mathematical sciences - mellon college of science at cmu ...int.pdf · 9zce751 :exubd3qxe:

97

Upload: others

Post on 13-Jul-2020

9 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

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

021'354687569�:<;>=@?�9�;(A(BDCE1�:FHGI9�4JIK56#CMLON�1'K5BDCPN�QSR�TVU'FH=W6#3IX<BD1'3YX<7H9�XP1�3H4#6ZX<756Z;56#X[9�B]\D6#;�^H\]9�3>1'_V9@`a1I1�b�BDC�=c9�;56�FX<756d`�1,1�b8BDC�9�\D=W1'CMX�e�:<BDXEX<6#3aF�9�35;@9�C�756d7H9�;f9�\g:<69�;(?Ze�:<BgXEX<6#3@9P_�6#eh`�1,1�b(C�756de�9�C�46i:EX[9�BD35\g?ZCM^�6j9�b(BD3HkP1'_(6#lI^a6i:<BD6354#6'mno6pkq9Aq6p=c6oX<756pB]=)^5:<6CECEBD1'3YX<7H9�X�7H6p41'K5\D;Ye�:<BgX<6p;HBg:<6#4#X<\g?r9@Aq6#:E?c:<6j9�CE1'3H9�`H\]6oX<6il,XF,eP7HBD4[7>756Zkq9#Aq6pX<1W9fCE6#4#:<6iX[9�:E?_�1�:�Xs?I^%BD35k5t(=c9?,`�6o756pk'9Aq6oX<756#3�4[7H9�^5X<6#:<CuX<1f1'356o1'_�75BDC�CMX<K5;56#3IX<C#F59�C�756o;HBD;rePBgX<7Y=W6o_�1�:�1�3H6o1'_�75BDCu`a1I1�b(C#v#F59�35;Aq6i:E?w_�6iexX<64[7535B]49�\�;56iX[9�BD\]CP7%9�;yX<1c`�6{z5l(6;am|noBDCP^H75B]\D1'CE1�^%7q?wCE6#6=W6;wX<1c`a68X<7%9�X|X<756#:<6@BDC|351Y3566#;yX<1rCM^�635;yX<1,1=@KH4[7yX<BD=W6Z^�1'\DB]CE75BD35kWX<7568X<6il,X|1�:|zH35;HBD35kcX<756{`�6#CMX|^�1'CECEBg`H\]68CMX[9�X<6#=c6#3qXF�9�CPX<756{k'1q9�\�BDCPX<1cX[9�b'6@=c9�3q?>:<69�;56i:<CPX<1X<756)_]:<1'3qX81'_-:<6#CE6j9�:<4}7aF�1�:ZX<1Y`�6)=W1�:<6f^5:<6#4BDCE6fX<1�1�3H6f_�:<1'3qX81�_-:<6CE69�:<4[7aF `�649�K5CE6)BD3~X<756f`�6#k'BD353HBD35kY7H6)CE6#6=WCoX<17H9Aq6P4[7%9�3Hk�6;WX<1�^HB]4#C-6iAq6i:E?fXse218?q69�:<C#m��oC-_�1�:d={?(CE6\D_�Fq0�7H9A'6|351�X-?q6iX-e�:<BgXEX<63c9p`�1,1�b�F,9�35;r9pz5:<CMXd:<69�CE1'3cB]C�X<7H9�X-09�=�JIK5BgX<6oK53H9�`H\D6|X<1{e�:<BgX<6|BD3�9�;5A�9�354#6p9{^5:<6#4BDCE6P^%\]9�3r1'_aeP7H9�Xu0u9�=!k'1'BD35k@X<18X[9�\gbr9�`a1�K5Xt59�\DCE15FI0d7H9A'6o3H6iAq6i:2`�6#63Aq6i:E?>k'1,1,;y9�XPe�:<BgX<BD3HkW6iAq63wBD3y={?>=W1�X<756#:PX<1'35k'K56�F%eP75BD4}7wBDC|�5:<6354[7 F�9�3H;�9�kq9�BD3�9�3H;�9�kq9�BD3w0�3H6#6#;wX<1W:<6j9�;>eP7H9�X0�7H9A'6{9�\g:<6j9�;(?Ye�:<BgXEX<6#3>KH3qX<BD\a0uzH3H;>X<756ZX<6#lIX|9�4#4#6#^5X[9�`H\D6't(X<7H9�X|351�X<BD1'3>1'_V9�4#46i^5X[9�`HBD\]BgXs?r6iAq1'\gAq6#C�ePBgX<7>X<BD=W6'FH9�35;>09�=�751�:E:<BgzH6#;�`I?�={?�CMXs?,\]6p1'_�Xse263qXs?r?q69�:<CP9�k'1(tH1�`IA(BD1'K5CE\g?rX<7HBDC2e�9?�1'_�e�:<BgX<BD35kfBDCPJIK5BDX<6oBD3H6i�c4#BD63qXP9�35;�e�:<BgX<BD3Hk)9`�1,1�bre21'K5\D;>`�6Z^5:<1'75Bg`HBgX<BDA'6\g?c\]1�3Hk(m

� 356WCE1'\DK(X<B]1�3~e21'K5\];�`�6W351�XZX<1�e�:<BDX<6f`�1,1�b,C#F�9�35;�eP7H6#3�0pk�1wBD3IX<1w9�\DBg`H:[9�:E?�0p9�=�9�\g:<69�;(?�9�=r9��#6#;�`I?�X<7H63IK5=@`�6i:u1'_�`a1I1�b(C�eP75B]4[7c7H9Aq6�`�66#3)e�:<BDXEX<6#3W1'3cCE1{=c9�3q?)CEK(`(�E6#4iX<C#F(9�35;ceP75BD4[7W0�7H9A'6|351�X-:<6j9�;c1�_�41�K5:<CE6�FI`�6#4j9�KHCE60f3H6iAq6i:c:<6j9�;�=@KH4[7am/0)9�=�6iAq63�=W1�:<6w9�=c9��6#;�`I?�X<756w3IK5={`a6i:r1'_|`�1,1�b,CWeP75BD4}7�9�:<6w3H1�XrBD3�X<756w\DBg`5:[9�:E?'F29�35;9�\gX<751'K5k'7)0�7H9A'6|9�4#46#CEC-X<1{9Zk'1I1,;cBD3qX<6i:ELO\]Bg`5:[9�:E?@\D1q9�3WCE6#:EA,B]4#6�Fq0�9�=�4#1'3546i:<356;)ePBgX<7W751je�;5Bg�c4#KH\gX-BDX-BDC-_�1�:u_�9�:[9e�9?CMX<K5;563qX<C�X<1)7%9#Aq6{9�4#46#CEC�X<1WCE4#BD63qX<BgzH4pb(351jeP\D6#;Hk�6W�O9�35;>02;51c4#1'35CEBD;56#:P�~9�X<756#=r9�X<BD4#C�9�C�^�9�:EXP1'_-�,4#BD6354#6��[m

0�3�X<756W�q^5:<B]35k~�j�'�'�IF�0P_�1'K535;yX<756@:<BDk'7qXpCE1'\DK(X<BD1'3�_�1�:p=c6�F�eP75B]4[7�BDC|X<1Yk�BDA'6)9r4#1'K(:<CE6)9�35;yX<1r^5:<6#^%9�:<6f\D6#4#X<K(:<6351�X<6#C�_�1�:�X<756uCMX<K5;563qX<C#F�XE:E?(BD35k�X<1�e�:<BgX<6d;51jeP3@9�_]X<6#:�6j9�4}7{4#1'K(:<CE6dX<756-Xse21|1�:�X<75:<6#6-^%9�k'6#C�;56#CE4#:<Bg`HBD35kPeP7H9�X�0 7H9�;��EK5CMXX[9�K5k'7qXt(_�1�:2CEKH4[7�CE751�:EX2X<6#lIX<Cu=@?W^5:<1�`H\]6#=WC�9�`�1'K(X2e�:<BgX<BD35k@9�:<6p351�XuX<1,1)9�4K(X<6�mV0d4#1'K5\D;Y7H9�:<;5\D?c7H9Aq6|b(351jeP3�9�XuX<7H6`�6k�B]3535BD35kf1'_�X<7568�,6#=W6CMX<6i:�7H1e�=@K54}7>9�3�BD3IXE:<1I;HK54iX<BD1'3rX<1 � 4#6j9�351'k�:[9�^H7I?Y={?r41�K5:<CE6oe21'K5\];Y`�6�FH9�3H;YeP7563>9�_]X<6#:9ZCE751�:EXuBD3qXE:<1,;5K54#X<BD1'3W9�35;)X<7H6�;56CE4i:<Bg^HX<BD1'3)1'_�CE1'=W6�4\]9�CECEBD49�\H=W6#X<751,;5C�_�1�:dCE1'\gA(BD35k8�o�,��Q��%��L��I��R����aU�6jJqK%9�X<B]1�3HC��SBD3X<756-1jA'6#:ELOCEBD=)^%\DBgzH6;pAq6i:<CEBD1'3ZeP75BD4[78=r9�X<756#=r9�X<B]4#B]9�35CaK5CEK%9�\]\g?o41�3HCEBD;56i:i�[F�BgX e�9�C X<BD=W6VX<1P;56#CE4#:<Bg`�6-CE1'=W6-1�_,X<756-=W1,;56#\]C4#1'35CEB];56i:<6;WBD3 � 4#6j9�351'k�:[9�^H7I?'FI0�:<6j9�\DBD�#6;fX<7H9�X-0�;5BD;c351�X-`�6\DBD6iAq6�X<1I18=fK54[7cB]3)X<756P;56i:<BDA�9�X<BD1'3)1'_�X<756#CE6P=c1,;56#\DC#FI9�35;0�^5:<6#_�6i:<6;{X<1|zH3HBDCE7@X<75624#1'K(:<CE62`I?{;56CE4i:<Bg`HB]35koCE1'=W621'_5X<7562k�6356i:[9�\(=c9�X<756#=r9�X<BD49�\qX<1I1'\DC�_�1�:�CMX<K5;5?,B]35k|X<7562351'35\DBD356j9�:^%9�:EX<B]9�\q;5Bg a6i:<6#3IX<B]9�\'6JIKH9�X<BD1'35C�1'_%¡u1'3qX<BD3,K5K5=x��6#4}7H9�35BD4C#F�CE1�=c6d1�_(eP75BD4}7Z0 7%9#Aq6u;56iAq6#\]1�^H^�6#;{={?(CE6#\]_�mV¢�756-:<6#CEK5\gX<B]35kCE6iX�1�_H\D64iX<K(:<6u3H1�X<6C�BDC�3H1�X-9�C�k'1,1,;f9�C�0 e21'K5\D;f7H9Aq62\DBgbq6;aF�`HK5X�9�3fBD=)^�1�:EX[9�3qX�^�1'BD3IX�e�9�C�X<1o=r9�b'6uX<75BDC�BD3qXE:<1,;5KH4iX<1�:E?4#1'K(:<CE6)9A'9�B]\]9�`H\D6{1�3�X<756@e26i`~^%9�k'6@1'_�X<756c¡u6#3qX<6#:p_�1�:p�o1'35\DB]3569�:o�o3H9�\g?(CEBDCo1'_VX<7H6@£p6#^%9�:EX<=W6#3IXo1'_���9�X<7H6#=c9�X<BD4j9�\�I4BD6#3546#C{9�XW¡u����T��a¤�Q��%LO���%¥,¥5R�T*¦o35BgAq6i:<CEBgXs?��S7qXEXE^ §©¨�¨je�e�e8m =c9�X<7am 4#=fKam 6;5K�¨�4#3%9q¨j^HK5`H\DBD4j9�X<B]1�3HC#m 7qX<=c\�ª@351�X<6#CY«E0�3(LXE:<1,;5K54#X<BD1'3YX<1 � 469�351'k�:[9�^H7q?(¬,�[m

0�3�X<756c�q^5:<BD35kw­�®�®'®(F�0�X[9�K5k'7qXZ9r41�K5:<CE6f;5BgA(BD;56;yBD3�Xse21r^%9�:EX<C#F�X<7H6{z5:<CMXp^%9�:EXp1'3��5RV¯ R�¥I�%�/CM^%9�4#6#C#F�9�35;~_�1�:X<756WCE6#41'35;�^%9�:EXZ0p4[7H1�CE6WX<1>X<69�4[7°9�`�1'K(X80�3qX<6i:E^a1�\�9�X<B]1�3�CM^%9�4#6C#mW0Z9�\DCE1>;564#BD;H6#;�X<1w9�;5;°CE1�=c6WBD35_�1�:<=r9�X<BD1'3~X<7H9�X1'356p:[9�:<6\g?rzH35;HCPBD3>41�K5:<CE6#CP1�_��~9�X<756=c9�X<BD4#C#F5CE1'=c6iX<75BD3HkW9�`�1'K(X�X<756p^a6#1�^H\D6'm

0u7H9�;�X<756o^5:<BgA(BD\]6#k'6PX<1)CMX<K5;(?YB]3Y±�9�:<BDC#F(X<1f7%9#Aq6Zk�:<69�X2X<6j9�4}756i:<C�\DBgbq6oN�9�K(:<63qXo�5²�³�´{���H��µ)9�35;>GI9�4JIK56#CMLON�1'K5BDCN�QSR�T�U�F�9�35;yX<1c7H9Aq6{=c6iX|=c9�3q?w_�9�=c1�KHCP=c9�X<756=c9�X<BD4#B�9�3HC#m�¢�75BDCP7H9�Cok�BDA'63w=c6{9c;5Bg �6#:<6#3qX|A,B]6iex1�_��~9�X<756#=r9�X<BD4#CX<7H9�3¶X<756Y1'356YX<7%9�XW4#1'=c6#Cf_�:<1�=·:<69�;5BD3Hky`a1I1�b(CW9�3H;�9�:EX<BD4\D6#C#F�eP75BD4}7h0{zH35;¶X<1,1~;(:E?'Fu9�35;¶0@7H9Aq6YXE:<BD6#;¶X<1�k'BgAq69�\DBgXEX<\D6c=c1�:<6Y\DBD_�6cX<1�=@?°CMX<1�:E?°`I?°X<6#\D\DB]35k�9�\DBgXEX<\D6Y9�`�1'K(X@X<756>9�4iX<1�:<C#tu_�1�:fX<751'CE6Y=r9�X<756#=r9�X<B]4#B]9�35C8eP7H1�=·0{7H9Aq6=W6#XF�0p7H9Aq6cK5CE6;�X<7H6#Bg:8z5:<CMX{3%9�=c6#C{BD3�X<756WX<6il,XF�9�35;�0p7H9Aq6WXE:<BD6;~X<1wk'BgAq6WCE1'=W6cCEBD=)^%\D6)`HBD1'k�:[9�^%75BD49�\-;H9�X[9>_�1�:9�\D\�^�6#1�^H\]6)JIK51�X<6#;�BD3�X<7H6fX<6il,XFaB]3~1�:<;56i:ZX<1�CEBgX<K%9�X<6fX<756=>Fa`�1�X<7~B]3~X<BD=W6)9�3H;�BD3�CM^%9�4#6'm@�H1�:8=r9�X<756#=r9�X<B]4#B]9�35Co1�_X<756P^%9�CMXF(98\�9�:<k'6�^�9�:EX21'_�X<75B]C-B]35_�1�:<=r9�X<B]1�3c4#1'=W6Cd_�:<1'=¸K5CEBD35ky«[¢�756|�~9�4¢�K(X<1�:unoB]CMX<1�:E?W1'_a�~9�X<756#=r9�X<BD4#Cu9�:<4[75BDA'6¬�S7qXEXE^ §©¨'¨je�e�e�LO75BDCMX<1�:E?'m =W4C#m CMXEL�9�35;am¹9�4�m K(b%¨�75BDCMX<1�:E?%�MºjF(`HK(X�_�1�:�3H9�=W6#C2eP75B]4[7�9�:<6p351�XZ�+?q6iXi�-BD354\DK5;56;YBD3YX<75BDC�9�:<4}75BgAq6�F0|;5BD;~CE69�:<4[7~X<7H6@e26i`�_�1�:pB]35_�1�:<=r9�X<B]1�3 F�9�35;�BgXpBDCo^�1'CECEBg`H\D6@X<7H9�XZCE1'=W6f1'_-={?�BD35_�1�:<=c9�X<BD1'3yB]CoBD3H4#1'=)^H\]6iX<6@1�:p6#A'63BD3H9�44#K(:[9�X<6�mf�y?~BD3qX<6#:<6#CMXZB]3�noBDCMX<1�:E?�BDCZ351�Xp:<64#63qXFa`%K(X8=@?�BD3IX<6i:<6#CMX8BD3�noBDCMX<1�:E?�1'_d�~9�X<756=c9�X<BD4#Cp7H9�CZB]354i:<6j9�CE6;:<6#46#3qX<\D?'F�BD3°^%9�:EX)_�:<1�=»zH35;5BD3HkyX<7H6Y9�`�1jAq6Y=c6#3qX<B]1�3H6#;¶9�:<4[75BDA'6'FV`HK(XW9�\DCE1~9�C)9w:<6CEK5\gXf1'_�CE66#BD35k~CE1�=r9�3q?°1'_�=@?

vwN�Q+R�T�U�GHm LON-mDF�¼2½H¾#¿©Ài½%¾}Á�Â�þ#Ä�Å5ÆÇI¾}ÁPÇI¾�ÈVþ}Á#Æ�¿¹½5Ä+ÉSÆ�Ê>ÇI¾iÁ�Ë�È[ÆqÌi¿i;iÂr¾}ÁPÎ�½qÏ{¿¹É�ÂfÉ+Ä�¾iÁ2Ê�Æ�Ê�¿¹É�Ê{þ#Î�É�È<¾iÁ<Fq£pKH351,;at(Ðp9�K(X<75B]6i:ELÑpBD\D\�9�:<C#F5±�9�:<B]C8��'Ò'�@l,l%ÓpÔ'Ô�Õ)^5^ mº 0W9�=ÖAq6#:E?¶X<7%9�35b,_�K5\�X<1°Gq1�7H3�GHm �{× ¡�R�T�T�R��Ø9�3H;�Ùd;5=fK535;/�2m-ÚpRV¯��%�5��UjR�TuFd_�:<1'=ÛX<7H6w¦o35BgAq6#:<CEBgXs?¶1'_{�qX

�o35;(:<6#ePC�B]3y�I41�X<\]9�35;aFH_�1�:P7H9A(BD35kW4i:<6j9�X<6#;>CEK54}7y9�3wBD3IX<6i:<6#CMX<B]35kW9�:<4[75BDA'6'm

Page 2: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

BD;56j9�CZ9�XEXE:<Bg`%K(X<6#;�X<1>1�X<7H6i:<C#F eP751�1'_�X<6#3�;51>351�XZ6#A'63�K535;56#:<CMX[9�35;~X<7H6#= e26#\D\St 0o7H9Aq6fXE:<B]6#;�X<1Y`a6W9�CZ9�44#K(:[9�X<6W9�C^�1'CECEBg`H\]684#1'354#6#:<35BD35kWX<7568e21�:Eb>1'_�1�X<756#:<C#F�9�35;w0�7H9Aq6ZXE:<BD6;>X<1c\D69�:<3w=W1�:<6{9�`�1'K(XPX<7568=r9�X<756#=r9�X<B]4#B]9�35C2eP751r7H9Aq6BD3qXE:<1,;5KH4#6#;wCE1'=W681'_�X<7H68BD;569�C�eP7HBD4[7>02e�9�CPX[9�K5k'7qXPeP756#3w0ue�9�Co9)CMX<KH;56#3qXoBD3w±V9�:<BDC|BD3>X<7568\]9�X<6{Ò�®�C#md0�751�^�68X<7H9�X02ePBD\]\�`�68k'BgAq63�X<7H684#1�:E:<6#4#XPBD3H_�1�:<=c9�X<BD1'3�`I?y9�3I?'1'3568eP751WzH3H;5C|1�3H681'_�=@?�=WB]CMX[9�bq6C#F%9�35;wX<7H9�X�0�ePBD\D\a`�68_�1�:<k'BgAq6#3_�1�:�X<7H6#CE6ZKH35BD3qX<63qX<BD1'3H9�\a6#:E:<1�:<C#m

0�e�9�C�`�1�:<3)BD3@�H:[9�354#6�_�:<1�= 98�q?,:<B]9�3f_�9�X<756#:�9�35;W9o�5:<6#3H4[7)=W1�X<756i:-9�35;@0�\D6#_�X�=@?{41'K53qXE:E?@_�1�:V^�1'\DBgX<BD4j9�\':<6j9�CE1'35C9�35;w0�6#3��E1j?>3H1exX<756{751'CM^HBgX[9�\DBDXs?�1'_�9�3y�o=c6i:<BD4j9�3>KH35BgAq6i:<CEBgXs?qF�9�3H;wX<75BDC|=c9?>6il,^H\�9�B]3>=@?>BD3qX<6#:<6#CMX|BD3y=W63qX<BD1'35BD35kX<7H9�X81�X<756i:<CZ7H9Aq6fe21�:Ebq6#;�BD3�9Y;HBg �6i:<63qXZ41�KH3qXE:E?yX<7%9�3~X<7H6)1'356feP756#:<6fX<756i?ye26#:<6)`�1�:<3at�0o9�= 351�X8BD3qX<6i:<6CMX<6#;�BD3X<756W^5:<6#4BDCE6W4BgX<BD�#635CE75Bg^°1'_dX<7H6)^�61�^%\D6W=c6#3qX<BD1'356;aF�`HK(X{0oe�9�3qX<6#;°X<1>41'3qAq6i?~X<756cBD;56j9�X<7H9�X8X<756W;56#Aq6#\D1�^H=c6#3qX{1�_�~9�X<756=c9�X<BD4#C�BDCd9�3WBD3IX<6i:<3H9�X<BD1'3H9�\(6#3H;569Aq1'K(:m � 356�=cBDk'7qX�1�`HCE6#:EA'6�X<7H9�X�X<756#:<6�7H9Aq6�`a6#6#3W6i�c4#BD63qX-CE4[7H1I1'\DC-BD3)CE1'=c69�:<69�Cp1�_d�~9�X<756#=r9�X<B]4#Co9�XpCE1'=W6@^%\]9�4#6#Cp9�35;�9�XpCE1�=c6f=W1'=W63qX<CoBD3yX<B]=W6�F�`HK(X89�\gX<751'K5k'7�X<756@41'35;5BgX<BD1'35Co=cBDk'7qX|`a6\D6CECp_�9Aq1'K(:[9�`H\D6)1'K(X<CEB];56@X<756#CE6WBD=)^�1�:EX[9�3qXZ4#63qX<6i:<C#Fa0Pe21'K5\];�\DBgbq6{X<1YX<75B]3(byX<7H9�X89Y\D1�XZ1�_uk'1I1,;~e21�:Eb�4j9�3~`�6f;H1�3H66#\]CE6ieP756#:<6�F(9�3H;cBgXd:<6=WBD35;HCd=W6|1�_�eP7%9�X�9�3c0sX[9�\DB]9�3W_�:<BD6#3H;WX<1�\];c=c6|98_�6ie/?q6j9�:<C29�k'15§V756Pe26#3IXdX<18X<6j9�4}7cB]3��I1'=r9�\]B]9(F_�1�:uCEBglW=W1'3qX<7HC-B]_�0V:<6#=c6#={`a6i:-e26\D\SFI9�35;Y98CMX<K5;56#3IXu49�=c6�1'356|;H9?)X<186#lI^%\]9�BD3)X<1{75BD=�X<7H9�Xd756|CE751'K5\D;c3H1�Xu`�6PK(^%CE6iXeP7563>CE1'=W6Z1'_�X<756pCMX<K5;563qX<CP_�6\D\a9�CE\D66i^w;5K(:<BD35kW75BDC�\]6#4iX<K5:<6#C#FH9�C2X<756p:<6j9�CE1'3�e�9�C�351�X�X<756Bg:�\]9�4<bY1'_�BD3qX<6i:<6CMX�_�1�:�X<7H6CEK(`(�M64iXF5`HK(X�X<7H9�XPCE1�=c6iX<BD=c6#C2X<756#?Y7H9�;w69�X<6#3>351�X<75BD35kW_�1�:|9fe26#6#b�m�0ue21'K5\D;>\DBDb'6pX<1fX<75BD3(bYX<7H9�X�={?�\D64iX<K(:<6Z351�X<6#C4#1'K5\];r9�:E:<BgAq6|_]:<66#\g?WBD3rCEK54[7r:<6=W1�X<6P^H\]9�4#6C-eP756#:<6PX<756#:<6|9�:<6oCEK54[7Y4#1'K(:[9�k'6#1'K5C2CMX<K5;563qX<CueP751f9�:<6PXE:E?(BD35k{X<1@9�4JIK5Bg:<6CE1'=W6Z^5:<6#4BD1'K5C�b(351jeP\D6#;5k'689�`�1'K(XP�~9�X<756=c9�X<BD4#C#F5;H6#CM^HBgX<6pX<75686#351�:<=c1�KHC�;5Bg�c4#K5\DX<BD6#C2X<7H9�X�X<756#?Y6#3541'K53qX<6i:|BD3�X<756#BD:6iAq6#:E?,;%9#?�\DB]_�6'm

0{751�^�6rX<7H9�X@=@?�\�9�4[b�1'_�1�:<kq9�35BD�9�X<BD1'3hCMb,B]\D\DC8ePBD\]\-3H1�Xf`�1�X<756i:@X<1,1�=@K54}7hX<756c:<6j9�;H6i:<C#m�084#1'35CEB];56i:fX<69�4[75B]35k4#1'K(:<CE6C�\DBDb'6�\D69�;5B]35k�k�:<1'K(^HC�1�_,356#e24#1'=W6#:<C�BD3qX<1�K53(b(351jeP3841�KH3qXE:<BD6#C#F�351�X�KH3(b(351jeP3ZX<1�=c6-9�C 0�7H9Aq6-1'_]X<63pe�9�35;56#:<6#;9�:<1'K535;at(CE1'=W6o=W6#=@`�6i:<Cd1'_�98k�:<1�K5^reP751@7H9Aq6o9�\g:<69�;(?):<69�;Y9�`�1'K(XuX<756P:<6#k'BD1'3r1�:27%9#Aq6|`�6#63cBD3r1�X<756i:26il,^�6;5BgX<BD1'35CePBgX<7Yk'K5BD;56C2=@KH4[7�=W1�:<6o1�:<kq9�35BD�#6;rX<7H9�3�=W6p=WBDk'7qX�_�66#\�;HBDCE1�:<BD6#3qX<6;r`I?r=@?r4[751'BD46o1'_a^%\]9�4#6#C2X<1@A,B]CEBgXF(9�35;�BD35;H6#6#;08=c9?�7H9Aq6Y_�1�:<k'1�XEX<6#3�X<1yCE7H1e�9w_�6ie�BD3IX<6i:<6#CMX<B]35k>^H\�9�46#C#F�`%K(Xf={?�k'1q9�\uBDC{X<1y_�9�=cBD\DB]9�:<BD�6fX<7H6W:<6j9�;H6i:<C{ePBDX<7�X<7H6CEK(`(�M64iX|9�35;w351�XPX<1fe�:<BgX<689);56#zH35BgX<BDA'689�44#1'K53qXm

¢�756#:<6u9�:<6d:<6#CEK5\DX<C eP75BD4}7{9�:<6u=W6#3IX<BD1'356#;ZePBgX<7H1�K5X ^H:<1I1'_sF�9�3H;{CE1'=W6iX<B]=W6#C X<756#?89�:<6-^5:<1jA'63{\]9�X<6i:�`HK5X�CE1�=c6iX<BD=c6#CX<756#?Z9�:<6-351�XF�9�35;@9�C�351P:<6#_�6i:<6#3H4#6#C�9�:<6-k'BgAq6#381'356dCE751'K5\D;Z:<6#=c6#=@`�6i: X<7%9�X�0a7H9Aq6-`�6#63ZXE:[9�B]356#;{9�C�9�=c9�X<756=c9�X<BD4B]9�3aF9�35;�X<756WCMX[9�X<6=W63qX<CpePBgX<7H1�K5Xp^5:<1,1'_�C87H9Aq6WBD35;566#;�`�6#63~^H:<1Aq63�BD3°9�=r9�X<7H6#=c9�X<BD4j9�\�CE6#35CE6'F�`�6#4j9�KHCE6)B]_-X<756#?�7H9�;351�X�0ue21'K5\];�7H9Aq6Z4j9�\D\D6#;YX<7H6#=�4#1'3��E6#4#X<K(:<6#C�BD35CMX<69�;at57H1e26#Aq6i:F50�9�= 9�\DCE1)7,K5=c9�3w9�35;>=@?Y=W6=W1�:E?YBDC�351�X�^�6i:<_�6#4#X9�35;>02=c9?�7H9Aq68=c9�;56Z=cBDCMX[9�bq6#C#mV0u`a6#\DBD6#Aq6oX<7H9�X�X<756p:<B]k�7IX|9�XEX<BDX<K5;56pBD3w�~9�X<756#=r9�X<B]4#C�B]C�X<1f`a689�`H\D6pX<1W6il,^H\]9�BD3w9�\D\X<756)CMX[9�X<6#=W63qX<CoX<7H9�XZ1'356f=r9�b'6C#Fa`HK(XZB]3�9Y4#1'K(:<CE6W1�3H6f7H9�CpX<1>9�CECEK5=W6)X<7H9�XoX<756f:<69�;56#:89�\g:<69�;(?�7H9�CZCE1�=c6@`�9�CEB]4b(351jeP\D6#;Hk�6Y1�_���9�X<7H6#=c9�X<BD4C#F�9�35;¶CE1'=W6r^5:<1,1'_�C@1'_P9�=W1�:<6r6#\]6#=W63qX[9�:E?�3H9�X<K(:<6�9�:<6r1'=cBgXEX<6#;am�no6i:<6�9�35;hX<756i:<6r0=W63qX<BD1'3w9f:<6#CEKH\gX�X<7H9�X�027%9#Aq6{7569�:<;w1'_�`%K(XP_�1�:�eP7HBD4[7�0�356#A'6#:�:<69�;y9f^5:<1,1'_�1�:P;5BD;w351�XP=r9�bq68K(^y=@?Y1jeP3>^5:<1,1'_�F9�35;y0�KHCEKH9�\D\g?>C<9?wCE15m��o4#X<KH9�\D\g?'F�9�3H;w0�X<75BD35b�X<7H9�X|=@?w9�;(A(BDCE1�:|=c6#3qX<B]1�3H6#;>X<7%9�X|X<1r=W6'F%BgX|B]CPKHCE6#_�K5\�X<1W:<6j9�;y1�3H\g?X<7568CMX[9�X<6=W63qX�1'_V9fX<7561�:<6= 9�3H;>1'3568CE751'K5\D;>:<69�;�X<7H6p^5:<1,1'_�1'35\D?YB]_�1�3H6Z49�353H1�X|CEK(^5^%\g?�1�3H6 � m

�o_�X<6i:{7569�:<BD35ky9�`�1'K(X)�(RV¯ R�¥,�%�*CM^�9�46#C{BD3�CE6=WBD3H9�:<CZ`I?�GI9�4JIK56CMLON�1'K5BDC{N�QSR�T�UW1�:{CE1'=W6c1'_u75B]C8CMX<K5;56#3IX<C{9�XÙd4#1'\D6f±V1'\g?,X<6#4}7535B]JIK56fBD3��j�'Ò'ÒIFa0o\D6j9�:<356#;�9�\DBgXEX<\D6{=c1�:<6fBD3�75BDCp41'K(:<CE6#C89�XpX<756)K535BgAq6#:<CEBgXs?yBD3~X<756f_�1�\]\D1jePBD35kr?q6j9�:<C#F9�35;¶08:<69�;¶75B]C���'Ò'­y4#1'K(:<CE6���BD3¶��1'3qXE:<6j9�\OFu¡29�3H9�;H9(Fd9�35;�9w`a1I1�b°`I?��I75=fK56#\2�Z¤��8R�T���m�0{7H9�;¶\]69�:<356#;�9�`�1'K(X

� ¢�756>�~9�4¢�K(X<1�:c9�:<4}75BgAq6>=W6#3IX<BD1'35C)9�3�BD3qX<6#:<6#CMX<BD3Hk�9�356#4;51�X<6�BD3¶X<75BDCf:<6#CM^�64iXW4#1'3546i:<35BD3Hk�9�A(BDCEBgX)1�_|�o3qX<1'35B����¤���T���X<1cX<7H6{¦o3HBgAq6i:<CEBgXs?w1'_� 2KH6#351'Cp�oBg:<6#C#F��|:<k'6#3qX<BD3%9,F�BD3¶�j��Õ��,ta�o\g`�6#:EX<1�¡u��¥��������R�T�e�9�Cp9rCMX<K5;563qXpX<756i:<69�35;W7H6�e�9�C-^HK5�#�\D6#;)`I?W98JIK56#CMX<BD1'3)X<7H9�X�����¤���T���7H9�;c9�CMbq6#;aF,9�35;c756�C<9�BD;)X<7H9�X�X<7H6P9�3HCMe26i:-e�9�C-BD3�����¤���T�� × C1jeP3�`�1,1�b*«[¢�:<BDk'1'351'=W6iXE:<B]4)CE6i:<BD6CE¬5F�`HK(X�����¤���T�� ;5BDC<9�k�:<6#6;at-9�_]X<6#:{;5BDCE4#KHCECEBD1'35C8X<756c=r9�XEX<6#:8`�6#4j9�=W6W4\D69�:FV9�C¡u��¥��V�����R�Th7H9�;�:<69�;w9fCMX[9�X<6#=W63qX�BD3�X<756p`�1,1�b�9�35;>CEK(^H^H\DBD6;�75BDC�1jeP3�^H:<1I1'_sF(eP75BD4}7Ye�9�C�=W1�:<6Zk'6#356#:[9�\�X<7H9�X2X<7H61'356pX<7H9�X�����¤���T��¶7H9�;�e�:<BgXEX<63at%¡u��¥������ �R�T × C2^5:<1,1'_�9�35CMe26i:<6;�X<7568JIK56#CMX<BD1'3�X<7H9�X����-¤����T��¶7H9�;@�EK5CMX|9�CMbq6;aF`HK(Xo¡u��¥��������R�T°7H9�;�351�X�\]1I1�bq6#;>9�X����-¤���T�� × Cu^H:<1I1'_�9�3H;�7H9�;�351�X2b(351jeP3Y`�6_�1�:<6oX<7H9�X�75BDC2^5:<1,1'_ e�9�C�;5Bg a6i:<6#3IX_�:<1�=�X<756Z1'3568BD3�X<756Z`�1,1�b�m� N�Q+R�T�UdGHm LON-mDF��2È<ÆqÌ#¿};}Âr¾iÁuÎ�½IÏp¿¹É�ÂfÉ+Ä�¾iÁuÇqÎ�ÊHÁu¿g¾iÁ�þ#À#½5Î'Ä+ÉOÆ�ÊHÁ2Î�½qÏZÇPþiÈ}É��uþ¾iÁ ËHÎ�ÈiÄ+ÉS¾#¿�¿g¾iÁ<F�£Z6#K(l(B! 6=W6#"6#;5BgX<B]1�3 F'�$"6#=cBD3H9�BgL

:<6P;56���9�X<7%"6#=c9�X<B]JIK56Cu�IK(^%"6#:<BD6#K5:<6#C#FI�o15m��8� "Ù-X&"6�F%�j�'Ò�­q��N�6#C-±-:<6CECE6#Cd;56P\ × ¦o3HBgAq6i:<CEBgX&"6�;56P��1'3qXE:&"69�\SFI��1'3qXE:<69�\SF&'oK56�m�j�'Ò�Ô,F��)(�Ò@^5^ m� �Z¤��8R�T°�%mDF&*�¾,+iÄ+½(È<¾iÁoÆ�Ê�¾i¿�¿ É Ë�Ä+É-+8Ì[Æ�½(Ê%ÇqÎ�È/.0��Î'¿¹½H¾dË�È[ÆqÌi¿g¾iÂ)Á<F�1 BD\D\DB]9�= �~9�:<CE7rÚPBD4#6P¦o35BgAq6#:<CEBgXs?'FIno1�KHCMX<1�3 F,¢�6#l�m

­

Page 3: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

;5BDCMXE:<BD`HK(X<BD1'35CdBD3rN�9�K(:<63qXP�(²�³�´8���5��µ × C24#1'K(:<CE6p9�XuÙd4#1'\D6|±V1'\g?,X<6#4[7H35B]JIK56PBD3y�j�'Ò'Ôq¨�Ò'Ò{9�35;rX<756#3c0�:<69�;r7HBDC-`�1,1�b � mV0z5:<CMX-:<69�;c9�`�1'K(X-B]3qX<6i:E^�1'\]9�X<BD1'3fBD3c9p`�1,1�b@X<7H9�X-GI9�4JIK56CMLON�1'K5BDC-N�QSR�T�U�e�:<1�X<62ePBgX<7WÙ-35:<BD4#1Z����¤V��T��aU���9�35;)X<7563)7H6kq9A'6�=W6�75BDC-9�:EX<BD4#\]6dePBgX<7WGI9�9�bf±d�%��� ���WX<1o:<6j9�;aFI9�35;W\]9�X<6i:�756P9�CMbq6#;W=W62X<1ZCE1�\DA'6�CE1'=W62^H:<1�`%\D6#=cCVBD3f0�3IX<6i:E^�1'\]9�X<BD1'3_�1�:2={?)X<756#CEB]CdBD3��j� (,��F(9�3H;Y9�:<1�KH35;cX<7H9�XdX<BD=W6�0�;HBD;W:<6j9�;r98_�6ie 9�:EX<BD4#\D6C-1'3W0�3IX<6i:E^�1'\]9�X<BD1'3am�0-9�\DCE1{\]69�:<356#;cB]3c1�X<7H6i:4#1'K(:<CE6C�`I?>GI9�4JIK56CMLON�1'K5BDC|N�QSR�TVU'F%9�35;wX<7568K5CEKH9�\ ^H:<1I46#CEC�e263qX|1'3aF%\D6j9�:<3HBD35k5FH_�1�:<k'6#XEX<BD35k5F5B]3qAq6#3qX<BD3HkW9W3H6ie�^5:<1,1�_9�35;):<6;5BDCE41Aq6#:<BD35k89p^5:<1,1'_�FqeP756#3c9�CMbq6#;c98JIK56#CMX<BD1'3)`I?W9Z_�6\D\D1je�:<6CE69�:<4[756#:d1�:d9ZCMX<KH;56#3qXFICE1ZX<7H9�X-_�1�:-=r9�3q?@:<6CEK5\gX<CBD3>X<756CE68\D6#4#X<K(:<6#C�024j9�3>7%9�:<;H\g?�C<9?>BD_�0�7H9Aq6Z:<6j9�;>X<756=�1�:PzH\D\]6#;�X<7568kq9�^HCPB]3>CMX[9�X<6=W6#3IX<C�X<7H9�X�0�7%9�;y7569�:<;amu�%1�:X<756Z^HK(:E^�1'CE681'_�X<7H6Z\D64iX<K(:<6C20�4#1'35CEK5\gX<6;�X<756p`�1,1�bY`I?�GHm� ��%��¤V³¶9�35;wGHmHN��R��%U� ���R����m

�y?)^�6#:<CE1�3%9�\H:<6j9�CE1�3r_�1�:u`�6#B]35k8BD3qX<6#:<6#CMX<6#;cB]3)X<756|CEK(`(�M64iXu1'_�X<756CE6|\D6#4#X<K(:<6#C-B]C�X<7%9�XdX<756CE6|JIK56#CMX<B]1�3HCd9�^5^a69�:uBD39f3H9�X<K(:[9�\�e�9?ceP7H6#3�1'356oe�9�3qX<C�X<1fCE1'\gAq6o^%9�:EX<B]9�\�;HBg �6i:<63qX<B]9�\�6jJqK%9�X<B]1�3HC2_]:<1'= ¡u1'3qX<BD3IK5KH= ��6#4}7H9�35BD4C21�:�±d7q?(CEBD4#C#m� k'1I1,;we�9?>X<1c\D6j9�:<3y=W1�:<68B]3wX<75BDC|;5Bg:<64iX<BD1'3wBDCPX<1c41'35CEK5\gX|X<7568`a1I1�b(C|1'_VÚP1�`�6#:EX|£Z� ������&��9�35;�GI9�4JIK56#CMLON�1'K5BDCN�QSR�T�U��m

±dBgXEX<CM`HK5:<k�7 F5�oK5k'K5CMXp­�®'®�®

N�K54@¡u7H9�:<\D6CP¢����H�(���¦o35BDA'6#:<CEBgXs?Y±�:<1'_�6#CECE1�:|1�_��~9�X<756=c9�X<BD4#C£p6#^%9�:EX<=W63qX�1'_��~9�X<756#=r9�X<B]49�\��I4BD6#3H4#6#C¡u����T���¤VQ��HLO����¥,¥5R�Th¦o35BgAq6i:<CEBgXs?±dBgXEX<CM`HK5:<k�7 F5±a�@F �jÔ'­(����L � �'��®,F%¦@m �%m �@m

�IK5=c=W6#:�0�35CMX<BgX<K5X<6p_�1�:P�o;(A�9�3546#;�Ð|:[9�;5K%9�X<6f�IX<K5;56#3IX<C#F��j��Ò��,m� �(²a³�´{���5��µfN-mDF���ÅoþÆ�È}ÉO¾{ÇI¾}ÁZÇ�É�Á}Ä+È}ÉOÌ}½5Ä+ÉSÆ�ÊHÁ[F5±dK(`H\DB]49�X<BD1'35C2;56o\ × 0�35CMX<BDX<K(X2;H6o�~9�X<7�"6=c9�X<B]JIK56o;56p\ × ¦o35BgAq6i:<CEBDX&"6P;H6

�qXE:[9�CM`�1'K(:<k5F5�o15mV0��|L��@m5�o1�K5A'6\D\D6 "6;5BgX<BD1'3aF(6#3IX<B! 6i:<6=W6#3IX241�:E:<B]k�"66�F,:<6#_�1'35;5K56Z6#X�9�K5k'=W6#3IX&"6#6�m-no6i:<=r9�3H3aF(±�9�:<BDCZ�j��Ò'Òl(BDBDB�ÓoÕI­�®@^5^ m�~N�Q+R�T�U{GHm LONdm�� ����¤V�%T��aU8ÙPmgF �2È<ÆqÌi¿i;}Âr¾}Á@Î�½qÏY¿¹É�ÂfÉ+Ä�¾iÁ8Ê�Æ�Ê~Å5Æ�ÂcÆ��-;iÊ�¾}Áf¾#Ä2Î}ËqË�¿ É!+<Î'Ä+ÉOÆ�Ê5Á[F%Ñ-1'\Smp��m�¢�:[9A�9�K(l>6#X

ÚP64[756#:<4[756C��~9�X<7�"6=c9�X<B]JIK56C#F5�o1(mP�)(@£pK53H1I; F5±�9�:<BDC{��'Ò �@l,lHÓ��(�­@^5^�m� �����¤�³*GHm�� N �R��%U� � �R���GHmgF��}ÊHÄ�¾iÈ�ËHÆ�¿©Î'Ä+ÉOÆ�Ê�ÁOË5Î�+[¾}Á����|ʶÉ�Ê%Ä+È<ÆÇ�½�+iÄ+ÉOÆ�ÊHF�Ð|:<K535;H\D6#7(:<63°;56#:{��9�X<7H6#=c9�X<BDCE4}756#3

1 BDCECE6#35CE4}7H9�_]X<63aFH�o15md­'­��,m2�q^5:<BD3Hk�6#:ELSÑ-6#:<\]9�k5F& 26#:<\DBD3(LO�o6#e �-1�:Eb�F �j��(�Ò,m�lHÓZ­�®�({^5^�m�£p� �� ���&�¶Ú8m!�ØN�Q+R�T�UpGHm LON-mDF�"�Î'Ä�Å5¾iÂWÎ'Ä+É-+<Î'¿ Î�Ê�Î'¿ .�Á[É�Á8Î�Ê�ÇWÊH½(Âc¾iÈ}É!+<Î'¿aÂc¾#Ä�Å5ÆÇ�Á�#iÆ�ÈZÁ +iÉS¾iÊ +<¾@Î�Ê�ÇYÄ�¾ +}ÅIÊ�Æ�¿gÆ$� .�F

Ñd1�\Om���m �uÅ�.�Á[É!+<Î'¿aÆ�È}É%��É�ÊHÁpÎ�Ê%Ç +#¿©Î�Á[Á[É!+<Î'¿%Âr¾#Ä�Å5ÆÇ�Á[F�1*BDX<7cX<756|41'\D\]9�`�1�:[9�X<BD1'3W1'_a±d75B]\DBg^5^�6� �"6#3HBD\]9�3aFI��BD4}756#\�¡u6#CECE6#3%9�XF�o35;(:&"6fÐo6i:EA'9�XF5�o\�9�B]3'&89Aq6#351�b,?>9�3H;>n "6\! 6#3H6pN�9�354}751'3aF%¢�:[9�3HCE\]9�X<6#;>_�:<1'=�X<7H6Z�5:<6354[7>`I?Y0M9�3w�@m��,356#;5;H1�3 m�1*BgX<79r^5:<6_�9�4#6@`,?�Gq6j9�3�¢�6BD\D\]9�4'm@�q^5:<BD3Hk�6#:ELSÑ-6#:<\]9�k5F �6#:<\DBD3(LO�o6#e(�-1�:Eb�F��j�'��®,m8l,A(BDBDB+ÓpÒ'�'Ô)^5^ mDFaÑ-1'\Smf­,m*)�½,Ê +#Ä+ÉSÆ�Ê�Î'¿-Î�Ê�Ç��Î�È}É+Î'Ä+ÉOÆ�Ê�Î'¿(Âr¾#Ä�Å5ÆÇ�Á[F 1 BgX<7@X<756�4#1'\D\]9�`�1�:[9�X<BD1'3f1'_%�~BD4[756\(�|:EX<1'\]9(F'�~9�:<4��oK(X<75BD6#:F'±d75BD\DBD^5^�6� �"6#3HBD\]9�3aF'��BD4[7H6#\�¡u6#CECE6iL3H9�XF�Gq6j9�3���BD4}756#\u¡u1'={`�6C#F�n "6#\! 6356)N�9�354[7H1�3 F �6i:EXE:[9�35;���6#:<4#BD6#:F�¡u\]9�K5;H6 1*B]\D;�9�35;¶¡u\]9�K5;56 ��K5BD\D?'m)¢ :[9�35CE\�9�X<6;_�:<1�= X<756Z�5:<6#3H4[7>`I?r0M9�3>�@m��I3H6#;5;51'3am��q^5:<B]35k'6i:ELSÑd6i:<\]9�k5F �6i:<\]BD3(LO�o6ie �-1�:Eb�F �j� � �,m�l,A(B�ÓpÔ'Ò(�|^H^ mgF5Ñd1�\Om+�,m-,'ËH¾,+#Ä+ÈEÎ'¿Ä�Å5¾[Æ�È/.rÎ�Ê�ÇYÎ}ËqË�¿ É!+<Î'Ä+ÉOÆ�Ê5Á[F&1*BDX<7YX<756Z41�\]\]9�`�1�:[9�X<B]1�3Y1'_���BD4[7H6#\a�|:EX<1'\]9)9�3H;>�~BD4[756\�¡u6#CECE63H9�Xmd¢ :[9�35CE\]9�X<6;�_�:<1'=�X<7H6�5:<6354[7Y`I?rGq1'753w¡|m(�o=WCE1'3amu�I^5:<BD35k'6i:ELSÑd6i:<\�9�k5F� �6#:<\DBD3aF��j����®(mVl%ÓZÔ,�jÔ8^5^ mDF,Ñd1�\OmVÕ5m��[Ê%Ä�¾.��ÈEÎ'¿ ¾EÀ#½5Î'Ä+ÉOÆ�ÊHÁpÎ�Ê�ÇWÊ5½(Âc¾iÈ0/É-+EÎ'¿%Âr¾iÄ�Å5ÆÇ�Á[F�1 BgX<7WX<756o4#1'\D\]9�`�1�:[9�X<BD1'3W1'_a��B]4[756\%�|:EX<1'\]9(FI±d75BD\DBD^5^�6 �"6#3HBD\]9�3aFI��BD4}756#\ �6i:<3H9�;51'KaF(��BD4[7H6#\ ¡u6#CECE6#3%9�XFGq69�3(L�¡u\�9�KH;56Z� "6#;�"6\D6#4'F,GI9�4jJqKH6#C�±d\]9�354[7%9�:<;w9�35; u:<KH351r�I4}756#K(:<6#:md¢ :[9�35CE\�9�X<6;�_�:<1'=�X<756Z�5:<6354[7>`I?YGq1'753�¡|m5�o=WLCE1'3am��q^5:<B]35k'6i:ELSÑd6i:<\]9�k5F� �6i:<\]BD3aFu�j�'��®(mYl%Ó|ÕIÒ'Ô�^5^ mDF�Ñd1'\SmwÔ,m21���Æ�¿ ½(Ä+ÉOÆ�Ê�Ë�È<ÆqÌ#¿g¾iÂfÁ����}F 1*BgX<7�X<756c4#1'\D\]9�`�1�:[9�X<BD1'3�1�_��BD4}756#\ �|:EX<1'\]9(F5�~BD4[756\�¡u6CECE6#3H9�Xo9�35;wn "6\! 6#356ZN�9�3H4[751'3amu¢�:[9�35CE\]9�X<6#;w_]:<1'= X<7H68�5:<6#3H4[7>`,?��o\]9�3�¡d:[9�B]k(m��q^5:<B]35k'6i:ELÑd6i:<\]9�k5F� �6#:<\DBD3aF��j�'�'­ImVl(BgA%Ó�(�®q�p^5^ mDFIÑd1'\Sm-Ò,m31���Æ�¿¹½5Ä+ÉSÆ�Ê)Ë�È<ÆqÌ#¿g¾iÂfÁ��4�5�}F�1 BgX<7WX<756o4#1'\D\]9�`�1�:[9�X<BD1'3W1'_�¡u\]9�K5;56� �9�:<;51'C#F��BD4}756#\2¡u6#CECE63H9�XF��o\]9�BD36&89Aq6#3H1�b,?qF�±V9�XE:<BD4[b~N�9�CE4j9�K5l�F� �6i:EXE:[9�35;���6#:<4#BD6#:F � \DBgA(BD6#:8±-BD:<1�3H3569�KaF� u:<KH351~�I4[756K(:<6i:9�35;�Ú "6=WBu�I6#3qX<B]C#mc¢ :[9�35CE\�9�X<6;�_�:<1'= X<756W�5:<6354[7�`I?��o\]9�3h¡d:[9�B]k(mr�I^5:<BD35k'6i:ELSÑd6i:<\�9�k5F �6#:<\DBD3aF-�j�'���ImWl,B]B�Ó|Õ��'Ôc^5^ mDF�PÊ�Î'¿ .�Á#¾WÂWÎ'Ä�ÅpþiÂWÎ'Ä+ÉSÀ#½H¾Y¾iÄ�+<Î'¿ +i½(¿VÊH½, þ}È}ÉSÀ#½H¾pËHÆ�½,ÈW¿g¾iÁ@Á +iÉS¾iÊ +<¾iÁr¾iÄP¿g¾iÁ)Ä�¾ +}ÅIÊHÉ+À#½H¾iÁ[F Ñd1�\Om#(Im Ã1���Æ�¿¹½(Ä+ÉOÆ�Ê�7�)�Æ�½(È}ÉS¾iÈ�8*�Î}Ë�¿©Î�+[¾}F5ÚP6#^5:<BD3qX�1'_ X<7H6)�j� �'Ô@6;5BgX<BD1'3amV0��8�,¢��@§%¡u1'\D\D64iX<BD1'3rÙd35CE6BDk'356#=c6#3qXm-�~9�CECE1'3aF5±�9�:<BDC#F �j� � �Im�l(\DBDAHÓ��Õ�Õ,Ó|l,BDl^5^ mDF(Ñd1�\Om$�,m Ã1���Æ�¿ ½(Ä+ÉOÆ�Ê97|Ái¾iÂ)É:/;��È<Æ�½�ËH¾<8���Î�È}ÉSÎ'Ä+ÉSÆ�ÊHÊ�¾i¿ F5ÚP6i^5:<B]3qX�1'_ X<756f�j� �'Ô@6#;5BDX<BD1'3am�0��8�,¢��@§H¡u1'\D\]6#4iX<B]1�3rÙd35CE6#B]k�3H6iL=W63qXm���9�CECE1'3aF%±V9�:<BDC#F �j� � �,m�^5^ m-B>=ql(\DBgA�F?��ÕIÔ@=��'Ô�ÕW9�35;>B>=ql(Bgl�FIÑd1'\Smd�,m Ã1 ��Æ�¿¹½5Ä+ÉSÆ�Ê�7PÊH½,Â�þiÈ}É+À#½H¾<8dÄ+È<Î�Ê5ÁOËHÆ�ÈiÄ+F5ÚP6i^H:<BD3qX1'_�X<7H6W�j� ��Ô)6#;5BDX<BD1'3amV0��8�,¢��@§�¡u1'\D\D6#4#X<BD1'3YÙd35CE6BDk'356#=c6#3qXm-�~9�CECE1�3 FH±�9�:<BDC#F��j� � �,m�^5^ m-B>=ql(\DBgAY9�35; �'Ô'Ô�=��@��®A�Im

Page 4: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®5��m-��1'35;H9?�GI9�3IKH9�:E?�� (IF�­�®'®'®(m0�3cX<7H6P=cBD; ��®�C#FH�I6i:<k'6Ba�(RV¯ R�¥,�%��vuBD3qXE:<1,;5KH4#6#;cCE1'=c6|_�K5354iX<B]1�3%9�\%CM^%9�4#6CdeP75B]4[7r7H9A'6|`�6#63WAq6#:E?WBD=)^a1�:EX[9�3qXuBD3

X<7568;56#Aq6#\D1�^H=c6#3qXP1'_�^%9�:EX<B]9�\ ;HBg �6i:<63qX<B]9�\a6JIKH9�X<BD1'35C#F%=W1'CMX<\g?YX<751'CE6Z:<6\]9�X<6#;�X<1�¡u1'3qX<BD3IK5KH= ��6#4}7H9�35BD4#CP1�:|±d7q?(CEBD4#C#m¢�756#?w9�:<68b(351jeP3�9�Cp�(RV¯ R�¥,�H��CM^%9�46#C|351je8FH`HK5XP0�7H9A'6{756j9�:<;wX<7%9�Xo1�X<7H6i:<CP7H9Aq6{4\]9�BD=W6;YX<1c7H9Aq6{7H9�;wX<7568C<9�=c6BD;56j9(F�\DBDb'6{�VQ+²a³�������º{9�35;��V��Q�� �V��QS²�³VU � m���CEBD=WBD\�9�:|BD;56j9We�9�CoKHCE6#;~9c\]BgXEX<\D6{9�_]X<6#:|`I?yGq6j9�3~N������&� � BD3y7HBDCoCMX<K5;(?1'_�e26j9�bYCE1'\DK5X<BD1'35C � 1'_��o�,�VQ���� L��,��R�����U � 6JIKH9�X<BD1'3am

¢�756W`�9�CEB]4cBD;569w_�1�:@;56iz%35BD35k�9~�(RV¯ R�¥,�H� CM^�9�46�F�41�3HCEBDCMX<C{BD3°K5CEBD3Hk>e269�b�;56i:<BgA'9�X<BgAq6#C#F�9�C)�I6i:<k'6B2�(RV¯ R�¥,�H�1�:oGq69�3yN���������;HBD;yBD3>X<7H68=cBD; ��®�C#t�BDXP41�3HCEBDCMX<C|BD3yk'BgA(BD35kc9)^H:<6#4#B]CE6{=W6j9�3HBD35k)X<1cX<756{CMX[9�X<6#=W63qXPX<7H9�Xo9c_�K5354iX<B]1�3� `�6\D1'35k'BD35kcX<1rX<756fN���¯���U�¤��� �8CM^%9�46����H���P���{4#1'K5\];y7H9Aq6)9�\D\�BgX<CP^%9�:EX<B]9�\�;56i:<BDA�9�X<BgAq6#C������� � `�6\D1'35k'BD35kcX<1����5���P�[F_�1�:p9W351'356=)^5Xs?w1�^�6#3yCE6iX�� BD3 �"!8m�¢�7H6i?w;5BD;w3H1�Xo;56#zH3568^�9�:EX<B�9�\�;56#:<BgA�9�X<BgAq6C�_�1�:o6iAq6i:E?>_�K5354#X<BD1'3wBD3#���(���P�[F%`HK(X1'35\g?>C<9�B];wX<7H9�X|CE1'=W6{1�_�X<756CE68_�K5354#X<BD1'35CP7H9Aq68^%9�:EX<B]9�\�;56#:<BgA�9�X<BgAq6#C�`a6#\D1'35k'BD35kr9�\DCE1WX<1$� � ���P�[F�9�3H;yBgX�e�9�C|N�9�K(:<63qX�(²�³�´8���5��µAPeP751f;56izH3H6#;Y=W1�:<6|k'6356i:[9�\�=r9�X<7H6#=c9�X<BD4j9�\H1�`,�E6#4#X<C#FIeP75BD4}7Y756|4j9�\]\D6#;r;5BDCMXE:<Bg`%K(X<BD1'35C#FIeP75B]4[7c^�6#:<=WBgXdX<1

v~�I6#:<k'6#B-N × A'1jA(BD4}7¶�(RV¯ R�¥,�H�2F ÚPK5CECEB�9�3°=r9�X<7H6#=c9�X<BD4B]9�3aF-�j��® ��L<�j� �'�Imc0oz5:<CMX{=W6#X875B]= eP7H6#3�0oe�9�C{9>CMX<K5;563qXFz5:<CMX|B]3w±�9�:<BDCPBD3°�j�'Ò��,FHX<7563y9�XPX<756Z0�3qX<6#:<3H9�X<BD1'3H9�\�¡u1'35k�:<6#CEC|1'_V�~9�X<756#=r9�X<B]4#B]9�35C�BD3w�oBD46ZBD3°�j��(�®,F�9�35;w41�3IA'6#:<CE6#;ePBgX<7y75BD= BD3w�H:<6#354}7aF%eP7HBD4[7y756@CM^�1�b'6{^�6i:<_�6#4#X<\g?��S=c1�CMXo6#;5K54j9�X<6#;wÙdK(:<1�^�6j9�35C|;5B];y\D69�:<3w�H:<6#354}7yB]3wX<756Z`a6#k'BD3535B]35k1'_VX<7HBDC8¡u6#3qX<K5:E?'F�eP75BD4}7�1'35\D?w6#35;HCp1'3�£Z6#4#6={`�6#:�(��F�­�®'®'®I�[mo0P1'35\D?w=c6iXp75BD= 1'354#6f=c1�:<6'F�eP7563y0PXE:[9Aq6#\D6;wePBgX<79~�5:<6354[7�k�:<1'K(^�_]:<1'=·0��oÚ�0�� BD3 �j� (�ÒyX<1��|b�9�;56#=ck'1�:<1,;51�b�F-�o1jAq1�CEBD`HBg:<CMb�F�eP7H6i:<6>756Ye21�:Eb'6;am�¢�756#:<6�BDC)351je 9�(RV¯ R�¥,�%��0�35CMX<BgX<K(X<6P1'_ �~9�X<756#=r9�X<BD4#Cd1'_�X<756Z�IBg`�6#:<B]9�3W`5:[9�354[7Y1'_�X<756|ÚPK5CECEB�9�3Y�o4j9�;H6#={?W1'_��,4#BD6354#6C#FI�o1jAq1'CEBg`HBg:<CMb�mº°Ðp9�6iX[9�351o�VQ+²�³������uF0sX[9�\DB]9�3{=c9�X<756=c9�X<BD4#B�9�3 FI�j�'­'­jL<�j�'�'Ò,m�no6de21�:Ebq6;@9�X�¦o35BDA'6#:<CEBgXs?Z1'_5ÚP1'=c6-0d�SN�9p�,9�^HBD635�9I�[m� &8K(:EX � XEX<1°����Q�� �V��QS²�³�U�F�Ðo6i:<=r9�3(LS`�1�:<3/=c9�X<756#=r9�X<BD4#B]9�3aFo�j��®5�}L<�j� �'­,t|756y6#=cBDk�:[9�X<6#;�X<1°¦o35BgX<6;��qX[9�X<6CcBD3

�j��� (IF5X<1o�E1'BD3y¡�R������T���mÚPB]4[7H9�:<;�¡�R������T���FVÐo6i:<=r9�3(LS`�1�:<3°=r9�X<7H6#=c9�X<BD4B]9�3aFd� � � �jL<�j��(�­,t-756c6=WBDk�:[9�X<6#;�X<1w¦o35BgX<6;��qX[9�X<6#C{BD3/�j����Õ(F

9�35;We21�:Ebq6#;r9�X-�o6#e �d1�:Eb)¦o3HBgAq6i:<CEBgXs?qmV¢�756�£Z6i^%9�:EX<=W63qXd1�_��~9�X<756=c9�X<BD4C�1'_��o6ie �d1�:Ebf¦o35BDA'6#:<CEBgXs?fB]C�351je�3H9�=W6;X<756f¡�R������T��~0�3HCMX<BgX<K(X<6Z1'_��~9�X<756=c9�X<BD4j9�\��I4#B]6#3546#C#m� Gq6j9�3�N��������uF��5:<6354[7�=c9�X<756=c9�X<BD4#B�9�3 F��j��®qÒ�L<�j�'� �,mPno68:<6#46#BgAq6#;wX<756 1�R�¥ �w^5:<BD�#6{BD3h�j��(��ImPno6{756\D;�9W4}7H9�Bg:

�O¢�7�"61�:<B]6@;56C "6jJIKH9�X<BD1'35Co;5Bg �"6#:<6#3qX<B]6#\D\D6C|6iXZ_�1'354#X<BD1'35356#\]\D6#C}�P9�X@¡u1'\D\! 6k�6@;56)�5:[9�3546�F ±�9�:<BDC#F��j��Õ�(jL<���( �,t 7H6@e�9�CZ9�\DCE1_�1�:PCE1'=c6pX<BD=c6Z9)=c6#={`a6i:�1'_�X<756p0�3HCMX<BgX<K(X<6Z_�1�:P�o;(A'9�354#6;��qX<K5;(?�BD3w±�:<B]354#6#X<1�3 F5�oGHm

ÚPB]49�:<;51 1�R�¥ � FdÐo6#:<=c9�3(LS`�1�:<3�;5BD^H\D1'=c9�XW9�35;�^H75B]\]9�3qX<7(:<1�^HBDCMXF|� � ��(jL<�� �(��t�756�6=WBDk�:[9�X<6#;¶X<1h¡uK(`�9�`�6_�1�:<61°1�:<\D;�1h9�:-0}tI_]:<1'= �j��Ò(��X<1c�j��( �p7H6�e�9�C�¡uK(`%9�3r�o={`%9�CEC<9�;51�:dX<1Z0�CM:[9�6\SFqeP756i:<6|756PCMX[9?q6#;Y9�_�X<6i:Ee�9�:<;5C#t,X<756%1�R�¥ �_�1�KH35;H9�X<BD1'3Ze�9�C�6#CMX[9�`H\DB]CE756#;{BD3r���(�Ò�ePBgX<7875BDC�ePB]_�6'F��H:[9�354#BDCE4j9p���¯�Q�����T���L!1�R�¥ �aFq�j��®'®�L<�j� �(��F�«MX<1P^5:<1'=W1�X<6dCE4#BD6354#69�35;y9�:EXP_�1�:�X<756Z`�6#356#z5XP1'_�=r9�3(b(BD35;5¬5m�~Gq6j9�3�N��%���&�hX<751'K5k'7qX�X<7H9�X�75BDC�351�X<BD1'3Y1'_�e269�brCE1'\DK5X<BD1'35Cue�9�C2:<6\]9�X<6#;rX<1@X<K5:E`HK5\D63qX&%H1jePC#t%9�\DX<751'K5k'7Y351�`�1,;(?

:<69�\D\D?rK53H;56i:<CMX[9�35;5CPeP7H9�X�X<K(:E`HK5\D6354#68BDC#FH75BDC�BD;56j9�C�e26i:<6Z\�9�X<6#:�:<6#^H\]9�4#6;�`I?YX<751'CE681'_ &@R�¥��{R�¤�R���R �2m�o35;5:<6#B �oBgbq1�\�9�6#A(BD4[7'&@R�¥��{R�¤�R���R �2F5ÚPK5CECEB]9�3w=c9�X<756=c9�X<BD4#B�9�3 F��j��®A��L<�� ��(ItH756Z:<6#4#6BgAq6#;�X<7561�R�¥ ��^5:<BD�6ZBD3

�j� ��®(m� ¡u\]9�K5;H6cN�1�KHBDC{�~9�:<BD6rno6#3(:<Bd�o�,�VQ����-F��H:<6#354}7h=c9�X<756#=r9�X<BD4#B]9�3aFu�)( �'ÔjL<� ���'Ò,t-756rBD3qXE:<1,;5KH4#6#;°X<756r6JIKH9�X<BD1'3°BD3

� �'­,��mÐo61�:<k'6PÐp9�`5:<BD6#\H�I��R�����U'F�0s:<BDCE7(LS`�1�:<3f=r9�X<7H6#=c9�X<BD4B]9�3aF,� �(�j�jL<�j��®A�,t'7H6uBD3IXE:<1I;HK54#6;{\�9�X<6#:�X<7H6u\DB]3569�:<BD�6#;8A'6#:<CEBD1'3

351jeØb(351jeP3�9�C8�,��R�����U × Co6JIKH9�X<BD1'3aF�eP756i:<6@BD3H6i:EX<B]9�\�6# �6#4#X<Co9�:<6@3H6#k'\D6#4#X<6#;amZ�I��R����aUf756#\];wX<756fN�K54j9�CEB]9�3y4}7H9�Bg:p9�X¡29�={`5:<B];5k'6�F�����ÕI��L<�j��®A�,m

no63(:E?�N �²��VU'FHÙd35k'\DBDCE7>4#\D6#:<k�?(=r9�3 F �Ò(�®�L<�jÒ�Ò��q�[m�~no63(:<B�N�"61'3YN���¯��aUj¤����F,�H:<6#354}7�=c9�X<756=c9�X<BD4#B�9�3 F�����(�Ô�L<�j��ÕH��m�no6o7H6#\D;�9@4[7%9�BD:p�S�~9�X<7�"6=c9�X<B]JIK56C}��9�XP¡u1'\D\- 6#k'6

;568�5:[9�354#6'FH±�9�:<BDC#F��j�'­,�iL<�j��ÕH��m�~0sXVe�9�C��5:<BDk�?q6#C��S�5:<6;56i:<BD4�� ÚPQ���U�µ5F�noK535kq9�:<B]9�3f=c9�X<756=c9�X<BD4#B�9�3 F(� � ��®�L<�j�'Ô'ÒIF'eP751pBD3qXE:<1,;5KH4#6#;@X<7H6'���5���P��CM^%9�4#6#C

_�1�:{�)(+*#(-,/m-no6Ze21�:Eb'6;wBD3 �K5;H9�^�6CMXm�N�9�K5:<6#3qX{�(²�³�´8���5��µ5F��H:<6#354}7�=c9�X<756=c9�X<BD4#B�9�3 F5`a1�:<3�BD3h�j�(�jÔImono68:<64#6BgAq6#;>X<7H6{�VQ��%¥��-Uf=W6;H9�\�BD3h�j��Ô�®(m|no6

e21�:Eb(C�9�X�Ùd4#1'\D6p±V1'\g?,X<6#4[7H35B]JIK56�F5±�9�\�9�B]CE69�KaF59�35;�0u7%9�;>75BD= 9�CP9@X<69�4}756i:PBD3��j��Ò'Ôq¨�Ò'ÒY�+eP756#3�Ùd41�\]6p±V1�\D?IX<64[753HB]JIK56e�9�C�B]3>±�9�:<B]C}�[m

Gq1'753~¡u7H9�:<\D6C���Q���¥��-U'F�¡29�3H9�;5B]9�3>=r9�X<756#=r9�X<B]4#B]9�3aFa� �'Ò ��L<�j���'­Im

Õ

Page 5: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

;56#zH356W9�Cp=c9�3q?�;56i:<BDA�9�X<BgAq6#Cp9�CZ1�3H6f=c9?we�9�3qXF _�1�:89�3I?y\D1,49�\D\D?>BD3IX<6#k�:[9�`H\D6@_�K5354#X<BD1'3amZN�9�K(:<6#3IX{�(²a³�´{���5��µre263qX_�K(:EX<756i:dX<7H9�3cX<756�X<7561�:E?);56#Aq6#\D1�^�6;)`I?Y�,6i:<k'6#Ba�(RV¯ R�¥,�H�2F'eP75B]4[7c756|;5BD;c3H1�Xdb(351je*9�`�1'K(XF(9�35;r756�^�1'BD3qX<Cd1'K(XuX<7H9�X PR�²�³�T���� v � 7H9�;�9�\DCE1@1�`5X[9�BD356#;YCE1�=c6P^%9�:EX<B]9�\%:<6#CEK5\DX<C#FIeP75BD4}7r7H6o9�\DCE1@1'35\g?c\D69�:<356;Y9�`�1'K(X2\]9�X<6i:m�ÚPK5CECEB]9�35C�9�\ge�9?(CJIK51�X<6YÐo�%¥�� �j��T��2v<vF _�1�:8;56#A'6\D1�^HBD35k>X<756)X<756#1�:E?~1'_2;5BDCMXE:<Bg`HK5X<BD1'35C#F `HK(X{N�9�K(:<6#3qX)�(²�³�´8���5��µ>X<1'\D;�=c6)X<7H9�XZeP7H9�XÐo��¥�� ���T���;HBD;°e�9�Cf=c1�CMX<\D?�X<1y^�1�^HK5\]9�:<BD�6cX<756rX<7561�:E?qm��,1�=c6#1'356r^�1'BD3qX<6;h1'K(XfX<1�=W6rX<7H9�Xfno6i:<=r9�353 1 ����¥�v�ºCE751'K5\D;�`�6)JIK51�X<6#;�_�1�:oX<756@X<756#1�:E?wX<1,15F�`HK(Xp0P7H9Aq6f351�Xp4}756#4[b'6;�X<7H9�XFa9�35;~N�9�K(:<6#3IX{�(²a³�´{���5��µrBDCp351�XZ9e�9�:<6f1�_75BDC�e21�:Eb�m

� 354#6>;5BDCMXE:<Bg`%K(X<BD1'35C@ePBD\D\d`a6�;56izH3H6#;�9�35;¶X<756#BD:f`%9�CEBD4Y^H:<1�^a6i:EX<BD6Cf1�`HX[9�B]356#;aFd1'356�ePBD\D\2;56izH3H6YX<756y�(RV¯ R�¥,�H�CM^%9�4#6C����� �5���P��_�1�:Z9Y351'35356k'9�X<BgAq6fBD3IX<6#k'6i:�Fa_�1�:W� ( * ( , 9�35;�_�1�:89�3~1�^�63�CE6#X)�x1'_ �)!{§|BgXpBDCoX<756fCM^%9�461'_P9�\D\u_�KH354iX<BD1'35C ��� ���(���P�8CEK54[7hX<7%9�X� �� ��� ���(���P�8_�1�:W9�\]\u=@K5\gX<BDLOBD35;5BD46#C�������� v���������� � ! �pePBDX<7�� ��� (�~FeP756#:<6����! ®�_�1�:#"$�Û� �������%�'& 9�35;(� �)�*�+� v Ó ����� Ó�� ! F29�3H;, �� � � �.-0/

��� -0// ����� �.-�1��� -�11� �S1'356w9�\DCE1�;56351�X<6#C

�324�5� v 2 ����� � ! 2DF,9�35;)X<756CE6�CEBD=)^%\DBD_�?,B]35ko3H1�X[9�X<BD1'35C-7H9Aq6�`a6#6#3WBD3IXE:<1I;HK54#6;f`,? 1*³�Q���T�����v � �[m � 356�=@KHCMX-`�6�49�:<6_�K5\X<7H9�Xr1�X<756i:�9�K(X<751�:<Cr\DBDb'6wN�9�:<Crn �R����Z��T��V�%�dv � K5CE66 X<1°;56351�X<6 v 7 ����

BD35CMX<6j9�; t|1'356yCE751'K5\D;/351�Xr`�6wCEK(:E^H:<BDCE6#;X<7563�BD_VXse21Y;5Bg �6#:<6#3qXo`�1,1�b,Co41'3qX[9�BD3yCEB]=WBD\]9�:|_�1�:<=@K5\�9�C|ePBDX<7y;5BD �6i:<63qXo41�3HCMX[9�3IX<C#Fa9�35;~1�3H6@CE751'K5\D;~4}756#4[b>eP7%9�XpBDCX<756�;56#zH35BgX<B]1�3¶1'_�X<756�CM?,=@`�1'\8 wFu9�35;�9�\DCE1�eP7H9�Xc9�:<6YX<7H6r^5:<64#BDCE6�41�3HCMX[9�3IX<CfK5CE6;�BD3�;56#zH35BD3HkyX<7H6���R����Q����-v �XE:[9�35CE_�1�:<=wm

¢�756W:<69�CE1'3°1�_��,6i:<k'6#B2�(RV¯ R�¥,�H� _�1�:{B]3qXE:<1,;5K54#B]35kYX<7H6WCM^%9�4#69� v:� º����P�[F�eP75BD4[7�BDC{9�\DCE1>;H6#351�X<6#;<;°vj���P�f�+`HK(XCE751'K5\D;y3H1�X|`�6{4#1'35_�K5CE6;wePBgX<7>X<7568no��� � ��v � CM^%9�4#6'v ��F%eP75BD4}7>ePBD\D\ `�68;H6#351�X<6#;�=~v}�[F%B]C�X<7H9�X|BgX|BDC|9c3H9�X<K(:[9�\�CM^%9�46_�1�:�CE1'\gA(BD35kf6jJIKH9�X<BD1'35C�1�_�X<756p_�1�:<=+>)? � ��@>ePBgX<7Y`�1'K535;H9�:E?Y41'35;5BgX<BD1'35C#FH9�3�6JIKH9�X<BD1'3Y3H9�=c6#;>9�_]X<6#:�N��BA%¥,��²���v �1�:�±uR�QSU�U�R�T v m,0sX�49�3Y`�6Z4#1'35CEBD;56#:<6#;>9�C2:<6\]9�X<6#;rX<1fX<756p£pQ���QS²�³�¥I��� º � ^5:<BD3H4#Bg^H\]6�FIeP75B]4[7�4#1'35CEB]CMX<C�BD3�351�X<BD4#BD3Hk@X<7H9�XBD_ 98_�K5354#X<BD1'3 � 1'_a4\]9�CEC8C8ºP=WBD35B]=WBD�6#C�X<756|_�K53H4iX<BD1'3H9�\�D~;56iz%356#;c`,?EDd�GF5�H��I0JE� KML.N4O��GF(��� ºPOMQR>w­3I0J�@)F3OMQ>9�=W1'35k9�\D\2_�K5354#X<BD1'35C)7H9A(BD35k�9~k'BgAq6#3¶`�1'K53H;H9�:E?hA�9�\DK56'F�X<7563 � C<9�X<BDCMz%6#C�>)? � �S@�BD3 �om�¢�756�^5:<BD3H4#Bg^H\]6ce�9�CW3H9�=W6;9�_]X<6#:f£pQ���QS²�³�¥,�%��`I?hÚPQ�� �Z��T�TVºiv#F�9�Cf756Y7%9�;�7569�:<;¶BgXf_�:<1'=»7HBD=>FV`HK(X)BDXf7H9�;h`�66#3¶K5CE6#;¶`�6_�1�:<6r`I?�Ðo� �U�Ujº<º

v � �,9�\D1'=c1�3 PR�²�³�T��%�dF%±V1'\DBDCE7(LS`�1�:<3�=c9�X<756=c9�X<BD4#B�9�3 F�� ���'��L<�j� ��­,ta756@6#=cBDk�:[9�X<6#;>X<1r¦o35BgX<6;��qX[9�X<6#Cp9�_�X<6i:f�j�����IF9�35;w756pe21�:Ebq6#;y9�XP±-:<BD3546iX<1'3�¦o3HBgAq6i:<CEBgXs?qF5�oGHmvEvw0��#:[9�B]\2�~1�B]CE6#6iA(BD4}7/Ðo�%¥�� �j��T��2F�Ú|K5CECEB]9�3�=r9�X<756#=r9�X<B]4#B]9�3aF�`�1�:<3�BD3 �j�(�@�Im�no6Y:<64#6#BDA'6;¶X<756 1�R�¥ ��^5:<BD�6�BD3�j��( �,m-no6Ze21�:Eb(C|9�X�Ú ���¤V�%�VU8¦o35BgAq6#:<CEBgXs?'F5�o6ie 2:<K535CMePBD4[b�F5�oGHm

no63(:E?�Ú ���¤�����U'FH�o=W6i:<B]49�3�4#1'\D1'356\Smvsº�no6#:<=c9�353 &Z\]9�K5ConoK5k'1 1*���V¥%F Ðo6#:<=c9�3(LS`�1�:<3�=r9�X<756#=r9�X<B]4#B]9�3aF�� � ��Ô�L<�j�'Ô'ÔIta756@6=WBDk�:[9�X<6#;wX<1r¦o35BgX<6;��qX[9�X<6#CBD3��j�����)9�3H;>7H6pe21�:Ebq6#;y9�X�X<756p0�35CMX<BDX<K(X<6Z1�_��o;(A'9�3H4#6#;~�qX<K5;(?�BD3>±-:<BD3H4#6iX<1'3aF5�oGHmv � np9�CECE\D6#: 1 ³�Q�� T����2F-�o=W6#:<BD49�3�=c9�X<756=c9�X<BD4#B�9�3 Fo�j��®�(jL<�j� �'�Im no6>:<6#46#BgAq6;�X<756�1�R�¥ ��^5:<BD�#6yBD3x�j� �'­,m no6e21�:Ebq6#;y9�X�X<756p0�35CMX<BDX<K(X<6Z_�1�:P�o;(A'9�3H4#6#;��qX<K5;5?�BD3>±-:<BD354#6#X<1'3aF5�oGHmv � N�9�:<CPn �R����Z��T��V�%�dF��Ie26#;5BDCE7w=r9�X<7H6#=c9�X<BD4B]9�3aF,`a1�:<3wBD3°����(��mdno6Z:<6#46#BgAq6;YX<7568�VQ��%¥��-U8=c6#;H9�\ B]3��j��Ò'­,F%9�35;X<756�1 R�¥ ��^5:<B]�#6ZBD3��j� � �Im-no6Ze21�:Eb,C|9�X�N�K535;w¦o35BgAq6i:<CEBDXs?'mv � Gq6j9�35L- �9�^5X<B]CMX<6fGq1�CE6#^H7~�-R����Q��%�dFa�5:<6#3H4[7~=r9�X<756#=r9�X<B]4#B]9�3aFV�)(�Ò ��L<� ����®(m{no6@e�9�Cp^5:<6#_�6#4#XpBD3°Ð|:<6#351�`H\D6fKH35;56i:�o�BA�R�¥��aR�TdtX<756d0�3HCMX<BgX<K(X��-R����Q��%��BDC�X<756d;56#^%9�:EX<=W63qX�1'_(�~9�X<756=c9�X<BD4C�1'_5¦o35BgAq6#:<CEBgXs?p1'_�Ð|:<6351�`H\D6d0}F�BgX<CE6#\]_,3H9�=W6;9�_]X<6#:PGq1'CE6i^H7w�-R����Q����-m

�p9�^�1'\ "61�3 PR�T��BA'���5� ��Fd�5:<6#3H4[7/k'6#356#:[9�\OFp�)(�Ò'�jL<� �'­(��t|7H6>^5:<1,4#\�9�B]=W6#;�7HBD=WCE6\D_o6#=W^�6i:<1�:F2K535;H6i:rX<756y3H9�=W6�o�BA�R�¥��aR�T°0}F � ��®�Õ�L<� �(�#Õw�O9�35;°�®'®f;%9#?(CPBD3�� �(�jÔ'�[mv � Ðo1,;5_�:<6i?)np9�:<1'\D;Wno��� � �uF'Ùd35k'\DBDCE7W=c9�X<756=c9�X<BD4#B�9�3 F5� ��( (L<�j��Õ�(ImVno6�7H6#\D;)X<756o�,9�;5\]6#Bg:<B]9�3)4}7H9�Bg:d1�_�±dK(:<6P�~9�X<7(L6#=r9�X<BD4#CP9�Xp¡29�={`5:<BD;Hk�6'F �j���(�iL<���ÕI­,m

N�9�;5?w�,���V¥,�%Q��°635;51je26#;>X<75684[7H9�Bg:PB]3��)(�®5���[mv ��¢�7568X<6i:<= CE66#=cC�X<1c7%9#Aq68`�6#63wBD3IXE:<1I;HK54#6;>`I?w�~9�:<4#6\ Ú|Q��aU�µ(F�noK535kq9�:<B]9�3(LS`�1�:<3w=r9�X<7H6#=c9�X<BD4B]9�3aF � � ��Ò�L<�j�'Ò'��+X<756Z?'1'K53Hk�6#:�`5:<1�X<756i:|1'_��H:<BDk�?q6#C�ÚPQ���U�µ,�[mHno6pe21�:Ebq6#;wBD3>N�KH35;aF��qe26;56#3 mv.��±dBD6#:E:<6iL��IBD=c1'3�N��TA�¥,��²���Fu�5:<6#3H4[7*=c9�X<756#=r9�X<BD4#B]9�3aFZ�)(�Õq��L<� �'­�(qmØno6ye21�:Eb'6;�B]3/±�9�:<BDC#F|9�XrX<756 �K(:<6j9�K�;56#CN�1'35k'BgX<KH;56#CP9�35;>X<7568±V9�:<BDC � `HCE6i:EA'9�X<1�:E?qm-no6p`�6#4j9�=c6Z4#1'K53qX|BD3�� ��®qÒf9�35;w=r9�:[JIK5B]C�BD3�� �(�)(Imv.¶�IB]= "6#1'3>£p6#3HBDC�±2R�Q+U�U�R�TdFH�5:<6354[7y=c9�X<756=c9�X<BD4#B�9�3 F��)( �(�iL<����Õq®(m-no6pe21�:Ebq6#;wBD3>±�9�:<B]Cº � Gq1'7H9�353�±V6#X<6i:8ÐoKHCMX[9#AyN���Ui� �T���£pQ���QS²�³�¥,�%��F Ðo6i:<=r9�3�=c9�X<756=c9�X<BD4B]9�3aF�� ��®'Ô�L<� �'Ô'�Imono68e21�:Ebq6#;�BD3 �6i:<\DBD3 F� �'­ ��L<� �'Ô'ÔIF�9�3H;�X<7H6#3>BD3~ÐWV1�XEX<B]35k'6#3amº}v Ðo61�:<kc�5:<BD6;(:<BD4}7 �6i:<3H7H9�:<;>Ú|Q�� �p��T�TuF�Ðo6i:<=c9�3>=r9�X<756#=r9�X<B]4#B]9�3aFa� �'­�Ò�L<� �'Ò'ÒIm-no6Ze21�:Eb'6;wBD3yÐWV1�XEX<BD3Hk�63amºEº�Gq1'7H9�353�¡29�:<\ �5:<B]6#;(:<BD4}7�Ðo� �U�U�FaÐo6i:<=c9�3>=r9�X<756#=r9�X<B]4#B]9�3aFa�)( ( (jL<� �'Ô'ÔIm-no6Ze21�:Eb'6;wBD3yÐWV1�XEX<BD3Hk�63am

Ô

Page 6: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

9�35;�`,?hÐo������TVº � m 1 ��Q�����U� ���VU�U�º � ^�1'BD3qX<6;�1'K(X@\]9�X<6i:8X<7H9�X8X<756c_�K5354#X<BD1'3H9�\�=cBDk'7qX{351�X@9�XEX[9�BD3�BgX<C8=WB]35BD=@KH=>t�0X<75BD35b>X<7H9�X|X<756f4#1'=)^%\D6iX<6{CE1'\DK(X<BD1'3�1�_�X<756@£pQ���QS²�³�¥I���h^5:<BD354Bg^H\D68e�9�Cp1'356@BD3yX<756@_+9�=c1'K5Co\DBDCMXo1'_V^H:<1�`%\D6#=cCPeP75BD4}7noQ�¥,¯����5� º �p^5:<1�^�1'CE6;yBD3h�j��®'®r9�XPX<75680�3qX<6#:<3H9�X<BD1'3H9�\V¡u1�3Hk�:<6CEC|1'_V��9�X<7H6#=c9�X<BD4B]9�35C�BD3y±V9�:<BDC#t�X<756{BD3qXE:<1,;5K54#X<BD1'3w1�_�(RV¯ R�¥,�%��CM^%9�4#6#C#F�eP7HBD4[7�9�:<6YnoQ�¥,¯����5�*CM^%9�4#6C[º � F�X<1'k'6iX<756#:@ePBgX<7hCE1'=c6r;H6iAq6#\D1�^H=c6#3qX<CfBD3¶�HK5354#X<BD1'3H9�\u�o3H9�\g?(CEBDC#F`I?���� ���²a³�����º$��Fd�umuÚPQ��aU�µh9�35; ���T���²�³�º5��eP75B]4[7�7H9�;�^%9Aq6#;�X<756we�9#?qF2^%9�:EX<\g?�CE1'\gAq6;�X<756>^H:<1�`%\D6#=wm/�oCWX<7H6^5:<BD3H4#Bg^H\]6@BDCZ9�\DCE1�3H9�=W6;~9�_�X<6i:8¢�³VR��{U�R�TVº �FaBgXpBDCo^�1'CECEBg`%\D6@X<7H9�X{�,6i:<k'6#Bd�(RV¯ R�¥,�%�/7H9�;�4#1'35CEBD;H6i:<6#;�X<756WJIK56#CMX<B]1�31'_�Ùd\D64iXE:<1'CMX[9�X<BD4#C#FH9)CEBD=)^H\]BgzH49�X<BD1'3�1�_���� � ´>��¥,¥ �:� 6jJqK%9�X<B]1�3 m

no�������p��� � � vYB]3qXE:<1,;5K54#6;/X<7H6�3H1�X<B]1�3*1'_Ze26#\D\|^a1�CE6;�^5:<1�`H\D6=WC>9�3H;*CE751je26#;*X<7H9�XYX<7H6i:<6�9�:<6�41'3qX<BD3IK51'K5C_�K5354iX<B]1�3HC @�_�1�:feP75BD4[7¶X<756�CE1'\DK(X<BD1'3 � BDC)351�X)1'_P4#\]9�CEC C8º�m � 356Ye�9?hX<1~CE1'\gAq6rX<75B]Cf;5BD�W4K5\gXs?°B]C@X<1�e21�:Eb°ePBgX<7X<756>_+9�=WBD\D?h1'_|CM^%9�46#CEC�� � � FdeP756#:<6���BDCc9~351'353H6#kq9�X<BgAq6>BD3qX<6k�6#:r9�3H;�®�� � (���F23%9�=c6#;�9�_�X<6i:cn �R�¥������ � ºY1�:N�Q AaUj²a³�Q���µ � � t�JqKH6#CMX<BD1'35C|\DBgbq6#@ � C�� � ��BD=)^H\]BD6#C � � C��� º � �y_�1�:p®�5���Ø�8e26i:<6@BD3qAq6#CMX<B]k'9�X<6;>`I?~�(²�³�� ������ � � F`HK(X)X<756�CEBD=cBD\]9�:fCMX[9�X<6#=W63qX)BDCf_+9�\DCE6�_�1�:�� �»��Fd9�35;¶X<756�41�:E:<6CM^�1'35;5BD3Hk~CM^%9�4#6C)CE751'K5\D;¶`�6Y:<6#^H\]9�4#6;h`I?¶CM^%9�4#6#CBD3qXE:<1,;5KH4#6#;>`I?>�o3qX<1'35B����-¤���T�� � ��mH�o3wK535;H6i:<\g?(BD35kcJIK56CMX<BD1'3>BDC�:<6#\]9�X<6#;>X<1WCEBD35k'K5\�9�:PB]3qX<6#k�:[9�\DC|9�4#X<BD35kW1'3wCM^%9�4#6#CC � � �/_�1�:Y®�� � �·��F�eP75BD4[7�e26i:<6w6#l,X<6#35;56;�X<1 ����_�1�:���� *�� , `I?��o\g`a6i:EX<1¶¡u��¥��V�%� �R�T �$� 9�35;/�o3qX<1'35B����¤���T��2FaCE1YX<7H9�XZ_�1�:W��� *�� ,/F @ � ���(���P�|BD=)^H\]BD6#C � � � º%� ������ ���P�[t X<756WJIK56#CMX<BD1'3�_�1�:p`�1'K53H;H9�:E?�4#1'35;5BgX<B]1�3HCe�9�C�B]3qAq6#CMX<BDkq9�X<6#;�`I?y�,75=@K56\ �Z¤��8R�T � ��FH�|A,:<1'3>£8R��¤�¥,QSU � �89�3H;>N�1'K5BDCP�oQ����%TV¯��%��¤ � �F5`HK(XPX<756849�CE6&* � ­@e�9�CK535;H6i:<CMX<1,1,;�69�:<\DBD6#:F `�649�K5CE6W1'356c49�3°K5CE6W�-R����Q��%�/XE:[9�35CE_�1�:<=wF�9�35;�X<756#:<6)e26i:<6WCEB]=)^H\D6#:8=c6iX<751,;5C8_�1�:8^5:<1jA(BD35k

º � Ðo61�:<k'6hÐo�����%TuF2Ùd35k'\DBDCE7*=c9�X<756#=r9�X<BD4#B]9�3aF8�)(����jL<� ��ÕH��m�no6�e�9�C>9h=cBD\D\D6#:Y9�35; 356iAq6i:>7H6#\D; 9�3q?*9�4j9�;56#=cBD4^�1'CEBgX<BD1'3amº � &{9�:<\�¢�756#1,;51�: 1 BD\D756\D= 1 ��Q�����U� ���VU�U'F�Ðo6#:<=c9�3>=r9�X<7H6#=c9�X<BD4B]9�3aFa� �,�jÔ�L<� �'� (Imdno6pe21�:Ebq6;>B]3 �6i:<\DBD3 mº � £Z9A(BD;>noQ�¥(¯��%�H��F�Ðo6#:<=c9�3>=r9�X<7H6#=c9�X<BD4B]9�3aF�� �'Ò'­jL<�j��Õ �,m-no6Ze21�:Ebq6;>BD3�ÐWV1�XEX<BD35k'63amº � ¢�756{X<6#:<=�e�9�C|41'BD356#;w`,?wG "9�351'Cf�SGq1�7H3��2A'1'3��o����p��T�TdF%noK535kq9�:<B]9�3(LS`�1�:<3�=c9�X<756=c9�X<BD4#B�9�3 F��j��®A��L<�j�'Ô (ImPno66#=cBDk�:[9�X<6#;YX<1W¦o35BgX<6;y�qX[9�X<6CP9�3H;>7H6pe21�:Ebq6#;y9�X�X<756p0�35CMX<BDX<K(X<6Z_�1�:P�o;(A'9�3H4#6#;~�qX<K5;(?�BD3>±-:<BD3H4#6iX<1'3aF5�oGHmº ���~9�K(:<B]4#6ZÚ|6#3�"68��� ���²a³�����FH�5:<6#3H4[7w=r9�X<756#=r9�X<B]4#B]9�3aF�� ��( ��L<���( �,m-no6Ze21�:Eb'6;>BD3>±�9�:<BDC#mº5�¶�qX<6_�9�3 ���T���²�³dF5±V1'\DBDCE7w=c9�X<756#=r9�X<BD4#B]9�3aFa� �'�'­jL<�j��ÕIÔ,m-no6pe21�:Ebq6#;>BD3wN Aq1jA~�S3H1exBD3>¦|b,:[9�BD356��[mº5 1 BD\D\DB�9�= ¢�³�R��8U�R�TuF50s:<B]CE7(LS`�1�:<3w^H7q?(CEBD4BDCMXF�� �'­�Õ�L<�j��®�(It�BD3h� ���'­W7568e�9�C|=r9�;56Z`�9�:<1'3 &Z��¥}�VQ�T�1�_VN�9�:<k'C#F�9�35;X<756#:<69�_]X<6#:�b(351jeP3y9�C�N�1�:<;�&Z��¥}��Q�Tum5no6Ze21�:Ebq6;>B]3�Ðo\]9�CEk'1e8F��I41�X<\�9�3H;am�:� GI9�=W6Co¡u¥,�%������� � ´>��¥I¥�F��I4#1�XEX<BDCE7�^H7q?(CEBD4BDCMXF � ���,�iL<� ��(��Im-no68756#\];YX<756Zz5:<CMXp¡u�,�V�%T��VQ+U³h±-:<1'_�6CECE1�:<CE7HBg^y1�_±d7q?(CEBD4#C|9�Xo¡29�={`H:<BD;5k'6�F�� � (,�iL<� ��(��,m

no63(:E?y¡u�,����T��VQSU³dF5Ùd35k'\DBDCE7>^H7q?(CEBD4#B]CMXF � ( �(�iL<� �,�®(m� v GI9�4JIK56C��,9�\D1'=W1'3Zno�������p��� �2F#�5:<6354[78=r9�X<756#=r9�X<B]4#B]9�3aFq� �'Ò�Ô�L<�j�'Ò��Im�no6�756#\];89�4[7%9�BD:-�S� "649�35B]JIK56-9�3H9�\g?,X<B]JIK566iXP= "6#4j9�3HB]JIK56p4 "6\D6#CMX<6��u9�Xp¡u1'\D\- 6#k'6Z;56Z�H:[9�3H4#6�F%±V9�:<BDC8�j��®q��L<�����(Im� º�Ù-:<35CMXZn �R�¥��V���-F�Ðo6#:<=c9�3~=c9�X<756#=r9�X<BD4#B]9�3at�0|1'354#6)C<9e 75BD= 9�X{9r=W66iX<BD3HkrB]3 � `�6i:Ee21'\D_+9�4[7aF�9�35;�0Pe�9�CoX<1'\D;X<7H9�XVBgXVe�9�C�75BDCV_+9�X<7H6i:�eP751pe�9�C�b(351jeP3)_�1�:VX<7562BD356JIKH9�\DBDXs?W�O9�35;fCE1'=c6u:<6#CEK5\DX<C�B]3@�o\Dk'6i`H:[9q�[F � XEX<1oN�K5;(ePBDkon �R�¥��V���-FÐo6i:<=r9�3>=c9�X<756=c9�X<BD4#B�9�3 F�� �'Ô'��L<�����(Imd¢�756i?Ye21�:Ebq6#;wBD3>N�6#Bg^%�#BDk5m� � ÚPKH;51'\D_ � XEX<1Y�IBDk'BDCE=@KH35;>N�Q AaUj²a³�Q���µ5F�Ðo6#:<=c9�3>=c9�X<756=c9�X<BD4B]9�3aFa�����'­�L<�j��®A�,m-no6pe21�:Ebq6#;wBD3 �1'353am� � GqK5\]BDK5CE�Z±�9e26#\V�(²�³�� ��V���-FH±V1�\]BDCE7�=r9�X<756#=r9�X<B]4#B]9�3aFa� �'����L<�j��Õ �Imdno6pe21�:Eb'6;wBD3>N Aq1jA��S351je�BD3>¦|b,:[9�BD356��[m� � �o3qX<1'35BV�I�#4�#6i^�9�3 ����¤���T��uF%±�1'\DB]CE7(LS`�1�:<3w=c9�X<756#=r9�X<BD4#B]9�3aFa�j��®'®�L<�j�'�'­,mdno686#=cBDk�:[9�X<6;YX<1W¦o35BDX<6#;��qX[9�X<6C�BD3�j��Õ'®(F%9�35;w756Ze21�:Eb'6;y9�XP¦o3HBgAq6i:<CEBgXs?Y1'_�¡u75B]49�k'1(m� � �o\g`a6i:EX<1w±�m-¡u��¥��V�%� �R�TdF��|:<k�63qX<BD35B�9�35LS`�1�:<3�=r9�X<756#=r9�X<B]4#B]9�3aFd�j�'­�®�L<��'� �,m>no6W:<64#6BgAq6#;�X<756 1 R�¥ ��^H:<BD�#6cBD3�j� ���,m@no6fe21�:Ebq6#;°9�X8¦o35BgAq6i:<CEBgXs?�1'_2¡u7HBD49�k'15Fa`HK(XZbq6i^5XZCMXE:<1'35k>X<BD6#CoePBgX<7~�|:<k'63qX<BD3H9(F�9�CZ49�3~`�6fePBgX<356#CECE6;�_]:<1'=X<756{\�9�:<k'6{3IK5=@`�6i:p1�_�=c9�X<756=c9�X<BD4#B�9�3HC�_]:<1'= �|:<k'6#3IX<BD3H9W7H9A(BD35krCMX<K5;5BD6;ynp9�:<=W1'35BD4{�o3H9�\g?(CEBDC#F�9�35;�1'_]X<63we21�:Eb,B]35k351jexBD3>¦o35BgX<6;y�IX[9�X<6C#m� �h�I7H=@K56\ �Z¤��{R�TdF(0�CM:[9�6\DBa=r9�X<756#=r9�X<B]4#B]9�3aF,`�1�:<3>BD3°��'­'­,m-no6pe21�:Eb(CP9�XPno6#`5:<6ie�¦o35BgAq6i:<CEBgXs?qF5Gq6i:<K5C<9�\D6=>m� ���|A,:<1'3�£8R��¤�¥,QSU'F��o=W6#:<BD49�3�=c9�X<756#=r9�X<BD4#B]9�3aF��j�,� ��L<�j�'��Ô,mZno6@e21�:Eb'6;�9�XZ¦o35BgAq6#:<CEBgXs?w1'_���9�:E?(\]9�35; F�¡u1'\D\]6#k'6±�9�:Eb�m� �N�1�KHBDC|�oQ�����T�¯�����¤2F�¡29�3H9�;5B�9�35LS`�1�:<3y=r9�X<7H6#=c9�X<BD4B]9�3aF5`�1�:<3�BD3h�j�'­�Ô,mPno6{:<6#46#BgAq6;>X<756)¡u��� �5R�R�����^5:<B]�#6{BD3�j� ��­,m-no6Ze21�:Eb(C|9�X�X<756f¡�R������T��~0�3HCMX<BgX<K(X<6Z1'_��~9�X<756=c9�X<BD4j9�\��,4#BD6354#6C#F5�o6ie �d1�:Eb�¦o35BDA'6#:<CEBgXs?'m

no1'\Dk'6i:8¡u�����(R�R����2F��qe26#;HBDCE7w`%K5CEBD356CE=c9�3aF��j��® ��L<�j� �'­IF�9�35;y7HBDCPePBD_�6{�o353H9�L�Ð|:<6#X[9Y¡u��� �5R�R�� �2F��j�(�#Õ�L<��'��Õ5F6#CMX[9�`H\]BDCE756#;�X<756>^5:<B]�#6wBD3��j� ��®~`,?�9�;51'3H9�X<BD1'3�X<1�X<756wÚ|1?I9�\p�qe26;5BDCE7/�o49�;56={?Ø«EX<1�^5:<1'=W1�X<6>`%9�CEBD4yCE4#BD63qX<BgzH4:<6#CE6j9�:<4[7rBD3��qe26#;H6#3Y9�3H;cBD3c1�X<756#:d^%9�:EX<Cd1'_�X<756Pe21�:<\];cBD3c�~9�X<756#=r9�X<BD4#Cd9�35;r�oCMXE:<1'351'={?qF5Ðo6#1'CE4#BD6354#6C#F� �BD1'CE4BD6#3546#CePBgX<7�^%9�:EX<BD4#KH\]9�:�6#=W^H7H9�CEBDC�1'3w6#41�\]1�k�?qFH9�3H;>ÚP756K5=c9�X<1'BD;w9�:EX<7(:<BgX<B]CE¬(m

Ò

Page 7: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

:<6#k'K5\�9�:<BDXs?h:<6CEK5\gX<CrBD3�X<7HBDCc49�CE6�F�`I?�N�1'K5BDCc�oQ�����T�¯�����¤2F�`I?�GI9'9�b�±-����� ��� � � m-0�3�X<756w\�9�X<6�Ô�®�Cr9�35;/69�:<\g?�Ò�®�C#F�(RV¯ R�¥,�%�*CM^%9�46#C8e26i:<6WKHCE6#;°BD3°9�=c1�:<6WCM?(CMX<6#=r9�X<B]4fe�9?~_�1�:{CE1'\gA(BD35k>\DB]3569�:Z^�9�:EX<B�9�\�;HBg �6i:<63qX<B]9�\�6JIKH9�X<BD1'35C8_]:<1'=¡u1'3qX<BD3IKHK5= ��6#4}7H9�35BD4#C|1�:o±d7q?(CEBD4C#F%ePBDX<7yCEK5BgX[9�`H\D68`�1'K535;H9�:E?w41'35;5BgX<BD1'35C#F%X<7H6{N�� � � v LO�~Q�¥5¤V������ º \D6=W=c9)`�6#B]35kX<756P4#1�:<356#:<CMX<1�3H6�_�1�:-X<756�6#\]\DBg^5X<BD4�49�CE6#C#FI`HK(Xd1�X<756#:<C-7H9�;W1�`5X[9�BD3H6#;)X<7H6�C<9�=c6�:<6CEK5\gXFI\DBgbq6��~9�:EbfÑpQSU³�Q�� � � tq6il,X<6#35CEB]1�3X<1@6#Aq1�\]K(X<BD1'3W^H:<1�`%\D6#=cCde�9�C2e21�:Ebq6;Y1'K(X2`I?cGI9�4JIK56CMLON�1'K5BDC2N�QSR�T�U � � FIeP751@ePBgX<7rGI9'9�bc±d���%� ���>BD=W^5:<1jA'6;cX<756|:<6j9�\=W6#X<751,;5CZ_�1�:{BD3qX<6i:E^a1�\�9�X<B]1�3~1'_d3H1�:<=c6#;�CM^%9�4#6C#F�CMX<KH;(?(BD35k�X<756c9�^H^H\DBD4j9�X<B]1�3�X<1��5RV¯ R�¥I�%��CM^�9�46#CZePBgX<7�3H1�3HBD3qX<6#k'6#:1�:<;56i:�ePBDX<7>Ùd3(:<BD4#1W����¤V�%T���U �/� t(X<7H6Z\]9�X<6{Ò�®�CPC<9e X<7H6Z6il,X<635CEBD1'3YX<1W351'35\]BD3569�:�^%9�:EX<B]9�\a;5BD �6i:<63qX<B]9�\a6JIKH9�X<BD1'35C#m

¢�756@_�:[9�=W6ie21�:Ebw1'_�X<756{X<7561�:E?w1'_V;5B]CMXE:<Bg`HK(X<BD1'35C|1'_VN�9�K(:<6#3qX{�(²�³�´8���5��µr=r9�;H68X<756@K5CE6@1'_u�(RV¯ R�¥,�H��CM^%9�4#6#C9�35;�X<756�CMX<K5;5?¶1'_PX<756#BD:f^5:<1�^�6i:EX<BD6Cf=c1�:<6�6j9�CM?qFu9�35;�BgXW4#1'K5\D;�BgX<CE6\D_�`�6�41'35CEBD;56#:<6#;�9~3%9�X<K5:[9�\�6#lIX<635CEBD1'3�1'_�X<7H6^5:<6#A,B]1�KHCE\g?);H6iAq6#\D1�^�6;WX<756#1�:E?W1'_aÚP���-R�T � � =W6j9�CEK5:<6#C#F(9�35;rCE1'=c6P1'_�X<756|3564#6CEC<9�:E?W:<6CEK5\gX<Cd1'_a�%K5354iX<B]1�3%9�\%�o3%9�\D?,CEB]C7H9�;f`�66#3);H6iAq6#\D1�^5^a6#;)_�1�:VX<7%9�X�^HK(:E^�1'CE6'tIBgX�BDC�4#6#:EX[9�B]35\g?{=@KH4[7W=W1�:<6�;5Bg�c4#KH\gXVX<1oX<75BD3(bf1�_HX<7%9�X-6il,X<6#3HCEBD1'3fBD_%1'356�7H9�;1'35\g?@41'35CEBD;56#:<6#;fX<756�9�`HCMXE:[9�4iX�X<7H6#1�:E?f9�35;fX<756 PR�����¥ � �u=c69�CEK(:<6C#F'\]Bgbq6u^5:<1�`%9�`HBD\DB]CMX<CV;515m � _�4#1'K(:<CE6�3H1�X<7HBD35kZ4#1'K5\D;7H9Aq6d`a6#6#3f;51'356dePBgX<751'K(X�X<756u;56#Aq6#\D1�^H=c6#3qX<C�1'_5N���¯���U�¤����F�9�35;f9�\gX<751'K5k'7@BD3qX<6k�:[9�X<B]1�3@BDC�=c1�:<6u69�CM?8X<1|K53H;56i:<CMX[9�35;X<7H9�3�;5BD �6i:<63qX<B]9�X<BD1'3~BD_u1'356c4#1'35CEBD;H6i:<CZX<7H9�X8�|:<4}75BD=c6#;56C � �W7H9�;�4#1'=)^%K(X<6#;�X<756c9�:<6j9Y`a6#\D1je¸9�^%9�:[9�`�1'\]9(F ePBgX<7H1�K5X6iAq63{7%9#A(BD3Hko9�X�7HBDC�;HBDCM^�1'C<9�\,9Z¡29�:EX<6#CEB]9�3@6JIKH9�X<BD1'3{1'_(X<756d^�9�:[9�`�1'\]9o9�C��o3H9�\g?,X<BD49�\(Ðo6#1'=W6#XE:E?pe�9�C�1'35\D?8BD3qAq63qX<6#;{`I?£p�aU�²����H� ��U�� F%9�35;w1'356{7H9�;>X<1We�9�BgX|9�\D=W1'CMX�Xse21WX<751'K5C<9�35;w?q69�:<C�X<1WCE66ZX<7568B]3qAq6#3qX<BD1'3w1'_�£pBg a6i:<6#3IX<B]9�\�¡29�\]4#K5\DKHC`I?¶�o�%´>��R�T � � 9�35;�N��%Q+¯�T�QSµ � vm��o\gX<751'K5k'7¶e26>K5CEKH9�\D\g?hX<69�4[7�z5:<CMX)X<756>ÚPQ�� �Z��T�T�BD3qX<6k�:[9�\OF�ePBgX<7�£p����¯ R� � � ºCEK5=cC#F%X<756#:<6@9�:<6@351�X|6351'K5k'7�ÚPQ�� �Z��T�T�LOBD3qX<6#k�:[9�`H\D68_�K5354#X<BD1'35C|BD3y1�:<;56i:oX<1r=c9�bq6{CE1�=c6@3H9�X<K(:[9�\�CM^%9�4#6#Co4#1'=)^%\D6iX<6'F9�35;yX<75BDC|49�3y`�6{;51'356{`I?wKHCEBD35kcX<756@N���¯���U�¤����BD3qX<6#k�:[9�\Sm��o\gX<751'K5k'7wX<756@CM^%9�4#6 ��v� �{��1'_�N���¯���U�¤���%LOBD3qX<6k�:[9�`%\D6_�K5354iX<B]1�3HC,� � 1'_�9{:<69�\�A�9�:<B]9�`H\D6|BDC24#1'=W^H\D6iX<6'F,BgX2BDC2351�X�9@;5K%9�\�`HK5X�1'356o49�3Y41'35CEBD;56#: � v � �{�d9�C�9@CEK(`HCE6#X�1'_aX<7H6o;5KH9�\1'_)C � � �{�[F�X<7H6cCM^%9�46c1'_241�3IX<BD3IK51'K5C@_�K5354iX<B]1�3HCZePBgX<7°41�=W^%9�4iX@CEK(^5^�1�:EX � � FV9�35;°X<756i:<6_�1�:<6W`�1'K53H;56#;hCE6JIK56#3H4#6#C@BD3��v� �{��=r9?y9�^5^5:<1q9�4[7~9rÚP���-R�T�=W69�CEK(:<6Y�SBD3yX<756{e269�b � X<1�^�1'\D1'k�?%�[t�_�1�:p6il59�=)^H\D6{X<756@CE6JIK56#3H4#6 ��� ;56iz%356#;�`I?

