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a2MiBK2Mi M HvbBb7Q`h2Hm;mG M;m ;2 - IIIT Hyderabadweb2py.iiit.ac.in/research_centres/publications/download/masters... · *QMi2Mib *? Ti2` S ;2 R AMi`Q/m+iBQM.....R RXR a2MiBK2Mi

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Page 1: a2MiBK2Mi M HvbBb7Q`h2Hm;mG M;m ;2 - IIIT Hyderabadweb2py.iiit.ac.in/research_centres/publications/download/masters... · *QMi2Mib *? Ti2` S ;2 R AMi`Q/m+iBQM.....R RXR a2MiBK2Mi

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#v i?2 mb2 Q7 కృతజత U;`�iBim/2VX

మం ,ఆయన ఎ న ం , పజల కృతజతవ కం UjXkV

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jXjXk am#D2+iBp2 a2Mi2M+2b

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న ంద చ త అత ంతప వవంతమన వ UjXeV

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అమ అధటం వరణఒప ందం ం అమవ ల ం

la S`2bB/2Mi .QM�H/h`mKT rBi?/`2r i?2la 7`QK i?2 S�`Bb *HB@K�i2 �;`22K2Mi

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h?2 *?B27 JBMBbi2`b�B/ iQm`BbK Bb �MBKTQ`i�Mi bQm`+2 Q7BM+QK2 7Q` i?2 bi�i2 Q7�M/?`� S`�/2b?

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j ఇం ఎవ అభ ంతరంఉండనవసరం

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8, మ క సం-

షమన ప న -ం

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eరత ప నమం న ం-

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SQbBiBp2 L2;�iBp2 lM+2`i�BM _2KQp2/

dఅ యడం వల ఐప ష దబ త ందధ సనం ం

h?2 #2M+? b�B/ i?�i #v/QBM; bQ- i?2 *"A rBHH?�p2 /�K�;2 iQ Bib `2T@mi�iBQM

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Ç SQbBiBp2 a2Mi2M+2b, � bm#D2+iBp2 b2Mi2M+2 r?B+? `2~2+ib TQbBiBp2 QTBMBQM- 722HBM;- 2KQ@iBQM Q` b2MiBK2Mi BM Bi Bb +�HH2/ � TQbBiBp2 b2Mi2M+2X 6Q` 2t�KTH2- b2Mi2M+2 UjXdV Bb �TQbBiBp2 b2Mi2M+2 #2+�mb2 i?2 r`Bi2` 2tT`2bb2b TQbBiBp2 2KQiBQM BM Bi #v ఇషపడ UHQp2iQVX aBKBH�`Hv b2Mi2M+2 UjX3V Bb � TQbBiBp2 b2Mi2M+2 �b Bi 2tT`2bb2b i?2 TQbBiBp2 722HBM;ఆనందంవ కం U2tT`2bb2/ ?�TTBM2bbVX

త ల డ ఇషపడ UjXdV

h`�MbHBi2`�iBQM, LĚMm i2Hm;m v­Fț�M bBMBK­H�Mm +ɟé�Ȫ­MBFB BțȪ�T�é�i­Mm1M;HBb?, A HQp2 iQ r�i+? h2Hm;m �+iBQM JQpB2b

శం నలధ ంచ న చర ల ం అధ ఆనందంవ కం UjX3V

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పజ వ ల ల భయ ం ల UjXNV

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అ ఒక అ భవం UjXRyV

h`�MbHBi2`�iBQM, �/B QF� +Ě/m �Mm#?�p�Kö1M;HBb?, Ai r�b � #Bii2` 2tT2`B2M+2

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రనం అర ఆక ం UjXRRV

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ఈప ఎ కలత త ర అ చనఉం UjXRkV

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jXjXj S2`bT2+iBp2

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అ తమమన రతసన ం ప సం ం UjXRjV

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jXjX9 1TBbi2KB+ JQ/�HBiv

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ట ల హత UjXR9V

h`�MbHBi2`�iBQM, >BȪH�` vɟ/mH�Mm ?�i�K­`+­ém1M;HBb?, >BiH2` FBHH2/ C2rb

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ట ల చం ఉండవ UjXR8V

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రదృషవ ట ఖ తం ల చం UjXReV

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ఈ హద ఫఓవ ం UjXRdV

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jXjXe AHHQ+miBQM�`v aT22+? �+ib

