Fraction-free Row Reduction of Matrices of Skew
Polynomials
Bernhard Beckermann
Laboratoire d’Analyse Numérique et
d’Optimisation
Université des Sciences et Technologies de Lille
France
ABSTRACT
.0/21
34/ 5$/76
8:9;6/ <=9>@?-A3$B@8DCEGFA3H34A<I34/ J%KL 8DB@A6MAFN9OEP98D3$BRQ
AF#5DS/ <M1A>@T6AE:B@9>U5 VHWNC/H9>@?-A3$B@8DCEXLY96[Z\/]K5$/ J;FA3N^
6JBR6?
FUK>R>#3496S_J/ LYAE21A-5DB@8DB@A65296J`A-8DC/43H3496
S[34/Ya/ 9>RBR6?O8D34965DFA3$b
Ec98DB@A65`AF:Ec98D3$B@L / 5OAF:5DS/ <d1\A>@T6AE2B@9>e5 VfWNC/`9>@?-A3$B@8DCE
B@5`BR6
8$/76J/YJgFA30LYAE21
K8$9-8DB@A6IBR6h/7Q
9L 8`93$B@8DCEc/ 8DB@LMJ
AEP9BR65
<NC/734/[8DC/[?34A<]8DCiAFHLYA/7jkL7B@/76
8$5PBR6iBR6
8$/734EP/ JB@98$/0L AE:1K8$9b
8DB@A65PB@5k90L /76
8D349>lL A6LY/73$6mVnWNC
B@5kL A/7jkL7B@/46
8;?34A<]8DCnB@5kL A6b
8D34A>R>@/ JoZ
TpK5DBR6?qFe349-L 8DB@A6
bFe34/ /`EP/ 8DCAJ
5 VIWNCB@5O9>R>@A<r5;K5k8$A
AZ8$9BR609s1\A>@T6AE2B@9>RbD8DBUEP/P9>@?A3$B@8DCE[t(FA3u96`vfwyxHEP98D3$BRQkAF
BR6
1
K8l5DS/ <M1A>@T6A-E2B@9>U5zAF{J/Y?34/ /:|X<NB@8DC_L A/7jkL7B@/46
8$5H<NCA-5$/
>@/46?-8DC5P934/PZAK
6J
/ JnZ
T~}8DC/k9>@?A3$B@8DCEGC95c9[<NA345$82L 9-5$/
LYAE21
>@/7QB@8DT`AF2v[x -|[}`
Z
B@8HA1/73498DB@A65 V
1.
INTRODUCTION
e60R mrZ349-EPAak96J[34A65$8$/7BR60? B@a/z926/ <MF9-5$8N9>@?-A3$B@8DCEdF*A3
J
/ 8$/734E:BR6BR6?P<NC9-88DC/ T:L 9>R>\9]3496
SP34/ a-/ 9>RBR6?:8D34965DFA34EP9-8DB@A65
F*A3O5DS/ <f1A>@T6A-E2B@9>cEP98D3$B@LY/ 5 VWuC/ 5$/`934/`8D34965DFA34EP9-8DB@A65
<NC
B@LC0LYA6
a/4348]9cEP98D3$BRQ_AF#5DS/ <o1A>@T6AE2B@9>e5HBR6
8$AkA6/z<NC/734/
8DC/N3496
SPB@5NJ
/ 8$/734E2BR6/ JO/76
8DBR34/7>@TPZ
T28DC/N>@/ 9JBR6?;A3N8D349BR>RBR6?cL Ab
/4jkL7B@/76
8(EP98D3$BRQV{WuC/ Tz5DCA<q8DC98#8DC/7BR39>@?-A3$B@8DCELY96PZ/(K5$/ J
F*A3r9z6KEzZ/73uAF#911
>RB@L 9-8DB@A65HBR6L7>RKJBR6?k8DC/lJ/ 5DBR6?K
>@93$BU 9-8DBeA6
AF#>RBR6/Y93:34/ L7K
3$34/76LY/25$T5$8$/ Ec5H96J[FA3HLYAE21
K8DBR6?[3498DB@A69>(5$Ab
>RK8DB@A65PAFN9k>@934?/kL7>@9-5$52AF>RBR6/ 93PFeK
6L 8DB@A69>r5$T5$8$/ EP5 Vce6~8DC/
F*A34EP/73c91
1
>RB@L 98DB@A6I8DC/7BR3:3496
S~34/ a/Y9>RBR6?~91
134A-9L$CMB@E21
34Aa/ 5
A6`8DC/2]b/7>RB@E2BR698DB@A6nEP/ 8DCAJ`AF(rZ349-EPAa@7V
e6`8DC/zL AEcEsK8$98DB@a-/2L 9-5$/z8DC/z9>@?A3$B@8DCEAFrR B@5]8DC/z5$9Ec/z95
8DC98#?B@a/76cZ
T2N/YLS/734EP96
6k96J:m9Z9C6` V{*6:ZA8DCkLY95$/ 5#8DC/
9>@?A3$B@8DCEP5O934/PF9-5$8sZ
K8c8DC/7BR3yEP/ 8DCAJ5234/YKBR34/`/7Q
9-L 8:93$B@8DC
b
Ec/ 8DB@L2<NC
BR>@/26A8]C96J>RBR6?`L A/4jkL7B@/76
8z?34A<r8DC0/7Q
LY/718l8DC34AK?C
LYA/7jkL7B@/76
8yzuL AE:1K8$98DB@A65 V`.nB@8DCAK8c8DC/kzNfL A-E21
Kb
8$9-8DB@A65H8DC/2LYA/7jkL7B@/76
8l?34A<r8DC`L 96[Z/:/7Q1A6/46
8DB@9>V
WNC/~EP9BR6=L A6
8D3$BRZ
K8DB@A6BR6h8DCB@5`191/730B@5`9q6/ <9>@?-A3$B@8DCE
<NC
B@LCi1/73$FA34EP5234A<34/YJ%KL 8DB@A65[A6i9`?B@a/46oEP9-8D3$BRQMAFl5DS/ <
1\A>@T6AE2B@9>e5PBR6
8$A`A6/PC9aBR6? 9[FUK>R>N3496S0>@/ 9-J%BR6? A3P8D349BR>RBR6?
Howard Cheng
George Labahn
Department of Computer Science
University of Waterloo
Canada
!#"$!Waterloo,
% &'!(*)+ *
,
+
#$-
L A/7jyL7B@/76
8rEP98D3$BRQVWNC/r34/ J%KL 8DB@A65lL 96OZ/]K5$/ J[8$Az^
6JO9sFeK
>R>
3496
S¡J/ LYAE21A-5DB@8DB@A6iAFu9[EP9-8D3$BRQ~AFN5DS/ <I1A>@T6AE2B@9>e5k9>@A6?
<NB@8DC 3496S`34/Ya/ 9>RBR6?08D34965DFA34Ec98DB@A65zK5$/ J0Z
T[]Z
349EcAa`96J
34A65$8$/7BR6V=WNC/OEP9BR68$AA>HK5$/ JMBR6o8DC/09>@?A3$B@8DCE¢B@5 £¤¦¥§ ¤
¨4©ª § ª<uC
B@L$C[J
/ 5$L73$BRZ\/ 5H9>R>m5$A>RK8DB@A65lAF9:?B@a/46`A34J
/73N1
34AZ
>@/ E[V
WNC/c9>@?A3$B@8DCEB@5z6A-8$/ <NA348DC
T[Z/ L 9K5$/PB@8HK5$/ 5zA6
>@TFU349L 8DB@A6b
Fe34/ /(93$B@8DCEP/ 8DB@Lr<NB@8DCAK8#L A/7jkL7B@/46
8(zunL A-E21
K8$9-8DB@A65 «<NC
BR>@/
98c8DC/O5$9EP/O8DB@EP/[L A6
8D34A>@5OL A/7jkL7B@/46
8;?34A<]8DCiAF]BR6
8$/434EP/ JBRb
98$/`LYAE21
K8$9-8DB@A65 VIWNC
B@5yB@5k5DB@E2BR>@9308$A 8DC/[1
34AL / 5$5PK5$/YJMZ
T
8DC/P5DKZ
34/ 5DK
>@8$96
8k9>@?-A3$B@8DCEGFA3:L AE:1K8DBR6?08DC/kzNfAFr8<NA
5$L 9>@93k1A>@T6AE:B@9>@5~ ¬«r
«]-7V~WNC/;9>@?A3$B@8DCEB@52Z95$/ JiA6
8DC/~®#®#®#Fe349L 8DB@A6bFe34/Y/~EP/ 8DCAJ¯K5$/ JgBR6°N/ L$S/434EP96
6I96J
m9Z9C6± ²#<NCB@L$Cn<N9-5HJ
/ a/4>@A1/ J0FA3lFe349L 8DB@A6bFe34/Y/PL AE:1K8$9b
8DB@A6cAFEP9-8D3$BRQz3498DB@A69>91
1
34A³QB@Ec96
8$5 «
EP98D3$BRQczNl5m96J:?/76
b
/7349>RB@ / JM´NB@L$C934J5$A6/7Q
8D3491\A>@98DB@A6o134ALY/ 5$5$/ 5 Vne6i8DC/k5$L 9>@93
L 9-5$/H8DC/H®#®#®#9>@?A3$B@8DCEµ?/46/7349>RB@ / 5l8DC/l5DKZ
34/ 5DK
>@8$96
8zzN
9>@?-A3$B@8DCEfR¶ DV. /9>@5$AH?B@a/r9rL AE:1>@/7QB@8TP969>@T5DB@5(AF\AK
3#9>@?-Ab
3$B@8DCE[V(®A3(96PvIwlxEc98D3$BRQcAFBR6
1K8(5DS/ <~1A>@T6A-E2B@9>U5r<uCA5$/
L A/7jyL7B@/76
8$5]C9a/H>@/76?8DC5NZAK
6J/ J`Z
Tk}8DC/H9>@?-A3$B@8DCEdC95r9
<NA345$8]LY95$/zL A-E21>@/4QB@8DT`AF2vOx -|[}
YZB@8lA1/43498DB@A65 V
WNC/z34/ EP9BR6J
/73:AFN8DC/s191\/73zB@5H9-5]FA>R>@A<r5 VzWNC/s6/7Q
8l5$/ L 8DB@A6
?B@a-/ 5k8DC/kZ9-5DB@L`J/4^6
B@8DB@A65[FA3y8DC/k134AZ>@/ E¢96J~BR6
8D34AJ%KL / 5
A34J
/73lZ9-5$/ 5HAF(5DS/ <o1A>@T6AE:B@9>@5 «#8DC/s1
3$B@EP934T[8$AA>{8DC9-8H<NBR>R>
Z/K5$/ J:8$Ar5$A>@a/AK
3#134AZ>@/ EOVN·/ L 8DB@A6k¸r?B@a-/ 5#9]>RBR6/ 939>@?-/7Z
349
FA34EsK
>@98DB@A6`8$A:AK
31
34AZ
>@/ Ef<NCBR>@/z8DC/NFA>R>@A<NBR6?P5$/ L 8DB@A6`?B@a-/ 5
AK
3mFU349L 8DB@A6bFe34/ /(34/ L7K
3$34/76LY/V·%/ LY8DB@A6y¹uJB@5$L7K5$5$/ 5(8DC/(5$8$A1
1BR6?
L73$B@8$/73$B@A6n96J`L A-E21>@/4QB@8DT0AFAK
3z9>@?-A3$B@8DCE[V:WNC/z191/73º/76J
5
<NB@8DC09PL A6L7>RK5DB@A6¡9>@A6?O<NB@8DC`9PJB@5$L7K5$5DB@A6nAF#FeK8DK
34/z<NA3$SV
2.