� � GI9'9�b�±-����� ����FHÙdCMX<1'35B]9�3(LS`�1�:<3>=r9�X<7H6#=c9�X<BD4B]9�3aF,`�1�:<3>BD3��j���'Ô,m-no6pe21�:Eb(CP9�XPN�KH35;w¦o35BgAq6i:<CEBgXs?qF%�qe26;56#3 m� vy±V6iX<6#:W£fm-N�� � F�noK535kq9�:<B]9�3(LS`�1�:<3�=r9�X<7H6#=c9�X<BD4B]9�3aFV`a1�:<3�B]3 ��'­'Ô,m�no6�:<6#4#6BgAq6#;¶X<756 1�R�¥ �¶^5:<B]�#6>BD3 �j� ��(qmno6@6=WBDk�:[9�X<6#;wX<1Y¦o35BDX<6#;��qX[9�X<6#C|`a6#_�1�:<601°1�:<\D; 1h9�:o0E0}t�756@e21�:Eb,CZ9�XoX<756W¡�R������T��°0�35CMX<BgX<K(X<6@1'_���9�X<7H6#=c9�X<BD4j9�\�I4BD6#3546#C#FH�o6ie �-1�:Eb�¦o35BgAq6i:<CEBDXs?'m� º��|:EX<7IK5:P�o1�:EX<1'3>�~Q�¥5¤V�����fFH�o=W6i:<B]49�3�=r9�X<756#=r9�X<B]4#B]9�3aFI`a1�:<3wBD3°��(�j­,m� � �~9�:Eb�0�1'CEBD_�1jA(BD4[7~ÑpQ+U³�Q���F ÚPK5CECEB]9�3�=c9�X<756#=r9�X<BD4#B]9�3aF�`�1�:<3�BD3��j�'­,��m)no6fe21�:Eb,C{9�XpX<7H6)Ú|K5CECEB]9�3��o4j9�;56#=@?�1�_�I4BD6#3546#C#FH��1'CE4#1je8m� � GI9�4JIK56CMLON�1'K5BDC8N�Q+R�T�U'F �H:<6#354}7°=c9�X<756#=r9�X<BD4#B]9�3aFa`a1�:<3°BD3��j�'­ �,mcno6):<6#46#BgAq6#;�X<756cGI9�^�9�3�^5:<BD�#6WBD3/��'�(��mcno6e21�:Eb(C@9�XW¡u1'\D\! 6#k'6c;56c�H:[9�354#6'F�±�9�:<BDC#F�eP756i:<6r756r756#\D;¶9w4[7H9�Bg:Y�S�o3H9�\g?(CE6c=c9�X<7�"6=c9�X<B]JIK56);H6#C@CM?(CMX� 6#=W6C86#X@;56r\D6#K5:4#1'3qXE:��1'\D6��u�j� ( ��L<�j�'� �,mV0�z5:<CMXu7H9�;W75BD=¸9�Cd9pX<69�4[7H6i:u9�XdÙ-41'\D6�±V1'\g?,X<6#4}7535B]JIK56�BD3w��'Ò'Òq¨�Ò (IF,9�35;W0V;5BD;):<6#CE6j9�:<4[7WKH35;56i:75BDC�;HBg:<6#4#X<BD1'3aF5K53qX<BD\ ={?YX<756#CEBDCPBD3��j� (,��m�,� Ùd3(:<BD41W����¤V�%T���U'F50sX[9�\DB]9�3�=c9�X<756=c9�X<BD4#B�9�3 FI`a1�:<3wBD3��j�'­ �,m-no6Ze21�:Eb(CPBD3>±�9A(B]9(m� � Gq1'7H9�353yÚP���-R�TuF�¡u�#64[7(LS`�1�:<3w=c9�X<756=c9�X<BD4B]9�3aFa��� ��(jL<�j��Ô'Ò,m-no6pe21�:Ebq6#;wBD3�ÑpBD6353H9(F5�oKHCMXE:<B]9(m� � �0"6#\]BglYÙd;51'KH9�:<;wGqK5CMX<BD3>Ùd=WB]\D6 PR�����¥�F5�H:<6#354}7>=r9�X<756#=r9�X<B]4#B]9�3aFa� ��(I�iL<�j�'Ô'ÒIm-no6Ze21�:Eb'6;wBD3�±�9�:<BDC#m� ���|:<4}75BD=W6;56#C#F%­ ��(0 |¡uÙ�Lu­(�j­ |¡uÙPF(e21�:Ebq6#;wBD3��q?,:[9�4#K5CE6'F5X<756#3y9rÐ|:<6#6#b�4#1'\D1'3q?'m

|¡uÙ � �6#_�1�:<6�¡u1�=c=W1'3/Ù-:[9,FP9�:<6i^H\]9�46#=W63qXr_�1�: P¡ �+eP75BD4}7/4#1'K5\D;�9�\DCE1°`�6wX[9�bq6#3�9�Cr=W6j9�35BD35k 26_�1�:<6¡u7(:<BDCMX<B�9�3>Ù-:[9)_�1�:�X<751'CE6peP751cBD35CEBDCMXPBD3>\DBD35b,B]35k)JIK56CMX<BD1'35C�1'_�;%9�X[9�X<BD1'3�ePBgX<7wJqKH6#CMX<BD1'35C�1'_�:<6#\DBDk'BD1'3��[m� �ÚP63�"68£p�aU�²����H� ��U'FH�5:<6354[7y=c9�X<756#=r9�X<BD4#B]9�3aFa�Ô'�'Ò�L<�jÒ�Ô�®(m� � �IBD:Y0�C<9'9�4~�o�%´>��R�TdF2Ùd35k'\DB]CE7�=r9�X<7H6#=c9�X<BD4B]9�3aF8�jÒ�Õ ��L<�)(�­ (Im�no6~756#\];/X<756�N�K54j9�CEB]9�3�4}7H9�Bg:>9�X�¡29�={`5:<B];5k'6�F�jÒ'Ò���L<�)(�®q­Im� v�Ðo1�XEX<_�:<BD6#; 1*B]\D756#\]=!Aq1'3>N���Q+¯�T�QSµ(F�Ðo6i:<=r9�3>=c9�X<756=c9�X<BD4#B�9�3 F��jÒ�ÕIÒ�L<� (,�jÒ,m� º�Gq6j9�3~Ðp9�CMX<1'3w£p����¯ R� � F5�5:<6354[7w=r9�X<7H6#=c9�X<BD4B]9�3aFa� ��ÕI­�L<�j�(� (Im-no6Ze21�:Ebq6;>B]3>±�9�:<B]C#m� � ¢�756#?�9�:<6r9�4iX<KH9�\D\g?~6JIK5BDA�9�\D6#3H4#6)4\]9�CECE6#C#F�9�C{1'356WBD;H6#3qX<Bgz%6#CZXse21w_�KH354iX<BD1'35CZeP7HBD4[7�1'35\D?~;HBg �6i:{1'3°9>CEK(`%CE6iX@1�_=W6j9�CEK(:<6p®(F(BSm 6�m-9@CEK(`HCE6#X�eP75BD4[7Y_�1�:�6iAq6i:E?�����®@4j9�3Y`�6o4#1jAq6i:<6;r`I?YBD3qX<6#:EA�9�\DCdeP751'CE6pCEK5=WC�1'_ \D635k�X<75C2BDC2\D6CECuX<7H9�3�qm� �°�%1�:r9~41�3IX<BD3IK51'K5C)_�K5354#X<BD1'3 @�;56#zH356;�1�3�9�X<1�^�1'\D1'k�B]49�\uCM^�9�46>9�35;�X[9�b(BD35k�A'9�\]K56#C)BD3�9�Aq64iX<1�:)CM^�9�46�F-X<7H6CEK(^5^a1�:EX|BDC�X<75684#\D1'CEK(:<681'_�X<7568CE6iXP1'_BQyCEK54[7wX<7H9�X@��GQa��*®(m

(

Page 8: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

��� �GQa�� � 1'3��O® � ��� � �o9�3H; ��� �GQa� �¸®>6#\DCE6#eP756i:<6)41'3qAq6i:<k'6#CZX<1�X<7H6)£ZQ�����² � � =r9�CEC � � 9�X{®(F�9�35;h9�C81'356W4}756#4[b,CX<7H9�Xp_�1�:p6#Aq6i:E?�� � C � � �{��1'356f7H9�CRI�� ��� �GQa���u�GQa�4O Q��d�O®I�[F�X<756@£ZQ�����²°=c9�CECp9�XZ®r41�:E:<6#CM^�1'35;5CoX<1rX<756@\]BD3569�:_�K5354iX<B]1�3%9�\�� ����u�O®q�[mh��1�:<6>k'6#356#:[9�\]\g?h9~ÚP���-R�T�=W6j9�CEK5:<6�� 1'3�9�3�1�^�63¶CEK5`HCE6iX �!1�_ � ! BDCW9~\DBD3H69�:)_�1�:<=1'3$C � ���P�[F�X<756rCM^%9�4#6r1'_�4#1'3qX<B]3IK51'K5C@_�KH354iX<BD1'35C{ePBgX<7°4#1'=)^%9�4#X{CEK(^H^�1�:EX@B]3 �oF��������� � ���[F�CEK54}7°X<7%9�X@_�1�:f6#A'6#:E?4#1'=W^%9�4iX���� �oF�X<756#:<6f6il(BDCMX<CZ9r4#1'35CMX[9�3qX�C)�����PCEK54[7~X<7H9�X������ � ����� (�CW�����,=c9�l � �"! � �u�GQa���5_�1�:89�\D\#� � C � ���P�7H9A(BD35k)X<756Bg:�CEK(^H^�1�:EXPBD3$�°m

� � ±�9�K5\��o;(:<BD6#3y��9�K5:<BD4#6{£ZQ�����²�F%Ùd35k'\DB]CE7w^H7q?(CEBD4#B]CMXF����®q­�L<�j� ��Õ5m|no68:<64#6BgAq6#;>X<7H6{�ZRV¯���¥�^5:<BD�#6@BD3y±d7q?(CEBD4C|BD3�j��� �,m-no68756#\D;�X<7568N�K54j9�CEB�9�3w4[7H9�Bg:|9�Xp¡29�={`5:<B];5k'6�F�����'­�L<�j��Ò'�,m

�o\D_�:<6#;°�ZRV¯��%¥�F��qe26#;HBDCE7°BD3H;5K5CMXE:<B]9�\DBDCMX@9�35;�^H7HBD\]9�3qX<7(:<1�^HBDCMXF2� ������L<���'�'Ò,m�no6c4#:<69�X<6#;h9w_�K535;�X<1w`a6cK5CE6#;¶9�C9e�9�:<;5CP_�1�:�^a6#1�^H\D6peP751'CE6Ze21�:Eb�=W1'CMX�`�6#3H6iz5X<6;>7IK5=r9�3HBgXs?'m� � ¢�756@BD3IX<K5BgX<BD1'3w1'_-9W^�1'BD3qXp=c9�CEC@�S1�:o4}7H9�:<k'6���B]CP1�`IA(BD1'K5Co_�1�:Z9�3I?'1'356fBD3qX<6#:<6#CMX<6#;yB]3y±-7I?,CEB]4#C#F�`HK(Xo£ZQ�����²�e263qX=@KH4[7{_�K(:EX<7H6i:�X<7H9�38;569�\DB]35k�ePBgX<7pX<7H6#CE6-1�`(�M64iX<C#F�9�C�756�e�9�C�351�X�9�_]:[9�BD;81'_IX[9�b(BD35k�;56#:<BgA�9�X<BgAq6#C 1'_,75B]C CMXE:[9�35k'6\g?p;56iz%356#;«<_�KH354iX<BD1'35¬5F 9�JIK5BgX<6@`�1'\D;�=W1jAq6@eP75BD4}7�e�9�Cpk'BgAq6#3�9r^5:<6#4BDCE6f=c9�X<756=c9�X<BD49�\�=W69�35B]35kr`I?�N�9�K(:<63qX@�(²�³�´8���5��µ�BD375BDC�X<7H6#1�:E?�1�_�;5BDCMXE:<BD`HK(X<BD1'35C#m

Page 9: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®q­,m�1°6#;53H6#CE;H9?�GI9�3IK%9�:E?��j�IF�­�®'®'®(m¢�756{N���¯��aUj¤���y=c69�CEK(:<6{1�3 �)!ØBDCPBD3qA'9�:<B]9�3qX�`I?�XE:[9�35CE\]9�X<BD1'3w9�35;w`I?�:<1�X[9�X<B]1�3 F5BSm 6�md`I?�:<BDk'BD;>;5B]CM^H\]9�4#6=W6#3IX<C

�O9�35;�9�\]CE1y`I?h=WBg:E:<1�:@CM?(=c=W6iXE:E?%�[§)\D6#X#N � �)!�9�35;�� ����� � & �[FVX<756rCM^�64#B]9�\o�SBSm 6�m�7%9#A(BD3Hk�;56iX<6#:<=WBD3%9�3IXWÓ@���1�:EX<751'k'1�3%9�\�k�:<1'K(^h9�4iX<BD35k�1'3 �)!{F X<756#3�B]_��¸BDCZN���¯��aU�¤���°=c69�CEK(:[9�`H\D6W9�35;� BDCpX<756)BD=r9�k�6f1'_��`I?�X<756f:<BDk'BD;;5BDCM^%\]9�4#6#=c6#3qXQ �� N@Ó�� Q�F�X<756#3�� BDCoN���¯��aU�¤����=c69�CEK(:[9�`H\D6)9�35;~7H9�C|X<756fC<9�=W6f=W6j9�CEK(:<6{X<7H9�3 �{m � 3H6f49�3«<4#1'35CMXE:<K54#X<¬)351'3>=c69�CEK(:[9�`H\D6pCE6#X<C�`I?�K5CEBD35k)X<75689�l(BD1'= 1'_�4}751'BD4#6'F,X<756Z4#\�9�CECEB]49�\a6il59�=)^H\]6|`�6BD35kfX<1)CMX[9�:EX�ePBgX<7YX<7H6K535BDX{4Bg:<4#\D6 � v#F�9�35;�X<1y;H6izH356r6JIK5BDA�9�\D6#3H4#6W4#\]9�CECE6C#F�CE1yX<7H9�X8Xse21w^a1�B]3qX<C@9�:<6c6jJIK5BgA�9�\D63qX8B]_21'356c4j9�3�`�6r1�`5X[9�BD356#;_�:<1�=�X<756Z1�X<756#:�`I?>9�^5^H\g?(BD35kc9f:<1�X[9�X<BD1'3�1'_V9�3wBD3IX<6#k'6i:|9�3Hk�\]6 � ��� t(X<7H6#3w1'356ZK5CE6C�X<756{9�l(BD1'= 1'_�4[751'BD46ZBD3w1�:<;H6i:X<1�9�CECE6i:EX)X<7%9�XWX<756i:<6>6il(BDCMX<CW9~CEK(`HCE6#X�� eP75BD4[7�4#1'3qX[9�BD35CW6il59�4iX<\g?h1'356>6#\D6=W6#3IX)BD3�6j9�4}7�6JIK5BgA'9�\D6#3546Y4#\]9�CEC#Fu9�35;;56351�X<BD35k�� � � � Ó��ØX<756@CEK(`%CE6iXp1�`5X[9�BD35BD3Hkc_]:<1'=��Ø`I?�9W:<1�X[9�X<B]1�3y1'_ � F�1'356{zH35;5C|X<7H9�X � v BDCP^%9�:EX<BgX<BD1'356#;yB]3qX<1X<756�� � � � ��� FaCE1YX<7H9�XZBD_�� e�9�CpN���¯��aU�¤����=W6j9�CEK(:[9�`H\D6�F 9�\D\�X<756�� � e21'K5\D;~7H9Aq6fX<756fC<9�=W6f=W6j9�CEK(:<6)9�35;�X<75B]C=W6j9�CEK(:<6�4#1'K5\D;)351�X�`�6 ��®|`�649�K5CE62X<7H62=c69�CEK(:<6�1'_ � vVB]C-­���F'`HK5X�BD_HX<756�=c69�CEK(:<62e�9�C-®,F � vVe21'K5\D;)`�6�9o41'K53qX[9�`H\D6K535B]1�3>1'_�CEK(`HCE6#X<C�1'_�=W6j9�CEK(:<68®)9�35;>e21�KH\D;�7%9#Aq68=W69�CEK(:<68®(F%9�3H;w9�C���49�3>7%9#Aq6Z356BgX<756i:|9@^�1'CEBgX<BgAq6p=c69�CEK(:<6Z351�:9)�#6#:<1)=c69�CEK(:<68BgXP1'35\g?r:<6=c9�BD35C�X<756p^�1'CECEBg`%BD\DBgXs?rX<7H9�XPBgX�7%9�CP351W=W6j9�CEK(:<6{9�XP9�\D\Sm

� =c1�:<6�CEK(`5X<\D6�4#1'35CMXE:<K54#X<BD1'3°e�9�C)49�:E:<BD6;¶1'K(X)BD3 � � `,?hno� �U���R�� �?�av#F�9�35;�CEBD=W^H\DBgzH6;°`I? ���T���²�³ 9�35;¢�����U��Q+º�FIk'BgA(BD35k8X<756|no� VU��-R�� � �(L- ���T���²�³�L�¢����VU��Q%^�9�:[9�;51jl�§�BD_�� 9�35;���9�:<6|Xse21{4#\D1'CE6;W`�1'K53H;56#;YCE6iX<Cu1'_ �)!ePBgX<7w351'356=)^5Xs?>BD3IX<6i:<BD1�:{�O9�35; & �q�[FHX<7563>X<756#:<686#l,B]CMX<CP9)^�1'CEBgX<BDA'68B]3qX<6#k'6i: �F%9)^�9�:EX<BDX<BD1'3>1'_��ØBD3qX<1y�S;5BDC��E1'BD3qXi�CEK(`HCE6#X<C�� v���������� � � F,9p^%9�:EX<BgX<BD1'3W1'_�� BD3IX<1c�S;5BDC��E1�B]3qXi��CEK(`HCE6#X<C�� v���������� � � FICEK54[7WX<7H9�Xd_�1�:�� �Ø� ��������� �X<756PCEK(`HCE6iX� 7 BDC�X<7568BD=r9�k�6Z1'_�� 7 `I?>9):<B]k�B];�;5B]CM^H\]9�4#6=W6#3IXt51'_�41�K5:<CE68CE1�=c681'_�X<7568CEK(`HCE6#X<C|9�:<68351�XP=c69�CEK(:[9�`H\D68BD_��Ø9�35; �7H9Aq6o;HBg �6i:<63qX2=W6j9�CEK(:<6"! 0d7H9Aq6|:<6j9�;cX<7H6oCMX[9�X<6=W63qXp�+`HK(X�351�X2CE66#3rX<756|^5:<1,1'_[�VX<7%9�X�_�1�: & � ­ZX<756#:<6o;51,6#C26il(BDCMX�9zH35BDX<6#\g?c9�;5;HBgX<BgAq6|=W69�CEK(:<6o;H6izH356;Y_�1�:�9�\]\%CEK(`%CE6iX<C�9�35;YBD3qA�9�:<B]9�3qXu`,?WXE:[9�35CE\]9�X<BD1'3Y9�35;r:<1�X[9�X<BD1'3aF,CE1@CEKH4[7�9{^%9�:[9�;51jl;51,6#CP351�XP751'\D;>B]3��8º$#Om

¦|^cX<189Z=@KH\gX<Bg^H\DB]49�X<BD1'3@`I?W9Z4#1'35CMX[9�3qXdX<756�N���¯��aU�¤���r=c69�CEK(:<6PBDC�X<756�1'35\g?)351'35�#6#:<18ÚP����R�T~=c69�CEK(:<6�eP75BD4}7BDC2B]3qA�9�:<B]9�3qXu`I?cXE:[9�35CE\�9�X<B]1�3 F(9�35;YX<756i:<6_�1�:<6oBgX�BDC2K535B]JIK56\g?c;56iz%356#;�BD_ e26p9�;5;YX<756|:<6jJIK5Bg:<6#=c6#3qXuX<7%9�X2X<756oAq1�\]K5=W6o1�_X<756fK535BgXp4K(`�6{`a6Y��mp�H1�:89�3I?y\D1,49�\D\D?�4#1'=W^%9�4iX � 41'=W=@K5X[9�X<BDA'6��{k�:<1'K(^~X<756i:<6f6il(BDCMX<Cp9r351'35�#6#:<1YÚP���-R�T�=W6j9�CEK5:<6eP75BD4}7rBDCuBD3IA�9�:<B]9�3qXd`,?WXE:[9�35CE\]9�X<BD1'3aF,K535B]JIK56oK(^YX<1@=@K5\gX<BD^H\DBD4j9�X<B]1�3)`I?r9@4#1'35CMX[9�3qXF59{no����� � =W69�CEK(:<6o1'_�X<756ok�:<1�K5^ m�H1�:cX<756wk�:<1'K(^ � F�9�no�����x=W69�CEK(:<6wBDC)X<756w4#1'K53qX<BD3Hk�=W6j9�CEK5:<6�t�_�1�:WX<756wk�:<1'K(^ �)!{F�9�no������=W6j9�CEK5:<6>B]CfX<7H6N���¯��aUj¤���y=c69�CEK(:<6�tH_�1�:�X<7H6Z=@KH\gX<Bg^H\DB]49�X<BgAq6|k�:<1'K(^h�O® � ,��[FH9)no�����°=W6j9�CEK(:<6ZBDC�O&% �'%}m�o\gX<7H1�KHk�7@X<75624#1'3qAq1'\DK(X<BD1'3{^H:<1I;HK54iXV4j9�3@`�62;56#zH356#;f_�1�:�9�3q?@\D1,4j9�\]\g?Z4#1'=W^%9�4iXVk�:<1'K(^W_�1�:VeP75BD4[7f1'35627H9�CV4}751'CE6#39yno�����*=W6j9�CEK(:<6�FVe26YCE7H9�\D\uK5CE6YBgXf=W1'CMX<\g?�_�1�: �)!{m��H1�:�@ � K � C � � �"!p�[FVX<756r41'3qAq1�\]K(X<BD1'3°^5:<1,;5K54#X�(,� @ � KBDC8;56#zH356;�`,?)(��GQa� � I�� 1 @��+*5� K��GQ�>�*5�4O,* � I � 1 @��GQ�>�*5� K��+*H�MO,*%F�CE751jePBD35k>X<7%9�X8X<7H6W4#1'3qAq1'\DK(X<B]1�3�^5:<1,;5K54#X{BDC4#1'=c=@K(X[9�X<BgAq6�m � 356Y7H9�C�@ � K � C � � �"!p�89�35;�- � * */. L�%i� @ � K5� �0- � *�*�. L�%i� @a��Ó1- � *�*�. L�%i� K5�[m�¢�756Y4#1'3qAq1'\DK(X<B]1�3^5:<1,;5K54#XPBDCP9�CECE1,4#B]9�X<BgAq6�F5BOm 6'm�_�1�:)N �320�34 � C � � �)!p�u1'35687%9�C{�GN � 2 � � 4 � N � � 2 � 4 �[m¡u1'3qAq1'\DK(X<BD1'3�1'_24#1'3qX<BD3,K51'K5C8_�K5354#X<BD1'35CZePBgX<7�41'=)^%9�4iX{CEK(^5^�1�:EX@C<9�X<BDCMzH6#C �8R��TV¤ � BD356JIKH9�\DBDXs?yeP75B]4[7°9�CECE6i:EX<CX<7H9�X�BD_��)(+* �654� L�9�35; v7 � v

�Ó v8 >@��F�X<756#3{_�1�:*@ � K � C � � �)!p�a1'356d7H9�C�� � @ � K � � 7 ((� � @ � � � � � K � � 8 F�eP756i:<6d_�1�:u�)(9-)(-,/F

v �%6#\DBgl�no� VU��-R�� � �aF�Ðo6i:<=r9�3w=c9�X<756#=r9�X<BD4#B]9�3aFa� �'Ò'�jL<�j��ÕI­,m-no6pe21�:Ebq6#;wBD3 �1'353amº~�o\D_�:<6#;�¢����VU��Q�F�±�1'\DB]CE7(LS`�1�:<3y=c9�X<756=c9�X<BD4#B�9�3 F��j��®q­�L<�j� � �,mPno6{6#=cBDk�:[9�X<6;>X<1r¦o35BgX<6;~�qX[9�X<6#Co9�35;�756{e21�:Eb'6;

9�XP¦o35BgAq6i:<CEBDXs?r1'3~¡29�\DBD_�1�:<3HB]9(F �6i:Ebq6#\]6i?'m� 027H9Aq68756j9�:<;wN�9�K5:<6#3qXp�(²a³�´{���5��µWC<9?YX<7H9�X�X<7H6p:<6#CEKH\gXPBDC�351�X�XE:<K568_�1�:|9�\D\ 351'354#1'=)^�9�4#XPk�:<1�K5^HC#m� 0�3�X<7H6p3H1�3H4#1'=W=fK(X[9�X<BgAq6p49�CE6Z1'356Z;5B]CMX<BD35k'K5BDCE756C2`�6#Xse26#63�BD3IA�9�:<B]9�354#6p`I?YXE:[9�3HCE\]9�X<BD1'3�1�3>X<756Z\D6#_�XP9�3H;>BD3qA'9�:<BDL

9�354#6Z`I?YXE:[9�35CE\�9�X<B]1�3>1'3�X<756Z:<BDk'7qXm� �o\D_�:&"6#;>no�����dF5noK535kq9�:<B]9�3>=c9�X<756=c9�X<BD4#B�9�3 F�� � �'Ô�L<������,m-no6pe21�:Ebq6#;y9�Xp�I�#6k'6#;am� 1 BD\D\DB�9�= no63(:E?*�8R��T�¤uFqÙd35k'\DBDCE7c=c9�X<756#=r9�X<BD4#B]9�3aF%� �'Ò��jL<�j��ÕI­,mVno6�e21�:Ebq6#;rBD3cN�9�K5C<9�35356'FH�qePBgX<�6i:<\]9�35; m�no6PBDC

C<9�BD;WX<1{7H9A'6|;5B]CE4#1jAq6i:<6#;cN���¯��aU�¤���YB]3qX<6#k�:[9�X<BD1'3)Xse218?q69�:<C-`�6_�1�:<6PN���¯��aU�¤����m 1 BgX<7W75BDC-ePB]_�6pÐ|:[9�4#6p¡u³�Q+U³�R�¥ �oL�8R��TV¤2F�Ùd35k'\DBDCE7)=c9�X<756#=r9�X<BD4#B]9�3aF5� �'Ò ��L<���Õ'Õ5F'X<756#?@e21�:Ebq6;)1'3)CE18=c9�3q?@^5:<1�`H\D6=WCVX<1'k'6iX<756#:F'CE1pX<7%9�X-BDX�BDC�;5BD�W4K5\gXX<1cb,3H1e BD_�9�3q?>:<6#CEKH\gXp9�XEXE:<BD`HK(X<6#;>X<1r75BD= B]C|9p�E1�B]3qX|e21�:Eb>ePBgX<7y75B]C�ePB]_�6{1�:|3H1�Xm|¢�7H6#Bg:|CE1'3aF�N�9�K(:<6354#6W¡u75BDCE751'\D=�8R��TV¤2F�`a1�:<3�BD3x�j��®qÔ�BDCcb(351jeP3�_�1�:Y75BDCr1jeP3�=c9�X<756#=r9�X<BD49�\�:<6#CEKH\gX<C#FP9�3H;�9�=W1'35k°X<756#=�X<756yB]3qXE:<1,;5K54iX<B]1�3/1�_�8R��TV¤�=W6j9�CEK5:<6#CPBD3>X<756f¡29�\D4#K5\]K5C�1'_�Ñu9�:<B]9�X<BD1'35C#t50u^HB]1�3H6#6i:<6;>X<756#Bg:PKHCE68BD3>^%9�:EX<B]9�\ ;5Bg �6#:<6#3qX<B�9�\ 6JIKH9�X<BD1'35CPBD3�X<7H6\]9�X<6 (�®�C#F�351�X{b(351jePBD3Hk�9�X@X<756WX<BD=W6WX<7H9�X@756r7H9�;°BD3IXE:<1I;HK54#6;�X<756= �O9�\DX<751'K5k'7�087H9�;°z5:<CMX@=W6#X{7HBD=�BD3/�j� (,�r9�X

Page 10: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

� � ( � � � =W6j9�3HC#� � ( � � ����� � 1�� md0�_�*�1�: 5 1�: LfB]C8��F%BgXPBDCV�MKHCMX|9�3y9�^5^%\DBD49�X<BD1'3>1'_�n �R�¥������hBD356jJIKH9�\DBgXs?�9�3H;wBgXPBDCP1�^5X<BD=r9�\OFeP75BD\]68_�1�:|1�X<756#:o4j9�CE6C|1'356@=c9?>^5:<1jAq6{BDXP`I?y9�^5^H\D?,B]35kr9c_�6iexX<BD=W6CPn �R�¥��V�%�¶B]356JIKH9�\DBgXs?qF%1�:pGq�%T�U�%T��8BD356jJqK%9�\]BgXs?'F`HK(X�X<756o4#1'35CMX[9�3qX�B]C2351�X21�^5X<B]=c9�\�9�35;YX<7H6|`�6#CMX�4#1'35CMX[9�3qX)CW� * � 5 ��_�1�:2eP7HBD4[7Y1'356p7H9�C�� � @ � K � � 7 ( CW� * � 5 ��� � @ � � � � � K � � 87H9�C�`�66#3y_�1'K535;yBD35;56#^�6#3H;56#3qX<\D?r`,? 1*BD\]\DB]9�= ��a²���T��%���peP751cK5CE6#;w^5:<1�`%9�`HBD\DB]CMX<BD4Z=W6iX<7H1I;HC#F%9�35;>`I?>Ùd\D\]BD1�X�N�Q���¯ 9�35; �����U�²���� A�v � eP751yKHCE6#;h351'3(^5:<1�`%9�`HB]\DBDCMX<BD4c=W6iX<7H1I;HCr�S6JIKH9�\DBgXs?�751'\D;HC{_�1�:@CE1'=c6W^%9�:EX<BD4K5\]9�:WÐp9�KHCECEB]9�35C}�[m � _4#1'K(:<CE6'F�KH35;56i:VX<756d^H:<6#4#6;5BD35k|:<6\]9�X<BD1'38`�6#Xse26#6#3 * �654� L,F�X<7H6u4#1'3qAq1'\DK(X<BD1'3{^5:<1,;5K54#X<C�6#l,X<6#35;5C�_�:<1�= ���(� �"!p�� � 8 � �)!p�BD3qX<1#� 7 � �"!p�oePBgX<7�X<756cC<9�=c6cBD356JIKH9�\DBDX<BD6#C#F�9�35;�X<75B]C84j9�3°`�6W^5:<1jAq6#3h6#BgX<756#:{;5Bg:<64iX<\g?~1'_u`I?�K5CEBD3Hk>X<756c;5635CEBgXs?�1�_C � � �)!p�uB]3 ���5� �)!p�u_�1�:@��(+* �-,�F%9�3H;�X<7H6pe269�b � ;H6#35CEBgXs?�BD_ * ��,�m�o1�X<6�§�0o9�;5=WBgXoX<7H9�XoX<75BDCo;5635CEBgXs?�7H9�C|`�66#3�^5:<1jAq6#3�eP7563�41'35CMXE:<K54iX<B]35krX<756fN���¯���U�¤����=W6j9�CEK(:<6�F 9�35;�9�\gX<751'K5k'7e26pCE7H9�\D\�CMX<K5;(?Y\]9�X<6i:�9�3�6il,^H\DB]4#BgXde�9?r1'_�9�^5^5:<1q9�4[75B]35kf_�KH354iX<BD1'35C�BD3����H� �)!p�-`I?r_�KH354iX<BD1'35C�BD3 C�� � �)!Z�[F,X<7H6|^5:<1,1�_ePBD\D\aKHCE6ZX<756Z_�9�4iX�X<7%9�X C � � �)!Z�uBDC�b(351jeP3>X<1f`a6Z;5635CE6�F%9�35;>ePBD\D\a351�X�`�6{9�3wBD3H;56i^�635;563qX�^5:<1,1'_�1'_�X<7H9�X�:<6#CEK5\gXm� ��%"-� j�D������ � §°�SB��80�_{� � * � ,�F8@ � ���(� �"!p�f9�35;$K � ���� �� �)!p�[FdX<7563 @ � K � C � � �)!p�[FdX<756>CM^%9�4#6w1�_4#1'3qX<B]3IK51'K5C{�+`�1'K535;56;��u_�KH354iX<BD1'35C�41'3qAq6i:<k'BD35k)X<1c®W9�X�B]3(zH35BgXs?qm

�SBDB+��0�_�@ � ��vj� �)!p��9�35;#K � ��>� �)!Z�[F�X<7563 @ � K � ���RC)� �"!p�[F�X<7H62CM^%9�4#6�1'_H`�1'K535;56;fKH35BD_�1�:<=c\g?841'3qX<BD3IK51'K5C_�K5354iX<B]1�3HC#m�2È<ÆjÆ5#[§��8R��T�¤ × C�B]356JIKH9�\DBgXs?rB]3YX<7H9�XP49�CE6Z_�1'\D\D1jePC�_�:<1'= � � @ � K5�}�GQa���.���� I � 1 @��+*5� K��GQ > *5�4O&*��� ((� � @ � � � � � K � � 8 _�1�:|9�\D\Q�`,?~n �R�¥�������BD3H6JIKH9�\DBgXs?qmf¢�756i:<6W6#l,B]CMX<CZ9>CE6JIK56#3H4#6�@ �!� C � � �)!p�o4#1'3qAq6i:<k'BD3HkYX<16@°BD3 ���5� �"!p�|CMXE:<1'35k5F�9�3H;�9CE6JIK56354#6)K � � C � � �)!Z�V4#1'3qAq6i:<k'BD35k8X<1RKfBD3$���� s� �)!p�VCMXE:<1'35k5F(9�35;Y9�C8@ � K�> @ � � K � ��@ � � K�> K � �,Ó�� @ > @ � � � K � F1'356f;56;5K54#6C|X<7%9�XE� � @ � K�> @ � � K � � � (�� � @ � � � � � KW><K � � � � Ó�� � @9> @ � � � � � � K � � � � � ®(F 9�3H;�X<756#:<6#_�1�:<6 @ � K>BDC|X<7H6K535B]_�1�:<=!\DBD=cBgX-1'_�X<756oCE6JIK56354#6R@ � � K � � C � � � ! �[F(9�35;rX<756i:<6_�1�:<6P`�6\D1'35k'C-X<18X<756o4#\D1'CEK(:<6|1'_TC � � � ! ��BD3 � � � ! �[FeP75BD4}7>BDC C � � �)!Z�[m

¦oCEBD3Hk@98CE6JIK56354#6 @ � � C � � � ! ��41'3qAq6i:<k'BD35k8X<1�@rB]3$� v � � ! �[F,1'356|7H9�CR� � @ � K)>�@ � � K � � ( � � @ >�@ � � � v � � K � � F9�35;yX<756i:<6_�1�:<6W@ � K�BDCPX<756@K535B]_�1�:<= \DBD=WBDXP1�_VX<756{CE6jJqKH6#3546W@ � � K%F�9�3H;yBgXoBDC|6351'K5k'7wX<1rCE751jexX<7H9�Xo69�4}7 @ � � K�BDC`�1'K535;H6#;{K53HBD_�1�:<=c\g?o41�3IX<BD3IK51'K5C#F'9�C�9PK53HBD_�1�:<=�\]BD=WBgX 1'_(CEK54}7{_�K5354iX<B]1�3HC�9�\DCE1�`�6\D1'35k'C X<1�X<756dC<9�=W6dCM^%9�4#6����RCW� �)!p�[m�oCcX<756y_�K5354#X<BD1'35@ � `a6#\D1'35k'CcX<1 C � � �)!p�)BgXYBDCrK535BD_�1�:<=W\g?�41'3qX<BD3IK51'K5C#FPCE1°X<7H9�X � @ � �GNH��>�@ � � 2 ��� ( � � � � N > 2 � �ePBgX<7*\DBD=���� � � � � %<�6� ®,m � 3H6~7H9�C�� @ � � K5�}�GQa� > � @ � � K5�}�GQ����6� I � 1�� @ � �GQ<>�*H� > @ � �GQ��H> *5����K��+*5�4O,*%F|`HK(XX<756>BD3qX<6k�:[9�\�=c9?h`�6�:<6#CMXE:<BD4#X<6#;¶X<1~X<7H6>CE6iXW1'_ *hCEK54[7�X<7H9�X � * >,QH�&( � � 9�35; � * > Q�� �&( � � BD_PX<756wCEK(^5^�1�:EX1'_ @ � BDCfBD354\DK5;56;hBD3hX<756�4#\D1'CE6#;¶`%9�\D\u46#3qX<6#:<6#;�9�XW®yePBgX<7h:[9�;5BDK5C � � tuX<756i:<6_�1�:<6<� � @ � � K5�}�GQa� >�� @ � � K5�}�GQ������ (I � !#"�� $ �&% � � !'" � � $ �&% � @ � �GQ >�*5�T>!@ � �GQ�� > *5����� K��+*5��� O&*�( �I� � Q > Q�� � ��� � K � � )(�N - � �Y�O® � � � � � F5CE7H1ePB]35k)X<7H9�X @ � � KBDC�K53HBD_�1�:<=c\g?r41'3qX<BD3IK51'K5C#m� _�4#1'K(:<CE6�FVX<756r^5:<1�^�6i:EXs?°1'_�4#1'=W=fK(X[9�X<BgA(BgXs?~1'_�X<756r41�3IA'1'\DK5X<BD1'3°^5:<1,;5K54iXf6#lIX<635;5C{X<1yX<756Y49�CE6reP756i:<6rBDX@BDC

;56#zH356#;°1�3 ���5� �)!p�*� � 8 � �)!p�f�SBSm 6�mWBD_�* �65 �c9�35; v�Ó v8 ���[F�9�35;°CEBD=cBD\]9�:<\g?yX<756)^5:<1�^�6i:EXs?�1'_29�CECE1I4B]9�X<BgA(BgXs?

1'_ X<756Z4#1'3qAq1'\DK(X<BD1'3r^H:<1I;HK54iX�6il,X<635;5C2X<1@X<7H6o4j9�CE6peP756#:<6oX<756p_�K5354#X<BD1'35Cu`�6\D1'35kfX<1��,+5� �)!Z� � �.-� �)!p� � � � � �)!p�|�SBSm 6�mN � 2 � 4 ��F v+ Ó v- ��F v- Ó v� ��FH9�3H; v+ Ó v- Ó v

� /­q�[F59�35;>49�3Y`a6o^5:<1jAq6#;>;5Bg:<6#4#X<\g?cK5CEBD35kW��V¯�Q�T�Q vEv × C�X<756#1�:<6=>m

no1je26iAq6#:F(1'356p=@K5CMX2`a6o4j9�:<6_�KH\%X<7%9�X�X<756i:<6p9�:<6p1�X<756#:24j9�CE6C2eP756i:<6oX<756p4#1'3qAq1'\DK(X<BD1'3r^5:<1,;5K54#X<C)@ v � @ º F @ º � @ � F� @ v � @ º � � @ � 9�35; @ v � � @ º � @ � �u=r9?>9�\D\a`�68;56iz%356#;aF%_�1�:P6il59�=)^H\D6ZBD_�BD3w69�4}7>41'3qAq1�\]K(X<BD1'3�^5:<1,;5K54#XP41'35CEBD;56#:<6#;w9�X\D6j9�CMX21'356|1'_aX<756|_�K53H4iX<BD1'35Cu7H9�Cu41�=W^%9�4iX2CEK(^5^�1�:EXF,`HK(XuePBgX<7y� @ v � @ º � � @ � � @ v � � @ º � @ � �[§V\D6#X�@ v �Ø��F�@ º � C � � �{�ePBgX<7 I���@ º �GQa�4OMQ9�*®(FH9�35;>\D6iX @ � `a6oX<756pno���,��QSUQ��V��vsºo_�K5354#X<BD1'3aF(;56#zH356;Y`I? @ � �GQa�H�*®f_�1�:�Q ��®)9�35;6@ � �GQa�H�Ø�_�1�:*Q �/®(tq1'356�CE6#6#C�B]=W=W6;5B]9�X<6#\D?pX<7%9�X @ v � @ º �*®p9�35;E@ � ��@ º � @ � � C � � �{�[FI9�3H;)1'356�7H9�CVX<1p4}756#4[b@X<7H9�X @ º 49�3@`a64[7H1�CE63�BD3YCEK54[7>9{e�9?cX<7H9�X I���@ � �GQa�4OMQ �*®(t5BD_ 1'356p4[751,1'CE6#C)@ º ePBgX<7YCEK(^H^�1�:EX�BD3! >8� � Ó@� #OF,ePBgX<7�I ��v" v @ º �+*5�4O&*��*®¦o35BgAq6#:<CEBgXs?Y1'_ 1 BDCE41�3HCEBD3aFH�~9�;5B]CE1�3 F5eP756i:<687H6Ze21�:Ebq6;��[F%9�C�0�7H9�;w7H69�:<;y9�`a1�K5X�X<756= 9�CP^%9�:[9�=c6iXE:<BD�6#;>=W6j9�CEK5:<6#CBD3>CE6=WBD3%9�:<C�1'3>41'3qXE:<1'\aX<756#1�:E?qm�~Gq1'7H9�3�N�KH;(ePBDk 1*B]\D\DB]9�= Ñu9�\D;56#=r9�:oGq��TVU��TdFa£Z9�35BDCE7�=c9�X<756=c9�X<BD4#B�9�3 F����'Ô'��L<�j��­'Ô,m{no6@3H6iAq6i:Z7H6#\D;�9�3q?~9�49�L

;56=WBD4p^�1'CEBgX<BD1'3aFH9�35;>e21�:Ebq6;>_�1�:|9fX<6\D6i^H7H1�3H6p41'=)^%9�3q?qm� 1 BD\D\DB�9�= ��a²���T��%�dF5�o=W6i:<B]49�3�=c9�X<756=c9�X<BD4#B�9�3 mVno6pe21�:Eb(C|9�XP¦o35BgAq6i:<CEBgXs?Y1'_V¢�6il59�C#F5�oK5CMX<B]3am�Ùd\D\DB]1�XEX�n@m5N�Q���¯�F5�o=c6i:<BD4j9�3�=c9�X<756#=r9�X<BD4#B]9�3amVno6pe21�:Eb(C|9�XP±-:<BD3H4#6iX<1'3>¦o35BgAq6i:<CEBDXs?'F5�oGHmv � no6#:<= GI9�3� �����U�²���� A�FH=c9�X<756=c9�X<BD4j9�\�^H7I?,CEB]4#BDCMXmvEv�ÐoK5B];51W��V¯�Q�T�Q�F50sX[9�\DB]9�3(LS`�1�:<3>=c9�X<756=c9�X<BD4#B�9�3 Fa� ��(���L<�j��Õ �Imdno686#=cBDk�:[9�X<6#;YX<1W¦o3HBgX<6#;��qX[9�X<6#C�BD3°�j���'�IF�9�3H;>7H6e21�:Ebq6#;wBD3>�o6ie �d1�:Eb�mvsº � \DBgAq6i:2no���,��QSUQ��V��F,Ùd35k'\DBDCE7Y6#3Hk�B]356#6#:F�� ��Ô�®�L<�j�'­�Ô,m-no6o;56#Aq6#\D1�^�6;Y9�3Y1�^�6i:[9�X<BD1'3H9�\�4j9�\D4#K5\]K5C#FIeP75BD4}7re�9�C�k'BgAq6#39f^5:<6#4BDCE6Z=r9�X<756#=r9�X<B]49�\�6il,^H\]9�3H9�X<BD1'3�`I?�N�9�K(:<6#3IXp�(²a³�´{���5��µ(m

�®

Page 11: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

9�35; I �Vv" v �E�8>)*5�:@ º �+*5�4O&* � ®(F(X<7563>1'35687H9�C@ � �GQa�H� I �" v @ º �+*5�4O,*%F%9�35; I ��@ � �GQa�4O Q9� I ��v" v �E��> *5�:@ º �+*5�4O,* �*®(m0�_{1'356~K5CE6C�1'35\D?/_�K5354#X<BD1'35CYB]3 ��v� ���� �{�y�SBSm 6�m�BD3qX<6k�:[9�`H\]6�1'3*6#Aq6i:E?/`�1'K535;56;*BD3qX<6#:EA�9�\��[F�eP75B]4[7�A'9�3HBDCE7*1'3

�:> , � ®I�[F(X<756#3YX<756p4#1'3qAq1'\DK(X<B]1�3r^5:<1,;5KH4iX�BDCue26\D\�;56izH3H6#;>9�C @ � KWA�9�35B]CE756#C�1'3��:> , � ®I�d9�35;�C<9�X<BDCMzH6CZ� @ � K5�}�GQa�H�I �� @��+*5� K��GQ�>�*H�MO,*~_�1�:�Q � ®(F�CE7H1ePB]35k>X<7H9�XW@ � K � ��v����� � �{�[mr�!X<7561�:<6= 1�_�¢�Q���²�³��p����U³dv � 9�CECE6i:EX<C8X<7H9�X{BD_X<756{CEK5^5^�1�:EXp1'_3@~CMX[9�:EX<Co9�X Nw9�35;yX<756{CEK5^5^�1�:EXp1'_BK�CMX[9�:EX<Cp9�X 2 F�X<7563wX<756@CEK(^5^�1�:EXo1'_ @ � K�CMX[9�:EX<Co9�X N{Ó 2 F�9�35;GI9�4JIK56CMLON�1'K5BDCpN�QSR�TVU)7H9�Cpk'6#356#:[9�\DBD�#6;wX<7HBDC|:<6#CEK5\DXoX<1 �)!{F�^5:<1jA(BD35kYX<7H9�XoX<7H6@4#\]1�CE6;~4#1'3qAq6il�7IK5\D\�1'_�X<7H6fCEK(^5^�1�:EX1'_ @ � KcBDC�6JIKH9�\�X<1fX<756W�+Aq6#4iX<1�:i�-CEKH= 1'_�X<756p4#\]1�CE6;�4#1'3qAq6il�7IK5\D\a1'_ X<7H6pCEK(^5^�1�:EXP1�_H@y9�35;�X<756p4\D1'CE6#;�4#1'3qAq6#lY7IK5\D\1'_�X<7H6ZCEK(^5^a1�:EX|1'_ K%m

�o3yBD=)^�1�:EX[9�3qX�^5:<1�^�6#:EXs?�1'_V4#1'3qAq1'\DK(X<BD1'3>B]C�X<7H9�XPBDXP41�=c=@K(X<6C�ePBDX<7�XE:[9�35CE\]9�X<BD1'3at5X<75B]CPBDCP1�_V41�K5:<CE6Z:<6\]9�X<6#;�X<1X<7568_�9�4iX�X<7H9�X�X<756ZN���¯��aU�¤����=W69�CEK(:<68BDC�BD3qA'9�:<B]9�3qX�`I?�XE:[9�35CE\]9�X<BD1'3am�o1�X[9�X<BD1'3a§V�H1�:|9@Aq6#4iX<1�:�N � �)!8FH9�35; @ � ��v� ��� � �)!p�[F � + @w;56351�X<6#C2X<7H6p_�KH354iX<BD1'3�;56#zH356;Y`I?�� � + @a�}�GQa� � @��GQ�> NH�_�1�:p9�\D=c1�CMX|6iAq6#:E? Q � �)!��+X<7H68k�:[9�^H7�1�_ � + @~BDCP1�`5X[9�BD356;w_�:<1'=�X<7H9�Xo1'_3@y`,?w9WXE:[9�35CE\]9�X<BD1'3w1�_VAq6#4iX<1�:@�GN � ®q�E�[m � _4#1'K(:<CE6'FH1'356Z7%9�C � - � � + @a�H� � +�&- @w_�1�:|9�\D\ N � 2 � �)!8m� ��%"-� j�D������ � §p�SB��u0�_�X<7H6{4#1'3qAq1'\DK(X<BD1'3w^5:<1,;5K54iXo1'_3@�9�3H;�K�BDC|;56#zH356#; F%X<756#3 � + � @ � K5� ��� � + @�� � KE�(@ � � � + K5�_�1�:P6#Aq6i:E?9N � �)!{m

�SBDB+�u0�_ � ®(F @ � C �� � � ! �[F�X<756fCM^%9�4#6f1�_-_�K5354iX<B]1�3HCo1�_-4#\�9�CECRC � ePBgX<7�4#1'=W^%9�4iXpCEK(^5^a1�:EXZ9�35;6K � � v��� � � � ! FX<75636@ � K � C��,� �)!p�29�35; ���� @ � K5�*�Ø�� E� @�� � KY_�1�:|9�\D\ ;56i:<BgA'9�X<BgAq6#C�1'_�\D635k�X<7 (���m�2È<ÆjÆ5#[§ � 3H6>7H9�C � � � + @a� � K � �GQa��� I � 1 � � + @a�}�+*5� K��GQ >�*5�4O&* �+I � 1 @��+*9> NH� K��GQ6>�*5�4O,*%FdeP75BD4[7�`,?�9~4[7H9�35k'61'_ZA'9�:<B]9�`H\D6�BD3*X<756~BD3IX<6#k�:[9�\oBDC I � 1 @��+*5� K��GQ<>5N6> *5�4O,*(� � @ � K5�}�GQ<>�NH�6� � � + � @ � K5���q�GQa�[F|CE751jePBD35k�X<7H9�X� � + @a� � KW� � + � @ � K5�[FH9�35;>X<7H9�XPBgXPBDCP9�\DCE1E@ � � � + K5�u_�1'\D\D1jePC�`I?�41�=c=@K(X[9�X<BgA(BgXs?r1'_�X<756Z41'3qAq1�\]K(X<BD1'3Y^H:<1I;HK54iXm0�_�( v���������� ( ! BDCdX<7H6|49�351'35B]49�\H`%9�CEBDC21�_ �)!{FIX<7563Y9{_�K53H4iX<BD1'3 (r7H9�C�98^%9�:EX<B]9�\%;56#:<BgA�9�X<BgAq6������� � 9�X Q�BD_ 9�35;Y1�3H\g?BD_ v�� (�> � ��� � (��-7H9�C�9@\DBD=cBgX29�X8Q�eP7H6#3 �ZX<635;5CuX<1)®��+ePBgX<7�� �*®(F(1'_�4#1'K(:<CE6��[m�0�_ @ � Cfv� � �)!Z�[FIX<7H6#3 v� � @�> � ��� � @��4#1'3qAq6#:<k�6CuK535BD_�1�:<=W\g?fX<1 ���

� � �9�3H;cX<756i:<6_�1�:<6PB]_a1�3H6�X[9�bq6#CdX<7H6P41'3qAq1�\]K(X<BD1'3W^5:<1,;5K54#XdePBgX<7Y98_�K5354iX<B]1�3�K � ��v� ��� � �)!p�[F

1'356)z%35;5C8X<7H9�X v�� @ > � ��� � @a� � K�4#1'3qAq6i:<k'6C8K53HBD_�1�:<=c\g?�1'3�41'=)^%9�4iX{CE6iX<C8X<1 ���� � � � K%tVBD_u1'356W;H6#351�X<6#C ( � @ � K%F

1'356r7H9�C v�� @�> � ��� � @�� � K � v

�� ( > � ��� � (��[FV9�35;hX<756#:<6#_�1�:<6WX<7H6r\DBD=cBgX8=fK5CMX{`�6 ������ � 9�35;hBgX@BDC{6jJIKH9�\-X<1 ������ � � K%m� :<6#BDX<6i:[9�X<BD1'3y1'_�X<75BDCp9�:<k'K5=W63qX)�O9�Xp=W1'CMX��wX<BD=W6C|BD_ @ � C����� � � �"!p�E��k�BDA'6C|X<7H6#3 ���� @ � K5���!�� ��P@a� � Kw_�1�:89�\D\

;56#:<BgA�9�X<BgAq6#C�1'_�1�:<;H6i:�� ��� (��%m0�_ @ � C�� � �)!Z�u�+eP75B]4[7WBD3)X<756�351�X[9�X<BD1'35C-1�_�N�9�K(:<6#3IX��(²�³�´8���5��µZB]C-;56#3H1�X<6;)`I?��� �)!p�E�[F'X<7563E@ � K{`a6#\D1'35k'CX<1 C��I� �"!p�8_�1�:W9�\D\ ��F-BSm 6'm @ � K � C �� �)!p�c�+eP75BD4}7hBD3hX<756Y351�X[9�X<BD1'35C@1'_PN�9�K(:<63qXc�(²a³�´{���5��µ�BDCf;56#3H1�X<6;h`I?

� �)!p�E�[mw�oC8ePBD\D\�`a6cCE751jeP3¶\]9�X<6i:F�X<756i:<6Y9�:<6c6351'K5k'7h_�KH354iX<BD1'35C@BD3 C � � �)!Z�p_�1�:f9�^5^H:<1'9�4}75BD35k~9�3q?�_�K5354#X<BD1'3°BD3���5� �)!Z��_�1�:W� ( *�� ,�Fa`HK(Xu�EK5CMXZ1'356f^%9�:EX<BD4K5\]9�:p_�KH354iX<BD1'3~BD3 C � � �)!p�P7H9�CpX<1r`�6)4#1'35CMXE:<KH4iX<6#;�6il,^H\DB]4#BgX<\g?qF�9�35;X<756Z^5:<1�^�6i:EX<BD6C�1'_�4#1'3qAq1'\DK(X<BD1'3�ePBD\]\�756\g^w_�1�:�X<7H6p:<6#CMXP1'_�X<75689�:<k'K5=c6#3qXm� & ����$ §d¢�7568_�KH354iX<BD1'3��Y;56#zH356;w1'3 �)!x`I?����GQa�3� (�Q * � " vv " � � � � �{BD_)� QH� ���{9�35;����GQa�*� ®)BD_)� QH�4 x�p`�6\D1'35k'C�X<1C � � �)!p�[m�2È<ÆjÆ5#[§d0sX|BDCP351'353H6#kq9�X<BgAq6�F�9�35;y41'3qX<BD3IK51'K5CP`�649�K5CE6{B]_)� QH� � �ZePBgX<7 � QH� �x��F%X<7563 " vv " � � � � � > , 9�35;����GQa����®,t1�`IA(BD1'K5CE\g?��/7H9�C>_�1�:�CEK5^5^�1�:EX�X<7H6�4\D1'CE6#;*KH35BgXY`�9�\]\Sm � 3H6�7H9�C ���

� � �� " º � �� v " � � � � � ���%Fp9�3H;�`I?�BD35;5KH4iX<BD1'3� �� �(�� -

� � �� v " � � � � � ��� - � �~_�1�:@9�^�1'\g?(351'=WB]9�\�� � F�9�35;�X<756#:<6#_�1�:<6)eP756#35� QH� � �fePBgX<75� QH� � ��F�X<756W6#lI^a1�3H6#3qX<B]9�\VePBD35CZ1jAq6i:8X<7H6X<6i:<= v� v " � � � � � ��� - � 9�35;�X<756i:<6_�1�:<6 E��� � ®(F�CE751jePBD35k°X<7H9�XY9�\]\�;56#:<BgA�9�X<BgAq6C)1'_���9�:<6y4#1'3qX<BD3,K51'K5CY9�35;�X<756i:<6_�1�:<6� � C � � �)!Z�[m

v � Ùd;(e�9�:<;>¡u7H9�:<\D6Cu¢�Q���²a³��Z���VU#³uFqÙd35k'\DB]CE7c=c9�X<756#=r9�X<BD4#B]9�3aF%���'�'��L<�j��Ò��,m�no6P756\D;WX<756p�,9A(BD\DB�9�3c4[7H9�Bg:u1'_�Ðo61'=)L6iXE:E?>9�X � l(_�1�:<;aF5X<7%9�XPno����� �¶7H9�;w1,4#4K(^HBD6;Y`�6_�1�:<68\D69A(BD35k)_�1�:p¡29�={`5:<BD;Hk�6'm

�IBD:Pno6#3(:E?w�,�,��Q�¥,��FHÙd35k'\DB]CE7�=r9�X<7H6#=c9�X<BD4B]9�3aFa�jÔ�ÕI��L<�jÒ'­�­,m

�'�

Page 12: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®A�,m-�5:<BD;H9?�GI9�3IKH9�:E?y­(��F%­�®'®'®,m� 354#621'35627H9�CV4#1'35CMXE:<K54#X<6#;f1�3H6u351'35�6i:<1p_�KH354iX<BD1'3@BD3WC � � �)!Z�[F�4#1'3qAq1'\DK(X<B]1�3{ePBD\]\I756#\D^@4#:<69�X<BD35kp9�K(X<1'=c9�X<BD49�\D\D?9c\D1�XP1'_VCEK54[7�_�K5354iX<B]1�3HC#FH6351'K5k'7y_�1�:p9�^5^5:<1q9�4[75B]35kr9�\D\�_�K53H4iX<BD1'35CPB]3 � � � � ! �2_�1�:f�"( * � ,�m2�%1�:p;51'BD35kWX<75B]C#F5X<7H6

4#1'3546i^5XP1'_V9)CE=W1,1�X<75BD35k)CE6jJIK56#35468BDC�K5CE6_�K5\Om� &������D������ §���CE=W1,1�X<75BD35koCE6JIK56354#6uBDC�9|CE6JIK56#3H4#6 � � � C � � �)!p��CEK54[7@X<7H9�X�- � * *�. L�%i� � � �#����®��qFMI � 1 � � � �GQa��� OMQBDCZ`�1'K53H;56#;¶9�3H; I � 1 � � �GQa�4OMQ � ��mr��CM^�64#B]9�\-CE=W1,1�X<75BD35kwCE6jJqKH6#3546WBDC{;56iz%356#;�`I? � � �GQa� � � ! � v � � Qa�PeP756#:<6� v � C�� � �)!p�uBDC�3H1�3H356#kq9�X<BgAq6�FH7H9�C�B]3qX<6#k�:[9�\��89�35;w7H9�CPBgX<C�CEK(^5^�1�:EXPBD3>X<756Z4\D1'CE6#;>K53HBgX�`%9�\D\ 4#63qX<6i:<6;w9�X|®(m

�qX[9�:EX<BD3Hk�_]:<1'= 9�3�9�:E`HBgXE:[9�:E?y351'35�6i:<1>_�KH354iX<BD1'3 � � C � � �)!Z�[F 1�3H6f=r9#?~9�CECEK5=c6fX<7H9�XZBgX8BDCp351'35356kq9�X<BDA'6f`I?:<6i^%\]9�4#BD35kyBgX@6#A'63qX<KH9�\D\g?~`I? � º F�X<7H9�X@BgXf7H9�C{BDX<C{CEK(^5^�1�:EXfBD3°X<756r4#\]1�CE6;°K53HBgX{`%9�\D\d4#63qX<6i:<6;h9�X)®>`I?�:<6i^H\]9�4BD35kwBDX6iAq63qX<KH9�\D\g?Y`I?�d� � Qa�2_�1�: �Y\�9�:<k'686#351'K5k'7aF�9�3H;>X<7H9�X|BgXP7%9�C|BD3qX<6#k�:[9�\-�Z`I?>=fK5\gX<Bg^%\g?(BD35kfBgXP`I?w9)CEKHBgX[9�`H\D684#1'35CMX[9�3qXm¢�75BDCpk'BgAq6#Cp9Y_�K53H4iX<BD1'3 � v FaeP75BD4[7~BDC|X<7H6#3�:<6#CE4j9�\]6#;y`I? � � �GQa�8� � ! � v � � Qa�[F�CE1rX<7%9�XpX<756fBD3qX<6#k�:[9�\�1�_ � � BDC)��F 9�35;BgX<C�CEK(^H^�1�:EXPBDC�BD3�X<75684#\D1'CE6;�`%9�\D\ 4#6#3IX<6i:<6#;y9�X|®fePBgX<7�:[9�;5BDK5C v� m� ��%"-� j�D������ §{�SB��20�_�� ( * � ,/F @ � ���5� �)!Z��9�35; � � BDCp9cCE=W1,1�X<75BD3HkcCE6jJqKH6#3546�F�X<756#3 @ � � � � @~BD3+���5� �)!p�CMXE:<1'35k5m

�SBDB+�80�_ @ � � >� �)!p�f9�35; � � B]Cc9�CE=c1I1�X<75B]35k�CE6jJIK56#3546�FuX<7563 @ � � � � @/BD3 � �� �)!p�@e269�b � F�9�35;�BD3� 8��� �� �"!p�uCMXE:<1'35kW_�1�:@��( 5 �-,�FHBSm 6�m�_�1�:P6iAq6i:E?�41�=W^%9�4iX���1'35687%9�CI ! � @ � � � > @ � 8 O Q �ۮ,m

�2È<ÆjÆ5#[§�¢�756i:<6�6il(BDCMX<C�9pCE6jJqKH6#3546 @ � � C � � �)!p��eP7HBD4[7)4#1'3qAq6i:<k'6CVX<1�@cBD3 ���5� �)!p��CMXE:<1'35k5m � 3562e�:<BDX<6#C3@ � � � > @ �� @�> @ � � � � � Óx� @ � � � � > @ � �VÓx� @ � > @��[F�CE1wX<7H9�X{BD_21�3H6c4[751,1'CE6#C�ÖCEK54}7�X<7H9�X � � @ >,@ � � � � ( �qFV9�35;°BD_8CBDC�9°`�1'K53H;�_�1�:�9�\D\|X<756+��vj� �)!Z�W351�:<=WCY1'_ � � F|1'356�7H9�C!� �¹� @ >�@ � � � � � � � � ( � � @ > @ � � � � � � � � � � v ( C �qF|9�35;X<756#:<6#_�1�:<6!� � @ � � � >�@ � � � ( C �8Ó � � @ � � � � >�@ � � � � Ó �qF29�35;�BgX):<6#=r9�BD35CfX<1�CE751je X<7%9�XW_�1�: zHl,6; @ � � � �4#1'3qAq6#:<k�6C|X<1 @ � BD3+���5� �)!Z��CMXE:<1'35k�9�C � X<635;5C|X<1YBD3(zH3HBgXs?'mP0�_ I�� 1 � � �GQa�MOMQ6��� � F�1'356@e�:<BgX<6#CR@ � � � � >$@ � �� @ � � � � >� � @ � ��ÓØ�� � > ���:@ � F-9�35;h`�6#4j9�KHCE6�� � � �cX<756rCE64#1'35;h^%9�:EX@X<6#3H;5C{X<1�®yB]3 ���5� �)!Z�ZCMXE:<1�3Hk(F-9�35;`�649�K5CE6p1�_ X<756pK535B]_�1�:<= 4#1'3qX<BD3IKHBgXs?c1'_ @ � F,X<7H6|z5:<CMX�^%9�:EX2X<635;5C2X<1)®@K535B]_�1�:<=W\D?'F(9�35;YX<756#:<6#_�1�:<6oBD3����5� �)!Z�-CMXE:<1'35k`�649�K5CE6{BgX<C�CEK5^5^�1�:EXoCMX[9#?(C�`a1�KH35;56#;�9�C|BgXPBDC|BD354#\]K5;56#;wBD3�- � * */. L�%i� @ � ��Ó�- � *�*�. L�%i� � � �[t%BD35;566#;aF%BD_�- � *�*�. L�%i� � � �2BDCBD354\DK5;56;~BD3�X<756@`%9�\D\�46#3qX<6#:<6#;�9�X8®rePBgX<7�:[9�;5BDK5C � � F�9�3H; � � BDC|X<7H6f=c1I;HK5\DK5Cp1'_-K535BD_�1�:<= 4#1'3qX<BD3,K5BgXs?y1'_ @ � F 1�3H67H9�C �� I�� 1�� @ � �GQ�>*5�B> @ � �GQa��� � � �+*5�4O,* �� ( I�� 1 � � � �+*H��� O,*u=c9�l � ! � $ � % � @ � �GQ�>*5�B> @ � �GQa��� ( C�� � � � � �[m�%1�:W@ � � � � ! �[F�1'356We�9�3IX<CZX<1>CE7H1e�X<7%9�XW@ � � � 4#1'3qAq6i:<k'6CZX<16@°BD3 � � � ! �|e269�b � F�9�35;�X<7HBDCZ=c69�35CX<7H9�Xu_�1�:26iAq6i:E?�K � ��vj� �)!p��1'356|7H9�C)I�� 1 � @ � � � �}�GQa� K��GQa�MOMQ�� I � 1 @��GQa� K��GQa�4OMQ�m � 356|351�X<BD46#CdX<7H9�XFq`I?c��V¯�Q�T�Q × CX<7561�:<6=>F51�3H6p7%9�C I � 1 � @ � � � �}�GQa� K��GQa�4O QE� I � 1�� � 1 @��+*H� � � �GQ >�*H� K��GQa�MOMQ O&*���I � 1 @��+*5�}� K ���� � �}�+*5�4O,*%F(eP756#:<6�� � �+*5�8� � � �:> *H�[F�CE1rX<7H9�X �� � BDCp9cCE=c1,1�X<7HBD35krCE6jJqKH6#3546@9�35;�`I?wX<756{z5:<CMXo^�9�:EX K ���� � 41�3IA'6#:<k'6#C|X<1 K>BD3+��vj� �)!p�CMXE:<1'35kf9�35;YX<756i:<6_�1�:<6RI � 1 @��+*5�}� K ���� � �+*H�MO,* � I � 1 @��+*5� K��+*5�4O&*�mV0�3Y1�:<;56i:2X<1@CE751je*X<7%9�X I ! � @ � � � > @ � 8 OMQ �·®_�1�:p9c4#1'=)^�9�4#X �Ö9�3H;h� ( 5 ��,�F%1'356{351�X<BD46#C�X<7%9�X|X<756{BD3qX<6k�:[9�\�1'35\g?>K5CE6CPA�9�\DKH6#CP1'_3@~BD3y9W`%9�\D\�46#3qX<6#:<6#;�9�Xp®ePBgX<7c:[9�;5B]K5C � � \]9�:<k'6|6351'K5k'7~�SCE1{X<7%9�XuX<7H6|`%9�\D\%41'3qX[9�BD35C � 9�35; � Ó)- � *�*�. L�%i� �� � �E�[F(9�35;YX<756i:<6_�1�:<6|BD_��@�4#1'BD354BD;56CePBgX<79@rB]35CEBD;56�X<7H6�`%9�\D\H46#3qX<6#:<6#;Y9�Xu®8ePBgX<7):[9�;5BDKHC � � 9�35;rBDC2®Z1'K(X<CEBD;H6PBDXF'X<7563WX<756|BD3qX<6#k�:[9�\HB]C I ! ���@ � � � >��@ � 8 O Q�FeP75BD4}7�;51,6#C�4#1'3qAq6#:<k�6pX<1)®@`a6#49�K5CE6 �@ � � � 4#1'3qAq6i:<k'6C2X<1 �@wBD3�� 8 � �)!p�[FH9�C�9f4#1'35CE6JIK56354#6Z1�_�X<756oz5:<CMX�^%9�:EX|9�3H;�1�_X<7568_�9�4iX�X<7H9�X �@w`�6#\]1�3Hk�C�X<1 � 8 � �)!Z�[m

� _�4#1'K(:<CE6�F @ � � � `�6\D1'35k'C8X<1 C � � ! �89�35; � � @ � � � ��� @ � �� � � � �p_�1�:)9�\D\d=@KH\gX<BgLOBD35;HBD4#6C��dFV`HK5X{X<7H6CEK(^5^a1�:EX|1'_H@ � � � BDC�351�XP41'=)^%9�4iXPBD3>k'6#3H6i:[9�\Sm�{%���.+.�$H��� §-�H1�:{�)(+* �-,/FHX<756ZCM^%9�46�C � � �)!p�uB]C�;56#35CE68B]3 ���5� �)!p�[m C � � �)!p�uB]C�e269�b � ;H6#35CE68BD3 ��>� �)!Z�[m�2È<ÆjÆ5#[§�0�_W� ( * � ,�F @ � � � � � C � � � ! �f`�649�K5CE6 @ � 7H9�CY4#1'=)^%9�4#XrCEK(^5^�1�:EXm 2649�K5CE6 @ >�@ � � � � �� @6>5@ � �dÓ � @ � >�@ � � � � �[F-X<756w9�:<k�KH=W6#3IXWK5CE6#;�BD3¶X<756�z5:<CMXW^%9�:EXW1'_PX<756>±-:<1�^�1'CEBgX<BD1'3�CE751jePCfX<7%9�XWX<756i:<6w9�:<6CE6JIK56354#6C�

�9�3H; � � CEK54[7wX<7H9�X @ ��� � � � � 4#1'3qAq6i:<k'6#C�X<1E@yBD3 ���H� �)!p�uCMXE:<1'35kr9�C � � ,�m

0�3�X<756)4j9�CE6E@ � ��>� �)!Z�[F 1'356W;56#zH356CK � `,?6K � �GQa� ��@��GQa�PBD_R� QH� (�Ö9�35; K � �GQa� ��®�B]_� QH� ���F�9�35;9�CR � , K � 41'3qAq6i:<k'6#CZX<1 @°BD3 ��>� �)!Z�Pe26j9�b � 9�35; � 8 ��� � � �)!Z�oCMXE:<1'35kw_�1�:c��( 5 � ,�t�1'356W41�3H4#\DK5;H6#CZ`I?351�X<BD4BD35kcX<7H9�Xp_�1�:�z5l(6#; K � � � � 4#1'3qAq6i:<k'6C|X<1 K � BD3+� �� �)!Z��e269�b � 9�35; � 8 ����� � �)!Z��CMXE:<1'35kY_�1�:)� ( 5 � ,�O9�35;�X<75B]CP9�:<k'K5=c6#3qXPKHCE6#C�X<7568_�9�4#X�X<7H9�XP1'3�`�1'K535;H6#;wCE6iX<CP1'_�� >� �)!p�dX<7H6pe269�b � X<1�^�1'\D1'k�?�BDC�=W6#XE:<BD�9�`H\D6��[m

�j­

Page 13: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®�Õ5m-��1'35;H9?�GI9�3IKH9�:E?y­�Õ5F�­�®'®'®(m�%1�:8:<6j9�CE1�3HCZeP75B]4[7�ePBD\D\�`�6#41'=W6)=c1�:<6W1�`IA(BD1'K5C8\]9�X<6#:F BgX8BDC8K5CE6#_�K5\�X<1>;56#zH356WB]3�9�=c1�:<6)k'6356i:[9�\�CE6iXEX<B]35kYX<7H6

XE:<K5354j9�X<BD1'3�CMX<6#^wKHCE6#;>^5:<6iA(BD1'K5CE\D?'m� &������D������ §-�xXE:<K53549�X<BD3HkfCE6jJqKH6#3546pB]C�9)CE6JIK56#3H4#6 � � � C � � �)!p�dCEK54}7�X<7H9�X � � �GQa��� �Z_�1�:�9�\D=c1'CMX�6#Aq6i:E?EQ�F � �BDC�`a1�KH35;56#;yB]3 � >� �)!Z�[F%9�35;y_�1�:|69�4}7w=@KH\gX<BgLOBD35;H6il9�¶ePBgX<7$� ��� x�ZX<756{CE6JIK56#3H4#6� �� � � X<6#35;HC�X<1r®WBD3 � >� �)!p�CMXE:<1'35k5m|� CM^a6#4#B�9�\�XE:<KH3549�X<BD35krCE6JIK56354#6@BDCo;56iz%356#;y`,? � � �GQa� � � v � �� � eP756#:<6 � v � C � � �)!p�[Fa®�( � v �GQa� ( �@_�1�:9�\D\ Q�FH9�35; � v �GQa�H� �Z_�1�:W� QH� ( ��m

¢�7H9�X|CEK54[7�9 � v 6il(BDCMX<C�_�1'\D\D1jePC�69�CEBD\D?Y_]:<1'=�X<756Z:<6#k'K5\]9�:<BD�j9�X<BD1'3Y`I?�41�3IA'1'\DK5X<BD1'3aF%9�35;w=W1�:<6p^H:<6#4#B]CE6#\g?Y1'35687%9�CX<7568_�1'\D\D1jePBD3Hk@:<6CEK5\gXm� & ����$ §PN�6iX8® � N � 2 � 4 FaX<7563�X<756#:<6f6il(BDCMX<C � � C�� � �)!p��ePBgX<7�® ( � �GQa� (��)_�1�:89�\D\ Q�FaePBgX<7 � �GQa��� �fBD_� QH� ( N�F � �GQa�H�*®fB]_�� QH�4 4 F%9�35; I � 1 � �GQa�MOMQE� I � � � $ - OMQ�m�2È<ÆjÆ5#[§WN�6#X @¶`�6rX<756Y4[7%9�:[9�4#X<6i:<BDCMX<BD4r_�K5354iX<B]1�3¶1'_�X<756r`%9�\D\u4#63qX<6i:<6;¶9�XW®wePBDX<7°:[9�;HBDK5C 2 td\]6iX �>C<9�X<BDCE_�?°®�� � �=WB]3 � 2 >,N � 4 > 2 �qF-9�35;�\D6#X � � � C�� � �)!p�Z`�6Y351'35356kq9�X<BDA'6'FVePBgX<7 I�� 1 � � �GQa�4O Q � ��9�35;¶CEK(^5^a1�:EX)B]3°X<756r`%9�\D\4#63qX<6i:<6;Y9�X�®8ePBgX<7c:[9�;5BDK5C �qmd¢�756#3 � ��@ � � � C<9�X<BDCMzH6#C29�\D\%X<756o;56#CEBg:<6;r^5:<1�^�6i:EX<BD6C#FqX<7H6|\]9�CMX21'356o4#1'=cBD35k{_�:<1'=�X<7H6_+9�4#X�X<7H9�XP_�1�:�Xse21W_�K5354iX<B]1�3HC @ � K � ��v� �)!Z�[F51'35687%9�CI�� 1 � @ � K5�}�GQa�4OMQ9� � I�� 1 @��GQa�4O Q � � I � 1 K��GQa�4OMQ ��m

� _�41�K5:<CE6�F�BD_�( � ���(� �)!Z�P9�35; � � BDCp9cXE:<K5354j9�X<BD35krCE6JIK56#3H4#6�F ( � � 4#1'3qAq6i:<k'6Cp9�\]=W1'CMXo6#A'6#:E?,eP756i:<6{X<1 (~9�35;BDC8`a1�KH35;56#;°`,?!CE� ( �¹FV9�3H;°X<756i:<6_�1�:<6c`I?�N���¯���U�¤��� × C{;H1�=cBD3H9�X<6#;°41�3IA'6#:<k'6#3546WX<7561�:<6#=>F�( � � � (hBD3 ���5� �)!p�CMXE:<1'35kWB]_u��(+* �-,�F%9�35;wBD3 � >� �)!Z�de269�b � 9�35; � 8 ��� � � �)!p�uCMXE:<1'35kc_�1�:{�)( 5 � , B]3�X<756Z4j9�CE6&* ��,�m

�o_�X<6i:dX<756CE6�^H:<6#\DBD=cBD3H9�:<BD6C#F'1'356|;56izH3H6#CuÚP���-R�T~=W6j9�CEK(:<6#C29�3H;r;5BDCMXE:<Bg`%K(X<BD1'35C-1'3Y9�3r1�^a6#3rCE6iX'��1'_ �)!*BD3WX<7H6_�1�\]\D1jePBD35kfe�9#?qm� &������D������ §VB����/ÚP���-R�Ty=W6j9�CEK5:<6 �>B]3 �¶BDC�9p\DBD3H69�:�_�1�:<= ;56#zH356;)1'3 C � ���P�d�+X<756�CM^%9�4#6�1�_%41'3qX<BD3IK51'K5C-_�K5354iX<B]1�3HCePBgX<7w4#1'=W^%9�4iX|CEK(^5^�1�:EX|BD3+�P�[F � �� ��� � ���[FHCEK54[7wX<7%9�Xo_�1�:|6#A'6#:E?�41�=W^%9�4iX � � ��X<7H6i:<6{6il(BDCMX<C|9W4#1'35CMX[9�3qXCW�����ePBgX<7 ����� � ����� ( C)������� � ��� � _�1�:u9�\D\ � � C � ���P��ePBgX<7 - � * *�. L�%i� ����� �hm � 356�e�:<BgX<6C�� ��� ���P�[F,9�35;WX<756P6#\D6=W6#3IX<C1'_ C � ���P�29�:<6Z4j9�\]\D6#;YX<6CMXP_�K53H4iX<BD1'35C#m

BDB+�2��;5BDCMXE:<Bg`%K(X<BD1'3 � B]3+��BDCp9r\DBD356j9�:o_�1�:<= ;H6izH356;�1'3 C�� ���P�8�+X<756@CM^%9�4#6@1'_3C _�KH354iX<BD1'35C|ePBDX<7y4#1'=)^%9�4#XCEK(^5^a1�:EX�BD3"�P�[F"� �� � � � ���[FjCEK54}7ZX<7H9�X�_�1�:�6#Aq6i:E?o41'=)^%9�4iX � � �~X<756i:<6�6#l,B]CMX<C�9241�3HCMX[9�3IXBCW������9�35;{9�351'35356k'9�X<BgAq6BD3qX<6k'6i:�������oePBgX<7�� � � � ������( C)�����,=c9�l � � � $ � � � � � � �� ��� � _�1�:)9�\]\ � � C � ���P�oePBgX<7�- � * *�. L�%i� ��� � �°m � 3H6e�:<BgX<6C�� � � �O���P�[FH9�35;>X<756Z6\D6#=c6#3qX<C�1�_HC � ���P�29�:<6849�\D\D6;YX<6#CMXP_�K5354#X<BD1'35C#mBDB]B��p0�_�1'356�49�3¶X[9�bq69������#� � _�1�:W9�\D\24#1'=)^%9�4#X � � �oF�X<7563hX<7H6Y;5BDCMXE:<BD`HK(X<BD1'3hBDCfC<9�BD;¶X<1~7H9Aq6�1�:<;H6i:( � m

ÚP���-R�Th=c69�CEK(:<6CP9�:<68;5BDCMXE:<Bg`%K(X<BD1'35C#t(X<756#?�9�:<686il59�4iX<\g?rX<756Z;HBDCMXE:<Bg`HK(X<B]1�3HC�1'_�1�:<;56#:"(�®(m� v��� ����P� ;H6#351�X<6#C�X<756�CM^%9�4#6�1'_%\D1,49�\D\g?8BD3qX<6k�:[9�`%\D6u_�K5354#X<BD1'35CVBD3 �oF'BSm 6�m�X<7H6p�S6JIK5BgA'9�\D6#3546d4\]9�CECE6#CV1'_}� N���¯��aU�¤���

=W6j9�CEK(:[9�`H\D6r_�K5354#X<BD1'35C{eP75BD4[7¶9�:<6YBD3qX<6k�:[9�`%\D6c1'3¶6iAq6i:E?°4#1'=W^%9�4iX � � �ot-BgXfBDC@351�X)9 ���T���²a³*CM^�9�46r`HK(XfBDX@BDC9w�V� ��a²�³��%�/CM^%9�4#6'vY�SBSm 6�mYBDX{BDC{\D1,49�\D\D?�41�3IA'6#l�F�=c6iXE:<BD�j9�`%\D6W9�35;h4#1'=W^H\D6iX<6��[F�9�3H;h9>CE6jJIK56#35469@ � 41�3IA'6#:<k'6#C{X<1y®BD3 ��v

��� ����P�PBD_d9�35;�1'35\D?yBD_-_�1�:Z6iAq6#:E?y4#1'=W^%9�4iX ��� �x1'356)7H9�C�I ! � @ � �GQa��� OMQ � ®(m � 356WBD;56#3IX<BgzH6#CZ9�3q?�_�K5354iX<B]1�3

@ � � v� ��� ���P�|ePBgX<7h9>ÚP���-R�T*=W6j9�CEK5:<6y�O9�3H;�X<7H6i:<6#_�1�:<6)ePBgX<7h9>;5BDCMXE:<BD`HK(X<BD1'3��[F�eP75BD4}7°1�3H6cK5CEKH9�\D\g?�9�\DCE1>e�:<BDX<6#C#@ F;56#zH356#;¶`I?�X<756Y_�1�:<=fK5\]9 ��@ � ����� I J @��GQa���d�GQa�4OMQ¶_�1�:W9�\D\ � � C � ���P�[my0sXfBDC@351�X@:<6j9�\D\g?�CEK54}7�9yk'1,1I;¶351�X[9�X<BD1'3aF`�649�K5CE6|BgXd:<6#\DB]6#CdK(^�1'3r7H9A(BD35k@CE6#\D64iX<6#;WX<7H6PN���¯��aU�¤����=W6j9�CEK(:<6 O Q>9�35;rBDX-e21'K5\];W`�6P`�6#XEX<6i:dX<1{4j9�\]\5X<75BDCu=W6j9�CEK5:<6�S1�:o;5B]CMXE:<Bg`HK(X<BD1'3�� @)OMQ�t�X<75BDCp9�`%K5CE6f1'_�351�X[9�X<BD1'3yBDCo1'_�351Y4#1'35CE6jJIK56#3546@_�1�:o1�^�6#3~CE6iX<Co1'_ �"!8Fa9�35;~41�:E:<6CM^�1'35;5C|X<1X<756oK5CEK%9�\aBD;563qX<BgzH4j9�X<B]1�3r1'_�� º ���P�VePBDX<7rBgX<C2;5KH9�\SF,`HK(X2eP7H6#3Y1'356p;56j9�\]CdePBgX<7�9@;5Bg �6#:<6#3qX<B]9�`H\]6P=r9�35BD_�1�\];r1'356oCE7H1�KH\D;:<6#=c6#=@`�6i:�X<7H9�X�X<756i:<6ZBDC�351W^5:<6#_�6i:<6;�Aq1'\DK5=W6Z_�1�:<= \]Bgbq6OMQ�m

0�_BN � �oF5X<7568£pQ�����²~=r9�CEC|9�X)N�F(eP75B]4[7>BDC�;H6#351�X<6#;�� + FHBDC�;H6izH356;>`I? ��� +4� ���H���d�GNH�d_�1�:|9�\D\ � � C � ���P�[FH9�35;BgX8BDC{9�ÚP����R�T�=c69�CEK(:<6c9�35;�X<756#:<6#_�1�:<6W9�;5BDCMXE:<BD`HK(X<BD1'3amf0�_29�CE6jJIK56#3546�N � � � 4#1'3qAq6i:<k'6#CZX<1>X<756)`�1'K535;H9�:E?� �v ¢�756-X<6#:<= ���T���²�³�CM^%9�4#6-e�9�C�BD3IXE:<1I;HK54#6;Z`I?8��� ���²a³�����tj0a;51P3H1�X�b(351jeheP751|BD3qXE:<1,;5K546#;ZX<756-X<6i:<=x��� ���²a³����

CM^%9�4#6'm

�@�

Page 14: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

1'_ � 9�35; 4 � BDCP9�3�9�:E`%BgXE:[9�:E?�CE6JIK56354#6'FHX<756#3 � � � � 4 � � + % BDC|9WÚ|���-R�T¶=c69�CEK(:<68BD3+��`�649�K5CE6{BD3>X<7H68_�1�:<=@KH\]9��� � ���H� � � 4 � �u�GN � �[FH1�3H\g?�9fz%35BgX<6Z3IK5=@`�6i:|1'_TN � `�6\D1'35k)X<1fX<7H6Z4#1'=W^%9�4iX�CEK5^5^�1�:EX��»1'_#�-m±d7q?(CEBD4BDCMX<C�K5CE6pX<756Z351�X[9�X<BD1'3 �(�GQ�> NH�dBD35CMX<6j9�;>1'_ � + FH9�35;�X<7H6i?Y;56izH3H6 �(�GQa�u9�C�X<756�«<_�KH354iX<BD1'35¬feP75BD4}7�BDC�®)_�1�:Q �*®f9�35;>7H9�C�BD3qX<6#k�:[9�\V��t51�_�4#1'K(:<CE6oX<7H6i:<6pBDC�351fCEKH4[7>_�K53H4iX<BD1'3�9�35;>BgX�BDC�9�4iX<KH9�\D\g?r9f=W6j9�CEK5:<6�F,`%K(XP9�_�X<6i:�CMX<K5;(?(BD35k

9�`�1'K(Xu=W6j9�CEK5:<6#Cu9�35;r;5BDCMXE:<BD`HK(X<BD1'35C-1'356|\D6j9�:<35C-eP75BD4[7c_�1�:<=@K5\�9�Cu9�:<6P:<BDk'7qX29�35;WeP7HBD4[7c1'356Cu9�:<6�e�:<1'35k@9�35;r1'356|49�3X<7563�;564#BD;H6ZJIK5BD4<b(\g?rBD_�9)_�1�:<=@KH\]9@KHCE6#;�`I?Y^H7q?(CEBD4#B]CMX<C�49�3�`�6p^5:<1jAq6#;>6j9�CEB]\g?'F(1�:�BD_�BgX�BDC�9WJqKH6#CMX<BD1'3H9�`H\D6p1'356�F56BgX<756#:`I?cCE751jePBD35k{X<7H9�X2BgXuBDCd_+9�\DCE6|1�:u`,?W3H1�X<B]4#BD35kZX<7%9�X2=r9�X<756#=r9�X<B]4#B]9�35C-;51@351�Xdb(351je�?q6iX2751je�X<1{=c9�bq6|CE6#35CE6o1'K(X21�_�X<7H6_�1�:<=r9�\a41�=W^HK(X[9�X<BD1'35C�K5CE6#;>`I?Y^H7q?(CEBD4BDCMX<C�BD3�X<7%9�X�^�9�:EX<B]4#K5\]9�:P49�CE6�m

� 356r49�3h4#:<69�X<6r9y\D1�X@1�_�;5BDCMXE:<Bg`%K(X<BD1'35C8`I?�X[9�b(BD35ky;56i:<BDA�9�X<BgAq6#C#FV9�35;hBgXfBDC{1'356r1'_2X<7H6cBD=W^�1�:EX[9�3qX8^H:<1�^a6i:EX<BD6C1'_P;5BDCMXE:<BD`HK(X<BD1'35C@X<7H9�XfX<756#?h7H9A'6>9�C)=r9�3q?h;56#:<BgA�9�X<BgAq6C@9�C)1'356Ye�9�3qX<C#F29�3H;�9�C)\]1I4j9�\D\g?�BD3qX<6k�:[9�`%\D6Y_�K5354#X<BD1'35C)9�:<6�S=W6j9�CEK(:<6#CP9�35;>X<756#:<6#_�1�:<6��d;5B]CMXE:<Bg`HK(X<BD1'35C#F51'35687H9�C�X<7563w9fe�9#?�X<1W;56izH3H6pX<756Bg:P;56i:<BDA�9�X<BgAq6#C#m� &������D������ §80�_8�5� ��� v���������� � ! �oB]C89>=@K5\DX<BgLOBD35;56#lyePBgX<7<���9 �®�_�1�:" � � ���������'& F�X<7563�_�1�:@9�3q?�;5BDCMXE:<BD`HK(X<BD1'3� � � �S���P��1'356�;56iz%356#CVX<7562;5BDCMXE:<BD`HK(X<BD1'3# E� �h`,?{X<7562_�1�:<=@K5\]9 � �� � � ���H�Ø�:>Z��� � � � ��� � �� ����_�1�:-9�\D\ � � C�� ���P�[m

� 356w=fK5CMXczH:<CMXr4[7564<b�X<7%9�XE �� � ;56#zH356#;/BD3�X<75BDC)e�9?�BDCr9�;5BDCMXE:<Bg`HK5X<BD1'3aFuBSm 6�m*_�1�:Y9�3q?�4#1'=W^%9�4iX���� �1'356�=fK5CMXY`�1'K53H; ��� �� � � �����V_�1�:$� � C � ���P�fePBgX<71- � * */. L�%i� �V�� �hF|9�35;�X<756y`�1'K535;*CE751'K5\D;*1'35\g?�BD3IA'1'\gAq6X<756hCEK(^ 3H1�:<= 1'_W9�z5l(6#;xzH35BgX<6h3,K5={`�6#:~1'_);56#:<BgA�9�X<BgAq6#Cy1'_ �dm 0�35;H6#6#; � � E� � � ������� �¹�:>8�j� � � � ��� � �� ����� (CW�����,=r9�l � ��� $ � � ! � � � � �� E� �V��� � F|9�3H;*9�C �� � � BDCY9h;56#:<BgA�9�X<BD1'3�1'_Z1�:<;56#: (+������2Ó�� ���¹F�X<75B]CrBDCc`�1'K53H;56#;`I? C)�����,=c9�l � � � $ � � ! � � � � � � � � ��� � F-CE1 E� ��BDC)9�;5B]CMXE:<Bg`HK(X<BD1'3am � 356�;56#;HK54#6C@X<7H9�X)BD_ ��BDC)9~;5BDCMXE:<Bg`HK5X<BD1'3h1�_1�:<;56i:"(, � X<7563� E� ��BDCP9);5BDCMXE:<Bg`HK5X<BD1'3�1�_V1�:<;H6i:"( � Ó�� ���¹m� 356o=@K5CMX2X<7563r4}756#4[bWX<7H9�X2X<756|_�1�:<=@K5\�98B]Cu41�=W^%9�X<Bg`H\D6PePBgX<7cX<756o351�X<BD1'3r1'_ ;H6i:<BgA'9�X<BDA'6|_�1�:�CE=W1,1�X<7r_�KH354iX<BD1'35C#FBSm 6�mYBD_�@ � Cfvj���P�[F�X<756#3 ������ � � C � ���P�[F�9�35;¶9�C8`�1�X<7$@¶9�35; ���� � � 9�:<6c\D1,49�\D\D?�BD3qX<6#k�:[9�`H\D6WX<756i?�9�:<6W;HBDCMXE:<Bg`HK(X<B]1�3HC9�35;>1�3H6p=@K5CMXP4[7564<brX<7H9�X�X<756o;H6i:<BgA'9�X<BDA'6o1'_�X<756p;5BDCMXE:<BD`HK(X<BD1'3�9�CECE1I4B]9�X<6#;YX<1�@��+eP75BD4}7�CE751'K5\D;�7H9Aq6p`�6#63�;56#3H1�X<6;@)OMQa�dBDC�X<756Z;5BDCMXE:<Bg`%K(X<BD1'3>9�CECE1I4B]9�X<6#;�X<1 ���

��� �F%9�3H;�X<75BDC�=W69�35C�X<7H9�X�1'3568CE751'K5\D;>4}756#4[brX<7H9�X�_�1�:�6iAq6#:E? � � C � ���P�

1'356W7H9�C#I0J � ���� � � ��Ó5@ ������ � � O Q �¸®(F `%K(X8X<75BDC8BDC�I0J � � ��� ���� �

O Q�F�9�35;�`a6#49�K5CE6E@ �/7H9�C841�=W^%9�4iX8CEK(^5^a1�:EXF�1'356W49�3BD3qAq1�bq6f��V¯�Q�T�Q × CpX<7561�:<6= 9�35;�1'356W=c9?�CMX[9�:EXp`I?~BD3qX<6#k�:[9�X<BD35k�BD3 Q ��F�9�35;~X<7563�BD3�X<756)1�X<756i:ZA'9�:<B�9�`%\D6#C#t 1'356)7%9�CX<1rBD3qX<6#k�:[9�X<6{1�3~9�3y1�^�63yCE6#X � 1�_ �¸9r_�K53H4iX<BD1'3>ePBDX<7y4#1'=)^%9�4#X|CEK(^5^�1�:EXFa9�35; � 41'K5\D;w`�6@=c9�;56@1�_-9c4#1'K53IX[9�`%\g?BD3(z%35BgX<6�3IK5=@`�6i:u1'_�1�^�63WBD3IX<6i:EA�9�\DC#Fq`HK(Xu1'35\g?W9pzH3HBgX<6�3,K5={`�6#:u1�_�BD3IX<6i:EA�9�\DC-7H9Aq6�X<18`�6�X[9�bq6#3cBD3qX<1@9�4#4#1'K53IXF,9�3H;c_�1�:69�4}7>1'_�X<756#CE68B]3qX<6i:EA'9�\]C�1'356ZB]3qX<6#k�:[9�X<6#C�X<7568;56i:<BDA�9�X<BgAq6p1'_V9 CfvP_�K5354#X<BD1'3�A�9�35B]CE75BD35k)3H69�:�X<7H6Z6#3H;5CP1'_�X<7H6ZBD3qX<6#:EA�9�\9�35;>X<756ZB]3qX<6#k�:[9�\aB]C2X<7563w®(m

¢�756Z;56#:<BgA�9�X<BgAq6o1�_�X<756pno���,��QSUQ��V�y_�K53H4iX<BD1'3~�+eP7HBD4[7��p_�1�:�Q ��®)9�35;>®)_�1�:�Q���®I�uBDC�� � F(X<7H6p£pQ�����²�=r9�CEC�9�X®(m�0�35;566#;aFHB]_B ;56#3H1�X<6C��� � FH1'356Z7%9�C � ,; � ���*�(> � ; � ���H�(> I � ��OMQ9���u�O®I�d_�1�:|9�\D\ � � C�� � �{�[m

N�6iX � � >Z�2Ó�­.;hFHeP75B]4[7yBDC�X<7H68CEBDk'3y_�K5354#X<BD1'3aF%CE1WX<7H9�X � ��­ � � t�3H1�X<B]4#BD35k)X<7H9�X � � � � 9�35; � º � ��F�1�3H6;5BDCE41jA'6#:<C|X<7H6@_�1�\]\D1jePBD35k°«M^�9�:[9�;51jl(¬5F�X<7H9�X y� � � �8� ­ � � `HK5X � � º% � � Ò � � m � _�41'K(:<CE6f1'356{e21'K5\D;~7H9Aq6@`�66#3�BD3XE:<1'K(`H\D6�ePBgX<7W4[7H6#4<b(BD3Hk8BD_� y� � ºj�[FqeP75BD4}7WBDCu®(FI4#1'BD354BD;56CVePBgX<7Y­ � � Fq`a6#49�K5CE6PX<756�=fK5\gX<Bg^%\DBD49�X<BD1'3f1'_ � `I? � � BDC-351�X;56#zH356#; tH1'356Z4j9�3y9�4iX<K%9�\]\g?r=fK5\gX<Bg^%\g?Y9�3q?�ÚP���-R�Th=W6j9�CEK5:<6p`I?w9f41'3qX<BD3IK51'K5CP_�K5354iX<B]1�3 F(`HK5X � BDC�3H1�XP41'3qX<BD3IK51'K5C#m

