSVF
A_EgYImSVgYIV
& @KMUH>LE AK> P

D
?
[email protected](> P
APQ=4A\S]^IJAcN A0TI > =dN A
7
N ?;[email protected]?hUHPQ=
@[email protected] IJA\[email protected] AbaLAODTQG [email protected]?;KN [email protected]?hUHPQ==4AIJUHP
b"
n
#
%
G IJUHKM=UHPOTVA> @
[email protected]\
TV=jTVUHI HP8UH\z?G I <
#
2
2
<
Z
q
P8AjTQG [email protected]?hA
r
[email protected]@wt?;I6> P
APQ=4A\S]^IJA
@AI ^>,A
[email protected]
p
p
2
1
2
1
2
P8UHP
3

! UCTViKbG @?hUHPQ= U
)*,+.-0/(10-0+
APQ=4A\S]^IJAFN A N ?;[email protected][email protected] F > P8AvgI [email protected]>VAvNG PQ=
D _vG IJUHKM= EST U N IQU E(F ,D) TVA>[email protected] @KMA KMiC=4UHI > AP O(nC)
n = maxp∈D (maxM ∈E p(M ) − minM ∈E p(M )) [email protected] C [email protected] G gCUH\ TIJAba!?;@i N A
IJAC=[gCUHPVE AbaLAC=
<
=?;\
TIJAC=
?
D
=4AIJUHP
[email protected]> P
µVµ„ƒ
8hw
M 8W
' &^
fw^
SVI‰K0BES4FEl E QHFEDN_iA4gˆKsILD5BED Z BEA4FEG0omA'lomA4GJGYD F LP± FGJD5H‹bDNQEFEDNBHImgJDflfKJImSVF
L
A
v
J
g
E
U
‰
I
s
K
s
g
v
A
m
I
J
g
D
I:Imo²DŒ‹bImGYKJD F ∈ F BEI ŸMfgJDfFzK5BED E A|{VA4FdKNomDfG5hehDG _EgYS\WYDlKsImS4FEG5}zQED E
p∈D
A4omSVgYG8ImoPDŒ‹bImGYKJD5QHFED5BHgJSVI‰KsDr_iA4gJA4omoL™oLD0‘ p KJDoLomDr}zQED F FED lSVFdKsILDFdKw_iA4G€QEFBEDGw_aSVImFdKsG€DŒ‹bKJgJefh'DfG
BED E GYQEgtlDfKYKsDBEgJS4ILKsD L:± F‡_aDQbKqBES4FElQHKsILomImGYDgNoxOyAvom¬VSVgYILKJžEh'D _.SVQHggJMGYSVQEBEgYDoLD _EgJSVUHom™hD
L
EST U N IQU E
2
EST U N IQU E(F ,D) SV´ F DGˆK’lfSVFdKsDfFXQHD5BiAvFEG2oLDG€lS4Fzc4D‹bDG’GJImh_EoLDG8GJDfomSVF
D ± F>lŠžHSVImGYILK
A4gJUHILKsgJA4ImgYD L
p∈D
# $ i ∈ {pmin . . . pmax} % & omDfG8_.SVILFdKsG%KsDfomG€}zQED [AB] = E ∩ {p = i}
A,B ←
wMfGJSVQHK
A|cVDfl (X (E)) u α = ∅ u β = Z \ {A}
(F ,D)
ªImo.{AQEFED GJSVoLQHKsILSVF gJDKsSVQHgJFED ± ® ®’“
,! wMfGJSVQHK
A|cVDfl (X (E)) u α = ∅ u β = Z \ {B}
(F ,D)
ªImo.{AQEFED GJSVoLQHKsILSVF gJDKsSVQHgJFED ± ® ®’“
2
i\[email protected] @?hUHP

)*,+.-0/(10-0+
€
q
q∈D
0
0
q
q∈D
0
0
q
€
€
)*,+.-0/(10-0+
,! "! €
gJDKsSVQHgJFED ®
€
q
€
®’“
2
€
q
2
QlŠžiA4_EI‰KsgJD 'SVFA'KsgYSVQHcVMQEFA4oL¬VSVgJI‰KsžEhDr_aSVQEgwgJDlfSVFEGˆKsgJQHImgJDQEF [ lfSVFdcVD‹bD‘'_iA4gYKJImgwBED
GYDG_HgJS\WYDflfKsILSVFEG L >A4ImGrcvA\[ Kˆ[ ILoefKJgJDQHKsILomImGJA4UEoLDNDF­_HgsAvKJIm}zQEDa¸ “ FjœAvILK omDt_EgYSVUEom™fh'Dth'A„WYDQHg DfGYK
oxO SVUHomIm¬VAvKsILSVF BPO A¯c4SVImg’BEDGr_EgYS\WYDlKsILSVFEG0GJA4FEG A4QElfQEFEDDfgJgJDfQEguŸDFD .DfK GYIªSVFhSXBEILiD omMf¬V™gYDhDFdK
oLDG’_HgJS\WYDflfKsILSVFEG’A4omSVgYG2oxOyAvom¬VSVgYILKJžEh'DrgYM_aSVFEBEgJAqGsAvFEG0BESVQbKsDN}zQPORImo¤FPO {ŽA_iA4G0BEDGJSVoLQHKsILSVF La± FFED
_aDQHK'BESVFHl>QHKJImoLImGJDfgNlDKJKJDhMfKsžHSbBED}zQED>BiA4FEGomDfGlA4GqS4´‡SVF°jœA4ILKtBHDGh'DfGJQEgYDGqGJA4FEGDgYgJDQHgu
lfSVh'hD_iA4g D‹bDh_EomD'DF*hImlfgJSVGYlSV_EILDtMomDflfKJgJSVFEIL}zQEDqSV´<SVF­_iA4gˆcXImDFdKN‘>lS4h'_HKJDg5BEImgYDlfKJDhDFdK
oLDFESVhUEgJDBPO AvKJSVh'DfG L A4ImG’BE™G }zQEDomDfG0SVUXWYDKsGrGYSVFdK BEDG BEILGJlgYMfKJImGJMfGrBPORSVUXWYDfKJGrlSVFdKJImFzQTImo jœAvQHK
A4QHGJGJI#lŠžHDgJlŠžHDg’BHDGwA4om¬4SVgJI‰KsžEhDGBESVFEFiAvFzKwBEDfG2GYSVomQbKsImS4FEG8A4_E_HgJSXlŠžEMDfG L
@#D>KJgsA|cvA4Imo€BEMflgJI‰KCBiAvFEGqlfDŽlŠžiA4_EI‰KsgYDŽA<MKsMgYM|A4oLImGYMA|cVDflopA,lSVoLopA4UaSVgsA\KsImS4F*BPO omUaDgˆKsS­]2Do
@#QEFE¬4SDK A4gsA ¹gYQEFEDKJKsI L
q
&
p
N€ € Œª€5
± FcvAh'A4ImFdKsDfFiA4FdK€GORImFdKJMgJDfGJGYDg2A4Q_EgJSVUHom™hD A4oL¬VSVgJI‰KsžEhIm}zQEDwGJQHILcvA4FdK#
]2SVFEFEMfD' (F* ,p,q) (p ) ,(p ) ,(q> ) ,(q )
SVgYKJImD
E∈F
X (i) ≤ p , q ≤ X (j) ≤ q
± F>_HgJDFHB>oLDG8FES4KsAvKsILSVFEG8GYQEILcvA4FdKsDfGP
)*,+
#
,58-
#
%
[email protected]>[email protected]?hUHP
j> G @KMAv=>L?;@AC=
i i∈Z
XY
Z PV?<
P APQ=4A\S]^IJA
p
0
i
GTTKCU4g ViCA
0
i i∈Z
<
j j∈Z
0
j j∈Z D
EHG IJA> KM=
_2=?m?;[email protected]_YN [email protected][email protected]?hUHPQ=
j
N
E iKW? [email protected]
[email protected]
: pi
: qj
: p0i
: qj0
> 0},
> 0},
> 0},
> 0},
pmax = max{i : pi > 0}
qmax = max{j : qj > 0}
pmax0 = max{i : p0i > 0}
qmax0 = max{j : qj0 > 0}
DK
lp = pmax0 − pmin0 + 1, lq = qmax0 − qmin0 + 1
∆ = {N ∈ Z2 | pmin0 ≤ p(N ) ≤ pmax0 qmin0 ≤ q(N ) ≤ qmax0 }.
µVµ¯Z
=>[email protected]
∀i,j ∈ Z pi ≤
0
j
q
pmin = min{i
qmin = min{j
pmin0 = min{i
qmin0 = min{j
NG PQ=
µVµ
f}
8 &^
fw^
x
# & ^
I
SVQ
A4oLSVgJG oLD_EgJSVUHom™hD
[qmin ,qmax ]
± FcvA>,pmax
FPOyA[pmin,pmax]
_iA4G5BED'GYSVomQbKsImS46⊂F L¤[pmin
MKsQEBEILD]g DF[qmin,qmax]
BEMfKsA4Imo:omDtlA4GSV6⊂´ [pmin,pmax]
DfK
= [pmin ,pmax ]
DFgJDfGYKJgJDIL¬VFiA4FdKwomDr_HgJSVUEoL™hD q
[qmin,qmax] = [qmin ,qmax ]
]’SVFEFEMfD' * (F ,p,q) (p ) ,(p ) ,(q ) > ,(q ) N
pmin = pmin ,pmax = pmax ,qmin = qmin ,qmax = qmax
SVgYKJImD
E∈F
∀i,j ∈ Z p ≤
X (i) ≤ p , q ≤ X (j) ≤ q
]’S4FEl KJSVQHKJDGwomDG€GYSVomQHKJImSVFHG8BED ¶SVFdKwQEF_aSVImFdK€BiAvFEG2lŠžEA4}zQEDlfS4KsM BEQ_iA4gsAvomomMŒ[
omSV¬VgJA4hh'D ∆ L
I²S4F<AQEFA4om¬VS4gJILKJžEhD_aSVQEg omDt_EgYSVUEom™fh'D ru#_.SVQHg gYMGYSVQEBEgYD ILo
GJQ tK BPO A4_E_EoLIm}zQEDgrlfDfK A4oL¬VSVgYILKsžHh'DA¯c4DltomDfGr_EgJS„WYDlfKJImSVFHG ((pp ),(pp ),(qq ),(qq )) SV´ pp DK qq
GJSVFdK2BEMfiFHImDG€_iAvgP
GJI i = ppmin SVQ i = ppmax
pp = max(1,p )
GJImFHSVF
pp = p
GYI i < ppmin SVQ i > ppmax
pp = 0
GJImFHSVF
pp = p
GYI
SVQ j = qqmax
qq = max(1,q ) j = qqmin
GJImFHSVF
qq = q
GJI j < qqmin SVQ j > qqmax
qq = 0
GJImFHSVF
qq = q
DfKwlDflI_.S4QEg%KsSVQbK ppmin,ppmax,qqmin,qqmax cVMfgJILiA4FdK
0
0
0
0
0
0
)*,+
,58-
o)*,+
0
[email protected]>,[email protected]?hUHP
j>QG @wKMAR=> ?;@AC=
i i∈Z
@Z AI;IJAC= ^>,A
P APQ=4A\S]^IJA
0
i
p
GTTKMU4g ViCAG E ACg\
0
i i∈Z
0
j j∈Z
\FAC=]UHKP8AC=
<
0
j j∈Z DEHG IJA> KM=cNG PQ=
0
0
[email protected]=> [email protected] XYPV?
0
_2=?m?;[email protected]_YN [email protected][email protected]?hUHPQ=E iKW? [email protected]
j
0
j
q
p)*,+
<
i
,58-
F)*,+
,58-
v)*,+
Q
i
i
0
,5x-
0
i
,58-
0
j
j
i
i
0
i
0
i
i
0
i
j
j
j
0
j
0
j
j
0
j
pmin0 ≤ ppmin ≤ pmin, pmax ≤ ppmax ≤ pmax0 ,
Po {ABESVFEl (lp lq ) _.S4GJGJILUEImoLILKJMG¹_aSVQEg ppmin,ppmax,qqmin,qqmax L
± F DFBEMfBEQEI‰K8}zQEDrGJIPS4FANQEFA4om¬VS4gJILKJžEhDwBHDrlSVh_EomDŒ‹bILKsM C _aSVQEg%gYMGYSVQEBEgYD A4omSVgYG²SVF'A QHFqA4oL¬VSVgYILKsžHh'DBED€lSVh_EomDŒ‹bILKsM (lplq) C _.S4QEg:gJMGYSVQEBEgYD udDfF'_EA4gYKJImlQHomImDfg
*DfK GJSVFdKw_aSVoL{XFESVhIpAvomDhDFdK¹M}zQEILcvA4oLDFdKsG L
@#D‚UHQHK­BED‡lD³lŠžEA4_EILKJgJD‡DGYK­BED‚_EgYMGYDFdKsDfg*QEF A4oL¬VSVgJI‰KsžEhD*_aSVoL{XFESVhIpAvoN_.S4QEg<gYMGJS4QEBEgJD
Lrn SVQEgŽlfDopA³S4F¶c4A³DfGJGJA¯{4Dg,BPORD‹b_EgJILh'DfgŽBEILgJDlKsDfh'DfFzKŽo~ORD‹bImGˆKsDFHlD
BPO QHFED5GYSVomQbKsImS4F (Q(p,q),p,q)
[ lSVFdcVDŒ‹bD_iA4g8QHFEDNƒ¯[ jkSVgYhNQEomD Li± FGJD H‹bD5BES4FEl _.SVQHg€KJSVQHKJD lfDfKYKsD GJDlKsILSVF>oLDG
}XQEAvKsgYD†_HgJS\WYDflfKsILSVFEG (p ),(p ),(q ),(q ) }zQEI4c4MgYILiDfFzK¤oLDG#žd{X_.S4KJžE™GYDG¤BED²BESVFHFEMD²BED L
qmin0 ≤ qqmin ≤ qmin, qmax ≤ qqmax ≤ pmax0 .
2
2
2
)*,+
,58-
)*,+
,58-
)*,+
)*,+
o)*,+
,58-
j)*,+
,58-
,58-
,58-
q
0
i
i
$-
j
0
j
% "% -'' Šl žiA4}zQED'_.S4ImFdK M BED ∆ SVF,c4AA4GYGJSXlImDfg cvA4gYIpA4UEoLDGrUaSbS4omMDfFEFEDfG (V (M )) SV´ K =
L @A cvA4gYIpA4UEoLD V (M ) D‹b_EgYImhDwoLD8jœA4ILK:}zQEDwopArGJS4omQHKJImSVF E c4ArcVMfgJI‰iDg R (M ) ∩ E = ∅ L
{0,1,2,3}
@:ORImBHMDTBPORQHKsILomImGYDgCBHDTKJDoLomDGCcvA4gYIpA4UHomDG DfGYKBEQED‘ š žEgJS4UiA ³DK] EgJg<›Y¢ š ]0¥V¥ ¦ JŒu8hqAvImGCILomGCFED
o~O A¯cvA4ILDFdK8jœA4I‰K8}zQED _.SVQHgwBHDG€BEImgYDlKsImS4FEG8žESVgYI S4FzKsA4omDfG8DfK8c4DgYKJImlA4omDfG L
}
k
k
k∈K
~
k
µVµ
n omQEG _EgYMlfImGJMfh'DfFdK‘QEFED_iA4gˆKsILDBED SVFc4AA4GYGJSXlImDfg o~ORImFEGˆKŠA4FElfIpAvKJImSVF V = Ψ(E) BEDfG
cvA4gYIpA4UEoLDG%UaSXSVomMfDFEFEDfG V (M ) BHMfiFEILD _iA4ZgP E
GYGJI V (M ) = R (M ) ∩ E = ∅ , A|cVDl k ∈ K, M ∈ ∆.
V = Ψ(E)
€MlIL_EgJSX}zQEDhDFdK‘2QEFEDImFEGˆKŠA4FHlIpA\KsImS4F V BEDG c4AvgJIpAvUEomDfG¤UaSXSVomMfDFEFEDfGªSVF cvA’A4GYGJSXlImDfgªomAw_iAvgYKsILD
BED Z E = Φ(V ) GJQEI‰c4AvFzKJD
GJGJI E = {M ∈ ∆ | ¬V (M ), ∀k ∈ K}.
E = Φ(V )
@A'jkSVgYhNQEoLD5DŒ‹H_HgJImh'A4FdK0oxO DŒ‹bImGYKJDFElfDBPORQEFEDNGYSVomQHKJImSVF [ lS4Fzc4D‹bDcvAqefKJgJDNlfSVFEGYKJILKJQEMDNBHDZCGJSVQHGˆ[
jkSVgYhNQEoLDG’DŒ‹b_EgJILhqA4FdKrgYDGY_.DflfKsI‰cVDfh'DfFzK omA [ lS4Fzc4D‹bILKJM4uPomACU.S4gJFEDNILFHjkMgYImDfQEgJDvuŸDfK DfFHiFTopA'U.SVgYFED
GYQE_.MfgJILDQEgYD L
^ J ^
2
k
k
k
2
k
q
q
$(
!# "-'
$%
-' - " )
± F_EgJDfFEBoLDG8FES4KsAvKsILSVFEG κ,δ BED opAGJDflfKJImSVF>ƒ L µ L @A [ lSVFdc4D‹bILKJM _aDQbK’AvomSVgYG%efKJgJD D‹b_EgYImhMD
‘o~O A4ILBED0BEDfGwjkSVgYhNQEoLDG%GYQEILcvA4FdKsDfGP
q
R0 =
^
((V0 (hi,ji) ⇒ V0 (i − δ,ji)) ∧
hi,ji∈∆
hi−δ,ji∈∆
^
hi,ji∈∆
^
(V0 (hi,ji) ⇒ V0 (hi,j + δi))
hi,ji∈∆
hi,j−δi∈∆
∧
R1 =
^
^
(V0 (hi,ji) ⇒ V0 (hi − u,j − vi))
0<u<δ
0<v<δ
v≡κu[δ]
hi−u,j−vi∈∆
((V0 (hi,ji) ⇒ V0 (i + δ,ji)) ∧
hi,ji∈∆
hi+δ,ji∈∆
^
(V0 (hi,ji) ⇒ V0 (hi,j − δi))
hi,ji∈∆
hi,j−δi∈∆
∧
^
^
(V0 (hi,ji) ⇒ V0 (hi + u,j − vi))
0<u<δ
hi,ji∈∆
0<v<δ
v≡−κu[δ]
hi+u,j−vi∈∆
R2 =
^
((V0 (hi,ji) ⇒ V0 (i + δ,ji)) ∧
hi,ji∈∆
hi+δ,ji∈∆
^
hi,ji∈∆
^
(V0 (hi,ji) ⇒ V0 (hi,j + δi))
hi,ji∈∆
hi,j+δi∈∆
∧
R3 =
^
^
(V0 (hi,ji) ⇒ V0 (hi + u,j + vi))
0<u<δ
0<v<δ
v≡κu[δ]
hi+u,j+vi∈∆
((V0 (hi,ji) ⇒ V0 (i − δ,ji)) ∧
hi,ji∈∆
hi−δ,ji∈∆
^
(V0 (hi,ji) ⇒ V0 (hi,j + δi))
hi,ji∈∆
hi,j+δi∈∆
∧
^
hi,ji∈∆
^
0<u<δ
0<v<δ
v≡−κu[δ]
hi−u,j+vi∈∆
(V0 (hi,ji) ⇒ V0 (hi − u,j + vi))
µVµ
8 &^
fw^
x
# & ^
± F*BEMfGJIm¬4FEDC_iA4g š ± opA>jkS4gJhNQEoLD
L @AjkSVgYhNQEomD š ± _aDQHK
GJDhUEomDfglfSVh_EomIL}XQHMD h'A4ImGNlvO DfGYKGJDfQEomDfh'DfFdK'RomA∧lSVRF\WYS4∧FElfRKJImSVF”∧ BERDfGtjkS4gJhNQEoLDG V (M ) ⇒ V (N )
A¯c4Dl N ∈ R (M ) DfK M DfK N A4GJGYD _EgJSXlŠžEDfG L
q
q
€c
0
E iK? XjA
2
k
? V [email protected] > P8Az?;[email protected] P8g?G @w?hUHP N AC=oEHG KW?G]^IJAC=p]U4UHIJiCAPVP8AC=OE
M,N ∈ ∆ @BAI ^>VA N ∈ Rk (M ) I G f, UHK\z> IJA
TVUH> [email protected]> =
€d
3
~
k
1
k
Rk
iKW? XyG [email protected]
V
G IJUHKM=
Vk (M ) ⇒ Vk (N ).
± FGYQE_E_aSVGJD _iA4g8DŒ‹HDfh'_HomD k = 0 DK M = hi ,j i L g 4lDromAjkSVgJhQEomD R SVF
GsA4I‰K%}zQEDr_aSVQEgKJSVQHK i,j ubSVF A V (hi,ji) ⇒ V (hi − δ,ji) DK V (hi,ji) ⇒ V (hi,j − δi) L ]’SVFEl’SVF
_.DfQHKwh'SVFdKJgJDg8_EA4g€gJMlfQEgJgYDFElfD V (M ) ⇒ V (hi − kδ,j − lδi) ui_aSVQEg%KJSVQHK k,l ∈ N L
S4ILK N QEF_aSVImFdK€BED R (M ) La S4ILK N KsDfoP}zQED
i\[email protected] @?hUHP
M
0
0
0
0
M
0
0
M
0
M
0
0
q(M ) − q(N 0 )
q(M ) − q(N )
=
δ
δ
p(M ) − p(N )
p(M ) − p(N 0 )
,
=
δ
δ
± FŽcXILDFdKWYQEGˆKsDBEDNhSVFdKsgYDg0}zQEDo~ORSVFA V (M ) ⇒ V (N ) DKrlS4h'hDomAjkSVgYhNQEomD
DGˆKwBiA4FEG R SVFAUEILDF V (M ) ⇒ V (N ) L
V (N )
@#D omDfh'hD’GYQEILcvA4FdKwh'S4FzKJgJDr}zQED š ± D‹b_EgYImhD UHImDFomA [ lSVFdc4D‹bILKJM
0
0
0
0
p
[email protected]
SVILDFdK
|
I
KsSVQbK
TQGH=SN
^>,A
,fUHK\z>
IJA
*
I ?;[email protected] G P8g?G @?hUHP
8_G IJUHKM=
IVTVA>
>;[email protected] >
C=4AzN A
$-
%
*jt
E⊂∆
TVUH> [email protected]> @
V = Ψ(E)
E = Φ(V )
=^G @?J= &
, G ?;@[email protected]>Lt
[email protected]+*jt gCUHPVE AbaLA
Lª± F­AŽA4omS4gJG
BHSVFElqlfSVhh'D
DfK€BES4FEl DF_EA4gYKJImlQHomRImDfg¹(NoLDG%) jkSV⊂gYhNRQEom(M
DfG R) L
DfK€BES4FEl BPOyAv_EgJ™fG8omDromDfh'hD ! L µ L µVuES4F V (N ) _aSVQEg
L
M,N ∈ ∆
N ∈ Rk (M )
V = Ψ(E)
Vk (M ) ⇒ Vk (N )
M 6∈ E
k
Vk (M )
N ∈ Rk (M )
Rk (M ) ∩ E = ∅
|
E⊂∆
q<
KsDfo}zQED
VS F>A
uHImoŸDŒ‹bImGYKJD KsDoŸ}XQHD
L “ KwBESVFEl
i\[email protected] @?hUHP
q
€c
APQ=4A\S]^IJA
efUH>LKM=IJAC=,fUHKW\z> IJAC=* 2<
? I ?;[email protected] P8g?G @?hUHP V =^G @?J= ,&G ?;@I G
=4AIJUHP
V0 (N 0 ) =⇒
2
0
q
UH> [email protected]>[email protected]
0
0
k
k
k
k
2
@y=4A\S]^IJAKz]^?4G KWKMA
^>VAoI G
TKMA\z? KMA
TQG [email protected]?hApN >
IJA\z\[email protected]?hAPVP8A
gCUHPVE Aba!?;@i=>LK
@c\[email protected],&G ?;@
I APQ=4A\S]^IJA
AbaLAN A
E _ \SG [email protected] P8UHKW\SG Iug G KnUHPvTVA>
Φ ◦ Ψ(E) [email protected] i<G I6DFI APVE AIJUCTTVA+*jt gCUHPVE
E<
' "' #"$ AvImFdKsDfFiA4FdK0SVF>c4A'D‹b_EgJILh'Dfg2}zQEDomDG2_EgJS„WYDlfKJImSVFHG0BEDG’GJSVoLQHKsILSVFEGwGJSVFdK0GYQE_aMgJILDQEgYDGrAvQb‹ L
± F”cvA,lfSVFEGYImBEMfgJDgQEFEDGYDQEoLDBHgJSVI‰KsD p = i L† SVILK qmin = min{j|hi,ji ∈ ∆} DfK qmax p=
L oajœA4QHK8gYDh'A4gJ}zQEDfg8}XQHDrGJI δ 6= 1 lDfGwFHSVhNUEgYDG%FEDrGYSVFdK€_iA4G%KJSVQ\WYSVQEgYG%M¬VA4Qb‹
max{j|hi,ji ∈ ∆}
‘ qmin DK qmax L
i
i
i
^ @ANjkSVgJhNQHomD
cvAtD‹b_EgYImhDg8}zQED
LB(p,i,l)
LB(p,i,l) = V RAI


