motivations for an arbitrary precision interval arithmetic ... · motivations for an arbitrary...

17
Motivations for an arbitrary precision interval arithmetic and the MPFI library Nathalie Revol, Fabrice Rouillier To cite this version: Nathalie Revol, Fabrice Rouillier. Motivations for an arbitrary precision interval arithmetic and the MPFI library. [Research Report] RR-4498, INRIA. 2002. <inria-00072090> HAL Id: inria-00072090 https://hal.inria.fr/inria-00072090 Submitted on 23 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non, ´ emanant des ´ etablissements d’enseignement et de recherche fran¸cais ou ´ etrangers, des laboratoires publics ou priv´ es.

Upload: others

Post on 27-Mar-2020

30 views

Category:

Documents


0 download

TRANSCRIPT

Motivations for an arbitrary precision interval

arithmetic and the MPFI library

Nathalie Revol, Fabrice Rouillier

To cite this version:

Nathalie Revol, Fabrice Rouillier. Motivations for an arbitrary precision interval arithmeticand the MPFI library. [Research Report] RR-4498, INRIA. 2002. <inria-00072090>

HAL Id: inria-00072090

https://hal.inria.fr/inria-00072090

Submitted on 23 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--44

98--

FR

+E

NG

ap por t de r ech er ch e

THÈME 2

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Motivations for an arbitrary precision intervalarithmetic and the MPFI library

N. Revol (ANO, U. Lille et Arenaire) and F. Rouillier (Spaces)

N° 4498

July 2002

Unité de recherche INRIA Rhône-Alpes655, avenue de l’Europe, 38330 Montbonnot-St-Martin (France)

Téléphone : +33 4 76 61 52 00 — Télécopie +33 4 76 61 52 52

��������������� ���������� �� ������������������� "!#������� �$���%�� &���' ()�������*�+, -����! ��./��*� 0��132546(�����������

798;:=<?>A@ABDCFE�7�GIH;J98LK"MNBOBN<�<?P=E�QR<TSLUVMOQR<XW∗UVS�Y[Z\8�:=@A]�MNBNBOMN<TQ�C_^�`LUVaT<cb�W

d&egfih\j kDl�mniogpqj�rts?uTptvwpqjirjwxyvwz{r|v�}~r�����hD�Vs?rtp|�T}~j�;��s{�Fjwx�����nwo~z?pq��j

�yz{�g�Vs?�_x��Xj��_jRv�egjw��v�egj3oV���T�T�T��l��?}~rq��k?�?�ck�l��R�%��z{u?jR�

�=���i�w���g ?�?¡ d&egp|�&�~z{�Vjw���F}~��x_pq¢~jR�&£"ec�¤z?o�z?�_�~p¥x���z?�_�\�g��jiv�p|�_pqsTo=ptoTx�jw��¦�z{r§z{��p¥x�egh¨j�x�ptv'pt�&o~jwji�gji�§©�x_s�~�_s�¦�p|�Xj%zTvwv�}~��z{x_j\�_jR��}grqx��iª�ptocx_jw��¦�z{rLv�sTh¨�g}Xx�z{x_pts?o~���ji�c}gpt�_j¨�_h¨z?rqr�ptog�g}Xx ptocx_ji�_¦�z{r|�i«§x_egp|��jw¬��~rtz?pqo~�£"e��¤�~pt�_jiv�x_pts?o�pt���_s¨s{­®x�jwo�jwh¨�grts��?ji��pto�ptocx_ji�_¦�z{r�z?rquTs?��p¥x�egh¯�w°d&e~j=±²�´³�µ�rtpq�g��z{���9e�z?� �Vjwjio¶�g}gptrqx%pto¶s?���Xjw� x_s²­·}grq¢~rtr-x_egp|� ogjiji�§©¨pto~�Xjiji�§ª�ogs¸jw¬Xpt��x_ptogu¸rtpq�~��z?�_�h¨jwx x_egj��_jR�c}gpq��ji�¹�_��jRv�pq¢�vwz{x_pts?o~�i°�d&egj¤h¯z{pto¹­·jiz{x_}g��ji�Ds{­&x_egp|� rtpt�g��z?�_�ºz{��j¯�g��pqjw»~�9uTpq¦Tjwo¶z{o��ºzvws?h¨�~z?�_p|��sTo¨£"p¥x�e�z�¢~¬�jR��¼½�g�_jRv�p|��pts?o¯ptocx_ji�_¦�z{rAz{��pqx_egh¨j�x�ptv?ª�s?o=z%���Vjivwp¥¢�v'�g��s?�grtjwh�ªcpt�#�g��ji�_jwocx_jR�§©�p¥x�_egs�£��#x_e�z�x"x_e~j�s�¦?jw��egjRz?�¤�X}~j�x_s¨x_e~j�hD}grqx_pt�grtj��g�_jRv�p|��pts?o�pt�yv�s?h¨�grtj�x�jwrt�=z?�gh\p|����pt�grtj?°¾´¦?jwocx�}~z{rtrq�Tª��_s?h¨j¯z{�g�grtp|vwz�x�pqsTo~��~z?�_ji��s?o¹±²�´³�µ�z{��j%uTpq¦Tjwo�©���s?�Vs{x�ptvi�wª§p|��sTrtz{x_pts?o�s{­&��sTrq��ogsTh\p|z{r��jiz?rA�_s�s{x���¿·���=z?o�z?rquTs?��p¥x�egh�v�sThD�gptogptogu¯����hD�Vs?rtp|vz?o~��oc}~h\ji�_p|vwz?r�v�sTh\�~}Xx�z{x_pts?o~��À-z{o~��z{�g�~�_sÁ¬Xpq¼h¯z{x_pts?o�s{­��_jRz{rA�_s�s?x��&£"pqx_e²z{���gpqx_��z{���¨zTvwvw}g��zTv��T°Â²ÃcÄAÅ�ÆDÇ ��È��?¡ z?�_�~p¥x���z?�_�¯�g��jivwpt�_pts?o=ptocx_ji�_¦�z{r�z?�_pqx_e~h\jwx_p|v{ªg��jwrtp|z{�gptrqpqxF�=z?o~��zTvwvw}g��zTv��

∗ ÉXÊ�ËTÌ�Í�Î�Ï#дÑ�ÒÁÓ Ô�ÕFÖO×OÓ Ø®ÙºÚ�Û'É�ÓÝÜÝÜÝÕ¤Ê�ÒÁÞ5ߧÎ�à�áÁâ�ã�Î�áiÉXâ�ä·Î�à�ä®ÍæåÁÖçÚ�èOÕFéêبÍ�ÖOÕFÒ�Ê�ÓÝÖçÕ�ÐyÉ�ä·ëgдì~éFÚ�ÜÝÕ�Î�Ú�ÖOí�Ê�ÜÝÕáiîÁå{ïFÖOÓÝÕFîÁÖçÕ#ÞRÕLÉcÙ�Ú�ÒcÐ�ð�ÖOÊ�ÒÁéFÕ�Ð{ñ�ò�ó�ôwò�õiö�÷Xøtùw÷�úwû�õ�ü�÷�ýRþwÿ_õ��wû_ý~ø����† ë�ÖçÚ�èOÕFéêØ�áiå�Ê�éFÕF×_Ð�ÉgÏ�à�ä®ÍLâ�ä·Î�à�ä®Í�â�É�ä·ë��RÐRð�ÖNÊ�ÒÁéFÕ�Ð�wò���Áö���÷Xøtùiû� �ö_õ�õiö�÷���ü�õ�û��Áö�òXø����ßAÕ�ØOÕ��wØOÕ�ÕF×Oاï���Ê�ÜÝÕFí"ÕFÒwØ�ÞÁÓÝ×Oå{Ú�ÒÁÓÝËÁÜÝÕ�×NÚ�îÁ×AÛqÚ�ÖOí"Õ�ÞÁÕ�ÖOÊ�åÁå{Ú�Ö·Ø�ÞRÕ�ÖçÕ�é��ÁÕFÖOé��RÕ´ÞRîyÉ�Ê�Ë?Ú�ÖOÊ�ØOÚ�ÓÝÖOÕ;ÞÁÕ�Ü�� ä®ÒRÛtÚ�Öçí�Ê�ØOÓ��iîÁÕÞÁî�ëgÊ�ÖOÊ�ÜÝÜÝï�ÜÝÓÝ×Oí"Õ�ÐRéêÛçÌXô�ó�ó��������������~ø®÷�ýRþwÿ_õ��wû_ý~ø���� ��!#"�$�Ì

���������'������ ��� ������������ �������*�+��-�������� �� � ���%�� &���' (w(� "� "������"!#�w����� � ����������� ���� -�������"�� "�%�������� ��.� (�,������(w������*�����

��13254

�� ����� ¡�� j�x�z{�_x_p|v�rtj"�F}~��x_pq¢~j\rqj%��jR��sTpqo¸��� }go~j\z?�_pqx_egh¨nwx_p|�T}~jD�~z?�yptoTx�jw��¦�z{rtrqjR��jwo9�g�_nRv�p|��pts?o¸z?��¼�~p¥x���z?pq��j¯©&��sT}g�"­·s?}~�_ogpt�'�gji����ni�_}grqx�z{x����~�_nRv�p|�wª�}go¸vwz?rtvw}gr��~z?�yptocx_jw��¦�z{rtrqjR�"��ji�c}gptjw�_x'�Xji�ypqocx�jw��¦Áz?rqrtji�jio�jiocx_��nwj��c}gp���sTpqjiocx#¢~o���«gv�� ji��x"��sT}g��v�jwx�x_j���z{p|�_s?o=�c}gj�r|z%�~pt����jRv�x�pqsTo�jR�Fx"}go��g��s�vwni�Xn��_p���sT}g¦?jiocxjih¨�grqs��Tn�~z{o~�&rtji��z?rquTs?��p¥x�egh¨ji�-�~z?�"pqocx_ji�_¦�z?rqrtji�i°��zD�~pq�grtpts{x_e~fi�c}gj�±²�´³�µ-z%n�x�n�gnw¦?jirqsT�g�Vnwj��sT}g�&�_ni��sTo~�X��j��\v�j��jR��sTpqoº©�jwo�j��Aj�xRªgz{}~vw}gogj�gpq�~rqpts{¼x�egfi�c}gj-j�¬Xp|�Fx�z{ocx_j#o�� s��A��z?p¥xL�Xj&���Vnivwp¥¢�viz�x�pqsTo~����z�x�pt��­çz{p|��z{ocx_jR�w°���ji��vwz?��zTv�x�nw��pt��x_p|�c}gji���Xj-vwj�x_x_j-�gpq�~rqpts{¼x�egfi�c}gj��_s?ocxL��z{�gp|�Xjwh¨jioTx;�XsTogogniji���g}gp|�L}go~j�v�s?h¨�~z?��z?pt�_s?o\zÁ¦?jRv´}go~j"�gpq�~rqpts{x�egfi�c}gj���� z{��pqx_egh¨n�x�pt�c}gj��z{�;ptocx_jw��¦�z{rtrqjR�Ljwo¯�~�_nRv�p|��pts?o\¢g¬Xnwj"jR�Fx#h\jiognwj'��}g�#}go¯�g��s?�grtfwh¨j��_��nRv�pq¢��c}gj ©�jwrtrqjyh\jwx´jwo¯ni¦�pt�Xjio~v�jrtj­çz{pqxy�c}gj�rqj��_}g��v�s��gx&rtptn��%r|z\u?jR�Fx�pqsTo��Xj�r|z\�g��niv�p|�_pqsTo=hD}~r¥x�pq�grtj�ji��x"x_sT}Xx �%­çz{pqxyz?viv�jw�gx�z{�~rqjT°��sT}g�;x�jw��h\ptogji�iª��c}gjwr|�c}gji�-z{�g�~rqp|vwz{x_pts?o~�´�~z?�_nwjR�;��}~�-±²�#³�µ#��sToTx#�g�_nR��jiocx_nwjR��©��_sT��s?x_p|�c}gj?ªcpt�_s?r|z�x�pqsTo�gji����z?vwpqo~ji�"��nwjirqrtji���gjD�Vs?rt�co"!?h¨ji� ¿ç�~z{�'}go9z?rquTs?��p¥x�egh¨j v�s?h%�gpto~z{ocxyvwz?rtvw}gr��_��hD�Vs?rtpt�c}gjDj�x'viz{r|v�}gro�}gh¨ni�_p|�c}gjRÀ#j�xyz?�g�g��sÁ¬Xpqh¯z�x�pqsTo=zÁ¦?jRv}gogj��g��niv�p|�_pqsTo�z?�_�gpqx_��z{pt��j�Xj$#inw��sT�-�_nijwr|�w°% Ç �R� Å  �& ��¡ z{��pqx_egh¨n�x�pt�c}gj��~z?�"pqocx�jw��¦Áz?rqrtji�-jio��~�_nRv�p|��pts?o�z{���gpqx_��z{pt�_jTªcviz{r|v�}gr§ucz{��z{ocx_pAj�x��~�_nRv�p|�

�������������� ��������� ��������������������! ��"����� #�� ��$�����%�&'�����)(+*,������ - �.,�0/2143 �

576�8:9�;-<=8:9>6@?BADCE6�FG<�?H<�FJIK9!8:FL<�FNM7OKF'PRQR9�A:9�6S?T9�?-8LPRF�;-<�U<�FJ9�84VBWXPY8:9>QZ<[?K\]8:VKP^5`_badceU�9�IKF:<fF�M

7�8;:¤<T>V@AB9UVS�Y Z\8L:¤@A]�MNBOBNMO<cQgV]�B�hjilk�k�i

monLprq$s:t'prqun�v=wyx!n2z|{�}=t:v=~�quv=~�t4z�q$pr}=���Nprqu{�ys�£-zT�gzÁ�X�wªAv�sTh¨�g}Xx�z{x_pts?o~�pto�¦?s?rt¦?j\h¨s?��j\z?o~�¸h¨sT�_j\s?�Vjw��z�x�pqsTo~��z{o~�ºv�sTo~��jR�c}gjwocx_rt��jw���_sT���i°Dd&egjrtpth\pqx��&s?­�z?�g�grtptviz{�gptrqpqxF�¯s{­���sTh\jo�}gh¨jw��ptviz{r§z{rtu?sT�_pqx_e~h¨�-z{��jogs�£ ��jiz?v�e~ji�§©�­·s?�&pto~�Fx�z{o~vwjyx�egjx_egjis{¼��j�x�ptviz{r~��x�z?�gptrqpqxF�Ds?­§zD�Xjwo~�_j�h¯z�x��_pq¬D­çzTv�x�s?��p #Rz�x_pts?o�¿ �'� sT����yÀ�p|�;jwo���}g��ji�¨}~o~�Xjw�;x_egjz?����}gh¨�Xx�pqsTox�e~z�x

n3u < 1ªX£"e~jw��j

npt�-x�egj��Xpqh¨jio~��pts?o�s{­�x_egj�h¯z�x��_pq¬�z?o~�

u = 1+ − 1ªX£"pqx_e

1+x_e~j��_h¯z{rtrqjR�Fx

»�sTz�x�pqo~u{¼½��sTpqocxyr|z{��u?ji��x_e�z{o¶�T«Ax_egp|�h\jRz{o~�yx_e�z�xnhD}~��x�Vj\rqjR�_�yx_e~z?o�k{�T�gª �?�T�gª~£"e~ptv�e¸p|��z{rth¨sT��x