��X-X<75BDCV^�1'BD3IX-1�3H6�CE751'K5\D;):<6#=c6#={`a6i:�X<7H9�X�^5:<1,;5K54#X<C-9�:<6�351�Xu9�\ge�9?,C-;56#zH356#; FI9�35;)X<7HBDCVePBD\D\,`�6�1'K(:d356#lIX�X<1�^HB]4�m

�#Õ

Page 15: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®qÔ,m�1°6#;53H6#CE;H9?�GI9�3IK%9�:E?y­'ÒIF�­�®'®'®(m1 BgX<7�X<756Z351�X<BD1'3>1'_�;5BDCMXE:<Bg`%K(X<BD1'35C#F5BgXPBDC�351je�6j9�CM?rX<1Wk'BgAq6pX<756Z;H6izH35BDX<BD1'3�1'_�eP7H9�Xp�(RV¯ R�¥,�H��CM^%9�4#6CP9�:<6�m

� &������D������ §w�H1�:r9~3H1�3H356#kq9�X<BgAq6�B]3qX<6#k'6i:��Fd_�1�:y� ( * ( , 9�35;�_�1�:r9�3�1�^�63�CE6iX$� � � ! FdX<756��(RV¯ R�¥,�H�CM^%9�4#6 ���� �5���P��BDC|X<756fCM^%9�4#6f1'_P�S6JIK5BDA�9�\D6#3H4#6{4#\]9�CECE6#Cp1'_[��_�K5354#X<BD1'35C �!� ���5���P��CEK54[7~X<7H9�X �� �!� ���H���P��_�1�:89�\D\;56#:<BgA�9�X<BD1'35C� ��y1'_�\D635k�X<7!� ��� (,�m�0sX�BDCP9W351�:<=W6#;wCM^%9�4#686JIK5BD^5^�6#;�ePBgX<7�X<756Z3H1�:<= � � � � � � � � � � $ � � � E� � � � � FH1�:X<756u6jJqKHBgA�9�\D63qX�351�:<= � � � � � �� � � � I J � � � � � $ � � E� � � � � OMQ � v � � _�1�:2�)(+* �-, 9�35;�� � � � � � � ��=c9�l � � � $ � � � �� � � � _�1�: * ��,�m� ��%"-� j�D������ §dB+�d�H1�:@��(+*#(-, 9�35;�+ �®fX<756f�(RV¯ R�¥,�H��CM^%9�4#6#���� �5���P�uBDCP9# ���T���²a³hCM^%9�4#6�m

BDB+�d�H1�: * � ­,FP���� º����P�uBDCP9)noQ�¥(¯��%�5�°CM^%9�46�FH_�1�:�X<7568CE49�\]9�:�^5:<1,;5K54#X{�E� � � F(�E�*� I J � � � � � $ � �� � � F O Q �'m¢�756oCM^%9�4#6 ���� º����P�VBDCu1�_�X<6#3Y;56#351�X<6; ;6�)���P�[F(983H1�X[9�X<BD1'3Y9�\DCE1{K5CE6;r_�1�:2no����� ��CM^%9�4#6#C#F(_�1�:ueP75BD4[7rX<756|351�X[9�X<B]1�3=�8ZePBD\D\�`a6ZK5CE6;��S_�1�:|® � 5 (-,��[m�2È<ÆjÆ5#[§cN�6iX ��� `�6�9°¡u� V²a³ ��vfCE6JIK56354#6�B]3,���� �5���P�[F-BSm 6'm�_�1�:W6iAq6i:E? � � ®yX<756#:<6�6il(BDCMX<C � � �'�8CEK54[7�X<7H9�X)_�1�:� � � �R � � ���{1'356>7%9�C � � ��� > ��� � �&( �qm�¢�75BDCWBD=W^H\DBD6C{X<7H9�XW_�1�:c69�4[75k�=fK5\gX<BgLOB]35;56il ��ePBgX<7�� ���&( 1'356�7%9�C� � � ��� >< � ��� � � � ( �IF%BSm 6�m8 � ��� BDC|9Y¡u� �²�³���CE6JIK56354#6{BD3 � � ���P�[F�9�35;�9�C � � ���P�2BDCP41�=W^H\D6#X<6W�+`�649�K5CE6{1�3H6K5CE6C�X<7H6�N���¯��aU�¤���Y=W69�CEK(:<6'Fq_�1�:-eP75BD4}7)X<756PÚPQ���U�µ�LO��QSU�²�³��%�Vº2X<7H6#1�:<6#=¸9�^5^H\]BD6#C}�[Fq1'356P;56#;5KH4#6#C-X<7H9�X E� ��� � @ �BD3 ���5���P�rCMXE:<1'35k5m � 3H6�=@K5CMXwX<756#3 ^5:<1jA'6; X<7H9�X @ � � ��P@ � 9�3H; X<7H9�X ��� X<6#3H;5C>X<1 @ � BD3����� �5���P�[m��%1�:X<75BDCY1'356�KHCE6#CrX<7H6�;56i:<BDA�9�X<BgAq6wBD3/X<7H6�CE6#35CE6~1'_Z;HBDCMXE:<Bg`HK(X<B]1�3HC#F�9�35;*1'356�7H9�C � �� @ � � ��� � �:>Z��� � � � ��@ � � �� ��� ��:>Z��� � � � \DB]= � � ��@ � � � ���H�*\DBD= � � � � @ � � ���V_�1�:�9�\D\ � � C � ���P�-9�3H;Y9�\D\%=fK5\gX<BgLOB]35;5BD46#C3�dF,`HK(XueP7563 � ��� ( X<756)\]9�CMXZ\DBD=cBgXoB]C ��@ � � ���[F CE751jePBD35k�X<7H9�XR ��P@ � � @ � F�CE1YX<7%9�X#@ � � ���� �(���P�|9�35; E� ��� � E� @ � B]3 ���5���P�P_�1�:� ��� ( �FHCE1)X<7H9�X�`I?YX[9�b(BD35kfX<7H6Z\DBD=cBgX � � � ,/F%9�35;w1�3H6ZzH35;5C�X<7H9�XW� � ��� >!@ � � � ( �qFHBSm 6�m ��� � @ � BD36� � � �5���P�[m

¢�756{^5:<1�^�1'CE6;�_�1�:<=fK5\]9c_�1�:|X<7H6{CE49�\]9�:|^H:<1I;HK54iXpBDC|BD35;566#;�\DBD356j9�:oBD3 � 9�35;�9�3qX<B]\DBD356j9�:oBD3 FHF 9�3H;y_�1�: FE� � BDXk'BgAq6#C�X<7568C<JIKH9�:<6Z1'_�X<7H6Z351�:<=>m

0�3�X<756@^5:<1,1�_sF�1'356f7H9�CoCE751jeP3�CE1�=c6{b(BD35;�1'_�41'3qX<BD3IK5BDXs?w_�1�:pX<756@;56i:<BDA�9�X<BD1'35C|1'3 � �O���P�[mo0�35;566#;yX<756#:<6@6#l,B]CMX<C9YX<1�^a1�\]1�k�?y1'3<C�� ���P�|_�1�:peP75B]4[7~X<756);5KH9�\VB]C�� �O���P�P9�35;�1'356WK5CE6#C81�3 � �O���P��X<7H6@e26j9�b�X<1�^�1'\D1'k�?'F�eP7HBD4[7�BDCp351�X=W6#XE:<BD�9�`H\]6�F,`%K(XPBDX�BDC�:[9�:<6#\g?Y3H6#4#6CEC<9�:E?YX<1)b(351je eP7H9�XPX<75BDC�X<1�^�1'\D1'k�?YBDC#tH3H6iAq6i:EX<756\D6#CEC�BDXPBDC�K5CE6#_�K5\aX<1Wb(351je X<7H9�X|9CE6JIK56354#6 � � 4#1'3qAq6#:<k�6CpX<1 � BD3�X<7H9�XpX<1�^�1'\D1'k�?yB]_d9�35;�1'35\g?�BD_ ��� � � ����� ��� E� ���P_�1�:89�\D\�� � C � ���P�{�O9�CoX<7H6X<1�^�1'\D1'k�?YB]CP351�X|=W6#XE:<BD�9�`H\]6�F(X<756Zb(351jeP\D6;5k'681'_�41�3IA'6#:<k'BD35kcCE6JIK56354#6CP;51,6#C|351�X|4[7H9�:[9�4iX<6#:<BD�#6ZeP7H9�XPX<7568X<1�^�1'\D1'k�?BDC}�[md�o3I?�;56#:<BgA�9�X<BD1'3 ��yBDCP\DBD356j9�:P41'3qX<BD3IK51'K5C|_]:<1'= � �O���P�uB]3qX<1WBgX<CE6#\D_sF(`HK(X|X<756{9�:<k�KH=W6#3IXPBD3>X<756Z^5:<6#46#;5B]35k)^5:<1,1�_1'35\g?�CE751jePC�X<7H9�XPBgXPBDC�CE6JIK563qX<B]9�\D\g?Y4#1'3qX<B]3IK51'K5C#md�o\gX<751'K5k'7�BDXPBDC�:[9�:<6#\g?Y3H6#4#6CEC<9�:E?YX<1WK5CE6ZX<756Z^5:<6#4BDCE6pX<1�^�1'\D1'k�?Y1'3C � ���P� 1�:�1'3 � � ���P�[F'BgXVBDCVK5CE6_�KH\,X<1o4[7564<b{X<7H9�X�9�\D\,X<75621�^a6i:[9�X<BD1'35C�eP75BD4}7f1�3H62;56iz%356#C-9�:<6�CE6JIK56#3IX<B]9�\D\g?84#1'3qX<BD3,K51'K5C#t9yCE6JIK56#3H4#6 � � 4#1'3qAq6#:<k�6C{X<1 � BD3$C�� ���P�ZBD_�9�35;¶1'35\g?�BD_P9�\]\�X<7H6 � � 7H9A'6rX<756Bg:@CEK(^H^�1�:EX<CfBD3h9y41'=)^%9�4iXfCE6iX� � �oF�9�3H;>B]_B �� � � 41�3IA'6#:<k'6#CPK535BD_�1�:<=W\g?rX<1 �� � _�1�:|9�\]\a=fK5\gX<BgLOBD3H;5BD4#6C8�dm

¢�7568356il,X�X<75B]35k)X<1);H6izH3568_�1�:P;5B]CMXE:<Bg`HK(X<BD1'35C�BDC�=@KH\gX<Bg^H\DB]49�X<BD1'3r`I?�CE=W1,1�X<7>_�KH354iX<BD1'35C#mN�9�K5:<6#3qXP�(²�³�´8���5��µ{7H9�CuCE751jeP3cX<7H9�X2BgXdBDCu351�X-^a1�CECEBD`H\D6PX<18;H6izH356P^5:<1,;5KH4iXu1'_a;5B]CMXE:<Bg`HK(X<BD1'35CdBD3r9�3Y9�CECE1I4B]9�X<BgAq6

e�9?'FP9�35;/=W1�:<6w^5:<64#BDCE6\g?�1'356y7H9�C�� * F v�Qa� � � � � � � � � � F2eP75B]\D6�*�F v� �GQ � � �9� * F v� ®,�»®(m no1je26iAq6i:F�CE1'=c6

^H7q?(CEBD4BDCMX<C�e�:<BgX<6Z_�1�:<=fK5\]9�C�\DBgbq6 �(�GQa� �(�GQa�T> v�v������(F5`%K(XPBgX�B]C�351�XP4#\D6j9�:�eP7H9�XPCEK54[7�9)_�1�:<=@KH\]9)=W69�35C#m

�o\g:<6j9�;(?°e26�b(351je X<7H9�X �(�GQa�{CE751'K5\D;�`a6�e�:<BgXEX<6#3 � � 9�35;�B]CW9~Ú|���-R�Tx=c69�CEK(:<6�`HK(Xc351�Xr9~_�K5354#X<BD1'3aFu9�35;9�C v�9�35; v

�� 9�:<6@3H1�Xp\]1I4j9�\D\g?>BD3qX<6k�:[9�`%\D6@_�K53H4iX<BD1'35C|`�6#4j9�KHCE6f1�_�X<756fCEBD35k'K5\]9�:<BgX<BD6C|9�XZ®(F�=r9�b,B]35kc;HBDCMXE:<Bg`HK(X<B]1�3HC|1'K(X

1'_�X<756=�:<6jJIK5Bg:<6#CfCE1'=c6�49�:<6�m�¡u� �²�³ �x7H9�;�9�\g:<69�;(?h;56#zH356#;¶X<756�3H1�X<B]1�3�1�_�^5:<BD3H4#Bg^%9�\uA�9�\DK56'F-9�35;�`I?¶9�3H9�\D1'k�?N�9�K(:<63qX��(²�³�´8���5��µ{;56izH3H6#;r98;5BDCMXE:<Bg`HK5X<BD1'3);H6#351�X<6#;W`,? * F v

�9�35;c49�\D\]6#;)X<756�^5:<B]354#Bg^�9�\5A�9�\DK56P1'_ v

�FI`I?� *�F v

� � ���H�\DBD= � � I � � � � /% � � � �� O Q�t@X<75BDC>;51,6#C>;56#zH356°9�;HBDCMXE:<Bg`HK(X<B]1�3 1�_@1�:<;56#: ( ��Fo`�6#4j9�KHCE6�BD_"- � *�*�. L�%i� ��� � ! >8N � Ó N,#SFX<75635I �

�� � /% � � � �� O Q � I /% $�� � � $ + � � � �

" � � � ��

O Q � I ��� $ + � � � �

" � � � ��

OMQ�F2eP75BD4}7�6#l(BDCMX<CW`a6#49�K5CE6 � �d�GQa��>��d�O®I��� (

vw�oK5k'K5CMX<B]3¶N�1'K5BDCr¡u� V²a³ �2F��H:<6#354}7�=c9�X<756=c9�X<BD4#B�9�3 F��)( �'��L<� �'Ô (Im¶no6re21�:Ebq6#;�B]3¶±�9�:<BDC#m¶¢�756�41�3H4#6i^HX)1'_|9¡u� �²�³��¶CE6JIK56#3H4#6pe�9�C�z5:<CMX|BD3qXE:<1,;5K546#;w9)_�6ie ?q69�:<C�`�6#_�1�:<6p7HBD= `I? PR�¥5µ���T�Rum

�6i:<3H7H9�:<;8±d\]9�4#B];5K5C�Gq1'7H9�353{�o6i^a1�=fK(b� �R�¥5µ'��T�RuF�¡u�6#4}78=r9�X<7H6#=c9�X<BD4B]9�3aFq�)( �(�iL<����Õ��,m�no6�e21�:Ebq6#;{BD38±-:[9�k'K56�mº~Ù-:<35CMX|�,BDk'BDCE=@K53H;Y�VQ+U�²�³����-F(�oK5CMXE:<B]9�3(LS`�1�:<3�=c9�X<756#=r9�X<BD4#B]9�3aF�� ��(�Ô�L<��'Ô�Õ5m�no6|e21�:Ebq6#;�BD3w¡u1'\D1'k'356�F%Ðo6#:<=c9�3q?'m

�jÔ

Page 16: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

� QH��� � ��� � t(X<7%9�X�X<75B]C�;5BDCMXE:<BD`HK(X<BD1'3YB]C23H1�X|9fÚP���-R�T°=W6j9�CEK(:<6Z49�3Y`a6pCE66#3�`I?Y6BgX<756i:�41�3HCMXE:<K54iX<BD3Hk)9f41'3qX<BD3IK51'K5C_�K5354iX<B]1�3*_�1�:�eP75BD4}7�X<756~;56iz%35BgX<BD1'3*k�BDA'6CwÓ),/F|1�:�`I?�41�3HCMXE:<K54iX<BD3Hk�9hCE6JIK56354#6�1'_8_�K53H4iX<BD1'35C$�� � C�� � �)!p�eP75BD4}7~CMX[9#?�KH35BD_�1�:<=c\g?>`�1'K53H;56#;aFabq66i^~X<756#BD:pCEK(^H^�1�:EXZBD3�9cz5l(6;~4#1'=W^%9�4iXpCE6iX{9�3H;~_�1�:peP75BD4}7 � *�F v� � � � � � Ó),/t

X[9�b(BD35k ��351'35356kq9�X<BDA'6PePBgX<7WCEK(^H^�1�:EX2BD3�! ® � �$#�9�35; �

��GQa�H��P_�1�: v

�( Q#( �B> v

�F,1�3H6P7%9�C�� * F v

� � � � � Iv " /�/� � �

�m

no�������p��� ��;56#zH356#;8=c1�:<6-k'6#356#:[9�\'zH35BgX<6�^%9�:EX<C�1'_ v� �F�9�35;8N�9�K5:<6#3qX-�(²�³�´8���5��µ�;H6izH356;Z`,?Z9�3H9�\D1'k�?Z9�;5BDCMXE:<BD`HK(X<BD1'3

eP75BD4}7>49�3�`a6Z4#1'35CEB];56i:<6;�X<7H6pzH35BDX<6o^�9�:EXP1'_ v� �m

� 356y;56iz%356#CcX<756>^H:<1I;HK54iXr1'_Z9�;5BDCMXE:<BD`HK(X<BD1'3�`I?/9!C Û_�K53H4iX<BD1'3aF�1�:r=W1�:<6wk'6#3H6i:[9�\D\g?�1'_p9°;5BDCMXE:<Bg`HK5X<BD1'3�1�_1�:<;56i: ( `I?/9 C � _�K53H4iX<BD1'3�9�Cr_�1'\D\]1ePC#m��oCY9°4#1'35CE6jJIK56#3546�F�1'356�7H9�C�Q * F v

�� ��9�35; Q � � �·®(F�CE1hX<7H9�X

Q�� *�F v�Ó C � � �*�Ø�Z_�1�:o9�\D\ CfF5`HK(X *�F v

�49�3�`a6ZCE751jeP3>X<1)`�6ZX<756Z1'35\D?YCE1'\DK(X<BD1'3�� 1'_BQ �5�*®feP7HBD4[7>B]C�1I;H;am

� &������D������ § 0�_ � � � � ���P�a9�3H; � � C ���P�[FjX<7563 � �/�S1�: � � ��B]CaX<756-;5BDCMXE:<Bg`HK5X<BD1'3p;56iz%356#;Z`I? � � � � ���H� ��� � � ���_�1�:|9�\D\ � � C�� ���P�[m

� _�41�K5:<CE6�FH1'3568=fK5CMXP4[7564<bYz5:<CMXPX<7H9�X � � BDCP9);5BDCMXE:<BD`HK(X<BD1'3aFH9�35;>X<75BDC�_�1'\D\D1jePC�_�:<1'=�N���Q�¯�T�Q+µ × C�_�1�:<=@K5\]9(mN��%Q+¯�T�QSµ × C�_�1�:<=@KH\]9|B]3f1'356�;5BD=c6#35CEB]1�3fCMX[9�X<6C�X<7H9�X�� @�K5� � � � � � ���� � � �� �.@ � � � K � � " � � F'eP756i:<6�( � � � ;56351�X<6#C�X<7H6;56#:<BgA�9�X<BgAq6V1'_,1�:<;56i: �o1'_ (aF�9�35;ZX<7H6V`HB]351'=WB]9�\�41I6#�c4#BD63qX � �� � BDC ���

� � � � " � � � t�BgX�BDC 6j9�CEB]\g?|^5:<1jA'6;Z`,?pBD35;5K54#X<BD1'3aF�CMX[9�:EX<B]35k_�:<1�=�� @8K5� �P��@�� K�ÓE@�K �Sm�1�:<BgX<BD3Hkp9�3)6il,X<635CEBD1'3f1'_HN���Q+¯�T�QSµ × CV_�1�:<=@K5\�9PX<1oX<756 & LO;HBD=W635CEBD1'3H9�\(49�CE6�BDCVCEBD=)^%\DBgzH6;{`I?K5CEBD3Hk{9Z351�X[9�X<B]1�3W_�1�:d=@K5\DX<BgLOBD35;5B]4#6#C#tM�32q=c69�35C3� v 2 ����� � ! 2DF � �� � =c69�35C � � /� / � ����� � � 1� 1 � � � �� � � � " � � � 9�35;�� (,��=c69�35C���)(,� �P_�1�:H"#�Ø� ���������:& tIX<7563cN���Q�¯�T�QSµ × Cd_�1�:<=@K5\]9Z7%9�CdX<756PC<9�=W6|_�1�:<= � � @�K5�*� � �$ � � �� �� � @ � � " � � K%F,9�35;BgXPBDC�69�CEBD\g?Y^5:<1jA'6;>`I?�BD35;5K54#X<BD1'3�1'3 & m0�_ - � * */. L�%i� �V��� �� �oF�1�3H6)7%9�C9� � ���� � �V��� � ( � � $ � � �� �4� � � � � � ���*� ! � � � E� " � �8� � F�CE1>X<7H9�X81�3H6)7%9�C=c9�l � � � $ � � � E��� � ����� � ( C)�����,=c9�l � � � $ � � � �� ��� � t21�3H6>9�\DCE1�;56#;5KH4#6#C)X<7H9�X)BD_ � BDCW9�;HBDCMXE:<Bg`HK(X<B]1�3�1�_|1�:<;H6i:( �F5X<756#3 � � BDCP9�\DCE1c9);5BDCMXE:<Bg`HK5X<BD1'3�1'_�1�:<;H6i:"(,~m

� 356|=@K5CMX�9�\]CE184}756#4[b)X<7H9�XdX<756|351�X[9�X<B]1�3cB]C-41'=)^%9�X<Bg`H\]6�ePBgX<7WX<756P4\]9�CECEBD49�\H=fK5\gX<Bg^%\DBD49�X<BD1'3aFqBSm 6'm�B]_ @ � � v��� � ���P�9�35; �ØBDCPX<756@4#1�:E:<6#CM^�1'35;HBD35kr;5B]CMXE:<Bg`HK(X<BD1'3h�+eP75BD4[7yCE7H1�KH\D;y`�6{e�:<BgXEX<63 @)OMQa�[F%X<7H6#3yX<756{;5BDCMXE:<BD`HK(X<BD1'3 � 9�CECE1,4#B]9�X<6#;X<1 � @ BDC�BD35;566#; � � 9�CYe�9�C{�EK5CMX�;56#zH356;am�¢�75BDC�_�1'\D\]1ePC�_]:<1'= X<7H6�;H6izH35BDX<BD1'3aF|9�C � � � ����� I J�� � @�����OMQ �I0J�@�� � �V�4OMQ9� ��� � � ���H� � � � � ���u_�1�:o9�\D\ � � C � ���P�[m¢�756Z=r9�^H^HBD35k>� � � �o���� � � _�:<1�= C � � � ���P�-B]3qX<1� � ���P�dBDC�CE6JIK563qX<B]9�\D\g?r41�3IX<BD3IK51'K5C#md¢�756ZCM^%9�46�C ���P�BDC89��V� ��a²�³��%��CM^%9�4#6c9�35; � � � � BD3<C�����P�|=c69�35CpX<7%9�X8_�1�:86iAq6#:E?�41�=W^%9�4iX �� � 9�3H;�_�1�:86iAq6#:E?�=fK5\gX<BgLBD35;H6il,�dF� �� � � � �� � K535B]_�1�:<=W\D?¶1'3 �hm ¢�756wX<1�^�1'\D1'k�?�1'_�� �����P�fBDCc=W1�:<6>X<64[753HBD49�\�X<1�;H6#CE4i:<BD`�6>`HK5XY9CE6JIK56354#6 � � 41'3qAq6i:<k'6#CcX<1 � B]3 � �O���P�fB]_Z9�35;/1'35\g?�BD_ ��� � � ��� � ��� E� ���[m �6#4j9�KHCE6wX<756yCM^�9�46 C ! ���P�)1�_X<756y_�K5354#X<BD1'35CcBD35C�� ���P�@eP7HBD4[7/7H9Aq6wX<7H6#Bg:rCEK(^H^�1�:EXYBD3 � BDCr9°�V� ��a²�³����xCM^%9�4#6'F�BDXr7H9�C ���Q���� � × Cc^H:<1�^a6i:EXs?'F_�:<1�= eP7HBD4[7 2��T���²�³�L��I� ��Q�T�³�� VU � X<7H6#1�:<6#= _�1'\D\D1jePCp9�35;~X<756i:<6_�1�:<6fBD_ � � � � FaX<7563�X<756#:<6f6il(BDCMX<CZ9Y4#1'35CMX[9�3qXCW�����c9�35; 9�3*B]3qX<6#k'6i: ������WBD35;H6i^�635;56#3IX>1�_ � CEK54[7*X<7H9�X � ��� � � ����( CW�����,=r9�l � � � $ � � ! � � � �� � � _�1�:w9�\D\ � C ! ���P�[m�¢�756i:<6_�1�:<6 ��� � � � � > � � � � ����� ��� � � > � � � � � ���dÓ � � � � � > � � � ���[FdCE1�X<7H9�XW1'356>7%9�C����� � � � � > � � � � ����� ( ����� � � � � � > � �������Ó�� ��� � >� 9� � �����¹F�9�35;yX<7568zH:<CMXoX<6i:<= X<635;5CPX<1�®W`�6#4j9�K5CE6@_�1�:69�4}7�=@K5\gX<BDLOBD35;56#l �uF �� � � � � > � �����p4#1'3qAq6i:<k'6C2KH35BD_�1�:<=c\g?WX<1)®{`,?rK5CEB]35kfN��%Q+¯�T�Q+µ × C2X<756#1�:<6#=wF59�3H;YX<756oCE64#1'35;X<6i:<= X<635;5C�X<1W®f`I?>;56iz%35BgX<BD1'3am� ��%"-� j�D������ §d�%1�: � � C ���P� � � � � � ���P�u9�35;y9�3q?��dF51'35687%9�C � � � �o�H� � � $ � � �� �q�� � � �}�� � " � �o�[m�2È<ÆjÆ5#[§ � 356Z^5:<1jAq6#C�X<7H9�X � ���� �

��� �� � ���� �

�¶Ó � � � � �

_�1�:8"��� ���������:& F%9�35;�X<7563>X<756Z_�1�:<=@KH\]9)_�1'\D\D1jePC�`I?�BD35;5K54#X<BD1'3am�H1�:oCE751jePBD35krX<75BDC#F%1'356@351�X<BD4#6C�X<7H9�X|_�1�:o6#A'6#:E?� � C�� ���P��1�3H6{7H9�C � � ���� �

��� � � ��� � > � � � � � �� � � � � > ��� � � � �� � � � ���� � > �

��� � ���� �

Ó � � ���� �

�*� ��� ��� � � � ��� �� � �

� � �� � ���H� � � �

��� ��¶Ó � �

��� � � ���[m0�_ � � C >���P��9�35;6N � �oF�X<7H6#3 � � + � � �GNH� � + F 9�C � � � +4� ���8� ��� +4� � ��� �!� � �V�}�GNH� � � �GNH� ��� + � ����_�1�:Z9�\D\� � C � ���P�[mV0�3�^�9�:EX<B]4#K5\]9�:)Q � � � �*®)_�1�: "��Ø� ���������:& m� 356W7H9�C�Q *�F v

�� ��F�`�649�K5CE6�� Q * F v

� � ���� � *�F v� � Q���� �¸\DBD= � � I � � � � /% ��� � � �� OMQ$� I � �d�GQa�4OMQ!� ��� � ���

_�1�:|9�\D\ � � C�� � �{�[m� ÚP63�"6iLON�1'K5BDC 2��Q�����F5�5:<6#3H4[7w=c9�X<756=c9�X<BD4#B�9�3 F�� ��(�Õ�L<����'­,m-no6pe21�:Ebq6#;wBD3>£ZB �M1'3aF5�H:[9�354#6'm��noK5k'1p£p?'1'35B]�i?W�I���%Q�T�³�� �U'F�±�1'\DB]CE7f=c9�X<756#=r9�X<BD4#B]9�3aF5� � ��(jL<���(�­,m�no62e21�:Ebq6#;)B]3fN�A'1jAY�S3H1e�B]3f¦|b,:[9�BD356�� K53qX<BD\

�j��Õ5��F%9�35;wBD3 1�:<1,4#\�9#ex9�_]X<6#:{���ÕIÔ,m

�jÒ

Page 17: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®qÒ,m-�5:<BD;H9?�GI9�3IKH9�:E?y­ �,F%­�®'®'®,mnp9A(BD35kr635\]9�:<k'6#;wX<756f4#\]9�CECo1�_-_�K5354iX<B]1�3HCP`I?wB]3qXE:<1,;5K54#B]35kr;5BDCMXE:<BD`HK(X<BD1'35C#F�CE1'=W6{^%9�:EX<B]9�\�;5BD �6i:<63qX<B]9�\�6jJqK%9�X<B]1�3HC

X<7H9�Xd7H9�;W`�6#63WCMX<K5;5B]6#;)`a6#_�1�:<6�=r9?f7H9Aq6Pkq9�BD356#;W356#e/CE1'\DK(X<BD1'35C-eP75BD4[7c4#1'K5\D;W351�X-`a6�41�3HCEBD;56i:<6;)`�6_�1�:<6'FI9�35;WCE1'=c61�X<756i:Z^%9�:EX<B]9�\�;HBg �6i:<63qX<B]9�\�6jJqK%9�X<B]1�3HCp;51�351�X8k'6iXZ356#e CE1'\DK(X<BD1'35C#m@�H1�:86il59�=)^%\D6�FaX<756f6jJIKH9�X<BD1'36Q @ ��®�9(m 6�m@BD3 �_�1�: @ � ��v��� � � �@�p1'35\D?�7H9�C@®�9�Cf9wCE1'\DK(X<B]1�3 F�`HK5X@_�1�:f;5BDCMXE:<Bg`HK5X<BD1'35C#@ � � �S� �{�[F @ � 4 � � B]C@9wÚ|���-R�T =W6j9�CEK5:<6eP75BD4}7~BDCp9YCE1'\DK5X<BD1'3¶�O9�35;�^H7I?,CEB]4#BDCMX<Cp;51Y4j9�\D\�BgXZ9Y£pQ�����²°_�KH354iX<BD1'3��[F�`HK5XZBgXpBDCpK5CE6#_�K5\�X<1rb(351je B]_VX<756#:<6)9�:<6f1�X<7H6i:;5BDCMXE:<BD`HK(X<BD1'35C � � � � � �{�uCE1'\DK(X<BD1'35C�1'_BQ � �*®(m� & ����$ §�0�_ � � � �O� �{�29�35;�Q �5�*®(F5X<7563�X<756i:<686il(BDCMX<C C � � CEK54[7>X<7H9�X �5��C � � m�2È<ÆjÆ5#[§�0�_ � � C � � �{��C<9�X<BDCMz%6#C �d�O®I�H�*®(F'X<7563 �u�GQa�H� Q � �GQa��9�35; � � C � � �{�[tIB]35;56#6;)X<756�¢�����¥5R�� v 6#l,^%9�35CEBD1'31'_ �>3H69�:-®oBDC��d�GQa�H��� � �O®I� QdÓ ����� Ó � �

% � � � ���� Q � Ó".,�GQ � �[Fq9�35;@X<7H6i:<6#_�1�:<6 � �GQa�H��� � �O®I��Ó ����� Ó � �% � � � ���� Q � " v Ó".,�GQ � " v �[F

CE751jePC�X<7H9�XP1'3568=@K5CMX�X[9�bq6 � �O®q�*��� � �O®I�[FH9�35;w=W1�:<6Zk'6#3H6i:[9�\D\g? � � � " v � �O®q�*� � �% � � � �� _�1�: � ��F%9�35;y9�CPN���Q+¯�T�QSµ × C

_�1�:<=fK5\]9fk'BgAq6#C�� � � � �GQa�H�5Q � � � � �GQa�aÓ � � � � " v � �GQa�[F(X<756Z;56i:<BDA�9�X<BgAq6#C21'_ � 9�:<684#1'3qX<BD3,K51'K5CP9�XP®(m � 356Z;56;5K546#C�X<7H9�X��� � ��� � ��� � Q � � � � Q � � � � ��®(m)N�6#X � � C � � �{�PePBgX<7 � �O®q� ����F X<7563�6#Aq6i:E?~_�K53H4iX<BD1'3 � � C � � �{�|=c9?y`a6e�:<BgXEX<63wBD3yX<756@_�1�:<= �u�GQa� � �d�O®I� � �GQa� Ó$Q � �GQa��_�1�:p9c_�K5354#X<BD1'3 � � C�� � �{�[F%`�649�K5CE6 � > �d�O®I� � A�9�35BDCE7H6#Co9�XZ®,F9�35;>X<756i:<6_�1�:<6 ��� � ���H� ��� � �u�O®I� � Ó Q � �H���u�O®q� ��� � � �[F5CE1)X<7H9�X �5��C � � ePBgX<7 C�� ��� � � �[m

� 3�X<756f1�X<7H6i:Z7H9�35;~BD_ �xBDCp9Y4#1'353564iX<6;�1�^�63~CE6iXZ1�_ �)! X<756f1'35\g?y;5B]CMXE:<Bg`HK(X<BD1'35C � � � �O���P��eP75BD4[7~C<9�X<BDCE_�?� ��� �

�*®)_�1�: "#�Ø� ���������'& 9�:<6pX<7H6Z4#1'35CMX[9�3qX<C#m� &������D������ §p¢�756@X<635CE1�:o^5:<1,;5KH4iX�@ v�� @ º 1'_d9r:<69�\2�S1�:Z4#1'=)^%\D6il%��_�KH354iX<BD1'3<@ v ;56#zH356;~1'3�9rCE6iX�� v 9�35;�9c:<6j9�\�S1�:p41'=)^H\D6#l%��_�K53H4iX<BD1'3<@ º ;56#zH356#;�1'3�9YCE6iX�� º BDC|X<7H6@:<69�\��S1�:84#1'=)^H\]6il%��_�K5354#X<BD1'3�;H6izH356;�1'3�� v ��� º `I?yX<7H6_�1�:<=fK5\]9�� @ v� @ º �}�GQ v�� Q º �*��@ v �GQ v �:@ º �GQ º �u_�1�:|9�\D\-�GQ v�� Q º � � � v �� º m� ��%"-� j�D������ §�N�6#X�� `�6@9c4#1'353H6#4iX<6;y1�^�63yCE6iXp1�_ �)!{m�0�_ � � � �O���P��C<9�X<B]CMzH6#C �

� � ���®c_�1�:)"��¸� ���������:& F%X<7563

� BDCP9W4#1'35CMX[9�3qXFHBSm 6�m�X<756i:<6Z6#l,B]CMX<C C CEK54}7>X<7H9�X ��� � ���H��C!I J �u�GQa�MOMQw_�1�:|9�\D\ � � C�� ���P�[m�2È<ÆjÆ5#[§� 2?¶9�4#1'353564iX<6;5356#CECc9�:<k'K5=W63qXWBgX)B]C)6#351'K5k'7�X<1�CE751je�X<756�:<6#CEK5\DX)ePBgX<7 �¸:<6i^%\]9�4#6#;¶`I?�9�3q?¶1�^�6#3�4#K(`a6� � � �om

¢�756�^5:<1,1�_ZBDCY1�`HX[9�B]356#;/`I?�BD35;5KH4iX<BD1'3/1'3/X<756y;HBD=W635CEBD1'3 & FP9�35;�1'356�CMX[9�:EX<CrePBgX<7�X<756�49�CE6 & � ��F � �`�6BD35kh9�3/BD3qX<6#:EA�9�\{�GN � 2 �[m � 356y3H1�X<B]4#6#CcX<7H9�XYBD_ � � C�� �GN � 2 �fC<9�X<B]CMzH6#C I -+ �u�GQa�4O Q(�·®(F2X<7H6#3 ��� � �� � _�1�:Y9_�K5354iX<B]1�3 � � C�� �GN � 2 �[F29�35; � BDCWk'BgAq6#3�6#l,^H\DBD4BgX<\g?�`I? � �GQa��� I �+ �u� %<�MO&%}m � 356w4}751,1�CE6C � � C � �GN � 2 �@CEK54[7X<7H9�X�I -+ ���GQa�4O Q<����F�9�35;~X<756#3�6#Aq6i:E? � � C�� �GN � 2 �P4j9�3~`a6@e�:<BgXEX<63�9�C �,��� I -+ �u� %<�MO&%<���@Ó � �� � _�1�:89�_�K5354iX<B]1�3� � C � �GN �32 �[F�`�6#4j9�KHCE6WX<756cBD3qX<6k�:[9�\-1'_ �<>*� I -+ �u� %<�4O %<����B]C{®(m�¢�756#:<6#_�1�:<6���� � ���� ��� � � I -+ �u� %<�4O %<���)Ó � �� � �R�� I -+ �d� %<�4O&%<� ��� � � � Ó ��� � �

�� � ��� C)� I

-+ �d� %<�4O&%<� > � � � � � � ��� C)� I-+ �u� %<�MO&%<��_�1�:Z9�\D\�� � C � �GN � 2 ��ePBgX<7 C � ��� � � �[F9�35;>X<7H9�XP=W6j9�3HC � ��C)m

1�:<BDX<BD35k�� � ��� �h�GN �32 ��eP756i:<6 �*BDCp9r4#K(`a6{B]3#�)! " vF�X<7H6#3�_�1�: � � C�� ���2��1'356@;56#zH356C � � � � �O�GN � 2 �2`I?��� � � � �)� ��� � � � � �P_�1�: � � C�� �GN � 2 �[F 9�35;�1'356)4[7H6#4<b(CZBD=c=W6#;HB]9�X<6#\g?>X<7H9�XoX<7HBDCpBD35;H6#6#;�;56iz%356#C89Y;5BDCMXE:<BD`HK(X<BD1'3� � 1'3��GN �32 �-`�649�K5CE6oX<7H6o`a1�KH35;5C�1'3�;H6i:<BgA'9�X<BDA'6C21'_�� � � 1�3H\g?rB]3qAq1�\DA'689@zH35BgX<6p3IK5=@`�6i:P1'_�;56i:<BgA'9�X<BgAq6#C21'_ � 9�35;X<756)CEK5^5^�1�:EX81'_ � � � BDCpX<756f^H:<1I;HK54iX{1�_dX<756)CEK(^H^�1�:EX<C81�_ �/9�35;�1�_ � 9�3H;~X<756#:<6#_�1�:<6)CMX[9?,C8BD3�9Yz5l(6;�4#1'=)^%9�4#XCE6iXPeP7563>X<7568CEK(^H^�1�:EX|1'_ � CMX[9?,C|BD3y9fz5l(6#;w41�=W^%9�4iX|CE6iXF �°`a6#BD35kWbq6i^5XPz5l(6#;am2¢�756#3 � � ��� � 1 �

� �*� > ��� � � ��� � 1

�3�> ��� � � � � �� � 1

�*�(> ��� � �� ��� � ���� 1

�H� �&� ��� 1 �

� � � �*�*®(F'CE1|X<7%9�X � � BDCV9o4#1'35CMX[9�3qX C � F'BSm 6�m ��� � � � � �H��C � I -+ � � %<�4O&%_�1�:�6iAq6i:E? � � C � ���u�W9�3H; � � C � �GN �32 �[m � 356�K5CE6CcX<7H6#3�X<7HBDCr_�1�:<=fK5\]9�X<1°CE751je X<7H9�X � �� C � ;56#zH356#C�9;5BDCMXE:<BD`HK(X<BD1'3 � 1'3���F,9�C-BgX-BDC-1�`IA(BD1'K5CE\g?)\DBD356j9�:d9�35;cBD3W1�:<;56i:-X<181�`5X[9�BD3fX<756P;56#CEBD:<6#;)`�1'K535;5Cd1�3H6�4}751,1�CE6Cd9Z_�K5354iX<B]1�3� � C � �GN � 2 � ePBgX<7�I -+ � � %<�4O&% �*®Z9�35;)X<7562`a1�KH35;5C-_�1�: � _�1'\D\D1je�69�CEBD\g?@_]:<1'= X<7562`�1'K535;5C-_�1�: �ZFI9�3H;fX<7H6i:<6#_�1�:<6�1�3H649�3we�:<BgX<6���� � � � � � � � � � ��� I -+ � � %<�MO&%�_�1�:o9�\D\�� � C � ���u��9�35;�9�\D\ � � C � �GN �32 �[m�¢�756#3y_�1�:8"�� � ���������:& >/�1'356)7H9�C8®9� ���

��� � � � � � � � > ��� � �� ��� � �� � �

�)� > ��� � ������ � � � � � > � � � ������ � � I-+ � � %<�4O&% � �����

��� � � ��� I-+ � � %<�4O&%|9�35;X<756#:<6#_�1�:<6 ���

��� �� ®(FuCE1�X<7H9�X)`,?¶X<756>BD3H;5K54iX<B]1�3�7q?,^�1�X<756#CEB]C � B]CW9�4#1'35CMX[9�3qXEC���FuCE1�X<7%9�Xc1'356w7H9�CWCE7H1eP3�X<7H9�X

v� 2:<1,1�b�¢��&�V¥5R��-FHÙ-3Hk�\]BDCE7>=c9�X<756=c9�X<BD4B]9�3aFa�Ò �'Ô�L<�)( �(��m

�)(

Page 18: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

��� � � � � � �(C���� I � �d�+*H�MO,*5�}� I -+ � � %<�4O %<� �(C�� I0J � � � � � �}�GQa�4O Q�F�9�35;yX<75BDCoCE751jePC|X<7H9�X � �(C��8B]_V1'356@K5CE6#C|X<7H6:<6#CEKH\gX�X<7H9�X�zH3HBgX<6Z41�=@`HBD3H9�X<BD1'35C�1'_�X<635CE1�:�^5:<1,;5K54iX<C|9�:<6Z;5635CE6c�SBD3�X<75689�;56jJIKH9�X<6pX<1�^�1'\D1'k�?H�[m

� 354#6p=fK5\gX<Bg^%\DBD49�X<BD1'3c7H9�C2`�66#3�;56#zH356#; FH9�3H;�N���Q�¯�T�Q+µ × C2_�1�:<=@K5\�9@7H9�Cu`a6#6#3�6#l,X<6#35;56;aF(1'356p49�3Y^5:<1jAq6p;5635CEBgXs?:<6#CEKH\gX<C#m� ��%"-� j�D������ §d�%1�:{�)(+* �-,/F%9�35;y9�3I?�BD3qX<6k'6i:�S �®,FHX<756ZCM^%9�46�C�� � �)!Z�uBDC�;5635CE68BD36���� �5� �)!Z�[m�2È<ÆjÆ5#[§pN�6#X � � `�6c9�CM^�6#4B]9�\VXE:<K5354j9�X<B]35k�CE6JIK56354#6'F�BSm 6�m � � �GQa� � � v � �� ��F�ePBgX<7 � v � C � � �)!Z�[F�®#( � �GQa�)(��)1'3� ! 9�3H; � �GQa� � �f_�1�:E� QH��(!��m@�H1�: �$� � �� � � � ! �[F 1'356);56iz%356#C ��� � � �� F�9�3H;�1�3H6f351�X<BD46#CoX<7%9�X ��� � � BD3���� �5� �"!p��CMXE:<1'35k5mV0�35;566#;c1�3H6P7H9�C � ��� �GQa����( � � �GQa����9�\D=W1'CMXd6#A'6#:E?,eP756i:<6'FI9�35; ��� �GQa�#� � �GQa��9�C � � ,/F(9�35;W`I?N���¯��aUj¤��� × Cu;51'=WBD3%9�X<6;c41'3qAq6i:<k'6#3546PX<7561�:<6#=!1'356|;56;5K546#CuX<7H9�X ��� � � B]3 ���(� �)!Z��CMXE:<1'35k5m-¢�7563r_�1�:�� ��� ( 1'356{7H9�C �� ��� � � �$ � � �� �� � � � E� " � � F�9�35;yX<756ZX<6#:<= _�1�: �!��®W41'3qAq6i:<k'6#C|X<1� �� � 9�kq9�BD3w`I?w9�3�9�^5^H\]BD49�X<BD1'31'_aN���¯��aU�¤��� × Cu;51'=cBD3H9�X<6#;c41�3IA'6#:<k'6#3546PX<7561�:<6=>FIeP75BD\D6PX<756PX<6i:<=WCu_�1�:�� �)� ��®{4#1'3qX[9�BD35Cu;56#:<BgA�9�X<BgAq6C-1'_ � � eP75BD4}74#1'3qAq6#:<k�68K535BD_�1�:<=W\g?rX<1c®(FHCE1)X<7H9�XP1�3H687H9�C) �� ��� � �� � B]3 ���5� �)!Z�uCMXE:<1�3Hk(m

¢�7563�1�3H6>9�^H^5:<1q9�4}756#C ��� `I?h_�K5354#X<BD1'35C)BD3 C � � �)!p�8`I?¶4#1'3qAq1'\DK(X<BD1'3h`I?�9�CE=c1I1�X<75B]35k~CE6JIK56#3H4#6 � � _�1�:W9CE6JIK56354#6P1'_ �o4#1'3qAq6i:<k'BD35kZX<1{®(F,9�3H;cK5CEBD35k{98;5B]9�k'1'3H9�\H9�:<k'K5=c6#3qX-X<756i:<6PBDCd9ZCE6jJIK56#3546 ��� � � � � � � � C � � �)!p��eP75BD4}74#1'3qAq6#:<k�6C�X<1 � BD3 ���� �H� �)!p�2CMXE:<1'35k5m�¢�756{4i:<K54B]9�\a^�1'BD3qX|BDC�X<1c351�X<BD4#6ZX<7H9�X|_�1�:�� �)�(, 1'356{7H9�C ���� � � � ��� �3�� � � � ��� F�eP75B]4[7�4#1'3qAq6i:<k'6#C8X<7563~X<1 � ��� BD3 � � � � ! �oCMXE:<1'35k5mW0�35;566#;�_�1�:@9�3I?�X<6CMX8_�K5354#X<BD1'3 � � C � � � ! �[F1'356w7%9�C�� E��� � � � ��� � � �����Ö�:>8�j� � � � � � � � ��� � �� �������:>Z��� � � � � ��� � �� � � �� ���E�Ö�:>Z��� � � � � ��� � E��� �� � � �V���E�� �� ��� � �� � � ���H� � � � � �� ��� � ���[F5eP756i:<6 �@wBDC�;H6izH356;>`I? �@��GQa�H��@��:>8Qa�[m

�%1�: *!� ,�F X<756)C<9�=W6W=W6iX<7H1I;�CE751jePCZX<7H9�X81�3H6)49�3°9�^H^5:<1q9�4}7�9�3q? � � ��� � �� �"!p�P`I?�9�CE6JIK56354#6 � �!�C � � �)!p�uCEK54[7>X<7H9�X�_�1�:�6#A'6#:E? � ��� ( �F� �� � � 4#1'3qAq6#:<k�6C2X<1� E� � B]3 � �� �)!Z�-e26j9�b � 9�35; � 8 ����� � �)!p�dCMXE:<1'35kW_�1�:6iAq6#:E?rz%35BgX<6 5 m

0�_��/BDC�9�3�1�^�6#3�CE6iX�1'_�� ! F(BgX�BDC2351�X2XE:<K56pBD3Yk'6#3H6i:[9�\�X<7H9�X�C � � � ! �-BDC2;5635CE6pBD3�� �� � ���P�[F(9�35;�1'356oB]C2\D6#;rX<1X<7568_�1'\D\D1jePBD3Hkf;56#zH35BgX<B]1�3 m� &������D������ §8� �� �� ���P�uBDC�X<756Z4\D1'CEK(:<6Z1'_HC�� ���P�uB]36���� �5���P�[m

0�_ X<756|`�1'K53H;H9�:E? � ��BDCuCE=c1,1�X<7Y6351'K5k'7aF(X<756#3YX<756o_�K5354iX<B]1�3HC21'_B� �� �� ���P�d9�:<6p®@1'3rX<7H6|`�1'K535;H9�:E?'FH9�CuePB]\D\H`a6CE6#63�\]9�X<6i:m�0�_ � ��BDC2X<1I1fCE=c9�\D\SF,X<756#3�BgX�=c9?r7H9�^5^�6#3YX<7%9�X � � � �� ���P�H������ �5���P�[t(X<75BDC2BDCu:<6\]9�X<6#;rX<1@X<756p_�9�4iX�X<7H9�XX<7568_�K53H4iX<BD1'35C�BD3 ���� �5���P�u9�:<6Z3H1�XP3H6#4#6CEC<9�:<BD\g?Y4#1'3qX<BD3IKH1�KHC#F%9�C�CMX[9�X<6#;>`I?YX<756ZBD={`�6;5;5BD3HkfX<756#1�:<6= 1'_��(RV¯ R�¥,�%�uFCMX[9�X<6#;>3H1eØ9�35;>^5:<1jAq6#;w\]9�X<6i:m��� &,%��&�� §2B��u0�_�� ( * � !� X<7563 ���� �5� �)!Z� � � 7 � �)!Z�2ePBgX<7 v7 � v

�> �!F%`HK(XR���� �H� �)!p��BDC|351�Xp9cCEK(`%CM^%9�4#6

1'_�� � � �)!Z�u_�1�: - � L,mBDB+�Z0�_ *�� !� X<7563 � �� � � � ! ��� ��8'� � ! �{_�1�:W6iAq6#:E? 5 � ! * � ,��[Fd`HK(XE� �� ! � � ! �{BDC)3H1�Xr9~CEK(`HCM^%9�4#6w1�_

� �� �"!p�uBD_ * � ��tH751je26iAq6i: � ! � vj� �"!p� � C � � �)!p�[mBDB]B��d0�_ !� � * � , X<756#3 ���� �5� �)!Z� ��C � � �)!p�[F%X<756{CM^%9�4#6{1'_V4#1'3qX<BD3,K51'K5C|_�K53H4iX<BD1'35C�X<635;5BD3HkWX<1r®c9�X�,�mu0�_

!�� * � !

�" v _�1�:Z9�3�BD3IX<6#k'6i:��%F�X<7563 � � � �(� �)!Z��� C � � � � �)!Z�[F%X<756fCM^%9�4#6f1�_-n �R�¥��V�%��4#1'3qX<B]3IK51'K5Cp_�K53H4iX<BD1'35Co1�_

1�:<;56i:���FHePBgX<7�� � �#> !�m

�%1�:26#l59�=W^H\D6PBD_ � �-�)!��WFIeP756#:<6�xBDCu98zH35BgX<6|3IK5={`a6i:u1'_�^�1'BD3IX<Cu9�35;)*#( !� F,X<756#3 � �� �� ���P�H������ �H���P�9�35;w4#1'BD354BD;56C2ePBgX<76���� �5� �)!Z�[F%9�C�ePBD\D\�`�6ZCE7H1eP3y\]9�X<6i:m

0sXPBDC�K5CE6_�KH\�X<1fb(351je X<7H9�X�9�3q?�4\D1'CE6#;�CE6iX C 1'_��)!�4j9�3Y`�6pX<756Z�6i:<1)CE6#X�1'_�9�C !_�K5354iX<B]1�3 F(`�6#4j9�KHCE6 �"! � C49�3>`�6Ze�:<BgXEX<6#3w9�C�X<756Z41'K53qX[9�`H\D68K53HBD1'3>1'_�1�^�6#3>`%9�\D\DC ����� � � L � �[F�9�3H;>B]_�� � C � � � ! �27H9�CPBgX<C�CEK(^5^�1�:EX|6jJIKH9�\�X<1X<756f4\D1'CE6#;~KH35BgXp`%9�\D\V9�35;~B]C|^�1'CEBgX<BDA'6fBD3�X<756)1�^a6#3~K535BgXp`%9�\D\SF�X<7563~1'356)4#1'35CEBD;H6i:<CoX<756fCE6#:<BD6#C � � 4 � � � � "�� %7 % �)9�35;1'356W4j9�3�4[751,1'CE6WX<756WCE6JIK56354#6 4 � CEK54[7�X<7H9�XZX<7H6WCE6i:<B]6#C84#1'3qAq6#:<k�6C8KH35BD_�1�:<=c\g?'F�9�CZe26\D\-9�C{9�3q?�1'_2BgX<CZ;56#:<BgA�9�X<BgAq6C#m¢�756#:<6#_�1�:<6�X<756>�6i:<1�CE6#XW1'_o9�CE=W1,1�X<7�_�K5354iX<B]1�3�49�3�`�6w9�CWBg:E:<6k�KH\]9�:W9�CW1'356>=c9?¶ePBDCE7 �O9�=c1�3Hk�4#\D1'CE6#;�CE6#X<C#Fu1�_4#1'K(:<CE6��[m

0sX�BDC�K5CE6#_�K5\qX<1�b(351jehX<7H9�X�X<756#:<6u9�:<6u1�^�6#3{CE6iX<C�ePBgX<78X<75BD4[bo`a1�KH35;H9�:E?'F�_�1�:�6il59�=)^%\D6-BD_(1�3H6d7H9�C�9P3IK5=c6i:<1�X[9�X<BD1'381�_X<756�^�1'BD3qX<C�ePBDX<7f:[9�X<BD1'3H9�\541,1�:<;HBD3H9�X<6#C-1'_ �"!8F�� v��������%� � � ������� F,9�35;c_�1�: ���/®Z1'356�41'35CEBD;56#:<C�� � ��� � ����� � � �V­

" � �[FX<7563�� � B]Cu1�^a6#3aF(7H9�C2N���¯���U�¤����=c69�CEK(:<6 ( �89�35;YBgX<C-`�1'K53H;H9�:E?cBDCuBgX<Cu4#1'=)^%\D6#=c6#3qX �)!�� � � eP75BD4[7Y7H9�C2BD3(zH3HBgX<6N���¯��aUj¤���y=c69�CEK(:<689�35;y9�3>6=)^5Xs?�BD3qX<6#:<BD1�:P9�CPBgXP4#1'3qX[9�BD35CP351)^�1'BD3qX�ePBgX<7�:[9�X<BD1'3H9�\a41,1�:<;HBD3H9�X<6#C#m

� �

Page 19: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®�(Im-��1'35;H9?�GI9�3IKH9�:E?2�,��F�­�®'®'®(m�%1�:Z9c4#1'3qX<B]3IK51'K5C|_�K5354#X<BD1'3 � _]:<1'= 9WX<1�^�1'\D1'k'BD4j9�\�CM^%9�4#6 �·BD3IX<1r9WAq6#4iX<1�:oCM^%9�4#6'F%X<7H6{CEK(^5^�1�:EXpBDCPX<756{4#\]1�CEK5:<6

1'_dX<756WCE6iX{1'_d^�1'BD3qX<CRQ � � CEK54}7�X<7H9�X � �GQa� �¸®(m � 356W49�353H1�X{;H6izH356WX<756WCEK(^5^�1�:EX{1'_29�ÚP���-R�T�=W6j9�CEK5:<6)1�:@9;5BDCMXE:<BD`HK(X<BD1'3>BD3>X<75BDC�e�9#?qF�9�3H;w1'3568K5CE6C|9W4[7H9�:[9�4iX<6#:<BD�9�X<BD1'3>1'_�X<75684#1'=W^H\D6#=c6#3qXP1'_�X<7H68CEK(^5^�1�:EX§dBgXPB]C�X<7568\�9�:<k'6CMX1�^�6#3wCE6iXP1'3>eP75BD4[7 � BDCP®(F%9�35;>X<75BDC�\D6j9�;HC�X<1fX<7H6Z_�1�\]\D1jePBD35k);56izH3HBgX<BD1'3am� &������D������ §�� ÚP���-R�TØ=W69�CEK(:<6 � � � ���P�@BDCcC<9�BD;�X<1�`�6y®�1'3/9�3�1�^�6#3�CEK5`HCE6iX � � � BD_ ��� � ����� ®�_�1�:9�\D\ � � C � ���2�[m�� ;5BDCMXE:<Bg`%K(X<BD1'3 � � � �����P�ZBDCfC<9�BD;¶X<1�`�6�®�1'3�9�3¶1�^�63¶CEK(`HCE6#X � � �!BD_ ��� � ���#� ®�_�1�:c9�\D\� � C � ���u�[m

0�3>1�:<;56i:�X<1);H6izH356pX<756ZCEK(^H^�1�:EX�1'_�9)ÚP���-R�Th=W69�CEK(:<6Z1�:P9);5BDCMXE:<Bg`HK5X<BD1'3aF(1'356Z=@KHCMX�;56;5K54#6ZX<7H9�X�`�6#BD3HkW®f1'39)_�9�=cBD\g?Y1�_�1�^�6#3wCE6iX<CPBD=)^H\]BD6#Cu`a6#BD35kc®)1'3wBgX<C�K53HBD1'3aF%9�35;>X<75BDC�BDC�;51'356Z`I?>K5CEBD35kc9f^%9�:EX<BgX<BD1'3>1�_�K535BDXs?'m� ��%"-� j�D������ §dN�6#X� `a6Z9)4\D1'CE6#;>CE6iX�1'_��)!Ø9�35;>\D6#X � 7 F � � �HF(`�689�3>1�^�63>4#1jAq6i:<BD3Hkf1'_ Wm-¢�7563>_�1�:P69�4[7 � � �X<756#:<6{6il(BDCMX<C � 7 � C �� �)!p�2ePBgX<7 - � * *�. L�%i� � 7 � � � 7 Fa®$( � 7 (Ø�f9�3H; � 7 ��� �

7 ���{1'3�9�3�1�^�6#3yCE6#X��¸4#1'3qX[9�BD35B]35kWF(X<756pCEKH=�`a6#BD35kf\]1I4j9�\D\g?WzH35BgX<6'F(BSm 6�m�_�1�:�69�4}79Q � � 7 ��� � 7 X<756#:<6p6il(BDCMX<C�9�3�1�^�6#3>CE6iX � � 41'3qX[9�BD35BD35kWQwCEK54[7>X<7H9�X1'35\g?�9)zH35BgX<6Z3IKH={`�6#:P1'_ � �89�:<68351�XPBD;H6#3qX<BD4j9�\D\g?r®)1'3 � � m�2È<ÆjÆ5#[§rN�6#X� v � C�� � �)!p�[F-ePBgX<7�- � * */. L�%i� � v � � �Y�O® � ���[F�� v �®�9�3H;,I � 1 � v �GQa�MOMQ5�»��Fd9�35;�_�1�: � ��®~\D6iX� � �GQa��� �

"! � v � � � ��m/�H1�:r6j9�4}7 Q � X<756i:<6w6#l,B]CMX<C��[�GQa� � �hCEK54}7�X<7H9�X Q � � 7 � � � 9�35;/®�� L(�GQa��(»�wCEK54[7

X<7H9�X"� � Q � Õ L(�GQa��� � � 7 � � � mh�H1�: � ��Fd\]6iX � � �0Q � Ö§ � >�� ( � QH� ( � �'F29�3H;�9�C � B]Cf41�=W^%9�4iXW9�35;4#1jAq6i:<6;¶`I?¶X<756�1�^�63¶`%9�\D\DC�� � Q � L(�GQa� � _�1�:�Q � � FdBgXWBDC)41Aq6#:<6#;¶`,?�9�zH35BgX<6�3IKH={`�6#:W1'_PX<756=>F-ePBgX<7�4#6#3IX<6i:<C* ��� � F � � `�6BD35kw9�zH35BDX<6)CEK(`HCE6#X{1'_ � m � 3H6)4}751,1'CE6#C � � � =WBD3 ! ��� % L(�+*H�[F�9�35;�_�1�: * ��� � 1'356W;H6#351�X<6#C�� !X<756W4[7%9�:[9�4#X<6i:<BDCMX<BD4W_�K53H4iX<BD1'3�1�_�� � * � ­.L(�+*5���c9�35; �

!� � � �

% � � ! F�CE1>X<7H9�X � !� � C�� � � � * � �.L(�+*5��� � � C�� � � 7 � ! � �9�35; � !� �!�{1'3�� � * � L(�+*H����Fa9�35;yX<756#:<6#_�1�:<6 � � � � ! ��� % � !� �{1'3yX<756@1�^�6#3�CE6iX� � � � ! �� % � � * � L(�+*5���Y�O9�35;� � /®)6#\DCE6#eP756i:<6��[F(eP75BD4[7w4#1'3qX[9�BD35C� � m�%1�:R" � �HF�\]6iX �.�W`�6)X<7H6cCEK5=�1'_�9�\D\ � !� _�1�:{eP75BD4}7)�<�+*5� � "5t�X<756#:<6c=WBDk'7qX8`a6r9�3°BD35zH35BgX<6W3IK5=@`�6i:f1�_�CEK54[7

*%F�`HK(XZ`�649�K5CE6 � >���( � * � ( � 9�35; �!� BDC8®�1'K(X<CEBD;H6����+* � Õ,�[F X<756WCEK5= BDCZ\]1I4j9�\D\g?wzH35BgX<6c9�3H; � � � C��� �T���@�SB]_

BDCZ4#1'=)^%9�4#XF�X<756#:<6)9�:<6)1�3H\g?�9rzH3HBgX<6f3,K5={`�6#:Z1'_-X<6i:<=WC89�3H;~X<756i:<6_�1�:<6 � � � C � � �T���PB]3�X<75BDCZ49�CE6��[m)�IB]=WBD\]9�:<\g?qF\D6#X�� � � � ��� � ��F�X<756fCEK5= `a6#BD35k�9�\]CE1r\]1I4j9�\D\g?Yz%35BgX<6f9�35;~6JIKH9�\�X<1 � � � � Fa9�35;�X<756i:<6_�1�:<6 �9 ��{1'3�� � � � � � m¡u751,1'CE6 � � C � � ! ��CEK54[7�X<7H9�X � �Ø®Y1�3 � 9�3H; � ��®r1'3#� ! � �Wmp�H1�:)" � �%Fa\D6iX � ��� � �

� � � F�eP75B]4[7�BDC C `�649�K5CE6��pÓ ��;51,6#CZ351�XpA'9�3HBDCE7��O9�C�� ¸�)9�35; � � ®Y1'3 ��9�35; � �x®�9�35;��� Ø®Y1�K5X<CEBD;56 �@�[F�9�35;~X<756i:<6_�1�:<6- � * */. L�%i� � �j� �9- � * *�. L�%i� �.�j� � � �'m � 3H6Z7H9�C � � ��� � � � �

� �� F(eP7HBD4[7>B]C8�p1'3���9�C � �*®)1'3��Wm

�{%���.+.�$H��� §�0�_V9WÚP����R�T�=W69�CEK(:<6�� ��� ���P�u1�:o9);5B]CMXE:<Bg`HK(X<BD1'3�� � � �O���P�uB]CP®W1�3 � 7 �-� _�1�: � � �%F5X<7563wBgXPBDC®)1'3 � 7 ��� � 7 m�2È<ÆjÆ5#[§�N�6iX�� ��� 7 ��� � 7 9�3H;� � C � ���u�-ePBDX<7�CEK(^H^�1�:EX��°md¢�7H6i:<6ZB]C�9fzH35BgX<6Z3IKH={`�6#:P1�_�_�K5354#X<BD1'35C � 7 � C�� ��� 7 �ePBgX<7 �57 � 7 �Ø�u1'3 � 9�35;@X<756#:<6#_�1�:<6 � � � 7 ��� �

7 �-Fq9�3H;)9�C � 7 � � C � ��� 7 � 1'35627H9�C ��� � � 7 ���H� ®o9�3H;@`I?{CEKH=W=WB]35kBD3 �u1�3H6{;56#;HK54#6C�X<7%9�X ��� � ���3��®(m20�_ � � C � ���u�[FHX<7563y_�1�:p9WCE=W1,1�X<75BD35kcCE6jJIK56#3546 � � 1'356{;56#zH356#C � � � � � � � F9�35;�_�1�: � \]9�:<k'6f6351'K5k'7~X<756)CEK(^H^�1�:EXZ1'_29�\]\�X<7H6 � � CMX[9?(CZBD3�9rz5l(6;�4#1'=)^%9�4#XZCE6iX � ��1'_&�ot�4#1'35CEBD;56#:<BD35k���9�C89;5BDCMXE:<BD`HK(X<BD1'3aF�X<756f^5:<6#46#;5B]35kY:<6#CEK5\DXpCE751jePCZX<7H9�X ��� � � � �)� ®Y_�1�: � \]9�:<k'6f6#3H1�KHk�7 Fa`HK5X89�C � � � � K535BD_�1�:<=W\g?y1'3� � 1'356Z7%9�C ��� � � �H��\]BD= � � ��� � � � �*�*®(m

±�9�:EX<BgX<BD1'35C-1'_�K535BDXs?fePBD\D\(`�6|K5CE6#_�K5\H_�1�:uCMX<K5;(?(BD35k8751je*_�K53H4iX<BD1'35C-B]3��(RV¯ R�¥,�H��CM^%9�4#6C-`�67H9Aq6P3569�:dX<756�`�1'K535;(L9�:E?�1�_V9�3w1�^a6#3>CE6#Xm

¢�756#:<6W9�:<6f^5:<1�^�6#:EX<BD6#Cp1'_��(RV¯ R�¥,�H�*CM^%9�4#6#CpeP75B]4[7�;51>;H6i^�635;�K(^�1'3~X<756)CE=c1,1�X<7H356#CEC81�_-X<756)`�1'K535;H9�:E? � �oF`HK(X2_�1�:uCE1'=W6|1�X<756#:u^5:<1�^�6i:EX<BD6C-X<756P`�1'K53H;H9�:E?)^%\]9?,Cu351{:<1'\D6�F(9�35;cX<756CE6P^5:<1�^�6#:EX<BD6#Cu9�:<6|C<9�BD;cX<1{`�6|\D1,49�\SF,9�35;cX<756#?=c9?Y`�686il,^5:<6#CECE6;w_�1�:P\]9�:<k'6i:PCM^%9�4#6#C#m� &������D������ §d�H1�:o9�3w1�^�63>CE6iX � � � ! FH9�3wBD3qX<6k'6i:�S �®W9�35;°�)(+*#(-,/FHX<756ZCM^%9�46#� � � �

��� ����P�dB]C�X<756ZCM^%9�4681�_

;5BDCMXE:<BD`HK(X<BD1'35C � � � �O���P�uCEK54[7>X<7H9�XP_�1�:P6iAq6i:E?$� � C�� ���P�u1'35687H9�C � � � ���� �H���P�[m� 356u4[7564<b(C�BD=W=c6#;5B]9�X<6\g?|X<7H9�X�X<756uCM^%9�4#6&� v

��� ����P�[F�eP75BD4}78e�9�C�;56CE4i:<Bg`�6;8^5:<6#A,B]1�KHCE\g?89�C�X<7H6dCE6iXV1'_a�S6jJIK5BgA�9�\D6354#6

4#\�9�CE6C�1'_[��N���¯��aU�¤���c=W6j9�CEK(:[9�`H\D6�_�KH354iX<BD1'35C ��CEKH4[7)X<7H9�X�_�1�:d9�3I?@41'=)^%9�4iX#� � �¶1'356�7H9�C ! � � ��vj���2�[F�eP756#:<6

�j�

Page 20: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

! B]C�X<7H6u4}7H9�:[9�4#X<6i:<BDCMX<B]4d_�K5354iX<B]1�3f1'_ �hF'BDC�BD3H;56#6;@BD;563qX<BD4j9�\IePBgX<7{X<7562CM^%9�4#62;56CE4i:<Bg`�6;@BD3@X<756u^5:<6#46#;5BD3Hko;56izH3HBgX<BD1'3aFeP75BD4}7>BDC�X<7568CM^%9�4#681'_�;5BDCMXE:<Bg`HK5X<BD1'35C�� CEK54}7>X<7H9�X � � � ��vj���P�u_�1�:P6#A'6#:E? � � C � ���P�[m� _�41�K5:<CE6�FP� �� ������

���P�dBDC�351�Xo9# 2��T���²�³hCM^%9�4#6'F5`HK(X|9)�V� ��a²�³��%�°CM^%9�46�m� 354#6y1�3H6>ePBD\D\�7H9Aq6>^5:<1jAq6#;�X<7%9�Xr_�1�:�� ( * � & 1'356y7H9�C9� v:� �5� �)!Z� � ��� � � �)!p�@ePBgX<7 v

� � � v�> v!F

X<7563�1'356@ePB]\D\�9�K(X<1'=c9�X<BD49�\D\D?>;56#;5KH4#6@X<7H9�XZ_�1�:89�3I?y1�^�6#3~CE6#X)� ���)! 1'356f7H9�C�� v'� �H���P� � � � ���������P�[mp0�35;56#6;~_�1�:

� � C � ���P��9�35; �<� � v'� � ���P�[F%X<756f_�KH354iX<BD1'3 � � 9�\DCE1r`�6\D1'35k'C|X<1 � v'� � ���P�P9�35;�B]Cp®c1'K(X<CEB];56{X<756@CEK(^H^�1�:EXp1�_ �-F9�35;�`,?�6il,X<6#35;HBD35kYBgXo`I?y`�6BD35k>®r1'K(X<CEBD;56 �oF 1'356@z%35;5C89r_�K5354#X<BD1'3��� �!� � v'� �H� �)!p� � ��� � � �)!Z�[FaCE751jePBD3HkYX<7H9�X1'356Z7%9�C�� � � ��� � ���P�[m� ��%"-� j�D������ §¶�SB��{0�_f� ( * � 54� L ( , 9�35; v7 � v

�Ó v8 FuX<756#3�_�1�: � � � v'� �H���P�f9�35;,F � � v'� 8 ���P�@1�3H6>7%9�C

� F � � v'� 7 ���P�o�O9�3H; � � � F�� � v:� 7 ( CE� � � � � v'� � � � F � � v'� 8 �[m�SBDB+��0�_��)(+* � 54� - � & 9�35; v� � v�Ó v8 > v

!FqX<756#3c_�1�: �6� � v'� �5���P�V9�35; F � � v'� 8 ���P��1'356P7H9�C � F � � v:� �

��� ����P�[m

�2È<ÆjÆ5#[§-¢�756oz5:<CMX�^%9�:EX�BDCP9f4#1'35CE6jJqKH6#3546p1'_�9�^H^H\g?(BD35k)n �R�¥�������BD3H6JIKH9�\DBgXs?cX<1@X<756Z_�1�:<=@K5\]9 � � ��� ���� �

� ������ �

FpÓ � ������ � m�H1�:�^5:<1jA(BD35k@X<756p_�1�:<=fK5\]9(F(1'356p=@KHCMX�CE7H1e X<7H9�X > I J � F ������ � OMQ9� I J � ������ � F|Ó � ������ � � ��OMQ>_�1�:�6iAq6i:E? � � C � ���P�[m� 3H6o4}751,1�CE6C � � C � ���P�VePBDX<7 � � �o1'3rX<756pCEK(^5^�1�:EX � 1'_��-F59�35;YX<756p_�1�:<=fK5\]98X<1@`�6o^5:<1jAq6#;�;51,6#C�351�X�4[7H9�35k'6pBD_1'356p:<6i^H\�9�46#C � `I? � � �O9�C�X<756Z;56i:<BgA'9�X<BgAq6o1�_ � A�9�35BDCE756C�1'3�X<756ZCEK(^5^�1�:EX�1'_����[F(`HK(XP9�C � � 6il,X<6#35;H6#;�`I?�®f1'K(X<CEBD;H6��B]C�9f_�K5354#X<BD1'3�1�_H� v'� � � � ! �[F51'356Z=c9?�9�^5^5:<1q9�4[7wBgX�`I?�9)CE6JIK56354#6�� � � C � � � ! �[FH9�3H;w9�C2X<756Z_�1�:<=fK5\]9fB]CuXE:<KH6_�1�: � :<6i^%\]9�4#6#;�`I?�� � 1'3562�EK5CMXP\D6iX � X<6#35;>X<1$, 9�35;w69�4[7>X<6#:<= 41'3qAq6i:<k'6#C�X<1)X<756p:<BDk'7qX|JIKH9�3qX<BDXs?'m

¢�756|CE6#4#1'35;W^�9�:EXuBDCdCEBD=WBD\�9�:u9�35;c4#1'35CEB]CMX<C-B]3WK5CEBD3HkZX<756|_�9�4iXdX<7H9�Xd1'3WX<756�CEK5^5^�1�:EX � 1'_ ��1'356P7H9�C � � ��� � �����9�35;�F � � 8 � �����[m

� ^5:<1�^�6#:EXs?weP75B]4[7�BDCoXE:<K56fBD3#�)! `%K(Xp3H1�XZBD3�6#Aq6i:E?y1�^�6#3�CE6#X)�xBDC|X<756f_�9�4#XoX<7H9�X�� v:� 4#1'BD354BD;56C|ePBgX<7wX<7H6CM^%9�4#681'_�N�Q A�U�²�³�Q���µ)_�K5354#X<BD1'35C#m� ��%"-� j�D������ §5� v:� >� �)!Z� � ��� *�� �)!Z�[m!0�_ � B]C>9�3 1�^�63 CEK(`%CE6iXy1'_"�)!{FoX<756#3 ��� *����P� � � v:� ����P�[Fp9�35;� v'� ���P� � � ���P�#��� * ����� ���P�[F�eP756i:<6 ��� * ����� ���P�pBDCZX<7H6y�S�V� ��a²�³��%�V�oCM^%9�46c1'_�\D1,49�\D\g?�N�Q A�U�²�³�Q���µw_�KH354iX<BD1'35C#tBD_ �$� � v'� >���P�|9�35; � � KML.N4O�� � ��� � ( �hFaX<756#3�1'356W7H9�C�� � �GQa�*> � �+*5��� ( ��O J �GQ � *5�[FaeP756#:<6#O J BDCpX<756fk'61I;H6#CEBD4;5BDCMX[9�3546Z_�:<1'= QwX<1"*YBD3 �oF5X<7568CE751�:EX<6#CMX|\D6#35k�X<7>1'_�9WCE=W1,1�X<7Y^%9�X<7w4#1'353H6#4iX<B]35kWQwX<1"*YBD3 �om�2È<ÆjÆ5#[§20�_ � � � v'� >� �)!Z�[F�9�35; � � BDCo9rCM^�64#B]9�\�CE=c1I1�X<75B]35krCE6JIK56354#6'F�X<7563 ��� � � � � �<� C��� �)!Z�[F3� � ��� � � (� � � � � 9�35;)_�1�:T"�� � ���������:& 1'356�7H9�C �� �� ��� %��� � �� �� ( �� �� ������ � �� �� ((� � KML.N4O�� � ��� � FI9�35;W9�CVX<75BDC�BD356JIKH9�\DBDXs?@9�^H^H\DBD6C�X<189�3q?;5Bg:<64iX<BD1'3fBgX�BD=W^H\DBD6C�X<7%9�X� KML.N4O�� ��� ��� ((� � KML.N4O�� � ��� � BD3��)!8FI9�35;fX<7H6i:<6#_�1�:<6�� ��� �GQa� > ��� �+*5��� ((� Q�> * ��� � KML.N4O%� � ��� � _�1�:29�\D\�Q � * � �)!{t(9�Cu98CEK(`HCE6jJqKH6#3546P1'_ ��� 4#1'3qAq6#:<k�6Cu9�\D=c1�CMXu6iAq6i:E?,eP756#:<6�X<1 � F,1'356P;56;5K546#C-X<7%9�XR� � �GQa��> � �+*5��� (� Q >�* ��� � KML.N4O�� � ��� � _�1�:f9�\D=c1�CMX{6#Aq6i:E? Q � * � �)!{FVBSm 6'm � B]C{N�Q A�U�²�³�Q���µw4#1'3qX<B]3IK51'K5C{ePBgX<7°N�Q A�U�²�³�Q���µy4#1'35CMX[9�3qX� � KML.N4O�� � ��� � mØ¡u1'3qAq6i:<CE6\g?�B]_ � � ��� *�� �)!p�W9�35; ��� � � � � � � C �� �)!Z�[F2X<7H6#3/_�1�:�9�3q?9( � �"! 1'356y7%9�C��� > � � � ��� � � � > � � � � � � � � FHX<7563w1'356{;56#;HK54#6CW� � ��� > � � � ��� � � ( � � � > � � � � � � ( � - � (B�¹FHeP756i:<6 �·BDC�X<7H6N�Q AaUj²a³�Q���µ@41�3HCMX[9�3IX�1'_ � F59�35;YX<756i:<6_�1�:<6p9�_]X<6#:2;5BgA(BD;5B]35k{`I? -o9�35;�\D6iXEX<BD3Hk�-|X<6#35;rX<1)®@1�3H6o;56#;5KH4#6#C2X<7H9�X�9�X26#A'6#:E?^�1'BD3qXpX<756f;56#:<BgA�9�X<BgAq6@1'_ ��� BD3�9Y;5BD:<6#4iX<B]1�3 (�BDCo`�1'K535;56;�`I?�� � (B�¹F BSm 6�mE� � KML.NMO�� ��� ��� � ( � 9�35;�X<7H6#3~\D6iXEX<B]35k �X<6#3H;�X<1 , k'BgAq6#C�� � KML.NMO�� � ��� � ( �hm

¢�756W^5:<6#46#;5B]35ky9�:<k'K5=c6#3qX{BDCZA�9�\DB];�BD_ �,� ��� *����P�[F�9�C � � � � BDCZe26#\D\-;56iz%356#;°9�X@9>CE751�:EX@;5BDCMX[9�354#6W_�:<1'= X<7H6`�1'K535;%9�:E?qm�¢�7568^%9�CEC<9�k�6@_�:<1'= 9)`�1'K53H;y1�3 � � KML.NMO�� � ��� � X<1r9W`�1'K535;y1'3 � � �GQa�B> � �+*5���,_�1�: � � C ����P�u:<6\DBD6#CP1'3X<756o_�9�4iXuX<7H9�XuX<756|CE6k'=W6#3IX ! Q *&#aBDCuBD354\DK5;56;cBD3 �oF(9�35;YBgXu4j9�3r`�6P:<6#^H\]9�4#6;c`I?WX<756|CEKH=�1�_aX<756o\D6#35k�X<75Cu1'_ CE6#k'=W63qX<C_�1�:P9{^�1'\g?(k'1'3H9�\%^%9�X<7{�E1�B]35BD35k�Q>X<1 *c9�35;�CMX[9?(BD35kfBD3HCEBD;56)�oF59�35;YX<756pBD3(zH=fK5=�1'_ X<756#CE6ZJIKH9�3qX<BgX<BD6CuBDC2X<756ok'61I;H6#CEBD4;5BDCMX[9�3546 O J �GQ � *5�[m

0�_ � BDC�X<7H681�^a6#3wCEK(`HCE6#X|1'_��8ºZ;56#zH356;wBD3>^�1'\]9�:P4#1,1�:<;5BD3%9�X<6C�`I? >��� � ���h9�35; L ����F5X<7H6#3>X<7568_�K5354iX<B]1�3� � � C<9�X<BDCMzH6C ���

���� > !

7 ��9�35; � �� ! � �7 �(F|CE1¶X<7H9�X � � � v:� ����P�[FP`HK(Xw_�1�: � �·®¶CE=c9�\D\|X<756~^�1'BD3qX<CYePBgX<7¡29�:EX<6#CEB]9�3�4#1,1�:<;5BD3H9�X<6#CZ�:>P­ � > ���d9�35;��:>P­ � Ó ���d9�:<6Z9�X�ÙdK54#\]BD;569�3�;5BDCMX[9�354#68­ �ZeP75BD\D6|X<7H6o;5BD �6i:<6354#6pBD3rA'9�\DK56#C�1�_ �BDC�41�3IA'6#:<k'BD35k)X<1r­'�h9�C ���Û®>�+X<756Zk'6#1,;56#CEB]4Z;5BDCMX[9�3546pX<6#35;5C�X<1r­�� �PÓ ��

� �[m

­�®

Page 21: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®��,m�1°6#;53H6#CE;H9?��%6i`5:<K%9�:E?y­,F%­�®'®'®�(RV¯ R�¥,�%� × C)BD=@`�6#;H;5BD35kyX<7H6#1�:<6#=·:<6jJqKHBg:<6#CfCE1'=W6Y:<6k'K5\]9�:<BgXs?°1'_PX<756Y`�1'K535;%9�:E? � �omh�H1�:W6#l(9�=W^H\D6�F-1'356�7%9�C

; v � � º �*��� v:� º � � º � � � � � � º �V_�1�:26iAq6i:E? * � ! ­ � ,��[FI`HK(XdX<756PCEBD=cBD\]9�:-^5:<1�^�6i:EXs?W;H1I6Cd351�Xu751'\D;c_�1�:dX<756�^�9�:EX<B]4#K5\]9�:1�^�6#3cCE6iX&�,���I�GQ � *5�d§I® � Q � � � ® ��*� Q�º��'m��H1�:-^5:<1jA(BD35kZX<75B]CV:<6=c9�:Eb�F'1'356P4[7H6#4<b(C-_�1�:�eP7HBD4[7)A�9�\DKH6)� � �*X<7H6_�K5354iX<B]1�3 � �GQa�H�5Q �>`a6#\D1'35k'C�X<1 ���5���P�29�35;y9�CP1'35683566#;5CI v� Q � � � ºPOMQ���I J Q � �HO Q O,* � ,/F%1�3H6ZzH35;5C�X<7%9�X�X<7H64#1'35;HBgX<BD1'3rBDC8� *8Ó�­ ��>8��t,X<756#:<6#_�1�:<6 �6� ;°vj���P�-BD_�9�3H;�1�3H\g?cBD_�­(���E>°���HÓ�­ ��>8��F(BSm 6�m*� ��> vº F(9�35; � � ���H���P�BD_�9�35;w1'35\g?YB]_B� � � � > vº F5BSm 6�m*; v ���P�uB]C�351�X|9fCEK5`HCE6iX|1'_�� � ���P�u_�1�: * �/Ò,m � 35684}756#4[b(C�6j9�CEB]\g?cX<7H6Z\DBD=cBgX[9�X<BD1'35C2_�1�:1�X<756i:P4K5CM^HCP1'3�X<7H6p`�1'K535;%9�:E?qFH\DBgbq6Z_�1�: � ���I�GQ � *5�d§I® � Q�� � � ® ��*� Q

��qF5ePBgX<7 � � ��m

��^%9�:EX{1'_P�(RV¯ R�¥,�H� × C{BD={`a6#;5;5B]35k�X<7561�:<6=�9�CECE6i:EX<C{X<7H9�X{_�1�:Y� ( *�� & 1'356c7H9�CW� v'� �5� �"!p� � ��� � � �)!p�ePBgX<7 v

� �� v�> v!F(1�: * � � ! �

!"�FH9�3H;�1'356p;H6#;5K546#C�X<756#3YX<7H9�X � v'� �H� �)!p� � � 8 � �)!Z�-_�1�:�6#Aq6i:E? 5 � ! * � *

� #a`I?cX<7H6_�1�\]\D1jePBD35kW9�^H^H\DBD4j9�X<B]1�3�1'_�n �R�¥������°BD356jJqK%9�\]BgXs?'m� & ����$ §f0�_8�#( * � � * � ��* v ( , 9�35; � � �O® � �j�ZBDCf;56iz%356#;h`,? v

���� v " �� � Ó �

� /FVX<7563h1'356�7H9�C � � � � � ��� (

� � � � � v"�

� � � � � � � �� / _�1�:|9�\D\ � � � � � ���P� �� � / ���P�[m�2È<ÆjÆ5#[§ � 356�9�^5^H\DBD6C�n �R�¥������ B]356JIKH9�\DBgXs? � � @�K � � v ( � � @ � � 8 � � K � � 8 ePBgX<7 @ � � � � � v " � � ���~9�35;5K(� � � � � ����F|ePBgX<75 � � �� v " � � ��� 9�35; 5 � � � /� ��� FHeP75BD4[7y9�:<684#1'3��EK5kq9�X<6Z6il,^�1'3563qX<C#m

¢�756Z^H:<6#4#6;5BD35k):<6#CEK5\gXPB]C�351�X�:<6#CMXE:<BD4#X<6#;�X<1fX<7H6ZN���¯��aUj¤���y=c69�CEK(:<6�F�9�3H;�X<7H6p:<6#CMXE:<B]4iX<BD1'3YX<7H9�X * � ��ZBDC�351�X3564#6#CEC<9�:E?��O9�\DX<751'K5k'7YX<7568351�X[9�X<BD1'3!� � F � � 7 BDC�3H1�X|9W351�:<=�_�1�:|® � L � �j�[m�(RV¯ R�¥,�%� × CPBD={`�6;5;5BD3Hk@X<7561�:<6#=�B]C�3H9�X<K(:[9�\aBD_�1'356841'35CEBD;56#:<C�X<7568JIK56CMX<BD1'3>1'_�CE4j9�\]BD35k5m

� ��%"-� j�D������ §�0�_��)( 5 �+*r1�:|��(+* � & 9�3H; 5 �+*� F'X<756#:<6�BDC-351pz%35BgX<6�41'35CMX[9�3qX C CEK54}7)X<7H9�X � � � � � 8 ( C � � � � � v'� �_�1�:|9�\D\ �6� C�� � �"!p�[m

�2È<ÆjÆ5#[§��%1�:�� � ®(F�1'356�9�^5^H\DBD6CWX<7H6yBD356JIKH9�\DBDXs?�� � � � � 8 ( C � � � � � � Ó�C � � KML.N4O�� � ��� � � X<1�X<756y_�K5354#X<BD1'3,F�;56iz%356#;`I?$F��GQa�E� � � �� �w_�1�: Q � �)!{m � 356w351�X<BD46#C)X<7H9�Xc_�1�:y� ( L�� , 1'356w7H9�C � � F � � 7 � � I�� 1 �� � � �� � �� 7 O Q � v � 7 �� I � 1 � � �+*H��� 7�� ! O&*� v � 7 ��� ! � 74� � � � � 7 F�9�35; � � KML.N4O��GF(��� � 7 ��� " v � ! � 74� � KML.N4O�� � ��� � 7 m�¢�756i:<6_�1�:<68BD_�X<7568B]356JIKH9�\DBgXs?Ye�9�CXE:<K56p1'356|e21'K5\];�;56#;5KH4#6oX<7H9�X�� ! � 8 � � � � � 8 ( C� ! � ��� � � � � � Ó<C�

" v � ! � � � � KML.N4O�� � ��� � � F(BOm 6'm�9�3�BD356JIKH9�\DBDXs?W1'_ X<756p_�1�:<=� � � � � 8 (�C��� � � � � � � Ó C�� � � KML.N4O�� � ��� � � _�1�:89�\D\� ��®(mp0�_-1'356f7H9�; � ��®Y9�35; � ��®(F�X<7H6#3�`I?y\]6iXEX<BD35k���X<6#35;�X<1

®>1'356We21'K5\];�;56#;HK54#6WX<756c41'3qXE:[9�;5BD4iX<B]1�35� � � � � 8 �!®>_�1�:@9�\D\ � � C�� � �)!Z�[t�X<75BDC{4#1�:E:<6#CM^�1'35;HC8X<1>X<756W4j9�CE6 5 �-*�m�IBD=cBD\]9�:<\g?�BD_P1'356�7H9�; � ��®~9�35; � � ®~1'356Ye21'K5\];¶;H6#;5K546YX<756�4#1'3qXE:[9�;5B]4iX<BD1'3h`I?¶\D6iXEX<B]35k���X<6#35;¶X<1 ,�tuX<75B]C4#1�:E:<6CM^�1'35;5C�X<1fX<7H6Z49�CE6 5 �+*

� m0�_�X<7H68BD356JIKH9�\DBDXs?rBDC�XE:<K568_�1�: 5 � *

� F5X<7H6#3>X<7568C<9�=W6{9�:<k'K5=W63qX|CE751jePC�X<7%9�X|1'35687H9�CW� � � � � � � ( C5� � KML.N4O%� � ��� � �_�1�:Z9�\D\ � � C�� � �)!Z�[mPno1je26iAq6i:F%X<7HBDC|BDC|351�Xp9W^5:<1,1'_�X<7%9�XoX<756{BD3H6JIKH9�\DBgXs?>BDCPXE:<K56�F�9�C|_�1�:o6#l(9�=W^H\D68X<756{BD356jJIKH9�\DBgXs?� � � � � ( C � � KML.N4O�� � ��� � ! B]=)^H\DB]6#CZ3H1w4#1'3qXE:[9�;5B]4iX<BD1'3�`I?~X<756W^H:<6#4#6;5BD35kwCE4j9�\]BD35kw9�:<k'K5=W63qXF�`HK(X@BgX{BDC83H1�X{XE:<K56c_�1�:& � ��m

� 356�:<6j9�CE1'3*eP7q?*1�3H6~49�353H1�X>;H6#;5K546�`I? 9¶CE49�\DBD35k�9�:<k'K5=c6#3qX�X<7%9�X�X<7H6�\]BD=WBgX<B]35k°4j9�CE6�1'_8X<756°�(RV¯ R�¥,�H�BD=@`�6#;5;HBD35krX<756#1�:<6= ;51,6#Cp351�XZ7H1�\];�_�1�:&* � & F�BDCoX<7H9�XpBD3~X<756f\]9�:<k'6i:Z4#1'3qX<6#lIXZ1'_�X<756)N�R����%T���µ5vpCM^%9�46#CZ9�\D\�X<7H6CM^%9�4#6C&� � � 8q� � ! ��CE4j9�\]6PB]3cX<756|C<9�=c6Pe�9?c_�1�:2;5BD �6i:<63qXdA'9�\]K56#Cu1'_ 5 � !©� � , #SmV0�_�9�\D\HX<7H6|^%9�:EX<B]9�\%;56i:<BgA'9�X<BgAq6#Cd1'_ � 9�:<66#CMX<B]=c9�X<6#;cBD3$� ! � vj� �)!Z�VBgXd^5:<1jA,B];56#Cu9Z`�1'K53H;c_�1�:dX<756PCEK5^Y351�:<=¸1'_ � FIeP7HBD\D6�_�1�:�9�3q? 5 � ��X<756i:<6|6il(BDCMXdK53I`�1'K535;56;_�K5354iX<B]1�3HC�FcePBgX<7w9�\]\�^%9�:EX<B]9�\a;56#:<BgA�9�X<BgAq6C2B]3 � ! � 8 � �"!p�[m

¢�7568=c6iX<751,;>1'_-�5RV¯ R�¥I�%��_�1�:P^5:<1jA,B]35k)X<7568BD={`a6#;5;5B]35kfX<7561�:<6=�_�1�:� v'� �5� �"!p�uBD3>X<756849�CE6c�"(+* � & e�9�C`%9�CE6#;w1'3>X<756ZKHCE681�_V9�3>6#\]6#=W63qX[9�:E?rCE1'\DK5X<BD1'3>1'_�X<7568N�9�^H\]9�4B]9�3�?rm� &������D������ §�0�_ �W� �I� � � � N � � ��BDCp9W^�1'\g?(351'=WB�9�\�BD3�� � �)! �+ePBgX<7y41'35CMX[9�3qXo41I6#�c4#BD63qX<C}�[F�9�35; �W�� ���;56351�X<6#CX<756);HBg �6i:<63qX<B]9�\V1�^�6i:[9�X<1�: �W�� �� � � � N � � F�9�3�6\D6#=c6#3qX[9�:E?yCE1'\DK5X<BD1'3�1�_ �W�� ��PBDC89�;HBDCMXE:<Bg`HK(X<B]1�3�� CEK54[7�X<7H9�X�W�� ����(� � � m

v Ðo61�:<k'6{ÐoKH3qX<756i:PN�R����%T���µ5F(ÚPK5CECEB�9�35LS`�1�:<3�=r9�X<7H6#=c9�X<BD4B]9�3aFI`�1�:<3�BD3��j�,�®(m�no6Z6#=WB]k�:[9�X<6;cX<1)¦o35BgX<6;w�qX[9�X<6#CBD3��j�'Ô��,FH9�35;w756Ze21�:Eb(CP9�XP¦o35BgAq6i:<CEBgXs?�1�_V¢�6il59�C#F5�oK5CMX<B]3am

­(�

Page 22: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

Ùd\D6=W6#3IX[9�:E?8CE1'\DK(X<B]1�3HCV9�:<6u351�XVKH35B]JIK56�F�`HK(X-9P^%9�:EX<BD4#K5\�9�:V6#\]6#=W63qX[9�:E?8CE1�\]K(X<BD1'3@=c9?{1'_]X<638`a6uCE6\D6#4#X<6#;{`I?@K5CEBD35kCM?(=W=c6iXE:E?r9�:<k'K5=c6#3qX<C#F%9�35;�B]3rX<756p4j9�CE6p1'_ ? � � !7 ��v �

������ F51'356pzH35;5C�:[9�;5B�9�\aCE1'\DK(X<BD1'35C21'_�X<756Z_�1�:<= � ��C ! L'º

"!

_�1�: & (�,F � � C º \]1�k3Lr_�1�: & �¸­Y9�35; � � ���º _�1�: & � ��m@�o3qX<BD4Bg^%9�X<BD35kY1'3�X<756f^5:<1�^�6#:EX<BD6#Cp1'_d4#1'3qAq1'\DK(X<B]1�3

ePBgX<7W;5BDCMXE:<BD`HK(X<BD1'35C#Fq1'356P7H9�C � � � � ? � � � !7 ��v ������ � � ������ � F,9�35;WX<756�:<6#CEKH\gX�e21'K5\];)`�6|9Z4#1'35CE6jJqKH6#3546�1�_9�8R��TV¤ × CBD356jJIKH9�\DBgXs?wBD_-1'356f7H9�; ������ � � � ! � �)!Z�[F�`HK(X89�CoX<756#CE6);56#:<BgA�9�X<BgAq6#Cp9�:<6f1�_-X<756f1�:<;56i:Z1'_*L,v

"! X<75BDCo_+9�B]\DC|X<1r`�6@X<7H6

49�CE6{�O9�C & /­q�[Fq`HK5X��5RV¯ R�¥I�%�~^H:<1Aq6;WX<7H9�X+�8R��T�¤ × CdBD356JIKH9�\DBDXs?@CMX<B]\D\5751'\D;5CuBD_�BD35CMX<6j9�;c1'_a98_�K53H4iX<BD1'3cBD3$� 8 � �)!p�1'356|K5CE6#CdX<7H6P_�K5354#X<BD1'3 L " ! � 8 m-¢�75BDCu\DBD356P1'_ 9�:<k'K5=c6#3qXu`I?Y�(RV¯ R�¥,�%��e�9�CuBD=)^5:<1jAq6#;r\]9�X<6i:d`I?cGI9�9�bW±d���%������FIK5CEBD35kX<756oX<756#1�:E?r1�_�BD3IX<6i:E^�1'\]9�X<BD1'3ceP75BD4[7Y756p7H9�;�;56#A'6\D1�^�6#;YBD3r^%9�:[9�\D\D6\HePBDX<7rGI9�4jJqKH6#CMLON�1'K5B]C2N�Q+R�T�U'FI`HK(X�X<756|^�9�:EX<B]4#K5\]9�::<6#CEKH\gXu1'_aBD3qX<6#:E^�1'\]9�X<BD1'3WBD3cNVR�����T���µ{CM^%9�4#6#Cu7%9�;r`�6#63c1�`5X[9�BD356;W`,? �{× �o�%Q�¥5º�FqeP751{e�9�Cu6il,X<6#35;HBD35k@9Z:<6CEK5\gX29�`�1'K(X351'35BD3H4i:<69�CEBD3Hkf:<6j9�:E:[9�3Hk�6=W63qX<C�1'_�no��� � ��9�3H;>N�Q���� ¥,�H´>R�R�� � m

� CE6#41�3H;>=c6iX<751,;>_�1�:�^H:<1A(BD3Hkr�5RV¯ R�¥I�%� × CPB]={`�6;5;5BD35k)X<756#1�:<6#=�e�9�CP;56iAq6\D1�^5^�6#;wBD35;56#^�6#35;H6#3qX<\g?Y`I?�Ùd=WB]\DBD1Ðo��¤V¥,Q���� �-R � 9�35;�`I?YN�1�KHBDC��oQ����%TV¯��%��¤2F,`HK(X�X<756pC<9�=W6pB];569f7H9�C�9�\DCE1f`a6#6#3�KHCE6#;�`I? � \]k'9fN���� �-µ'³���TVU�����%� � m� 3H6�zH:<CMXW^5:<1jAq6#CcBD356JIKH9�\DBDX<BD6#Cf_�1�:cCE=W1,1�X<7�_�K5354iX<B]1�3HCfePBDX<7�4#1'=W^%9�4iXWCEK(^H^�1�:EXF�9�3H;�X<7563�1'356w6il,X<6#35;HC)X<756=·X<1�(RV¯ R�¥,�%�¶CM^%9�46#C�`I?�;5635CEBgXs?qm� & ����$ §V�H1�: @ � C�� � �{�o�O9�35;�`,?�;5635CEBgXs?Y_�1�:@ � � v'� vj� �@�E�[F51�3H687H9�C|­�� @��GQa��� ( I � �� � �� � �� OMQy_�1�:|9�\D\ Q � �fm�2È<ÆjÆ5#[§>�H:<1'= @��GQa��� I �" � �� � �+*5�4O&* � > I

�� �� � �+*H�MO,*%F21'356w;56;5K54#6CWX<7H9�X � @��GQa���'(+I � �� � �� � �� O&*�9�3H; � @��GQa���'(I

��� � �� � �� O,*%FH9�3H;>9�;H;5BD35k@X<756CE6|Xse21fB]356JIKH9�\DBgX<B]6#Cdk'BgAq6#CuX<756|:<6CEK5\gXm��oC�C � � �{�-BDC2;5635CE6oB]3 � v'� v � �@�[F(BgX2CE751jePC�X<7H9�X

� v'� vj� �{� � C � � �{�[m� & ����$ §c�H1�: & ­,Fu9�35;����»� ���������:& Fd\D6iX�@ 7 `�6>9�=W6j9�CEK(:[9�`H\D6Y_�K5354#X<BD1'3¶BD35;H6i^�635;56#3IX)1'_)Q 7 Fu9�35;�9�CECEK5=c6X<7H9�X9@ 7 � � !

" vy�+eP7563�1'356>:<6#CMXE:<BD4#X<CfX<1 Q 7 � ®~_�1�:c6#l(9�=W^H\D6��[FdX<756#3 ��� !7 �Vv @ 7 `�6#\]1�3Hk�C)X<1 ��vj� �)!p�f9�35;� � 9� � v (�� !7 �Vv � � @ 7 � � ! " v m�2È<ÆjÆ5#[§@¢�7H6c4j9�CE6 & � ­>BDC{1�`IA(BD1'K5C@9�C c�GQ v�� Q º �R� @ v �GQ º �:@ º �GQ v �Z9�3H;�� � 9� � v � � � @ v � � v � � @ º � � v m>�H1�: & !­>1�3H6K5CE6CP9�3wBD35;5K54#X<BD1'3�1'3 & m-�%1�:��B� � ���������:& >���FH\D6iX8K 7 � � I � � @ 7 � ! " v'OMQ ! � v � � !

" º � FHCE1)X<7H9�X8K 7 BDC�BD35;56#^�6#35;H6#3qXP1�_Q 7 9�35;6Q ! 9�35; K 7 � � !

" º8B]3yBgX<C & >¶­c9�:<k'K5=W63qX<C#F�ePBgX<7 � � K 7 � � ! " º ( � � @ 7 � � � ! " v � � � ! " º �!" v F�9�35;y`,?>X<756{BD3H;5K54iX<B]1�3

7q?,^�1�X<756CEBDCc1'356y7%9�C � ��� ! " v7��v K 7 � ��vrBD3�X<756�9�:<k'K5=W63qX<C�Q v���������� Q ! " v m 2?�BD3qX<6k�:[9�X<BD3Hk�z5:<CMXYBD3 Q ! 9�35;

K5CEBD3Hk>n �R�¥�������BD3H6JIKH9�\DBgXs?qFa1'356W7H9�C#I���� 9� O Q ! (�� !" v7��v K

� ! " º � � � ! " v �7 @ ! � �� ! " º � � � ! " v � @ ! F�9�35;�X<756#:<6#_�1�:<6�F

9�CdX<7H6|4#1'3��EK5kq9�X<6|6il,^�1'356#3IX21�_ & >��|BDC ! " v!" º F,1'356o7H9�C�I � 1 � 9� OMQ#( I � 1 / � � � � ! " º � � � ! " v � � @ ! � OMQ v ����� O Q ! " v (

� I�� 1� / � � � OMQ v ����� OMQ ! " v � � !" º � � � ! " v � � � @ ! � � ! " v (�� !7 ��v � � @ 7 � � ! " v m� ��%"-� j�D������ §d�%1�:)N ���Z9�35;°���+* �-,/FH1'35687H9�C

�� � 1

� � � !�+ � � ! " v � O Q�� � ! " v � � ! ( N ­ �!�7��v

��� ���� ���Q 7

��� ��� � �v � ! ��

� 1� � � � + " v � � OMQ�� v � ��

_�1�:|9�\D\ �6� C�� � �"!p�[m-�H1�: * �Ø�81'356Z7H9�CW� � � � � v � ( vº � � !7 �Vv �� �� � �� � � �� �� v � v � ! _�1�:o9�\]\ � � C � � �)!Z�[m�2È<ÆjÆ5#[§a�|^5^%\g?(BD35k2X<7H6Vz5:<CMX�\D6=W=r9dePBDX<7�@9� � � � +�_�1�: �6� C � � �)!p�V�+`�6#4j9�KHCE6 � � � +�BDC 1'_,4#\]9�CECBC)v�9�35;ZX<756�z5:<CMX�\D6=W=c91'35\g?8:<6jJqKHBg:<6#C�X<7562_�K5354#X<BD1'3{X<1o`�6#\]1�3Hk|X<1R� v'� vj� �{�E�[F'1'356�1�`5X[9�BD35C � � �GQa��� + ( + º I���� � � + " v �� ������ � �� OMQ 7 F'CE1pBD_H1'356�;56351�X<6#C@ 7 � � + º I ��� � � + " v �� � �� � � �� OMQ 7 � v � !