LB(p,i,l) = 

SV´
G
µVµ !
J ^
Xp (i) ≥ l

l=0

^
qmin≤j≤qmax−δl
j≡κi[δ]

(L1 (j) ∧ L2 (j) ∧ L3 (j) ∧ L4 (j))

∧ (L01 ∧ L02 ∧ L03 ∧ L04 )
! !
L1 (j) = V0 (hi,ji) ⇒ ¬V2 (hi,j + δli)
L2 (j) = V0 (hi,ji) ⇒ ¬V3 (hi,j + δli)
L3 (j) = V1 (hi,ji) ⇒ ¬V2 (hi,j + δli)
L4 (j) = V1 (hi,ji) ⇒ ¬V3 (hi,j + δli)
L01 = ¬V2 (hi,qmini + δ(l − 1)i)
L02 = ¬V3 (hi,qmini + δ(l − 1)i)
L03 = ¬V0 (hi,qmaxi − δ(l − 1)i)
L04 = ¬V1 (hi,qmaxi − δ(l − 1)i)
E [email protected] *jt gCUHPVE AbaLA=4AIJUHP p [email protected] q [email protected] E iKW? XjA Xp E(i) ≥ l
IJUHKM= V = Ψ(E) =^G @?J= ,&G ?;@ LB(p,i,l) <
? V =^G @?J= ,&G ?;@ QCON V ∧ LB(p,i,l) _ G IJUHKM= E = Φ(V ) =^G @?J= &
, G ?;@ Xp E(i) ≥ l <
?R> P
APQ=4A\S]^IJA
@#DrlA4G l = 0 DGˆK€KsgJI‰cXIpA4oxubBES4FEl SVF>_aDQbK2GYQE_E_aSVGJDfg l ≥ 1 L
SVILK j KsDo#}zQED qmin ≤ j ≤ qmax − δ La SVILK A = hi,ji,B = hi,j + δli La± F>A A,B ∈ ∆ L
IŸSVFA V (A) ∨ V (A) A4oLSVgJGImo.FPOR{qAN_iA4G8BHD0_.S4ImFdK8GJQHg%opAoLIm¬VFHD p = i ‘N¬VA4QElŠžEDrBED A L o
{CANGJDfQEomDfh'DfFzK l − 1 _aSVImFdKJG%DFdKsILDgYG8GJQEg€omD’GJD¬4h'DfFzK ]AB[ uiDK E A l _aSVILFzKJG8GJQEg8omANBEgJS4ILKsD
BESVFHlCImo²{AQEF­_aSVILFzK M = hi,j i ∈ E c4MgYIL A4FdK q(M ) ≥ j + δl L ]’SVFElCS4F­A
p = i
L ± F>ABES4FEl _EgJSVQbcVM (V (A) ∨ V (A)) ⇒ (¬V (B) ∧ ¬V (B)) lDr}zQEI
¬V (B) ∧ ¬V (B)
DfGYKwM}zQEI‰cvA4omDfFzKw‘omAjkS4gJhNQEoLD L (j) ∧ L (j) ∧ L (j) ∧ L (j) L
€DGˆKsDfFzK2‘_EgYSVQHcVDfg2oLDG8ILh'_EoLImlAvKsILSVFEG†_.SVQHg8omDG€UaSVgJBHG L ± g€SVFANjkSVgJlfMhDFdK
l|A4g8ILo
{'AGYDQEoLDhDFdK l − 1 _.SVILFdKsG M c4MgYIL A4FdK p(M ) = i DfK q(M ) ≤ qmin + δ L L]’Dr∧hLehD2SVF
A L ∧L L
SVILK A = hi,qmin i DfK B = hi,qmin + li L ]O A4_EgJ™fG'oLD omDfh'hD ! L µ L µDfK L ,L uSVF°A
u _aSVQEg%KJSVQHK M ∈ [A,B[
¬V (M ) ∧ ¬V (M )
QE_E_aSVGJSVFHG2}zQED
DK
uilS4h'hD GJAvKsILGYjœA4I‰K š ± DfK2BPO A4_EgJ™fG€omD omDfh'hD
! L µ L µVu#SVFGsA4I‰K }XQH¬VD ¬V(A)(M )¬V∧ ¬V(A)(M ) _aSVQEg VKsSVQbK M ∈ [AB) ∩ ∆ L ]’S4FEl E = Φ(V )
lfSVFdKsILDFdK5KJSVQEG5omDG _aSVImFdKsG DFdKJImDgYG5BHD [A,B[ u ILo{<A l _.S4ImFdKsG GJQEg [AB[ u S4F­A>BHSVFElUEImDfF
L
X E(i) ≥ l
oŸgJDGˆKsD5‘NcVSVILg8omDrl|AvG V (A) SVQ V (A) L SVI‰K
SVQ V (hi,ji)}
j = max{j : V (hi,ji)
i\[email protected] @w?hUHP
|
0
1
M
2
3
0
1
2
1
2
3
3
4
0
2
0
1
i
|
0
3
0
4
i
2
0
1
i
3
q
0
1
0
1
p
0
1
0
0
1
€d
0
2
µVµ
4
8 &^
fw^
x
# & ^
VS I‰K C = hi,j i DfK D = hi,j + δli L ]5OyA4_HgJ™GomDfGjkS4gJhNQEoLDG L SVQ L SVF GsA4I‰K>}zQED
DfK'BESVFHl D ∈ ∆ L ]O A4_EgY™G'opAThqA\‹bILhqA4oLILKJMCBED j SVF‚GJA4ILK'}zQED_aSVQEg
j ≤ qmax − δl
KsSVQbK M ∈]CD) ∩ ∆ SVF*A ¬V (M ) ∧ ¬V (M ) L š SVhh'D V (C) ∨ V (C) oLDG5jkS4gJhNQEoLDG
h'S4FzKJgJDfFzKw}zQED ¬V (D) ∧ ¬V (D) L ± F>DfF>BHMBEQEI‰Kw}XQHD
L (j ) ∧ L (j ) ∧ L (j ) ∧ L (j )
_.S4QEgqKsS4QHK M ∈]CD] S4F A ¬V (M ) _.S4QEgqKsS4QHK k DfKBHSVFEl E lfSVFdKsILDFdK ]CD] }XQHIrA
D‹HA4lKsDhDFdK l _aSVImFdKsG8DfFdKsImDfgJG L
2
± F_.DfQHKwBEMfEFEImg€BED opANhehD0jœA fSVF LB(q,j,l) L
0
0
0
0
3
i
0
1
2
0
3
0
4
0
0
1
0
2
0
4
0
1
3
k
$-
%
' !
" #
AvImFdKsDfFiA4FdKwSVF cVDQbK’DŒ‹H_HgJImhDg8}zQED X (E)(j) ≤ l L
š SVhhD _.SVQHgo~O A4om¬VS4gJILKJžEhDqBEQ°lŠžiA4_EI‰KsgYD Xu²SVF°cvAefKJgJDSVUEoLIm¬VMCBHDH‹bDfgBHDQb‹‡_EILDBEG L†± F
lSVFEGYImBE™fgJD BESVFEl BEDfQb‹_aSVImFdKJG A,B ∈ ∆ KsDfomG€}zQED p(A) = pmin,p(B) = pmax L
± FlSVFHGJImBH™gJD A4oLSVgJG8QHFED BEgJSVI‰KsD q = j L @¤ANjkSVgYhNQEoLD UB(q,j,l,A,B) cvAefKJgJD BEMiFEImD _iAvgP
q
UB(q,j,l,A,B) = IN(A) ∧ IN(B) ∧
^
U (i)
pmin≤i≤pmax−δl
j≡κi[δ]
SV´r
IN(M ) = ¬V0 (M ) ∧ ¬V1 (M ) ∧ ¬V2 (M ) ∧ ¬V3 (M )
U (i) = ¬V0 (i,j) ⇒ V1 (i + δl,j)
U (i) = ¬V3 (i,j) ⇒ V1 (i + δl,j)
U (i) = ¬V0 (i,j) ⇒ V2 (i + δl,j)
U (i) = ¬V3 (i,j) ⇒ V2 (i + δl,j)
?j> P APQ=4A\S]^IJA E [email protected] [email protected] A,B [email protected] *jt gCUHPVE AbaLAO=4AIJUHP p [email protected] q [email protected]=?
Xq E(j) ≤ l _RG IJUHKC=I ?;[email protected] G P8g?G @?hUHP V = Ψ(E) =^G @w?J= ,&G ?;@ QCON V ∧ UB(q,j,l,A,B) <
? V =^G @?J= ,&G ?;@ QCON V ∧ UB(q,j,l,A,B) _ G IJUHKM= E = Φ(V ) [email protected][email protected]= TVUH?;[email protected] = A,B _
[email protected] iKW? XjA Xq E(j) ≤ l <
I j ≤ min{q(A),q(B)},
I q(A) ≤ j ≤ q(B)
I q(B) ≤ j ≤ q(A)
I j ≥ max{q(A),q(B)}
VS I‰K C = hi,ji ∈ ∆ KsDfo#}XQHD pmin ≤ j ≤ pmax − δ Li± FlfSVFEGJILBE™gYD D = hi + δl,ji ∈ ∆
QE_E_aSVGYSVFEG}zQEDo~ORSVFTDGˆKBEA4FEG omDtl|A4Gq›xAV j ≤ min{q(A),q(B)} L £¤SVQHK5_.SVILFdK M BEDopA
BEgJS4ILKsD q = j c4MgJI‰iD A4omS4gJG ¬V (M ) ∧ ¬V (M ) L
I SVF'A¯cvA4I‰K
A4oLSVgJG lSVFdKJImDFHBEgsA4I‰K:KsSVQEG:oLDG†_.SVILFdKsG²BED [CD] }zQEIaA l + 1
_.S4ImFdKsG8DfFzKJImDf¬VgJG L ]’(C)S4FE∧l ¬V
SVFUE(D)
ImDfF>omAjkSVgJhNEQHomD U (i) L
± F>_EgYSblf™BED BED omAthehD’jœA fSVF_.SVQHg8omDG8lA4G5›œU Œu¤›œl|8DK›œB  L
š SVhh'D V GsA\KsImGˆjœA4ILK IN(A) ∧ IN(B) uioLDG8_EILDBEG A DfK B GJSVFdKwBiA4FHG E L
š SVFEGYImBEMfgJSVFEGTDfFElSVgYD”omD*l|AvG°›xAd. j ≤ min{q(A),q(B)} L @#DGcvA4gYIpA4UEoLDG V GJAvKsILGYjkSVFdK
š ± Nu€BESVFHlBPO A4_EgY™GomDTomDfh'hD ! L µ L µ,KJSVQHK_aSVImFdK M BED,opA*BEgYSVILKJD q = j cVMfgJI‰iD
L
DfK
¬V (M ) ¬V (M )
i\[email protected] @?hUHP
|
2
0
1
|
q
€c
2
3
3
µVµ¯¥
I
_.SVQHg’KsSVQbK M KsDoª}XQHD p(M ) = i AvomSVgYG X (E)(i) = 0 DfKrBES4FElomDNoLDhh'D DGˆK
UHImDVF c4(MMgJI‰)iM L
ImFESVFSVF­lfSVFEGJILBE™gYD i = min{i : ¬V (hi,ji)} Lª SVILK C = hi ,ji DfK D = hi + δl,ji L
]5OyA4_HgJ™G S4F GsA4I‰K%}zQEDro~ORSVF A (D) LH± F DF BEMfBEQEILK%}XQHD0KsSVQEG¹omDG_.S4ImFdKsG%BHD E GJQHg%opA
BHgJSVI‰KsD p =U (i)i GYSVFdK8BiA4FHG¹omDwGJDf¬VhDFdVK [CD[
}XQHIPA5DŒ‹HA4lfKJDhDFdK l _.S4ImFdKsG†DfK¹BESVFEl X (i) ≤ l L
± F_EgYSblf™BED BED opAhehD0jœA S4F_aSVQEg8oLDG€l|AvG›kU u#›œl¯%DKN›kB  L
2
± F_.DfQHKwBEMfEFEImg8BED opANhehD0jœA fSVF UB(p,i,l,A,B) L
^ J ^
0
q
0
0
0
0
1
q
$( %- " )
& #
-' ) )"('
SVI‰K (P,P ,Q,Q ) = ((p ),(p ),(q ),(q )) QHFED ImFEGˆKŠA4FElfD BED L
± FH‹bDqA4omSVgYG }XQEAvKsgYDq_aSVImFdKsG A,B,C,D KsDfomG }XQHD p(A) = pmin u p(B) = pmax u q(C) =
u
_EQEILG€SVFlSVFEGˆKsgYQEILKwopAjkS4gJhNQEoLD0UaSbSVoLMDfFEFED
qmin q(D) = qmax
0
0
0
i
i
)*,+
0
j
j
RAPP(P,P 0 ,Q,Q0 ,A,B,C,D) = QCON V ∧
,58-
LB(p,i,pi ) ∧ UB(p,i,p0i ,C,D)
^
LB(q,j,qj ) ∧ UB(q,j,qj0 ,A,B)
pmin≤i≤pmax
∧
“ FgJMQHFEImGYGsA4FdKwomDfG8omDhhDG ! L µ L ƒbu ! L µ L DK ! L µ L buHSVFSVUHKsILDFdK#
qmin≤j≤qmax
}
!
^
!
7 G ,fUHKW\z> IJA RAPP(P,P ,Q,Q ,A,B,C,D) [email protected] =^G @w?J= ,&G ?J=^G]^IJA [email protected] =4A> IJAt
\[email protected]$=?s?;I
[email protected]
> P
*jt gCUHPVE AbaLA [email protected] [email protected] A,B,C,D =^G @w?J= ,&G ?J=^G [email protected] IJA TKCU ]^I \FA
)*,+ ,58-
A TI >=
=? V =^G @?J= ,&G ?;@
(Q(p,q),p,q) G E ACg gCUH\z\FA N UHPVP8iCA (P,P ,Q,Q ) <
)*,+ ,58-
RAPP(P,P ,Q,Q ,A,B,C,D) G IJUHKM= E = Φ(V ) [email protected]`>LP8A =4UHI >[email protected]?hUHP N A
š DlfIªBES4FEFEDNILh'hMBEImAvKsDfh'DfFdK€QEFA4oL¬VSVgJI‰KsžEhDr_aSVQEg’gYMGYSVQEBEgYDn)*,+ ,5x- .SVF>< H‹bDNomDfG }
0
0
0
0
0
0
_aSVImFdKJG A,B,C,D uXSVF lA4omlfQEomD2opA jkS4gJhNQEoLD RAPP ub_EQEImGSVF lŠžEDfgJlŠžEDroLDG%UaSbS4omMDfFEG V GsAvKJImGYjœAvImGsAvFzK
LEn SVQEgGJD’jœA4ILgJDwSVFCQHKJImoLImGJDwQEF Avom¬VSVgYILKJžEh'D€oLImFEMA4ImgYDwgYMGYSVoLcvA4FdK€ƒ £ ›Y¢ 0n £ ! ¥ ¦  LH IaImo {qA
RAPP
QEFHD5GYSVomQHKJImSVFSVF>gJDKsSVQEgYFED E = Φ(V ) GJImFHSVF>S4FlŠžESVILGJI‰K2QHFEDAvQHKsgYD5_aSVGJI‰KsILSVFBED5_HImDBHG L. ISVF>A
MKsQEBEILM0KsSVQbKsDG€oLDG€_.S4GJILKJImSVFHG%BED _EImDfBEGwA4omS4gJG%ILoPFPO {A_iA4G€BED GJSVoLQHKsILSVF L
o¹{<AA4Q”_EomQHG lp lq _aSVGJGYImUEILomI‰KsMG BED _EILDBEG L ]O A4QHKJgJD _iA4gˆKopAjkSVgJhNQHomD RAPP _.DfQHKtefKJgJD
lA4omlfQEomMfDrDfK€gJMfGJSVoLQED DF O(lplq) L ± F>DfF>BHMBEQEI‰K#
2
& UCTViKbG @?hUHPQ=^<
I
2
[email protected]
7
>LP
G I
<
UHKW?;@'> \FAvKMiC=4UHI EHG [email protected]
GTTI ?hg G @w?hUHP
Ψ◦Φ
\
\FA
)*,+
,5x-
[email protected]?;[email protected] >a
[email protected]>[email protected]?;I ?J=^G [email protected]
O(lp3 lq 3 )
V E iKW? XyG [email protected]
p = x_ q = y _
EHG KW?G]^IJAC=
RAPP P [email protected] TQGH= i<G IJA D I ?N [email protected]?;@Bi^< P TVA>[email protected] TKMAPN KCAoTQG K AbaLA\ TIJA
(pi ) = (p0i ) = (1,0,1),(qi ) = (qi0 ) = (1,0,1) < E UH?;K0X < > KCA <
G [email protected] PQ=IJA g GH=?;PQ=4iCg G]^IJA ∀i p > 0,q > 0 I GTTI ?hg G @?hUHP Ψ ◦ Φ [email protected]]^?hAP i<G IJAD
i
i
r
I ?N [email protected]?;@iy=> KjIJAC= EHG KW?G]^IJAC=yE iKW? XyG [email protected] RAPP <
P A [email protected]\z\FA Φ(V ) [email protected]*jt gCUHPVE AbaLAcUHPFG
Φ◦Ψ◦Φ(V ) = Φ(V ) KMA\SG K ^>,A < <-9 < KdgCUH\z\FA Φ(V ) [email protected]?;PQ=4iCg G]^IJAC_TVUH> [email protected]> @QTVUH?;[email protected]
UHP8g
M ∈ ∆ \ Φ(V ) ?;[email protected]>LP
^>QGN KbG [email protected] Ri (M ) @BAI ^>,A Ri (M ) ∩ Φ(V ) = ∅ <
Φ(V ) = Φ(V 0 ) ⇒ V = V 0 N U APnTKMAPG [email protected] V 0 = Ψ ◦ Φ(V ) UHP G ]^?hAP V = Ψ ◦ Φ(V ) <
QEFEIL}zQE D µ„ƒ4•
8 &^
fw^
x
# & ^
V0
V1
V2
V3
V2
V3
Φ
Ψ
V0
3`:'465
$-
! Lµ
)#- &%&#
V1
|
r
aLA\
Ψ ◦ Φ 6= Id <
TIJA\[email protected] [email protected] ^>,A
-' ) )"$-'
%
)-
L
“ Fq_HgJDFEA4FdK
DfK
u ,gYDBEDcbILDFdK%oLD€_EgJSVUHom™hD
L
’] SVFEl o~O A4om¬4SVgJI‰KsžEhP D2=_HgJPMlfMBEDfQFdK’=_aDQgYh'DKwA4QEGJGYIBED gJMGYSVQEBEgYD
“ FjœA4ILK8BEA4FEG€lD l|AvG8SVF>_aDQbKwA4h'MfomILSVgJDfg¹o~O A4oL¬VSVgJI‰KsžEhDwl|A4g8lfSVh'hDrBiA4FHGwoLD0lŠ(Q(p,q),p,q)
žiA4_HILKsgYD S4F>A
U.DfGJSVILF>BEDrH‹bDfgwGYDQEoLDhDFdK’ƒN_EImDfBEG L
“ FD .DfKlSVhh'D8S4F'BES4ILK†A|cVSVILg:BED8KsSVQHKJD8jœA SVF P p = P q udSVFqA GJDfQEomDfh'DfFdKUaDGYSVImF
BPO DŒ‹b_EgJILh'Dfg omA€UaSVgYFED:ImFHjkMfgJImDfQEgJD:GYQEg QHFEDBHImgJDflfKJImSVF DfK opA8UaSVgJFHDGYQE_.MfgJILDQEgYDGYQEg oxOyAvQHKsgYD L„n omQEG_EgYM[
lImGYMhDFdKGJI GYSVFdK#KsDfomGP}zQED p(A) = pmin DK p(B) = pmax A4omSVgYG
(Q(p,q),p,q)
c4AA|cVSVILg8QEFEA,B
D GJS4omQHKJImSVF lfSVFdKsDFEA4FdK A DK B GYI#DfKwGJDfQEomDfh'DfFzKwomAjkSVgJhNQHomD
0 )*+
0
5x-
)*,+.-0/214365872+936:;-0/
p)*,+.-0/21 365x7(+936:;-m/
&
n
i=1
n
j=1 j
i
2)*,+.-0/(14365x7(+936:;-0/
^
REXACT(P,Q,A,B) = QCON V ∧
LB(p,i,pi ) ∧
^
UB(q,j,qj ,A,B).
D GYK GJAvKsILGYjœA4ILGsA4UEoLD L š DlfI²_.DfgJhDfK BEDKsgYSVQHc4DgrQEFA4om¬4SVgJI‰KsžEhD5DfF O(lplq min{lp ,lq }) _aSVQEg’gJM[
L š ORDGˆK'KJžEMS4gJIm}zQEDfh'DfFdK'hImDfQb‹<}XQHD>oxOyA4oL¬VSVgYILKsžHh'DBEQ
GJSVQEBHgJD
(Q(p,q),p,q)
lŠžiA4_EI‰KsgJD Nh'A4ImG%S4Fc4DgJgJA}zQED BiA4FHGwomAN_EgsA\KsIm}zQEDrlfD5FŸO DfGYKw_iA4G8cXgsA4ILh'DfFdK€omDrl|AvG L
pmin≤i≤pmax
qmin≤j≤qmax
2
2
)*,+.-0/214365872+936:;-0/
$-
-'' " )
I F DGYK omAClopAvGJGJDtBEDfG· 2[ lSVFdcVDŒ‹bDG „[ lS4FEFED‹bDfGA4oLSVgJGrBŸOyA4_HgJ™G oLDNKsžEMfSVgJ™fh'D'ƒ L L Z F DGˆK
opA'lfopA4GYGJD5BEDG [ lSVFdcVDŒ‹bDGrImFHGJMlA4UEoLDGwGJDoLSVF {x,y} Lan SVQEg2gJMfGJSVQEBHgJD (F ,{x,y}) ILo
GJQ tK2BED gJDh_EomA4lDfg (p ) _iA4g (max(p ,1)) DK (q ) _iA4g (max(q ,1)) Li± FDFBEMfBEQEILKP
c
q
Q
n)*,+
& 7
i
)*,+
i
,58-
j
,58-
j
KsQHBEImSVFHGh'A4Im FdKsDfFiA4FdKNomA v[ lO(lx
SVFHFED‹blyILKJM )L @A_EgJS4_.SVGYILKJImSVF<ƒ L L BESVFEFED'QEFEDClA4gsAvlfKsMfgJILGsAvKJImSVF
BEDG²· 2[ lSVFdcVDŒ‹bDG v[ lSVFEFHD‹bDG²A|c4Dl%oLDGª}zQiAvBEgsA4FdKJG LVn oLQEG _HgJMlfImGYMhDFdK²GJI E DGYK [ lSVFdc4D‹bD%GJDfomSVF
!t gCUHPVP8AbaLAC=RTVA> @
c
A.TKMU ]^I \FA
@KCAKMiC=4UHI >
u}
AP
}
3
3
(F ,{x,y})
U
F
[email protected] GgI GH=C=4AyN AC=
UCTViKbG @w?hUHPQ=^<
q
6t gCUHPVE AbaLAC=
{x,y}
µ„ƒbµ
4
J ^
4w4
A4omSVgYG E DGYK v[ lSVFHFED‹bD GJI#DK2GYDQEoLDhDFdKwGJIV
d}
SVQ
SVQ
E ∩ (R0 (x,y) ∪ R2 (x + 1,y + 1)) = ∅ =⇒ ((x,y) 6∈ ∆
(x + 1,y + 1) 6∈ ∆)
E ∩ (R1 (x + 1,y) ∪ R2 (x,y + 1)) = ∅ =⇒ ((x + 1,y) 6∈ ∆
(x,y + 1) 6∈ ∆)
± F_.DfQHKwBESVFEl DŒ‹H_HgJImhDg8omA v[ lfSVFEFEDŒ‹bILKsM _iAvg8opANjkSVgJhQEomD
}
^
4CON N =
(¬V0 (x,y) ∨ ¬V2 (x + 1,y + 1)) ∧
(x,y)∈∆
(x+1,y+1)∈∆
& 7
v%
$
)*+
A9TKMU ]^I \FA
3
3
O(lx ly )
AP
5x-
(F ,{x,y})
U
F
[email protected] GgI GH=C=4AyN AC=
UCTViK G @?hUHPQ=^<
.t gCUHPVE AbaLAC=
€ Œ €f5
#
#
#
-' )#
BHD'¢ š ]0¥V¥ ¦ L  ± FDFBEMfBEQEILKP
yUHP
Vt gCUHPVP8AbaLAC=yTVA>[email protected] @wKMAKMiC=4UHI >
s§
(¬V1 (x + 1,y) ∨ ¬V2 (x,y + 1))
(x+1,y)∈∆
(x,y+1)∈∆
›kSVFgJDfKJgJSVQbcVD omDG%jkS4gJhNQEoLDG
^
)#"-'
v%
%! ! ± FcvAMfKJDFEBHgJD omDr_EgYSVUEom™fh'D ‘tQEFFESVhNUEgYDr}zQEDoLlSVFE}zQED BED BEILgJDlKsILSVFEG
(F ,D)
]2SVFEFEMfD' 2 (r ) ,(r )
N
= rmax
p∈D
SVgYKJImD rmin = rminE ∈,rmax
F
∀p ∈ D∀i,j ∈
Z r ≤ X (i) ≤ r
± F FED²_.DfQHK_iA4G#¬4MFEMfgsA4oLImGJDfgPo~O A4oL¬VSVgJI‰KsžEhDBED²opA%GYDlKsImS4Fr_EgJMflMBHDFdKsD_.SVQHgPgJMfGJSVQEBHgJD (Q(D),D)
_aSVQEg2BEDGrDfFEGJDfhNUEomDfG D ‘q_EoLQEG0BEDBEDfQb‹BHImgJDflfKJImSVFEG LŸ“ FjœA4I‰K’GYSVFD‹XKsDfFEGJILSVFTFiAvKJQEgJDfomomD cvAC_.Dfgˆ[
hDfKYKsgJDrBHD gYDlfSVFEGYKJgJQEILgJD BEDfG [ lSVFdcVDŒ‹bDG , L
@#D²UEQHKBED²lfDfKYKsD:_iA4gˆKsImDªDGYKBHSVFEl²BED²lŠžEDfgJlŠžEDfg QHF5Avom¬VSVgYILKJžEh'D¤_aSVQEgPgYMGJS4QEBEgJD (QF (D),D)
SV´ QF (D) BEMfGJIL¬VFED opANlfopA4GYGJD BEDfG [ lfSVFdcVDŒ‹HDfG2jkSVgˆKsG8GYDomS4F D L
± F>GYDrH‹bD BHSVFEl QEFED GJQEI‰KsDrBED _EgYS\WYDlKsILSVFEG (r ) La SVILK ∆ omDr_aSVoL{X¬VSVFHDrBESVFdKwo~ORM}zQiAvKJImSVF DfGYK#
)*,+
)*,+
,58-
<
p
i i∈Z,p∈D
0p
=> ?;@BAC=
Z
p
P
0p
D EHG IJA> KC=oNG PQ=
i i∈Z,p∈D F
p
0p
[email protected]=> [email protected],[email protected];IJAC=^>VA
TVUH> [email protected]> @
<
_(=?6?;[email protected]_0N H
U [email protected][email protected]?hUHPQ=
APQ=4A\S]^IJA
p
i
,58-
E iKW? [email protected]
0p
i
p
j)*,+
q
[email protected];=
j)*,+
,5x-
,58-
q
p
i
∆ = {M ∈ Z2 | ∀p ∈ D; rminp ≤ p(M ) ≤ rmaxp }
“ F*jœA4ILKSVF­cvATQbKsImoLImGYDg D‹HA4lfKJDhDFdKomDfGNh'efh'DfGKsDflŠžEFEIL}XQHDG}XQHD _EgJMflMBHDhh'DfFzKh'A4ImG5DF
gYDh_EopA A4FdK€omDfGw}zQiAvBEgsA4FdKJG2_iAvgwBHDG rHnLa± FBEMfGJIm¬4FED5_EA4g A = A(D) o~ORDFHGJDhUEomD BEDfGwILFEBEILlDG
BEDfG rHnL
]’SVFHl _aSVQEg8KJSVQHK M ∈ ∆ SVF>A4GYGJSXlImDroLDG8cvA4gJImA4UEoLDG%UaSbS4omMDfFEFEDfG (V (M )) L
± F>gYDBEMiFEI‰K€omDG8jkSVFHlfKsILSVFEG Φ DfK Ψ lfSVhh'D0GYQEILKP
GJGYI F = {M ∈ ∆ / ∀(k,p,q) ∈ A ¬V (M )};
E = Φ(V )
GJGYI V (M ) = R (M ) ∩ F = ∅ .
V = Ψ(F )
± FBEMfEFEILKwopAjkS4gJhNQEoLD š ± _iA4g
pq
k
q
pq
k
€d
pq
k
SQCON V =
pq
k
^
(i,p,q)∈A
Rpq
i
(k,p,q)∈A
„µ ƒVƒ
#
SV´ R DfGYK2BESVFEFHM _EA4g8omDfGwhehDG%jkS4gJhNQEoLDG R }zQED _EgYMlMfBEDhhDFdK L
± FA'AvomSVgYG%o~ORM}zQEI‰c4AvomDFdK€BEQoLDhh'D ! L µ L ƒ¹
8 &^ fw^
pq
i
x
& ^
k
UH> [email protected]> @
E ⊂∆
IJAC=F]U4UHIJiCAPQ=
V = Ψ(F )
,fUHK\z>
E iKW? [email protected] G
IJA
*jt
± F'cvA BEMlfSVQE_aDglŠžiA4}zQED0BEgYSVILKJD€BED2BHImgJDflfKJImSVFBiA4FEG D DfFqBEDfQb‹qBEDhI‰[ BEgJSVI‰KsDfGPV_aSVQEg†lDomA S4F
H‹bDQEFcVDflfKsDfQEg0BEILgJDflfKsDfQEg p~ _aSVQEg€lŠžiAv}XQHDBEILgJDlKsILSVF p ∈ D L @#DG2BEDQX‹BEDhI‰[ BEgJSVI‰KsDfG8BPO S4gJIm¬4ImFED
DfKwBED BEILgJDflfKsILSVF p GJS4FzK’A4oLSVgJGP
M
DK −−→ −−→
s (M ) = {N : p(N ) = p(M ) p~ · ON ≥ p~ · OM }
DK −−→ −−→
s (M ) = {N : p(N ) = p(M ) p~ · ON ≤ p~ · OM }
SV´ · BHMGJIL¬VFED omDr_HgJSXBEQEILK%GJl|AvopA4ILgJDrDfK O DGYKwQEFHD5SVgYIm¬VILFED0AvgJUEI‰KsgsAvImgJD L
š DflIPcvAFHSVQEG€_.DfgJhDfKYKsgJDrBHD _EA4gYKsA4¬VDg A _.S4QEg8lŠžiA4}zQED BEImgYDlKsImS4F p BHD D L
?
V
E iKW? XjA
*
E = Φ(V )
G IJUHKM=
[email protected]
*jt
gCUHPVE AbaLA f
, [email protected] <
p
+
p
−
± F>A
A = Ap+ ∪ Ap−
Ap+ = {(q,r,k) |sp+ (M ) ⊂ Rkqr (M )}
Ap− = {(q,r,k) |sp− (M ) ⊂ Rkqr (M )}
L
sp− (M )
0 0
sp+ (M )
Rqk0r (M )
3`:'465
! Lƒ
|
M
Rqr
k (M )
(q 0 ,r 0 ,k 0 ) ∈ Ap−
(q,r,k) ∈ Ap+
G [email protected]?;@w?hUHPVP8A\[email protected] AI APQ=4A\S]^IJAON AC=N ?;[email protected]?hUHPQ=ON
cVDfl5lfDGwFES4KŠA\KsImS4FEG%SVFcvAt_aSVQHc4SVImg8BHSVFEFEDfgwomAjkSVgJhQEomDrDŒ‹H_HgJImh'A4FdK€}zQED