��jizTv�egji������h¨sX�Xjw��o��_pqh%}gr|z�x_pts?o��w°�d&egj\oc}~h\ji�_p|vwz?r��c}~z?rqpqxF��s{­#�_s?rt¦?ji���ypt�'ogs�£ z{o9pt����}~j?ª�z{o��²ogs{xsTogrt�Dx�egjwpt�#h¯z{x_egjih¨z{x_p|vwz?rV�c}~z?rqpqxF�?°���jwx#}~�-v�pqx_j�Fx�}~�Xptji�#��ji��­·sT�_h¨ji�¨���\x_e~j � ¾´�6¿ç³~�_jio~v�e���}�v�rtjiz{��yu?jio~v��gÀ�s?o¨x�egjy�_pqh%}gr|z�x_pts?o¨s?­Ao�}~vwrqjRz{�;�~rtz?oTx´zTvwvwpt�Xjiocx��´z?o~�¨z{r|��s �_s{­®xF£&z{��ji�Lvws?ocx_��s?rtrtpqogu�z?o~�¨��sc�F¼�_pt�grq��v�s?����jiv�x�pqo~u%o�}gh¨jw��p|vwz{r§�~�_sTu?��z{h¯�wª��_}~v�e�zT� � z?�Xo�z0� �%��sT� � jwo~z��Ýk���½°

�yogs{x�egjw�&z{�g�~�_scz?v�e¨v�s?o���p|�Fx��;pto=v�sTh¨�g}Xx_ptogu%£"p¥x�e=v�ji��x�p¥¢~jR�¨jwo~vwrqsc��}~�_jR�wªTo~z{h¨jwrt�¨pqocx�jw��¦Áz?r�z?�_pqx_eX¼h¨jwx_p|v0�ÝkE��ª�kXª�kX���N°¯d&egj\­·}go~�~z{h¨jwocx�z?rL�g��pqo~vwpq�~rqj¨s{­#x_egp|��z{��pqx_egh¨j�x�ptv¨vws?o~�_pt��x��pto���jw�~rtzTv�ptogu�ji¦?ji�_�o�}gh%��ji���c�¸z?o9ptocx_jw��¦�z{r�jwo~vwrqsc��ptogu�p¥xR°%³gsT�'pto~��x�z?o~v�jTª

πvwz?ogogs?x'�Vj\jw¬gz?v�x�rq�²��jw�~�_jR��jioTx�ji�²}���ptogu�z

�~pqo~z?�_��s?� �XjRv�pth¯z{r#z?�_pqx_e~h\jwx_p|v{ª��g}XxDpqxDp|� v�ji��x�p¥¢�ji�ºx_e�z�xπ��jirqsToguT��x_s

[3.14159, 3.14160]°%d&egj

zT�X¦�z{ocx�z?u?jR�&s{­Lpqocx�jw��¦Áz?r�z?�_pqx_egh¨jwx_p|v z{��j�o�}gh¨jw��s?}~�i°=�'o�x_egj%s?ogj e�z{o~�§ª�p¥xj�¬Xegpt�gpqx��yx_egj �~�_sT��ji��xF�s?­L�%�&���.�����!.ys?�� !������� ����.� !�*=�l������ ��?°��'o¨x_egj�s?x_egji�´e~z?o~�§ªcv�s?h¨�g}gx_jw�#pqh¨�grtjwh¨jioTx�z�x�pqsTo~�´z?�_j"�~zT��jR�sTo)sT}XxF£&z{���¶��s?}go~�gpqoguc�¨z{o~�)x_e�}~�¯vws?h¨�g}Xx�ji� ��ji�_}gr¥x��\x�z��?j�pqocx_s¶zTvwvws?}gocx¨�_sT}go~�Xptogu¹jw���_sT���\z{o��vws?o~��x_pqx_}gx_j�z¸£&zÁ�9x�s�jR�Fx�pqh¯z�x�j¤x�egji�_j�ji�_��s?���i°�� rtzT�Fx¨z{o~�5¦?jw���ºpqh¨�Vs?�_x�z{ocx¨z?�X¦�z{ocx�z{u?jTª�jw¦?jio¶pq­pqx p|��s{­®x�jwo¹rqjR�_����ogs�£"o�ª�p|��x_e~z{x�x_egp|� z?�_pqx_egh¨jwx_p|v¨�g�_s�¦�p|�Xji�uTrqsT�~z{rLptoX­·s?��h¯z�x�pqsTo�©­·s?� pto~��x�z{o�v�j?ª�p¥x�~�_s�¦�p|�Xji�;x_egj'��z?ogu?j�s?­�z�­·}go�v�x_pts?o�s�¦?ji�´z £"egsTrqj��jwx

Sª�£"egp|v�e¤p|�#vw�_}�v�p|z{rg­·sT�#uTrqsT�~z{r�s?�Xx�pqh¨p #iz{x_pts?o�«

­·}~��x�egjw��h¨s?��j?ªcp¥­�x_egp|�-��z?ogu?j'pt�&zD�_}g�~�_j�x&s{­Sª�x_egjio0�-��s?}�£#ji��� �Lx�egjwsT�_jih �Fx�z�x_jR�#x_e~z{x#x�egpt�#­·}go�v�x_pts?o

e�z?�&z%¢g¬Xji��¼½��sTpqocx�z{o��¯x_e~pt��viz{o���j�}~�_ji��������ji£&x_s?o�� �"z?rquTs?��p¥x�eghæ­·s?�"pto~��x�z{o�v�j?°2��}�v�e��g��s?�Vjw�_x_ptji�viz{ogo~s{xy�Vj%��jizTv�egji��£"pqx_e~s?}Xx���jwxv�sTh¨�g}Xx_ptogu�ªAz?o~�²ptocx_jw��¦�z{r�z{��p¥x�egh¨j�x_p|vDvws?h¨�g}Xx�ji�'£"p¥x�e9�_j�x��z{o��p|�&jizT��ptrt�=zÁ¦�z{ptrtz?�grqjT°�ys�£#ji¦?jw�RªApqoI�_�gpqx_j¨s{­#x_egj¨pth\�~�_s�¦Tjwh¨jwocx��'ptoºptocx_ji�_¦�z{rLz?o~z{rt�X��p|�wªAx�egj¨�g��s?�grtjwh s?­´s�¦?ji�_jR�Fx�pqh¯z{¼

x�pqsTo�ª��������;s?­�jio~v�rtsT�_}g��ji�-£"egp|v�e²z{��j'­çz?�&x_s�s¨rtz?�_uTjz?o~��x_e�}~�"pto~zTvwv�}~��z{x_j?ªX�_jwjih¨�-x�s¯��j�x�egj �XjR�Fx�pqo��s?­"pqocx_ji�_¦�z?r#v�sTh¨�g}Xx�z{x_pts?o¶£"e~jwoIpqx%p|�Dpth¨�grqjih¨jwocx_jR�I}~�_pqogu¸¢g¬Xji��¼½�g��jiv�p|�_pqsTo�»�sTz�x�pqo~u{¼½��sTpqocx%z?�_pqx_eX¼

à�à�Ò r¡�¡�¢�£

� �����=���%��&' �.�1����=������&�&������

h¨jwx_p|v{° � sTh¨�g}Xx_ptogu¸£"pqx_e)pqocx�jw��¦Áz?rt�D�g��s�¦cp|�XjR��u?}~z?��z?ocx_jwjR�º�_jR��}grqx��iª��g}Xx\x_egj���sT}go~�g�\vwz?oI��j=­çz{�z?�~z{�_x&jw¦?jio�£"e~jwo�x_egj�ptog�g}Xxy�~z�x�z¨z?�_j�g��s�¦cp|�XjR�¤£"pqx_e�x_egj�h¯zTv�egpqo~j�~�_jRv�p|��pts?o�«Xz\��jwh¨ji�X�¤­·s?�&x_e~pt��~egjwogsTh¨jwogsTo5v�sTo~��p|��x��%pqo v�sTh\�~}Xx_ptogu9£"pqx_e z¸e~pquTegjw�¨�g��jivwpt�_pqsTo�°Id&egp|�%�~�_sT��sc�_z?r´p|�Dx�egj�v�s?��j=s?­x�egj=±²�#³�µ�rtpq�~��z?�_�5¿ �B��&���� ��& �,/����! ��"�!���� 1�& ������ ���� ����� �� 3� ��$�����& �����)(+*,������ 0&����������wÀ�ªLz�rtpq�~��z?�_�pth¨�grtjwh¨jwocx_ptogu¤z?�_�gpqx_��z{���¨�g��jiv�p|�_pqsTo�ptocx_jw��¦�z{r§z{��pqx_egh¨j�x�ptv�£"e~ptv�e�pt�y�Xji��v���pq�Vji��pto�x�egp|�"�~z{�Vjw�R°

d&egp|�"�c}gji��x"­·s?�&jw¬cx���z¨zTvwv�}~��zTv��¯vwz?o��Vj�­·s?}go��=pto�s{x_e~jw�&£#sT���X�&�_}~v�e�zT�[� �%��£"egjw��j���sTrq��ogsTh\p|z{rjw¬X�g�_jR�_�_pts?o~��z?�_j¯�_�ch%��sTrqp|vwz?rqrt�9��jw£"��pqx�x_jio¹��jw­·s?��j¯��jipqogu9jw¦�z{rt}~z�x�ji�§ª��_s¸z?��x�s²�_jR�X}~v�j¨x�egj¤s�¦Tjw��ji��¼x�pqh¯z{x_pts?o �X}gj�x_s¶�gjw�Vjwo~�Xjio~v��Tª&s?�¯���j� ��y£"egji�_j�egptu?eg¼NsT���Xji�\d�zÁ��rqsT�¨j�¬X�~z{o���pts?o~�¯z?�_j�}~�_ji��° µêox�egp|�yr|z�x_x_jw�'£#sT���Aª~x_egj%x_pth¨jDs�¦?ji�_e~jiz?��p|�yz?��sT}Xx%���{�?�\­·sT�'z���ptogu?rtj�­·}go�v�x_pts?o9jw¦�z{rt}~z�x�pqsTo�ªVegs�£#ji¦?ji�pqx¤p|�¤v�sTh\�Vjwo��_z{x_ji�)���¶x�egj²�_jR�X}~v�x_pts?o pqo x_e~j²oc}~hD�Vjw�¤s?­���x_ji�~�¨��ji��­·sT�_h¨jR�5���¶x�egj¸z{rtu?sT�_pqx_e~h�°��jiz{rq¼½£#sT�_r|�¯z{�g�~rqp|vwz{x_pts?o~�-£"egji�_jj�¬�x_��zDzTvwvw}g��zTv��¨p|�#��ji�c}gpt��ji�=z?�_jyx_s\��j­·sT}go~��pqo�z{}Xx�s?h¯z�x�ptvi�'¿ç£#je�zÁ¦?j-��jijwo¯z?���?jR� x�s�pqocx_jiu?��z�x�j#rtpqo~jiz{�;�_�X�Fx�jwh¯��£"pqx_e¯egptu?e¨z?viv�}g��z?vw�XÀ�s?�;v�e~jwh¨pt��x_���A©��Xjwx_ji�_h¨ptogpqo~u�zh¨sTrqjRv�}gr|z{�Lv�sToX­·s?��h¯z�x�pqsTo jwocx�z?pqr|��x_e~j-h¨ptogpth\p #iz{x_pts?o\s{­Az{o\jwogji�_uT�'­·}~o~v�x�pqsTo¨z?o~�%�_jR�T}~pq��ji��z?viv�}g��z�x�jji¦�z{rt}~z�x�pqsTo~�-s{­�x_egp|�"jwo~jw��u?�¨­·}go~v�x_pts?o�°

±²sc�Fx%}~��}�z{r´ptocx_ji�_¦�z{r#z{��p¥x�egh¨j�x_p|v¯rtpq�~��z?�_ptji�¯¿ç�;�_s?¢~r���-µF��� �Ýk{�%�½ª�µêoTx�rqpt�j�t�%�>�½°w°i°�À s?�Dvws?h¨�gptrqji���¿���}go²³gsT��x�j?ª'��� ��� �

++� ��%�®À&z?�_j��~zT��jR�=sTo .E�����& �&»�sTz�x�pqo~u{¼½��sTpqocx&oc}~hD�Vjw���"z{o~���Xs¨ogs?x��g�_sT��sc��j

z?�_�~p¥x���z?�_�I�g��jivwpt�_pts?o5ptocx_ji�_¦�z{r'z{��p¥x�egh¨j�x_p|v{° d&egj�[� � r|z{oguT}~z{uTji��� kg���ypto~v�rt}~�Xj¸z¹rts?o~u¹zTvwvw}ghD}X¼r|z�x�s?��xF���Vj¯­·s?�\z?viv�}g��z�x�j¤�Xs{xD�~�_sX�X}~v�xiª�£"egp|v�eIe~z?�%z{r|��s²�Vjwjio¶}~�_ji�º­·sT����sT�_o~jw� ji¦Áz?rq}�z�x_pts?o¹s?­"z�Vs?rt��ogs?h¨p|z{r"pto x_egj²ptocx_jw��¦�z{r �yjw£&x_sTo z?rquTs?��p¥x�egh �Ýk?k��½°j��s�£-jw¦Tjw�RªLx_e~pt�¯rts?o~u¹zTvwvw}ghD}gr|z�x�s?�\xF����jp|�\ogs{x¨ptoTx�jwo~�gji�I­·sT�¯z¸u?jiogjw��z{r-}~�_j=x�eg��s?}guTegs?}Xx%x_egj²vws?h¨�g}Xx�z�x_pts?o�ª;z{o~�¶­·s?�¨pqo��Fx�z{o~vwj�pqx\p|�\ogs{x�VsT���_pq�grtj\x_s¸vws?h¨�g}Xx�j¤z²�Xpq¦�p|��pts?oºs?� z�rts?ucz{��p¥x�egh }~�_ptogu�rqsTogu²z?vwvw}ghD}~rtz{x_s?���i°¨³~}g�_x_egji�_h¨s?��j?ª�x_egjh¯z{¬�pth¯z{r&�g��jivwpt�_pqsTo5p|�¯rtpqh¨pqx_ji��ª-�_pqo�v�j�x_egj�rts?oguIzTvwv�}~hD}gr|z�x�s?�%xF�c�Vj²��jwrtpqjR�%sTo5x�egj¸�XsT}g�grtj�xF���Vj?ª£"e~sT�_j��z{o~u?j�s{­Lj�¬X�Vs?ogjiocx��&p|�

{−1022, · · ·1023}°���jw¦Tjw��z{r§h%}gr¥x�pq�~rqj �~�_jRv�p|��pts?o�pqocx�jw��¦Áz?r��~zTv!�Áz?u?jR�

z?�_j�zÁ¦�z{ptr|z{�grtj?° ��j�x�}~�'�c}gs{x�j�­·s?�"pto~�Fx�z{o~vwj�pqocx��~z����q�R�%��z?o~��pqocx_��z��� �q�Ák���­·sT��±¸z{�~rqj�sT�yz¯��pth¨pqr|z{���z?v!��z{uTj�­·s?��±¸z�x�egjwh¯z�x�ptvizB�Ýk{��½°� '}gj%x_s�}go�¦?jw��pq¢~ji�²zT�_�_}gh¨�Xx_pts?o��ysTo�x_egj\�_sT}go~�Xptoguc��s{­;jirqjih¨jwoX¼x�z{����­·}go~v�x_pts?o~�¯¿O�g° ��}~rq��­·s?��ptoTx��~z��9pqo¹±¸z{�~rqjTª;�¨}grq��­·s?��±¸z{x_egjih¯z�x_p|vwzcÀ�ª§x�s��g}gu?uTji�9�_sT}go~�XptoguT�¿·­·s?��pto~��x�z{o�v�j?ª�£"p¥x�e¶���Xjivwpqh¯z{r#�XpquTp¥x��wª�x_egj¯��s?}~o~�Xptogu�x_s�£&z{���g����s?­