" v 1'356�=c9?@9�^5^%\g?8X<7562CE64#1'35;f\D6=W=c9o9�35;f1'356�;56#;5KH4#6#C I � 1 � � �GQa��� !�+ � � ! " v � O Q+(� ��� !7 �Vv @ 7 � � v (�� !7 ��v � � @ 7 � � ! " v � � + º � ! �

� ! " v � � !7 ��v � I � 1 � � � + " v �� ������ � �� OMQ � v �� ! " v � F%eP7HBD4[7w`I?yn �R�¥��V���¶BD356jJIKH9�\DBgXs?

º~ÚPB]4[7H9�:<;~¡|m �@× �o��Q�¥%FH�o=W6#:<BD49�3�=r9�X<7H6#=c9�X<BD4B]9�3amVno6pe21�:Eb(CP9�Xp�qX[9�X<6Z¦o3HBgAq6i:<CEBgXs?Y1'_��o6#e �-1�:Eb�F5�o\D`%9�3q?'m� Gq1'753�Ùd;5635CE1�:rN�Q�� � ¥,�%´�R�R��2F�Ùd35k'\DB]CE7�=r9�X<7H6#=c9�X<BD4B]9�3aF|��� �'Ô�L<�j� ( (Im/no6w756\D;�X<7H6>356ieP\D?¶_�1'K535;56;�ÚP1'K5CE6

���¥,¥Y±-:<1'_�6CECE1�:<CE7HBg^�9�Xp¡29�=@`5:<BD;5k'6�F��j�'­ ��L<�j�'Ô�®,m1h9�\gX<6#:%1 BD\D\DB�9�=�Ú|1�KHCE6 2��¥,¥�F5Ùd35k'\DBDCE7w=c9�X<756#=r9�X<BD4#B]9�3aFa���'Ô�®�L<�j��­'Ô,m

� Ùd=WB]\DBD1cÐo��¤V¥,Q���� �-RuF50sX[9�\]B]9�3�=c9�X<756#=r9�X<BD4#B]9�3amVno6pe21�:Eb(CPBD3�±�9A(B]9(m� � \Dkq9>�o\D6#b,C<9�35;5:<1A(3H9wN���� ��µ'³��%T�U���&�%�uF�ÚPK5CECEB]9�3h=c9�X<756=c9�X<BD4B]9�3aF `�1�:<3°B]3��j�'­'Ô,my�I756ce21�:Eb(C@9�X@ÚPK5CECEB]9�3

�o49�;56={?Y1'_-�I4#B]6#3546#C#F��qXm-±V6iX<6i:<CM`%K(:<k5m

­'­

Page 23: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

BDC)( � + º � ! �� ! " v � � !7 ��v �� �� ������ � �� �� v �

� ! " v �� � I�� 1 � � � � + " v � �� OMQ � ! � � !

" v � �� F%9�3H;�X[9�b,B]35k@X<7H6p^�1je26i:{� & >��j��� & k'BgAq6#C2X<7H6:<6#CEKH\gXm��%1�: * �Ø��FH1'356pX[9�bq6#C�N �Ø�89�35;w1'356ZCMX<1�^HC�`�6_�1�:<689�^5^%\g?(BD35k)n �R�¥��V�%�°BD3H6JIKH9�\DBgXs?qm

�(RV¯ R�¥,�%� × CdB]={`�6;5;5BD35kZX<7H6#1�:<6#= � v:� �5� �)!Z� � ��� � � �)!Z��_�1'\D\D1jePCdBD3WX<756P4j9�CE6{�)(+* � & `I?W4}751,1�CEB]35kRNWCEK54[7X<7H9�X !�+

!" v ��GN#>���� * � F%9�35;>X<75BDC�4#1'=c=W1'3YA'9�\DK5689�^5^�69�:<C�X<1)`�6&* � m

­��

Page 24: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®q�,m-�5:<BD;H9?��H6#`5:<KH9�:E?�Õ5F�­�®'®'®(m0�_V1'356{;56#351�X<6C � � � � !7 ��v �� �� ������ � �� �� � � v � ! F�9�35;y1'3568;H6izH356C���`I?��o�GN5�3��I � 1 � � � !�+ � � ! " v � O Q�F%X<756#3wX<75684j9�CE6

*�� & 1'_�X<756Z^5:<6#46#;5BD3Hk)B]356JIKH9�\DBgXs?YBDC��o�GNH� ( � +��º � ! � � ! " v � �o�GN�>����2_�1�:|9�\D\ N ����m � 3568;H6#;5K546#C�X<7%9�X|9)`�1'K535;1'3,� � � � � v'� ! F%eP75B]4[7�k'BgAq6#Co9c`�1'K535;~1�3��o� & >/���8� � � � � � !! 9�35;~9W`�1'K53H;�1'3 � F�BD=W^H\DBD6CP9c`�1'K535;~_�1�: � � � � � 8 _�1�:Z9�\D\5 � & m���1�:<6�^5:<6#4BDCE6#\g?qFq_�1�:u9�3I?)BD3qX<6#k'6#:*N9 & FI1'356P7H9�C��o�GNH� ( � + �� ! " v � � � ! �

� ! " v � � � º �� + ��v " ! � ! � � ! " v � �o� & >w���[m

�o_�X<6i:ZX[9�b(BD35k�X<756c\D1'k'9�:<BgX<7H= 1'356WzH35;5C{\DBD=�CEK5^ + � v+ � �o�GNH���� ! " v � � � !�+ � ( �º � F�1�:8\DB]=�CEK(^ 8 � � �

�� �8 (

� ! " v � �º ! � F

9�CP9)41�3HCE6JIK56#3H4#6Z1'_-�,� Q���¥,Q�T�¤ × C�_�1�:<=fK5\]9(v#F�N 2� � + � � + � ­�� N�m�o351�X<756#:V^5:<1�^�6#:EXs?@1'_HX<756�49�CE6�* � & B]C�X<7H9�X�_�1�: �6� � v'� !@� �)!p��9�35; ��^�1'CEBgX<BDA'6�9�35;)CE=r9�\]\(6#351'K5k'7>�+`HK(X�7H1eCE=c9�\D\�;56#^�6#35;HCpK(^�1'3 � �[F ( � � � � BDCp\D1,49�\D\g?>B]3qX<6#k�:[9�`H\D6'mo0�35;56#6;~1'3546@1'356@b(351jePCoX<7H9�X�� � � � � 8 ( � 5 _�1�: 5 5 � 9�35;5 � \�9�:<k'6f6351'K5k'7��O9�35; � �

� ! " v � �º ! � �[F 1�3H6f7H9�C�I�� 1 �

8 � 8 �� � � � � � 8 � O Q<� �

8 � 8 � I�� 1� � � � � � 8 � O Q ( �

8 � 8 �� � ! 8 � 8 � F

eP75BD4}7wBDC ��, B]_ � � ( �x��m�¢�756{C<9�=c6Z:<6CEK5\gXPe�9�Co9�\DCE1c1�`5X[9�BD356#;yBD3y9c;5Bg a6i:<6#3IX�e�9?>`I?>N�1'K5BDCP�oQ�����T�¯�����¤�9�35;�5:<BDX<�@G,R�³�TVº@9�CZ9r^5:<1�^�6i:EXs?�1�_�X<756fCM^%9�46 �� � �! �1'K535;56;~��69�3 � CE4BD\D\]9�X<BD1'3 � �[F�eP75B]4[7yX<756#?y7H9�;~BD3qXE:<1,;5KH4#6#;~BD3^%9�:EXP_�1�:PCMX<KH;(?(BD35k)X<756Z\]BD=WBgX<B]35k@4j9�CE6'* � & 1�_-�(RV¯ R�¥,�%� × C�BD={`a6#;5;5B]35kfX<7561�:<6=>m

�%1�:8X<756)49�CE6 * � & F�1'356)351�X<BD46#CoX<7H9�XpeP7H6#3��GN�>*��� *��-9�3H; !�+!" v 9�:<6)6JIKH9�\�X<756#?yX[9�b'6fX<756)3H6#kq9�X<BgAq6@A�9�\DK56> !"�

�"!F�9�3H;~X<756#:<6#_�1�:<6fBD_u1'356W;56#351�X<6C 5 � � > !"�

�"!Ó � � �cePBDX<7 � � ! � � " v �

� � ! " v � ����F�X<756#3~X<756W4[751'BD46)1'_3N�k'BgA(BD35k�GN�>���� *�� � 5 � k'BgAq6#C !�+

!" v � 5 ���v tH1'356Z4[751,1'CE6#C � � �

��"!CE1fX<7%9�X 5 � � *�m-¦oCEBD35kWN�( 8 �� ( �

� � �IFH1'356pz%35;5C�X<7H9�X � 1

� � � 8 ��� / OMQ ( � � � � �­ * � � ! � � ! " v � � � 1 � � � 8 � O Q����

¢�75BDCZk'BgAq6#C89�3�6CMX<BD=c9�X<6f1�_R� � � � � 8 � _�1�:89�\D\ ��BD3�X<6#:<=WCp1'_ � � � � � v'� � �+eP75BD4}7~k'BgAq6#CZ9Y`�1'K535;�1'3 � � � � � � 9�35;�1'3 � �|9�35;��t�9�C6� � � � � � \DBD= 7 � � � � � � 7 1�3H6�=@K5CMX)CE751je X<7%9�X6� � � � � 8 � BDC@`�1'K53H;56#;�BD35;H6i^�635;56#3IX<\g?°1'_ ��m u?h751'=c1�k'6356#BgXs?1'_PX<756w_�1�:<=@KH\]9(Fd1'356w7H9�C 5 ���v � !

!" v Ó 5 � � 9�3H;�X<756#:<6#_�1�:<6>BD_|1'356�^HK(X<C � � �3� ���º � � F � FuX<7H6>_�1�:<=fK5\]9�`�64#1'=W6CI � 1 � F�� 8 ��� / OMQ ( � � ! � � ! " v � � I � 1 � F�� 8 �TOMQ �

�m�£p6#351�X<BD3Hk@�� �5�3��\D1'k � I � 1 � F � 8 �TO Q �'F�1'3562;56;5K54#6C*@�� ��ÓY��� ( � � Ó

��@�� �H� ePBgX<7 ��� !�� ��� �!" v Fq9�35;@`I?@BD3H;5K54iX<B]1�3{X<75B]C�k'BgAq6C*@�� �H�&(�� � � �H>r���jÓ>� �H>f­'� �|Ó ����� Óc­ � � " � Ó � � " º �HÓ � � @��O®I�_�1�:��6 x­,Fa9�35;�9�C 5 � � > !"�

�"!Ó � � �)1'356@zH35;5CoX<7H9�X � � � �8 � � v� � @��O®I��Ó �� �" v � � � Fak'BgA(BD35kY9c`�1'K535;~_�1�:�� � � � � BD3

X<6i:<=cC�1'_�� � � � � v:� � m�/;5Bg a6i:<6#3IX�e�9?{X<1o1�`5X[9�BD3)9P`a1�KH35;fB]3����� �"!p�[F'_�1'\D\]1ePB]35kZ�(RV¯ R�¥,�%� × CV=c6iX<751,;fBDC�X<1|:<6i^H\]9�46dX<756�6#\D6=W6#3IX[9�:E?

CE1'\DK(X<BD1'3 ��1'_�?*`I?{9�^%9�:[9�=W6#XE:<Bgl ��F�eP75B]4[78BD35CMX<6j9�;81'_(CE1'\gA(BD35k ? � � � � C<9�X<BDCMz%6#C ? (� � � Ó � ePBgX<7 � � C � � � ! �[t1'3568X[9�bq6C � � �¸ePBgX<7 � � C�� � �)!p�26jJIKH9�\ X<1w�{BD3y9cCE=c9�\D\a`%9�\D\�9�:<1'K535;�®(F�9�35;�9�\gX<7H1�KHk�7wX<756{;56i:<BgA'9�X<BgAq6#C ������ �9�:<6 � �GL,v"!o��3H69�:Z®�9�Co_�1�:oX<756@^%9�:EX<B]9�\�;H6i:<BgA'9�X<BDA'6C|1'_ �WF 7H9�Co41'=)^%9�4iXpCEK5^5^�1�:EXZ9�35;�X<756#:<6#_�1�:<6� � � 8 � �)!p�

_�1�:)� ( 5 � !!" v F�BD3w^%9�:EX<BD4K5\]9�:|_�1�: 5 � *���BD_ * � & mP¦oCEBD35k � � � � � ? �> � � � � !7

��v � �� � � � ������ � > � � � 1�3H6;56;5K54#6C�X<7%9�XW� � � � � ( � �7 ��v �� �� ������ � �� �� � �� �� ������ � �� �� � Ó � � � � � � � � � � � � m1 7H9�X<6iAq6i:�X<756pe�9?Y1'35687H9�C�1�`5X[9�BD356#;w9f`�1'K535; � � � � � (��,� � � � � � Ó ��� � KML.N4O%� � ��� � � eP7563 * � & F5X<756ZCE49�\DB]35k9�:<k'K5=W63qX2BD=)^%\DBD6#C-X<7H9�X�1'356|7H9�C�� � � � � ( C � � � � �¹v " � � � KML.N4O%� � ��� � �� ePBgX<7 � � ! � mV0�35;H6#6#;�9�^5^H\D?,B]35k8X<756oBD3H6JIKH9�\DBgXs?fX<1

vwGI9�=W6Cp�I� Q���¥,Q�TV¤2Fa�I4#1�XEX<BDCE7w=r9�X<7H6#=c9�X<BD4B]9�3aF �jÒ'��­�L<�)( (�®,F�1'35\D?�BD=W^5:<1jA'6;y9W_�1�:<=@K5\]9)e�9�CP7%9�;w`a6#6#3y1�`5X[9�BD356#;`I? ������R�Q�������m

�|`5:[9�7H9�= ������R�Q�������FH�5:<6354[7w=r9�X<7H6#=c9�X<BD4B]9�3aFa�jÒ�Ò�(jL<�)(�Ô�Õ5mº~�H:<BgX<�ZG,R�³�TuF�Ðo6i:<=r9�35LS`�1�:<3�=r9�X<7H6#=c9�X<BD4B]9�3aF��j�(�®�L<�j�'��Õ5m-no6p6=WBDk�:[9�X<6#;YX<1W¦o35BgX<6#;y�IX[9�X<6C�BD3��j���'Ô)9�3H;w9�_]X<6#:

�j��ÕqÒf756Ze21�:Ebq6#;y9�X�X<756f¡�R������T��~0�35CMX<BgX<K(X<6Z_�1�:P�~9�X<756=c9�X<BD4j9�\��I4#B]6#3546#C#FH�o6ie �d1�:Eb�¦o35BgAq6i:<CEBgXs?qm� �� � BDC@X<756rCM^%9�4#6�1'_�\D1,49�\D\g?�BD3qX<6k�:[9�`%\D6r_�K5354iX<B]1�3h_�1�:@eP75B]4[7°X<756#:<6r6#l(BDCMX<CWC CEK54[7hX<7%9�X I�� � � > � � � O Q�(

C )(�N -'���@�u_�1�:P6#A'6#:E?Y4#K5`�6��fFH;56351�X<BD35k)`I? � � X<756{9A'6#:[9�k�681'_ � 1'3��fF%BSm 6'mH (�N -'���{� � � ��I�� � �GQa�4OMQ�m� ¢�756Ze21�:<;>7%9�C�`�66#3w4#1'BD356;Y`,?�no�������p��� �2m

­�Õ

Page 25: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

F��GQa��� � � �� �fk'BgAq6#C � � � � � (1�,� � � ! � ��� � � � � � Ó�� � � � " v ��! � � � � KML.N4O%� � ��� � � F�9�3H;�4}751,1'CEBD35krX<7H6{`�6CMX � ��®rk'BgAq6#C|X<7H6;56CEBg:<6#;�`a1�KH35;am� 356|;56#;5KH4#6#CdX<756#3WX<7%9�Xu_�1�: * � & 1'356|7H9�C � � C � �

� ePBgX<7 ����T> !�`I?W9�^5^%\g?(BD35kZX<756�^H:<6#4#6;5BD35k{BD356jJIKH9�\DBgXs?

X<1 � � � > � md0�3qX<6k�:[9�X<BD3Hk �� � � �GQ�> % (��u_]:<1'= ®)X<1y�Z1'35681�`5X[9�BD35CW� � �GQ�> (��B> � �GQa��� ((� ( � I v� � KML.N4O�� � ���¹�GQ)Ó % (��MO&%}F9�35;�X[9�b,B]35k�X<756c351�:<= BD3 ���5� �)!Z�o1�_u`�1�X<7�CEBD;H6#Cc�O9�35;�K5CEB]35kwGq��TVU��T × C{BD356JIKH9�\DBDXs?H�|1�3H6W1�`5X[9�BD35C9� � � � � > � � � � (� (B��� � KML.N4O%� � ��� � � F%9�3H;w9�CW� � KML.N4O%� � � � > � ��� � � (/­�� � KML.N4O�� � ��� � � 1'35681�`HX[9�B]35C#� � � � � > � � � ( C��4� (B�¹v " ! � � � � KML.N4O�� � ��� � � m¢�7H9�X��(RV¯ R�¥,�%� × CWBD={`�6;5;5BD3Hk~X<756#1�:<6#=Û4j9�35351�XW`�6>BD=W^5:<1jA'6;�4j9�3�9�\DCE1�`�6>CE66#3�`I?�4#1'35CMXE:<KH4iX<BD35k�4#1'K53IX<6i:EL

6il59�=)^%\D6#C#m��%1�:cBD35CMX[9�3546�FuBD_�� � C � � �)!Z�{B]C)6jJIKH9�\2X<1���BD3�9�CE=c9�\D\2`%9�\D\P9�:<1�KH35;�®(FuX<756#3 L � � � ���5� �)!Z�{BDC6JIK5BDA�9�\D6#3IX)X<1�* �°Ó & >Ø� � >8��F2BSm 6'm � � > !

�9�35;,L � � � � v:� � � � ! �fBDCW6jJIK5BgA�9�\D63qX)X<1<� >x� � > !

�tPBD_

��(+* � & 9�35; 5 �+*� 1'356o49�3Y4[751,1'CE6��CEK54[7YX<7%9�X8�E>°� ��> !

�`%K(X8����> ! 8 F(k'BgA(BD35kf9{_�K5354iX<B]1�3YBD3 � v'� �H� �)!p�

eP75BD4}7>;51,6#CP351�X�`�6\D1'35kfX<1 � 8 � �"!p�[m�%1�:�X<756o49�CE6 * � & F51�3H6o41�3HCEBD;56i:<C�_�K53H4iX<BD1'35C���\D1'k L��

� �dF(9�35;�1'356|zH35;5C2X<7H9�X���\D1'k L�� � � � � v'� !@� �)!Z��B]_ 9�35;1'35\g?wBD_ �s\D1'k L�� � " v'L " v � � � !{� �"!p�[F�BSm 6�mPB]_�9�35;�1'35\g?wBD_ & � � >/��� � >8��F�9�35;yX<7H6i:<6{6#l,B]CMX<C � � ®cC<9�X<BDCE_]?(BD3HkWX<75B]CBD356jJIKH9�\DBgXs?rBD_ & � �W�S_�1�: & �Ø��F%9f^5:<6#46#;5B]35kW\D6#=c=c9f7H9�C�CE751jeP3wX<7H9�X � v'� vj� �{� � C � � �{�E�[m

�oCW=W63qX<BD1'356;¶`�6_�1�:<6�FP�(RV¯ R�¥,�H� × C)B]={`�6;5;5BD35k~X<7H6#1�:<6#=Ö49�3�`�6>=r9�;56>=W1�:<6�^5:<6#4BDCE6�`I?�K5CEBD3Hk�N�R����%T���µCM^%9�4#6C#F ��� � 8 F-eP75BD4}7�BD354i:<6j9�CE6rePBgX<7 5 F-ePBgX<7 ��� � �!� ����9�3H; ��� � Û9~CM^%9�4#6�BD3qXE:<1,;5KH4#6#;¶`I?¶������²�Q�T���Q��H´>QS² µ � FeP75BD4}7�BDCpX<756)CM^%9�46)1'_o�S6jJqKHBgA�9�\D6354#6f4#\]9�CECE6#CZ1'_[�|=c69�CEK(:[9�`H\D6)_�K5354#X<BD1'3~C<9�X<BDCE_�?(BD35k I � � @ � O Q ( C (�N -����2�[v � �� d_�1�:6iAq6#:E?�=W69�CEK(:[9�`H\]6cCE6#X ��mwGI9'9�b°±-����� ����CE751je26#;hX<7H9�X@_�1�:Y� ( *�� & 1'356r7H9�C�� v'� �H� �)!p� � ��� � � � � �)!{m>�%1�:*�� & F-X<756Y:<6CEK5\gXW1'_P�H:<BgX<�YG,R�³�Tx9�35;�N�1'K5BDC)�oQ�����TV¯���¤xK5CEBD35k �� � e�9�C)B]=)^5:<1jAq6#;¶`I?¶�o6BD\2¢�� ��VQ�T�¤V�%� � FeP751rCE7H1e26;yX<7H9�XoBD_ � � � v'� !@� �)!Z�[F%X<756#3�_�1�:o6iAq6#:E? C � ®c1�3H6{7%9�C�( � � � � 1 � � 1 / � � ��v��� � � �"!p�[m|¢�7568:<6#CEKH\gX|e�9�C6il,X<635;56#;y`I?ynp9�V �D= ����aµ'Q+U��{9�35;°�qX<6#^H7563 1*��Q�TV¤V�%���ZeP751YCE751je26#;~X<7H9�XoBD_ � 7%9�Cp9�\D\�BDX<CP^%9�:EX<B]9�\�;56#:<BgA�9�X<BgAq6#C|BD3X<7568CM^%9�4#6"� ! � 8 � �)!p�[F5X<7563 ( � � � � � ��v����� � �)!Z�[m

'oK56#CMX<BD1'35Cf9�`a1�K5X8X<756W`�6#CMX@4#1'35CMX[9�3qX<C@BD3��(RV¯ R�¥,�H� B]={`�6;5;5BD35k>X<7561�:<6#=WC87%9#Aq6c`�6#63°BD3qAq6#CMX<BDkq9�X<6#; F�BD3�^�9�:ELX<BD4K5\]9�:f`,?�¢�75B]6i:E:E?¶��V¯�Q�T 9�35;�`,?�ÐoBD1�:<k'BD1�¢���¥,�%T�� Q v � t-9�k'1,1I;�4\]9�CEC)1'_|_�K5354#X<BD1'35C)_�1�:WzH3H;5BD35k�X<756>1�^5X<BD=r9�\4#1'35CMX[9�3qX<C|9�:<6pX<751'CE681'_�X<756Z_�1�:<= v� v � + 7 � � � m¢�756Z^H:<6#4#6;5BD35kW:<6#CEK5\DX<C�49�3>`�686il,X<6#3H;56#;>X<1W_�K5354#X<BD1'35C�7H9A(BD35kc;56i:<BgA'9�X<BgAq6#C ���

��� � � � � � � � ! �[F%351�X|9�\D\* �Z`�6#B]35k6JIKH9�\��SBgXu1I44#K(:<Cu3H9�X<K(:[9�\D\g?)BD_a1'356|4#1,1�:<;5BD3H9�X<6P;56351�X<6#CdX<BD=W6|9�35;WX<7H6P1�X<756#:<Cd;56351�X<6|CM^%9�4#6�F,_�1�:26il59�=)^H\]6��[m�0�3y�j��( �IF0uA(BDCEBgX<6;>¢ :<63qX<1c9�3H;>7569�:<;y9@X[9�\gbY1'3�X<7HBDC�CEK(`(�E6#4iX�`,?Y�o\]1�B]C-&���5T��%�dvEvF(eP7H1)_�1'\D\D1je26#;�X<756Z3H9�X<K(:[9�\�9�^H^5:<1q9�4}7w1�_Ùd=WBD\]BD18Ðo��¤V¥IQ���� �-Ruvsºu1�:-N�1'K5B]C��oQ����%TV¯��%��¤2F�9�C�X<756�=W6#X<751,;f1'_ �(RV¯ R�¥,�H�~49�35351�X�`�6�K5CE6;WBD3@X<7HBDC�49�CE6Z�O9�X-\D69�CMXF0u;51)351�X�CE6#6p7H1e 1'356p41'K5\D;YKHCE6pBgXi�[F,`HK(X�0d\D6j9�:<356#;>9�_�X<6i:Ee�9�:<;5C2X<7H9�X�BDX�7H9�;Y`�6#63Y1�`5X[9�BD356;Y6j9�:<\]BD6i:2`I?�¢���R�Q+UQsv � m,07H9Aq6|1�`5X[9�BD356;r9�3r6#l,X<6#35CEBD1'3c1'_ 9�\D\HX<756CE6|=W6#X<751,;5Cd_�1�:dX<756|4j9�CE6|eP756i:<6PX<756P^%9�:EX<B]9�\%;56#:<BgA�9�X<BgAq6C�`�6#\]1�3Hk8X<1{;5Bg a6i:<6#3IX

� G "1'�#6_d������²�Q�T���Q��%´>Q+² µ5Fa±V1'\DBDCE7�=r9�X<756#=r9�X<B]4#B]9�3aF��j�(�#®�L<�j��Õq®,m)no6W;5BD6#;�;HK(:<BD35k 1h1�:<\]; 1h9�:p0E0}Fa^H:<6#CEK5=r9�`%\g?6il(64#K(X<6;Y`I?�X<756@�,1A(BD6#X<C2ePBgX<7>X<751'K5C<9�35;5CP1'_�1�X<7H6i:P±V1'\DBDCE7>1��c46i:<C#m� �o6BD\��%m,¢�� ��VQ�T�¤V�%�dFq�oK5CMXE:[9�\DB]9�3c=r9�X<7H6#=c9�X<BD4B]9�3aF'`�1�:<3cB]3y���ÕI­,m�no6�e21�:Eb(C29�Xu�oK5CMXE:[9�\DB]9�3c�p9�X<BD1'3H9�\H¦o35BgAq6#:EL

CEBgXs?qF�¡29�3q`�6#:E:[9(m�~np9�V �D= ����aµ�QSU'F��5:<6#3H4[7�=r9�X<7H6#=c9�X<BD4B]9�3aFH`�1�:<3yB]3h���Õ'Õ5mPno6{e21�:Eb(Cp9�Xo¦o3HBgAq6i:<CEBgXs?>±�9�:<BDCPÑo08�S±dBD6#:E:<686#Xo�~9�:<BD6

¡ ���Q��a�[F5±�9�:<BDC{�O9�3H;>BgX�CE66#=cCP9�X�Ú����¤V����U8¦o3HBgAq6i:<CEBgXs?qF(�o6ie 2:<K53HCMePBD4<b�FH�oGHm �[m±dBD6#:E:<6{¡ ���Q���F5�H:<6#354}7>^H7q?(CEBD4BDCMXF � �'Ô���L<�j��®qÒIF�9�35;>7HBDC�ePB]_�6Z�~9�:<BD6@�I¥5R��-Ra´yU����¡ ���Q���FH±�1'\DB]CE7(LS`�1�:<3>^H7q?(CEBgL

4#B]CMXF ���'Ò�(jL<�j� ��Õ5F��E1'BD3qX<\g?r:<6#46#BgAq6#;�X<7568�ZRV¯��%¥c^H:<BD�#68BD3�±d7q?(CEBD4C�B]3��j��®A�IF�9�3H;>CE756{9�\DCE1):<6#46#BgAq6#;�X<756Z�ZRV¯���¥c^H:<BD�#6BD3�¡u7H6#=WB]CMXE:E?rBD3°��(�'��m�h�qX<6#^H7563 1*��Q�T�¤V�%�dFd�o=W6#:<BD49�3/=c9�X<756=c9�X<BD4#B�9�3 Fu`�1�:<3�BD3x�j���'ÒImØno6we21�:Eb(C>9�X�¦o35BgAq6i:<CEBgXs?�1'_1 BDCE41�3HCEBD3aF

�~9�;5BDCE1'3am�¢�75B]6i:E:E?@��V¯�Q�TuF��H:<6#354}7W=c9�X<756#=r9�X<BD4#B]9�3am�no62e21�:Eb(C-9�X-¦o3HBgAq6i:<CEBgXs?@±�9�:<B]C�Ño02�S±dBD6i:E:<6�6iX���9�:<B]6P¡ ���Q�� �[F�±V9�:<BDC#mv � ÐoBD1�:<k'BD1rÐ@m�¢���¥,��T�� QsF50sX[9�\]B]9�3�=c9�X<756#=r9�X<BD4#B]9�3amVno6pe21�:Eb(CPBD3��V\D1�:<6354#6'mvEvy�o\D1'BDC &���5T��%�dF�¡u�6#4[7w=c9�X<756#=r9�X<BD4#B]9�3amVno6pe21�:Eb(C|9�Xp¡u�6#4}7>�o49�;56={?Y1'_-�I4#BD6354#6C#F5±-:[9�k'K56�mvsº�GqK5CMX�9�_]X<6#:�X<756dX[9�\gb�Fj0 =c6iX�Ù-=cBD\DBD1oÐo��¤�¥,Q�������RuFeP7H1�=�0�7H9�;8z5:<CMXV=W6#X�X<756-e26#6#bZ`�6#_�1�:<6dBD3{±�9#A(B]9(F'9�3H;{\]69�:<356#;X<7H9�X|756{e�9�Cp9�\DCE1WX<6j9�4[75BD3HkcBD3y¢�:<6#3qX<15t�7568e�9�C|3H1�X|X<756{\D6j9�CMXoBD3qX<6i:<6CMX<6#;w`I?�X<7H6{BD;56j9�CPX<7H9�X|7H6{7H9�;yBD3qXE:<1,;5K546#;yBD3X<756)^%9�CMXF�9�35;�7H6W4#1'3qX<B]3IK56#;�X<756W;5BDCE4K5CECEBD1'3�X<7H9�XZe26W7H9�;°9�_�6ie¸;H9?,CZ`a6#_�1�:<6W1�3°75BDCZ_+9Aq1�:<BDX<6)CEK(`(�M64iXF�9�^5^H\D?,B]35k�~9�X<756=c9�X<BD4#C�X<1W��K5CEBD4'mv � �~9�:<BD1c¢���R�QSUQ�F50sX[9�\DB]9�3�=c9�X<756=c9�X<BD4#B�9�3 mVno6pe21�:Eb(CPBD3��,9�\D6#:<3515m

­'Ô

Page 26: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

N�R����%T���µfCM^%9�4#6C#F(`,?Y9f;5Bg �6#:<6#3qX�=W6iX<7H1I;��+X<756p=W6iX<7H1I;HCuX<7H9�X�7H9A'6p`�6#63�;56#CE4#:<Bg`�6#;�;51)351�X�CE66#=�X<1@`a6pCEK(�c4#BD63qX_�1�:�^H:<1A(BD3Hk)CEKH4[7y9)k'6#3H6i:[9�\�:<6CEK5\gXi�[m

­'Ò

Page 27: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�®(m-��1'35;H9?��%6i`5:<KH9�:E? (qF�­�®'®�®(m�%1�:{�)(+* �-,/FH1'356pX[9�b'6C2X<756Z351�:<= 1'_�X<756ZCM^%9�46�� v'� �H� �)!p�dX<1f`�6 � I�� 1 � � � �P� OMQfÓ I�� 1 � KML.N4O�� � ��� � OMQ � v � � F9�35;yBgX|BDC|K5CE6#_�K5\�X<1c351�X<BD4#6ZX<7H9�Xo9�;5;HBD35k I � 1 � � � � OMQ�9�35; I � 1 � KML.NMO�� � ��� � OMQ�B]C|9WCMXE:[9�35k'68^H:[9�4#X<BD4#6'F5eP75BD4}7y=c9�X<756#L=c9�X<BD4B]9�35Cu_�1�\]\D1je 9�\]=W1'CMX�9�\D\�X<756|X<B]=W6�F59�3H;YeP75BD4[7Y=r9�bq6Cu^H7q?(CEBD4BDCMX<Cue21'35;56#:�B]_ =c9�X<756=c9�X<BD4#B�9�3HC-b(351je*eP7H9�X2X<756#?

9�:<6pX[9�\gb(BD35kW9�`�1'K(Xm¢�7568bq6i?�^�1'BD3IXPBDCo9cJqKH6#CMX<BD1'3w1'_�KH35BgX<C#md0�3w:<6j9�\a^H:<1�`%\D6#=cC�Q�KHCEKH9�\D\g?�;H6#351�X<6#CPX<7568CM^�9�46ZA'9�:<B]9�`H\D6#C#FHeP75BD4[7�9�:<6

=W6j9�CEK(:<6#;�B]3yK53HBgX<C|1'_�\D635k�X<7h�S351�X<6;#�2�[F�eP75B]\D6 %�;H6#351�X<6#C|X<7H68X<B]=W68A'9�:<B�9�`%\D6�F�=c69�CEK(:<6#;�BD3�K535BgX<Co1'_VX<BD=c6r�S351�X<6#;�o�[F59�35;YBD_ 1'356p4#1'35CEBD;56#:<CuX<756|e�9Aq6o6jJIKH9�X<BD1'3 �

��� � � > 4 º � !� ��v �

��������*®(F 4 B]C29@4[7H9�:[9�4iX<6#:<BDCMX<BD4PAq6\D1,4#BgXs?qFI=W69�CEK(:<6;YBD3

K535BDX<C&� � " vF59�35;rX<7H6o6JIKH9�X<BD1'3rBDC241'35CEBDCMX<6#3IX�9�C26j9�4}7Y1'_aX<756oX<6i:<=cCu1'_aX<7H6o6JIKH9�X<BD1'3rBDC2=W6j9�CEK5:<6#;YB]3rKH35BgX<C � � " º�FeP7H9�X<6iAq6#:PX<7H6{KH35BgX ��_�1�: � BDC@� � 4#1'K5\];w`�6f9WAq6i:EX<BD4j9�\�;5BDCM^H\�9�46#=c6#3qX|B]_V1'356@\D1,1�b,Co9�XoCE=r9�\D\ e�9Aq6#Co1'3yX<756@CEK(:<_�9�461'_�9W\]9�b'681�:o9WCMePB]=W=WB]35k)^�1,1�\�9�35; & ��­)BD3>X<7H9�X|4j9�CE6'F%1�:o9)^5:<6CECEK(:<6{BD_V1�3H68\D1,1�b(CP9�XP^H:<1�^�9�kq9�X<BD1'3w1'_�CE1'K53H;yBD3X<756f9�X<=W1'CM^H756#:<6�F�1�:pBD3wX<7H6{1,469�3aF�1�:pBD3yX<756@k�:<1'K53H;aF 9�3H; & �(�cBD3yX<7H9�Xo49�CE6��[mo�H1�:p351'3�\DBD3H69�:o6JIKH9�X<BD1'35C#F�\DBgbq6X<756 ����¤V�%�VU�v × C86jJIKH9�X<BD1'3 � �

� � Ó � � ���� �!®,F�X<756W;5BD=c6#35CEBD1'3�1'_ � =fK5CMX8`�6)X<7%9�X{1'_29�Aq6#\D1,4BgXs?+� �" v#F�`HK(X{CE1'=c6

^H7q?(CEBD4BDCMX<C�^5:<6#_�6i:PX<1WBD3IXE:<1I;HK54#6{9W4[7H9�:[9�4iX<6#:<BDCMX<BD4pAq6#\D1,4#BDXs? 4 9�3H;>e�:<BgX<6ZBgX����� � Ó 4 � ������ � ®c9�3H;wBD3>X<7H9�XP49�CE6 � 7%9�C351c;5BD=W635CEBD1'3am�~9�X<756=c9�X<BD4B]9�35CrCMX<K5;(?(BD3Hkh6jJqK%9�X<B]1�3HCr_�:<1'= ¡u1'3qX<BD3IKHK5= ��64[7H9�35BD4CY1�:�±d7q?(CEBD4CYCE751'K5\];/`�6~4j9�:<6_�KH\o9�`�1'K(X

X<756rJIK56CMX<BD1'3�1'_2K535BgX<C#FV9�35;¶9�Cf�(RV¯ R�¥,�%� CM^%9�4#6C8e26i:<6c1�:<BDk'BD3%9�\]\g?�B]3qXE:<1,;5K54#6;°_�1�:@CMX<K5;(?(BD35kwCE1'\DK(X<BD1'35C{1'_u^%9�:EX<B]9�\;5Bg a6i:<6#3IX<B]9�\�6JIKH9�X<BD1'35C8_�:<1�=·¡u1'3qX<BD3,K5K5=�±d7q?(CEBD4#C#F�X<75BDC{JIK56CMX<BD1'3�;51,6#C{1,4#4#K5:{3H9�X<K(:[9�\D\g?~BD3�CMX<K5;(?(BD3Hk�X<7H6#=>mY¢�7H6JIKH9�3qX<BgX<BD6C I � 1 � � � � O Qx9�35; I � 1 � KML.N4O%� � ��� � O QØ9�:<6�3H1�Xw=W6j9�CEK(:<6#; BD3�X<7H6�C<9�=c6�K535BgX<C#F|X<756~zH:<CMX�X<6#:<= 7H9A(BD35k;5BD=c6#35CEB]1�3 � � � !*9�35;WX<756PCE6#41'35;c1'356P7H9A(BD35k8;5B]=W6#3HCEBD1'3 � � � ! " �(F,9�35;cBgX-e21'K5\D;W`�6P=W1�:<6�3%9�X<K5:[9�\HeP756#3c;569�\DB]35kePBgX<7W^H7I?,CEB]49�\H^H:<1�`%\D6#=cC-X<1{K5CE6p9{351�:<=¸\]Bgbq6 � I � 1 � � � �HO QoÓ#� � � I � 1 � KML.N4O�� � ��� �HO Q � � FIeP7H6i:<6 � � BDC2984}7H9�:[9�4iX<6i:<B]CMX<BD4\D635k�X<7aF�`HK(X>9�Cce�9�C>9�\D:<69�;(?�3H1�X<B]4#6#;/eP756#3/X<756~9�:<k'K5=W63qXY1'_ZCE49�\DBD3Hk°e�9�CYK5CE6;�B]3�:<6#\�9�X<B]1�3�ePBgX<7 �(RV¯ R�¥,�%� × CBD=@`�6#;5;HBD35k>X<7H6#1�:<6#=wF�1�3H6c4j9�3hCMX[9�:EX@_]:<1'=·9�3°BD356jJIKH9�\DBgXs?�e�:<BgXEX<63�ePBgX<751'K(X{^%9?,B]35k�9�XEX<6#3qX<BD1'3�X<1wKH35BgX<C@9�35;°X<7563;56;5K54#68_]:<1'= BgXP9f`�6#XEX<6i:�1'356ZeP7HBD4[7>;H1I6C�X[9�b'68BD3qX<1c9�4#41�KH3qX�X<75BDCPJIK56CMX<BD1'3am

�o3�BD=)^a1�:EX[9�3qX|:<6=c9�:EbwBDC|X<7H9�Xo_�1�:pCE1'=W6@1�^�6#3�CE6#X<C �x9�35;�CE1'=c68^�9�:EX<B]4#K5\]9�:oCEK(`HCM^�9�46#Cp1'_ � v'� �5���P��1'356f49�39Aq1�B];w9�;5;5BD35k)X<756ZX<6i:<=cC I J � � � �HOMQ�9�35; I J � KML.N4O�� � ��� � OMQw`�649�K5CE68±uR�Q�T�²��������ºpBD356jJqK%9�\]BgXs?r751'\D;5C#m� &������D������ §{0�_p� (-*-( , 9�3H; � BDC{9w351'356#=W^5Xs?�1�^a6#3°CEK(`HCE6#X{1'_&� ! F�1'356cC<9?(CZX<7H9�X@±2R�Q�T�²���� ��hBD356jJIKH9�\DBgXs?751'\D;5C|_�1�:o9cCEK(`HCM^�9�46 �¸1'_3� v'� �5���P�2B]_�X<7H6i:<6{6il(BDCMX<C|9c4#1'35CMX[9�3qXRC CEKH4[7yX<7H9�X|1'356@7H9�CW� � � � � � (�C � � KML.NMO�� � ��� � � _�1�:9�\D\ � � �)m

� _-41'K(:<CE6�FTC!7%9�CpX<7563�X<756f;HBD=W635CEBD1'3�1'_u9Y\D6#35k�X<7aF 9�35;�BD_�X<756#:<6fBDCp351�4}7H9�:[9�4#X<6i:<BDCMX<B]4@\D635k�X<7yX<7%9�XZ1'356)49�39�XEX[9�4[7�X<1$�oF5X<756#3w1'356Z6#l,^�6#4#X<C2X<7H9�XP±2R�Q�T�²���� ��yBD3H6JIKH9�\DBgXs?r;51,6C�351�XP751'\D; m� ��%"-� j�D������ §2B��d0�_ (�N -'���P� � , 9�35;wB]_�X<75684#1'35CMX[9�3qXo_�K5354#X<BD1'3��Z`a6#\D1'35k'C�X<1r9WCEK(`HCM^%9�4#6 ��1'_*� v'� �H���P�[F5X<7563±2R�Q�T�²���� ��yBD3H6JIKH9�\DBgXs?r;51,6C�351�XP751'\D;w1�3��)m

BDB+�Z±2R�Q�T�²���� ���BD3H6JIKH9�\DBgXs?°;51,6#C)351�X)751'\D;�1'3,� v'� �� ���P�8BD_ �¸41'3qX[9�BD35CW9�:E`%BgXE:[9�:<BD\g?�\]9�:<k'6r`%9�\D\]C#F-BSm 6'm�BD_�X<756i:<66il(BDCMX<CP9)CE6jJqKH6#3546 L � � , 9�35;>^�1'BD3qX<C�Q � � �*CEK54}7>X<7H9�X��Y�GQ � � L � � � �omBDB]B��P0�_ ��BDC{BD354\DK5;56;°BD3h9wCMXE:<Bg^h1�_2ePBD;5X<7 O%FVBSm 6'mYX<756#:<6c6il(BDCMX<C � � �)!!ePBgX<75� � ��� �r9�35; � � ��Q � �)! §� ��� � � Qa� � � �W9�35; O � � >!�dFaX<756#3,� � � � � � ( C � O � � KML.N4O�� � ��� � � _�1�:89�\D\ �!� � v'� �� ���P�[F�eP756#:<6�C � BDCp9YK535BgAq6#:<C<9�\4#1'35CMX[9�3qXFIBOm 6'mVBD35;56#^�6#3H;56#3qXd1'_�eP75BD4[7 ��BDC-K5CE6;amV0�_�* ��,/FI±2R�Q�TV²a��� ��rBD356jJqK%9�\]BgXs?f751'\D;5C-1'39� v'� � ���P��BD_a9�3H;c1�3H\g?BD_�X<756i:<6p6#l(BDCMX<C)C � ,�CEK54}7�X<7H9�X�_�1�:P9�\]\PQ � ��1'356p7%9�C�O,� - %i�GQ � ���P� ( C)F(eP756i:<6 O&� - %2BDC2X<756ZÙ-KH4#\DBD;H69�3�;5BDCMX[9�3546�mBgA%�)0�_�)(�N -'���P� � ,�FPX<7563*±2R�Q�T�²���� �� B]356JIKH9�\DBgXs?�7H1�\];5C�_�1�:6� v'� �� ���P�c_�1�:��-( * ( ,�Fo9�35;*1'356~7%9�C� � � � � � ( C)� * � )(�N -'���P�[v � ! � � KML.NMO�� � ��� � � _�1�:|9�\D\ �6� � v'� �� ���P�[m

vwGq1'7H9�35356CZ�~9�:EX<BD3,K5C %���¤V����U'F�£pK(X<4}7(LS`�1�:<3�=c9�X<756=c9�X<BD4B]9�3aF�� �'�'Ô�L<�� �(��m{no6f6=WBDk�:[9�X<6#;�X<1�¦o35BgX<6#;°�qX[9�X<6#CBD3��j�'Ô'Ô)9�3H;�e21�:Ebq6#;y9�XP¦o35BgAq6#:<CEBgXs?r1'_��~9�:E?(\]9�35;aF�¡u1�\]\D6#k'6p±�9�:Eb�mº~GqK5\]6#C{no63(:<Bd±2R�Q�TV²���� ���F��5:<6354[7h=r9�X<756#=r9�X<B]4#B]9�3aFu� �'Ô�Õ�L<�j�(�j­Imyno6We21�:Ebq6#;hBD3°±�9�:<BDC#my¢�756i:<6rB]C@9�3�0�35CMX<BgX<K5X

no6#35:<B�±2R�Q�T�²���� ����+0�no±��[F�;56;5BD49�X<6;¶X<1��~9�X<756=c9�X<BD4#Cc9�3H;�¢�756#1�:<6iX<B]49�\2±d7q?(CEBD4#C#Fu^%9�:EXc1�_o¦o35BgAq6i:<CEBDXs?h±�9�:<BDC)Ño0�S±dBD6i:E:<6�6#Xw�~9�:<BD6¶¡ ���Q�� �[m20�7H9Aq6�`�66#3 X<1'\D;*X<7H9�X>X<75BDC>b,B]35; 1'_@B]356JIKH9�\DBgXs?/eP75BD4[7 B]C>351je�3H9�=W6#;�9�_]X<6#:w75BD=9�^5^�6j9�:<6;>BD3>7HBDC�e21�:Eb�1'3�X<BD;56C#m

­�(

Page 28: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

A%�o0�_�X<756rBD3��E6#4#X<BD1'3°1�_ � BD3qX<1#���5���P�ZBDC@4#1'=)^%9�4#XF�X<7563h±2R�Q�T�²���� ���BD356jJqK%9�\]BgXs?�751'\D;5C@1'3¶9yCEK(`HCM^%9�46 � 1�_� v'� �5���P�uBD_�9�35;w1'35\g?�BD_�X<756Z41'35CMX[9�3qXP_�K5354#X<BD1'3��8;51,6#C�351�X�`�6\D1'35k)X<1 �)m�2È<ÆjÆ5#[§�B��a0�_�)(�N -'���P� �-,�F�X<7563�� � � v'� � ���P�[F�`HK(X-9�CBKML.N4O%�E���*�*®|1'3562=fK5CMX�7H9Aq6�C(�*®,F'eP75BD4[7fB]C�BD3541�=W^%9�X<Bg`H\D6ePBgX<7�� � �Wm

BDB+�|N�6#X � � C � � �"!p�|ePBgX<7 � �!®w9�35;�- � * */. L�%i� �V� � �Y�O® � �j�[F�X<756#3°1�3H6c;56izH3H6#C ��� `I? ��� �GQa� � � � � " � %7 % �`�6\D1'35k'C�X<1�C � ���P�V9�35;c1'356|7H9�C � � ��� � � � �5L ! � �� � � �8� � � 9�35; � � KML.N4O%� ��� ��� � � � L

" v ��! � �� � � KML.N4O�� �V��� � � t,BD_�X<756PBD356jJIKH9�\DBgXs?e�9�C{XE:<K56Y1�3H6ce21'K5\];h7H9A'6~�#( �7 % m�¢�756#:<6#_�1�:<6rBD_�±2R�Q�T�²���� ���BD356jJqK%9�\]BgXs?�751'\D;5C@1'3,� v:� �� ���P�pBgXfk�BDA'6Cf9�3¶K(^5^a6i:`�1'K535;w_�1�:�X<756ZCEBD�6Z1'_�`%9�\D\DC�B]354#\DKH;56#;>BD3 �om

BDB]B�� � 356�CMX[9�:EX<C)_�:<1�=·X<756Y4j9�CE6 & �»��F�eP7H6i:<6Y1'356�7%9�C)CE751jeP3¶X<7%9�X)=r9�l � � � � F%�GQa��� ( vº I�� �� � �� � �� O Q�_�1�:c9�\D\F � C�� � �@�[F%CE1rBD_V_�1�:p9�3yBD3qX<6i:EA'9�\ ������� � ���21'356{7H9�C � � C � � �5�[F%1'356{;56;5K54#6CPX<7H9�X�� � � � � � ( � � � � � OHv � �r9�3H;�9�CI��� � �� � �� OMQ ( �� �� � �� � �� �� � O /� F 1�3H6f;56#;5KH4#6#CpX<7H9�XE� � � � � � ( � º �� �� ���� � �� �� � _�1�:Z9�\D\ � � C�� � �5�[m � 356);56#;HK54#6CoX<7H6f49�CE6f1'_�X<7H6

CMXE:<Bg^�`I?�9�^H^H\g?(BD35kYX<756)^5:<6#46#;5BD3Hk�BD356JIKH9�\DBDXs?yBD3�9�3�1�:EX<751'k'1�3%9�\V`�9�CEB]CoeP7H1�CE6W\]9�CMXZAq6#4#X<1�:8BDC�( ! � �,F�CE1�X<7H9�XpX<7H6CMXE:<Bg^�BDCZ;56izH3H6#;�`I? � ��Q ! � �uF�9�35;�_�1�:869�4[7�4}751'BD4#6)1'_3Q��H���GQ v���������� Q ! " v �P1'356W7H9�C�I �� � � �GQ�� � Q ! ��� � OMQ ! (­ " � O �3I �� �� ������ 1 �GQ�� � Q ! ��� �HO Q ! FP9�3H;/1'356yBD3qX<6k�:[9�X<6CWX<756#3/X<75BDCrBD356jJqK%9�\]BgXs?¶B]3,Q��|BD3/1�:<;H6i:rX<1h1�`HX[9�B]3�±2R�Q�T�²���� ��BD356jJIKH9�\DBgXs?�BD3�X<7H6W49�CE6y��(-* � ,/mc0�3�X<756c49�CE6 *$� ,�F�X<756c4#1'35;5BDX<BD1'3�BDC83H6#4#6CEC<9�:E?~`�649�K5CE6W1'_2B]B��[F�9�35;°BgX{BDCCEK(�c4#B]6#3qX|`�649�K5CE6{_�1�:|6j9�4[76Q � � X<756i:<6{6#l(BDCMX<C�� � � � ePBgX<7$� QE> � � (�CfF�9�35;yB]_ � � C � ���P�uX<756#:<6{6il(BDCMX<C�*>1'3X<756cCE6k�=c6#3qX ! Q � � #29�35;°1'K(X<CEBD;56WX<7H6cCEK(^5^�1�:EX@1'_ � CE1wX<7H9�X � � �GQa���P� � � �GQa�3> � �+*5����( C � � KML.NMO�� � ��� � tVX<7563�X<7H6C<9�=W68BD356JIKH9�\DBDXs?r6il,X<6#35;HC�X<1�� v'� � ���P�[m

BgA%�|0�_�*$� ,/F�BgX{_�1�\]\D1jePCZ_�:<1�=�BDBDB��[mc0�_o� (-* � ,�F�1'356c4}751,1'CE6#C 5 � & CEK54[7°X<7H9�Xr��( 5 ( *�� 5� F�9�35;

1'356|KHCE6#CP�(RV¯ R�¥,�H�°BD={`a6#;5;5B]35kZX<7H6#1�:<6#=S� � � � � 8 � ( C � � KML.N4O�� � ��� � 8 _�1�:�9�\D\ �6� C � � �)!p�[F59�3H;rn �R�¥��V����B]356JIKH9�\DBgXs?�§� � � � � � (S� � � � � 8 � (�N -'���P� ��ePBDX<7 � � v�> v8 � 9�35; � � KML.N4O�� � ��� � 8 (S� � KML.N4O%� � ��� � � (�N -����P� � ePBgX<7 �(� v8 > v

�F�9�35;

X<756#:<6#_�1�:<6)�ZÓ � � v!m�1 BgX<751'K(X�X<756�^5:<64#BDCE6�6#CMX<BD=r9�X<62_�1�:�X<7H6�4#1'35CMX[9�3qXdBgX-4j9�3W9�\DCE1Z`�6�^5:<1jAq6#;W`I?@X<756�41'=)^%9�4iX<3H6#CEC

9�:<k'K5=W63qXPKHCE6#;wB]3�A%�[m2¢�756#:<68BDCP9W;HBg �6i:<63qX�^5:<1,1'_�_�1�:|X<756849�CE6'*�� ­f`�9�CE6;w1'3w�-R����Q��%�°XE:[9�35CE_�1�:<=wF%9�35;�9�\DCE1c9^5:<1,1'_�1'_�X<756Z41'=)^%9�4iX<3H6#CEC�^5:<1�^�6#:EXs?YKHCEBD35kW�-R����Q��%��XE:[9�35CE_�1�:<=>F%9�3H;�X<7H6i?YePBD\D\�`�68CE751jeP3w\]9�X<6i:m

A%�f¢�756w356#46#CECEBgXs?¶X<7H9�Xy�wCE751'K5\D;�351�XW`�6\D1'35k~X<1 � _�1'\D\]1ePCW_]:<1'=ÖB��[m/¢�7%9�XWX<75B]CW4#1'35;HBgX<BD1'3�BDCWCEK(�c4BD6#3qXrBDCX<756y41�3HCE6JIK56#3H4#6y1'_oeP7H9�Xr0)4j9�\]\�X<756y6jJIK5BgA�9�\D6354#6w\D6#=c=c9(F2CE7H1eP3�\]9�X<6i:F2`I?�X[9�b(BD35k � v � �fF�� �SKML.N4O�9�35;� º � � ���(���P� � ! 9�3H; �!X<756ZB]3��M64iX<BD1'3�BD3IX<1 � � � ���5���P�[m

� _�4#1'K(:<CE6'FIBD_�� v � � º 9�3H;)±2R�Q�TV²a��� ��cBD3H6JIKH9�\DBgXs?@7H1�\];5C�_�1�: � v'� �� ��� º �[F'X<7H6#3WBgX-751'\D;5Cd_�1�:8� v:� �� ��� v �[Fq`�6#4j9�KHCE669�4}7>_�K5354#X<BD1'3�1'_ � � � v:� �� ��� v �u4j9�3>`�686il,X<6#3H;56#;�`I?>®W9�35;wk'BgAq6#CP9)_�K5354iX<B]1�3 �� � � v'� �� ��� º �[m

­ �

Page 29: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�'��m�1°6#;53H6#CE;H9?��%6i`5:<K%9�:E?y�,F%­�®'®'®,m'oK56#CMX<BD1'35C�1�_�6JIK5BgA'9�\]6#3546P1'_ 3H1�:<=cCd^%\]9?r9�3YBD=W^�1�:EX[9�3qXu:<1'\D6|BD3rX<756|X<7H6#1�:<6iX<BD4j9�\H^%9�:EX�1'_ �oKH=W6i:<B]49�\%�o3H9�\g?(CEBDC#F

`�649�K5CE6pBD3qX<6i:E^a1�\�9�X<B]1�3Y_�1�:<=@KH\]9�C21�:�JIKH9�;(:[9�X<K(:<6p_�1�:<=fK5\]9�C�9�:<6pK5CE6#;�1'3YXE:<B]9�35k'K5\]9�X<BD1'35C2=c9�;56|ePBDX<7YCE=c9�\D\�6#\D6=W6#3IX<C9�35;�BgX8BDC8BD=)^a1�:EX[9�3qXZX<1�b(351je¸751je�X<756W6#:E:<1�:<CZ`a6#7H9Aq6WBD3�X<6i:<=cCZ1'_dX<756WCEBD�6)1'_dX<756CE6W6\D6#=c6#3qX<C#t�BD3��j� (�Õ5F�9�_]X<6#:Z0e�9�CuX<1'\D;Y9�`a1�K5X�9@4#\]9�CECEBD4j9�\%:<6#CEKH\gX23H9�=W6;Y9�_]X<6#:2GI9�=c6#C� ������Z¯�¥,� v 9�35;YnoQ�¥(¯��%�5� º FIeP751'CE6|X<7561�:<6iX<BD4j9�\H^�9�:EX2:<6#\]BD6#C1'3>:<6#CEK5\gX<CP1'_�GI9�4JIK56CP£p��T � � 9�3H;wGI9�4JIK56CMLON�1'K5BDCPN�QSR�TVU'F50�;56#A'6\D1�^5^�6;w9W=W1�:<6Zk'6356i:[9�\ _�:[9�=W6ie21�:Eb�F5eP75BD4[7y9�\DCE1k'6#356#:[9�\DBD�#6Co9Y;5Bg �6#:<6#3qXoXs?I^a6@1'_�:<6#CEKH\gX<C|`I?yGI9'9�by±d���%������F�eP75BD4}7y0P7%9�;~CE66#3�`�6#B]35kYK5CE6#;~BD3~9r`�1,1�bw1'_-GI9�4JIK56#CMLN�1'K5B]CdN�QSR�T�Uo9�35;cÙd3(:<B]4#1{����¤���T��aU|BD3r1�:<;56i:uX<18^5:<1jAq6PX<756|�������V³�R�¥ � � 9�\gX<6#:<3H9�X<BgAq6�_�1�:u6\D\DBg^5X<B]42`�1'K535;%9�:E?WA�9�\DK56^5:<1�`H\D6=WC#m�0�4j9�\]\,={?f_]:[9�=W6#e21�:Eb{X<756�6JIK5BgA'9�\D6#3546u\D6#=c=c9(F'9�35;W9�X-9oX<7561�:<6iX<BD4j9�\,\D6#Aq6#\,BDXVB]CVK5CE6_�K5\5BD3)1�:<;56i:�X<1p1�`5X[9�BD3=c9�3q?�A�9�:<B]9�3qX<C@1�_�±2R�Q�T�²���� ���BD356jJqK%9�\]BgX<BD6#C8BD3�A'9�:<BD1'K5C@CEK(`HCM^%9�4#6Cf1'_|�(RV¯ R�¥,�%� CM^�9�46#C#FV`HK(XfBgX@:<6JIK5Bg:<6C{6#351'K5k'7:<6#k'K5\�9�:<BDXs?r1'_�X<756p`�1'K53H;H9�:E?�BD3>1�:<;56#:�X<1WC<9�X<B]CE_]?YX<75687q?,^�1�X<756#CEBDC�1'_�4#1'=)^�9�4#X<356#CEC#m� ��%"-� j�D������ §-N�6iX � v `�6o9 ���T���²�³�CM^%9�46�F � º � � � 351�:<=W6;rCM^%9�4#6Co�+ePBgX<7 � � � � � �|;56#3H1�X<B]35k{X<756|351�:<=!1'_ �8���[F,9�35;\D6#X�� � � � � v�� � º �[F � � � � � v%� � � �uCEK54}7�X<7%9�XP1'35687H9�C�O9I�R� � � � � v � � � � � � º Ó�� � � � � � ��+`�����BDC�4#1'=W^%9�4iXm

¢�7563w1�3H6Z7H9�C�X<756Z_�1'\D\D1jePBD35kf^5:<1�^�6i:EX<B]6#CB��2¢�756pbq6#:<356#\ 1'_��xBDC�zH35BDX<6p;HBD=W635CEBD1'3H9�\SmBDB+�d¢�7H6Z:[9�3Hk�681'_ �xBDC�4\D1'CE6#;amBDB]B��p¢�756i:<6Y6il(BDCMX<C@9y41'35CMX[9�3qX�C � CEK54}7hX<7H9�X@BD_ BDCf9y351�:<=W6#;hCM^�9�46�9�3H; � � � � � v�� f�ZC<9�X<BDCMzH6#C � � ��®

eP756356iAq6i: � � �*®(F(X<7563w1�3H6Z7H9�CW� � � � � � � ( C � � � �� ��� � � � � � º _�1�:|9�\D\ � � � v mBgA%�p0�_ � BDC)9�351�:<=W6#;�CM^%9�4#6�9�35;�� � � � � v�� � �8C<9�X<BDCMzH6#C�� � � ®yeP756#3H6iAq6i:"� � � ®~9�35; � � ®(F�X<7563� � � � � v � � � � � � º Ó � � � � � � � m�2È<ÆjÆ5#[§VB�� � 3 �+� � (�L(� �Z�[F,X<7H6p4#\D1'CE6;�K535BgX�`%9�\D\a_�1�:#� � � � � v B]C241'=)^%9�4iXmV0�35;H6#6#;>BD_8� � ��� � � v ( ��F(X<7H6#3 � ��� CMX[9#?(C�BD39c4#1'=)^%9�4#X|1'_� � `I?�`a�2CE1�9WCEK(`%CE6JIK56#3H4#6 � � � 41'3qAq6i:<k'6#CoBD3 � � 9�35;yB]C�X<7H6i:<6#_�1�:<6{9�¡u� �²�³���CE6jJIK56#3546{BD3 � � F9�35;w9�C�� � � �*®)BgX�BDCP9c¡u� �²�³��¶CE6JIK56354#6ZBD3 � º 9�35;>X<756i:<6_�1�:<6Z9I�uBD=W^H\DBD6CuX<7H9�X � � BDCP9c¡u� V²�³��hCE6jJIK56#3546ZBD3� v F5eP7HBD4[7>41'3qAq6i:<k'6#C|9�C�� v B]CP9 ���T���²�³hCM^%9�4#6'm$ u?>9fX<7561�:<6=�1�_V�umHÚPQ���U�µ5F ��=fK5CMX�`�6pz%35BgX<6Z;5BD=c6#35CEBD1'3H9�\Sm

BDB+�-�oC|9)4#1'35CE6jJqKH6#3546Z1'_�no��³�T � L- 2��T���²�³°X<7561�:<6=>F ��`�6#B]35kfzH35BDX<6Z;5BD=c6#35CEB]1�3%9�\a7%9�C|9fX<1�^�1'\D1'k�B]49�\�CEK(^H^H\D6iL=W63qX��rF%BOm 6'm � �� ���j®��f9�35;wX<756#:<686#l(BDCMX ��� � � � � v�� ���29�35; ��� � � � � v�� �W�2CEK54}7wX<7H9�X ( � ���@� (���Ó ���|� (��_�1�:|9�\D\�( � � v F%9�35;wBD3�^%9�:EX<BD4K5\]9�:��!BDC�4\D1'CE6#;w9�CPBgX�BDC�X<7H6pbq6i:<356\a1'_ ���{F�9�3H;�X<7H6i:<6#_�1�:<6�!BDCP9# ���T���²�³hCM^�9�46�m

� 356{CE751jePC|X<7563>X<7H9�X|X<756i:<6{6il(BDCMX<C � �*®WCEKH4[7yX<7H9�X�� � � � � � º 5��� � � � � v _�1�:o9�\D\ � � �rm�0�35;56#6;aF�BD_VBgXPe�9�C|351�XXE:<K562X<756#:<6ue21'K5\];f6#l,B]CMX�9pCE6JIK56#3H4#6 * ��� ��ePBgX<7 � � * � � � v �Ø��9�35;"��* � �Û®,FI9�35;W9�kq9�BD3fX[9�b(BD3Hkp9oCEK5`HCE6JIK56354#6�CEK54[7X<7H9�X���* � 41'3qAq6i:<k'6#CuBD3 � � 1'356PzH3H;5CdX<7%9�X* � e21�KH\D;W`a6o9)¡u� �²�³ ��CE6jJIK56#3546|BD3��9�35;rBgX<Cu\DB]=WBgX�* � � e21�KH\D;C<9�X<BDCE_�? ��* �*®(FHBOm 6'm�* � �·9�3H;�X<756i:<6_�1�:<6 * �*®(FH41'3qXE:[9�;5BD4iX<B]35k@X<7H6Z_+9�4#X�X<7H9�XW� � * � � v ��\DBD= � � � * � � � º �Ø��m

¢�7563wBD_H@ ��� �W� �Z�uC<9�X<B]CMzH6#C@ � � @ BD3 � º FH1'35687%9�C@ � ��� ( � ���@� ��� ( � Ó)��� ( � �*��� ����( � t5X<756i:<6_�1�:<6BD_�1�3H6{;H6#351�X<6#C * � �1����( � F�1'356@7H9�C��� � * � >)* � � � v ( � � � * � >���* � � � º � � � @ � > @ � � � º CE1rX<7H9�X * � BDCo9�¡u� V²�³��CE6JIK56354#68BD3�� 9�3H;>BgX<C�\DBD=cBgX * C<9�X<BDCMz%6#C � * ��@ FHCE751jePBD35kWX<7H9�X �)� �Z�uB]C�4#\D1'CE6#; mBDB]B��V�oC� BDC298`%B �M64iX<BD1'3r_]:<1'=���1�3IX<1 �)� �Z��BgX27%9�C�9�3YBD3IA'6#:<CE6 wF59�35;�9�C�1�3H6o4#1'35CEBD;56#:<C �W� �Z��6JIK5Bg^5^a6#;cePBgX<7X<756o351�:<=¸1'_ � º F � � � �)� �Z� � � �PePBgX<7 � � � � ( v� `,?)X<7H6P^5:<6#A,B]1�KHCE\g?)1�`5X[9�BD3H6#;rBD3H6JIKH9�\DBgXs?>�SBgXuCE751je*X<7H9�X �)� �Z��BDC9 2��T���²�³°CM^%9�4#6'FH9�\DX<751'K5k'7�1�3H6p7H9�C�351�XP9�CECEK5=W6;YX<7%9�X�� º BDC�9 ���T���²�³°CM^%9�4#6'FH9�35;YX<7H6p4#\D1'CE6;Yk�:[9�^H7YX<7H6#1�:<6#=7H9�Cd351�X-`�66#3WK5CE6#;��[m�1 BgX<7fX<7HBDC-;56#zH35BgX<BD1'3)1'_� ¸1'356P7H9�C�*��5 9� *f_�1�:u9�\D\ * � �rFI9�35;cB]3f^�9�:EX<B]4#K5\]9�:3 9� ( � ��� (vwGI9�=W6C�n@m& ������Z¯�¥,��F5�o=c6i:<BD4j9�3�=c9�X<756#=r9�X<BD4#B]9�3amVno6pe21�:Eb,CP9�X|¢�6#l(9�C�� �Ø�»¦o35BDA'6#:<CEBgXs?'F�¡u1'\D\D6#k'6{�qX[9�X<BD1'3amº°�qX<6#^H7563>Ú{m5noQ�¥(¯��%�5�-F5�o=c6i:<BD4j9�3�=r9�X<756#=r9�X<B]4#B]9�3amVno6pe21�:Eb(CP9�X�0sX<7H9�4j9c¡u1'\D\]6#k'6�F(0sX<7H9�49(F5� �fm� GI9�4JIK56C{£p��T �2F��5:<6#3H4[7h=c9�X<756=c9�X<BD4B]9�3amcno6We�9�C@={?�41'\D\D69�k'K56WBD3 � :<C<9?�_]:<1'= �j��(�Ô�X<1°�j� �'­IF�1'356c1'_uX<7H6

Aq6i:E?c_�6ie*eP751@6#lI^H:<6#CECE6#;Y75B]CuCEK(^H^�1�:EX�_�1�:2=@?y�S\D1'CMXi��z%k�7IX�9�kq9�BD35CMX2X<756|=c6iX<751,;5C21'_a_+9�\DCEBgzH4j9�X<B]1�3HCd1�:<kq9�3HBD�#6;c`I?r1'K(:4#1'=c=@K535B]CMXo4#1'\D\]69�k'K56#Cp9�3H;�X<756Bg:o_�:<BD6#3H;5C|eP7H1r41'3qXE:<1'\D\D6#;yX<756f¦o35BDA'6#:<CEBgXs?�t�756f=fK5CMXp7%9#Aq6@`�66#3y`a1�:<3~BD3���(� �,F 9�C9�=W6#6#X<BD35k�1�:<kq9�35BD�6#;�BD3��j� ���YBD3~7HBDCZ751'351'K(:8=fK5CMX87H9A'6)`�66#3~:<6#\]9�X<6#;~X<1�75BDCp`�6BD35kwÒ'Ô,F X<756)=c9�35;H9�X<1�:E?~9�k�6W_�1�::<6iX<BD:<6#=W63qX�BD3>�H:[9�3H4#6�m��Ù-:<Bgbr0sA'9�:P�������V³�R�¥ �fF��qe26#;5B]CE7>=r9�X<756#=r9�X<B]4#B]9�3aFa� �'Ò�Ò�L<�j�'­�(qm-no6Ze21�:Eb'6;wBD3y�IX<1I4[b(751'\D=>m� np9�35CPno��³�TuF5�oK5CMXE:<B]9�3>=c9�X<756=c9�X<BD4B]9�3aF�� ��(���L<�j����Õ(mdno6pe21�:Eb'6;wBD3�ÑpBD6353H9(m

­'�

Page 30: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

_�1�:|9�\D\ ( � � v `�649�K5CE6 � ( ��� ��� (�m-�5:<1'= X<756Z7q?,^�1�X<756#CEB]C � � �*®f_�1�: �6� �hF51'356Z7H9�C � ( � ����� ( � �� � (_�1�:|9�\D\ ( � � v FH9�35;�X<7H6i:<6#_�1�:<6�� � � (4� � � ( � � �� ��� � � ��� � � ( � � º FH9�35;>X<756i:<6_�1�:<6RC =c9?r`�6pX[9�bq6#3�X<1)`�6pX<756Z351�:<=�1'_B BD3 � � �W� �Z� � � �'mBgA%� � 3H6|7H9�CR� � � (4� � º Ó � � � (4� �

�( � � � ��� �sÓ � � � � � ��� � (4� � v F(9�35;rBD_�X<7H6P3H1�:<=cCde26i:<6|351�X26JIK5BDA�9�\D6#3IXd1'356|4#1'K5\D;czH35;9YCE6JIK56354#6�( � � � v ePBgX<7,� � ( � � � v ���)9�35; � � � ( � � � º Ó(� � � ( � � � � � ®(m8�oCo`�6_�1�:<6'Fa9YCEK5`HCE6JIK56354#6 ( � e21'K5\];y`a6

CEK54}7wX<7H9�X � ( � BDC|9�¡u� �²�³ ��CE6jJqKH6#3546{BD3�� � F�9�3H;�9�C � ( � ��®cBD3 � º F ( � e21�KH\D;w`�6@9Y¡u� �²�³���CE6JIK56#3H4#6{BD3� v F�4#1'3qAq6i:<k'BD3HkYX<1w9Y\]BD=WBgX ( F eP75BD4}7~e21�KH\D;�C<9�X<BDCE_]?yX<7H6)4#1'3qXE:[9�;5BD4#X<1�:E?y^5:<1�^�6i:EX<BD6C�� � ( � � v � ��F � ( ��®�9�35;� ( �*®,m

� X<756i:89�^5^H\DBD4j9�X<BD1'35C|1�_-X<756@6jJqKHBgA�9�\D6354#6{\]6#=W=r9WePBD\D\�`a6@6#3H4#1'K53qX<6#:<6#;~\]9�X<6i:F�`%K(XZ9r4#:<K54#B]9�\�7q?,^�1�X<756#CEB]CoBDC|X<7H64#1'=W^%9�4iX<356CEC)9�CECEKH=)^5X<BD1'3¶ePBgX<751'K(XfeP75BD4}7hX<756r:<6CEK5\gX)=r9#?°`�6�_�9�\DCE6��+`HK(X)BgX)BDCf351�XW9�\ge�9?(Cf_+9�\DCE6��[tu_�1�:)6#l59�=W^H\D6'FX[9�b(BD35kW���+* �-,Û9�35; � v ��� v:� �5� �{�[F ��� �� � F � º � � � � ���5� �{��9�3H;��ØX<756PBD3��M64iX<BD1'3)1'_ � v:� �H� �{��BD3qX<1"���5� �{��+eP75BD4}7yBDC|3H1�Xp4#1'=)^�9�4#Xi�[FHX<7563wX<7H68:[9�35k'6{1'_�� BDC|351�Xo4#\]1�CE6;aFa9�3H;yBDCo;56#3HCE6r�S_�1�: *6�¸�{BgXoBDCo4#\D1'CE6#; F�6JIKH9�\�X<1WX<7H6CEK(`HCM^�9�4681'_�_�KH354iX<BD1'35C�B]3 ��vj� �{�dePBgX<7>BD3qX<6k�:[9�\�®I�[m

� 41�=W^%9�4iX<356CEC@:<6#CEK5\DXFd9�XEXE:<BD`HK(X<6#;hX<1~ÚP��¥I¥,QS²�³ � 9�35;¶X<1 &{R�T������ �U�R � � F�9�CECE6#:EX<CfX<7H9�XfB]_ �¸BDC)9y`�1'K53H;56#;1�^�6#3yCE6#Xo1�_ �)!xePBDX<7y9cCE=W1,1�X<7w6#3H1�KHk�7y`�1'K535;%9�:E? � �oF�X<756#3wX<7H68BD3��M64iX<BD1'3w1'_3� v'� �5���P�2B]3qX<1$���5���P�2BDC|4#1'=W^%9�4iXF9�35;>BgX�49�3Y`�6Z;56#;5KH4#6#;>_�:<1�=�9@:<6#CEK5\gXP9�CECE1,4#B]9�X<6;rePBgX<7YX<756p3H9�=W6C�1�_��hm5ÚPQ���U�µ5F5�V� ��a²�³�����9�35; &{R�¥ �8R�¤-R���Ra�2F`HK(XZBgXp4j9�3~`�6{^H:<1Aq6;~69�CEBD\D?w_�:<1'= X<756@:<6CEK5\gXZ_�1�:�� v:� �� ��� ����^5:<1jAq6#;~`�6#\]1e8Fa1'3546f6il,X<6#35CEB]1�3�^5:<1�^�6i:EX<BD6C|ePBD\D\�7H9Aq6`�66#3�CMX<K5;5BD6;amW�H1�:{K53I`�1'K535;56;�1�^�63�CE6#X<C#F�1�:{_�1�:Z`�1'K53H;56#;�CE6#X<CZePBgX<7�351'35CE=c1I1�X<7�`a1�KH35;H9�:<BD6#C#F�X<756WCEBgX<K%9�X<B]1�3�BDC351�X|9�C�CEB]=)^H\D6'F(`HK(XP_�1�:�X<756Z4j9�CE681'_*� v:� �� ���P�-X<7H6ZCE=W1,1�X<75356CEC�1'_�X<756p`a1�KH35;H9�:E?�BDC�3H1�XPB]=)^�1�:EX[9�3qXm

��1'CMXc4#1'=W^%9�4iX<356CECfX<756#1�:<6=WC)K5CE6wBD3�CE1'=W6�e�9?hX<7H6Y`%9�CEBD4>:<6#CEK5\gX<CW1�_|�o�Vµ���¥ ����>9�35;��ZU�² R�¥,Q;�§�BD_ ��� BDCW9`�1'K535;H6#;YCE6jJIK56#3546o1'_ :<69�\�41'3qX<BD3IK51'K5C�_�K5354#X<BD1'35Cu1'3>9@CE6i^%9�:[9�`H\D6p4#1'=)^�9�4#X2=W6#XE:<BD4|CM^%9�46 �hF,X<756#3�_�1�:�6j9�4}7EQ � �1'356y49�3�6#l,XE:[9�4#Xr9°CEK(`HCE6JIK56354#6 � � CEK54[7�X<7H9�X � � �GQa�f4#1'3qAq6i:<k'6#C#FP9�3H;�`,?�9�;5B]9�k'1'3H9�\|9�:<k�KH=W6#3IXcX<75BDCc49�3�`a69�4[75B]6iAq6#;�_�1�:{9�\D\ QhBD3�9�41�KH3qX[9�`H\D6);5635CE6WCEK5`HCM^%9�4#6'F�9�35;�X<75BDCZBDCZ6il,X<6#3H;56#;�X<1�1�X<756#:Z^�1'BD3IX<CpB]_d1'356c9�CECEK5=W6CZX<7H9�XX<756oCE6jJqKH6#3546oBDC�6JIK5BD41�3IX<BD3IK51'K5C�9�X�6iAq6i:E?c^�1'BD3qXF598e�9?rX<1@C<9?rX<7H9�X�9�X�9�3I?c^�1'BD3qX *WX<756o_�KH354iX<BD1'35C�9�:<6o41'3qX<BD3IK51'K5CBD3>X<7568C<9�=W6Ze�9?qF%BSm 6�md_�1�:P6#Aq6i:E? � ��®)X<756i:<686#l,B]CMX<C � �/®(F%;56i^a6#35;5B]35kWK(^a1�3 *�9�35; �{`HK(X|351�X|K(^�1'3 � F%CEK54}7wX<7H9�XO��+* � �,� ( �8B]=)^H\DB]6#C � ��� �+*5�P> ��� ���I��� ( �Z_�1�:�9�\D\ � mV0�3Y1�:<;56#:2X<1@41jA'6#:2=r9�3q?Y9�^5^H\DBD4j9�X<BD1'35C-X<1@e269�bc41�3IA'6#:<k'6#3546o1�:e269�b � 41�3IA'6#:<k'6#3546o\DBDb'6|X<756 2��T���²�³�LO�o¥I��R�¤�¥�uv � X<756#1�:<6#=wF(1�3H6p9�\DCE1fK5CE6C�9@=c9�l(BD=c9�\DBgXs?W9�:<k'K5=c6#3qXF575BD;5;563YBD3¢�Q�³VR�T�Ra��v<v�X<7561�:<6#=>F(X<7%9�X|9�3q?Y^5:<1,;5KH4iXP1'_�41'=)^%9�4iXPCM^�9�46#CPBDC�4#1'=W^%9�4iXm

�%1�:�_�K53H4iX<BD1'35CrBD3 �(RV¯ R�¥,�H� CM^%9�4#6C#F�eP75B]4[7�9�:<6y351�Xr3564#6CEC<9�:<B]\g?�4#1'3qX<BD3IKH1�KHC#F�`HK(XY49�3�`�6�9�^5^5:<1q9�4[7H6#;/`I?CE=W1,1�X<7Y_�KH354iX<BD1'35C#F(1'356o3566#;5C2X<1@4#1'3qXE:<1'\�^5:<64#BDCE6\g?)X<756o6#:E:<1�:F59�35;�CE1�=c6oCE=W1,1�X<75356CECu^5:<1�^�6i:EX<B]6#C21�_ X<756|`�1'K535;H9�:E?ePBD\D\�`a6Z3566#;56;>BD_�1'356Ze21�:Eb(C�ePBgX<76� v'� �5���P�[F5`HK(X�X<7568_�1'\D\]1ePB]35k@:<6CEK5\gXPBDC�1'35\D?Y4#1'3546i:<356;�ePBgX<76� v'� �� ���P�[m� ��%"-� j�D������ §rB��Z0�_ �¸BDCf9�3¶KH3q`�1'K535;56;�1�^a6#3¶CEK(`%CE6iX)1'_ �)!�CEK54[7¶X<7H9�X@X<7H6i:<6Y6il(BDCMX<C#L � ��®�9�35;�9yCE6jJqKH6#3546Q � � �*4#1'3qAq6i:<k'BD3HkfX<1WBD35zH35BgXs?rePBDX<7 ���GQ � � L � � � �oF5X<756#3�X<7H6ZBD3��E6#4#X<BD1'3�1'_H� v:� �� ���P�uBD3qX<1 ���5���P�uBDC�351�XP4#1'=W^%9�4iXm

BDB+�d0�_2� ( *+( , 9�3H;+� BDC|9�3y1�^�6#3yCE6iXPePBDX<7 (�N -'���P� � ,�FHX<7563wX<7568B]3��M64iX<BD1'3>1'_3� v'� �� ���P�2BD3qX<1$���5���P�2BDC4#1'=W^%9�4iXm�2È<ÆjÆ5#[§PB���N�6#X � � C � � �)!p�[FaePBgX<7- � * */. L�%i� �V��� �Y�O® � L � �[Fa9�35; � � ®(F�X<7563 ��� � � �

% � � � v'� �� ���P�[F � � ��� � � v'� � BDC4#1'35CMX[9�3qXF�`HK(Xp351YCEK(`HCE6jJIK56#3546@41�3IA'6#:<k'6#CoCMXE:<1'35k'\g?wBD3+���5���P�2`�6#4j9�K5CE6 ��� 41�3IA'6#:<k'6#C|X<1Y®rBD3+���5���P�2e269�bh�+e269�b� BD_�*9� ,��29�3H;>BDX�49�353H1�X|4#1'3qAq6#:<k�68CMXE:<1'35k'\g?YX<1W®W9�CPBgX<C�351�:<= CMX[9#?(CP4#1'35CMX[9�3qXF%9�35;w351�X|®(m

BDB+� � 356cCMX[9�:EX<CZ`,?~X<756c49�CE6WeP756i:<6 � BDC8`�1'K535;56;amr�%1�:@9�`�1'K535;H6#;°CE6jJIK56#3546 ��� � � v'� �� ���P�o1�3H6We�9�3IX<CZX<1CE751je X<7H9�XWBgX@`a6#\D1'35k'CfX<1�9�41'=)^%9�4iX)CE6#X)1'_ ���5���P�[F-BOm 6'm�X<756#:<6YBDCW9�CEK5`HCE6JIK56354#6�eP75BD4}7¶41'3qAq6i:<k'6#C)CMXE:<1'35k'\g?hBD3���5���P�[mV�%1�:2^5:<1jA(BD35k8X<7%9�Xu^H:<1�^a6i:EXs?WBgX2BDCu6#3H1�KHk�7rX<1{CE7H1e*X<7H9�Xu_�1�:26iAq6i:E? ����®{1'356o49�3czH3H;Y9{4#1'=)^%9�4#XuCE6iX � � 1�_���5���P�2CEK54[7wX<7H9�X|6j9�4[7 ��� BDCo9�Xp9W;5BDCMX[9�354#6@9�X|=W1'CMXC �f1�_�� � F�BSm 6�m21'356{49�3y;56#41'=)^�1'CE6 ��� ��F � � � Ó � � � � FHePBgX<7� � � � � � � � � ( C �f9�35;�F � � � � � � tHBD35;H6#6#;w_�1�:|9WCEK(`HCE6JIK56354#6{1'35687H9�CP\DB]=�CEK(^ �� � � � � � � > � � � � � (/­ C �qF�9�35;y9;5B]9�k'1'3H9�\aCEK5`HCE6JIK56354#68BDC�X<7H6i:<6#_�1�:<689r¡u� �²�³��¶CE6JIK56#3H4#6�m

� �H:[9�3H�8ÚP�%¥,¥,QS²�³dFaÐo6i:<=c9�3>=r9�X<756#=r9�X<B]4#B]9�3amVno6pe21�:Ebq6#;>B]3�ÐWV1�XEX<BD35k'6#3am��Ñfm50}m &{R�T��V��� �U�Ra�2F5ÚPK5CECEB]9�3w=c9�X<756=c9�X<BD4B]9�3am�h¡u6C<9�:<68�o�Vµ���¥ ��dF(0sX[9�\DB�9�3>=c9�X<756=c9�X<BD4B]9�3aFa����Õ�(jL<�j�,�j­,m-no6pe21�:Ebq6#;wBD3 �1'\D1'k�3%9,mhÐoBDKH\DBD1)�ZUj²�R�¥IQsF50sX[9�\]B]9�3�=r9�X<7H6#=c9�X<BD4B]9�3aFa� ��Õ ��L<� �'��Ò,mv � N�6#1'35BD;H9�C��o¥,��R�¤V¥�dF5�o=c6i:<BD4j9�3>=c9�X<756=c9�X<BD4B]9�3amvEv �o35;5:<6�� �I�oBgbq1�\�9�6#A(BD4[7{¢�Q�³�R�TVR �2FÚPK5CECEB�9�3@=r9�X<7H6#=c9�X<BD4B]9�3aFI�j��®'Ò�L<�j�'���ImVno6-e21�:Eb'6;@9�X���1'CE41e/�qX[9�X<6d¦o35BgAq6i:<CEBDLXs?'m

��®

Page 31: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

�%1�:�;51'BD3Hk°X<7H9�XF�1'356�6#lIX<635;5CcX<7H6y_�K5354#X<BD1'35C ��� `I?�®°1'K(X<CEBD;H6+� �SCMX<B]\D\P49�\D\]BD35k�X<756#= ��� BD3HCMX<69�;�1'_ ���� �[FCE1°X<7H9�XY1�3H6y7H9�CY9�`a1�KH35;56#;�CE6JIK56354#6 ��� � � v'� �5� �"!p�fePBgX<7�CEK(^H^�1�:EXYBD3�9�z5l(6#;�`�1'K535;H6#;�CE6iXY1'_ �)!{m �%1�:9yCM^�6#4B]9�\-CE=W1,1�X<75BD35kyCE6jJIK56#3546 � � �GQa�R� v

� 1 � v � � � ���+ePBDX<7 � v � C � � � ! �[F�- � * *�. L�%i� � v � � �Y�O® � �j�[F � v ¸®y9�35;I�� 1 � v OMQ9�Ø���[F'1'3562X[9�bq6#C F � � � � ��� � � � m�¢�7HBDCVk�BDA'6C � � F � � � � � ((� � ��� � � � � � � � � � � 9�35; �� �� ��� % � ���� �

�� �� ( �� �� ��� %� � � �� �� � � � � � � � � FBSm 6�mW_�1�: � � ®�z5l(6; F � � � CMX[9?(C8BD3°9�`�1'K535;56;�CE6iX{1'_2N�Q A�U�²�³�Q���µ>_�KH354iX<BD1'35C#F�9�35;�bq66i^�X<756Bg:8CEK5^5^�1�:EX{BD3°9YzHl,6;4#1'=W^%9�4iXPCE6iX|1'_�� ! F�9�3H;>X<756#:<6#_�1�:<6Z`I?>�o��µ'��¥I��LO�ZU�² R�¥,Q�X<7561�:<6=>F%9)CEK(`%CE6JIK56#3H4#6{4#1'3qAq6i:<k'6#C|K535BD_�1�:<=W\g?Y1'3 � ! F9�35;¶X<756i:<6_�1�:<6rX<756YCE6JIK56354#6�1'_�:<6CMXE:<BD4iX<BD1'35C{X<1 �¸4#1'3qAq6#:<k�6CfCMXE:<1'35k'\g?hBD3 � >���P�[F-9�35;hX<7H6i:<6#_�1�:<6YBD3 ���5���P�[m~0�31�:<;56i:ZX<1Y6#CMX<B]=c9�X<6�� � ��� >!F � � � � � � Fa1'356)351�X<BD4#6C|X<7%9�XW� ��� > ��� � � � �}�GQa�)� I � 1 � � �+*H� � ��� �GQa�H> ��� �GQ9> *H� � O,*%F BSm 6�m��� > ��� � � � ��I�� 1 � � �+*5�}� ��� > � ! ��� �4O&*%F�9�3H;�X<7H6i:<6#_�1�:<6E� � ��� > ��� � � � � � � (5I�� 1 � � �+*H��� � ��� > � ! ��� � � � O&*%FH`HK5X|9�C1'356)7H9�C�� � ��� > � ! ��� � � � (�� * ��� � KML.N4O�� ��� ��� � � 9�35; I � 1 � * � � � �+*5�4O&*�� � �IFa1'356f;56;5K54#6C�� � ��� > ��� � � � � � � ( ��� �qFeP756#:<6 ��BDCP9�3>K(^H^�6i:P`�1'K535;w_�1�:W� � KML.NMO�� ��� ��� � � _�1�:|9�\D\ � m0�_ ��BDC-K53q`�1'K53H;56#;W`HK5Xd7H9�C�zH35BDX<6�=W6j9�CEK(:<6�FI1�3H6�4[751,1'CE6#C*L � � L v �-,»CEK54[7WX<7%9�X-X<756�=c69�CEK(:<6�1'_ � �/�Y�O® � L � �BDC���%F�9�35;w1'356{4[751,1'CE6#C � � C�� � �"!p�2CEK54[7wX<7%9�X � � �81'3 �Y�O® � L � ��9�35;�- � * */. L�%i� � � ���Y�O® � L v �[m2¢�7H68CE6jJqKH6#3546��� > � ��� B]C�`�1'K535;56;yBD3 � v'� � � � ! ��9�3H;yBDC|®W1'K(X<CEBD;56 � ��Y�O® � L � �p�S1'356{CE751'K5\D;�9#Aq1'BD;yK5CEB]35kcBgX<CPCEK(^H^�1�:EXFHeP75BD4}7BDC�4\D1'CE6#;w9�35;w4#1'K5\];Y`�6ZAq6i:E?Y`HBDkWBD_ � � BDC�X<75B]4<b%�[m

�%1�: * � ,/F�X<756@=c9�l(BD=@KH= ;5BDCMX[9�354#6@_�:<1'= 9W^�1'BD3qXo1'_ � � ���O® � L � �2X<1rBgX<CP`�1'K53H;H9�:E?wBDCo9�Xp=W1'CMX C)� & ����v � !8F9�35;�9�Cp1'356)4j9�3~X[9�b'6W9Y4#1'=c=W1'3~N�Q A�U�²�³�Q���µY41'35CMX[9�3qX8_�1�:89�\D\�X<7H6f_�K5354#X<BD1'35C ��� > � ��� F 1'356f;H6#;5K546#CpX<7H9�XpX<756#?9�:<6cKH35BD_�1�:<=c\g?~CE=r9�\]\-BD3 � ����P�[FV9�35;¶9�C � ��� CMX[9?(C@BD3h9>`�1'K535;56;°CE6#X@1'_)� v'� �� � � ���O® � L v ����FVBgX{:<6#=c9�BD3HC8BD3h94#1'=W^%9�4iX�1'_ � � � � ����O® � L v ���Z`I?>9�^5^H\g?(BD3HkfX<756p:<6CEK5\gXP_�1�:�X<756Z4j9�CE681'_�`�1'K535;56;w1�^a6#3>CE6#X<C#m�%1�:8�)(+* �-,�F(1'356o`a1�KH35;5C2X<7H6o351�:<= 1'_8� � ��� > � ��� � � � `I?YK5CEBD35kc�(RV¯ R�¥,�%�hBD=@`�6#;H;5BD35k@X<756#1�:<6#=wF(4[751,1'CEBD35k5 � & CEKH4[7>X<7H9�X{��( 5 (+* � 5

��-,�F%9�35;y9�C ��� > � ��� BDC�`�1'K535;56;wBD36� v'� 8 � �"!p�29�35;�X<7H6i:<6#_�1�:<6ZBD3 � 8 � � �)!p�[F

1'356|7%9�CR� � ��� > � ��� � � � ((� � ��� > � ��� � � 8 � )(�N - � � � �Y�O® � L � � � � ePBgX<7E� � v�> v8 � ��®(FI^H:<1A(BD3Hk8X<756|;H6#CEBg:<6;cK53HBD_�1�:<=CE=c9�\D\�`a1�KH35;w_�1�:�� � ��� > � ��� � � � m¢�756#:<6ZBDCP9W;5Bg �6#:<6#3qX�^5:<1,1'_�BD_ * � ­feP75BD4}7�:<6#\DB]6#C�1'3>�-R����Q��%�°XE:[9�35CE_�1�:<=wm

0sX|BDC�X<756#3wX<BD=W6Z351je�X<1cCMX[9�:EX|CMX<K5;(?(BD3Hk)X<7568=r9�3I?>JIK56#CMX<B]1�3HC�eP756i:<6ZX<7H6Z:<6#k'K5\]9�:<BgXs?Y1'_�X<756Z`�1'K53H;H9�:E?�^H\]9?(CP9:<1'\D6�F'9�^5^H:<1l(BD=r9�X<BD1'38`,?{CE=c1I1�X<7@_�K5354iX<B]1�3HC#F�41�=W^%9�4iX<356CEC#F�6#l,X<6#35CEBD1'3@X<1|X<756deP7H1�\]6uCM^%9�46�F�XE:[9�46#C�1'3@X<756u`�1'K535;%9�:E?qm

�(�

Page 32: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�j­,m-�5:<BD;H9?��H6#`5:<KH9�:E?��'��F�­�®'®'®,m�%1�:Z9�^5^5:<1q9�4[75BD3Hkr_�KH354iX<BD1'35C|BD3 � v:� �5���P�2`HK(Xo351�XoBD3 � v'� �� ���P�[F�1�3H6{3H6#6#;HCPCE=c1,1�X<7y_�K5354#X<BD1'35CPeP751'CE6@CEK(^5^�1�:EX

BD3qX<6#:<CE6#4#X<C�X<756p`�1'K53H;H9�:E?����om� &������D������ § ��� ���u;56351�X<6#C�X<756p:<6CMXE:<BD4iX<BD1'35C2X<1$�*1'_�_�K5354iX<B]1�3HC�BD3 C � � �)!Z�[m

�%1�:)CE1'=c6r3HBD4#6r1�^�63hCE6#X<C��oF��Y� �P�ZBDCf;56#3HCE6rB]3 � v'� �5���P�Z_�1�:��#( * � ,/F�`HK(X@X<7HBDC@BDCf351�X@XE:<K56Y_�1�:fCE1'=c61�^�6#3>CE6iX<C#F5\]Bgbq6p9f;HBDCE4ZBD3YX<756o^H\�9�3H6p_�:<1'=�eP75BD4}7�1'356p:<6#=W1jAq6#CP9f4#\]1�CE6;�CE6#k'=W63qX ! N � 2 #SF(X<7H6pBD3qX<K5BDX<BgAq6|:<69�CE1'3�`�6#B]35kX<7H9�Xf_�K5354#X<BD1'35CfBD3���� ���89�:<6�4#1'3qX<BD3IKH1�KHC)9�4i:<1'CECfX<756YCE6#k'=W63qXFVeP7HBD\D6cX<756#:<6�9�:<6Y_�KH354iX<BD1'35CfB]3 � v'� �5���P�ZeP751�9�:<6;5BDCE41'3qX<BD3IK51'K5C�9�4#:<1'CECuBgXp�O9�35;Y_�1�:2k'BgA(BD35kf9{=c9�X<756=c9�X<BD4j9�\%=W6j9�3HBD35k8X<1@X<75BDCuBD;56j9(F,1'356|ePBD\D\�7H9Aq6|X<1@;56#zH356p9{3H1�X<B]1�31'_-XE:[9�4#6W1�3~X<7H6f`�1'K535;H9�:E?%�[m)¢�7H6f^5:<6#46#;5B]35k�6#l59�=W^H\D6fBDCZ1'356feP7H6i:<6fX<756W1�^a6#3�CE6iX{BDCp351�X{\D1,49�\D\g?w1'3�1'356)CEBD;56W1�_X<756@`�1'K535;H9�:E?'F 9�35;yePB]\D\�`�6{:<K5\D6;�1'K(Xp_�1�:oX<756@=c1�=c6#3qXF�`HK(Xp1'356@CE7H1�KH\D;y:<6#=c6#=@`�6i:oX<7H9�X|X<756#:<6f9�:<6f9�^5^H\]BD49�X<BD1'35CeP756#:<6W1'356W=fK5CMX841'35CEBD;56#:{1�^a6#3�CE6#X<C81�_uX<75BDCZXs?,^�6�F�B]3�X<756WCMX<K5;(?�1'_u4#:[9�4<b�^5:<1�^%9�kq9�X<BD1'3aF�1�:{BD3~X<7H6WCE49�XEX<6#:<BD35k>1�_e�9Aq6#C�`I?>9fX<75BD3>^H\]9�X<6�F5_�1�:P6il59�=)^H\D6��[m� &������D������ §-B��d�o3Y1�^�6#3�CE6#X �/1'_��)!�BDC2C<9�BD;YX<1)7H9Aq6Z9f4#1'3qX<B]3IK51'K5C2`a1�KH35;H9�:E?'F5BD_�_�1�:�6#Aq6i:E?�� � � �oF(X<7H6i:<6p6#l,B]CMX<CL � � ®(F29�3�1�:EX<7H1�3H1�:<=r9�\2`%9�CEBDC ( v���������� ( ! F�9�35;�9~41'3qX<BD3IK51'K5Cc_�KH354iX<BD1'3 1'_ Q���·�GQ v���������� Q ! " v �{CEK54[7�X<7H9�X�0Q � � � � Q > �P� �,L � � ����Q � �)! � � Q > � � � L � � Q ! � c�GQ���� �qmBDB+���o3r1�^�6#3YCE6iX'��1'_ �)! BDCuC<9�BD;cX<1@7H9A'6p9{N�Q A�U�²�³�Q���µ8`�1'K535;H9�:E?'F(BD_�X<756|C<9�=W6P^H:<1�^a6i:EXs?W751'\D;HCdePBgX<7 x`�6#B]35k9)N�Q A�U�²�³�Q���µ)41'3qX<BD3IK51'K5C|_�K53H4iX<BD1'3am

� _u41�K5:<CE6�F�( v���������� ( ! 9�35;��A�9�:E?�ePBDX<7~X<756)^�1'BD3IX��(F�9�35;�X<756W1�:<B]k�B]3�1'_uX<756W4#1,1�:<;5BD3H9�X<6WCM?(CMX<6#= =c9?�9�\DCE14[7%9�3Hk�6ZePBDX<7 �5m-�%1�:�X<7HBDC�4#\]9�CECP1'_�1�^�63>CE6iX<C#F �*B]C�\D1,49�\D\g?r1'3>1'35\g?�1'3568CEBD;56Z1'_�X<756p`a1�KH35;H9�:E?'m

1 BgX<7>X<7H68^5:<6#46#;5B]35kc;H6izH35BDX<BD1'3aF�9�CECEK5=cBD35k�N�� 2 F�X<756{1�^�63yCE6iX �I�GQ � *5� � �8º � Q �*® � N Q�º���* � 2 Q�º��fBDCo9�31�^�6#3wCE6iX�ePBgX<7>41'3qX<BD3IK51'K5C�`�1'K535;%9�:E?>BD_TN 2 ��®(F(`HK5XP351�XPBD_TN 2 �®(m� ��%"-� j�D������ §PN�6iX�� `�6f9W`�1'K535;H6#;�1�^�63�CE6iXoePBgX<7y4#1'3qX<BD3,K51'K5C|`�1'K535;H9�:E?qmP0�_���( * � ,�F�X<756#3 ��� ����BDC|;5635CE6BD36� v'� �H���P�[m

0�_ �5� � v:� ����P�[F�X<7H6i:<6r6il(BDCMX<C@9wCE6JIK56#3H4#6 ���,� ��� �P�pCEK54[7hX<7H9�X ��� � � BD3 � ����P�oe269�b � 9�35; � 8 ���P�CMXE:<1'35kW_�1�:{��( 5 �-,�F%9�3H;>_�1�: "�� � ���������:& F ���

%��� � �

������ �

B]3 � ����P�de269�b � 9�35; � 8 ���P�uCMXE:<1'35kW_�1�:{��( 5 �-,�m�2È<ÆjÆ5#[§��~`�6BD35k�`�1'K535;56;aF ����BDC�4#1'=)^%9�4#XF�9�35;{9�C�BgX�BDC�41Aq6#:<6#;8`I?pX<756d1�^�6#38`%9�\D\DC ����� � L � ��_�1�: � � ���oF�BgX�B]C�4#1jAq6i:<6;`I?~9YzH35BgX<6f3,K5={`�6#:Z1'_-X<756#=wF�ePBgX<7~4#63qX<6i:<C � v���������� � � m@¢�756i:<6)6il(BDCMX<C � �x®YCEK54}7~X<7H9�X�� �7 �Vv �Y��� 7 � L � � �P41'3qX[9�BD35C9�\D\�X<756p^�1'BD3qX<CP9�X�;5BDCMX[9�3546�( �{_�:<1'= ���oF5CE1fX<7%9�X ��BDC�4#1jAq6i:<6;�`I?rX<756Z1�^�63>CE6iX<C �Y��� v�� L � / � ��������� �Y��� �#� L

��

�u9�35;� ����Q � � � O&� -$%i�GQ � � �P� � � �'mZN�6#X � v���������� � ��� �c`�6)9c^%9�:EX<BgX<BD1'3�1�_-K535BDXs?y9�CECE1,4#B�9�X<6;yX<1rX<7HBDCo4#1jAq6i:<BD3Hk(FaCE1YX<7H9�X� 7 � C � � ����� 7 � L � � � � � �)� � ��������� ~F � � C � � �{�o9�3H; � �7 ��v � 7 Ó � � �)1'3 �oF�CE1�X<7H9�X86iAq6i:E? � � � v:� �5���P�|49�3`�6);H6#4#1'=W^�1'CE6#;°9�C � �7 ��v � 7 � Ó � � mW�oC� � 7H9�CZCEK5^5^�1�:EX8BD3 �fF�BgXZ4j9�3�`�6c9�^5^5:<1q9�4[756;~`,?~_�KH354iX<BD1'35C8BD3<C � ���P�`I?�CE=W1,1�X<75BD35k�`,?�4#1'3qAq1'\DK(X<B]1�3 m��H1�:Y6j9�4[7��EF�BD3�1�:<;56i:rX<1h9�^5^5:<1jl(BD=c9�X<6�F 7 � � 7 � `I?�_�K5354#X<BD1'35CcBD3 ��� �P�[F�1�3H64[7H1I1'CE6CoX<7H6fCE6#XZ1'_-1�:EX<751'k'1'3H9�\V;5Bg:<6#4#X<BD1'35CoeP75BD4[7�k'BgAq6#CZ9Y41�3IX<BD3IK51'K5CZ6JIKH9�X<BD1'3~_�1�:ZX<756@`�1'K535;H9�:E?qF�9�35;�1'356)K5CE6#CX<756W_+9�4#XZX<7H9�X8_�1�:@9Y:<BDk'BD;�;5BDCM^H\]9�46#=W63qX#@/�SBSm 6'm�@��GQa� � �6QrÓ,N~_�1�:{9�3�1�:EX<751'k�1'3H9�\-=c9�XE:<Bgl��!9�35;°9YAq64iX<1�:RNH�[F9�35; @����2� � �,�SF%X<7H6#3wB]_#� � � v'� �H��� ����9�35; � BDC|;56#zH356;w`I? � �GQa� � � � @��GQa� � X<7563w1'356{7H9�C � � � v'� �5���2�E�[m � 3H6CMX<K5;5B]6#CoX<756@4j9�CE6f1'_d9YCM^�6#4B]9�\�;H1�=r9�BD3 � � ���0Q � � ! § Q ! ��c�GQ � � �@ePBDX<7 �SKH35BD_�1�:<=c\g?%�24#1'3qX<B]3IK51'K5C)�O9�Cp1�3H69�4iX<KH9�\D\D?Y1'35\D?�K5CE6C|_�K5354iX<B]1�3HC�eP75B]4[7w7%9#Aq68X<7H6#Bg:|CEK(^5^�1�:EXoBD3�9)z5l(6#;w41'=)^%9�4iX|CE6iXF�1'356{1'35\g?>356#6;5C�¸4#1'3qX<BD3,K51'K5C#F9�35;°`�649�K5CE6rBgX@BDC{KH35BD_�1�:<=c\g?�4#1'3qX<BD3,K51'K5C@1�3h41'=)^%9�4iX@CE6iX<C#F�1�3H6c=r9#?�4}7H9�35k'6 _�9�:f9e�9?°BD3°1�:<;56#:{X<1y=r9�b'6cBDXK535B]_�1�:<=W\D?Y4#1'3qX<BD3,K51'K5C}�[m-0�_BF � � v:� �5��� � ��9�3H; FY7H9�CP4#1'=)^�9�4#XPCEK(^5^�1�:EXF%1'356Ze�9�3qX<C�X<1r9�^H^5:<1q9�4}7wBgX�`I?>_�K5354iX<B]1�3HC_�:<1�= �Y� � � �[F�9�35;y_�1�:|X<75B]CP1'356{XE:[9�3HCE\]9�X<6#C|BgX|;51jeP3aFaBSm 6'm�_�1�: ( �*®c1'356{;56#zH356#C F � �GQ�� � Q ! � ��F��GQ�� � Q ! Ó�(��[F�9�35;X<7563>1'356ZXE:<K5354j9�X<6C�F � 9�35;>1'356Z:<6k�KH\]9�:<BD�#6C2X<756Z:<6#CEK5\gX�`I?�41'3qAq1�\]K(X<BD1'3am�%1�:r;H1�B]35k�X<756�XE:<KH3549�X<BD1'3aF21�3H6�:<6k'K5\]9�:<BD�#6C `I?�41'3qAq1�\]K(X<BD1'3aF29�35;�`�6#4j9�KHCE6 �BDCWKH35BD_�1�:<=c\g?h41'3qX<BD3IK51'K5C1'356y49�3�1�`5X[9�BD3/BD3�X<75BDCWe�9?/9�_�K5354iX<B]1�3 � � C �� �)! " vi�fCEK54[7/X<7H9�X �> � � ( � ( �Ó � � 1'3 �)!