PRELIMINARIES
m/Y82» ¼fZ/P96`BR6
8$/ ?349>rJA-EP9BR6p<uB@8DC ½gB@8$5zKA8DB@/76
8z^/7>@J 96J
>@/ 8 ½zR¾z¿DÀ(Z/P8DC/kÁ]34/PJ
AEP9BR6MAFN5DS/ <I1A>@T6AE:B@9>U5;Aa-/73 ½
<NB@8DCn9K8$AEcA3$1
CB@5$EÀn96J~ÂOÃVkWNCK528DC/P/7>@/YEP/76
8$5PAF ½
BR6
8$/7349-L 82<NB@8DCn8DC/k5DCBRF*8P¾=aB@90¾]Ä0ÃÀ#Ä
$¾sVkr6n/7Q
9E:1>@/kAF
5DKLC±9 J
AEP9BR6oB@5`» ¼ÃG» Åk ÆDÇ ½µÃG» ÅkÆ2<NB@8DC±¾d8DC/`5DCBRF*8
A1\/73498$A3H96J`À#ÄÆ$(ÃMÄÈÆyÉ Ê
FA3r5$A-EP/HBR6
8$/ ?-/7345rÈ296JOÊ
<NB@8DCPÈOÃM
Ë
V#WNC/L 95$/N<NC/76cÀÌÄÆ$(Ã¯ÄÆ(ÉkÍB@5#19348DB@L7K>@93$>@T
B@E21A348$96
8kBR6o91
1
>RB@L 98DB@A65 VI.0/;34/ Ec93$Si8DC9-8 «N9-5yBR6°R D«<N/
L 96~C96J>@/[8DC/21
34AZ
>@/ EÎAFrL A-E21K8DBR6?349-8DB@A69>l5$A>RK8DB@A65PAF
>RBR6/ 93kFeK
6LY8DB@A69>z5$T5$8$/YEP5PL A6
8$9BR6
BR6?nJBRÏ/734/76
8DB@9>«HJBRÏ/734/76L /
96J[bJBRÏ/734/76L /PA1/7349-8$A345]Z
TkEP911
BR6?;8DC/ EBR6
8$AP8DC/z9Z\Aa-/
J
AEP9BR65 V
]B@a/46n9kEP98D3$BRQ¡AF(1A>@T6AE:B@9>U5PAF5DS/ <I1A>@T6A-E2B@9>@5k<N/:934/
BR6
8$/734/ 5$8$/YJpBR6p91
1>@TBR6?~34A<=A1/73498DB@A65k<NC
B@LCn8D34965DF*A34E8DC/
E 98D3$BRQoBR6
8$AM9nEP9-8D3$BRQIAF:5DS/ <1A>@T6A-E2B@9>@5¡<uC
B@L$CgC95`8DC/
c
1
34A1\/7348DT08DC9-8z8DC/z3496
S0B@5H34/ a/ 9>@/ J Z
T`/7B@8DC/73:8DC/P8D349BR>RBR6?0A3
>@/Y9JBR6?oL A/4jkL7B@/76
8 Vµ.0/ <NBR>R>:FAL7K5[A68DC/~L 9-5$/ AFP8D349BR>RBR6?
LYA/7jkL7B@/76
8$5 Vu*6_8DCB@5l5$/ L 8DB@A60</]1
34AaB@J
/:8DC/H1
34/7>RB@E2BR6934T0J/4^b
6
B@8DB@A65z96J`8$AA>@5l<NCB@L$C`FA34Ef8DC/HZ95DB@5]F*A3lAK
3l91
134A-9L$CmV
.0/`95$5DKEP/`8DC98k<N/`934/[?B@a-/76 H¾H7«9 34/ L 8$96?K>@93`v¢wnx
½ ¾s¿À
Ec98D3$BRQ`AF(5DS/ <o1A>@T6A-E2B@9>@5:<NB@8DC0/76
8D3$B@/ 5]BR6 s
H¾l#Ã
¾ Çr<NB@8DC
» ¼
.0/:9J
918z8DC/:L A6
a-/76
8DB@A6pAFJ/76A8D BR6?[8DC/:
/7>@/ EP/76
8$5zAFH¾l
V{®A3l96
TOa/ LY8$A3
Z
TH¾H 7«96J`8DC/z/7>@/ Ec/76
8$5HAF
Z
T
AFBR6
8$/Y?/7345 Ã Ç!!"m
Ç# 7«<N/z >@/ 8:¾%& $ J/76A8$/c8DC/:EP98D3$BRQ
AFr5DS/Y<g1A>@T6AE:B@9>U5yC9aBR6?0¾'&$ ( A6~8DC/PJB@9?A69>r96J~k/ ab
/434T<uC/734/l/7>@5$/VNoEc98D3$BRQOAF5DS/Y<M1A>@T6AE2B@9>e5HB@55$9B@J[8$AzC9a-/
34A<oL A>RKE26\J
/ ?34/Y/)* BRF8DC/e+ b-,/2
. 34A<oL A>RKE26\ÌC95(EP9QB@EP9>
J
/ ?34/ /0* 1 V(WNC/za-/ L 8$A3%2 B@5r8DC/za-/ L 8$A3ºL A65DB@5$8DBR6?[A6
>@T[AFrV
WNC/N?A-9>
AF8DCB@5m191\/73B@5#8$AlL A65$8D3$KL 8#8DC/8DT1/NAF3496Sz34/Ya/ 9>Rb
BR6?08D34965DFA34EP9-8DB@A6526/ / J
/ J Z
T0]Z
349EPAa096J~34A65$8$/7BR6 ³
F*A3r8DC/7BR3l91
1>RB@LY98DB@A6\5 Vrm/ 8 ½zR¾z¿DÀR¾
¿¦À
Z/z8DC/]B@8$/7349-8$/ J
J
AEc9BR6¡<uC/734/z<N/rC9a-/28DC/HB@J/46
8DB@8DB@/ 5
5 ¾ 3 Ão
¾ 3 @5 ¾oÃgÇO¾5 Ä65@¾ 3 ¾
F*A3l9>R>{Ä
3 43 ÃoÀÌÄ
7ÇO¾
3 5RÄ65 ¾MÃoÀ 3 Ä
½HV
WNC/s3496S`34/Ya/ 9>RBR6?`8D34965DFA34EP98DB@A65:AF]Z
349EcAa`96J034A6b
5$8$/4BR6[LY96kZ/NFA34Ec9>RB@ / JO95#FA>R>@A<r5 V]B@a/76O8DC/HEP9-8D3$BRQkAF\5DS/ <
1\A>@T6AE2B@9>e5 H¾H
» ¼[ ¾s¿À
e1\A5$5DBRZ
>@T[9F*8$/73N9:5DC
BRF8r<uB@8DC
¾ 47«-<N/N<NB@5DCk8$AH^
6J y¾
» ¼k ¾
¿$À
5DKL$Ck8DC98
8 3 7 3 3 9
8 ¾ 3 45:H¾H#Ã<;ξH » ¼OR¾z¿$À=#7Ç
P
<NB@8DC=8DC/ 6KEsZ\/73nAFc6A6 /434AM34A<r5)i
> AF';ξl;L ABR6L4B@J%BR6?
<NB@8DC_8DC/H3496S_AF#8DC/z8D349BR>RBR6?[LYA/7jkL7B@/76
8@? «
96J;C/76L /:<uB@8DC
8DC/2349
6S0AFAÎ
; ¾l7V:*6¡9J
JB@8DB@A6i<N/z34/ KBR34/k8DC/2/4QB@5$8$/76LY/;AF
B¾l ½zR¾z¿DÀ 9 5DKL$C08DC9-8
B ¾H65:y8 ¾ 3 (ÃDC
ElA8DB@L /c8DC9-8z8DC/P5$/YL A6J FA34EsK
>@9[8$/7>R>@5zK5z8DC9-8z8DC/21
34AL / 5$5zAF
/4>RB@E2BR69-8DB@A6nFA3:?-/ 8$8DBR6?); ¾HB@5HBR6
a/7348DBRZ
>@/VG`
F A34/21
34/ L7B@5$/4>@T«
</zAZ8$9BR6`FA3H¾l8DC/HIJK
J ¤¦©ML#;N ¥§4O £MPRQ £ªS=TKS £L
H¾H#ÃU
; ¾HÃYX#B ¾HZ5G
; [ ¾l
$
B ¾HV5WÎ
<NB@8DC\
;[ ¾H » ¼OR¾z¿$À=] _^ AZ8$9
BR6/ JZ
T/7Q
8D349-L 8DBR6?X8DC/
6A6 /734AP34A<r5HAF`Î
; ¾H7«96JaBX ¾l ½HR¾z¿$À H] Z
T;/4Q
8D349L 8Db
BR6? Fe34A-Eb
B ¾lH8DC/OL A3$34/ 5D1\A6J%BR6?ML A>RKE:65 Vc7`F A34/ Aa-/73«N8DC/
3496S¡AFu8DC/P8D349BR>RBR6?¡L A/7jkL7B@/46
82AF G
; [ ¾l]B@52AFFUK>R>N34A<I3496
S
> «96J~8DC
B@5zK96
8DB@8DT~L ABR6L4B@J/ 5c<uB@8DC¡8DC/23496
S0AF
; [ ¾H7V:®mBRb
69>R>@T-«mFe34AEµ8DC
B@5H>@95$8H/YK9-8DB@A6n<N/:5$/ /:8DC9-8H8DC/23496
S`AFH¾l
B@5:ZAK
6J
/ Ji9ZAa/kZ
Td
> «(<NC/434/ 95:8DC/P^
345$8P/ K98DB@A6o8$/7>R>@52K5
8DC98k3496
SeH¾H)X
f 3496
SÎ
; ¾lkÃg%> VWNCK5[<N/[C9a/[FAK6J
> ñ3496SRº¾l7V
º
3
7
2.1
Order Basis
e6P8DC
B@5(5DKZ5$/ L 8DB@A6k<N/BR6
8D34AJKLY/]8DC/6A8DB@A6[AFA34J/43(96JPA34J
/73
Z95$/Y5FA3u9z?B@a/46`EP9-8D3$BRQ[AF5DS/ <M1A>@T6A-E2B@9>@5 VlWNC/Y5$/]934/H8DC/
1
3$B@EP934T[8$AA>@5l<NCB@L$C0<NBR>R>#Z/HK5$/YJ[FA3lAK
3l9>@?-A3$B@8DCE[V
h ikj#lnmVlpoqlKrVmf%V@Ves§WTt2¾H ½s ¾s¿À\ 9 ¨7§O©du-§vOWT£¤k£H
ª-N%§!wUQ£MJyxL£MP7S©MJRªO©ML¥U ©zP7IJnTKSK{|SKL¥§}~£H~SLVT§§ ¤7ª"V
§!L
t2¾HeS ªHªY©MS ¥T £7
©M-u §NA34J
/73 S H
-
2t ¾H Z5WH¾H#ÃU[ ¾HZ5 ¾ & $
wS=Tn
a_ ¾l ½HR¾z¿$À # [ ¾lSKLz-W'eS ª'4O ©MJK@J §4¥[©`¤$§ ªv*S ¥MI ©MJn
*6cL A6
8D3495$8(8$A]1
34/ aB@AK5191/7345J/ 9>RBR6?:<NB@8DCkA34J
/73#Z95$/ r
5 «¹%«
«²«<N/zC9a-/2L$CA5$/760960A34J
/73zL A6JB@8DB@A6pA608DC/s3$B@?C
8 r
V WuC
B@5
C95H8DC/z9J
a96
8$9-?/:8DC9-8]<NC
BR>@/2<N3$B@8DBR6?