1 − 9.10−5uTpq¦Tji�¨�¨pto~�Fx�jizT�

s?­��~° �T�?�%pqo²±¸z?�grqj�¦��¨z{o��=¦��{À�ª~z{o��¤x�s¤��ji¦?jw��z{rAs?x_egji�"}go~�X}gj zT�_�_}gh¨�Xx�pqsTo~�wª�x�egji�_j��~zTv!�Áz?u?jR�-viz{oX¼o~s{xD�Vj=v�sTo~�_pt�Xji�_jR�¹z?� ��jwrtp|z{�grtj?°��yx_e~jw� £-s?���X��pqo�v�rt}~�Xj¯x�egj�����z?ogu?j¯z?�_pqx_egh¨jwx_p|v��K�q���-z{o��¹µêoTx ��z?�� �?��N°��� +�E��p|� z�h%}gr¥x�pq�~rqj¤�~�_jRv�p|��pts?oºrqpt�g��z{���¸£"egp|v�eIz{pth¨�Dz�x pto~�Xp|vwz�x�pqo~u�x_egj¯o�}gh%��ji� s{­"v�sT�_��jiv�x�gpquTp¥x��'��z�x�egjw�x�e~z{o¹z�x��Vjw�_­·s?��h\ptogu�pqocx�jw��¦Áz?rLz{��p¥x�egh¨j�x_p|v{ª�z?o~��£"egptv�eº}~�_ji�r|z{��u?jDs�¦Tjw��ji��x_pth¯z�x_pts?o��x�s\vws?h¨�g}Xx�jx_egj���z{oguTjys{­�zD��ji�_}grqxi«Xz?viv�sT���XptoguDx_s¨p¥x��&z{}Xx�egs?�Rª���s{­�v�s?}~���_j?ª���z{oguTj'z{��pqx_egh¨j�x�ptv'pqx��_jwrq­p|�&z ­·sT�_h s{­�ptocx_jw��¦�z{r§z{��pqx_egh¨j�x�ptv?ªX�g}Xx"egji�_j'x_egj�ptocx_jw��¦�z{rAp|�&s?ogrt�¤v���}~�Xjirq�¯��jw�g��ji�_jwocx�ji���{°�d&egjµêocx�¼��z{�¨rtpq�g��z{��� �g��pth¨z?�_ptrt� pqh¨�grtjwh¨jioTx��;j��¯vwpqjiocx_rt�DptoTx�jw��¦�z{r~z?rquTs?��p¥x�egh¯��}~��ptogu ±¸z{x ��z{��ªcz{o~�§ªT��jR��p|�XjR�wªh¯z?pqogrt�¶�g��s�¦�pt�gji�¯z�xF����j�­·sT�¤z{���gp¥x���z?�_�I�~�_jRv�p|��pts?o v�sTh\�~}Xx�z{x_pts?o~�¨�g}gx¤pqh¨�grtjwh¨jwocx��¯­·jw£���jwr|z�x_jR�­·}~o~v�x�pqsTo~z{rtp¥x�pqjR�w©�pto=¦Tjw����pts?o¤�~ªcsTogrt�%x�egj�j�¬X�Vs?ogjioTx�ptz?r~­·}~o~v�x�pqsTo�z{o��¯x_e~j

πvws?o~��x�z?ocx-z{��j'zÁ¦�z?pqr|z{�grtj?°

�y�_�~p¥x���z?�_���g�_jRv�p|��pts?o²ptocx_ji�_¦�z{r�z{��p¥x�egh¨j�x�ptvD£-zT�yz{r|�_s¯h\jiocx_pts?ogjR�¸z?�'z{o¸jRz?�_�T¼Nx_s?¼Npth¨�grtjwh¨jwocx�j�¬�x�jwoX¼�_pts?o�x_s��-�_jioTx � �yhD}grqx_pt�grtj%�~�_jRv�p|��pts?o²�~zTv!��z{u?j ±¸� z?�'jiz?�_rt��zT� �R�?�g�,�����½«§egs�£-jw¦?ji�iª��#��jwocxye~pqh¯�_jwrq­zT�X¦�pt�_ji�yx_s�}~�_j\z?ogs{x�egjw��~z?v!��z?u?j�x�e~z{oº±²� o~s�£�©���±²� pt�ogs�£3sT�~�_s?rtji��v�jwocxR°��Lji�_��­·ji£ v�e~z{o~u?ji�x�s=x�egj¨v�sX�Xj\s?���XsXv�}gh¨jwocx�z�x_pts?o9e~zÁ¦?j%��jijwo9h¯zT�Xj\�_pto~v�j��i�T�g�����t°q°t° �¤µêoºu?jiogjw��z{rNªV£#j\�_jRv�sTh\h¨jio~�x�egj¯}~�_j¯s{­&z�h\sT�_j¨h¨sX�Xjw��oº�~z?v!��z{uTj?ªA­·sT� j�¬gz{h¨�grtj� 'zÁ¦�p|���&zÁ��rtjw� � ��±¸�;�3�~z?v!��z?u?j\s?� ±²�´³�� �¿Ov�­F°"!�#$#$%'&)(�(+*-,�.'/�021�3�4$5+.'/6178/95-0"/;:�<-(�1�:=024�(+*-12>�<-(+>=?0�!@5�>�AB/;.�>@,2C�#-(�%�:$.D(+%$:�.�EF�G8/H!�#23I4cÀ�°l��pto~v�j

ä·Î�à�ä®Í

�������������� ��������� ��������������������! ��"����� #�� ��$�����%�&'�����)(+*,������ - �.,�0/2143 �

o~s?ogj¤s?­"x_egj�z�­·sT�_jih¨jwocx_pts?ogjR�º�~z?v!��z{uTji�Dpqh¨�grtjwh¨jwocx��%z¸vws?h¨�grtj�x�j�z{o��I�_jRz{rtrq�º��jwrtptz?�grqj�z{���gp¥x���z?�_��~�_jRv�p|��pts?o�pqocx_ji�_¦�z?r�z{��p¥x�egh¨j�x_p|v{ªgx_egp|�"rtji��}~�&x_s¨pth¨�grqjih¨jwocx�s?}g�"s�£"o�rtpq�g��z{���?°

µêo¨x�egj�ogjw¬cx-��jRv�x_pts?o�ª{x_e~j�ogjwjR�\­·s?�´z?�_�~p¥x���z?�_� �~�_jRv�p|��pts?o\pqocx�jw��¦Áz?r~z{��pqx_egh¨j�x�ptv�pt���F}~�Fx�p¥¢�ji�\­·��s?h zx�egjwsT�_jwx_p|vwz?r��Vs?ptocx&s?­�¦�ptjw£ z?o~�=z?rt�_s%���¤vws?o~�_pt�gjw��pqoguDx_egj�}~�_j�s{­��~pt�_jiv�x_pts?o�pqo�pqocx_ji�_¦�z?rAz?rquTs?��p¥x�egh¯�w°d&e~jwo�x_egj�±²�´³�µ�¿$�B��&���� �l& �f/����� ��"����� B12& �>����� +�2� ����� ��:3� ��$�����&'�������(�*-������ @&�����������wÀ&pt�&�~�_jR��jioTx�ji�§©¢�����x_rt�¤x�egj ��ji�c}gpt�_jih\jiocx��&s?o�x�egj }go~�gjw��rq��ptogu¨»~sTz{x_ptogu{¼½�Vs?ptoTx�z?�_pqx_egh¨jwx_p|v z{��j�j�¬X�gr|z{ptogji��z{o��=x�egjwox�egj ­·}~o~v�x�pqsTo~z{rtp¥x�pqjR�'z?o~�²pth¨�grqjih¨jwocx�z{x_pts?o9v�egsTptvwji��z?�_j%�Xjw¦Tjwrts?�Vji�§°���s?h¨j �Vjwo~v�e~h¨z?���X�ys?o�m�z?}~�F¼�_p|z{o�jwrtpqh¨pto~z�x�pqsTo���ji��­·sT�_h¨ji��sTo�±²¼Nh¯z�x��_p|v�jR��z{rtrts�£ x_s¯h¨jRz?�_}g�_j�x�egj s�¦Tjw��egjizT�¯­çz?v�x_sT�y�X}gj�x�s¯x_egje�z{o~�Xrtptogu¯s{­�hD}grqx_pt�grtj �g��jivwpt�_pqsTo�o�}gh%��ji���iªg£"egp|v�e�pt���¨v�sTh\��z{��ji�=x_s¨x_e~j�¢g¬Xji��¼½�g��jivwpt�_pqsTo�ptocx_ji�_¦�z{rrtpt�g��z?�_���I?���4-? .§«cx_e~j�j��Ajiv�x���s?­�pto~v���jizT��ptogu\x_egj��~�_jRv�p|��pts?o�z{��j�z{r|��s¯�_egs�£"o�° ��zT�Fx�rq�Tª~�_s?h¨j�z{�g�grtp|vwz�¼x�pqsTo~�"}~�_ptogu¤±²�´³�µ�z{��j��g��pqjw»~���~�_jR��jioTx�ji��pto�sT���gjw�"x�s¯pqrtrq}��Fx���z{x_j�x_e~j ¦�z{��pqsT}~�"ogjiji�g��z?o~����jiogj�¢~x��x�z��Tjwo�­·�_sTh x_egp|����pqo���s?­-z?�_pqx_egh¨jwx_p|v{©�u?}~z?��z?ocx_jwjR�9�_s?rt}Xx_pts?o��'s?­&z��g��s?�~rqjih ��sTrq¦�z{�~rqj¨ogjip¥x�egjw�����jw¬gz?v�x\ogs?�¨���¹}~�_}~z{r&o�}gh¨jw��ptviz{r-h¨j�x_e~s��~�wª´zTvwvwjwrtjw��z�x_pts?o5s{­'z?o¶j�¬gz?v�x\h¨jwx_egsX�5z?o~�¶e�z{o~�Xrtptogu�s?­pto~zTvwv�}~��z{x_j��gz�x�zgªg�_s?rt}Xx_pts?o�s{­Lz\�g�_sT�grtjwh�£"pqx_e²z{���gpqx_��z{���¤z?viv�}g��z?vw�?°

� } �Jn2zr�Nprqu{�t����=tL{ ~2zrn��=v� d&e~j�x�egjwsT�_jwx_p|vwz?r-��ji�_}grqx¯}go~�Xji�_rt��pqogu�x_e~pt�¨p|�Xjiz¹viz{o ��j�­·s?}~o~� pqo � k?�%�½©�rtj�x¯}~�¯�gjwogs?x_j����

Xz?o

ptocx_ji�_¦�z{rNª{���fzy­·}~o~v�x�pqsTo¤z{o~�%���

Fz?o%ptocx_jw��¦�z{rXjw¬cx�jwo~�_pts?o\s{­

fª{£"egjw��j

Fp|��u?pt¦?jio%���Dz$��pq�~��v�egpqx #

jw¬X�g�_jR�_�_pts?o�ª;rtj�xεvws?���_jR���Vs?o��¹x_sºx_e~j�vw}g���_jioTx¯v�sTh¨�g}Xx_ptogu¹�g��jivwpt�_pqsTo

ε = 2−pª´x�egjwo

F (X)s�¦Tjw��ji��x_pth¯z�x_jR�f(X)

z{o~��x_e~j�s�¦?jw��ji��x_pth¯z�x�pqsTo¤pt�&�Vs?}~o~�Xji�����

q(f(X), F (X)) ≤ c1w(X) + c2ε¿F�RÀ

£"e~jw��jqp|�%x_egjB�'z{}~���Xs?� � �Xp|�Fx�z{o~vwj?ª

w(X)pt�\x�egj�£"p|��x�e)s?­

Xz?o~�¶x�egj�v�s?o��Fx�z{ocx��

c1

z{o��c2�gjw�Vjwo~�9s?o

F°%d&egpt�h¨jiz?o~�yx_e~z{xx�egj¨v�s?h¨�g}gx_ptogu��~�_jRv�p|��pts?o9viz{o���jRv�s?h¨j¨z¤rtpqh¨pqx_ptogu�­çz?v�x_s?��z{o��

x�e~z�x��Vjwptogu¯z{�~rqjx_s¯pto~vw�_jRz?�_j'pqxyvwz?o��Vj z{o�p|�_�_}gj?°³~}g��x�egjw��h¨s?��j?ª�zDvwrtzT�_�_ptviz{r��g�_sXvwji�X}g��jypto=ptocx_ji�_¦�z{rVz{o~z?rq�X�_pt�#p|�;x�egj'�~pt�_jiv�x_pts?o¤sTogj?©�pq­�x_egj's?}Xx��g}Xx

£"p|��x�e²pt��x_s�s¤r|z{��u?j?ª�x�egjwo¸x_egj ptog�g}gx��yz?�_j �_�grtp¥xpqo�xF£-s¸¿·sT��h¨s?��jRÀ-�~z?��x��yz?o~��x_egj\v�s?h¨�g}gx�z�x�pqsTo�pt���jw�Vjiz{x_jR��s?o²jizTv�e²�~z{�_xi° �-p|��jRv�x�pqsTo²pt�ys{­®x_jio²x_e~jDr|z?��xy�_jR��sT��x"x�s¤u?jwx'h¨s?��j z?viv�}g��z?vw�?ªg������ji�X}~vwpqo~upto ­·sT�_h%}grtz ¿F�ÁÀ¨x_e~jº�c}~z?oTx�p¥xF�

w(X)°6µêo viz?�_ji�¤£"e~jw��j

w(X) = u¿ç£"egptv�e e~z{�~��jiogji� pto sT}g�

jw¬X��ji�_pth¨jwocx���s?o\u?rts?��z{r�s?�Xx�pqh¨p #iz{x_pts?o�À�ª{s?ogrt� z{o\pto~v���jizT��j#pqo\x_e~j&vws?h¨�g}Xx�pqo~u�zTvwvw}g��zTv��TªR�����_z?�~�Xpqo~uo~jw£æ»~scz�x_ptogu?¼N�Vs?ptocxDo�}ghD�Vjw��� �¤�Vj�xF£-jwjioIx_egj�jwo��X��sTpqocx��Ds?­

Xª�£#sT}grt�¶e~zÁ¦?j¤��ptjwr|�Xji�5z¸��sTrq}gx_pts?o�°

�-p|��jRv�x�pqsTo�viz{o�h¯z{ptogrq����j �_jwjio²z?�"z¨£&zÁ�¨x_s¯ji��vwz?��j�x�egj�£"��z{�g�gptogu¨j��Ajiv�x����=�g��s�¦�pt�Xptogu¨z¨�~zÁ¦�pqo~us?­Lx_egj¯�_s?}guTecx'��jwxi©"x�egj\�~zÁ¦�pqogu�����z�}gogpts?o9s{­#�_h¯z{rtr��VsÁ¬Xji�vwz{o9��j¯z?�z?viv�}g��z�x�jDz?��XjR��pt�_jR�¸z{o��vwpq��v�}~hD¦?jiocx��"x�egj\s�¦?jw��ji��x_pth¯z�x�pqsTo²pqo~�g}~v�jR�¸����x_egj\pto~v�rt}~�_pqsTo¸pqocx�s�sTogrq��sTogj%�VsÁ¬§°f�#p|�_jiv�x�pqsTo9z{r|�_sz?rqrts�£��yx_s��_jR�X}~v�j%x_e~j¨�gjw�Vjwo~�Xjio~v��9�g�_sT�grtjwh�ªAjw¦?jio9pq­Fª�pto9x�e~z�x���ji�_��jRv�xiª§o~s{x_e~pqogu��_}g��ji���_ji�gji�'x_egj}���j�s{­Lz\u?s�sX�¤­·sT�_h%}gr|z�x_pts?o�°

µ½x#viz{o¨�Vj�ogs{x�ptvwji�%x_e�z�x;x�egj���}grqj"s?­�x�e�}ghD� ���$�������'*,����f.�� ���������L�� l�R(r�=�$�f�� l ����!%���[�)(r�� !�*���������� +� �����! ��"������ B���@��������(�&��S�)(r� ��*,�� ���*,�����=� �[.� �������H�yvwz?oD­çz?pqrNª{­·sT�;pto~�Fx�z{o~vwj"£"egjwo¯vws?h¨�g}Xx�pqoguzº���T}�z{��j=��scs?x%sT�\h¨sT�_j�u?jwo~jw��z{rtrq�Iz