 ^

LB(p,v,l) = 

 M ∈∆
M +l~
p∈∆
p(M )=v


^
(p,q,k)∈Ap+
^
(Vkqr (M )
⇒
¬Vkqr (M
(p,q,k)∈Ap−
(p0 ,q 0 ,k 0 )∈Ap+


¬Vkpq (Mmin + (l − 1)~
p) ∧ 
<
Xp (v) ≥ l




+ l~
p))
∧

^
(p,q,k)∈Ap−

¬Vkpq (Mmax − (l − 1)~
p) 
„µ ƒ4Z
SV´ M ›œgYDGY_ M 2DfGYKroLD_aSVImFdK2BED ∆ KsDfoª}zQED p(M ) = v }zQEI h'ILFEImhImGYD›œgJDfGJ_h'A\‹bImhImGYD¯
−−→ L
OM · p~
š SVhh'D A|c4Dl BEDQb‹ BEImgYDlfKJImSVFHG8SVF>_aDQbK€h'SVFdKJgJDg€oLDromDhhD’GJQHILcvA4FdK#
4
min
J ^
4w4
max
Ψ(E)
? V E
? E [email protected] *jt gCUHPVE AbaLA ,[email protected] =4AIJUHP D [email protected] E iKW? XjA Xp E(v) ≥ l
LB(p,v,l) <
iKW? XjA SQCON V ∧ LB(p,v,l) G IJUHKM= E = Φ(V ) E iKW? XjA Xp E(v) ≥ l <
Xp E(v) ≤ l
n−1
(Aq ,Bq )q∈D\{p}
D
p
G IJUHKM=
V =
E iKW? XjA
L n SVQEg€lDfopASVF cvAeKsgJD
o#gJDfGYKJD omANU.S4gJFED GJQH_.MfgJImDfQEgJD La± Fc4DQHKwBESVFHl DŒ‹b_EgJILh'Dfg
VS UHomIm¬4MrBED b‹HDfg _iA4ImgYDGwBED _EImDfBEG
_.SVQHg’A|cVS4Img€omDfG2_aSVImFdKJGw}zQEI¤D‹XKJgJMhImGYDFdK
KJSVQHKsDfG€omDG8BHImgJDflfKJImSVFEG8BHD GJA4QHj L
@AGJDQHomD lŠžESVGYD‘_EgYSVQHcVDfg2DfGYKwomDroLDhh'D’GJQEI‰c4AvFzKP
UH>LK
(q ,r ,k ) ∈ Ap+ N ? [email protected]
p
qr
[email protected] W+ (M ) = Vk <
0
0
M ∈ ∆ ?;I
(q,r,k) I @BUH> @
0
N A
[email protected] >
TI >=
(q,r,k) ∈ Ap+ @AI ^>,AzTVUH> [email protected]>
P N AC=dTVUH?;[email protected];= (Aq ,Bq )q∈D\{p} <
>LP
0 0
Rkp0q [email protected][email protected]>
@
P
L
S4ILK omDr_aSVILFzK%BED }XQHI#hqA\‹bILh'ILGJD −−→ BiA4FEG
š SVhh'D_aSVQEgtKsSVQbMK (q,r,k) ∈ A SVF‚Q A R (M ) ⊃ ROM(M· p~ ) ILo8GJconv(∆)
Q tKqBED>_HgJSVQHc4Dg'opA
_EgYSV_EgYImMfKJMr_.S4QEg M = M L
A4ILG M DfGYK8GYQEg%QEF lfS4KsM c BEQ _aSVo‰{b¬4SVFED ∆ BED’BEImgYDlfKJImSVF q 6= p LE± F ABESVFHl0QEF _EILDB A
GYQEg€lD lS4KJM L š SVFEGYImBEMfgJSVFEGwomD lfS4KsM c lfSVFEGJMflQHKJILj²BHD c BiAvFEG ∆ }XQHI¤FŸO DfGYK2_iA4GwBEQlfS4KsM BED A _iA4g
gJA4_E_aSVgYKw‘ M uilfD lS4KJM DfGYKwBED BEILgJDlKsILSVF q ∈ D L
QE_E_aSVGJSVFHGrBPOyAvU.SVgYB q 6= p L# SVILK0A4omSVgYG B omD5_EImDfBŽH‹bMGJQEgrlfDlSvKsM L oD‹bILGYKsD k KJDo }XQHD
DGˆKNBiAvFEG A ›œl|Avg c DfK c GJS4FzKlSVFHGJMlfQHKsI‰jkGŠ L± FcvAh'SVFdKJgJDg }zQEDC_aSVQEgrKJSVQHK
(q ,q ,k)
BHI .MgJDfFdK2BHD (q ,q ,k) oxO 0Hn R (M ) lfSVFdKsILDFdK A SVQ B L
(q ,q ,k ) ∈ A
š S4FEGJILBEMgYSVFEG€BESVFEl (q ,q ,k ) ∈ A KsDfoP}zQED Π = R (M ) FED lfSVFdKsImDfFdK2FHI A FEI B L
± FBEMiFEILKwAvomSVgYG8}XQEAvKsgYD BHDhI‰[ BHgJSVI‰KsDGŸ
DGˆKqomABEDhI‰[ BEgJSVI‰KsD'BED>BHImgJDflfKJImSVF p u²BPO S4gJIm¬4ImFED M }zQEI8FPORDGYK'_iA4GtlfSVFdKsDFzQED
D
BiA4FEG ∆ u
DGˆKwopABEDhI‰[ BEgJS4ILKsD2BED BEImgYDlKsImS4F q uiBPORSVgJIL¬VImFHD M }zQEI#FED lSVFdKJImDFdKw_iA4G A u
D
DGYK2opAtBEDfh'I [ BEgYSVILKJDrBED5BHImgJDflfKJImSVF q u BPORSVgJIL¬VImFHD M }zQEI¤FPORDGYK’_iA4GwBEQlfS4KsM5BED
D
›k_iA4g8gJA4_E_aSVgYKw‘opABEgYSVILKJD D u
A
DGˆKwo~O A4QHKsgYDrBEDhI‰[ BEgJS4ILKsD2BED BEImgYDlKsImS4F q uiBPORSVgJIL¬VImFHD M L
D
›¡cVSVILg%i¬VQEgYD ! L Zd
n A4g’BHMfiFEI‰KsILSVF Π lfSVFdKsILDFdK D L @²O rHn Π lfSVFdKsImDfFdK D l|A4g’Imo¤FEDNlS4FzKJImDfFzK’_iA4G A DfK0DfGYK
QHF 0HnUL
@#D'}XQEA4BEgsAvFzK Π DGˆKQEF rbn BES4FElqILolfSVFdKsImDfFdK D SVQ D L² I Π lfSVFdKsILDFdK D uAvomSVgYG5ILo
lfSVFdKsILDFdK¹omA gYM¬VILSVFDFdKsgYD D DK D DK%BES4FEl2ILoilSVFdKJImDFdK B uzImh_.S4GJGJILUEomD L ]’SVFHl Π lSVFdKJImDFdK
DfK€BES4FEl Π = R (M ) L
D ,D ,D
I q = p QEF 0Hn BED BHImgJDflfKJImSVF*BiAvFEG A }zQEI%FHDlfSVFdKsILDFdKt_iA4GtomDq_.SVILFdK A lSVFdKsILDFdK
jkS4gJlMfh'DfFdK D DK D DK¹DfGYK¹BESVFElwomD€}zQiA4BEgJA4FdK%BED€jkgJS4FzKJIm™fgJD D ∪ D }zQEIaDGˆK¹QHF rHn lA4g
DK p GJSVFdK2A4omS4gJG8lS4FEGJMflQHKJILcVDfG L
q
2
± FBEMfEFEILKwBED opANhehD0jœA fSVF W (M ) L
i\[email protected] @w?hUHP
2
max
qr
k
+
Q
qr
k
max
max
1
max
0
2
max
|
1
2
3
p
+
0
4
2
p
+
‚
1
3
|
0
4
0
q3 q4
k
2
p
+
max
q3 q4
k
max
1
|
max
2
|
3
1
max
2
max
1
|
4
2
max
1
2
3
1
|
2
3
q1 q2
k
4
4
3
2
max
p
+
2
1
2
1
1
p
−
2
µ„ƒ
}
8 &^
fw^
x
# & ^
q1 = cste
∆c
A
D4
p = cste
D1
Mmax
D3
B
D2
! LZ
q2 = cste
3y:'465
|
UH>LK ^>VUH?u>LP=4A> I
[email protected][email protected]?
A
PV?
B
@ANjkSVgJhQEomD0DŒ‹b_EgJILhqA4FdK€omANU.SVgYFED GJQE_aMgYImDfQEgJD DGˆK’omANGJQEI‰cvA4FdKsD
^
UB(p,v,l,(Aq ,Bq )q6=p ) =
¬W−p (M ) ⇒ W+p (M + l~
p)
M ∈∆
M +l~
p∈∆
u(M )=v
q∈D\{p}
SV´
^
(IN(Aq ) ∧ IN(Bq )) ∧
L @#D omDhhD
GJD ¬VMFHMgsAvomImGYDrBESVFEl BED omAjœA SVFGYQEILcvA4FdKsDª
¬Vkpq (M )
7.1.5
? E [email protected] *jt gCUHPVE AbaLA,[email protected]=4AIJUHP D [email protected]@[email protected] (Aq ,Bq )q6=p
Xp E(v) ≤ l G IJUHKM= V = Ψ(F ) E iKW? XjA UB(p,v,l,(Aq ,Bq )) <
? V E iKW? XjA UB(p,v,l,(Aq ,Bq )) G IJUHKM= Xp E(v) ≤ l U E = Φ(V ) <
(Aq ,Bq )
IN(M ) =
V
(p,q,k)∈A
GJDfomSVF,KsSVQbKsDG omDfG5BHImgJDflfKJImSVFEG L
AvImFdKsDfFiA4FdKS4FGYQE_E_aSVGJD'}XQHDqo~ORSVF,Ab‹HMBEDfG_EILDBEG
± FBEMiFEILKwAvomSVgYG
RAPP((r),(r 0 ),(Aq ,Bq )) = SQCON V ∧
^
@DfG2oLDhh'DfG%_EgYMlMfBEDFdKJG’hSVFdKsgJDfFdK#
$ [email protected] iK? XjA
p
LB(p,v,rvp ) ∧ UB(p,v,r 0 v ,(Aq ,Bq )q6=p )
p∈D
rminp ≤v≤rmaxp
V =^G @w?J= ,&G ?;@ RAPP((r),(r 0 ),(Aq ,Bq )) G IJUHKM= Φ(V ) [email protected] >LP8Ap=4UHI > t
@w?hUHP N A
(QF (D),D) [email protected] [email protected] (Aq ,Bq ) <
)*,+
,5xciCg?TKCU ^>VA\[email protected];_9=? E [email protected]>LP8A =4UHI > @?hUHP N A
(QF (D),D) G IJUHKM= V = Ψ(E)
0
E iKW? XjA RAPP((r),(r ),(Aq ,Bq ))
RAPP
)*,+
š SVh'hD
,58-<
& )*,+
UCTViK G @?hUHPQ=
?
DfGYK QEFHDqƒ¯[ jkSVgYhNQEomD5SVFTDFTBEMfBEQEILK QHFAvom¬VSVgYILKJžEh'D _aSVoL{XFESVhIpAvo¤_aSVQEgrgYMGYSVQEBEgYD
7 A
d = |D|
U
,58-
TKMU ]^I \FA
[email protected]
)*,+
,58-
n = maxp∈D (lp) <
(QF (D),D)
TVA> @
@KMA
KMiC=4UHI >
AP
O(n2d+2 )
J ^
4w4
-' )#
µ„ƒ
4
)#"-'
%
)
€ DfcVDfFESVFEG8h'A4ILFzKJDFiAvFzK%‘opA gJDflSVFEGˆKsgYQElfKJImSVFCDŒ‹HA4lfKJD L š SVhhD’A|cVDflrBEDQX‹CBEILgJDflfKsILSVFEGImo GJQ tK
BPORImh_aSVGJDfg²opArU.S4gJFED€GJQH_.MfgJImDfQEgJD€_aSVQEg:GJDfQEomDfh'DfFdK¹QEFHDwBHImgJDflfKJImSVFtDfK†omA UaSVgYFED€ImFHjkMfgJImDfQEgJD€_aSVQEg²oLDG
A4QbKsgJDrBHImgJDflfKJImSVFEG L
n S4QEg:D‹b_EgJILh'Dfg²opArUaSVgJFED8GYQE_aMgJILDQEgYD€BiA4FEG†opA0BEILgJDflfKsILSVF p SVFtDGˆK¹SVUHomIm¬4M%BED8H‹bDg:omDfG 2(d − 1)
_EILDBEG8}zQEIGYSVFdK2GYQEg€omDfG8BEgJSVI‰KsDfGwDŒ‹XKsgJMfhqAvomDG8BHD BHImgJDflfKJImSVF BEI ŸMfgJDfFzKJD BED p L
A4ILG¹SVF _aDQbK%jœA4ImgYD2oLDG%S4_.MfgsAvKJImSVFEG†BED0lfSVh_EomMKsImS4FqBEMflgJI‰KsDfG%BiA4FEG¹omD’lŠžEA4_EILKJgJD Xub_EoLQEG%DŒ‹EAvl[
KJDhDFdKSVF‡GJA4ILKt}zQPO A4_EgY™G'A|cVSVILgNH‹bM>BEDfQb‹°_EILDBEGtBiA4FHG'QEFHDBHImgJDflfKJImSVF q 6= p DK'D ŸDflfKJQEM>oLDG
SV_aMgJAvKsILSVFEGBEDrlSVh_EoLMfKsILSVFqBHMlgYILKJDrBiA4FEG8oxOyA4oL¬VSVgYILKsžHh'D uHA4omSVgYGKJSVQHKsDfG8omDfG%BEgJS4ILKsDfG%BEDrBEILgJDflfKsILSVF
BiAvFEG D c4SVFdKwefKsgYD BEDrK {b_aD ti L ›kc4SVImg8gYDh'A4gJ}zQED L ƒ L z L
± _aMgJAvKsILSVFEG BEDqlfSVh_EomMKsImS4FFEMlfDGJGJA4ImgYDGN_aSVQEgrjœA4ILgJDtBEILh'ILFXQHDg omDtFESVhUEgJDBED
_EILDBEG8BEA4FEGwopANgJDflSVFHGYKsgYQElfKJImSVFBEDfG [ lSVFdcVDŒ‹bDG2jkS4gYKsG L
A4ImgYD0SV_aMgsA\KsImS4FEG ⊕ , L
' KsSVQbKsDrBEgYSVILKJD D BHD BHImgJDflfKJImSVFBiA4FHG D % A4ImgYD0SV_aMgsA\KsImS4FEG ⊕, , ⊗ , , , GJQEg€opANBHgJSVI‰KsD D
"! @#DG€SV_aMgJAvKsILSVFEG%FPORSVFdKwgJILDFlŠžiA4FE¬4M
“ FŽ_iA4gˆKsILlQEoLImDg8oLDGwBEgJS4ILKsDfG2DŒ‹XKsgJMfhqAvomDGw_aSVQEgwQEFEDBHImgJDflfKJImSVF p ∈ D GJSVFdK’BED5K {X_aD5K|µ4u KŠƒSVQ
KsZ L± FTlSVFEFEA mK BES4FEltGJQEg lŠžiA4lfQEFEDBEDtlDfG5BEgYSVILKJDGrBEDfQb‹T_aSVImFdKJG A DK B BESVFdK5A4QTh'SVILFEGro~ORQEF
BPORDFdKJgJDqDfQb‹,DGˆKNBiAvFEG KJSVQHKJDG omDG GJS4omQHKJImSVFEGrlŠžHDgJlŠžHMDG L ›kc4SVImgri¬4QEgJD ! L z ± F_.DfQHK5BESVFEltjœA4ImgYD
Q


}
q
0
0
0
00
{
}
!L
3y:'465
}
|
\FUH?;PQ=I > P
A
α
β−α
t1
β
t2
AB
A
t3
UH>LKnIJAC=
N
@KMUH?J=
[email protected]> a
@;WTVAC=
N A
N KMUH?;@BAC=C_UHP
B
TVA>[email protected]@KMUH>LE AK
N A> aFTVUH?;[email protected];=
A
[email protected]
B
N [email protected] >
[email protected] [email protected]>[email protected]=OIJAC==4UHI > @w?hUHPQ=^<
GYDQEoLDhDFdK ƒ'lŠžHSVI‰‹_aSVQEg2lDG’_EImDfBEG LŸ± FAqBESVFHlNA4QKJS4KŠAvo 2 lŠžESVI‰‹>_aSVQEg2omDG2_EImDfBEG’GYDoLSVFoLDG
BEILgJDlKsILSVFEG L
d−2
š A4oLlQEoLSVFEG8h'A4ImFdKsDfFiA4FdK€opANlfSVh_EomDŒ‹HI‰KsM0BHD oxOyAvom¬VSVgYILKJžEh'D› n BEMfGJIL¬VFEDrKsSVQ„WYSVQEgJG max (lp) 
o.{A'AvQ_HomQEG n _aSVGJGYImUEILomILKJMG_.SVQHg8omDG€_EILDBEG8oLImMfG€‘tQHFED _EgJDfh'IL™gJDrBHImgJDflfKJImSVF L
_EgJ™fGNA|cVS4Img D .DlfKJQEMqoLDG SV_aMgsA\KsImS4FEG5BHD'lfSVh'_HomMfKJImSVFTBEMflgYILKsDfGBiAvFEGoxOyA4oL¬VSVgYILKsžHh'D }zQEI
_HgJDFHFEDFdK O(n ) SV_aMgJAvKsILSVFEG Imo²gYDGYKJD A4Q_EomQEG5QEF<FHSVhNUEgYDqlS4FEGYKsA4FdKNBED'lŠžESVI‰‹BED'_EImDfBEG
› 2 %_.S4QEg8omDfG d − 2 A4QbKsgJDfG€BEImgYDlfKJImSVFHG8BESVFdKwomDG8_HImDBHGwBHSVILc4DFdKwefKsgYDrH‹bMG L
n SVQEg8lŠžiAv}XQHDlŠžESVI ‹ BED _EILDBopAgYMBEQElKsILSVF‘'ƒ¯[ £ _.DfQHK€efKJgJD D .DlKsQEMfD5DfF O(|∆|) SV_aM[
gJAvKsILSVFEG%GYSVILK O(n ) SV_aMgsA\KsImS4FEG L
š DflIªBHSVFEFEDNiFEA4omDfh'DfFzK’QEFTA4oL¬VSVgJI‰KsžEhDrBEDgYDlS4FEGYKJgJQElKsImS4FBEDNlS4h'_EoLD‹bILKJM O(n (n + n )) =
L
O(n )
d−2
p
|
2
&
|
4
2d−2
&
|
2
2
6
4
2
µ„ƒ
& 7
8 &^
fw^
)*+.-m/(14365872+936:;-0/
AdTKMU ]^I \FA
x
(QF(D),D)
# & TVA> @
@KMAoKMiC=4UHI >
& C [email protected] G
O(n6 )
=?
AP
D
?
P8A=^G @?J= &
, G ?;@2TQGH=
N >@'>ViCUHK \FA
(C,D) TVA>[email protected]=4A
G > agCUHPN ?;@?hUHPQ=zi ^>L?;EHG [email protected]=
)*+.-m/(14365872+936:;-0/
gI GH=C=4AN [email protected]@hG IJA\[email protected] AbaLAC=G IJUHKC=
^
O(n6 )
AP
± FABHSVFElrKsgYSVQHcVM QEFhDImoLomDfQEg%A4oL¬VSVgYILKsžHh'Dw_.S4QEg€omDG%KJS4KŠAvomDhDFdK8lfSVFdcVDŒ‹HDfGP
UCTViK G @?hUHPQ=^<
<
<
[email protected]
KMiC=4UH> N KMA
UCTViKbG @?hUHPQ=^<
€ ¹5 ##ª '² ©8 N€ f% rf5 f ]0AvFEGoLD_HgJSVUEoL™hD ‡SVFImh_.SVGYDtA4Qb‹GJS4omQHKJImSVFEGrBŸO eKsgJD'lSVh_EgYImGJDfG5DfFdKsgJD
BEDQb‹_EA4gYKJImDG α DfK β Li“ FjœAvILK8SVF>A%
"
v%
)*,+
! 0
^%
0
7
,58-
0-0+
)*,+
A.TKMU ]^I \FA
)*,+.-0/21 0-0+
,58-
0-0+
O(n2d+2 )
“ F”D ŸDKtImo†GJQ tKtBEDCgJA„WYSVQHKJDgNBEA4FEGopA(QF
jkSVgJhN(D),D)
QHomDqUaSXSVomMfDFEFE DiFiA4oLDComDfGjkSVO(n
gYhNQEom)DfG IN(M ) _.SVQHg
KsSVQHK M ∈ α DK2oLDG%jkSVgYhNQEoLDG W (M ) ∨ W (M ) _.S4QEg%KsSVQbK M ∈ ∆ \ β L
a ©€ €¤ @²O A4om¬VS4gJILKJžEhD’A|cVDfl _EgJS\WYDflfKJImSVFEG
KsDoŸ}XQŸO ILoPDGˆK2BHMlgYILK8BEA4FEG€lDrlŠžiAv_EILKJgJDrDGˆK€BiA4FEG
>
L
“
opA>_HgsAvKJIm}zQEDtKsgJ™fGoLDFdK F­D .DfKlSVFdKsgJA4ImgYDhDFdK‘>o~O A4oL¬VSVgJI‰KsžEhDNBEQ<lŠžiA4_EI‰KsgJD buSVF<DfGYKS4UEomIL¬VM
BED>jœA4ILgJDCKJSVQEGomDGlŠžESVI ‹”BEDfGq_EILDBEG L >A4ImGtlS4h'hDomDG [ lfSVFdcVDŒ‹HDfGqjkSVgˆKsGGJSVFdK'DF‚_EA4gYKJImlQHomImDfg
[ lSVFdc4D‹bDGfuESVFq_aDQbK¹QHKJImoLImGJDfg²omDfGSV_aMgJAvKsILSVFEG:BED2lSVh_EomMKsILSVFtBEQqlŠžiAv_EILKJgJD LXn A4g†lSVFdKJgJD’ILoEjœA4QHK
omDGwA4BEA4_HKsDfg€A4Qb‹lA4G€BED _EgJS\WYDflfKJImSVFEG€Av_E_EgJSXlŠžEMfDG L
n omQHG²_EgJMflImGYMhDFdK†omDfG²SV_.MfgsAvKJImSVFHG ⊕, , FPORQHKsILomImGYDFdK²_iAvG²omDG:_EgYS\WYDlKsILSVFEG²DfK:_aDQbcVDFdKBESVFEl
A4QEGJGYI efKsgYD€QHKsILomImGYMDfGu4KJDomoLDG²}zQEDfomomDfGudBiAvFEG:omD€l|A4G†A4_H_EgJSXlŠžEM Lb± Ft_.DfQHKA4QEGYGJI D .DlKsQEDfgoxO SV_aMgJAvKsILSVF
A|cVDflomDfG’_EgYS\WYDlKsImS4FEG’hImFEILhqAvomDG8DKromDfG’SV_aMgJAvKsILSVFEG , A¯c4DloLDG’_HgJS\WYDflfKsILSVFEG2hqA\‹bILhqA4oLDG L
⊗
n SVQEgQbKsImoLImGYDgoLDGNS4_.MfgsAvKJImSVFEG DK BEA4FEGomDql|A4GAv_E_EgJSXlŠžEM SVF”_aDQHKKsSVQ\WYS4QEgJGQHKJImoLImGJDfg5oLDG
lSVFEBHILKsILSVFEG S (M )+S (M ) ⊕> A h'A4ImG#ILo4jœA4QHK¤l|AvomlQHomDgoLDG S A|cVDfl¹oLDG_EgJS„PWYDlfKJImSVFHG¤hImFHImh'A4omDfG
KŠA4FEBHImG€}zQED A DfGYKwl|AvomlQHomM A|cVDfl5oLDG8_EgYS\WYDlKsImS4FEG€hqA\‹bILhqA4oLDG0› A = min p  L
@#DGwKsDfh'_EG2BEDlA4omlfQEomG€BED5lDKJKsD5cVDgYGJILSVFBED5o~O A4om¬4SVgJI‰KsžEhD0GJS4FzK’BESVFEFHMG0DfFA4FEFEDŒ‹bD L ƒ LŸ± F
cVSVI‰K }zQEDopA,lSVh_EomDŒ‹bILKsM>D‹b_aMgJILh'DfFdKŠA4oLDDGˆK _EoLQHK 4KqDF O(n ) GJI0S4F³_EgJDfFEB BEDG _HgJS\WYDflfKsILSVFEG
ImFHjkMfgJImDfQEgJDfG€DfK€GJQH_.MfgJImDfQEgJDfG’AvGJGJD _HgJSXlŠžEDG L
UCTViK G @?hUHPQ=C_YIJA TKMU ]^I \FA

