close

Вход

Забыли?

вход по аккаунту

- Администрация Мичуринского Сельского Поселения;docx

код для вставкиСкачать
!∀#∃∀%&∀#∋∀(∀)
&
∀∃)∗+,,−./
&&0&
01
%
2&)
3∀−∗. 45 1((2,+4+,6
7
0,,,+ 8+
9
White Rose Consortium ePrints Repository
http://eprints.whiterose.ac.uk/
This is an author produced version of a paper published in International Journal
for Numerical Methods in Fluids.
White Rose Repository URL for this paper:
http://eprints.whiterose.ac.uk/1756/
Published paper
Gaskell, P.H., Jimack, P.K., Sellier, M. and Thompson, H.M. (2004) Efficient and
accurate time adaptive multigrid simulations of droplet spreading. International
Journal for Numerical Methods in Fluids, 45 (11). pp. 1161-1186.
White Rose Consortium ePrints Repository
[email protected]
$&%('*)+,).-
/1032546)7825':9,);< />[email protected]?BACBDBDBD1EGFGF.HJILK>M
!"#
NO42QP>RS42L?T/1AL=U%>VXWYSZS[\:]1^ _a` bc5dG254&AL=:eB%gfh>iijhakliBmakgnSoqpjn)UiSnQr
sutwvx5yz!{}|3zq~|€vv‚ƒ|„{y…{x5†‡yˆ|€~8|€‰{ƒx6Š‹yˆ†Œqa{x6Ž‚x~’‘x5†“q|„{ƒx”zq‘•”3–
~‚”3‰86y{—‘‰q‚y|3~qxzŽ
˜3™š›™œ#žBŸ# G¡¡ I¢¤£ ˜€™¥™¦§¨©#ªSŸ C.£«
™¬ G¡¡§* (­ I
#®¯°š›™
«
™±8²O³¨´žB³®
I
µ€¶ %>VS=U%G2L264&=:%>V;< /1=:?‹7·2¹¸a91RB%([email protected]¸6Aº25AL2LRB4Q¸&9¼»O4eB/P,½¾l¸&9,e6eB<e¹¿7·2L¸&91RS%(=:¸LRB< ¶ %>VB=:%G2¹254&=U%>Vj½
À ¾l¸a91e6eS<eQ¿ÁƒeB0Pg/>'Â=U%>Vj½
ä9128Ä%(=Upj264&AL=U'ÂŋeQ¿Æ¤2L2L?BAQ½ƒÆ¤2L2L?BAÇÆ#¾ghÈmB+lÃ.½„Ä.É3)
ÊGˋ̼Ì̈́ÎÏ
̀ÐqѐÒÓjÔÕÑjÐ(Ö
פØlÙUÙ.̀ڤÚgÛaÜBÝlÔÕÞ8ßSÖaÔUÜ>зÊGÖaÜÛ6ß>à1ÑXá×G͏ÊlâÌãؤÙJÖaÔÕà>ÛaÔUä·ß>ÙÕà1ÜÛaÔJÖaå¤ÞæÔÕçؤç&ÑBäÖaÜ!ç&Ü1ÙÕè,Ñ3ߋÛ6ßФà>рÜéäGÛaÜ1Ú¤ÙÕѐÖ
ç&ÚlÛaÑBß>älÔÕФà·ê.ÜBëìç‹ÞqÜGälѐÙUÙÕÑBäíß>ç!ß·ÓjÜ1ØlÚ¤ÙÕÑBäîç&ѐÖ!Ü>éФÜ>ФÙÕÔÕФÑBßSÛ!ÙUØlïlÛaÔÕÓBßSÖaÔUÜ>ÐðÑBñ(ؤßÖaÔÕÜ1ÐlçjòóålÑßÙUà>Ü>ÛaÔJÖaå¤ÞôÔÕçé@ؤÙÕÙJõ
ÔÕÞqÚ¤ÙÕÔÕÓjÔJ֋ß>Фäíå¤ß>çÑÞÈï#ÑBälälÑBäëìÔJÖaålÔUÐíÔJÖ!ß>Лß1älß>ÚgÖaÔUè1ÑXÖaÔÕÞqѐöçLÖaÑjÚlÚ¤ÔÕФàqç&Ó6å¤ÑjÞqÑ!Öaå.ßS֋ÑjФß>ï¤ÙÕÑjç€Öaå¤ÑXçaß>ÞqÑ!ÖaÜï#Ñ
Ü1ÚgÖaÔUÞqÔÕç&ÑBä¼ÔÕÐí߷ӐÜ1Ð(Ö&ÛaÜ1ÙÕÙÕÑBäîÞ8ß>ФÐlѐۀç&ؤïG÷LÑjӐÖ3ÖaÜãß8ç&Ú#ÑjÓjÔJø.ÓXѐÛ&ÛaÜۏÖaÜ1ÙÕѐÛ6ßФÓjÑ1òƒóålÑÈÞqѐÖaålÜGäÔUç€ø¤ÛaçL֏è>ß>ÙÕÔUä¤ßSÖaÑBä
ß>à1ß>ÔÕФçLÖìߏÛ6ßФà1фÜ>éƒßÐ.ß>ÙJõGÖaÔÕÓBß>ÙßÐ.äѐÝgÔÕçLÖaÔÕФà!ÐGؤÞqѐÛaÔÕÓBßÙ¤ÚgÛaÑBälÔÕӐÖaÔÕÜ1ÐlçÓjÜ>ÞqÞqÑjФç&ØgÛ6ßÖaÑëìÔJÖaå·ägÛaÜ>Ú¤ÙÕѐÖç&ÚgÛaÑBß1ägÔÕФà
ß>ФäðÖaå¤ÑjÐùØlç&ÑBä›Öaܼç&ÔÕÞÈØlÙUßÖaÑ·ßç&ѐÛaÔÕÑjçÜéìФѐë‹úÖaågÛaÑjѐöQälÔÕÞqÑjÐlç&ÔÕÜ1Ð.ßÙê¤ÜBëìçÈÓjÜ>Фç&ÔÕçLÖaÔÕФàîÜ>é€ägÛaÜ>Ú¤ÙÕѐÖTÞqÜ>ÖaÔÕÜ1ÐðÜ1Ð
ç&ؤïlçLÖ&Û6ßÖaÑjçXÓjÜ1Ð(Ö6ßÔUÐlÔÕФà¼ÖaÜ1Ú#Ü>à>Û6ß>Úlå¤ÔÕÓß>Ð.äðëѐÖ&ÖaÔÕФàîå¤Ñ5ÖaѐÛaÜ1à>ÑjФÑjÔJÖaÔÕÑjçjòóålÑ·ÙUßSÖ&ÖaѐÛqßSÛaÑ·Üé„Ú.ßÛ&ÖaÔÕÓjØlÙUßÛÔÕÐ(ÖaѐÛaÑjçLÖ
ß>ФäÛaѐè,ÑBß>Ù
ålÜBëûägÛaÜ>Ú¤ÙÕѐÖaçÓBßÐ"ï#ÑãÞ8ß1ägÑqÖaܛç&ÚgÛaÑBß1ä"ÚlÛaѐé@ѐÛaÑjÐ(ÖaÔUß>ÙÕÙJõ"Ü1Щç&ؤïlçLÖ&Û6ßÖaÑjçÜBëìÔÕФàîÖaܛß>ЩÔUÐ(ÖaѐÛaÚlÙUßBõ
ï#ѐֹëÑjÑjÐîägÔÕüÑ5ÛaÑjÐ(քÖaÜ>Ú#Ü1àÛ6ß>Ú¤ålÔÕÓÈßÐ.äç&ØlÛ&éÂß>ÓjÑ!ëÑ5Ö&ÖaÔUÐlà8Ó5å¤ßÛ6ß>Ó5ÖaѐÛaÔÕçLÖaÔUӐçjòýÜ>ÚGõGÛaÔÕà1å(Ö
ÊgÜ>Фçjú
Ö6äò
Ó
þ
ÿ Ü>å¤
Ð °ÔÕÙÕѐõ
äGÛaÜ1Ú¤ÙÕѐÖaç1
ç&ÚlÛaÑBß>älÔÕФà,ÞTؤÙJÖaÔÕà>ÛaÔUä>ß1älß>ÚgÖaÔUè1ÑÖaÔUÞqÑ5öQçLÖaѐڤڤÔÕÐl
à ÙÕؤïlÛaÔÕÓBßSÖaÔÕÜ1Ð8ß>ÚlÚlÛaÜBÝgÔÕÞ8ßÖaÔÕÜ1Ð>é@ؤÙÕÙJõ
ÔÕÞqÚ¤ÙÕÔÕÓjÔJÖBò
"!#%$'&)()*)+,# -&)!
# .0/)1324657%281:9-.0;<7%=>=/?%;@7 9A.0/B/6C46;@/ED
9'DF50G/E?A;<4E>[email protected];J2DK21ML2N'7 ;@[email protected];JDPRQ?A2=HJ/E9%7-=0?A/>Q;JDP
9A.0?-250P.S9A.0/T57-/T281>[email protected]
WX;@G=0HJ;<46;J9YZ9-;@G/6W[>Q0>=9-;@O/T\]50H^9A;JP?-;<Q_>[email protected]?-;J9-.G`a54.bL2N'7]>?-/c21
/D02?AG257d7-;JPD0;Je4E>D46/f;@DSGg>D
Vih0?>8D4.0/7j281g7-4E;J/D46/c>DQb462GG/E?46/Y%;JDQ57k9-?A;@>Hl>8=0=HJ;<4E>89-;@2D7
;@[email protected];@D0PYm132?g/6n0>GR=HJ/Y:9A.0/]Q/=2
7k;J9-;@2Do281)4E2>89-;@D0P7>8DQp;@D0q7EYrQ0;J?A/469=>9k9A/E?AD0;@D0Ps211350D4t9-;@2D>8H
H<>uV/?A7cQ5?-;@D0PvG;<46?A27-.0;J=w=0?A2Q5469-;@2DIYR7k=0?A/>Q;@D0P_281d=x/7k9-;<46;<Q/7o>8DQy9A.0/zL2N{281d2;JHK;JD|.0/>9
/En4.>8D0P/E?7l}J8YI~u€# ./‚7A>8G/R>[email protected]ƒ>?-;<7k/‚;JDs289A.0/E?BQ;@O/E?7-/‚>8?A/>7'21„7-4E;J/D46/R?A>D0P;@D0Pg13?A2G…h;J2HJ2PVY
N%.0/?-/)9A.0/EVg132?-G†GR/Glh0?>8D/7,2D`Gg>8GGg>[email protected];@>[email protected]
7%>8DQƒ9-/>8?,[email protected]';JDK9-.0/‡/V/YF9-2P/[email protected]%.0/?-/
9A.0/EVƒ13/>9A50?A/);@[email protected]>uO>‚L2N'7B} ˆ€
‰ [email protected]/EGg7Š>8?A/'G2FQ0/[email protected]/Qg57-;JD0P‚>[email protected] >uO/Y2?‹ŒŽEt’‘“”X’•–Y>=0=0?A2unF;@Gg>9A;J2DRN%;J9-.9-.0/B>QO>D9>8P/
9A.>9j=0?A2F4E/E/Q;JDPo;@Dz9A.0;@7jN >uVU?-/Q54E/7ƒ9-.0/cQ;@G/ED7-;J2D>[email protected];J9"V—hFV˜2D0/™>DQzVF;@/EH<Q07`G2?A/]9-?>469A>h0HJ/
š œ
›tžXŸ" X¡8›t¢u£Ÿk¢¤kŸj¥ž,
› HŠ¦§ r§ ¨m© žªtŸ"«¬«<­x¢®t¯¬¢uŸ"Ÿkž¯¬¢u®©«¬°¯¬£ BŸ"¤€±u©¢u¯¬¤" ›IŸ" XŸk©X¤X±U¨²ž›t°u¡
­¤€±u›E›t«)›³8BŸ"¤€±u©¢u¯¬¤k©«
x¢u®t¯¬¢ŸkŸ"ž¯¬¢u®u­(I±uŸì:¢¯ ´6Ÿkž X¯ ¥¶µ'›³FŸ"Ÿ"£u -­,FŸ"Ÿk£ 1C%·BB¸(
­ ²¹,§Ÿ[º¶»'©¯¬Â« HE¡
§ ±
§ ®6© XªtŸ"«¬«^¼²Ÿ[³J»B§ «¬ŸkŸ"£u k§ ©¤t§ °uª
ýÜ1Ú(õGÛaÔÕà1å(Ö
þÓ
ÿE‡ Ü1å¤Ð‚°ÔÕÙUÑ5õ
ÊGÜ1Фçjú
Ö6äò
½r¾[¿k¾-À<Á¾[ÂlÃ8ÄgÅr¿AƒÇuÈk¾-É'ÃÊuÊEÃ
½r¾-ÁtÀ<ËA¾[ÂÃÌlÍuÎ8Ï0¾%ÃÊuÊÐ
~
¦§,„§8¨
¹¤x­¦0§¹,(
§ B
œ¹,­(R§(>.¤*¸(
­ „§,R§,B¦,
13250?k9A.W€2?Q/E?=>8?-9-;<>8H Q; /E?A/ED
9A;@>H„/ 5>89-;@2D73=M Qœ / 7B
9-.>8Dp/D46250D
9-/?-/Qf;JDp>][email protected] !Z>uOF;J/?kW"a
9-2q/7
132?-G‚[email protected]>89-;@2DI(/7k=0;J9-/r9-.0;<7M7-;[email protected];Je4E>89-;@2DIY7k;@PD0;Je4>8D
9I2h7k9A>[email protected]/7?A/EGg>8;@DB9-2%hx/r2O/E?462G/Yt=0?-;@D4E;J=>[email protected]
Q5/9-2ƒ9-.0/R7"9A; D0/7-7Z;JD
9A?-2Q54E/QdhFVj7k5?k1’>46/‡9A/ED7-;J2D >8DQ`9-.0/RD0/E/Qj9-2ƒ?-/7k2HJO/7k.02?k9ZHJ/D0P89A. 7-4>[email protected]/7
4EHJ2
7k/Z9-2N,/69k9A;[email protected];@D0/7aF;@G;JH<>8?AHJVYF/[email protected];@4E;^9%9A;JG/‡7"9A/E=0=;JD0PgDF50G/?-;<4E>H7A4.0/GR/7 >8?A/Z21²[email protected];JG;J9-/QƒO>[email protected]/
7-;JD46/)9-.0/VK7-
5 /E?'>8Dj;@G=0?A>4t9A;@4>8Hœ?A/7k9-?A;@469-;@2DK>=0?A2=x27Š9-.0/B9A;JG/7k9-/=I
Y tY0N%.0;<4.KG‚57k9'h/>89'GR2
7"9
tYN%.0/E?A
/ |;<7 9-.0/7-=>89-;<>8Hœ?A/[email protected];@2DIY
9A2>uO2;<QDF50G/?-;<4E>Hœ;JD7k9A>h0;@HJ;J9"V } 8€
aF/EO/E?>8H >59-.2?7˜.>uO/[email protected]/Q†9A.0/v9A;JG/EW€Q/=/DQ/D9SHJ5h0?-;<4E>89-;@2D / 5>89-;@2D7 7k5446/[email protected] ;JD
>yN%;<Q/bOu>?-;@/69"V†281s=.
V7-;@4>8Hd462D
9-/6nF97E ‰ ?-2GR;@D0/D9˜>8G2D0P 9A.0/7-/_.>uO/ihx/E/D†9A.0/v7"9A5Q;@/7U21
a4.FN >8?-9 R>8DQs4628W€N,2?Aq/E?7'N%.2K.>uO/l/>7-/Qd9A.0/g7k/O/?-/‚?-/7"9A?-;<4t9A;J2D 2D 9-;@G/7k9-/=sD289-/Q >h2O/‚hFV
/GR=HJ2VF;@D0PR>H^9A/E?AD>9A;JDP8W[Q;J?A/469-;@2DK;[email protected];@4E;^
9 (Z %>[email protected]?-;J9-.G7,N%.0;<4.IYF132? 9-.0?A/E/EW€Q;@G/ED7-;@2D>HL2N'7EY
57-/f>8HJ9-/E?AD>89-;@D0Po7-NŠ//E=7K;@D />4. Q;@?-/4t9A;J2Di7-2T9-.>[email protected]>Th>DQ/Q 7-VF7k9-/G 281l
/ 5>9A;J2D7ƒD/E/Q
hx/[email protected]/Q™>89l/>4.f9-;@G/K7k9-/E=z} u€m# .0/7-/ƒ7A4.0/EG/7YI21¶9-/EDp?-/E13/E?A?-/Q 9A2s>7g”ž "!#‹¶”X”ž¶–%$8Y } &8XY².>uO/
hx/E/D˜57k/Qo9-2T>8D>[email protected]/d>f?A>D0P/j21)=0?A2hHJ/G7;JD4EHJ5Q;JDPY¸132?/6n0>[email protected]/Yr9-.02
7k/]Q5/j9-2f9-./]'/ /469A7
21 7-50h7k9-?>9A/4E50?AOu>89-50?A/g>8DQf7-50?-1’>46/R9A/ED7-;J2DfP?>Q;@/ED
9A7)2Ds9A.0/9-.;JD0D;JD0P 281,4E2>89-;@D0P7g})(€[email protected]/69
G289A;J2D7I2D‡.0/69A/E?A2P/ED0/257œ7-50h7k9-?>9A/7IN%;J9-.‡Gg>89-/E?A;<>8H<7I.>uOF;@D0P%N%;@Q0/[email protected])Q;*/E?A/ED
9:/5;[email protected];Jh?-;@50Gb462D
9>4t9
>[email protected]/7)} +0%
Y , œ>DQP?A>uOF;J9"V
W:>8DQK7k5?k1’>46/7-.0/>?,7k9-?A/7A7kW€Q?A;@O/ED9A.0;@DK462
>9A;JD0PlL2N'7}J-8X
[DK>HJH21œ9-.0/7k/
7k9-5Q0;J/7EY07-=0?A/>Q0;JD0PRG289A;J2DK;<7,1’>4E;[email protected];^9>9A/Q`hFVƒ7k=x/4E;^13VF;@D0Pg>‚9-.0;@Dj/ED0/?-P/69-;<4E>[email protected]>8hHJ/BN,/69k9A;JDPRH<>uV/?Y
2.
? #x ‘EŒ !E•
 /„0‹ ‚YŠ281B9-.;@4qFD0/7-
7 13d
2
2O/E?9-.0/ 7-50?-1’>4E/]281B9-./ 7k5h7"9A?A>89-/YŠ>8Dz>=0=0?A2>4.p9A.>9`.>7
7-;JPD0;Je4E>D9'4E2G=059>9-;@2D>8HI>Q0Ou>D
9A>8P/7Š2O/? >H^9A/E?AD>9A;JO/Y07-HJ;@=`GR2Q/[email protected]}^8YI~uX
[Dg462D
9A?A>7"9m9-2B9A;JG/EW€[email protected];^9-9-;@D0PY
\d50HJ9-;@P?A;@QRG/69A.02Q07r>8?A/ 1’>7k9„[email protected]/E?7Y2?-;@P;@D>[email protected]/O/[email protected]=x/Ql132?r9-.0/
[email protected];@2D‡210/[email protected];@=9A;@4Š=I Qœ / 7u# .0/E;@?M/7-7-/ED4E/„;<7œ9A257-/Š>'7-;@GR=HJ/,;^9A/E?>9A;JO/¸9-/4.0D04; 5/„>7²>'7-GR2F29-.0/?YD0289
>7²>)[email protected]/E?Y2D‚>)7-/
50/ED46/,2814E2G=059>9-;@2D>8HFP?A;@Q7I9A2)?A/Q0546/,.0;@P.13?A/ 50/D46V‡/?-?A2?m462G=2D0/ED
9A7²2D
>=>8?-9-;<4650H<>8?²P?A;<Q}^ˆu€u# ./EVB.>uO/Šh//EDR>[email protected];@/Q‡9A2;@D46?A/>7k;@D0PHJVBQ;JCg4E50HJ9¸7-V7"9A/EGg7:281=I Qœ / 7m>8DQ>?-/
[email protected]/Q‡[email protected]/„9-.0/ Q;<7-4E?-/E9-/Š>D>[email protected]/7:>8?A;@7-;@D0P%13?A2Gv>'N%;<Q/,O>8?A;J/E9"[email protected]/EGg7
132?ŠN%.0;<4.ƒGg6
> "5 2?,;JG=0?A2O/GR/D
9A7r;@Dƒ/EC46;@/ED4EV>8DQƒ?-2h057k9-D0/7-7Š>8?A/hx/E;@D0PR>4.0;@/EO/Q]}J 0Y
Yx
&Y7(€
# .09
/ [email protected]:
H =0=0?A2un;JGg>9A;J2Da
9-2?A>P/;<8)
a=MOu>?-;<>8D
9I21
9A.0/Š\]50H^9A;JP?-;<QBG/69A.02QB132?:[email protected];@D0/>?I/
5>89-;@2D7
}J +um;<7;@G=0HJ/G/ED
9-/Qj9-2`7k2HJO/l>7-50;^9>[email protected]/eD0;^9A/RQ0; /E?A/ED4E/l>8==0?-2un;@Gg>9-;@2D`9A29-./l9A;JG/6W[Q/=/DQ/ED
9
[email protected]?A;@4>9A;J2Ds
/ 5>9-;@2D7132?B9-.0/L2N 281 [email protected]/697B2O/E?B7-50h7k9-?>9-/7B/E;J9-.0/?‡N%;^9A.f2?‡N%;J9-.0259‡NŠ/E9k9-;@D0P
>D
Q >u2?g9-./ =0?A/7-/ED4E/]281B9A2=x2P?>8=.0;@4j./69-/?-2P/D0/E;J9-;@/79
?„Glhx/Q0Q/QUN%;^9A.0;@Dz9-./s>[email protected]?-;J9-.G ;<7K>D
/ECg4E;J/D
9Y%>Q0>=9-;@O/ 9-;@G/6W[7"9A/E=0=0;@D0PU7-4./EG/Y,h>7-/Q 2A
D @/E50DCB 7d7-/4E2DQFW€2?Q/?=0?A/Q;<4t9A2?-W€4E2?A?-/4t9-2?
G/69A.02QT}J ,u€YN%.;@4. >59-2Gg>9-;<4E>[email protected]>7
Q 5"57k9A79-.0/l9-;@G/‚7k9-/E=s7k50h=5"/4t99-2K>ƒ7k=x/4E;^e/Qd/E?A?A2?%[email protected]/E?>8D4E/
/D>[email protected];@D0PDŠ9-2hx/‡2=9A;JG;<7k/Q`;JDd>[email protected]/QƒGg>8D0D0/?
4E2G=>469,Q/?-;@O>9A;J2Dg281œ9-./P2O/E?AD0;@D0P‡/ 5>89-;@2D7r9-2lhx/[email protected]/Q;@7Š=0?A/7-/ED
9-/QR132? 462GR=HJ/E9-/ED/7A7EY
[email protected]Š/QdhFVdQ/69>8;@[email protected])281„9-.0/R50DQ/?-=;JD0D;JD0P`DF50G/E?A;@4>8H²G/69-.2FC
Q ’7 >Q2=9A/Q]132?Z9-.0/;J?‡[email protected];@2DI# .0/
\d5H^9A;JP?-;<Q [email protected]/?s;<7se?A7k9fOu>HJ;<Q0>89-/Q >8P
>8;@D7"9f> ?>8DP/p21K>8D>[email protected];@4>8H>8DQ =0?A/EOF;@257-HJV
W€[email protected];<7k.0/Q
DF50G/E?A;<4E>8Hl=0?-/Q;<4t9-;@2D7j132?™[email protected]/69 7-=0?-/>Q;@D0P˜=0./ED02GR/D>˜>8DQS9-.0/D_/6nF9-/DQ/QS9A2 7-;JG‚[email protected]>89-/p>
O>8?A;@/69"V217-=0?A/>Q0;JD0PZL2N'7m2DR7-50h7k9-?>9A/,N%;J9-.Q0;/E?A/ED
9rN,/69k9>8h0;@[email protected];^9"Vl4.>8?>4t9A/E?A;@7k9-;<4E7m>DQ‚4E2D
9A>;JD;JD0P
9A2=x2P?>8=.0;@413/>89-50?A/7
~%\E¸
#[email protected]?„\E¸
# -+HGIc\s&)G
( ?9I
8²;@P50?A/s`;<7g>s7A4.0/EGg>89-;<4`7-D>=W€7-.029‚281Z9-.0/`L2N†281)>™[email protected]/69K
Y JLNMPOQ9ORG6Y:2O/E?> 9A2=x2P?A>=0.0;<4
13/>9-5?-/YFS;NMEOTQtY%21R>[email protected];J9-5Q/LSVUf>8DQ [email protected]/ED0P9-.7XW9Y >DQ[Z\y
Y
N%;^9A.i?A/7-=x/4t9`9-2o9-./™?A/613/?-/D46/
4E2
2?AQ0;JD>89-]
/ NMEOTQ‚2D—>™7-50h7k9-?>9A/K;@D4EHJ;@D0/Q—>9g>8D—>8D0PHJ/X^S9-2™9A.0/j.02?A;*2D
9A>Hž²€9A7g>7k=x/469R?>9A;J2
_a`
WbYdc6Z\r
Y :# ./gL50;<Qf;@7‚!'/N 9-2D;@>Df>DQf;@D462GR=?-/7-7-;JhHJ/YI281%Q/ED7k;J9"\
V e™>8DQ™OF;<7-4E27-;^9"V\f¸YM>8DQ
›t¡EµX¯¬®t±6¥
g
¤
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjÇ
i
›t¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
ˆ
î
:¦íq>’¨°q.