H¾H#à ¾ Çt2¾là ¾ Ç
<N/HC9a/
t2¾HZ5H¾H#Ã
¾ Ç
Ã
À 3 v
¸
H/46L /8DC/NK
6S6A<u65 L 962Z/AZ8$9BR6/ JPZ
TsZ
KBR>@JBR6?P9]>RBR6/ 93
5$T5$8$/ EµAZ8$9BR6/ J`Z
T;1
K8$8DBR6?[8DC/sK
6J/ 5DBR34/YJ L A/4jkL7B@/76
8$5zAF
/ K9>8$AH /434AV(e6:A34J
/73(8$AH5D1/YL7BRFT:8DC/ 5$/L A/7jkL7B@/46
8$5]5$/ // K9b
8DB@ A6¹-]Z/7>@A<l7«Ì>@/Y82K5:<N3$B@8$/kEPA34/k/7Q1
>RB@L7B@8D>@TD |2
t ¾l$2tRÃ
FA3H8DC/zL A/7jyL7B@/76
8$5HALYL7K3$3$BR6?;BR6n¸7V
EHA-8DB@L /N8DC98 «8DCAK?Cy8DC/N9>@?/7Z
349
½s ¾s¿ÀB@5m6A6
bL AEcEsK8$98DB@a-/«
8DC/PEP98D3$BRQ0AFrL A/7jyL7B@/76
8$5:<uBR>R>(C9a-/P/7>@/ Ec/76
8$5HBR6n8DC/s^/7>@J ½zV
WNCK5#</EP9THZ
KBR>@JkJ
/ 8$/734E2BR696
8$5(96JP911
>@T2A-8DC/43(8$/ LC
6
B@K/ 5
S6A<N6PFU34AEoFe349L 8DB@A6 -Fe34/ /r9>@?-A3$B@8DCEP5N<NC
B@LCk/469Z>@/rK5#8$AlL A6
b
8D34A>(8DC/:5DB@ /2AFBR6
8$/434EP/ JB@98$/kK96
8DB@8DB@/ 5:96J`8$Ay1
34/ JB@L 8zL A-E2b
EPA6[F9-L 8$A345 V
9
*6y<uC98mF*A>R>@A<r5N<N/N<NBR>R>L A65$8D3$KL 8N/4>@/ EP/76
8$5FU34AE ½H ¾s¿$À
96JkEPA34/N1
34/ L7B@5$/7>@TkFe34AEf» ¼OR¾z¿DÀ
#<NCB@L$CO<uBR>R>/769Z
>@/HK5
8$APJ
/ 5$L73$BRZ/c8DC/z/76
8DBR34/:5$/ 8rAF{a/ LY8$A345HAF#9P?B@a/76¡A34J
/73V
9
hikj#lnmVlpoqlKrVm es WT
MP P7I#JnTS{SKL }9P 9T Sn} H
-N !w'Q MJyxML MP7S MJ V S MS eT ML
H a ML | a* S H'Tp
_ }MS "T )PI#JnTKSK{SL }
* * !np _* IO
Tn
9T
©!¡
§!u § ¤Px ¤$£M±w £H
¾H
©ªº£¤$¥§ ¤'`
¨H £¤z§!-u § ¤vx2
t ¾H ½s ¾s¿À\ £zH £¤¦¥§³¤ Tn
§ ¤$§z§}MSeª"T*ªl©
[¾H ½zR¾z¿DÀ 9 ªvIqO
)Tp9
©9T
t ¾lÃ<[ ¾l45W=
2
¾H7Ç
w
§ ¤$§HJ
/ ?¡_ ¾l ¢ J
/ ?A
2t ¾lZ£¤* H £¤z©MJJk¥
OTn
§³¤¦§2§}MeS ª"eT ªH©L £MLk¦ § ¤$£lÊ ½ªIO
Tn
©9T
=¾H(ÃMÊ5¾ § $ É©¨]¾l
w
§ ¤$§HJ
/ ?¡]¨ ¾H ¢¤ª E2E2BRBR66 ** Ç#M£ * -M£ Ç#* & M£ &¬«« fc® Í6Ç
f%V %V §
¨7§HªY£ P§
¥§
P© e¤ P£
ª %§ £ £ © Rª =¾H
½s ¾s¿À\
eªª © ¥ £2¨4§r© 2A34J/43
Z95DB@50£ k£¤$¥§ ¤
© ¥`¥§ ¤$§4§
§ ¤¦§;§ eª eªP©
¥§
Ãg Ç @Ç
ª
s
©
i
9T Tn
9T w
!L R5"2
H eMSKu :L Tn
§!L
!u vx Mw H $
' !LVO Q "TA H LSTS*£ML
#ndSKPQVJyS 'Tn
9TTn
}_S !T )
79 ªIO
Tn
9T
¾lZ5¾H#Ão¾ $ Ç J
/ ?A ¾l ¢ '£¤* Ç
Tn
©9T US ª ©TKxvQ §:£sH ª-
S H!T §4¥SKLu-§ ¤7ªY§WV
§k§v}MSeª"T§!LVO7§l£H2©[ª-
S H!T§4¥
J@§=H!TZSL-u §³¤4ªY§ ¾lRwSKJKJ §!L ©¨!J@§AIªeT£§!L§ ¤$©MJySn¦§ATn
§L£9TKS£MLk£H
I#LSKcP £¥MI@J ©¤¡eT ¤$©ML ªH £¤PP ©9TKS £ML ª(£]H £¤¦¥§ ¤r¨7©ªY§ ªT £Tn
§e7O ©ªY§H£HTn
§
L\£ML{4O £MPP7IqT ©9TKSK-u §r©MJ §¨³¤¦© ½HR¾z¿$À
fV ¸
V
z£ § ©
§
Ã
£ ¤r©
-§ § -§ ¤ c¤$£ £ r¾ ©ª:£¤$¥§ ¤ z§ 4§ ©¤ $¨ k£ z§
§ ª © § ¤$§O§ Uª eªc© ¾H
½s s
¾ ¿À
©
r6k/ 5$5$/76
8DB@9>B@E21
>RB@L 98DB@A60AFl/7^
6B@8DB@A60%V %eZ\mB@58DC/FA>R>@A<NBR6?t
Vi
r Vi %V VIWQ9Q£ªY§Tn
©9TTn
§ ¤$§`§}MSeª"T*ªO©ML±£¤$¥§ ¤`¨4©ªvSUª
=¾HP£HP£¤¦¥§³¤ ©ML¥O¥§¤$§4§)* 0V
§!L©Tn
§ ¤$§P§}MSeª"Teªz£MLJyxGTn
§
eT ¤SKuWS ©M
J ¤¦£Mwz-u §OWT £¤4t2¾H(ÃM~wSTp
GO7£MJyI#PL_¥§¤¦§7§ ¢ * £ 2 ©ML¥
£¤¦¥§ ¤Af V
I ª HY£¤N©MLx¡ ©H¤$£Mw0u-§OWT£¤wSTp
'O4£MJyIP7Lk¥§|¤¦§7§
¢ * £2 ɤ2 ©M\L ¥[£¤$¥§ ¤f eS ªI#LS !I §IWQ©T £P7IJnTKS QVJyS7O ©9TKS £ML
wSTpo
©M±L §:@J §!cP §!LVT7H ¤$£MP ½ L0Q ©¤"TKS=O!I@J ©¤ ©M±L £¤$¥§ ¤O¨7©ªeS ª'eS ª
I#LS!I§IQT£PI#JnTKS QVJyS=O4©9TKS£LO¨!x'O4£MLª!T©MLVTeªqH7¤$£MP ½
H
!(¤$££ #".0/HA6>@TP6/ / JO8$A25DCA<g8DC/]^
345$8N19348NL A6L /73$6
BR6?;34A<
t
U* £ 2 a-/ L 8$A345 2¾l2<NB@8DCiL A>RKE26J
/ ?34/ /
96JiA34J
/73
«N8DC/
-A 8DC/43`19348$5;Z/7BR6?o96oB@EcEP/ JB@98$/nL A65$/ K/76L /nAFc8DC/`^345$8 V
·K
11A-5$/k8DC9-8 2¾HºÃg
Ë «ÊOÃ¥§ 2¾H7«96J`>@/ 8 [¾HNZ/:95
BR6Ol/7^6
B@8DB@A6p%V %eZ\7V#.0/]^
345$8L4>@9B@Ef8DC98N8DC/734/H/4QB@5$8$5N98#>@/ 95$8
A6/zBR6J/7Q P<NB@8DC
G¥
t
|Vt
|V ïÊ
=
ÁH8DC/734<NB@5$/«r¥§|© [¾H 5
¾H W ¢ * ÉI¥§|V [¾H ® Ê
F*A3l9>R>``96J « 5DBR6L /_¥§V¾l ¢ * Z
T[H/4^6
B@8DB@A6iVRL7«
BR6yL A6
8D349-J%B@L 8DB@A6`<NB@8DCk8DC/rJ/7^
6
B@8DB@A6`AFÊV#/ 8Z¥rZ/r8DC/>@934?/ 5$8
BR6J
/7Q0a/73$BRF*TBR6?0=7V#WNC/7609?-9BR6`Z
TkH/7^
6
B@8DB@A6iVRL
¥§ ¾l =
[
¾H $ ¾H ¢ Ên£ Ç
¢ Ê£¤* M£ ÌÉg¥§|6=
$ 3
¾l ¥§ _ ¾l
¢ 3 ¥§|Z [¾H ɤ * £n ¢ Ê£nÇ ¥ § [ ¾H
¾H Ã=¥§V[ ¾H É * ÃMÊ
WNC
B@5sB@E:1>RB@/ 5k8DC98k¥§|V2
t ¾H ÃÊdf * «#<NC
B@LCpL A6
8D349-J%B@L 8$5
8DC/z95$5DKE218DB@A6¡AF(8DC/zJ/Y?34/ /:AF¡2
t ¾l7V
*
Ég¥§ [¾H
3. DETERMINANTAL REPRESENTATIONS
e6O<uAND
C98(FA>R>@MAHLER
A<r5r<N/H<NBR>R>134SYSTEMS
A1A5$/z96`9>@?A3$B@8DCEfFA3L AE:1K8DBR6?
`
34/YL7K345DB@a-/7>@TPA34J/73Z9-5$/ 5 ¾lF*A3mBR6L734/ 9-5DBR6?:A34J
/73{a/ L 8$A345 V(*6
A34J/43(8$Au1
34/ JB@L 8#8DC/5DB@ /AF8DC/ 5$/(AZ&%7/YL 8$5#96JH134/YJ%B@L 8(L A-EPEPA6
F*9L 8$A345 «m<N/:J
/73$B@a/PBR6¡8DCB@5z5$/ LY8DB@A6n9OJ/ 8$/434E2BR696
8$9>N34/4134/ 5$/46b
8$9-8DB@A6i8$A?-/ 8DC/73P<NB@8DCi9[19348DB@L7K
>@93OL$CAB@L /[AFr8DC/kL A65$8$96
8PÊV
4t
' t
~ « !!! ¥ !!! q £
ºÃY |2
t ¾H$(Ã (*) +& , -/$ .10 2 À 2 3 2 v
2
WNCB@5r>@/ 9J
5H8$Ac5$AEP/z5$T5$8$/ EfAF#>RBR6/Y93:/ K98DB@A65zAF(8DC/HFA34E
Ç!pn@ Ç Ç"pn Ç Ç"pn Ç 5 }~W' É2 Ç# (ï
Ç
¹-
-4$ 3 -4$ 5
<NC/434/z8DC/z?-/76/4349>RB@ / Jp·%T>@a-/ 5$8$/73zEc98D3$BRQ[B@5lAF{8DC/zFA34E
}p:' É2 Ç# #ÃI} :' ÉM-Ç $ 6 6 6 Ç
666
<NC/434/z8DC/ ' w 5DK
ZEP98D3$BRQ}d ' Ç N/ K9>@5
78 À6 ÀV
8
À 5!5!5!5!55 5!5!5!5!55 ÀVÀ & 3 : ;;;
8
VV
VV & 3
V V
VV
V
V
V
V
9
<
5!5!5 À - 3 5!5!5 À - 3 & 3 -
8DCK5:}~W' ɤ2 Ç# /=mrEP9T;Z/
>@/ 93$>@T-«#}a :' ɯÇq /=M96J <N3$B@8$8$/76`9-5r5$AEP/z5$8D3$BR1/YJ?>H34T>@AaOEP98D3$BRQ0 ²«
8DC9-8uB@5 «92Ec98D3$BRQ
·K1
1A5$/r8DC98#<N/(934/#>@AASBR6?zFA3{9]34A<~a/YL 8$A3 2¾HmAFL A>RKE26
J
/ ?34/ / ' C9aBR6?cA34J
/73 VÌuA-E2193$BR6?P<NB@8DC`¸7«</S6A<i8DC9-8
2¾HC95rA34J
/73
BRÏ`F*A3l9>R> Ãh-Ç ÇYxÇ ºÃ¯
Ç Ç
nt
G
AF(8DC/HFA34E
7
9
@&1BC A 5!5!5EDd&F-4$ 3C 3 @&1BC A 5!5!5 @1& C A "5 5!5GDd&F- $ 5C 3 @F& C A :< HA<N/Ya/73«Z
T[5$8$/71
1
BR6?`Fe34AEµA6/PL A>RKE:6n8$A[8DC/26/7Q
8z<N/z6A-8
A6
>@TMEzK
>@8DBR1>@Ti<uB@8DC±9>@A<N/43[5DC
BRF8kEP9-8D3$BRQqZ
K8P9>@5$An91
1
>@TpBR6
9J
JB@8DB@A6°8DC/O91
1>RB@L 9-8DB@A6°À#VnWuCK5 «BR6±L A6
8D349-5$8k8$Aq ²«#C/734/
<N/zAZ8$9BR6n9k5$8D3$BR1/ H
J >H34T>@Aa`Ec98D3$BRQ0<NB@8DC09kEP98D3$BRJ
Q IgC9aBR6?
A1\/73498$A3$bDa9>RK/ J0/7>@/ Ec/76
8$5 V
HA<L 960</z/7Q1>@AB@8z8DCB@5r34/71
34/ 5$/76
8$98DB@A6~BR60A34J
/73º8$APJ/43$B@a/P9
J
/ 8$/734E2BR696
8$9>N34/71
34/ 5$/76
8$98DB@A6iAFrA34J/73zZ95$/Y5LKgHL LYA34JBR6?`8$A
¹7«-B@8FA>R>@A<r5Fe34A-E=WuC/ A34/ E%V 2
8DC98#BRF8DC/734/l/7QB@5$8$5N96[A34J/43
Z95DB@5
¾HHAFrA34J
/73 96J~J
/ ?34/Y/D* 8DC/76p}p* Çq NC9-5sFeK
>R>
34A<¯3496S«96J`EPA34/H1
34/ L7B@5$/4>@T
ÃgÇWnp@ ÇDvµt 3496
S°}p* Ç# ÃM3496
S°}~* É2 Ç# (ÃNM|* My
k
G 'P7IJnTKSn M* S :L T * # k
* # W
J vx !T !P * k
7
~ * ¾l#ïÊ5¾ § $ É >@A<N/43HA34J/43HLYA>RKE26~J
/ ?34/ / 5W
·K1
1A5$/cEPA34/z?-/76/4349>R>@T`8DC98
96J
934/zEzK>@8DBRbBR6J%B@L /Y52a-/73$b
BRFTBR6?`7V(. /zLY9>R>#9
¤$©¥ *§ %Ê:ÃoÊ Ç 96
T;LYA65$8$96
8
O N8DB@Ec/ 5#8DC/J/ 8$/434E2BR696
8AF9]34/Y?K
>@935DKZEP98D3$BRQy} Ç ÍAF
EP9QB@EP9>A34J
/73AF{}~ Ç 7«96JPQ
9 P_© @§ ¤Nª -ª § iL A3$34/ 5D1\A6J%b
BR6?`8$A~ Ç H9`EP98D3$BRQ01A>@T6AE:B@9> =¾H]<NB@8DC~34A<r5HC9aBR6?