Q
TVA> @
p
−
0
(QF (D),D) TVA>
@KMAKCiC=4UHI >
p
+
AP
@ @KMA`KCiC=4UHI >
6
AP
UCTViKbG @?hUHPQ=^<
0
GTTKMU4g ViCAC=
&
q
q
0
&
00
0
k
k+1
0
k
p∈D
~
4
0
i

5 f #
@²O S4UXWYDlKsILj€ILFEILKJIpA4oªBEDqlfDfKJKJDqKsžH™GJD MKŠA4I‰KoxO MKsQEBHDqBEDfGNlSVFdc4D‹bDGtBED DK DfKNBEDGN_aSVILFzKJG
A4GYGJSXlILMGwA4Qb‹lfSVFdcVDŒ‹HDfG’lfSVh'hDromDfG8_.S4ImFdKsG8GJA4ImoLopA4FdKsG¹SVQh'MfBEIpA4FHG L Z Z
® FHD0BEDG8BHI lQEo‰KsMfG%MfKsA4ILK%A4omSVgYGoLD’FES4hNUEgJD’BEDG8BEMiFEI‰KsImS4FEG¹_aSVGJGYImUEoLDG%_aSVQEgopAlfSVFdcVDŒ‹HI‰KsMHopA
lfSVFdcVD‹bI‰KsMKJS4KŠA4oLD4uiomA'· 2[ lfSVFdcVDŒ‹HI‰KsMvuŸopAtlS4Fzc4D‹bILKJMNGJILh'_HomDrGJDfomSVF>_EomQHGJImDfQEgJG2BEImgYDlKsImS4FEG L ]O A4QHKJgJD
_iAvgYK|u#ILoMfKsA4ILK GYSVQHcVDfFdKNFEMflDGYGsA4ILgJDBEDgsA„WYSVQbKsDg BEDfG5žd{X_.SvKsžE™fGJDGBEDtlSVFEFHD‹bILKJM'_aSVQEg A|cVSVILg BEDG
_EgYSV_EgYImMfKJMG0ILFdKsMgYDGYGsA4FdKsDfG Ln A4grD‹bDh_EoLD4uŸoLDGr_aSVoL{4SVhImFESVG2· w[ lfSVFdcVD‹bDfG5lfSVFdKsImDfFEFEDfFzKrKJSVQ\WYSVQHgJG
oLDQEgYG8_.SVILFdKsG8hMBEImA4FEGuEA4omSVgYG8}zQED lD FPORDGYK2_iA4G€oLDrl|A4G€_aSVQEg8omDfGw· 2[ lSVFdc4D‹bDG L
± g'oLDG_.S4oL{VS4h'ILFESVGt· w[ lfSVFdcVD‹bDfG SVFdKqQEFHD_EgYSV_EgJILMfKJMDfGJGYDFdKsILDomoLD†GJI2A4QHKsS4QEgtBPO QHF³_.S4ImFdK
lŠžiAvlQEFBEDG†}zQiA4BEgJA4FdKsG:lS4FzKJImDfFzK†QEF_aSVImFdK²BEQ_aSVoL{4SVh'ILFESrA4oLSVgJG²oLD%_aSVoL{4SVh'ILFES0lfSVFdKsImDfFdKoLD8_.S4ImFdK
lfDFdKsgJA4o ›œGYDFEGBHImgJDflfKBHQ­KsžEMfSVgJ™fh'Dƒ L L ZHu BEM^WJ‘Ž_EgYMGYDFdKtBiA4FEG¢ %][email protected] n ¥ ¦  L š DfKYKsDC_HgJSV_EgYImMKsM
BEDfGr_.S4oL{VS4h'ILFESVG2· 2[ lSVFdc4D‹bDG5Aq_aDgJhImG2BPORImFdKsgYSbBEQHImgJD5QEFEDFES4QHcVDfomomDFES4KsILSVFŽBEDlSVFdcVDŒ‹bILKsMª#opA
[ lfSVFdcVDŒ‹HI‰KsM'›œlfSVFdcVDŒ‹HI‰KsM _iA4g8}zQiA4BHgsA4FdKsGs L
š DfKYKsD%FHSVQHcVDfomoLDFESvKsImS4FNA2BHD¹UaSVFEFEDfG_EgJS4_EgJILMfKsMfGªh'AvKsžHMh'AvKsIL}XQHDGP„o~ORImFdKJDgJGYDlKsImS4FBEDBEDQb‹ [
lfSVFdcVD‹bDfG0DGˆK [ lfSVFdcVD‹bD5DfK2QEF [ lfSVFdcVDŒ‹HD iFEIDGˆK0lfSVh'_Hom™fKJDhDFdK€l|AvgsA4lKsMgYImGYM5_iAvg2GYDG’_aSVILFzKJG
GJA4ImoLopA4FdKsG L ]’DŽ_EoLQEGqDfomoLD>_aDgYh'DKCBED>¬VMFEMfgsA4oLImGYDg›kQEF³_aDQ tomDKsžEMfSVgJ™fh'D>BPOyAv_E_iA4gˆKsDFEA4FElDBEDG
hMBEImA4FEG L
A4ILGto~ORImFdKsMfgJefKtBHDlfDfKYKsDBEMfiFHILKsILSVF”A4_E_EA4gsA LKtlomA4ImgYDhDFdKBiA4FHG'oLD _EgJSVUHom™hD BEDopAŽKJSVh'S\[
¬VgJA4_EžEILD L8“ F D .DfK>o~O A4om¬VS4gJILKJžEhD>BED,gJDflSVFHGYKsgYQElfKJImSVF BEDG [ lfSVFdcVD‹bDfG*›œlŠžEA4_EILKJgJD V A4_E_iAvgsA LK
iFEA4omDfh'DfFzK€lfSVhh'D QEFED UaSVFEFED ¬VMfFEMgJA4omILGsAvKJImSVF‘tBEDGwBEILgJDlKsILSVFEG8}zQEDoLlSVFH}XQHDG’BHD5oxOyA4oL¬VSVgYILKsžHh'D
BEM WJ‘lSVFEFzQ>_.SVQHg8omDG8_aSVo‰{VSVhImFES4G8· w[ lS4Fzc4D‹bDG L
4QEGY}XQŸO ILlIdSVF A|cvA4ILK#QbKsImoLImGYMopA%GYImhImomI‰KsQEBHD BEDfG [ lSVFdc4D‹bDG A|cVDl†omDfG#_.S4oL{VS4h'ILFESVGP· 2[ lSVFdcVDŒ‹bDGu
h'A4ImG€oLDG [ lSVFdc4D‹bDGrGYSVFdKrA4QEGYGJI¤KsgJ™fG’_EgYSblŠžHDG0BEDfG’KsSvKŠA4oLDhDFdKwlSVFdcVDŒ‹bDGP#omAlfSVFdcVD‹bI‰KsMKJS4KŠA4oLD
DfGYKDfF°}zQEDfom}zQEDCGYSVgYKJD opA [ lSVFdc4D‹bILKJM GJDoLSVF­KsS4QHKsDfGNomD'BEImgYDlfKJImSVFHG ›œ_HgJSV_aSVGJI‰KsILSVF*ƒ L L µ„ L o¹DfGYK
A4oLSVgJG FiAvKJQEgJDfo¹BEDq_.DfFEGJDfgN}zQED4uoLSVgJGY}zQEDqoxO SVF­AŽA4GJGYD BEDqBEImgYDlfKJImSVFHGu¤opA [ lSVFdcVDŒ‹bILKsMCA4QEgsA>BEDG
_EgYSV_EgYImMfKJMGGJILh'ILopA4ILgJDGt‘,opATlSVFdc4D‹bILKJMKsS4KsA4omDª†lvO DfGYK lfD}zQEI2GJD_HgJSXBEQEILK'DF‡KsSVhSV¬VgJA4_EžEILD4uoLDG
gYMGJQHoLKŠA\KsG¹BŸO QEFHImlI‰KsMfGBED A4gYBEFEDfgDK rgJILK h'A4FEFGORMfKJDFEBEDfFdKwA4Qb‹ [ lS4Fzc4D‹bDG L @A5lS4FEGJMf}XQHDFElfD
ILh'_aSVgYKsA4FdKsD:BEDlDfKYKsD%_HgJSV_EgYImMKsM4uvA4GYGJD BEI lILomD†‘’BEMfh'S4FzKJgJDfg8›kc4SVImg L ƒ L ƒVu\DGˆKopAwgJMfGJSVoLQHKsILSVFBPORQEF
_EgYSVUEoL™hDN_.S4GJM_iA4g rgJI‰K hqA4FHFr.SVFTGJA4ILKrgYDlfSVFEGYKJgJQEILgJDtDFKJDh_EGr_.S4oL{XFESVhIpA4o¤omDfG’KsSvKŠA4oLDhDFdK
lfSVFdcVD‹bDfG’omS4gJGJ}zQED oxO SVFA_EoLQEG8BED GJDf_HK’BHImgJDflfKJImSVFEG ›kSVQ}zQiAvKsgYD BEImgYDlfKJImSVFHG8UEImDfFlŠžHSVImGYImDGs L
µ„ƒ !
%
2
3
‚
c
d
d
d

6€
d
q
q
q
q
{
q
&
{
d
q
c
q
q
q
~
c
~
~
~
‚
q
µ„ƒ
8hw
` 4
4
8
Mf
]0AvFEGopA2GJQEI‰KsD¹GYSVFdK²gJMfl|A4_HILKsQHomMGªomDG gJMfGJQEo‰KŠAvKJGlSVFHFXQHG²BiA4FEGoLDG _EgJSVUHom™hDGªBED%gJDflSVFHGYKsgYQElfKJImSVF
DfKNBŸO QEFHImlI‰KsMtBEDfGDFHGJDhUEomDfGBEDlDfgYKŠAvImFEDfG5lfopA4GYGJDG‘>_iAvgYKsILgrBEDqoLDQEgYG _EgJS\WYDflfKJImSVFEG L @#DfG5_iAvgYKsILDG
DFElA4BEgJMfDG2GJSVFdKwomDfG€A4_E_aSVgYKJGwBHD lfDfKYKsDrKsžE™fGJD L
$-
(
! " )
]’SVFEFEMfD' |D| (F ,D)((r ) ,p ∈ D)
N
SVgYKJImD
E∈F
∀p ∈ D∀i ∈
@#D FESVhNUHgJZD Xn gJ(i)Df_EgJ=MfGJrDFdKJD max lp SV´ lp = max{i : r > 0} − min{i : r > 0} L
]’ImgYDlKsImS4FEG
|D| ≥ 4
!#"%$#& '(*)#),+*'.š opA4GYGJDfG
{x,y}
|D| = 2
{x,y,x − y}
|D| ≥ 3
n [ lS4h'_EoLDfK
n [ lfSVh'_HomDfK
}XQHDomlfSVFE}zQED
O(n )
¢ 8{XG 0/Hui8{XG21VZ ¦
¢ n ¥4¥ ¦
¢ n ¥V¥ ¦
n [ lSVh_EoLDfK
_.S4oL{VS4h'ILFES
¸
¸
¸
¸
¢43<SXD|¥!1 ¦
n [ lS4h'_EoLDfK ¸
n [ lSVh_EoLDfK
ždcd[ lfSVFdcVD‹bD
¸
¸
¢ 5†Qi¥01 ¦ u
4¢ 3<SXD|!¥ 1 ¦
¢ %]’@ 2•V• ¦
_.S4oL{VS4h'ILFES
¢ %]’O(n
@ n )0¥ 1 ¦ u
¸
¸
¸
¸
ždcd[ lfSVFdcVD‹bD
¢ %]’@ n ¥ ¦ u
¢ š ]0¥V¥ ¦
1\[ lfSVFEFEDŒ‹HD
¸
¸
¸
lS4Fzc4D‹bDGYDoLSVF
¢ %O(n
]’@ 2)•V• ¦ ¸
h,v,x + y
[ lfSVFdcVD‹bDfG
O(n )
GJDfomSVF D
s
K
H
ž
h 8 L Lµ
9
O(nL )L
O(nL )L
O(nL )L
f
l
V
S
6g / ƒ /
f
l
V
S
6g / ƒ /
l
V
S
2g / ƒ /
[ lfSVFdcVD‹bDfG
7 _EgYSzƒ L / L 1
jkSVgYKJG
lO(n
SVg2/ L ƒ )L /
KJS4KŠAvomDhDFdK
¸
¸
¸
¸
lfSVFdcVDŒ‹HDfG
lO(n
SVg2/ L ƒ )L
)*,+.-0/214365872+936:;-0/
Z
<
p
i i∈Z
=> ?;@BAC=
P[APQ=4A\S]^IJA
DFEHG IJA>LKM=ONG PQ=
[email protected]=> [email protected],XYPV?<
_Y=?`?;[email protected]_cN [email protected][email protected]?hUHPQ=OE iK? [email protected]
p
i
p
i
p
p
XXX
XXX
XX
XX
i
p
XX
4 , 3 ,2,3,4}
6∈ { 3
2
€
€
2
O
€
€
€
j
2€
j
2€
4

2€

2€
!
5
q
7
q
4
4
6
}
6
6
` x
$(
µ„ƒ4¥
b '" " )
’] MflgJI‰cVSVFHGrBPOyAvU.SVgYBŽomDfG0l|AvG0SV´ŽILo¤{AqQEFEILlI‰KsM_aSVQEgwKsSVQHG0omDfG’DFHGJDhUEomDfGrBEDNopAlomA4GJGYDŸ_EoLQEG
_EgYMlfImGJMfh'DfFdK5S4F,BHILK }zQEDo~ORSVFTA QEFEILlILKJM_.S4QEgropA lomA4GJGYD F DfK oxO DfFEGJDfhNUEomDtBED
BEILgJDflfKsILSVFEG D8GYI#_.S4QEg%KsSVQHG E ,E ∈ F SVF(FA%,D)

/(:;+.: 3
1
2
(∀p ∈ D Xp E1 = Xp E2 ) =⇒ E1 = E2 .
]’ILgJDflfKsILSVFEG JK SVQEG€omDfGw|D|UHImgs≤Av_E3_.S4SVgYQ KsG¹BiA4FEG
š omA4GJGYDG
{ , ,2,3,4}
±
}zQEDoLlSVFH}XQHDG
¢ ¹][email protected]
±
_aSVoL{4SVhImFES
·€[ lfSVFdcVDŒ‹HD
¢ ¹][email protected]
±
· w[ lfSVFdcVD‹bD
¢ ¹][email protected]
[ lS4Fzc4D‹bDG
GJDfomSVF D ⊂ D
±
A|cVDfl |D | = 2
¢ %]’@ n ¦ u KsžEh 1 L ƒ L
[ lSVFdcVDŒ‹bDG’GYDoLSVF D
KsSvKŠA4oLDhDFdK€lSVFdc4D‹bDG
XXX
X
XXX
XX
4 3
3 2
XXX
|D| ≥ 4
#"%$ & '(*)#),+*'.-
6€
€
n¦
€