Ȳ
¦.ƒ"
¦ƒ
²¨
g
Z
Y
H(X
,Y,T
(0,0)
)
X
S(X
(X ,Y )
1 1
,Y)
S0
LT
WT
α
×ÔÕà1ØlÛaÑ>òÊGÓ5ålÑjÞ8ßÖaÔÕӏç&Фß>Úlöç&ålÜ>քÜ>éÖaålяê.ÜBë
Ü>éßqäGÛaÜ1ÚlÙUÑ5ÖBúùá
3â5ú¤ÜSè,Ñ5ۄßÈÖaÜ1Ú#Ü1àÛ6ß>ÚlåGõ(úáÈ
â5ú¤Ü>мß
ç&ؤïlçLÖ&Û6ßÖaяÔÕФӐÙUÔÕÐlÑBäßքßÐßФà1ÙÕÑùÖaÜXÖaå¤ÑålÜ>ÛaÔjÜ1Ð(Ö6ßÙú¤ç&å¤ÜBëìÔÕФàqßXç&ÑjӐÖaÔÕÜ1ÐÖaålÛaÜ1Ølà1å·Öaå¤Ñ‹äGÛaÜ1Ú¤ÙÕѐÖ
ÜéÓjÙUßÛaÔJÖ¹õ(úlÔÕÐÖaålÑ
é@Ü>ÛìÖaå¤Ñ3ça
ß ,Ñ
ÚlÙUß>ФÑ!ßÐ.ä8Öaå¤ÑÚ¤ßÛ6ß>ÞqÑ5ÖaѐÛaç„älѐø¤Ð¤ÔÕФàXÖaå¤Ñ3ÖaÜ>Ú#Ü1à>Û6ßÚ¤å(õ(ò
\]>?A>D0P2D0;I/ x/4t9A7B>?-/;@PD2?A/Q]7-2g9-.>9)9-.0/7k5?k1’>46/‡9A/ED7-;J2DI
Y mY;@7)>8H<7k2K462D7"9>8D
9# .0/lG29-;@2D]21
9A.0/[email protected]/69%;<7%P2O/E?AD0/QghFVg9A.0/B9-;@G/6W[Q/E=x/EDQ/ED
9Z!Z>uO
;@/E?-W"a
9-2q/7Š/
5>89-;@2D7YOF;*
e
"!#
!
R%$&#
' ()#
(
`
+
`
-=O
*
'#
f
(-,.$
(-/0#
$
e1 O
k
X~ `
N%.0/?-/
O32bOTZ f>DQ
>8?A/—9-.0/UL50;@QwO/[email protected];J9"V >8DQ =0?A/7A7k5?-/U?A/7-=/4t9-;@O/HJV >8DQ1 `
#
#
,
’7-;JD;^9O- O + 462
7^ 6YN%.0/E?A/41 ` , +6587:9
;<7Z9-.0/>446/[email protected]/E?>9A;J2D]Q50/‚9-2`P?>uOF;^9"V#²>8qF;JDP J U >79-.0/
'
/
4.>?A>4t9A/E?A;@7k9-;<[email protected]/69'9A.0;@4qFD0/7-7>8DQ W U >7'9-.0/H<>9A/E?>8HI/EnF9-/ED
921¸;^97'132F289A=0?A;JD
92DK9A.0/‚7-50h7k9-?>9A/Y
` [email protected]? ;<7 7kGg>HJHœ7-29-.>9 9-.0/‡!Z>uO
;@/E?-W"a
9-2q/7r/
5>9A;J2D7 4E>Dh/‡7-;[email protected];Je/QK57-;JD0P
;J9%;@7 >7A7-50G/Q9A.><
9 ; >
A ?
9A.0/'[email protected] >uO/Y2?'‹ŒŽEt’‘“”X’•–YF>8==0?-2un;@Gg>9-;@2DM6'[email protected]>8D>[email protected]/7„>8?A/%=/?k132?-G/Q>DQ?-/7k5H^97¸=0?A/7-/ED
9-/Q
;@D`9-/E?AGg7 281¸462?-?A/7-=2DQ;@D0PlD2DW[Q;@GR/D7-;J2D>8HK’HJ2N,/E?%4E>7k/7ŠOu>?-;<>8hHJ/7EYF9A.0/‡q/Vg2D0/7 h/;JD0PCB
1
`
1 ED OGFVO
JL<MEOQ9ORj
`
O 7:ED GO =
H
F J U
WbU
`
,
O EM OG3
N OPO L
;
;
SFNMEOTQ
O
J U
`
ED OGF=
O32bOTZ
#
RVU
W U
O
`
NMEOTQ
OJI `
W U
R
`
OARVU `
R U
K
J U
f WKU
;RQ
O
žˆ O
N%.0/?-/Š9-./Š9A;JG/'7A4E>HJH
/ R U ;<7²9-.>89rQ/E?A;JO/Qlh
V&Z?A4.>?AQƒ} ~ -u132?²9-./,[email protected]/EO/[email protected];@D0P)2817k50?-1’>4E/,Q0;@7k9-50?Ah>D46/7E
!289-/l9-.>89B7-;@D46/l9-.0/R7k50h7"9A?A>89-/‚7-50?-1’>4E/;@7ZP;@O/D]hFVƒ9A.0/‚9-2=x2P?A>=0.FV1350D4t9-;@2S
D 7:EDCOTF=7-2g9-.>9)9-.0/
Q?A2=HJ/E9,[email protected];J/7 h/E9"NŠ//EDI ` 7B>DQUI ` 1
70*DQ/E?,9A.0/‡>8hx2O/)D02DW[Q;@G/ED7-;J2D>[email protected];<7->89-;@2DIY
D0/EPHJ/4t9A;JD0P
$
9A/E?AG7 21 V ; ,2?,.;JP.0/E?YF9-.0/!Z>uOF;J/?kW"a
9-2q/7„?A/Q546/B9-2CB
/
!
/
!
!
!
!
g¤
`
!
/
N
`
I
I
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
!
!
/
L
L
!
I
/
!
›t¡EµX¯¬®t±6¥
M
`
+
+JWYX
;
D
L
F
7k;@;
D ^bO
O
W4X
<:
462
7=^9O
& $&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
¦§,„§8¨
¹¤x­¦0§¹,(
§ B
œ¹,­(R
§(>.¤*¸(
­ „,
§ R,
§ B¦,
A?
`
N%.0/?-/
;@7Z9-./ ,2DQ]DF50G‚h/?)G/>7-50?-;@D0Pƒ9-.0/?A/EH<>9-;@O/‚;JG=x2?-9A>D46/‚281ŠP?A>uOF;J9A>89-;@2D>Hœ9-2
7-50?-1’>46/9A/ED7-;@2Df132?46/7E:# .0/7-/g/ 5>89-;@2D7>?-/[email protected]/Qf7k50h=5"/4t9l9-2j9-./K462DQ;J9-;@2Df281%D02W€[email protected];J=c>9l9-.0/
7-50?-1’>46/B21M9-./7k50h7"9A?A>89-/‡>8DQ /E?A2l9>8D0P/ED
9-;<>8HI7k9-?A/7A7 >9%9-./B13?-//B7-50?-1’>4E/)21:9A.0/[email protected]/69YD>8G/HJV
W4X
EM GO N
`
I
-=O-:
>89
7
`
! MI
!
>8DQ
`
I
! NI
!
`
I
-
`
>89
$ 7
1
I
™} 7
?95>9-;@2D7H<:²>8DQ :;JD
9-/P?>9A/Q‡9"N%;<46/ N%;J9-.R?-/7k=x/469m9-2 Z2O/E?²9-./ Q0?-2=0HJ/E9:9A.0;<4q
D/7A7
>DQœY07-50h5"/469%9-29A.0/>8hx2O/)h250DQ0>?-VK462DQ;^9A;J2D7YV
;@/EH<Q
M
! LD + W4; X 7-;JDF^ * I + 77 ~ I + 77 + 1 *
!
! L F * I + 77 ~ I + 77 + 1 * '
!
`
N
`
O
"(6
OT1
$ 7E
+ , # .0/'9-;@GR/EW€Q0/E=x/EDQ/D
9Š[email protected]?A;@4>9-;@2Dg
/ 5>9A;J2D132?r9A.0/'/[email protected];J2Dg2819-.0/)[email protected]/69r9A.0;<4q
D/7A7¸;@7Š2h09A>8;@D0/Q
hFVƒ4E2D7-/E?AOF;JD0PRGg>7A7,9-.0?A25P.0259 9-.0/BL2N‡YOF; !
I
EM GN
1
!
+ ( '
`
O
k - Y
`
O [D
9-/P?>9-;@D0P‡/
5>9-;@2D7 <+:¸>8DQ <, ¸9-2‡132?-G
>8DQ7k5h7"9A;^9A59-;@D0P‚;JD
9-2
N .0/?-/
%
/5>9-;@2D "- ŠVF;J/[email protected] -9 .0/[email protected]%;@D0P‚/
5>9A;J2DK;@D`9-/E?AGg7 281²9-.0/‡=?-/7-7-50?A/e/[email protected]
!
1
!
`
!
!D
1
ˆ
Q ! L + 4
W;X
!D
7-;JDF^
E5 +
4E27
* $ ! ! F
1
ˆ
Q " ! L
! F * '
L
k
[D 4E2GG2D]N%;J9-. 9-.0/‚NŠ2?-q`281r29-.0/?A7R} +Y ,ur>ƒQ;<7N5"2;JD;JD0PK=0?A/7A7k50?A/B9-/?-G…;<7Z/[email protected]/Q`9A2K>[email protected]/EOF;<>9-/
9A.0/ 7-;JD0P50H<>8?A;^9"Vp>89NŠ/E9k9A;JD0PpHJ;@D0/7„# .0;<7G2Q/HZ>7A7-50G/7g>™9-.;JD˜=0?A/4E50?7k2?eHJG{281)9-.;@4qFD0/7-7 1V2
>.0/>Qp21Z9A.0/]7->G/j>DQo?-/[email protected]>89-/7R9A.0/d2h7k/?-O/Qp4E2D
9A>4t9>[email protected]/K132?g=>8?-9-;<>[email protected]@VTNŠ/E9k9A;JD0Pc7-V7"9A/EGg7R9-2
;@D
9-/[email protected]/[email protected]>?¸132?46/7¸9A.>9,h/462G/Z;JG=x2?-9A>8D
9Š132?„[email protected];45;@Q07Š281M7-50h0G;<46?A27A462=0;@4%Q0;JG/ED7k;@2D7)} ,uXF# .0/
Q;<7<"5 2;@D0;JDPR=?-/7-7-50?A/Z9A/E?AGjY 13t Y;@7 P;JO/EDf} ~0tMhFV
13 `
$
5
+
7'
1 2 7'k
+
+ 5 ;
/
1V2
1
*"! + 1V2
%
1
*"# O
CD OTF %;@7
k~ N%.0/?-/ f>8DQ
>8?A/Z9A.0//EnF=x2D/ED
9A7 21²9A.0/‡;@D9A/E?>469-;@2D`=29-/ED
9A;@>HM>8DQ
9A.0/‡/
50;@[email protected];Jh0?A;@50G
4E2D
9A>4t9Z>[email protected]/!'29-/9A.>9)9-./‚Q;<N7 5"2;JD;JD0PK=0?A/7A7k50?A/‡9A/E?AG ;<7GE/E?A2g132?)>[email protected]Š/E9k9A;JD0Pj7-VF7k9-/G…7k;@D46/
` -
?¸n=x/E?A;JG/ED
9>8H/EOF;<Q/ED4E/‡7k50PP/7"97„9A.>9Y0;@DK=0?>469-;<46/Y 1 2 [email protected];J/7 ;JD`9A.0/B?A>D0P/R6W - -D0G } ~~M7k2R9A.>9
4E50?-?A/ED
9AHJV;J9;<7%D029%13/>7-;@h0HJ/)9A2g57k/‡?A/>HJ;<7k9-;<4132)O>HJ50/7 ;JD]4E2G=059>9A;J2D7E0# .;@7%?A/7k9-?A;<4t9-;@2Dj.>7,9"NŠ2
?>8G;Je4E>89-;@2D7:# .0/ƒe?7"9‚>DQcG27k9‚;@G=2?k9>8D
9l;<7‚Q050/9A2]9-./1’>469l9A.>9>4E4E50?>9-/ƒ?-/=0?-/7k/D
9A>9A;J2D
21Z> N,/69-9-;@D0P™HJ;@D0/d>8DQIY²;@D—=>8?-9-;<4650H<>8?Ym>uO2;<Q0>D46/21D0/P>9A;JO/[email protected] 9A.0;@4qFD0/7-7-/7R;@DT9A.0/E;@?O
;<46;@D0;J9"V
?A/
50;@?-/7Š9-.>89,9A.0/7-=>9A;@>Hœ?-/7k2HJ59A;[email protected]/‡21M9A.0/‡7->GR/B2?AQ/? 281:Gg>PD0;J9-5Q0/B>7;13280# .0;<7%>8DQ
?A/EH<>9A/Qd;<7-7-50/7>8?A/‚Q0;@7A4657A7-/Qd;@D Q/E9A>8;@H²h
V ,/E?-9-2 ;,} r>8DQ].>uO/[email protected]/Qd9-2K9-.0/RQ/EO/[email protected]=0G/ED
9)281rD0/N
DF50G/E?A;<4E>8H 7A4.0/EG/7RN%.0;<4.p/D7-50?-/K9-.>99-.0/`[email protected] 9-.;@4qFD0/7-7R?A/EGg>8;@D7R=2
7k;J9-;@O/`N%;J9-.0259g?-/
5;J?A;JDP
/En4E/7A7k;@O/%G/7k.?A/6eD0/EG/ED
9„D/>8?„NŠ/E9k9-;@D0P‚HJ;@D0/7Y} ~8ˆXF# .0/Z7-/4E2DQ=?-2h0HJ/G ?A/EH<>9A/7m9A29A.0/%1’>469„9A.>9
1 2 ;@7g9-.0/j=0?A;JD46;@=>8H'Q/E9-/?-G;@D>8D
9281B7k=?-/>Q;@D0P™?A>89-/c} ,[email protected]Š/O/?Ym>8HJ9-.0250P.—;@D>8h0;@[email protected];^9"Vc9A2c57-/j>
7-5C46;@/ED
[email protected]>HJH 132l;@Ds4E2G=059>9-;@2D7'[email protected]/>Q7'9A2`>K7kV7k9-/EGg>89-;<42O/?kW€=0?A/Q;<4t9A;J2DK21Š7-=0?-/>Q;@D0P?>9A/Y
9A.0;@7Q;^C4E50H^9"V`4E>DKD02?-Gg>[email protected]/4E2G=x/ED7A>9A/QK132?%hFV57-;@D0Pg>8Dd>8D>HJV
9A;@4B462?-?A/469-;@2D™} +u€
&
›t¡EµX¯¬®t±6¥
g¤
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
î
:¦íq>’¨°q.
Ȳ
¦.ƒ"
¦ƒ
²¨
# .0/r=0?A/7A7-50?-/²e/[email protected])9A.0?A250P.0259œ9A.0/ŠQ0?-2=0HJ/E9I;@7M2h9>8;@D0/Q)hFV);JD
9-/P?>9A;JD0P / 5>9-;@2
D & N%;J9-.?A/7-=x/4t9
9A2YIN%./E?A/%9-.0/)462D7"9>8D
9„21I;JD
9-/P?>9A;J2DXž7k/E9k9A;JD0P‚9-./=0?A/7A7k5?-/'Q0>89-50Gy9A2‚>9AG27-=0.0/E?A;<4m;<7„P;JO/EDghFV
N%.0/?-/ 9A.0/
1
( /
[email protected]@/>Q0;JD0PR9A2
+
`
L
(
1
/
+
77
$
2D8I
1
`
kˆ
7 O
1
$
77²9-/?-Gy;<7¸9-./7kGg>[email protected]=x/'>8=0=?-2un;@G>89-;@2D‚9-2‡9A.0/7-50?-1’>46/'4E50?AOu>89-50?A/%2819A.0/'9A.0;JD
$
+
`
L
1
/
(
+
7
$
13
4627
WYX
$
^91
+
7
$
I:
k
'
:
€9;<7g;@GR=x2?-9A>D
9R9A2cD0289A/`9-.>89ƒ7k;@D46/j9A.0/[email protected]?A;@4>9-;@2DU>8==0?-2un;@Gg>9-;@2Do;@7g132?AG>[email protected]>[email protected];<[email protected]?
L2N'7‚132?gN%.0;@4.—9A.0/`13?-//]7k50?-1’>4E/`[email protected]=x/`;<77-Gg>[email protected]žY¸;J9A7g>4E465?A>46VT46250H<Qphx/]
> /469-/QohFVT9A.0/d7"9A/E/E=
7-HJ2=/7B281Š9-.0/9A2=x2P?>8=.0;J/7B462D7k;<Q/E?A/Qs.0/?-/ @'2N,/EO/?Y?A/4E/ED
97"9A5Q;@/7g} ~ 0Y:~ Š>DQs4E2G=>?-;<7k2D
N%;J9-./6n=x/E?A;JG/D9} ~FYˆ [email protected]?A;<4E>9A;J2Dl9A.0/E2?-VB9A2‡h/%?A/EGg>?-q>8hHJV‡?A2h057"9¸;@D7-54.46;@?A4E50Gg7"9>8D4E/7
=0?A2OF;@Q0;JD0P)9A.0/>7A7-2F4E;@>89-/Q‚$%/VFD02H<Q07r>DQ+,>8=0;@HJH<>8?AVDF50G‚h/?A7r>?-/%D289¸9-2F2‡H<>8?AP/8!'289A/>8H<7-2Z9A.>9„9-.0/
132?-G 21²
/ 5>9A;J2D]k
Š/D7k5?-/7,9-.>89%9-.0/ IgQ/=/DQ/D46/;@D`/
5>9A;J2D " %Š.>7 D02g;@DL5/ED4E/‡2DK9-.0/
/[email protected];J2DK21M9-./BeHJG†9-.0;<4qFD0/7A7%>8DQK;J9';<7,9-.0/?-/E132?A/)2G;^9-9-/QK13?A2G†7-50h7-/
50/D
9>8D>HJV7-;@7
‰ ?A/EOF;@257 N,2?Aq/?A7 .>uO/7k;@[email protected]`7-50h7k9-;J9-59A/Qd/
5>9A;J2L
D " %,;JD
9A2/ 5>89-;@2DL"
,9A2gVF;J/[email protected]?-9-.W
2?AQ/? 9-;@G/6W[Q/E=x/EDQ0/ED
[email protected]?A;@4>9A;J2D`
/ 5>9A;J2D7%=050?A/[email protected]`;JDj9A/E?AG7281²9-.0/[email protected]†9A.0;@4qFD0/7-7
Y 1M
@2NŠ/O/E?
;@Dd9-./l=0?A/7-/ED
9Z7k9-5Q0V`;J9B;@7'13250DQ >QO>8D
9>8P/257%[email protected]/B9A.0/‚4E250=HJ/QdD02DHJ;@D0/>?
/ 5>9-;@2D]7-/69Z132?
1U>DQ
/ 5>89-;@2D7 k7
‚>DQ[" %‚?-/7k=x/469-;@O/[email protected];@D46/`9-./]7k/462DQFWX2?AQ0/E?Q; /E?A/ED
9A;@>H%2=x/E?>9-2?A7
L Y¸
>?-/g7-;[email protected]/E?‡9A2]Q;<7-4E?-/E9-;<7k/g>8DQ™>j7-50h7k9-?>9A/gNŠ/E9k9A>h0;@HJ;J9"V]1350D4t9-;@2D
7 DCG
O F=6YI132?lOu>?-;<>8hHJV]N,/69-9A>h0HJ/
7-50h7k9-?>9A/7Y04E>8D`hx/‡;JG=x27-/Q`N%;^9A.02509';JG=x27-;JD0Pg4E2D7k9-?>8;@D
9A7,2D`;^97'Q*; x/?-/D
9-;<>8h0;@HJ;J9"V } +€
?b
5>89-;@2D
7 k 7
%>8DQ " %%>8?A/[email protected]/Qd7-50
h 5"/4t9Z9-2ƒ7-VFGRG/E9-?AVKhx250DQ0>8?AV`4E2DQ0;^9A;J2D7>89'9-.0/‚/QP/7
21:9A.0/‡462G=0509A>9A;J2D>8HIQ02Gg>8;@DIY0D>G/[email protected]
!
!
L
`
!
!
1
`
k
-=O
N%.0/?-/ ™;@7,9A.0/‡D02?-Gg>8H9A2R9-.0/7-50h7k9-?>9A/)hx25DQ0>8?AV
ˆ0%!Z*\EŠ
? $%-+H c
I
E
\ ?Š#[email protected])&)(
6 #“”X’“‹
4!E‘E6 ”X4!E“”X’•–
HJHr=?-2h0HJ/Gg7‡>8?A/[email protected]/Q
2D™>d7T
5>8?A/462G=0509A>9A;J2D>8HrQ02Gg>8;@DsN%;J9-. DCOTF ` -=OE7 <- OE
57-;@D0Pƒ>g\d50HJ9-;@P?A;<Qd>8=0=0?A2>4.I# .0/eD0/7k9P?-;<QjHJ/O/Hb3;X /l9-.0/‚>4t9A5>8H:P?A;@Q`2DdN%.0;@4.j9A.0/‚[email protected];J2D`;<7
Q/7k;@?A/Q m.>j
7 ž
~ mD2FQ0/7r;@D/>4.Q;@?-/4t9A;J2Dƒ7k29-.>89„9-./Z7-=>9A;@>H462WX2?AQ0;JD>89-/7¸281œ;J9A7ŠP?A;@Q=x2;@D
9A7
$
>?-/P;JO/EDThFVSD ` 9 >8DQ F ` 9 ;JDc9A/E?AGg7‚21%9-.0/KeD0/KP?A;@QpQ/ED7-;J9"V™=>8?>8G/69A/E?:[D
9A.0/‚=?-/7k/D9B7"9A5QV / ˜} &=OE-m462?A?A/7-=/ 2DQ;@D0Pg9-2`P?A;<Q07>89Z9A.0/leD0/7k9)[email protected]/EO/EHmN%;J9-. hx/69"N,/E/D\&:g>8DQ
-~ D02Q/7¸;JD/>4.Q;J?A/469-;@2DI ?b
5>9A;J2D7F"²
>8DQk %m>8?A/ >=0=0?A2un;JGg>9A/Q‚7k=>89-;<>[email protected]‚57-;JD0P‡4E/ED
9-?>8H
Q*; /E?A/ED4E/7‡} Y~ &uIHJ/>Q;@D0PR9-29A.0/[email protected]@2N%;JD0P7-/462DQFW€2?Q/E? >44650?>9A/‡7k=>89-;<>8HIQ;<7A46?A/69-;<7A>9-;@2D7
!#" 1
!
!