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8XkXR �+iBp2 G2�`MBM;

*QMbB/2` r2 ?�p2 � HBKBi2/ MmK#2` Q7 H�#2H2/ b�KTH2b UDLV- �#mM/�Mi mMH�#2H2/ b�KTH2bUDU V �M/ �M Q`�+H2X AM �M �+iBp2 H2�`MBM; b2iiBM;- �M �+iBp2 H2�`M2` K�v TQb2 [m2`B2b 7`QKmMH�#2H2/ /�i� BMbi�M+2b UDU V iQ #2 H�#2H2/ #v �M Q`�+H2 �M/ i?2M �//2/ iQ H�#2H2/ /�i�BMbi�M+2b UDLVX

8XkXk AMTmi iQ i?2 avbi2K

h?2 b2Mi2M+2 p2+iQ`b UDV ;2M2`�i2/ BM b2+iBQM 8XRXR Bb ;Bp2M �b BMTmi iQ i?2 bvbi2KX 1�+?b2Mi2M+2 p2+iQ` ?�b � H2M;i? Q7 Ryy /BK2MbBQMbX AM i?2 BMBiB�H b2imT- r2 bTHBi i?2 2MiB`2 /�i�UDV BMiQ i?`22 T�`ib r?B+? �`2 mMH�#2H2/ /�i� BMbi�M+2b UDU V- H�#2H2/ /�i� BMbi�M+2b UDLV �M/i2bi /�i� UDT VX

8XkXj Zm2`v a2H2+iBQM ai`�i2;B2b

AM �+iBp2 H2�`MBM;- iQ ;2i i?2 BM7Q`K�iBp2 b�KTH2b 7`QK mMH�#2H2/ /�i� BMbi�M+2b- i?2`2 �`2/Bz2`2Mi [m2`v b2H2+iBQM bi`�i2;B2b T`QTQb2/ BM i?2 HBi2`�im`2 U`272` b2+iBQM kX8VX q2 mb2/ i?27QHHQrBM; bi`�i2;B2b 7Q` Qm` rQ`FĜ

lM+2`i�BMiv a�KTHBM;

S2`?�Tb i?2 KQbi +QKKQMHv mb2/ [m2`v b2H2+iBQM bi`�i2;v Bb mM+2`i�BMiv b�KTHBM;X AM i?Bbbi`�i2;v- �M �+iBp2 H2�`M2` [m2`B2b i?2 BMbi�M+2b r?B+? Bi Bb H2�bi +QM}/2Mi Q` KQbi mM+2`i�BM?Qr iQ H�#2HX 6Q` T`Q#�#BHBbiB+ H2�`MBM; KQ/2Hb- mM+2`i�BMiv b�KTHBM; [m2`B2b i?2 BMbi�M+2 r?B+?Bb ?�pBM; � TQbi2`BQ` T`Q#�#BHBiv Q7 #2BM; TQbBiBp2 M2�`2bi iQ yX8 7Q` #BM�`v +H�bbB}+�iBQMX 6Q`i?`22 Q` KQ`2 +H�bb +H�bbB}+�iBQM- mM+2`i�BMiv b�KTHBM; [m2`B2b i?2 BMbi�M+2 r?Qb2 T`2/B+iBQMBb H2�bi +QM}/2MiX Ai }M/b i?2 H2�bi +QM}/2Mi T`2/B+iBQM BMbi�M+2 #v T`2/B+iBM; �HH i?2 BMbi�M+2bQM i?2 +H�bb rBi? ?B;?2bi TQbi2`BQ` T`Q#�#BHBiv mTQM r?B+? i?2 BMbi�M+2 rBi? H2�bi T`2/B+iBQM Bbi?2 `2[mB`2/ QM2X �MQi?2` K2i?Q/ UKQbi +QKKQMHv mb2/V 7Q` mM+2`i�BMiv b�KTHBM; Bb #v mbBM;2Mi`QTv �b i?2 mM+2`i�BMiv K2�bm`2 (9N)X q2 mb2/ i?2 H2�bi +QM}/2Mi K2i?Q/ 7Q` Qm` �TT`Q�+?X

