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Page 1: math.univ-lyon1.frmath.univ-lyon1.fr/homes-www/gentil/logsob-poincare.pdf · y ,% ´| ; > %- ~ ' -- ;- < % ´| [[\ [[] ^ £ ^ ^ . ; » =! @; /| | =! > # ; | # ; > # */}

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½ z� ?Dcn� ½ �]|P22 z £�^ � ½ �Q ^ S_2 z QO����HQ ?Dcn� ½ �]»P2�=<P&� º"`� �S-j ?Dcn� ½ �]»P2�=<P&� º � �a>��P \]HI[eC:?D9�PO� � H-Y�?D^<HG�LH_U@P&?DC:?B9�P±=<P@�oY<HIPLH_8"=<A�6@8:9<6*H_8:C:?DHIJg9<c2C:ZLH-c�9�ABAD9 � ?BPLY�?BPLHI`]K&=<AB?DCEsutwv x y p z_{ |}f ~ W =AX ¼�8 �/pp ;Np z � M?> �3bdc.0 =AX ¼¹� fª�V���P�6@=>8:C:?T[_K&AB=>8 � H­6&8"9�^<H­C:Z@=>CG?BPLHI`]K&=<AB?BCbE�bdc.0 =AX ¼ J:=>C:?BJVU@HIJGJV9�F/H­9<cgC:ZLH­6@8:9<6*H_8:C:?DHIJ-J:Z@=>8:HI��;]E��29�?BP&[_=>8�H9<8 � 8:9�J:JgAD9<Y�=>8�?DC:Z&F/?B[u\]9<;*9�ADH_^­?BP&HI`aK&=<AT?DC:?DHIJN�V������9<8ª�����:��mLP&=<F/HIADE/CVHIP&JV9<8�?BJ:=>C:?D9�P±=<P&�±6*H_8:C:KL8:;@=>C:?B9�P��CL49<CVHC:Z&=>C�?BPkC:ZLHq[_=<J:H º®� �>m � H¬U@P&� ?TPLHI`aK@=<AB?DC:?DHIJ�K&JVHI�k;aE ³´9<;&µ<9�^��b©¢HI�L9�K�d7?BP � ³g©2·����-=<P&� c�9<8 º®�l�?BPLHI`]K&=<AB?BCbEeb4c.0 =YX ¼¹� fG�´?BJ�HedL=<[eC:ADE�C:ZLH � 8:9�J:J4AD9<Y�=>8"?DC:Z@F/?B[-\]9<;*9�ADH_^�?TPLHI`aK@=<AB?DCbE<�f H�6&8:HIJ:HIP�C�=<ABJV9o=o[e9�P&[eHIPaCV8"=>C:?D9�P�6&8:9<6*H_8:CbE � Z&?B[�Zq?TJ(=<�@=>6&CVHI�¬CV9oC:Z&?BJu?BPLHI`]K&=<AB?BCbE<�hg 9<8:H16&8:HI[_?BJ:HIADE<m¹?DcW=F�HI=<J:K&8:H�M«J:=>C:?BJVU@HIJ�C:ZLH­?BP&HI`aK&=<AT?DCbEibdc.0 =AX ¼ � fG��m � H­Z&=�^<H­C:Z&=>Cª?Bc´p�?BJ-= ©¢?D6@J"["Z&?DC,+1c�K&P&[eC:?D9�P¬9�P�Q � ?DC:Zj p jlk6m : | ��C:ZLHIP�mLC:ZLH_8"H-?BJ�noOP��J:K&[�ZoC:Z&=>Cnc�9<8NH_^<H_8"ETpqOP��9�P&H-Z&=<J

M ¼�8 p�� ~ p � M ¼ »rp ; | Hed�6 y ��n�F/?BP y p ¼ �sp zI{I{ > � ���

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� Z&?TJn?BPLHI`]K&=<AB?DCE � =<J(6&8:9�^<HI�q;aE � =<AB=>Y<8"=<P&�¬?BP � � =<A�·��<m � =<AT·<�°�S=<P&�q=<ABJV9��&HIJ:[e8"?D;*HI� ;]EqgO=<KL8:H_E � ?BC:Z C:ZLHJV9/[_=<ABABHI��6&8:9<6*H_8:CbE¬���@�´?BP � g =<K ·��_� m&³´9<;&µ<9�^a�b©¹HI�&9�K�d ?BP�� ³g©r·����¢JVC:K@�LE­C:ZLH�6 =>8:C:?B[_K&AB=>8�[_=<JVH­� º � �����´©¹H_CK&J�PL9<CVH­C:Z&=>C-C:ZLH­[_=<J:HIJGº�» �o=>8"H/JVC:K&�&?DHI�¬;]E¬³�9<;@µ<9�^a�b©¹HI�L9�K�d�?TP«� ³´©����>� m¢8"HIADE]?TPLYoF/=<?BP&ADE 9�P�³´8"K&P&P��gO?BPLµ<9 � JVµ]?L?TPLHI`aK@=<AB?DC:?DHIJ_m�=<P@��;aE1³´9<;&µ<9�^a���wH_Y�=>8"AT?BP&JVµ�?@?BP � ³��N�>¸�� � Z&?B[�Z18:H_U@PLH�C:Z&Hn8:HIJ:K&ABC:J�6&8:HIJVHIPaC:?BPLYLma^]?B=� =>8"�LE F JG?BP&HI`aK&=<AT?DCbE<m�JV9�F�H�PLHI[eHIJ:J:=>8"E�=<P&� J:K��­[_?DHIPaCª[e9�P&�@?DC:?D9�P�c�9<8�F�HI=<J:KL8"HIJª9�P�C:ZLH±8:HI=<A�AT?BPLH<� ©¹H_C1K&JPL9<CVH-CV9­U@P&?BJ:Z±C:Z&=>CNC:ZLH_E±K@JVH<m&c�9<8nC:ZLH-[_=<JVH1º�» �]m W ^ � ½ ���i� ½ � ^ � ?DC:Z �R�>ºTS �R�UQ¬� �>���P}\]HI[eC:?D9�P7�]� �]m � HOHed�CVHIP&� )(CVCV9>��u?BABAB=<P&? F J�C:ZLH_9<8:HIF ��JVH_H � )�����<���/c�9<8�C:ZLHq8:HIAB=>C:?D9�P � ?DC:Z AB9<Y�=>8"?DC:Z&F/?T[\]9<;*9�ADH_^ ?BP&HI`aK&=<AT?DCbE =<P&�qCV8"=<P@JV6*9<8:C:=>C:?D9�P¬?BPLHI`]K&=<AB?DCE<�-©¹H_C�K&Ju�LH_U@P&H b =YX ¼O;aEqb =YX ¼ � W &=YX ¼ m*C:ZLH��LHIP&[�ZLHIA �©¹H_Y<HIP@�L8:HNCV8�=<P&JVc�9<8"F 9<c W =YX ¼w� f H46@8:9�^<HuC:Z&=>C´?Dc¹=-6&8"9<;@=>;@?BAB?BCbE1F�HI=<J:KL8:H4M 9�P±Q�R J:=>C:?BJ:U&HIJ�C:ZLHu?BPLHI`]K&=<AB?BCbEbdc?0 =YX ¼ � fª�NC:ZLHIPqC:ZLH_8:H�=>8:H 2� ?O ��=<P&��� O �±J"K&["ZqC:Z&=>C�?DCuJ:=>C:?BJVU@HIJ(=<ATJV9o=�CV8�=<P&JV6*9<8:C:=>C:?D9�P¬?BPLHI`]K&=<AB?BCbE��c�9<8N=<ABA*c�K&P&[eC:?D9�P�� 9�PoQ R m@�&HIP&J:?DCE�9<c26&8:9<;@=>; ?BAB?DCE � ?DC:Z�8:HIJ:6wHI[eC�CV9�MSm� k ����� ¿ ��� � MS� � Mr�g|�� s�twvIx ��G� � � � = � X ¼ ���o�V�� ZLH_8"H � k ����� ¿ ��� � MS� � Mr�W�7?BPLc��W~ob = � X ¼ � ½ ���L� � ¡�� ½ � �&�"!��� ZLH_8"HNC:ZLH(?BPLU@FXK&Fi?BJ�C:=>µ<HIP±9�^<H_8�C:ZLHuJVH_C´9<c�6&8:9<;@=>; ?BAB?DC:?DHIJSF/HI=<J:KL8:HIJ´¡ 9�P�QSR$#1QSR J"K&["Z�C:Z&=>Cg¡OZ&=<J�C � 9F/=>8:Y�?TP&J%� � M«=<P@� � MS� � Z@?BJu?BPLHI`]K&=<AB?DCE � =<JG?BPaCV8:9]�@K&[eHI�¬;aE � =<AB=>Y<8"=<P&��?TP�� � =<AT·��°��c�9<8-C:Z&H/[_=<JVH�ºk� �=<P&�oº � �>�W©¹H_C�K&JgPL9<CVH�C:Z@=>C´C:Z&Hu[_=<J:Hªºq� � � =<J�=<ABJV9�JVC:K&�@?DHI��?TPq� ³ � ©��L�_� � ?DC:Z�Hed�=<[eC:ABE�C:Z&?TJ�c�9<8"F¯=<P&�C:ZLH-[_=<JVH1º�»k� � =<J4JVC:K&�&?BHI�o?BP�� � HIP��L�e� ���Pk\]HI[eC:?D9�PE� � HO6&8:9�^<H<m�=<J�?BP�� ©C)����<� m´C:Z@=>C/C:ZLH F�HI=<J"KL8:H M¹¼ �LH_U PLHI� ?BP � ���/J:=>C:?BJVU&HIJ/C:ZLH ?BPLHI`]K&=<AB?BCbEbdc?0 =YX ¼�� fª���7g 9<8:H±6&8:HI[_?BJ:HIADE � H±6&8:9�^<H±C:Z&=>CGC:ZLH_8:H�?BJ'&G� n O �qJ:K&[�Z�C:Z&=>CXM¹¼�J:=>C:?BJ:U&HIJªc�9<8G=<ATAWJ:F�9]9<C:Zc�K&P&[eC:?D9�PoJ:K@["ZoC:Z&=>C(pO»E�/=<P&� � p z � M¹¼­� �>m

s�twv x ¿ y p ze{ |(& ��� x ¿ � p¹� S_n ~�)+* z-,,,,p� p ,,,,^ p z � M¹¼N>

. KLH-CV9±C:ZLH-c�=<[eC4C:Z@=>CNM ¼ HIPHGV9�E�J �r9�?TP&[_=>8B�HX?TPLHI`aK@=<AB?DCbE<mLM ¼ J:=>C:?BJVU&HIJ4=<ABJV9±?BPLHI`]K&=<AB?DCE bdc?0 =YX ¼ � fª�gc�9<8uJV9�F�H[e9�P&JVC:=<PaC:J(f9OP�/=<P&� 2%OP���)uKL82F�H_C:ZL9��X8:HIAB?DHIJr[e8"K&[_?B=<ABABE(9�P � =>8"�LE F J�?BPLHI`]K&=<AB?DCE � H´8:HI[_=<ABA�PL9 � �rADH_CrMS�0/ª;*H�C � 9�U@P&?DCVHW³´9<8:HIA�F�HI=<J"KL8:HIJ9�PoQ � � � Z&HIPoC:ZLH�;*HIJVCn[e9�P&JVC:=<PaC1& JV9/C:Z@=>CnH_^<H_8:EoJ:F�9]9<C:Zoc�K&P&[eC:?D9�P p¬J:=>C:?TJVU&HIJ~324 � pS� ½ �2��pS�3���V� z � MW� ½ �g|�&¬~324 p z � / �����

?BJgU@P@?DCVH�?Dc2=<P@�o9�P&ADE±?Dc n � J:KL6§ * 4 MW�V� ½ ��j�� � ~ §4 8 � / =05�76 ; £98 �:6 � �<�?BJ(U@P&?BCVH<m � ZLH_8:H�/ =05 ?BJ(C:ZLH�=>;@JV9�ATKLCVHIADE [e9�PaC:?BP]KL9�K&J(6@=>8:Cu9<c;/ � ?DC:Z 8"HIJV6*HI[eC(CV9oMS� g 9<8:H_9�^<H_8Im � ZLHIP�& ?BJU@P&?BCVH � HªZ&=�^<H n¯|(& |«¸/n >� Z&?TJN?BPLHI`]K&=<AB?BCbE � =<J(6&8"9�^<HI�q;aEig K&["µ<HIP&ZL9�KL6@C1� gOK&[��>�>� m*9�PLH1[_=<P¬JVH_H1=<ABJV9 � 54³����.����m*³ � �/�>��c�9<8(?BPaCVH_8V�HIJVC:?BP&Y�8:H_^]?BH � =<P&� =>6&6@AT?B[_=>C:?D9�P�9<c2C:Z&?BJ�8:HIJ"K&ADC_���?BP&=<ABABE ?BP¬\�HI[eC:?D9�P¬¸ � H � ?BATA¢6&8"HIJVHIP�C�JV9�F/H1?TPLHI`aK@=<AB?DC:?DHIJ4J:=>C:?BJVU&HI�q;]E 9<C:ZLH_8-F/HI=<J:KL8:HIJ_� g 9<8:H�6@8:HI[_?BJVHIADE<mADH_C1<�;*H�C � ?B[eH-[e9�P�C:?TPaKL9�K@J:ADE��&?>=*H_8:HIP�C:?T=>;@ADHu=<P&� PL9<CVH-C:Z&H-6&8:9<;@=>; ?BAB?DCE­F�HI=<J:KL8"H�M@?±;aE<m

M@?�� ��½ ��� �¾ ¥ £ ?BA §"C ��½ > ��@��5(F�9�PLY�C:ZLHIF ?BJ�[e9�P&J:?B�&H_8:HI�<´� ½ ���i� ½ � ¼ ��AD9<Yw� ��S�� ½ ���V� ^ � � ?DC:Z ºI�*�b�>�"�L� � QM�oQ��

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� Z&?T["Z�HedLZ&?D; ?DC:J-=qF�9]�@?DU&HI��AB9<Y�=>8"?DC:Z&F/?T[�\]9<;*9�ADH_^ ?BPLHI`]K&=<AB?DCE¬9<cnc�K@P&[eC:?D9�P W ���@?>=wH_8"HIP�CX?BP�P&=>C:K&8:H�c�8:9�FW =AX ¼L��mL=<P&� � Z&?T["Z­?TJ�PL9<Cg[e9�^<H_8"HI��;aE­©r=>C:=�( AB=°�*)uADHIJ:µ]?DH � ?B[:µB+4?BP&HI`aK&=<AT?DCbE<� f H�=<ABJV9G6&8:HIJVHIPaC�HedL=<F�6@ABHIJ � Z@?B["Z=>8:H�K&Pa;*9�K&P&�LHI�O6*H_8:C:KL8:;@=>C:?B9�P 9<c�M ¼ � f H�C:ZLHIP �LH_8"?B^<H1P&H � [e9�P@[eHIP�CV8"=>C:?B9�P�?BPLHI`]K&=<AB?BC:?DHIJN?BPqC:ZLH�JV6 ?D8"?DC49<c� =<AB=>Y<8"=<P&� =<P@�qg =<K&8:H_E�9<84³�9<;@µ<9�^a�b©¹HI�L9�K�d©¹H_CuK&JNU@P&=<ABADE�[e9�F/F�HIPaCuC:ZLHG[_=<J:HX9<cSY<HIP&H_8"=<A�[e9�P�^<Hedq6w9<CVHIPaC:?B=<A < =<P&�OMW� ��½ �´� ¥ £ ?BA §"C ��½ � � ZLHXP&=>C:KL8"=<AHed�CVHIP&J:?D9�PG9<c&9�KL8�F�9��&?DU&HI�GAD9<Y�=>8"?DC:Z&F­?B[�\]9<;*9�ADH_^-?BP&HI`aK&=<AT?DCbE � 9�K&AB�-;*H�CV9u[e9�P@J:?B�LH_82=Nc�K&P&[eC:?D9�P W ;*HIZ&=I^�?BPLY`]K&=<�L8"=>C:?B[_=<ATADE«PLHI=>8oC:ZLH 9<8"?BY�?BPk=<P@� AB?Dµ<H <�&���C:ZLH�©¢H_Y<HIP&�L8:H¬CV8"=<P@JVc�9<8�F 9<c�<���c�9<8 AB=>8"Y<Hq^>=<ABKLHIJI� � ZLHHed�CVHIP&J:?D9�P ?TJ�ZL9 � H_^<H_84;]E�PL9 � =�EoCV8"?D^�?B=<A*=<P&�o8:HI`]K&?D8"HIJ�=>6&6&8:9<6&8�?B=>CVHuCVHI["Z&P&?T`aKLH-[_K&8:8:HIPaC:ADE±K@P&�LH_8nJVC:K&�&E<�� � +-,/3����W, �:+���)¹3�' ��� 3"0��ª+���+������ 3"%����G.���"3�'�3������l,����­%�3�'*3"+4% n%1, ���W%��

��) ��! 1)�+ "��)w'�3����#�$&% ')(+*-,/.102.&34,/576 ,98;:+<16 5=5=.&:>6 <�?9@A3B?C(D@20A.&(E5©¹H_Cuº_� � �>�"���S=<P&�qQ�»}�/J:=>C:?TJVc�E�?BPLY �R�>º7S �R�UQ��i�G=<P&� ADH_C 2 Or����©¢H_C4�LH_U@P&H-C:ZLHªc�K&P&[eC:?B9�P&J�b =AX ¼ =<P&�W =AX ¼ �� c�º �*�b�>�"��� � HªPL9<CVH

b =AX ¼*� ½ ��� Z[\[]½ z� ?Dc�� ½ ��|E22az £ ¼ � ½ �º ¼ S_2az ºq����<º ?Dc�� ½ ��»E2

=<P&�W =AX ¼*� ½ ��� Z[[\

[[]

½ z� ?Dcg� ½ ��|P22 z £�^ � ½ �Q ^ S_2 z Qq����HQ ?Dcg� ½ ��»P2

� c�º¬� � � HªPL9<CVHb =AX 8 � ½ ��� Z[\

[]½ z� ?Dcg� ½ ��|P22*� ½ �>� 2 z� ?Dcg� ½ ��»P2 =<P&� W =AX 8 � ½ ��� Z\ ]

½ z� ?Dc�� ½ ��|P2j ?Dc�� ½ �OP2©¹H_C # �T$'&�=<P&� ½ � � ½ 8 �GFGFGF¢� ½ R �4� QSRrm � HªPL9<CVH

b A R C=AX ¼ � ½ ��� RH mJI 8 b =YX ¼¹� ½ m �g=<P@� WA R C=YX ¼ � ½ ��� RH mJI 8 W =YX ¼�� ½ m �s>L49<CVH/C:Z&=>C � ZLHIP¬C:Z&H_8:H/?BJuPL9 =<FG;@?DY�K&?DCE � H � ?BABA2�L8:9<6 C:Z&H/�LH_6*HIP&�LHIP&[eH�?BP # =<P&��PL9<CVH b =YX ¼¬?BP&J:CVHI=<�¬9<cb A R C=AX ¼ �©¹H_C4K&J��LH_U@PLH�C:Z&H-AD9<Y�=>8"?DC:Z&F­?B[-\]9<;*9�ADH_^±?TPLHI`aK@=<AB?DCbE�9<crc�K&P&[eC:?B9�P W =YX ¼w�

KMLON tQP�vRPTS tVUXWZYV[]\G^ M`_ \ba�cedgf _ a _ihTjkh ^ml!no\�aqpsrtd�\7f=u QSR4v�M pia2^ h pxwy\ipoa j f�z{a2d h ^T|{n hm}M~ f _ f j \G� h ue\��GrDa jkh ^mlfg���sr>u } ^ h f=u W =AX ¼�� h ^T| } f=uDps^1a=uE^ f�v u�f2^&\�� bdc.0 =AX ¼ � fª��v/h ���Gf=d�\G�=\idsl f 8 �sr>u } ^ h f=u p�OE� p�r } |M^T|>a2^�\i�=\Gd�lh uE^&\1zAdga j p�\�� h ps^Tp�f=ue\�|>aqp

sutwv x y p z {-|}f�~ W =AX ¼ 8 �/pp ; p z � MS� �3bdc.0 =AX ¼¹� fª�V�� |>\Gd�\ W =YX ¼�8 �/pp ;7� RH mJI 8 W =AX ¼�8��

p� ½ m

�p ; >f Hª�LH_C:=<?TA¹JV9�F�H-6@8:9<6*H_8:C:?DHIJ´9<c?b =YX ¼o=<P@� W =YX ¼o?BP�C:Z&H�c�9�ABAB9 � ?BP&Y1ABHIF/F/=��

¸

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�yL����M� UXW U��Xr>u } ^ h f=uDp b =AX ¼ a=u�� W =AX ¼ pGa2^ h pTwy\sp��h ��x� ºI�*�b�>�"���Zv.b =YX ¼ a=u�� W =AX ¼ a=d�\ f 8 f=u Q�hTh � b &=YX ¼ � W =AX ¼�v � |>\id�\ b &=YX ¼ h p7^T|>\��]\iu } |>\ j� [4\1zt\iu��2d�\ ^mdga2u+px�Gf=dsn fg� b =AX ¼��� � } f2rtd�pG\ � \"|>a=�=\ a j pGfW &=AX ¼ � b =AX ¼�hxhTh ���4f=d�a jTj 6 OP� f2ue\�|taqp/�Gf=d�a jxj ½ �oQ

b =YX ¼ � 6½ ��� 6 z b � � X ¼ � ½ ��� W =YX ¼ � 6 ½ ��� 6 z W � � X ¼ � ½ �s>h ����[]\G^ ��|P2�|E2 v f=ue\�|>aqp/�Gf=d�a jTj ½ � Q

b =AX ¼�� ½ �g|Pb = � X ¼ � ½ ��� W = � X ¼ � ½ �g| W =YX ¼�� ½ �s>����x� ºI�*�b�>�"���Zv.b =YX ¼ a=u�� W =AX ¼ a=d�\�ps^md hm} ^ j l } f2uD�=\��Ma=u��opia2^ h pxw-\sp

AB?BF¦ §�¦ � 2W =AX ¼�� ½ �½ � AB?BF¦ §�¦ � 2

b =YX ¼¹� ½ �½ ��j">� ZLH�=<J:J"K&F�6&C:?D9�P@JrY�?B^<HIP�9�P/º =<P@�1Q =>8:HnJ:?DY�P@?DU@[_=<PaC29�P@ADEªc�9<8�[e9�P&�&?DC:?B9�P����@ma=<P@��[e9�P&�&?DC:?D9�P��X?BJ�J:?DY�P&?BU@[_=<P�Cc�9<8n³�8"HIP&?DH_8V� gO[R�g=<P&P�� � =<PLY<;*9 F J�C:ZLH_9<8:HIFom � Z&?B[�Zo?BJ�[e8"K&[_?B=<A*c�9<8�C:ZLH-JVC:K@�LE�9<c2C:ZLH-AB?TPLµ/;*H_C � H_HIP F�9��&?DU&HI�AD9<Y�=>8"?BC:Z&F/?B[-\]9<;*9�ADH_^±?BPLHI`]K&=<AB?DC:?DHIJg=<P&� CV8"=<P&JV6*9<8:C:=>C:?D9�Po?TPLHI`aK@=<AB?DC:?DHIJ´9<c�C:ZLH-PLHed�CNJVHI[eC:?D9�P¹�� H_8:Hª=>8:H-JV9�F/H-6&8:9<6*H_8:C:?DHIJg9<c2C:ZLHª?TPLHI`aK@=<AB?DCbE�bdc.0 =AX ¼ � fª����� S���S � P�vRPTS t UXW"! h#%$ | h p-c�d�f�c+\Gds^ml h p'&Ru�f � u)r>u��O\id�^T|>\�uea=no\�fg��^&\iuDpif=d h pGa2^ h f=u

[]\G^ M 8 a2u�� M z ^ � f�cedgf _ a _ihTjkh ^mlMno\�a=p�r>d�\ip�f=u Q R)( a2u�� Q R ¨ ~ rRcAc>fqp \�^T|ta2^ M 8+* d�\sp&c 4M z-, pGa2^ h pxwy\ip^T|+\ h ue\�� rDa j h ^ml b4c.0 =YX ¼�� f 8 � * d�\ip1c 4b4c.0 =YX ¼�� f z � , ^T|>\Gu ^T|>\Cc�d�f _ a _ihTj h ^ml M 8�. M z f=u Q�R/( �wR ¨ v pGa2^ h pxwy\iph ue\��GrDa jkh ^ml bdc.0 =AX ¼ ������v � |>\Gd�\ �¯�7F­=°d�0°f 8 ��f z�1 hxh#%$ | h p-c�d�f�c+\Gds^ml h p&Ru�f � u)rtue�{\id�^T|+\�u�a=no\�fg�-c+\Gds^mr>d _ a2^ h f=u

[]\G^ M a"no\�aqpsrtd�\�f2u QSR piaA^ h pT��l h uOz b4c.0 =YX ¼ � fª�2 []\G^43 a _ f=r>u��{\�� ��r>u } ^ h f2u�f=u QSR a=ue�M�{\mw�uX\��65Ma=p � 5MO� ¥87¾ � MS�� |+\id�\ ¾ � � ¥ 7 � M9$ |+\iu ^T|>\7n�\�aqp�r>d�\65M pGa2^ h pTwy\sp7^T|>\ h ue\��GrDa jkh ^ml bdc?0 =YX ¼¹���o� � h ^T| � � f ¥�z2:<;#= A 7 C v � |>\Gd�\ fqp }�� 3 �1�J"KL6¢� 3 ����?BPLcI� 3 �2

hThxh# [ h u>& _ \G^ � \�\iu bdc?0 =YX ¼�� fª� h uX\��GrDa jkh ^ml � h ^T|�?�f h u } a=dA@\ h ue\�� rDa j h ^ml []\G^ M a n�\�aqp�r>d�\of2u Q�RA �x� M pGa2^ h pxwy\ip b4c.0 =YX ¼ � fG��v ^T|+\iu M pGa2^ h pTwy\sp�aB?�f h u } a=dA@\ h ue\��GrDa jkh ^ml � h ^T|^T|+\ } f=u+pi^1a=uE^ f��>�C [4\ ^/rtp d�\ } a jxj ^T|ta2^ M pGa2^ h pxwy\ip�a�?�f h u } a=dA@\ h ue\�� rDa j h ^ml � h ^T| } f=uDps^1a=uE^ f��>� h �

��� x � p¹�´| f � ~ � �/p�� z � MS� � ·���Gf=d�a jxj psn�f f2^T|��sr>u } ^ h f=u p4

D�EGF�F�HI )uP&H�[_=<P�U@P&��C:ZLH��LH_C:=<?BABJG9<cnC:ZLH±6&8:9]9<c�9<cnC:ZLH±6&8:9<6*H_8:C:?DHIJG9<cnCVHIP&J:9<8"?BJ:=>C:?D9�P =<P&��6*H_8:C:KL8:;@=>C:?B9�P�=<P&�C:ZLH�?BF�6 AB?B[_=>C:?D9�PO9<cWC:ZLH �29�?BP&[_=>8�H�?BPLHI`]K&=<AB?DCE ?BPq["Z&=>6&CVH_8�J/��=<P&� �±9<c(� 5(³����.���>�N� \]HI[eC:?D9�P �>� �]���]�Dm � ZLH_9>�8:HIF �]� �]�B�G=<P&� � Z&H_9<8:HIF �]��¸L�������KJL49<CVH-C:Z&=>CnC:ZLHªCVHIP&JV9<8"?BJ"=>C:?D9�Po9<c2C:ZLHª�WPaCV8:9<6]E±?TJ � HIATA*µ]PL9 � Po6&8"9<6wH_8"CbE��&?BJ:[_K&J"JVHI�±;aE�©¢?DH_;O?BP�� ©¢?DH��>��� �

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�"L�� �� �� U W���� \on�a=l fg� } f2rtd�pG\7�O\mw�ue\ j f�zOa=d h ^T|{n hZ}7~ f _ f j \G� h ue\�� r+a j h ^ml!fg�C�sr>u } ^ h f=u W v � |>\id�\ W � ½ �h p � rDaA�=d�a2^ hm} �if=d p�n�a jTj �qa j rE\sp f�� � ½ � a=u�� � h ^T| } f=uD�2\�� v �Gaqpi^&\id ^T|ta2u �GrDaA�=d�a2^ hZ}iv z2d�f � ^T| �Gf=d j a=dZzO\ � ½ � ~ \�\ ~ \ } ^ h f=u��o�Gf=d�p�r } |�\��Oa=nc j \sp fA^&\�^T|taA^ ?Cdgf�c+fqp h ^ h f2u�� � h p�fg� } f=r>d�p \�pi^ hxjTj �qa j h � �Gf=d�^T| h p & h u�� f��h ue\�� r+a j h ^ml $ |>\sp \ h ue\��GrDa jkh ^ h \sp�a2d�\�a j pifbpi^mrD� h \�� h u)a�zO\Gue\id�a jQ} aqpG\ h u��T[]\�������� h u ?Cdgfsc>fqp h ^ h f=u�� � 5(J2?TP�� ©C) ���Lm°³g©2·���� m<;aEGK&J:?BPLY � H_8:; JVCYF J�=>8:Y�K&F�HIPaC_m<9�PLHg[_=<PGY�?B^<H´6@8:HI[_?BJVH�HIJVC:?BF/=>CVHIJ�=>;*9�KLC�[e9�P&[eHIP�CV8�=>C:?D9�P���� S���S � P�vRPTS t UXW���� p�p�r>no\"^T|ta2^ ^T|>\ c�d�f _ a _ihTj h ^ml no\�aqpsrtd�\ M f=u Q pGa2^ h pxwy\ipb^T|>\ h ue\�� r+a j h ^ml b4c.0 =YX ¼�� fG�2[4\ ^ � _ \�a�sr>u } ^ h f=u)f=u Q�R p�r } | ^T|taA^�� ��v j � m � j |��>v ^T|>\Gu � \ zt\G^E�if=d p »E�tvM R � ,,

��� M R ���G� ,,»rpw�g| Z[[\

[[]�2Hed]6 8 � � ¼# ¼ £98 � ¼ � p­��2Lf # �@� �(��º��V� ¼ ��2]z ����º�<º ; h � p » = � R�!z ��2Hed]678¢� �/p z# f"� z ; f2^T|>\Gd � h pG\ v

� |>\Gd�\ � ¼­� � ¼ � ºq�«���08V£ ¼ 2 z £ ¼º2f ¼ £98 D�EGF�F�HI ©¹H_CuK&JnU@8"JVCn6&8"HIJVHIP�C4C:ZLHX6&8:9]9<c � ZLHIP # �i�>�454J:J"K&F�H<m � ?DC:ZL9�KLC4AD9�J:Jn9<cSY<HIP&H_8"=<AB?DCE<m C:Z&=>C � � � M � ���©¹H_CGK&J�8:HI[_=<ABA�;&8"?DH$#&E � H_8:; JVCYF Jª=>8:Y�K@F�HIP�C±��JVH_Hq� 5(³�� � ���>�Wc�9<8ªF�9<8:H­�&H_C:=<?BABJ���� . HIPL9<CVH&%�� 6 �(� � ¥('*) � MSm=<P&��8:HIF­=>8:µ�C:Z&=>C bdc?0 =YX ¼�� fª�´=>6&6 AB?DHI��CV9�p z � ¥ '*) m K&J:?BPLYX;@=<J:?T[u6&8:9<6*H_8:C:?DHIJg9<c W =AX ¼ mLE�?DHIAB�@JgCV96 % � 6 ����%�� 6 �]AD9<Y+%�� 6 �g|kf W =YX ¼ 8 6 �� ;�%�� 6 � ���Y���� Z&?T["Z�mL�LHIPL9<C:?TPLY � � 6 �S� ���R� 6 �]AD9<Y,%�� 6 ��m&HIP�C:=<?TABJ

� � 6 �´| f6 z W =AX ¼%8 6 �� ; >� ZLHIP¹m�?TP�CVH_Y<8"=>C:?TPLYLm@=<P&� K@J:?BPLY � �3����� � � � Mq� ��m � H-9<;&C:=<?TP%-� 6 �g|«Hed]678wf 6 ~ '4 �- z W =AX ¼ 8 - �� ; � - ;M> ���<���� ZLH-©r=>6@AB=<[eH�CV8"=<P@JVc�9<8�F�9<c�� ?BJgC:ZLHIPo;*9�K&P@�LHI�±;aE

%�� 6 �´| Z[[\[[]Hed]6 8 f 6 ^ � ^ 2 z £�^� ^ Q��VQq�«��� S«f 6 2.� Q ����L�VQq�«��� ��f�2az Q ����HQ ; ?Dc 6 » z =! �Hed]6 8 f � z 6 z@ ; ?Dc��/| 6 | z =! >�L9<84C:ZLH # �b�&?BF�HIP@J:?D9�P&=<AwHed�CVHIP&J:?D9�P�m&K@JVH�C:ZLH-CVHIP&J:9<8"?BJ:=>C:?D9�P�6@8:9<6*H_8:CbE±9<c?bdc.0 =AX ¼±=<P@�RH mJI 8 W =YX ¼�8

6� � m ��; | # W =YX ¼48 6 �� ; >� ZLHIP � Hª[_=<POK&JVH�C:ZLH-[_=<JVHG9<c2�@?BF�HIP&J:?B9�Pq� � ?DC:Z�C:ZLH-[e9�P&J:C:=<P�C�f 8:H_6@AT=<[eHI�o;aE f # ��J

�"L�� �� �� U W*/`[]\G^ rtp]c�d�\ipG\GuE^Qa p h nc j \�a�c{c j hZ} aA^ h f=u f��-^T|+\�ced�\ } \�� h uOz ?Cdgfsc>fqp h ^ h f=u ^1f��{\G� h a2^ h f=u h ue\��GrDa jkh ^mlfg�)^T|>\ \inc h d hm} a j no\�a=u;fg��a �srtu } ^ h f=u 10 f=uDp h �{\id�^T|+\ d�\�a j �qa j rE\�� ��r>u } ^ h f2u p]v � h ^T| � p�  ��| �qv a2u���­� ½ 8 �Y> > >D� ½ R ��� 8R32 R 4 I 8 pS� ½ 4 ��v � | hZ} | h u+|>\id h ^Tp-^T|>\�c�d�f�c+\Gds^ml�^T|ta2^ � � m ����| �R� # v � \�^T|{rtp�zO\ ^B^T|>\B�if jTj f � h utz5 f \76� h uOz"^ml�c+\ h ue\��GrDa jkh ^ml v´�98 m � 8�: m :&R _ \ h uOz h u��{\ c+\iu��O\iuE^Ca=u�� h �{\GuE^ hm} a jTj lM� h ps^md hZ_ r>^&\��Ma }�} f=d�� h utzM^1f M¹¼ev;�< �# ,,,,,

RH4 I 8 pS�98 4 �2��M¹¼*� p¹� ,,,,,Orp>= | Z[[\

[[]�2Hed�6 8 � # � ¼�� p�� 2Lf/� ����º��V� ¼ ��2 z ����º�<º ; h � pO» = �z ��2Hed�6 8 � # �/p zf ; f2^T|>\Gd � h pG\

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�"L�� �� �� U W�� f2^&\ ^T|taA^�^T|+\)f _ ^1a h ue\��7�Gf=d�n F/?BP¹� p ¼ �sp z �Mh p u�a2^mr>d�a j�h u d�\1zOa=dg� ^1f��Ca=rtp�p h a=u ascAc�d�f � h n�aA^ h f=u ��u��O\�\�� v"} f=uDp h �{\id v �if2d \��Oa=nc j \ v �­� ½ 8 �Y> > >D� ½ R �o� 8� R 2 R 4 I 8 pS� ½ m � � |>\id�\ � p�  �u| � � \ |ta=�2\� � m ���L| # £98¤ z � | hZ} | \Gu�a _ij \sp rtp�^1fbd�\ } f2�=\id�^T|>\��Ca=rtp�p h a2u } f=u } \iuE^md�a2^ h f=u

;�< �� # ,,,,,RH4 I 8 pS�98 4 �2� M¹¼�� p¹� ,,,,,

Orp = | ¥ £ z�� ¨ ¤ ����Gf=d�a jTj # » ¸6p z ���32I� z �"L�� �� �� U W����4f=d�zO\iuX\id�a j-j f�zOa=d h ^T|{n hZ} ~ f _ f j \G�!fg����r>u } ^ h f2u W v � \Mn�a=l f _ ^1a h u } d�rD�{\ \spi^ h n�a2^ h f=u f��^T|>\ } f=u } \iuE^md�a2^ h f=u v a2^ j \�a=ps^C�Gf=d j a2dmzt\ p ��ue�{\�\�� v rtp h uOz h ue\��GrDa jkh ^ml *� � , v � \ |ta=�=\ � h d�\ } ^ j l ^T|ta2^b^T|>\} f2u } \GuE^mdgaA^ h f=u _ \i|ta=� h f=d h pQz h �=\Gu _ l�^T|+\ �4\Gu } |>\ j [4\&zO\Gu��=d�\C^md�a=u+px�Gf=dsn fg� W �Gf=d j a=dZzO\-�=a j rD\ip v pG\�\ ~ \ } ^ h f=u ��Gf=d n�f=d�\��{\G^1a hxj p

#�$x# ��. ,����C(D0��"(E(E, .&,/(�����6 <&.10������! =YX ¼#"%$'& 6 ,/8`02@A6 ,95R?C34@2026�0A.&34, .&,�((�)�/6 <&.Z0*�KMLON tQP�vRPTS tVUXW�+`[]\G^ M _ \)a c�d�f _ a _ihTj h ^ml no\�aqpsr>d�\ f=u QSR]vuM piaA^ h pTwy\sp)a ^mdga=uDp1c>f=di^1a2^ h f=u h ue\�� r+a j h ^ml f���srtu } ^ h f=u b =YX ¼ � h ^T| } f=u+pi^1a=uE^ f�v u�f2^&\��$� =AX ¼¹� fª��vh ���Gf=d7\G�=\Gd�l �sr>u } ^ h f=u ��v �{\iuDp h ^ml fg� c�dgf _ a _Ghxjkh ^ml � h ^T|d�\sp1cD\ } ^-^1f M�v f=ue\�|>aqp � k � � ¿ ��� � MS� � Mr�g|kf s�twv_x ���G�w� � � =AX ¼ � fª�V�� |>\Gd�\ � k � � ¿ � � � MS�VMr���7?BP&c ��~ b =YX ¼�� ½ � �L� � ¡�� ½ � �&� ! �� |>\Gd�\�^T|>\ h u w�n�rtn h p ^1a�&2\Gu!f=�2\id ^T|>\ pG\ ^/fg�9c�d�f _ a _Ghxjkh ^ h \ip no\�aqp�r>d�\sp ¡ f=u Q R #/Q R p�r } |7^T|ta2^ ¡ |taqp ^ � fn�a=dZz h u+p � � M a=u�� M9)(CVCV9 =<P&���u?BABAT=<P&?¹6&8"9�^<HI� C:Z&=>C-=oAB9<Y�=>8"?DC:Z&F/?T[1\]9<;*9�ADH_^O?BPLHI`]K&=<AB?DCE ?BF�6@AT?DHIJN=�CV8"=<P@JV6*9<8:C:=>C:?D9�P ?BPLHI`]K&=<AB?BCbE� ?DC:Z�=ª`aK&=<�&8"=>C:?B[�[e9�JVCu��C:Z&?BJ�?BJ�C:Z&HN[_=<JVH(º �"Q¬� ����m]J:H_HG� )�����Lm�³ � ©��L�_� � � ZLH_EX6&8:9�^<HNC:Z&=>C�?Dc@M J"=>C:?BJVU&HIJC:ZLH­?TPLHI`aK@=<AB?DCbEibdc.0-, X z � fG��m´� � ZLHIP º}�¯� C:ZLH­[e9�P&J:C:=<P�C 2q?TJ-PL9<CG=<PaE F�9<8:H­= 6@=>8"=<F�H_CVH_81?BP C:Z&?TJ-[_=<JVH���mC:ZLHIP�M J"=>C:?BJVU&HIJWC:ZLHN?BPLHI`]K&=<AB?BCbE � , X z ��¸afª���?��P � ³ � ©!�L�I� =<PL9<C:ZLH_8W[_=<J:H4?BJ�J:C:K&�&?DHI��m � ZLHIP­º � �n=<P&� Qq��j}���P�C:Z&?BJ�U&8�JVCgC:ZLH_9<8:HIF � H-Y�?D^<HG=<PoHed]CVHIP@J:?D9�Poc�9<8nC:ZLH-9<C:Z&H_8N[_=<JVHIJ_m � ZLH_8:Hªº ��� �>�"��� �.0/ L S qL�� UXWZY1 [4\ ^ M _ \ a"c�dgf _ a _Ghxjkh ^ml�n�\�aqp�r>d�\ f2u Q R a=ue� p�rRcAc+fqpG\ ^T|ta2^ M pGa2^ h pTwy\spM^T|>\ h ue\��GrDa jkh ^mlbdc?0 =YX ¼�� fª�2$ |>\Gu M pia2^ h pxw-\sp�^T|>\�^md�a=u+p&c>f=di^1a2^ h f2u h ue\��GrDa jkh ^ml'� �32¨ X ¼ � f���¸a�2D�EGF�F�HI 5(J-?BP«� ³ � ©��L�e� m � H­K&JVH � =<F/?BABCV9�P��%4�=<[e9<;@?�HI`]K&=>C:?D9�P@J_�/©¹H_CGp�;*H/=o©¢?B6@J:["Z@?DC,+1;w9�K@P&�LHI�qc�K@P&[eC:?D9�P¬9�PQ�Rrm =<P&� JVH_C