6€
€
d
€
n¦
€


6€
4 , 3 ,2,3,4}
6∈ { 3
2
€
n¦
q
0
q
0
€

2€
}
¸„¸„¸
lfSVF\WYDflfKsQHgJD L ƒ L ƒV
±®
KsžEh 1 L ƒ L 1
±r®
¢ ¥/¦
O
€
O
µ V•
8hw
` 4
4
8
Mf
]’SVFEFEMfD'
E∈F
QHDGYKJImSVF,
F ∈F
F 6= E
∀p ∈ D∀i,j ∈ Z X E(i) =
X F (i)
@#DG¹lSVh_EomDŒ‹bILKsMfG¹lfSVFEFzQEDG8BHD0lD0_HgJSVUEoL™hD’_aSVQEgBEI .MgYDFdKsDfG%lomA4GJGYDG8GJS4FzK%omDGGJQEI‰cvA4FdKsDGŸ n
DGYK2omDrhehD0}zQED _EgJMflMfBEDhh'DfFdK2lA4omlfQEomM A|c4Dl omDG8_HgJS\WYDflfKsILSVFEG€BED E 
]’ImgYDlKsImS4FEG
|D| ≥ 4
0#" $ & '(*)#),+*'.š opA4GYGJDfG
{x,y} |D| = 2
|D| ≥ 3
n [ lS4h'_EoLDfK
}XQHDomlfSVFE}zQEDfG
¢R€O(n{XG21 )¦
¢ n ¥4¥ ¦
· 2[ lSVFdcVDŒ‹bDG
¸
¸
\[ lS4FEFED‹bDfG O(n ) ¸
[ lfSVFdcVD‹bDfG
_EgYO(n
SV_ / L )L µ _HgJSVO(n
_ 8 L 1 )L µ 7 KsžEh µ1 L ƒ L 1
_EgJS4_ 1 L L µ
7 _HgJSV_ 1 L L µ
[ lSVFdc4D‹bDGwjkSVgYKJG
_EgYO(n
SV_ / L )L µ
_EgJO(n
S4_ / L )L µ KsžEh µ1 L ƒ L 1
7 _HgJSV_ 1 L L µ
7 _EgYSV_ 1 L L µ
µ
KJS4KŠAvomDhDFdK8lfSVFdcVDŒ‹HDfG ¸
¸
¸
¢ ¥/¦
EST U N IQU E(F
,D)
Z
q
sr
P8A
<
TQG [email protected]?hA
[email protected]@t?;I6>LP
p
APQ=4A\S]^IJA
@BAI ^>,A
[email protected]
p
‚
XX
XXX
XXX
XX
XX
4 , 3 ,2,3,4}
6∈ { 3
2
2
€
c
5
}
q
q
5
8
5
7
O
s§ µ Hµ
C^
&'
c%
2%
¤£ DgJhImFHSVFEG#lDKJKJD†KJžE™GYD_iAvg¤oLDG#}zQEDfGYKsILSVFEG¤lSVFHlDgYFiA4FdK opA%KsSVhSV¬VgJA4_EžEILDªDKª}zQEIdgYDGYKJDFdK DFElfSVgJD
FES4FgYMGYSVomQEDfG L
$
-' )#
)#"-'
ro gJDfGYKJD<DFHlSVgYDUaD|A4QHlSVQE_ BEDTl|A4GSV´ opA*lS4h'_EoLD‹bILKJMBED
ILFElSVFHFXQHD L
)*,+.-0/214365872+936:;-0/
(F ,D)
DGˆK
" zo GO A4¬VI‰K BHD¹opA€gJDflSVFHGYKsgYQElfKJImSVF5BEDG _.S4oL{VS4h'ILFESVG¤· w[ lS4Fzc4D‹bDG‘2_iA4gˆKsILg¤BHDGª_HgJS\WYDflfKsILSVFEG¤GJDfomSVF
L
{x + y,x − y}
@#DGlDFdKJgJDG¹GJSVFdK%GJA4FEGBESVQHKJDG¹oLDG†U.S4FEG%A4FEA4omSV¬4QEDG:BEDfG¹_aSVImFdKsG†hMBEImA4FEG†_.SVQHglfD’_EgYSVUEom™fh'Dª
d
,! ! \z?J=4AI G
P
N ?;@
^>VA
C ∈ Z2
[email protected]> P
E
[email protected] A
=4AIJUHP
d
> P8AoN [email protected] G P8gCA
=?u?;I.\z?;PV?;t
^> G [email protected]?;@Bi
D 0 (C) = max d(C,N ).
€I SVF°_EgYDFEB
A4omS4gJGtomDfGlfDFdKsgYDGCBHD FED
BEMf_.DfFEBEDfFzK}zQEd(M,N
DBEDfG_H)gJS\=WYDfdlfKsIL(M,N
SVFEGGYD)omS4=F {x|x +−y,xx −| +y}|yLª“ oLom−DfGNyFED | BEMf_.DfFEBEDFdK'DF*jœA4I‰K}zQEEDBEQ
gYDlfKsA4FE¬VoLDrBESVFdKwomDfG8lS4KJMG€GJSVFdK2_iA4gJA4omoL™omDfG%A4QX‹BEIpAv¬VSVFiA4oLDG%DKw}XQHI#lSVFdKsILDFdK E L
N ∈E
1
M
N
M
N
! ! ± FCGsAvILKhqA4ILFdKsDFEA4FdK¹gYDlS4FEGYKJgJQEILgJD’DfFCKsDfh'_EG_.S4oL{XFESVhIpA4oHomDG†KsS4KsA4omDfh'DfFdKlfSVFdcVDŒ‹HDfGw‘o~O A4ImBHD
BEDfG5_HgJS\WYDflfKsILSVFEG GJDfomSVF,lDfgYKsA4ImFEG DfFEGJDfhNUEomDfGBED BEImgYDlKsImS4FEG lSVhh'D {x,y,x + 2y, − 2x + y} L
AvImG%_iAvg8D‹bDh_EomD _aSVQEg8oLDG8BEILgJDlKsILSVFEG
uElfDfKJKJD h'MKsžESXBED0FEDrjkSVFHlfKsILSVFEFEDr_EA4G
lA4g8ImoPFŸOR{A_iA4G€QEFHImlI‰KsMr_aSVQEg€omDfG [ lS4Fz{x,y,x
c4D‹bDG L +>y,x
A4ImG€GJ−I#_ay}SVQHcvA4I‰K8hSVFdKsgJDfg€_iA4g8D‹bDfh'_EoLD }XQŸO ILoP{
AQEFEImlfILKJM5QEFHDjkSVImGw}zQEDNoxO SVFA'H‹bMNoLDGw_EImDfBEGu.SVFŽA4QHgsA4I‰K’QEFŽA4om¬VS4gJILKJžEhDr_.SVQHg’gJDflSVFHGYKsgYQEImgYDoLDG
KJS4KŠA4oLDhDFdK8lfSVFdcVD‹bDfG L
p}
q
-' )#
)#"-'
%! ! ²@ OyAvom¬VSVgYILKJžEh'D _HgJMGYDFdKsMBEA4FEGqlfDŽhMhSVImgYD_aSVQEggJMfGJSVQHBEgJDŽoLD>_HgJSVUEoL™hDAv_E_EgJSXlŠžEMA,QEFED
lfSVh'_HomD‹bI‰KsMMfomDcVMD O(n ) LPn DQHK SVFŽKJgJSVQbcVDgrQHFAvom¬VSVgYILKJžEh'D _aSVoL{XFES4h'ImA4o#DFjkSVFElKsILSVFŽBEQ
FES4hNUEgJD d BED BEILgJDlKsILSVFEG4¸
® FCA4QHKsgYD2_HgJSVUEoL™hD2lfSVFElfDgJFEA4FdK8opA gJDflSVFEGˆKsgYQElfKJImSVF Av_E_EgJSXlŠžEMfD’DGˆK8opA gJDflŠžEDgYlŠžEDrBED2opA GJSVoLQb[
KJImSVF SV_bKsImh'A4oLD Lbn oLQEG€D‹HA4lKsDhDFdK2Imo.GO A4¬VILK€BHD gYDlfSVFEGYKJgJQEILgJD QEFDFEGYDhNUHomD E ⊂ Z hImFEILh'ILGsA4FdK
2d+2
2
XX
|Xp E(i) − pi |
0] AvFEG ¢ ¹] ¦ oLDG€A4QbKsDQHgJG8BESVFHFEDFdK€QEFA4oL¬VSVgJI‰KsžEhDw}XQHIŸjkSVQEgYFEILK¹QEFEDrGJS4omQHKJImSVFCS4_HKsILhqA4oLD2_aSVQEg¹opA
lfopA4GYGJDqBHDq_aSVoL{4SVh'ILFESVG · 2[ lfSVFdcVDŒ‹HDfGDfK D = {x,y} u¤h'A4ILG lfDfKNA4om¬VS4gJILKJžEhDFPO DfGYK_iA4G_HgJSVQHc4M
lfSVh'hD0_aSVoL{XFES4h'ImA4o.lA4g€Imo.QHKsILomImGYD0QEFGJSVo‰cVDfQEg€BED \[ £ L
p∈D i∈Z
 €
c
µ dƒ
8hw
` 4
4
8
Mf
! ! ! ! @ AgYDlS4FEGYKJgJQElKsImS4F­BEDGlfSVFdcVDŒ‹HDfGtBED R ‘_iA4gˆKsILgBED'omDQHgJGN_EgYS\WYDlKsILSVFEGDfGYKQHF*_EgJSVUHom™hD
}XQHI2AMKsM>A4GJGYD MfKsQHBEImMvuhqAvImGtA4QElfQEF‡A4om¬VS4gJILKJžEhDqGsA\KsImGˆjœA4ImGJA4FdKFPO A,MKsM KsgYSVQHc4M cVSVILg¢ A4gs¥08Xu
_ 8Vƒ„,[ 8 ¦ um¢ V¥ ¦  LE® FEDrGYSVomQHKJImSVF lfSVFEGYImGYKJDr_.DfQHKY[ efKJgJD ‘NKsgYSVQHcVDfg€QEFED0cVDfgJGYImSVFlSVFdKJImFzQEDrBEDroxOyA4o [
¬VSVgJI‰KsžEhDBHDgYDlfSVFEGYKJgJQElKsILSVFTBEImGYlgJ™KsDBEDfG [ lSVFdc4D‹bDG L @²O A4FiA4oLSV¬VQEDlSVFdKsILFzQ,BHDopAClomA4GJGYDBEDG
[ lSVFdc4D‹bDGDfGYKNGJA4FEGNBHSVQHKsDlDfomoLD'BEDfGlfSVFdcVD‹bDfGGYImh_EomDfG lSVFEFHD‹bDG L²“ F<D .DfKomDfG [ lfSVFdcVD‹bDfG
jkSVgYKJGwBHImGJlfgJDKsG€GJSVFdKwomDfG8KsgsAvlDG€GYQEg Z BEDfG2lfSVFdcVD‹bDfG’GJILh'_HomDG8lfSVFEFEDŒ‹bDG cVS4Img ¢R]0A4Qi•V• ¦  L
2
~
q
q
q
&
2
'" " )
! $ ® FED ¬VMFEMfgsA4oLImGJAvKsILSVFCBEQKsžEMfSVgJ™fh'D 1 L ƒ L 1_.SVQHgJgsAvILK8eKsgJDª
! ? D =^G @w?J= ,&G ?;@6G >aogCUHPN ?;@w?hUHPQ= i ^> ?;EHG [email protected]=
& ( !
/(:;+.: 3
N >[email protected] V
> iCUHK \FA
(Q(D 0 ),D) <
D 0 = {x,y} D = {x,y,2x + y, − x + 2y}
<
< nG IJUHKM=2TVUH> K
š DfKJKJD2lfSVF\WYDflfKsQHgJDwDGYK†cXgsA4ILD8_.S4QEg
u
_.S4QEg²omDfG†DfFEGJDfhNUEoLDG
ImFElfomQEG8BiAvFEG€QEF>lA4gJgYM 13 × 13 cVSVILg8lSVF„WYDlfKJQEgJD 1 L µ L ƒV L
“ F_EgJDfFiA4FdK
SVFcVS4ILK:}zQEDwlDfKYKsD€lS4F\WYDlKsQEgYDwDfGYK_EomQHGjkSVgYKJD8}XQHD2omA0lSVF\WYDflfKJQEgJD L 8
BED¢ %]’@ n ¦ L DAvImG8=GJI#{x,y}
lfDfKYKsD lSVF„WYDlfKJQEgJD DfGYK€cXgsA4ILD4uEGJAt_HgJDQbcVD GJDhUEomDr_EoLQEGwA4gJBHQEDr}zQED lDoLomDrBEQ
KsžEMfSVgJ™fh'D 1 L ƒ L 1Nl|A4g¹SVF FEDr_aDQHK€_iAvG%_iA4gˆKsILKJImSVFHFEDgomDGG €ILKJlŠžEImFE¬\[ lS4h'_aSVFEDfFzKJG CBEDropAhehD
jœA SVF c4SVImg8oLD NG€ILKJlŠžEImFH¬v[ lfSVh'_aSVFEDfFdK BHD omANi¬VQEgYD L µ„ L
D0 ⊂ D
@BUH> @
@AI ^>VA
|D 0 | = 2
UHP
G
c}
0

2€
O
E+
D0
E−
D
M2
3y:'465
= u?;@Bg
Lµ
> ?;P<
Z
|
t gCUH\
P
= u?;@g
> ?;P<
[email protected]=4AIJUHP
N A
D
0
D 0 t *jt
gCUHPVE AbaLA
=4AIJUHP
D _z=4UHP
?;\SG
<
A
M2
TQG K
P
[email protected] >=
> P
± F _.S4QEgJgJA4ILKTAvQEGJGYItA PAvImUEoLImgopA³lS4Fzc4D‹bILKJM‡DfF lfSVFEGYImBEMfgsA4FdKT_iA4gDŒ‹bDh_EomD”BEDfG,lfSVFdcVDŒ‹HDfG
KJgJSVQEMfG *lvO DfGYKq‘,BHImgJD BHDGtDFEGYDhNUHomDG E \ F A|cVDfl E DK F lSVFdc4D‹bDG L c4SVImg_HgJSVUEoL™hDƒ L 1
BED¢ A4gs¥08 ¦ 