Zm2`v@"v@*QKKBii22

Zm2`v@"v@*QKKBii22 UZ"*V Bb � KQ`2 i?2Q`2iB+�HHv@KQiBp�i2/ [m2`v b2H2+iBQM bi`�i2;vX AMZ"*- � +QKKBii22 Q7 KQ/2Hb `2T`2b2MiBM; +QMi2M/BM; ?vTQi?2bBb- r?B+? �`2 �HH i`�BM2/ QMi?2 +m``2Mi H�#2H2/ /�i� BMbi�M+2b UDLV Bb K�BMi�BM2/X 1�+? +QKKBii22 K2K#2` i?2M H�#2Hb�HH i?2 [m2`v BMbi�M+2b �M/ i?2 [m2`v BMbi�M+2- r?B+? i?2v KQbi /Bb�;`22 QM Bb +QMbB/2`2/i?2 KQbi BM7Q`K�iBp2 BMbi�M+2 Ui?2 #2bi [m2`vV 7Q` �+iBp2 H2�`MBM; (dR- k3)X h?2 7mM/�K2Mi�H#�bBb #2?BM/ i?2 Z"* bi`�i2;v Bb KBMBKBxBM; i?2 b2i Q7 ?vTQi?2bBb i?�i Bb T2`bBbi2Mi rBi?

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Zm2`vBM; AM7Q`K�iBp2 �M/ _2T`2b2Mi�iBp2 1t�KTH2b

J�Mv [m2`v b2H2+iBQM bi`�i2;B2b 7Q` �+iBp2 H2�`MBM; b2H2+i 2Bi?2` BM7Q`K�iBp2 Q` `2T`2b2Mi�iBp2b�KTH2 BMbi�M+2b �b i?2B` [m2`B2bX Zm2`vBM; AM7Q`K�iBp2 �M/ _2T`2b2Mi�iBp2 1t�KTH2b UZlA_1VT`QpB/2b �M Q`;�MBx2/ r�v Q7 +QK#BMBM; #Qi? BM7Q`K�iBp2 �M/ `2T`2b2Mi�iBp2 b�KTH2 BMbi�M+2b#v K2�bm`BM; BM7Q`K�iBp2M2bb �M/ `2T`2b2Mi�iBp2M2bb Q7 � b�KTH2 BMbi�M+2 #�bBM; QM KBM@K�tpB2r Q7 �+iBp2 H2�`MBM; (j3)X

.2MbBiv q2B;?i2/ lM+2`i�BMiv a�KTHBM;

.2MbBiv q2B;?i2/ lM+2`i�BMiv a�KTHBM; U.qlaV Bb � /2MbBiv@r2B;?iBM; �TT`Q�+?X aBKTH2`[m2`v b2H2+iBQM bi`�i2;B2b HBF2 mM+2`i�BMiv b�KTHBM;- Z"*- 2i+X �`2 KQ`2 HB�#H2 iQ [m2`vBM;QmiHB2`b #2+�mb2 i?2v i�`;2i QM HQM2 BMbi�M+2b `�i?2` i?�M QM i?2 2MiB`2 BMTmi bT�+2X h?2vbT2M/ iBK2 [m2`vBM; TQi2MiB�H QmiHB2`b #2+�mb2 i?2v �`2 +QMi2MiBQmb- Q` �`2 �MiB+BT�i2/ iQ T�bbQM � bB;MB}+�Mi +?�M;2 BM i?2 KQ/2H- #mi i?2 BM7Q`K�iBp2 BMbi�M+2 Bb MQi Ǵ`2T`2b2Mi�iBp2Ǵ Q7 Qi?2`BMbi�M+2b BM i?2 /Bbi`B#miBQM (dy- ed)X q2 +�M �pQB/ i?2b2 T`Q#H2Kb #v 2KTHQvBM; mMH�#2H2/ /�i�BMbi�M+2b UDU V r?BH2 +�H+mH�iBM; 7mim`2 2``Q`bX AM .qla- 7Q` }M/BM; i?2 BM7Q`K�iBp2 BMbi�M+2-r2 MQi QMHv /2T2M/ QM i?2 mM+2`i�BM BMbi�M+2- #mi �HbQ QM i?2 BMbi�M+2b r?B+? �`2 `2T`2b2Mi�iBp2Q7 i?2 mM/2`HvBM; /Bbi`B#miBQMX q2 +QK#BM2 mM+2`i�BMiv rBi? i?2 /2MbBiv Q7 i?2 mM/2`HvBM; /�i�iQ ?2HT `2/m+2 i?2 2``Q` [mB+FHvX q2 mb2/ F@K2�Mb +Hmbi2`BM; �b � K2i?Q/ 7Q` mM/2`HvBM; /�i�7Q` #mBH/BM; �M BMBiB�H i`�BMBM; b2iX h?2 iBK2 `2[mB`2/ 7Q` b2H2+iBM; i?2 M2ti [m2`v BM .qla +�M#2 7m`i?2` `2/m+2/ B7 i?2 /2MbBiB2b �`2 T`2@+QKTmi2/ �M/ +�+?2/ �M/ i?2 iBK2 Bb �HKQbi bBKBH�`iQ i?�i Q7 mM+2`i�BMiv b�KTHBM;X