5 ' pS� ½ �S�®?BPLc6�7�8 � pS� �L� S 6 b ��2¨ X ¼ 8 ½ � �6 ; ! � 6 OP��� ½ ��Q R � ���I���=<P&� 5 4 p���p2� � ZLHgc�K&P@[eC:?D9�P 5 ' p�?BJ�µ�PL9 � P/=<JSC:ZLH � 9<6&c��b©¢=°d/JV9�ATKLC:?D9�P19<cwC:ZLH � =<F/?BADCV9�P��%4�=<[e9<;@?�HI`]K&=>C:?D9�P

9 � �� 6 � 6 � ½ � � W �32¨ X ¼ � ���L�e� 6 � ½ ��� 6 OP��� ½ � Q R ��w�3��� ½ � � pS� ½ ��� ½ ��QSRr�

JVH_H-c�9<8nHedL=<F�6@ADH­� ³g=>8�·°¸@m@��^>=�·�@°� ��L9<8 6 »P��m �LH_U@PLHuC:ZLH-c�K&P&[eC:?B9�P�:k;]E :(� 6 ��� ~ ¥<; �2�= � ) � M?>

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\�?BP@[eH�p¬?TJn©¢?D6 J:["Z&?BC,+u=<P@�o;*9�K&P&�LHI��c�K&P&[eC:?D9�P�9�P&Hª[_=<P 6&8"9�^<H-C:Z&=>C 5 ' p¬?BJ�=<ABJ:9­=/©¢?B6@J:["Z@?DC,+�=<P&�o;*9�K&P&�LHI�c�K&P&[eC:?D9�P�9�P 6 c�9<8N=<ABF�9�JVCNH_^<H_8:E ½ �oQSR2mLC:ZLHIP :7?BJ�=�f 8gc�K&P&[eC:?B9�P�9�P Q �(�d)uPLH�Y<H_C:J: � 6 ����~ ¸f 5 ' p ¥<; �2�= � ) � M ��~ ¸ 6f W �32¨ X ¼ � � 5 ' p¹� ¥<; �2�= � ) � M

� � 6 s�twv x � ¥ ; �2�= � )�� S �6 :u� 6 �]AD9<Y :(� 6 ��� ~ ¸ 6f W ��2¨ X ¼ �b� 5 ' p¹� ¥ ; �2�= � ) � M©¹H_C4K&JVH-?BPLHI`]K&=<AB?BCbEeb4c.0 =YX ¼ � fG��CV9/C:ZLH-c�K&P&[eC:?D9�P�Hed�6 y z '� 5 ' p*{�CV9/Y<H_C

: � 6 �´| �6 :u� 6 �]AD9<Y :(� 6 �!S f 6 8 ~ W =AX ¼�8 � 6f � 5 ' pN; ¥ ; �2 = � ) � M � ~ ¸ 6 zf z W �32¨ X ¼ � � 5 ' p¹� ¥ ; �2 = � ) � M<;�>. KLH�CV9/C:ZLHª6&8:9<6*H_8:CbE±9<c W =YX ¼���JVH_HG©¹HIF/F/=��]� ����m

W =YX ¼ 8 � 6f � 5 ' p ; � ¸ 6 zf z W �32¨ � X ¼ � � 5 ' p¹�s>� ZLHIP�c�9<8N=<ABA 6 ��� ���I� � mw9�PLH-Z&=<J

W =YX ¼�8 � 6f � 5 ' pN;�| ¸ 6 zf z W �32¨ X ¼ � � 5 ' p¹�s>� ZLHIP � 6 ��� ���I� � � 6 : � 6 �2� :(� 6 �]AD9<Y :(� 6 �g|P�54c�CVH_8N?TP�CVH_Y<8"=>C:?B9�P 9�P�� ���I� � m � HªZ@=I^<H :u������| Hed�6 :  �3���:(�3��� �c�8:9�F � ZLH_8"H ~ ¥ ;2)= ( ) � M�| ¥�� ;2 )�� x > ���A���\�?BP@[eH s�twv x � �G�S�7J:K&6 � ~ �� � MS�u~ ¥� � M�|�� ! �� HªZ&=I^<H � ?DC:Z���� �� 5 8 p�� � �� p � MSm

~ � 8 5 8 p�� ~ p � M ; � M�| f ¸ s�twv_x ���G� >©¹H_CªC:=>µ<H­C:ZLH/J:K&6&8:HIFXK&F 9�P C:ZLH­J:H_C-9<c�©¢?D6@J"["Z&?DC,+1c�K&P&[eC:?D9�P�p2m¢C:ZLH��-=<PaCV9<8:9�^�?B[�Z�� � KL; ?BP&JVCVHI?BP<F JuC:ZLH_9<8:HIF=>6&6@AT?DHI�±CV9/C:ZLHª�&?TJVC:=<P&[eH � k � � ¿ ��� � MS� � Mr��mwJVH_H�� �u?BA �/��� m@?BF/6@AB?DHIJ´C:Z&=>C� k �32¨ X ¼���� � MS� � Mr��| f ¸ sutwv x ���G� >J5(J­?DC­?TJ­=<ABJV9�C:ZLH [_=<JVH¬?BP«`]K&=<�L8�=>C:?B[ [_=<JVH<m � ZLHIP«C:ZLHqF�HI=<J:KL8"H ?BJ­AB9<Y>�b[e9�P&[_=I^<H¬9�PLHq[_=<P 6&8"9�^<HqC:Z&=>C�=CV8"=<P&J:6w9<8"C:=>C:?D9�Po?BPLHI`]K&=<AB?DCE�?BF/6@AB?DHIJg=�AD9<Y�=>8"?BC:Z&F/?B[-\]9<;*9�ADH_^�?BPLHI`]K&=<AB?DCE<��Bf2d7^T|+\Mue\��A^ ^T|>\�f=d�\in � \Mp�rRcAc>f=pG\ ^T|ta2^ ^T|+\ �sr>u } ^ h f=u fg�7^md�a=u+p1c+f=ds^ z h �2\iu _ l ^T|+\M^T|>\�f=d�\Gn fg����d�\Gu h \id �Ca=uOz _ f �� } 0 a=uDu h p"a�� z ��r>u } ^ h f=u ~ r } | a d�\1zAr j a=d h ^ml d�\spsr j ^ h p7f=r+^Tp h �{\7^T|+\"p } f�c+\ fg�b^T| h p�c>a�c+\GdMa2u��� \ d�\m� \id ^1f�� hxjTj a2u h ��� hTj�� � �Gf=d��sr>ds^T|>\Gd � h p } rOp�p h f=u+p!a2dgf=r>u�� ^T| h p�cedgf _Gj \Gn 5 f � \i�2\id � \ p�|tf � |>\Gd�\ v^T|ta2^yf=u } \�^T| h p d�\spsr j ^9a=p�p�r>no\�� v ^T|>\�no\G^T|tf �{f j f�z2l�c�d�\spG\GuE^&\�� h u���f _ &=f=� ��\GuE^ hTj [4\��Af=rA� ���0��[ � � v �if2d ^T|>\\��ic>f=ue\GuE^ h a j no\�a=p�r>d�\ v pi^ hxjTj � f=d & p

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.0/ L S qL�� UXWZYEY [4\ ^ M _ \�a c�d�f _ a _ihTj h ^ml7n�\�aqp�r>d�\�f=u Q R � p�psr>n�\�^T|taA^MW� ��½ ��� ¥ £ ? A §"C ��½

� |>\Gd�\ < h p�a } f2uD�=\�����r>u } ^ h f=u f=u QSR4�x� M pia2^ h pxwy\ip ^T|>\ h ue\��GrDa jkh ^ml � =YX ¼ � fG� ^T|>\iu��if2d a jxj pqOkf�v*M piaA^ h pTwy\sp ^T|>\ j f�zOa=d h ^T|{n hZ}�~ f _ f j \G� h ue\��GrDa jkh ^mlbdc?0 �¨�� X ¼ � � � ¨� £ � � D�EGF�F�HI ©¹H_C4P&9<CVH � �LHIP&J:?DCE�9<cS6@8:9<;@=>;@?BAT?DCbE � ?DC:Zo8:HIJ:6wHI[eCNCV9­MS��54J"J:K&F�H-C:Z@=>C1� ?TJ � z m&C:ZLHªY<HIP&H_8"=<A¢[_=<JVH1[_=<P8:HIJ:K@ADCg;aE��LHIP&J:?BCbE<�³´E­C:ZLH-³´8:HIP@?DH_8V� � =<PLY<;*9>� gO[R�g=<P&P F JgC:Z&H_9<8:HIFom@JVH_H­� ³´8:H�·���m � gO·��<� mLC:ZLH_8:HuHedL?BJVC:J�=1c�K&P&[eC:?D9�P�%}J:K&[�ZoC:Z&=>C

c��"������� W =YX ¼���� %(�CV8"=<P&J:6w9<8"C:J� � M�CV9­C:ZLH-F�HI=<J:KL8"H-MSmLc�9<8NH_^<H_8:EoF�HI=<J:KL8"=>; ADH�;*9�K&P&�LHI�±c�K&P&[eC:?D9�P��

~ �w� c���� � Mq��~ � � M?>� ZLH±c�K&P&[eC:?D9�P�% ?BJ1=ib =YX ¼a�b[e9�P&[_=�^<H c�K&P&[eC:?D9�P =<P&��?Dc % ?BJ � z m�=q[_AT=<J:J:?B[_=<Ag=>8:Y�K&F/HIP�C19<c4[e9�Pa^<HedL?DCbE ��JVH_H["Z@=>6&CVH_8±�q9<c/� �u?BA �/������m´9�PLH Z&=<J � ��� W =AX ¼���� %�� ½ �b�4?BJ��&?B=>Y<9�P@=<AB? +I=>;@ADH � ?BC:Z�8:HI=<A�HI?DY<HIPa^°=<ATKLHIJ_mg=<ABA�ADHIJ:JC:Z&=<P��>�5([_[e9<8"�&?BPLYGCV9GC:Z&H4=<J:J:K@F�6&C:?D9�P�F/=<�LHN9�P�c�K&P&[eC:?D9�P %(ma9�P&H4[_=<P�=<J:J:K&F/HNC:Z&=>C�c�?TJWJ:K���[_?DHIP�C:ABE1J"F�9a9<C:Z±=<P&�� H-9<;&C:=<?BP�c�9<8 ½ �oQ R m ��� ½ � ¥ £ ? A §"C � ¥ £ ? ��� A §"C �LH_C¢� �ec�� ½ �V� > ���e¸a�g 9<8:H_9�^<H_8(C:Z&?BJgc�K&P&[eC:?D9�P Y�?D^<HIJ�C:ZLH-9<6@C:?BF/=<A*CV8"=<P&J:6w9<8"C_m&? ��H<�

� k � � ¿ ��� � MS� � Mr��� ~ob =AX ¼¹� � W =YX ¼���� %n��� � M?>� ZLHIP�;]E����e¸a� m&9�P&H-Z&=<Jnc�9<8 ½ � Q2R2m

AD9<Y �­� ½ ��� <´� ½ ��� <´� ½ ��� W =YX ¼���� %-� ½ �V�<S�AB9<YW�LH_Cr�V����� � � � W =AX ¼��g� %�� ½ �b���s>� ZLHIPoJ"?BP&[eH � ��� W =YX ¼���� %�� ½ �b�¢?BJN�&?B=>Y<9�P&=<AB? +I=>;@ADH � ?DC:Z�8:HI=<A¹HI?DY<HIP�^>=<ABKLHIJIm&=<ABA�ADHIJ:J�C:Z@=<P��>m � H-Y<H_C

AD9<Y��LH_C¢�V���/� � ��� W =AX ¼���� %�� ½ �b���g|��n�&?B^ � � W =AX ¼��g� %�� ½ �V�->\�?BP@[eH < ?TJ�[e9�P�^<Hed � HªZ&=�^<H <´� ½ ��� <�� ½ ��� W =YX ¼ ��� %�� ½ �V�´| � W =AX ¼ ��� %�� ½ �]F�� <´� ½ �g=<P@� � H�9<;@C:=<?BP

sutwv x � �G�´|k~ 0°� W =AX ¼��n� %�� ½ �]F��$<�� ½ �����&?B^w� � W =AX ¼���� %�� ½ �V� 1 �­� ½ � � MW� ½ ���=>c�CVH_84?BPaCVH_Y<8"=>C:?D9�Po;]E±6@=>8:C:J s�twv x � �X�g| ~ ��� F�� W =YX ¼���� % � M?>©¹H_C�pTOP�­=<P&�oADH_CNK&J:H�9�K&PLY­?BP&HI`aK&=<AT?DCbE/c�9<8nC:ZLHG[e9�Fª;@?TPLHI�oc�K&P&[eC:?D9�P&J4b =YX ¼o=<P&� W =YX ¼

p � �� F�� W =AX ¼��n� %7| W =AX ¼�8 p ���� ;IS b =YX ¼�� � W =YX ¼���� %n�s>� Z]K&J

sutwv x ���G�g| �p ~ W =AX ¼�8 p ���� ; � � M S �p ~ b =AX ¼¹� � W =AX ¼��n� %n��� � M|rp(~ W �

�X ¼ 8 ���� ; � � M�S �p � k � � ¿ �� � MS� � Mr� >

·

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� Z]K&J�?DcrM�J:=>C:?BJVU&HIJnC:ZLHª?BPLHI`]K&=<AB?BCbE � =YX ¼�� fG� � HªY<H_Cnc�9<84=<ABA!p7Okfs�twv x ���G�g| p zp���f ~ W �

�X ¼ 8 ���� ; � � M?>

©¹H_C4K&J�PL9<CVHªPL9 � p z � ��m � H-Y<H_Cs�twv x y p*z_{-| p zp­��f ~ W �

�X ¼ 8 � �/pp ; p*z � M

| ¸6p zp­��f ~ W �¨�� X ¼ 8 �/pp ; p z � M?>� ZLHIP�M�J"=>C:?BJVU&HIJ_mLc�9<8N=<ABACpTOkfi?BPLHI`]K&=<AB?DCEebdc?0 �¨�� X ¼ � � � ¨� £ � � � J

�"L�� �� �� U WZYtU � ue\ } a=u psr>n�n�a=d h�� \sp $ |>\�f2d�\Gn � � a=u�� � � _ l ^T|>\�Gf jTj f � h uOz � h a z2d�a=n * r>u��{\Gd"aqp p�r>nce^ h f=u�fg� $ |>\�f=d�\in � � , � bdc.0 =AX ¼�� fª��� � �32¨ X ¼ � f���¸a�� =AX ¼�� fª��� � bdc?0 �¨�� X ¼ 8 ¸6p zp­��f ; ! ��� � > f2^ hm} \ v aqp h ^ h p�^T|>\ } aqpG\9�Gf=d�^T|+\�^mdgaA� h ^ h f=u�a j�j f�z{a2d h ^T|{n hm}�~ f _ f j \G� h uX\��GrDa jkh ^ml v ^T|ta2^-^T|>\id�\ h p�a j fqp�p�a2^y^T|>\j \i�=\ j f��-^T|+\ } f=u+ps^1a2uE^Tp h uo^T|+\�� h d�\ } ^ h f=u7^mdga2u+p1c>f2ds^1a2^ h f=u h ue\��GrDa jkh ^ml h nc j h \sp j f�z{a2d h ^T|{n hm}~ f _ f j \i� h ue\�� r+a j h ^ml �7|>\Gu º �"Qq� �Ov � \zO\ ^�aqp h u � � � � � � v � , X z � fª��� bdc.0-, X z ���A��fG�� � p h u � � � � � � v $ |>\�f2d�\Gn � � } a=u _ \n�f � h wy\�� h u)^T|>\ } a=pG\ 5 \sp�p �<��g»rp �i� v � |>\Gd�\ pq�oQ � j pif j \G^BrOpyu�f2^ hZ} \�^T|ta2^Qaqp h ub^T|+\�� r+a{�=dgaA^ hm} } a=pG\ � \��Af�u�f2^�&qu�f � h �-^T|>\sp \�^ � f h uX\��GrDa jkh ^ h \ipCa=d�\�\��Gr h �qa j \iu�^ 5(Jn?BP ��8"9<6w9�J"?DC:?D9�Po�]���]m ZLH_8"H-=>8:HªJV9�F�Hª6&8:9<6*H_8:C:?DHIJg9<c2C:ZLHª?BP&HI`aK&=<AT?DCbE � k � � ¿ � fª����� S���S � P�vRPTS t UXWZY�! h#0 f=u } \GuE^md�a2^ h f2u h ue\�� r+a j h ^ml � p�psr>n�\ ^T|ta2^ M pia2^ h pxwy\ip)a ^mdga2u+p1c>f2ds^1a2^ h f=u h ue\��GrDa jkh ^ml�� k � � ¿ � fª� ^T|>\Gu M piaA^ h pTwy\sp)^T|>\��if jTj f � h utz

} f=u } \iuE^md�a2^ h f=u h ue\�� r+a j h ^ml� &�� R � � h ^T| MW��&u�g» �� � MW�V� &��I� 5 �g| � ¥ � £ (2 k � � ¿ A � C � �

� |+\id�\ ��&��I� 5 � 0 ½ �oQ R � � ��&G� ½ ��»� 1 hxh# � p h u�?-d�f�c>fqp h ^ h f=u� =v ^T|>\]c�d�f�c+\Gds^ h \sp�fg�y^&\iu+pGf=d h pGa2^ h f2u7a=d�\�a j pGf �qa jkh ���if2dy^md�a=u+p&c>f=di^1a2^ h f2u h ue\��GrDa jkh ^ml� =YX ¼¹� fG�2

[]\G^ M 8 a=u�� M z _ \)^ � f!c�dgf _ a _Ghxjkh ^ml no\�aqpsrtd�\sp f=u Q R ( a=u�� Q R ¨ ~ r c{c>fqpG\�^T|ta2^ M 8 * d�\ip1c M z ,pGa2^ h pxwy\ip)^T|+\ h ue\��GrDa jkh ^ml�� =YX ¼�� f 8 � * d�\sp1c � =YX ¼�� f z � , ^T|>\iu`^T|>\Mc�dgf _ a _Ghxjkh ^ml M 8 . M z f=u QSR/(s�wR ¨ vpGa2^ h pxwy\ip h ue\��GrDa jkh ^ml � =YX ¼�������v � |>\id�\ ���7F/=°d�0>f 8 ��f z 1 hThxh# �x�y^T|>\-no\�aqp�r>d�\ M �=\Gd h wy\sp@� =AX ¼¹� fª��v ^T|>\iu M pGa2^ h pTwy\spCa ?�f h u } a=d @\ h ue\�� r+a j h ^ml � ·�� � h ^T|�^T|>\ } f=u+pi^1a=uE^ f�

D�EGF�F�HI � ZLH��LHIF�9�P&J:CV8"=>C:?D9�PX9<c m � 9<c@C:ZLHIJ:H´8"HIJ:K&ADC:J�?BJ�=�J:?BF/6@ADH�=<�@=>6&C:=>C:?D9�P19<c@C:Z&H�CV8"=<�&?DC:?D9�P@=<Aa[_=<JVHn?TP�CV8:9��&K&[eH;]E_gO=>8:CV9�P}?BP�� g =>8 ·��>� m f H 8:H_C:K&8"P�CV9�C:ZLH 8:H_c�H_8"HIP&[eHIJ/c�9<8­6&8"9a9<c�J ��c�9<8­HedL=<F�6@ADH [�Z&=>6&CVH_8"J��]mn��=<P&�E@9<c�� 54³����.���>������ ZLH­6&8:9]9<c�9<c" ?BJª=<P�=<�&=>6@C:=>C:?D9�P�9<c�C:ZLH�`aK&=<�&8"=>C:?B[/[_=<JVH<�O\LKL6&6*9�JVH�C:Z&=>CGM J:=>C:?BJVU@HIJª= � =YX ¼¹� fG����³´E�=[_AB=<J:J"?B[_=<A¢=>8:Y�K@F�HIP�C(9<cW³´9<;&µ<9�^�� ���9<C,+_H<mrC:ZLH1F�HI=<J:KL8:HXM�J"=>C:?BJVU&HIJ4C:ZLH1�&K&=<A¹c�9<8"F 9<c � =YX ¼�� fª� � Z&?B[�Z ?BJ4C:ZLH?BPLHI`]K&=<AB?BCbE ���A��� m ~ ¥ (2)= ( ) � M�| ¥�� (2 )�� x � ���I���

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� ZLH_8"H 5 8 pq?BJn�LH_U@P&HI�o=<JN?BP����I��� � ?DC:Z�C:ZLH�c�K&P&[eC:?D9�P b =YX ¼*�©¹H_C4PL9<CVHGpo��� � � ?DC:Z��@m � 8 mL;*9�K&P&�&HI�o=<P&� � ?DC:Z � ��=<ABJV9�;*9�K&P&�&HI��m � H-Y<H_C5 8 pS� ½ �S� 5 8 ��� ���e� ½ �����O?BPLc� 7�8�� � �w� ½ ����a�<S�lb � � X ¼ ���]��

��� �*� ½ ��� � z� � � �*� z S��L��� z �� ZLHIP � H-9<;&C:=<?TPo;aE����I���_m� S �f ~ � � M � � z��f ~���� �*� z_� M S � z��f z ~ �az � M%S��L���:zI�g|��'S �f ~ � � M�S � z��f z 8 ~ � � M ; z S��L���:z����

?BF�6 ADE/C:Z&=>C �� x � ����| f�~�� � �*� ze� M?>J� PLc�9<8"C:K&P&=>CVHIADE<m]=<J�?BP­C:ZLHNCV8�=<�&?DC:?D9�P&=<A@[_=<JVHN9<c�C:ZLH4CV8"=<P&JV6*9<8:C:=>C:?B9�P­?BPLHI`]K&=<AB?BCbE<m � H4�&9GPL9<C�µ]PL9 � ?Bc C:Z&?TJ�9�PLHZ&=<J�6@8:9<6*H_8:CbE±9<cr6*H_8:C:K&8:;@=>C:?D9�P�=<Jnc�9<8N?BP&HI`aK&=<AT?DCbE�bdc.0 =AX ¼*� fG���3��8"9<6w9�J"?DC:?D9�Po�]�������� � % 3�� ±+4)w'en%(' ������ ���� +(%����o' ��� � �/C�w.1) ��� ©¹H_C(º�»��ª=<P&� �LH_U@P&H(C:Z&H-6&8:9<;@=>; ?BAB?DCE­F�HI=<J:KL8"H�M ¼ 9�PoQ�;]E

M¹¼�� ��½ ��� �¾ ¼ ¥ £¢¦ §�¦ ¿ ��½ �� ZLH_8"H ¾ ¼/� � ¥ £¢¦ §�¦ ¿ ��½ ��L9<8(\�HI[eC:?D9�Pq��=<P&��¸ � H � ?TABA*PL9<CVH�;aE p�n�f fA^T|���r>u } ^ h f2u =�AD9�[_=<ABADE�=>; JV9�ABKLCVHIADE±[e9�P�C:?TPaKL9�K@J�c�K@P&[eC:?D9�P±9�PoQ��.0/ L S qL�� ! WZY [4\G^ º9�*�b�>�"��� $ |>\Gd�\!\�� h pi^Tp � | &ª� n h¯j psr } | ^T|taA^ ^T|>\7no\�a=p�r>d�\ M¹¼ pia2^ h pxw-\spM^T|>\�Gf jxj f � h uOzMn�f � h wy\�� j f�zOa=d h ^T|{n hZ}o~ f _ f j \i� h ue\�� r+a j h ^ml v �Gf=d�a=uDl7p�n�f fA^T|��sr>u } ^ h f=u p f=u Q psr } | ^T|ta2^ p�» �a=u�� � p z � M¹¼­� � � \�|ta2�=\

s�twv x ¿ y p ze{ |(& ��� x ¿ � p¹� S_n ~�)+* z-,,,,p� p ,,,,^ p z � M¹¼*� ���A���

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s�twv x ( y p z_{ |(& ��� x ( � p¹� > �����<�� S S���� �� ! ! W U`[4\ ^ º9�*�b�>�"��� a=u���aqp�psrtno\ ^T|ta2^ p h pMa p�n�f f2^T|7��r>u } ^ h f2u f=u Q $ |>\Gu � \ f _ ^1a h u ^T|>\�Gf jxj f � h uOz \spi^ h n�a2^ h f=u

s�twv x ¿ y p*z_{-|3& ��� x ¿ � p¹� S_n ~#" ,,,,p p ,,,,^ p*z � M¹¼*� ���A@��

� |>\Gd�\ $� 9 p � »k� ~ p z� � M¹¼&%(' 9 p £ » � ~ p z£ � M¹¼&%4�

p � �7F/=°d¢� pw� ��� a2u�� p £ � F/=°d����4pw� ���2

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D�EGF�F�HI f HXZ&=I^<HGp z � p z� S p z£ � � Z&HIP

s�twv x ¿ y p z { � J:KL6 � ~ p z � � M¹¼ � ?DC:Z ~ ¥ � M¹¼�|��B!� J:KL6 ��~ p z� � � M¹¼hS ~ p z£ � � M¹¼ � ?DC:Z ~ ¥ � M¹¼o|�� !| s�twv x ¿ y p z� { S s�twv x ¿ y p z£ { >³´E � ZLH_9<8:HIF �]�B�ªC:ZLH_8:H�HedL?BJVC:J &G� noOP�/?BP@�LH_6*HIP&�LHIPaC�9<cSp¬J:K&[�ZoC:Z&=>C

s�twvIx ¿ y p*z� {ª|(& ��� x ¿ � p � � S n ~ "� ,,,,

p� �p � ,,,,^ p*z� � M ¼ �

s�twv x ¿ y p*z£ {ª|(& �� x ¿ � p £ � S_n«~#"� ,,,,

p� £p £ ,,,,^ p*z£ � M¹¼w�

� ZLH_8"H$� � � p � » ��� � p z� � M¹¼ O=<P@�

$£ � � p £ » ��� � p z£ � M¹¼ r�� 9­[e9�P&[_ABK&�&H<m&?DC�?BJ�HIPL9�KLY�Z CV9­PL9<C:?B[eH-C:Z@=>C

�� x ¿ � p � �<S �� x ¿ � p £ � � ~ p z � M ¼ � < 8 ~ p � � M ¼ ; z S 8 ~ p £ � M ¼; z =| ��� x ¿ � p¹� �=<P&�

~ "� ,,,,

p� �p � ,,,,^ p z� � M¹¼ S ~ "