`
Q
1
ˆ
WYX
;
›t¡EµX¯¬®t±6¥
g
¤
/
$
*" 1
Q
"
ˆ%$ '&
&
7-;@;
D ^
*" L
1
/
*" CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
"
L
1
+
L
"
+
#" +
~
#" L
ˆ
1
ˆ($
Q
1
+
)
"
9 Q
1
+
$
*" *
L
&
9
$
"
&
9
#" L
/
+
#" L
'O
+
)
" 9 L
#" )
9 $
+
k
& $&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
¦,
§ „§8¨
¹¤x­¦0§¹,(
§ B
œ¹,­(R§(>.¤*¸(
­ „§,R§,B¦,
&
1
"
7 "
$
*" 7 #" $
$
+ ' 5 + " + 4627 +
*" 9
7 #" )
9 1 #" $ 7 *" $
+ 5 ;
1 2 $
/
1V2
1V2
+ +
"
1 *
7 #" 4E27=^
`
-=O
W X
Y
1#" "
1#" * #&
$
$
/
* !
L #" $
1
/
9 " 1
$
$
7 9) " $
1
k
7(6
`
132?ƒ/>4. 3;XY 5;@D˜9-./™4E2G=059>9-;@2D>8HZQ02Gg>8;@DIY,N%.0/E?A/\{
;<79A.0/s7-=>9A;@>H;@D46?A/EG/D9 ~9 31 2?Z9A.0/‚eD0/7k9‡P?A;@Q HJ/O/H4Z>8DQ]9-.0/ " Y #" 9A/E?AG7B>?-/‚9-.0/#xT '6“‘6”[• B
! >7A7k246;<>9A/QdN%;J9-.
Q $ &
Q $
&
9A.0/‚Q;<7A46?A/69-;<7A>9-;@2DM @'/?-/?A/7-50HJ9A7'>?-/‡2h9A> ;JD/Qj57-;JD
P x.02?-D0;J9A7-q>uV>>8DQ&Š/?k9A2 E;B 7} ~8ˆ
 %
• !t¶”ž¶” ,T ! E ¶%
– $ œ‘ ',
132?%N%.0;@4.K9-./‡=0?-/EWž1’>4t9A2?7Š;@Dj/
5>89-;@2D
"&:,4E>8D`hx/7k.02N%DK9-2ghx/‡P;@O/DKhFV
Q
1
ˆ
N%;J9-.d7-;JG;@[email protected]>?%/6n=0?A/7A7k;@2D7Š132? 9-.0/
$
&
"
`
*" 1 " / /
ˆ !
1 "
1*" $
~
Y
Q $ 9 & " Q $ *" 1
&
>8DQ
k
O
+ 9-/?-Gg7
Q $ * " 9 &
0•8-“
‹ 4!E‘E6”X4 !E“”X’•–
#
"Q/>[email protected]²9-;@GR/EW€7k9-/=0=0;@D0Pf7-4.0/G/7l132?R9-.0/`DF50G/E?A;@4>8H 7k2HJ509-;@2Dp21'9-?>8D7k;@/ED
9lL2N'[email protected]/jhx289-.
—‘E<
 E –”>8DQ]“‘A‘EŒ-“8"” E# .0/'132?AGR/?„/D7k5?-/7m9-.>89Š7-G>HJH9-;@G/Z7k9-/E=7„>8?A/'>uO2;@Q0/QRN%.0/ED9A.0/[email protected];J2D
O>8?A;@/7 [email protected]%[email protected]%.;[email protected]/)9-.0/BH<>9k9A/E?%7-.0250H<QK/ED>h0HJ/Z9-./)/?-?A2?Š9-2Rhx/462D
[email protected]/Q9A.0?A250P.0259,9-./[email protected];J2D
=0?A246/7-7\d2
7"9,=0?A/EOF;J257rDF50G/?-;<4E>H7k9-5Q;@/7Š[email protected]?A;<4E>9A;J2DWX9"V
=x/'L2N'7r.>uO/%1324657A7k/Qg2Dƒ;JD4E?-/>7-;JDP
9A.0/„/ECg4E;J/D46V)281F9A.0/„[email protected];@2D>DQœYE;@D=>?k9A;@4E50H<>8?Y62O/?A4E2G;JDP„9A.0/Š7k9A>h0;[email protected];J9"VZ?A/ 50;@?-/G/ED
9œ132?:/[email protected];@4E;^9
7A4.0/EG/79-.>89I9-./¸9-;@G/Š7k9-/E
= œ7-.0250H<Q)h/ 6xW>8D‡;@G=0?>4t9A;@4>8H8?A/7k9-?A;<4t9-;@2DBN%.0/D‡/EO/EDBG2FQ0/E?>9-/
7-=>9A;@>H¸?-/7k2HJ509-;@2D™;@7B/GR=HJ2V/QœMaF/O/?A>Hm7k9-5Q;J/7‡.>uO/R>4.0;J/O/Q]9A.0;@757k;@D0Pd>8HJ9-/?-D>9-;@D0PW€Q;@?A/4t9A;J2D
;@G=0HJ;<46;J
9 <Z( ,>[email protected]?-;J9-.G7Š/[email protected];JDP‚>8HJ9-/?-D>9-/)7kN,/E/=7Š;JD`/>4.ƒQ;@?-/4t9-;@2DK7k2‚9-.>89,[email protected]>h>DQ/Q
7-V7"9A/EG_281/ 5>9A;J2D7:D0/E/Qlhx/,[email protected]/Ql>89²/>4.‡9A;JG/%7"9A/E=`} X# .0/7k/)”ž "!#‹¶”X”ž¶–%$ >[email protected]2?A;J9-.0Gg7:/ED>h0HJ/
9A;JG/7k9-/E=7'7k/O/?A>H2?AQ/?A7,21²Gg>8PD0;J9-5Q/BH<>8?AP/?Š9A.>8D Š9-2ghx/‡57-/Q`7k54E46/[email protected]
80/Nz7"9A5Q;@/7r.>uO/'?A/E=x2?-9-/QR>89k9A/EG=9A7„9-2R462D9A?-2H9-./Z>44650?>4EVl21x9A;JG/EW€7k9-/=0=0;@D0P‚7A4.0/EG/7„>8DQ;^9
;<7r2D0HJVO/E?AVl?A/46/D
[email protected]>9 >8Dg/7k9-;@Gg>9A/2819A.0/[email protected]>8H9-?A50D4>9-;@2Dg/?-?A2?„.>7„h//ED57-/QN%.0/[email protected];JD0P
/ 5>9-;@2D7 281²9-.0/‡132?-G "7
>8DQ k :Š9A29-/7"9'N%.0/E9-.0/?%9-./EVK>?-/Bhx/E;@[email protected]/Qƒ9-2>g7-=x/46;Je/QK/?-?A2?
[email protected]/E?>8D4E/‚} 6
~ &€# .0/[email protected]>89k9A/E
? B 7 7A4.0/EG/Y.2NŠ/O/E?YQ02
/7 D0289 2=09-;@GR;<7-/Z9A.0/)9-;@G/B7k9-/=ƒ132?%>lP;JO/EDƒ/?-?A2?
[email protected]/E?>8D4E/# .0/‚=?-/7k/D9BN,2?Aq`>Q0Q?A/7A7k/7'9A.0;@7);<7-7-50/RhFVd57k;@D0PK9-;@G/6W[7"9A/E=0=0;@D0P`h>7-/Q 2DsHJ24E>H:/?-?A2?
/7"9A;JGg>9A/7 13?A2G >8Dj;@G=0HJ;<46;J9Y7-/462DQFW€2?Q/E? G/69A.02QjN%.0;<4.d?-/Q546/7 9-2 @'/E5D B 7G/69-.2FQc}J ,:;@Dj9-.0/
4>7-/‚21,>ƒe0n/Q 9-;@GR/7k9-/=II# .0;<7B57-/7B>8D™/7k9-;@Gg>9-/21„9-./[email protected]>Hm9-?A50D4E>89-;@2Ds/E?A?-2?Yx2h9>8;@D0/Q 13?A2G
9A.0/gQ; /E?A/ED46/h/E9"NŠ//EDs9-.0;<7‡[email protected];@2Df>8DQ™>8D™/[email protected];<46;J9B=?-/Q;@469-2?Yx9-2j;@D46?A/>7k/l9-.0/9A;JG/g7k9-/E=™;@Df>
4E2D
[email protected]/Q`G>D0D0/?N%.0;@[email protected])>899A.0/‚7A>8G/‡9A;JG/lG;@D0;@GR;<7-;JD0P9-.0/R462G=0509A>9A;J2D>8H:/6n=x/ED7-/‚>7-7-246;<>9-/Q
N%;J9-.j?A/E=x/>9A/Qƒ9-;@G/7"9A/E=`1’>8;@HJ5?-/
# .0/j>8=0=0?A2>4.f?-
/ 5;J?A/7l/
5>9A;J2DA"& 9-2sh/`?-/N%?-;J9k9A/EDp>7‚>s7kV7k9-/G 21ZQ*; x/?-/D
9-;<>8HJW€>HJP/Eh0?>8;<4
/ 5>9-;@2D7 ;JD`9-./B132?AG
!
*" !
1
`
!#" O L *" 6O1! " O L " O1!#" O L *" 1
HO
k
, 132?,/>4.PO%Š;JDK9-.0/‡4E2G=059>9-;@2D>8HQ2Gg>8;@DI# .0/B=0?A/Q;<4t9A2? 7"9>8P/Z;<7Š[email protected]/EnF=HJ;<46;J9Y07-/4E2DQFW€2?Q/?
›t¡EµX¯¬®t±6¥
g
¤
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
î
:¦íq>’¨°q.
Ȳ
¦.ƒ"
¦ƒ
²¨
(
>4E465?A>89-/);JD`9-;@G/>8DQ`=0?A246//Q07,[email protected];JD0P
1
*" !
`
*" 9)
1
/
$
!
+
#" 1
/
!
$
!
" $
#" 1
*" O L
!
O1
!
" " O L
!
!
O1
#" !
O L
*" !
O
X~
6- N%.0/?-/ ` & >DQd9A.0/7k50=x/E?7A46?A;J=9 TQ/ED289-/7ZQ>9A>K>9)9-.0/ 9-.]9-;@GR/7k9-/E=Mx€9‡4E>D]hx/R7-.02N%DIY
OF;<>K>`#:>[email protected]?)7k/ ?-;@/7)/6n=>D7-;J2D]281„
/ 5>9A;J2D ž~ - 6Y9A.>9B9A.0/HJ24E>HM9A?-5D4E>89-;@2D /?-?A2?.<I#F?H%
132?)9-.0/
=0?A/Q0;@469-2?'7k9A>P/);@7'P;@O/EDKhFV
WKR
! "
#" `
!
/
k
!
&
$
*" Q 1
!
X~
O
$ #$&%
Q
!
N%.0/?-/)9A.0/B9-.0;@?Q`Q0/E?A;JO>9A;JO/)9-/E?AG ;<7%/EO>HJ5>89-/QK>89%>R9A;JG/ '
N
O
t
# .0/[email protected];@2Dj7"9>8P/Z;<7%;@GR=HJ;<46;J9>8DQ`;<7%>8H<7k2g7-/4E2DQFW€2?Q/?,>4E465?A>89-/);JD`9-;@G/Y0P;JO/ED`hFV
#" 1
!
1
*" !
+
!
~
~
$
!
1
#" 1
#" 132?%N%.0;@4.`9A.0/
I #F?—;@7
<WbR(
#" +
`
!
!
Q
$ #
!
Q
! !
#" *" !
!
!
O\)"
O L
#" OT1
!
#" Q 1
!
O1
!
" O L
!
" O L
!
" O1
" OT1
!
!
~
*" O L
!
*" O L
O L
#" N
O
! !
WKR
#" `
~+* -,
$
#" 1
!
+
,/.
1
*" !
'
X~~
X~8ˆ
!
Z7ZQ/7-4E?-;@hx/Qd;@Do}J ,u€Y9-.0/>7A7k5GR=09-;@2Dd9A.>9Z9-.0/‚9-.0;@?QFWX2?AQ0/E?Q/?-;@O>9A;JO/9-/?-G
>GR250D
9%2O/E?,9A.0/B9-;@G/7"9A/E=j/ED>[email protected]/7,9-.
/ WKR(v9-2h/‡/7"9A;JGg>89-/Q`hFV
`
'
O>8?A;@/7'[email protected]>K7kGg>HJH
X~
:
[email protected]%;@D0Pj} :
~ (XYF9A.0;<7%/6n=0?A/7A7k;@2Dƒ;@7 57-/Qƒ9-22h9A>;JDj>8D`/7k9-;@Gg>9-/B21M9-./)2O/E?>[email protected]?-5D4E>89-;@2D`/E?A?-2?ŠhFV
eDQ;JDP;J9A7C?„[email protected];<Q/>D‡D02?AGj1
Y 0'Wb(
R 208YtN%.0;@4.‡;<7M9-.0/D57-/Q)9-2Z7k=x/4E;^13V)9-./„D0/En
9:9-;@G/Š7k9-/E=B9A.0?A250P.0259
9A.0/Q2Gg>;JDIY YOF;<>
!
/
!
/
`
-
'
,:
!
R
0
&
W
WKR(20*
O
X~
;J3
1 0'WbR(2054 R WYN%./E?A/>7;^160'WKR2087[R WU9A.0/‚;J9-/?A>89-;@2D];<7Z?-/7"9>8?-9-/Q]N%;^9A. .>H^1¸9A.0/R4E50?A?-/D9
9A;JG/7k9-/E=M
# .0/B;[email protected];<46;J97kV7k9-/EG†281²>HJP/Eh0?>8;<4Z/ 5>9A;J2D7 "(6,>8DQ]ž~~ ,>8?A/B7k2HJO/Qƒ>89%/>4.K9-;@G/‡7"9A/E=j57-;JD0P
9A.0
/ 8ZaKGl5H^9A;JP?-;<QƒG/E9-.02QjQ/7A46?A;@h/QKhx/[email protected]‡
=+9dŒ‹”ž $t;:
•‹ 7E
# 0
. /?-/l;@7ZGl54.d;@Dd9A.0/[email protected];^9A/E?>9A50?-/‚7k50?A?A250DQ0;JD0P\]50H^9A;JP?-;<QdG/69A.02Q07 3/ Pm}JˆY: 0Y²F
Y²&0Y:(
Y:+uN
 tY
7-5C46/K;^9‚9-2s7A>uVs9-.>89‚9A.0/EVc>8?A/g1’>7"97k2HJO/E?7EYœ9A.0/`/7A7k/D46/21'N%.0;<4.T;@7l9-2s57-/K> 7-;@GR=HJ/K;^9A/E?>9-;@O/
9A/4.0D04; 50/'>7„>‡7kG2F289A.0/E?Y8D029„>7r>‡[email protected]/?Yu9A2‡?-/Q546/%.;JP.13?-
/ 5/ED4EVl/E?A?A2?7:2DR9-.0/4E2G=059>9A;J2D>8H
P?-;<QjN%.0;@HJ/‚HJ2N,/E?13?-
/ 5/ED4EVK/?-?A2?7'>8?A/l7-G2
29-.0/Qd2509Z2Ds>ƒ7-54E4E/7A7k;@2Dj281„4E2>?A7-/E?4E2G=059>9A;J2D>8H
U
P?-;<Q076
U.0;J/?A>?A4.FV‡21xP?-;<Q07m;<7¸Q/6eD/Qœ1
Y < =
Y <[email protected]@JY < A 7A>uVYN%./E?A/Š9A.0/%P?A;<QR7-=>46;@D0PZ.>[email protected]/7²13?A2G 2D0/
›t¡EµX¯¬®t±6¥
g
¤
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
+
¦,
§ „§8¨
¹¤x­¦0§¹,(
§ B
œ¹,­(R§(>.¤*¸(
­ „§,R§,B¦,
U
P?-;<QK9-29-./D0/6nF9>DQ`N%.0/E?A5
/ <
Q/D0289A/7 9A.0/l462
>8?7k/7"9%P?-;<QK>DQ < A 9-.0/‡eD0/7k9x!'289A/B9-.>89Y132?>HJH
U
[email protected];@2D7 =0?-/7k/D
9-/QKH<>9-/?%9-./462>?A7-/7k9 P?A;@QKHJ/O/H <
;@7>K(nœ7
(2D/7k2R9-.>9%9-./BeD0/7k9P?-;<Qƒ[email protected]/EO/EH
< A Y8;X /89A.0/'2D0/%2DN%.0;<4.R9-.0/[email protected];J2DR;<7r>4t9-5>[email protected]
9Y.>7„~ A %D2FQ0/7¸;@D/E;J9-.0/?ŠQ;@?-/4t9A;J2D
$ \d5H^9A;JP?-;<Qd7k2HJ509-;@2D7'>?-/
2?Y0;@Dj9-/?-Gg7281²9-.0/leD0/P?-;<Q`=>8?>8G/69A/E?%;@D
9-?A2FQ0546/Qd>8hx2O/!
Y ` W
$
?A/E=x2?-9-/QdN%.0/E?A/9A.0/‚eD0/7k9BP?A;@[email protected]/EO/EH²?A>D0P/7%13?A2G <
N%;^9A.]&:un=& RD02Q/7'9A2 < N%;J9-.p-~ unœ-
~ /
D02Q/7E7
802?:/En0>[email protected]/YE;J1>Z\d50HJ9-;@P?A;@[email protected];J2D‡;@7²Q/7-;@?-/Q‡2Dl>l~ ,nœ~ ,ŠP?-;<Q)9-./ED9A.0;<7:4E2?A?-/7k=x2DQ79-2
U
P?-;<Q < Q N%;J9-.;@D
9-/E?AG/Q;<>9A/[email protected];@2D7„2h9>8;@D0/Q2DgP?A;@Q7 <
k7(nœ(6tY < ’ˆˆn0ˆˆ:m>8DQ <
<&:un=& /
hx/6132?-/ 9A.0/%eD>[email protected];@2D2D < Q ;@7Š>4t9A5>[email protected]@Vl2h09A>8;@D0/Qœ!'29-/%9A.>9„9-.0/'P?-;<Q7k=>46;@D0P>7-7-246;<>9-/QlN%;J9-.
/>4.`P?A;@QKHJ/O/H < ;<7 50D0;J132?AG >DQK/
5>8Hœ;@Djh29-.dQ;@?-/4t9A;J2D7%>8DQK;<7'7k;@[email protected] `
# .0/'[email protected];JD0/>8?,Q;@7A46?A/69A;@7-/QR
/ 5>9-;@2D7 "(6¸>DQ ž~~ ¸>?-/'[email protected]/QR57k;@D0Pl>l462G‚h0;@D>9A/ ;J2Dg
2819-.0j
/ 0
8 [email protected]
=0=0?A2unF;@Gg>9A;J2Dfa
9A2?>8P/8Z
a=BG/E9-.02Qc>8DQ][email protected]„\d50HJ9-;@P?A;@Qs9A/4.0D04; 50/]}JˆYr0Y„7FYr
& Yr7(FYr+u€
8²;@P50?A/)~;[email protected]"9A?A>89-/7„9-./Z7k9-?A54t9A50?-/Z281I9A.0/[email protected];J2D=0?A2F4E/7A7r2O/E?r9A.0/Z13250?,P?A;@QHJ/O/EH<7¸9A.>9%>8?A/'57-/Q
N%.0/DR>)[email protected];@2D‚2DIY7A>uVYP?A;@QlHJ/O/EH < Q Y;ž /u~6,nœu~6,0YE;@7²2h09A>8;@D0/Qœ6'DR;JG=x2?-9A>D9¸>Q0Ou>D
9A>8P/„2109-.0;<7
G 3, 0
G3
G 3, 1
G 3, 2
G 3, 3
G2
G1
G0
Ncyc
: smoothing
: exact solution
: Full Multigrid interpolation
×ÔÕà1ØlÛaÑ
ÿ ò Ñjç&ӐÛaÔÕÚlÖaÔÕÜ>зÜéÖaå¤Ñ„פؤÙÕÙ.ÌãؤÙJÖaÔÕà>ÛaÔUäÖaÑjÓ5ålФÔUñ(ؤÑ
ФÜGägÑjçÔÕÐ8ÑjÔJÖaå¤ÑÛälÔJÛaÑjӐÖaÔÕÜ>Ѓò
ëìÔJÖaå
µ
ëìå¤ÑÐqç&Ü1ÙÕØlÖaÔÕÜ1ÐlçßSÛaрÜ>ïlÖ6ß>ÔÕÐlÑBäqÜ1Ðqà>ÛaÔUä
ÛaÑjÚlÛaѐç&ÑjÐ(ÖaçÖaå¤Ñ„ÔÕФÔJÖaÔUß>Ù#ç&Ü>ÙUØgÖaÔÕÜ1зÜ1Ð
€ßSÖÖaålрÚgÛaÑjèGÔUÜ>ؤçÖaÔÕÞqрçLÖaÑjÚú
À Öaålяç&Ü1ÙÕØgÖaÔUÜ>Ðîßé:ÖaѐÛìÖ¹ëÜ €öӐõgÓjÙÕÑjçjúlѐÖaÓ1ò
ç&Ü>ÙÕØlÖaÔÕÜ1мßSé@ÖaѐۄÖaå¤Ñ€ø¤ÛaçLÖìø.Ðlяà>ÛaÔUä „öQÓ5õlӐÙUÑ>ú
ÿ Öaå¤Ñ
9A/4.0D0;4
50/l;<79-.>89)>8Dd;@D0;J9-;<>8H²P50/7-7'9-2ƒ9-.0/R7k2HJ59A;J2D >9)>P;@O/DdHJ/O/HM;<7=0?A2OF;@Q0/QjhFVK;@D
9-/?-=x2H<>9A;J2D
21,9A.0/[email protected];J2Dc2Df9-./D0/6nF9462>?A7-/7k9‡HJ/O/Hž ,9R>[email protected]„h59l9A.0/ƒ462
>8?7k/7"9P?-;<[email protected]/EO/H„>je0n/QcDF50Glhx/E?
2
1 'W[46V4EHJ/7B;@7‡>[email protected];J/QœYœ9-./7A>8G/DF50Glhx/E?‡2
1 W€4EVF4EHJ/7EY " " YIh/;JDPj>[email protected];J/Qs>89l>[email protected]¸;JD
9A/E?AGR/Q;<>9-/
P?-;<Q [email protected]/EO/EH<8
7 < ƒ132?‡N%.0;<4. - W ,9‡9-.0/K462>?A7-/7k9BHJ/O/EHr>Df/6n0>[email protected];JD/>[email protected]/?BG>uV h/
>[email protected];J/Q‚2?¸>[email protected]>?-P/ŠDF50G‚h/?¸281œ7kG2F289A.0;JDP);J9-/?A>89-;@2D7m=x/E?-132?AG/
Q B;@DR9-.0;<7¸N,2?AqB9-.0/ 132?-G/?r>8=0=0?A2>4.
;<7>Q02=9A/QœYIh>7-/Qs50=2Dc!'/EN 9A2D B 7‡G/69A.02Qœ ,9‡9A.0/O/E?AV]h/P;@D0D0;@D0Pd281,9-./1350HJH,\d50HJ9-;@P?A;@Qf4EVF4EHJ/
9A.0/[email protected];@2D13?A2G 9-./=0?A/EOF;@257m9A;JG/)7"9A/E=g;<7„57-/QR9A2=0?A2OF;@Q/Z>8Dg;@D0;^9A;@>H?A/7-;<Q5>8H>8DQg>7„>7k9A>?k9A;JD0P
=x2;@D
9%132? 9-./\d50HJ9-;@P?A;@QK=0?A2F4E/Q5?-/
[Dj2?Q/E? 9A2g/6n=0H<>8;@D`4EHJ/>8?AHJVg9A.0/l7k9-/=7 9A>q/D`Q5?-;@D0Pg>g7k;@D0PHJ
/ W€4EVF4EHJ/YF9-.0/lP2O/?-D;JD0P‚/
5>89-;@2D7
k7(6,
>DQ ž~~ tY0>89,9A.0/‡=>?k9A;@[email protected]>?%P?A;@[email protected]/EO/H YGg>uVhx/‡?-/N%?-;J9k9A/ED`;JD`9-./[email protected]%;@D0PN,>uV B
'
*
1
OL
*71
!
OL
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
`
.
.
-FO
`!
X~6& *71
O
O
L .
X~ (6
!
!
g¤
!
›t¡EµX¯¬®t±6¥
!
!
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
,
î
:¦íq>’¨°q.
Ȳ
¦.ƒ"
¦ƒ
²¨
! ;JO/ED ! 1 U
!
U ! >7‡>8D™;JD;^9A;@>H¸P50/7-7)9A2K9A.0/[email protected];@2Ds281
N%.0/?-/‚9-./[email protected]…9A.0;@4qFD0/7-7-/7B>DQ]=0?A/7A7-50?-/R>9)9-.0/
9-;@G/g7"9A/E=P >8?A/lqFD02N%D™>8DQs>8?A/lP;@O/D
`
`
1
1 >8DQ L
Y>DQ
4E2?A?-/7k=x2DQ07Š9-29-./)?A;@P.