A34J
/73
96JOJ/Y?34/ /:5$8D3$KLY8DK34/
*6¡A34J
/73º8$A[5DCA<°8DC9-8z5DKL$C~9O5$T5$8$/ E/7QB@5$8$5 «<N/:<N3$B@8$/PJA<N6
/7Q1
>RB@L7B@8D>@T08DC/H>RBR6/ 93º5$T5$8$/ EfAF{/ K98DB@A65]6/ / J
/ J`8$A:L AE:1K8$/
8DC/;K
6S6A<u6iL A/4jkL7B@/76
8$5kAFl8DC/ 8DCn34A<AF ¾l7tPJ
/76A-8$/
Z
S
T R Ç P8DC/[34A<f9J
J/ Jo<NC
BR>@/195$5DBR6?qFU34AE}~ Ç 28$A
}~ É
Ç 7V_WuC/76«mZ
TM¹-7«#8DC/Oa/ LY8$A3cAFlL A/7jkL4B@/76
8$52B@5P9
5$A>RK8DB@A6¡AF(8DC/PAa/734J
/ 8$/734E:BR6/ JH5$T5$8$/ E
#
* * 2 #
5 * # 5 * # 5 * # 5 * # @
* q T ³}p Ç ÃoÊ #R Ç
<NCB@L$C`Z
T`B@5H/YKB@a9>@/46
8z8$AP8DC/z5$T5$8$/YE
T -} Ç #Ã¯Ê UR Ç 7Ç
¶-
* # #
* q eE * 2 F
<NC/734/;BR6 R Ç :96JpBR6¯} É
Ç 2<N/PS/Y/71o8DC/[5$9-EP/
LYA>RKE265:95rBR6n} Ç 7V HA8DB@LY/28DC98 «Z
T0349Ec/73 5N3$K
>@/«(¶
>@/Y9J
5N8$AH9z5$A>RK8DB@A6[<NB@8DCkLYA/7jkL7B@/76
8$5(BR6`» ¼[V `A34/ Aa/73«<N/NEP9T
F*A34EP9>R>@T~<N3$B@8$/;J
A<N6n9`J/Y8$/734E2BR696
8$9>r34/71
34/ 5$/76
8$9-8DB@A6±AFr8DC/
/4>@/ EP/76
8$5lAF(96_J/Y8$/734E2BR696
8$9>A34J
/73rZ95DB@5 «69EP/7>@T
¾H W * É2 Çq § $ C 3 $ C . ¾H
²
<NB@8DC
- ¾lÃg Ç"!!mǦMRÇ4¾]ÇW!Ç4¾ - M
Ç!!!Ç$ = Ç ¬
8DC/26A6 /734AO/76
8D3$B@/ 5HBR6
¾lNAL L7K
3$3$BR6?[BR6¡8DC/ « b8DC05$8D3$BR1/V
e6`9-JJB@8DB@A6«m
</HC9 a/z8DC98 ^
¾l º¾l à O ¥§WT } * É2 Ç# § $ $ /$ . ¾H Ç
Ã
T
O ¥§ }
$Y
<NC/734/
- $ ¾H
Ã
y
Ç!!!Ç4¾ -4$ 3 3 º¾l Mv""!! M
H¾H Ç!!"Í
Ǿ - $ 5 3 H¾H = H¾H
e6kZA8DC¡²96J0$ (8DC/lEP98D3$B@L /Y5uC9a/zL A-EPEzK8$9-8DB@a/z/76
8D3$B@/ 5
BR6¡9>R>#Z
K8H8DC/s>@9-5$8zL A>RKE26V:8NB@5]K
6J/7345$8$AAJ¡8DC9-8H8DC/2J
/ 8$/73$b
E:BR696
8]BR6`ZA-8DC0L 95$/ 5NB@5l/7Q196J
/ J09>@A6?k8DC/H>@95$8HLYA>RKE26mV
.0/z^69>R>@T~Ec/76
8DB@A6¡8DC9-8 «Z
T`8DC/zK
6B@K/76/ 5$5s34/Y5DK>@8zAFNWNC/ Ab
34/YE%V «96
T`A34J
/73zZ9-5DB@52AFNJ
/ ?34/ /
96J A34J/43
L ABR6L7B@J
/ 5
K
1`8$AcEsK
>@8DBR1>RB@LY98DB@A6n<NB@8DC05$AEc/z/7>@/ EP/76
8NBR6 ½o<NB@8DC`9 `9C>@/73
5$T5$8$/YE9-5$5$AL7B@98$/YJ;8$A: Ç 7«-<NCB@L$Ck8DC/734/7FA34/NB@8$5$/7>RFB@596kA34J
/73
Z95DB@5rAF#8DC/]5$9-EP/HJ
/ ?34/ /H96JOA34J
/73V(WuC/rL A6
a-/7345$/z5$8$98$/YEP/76
8
B@5`?-/76/7349>R>@Tg6A8`8D3$K/V lA<N/ a/43«]Z
To9~19348DB@L7K
>@93~1
B@aA-8DBR6?
8$/YLC
6B@K/:<N/HEP9TP34/ L Aa-/73lA34J
/73rZ9-5DB@5]Z
TkL AE:1K8DBR6? `9C>@/73
5$T5$8$/YEP5 V
c* *
F
'F
4.
e6i8DCTHE
B@5P5$/YL 8DB@AALGORITHM
6±<N/k5DCA<CA<8$A34/YL7K345DB@a-/7>@TnL AE:1K8$/`A34J
/73
)
Z95$/Y5yBR6±9 Fe349-L 8DB@A6
bFe34/ /`<N9TVn®A3O96iA34J
/73kZ9-5DB@5 ¾lsAF
9O?B@a-/76n8DT1\/[ Ç ]C9aBR6?`9 `9C>@/73c5$T5$8$/ EX6A34Ec9>RB@ 9-8DB@A6#«
</y>@AASn98:8DC/;^
345$828$/734Ec52AFH8DC/;34/Y5DB@J%K9>@5 Vp*Fl8DC/YT~934/k9>R>
/YK9>#8$AP /434A28DC/76`</]C9a-/296OA34J
/73rZ9-5DB@5HAF(9zC
B@?C/73zA34J/73V
ÁH8DC/734<NB@5$/«
<N/r?B@a-/z9H34/ L7K
345DB@a/HFA34EsK
>@9zFA3(ZK
BR>@JBR6?k96OA34J
/73
Z95DB@5`AF2C
B@?C/730A34J/73096JoJ
/ ?34/ /V lA</ a/73«l9~13$B@A3$B:8DCB@5
6/ <5$T5$8$/ EC95rL A/7jkL4B@/76
8$5]Fe34A-E ½¯5DBR6L /:<N/HJB@aB@J
/P8DC34AK?C
5$A-EP/zF9L 8$A345 Vl*60AK3ºL 95$/«CA<N/ a-/73«8DC/z6/ <I5$T5$8$/ Eµ<NBR>R>#Z/
9 `9C
>@/73:5$T5$8$/ Ef9LYL A34JBR6?O8$AP8DC/z/7QB@5$8$/76L /P96J[K
6B@K/76/ 5$5
34/Y5DK>@8$5O/ 5$8$9Z
>RB@5DC/YJMZ/7FA34/«r96JqC/76L /`<N/k<NBR>R>HS/Y/71oAZ %4/ L 8$5
<NB@8DC0L A/4jkL7B@/76
8$5rBR6n» ¼;V
* # F
F
e6k8DC/uF*A>R>@A<NBR6?:8DC/YA34/ E<N/N?B@a-/H9r34/ L7K
3$34/76LY/]34/7>@9-8DB@A6O<NCB@L$C
L4>@A5$/7>@T`FA>R>@A<r5H8DC/:L AEPEzK8$98DB@a/cL 95$/:AFr ²«WuC/ A34/ EX
V -L
<NB@8DCn8DC/234/ 5DK
>@8DBR6?~A34J
/73:Z9-5$/ 5sC9aBR6?`1
34A1/7348DB@/Y5P5DB@E2BR>@93y8$A
²«
WNC/ A34/ E ¶%V ³96J~ ²«
WNC/YA34/ Eµ¶V ¸DV
Vi
r Vi gV@Ves§WT¾Hk¨4§`©MLM£¤¦¥§ ¤[¨4©ªSeªO4£¤4¤$§ ª/Q£M L¥M{
S L# d T£ * Ç# X t Ã É 2 I
¤!Tn
§ ¤vPc£¤$§ ¥§!L£9T§k¨:xG> Ã
& $ |
¾H$ Tn
§¤4ª!TeT§³¤P £H'Tp
§k¤$§ ªS¥MI©MJ#HY£¤'Tn
§¥#{Tn
~¤$£Mw
©M\
L ¥V{Tn
)7O £M JyI#PL_£H¾H!
nH> Ãnpà > Ãf©Tn
§!Ld
[ ¾H2t à ¾lSeªy©MLn£¤$¥§ ¤
¨7©ªeS ªl£Hz¥§|¤¦§7§' tRà * ©ML\¥;£¤¦¥§ ¤ X
Tn
§ ¤wSeª § J@§WTo¨7§7Tn
§sªPc©MJKJ@§ ª!TZSKL\ ¥§v}wSTp
>_Ã¯Ë 0©ML¥
* PÃME2 BR6 * t> ÃoË & V
§!nL ©M¡L £¤$¥§ ¤:¨7©ªeS ª
[ ¾H:£2H ¥§| ¤$§4§' t Ã* É2 n©ML¥
£¤$¥§ ¤ X wS=Tn
07O £§DO!S §!LVeT ªSKL ½ eS ª:£¨WT ©MSK
L §4¥u*S ©Tn
§AH £¤{
PI#@J ©ª
5W
[ ¾H ÃY> 5:
¾H £0> 5
¾l $³
H £¤ « ÇP ÃIÇ%Ç!np@ Ç$v « ÃË ©ML ¥
À# 5
[ ¾l ÃD> 4U5 ¾5K=
¾H £ À# 5 h
[ ¾l $-
H £¤'$#° %$ à - Ç4%Ç!nn Çv w
§ ¤¦§ ÃÎLYA/7jkL7B@/76
8|
¾l Ç
! 3 !
¾%"§ $ !
L ©¥¥MS=TKS £MLG
¾HUS ªP© P_©W
@J §³¤:ªx ª"T §!P wS=Tn
¤¦§³ª=Q §vOWTAT £
&' nHSK¡
* Ç# Tn
§!L
[ ¾l eS ªu©MRJ ª £z©
P`©W
@J § ¤(ªv-x ª!T §!P wSTpP
¤$§ ª=Q §vOWT
T £;W' Çv X !
LQ ©¤"TSO:I#@J ©¤ [
¾lR
©ª@7O £§DO!S §!LVeT ª`SK`L » ¼)
!(¤$££#H "
(#9348`92B@5kL7>@/ 93[Fe34A-E8DC/[F9-L 8k8DC9-8k8DC/[34A<r5PAF
¾lÌC9a-/2A34J/73X <NC/46a> Ãnp- Ã<> ÃoV
®A319348reZm6A8DB@L /r^345$8(8DC9-8 =
[ ¾HÍC95NA34J/73 X Z
T2L A65$8D3$KL7b
8DB@A6«l95P34/ K
BR34/ JMBR6H/7^
6
B@8DB@A6h%V %97Vr>@5$A«ra/43$BRFTBR6?i8DC/
6/ <MJ
/ ?34/Y/HL A65$8D349BR6
8$5]AFH/7^
6B@8DB@A6¡%V %LN<NB@8DC * Z/7BR6?z34/7b
1
>@9L / J`Z
T© ÌFA3l8DC/:EP98D3$BRQ
[ ¾H#B@5H5$8D349B@?C
8DFA34<934J096J[B@5
'
8DC/25$9-EP/29-5NBR6~8DC/2LYAEPEzK8$9-8DB@a/PLY95$/«5$/ /; ²«WNC/ A34/ E ¶%V ³V
]>@5$A
«6A8DB@L /r8DC98#8DC/>@/ 9JBR6?PL A/7jkL4B@/76
8AF9>R> ¾l
/ K9>@5
Z
T;LYA65$8D3$KL 8DB@A6n8DCAK?C`FA3u8DC/zEPA-EP/76
8r<N/l934/H6A8r5DK34/
8$APAZ8$9BR60926/Y<gA34J/73lZ9-5DB@5H<NB@8DC0L A/7jyL7B@/76
8$5]BR6n» ¼c7V
V[
>
.0/ 6A<FAL7K5`A6°8DC/ 134A1/7348DB@/ 5¡AFkl/7^6
B@8DB@A6 VReZ7V *F
2¾H
½H ¾s¿$À
C95kA34J
/73
8DC/46MB@8PC9-5kA34J
/73
96J
5$AP8DC/734/z/7QB@5$8$5l96 [¾H
½H ¾s¿À
5DKL$C`8DC98
t
9 )X U _ t2¾H#Ã [¾H 5W¾l )
<NB@8DCn¥§|d [¾l ¢ ¥§at2¾HZ£* «<NC/734/7=¾H ) J
/76A-8$/ 5
8DC/¥b8DC[34A<IAF¡¾H7Vrr11
>@TBR6?`8DC/z^345$8l5$/ 8HAF#34A<°A1/7349b
8DB@A65]BR6n$-N8$A234A<r5 « Ã
Ë 34/Y5D K>@8$5rBR6
t2¾l#Ã + [* ¾H 5
[ ¾l ) É- , ¾l 5=
¾H ) $ ¸
<NC/434/
, ¾l à [ ¾H 5 > FA3H9>R>Z;¥ ÃË 96J
[
, ¾H Ã [ ¾H 1 5 > > 1 `
1 EHA-8$/28DC98k¥§| [ , ¾H ¢ ¥§| © 2t ¾H4£ * ÃX¥§|02t ¾H £' FA3N9>R>6¥yÃ
Ë [<NC
BR>@/k¥§.