� ,,,,p� £p £ ,,,,

^ p z£ � M¹¼/� ~ " ,,,,p� p ,,,,^ p z � M¹¼N>

J� C-?BF�6@AB?BHIJNC:ZLH­Hed�?BJ:CVHIP&[eH/9<c42a¼IO �o=<P&���¬|if´¼�hij}mrJ"K&["Z C:Z@=>C-M¹¼¬J:=>C:?BJVU&HIJª=oAD9<Y�=>8"?DC:Z&F­?B[/\]9<;*9�ADH_^?BPLHI`]K&=<AB?BCbE 9<cNc�K&P&[eC:?B9�P W = ¿ X ¼ � ?BC:Z�[e9�P&J:C:=<P�C�f ¼ � ��P&�LH_HI��mSC:Z@?BJ1?BJ1[_ADHI=>81C:Z&=>C�M ¼ J"=>C:?BJVU&HIJ1=��29�?BP&[_=>8�H?BPLHI`]K&=<AB?BCbE<m¹��JVH_Hª[�Z&=>6&CVH_8 ��9<c�� 5(³�� � ���>����m � ?BC:Zo[e9�P&JVC:=<PaC��/|rp@¼�hkj}m

��� x ¿ � p¹�´|rp@¼N~�p z � M¹¼N>� ZLHIP¹m];]Eo?BPLHI`]K&=<AB?DCE¬���A���em � Hª9<;&C:=<?BP�c�9<84=<PaE±J:F/9a9<C:Zoc�K&P&[eC:?B9�P piOP��9�PoQ�m

s�twv x ¿ y p z_{ |(&�p@¼ ~ p z � M¹¼�S_n ~ ,,,,p� p ,,,,^ p z � M¹¼N>

©¹H_C´K&JSY�?D^<H(=-c�H � Z&?TP�C�9�P­C:ZLHN6@8:9a9<c*9<c*C:ZLH � ZLH_9<8:HIF �]�B�>m � Z&?B["Z � ?BABA�HIP&=>; ADHNK&JWCV9ª6&8:HIJ:HIP�C�µ<H_E�=<K�d�?TAB?B=>8:EADHIF/F­=<J_� f H-U&8"J:CnK&JVH�C:ZLH-c�9�ABAD9 � ?BPLY1?BPLHI`]K&=<AB?BCbE­c�9<84p¬»P�­J:K&["Z C:Z&=>C � p z � M¹¼­� �>m~�p*z*AD9<Ygp*z � M ¼ | ��~�� p��«��� z � M ¼ S ~ � p������ z � AD9<Y*� p±����� z � � M ¼ ���I·��

� ZLH_8"H-?DC�?BJ�9<;]^]?D9�K@J�C:Z&=>CnCV8"K@P&[_=>C:?D9�Po=>8:Y�K&F/HIP�C:J4=>8:Hª[e8"K&[_?T=<A � f H � ?BATA C:Z&HIP PLH_HI�oC:ZLH�c�9�ABAD9 � ?BPLY�ADHIF­F/= ��yL����M� ! W"! [4\ ^ M _ \ a�c�d�f _ a _ihTj h ^ml�n�\�aqp�r>d�\ f=u Q a=u�� j \G^ pO»E� psr } |�^T|ta2^ � p z � MO� � ^T|>\Gu � \ f _ ^1a h u

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h � ~ � p±�«��� z � M�| � ��� x � p¹� >hTh � ~ )�� z p*z � M�|E@ ��� x � p¹� >hxhTh � ~ )+* z p*z*AD9<Y´p*z � M�| AD9<Y�¸AD9<Y�¸-�«� s�twv x y p*z {ªh ¸ s�twv x y p*z_{ >D�EGF�F�HI h�� f HªZ&=I^<H � � p±� ��� z � MO� ��� x � p¹�NS y �N� � p � M { z ��\L?BP&[eH � p z � MO� ��=<P@� p¬»P� � H�9<;@C:=<?BPoC:Z&=>C��| � p � M�|��>m&C:ZLHIP y � p � M { z | � p � MS� � ZLHIP

~ � p�� ��� z � M�| �� x �bp¹� S7� ��� x � p¹�V� z �;@KLC�J:?TP&[eH � p z � MO� �>m ��� x � p¹�´|��>m&C:ZLHIP � � p±�«��� z � M�|k� ��� x � p¹�"�hTh�� )uP&H�^<H_8"?DU&HIJ�CV8�?D^]?T=<ABADE�C:Z&=>C � ZLHIP ½�� �]m ½ z | ¸@� ½ � ��� z =<P&� =>6&6@ADE h��hThxh���©¹H_CNK@J�Y�?D^<H�C:ZLH-6@8:9a9<c2Y�?D^<HIP ?TP�� � � �>¸<� � � c ½ OP� � HªZ&=�^<H ½ AB9<Y ½ S �n� ½ »E� � Z&?B[�Z�E]?BHIAB�&J

~ ) : z p z AD9<Ygp z � M�S MW� pq| ������~ ) : z p z � M�»P���ZLHIP&[eH s�twv x2y p z { » � )+* z p z AD9<Y´p z � MO� � ) * z p z � MS�´\�?BP&[eH

~�)+* z p*z � M�| �AD9<Y�¸ ~ ) * z p*zwAB9<Y�p*z � MS�� H-9<;&C:=<?BP s�twv x y p z { » � �n� 8����� � � � ) * z p z AB9<Y�p z � MS� J� HI[_=<ATA C:Z&H � =>8"�&E F J�?BP&HI`aK&=<AT?DCbE�6&8"HIJVHIP�CVHI�o?TP�C:Z&Hu?TP�CV8:9��&K&[eC:?B9�P��S©¹H_CgMS�0/±;*Hu³´9<8:HIA�F/HI=<J:KL8:HIJg9�P�Q �(mLC:ZLH;*HIJVCn[e9�P&JVC:=<PaC & JV9�C:Z&=>CNH_^<H_8:E J:F�9]9<C:Zoc�K&P&[eC:?D9�P pqJ:K&[�ZoC:Z&=>CN?TP�CVH_Y<8"=<ATJn=>8:H � HIABA��LH_U@P&HI��m&J:=>C:?BJVU@HIJ

~ 24 � pS� ½ �2��pS�3���V� z � MW� ½ �g|�&¬~ 24 p z � /L� � �a���?BJgU@P@?DCVH�?Dc2=<P@�o9�P&ADE±?Dc n � J:KL6§ * 4 MW�V� ½ ��j�� � ~ §4 8 � / =05� M ; £98 �:6 � �����?BJgU@P@?DCVH<��54P@� � ZLHIP & ?BJ�U@P@?DCVH � HªZ@=I^<H-C:Z&?TJ�HIJVC:?BF/=>C:?B9�Pn¯|(& |«¸/n >5 �&?D8"HI[eC�6&8"9a9<c�9<c�C:Z&?TJ�?BP&HI`aK&=<AT?DCbE � ?DC:Z­C:ZLHIJVH(6&8:9<6*H_8:C:?DHIJ�9<c�8:H_Y�K&AT=>8"?DC:?DHIJWc�9<8np [_=<P±;*HNc�9�K&P&�­c�9<8gHedL=<F�6@ADH?BP � Z&H_9<8:HIF �]� �]�B�-9<cn� 5(³�� � ���>� �f H�C:ZLHIPo6@8:HIJVHIPaCg�@?>=wH_8"HIP�C�6&8"9a9<c¹9<c2C:ZLH-�LHIJ"?D8:HI��?BPLHI`]K&=<AB?DCE<mLJVC:=>8:C:?BPLY1c�8:9�F����I·���m =<[_[e9<8��&?BPLY�CV9�C:ZLH�^>=<ABKLH9<c s�twv x y p z { m�?BP � Z&?B["Z � =>8��LE F Ju?BP&HI`aK&=<AT?DCbE�6@AB=�E]Ju=±[e8"K&[_?B=<Ar8:9�ADH<� ��?D8"JVC_m � Z&HIP C:ZLHXHIPaCV8:9<6aEq?BJ(AB=>8:Y<H � H� ?BATA PLH_HI��yL����M� ! W�� [4\ ^ 3 �{\Zw�ue\��Maqp��if jxj f � v

3 � ½ �W� � � h � � ½ �]|��>�� ½ � z £ ¼ h � � ½ �]»��a>$ |>\Gu ^T|+\id�\�\�� h ps^Tp �­|kf 7 hkj p�r } | ^T|ta2^e�Gf=d�\G�=\idsl�psn�f f2^T|��srtu } ^ h f=u � � \�|>a=�=\

s�twv x ¿ y � z {-|}f 7 ~ � z 3 � M¹¼N> � �<���

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Page 14: math.univ-lyon1.frmath.univ-lyon1.fr/homes-www/gentil/logsob-poincare.pdf · y ,% ´| ; > %- ~ ' -- ;- < % ´| [[\ [[] ^ £ ^ ^ . ; » =! @; /| | =! > # ; | # ; > # */}

D�EGF�F�HI f HGK&J:H � ZLH_9<8"HIFD��9<cN� ³ � �/�>� � Z@?B["ZO?BJN=­8:H_U@PLHIF�HIPaCn9<cSC:ZLHX[e8"?DCVH_8"?B9�Po9<c�=­³´9<;&µ<9�^�� ���9<C,+_H­C:ZLH_9<8:HIF��JVH_H � ZLH_9<8:HIF²�]���/9<cg� ³ � ·<·<������ ZLH-[e9�P@JVC:=<P�Cuf 7 J:=>C:?BJVU&HIJNF/=°d���� £ ��� � �g|kf 7 |«F­=°d��3n £ � n � � � ZLH_8:H

� � �}J:K&6§ * 4 M¹¼¹�V� ½ �sS-j � �]AB9<Y 8 �'S ��°M¹¼¹�V� ½ �sS-j�� � ; ~ §4 ¾ ¼ ¥ ¦ ' ¦ ¿3 � 6 � �:6 �� £ � J:KL6§ : 4 M¹¼¹���@��j}� ½ ���]AD9<Y 8 �'S ��°M¹¼¹��� � j}� ½ ��� ; ~ 4

§ ¾ ¼ ¥ ¦ ' ¦ ¿3 � 6 � �76 �n � �7J:KL6§ * 4 M¹¼��V� ½ �sSªj � �]AD9<Y 8¹� S ¥�zM¹¼��V� ½ �sS-j � � ; ~ §4 ¾ ¼ ¥ ¦ ' ¦ ¿3 � 6 � �76 �n £ �7J:K&6§ : 4 M¹¼*���w��j}� ½ ���]AB9<Y 8 �'S ¥IzM¹¼��V� �4j}� ½ � � ; ~ 4

§ ¾ ¼ ¥ ¦ ' ¦ ¿3 � 6 � �:6 >5(PoHI=<JVEo=>6&6&8:9�d�?TF/=>C:?D9�P�6&8:9�^<H-C:Z&=>Cnc�9<8NAB=>8"Y<H-6*9�J:?DC:?D^<H ½M ¼ �V� ½ ��j�� ����~32§ �¾ ¼ ¥ £¢¦ ' ¦ ¿ �:6�� 2 �¾ ¼Lº ½ ¼ £98 ¥ £&§ ¿ � �����

~ §4 ¾ ¼ ¥ ¦ ' ¦ ¿3 � 6 � �:6�� 2 ¾ ¼º ½ ¥ § ¿ �=<P&��9�PLH-F­=IE�6&8:9�^<H-J:=<F�H-HI`]K&?D^>=<ADHIPaC´c�9<8NPLH_Y�=>C:?B^<H ½ ��5 J:?BF�6 ADH([_=<AT[_K&AB=>C:?D9�P�C:ZLHIP E]?DHIAT�&J´C:Z&=>Cn[e9�P&JVC:=<PaC:J� � m�� £ m n � =<P&�Tn £ =>8:H-U P&?DCVH�=<P&�oC:ZLHGADHIF/F/=�?BJg6@8:9�^<HI��� JL49<CVH(C:Z@=>C´C:Z&H4c�K&P&[eC:?D9�P 3 ?BJ�C:ZLHuJ:F/=<ABADHIJ:CWc�K&P&[eC:?D9�P±J:K&[�Z­C:Z&=>CgC:ZLHu[e9�P&JVC:=<PaCNf 7 ?TP­C:ZLHu?BPLHI`]K&=<AB?BCbEO� �<����?BJU@P&?BCVH<�?g 9<8:H-6@8:HI[_?BJVHIADE<m&?Bc 53 J:=>C:?BJVc�E

AB?BF§8� 253 � ½ �3 � ½ � ��j

C:ZLHIPoC:Z&H-[e9�P&JVC:=<PaCuf��7 ��j}���P�C:ZLH-[_=<JVH-9<c�J:F/=<ABAwHIPaCV8:9<6]E<m � H � ?BABAwK&JVH-JV9>�b[_=<ABABHI� %�� \]9<;*9�ADH_^­?TPLHI`aK@=<AB?DC:?DHIJ���H_^<HIP ?Dcr9�KL8n[e9�PaCVHed]C(?BJ�ADHIJ:JY<HIPLH_8"=<A���m]JVH_H �gZ&=>c�= � � � �gZ&=��>¸�� c�9<8�=ª[e9�F/6&8:HIZLHIP&J"?D^<Hn8:H_^�?DH � m]=<P&�­³g=>8:C:ZLHe�*��=>CVC:?B=<K�d]� � 9<;*H_8:CV9o� ³�� � �>¸<�@c�9<8=1Y<HIPLH_8"=<Ar=>6&6&8:9�=<[�Z ?BPoC:Z&H-[_=<JVHª9<c�F�HI=<J:KL8"H�M¹¼*��yL����M� ! W��`[4\ ^ � 8 h � z v � ��� � 8 � � z � a=u�� � _ \�aop�n�f fA^T|���r>u } ^ h f2u �{\Zw�ue\��Mf=u � � ��j � � p�p�r>no\�^T|ta2^