! !
± F FED*lS4FEFiA LKTA4QElfQEF gJMGYQEoLKsAvKTBPORQEFEILlILKJM­BEDfG,GYSVomQbKsImS4FEGlfSVFdcVD‹bDfG hehD<KJS4KŠA4oLDhDFdK
lSVFdcVDŒ‹bDGŠA4Q_EgJS4UEom™fh'DBEDqomAgYDlfSVFEGYKJgJQElKsILSVF­DF<BEImhDFEGYImSVF GsA4QHj%BiA4FEG5omDl|A4G5SV´­omDfG5BHI‰[
{
µ
gYDlfKJImSVFHGwGYSVFdK’lfSV_EomA4FiA4ILgJDG€SV´o~ORSVF_aDQHK’A4_E_EoLIm}zQEDfgwoLDG€gJMfGJQEo‰KŠAvKJG0ƒv]r L “ F>jœAvILK€hehD _aSVQEg€oLDG
lfSVFdcVD‹bDfG’BED R uESVF>AAvQElQEFgJMfGJQEo‰KŠAvK L c4SVImg8_HgJSVUEoL™hDGwƒ L µf[^ƒ L BED¢ Avgs¥ 8 ¦ 
C^
3
µ
}
8hw
` 4
4
8
Mf
"! #$&% # ')($'+*-,#"./*-,#"! .)'/!0%1#$!32#546!078')% (:9'/;=<>([email protected]?D#"-% !FEG(IHA.)($,.)$!J;D KA($#LHM.ONPHQ./;D?Q<
R')%[email protected]#$#!8S($./ HA$! !R!R#$#$&!X%1THA,ZHAYZ, *-(:,9U#$HA.)./#"'V!X%1RHQ# U1 LHA0%1'/#"#"-%8.)#$!8!R78'V%1(:9');D!8! ?A-%XHA,S! ! 'S $("'/!R$#$!\[
W
] 434b%*-?A($%:?AHA-.`cE^9&`dC?DEe,f8?D.),S#$!+F%1.)#$?Q!F%:HAgX.)#$<>@ ("?D#$&&`D%_#hYS($?A#"-!i`A!R#a#$YZ./?A#$! L L
] 4b*-($?A-`-% ?D,!iELC?D,_.)#$!FgX<>("?D&`D#aYZ#"!i! #"./?DL{x,y}
%1?A,S%1#"!F.)#$!_2') #"("%1')?D!_2#jR?=kR#$(a%1')?D L
] 4b*-($*^ELC?D,_.)#$!FHA,S% #$! L
]
lnm$o prq ts8u qavTw x {x,y}
y NzHA{C?D 2 ./#"!_!R78')% (:9'/S; 2\Nz#"! #"@b{./#"!X*-,#"./("?D*-,#$!F'/S($./,S!_2UHA!8, (IHAR 3 × 3 [
 L *-("*
 L_} ?D0%
| L8} ?AR%
| „ L+} ?D0%
~ L *-("*
L8} ?AR%
L+} ?AR%
‚ L_} ?D0%
€ L *-("*
ƒ L *-($*
8
|
8
|€
c78'V%1(:9');D!_2#TgX<>("?D&`D#hYS#"!i!R#$./?A x,y ')($.),!_2UHA!8, (IHQ | L+} ?DR%
| „ L *c("?D&`-%1?D,!
| ƒ L8} ?AR%
-
4
!#"%$'&)(*$'+,
4×4
[
~A‚ L+} ?AR%
~ L+} ?DR%
|D| L *c("?D&`-%1?D,!
~A„ L8} ?AR%
L+} ?DR%
| ~ L+} ?DR%
~Z| L *-("?D&`c% ?D,!
.
 L+} ?DR%
|0. L+} ?DR%
~D~ L8} ?AR%
.
L *-($?D&`-%1?D,S!
|  L+} ?DR%
~1. L8} ?AR%
.
€ L *-($?D&`-%1?D,S!
|'/ L+} ?DR%
~  L *-("?D&`c% ?D,!
.2.
 L+} ?DR%
| € L *c("?D&`-%1?D,!
~2/ L8} ?AR%
.
‚ L+} ?DR%
|  L+} ?DR%
~A€ L8} ?AR%
.3/
ƒ L+} ?DR%
| ‚ L *c("?D&`-%1?D,!
~  L8} ?AR%
.
.
/
~Aƒ L *-("?D&`c% ?D,!
D„ L *-("?D&`c% ?D,!
S| L+} ?AR%
&~ L+} ?AR%
L *-("?D&`c% ?D,!
 L+} ?AR%
L+} ?AR%
D€ L+} ?AR%
E
2 %0 ,$0
{x,y}
 € L+} ?AR%
/2/
.
D‚ L8} ?AR%
- L+} ?AR%
/
.
Dƒ L8} ?AR%
 ‚ L+} ?AR%
/
 „ L8} ?AR%
 ƒ L+} ?AR%
 | L8} ?AR%
/
A„ L *-("?D&`c% ?D,!
 ~ L *-("?D&`c% ?D,!
 . L8} ?AR%
/
D L8} ?AR%
/1.
 / L *-("?D&`c% ?D,!
/
 L8} ?AR%
.
L_} ?D0%
€  L+} ?D0%
A€ L_} ?D0%
€3/ L+} ?D0%
 L_} ?D0%
€D€ L+} ?D0%
/
A‚ L_} ?D0%
€  L+} ?D0%
/
Aƒ L *-($?A-`-% ?D,!
€D‚ L+} ?D0%
/
Z| L+} ?AR%
€D„ L *-($?A-`-% ?D,!
€Dƒ L+} ?D0%
D~ L+} ?AR%
€S| L *-($?A-`-% ?D,!
 „ L *-($?A-`-% ?D,!
L+} ?AR%
€&~ L_} ?D0%
 | L *-($?A-`-% ?D,!
 L *-("?D&`c% ?D,!
€2. L_} ?D0%
 ~ L+} ?D0%
|0. 
4
|0.D‚
 . L *-($?D&`-%1?D,S!
‚&~ L+} ?DR%
= L+} ?DR%
‚2. L *c("?D&`-%1?D,!
ƒ&~ L8} ?AR%
| „S| L+} ?AR%
 / L+} ?DR%
‚  L+} ?DR%
ƒ2. L8} ?AR%
| „&~ L+} ?AR%
 € L *-($?D&`-%1?D,S!
‚3/ L+} ?DR%
ƒ  L8} ?AR%
| „2. L+} ?AR%
A L *-($?D&`-%1?D,S!
‚D€ L *c("?D&`-%1?D,!
ƒ3/ L *-("?D&`c% ?D,!
| „  L+} ?AR%
 ‚ L+} ?DR%
‚  L+} ?DR%
ƒD€ L8} ?AR%
| „3/ L *-("?D&`c% ?D,!
 ƒ L+} ?DR%
‚D‚ L+} ?DR%
ƒ  L8} ?AR%
| „D€ L+} ?AR%
‚A„ L+} ?DR%
‚Dƒ L *c("?D&`-%1?D,!
ƒD‚ L8} ?AR%
| „  L+} ?AR%
‚Z| L *-($?D&`-%1?D,S!
ƒD„ L+} ?DR%
ƒDƒ L8} ?AR%
| „D‚ L+} ?AR%
ƒS| L8} ?AR%
!#"%$'&)(*$'+,
| „D„ L *-("?D&`c% ?D,!
E
2 %0 ,$0
| „Dƒ L8} ?AR%
{x,y}
|0.Dƒ
|„
l m
4
!#"%$'&)(*$'+,
rp q u q"vTwLx {x,y,x + y,x − y}
-c78'V%1(:9');D!_2#TgX<>("?D&`D#hYS#"!i!R#$./?A x,y ')($.),!_2UHA!8, (IHQ 7 × 7 L
| L *-($?D&`c9&`
€ L_} ?D0%
A| | L+} ?DR%
~ L+} ?DR%
 L *-($?A-`-% ?D,!
| ~ L+} ?DR%
L *-($?D&`c9&`
‚ L_} ?D0%
|. L+} ?DR%
 L *-($?D&`c9&`
ƒ L *-($?A-`c9&`
|  L *-($?D&`c9&`
L *-($?D&`c9&`
| „ L *-($?A-`c9&`
|0/ L+} ?DR%
.
/
E
%0 ,$0
| € L8} ?AR%
|  L8} ?AR%
{x,y,x + y,x − y}
| ‚ L *-($?D&`c9&`
| ƒ L+} ?D0%
~A„ L *-("?D&`Z9&`
||
|~
l m
4
!#"%$'&)(*$'+,
rp q u q"vTwLx {x,y,2x + y,x − 2y}
-c78'V%1(:9');D!_2 N #$S! #[email protected]{./#"!X*-,#"./($?A*-,#$!F'/("./,!_2HA!8, (IHAR 6 × 6 L
| L *-($*
. L *-($*
/ L *-($*
~ L *-($*
 L *-($*
€ L *-($*
E
r r
m"o
u vTwLx "! u qav3w # 8x $&% u vTw(' *) 8x +,' u -/. + -10 v q 324
#! u52 +76Lq 8
W NPHQ./;D?DR')% [email protected]#_2S,e(:9UHA'V%1R# / H3"% i'/@ .)[email protected] #$&%18#$:9<;3=>;&[email protected] | L ~<B !R, ,C;5DFE W HG /A„D„ L
?A, #hYSC$R'/@ #$&%1#" .ONPHA.);D?DR')%[email protected]# ?D^HT;A$" i2S#$!_gX< ($?D&`D#hYZ#$!_#$ #"UHA&%+.ONz#"-`A#$.)?DC# gX<
("HN?DM&&HQ`D-I #a%FYZ("#J#$!F2#3 f?=?DkR#"')&("%1% !X'/?DHA,M! L 9UHA! HA 2KJ\?DM($HA./(",./# ./#", !X ? kR#$("% '/?DS!LJC#"%?AM #"($?D!0%1R,')%i,#$!R#[email protected]{S./#
I ?A, D = {x + y,x − y} .)#$!_%1#"@BS!_2#j(IHA.)($,.!R?D&%X.)#$!_!R,')`QHA&%1!\[
#"! ! HA'/! %[email protected] ?OMP? ! B @ H=YQ? ! B RTSVUXWZY\[L] „ | ~ |~QI„„ || „D„D„„
„„ LL „A„A„D‚S‚| „„ LL „| & ƒ ~/ ƒSQ| 2~ ƒ / „|
„ LL |I‚Dƒ „ LL  |I„ ~ ƒD€  „
.A„„ || „D„D„„
„ L ‚ | L „D„A€ | ƒD‚ ~ „
/Q„
| „D„
„ L 2‚ /A„ | L ‚ ~ | ƒDƒ | „
€A „ | „D„
| ~Aƒ ~ /D~ „ ƒDƒ | „
„‚A„ || „D„D„„
~. LL .D|I3€2~ // ./ LL /1~A.A‚Dƒ~ „„ | ƒD„Dƒ „ „| „„
 LL ~Q ƒ3/ | „ LL /2/  | ƒ  . „
ƒ|IA„D„ „ || „D„D„„
/ |
ƒ
/1.D~
„
|
D
„
„
„
„
L
L
|A| „ | „D„
€ ƒZ| € ~Aƒ /A„A‚ | | „D„ „ „
W!R#$H @b|N{^`.)_b#$a !J("("?D?D./?A! ')2#c$R? $n!JB !R?D')&2%J'/*-%1,?D,=#LkR?A./H, %1!JHA'/2U.)./HQ#e!J2S#$,! #"(IHA! #"[email protected] {.)#$! R#$(:L 9W #"HM (:92S"#$!L,ZJ YZ'ed"./@B,! #^#a("YS?DHA./(a?A%1#"@B#e#"')-% 2'/./*-#"!,# #"Z./# <
[email protected]{ # 2 N #$!R!1HA')!LJt./H % ?D')! 'ed"@B#6("?D./?DL S#B.)# % #[email protected] [email protected] n?O×MD#" n f?D,ST#1f\#"("%1,S#$b.ONPHA.);D?DR')%[email protected]#gJ\. H  #
("?D?Dd./?DHBSa%1# 3')?A2{')./*c');D,S # 2 ./# N ,S%1%1#"')@B./'/!RS#$! @ <H -iYS;5')@DHQ? . ("#3W !RH ?D/T&^`%h7./a#"!X("?D($./HA?D!iS?<jd# 2'/.?AR#$!0#+%1#b.)# 2#"[email protected]{?D'/&# % 2 !XN #$')! ! 2HA"'/% !t#[email protected]?D,S')t$./#"!X! HA*-,k#"d$./!!
~
.)#$!F?Df"1HQ% '/?DS!X2#T([email protected] .)"%1')?D B L W H €<^`h7a ($?D.)?D#T2?D#T./# [email protected]{S #32\Nz#"! !1HQ'/!iC?D,X./#"! *-,#$.)!i').
|.
| D $'&)(, 1( & + ! $ & %$' ',,$' "%$'0
He"%1J')-,S%1')./# 2#SYZ#$ 2#"! '/#"2! L W #"!  ^`h7a Y #"% ‚<^`h7a Y ($?D.)?D#"!'/S2'/*-,#"-%L #$!Rf#"("% ')`D#"@B#"&%T.)#$!
[email protected]{R#$!82 N #$!R!1HA')!XC?D,_.)#$! *-,#"./!8?DL#$!0%i?A{./');Di2#SYZ#$8, ?D,L2#",ZY '/#"2!
}
2#$,ZIY ?D} ,?A'/!_D.),=!_{x,y,2x
;D1HQ2! B L + y, − x + 2y} ?D5H^2#$! % #[email protected] ! 2#J(IHQ./($,S./! 2, @[email protected] #b?D 2S #? #" HA')%
#$!R!1HA')! % [email protected]?OMQ? ! B @ H YP? ! B RgSVUXWZY\[L] „ | ~ | „ | „D„
„ LL „S'| / „ LL „  . „ | „A„ „ „
~A„ | „D„
„ L '| / „ L . . „ | „A„ „ „
„| L | €Dƒ| „~ L ‚Z„D2~| „/ „„ || „A„A„„ „„ „„
.D„„ || „D„D„„
L  ƒS| . L €2/A„ „ | „A„ „ „
/A„
|
D
„
„
|
L  | € „ | „A„ „ „
€D „ | „D„
. L .2. 
€
 LL €D„Dƒ  LL .D~  „ | „A„ „ „
„‚D„ || „D„D„„
 Q
ƒD„ | „D„
ƒ LL „.&~A‚ |A| .| LL „D/Q€D~Z‚| „„ || „A„A„„ „„ „„
| „D„ | „D„
| ~ €D€D€ |Iƒ €2/ „ | „A„ „ „
m
u v3wLx "! u qavTw +766 v u52 # 8x$ % u v3w ' *) 8x=v x +F' u
- . + -10 v tq 324 #! u52 + 6Lq ?D, %1#$!0%1#" .)# 2S#$,ZYZ'ed"@B# HA.);D?D 'V%[email protected] #t?A3,S% '/./')! # ./[email protected][email protected] # @Ba%19?c2# *-,# "($$2S#[email protected] @B#"-%t!1HA, }
I
*cH `A,\#$NPHA( ,./#".)!F'/#$S, ?=2kR#e#"("%1')R?D#$!_2S #"#^(I./HA#"!32S1HAR&?=%1kR#"#$! (a%1')?D!b2 2SN , U#"')#$!8! #"[email protected]{ ./#^gi< ($?D&`A#aYZ#^?D #Xff#$(a%1,#^.ONPHQ./;D?DR')% [email protected]#
p = max(0,X
L W #"(i)! %1#"−@Bd),p
!1HA, } C?D, i ∈ {pmin,pmax} ? j^?D(pe,pR#$)2 p = max(1,X
S!+2#=($HAX./(",(i).f!R+?D&d%
(i) + d)
HA./?DR!_./#"!_! ,'V`AHQ-% !8f?D,S D = {x + y,x − y} #a% d = 2 [
#$!R!1HA')! % [email protected]?OM ? ! B @BH=YP? ! B -{ ~ < !1HQ% „ | ~ .  | „ | „A„
„ LL D2„ -.S| „ LL '| /D~ ‚D‚ | € |  €&~ | „
~A„ | „A„
~L
.3/ /1.&~
ƒDƒ ~ ~ € /A„
L 2€ .&~ ƒDƒ | „ ƒ  |  ƒ
.D„ | „A„
A
~
‚
D
€
D
€
‚
|
A
€
D
ƒ
‚
 „ | „A„
/A€ L .  ~ ~AƒA€  L .D‚ 
| ƒD„Dƒ „ „| „„ |Iƒ „ .3. / /A/D€~
/A„ | „A„
/D~ L  ‚3/ €A„Dƒ L ~AƒS|
I ?D, D = {x,y,2x + y, − x + 2y} #"% d = 2 [
#$!R!1HA')! % [email protected]?OMQ? ! B @BH=YP? ! B c{ ~ < !1HQ% „ | ~ .  / €  ‚ | „ | „D„
„ LL „  | „ LL '| /D~ ‚D€ /Z| .D~ | „ € |  „ „ „ „
~.DA„„ || „D„D„„
„. L ‚D~2‚D/2‚/ ƒ~ L €D/A„S„D„| ƒDƒD‚€ ~A|D„| ~Q~A‚~ ~1| . ~Z| ‚| ||'/ ‚~ .| „„ „„
L  0| . .Z| L .A€Dƒ ƒD‚ | ‚ ~Q‚ .D„ |   . „ „ „
 „ | „D„
ƒ
/A„ | „D„
| ‚ L  ƒD„  ƒ L .  „ | „D„ ~D~ ~A~ ~2/ | ‚ | „ . „ „ „
i
0
i
i
i
p
p
0
i
p
|/
}
2I #"?D!X,giD< ($?D=&`A{x#aYZ#$+! }y,x?DR% !_−2y}, ("#"?D% ?Ad./. =HQ'/ 0#  ? ?DL ~ dL  % B #$!R[ % #b#$ HA')%F. NzHA.);D?D 'V%[email protected] #2#3 #$("?D!0%1 ,S("%1')?DM#hYHQ("%1#
#$!R!1HA')! % [email protected]?OMQ? ! B @BH=YP? ! B c{ ~ ! HQ% „ | ~ | „ | „D„
„ LL „S| „ „ LL |.  | €  .&~ |
~A„ | „D„
„ L | „ | L .1.Dƒ . ‚ 0|  . „
.D„
|
D
„
„
„
~
€
|
A
|
|
„
2.
ƒ
„
L /A‚&~ | L |I€  „ ƒD‚ ~ „
 „ | „D„
„
/A„
| „D„
„ LL ƒD ƒD€ | LL ‚2/A‚ „ ƒD€  „
€D „ || „D„D„„
~~ L /2€3/A„/ ƒ / L ./Q€3„/ „| | ƒD„Dƒ „ „| „„
 LL .  € €A€ LL ~1/D~ | ƒD‚ ~ „
‚D„„ | „D„
 L ‚D ‚ ƒ L /Q „D„ „ | „D„ „ „
ƒ| D„D„ „ || „D„D„„
€ . |. „Dƒ „ ƒDƒ | „
LL }
HA.),S;D'/?D! D !R?DHA')%1,S9S%1("@B#"#!+#"?D!($^L#"! C2#$,S?A% 2$$#"2!_,S! '/?D #i&% 2#i1HA%1!R?A! ,S#[email protected]%1#"{!+./($"#"#$!_!+22U?DHQ! "./#$H!_U./;AHT,($ ?A# @B |.)#aYZ')% [email protected] HA')?O!1MDHA#"&% ,##("?DR$./HQ%1i2')?D6#U("HA#$!
%$'% ',,$' %(( $ % "$'' + $ % ' !$ & %( |  €1
$'&)(, 1( & + ! $ & %$' ',,$' "%$'0
tps moy
23us*n^2,69
tps max
37us*n^2,76
4
tps moy
46us*n^2,72
tps max
177us*n^2,52
4
3
2
ln(temps en secs)
ln(temps en secs)
2
0
1
0
-1
-2
-2
-3
-4
| #$8HA.);D?
2.5
-4
3
3.5
ln(n)
4
4.5
2.5
D = {x + y,x − y}
3
3.5
ln(n)
4
4.5
D = {x,y,2x + y, − x + 2y}
6
tps moy
124us*n^3,44
tps max
3026us*n^3,47
10
| #$FHA./;A?
tps moy
37us*n^3,36
tps max
28us*n^3,78
5
8
4
3
ln(temps en secs)
ln(temps en secs)
6
4
2
2
1
0
0
-1
-2
-2
~ #jHA./;A?
2.4
~ #jHA.);D?
-3
2.6
2.8
3
3.2
3.4
3.6
3.8
ln(n)
D = {x + y,x − y},d = 2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
ln(n)
D = {x,y,2x + y, − x + 2y}d = 2
tps moy
33us*n^2,67
tps max
1053us*n^2,18
6
ln(temps en secs)
4
2
0
-2
-4
~ #"8HA./;A? D = {x + y,x − y} #hYSHA("% ? d = 0B
! #" $ &% L | ] > " $ &% (') +* &, -% log
/log
. !0 1 % 2 $ 3 45!67 08:9 ;5<* = 2.5
3