1/n¼Nx_e¶�Vs�£-jw�¨v�rtsT�_j=x�sº�~° ��s�£-jw¦Tjw�Rª�x_egj��_}~rqj�s{­'x_e�}ghD�

�Vjivws?h¨ji�-pto��_}~v�e�vwzT��jR� ���$�@�����α*,�����.� ������������ l��(�%�f�$�-�� l ����!%���f�)(r�� !�*=�l������ ��[�����! ��"������ #���

������%(+&��nα

.� ������� ��ª�zT�&rqsTogu¯z?�-x�egjDv�sTh¨�g}Xx_ptogu¨�g��jivwpt�_pqsTo�jio~z{�~rqjR�&x_s¨uTz?pqo²�Xptu?pqx��&sT�"£"p¥x�e¶¿·p|�Xjiz?r|À

à�à�Ò r¡�¡�¢�£

� �����=���%��&' �.�1����=������&�&������

ptoX¢~o~p¥x�j-�~�_jRv�p|��pts?o�°�µêo%s?x_egji��£#sT���g�iª�pto%h¨sT��x;vwzT��jR��z?o%pto~v���jizT��j#pqo\x�egj"v�s?h¨�g}gx_ptogu'�~�_jRv�p|��pts?o%�cptjwr|�g�z?o�pth¨�g��s�¦?ji��z?viv�}g��z?vw�¯s?o�x_egj���ji�_}grqx��i°

d&egp|�¤p|�=z?rt�_sIx�egj9��x�z{�_x_ptoguI�Vs?ptocx�s{­ ±��~rqrtjw� � �¤£#sT���)s?o z{o j �VjRv�x�pq¦Tj9��pthD}~rtz{x_pts?o s{­DzI�"jRz{r�y�y± �Ýk{��Nª�­·sTrqrts�£"pqo~u²x_egj�x_e~jws?��j�x�ptviz{r#�_jR��}grqx��%��� �#��z�x_x���z¸z?o~� �yjw�_x_rtpqogu � ���&sToIx_e~j¤­·jizT��pt�gptrqpqxF�s?­;z¤�"jiz?r����'±I°Vµêo9±��grtrqji��� ��£-s?���Aª~z¤vws?h¨�g}Xx�z�x_pts?o²p|�'��ji��­·sT�_h¨ji�²z?o~�§ª�p¥­Lx�egj ¢~o�z{r�zTvwvw}g��zTv���pt�o~s{x���}��¤vwpqjioTxRªAx_egjio9x�egj\£"egs?rtj\vws?h¨�g}Xx�z�x�pqsTo9p|�'��ji��x�z?��x�ji�²£"pqx_e¹z{o�pqo�v���jiz?�_ji�¸�g�_jRv�p|��pts?o�«Ax_egp|�pt���jwpqx_ji��z{x_jR�¤}gocx_ptr§x�egj�s?}Xx��g}Xx���z{��j�z?vwvw}g��z�x_jjwo~s?}guTe�°

� } � m������ �uq ��zrt:z�µêo%s?���Xjw��x_spth\�~rqjih\jiocxLz?oDz{���gpqx_��z{���'�~�_jRv�p|��pts?o ptocx_ji�_¦�z{rXz?�_pqx_egh¨jwx_p|v{ª{z'hD}grqx_pt�grtj-�~�_jRv�p|��pts?o�»�sTz�x�pqo~u�Vs?ptocx#rtpq�~��z?�_�¨£&z?�;o~jwji�gji�§°L�-�¯hD}grqx_pt�grtj'�~�_jRv�p|��pts?o�ªcpqx-p|�´h¨jRz{ocx´x�e~z�x-x_egj�v�sTh¨�g}Xx_ptogu\�g��jiv�p|�_pqsTo¯pt�o~s{x&rqpth¨p¥x�ji�=x_s%x_egj��_pqoguTrqjs?�&�XsT}g�grtj'�g��jivwpt�_pts?o=s?­�h¯z?v�egptogj'»~sTz{x_ptogu{¼½�Vs?ptoTx-o�}ghD�Vjw���w«�sTo¤x_e~j�vws?oX¼x���z?�_�TªXz{���gp¥x���z?�_�¨�g��jivwpt�_pts?o~�&�_egs?}gr|����j zÁ¦�z{ptr|z{�grtj?°�³g}g�_x_e~jw��h\sT�_jTªcx_egp|��v�sTh¨�g}Xx_ptogu¨�g��jivwpt�_pqsTo=h%}~�Fx�Vj=�X��o~z?h¨ptviz{rtrq�ºz?�Á�F}~��x�z?�grtj¯x_s�­·}grq¢~rtr´x�egj=zTvwv�}~��zTv��¸ogjiji�g�i°²��ogs?x_egji� ��ji�c}gpt�_jih\jiocx�­·s?�Dptocx_ji�_¦�z{rz?�_pqx_e~h\jwx_p|v-pt��x�egj"s?}gxF£-z?���%�_sT}go~�Xptogu­çz?vwpqrtpqxF�V©�x_egp|�Ljio~��}~�_jR��x_e~z{xL­·s?�;jRz?v�e\s?�Vjw��z�x_pts?o�ªÁx_egj�ptocx_ji�_¦�z{rvws?h¨�g}Xx�ji��}~�_ptogu¯»~scz�x_ptogu?¼N�Vs?ptocx'z{��p¥x�egh¨j�x�ptv vws?ocx�z?pqo��&x_egj%pqocx�jw��¦Áz?r�sT�Xx�z?pqogjR��p¥­Ljw¬XzTv�xy��jiz?r�z?�_pqx_eX¼h¨jwx_p|vD£-jw��jD}~�_ji��°�¾;¦?jio9h¨s?��j%�XjR��pt��z{�grtjDp|�,��+� ��L�Xpt�_jRv�x_jR�¸��s?}go��Xpqo~u¤x_s�zÁ¦?sTpt�¸rqsc��ptogu�zTvwvw}g��zTv��Tª��������x�egjyptocx_ji�_¦�z{rVv�sTh\�~}Xx_jR�¨}~��ptogu�»�sTz�x�pqo~u{¼½��sTpqocx#z?�_pqx_egh¨jwx_p|v"pt�;x_egj��h¯z?rqrtji��x´s?o~jD¿®­·sT�´pto~v�rt}~�_pqsTo�À�«e~s�£#ji¦?jw�RªRp¥xLp|����z?�_jirq�'­·}grq¢~rqrtji�%­·sT��jirqjih¨jwocx�z?�_�­·}go~v�x_pts?o~�i°�d�s���}~h }g��ª?vws?h¨�grtptz?o~v�j-£"pqx_e%x_egj-µê¾;¾;¾�����¨�Fx�z{o~�~z{���¯­·s?��»~sTz{x_ptogu{¼½��sTpqocx"z{��p¥x�egh¨j�x�ptv?ªXj�¬�x_jio~�Xji��x�s¯jwrtjwh¨jwocx�z?�_�¨­·}~o~v�x�pqsTo~�wª~pt�&£-jwr|v�sTh\jT°

d&egj �y�_pqx_e~h\sc�"�g��s{�FjRv�x"s{­�x_egj � ���yd x_jRz{h�ª�� °���ocxF£#ji�_�Vjwo�ª��-jwrtu?pt}gh � ��½ªgs?�&x�egj ±²�´³�� rtp¥¼�~��z?�_�D¿ �B��&���� ��& ��/����� ��"����� �1�& ������ ���� �r���� ��D����& �����& � �����)(+*,������ & ����������wÀ�ª{�Xji¦?jirqsT��jR���c�yx_egj ���~zTv�ji�x�jiz?h�ª�µ �'�"µF� ��s?����z?pqo~j?ª?³~��z?o~v�j��q�T���NªXz?�_jy�_}~v�e¤rtpt�g��z?�_ptji�i°�³gs?�-�Vs?�_x�z{�~pqrtp¥xF�¯z?o~�¯j �¤vwpqjio~v��¨��jiz?�_s?o��¿O±²�´³�� pt����z?�_ji� sTo¨m�±¸�¹z?o~� j��¤v�ptjwo~vw� pt�Lz'h¨s{x_x_s­·s?�Lpqx��L�Xji¦?jirqsT��ji����À�z?o~�Dz?rt�_s��jRvwz{}���j&s{­�x_egjzÁ¦�z?pqr|z{�gptrtp¥xF�²s?­´x_e~j¤��sT}g��vwj¯v�sX�Xj?ª�£#j¯v�egsc��j¯±²�´³���°�d&egj¤v�sT�_��ji�_��sTo~�Xptogu=rtpt�g��z?�_�Tª�o~z?h\jR��±¸�´³�µ­·sT���B��&���� �l& ��/ ���! ��"����� 1�& ������ ���� �r���� ��=3� ��$�����& �����)(+*,������ �&����������|� �g���½ª�pt�%z²��sT��x�z{�grtj¤rtpq�~��z?�_�£"��pqx�x_jio¤pto � ­·s?�&z{���gpqx_��z{���D�g��jivwpt�_pts?o¯ptoTx�jw��¦�z{rAz{��p¥x�egh¨j�x_p|vD¿O��sT}g��v�jyvws��gj'z{o��¤�XsXv�}gh¨jioTx�z�x�pqsTo=vwz?o�Vj ­·��jwjirq���Xs�£"ogrtsTzT�Xji��À�°�µ½xpt�'�~z?�_ji��sTo�x_egj¨m��=�6±²� rtpq�~��z?�_��z{o~��sTo²x_e~j%±²�´³�� rqpt�g��z{����z{o��p|�"�~z?��x"s?­�x�egj�rtz{x�x�jw�R°´d&egj�r|z{��u?ji��x"z?v�egptjw¦�z{�~rqj�v�sTh¨�g}Xx_ptogu¯�g��jivwpt�_pts?o�pt�"�~�_s�¦�p|�Xji�����=±²�´³��3z{o���gjw�Vjwo~�g� pqo¹�g��z?v�x_p|v�j¨s?o�x_egj¤vws?h¨�g}Xx�jw� h¨jih\sT�_�T°¯d&egj¯s?ogrt�²x�egjwsT�_jwx_p|vwz{rLrtpth\pqx�z{x_pts?o¹p|��x_e~z{x�x_egjjw¬X��sTogjwocx#h%}~��x´¢gx&pqo�z{o¤ptocx_jiu?jw�R°���j�x-}~���F}~��x-��zÁ�%x�e~z�x-p¥x&p|�´�VsT����pt�grtj�x�s%v�sTh¨�g}Xx_j£"pqx_e=o�}gh%��ji���s?­��_jw¦Tjw��z{rAh¨pqrtrqpts?o��&s{­��gpqo�z{���=�Xptu?pqx��&pq­�ogjiji�XjR�§°

µêocx_ji�_¦�z?rt��z{��j\pqh¨�grtjwh¨jwocx�ji��}~��ptogu�x_egjipq��jio~�X�Vs?ptoTx��wª�£"egptv�e¹z{��j%±¸�´³�� »�sTz�x�pqo~u{¼½��sTpqocx'o�}gh\¼�Vjw���i©¤x_e~pt�\p|�¨ogs{x¨¦�pt�_pq�~rqj�­·s?�\x_e~j�}���ji�%�g}gx¨jio~�_}g�_jR�%x_e~z{x%x�egj²��£-jwrtrqptoguºs{­yptocx_ji�_¦�z{r|����£"p|��x_e��¨pt�rtji���;pth¨��sT��x�z{ocx´x�e~z{o=£"p¥x�e¯x_egj'h¨pt�X�Vs?ptocx�¼½��zT�Xpt}~�;��jw�g��ji�_jwocx�z�x_pts?o¤�_}~v�e¤zT�;pth\�~rqjih\jiocx_ji�=���\�"}~h\�ptoºµêoTx ��z?� � �?�Xª§�{���N°%µêo~�XjwjR�§ª��_£"p¥x�v�egptogu=x�egj¨�_sT}go~�Xptogu�h\sX�XjR��pqo�v�}g���ogs��Vjwo�z{rqxF�²£"p¥x�e�h%}gr¥x�pq�~rqj�~�_jRv�p|��pts?o9z?�_pqx_e~h\jwx_p|vDz?o~�²x�egj%h¨s?x_pt¦Áz{x_pts?o9­·s?�'x_egp|��v�egs?p|v�j%pqo�µêoTx ��z?���Xs�ji�'ogs?xegsTrt�¸­·s?��±²�´³�µ�©ji¦?ji�_�¯h%}gr¥x�pq�~rqj��g��jivwpt�_pts?o�s?�Vjw��z�x_pts?o�pt�yz\�_s{­®xF£&z{��js?ogjT°

d&egj �_�Vjiv�pq¢�viz�x_pts?o��&}~��jR�¤­·sT�&x_egj�pth¨�grtjwh¨jwocx�z{x_pts?o²z{��j�~z?�_ji��s?o�x_e~jIEEE

��x�z?o~�gz{���K�t� �%�N©� z{o�ptoTx�jw��¦�z{r§p|�"z\vws?ogo~jiv�x�ji��v�rtsT�_ji����}~�~��jwx�s{­ IR

«

ä·Î�à�ä®Í

�������������� ��������� ��������������������! ��"����� #�� ��$�����%�&'�����)(+*,������ - �.,�0/2143 �

� p¥­ opp|��z{o

n¼êz{���¤sT��ji��z{x_pts?o�z{o~�

X1, . . . , Xn

z{��j�pqocx�jw��¦Áz?rt�iªXx�egj ��ji�_}gr¥x�s?­op(X1, . . . , Xn)pt�yz{o�pqocx_ji�_¦�z?r���}~v�e�x_e�z�xi©