_�M/QK a�KTHBM;

AM `�M/QK b�KTHBM;- �M �+iBp2 H2�`M2` [m2`B2b i?2 BMbi�M+2b `�M/QKHv (9R)X h?2`2 Bb MQ b2@H2+iBQM bi`�i2;v 7QHHQr2/ BM T�`iB+mH�` iQ ;2i i?2 BM7Q`K�iBp2 b�KTH2b 7`QK mMH�#2HH2/ BMbi�M+2bX

8XkX9 >v#`B/ Zm2`v a2H2+iBQM �TT`Q�+?

>v#`B/ [m2`v b2H2+iBQM �TT`Q�+? 2K�M�i2/ 7`QK i?2 KmHiB@�`K #�M/Bi T`Q#H2K (8j)X JmHiB@�`K #�M/Bi T`Q#H2K /2�Hb rBi? `2r�`/BM; i?2 �`KUbV #�b2/ QM Q#b2`p2/ 722/#�+FX �i Bi2`�iBQMn- +?QQb2 �M �`K an- Q#b2`p2 i?2 722/#�+F on(an)- �M/ Q#i�BM i?2 `2r�`/ rn(an)X AM Qm`�TT`Q�+?- 2�+? �`K ai `2T`2b2Mib � [m2`v b2H2+iBQM bi`�i2;vX A7 �M �`K ai Bb b2H2+i2/ �i Bi2`�iBQMn- i?2M �`K ai Bb `2r�`/2/ Q` MQi `2r�`/2/ #�b2/ QM i?2 T`2pBQmb Q#b2`p2/ 722/#�+F on−1(ai)

�M/ i?2M i?2 Q#b2`p2/ 722/#�+F on(ai) �M/ `2r�`/ rn(ai) �`2 mT/�i2/X

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6B;m`2 8X8 avbi2K �`+?Bi2+im`2

�b b?QrM BM 6B;m`2 8X8- i?2 ?v#`B/ [m2`v b2H2+iBQM �TT`Q�+? i�F2b #Qi? mMH�#2H2/ /�i�BMbi�M+2b UDU V �M/ H�#2H2/ /�i� BMbi�M+2b UDLV �b �M BMTmiX � /2+BbBQM K�FBM; KQ/2H Bb #mBHi�HQM; rBi? i?Bb �TT`Q�+? iQ T`2/B+i i?2 TQbi2`BQ` T`Q#�#BHBiv 7Q` 2�+? BMbi�M+2 Q7 DU X �7i2`+�H+mH�iBM; i?2 b�KTHBM; [m2`v /Bbi`B#miBQM φ(D(n))- #�b2/ QM KmHiB@�`K #�M/Bi �TT`Q�+?- �#2bi b�KTH2 BMbi�M+2 xi ∈ D Bb b2H2+i2/ 7Q` [m2`vBM;X A7 xi ∈ DU - i?2M i?Bb b2H2+i2/ b�KTH2BMbi�M+2 UxiV Bb H�#2H2/ rBi? �M Q`�+H2fH�#2H2` �b yi �M/ �//2/ iQ DLX LQr i?2 +H�bbB}2`UCnV Bb i`�BM2/ QM i?2 mT/�i2/ DLX h?Bb T`Q+2bb Bb `2T2�i2/ mMiBH DU #2+QK2b 2KTivX h?2`2r�`/ Urn(aiVV �M/ Q#b2`p�iBQMUon(ai)V �`2 mT/�i2/ #v +QKT�`BM; i?2 H�#2H yi ;Bp2M #v i?2Q`�+H2fH�#2H2` rBi? i?2 +H�bbB}2` (Cn(xi))X