, ¾H ¢ ¥§|)t2¾H£c* sZ/ LY9K5$/HAF
8DC/]E:BR6B@EP9>RBe8T[AF* V(·BR6L /@2
t ¾H#96Jk9>R>8DC/
[ ¾l )8$/734Ec5
C9a-/NA34J
/73 X 8DCB@5#EzK5$8#9>@5$A]Z/(8DC/NL 95$/#FA3/
, ¾l 15 0h¾l ) V
/46L /23 , 5> ÃMz96Jk5$AHZ
TP95$5DKE218DB@A6`A64`<N/C9a/l8DC9-8
H
ÃMV(.3$B@8DBR6? [ , ¾l à [ ¾H 5³¾g?B@a-/ 5
t2¾l#Ã [, ¾H 5
[ ¾l ) É ¾l 5-¾©5W¾l )
$"
<NB@8DC°¥§| _
¾l ¢ ¥§<2t ¾l £o ÉIPÃ¥§|<t2¾HA£
' \VduA-E21
>@/ 8DBR6?o8DC/`34A< A1/7349-8DB@A6* 5~<NCB@L$C6A34EP9>RB@ /n8DC/
J
/ ?34/ / 5zAF¡=
[ ¾HBR6M$³N?B@a-/ 5z9z[ X ¾HN<NB@8DC2t ¾HNÃ [ X ¾l45
[=¾HÌC9aBR6?O8DC/lL A3$34/ LY8]J
/ ?34/ /HZAK
6J5 VlNA65$/ K/76
8D>@T«8DC/
8 CBEDF7G&H$IKJL1LNM>J>OQPKR>JSUTWV7XQMAY[ZEI]\
^1_0`babcedgfihijlk.kim[npoiqirsn
t2uivewif7xzyU{>|'m'}W~0.'>ns
Q>WAi}W{> /? Eb #0
1wifivewif7KyU{>iWnpmK'A'|'ns, 1EQ 79
$ns'}Woiqi{>b 2Q 7
*np¡>m'nsn£¤ ¢ >m[oinpm¥¦ ¢ m'{>i¨§
t2uedgfedg©i_pdgª0©>fedgabu7x«¬® ¬¯/ e° ¬ )>±
¤ ¢ ¬ ( ¢ ± ¦ ¢ ¬ ( ¢ §¬«(
²5i}WWn¨EAqi¨³bnpm.0.´snpm[¨m[0C0K # §?:µ ¶·¸
§¹¬«(
/»ºi¼½ )>± @W@ ±[¾1¿ º
À{>WrsqiW{>|'nCE>mÁ$ )>± @W@ ± ¶?
Âbm['||[npm[0.m[np[}Woiqi{>W»Ã ¬«Ä *n¸Âbin1[np|CÅÆÇpÁ1·Ç )>± @W@ ± ¶· ÈÉ0Ã
µ (AÈ
ËÊ$ÅÌ
µ Ç>ÈÉÍsÎ.ÏeÐ
Àie>'n$Ñ,Å[qir|'i{>|p
Ñ?}WbÒ0Á1£ÅK¤ ¢ }W -ÓAÔ Ç.¤ ¢ - ÈÖÕ
À{>WrpqiW{>|'nC×mÁN )>± @W@ ± ¶ ÁÉ:µ Ñ Ø ¬ÚÙ'ÛsÜ"Ý5Ù[ÞËÜ[ßQà' ± § $ C $ $ C 3
á[irpm[np{>'n$>m'oiâ nsm]×>mÁ$ )>± @W@ ± ¶ ÁÉ :µ ÑK
) ¬ Ã -ã ) = Ã -ã )
) ¬ â Ã -ã ) = Ã -ã )
á[irpm[np{>'n$>m'oinsm{>io{>osä¸qi[|oins¡>m'nsnrs>i'|'m[{>}We|'7×>m
m[0: ÑK â
) ¬ å @ A Ã -ã:-¸ ) =?æ Ø ]-s )
) ¬ å @ â A Ã -ã&-s )Q=?æ K Ø ]-s )
ç
boi{>|'n1qiW|'}W¡>m'{>oi} npA| oinp¡>m[npn${>io§
° à ¤ ¢ ¬¤ ¢ # è ¢ §¬§ # )
ÏAÐ ¿é Ê
ê$osäqi'|C m'ns'}Woiqi{>ib}Wë rp>Wqi½77E>mÁN )>± @W@W@ ± ¶
¬¯ ìgE>m'{>WWÖ
¢ «¢ # ¢
ÏeÐ ¦ ¿ Ê2ºi¦ ¼ è"
3,
1
34A1\/7348DTOAF(H/4^6
B@8DB@A6iVReZCA>@J
5 V
®mBR69>R>@T-«ÍFA3r/ 5$8$9Z>RB@5DC
BR6?;19348zL(<N/]S6A<g9>R34/ 9J
T;Fe34A-Ef·/ L7b
8DB@A60¸P96J[8DC/z/7QB@5$8$/76L /PAF#A34J/73lZ9-5$/ 5rAF(9P5D1\/ L7BR^/ J J
/ ?34/ /
96JiA34J
/73k8DC982Z\A8DCI Ç :96Jg ' Ç 25$98DB@5DF*T¯7VnNT~8DC/
K
6
B@K/76/ 5$5r34/ 5DK>@8rAFmWNC/ A34/YEf%V z<N/rA6
>@TP6/Y/ Jk8$Az5DCA<o8DC98
8DC/ >@/Y9JBR6?`L A/4jkL7B@/76
8 ÊkAF ¾lÌBR60H/4^6
B@8DB@A6iVRLB@5r9
EzK
>@8DB@?349JB@/768PAFl ' Ç 7«8DC/:>@98$8$/73zB@E21
>@TBR6? 8DC98 I¾HNB@5z9
`9C>@/43z5$T5$8$/ Eµ96J;BR6`19348DB@L7K
>@93zC95lL A/7jkL4B@/76
8$5]Fe34A-E» ¼[V
F
W X * # W vX ©X A[
[
l/76A8$/IZ
TfÊf8DC/=L A3$34/ 5D1A6JBR6? >@/Y9JBR6?LYA/7jkL7B@/76
8 fAF
=¾H7V *68DC/~L 95$/0JB@5$L7K5$5$/YJgBR6g19348~97«H</ J
A~6A-8[BR6b
L434/ 95$/P8DC/23496S`Z
T`?-ABR6?[Fe34AE }p Ç r8$AO}p ' Ç s</ %K5$8
9-JJ`A6/zLYA>RKE26096J[S/ /41[FeK
>R>m34A<o3496
S7«C/76LY/zÊPÃ ÊPZ/7BR6?
9zEzK
>@8DB@?349JB@/76
8r<NB@8DCP34/ 5D1/YL 88$AP Ç ÍB@5(9>@5$A:9HEzK
>@8DB@?349JB@/768
<NB@8DCk34/ 5D1/YL 88$AP ' Ç 7V(*6y8DC/N^
69>L 95$/rJ
/ 5$L73$BRZ/YJ;BR6k19348]eZ\
</sC9a/ Êkà V l5DBR6?kFA34EzK>@90$ uFA3z8DC/234/ 5DB@JK9>NAFN8DC/
8DC034A<gAF ¾l]<N/H>@/Y93$6n8DC98 L ABR6L7B@J
/ 5[eK
108$Ak9;5DB@?6
<NB@8DCO8DC/HJ
/ 8$/734E2BR696
8rAF9:5DKZEP9-8D3$BRQ[AFA34J
/73 M ' MAF{}p ' Ç 7V
·BR6L / ÃM
Ë zZ
T2LYA65$8D3$KL 8DB@A6m«B@8#FA>R>@A<r5r8DC98 ÊPÃ B@5N9H6/ <
EzK
>@8DB@?349JB@/768 «m9-5N34/ KBR34/YJFA3r8DC/zL A6L7>RK5DB@A6V
>
dX >
: vX * # * # e>
: vX X
= : X X D>
WNC/ A34/YE V k?B@a/ 5k9`L A-E21
K8$9-8DB@A69>r134ALY/ J%K
34/`8DC98z34/ 5DK>@8$5
BR6p8DC/P®#®
34/ J%KL /k9>@?-A3$B@8DCEÎ?B@a/76~BR6pW#9Z
>@/[VcWuC/P5$8$A11
BR6?
L43$B@8$/73$B@A6_96JP8DC/L AE21
>@/7QB@8T;AF8DCB@5(9>@?-A3$B@8DCEB@5?B@a/76kBR6k8DC/
6/7Q
8l5$/ L 8DB@A6V
_i
Æ
H¾H#Ã
VRV £¤7Tn
§z¥£MPc©MSKL`» ¼=Ão¾z Æ J@§WT
¸ z
£
Én
£ É ¾nÉ ¸Æ ¾
z
T'Tn
§Q¤4ª!T7S=T §³¤¦©9TKS £ML (w §d
©M-u §< à * à Ç$ ©ML ¥©Tn
§
O4 £M
L£ª"£Tª©MSKLVL#T O7_£§ÃgDOO!S©M§!LVL Te¥ªQ SK§Lc¤H Tn£
¤§PSKLk¤4ª"TTp
O7£M§zJyI#¤$§4P¥MLiIqO©TKS ¤¦£M§;L Æ (w
§PÉ£¨W%T ÇW©M£zSKL 7 =
.-0/12$3
" !
$#&% '
(
!
*#&%,+
)
(4(
(4( +
# ! #5% !76 #&< 8 % !
#:9 ! #,; = ( ) + # < % ! !
1
$#&#,% ;>9 < % = ! ! 6
#&=?;>% 9 ! +
A@
z£9T §7Tp
©9T`Tn
§O4£MLª"T©MLVTZO4£§ cO!S§!LVTeªASKLaTn
§sªY§O7£ML¥O7£MJyI#PL0£H
[¾lP©¤$§7¦§ ¤¦£ ª £L\£`£vQ§ ¤¦©9TS*£MLªl©¤$§P¤$§L!ISe¤¦§7¥GT£SKLVO ¤$§4©ª §G T £P$ÇY!
º£Mw Ãg$Ç ³ * Ãh$-Ç$ ©M
L ¥]Ê:ÃoÆ
É[#V
§4O £§DO!S §!LVeT ª
£º
H ¾ SKL©Tn
§¤4ª!T¡7O £MJyI#PLaeS ª: ÆrÉoÆ 6 Éo¹Æ
rÉD- Æ[ÉM
Ç$¸ =A
V
££ªSKL# nÃ= ©ML¥Q§ ¤H £¤PSKLkTn
§2¤$£Mw¤$§4¥MIOWTKS£MLª w(§PªY§7§
Tn
©9TV¾HSeª
£ Æ£~¸ÍÉn¸-¾ £l¹Æ
£0-Æ£0Æ £¡Æ 6 £0
r£ Æ
¡
Æ
ÉM-Éq¸-¾
w
SKJ@§=¾Hv5 º¾l SeªMSKu-§!LSKL
SnMI
¤$§$í w
§ ¤¦§ w(§`
©Mu§¥MSKuWS¥§7¥
¤$£M
w ík¨:ºx Ê~©ML ¥2¤¦£Mw<O¨:xºÀÌÊ
(ÃMÆ
ÉnÆkÉq¸q
®mBR69>R>@T«l<NB@8DCq34/Y5D1/ L 8P8$A08DC/y3496Sn34/ a-/ 9>RBR6?~8D34965DFA34EP98DB@A6
EP/76
8DB@A6/YJBR6¡·/ L 8DB@A6n«</]C9a-/28DC/HFA>R>@A<NBR6?V
î r r»e&bï V ¸Ves§WTV¾Hl¨7§eTp
§*`L©MJ
£¤$¥§ ¤l¨7©ªSeª£HH£¤{
c à `2 ©ML¥n¥§¤$§4§<* ©ML¥0J@§WT¾H~¨4§0Tn
§ª-
S H!T§4¥
J =H!TSKLu-§ ¤7ªY§z£H=¾Hz©ª:§}"QVJ@©MSL§7¥SKL&ð§!PP©¤-N%#"ñ#7V
§!LaTp
§
¥ § ¤
@§
É
¸
¾~É £
ÆÍ
ÉMÆPÉn¸
¾ 6 WÆOÉM Æ
ÉqÆkÉq¸
z
£rÆ £¡Æ 6 £n¹Æ
£aÆkÉo
² £r¸
Æ
Én
>DF7G¾HZ5WH¾l
!I MLVTKS=TKS§ ª
©
3 $ 5[¾HZ5¾ $
8y¾ 3 Ã ¾ 3 $ 5
¾H
B¾H¢Ã ¾ 3 $ 5 ¾HZ5¾ $
ªY£MJyu-§Tn
§ HvI#JKJ¤$©MLkNP¥§vO4£MPRQ£ªS=TKS £ML
Q ¤¦£¨:@J §!P ¸ í!