{op(x1, . . . , xn), xi ∈ Xi} ⊂ op(X1, . . . , Xn)«

� pqoIviz?�_j op(x1, . . . , xn)pt��o~s{xD�Xjw¢~ogji��ª�x_e~jwoIz���5��5¿ �u�ys{xDz ��}~hD�Vjw�9��ª�£"egptv�e¹��x�z{o��g�­·s?�

z{o�pqo�¦�z{rtpt�9s?�Vjw��z�x�pqsTo�À"p|�yuTjwogji��z{x_jR�§ª:�����>� x_egj\ptoTx�jw����jRv�x�pqsTo¸£"p¥x�e9x�egj¨�Xs?h¯z{pto¸s?­opp|�ogs{x

x�z��?jwo��g��pqsT�&x_s\x_e~j�s?�Vjw��z�x_pts?o�«� jizTv�e5jwo��X��sTpqocx¯vwz?�_��pqjR� pqx��\s�£"o5�~�_jRv�p|��pts?o ¿O��jwx¨z{x\ptogpqx_p|z{rtp #Rz�x�pqsTo5s?�\h¨sX�Xp¥¢�ji�5�g}g�_ptogu�x_egjv�sTh\�~}Xx�z{x_pts?o~��À�°d&egj²z?�_pqx_egh¨jwx_p|v=sT��ji��z{x_pts?o~�¨z{��j=pth¨�grtjwh¨jwocx_jR�)z?o~� z{rtr#­·}~o~v�x�pqsTo~�¨�g��s�¦cp|�XjR�I���¶±²�´³�� z{��j

pto~vwrq}~�gji��z?��£#jirqrV¿®x_��ptu?s?o~s?h¨j�x��_p|v�z{o~��e�����ji�_�Vs?rtp|v�x_��pquTs?ogsTh\jwx_��ptv�­·}go�v�x_pts?o���z{o��x_egjipq���_jRv�pt�g�_sXviz{r|��À�°� sTo�¦?jw���_pqsTo~��x�s¸z{o~��­·��s?h }���}~z?r´z{o��¹m�±¸� �gz�x�z=xF���Vji� z?�_j¯zÁ¦�z{ptr|z{�grtj¯z?��£-jwrtr´z?����}~�Xpth¨jwocx�z?�_�ptog�g}gx �ÁsT}g�g}Xx\­·}go~v�x_pts?o~�i°¹d&egj�vwsX�Xj=p|�D£"��pqx�x_jio)zTvwv�sT���gpqogu¸x_sºm�±¸� ��x�z{o��gz{���g�=¿·­·}go~v�x_pts?o~�¨z{o��z?�_uT}gh¨jwocx��-o~z?h¨ji�iªXh\jih¨s?���¤h¯z{o~z?u?jwh¨jioTx�À�°

d&egjy�grtz?ogogjR�D­·}~o~v�x�pqsTo~z{rtp¥x�pqjR�wª?x_e~z{x´£"ptrqrg�VjyzT�g�Xji�\pto¤z�ogjiz?��­·}Xx_}~�_jTªTpto~v�rt}~�Xj�z �++

pqocx�jw�_­çz?v�j�¤rtz¯�´�_s?¢~r ���-µF����� k?�%��­·s?��jRz?�_j s{­L}~�_j?ªV�~z?�_p|vx_s�s?r|�"­·sT�yrtpqogjRz{�yz?rquTjw�g��z²¿·¦Tjiv�x_s?�'z{o~��h¯z�x��_pq¬��~z�x�zxF���Vji�iª�zT�g�Xpqx_pts?o~�'z{o~��h%}gr¥x�pq�~rqp|vwz{x_pts?o~��À"z{o��²z{}Xx�s?h¯z�x�ptv��Xp �Ajw��jwocx_p|z�x�pqsTo¶¿®­·s?��£&z{�����Xp �Ajw��jwocx_p|z�x�pqsTo���¤s�¦Tjw��rqscz?�Xptogu%s?�Vjw��z�x_sT���-z{o��=­·}~o~v�x�pqsTo~��À�°

�y�'v={�}���t4z w�"viv�sT���Xptogu5x_s s?}g��j�¬X�Vjw��pqh¨jwocx��wª'x_egjI±²�#³�µ�pqh¨�grtjwh¨jioTx�z�x�pqsTo p|��zT��j �¤v�ptjwocx²z?��x_egjis?��j�x�ptviz{rtrq�jw¬X��jRv�x_jR�§©¨­·s?�Dx_egj�m�z{}~���_ptz?oIjwrtpqh¨pto~z�x�pqsTo¶��ji��­·sT�_h¨jR�¹s?o5z ����z{o~�XsTh��¯±�¼½h¯z�x��_pq¬ºs?­y�Xpth\jio~�_pqsTo�T�?�¯z?o~��­·s?rtrts�£#jR�����²z¤vws?h¨�g}Xx�z�x_pts?o9s{­�x_egj\��ji�_pt�X}�z{r§x�s�v�egjRv!�=x�egjD��ji�_}grqxiª�x_egj%s�¦?jw��egjRz?��­çz?v�x_s?�p|�"v�rtsT�_jx_s

2ª�v�s?h¨�~z?�_jR�¯x�s¨±¸�´³���°

d�s)uTpq¦Tj�z5�g�_jRv�p|��jºp|�Xjiz5s{­ x�egjIz?�g�Xpqx_pts?o�z{r�v�sT��x�viz{}~�_ji� ���)x�egj¹}~�_j9s{­\z5h%}gr¥x�p¥¼½�g��jivwpt�_pqsToz?�_pqx_e~h\jwx_p|v{ªX£-jvws?h¨�~z?�_j�±²�´³�µ-£"p¥x�e �I?���4@? .§°�d&egj�rqpt�g��z{��� �I?���4-? .B�Ýk�����p|�&�~zT��jR�¤s?o�x_e~j�o~z�x�pq¦Tj.E�����& ��xF����j¸­·s?�¤»�sTz�x�pqo~u{¼½��sTpqocx=o�}gh%��ji���i«'pqx��=h¯z{pto ­·jiz{x_}g��ji��z{��j²s?o x�egj¸s?o~j9e~z?o~� x_s5zÁ¦Ts?p|��_£"pqx�v�egptogu¸��s?}go��Xpqo~u¸h¨s��gji�iªL�c�IzT�g�Xptogu¸s?�\��}~�Xx_��z?v�x_ptogu¸s?o~j=}grt�Ix_s�jizTv�eIjwo~�X�Vs?ptocx\z{­®x_jw�\jizTv�esT��ji��z{x_pts?o5¿çegs�£-jw¦?ji�iª§z=­çz{pqx_eX­·}~r;�_sT}go~�Xptogu�pt��zT�_�_}gh¨ji�~À�ª�z{o~�9s?o�x_egj¨s?x_egji�e�z{o~�¸x_s�pqh¨�grtjwh¨jwocx�_s{­®xF£&z{��j¤ji¦Áz?rq}�z�x_pts?o��%s?­yjwrtjwh¨jioTx�z{���º­·}go~v�x�pqsTo~�\x_e�z�x¨�_jwx_}g��o5jio~v�rtsT�_pqo~u�ptocx_jw��¦�z{r|�%z?o~�Ix�sºzÁ¦Ts?p|�x�egj�}~��j�s{­yjirqjih¨jwocx�z?�_��­·}go~v�x_pts?o~�\�g��s�¦�pt�XjR�I���ºx_egj�v�sTh\�~}Xx_ji�%sT�Dx�egj�vws?h¨�gptrqji�iªL£"e~ptv�e¶��j�x�}g��o��ji�_}grqx��-£"pqx_e�}gor��ogs�£"o�z{o~����sc�_�_pt�grq�¯r|z{��u?jjw����s?���w°�d�s%x�z��?j�pqocx�s\zTvwv�sT}gocx#x�egj��Vs?ptoTx"s?­�¦�pqji£ z{o��x�egj¤v�sTo~��x_��z{ptoTx��s{­"z�}���ji�iª�£-j¯pth\�~rqjih\jiocx_ji�¹z��_pqh¨�grtj=m�z{}~���_ptz?o�jirqpth¨pqo~z{x_pts?oºh¨j�x_e~s��ºx�s²��sTrq¦Tjrtptogjiz?�-�_�X�Fx�jwh¯�wªX}���ptogu\��s?x_e=��z?v!��z{uTji�i°�d&egj­·}go�v�x_pts?o��-z?o~�¯x�egj��gz�x�z%��x_��}~v�x_}g��ji�-pth\�~rqjih\jiocx_ji��pto�I?���4-? .¤z?o~��±¸�´³�µ#e~zÁ¦Tj'�Vjwjio�egs?h¨s?uTjwogp #wjR�9¿çjwhD�Vji�~�Xji��ptocx_s¯v�r|z?����jR�-e~zÁ¦�pqogu\j�¬gzTv�x_rt�¯x�egj �_z?h¨j�_�Vjiv�pq¢�viz�x_pts?o��\¼ £#j¸�Xpt��jiv�x�rq�5}~��jR� s?ogrt�Ix_e~j � ­·}go�v�x_pts?o��¤z{o~� �gz{x�z¹��x_��}~v�x�}g�_jR�\­·�_sTh �I?��$4-? .~À�_s�x_e~z{x\x�egj²z{rtu?sT�_pqx_egh¯�%}~�_pqogu�x_e~ji�_j�z?�_pqx_e~h\jwx_p|v�rqpt�g��z{��pqjR�%z?�_j��Fx��_p|v�x�rq�Ip|�XjioTx�ptviz{rN°5µêo s?���Xjw�%x_sh¨jRz?�_}g�_j´x_egj&�c}~z{rtpqxF�s?­gx_e~j#pth¨�grqjih¨jwocx�z{x_pts?o�ª�£-j#jihD�Vji�g�gji� pto x�egj&�_z?h\j-£&zÁ�'e~z?���g£-z?�_j"�_�XsT}g�grtj �vws�j �¤v�ptjwocx���¿·»~sTz{x_ptogu{¼½�Vs?ptoTx"o�}gh%��ji���iªXogs{x"ptocx_jw��¦�z{r|��À-zT�&£#jirqr�z?o~����z?o=x�egj��_jR��}~r¥x�pqogu¨z{rtu?sT�_pqx_e~h�°

�'o�x_egj�sTogj�e~z{o~��ª�x_egj�v�sTh¨�g}Xx�z{x_pts?o=x_pth\j�pto~vwrq}~�gji�-x_e~j�vwsT��x-s?­�x�egj�¦Áz?�_pts?}��#rts�s?�~�&z?o~�¤­·}~o~v�¼x�pqsTo²vwz{rtr|�wªX�g}gxys?o�x_egjDs{x_e~jw�"e~z?o~��p¥xyz?rt�_s¨pqo~vwrq}��Xji�&x�egj�h¨jwh¨s?���¤h¯z{o�z{u?jih¨jwocx�z?�_��jRv�x���pqo~�g}~v�jR�

à�à�Ò r¡�¡�¢�£

� �����=���%��&' �.�1����=������&�&������

���¯x�egj ±²�´³�µ-�~z?v!��z?u?j¨¿®x_e~j o�}ghD�Vjw���"z{��jz?rq£&zÁ�X�&�X��o~z{h¨p|vwz{rtrt�¤z{rtrqsXvwz{x_jR�~À�°d&egj�x_pth\ptoguc�;pto¤��jRv�s?o��g��s{­§s?}~�;�g��s?u?��z{h¯��z{o��¨��z{x_ptsT�Lv�sTh¨�~z{��ji�%x_s �I?���4-? .Dz?�_j&x�egj"­·sTrqrts�£"pqo~u

sTo�z?o��-x�egrts?o¹�Rm�� #�~�_sXv�jR�_�_s?�R©�XsT}g�grtj �I?��$4-? . ±²�´³�µ �{�\�gpqx���~° ��k �~° �?� kX�T° ���

��° �$�±²�´³�µ �?�%�gpqx�� ±²�#³�µ�Rk����~p¥x�� ±¸�´³�µ"k���D�gpqx�� ±²�´³�µ'�g�?��gpqx��

k?kg° �$� k{�~°t�?� �{��° �T� �X�T° �~��g° �T� �X° �T� �X° �2� �?�T° ���

±¸�´³�µ�R�Tk?� �gpqx�� ±²�´³�µ"k{�{�����~p¥x�� ±²�#³�µ-�T�T��D�gpqx���?�ÁkX° �?� k?��X°��E� �g�%�X° �$�k��g° �?� �$�X° �� �i���~°q�T�

� Ç � � Ã�� �i� Ç�� ��� à ��� ��� �� �d&e~j¯¢~���Fx ��jwh¯z{���9p|��x�e~z�x%x_egj¤s�¦Tjw��egjizT�9pto~�X}�v�ji�¹���º±²�´³�µ�pqo �?���gpqx��%h\sX�Xj¯p|� ��jizT��sTo~z{�~rqj²¿çz­çzTv�x�s?��rqjR�_��x�e~z{o �TÀ�v�s?h¨�~z?�_jR��x�s�z�~zTv!��z{u?j#x_e�z�x;}~�_ji��e~z?���g£-z?�_j#»~sTz{x_ptogu��sTpqocx´z{��pqx_egh¨j�x�ptv?°�d&e~pt�s�¦Tjw��egjizT�²p|� �X}gj¨x�s�x_egj¤�X��o~z?h¨ptv¨h¯z{o�z{u?jih¨jwocx�s{­&h\jih¨s?���5¿çz?rqrtsXvwz�x�pqsTo�s?­#o�}gh%��ji����Àpqo¶m�±²� ±²�#³�µ�p|���g}gptr¥x%}g��sToI±²�´³��æ£"egp|v�e�p|���~zT��jR�9sToIm�±¸� ª�£"egp|v�e¹pt�Dv�s?h¨�~z?��z?�grtjDx�s�x�egj¤v�sc�Fxs?­´z{��p¥x�egh¨j�x�ptv%s?�Vjw��z�x�pqsTo~�"­·sT���_h¯z{rtr��~�_jRv�p|��pts?o~�i°���s�£-jw¦?ji�iª�egptu?egrt�²s?�Xx�pqh¨p #wjR�²­·}go~v�x_pts?o~�£"��p¥x_x_jiopto)zT�_�_jwh%�grt��r|z{oguT}~z{uTj=z{rtrts�£ x�s��Vjw�_­·s?��h j �¤v�ptjwocx�rq�¹s?�Vjw��z�x�pqsTo~� £"pqx_e �gz�x�z¸jio~v�sX�XjR�¶£"pqx_e¶­·ji£h¯zTv�egpqo~j£-s?���g�"z?o~��x_egp|�"�g��jw¦Tjwocx��-x�egpt��s�¦?jw��egjRz?�¯­·��s?h ��jipqo~u¯rtz?�_uTjw�-­·s?���_h¯z{rtr��g��jiv�p|�_pqsTo~�w°;d&e~pt�jw¬X�grtz?pqo��;£"e��Dx�egjs�¦?jw��egjRz?�%pqh¨�grtpqjR�¤���¨z �Xs?}~�grqptogu%s{­§x_e~jy�g��jivwpt�_pqsTo¯pt�´ogs{x�ptvwjiz?�grqj'�Fx�z{�_x_ptogu�­·��s?h�R�Tk?� �gpqx�� ¿O�Tk%h¯z?v�egptogj�£-s?���g��À#sTogrq�T°��j�x"}~�"z{r|�_sDegptu?egrtptu?ecx#x�egj�s�¦?ji�_e~jiz?�¨pto~vw}g�_��ji�=���¤��}~viv�jR�_�_pq¦Tj�gs?}g�grtptoguT�-s{­�x_egj��g��jivwpt�_pqsTo�©�x_egj

s�¦Tjw��egjizT��­çz?v�x_sT�p|�rqjR�_��sT��ji�c}~z{r�x�s��~ª�£"egp|v�e�p|���X}gj¨x�s=x�egj¤v�sc�Fx�s{­���z{��z�x��_}g�~z=hD}grqx_pt�grtptviz�x�pqsTopto=h%}grqx_pt�grqj�g��jivwpt�_pqsTo�ª�z{o~�¨x�egj'­çz?v�x�s?�-��jwxF£#jijwo=±¸�´³�� z?o~�¤±²�´³�µ#�_jih¨z?pqo��´v�sTo~��x�z{ocx&z{o~�¯jR�c}~z{rx�s\k­·sT�´x�egp|�´�g��s?uT��z?h sTo=±�¼½h¨z{x_��ptvwji�'¿·pqx-p|�´}~�g��ji�#�Vs?}~o~�Xji�=���\�D£"egjwo�u?jwo~jw��z{rgh%}gr¥x�pq�~rqp|vwz{x_pts?o~�sXviv�}g��À�°� Ç � � Ã�� �i� Ç�� ��� à �~ ? ������g  Ä Ç� ��� à � à ����&·�R�µêo �?�'�~p¥x��Lh¨s��gj?ªTx_egj"ptocx_jw��¦�z{r|���g��s��g}~v�jR�%���D±¸�´³�µ�z?�_j"z?rq£&zÁ�X����x_��p|v�x_rt�Dpto~vwrq}~�gji�¨ptoTx�sx_egsc��j"�g��s{¼¦�p|�Xji�¤�c�\x�egj �I?���4-? .\�~z?v!��z?u?j?ªT£"egj�x�egjw�#x_e~j�_s?rt}Xx_pts?o=s?�#x_egj'�_jR��p|�X}~z?r�p|�#vws?o~vwjw��ogji��°�d&egp|�#pt�&�X}gjx�s¯x_e~j }~�_j?ª~pto �I?��$4-? .§ªXs{­;z¯�_s{­®xF£&z{��j����grts�£"pqogu=�'s{­�x_egjDpqocx_ji�_¦�z?rt�&x�s=zÁ¦?sTpt�=­·��ji�c}gjwocx'�_£"pqx�v�egptoguT�s?­���s?}go��Xpqo~u\h¨sX�XjR�w°Ld&egj���ji�_}grqx��&s?�gx�z{ptogjR�=��� �I?���4@? .¤z{��j�j�¬�x_��jwh¨jirq�=vwrqsc��j'x_s¨x�egsT�_j�v�sTh\�~}Xx_jR����=±¸�´³�µ#}~�_ptogu �?kD�gpqx��i°�-jwptogu²z{�~rqj\x_s²£-s?���²£"pqx_e �X�T�%�gpqx���s?­-�~�_jRv�p|��pts?o ¿ �������9�R�¤x�pqh¨jR��h¨s?��j%x�e~z{oº£"pqx_e¹e~z{���X£&z{��j