�H;Q`Bi?K R /2b+`B#2b � ?v#`B/ �TT`Q�+? 7Q` �MMQi�iBM; i?2 mMH�#2H2/ b�KTH2bX h?Bb �H;Q@`Bi?K i�F2b /�i� . 4 UDU , DLV- [m2`v b2H2+iBQM bi`�i2;B2b Za 4 &qs1, qs2, ..., qsK' �M/ MmK#2`Q7 [m2`B2b iQ H�#2H UhV �b �M BMTmiX q2 BMBiB�HBx2 i?2 KBMBKmK T`Q#�#BHBiv 7Q` `�M/QK b2H2+iBQMQ7 [m2`v bi`�i2;v qsi 7`QK Za �b pmin = 1/len(QS)- Bi2`�iBQM UMV 4 R- r2B;?i p2+iQ` qUMV 4&w1, w2, ..., wK' r?2`2 wi 4 Rc R I4 B I4 E �M/ � +H�bbB}2` UCnV

h?2`2 �`2 8 K�DQ` bi2Tb BM Qm` �H;Q`Bi?KX AM ai2T R UGBM2 9V- r2 mb2/ KmHiB@�`K #�M/BiT`Q#H2K iQ K�F2 � +?QB+2 iQ b2H2+i � qsi rBi? i?2 T`Q#�#BHBiv p2+iQ` SUMV4&p1, p2, ..., pK'∈ [pmin, 1] r?2`2 pmin=y b+�H2/ 7`QK r2B;?i p2+iQ` qUMVX AM ai2T k UGBM2 eV mbBM; KmHiB@�`K#�M/Bi T`Q#H2K- b2H2+i qsi `�M/QKHv #�b2/ QM SUMV X AM ai2T j UGBM2 dV 7Q` �HH xi ∈ D- 7Q`K �T`Q#�#BHBiv /Bbi`B#miBQM φi mbBM; qsi ∈ Za �M/ }M/ i?2 [m2`v p2+iQ` qi mbBM; 2[m�iBQM 8XRX

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�H;Q`Bi?K R �+iBp2 G2�`MBM; mbBM; >v#`B/ Zm2`v a2H2+iBQMR, ALSlh . 4 &G�#2H2/ .�i�, DLc lMH�#2H2/ .�i�, DU}c Za 4 &qs1, qs2, ..., qsK', [m2`v

b2H2+iBQM bi`�i2;B2bc h , LmK#2` Q7 [m2`B2b iQ H�#2Hk, AMBiB�HBx2 pmin = 1/len(QS) , KBMBKmK T`Q#�#BHBiv iQ b2H2+i qsi 7`QK Zac M 4 Rc h`�BM �

+H�bbB}2` C0 QM DL

j, r?BH2 i = 1 → T /Q9, ai2T R, lbBM; JmHiB@�`K #�M/Bi T`Q#H2K- �/QTi � r2B;?i p2+iQ` qUMV r?B+?8, Bb b+�H2/ iQ T`Q#�#BHBiv p2+iQ` SUMV ∈ [pmin, 1]e, ai2T k, lbBM; JmHiB@�`K #�M/Bi T`Q#H2K- b2H2+i qsi `�M/QKHv #�b2/ QM SUMVd, ai2T j, 6Q` �HH i?2 b�KTH2b xi ∈ .- +�H+mH�i2 qi(n)4