" vto1�3H64[7H1I1'CE6C � � C �� �{�@CEK54}7�X<7H9�X ��� %<����>_�1�:"% " v

� 9�35; ��� %<��� ®�_�1�:"% ( " º� F�9�35;�1'356>XE:<K5354j9�X<6#C�F � `I?

;56#zH35BD35k � � �GQa�*� F � �GQa��� � � 1"�� � �

��'F5CE1WX<7H9�X � � �GQa�H� F � �GQa�dBD_BQ ! c�GQ����B> � � `�6#4j9�K5CE6 Q ! � �GQ����B> � � 9�35;

� � �GQa�3� ®cBD_BQ ! ( c�GQ����B> � �� `�6#4j9�K5CE6RQ ! ( � �GQ���� > º �� m% �6#4j9�KHCE681'_�X<7568XE:<K53549�X<BD1'3y1�3H687H9�C �������� � �GQ�� � Q ! � ��'­

Page 33: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

������ �

�GQ�� � Q ! �(���� � � 1"�� � �

��)> F%�GQ�� � Q ! �(���� � � � 1

"�� � �

�� ������ �

� � ��

_�1�: " � & 9�35; ������ � 1

�GQ�� � Q ! �*� ������ 1

�GQ�� � Q ! Ó(���� � � 1

"�� � �

�� Ó!F��GQ�� � Q ! Ó�(���� � � � 1

"�� � �

�� v�F5`�649�K5CE6pX<7H6o^�9�:EX<B�9�\a;H6i:<BgA'9�X<BDA'6C21'_TF � =WB]k�7IX�7%9#Aq6{9�351�X<756#:�^%9�:EX

eP75BD4}7°BDC{CEK(^5^a1�:EX<6;h1'3<Q ! � c�GQ����3> (�`%K(X{X<75BDC8^%9�:EX@BDC{b,B]\D\D6#;�`I?�X<756cX<6i:<= � � � 1"�� � �

���my¢�756i:<6_�1�:<6cBD_�1�3H6

49�\D\]CR� � X<756f:<6#CMXE:<B]4iX<BD1'3�1'_ � � X<1 � � F 1�3H6@zH3H;5CpX<7H9�X ��� �� � � �GQ�� � Q ! � � � ���� �

�GQ�� � Q ! Ó (��PB]3 � � _�1�: " ( & F�9�35;X<756#:<6#_�1�:<6�� � 41�3IA'6#:<k'6#C2X<1#FcB]3 � v:� �5��� � �dCMXE:<1�3Hk)BD_ * �-,/m�0�3�1�:<;56#:�X<1)9�^5^5:<1q9�4[7 � � `I?Y_�1�3H4iX<BD1'35C�BD3 ��� � � �[F1'35689�^5^H:<1'9�4}756#C � � `,?�4#1'3qAq1'\DK(X<B]1�3HC�`I?�CE=W1,1�X<7>_�KH354iX<BD1'35CP9�35;w1'356Z:<6#CMXE:<BD4#X<C2X<756= X<1$� � m

¢�756#:<6ZBDCP9�351�X<756#:�e�9?YX<1);51WX<756p^H:<6#4#6;5BD35k)Xse21WCMX<6i^%C�1'_�XE:<K53549�X<BD1'3w9�35;�:<6#k'K5\�9�:<B]�9�X<BD1'3�BD3>1�3H68CEBD35k'\D6ZCMX<6i^ F9�35;>X<7568BD;56j9fBDC�X<1W;51c9W4#1'3qAq1'\DK(X<BD1'3>1�_ FcePBgX<7w9)CE6JIK56354#681'_�351'35356k'9�X<BgAq68CE=W1,1�X<75BD35kW_�K5354iX<B]1�3HC � � � C�� � �)!p�ePBgX<7fBD3qX<6#k�:[9�\��deP751'CE62CEK(^5^a1�:EX�CE7(:<B]3(b(C�X<1 ��®��'m�no1je26#A'6#:F'K535\]6#CEC *BDC�N�Q AaU�²�³�Q���µp4#1'3qX<BD3,K51'K5C#F'1'3562=fK5CMXV351�X�K5CE6�9CM^�64#B]9�\,CE=c1,1�X<7HBD35kpCE6JIK56#3H4#6�m�9�35;)1'356�\D6#X<C�X<7H62CEK(^5^�1�:EX-1'_ � � CE7(:<BD35b{X<1 ��®��PBD3)9oCM^�64#B]9�\,e�9#?qm�0�_ �+�1- � *�*�. L�%i� �H�[FX<756f41�3IA'1'\DK5X<BD1'3¶�GF � �H�}�GQa��� I � 1 F%�GQ > *5� ���+*5�4O&*yBDCp1�3H\g?y;56#zH356;~BD_*Q9> � � � � F 9�35;�1'356@e�9�3qX<CpX<75B]CoCE6#XoX<14#1'3qX[9�BD3+� � F�1�:P`�6iXEX<6#:PX<1c4#1'3qX[9�BD3+� � " � _�1�:|CE1'=c6 � ��®(t�_�1�:|;H1�B]35kWX<75BDC#F�1�3H6@9�CECEK5=W6C�X<7%9�X � * � �( �f_�1�: * � �°F9�35;>X<7H6i:<6#_�1�:<69� c�GQ����B>�c�+* �����( ��� ���2eP756i:<6��/BDC�X<7568=c1,;5K5\DK5C�1'_VK535B]_�1�:<=�41'3qX<BD3IK5BDXs?Y1'_ WF�9�35;w1'356{9�CMb(CPX<7H9�X* � �»B]=)^H\DB]6#C* ! (�>��#> ��� ���[m¢�75B]CZ=W6#X<751,;�9�\DCE1y9�^H^H\DBD6CpB]3�CE1'=W6W;5BD �6i:<63qX8CEBgX<K%9�X<B]1�3HC#F \DBDb'6W_�1�: � � �q�GQ � *5� � Q � ® � ® � * �(Q º �qF�eP75BD4}7BDCp351�XZ9�3~1�^�6#3~CE6#XpePBgX<7�41�3IX<BD3IK51'K5Co`a1�KH35;H9�:E?wePBgX<7~1�K5:Z;56iz%35BgX<BD1'3aF�`%K(X���� �P��BDCp;5635CE6�F�9�35;~BD3~1�:<;56i:pX<1YCE7H1eX<7H9�X2X<756p4#KHCM^w9�X�®fBDC2351�XP9{^5:<1�`H\D6=�1'356Z351�X<BD4#6CdX<7H9�X�BD_ 1'356pXE:[9�3HCE\]9�X<6#C �/`I?Y9@Aq6#4iX<1�:Z�:>�N � > 2 ��ePBgX<79N ��®f9�35;2 �5N5º�F�X<7563y1'356f1�`5X[9�BD35Co9�3�1�^�63�CE6iX)� ��eP75BD4}7�4#1'3qX[9�BD3HC �oF�k'BgA(BD35kc:<1,1'= _�1�:|XE:[9�35CE\]9�X<BD3Hk��+`I?wCE=r9�\]\�9�=c1'K53qX<C}�9�35;>:<6#k'K5\]9�:<BD�BD35kf`I?�4#1'3qAq1'\DK(X<BD1'3am� ��%"-� j�D������ §VN�6#X �h`a6�9o`�1'K535;56;)1�^�6#3)CE6iX�ePBgX<7fN�Q A�U�²�³�Q���µo`�1'K535;H9�:E?qm�¢�756#3fX<756#:<626#l(BDCMX<C�9o\]BD3569�:V41'3qX<BD3IK51'K5C6il,X<635CEBD1'3 ��_�:<1�= � v'� �H���P��BD3qX<1#� v'� �H� �)!p�V_�1�:|�)(+*#(-, �O9�3c6il,X<6#35CEB]1�3cBDCd4[7H9�:[9�4iX<6#:<BD�#6;)`I?)X<756�^H:<1�^a6i:EXs?)X<7H9�X� � �� J � � _�1�:|6iAq6i:E? �6� � v'� �H���P�E�[m�2È<ÆjÆ5#[§ � 356P4#1'35CMXE:<KH4iX<C-X<756P6il,X<6#35CEB]1�3W_�1�:-X<756P;56#35CE6|CEK(`HCM^%9�4#6 ��� ���[F,9�35;cKHCEBD35k@9p^%9�:EX<BgX<BD1'3)1'_�K53HBgXs?'FIBgX-B]C-6#351'K5k'7X<1@41'35CMXE:<K54iX�X<756o6#l,X<6#35CEBD1'3r_�1�: � � F,eP7H6i:<6 BDC2N�Q A�U�²�³�Q���µ@41'3qX<BD3IK51'K5C#m � 356p;56#zH356#C � � �GQ�� � Q ! �*� � �GQ�� � Q ! �-BD_Q ! � c�GQ�����9�35; � � �GQ�� � Q ! �H� � �GQ�� � ­ c�GQ���� >�Q ! ��BD_MQ ! � c�GQ����[m�0�3ZX<7H9�X e�9? � � BDC 4#1'3qX<B]3IK51'K5C�9�X X<7H6�BD3qX<6i:<_+9�4#6� � � 9�35;>1�3H6o7H9�C � � ���� � �GQ � � Q ! �H� ���

��� ��GQ � � Q ! �-BD_ Q ! � c�GQ � �d9�35; " ( & FH9�35; � � �� � � �GQ � � Q ! �H� ���

� � ��GQ � � ­ c�GQ � � >

Q ! ��Ó¶­ ������ � �GQ�� ���� � 1 �GQ�� � ­ c�GQ���� > Q ! �uBD_ Q ! � c�GQ����u9�35; "�� & FH9�35;-� � ���� 1 �GQ�� � Q ! �*�(> � ���� 1 �GQ�� � ­ c�GQ���� >6Q ! �dBD_Q ! � c�GQ � �[FH9�35;w1'356ZA'6#:<BgzH6C�1�3>X<756CE6Z_�1�:<=@K5\]9�C�X<7H9�X � BDC�B]35;56#6;>\DBD3H69�:�4#1'3qX<BD3IKH1�KHC#m¢�75686#l,X<6#35CEBD1'3>41'35CMXE:<K54iX<6;wBDC�X<7H68C<9�=c6ZeP7H9�X<6iAq6i: *�B]C#F5`HK(XPX<75BDCP=W6#X<751,;>;51,6#CP351�Xo9�^5^H\g?�_�1�:PCE751jePBD3HkWX<7H9�X

X<756#:<6u6#l,B]CMX<CV9|41'3qX<BD3IK51'K5C�6il,X<6#3HCEBD1'3@_�:<1�= ���� �5���P��BD3qX<1R���� �5� �"!p� _�1�:H+ /­�`a6#49�K5CE6�75BDk'756i:�1�:<;H6i:�;56i:<BDA�9�X<BgAq6#C1'_ =WBDk'7qXZ3H1�X{6il(BDCMXmc�I� ��Q�Tdv87H9�CZ41'35CMXE:<K54iX<6;�9�;5Bg a6i:<6#3IXZ6#l,X<6#35CEBD1'3~eP75BD4[7�=r9�^HC����� �H���P�PB]3qX<1 ���� �5� �)!p�_�1�:89�\D\B x®Y9�35;�9�\D\ * � !©� � , #SFa`HK(Xpe26)CE7H9�\D\�1'35\D?y4#1'35CEB];56i:89YCEBD=)^%\D6i:p1'356feP75BD4}7�4j9�3�`�6fK5CE6;�_�1�:Z1�^�6#3�CE6iX<CePBgX<7>CE=W1,1�X<7�`�1'K535;%9�:E?qF%9�35;>X<756ZB];569fB]C�CE751jeP3w_�1�: � �-� !

����0Q � � ! � Q ! ��®��qm

� & ����$ §�¢�756i:<6°BDC�9�41�3IX<BD3IK51'K5C�6il,X<6#3HCEBD1'3 _]:<1'= ���� �(� �)!���BD3qX<1���� � �5� �)!Z�[F8;56izH3H6#;�`I? � � �GQ�� � Q ! �!�

� �GQ�� � Q ! �pB]_8Q ! ��®~9�3H; � � �GQ�� � Q ! ��� � �� ��v � � � �GQ�� � >3"HQ ! �ZB]_8Q ! ��®(FVePBgX<7hCEK5BDX[9�`%\D6r4#1,6#�W4BD6#3IX<C�� � � "<�� ��������� �m�2È<ÆjÆ5#[§V¦oCEBD35k)X<756oX<6#4}7535B]JIK56C�9�\D:<69�;(?r^5:<6#CE63qX<6#;aFH1'356ZCE751jePC�X<7H9�X ��� � !

��uBDC�;56#35CE6ZBD3IX<1����� �5� �)!

��[m�0�3�1�:<;56#:�X<1

4[7H6#4<byX<7%9�XpX<756);56iz%35BgX<BD1'3�;56#zH356CZ9Y41'3qX<BD3IK51'K5CZ1�^�6#:[9�X<1�:F 1�3H6f=fK5CMXZCE751je X<7H9�Xp;H6i:<BgA'9�X<BDA'6CoK(^~X<1�1�:<;56i: >��9�:<6)4#1'3qX<B]3IK51'K5C81'3 Q ! �¸®(m)�oCp_�1�:8CE=c1I1�X<7�_�K53H4iX<BD1'35CpX[9�b(BD35k�X[9�3Hk�63qX<B]9�\V;56i:<BDA�9�X<BgAq6#C)�SBOm 6'm@3H1�X8BD3IA'1'\gA(BD3Hk �

��� 1�

4#1'=c=@K(X<6C|ePBgX<7�:<6CMXE:<BD4iX<BD3HkcX<19Q ! � ®(F BgXpB]Cp6#351'K5k'7~X<1�4[7564<byX<7H9�X#� � � ���� �1

BDCp41�3IX<BD3IK51'K5CZ_�1�: � � ® �������%� >*��m� 3H6wzH35;5CcX<756w41'35;5BgX<BD1'3�X<1�`�6 � �� �Vv ���'�:> ",� � ���w_�1�: ����® ��������� >Ø��F�9�35;�9�CcX<75BDCc\DBD3H69�:cCM?,CMX<63/7H9�Cr9�5��T��V�����8R�T��V��ºo=r9�XE:<BDl�F5BDX�BDC�BD3IA'6#:EX<Bg`H\D6Z9�35;>X<756Z41,6i�c4#BD63qX<C)� � � "��Ø� ��������� 9�:<6Z;H6izH356;wBD3w9)K535B]JIK56Ze�9?'m

¢�756@6il,X<6#35CEB]1�3w^H:<1�^a6i:EXs?w;51,6#C|351�Xo356#46#CEC<9�:<BD\g?>751'\D;y_�1�:o1�^a6#3�CE6iX<CPeP75BD4}7y1�3H\g?>7%9#Aq6f9cn �R�¥��V���h41'3qX<BD3IK51'K5C`�1'K535;%9�:E?�1'_21�:<;56#: � ����m>¢�75BDC{49�3�`�6c4[7564<bq6#;hBD3�X<756W^H\]9�356c_�1�:8X<7H6c1�^a6#3°CE6iX�� � �q�GQ � *5� � ® � Q � >8Q�v � � �

vwÙd\DB]9�C��hm,�I� ��Q�TdF'�o=W6#:<BD49�3)�~9�X<756#=r9�X<B]4#B]9�3aF�`�1�:<3WBD3>�j���(��mVno62:<6#4#6BgAq6#;fX<7H6�1�R�¥ �)^5:<BD�#6�BD3w��'�'�,mVno62e21�:Eb(C9�XP±-:<BD354#6#X<1'3�¦o35BDA'6#:<CEBgXs?'F5�oGHmº~�o\D6#l59�3H;(:<68¢�7�"6#1�^H75B]\D6pÑp��T��V�%���8R�T�����F(�H:<6#354}7w=r9�X<7H6#=c9�X<BD4B]9�3aF��)( �'ÔjL<�)(��'Ò,m

���

Page 34: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

*� Q�v � � �qF(`I?�CE751jePBD35k)X<7H9�X�;°vj���P�dBDC�351�X8�S4#1'3qX<B]3IK51'K5CE\g?%�dBD={`a6#;5;56;�BD3w9�\]\ ���5���P�d_�1�:o­ ( *�� ,/F(eP75B]4[7�e21�KH\D;`�68X<756849�CE6{BD_V9W41�3IX<BD3IK51'K5C|6il,X<635CEBD1'3w6il(BDCMX<6#; F5`�6#4j9�K5CE6f�5RV¯ R�¥I�%��BD=@`�6#;H;5BD35k)X<7561�:<6= 9�CECE6i:EX<CPX<7H9�X);hvj� �8ºj�2BDC�S4#1'3qX<B]3IK51'K5CE\g?%�dB]={`�6;5;56#;>BD3w9�\D\�� � � � º �u_�1�:o­ (+* �-,�m

�%1�: � � C � � �8ºj�{ePBgX<7 �d�O®I���Ö�w1�3H6>;H6izH356C � `I? � �GQa���+Q � �u�GQa�[F29�35;�1'356w4}756#4[b,Cc_�1�:WeP7H9�XcA�9�\DK56C1'_)� X<756�_�KH354iX<BD1'3 � ;51,6#Cf`�6#\]1�3HkyX<1 ���H���P�81�:fX<1 ;hvj���P�[m°¢�756Y_�K5354#X<BD1'3 � `�6\D1'35k'C@X<1+���5���P�8BD_|9�3H;¶1'35\D?°BD_I v� Q� � ��v � � OMQ � ,/F�BSm 6�m * �hÓ v

�� >8��FP9�35; � `a6#\D1'35k'CcX<1!;°vj���P�fBD_89�3H;/1'35\g?�­,���,>Ø�j�2Ó v

�� >8��tPX<7H6

�S6il(4\DK5;56;��p4i:<BDX<BD49�\-A�9�\DK56E� � � vº > vº � ��" vº � �+eP75BD4}7hBDC� ®q�Z41�:E:<6CM^�1'35;5C@X<1yX<7H6��S6#l(4#\DK5;H6#;��p4#:<BgX<BD4j9�\-A�9�\DK56

* � �(> v� �

> v� � �

� º � v � � �v " � F%9�3H;�X<7H6i:<6#_�1�:<6 ;hvj���P�dBDC�3H1�XPB]={`�6;5;56#;wBD3 ���5���P�u_�1�: * � * � m

��Õ

Page 35: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�@�,m-��1'35;H9?��%6i`5:<KH9�:E?��iÕ5F�­�®'®'®�%1�:Z9�3�1�^�6#3�CE6iXoePBgX<7y9c41'3qX<BD3IK51'K5C|`a1�KH35;H9�:E?'F�X<756i:<6@BDCo9c3H1�X<B]1�3�1'_�:<6CMXE:<BD4iX<BD1'3wX<1cX<7568`�1'K53H;H9�:E?°�S49�\D\D6;y9

XE:[9�4#6��d_�1�:P_�K5354#X<BD1'35C�1'_H� v'� � ���P�[F(eP75B]4[7>BDC�6j9�CEBD\g?Y;56#:<BgAq6#;>_�1�:�X<756849�CE681'_ � � FHePBgX<7 �41'3qX<BD3IK51'K5C#m� & ����$ §V�H1�: � � ��� � � �29�35;°��(+* �-,�FH1'35687H9�C �� �� � � Q�� � c�GQ������ �� �� � �

� � 1 / � (+*�v � � �� �� ������ 1 �� �� v � �� �

� J � � � � � � �� � " v � � ��

� J � � m�2È<ÆjÆ5#[§�0�_3F � C � � �{�P1�3H6f7H9�C�� F��O®q��� �E��> I � � � � � � � �

� � OMQ ( * I � � F�� � " v � F � � OMQ ( * � � F�� � � � " v � � ���

� � � � � � F � � � v � �� �

� � � � m � 3H69�^5^H\DB]6#C2X<7HBDC�BD356jJqK%9�\]BgXs?cX<1�F%� %<�*� � � Q�� � %�Ó c�GQ������{9�35;�X<7563w1�3H6ZBD3qX<6k�:[9�X<6C�BD3 Q��SF5KHCEBD35kWn �R�¥��V����B]356JIKH9�\DBgXs?qm�%1�: * ��,�F'X<756�_�K5354#X<BD1'35C�1'_�� v'� >��� � ��9�:<6�\D1,49�\D\g?{K53HBD_�1�:<=c\g?@4#1'3qX<B]3IK51'K5CP�S69�4[7W_�K5354iX<B]1�3WBDC-9�3)6jJIK5BgA�9�\D6354#64#\�9�CECW9�35;�1'356�6\D6#=c6#3qX)1'_�X<756�6jJIK5BgA�9�\D6354#6Y4#\]9�CEC)BDCf41'3qX<BD3IK51'K5C}�[Fu9�35;¶X<7H6rXE:[9�4#6�BDCP�EK5CMX)X<756Y:<6#CMXE:<B]4iX<BD1'3hX<1yX<7H6

`�1'K535;%9�:E?qm� H�j$5������ §�¢�756>\DBD3H69�:W4#1'3qX<B]3IK51'K5CW1�^�6i:[9�X<1�:c1�_|XE:[9�4#6>1�3�X<7H6�`�1'K535;H9�:E?qFu;56#zH356#;�`,?�6il,X<6#3HCEBD1'3¶`I? �SK535BD_�1�:<=Y�4#1'3qX<B]3IK5BgXs?Y1'_�X<756Z1�^�6#:[9�X<1�:P1'_�:<6#CMXE:<BD4#X<BD1'3�;56iz%356#;w_�1�: ��� �P�dePBD\D\�`�68;56#351�X<6; � � m

�o1�X<BD46ZX<7H9�X�� � BDC|351�Xo;H6izH356;~9�CPX<756{:<6CMXE:<BD4iX<BD1'3wX<1WX<756{`�1'K53H;H9�:E?'F�`�6#4j9�KHCE68X<7H68`a1�KH35;H9�:E?w7H9�Co=W6j9�CEK(:<6f®,F9�35;>X<756p:<6CMXE:<BD4iX<B]1�3YX<1c9)CE6iX|1'_�=c69�CEK(:<6{®fB]C�351�XP;56izH3H6#;w_�1�:P_�K5354#X<BD1'35C�eP75BD4}7w9�:<6Z351�X|CE=W1,1�X<7�6351'K5k'7am� & ����$ §a0�_a�)(+* �654� L (-, 9�35; v7 � v

�Ó v8 F�X<756#3{_�1�: �6� � v:� �5��� � � 9�35;�F � � v'� 8 ��� � �a1'356u7H9�C � F � � v'� 7 ��� � �9�35; � � � � F(�*� � � � � � F%m

�2È<ÆjÆ5#[§8¢�756c_�1�:<=@KH\]9�BDCZXE:<K56WBD_ � � F � ��� � � �p9�35;°9�CZ`�1�X<7�CEBD;H6#C81'_dX<756W6jJIKH9�\DBgXs?�K5CE6c41�3IX<BD3IK51'K5C8=r9�^H^HBD35k'C81'3� v'� �5��� � � �6� v'� 8 ��� � �[F(X<7H6Z_�1�:<=fK5\]9)BDC�XE:<K56Z`I?�;56#3HCEBgXs?'m¦oCEBD3Hkr9W^%9�:EX<BgX<BD1'3w1'_�K535BgXs?>1'356{4j9�3wX<7563y;56iz%356f9W351�X<BD1'3y1'_�XE:[9�4681'3yX<7568`�1'K535;%9�:E?w_�1�: � � � v'� �H���P�2BD_ �

7H9�C@9w4#1'3qX<BD3IKH1�KHC8`�1'K535;H9�:E?qF�`HK(X@1�3H6cCE751'K5\D;�`a6c49�:<6#_�K5\�X<7H9�X8X<756c;H6izH35BDX<BD1'3�;51,6C{;56i^a6#35;°K(^�1'3�X<756c4}751'BD4#6c1�_X<756@^%9�:EX<BgX<BD1'3y1'_-K535BgXs?�9�35;�X<756f4[751'BD46@1'_�\]1I4j9�\�1�:EX<751'351�:<=c9�\�`%9�CE6#C#m � 356fCE751'K5\D;�`�6)9e�9�:<6@X<7%9�XoX<7H6@K5CEKH9�\�9�:<69=W6j9�CEK(:<621'3@X<756u`�1'K535;H9�:E?'FqBSm 6'm�X<756p� & >r����LO;5BD=c6#35CEBD1'3H9�\,no� �U���R�� �?�@=c69�CEK(:<6'F�BDCV351�XV;H6izH356;f_�1�: 4#1'3qX<BD3,K51'K5C#t_�1�: �N�Q A�U�²�³�Q���µ)4#1'3qX<BD3,K51'K5C�BgX�B]C�� �uÓ�� � c�GQ � ��� º O Q��OFH9�3H;>BDX�7%9�C�X<7H6pB]=)^�1�:EX[9�3qX�^5:<1�^�6i:EXs?YX<1f`�6ZB]3qA�9�:<B]9�3qX�`I?:<BDk'BD;);HBDCM^H\]9�4#6=W63qX<C��+:<1�X[9�X<BD1'35C-9�35;WXE:[9�3HCE\]9�X<BD1'35C}�[m�¦oCEB]35kpX<756PBD3qA'9�:<B]9�354#62`I?f:<1�X[9�X<BD1'3W1�_�X<7568� & >y����LO;5BD=W635CEBD1'3H9�\no� �U���R�� �?�r=W69�CEK(:<6'F,1'356|4j9�3rCE751je�X<7%9�X2_�1�:�98`�1'K535;H6#;r1�^�63rCE6iXuePBgX<7r9{N�Q AaU�²�³�Q���µ{`�1'K535;H9�:E?WX<756|XE:[9�46|;51,6#C351�XP;56#^�6#35;wK(^a1�3>X<756Z4}751'BD4#681�_�X<756Z^%9�:EX<BgX<BD1'3�1'_�K53HBgXs?�1�:�X<7H6Z4[7H1�B]4#6Z1'_�\D1,49�\a1�:EX<751'351�:<=c9�\a`%9�CE6#C#m

�I1'=c6P3H1�X<B]1�3r1'_�XE:[9�46P4j9�3W`�6o;56iz%356#;r_�1�:21�X<756i:21�^a6#3rCE6#X<C#F,_�1�:26il59�=)^H\D6|CE1�=c6PeP75BD4[7�9�:<6|351�X26#A'63r\D1,49�\D\D?f1'31'356{CEBD;H681'_�X<756#BD:P`�1'K535;H9�:E?qm��H1�:o6il59�=)^H\]6�FH\D6#X � `�68X<756{1�^�6#3yCE6#X|1'_ �8ºZ;H6izH356;yBD3>^a1�\�9�:o4#1,1�:<;5BD3H9�X<6#CP`I? L��x�9�35;°®�� � � ­'�-F�BSm 6'm{X<756)1�^�63�K535BgX8;5B]CE4)CE\DBgXZ1'3~X<756)351'35356kq9�X<BDA'6�Q°9�l(BDC#m � 356W49�3�9�^5^H\g?yX<7H6)N�6=W=c9rX<1YX<7H61�^�6#3~CEK(`HCE6iX<C)�

�;56#zH356#;�`I? L � �@9�35;�® � � �9��9�35; � " ;56iz%356#;�`I? L � �f9�35; � � � � ­���F 9�3H;yX<756i:<6_�1�:<6

1'356p49�3Y;56izH3H6|Xse21@XE:[9�4#6Cu1'3YX<756|^HBD64#6o1'_aX<756|`�1'K53H;H9�:E?r4#1�:E:<6#CM^�1'35;HBD35k{X<1 *��*®(F5® � Q�� ��F51�3H6o_]:<1'=!X<756oCEBD;H61'_��

�9�35;�1'356o_�:<1'=!X<756pCEBD;56p1�_�� " t(X<756#CE6oXse21@XE:[9�4#6C�9�:<6o351�X�3H6#4#6CEC<9�:<BD\g?WX<7H6oC<9�=W6o_�1�: �6� � v:� �H���P�[F(9�\gX<751'K5k'7

X<756#?y9�:<6{X<756fC<9�=c6@_�1�:p_�K53H4iX<BD1'35CoBD3 ��� ���8�O9�35; ��� �P��BDCo351�Xp;56#3HCE6fBD3 � v'� �H���P��BD3wX<7HBDCo49�CE6��[t�X<756#:<6@BDCp9�4iX<KH9�\D\g?CE1'=W6Zb,B]35;>1'_�9)41�=W^%9�X<Bg`HBD\]BgXs?c41�3H;5BgX<BD1'3w9�X|®f`�6#Xse26#63�X<756pXE:[9�46#C�1'3>X<756pXse21WCEBD;56C8�S_�1�: * � ���[m

�o3>B]=)^�1�:EX[9�3qX�:<6#CEK5\DX�BDC�X<1)BD;563qX<BD_�? � v'� �� ���P�[F(eP7HBD4[7>B]C2`I?�;56#zH35BgX<BD1'3�X<756Z4#\D1'CEK(:<6Z1'_HC�� ���P�uBD3 � v'� �H���P�[FH9�CX<756Zb'6#:<356#\ 1'_ � � F%9�3H;�X<7HBDC�=c9�bq6#C�K5CE681'_�no����� � × C�B]356JIKH9�\DBgXs?qm� & ����$ §h�H1�:�* � ��9�35;�@ � ���5�O® � ,��[F|1'356�;H6izH356C9K�`I? K�� %<� � v� I �� @�� -��MO -�m � 356�7H9�C � � K � � �

� � � � (��" v � � @ � � � �

� � � � m�2È<ÆjÆ5#[§ 2?~;H6#35CEBgXs?qF�BgX{B]C86351'K5k'7�X<1w^5:<1jAq6WX<756W:<6#CEK5\DX{_�1�:W@ � C � �O® � ,��[F�_�1�:8eP75B]4[7 K�B]C{®w3569�:f®w9�35;°;H6#49?(CBD3 � � _�1�:r\]9�:<k'6 %>�SCE1�X<7H9�X�K¶;51,6#Cc351�Xc`�6#\D1'35k�X<1 � v �O® � ,��{eP756#35@ � � v �O® � ,��)9�35; I � @�� %<�4O % � ®I�[m � 3H67H9�C�%4K �O� %<�2Ó�K�� %<� � @�� %<�[F|9�35;*1'356~=@K5\DX<Bg^H\DBD6CW`I? *�� K � � " v$- �GK � � K5�c9�35;*1�3H6�B]3qX<6#k�:[9�X<6#CY1'3Ø�O® � ,��[FP1�`%CE6i:EA(BD35kX<7H9�X I � %i� � K � �,� �%O&%#�S> I � � K � �HO&%Z`�6#4j9�K5CE6�% � K�� %<��� ��� ®�9�C % � , �+`a6#49�K5CE6"* � ���89�35;hX<756i:<6_�1�:<6r1'356r7%9�C� *�>��j��I � � K � � O&%3�(>�*RI � @�� K � � " v - �GK � � K5�4O %}F%9�35;w1'356Z41�3H4#\DK5;H6#C�`I?>K5CEBD35kcn �R�¥��V���°BD356JIKH9�\DBDXs?'m

�o351�X<756#:o^5:<1,1'_�1�_-no��� � � × C|BD356jJIKH9�\DBgXs?>K5CE6C �8R��TV¤ × C|BD356jJIKH9�\DBgXs?>_�1�:p41'3qAq1�\]K(X<BD1'3aF�351�X<BD4#BD3HkWX<7H9�X)�O® � ,���BDC9y=@K5\gX<BD^H\DBD4j9�X<BDA'6Wk�:<1'K(^¶ePBgX<7hno������=c69�CEK(:<6 � �� FVX<756#3h1'356r7%9�C�K�� %<��� I �� @�� -�� � � � �� FVBOm 6'm K�BDC{X<756r4#1'3qAq1'\DK(X<B]1�3�'Ô

Page 36: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

^5:<1,;5K54#X81'_8@°ePBgX<7�X<7H6)_�K5354#X<BD1'3)(�;56#zH356;�`I?(�� %<� �¸®�_�1�:{® � %����W9�35; ( � %<� � v � _�1�:r� � %�� , �SCE1>X<7H9�X( � ��v � ® � ,�t � �� ���[m� & ����$ §8�H1�:� 4#1'3qX<BD3IKH1�KHC#FV9�35; * ����FV1�3H6c7H9�C �� �� � � � � � 1 � " � � � � � �� 1

"� � � � �� �� �

� J � � ( ��" v �� �� ������ 1 �� �� �

� J � � _�1�:f9�\D\ � �� v'� �5��� � �[m�2È<ÆjÆ5#[§ � 356r9�^H^H\DBD6C8no����� � × C8BD356JIKH9�\DBDXs?yX<16@�� %<�� ���

��� 1�GQ�� � c�GQ����VÓ�%<�[F�1�3H6WX[9�bq6CZX<756)^�1je26#: *¶9�35;�X<756#3°1�3H6BD3qX<6k�:[9�X<6C�BD3�Q��Sm

� ��%"-� j�D������ §{0�_ BDCZN�Q AaU�²�³�Q���µw4#1'3qX<B]3IK51'K5C{9�35; * ����F�X<756#3 � v:� �� ��� � �|BDCZX<756WCEK(`%CM^%9�4#6c1'_ � � � v:� �H��� � �C<9�X<BDCE_�?,B]35k � � � �*®(m�2È<ÆjÆ5#[§V0�_ � � � v'� �� ��� � �-X<7563YX<756#:<6Z6il(BDCMX<C�9)CE6JIK56#3H4#6 � �9� C�� ��� � �uCEK54[7�X<7%9�X � � � � BD36� v:� �5��� � �[tH9�C � � BDC4#1'3qX<B]3IK51'K5Co_�:<1�=+� v'� �H��� � �2X<1$���5� �)!

" vi�[F � � � B]CPX<756{\]BD=WBgXP1'_ � � � � F�9�35;�BDCo®W`�649�K5CE6@69�4[7 � � BDCp®W3H69�: � � �9�35; � � B]C�X<756p:<6CMXE:<BD4iX<BD1'3YX<1)X<756Z`�1'K535;H9�:E?�_�1�:P_�K5354#X<BD1'35C�BD3���� � � �[m¡u1'3qAq6i:<CE6\g?'F�_�1�: � � � v:� �5��� � �PC<9�X<BDCE_�?,B]35k�� � � ��®,F�1'356)=fK5CMX{9�^H^5:<1q9�4}7 � `I?�9YCE6jJqKH6#3546)_]:<1'= C�� ��� � �[m�VBg:<CMXo1'3568XE:<K5354j9�X<6#Co9�X�,�F�BSm 6'm�1�3H6{4[751,1'CE6#C � � C � � �)!Z��CEK54[7yX<7H9�X � �GQa� ���{_�1�:�� QH� (Ø�{9�35;�1'356f9�^H^5:<1q9�4}756#C� `I? ��� ;56#zH356#;°`I? ��� �GQa� � � �GQa� � � �� � F�9�35;°1�3H6c7H9�C � � ��� � ®w9�35; ��� 41�3IA'6#:<k'6#C8X<1 � �O9�35;�X<756W^5:<1,1'_2K5CE6#CN���¯��aUj¤���y;51'=cBD3H9�X<6#;>4#1'3qAq6i:<k'6354#6ZX<756#1�:<6#=��[m � 3568=r9#?YX<7563w9�CECEK5=W6ZX<7%9�X�X<7H6pCEK5^5^�1�:EXP1'_ � B]C�`�1'K535;56;am

¢�7563Z1'356�e�9�3qX<C X<1�XE:<K53549�X<6�356j9�: X<756�`�1'K535;H9�:E?'F�9�3H;8_�1�: X<75B]Ca1'356-K5CE6#C X<756�^5:<64#6#;HBD35k�N�6#=c=c9(FjBSm 6 �� 1

"� � � � ����5��� � �[m�N�6iX�� � C��� �@�dePBgX<7���� %<�H�*®W_�1�: %'( �Z9�35; ��� %<�H� �p_�1�: % /­,m � 35689�^5^5:<1q9�4[7H6#C � `I? ��� ;56iz%356#;>`I?

��� �GQa�*� � �GQa��� � � � Q ! >�c�GQ���� ��� m�¢�7H6PCE6JIK56354#6 ��� 41'3qAq6i:<k'6#CdX<1 � BD3 ���(��� � ��CMXE:<1'35k8`I?WN���¯��aU�¤���Y;H1�=cBD3H9�X<6#;4#1'3qAq6#:<k�6354#6YX<756#1�:<6= �SBD_�*5� , X<756Y4#1'3qAq6i:<k'6354#6YBDCfBD3 � ��� � �pe26j9�b � 9�35; � 8 ��� � CMXE:<1'35k~_�1�:W9�\D\ 5 � ,/F-1�_4#1'K(:<CE6��[m��,BD=WBD\�9�:<\D? � � %��� �

7H9�Cc9�X<6i:<= ������ �� � � � Q ! > c�GQ������ ��eP75BD4}7�41�3IA'6#:<k'6#C)X<1 ������ � Fd`HK(Xc9�\DCE1�9�351�X<756#:)X<6i:<=

��� � � � � � Q ! > c�GQ������ ��� �oePBgX<7 � � � � ���� � �[FH9�C2BDX�BDC > ������ � BD_�" � & 9�35;��pBD_P"#� & md¢�75BDC�\]9�CMX�X<6i:<=�X<6#35;5C2X<1®@`I?�N���¯���U�¤���y;51'=WBD3H9�X<6;�4#1'3qAq6i:<k'6#3H4#6pX<756#1�:<6=!`�649�K5CE6Z1'356Z=c9?re�:<BgX<6pBgX �

� 1"� � � � � � � Q ! >c�GQ���� � � �oePBgX<7

� � � %<� � � % � �S� � %<�[F�9�35; � � BDCZ`a1�KH35;56#;°`I?�CEK5^ � � � v'� º � %0� � ��� %<����9�35; � � � %<�|X<635;5C8X<1y®>_�1�:@6iAq6#:E?�% ��®(m � 3H6c=c9?X<7563w9�CECEK5=W6ZX<7H9�X � 7H9�CPBgX<C�CEK(^5^a1�:EXP`�1'K535;H6#;y9�35;>`�1'K535;56;w9e�9?�_�:<1'= X<756p`�1'K53H;H9�:E?'m

¢�7568\]9�CMX�^%9�:EXPBDC�X<1):<6#k'K5\]9�:<BD�6o`I?>4#1'3qAq1'\DK(X<BD1'3am�{%���.+.�$H��� §)0�_ ��BDC@`�1'K535;56;hePBgX<7hN�Q AaU�²�³�Q���µy`�1'K535;H9�:E?¶9�35; * � ��FVX<7563 � v'� �� ���P�pB]C{X<756YCEK(`HCM^�9�46Y1'_ ���� v'� �5���P�uC<9�X<BDCE_�?(BD35k � � � �*®(m�2È<ÆjÆ5#[§ � 356{K5CE6#Co9)^%9�:EX<BgX<BD1'3>1'_�KH35BgXs?>9�35;�9W\D1,49�\ 4}7H9�35k'681'_V1�:EX<751'351�:<=c9�\a`%9�CEBDC|9�35;y1'356{9�^5^%\DBD6#C�X<7568^5:<6#46#;5B]35k:<6#CEKH\gXm

�'Ò

Page 37: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�#Õ5m�1°6#;53H6#CE;H9?��%6i`5:<K%9�:E?��jÒIF�­�®'®�®�o3�BD=W^�1�:EX[9�3IX@:<6#CEKH\gX)BDCrÐo�����%T × C)_�1�:<=fK5\]9(F-1�:)CEB]=)^H\g?hBD3qX<6#k�:[9�X<BD1'3h`I?°^%9�:EX<C#Fu9�35;�_�1�:fX<75B]Cf1�3H6�356#6;5C@X<7H6

`�1'K535;%9�:E?YX<1f`�6ZCE=W1,1�X<7�6#351'K5k'7>CE1)X<7H9�XP9f351�:<=r9�\aB]C�;56#zH356#;w9�\D=W1'CMX�6iAq6#:E?IeP7H6i:<6)�S_�1�:�X<756)� & >�����LO;5BD=W635CEBD1'3H9�\no� �U���R�� �?�>=W69�CEK(:<6��[m� H�j$5������ §u0�_ � B]C|9�3w1�^�6#3yCE6iXPePBDX<7>N�Q AaU�²�³�Q���µW`�1'K53H;H9�:E?'F � ;56351�X<6#CPX<7568K535BDX|6il,X<6i:<B]1�:P3H1�:<=r9�\Smu�%1�:|X<75684j9�CE61'_ � � ���I�GQ�� � Q ! � � Q ! ��c�GQ���� �W_�1�:89YN�Q A�U�²�³�Q���µY_�KH354iX<BD1'3 WF 1'356f7H9�C � � � Q�� � c�GQ������#� v

� v � � � � � � � � � ������ � �GQ����_�1�: "�� & 9�35; � ! � Q�� � c�GQ������ �(> v� v � � � � � � � � � m

� & ����$ §)0�_ BDCfN�Q AaU�²�³�Q���µ~4#1'3qX<BD3,K51'K5C#Fu9�35; � � � v:� �5��� � �[F*F � � v:� �� s��� � �[F�X<7563¶1'356�7H9�C I0J � � � ������ 1 Ó������ 1

F � O Q ��I�J�� � � � � F � ! O ; !

" vFHBOm 6'm�>6I � 1� / � � � � � F3OMQ��Sm�2È<ÆjÆ5#[§W�H1�: � � F � ��� � � �89�35;¶6j9�4}7 Q � � � ! " v F�1'356Y7%9�C I � � � � � � � �� � 1 Ó ���

��� 1F � O Q ! � I

� � � � �� � � ���� 1

OMQ ! �> � � Q�� � c�GQ�������F � Q�� � c�GQ�������m80�3qX<6k�:[9�X<B]35kYX<75BDCp6jJIKH9�\DBgXs?yBD3 Q��VCE751jePCZX<7H9�XpX<756)_�1�:<=@KH\]9rBDCpXE:<K56f_�1�: � � F � ��� � � �[F9�35;/9�CW`a1�X<7�CEB];56#Cc1�_|X<7H6>6JIKH9�\DBgXs?�9�:<6>41'3qX<BD3IK51'K5Cc_�KH354iX<BD1'3H9�\DCWBD_o1'356>K5CE6CWX<756>X<1�^a1�\]1�k'BD6Cf1'_R� v'� �5��� � �f9�35;� v'� �� ��� � �[F�X<756c\D6#=c=c9YBDCZ^5:<1jAq6#;/�SBD3�X<756W49�CE6WeP756i:<6�* � �W1�: *$� ,/F�1'356)zH:<CMX@9�^5^5:<1q9�4[7H6#C8X<756W_�K5354#X<BD1'3�BD3� v'� >��� � �V9�35;rBgX<C-^%9�:EX<B]9�\H;H6i:<BgA'9�X<BDA'6C�B]3$��>��� � ��e269�b � F(9�35;cX<756#3cX<756P1�X<756#:d_�K5354#X<BD1'3cBD39� v'� vj��� � �VCMXE:<1'35k,�[m� & ����$ §�0�_� BDC2N�Q AaUj²a³�Q���µf4#1'3qX<BD3,K51'K5C#F59�35; �6� � v'� �5��� � �[F F � � v'� �� s��� � �[F,X<7563Y_�1�:�6iAq6i:E?W"��Ø� ���������'& 1�3H67H9�CI0J

� � � ���� � � Ó ������ � F � OMQ ��I � J � � � � � � F � ��OM; !" v m

�2È<ÆjÆ5#[§)¢�756Y4j9�CE6�" � & 7%9�C@`�6#63°^5:<1jAq6#;amy0�_ " � & 9�35;¶BD_�1'356Y=c9�bq6#C@1'35\g? Q �cA�9�:E?qF�X<756YBD3qX<6#:<CE6#4iX<B]1�3�ePBgX<7� � 4j9�3¶`�6>9�3�9�:E`%BgXE:[9�:E?°1�^�63�CEK(`HCE6iXc1�_ �fFdBSm 6�mh9~41'K53qX[9�`H\D6�K53HBD1'3¶1'_P1�^�63�BD3qX<6i:EA'9�\DC#F-CE1�BD3�1�:<;56i:)X<1�9Aq1'BD;X<75BDC@X<64[753HBD49�\2;5Bg�c4#K5\DXs?°1'356�K5CE6#CW9�3H6ie�1�:EX<7H1�k'1'3H9�\u`%9�CEBDC#F�ePBDX<7¶\]9�CMXfAq6#4iX<1�: ( �

!� v� v � � � � ( ! Ó ��( ���r�+eP756i:<6( v���������� ( ! B]C2X<7568BD35BgX<B]9�\�`%9�CEBDC}�[m � _�41�K5:<CE6 ����®)BDC�X[9�b'63>CE=c9�\D\a6351'K5k'7aFHCE1)X<7H9�XPK5CEBD3Hk�*rX<1);56#3H1�X<684#1,1�:<;5BD3H9�X<6CBD3�X<756w356ie `%9�CEBDC#FuX<756>1�^�63�CE6#Xc49�3�`�6>e�:<BgXEX<63�9�C"* ! � � �+* ���{ePBgX<7 � N�Q AaU�²�³�Q���µ�41�3IX<BD3IK51'K5C#m�¢�756i:<6_�1�:<6

X<7568^H:<6#4#6;5BD35krN�6=W=r9WCE751jePC|X<7H9�X I J� � � ���� ! 1 Ó ���

�!1F � O,*E� I

�J�� � � � � F|� � � ( � ! �MOM; ! " vm � 356@1�`HCE6i:EAq6C|X<7H9�X

����!1� v� v � � � ������ 1

Ó �� v � � � ������ � 9�35;cCEBD=WBD\�9�:<\D?r� � � ( � ! �H� v� v � � � � ! Ó�� v � � � � ��F,9�35;)X<756#:<6#_�1�:<6P9�_�X<6i:d=@KH\gX<Bg^H\DB]49�X<BD1'3`I? � �uÓ � º FI1'356P1�`5X[9�BD35Cu9Z:<6#\]9�X<BD1'3W1'_�1�:<;56#:|�PBD3 �qtIX<756P6jJIKH9�\DBgXs?f_�1�: � �*®8BDC-XE:<K56|9�XdBDXdBDC-X<756�^5:<6#46#;5BD3Hk8N�6#=W=r9(F

9�35;>X<756i:<6_�1�:<6pX<75686JIKH9�\DBgXs?Y1'_�X<75684#1,6i�c4BD6#3qX<C�1'_ �{k'BgAq6#C�X<7568;56#CEBD:<6#;�:<6\]9�X<BD1'3am�{%���.+.�$H��� §x�%1�:°9�3I?�`�1'K535;56; 1�^�6#3 CE6iX ��ePBgX<7ØN�Q AaUj²a³�Q���µ�`�1'K535;%9�:E?Ø1�3H6�7H9�C I J � � ������ � Ó ���

��� �F � OMQS�

I�J � � � � � F � ��OM;+!

" v|_�1�:8"��Ø� ���������:& FH_�1�:|9�\D\ �6� � v:� �5���P�[F F � � v'� �� s���P�[m�2È<ÆjÆ5#[§ � 356�K5CE6#Cu9p^%9�:EX<BgX<BD1'3f1'_�K535BDXs?f9�35;WX<756�_+9�4#X�X<7%9�X-1'356P7H9�Cd9Z_�1�:<=@KH\]9�X<BD1'3)BD3qA'9�:<B�9�3IX�`I?@:<BDk'BD;W;5BDCM^H\]9�46#=W63qX<C#F_�1�:o6il59�=)^H\D6Z`,?�e�:<BgX<B]35k)X<756{_�1�:<=fK5\]9EI0J � � � �$F � (���Ó*� � � � (��4F � OMQ ��I � J � � � � � F|� � � (��4O ; ! " vo_�1�:o9�\D\ A'64iX<1�:<C (�S1'_�41'K(:<CE6�F5X<7H6Z_+9�4#X�X<7H9�X�X<7568N���¯���U�¤����=W6j9�CEK(:<6 OMQyBDCP9�\DCE1WBD3qA'9�:<B�9�3IX�`I?�:<BDk'BD;�;HBDCM^H\]9�4#6=W63qX<C�BDCP9�\DCE1WK5CE6;��[m

0sXPCE751'K5\D;>`�6Z351�X<BD4#6;YX<7%9�XP6iAq6#3w_�1�:|9 C��_�K5354#X<BD1'3 WF(X<7H6ZCE6iXP1'_TQ�� CEKH4[7>X<7H9�X�c�GQ����H� �y49�3�`�689)k'6#356#:[9�\4#\]1�CE6;@CE6#Xtq9�4#X<KH9�\D\g?'F�_�1�:�9�3q?@4#\D1'CE6#;fCE6#X�� � �)! " vF�X<7H6i:<626#l,B]CMX<C � � C�YF�ePBDX<7 � �®o9�35; �0Q�� � � �GQ����H�*®�� � �°m�H1�:p41�3HCMXE:<K54iX<BD3Hk � F�1'356@3H1�X<B]4#6#CPX<7H9�XoX<756@4#1'=)^H\]6#=W63qX|1'_��ÛBDCo9r41�KH3qX[9�`H\D6@K535BD1'3�1'_�1�^a6#3y`%9�\D\]C � ! " v � � �� � �Y� � � � L � �8�S_�1�:p6#l(9�=W^H\D6@_�1�:p6j9�4[7�^a1�B]3qX � �� � 9�3H; � ePBgX<7y:[9�X<BD1'3H9�\�4#1,1�:<;5BD3%9�X<6C#Fa1�3H6@bq6#6#^HCoX<756f\�9�:<k'6CMX1�^�6#3�`%9�\D\-4#63qX<6i:<6;h9�X � eP7HBD4[7°;51,6#C{351�X@BD3qX<6i:<CE64iX ���[m � 356c4}751,1�CE6C{9w_�KH354iX<BD1'3 � � C�� � �)! " vi�pCEK54[7°X<7H9�X�0Q � �u�GQa� �!®��E� �Y�O® � ���[F�9�35;�1'356c=r9#?�9�CECEKH=W6�� ®(m � 356W;H6izH356C � `I? � �GQa� � � � 4 � � � � "�� %7 % � ePBgX<7°9�\D\4 � �Ø®(F�9�35; � C<9�X<BDCMz%6#CpX<756);56CEBg:<6#;~^H:<1�^a6i:EXs?~BD_d1�3H6)4[751,1'CE6#CZX<756 4 � 41�3IA'6#:<k'BD35k�X<1>®�_�9�CMX86#3H1�KHk�7�CE1�X<7H9�XpX<7H6CE6i:<B]6#Cp4#1'3qAq6i:<k'6#CZK53HBD_�1�:<=c\g?'Fa9�Coe26\D\V9�C89�3q?�1'_-BgX<Cp;56i:<BgA'9�X<BgAq6#C@�SCE1YX<7%9�XR �� � � C � � �)!

" vi��_�1�:89�3q?�=@KH\gX<BgLOBD35;H6il���[m

no1je26iAq6#:FHX<756#:<68BDCo9):<6CEK5\gX|1'_d�I��� �2v�eP7HBD4[7yC<9?(CPX<7H9�X|_�1�:|=c1'CMX �yX<756{CE6iX|BDC|351�XPX<7%9�XP`%9�;at�_�1�:|6#l59�=W^H\D6'FHBD_ � C v � �{�[FHX<7563w6il(4#6#^5XP_�1�:|9WCE6#X|1'_�wePBgX<7w=W6j9�CEK5:<6{®,F�9�X|9�3q?�^�1'BD3IX)QyePBgX<7 c�GQa�*� ��1�3H687H9�C � �GQa� � ®(F%CE1X<756CE6p^�1'BD3qX<CP9�:<68BDCE1'\]9�X<6;am

vw�|:EX<7IK5:p�I�����2F5�o=c6i:<BD4j9�3>=c9�X<756=c9�X<BD4B]9�3aFa���®q��L<�j� ��®(m-no6pe21�:Ebq6#;y9�X'oK56#635Cp¡u1�\]\D6#k'6�F5�o6#e �d1�:Eb

��(

Page 38: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

¢�756)356#lIX8JIK56#CMX<B]1�3~BDCpX<1�BD;56#3IX<BD_]?wX<756@:[9�35k'6)1'_ � � F _�1�:89r`�1'K535;H6#;�1�^�6#3~CE6#X"��ePBgX<7~N�Q AaUj²a³�Q���µY`�1'K535;H9�:E?_�1�:�6il59�=)^%\D6�m�¢�7HBDCde�9�C2;H1�3H6|`I?rÙd=WBD\DB]1)Ðo��¤V¥IQ�� �-RuFI`HK(X�9�\gX<7H1�KHk�7 � � BDC2CEK(:+�M64iX<BgAq6o_]:<1'= � v'� vj���P��1'3qX<1���v� � �P�[FBgXZB]CZ351�X8CE1w_�1�:'* ����F�9�3H;~X<756WBD=c9�k'6f1'_�� v'� � ���P�P`I?� � BDCZ351�X"� � � ���P�P_�1�:'* �!��m)�o4#X<KH9�\D\g?y_�1�:'*!� , BDXZBDC� v'� >� � �P�[F(X<7H6ZCM^%9�4#681'_�N�Q AaU�²�³�Q���µW4#1'3qX<BD3,K51'K5C�_�K5354#X<BD1'35C�1'3�X<7H6p`�1'K535;%9�:E?qF%9�35;y9�C *wA�9�:<BD6C�_�:<1'= �pX<1 , 1�3H6k'1,6#CP_�:<1'=�XE:[9�46#CP7H9A(BD35kW3H1c;56i:<BgA'9�X<BgAq6#C�BD3 ��v�X<1WXE:[9�4#6#C|7H9A,B]35kW1'356{;56i:<BDA�9�X<BgAq6ZBD3 ���F�9�35;w1'356{49�3wk'K56CECPX<7H9�X_�1�:f� � * ��, X<7568XE:[9�4#6C|7H9Aq6Y�)> v

�;56#:<BgA�9�X<BgAq6#CPBD3#���(F�BD_�1'3568zH35;HCp9We�9?>X<1c6#l,^5:<6#CEC|eP7H9�XPX<7HBDC|=W69�35C#m|¢�7H6

4[7%9�:[9�4#X<6i:<BD�j9�X<B]1�3~1�_-X<756@XE:[9�46#CpBDCZCEBD=)^H\]6i:pBD3~X<756f4j9�CE6 *<��­c`�649�K5CE6)1�3H6f4j9�3�KHCE6f�-R����Q��%��XE:[9�35CE_�1�:<=wF�eP75BD4}7ePBD\D\�`a6ZCMX<K5;5B]6#;>_�1�:�X<7H9�X�:<69�CE1'3am

�6#_�1�:<6o;H1�B]35k@X<7%9�XF51'356Z7H9�C�351je�9fCEBD=W^H\D6oe�9#?rX<1f^5:<1jA'6pX<756Z4#1'=)^%9�4#X<356#CEC�1'_ X<756ZBD3��E6#4iX<B]1�3Y1'_ � v'� � ���P�-B]3qX<1���5���P��BD3�X<756f49�CE6f1'_�`a1�KH35;56#;�1�^�6#3~CE6iX<CoePBgX<7�N�Q AaU�²�³�Q���µY`�1'K535;H9�:E?'FaX<756f49�CE6f1'_-41�3IX<BD3IK51'K5Co`a1�KH35;H9�:E?y`�6#B]35k\D6_]XP_�1�:P\]9�X<6i:m� ��%"-� j�D������ §h0�_ � BDCY`a1�KH35;56#;*ePBgX<7�N�Q AaU�²�³�Q���µh`a1�KH35;H9�:E?'FPX<7H6#3�X<756~BD3��M64iX<BD1'3/1'_W� v:� �5���P�WB]3qX<1 ���5���P�WBDC4#1'=W^%9�4iXm�2È<ÆjÆ5#[§d¢�7568`%9�CEBD48BD;56j9fB]C�X<7H9�XPX<756i:<686#l(BDCMX<CP9W4#1'3qX<BD3,K51'K5CP6il,X<635CEBD1'3 � _�:<1'= � v'� � ���P�dX<1E� v'� � � � ! �[F�9�35;>X<7HBDC�BDCCE6#63)`I?f\D1,49�\DBD�j9�X<B]1�3 F�K5CEBD3Hk89o^%9�:EX<BgX<BD1'3)1�_�K535BDXs?'m�1�:<BgX<BD35k � � � 7 � 7 � Fq6j9�4}7 � 7 � `a6#\D1'35k'CVX<189pCM^%9�4#6 � v'� �5��� � � ��BD39�3w1�:EX<7H1�3H1�:<=r9�\a`%9�CEBDCP;56i^�635;5BD3Hk)K5^�1'3 �<FH_�1�:PCE1'=W68N�Q AaUj²a³�Q���µW4#1'3qX<BD3IKH1�KHCP_�KH354iX<BD1'3 7 md¢�756#:<68BDCP9)41'3qX<BD3IK51'K5C6il,X<635CEBD1'3 � 7 _]:<1'= � v'� �5��� � � ��X<1�� v:� �H� �)!p�[F 9�35;~X<756i:<6_�1�:<6{X<7H6i:<6fBDCZ9r41�3IX<BD3IK51'K5Cp6#l,X<6#35CEBD1'3 ��_�:<1�= � v'� �H���P�X<1E� v:� � � � ! �[F5k'BgAq63�`I?�� � � � 7 � 7 � � 7 � �[m

N�6iX � � C�� � �"!p��CEKH4[7WX<7H9�X ���GQa�H� ��_�1�:3Q � �oF,9�35;W\]6iX&� �5`�6|9p`�1'K535;56;c1�^�6#3cCE6iXd41�3IX[9�B]35BD35k - � *�*�. L�%i� �H�[m¢�7563cBD_a98CE6jJIK56#3546 ��� BDC-`�1'K535;H6#;rBD39� v:� �5���P�[FIX<756PCE6JIK56354#6|1�_a6#l,X<6#35CEBD1'35C � ��� BDC-`�1'K535;56;rBD39� v'� �5� �"!p�[F,9�35;X<756@CE6JIK56354#6@1'_VXE:<K5354j9�X<6#;�_�K53H4iX<BD1'35C ��� � ��� ��BDCP`�1'K53H;56#;�B]3 � v'� �� ��� ���[Fa9�3H;yX<756i:<6_�1�:<6@9rCEK(`%CE6JIK56#3H4#6 ��� � � � �`�6\D1'35k'CuX<1)9f4#1'=W^%9�4iX�1�_�� � ��� � �[FH9�35;YX<756pCE6jJqKH6#3546o1'_ :<6CMXE:<BD4iX<B]1�3HCdX<1 �oF,eP75B]4[7�BDC � � F(`�6\D1'35k'CuX<1)9f4#1'=W^%9�4iX�1�_���5���P�[m

� �

Page 39: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�jÔ,m-�5:<BD;H9?��H6#`5:<KH9�:E?�� �,F�­�®'®'®� ;H6#4#1'=W^�1'CEBgX<BD1'3°BD3°�-R����Q��%�*CE6i:<BD6C@1�_�9yCE49�\]9�:@_�K5354#X<BD1'3$@¶BD3h1'356cA�9�:<B]9�`H\]6c4#1'35CEBDCMX<C@BD3°e�:<BgX<BD35k @��GQa���

� � � � 4 � ( º 7 � � _�1�:ZCE1'=c6>�S4#1'=)^%\D6il%��4#1,6#�W4BD6#3IX<C 4 � � � ��� m@0�_-X<7H6fCE6#:<BD6#CZ4#1'3qAq6#:<k�6CpX<756)_�K5354iX<B]1�3!@�=@KHCMXp`a6^�6#:<BD1,;5BD4oePBgX<7�^�6#:<BD1,; �Zm� ;564#1'=)^�1'CEBgX<B]1�3yBD3y�-R����Q����¶BD3IX<6#k�:[9�\�1'_-9WCE4j9�\]9�:o_�K5354iX<B]1�3 @~BD3y1'356{A�9�:<B]9�`H\D6{4#1'35CEBDCMX<CoBD3we�:<BgX<BD3Hk9@��GQa� �

I � �@�� �I� (jº 7 ��� O �f_�1�:PCE1�=c6Z_�K5354#X<BD1'3 �@�m¢�75686il,X<6#3HCEBD1'3�X<1W_�K53H4iX<BD1'35C�1�_ & A'9�:<B�9�`%\D6#C�\D69�;5C�X<1)X<7568_�1'\D\D1jePBD3Hkf;56#zH35BgX<B]1�3 m

� &������D������ §-�%1�:@ � ��vj� �)!Z�[F,X<756Z�-R����Q�����XE:[9�35CE_�1�:<= 1'_ @wBDC2X<756p_�K5354#X<BD1'3���@>1�: �@>;H6izH356;�1'3��+X<756p;5K%9�\a1'_}��)!�`,?���@�� �q�*��I � 1 @��GQa� ( " º 7 � ��� � � O Q�m � 35689�\DCE1W;56#zH356C ��`,? �E@�� �H�*��I � 1 @��+*H� ( � º 7 � ! � � � O&*%m

np9A(BD35kr\D69�:<356;wX<756{X<756#1�:E?w_�:<1'= N�9�K(:<63qX8�5²�³�´{���H��µ(F%0PK5CE6@7HBDC|351�X[9�X<BD1'35C#F%`HK5Xo=W1'CMXp=c9�X<756#=r9�X<BD4#B]9�35CP;51351�X2^HK(X2X<756o4#1,6i�c4BD6#3qX�­��~B]3cX<756oBD3qX<6k�:[9�\OF,9�35;�CE1�=c6o;5Bg �6#:<6#3qXu^a1e26#:<C21'_��~1,4#4K(:2X<756#3YBD3cX<7H6#Bg:2_�1�:<=@K5\]9�C#F(BD3cX<7H69�:<k'K5=W63qXo1�_VX<7H6{6il,^�1'3563qX<B]9�\DC|9�35;�=@K5\DX<Bg^H\g?(BD35k)X<756@BD3IX<6#k�:[9�\DC#m � 356{CE751'K5\D;�`�6@9e�9�:<6{1'_VX<756@_�9�4iX|X<7H9�Xo;5Bg a6i:<6#3IX`�1,1�b(C�=c9?�K5CE68;5Bg �6#:<6#3qXP4#1'35CMX[9�3qX<C#m

¢�756@�-R����Q��%�hXE:[9�35CE_�1�:<=�� =r9�^HC ��vj� �)!Z��BD3qX<19C � � �"!p�[F%`HK5Xo1'356{BD=W^�1�:EX[9�3qXP^5:<1�^�6#:EXs?wBDCPX<7H9�XoBgX|6il,X<635;5C9�CP9�3>B]CE1�=c6iXE:E?Y_�:<1'= �dº�� �"!p�dX<1WBgX<CE6\D_�F(ePBgX<7>BD3qAq6i:<CE6 ��m

�o351�X<756#:wBD=W^�1�:EX[9�3IX�^5:<1�^�6#:EXs?*BDC>X<7H9�XwBgX>XE:[9�35CE_�1�:<=cC>;H6i:<BgA'9�X<B]1�3 BD3qX<1�=@KH\gX<Bg^H\DB]49�X<BD1'3aF|1�:y=W1�:<6�k'6356i:[9�\D\g?4#1'3qAq1'\DK(X<B]1�3>BD3qX<1)^5:<1,;5KH4iXF%9�3H;>1'35684[7564<b(CP69�CEBD\g?rX<7H6Z_�1�\]\D1jePBD35kf^5:<1�^�6i:EX<BD6C#m

0�_*@ � K � � v � � ! �[FHCE1)X<7H9�X@ � K � � v � � ! �[F5X<7563>1'35687H9�C�>� @ � K5�H���@��#K%m0�_*@ � ��vj� �)!Z�29�3H; Q � @ � ��vj� �)!Z�[F(X<7563>1'356Z7%9�C �

� � � ���� � ��>�:>P­ �+��Q � @��[m

0�_*@ � ��vj� �)!Z�29�3H; ������ � � ��v� �"!p�[F5X<756#3�� ������ � � ­ � � � ����@�m0�3�1�:<;H6i:ZX<1Y;56#zH356fX<7H6f��R����Q�����XE:[9�35CE_�1�:<= _�1�:ZCE1'=W6);5BDCMXE:<BD`HK(X<BD1'35C)�+X<756fBD35BDX<B]9�\�;56izH3HBgX<BD1'3~6il,X<6#35;HCpBD=W=c6iL

;5B]9�X<6\g?cX<1cÚP����R�T¶=W6j9�CEK(:<6#C�ePBgX<7�zH3HBgX<6pX<1�X[9�\a=r9�CEC�`I?��-� �I�H� ��� � ( " º 7 ��� � � � ��9�35;wk'BgAq6#C � � C - � �)!p�E�[FHN�9�K(:<63qX�(²�³�´8���5��µ>B]3qXE:<1,;5K54#6;~X<7H6)CM^�9�46W1'_d:[9�^HBD;5\D?�;5649?(BD35k>CE=W1,1�X<7�_�K5354#X<BD1'35C��o� � ! � � � �,� C � � ! � � � � � �� �� �"!p�@_�1�:r9�\D\�^�1'\g?(351'=WB]9�\DC �)F29�35;/9�\D\�=fK5\gX<BgLOBD3H;5BD4#6CW� �h�+eP75BD4}7�BDCc9��V� ��a²�³���� CM^%9�4#6��[m 2?¶BgX<6#:[9�X<BD35k�X<7H6^5:<64#6#;HBD35k{:<6=c9�:Eb(C#F,1'356p4[7564<b(C269�CEBD\D?)X<7H9�X���=r9�^%C��o� �)!Z��B]3qX<1@BgX<CE6\D_stI1'356Z9�\DCE1{zH3H;5C2X<7H9�X2BgX<CuB]3qAq6i:<CE6oBDC ��F59�35;X<7H9�X21'356o7H9�C2±d¥,��T�²�³��%���%¥�v2_�1�:<=@KH\]9WI�� 1 ����@�� K OMQ ��I�� 1 @����#K5�MOMQ>_�1�:�9�\D\ @ � K � �o� �)!p�[m-¢�75B]CuBDCuX<756|:<6#\]9�X<BD1'3eP75BD4}7fN�9�K(:<63qXu�(²a³�´{���5��µpK5CE6#;)_�1�:-;56izH3HBD35koX<756���R����Q�����XE:[9�35CE_�1�:<= 1'3fX<7562;5K%9�\�� � � � ! ��1�_��o� � ! �[Fq49�\D\D6;8X<7H6CM^%9�4#6{1'_�X<6#=)^a6i:[9�X<6Z;5B]CMXE:<Bg`HK(X<BD1'35C{�S1'356{49�35351�X|;56izH3H68X<756{��R����Q����°XE:[9�3HCE_�1�:<= 1�_�9�3�9�:E`HBDXE:[9�:E?>;5BDCMXE:<Bg`%K(X<BD1'3>1'3�)!{F'1�:�6#A'63f1'_�9�3W9�:E`HBgXE:[9�:E?{CE=W1,1�X<7f_�K5354iX<B]1�3@eP7%9�X<6#Aq6i:�BgX<C�k�:<1je�X<7)9�X ,�BDC#Fq9�35;fbq6#6#^W9�\]\IX<756ub(351jeP3f^5:<1�^�6i:EX<B]6#C}�[m�H1�: � � ���O� �)!p�[F�1'356@;56iz%356#C�� � � ���O� �)!Z�2`I? ��� � � ��� � ��� � � ����_�1�:o9�\D\�� � �o� �)!Z� � � � �.��� �)!p�[F�9�35;~1�3H6zH35;HCp69�CEBD\g?>X<7H9�Xp1'356f7H9�C�� �

��� �� ­�� � � ��� ��9�3H;��>�:>|­ � �RQ � �o��� ���

��� � _�1�:89�\D\ � � � � � � ! �P9�3H; "E�!� ���������'& F9�35;>X<7H9�X�� B]CP9�3wBDCE1'=W1�:E^H75BDCE=�_�:<1'=����S� �"!p�u1'3qX<1cBgX<CE6#\D_sF,ePBDX<7�BD3IA'6#:<CE6 ��m

�%1�:�6#l(9�=W^H\D6�Fj4#1'=)^%K(X<BD35k�X<756��-R����Q��%�WXE:[9�35CE_�1�:<= 1'_ � �GQa�*� ( " � � � � �+eP75BD4}7p`�6#\D1'35k'CaX<1�o� �)!p�E�[F�1'356-351�X<BD4#6CX<7H9�X ���

��� �� >|­��RQ � � F�CE1rX<7H9�Xp1�3H6{7%9�C � � ���� � � >P­�� � ��� � _�1�:�"��¸� ���������:& Fa9�35;yX<756#:<6#_�1�:<6 � � � �q�8�(C ( " � � � � t

1'356dzH35;HCHC � �-`I?8KHCEBD35kPX<7H6d_�1�:<=fK5\]9P_�1�: ���*®|9�35;@K5CEBD35kRI � ( " � � OMQ �Ø�o�+X<756d4\]9�CECEBD49�\q=W6#X<751,;{_�1�:�41'=)^HK(X<B]35k� � 4#1'35CEBDCMX<CPBD3>=c1A(BD3HkW9f^%9�X<7>1'_�BD3qX<6k�:[9�X<BD1'3�BD3>X<756Z41�=W^H\D6#lc^H\�9�3H6��[m

�o3>1�X<756i:�6il59�=)^%\D6oB]C2X<1)CE751je X<7H9�X�>� � � � F(`,?Y351�X<BD4#B]35k@X<7H9�X8� � � >� �)!p� �!���S� �"!p�[F5CE1)X<7H9�X�>�Z6il(BDCMX<C#F9�35;>_]:<1'= � v

� � ��*®@1'356Z;56;5K546#C2X<7%9�X � �"�>� �*®)_�1�: "��Ø� ���������:& FH9�35;�X<756i:<6_�1�:<6��>� ��C � � t51'356pzH35;5C C��Ø�o`I?

K5CEBD3Hk � �GQa�*� ( " � � � � BD3�X<7568;56iz%35BgX<BD1'3aF5CE1WX<7H9�X C � ���>� � � �*� ��� � � � �H���m�H:<1�=�9{CE4j9�\DBD35k@^�1'BD3qX�1�_ A,B]6ie8F,1'356pCE751'K5\D;Y:<6#=W6={`�6#:uX<7H9�X2KHCEBD35k��hX<1f;56#3H1�X<6C�9@\D635k�X<7rK535BgX�_�1�:�=W69�CEK(:<B]35k

Q�9�35; �ØX<1W;56#3H1�X<6{9)K535BgXP_�1�:P=W69�CEK(:<B]35k � F(X<7563��)CE49�\D6#CP9�C �" v|9�35;#� � CE4j9�\D6#CP9�C � � !8m

v ��B]4[756\|±d¥,��T�²�³��%���%¥�F��qePBDCECY=c9�X<756=c9�X<BD4B]9�3aFp� � �'Ô�L<��'Ò�(ImØno6ye21�:Eb'6;*9�XYÙu¢�n �3VK(:<BD4[7Ø�SÙdBD;Hk�63 V1'CECEBDCE4[7H6¢�6#4[7H35BDCE4[7H68no1I4}75CE4[7,K5\D6��[m

�'�

Page 40: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

�6#4j9�K5CE6 � BDC89�3�BDCE1'=W6#XE:E?�_]:<1'= �dº�� �"!p�|1'3qX<1>BgX<CE6\D_sFa1'356W49�3�BD;563qX<BD_�?wX<756WBD=c9�k'6@`,? � 1'_-X<7H6r�(RV¯ R�¥,�H�CM^%9�4#6R;hvj� �)!Z�H��� v'� º�� �"!p�[m-�oC� � �

��� �� ­ � � � �"� � FH1'356pz%35;5C�X<7H9�XI�� 1 � KML.N4O�� � ��� º�O Q ��I�� 1 Õ ��ºM� � � º � � � � º O �(m

� &������D������ §d�H1�:o9@:<6j9�\ - �®(F�; � � �"!p�H� � � � �uº�� �)!p� � � � � � � �6� �uº�� �)!Z� �qm�%1�:o9f:<69�\ -���®,F�; � � �)!Z�H��� �6� ���S� �)!Z� � �E�uÓ�� � � ºj� � � º�� �6� �uº�� �)!Z� �qm0�_�-pBDC�9@351'35356kq9�X<BDA'6pBD3qX<6k'6i:8~F(X<756#3YX<75B]C2;56iz%35BgX<BD1'3Y1'_ ; � � �)!p�d;51,6#C�k'BgAq6|X<756pC<9�=W6pCM^%9�4#6Z9�C ���� º�� �)!p�[m�%1�:�- � ®(F-X<756YCM^�9�46E; � � �)!Z�8BDCf351�XW9�CEK(`HCE6#XW1'_ �uº�� �"!p�[Fd9�35;�BD3¶1�:<;56#:fX<1~K5CE6��»1'356�CMX[9�:EX<Cf_�:<1'=Û9�3

6#\]6#=W63qX�1'_ ���S� �)!Z�[F5`HK(XP1'356849�353H1�XP;H6izH356c�E�uÓ�� � � � � � � `a6#49�K5CE6c�E�uÓ�� � � � � BDC�3H1�X|9�C��_�K5354#X<BD1'3am� _�41�K5:<CE6�F�; " � � �)!Z�uBDC�X<756Z;HKH9�\ 1'_T; � � �)!Z�[F(`%K(X�X<756Z3H1�X[9�X<BD1'3>_�1�:|9�3w1�^�6#3>CE6#X �*B]C�;5Bg �6#:<6#3qXm

� &������D������ §o�H1�:89�3�1�^a6#3~CE6iX"� ���)!{F�9�35;�9r^�1'CEBgX<BgAq6@BD3IX<6#k'6i:�Fa1'356f;56351�X<6#C; " �)���P��X<756f;5KH9�\V1'_*; �� ���P��+eP75BD4}7>BDC�X<75684#\D1'CEK(:<6Z1'_HC � ���P�dBD3�; �)���P�E�[m� ��%"-� j�D������ §#; " vj���P� � � � � � �O���P� � � � @ � > � !� �Vv ��� ���� � � @ � ��������� @ ! � �uº����P� �'mr0�_2±2R�Q�TV²���� ��¶BD356jJIKH9�\DBgXs?751'\D;5C�_�1�:);hv� ���P�[F(X<7563>6iAq6i:E?EK � � �uº����P�u49�3�`�6Ze�:<BgXEX<6#3w9�C � !� �Vv � � �

��� �F(ePBDX<7EK v���������� K ! � �uº����P�[m

�2È<ÆjÆ5#[§�¢�7562=r9�^5^HBD35k � �� � � � ������ / ��������� ������ 1 � B]CV9�3fBDCE1'=c6iXE:E?81'_�;°v� ���P� 1'3qX<189|4\D1'CE6#;fCEK(`HCM^�9�46�1�_ �uº����P� ! ��vF'CE1Z9\DBD3H69�:�4#1'3qX<BD3IKH1�KHC�_�1�:<= �y1'3�;hv� ���P��BDC XE:[9�35CM^a1�:EX<6;81'3qX<1PX<75B]C CEK5`HCM^%9�4#629�35;{6#l,X<6#35;56;ZX<1|9�\DB]3569�:�4#1'3qX<BD3,K51'K5C�_�1�:<=1'3��uº����P� ! ��v#FI9�3H;@X<756#:<6#_�1�:<6uX<756i:<6�6il(BDCMX3@ � ��������� @ ! � �uº����P��CEK54}7@X<7%9�X �P� �V�H��I J � @ � �PÓ � !� �Vv @�� ������ � � O Qc_�1�:-9�\D\� � ;hv� ���P�[F51�:P6jJqKHBgA�9�\D63qX<\g?c_�1�:|9�\D\ � � C�� ���P�[F5`HK(X�X<75B]C�=W69�35C�X<7%9�X �P� ���H� ��� � ���-ePBDX<7 �5��@ � > � !� ��v ��� ���� � F9�35;>`�6#4j9�K5CE6�C � ���P�uBDC�;5635CE68BD3 ;°v� ���P�[FH1'356Z7H9�C �$� �Zm0�_8±uR�Q�T�²������� BD356JIKH9�\DBDXs?�7H1�\];5CY_�1�:9;°v� ���P�[F�X<7H6#3/X<756�=c9�^5^HB]35k � �� � ������ / ��������� ���� � 1 �~BDC�9�3�B]CE1�=c6iXE:E?�1�_;hv� ���P�W1'3qX<1�9¶4#\D1'CE6#; CEK(`HCM^%9�4#6�1'_ �uº����P� !8Fo9�35;*6iAq6#:E?/\]BD3569�:�4#1'3qX<B]3IK51'K5C>_�1�:<= � 1�3�;°v� ���P�c7H9�CYX<756�_�1�:<=�P� �V� � � !� ��v K0� ������ � O Q~_�1�:o9�\D\�� � ; v� ���P�[FHePBgX<7�K v���������� K ! � � º ���P�[F�9�3H;yBD3>^%9�:EX<BD4K5\]9�:|BD_TK � � � º ���P�21'356@49�3e�:<BgX<6 I J K � ��OMQyBD3YX<7HBDC�e�9?'m� ��%"-� j�D������ §d�%1�:|9�3q? - � �fF C � � �)!p�uB]C�;56#35CE68B]39; � � �)!Z�[m�2È<ÆjÆ5#[§�¡u1'35CEB];56i:<BD3HkPX<756uCM^%9�46��W; � 1'_H_�KH354iX<BD1'35C�BD3 �uº�ePBDX<78X<756de26#B]k�7IX2�E�IÓ�� ��� ºj� � F'BSm 6'm I � 1 �E�IÓ�� � � ºj� � � � � �I��� ºPO � �,�m � 356@49�3�9�^5^H:<1'9�4}7~9�3q?y_�K53H4iX<BD1'3yBD3 �W; � `,?w_�K5354#X<BD1'35CPePBgX<7y4#1'=W^%9�4iXoCEK(^5^�1�:EXo`I?wXE:<K53549�X<BD1'3aF�;H6izH35B]35k ���`I? ��� � �q�H� � � �I��BD_ � � � ( � 9�35; ��� � �I�H�*®@BD_8� � � � � FH9�35;Y`I?rN���¯��aUj¤���>;H1�=cBD3H9�X<6#;Y41�3IA'6#:<k'6#3546�F ��� 4#1'3qAq6i:<k'6#CX<1 � BD3 �W; � mP�o3q? � � �W; � 7H9A,B]35kr4#1'=W^%9�4iX|CEK(^H^�1�:EXp49�3w`a6@9�^5^5:<1q9�4[756;y`I?w_�K5354#X<BD1'35C|BD3 C � � �)!Z�2`�6#4j9�KHCE6_�1�:Z9rCE=W1,1�X<75BD35krCE6jJIK56#3546 � � 1'356@7H9�C � � � � � � BD3+�uº�� �)!Z�[Fa9�35;y`�649�K5CE6@X<756@CEK(^5^�1�:EX<CpCMX[9#?yBD3~9W`�1'K53H;56#;CE6iX)1'356Y7H9�C � � � � � � BD3 �W; �jm�¢�756#:<6#_�1�:<6 C � � � ! �ZB]C@;5635CE6YBD3 �W; ��F-9�3H;hX<756i:<6_�1�:<6 �o� � ! �ZB]C@;5635CE6YBD3�W; � m�¦oCEBD3Hkf��R����Q����~XE:[9�35CE_�1�:<=�1'356p;56#;HK54#6CuX<7%9�X�|� �"!p�-B]C2;56#3HCE6pBD3E; � � �)!p�[m�N�6iX8+ 9-|`a6p9f351'35356k'9�X<BgAq6BD3qX<6k'6i:2X<756#3�1�3H6o4j9�3>9�^5^5:<1q9�4[7>9�3q?Y_�K53H4iX<BD1'3EF � �o� �)!Z��`I?Y9@CE6jJqKH6#3546oBD3 C � � �"!p�[F,X<756o41�3IA'6#:<k'6#3546o`�6#B]35k@BD3;6�)� �)!Z�-CMXE:<1'35k5FH9�35;�X<756i:<6_�1�:<6Z9�\DCE1)BD3 ; � � �)!Z�dCMXE:<1�3Hk(F%9�35;�X<75BDC�BDC�;51'356p`I?�9�^5^5:<1q9�4[7HBD35kWFW`I?9F%�GQa� � � �� � ePBgX<7� � C � � � ! �dePBgX<7 � �GQa� � �p_�1�:W� QH� ( ��m

0�3~X<756f4[7H9�:[9�4iX<6#:<BD�9�X<BD1'3�1'_�X<756@XE:[9�4#6Co1'_d_�KH354iX<BD1'35Cp_�:<1�= ; � � �)!p�{�S_�1�: - � vº �[Fa1'356@ePB]\D\�KHCE6@X<756f_�1'\D\D1jePBD35k:<6#CEKH\gXm� & ����$ §p0�_ � � �o� �)!Z�Z9�3H; F � �o� �)!

" v}�pBDCZX<756W:<6CMXE:<BD4iX<B]1�3�1'_ � 1'3�X<756c7q?,^�6i:E^%\]9�356�Q ! ��®(F�BSm 6�mEF��GQ���� �� �GQ�� � ®I�u_�1�:)Q�� � �)! " vjF5X<756#3>1'35687%9�C��WF�� �#���H� I � � � � � � � � ! �4O � ! _�1�:��#� � �)! " vm�2È<ÆjÆ5#[§ �649�K5CE6�� � � �Ø��F�1'356u7H9�C��u�O®I�H��I�� � �u� �q�4O �,m � 3H6uK5CE6#C�X<7HBDC�:<6#\]9�X<BD1'3{_�1�:�X<756u_�K5354#X<BD1'3RQ ! �� � �GQ�� � Q ! �[F9�35;>X<756#3w1�3H6pX[9�bq6#C�X<756Z�-R����Q��%��XE:[9�35CE_�1�:<= BD3 Q�� 1'_�`�1�X<7>CEBD;56C�1'_�X<756Z6jJIKH9�\DBgXs?'m

Õq®

Page 41: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�jÒ,m-��1'35;H9?��%6i`5:<KH9�:E?y­,��F�­�®'®'®� ��%"-� j�D������ §°�%1�: - � vº FP_�K5354#X<BD1'35CY1�_R; � � �)!Z�W7H9A'6�9°XE:[9�4#6�1'3/X<756�7I?I^a6i:E^H\]9�356 Q ! �Û®(F�`�6#\]1�3Hk�B]35k�X<1; � " /� � �)! " v}�[md¢�7568=c9�^5^HBD3Hk � � BDC�CEK(:+�E6#4#X<BgAq6Z_]:<1'= ; � � �)!p�u1'3qX<1�; � " /� � �)! " v}�[m�2È<ÆjÆ5#[§Y�H1�:W^5:<1jA(BD35k~X<756�z5:<CMXW^%9�:EXFdBgXWBDC)6351'K5k'7�X<1�CE751je�X<7H9�X)X<756i:<6>6il(BDCMX<C C CEK54[7�X<7H9�XW_�1�:c9�\D\ ��� �o� �)!p�9�35;5F�;56iz%356#;�`,? F��GQ � � � � �GQ � � ®I�W_�1�: Q � � � ! " v 1�3H6�7%9�C!� � F�� � � � � / �

� � � � 1 / � ( C � � � � � � � � � 1 � m � 3H6�7%9�C�WF�� � � �H� I � � � � � � � � ! �4O � ! F,9�3H;c1�3H6�7H9�C� �WF�� � � ��� º ( � I �u�E��Ó!� � � º � � � � � � � � � � ! ��� º O � ! � � I �2�E��Ó!� � � º � " ��O � ! ��`I?¡u� �²�³���L��(²a³�´{���5��µpBD356JIKH9�\DBDXs?'m�¦oCEBD35koX<7H624}7H9�35k'6�1'_HA�9�:<B]9�`H\D6 � ! � % � �uÓ � � � � º Fq1'356�7H9�C I � �E�HÓ � ��� ºj� " � O � ! �� � �uÓ�� � � � º �[v " º ��I�� � �� v � � � � � � CW� -��}�E�@Ó � �#� � ºj� � v � º � " � Ff9�35;xX<756#:<6#_�1�:<6h1'356¶7H9�C¶�E�)Ó � �#� � ºj� � " � v � º � � �WF%� � ����� º (CW� -j� I �2�E�dÓ�� � � º � � � � � � � � � � ! ��� º O � ! F5eP75BD4}7>k'BgAq6#C2X<7H6p;H6#CEBg:<6;�:<6#CEK5\DX�`I?�BD3qX<6k�:[9�X<B]35kfB]3 � � F5`�649�K5CE6�C)� -�� �-, BD_9�35;w1'35\g?YBD_�- � vº m0�3y1�:<;H6i:|X<1)^5:<1jAq68X<7568CEK5:+�M64iX<BgA(BgXs?qF51'356{=@K5CMXoCE751jexX<7H9�X|BD_BF � ; � " � v � º � � � ! " v �uX<756#:<686il(BDCMX<C � � ; ��� � ! �CEK54}7>X<7H9�X�WF�� � ���H��I � � � � � � � � ! �4O � ! _�1�:|9�\D=W1'CMX|9�\]\ � � � �)! " vm

� 356�;56#zH356C � � � �#� � � ! �,� �WF�� �#����� � � 1� v � � � � � � v� v � � � � � F@ePBgX<7 � � C � � �{�y9�35; I�� �u� %<�4O % � ��F)9�35;

1'356°=@K5CMX�4[7564<b*X<7%9�X � � ; � � �)!Z�[m � 356°CE751jePCwX<7H9�XwX<756#:<6�6il(BDCMX<Cy9�41�3HCMX[9�3IX CÛCEK54}7 X<7H9�Xy_�1�:y9�\D\ � � ��)!

" v)1'356�7%9�C I �u�E�oÓ � � � º� � � � � � � � � � ! ��� º O � ! � CW�E�|Ó � � � � ºj� � " � v � º � � �WF�� � ����� º�Fd9�35;¶X<75B]C)9�=c1'K53qX<CfX<1 I �2�E�oÓ� ��� º� � �� � � � 1� v � � � � � � �� º vv � � � � �HO � ! � CW�E�uÓ � � � � º� �

" � v � º � _�1�:o9�\D\ � � � �"! " vF%eP75BD4[7yBDC�^5:<1jAq6#;w`,?>K5CEB]35kWX<756{4[7H9�35k'61'_�A'9�:<B]9�`H\D6 � ! � % � �uÓ�� � � � º 9�3H;>1'35681�`5X[9�BD35C C���I��d�E�uÓ % º � � �d� %<� º O&%}m

¢�75624#1'35;HBgX<BD1'3�- � vº 7H9�C�3H1�X-9�^H^�69�:<6#;fBD3{^5:<1jA(BD35kPX<7H6uCEK(:+�E6#4#X<BgA(BgXs?'F�`HK5X�1'3562CE751'K5\D;f351�X<BD4#6dX<7H9�X�X<756u_�K5354iX<B]1�3� 4#1'35CMXE:<KH4iX<6#;wBDC�3H1�XP1'35\D?YBD3 ; � � �"!p�d`HK5XPBDC�CEK54[7wX<7H9�X�� � 7%9�CPBgX<C�CEK(^H^�1�:EXPBD3w9f:<6k'BD1'3!� � ! � ( C � �uÓ�� � � � º m

�%1�:Y9�_�K5354iX<B]1�3 � � ;~º�� �"!p�@_�1�:r6il59�=)^H\]6�Fd1'356w49�3�;56iz%356>X<756�XE:[9�4#6 � � � � ; � � º�� �"!" vi�{`HK(XY9�\DCE1~X<7H6

351�:<=c9�\�;56i:<BDA�9�X<BgAq6 � v � � � � ������ 1 �;°v � º�� �)! " vi�Z�SBD3�k'6#356#:[9�\�1'356{X[9�b'6C�� v � X<1r`�6{X<756@351�:<=c9�\�;H6i:<BgA'9�X<BDA'6'FHePBgX<7X<756>351�:<=c9�\2^a1�B]3qX<BD35k~X<1~X<756>1'K(X<CEBD;56��[Fu9�35;/9~=W1�:<6Y^H:<6#4#B]CE6Y:<6CEK5\gXWBDC)X<7H9�X � �� � � � � � � v � �{BDCWCEK(:+�M64iX<BgAq6>_]:<1'=;�º�� �)!p�u1'3qX<1�; � � º'� �)! " v}� � ;°v � º�� �)! " vi�[FH9�35;w=W1�:<68k�6356i:[9�\D\D?r1'35687H9�C�X<7568_�1'\D\]1ePB]35kfCEK5:+�M64iX<BgA(BgXs?r:<6#CEK5\DXm

� ��%"-� j�D������ §�0�_ ØÓ vº �9-�� ØÓ���Ó vº F59�35;�_�1�: � �*® ��������� 1�3H6o;56351�X<6#C �� � X<7H6|XE:[9�4#6o1'39Q ! �*®@1'_ �

� ���� �1

FX<7563 � �� � � � � ��������� � � � �uBDC�CEK(:+�E6#4#X<BgAq6p_�:<1'= ; � � �"!p�u1�3IX<1�; � " � v � º � � �)! " vi� � ����� �9; � " � " � v � º � � �)! " v}�[m�2È<ÆjÆ5#[§V0�_ �6� �o� �)!Z�29�3H; F � � � � � F5X<7563>1'356Z7%9�C��WF � � �#���*��I � ��­ � � � ! � ��� � � � � � � ! �4O � ! m

0�_PF� � ; � " � " � v � º � � �)! " v}�[FqX<7563W1'356P\DBD_�X<C�BgX�`,?fX[9�b(BD3Hk � ;56#zH356;W`I? � � � �#� � � ! �H��WF � � � ����� � � 1� v � � � � � � �E��Ó� �#� � ºj� " � � �Vv � � º�F59�35;Y1'356oBD=W^�1'CE6#C I�� ��­ � � %<� �"�d� %<�4O %*� ��F,`HK(X�9�\]CE1 I�� ��­ � � %<� � �u� %<�MO&%*� ®{_�1�:3"��*® ��������� �>�� � �PÓ� ��������� �F5CE1WX<7H9�XP1'356Z7H9�C �6� ; � � �"!p�[F �

� � � FY9�35; � � � �*®)_�1�: "�� ® �������%� �#>�� � �{Ó�� ��������� ~m0�3y1�:<;56#:PX<1r;56#zH356�; � ���P��_�1�:�� ���)!{F�BgX|BDCPKHCE6#_�K5\�X<1c;56#;HK54#6@9W^5:<1�^�6#:EXs?>1'_ ; � � �)!p�2eP75BD4[7y;H1I6C|351�X|K5CE6

6il,^H\]BD4#BgX<\D?c�-R����Q�����XE:[9�35CE_�1�:<=wm� & ����$ §V�H1�:o® ��- � ��F � � ; � � �)!Z�uBDC�6JIK5BDA�9�\D6#3IX2X<1 �6� �uº�� �)!Z�29�35;6I�� 1 I�� 1

�� � � � " � � ! � � ���" ! � 1 � � � OMQ O&* �-,/m

�2È<ÆjÆ5#[§8�%1�:�( � �)!@F�9�35; � � � �GQa�� � �GQ >�(��[F�1'356c7H9�C � � � � � �I�� (jº 7 � � � � � � � � �I�[F�9�35;�X<756i:<6_�1�:<69I � 1 � � � � >� � ºPO Q���I � 1 �¹��> (jº 7 � � � � � � º � � � � �q��� ºPO �,Fq9�35;)9�C �¹��> (jº 7 � � º ��Õ�CEBD3 º �dF'1�3H6u;56;5K546#C�X<7H9�X3I�� 1 v�

�� 1 � � � � I�� 1 � � � � >

� � ºPO Q � O (�� I � 1 � � � � �q��� º � I � 1 ����� �� � � � � ��

�� 1 � � � O (�� O �(t%BD3w1�:<;56#:PX<1c41'=)^HK(X<6 I � 1 ����� �

� � � � � ��

�� 1 � � � O (y_�1�: � � ®(F%1'356{K5CE6#C

X<756yBD3IA�9�:<B]9�354#6w`I?�:<1�X[9�X<BD1'3�9�35;�X<756y4}7H9�35k'6y1'_oA'9�:<B]9�`H\D6�(�� � ��� �h9�35;/X<756yBD3qX<6k�:[9�\|BDC � � � º � I � 1 ����� �� �/� �'� 1 � � � O �5F

9�35;°X<756#:<6#_�1�:<6 I � 1 I � 1�� � � � " � � ! � � ���"�! � 1 � � � OMQ O&*!� C I � 1 � ��� º � � � � � �q��� ºPO � � ,/F�9�35;$C � I � 1 ����� �

� �/� �'� 1 � � � O ��BDC{zH35BgX<6`�649�K5CE6E�sCEBD3 � � v � (��9� � �,_�1�:��W356j9�:|®)9�35;�-�� �89�35;$��CEBD3 � � v � (��p_�1�:��W3569�: , 9�35;�- ��®,m

�%1�:f9�3°1�^a6#3�CE6#X �oF�1'356W41'K5\D;�;56#zH356 ; �j���P�o_�1�:@® � - � �WBD3°9�X{\D69�CMX8X<7(:<6#6c;5Bg �6#:<6#3qX8e�9?(C#t�9�z5:<CMX@1�3H6BDCpX<1>;56#4BD;56fX<7H9�X �$� ; � ���P�|=W6j9�35C � � �uº����P�o9�35; I0J�I0J �

� � � � " � � ! � � ���"�! � 1 � � � OMQ O&* � ,�t�9�CE6#41'35;�1'356WBDCpX<1>;56#4BD;56X<7H9�X � � ; � ���P�o=W69�35C8X<7H9�X � � � �� J ePBgX<7 � � ; � � �)!Z�[t�9�X<75BD:<;�1'356cBDC8X<1>;H6izH356 ; � ���P�|`I?�BD3qX<6#:E^�1'\]9�X<BD1'3

ÕH�

Page 42: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

�S_�1�:�9Z`�1'K535;H6#;r1�^�63cCE6#XdePBgX<7cN�Q AaU�²�³�Q���µ8`�1'K53H;H9�:E?'FIX<756PX<7(:<6#6|;56#zH35BgX<BD1'35Cdk'BgAq6�X<756|C<9�=W6|CM^%9�46PePBgX<7c6jJqKHBgA�9�\D63qX351�:<=WC}�[m

0�_ �<� � v:� �� �"!p� � ��� *�� �)!Z�[F%X<7H6#3yX<756{:<6#CMXE:<BD4#X<BD1'3w1'_ � X<1�Q ! �x®cBDCp9cN�Q AaUj²a³�Q���µr4#1'3qX<B]3IK51'K5Co_�K5354iX<B]1�39�35;y4#1'3qAq6#:<CE6#\g?�t%BD_BF�BDCP9cN�Q AaUj²a³�Q���µW41�3IX<BD3IK51'K5C|_�K5354iX<B]1�3w;56#zH356;y1�3�9W4\D1'CE6#;wCE6iX �xePBgX<7wN�Q A�U�²�³�Q���µc4#1'35CMX[9�3qX��F�X<7563�1'356W49�3�6#lIX<635;�BgXZX<1 � ! BD3qX<1w9�N�Q AaU�²�³�Q���µw4#1'3qX<B]3IK51'K5C8_�K5354iX<B]1�3 � ;H6izH356;�1'3 � ! 9�35;�7H9A(BD35k�X<7H6C<9�=W68N�Q A�U�²�³�Q���µ)41�3HCMX[9�3IX � `I? � �GQa�H�1- � * + � � � F��GN5�B>)� � N#> QH� ��m

0�_ � � ;hvj� �)!Z��� � v'� º�� �)!Z�[F�X<756�XE:[9�46 F�`�6#\]1�3Hk�CrX<1 ;°v � º�� �)! " vi�[F�eP75BD4}7/e�9�C8�EK5CMX�CE751jeP3/X<1¶=W6j9�3F � �uº�� �)!

" vi�p9�35;$I�� 1� / I�� 1 /�� � � � " � � ! � � ���"�! � 1 OMQ O&*�� ,�mr�o4iX<KH9�\D\D?'F�BD_ � � � v:� �5� �)!Z�p9�3H;/� �-*�� ,�F�X<7563

Ùd=WBD\]BD1@Ðo��¤V¥,Q���� �-R�CE751je26#;rX<7H9�XdX<756PXE:[9�4#6 F)C<9�X<BDCMzH6#C F � ���5� �)!" vi��9�35; I � 1� / I � 1 /

�� � � � " � � ! � � ���"�! � 1 � � �)OMQ O&* �-,/F9�35;>X<7H9�X�X<75BDC�4[7H9�:[9�4iX<6#:<BD�#6C�X<756ZCM^%9�46Z1'_�XE:[9�46#C#m

0�_ �6� � v'� vj� �)!Z�[F(X<756#3wÙ-=cBD\DBD1cÐo��¤V¥IQ���� �-R × C�4[7H9�:[9�4iX<6#:<BD�9�X<BD1'3�BDC2X<7H9�X)F � ��vj� �)!" v}�[m-GI9'9�bY±d�%��� ���y7%9�C

CE751jeP3>X<7%9�X�X<756i:<68BDC�351c\DBD356j9�:P4#1'3qX<BD3,K51'K5C�\]BD_]X<B]35kf_]:<1'= ��vj� �)! " vi�dX<1�� v'� vj� �"!p�[vm

v>027H9Aq6Z:<69�;>X<7H9�X�CMX[9�X<6=W6#3IX�`HK(X�027H9Aq68351�XP\D1,1�b'6;>_�1�:�X<7H6p^5:<1,1'_�m

ÕI­

Page 43: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�)(Im�1°6#;53H6#CE;H9?��%6i`5:<K%9�:E?y­��IF�­�®'®�®� & ����$ §�0�_ � � ; v � � ! �29�3H; & �,F5X<756#3 ��

�� � � º � � ! �[m

�2È<ÆjÆ5#[§ � 356°^5:<1jAq6#CyX<7H9�X �� �� �� � � �� �� º ( CE� � KML.N4O%� � ��� � º _�1�:~9�\D\ �+� C � � � ! �[m �%1�:�9�3q? � � ��1�3H6°7H9�C~® (I � 1 � !� �Vv �� ������ � Ó � � � �

7 � �� º OMQS� I � 1 � KML.N4O�� � ��� ºPOMQhÓ � !� ��v I � 1 ­ � ������ � � � �7 � Ó I � 1 ����

7 ��OMQ�m �649�K5CE6 � �7 � �� v'� v������ �)!Z�Y�S_�1�: & �q�[F�9�3H; � !� ��v ���� � � � �7 � � � !

" º7 ��F21'356w;56;5K546#CWX<7H9�X9I � 1 � KML.NMO�� � ��� º OMQ � � & >*­q� � >� º � I � 1 �

�7 � O Q�F%9�35;w1'356ZX[9�b'6C)� � !