;
¾H¢Ã
D
D
¾
CBED!"
STOPPING CRITERIA AND COMPLEX*6±8DCITY
B@5k5$/ LY8DB@A6«r<N/[5DCA<X8DC98k8DC/[5$8$A1
1
BR6?iL73$B@8$/73$B@9nAFºAK
3
9>@?-A3$B@8DCE/765DK
34/ 5l8DC9-8H8DC/s34/ 5DK
>@8HB@5HLYA3$34/ L 8l96J JB@5$L7K5$5z8DC/
<NA345$8PL 95$/`LYAE21
>@/7QB@8DTiAFº8DC/`9>@?A3$B@8DCE[V®A3kL A6
a-/76
B@/76LY/«
<N/zL 9>R>(8DC/:L A-E21K8$98DB@A6n8$APBR6L434/ 95$?
/ M M%Z
T0:95H9k5$8$/71«96J
8DC/[L A-E21K8$98DB@A6I8$A`BR6L734/ 95$/
Ã
8$A 43 ÃG OÉh³ 96
B@8$/7349-8DB@A6«#5$A[8DC98z8DC/734/:934/kx25$8$/415HBR6¡/ 9L$C0B@8$/73498DB@A6mVcH5]BR6
8DC/H9>@?-A3$B@8DCEdB@8$5$/4>RF(<N/rJ%34A1`8DC/]6/ / Jk8$A25D1\/ L7BRFTk8DC/Ha93$B@9Z>@/
¾sVºm/ 8
Z/c8DC/ `9C>@/43P5$T5$8$/ E AFJ/ ?34/ /
96J
Z/
8DC/u34/Y5DB@J%K9>9F*8$/73(8DC/ b8DC2B@8$/7349-8DB@A6_<uB@8DC
mJ
/76A8DBR6?P8DC/
8D349BR>RBR6?`L A/7jyL7B@/76
8HAF
V
|
U `2
7 ;[
w
SOv
DSeªaTn
§`ª ©MPc§~©ªTp
§¡§v}©MPQVJ@§H7¤$£MP,+=.- §}9O7§QTTn
©9TS=T
SUª)PI#JnTKS QVJyS §4¥© £MLUTn
§ ¤SnW
Ty~ ¨!0x ¾ 0
/mªSKL#n £M
I ¤`©MJ £¤vSTp
P
w(§aT § ¤vP7SKL ©9T §k©9T' ÃGÇ$SKLDw
S=O
©Tn
§[¤$§ ªv*S ¥MI ©MRJ ªc©¤$§L £9T
©MJJV¦§³¤¦£SKL0Tn
§J@©ª"T`Tw(£kª!T§Qª"V
§Te¤$©MSKJySL#%O4£§DO!S§!LVT#£HTn
§
¤¦§ ªvS ¥M
I ©MJ[ ¾Hr£¨WT ©MSL §7¥l£ML §RST § ¤¦©9T*S £MLRQ ¤¦§:uWS £M
I ªJyxTn
©9TeS ª TK(w £
ª"T §
Q ªr©W £!z ©9T Ãg¹Ç4¹
©ªr©k¥§WT § ¤vP7SKL ©MLVTqTp
©9TZUS ªH©ML'SKL T §| § ¤
4O £M
L ª"T ©MLVTZTKSKcP § ªr ^ ÆkÉM~£ ²-
1k¤S=TKSKL#Î
; ¾HÃg¾ 3 @ ! A 5W[ ¾HZ5 ¾ @ ! A Tn
§:¥§WT § ¤PSL ©MLVmT £H
Tn
§Te¤$©MSKJySL#O7£§DO!S§!LVT
£HZ;ξl Seª¡Tn
§Nª ©MPc§H©ª¡Tn
©9T SL2+=.- IWQ
T £k©4O £M
L ª"T ©ML T/ 1[§z¤$§!cP ©¤NTn
©9TTn
§ Q ¤$£¥MIqOWmT £zH ©MJKJVTn
§H ©9OWT £¤4ª
¤¦§!P
P £M-u §4¥;¥M
I ¤SKL#Tn
§7O £MPQV@J §WT § Q ¤$£W7O § ª4ªeS ª
¸¹³¹¬³¶%Y²²-¸¸M
9- --zÍ
^ ÆPÉoZ~£ ²-
.0/ 34/ Ec93$S8DC98[8DC/~9>@?A3$B@8DCE AFOR :9>@5$Ai9a-AB@J
5 8DC/ K5$/
AFHFe349-L 8DB@A65;Z
Tn8$9SBR6?~9-Ja96
8$9?/0AFlFe349LY8DB@A6
bFU34/ /0H9K5$5DB@96
/4>RB@E2BR69-8DB@A6iAFrN934/7B@5$5k ¸#FA3:8DC/2S/43$6/7>L A-E21K8$98DB@A65:K5$/ J
BR608DC/7BR3º9>@?A3$B@8DCEOVHWNC/z5DB@ /2AF(8DC/zLYA/7jkL7B@/76
8$5HBR608DC/HS/43$6/7>
a-/ L 8$A345PL 96nZ/k34/ JKL /YJpZ
T034/ EPAaBR6?¡8DC/O?34/ 9-8$/ 5$8PL A-EPEPA6
F*9L 8$A3(9EPA6?l8DC/NLYAE21A6/46
8$59-5#JA6/NBR6c8DC/B@E:1>@/ Ec/76
8$98DB@A6
AF:R BR6 `91>@/O²V HA<N/ a-/73«#/7Q
8D3496/YAK52F9LY8$A345zBR6
8D34AJ%KL / J
BR6`1
34/ aB@AK5HB@8$/7349-8DB@A65z934/H6A8N34/ EPAa-/ J[Z
Tk5DKL$C0921
34AL / 5$5 V
F
` =
M2 c % * .0/s^
345$8]1
34Aa/P9P>@/ EPEc9[<NCB@L$C`34/7>@9-8$/ 5z8DC/z6KEsZ/43:AF(1B@a-A8$5
K5$/ J`J%K
3$BR6?;8DC/k zɯ³b5$8B@8$/73498DB@A6~96J`8DC/H3496
S`AF
7V
=
` 5i=¹V -Vs§WTfi ©ML¥R>O¨4§Tn
§u-©MJyI§r£H6O©9TZTn
§z§!L¥
£HS=T§ ¤$©9TKS£ML7:ɯ_V
§!L'>ºÃh¤¦©ML#N0 !
!(¤$££H#"Xl/76A8$/2Z
T87 » ¼ # « OÃÇWnp@ Ǧx«8DC/:L A/7jyL7B@/76
8
AF{¾ AF 5|qJK3$BR6?:8DC/bD8DCcB@8$/73498DB@A6`98(8DC/uZ\/ ?BR6
6BR6?k8DC/
5DBR6?>@/k5$8$/71:
9 ɤ2 8DCK5;7 B@5z8D34965DFA34EP/YJBR6
8$A87 $ JK3$BR6?P8DC
B@5N5$8$/71\7V®mBR345$8 «
<N/rL7>@9B@Ef8DC9-8<NC/76k34A<;B@5LCA5$/76
_i
V
Vs§WT$#Ão¾z Æ#Ç4 ^ l©ML¥O7£MLªS¥§ ¤
( = ) #
(
=
% (
/12$
( = ) % < 80; +
=
&'%(%>9'% ( =
% ( %& + # = ) (= <*< )'% )e) % ) (8 ( +
# = ) &>( ;'&e%%4= < ( >( ( + 6
# (( ) %) ( + # % ^ "! ( # ) < ( )') g( ! # ) + ±
G
z
5.
¾
e 6:8DC/(L 9-5$/<NC/734/A6/#B@5mA6>@THBR6
8$/434/ 5$8$/ JzBR6PJ
/ 8$/734E2BR6
BR6? G¾l
BR690FeK>R>l3496
SiJ/ LYAE21A-5DB@8DB@A6f$³7«N8DC/76A6/`L 96oJ
A~9~5DB@E2b
1
>@/PEPAJBR^L 98DB@A6nAF(AK3º9>@?A3$B@8DCE8$APAZ8$9BR6~960965$<N/73º<uB@8DC
5$Ec9>R>@/730L A/4jkL7B@/76
8$5 V°*6J
/ / J«]B@8k<NBR>R>zZ/`5DCA<N6iBR68DC/[6/7Q
8
5$/YL 8DB@A6´r/ EP93$Sn¹V H8DC9-82A6/PLY96~BR6~F*9L 8zK5$/c8DC/P34/ 5DB@JK9>
A6/NB@8$/7349-8DB@A6OZ/7FA34/]LYAE21
>@/ 8DB@A6OFA3AK
3(3496
S234/ a-/ 9>RBR6?P8D34965Db
F*A34EP9-8DB@A65 VlÁ]6/z5DB@E21
>@T`K5$/ 5r8DC/s34A<r5rL A3$34/Y5D1A6J%BR6?`8$AP8DC/
1
B@aA-8]34A<r5NBR608DC/H>@9-5$8]B@8$/7349-8DB@A6V
©F
6
Æ ÉnÆ Én¹Æ É Æ
É~ÆkÉM
¸Æ
ÉnDÆ
959H1B@a-A8NFA3L A>RKE26 ͫ8DC/H5DK
Z5D19LY/z?/76/73498$/YJyZ
Tc8DC/N34A<r5
A<
F 7 B@5N8DC/H5$9-EP/H9-58DC/H5DK
Z5D19LY/H?/46/7349-8$/ J;Z
T:34A< [AF
7 L 9>R>@/ Jn9;1
B@aA8z34A<l]96J~8DC/234A<r5zA=
F 7 $ VcWuC
B@5zB@5zL7>@/ 93$>@T
8D3$K/]9F8$/738DC/lA34J
/73NC95#Z/ /76kBR6L734/ 9-5$/ J;FA334A<r5 Ã Ë `95(8DC/
34/ L7K
3$34/76L /k$NB@5]BR6
a-/7348DBRZ>@/-V [K
>@8DBR1
>@TBR6?[34A< ~AF
Z
T_¾
1
34AJ%KL / 5l /734A5NBR6O34A< [BR608DC/HK1J
98$/YJ`EP98D3$BRQ«5$AP8DC9-834A<
MA=
F 7 EsK5$8HZ/PS/418 VO®mBR69>R>@T«r8DC/;9-#J %$K5$8$EP/46
8sFe34A-EX34A<r5
Ã Ë [B@5r9?-9BR6`BR6
a-/7348DBRZ>@/-VrWuCK5 «
8DC/z5DK
Z5D19-L / 5H934/H8DC/H5$9-EP/V
F
«
)
« ¡
*6n19348DB@L7K
>@93«lB@82FA>R>@A<r5[8DC98 «9F8$/43k8DC/ b8DCnB@8$/73498DB@A6m«l8DC/
34A<r5HAF>7
Ã
r5D196n8DC/:5$9Ec/P5D19-L /P9-5z9>R>(1B@a-A8H34A<r5
1
>RK5z8DC/s34A<r5rAF?7 $ Vr´r/ LY9>R>RBR6?`8DC/z^345$8 izL A>RKE265zAF
7 934/: /734APZ
T`8DC/:A34J
/73ºL A6J%B@8DB@A6nFA3 `9C
>@/73:5$T5$8$/ EP5 «<N/
5$/ /l8DC9-
8 7 $ ÃMV·BR6L /HBR6_9J
J%B@8DB@A6~8DC/ %b8DCOL A-E21A6/76
8rAF
8DC/H1B@a-A8]34A<o9-8]5$8$9-?/ [/ K9>@5 ÃM
Ë «8DC/H1
B@aA-8]34A<r5FA34E
9zFeK>R>34A<M3496
SkK1
1/73l/ L$C/7>@A60EP9-8D3$BRQ«
96
J 7 $ ÃMV H/46L /
3496
S
à V
¤ e <>
>
@F
£
`
.0/u6/7Q
81
34Aa/r9z>@/ EPEP9zA6[8DC/N1B@a-A8$5K5$/ JPBR6O/ 9L$CkB@8$/73498DB@A6mV
i=¹VRV'V
§4QVSKu-$ £9 Teª`IªY§7¥SL[£ML\§ ST§ ¤¦©9TS*£MLk£H ¤$§4¥MIO4§
MS !TKSKL OT Tn
©9T4SUª rÉ
P`£¤¦§7£Mu-§ ¤ ¤$©ML#N©
S eSKLVO vSL#SKL' * ¢ * 2
!(¤$££#H "XTPH/7^
6
B@8DB@A6¡%V %eZ\7«
8DC/434/H/7QB@5$8$5N9z1A>@T6A-E2B@9>¡ ½HR¾z¿$À 79 5DKLC08DC98