�w� � �W���]� �±» � a2u�� ~(2

� z � M¹¼o| �A�]�^T|>\Gu ~ 2

� �X����� z %�� � z � � M ¼ |kf ~ X 2

� z � M ¼ � � �°¸a�� |>\Gd�\ %-� ½ ����AD9<Y ¨�� ¿ � (� ¿ � ½ �2B$ |>\ } f=u+ps^1a2uE^ f �O\ c+\Gu��Mf=u)^T|>\�no\�a=p�r>d�\ M ¼ _ r+^-�Af \ip�u�f2^C�{\ c+\iu��Mf2u ^T|>\�qa j rE\�fg�� ��� � 8 � � z � D�EGF�F�HI ©¢H_C/K&JVH � =>8"�LE F J/?BPLHI`]K&=<AB?DCE�=<J�Hed�6@AB=<?TPLHI� ?BP�C:ZLH ?BPaCV8:9��&K&[eC:?D9�P¹� f HqZ&=I^<H �*� � ��� �]� f HO=>6@6@ADE?BPLHI`]K&=<AB?BCbE�� �a���N9�P � � ��j �w=<P&� � ?DC:ZOC:ZLHXc�K@P&[eC:?D9�P¬p¬� �����q��PL9<CVH1C:Z&=>C-pS� � ��� ���4=<P&�OC:ZLH1c�9�ABAB9 � ?BP&YF�HI=<J:K&8:HIJ � MO� y AD9<Y � zI{ ¨�� ¿ � (� ¿ � M¹¼o=<P&� /­�}M¹¼ >

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� ZLHIP�C:Z&Hª[e9�P&JVC:=<PaC(f ?TPo?BPLHI`]K&=<AB?DCE¬� �°¸a�n?BJgU@P&?DCVH-?Bcr=<P&�o9�P@ADE�?Dcni� J:KL6§ * ~ §

¾ ¼ ¥ ¦ ' ¦ ¿ �76 ~32§ y AD9<Y � z�{ ¨�� ¿ � (� ¿ � M¹¼*�

?BJgU@P@?DCVH<�\�?BP@[eHG�L� ºq�«��� �>º«h �ªC:ZLHGc�K&P&[eC:?B9�P ½ � � AD9<Y ½ � ¨�� ¿ � (� ¿ ?TJN[e9�P&[_=�^<HX9�P�� ¸L��j � �(³�E 4<HIP&JVHIPO?BPLHI`]K&=<AB?DCE � H9<;&C:=<?BP c�9<8N=<ATA ½ » � m~32§ y AB9<Y � zI{ ¨�� ¿ � (� ¿ � M¹¼o|«AD9<Y ¨�� ¿ � (� ¿ 8 � 2§ � z � M¹¼M¹¼��V� ½ ��j � � ;nM¹¼¹�V� ½ ��j � � >

� ZLHIP�;]E±C:ZLH-6&8:9<6*H_8:CE�9<c � � HªZ&=�^<Hn | J:KL6§ * ~ §

¾ ¼ ¥ ¦ ' ¦ ¿ �76 AB9<Y ¨�� ¿ � (� ¿ 8 �A�M¹¼*�V� ½ ��j � � ;NM¹¼*�"� ½ ��j � �

| J:K&6§ * ( ~ § (¾ ¼ ¥ ¦ ' ¦ ¿��76 AB9<Y ¨�� ¿ � (� ¿ 8 �A�M ¼ �V� ½ ��j � � ; M¹¼*�"� ½ ��j � � >

� J:?BPLY1C:ZLHª=>6@6&8:9IdL?BF/=>C:?B9�P ~ §4 ¾ ¼ ¥ ¦ ' ¦ ¿ �76 � 2 ¾ ¼º ½ ¼ £98 ¥ § ¿ �=<P&�±C:Z&=>CgY�?D^<HIPo?BP�HI`]K&=<AB?DCE¬� ����� � H�6&8:9�^<H(C:Z@=>C4n ?BJ´U@P&?BCVH<ma;*9�K&P&�&HI�/;]E�=1[e9�P&JVC:=<PaC4f � Z&?B[�Z±�&9aHIJgPL9<C�LH_6*HIP&�±9�P � ��Jf H �&?B^]?B�&H±C:Z&H 6&8:9]9<cN9<c � ZLH_9<8"HIF �]�B� ?BP�C � 9�6@=>8:C:J �OAB=>8:Y<H =<P&� J:F/=<ABAgHIPaCV8:9<6]E<m´;*9<C:Z«?TP C:ZLH [_=<JVHO9<c6*9�J:?DC:?D^<Huc�K&P&[eC:?D9�P¹��©¢H_CNK&JnP&9 � 6&8"HIJVHIP�CNC:ZLH�6&8:9]9<c2?TP�C:ZLHª[_=<JVH-9<cSAB=>8"Y<H�HIP�CV8:9<6]E<�©¢=>8"Y<H-HIP�CV8"9<6aEo[_=<JVH ��� S���S � P�vRPTS t ! W*/ ~ rRcAc>fqp \b^T|ta2^ º"�*�b�>�"��� $ |+\id�\b\�� h ps^Tp � | & k � n k h j psr } |!^T|taA^ �if2doa=uDl7p�n�f fA^T|�srtu } ^ h f=u+p pq»P� pia2^ h px�sl h uOz ~®p*z � M¹¼­� � a=u��-s�twv x ¿ y p*z_{-»��>� � �<���� \ |ta=�2\

s�twv x ¿ y p z_{ |(& k ��� x ¿ � p¹� S_n k ~�)+* z ,,,,p� p ,,,,^ p z � M¹¼N> � �����

�x� º � �qv ^T|>\Gd�\!\�� h pi^Tp � | &1 k h®j a=u�� p Oo� psr } | ^T|taA^��Gf=d \i�=\Gd�l!psn�f f2^T|"�sr>u } ^ h f=u p » � � h ^T|� p z � M 8 � �qv � |>\iu � p �]|rpXv ^T|>\iu � \ zO\ ^s�twv x ( y p*ze{-|(& k ��� x ( � p¹� >

D�EGF�F�H FGH D�EGF � F�������� F��� ��I ©¢H_C4pO;wH-=­J:F�9]9<C:Zoc�K@P&[eC:?D9�PoJ:=>C:?BJ:c�E�?BPLY/p¬»P��m � p z � M¹¼­� �-=<P&� � p � � M¹¼ohkj}�5 [_=>8:H_c�K@A¹JVC:K&�LE�9�P Q �79<c2C:Z&?BJgc�K&P&[eC:?D9�P

½ � � ½ z AD9<Y ½ z S �L� ½ � ��� z S ½ z �«�dS7� ½ ����� z � AB9<Y � ½ ����� z �6&8:9�^<HIJnC:Z&=>CNc�9<8nH_^<H_8"E ½ »P�½ z AD9<Y ½ z | �L� ½ � ��� z S ½ z �«�dS � ½ ����� z � AB9<Yw� ½ ����� z � >

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Page 16: math.univ-lyon1.frmath.univ-lyon1.fr/homes-www/gentil/logsob-poincare.pdf · y ,% ´| ; > %- ~ ' -- ;- < % ´| [[\ [[] ^ £ ^ ^ . ; » =! @; /| | =! > # ; | # ; > # */}

� ZLHIP � H-9<;&C:=<?TPo;aE�©¹HIF­F/= �]���]� ? m@8"HI[_=<ABAB?BPLY1C:Z&=>C � p z � M¹¼­� �-=<P&� p¬»E��m~�p z AD9<Ygp z � M¹¼ | ��~ � p��«��� z � M¹¼ S«~ � p z �«��� � M¹¼hS«~7� p������ z � AD9<Y*� p±����� z � � M¹¼

| �Y� ��� x ¿ � p¹� S«~�� p������ z � AD9<Yw� p������ z � � M¹¼� Z&?T["Zo?BJgC:ZLHG=<P&PL9�K&P&[eHI�oJ:C:=>8:C:?BPLY�6*9�?BPaCn?BPLHI`]K&=<AB?DCEq���I·����\�?BP@[eH � p z � M¹¼/� �>m&9�PLHª[_=<P HI=<J:?BADE�6&8:9�^<H-C:Z&=>C

~ � p������ z � � M¹¼o|��>�C:ZLHIP � � p������ z � AD9<Yw� p±����� z � � M¹¼o| s�twv x ¿ y � p������ z � { �@=<P&�

s�twv_x ¿ y p*z_{ª| �Y� ��� x ¿ �bp¹� S sutwv�x ¿ y � p������ z � { > � ���<�� =>8"�LE F Jn?BPLHI`]K&=<AB?DCE­9<c�©¹HIF/F­=e�]��¸ � ?DC:Z��/� � p������ � Y�?D^<HIJ

sutwv x ¿ y � p������ z � { |}f 7 ~ � p������ z� 3 � M¹¼­��f 7 ~ )+* z p z 3 � M¹¼N> � ��@���L9<8'5¢���1O��-J:K&[�ZoC:Z&=>CN=<P@���R�l51S7�R���ª� � � HªZ@=I^<H-c�9<8NH_^<H_8:E ½ � ��OP�/;aE �9�K&PLY­?TPLHI`aK@=<AB?DCbE<m

½ �±| ½ :5 S ����> � �<·��

�´9�P&J"?B�LH_8�C:ZLHIPoº �*�b�>�"���¹=<P&� Q¬� º.��� º­�������W©¹H_C�5��"QC�>�G=<P&���-�"QC���VQ��O������©¹H_C�� OP�X=<P&�±ADH_C´=>6@6@ADE?BPLHI`]K&=<AB?BCbE � �<·��gCV9/C:ZLH�8�?DY�Z�C�CVH_8�F®9<cª� ��@��em � H-9<;&C:=<?TPo9�P 0�p¬» � 1 m�

� A ^a£ z CT¤*^ 8 p p ; z � A ^a£ z CT¤*^ 3 | �Q�� A ^�£ z C�¤ z ,,,,p p ,,,,^ S QO���Q � 3 ^<¤ A ^a£ z C �

C:ZLHIPs�twv x ¿ � � p±����� � z � | ��f 7Q�� A ^a£ z CT¤ z ~ )+* z,,,,

p� p ,,,,^ p z � M¹¼ S QO���Q f 7 � ~�)+* z 3 ^>¤ A ^�£ z C p z � M¹¼©¹H_C�M�=16@8:9<;@=>;@?BAT?DCbE/F�HI=<J"KL8:H<mLC:ZLHIP � H-Z&=�^<H-c�9<8NH_^<H_8:E�c�K&P@[eC:?D9�PopOJ"K&["ZoC:Z@=>C � p z � MO� ��=<P&� c�9<8�H_^<H_8:EF�HI=<J:K&8"=>;@ADHuc�K&P&[eC:?D9�P���J:K&["Z C:Z&=>C � p z � � M�Hed�?TJVC:J � H-Y<H_C

~ p z � � M�| s�twv x y p ze{ S AD9<Y ~ ¥ � M?>� Z&?TJ´?TPLHI`aK@=<AB?DCbE±?BJ�=<ABJV9­CV8"KLH�c�9<8N=<ABA*c�K&P&[eC:?D9�P���»P��H_^<HIP ?TP�CVH_Y<8"=<ATJn=>8:Hª?BP&U@P&?DCVH<�©¹H_C�OP�­=<P&� � Hª=>6&6@ADE�C:ZLH�6&8:H_^�?D9�K&J�?TPLHI`aK@=<AB?DCbE � ?DC:Z��/�� 3 ^>¤ A ^�£ z C(� �±»E����m � H-9<;&C:=<?BP�C:ZLHIPsutwv x ¿ � � p������ � z � |

��f 7Q�� A ^a£ z CT¤ z ~�) * z,,,,p p ,,,,^ p*z � M ¼ S �VQO�����Vf 7 �Q� 8 s�twv_x ¿ y p*z_{'S�AD9<Y�~ Hed]6 � 3 ^<¤ A ^a£ z C � � M ¼ ; >����a���

\�?BP@[eH Q �7ºC��� º¬� ����m 3 � ½ � ^>¤ A ^�£ z C � ½ ¼ ?Dc�� ½ �]»��>m&C:ZLHIP � H-U&d�­� �R�>�]�´54P&�oP&9<CVH �7AD9<Y ~ Hed]6 8 �� 3 ^<¤ A ^a£ z C ; � M ¼ hkj >

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��? doPL9 � ���7?BP&c�0RQ.��� �VQO�����¸af 7 ���,QC���V�VQ �����¸af 7 � 1 =<P&� PL9<CVH � � f 7 ��� A ^a£ z CT¤ z f Hª9<;&C:=<?TPs�twv x ¿ � � p������ � z � | � ~�)+* z-,,,,

p p ,,,,^ p*z � M¹¼ S �¸ s�twv x ¿ y p*ze{dS �¸ >

5(J s�twv x ¿ y p z {-» �>m@?BPLHI`]K&=<AB?DCE¬� ���<�n?BF�6@AT?DHIJs�twv x ¿ y p ze{ | �a� �� x ¿ � p¹� S � � ~ )+* z 8 p� p ; ^ p z � M¹¼ �

� Z&?T["Z�6&8:9�^<HIJN?BP&HI`aK&=<AT?DCbE � ����� � ?BC:Z�& k � �a�­=<P&� n k � � � �5(J:J:K&F�HuPL9 � C:Z@=>C�º � �u=<P&�±C:=>µ<Hªp J:K&["Z�C:Z@=>C-� p� �]|rp¹� f H-=>6&6@ADE�C:Z&H(AB?TF/?DC�[_=<J:H(9<c �9�K&PLY�?BPLHI`]K&=<AB?BCbECV9/Y<H_Cn9�P 0�p¬» � 1 m 8 p� p ; z 3 p z | 8 p � ; z 3 p z >� ZLHIP¹m � ?DC:ZqC:ZLH/J:=<F�H/[e9�F/6@KLC:=>C:?D9�P�=<Jª?BP¬C:ZLH/[_=<JVH�º �*�b�>�"��� m¢9�P�[_=<P�c�9�K&P@� p O � =<P@� �O| & k h jJ:K&[�ZoC:Z&=>Cnc�9<8N=<ABA�J"F�9a9<C:Z c�K&P@[eC:?D9�P p¬J:=>C:?BJVc�E�?BPLY�Z�E]6*9<C:ZLHIJ:?BJg9�P�� �<����=<P&��� p �]|rp¹ms�twv x ( y p ze{ |(& k ��� x ( � p¹� >