!Kl=1 pl(n) ∗ φl(xi)

3, ai2T 9, *?QQb2 � b�KTH2 BMbi�M+2 xi #�b2/ QM qi(n) �M/ `2+Q`/ BiǶb r2B;?iN, Wn4Rfqi(n)

Ry, B7 xi ∈ DU i?2M UBX2X- 7`QK mMH�#2H2/ /�i�VRR, �, H�#2H i?2 xi �b yi �M/ KQp2 Uxi, yiV 7`QK DU → DL

Rk, #, h`�BM i?2 +H�bbB}2` Cn QM mT/�i2/ DL

Rj, +, i → i+ 1R9, 2M/ B7R8, ai2T 8,*�Hm+mH�i2 i?2 `2r�`/ r 4 Wn UCn(xi) 44 yiV �M/ �// r iQ �HHRe, [m2`v b2H2+iBQM bi`�i2;B2b r?B+? bm;;2bi xiRd, ai2T e, n → n+ 1

R3, 2M/ r?BH2

qi =K"

l=1

pl(n) ∗ φl(xi) U8XRV

AM ai2T 9 UGBM2 3V- b2H2+i � b�KTH2 BMbi�M+2 xi #�b2/ QM qi �M/ `2+Q`/ BiǶb r2B;?iX GBM2b RyiQ R9 /2�H rBi? i?2 +�b2b r?2M b�KTH2 BMbi�M+2 xi ∈ DU BX2 MQ H�#2H 2tBbib 7Q` xiX LQr mbBM;Q`�+H2fH�#2H2` �bbB;M � H�#2H 7Q` i?2 BMbi�M+2 xi �b yi �M/ KQp2 Uxi-yiV 7`QK DU → DLX lbBM;i?2 M2rHv mT/�i2/ DL- � +H�bbB}2` Bb i`�BM2/ iQ #mBH/ � KQ/2HX AM ai2T 8- i?2 `2r�`/ Bb 72/ #v+QKT�`BM; T`2/B+i2/ H�#2H UCn(xi)V �M/ yiX AM ai2T e- i?2 Bi2`�iBQM MmK#2` UnV Bb BM+`2K2Mi2/Xh?Bb T`Q+2bb UBX2X- ai2Tb R@eV Bb `2T2�i2/ mMiBH hX

8XkX8 *H�bbB}+�iBQM KQ/2H 7Q` h2Hm;m b2MiBK2Mi �M�HvbBb

6`QK i?2 b2+iBQM 8XkX9- i?2 mT/�i2/ H�#2H2/ /�i� b2i UDLV Bb mb2/ 7Q` i`�BMBM; i?2 +H�bbB}@+�iBQM KQ/2H 7Q` b2MiBK2Mi �M�HvbBbX q2 #mBHi i?2 ?v#`B/ [m2`v b2H2+iBQM �TT`Q�+? rBi? i?`22KQ/2Hb pBxX- aoJ (RN)- :"h- �M/ s:"QQbi (R9) �M/ TB+F2/ i?2 #2bi mM/2`HvBM; KQ/2H r?B+?;Bp2b i?2 HQr2bi 2``Q` `�i2 QM i2bi /�i� UDT VX lbBM; i?2 i`�BM2/ KQ/2H- r2 T`2/B+i2/ i?2 H�#2HbU�b TQbBiBp2 Q` M2;�iBp2V 7Q` i2bi /�i� UDT V �M/ 7QmM/ i?2 2``Q` `�i2 #v +QKT�`BM; i`m2 H�#2HbrBi? T`2/B+i2/ H�#2HbX G�i2`- mbBM; i?Bb KQ/2H- r2 �MMQi�i2/ MmK2`Qmb h2Hm;m b2Mi2M+2b �M/K�/2 i?2 b2MiBK2Mi �M�HvbBb 7Q` h2Hm;m 7�+BH2X

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h�#H2 8X9 S�`�K2i2`b mb2/ 7Q` 2�+? +H�bbB}2`

*H�bbB}2` S�`�K2i2`b mb2/

s:" Ln2biBK�iQ`b 4 Ryy- G2�`MBM; `�i2 4 RXy- J�tn/2Ti? 4 j

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