" ºº m¢�756Z:<6#CEK5\DXPBDC�1�`IA(BD1'K5CE\g?Y_+9�\DCE6Z_�1�: & � ­,FH6iAq6#3w_�1�: � CE=W1,1�X<7�ePBgX<7 � �O®q�� ®W9�C v7 �� �dº����� � �8ºj�[m

�{%���.+.�$H��� §�¢�7H6pCM^%9�4#6p1'_�_�K5354iX<B]1�3HC2BD3 C � � �)!Z��eP75BD4}7�9�:<6Z®@BD3>9@CE=c9�\D\�`%9�\D\a9�:<1'K535;>®@BDC2;H6#35CE6pB]3E;hvj� �)!p�d_�1�:& ���O9�3H;w9�\DCE1W_�1�: & � ­q�[m�2È<ÆjÆ5#[§��oC C�� � �"!p�{B]C);56#3HCE6>B]3 ;hvj� �)!Z�{_�1�:r9�\D\ & Fu1'356w=@K5CMXc9�^5^H:<1'9�4}7�9�3q? �(� C�� � �"!p�8`,?�9~CE6jJqKH6#35461'_p_�K5354iX<B]1�3HCWeP75B]4[7�9�:<6�®�BD3�9�CE=r9�\D\�`%9�\D\|9�:<1'K535;*®(F�9�35;�X<75B]CcBDCr;51'356y`I?�X[9�b(BD35k ��� �GQa� � � �GQa� � � � Qa�{ePBgX<7� � C �� �)!Z��6JIKH9�\�X<1�®r_�1�:�� QH� (��f9�35;~6jJqK%9�\�X<1~�@_�1�:�� QH�� x­,m � 356f7%9�C ��� � � BD3+�uº�� �"!p��`I?�N���¯��aU�¤���;51'=WB]3H9�X<6#;/4#1'3qAq6i:<k'6354#6yX<7H6#1�:<6#=wF�9�35;�CEB]=WBD\]9�:<\g? ���

��� �� � � Qa� � � �

��� �B]3-�uº�� �)!Z�[F�9�35;�B]3/1�:<;56i:YX<1hCE751je X<7H9�X

��� � ���� � � � Qa� � ®cB]3#�dº�� �"!p�[F�1'356f9�\DCE1Y9�^5^%\DBD6#C|N���¯��aU�¤����;51'=WBD3%9�X<6;y4#1'3qAq6i:<k'6354#6@X<756#1�:<6#= X<1 ���� @�� � Qa�2ePBgX<7

@��GQa�H� � QH� � ���� �

m� & ����$ §�¢�7568CM^%9�4#681'_�_�K5354#X<BD1'35C�BD3 C � � �8ºj�deP75BD4}7w9�:<68®)BD3w9)CE=r9�\]\�`%9�\D\�9�:<1'K535;y®)BDC�;56#3HCE68BD3 ;hvj� �8º�[m�2È<ÆjÆ5#[§ � 356o49�35351�X�9�^H^H\g?)X<756|C<9�=W6p9�:<k�KH=W6#3IXuK5CE6;rBD3cX<756|4#1�:<1'\D\]9�:E?)_�1�: & �,m � 356P^5:<1,1'_a41'35CEBDCMX<CdB]3r9�^5^H\D?,B]35kno��³�T�L- ���T���²�³>X<7561�:<6=>Fq9�35;)CE751jePBD35kpX<7H9�X-BD_ � � ;

" vj� �8ºj�H� � ;°vj� �{ºj��� � FI9�3H; ��� � ���H�*®o_�1�:-9�\D\ � � C�� � �{º�eP75BD4}7�9�:<6W®YBD3�9rCE=r9�\D\�`%9�\D\V9�:<1'K535;�®(FaX<756#3 � � ®(m �6#4j9�KHCE6 ��� � ����� ®Y_�1�:Z9�\D\#� � C � ���u�P_�1�:89�3q?�1�^�63~CE6iX��CEK54}7°X<7H9�X@® �� ��F�1'356WzH3H;5C8X<7H9�X8X<756cCEK(^H^�1�:EX@1'_ �¸49�3°1'35\D?�`a6���®����SBD_ ��B]C8351�X@®I�[mY�oC8ePBD\D\V`�6cCE66#3aF�B]_29;5BDCMXE:<BD`HK(X<BD1'3��x7H9�CoCEK(^5^�1�:EX ��®��qF%X<756#3 � � � � 4�� �� � � �+z%35BgX<6{CEK5=Y�[F%`HK5XoBD_VCE1'=W6 4�� ��®WX<7H6#3 � �� ;

" vj� �8º�[F`�649�K5CE6�� ��� � � 4�� ��­�� � �q� �y9�3H;>3H1W351'35�6i:<1)^�1'\g?(351'=WB]9�\ � C<9�X<B]CMzH6#CI�� � � � � � � � �v � � � � � O � �-,�m

� 3562;56;5K54#6C�X<7H9�X�BD_�� � � ! � WF�eP756i:<6 *B]CV9|zH35BDX<6u3IKH={`�6#:V1'_H^�1'BD3qX<C�9�35; & /­�X<7563#; v� ���P�H�5; v ���P�H�;hvj� �)!p�[m-¢�75BDCdB]Cd351�XdXE:<KH6P_�1�: & �Ø�|9�CdX<756P_�K5354#X<BD1'35CdB]3 ;hvj� �{��9�:<6P41'3qX<BD3IK51'K5C#m�1*BDX<7cCE1�=c6PX<6#4[7H35BD49�\%4[7H9�35k'6#CX<756pC<9�=c6o^H:<1I1'_�C�9�;H9�^5X�X<1�� v'� �5� �"!p�dBD_u� �+*#( & m2�IBD=WB]\]9�:2BD;H69�C�CE751je X<7H9�X�1'356Z4j9�3>9�^5^H:<1'9�4}7>6iAq6i:E?Y_�K5354iX<B]1�31'_ ;°vj� �"!p�|_�1�: & �r`I?�_�K53H4iX<BD1'35C8BD3<C�� � �)!p�|eP75BD4[7�A�9�35B]CE7�BD3�9>356#BDk'7q`a1�K5:<751,1I;�1'_29�k'BgAq6#3�CE6k�=c6#3qXF�`HK(XX<7H9�XPBDC�351�X�XE:<K568_�1�: & � ­)9�C�X<7568_�KH354iX<BD1'35C�B]3 ;°vj� �{º�u7H9Aq6ZXE:[9�4#6#C�1'3>X<7568CE6#k'=W63qXm

N�Bgbq6�_�1�:-X<756P\DBD=cBgX<BD35kp49�CE6P1'_ �5RV¯ R�¥I�%��BD={`a6#;5;5B]35kpX<756#1�:<6#=wF'X<756#:<6P9�:<6P351�:<=WC�eP7HBD4[7cCE49�\D6�BD3WX<756�C<9�=c6�e�9#?qF`HK(XPeP75BD4}7y9�:<683H1�X|4#1'=W^%9�:[9�`%\D6�tH_�1�:|6il59�=)^H\]6pB]_ & � ­fX<756#3$� � KML.N4O�� � ��� � º 9�35;$� � � � � CE49�\D68BD3�X<7568C<9�=W6Ze�9?�`HK(X_�K5354iX<B]1�3HCZBD3 ;hvj� �8ºj�o9�:<6)3H1�X{3H6#4#6CEC<9�:<BD\g?y`�1'K535;H6#;aF�9�35; �� �� � 7 �� �� º 9�\]CE1>CE49�\D6CpB]3~X<756WC<9�=W6)e�9?�`HK5X � � ;°vj� �{º�;51,6#CP351�XPBD=W^H\g? � 7 � �uº�� �{ºj�[m�no1je26iAq6i:|1'356Z7%9�C�X<7568_�1'\D\D1jePBD3Hk@:<6CEK5\gXm� & ����$ §a0�_�� ���Y�O® � � � � � �8º�F�X<7H6#3{X<756u6il(BDCMX<C C�CEK54[7@X<7H9�X �� �� �7 � � � � 7 � � � � �� �� º ( CE� � KML.N4O%� � ��� � º _�1�:V9�\D\ � � ;hv� ���P�[m�2È<ÆjÆ5#[§ � 356>^5:<1jAq6#CrX<756wBD3H6JIKH9�\DBgXs?¶_�1�: � � C � � ���O® � � � � � 9�3H;�X<7H6i:<6#_�1�:<6wBgXrBDCWXE:<K56y_�1�: � � C�� ���P�)9�3H;�BDX6il,X<635;5CdX<7563cX<1�;°v� ���P�[mV�%1�:)@�CE=c1I1�X<7aF(1�3H6|;56#A'6\D1�^HC�I �

� � � � � � � !� ��v �� ������ � Ó6Q � @��GLq� � �� º OMQ �®(F59�35;Y1'356|K5CE6CdX<7H6BD3qX<6k�:[9�X<B]1�3W`I?W^%9�:EX<C8I

�� � � � � � ­ � ������ � Q �0@��GLq�MOMQ9� >6I �

� � � � � � � � � º � @��GLq�(Ó�Q�º� � � 7 �7 � OMQ�FIeP75BD4}7cBDC-A'9�\]BD;WBD_�L3@��GLq�V9�35;L'º�@��O�GLq�u`�6#\D1'35kWX<1 ��v�O® � � � > ����_�1�:o6#A'6#:E? � �*®(F�9�35;�1'356{;56;5K54#6C I �

� � � � � � � KML.N4O�� � ��� ºPO Q �I �� � � � � � � � � º���­ @WÓL*@ � > L º @ º �4OMQ�t'BD_51'356uX[9�bq6#C*@9� �

7 � F'1'356u7H9�C-­ @�Ó�L*@ � > L º @ º � � 7 > �

�7 � F'9�35;f1'35629�^5^H:<1'9�4}756#C�X<7563�K��

" vº � � � � 7 � � � � FeP75BD4}7>4#1�:E:<6CM^�1'35;5C�X<1)=fK5\gX<Bg^%\g?(BD35k � � � ºo`I? v�67 � � � ��� � 7 � � � � � � m

�6#4j9�K5CE6oX<756p\]1�kq9�:<BgX<=!A'9�3HBDCE756#C�_�1�:�� QH� � � � F5BgX�BDC�BD=W^�1�:EX[9�3qXuX<1)7H9Aq6"��`a1�KH35;56#; F(`HK5X2X<7H6pC<9�=W6Z9�:<k'K5=W63qXe21�:Eb(C-BD_ �¶BDC�KH3q`�1'K535;56;r9�35;WBDC�1'K(X<CEB];56P9o`%9�\D\ �Y�O® � � � �[F'`HK(XdX<756i:<6�BDC-9p^5:<1�`H\D6= ePBgX<7fX<7H6�6#3IX<Bg:<6�CM^%9�4#6't,9�4#X<KH9�\D\g?qFGI9�4JIK56CMLON�1'K5BDC�N�QSR�TVU29�35;fGI9�4jJIK56#C�£p��T �w7H9Aq6uCE751jeP3{X<7H9�X�X<756u4#1'=)^H\]6iX<BD1'381'_�C � � �8º� _�1�:�X<756u3H1�:<= � � KML.N4O%� � ��� � ºBDC�351�X|9WCM^%9�4#6Z1'_�;5BDCMXE:<Bg`%K(X<BD1'35C�1'3 �8º�m

Õ �

Page 44: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

� �,m-�5:<BD;H9?��H6#`5:<KH9�:E?w­'Ô,F�­�®'®'®0�3w1�:<;56i:�X<1WBD;H6#3qX<BD_�?Y;5BDCMXE:<Bg`HK5X<BD1'35C2ePBgX<7>CEK(^5^a1�:EXo9�X|9f^�1'BD3IXF51�3H68K5CE6#C�X<756Z_�1'\D\D1jePBD35kf:<6#CEKH\gXm

� ��%"-� j�D������ §oN�6#X � � � � ���P��7H9Aq6fBgX<CoCEK(^5^a1�:EX8BD3~9r41'=)^%9�4iX�� � � �oF 9�35;�9�CECEK5=c6@X<7H9�X � BDCZ9r;5BDCMXE:<BD`HK(X<BD1'31'_21�:<;56i:#~mw¢�756#3hBD_ � � � C�� ���P�pC<9�X<BDCMzH6C� �� � � �GQa� � ®>_�1�:)9�\]\*Q � � � 9�3H;h9�\D\-=fK5\gX<BgLOBD3H;5BD4#6CR�*CEK54}7°X<7H9�X� ��� ( �FH1'356Z7H9�C ��� � � � �H�*®(m�2È<ÆjÆ5#[§u�oC �xBDCo9�CECEKH=W6#;wX<1W`a681'_V1�:<;56#: ~F�_�1�:|6#A'6#:E?>4#1'=)^�9�4#X � � �*X<7H6i:<6{6il(BDCMX<C CW�����uCEK54}7wX<7%9�X � ��� � �����(CW�����,CEK(^

� �"! � � � � $ � � �� �d�GQa���V_�1�:w9�\D\�� � C � ���P�WCEK54[7�X<7%9�X�- � * *�. L�%i� ����� �hm N�6#X � � �»®°`�6~CEK54[7*X<7H9�X�0Q � �)! � O��GQ � � � � ( � � � � �om/�H1�:r®�� � � � � Fu\D6#X � � � �0Q � �)! � O��GQ � � � � ( � �qF�9�3H;�\D6iX � `�6�X<7H64[7%9�:[9�4#X<6i:<BDCMX<BD4@_�K53H4iX<BD1'3�1�_�� � mpN�6iX�� v � C � � �)!p��ePBgX<7- � *�*�. L�%i� � v ��� ���O® � ����9�35; I � 1 � v �GQa�4OMQ �!��Fa9�35;�9�CK5CEKH9�\ �

��GQa�H� v

� 1 � v � � � �8_�1�: � ��®(m0�_3� � � � � Fa\D6#X � �� �º � � � �

FaCE1rX<7H9�X �� � C � ���P�[F � �

�GQa�8���{B]_HQ � ��9�3H;- � *�*�. L�%i� � �

��� � � ���� � � m� 3H6{7H9�C ��� � � � � � ��� � � �

� � �[F%`�6#4j9�K5CE68X<756{;5Bg �6#:<6#35468BDC ��� � �E��> � ���� � ��9�35;wX<756{CEK(^5^a1�:EXo1'_�� > � �

BDC|BD354#\]K5;56#;BD3+� � � � F�BSm 6'muX<7H68\]9�:<k'6#CMXo1�^a6#3yCE6iX|eP756i:<6 �xBDCo®,m � 3568^H:<1Aq6CPX<7H9�X ��� � � �

� � ����®c9�C � �Ö®)`I?wCE751jePBD35krX<7H9�X_�1�:|9�3q?>=@K5\DX<BgLOBD35;56#lE�¶CEK54[7>X<7H9�XW� ��� ( 1�3H687H9�C�CEK(^

� � !� �� E��� �

�� � �}�GQa��� �·®W9�C � �·®(m

� 356@7H9�C�� � ���GQa��� ( C � "�� ��� _�1�:p9�\D\BQ�9�35; � ����(�~m 2649�K5CE6f1�_-¢�����¥5R���_�1�:<=@KH\]9c9�35;�X<756{A'9�3HBDCE75BD35kY1�_

X<756f;56#:<BgA�9�X<BgAq6C|1'_ � � 1'3 ��K(^~X<1�1�:<;H6i: �Fa1'356f7H9�CE� � � � �GQa��� (�O%�GQ � � � � �" � � �� � O%�GQ � � � ���)_�1�:E� �*��(�·9�35;

��� %<��� ®y9�C % � ®(m u?�N��%Q+¯�T�QSµw_�1�:<=@K5\]9(FV9�3H;�KHCEBD35kwX<756c_�9�4iX{X<7H9�X�O��GQ � � � �"( � ��_�1�:�Q � - � *�*�. L�%i� � ��[F�1�3H6

;56;5K54#6C�X<7%9�XW� ���� ��� � �}�GQa��� ( C � � O��GQ � � � � � m

�{%���.+.�$H��� §P0�_ � 7H9�CpCEK(^5^a1�:EX{9�X{9c^�1'BD3qX N � �oFaX<756#3 � BDCZ9rzH35BgX<6f4#1'={`HB]3H9�X<BD1'3�1'_-;H6i:<BgA'9�X<BDA'6C|1'_�X<756)£pQ�����²=c9�CEC|9�X)N%m�2È<ÆjÆ5#[§20�_#� � �Y�GN � Lq�����oF�X<7563 �Ø7H9�C|zH3HBgX<6{1�:<;56i:�1'3��hFa9�35;�`I?wX<756{^5:<6#46#;5BD3Hkc:<6#CEK5\DXFP � � � �GN5�8�x®r_�1�:9�\D\ � ���( BD=)^H\]BD6#C ��� � � � �*� ®,mu�x:<6CEK5\gXP1'_�N�B]3569�:P�o\Dk'6i`H:[9fC<9?(C�X<7H9�X|1'3w9�3q?�Aq6#4iX<1�:|CM^%9�4#68BD_�_�1�:|\DBD3H69�:�_�1�:<=WC� � ��������� � � 6#Aq6i:E? � C<9�X<BDCE_]?(BD3Hk�� v � � ����� � � � � � ®>9�\DCE1>C<9�X<B]CMzH6#C)� � � � ®rX<7H6#3~X<756#:<6)9�:<6)CE49�\]9�:<C � v���������� � �CEK54}7WX<7H9�X � � � � � � ��v � ��� ��m�¢�7H6i:<6#_�1�:<6�X<756i:<6P9�:<6�CE4j9�\�9�:<C � � _�1�: � ��� ( CEK54}7WX<7H9�X ��� � ���H� � � � � $ � � � �� �u�GNH�_�1�:|9�\D\ � � C�� ���P�-ePBgX<7�- � * */. L�%i� �V��� �hFHBOm 6'm �5� � � � � $ � �:>Z��� � � � � � �� � + m

��1'CMX�4#1'=W^%9�4iX<356CECu:<6CEK5\gX<Cu:<6\g?c1'3YX<756pX<756#1�:<6#=cC21'_��o��µ'�%¥ ��h9�35;��ZU�² R�¥,Q�FH9�3H;YX<756|`�9�CEB]4o:<6#CEK5\DX�1'_ B]3qX<6i:<6CMX756#:<6oBDCdX<7H9�X2B]_a1'356|e21�:Eb(Cu1'3�9{41'=)^%9�4iX2CE6#X � 1'_ �)!�9�35;YBD_ 1'356o7H9�C�9{CE6JIK56354#6 ��� � C)�����V1'_ _�K5354#X<BD1'35CdeP75BD4}77H9Aq6fX<756)C<9�=W6)=W1,;5K5\DKHCp1'_-KH35BD_�1�:<= 41'3qX<BD3IK5BDXs?'FaBOm 6'm � ��� �GQa�H> ��� �+*5����( �P� � Q >�* � �P_�1�:{9�\]\BQ � * � � 9�35;�9�\D\ � FePBgX<7 ��� %<�#�·®89�C�%#�·®(FIX<756#3WX<7H6i:<6P6#l,B]CMX<Cd98CEK(`HCE6jJIK56#3546 � � eP75BD4[7c41'3qAq6i:<k'6#CuK535BD_�1�:<=W\g?f1'3 �hFIX<1 � � CW������SK5CEBD3HkW9f;5B]9�k'1'3H9�\ 9�:<k�KH=W6#3IX�1'35686il,XE:[9�4iX<C�9fCEK(`%CE6JIK56#3H4#6peP75BD4}7�41�3IA'6#:<k'6#C�1'3>9)4#1'K53qX[9�`H\D6Z;5635CE6ZCE6iXP1�_ �hFH9�35;X<756�CEK(`HCE6JIK56354#6>9�\DCE1~41�3IA'6#:<k'6#CW9�XfX<756Y1�X<756#:f^�1'BD3qX<C@`I?¶6JIK5BD41'3qX<BD3IK5BDXs?'F�9�35;¶X<756�\DBD=WBDX{BDCf41'3qX<BD3IK51'K5C)_�1�:fX<7H6C<9�=W6Z:<69�CE1'3��[m

�%1�:@^5:<1jA,B]35kw4#1'=)^%9�4#X<356#CEC{BD3 ���5���P�o_�1�:r� ( * � ,/F�1'356c6#l,X<6#35;5C8X<756c_�K53H4iX<BD1'35C8`I?�®y9�35;°1'356r9�^5^H\]BD6#C{94#1'=W^%9�4iX<356CEC�:<6#CEK5\gXPB]3 ���(� �)!Z�[FHK5CEKH9�\D\g?�9�XEXE:<Bg`%K(X<6#;�X<1 &{R�¥ �8R�¤-R���Ra�2m� & ����$ §�0�_�9WCE6JIK56354#6 ��� BDC�`a1�KH35;56#;wBD3 ���5� �)!Z�29�35;wC<9�X<B]CMzH6#C

B��u�H1�:|6iAq6i:E? ����®(F5X<756i:<686il(BDCMX<C �)� ���2CEK54[7>X<7H9�XI ��� � � � � � � ��� � �HO Q#( �@_�1�:|9�\]\ � mBDB+�-�%1�:P6#A'6#:E? � �/®fX<756i:<686il(BDCMX<C � ��®)CEK54[7wX<7H9�X�BD_)� ( � ( �@X<7563 I � 1 � ��� �GQ)Ó�(��B> ��� �GQa��� �BOMQ#( �qm¢�7563>X<756i:<6Z6il(BDCMX<CP9)CEK(`%CE6JIK56#3H4#6 � � eP75B]4[7>41�3IA'6#:<k'6#CPCMXE:<1�3Hk�\D?rX<1 � � ���5� �)!Z�[m

�2È<ÆjÆ5#[§~0sXYBDCr6351'K5k'7/X<1hCE751je X<7H9�XY_�1�:Y6iAq6i:E? � ��®°1'356�4j9�3�e�:<BgX<6 ��� � F � Ó � � CEK54[7�X<7H9�X<� � � � � � � (S�9�35;*X<7H9�X�_�:<1'= F � 1'356�49�3*6#l,XE:[9�4#X>9h41'3qAq6i:<k'BD35k�CEK(`HCE6jJIK56#3546 F � t8_�1�:�X<756�CEK(`HCE6jJqKH6#3546 � � 1'356�7H9�CYX<7563\DBD=/CEK(^ �� � � � � � � > � � � � � (�­ �dmP�IX[9�:EX<B]35kW_]:<1'=�X<756{CE6#\D64iX<6;>CEK(`HCE6jJqKH6#3546�F�1'3568:<6i^�6j9�X<C�X<7563>X<756{9�:<k'K5=W63qXePBgX<7��°:<6i^%\]9�4#6#;>`I? � º FH9�35;wCE1c1�3 F%9�35;w9);HB]9�k�1'3H9�\ CEK(`HCE6jJqKH6#35468BDCP9r¡u� V²a³ ��CE6JIK56354#6'm¦oCEBD3Hk � � C�� � �)!Z�{CEK54[7�X<7H9�Xc® ( � ( �>9�3H; � �GQa���»��_�1�: � QH� ( �)� �'�[Fd1'356�4}751,1'CE6#C�F � � � ��� 9�35;� � � �E� > � � ��� F,9�35;c1'356P3H1�X<B]4#6#C�X<7H9�X � � � � � � � ( �|`,?)B��[F,9�35;W`�6#4j9�K5CE6 � � F � > F � �Ø� � � � > � � � � ��� Ó � � � � ��� > ��� �9�35; � � � � � > � � � (1� � ( � Fa1'356{z%35;5CoX<7H9�X F � BDC|`�1'K535;56;~BD3#� � � � ! �[F�7H9�CoBDX<C|CEK(^5^a1�:EXZBD3~9cz5l(6#;y`�1'K53H;56#;~CE6iXF9�35;wC<9�X<BDCMzH6#C�B]B��[m

Õ'Õ

Page 45: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

�oCECEK5=cBD35k�X<7%9�XcX<7H6>_�K5354#X<BD1'35C ��� C<9�X<BDCE_�?�BDB��@9�35;/7H9Aq6>X<7H6#Bg:rCEK(^5^�1�:EXrBD3/9�z5l(6#;�`�1'K53H;56#;�CE6#XF21'356yK5CE6#CF � � ��� � � �

_�1�:w9hCM^�64#B]9�\oCE=W1,1�X<75BD3HkhCE6jJqKH6#3546 ��Fo9�35; � � � � ��� > ��� � � �

�[t81'356~49�3 9�^5^H\g?/�o��µ'��¥I��L�ZU�² R�¥,Q�X<1�X<756YCE6JIK56#3H4#6EF � F-9�CfX<756i?h_�1�:<=Û9w`�1'K535;H6#;�CE6JIK56354#6Y1'_�N�Q AaU�²�³�Q���µ�41'3qX<BD3IK51'K5C)_�K5354iX<B]1�3HC@7H9A(BD35kX<756Bg:pCEK(^5^�1�:EXZBD3~9rz5l(6#;�41�=W^%9�4iXpCE6#Xta`�6#4j9�KHCE6 � � �GQa�8� I � 1 � �

�+*H� � ��� �GQa�H> ��� �GQ9> *5��� O,*%Fa1'356f7%9�C�� � � � � � � (I � 1 � ��+*5��� � ��� > � ! ��� � � � O,* ( �qm

0�_ �*B]CP9)`�1'K535;56;>1�^�63wCE6iXPeP75BD4[7wBDC�CE=c1,1�X<7w6#351'K5k'7wCE1WX<7H9�X�X<756i:<686il(BDCMX<CP9W4#1'3qX<BD3IKH1�KHCP6il,X<6#3HCEBD1'3���_]:<1'=� v'� �5���P��X<1�� v'� �5� �"!p�[FqX<756#3)X<756�BD3��M64iX<BD1'3)1�_ � v:� �5���P��B]3qX<1)���5���P��BDC-4#1'=W^%9�4iXF'`�6#4j9�K5CE6�_�1�:d9p`�1'K535;56;WCE6jJqKH6#3546��� � � v:� �H���P� 1'356�4#1'35CEBD;56#:<C�X<756�CE6JIK56#3H4#6 � � ��� ePBDX<7 � � C�� � �)!Z��9�35; � �Ø�21'3 �oFI9�3H;@X<7562^5:<6#46#;5BD3Hk|:<6CEK5\gX<CCE751je�X<7H9�X|9WCEK(`HCE6JIK56354#6 � � � � 4#1'3qAq6#:<k�6C�CMXE:<1'35k'\g?�BD3 � � � � ! ��9�35;>X<756i:<6_�1�:<6ZX<756Z:<6#CMXE:<BD4#X<BD1'3 � � � � � � � �� J4#1'3qAq6#:<k�6C�CMXE:<1'35k'\g?�BD3 ���5���P�[m� ��%"-� j�D������ §o0�_ �ØBDCZ9c`a1�KH35;56#;�1�^�63~CE6#XpePBgX<7~9Y41�3IX<BD3IK51'K5Cp`�1'K535;%9�:E?qFaX<756#3~X<756fBD3��E6#4#X<BD1'3�1'_ � v'� �H���P��B]3qX<1���5���P�dB]C�4#1'=)^%9�4#Xm�2È<ÆjÆ5#[§{¢�756W^5:<6#46#;5B]35ky9�:<k'K5=c6#3qX@;51,6#C{351�X@9�^5^H\g?qF�9�35;°1'356c=@KHCMX8zH3H;°9w;5Bg �6#:<6#3qX8^5:<1,1'_�mY¦oCEBD35ky9�^�9�:EX<BDX<BD1'3�1�_K535BDXs?~1'356c7%9�C{X<1>41'35CEBD;56#:8X<756c49�CE6c1'_ � � ePBgX<7 K535B]_�1�:<=W\D?�41�3IX<BD3IK51'K5C#FV_�1�:@9wCEK(`HCE6JIK56354#6r7H9A(BD35kyCEK(^5^�1�:EXBD3�9r`�1'K535;H6#;�CE6iXm � 356)351�X<BD4#6C|X<7H9�XZB]3�;HBD=W635CEBD1'3��f1�3H6f7H9�C�� v'� �5�O® � ,�� ��C - �O® � ,��[F 9�35;�X<7H6i:<6#_�1�:<6f1'356f7%9�CI � � � � � � �� � � � � � �GQ�� � Q ! ��� �HO Q#( C �WI � � � � � � � �GQ�� � Q ! ��� ��Ó �� ������ 1 �GQ�� � Q ! � �� � � O Q ! F�eP75BD4}7@1'3562=r9?8X<756#3fBD3IX<6#k�:[9�X<6uBD3�Q��Sm¦oCEBD35kpX<756�K53HBD_�1�:<= 4#1'3qX<B]3IK5BgXs?@1'_� 1'356�49�3)41�3HCMXE:<K54iX � � C �� �"!p��ePBgX<7)® ( � ( ��F � �GQa�H�*®pBD_�Q ! � c�GQ�����Ó �9�35; � �GQa�H�Ø�pB]_TQ ! � c�GQ � � Ó�­ �qm � 3568K5CE6#C�X<7563 F � � � ��� 9�35; � � �Ø�E��> � � ��� m

0sX{BDCZCE1'=W6#X<BD=W6CpK5CE6_�K5\�X<1�b(351je¸;5Bg a6i:<6#3IXp^H:<1I1'_�C81'_dX<756WC<9�=c6):<6#CEK5\gXF�9�3H;�K5CEBD35kw��R����Q�����XE:[9�35CE_�1�:<= 1�3H649�3�CE751je X<7%9�X8BD_ �ØBDCZ9�3~1�^�63~CE6iXZePBgX<7�zH35BgX<6f=c69�CEK(:<6@X<7563~±2R�Q�T�²���� ���B]356JIKH9�\DBgXs?y751'\D;5Cp_�1�: ;°v� ���P�|9�35;�X<7H6BD3��E6#4#X<BD1'3�1'_T;hv� ���P�uBD3qX<1 �uº����P�uBDC�4#1'=)^�9�4#XmN�6iX � � ;hv� ���P�[F�6il,X<6#3H;56#;~`I?�®�1'K(X<CEB];56$�oF�9�35;�\]6iX � � � � � � � º 9�3H;� � � � KML.N4O%� � ��� � º m� 2649�K5CE6 � � � �I� �I0J � �GQa� ( " º 7 � ��� � � OMQ/1'356y7H9�C � � � � �I���'(�� (�N -'���P�[v � ºY_�1�:r9�\D\ � � �"!8F�9�35;�X<756#:<6#_�1�:<6 I � � � $ � � � � � �I��� º�O �-(�pº (�N -'���P� � ! �!{F�eP756i:<6 � ! BDC�X<756dAq1'\DK5=c6-1'_(X<7H6dK535BDX�`%9�\D\Sm��oCHI � � � � � � � � � �I��� ºPO ��( I � � � � � �

� � � � ��� ��� � � � � �I��� ºPO � (

��� ���(FH1'3568;H6#;5K546#C�`I?>9�;5;HBD35k)X<7H6#CE6ZXse21WB]356JIKH9�\DBgX<B]6#C2X<7H9�X �pº)� I � 1 � � � � �q��� ºPO � (��pº%)(�N -'���P� � ! �!�Ó �

��� ���

_�1�:f6#A'6#:E? � �!®(F�9�35;°X<7H6i:<6#_�1�:<6WX<7H6r4[751'BD46 (�N -'���P� � ! �! � vº k�BDA'6C � ( 4�! (�N -'���P�[v � !�� _�1�:)9yK535BgAq6#:<C<9�\4#1'35CMX[9�3qX 4 ! �SBSm 6�m�BD35;56#^�6#3H;56#3qXP1'_�X<756Z1�^�63wCE6iXi�[m

0�3�1�:<;56#:cX<1�^5:<1jA'6wX<7H9�XWX<756yBD3��E6#4#X<BD1'3�1'_ ;hv� ���P�@B]3qX<1 �uº����P�@BDCc4#1'=)^�9�4#XF21'356y9�CECEK5=c6#CcX<7H9�X ��� � ®�BD3; v� ���P��e26j9�b�F�9�35;�1'356fe�9�3qX<CpX<1Y^5:<1jAq6fX<7H9�X ��� � ®YBD3 � º ���P�|CMXE:<1�3Hk(m{0�35;566#;�1'356)=c9?yX[9�bq6 � � ��� � � º ( ��9�35;� � KML.N4O�� ��� ��� � º (��YFH9�35;>`�6#4j9�KHCE6�� ��� � �q�dBDC�X<756)�uºoCE4j9�\]9�:�^5:<1,;5K54iXP1'_ ��� `I?>9@zHl,6;>_�K5354iX<B]1�3>BD3 �uº����P�[F51'356Z7%9�C� ��� � �q� �Ö®W_�1�:|6#Aq6i:E?�� � �)!8m 2649�K5CE6 � � � � �I��� (9�6)(�N -'���P�[v � ºp_�1�:p9�\D\ � � �)! 1'3568;H6#;5K546#C|`I?>N���¯��aU�¤���;51'=WB]3H9�X<6#;�41�3IA'6#:<k'6#3546@X<7561�:<6#= X<7H9�X�I � � � $ � � � ��� � �I��� º O � � ®Y_�1�:89�3q? � �x®(m �6#4j9�KHCE6 I � � � � � � � ��� � �I��� º O � (

��� � �

�(FH1'356Z;56;5K546#C�X<7H9�XP\DBD=�CEK(^ � � � � ��� � � º ( �

º � F%9�35;w\D6#XEX<BD35k@X<7563�� � , 1�3H687H9�C#� � ��� � � º �·®(m

ÕIÔ

Page 46: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�j�,m-��1'35;H9?��%6i`5:<KH9�:E?y­ �,F�­�®'®'®¢�756Z=r9�B]3Y:<69�CE1'3�eP7q?y�(RV¯ R�¥,�H�¶CM^%9�4#6CP9�:<6ZBD=)^a1�:EX[9�3qX�BDC2X<7H9�X�X<756i?�9�:<6oX<7H6p3%9�X<K5:[9�\ _�K5354iX<B]1�3%9�\aCM^%9�4#6C�_�1�:

CE1'\gA(BD35kWX<756{`�1'K53H;H9�:E?>A�9�\DK568^5:<1�`H\D6=WC|1'_u¡u1'3qX<BD3IK5KH= ��6#4}7H9�35BD4Co9�35;�±d7q?(CEBD4C@�O9�X|\D6j9�CMXoK(^�X<1r351jep�[t�X<7H6i?w=c9?`�6@6#\]\DBg^5X<BD486JIKH9�X<BD1'35Co\DBgbq6�? � � @~_�1�:oeP75B]4[7�1�3H6{B]3qAq1�bq6C|X<756@3H9�=W6#Co1'_�N��BA%¥,��²���1�:p±uR�QSU�U�R�TdF%^%9�:[9�`�1'\DBD4@\DBgbq6X<756{7569�X|6jJIKH9�X<BD1'3-���

� � > � ? � ��@�_�1�:|eP75BD4[7y1'356{BD3qAq1�bq6#CPX<756{3H9�=W681'_V�-R����Q����-F%1�:o7q?,^�6i:E`�1'\DB]4Z\DBgbq6ZX<756Ze�9Aq66JIKH9�X<BD1'3-���� � �E> 4 º ? � ��@y_�1�:�eP75B]4[7>1'3568BD3IA'1�bq6#C�X<7H6Z3H9�=W6C�1'_�£ × �o¥I� �Z¯����5��vP1�:P£fm� ��%��TVR��¥,¥,Q+º�m¢�756��(RV¯ R�¥,�%� CM^%9�4#6!;°vj���P�YBDC�9�;%9�^HX<6#; X<1�^5:<1�`H\D6=WCw1'_@X<756°_�1�:<= > � 7 � � ���� � � � 7 � ������ � � � @�Fpe�:<BDXEX<6#3

>8O,��F � � KML.N4O%� � � � ��@YeP756#3YX<756o=c9�XE:<Bgl����+eP75BD4[7YBDCuKHCEKH9�\D\g?cCM?(=W=W6#XE:<BD4|BD3Y9�^5^H\]BD49�X<BD1'35C}��7H9�Cu`�1'K535;56;Y=c69�CEK(:EL9�`H\D6{41,6i�c4#BD63qX<Co9�35;�C<9�X<BDCMzH6CPX<756{6\D\DBg^HX<BD4#BgXs?�4#1'35;5BgX<B]1�3wX<7H9�X|X<756#:<6{6il(BDCMX<C � �*®cCEKH4[7yX<7H9�X � 7 � � 7 � � 7 � �# ���� � � º_�1�:|9�\D\ � � �)!{FH_�1�:|9(m 6�mHQ � �omd¢�756#:<689�:<6pA'9�:<BD1'K5C�^H7q?(CEBD4j9�\ BD3qX<6#:E^5:<6iX[9�X<BD1'35C�^�1'CECEBg`H\D6'm��xz5:<CMXP1'3568B]C2X<1W4#1'35CEBD;H6i:X<756rCMX[9�X<BD1'3H9�:E?�756j9�Xf6JIKH9�X<BD1'3aFVCE1wX<7H9�X � BDC8X<756cX<6#=W^�6i:[9�X<K(:<6r9�35;�� KML.N4O%� � �oB]C8X<756r7569�X %HK(l�mw� CE6#41�3H;h1�3H6BDC8X<1w41�3HCEBD;56i:@Ùd\D6#4#XE:<1'CMX[9�X<B]4#C#F�eP75BD4}7�B]C{9wCEBD=)^H\]BgzH49�X<BD1'3�1'_2��� � ´>�%¥,¥~6jJIKH9�X<BD1'35CZeP7H6i:<6WX<756#:<6WBDC{351y=c9�k'356#X<BD4zH6\D;h9�35;¶351yX<BD=W6c;56i^�635;56354#6'FVCE1yX<7H9�X � BDC8X<7H6r6#\D64iXE:<1'CMX[9�X<BD4)^�1�X<6#3qX<B�9�\OF �+�+>8KML.NMO�� � �pBDC{X<756r6#\]6#4iXE:<B]4fz%6#\D;aF � � KML.N4O�� � �{BDC)X<756�^�1'\]9�:<BD�j9�X<BD1'3hzH6#\];aF�@/BDC)X<756>;H6#35CEBgXs?�1'_|6\D6#4#XE:<BD4�4[7H9�:<k'6��SK5CEK%9�\]\g?¶;56#3H1�X<6; �H�[F29�35;�X<7H6;5635CEBgXs?�1'_�6#\]6#4iXE:<B]4)6#356#:<k�?�BDC (9� vº � � � ��[tVKHCEBD35kwX<756��(RV¯ R�¥,�%� CM^%9�4#6�;hvj���P�o_�1�: � 41�:E:<6#CM^�1'35;5C{X<1y7H9A,B]35ky9zH35BDX<6o6#\D64iXE:<BD4o6#356#:<k�?rCMX<1�:<6;�BD3 �x� ��BDC249�\D\]6#;cX<756o^�6#:<=WBgXEX<BgA(BgXs?)X<6#3HCE1�:�BD3rX<75B]C249�CE6��[m-� X<75Bg:<;Y1'356pB]CuX<1f4#1'35CEBD;H6i:9�;5BD �6i:<63qX8CEBD=)^H\]BgzH49�X<BD1'3~1'_d��� � ´>�%¥,¥�6jJqK%9�X<B]1�3HCoePBDX<7�351>=c9�k'356#X<BD4@z%6#\D;°9�35;�3H1�X<B]=W6f;56#^�6#35;H6#3546W9�\DCE1>`HK(XeP756#:<6)1'356W4#1'35CEBD;56#:<CpX<756W6#\D64iXE:<BD4f4#K(:E:<63qX "��+eP7HBD4[7�BD3~^5:<BD354Bg^H\D6)=@KHCMXZC<9�X<BDCMz%6#CpX<7H6)6JIKH9�X<BD1'3 � �

� � Ó,O,��F%� ",� �¸®,FeP75BD4}7@6#lI^H:<6#CECE6#CVX<75624#1'35CE6i:EA'9�X<BD1'3{1'_5X<1�X[9�\I6\D6#4#XE:<BD4d4}7H9�:<k'6�� BDC�:<6#\]9�X<6#;@X<1|X<75626#\D64iXE:<BD4dzH6#\D;{`I? � ³�� × C � \]9e<"�� � ��SCE1��xBDC�X<75684#1'35;5K54#X<BgA(BgXs?rX<6#35CE1�: � BD3�X<75BDC�4j9�CE6��[m

1 7H9�X<6iAq6i:|X<756{^H7I?,CEB]49�\�BD3qX<K5BDX<BD1'3wBDC#F�X<756{6jJqK%9�X<B]1�3!>�O&� F � � KML.N4O�� � ���R�(@�F�X<1'k'6iX<756#:PePBDX<7w`�1'K535;H9�:E?w41�3H;5BgLX<BD1'35C#F�B]Co;569�\gXoBD3~9r=r9�X<7H6#=c9�X<BD4j9�\ae�9?y`I?yK5CEBD35kYN�� � LO��Q�¥5¤V�����¸\D6=W=r9w�+eP75B]4[7ye�9�CZ9�\DCE1Y;5BDCE41Aq6#:<6#;y`I?y�~9�:EbÑpQSU#³�Q��u�[Fq1�:uCE1�=c6PA�9�:<B]9�3qXm � 356|4#1'35CEBD;56#:<Cu98noQ�¥(¯��%�H�yCM^%9�4#6��!�+eP75BD4[7cB]C-KHCEKH9�\D\g?c984#\D1'CE6;cCEK(`HCE6iX21'_�;hvj���P�V4#1'3(LX[9�BD35BD3Hk);°v� ���P�E�[F'9�35;f9�`HBD\]BD3569�:�4#1'3qX<B]3IK51'K5C�_�1�:<=�N�� � � F5�a1'3 � � �Ø�+eP75B]4[7{BDC�KHCEKH9�\D\g?#I J � ��KML.N4O�� � � � KML.N4O��GF(����O Qa�[F9�35;>9f\DBD356j9�:�4#1'3qX<B]3IK51'K5C�_�1�: �P�GF5�-1'3 ���+eP7HBD4[7�BDC2K5CEKH9�\D\D?r1�_ X<756p_�1�:<= I J�@ F3OMQ{Ó I

�J K � � F O ; !

" vi�[F59�35;�X<756i:<6BDCr9�K535B]JIK56wCE1'\DK(X<B]1�3 � 1'_|X<756>A�9�:<B]9�X<BD1'3H9�\�_�1�:<=@K5\]9�X<BD1'3 N�� � � F(�E� �P�GF(�f_�1�:r9�\D\�F � � KH35;56i:cX<7H6��SCEK5�W4BD6#3IXi�4#1'35;HBgX<BD1'3fX<7H9�X-X<756�`HBD\DB]3569�:�_�1�:<=�BDC �pLO6#\]\DBg^5X<BD4'F�BOm 6'm�X<756#:<6�6il(BDCMX<C*� �/®pCEK54}7WX<7H9�X*N�� � � � � ,��� � � � � º ��_�1�:u9�\D\ �6� �)m�pLO6\D\DBg^HX<BD4#BgXs?c751'\D;HC�BD_�9�35;w1'35\g?�BD_�±2R�Q�T�²���� ��yBD356jJIKH9�\DBgXs?Y751'\D;5C�_�1�: �Wm

�%1�:)X<756Y49�CE6�1�_P751'=W1'k'6#3H6#1'K5Cf£pQ���QS²�³�¥I���*4#1'35;5BgX<B]1�3 FVBOm 6'm � � � � ®�1'3 � �oF-1'356rX[9�bq6#C�� � ; v� ���P�89�35;±2R�Q�T�²���� ��rBD356jJqK%9�\]BgXs?@751'\D;5C-BD_ ��7H9�C�zH3HBgX<6�=W6j9�CEK5:<6�1�:dBDC-BD354\DK5;56;)B]3W9ZCMXE:<Bg^WePBgX<7fzH3HBgX<6�ePBD;(X<7aFI9�35;WX<756i:<6�6#l,B]CMX<C9YK535B�JqKH6fCE1'\DK(X<B]1�3~_�1�:�@ � ;

" vj���P��1'_3N�� � � F5�)� �P�GF5�P_�1�:Z6#Aq6i:E?6F � �WF 1�:86JIK5BgA'9�\]6#3qX<\g?y_�1�:Z6#A'6#:E? F � C � ���P�`I?°;H6#35CEBgXs?qF-9�3H;hX<7H9�XfBDC@6#l(9�4#X<\g?~X<756Y6jJqK%9�X<B]1�3 my�H1�:fX<756Y49�CE6Y1'_�3H1�3H751'=W1'k'6#3561�KHCf£pQ���Q+²a³�¥,����4#1'35;5BgX<B]1�3 FVBSm 6�m� � � � Ky1'3���oF�1'356c9�CMb(CZX<7H9�X Kw`�6\D1'35k'CpX<1 � � ; v ���P�@�+eP75BD4[7�B]C ; v � º � � �P�|BD_'�ØBDCZ`�1'K535;56;~ePBgX<7�N�Q AaU�²�³�Q���µ`�1'K535;%9�:E?%�[FaCE1YX<7H9�XoX<756#:<6@6il(BDCMX<C � v � ;hvj���P��ePBgX<7 � � � v � K%t 1'356f\]1I1�b(CPX<7H6#3~_�1�:89rCE1'\DK(X<B]1�3 � � � v Ó ��ePBgX<7� � ;hv� ���P� C<9�X<BDCE_�?(BD35k|X<75626jJIKH9�X<BD1'38ePBDX<7�@f:<6#^H\]9�4#6;{`I?�@-ÓWO&��F � � KML.N4O%� � v ����Fq9�35;@X<756#:<626il(BDCMX<CV9oK535B�JqKH6uCE1'\DK(X<B]1�3_�1�:@ � ;

" vj���P�u9�35; K � � � ;hvj���P�[m0�_�� � ; v ���P�rX<756#3 ±2R�Q�TV²���� ��xBD3H6JIKH9�\DBgXs?�;51,6#Cw351�Xy751'\D; BD3 k'6#356#:[9�\OFo`HK(X�9�CECEK5=WBD3Hk�X<7H9�XwX<756#:<6�BDCy9

CE1'\DK(X<BD1'3 � � ;hvj���P�|1'_3N�� � � F5� � �P�GF(�|_�1�:{6iAq6i:E? F � ;hvj���P�o9�35; ��7H9�CZX<756WCEBD=)^H\]6f_�1�:<= �P�GF(� ��I0J�@)F3OMQYÓI�J K � � F3OM;+!

" vfePBDX<7,@ � �dº����P�{9�35; K � �dº�� � �P�[F-BgX)BDCfKHCE6#_�K5\uX<1~4}7H9�:[9�4iX<6i:<B]�#6reP7H9�XW9�CE1'\DK(X<B]1�3�49�3h`�6�BD3X<75BDCY4j9�CE6'm ¢�9�b,B]35k¶9�\D\ F � C�� ���P�Wk'BgAq6#CrX<756~6JIKH9�X<BD1'3�>�O&��F � � KML.N4O%� � ���$� @*B]3 �om ¢�7563*9�CECEKH=WBD35k¶X<7H9�XX<756�41I6#�c4#BD63qX<C#N 7 ��9�:<6�N�Q A�U�²�³�Q���µ~4#1'3qX<BD3,K51'K5C)9�35;�X<7H9�XfX<756Y`�1'K535;%9�:E?¶1'_ �¸B]CfCE=W1,1�X<7aF-1'356�4j9�3¶CE7H1e�X<7H9�X�6� ;~º����P��9�35;cX<7H6#3Y9�3rBD3qX<6k�:[9�X<B]1�3W`I?)^%9�:EX<CuCE751jePCuX<7H9�Xu1�3H6P7%9�Co� � � � KML.NMO�� � � � � �H� Kf1'3 ���oFIX<756|�o����p��T�T �

vwGq6j9�3wN�6ZÚP1'35;w£ × �o¥,� �8¯��%�5�-F5�H:<6#354}7>=r9�X<756#=r9�X<B]4#B]9�3aFa�)(,� (jL<�)( ���Imº~£Z9�35B]6#\ 2����T�R��¥I¥,QsF��qePBDCEC�=r9�X<7H6#=c9�X<BD4B]9�3aFa�)(�®'®�L<�)( ��­,mdno6pe21�:Ebq6;>B]3 �9�CE6\Sm� Ðo61�:<k��IBD=c1�3 � ³��fFaÐo6i:<=c9�3>=r9�X<756#=r9�X<B]4#B]9�3aF��)( �'��L<���'Ô�Õ5m-no6Ze21�:Eb'6;>BD3>�~K535BD4}7am���H:[9�3H�yÙ�:<3HCMXr�o� ��Z��T�TuF2Ðo6i:<=c9�3�=r9�X<756#=r9�X<B]4#B]9�3aFo�)(�� ��L<� �'�'ÔIm*no6we21�:Eb'6;/BD3 &�V1'35BDk'CM`�6i:<k/�S351je &89�\DBD35L

BD35k�:[9�;aFHÚPK5CECEB]9I�[m

ÕIÒ

Page 47: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

4#1'35;HBgX<BD1'3am�0�_�X<756|`�1'K535;H9�:E?cBDCu351�X2CE=c1I1�X<7r6#351'K5k'7cX<7H6|CE1'\DK(X<BD1'3r=c9?W351�Xu`�6\D1'35k8X<1�;�º����P�[F,9�35;cX<7H6|4#1,6#�W4BD6#3IX<C=c9?>351�XP`�68CE=W1,1�X<7w6#BgX<7H6i:F%9�35;�9�3yBD3qX<6#:E^5:<6iX[9�X<BD1'3>1�_�X<756Z`a1�KH35;H9�:E?>4#1'35;5BDX<BD1'3�ePBD\D\a`�6{CMX<K5;5BD6;>\�9�X<6#:F5`HK(X|X<756i:<6BDCP9�351�X<756#:PBD=)^�1�:EX[9�3qX�^�1'BD3qX�X<1W;5B]CE4#K5CEC#F%4#1'3546i:<35B]35k)6il(BDCMX<6354#6'm

0�_ � 7%9�C@zH35BgX<6r=W6j9�CEK(:<6�F�� � ;hvj���P�Z9�35;aFV`�649�K5CE6�N�� � � ����� ®y_�1�:f6#A'6#:E? ��� �+� ;°vj���P�[F�9y3564#6#CEC<9�:E?4#1'35;HBgX<BD1'3y_�1�:oX<7H6f6il(BDCMX<6#3H4#6@1'_d9rCE1'\DK(X<B]1�3~BDC|X<7H9�X��P�E�����Ø®(m1 BgX<7yX<7H6@^H7q?(CEBD49�\�B]3qX<6i:E^5:<6#X[9�X<BD1'3y1�_u9rCMX[9�X<BD1'3H9�:E?756j9�Xd6JIKH9�X<BD1'3WBgX-=c69�35C�X<7H9�X-X<756�X<1�X[9�\H9�=W1'K53qXd1'_�7H69�X-B]Cd®(FI9�;5;5BD3HkZX<756�CE1'K(:<4#6P1'_�7569�XdB]35CEBD;56 �¶eP75BD4}7)B]C I J @)OMQ9�35;YX<756p7H69�X&%HK(lYBD=W^�1'CE6#;Y1'3YX<756o`�1'K535;%9�:E?�� �oF(eP75B]4[7YB]C I

�JK3OM; ! " vjt(BD_ X<75B]C24#1'35;5BDX<BD1'3rB]C23H1�X�C<9�X<B]CMzH6#;YX<7563

X<756PCE1'\DK(X<BD1'3W1'_%X<7H6�6iAq1'\DK(X<BD1'3W7569�Xd6jJIKH9�X<BD1'3fePBD\D\53H1�Xu4#1'3qAq6i:<k'6�X<1{9Z\DBD=WBDXF'9�35;cBgXu9�4iX<KH9�\D\D?{X<635;5C�X<18B]3(zH35BgXs?>�+ePBgX<79yCEBDk'3h;56i^�635;5BD3HkyK(^a1�3hX<756rCEB]k�3h1'_�X<756cX<1�X[9�\d756j9�XfBD=W^�1'CE6#;at-1'_�4#1'K(:<CE6'FVeP7563°X<756cX<6=)^�6#:[9�X<K5:<6W`�6#41�=c6#C{X<1,1\]9�:<k'6�F'X<756�=W1,;56#\]BD�9�X<BD1'3{`,?W9p\DBD3H69�:�6jJqK%9�X<B]1�3)BDC-351�X�Aq6#:E?@k'1,1,;aFI9�35;WBD3c9o:<69�\(^H:<1�`%\D6#= X<756�9�`HCE1'\DK(X<6�X<6=)^�6#:[9�X<K(:<649�353H1�X�`a6#4#1'=c6p3H6#kq9�X<BgAq6Z9�3q?,e�9?%�[m

0�_aX<756o3564#6#CEC<9�:E?r4#1'35;5BgX<B]1�3 �P�E���H�*®{BDCuC<9�X<BDCMzH6;�9�3H;rB]_�X<756oBD3��E6#4#X<BD1'3c1�_ ;hvj���P��BD3qX<1 �uº����P�VBDC24#1'=)^�9�4#XdX<75639yCE1�\]K(X<BD1'3h6il(BDCMX<C#F�`%K(XfBgX@BDC@351�XfK535B]JIK56�9�Cf1'356r=c9?h9�;5;¶9�3�9�:E`HBgXE:[9�:E?�4#1'35CMX[9�3qXfX<1wX<756YCE1�\]K(X<BD1'3/�O9�3h6il59�=)^%\D61'_u9Y�V��� �V³VR�¥ � 9�\gX<6#:<3H9�X<BgAq6��[mZ0�_d1'356);56351�X<6#Cp`I? � J X<756W9Aq6i:[9�k'6)1'_ � 1'3 �oF�9�35;~`I? � � J X<756W9Aq6i:[9�k'6)1'_ � � �1'3 � �oF5X<7H6#3�X<756Z41�=W^%9�4iX<356CECP9�CECEK5=)^5X<B]1�3>BD=W^H\DBD6CuX<7H9�XP±2R�Q�T�²���� ��wB]356JIKH9�\DBgXs? � � � � � º ( CE� � KML.N4O%� � ��� � º 751'\D;HC�_�1�:9�\D\ �(� ;°vj���P�{C<9�X<BDCE_�?(BD35k � J � ®,F29�3H;�X<7H9�X)BgXc9�\DCE1�751'\D;5CW_�1�:c9�\D\ �(� ;°vj���P�@C<9�X<B]CE_]?(BD35k � � J � ®,m¶Ù-Aq6#3�BD_X<756{4#1'=)^�9�4#X<356#CEC|41�3H;5BgX<BD1'3w;51,6#C|351�Xo751'\D;>`%K(Xo1�3H6{1'_�X<7H6#CE6{±2R�Q�TV²���� ��~BD356JIKH9�\DBDXs?YB]C�XE:<K56�F%X<756#3wX<7H6i:<6{6il(BDCMX<C|9CE1'\DK(X<BD1'3am

¦oCEBD3Hkf±uR�Q�T�²�������>BD356jJIKH9�\DBgXs?c_�1�:�9�\D\ �6� ;hvj���P�-C<9�X<BDCE_]?(BD3Hk � J �*®(F51'356p4[7H9�35k'6C �ØX<1f;56#351�X<6oX<756pCEK(`%CM^%9�4#61'_ � � ;°vj���P�rCEK54[7 X<7%9�X � J � ®(FZ9�35;*X<7H6~`HBD\DBD3H69�:>_�1�:<= BDC�X<7563 �pLO6#\D\]Bg^5X<BD4~9�35;�9�CE1'\DK(X<BD1'3 6il(BDCMX<C#m � 3H6;51,6#C@351�X{7H9Aq69C � ���P� � �)F�`HK(X�N�� � � F5� � ���GF5�p1�3H\g?�_�1�:�F � C � ���P�pC<9�X<B]CE_]?(BD35k I J F3OMQ ��®(FV9�35;°X<756i:<6_�1�:<6X<756#:<6�6il(BDCMX<CW9~N���¤V����TV¤V� � =@KH\gX<Bg^H\DB]6i: ��CEK54}7�X<7H9�XWN�� � � F(��� �P�GF5�-Ó��I�JF O Q/_�1�:c9�\D\ F � C�� ���P�[FdCE1�X<7H9�X>8O,��F � � KML.N4O%� � ����� @cÓ �~B]3 �otaX<756#3~1'356f1�`5X[9�BD35CoX<756@`�1'K535;H9�:E?y41'35;5BgX<BD1'3~9�35; ��BDCpCEK54[7~X<7H9�XoX<756f3564#6#CEC<9�:E?4#1'35;HBgX<BD1'3�=fK5CMXP751'\D;w9�35;wBgXPBDC�X<756#:<6#_�1�:<6Z6jJqK%9�\ X<1W®(m

¦oCEBD3HkZ±2R�Q�T�²���� ��rB]356JIKH9�\DBgXs?f_�1�:u9�\D\ �6� ;hvj���P��C<9�X<BDCE_�?(BD35k � � J � ®(Fq1'356P4[7H9�35k'6#C � X<18;56#3H1�X<6�X<7H6�CEK(`%CM^%9�4#61'_ �6� ; v ���P�-CEK54}7YX<7H9�X � � J � ®(F59�3H;rX<756|`HB]\DBD356j9�:2_�1�:<=�BDCuX<756#3 �pLO6\D\DBg^HX<BD4|9�35;�9@CE1'\DK5X<BD1'3r6il(BDCMX<C#m � 3H6o7H9�C23H1eC � ���P� � �)F,CE18X<7%9�X>8O,��F � � KML.NMO�� � ��� ��@rB]3 �otIX<7H6#3r1�3H6P1�`5X[9�BD3HC-X<756�`a1�KH35;H9�:E?c4#1'35;5BgX<B]1�3r9�35;Y98N���¤�����T�¤V�=@KH\gX<Bg^H\DB]6i:d9�^5^�6j9�:<CuBD3cX<756�`�1'K535;%9�:E?c4#1'35;5BgX<B]1�3W`%K(XdX<756|356#46#CEC<9�:E?W4#1'35;HBgX<BD1'3W=fK5CMXu751'\D;Y9�35;rBgXuBDC-X<756i:<6_�1�:<6P6jJqK%9�\X<1c®(m

� Gq1'CE6i^%7(LON�1'K5BDC|N���¤V����TV¤V��F,0sX[9�\DB]9�3(LS`�1�:<3>=c9�X<756=c9�X<BD4#B�9�3 F��)( �'Ò�L<���(�@�,m-no6pe21�:Ebq6#;wBD3>±�9�:<BDC#m

Õ�(

Page 48: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� �������������! #"�$H�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

­�®(m�1°6#;53H6#CE;H9?���9�:<4}7°��F�­�®'®'®N�6iX&�¶`a6P9p`�1'K53H;56#;c1�^�63cCE6iXd1'_��)!/ePBgX<7WN�Q AaU�²�³�Q���µZ`�1'K53H;H9�:E?'F,9�35;c\D6#X �6� ;hvj���P��C<9�X<BDCE_�?WN�� � � F5�H� ���GF5�_�1�:r9�\]\ F � ;hvj���P�[F-ePBgX<7$N�� � � � ��� I J � �57 � � � 7 � ������ � � ���� � � O Q/_�1�:c9�\D\ � � � � ;hvj���P�{9�35; �P� � ��� I J�@ � O QyÓ

I�J K � � � OM;+!

" v@_�1�:@9�\D\ � � ;hvj���P�[F ePBgX<7 @ � �uº����P�o9�35; K � �uº�� � �P�[mc¦oCEBD35ky9�\]\HF � C � ���P�|1'356c;56#;5KH4#6#CX<7H9�X�> � 7 � � ���� � � � 7 � ���� � � � � @�BD3#�oF�9�35;wX<756@JqKH6#CMX<BD1'3wBDC�X<1cK535;56#:<CMX[9�3H;weP7H9�X|BDC�X<7568=c69�35BD3Hk)1'_�X<7568`�1'K535;H9�:E?4#1'35;HBgX<BD1'3YX<7%9�X � C<9�X<BDCMz%6#C#m

0�_21'356c9�CECEK5=c6#C8X<7H9�X � 7 � � � v:� ����P�o_�1�: � � " � � ���������:& 9�3H; �,� ;~º����P�[F�X<7563� 7 � ���� � � � ;°vj���P�o_�1�:@9�\D\� � " �·� ���������:& Fu9�35;�9�3�B]3qX<6#k�:[9�X<BD1'3¶`I?h^%9�:EX<CWk�BDA'6C I J�@)F3OMQ�� N�� � � F5� >�I � J � � 7 � � � 7 � � � ������ � � � F � 7 � O ; ! " v_�1�:y9�\D\F � ;hvj���P�[Fp9�35;*X<756#:<6#_�1�:<6�1�3H6�7H9�C � 7 � � � 7 � � � ������ � � � F �

7 � K/1'3 � �·�SCE1�X<75BDC>49�3 1�3H\g?�7H9�^5^�6#3 BD_K � ;°v � º�� � �P�E�[m

0�3w9�^5^H\DBD4j9�X<BD1'35C#F(1'356Z;51,6#C�351�XP9�\ge�9?(C�7H9Aq6ZN�Q AaU�²�³�Q���µ)41,6i�c4#BD63qX<C � 7 ��F5`�649�K5CE6Z1'356Z1�_�X<6#3�=cBgl(6#C�;5Bg a6i:<6#3IX=c9�X<6#:<B]9�\DC|9�35;yX<756i:<6f9�:<6@BD3qX<6i:<_+9�4#6CP1'_�;5B]CE4#1'3qX<BD3IKHBgXs?>_�1�:|X<756@41I6#�c4#BD63qX<C#m�0�3�9�^5^%\DBD49�X<BD1'35C#F�1'356@;51,6C|351�Xp9�\ge�9?(C7H9Aq6�CE=c1I1�X<7f`a1�KH35;H9�:<BD6#C#FI9�35;)41�:<356i:<C-BD3fX<7562`�1'K53H;H9�:E?{^%K(Xd9p\DBD=WBDX�_�1�:�X<7562:<6#k'K5\]9�:<BgXs?@1'_HX<756�CE1'\DK(X<BD1'3am��o\gX<7H1�KHk�7BgXoBDC|XE:<K568X<7H9�X|X<7H6{CE1'\DK5X<BD1'3y_�1�:R@ � �uº����P��9�35;6KE�x®r;51,6#CoC<9�X<B]CE_]? �<� ;�º����P��BD_ ��BDC|41'3qAq6il�F�BDX|B]C|351�Xp9�\ge�9?(CXE:<K568BD35;56#^�6#35;H6#3qX<\g?�1'_�X<756ZCEBD�6Z1'_�X<75684#1�:<356#:<CP1'3�X<7H6p`�1'K535;%9�:E?qF�9�C�X<7568_�1'\D\D1jePBD3Hk)6il59�=)^H\]6pB]3�X<756p^%\]9�356ZCE751jePC#mN�6#X)� `a6{X<756fCE64iX<1�:Z® � � � � � ePBgX<7 ��� � � �x­���F 9�35;~\D6iX � �(L ��4#1'Cj��� � ���¶ePBgX<7 � � C � � �8º��ePBgX<7 � �!�356j9�:dX<756�1�:<BDk'BD3 m� �6#4j9�K5CE6 � � � L �P4#1'C��� � ��BDCd7H9�:<=W1'35BD4'F'BOm 6'mVC<9�X<BDCMzH6#C ? � � �*®W�+`�6#4j9�K5CE6�BDX-BDC�X<756�:<69�\5^%9�:EXd1'_ � �_�1�:P6il59�=)^H\]6��[F,1'356ZCE6#6C�X<7H9�X ? � BDC�®f3569�:�X<756p1�:<BDk'BD3at(X<756Z351�:<=c9�\�;56#:<BgA�9�X<BgAq6o1'_ � 1'3�X<756pCEBD;H6 � � ®@B]C�®(FH9�35;�BDXBDC|9�\DCE1c®)1'3>X<7568CEBD;56 � � � � BD_ � � � �9��FHeP75BD4[7wk'BgAq6#C vº � ������F%9�35;>X<756#:<6#_�1�:<6Z1'3568;51,6#CP351�X|7H9A'6 � � ; º ���P�[FeP75BD4}7�:<6JIK5BD:<6#C)� � ��m-�o4iX<KH9�\D\g?qF(_�1�:o9f41'3qAq6il�;51'=c9�BD3 �oF �6� ;°v� ���P�29�35;6? �6� �uº����P�dBD=W^H\g?E;~º����P�[m

� ;HBg �6i:<63qXPe�9?>X<1WXE:<6j9�XPX<75BDCP^5:<1�`H\D6#= 1'_Vk�BDA,B]35kc9c=W6j9�3HBD35kWX<1WX<756{�o����p��T�T¶41'35;5BgX<BD1'3aF%BDCPX<7568_�1'\D\D1jePBD35k9�:<k'K5=W63qXvP1'_�GI9�4JIK56CMLON�1'K5BDC�N�Q+R�T�U'm

� &������D������ §3;h�GO,��F%t �P�*��� �6� � � º ���P��� ! � O,��F�� � � � � º ���P� �qm� _�41�K5:<CE6�F�;¶�GO&� FHt��P�uBDCP9)noQ�¥(¯��%�5�°CM^%9�4#6�m� 3568\D1,49�\DBD�6#C�`I?�=fK5\gX<Bg^H\D?,B]35k)9�\D\aX<756841�=W^�1'356#3IX<C�1'_ � `I?YX<7H68C<9�=c68_�KH354iX<BD1'3 � FH3H1�X<B]4#BD35k)X<7H9�XPBD_BF0�� � � �

_�1�: "#�Ø� ���������'& F5X<756#3w1'35687H9�C�O,��F��GF(�*� � O,��F�� � ��Ó � � � KML.N4O�� � ����m0�_ �ØBDC�9�3�B]3qAq6i:EX<Bg`H\]6P=r9�XE:<Bglr9�3H; � � � � �oF(1'356oXE:[9�35CM^�1�:EX<C�9@CE49�\]9�:�_�K5354#X<BD1'3 ��;56#zH356;�1�3 �/X<1@X<756oCE4j9�\]9�:_�K5354iX<B]1�3 � ;56#zH356;°1'3 � � `,? � � � Qa�R� �u�GQa�o_�1�:�Q � �oFV9�35;°1�3H6We�9�3qX<C8X<1wXE:[9�35CM^�1�:EX � � ;¶�GO,��FHt��P�oX<1�F �;¶�GO&��F%t � �+�uBD3�CEK54[7y9fe�9?rX<7H9�X�1'35687H9�C I J � � � KML.N4O�� �V����OMQ ��I J � F � KML.N4O%� � ��� O Q��SF5`HK(X|9�C8� KML.NMO�� � �}� � Qa� � � *5�*�� KML.N4O%� ���}�GQa� � *H�)1'356�7H9�CEKML.N4O�� � �}� � Qa� � �

" KML.NMO�� �V�}�GQa�w�S;56351�X<BD35k � " � � � � " v}�[Fp9�3H;*1'356�9�CMb(C�X<7H9�X� � �GQa� � KML.N4O�� �V�}�GQa� � � � F�� � Qa� � KML.NMO�� � �}� � Qa� � � O( %i� �@���¹F2eP7HBD4[7/k'BgAq6#C�F%� � Qa� � � O( %i� �@��� " v�� � �GQa�[m 0�_ ��BDCY9�31�:EX<751'k'1�3%9�\�=c9�XE:<BglrX<756#3 F�� � Qa�*� � � �GQa�[m

� 354#6�1�3H6Ye21�:Eb(Cf1'3 � � _�1�:W9�N�Q A�U�²�³�Q���µ�41�3IX<BD3IK51'K5C)_�KH354iX<BD1'3 WF-1'356r^5:<1jAq6#CW69�CEBD\g?�X<7H9�X � ��� � � ��� ! BDC;5635CE68BD3 ;¶�GO&� FHt�� � �[m

�o\D\�X<75BDC)^5:<1jAq6#CWX<7H9�X � �Y� �P��� ! B]C);H6#35CE6wBD3 ;¶�GO&� FHt��P�[m/¢�756w356#l,XWCMX<6#^�B]CfX<1�^5:<1jA'6>X<7H9�Xc1'356w49�3�;56izH3H6X<756c351�:<=r9�\-XE:[9�46y� � � � �[t�_�1�:@CE=W1,1�X<7�_�K5354#X<BD1'35C{BgX{=W6j9�3HC � � � � � � � ��F�`HK(X@_�1�:�;¶�GO&� FHt��P�|X<756c;H6izH35BDX<BD1'3�KHCE6#C@94#1'=W^H\D6iX<B]1�3>9�:<k'K5=c6#3qXm

� ��%"-� j�D������ §Z¢�756)=c9�^5^%BD35k � �� � � � � �)� � � � � � � � ��F ;56iz%356#;�_�:<1'= � ��� ��� � ! BD3qX<1 ��>� � �P�[F 6#lIX<635;5CpBD3qX<1>9\DBD3H69�:-4#1'3qX<BD3,K51'K5C-=c9�^r_]:<1'= ;¶�GO&��F%t �P�VBD3qX<1 � � � ; v ���P��� � FqX<756�;5KH9�\H1'_�X<756�CM^%9�4#6P1'_%XE:[9�4#6#Cd1�_�_�K53H4iX<BD1'35C-1'_ ; v ���P�[FBSm 6�m*; " v � º�� � �P�o�O9�C � � 7H9�C�3H1)`�1'K535;%9�:E?qF�; v � º� � � �P�H�5;°v � º�� � �P�E�[m��~1�:<61jA'6#:�X<756Z=r9�^5^HBD35kWBDC�CEK5:+�M64iX<BgAq6�m

v>0o`�6\DBD6#A'6)X<7H9�X8756W^5:<1jA'6;°BgXZeP75B]\D6f0oe�9�C{9>CMX<K5;H6#3qXF�`�6#4j9�KHCE6WBD3�X<756)zH:<CMX{4#1'K(:<CE6#C8X<7H9�XZ0p_�1'\D\D1je26#;�756cK5CE6#;X<756W9�:<k'K5=c6#3qXZePBgX<7~X<756�; º ���P�|7q?,^�1�X<756CEBDC#F�9�35;�\]9�X<6i:8756WCMX[9�:EX<6;~X<69�4[7HBD35k�X<756)356#e¸9�:<k'K5=W63qXZePBgX<7�X<756WCM^%9�46;¶�GO&��F%t �P�[m

��

Page 49: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

�2È<ÆjÆ5#[§��H1�: �6� � ��� �P��� ! 9�35;�F � ; v ���P��1'356�7H9�C I0J � � � � � � � ���� � Ó�O&��F�� � �4F � O Q���I � J-� � � � � � � � ��� � � F3OM; ! " v F9�35;�9�CuX<7H6o\D6#_�XuCEBD;H6o1�_ X<756oB];56#3qX<BDXs?WBDC24#1'3qX<BD3IKH1�KHC21'3E;¶�GO&��F%t �P�,��;°vj���P�[F(CE1fBDCuX<756|:<BDk'7qX�CEBD;56'FIeP75BD4}7Y1'356|e�:<BgX<6C��� � � � � � � � F �d9�C�9@\DBD356j9�:241'3qX<BD3IK51'K5C�_�1�:<=�1'3 � � ; v ���P�[t(351�X<BD46PX<7H9�X�BD_ 1�3H6oCMX[9�:EX<C2_�:<1'=�9�3Y6\D6#=c6#3qX21'_ � � ; v ���P��BDX;51,6#C|351�Xo=c9�XEX<6#:�eP75B]4[7�F�1'356{4[7H1I1'CE6CPeP75BD4}7w7H9�C|X<75BDCP6\D6#=c6#3qXo9�CPBDX<C�XE:[9�4#6�F�9�C�X<7H6{\D6#_�X|CEBD;568ePBD\D\ k'BgAq68X<7568C<9�=c6A�9�\DKH6oeP7%9�X<6#Aq6i:�X<75684[751'BD46pB]C#m

0�3 1�:<;56i:�X<1�CE751je CEK5:+�M64iX<BgA(BgXs?qFP1�3H6~X[9�bq6CEK � � � � ;hvj���P��� � 9�35; 1'356�CE1'\gAq6#C I J � KML.N4O%� � �j� � KML.NMO��GF5����O Q�ÓI J � ��F3OMQ9� � K � � � F ��_�1�:-9�\D\ F � ; v ���P�[F�eP75B]4[7)7H9�C-9oKH35B]JIK56�CE1'\DK(X<BD1'3 � � � ; v ���P�[F�eP75B]4[7)C<9�X<BDCMzH6C8> ? � �5Ó � � �®cBD3+��9�3H;wX<756#:<6#_�1�:<6 � ���KML.N4O%� � �j�2`�6#\D1'35k'CPX<1 ;¶�GO&��F%t �P�[F�C<9�X<BDCMzH6C O,��F%� � �j� � � ��F�9�3H;wX<756{^5:<6#4BDCE6ZA'9�:<B]9�X<BD1'3H9�\_�1�:<=fK5\]9�X<BD1'3�C<9?(C�X<7H9�X8� � � � � �H� K%m

¢�756f6il59�=)^H\]6{1'_ �8º8ePBDX<7 � v � @ v �GQ v �:@ º �GQ º �P9�3H; � º � K v �GQ v � K º �GQ º ��CE751jePCoX<7H9�Xo1'356@7%9�C �<� ;¶�GO,��FHt �8ºj�BD_ @ v�� K º � ;°vj� �@��9�35; @ º.� K v � �uº�� �{�[F�9�35;yX<756i:<6_�1�:<6 � v 4j9�3y`�6{;HBDCE4#1'3qX<BD3,K51'K5Cp9�\D1'35kWX<7H6{\DBD356�Q º ��®WeP7HBD\D6 � º=@KHCMX�`�6Z41�3IX<BD3IK51'K5C#F�9�35;°� � � � �H�(> � º BD_ � �-�8º� m0�3�9~^5:<1�`H\D6=Û1'_|Ùd\D64iXE:<1'CMX[9�X<BD4#C#F-X<756�^�1�X<6#3qX<B�9�\ � BDCWBD3 ;°vj���P�@9�35;�7H9�Cc9�XE:[9�46>1'3�X<756�`�1'K535;%9�:E?�t�=W1�:<6

k'6#356#:[9�\D\g?'Fd1'3�9�3q?�BD3qX<6#:<_�9�46 � X[9�bq6#C)X<756wC<9�=c6YA'9�\DK56>1�3�`a1�X<7�CEBD;56C)1'_|X<756>BD3qX<6#:<_�9�46�m�¢�756�^�1'\]9�:<BD�j9�X<B]1�3¶zH6#\D; C<9�X<BDCMzH6#CEO&��F��� �� � ��9�35;�X<756i:<6_�1�:<66 � ;h�GO,��F%t �P�)BD_ � � �uº����P�[F�9�3H;�X<756y351�:<=r9�\|4#1'=)^�1'3563qXr1'_ BDC4#1'3qX<B]3IK51'K5Cc9�XY9�3I?�BD3qX<6#:<_�9�4#6��SBD_|BgXc;51,6#CW351�XrCEK(^5^�1�:EXr9�351'35�#6#:<1�4[7H9�:<k'6��[m��%1�:WX<756>6\D6#4#XE:<BD4YzH6\D; �cFuBgXWB]CfX<7H6X[9�35k'6#3qX<B�9�\�41'=)^�1'3563qXo1'_ �!eP75B]4[7�BDC|41�3IX<BD3IK51'K5C#F 9�35;�BgX<C|A�9�\DK56@BDCPX<756@X[9�35k'6#3qX<B]9�\�;56i:<BgA'9�X<BgAq681'_VX<756@XE:[9�46{1'_ � t1'356{49�3�9�4iX<KH9�\D\g?�;56izH3H68X<756{CM^%9�4#6�;h� 4 � L � t��P�*��� � � � � º ���P��� ! � ��� ���� � > ��� ���� � � � º ���P�u_�1�:o9�\D\ � � "W� � ���������:& �'F9�35;>^5:<1jAq6{9�3y9�3%9�\]1�k'1'K5C�X<756#1�:<6=>F(X<7H9�X�X<756ZX[9�3Hk�63qX<B]9�\�XE:[9�4#68BDC�;56iz%356#;am

ÕI�

Page 50: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®5��m�1°6#;53H6#CE;H9?���9�:<4}7 �IF�­�®'®'®(m�o\gX<7H1�KHk�7rX<756|X<6i:<=�BD3qX<6i:E^a1�\�9�X<B]1�3cCM^�9�46o1'35\D?c9�^5^a69�:<6#;Y=@KH4[7Y\]9�X<6i:FIX<7H6oCEK(`,�E6#4#X27%9�C�BgX<Cu1�:<BDk'BD3rBD3YJIK56CMX<BD1'35C

CMX<K5;5B]6#;c`I?c�hmIÚPQ���U�µ5F,9�35;cX<7H6#3c`I?r¢�³VR���Q�Tuv#F,9�3H;Y9�\DCE18`I?c������²�Q�T���Q��H´>QS² µ5t'X<756#?)=cBDk'7qXu7H9Aq6P`�66#3r=W1�X<BgA�9�X<6;`I?>CMX<K5;(?(BD35k)X<756Z^5:<1�^�6i:EX<BD6C�1'_�X<756ZnoQ�¥(¯��%�5��XE:[9�3HCE_�1�:<=>m

� 751'\D1'=W1�:E^H75B]4W_�K5354iX<B]1�3°BD3h9�3°1�^�6#3hCE6iX@1�_2X<756c41'=)^H\D6#l~^H\]9�356cBDC@9>41�=W^H\D6#l�A'9�\DK56#;°_�K53H4iX<BD1'3�eP75BD4}7�7%9�C9h;56i:<BgA'9�X<BgAq6yBD3/X<756�41�=W^H\D6#l�CE6#3HCE6�F|BSm 6�m � � � � " � � � � �� "�� � 7%9�C>9°\]BD=WBgXY9�C �°X<635;5CYX<1 � � F|9�35;*BD_ �5� Q�Ó �&*�9�35;@����,� � �W�GQ � *5��Ó����)�GQ � *5�PBgXZ\D69�;5CpX<1YX<756r¡u� �²�³���LOÚPQ�� �p��T�T�6jJIKH9�X<BD1'35C � ���� � �

��! 9�3H; � �

�! � > � �

� �F CE1>X<7H9�X

`�1�X<7 �x9�35; �Ø9�:<6|7H9�:<=W1'35B]4�FIBSm 6�mVC<9�X<BDCE_�?�? ����? �(�*®ZeP756#:<6PX<756|N�9�^H\�9�4B]9�3E?�BDC ������-Ó �

��! �(mV0�_a1'356|e21�:Eb(CuBD3

X<756|K(^5^a6i:u7%9�\]_�^H\]9�356 * ��®(F,9�35;Y1�3H6PB]=)^�1'CE6#C-X<756P:<6j9�\H^%9�:EXu1'_T@Y1'3WX<756P`�1'K53H;H9�:E?'FIX<7563 ��BDCu;56iX<6#:<=WBD3H6#;w�SB]_%X<7H6k'BgAq6#38XE:[9�46dBDC�35BD46d6#3H1�KHk�7�� 9�35;{X<75638X<7H6d^%9�:EX<B]9�\q;56#:<BgA�9�X<BgAq6C 1'_ �*9�:<6db(351jeP3aF�CE1|X<7%9�X ��BDC�;H6izH356;@K(^@X<1o9�;5;5BgX<B]1�31'_�9�3w9�:E`%BgXE:[9�:E?c:<6j9�\a4#1'35CMX[9�3qXm�0�3�X<75BDC2e�9?Y1'356ZBDC�\D6#;YX<1)CMX<K5;(?YX<756Z_�1'\D\D1jePBD3Hk{XE:[9�35CE_�1�:<=�3H9�=W6#;w9�_]X<6#:�noQ�¥,¯����5��F; � � v

* F v� � � FIBSm 6�m ; � �GQa�H�*\DBD= � � � v I � !#" � � � � � �! ��" ! O&*�FqeP75B]4[7):<6\]9�X<6#C�X<756�:<69�\5^%9�:EX-X<1ZX<756PBD=c9�k'BD3H9�:E?@^%9�:EXd1'3

X<756�`�1'K53H;H9�:E?'m�¦oCEBD35k8��R����Q����wXE:[9�35CE_�1�:<=�1�3H6�4j9�3cCE751je/X<7H9�X*; BDCd9ZCEK5:+�M64iX<BgAq6�B]CE1�=c6iXE:E?f_�:<1�= �uº�� �{��BD3qX<18BgX<CE6#\]_9�35;9;�º)� > �%m-¢�75B]Cue�9�C�;51'356Z9f\D1'35k@X<BD=c6|`�6#_�1�:<6oN�9�K(:<6#3qXo�(²a³�´{���5��µ@6#l,X<6#35;56;��-R����Q����~XE:[9�35CE_�1�:<=!X<1fCE1'=c6;5BDCMXE:<BD`HK(X<BD1'35C#F�`HK(X@75BDC8^5:<1,1'_2BDC{9�C{_�1'\D\D1jePC#§{1'356c351�X<BD4#6CZX<7H9�X * F v

� � ���O� �{�p9�C8X<7H6WCEK5=�1'_�9w;5BDCMXE:<Bg`HK5X<BD1'3~ePBgX<74#1'=W^%9�4iX�CEK(^5^�1�:EX�9�35;>9{`�1'K53H;56#;�_�K5354#X<BD1'3aF59�3H;r`�649�K5CE6Q *�F v

�� ��F(1'356p;H6#;5K546#CuX<7%9�X �� � � �>� *�F v� ��� �(>P­ �+� � � FBSm 6�m��>� * F v

��3� > � � - �GK � � �I�aÓ,C)F%9�35;y1'356@;56#;5KH4#6#CC � ®c_]:<1'=�X<756{_�9�4#XPX<7H9�X * F v

�BDC|1I;H;wCE1cX<7H9�X|BgX<CP�-R����Q��%�

XE:[9�35CE_�1�:<= =@K5CMX8`�6)1,;5;��S1'_d41�K5:<CE6)7H6)7H9�;�;H6izH356;�B]3�9�3H9�X<K(:[9�\�e�9?�eP7H9�X{BgXZ=W6j9�35CpX<1Y`�6W6iAq6#3�1�:Z1,;5;�_�1�:{9;5BDCMXE:<BD`HK(X<BD1'3��[m-¢�7563>1'35687H9�C�>�G; � �}� �q�H�(> � - �GK � � �q� � � � �I�u9�35;>X<756i:<6_�1�:<6E� � ; � � � º � � � � � � º 9�35;�;�º)�(> �Hm0PX<7HBD3(bwX<7%9�XZBgXpe�9�Cp�¶m�Ú|Q��aU�µYeP751�CE751je26;~X<7H9�XpX<756fnoQ�¥,¯����5�¶XE:[9�35CE_�1�:<= BDCp41'3qX<BD3IK51'K5CZ_]:<1'= ���5� �{�PB]3qX<1BgX<CE6\D_�_�1�:o� � *�� ,/Fq`HK5X-X<7H6�:<6CEK5\gXdBDCd351�X-XE:<KH6P_�1�: * �Ø�P1�:u_�1�:�*9� , �S1�3H6PK5CEKH9�\D\g?f:<6#^H\]9�4#6C ��v-`I?)X<756Pno����� �CM^%9�4#68=~v#FI9�35; � Ø`I? ��� � �[m�0�CEK(^5^�1'CE6�X<7H9�X-BgXVe�9�C-B]3f:<6#\]9�X<BD1'3@ePBgX<7fX<756�^5:<1�^�6#:EX<BD6#C�1'_HX<756�noQ�¥(¯��%�H�YXE:[9�35CE_�1�:<=X<7H9�Xo�¶m�ÚPQ��aUjµr^5:<1jAq6#;yX<7H6{_�1'\D\D1jePBD35k�«<BD3IX<6i:E^�1'\]9�X<BD1'35¬W:<6CEK5\gXoB]3h��'­'Ò,F�BD3yX<756@49�CE6 * � ( 5 � t�X<7HBDCP:<6CMXE:<BD4iX<BD1'3we�9�C:<6#=c1jA'6;�`I?w¢�³VR���Q�TuF5BD3��j��� �Im� ��%"-� j�D������ §V0�_V�)(+* � � * v�� 5 � �65�v (-,�F,9�35;Y9Z\]BD3569�:d=r9�^���B]C-41'3qX<BD3IK51'K5Cu_]:<1'= � � � ���P�VBD3qX<1 ��8 � ��� � ��9�35;r_]:<1'=��� / ���P�dBD3qX<1 � 8 / ��� ����X<7H6#3>_�1�:P® ( � ( �pBDX�B]C241'3qX<BD3IK51'K5C�_�:<1'= ���������P�dBD3qX<1 � 8 ����� ���[F(eP756i:<6 v

� �� v " �� � Ó

�� /9�35;

v8 � � v " �8 � Ó �8 /m|��1�:<6#1jAq6i:p1'356@7H9�C � � ��� � � ( � � ��� � v

"�� � � ��� � � v F%eP756#:<6�� � ��� � � ;56351�X<6#CPX<7H6{3H1�:<= 1�_���9�Cp9c=c9�^5^HB]35k

_�:<1�= ��� �����P�dB]3qX<1 � 8 ����� ���[m0�_aX<756onoQ�¥,¯����5�yXE:[9�35CE_�1�:<=¸e�9�C2=c9�^5^HBD3Hk ��vj� �{��B]3qX<1@BgX<CE6#\D_sFqX<7563c`,?WX<75BDCuB]3qX<6i:E^�1'\]9�X<BD1'3W:<6#CEK5\gX2B]_�e21'K5\D;Y=c9�^

� � � �{�2BD3qX<1cBgX<CE6#\D_�_�1�:{�"( *+(*­W9�3H;>`I?�XE:[9�3HCM^�1'CEBgX<BD1'3w_�1�:p­�( * ( ,/FH`HK5XPBgXP;H1I6CP351�X|=c9�^#� v � �@�uBD3IX<1 � v � �{�[mno1je26iAq6i:oX<7H6i:<6@6#l(BDCMX<Co9Y4#1'35CMX[9�3qXRC!CEK54[7~X<7H9�XpBD_ �<� ��vj� �{��1'356f7H9�C (�N - ��Q°§B� ; � �GQa������% �$( � � � � � � /� _�1�:Z9�\D\% ��®(Fa9�35;~_]:<1'= X<7H9�Xo:<6#CEK5\DXF�X<756@41'3qX<BD3IK5BDXs?wBD3+� º � �@�P9�3H;yX<756@CM?(=c=W6iXE:E?w1'_VX<7H6@noQ�¥,¯����5�hXE:[9�35CE_�1�:<= 1'356f49�3;56;5K54#6ZX<7H9�XPBgX�=r9�^HC ���5� �@�dBD3qX<1WBgX<CE6\D_�_�1�:{���+*�� ,/m

¢�³�R���Q�T × CP^H:<1I1,1'_VKHCE6#;~9W^H:<1�^a6i:EXs?w1'_�X<7H6{=c1,;5K5\DK5Co1'_V751'\D1'=W1�:E^H7HBD4{_�K5354iX<B]1�3HC#F%X<756{X<7(:<6#6@\DBD3H6#C�X<7561�:<6#=»�O9A�9�:<B]9�3qX�1'_Hno�������p��� �wX<75:<6#624Bg:<4#\D6C�X<7561�:<6=Y�[F'CMX[9�X<BD35koX<7H9�XVB]_P@����,��BDCV751'\D1'=c1�:E^%75BD42BD3@X<7562CMXE:<Bg^W® ��� ���5Q � ��F4#1'3qX<B]3IK51'K5Cp1'3�X<7H6f4#\D1'CE6#;~CMXE:<Bg^�® (�Q (��)9�3H;�CEKH4[7~X<7H9�XE� @��+�&*5����( � � 9�3H; � @��E��Ó��&*H��� ( � v _�1�:89�\D\ * � �fFX<7563>1'35687H9�C#� @�� � Ó �&*H��� (�� v " �� � � �

7 !v _�1�:o9�\D\ � � �O® � ���u9�35;y9�\D\�* � �)mÐo6356i:[9�\DBD�BD35k�X<756dBD;H69�1�_5¢�³�R���Q�TuF�9�4#1'=)^%\D6ilpBD3IX<6i:E^�1'\]9�X<BD1'3Z=c6iX<751,;Ze�9�C�;56#A'6\D1�^�6#;Z`,?Z�o\D`�6i:EX<1o¡u��¥��������R�TuF

`I?>GI9�4jJIK56#CMLON�1�KHBDC�N�QSR�TVU{9�35;>`I?>�hm?&8����Q�T�º�m0�_*@ � � � ���P�[F(X<7563>n �R�¥������°BD356jJqK%9�\]BgXs?rk'BgAq6#C I � � @H� OMQ ( � � @H� � � (�N -�� �)� v � � _�1�:|9�\D\a=W6j9�CEK(:[9�`H\D6ZCEK(`HCE6#X<C �

1'_ �oFq9�35;f�~����²�Q�T���Q��%´>Q+²�µ|B]3qXE:<1,;5K54#6;)9|CM^�9�46�CE1�=c6iX<BD=c6#C�49�\D\D6;{e269�b ���f�+eP75B]4[7f1'3562CE751'K5\];f351�X�=WB]CMX[9�bq6dePBgX<7���P6JIK5BD^5^�6#;8ePBgX<78X<756-e269�bZX<1�^�1'\D1'k�?%�[F�9�35;@;56351�X<6#; ��� � �BD38X<7H6-CE4j9�\D6d1'_(N�R����%T���µ|CM^%9�4#6C#F�eP75BD4[7@BDC X<756uCM^%9�4#6u1�_�S6JIK5BDA�9�\D6#3H4#6�4\]9�CECE6#Cu1'_[�V=c69�CEK(:[9�`H\D6P_�K5354#X<BD1'35C*Kf_�1�:deP75BD4}7WX<7H6i:<6P6#l,B]CMX<C C�CEK54[7rX<7H9�X�I � � K � O Q+( C (�N -'� �)�[v � �� _�1�:�9�\D\H=c69�CEK(:[9�`H\D6|CEK(`HCE6#X<C �Ø1'_��ot,BgXu4#1'3qX[9�BD3HC ���5���P��`HK(X2BD_ � � �"! 9�3H;���(+* �-,ÖBgX29�\DCE1841'3qX[9�BD35Cu_�K5354iX<B]1�3HC

v Ð@m � \D1'_�¢�³VR���Q�TuF��qe26;5BDCE7w=c9�X<756=c9�X<BD4#B�9�3 t,7H6pe�9�C|9)CMX<K5;563qXP1'_��~9�:<4#6\aÚPQ��aUjµcBD3>N�KH35;aF��qe26;56#3 mº~�~9�:EbyÐ|:<BDk'1�:<BD6iA(BD4}7'&Z���%Q�TdFH¦|b,:[9�BD35B]9�3>=c9�X<756#=r9�X<BD4#B]9�3aFa���®�(�=��� �'�,m-no6pe21�:Ebq6#;wBD3 &8BD6iA�m

Ô�®

Page 51: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

\DBgbq6 v��� 1 � �

mr0�3��j���'�,F�������²aQ�T���Q��%´>QS² µ�^%K(`H\DBDCE7H6#;�X<756c_�1'\D\D1jePBD35k>:<6#CEK5\gXF�9�C@9w351�X<6)ePBgX<751'K(X8^H:<1I1'_sF�9�35;°^5:<1,1�_�Ce26i:<6Ze�:<BgXEX<6#3>\]9�X<6i:�`I?���BDCE4}7H9r¡�R�� ¥,��� � 9�3H;�`,?��o3qX<1'35B����-¤���T��2m� ��%"-� j�D������ §|0�_P� ( * � � * v�� 5 � � 5�v ( ,�F 9�35;�9Y\DBD356j9�:p=c9�^� BDCp4#1'3qX<BD3IKH1�KHCo_�:<1'= � � � ���P��BD3qX<1���8 � � ��� � �P9�35;_�:<1�= ��� / ���P�VBD3qX<1"� 8 / � ���� ����X<7563c_�1�:u® ( � (���BgXdBDCd4#1'3qX<BD3IKH1�KHC-_�:<1'= ���������P��BD3qX<1"� 8 ����� ����K535;H6i:dX<756P41�3H;5BgX<BD1'3X<7H9�X * � ( 5 � FHeP756i:<6 v

� �� v " �� � Ó

�� /9�3H; v8 � � v " �8 � Ó �

8 /m

¢�756w:<6CEK5\gX<Cr1'_pÚPQ���U�µ5Fu¢�³�R���Q�TuFd9�35;�������²aQ�T���Q��%´>QS² µ(F-e26i:<6yk'6356i:[9�\DBD�6#;/9�CcX<756#1�:<BD6Cc1'_o0�3qX<6i:E^a1�\�9�X<B]1�3 F9�35;yX<756{=c9�BD3y4#1'3qXE:<Bg`%K(X<1�:<CPe26i:<6@�p9�4}75=c9�3y�o��R�T�U�µ��BUiT � F��o\g`�6#:EX<1Y¡u��¥��������R�TuF%Ùd=cBD\DBD1YÐo��¤V¥,Q���� �-RuF!&Z���%Q�TdFGI9�4JIK56CMLON�1'K5BDCpN�Q+R�T�U)9�35;~GI9'9�b�±d�%��� ����F�`HK5XpCEBD=cBD\]9�:|X<64[753HB]JIK56#Co7H9Aq6f`�6#63�KHCE6#;�`,?�CM^�6#4B]9�\DBDCMX<C|1'_-np9�:<=c1�3HBD4�o3H9�\g?(CEBDC#F�\]Bgbq6)�,� �%Q�Tum�±�9�:EXZ1'_�X<756f=c1�X<BDA�9�X<BD1'3y1'_�GI9�4JIK56CMLON�1'K5BDCZN�QSR�TVU@e�9�CoX<756)JIK56CMX<BD1'3�1'_-BD;563qX<BD_�?,B]35kWXE:[9�46#C1'_��5RV¯ R�¥I�%��CM^%9�4#6#C|9�35;>X<756#BD:�A�9�:<B]9�3qX<C#m� &������D������ §�N�6iX � � 9�3H; � v `�6{351�:<=c6#;yCM^%9�4#6C#F�4#1'3qX<B]3IK51'K5CE\g?wBD={`a6#;5;56;yBD3qX<1Y9)X<1�^�1'\D1'k'BD4j9�\ Aq6#4#X<1�:|CM^%9�4#6 CE1X<7H9�X � � � v 9�35;�� � Ó � v 9�:<68;56izH3H6#;am�o3�B]3qX<6i:<=c6#;5B]9�X<6{CM^�9�46@`�6iXse266#3 � � 9�35;�� v BDCp9�3q?y3H1�:<=c6#;�CM^�9�46 � CEK54[7~X<7H9�X � � �� v � � ��� � Ó � v�+ePBgX<7>4#1'3qX<BD3,K51'K5C�B]={`�6;5;5BD35k'C}�[m

�o3�BD3IX<6i:E^�1'\]9�X<BD1'3rCM^%9�4#6o`�6#Xse26#63�� � 9�3H;�� v BDC�9�3q?YBD3qX<6i:<=c6#;5B�9�X<6oCM^%9�4#6 � CEKH4[7�X<7H9�X�6iAq6i:E?r\]BD3569�:�=c9�^5^HB]35k_�:<1�= � � Ó � v BD3qX<1�BgX<CE6#\D_�eP7HBD4[7�BDC)4#1'3qX<BD3,K51'K5CW_]:<1'= � � BD3qX<1~BgX<CE6\D_P9�35;�_�:<1'= � v B]3qX<1~BgX<CE6\D_�BDCW9�K(X<1'=c9�X<BD49�\D\D?4#1'3qX<B]3IK51'K5Cp_]:<1'= � BD3IX<1rBDX<CE6#\D_sm|0sXpBDCoC<9�BD;�X<1c`a6@1'_-6il,^�1'3563qX � �+ePBgX<7~®�( � ( ���[FaBD_VX<756#:<6@6#l,B]CMX<Cp9r4#1'35CMX[9�3qXRCCEK54}7>X<7H9�X�1'35687%9�CW� � ��� � � � � � � � ( C � � ��� � v " �� � � � � � � � � � ��� � �� � � / � � / � _�1�:|9�\D\ � � � � � � � � � � � � � v%� � v �[m

� 356{BDCPBD3qX<6#:<6#CMX<6#;yBD3wk'6356i:[9�\�=W6#X<751,;5C{�S1�:P_�K5354#X<1�:<C}�ueP75BD4}7w4#1'35CMXE:<KH4iXoBD3qX<6#:E^�1'\]9�X<BD1'3�CM^%9�46#C|_]:<1'= Xse21r9�:E`HBgLXE:[9�:E?�351�:<=W6#;wCM^%9�4#6#C{�S1�: ���T���²�³hCM^%9�46#C#FH1�:PnoQ�¥,¯����5��CM^�9�46#C}�[m

�%1�:8Xse21 ���T���²�³/CM^%9�46#C � � � � v F X<756)41'=)^H\D6#l�=W6#X<751,;~4#1'35CEB]CMX<CZBD3�\D1,1�b(BD35k>9�XZX<756)CM^%9�46)1'_d:<6j9�\-9�3H9�\g?,X<BD4_�K5354iX<B]1�3HCW@¶ePBgX<7�A'9�\]K56#C{B]3 � � Ó � v F�;56izH3H6#;¶1�3°X<7H6r1�^�6#3hCMXE:<Bg^�® � Q�� ��FV4#1'3qX<BD3,K51'K5C@1'3°X<756r4\D1'CE6#;hCMXE:<Bg^® ( Q (���F�9�3H;�CEK54}7�X<7H9�X9@��+�&*5�fBDC)`�1'K535;56;�B]3 � � 9�35;5@��E�pÓ �&*5�@B]Cf`a1�KH35;56#;�BD3 � v Fu6jJIK5Bg^5^�6;�ePBgX<7¶X<7H6351�:<= � � @ � ��� =r9�l �jCEK(^ ! � � @��+�&*H��� � � � CEK(^ ! � � @��E��Ó��&*5��� � v �qF�9�35;�1'356);56iz%356#C�! � � � � v # � _�1�:8® � � �¸�)9�CoX<7H6fCM^%9�461'_ N � @�� � �[F ePBDX<7~X<756W351�:<= � � N � ��� � � � � /�� � �¸BD35_ � � � � � + � � @ � � m � _u4#1'K(:<CE6cCEK54}7h9�CM^�9�46W41�3IX[9�B]35C � � �� v FV9�C81�3H649�3yX[9�b'6�@�X<1c`�6f9c4#1'35CMX[9�3qXp_�K53H4iX<BD1'3wX[9�b(BD35krBgX<CPA�9�\DK56{B]3 � � �� v mo¢�756@BD3qX<6#:E^�1'\]9�X<BD1'3>^5:<1�^�6i:EXs?w_�1'\D\D1jePC|69�CEBD\g?_�:<1�=!X<756|_+9�4iXuX<7H9�X2BD_ � � � � � � � � � � � � v�� v �[FIX<756#3 � @��GQpÓ��&*5�-C<9�X<BDCMzH6C29@CEBD=WBD\�9�:d^5:<1�^�6#:EXs?WePBgX<7cX<756oCM^%9�4#6#C � 9�35;� v F�9�35;�X<756i:<6_�1�:<6)1'356)7H9�CE� � �6N � ��� � � � � / � � (x=c9�l � � � ��� � � � � � � � � � � � � ��� � � � � / � � / � � � � N � ��� � � � � / � � F�9�35;�1'356W=c9?9�4iX<KH9�\D\D?c:<6#^H\]9�4#6Z=c9�l � � � ��� � � � � � � � � � � � � ��� � � � � / � � / � �8`I? � � ��� � v

"�

� � � � � � � � � � ��� � �� � � / � � / � m��X8\D6j9�CMXZ_�1�:ZX<756W49�CE6)1'_dGI9�4JIK56CMLON�1'K5BDCZN�Q+R�T�U'F�1'356W=W1�X<BgA�9�X<BD1'3�_�1�:8BD3qXE:<1,;5K54BD35kYB]3qX<6i:E^�1'\]9�X<BD1'3�CM^�9�46#CZe�9�C

X<756)JqKH6#CMX<BD1'3�1�_VXE:[9�46#Co_�1�:oA'9�:<B]9�3qX<C|1'_2�(RV¯ R�¥,�H��CM^%9�46#C#mp�H1�:p6#l(9�=W^H\D6�F�BD_ ��� �)!�� �0Q � �)! � Q ! � ®��qFa9�35;

1'356re�9�3qX<C@X<1y;H6#CE4i:<BD`�6cX<756rXE:[9�4#6r1'3hX<756r`�1'K535;H9�:E?°1'_P9y_�K5354#X<BD1'3 ��� � v:� �5���P�[FV1'356Y=c9?°41�3HCEBD;56i:@X<7H9�X �5���� � ® � ,/t%� v:� �5� �)! " vi� � 9�35;wX<7H9�X � �� � 1 �

��� � ® � ,�t ���5� �)! " vi� � F�9�35;y756{BD3qXE:<1,;5K546#;y9c=W1�:<68k'6#356#:[9�\�_�:[9�=W6ie21�:Eb�FeP75BD4}7�GI9'9�bw±d�%��� ���°9�\DCE1Y;5BD;�BD35;56#^�6#3H;56#3qX<\D?>CE1YX<7H9�XoBgXokq9A'6)9Z�M1'BD3qXZ9�:EX<BD4#\D6'F%eP756i:<6@X<756i?y4#1'35CEBD;56#:<6#;¶�SCMXE:<1'35k'\g?=W6j9�CEK(:[9�`H\D6��8_�1'354#X<BD1'35C@;H6izH356;¶1'3*�O® � ,��pePBgX<7hA�9�\DKH6#C@B]3 � � Ó � v 9�35;�CEK54[7¶X<7H9�X % � � ��� ��� � �O® � ,�t � � �89�35;% � / ���� � � ��� / �O® � ,�t � v �[F�9�35;°\D1,1�bq6#;�_�1�:8X<756cCM^%9�4#6cCM^%9�353H6#;�`I? � �O®I�o_�1�:@9>CM^a6#4#B�9�\V:[9�35k'6c1'_u^%9�:[9�=c6iX<6#:<CpeP756#:<6� �O®I�VBDCu9�K(X<1'=c9�X<BD4j9�\D\g?f;56iz%356#;amV0sX2CE6#6#=cC-X<7H9�XdX<75BDCdBDC298_�1�K5:d^%9�:[9�=W6#X<6i:<Cd_+9�=WBD\D?f1'_aCM^%9�4#6C#FI`HK(X24[7H9�35k'BD35k %�BD3qX<1 %

�CE751jePCoX<7H9�XoX<7(:<66{^%9�:[9�=W6#X<6i:<Cp9�:<6f6#3H1�KHk�7 taBgXoe�9�CpGI9�9�b�±-����� ����eP751rz%3H9�\D\g?wCE751je26;�X<7H9�XoX<756f_�9�=WB]\g?w9�4iX<KH9�\D\g?;56#^�6#35;HC�1'35\g?�K(^�1'3>Xse21f^�9�:[9�=c6iX<6i:<C#F%9�35;y9�_]X<6#:�CEBD=W^H\DBgzH4j9�X<BD1'3YBgXP\D6#;�X<1)X<756 �yLO=W6#X<751,;>9�35;>X<756#D LO=W6iX<7H1I;>X<7H9�Xe2689�:<68k'1'BD35kfX<1WCMX<K5;5?'m

� ��B]CE4[7H9W¡�R�� ¥,���dF,�|:<k'6#3qX<BD3HB]9�3(LS`�1�:<3r=c9�X<756=c9�X<BD4B]9�3aF%�j�(�@�Im-no6Pe21�:Eb(C�9�XP¡u6#3qXE:[9�\�¦o35BgAq6#:<CEBgXs?W1'_�Ñd6#3H6#�#KH6#\]9(F¡29�:[9�49�C#m���p9�4[7H=c9�3��o��R�TVU�µ'�TU#TdF5±�1'\DB]CE7(LS`�1�:<3�=c9�X<756=c9�X<BD4#B�9�3 F��j��®�(L<�j� ��®(m-no6o6=WBDk�:[9�X<6#;YX<1f¦o35BgX<6#;y�qX[9�X<6#C�BD3��j��Õ��IF

9�35;w756pe21�:Ebq6#;wBD3wN�9�K(:<6#3H4#6�F &89�35C<9�C#FHeP756#:<6p0uA(BDCEBgX<6;�75B]= ;HK(:<BD35kW={?Yz5:<CMX�A(BDCEBDX�X<1W¦8�%FHBD3��j��(I��m

Ô(�

Page 52: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K®q­,m-�5:<BD;H9?��~9�:<4[7°�®(F�­�®'®�®(m� &������D������ §pN�6iX � � 9�35; � v `�6@Xse21�3H1�:<=c6#;�CM^%9�4#6#C#F 41'3qX<BD3IK51'K5CE\D?yBD=@`�6#;5;H6#;~BD3qX<1>9rX<1�^a1�\]1�k'BD4j9�\�Aq6#4#X<1�:pCM^%9�46 CE1>X<7H9�X{1'356W49�3�;56#zH356 � � �� v F�6JIK5Bg^5^H^�6#;�ePBgX<7~X<756c351�:<= � � N � � � � � � / �¸=c9�l � � � N � � � � � � N � � v �'F�9�35;� � Ó � v F6JIK5BD^5^5^�6;�ePBgX<7�X<756Z351�:<= � � N � � � � � � / ��B]35_ + � + � � + / � � N � � � � Ó � � N v � � v m�%1�: N � � � Ó � v 9�35; % ��®W1�3H6Z;56#zH356#C��¶� % � NH�H��BD35_ + �&+ � � + / � � N � � � � Ó)%T� � N v � � v m�%1�:-® � � � �29�35;���(+*#(-,�F�1�:V_�1�: � �*® � �dePBgX<7 *9� ,/F�1'356u;H6izH356C�� � � � � v � � � � 9�C�X<756uCM^�9�46u1'_ N � � � Ó� v CEK54}7>X<7H9�X %

"� �h� % � NH� � ��� � ® � ,�t � �� �'FH6JIK5Bg^5^a6#;�ePBgX<7�X<7568351�:<= � � N � � � � � � � / � � � �

� � � % " � �¶� % � NH��� � � �

� � � �� � � � � � m�o3WBD;56j9p1�_�Ùd=WBD\]BD18Ðo��¤�¥,Q�������R�B]CVX<1Z4#1'35CEB];56i:d9p^H\]9�3562ePBDX<7)4#1,1�:<;5BD3H9�X<6CHQ � � Q v 9�35;c_�1�:d9Zk�BDA'63WN � � � Ó � vX<1)=r9�:Eb>9�\]\�X<756p^�1'BD3qX<C�CEKH4[7�X<7H9�X�X<756i:<6Z6il(BDCMX<CP9f;564#1'=)^�1'CEBgX<B]1�39N��5N � Ó<N v ePBgX<7!� � N � � � � ( Q � 9�35; � � N v � � v ( Q v m¢�75BDC�CE6iX�BDC�4#1'3qAq6ilp`a6#49�K5CE6uBD_4N�� 2 � Ó 2jv ePBgX<79� � 2 � � � � (�* � 9�35;�� � 2jv � � v (�* v F�X<756#3{_�1�:V® � � � �-1'356d7H9�CTN�� 4 � Ó 4�vePBgX<7 4 � � �E� >��H� N � Ó � 2 � 9�35; 4�v �Ø�E� >��H� N v Ó � 2�v 9�35;rX<756PXE:<B]9�35k'\D6PBD3H6JIKH9�\DBgXs?)k'BgAq6#CR� � 4 � � � � ( �E� >��H� Q � Ó ��* �9�35;$� � 4�v � � v (��E�)> �H� Q v Ó ��* v mu¦oCEB]35k)X<7568_�K5354#X<BD1'3 % �� �¶� % � NH�uBDCP1'3568e�9#?>1'_�;H6#CE4i:<BD`HBD35k)X<7568`�1'K535;H9�:E?>1�_�X<75B]C4#1'3qAq6#lYCE6#Xm�%1�: % �!®(FHN �� �h� % � NH�pBDCf9y351�:<=»6jJqKHBgA�9�\D63qX8X<1yX<756Y351�:<=»1'3 � � Ó � v m��h� % � NH�pBDC@351'35;564i:<69�CEBD3Hk�BD3)%

9�35; ! � � � + �� BDCf351'35BD3H4i:<69�CEBD3Hk�BD3 %}Fd9�35;¶=c1�:<6#1jAq6i: �¶� % � NH�ZB]Cf4#1'3549Aq6YB]3 %}Fd9�C)9�3�BD35zH=@K5=·1'_|9��c356�_�KH354iX<BD1'35C#F9�35;¶X<7H6i:<6#_�1�:<6Y41�3IX<BD3IK51'K5C#m � 356>49�3�k'BgAq6�9~;56#zH35BgX<BD1'3¶1'_�X<7H6�CM^%9�4#6>BD3qAq1'\gA(BD35k~9~CEK5=ÛB]35CMX<69�;�1'_|9�3�BD3qX<6k�:[9�\O§1'3¶9�3hBD3qX<6i:EA'9�\�( � ( % ( ( � ��v 1'356r7H9�C �¶� ( � � NH��( �¶� % � N5� ( ( �¶� ( � � NH�o_�1�: ��� � F�9�35;¶9�C{X<756r=c69�CEK(:<6r1�_� ( � � ( � ��vi�o_�1�:8X<756c=W6j9�CEK5:<6 � �� BDCc��F�1'356cCE6#6C8X<7H9�X�N � � � � � � v � � � � BD_29�35;°1'35\g?�BD_ ( " � � �¶� ( � � NH� � � �5� � �[F�9�35;� � ( " � � �¶� ( � � NH��� � � �

� � � BDCP9�3w6JIK5BgA'9�\D6#3qX�351�:<= 1�3°� � � � � v � � � � m� & ����$ §�0�_�® � � ���89�3H;��)(+* � 5 (-,�F51'35687H9�C{� � � � � v � � � � � � � � � � v � � � 8 �+ePBgX<7�41'3qX<BD3IK51'K5CPBD=@`�6#;5;HBD35k,�[m�2È<ÆjÆ5#[§�¦oCEBD35k)X<756Z;H6izH35BDX<BD1'3�K5CEB]35kWCEK5=WCPBD35CMX<6j9�;w1�_�BD3qX<6k�:[9�\]C#F51'3568351�X<BD4#6C2X<7H9�X � �)BDC�BD3H4i:<69�CEBD3HkfePBgX<7 *�m

�o351�X<756#:8e�9?�X<1w^5:<1jA'6cX<756cC<9�=W6W:<6CEK5\gX@BDC8X<1w351�X<BD46)X<7H9�X@BD_o� ( * � ,�FV9�35;)% � ��®w1'356c7H9�C �¶� % � NH�� �¶� % � � N5�u_�1�: % ��% � 9�35;>X<756#:<6#_�1�:<6E� � N � � � � � � � � / � � � �

�¶� % � � NH� � I � � %"� � � �� � �h� % � � NH� �

� � �� � � FHk'BgA(BD35k�%"�� �¶� % � � N5�'(

C � � N � � � � � � � / � � � �

Fd9�35;�X<756i:<6_�1�:<6!� � % " � �¶� % � NH��� � � � � � � #� � � � � � ( C � � N � � � � � � � / � � � �

Fu9�35;¶`I?hn �R�¥��V��� BD356jJIKH9�\DBgXs?°1�3H61�`5X[9�BD35C#� � N � � � � � � � / � � �

� � � % " � �h� % � NH��� � � � � � #� � � � � � ( C��4� � N � � � � � � � / � � � �

_�1�: *#( 5 (-,/m �6#4j9�K5CE6�_�1�:*N � � � Ó � v 1'356�7H9�C#�¶� % � N5� �=cBD3 �q� � % � � � N � � � � � � / F'1'356�CE66#C�X<7H9�X�BD_�� � � � � v � � � � BDC�351�X�:<6;5K54#6;X<1W®f1'356Z=fK5CMX�7H9Aq6 % " � =WB]3 �I� � % � � ��� � ® � ,/t � �� �qFH9�3H;�X<756i:<6_�1�:<6oX<7H6pCM^�9�46W� � � � � v � � � � BDC�:<6#;5K546#;�X<1W®fBD_ � ��®1�:PBD_ � ����FH9�35;y9�\DCE1WBD3�X<7H6Z49�CE6C � � ®f1�: � �Ø��F%BD_ * �-,/m �6#4j9�K5CE68_�1�: N � � � � v 1'35687H9�C�X<7H68;56#41�=W^�1'CEBgX<BD1'35C�N �5N8Ó¶®c9�35;�N � ®|Ó$N%F%1�3H6ZzH35;5C�X<7%9�X �¶� % � NH�&(=WB]3 �I� � % � � � N � � � � � � / FH9�35;>X<756i:<6_�1�:<6p_�1�:P9�\D\�X<7H6o^%9�Bg:<CZ� � � *a��eP75B]4[7w9�:<6Z4#1'35CEBD;56#:<6#;>1'356Z7H9�C�� � � v � � � � � � v � � � ��+ePBgX<7>4#1'3qX<BD3,K51'K5C�B]={`�6;5;5BD35k,�[m�o\gX<7H1�KHk�7~BgXoePBD\D\�`�6fBD=W^�1�:EX[9�3IX|X<1Y4}7H9�:[9�4iX<6i:<B]�#6f9�Cp=fK54[7�9�Co^�1'CECEBg`%\D6{eP7H9�XpX<756#CE6)BD3qX<6#:E^�1'\]9�X<BD1'3yCM^%9�4#6CZ9�:<6

BD3>6j9�4[7>41�3IX<6il,XF(X<7H6ZBD3qX<6#:E^�1'\]9�X<BD1'3r^5:<1�^�6#:EXs?�4#1'=W6CP9�K5X<1�=r9�X<BD49�\D\g?qm� ��%"-� j�D������ §u0�_��xB]C�\DB]3569�:�_�:<1'= � � Ó � v BD3qX<1 � Ó v 9�35;w=r9�^%C � � BD3qX<1 � ePBgX<7$� � �6QH� � � � ( � � � � QH� � � � _�1�:9�\D\ Q � � � 9�35;y=c9�^HC � v BD3IX<1 v ePBgX<7$� � �6QH� � � / (9� v � � QH� � � / _�1�:p9�\]\ Q � � v F%X<7563��ØBDC|\DBD3H69�:P41'3qX<BD3IK51'K5Co_]:<1'=� � � � � v � � � � BD3IX<1�� � � v � � � � _�1�:�9�\D\ � � *�F(9�35;�_�1�:�® � � � �o1'356p7H9�C�� � �6N � � � � � � � / � � � �

(�� v " �� � �v � � N � � � � � � � / � � � �

_�1�:9�\D\ N � � � � � � v � � � � m�2È<ÆjÆ5#[§a�%1�:�6j9�4}78;56#41'=)^�1'CEBgX<BD1'3N � N � Ó N v ePBgX<7 N � � � � 9�35;RN v � � v F�1'356-7H9�C �6N����6N � Ó �6N v F�9�35; � N � � �ePBgX<7�� � �6N � � � � � ( � � � � N � � � � � 9�3H;��6N v � v ePBgX<7�� � �6N v � � � / ( � v � � N v � � � / m � 356�;56;5K54#6C)X<7H9�X �¶� % � �6NH�$(� � �6N � � � � � Ó %T� � �6N v � � � / ( � � � � N � � � � � Ó % � v � � N v � � � / � � � � � � N � � � � � Ó � � /� � � � N v � � � / � Fo9�35;�X<7H6i:<6#_�1�:<6~1�3H6�7%9�C�¶� % � �6NH� (1� � � � � � /� � � N��mp¢�7563�K5CEBD3Hk -�� � � /� � 1'356@;56;5K546#C|X<7H9�X %

"� �¶� % � �6NH� (1� v

"�� � �v -

"� �h� - � N5��9�35;

9�C � �� � � �� 1'356Pz%35;5CuX<7H9�XR� � �6N � � � � � � � / � � � �

� � � % " � �¶� % � �6NH��� � � �

� � � W� � � � � � ( � v " �� � �v � � -"� �¶� - � NH��� � � �

� � � #� � � � � � �� v " �� � �v � � N � � � � � � � / � � � �

_�1�:|9�\D\�N � � � � � � v � � � � m�o3�BD=)^�1�:EX[9�3qX26#l(9�=W^H\D6|B]CuX<756p49�CE6 � � � � v ���P�[F � v � � ���P�[F(_�1�:2eP7HBD4[7YX<756o41�:E:<6#CM^�1'35;5B]35kfBD3qX<6#:E^�1'\]9�X<BD1'3CM^%9�4#6C�9�:<6uX<7562N�R����%T���µoCM^%9�4#6C#vF�_�1�:�� � *��-,»9�3H;���( 5 (-, 1'3562;56#351�X<6C ��� � 8 ���P�H� � ��vj���P� � ������P��� v � � � 8 F

vwNVR�����T���µW7H9�;wBD3qXE:<1,;5K546#;�X<756#CE68CM^�9�46#C�`�6#_�1�:<6pX<756p0�3qX<6#:E^�1'\]9�X<BD1'3YX<7H6#1�:<BD6#C�e26i:<68;56iAq6#\]1�^a6#;am

Ô'­

Page 53: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

9�35;r1'356|ePBD\D\5z%35;rX<7H9�X&��� � �H���P�H� ���5���P���+ePBgX<7c6jJIK5BgA�9�\D63qXu351�:<=WC}�[mV�%1�:�9{_�K5354iX<B]1�3 @ � ��v���P�5Ó#� ����P�V1'356o49�349�\D4K5\]9�X<6p6#lI^%\DBD4#BDX<\g? �h� % � @��[F%9�35;>X<756Z_�1�:<=@K5\]9)=c9�bq6#C�K5CE681'_�X<756Z351'35B]354i:<6j9�CEB]35k):<69�:E:[9�35k'6#=W63qX�1'_*@�m

�%1�:-9�=c69�CEK(:[9�`H\D6uCE49�\]9�:V_�K53H4iX<BD1'3W@f1'3���CEKH4[7{X<7%9�X�_�1�:�6#Aq6i:E? � ��®P1'356u7H9�CB)(�N - �0Q � � � � @��GQa��� � � � �-,/F1'356y49�3/;56#zH356yX<756y351'35BD354#:<69�CEBD35k�:<6j9�:E:[9�35k'6=W6#3IXr1'_�@�F�;56#3H1�X<6;�@ � �O9�35;/6il,X<6#35CEBDA'6\g?�K5CE6#;/`I?�no����� � 9�35;N�Q�� � ¥,�%´�R�R��d�[mq0sXdB]C-X<756|1'35\g?w�+:<69�\���3H1�3HBD354i:<6j9�CEBD35k{_�K5354#X<BD1'3c;56izH3H6#;r1'3 � ® � (�N -����P����eP75B]4[7rBDCd6jJqKHBD=W6j9�CEK5:[9�`%\D6X<1 � @ �¹F�9�35;~BgXo49�3w`a6{;H6izH356;w`,? � � � @�� %

�� � @�� % " ���8BD_�9�35;�1'35\D?>BD_H)(�N - �0Q � � � � @��GQa��� � � � (9% (5 (�N -��0Q �

� � � @��GQa��� � �'m%1 756#3y3H6#4#6CEC<9�:E?'F�1'356{6#lIX<635;5C @� � %<�uX<1W`�6@®c_�1�: % � (�N -����P�[m � 3568`�9�CEB]4Z^5:<1�^�6#:EXs?>B]_�X<7%9�Xo_�1�:

9�3q?Y^%BD6#46iePBDCE6p41'3qX<BD3IK51'K5CP_�K5354iX<B]1�3���;56izH3H6#;w1'3 ! ® � ,��u1'35687H9�C I J �o� � @��GQa��� �4O Q9� I � � + � � J �� � � @ � � %<��� O&%}m� & ����$ §80�_ � � � � v ���P�89�3H; � v � � ���P�|X<7563°_�1�:f9�3q?�_�KH354iX<BD1'3$@ � � v ���P��Ó-� ���P�p1�3H6c7H9�C �¶� % � @a�R�I �� @ � � -��4O -Z_�1�:|9�\]\/% �/®>�S6il,X<6#35;HBD35k�@ � `I?>®)_�1�: % � (�N -'���P�E�[m�2È<ÆjÆ5#[§�0�_ 1'356p;H6#4#1'=W^�1'CE6#C)@9��@ � Ó!@ v ePBgX<7�@ � � ��vj���P�d9�35;!� � @ v � � � �*� J � ( ���O9�35;�� ��®I�[F,X<756#3YX<756oB]3(zH=@KH=�1�_� � @ � � � � / � J � BDCu1�`5X[9�BD356;c`I?WX[9�b,B]35kW@ v �GQa�H��@��GQa��eP7H6#356#A'6#:�� @��GQa��� ( � F59�35; @ v �GQa�H� � � � � ��

� � � � � eP756#356#Aq6i:�� @��GQa��� � � F9�35;{X<7HBDC�CE751jePC�X<7H9�X��h� % � @��*��BD35_ � � � � I � � � � � � � � � � @��GQa��� > �a�MOMQVÓ % ��� �*BD35_ � � � � I � � � � � � � � @ � � -j� > �a�MO -IÓ % ����mV¢�7H6BD3(z%=@K5=�BDCu9�XEX[9�BD3H6#;W_�1�:u9�3q? �cB]3)X<756PBD3qX<6#:EA�9�\ ! @ � � %

�� � @� � % " � #�9�35;cBDC I �� @ � � -j�4O -8�S1'356�6#l,X<6#35;5C8@W`I?c®Z1'K(X<CEBD;56 �9�35;6@ � `,?�®@_�1�: % � (�N -'���P�E�[m�0�3H;56#6;�\D6iX � `a6oCEK54}7�X<7H9�X�� � ! @ � � � � � � @ � � � " � #OF(X<7563 I � � � � � � � � @ � � -�� >����4O -�Ó% �9��I��� @

� � -��MO --Ó ��� %B> � �[FH9�3H;>BgX�BDC�6#3H1�KHk�7>X<1)4[7564<bYX<7H9�X I��� @ � � -j�4O -dÓ ��� %B> � � �®)_�1�:|9�\]\ � ��®(t(X<75B]C�BDC�94#1'35CE6jJIK56#3546Z1'_*@ � � -�� �w_�1�: -���%�9�3H;6@ � � -��&( �w_�1�: - ��%}m

0�3w1�:<;56i:�X<1W41�=W^%9�:<6pXse21W;56#zH35BgX<BD1'35C�1'_�N�R����%T���µWCM^%9�4#6#C#FHe26ZCE7H9�\D\ KHCE68no����� �¶BD356jJIKH9�\DBgXs?'m� & ����$ §�N�6#X°��( 5 ( , 9�3H; � � ��m�0�_ % � � � � 8 � ® � , � � �� ��9�35; � � %<� � v� I �� �d� -��4O -~X<7563 1'356�7%9�C% � � � � 8 � ® � , � � �� �{9�3H; � � � � � � � � � #� � � � � � ( vv " � � � �8� � � � � � #� � � � � � m�2È<ÆjÆ5#[§-¢�756849�CE6 5 ��, BDC�1�`IA(BD1'K5C#FH`�6#4j9�KHCE6E� �d� %<��� (�� % " �w_�1�:|9�\]\/% ��®)BD=)^H\]BD6#C�� � � %<��� ( � � -v " � _�1�:|9�\D\�% ��®(m

�%1�:8��( 5 �-,�F(1'356oK5CE6C2X<756o_�9�4#XuX<7H9�X�C � �O® � ,���BDC2;5635CE6oB]3cX<7H6oCM^%9�4#6p1�_���CEK54}7rX<7H9�X% � � � � 8 � ® � , � � �� � FCE18X<7H9�Xu1'356|=c9?c9�CECEKH=W6PX<7H9�X � � C � �O® � ,��[FIBD3WeP75BD4}7c4j9�CE6 � A'9�35BDCE756C-3H69�:2®{9�35;c`�67H9A'6Cu9�C � � _�1�:�%�\]9�:<k�6'mV�oC� B]Co1�_d4#\]9�CEC C)vp9�35;�% � �O� %<��Ó � � %<��� �d� %<�[F�1'356f=@K5\gX<BD^H\DBD6C�`I? % � 8 � � � 8 " º � 9�35;~1'356fBD3qX<6k�:[9�X<6Co9�kq9�B]35CMX � �� t 1�3H6zH35;HC I � % � � % � 8 � � � 8 " º � � �� � v

8 I� % � 8 O � � � 8 �(>)��I � % � 8 � � � 8 � �� F'`�6#4j9�KHCE6�% � � � %<�aX<635;5CVX<1p®p9�X ,�m�¢�75BDCVCE751jePC

X<7H9�X{�E��><����I � � % � � � 8 � �� � I � � % � � � 8 " º$% � � % � � � �� FH9�35;yn �R�¥������°BD356jJqK%9�\]BgXs?rB]=)^H\DB]6#CZ�E��> �-��� � % � � � � ( � � % � �8� � FeP756#:<6pX<7568351�:<=�BDC�X<7H9�XP1'_ � 8 � ® � , � � �� � m� ��%"-� j�D������ §@�%1�:r� �-*�� , 9�35;/��( 5 ( , 1'356W7%9�C � � � 8q���P�� � � v ���P� � � ���P��� v � � � 8 � �4@ � � v ���P�VÓ� ����P� � %}v � � @ � � %<� � � 8 � ® � , � � �� ��� F 9�3H;,� � %}v � �M@ � � � � � � � W� � � � � � B]Cp9�3~6jJqKHBgA�9�\D63qXo351�:<=·�O9�35;yX<756#:<6#_�1�:<6 ��� � �5���P������5���P�oePBgX<7¶9�3h6JIK5BgA'9�\]6#3qX{351�:<=��[m ���.� >���P�pBDC{X<756ce269�b ���wCM^%9�4#6Y1'_�������²�Q�T���Q��H´>QS² µh�+ePBgX<7h9�3h6jJqKHBgA�9�\D63qX351�:<=Y�[m�2È<ÆjÆ5#[§�¢�7H6{;H6izH35BDX<BD1'3y1�_VX<7H6{BD3qX<6#:E^�1'\]9�X<BD1'3wCM^%9�4#6{e21'K5\D;�7H9Aq6 % " � �h� % � @�� � � 8 � ® � , � � �� �qF%ePBgX<7 � � v

� Fa9�35;~9�C�¶� % � @a� ��I �� @ � � -��MO -E %�@ � � %<�P`�649�K5CE6E@ � BDCZ3H1�3HBD354i:<6j9�CEBD35k5F�BgX8BD=W^H\DBD6C % " � �h� % � @��R %}v " � @ � � %<� � %}v � �M@ � � %<�[m¡u1'3qAq6i:<CE6\g?'F�BD_,%}v � �M@ � � � 8 � ® � , � � �� � F�X<7563{no����� �>BD356JIKH9�\DBDXs?pBD=)^H\]BD6#C�%}v � � v� I �� @ � � -j�4O - � � 8 � ® � , � � �� � F�`�6#4j9�KHCE6� � v

�� ��FH9�35; %}v � � v� I �� @ � � -��4O - �9% " � �h� % � @��[m¢�756w;56izH3HBgX<BD1'3�1�_PX<756�e269�b ���~CM^%9�4#6w1�_|������²�Q�T���Q��%´>Q+² µ�BDCfX<7H9�X)X<756#:<6�6#l(BDCMX<C"� CEK54}7�X<7H9�XW_�1�:c6#A'6#:E?

=W6j9�CEK(:[9�`H\D6�CEK(`%CE6iX��1'_ � 1'356~7H9�C I � � @ � OMQ ( �S (�N -'���u� v � �� m�¢�756~CMX[9�X<6=W63qX�BDCrX<7563/X<7H6�4#1'35CE6jJqKH6#35461'_PX<756�_�9�4#XfX<7H9�XW_�1�:�% � ®~1'356>7H9�C)CEK(^ � � + � � � � � � I � � @H� OMQ�� I �� @ � � -��4O -�Fu9�3H;¶X<75BDC)BDC)CE66#3�`I?¶4[751,1'CEBD35k� �! @ � � %

�� � @� � % " � #|9�35;�;H6izH35B]35k � � � �0Q � � @��GQa��� � � ��9�35; � v � ��Q §R� @��GQa���8 � F2CE1�X<7H9�X )(�N -'��� � � ( % ( (�N -���� v ���O9�35; � @��GQa��� � %-1'3 � v � � � �[m�0�_ �hBDCu351�X29{CEK(`HCE6#X21'_ � v F(1�3H6PB]354i:<6j9�CE6C-X<756|BD3IX<6#k�:[9�\%1�_ � @ ��`I?W:<6i^%\]9�4#BD35kX<756p^%9�:EX�1'_ �¶eP7HBD4[7>BDC�3H1�X�BD3� v `I?>9@^%9�:EX�1�_�X<7H6pC<9�=c6p=W6j9�CEK5:<6pB]3 � v � ��tHBD_ ��B]C�9fCEK(`%CE6iXP1'_ � v `%K(X�;51,6C�351�X4#1'3qX[9�BD3 � � F�1'356{BD354#:<69�CE6#C|X<756{BD3IX<6#k�:[9�\�1'_ � @H�,`I?>:<6#^H\]9�4#BD3Hkr9W^%9�:EXo1�_�/eP75B]4[7yBDC|351�XoB]3 � � `I?�9c4#1�:E:<6#CM^a1�3H;5BD35k^%9�:EXp1'_�X<756fC<9�=c6@=c69�CEK(:<6@B]3�� � � ��FaCE1YX<7H9�XozH3H9�\D\g?>X<7H6@CEK(`HCE6#X<Cp1'_�=c69�CEK(:<6 %P_�1�:oeP75BD4[7�X<756@BD3IX<6#k�:[9�\�1'_� @H�5BDC

=c9�l(BD=fK5= =@KHCMXP4#1'3qX[9�BD3� � 9�35;>`�6Z41'3qX[9�BD356#;wBD3� v m

��

Page 54: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®A�,m-��1'35;H9?��~9�:<4}7��@�,F�­�®'®'®,m1°6)e�9�3qX8X<1>BD3qX<6#:E^�1'\]9�X<6f351je�`�6#Xse26#63h�(RV¯ R�¥,�%�*CM^%9�4#6C8BD3�1�:<;56i:8X<1>;56iz%356WCM^%9�46#C8\DBgbq6W; � ���P��eP7H6#3�-)BDC

351�X89�3�BD3qX<6k�6#:mp0�3�X<756f49�CE6feP756i:<6 ��� �)!{F�eP756#:<6@X<7H6f;56izH3HBgX<BD1'3�BDCo`%9�CE6#;�1�3~�-R����Q��%��XE:[9�35CE_�1�:<=>F�X<7H6i?�9�:<6BD35;H6#6#;>BD3qX<6#:E^�1'\]9�X<BD1'3YCM^%9�46#C�ePBgX<7YX<756p^%9�:EX<BD4K5\]9�:�4[751'BD46&*9� ­IF%9�35;>_�1�: * � ­{X<756#?r`�6\D1'35kfX<1W9f\�9�:<k'6#:�_+9�=cBD\g?r1�_CM^%9�4#6C�3%9�=c6#;w9�_�X<6i: � \D6#k# ��aU�Ra��v#m

�%1�:�- � �)F�\D6iX � ;56#351�X<6WX<756cCM^�9�46c1'_p�S6jJIK5BgA�9�\D6354#6W4#\]9�CECE6#C@1�_}�o=W6j9�CEK5:[9�`%\D6c_�KH354iX<BD1'35C�F~CEK54[7°X<7H9�Xr�E�|ÓÕ ��ºM� � � º � � � � º%F � �uº�� �"!p�[F�X<7563�X<756y�-R����Q��%�xXE:[9�35CE_�1�:<= ��B]Cr9�3/B]CE1�=c6iXE:E?�_�:<1�= ; � � �)!p�W1�3IX<1 � 9�35;�X<7H6BD3qAq6#:<CE6f�-R����Q��%��XE:[9�35CE_�1�:<= � BDCZ9�3~BDCE1'=c6iXE:E?y_�:<1'= � 1'3qX<19; � � �)!p�[m@¢�7H6i:<6#_�1�:<6@X<7H6fBD3IX<6i:E^�1'\]9�X<BD1'3y^5:<1�^�6i:EXs?BD=W^H\DBD6CYX<7H9�X�� =c9�^HC>41�3IX<BD3IK51'K5CE\g? � ; ��� �)!p� � ; � � �"!p��� � � � BD3qX<1 � � � � � � � � 9�35; � =c9�^HC°� ��� � � � � � B]3qX<1� ;6��� �)!Z� � ; � � �)!p� � � � � 9�35;�X<756i:<6_�1�:<6 � ;6��� �"!p� � ; � � �)!p� � � � � 41�B]354#BD;H6#CfePBgX<7�X<756°�+X<6#=W^�6i:[9�X<6��8;HBDCMXE:<Bg`HK(X<B]1�3HCeP751'CE6p��R����Q����~XE:[9�3HCE_�1�:<=!`�6\D1'35k'CdX<1�� � � � � � � � �O9�35;�1�3H6o;56#;5KH4#6#C�BD3rX<756oC<9�=c6|e�9#?cX<7%9�X�BgX2BDC�9�3YBDCE1'=c6iXE:E?cBD_1'356ZKHCE6#C�X<75684#1�:E:<6#CM^a1�3H;5BD35kW351�:<=WC}�[m

0�;563qX<BD_�?(BD35k�B]3qX<6i:E^�1'\]9�X<BD1'3*CM^%9�4#6C�`a6iXse26#63Ø�(RV¯ R�¥,�H��CM^%9�4#6#C ; � � �)!Z�rBDC�X<7H6#3*X<7H6�C<9�=W6°JIK56#CMX<B]1�3 X<7H9�3BD3qX<6#:E^�1'\]9�X<BD35kr`�6#Xse26#6#3�CE1'=W6 �dºfCM^%9�4#6#CpePBgX<7~e26#BDk'7qX<C#F�9�35;~X<7HBDCp356#e�JIK56#CMX<BD1'3�49�3~`�6)CE6iXEX<\D6;�6j9�CEBD\g?yBD3�9�=W1�:<6k'6#356#:[9�\aCE6#XEX<BD35k5m� ��%"-� j�D������ §h�H1�:w9��S=c69�CEK(:[9�`H\D6��f^�1'CEBgX<BgAq6�_�KH354iX<BD1'3 � 1'3 �oF|\D6iX��c� �Z� � � � § I J � � �GQa��� º �@�GQa�4O Q � , �ePBgX<7 � � � � �

� � I J � � �GQa��� º �{�GQa�4OMQ � v � º m 0�_ � � � � v 9�:<6°Xse21�CEK54[7Ø_�K53H4iX<BD1'35C#F8X<7563x_�1�:�® � � � �°1'356h7%9�C� �c� � � � � �r� � v � � � � º � �r� � � �dePBgX<7�^5:<1�^�1�:EX<BD1'3H9�\a351�:<=cC#F5eP756i:<6 � � � � v

"�� � � v m

�2È<ÆjÆ5#[§ � 3568K5CE6CP9)A�9�:<B]9�3qXP1�_�X<756 �»_�K5354iX<B]1�3%9�\ 9�;H9�^5X<6;>X<1 �uºpCM^%9�46#C#FH3H9�=c6#\g? � º � % � NH�H��BD3H_ + � + � �&+ / � � � N � � � º� Ó%Eº � � N v � � º v � v � º F,9�3H;c1'356P4}756#4[b(C-B]=W=W6;5B]9�X<6#\D?8X<7H9�X � º � % � NH� ( �¶� % � NH� ( � ­�� º � % � N5�V_�1�:u9�\D\�N � � � Ó�� v FIeP7H9�X<6iAq6#:X<7568351�:<=W6#;wCM^%9�4#6#C � � � � v 1'_�X<75689�`HCMXE:[9�4#X�X<756#1�:E?�9�:<6�m�%1�: � � ���r� � � ��9�35; � v � �r� � v �[F%_�1�:o9�3q?�N � � � Ó � v 9�35; % ��®W1'356@49�3w4j9�\]4#K5\]9�X<686il,^H\DBD4BgX<\g? � º � % � NH�[m0�35;566#; � º � % � NH�Mº��*BD35_ + �&+ � � + / I J � � N � �GQa��� º � � �GQa��Ó %Eº � N v �GQa��� º � v �GQa��� O Q�FH9�35;>1'356pBDC�\D6;rX<1)4[751,1'CE6Z_�1�:�N � �GQa��X<7H6A�9�\DKH6 ��eP7HBD4[7�=WBD3HBD=WBD�6#C�� �H� º � � �GQa��Ó�%EºM� N��GQa�*> � � º � v �GQa�[F�9�35;h9�C �°BDC84[7H9�:[9�4iX<6#:<BD�#6;~`I? � � � �GQa�*> %Eº��GN��GQa�*>��� � v �GQa��¸®(F�1'356)zH3H;5CRN � �GQa� �

� �� / � � �

� � � � � � � � � / � � � N��GQa�o9�35;<N v �GQa� � � � � � �� � � � � � � � � / � � � N��GQa�f�+eP75BD4[7°9�:<6W=W69�CEK(:[9�`H\]6��[F9�35;{X<7HBDC�1�^5X<B]=c9�\q4}751'BD4#6uk'BgAq6C�� N � �GQa��� º � � �GQa�Ó %Eº � N v �GQa��� º � v �GQa�H�

� �� � � � � � / � � �

� � � � � � � � � / � � � � N��GQa��� º�Fq9�3H;{X<756#:<6#_�1�:<6 � º � % � NH�H�� I0J � �� � � � � � / � � �

� � � � � � � � � / � � � � N��GQa��� º O Q � v � º m�%1�:w® � � ����1'356�7H9�C!� � % " � � º � % � NH��� � º� � � � � W� � � � � � � I � I J % " º � � �� � � � � � / � � �

� � � � � � � � � / � � � � N��GQa��� ºPOMQ ��� F�eP75BD4}7�1�3H6

4#1'=W^HK(X<6#C�`I?>BD3IX<6#k�:[9�X<BD35k)BD3 %2zH:<CMXF5`,?����¯�Q�T�Q × C�X<7561�:<6=>m � 3568=c9�bq6#CPX<75684[7H9�35k'6{1'_�A�9�:<B]9�`H\D6 %Eº � � � � � �� / � � � -º�F%CE1X<7H9�X � �� � � �� 9�3H;�1�3H6@zH3H;5C I � % " º � � �

� � � � � � / � � �� � � � � � � � � / � � � �

�� ��� � �GQa�[v"� � v �GQa� � I � � / � �

v � � � O&%}F�9�35;~X<756#:<6#_�1�:<6f1�3H6fzH35;5C� � % " � � º � % � N5��� � � � � � � #� � � � � � ��C � I J � N��GQa��� º � � �GQa�4O Q � º ePBDX<7�C{º)��I � � / � �

v � � �RO %p�SBSm 6�m C8º)� º ��� � � � � �[m¦oCEBD3Hk)X<756@¢�756#1�:E?>1�_�0�3qX<6#:E^�1'\]9�X<BD1'3aFH1�3H684j9�3yBD=)^H:<1Aq6ZX<7568no� ��-R�� � �(L �8R��T�¤¶BD356jJIKH9�\DBgXs?%ºjF%eP75BD4[7y9�CECE6i:EX<C

X<7H9�XuX<756o�-R����Q��%�~XE:[9�3HCE_�1�:<=!=r9�^HC&� � � � ! ��BD3IX<1 � �� � � ! ��BD_-��(+*#(/­,F59�35;rX<75BDC2BD=)^H:<1Aq6=W6#3IXuK5CE6C2N�R����%T���µCM^%9�4#6C#m�0�35;566#; ��@�� �q�H� I�� 1 @��GQa� ( " º 7 � ��� � � OMQ)k'BgAq6C�BD=W=c6#;5B�9�X<6\g?E� � ��@ � � ((� � @ � � v F�eP756i:<6�� � � � � � ;H6#351�X<6#C�X<756'���351�:<=>tH1'3�X<756Z1�X<756i:P7H9�35;$� � ��@ � � º � � � @ � � º 9�3H;�X<756i:<6_�1�:<6pX<756ZBD3qX<6#:E^�1'\]9�X<BD1'3r^5:<1�^�6#:EXs?�9�CECE6i:EX<C�X<7H9�X�X<756Z�-R����Q��%�XE:[9�35CE_�1�:<=·=c9�^HC � ��v� �)!Z� � �uº�� �)!p��� � � � BD3qX<1 � � �� �)!Z� � �uº�� �"!p��� � � � m�¢�756rB]=)^�1�:EX[9�3qX{:<6BgX<6i:[9�X<BD1'3�X<7561�:<6#= �S1'_GI9�4JIK56CMLON�1'K5BDCoN�QSR�TVUf9�35;�GI9'9�b>±d���%� ���a�2ePBD\]\ CE7H1eØX<7%9�X|X<756CE6@CM^%9�4#6#Cp9�:<6{B]3wX<756@_�9�=WB]\g?�1'_VNVR�����T���µcCM^%9�4#6C#F9�35;wX<7568:<6CEK5\gXPePBD\D\aX<7H6#3w`�68X<7H9�X|_�1�:f� � * �*­c9�35;h� ( 5 ( , X<756{��R����Q����°XE:[9�35CE_�1�:<= =r9�^%C ��� � 8 � �"!p�2B]3qX<1���� � 8 � �"!p�[F%9�35;wBD3�^%9�:EX<BD4K5\]9�:�=c9�^HC ���5� �)!Z�uBD3qX<1 ���� � �5� �)!Z�[F5eP75BD4[7wBDCP9)CEK(`%CM^%9�4#681'_ ���� �� �"!p�d`�649�K5CE6'* �+*��Smv � \D6#kwÑfm ���U�R �uF�Ú|K5CECEB]9�3°=r9�X<7H6#=c9�X<BD4B]9�3amrno6We21�:Eb(C@9�X8X<756��,� �%��¥5R �*0�35CMX<BgX<K5X<6W1'_2�~9�X<756=c9�X<BD4#C81'_uX<7H6

ÚPK5CECEB�9�3w�o4j9�;H6#={?�1�_-�I4BD6#3546#C#FH��1'CE4#1je8mÑp\]9�;5B]=WBg:��o3H;(:<6#6#A,B]4[7��I� ����¥(R �2F5ÚPKHCECEB]9�3>=r9�X<756#=r9�X<B]4#B]9�3aFa� �'Ò�Õ =��j�'­�Ò,m

º �8R��TV¤�7H9�;W^H:<1Aq6;cX<756P:<6#CEK5\DXdeP7563�* �%BDCu9�3r6#A'63cB]3qX<6#k'6i:F,9�35;Yno� VU��-R�� � �c7H9�;c^5:<1jAq6#;cX<756|k'6#3H6i:[9�\%49�CE6�mÔ�Õ

Page 55: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

ÚP6CEK5\gX<Co41�3H4#6i:<3HBD35kr41�3IA'1'\DK5X<BD1'3y49�3~9�\DCE1c`�6@BD=W^5:<1jAq6#;�K5CEBD35krX<756f¢�7561�:E?y1'_V0�3qX<6#:E^�1'\]9�X<BD1'3�9�3H;�N�R����%T���µCM^%9�4#6C#F�9�35;~BD3�^%9�:EX<BD4K5\]9�:oX<756c�(RV¯ R�¥,�%�/BD=@`�6#;5;HBD35krX<756#1�:<6= 49�3�`�6fB]=)^5:<1jAq6#;aF 9�Cp351�X<BD46#;�`I?�GI9'9�by±d���%� ����m¢�756o4#\]9�CECEBD4j9�\%:<6#CEKH\gXuBDCuX<7H9�Xu_�1�:p�)(+* � & 1'356|7%9�C � v:� � � � ! � � � � � � � ! �VePBDX<7�* � � !"�

!"�1�: v� � � v

�> v!F(9�35;

X<75BDC�ePBD\]\q`�62BD=W^5:<1jAq6#;@_�1�:����+*�� & BD3qX<1�� v'� �5� �)!Z� � ��� � � �5� �)!Z�[F�eP75B]4[7fBDCVBD354\DK5;56;@BD3���� � � �)!Z�[mV¢�75621�:<BDk'BD3H9�\^5:<1,1'_H1�_a�(RV¯ R�¥,�H�yK5CE6;)9|41'3qAq1�\]K(X<BD1'3@_�1�:<=@K5\]9 � � � � ������ � � ������ � _�1�:�9�3f6\D6#=c6#3qX[9�:E?8CE1'\DK(X<BD1'3 � 1'_ ?YFq9�35;)9�CV1�3H649�3yX[9�b'6 � � � 1�

�� 1 � _�1�: & �W9�35; � ��C º \D1'kH� � QH� ��_�1�: & �Ø­,F�1�3H68z%35;5CPX<7%9�X ���

��� � � � ! � �� �)!Z�[mP¢�1'k'6iX<7H6i:ePBgX<7cX<7H6|:<6BgX<6i:[9�X<BD1'3cX<756#1�:<6#= 9�35;�9@;5K%9�\]BgXs?WX<756#1�:<6#= �O9�\DCE1@1'_ GI9�4JIK56CMLON�1'K5BDC�N�QSR�TVUZ9�35;�GI9�9�br±-����� ��� �[FIeP75BD4}79�CECE6i:EX<C � X<7%9�X'� ! � � � ! ��BDCuX<756o;5KH9�\�1'_ � ! � v � � ! �[F,1'356ozH35;5CuX<7%9�X�_�1�:p���+* � & 9�3H;~��( 5 (-,/F(4#1'3qAq1'\DK(X<B]1�31'_���� � 8 � �)!Z�d`I? � ! � �� �)!Z�uk'BgAq6#CP9f:<6#CEKH\gXPBD3 ��� � � 8 � �)!Z�[m

¦o35_�1�:EX<K53H9�X<6#\D?'F'X<75BDCd9�:<k�KH=W6#3IXd;51,6#C-3H1�Xdk'BgAq6o�(RV¯ R�¥,�H��BD=@`�6#;H;5BD35kpX<7561�:<6=�_�1�: * �Ø��FI1�:dX<756�B]=)^5:<1jAq6#=c6#3qXX<7H9�X�� v:�¹v � � ! �VBDCd4#1'3qX<B]3IK51'K5CE\g?WB]={`�6;5;56#;cB]3$� v � � v � � ! �[FIeP75BD4}7cBDCdBD3H;56#6;WXE:<K56�m-¢�756P:<69�CE1'3rBDC-X<7H9�Xu4#1'3qAq1'\DK(X<B]1�31'_���vj� �)!Z��`I?�9�3I?rN�R�����T���µfCM^%9�4#6��,+ � -� �)!Z�-k'BgAq6#C�9{:<6#CEK5\gX�BD3��,+ � -� �)!Z�d9�35;�351�X2`a6iXEX<6i:F59�C21'356p4j9�3�9�^5^5:<1q9�4[7X<7568£pQ�����²~=r9�CECP9�X|®)`I?>9f`�1'K535;H6#;wCE6JIK56354#68BD3 ��vj� �)!Z�[m

¢�75B]C2B]C2CE1'=c6iX<75BD3Hk@X<7H9�X�1'356ZCE751'K5\D;�`�6Z9e�9�:<6p1'_sF,X<7H9�X�;5Bg a6i:<6#3IX2e�9?,C�1'_�K5CEBD35kfX<756Z¢�7561�:E?r1'_ 0�3IX<6i:E^�1'\]9�X<BD1'3=c9?�\D6j9�;�X<1):<6#CEK5\DX<C�BD3>;5Bg a6i:<6#3IX�BD3qX<6#:E^�1'\]9�X<BD1'3�CM^%9�4#6C#FHK5CEKH9�\D\g?�;5BD �6i:<BD3Hkf1'35\g?�BD3�X<7568CE6#4#1'35;>^%9�:[9�=W6#X<6i:m

¢�756YK5CEK%9�\2CE49�\DB]35k�9�:<k�KH=W6#3IX<C#FV_�1�:f6#l(9�=W^H\D6�F�9�:<6YBD35CE6#3HCEBgX<BgAq6WX<1yX<756rCE64#1'35;h^%9�:[9�=W6iX<6#:f_�1�:@X<756YN�R����%T���µCM^%9�4#6C#F�9�35;y49�35351�XP`�6{K5CE6;>X<1c4[7H6#4<b�X<7H9�Xo9Wk'BgAq6#3>:<6#CEKH\gXPB]C�1�^5X<BD=r9�\Smd�H1�:|6#l59�=W^H\D6'FHBD_ � � ��vj� �)!p��Ó � >� �)!p�9�35;�_�1�: � �x®r\D6iX ��`�6@;56izH3H6#;y`I? �)�GQa�8� � � �)Qa��_�1�:Q � � ! F�X<7563�9�3q?y;564#1'=)^a1�CEBDX<BD1'3w1'_ � � N � Ó$N v ePBgX<7N � � ��vj� �)!Z�89�35; N v � ��>� �)!Z�Zk'BgAq6#C)9y;564#1'=)^�1'CEBgX<B]1�3 �+� � � Ó � v ePBgX<7����'�GQa��� N ��� �)Qa�Z_�1�:WQ � �)!9�35;E"�� � � ­,md¢�756#3w1'35687H9�CW� � � � � � � / � � 1 � � � �H� " !W� � N � � � � / � � 1 � 9�35;$� � � v � � � �*� � 1 � � � � N v � � � �*� � 1 � F%9�35;>X<756i:<6_�1�:<6�¶� % � �{��� � � � " ! �¶� % � � � ! � � �[Fa_�:<1�= eP7HBD4[7~1'356f;56;5K54#6CoX<7H9�XE� � � � � �

� � � � 1 � � � � � " ! � ��� � � � � �� � � � 1 � FaCE1YX<7%9�XoX<7H6^%9�:[9�=W6#X<6i: 5 ;51,6#C�3H1�X|9�^5^a69�:P6#l,^H\DBD4BgX<\g?cBD3�X<756Ze�9?YX<7568351�:<=�4[7%9�3Hk�6C#m

� ¢�756y:<6#CEKH\gXY1'_pGI9�4JIK56CMLON�1'K5BDCYN�QSR�T�U�9�35;�GI9'9�b�±-����� ��� BDCcA'9�\DBD;�_�1�:Yk'6#356#:[9�\� ���T���²a³ØCM^%9�46#C#F|9�35;�X<7H6^%9�:EX<BD4K5\]9�:�:<6CEK5\gX�_�1�:PN�R����%T���µWCM^%9�4#6#CP=c9?�7H9Aq6Z`�6#63�b(351jeP3>`�6#_�1�:<6�m

Ô'Ô

Page 56: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®�Õ5m�1°6#;53H6#CE;H9?���9�:<4}7°�Ô,F�­�®'®'®(m�o351�X<756#:cBD3IX<6i:E^�1'\]9�X<BD1'3�=c6iX<751,;�BDCWX<756 D LO=W6iX<7H1I; m�0sXcBDCcBD3�CE1�=c6>e�9#?�9�;5KH9�\P=W6#X<751,;�4#1'=)^%9�:<6;�X<1�X<7H6

�yLO=W6#X<751,;am¢�756 �yLO=c6iX<751,;¶BDC@X<7H6Y3H9�X<K(:[9�\u:<6CEK5\gX)1'_�B]3qAq6#CMX<BDkq9�X<BD1'35C{eP7HBD4[7¶1�:<BDk'BD3%9�X<6;hB]3�JIK56#CMX<BD1'35C)1'_�XE:[9�46#C#§rBD_ ���

��� � �O® � ,�t � � ��9�35; � � � ��� / �O® � ,/t � v ��ePBgX<7¶��( * � � * v ( ,/F�X<7H6#3 �!� C � �3! ® � � #St � � Ó � v �P9�35;�X<756i:<6_�1�:<6 � �O®I�6il(BDCMX<Cf9�35;hX<7H6YJIK56CMX<BD1'3hBDC{X<1~4[7H9�:[9�4iX<6#:<BD�#6cX<7H6rCM^�9�46Y1�_PCEK54}7hA�9�\DK56Cf9�XW®h�+XE:[9�4#6C}�[m��oCf1�3H6Y49�3¶4[7H9�35k'6 � � %<�BD3 � � % � �@ePBgX<7 � � ®h9�35;/351�Xr4[7H9�35k'6 � �O®I�[F21'356wX<756#3/;5BDCE4#1jAq6i:<Cr3H9�X<K(:[9�\D\g?�X<7H9�Xr1�3H6w49�3/4#1'35CEB];56i:rCM^%9�46#Cr1�__�K5354iX<B]1�3HC{CEK54[7°X<7%9�X % � � �5� ��� � �O® � ,�t � � �Z9�35;)% � / � � � ��� / �O® � ,/t � v �[F�9�35;h_�1�:fCE1�=c6cCE6#X@1'_2^%9�:[9�=c6iX<6#:<C � �O®I�6il(BDCMX<C#mP¢�756#CE6{B];569�C|=c9?>7H9Aq6{CMX[9�:EX<6#;wePBgX<7yÙd=WBD\DB]1rÐo��¤V¥IQ���� �-RuF�9�3H;w0�;51r351�X|b,3H1eØBD_V756{7H9�;yz5:<CMXoBD;56#3IX<BgzH6#;X<756ZXE:[9�4#6C�1'_�_�KH354iX<BD1'35CP_�:<1�= � v:� �5� �)!Z�u1'3y9�3w7q?,^�6i:E^H\�9�3H6Z`�6#_�1�:<6Z1�:|9�_�X<6i:�X<75B]3(b(BD35k)1'_�X<7568k'6356i:[9�\ _�:[9�=W6ie21�:Eb�F`HK(Xp46i:EX[9�BD35\D?>GI9�4jJIK56#CMLON�1�KHBDCoN�Q+R�T�Uf9�35;�GI9'9�by±d�%��� ����^�6#:<_�64iX<6#;yX<756@_�:[9�=c6ie21�:Eb�F�9�35;�X<756 �yLO=W6#X<751,;yB]CoGI9�9�b±d�%��� ��� × C2_�K(:EX<756i:�CEBD=W^H\DBgzH4j9�X<BD1'3aFqeP75BD4}7YCE751jePC2X<7H9�XuX<7H6|^5:<64#6;5BD35kf_�9�=WB]\g?)1'35\D?c;56#^�6#35;HC2K(^�1'3rXse21@^�9�:[9�=c6iX<6i:<C#m

0�_�1'356�e�9�3qX<CVX<184[7H9�:[9�4iX<6#:<BD�#62X<756�;5KH9�\DC�1'_%X<756�CM^%9�4#6C�1�`5X[9�BD356;aF'X<756#3W1�3H6�zH35;5C-6j9�CEBD\g?{X<7H9�X�X<756CE6�;5K%9�\HCM^%9�4#6#C9�:<6r3H9�X<K(:[9�\D\g?�;56iz%356#;�9�C@BD3qX<6k�:[9�\]C#FV9�35;¶1'356r4#1'35CEB];56i:<C@X<756#3¶JIK56CMX<BD1'35C@\DBgbq6WX<7H9�Xf1�_�BD;563qX<BD_�?,B]35k>eP75BD4[7¶9�:<6cX<7H66#\]6#=W63qX<C N � � � Ó � v eP75B]4[7{4j9�38`�6de�:<BgXEX<6#3{9�CHI � F�� %<�4O %aeP756i:<6�% � � F � � 8 � �O® � ,�t � � �a9�35; %

�/ F � � 8 / �O® � ,/t � v �[F_�1�:oX<7568:[9�35k'6@1'_V^%9�:[9�=c6iX<6#:<CPeP756#:<68X<756@BD3qX<6k�:[9�\�BDC|;56#zH356;amP�okq9�BD3aF�\]1I1�b(BD35kr9�X F�� % � ��CE751jePCoX<7H9�X|X<756i:<6@9�:<6@351�X

:<69�\D\D?w_�1'K(:p^%9�:[9�=W6#X<6i:<C#F�`HK(XZ1'356fBD=)^a1�:EX[9�3qXp1�`HCE6i:EA'9�X<B]1�3~BDC|X<7H9�XoX<7H6#CE6fCM^%9�4#6CZ9�:<6Y�O9�\D=c1�CMXi��X<756fC<9�=W6@X<7H9�3yX<7H61'356#CW;56#zH356;¶`I?¶XE:[9�4#6#C#F29�3H;�0@;51�351�X)b(351je BD_PÙd=WB]\DBD1�Ðo��¤�¥,Q������-R�7H9�;�BD3IA'6CMX<BDkq9�X<6#;�CEK54[7�JIK56CMX<BD1'35Cf`�6_�1�:<6X<756o`�9�CEB]4oe21�:EbY1'_�GI9�4jJIK56#CMLON�1�KHBDC�N�QSR�TVU89�35;>GI9�9�bY±d���%������mH¢�756�DaLO=c6iX<751,;�BDC2X<7563YX<7H6pCEB]=)^H\DBDzH49�X<BD1'3c`I?�GI9�9�b±d�%��� ���y1'_�X<756p^5:<64#6;5BD35kW_�:[9�=c6ie21�:Eb�m� &������D������ §d�H1�: F � � � � v 9�35; % ��®(FH1'356Z;56351�X<6#CDd� % � F5�H��=r9�l � � � F�� � � � %0� � F � � v �qm

¢�756@4j9�CE6 %8�¸�{41�:E:<6CM^�1'35;5C|X<1cX<756@K5CEKH9�\�351�:<= 1'3 � � �� v F�eP75BD4[7�=c9�bq6#CP`�1�X<7�BD3��E6#4#X<BD1'35CPBD3qX<1 � � 1�: � v4#1'3qX<B]3IK51'K5C|9�3H;�ePBDX<7�351�:<=cCP9�XP=W1'CMX{��m Dd� % � F(�uk'BgAq6#C�X<7H6#3w9W_�9�=WB]\g?r1'_�6jJqKHBgA�9�\D63qX�351�:<=WC�1'3�� � � v m� &������D������ §2�H1�:o®�� � ���{9�35;h�)( *+( ,�F%1�:o_�1�: � � ® � �{9�35; *�� ��F%1'356{;56iz%356#C@� � � � � v � � � � � � 9�C�X<7H68CM^%9�461'_PN � � � Ó � v eP7HBD4[7c49�3W`�6�e�:<BgXEX<63c9�C N���I � F%� %<� � �� ePBgX<7 F%� %<� � � � � v _�1�:29�\]=W1'CMXd9�\D\ % ��®{9�35;rC<9�X<B]CE_]?(BD35k% " � D � % � F�� %<��� � ��� � ® � ,�t � �� ��m � 356);56#zH356#CE� � N � � � � � � �

� BD35_ � � � %"� D-� % � F(��� � � �

� � � �� � � � � � FaX<756WBD3(zH=fK5= `�6#B]35kYX[9�b'639�=W1'35kc9�\D\a^�1'CECEBg`H\D6 FceP751'CE6841�:E:<6CM^�1'35;5BD3Hk)B]3qX<6#k�:[9�\ak'BgAq6C�N%m

�oCP6iAq6i:E?9N � � � � v 4j9�3>`�6pe�:<BgXEX<63>9�C)N�� I � �u� %<� N � �� ePBDX<7�°7H9A(BD35kW41�=W^%9�4iXPCEK(^5^�1�:EX|BD3��O® � ,��u9�35;C<9�X<BDCE_�?,B]35kWI � �u� %<� � �� �Ø��F(1�3H6o4#1'K5\D;r41'35CEBD;56#:21�X<7H6i:uA�9�\DKH6#Cu1'_ � � *�Fq`HK5XuX<756oBD3(zH=fK5=!1'_ � � % " � Dd� % � F(��� � � �

� � � �� � � � � �e21'K5\D;�`a68®)B]3YX<756CE6849�CE6#C#m�0�3H;56#6;>1'3568=c9?Y:<6i^H\�9�46 ��`I?�d� � %<�u9�35;w\D6#X �>X<635;�X<1 ,�FH9�35;>X<7568BD3(zH=fK5= X<635;5CX<1Y®WBD_ � �*®W1�:oBD_ � ��®r9�35; * �Ø��t�CEBD=cBD\]9�:<\g?�\D6#XEX<BD35k �yX<6#35;wX<1Y®WX<756{BD3(z%=@K5= X<635;5CPX<1Y®WBD_ � �Ø�81�:oBD_ � ���9�35;�* � ��m

¢�756>BD=W^�1�:EX[9�3IXf^5:<1�^�6i:EXs?¶BDC)X<756�_�1'\D\]1ePB]35k~6JIK5BgA'9�\]6#3546c:<6CEK5\gXF-eP7HBD4[7�C<9#?(C)X<7H9�Xc9�^�9�:EXc_]:<1'=·X<756>6il,XE:<6#=c649�CE6C � � ® � �ZeP756i:<6ZX<7H6ZXse21c=W6iX<7H1I;HC�K5CE6@;5Bg �6#:<6#3qX�A'9�\]K56#CP1'_�*~9�3q?,e�9?'FHX<756WD LO=W6#X<751,;>k'BgAq6C�X<7568C<9�=c68CM^%9�4#6#CX<7H9�3�X<756 �yLO=W6iX<7H1I; m� ��%"-� j�D������ §-�H1�:�® � � � �o9�35;~��( * ( ,�FIX<7H6 D LO=W6#X<751,;rk'BgAq6#CdX<756oC<9�=W6oCM^%9�4#6CuX<7H9�3rX<756 �yLO=W6#X<751,;aFIePBgX<76JIK5BDA�9�\D6#3IX�3H1�:<=cC#m�2È<ÆjÆ5#[§�N�6iX)N � � � � � � v � � � � � �

F%CE1)X<7H9�X N�� I � � � -�� � �� ePBgX<7�- " � D � - � � � -�� � � ��� � ® � , � � �� � m-�oC)N �� �¶� % � NH�uBDCP9351�:<=>F�1'356-;H6#;5K546#C�X<7H9�X �¶� % � N5� ( I � � � % � � � -j��� � �� ( I � =WBD3 � � � � � -���� � � � %T� � � � -���� � v � � �� F�`�649�K5CE6-_�1�: �6� � � � v1'356�7H9�C)X<756�;564#1'=)^�1'CEBgX<B]1�3HC � � � Ó ®!� ®@Ó � 9�3H;¶X<756#:<6#_�1�:<6 �¶� % � � �$( =WB]3 � � � � � � � � %T� � � � � v �qm �6#4j9�KHCE6�_�1�:�(� � � �� v 1'356>7H9�C � � � � � � ( Dd� - � � �@9�3H;(� � � � � v ( v� Dd� - � � �[Fu1�3H6�;56;5K546#C)X<7H9�XW=WB]3 � � � � � -���� � � � %T� � � � -���� � v � (=WB]3 �H� �

�� �4D � - � � � -j����F�9�35;�X<756i:<6_�1�:<6 % " � �¶� % � NH� ((I � =cBD3 � � �� �"�� � �� � v

"� � - " � D � - � � � -���� � �� m8¢�75B]CoBDCZ9r41'3qAq1�L\DK(X<B]1�3>^5:<1,;5K54iXo_�1�:PX<7568=fK5\gX<Bg^%\DBD49�X<BgAq6ok�:<1'K(^¶�O® � ,��dePBgX<7>X<756{no�����°=c69�CEK(:<6 �

�� F%1'_�X<7H68_�K53H4iX<BD1'3 % " � D � % � � � %<���eP75BD4}7�`�6\D1'35k'CZX<1 ��� � ® � ,/t � �� �r9�35;�X<7H6W_�K5354iX<B]1�3�=WB]3 � % " � � %}v " � �qF�eP75BD4}7�`�6\D1'35k'CpX<1 ��v � ® � ,�t � �� ��F�9�35;�X<756#:<6iL_�1�:<6�1'356P7H9�C� � % " � �¶� % � N5��� � � �

� � � �� � � � � � ( C � � % " � D-� % � � � %<�E��� � � �

� � � �� � � � � � Fq^H:<1A(BD3HkpX<7H9�X|� � � � � v � � � � � �� � � � � � v � � � �ePBgX<7>4#1'3qX<BD3,K51'K5C�B]={`�6;5;5BD35k5m

Ô'Ò

Page 57: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

0�3r1�:<;56i:uX<1Z^5:<1jAq6PX<7H6P1�^5^�1'CEBgX<6P41�3IX<BD3IK51'K5CuBD=@`�6#;H;5BD35k5Fq1'356|=fK5CMXuCMX[9�:EXu_]:<1'= N � � � � � � v � � � � 9�35;r4#1'35CMXE:<KH4iX� � %<� � � � � v CEK54[7cX<7H9�X N���I � � � %<� � �� 9�35;"% " � D � % � � � %<� � � ��� � ® � ,�t � �� � F,9�35;r_�1�:dX<7H9�XuBgXdBDCd6351'K5k'7WX<18635CEK(:<6X<7H9�X|1�3H6849�3w41�3HCMXE:<K54iX|CEK54[7�9 � C<9�X<BDCE_�?,B]35kED � % � � � %<� � ( C �¶� % � NH�u_�1�:|9�\D\�% �/®(m2¢�75BDCP_+9�4#XPB]C�XE:<K568B]3w9WCE\DBDk'7qX<\g?=W1�:<6Zk'6#356#:[9�\a41�3IX<6il,XF51'35684[751,1'CE6#C � � %<�H� ��� _�1�:*( � ��% � ( � �VvF5CE1)X<7H9�X)N���I � � � %<� � �� =c69�35C�N � � � " ��� F9�35;�X<756)`%9�CEBD4c4#1'35CMXE:<K54#X<BD1'3�BDC8CE751jeP3�BD3�X<756c_�1'\D\D1jePBD3Hk�N�6#=c=c9(m� 2649�K5CE6�N � � � � � � v � � � � ��� � � � � v � � � F�1�3H67H9�C��h� % � NH� ( C�% � 9�35;>X<756Z7I?I^a1�X<7H6#CEBDC�1'_�X<756ZN�6#=W=r9fBDC�BD3H;56#6;�C<9�X<BDCMzH6;am� & ����$ §�0�_�N � � � Ó � v C<9�X<BDCMzH6C �h� % � NH���·®@9�C�%��·®@9�35; ! � � � + �� �·®@9�C�%�� ,/F,X<7563r_�1�: � ��� X<756#:<6|6#l,B]CMX<C���6� � � �� v CEK54}7yX<7H9�X|X<756{_�K5354iX<B]1�3 � ;56#zH356;w`I? � � %<� � ��� _�1�:�( � � % � ( � �Vv|C<9�X<BDCMz%6#C N9� I � � � %<� � �� 9�35;D � % � � � %<����( C �¶� % � NH�d_�1�:|9�\]\/% ��®>�S_�1�:|9)K535BgAq6#:<C<9�\�4#1'35CMX[9�3qX C@�[m�2È<ÆjÆ5#[§�N�6#XHC � � ��F'9�35;f_�1�:�6j9�4}7 � ��� \D6#XBN � N � � � Ó#N v'� � ePBgX<7�N � � � � � � F N v'� � � � v 9�35; � � N � � � � � � Ó�( � � � N v'� � � � v (C � �¶� ( � � NH�[mV0�3>^%9�:EX<BD4K5\]9�:#� � N � � � � � � �Û®W9�C � � > , 9�35;$� � N v:� � � � v �·®c9�C � �ÖÓ),/m

N�6iX ��� � N � � � ��v >$N � � � � N v'� � >!N v'� � �Vv F CE1�X<7%9�X ���!� � � �� v F�9�3H;�_�1�: � ��">1'356)7H9�C � 7 Ó ����� Ó � ���N � � � ��v >WN � � 7 � N >WN v:� � ��v >WN � � 7 FqeP75BD4[7W41�3IA'6#:<k'6#C�X<1 N@B]3 � � Ó � v 9�C���� > ,Û9�3H;�" ��Ó�,�`�649�K5CE6�N v'� � ��v �·®BD3 � v 9�CR" � Ó), 9�35; N � � 7 � ®yBD3 � � 9�C � � > ,�m �6#49�K5CE6 �¶� % � NH�ZBDC@351'35;564i:<6j9�CEB]35k~BD3)%{9�35; ! � � � + �� BDC351'35BD3H4i:<69�CEBD3HkfBD3�%}F51'356p7H9�C �¶� ( � � NH� ( �¶� % � NH� ( �¶� ( � �Vv � NH�u9�35; �

�% � / �¶� ( � �Vv � NH� ( �¶� % � NH� (

��% �¶� ( � � N5�d_�1�:( � � % � ( � �Vv#m � 356W7%9�C � � ��� � � � ( � � N � � � ��v � � � Ó � � N � � � � � � ( C � �¶� ( � ��v � N5�VÓ C � �¶� ( � � N5� ( C � �E�PÓ (�� �¶� % � NH�[F9�35;�% � � ��� � � v (�%T� � N v:� � ��v � � v Ó %T� � N v:� � � � v ( C �

��% � / �h� ( � ��v � NH�,Ó6C � �

�% �¶� ( � � NH� ( C � �E� Ó (�� �¶� % � NH�[FI9�35;cX<756i:<6_�1�:<6D � % � � � %<����( C � �E�dÓ (�� �¶� % � NH�[m

Ô�(

Page 58: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®qÔ,m-�5:<BD;H9?��~9�:<4[7°�)(IF�­�®'®�®(m� &������D������ §��H1�:�®�( � ( ��F,1'356|C<9?(C-X<7H9�X�98351�:<=W6;cCM^%9�4#6 �xB]Cd1'_�4\]9�CEC � � � ��B]_ � � � v � � ��� � � � � v � � � �� ! FX<7H9�X � B]Co1�_d4#\]9�CEC��w� � ��BD_P� � � � � v � � �¹v � �

��� ��� � Ó � v F�9�3H;yX<7%9�X � BDCp1�_-4#\�9�CEC =~� � ��BD_P� � � � � v � � �¹v � ���� �

� � � � � v � � � �� ! m� _-41'K(:<CE6�F�_�1�:8®�� � �¸�fX<756)BD35;5B]4#6#C�D�9�35; � =c9?w`�6);(:<1�^5^�6;�9�CoX<7H6@Xse21>BD3qX<6#:E^�1'\]9�X<BD1'3y=c6iX<751,;5Cpk'BgAq6

X<756oC<9�=W6pCM^%9�4#6C#F,`HK5X�_�1�:2X<7H6o6il,XE:<6#=c6PA�9�\DKH6#C � �*® � �|X<756o1'35\D?cBD3qX<6#:E^�1'\]9�X<BD1'3cCM^�9�46#C2X<7H9�X2e26oK5CE6p4#1�:E:<6CM^�1'35;rX<1* �Ø�Z_�1�:�X<756#D LO=W6iX<7H1I;w1�: * ��, _�1�:�X<756 �wLO=c6iX<751,;am

¢�756W:<6BgX<6i:[9�X<BD1'3�X<756#1�:<6= ePBD\D\�CMX[9�X<6WX<7H9�X{BD_ � BDC{1'_24#\�9�CECR=~� � � �Z9�35; v BDC81'_24#\]9�CECR=~� � v �oePBgX<7 � � � � v FX<7563~_�1�:{® � � �¸�)9�35;��$( * ( , 1'356)7H9�C)� � � v � � � � ��� � � � � v � � � � ePBgX<7 � � �E� > � � � � Ó � � v m@¢�756i:<6_�1�:<6BD_ � � � � � � � v � � � � � � 9�35; v � � � � � � v � � / � � / F�X<756WBD3qX<6#:E^�1'\]9�X<BD1'3~CM^%9�4#6w� � � v � � � � BDCZX<756WC<9�=c6�F�eP7H9�X<6iAq6#:pX<7H6^5:<64#BDCE68A�9�\DK56C * � � * v 9�:<6'F%BD_ � � � � v F%`HK(X|X<756{BD3qX<6#:E^�1'\]9�X<BD1'3�CM^%9�46#C|;H1c;56i^�635;yK(^�1'3 * � � * v B]3�X<7H684j9�CE6 � � � � v F9�35;wBD3�X<7H9�XP49�CE6ZX<756i?�=c9?Y`�6Z356#e�CM^%9�4#6C#FHBOm 6'm�351�XPBD354\DK5;56;�B]3�X<756Z_+9�=WBD\D?rBD35;56#l,6;�`I? � � *�m� ��%"-� j�D������ §P��351�:<=c6#;�CM^%9�4#6 � BDCo1'_�4#\]9�CEC � � � ��BD_-9�35;�1'35\D?wBD_� � �� v � � 9�35;yX<7H6i:<6@6il(BDCMX<C C¸CEK54}7�X<7H9�X�¶� % � NH� ( C�% � � � N � � � _�1�:|9�\D\�% ��®c9�35;w9�\D\ N � �cm��351�:<=W6#;�CM^�9�46 � BDCZ1'_u4#\]9�CEC��y� � �|BD_u9�35;�1'35\g?~BD_ � � � � Ó � v 9�35;�X<756#:<6)6il(BDCMX<C�C CEK54}7�X<7H9�X9� � N � � � (C�% " � Dd� % � NH��_�1�:u9�\D\ % ��®89�35;r9�\D\�N � � � � v FI1�:dBD_a9�35;W1'35\g?)B]_%X<756i:<6P6il(BDCMX<C C CEK54[7WX<7%9�XR� � N � � � ( C � � N � � v " �� � � N � � � v_�1�:|9�\D\ N � � � � v m�2È<ÆjÆ5#[§ �oC�1�3H6-=@K5CMX�7H9Aq6 � � N � � � � � � � / � � �

��� � ( C � � N � � � _�1�:�9�\D\ N � �cF�9�35; � � N � � � � � � � / � � �

��� � � � � %"� �h� % � NH��� � � � � � � � FX<75684#1'35;5BgX<B]1�3>BDC�X<7568C<9�=W6pX<7%9�3$�h� % � NH� ( C�% � � � N � � � _�1�:|9�\D\/% ��®(m� 356r=@KHCMX@7H9Aq6 � � N � � � ( C � � N � � � � � � � / � � � / � � _�1�:)9�\]\*N � �cFV9�35;°X<756r3564#6#CEC<9�:E?�41�3H;5BgX<BD1'3°_�1'\D\]1ePC{_�:<1�= X<7H6

_+9�4#XfX<7%9�XW_�1�: N � � � � v 1'356>7H9�C6� � N � � � � � � � / � � � / � � ( %"� Dd� % � NH�8_�1�:r9�\]\�% � ®(mh0�35;566#;�B]_ � � ��v � ® � ,�t � �� �9�35; I � �d� %<� � �� � ��F�X<7H6#3�6iAq6#:E?,N � � � � v 49�3/`�6�e�:<BgXEX<6#3*9�CEN � I � � � %<� � �� ePBgX<7 � � %<� � �d� %<� N�_�1�:

%�*®(Fa9�3H;y1'356@7H9�C�� � N � � � � � � � / � � � / � � (�I� % " � � �d� %<��� D-� % � NH� � �� mP�H1�: % � �*®(F�1'356{X[9�bq6#Co9cCE6JIK56354#6 � � 41'3qAq6i:<k'BD35k

X<1 % � � � � �S_�1�:�6#l59�=W^H\D6 � � � %<��� � BD_ % � � % � � ��v� % � 9�35; � � � %<���֮h1�X<756i:EePB]CE6��[FP9�35;*1'356~1�`5X[9�BD35C�9�XrX<7H6\DBD=cBgX�� � N � � � � � � � / � � � / � � ( %

"�� Dd� % � � NH�[mfnp9A(BD35k>CE751jeP3�X<7H9�X9� � N � � � ( C %

"� Dd� % � N5� � C =c9�l � � � N � � � � % � � N � � v �c_�1�:{9�\D\

N � � � � v 9�35;�9�\]\ % � ®(F�1'356wX[9�b'6CWX<756w=cBD35BD=fK5=ÖBD3�%}F2eP75B]4[7�BDCr9�XEX[9�BD356;�_�1�:�%9� � � + � � �� � + � � / 9�35;�X<756i:<6_�1�:<6� � N � � � ( C % " � D-� % � NH�)_�1�:>9�\D\ N � � � � v 9�35; 9�\]\�% �»®hBDCY6jJqKHBgA�9�\D63qXcX<1�� � N � � � ( C � � N � � v " �� � � N � � � v _�1�:>9�\D\N � � � � v m ¡u1�3IA'6#:<CE6#\g?qF�9�CECEK5=W6wX<7H9�XWX<756#:<6>6#l(BDCMX<Cc9�4#1'35CMX[9�3qX9C CEKH4[7�X<7H9�X � � N � � � (+C�% " � Dd� % � NH�@_�1�:Y9�\D\N � � � � v 9�35;>9�\D\�% �/®(t(_�1�: 2 � � � � � � v � � � v � � 9�35;w9f;56#41�=W^�1'CEBgX<BD1'3 2 ��I � � � %<� � �� ePBgX<7 � � %<� � � � � v 9�35;I � % " � D � % � � � %<� � � �� � ,/F 1�3H6f7H9�C�� � 2 � � � ( I � � � � � %<��� � � � �� ((I � C�% " � D � % � � � %<� � � �� F 9�35;�X[9�b(BD35kYX<756fBD3(z%=@K5=1'3y9�\]\a;56#41�=W^�1'CEBgX<BD1'35C�1�_ 2 1�3H68;56#;5KH4#6#C�X<7H9�XW� � 2 � � � ( CE� � 2 � � � � � � � / � � � / � � m¢�756p^5:<64#6;5BD35k)1�`HCE6#:EA�9�X<BD1'3�1'_�GI9�4JIK56#CMLON�1'K5BDC�N�Q+R�T�U89�35;>GI9'9�bY±d���%� ���yBDC2Aq6#:E?YK5CE6#_�K5\SFH9�35;>=@KHCMXP1�_�X<6#3Y`a6

K5CE6;yePBDX<7yX<756{:<6BgX<6i:[9�X<BD1'3wX<7561�:<6#=>mp�%1�:p6#l(9�=W^H\D6�F��(RV¯ R�¥,�%�/CM^%9�4#6#;hv � º�� �{��B]Co351�XpBD=@`�6#;H;56#;�B]3+� �� �{�[F�`HK(XX<756PCE\DBDk'7qX<\g?fCE=r9�\]\D6i:dBD3qX<6#:E^�1'\]9�X<BD1'3fCM^%9�46 � ; v � �{� � � º � �{��� v � º%� v BDC-BD=@`�6#;H;56#;WBD3qX<1�C � � �{�[FqX<7H6�CM^%9�46P1'_�41'3qX<BD3IK51'K5C_�K5354iX<B]1�3HCdX<635;5BD35k{X<1)®@9�X ,/F,`�649�K5CE6o1'_aX<756o_+9�4iXuX<7H9�X8;°vj� �@� � C � � �{�VePBgX<7rX<756|^5:<6#4BDCE6|6CMX<BD=c9�X<6�� � � � � ���*� � � (� � � � � v � º� � � � � � � � � � � v � º� � � � � t%X<7H6#3w`I?wK5CEBD35krX<7568:<6#BDX<6i:[9�X<BD1'3�X<7561�:<6=>F�1�3H68zH35;5C|X<7%9�Xo_�1�:p® � - � vº X<756{CM^%9�4#6�; � � �@�2BDC4#1'3qX<B]3IK51'K5CE\g?�BD=@`�6#;5;H6#;>BD3�X<7568N�R�����T���µ)CM^�9�46)� � � � � � º ePBgX<7 v

� � � � � vº >�-�m¢�756#:<6#_�1�:<6�Fa1'356fCE7H1�KH\D;�`�6)9e�9�:<6fX<7H9�XZCE1�=c6@:<6#CEK5\DX<C|eP75B]4[7�9�:<6f351�XoXE:<K56f_�1�:p\]BD=WBgX<B]35kc49�CE6#C#F \DBgbq6W�(RV¯ R�¥,�H�BD=@`�6#;5;HBD35kcX<756#1�:<6#=cC#F�=r9?>`�6@1�`5X[9�BD356#;y`I?wX<756f¢�7561�:E?y1'_V0�3qX<6#:E^�1'\]9�X<BD1'3>`�649�K5CE6{X<7H6{\]BD=WBgX<B]35kW4j9�CE6@BDCp9�4iX<KH9�\D\g?XE:<K56yBD_o1'356yKHCE6#CY9�CE\DBDk'7qX<\D?�;5Bg �6#:<6#3qXrCM^%9�4#6'F�9�35;�X<756y;5Bg �6#:<6#3546w;51,6#Cr351�Xr=r9�XEX<6#:WX<7H9�Xr=fK54[7�`�649�K5CE6y1'_|X<7H6:<6#BDX<6i:[9�X<BD1'3YX<756#1�:<6=>m

0�3qX<6#:EAq6i:EX<BD35k)X<756Z1�:<;56#:P1'_�X<756ZCM^%9�46#CPBDCP9)CM^�6#4B]9�\a4j9�CE681'_�X<756Z:<6#BgX<6#:[9�X<BD1'3YX<756#1�:<6#= `HK5XP4j9�3>`�6ZCE66#3w69�CEBD\g?qm� & ����$ §�0�_ � � � v 9�35; v � � � F5X<7H6#3�� � � v � � � � �Ø� � � � � v � v " � � � m�2È<ÆjÆ5#[§�£p6#351�X<BD3Hk � � � % � NH�dX<756 �»_�K5354iX<B]1�3%9�\ K5CEBD3Hk)X<7568CM^%9�4#6#C � � v FH_�1�:|9�3q?>;56#41�=W^�1'CEBgX<BD1'39N�� N � Ó!N v ePBgX<7N � � � � 9�35; N v � � v F�1'356)7%9�C � � � % � NH� � BD35_#� � � N v � � v Ó�%T� � N � � � � � � %2B]35_ � � � N � � � � Ó v � � � N v � � v � � % � � v � � N�'F�9�35;X<756#:<6#_�1�:<6pX<75684[7H9�35k'681'_�A�9�:<B]9�`H\D6 % � v� k'BgAq6#C#� � % " � � � � % � N5��� � � �

� � � �� � � � � � � � � -�v " � �¶� - � NH��� � � �

� � � �� � � � � � mÔ �

Page 59: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®qÒ,m-��1'35;H9?��~9�:<4}7y­�®(F�­�®'®'®,m¢�756ZX<756#1�:<6=�1'_�:<6BgX<6i:[9�X<BD1'3�BDC�^5:<1jAq6#;wBD3�Xse21cCMX<6i^HC#m

� ��%"-� j�D������ §�0�_ � � ��� � � � � v � � � � 9�3H; � v ��� � � � � v � � / � ePBgX<7 � � � � v �O9�35;Y4#1'3qX<BD3IKH1�KHC2BD={`a6#;5;5B]35k'C}�[FqX<7563_�1�:|® � � ���89�3H;��)(+*#(-, 1'35687%9�C{� � � � � v � � � � � � � � � � v � � � � ePBDX<7�� �Ø�E��> � � � � Ó � � v m�2È<ÆjÆ5#[§ � 3568K5CE6C�X<756 �wLO=c6iX<751,;w9�35;>X<756Z_+9�4iX�X<7H9�XP_�1�: % �/®f1'35687%9�CW� � % " � � �¶� % � NH��� � � �

� � � �� � � � � � ( C � � � N � � � � _�1�:9�\D\BN � � � F�9�35;~BD3�^%9�:EX<BD4K5\]9�:p1'356f7H9�C �¶� % � K � � ((C � % � � � � K � � � � � _�1�:{9�\]\TK � � � � t�CEBD=WBD\�9�:<\D?>1'356f7H9�C �¶� % � K v � (C v % � / � � K v � � � / _�1�:|9�\D\PK v ��� v m�%1�:WN � � � Ó � v F�\D6iX ��� % � NH����BD3H_ + � � � � � / � � K � � � � � Ó %T� � K v � � � / F�X<756#3 �¶� % � NH� ( �¶� % � K � �-Ó �¶� % � K v � (C � % � � � � K � � � � � Ó�C v % � / � � K v � � � / F�9�35;/=WBD3HBD=WBD�BD35k�9�=c1'35kh9�\]\�;564#1'=)^a1�CEBDX<BD1'35CW1�_ N¶1'356y;H6#;5K546#CcX<7H9�X��¶� % � NH� (C � % � � � � � � /� � % � /

"� � � N�>9�35;¶X<7H6i:<6#_�1�:<6�% " � �¶� % � NH�$( C � % � �

"� � � � � /� � % � /

"� � � N�'m°�oC � v � � � 1'356>=c9?hK5CE6YX<7H6

4[7%9�3Hk�681'_�A'9�:<B�9�`%\D6 - � % � /"� � �+eP75B]4[7>k�BDA'6C � �� �� � v > � � � �

�� �[F%9�35;w9�C - " � � % " � � � / " � � � � % � � " � 1'356pz%35;5C�X<7H9�X% " � �¶� % � NH� (�- " � � � � � /� � - � N�'FH9�35;>X<756#:<6#_�1�:<6 N � � � � � � v � � � � B]=)^H\DB]6#C8N � � � � � � v � � � � m� ��%"-� j�D������ §�0�_�� � � � � v � � � � v �5; � 9�35;�� � � � � v � � / � v �5; v ePBgX<7 � � � � v �O9�35;�4#1'3qX<B]3IK51'K5CoBD=@`�6#;5;HBD35k'C}�[FHX<7563_�1�:|® � � ���89�3H;��)(+*#(-, 1'35687%9�C{� � � � � v � � � � � �G; � � ; v � � � � ePBDX<7�� �Ø�E��> � � � � Ó � � v m�2È<ÆjÆ5#[§ � 356�K5CE6CVX<7H6 D LO=W6#X<751,;W9�35;)X<756�_+9�4iXVX<7%9�X-_�1�:�% �/®p1'356�7H9�C� � � � � � � ( C � � � � � � � � � � � / � � � � / ( C � %

"� � Dd� % � � �

9�35;$� � � � � � / ( C v � � � � � � � � � � / � � / � / ( C v %"� / Dd� % � � �u_�1�:|9�\D\ � � � � � v m

�%1�:�N � � � � � � v � � � � 1'356w7H9�C N5� I � � � %<� � �� ePBDX<7 � � %<� � � � � v 9�35;�%"� D � % � � � %<��� � � � � ® � ,�t � �� ��m� 3H6�4}751,1�CE6C � � � v > � � FZ9�3H; `�6#4j9�KHCE6 � � %<� � � � � v � ; � !; v Fp1�3H6�4j9�3 6#CMX<BD=r9�X<6<D � � % � � � � %<� � �=c9�l � � � � � %<��� � � � � % � � � � � %<��� � � / �qFq`HK(X�9�C� � � � %<��� � � � ( C � %

"� � D � % � � � %<���|9�3H; � � � � %<��� � � / ( C v %

"� / D � % � � � %<����F,1�3H6�zH35;5CX<7H9�X|=c9�l � � � � � %<��� � � � � % � � � � � %<��� � � / � (5C�%

"� � D � % � � � %<� � FHePBgX<7 C �*=c9�l �.C � � C v �qm � 35687H9�C@� % � � " � D � � % � � � � %<� � (C�% " � D � % � � � %<���d`a6#49�K5CE6 >�� � > � � �(>8� � v > � � � � > � � �(>8�E��> � � � � > � � v �(>���m�0�_H1'3562;H6izH356C F%� % � �*� � � %<�[F'1�3H6zH35;HC�X<7H9�X�N � I � � � %<� � �� � � I � F%� %<� � �� F%9�35; % " � D � � % � F�� %<��� � ��� � ® � ,�t � �� ��FHCE7H1ePB]35k)X<7H9�X)N � �G; � � ; v � � � � m

�{%���.+.�$H��� §@0�_ � v � � � F�� � � � � v � � � �¹v � � � � � � � � v � � � � 9�35;�� � � � � v � � / �¹v � v � � � � � � v � � / � FVX<756#3h_�1�:® � � � �P9�35;w�)(+*#(-,»1'356�7%9�C|� � � v � � � � �Ø� � � � � v � � � � ePBDX<7 � �Ø�E� > � � � � Ó � � v F'ePBgX<7W6jJqKHBgA�9�\D63qX�351�:<=cC#m�oCV9�3)9�^5^H\]BD49�X<BD1'3aF�\D6#X�K5CV41�3HCEBD;56i:VX<756u\DBD=cBgX<BD35kP49�CE621'_��(RV¯ R�¥,�%�yBD={`�6;5;5BD3HkPX<7561�:<6=ØBD3 �8º�F�eP756i:<6 ;°vj� �{º�

BDC8351�X{BD={`�6;5;56;�BD3 � >� �8º�|`HK(X{356#Aq6i:EX<756\D6#CEC8_�1�:{® � -���)X<756WCM^%9�46�; � � �8º�� � ;hvj� �8º� � �uº�� �8ºj� � v " � � º BDC9�4iX<KH9�\D\D?�B]={`�6;5;56#;�BD3qX<1 � � �� �8ºj� � �uº�� �{º� � v " � � º F�eP75B]4[7/B]C � + � � � � º�� �{º�fePBgX<7 v+ � � � � v " �º `I?�X<7H6y:<6#BgX<6#:[9�X<BD1'3X<7561�:<6=>m

¢�7562:<6#CEK5\gX�_�1'\D\D1jePC�_�:<1�= X<7562_+9�4#X�X<7H9�X � � � ;~º�� �8ºj� � �uº�� �{º� � v � º%� v � � >� �8º�d�O9�35;)9�4iX<KH9�\D\D? �!� ��vj� �{º� �C � � � º �[F'X<756�CM^%9�4#6P1'_�4#1'3qX<B]3IK51'K5C-_�K53H4iX<BD1'35C-4#1'3qAq6i:<k'BD3HkpX<18®89�X&,��[m��oC�`�1�X<7 � 9�35;�; v � � º ��9�:<6�1'_�4#\�9�CECH=~�E����­q�_�1�: � � � ;�º�� �8º�|9�35;� v � �uº�� �8ºj�[F 1'356)7%9�C ; � � �8ºj�)� � ;°vj� �{ºj� � �uº�� �8ºj��� v " � � º � � � � �uº�� �8ºj��� v " � � º `I?yX<7H6:<6#BDX<6i:[9�X<BD1'3YX<756#1�:<6=>FH9�35;>X<756i:<6_�1�:<6 � � � �� �8ºj� � �dº�� �8º��� v " � � º � � + � � � � º�� �8º�[m�%1�:�- � ! º 1'356u7H9�CT; � � �)!Z� � � ��vj� �"!p� � C � � �)!p�[F�`�6#4j9�K5CE6 � � ; � � �"!p�aBD=W^H\DBD6Cd�E�qÓ � � � � � � �6� �uº�� �)!p�9�35;h9�Cr�E�|Ó � � � � " � � � º � � ! �[F�1'356r;56#;5KH4#6#C{X<7H9�X � � � � v � � ! �[m �6#4j9�KHCE6 ; º � � º � � � � � º �[F�1�3H6c;56#;5KH4#6#CX<7H9�X6� � � � � � � � � � � ( C � � Yº � � � � � � � � � Ó�C � � � � � � � � � � � _�1�:W9�\D\ � � ;�º�� �8ºj�[Fd9�3H;h`,?°:<6#CE4j9�\DBD35k5F�BOm 6'm°9�^H^H\g?(BD35kyX<7H6BD356jJIKH9�\DBgXs?fX<1 � �GQa�H� F�� �)Qa�V1'356o;H6#;5K546#CuX<7H9�X�� � � � � � � � � � � ( C � � ��� � Yº � � � � � � � � � Ó �� � � � � � � � � � � � � � F(9�35;rX[9�b(BD35k8X<7H6`�6CMX �Wk'BgAq6C � � � � � � � � � � � ( C � � � º � � � v � º� � � � � � � � � � � v � º� � � � � � tqX<7HBDC�BD=)^%\DBD6#C � � � � � � � � � � � ( C � � � � � � v � º�

� � � � � � � � � � v � º� � � � � � eP75BD4}7BDCP6JIK5BgA'9�\D6#3qX�X<1 � ;�º�� �8ºj� � �uº�� �8ºj��� v � º �¹v � � >� �8º�Z�+X<7568C<9�=W68CE49�\DB]35kc9�:<k'K5=c6#3qX|e21�:Eb(C�ePBDX<7 � �� �8ºj�u:<6i^%\]9�4#6#;`I?#� ��vj� �{º�E�[m�o351�X<756#:2e�9?cX<1@1�`5X[9�BD3YX<756oC<9�=W6o:<6#CEK5\gX�BDCuX<1f\D6iX ��� � � � � � � � � � � � F � � � � � � � � � � � � � F59�3H;�_�1�: � ��®{X<1@`�1'K535;

I � � � $ � � � � � O �p`I?�� � I � � � $ � O � � v � º � � � ���)9�35;9I � � � � � � � � � O �p`I?w� I � � � � � � ��� � � � � � º�O � � v � º � I � � � � � � � � " � � O � � v � º (� �

�F%9�35;>X<756i:<6_�1�:<6E� � � � � � � / � � � � ( C � � �ZÓ �

��{9�35;>X<756Z`�6#CMX �Yk'BgAq6#C#� � � � � � � / � � � � ( C�� � ���rm

Ô'�

Page 60: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®�(Im�1°6#;53H6#CE;H9?���9�:<4}7�­�­,F�­�®'®'®(m �6#_�1�:<6c9>k'6#356#:[9�\d¢�756#1�:E?�1'_d0�3IX<6i:E^�1'\]9�X<BD1'3�7H9�;�`a6#6#3°;56iAq6#\]1�^a6#;aF�_�1�:8eP75BD4}7�X<756cBD3qX<6i:E^a1�\�9�X<B]1�3~^H:<1�^a6i:EXs?�BDC

^5:<1jAq6#;�_�1�:c\DB]3569�:c4#1'3qX<BD3,K51'K5Cc=c9�^5^HB]35k'C#F2CE1�=c6w351'35\DBD356j9�:WB]3qX<6i:E^�1'\]9�X<BD1'3�=W6iX<7H1I;�7%9�;�9�\D:<69�;(?¶`�6#63�K5CE6;aF�_�1�:6il59�=)^%\D6{_�1�:o^5:<1jA(BD35kcX<7H9�XoX<756@=c9�l(BD=c9�\�_�KH354iX<BD1'3�=c9�^HC ���H� �)!p��BD3qX<1rBDX<CE6#\D_V_�1�:)� � * ( ,/m|±-:<1�`%9�`H\g?w`�6#4j9�KHCE6X<75BDCZ4\]9�CECEBD49�\V^5:<1,1'_dBDCZe26#\D\Vb(351jeP3�X<1�CM^a6#4#B�9�\]BDCMX<Cp1'_unp9�:<=W1'35BD4)�o3H9�\g?(CEBDC#F�1�:{6il,^�6#:EX<CZBD3~X<756)X<7561�:E?�1'_��IBD3Hk�KH\]9�:0�3qX<6k�:[9�\DC#F5X<756#?r:[9�:<6\g?Y=W63qX<BD1'3�X<7568¢�7561�:E?Y1'_�0�3qX<6i:E^a1�\�9�X<B]1�3YeP7H6#3>X<756i?�K5CE6ZX<75BDC�Xs?,^�681�_V9�:<k'K5=W63qXm

� &������D������ §W�H1�: @ � ��v����� � �)!p�[F�X<756Y=c9�l(BD=c9�\-_�K5354#X<BD1'3 � @¶B]C{;H6izH356;°`,?)� @��GQa�R��CEK(^ 7 � �I

� � �� ���� � ! � � � !�

�� � � 7 � � F

eP756#:<6 �Y�GQ � Lq�dBDC�X<7H6p`%9�\D\ 4#63qX<6i:<6;w9�X)Q�9�3H;�ePBDX<7Y:[9�;5BDKHC�L,F%9�35;$� ���GQ � Lq���qBDC�BgX<C�Aq1�\]K5=W6'm¢�756�41�3H4#6i^HX�e�9�C�BD3IXE:<1I;HK54#6;@`I?@no��� � �~9�35;fN�Q���� ¥,�H´>R�R��2F�eP751o^H:<1Aq6;fX<7562_�1'\D\D1jePBD35k|:<6CEK5\gX�BD3f;5BD=c6#35CEB]1�3

��F(X<7H6Zk'6#356#:[9�\ 49�CE6p`a6#BD35kW;5KH6pX<1#1*Q���T����-v� ��%"-� j�D������ §80�_o� � * ( ,/F�X<7563!@ � ���5� �)!Z�|B]=)^H\DB]6#C � @ � ���H� �)!p�[F�ePBgX<7 � � � @ � � � ( C)� * ��� � @ � � � _�1�:@9�\D\@ � ���5� �)!Z�29�35;6C)� * ��� , 9�C * � ��m

¢�75B]CuePBD\D\%`a6|^5:<1jAq6#;Y`a6#\D1je8F,`HK(X2X<7H6o_+9�4iX2X<7H9�X2X<7H6|:<6CEK5\gX�BDC2351�X2XE:<K56p_�1�: * �Ø�p9�35;YX<7H9�X)C)� * ��=fK5CMX2X<635;rX<1,·9�C *cX<635;5C�X<1W��B]C�CE66#3W6j9�CEBD\g?{`I?)41'35;5BD;56#:<BD35kZ_�1�:8@WX<756�4}7H9�:[9�4#X<6i:<BDCMX<B]42_�K5354#X<BD1'3)1'_%X<756�K53HBgX�`%9�\D\SFq_�1�:-eP75BD4}7W1�3H67H9�C � @��GQa� v� v � � � � �G1 `�649�K5CE68_�1�: L�� �uÓ � QH�I1'35687%9�C I

�� � � 7 � � @��+*5��� O&* � � ���O® � ����� FH`�6#4j9�KHCE6 �Y�GQ � Lq�u41'3qX[9�BD35C�Y�O® � �j�[F�9�35;5� �Y�GQ � Lq���P� L !�� ���O® � ����� mr¢�756#:<6#_�1�:<6�� @h;51,6#C8351�X8`�6\D1'35k�X<1 ��v� �)!Z�[F�9�3H;°9�C9� � � @H� � � � , 9�C

* � ��FH1�3H68=@K5CMXP7H9Aq6�CW� *a��� ,�m¢�756oC<9�=W6o9�:<k�KH=W6#3IXuCE751jePCuX<7H9�XuBD_T@ � ��vj� �)!p��9�35; @ �*®(FIX<7563�� @�BDCd3H6iAq6i:2BD3$��vj� �)!p�-9�CuBgXuBDC-`�1'K53H;56#;

`�6\D1je°`I? ���� 1 _�1�:)� QH�i\]9�:<k�6'F�9�3H;{BD38X<75B]C�49�CE6u1�3H6d7H9�C�9�:<6#CEK5\DX�BD3qAq1'\gA(BD35k�X<756-e269�b"��v�CM^%9�46�F�eP7HBD4[7{7%9�CV9�;56#zH35BgX<B]1�39�3H9�\D1'k'1�KHC�X<1)X<7H9�X|1'_�X<7568������²�Q�T���Q��H´>QS² µWCM^%9�4#6CP_�1�: * ����F5eP7HBD4[7w4#1'BD3H4#BD;56ZePBgX<7�X<7568NVR�����T���µWCM^%9�4#6C � �.�

_�1�:�* � ��Fd`HK(XWX<756Ye26j9�b ��vWCM^%9�46>BDC)351�XWB]354#\DKH;56#;�BD3¶X<756>_+9�=cBD\g?°1'_oBD3qX<6i:E^a1�\�9�X<B]1�3¶CM^%9�4#6C)`�6#Xse26#63 ��vj� �)!p�9�35;+� >� �)!p�[Fa9�CoBDX|B]Co351�XoBD3+��vj� �)!p��Ó � >� �)!p�8�+`HK(XpBgX|e�9�Co^5:<1�`%9�`H\g?wBD3H4#\DK5;H6#;�BD3wX<7H6{1�:<BDk'BD3%9�\�;H6izH35BDX<BD1'3y1�_N�R����%T���µ@CM^�9�46#C}�[md¢�756|^H:<1I1'_�1'_ X<756|:<6CEK5\gX�_�1�:�* �Ø�oKHCE6#C�9@4#1jAq6i:<BD3Hk@\D6=W=c9(FIeP7HBD4[7YCE6#6=WC�JIK5BDX<6o4#\]9�CECEBD4j9�\OF,9�35;e�9�C�^H:<1�`�9�`%\g?Yb,3H1eP3>X<1W6#BDX<756i:�ÑpQ��(��¥,Q º 1�: ��aUQ+² R �VQ���²a³ � eP751W7H9A'6Z^H:<1Aq6;>=c1�:<6p:<6iz%356#;w4#1jAq6i:<B]35kf:<6#CEKH\gX<C#m� & ����$ §�N�6iX � `�6)9r=W69�CEK(:[9�`H\]6@CEK(`HCE6#XZ1'_ �"!8Fa4#1jAq6i:<6;�`I?�9r_+9�=cBD\g?w1'_��S4\D1'CE6#;���`%9�\D\]C � 7 � �Y� C 7 � L 7 � � � � �HFeP751'CE68:[9�;5BDB�C<9�X<BDCE_�?w® � L 7 ( � � � , _�1�:o9�\D\ � � �%m�¢�7563y_�1�:|6j9�4[7 � �*®)X<756#:<686#l(BDCMX<C|9cCEK(`H_+9�=cBD\g? D � ��CEK54[7X<7H9�X�X<756Z`%9�\D\DC � �89�:<68;5BDC��E1'BD3qX�_�1�: " � D-F%9�35;$� ��� ( �.�PÓ ��� ! � � � �

� �)�4�¹m�2È<ÆjÆ5#[§��%1�:�® �,��� �|1'356o4}751,1'CE6#C�9{z5:<CMXu`�9�\]\ � � / ePBgX<7r:[9�;HBDK5C L%� / �E�*>����,CEK(^

7 ��� L 7 F59�3H;r1'356o;HBDCE49�:<;5C�9�\D\HX<7H6`%9�\D\DC|eP7HBD4[7~BD3qX<6#:<CE6#4#X �)� / F 9�35;~1'356@:<6#^�69�X<CoX<756@^5:<1,4#6CECp9�Cp\D1'35k�9�CoX<756#:<6f9�:<6)9�3q?y`%9�\D\DCo\]6#_�Xmo0�3yX<7%9�Xpe�9?'F 1�3H67H9�C|CE6#\D64iX<6;w9WzH35BDX<681�:oBD35zH35BgX<68CEK(`%_�9�=WB]\g? DhCEK54[7yX<7H9�XPX<7568`%9�\D\DC�� �{9�:<6{;5BDC��E1�B]3qX|_�1�:�" � D°`,?>41�3HCMXE:<K54iX<BD1'3amu0�_� � � �

� �)�4� � Ó),�X<756Z:<6CEK5\gXPBDC�^5:<1jAq6#;am-0�_ � � � �� � �4� �-, 9�35;wBD_�X<756Z_+9�=WBD\D?rBDC�B]3(zH35BgX<6Z1'35687H9�C#� �)� % �"�·®W9�35;