¾z5
à 5 $ Ç ¥§|6 ¢ * ɱ£* $ FA3l9>R>Z¥-Ç « V
5
© ¤$§c¥ eª
eª
¤$4§ ©ª
§$ K¥ «
A-E2193$BR6?n8DC/[L A/4jkL7B@/76
8$5O98k¾ ! . $ BR6M1\A5DB@8DB@A6= -Ç 7«N<N/
N
C9a/lA6O8DC/r>@/7F8N9z6A65DBR6?K>@93H>@A<N/73r8D3$B@96?K>@93zEP9-8D3$BRQ0<uB@8DC
À#Ê
HA6~JB@9?-A69>U7«N96J0A6p8DC/23$B@?C
8z8DC/:>@/ 9JBR6?034A<°L A/7jPb
$
L4B@/76
8#EP98D3$BRQ MAF <NB@8DCcL A/7jkL7B@/46
8$59-8Í1A<N/73
ÉP
EzK
>@8DBR1>RB@/ J[Z
T:9H>@A<N/738D3$B@96?K
>@93ºEP98D3$BRQ 2V(·BR6L /H<N/934/N6A<
BR6±8DC/kKA8DB@/76
8P^/7>@J« fEsK5$82Z/k6A65DBR6?K
>@93«l96Jn5$A XB@5
9>@5$A;6A65DBR6?K>@93P96J[C/76L /2>@A<N/73l8D3$B@96?K>@93V H/76L /:8DC/zJ
/7b
?34/ / 5#A6P8DC/NJB@9?A69>L 966A8#Z/N5$EP9>R>@/738DC96P
«5DCA<uBR6?z8DC98
$ «A3«BR6OA8DC/73r<NA34J5 « $
É~
#É V#WNCK5 «
8DC/
1
B@aA-8$5NBR6`A6/]B@8$/43498DB@A60934/HJB@5$8DBR6LY8 VN]>@5$A
«J
/76A8DBR6?2Z
T Ig8DC/
8D349BR>RBR6?¡L A/7jkL7B@/46
82AF [«m<N/c/ 95DBR>@T0AZ8$9BR6i8DC98 I
$
LYABR6L7B@J/Y5z<uB@8DC_8DC/lEP98D3$BRQOAZ8$9BR6/ J`Z
TP911
>@TBR6?OÀ 8$A:9>R>{/7>@/7b
Ec/76
8$5NAF
r<uC
B@L$C;C95r8DC/H5$9-EP/]3496SO95
$7V l/76L /
8DC/H3496
S`AF
B@5NBR6L734/ 95DBR6?V
* f * 4 ¡ * 9£)*
¡* ¢ * D2 5_
e .0/H934/N6A<M34/ 9-JTP8$AH1
34Aa/l8DC/lL A3$34/ LY8D6/ 5$5AF8DC/l9>@?-A3$B@8DCE[V
Vi
r Vi ¹%V ¸VV
§PP ©9Te¤Sn}' ¤¦§TKI
¤vL\§7¥O¨:x ¤$§4¥MIqO7§2ªY©9TK{
SUª N§ ªr¤$©MLkN[(Ãg¤$©MLkN
!(¤$££#H "·BR6L /P8DC/s3496S0L 966A8HBR6L734/ 9-5$/P9F*8$/73zEzK>@8DBR1
>RB@L 9-8DBeA6
<NB@8DCi9`5$K934/[Ec98D3$BRQ«N</;?-/ 8zFe34AE
5#Ã 5 ¾ 8DC/
34/4>@98DB@A6n3496Sefo3496
S:|
5 ¾ ]Ãg3496Se V:Á]6¡8DC/PA8DC/73
C96J«%K5DBR6?28DC/r5DCBRF8$/YJy>@/4F8mBR6
a/7345$/@ AF6 AF´r/ Ec93$Sy%V ¸
</HC9a/z8DC98
3496SRqÃM3496
Sz¾eR5r Ã
3496
Sz| 5
-5 ¾
¢ 3496
Sz| 5¾ #ÃM3496
Se Ç
5DCA<NBR6?08DC9-8H3496
S@IÃh3496Se
FA3:9>R>¡Í VOeFr8DC/P9>@?-A3$B@8DCE
5$8$A15l9F8$/43r8DC/k=zÉob5$8B@8$/7349-8DB@A6«m8DC/76
> ñ3496Se ¢ 3496Se $ ¢ 3496
Se $ ¢ >
º
Z
Tk/ EPEP9c¹%V@«
m/YEPEP9k¹VRz96J[8DC/]F9-L 8N8DC98e $ L A6
8$9BR65
> 6A6Y/734AH34A<]5 VÌuA65$/YK/76
8D>@T-«</C9a-/]/YK9>RB@8DTk / a/434T<uC/734/«
r
96J)º
> Ãq3496
S@ $ ÃM3496
SeHV
if¹V
V1O§A
©Mu-§ª-
£MwLSKPQVJyS=O!S=TKJyxRSLTn
§ZQ¤$££HATn
©9T
S Hw(§sª"T£vQ`©H!T§ ¤S=T§ ¤$©9TKS£MLk=lÉq Tn
©9TZTn
§Te¤$©MSJySKL#O4£§DO!S§!LVT
£H
©MJR¤¦§7©¥Mx
©ªHvI#JKJ¤¦©ML#N%># SLVO7§k©MJKJ¡Tn
§QVSKu-£9Teª7SKLo£ML§
ST § ¤¦©9T *S £MnL ©¤$§`¥MeS ª"TKSKLVOWrT ¨!xds §!P7PP ©9p Tn
§;ªY§WzT £HQVSK-u £9NT ¤$£M#w ª
SL0 k©MRJ ª £'
©ª:¤$©ML#N>#DV
§ ¤¦§/YH £¤$§ Tn
§PªvI ¨!PP ©9eT ¤Sn`} £H O4£MLªSeª"TSL#P£HTn
§RQVSKu-£9T#¤$£Mw#ªk©MJR¤¦§7©¥MxJ@§4©¥ª7T£[©HvI#JKJ¤$©ML#N[¥§!{
4O £MP
Q £ªS=TKS £MLI ªSKL#7Tp
§74O £¤7¤¦§³ª=Q £ML ¥MSKL#4O £MJyIP7L ªl£HTn
§sª-
S H!T §4¥
J@§=H!TSKL-u § ¤4ª §z£HTn
§ Q ¤$§!uWS £M
I ªP`©W
@J § ¤Hªx ª"T §:Pe
e6kA34J
/738$AzJ/Y8$/734E2BR6/z8DC/NZ
B@8L A-E21
>@/7QB@8DTkAF8DC/r®#®34/ JKL /l9>Rb
?-A3$B@8DCEf<N/rEP9S/H8DC/H9-5$5DKE218DB@A6_8DC9-8N8DC/]>@/76?8DC5r96J;LYA5$8$5
AF(L A/7jkL4B@/76
8H93$B@8DCEP/Y8DB@LP5$98DB@5D^/ 5
2 Æ
#,=.mÄ5#RY
xW,Ä5#RY
Ã
Ã
2 Æ
#,=.mÄ
mÉ 2 Æk,/.mRY7Ç
2 2 Æ
#
,=.mÄ
5 2 Æ
# ,=.mRY$v
l/734/©J@§:LkkTn
`EP/ 9-5DK34/ 5`8DC/08$A8$9>º5$8$A349-?/ 6/ / J
/ J<uC
BR>@/<O4£ª!T
Ec/ 95DK
34/ 5:8DC/z6KEzZ/73cAF(ZAA>@/ 96nA1/7349-8DB@A65 Vk.qB@8DCn8DC
B@5z95Db
5DKE218DB@A6¡<N/H?B@a/:AK
3zL A-E21
>@/7QB@8DT[BR608DC/HFA>R>@A<NBR6?V
VirVi¹%V ¹%VnHH¾H
©ª`T£9T©MJ¥§|¤$§4§m|Tn
§!LJ £¤S=Tn
P
MIO4§]¤¦§4!I#Se¤$§ ª©9TZPP£ª"Tv0|oÉ RS=T§ ¤$©9TKS£MLª!*P`£¤¦§7£Mu-§ ¤ S H
¤$§4¥
S MIL MLaTp
§J@§!L# 9$ Tn
BA ªH$ £ H7Tn
§O4£§DO!S§!LVTeªr©vQ9Q§7©¤SKL#SKL
5y v!"mÇ4¾ 5yH¾l Tn
§:LTn
§N¨!S=TO4£MPRQVJ@§}MS=TKx
HTn
MJ S=Tn
P¤Seªr2v @ x | }
!
}
Uª:©[¨4£ ¥c£
H¾H7Ç4¾ H¾l7Ç
£ §2© £¤
H
!(¤$££ #"
¡ £0 M2 ®34A-Ef8DC/zJ
/7^
6B@8DB@A6nAF
</z5$/ /28DC98z|
É
@B 5[96iK1
1/73`Z\AK
6JoFA3O8DC/`5DKE¢AF:8DC/`34A<µJ
/ ?34/ /Y5[AF
V(]F*8$/73NB@8$/7349-8DB@A6 :ɱ-«<N/HS6A<¯FU34AEfm/YEPEP9k¹VR28DC98r9
L A-E21A6/76
8rAF{8DCB@5NK
11\/73rZAK
6J`/4B@8DC/73rB@5NLYA65$8$96
8zeFA31B@a-A8
34A<r54]A3lA-8DC/734<NB@5$/PJ
/ L734/Y95$/ 5HZ
TOA6/Vl·BR6L /«Z
T[/ EPEP9O¹V -«
FA3k9>R>rZ
K8k8DC/;>@95$8PB@8$/7349-8DB@A68DC/734/;B@5k9-82>@/ 95$8kA6/k6A6 /434A
34A<IBR6
<NCB@L$C0B@5s6A8H1
B@aA-8;eZ
K8:<NCB@L$C~1A-8$/76
8DB@9>R>@TnL AK>@J
Z/ LYAEP/k9[Y/734A;34A<IBR6
$ 7«<N/PEP9T`L A6L7>RKJ
/[8DC98z/7B@8DC/73
8DC/25DKEfAF(8DC/2J
/ ?34/ /sZ\AK
6J
5zAF#6A6
8D3$B@aB@9>34A<r5NBR6
$ B@5
>@A<N/734/ JkZ
Tk98N>@/ 95$8z-«A3l<N/]C9a/zL734/ 9-8$/ J[96`9-JJB@8DB@A69> /734A
34A<zV`W#9SBR6? BR6
8$A09L L AK6
8c8DC9-828DC/;BR6
B@8DB@9>º5DKEB@52ZAK6J
/ J
9Z\Aa-/]Z
TPv_| «</]LYA6L7>RKJ
/z8DC98N8DC/734/H934/r98(EPA5$8Nv0|MÉ ³
B@8$/7349-8DB@A65 V
* c
c © ®A3;Z\AK
6JBR6?o8DC/`ZB@8OL AE21
>@/7QB@8T«:<N/[FA>R>@A<8DC/0L AE21
>@/7Qb
B@8DT`969>@T5DB@5sBR6i ²(<NC/434/28DC/sK
1
1/73:ZAK
6J Pv M M }
NC9-5
Z/ /46[/Y5$8$9Z
>RB@5DC/ JVrWNC/NS/ T2AZ5$/434a9-8DB@A6OB@58DC98(9>R>L A/7jyL7B@/76
8$5
L 96IZ/~<N3$B@8$8$/76=9-5`J/Y8$/734E2BR696
8$5 Z
T=²[96J$Y7V WuCK5 «
H9-J9-EP934
J 5;BR6/ K9>RB@8DTL 96oZ\/ 911
>RB@/ J8$AqAZ8$9BR69~ZAK
6J
A6_8DC/z>@/76?8DC5HAF(8DC/zL A/4jkL7B@/76
8$5 V
|
i f¹%V V1O§N¤$§!cP ©¤NTn
©9T ªSKLV4O § Tn
§4O £ª!T £(H ©¤S=Tn
PP §WTKS=O
SKLn» Åk ÆzªY©9TKSeª N§ ª:£MI¤'O4£MPQVJ@§v}MS=TKxGPP£¥§!Jw
§!LDw(§[©¤¦§SKLVT§ ¤v{
§ ª!T §4¥SK?
L §:@J ¥0£vQ § ¤$©9TKS £ML ªSKnL » Å Tn
§kª ©McP §)7O £MPQV@J §}MS=TK[x ¨4£MI#L ¥
Seª:©vQ9QVJyS=4O ©¨!@J §ZH £¤:» ¼ û ÅP Æ\wSTp `
}¢¨7§!SKLk%Tn
§cP ©W}MSKPI#µ
P ¥§!{
¤$§4§G-SK_L Æ6_ £PH ©MJKJ¡B TpA
§' 4O £§ cO!S §!LVeT ªRSML ©vQ9Q §4©¤vSL#SLaH¾H7Ç4¾<5
H¾H7Ç!"!