9kW€.>8DQK7-;@Q0/)21²/
5>9A;J2]
D X~~ 6
L !
!
>DQ
L
!
hFV
/
5>9A;J2D.
7 ž~ & )>8DQ
ž~:(66Y9-.0/
4E2G=059A/Q]O>HJ50/7B>1¶9-/?‡[email protected]/69A;J2D 21r9-.0/ W€4EVF4EHJ/>8?A/PB 1
>DQ L ?„>4. 'W[[email protected]/RN%;^9A.0;JD
9A.0/g7k2HJ59A;J2Ds=?-246/7-7)Gg>uV]hx/gQ/6eD/Q™;JDf>`?-/4650?7-;JO/lGg>8D0D /E?‡57-;JDP`=?- /46;<7k/HJVj9A.0/g7->GR/=7-/E5Q28W
4E2FQ0/'132?AGg>[email protected];@7-G|>7Š;JDK#M?A289-9-/EDFhx/E?APƒ} ~6+ B
[D1350D469-;@2D>HD289A>89-;@2Dg9A.0;<7W€[email protected]/'13?A2GwHJ/O/HqGg>uVh/
N%?A;^9-9-/Dj>7
71
_ S Q L
`
VO1
U
OL U
!
O
O
O O
'
!
!
!
!
.0/B135D4t9A;J2DX8)a+
‡+zGg>uVhx/‡h0?-2q/DKQ2N%D`;JD
9A29-.0?A/E/‡Q;<7"9A;JD4t97k9A>8P/7
*
#
O
!
*
.
'
'
'
.
B
!"$#%
& /=05=0>[email protected] ;@2 D7 ž~6&:9-;@,G>8/D7Q >fž:~ $%(6/,QF7-W 2‚Š9AH<>.4>q 9Y0 ;@DK>851357AD7"W"4taF9A/;J2;@QD/>8H'Hœ7-D042.9A/E>8G9-;@/c2D } ~ ,u'9-2c>™[email protected];JD0/>8?A;@7-/Qp132?AG
"'1 U O( U `*) W _ M 1 U O U O O !
!
!
!
!
~0,+-#-./1032%.%214536#%
& +Š2GR=59-/B9A.0/Q/613/4t97
` "'1 U O7( U O `
'1 U O8( U B
'
21
B
'
'
L
*
L
*
.
'
.
B
6B
'
+
ž !'289A/)9-.>89
P/ED0/?A>H132?
&
!
)
'
'
+
L
!
;@Dj250?%4>7-/Y
13?-2G|/
5>89-;@2D
*
!
*
.
ž~6&:tY
`
'
A
WY0>7%7-.02N%DKhx/[email protected]‡)
L
!
'
.
-Y.02NŠ/O/?„9A.0;@7 ;<7 D029,9A.0/‡4E>7k/);JD
$%/7"9A?-;<4t9)9-./RQ/E13/469Z9A29-./RD/6nF9)4E2>8?7-/P?A;@Q]HJ/O/EH²57-;JDP`>ƒ.>8HJ1¶WXN,/E;@P.
9A;JD0Pƒ?-/7"9A?-;<4t9A;J2D
2=/?A>89-2?Y 9) }JˆuB
`9) ) O `9) ) )
>8DQ9( U ;@D`9-.0/7A>8G/BN,>uV
U
!
!
'1 U ) `*) '1 U O:( U `*) ;( U !
!
!
!
'
'
&
$%/7"9A?-;<4t9 1
9
9
9
'
B
L
9
&
'
9
9
L
*
9
*
9
.
L
.
'
+Š2GR=59-/B9A.0/‡?-;@P.
9-WX.>DQƒ7-;<Q/>9 9-./‡D0/6nF9462
>8?7k/?Š[email protected]/EO/EHB
"'1=U ) O( U ) O ) ` '1 U O8( U !
!
!
!
Š
+
2
R
G
=
5
9
%
/
8
>
D
8
>
=
0
=
?
u
2
n
@
;
g
G
>
9
,
/
k
7
2
J
H
0
5
9
@
;
2
=
D
<
>
O
<
2
1
9
.
/
E
4
2
>
A
?
7
„
/
P
?
<
;
Q
/
5>9-;@2DR2D
&
) )
9
`
)
)
9
$
'
*
9
9
L
'
'
L
*
9
9
.
9
9
$
9
9
'
.
)9 '
)9 O
'
)9 O
) O
) O
9
9
O
`
'
9
O
9
O
O
'
9)
9)
`
9)
9)
O
X~6, '
'
€1xq ‡ B7k2HJO/„/ F7=ž~ + :>8DQ ž~6,::h
V‡=x/E?-132?AG;JDP9A.0/F8Za‚[email protected]/Š57k;@D0P 1=U
>7%>8D`;JD0;J9-;<>8H:>8=0=0?A2un;JGg>89-;@2D B
›t¡EµX¯¬®t±6¥
g¤
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
X~6+ '
€1mq ` PBFeDQK9A.0/‡/6n0>469'7k2HJ59A;J2DK57k;@D0PR9-.0/4E2>?A7-/ZP?-;<Q`7k2HJO/E?
7
<
! )9 =( U ) >8DQ
L
!
9
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
B
-
¦,
§ „§8¨
¹¤x­¦0§¹,(
§ B
œ¹,­(R§(>.¤*¸(
­ „§,R§,B¦,
* < )9 O/< '
9 .
O
O &
_ S Q
`
*
O '1 U
+
8O ( U
L
)9 !
)9 !
+Š2GR=59-/B9A.0/462?-?A/469-;@2D7 B
<N 9 &
< 9 O )
`
'1
+
U
O <N '
!
9 O
+
'
O
9)
< ' 9
O `
)9 (U
L
O
)9 !
)9 '
O &
)9 ˆ51!"$#%
[email protected]
' !
B
'1
`
U
!
<N $
'
<N 9 )9 `
< +Š2GR=59-/B9A.0/462?-?A/469-/QK>8=0=?-2un;@G>89-;@2DK2D
1 U
&
'
* <N O><N
)9 9 .
`
O
}JˆuB
'
B
U
L
'
'
[D
9-/?-=x2H<>9A/)9-.0/4E2?A?-/4t9-;@2D7Š57k;@D0PR9-.0/‡h;[email protected];JD/>8?%;@D
9-/?-=x2H<>9A;J2DK2=/?A>89-2?Y
<N ' .
!
`
(U
L
!
<N ' $
9-;@G/7 9A.0/‡?A/EH<>n0>9A;J2DK7A4.0/EGP
/ B
*1 O
` ) W
. 9
L
!
' *
_ M
!
1 U
!
!
OL U
!
O
O
'
.
# .0/>7A7k246;<>9A/Qj?A/EH<>n0>9A;J2D]7-4./EG/‚4E2D7-;@7k9A7Z281„>`7-;JD0PHJ/R!'/N 9-2D ;^9A/E?>9-;@2Ds7k2HJO/QdN%;J9-.™>ge0n/Q
DF50G‚h/?,21²7-NŠ//E=7 ' %2? '" Š281²$%/QFW ,[email protected]>4q >857A7kW[aF/;@Q0/EHXF# ./[email protected];JD/>8?A;<7k/Qg132?AG 21M/
5>89-;@2D7
X6
~ & „>8DQ\ž~:(6¸;<7,2h9>8;@D0/Qh
V4E2G=059A;JDP9-./[email protected]>8H
>462h0;<>8D2D0HJVY
7-29A.>9 />4.?A/EH<>n0>9A;J2D7kN,/E/=
’;JD`9-./‡=0?-/EW€7-G2F289-.j7k9A>8P/132?%/6n0>[email protected]
/ ,=?-246//Q07,[email protected];JDP B
!
1
!
!
!
.1
!
'
1
!
$
L
!
.1
L
!
!
`
L
'
!
!
$
+
`
L
'
*
+
1 U
!
' *71
!
!
OL U
!
U
OL U
!
O
.
žˆ - O
.
žˆ0
D/EN
>=0=0?A2un;JGg>9A;J2D|9-2 9-./[email protected];J2D ;<7—P;@O/ED hFV B1 U *" `
L [email protected]'/ED46/_>
1 U # " .1 O L U #" ` L U # " L
# ./o!'/50Gg>8D0D_hx25DQ0>8?AVb462DQ;^9A;J2D7 >?-/T9-?A/>! 9A/QbhFV
$
$
'
;@D
! 9-?A2FQ0546;@D0PcP.27k9! D2FQ0/7g25! 9A7-;@Q0/j9-.0/ 4E2G=059>9A;J2D>8HQ02Gg>8;@D˜>8DQ—hFVp;@G=2
7k;@D0Pf9-./dOu>HJ5/7g21
Q?A2=HJ/E9M9-.;@4qFD0/7-7-/7²>8DQ=0?-/7-7-50?A/„>9²9-./7-/„=x2;@D97:9-2hx/Š9-.0/ 7->G/„O>[email protected]/7²>7M9-.27-/,>89M9-./Šhx250DQ0>8?AV
U
,9‚9A.0/j462>?A7-/[email protected]/EO/EH ; < tY²9-./`Q;<7A46?A/69-;<7-/Qc/
5>9-;@2D7‚>?-/`7k2HJO/Qc57-;@D0P™>s!'/N 9-2Do;^9A/E?>9A;J2D
7A4.0/EG/0# .0/‡;^9A/E?>9A;J2DK;<7%Q/6eD0/Q`;@DK9A/E?AGg7 281²9-.0/
>4E2h0;<>8DKG>89-?A;@4E/7B
132?
.1
>DQ
_
'
`
'
!
!
U
L
U
O
'
`
W
!
!
!
1 U
U
O
`
U
!
!
!
L
U
O
`
U
!
!
!
1 U
'
žˆ~ !
;@O/D
> 7Ie?7k9²>8==0?-2un;@Gg>9-;@2DB9A2'9A.0/,4E2>?A7-/¸P?A;@Q/
5>9A;J2Db
YE9A.0/ED‚;JD4E?-/G/ED
9A7
U O U ' ?A/7-=/4t9-;@O/HJVY>?-/%2h9A>;JD0/[email protected];JDP‡7 9-.0X ~6/Z+:HJM;@D0>/D>8QD?A;@ž 7-~ /, Q6!'
O ' O
9A2
/N 9-2D;^9A/E?>9A;J2D
U O U Y
U >DQ U
O
O
O
O
/5>9-;@2D76B
_
›t¡EµX¯¬®t±6¥
g¤
'
O
O
U
'
W
U
$
$
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
U `
O
O
U `
U
'
'
+
U
+
U
U
U O
O
O
U O
'
O
O
U
'
U
O
'
žˆˆ žˆ6:
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
î
:¦íq>’¨°q.
Ȳ
¦.ƒ"
¦ƒ
²¨
aF2HJ509-;@2D
`
21g9-.0/7k/—/ 5>89-;@2D7s;<7 2h9A>;JD0/Q >7][email protected]'7E‡aF/E9k9-;@D0P ;@D žˆˆ 6Y‡>8Dv;@D0;J9-;<>8H
U
O
'
'
'
'
'
)
9
`
_
+
;<7 B9
Y¸N%.0;@[email protected]/j9-.0/d/
5>9-;@2Dp132?
>=0=0?A2unF;@Gg>9A;J2Dc9-2 ;<7
U O
U
U
U
U
U
' O U
O
VO
O
O
2h9A>;JD0/QƒhFVK/[email protected];JG;@D>9A;JD0D
P 13?-2G /
5>89-;@2D7 žˆˆ ,>DQ ’ˆ6%tYOF; U
O
5>9-;@2D
'
>DQ
U žˆ
?9
O
+
9)
_
W
';@7Z7k2HJO/QK9-2KVF;J/[email protected]
'
'
O
U
$
U
O
+
9)
_
U
$
'
W
U
O
+
U
O
'
U
O
`
U
>8DQ
U
O
O
'
'
+
U O
U
U
O
O
O
'
'
8
>
A
?
/
0
5

=
0
Q
>
A
9
/
]
Q
>
E
4
E
4
2
?
Q
J
;
D
P‚9A2
U
+
U
O
žˆ
O
U
O
U
Y0?-/7k=x/469-;@O/HJVF# ./‡;^9A/E?>9-;@2D`=0?A246/E/Q07 50D
9-;@Hœ9A.0/‡D02?AG
U
U
VO
'
O
O
U
$/
U
$
$
+
U
U
VO
O
'
O
U
$
O
U
$/
N%.0/?-/ DR;@7²9A.0/%DF50Glhx/E?¸2150Dq
D2N%D7²;JD2D/ Q0;J?A/469-;@2DIYu;<7¸7k5C4E;J/D9AHJVR7kGg>HJHX[Dl9A.0;<7mNŠ2?-q‡N,/,13250DQ
9A.>9 >‡?-/[email protected]>89-;@O/%?A/Q54t9-;@2Dg;@Dg9A.0/'?A/7-;@Q5>8H2%
1 5"57"9)- 9 Q ž462G=>?-/QR9-2‡9A.0/;@D0;J9-;<>8H?-/7k;<Q5>H013?A2G 9-.0/
[email protected];@2D>89„9A.0/=0?A/EOF;@257r9-;@G/Z7k9-/E3
= m
9-2‚h/)7k50Cg4E;J/D
9Y
.02NŠ/O/?rGl54.gP?-/>9A/E?„>4E465?A>46;@/7r>?-/'/>7-;[email protected]
2h9A>;JD>[email protected]/);J1m?-
/ 5;J?A/Qœ
%$;?Ša0* I #'a
# 0
. //6C46;@/ED4EV‚>DQg>44650?>4EV2819-./\d50HJ9-;@P?A;@QG/E9-.02Q;<7¸e?7"9„Q0/EG2D7k9-?>9A/QRhFVR4E2G=>?-;<7k2D‚N%;J9-.
>™7k/?-;@/7‚281)>8D>HJV
9-;<4E>H,>DQT=?-/O
;@[email protected]
WX?A/E=x2?-9-/QsDF50G/?-;<4E>H,?A/[email protected]/697k=?-/>Q;@D0P]L2N'7E
!/ENU?-/7k5H^97r>8?A/ 9-.0/Dg=0?A/7-/ED
9-/Q‚N%.;@4.g7-.02No.2Np9A.0/'G/69A.02Qg4E>8Dghx/'?A/>Q;[email protected]‚>[email protected];J/Q‚9-2l7-;JG‚50H<>9-/
Q?A2=HJ/E9 G29-;@2D`2Dd7k5h7"9A?A>89-/7ŠN%;J9-.j9A2=x2P?A>=0.0;<4Z>D
Q >u2? NŠ/E9k9A>h0;@HJ;J9"V;@D0.02G2P/ED/E;J9-;@/7
‘‘EŒ-“‘
ƒ• B” =:•‘“
‹ ,t-•8
9!t”žg
“”
# .0/Z>4E4E50?>46Vl2819-.0//7k9-;@G>89-/'21 I#F?'Y8/
5>9-;@2DEX~%tY8;<7„Q/G2D7k9-?>9A/QRhFVR4E2G=>8?A;@D0PB;^9,N%;^9A.9-.0/
9A?-50/[email protected]>8Hœ/?-?A2?,132? 9-.0/‡2?AQ;@D>?-VQ*; /E?A/ED
9-;<>8Hœ/
5>9-;@2D
1
$
1
`
-
O
1
`
N U 1
U
žˆ
'
& # .0;<7%
/ 5>9-;@2D`;<7%4.027-/EDj7k;@D4E/);J9A7>8D>HJV
9-;<4E>Hœ7k2HJ59A;J2D
1
N
`
1
U
U
U
/
*
žˆ
O
:(6
;<7m281x/6n0>[email protected]‡9-.0/'7A>8G/Š132?AG >7¸#²>8DD0/E?B 7 } ˆ -u[email protected]/Q‡132?-G 7k2HJ59A;[email protected]_9-.0;<4qFD0/7A7¸>9m9A.0/'46/D9A?-/
21 >Df>8n;@7-VFGRG/E9-?A;@4>[email protected]]7-=0?A/>Q0;JD0PdQ?A2=HJ/E92O/E?‡>j.02G2P/D0/E257B7k5h7"9A?A>89-
/ 7-/E/gH<>9-/?M€1F1
;<7
9A.0/DF50G/E?A;@4>[email protected];@2Dj21m
/ 5>9A;J2D žˆ &:,2h9>8;@D0/Qj>99-;@GR/ 57-;@D0Pg9-./B9-/EG=x2?>8HMQ0;@7A46?A/69A;@7A>9A;J2D
7A4.0/EG/%Q/7-4E?-;@h/Q‚>h2O/Y9-./EDR9-.0/ 9A?-5/,[email protected]>H/E?A?-2?²;<7²9-./'Q;/E?A/ED46/ h/E9"NŠ//EDD1
Yu9A.0/%DF50G/E?A;@4>8H
[email protected];@2Dj2h9A>;JD0/Qj>9%9-;@G
/ Y>8DQƒ9A.0/l4E2?A?-/7k=x2DQ0;JD0P>8D>[email protected];@4>[email protected];@2D`132?%9-.0/7A>8G/‡;@D0;J9-;<>8H
U
4E2DQ;J9-;@2D 1 N ` 1 YD>8G/[email protected]
V 1 N c
/
8²;JP50?-
/ @‚7-.02N'7Z.02Nb9-.0/?>9A;J2K281¸9-./R/7"9A;JGg>89-/Q HJ24>8H:9-?A50D4E>89-;@2Ds/E?A?-2?Y3?b
I #F?'d
Y ž6
~ :%9-2K9-.0/
9A?-50/lHJ24E>HI/E?A?A2?Y0F
# IC?'YOu>?-;@/7'N%;^9A.d9-;@G/l>8DQj=?-/7-4E?-;@h/Q`/E?A?-2? [email protected]/E?>8D46/Y#&jr
I 0€9ZQ0/EG2D7k9-?>9A/7
9A.>9B9-./R/7"9A;JGg>89-/462D7k;<7"9A/ED
[email protected]/?Z=?-/Q;@469A7Z9-./‚9A?-5/‚/?-?A2?B>8DQ]9-.>89Q/4E?-/>7-;JD0Pƒ9-.0/[email protected]/E?>8D4E/
;@D46?A/>7k/7Š9-.0/‡>44650?>4EVg281:9-.0/‡/7"9A;JGg>9A/)9-2R9-.0/‡/EnF9-/ED
9 9-.>9%132? R W i - 9
)9-.0/‡/7"9A;JGg>89-/[email protected]>8H
/?-?A2? ;<7 N%;^9A.0;JDdˆ -y21²;J9A7 9A?-5/)O>HJ50/
!
!
!
!
!
!
!
!
!
›t¡EµX¯¬®t±6¥
g
¤
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
u~
¦,
§ „§8¨
¹¤x­¦0§¹,(
§ B
œ¹,­(R§(>.¤*¸(
­ „§,R§,B¦,
3
−1
TOL = 10
−2
TOL = 10
−3
TOL = 10
−4
TOL = 10
−5
TOL = 10
−6
TOL = 10
ELTE/TLE
2.5
2
1.5
1
×ÔUà>ØlÛaÑ
0
1
2
3
t
4
5
6
gòl΀ßÖaÔÕÜÜ>éMgó©ÖaÜqó¸ ©éÂÜÛ3ÑjñGؤßÖaÔÕÜ1Лá1âìßç3ßXéÂؤÐlӐÖaÔÕÜ1ÐÜ>éOÖaÔÕÞqÑ1úGÖBú.ßÐ.ä
ó U‘6<E–x‘ K• ”
Œ.Et’‘“‹$9PE”
•
ò
:
# .0/Z/6C46;@/ED46VR2819-.0/)\d50HJ9-;@P?A;@Qƒ>8DQR9A;JG/6W[7k9-/E==0;JDPl=0?A246/Q050?-/7r>8?A/Q/G2D7k9-?>9A/QRhFV4E2D7-;<Q/E?A;JDP
9A.0/ [email protected]/[email protected]/69¸7k=0?A/>Q;@D0PZ2D‚>Z.02?A; E2D
9>8HF7k50h7"9A?A>89-/Š132?¸>=>8?-9-;<>[email protected],/69k9A;JDPB7-V7"9A/EG_N%.0/?-/
` -:t 82?
9A.0//
50;@[email protected];Jh0?A;@50G 4E2D
9A>4t9>[email protected]/D 7 - >DQ`9-.0/l'/ /4t97%281¸P?A>uOF;J9"V>8?A/‡D0/EPHJ/4t9-/Q W4` X
9A.0/Z=>8?-9-;<4650H<>8? 4E>7k/)281:>‚7k=?-/>Q;@D0P‚>8n;@7-VFGRG/E9-?A;@[email protected]/69Y4E/ED
9-?A/Qƒ;JD0;J9-;<>[email protected]@V>9 E DCTO F=
- OT- tY
'
'
9A.0/[email protected] 9-.;@4qFD0/7-7%>89%9-.0/Q?A2=HJ/E9%4E/ED
9-?A/Y 1 U <tY0;<7%P;JO/EDs} ˆ -MhFV
1=U:N `
,
132F
? 7
-Y
žˆ + N%.0/?-/
>DQ ` - ~ ,>8?A/7-=0?-/>Q;@D0P462D7"9>8D
9A7Y>R?A/7-50H^9'O/E?A;^e/Q`hFV II/[email protected]>.`>DQj\]>8?AGl5? B 7‡} ˆt
'
/EnF=x/E?A;@GR/D
9A7‡N%.0;<4.f1325DQ - & 4 4 - ˆ
~M# .0/K>n;@7-VFGGR/E9-?A;@[email protected]?A;<4E>9A;J2Df/
5>9A;J2D7B132?‡9-.0;<7
'
'
=0?A2hHJ/G NŠ/?-/‡7-50h7-/ 50/D9AHJVK7k2HJO/QDF50G/?-;<4E>[email protected] >8?-9B>[email protected]/EVs} +u:N%.02g;JDFO/7k9-;@P>9A/Q9-.0/
;@DL50/D46/d21Z9A.0/d=0?A/[email protected] 9-.0;<4qFD0/7A7D132j2D—9-.0/]7-=0?A/>Q;JDP™?>9A/j=>8?>8G/69A/E?7
>8DQ ;JD
/5>9-;@2D]’ˆ + [email protected]/E?A/Y
9-.0;<7ŠL2Nz;<7 7"9A5Q;@/QKh
[email protected];JDP9-./)[email protected]?A;<4E>9A;J2D/
5>89-;@2D7 kŠ>8DQP" :r
132?
9A.0/‡=>8?-9-;<[email protected]>? 4>7-/B281 ; ` - - - 0YV GO 5 ` ž=
ˆ O~ tYF9A.0/‡;@D0;^9A;@>[email protected]/69%=0?A28eHJ/B;<7 P;@O/ED`hFV
'
1 6 `
5D +
ˆ
~6,
/ *
O-
O
*
žˆ , N%.0/?-/ T;<7ƒ9A.0/™?A>Q;J57ƒ219-.0/s132F289A=0?-;@D
9`219-./™Q?A2=HJ/E9`2D˜9-.0/f7-50?k1’>46/s2819A.0/c7k50h7"9A?A>89-/ 'D
/5;[email protected];Jh?-;@50G 462D9>469l>8D0PHJ/21‡ 8ˆ
N,>74.027-/EDT7-2j9-.>9.1 U <
ƒ>[email protected]>?-P/9-;@G/7:!'29-/ƒ9-.>89Y
'
>=>8?-9 13?-2G|9-.0/‡;@D0;J9-;<>8HI=?-2eHJ/BD02g>8nF;<7-V
GG/69A?-Vg;<7%;@G=2
7k/QK;@DK9-.0/‡DF50G/E?A;<4E>[email protected];J2D7E
# .0/[email protected]%;@D0P 7-/
50/ED46/ƒ21%eP5?-/7/[email protected];@Q0>89-/ƒ9-.0/`/'/469R21%9-.0/`\d50HJ9-;@P?A;@Qp=>8?>8G/69A/E?7‡2DT9-.0/
4E2DFO/?-P/ED4E/‚21 [email protected];@2D7Bh
[email protected];JDPd9-.0/g?A/EH<>9A;JO/?-/7k;<Q5>[email protected]>89‡9-.0/eD0/7k9lP?A;<Q [email protected]/EO/EHXYIN%.0;<4.f>?-/
Q/EeD0/QR>7m9-./'?A>89-;@2B281?-/7k;<Q5>H0>9¸9-./%eD0/7"9rP?-;<QRHJ/O/EH0>81¶9-/?„>BP;@O/D‚DF50G‚h/?r281xeD0/'P?-;<Q 'W[[email protected]/7
4E2G=>8?A/Q9-29A.0/‡;JD;^9A;@>HI?-/7k;<Q5>Hœ>9%9-./‡h/P;@D0D0;@D0P281²9-.0/B9A;JG/7k9-/E=M
# .0// x/4t9Z21m9-.0/DF50G‚h/?21 W€[email protected]/7 " " '>89;JD
9A/E?AGR/Q;<>9-/‡P?-;<[email protected]/EO/[email protected]<
W9 2Dj9-.0/
?A/EH<>9A;JO/g?-/7k;<Q5>[email protected]%.0/?-/g9A.0/eD0/7k9‚P?-;<Q™HJ/O/EHŠ;@7ƒ~ ,nœ~ ,]’;ž /dW ` ˆ l>8DQ™9A.0/e0n/Q ¶132?l9-.0;<7
4E2G=>8?A;<7k2D
¸9A;JG/l7k9-/=j;@7‡
- 9 Y0;<7%7k.02N%Dj;@D 8:;@P5?-/ ž> 60€[email protected]/>[email protected]ƒQ/GR2D7k9-?>9-/7Š9-.0/hx/ED/6e09'21
›t¡EµX¯¬®t±6¥
g¤
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
ˆ
î
:¦íq>’¨°q.