=! [email protected]( BACD
?
3.5
ln(n)
9 <* =
4
4.5
0 E =! GF
; I D  ƒ
W ƒ&~!
y „D!„ yW y Dƒ ƒ
+
yW G I yW G I ƒD€!
yW G I ƒD‚!
yG
#$ ƒD„
05 2ƒ /!
y ƒ
y ƒDƒ i; !R&`AHQ./. J} L I . HA!R!LJ #a% L D HA k HA L ; } ./')#IHA < % '/[email protected] #LHA./;D?A ')% [email protected] } ?D6%1#"!R% '/;
% 9#B % ,S1J % 9 ? ? J ($#"R%1HA'/L *c,HA&%1' #$2 {C?Z?D.)#IHA [email protected],. HQ! )') 2 *
‚ . B |Iƒ  ƒ
nL +HAR($,("($' J - L R,#"%R%1' J"; L y #$. W ,;D? J #"% L GF')`QHQ% L F#"($?DS!R%1R,("% '/?D ? }
2')! (" #"% #i!R#"%1! } [email protected] % 9 #"#?D @B?DR#F<> HOMc! L y HA! L /' L ATJ&`D?A./,@ L #
|  €  2# B 5 1JS |IƒDƒ ] ~Z|I„ -ZS '/S;D#$XJ ~A„D„D„
9 L +HQR%19S#$./#"@ M #J " L ?D9#" J #"% ; L W ?D{!0%1#"'/ L &% / <* = L dHQ!0<
!R?D J | ƒDƒD~ L
- L R,#"%R%1'C#"%*; L y HA, HQ% L 8#$($?A!R% ,(a%1'/?A ? } 2')! (" #"% #i!R#"%1! } [email protected] %>7 ?J?D @B?DR#
F<>1HNMc! '/ HA M 2'/R#$(a%1'/?A L y HA$
!
* /'E= 4 $ = '& = ( *) ,+ .- % 0/ 21TJ L ~  | ] 2~ /Q‚ L#3 'V`D#$R! 'V%1
2% 4# +HA#" J ~A„D„A„ L
- L ,#a% % ' JA; L #$. ,;D? 6J 5 L #"7. 8'/!0%1?DR?AJ ; 6L 8 ,{UH JU#a9
% L GX')`QHQ% L F#"($?DS!0<
% R,?=(akR%1#$'/(a?A%15')?D? }! L  <XRHA$Sy 2 ,{S‚ W./< '/($($?DHQS%1')?D#$ (aJ%1#$2 y ($?DL &`A#aY 2'/!R($R#"%1# ! #a%1! } [email protected] ? 7 HAS2 ("?D./,@ I
| ƒDƒDƒ
nL +HAR($,("($' JA; L #$. ,;D?A:J L GX'V`QHQ%NJ#a% L ')< ;IHQ' L F<> HOMc!8(:9UHA HA("% #$R=' ;$');
[email protected]# (". HA!R! #"!X? } 2y ')! ($RW#"% # !R#"% ! L I $S,{./')(IHQ% '/?D I L
nL +HAR($,("($' J ; L #". ,;D?A>J L GX')`QHQ%LJA#a% L '/ ;IHA' L F#"($?D!0%1R,("% '/; ($?D&`A#aY
C?D. [email protected] '/?c#$! } ?Dy @ 9W?AL =' ;"?D&%:HA. HA2M`A#$0%1'/($HA.tSI ?=kR#"("%1')?D! [email protected]? = A BC DeJ
'| /1/ ? ~ B [ .D~Z| ] . & J |IƒDƒD€
nL HA (",($("' J*; L y #". W ,;D? J L GX')`QHQ%LJ #a%E L I '/<;$HA' L #$2S' HA! ? } f?A. M&<
?D? @ '/=S#?Ze#"J !$[K[ ; R?DCJ #$R% M L} ?D%19#JR#$("?D!R% ,(a%1')?D FL GIH 2 * *6+ 3 ‚ €Dƒ ] A | ƒAƒD‚
J L ?D, } @-9UHQ2 J8 L y ,{C?D')!LJ+#a% L GX'V`QHQ% L F#"($?DS!R%1R,("% '/; ? 9 J ` B <>($?A-`A#aY {' <
2')[email protected]?=kR#"#$(a%1! ')')?D?DU!HAL.tUHQR$% %#$,R{S./!T'/($? HQ} %1')?D?D{c kRL #"("% ! } [email protected] ?IYZ'/@BHQ%1#69?DR=' ;"?D&%:HA. HQ2 `A#$R% '/($HA.
I
L K L GiHQ% 9UHA J |IƒDƒD„ L
L #$ ;A#$ L
L
L L L
}
}
. MD?DJ @ '/?A,?c! #$*-L !",[=#a% %R9M< #_ $#"./! ?A?D,e.),S#"%1% ')?D9 bN< ? } H 5tq"<>22?D' f\, #" D_#"-9% #X' HA;D.-#$!\SMZ#$!01%1H=#"@%1'/SPL ; O ,S ("%1')?DRQ ?? ("=?DeJ&0|`D.#aY [ /C. ?Q] <
 / | ƒDƒ3/
= L 9#"fSL ?A'\#"% 5 L y 1HA;&HQ L [email protected],Z%1'/S;[email protected] #$2L '/HA C?D')-% ? } H3!R'/@ ./#F #"("% '/./')#IHQ
C?D. Mc;D?A )'TP UeJ  ƒ [ ~A‚Z| ] ~A3‚ / J | ƒAƒ 
L 9R?D{UH<@L#"
% L y VIL L 8#$($?A!R% ,(a%1'/S; E,=F<>("?D&`D#aYMf?A. [email protected] ')?Z#"! } [email protected] ?D <
% 9?D;D?DHA. R?=kR#$(a%1'/?A! )'TW UeJ €Aƒ ? € B [ ~Q2‚ . ] ~A‚Aƒ J |IƒDƒDƒ
| -
L
yW G „D„
|‚
$0! %( 9 L M< L 9UHQ! ! #"kM #"% ; L ?D&%:HA&`D#"R% L9K E ,+ / 2 * 1J
(:9HA')% #  < / L EX#" @d$!XJ | ƒDƒS| L
L " L y HA {C?D,ZY L -Z, ,S ?A{.ed"@B#L2#M;[email protected] "% '/#L"./[email protected] #$&%1HA'/R# L 9 D
QE=eJ ~ [ ~AƒD‚ ] .A„  J |I‚  ‚ L
; 7L y / HA, ? HQA% LTJX `A?A?D./,#[email protected] YZ# 'V%1 J#a% L ("?D&`D#aYZ'V%1 L 2SR'/#$ #"! !R("#$% !'/?D3 S#$'V./`D.)# #$R! 2U'V%:HAHA')! #$!Z2\L N .)yIHAHAS!L! J
| .  | ] .2/A„ I
~Q„D„D„ L
yyiW G „D„
;HQ($L ($?Ay 2HA,S')1;JHQ%L%1J ? ; HJL ./'/y S#$#I.HA+W 2S,'/!R%1;DHA? J+("#"# % L O L GF ')`QHQ% L #"B2' HQK C *?D')e-J % !^? [ } 2'/!R($R#"%1J # !R#"%1! L
1~ .  3€ / ]  2‚ . ~A„D„D„
yW G ƒDƒ
;S L ?=kRy #"("#"%1. ')?DW ,! L ;D? HA#a% ! E L 8 GX'V`QHQJU% (:L9UHA8'V#$%1($R?A# !RJU% ,L (a%1'/?A ? } L($?DS#$("L % #$2 ! #a%1! } [email protected] %>7 ?
y
 | 2€ . ] | ‚D‚ | ƒDƒDƒ
ƒD!ƒ yW G I - ƒD‚ ; L y #". W ,;D? JP L GX')`QHQ%LJB L I ')<;IHAS' JC#"% W L -c?D R' L D_9#[email protected] #$2'/HA!X? } 2'/!R($R#"%1#
!R#"% ! RL )'TW ZJ 3€ / [ ~A2‚ . ] ~AƒDƒ J | ƒDƒD‚ L
"jHA ƒ2/!
L K ? * & =>+gJ-`D?A./,@ # /A‚ 2# 5 + /' QE L 9 L "jHAR2#$ = 2 ! L [email protected]{R'/2;A# 3 'V`D#" ! 'V% M I #$!R!LJ | ƒD2ƒ / L
" $# { ƒA‚
L " #${UHQ. H L D_9#i #$("?D!0%1 ,S("%1')?D^? } ("?D&`D#hY^C?D. [email protected]')?c#$! } [email protected] 9?A =' ;"?D&%:HA.fHA2
`A#$0%1'/($HA.R?=kR#$(a%1')?D! L y HA!0 /' [email protected] = )' 5 ? ? = + & /' )') 2 1J `D?D.),@ # '| /D~c| 2# B 9 5 1JS L .3/Q„ ] .3/Aƒ L -ZR'/;A#$LJ | ƒDƒD‚ L
"L" ƒ  L 9 L "jHAR2#$J#a% I L " 'V'% ;[email protected] L y ')! (" #"% #^%[email protected] ?D;D HA9 MC[ y #a%1#[email protected]')UHQ% '/?D ? }
')% # ! #"% !X{ .
M F<> HOMc! L ? &! *WQE =7 eJ .  ƒ ? € B [ ~D~  | ] ~D~A3ƒ / J | ƒAƒ  L
"L" ƒAƒ
L 9 L "jHAR2#$ #"% I L "jR')% ;[email protected] L 3 '/*-,#"#$!R!^HA2("[email protected] ./#hYS'V% M '/ 2'/!R($R#"%1#
% [email protected] ?D;D1HQ9 M L y HQ! E 8 ƒDƒ JU(:9HA')% #  JU L ‚D‚ ] ƒD„ L | ƒDƒAƒ L
"L" I ƒDƒ!
L 9 L " HAR2#"LJ L " ')% ;[email protected] J#a% L 1HQ;D#$-{C#$R; L %19S# ([email protected],Z%:HQ% '/?DUHQ.
("[email protected] [ ./#hYS'V% M J ? } #"IL ($?DS!R%1R,("% '/; . HQ%R%1')y ($#e!RI #"%1! } [email protected] %19#"'/3Y-<> HNMZ! L O AQ?=eJ
~Q„&~  / ]  | | ƒDƒAƒ
"L ‚D„
L 9 L "jHAR2#$3#"% I L ( ,.)./#" L [email protected] @ #$$N !F<>1HNMG ?D{S./#[email protected] L H< QE=7 6 TeJ ~Z| [ |  | ] |  / J | ƒD‚D„ L
" 'ƒ  Iy L HA" ! ')L% ;[email protected];XHA9 ?DL S? `c6'V'% % ;69#_#"% #" ($?DL S5t!R%1'/?DR,R'/("?A% JC'/?D$62S')? %1}#",U S!X')J %1# O . HQ%R%1 '/("K # !R #"*%1! } R')?D @ % 9 #$') BX < 12HNMc*! L
- O K :/ F 1TJZ`A?D./,[email protected]# 0| . - 2#
B 5 1JS L | ƒ ] .D~ L
-c ');D#$XJ | ƒDƒ  L
" - ƒ2.
L " V -{UHA,[email protected] #a% " L L -Z9S#$9UHQ 2 L I ')( @ Nz!+% 9#$?DR#[email protected] CL EQ?= 2 Q =,+gJ | „A„ ? ~ B [ 0| /A„ ] | €Z| J |IƒD2ƒ . L
" $ ƒDƒ
* L D_9Ad"! #2# y ?c("% ?D1H=%NJ
J L " $1HQ 2 L D) W W ; M | J 3 'V`D#$R! 'V%1 2 N ;i,S`A#$R;D#<J | ƒDƒAƒ L X
E [email protected] €1. I L L [email protected] @B#" IL 3 S! ?D.V`D#$2 ?A{./#"@B! L y HA! 6+ BW Q?= 6$(F
$ &% D+TJ L  ƒA‚ ]  ƒDƒ L ;[email protected] #$ L dHQ% 9 L -Z?c( L J | ƒD2€ . L
E 8 ƒAƒ
" L D L EF#[email protected] HAB#"%*; L>8 ,{UH Jc$2'V%1#", ! L O / * & =>+ L ') @ A,!R#$LJ | ƒDƒAƒ L
8 '/@ ‚S|
L 8 '/@ L %19S#d($#"./.),. HA6("?D&`D#aYZ'V% M? } ([email protected] ./#hYZ#$! L
? &! R
,+ QE = * 5 1J I ; < . ? € B [ €S|  ] €&2~ / J GX? ` L | ƒD‚S| L
8 8 = ‚Dƒ!
y{C?c:L 28 '/#"! . ;$} ? [email protected] J ; %19:L #"8 '/_,{UHR?=JSkR#"#$%F(a%1; '/?AL = ! ?DL . O"' L ;i HA.);D?DR ')%[email protected] K } ?A+ * #"e($J ?D[ !0%1R,("% '/;6J ($?A-`A#aL Y
 ~A3„ / ] 1~ .  | ƒD‚Dƒ
ƒZ|
y HQ  ‚ y HQ, „D„ 2
8
,{ ƒDƒ
ƒD€
F?A 3‚ /!
F?A ‚D€ FMc! /  FMc! 2€ .
- !
F" /D~!
-Z(:9 ƒ&~ -Z?D ƒ  D ? } ƒD€ ?c# ƒA€ J HA? ƒD!„ J
, ƒD€
|ƒ
; LL 8 ,{UH L F#"($?DL !0%1R,("% '/?D')L2' f\#" #$&%^(". HA!R! #"!e? } ~ O y 2')! ("K #"% #M! #a%1! L y HA!
" #"R% 1HA2KJ: ?A, ')#<JZ#"% W I #$ R?A%1?A J$2S')%1#", !XJ * ') B
2 *A- O K :/ 1TJ `D?D.),@B# |'/A€D‚ 2# B 4 5 1JC L
|'/ . ] | €2. L -ZR'/;A#$LJ | ƒDƒDƒ L
J L a%1')`c')#$#a%7G L -SHA9#"{ L #$2'/HA! HAS25("#$&%1R#$!j? } C?D. [email protected] '/?c#$! L )'T GeJ /  [ |  / ] |I‚S| J | ƒDƒD€ L
L 8?D! # L y #')% '/?D! ? } ($?D&`D#hYZ')% M HA2 ("?D&`D#hYe9-,.)./! '/ 2'/;A')%:HQ.'/@BHA;D#"! L WQE= <*- 1TJ .  ? ~ B [  | ] ‚3/ J | ƒD‚2/ L
L F?A! # L ;r!0%1R?D; (:9?A 2R?DC#$R% M } ?D  <>("?D#"("% #$2 ($?A-`A#aY2'/;D'V%:HA._!R#"% ! L
U K = 2 *
eJ .3/ [ 2~ /Aƒ ] ~A€Dƒ J | ƒD‚D€ L
E L 9 L Mc! #" L [email protected]{'/HQ%1?DR' HA.f ?AL f#"R%1')#$!8? } @ HQ% ')($#$!_? } ;$#" ?c#$!XHA2 ?D#$! L ( H QE =eJ ƒ [ .  | ] . D J | 3ƒ / 
E;[email protected] 9 #"L '/($FH McJ ! #" L L D 1*)HA2,S("%1 ')?D } 1 HQQE HA')= ! #g2<J d H=%19L "@ dHQ%HQ'/%1*-9S,#$#"@B!_HQ($%1?D')@b(IHA{S.F'/U;iHQ!R% ! ?D?c'/($R'/#$HQ! %1U')?DHA ? } L
I
|
D
ƒ
2€ .
[email protected] '/?A J | ƒA€Dƒ J y ,S?Z2 L
; L F" MZ' L 5S ?=kR#"("%1')?D! ? } R?D{UHA{')./'V% M 2S'/!R% '){,S%1')?D! L SQE= D*eJ . [ | .S| ] |  ~ J | 2ƒ /D~ L
W L -Z(:9&7 HAR%; L ,+ 1JS`A?D./,[email protected]# J(:9UHA'V%1R# . L ~ JS L /D~ L EF#$ @BHAKJ | ƒDƒ&~ L
L L
L
2W #j-Z./HA?D, R '$HQ%N J #"3 -% 'V?D`D')#$2R#i! 'V2S%:HJ'f,2^W' 5tC')?D .)#$'/[email protected];$#<'/J | ?iƒD($ƒ ?D &L `D#"! !R?B.),;D? 2,#2S'/ T# ;$')?D' [email protected] ?D'/R#
Iy L ?c("D % ? ?D} 1% H=L %NJA? D = #$(:9 S'/ (I HQ. 3 ? ')&`A#$!R! ')')% Me*? } y ? #$= @B HA+ @ J L L D_9Ad$!R#2#
| ƒAƒD€
" L E L ?c#$;D');D#$ L D_9S#b #"($?D!0%1R,("% '/?DM? } f?D. [email protected] '/S?Z#"! } [email protected] 9?DR=' ;$?A-%1HA. HA2
`A#$R% '/($HA. ? kR#$("% '/?DS! L D #$(:9')(IHA. R#$C?DR% -5 < 3€ / J D 3 "j H ;<6J " 1H ;<J | ƒAƒD€ L
5 L 5 L J HQ? L [email protected] ,S%:H=%1'/?AUHA. ;D#[email protected] #"% kM L y HA! 9 L ` L W #"#$,&7 #" J$2S')%1#",L
J
) & /' = 5 1J `D?D.),@ #*; ;i.);D?DR')%[email protected]! HA*
2
[email protected] .)#aYZ')% M< J
(:9UHQ')% #  JU L .  . ] .D‚Aƒ L ./!R#"`c'/#"LJ |IƒDƒD„ L
L J , L ? = 2 = > = ) = + = 2 = ) L D #$(:9S'/!R(:9# 3 'V`D#$R! 'V%1#$ 'V% ')29? `D#" J '/S2Z< 9? `A#$ L J |IƒDƒD€ L y '/!R! #"R%:H=%1'/?A J D #"(:9'/!R(:9# 3 'V`D#" ! 'V%1#"')% ')29? `A#$ J ')29? `D#" J
| ƒAƒD€
|'/A„
$0! %( J ~Z|
A
;- I J J .A.D€ €
')1HASf?D0%NJ ƒS|
J ‚Dƒ
C
?DS#aYZ')% <J | 
?D&`A#aYZ')% << ($($?D?D&&`A`A#a#aYZYZ')')%% <<JJ |0.
2
0.
|
k
E,=F<>("?D&`D#aYZ'V%1<J
UHA} _?AR*-%1,U#gHQJ 21HA&% !LJ | ~  
.D€
!--#"< ./($?D?D &`D-Z#h?AYZ ') %1' gJ J € €  
! ')@BS./#<J
%1?Q%:HA.)#<J |1~ . .
J /Qƒ
d
J') #"("€D%1„ ')?D J
d
y 2'/HA;D?DUHQ~Z./| #<J
yy ')R!R?D%1')HA% # ($#j ?.)'/kR#$$("HA% ')'/€D`AR#<„#<J J /2/
ƒS|
J.
E(α,β)
');D1HAS9#<J  ƒ
J„
(F ,D)
5U?D(a%1')?D!
($?A-`A#aYZ#$!XJ &
!R% ')("%1#"@B#"&%8($?D&`D#hYZ#$!LJ /A„
EF=X<>("?D&`D#hYS'V%1gJ | 
>!R$($HA{'/.)')% <J .A„
J ~D~
L
D
D
pq
1
∞
i
" 2' HAKJ /2/
f"1HQ% '/?D^2S# ("[email protected] ./a%1'/?A J  .
I ?A'/#"'/2&%J ~ 
I @B"2' HQ J
/2/
!1HA')././HA&%NJ
I ?A'/&%8#aYc%[email protected] HA~ . J | 
I R?A#$. !MD*-?A,@B#h')<>2?A#"@BJ | ' <>./HA J .A€
I ?=kR#"("% '/?D!XJ /A€
J‚
Q(D)
Q(p,q)
gX<>("?D&`DJ #aYZ .'V%1<J ~  J .2.
} ?DR% #<J .A€
J.
QCON V F ORT (E))
J
~
1
/
QCON V (E)
J | ~Z|
QF (D)
J | ~Z|
(F ,D)
J |D| 
(F
,p,q) J | ~A€
(F ,D)
J
!