»�sTz�x�pqo~u{¼½��sTpqocxDo�}gh%��ji����À rts�sT�_pqo~u¸s?ogrt�¶z²­çzTv�x�s?�¨z{�Vs?}Xx��Rk9v�s?h¨�~z?�_jR�ºx_s¹z¸h¨s?��j�vws?o�¦?jiocx_pts?o~z?rptocx_ji�_¦�z{r�z{��p¥x�egh¨j�x_p|v�rtpq�~��z?�_�¤��z?�_ji��s?o�e~z?���g£-z?�_j��XsT}g�grtji�"�grtjizT�g��pto�­çzÁ¦?sT}g�"s{­;±²�#³�µ�©Vpqxypt�'v�sTh%¼­·sT��x�z{�grtj#x_s �Xji¦?jwrts?�\z{o¨z{rtu?sT�_pqx_egh z{o~�Dx_s�v�sT�_��jiv�x�zr|z?v!��s?­AzTvwv�}~��zTv������ pqo�v���jiz?�_ptoguyx�egj"�g��jivwpt�_pqsTos?­§x_egj}go~�gjw��rq��ptogu »~scz�x�pqogu?¼N�Vs?ptocx#z{��pqx_egh¨j�x�ptv?«cx_egp|�-rtzTv!�\s?­�z?viv�}g��z?vw�\viz{o¤�Vjz{x�x��_pt�g}Xx�ji�¤jip¥x�egjw�#x_s

ä·Î�à�ä®Í

�������������� ��������� ��������������������! ��"����� #�� ��$�����%�&'�����)(+*,������ - �.,�0/2143 �

zD�T}~ptv!�¨z{o~�¨o~s{x´¦Tjw���%vwz?�_jw­·}gr~jw¬X�g�_jR�_�_pts?o¨s{­§x�egj��g��s?�grtjwh ¿·pto¯x�egj�¢~����x#�Fx�z{uTj"s{­§x_e~j'�Xji¦?jwrts?�~h\jiocx­·sT�"pqo��Fx�z{o~vwjRÀ#s?�-x_s\x�egjwsT�_jwx_p|vwz{r§��jizT��sTo~�-��}~v�e�z?�-x�egsT�_j�h¨jwocx_pts?ogjR�=pto�x_egj �_jivws?o~����jRv�x�pqsTo=s{­�x_e~pt���z{�Vjw�R°

����� ��qu{�t4prq n�v�wd&e~jw��j�z?�_j��_jw¦?ji��z?rAh\s?x_pt¦?jR�´x�s¨}~��j�ptocx_ji�_¦�z{r�z?�_pqx_e~h\jwx_p|vpqo��_vwpqjiocx_pq¢�v�vwz?rtvw}grtz{x_pts?o�°2�'ogj�vwz?o=x�egptor�­·sT��j�¬gz{h¨�grtj¨s{­-x_egj¨­·sTrqrts�£"ptogu���sTpqocx��i©�¦�z{rtpt�~z�x_pts?oºs{­&o�}gh¨jw��ptviz{rLz{rtu?sT�_pqx_e~h¨�¯¿O�Fx�}~�X��s{­-x_egj¯�g��j�¼vwpt�_pts?o9s?­;x_e~j\sT}Xx_�~}Xx�£"egjwo�p¥x�p|�'o~s{xf��ogs�£"o¸­·��s?h x�egjDx�egjwsT�_�gÀ�ª§j�¬gz?v�x�viz{r|v�}gr|z�x�pqsTo~�%¿ç£"egjw��j\x_egj�c}~z?rqpqx�z{x_pt¦?jz?�_��jRv�x"s{­�x�egj��_jR��}~r¥x"h¯zÁ�¨��j�h¨s?��j'pth¨��sT��x�z{ocx-x_e~z?o=x�egj��T}�z{ocx_pqx�z{x_pt¦?j�z?�_��jRv�x�À�ªgz{rtu?s?¼��pqx_egh¯�#�_jR�T}~pq��pqo~u u?rts?�~z?r~ptoX­·s?��h¯z�x�pqsTo¤z{o~�¨x�e�}~�#�gji�_pquTogji�¨x�sD£-s?���%£"pqx_e=pqocx_ji�_¦�z?rVz?�_pqx_egh¨jwx_p|v{°���j�_egs�£ pto¸x_e~jD­·ji£3­·s?rtrqs�£"ptogu¤jw¬gz{h¨�grtji�yx_e~z{xyx_e~j%}~�_jDs?­´hD}~r¥x�p¥¼½�g��jiv�p|�_pqsTo�»~sTz{x_ptogu{¼½��sTpqocxpt�v���}~v�p|z{rpto�h¯z{o��=�_p¥x�}~z�x�pqsTo~�w°

�'�¨h\jiocx_pts?ogjR�¶pto¶x_e~j�ptocx_��s��g}~v�x�pqsTo�ª;e~zÁ¦�ptogu�z��g��jiv�p|�_j=ptoTx�jw��¦�z{r"z{��p¥x�egh¨j�x�ptv�h¯zÁ�º�Vj�}~�_j�­·}gr­·sT�y��x_}~�X��ptogu¸°i°w°�s?x_egji�yz{��pqx_egh¨j�x�ptvi�wªVz{o~��o�z{h¨jwrt�¤x�egj%zTvwv�}~��zTv��¤s?­�x�egj ��ji�_}grqxi°=�'ogj uTscsX��j�¬gz?h%¼�~rqj¯p|����ji�_e�z{�~�x�egj¤��sTrq}gx_pts?o¹s{­#x_egj¤�Xpt��jiv�x@��ptogjwh¯z{x_p|vw���g��s?�grtjwh ­·s?���~z?��z?rqrtjwrL��s?�Vs{x��-� k��%�N°¤d&e~pt��~�_sT�grqjih viz{o¹��j=j�¬X�g��ji����jR�ºzT� z{oIz?rquTjw�g��z{p|v¨���X��x_jwh �Xji��jio~�Xptogu¸s?o ��x�sI�Rk�¦�z{��p|z{�grtji�¯¿O��ji¦?jw��z{r­·sT�_h%}gr|z�x_pts?o���jw¬Xpt��x�À�z{o��Dogsf��ogs�£"o%o�}gh¨jw��ptviz{rXh¨j�x�egsX�%z?rqrts�£��wª�o~s�£-zT�gzÁ�X�wªRx_s��_s?rt¦?j&pqx�pto\x_egj"uTjwoX¼ji��z?r�vwzT��jT°��ºjDz?o~z{rt� #wji���_s?h¨jDvws?h¨�g}Xx�z�x_pts?o�z{r���x_��z�x�jwu?ptji�&£"e~ptv�e¸z?�_j�}~�_}~z?rqrt��jih¨�grqs��Tji�¹¿·h¯z?pqogrt�jipquTjwo�¦�z{rt}gjyvws?h¨�g}Xx�z�x_pts?o���À;�g}Xx´x�e~z�x-­çz{ptrVs?o=�_s?h¨j�jw¬Xz?h¨�grqjR�wªc���¯��pth¨�grq�¨��jw�gr|z?vwpqo~u�x_egj'e~z{���X£&z{��j»�sTz�x�pqo~u{¼½��sTpqocx\z?�_pqx_e~h\jwx_p|v=���¶±²�#³�µ�°;d&e~j���ji�_}grqx\£&z?�¨�_}g�_�~�_p|��ptogu�©¯}~��ptogu¹v�r|z?����p|vwz?r-z?�_pqx_e~h\jwx_p|v{ªo~s?ogjs{­�x�egj���sTrq}gx_pts?o~�&v�sT}gr|�¤�Vj�z?�g�g��sÁ¬Xpqh¯z�x�ji�§ªc£"egji�_jRz?�-z{rtrAx_egj��_s?rt}Xx_pts?o~�#£-jw��jvws?h¨�g}Xx�ji�¤£"pqx_ez\uTscsX��z?vwvw}g��z?v��¨}~�_ptogu��Rk{�%�gpqx��"s?­��g��jivwpt�_pts?o=£"pqx_e²±²�#³�µ�°�³g}g�_x_egji�_h¨sT�_jTªcsTogrt�¯­·jw£ j�¬gz?v�x&h¨jwx_eX¼sX�g��vwz?o ��sTrq¦Tj²j �¤v�ptjwocx_rt� x_egp|�¤�~�_sT�grqjih z{x��g��ji�_jwocxR° ³~s?�=x_egj�j�¬X��ji�_pth¨jwocx����Xji��v���pt��jR� z{�Vs�¦?j?ª£-jDe�z?�²x�s�vws?o~��x_��}~v�x�_s?h¨j%h¯z{x_��ptvwji�'}~��ptogu�j�¬gz?v�xvws?h¨�g}Xx�z�x_pts?o��%¿Nm���T�gogjw���~zT��jR��À"x_s�zÁ¦?sTpt�²z?ozTvwvw}ghD}gr|z�x�pqsTo s{­�oc}~h\ji�_p|vwz?rTji�_��s?���i°�d&egp|��p|��s?ogj-£&zÁ�'x�s'h¨pq¬Dj�¬gzTv�x�z?o~� o�}gh¨ji�_p|vwz?r�vws?h¨�g}Xx�z�x�pqsTo~�w°

�yx�egjw�#j�¬X��ji�_pth¨jwocx��´e~zÁ¦?j"�~}Xx#pto¤jw¦�p|�Xjwo�v�j"x�e~z�x´ptocx_ji�_¦�z{rVz{��p¥x�egh¨j�x�ptv"h¯zÁ�\�_ji�gr|z?v�jTªTz{­®x_jw�-��sTh¨jzT�g�Xpqx_pts?o~z?rgx�egjwsT�_jwx_p|vwz{r�£#sT���AªThD}grqx_pq¼N�~�_jRv�p|��pts?o¯��z�x�pqsTo~z{r�oc}~hD�Vjw���;pto¤j�¬gz?v�x-vws?h¨�g}Xx�z�x_pts?o��w°���jwx#}~�ptrtrq}~��x_��z�x�jx_egp|�"£"pqx_e�h¨j�x_e~s��~�"�~z?�_ji��s?o� 'ji��vwz{�_x_jR� � �-�_}~rqj�s{­L�_pquTo~�w°�&z?�_p|vwz{rtrt�¸¿ç�_jwj,� �Tk���­·sT�"j�¬gz{h¨�grtjRÀ�ª�x_egj�jw¬gz?v�x"¦Tjw����pts?o�s{­�x_egj �_s{¼êvwz?rqrtji�0�y�_��jio~�u��� � �"z{rtu?s?��pqx_egh

p|�´z?rq��jizT�X�S��ogs�£"o¨x�s �Vjy¦?ji�_�\j �¤vwpqjioTx´­·s?�´p|�_s?r|z�x_ptogu�x�egjy��jiz?rg��scs?x��´s{­§}gogpt¦�z{��ptz{x_j��Vs?rt��ogs?h¨p|z{r|�w©�p¥xviz{o9�gjiz{r�£"pqx_e¸sT��x�egs?uTs?o~z?r���sTrq��ogsTh¨ptz?rt�'rqp �?j� pqr ��pqo���sTo�� �'s?� � egjw���X�_egjw¦ � ��s?o~ji�iªVjwx�v?°'s?­´�XjwuT�_jijuT�_jRz�x�jw�#x_e~z?o�k?�?�?�~°�d�s¯��}gh¨h¯z{��p #wj?ªcx�egj�z{rtu?s?��pqx_egh p|�-�~z?�_ji��s?o yjR�_viz{�_x_ji��� �#��}grtj's?­��_pquTo~�-£"egp|v�ep|�"�Xjw¢~ogji���Vjwrts�£�°²Ç �i�g��� Ç����� ��-.��� l�����-���

sign(a)���)(r��_pquTo �u�� ��& ��*,�� ��

a ∈ IR�R%���!���� ��

0� �

a = 0�1� �

a > 0 �.

−1� �

a < 0��� l.f.E���� l� �)(r��o�}gh%��ji��s?­X�_ptu?o v�e~z?ogu?jR�

V (a)�� -��(�� &��"���

a = (a1, · · · , ak)�u�@��& ��*,�� ����S�u�IR \ {0}

���,�� l.��� ������ y�����k �

V (a1) = 0, V (a1, · · · , ak) =

{

V (a1, · · · , ak−1) + 1� �

sign(ak−1ak) = −1,V (a1, · · · , ak−1)

��)(r�����L�"���.

à�à�Ò r¡�¡�¢�£

�R� �����=���%��&' �.�1����=������&�&������

-�,��E�$�� �.��)(+�"�� l����������� #�$� �&��"���=� �-��& ��*,�� ������� IR��(��R*,�B ��� ��$�� ��������>���

�� �

b�"�S��(��-&��"���

�E���$�� ��!.0���,����*-���>�� +���������>���f�� a� ����.E���� l�

V (a) = V (b)�

�'��ptogu\x_e~j z{�Vs�¦?jogs{x�z�x_pts?o��wª@ 'ji��vwz?��x�ji��� �-��}grqj�s?­��_pquTo~�&p|�-x_egj�­·sTrqrts�£"pqo~u~©� � Ã�Ç � à � ������à �R ?�X�Á� à ��q�\� ��& òÇ� ��� ��� ���� ���

P =∑d

i=0aix

i�!�#B���&�� ��*���&[��

IR[x]�

3�� ���B.E�� ���$�B���V (P )

�)(r�� ���*,�!����� �,��� �� j �(� ������0�� |��(��0&��"���(a0, . . . , ad)

�.pos(P )

��(�� ���*-�!�����u�f���%���������%�����!& ���������� �

P !��� ��$�!. �L����( *���&���� �l&��� ��������������)(r��

pos(P ) ≤ V (P )�� �.