Ȳ
¦.ƒ"
¦ƒ
²¨
4
4
10
10
0 V−cycle
1 V−cycle
2 V−cycles
3 V−cycles
4 V−cycles
5 V−cycles
0
Relative Residuals
10
2
10
−2
10
−2
10
−4
−6
10
−6
−8
10
−8
10
−4
10
10
10
10
−10
0
1
2
3
4
5
6
V−cycles on finest level
7
65x65
129x129
257x257
513x513
0
10
Relative Residuals
2
10
8
9
10
10
©
Ÿ"¤"¥„›³¥ž±uŸ ¢E°u»Ÿ"m›³ ²º¶¤[µ¤k«¬Ÿ" ²©A¥
¥ž±uŸr¯¬¢E¥žŸ"X»%Ÿ"£u¯©A¥žŸ¸®tX¯¬£«¬Ÿ"´6Ÿ"«¬ M›t¢Z¥ž±uŸr¤"›t¢E´6Ÿ"X®tŸ"¢u¤"Ÿ
±u¯¬ ’¥X›t’µ ›³F ž›t«¬°¥ž¯¬›t¢u -§ I±uŸu¢uŸ" ž¥œ®tž¯¬£%«¬Ÿ"´6Ÿ"«<­ ­6¯¬ I6Ct·g
I6Ct·©¢£ µ I6D
§
−10
0
1
2
3
4
5
6
V−cycles on finest level
7
8
9
10
•
Ÿ"¤[¥o›³d¥X±uŸ!¢Ÿk ’¥p®tX¯¬£ «¬Ÿ"´6Ÿ"«j›t¢ X¥ ±Ÿ
¤"›t¢E´6Ÿ"X®tŸ"¢u¤"Ÿ ±¯¬ ž¥ž›tžµZ›³ ž›t«¬°¥ž¯¬›t¢u M
" ¯ ¥ž± $#
©¢u£% µ I6D & §
1000
10
4
10
2
10
0
∆t=10
−9
∆t=5*10
−8
∆t=10
−8
∆t=5*10
10
−2
10
−4
10
−6
10
−8
100
CPU time (s)
Relative Residuals
−9
65x65
129x129
257x257
513x513
1025x1025
10
10
−10
0
1
2
3
4
5
6
V−cycles on finest level
7
8
9
¤'ð
Ÿ"¤[¥]›³¥ž¯¬»%Ÿs ’¥XŸ"¡ t› ¢z¥ž±uŸf¤"›t¢6´6Ÿ"X®tŸ"¢u¤"Ÿ
± ¯¬ ž¥ž›tžµ¸›³ X›t«¬°¥ž¯¬›t¢u &"M¯ ¥X± ' (# ©¢u£r¥ž±uŸ¢Ÿk ’¥
u
®tX¯¬£«¬Ÿ"´6Ÿ"«<­ ­u¯¬ I6Ct·lI6tC ·§
10
1 3
10
10
4
5
10
Number of Unknowns
6
7
10
10
£)¤:Ÿk¡8Ÿ"¢u£uŸ"¢u¤"Ÿœ›³,
¦ ‚¥ž¯¬»%Ÿ© ©I³J°u¢u¤[¥ž¯¬›t¢r›³¥ž±uŸ
¥ž›¥X©«¢°u»8ŸkI›³0°u¢ª¢u*› "M¢u -­+"M¯ ¥ž±-,/.10 I6D &2 §
×ÔÕà1ØlÛaÑ43lòá65#â5ú
á67â5úá8(â'
9 ѐüѐӐÖÜé€ÞTؤÙJÖaÔÕà>ÛaÔUäùÚ.ßSÛ6ß>ÞqѐÖaÑ5ÛaçÈÜ1ÐùÖaålÑ8ÓjÜ1ÐGè,ѐÛaà>ÑjФӐÑålÔUçLÖaÜÛ&õÜ>é„ç&Ü>ÙUØgÖaÔÕÜ1Фçjòìá :â'9
älÑjÚ#ѐÐ.älѐФÓjрÜ>éý
;Ë
ÖaÔÕÞqÑìéÂÜÛ3ߋֹõgÚlÔUÓjß>Ù#ÖaÔÕÞqÑ3çLÖaÑjÚß>çìß!éÂؤÐlӐÖaÔÕÜ1ÐÜ>éÖaålрÖaÜÖ6ß>ÙÐGؤÞÈï#Ñ5ÛÜ>éؤРgÐlÜBëìФçjòg×ÙÕÜBë
ÓjÜ1ФälÔJÖaÔÕÜ1Ðlç<9GÑjè,Ü1ÙÕØgÖaÔUÜ>Ð8Ü>éƒß‹Ú.ßÛ&ÖaÔUßÙUÙJõqëѐÖ&ÖaÔÕФàXägÛaÜ>Ú¤ÙÕѐÖëìÔJÖaå4=?>[email protected]
á6LNM
â @—á : ÿ â
›t¡EµX¯¬®t±6¥
g¤
CBDBDBD‹t›t±u¢
GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
ú)A/@
)B +C E
ú [email protected]
B C IHgú)J š @
KB 3ß>Фä
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
¦,
§ „§8¨
¹¤x­¦0§¹,(
§ B
œ¹,­(R§(>.¤*¸(
­ „§,R§,B¦,
57-;@D0P]>[email protected]/>7"92D0/ W€4EVF4EHJ/>899-./;JD
9A/E?AGR/Q;<>9-/[email protected]/EO/EH,>7B9A.0/?A/EH<>9-;@O/g?A/7-;<Q5>8H<7‡>8?A/g?A/Q54E/QfhFV
>HJG27k9 9"NŠ2g2?AQ0/E?7Š21²Gg>PD0;J9-5Q0/l462GR=>8?A/Qg9A2g9-./4E>7-/BN%.0/E?A/‡D02g;@D
9-/E?AG/Q;<>9A/ 'W[[email protected]/B;@7'57k/Qœ
H^9A.0250P.KG2?-/)9-.>8D`2D0/ 'W[46V4EHJ/‡Q2F/7 D289%;JG=0?A2O/)D029-;<46/>[email protected]/4E2DFO/?-P/ED4E/281:9-.0/[email protected];J2D
>89%9-.0/BeD/7k9'HJ/O/EHœh5957-;JDP>R.0;@P./E?'2?AQ/? ;JD
9-/?-=x2H<>9A;J2D`7A4.0/GR/‡7-54.d>7 h0;<465h0;@4B;@D
9-/?-=x2H<>9A;J2D
132?'9-./805HJH¸\d50HJ9-;@P?A;<Qj;JD
9-/?-=x2H<>9A;J2Dd;JD7"9A/>Q]281rh0;@HJ;@D0/>8?;@D
9-/?-=x2H<>9A;J2DdG>uV`;@D46?A/>7k/B9A.0/lhx/ED0/Ee09
2157-;JD0P™G2?A/ƒ;@D
9-/E?AG/Q;<>9A/ W€4EVF4EHJ/7E¸# .0/`;@D0;^9A;@>H ;JD4E?-/>7-/ƒ;@Dp?A/EH<>9A;JO/ƒ?A/7-;@Q5>8H<7l;<7Q50/K9-2s9-.0/
1’>4t99A.>9‚>89‡9-.0/ƒh/P;@D0;JDPd281 9-.0/K7k2HJ59A;J2Df=0?-24E/7A7B9-.0/ƒ;JD0;J9-;<>8H„P50/7-7YIN%.0;<4.T462?-?A/7-=2DQ07Z9-2]9-.0/
[email protected];@2DU>9g9A.0/d=0?A/EOF;@257R9A;JG/]7k9-/E=MY„7->89-;<7"e/7 ž~ & ‚/6n0>[email protected];JD46/d;J9A7g?A;JP.
9kW€.>8DQo7k;<Q/]Q2F/7D0289
4.>D0P/:# .F57Y²>81¶9-/?‚>de?A7k9R4EVF4EHJ/Y²Q0/7-=0;^9A/K>d?A/Q054t9A;J2DT281%9A.0/ƒ?A/7-;@Q05>8H<7‡132?‚
/ 5>9-;@2D X~ ( tY:9-.0/
?A/7-;@Q05>8H<7 281m/
5>89-;@2D ž~ & ,4E>D`[email protected]ƒ;@D46?A/>7k/0# .0/‡Gg>8PD;^9A5Q/‡281²9-.;@7;JD0;J9-;<>8HM;JD4E?-/>7-/B;JDd?A/EH<>9A;JO/
?A/7-;@Q05>8H<7 ;@7Z46?A;^9A;@4>8H9A2g9-.0/‚7k5446/7-7 21²9A.0/[email protected];@2Dj=0?A246/Q050?-/>DQK9-.0/?-/E132?A/Y>H<>8?AP/? D
5Glhx/E?'21
`
=0?A/7-G2F289-.;JD0P]7-N,/E/E=7‡;@7‚=x/E?-132?AGR/Qs132?9A.0/ge?7k9 'Wk+Š[email protected]/P %BN%.0;@HJ/9"NŠ2d=0?A/7-G2F289-.;JD0P
7-NŠ//E=7,2D0HJV>?-/Z1325DQK9-2Rh/7-5C46;@/ED
9 132?,9-./[email protected]%;@D0P W€[email protected]/7B "
;<7%7-/69 9-2g~‚?-/P>[email protected]/7A7r21
9A.0/[email protected]/BDF50Glhx/E?
# .0/r/ x/4t9M21
9A.0/rP?-;<QB7k; E/r>89I9-.0/reD0/[email protected]/EO/EH8;<7I/EnF=HJ2?-/QD0/EnF9I132?²>,e0n/QBDF50Glhx/E?M281;JD
9A/E?AGR/Q;<>9-/
` ˆ >8DP
` - 9 V8²;JP50?A/
’h
'7-.02N'7'9-.>89Z132?Z>[email protected]²P?A;@Q07YN%;J9-.]Wz?>8D0P;JDP
W€4EVF4EHJ/
7 Q 13?A2G|ˆ k~ ,nœ~ , :9-2D
ˆn ˆ
t
Y89-.0/)?A>89-;@2l21œ9-.0/)?-/7k;<Q5>[email protected],>1¶9A/E?Š9-.0/B462G=0HJ/E9-;@2D21I9-./ W€4EVF4EHJ/
!
;<7'/ x/4t9-;@O/HJV`;@DQ/E=x/EDQ/ED
9Z21m9-.0/leD0/7k9P?-;<Q`HJ/O/EHX+Š[email protected]/E?Z>8D>HJV7-;@7%21m9-.0/RQ0>9>?-/O/>[email protected]%9-.>89'9-.0;<7
?>9A;J2`9A>q/7)>`O>[email protected]/g21 >=0=0?A2unF;@Gg>9A/[email protected]™
>8~ -0 8²;JP50?-/.3’4B7-.02N'7Z9-.>9 ¶enF/Q )9A;JG/7k9-/E=c7-;*/.>7
` c- [email protected];^[email protected]//'/4t9B2D 9-.0/462DFO/E?AP/D46/l.0;@7k9-2?-Vj50D
9A;JHr>9-.0?A/7-.02H<Q]O>HJ50/
9 ‚;@7)?A/>4.0/Qœ
# .0/;@D0;J9-;<>8Hr;@D46?A/>7k/R;@Ds?A/EH<>9A;JO/‚?A/7-;@Q05>8H<7)h/462G/7‡7"9A/E/E=x/E?>DQ™>8hx2O/‚9-.0;<7)9-.0?A/[email protected]>[email protected]/R9-.0/
[email protected];@2Ds=0?A2F4E/Q5?-/l1’>8;@[email protected]# .0;<[email protected];@GR;J9-;@D0PjO>[email protected]/R281„9A.0/R9-;@G/7k9-/=sQ/=/# DQ07B2D hx289-.s9-./RG/7k.f7k; E/
>DQ .2N >QO>D46/Q]9A.0/g7k2HJ59A;J2Ds;<7BI>7ZeD0/?>8DQ]eD0/E?‡P?-;<Q07B>8?A/R57k/QœC
Y Q/4E?-/>7-/7B>8DQs>9
H<>9A/E?‚9-;@GR/7EY ;@D4E?-/>7-/7M!'29-/K9-.>89l9-./`4>8H<4650H<>9A;J2D7l>?-/ƒ132?lO/E?AVs/>8?AHJV™9A;JG/7‚N%.0/?-/ƒ9-.0/
7-=0?A/>Q0;JD0P‚G289-;@2DK;@7,G27k9%>4t9A;JO/Z>DQg9A.0/‡Q/EGg>8DQ07,2D9A.0/)Gl50HJ9-;@P?A;@QK7k2HJO/E#? >8?A/G27k9%7k/O/E?A/%'D
#
>Q0Q;J9-;@2D>Hr>QO>8D
9>8P/R9A.0/E?A/6132?-/281,9-./>Q0>=9-;@O/R9-;@G/6W[7"9A/E=0=;JD0P]7A4.0/GR/Q0/7A46?A;Jhx/Q™=0?A/EOF;@257-HJVd;<7
9A.>9Z9-.0/‡9A;JG/EW€7k9-/= 7k/HJ/4t9A/Qj;<7Z>HJN >uV7%7kGg>[email protected]@/E?'9A.>8Dj9A.0;<[email protected];JG;J9-;@D0P9A;JG/R7"9A/E=]>7D02?A/7k9A>?k921m9-.0/
[email protected];@2D`=0?A246/Q050?-/B;<7 /EO/E?%D0/46/7A7A>8?AV
[DcHJ;@P.
921Š9A.0/ƒ>8hx2O/YM>[email protected]„7-50h7-/ 50/D9?A/7-50HJ9A7‡N,/E?A/2h9>8;@D0/Qf57k;@D0PP 'W[46V4EHJ/7>9/>4.fHJ/O/Hž
H^[email protected]/>[email protected]/E?-W€4>859A;J257EY9A.0;@7„.>7„[email protected]‚>l7kGg>[email protected]>Q0Q;J9-;@2D>H462G=059A>89-;@2D>H462
7"9,7k;@D46/GR2
7"9
21:9A.0/‡462G=0509A>9A;J2D>8H9A;JG/‡;<7'7-=/D9'=x/E?-132?AG;[email protected]/7%>9 9A.0/BeD0/7"9'[email protected]/EO/EHX
8²;JP50?-/
žQ 7-.02N'7Z.02Nb9-./g+ ‰ *v9A;JG/‚132?‡>g9"VF=0;<4E>H:9A;JG/7"9A/E=™Q/E=x/EDQ7Z2D P?-;<Q]Q/D7k;J9"Vd>8DQ
Q/G2D7k9-?>9A/7g9A.>9j2D/ 281l9-.0/™G27k9K;@GR=x2?-9A>D
9K=x289-/D
9-;<>8HB>QO>8D
9>8P/721‚\d50HJ9-;@P?A;@QzGR/E9-.02Q07Y
D>G/[email protected]>9j+ ‰ * 9-;@GR/ Ou>?-;@/7HJ;@D0/>[email protected]’;ž /„9-.0/ [email protected]=x/d281B9-.0/[email protected];@D0/];@D 8:;@P5?-/ 3’Q N%;J9-.
9A.0/]9A289A>HDF50G‚h/?ƒ2815D0qFD02N%D7Y„;<7ƒ>4.;J/O/QUhFVo9-.0/™7k2HJO/E?b
7KD029-/Q˜>h2O/YrN%./ED >[email protected];J/Q—9-2
/O/E?-W€;JD4E?-/>7-;JDP462G=0509A>9A;J2D>8Hx=x2NŠ/?Y9-.0;<7Š13/>9A50?A/Z21I9-.0/[email protected]/E?Š;@7Š=>?k9A;@[email protected]>[email protected];JG=x2?-9A>D9,N%.0/ED
7-Gg>[email protected]^W[7-4>[email protected]/Z=.0/ED02G/ED>RD0//QK9-2hx/7k;@Gl50H<>9A/Q`/6C4E;J/D9AHJVK>8DQj>44650?>9A/[email protected]
8²;JP50?-/ =’>: ;@[email protected]?>9-/7'.02NS9A.0/l9A;JG/R7"9A/E=]O>8?A;@/7'>7>1350D4t9-;@2Ds281b1V2Y7k.2N%;JD0P9-.>89)>1¶9A/E?9-.0/
[email protected];@2Dd.>7'7-/[email protected]/Q]Q2N%D`9A.0/9A;JG/6W[7k9-/E==0;JDPƒ7-4./EG//ED>[email protected]/7'7-50h7k9A>D
9-;<>8HM/EC46;@/ED4EVKP
>8;@D7 9-2ƒh/
?A/>HJ;<7k/Q>9,[email protected]>89-/?„9A;JG/7 >DQg9A.>9,9-.0/)O>[email protected]/21 1V2'.>7Š[email protected]>lGg>8?AP;@D>H;@DL50/D46/);JDƒ9-.0/B7-/ED7-/'9A.>9
H<>8?AP/F
? 1V2ZO>[email protected]/7'=/?-G;J9'[email protected]>?-P/E?Š9-;@GR/7k9-/=7 9A2Rhx/‡57-/Q`D0/>8? 9-./‡/ 50;@HJ;@h0?A;J50G 7"9>9-/
# .0/l>4E465?A>46V21²9A.0/[email protected];@2D7%;@7Z462D7k;<Q/?-/QKD/6nF93
8:;@P5?-/ 3h
;@DFO/7"9A;JP
>9-/7 9-./'/ /469281¸P?A;<Q
7-;* /ƒ2DT9-./`Q?A2=HJ/E9l9-.;@4qFD0/7-7.1=U:Nl
132?7-;JG‚[email protected]>89-;@2D7‚N%.0/?-/X132 ` - - >8DQ R W ` —- 938
'
€9j7k.02N'7g9-.>89ƒ9-./[email protected];@2Dz4E2DFO/?-P/7N%;J9-.z?A/7-=/4t99A2T9A.0/s7-=>9A;@>HZQ;<7A46?A/69-;<7A>9-;@2D˜>7g9-./ P?A;<Q
;<7=0?A2P?A/7A7-;JO/[email protected] ?A/6eD0/Qo>DQp9-.>89R9A.0/d7k2HJ509-;@2D7R2Dp9-.0/j9A.0?A/E/KeD0/7k9gP?A;@QpHJ/O/[email protected];*x/[email protected]
>=0=0?A2unF;@Gg>9A/[email protected] ƒ # .0/];@DL50/D46/s281B9-.0/sO>[email protected]/]281B9A.0/ =0?A/4E50?7k2?eHJG{9A.0;@4qFD0/7-
7 132d;@7K7k.2N%D
"
"
'
'
›t¡EµX¯¬®t±6¥
g¤
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[j
µ iǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
î
:¦íq>’¨°q.
Ȳ
¦.ƒ"
¦ƒ
²¨
−4
10
h* = 0.05
h* = 0.04
h* = 0.02
h* = 0.01
h* = 0.005
−5
10
5
4
−6
h0(t)
∆t
10
−7
10
3
−8
2
−9
1
33x33
65x65
129x129
257x257
513x513
1025x1025
10
10
−10
10
1
10
100
Number of time step
1000
0
10000
0
2.5e−06
5e−06
7.5e−06
1e−05
t
©O
Ÿ"¤"¥:›³0¡užŸk¤"°už X›t «¬»p¥X±u¯¬¤Xª¢uŸ" ž
š ›t¢¥X¯¬»%Ÿ
’¥XŸ"¡jŸ[´6›t«¬°¥ž¯¬›t¢
§OI±uŸu¢uŸ" ž¥®tX¯¬£ƒ«¬Ÿ[´6Ÿk«:¯¬ - 2 ­¯<§ Ÿt§
C'1
C§
)…B°u« ¥ž¯¬®tX¯¬£ ž›t«¬°¥X¯¬›t¢ _›³U¥X±Ÿw£uX›t¡«¬Ÿ"¥
¥ž±u¯¬¤€ªE¢uŸ" X x©A¥¥ž±uŸM¤"Ÿk¢6¥žXŸI›³©²¡©ž¥ž¯©«¬« µ "œŸ[¥ž¥ž¯¬¢u®
£X›t¡u«¬Ÿ[¥)© B©R³J°u¢u¤[¥X¯¬›t¢d›³r¥ž¯¬»%Ÿt­ €­M©¢u£j®tž¯¬£
£Ÿk¢ X¯ ¥¶µ ":¯ ¥ž
± š D D
u
§
10
h0(t)
h* = 0.05
h* = 0.04
h* = 0.02
h* = 0.01
h* = 0.005
1
10
−7
−6
−5
10
10
10
−4
−3
−2
10
10
t
¤' B°u« ¥ž¯¬®tX¯¬£ X›t«¬°¥ž¯¬›t¢u ›³ ¥ž±uŸ £uX›t¡«¬Ÿ"¥
¥ž±u¯¬¤€ªE¢uŸ" X ²©A¥I¥X±uŸr¤"Ÿ"¢E¥žXŸ¸›³©„¡©ž¥ž¯©«¬« µ?"œŸ[¥ž¥ž¯¬¢u®
£X›t¡u«¬Ÿ[¥© ©M³J°u¢u¤[¥X¯¬›t¢„›³¥ž¯¬»%Ÿ6­'€­t©¢£Š¡užŸ"¤k°X X›t
u«¬»T¥ž±u¯¬¤€ªE¢uŸ" X
š 1
§ I±u
Ÿ ¢uŸ" ž¥I®tž¯¬£«¬Ÿ[´6Ÿ"«¯¬ 2 ­
¯<§ Ÿt,
§ C',
C§
×ÔÕà1ØlÛaÑ
C
òÊgÔÕÞTؤÙUßÖaÔÕÜ1ÐãÚ.ßÛ6ßÞqѐÖaѐÛaçìéÂÜÛÈá65.â5úá67â5úá8('
â 9&=?>
(@
›t¡EµX¯¬®t±6¥
g¤
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
@
ú A
@
&
)&2
)B +C
úá6L M¼â/@}áP ÿ â5ú
DGF @
BC
H
ß>Фä
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
&
¦,
§ „§8¨
¹¤x­¦0§¹,(
§ B
œ¹,­(R§(>.¤*¸(
­ „§,R§,B¦,
óß>ïlÙÕÑ
T
J š
6òOóå¤Ñ·ÑüÑjÓ5ÖÈÜé€ÚgÛaÑjÓjØgÛaç&Ü>ÛÈø¤ÙÕތÖaålÔÕÓ GФѐç&ç
C
ÿ
C
Ý ÿ
ú¤ëìÔJÖaå
132
-0 - -0 -6
-0 -~
-0 -0
-0 - - @
&21ò
- ˆ:8ˆ
- ˆ &~
- ˆ + - :-:(
- <9
Ü1Ðç&ÚlÛaÑjß1älÔÕÐlàîÛ6ßSÖaÑÚ¤ßÛ6ßÞqѐÖaѐÛaç ø¤Ð¤ÑjçLÖTà>ÛaÔUäÙÕÑjè,ÑjÙÔUç
-
$%/7A4E>HJ/Q
-0 &0
-0 &~ (
-0 & &0
-0 & ,0
-0 & , &
[email protected], &
[email protected], [email protected],
[email protected]+%(
[email protected]+%(
9
;@DL8²;JP50?-/=’4tYM132?‚[email protected];J2D72h9A>;JD0/Q™N%;^9A.p>jeD0/7"9‚P?A;@[email protected]/EO/EH„21~ (un~ (YVR W ` - 3>8DQ
>Q0>8=9A;JO/9A;JG/j7"9A/E=0=0;@D0P²€9;@[email protected]?>9-/79A.>9g>1¶9A/E?>8DT;@D0;J9-;<>8H =0.>7k/ƒ9-.0/[email protected]/69‚9-.0;<4qFD0/7A
7 1 U N
2h/VF7'9-.0/‚/6n=x/4t9A/Qd=x2N,/E?H<>uNb?-/[email protected]>89-;@2D7-.0;@P
= 1 U N `
c7 , YN%./E?A/‡9-.0/R462F/6C4E;J/D97
>8DQ >?-/
;@Dc#:>h0HJ/dM!'29-/9A.>9YM2N%;@D0P`9-2j9-./9-;@GR/K7-4>[email protected];JD0Pj/GR=HJ2V/Qs;JDf9A.0/g=0?A/7-/ED
9l7"9A5QVYI;JDc2?AQ0/E?)9-2
4E2G=>8?A/%9A.0/7-/?A/7-50HJ9A7ŠN%;^9A.9-.27-/281ma4.FN,>?kT9 '>8DQ [email protected]/EVj} +89A.0/
O>[email protected]/7ŠGl57k9 h/ZGl50HJ9-;@[email protected];J/QƒhFV
9A.0/‡1’>469-2? - - , , # .0/ Ou>HJ5/7'>?-/B13250DQ`9-2ghx/l4EHJ2
7k/B9A2g9-.0/l/6n=/4t9A/QjO>[email protected]/‡281K- ~0YM} ˆ -u€Y>8DQ
` - &0132 ` - -: ¸>DD
9A.0/O>[email protected]/7„21
>8H<7-2l462GR=>8?A/'NŠ/HJHxN%;J9-.9A.0/O>HJ50/7
Q -0 & + 1V2 ` - - - 2h9A>;JD0/QƒhFV™} +u€
9
'
M“8 $:Z•–-:
'
Œ Ž E
4! #x “ : ¶–:$j• “–
'
'
6”Xt’‘ l-•#x*‹ E”
+Š2GR=>8?A;@7-2D;<7„Gg>Q0/'h/E9"NŠ//EDDF50G/?-;<4E>H=0?A/Q;<4t9A;J2D7„>DQ4E2?A?-/7k=x2DQ0;JD0P>D>[email protected]
9-;<4E>[email protected];@2D7r132?