‚
(F
,D)
8#$($?A!R% ,(a%1'/?A J  „
J
" $# 
„
(F ,D)
8$! #$HA, J ~D~
J~
R (M )
-SHA')././HA&%NJ 2~ / J  ~
@b,.V%1'/S./'/("')% <J ~A€
*-,UHA2R<J ~A€ }
*-,UJ HA2R$!F($?D!R$(",S%1' !LJ ~A€
/
S (k)
J/
S (M )
[email protected] @ #"%NJ | 
-Z?D,S!0<>R$!R#IHA, J ~D~
-Z,Sf?D0%1#"LJ | „2.
h·,·i
i
i
i
|'/Z|
'| /D~
-c78')% (:9'/;=<>([email protected]?D#"-%LJ ‚Dƒ
D HA! } ?A @BHQ%1')?D
.)S'/?=$kRHA#"'/("R%1#<'VJ `DƒS#g|J
ƒZ|
J ‚Dƒ
(F ,D)
= ?D'/!R'/UHA;A#<J /Q€
J /A€
X E(i)
p
|'/1.
!" #%$& '()*"+,-./.0
1324658792;:
=[ ]
] ] ] C Y Y ] R/Y\[
] M YHNO6PRQ 9 S9UT [LR/Y = [LR Y ]
<[=
MD W= I /M YY_bX a`F h?R `F > ^`_ [_ba I a`h7a _ RT>] [email protected] aBZ;A [ a I a]a \V> Y ] ^ aHa*a R_a ^ [ >AE]YD = I M/A`Y >_a F M aH_ GY IKR JLFD ] >`D @ R/aHY @ @ a a?AED Rba a]\V> ] F%@ @V> Y A`_bI ^ a ] adI ceI [ @VAE> D RbA`_ba a a]\V>Z] F
Y p'> h q N= O6IPRa Q Y afI`D R A Y a`] _ ]\[R > R a Y @VM > R _baEA R] >`D MvR/a Yhg I [ AED [NRbYbaY a]\V> ]M F X ] D R]\[ I aj[ ikIY [ AED Rb[?a a]^ \V> Y ] F M/Y Y ] M af]\Il[ I [email protected] M RTAE] D RT] >RnRna Mo]g aY ] A ] Z < [ Y
9S6rksut
= D6IlyzND h?> R D Y?{Ba | W a AED AERnDNa _ba]a \ a Y Zda } R D F ] afM [email protected] aw> a%AfII a Y IlaY> a x R/Y > a RTF ] _bF`a _ ] IlD > M a] a GY I A a Y [email protected]> >~€I aF`_ba RTD ] a Y I a`R h7D a] >`D R/AEYhDg a ] a]D\ R a @va D R I R a a
@Yk[ a Y[L= RT_D]bY = _ >`F ] F Y @ a YY= D6> Rg]bYY= [ _ ] >`A M Il> a`_ Y @ a‚A a Y a R/Y a`h S I a Y AEND hHh7a%I a Y = D6> Rg]bY h‚[email protected]> [LR/Y a ] I a Y = D6> RT]bY
>lIlI Z
ƒ D M ] aI [ @ a M \„> ^`h7a = [ _ ] > a,a Y ] @[email protected]>`F a [ M†= _D S I ^`[email protected] a I [ ] ND h‚D X _ [ =vC > [email protected] [LR/Y I a = I [LR @V> Y `A _ba ] Z Z
Y [ X ] Y = RT] R/Y ] M M R R/Y S = [ ] M R S = RT]bY [LR/Y Y ] Y
=[ I [ _ [J IlI ^f> I a Y > h [email protected] I a a`[email protected] > _baEA @ ] a >`D _bR/aEY AED @vD R R_ F >a _bY a Z < J [ aI X ND _ a`> h ] C h7I a‚a G = D6Ily _ R > ND _ [email protected] > [ I g ND @vh Fˆ‡G‰_ba?AED @ R a RnMŠD6= > D M [email protected] I a Y= I D6a IlyzND @ h?_D6> >R Da Y
{B| W AED Rba a]\ a Y [Ea aEABI a Y @V> _baEA ] >`D R/Y‹C ND _ >lŒhD Rg]\[ I a Y a ]8a a`_ ] >`A [ I a Yg<Y a X F R `F _ [ Il> Y a [ M \Ž W AED Rba a]\ a Y'= D M _‹@ a Y
@V> _baEA ]] >`D R/R/YY Y x MVM ’ afI`Y AED RTR ];x = M a M Y Z! J ] [ M ] _ba R = [ _ M ]hRFg I] a ]\] [ C FEND _b^`Rgh7] [email protected]€RnJ Ma R >`Af> ] Y F‘X x M R > h‚[ D RTY ] _ba[ M/a YbRWY = [ [M _ ] >`A M W Il> a`Rb_?a x M a Y Y a = R ]
@VR > _baEA >`DM ] M a Y M D [email protected] F [*a`[N_ Y\h?Y > a`_ D ] I a`R/h7Yda = M AED [ha a]\ a M R a F ] `F _ Il> [*a Y M ] > Reg \w[ Ž AED R/Y a]] \ Ma Z] } R
[email protected] a [email protected]] FED @ ]\[ > I a`h7x a a‹RT] I`ND AE_ D Rbx a a]a'\ IKa JLDY“= a M ] Y aafŒ.^ [ @ > a._ba @Va > _bR7aEA] a`>`h D =/Yd= D D6Ily _ R ND h?D6> > _ [ I >`AfR > [[email protected][ M/a'YbI Y > M D6R I [ I >`X D ND _ > ]I C h7_baEa AED = D6Ily _ R ND A h?>`D> [ I
ZV}
@ aF_baEAED R/Y ] _ M A ] >`D R•”U–b– ; ON]— /t 9 @ a Y Ž W AED Rba a]\ a Y Z
˜W™u46šv™K›‹œu™u“ž:eŸ I X ND _ > ] C h?>`x M a
¢¡€£z4h¤EšvœK24B:!¥ D Rna a]\„> ] [email protected]> Y `A _b^ ] a g ] ND h‚D X _ [ =vC > a@V> Y `A _b^ ] a g _baEAED R/Y ] _ M A ] >`D R @€J > h [ X a Y S > R [ > _ba Y
¦§/¨.¡e©z§"£ª¡;™K©zž4: <e< Ÿ [ ¥« g [­¬ ƒ ¥ I a`_ h‚D RT] Wu® a`_ _ [LR @ g/¯'°“±ª²‚²ª³ «´ U Ÿ M S > ^`_ba ¥ [email protected] a]\
< [ Ÿ ® Ÿ g ¬ R > a a`_ Y > ] F  a R > Y  >`@ a`_D ]hg ´6µ U µV« °K[ _ > Y ¥ [email protected] a]\†¶ µ
2
n
2
Y·n¸¹»º¼-·†½¾#$&ww!.)º¼!" ¿‘+,-."w0¸
À ¨‹4Á£6©b§/š„£v:
ƒ C a‚/ _ Yb]= [ _ ] D ^ ] C a ] C a Y > Y > Y @ [email protected]>`A [L] [email protected] ] D ] C amAED Rna a]\„> ] yW> R] C a@V> Y A`_ba ] a = I [LR a Z DN_,h‚DN_ba
X a R a`_ [ IlIly Z Zv[ RÃ^ [ A ] ] C a`_baFa]\„> Yb] h [LR y R D ] >`D R/Y D ^ @V> Y A`_ba ] aAED Rba a]\V> ] yHc ] C a Y > h = I aYAED Rba a]\V> ] y [ I`D RvX
Y ND W h7a Rn= a _ba Y `A] _ > S [email protected]‘[ @VR > _baEA ] >`D [NR/YbYhY g ] C^ a ] RbD ]\a [ I€AE] D Rba a]C \V> ] CÅy_Xi ] C R a M/[ Y M [ I/Y,AESD Rba] C a]\V] > C] y> ] R ]\] [ C aHAED Rg]Rb> a RbM D M/Y YUo]]bgYFa [L] R A Z ƒ ] CC a
{BŽ | AEWD AED Rba]a \„a]> \ y_= > D6IlyzND h?af> ļR D AfY I Z‹Æ a D Y ]AEM D @Vy a]\V] C > ajy3IlÄ > RvÇ<>`Y A S a ]a >Ä a`a _a R Il>lŒ [ a IlI ] C D a Y a R D a ] >`D D R/Yhg5IlIly_[LR [email protected] D ] C a]a \ = _a D = a`_ ] > @ a Y D a ^
Y= aEAf> R [ I [ = D6] > CRT]bY Y D K^ ] R C a Y = a [ Y a ] ]bY Y M [ A (C ] CRg]a h7aEY ]@V> [LRÈR = D6] C > RT]bY [L[ R [email protected] = ] C a S Y\[ Il> a RT]^ = D6Y > Rg]bY ] Z ] X [ =vC R/Y W
] _ M A [ ] ]> Rv:X R/IlI Y [(ƒYaM C S/aEY aAE= ]D D @ ^ R Z _ ^ [ _Ä ND ha[ X ] _bC aa >] C RnM a`_bh a [ S a`[email protected][ D > ^ > Çg]bRYa = h D6R‚> Rg^> ]bY > ] R _C D a [ {BI a`A h|C W IlD > R Rna @Va > = `A[ _b_ =a [ IlaI afI ND ] h‚D D R Y ND _ h7Y>a [L=R y_b<ca Y_b] `AaEC _AE> D S C [email protected] W
@V> _baERgA ]\>`[ D [LR Z a a ] D6Il[ y ND h?> ] I R/I Y ND _ [L> RHSh X I _bR a @V[ y DÄ ] ND _ Ç a ] C AEW D Rnaa]\ Y D6Il]byzY ND h?[LR > <D [La R @ ] a R/D Y
_ >lŒhR(D [LR D I ] C a`[email protected] C [La`R_ @ >`A ] C Ia @VM> _bR aE>`A x M>`D a R a Y\A Y _ba YM a I ] a Ä Ca`_>`A C(Il>lŒ Y aEC @ DhÄ DHY"ċ] C ND [L_ ] ´Ä.>@V> _baEŽ A ] >`AED D R/Y a][ \_ba aY MV’ Af> [email protected] RT]5] [email protected]@ > a _b]aEa`A _ h?>`D > R a Z
}
AEND h M = I a ] ]afC Ily [L] [ ] D C ]\[ RIlIly‚] C AED Rba [a]\ Y a ]ÉR ^ _MvND XªhÊC > ]bY“= _] D ‡ aER/A Y ] >`] D R/Y C A[h[La R [M I R Y D MS a R X a Y\R Y a`] _ C [ Il>lŒ [email protected] ] R/DwY ]Ž WM AED ] Rba R a]\ Y^ a ] ]bY ]\Zª[ Æ a
@ [email protected] a A a Y ]bY Ä [LRa S a`[ _ba _bR_aHa= D R @V>[ _baEA ] >`D D [ Y a C [ha >`x [‘a = a R a _baE[ AED [ X _ ]A C >`D ] D R D IlIly W
AED Rna a]\ Y a ]bY'^ A ] a C h @[ =va?= > D6Ily [L] ND a h?> = I > h7] a Z€R/Æ Y a I D a D6Ily ND h?> I I ND _ > h D‘Â @ˎ
AED a]\ a _ND h af> _ _DE\V> h > a _D ‡ aEA >`D Z
Ì˞vÍÏΑ¡e©bГ4„:  > Y `A _ba ] aYAED Rba a]\V> ] y g  > Y `A _ba ] a ƒ ND h‚D X _ [ =vC y g¯ > R [ _ y [ h [ X a Ñ aEAED R/Y ] _ M A ] >`D R
2
n
2