V (P ) − pos(P )�"�@����� ��

��j�x#}��´z?���_}gh¨j"x_e~z{xPp|�´z ���c}~z{��j�¼N­·��jwj"�Vs?rt�co~s?h¨ptz?rgs{­��gjwu?��jwj

dpqo

IR[x]£"p¥x�e=z{rtrVp¥x��;��scs?x��´pqo

]0, 1[° 'j�¢�ogpqo~u

Pk,c = 2kdP (x+c2k )

ª~p¥­ yjR�_viz{�_x_ji��� �#��}grtj�s{­L��ptu?o~�-uTpq¦Tji�1¿ç�_jR����°��cÀ-£"egjwo�z{�~�grqptji�

x�s�x_egjy��sTrq��ogsTh\p|z{r(x+1)dPk,c(1/(1+x))

ª�x�egjwo¯x�egj��Vs?rt��ogs?h¨p|z{rPe~z?�;j�¬gz?v�x_rt�Ds?o~j"�_s�s{xy¿ç�_jR����°

o~s���s�s{x�ÀDpto5x�egj�ptocx_ji�_¦�z{r] c2k , c+1

2k [° d&egjwsT�_jih¯�%�X}~j�x�s � sTrqrtpto~���{� ����p¥x�z?�0� �%�yz?o~� �'pqo�v�jwocxB� ���%�

��x�z{x_j=x_e~z{x%­·sT�\�_}��¤v�ptjwocx_rt�Ir|z{��u?j¤¦�z{rt}gjR�Ds?­kª8 'ji��vwz{�_x_jR� � � ��}grtj=s{­��ptu?o~�%£"egjio)z{�~�grqptji�Ix_s�x_egj

�Vs?rt��ogs?h¨p|z{r(x + 1)dPk,c(1/(1 + x))

£"ptrqr�z?rq£&zÁ�X�-u?pt¦?j0sT�

d&egj�h¯z{pto��_jih¨z?���¯p|�-x_e�z�x"£#j�z?�_js?ogrt�¤ptocx_jw��ji��x_jR�=pto�x_egj��_pquTo~�&s{­�v�s�j �¤v�ptjwocx��wªgz?o~�¯x�ec}��&x_egjzTvwvw}g��zTv��¯ogjiji�XjR��­·s?�"x�egj���ji�_}gr¥x�s?­;v�s?h¨�g}gx�z�x�pqsTo~�"vwz?o���j �Vs�s?�R°Y�ys�£#ji¦?jw�Rªcx_egj x_��z{o~��­·s?��h¯z�x_pts?o��x → x/2

}~�_ji� �X}g��ptogu¹x_egj¸vws?h¨�g}Xx�z�x�pqsTo~�\­·s?���gp|�5x�egj²}~�_j²s{­�e~z{���X£&z{��j=»�sTz�x��²¿·ji¦?jwo ptocx_jw��¦�z{rz?�_pqx_e~h\jwx_p|v²�~zT��jR�)sTo e~z{���X£&z{��j�»~sTz{x���À%­·s?���_s?rt¦�pqo~uI��sTrq��ogsTh\p|z{r|�¯s?­�e~pquTe �XjwuT�_jij?ª-��jRvwz{}���j¸s?­rtpth\pqx�z{x_pts?o~�¨sTo x_egj²jw¬��Vs?o~jwocx����L��z?ogu?jT° ±²s?��jws�¦Tjw�Rª�x_egj¸z?rquTs?��p¥x�egh £"ptrqr"x�jw��h¨pqo~z{x_j�s?o~rq�¶pq­�x_egj�Vs?rt��ogs?h¨p|z{r

Ppt�y�_�c}~z?�_jw¼O­·��jwjT©�x�egp|�yv�sTo~�Xpqx_pts?o�e~zT�&ogs¯�g��jiv�p|�_j�h\jRz{ogptogu¨£"egjio�x_egj vwscj��¤v�ptjwocx��"s?­

Pz{��j��_ji�g�_jR��jiocx_ji�=���¯»~sTz{x��&sT�"pqocx_ji�_¦�z?rt�i°±²sT�_jis�¦?ji�iªc£"egjwo���jw�gr|z?vwpqo~u%��z�x�pqsTo~z{r§v�s�j��¯vwpqjiocx��&£"pqx_e�pqocx�jw��¦Áz?rt�iª�xF£-s\�~�_sT�grqjih¯�&h¨zÁ�=s�viv�}g�R©� z�r|z?v!��s{­go�}gh¨ji�_p|vwz?rTz?viv�}g��z?vw�'h¯zÁ��pto~�X}~vwj´�g��s?�grtjwh¯��pqo%��ptu?oD�gj�x_ji�_h¨pto~z�x�pqsTo~�-¿ 0 viz{oDz?�g�Vjiz{�pqo�ptocx_jw��¦�z{r|��À�«� x_egj²�Vs?rt��ogs?h¨p|z{r"h¯zÁ�¶ogs?x¤��j¸���c}~z{��j�¼N­·��jwj¹¿ç��zÁ�5zº��sTrq��ogsTh\p|z{r"£"pqx_e ptocx_jw��¦�z{r�vwscj��¤v�ptjwocx���_ji�g��ji�_jwocx��&z¯�_j�x"s{­���sTrq��ogsTh¨ptz?rt�#x_e�z�x�h¯zÁ�=v�sTocx�z{pto�z%o~s?o��_�c}~z?�_jw¼O­·��jwjs?ogjÁÀ�°�'�\�_egs�£"oIptod� �Tk��½ª�x_e~ji�_j¤xF£-s��_p¥x�}~z�x�pqsTo~� rtjizT�¹x_s9x_egj��_z?h¨j¤�g��s?�grtjwh�©¤�_s?h¨j��_pquTo¶�Xj�x�jw��h¨p¥¼

o�z�x_pts?o���£"ptrqr�ogs?x'�Vj%�VsT����pt�grtj?ªA£"egp|v�e¸h¨jiz?o~��x�e~z�xRªAp¥­#z���ptu?o��Xj�x�jw��h\pto~z{x_pts?o�­çz{ptrt}g�_j%pt�'}~�_ji��z?�yz��x_sT�g�gptogu v���pqx_jw��pts?o�ª?x_egjio¨x_egjz{rtu?s?��pqx_egh £"pqrtr�z?rq£&zÁ�X��x_jw��h¨pqo�z�x_jT°�d&e~j"¢~o~z?r~��ji�_}gr¥x#£"pqrtr~�Vjyz �_j�x´s?­p|�_s?r|z�x_ptogu¨ptoTx�jw��¦�z{r|�"z{o~�²z¯�_j�x�s?­�ptocx_jw��¦�z{r|�&­·s?�y£"egptv�e�ogs¤�gjiv�p|�_pqsTo�p|�"�VsT����pt�grtj¤¿®x�egjw��h¯zÁ�¤s?��h¯zÁ�o~s{x�v�s?ocx�z{pto9��jiz{r���s�s{x���À�°-�'ogj¯vwz{o9x_egjio�pto~vw�_jRz?�_j x_e~j\�~�_jRv�p|��pts?o�s{­;x�egj¤z{��p¥x�egh¨j�x�ptvDx_s²v�sTocx_ptoc}~jx�egj%vws?h¨�g}Xx�z�x�pqsTo~�w©-s?o~j pqh¨�Vs?�_x�z?oTxy­·jiz�x�}g��jDp|�"x_e�z�xyx�egj%vws?h¨�g}Xx�z�x�pqsTo¸vwz{o²�VjD��ji��x�z?��x�ji��jw¬XzTv�x�rq�£"e~jw��j�p¥x'­çz{ptrqjR�§ª'��������x_egji�_j p|�'ogs¤ogjiji��x_s=�_jR�Fx�z{�_x"x_egj\v�s?h¨�g}gx�z�x�pqsTo�­·�_sTh x�egj �VjwuTpqogo~pqogu�°"µ½­Lx_egjvws�j �¤v�ptjwocx��Ds{­&x_e~j=ptogp¥x�ptz?r´�Vs?rt��ogs?h¨p|z{r#z?�_j,��ogs�£"oIjw¬gz?v�x�rq�TªL��}~v�e¶z¸�g�_sXvwji����£"pqrtr#z?rq£&zÁ�X��u?pt¦?j¤zvws?h¨�grtj�x�j�z?o~��j�¬gz?v�x"�_jR��}grqxi°

d&egp|��z{rtu?sT�_pqx_egh p|��z¤¦?ji�_��uTscsX�9j�¬gz{h¨�grtjDs?­#����h%��sTrqp|v ��oc}~h\ji�_p|v vws?h¨�g}Xx�z�x_pts?o��w° d&egj\hD}~r¥x�p¥¼�~�_jRv�p|��pts?o z{��p¥x�egh¨j�x_p|v²�_��jiji�g�¨}g� x�egj¸v�sTh\�~}Xx�z{x_pts?o~�¨pto u?jwo~jw��z{r"�g}Xx=z?rt�_s¹z{rtrqs�£��\x�s¹�gjiz{r"£"pqx_e�Vs?rt��ogs?h¨p|z{r|�"£"egsc��jDvws�j �¤v�ptjwocx��z{��j�ogs?x'j�¬gzTv�x_jirq����ogs�£"o¶¿çz?�g�g��sÁ¬Xpqh¯z�x�j v�s�j �¤vwpqjioTx��wªV��jiz{r�z{rtu?j�¼�~��z?ptvo�}ghD�Vjw���iªXj�x�v?° À�°

ä·Î�à�ä®Í

�������������� ��������� ��������������������! ��"����� #�� ��$�����%�&'�����)(+*,������ - �.,�0/2143 �T�

�"ji¦?sTr�� �?�%��e~zT��pth¨�grqjih¨jwocx_jR��pqocx�jw��¦Áz?r��yjw£&x�s?oDz?rquTs?��p¥x�eghb�t�w����z?o~��e~zT��zT�gz{�Xx�ji��pqx�x�syh%}gr¥x�pq�~rqj�~�_jRv�p|��pts?o²v�sTh¨�g}Xx�z{x_pts?o~�i°&µêo²x_e~pt�yj�¬gz{h¨�grtj?ª�x_egjDh¨z?pqo�z?�X¦�z{ocx�z{u?j�s?­�}���ptogu=±²�#³�µ"p|�yx_e~z{xys?ogjDpt�o~s\h¨sT�_j�rtpqh¨pqx_ji�����¤x�egj v�sTh¨�g}Xx_ptogu¨�g��jivwpt�_pqsTo�©�pqo��XjwjR�§ªgs?o~j�viz{o�pqh¨�VsT�_j z{���gpqx_��z{���¨zTvwvw}g��zTv��¯sTo�Vs{x�e¹x_egj=�_s�s{x � �Dz{�g�g��sÁ¬Xpqh¯z{x_pts?o¹z{o~�ºx�egj¤��ji�_pt�X}�z{r;pto ��ji£&x_s?o�� �%z{rtu?s?��pqx_egh �q�R�%�N° ��j�xD}~�Dpqo���p|�FxsTo¹x_egj¤­çz?v�x%x_e~z{x x_e~j¤xF£-s9z�­·sT�_jih\jiocx_pts?ogjR��pth¨�grtjwh¨jwocx�z{x_pts?o~�Dh¯z{o~z?u?ji��x�s9z?�gz?�Xx%�X��o~z?h¨ptviz{rtrq�x�egj��g��jiv�p|�_pqsTo=x�s%x�egj v�sTh¨�g}Xx_ptogu¨ogjiji�g�&£"pqx_egsT}Xx���ji��x�z{�_x_ptoguDx_egj�£"egsTrqj��g��s?uT��z?h�°�d&egp|�"�XjR��pt��z?�grtj­·jRz�x�}g�_j�£"ptrqr���j �_s?}guTecx&z{­®x_jw��­·s?�&­·}Xx�}g�_j�pth¨�grtjwh¨jwocx�z{x_pts?o~�&s?­�s?x_egji��z{rtu?s?��pqx_egh¯�i°

� n�v={ � �=w+q n�v±¸�´³�µ;pt�-z rtpt�g��z?�_�%­·s?�-hD}~r¥x�pq�grtj�g�_jRv�p|��pts?o¯ptocx_ji�_¦�z{rAz{��pqx_egh¨j�x�ptv'£"egp|v�e¯­·}grtrq�¤��z�x�pt��¢~jR�Lx_e~jvws?ocx�z?pqoX¼h¨jioTx%�_jR�T}~pq��jwh¨jwocx%z?o~�Iz{rtrts�£���x_s9vws?h¨�g}Xx�j¤£"pqx_e5z?�_�gpqx_��z{���¸�g��jivwpt�_pts?o�°²µ½xDjw¬Xegpq�~p¥x��%zTvwv�ji�Xx�z?�grtjs�¦Tjw��egjizT�g��v�sTh\��z{��ji�9x_s²¢g¬XjR��¼N�~�_jRv�p|��pts?oºpqocx�jw��¦Áz?r;rtpq�g��z{��ptji��z{o���pqx��D��ji��­·sT�_h¯z?o~v�jR���Xs²ogs{x\�X�_sT��g��z?h¨z{x_p|vwz?rqrt�¨£"egjwo=x_egj�v�sTh¨�g}Xx_ptogu\�g��jiv�p|�_pqsTo¤pqo�v���jiz?�_ji�i°�±²�´³�µ´p|�-£"�_pqx�x�jwo�pqo � z{o��¤�g}gptrqx&}~��sTo±¸�´³�� z{o~�¯m�±²�)z{o~�¯viz{o¨��j�­·�_jijwrt�\�Xs�£"o~rqscz?�XjR�§°�µ½x´p|�´��x_ptrtr~}go~�gjw�#�gjw¦?jirqsT�gh¨jwocxi©�ogjw£ ­çzTv�ptrqpqx_ptji��_}~v�e�z?�"z?}Xx_sTh¨z{x_p|v��Xp �Ajw��jwocx�ptz{x_pts?o�z{o~��rqptogjRz{��z{rtu?ji�g��zD£"ptrqr���j zT�g�Xji��pto�x�egj�ogjRz{�&­·}Xx�}g�_jT°

µ½xe~z?�"jio~z{�~rqjR��}~�"x�s¤pqh¨�grtjwh¨jioTx�z{o~��x�ji��x'�_s?h¨jDz{rtu?sT�_pqx_e~h¨�iª~�_}~v�e¸z?�y�Xj�x�jw��h\ptogptogu=z?rqr���sTrq}X¼x�pqsTo~�s{­#x_egj¨u?jiogjw��z{r�vwzT��j\s{­#x_egj¯�Xpt��jiv�xf��ptogjwh¯z�x�ptv%�g��s?�grtjwh ­·s?���~z{��z{rtrtjwr��_sT��s?x��iª§pt�_s?r|z�x�pqogu���jiz{r��s�s{x��´s?­���sTrq��ogsTh¨ptz?rt�#s{­�r|z{��u?j'�XjwuT�_jij'z{o��¤£"pqx_e=�VsT���_pq�grt�¯pto~z?viv�}g��z�x�jwrt����ogs�£"o�v�s�j �¤v�ptjwocx-s?�"z{�X¼�~�_sÁ¬Xpth¨z{x_ptoguD£"pqx_e�z{���gpqx_��z{���¯z?vwvw}g��z?v�� #wji�_s�jR�#s?­�z%­·}go~v�x_pts?o�}~��ptogu¨z{o�z?�gz?�Xx_jR�¤ptocx_jw��¦�z{rN��ji£&x_sToz?rquTs?��p¥x�egh�°²d&egj=v�sTh¨�g}Xx_ptogu��g�_jRv�p|��pts?o¹e~zT� �Vjwjio5�X��o~z{h¨p|vwz?rqrt��zT�gz{�Xx�ji�¹pto¶s?���Xji��x_s²­·}~r¥¢~rtr-x_egjvws?h¨�g}Xx�z�x�pqsTo~z{r&ogjiji�g�iª´£"pqx_egsT}Xx¯�_jR�c}gpq��ptogu¸x�sº��ji��x�z?��x\x_e~j��~�_sTu?��z{h0­·�_sTh x_egj�¦Tjw���I��jiu?ptogogptogu~°d&e~pt�%£"pqrtr-��j��g}g����}gjR�I£"pqx_e)z9vwz{��j�­·}~r-��x_}~�g�ºs?­"x_egj���sTrq}gx_pts?o¶s?­�rqptogjiz?�%�_�X�Fx�jwh¯�Dz?o~�Is{­�uTrqsT�~z{rsT�Xx_pth¨p #Rz�x�pqsTo=s{­�v�s?ocx�pqo�}gsT}~�#­·}go~v�x�pqsTo~�[�q�R�Xª§�i��N°;�y�g�grtptviz�x_pts?o��-�_}~v�e�z?�-�~z{��z{h¨jwx_jw�-ji��x_pth¯z�x�pqsTo=ptoz?}Xx_sTh¯z�x_p|vw�[�t� �%��£"ptrtr§s��Ajw�-x_egj�sT�g��sT��x�}gogpqxF�¯x_s\uTz?pqo=­·}g�_x_egji�"pqo~�_ptu?ecx"pqo�x_egj��Xji¦?jirqsT�gh¨jwocx-s?­�ogji£z?rquTs?��p¥x�egh¯�w°