9A.0/g4E>7-/R281Š>8nF;<7-V
GG/69A?-;<4R7k=0?A/>Q;@D0PL2N 281,>`462G=0HJ/E9-/HJV]N,/69-9-;@[email protected]/69YI4E/ED
9-?A/Q ;JD;^9A;@>[email protected]]>9
CO = ` <- =O- 6Y2O/E?r>.2?A;*2D
9A>H07k50h7"9A?A>89-/'N%.0/DP?>uOF;^9>9A;J2D>8H/ x/4t9A7,Q2GR;@D>89-/'2O/E?„7-50?-1’>46/
9A/ED7-;J2Dj2D0/7%7k2R9-.>9%9-./ Š2DQKD
5Glhx/E?Y
Y0;<7%H<>8?AP/
[D`9A.0;@74E>7-/)9-.0/‡[email protected]?A;<4E>9A;J2DK/
5>9A;J2D7,132?
9A.0/RQ0?-2=0HJ/E9'9-.;@4qFD0/7-7 1 4>8D]hx/R?-/Q546/Q`9-2ƒ9-.0/‚[email protected]%;@D0Pƒ7-/4E2DQWX2?AQ/?'=I Qœ /x132?)>n;@7-VFGGR/E9-?A;@4
L2N
D GF
'
'
W4X
!
1
!
WYˆ X ! !
`
Q!
61
1
!
O
< - ;@D`9-/E?AGg7 281 FY9-.0/‡?>Q0;@>HIQ;@7k9A>D46/)13?A2G|9A.0/46/D
9-?A/)21:9A.0/7-50h7k9-?>9-/
# .0/;<Q/>HJ;<7-/Qj4E>7k/‡N%.0/?-/9-.0/;@D0;J9-;<>8H²[email protected]/69=0?A28eHJ/‡;<7';@Dj9-.0/132?AG
P;JO/ED`h
V
1 O `
`
- -
132?
7
U ->DQ
2
~
1
21¸>g(Z;@?A>4Q/EHJ9A>g1350D4t9A;J2DIY
`
2bO
<
N%.0/?-/ v;<7Š9-.0/BQ;@G/ED7-;[email protected]/7A7Š[email protected]/21I9-.0/[email protected]/69YF.>7Šh//ED`>8D>HJV7-/Qg=0?A/EOF;@[email protected];@Ds} ˆ~uX# .0/
H<>9-9-/E?Z7"9A5QVK7-.02N,/Qƒ9-.>89'/ 5>89-;@2D N:- 46250H<Qƒhx/[email protected]/Q`>8D>[email protected];@4>[email protected]ƒh
V`Q/6eD0;JDP9-.0/7-;JG;@[email protected]>?-;J9"V
`
O>8?A;<>[email protected]/
Q &
Y09-2VF;J/[email protected]
1 `
- (6ˆ
'
2 *
WYX
&
- (, ,
'
+ /
&
<
~ '
8²;JP50?-/G& ’>:¸462G=>8?A/7¸DF50G/?-;<4E>H0=0?A/Q;<4t9A;J2D7r21x9A.0/ZQ0?-2=0HJ/E9r9-.;@4qFD0/7-7Š>9„;J9A7,46/D9A?-/:
Y 1 U <tYF>7„>
` - & &R>8DQ
` - ~6&# .0/
1350D4t9-;@2Dd21M9A;JG/N%;J9-.`9-./l7k;@G;JH<>8?A;J9"Vƒ7k2HJ509-;@2D N
~ Š132? 9A.0/l4E>7k/
WYX
›t¡EµX¯¬®t±6¥
g¤
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
82
'
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
7(
î
:¦íq>’¨°q.
Ȳ
¦.ƒ"
¦ƒ
²¨
10
3
−6
t=3x10
−5
t=4x10
−4
t=3x10
−4
t=9x10
2.5
Similarity Solution
Numerical Solution
1.5
h
h0(t)
2
1
1
0.5
−8
−7
10
10
−6
10
10
t
−5
10
−4
0
−3
10
©
²X›t¡«¬Ÿ"¥`¥X±u¯¬¤Xª¢uŸ" ž s©A¥d¥ž±uŸf¤"Ÿ"¢6¥XžŸc›³©
¤"›t»%¡u«¬Ÿ[¥žŸ"« µ1"œŸ[¥ž¥ž¯¬¢u®²£X›t¡u«¬Ÿ[¥
&HA¤k›t»%¡u©X¯¬ X›t¢
8Ÿ[¥6"œŸ"Ÿ"¢ ¥ž±uŸl¢°»%Ÿkž¯¬¤k©«m¡uXŸ"£u¯¬¤[¥ž¯¬›t¢u ‡©¢u£j¥ž±uŸ
ž¯¬»%¯¬«©X¯ ¥¶µ% ž›t«¬°¥ž¯¬›t¢ SC [ §
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
)
ž›t X žº< XŸ"¤[¥X¯¬›t¢©« ¥ž±u¯¬¤€ªE¢uŸ" X ‚¡uX›<u«¬Ÿk R©¤"X›t X © ¤"›t»%¡u«¬Ÿ[¥žŸ"« µ "œŸ[¥ž¥ž¯¬¢u®b£už›t¡u«¬Ÿ[¥5HR¤"›t»%¡©X¯¬ ž›t¢
i
8Ÿ[¥6"œŸ"Ÿk¢)¢°u»%Ÿ"ž¯¬¤-©«¡uXŸ"£u¯¬¤[¥ž¯¬›t¢u žµ
» 8›t«¬ ©¢u£
¥ž±uŸ¸ X¯¬»%¯¬«©ž¯ ¥¶µ X›t«¬°¥X¯¬›t¢ SC€§
4
3.5
−7
t=2.6x10
−6
t=6.9x10
−5
t=8.2x10
−4
t=7.1x10
3
h
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
¤'X
œž›t X ’º¶ XŸ"¤[¥X¯¬›t¢u©«²¥X±¯¬¤€ª¢Ÿk ž K'¡už›<«¬Ÿ" ³J›t
© ¤"›t»%¡u«¬Ÿ[¥žŸ"« µ "œŸ[¥’¥X¯¬¢u®‚£uX›t¡«¬Ÿ"¥,›-´6Ÿ"'©‡¤"¯¬X¤"°u«©
¥X›t¡8›t®t€©¡±EµgH%¤"›t»%¡©X¯¬ ž›t¢ 8Ÿ"¥6"ŸkŸ"¢U¢E°u»%Ÿ"X¯¬¤k©«
¡uXŸ"£u¯¬¤[¥ž¯¬›t¢u žµ»8›t«¬ v©¢£ ¥ž±uŸ X¯¬»%¯¬«©X¯ ¥¶µ
X›t«¬°¥ž¯¬›t¢ I[§
×ÔÕà1ØlÛaÑ
gòÊgÔÕÞÈØlÙUßÖaÔÕÜ1ЩÚ.ßSÛ6ß>ÞqѐÖaѐÛaçé@Ü>Ûîá65.â5úìá67â5ú„á 8(â'9=
ÿ
›t¡EµX¯¬®t±6¥
g¤
Ý ÿ
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
ß>Ð.ä4
@
>(@ ÿ B C
)
gú
@
KB ÿ
:1úø.ÐlÑjçLÖqà>ÛaÔUä©ÙUѐè,ÑjÙìÑBñ(Ø.ßÙUç
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
+
¦,
§ „§8¨
¹¤x­¦0§¹,(
§ B
œ¹,­(R§(>.¤*¸(
­ „§,R§,B¦,
132?-G/E?,>8?A/'2h09A>8;@D0/QgN%;^9A.ƒ>BeD/7k9,P?-;<QHJ/O/H281ru~6,8nu~6,B>8DQR W ` ~ '`- 9 Q F# .0/Z7-;@Gl50H<>9A;J2D
'
57-/7‡9-.0/;@D0;J9-;<>8Hr=?-2eHJ/Ež ˆ ,:B7k;@D46/g;J9l;<7‡hx289-.T7-.>[email protected] =x/>q/Qs>DQfG;JG;<4E7Yœ9-2 7-2G//6nF9A/ED
9YM9-.0/
;@D0;J9-;<>8HŠ4E2DQ0;^9A;J2D7‡;@Df9-.0/ƒ>D>[email protected]
9-;<4E>Hr7k2HJ59A;J2DcN%.0;@[email protected]‚>9‡9A.0/ƒ7A>8G/9-;@G/P5>8?>8D
9-//E;@D0P`9-.>89 1 U N
4.>D0P/7„7-50h7k9A>D9A;@>[email protected]/E132?A/9A.0/[email protected]/69„L2N'7„259Š21I9-.0/B462G=059A>89-;@2D>HxQ02Gg>8;@DP<
t
/
7%/En=/4t9-/QœY9A.0/>8P?-//EG/ED
9 h/E9"NŠ//ED`9-.0/7-;@Gl50H<>9A;J2D`>DQj7k;@G;JH<>8?A;^9"[email protected];J2D`;<7 ;@D0;^9A;@>[email protected]ƒ=x/ 2F2?
3132?d
U
ˆ - 9 6 Y8Q50/,9-2Z9-.0/%Q*; /E?A/ED4E/7mhx/69"N,/E/EDl9-.0/ ;@D0;J9-;<>8H0462DQ;J9-;@2D7²;@Dl9A.0/,9"NŠ[email protected];@2D7Yuh059
;@G=0?-2O/7Š>9ŠH<>9A/E?Š9-;@G/7 >8DQ/EO/D
9-5>[email protected]/462G/7„O/E?AV‚P2
2Qghx/EV2DQg9-.0;<7,;JD0;J9-;<>8H=0.>7-/N%.0/?-/9-.0/
Gg>n;@Gl50G [email protected];@4qFD0/7-K
7 1 U <¸Q/4E?-/>7-/7²;@DR=?-2=2?k9A;J2D9-2 9 & >9rH<>9A/E?m9-;@G/7aF;@G;JH<>8?„>P?A/E/GR/D
9
;<7>8H<7-2j7-/E/Df;JL
D 8²;@P50?AD
/ & ’h
tYIN%.;@4.T7-.02N'7)9-./ƒQ/EO/[email protected]=0G/D9‡21Š9A.0/g9-.;@4qFD0/7-7‡=?-2eHJ/>46?A27A7Z9-.0/
Q?A2=HJ/E9YN%./E?A/‡9-.0/R>8P?-//EG/ED
9'>9)[email protected]>89-/E?'9-;@GR/7;<7'O/E?AVƒP2
2Qj/O/E?AVFN%.0/E?A/B/6n046/=9ZD/>8?'9-.0/R462D
9>4t9
[email protected];JD/„# .0/ [email protected]>89k9-/?Q;<7-4E?-/=>8D4E;J/7g>8?A/]4E>57k/QphFVp9-.0/]7-Gg>[email protected]žYrh59ƒD02DW /E?A2Yr7-51’>4E/j9-/D7k;@2DU;@D—9-.0/
7-;JG‚50H<>9-;@2D7 N%.0;@4.j>469A7,9A2g?-/Q546/‡4E50?-O>89-50?A/B;JDj;J9A7 OF;@4E;JD;^9"V
4
'!tŒ‹”<! 6• l
-•#x‹*E”
#x“ ¶:
– $
:
4
6 •T#•$-“T# F<'!
# 0
. / ?-/G>;JD;JD0Pc?A/7-50HJ9A74E2D4E/E?ADU/6n0>[email protected]/7g21Z9A.0/]L2N 21BQ?A2=HJ/E9A72O/E?9A2=x2P?>8=.0;@4K13/>9-5?-/7E
+Š2D7k;<Q/?Ze?A7k9Z9A.0/=0?A/EOF;J257)[email protected]/EG…21,>n;<7kVFGG/69A?-;<4L2Nv281 >[email protected]/69-/HJVdN,/69-9-;@[email protected]/69‡>9
.0;@P. Š2DQDF50G‚h/?Š/EnF9-/EDQ/Q9-29A.0/B4E>7-/21M>‚[email protected]/69A/[email protected]/DFO/[email protected]=x/Qƒ46;@[email protected]>?r9A2=x2P?>8=.
V€9%4E>D
hx/l7-.02N%Dƒ9A.>9%9A.0/l7-;JG;@[email protected]>?-;J9"Vƒ[email protected];@2D <
~ Š132?%9-./BL2Ni;@D`9-.0/‚>8h7-/ED4E/B281r>‚9-2=x2P?A>=0.FV;<7%[email protected]
7-HJ;@P.
9AHJVG2Q;Je/Q`hFV;J9A7%=?-/7k/D46/‡7-;JD4E/)9-.0/‡P2O/E?AD0;@D0Pl
/ 5>9A;J2]
D < - 7-;@GR=HJVhx/4E2G/7
!
1
!
W4X !
ˆ !
`
1
Q !
1
$
!
7
O
<ˆ N%.0/?-/ 7 ` 7 6r;<7¸9A.0/'9A2=x2P?>8=.0;@4 =?-2eHJ/
aF;@D4E/ 7:6
ŠQ2F/7ŠD0289„O>?-VRN%;^9A.g9-;@G/;J9Š;<7„/>7-Vl9A2l7-.02N
9A.>9%9A.0/7k;@G;JH<>8?A;J9"V7k2HJ59A;J2DjN%;^9A.j>‚9A2=x2P?>8=.
Vg;<7 P;@O/DKhFV
- (68ˆ 1 `
'
2
*
W4X
Q
& N%.0/?-/Y2D4E/‡>8P>;JDMY ` # .0/B9-2=2P?>8=0.FV 7:6 <; 7%P;@O/DKhFV
&
- (6, , +
'
&
/
+
7:6 ;O
< :
7: `
7 U
R 6
+
R
R- +
R / /
/
O
<: N%.0/?-/ 7 U ;<7 ;^97%.0/E;@P.
9'>8DQœ[email protected]@2N%;JD0Pd} ˆˆu€YR ,
;<7'>D`>?A469A>8/ DP/ED
9,1350D469-;@2D
R6
`
9) ~
I +
*<*
O
< & ;<7%;^97 ?A>Q;@57E
N%.0/?-/)9A.0/‡=>8?>8G/E9-/E?
4E2D
9-?A2H<7Š9-.0/7k9-//E=0D0/7-7'281M9A.0/B9-2=2P?>8=0.FV>DQ `
A ?
8²;JP50?-/&
’4 7k.02N'7%>4E2G=>8?A;<7k2DKhx/69"N,/E/D`DF50G/?-;<4E>Hœ=0?-/Q;<4t9-;@2D7%281²9-.0/Q0?-2=0HJ/E9%9-.0;<4qFD0/7A71
=0?A28eHJ/‡>4E?-2
7-7,9A.0/[email protected] >DQK9-.0/‚7k;@G;JH<>8?A;^9"VK7k2HJ59A;J2D N %Š
132? 9A.0/=>?k9A;@[email protected]>?'4E>7k/B281¸>R9-2=2P?>8=0.FV
` - ~>DQ ` - Y09A.0/132?-G/?'.>uOF;JDPgh//ED]2h9A>;JD0/QjN%;^9A. >eD/7k9P?A;@[email protected]/EO/EH
N%;J9-. 7 U ` - ~Y
' Q R W ' ` ~ c- 9 ' Q Z7B;JD 9-.0/R?-/[email protected]>89-/Q 4E>7k/‚N%;^9A.02509‡>9-2=2P?>8=0.FVYx?A/7-50H^97Z>?-/
21~ ,nœ~ ,g>8DP
' Q 2 ` - ~6&YFN%.0;@HJ/B9A.0/[email protected]/69 9A.0;@4qFD0/7-7 =?-2eHJ
` - & &‚>D8
2h9A>;JD0/Q132?
/ žˆ , Š;<7%G2Q;^e/Qj7k2
W4X
'
9A.>9%9A.0/‡;JD;^9A;@>Hœ13?-//‡7k50?-1’>4E/[email protected]/);<7,9-./7->GR/B;@Djh29-.d4E>7k/7EY;X /
1
›t¡EµX¯¬®t±6¥
g¤
$
7 ` 5 D  +
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
ˆ
~6,
/*
OR7:6
* '
<%(6
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
,
î
:¦íq>’¨°q.
Ȳ
¦.ƒ"
¦ƒ
²¨
# .0/`eP50?A/`7-.02N'7R7k;@G;JH<>8?ghx/E.>uOF;@250?‚9-2™9A.>9g?A/E=x2?-9-/Qp;JD 8²;JP50?A/&
3
h l132?R9A.0/]4E>7k/`N%;^9A.02509g>
9A2=x2P?>8=.
V
!'/En
9
[email protected]/69
7-=0?A/>Q0;JD0P
2O/E?
7-50h7k9-?>9A/
462D
9>8;@D0;JDP
9-.0?A/E/EW€Q;@G/ED7-;@2D>HžY ?A/469A>D0P50H<>8?
9A2=x2P?>8=.0;J/7œ;<7M/EnF=HJ2?-/Qœ# .0/,/6nF9A/ED
9:219-.0/„9-2=x2P?A>=0.FV;<7:7-=x/46;Je/QN%;J9-.?A/7-=/4t9:9-2'9A.0/„?A/613/?-/D46/
=x2;@D
9 ED OTF ` NM OTQ c6W U >7';@DQ;<4E>89-/Q];@P
D 8²;JP50?-/KV
[email protected]%;JDP } ~u²9A.0/9A2=x2P?A>=0.F
V 7:ED OGF=%;<7
Q/EeD0/Qo;@D—9-/?-Gg7g219-?>8D7"132?-G/Q—462WX2?AQ0;JD>89-/7EF
Y ED 2 OGF 2'g7A>uVYmN%;J9-.˜2?-;@P;@Do>89g9-.0/]4E/ED
9-?A/j281)9-.0/
9A2=x2P?>8=.
V>DQKP;@O/ED`hFV
D
# .0/B9A2=x2P?A>=0.FV)7:
7 U
`
U
N%.0/?-/
D
+
-
F 2
'
+
F
`
F
+
-
'
O
< + '
;<7,9A>q/D`>7
DCOGF=
7:EDCOTF=
+
D
2[`
)
9
)
9
F 2
+
DV2
+
8~
c
O
O
$
*
)
9
+
DV2
~
c
*
$
9
+
F 2
+
+
O
O
O
*<*
~
c
O
8~
c
*<*
O
< , 4E2D
9-?A2H<7,;^97'7"9A/E/=0D0/7A7%>8DQ
r9A>8D
`
U
9
~
9A>D
*
)
9
_
~
*
6- '
[D > HJH²281¸9A.0/l?A/7-50HJ9A7N%.0;@4.][email protected]/‚;@D0;J9-;<>[email protected]/69=0?A28eHJ/7>8?A/21¸9-.0/7->G/‡132?AG…>7'/
5>9A;J2D
ž ˆ , ŠG2Q;Je/QK9-29>8q/B>4E4E250D
9 281²9-.0/‡=0?A/7-/ED46/B281:9-.0/B9A2=x2P?>8=.
VYFD>8G/[email protected]
1
$
7
`
5
D
)
ˆ~
+
,
/6*
OR7 DCOTF
*
O
N%.0/?-/ ‚;@7 9A.0/‡?>Q;<>8HœQ;<7"9>8D4E/B13?-2G|9-./46/ED
9A?-/B281:9-.0/7-50h7k9-?>9A/
8²;JP50?-/7K(>DQ+B7-.02Nc9A.0/%/[email protected];@2D‚2819A.0/,13?-//%7k50?-1’>4E/j1
77:[email protected]/7m>8DQ‚>7-7-246;<>9-/Ql4E2D
9-250?A7
$
`
132?9-.0/dL2N†281‡>f462GR=HJ/E9-/[email protected],/69-9-;@[email protected]/69Yr46/D9A?-/Q—;JD0;J9-;<>[email protected]@Vo>9E DCOG=
F <=O 6Y¸2O/?>
'
'
`
9A2=x2P?>8=.0;@4j=x/>q V7U ` - g>DQU9A?-/D4. V7 U `&+ - g?-/7k=x/469-;@O/[email protected]—N%;^9A
.
~Y _ ` Y
'
'
'
`
D
- -c[Ds9-.0/R132?-G/E?‡4>7-/Yx9-.0/gQ?A2=HJ/E9B7-=0?A/>Q7)>8n;@7-VFGRG/E9-?A;@4>[email protected]
OGF ` <- ' ˆ O- ' ˆ Z>8DQ
'
50D
9A;JHI;J9%G/E/697,9-.0/‡/QP/B21M9-./Z9A2=x2P?A>=0.FVgN%.0/ED`;^9'/En=/?-;@/ED4E/7 >8Dj>QQ;^9A;J2D>8Hœ=?-/7-7-50?A/ZP?A>Q;J/D
9
9A.>94E>57-/7R;^99A2f7-=0?A/>QT=0?A/613/?-/D9A;@>[email protected]™13?-2G 9-./K9A2=x2P?A>=0.FVs9A2™9-.0/d7-50h7k9-?>9-/`;@D—>fQ;@?-/4t9A;J2D
D02?-Gg>H9A2l9A.0/B/QP/B281I9A.0/B9-2=2P?>8=0.FVaF;@D46/Z9-./[email protected]/69 GR//697,9-.0/Bhx289k9A2G [email protected]/61¶9%.>DQ`462?AD0/?Š21
9A.0/'9A2=x2P?>8=.