J�"L�� �� �� !BW�� � h ^T|�^T|>\ypia=no\�n�\ ^T|tf ��aqp-�{\G�=\ j f�cD\�� h u ?Cdgf�c+fqp h ^ h f2u /�� � \ } a=u c�d�f=�2\�^T|>\ h uX\��GrDa jkh ^ml � ������ h ^T|tf=r+^ �� x ¿ �bp¹� ~ rRcAc+fqpG\7^T|>a2^ º �*�b�>�"��� $ |>\Gd�\M\�� h ps^Tp & O � p�r } | ^T|ta2^]�Gf=dba=uDl���r>u } ^ h f2u+p pPO��piaA^ h pT��l h uOz ~®p z � M¹¼­� � a=u��Gs�twv x ¿ y p z {�» �� \ |ta=�2\

s�twv x ¿ y p*ze{-|(& ~ ,,,,p p ,,,,^ p*z � M¹¼ >

\�F/=<ATA HIPaCV8:9<6]Eo[_=<JVH ��� S���S � P�vRPTS t ! W�� [4\ ^ ºI�*�b�>�"��� �$ |>\Gd�\�\�� h ps^Tp &

�� n�OE� psr } |�^T|ta2^=�Gf=d-a=uDl���r>u } ^ h f2u+p pq»P� piaA^ h pT��l h uOz

~®p*z � M ¼ � � a=u��-s�twvIx ¿ y p*z_{-|��>�� \ |ta=�2\

s�twv x ¿ y p z_{ |(&���� x ¿ � p¹� S_n � ~ )+* z ,,,,

p� p ,,,,^ p z � M¹¼N> �������

�x� º � �qv ^T|>\Gd�\�\�� h ps^Tp �/|�& �h}j psr } | ^T|ta2^ v

s�twv x ( y p ze{ |(& ���� x ( � p¹� >

�Gf=d�a jTj p psr } | ^T|ta2^ ��p �L|�� D�EGF�F�H FGH D�EGF � F�������� F��� ��I ©¹H_Cªp » �oJ:=>C:?BJVc�E�?BPLY � p z � M¹¼O�²�>�G5(J-?BP���8:9<6*9�J:?BC:?D9�Pi�]���]m � H/JVC:=>8"C � ?DC:Z¬?BPLHI`]K&=<AB?DCE����I·���m � Z@?B["Z8:HI=<�&?TADE­?BF�6 AB?DHIJ

s�twv x ¿ y p z_{ � ~ p z AD9<Y�p z � M¹¼o| �Y� ��� x ¿ � p¹� S ~ � p������ z � AB9<Y�p z � M¹¼ > ���<���f H � ?BABA�P&9 � [e9�P�CV8"9�A�C:Z&H�JVHI[e9�P&� CVH_8"F 9<cnC:ZLH±8"?DY�ZaCGZ@=<P&��J:?T�LH­9<c�C:Z&?TJGAB=<J:CG?BPLHI`]K&=<AB?BCbE¬^]?T= C:ZLH�K&JVH�9<c%�� \]9<;*9�ADH_^±?BPLHI`]K&=<AB?DC:?BHIJ_m]P@=<F�HIADE�©¹HIF/F/= �]� �]�� ZLH_8"H_c�9<8:H � H-Z&=I^<HªCV9/[e9�P&JVCV8"K&[eCn=1c�K&P&[eC:?B9�P��±�LH_U@PLHI��9�P�� � ��j�����c�9<8N= � HIABA*["Z&9�JVHIP � � � ?BC:Z���» �1=<P&��w� � �W� ��=<P&� � Z&?B["ZoJ"=>C:?BJVU&HIJ_m

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��Y&��� ~ 2

� z � M¹¼o|��A�����Y����N~ 2

� �X����� z %�� � z � � M¹¼ � f�~ 2

� p������ z � AD9<Y´p z � M¹¼��

��Y ���V~ 2

� z � M¹¼o|kf�~ X 2 ���

)�� z�� � 8 ,,,,p� p ,,,,; p z � M¹¼ S � s�twv x ¿ y p z {>m

� ?DC:Z %-� ½ �S�7AD9<Y ¨�� ¿ � (� ¿ � ½ ��m �/h(� |��R�>��=<P&� � � ½ ��� ½ ^ >©¹H_C4PL9 � �&H_U@PLH � 8 hE�/=<P&� � z OP��J:K&[�ZoC:Z&=>C

M¹¼���� ��j7� � 8 ����� � @ �nM¹¼��V� � 8 � � z ����� �¸ =<P&� M¹¼*�"� � z �sS-j�� ��� � @ >\�?BP@[eH � p z � M¹¼/� ��C:ZLH_8:HªHed�?BJ:C:J � ��� � 8 � � z �rJ"K&["ZoC:Z@=>C4pS� � �´| �]���P�CV8:9��&K&[eH-P&9 � �±9�P�� � ��j��r=<JNc�9�ATAD9 �

������S � p������ � AD9<Y��p z �� ZLH_8"H�o� � ����º2�,��� �<º����. KLHªCV9±C:ZLHXc�=<[eC(C:Z&=>C�p�?BJ4=�J"F�9a9<C:ZOc�K&P&[eC:?D9�P ��AD9][_=<ATADEo=>;@JV9�ABKLCVHIABE [e9�P�C:?BP]KL9�K&J4c�K&P&[eC:?D9�P �gC:ZLHIP � ?BJ4=<ABJV9= J:F/9a9<C:Z c�K&P&[eC:?B9�P��eg 9<8"H_9�^<H_8 �qJ:=>C:?BJ:U&HIJ�w� � ���¯��=<P@� �w� ½ �X»i�oc�9<8ª=<ATA ½ » � �/©¹H_CGPL9 � [e9�F�6@KLCVH� 2 � z � M¹¼*� f HªY<H_C

~32

� z � M ¼ | � ~�2 (

¸ � M ¼ S � ~32 (

� p������ z � AD9<Y z ��p z � M ¼| � S«� ~

( X 2 ���)+* z�� p z AD9<Y z � p z � M¹¼N>\�?BP@[eH�� ��� ���I� � � HªZ@=I^<HªAD9<Y z � p z |«AD9<Y�p z 9�P 0°pq» � 1 � � ZLHIP � H-9<;&C:=<?BP ;aE�©¹HIF/F­=e�]���]� ?B?B?

~ 2

� z � M ¼ | � S ��~ )+* z p z AD9<Y z ��p z � M ¼| � S @ s�twv x ¿ y p z {| �A�]�J:?BP@[eH s�twv x ¿ y p z {-|��>�5(J:J:K&F�6@C:?D9�P&J´9�P ©¹HIF­F/=e�]� ��=>8:HGJ:=>C:?BJVU&HI�¹m � H-9<;@C:=<?BP�;aEo?BP&HI`aK&=<AT?DCbE � �°¸a�

~(2

� ������� z � AD9<Y ¨�� ¿ � (� ¿ �az � M¹¼o|}f ~ X 2 ��� * z�� � z � M¹¼N>©¹H_C4K&J�[e9�F�6@=>8:HªC:ZLH-^>=>8"?D9�K&JgCVH_8"F­JnPL9 � ���?D8"JVC_m&�&HIPL9<CVH������L� ºO� ��� �>º�m � H�C:Z]K&J�9<;@C:=<?BP� �X����� z � AB9<Y�� � z � � p������ z � AB9<Y z ��p z AD9<Y�� y ��S � p������ � AD9<Y���p z_{ z >)uP60°pq» � 1 m � H/Z&=�^<H­� S � pO� ��� � AD9<Y � p z » � S � pO� ��� � � � � ZLH_8:H � � AD9<Y � ¸L�/\�?BP@[eH � » ��=<P&�

�1S«� � �>m@9�PLH-Z&=<J� �1����� z � AB9<Y�� � z » � p������ z � AB9<Y z � � �Wp z � � p������ z � AD9<Y´p z >� ZLHIP � H-9<;&C:=<?TP ~32

� p������ z � AD9<Y´p z � M¹¼o| ~32

� �X����� z � AB9<Y ¨�� ¿ � (� ¿ � z � M¹¼N> �������

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\]HI[e9�P&�@ADE­9�PLHGZ&=<J�9�P 0°pq» � 1 m� � p AD9<Y��p z 8¹�'S �� � p����p(AD9<Y´p z ;-�

C:ZLHIP K@J:?BPLY�AD9<Y�p z »«AB9<YW¸�9�P 0�p¬»k� 1 9�PLH-9<;&C:=<?TP

,,� ,,z | ,,

p ,,z AD9<Y z � p z 8¢� S *� �AD9<Y�¸ ; z >

. HIPL9<C:?BPLY ��� �V�'S *� � �°AD9<Y�¸]� z mL9�PLHªZ&=<J~ X 2 ���

)+* z�� � z � M ¼ |(� ~ X 2 ���

) * z�� p z*AD9<Y]z � p*z � M ¼ � ���°¸a�9�P�� � ��j�� � 0°pq» � 1 >� ZLHIP¹m�K@J:?BPLY�?BPLHI`]K&=<AB?BC:?DHIJ���������=<P&�����°¸a�em&C:ZLH_8:H�HedL?BJVC:J4fi»P� ��?BP@�LH_6*HIP&�LHIPaC�9<c � ��� � 8 � � z ����m@J:K&[�ZoC:Z&=>C

~32

� p������ z � AD9<Y�p*z � M¹¼o|kf�~ X 2 ���

)+* z�� p zwAB9<Y�z � p*z � M¹¼ > ���<���f ZLHIPOºI�*�b�>�"��� m � HX=>6&6@ADE���PLHI`aK@=<AB?DCbE�� �<·�� � ?DC:Z �ª� ºC��� �ª��º���=<P&� 5o��º.��� �L� º �k���V��� f HG9<;&C:=<?BPoc�9<8H_^<H_8:E � OE��m~ X 2 ���

) * z�� 8 p p ; z y AD9<Y]z � p*z {�p*z � M¹¼o|�L� º¬�«���º � ¨ � ¿¨�� ¿ � (� ~ X 2 ���

)+* z��;,,,,p p ,,,,^ p*z � M ¼ S � ����ºº ~

X 2 ���)+* z�� p*z*AD9<Ygp*z � M ¼ >��? d �ªJ:K&[�ZoC:Z&=>C�>f z £ ¼¼ h �R�]�A�]m@C:ZLHIPoC:Z&H_8:H-HedL?BJVC:J�& OP�/J:K@["ZoC:Z&=>C

~ 2

� p������ z � AB9<Y´p z � M¹¼o|(& ~ X 2 ���

)+* z�� ,,,,p� p ,,,,^ p z � M¹¼ S ��A� ~ X 2 ���

) * z�� p z AB9<Y�p z � M¹¼N>� J:?BPLY�©¹HIF­F/=e�]���]� ?B?B? � HªZ&=�^<H<m~ 2

� p������ z � AD9<Ygp z � M¹¼o|(&¬~ X 2 ���

) * z��;,,,,p� p ,,,,^ p z � M¹¼�S �¸ s�twv x ¿ y p z {?>� ZLH-J"=<F�H-F�H_C:ZL9�� [_=<P ;*H-K&J:HI��9�P�� ��j}� � �¢=<P&�oC:ZLHIP � H-Y<H_C

~

£ 2 � p������ z � AD9<Y�p z � M¹¼o|(& ~ £ 2 X �� ��� )+* z�� ,,,,

p� p ,,,,^ p z � M¹¼ S �¸ s�twv x ¿ y p z { >

5(P&��C:ZLHIP�m � H�Y<H_C~ � p������ z � AB9<Y�p*z � M¹¼o| � & ~�)+* z ,,,,

p� p ,,,,^ p*z � M¹¼ S �� sutwv x ¿ y p*z_{?>L49<CVH-C:Z&=>CnC:ZLHG[e9�P&JVC:=<PaC1& �L9aHIJNPL9<CN�LH_6*HIP&�±9�P � �� ZLHIP¹m];]Eo?BPLHI`]K&=<AB?DCE¬���<���em@?BPLHI`]K&=<AB?DCE¬�������n?BJg6&8:9�^<HI� c�9<84ºI�*�b�>�"��� m � ?DC:Z�&

�� �°¸­=<P&�Tn

��7¸:&ª�5(J:J:K&F�H-P&9 � C:Z&=>C(º¬� �>� � P�C:Z@?BJn[_=<JVHG� � �>m&C:Z&HIP K&J:?BP&Y1?TPLHI`aK@=<AB?DCbE ���<��� � H-9<;&C:=<?TP

~ 2

� p������ z � AD9<Y�p z � M 8 |��7~ X 2 ���

)+* z�� AD9<Y´p z � M 8 |E@ � ~ X 2 ���

) * z�� p z � M 8 �

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c�9<8´JV9�F/H([e9�P&J:C:=<P�CNf � � ZLHIP�;aE/©¢HIF/F/=1�]���]� ?B? m � H(9<;&C:=<?BP�C:ZLH48:HIJ:K&ADC � Z&?B[�Z­[e9�P&[_ATK&�LHIJ�C:ZLHN6&8"9a9<c¹?BP­C:Z&?BJ[_=<JVH � ?DC:Z�& �� �Y��S @ �O� J©¹H_C4K&JgY�?D^<HªPL9 � =�6&8"9a9<c29<crC:Z&H-C:ZLH_9<8:HIFo�

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� 1 =<P&� n � F/=°d 0Rn k � n� 1 =<P&�¬c�9<8ªº � �19�PLH�U@P@� p O �=<P&� �/|�& @�7F/=°d 0"&1

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