�� "�� &�� "��!- "��q���S�%°§�y��ji��x�e�z{o��¸±I° �~°N�Xv�e~z?j�­·jw�R°�;�_jRv�p|��j\v�sTh\�~}Xx�z{x_pts?o¸}~�_ptogu=��z{oguTjDz{��pqx_egh¨j�x�ptv?ªA¦cp|z ����� °��� ����� �Vª��i��¿·��À�© �T�g� �c�g�?ª$ yjRv�jihD�Vjw���i�T�Tkg°

� k� m\°���rtj�­·jwr|��z{o~����°r��jw� #w�Vjw��u?ji�i°:3� �������.�� ������ K�$�,�� ��������%�&' �&��%���"��°;�"vwzT�Xjwh¨p|v�;�_jR�_�iª§�i�T�?�~°� ����±¶°��-jw� #�z?o~���~°���s�j�­)�Tjwo~�i° �Ljw��p¥¢�ji��egptu?eX¼½s?���Xjw�-ptoc¦Tjw����pts?o�s{­�­·}go~v�x_pts?o~z?r��Xjw�Vjwo��Xjwo~vwpqjR��z{o��

ptocx_ji�_¦�z{r��yjw£&x�s?o�h¨j�x_e~s��~�w°8����&���E��& � � ��* �������� +�{ªl�X©q� k?�gªXk{�?�~�?°� ��� �%°��-��z{x�x���z�z{o~�9��°���jw�_x_rtptogu~° ³~jiz?�_pt�grqj\��jiz{r���z{o~�gs?h z?viv�ji���'h¨zTv�egptogji�i°����Y�u� � ��* ��& ��E������ª

�i��¿ç�cÀ�© �T�T� �?k2�~ª~�R�?�?�~°� ���� °V��°��-��jwocxi°�� ³~s?�_x_��z{o²h%}gr¥x�pq�~rqjw¼N�g��jivwpt�_pts?o9z?�_pqx_egh¨jwx_p|v �~zTv!�Áz?u?jT° ��� ���������ªA�~©��E� ���~ª±9z{��v�e��R�����~°

à�à�Ò r¡�¡�¢�£

�Ák �����=���%��&' �.�1����=������&�&������

� ��� � ���yd6�"ji�_jiz?��v�eºm�_sT}g��°%�y�_pqx_e~h\sc�w©�z��_jirqp|z{�~rqj\pqocx�jwu?��z�x�ji��v�s?h¨�g}gx�z�x�pqsTo~z{r�jioc¦�pt�_sTogh¨jwocxi°�yogpt¦?jw���_p¥xF� s{­���ocxF£#ji�_�Vjwo�ª��-jwrtu?pt}gh�ª�!�#�#$%'&)($(2*�? C�� *�*�*'/ :�?25B/65@0 / .-,�(+:-(�0�5+C�#-(5�>I?�#$!+3=1 � ªk?�?�~�?°

� �%��±¶° � jw�Vjw��pqs z?o~� �;°Tm��z{o�¦�pqrtrtpqji���i°���s?rt¦�pqo~u��ys?ogrtptogjiz?�����X�Fx�jwh¯�L��� � s?o��Fx���z?pqocxLµêo�¦?jw���_pqsTo%z{o��µêocx�jw��¦Áz?r�����p¥x�egh¨j�x�ptv?°�µêoB3� ���� � � ����2� � ����� �� ����&�3� ��$��&�&�� �E�� l !�� �. ���*,�!�&��� � �*=�l���$����� � � 3 � ��� ������ � � � � 3 ��� � °

� ������° ¼½±I° � egji�_ogjiz?}X¬§ª��A°�m}~pqr|z{pto�ª~z?o~���~°@�'pquTogji�i° � 0������&�����)(�������� � ��� � ������������ ��$����� y���

������&��"������� �° !�#�#%'&)(�(2*$*�*��25�C$%'/)4@? %��8/ �$>-(�025AC-5gªX�R�?���~°� ��� m\° � s?rtrqpto~�iª?z{o��\�D°?� ����p¥x�z?�i°~��s?rt��ogs?h¨p|z{rX��jiz?rX�_s�s{xLp|��sTrtz{x_pts?o\}~�_pqogu 'ji��vwz?��x�ji�����_}grtj"s{­§�_pquTo~�w°

µêo ������� ��� ¿F�R���+�cÀ�ªX�g��°�kE�?k k��2�g°�t�i�����D° ¾y° � sTogogjirqrVz{o~�¯��° ±I° � s?��rqjR�_�i°A��o¯j�¬X�Vjw��pqh¨jioTx�z{r�pqocx_ji�_¦�z?r�z{��pqx_egh¨j�x�ptv��~z?v!��z{uTj"pqo¤±9z{�grtj?°

µêo �@��*-� � l�&��%���"� �L����( � ������*-����� ��=������&���������� �Y !������� ~ª��R�?�?�~°�t�?��� %°2 z{ogji�?ª?m\°�yz?og�_s?xiª��D° ��j�­·fw¦���j?ª{³-°��"s?}~pqrtrqptjw�Rª{z{o�����°��Vpqh¨h¨jw��h¯z{ogo�°cd&egj&±²�´³��5rqpt�g��z{���V°

!�#�#$%'&9(�(2*�*�* / 3-%�$> /91�>���ª?k?�?�g�T°�t�Rk�µ�°Vmjw}grtpqu¤z?o~� �3° ���� {h¨jw�R°�µêoTx�jw��¦�z{rtrq��jiv�ego�}go~u%pto²±¸z{�grtj¼ 'pqj�¾;��£-jwpqx_jw��}gogu¨ptocx_�~z��$ #i}gh

�Lz��?j�x"ptocx_�~z����Xji�=��e~z{��j�¼ ��pt�g��z{���A°�d�jiv�egogp|vwz?r��"jw�Vs?�_x��?���{kXª��yogpq¦Tjw����pqx! {x ��z{��rt�_�_}~egj?ªV�i�T�?�g°�t�i���¾y°��yz?o~�_jwo�°#" & ���!�&4��������*�� ������� y�+���� ����� ��$�����&'� l�&��%���"��°-±¸z{��v�jir� yj>���Tjw�Rª§�i�T�TkX°�t�w���¾y°l�'z{o~�_jwo¸z?o~�¸��° µ�°Vm��jwjioc�Vjw��u~°"�yo¸pqocx�jw��¦Áz?rJ��ji£&x_sTo¸h¨j�x_e~s���° �+�Y� � � ����&����!.0���)(r��� l.

� ��* �������� +�{ª��RkX© �?� �T�gª~�R�?�T�g°�t� ���d�° ¼½��°+��p|v!�?jw�¤z?o~���%°X�T}�z?o~�=±I° ¼$� °��;z{o�¾;h¯�Xjwo)µêoTx�jw��¦�z{rA����p¥x�egh¨j�x_p|v{©�­·��s?h �´�_pto~v�pt�grtji�#x_s

µêh¨�grtjwh¨jioTx�z�x�pqsTo ���L� � ��� �²ª�k?�?�Tkg°�t� ���$�´°L�Tz?}grqpto�ªL±I° ��ptj��Ajw�Rª2�%° 'pt�X��pqxiª´z?o~�I¾�°��¹z{rqx_jw�R° � ����&����!.y�� ��$�����&R �&��%���"��° ���g��pqo~u?jw�

�Lji�_r|z{u�ª~k{�T�g�?°�t�%�%��� ° �° ��jiz?��­·s?x�xiª±I° 'zÁ£&z{o��Xj?ª �¨° ¼ �A° '} z{o~� � ° ¼%$%°R��}�°��yrquTs?��p¥x�egh ���+��©Iµ �'d���µ �©-z

�Vs?�_x�z?�grtj�³�s?�_x_��z{o �E��ptoTx�jw��¦�z{r#��x�z?o~�gz?����¼N­·}go~v�x_pts?o¶rtpt�g��z?�_�T° ��� � �������ªLk{�~¿ç�cÀ�© �T�+� �$�?�gª�R�?�?�~°�t�i����� ° �° ��jRz{�_­·s{x�xR°B�=� �������+�=��& ���!&J���!��� �(

� ��� ����� ������+����������& ��*S��° ��rt}�£#ji�iªA�i�T���~°

�t�i����� ° �° ��jiz?��­·s?x�x"z{o~��m\° �3° �ºz?rt��x_ji�i°2�'o���x_sT�g�gptogu¯v���p¥x�jw��ptzDpqo�¦?ji�_pq¢~ji�=ogs?o~rqptogjiz?�"���X��x_jwh¯�-s?�sT�Xx_pth¨p #Rz�x�pqsTo�z?rquTs?��p¥x�egh¯�w° ��� �� �����ªVk2�~¿O�TÀ�© �+��� �?�T�gªE��ji�Xx_jihD�Vjw�'k?�?�T�g°

�Ýk{������° ��jipq�Vjw�R°�µêoTx�jw��¦�z{r�z{��pqx_egh¨j�x�ptvpto²±¸z�x�egjwh¯z�x�ptvizg°23� ��������& � �*=�l���$����� ���ª�¿ç�cÀ�ª§�R�?�?�~°�ÝkX����� ° ��rtz{x�x�j?ªY�D° ��}grtpt��v�e�ª � ° ��zÁ£#s�ª;±I°;��z?}~v�e�ª#z?o~� � ° � ptj�x_e~s���° � �'& � � �

++ �& �!�

& ����������=�������E�$�� �.��!.-�� ����� ���� �� � !�*=�l������ ��?°����g��ptogu?ji� �Lji�_r|z{u�ªA�i�?�T�g°

ä·Î�à�ä®Í

�������������� ��������� ��������������������! ��"����� #�� ��$�����%�&'�����)(+*,������ - �.,�0/2143 �R�

�Ýk?k� �3° ���� {h¨jw�Rª�� ° ��}grtpt��v�e¸z{o~����° ��s?e~ogjw�R° �@��*-������ !&:�$�>��& �!� f������������ ���!.K !�*=�l������ ��,3u3��� .� � !��.��f��*,������ ��&l/����E��& ��*���°�x_s¤z{�~��jRz{��¿F�R�?�?�cÀ�°

�Ýk{���@�%° ��o�}gji�g��jirO°¹�´� �³�µ � ���-µF���º¼�z²­çzT�Fx¨pqocx�jw��¦Áz?r#rtpt�g��z?�_�A° � �*=�l������ ��?ª �{��¿ç��¼½�cÀ�© k���� k{�+��ª�R�?�?�~°�Ýk������°���z{oguTrqsTpt�i°���}Xx�s?h¯z�x�ptv-rtpqogjRz{�Lv�sT�_��jiv�x_pts?o s?­~��s?}go~�gpqogujw����s?���w°���3 � �f��*,������ !& � & �E��������(�*S��ª

���T¿O�TÀ�©��X��� �{�?�~ªck?�?�g�T°�Ýk����±¶° ��jw��v�e�ªVm\°Vd&pt��v�egrtjw�Rª���° �ºs?r ��¦?sTo9m}~�Xjioc�Vjw��u~ª �3°��ys{­ç��v�ec}��Fx�jw�yz?o~� �3°����� {h¨ji�i°���(r�

�� ��������&�&����������-�2&�������� � � � � � ����� �� �2���!���������0� l.���*=�l& �S�����������*S��°��;�_ji�g��pqocx¨k?�?�~���Á��ª!�#�#$%'&9(�(2*�*�* / 3=5#$! /;:$C�?�� *$:$%�%@,>#@5�4 / �>-(�1>���(������-( � 1 �#$*�5>@,$( �D?�4�? .'/ !�#+3�4gª���o~pq¦Tjw����pqx! {x�º}g�g�Vjw�_x�z?rOªVmjw��h¨z?o���k{�T�g�?°

�Ýk2������° ��°�±²jw��rqjwxi° � ���f���E�!����������&�& ��& ����°���jw��h¨fi�&�Lz{��p|�wª§�R�?�?�~ªX�"s?�Vs{x�pt�c}gjT°�ÝkE�%��� ° ¾�°V±²s�s?��j?°L3� ��������%�&' �&��%���"��°´�´�_jioTx�ptvwjf�yz{rtrNª��i����g°�Ýk{�����D°�± �grqrtjw�R°²d&egj¤p|�����'±I©�¾;¬XzTv�x%�y�_pqx_e~h\jwx_p|v¤pto ����� °�µêo ��� ���(r�!�d�� � �� r�������� ������>�����

� l. � �*=�l& � E�������� � l& �%�!�"��� � �� ����! � � ���� °�Ýk{�����D°l�yjw}gh¯z?pqji�i°R3� ��������%�&J*,���)(r��.%�������@���%������*���� �S����������� r��° � z{h%�g��pt�XuTj@��ogpt¦?ji���_pqxF�¤�;��ji���wª

�R�?�T�g°� �?�����D°��"ji¦?sTrO°��yjw£&x�s?o�� � z{rtu?sT�_pqx_e~h }~�_pqogu²h%}grqx_pt�grqj=�g�_jRv�p|��pts?oºptoTx�jw��¦�z{r´z?�_pqx_egh¨jwx_p|v{°���ji�_jiz{��v�e

��jw�Vs?�_x"�T�T�{�~ª�µ �'�"µF� ª�¿O��}~�gh¨p¥x_x_ji��x�s,��}gh¨jw��p|vwz{r��yrquTs?��p¥x�egh¯��À"k?�?�~�?°� �g�����D°;�"jw¦Ts?r"z{o��5³-°;�"s?}~pqrtrqptjw�R°5d&egj²±²�#³�µ%rqpt�g��z{���A° !�#$#$%'&)(�(+*�*�*'/9,2C � �24���1�C / �$>@(�� C�>@,��@1�4gªk?�?�~�?°� �Tk��³-°��"s?}~pqrtrqptjw�#z{o~�¨��°��Vpqh¨h¨jw��h¯z{ogo�°�¾ �¤vwpqjioTx-pt�_s?r|z�x�pqsTo¨s{­§��sTrq��ogsTh¨ptz?rg�_jRz{rg��s�s{x��w°�x_sDz?�g�Vjiz{�

pto ���L� � � �*=�l���$����� �&: l. � �E�l&����!.��K���(��tªVk?�?�Tkg°� �?���f�§°���}gh¨��° � ���%��& ����*-�� ����0�� ����&���E��& � � ��* �������� +� � �4� � ���� l.E��� ��.E�tª´v�e~z?�Xx_ji�%µ �'d����=� ¼

µêocx�jw��¦Áz?r���z?��sT��z{x_s?���AªT��z{u?jR�=��� �i�?�~° ��rq}�£-jw�RªV�R�?�T�g°� �{���f�§°~�"}gh¨��°´³�z?��x�z{o~���~z{��z{rtrtjwr§ptoTx�jw��¦�z{r�z{��pqx_egh¨j�x�ptv?°���3 ��ªV�?��¿ç�TÀ�© �?�{� ���{�~ªg�R�?�T�g°� ���f�X}go²±²ptvw�_sc���X��x_jih¨�iªXµêo~v?° � ���d3� ��������%�& � �����)(+*,������ [/ ���������*�*��� ����=���������� � ��w°´k{�T�?�~°� �������D° ¼$� °-�ypto~v�jioTxR°���}g�"r|z\�_nR��sTrq}Xx�pqsTo��XjR�&ni�c}~z{x_pts?o~�-o�}gh¨nw��pt�c}gjR�w° ������� �&'.E�f���)(���*,���'�������

/ �������S��� � ���l& �'��� �!���=¿F�i�T���cÀ�ªX�{��� �+�{kg°

à�à�Ò r¡�¡�¢�£

Unité de recherche INRIA Rhône-Alpes655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)��������� ���� ���������� ��� ���

ISSN 0249-6399