[email protected]>7"9„21M>[email protected]žYHJ;4
50;<Qg;JD9-.0;<7r?A/EP;J2DR;<7„9-.0/[email protected]>7"9r9A2/6n=x/E?A;J/D46/'9-.0/)>Q0Q;J9-;@2D>8H=?-/7-7-50?A/
P?A>Q;J/D
9Y4>857-;@D0Pƒ>7-Gg>[email protected]²4E57k= ;JDd9A.0/‚4E2D
9A>4t9)HJ;@D0/l;@Dd9A.0;@7Z?A/EP;@2DM[Dd9A.0/‚4>7-/281¸9A.0/R7-=0?A/>Q;JDP
2O/E?g>s9-?A/ED4.I9
Y 8:;@P5?-/ +0Ym9A.0/d/Q0P/j281)9-.0/j9A2=x2P?A>=0.FVc=0?-/7k/D
9A7g>8DU>Q0O/E?7-/K=0?A/7A7-50?-/jP?A>Q;J/D
9
N%.0;<4.f?-/Q54E/7B9A.0/7k=x/E/Qf281%462D9>[email protected];@D0/ƒ>Q0Ou>D46/G/ED
9>46?A27A7B;J9A7‡/QP/7>8DQIYM7k;@D4E/gHJ4; 50;@Q™D0/>8?
9A.0/Bh29k9-2G [email protected]/61¶9%.>8DQK4E2?AD0/E?,;@7%>P>;JDg9A.0/[email protected]>7"9,9-2/EnF=x/E?A;@/ED4E/9A.0;@7%>QO/?A7-/=?-/7-7-50?A/P?>Q0;J/D9Y
9-.0/
Q?A2=HJ/E9%7-N,/[email protected]@7%7-HJ;@P.
9AHJV;@Dj;^97%hx289k9A2G HJ/E1¶9'.>8DQK?A/EP;@2DM
8²;JP50?-/b
7 ,>DQd
-7k.2No13?A/E/)7k5?k1’>46/'=0?A28eHJ/7„>8DQ>7A7k24E;@>89-/Q4E2D
9-250?A7¸132?Š9-.0/L2NU281M>lQ0?-2=0HJ/E9Y
`
`
4E/ED
9-?A/Q`;@D0;^9A;@>[email protected]>
9 DCOG=
F `
OT tY2O/E?'>R9A2=x2P?A>=0.FVƒQ/6eD0/QjhFU
V 7U ` - Y
Y _
Y
'
'
'
'
`
`
D
OGF
-=OT- )>8DQ
- -YN%.0/E?A/R9-.0/gN,/69-9A>h0;[email protected];J9"V]2Ds9-.0/9A2=x2P?A>=0.FV]Q
; /E?7)13?A2G 9A.>9‡21
'
`
9A.0/ƒ?A/7k9l281%9A.0/ƒ7-50h7k9-?>9A/ [DT9-.0/ƒ132?AGR/?l4E>7k/9-.0/KHJ;4
50;@[email protected]Š/E9A79-.0/`7k50h7"9A?A>89-/P
`
N%.0;@[email protected]/s2D0HJV˜=>8?-9-;<>[email protected]—NŠ/E9k9A;JD0Po9-./s9A2=x2P?A>=0.F[
V t [D 8²;@P50?A]
/ ,p;^9j;<7K/OF;@Q/D
9K9A.>9
'
9A.0/`Q0?-2=0HJ/E9‚?A/4E/Q/713?A2G 9-./ƒ9A2=x2P?>8=.
Vs;JDo2?Q/E?l9-2™;JD46?A/>7-/;J9A7462D
9A>469R>[email protected]/g9-2N >8?Q079-.0/
/ 5;[email protected];Jh?-;@50G…O>[email protected]/l21 N%.0/?-/>7Z;^9‡>QO>D46/72Dd9-.0/7-50h7k9-?>9-/‚;JDs2?Q/E?9-2`HJ2N,/E?Z;^97B462D
9>4t9
'
›t¡EµX¯¬®t±6¥
g
¤
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
~
-
¦§,„§8¨
¹¤x­¦0§¹,(
§ B
œ¹,­(R§(>.¤*¸(
­ „§,R§,B¦,
1.0
0.9
0.8
0.7
1.3
5
0.6
2.70
5
1.3
5
0.4
4.05
(a)
4.5
2.25
0
0.5
0.4
0.90
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0
0.9
0.8
0.7
0.70
05
2.10
05
0.35
50
0.4
1.
1.
2.80
3.
2.45
0.5
(b)
35
0.
0.6
0.70
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0
0.9
0.8
0.7
(c)
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
×ÔÕà1ØlÛaÑ
@
òO×lÛaѐÑç&ØlÛ&éÂß>ÓjÑîáJ
›t¡EµX¯¬®t±6¥
úá*ï#â
g¤
CBDBDBD‹t›t±u¢
@
úá*ÓSâ
@
)B C
GhK¯¬«¬Ÿ[j
µ iǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Sâ3ÚlÛaÜø.ÙÕÑjç8á*ÙÕѐé@Ö5â!ß>Фäðß>ç&ç&Ü(ӐÔ:ßSÖaÑBäÓjÜ1Ð(ÖaÜ>ØlÛaç8áÂÛaÔÕà1å(Ö5â€éÂÜÛßäGÛaÜ1ÚlÙUÑ5ÖTç&ÚgÛaÑBß1ägÔÕФà
ÜSè,ѐ۷ߛÖaÜ>Ú#Ü1à>Û6ßÚ¤ålÔUÓîÚ#ÑBß6ëìÔÕÖaå
@
0.1
[email protected]
ú
)B ÿ ú
@ 1
úá µ µ â w
@ á KB P KB 1â8ßÐ.ä [email protected] KB > òáß,â
,ògóå¤Ñ‹ägÑjФç&ÔJÖ¹õ8ÜéÖaå¤Ñ„ø.ÐlÑjçLÖ ÞqÑjç&å·ÔÕç ÿC Ý ÿC ú&J š @ KB ,ÿ
ßÐ.äY(@
&2 ò
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
~
î
:¦íq>’¨°q.
Ȳ
¦.ƒ"
¦ƒ
²¨
1.0
0.9
0.8
0.7
0.50
00
1.
3.0
0
3.
(a)
0.00
4.50
0.5
2.50
50
0.6
0.4
00
1.
0.50
0.3
0.00
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.7
0.8
0.9
1.0
0.9
1.0
1.0
0.9
0.8
0.7
0.70
5
2.8
0
0.35
35
2.4
0.5
2.10
3.15
(b)
0.
05
1.
0.6
0.4
05
1.
0.70
0.
00
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
1.0
0.9
0.8
0.19
0.28
8
9
05
0.0
7
1.
0.5
7
1.14
1.
0.57
0.0
9
0.19
0.66
0.19
0.4
7
05
76
0.
8
9
0.95
0.3
0.0
0.4
0.86
1.33
1.24
0.5
0.2
0.4
0.76
0.28
0.57
0.19
0.95
1.14
8
0.6
0.0
9
0.66
0.3
0.7
0.3
(c)
0.3
0.28
0.1
0.0
0.0
@
@
GòO×lÛaѐÑç&ØlÛ&éÂß>ÓjÑîáJ
×ÔÕà1ØlÛaÑ
›t¡EµX¯¬®t±6¥
g¤
1úá*ï#â
@
CBDBDBD‹t›t±u¢
>úá*ÓSâ
KB C
GhK¯¬«¬Ÿ[j
µ iǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Sâ3ÚlÛaÜø.ÙÕÑjç8á*ÙÕѐé@Ö5â!ß>Фäðß>ç&ç&Ü(ӐÔ:ßSÖaÑBäÓjÜ1Ð(ÖaÜ>ØlÛaç8áÂÛaÔÕà1å(Ö5â€éÂÜÛßäGÛaÜ1ÚlÙUÑ5ÖTç&ÚgÛaÑBß1ägÔÕФà
ÜSè,ѐÛTßãÖaÜ1Ú#Ü>à>Û6ß>Úlå¤ÔÕÓ8Ö&ÛaÑjФÓ6å›ëìÔJÖaå
@
0.1
ú
@
KB ÿ ú
@%1úá µ µ â @ á )B P KB 1âTß>Фä
@
ògóå¤Ñ‹ägÑjФç&ÔJÖ¹õ8ÜéÖaå¤Ñ„ø.ÐlÑjçLÖ q
Þ Ñjç&å·ÔÕç ÿC Ý ÿC &
ú J
(@
ßÐ.äY
K&2
KB
š @
>ò
áß,â
KB ,ÿ
ò
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
~~
¦§,„§8¨
¹¤x­¦0§¹,(
§ B
œ¹,­(R§(>.¤*¸(
­ „§,R§,B¦,
1.0
0.9
0.8
0.7
0.60
1.80
0
2.7
2.1
0
3.30
2.40
0.5
30
0.9
0.
0
0.6
1.20
0.30
(a)
0.4
2.10
90
0.
0.60
0.3
0.30
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0
0.9
0.8
0.7
1.00
1.2
0.7
5
5
0.6
5
2.50
2.00
0.5
2.2
0.25
0
(b)
0.5
0.4
0.7
5
5
1.2
1.0
0
0.50
0.25
0.3
0.2
0.1
0.50
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.9
1.0
1.0
0.9
0.15
0.60
0.8
0.45
1.05
30
0.75
5
1.50
1.0
0.
5
1.20
90
0.45
0.4
0.75
0.45
0.3
0.15
0.45
0.2
5
0.60
5
0.1
0.5
(c)
0.4
0.6
0.1
75
0.
1.20
0.7
0.
30
0.
0.1
0.45
0.0
0.0
×ÔÕà1ØlÛaÑ
ßjë
0.2
0.3
0.4
0.5
0.6
0.7
0.8
òO×lÛaѐÑç&ØlÛ&éÂß>ÓjÑîáJ
Sâ3ÚlÛaÜø.ÙÕÑjç8á*ÙÕѐé@Ö5â!ß>Фäðß>ç&ç&Ü(ӐÔ:ßSÖaÑBäÓjÜ1Ð(ÖaÜ>ØlÛaç8áÂÛaÔÕà1å(Ö5â€éÂÜÛßäGÛaÜ1ÚlÙUÑ5ÖTç&ÚgÛaÑBß1ägÔÕФà
6
@ )B C ú @ KB C ú @ 1ú á â @“á )B KB âTß>Фä @ KB 1ú
B CIH Ü1ÐãÖaålÑ3ÖaÜ1Ú#Ü>à>Û6ßÚ¤å(õßÐ.ä DGF @ H Ü>ÐãÖaå¤ÑÛaÑjçLրÜ>éÖaå¤Ñ‹ç&µ ؤïlçLµ Ö&Û6ßÖaÑ1òáß,â @ B ÿ >úá*ï#â
K úá*ÓSâ @ C )&2 òGóålÑ3ägÑjФç&ÔJÖ¹õÜ>éÖaå¤Ñ ø.ÐlÑjçLÖÞqÑjç&åqÔÕç ÿ C Ý ÿ C úEJ š @ KB ,ÿ ß>Фä (@ K2 ò
ßjõùé@ÛaÜ>އßÖaÜ>Ú#Ü1à>Û6ßÚ¤ålÔUÓ·Ú#ÑBß ðëìÔJÖaå
ëìÔJÖaå
0.1
@
C
DGF
›t¡EµX¯¬®t±6¥
@
g¤
CBDBDBD‹t›t±u¢
GhK¯¬«¬Ÿ[j
µ iǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
~8ˆ
î
:¦íq>’¨°q.
Ȳ
¦.ƒ"
¦ƒ
²¨
1.0
0.9
0.8
0.7
0.80
2.4
0
3.2
0.40
1.
20
(a)
0
4.0
0.5
2.80
0
1.2
0
0.6
0.4
0
0.40
0.8
0.3
0.40
0.2
0.1
0.40
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.6
0.7
0.8
0.9
1.0
0.6
0.7
0.8
0.9
1.0
1.0
0.9
0.8
0.7
0.90
0.
30
0.6
20
1.
0.30
3.30
1.50
3.00
0.5
(b)
0.4
2.40
2.70
20
1.
0.3
0
9
0.
0.60
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
1.0
0.9
0.8
0.7
0.6
(c)
0.30
1.20
0
2.70
1.80
3.00
0.60
1
2.
0.4
2.40
0.6
0
0
0.9
0.5
0.2
2.10
0.6
0
0.1
1.5
0
0.30
0.3
0.90
0.0
0.0
×ÔÕà1ØlÛaÑ
òg×lÛaÑjфç&ØlÛ&é*ßÓjÑÈá
J
0.3
0.4
›t¡EµX¯¬®t±6¥
g¤
CBDBDBD‹t›t±u¢
@ )B C
GD F @
0.5
ç&ÚgÛaÑBß1ägÔUÐlàTÜéß!äGÛaÜ1Ú¤ÙÕѐÖ
KB âÈßÐ.ä @ )B 1úëìÔJÖaå
@ KB C ú @ >úìá
â@
á KB
H Ü1ÐÖaå¤Ñ8ÖaÜ1Ú#Ü>à>Û6ß>Úlå(õ"ß>Фä
B CIH Ü1ÐÖaå¤ÑqÛaÑjçLÖqÜéìÖaµ å¤Ñ·µ ç&ؤï¤çLÖ&Û6ßSÖaÑ1òìáß,â @ B ÿ
1ú á*ï#â
K2 úá*ÓSâ @ C )&2 òGóålÑ3ägÑjФç&ÔJÖ¹õÜ>éÖaå¤Ñ ø.ÐlÑjçLÖÞqÑjç&åqÔÕç ÿ C Ý ÿ C úEJ š @ KB ,ÿ ß>Фä (@ K2 ò
0.2
SâÚlÛaÜø.ÙÕÑjç€á*ÙUÑ5é@Ö5âßÐ.ä8ß>ç&ç&Ü(ÓjÔUßSÖaÑBäÓjÜ>Ð(ÖaÜ1ØlÛaç€áÂÛaÔÕà1å(Ö5âé@Ü>Û
Ü1Ð(Öaܛ߼ÖaÜ>Ú#Ü1àÛ6ß>Ú¤ålÔÕÓÚ#ÑBߛëìÔÕÖaå
DGF @
@
0.1
>
GhK¯¬«¬Ÿ[j
µ iǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
ú
>
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
~6
¦§,„§8¨
¹¤x­¦0§¹,(
§ B
œ¹,­(R§(>.¤*¸(
­ „§,R§,B¦,
>[email protected]/r9A2N,>?AQ07I9-.0/ /5;[email protected];Jh?-;@50G_O>[email protected]/,28
1 - 8#:>q/D‡9-2P/69A.0/E?:9-./7-/Š/ x/4t9A7¸4>857-/r9A.0/ [email protected]/69²9-2ZL2N
2 c9-.0/R9-2=2P?>8=0.FVYx>7)7-.02N%Ds;JD 8²;JP50?-/,
’46 8²;JP50?-/`-`7-.02N'7Z9-.0//'/469‡281Š;@D
9-/?A4.>D0P;@D0Pƒ9-.0/
N,/69-9A>8h;[email protected];^9A;J/7 281:9-.0/)9-2=2P?>8=0.FVg>8DQK7-50h7k9-?>9A/F[Dƒ9A.0;<7%4E>7-/Z9-.0/‡462D
9A>469,[email protected];@D0/B2DK9-.0/)9-2=2P?>8=0.FV
>QO>8D4E/7%135?k9A.0/E?B>4E?-2
7-7%;J9)7-2ƒ>7%9-2K?-/Q54E/9-./‚462D
9A>469Z>[email protected]/‡9-2N >8?Q07 9A.0/‚/
50;[email protected];@h0?-;@50G O>[email protected]/21
- YN%.0;@[email protected]/B9-.>892D`9-./l7k5h7"9A?A>89-/B?A/46/Q/7Y9A.0/‡D0/69Z/'/4t9Z281mN%.0;@4.j;<7 9-2K4E>857k/)9-./[email protected]/69%9A2L2N
50= ž[email protected];JG‚h
,2D
9-2R9A.0/B9-2=2P?>8=0.0;<4Z=/>8qY=8²;JP50?-/- ’4t
# .0/)eD>HeP50?A/Y=8²;@P50?A/RY04E2D7-;<Q/E?7Š9-.0/)L2Ni281²>‚[email protected]@VNŠ/E9k9A;JD0PQ0?-2=0HJ/E9YF;@D0;^9A;@>[email protected]/ED
9-?A/QK>9
DCOGF= ` <- =O- tYœQ2N%Df>ƒ=0H<>8D/R;@D4EHJ;@D0/Qf>89~ + & 9-2`9-./g.02?A;*2D
9A>H:2O/E?‡>K9-2=2P?>8=0.0;<4‚=/>8q
'
'
'
` - ~Y _ ` Y ED OGF ` <- & O- %‡>8DQ
` - -0 7R/EnF=x/469-/QœY²9-.0/
Q/EeD0/QThFV 7 U ` - 0Y
'
'
'
'
'
P?A>uOF;J9A>9A;J2D>8Hr132?46/K4>857-/79-.0/[email protected]/69‚9-2™7k=0?A/>Qf2O/?‚>8DQc/EO/ED
9-5>[email protected]/ED0P50HJ1%9-.0/K9-2=2P?>8=0.FV
# .0/?-/7k50HJ9A7„>8H<7k27-.02No9-.>89r9-.0/)[email protected]/69„7-=0?A/>Q7¸G2?A;
/ 5;@[email protected]‚>[email protected]/'/QP/7¸2819A.0/%9-2=2P?>8=0.FV
4>857-;JDP9A.0/%462D
9>[email protected];JD0/%Q2N%D7"9A?-/>8Gb281x;^9m9-2Bhx/4E2G/ .0/>?k9-W€7-.>8=x/QIY8²;@P50?A/Z ’h
tY;@DQ//QR7k;@G;JH<>8?
13?A/E/B7k5?k1’>46/=?-2eHJ/7„.>7,h//EDK?-/=2?k9A/Qg;@DK?A/4E/ED
9Š/En=/?-;@G/ED
9A>H>DQg9A.0/E2?-/E9-;<4E>H;@DFOF7-/69A;JP
>9A;J2D7r21
P?A>uOF;J9"VW[Q?A;JO/ED‚L2N'7r2O/E?¸[email protected]
W€7-50h0G/E?AP/QR9-2=2P?>8=0.0;@/7} ˆ 0Y0~ €:,9,[email protected]>89-/?r9-;@G/7„9-.0/)h0?>8D4./7¸21
9A.0/'462D9>469rHJ;@D0/%hx/E.0;@DQR9-./ 9-2=x2P?A>=0.FV462
>[email protected]/7A46/ >8DQ‚132?AG >B7k9-?>8;@P.
9r462D9>[email protected];@D0/'=x/E?A=/DQ;<4650H<>8?
9A2‚9A.0/Q;@?-/4t9A;J2D`21²7-=0?A/>Q0;JD0PY:
8 ;@P5?-/:’4t 0+%&)!)+HIM*)aF-&)!Za
# 0
. /`DF50G/E?A;@4>8H%[email protected];@2DT21'9-./K9A;JG/EW€Q/=/DQ/[email protected]?A;@4>9A;J2Dp/ 5>89-;@2D7l132?R9A.0/`4>7-/[email protected]/69
7-=0?A/>Q0;JD0Pƒ;@7)462GR=HJ;<4E>89-/Q]h
VK9-./‚D0//Qj9A2ƒ2O/E?462G/Y0;JDs>8D]/ECg4E;J/D
9ZN >uVY09A.0/7k/O/E?A/?A/7k9-?A;@469-;@2D7
/D46250D
9-/E?A/Qp;JDU?A/EH<>9A;J2DT9-2™9A.0/j57k/j21Z=x/E?AGR;<7A7k;@[email protected]/K9A;JG/j;@D46?A/EG/ED
97g462D7k/5/ED
9R2Dp9"N,2™G>;JD
1’>4t9-2?A7B87k9-;*xD0/7-7„;JD
9A?-2Q54E/QRhFV‚7-50?-1’>46/ 9A/ED7-;@2Dg>8DQR9A.0/'D0//Q‚9-2l?-/7k2HJO/,[email protected]/ED0P9-.7A4E>[email protected]/7¸4EHJ2
7k/
9A2RN,/69-9-;@[email protected];JD0/7E
# .0/'>8D7kN,/E?m.0/?-/Y.>7mhx/E/D‚9A2Q/EO/[email protected]=R>8DQR/[email protected]>D‚/ECg4E;J/D
[email protected]@Vl;@GR=HJ;<46;J9YGl50HJ9-;@P?A;@[email protected]/E?
/Glhx2FQ0V
;@D0P>Q0>=9-;@O/9-;@G/6W[7"9A/E=0=0;@D0P7k/HJ/4t9A;J2DIY
9-.0/BH<>9k9A/E?,2=9A;JG;<7k;@D0P‚9-.0/‡4.02;@4E/21I9-;@G/B7k9-/=ƒ;@D`>
4E2D
[email protected]/QlGg>D0D0/E?,7k5h"5 /469r9-2>‡7-=/46;Je/QR9A/EG=x2?>8H/?-?A2?²[email protected]/E?>8D4E/ ˜N%;<Q/'O>?-;@/69"V‚281œ=0?A/EOF;J257-HJV
?A/E=x2?-9-/Q >8D>[email protected];@4>8H¸>8DQ]DF50G/E?A;@4>8Hm?A/7-50H^97EYx9A2P/E9-.0/?BN%;^9A.™>`7k/?-;@/7)21ŠD0/Nv2D0/7‡462D46/?-D0;@D0Pƒ9-.0/
G289A;J2Dc281%[email protected]/6972Df.0/E9-/?-2P/ED/E257‡7k50h7"9A?A>89-/7YI.>uO/gh//EDc57-/Q™9A2]O>[email protected];@Q>9-/7-5446/[email protected]/
>=0=0?A2>4.M
# .0/'9A/[email protected];<4E>H0?A/
50;@?-/GR/D
9r132?„9-.0/)7k;@Gl50H<>9A;J2Dg281œL2N'7„=>7k9Š/O/?„7kGg>[email protected]/E?„9-2=x2P?A>=0.0;<4%>8DQ
N,/69-9-;@D0PZ./69-/?-2P/D0/E;J9-;@/7Y>DQ4E2Glh;JD>89-;@2D7:2819-.0/%7A>8G/YGg>8q/7I9A.0/,/6C46;@/ED4EV‡2817-54.R>'DF50G/E?A;@4>8H
132?-G‚[email protected]>89-;@2DR/6nF9-?A/EG/HJVR>9k9A?A>4t9A;JO/Y8=>?k9A;@4E50H<>8?AHJVN%./E?A/%.0;JP.?-/7k2HJ59A;J2Dg>8DQR.0/D46/ eD/'G/7-.0/7r>?-/
/7-7-/ED
9-;<>8HX 8²;J?7"9AHJVYm9-;@G/6W[7"9A/E=U7k/HJ/4t9-;@2Do;@7/EC46;@/ED
9>8DQo/ED>[email protected]/7‚9-.0/j?A/EH<>9A;JO/[email protected]>[email protected],9-;@G/j7k9-/=7
?A/
50;@?-/QœYx132?‡/En>G=0HJ/YI>9B9-./Rhx/EP;JDD0;JDPd281,>j7-;JG‚50H<>9-;@2Ds2?‡N%.0/D 9A.0/L2Nv;<7B/[email protected];JD0PX
50;<4q
[email protected]
9A2ch/];@D4E?-/>7-/Qo;JDz>c4E2D
[email protected]/Qo>DQU>44650?>9A/`Gg>D0D0/E?ƒN%.0/EDz4E2DQ;J9-;@2D7g=/?-G;J9 aF/4E2DQœYr9-.0/
hx/ED0/Ee09„21x9A.0/'\d50HJ9-;@P?A;<Qg7k2HJO/E?m;@7Š7k54.‚9A.>9Š;^[email protected];[email protected];<46;J9„D>9A50?A/%/ED7-50?A/7m9A.>9r9A.0/%H<>8?AP/E?m9A;JG/
7k9-/=s/7"9A;JGg>9A/7)P;@O/D hFVj9-./>Q0>8=09-;@O/9A;JG/EW€7k9-/=0=0;@D0Pd7k/HJ/4t9-;@2Ds=0?A2F4E/Q5?-/R4E>8Df>469-5>[email protected]/57-/Q
;@DT=0?>469-;<46/Y:N%.0;@HJ/`>8H<7k2]2 x/?-;@D0Pd9A.0/K>89k9A?A>4t9-;@O/13/>9A50?-/ƒ9-.>89l9A.0/j+ ‰ *w9-;@G/ƒ9>8q/Df9-2 [email protected]/9-.0/
Q;<7A46?A/69-;<7-/QR/
5>89-;@2D7k/E9Y>89r/>4.l9A;JG/7k9-/=IY8;<7„7-;[email protected]
’!G¸N%.0/E?A/%!˜;@7¸9-.0/'DF50G‚h/?r28150D0qFD02N%D7E
›t¡EµX¯¬®t±6¥
g¤
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjÇ
i ›t¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
~
î
:¦íq>’¨°q.