Ǿ $ $ 5 º¾l! z£9T §ATn
©9T £M
I ¤A4O £MPRQV@J §}MS=TKrx ©ML ©MJy{
x ªeS ª7T ©W
N § ªSKLVT £O@ ©9Ov4O £MIL TTp
§7O £§DO!S §!LVTV ¤$£Mw`Tn0
¥MI ¤SKL#'Tn
§P©MJy{
£¤vSTp
P
6.
CONCLUSION
*68DC
B@5;191\/730<N/[C9a-/ ?B@a/76h9 Fe349LY8DB@A6
bFU34/ /~9>@?-A3$B@8DCE
FA3
8D34965DFA34E2BR6?¡9`?B@a/46oEP9-8D3$BRQnAFl5DS/Y<1A>@T6AE2B@9>@5;BR6
8$A~A6/
<NC/434/H8DC/]3496
SPB@5NJ
/ 8$/734E:BR6/ JOA6
>@TPZ
TP8DC/l8D349BR>RBR6?kA3N>@/ 9-J%BR6?
L A/7jyL7B@/76
8zEP9-8D3$B@L / 5 VrWNC/:9>@?-A3$B@8DCEB@5H9PEPAJBR^L 98DB@A6MAF(8DC/
®#®#®#f9>@?-A3$B@8DCEÎAF2 ²(BR6n8DC/kL AEPEzK8$98DB@a/OL 95$/-V[.0/PC9a/
5DCA<N6°8DC9-8[AK309>@?A3$B@8DCE 3$K65[BR61A>@T6AE2B@9>O8DB@EP/«z<NB@8DC
6/ 93$b>RBR6/ 93º?34A<r8DC;BR6O8DC/H5DB@ /Y5NAFL A/7jyL7B@/76
8$5NBR6`8DC/NBR6
8$/734EP/7b
JB@98$/z34/ 5DK>@8$5 V
WNC/434/P934/P9[6KEzZ/73cAF8$A1
B@L 52FA3zFeK8DK
34/234/ 5$/Y934L$CmVOÁ]K
3:91
b
1
34A9L$CoBR6L734/ 95$/Y5[8DC/`J/ ?34/ /`AF:8DC/[34A<r5PAF:96oA34J
/73kZ95DB@5
Fe34A-E^345$8r8$Az>@95$8 Vre6kF9L 8NB@8NB@5r/ 95$TO8$A25$/Y/z8DC9-8]A6/zLY96`9>@8$/73
8DC
B@5]A34J/43«%F*A3/7Q
9E21
>@/HZ95$/ JkA6O8DC/HE2BR6
B@EzKEfJ
/ ?34/Y/zAF8DC/
34A<r5]AF#8DC/H34/ 5DB@JK9>«m<NC
BR>@/25$8DBR>R>{EP9SBR6?PK5$/:8DC/rFe349LY8DB@A6
bFU34/ /
34/ L7K
345DB@A6±?B@a-/76oBR6i8DCB@5P191/73VnWNC
B@5P<NBR>R>HZ/OJ%B@5$L4K5$5$/ JMBR6i9
L A-E2BR6?P191/43V
.0/]1
>@9608$AzBR6
a/ 5$8DB@?-98$/:EP/ 8DCAJ5NFA3NB@E21
34AaBR6?k8DC/z/4jkL7B@/76L T
AFcAK
3`Fe349-L 8DB@A6
bFe34/ /~911
34A9-LCVµ]8P1
34/ 5$/76
8 «:AK
3`9>@?A3$B@8DCE
J
A/ 5]6A8r91
1/ 93:8$APJ
AP95l<N/7>R>#9-5r8DC/z9>@?A3$B@8DCEAF(rZ349-EPAa
96J~34A65$8$/4BR6I ³l95HB@E21
>@/ EP/76
8$/YJqBR6 `91
>@/[²]K6
>@/ 5$5kxPB@5
5$EP9>R>VoWNCB@52B@5kJK/O8$A 8DC/[F9-L 828DC98k8DC/[L A/4jkL7B@/76
8k?34A<r8DC
BR6nAK3:9>@?A3$B@8DCEGB@5z5DB@?6BR^L 96
8P/ a/46n8DCAK?C~B@8HB@5zL A6
8D34A>R>@/ JV
.0/P<NAK
>@J~>RBRS/k8$A;^
6J Fe349LY8DB@A6
bFU34/ /kEP/ 8DCAJ5:<NC/734/:>RBR6/ 93$>@T
BR6J/71\/76J
/76
834A<r5mBR6c8DC/>@/Y9JBR6?:A3(8D349BR>RBR6?:L A/7jkL4B@/76
8Ec98D3$BRQ
cF
934/]6A8NEcAJ%BR^/ J[JK
3$BR6?k96kB@8$/7349-8DB@A6«9z134A1/7348T;5$9-8DB@5D^/YJ;Z
T
8DC/z91
1
34A9L$C0BR6nR«DV
®
K348DC/734EPA34/«B@8]B@5l<N/7>R>#S6A<N608DC98lEPAJ%K
>@93:9>@?A3$B@8DCEP5zB@E2b
1
34Aa/`A6qFU349L 8DB@A6bFe34/ /OEP/ 8DCAJ5PZ
T~96iA34J
/73kAFºEP9?6B@8DKJ
/V
r6n9J
a96
8$9-?/[AFlAK
32Fe349-L 8DB@A6
bFe34/ /O91
134A-9L$CqB@5:8DC98:8DC/k34/7b
5DK
>@8:L AE21
K8$/ JZ
T`AK3:9>@?A3$B@8DCEB@5z<N/7>R>RbJ/4^6/ Jn96J LY960Z/
K5$/ J~9-529O8$934?-/ 8sF*A3z34/ L A65$8D3$KL 8DB@A6MBR6nEPAJK
>@93k9>@?A3$B@8DCEP5 V
.0/P/7Q1\/ L 8:8$A[8$9S/k9J
a96
8$9?-/;AFr8DC/P91
134A-9L$CiAFrBH@ (8$A
34/YJ%KL /[8DC/[LYA/7jkL7B@/76
8$5:BR6 ½z Æ8$A
FA3P5$A-EP/;1
3$B@EP/ n8$A
AZ8$9BR6`96[/4jkL7B@/76
8rEPAJK>@93l9>@?A3$B@8DCEF*A3rAK3N3496Sc34/ a/ 9>RBR6?
LYAE21
K8$9-8DB@A65 V
.0/z9>@5$AP1
>@9608$AP5$/Y/]CA<AK
3l9>@?-A3$B@8DCEL 96OZ/HK5$/ J`8$A2LYAE2b
1
K8$/`zNH5PAFz5$L 9>@93O5DS/Y<1A>@T6AE2B@9>@5`96JML A-E21934/;B@8c8$A
8DC/[5DK
Z
34/ 5DK>@8$96
8k9>@?A3$B@8DCEc5[AFlBz ³³V0®mBR69>R>@T«r<N/k934/kBR6b
8$/434/ 5$8$/ J;BR6_/7Q
8$/76J%BR6?[AK334/ 5DK>@8$5l8$As6/ 5$8$/ JO5DS/ <o1A>@T6A-E2B@9>
J
AEc9BR65 «m9>R>@A<NBR6?[FA3lL A-E21
[email protected]/ T>(9>@?-/7Z
3495 VlWuC
B@5
B@5z9kJBRjkL7K
>@82/4Q
8$/765DB@A6i5DBR6L /c8DC/76¡8DC/2L A3$34/ 5D1A6JBR6?095$5$AL4BRb
9-8$/ J[>RBR6/ 93c5$T5$8$/ EP5lJAP6A8NC9a-/PL AEcEsK8$98DB@a-/P/7>@/ EP/46
8$5 VrH5
5DKL$CI8DC/ 5$8$96J
934J8$AA>@5`8DC98`</[K5$/ FU34AEÎ>RBR6/ 93¡9>@?-/7Z349
«
69Ec/7>@T[J
/ 8$/734E2BR696
8$5l96J`349Ec/73 5(3$K>@/«JAz6A8r/7QB@5$8NBR6`8DC/
L4>@95$5DB@LP5$/765$/V
MI L MJ H eS !LVO
7. 4c·\REFERENCES
V
rZ349-EPAaVN]b/7>RB@E2BR698DB@A65V£ ¤ © £ H
§ ¤$§ 4§
! I©9TKS£MLª©ML¥7 Q9QVJyS=O4©9TKS£MLª4«-¹%t ¸¬-¸_¸¸
«# ¬-¬¬V
)F
WO MSKL# H #U0ií
s ML ML
v AF
¸:]V
N934/7B@5$5 V·T>@a/ 5$8$/73 5]B@J
/76
8DB@8T`96J`EsK
>@8DB@5$8$/71
BR6
8$/ ?/43$b1
34/ 5$/734aBR6?`H9K5$5DB@960/7>RB@E2BR698DB@A6VP_©9Tn
#£MPQ «
$ ¸7tR¹¹!
¹¶²«#Y¬²
V
R c·\V
rZ349-EPAa[96J ~V
34A65$8$/7BR6V#Á]6_5$A>RK8DB@A65zAF#>RBR6/ 93
FeK
6L 8DB@A69>(5$T5$8$/ EP5 V\*6 !(¤$£ 4§7§4¥ ªr£ / £ ¥£ «19?/Y52 VH -(#34/ 5$5 «\
V
ºHV/ LS/734EP966096J02Vm9Z9C
6VgK
6BRF*A34Eµ91
134A-9L$C
FA3 l/734E2B@8$/ (9-J /z96J`5DB@EzK>@8$96/YAK5((9J /
91
134A QB@EP96
8$5296JO8DC/4BR3zEP9-8D3$BRQ`?/46/7349>RB@ 98DB@A6#V
P§ ¤ 4©
%£¤
:ª7«¸
t ¹ %¹ « ¬-¬%V
e
I#P S=O MJ#J =S Tn
P
:
¹³ºHV/ LS/734EP966096J02Vm9Z9C
6VgK
6BRF*A34Eµ91
134A-9L$C
FA3l8DC/HF95$8 «
34/7>RB@9Z
>@/PL A-E21
K8$9-8DB@A6¡AF(EP9-8D3$BRQbD8T1/ (9-J /
91
134A QB@EP96
8$5 V P
*P`© e¤
©
«
¹t ²- ²¸«# ¬-¬ V
M
M
| 9 9T Sn}L MJnq Q9QVJn
ºHV/ LS/734EP966096J02Vm9Z9C
6V´r/ L7K
345DB@a/76/ 5$5zBR6
EP98D3$BRQO3498DB@A69>#BR6
8$/73$1A>@9-8DB@A6~1
34AZ
>@/ EP5 V
#£
P`©
«¶¶tR¹ ¸ «mY¬¬¶V
9 MPRQVIqT=
Q9QVJn 9Tn
! M
¶³ºHV/ LS/734EP966096J02Vm9Z9C
6V#Ï/ L 8DB@a-/PL A E:1K8$98DB@A6
AF#349-8DB@A69>(91
1
34A³QB@Ec96
8$5z96J[BR6
8$/73$1A>@96
8$5 V N§ ©¨ @§
#£
«
t ¸-¹ ¸¬-«
V
MPQVITKSKL#
$ð
!
!JyS !J
²ºHV/ LS/734EP966096J02Vm9Z9C
6V®349-L 8DB@A6
bFe34/ /
L AE:1K8$98DB@A6~AF(EP98D3$BRQzu 5r96J[349-8DB@A69>
BR6
8$/73$1A>@96
8$5 V P
P`© e¤
\©
@«
$7t@ «-V
!W "M
| 9 9T Sn}7@L MJn Q9QVJn
¼.gV34A<u6096J V
Wm349K
ZmV\Á]6`#KL7>RB@J 5z9>@?A3$B@8DCE 9 6J
8DC/:8DC/ A34TkAF(5DK
Z
34/ 5DK>@8$96
8$5 V
P[«Y²t ¹¹ ¹ «
¦%V
9U
! !
@ s2VNA>R>RBR65 V{·K
Z34/Y5DK>@8$96
8z96J[34/ J%KL / J[1\A>@T6AE2B@9>
34/ EP9BR6J
/73:5$/ K/76L / 5 V
P[« t ³² %«# ¬-¶V
9·
! !
@7 >PVH/ J
J
/ 5 «·VN 91A3«96J02V9Z9C
6mVJ £¤S=Tn
P2ªH £¤
#£MPQVIT§ ¤J §¨³¤¦©V >H>RK
<N/73«NA5$8$A6m«q`
F 2« ¬¬%V
@³mVÍBV I¨ ¤¦§³ªI#JnT©MLqT
§7£¤xH £¤esZSK
L §4©¤S l§ ¤$§!LVTKS ©MJ
sZSKL\§7©¤S l§³¤¦§!LVO7§l©ML¥ ¤¦§ !N£MJyxL £MP7S ©MRJ ª wS=Tn
Q9QVJyS=O4©9TKS £ML ª$V(ÌCI8DC/ 5DB@5 «
AC96
6/ 5>z/71
>@/73¥]6
B@aVÍBR6«
]K5$8D3$B@9«#Y¬¬
V
© Copyright 2026 Paperzz