Ȳ
¦.ƒ"
¦ƒ
²¨
1.0
0.9
0.8
0.7
60
0.
0
2.1
0.6
2.7
0.30
(a)
2.40
3.30
0.90
0
0.5
0.4
90
0.
1.80
0.60
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.9
1.0
1.0
0.9
0.8
0
0.
20
20
1.60
2.00
1.80
1.20
0.6
0.6
0.
0.80
0
0.4
1.00
0.7
0.5
(b)
1.0
1.40
0
0.60
0
0.
0
0.2
0.3
80
0.4
0.4
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1.0
0.9
0.8
0.30
0.45
0.1
5
0.9
0
7
0.
0.60
5
5
1.
1
0.
05
0.7
0.6
0
1.6
5
1.20
1.50
0.3
0
0.6
(c)
0.45
0.5
0.
45
0.1
5
0.7
5
0.3
0
0.1
5
0.3
1.0
5
0.4
0.60
0.2
0.1
0.0
0.0
×ÔÕà1ØlÛaÑ81ò
×gÛaÑjѼç&ØlÛ&éÂß>ÓjÑ"á
J
âÚgÛaÜ>ø.ÙÕÑjçîá*ÙÕѐé:Ö5â·ßÐ.ä
éÂØlÙUÙJõGöëѐÖ&ÖaÔÕÐlà8ägÛaÜ1ÚlÙÕѐÖBú.ägÜBëìÐîßÈÚ¤ÙUß>ÐlÑ!ÔUÐlÓjÙÕÔÕФÑBäßSÖ
@ )B C
@ KB [email protected]
@
KB KB 3
×ÔÕФѐçLրàÛaÔUäälÑjÐlç&ÔJÖ¹õ·ÔÕç
›t¡EµX¯¬®t±6¥
g¤
CBDBDBD‹t›t±u¢
GhK¯¬«¬Ÿ[j
µ iǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
ÿ
@
C
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Ü>éXß
ÖaÜÈÖaålыå¤Ü>ÛaÔjÜ>Ð(Ö6ß>Ù*ú#ÜSè,ѐۀßÈÖaÜ1Ú#Ü>à>Û6ßÚ¤å¤ÔÕÓ!Ú#ÑBß6·ëìÔJÖaå
Ý
0.2
ßç&ç&Ü(ÓjÔUßÖaÑBä°ÓjÜ>Ð(ÖaÜ1ØlÛaçíáÂÛaÔÕà1å(Ö5âÈéÂÜÛÖaålÑÞqÜ>ÖaÔÕÜ1Ð
ÿ ú ú J>ú
á â …á
:
(âßÐ.ä
µ
µ
Öaå¤Ñ„ç&ؤïlçLÖ&Û6ßÖaÑìëìÔÕÖaå8ÛaÑjç&Ú#ÑjÓ5ÖÖa܋Öaå¤Ñ„å¤Ü>ÛaÔjÜ>Ð(Ö6ß>Ù*òáß,â B H
ÿ
0.1
@ )B 1ò ÜÖ&ÖaÑBä¼ÙÕÔUÐlÑjç‹ÔÕÐ.ägÔUÓjßÖaÑTÖaålÑÈÔÕФӐÙUÔÕФßÖaÔÕÜ1ÐíÜ>é
ÿ)B
K ú#á*ï#â @
B
K2 úá*ÓBâ @
)&2 ò
ÿ C ß>Фä4 @ &21ò
B
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
~ &
¦§,„§8¨
¹¤x­¦0§¹,(
§ B
œ¹,­(R§(>.¤*¸(
­ „§,R§,B¦,
œ¹
›
h .¸
¨ ;å¤ÔÕÙÕÔUÚlç
óå¤ÑXß>ØgÖaå¤ÜÛaç„ëìÔÕç&åÖaÜÛaÑjÓjÜÛ6äãÖaå¤ÑjÔJۀà>Û6ßSÖaÔÕÖaؤälыÖaÜ
ß>ФäqÖaÜÈÌãÔÕÓ5ålÑjÙ ÑjÓ5Û Ñ€ÔÕÐ8Ú.ßSÛ&ÖaÔUӐؤÙUßÛ
é@Ü>Û
ålÔUç
ÙÕÑjӐÖ&ÛaÜ1ÐlÔÕÓjçjú ÔUФälÜSè,ÑjÐé@Ü>ۀç&Ú#Ü1Ðlç&Ü>ÛaÔÕФàqÖaå¤ÔÕçìëÜÛ ,ÑjÑjÐ8ÔÕÐ(ÖaѐÛaÑjçLÖ
ÔÕÐqÖaå¤Ñ„ç&ؤïG÷LѐӐÖ
Þ8ßÖ&ÖaÑ5ÛBú(ÔÕФç&ÔÕà1å(Ö&éÂØlÙ#ÓjÜ1ÞqÞqѐÐGÖaçß>Фä
ÐGؤÞqѐÛaÜ>ؤçålÑjÙÕÚléÂØlكç&Ølà1à>ÑjçLÖaÔÕÜ1Фçjò
ƒO
tI §:
Ÿ„¨²Ÿk¢¢uŸ" ²¦œ¨Š§hlŸ"¥’¥ž¯¬¢u®,H8 ž¥X©A¥X¯¬¤" :©¢u£B£µ¢©»%¯¬¤" -§(º26pB).7·e5?)¤N9>ÅBAa).Ik· 1E
¤H BC§
C§x¦0Ÿ"°užX°u¢u®
#R­l¨²€©"´6Ÿ" Ç ²§q¡u¯¬¢b¤"›6©A¥ž¯¬¢u®˜›-´6Ÿ"s¥ž›t¡8›t®t€©¡u±6µ6§ã$ ¶ƒ¶¶
Ã(4RB%(A&RS¸5'Â=:eB%(A°eB%æ¾l2503=:¸LeB%G?B/,¸6'*eB4
78RS%(/&¿aRS¸6'Â/14&=U%>V>).k
I ·t· # E
I'&H jC5Kj
M §
# §O²ž›t€
# LH · # ILK
¢ „
­Bm
©"´¯¬ 1r­ œ©¢uªt›<Èu
¨Š§F
›t¢u®º¶ ž¤-©«¬ŸIŸ[´6›t«¬°¥ž¯¬›t¢,›³8¥ž±u¯¬¢,«¬¯ E°u¯¬£ «¬»% -§jº26pB),7·e5?)(N9>ÅBAa),Ik·t·>E
· B
D §
u§ Ÿ"’¥X› "¯„
§I
±uŸB»'©A¥ž±uŸ"»'©A¥ž¯¬¤k %›³m»%›-´¯¬¢u®¤k›t¢6¥X©¤"¥Z«¬¯¬¢uŸ" %¯¬¢g¥X±¯¬¢K«¬¯ E°u¯¬£ «¬»% -§- eB'Â[email protected]¸¹25AeQ¿È':912 7¾¤)Ik·t· 1E
M &
H M t·5K>tM ·§
§
hlŸ"¯¬£u¢uŸ"·M
­ ¤X±I"I©’¥ ¼ h ­XŸ" 8q„
§
¯¬»Š°u«©A¥X¯¬›t¢d›³ ¤k›6©A¥ž¯¬¢u®j«©"µ6Ÿ"‡Ÿ"´6›t«¬°¥X¯¬›t¢™©¢u£s£X›t¡d³@›tž»'©A¥X¯¬›t¢]›t¢
±›tX¯ "›t¢E¥X©«F¤[µ«¬¯¬¢u£uŸ"X k§¤+1)ÁeB<:< eS=:?‹$L%('*2646).¾l¸6=Â)¤kI ·t·>E
'I LH C # K>C §
M §O±už¯¬ ž¥ž›-
´ O€­M¦0›t¢6¥XŸ" 8u
­ h‚©«¬®t€©Ÿ[³!
¨Š­Ÿk«©ž£uŸ
¨Š§¶»%¡u«¬¯¬¤"¯ ¥Z¥ž¯¬»%Ÿ X¡u«¬¯ ¥’¥X¯¬¢®³J›t)³J›t°už¥ž±º¶›tž£uŸ")¡©X©<8›t«¬¯¬¤
Ÿ E°©A¥ž¯¬›t¢u -§ÁeB0PG/>')7·26':91e5?BA PSPG<J).7·2L¸a9,) ¶ %>V>)¤kI ·t·>E
.H CBt
D ·5K>CBC
u§
§
hlŸ"¯¬£u¢uŸ"3:
­ E¤€I± "œ©ž¥ X h ­x
«¬Ÿ[µ¸
§I
›t«¬Ÿ%›³M X°u’³<©¤"Ÿ ¥žŸk¢ X¯¬›t¢®tX©£u¯¬Ÿ"¢E¥ž ,¯¬¢‚¤k›tžXŸ"¤[¥X¯¬¢®‡¤k›6©A¥ž¯¬¢u®£uŸ[³JŸk¤[¥ž ,¯¬¢
¤"›tž¢uŸ"X -§l+,)ÁeS<U< eB=:?‹$L%(')¤¾l¸[email protected])¤Ik·t·BM1E
.H MBM5K
§
§OE¤€I± œ
" ©ž¥ h ­Bx
«¬Ÿ[µ€r
§B¯¬»,°u«©A¥ž¯¬›t¢„›³£X›t¡u«¬Ÿ[¥»%›¥X¯¬›t¢Š›t¢,«¬›*"Mº<Ÿk¢Ÿkž®µŠ©¢u£ ±uŸ[¥žŸkž›t®tŸ"¢uŸ"›t°u  X°ž³@©¤kŸ" k,
§ +,)ÁeS<U< eB=:?
$L%('*2646).¾l¸6=Â)¤k
I ·t· 1E (F ¤HJI # KGI §
·§OE ¤€±I"œ©ž¥ hs5§ Iµ ’¥XŸ"XŸ[¥ž¯¬¤xŸN
Ÿ"¤"¥ž ¯¬¢¸£už›t¡u«¬Ÿ[¥F»%›¥ž¯¬›t¢¸›t¢¸±uŸ[¥XŸ"X›t®tŸ"¢uŸ"›t°u 0 ž°+u ž¥ž€©A¥žŸ" 5-H £u¯¬XŸ"¤[¥F¢E°u»%Ÿ"X¯¬¤k©«t ž¯¬»,°u«©A¥ž¯¬›t¢
§
ƤRB%>VS0€/1=U4!k
I ·t· 1E
I6
C &H # SD5K # # §
I6
D §XŸ"
qr
­¤€±I
" ©’¥ '
h ­M
›kµ Š§O0
¯¬¢u®tŸ"X¯¬¢u®¡u±Ÿk¢›t»%Ÿk¢u©g³J›t)£X¯ ´6Ÿ"¢j¤"›6©A¥X¯¬¢® «¬»% k§N9>ÅBAa);< /1=:?BACBDBDBD1E
M &
HJI6
C 5KGI6t
C ·§
IBt
I §M
›¤Xª¯¬¢u®3#R
§(I
¯ ´t©«¤k›t¢6¥X©¤"¥’º3©¢u®t«¬ŸŠ»%›£Ÿk«¬ I©¢£Z¥ž±uŸr X¡užŸ-©£¯¬¢u® ›³0£už›t¡u -§l+,)¤;< /1=:?!7·2¹¸a9,).k
I ·t·BC1E
.H M >ILK>M 1t
I §
I6
C §:
¯¬Ÿ ìB
„­¹m›t¢u£u¯¬¤ ­ Ÿ"ž¥ž› k¤
¯ Š§8¨²«¬› u©«F»%›£Ÿk«¬ œ³J›tI»%›-´¯¬¢u®,¤"›t¢6¥€©¤[¥m«¬¯¬¢Ÿk k§(N9>ÅBAa)#ºO25pB) ¶ CBDBDBD1E .H D1IBI6CBD §
I # §
hlŸ" X žŸ"«¬¯¬¢u® ¦0§G$L%('Â4e5?B/,¸6'Â=:eS%'*e378/1< 'Â= VS4&=:?X0325':91e6?SX
A §h`¯¬«¬Ÿ[µgH(:
ŸN" ›tXª8­¤k
I ·t·B
C §
I u§OE
¡u¯ ¥€©«¬Ÿ"ž¯R
­
›tX¯¬¢©«¬£Ÿk ž¯x
§¤o
»Š°u« ¥X¯¬®tž¯¬£ XŸ"»%¯ º¶¯¬»%¡u«¬¯¬¤"¯ ¥¢u¯ ¥žŸ%£u¯ ŸkžŸ"¢u¤"Ÿ»%Ÿ[¥ž±u›E£‚³J›tr¥ž±uŸ%¥6"œ›º¶£¯¬»%Ÿk¢ X¯¬›t¢©«
ž±©«¬«¬›'" "I©A¥XŸ"MŸ °©A¥ž¯¬›t¢u k§l$&%(')#+1)#- />02646)#7·26':9,)¤;< />[email protected]?BAkI ·t·>E ¤HJI6CBCt·5KGI6C
S
D §
I §O
±u›t°ã)
º„
l
§ T»,°u« ¥ž¯¬®tX¯¬£‡£¯ Ÿ"žŸk¢¤kŸ'©¡¡uX›6©¤€±‚¥X›B ’¥XŸk©£µ ›*" 8Ÿ"¥6"ŸkŸ"¢Ÿ"¤"¤"Ÿ"¢E¥žX¯¬¤'X›¥X©A¥ž¯¬¢u®B¤[µ«¬¯¬¢£uŸ"X -§#$&%('*)+1)
-/1032546)#7825':9,)l;< /1=:?BAìCBDBDBD1E
H 5
· K 6·
u§
I6
M §
¯©›E
*º §1‡
© X±u©kµ6Ÿ"ª!I
§>d
»,°« ¥X¯¬®tž¯¬£'©¡u¡uX›6©¤X±³J›tœ ž¥žŸk©£µ ž¥X©A¥XŸm«©»%¯¬¢u©x´¯¬ ž¤"›t°u ›'"M -§,$L%(').+,)¤- />02546)¤7·26':9,)
;< /1=:?SA CBDBD1IBE
.HJI6
D 6KGI6C # §
I §I
±u›t»%¡ X›t¢€
¦­6F
Ÿ "Ÿk©°,¦0§:
¡u¡u«¬¯¬¤k©A¥X¯¬›t¢r›³u¥ž±uŸx³J°u«¬«E©¡u¡už›'¯¬»'©A¥X¯¬›t¢r ’¥X›tX©®tŸ»%Ÿ[¥X±u›E£r¥X›I¥X±Ÿ¢°»%Ÿkž¯¬¤k©«E ž¯¬»,°u«©A¥ž¯¬›t¢
›³¥6"›º¶£u¯¬»%Ÿ"¢u X¯¬›t¢u©« ’¥XŸk©£µ„¯¬¢u¤"›t»%¡užŸk ž X¯ u«¬Ÿ´¯¬ ž¤"›t°u 0»,°u« ¥ž¯¬¡u±© žŸ ›*": k§S$L%(')(+,),- /1032546)17·25'U9()>;< />[email protected]?BAE #HJI6C1I 6K
I6C # ·§
I § X©¢u£¥íŠ
§'¨²°u¯¬£uŸj¥ž›c»,°u« ¥ž¯¬®tX¯¬£T£uŸ[´6Ÿ"«¬›t¡u»%Ÿ"¢E¥k§‹¶¢
7q/1< '@=JVB4&[email protected]?Ç0325':91e6?SA&fqƤ2L¸5'@/142©%GeB'*26Aù=U%
7·RB':912503RS'@[email protected]¸6AQ½
©¤€ªIu°u X¤X±jhs,
:
­ X›¥’¥XŸ"¢I
Ÿ"ž®! Ÿ"£u N€§l
¡už¯¬¢u®tŸ"žºŸ"X«©®u­ Ÿ"X«¬¯¬¢¤E,CBCBD5K # I6
C §
kI ·§O
±©¡€©‹,²(
­ I
©¢u©«¬ŸìM¦0§l- />0264&[email protected]¸LRB<.0325':91e6?BA¿&eB4„25%>VS=U%G2L264&€
A §(B
¤-¨²€© "Iº:
¯¬«¬«<­Gk
I ·t· §
CB
D §O²
ž¤€±©ž£>M
§g²
¢ X°u’³<©¤"Ÿ¸«¬Ÿ[´6Ÿ"«¬«¬¯¬¢u®,¯¬¢´¯¬ ž¤k›t° :«¬¯ E°u¯¬£u I©¢u£)®tŸk«¬ k§ OPPG< )#¾l¸[email protected])¤º25A&)
E
GH 1t
I §
C1t
I §B
¯ ¥ž«¬¯¬¢ì§G²
¢Z£uŸN"œŸ[¥’¥X¯¬¢®¤"›t¢u£¯ ¥X¯¬›t¢u k§ÁeB<:< eS=:?BA¾g/14:¿&R¸¹25A €flN9>ÅBAL=:¸Le6¸a91250) ¶ %>V)
AÂP12¹¸5'ÂA€Ik·t·
,E
.H ·6KGI6D1t
I §
CB
C §O6
¥€©ž›-´< R
­u¹¸©«¬¯¬¢u¯¬¢
,­(œ
±Ÿk¢Tœ
ºŠ
§ ?BpB)ÁeB<U< [email protected]?$&%('Â254:¿&R¸¹2„¾l¸[email protected])¤Ik·t·
,E (F.HJI §
C # § 8±u›tX¢¯ ¥X žª©kµE
© x­ Ÿ"ž¥ž› k¯ §-¦0›t ž¯ ¥X¯ ´¯ ¥¶µº<¡uXŸ" XŸ"’´¯¬¢u®I¢E°u»%Ÿ"X¯¬¤k©«t X¤X±uŸ"»%Ÿk ³@›tF«¬°uX¯¬¤k©A¥ž¯¬›t¢º<¥¶µ¡ŸŸ °©A¥ž¯¬›t¢u k§S¾,$ 7
+1)#-/1032546) %GRB< )lCBDBDBD1E
C &
H BC # K §
C u§‡
© "›t°u¤X±u¯GŠ
­SM
›t»% žµ,¨R
§B
XŸ"ŸM X°u’³<©¤"ŸE
¥ž›tªtŸ" ›'"ƒ›k´6Ÿk¥X›t¡8›t®t€©¡±Eµ6§1N9>ÅBAa)G;< />[email protected]?BACBDBD1IBE
I6
D LH
C 1
ILK>
C jM1t
I §
C §œ¨m© žªtŸ"«¬«F¦r
­Gt¯¬»'©¤€ª'¦¹,g
­ Ÿ"«¬«¬¯¬Ÿ"R
­,I
±u›t»%¡u ž›t¢R
­7h`¯¬«¬ ž›t¢8¸§8¨²X©k´¯ ¥¶µº¶£X¯ ´6Ÿ"¢ ›*"s›³¤"›t¢6¥X¯¬¢E°u›t°u ²¥ž±u¯¬¢
«¬¯ E°u¯¬£?
«¬»% œ›t¢)¢›t¢º¶¡8›tX›t°u M X°u
ž¥ž€©A¥žŸ" "M¯ ¥ž±Z¥ž›t¡›t®tX©¡u±6µE§#A¹/ 60€=:'@'*2L?!'*eX+1).;< /1=:?‹7·2¹¸a9,)GCBDBD # E
CB
M §¹m›t¢£u¯¬¤ãx
­ ²
¯¬Ÿ u
§²¦©A¥’¥XŸ"X¢d³J›tX»'©A¥ž¯¬›t¢d¯¬¢j¥X±Ÿ ›*"b›³,¥ž±u¯¬¢ «¬»|£u›*"M¢s©¢]¯¬¢u¤"«¬¯¬¢uŸ"£d¡u«©¢ŸS:
H ¤"›t¢u ž¥X©¢6¥ u°
¤"›t
¢ u®t°u€©A¥ž¯¬›t¢
§lN9>ÅBAa)¤;< /1=:?BAìCBDBD1IBE
IB'
I LH # I6M 5K # I u
§
C §:
›tX»'©¢u£‹S¸
§1- />0254&=:¸LRB<g025'U9,e6?BA¿&eB4?B= 25425%('Â=:RB<g2 5/1RS'@[email protected]%(A R¸LeB0PG/>'*RB'Â[email protected]%GRB<.R&PSPG4e5RS¸a9§(g
¦xžŸ" X -­(k
I ·t·B
M §
C §ž›¥ž¥žŸ"¢ 8Ÿ"X®ír
§78/>< 'Â= VS4&=:?&kÄ)TÃ(4eB'Â'*25% [email protected]½TÁƒ) q) eSA¹'*eB4 ¹2ãRB%G? €)¾l¸a9>/1<U< 2646)X»O/,26AL'ÁeB%('Â4&= 6/1'@[email protected]% 6Å „)
4RB%G?B':½ƒN) eBA ƒRB< ?ÈRB%G?‹É3).¾g'Â/ ¹25[
4 §(
©¢:
¯¬Ÿk®t›‹œ
›t¢³@Ÿ"XŸ"¢u¤"ŸìF
›t¢u£›t¢¤H(²
¤k©£uŸ"»%¯¬¤t­(CBDBD1t
I §
tC ·§:
©¤€ªIu°u X¤X±jhs§g78/1< 'Â= a»O4&=:?‹025'U9,e6?BA„RS%G?TR&PPG< [email protected]¸LRB'Â=:eB%(€
A §l
¡už¯¬¢u®tŸ"-­ Ÿkž«¬¯¬¢
­gk
I · §
# D §0©¢¢uŸ"x
1
§ I
±uŸ¸ X¡užŸk©£u¯¬¢u® ›³ X¯¬«¬¯¬¤"›t¢£uX›t¡ M›t¢Z±u›tX¯ "›t¢6¥€©« ž°už³@©¤"Ÿ" -§¤+1)lN9>ÅBA €)¤Ik··1E
¤HJI # §
# tI §
Ÿk«©±ÈqŠ
­(‡
©ž»,°uŠ
§GE
¡uXŸk©£u¯¬¢u®,ª¯¬¢uŸ[¥ž¯¬¤" ²›³0£už›t¡u :›t¢Z®t«© X k§¤+1)ÁeB<:< [email protected]?$&%(').¾l¸[email protected])¤k
I · 1IBE
¤H 1I 5K B
C §
"
(&
)&
!
"
#
>8
g¤
5?
1
1
CBDBDBD‹t›t±u¢GhK¯¬«¬Ÿ[µjÇ
i ›t¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
/&
*#
@6
+
=
›t¡EµX¯¬®t±6¥
-
'&
# #
#
%$
(,
#
!
3
54
7 98
1
6
0&
1
2&
* *
*
.
#
'&
:
;1
A
*!
<1 '
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
î
:¦íq>’¨°q.
Ȳ
¦.ƒ"
¦ƒ
²¨
~
(
# C§:
©ª©"µE©!m
§(¡užŸ-©£Z›³«¬°¯¬£È²
ž›t¡u I›-´6Ÿ":©:
›tž¯k›t¢6¥X©«F¦x«©¢uŸt§l+,)¤N9>ÅBAa)#¾le6¸j)+1R&P>RB%g)k
I · ,E#¤H # ·§
# # §O6¥X¯¬«¬« "I©®t›t¢T.:
­1©X ž›t¢I¨Š§1°u¢u£©»%Ÿ"¢6¥€©«¬ M›³0¥X›t¡8›t®t€©¡±u¯¬¤r X°u’³<©¤"Ÿ¸«¬Ÿ[´6Ÿ"«¬«¬¯¬¢u®u§¤+1)OPSPg< )#N9>ÅBAa).Ik·1E#.H BC1It§
# u§:
Ÿk¤"Ÿìq
­ œ©XŸ[¥º²§8¨²€©"´¯ ¥¶µ£už¯ ´6Ÿ"¢ &›*M
" :›³´¯¬ X¤"›t°u M«¬¯
°¯¬£u I›-´6Ÿ"M¥6"›º¶£u¯¬»%Ÿ"¢u X¯¬›t¢u©«¥ž›t¡›t®tX©¡u±u¯¬Ÿ" -§¤+,)¤;< /1=:?
782L¸a9,)(CBDBD # E .HJI 6KGI6MB
M §
›t¡EµX¯¬®t±6¥
g¤
CBDBDBD‹t›t±u¢
GhK¯¬«¬Ÿ[µjiǛt¢u -­1
¥ž£
§
N42QP>RB42L?È/1A¹=:%>VXWSYSZS[\:],^ _a` b
$&%(')+1)#-/1032546)7·26':9,)O;< /1=:?BACBDBDBD1EGFGF#HJILK>M
1/--страниц
Пожаловаться на содержимое документа