X<756#:<6#_�1�:<6�L%� % � ®>9�C � � ,/F�9�3H;�X<756i:<6_�1�:<6W9�\D\VX<756)`%9�\D\DCp7%9#Aq6W`�6#63�;5BDCE49�:<;56;�9�X{CE1�=c6fX<BD=c6@`�649�K5CE6WBD_u1�3H67H9�C@�E��>!��� L 7 � L%� % X<756#3wX<7568`%9�\D\ � 7 =@KHCMX|7H9Aq68`�6#63w;5B]CE49�:<;56#;w`�6#_�1�:<6ZX<7H68CMX<6#^ � F%1�:|X<756Z`%9�\D\ � � % 4#1'K5\];w351�X7H9Aq6{`�66#3�CE6#\D64iX<6;y9�XoCMX<6#^ � t�B]_�X<756@_+9�=cBD\g?>BDCPzH3HBgX<68X<7563�9�\D\�X<756{`%9�\D\]CP7%9#Aq6{`�66#3�6#BgX<756#:oCE6#\D64iX<6#;y1�:p;5BDCE49�:<;56;9�_]X<6#:W9�z%35BgX<6�3IK5=@`�6i:c1�_|CMX<6#^HC#m¶�o3q?h`%9�\D\� 7 7H9�Cf`�66#3�;HBDCE49�:<;56;¶`�649�K5CE6>BgX)BD3qX<6#:<CE6#4iX<Cc9~CE6#\D64iX<6#;¶`%9�\D\ �)�

�9�35;yX<756i:<6_�1�:<6{1'356@7H9�C L%��

�E�)>!��� L 7 F%eP7HBD4[7yBD=W^H\DBD6C�X<7H9�X � 7 �9��� C*��� � L%�

�2ePBgX<7 � �Ø��Ó ºv " � t�X<756i:<6_�1�:<6� � � � � ��Y� C*� � � L%���[F5CE1WX<7H9�XW� ��� (�� ! � � � �

� � �4�,9�35;>X[9�b,B]35k��hCE=r9�\]\a1�3H6849�3>4[7H1I1'CE6 ��( �PÓ �qm� ��%"-� j�D������ §d�%1�:@ � � v � � ! �u1'35687H9�C) (�N -��0Q � � ! � � � @��GQa���M �% �"( � 1 � � � � � /� _�1�:P6#A'6#:E?�% ��®(m

vw�o1�:E`�6#:EXpÐ@m&1*Q���T����-FH�o=c6i:<BD4j9�3�=c9�X<756#=r9�X<BD4#B]9�3aF � ����Õ�L<�j�'Ò�Õ5muno6pe21�:Ebq6#;y9�X|�y0M¢pF�¡29�={`5:<B];5k'6�FH�����S��9�CMLC<9�4[7IKHCE6iXEX<C�0�35CMX<BDX<K(X<6Z1�_V¢�6#4[7H351'\D1'k�?%�[mº°ÐoBDKHCE6i^5^�6PÑpQ��(��¥,QsF'0sX[9�\DB]9�3c=c9�X<756#=r9�X<BD4#B]9�3aF%����(�Ô�L<�j� �'­,m-¢�7H6P£p6i^�9�:EX<=c6#3qXu1'_a±dK(:<6o9�35;c�|^%\DBD6#;c�~9�X<756#=r9�X<BD4#C

1'_�X<7H6Z¦o35BgAq6#:<CEBgXs?r1'_���1,;56#3%9)BDC�3H9�=c6#;�«iÐoBDK5CE6#^5^�6ZÑpQ��(��¥,Q�¬5m� �|`5:[9�= �(9�=c1'BD\D1jA(BgX<4[7 ��aUQ+² R �VQ���²a³dF5ÚPK5CECEB]9�3(LS`�1�:<3>=c9�X<756=c9�X<BD4#B�9�3 F�� ���(�iL<�j��(�®(m-no6p756#\];YX<756pÚP1'K5CE6� ���¥,¥

±-:<1'_�6CECE1�:<CE7HBg^°9�X@¡29�={`5:<BD;Hk�6'F��j�'Ô�®�L<�j�'Ô �,F CEK544#66#;5BD3Hk�N�Q�� � ¥,�H´>R�R��2Fa9�35;�756fA(BDCEBgX<6#;~¦o35BDX<6#;��qX[9�X<6Cp_]:<1'=Ö�j��Ô �X<1w�j��Ò'Ò,m

Ò�®

Page 61: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

�2È<ÆjÆ5#[§hN�6iX � � � �0Q � �)! � � � @��GQa��� � -��qF|CE1�X<7H9�X>_�1�:>6#A'6#:E? Q � � � X<756#:<6~6#l,B]CMX<CEL(�GQa� �·®¶CEK54[7 X<7H9�XI

� � �� �� � � �

�� � ! � � � !�

�� � � 7 � � � � � � -�m � 3H6)KHCE6#C8X<756W\D6#=c=c9Y_�1�:8X<756W4#1jAq6i:<BD3Hk>1'_'� � `I?�9�\D\VX<756)`%9�\D\]C � � Q � L(�GQa� � ePBgX<7 Q � � � F9�35;�X<7H6w:[9�;HBDB|9�:<6w`�1'K535;56;�`�649�K5CE6 - � ���O® � ����� L(�GQa� ! � - �� � � Q � L(�GQa��� �� � I

�� � � 7 � � � � � @��+*5��� O,* ( � � @ � � v m � 3H6

zH35;HC)9y;HBDC��M1'BD3IXf_+9�=cBD\g?�1�_�`�9�\]\DC{ePBgX<7h46#3qX<6#:<C#Q � � � � � CEK54}7hX<7H9�X � � � � ( �.�ZÓ ��� ! �� � � �� � � Q � L(�GQa� � �� (

�.�fÓ ��� ! �� � �

I� � �

� �� � � �

�� � ! � � � !� (

� � � � � 1� � � @ � � v mxN�6#XEX<BD35k �~X<635;�X<1¶®hk'BgAq6#C � � � � ( � 1� � � @H� � v 9�3H;�4[751,1'CEBD35k- � %H> ��_�1�:�� ��®W9�35;w\D6iXEX<BD3Hk �YX<6#35;�X<1c®)k'BgAq6#C�X<7568;56#CEBD:<6#;�`�1'K53H;am� H�j$5������ §R� � @ � � � v ;56351�X<6#C�X<756ZCE=r9�\D\D6#CMX�41�3HCMX[9�3IX C �®)CEK54}7>X<7H9�X))(�N - �0Q � � @��GQa���M �% � ( � � _�1�:|9�\]\/% ��®(m

� 356W7H9�C9� � @ � � v ( � � @H� � � v _�1�:@9�\D\3@ � ��vj���P�[m)£p6#CM^HBDX<6fX<756W351�X[9�X<BD1'3aF � � � � � � v BDCZ351�X@9�351�:<=>t�1�3H6);H1I6C87H9Aq6� � �@H� � � v � � �H��� � @H� � � v _�1�:Y9�\D\PCE49�\]9�:<C � F2`HK(XcX<756wXE:<B]9�35k'\D6>BD356jJIKH9�\DBgXs?¶;51,6#Cr351�XY9�\ge�9?(Cc751'\D;am��%1�:r6#l(9�=W^H\D6�F2BD__�1�: � � �O® � ���Z1'356rX[9�bq6#C @�� %<�#� v� 9�3H;!K�� %<��� vv " � Fd9�35;�(,� @�Ó K%F-CE1yX<7%9�X6� � @ � � � v � � � K � � � v � ��F-1�3H6r7%9�C(�� %<�H� v� � v " � � 9�35;�( � � %<�*� ( � �º �8_�1�:|® ��% � ��FH9�35;$� � ( � � � v ��Õ(m

�o4#X<KH9�\D\g?W1�3H6p9�\ge�9?(Cu7%9�C�� � @ v Ó<@ º � � � v ( � � � � @ v � � � v Ó � � � @ º � � � v � º _�1�:�9�\D\ @ v�� @ º FI`a6#49�K5CE6o_�1�:�® �9- ��% �-,1'356w7%9�C �0Q §�� @ v �GQa�dÓ�@ º �GQa���8 % � � �0Q §�� @ v �GQa���8 - � � �0Q §R� @ º �GQa��� %)> - �~9�3H;�X<756i:<6_�1�:<6 (�N -'� �0Qا� @ v �GQa�dÓ�@ º �GQa��� % ����(

� �� /� �� /� Ó

� �� �� �� /� " � t�X[9�b(BD35k~X<756>A�9�\DK56>1'_ ->_�1�:WeP75BD4}7�X<756>:<BDk'7qXWCEBD;56wBDCW=cBD35BD=fK5=>FdBSm 6�m

-6� � � � � @ v � � � v 9�3H;�%�> -6� � � � � @ º � � � v ePBgX<7 ��;56#zH356#;�`,? %�� �2� � � � @ v � � � v Ó � � � @ º � � � v �{k'BgAq6#CW (�N -'� �0Q §� @ v �GQa� Ó @ º �GQa���M �% ���&(

� � �� /� �� / � �

� �� �� �� /� � � � � �

� /� �� / � �

� �� �� �� / � �� m

� H�j$5������ §{0�_�@°B]C89>=W6j9�CEK5:[9�`%\D6W_�K53H4iX<BD1'3�1'3 � _�1�:8eP7HBD4[7�X<756i:<6W6#l,B]CMX<C�C CEK54}7�X<7%9�X�)(�N - �0Q � �!§3� @��GQa��� - �"( � � _�1�: - �9- � �O9�35;�- � ��®I�[F(X<7563>1'3568;56#zH356#C����� % � @��H�*BD35_ � � � � � � � K � � � v Ó)%T� � ( � � _�1�: % ��®,m

�%1�: @ � ��vj���P� Ó � ����P�u1'356Z7%9�C����� % � @a� ( �h� % � @��[F5`�6#4j9�K5CE6E� � K � � � v ((� � K � � v _�1�:P6#Aq6i:E?EK � ��vj���P�[m� & ����$ §a0�_(X<756i:<6u6il(BDCMX<CHC�CEKH4[7{X<7%9�X (�N -��0Q � �*§�� @��GQa��� �- �"( � � _�1�:�- �9- � F�X<756#3@1�3H6u7H9�C %�@ � � %<� ( ���� % � NH�_�1�:|9�\D\�% ��®,m�2È<ÆjÆ5#[§� �6#4j9�K5CE6W� @ v � ((� @ º ��9(m 6'mVBD3 ��B]=)^H\DB]6#C � � @ v � � � v ((� � @ º � � � v FI1'356P;H6#;5K546#C-X<7H9�Xu9�=W1'35k@9�\D\5X<756P;56#41�=W^�1'CEBgX<BD1'35C@$� KfÓ (�ePBgX<75� � ( � � ( � F�X<7H6c1�3H6c_�1�:8eP75B]4[7�� � K � � � v BDC{\D1je26#CMX@4#1�:E:<6CM^�1'35;5C{X<1$� K ��� � � @H� > ���

��O9�35;�� ( � �

=WB]3 � � @ � � � ���[m��%1�: ���/®2X<7H6i:<6-6il(BDCMX<C�� �®�CEK54[78X<7H9�X8� �¹� � @ � > ��� � � � � v Ó % � ( �E��Ó ��� ���� % � @��[t�BD_MKW�Ø� � @ � > ����FjX<7563

9�CuX<756o351'35B]354i:<6j9�CEB]35k{:<6j9�:E:[9�35k'6=W6#3IX21�_ KWBDCp� @ � > ����F(1'356o7%9�C%�� @ � � %<�P> ���

�( � � K � � � v ( �E��Ó ��� � ��� % � @a�P> % � m0�_�#(5@ � � %<�2BgXP=c69�35C %�� @ � � %<� >���'( � � K � � � v (x�E�uÓ ��� ���� % � @��B> % � FHeP75BD\D6ZBD_� �5@

� � %<�uBgXP=c69�35C|® ( � � K � � � v (�E��Ó ��� ���� % � @��H>)% � F 9�35;~BD3�`�1�X<7�49�CE6#CZ1�3H6f7H9�C %�@� � %<� (��E��Ó ��� ���� % � @��[Fa9�35;�\D6iXEX<BD3Hk �WX<6#35;�X<1�®rk'BgAq6#C|X<7H6;56CEBg:<6#;�`a1�KH35;am

�%1�: @ � ��vj���P� Ó � ����P�u1'356Z7%9�C�X<7563 %�@ � � %<� ( ���� % � @��&( �¶� % � @��u_�1�:|9�\D\�% ��®(m�{%���.+.�$H��� §@N�6iXf® � � � �r9�35;/� ( 5 ( ,/m�0�_2X<756#:<6c6il(BDCMX<C#C CEK54[7°X<7H9�X� (�N - �0Q � � §8� @��GQa���B -��#( � �_�1�: - � - � 9�35;�%

"� � ��� % � @a� � � 8 � ® � ,�t � �� ��X<7563 @ � ��vj���P�-Ó � ����P�@9�35;�% " � �¶� % � @�� � � 8 � ® � ,/t � �� �'FdBSm 6�m@ � ��� � 8 ���P�u_�1�: * � vv " � F%9�35;$� � % " � �¶� % � @a��� � � � � � �� � � � � � ( v

�� � % " � ���� % � @a��� � � � � � �� � � � � � m

�2È<ÆjÆ5#[§��oC ���� % � @��dBDC�3H1�3H;56#4#:<69�CEBD35k)B]3�%}F %"� � ��� % � @�� � ��8 � ® � ,/t � �� �ZBD=W^H\DBD6C�% " � � �'� % � @�� � � � ® � ,�t � �� ��FHBSm 6�m���� % � @��&( C�% � _�1�:�% ��®(F(9�35;WX<7H6i:<6#_�1�:<6 @

� � %<� ( C�% � " vd_�1�:�% ��®(F(9�35;cX<756#:<6#_�1�:<6 @ � ��vj���P�(Ó � ����P�[m � 3H6P7%9�C%}v " � @ � � %<� � � 8 � ® � ,�t � �� ��F�9�35;>`,?>no����� ��BD3H6JIKH9�\DBgXs?Y1'3568;H6#;5K546#C�% " � �¶� % � @�� � � 8 � ® � ,/t � �� �8ePBgX<7�X<7568^5:<64#BDCE66#CMX<B]=c9�X<6pCE751jeP3am

� 356p49�3�351je zH35BDCE7YX<756p^5:<1,1�_�1'_aX<7H6pno����� ��LON�Q�� � ¥,�%´�R�R���¨�1 Q���T��%��:<6#CEK5\DX2X<7H9�X�X<756o=r9�l,B]=c9�\�_�K5354iX<B]1�3=c9�^HC ���5� �"!p�uBD3IX<1)BDX<CE6#\D_�_�1�:{� �+*#(-,�F%9�35;w1�`5X[9�BD3�X<756ZC<9�=W6Z:<6#CEK5\DXP_�1�:PN�R����%T���µWCM^%9�4#6C#m� ��%"-� j�D������ §d�H1�:8���+* �-, 9�35;���( 5 (-,�F @ � ��� � 8 � �)!p�dBD=W^H\DBD6C� @ � ��� � 8 � �)!Z�d9�35; � � � @H� � �

� � � � 1�� (� 1 � �

�" v � � @H� � � � � � � 1 � m

�2È<ÆjÆ5#[§��H1�: @ � ��v� �)!Z�jÓ ��>� �)!Z� 1'35627H9�C#� ��� % � � @a� ( � ! � � �� 1 � @���_�1�: % �/®(m�0�35;566#;fX<756u=r9�l,B]=c9�\I_�K5354iX<B]1�3

BDC2CEK5`%9�;5;5BgX<BgAq6'F,`�649�K5CE6 I � � �� ���

�� ! � � � � ! � � � !�

�� � � 7 � � (

I� � �

� ���

�� ! � � � !�

�� � � 7 � � Ó I � � �

� ���� � ! � � � !�

�� � � 7 � � (�� K��GQa�%Ó)� (��GQa�d9(m 6�mHQ � �)!

9�35;�_�1�:p9�\D\ L � ®(F�1�3H6{;56#;HK54#6CPX<7H9�X �¸� K{Ó�(�� (��SK8Ó�� (~9,m 6�mPBD3 � ! m��%1�:p69�4[7y;H6#4#1'=W^�1'CEBgX<BD1'3 @ ��K{Ó�(ePBgX<7<K � ��v� �)!Z�89�35;�( � � �� �)!p�[F�1�3H6r7H9�C{X<756#3�� @ ( �+KWÓ9� (¶9�35;hX<756i:<6_�1�:<6�� @,��K � Ó�( � ePBgX<7

Ò(�

Page 62: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

® (�K � ( �SKy9�3H;~® ( ( � ( � (�9�35;~X<756i:<6_�1�:<6 ���� % � � @�� (�� � K � � � � v Ó�%T� � ( � � � (�� � �SK � � � v Ó�%T� � � ( � � (� !W� � K � � v Ó�% � � (B� � F�9�35;�X[9�b(BD3HkrX<756)BD3(zH=fK5= 1�3�9�\D\�X<756W;56#41�=W^�1'CEBgX<BD1'35C#FaX<756@:<B]k�7IXZCEBD;56)4j9�3~`�6f=r9�;56W9�CZCE=c9�\D\9�C � ! BD35_ � � � � � � � � K � � v Ó �

� 1 � � ( � � � � � ! � � �� 1 � @��'m�0�_*@ � � �.� 8'� � ! �dX<7563 % " � �¶� % � @a� � ��8 � ® � ,�t � �� �pePBDX<7 � �v

� FV1'356r;56;5K546#C{X<7H9�X�� � %"� � ��� % � � @���� � � � � � �� � � � � � ( � ! � v " � � � � % " � �¶� % � @���� � � � � � �� � � � � � F�9�35;°X<7563°X<75BDC{BD=W^H\DBD6C

� � % " � �h� % � � @a��� � � � � � � � � � � � ( � 1 � / � ��

� � % " � �¶� % � @���� � � � � � � � � � � � m

Ò'­

Page 63: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®��,m-�5:<BD;H9?��~9�:<4[7�­�Õ5F�­�®'®�®(m�o351�X<756#:c_�9�=WB]\g?h1'_o351'35\DB]3569�:cBD3qX<6#:E^�1'\]9�X<BD1'3h:<6CEK5\gX<CWBDCW`%9�CE6#;�1�3�X<756>=W6#X<751,;�1'_PXE:[9�4#6CW1'_oGI9�4JIK56#CMLON�1'K5BDC

N�QSR�T�U�9�35;)GI9'9�b@±d���%������m � 356�41�3HCEBD;56i:<CVX<7H62CM^�9�46�1'_��+e269�b(\g?%� ;5Bg �6#:<6#3qX<B�9�`%\D6u_�K5354#X<BD1'35CV_�:<1'=��O® � ,���X<1 � � Ó � vCEK54}7�X<7H9�X % � � � � ��� � �O® � ,/t � � �p9�35;% � / � � � ��� / �O® � ,�t � v �p9�3H;�_�1�:@CEK5BgX[9�`H\D6)A'9�\]K56#C81�_8� � � * � � � v�� * v �S3%9�=c6#\g?� � Ó v� � ��®p9�35;W� v Ó v

� /� ��� X<756CE6u_�K5354#X<BD1'35C�9�:<6�9�K5X<1�=r9�X<BD49�\D\g?Z4#1'3qX<B]3IK51'K5CV1'3 ! ® � � #(ePBgX<7{A�9�\DK56C�BD3 � � Ó � v 9�35;X<756{CM^%9�4#6@CM^%9�35356;w`,? � �O®I�2BDCo9�3yB]3qX<6i:E^�1'\]9�X<BD1'3>CM^�9�46�m�¦oCEBD35kWX<7H6{4[7H9�35k'6{1'_�_�K53H4iX<BD1'3�F%� -�� � � � -

� �2ePBgX<7 � ��®9�=W1'K53qX<C-X<1p:<6i^%\]9�4#6�� � � � v `I? � � � � v ;56#zH356#;W`I? � � � �� � Ó

� " v� � 9�35;�� v � �� v >

� " v� / FI1�: � � Ó v

� � � � � � � Ó v� � �9�35; � v Ó v

� /� � � � v Ó v

� /��Ó � > ��Fa9�35;�X<756#:<6#_�1�:<68X<7H6@_+9�=cBD\g?w1'_�BD3IX<6i:E^�1'\]9�X<BD1'3wCM^�9�46#Cp;H6i^�635;5CpK(^�1'3�9�Xp=c1'CMX

X<7(:<66@^%9�:[9�=W6#X<6i:<C#F `HK(X8GI9'9�b�±d���%� ���°CE751je26#;�X<7H9�XpX<7H6)4#1�:E:<6#CM^a1�3H;5BD35k�CM^%9�46)B]Cp6JIKH9�\�X<1�� � � � � v � � � � F�9�35;�1�3H649�3>4}751,1'CE6 ��CEK54}7>X<7H9�X � � Ó v� � � � 9�35; � v Ó v

� /� � FH9�35;�*yB]C�;56izH3H6#;>`I? v

�� v " �� � Ó

�� /m

¢�75B]CPePBD\D\�`�6{CE751jeP3~\]9�X<6i:F%`%K(Xp9�CECEK5=cBD35kcX<7H9�XoX<756@4[7H9�:[9�4iX<6#:<BD�9�X<BD1'3w7%9�C|`a6#6#3�1�`5X[9�BD356;aF�1'356@49�3�;56;5K54#6f9_�6ie ^5:<1�^�6#:EX<BD6#C#m

¢�756|BD3qX<6i:E^a1�\�9�X<B]1�3)^5:<1�^�6i:EXs?)_�1�:u9Z\]BD3569�:d1�^�6#:[9�X<1�:� � � � � � � � � � � � v%� v �V_�1'\D\D1jePC�B]=W=W6;5B]9�X<6#\D?'F�`�6#4j9�KHCE6F�� %<�H��� � � %<�ak'BgAq6#C % � � F � ��� � �O® � ,/t � � 9�35; % � / F � � ��� / �O® � ,�t v �[m��o4iX<K%9�\]\g?'F�9�C�e�9�C�3H1�X<B]4#6#;{`I?{GI9�4JIK56#CMLON�1'K5BDCN�QSR�T�U�FH1'356Z4j9�3>;H6#;5K54689W351'35\DBD356j9�:�BD3qX<6#:E^�1'\]9�X<BD1'3rX<7561�:<6#=>m� ��%"-� j�D������ §V0�_ � � � � v F � � v F,9�35;���B]Cd9p^�1'CECEBg`H\g?)351'35\DBD3H69�:-1�^�6#:[9�X<1�:u_]:<1'= � v BD3qX<1 v eP75B]4[7WB]C-k'\D1�`%9�\D\g?N�Q AaUj²a³�Q���µ{4#1'3qX<B]3IK51'K5C#F,BSm 6�m � � �@� � ��> �@�GF5��� � v (�� v � � � >EF � � v _�1�:29�\D\ � � F � � v F(9�3H;ceP75BD4}7c=r9�^%C � � BD3qX<1 � ePBgX<7� � �@� � ��� � � (�� � � � � � � � _�1�:p9�\D\ � � � � F%X<756#3y_�1�:o® � � �Ø�{9�35;h� ( * (�,/F/� =c9�^HC@� � � � � v � � � � BD3qX<1y� � � v � � � � F9�35;$� � �@� � ��� � � � � � � / � � � �

( CE� � � � � � � � � � / � � � �

_�1�:|9�\]\ �6� � � � � � v � � � � m�2È<ÆjÆ5#[§�£p6izH3HBD35k�F%� %<�H� � � � � %<���'F51'35687H9�C#� � F�� %�Ó�(��T> F�� %<��� � � / � � � � � � � % Ó�(�����>� � � � %<���4� � � / (�� v � � � � % Ó�(��B>� � %<��� � � / F�9�3H;y;5BgA(BD;5BD3Hk)`I?$� ( �(9�35;y\D6iXEX<B]35k�(wX<6#3H;>X<1Y®W1�3H6{;56#;HK54#6C�X<7%9�X � � F ��� %<��� � � / (9� v � � � �O� %<��� � � / _�1�:p9(m 6'm�% ��O® � ,��[m-¢�756#:<6#_�1�:<6�F%9�CP_�1�:�X<7H6Z\DBD3H69�:�49�CE6�FH1'3568;56#;HK54#6C�X<7H9�X % � � F � ��� � �O® � ,/t � �29�35; % � / F � � ��� / �O® � ,/t v �[m

0�3h�j��(�®,F%GI9�4JIK56CMLON�1'K5BDC|N�Q+R�T�U{7H9�;�9�CMb'6;w=c6ZX<1c41�3HCEBD;56i:|X<75684j9�CE6ZeP756#:<6 � BDCP1'35\g?>n �R�¥��V���h4#1'3qX<BD3,K51'K5C#FeP756#:<6P7HBDCdBD;H698;51,6#Cd3H1�Xue21�:Eb�F59�3H;W0-351�X<BD4#6;WX<7H9�Xu75BDCd:<6#CEK5\DXd4j9�3W`�6P^5:<1jAq6#;Y;5Bg:<6#4#X<\g?@`,?)X<756 �yLO=W6iX<7H1I;cB]3r9Ze�9?eP75BD4}7c49�3c`�6P6il,X<635;56#;WX<18X<756P4j9�CE6P1'_�n �R�¥��V�%��41'3qX<BD3IK51'K5Cu=c9�^5^HB]35k'Cd9�CdePBD\D\5`�6PCE751jeP3r\�9�X<6#:FI9�35;cX<75BDC-e�9�Cu9�\DCE1351�X<BD46#;�`I?�GI9'9�b�±-����� ����m

� 3562�MKHCMXP351�X<BD4#6C2X<7H9�XP_�1�:P6#Aq6i:E?�;56#41'=)^�1'CEBgX<BD1'39N��5N � Ó N v 1'35687H9�C��@�GNH�H� 2 � Ó 2�v ePBgX<7 2 � ���@�GN � �u9�35;2�v � �@�GNH�H>��@�GN � �|9�35; � � 2 � � � � ( � � � � N � � � � 9�35; � � 2jv � � v � � � �@�GN5�*>��@�GN � ��� � v ( � v � � N >!N � � � v � � v � � N v � � v F 9�35;X<756#:<6#_�1�:<6 � � % � �@�GNH����(�� � � � � � /� � � N��m

¢�756#:<6Ye26#:<6�1�X<756#:WBD3qX<6#:E^�1'\]9�X<BD1'3hX<7561�:<6=WC#Fd_�1�:W6il59�=)^%\D6r`I?�GI9'9�b¶±-����� ���/1�:)`I?��H6\DBgl ���Ra´ �V���-v)`HK(XK535;H6i:@X<756r9�CECEK5=)^HX<BD1'3�X<7H9�X{X<756c=r9�^5^HBD35kyB]C8N�Q AaU�²�³�Q���µy4#1'3qX<B]3IK51'K5C@_]:<1'= � � X<1 � 9�35;h_�:<1'= � v X<1 v m>�o39�^5^H\DB]49�X<BD1'3�4#1'35CEBD;56#:<6#;�`I?�GI9�4jJIK56#CcN�1'K5BDCWN�QSR�T�U>e�9�C)X<1�BD3qX<6i:E^a1�\�9�X<6YX<7H6�:<6#k'K5\]9�:<BgXs?¶1'_|X<756wCE1�\]K(X<BD1'3�1'_|CE1'=c6A�9�:<B]9�X<BD1'3H9�\�BD3H6JIKH9�\DBgX<BD6C#F�9�C{7H6c7H9�;°;51'356r_�1�:@\DBD356j9�:r�S6#\D\DBD^5X<BD4f1�:{^%9�:[9�`�1'\DBD4��|6JIKH9�X<BD1'35CZePBDX<7�Ùd3(:<B]4#1w����¤V�%T��aU�F`HK(XPB]3�75B]C�6il59�=)^%\D6oX<756Z=c9�^5^%BD35k)41�3HCEBD;56i:<6;>BDC�351�XPN�Q A�U�²�³�Q���µ)4#1'3qX<BD3IKH1�KHC�_�:<1'= � � X<1 � F%9�35;�02CEK(^H^�1'CE6ZX<7H9�XBgX�e�9�C�X<756Z:<69�CE1'3>_�1�:P75BDC�^%9�:EX<BD4#KH\]9�:�7I?I^a1�X<7H6#CEBDC#m

¢�756�C<9�=W6�BD;56j9�1'_ZGI9�4jJIK56#CMLON�1�KHBDCrN�QSR�T�U�9�^5^H\DB]6#CcX<1¶9�`HBD\DBD3H69�:rCE6iXEX<BD3Hk��O9�35;/0W7H9Aq6�k'6#356#:[9�\DBD�#6;�BgXcX<1¶9351'35\DB]3569�:�CE6#XEX<BD35k,�[m� ��%"-� j�D������ §VN�6iX��x`a6�`HBD\DBD3H69�:-_]:<1'= � � � Ó � v �&�c� � Ó v ��BD3qX<1 � � Ó � v FqC<9�X<BDCE_�?(BD35kpX<756P_�1'\D\]1ePB]35kp4#1'35;HBgX<BD1'35C#mB����¸BDC2`HBD\DB]3569�:24#1'3qX<B]3IK51'K5C�_�:<1'= � � � � B]3qX<1 � � FH9�35;!� � �Y� ( � � @ � ��� � � � (�� � � � ( � � � � � � � @ � � � � � _�1�:P9�\D\ ( � � � �9�35;y9�\D\T@ � � � m

BDB+� � BDCo`%BD\DBD356j9�:p41�3IX<BD3IK51'K5C8_]:<1'= � � � v BD3qX<1 � v 9�35;�`HBD\]BD3569�:Z4#1'3qX<BD3,K51'K5CZ_�:<1'= � v � � BD3IX<1 � v F�9�35;� � ��� ( � � @ v ��� � � / (�� v � � ( � � � � � � � @ v � � � / _�1�:�9�\D\ ( � � � � 9�35;Y9�\D\�@ v � v F(9�35; � � ��� ( v�� @ � ��� � � / (�� v � � ( v � � � / � � @ � � � � � _�1�:9�\D\�( v � � v 9�35;y9�\]\T@ � � � mv �%6#\DBgl 2��R ´ �V�%�dF2�o=W6#:<BD49�3�=c9�X<756=c9�X<BD4B]9�3aF�`�1�:<3*BD3 �j�'­ �,m¸no6�e21�:Eb(C>9�X>Ú ���¤V�%�VU~¦o3HBgAq6i:<CEBgXs?qFP�o6#e

2:<K535CMePBD4[b�F5�oGHm

��

Page 64: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

¢�7563wBD_�® � � � � �x�ZePBgX<7 � Ó � ���{9�35;h�"( * � 54� L ( , ePBgX<7 v7 � v�Ó v8 F � BDC�`%BD\DBD356j9�:�4#1'3qX<B]3IK51'K5C|_]:<1'=

� � � � � v � � � � ��� � � v � � � 8 BD3qX<1°� � � � � v � � � � � 7 FV9�3H;5� � �Y� ( � @���� � � � � � � / � ��� � � �

( C � � (4� � � � � � � / � � � �

� � @ � � � � � � � / � � � _�1�:f9�\D\( � � � � � � v � � � � 9�3H;w9�\D\T@ � � � � v � � � 8 m�2È<ÆjÆ5#[§8�o3q? ( � � � � � � v � � � � 49�3�`�6We�:<BgXEX<6#3�9�C (�� � �O®I�|ePBgX<7�% � � � � � %<��� � � � � � � � ® � ,/t � �� �r9�35;% � � � � � � %<��� � � / ���� � ® � ,/t � �� �'F|9�35; 9�3I?�@ � � � � v � � � 8 49�3/`a6ye�:<BDXEX<6#3*9�C�@ � F%�O®I�WePBgX<7 % � � � F%� %<��� � � � � � 8 � ® � ,/t � �� ��9�35;% � � � F ��� %<��� � � / � � 8 � ® � ,/t � �� � m � 356f;56iz%356#C �@� %<� � � � � � %<� � F�� %<� � F 9�3H;��Y� ( � @�� � �@�O®I�[t�1'356@7H9�C % � � � � � �@� %<��� � � � �

� 7 � ® � ,�t � �� �'F(9�35;Y`�6#4j9�KHCE6 �*�O� %<�*�9� � � � %<� � F �S� %<����Ó�� � � �O� %<� � F%� %<����F(1�3H6o7H9�C�% � � � � � �*�O� %<��� � � / � � 7 � ® � ,�t � �� �'F(9�35;X<756#:<6#_�1�:<6 ��� ( � @a� � � � � � � v � � � � � 7 ePBgX<7>4#1�:E:<6CM^�1'35;5BD35k)`�1'K53H;5C#m¢�756cC<9�=c6W:<6#CEK5\gX{e�9�C{6#CECE63qX<B]9�\D\g?�1�`HX[9�B]356#;�`I? �@× �o��Q�¥�F eP7H1w;56#:<BgAq6#;�^5:<6#4BDCE6W`�1'K535;HC{_�1�:{X<756c4#1'3qAq1'\DK(X<B]1�3

^5:<1,;5K54#X|9�3H9�\D1'k�1'K5C�X<1)X<751'CE6Z_�1�:�X<756p^5:<1,;5KH4iXv#m¢�756r^5:<1,;5K54iXf4#1�:E:<6#CM^�1'35;HC{X<1wX<756r4[751'BD46 � � � � � � � � � ����P�Z9�35; � v � v � � v � ��vj���P�)�+ePBgX<7� � ��� v �Ø���[F%9�35;wBD3�X<75BDC�4j9�CE6ZX<756p:<6CEK5\gX�CMX[9�X<6C�X<7H9�XPBD_u���+* �654� L �-,�ePBgX<7 v7 � v

�Ó v8 9�3H;��)( N �320�34 (-,9�35; v

�� v+ Ó v- FqX<756#3WX<756�^5:<1,;5K54#XdBDC-41'3qX<BD3IK51'K5Cd_�:<1'= ��� � +5���P�&�"� 8 � -#���P��B]3qX<1)� 7 � � ���P�u�+X<756P\DBD=cBgX<BD35kp49�CE6C�ePB]\D\,`a6;5BDCE4K5CECE6#;~eP756#3~CMX<K5;5?,B]35krX<756f;HKH9�\DCo1'_-B]3qX<6i:E^�1'\]9�X<BD1'3yCM^%9�46#C}�[t�9�CZ9c^�9�:EX<B]4#K5\]9�:p49�CE6�FaX<756@^5:<1,;5K54#XpBDCo41'3qX<BD3IK51'K5C

_�:<1�= ���(���P� � � 8 ���P�dBD3qX<1 � 7 ���P�[FH9)CEBD=W^H\D6Z4#1'35CE6jJIK56#3546Z1'_�n �R�¥��V���°BD356JIKH9�\DBDXs?'m¢�756r4#1'3qAq1'\DK(X<B]1�3�^5:<1,;5KH4iX@41�:E:<6CM^�1'35;5C{X<1wX<756r4[751'BD46 � � � � � � � � ��vj� �)!p�89�3H; � v � v � � v �

� �� �"!p�o�+ePBgX<7 � � � � v � ���[F%9�35;yBD3>X<75BDC�49�CE68X<756Z:<6#CEKH\gXPCMX[9�X<6#C�X<7H9�XPBD_2��� * � 54� -�� , ePBDX<7 v� � v�Ó v8 >��9�35;h� ( N � 2 � 4 (�, 9�35; v

�� v+ Ó v- F%X<7H6#3>X<7H6{4#1'3qAq1'\DK(X<BD1'3>^5:<1,;5K54#XoBDCP4#1'3qX<B]3IK51'K5C|_�:<1'= � � � + � � ! � � ��8 � - � � ! �BD3qX<1 � 7 � � � �)!p�[mh�oC)9�^%9�:EX<BD4K5\]9�:f4j9�CE6�F-X<756�41'3qAq1�\]K(X<BD1'3h^5:<1,;5K54#X)B]Cf41�3IX<BD3IK51'K5C)_�:<1'= ���5� �)!Z� � � 8 � �)!Z�8B]3qX<1

� � � vj� �)!p�[F%9�3>BD=)^H:<1Aq6=W6#3IX�_�:<1�= X<756 �8R��TV¤¶B]356JIKH9�\DBgXs?�t(1'356Z4j9�35351�X�X[9�bq6 N � *y9�35; 2 � 5 FHeP75BD4[7�e21'K5\D;>k'BgAq6vº � 4 ����F,`HK(Xu1'356o=c9?W4}751,1�CE6 NE +*>9�35; 2 5 CEKH4[7rX<7H9�X 4 �Ø�@�S1'356|49�3r9�4#X<KH9�\D\g?W;56izH3H6PB]3qX<6i:E^�1'\]9�X<BD1'3cCM^%9�4#6#CePBgX<7w® � � � �89�35;y® �+*#(-,�F5`HK(XP_�1�:|® � *����pX<756i?>9�:<6Z1'35\g?>JqK%9�CEBDLO351�:<=W6#;wCM^%9�4#6#C}�[m

¢�756#:<6�BDCd9�351�X<756i:�`%BD\DBD356j9�:VB]3qX<6i:E^�1'\]9�X<BD1'3@:<6CEK5\gXFq;5K56�X<1ZGI9�4JIK56CMLON�1'K5BDC�N�Q+R�T�UP9�3H;)GI9'9�bf±d���%������F�ePBgX<7cJqKHBgX<6;5Bg a6i:<6#3IXP9�CECEK5=)^HX<BD1'35C#m� ��%"-� j�D������ §VN�6iX��x`a6�`HBD\DBD3H69�:-_]:<1'= � � � Ó � v �&�c� � Ó v ��BD3qX<1 � � Ó � v FqC<9�X<BDCE_�?(BD35kpX<756P_�1'\D\]1ePB]35kp4#1'35;HBgX<BD1'35C#mB����¸BDC2`HBD\DB]3569�:24#1'3qX<B]3IK51'K5C�_�:<1'= � � � � B]3qX<1 � � FH9�35;!� � �Y� ( � � @ � ��� � � � (�� � � � ( � � � � � � � @ � � � � � _�1�:P9�\D\ ( � � � �9�35;y9�\D\T@ � � � m

BDB+� ��B]C-`HBD\]BD3569�:u4#1'3qX<B]3IK51'K5C2_]:<1'= � v � v BD3qX<1 � v F(9�35;<� � �Y� ( v�� @ v ��� � � / (�� v � � ( v � � � / � � @ v � � � / _�1�:�9�\D\ ( v � � v9�35;y9�\D\T@ v � v m¢�7563wBD_V®�� � �x�{9�3H;°�"( * � 54� L�( , ePBgX<7 v7 � v�Ó v8 >/��F � BDC�`HBD\DBD3H69�:�41�3IX<BD3IK51'K5CP_�:<1'=�� � � � � v � � � � �� � � v � � � 8 B]3qX<1>� � � � � v � � � 7 FH9�35; � � ��� ( � @a��� � � � � � � / � � � �

( C � � (4� � � � � � � / � � � �

� � @H� � � � � � � / � � � _�1�:|9�\D\ ( � � � � � � v � � � � 9�35;9�\D\B@ � � � � v � � � 8 m

0sX-CE751'K5\D;)`�6�351�X<BD4#6;@X<7%9�X�X<756i:<6�9�:<6�CEBDX<KH9�X<BD1'35C�eP756i:<6�`�1�X<7fX<756#1�:<6=WCV4j9�3f`�6�K5CE6#;)`HK5X�k'BgAq6�;5Bg �6#:<6#3qX�:<6#CEK5\gX<C#FBD3WX<756PCE6#41'35;W^%9�:[9�=W6iX<6#:|�+X<756�z5:<CMXu1'356|BDC-K5CEKH9�\D\D?fX<756�C<9�=c6�F,4#1'=)^%9�X<Bg`%\D62ePBgX<7cCE49�\DBD3Hkp^5:<1�^�6#:EX<BD6#C}�[mV�H1�:u6#l59�=W^H\D6'F9�^5^H\g?(BD3Hk�X<7HBDC8\�9�CMX{`HBD\]BD3569�:8X<7561�:<6#= X<1wX<756W^5:<1,;5K54#X{ePBgX<7� � � � � � � � � ����P�[FV9�35; � v � v � �uº����P�9�3 � v � ��v���P�[F�1'356@1'35\g?w1�`HX[9�B]35CPX<7H9�X|X<756{^5:<1,;5K54#Xo=c9�^HC ��� � +H���P� � ��� � -j���P��BD3qX<1 ��� � º%� � ���P�uePBDX<7~­ � * ��,9�35;°�)( N �320�34 ( , 9�35; v

�� v+ Ó v- >���F5eP75BD\]6pX<756Zz5:<CMXP`HBD\]BD3569�:�X<756#1�:<6#= k'BgAq6#C�X<756Z:<6CEK5\gXPBD3 ��� � º%� vj���P�p�+`HK(XPBDX

7H9�CZKHCE6#;�9�=W1�:<6fk'6356i:[9�\VB]35_�1�:<=r9�X<B]1�3 F�X<7H9�XpX<756f^5:<1,;5K54#X81�_29Y_�K5354#X<BD1'3~BD3 ��v���P�|9�35;°9Y_�K5354#X<BD1'3~BD3 � ����P�PBDC;56#zH356#;��[m

�*CEBD=WB]\]9�:�CEBgX<KH9�X<BD1'3W9�:<BDCE6C-_�1�:-X<756�CE184j9�\]\D6#;)Ú|Q��aU�µ�L�¢�³�R���Q�T�X<756#1�:<6#=wF'eP75BD4[7WCMX[9�X<6#C-X<7H9�X-BD_�9Z\DBD3H69�:-=c9�^5^HB]35kBDC{4#1'3qX<BD3H1�KHC{_]:<1'= ��� � ���P�oBD3IX<1 � 8 � ��� ���Z9�35;°_�:<1'= ��� / ���P�pBD3qX<1#� 8 / ��� ���[F�X<7563�_�1�:f® � � ���cBgX@BDC841'3qX<BD3IK51'K5C_�:<1�= ���������P�8BD3qX<1+� 8 ����� ���[F�eP756i:<6 v

���� v " �� � Ó �

� /9�35; v

8 � � v " �8 � Ó �

8 /m�ÚPQ��aU�µ�7%9�;�1�3H\g?�^5:<1jAq6#;¶X<75BDC@:<6#CEK5\DX

K535;H6i:PX<756{9�;5;5BgX<BD1'3H9�\�9�CECEK5=)^5X<B]1�3>X<7H9�X * � ( 5 � F%9�35;>X<75BDCP4#1'35;5BDX<BD1'3Ye�9�C�:<6=W1jAq6#;�`I?w¢�³VR���Q�Tum%¢�756 �yLO=W6iX<7H1I;_�1�\]\D1jePCoÚPQ��aUjµ × C89�^5^5:<1q9�4[7�9�35;�BD=W^H\DBD6C|X<7H9�XpX<756)=c9�^5^HB]35k�BDCp41�3IX<BD3IK51'K5CZ_�:<1�= ����� � �H���P�PBD3qX<1 � 8 �%� �5��� ���P_�1�:{9�3q?* � !©� � ,)#OF%`HK(XoBD_V1'356{4}751,1'CE6#C *6� * � F%X<7568CM^�9�46 � 8 � � � �'��� ���2B]CP1'35\D?>BD354#\]K5;56#;yBD3 � 8 ����� ����BD_ * � ( 5 � mP¢�³VR���Q�T × C=W6#X<751,;�41�:E:<6#CM^�1'35;5C8X<1>X<756W4#1'=)^%\D6il~BD3qX<6#:E^�1'\]9�X<BD1'3~=W6#X<751,;aF�eP75BD4}7�1'3�X<75BDC86il59�=)^H\D6WBDC8=W1�:<6)^5:<6#4BDCE6�F�`HK5X{_�1�:1�X<756i:|JIK56CMX<BD1'35C�7H9�C�X<756Z;5B]C<9�;5A�9�3qX[9�k'6Z1'_�7%9#A(BD3Hk)1'35\D?Y1'356Z^%9�:[9�=W6iX<6#:m

vwno��� � ��9�3H;>N�Q���� ¥,�H´>R�R��¶7H9Aq68CE751jeP3>X<7H9�XI �� � @8K5� � O -)( I �� @ � K � O -Z_�1�:|9�\D\/% ��®(mÒ�Õ

Page 65: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

®q�,m-��1'35;H9?��|^5:<B]\ �IF�­�®'®'®(m¢�9�\gb(BD35kc9�`�1'K(X|A'9�:<B�9�3IX<CP1'_VBD3qX<6#:E^�1'\]9�X<BD1'3>=W6#X<751,;5C#F%BDXPBDC|K5CE6_�KH\�X<1c1�`HX[9�B]3 ���5���P��9�Co9�3yBD3IX<6i:E^�1'\]9�X<BD1'3>CM^%9�46

`�6#Xse26#6#3 ��vj���P�29�35; ��>���P�[F(`HK(XPePBgX<7�X<756Z6#l59�4#X ���)351�:<=wm� H�j$5������ §¶�%1�: N � � � Ó � v F|1'356�;56iz%356#C �

� � - � NH�W9�CYX<756�BD3(zH=fK5= 1'_ � � N � � � � 9�=c1'35k¶X<756~;56#41�=W^�1'CEBgX<BD1'35CN�� N � Ó!N v ePBgX<7 N � � � � � N v � � v 9�35;$� � N v � � v (9-�m0�3°:<6#\]9�X<BD1'3�X<1>X<756>Ðo��¤V¥,Q���� �-R �S4#1'3qAq6il%�pCE6iX)9�CECE1I4B]9�X<6#;�X<1 N�F�BOm 6'mYX<756rCE6iX@1'_Z�GQ � � Q v � � ! ® � ,�� ��! ® � ,��CEK54}7wX<7H9�XPX<7H6i:<6{6#l,B]CMX<C|9W;56#41'=)^�1'CEBgX<BD1'3�NE��N � Ó$N v ePBgX<7�N � � � � � N v � � v 9�35; � � N � � � � ( Q � � � � N v � � v ( Q v FHX<7H6`�1'K535;%9�:E?>1'_�X<75B]C�CE6iXP7H9�C�X<756Z6jJqK%9�X<B]1�3 Q � � � � �GQ v�� NH�[m

� & ����$ §��%1�: � � � ��vj���P�c9�35; � v � � >���P�[F|9�35; � � * � , 1'356�7H9�C I � *�� *<> ���6- � " º � � � - � N5�4O -$�I J�� N��GQa��� �HO Qy_�1�:P6iAq6i:E?9N � ��v���P� Ó � >���P�[m�2È<ÆjÆ5#[§¶¢�756�1�^5X<B]=c9�\o;H6#4#1'=W^�1'CEBgX<BD1'3�4#1'35CEBDCMX<C>BD3*X[9�b(BD35k N v �GQa� � N��GQa�cBD_9� N��GQa���)( -�9�35;�N v �GQa� � � + � � �� + � � � � BD_� N��GQa��� � -�F�CE1wX<7H9�X � � � - � NH� � I � + � � � � � � � � N��GQa��� >9-j� � O Q�mw¢�756#3aFVK5CEBD3Hky��V¯�Q�T�Q × C{X<756#1�:<6#=wF�1'356r7H9�C�I � *�� * >���6- � " º � � � - � NH�MO - � I � *�� * >W���6- � " º � I � + � � � � � � � � N��GQa��� > -�� � O Q � O - � I J � I

� + � � � �� *�� * >c���6- � " º�� � N��GQa��� > -��4O Q ��O - �I0J�� N��GQa��� �HO Q�m

0�_ �°BDCV4#1'3qAq6#l@1'_H4#\�9�CEC3C8ºu1'3��*9�35; �o�O®I�H� ���O�O®I�H�*®|X<756#3)1'35627H9�C I � ��� ��� -j� � � � - � NH�4O -)� I J��o� � N��GQa��� �4O Q_�1�:u6iAq6i:E?WN � ��v���P�IÓ ��>���P�2�S1'356PK5CE6#C-X<756o¢��&�V¥(R��y_�1�:<=@K5\�9pePBgX<7):<6#=c9�BD3H;56i:F �o� (��H� �o�O®I�IÓ����O�O®I� (�Ó I �� � ( >%<� ��� �S� %<�4O&%<�[mr0�_ � BDC84#1'3qAq6#l~ePBgX<7 �o�O®I� � �����O®q� ��®>X<7563 ��� �dBDC{9w351'353H6#kq9�X<BgAq6W=W6j9�CEK(:<6r9�35;°1'356c=fK5CMX{KHCE6r9�I� Q���¥� U#��U v BD3qX<6#k�:[9�\Sm-¢�756#:<6#_�1�:<6'F(X<7H6ZC<9�=W6{9�^H^5:<1q9�4}7w49�3w;569�\aePBgX<7 � ��¥,Q+² µ º CM^%9�4#6C#m� & ����$ §f0�_���B]Cf9y\DB]3569�:f=c9�^5^HB]35ky_�:<1'= ��vj���P��Ó � ����P�ZBD3qX<1+��vj��� ���-Ó � ���� ���peP75BD4}7hBDCf4#1'3qX<BD3,K51'K5Cf_]:<1'=� v ���P�PBD3qX<1 � v ��� � ��ePBgX<7�3H1�:<= � v 9�3H;�eP75BD4}7~BDCo41'3qX<BD3IK51'K5CZ_]:<1'= � ���P��B]3qX<1�� ��� � ��ePBgX<7~351�:<= � F�X<7563_�1�:{���+* �-, BgXPBDC�4#1'3qX<BD3,K51'K5C�_�:<1'= ���5���P�uBD3qX<1 ���5��� ���dePBgX<7>351�:<= ( � v � �v � v � �� m�2È<ÆjÆ5#[§@�H1�:f6iAq6#:E?�;564#1'=)^�1'CEBgX<B]1�3<N ��N � Ó N v ePBgX<7�� � N v � � v ( -c1'356r7H9�Cf9w;56#41�=W^�1'CEBgX<BD1'3)�6N � �6N � Ó�� N vePBgX<7 � � �6N � � v ( � -~9�35; 9�C!� � �6N � � � ( � v � � N � � � � 1'356~;56#;HK54#6CYX<7H9�X � � � � - � �6NH�+( � v � � � - � NH�[m ¢�7563I � - � " º � � � - � �6NH�MO - ( I � - � " º � v � � � �� � � N � O -!� � v � � " v I � � �

" º � � � � � NH�4O � F�BSm 6�m(I J � �6N��GQa��� � O Q (� v � �" v I J�� N��GQa��� � OMQ�m

0�3)1�:<;56#:�X<1p;56CE4i:<Bg`�6�CE1'=W6�X<6#4}7535BD4j9�\(B]=)^5:<1jAq6#=c6#3qX<CV4#1'3546i:<35B]35kpBD=@`�6#;5;HBD35koX<7561�:<6#=WC�1'_%CM^%9�4#6#C-1'_a�(RV¯ R�¥,�H�Xs?,^�6yBD3qX<1hNVR�����T���µ°CM^%9�4#6C#FPBgXrB]CrK5CE6#_�K5\|X<1°;56#:<BgAq6y6jJqKHBgA�9�\D63qXWe�9?(CrX<1h4[7564<b�X<7H9�X�9°_�K5354#X<BD1'3�`a6#\D1'35k'CcX<1¶9N�R����%T���µWCM^%9�4#6"��� � 8 ���P�[F(ePBgX<7����+* �-, 9�35;°��( 5 (-,�m¢�756f;H6izH35BDX<BD1'3yX<7H9�Xp7%9�Cp`�6#63�KHCE6#;�e�9�CoX<7%9�X�@ � ��� � 8 ���P��=c69�35CoX<7H9�X %

" v � �� �h� % � @�� � � 8 � ® � ,�t � �� � F 9�35;9�C��¶� % � @a��49�3�`�6f6#l,^5:<6#CECE6;~BD3�X<6i:<=cCo1'_�X<7H6f351'35BD354#:<69�CEBD35kY:<69�:E:[9�35k'6#=W63qX�@� 1'_ @~`,?��h� % � @���� I

�� @� � -��4O -�F

@ � ��� � 8 ���P�uBDC�6JIK5BgA'9�\D6#3qX�X<1 %" v � �� � I �� @ � � -��4O - � � � 8 � ® � ,�t � �� �'m

�6#4j9�K5CE6�%�@ � � %<� ( I �� @ � � -��MO -�FR@ � ��� � 8 ���P�cBD=W^H\DBD6C�%}v � �M@ � � %<� � � 8 � ® � ,/t � �� � Fp1�: % � v � � � " � v � 8 � @ � � %<� �� 8 �O® � ,��[FVeP75B]4[7°e�9�C@X<756r;56#zH35BgX<B]1�3hK5CE6#;¶`I?°N�R����%T���µ5m�0sXfBDCfBD35;566#;h6JIK5BDA�9�\D6#3IX{BD_Z� � * � , `I?°no��� � � × CBD356jJIKH9�\DBgXs?'F(`HK(X�X<75B]C�;56izH3HBgX<BD1'3�4j9�3w9�\DCE1)`�6ZK5CE6;w_�1�: * �Ø��m

¢�756@351'35BD354#:<69�CEBD35kW:<6j9�:E:[9�35k'6=W6#3IXPBDC|;56#zH356;y1�3 � ® � (�N -'���P� � F�9�35;y1'356@6il,X<6#35;HCPBgXP`I?w®cBD3y1�:<;H6i:PX<1r7H9Aq6BgXP;56#zH356;w1'3°�O® � ,��[mu�H1�: �<��� F%1'35684[751,1'CE6#C N ��� ! @

� � ( ��� � @� � ( � " � #SF%CE1)X<7%9�X|1'35687H9�C) (�N - ��Q � � @��GQa��� �,N � � (( � ( )(�N - �0Q � � @��GQa��� N � �w_�1�:W6#Aq6i:E? � � � m�¢�7H6#3�1'356>7H9�C9@ � ��� � 8 ���P�{BD_|9�35;�1'35\g?¶BD_ ( � � � N ��� � 8 � � �[m

0�35;566#;w_�1�: ( � ��% � ( � ��v|1�3H687H9�C @ � � ( � ��v" �&(5@ � � %<�&( @ � � ( ���[F%9�35;>X<756#:<6#_�1�:<6 N � ��v ( @

� � %<�&( N � t5X<75B]C�BD=W^H\DBD6C��� � ( � � � N � � � � � � � ((� � %}v � �M@ � � %<��� � � � � � �� � � � � � ( ��� � ( � � � N � � � � � � � FH_�1�:�Xse21)^�1'CEBgX<BgAq6Z4#1'35CMX[9�3qX<C � � �umvy¢�751'=r9�CPGI9�3��I� Q���¥� U#��U'F%£pK(X<4}7(LS`�1�:<3w=c9�X<756#=r9�X<BD4#B]9�3aFa� �'Ô'ÒjL<� �'��Õ5m-no6pe21�:Ebq6#;wBD3w¢�1'K5\D1'K5CE6'FH�5:[9�3546�mº 1

\]9�;(?(C�

\]9e � ��¥IQS² µ5Fa±�1'\DB]CE7�=r9�X<756#=r9�X<B]4#B]9�3aF��j��®A�jL<�j�'��®(m8no6{e21�:Ebq6#;�9�XoX<756f±V1'\DBDCE7~�o4j9�;H6#={?y1'_��I4#B]6#3546#C#F±V1'�#3H9�3am

Ò'Ô

Page 66: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

0�_�_�1�:P6iAq6i:E?�� �/®)1'35687H9�C) (�N - �0Q � � @��GQa��� � � � � ,/F5X<7H6#3 @ � ��� � 8 ���P�uBD_�9�35;y1'35\g?�BD_,( � � �5�GN � > N � ��v � �� 8 � � �[mV¢�756�9�;5;56;)4#1'35;HBgX<BD1'3fBDCV6jJIK5BgA�9�\D63qX�X<1p\]BD= � � N � �*®(mVN�6iX 2 � � N � >�N � �Vv FqCE1pX<7H9�XHN � � 2 � Ó 2 � ��v Ó ����� F9�35; ( � N � � ( � 2 � Ó ( " v ( � ��v 2 � ��v Ó ����� F5CE1WX<7H9�X ( � N � B]C�1�`5X[9�BD356;>_]:<1'= ( � 2 � `I?>9)4#1'3qAq1'\DK(X<BD1'3�ePBgX<7 4 � ;56iz%356#;`I? 4 � �*®W_�1�: � ��®W9�3H; 4 � � ( � _�1�: � (�®,F%9�35;>X<756#3y9�3y9�^5^H\]BD49�X<BD1'3�1'_ �8R��T�¤ × C�BD356jJqK%9�\]BgXs?rk'BgAq6#C�X<756Z:<6#CEK5\gXm

Ò'Ò

Page 67: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K�®(m�1°6#;53H6#CE;H9?��|^H:<BD\�Ô,F�­�®�®'®(m

� 356f4j9�3~1�`5X[9�BD3�9Y3H1�3H1�^HX<BD=c9�\�BD=c;56#;H;5BD35krX<7561�:<6= � v:� �H� �)!p� � � 8 � �)!Z��`I?�;564#1'=)^�1'CEBD3Hk � ��� � > � �� � ��Ó � � � � F(eP756i:<6�� � BDC�9fCM^�6#4B]9�\�CE=W1,1�X<75BD35kfCE6JIK56#3H4#6�m-¦oCEBD35k � � � � � > � � � � ((� (B��� � KML.N4O%� � ��� � � F59�35; � > � � � � �I2� � > � ! � � � � �+*5�4O&*/1'356�7H9�C!� � � > � � � � � � � ( I � * ��� � � �+*5��� O&*R� � KML.N4O%� � ��� � � � C ��� � KML.N4O%� � ��� � � t{1'356°9�\]CE1¶7%9�C� � � � � � � � ( � � � � � � � � � � � � � ( C � � � � � � �

"! � �� m ¢�75BDCY=W69�35CYX<7H9�XYK5CEBD3Hk � � � � � � � ! �W9�35; � v � � � � ! �[F1'356Y7%9�C �h� % � � � ( C �8Ó�C�% � " ! � �� P_�1�:W9�\D\ � � ®(Fd9�35;¶X[9�b(BD3HkyX<756r`a6#CMX �>k'BgAq6#C �¶� % � � �$( C�% � � � ! � �� � F-BSm 6�m

� � � � � � � v � � � � ��8 �%� � � ! �[FHePBgX<7 � ��®c9�3H;>X<756#:<6#_�1�:<6 5 � � *�9�3H;w4[7H1I1'CEBD3Hkr9�3I? 5 � � * � 5 � �21'35687%9�C|CE751jeP3X<7H9�X2X<756#:<6o6#l(BDCMX<C 5 �+*wCEK54[7�X<7H9�X�� � � � � 8 (�� � � � � � � Ó� � � KML.N4O�� � ��� � � _�1�:�9�\D\ � � � v'� �H� �)!p�[md¢�756o^5:<64#BDCE6oA�9�\DKH6o1�_5 1�`5X[9�BD356;cBDCd351�X2BD=W^�1�:EX[9�3qXF,9�Cd\D1'35k@9�C 5 �+*�F,9�35;Y_]:<1'=¸X<75BDCd351'351�^5X<BD=r9�\%BD={`�6;5;5BD3HkZX<7561�:<6#=>FqX<7H6�`a6#CMXub(351jeP3BD=@`�6#;5;HBD35k'C2ePBD\]\�`�6Z;56#:<BgAq6#;am

¢�756)z5:<CMX{CMX<6#^�B]CpX<756WK5CEKH9�\-CE49�\DBD3Hk>9�:<k'K5=W63qXt�1'356WCE751'K5\D;�3H1�X@9�;5; � � � � � � 9�35;5� � KML.N4O�� � ��� � � eP75BD4}7�9�:<6W351�X=W6j9�CEK(:<6#;>B]3�X<756ZC<9�=c6ZK535BDXFH9�35;w1'35689�^5^%\DBD6#C�X<756ZB]356JIKH9�\DBgXs?rX<1 � � ;56#zH356#;>`I? � � �GQa�H� � � �� � m� 356@1�`5X[9�BD35C � � � ! � 8 � � � � � 8 (1�,� � � ! � � � � � � � � Ó�� � � �� ! � � � " v � � KML.N4O�� � ��� � � _�1�:Z9�\D\ �<� � v'� �H� �)!p�[m � 356f4[751,1'CE6#C

X<756y`a6#CMX � F|9�35;�1'356~;56#;HK54#6CrX<7H9�XrX<756#:<6�6il(BDCMX<C � � �O® � � #oCEK54}7/X<7%9�X!� � � � � 8 ( C5� � � � �¹v"�

� � � KML.N4O�� � ��� � �� _�1�:>9�\D\�6� � v:� � � � ! ���SB]3WX<756o4j9�CE6f��(+* � & 1'356|zH35;5C2X<7H9�X21�3H6o=@K5CMX�7H9Aq6 5 (+*

� � !"�!"�F(1�:�9{4#1'3qXE:[9�;5B]4iX<BD1'3ce21�KH\D;

`�6);56i:<BDA'6;�`I?~\D6#XEX<BD35k ��X<635;�X<1>®I�[mf¢�756f3H6ie�BD356jJqK%9�\]BgXs?wBDCp3H1e�BD3IA�9�:<B]9�3qXp`I?�CE4j9�\DBD35k5F�eP7HBD4[7�=W69�35CpX<7H9�X � BDCCEK54}7>X<7H9�X ! 8 �Ø�E��> � � !

�Ó � � ! � >�� �� ! � > � FHBSm 6'm v8 � v " �

�Ó �� � ePBgX<7 v

� � � v�> v!�+ePBDX<7 * � ��, _�1�: * � &9�35;�* � ��®)_�1�: * � & �[m

¢�756|CE6#4#1'35;rCMX<6i^YBDC�X<1@9�^5^H\g?)X<756PB]356JIKH9�\DBgXs?@X<1@9ZCE6jJIK56#3546P1'_�_�K5354#X<BD1'3�� � � � �[Fq`%K(XuBgXdBDCdBD=W^�1�:EX[9�3IX-X<184}751,1'CE69)CE6JIK56354#6ZeP75B]4[7>BDCP9�;H9�^5X<6;�X<1 � F%9�35;>BgXPK5CE6#CPX<756Z\D6#Aq6#\DC�N � B]3qXE:<1,;5K54#6;�^5:<6#A,B]1�KHCE\g?'m

� 356Y;56#zH356C � � `,? � � �O®I��� ®~9�3H; ���O�GF(�#� �rB]C�N � � � F � ��N � " v 9�35; ���O�GF(�#� ®y1�X<756#:EePBDCE6�m � 3H6r7%9�C�� �� KML.NMO � � � � � ��� �� �� � � � � � �H� � �[m��H1�:-9�3I?�L � , 1�3H627H9�C3I � 1 � � � � � ��� 7 OMQ#((� N � >RN � " v � 7 (�N -��0Q � � � �GQa��� � N � � (� N � > N � " v � 7�( � 9�35; I � 1 � � � � � ��� 7�OMQ6 (� N � > N � " v � 7� (�N - ��Q � � � �GQa���M N � " v �� (� N � > N � " v � 7 ( � " v m � 3H6o;56#;5KH4#6#CX<7H9�XR� N � >9N � " v � ( � � " v � � 8 ( C � � N � >9N � " v � ( � � � � v

"� � �� F59�35;cX<7H6i:<6#_�1�:<6�� N � >9N � " v � � ( � � � 8 � " � � � v " � � � � � ( C (�v � 8 � �� F

9�35;rK5CEBD35k v8 > v"��� �� � FI1'356|;56;5K546#C-X<7%9�XR� N � >�N � " v � ( � � � � ( Cfv � � (�v � � 8 � � F,9�35;cX<7H6i:<6#_�1�:<6#� N � >EN � " v � ( � � � � �� �5� � �[m

0�3�X<756y4j9�CE6h� ( *�� & 1'356y7H9�C)* � � !"�!"�� ,�FP9�3H;�1'356y7%9�CrCE751jeP3/X<7H9�X � � � v:� �5� �)!Z�fBD=W^H\DBD6C

�6� ��� � � �5� �)!Z�[F(X<7H6ZBD=W^5:<1jA'6=W63qX�`I?�GI9'9�b�±d���%� ���y1'_�X<756Z1�:<BDk'BD3%9�\�:<6CEK5\gXP1�_d�I6i:<k'6#B��(RV¯ R�¥,�%�um0�3~X<756W49�CE6 *<� & 1'356)7H9�C'* � ��®>9�35;~X<7H6i:<6#_�1�:<6 � N � > N � " v � � � !@� � �[m@N�6#X 2 � � N � > N � ��v Ø®(FaX<7563

9�C � X<6#35;HC8X<1!> , 1�3H6r7H9�C�N � � 2 � Ó 2 � �Vv Ó ����� Ó 2 � " v Ó N � FV9�35;¶9�3¶9�^5^H\]BD49�X<BD1'3�1'_�n �R�¥�������BD356jJIKH9�\DBgXs?k'BgAq6#C�� N � � ( � N � �Ó � � 2 � � !�Ó ����� Ó � 2 � " v � !Z�[v � !W� � >!!�¹v � ! �Fa9�3H;wX<756#:<6#_�1�:<6{_�1�:o6iAq6#:E? � �*®c1'356@49�3y4}751,1'CE6�CEK54}7�X<7%9�X8_�1�: � �(Û1'356c7H9�C9� N � ��( ��� � > �¹v � ! �Ó � N � �¹F�BSm 6�mW_�1�:{6iAq6#:E? � � ®YX<7H6i:<6W6il(BDCMX<C�CW� ���|CEK54}7�X<7H9�X� N � � ! ( ��� � �#Ó C)� �'�d_�1�:|9�\D\ � (�®,m

�%1�: � �/®f1'356Z4[751,1'CE6#C � � v�

FH9�35;�1'356Z41�=W^HK(X<6C2X<756pB]3qX<6#k�:[9�\a1'_ ( �

��� 1 1'3�X<756ZCE6iX�eP756i:<6�� � �GQa���M N � tH1'3

X<756ZCE6iX�eP756#:<6 N � ��v ((� � �GQa��� � N � F(eP75BD4}7�7%9�CP9)=W69�CEK(:<6 ( ( � ��vF51'356Z7H9�C#� � �GQa��� ( N � 9�3H;YX<756#:<6#_�1�:<6 �#� � � ! �(� � N � � ! ( � ��� � ��Ó�� C)� �'�|_�1�:f9�\D\ � (�®,FV9�35;�X<756i:<6_�1�:<6EI �

� � � � � � + � ( �

��� 1 O Q ( � �� � ( � ( � �

� � � � � � � � � � ,/m¢�756#:<6#_�1�:<6 ( �

��� 1 � ��v� ��� � �)!Z�[F5eP75B]4[7wBDC�X<7568BD=W^5:<1jAq6#=W63qX�`,?��o6#B]\ ¢�� ��VQ�T�¤����°1'_V9f:<6CEK5\gXP1'_��5:<BDX<�ZG,R�³�T¶9�35;N�1'K5B]C)�oQ����%TV¯�����¤x4#1'3546i:<35BD3Hk 2� � FVeP7HBD4[7�BD=)^5:<1jAq6#;�X<756�:<6CEK5\gXW1'_Z�I6i:<k'6B|�(RV¯ R�¥,�%�x_�1�:)X<756>\]BD=WBgX<B]35ky4j9�CE6

* � & m�%1�: * � & 1'356p7H9�C�* � ��®f9�35;YX<7H6i:<6#_�1�:<6|_�:<1'= � N � > N � " v � ( � � � � � � �5� � �[F(1'356o;H6#;5K546#CuX<7%9�X � �� � " � N � >

N � " v � � ,/F59�35;rX<756i:<6_�1�:<6|X<7H9�X�� N � � (�� _�1�:�9�\D\ � ��� m�np9A(BD35k{^5:<1jAq6#;rX<7%9�X)� v'� �H� �)!p� � � �� �"!p�[F,1'356o=fK5CMX7H9Aq69� � � � � ( C5� � � � � v " �� � � KML.N4O�� � ��� � �� F%9�35; v� >

�!�*®,F%BSm 6'm � � !

�m

� 35689�^H^H\DBD6CuX<756#3YX<7H6o:<6#CEK5\DX2X<1#F�� � � � > � F5_�1�:�eP75B]4[7�1'356p7H9�C#� � F�� � � ((� (B��� � KML.N4O%� � ��� � � 9�35;!� � KML.NMO��GF5��� � � (­�� � KML.N4O%� � ��� � � F�9�35;�1'356w1�`HX[9�B]35C)X<756w6#CMX<BD=r9�X<6 � � � � � > � � � ( C � ( �¹v " � ! � � � � � KML.N4O%� � ��� � � FuBSm 6�m�X<756w_�1'354#X<BD1'35CWBD3� v'� �5� �)!Z�29�:<6Zn �R�¥��V�%�h4#1'3qX<BD3,K51'K5C�ePBDX<7�6#lI^a1�3H6#3qX � �Ø��> !�F%9�Co�,6i:<k'6#B��(RV¯ R�¥,�H��7H9�;>^5:<1jAq6#;am

Ò�(

Page 68: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�'��m-�5:<BD;H9?��|^5:<BD\�(IF%­�®'®'®(m¢�756�1�:<BDk'BD3H9�\(=W6iX<7H1I;f1'_H^5:<1,1�_%1'_a�I6#:<k'6#B%�(RV¯ R�¥,�%�y4#1'35CEB]CMX<6#;)BD3@e�:<BgX<BD35k � � � � ������ � � ������ � _�1�:-9�3)6#\D6=W6#3IX[9�:E?CE1'\DK(X<BD1'3 �¸1�_H?YF%9�35;wBgX�B]C�351�X|9�;%9�^HX<6#;�X<1)X<756849�CE6ZeP756i:<6ZX<756Z;H6i:<BgA'9�X<BDA'6C�9�:<68BD3>;5BD �6i:<63qX�CM^%9�46#C#m¢�756y;HBg �6i:<63qXr^5:<1,1�_Z1�_ZÙ-=cBD\DBD1¶Ðo��¤V¥,Q�����R 9�35;�1'_pN�1'K5BDCr�oQ�����T�¯�����¤ 4j9�3�`a6yK5CE6#;�_�1�:rX<756y49�CE6weP756#:<6

������ � � ��� � � �)!Z�u_�1�: "#�Ø� �������%�'& F�9�3H;�X<7HBDC�e�9�C�;51'356Z`,?�¢���R�QSUQ�9�35;>`I?>�o\D1'BDC-&���5T��%�dm

¢�756@4j9�CE6@eP756i:<6{X<756@;56#:<BgA�9�X<BgAq6C|9�:<6{B]3wX<756@C<9�=c6@N�R�����T���µrCM^%9�4#6@49�3y`a68^5:<1jAq6#;�`I?wK5CEB]35kcX<756f¢�7561�:E?y1�_0�3qX<6#:E^�1'\]9�X<BD1'3aF�9�C�e�9�C�;51'356�`I?8GI9�9�bZ±d�%��� ����F#`%K(X�X<756d\DBD=WBDX<BD35k24j9�CE6-eP756i:<6 ���

��� � � � ! � �5� �)!Z��_�1�:�"��Ø� ���������:& e�9�CXE:<69�X<6;c`,?rnp9�V �D= 2����µ'QSUZ9�3H;y�qX<6i^H7H6#3 1 ��Q�T�¤����~`I?�9�3H9�\g?(�#BD35kf9@_�1�:<=@K5\]9@1'_ �{× �o��Q�¥r9�`�1'K(X2X<7H6o351'35B]354i:<6j9�CEB]35k:<69�:E:[9�35k'6#=c6#3qXP1'_V9)4#1'3qAq1'\DK(X<BD1'3�^5:<1,;5K54iXm

�oC)_+9�:c9�C)0{b(351je8F-X<7H6#CE6>4#\]9�CECEBD4j9�\�=c6iX<751,;5C);51�351�X)^�6i:<=cBgXfX<1~XE:<69�X)X<756>49�CE6YeP7H6i:<6�X<756�;H6i:<BgA'9�X<BDA'6C)9�:<6BD3�;HBg �6i:<63qX8NVR�����T���µ>CM^%9�46#C#t�1'_u41�K5:<CE6�F�X<7HBDC{JIK56#CMX<BD1'3�B]C{JqKHBgX<6c9�4j9�;56#=cBD4�F `%K(X{CE6#:EAq6#C@9�C8XE:[9�BD35BD3Hk>k�:<1�KH35;°_�1�:CEBgX<KH9�X<BD1'35C�eP75BD4}7W1�_�X<6#3W1,4#4K(:-eP756i:<6P1'356�7%9�Cd;5Bg �6#:<6#3qXdBD35_�1�:<=c9�X<BD1'35C�BD3W;5Bg a6i:<6#3IX-;5BD:<6#4iX<B]1�3HC#F'_�1�:-6#l59�=W^H\D6�`�6#4j9�KHCE6CE1'=W684#1,1�:<;5BD3H9�X<6#C�:<6i^5:<6CE6#3qX|CM^%9�4#6{9�35;y9�351�X<756#:P1�3H6p:<6i^H:<6#CE6#3IX<C�X<BD=W6'm

�VBg:<CMXF5BgX2B]C2K5CE6#_�K5\�X<1@1�`HCE6i:EAq6oX<7%9�Xo�(RV¯ R�¥,�H�hBD=@`�6#;5;HBD35k{X<756#1�:<6=�_�1�:�*9� �oBDC26CECE6#3qX<B]9�\D\D?)X<7H6oBDCE1�^�6#:<BD=W6#XE:<BD4BD356jJIKH9�\DBgXs?'my¢�756Y4#\�9�CECEB]49�\dBDCE1�^�6i:<BD=c6iXE:<BD4cBD356jJqK%9�\]BgXs?�C<9#?(C@X<7H9�Xf9�=c1�3Hk�=W6j9�CEK5:[9�`%\D6cCE6#X<C � 1'_'�)!�ePBDX<7h9yk'BgAq6#3Aq1'\DK5=W6'FqX<756@� & >~����LO;5B]=W6#3HCEBD1'3H9�\%=W69�CEK(:<6|1'_�X<756|`�1'K535;H9�:E? � �*BDCu=WBD3HBD=@K5=�eP7563�� BDC298CM^H756#:<6�t56JIK5BgA'9�\]6#3qX<\g?qF_�1�:|9Wk'BgAq6#3>=W69�CEK(:<681'_�X<756Z`�1'K535;H9�:E?qF5X<756pAq1'\DK5=c6ZBDC�=c9�l(BD=@KH= _�1�:|9)CM^H756#:<6�m

�o3H9�\g?,X<BD4j9�\D\g?�BDXo=W6j9�35C|X<7%9�X (�N -'� �Z� (�C � � (�N -�� � �Z� � ! � � ! " v � F 9�35;~BgXoX<6#\D\DC|eP7H9�XoX<7H6{`�6CMXp41�3HCMX[9�3IXRC �BDC#F�eP75B]\D6Y�(RV¯ R�¥,�%� BD=@`�6#;5;HBD35k>X<756#1�:<6=�_�1�:�� v:�¹vj� �)!p�pk'BgAq6#CWI�� 1 � � � v � OMQ-( C v � � KML.N4O�� � ��� � !

� � ! " v �v F�`HK(Xf;51,6#C351�XPBD35CEB]CMXPBD3�B];56#3qX<B]_]?(BD35kfeP7H9�X�X<756Z`�6#CMXP41�3HCMX[9�3IX C v BDC#m¢�756w:<6#\]9�X<BD1'3�`�6iXse266#3�X<756>Xse21°BD356jJIKH9�\DBgX<BD6CfBDCWX<7H9�Xc1'356y49�3�9�^5^%\g?¶X<756w\]9�CMXcB]356JIKH9�\DBgXs?hX<1 � � � FuX<7H64[7%9�:[9�4#X<6i:<BDCMX<BD4|_�K5354#X<BD1'3r1'_ �{F,eP75B]4[7YBDC2351�X�BD3�� v:�¹vj� �)!Z�V`HK5X27H9�C�BgX<Cu^%9�:EX<B]9�\%;H6i:<BgA'9�X<BDA'6C � ���

��� �eP7HBD4[7�9�:<6oÚ|���-R�T

=W6j9�CEK(:<6#C#F�9�35;w1'356{49�3�9�^H^H\g?YX<7H68BD356JIKH9�\DBDXs?cX<1 � � � � 9�3H;>X<7563>\]6iX �{X<635;>X<1r®(t%BD3>X<75BDC�e�9?>1'3568\D69�:<35CPX<7H9�XC � ( C v m�¡u1'3qAq6i:<CE6\g?'Fdb,3H1ePB]35k~X<756�BDCE1�^�6#:<BD=W6#XE:<BD4rBD356jJIKH9�\DBgXs?'F-1'356>49�3�9�^5^5:<1q9�4[7�9~_�K5354#X<BD1'3 � `I?�9~CEK5=Ö1�_4[7%9�:[9�4#X<6i:<BDCMX<BD4|_�K53H4iX<BD1'35C#F,KHCEBD35k � � ���0Q � � � ( � �GQa� � � � Óh��� � �Z9�35;Y;56;5K54#68�5RV¯ R�¥I�%�°BD={`�6;5;5BD3Hk8X<756#1�:<6=>F,CE1X<7H9�X C v ( C � 9�35;�X<7H6oXse21WBD3H6JIKH9�\DBgX<BD6C29�:<686#CECE6#3IX<B]9�\D\g?cX<7568C<9�=c6�m-no1e26#Aq6i:F5X<756p^5:<1,1�_�1'_�X<756Z\]9�CMX�^%9�:EXPBD3IA'1'\gAq6CX<756ZX<6#4[7H35BD49�\aCMX<K5;5?�1'_�_�K5354#X<BD1'35C�1'_�`�1'K535;56;>A�9�:<B]9�X<BD1'3��S;56#3H1�X<6; 2Ñ{�[F(eP75BD4}7>BDC�4\]9�CEBD49�\aBD3w1�3H6Z;5BD=c6#35CEB]1�3 F(`HK(X1je26#CWX<1�X<756>e21�:Eb¶1'_oÙd3535BD1�£p� ÐoQSR���¤�Qsv#F-�V� �V�%���%�Vº�9�35; 1°635;56#\]\��V¥,� �ZQ�T�¤ � _�1�:WX<756>;H6iAq6#\D1�^H=c6#3qXc1'_PX<7H6& LO;5BD=W635CEBD1'3H9�\ 49�CE6�m

�qX[9�:EX<BD3Hk@_�:<1�= �(RV¯ R�¥,�%�°B]={`�6;5;5BD35k{X<7561�:<6#= � v:�¹vj� �)!p� � ��v � � �)!p�[F51'356o4j9�3�69�CEBD\g?c;56#:<BgAq6p9�\D\%X<7H6|:<6CEK5\gX<C9�\g:<69�;(?)1�`5X[9�BD356#; m�¦oCEBD3HkZX<756�_�K5354#X<BD1'35C � � 9�;H9�^5X<6#;WX<1 � Fq1'356�e�:<BgX<6C � � � � � � ��� � v � ( C � � � ���� � � � KML.NMO�� � ��� � v F,9�35;W`I?n �R�¥��V�%�yBD3H6JIKH9�\DBgXs?@1'356�7%9�C � � ���� � � � KML.N4O%� � ��� � v ((� � � �� � � � KML.N4O%� � ��� � � ( � � �� �F,9�35;W1'356P;56;5K54#6C�X<756PC<9�=W6�BD356jJIKH9�\DBgXs?X<7H9�3�`�6_�1�:<6'FB� N � " v > N � � ( � � � � � � �5� � �[m

¢�756|49�CE6�1'_�;56#:<BgA�9�X<BgAq6#C-BD3w�S;5Bg a6i:<6#3IXi��NVR�����T���µ8CM^%9�46#C-ePBD\D\(`a6�1�`5X[9�BD356;)`I?WK5CEBD3Hk{9Z=fK5\gX<Bg^H\]BD49�X<BgAq6uA�9�:<B]9�3qXm� & ����$ §�¢�75629�;5;5BgX<BgAq6-Aq6i:<CEBD1'3 � � � � � v � (�� � � �� �� ������ � �� �� v _�1�:V9�\D\ � � � v'� vj� �"!p�aBDC�6JIK5BDA�9�\D6#3IX X<1PX<756u=@K5\DX<Bg^H\DBD4j9�X<BgAq6Aq6i:<CEBD1'3 � � � � � v � ( & � � � � �� �� ������ � �� �� v � v � ! _�1�:|9�\D\ �6� � v'� v � � ! �[m�2È<ÆjÆ5#[§ � 356u:<6#CE49�\D6C�ePBgX<7@9|;HBg �6i:<63qXVCE49�\DBD35k|BD3f;5Bg �6#:<6#3qX�;HBg:<6#4#X<BD1'35C#F�BSm 6�m�1'356�9�^5^H\DB]6#C�X<756�9�;5;5BgX<BgAq6dA'6#:<CEBD1'3{X<1 FZ;H6iLzH356;w`,? F��GQ v���������� Q ! � � � � � /� / ��������� � 1� 1 ��F�9�35;y1'356{1�`5X[9�BD3HC{� � v ����� � ! �[v � v � � � � � � v � (���� v ����� � ! � � v� � �� �� ������ � �� �� v mvwÙd3535B]1)£p��ÐoQ+R���¤VQ�F(0sX[9�\DB�9�3�=r9�X<756#=r9�X<B]4#B]9�3aF��j�'­ ��L<��'�'Ò,m-no6p:<64#6#BDA'6;YX<7561 R�¥ ��^H:<BD�#6ZBD3��j����®(m�no6Ze21�:Eb'6;

9�Xp�I4#KH1�\�9f�o1�:<=r9�\]6{�IK(^�6#:<BD1�:<6�FH±dBDC<9º~no6#:E`�6i:EXpÐ@m5�V� ���������-FH�o=W6#:<BD49�3>=c9�X<756=c9�X<BD4#B�9�3 FI`�1�:<3>BD3��j�'­�®(m-no6oe21�:Eb(C|9�X ���R ´>Th¦o35BgAq6i:<CEBDXs?'F(±-:<1jA(BgL

;56354#6'F5Ú�0}m�oBD4}751'\]9�C% ���R ´>ThG':mDFH�o=W6#:<BD49�3�=c6i:<4[7%9�3IXF �)(�Ò'�jL<� ��ÕH��m

� 1°635;56\D\an@m%��¥,� �pQ�TV¤2FH�o=c6i:<BD4j9�3>=c9�X<756=c9�X<BD4B]9�3aF,`�1�:<3>B]3��j��­ �,mdno6Ze21�:Eb(C|9�X ���R ´>Th¦o35BDA'6#:<CEBgXs?'F5±-:<1jA(BgL;56354#6'F5Ú�0}m

Ò �

Page 69: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

¢�7563>1'35683H1�X<B]4#6#C�X<7H9�XPBD_� v ����� � ! � � ��®WBDC�k'BgAq6#3�X<7H6#3�X<7568=cBD35BD=fK5= 1'_ � � � �� � BDCP9�XEX[9�BD356#;�eP7563�� � � �#� �_�1�:d9�\D\M"Z_�1�:-CE1'=W6�N���¤�����T�¤��c=@K5\DX<Bg^H\DBD6#: �uFI9�35;)X<756�4#1'35CMXE:[9�BD3qX-k'BgAq6#C � ! � v ����� � ! � ��FqCE1pX<7%9�X�X<7H6�=WBD35B]=@K5= BDC!���� v ����� � ! �[v � !8tH1'35689�^H^H\DBD6C2X<75BDC�:<6#=r9�:EbrX<1fX<756Z4j9�CE6���� � �� �� � �� � � �� �� v 9�3H;>1'356pzH3H;5C�X<7H6p=fK5\gX<Bg^%\DBD49�X<BgAq6PAq6i:<CEBD1'3aF9�C|X<756{^�1je26i:<Co1'_���9�:<6@BD;56#3IX<BD49�\�1'3y`a1�X<7�CEBD;56C|1'_VX<756@BD356jJIKH9�\DBgXs?��+`�649�K5CE6{X<756@BD356jJqK%9�\]BgXs?�BDCp9�\g:<69�;(?wBD3qA'9�:<B�9�3IX

eP7563>1'356Z:<6#CE4j9�\]6#CP9�\D\aX<756Z4#1,1�:<;5BD3H9�X<6#C�B]3�X<756ZC<9�=W6Ze�9?%�[m¢�756�=fK5\gX<Bg^H\]BD49�X<BgAq6�Aq6i:<CEB]1�3ØB]=)^H\DB]6#CyX<756�9�;5;5BgX<BDA'6hAq6i:<CEBD1'3x`I?xX<756�k�61'=W6iXE:<B]4iL�9�:<BgX<75=c6iX<BD4�B]356JIKH9�\DBgXs?qF{BSm 6�m

�GN v ����� N ! � v � ! ( + / � � � � �&+ 1!_�1�:r9�\D\2^�1'CEBgX<BgAq6�3IK5=@`�6i:<C�N v���������� N ! t�^HK(XEX<B]35k N � � ( � � X<75BDCWBDCo�MK5CMXWX<756w4#1'3qAq6il,L

BgXs?YB]356JIKH9�\DBgXs?r_�1�:�X<756Z6#l,^�1'356#3qX<B�9�\ _�K5354iX<B]1�3 m� ��%"-� j�D������ §@N�6#X � C<9�X<BDCE_�? ������ � � ��� � � 8 � � �)!p�|eP756i:<6y� � *��� , 9�3H;���( 5 ��( , _�1�: " � � �������%�'& mrN�6#X* � ��� 9�3H; *

�� ���

`a6@;56#zH356#;�`I? v�������

� v!� � v� � 9�35; v

� ������� v�������

> v!t CEBD=WB]\]9�:<\g?'F 5 � ��� 9�35; 5

�� ���

9�:<6f;56iz%356#;�`I?v8 ����� � v

!� � v8 � FH9�35; v

8 ������� v8 ����� > v

!md¢�756#3w1'35687H9�CW� N � " v > N � � ( � � � ������ � � 8 �����q� � �[m

�2È<ÆjÆ5#[§�N�6iX9@%� � � ���� �

_�1�:#"$�·� ���������:& m � 3H6w9�^5^H\DBD6CfX<756>=fK5\gX<Bg^%\DBD49�X<BgAq6cAq6i:<CEBD1'3�X<1 � � � � �[F29�35;�1'356>7%9�C)X<16#CMX<B]=c9�X<6 � � ���� � � �:@%�4� � v m��»4#\�9�CECEB]49�\o:<6#CEKH\gX>1'_@no��� � ��9�35; N�Q���� ¥,�H´>R�R��!CMX[9�X<6#C>X<7H9�X>_�1�:y9�\D\�@ � ��v���P��Ó� ����P�)9�35;*9�\D\�=c69�CEK(:[9�`H\D6yCEK(`HCE6#X<C � � ��1'356y7H9�C I � � @��GQa��� O Q ( I � � + � � � �� @ � � -��4O -�m �oCcX<756y=W6j9�CEK5:<6y1�_X<756>^�1'BD3qX<C)eP756#:<6�� �� � � � � ®�BDCr9�Xr=W1'CMX ( � F21�3H6>;H6#;5K546#CWX<7H9�X � � � �� � � �:@%�4� � v ( �¶� ( � � @%���f_�1�:#" �Ö� ���������'& m �6#4j9�KHCE6 ( " � � � �¶� ( � � @%��� � � 8 � � � �oePBgX<7 � �9� v

� � _�1�: " � � ���������:& FV1'356r;56#;HK54#6C8`,?�n �R�¥��V����B]356JIKH9�\DBgXs?~X<7H9�X( " � � � ����� � � � � � � ��� � v � ( & � � � � ( " � � � �¶� ( � � @������ v � ! � � 8 ����� � � �[FH9�35;>X<75BDC�k'BgAq6C2X<7568;56#CEBg:<6;wBD356JIKH9�\DBDXs?'m0�_�* � ��� � & X<7563>BgXP=W69�35C �6� ���������0� 8 �����,� �)!Z�[m0�_ * � ��� � & 9�35; 5 � ��� � ��FHeP75B]4[7w=W6j9�3HC�X<7H9�X 5 �� �8_�1�: "W� � ���������:& FHX<7563>1'356{7H9�C�� N � " v > N � � � � vj� � �9�35;�X<756i:<6_�1�:<6)1'356);H6#;5K546#C{9Y`�1'K535;�_�1�:RN � F�BSm 6�m � � � � � ! �[t�K5CEBD3Hk�X<756);5635CEBgXs?~1'_ C � � � ! �|BD3 � � � � v � � ! �[F1'356Z;H6#;5K546#C�X<7H9�X � � C � � �"!p�[m0�_�* � ��� � & 9�35;°� � 5 � � � �-,/F5X<7H6#3>_�1�:P6iAq6i:E? � ��®)1'35687H9�C ( �

��� ����� � ��v� ��� � �)!Z�[m0�_�* � ��� � & 9�35; 5 � ��� � ,�F X<7563�1'356c;56#;HK54#6CZX<7H9�X9� N � ��( ��� � ��Ó �*9�35;�X<756i:<6_�1�:<6)X<756i:<6W6il(BDCMX<C � � � ®

CEK54}7YX<7H9�X ( � � � � � � ��v����� � �)!Z�[m-¢�75BDC2BDCuX<756p49�CE6p_�1�:�6il59�=)^H\D6|eP756#3>9�\D\%X<756p;56i:<BDA�9�X<BgAq6#Cd`�6#\D1'35k@X<1�� ! � >� �)!p�[F59�35;`�649�K5CE6f\D1'k%� � QH� ��BDCpCEK54}7�9r_�K5354#X<BD1'3aFaBgXpBDCp351�XZ9�\ge�9?(CoXE:<K56@X<7H9�X ( �

���� ��v����� � �)!Z��_�1�:89�\]\ � ��®(m{¢�756fCM^%9�46f1�__�K5354iX<B]1�3HC{4#1'35CEBD;H6i:<6#;¶9�:<6cBD3 �� � F�9�35;h�H:<BgX<�cG,R�³�T 9�35;hN�1'K5BDC@�oQ����%TV¯��%��¤ 7H9A'6rCE751jeP3hX<756cCMXE:<1'35k'6i:@:<6#CEK5\DX

X<7H9�XP_�1�:P6#Aq6i:E?�_�KH354iX<BD1'3>BD3 �� � X<756i:<686il(BDCMX<C � � ��®WCEK54[7>X<7H9�X ( � � � � � � ��v����� � �)!Z�[m0�_ * � � � � & X<756#3~1'356)7H9�C � � � �� �)!p�[m � _d4#1'K(:<CE6'Fa1'356)49�3~;H6#;5K546@X<7%9�X � BDCp4#1'3qX<B]3IK51'K5C#Fa`HK(XZ`�6#4j9�KHCE6X<756#:<6�9�:<62;5Bg �6#:<6#3qX�;56i:<BgA'9�X<BgAq6#C�BD3@;5BD �6i:<63qXV;5Bg:<64iX<BD1'35C#F�X<756#:<6uBDCV351Z9�K(X<1'=c9�X<BD4uCMX[9�X<6#=c6#3qXVX<7H9�X � =fK5CMXV`�62n �R�¥��V�%�4#1'3qX<B]3IK51'K5C#m

np9A(BD35kr;5Bg �6#:<6#3qXoBD35_�1�:<=r9�X<BD1'35CP1'3y;H6i:<BgA'9�X<BDA'6C�B]3y;5Bg �6#:<6#3qXo;5Bg:<64iX<BD1'35C|BDC|K5CEKH9�\�_�1�:|^%9�:[9�`a1�\]BD486jJIKH9�X<BD1'35C|\DBgbq6X<75627H69�X�6JIKH9�X<BD1'3am��H1�:-6il59�=)^H\D6'F�\D6iX �°`�6�9�3)1�^�6#3)CE6iX�1'_��)!{F'k'BgAq6#3 � � � �uº����P�[F'1�3H6249�3)CE751je�X<7H9�X�X<7H6i:<626#l,B]CMX<C98K53HB]JIK56|CE1�\]K(X<BD1'3 � 1'_ ���� � > ? � �*®{BD3 � ���O® � �o�VC<9�X<B]CE_]?(BD35k8X<756|BD35BDX<B]9�\54#1'35;5BDX<BD1'3 � �GQ � ®I�H� � � �GQa�VBD3 �/9�35;WX<7H6751'=W1'k'6#3H6#1'K5Cd£pQ���QS²�³�¥,���>`�1'K535;H9�:E?W4#1'35;5BgX<B]1�3 � � � �*®Z1'3 � � �c�O® � �o�[FqBD3WX<756�CE635CE6�X<7H9�X �6� C � ! ® � � #St �dº����P����F�6� � º � ® � �Zt'; v� ���P���89�35; ���� � � � º � ® � �Zt'; " v ���P����m

0�_ � � � ; v� ���P�fX<7563/X<756�CE1�\]K(X<BD1'3�9�\DCE1¶C<9�X<BDCMzH6#C ��� C � � ! ® � � #Ot:; v� ���P���'F ? � � ���� � � � º � ® � �Zt � º ���P���$��uº � � �w�O® � �o���'t(B]_�X<756o`�1'K535;%9�:E?rB]Cu1'_�4#\]9�CEC�C)v�1�:�BD_aX<7H6o1�^�63rCE6#X �/BDC24#1'3qAq6#l��S1�:�BD_�9�3�BD356JIKH9�\DBDXs?W751'\D;5C2_�1�:uX<7H6X<1�X[9�\a4K(:EA�9�X<K(:<6Z1'_�X<7H6p`�1'K535;%9�:E?%�[F5X<7H6#3>1'35687H9�C �6� �uº � ® � �Zt:;�º����P� � m0�_ � � `�6\D1'35k'C8X<1y9�3°BD3qX<6#:E^�1'\]9�X<BD1'3�CM^%9�46W`�6#Xse26#63 ;hv� ���P�Z9�3H; �uº����P�|X<756#3h1'356c7H9�C{BD3qX<6#:<=W6;5B]9�X<6):<6#CEK5\gX<C#F`HK(X�X<7HBDC�:<6JIK5Bg:<6C�6#351'K5k'7wCE=W1,1�X<75356CEC�_�1�:PX<756p`�1'K535;H9�:E?qm�oCf9�3°6#l(9�=W^H\D6�F�41'35CEBD;56#:f9w_�K5354#X<BD1'3 � �GQ � %<�o;56#zH356;h1�3 �"! � � 9�3H;hC<9�X<B]CE_]?(BD35k � � ���� � � ? �5� �uº�� �"! ��vi��+X<75BDC�BD=W^H\DBD6CuX<7H9�X ���

��� � � �uº�� �"! ��vi�d_�1�: "#�Ø� ���������:& `,?YKHCEBD35k)�-R����Q��%��XE:[9�35CE_�1�:<=��[m�£p6#351�X<BD3Hk>� � � � ��X<756Z;5KH9�\A�9�:<B]9�`H\]6#C#F�X<7H6)BD35_�1�:<=c9�X<BD1'3�BDCp6jJIK5BgA�9�\D63qXoX<1#� � � � � � � � � � º�� � � �uº�� �)! ��vi�@�O9�3H;~X<756i:<6_�1�:<6 � �"� � � �uº�� �"! ��vi�_�1�: "#�Ø� ���������'& �[m

�6#4j9�K5CE6@�E��Ó,� � ��Ó,� ��� ºj� � �6� �uº�� �)! �Vvi�[F,BD_ 1�3H6|CE751jePCuX<7H9�X vv � � � � � � � � � � ��� � >� �)! ��vi�-_�1�:2CE1'=c6 * � ��­ � ,��[F1'356c;56;5K54#6C8X<7H9�X � � � � 8 � º�� �)! ��vi�oePBgX<7 v8 � vº Ó v�9�3H;�`�649�K5CE6y� � 5 �¸­w9�35; � =r9�^HC"��vj� �)! ��vi�oB]3qX<1

� �� �"! ��vi�29�35; �uº�� �)! �Vvi�uBD3qX<1WBgX<CE6#\]_�F51�3H6pzH35;HC�X<7H9�X � � � 8 � º�� �)! ��vi�[mÒ'�

Page 70: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

� 356w7H9�C vv � � � � � � � � � � � �� �"! ��vi�[F29�35;�BDXWBDC)X<7H6�`�6#7%9#A(BD1'K(:r9�X�, X<7H9�XcBDCWBD3qX<6#:<6#CMX<BD3Hk�_�1�:c1�`5X[9�BD35BD35k~X<7H6CE=c9�\D\]6#CMX8A�9�\DKH6c1�_ *�F�9�35;°X<756i:<6_�1�:<6c1'356r4}756#4[b(C{_�1�:{eP7H9�X{A'9�\]K56c1'_�*�1'356r7H9�C v�

���� � � � � ��� � >� �)! ��vi�89�3H;h1�3H61�`5X[9�BD35C8X<7H6cC<9�=W6cB]35_�1�:<=r9�X<B]1�3°_�1�:{X<756rCE=c9�\D\]6i:{_�K5354iX<B]1�3 vv � � � � � � � � �5m � 356rK5CE6#C{X<756r751'=W1'k'6#356BgXs?~^5:<1�^�6i:EX<B]6#C{1�_X<756)_�KH354iX<BD1'3aF�9�35;�_�1�: � �x®�1'356f41'=)^HK(X<6C )(�N - �I� � � � � � v�

���� � � � � �qt `I?�=r9�b,B]35krX<756W4[7H9�35k'6)1'_d4#1,1�:<;5BD3H9�X<6C

� � �" v � �V9�35; � � �

" v � º �#�SFaBDXoBDCRC� " v " � ! � º � ePBgX<7 C �( (�N -��I� � � � � � v����� � � � �� �'F�9�35;�X<75BDCo4#1�:E:<6CM^�1'35;5CoX<1

* �Ø�-Ó ! º � ! � ºº F(eP75BD4}7�k'BgAq6C 5 � º � ! � º �! � � 9�3H; 5 �P� º � ! � º �

!" º BD_ & �Ym��H1�: & � ­@1'356pzH35;HC�X<7H9�X � � � 7 � � � �

_�1�:|9�\D\ L � ,/m

(�®

Page 71: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�j­,m-��1'35;H9?��|^5:<B]\��#®(F�­�®'®'®(m¦oCEBD3Hkp�-R����Q����>XE:[9�3HCE_�1�:<= 1'356�4j9�3W1�`HX[9�B]3)9�3)BD=@`�6#;H;5BD35ko:<6#CEK5\DX�_�1�:-CM^�9�46#CH; � � �)!Z��BD3IX<1pN�R����%T���µpCM^%9�4#6C#m

� ��%"-� j�D������ §d�%1�: - � ! º 1'35687H9�C�; � � �)!Z� � � ��vj� �"!p� � C � � �)!p�[m�%1�:o® �9-�� ! º 1'35687H9�C�; � � �)!p� � ��� � � � � º�� �)!Z�dePBgX<7 v

� � � � � vº > �!m

�2È<ÆjÆ5#[§ � 356f7H9�C�� � � �I�8�¸�E��Ó(� � � � � � � � �I� vv � � � � � Fa9�35;�9�C �<� ; � � �)!p��=c69�35Cf�E�2Ó(� � � � � � � � �I� � �uº�� �)!Z�[F�9�35;1'356Z=fK5CMX|4[7564<b�BD3�eP75B]4[7>NVR�����T���µWCM^%9�4#6ZX<756Z_�K5354#X<BD1'3 vv � � � � � BDC#m�%1�:�-�� ! º 1'356W7H9�C vv � � � � � � �uº�� �)!p�o9�35;�X<756i:<6_�1�:<6 � � � ��v� �)!Z�[F�9�35;�1'356)K5CE6CpX<756W_�9�4iXZX<7H9�X�� 1�: �=c9�^ ��vj� �)!p�uB]3qX<1�C � � �"!p�[m

�%1�:�® �9-�� ! º 1'356P7H9�C vv � � � � � ( v� � � � � � ! � � � �� �"!p�[F,9�35;cX<756i:<6_�1�:<6� �6� �,+ � � � � º�� �)!Z��ePBgX<7 v+ � � � � vº Ó �!m

�6#4j9�KHCE6 � " v � � =c9�^HC)� v � � ! �PBD3qX<1 � � � ! �o9�35; � º � � ! �PB]3qX<1�BDX<CE6#\D_sFaBgX8=c9�^HC � � v � � ! � � � º � � ! ��� � � º B]3qX<1� ��>� �)!Z� � �uº�� �"!p��� � � º tVX<756rz5:<CMX@CM^%9�4#6YBDC ��� � � � � º�� �)!p�ZBD_ v� � � � � v " �v Ó �

º F�9�35;°X<756r\]9�CMX@CM^%9�4#6YBDC � 8� � � � º�� �)!p�

ePBgX<7 v8 � � � � v " �º Ó � � �)> v� � � � F%CE1cX<7H9�X 5 � � � � *�� � � �+m�¢�7H6i:<6#_�1�:<6 � � � � + � � � � º�� �)!Z�2BD=W^H\DBD6C � � � - � � � � º�� �)!p�ePBgX<7 2 � -��*� N�� -j� � BSm 6�m v- � � � � vº > �

!m

�%1�:{�(RV¯ R�¥,�H��CM^�9�46#Co4#1�:E:<6#CM^�1'35;HBD35kcX<1)* �Ø­,F�1'356@4j9�3H351�X|KHCE6@�-R����Q����hXE:[9�35CE_�1�:<=>F�9�35;y^5:<1,1'_�CP=fK5CMX|`a61�`5X[9�BD356;>BD3w9);5Bg a6i:<6#3IX�e�9?'m� &������D������ §>�H1�:y� ( * ( , 9�35;�®�� - � ��F-X<756��(RV¯ R�¥,�H�ØCM^%9�4#66� � � �5� �"!p�{BDC);56#zH356;�9�CE� � � �5� �)!p� �� � v:� �5� �)!Z� � ���(� �"!p��� v " � � � m�%1�:@��(+* �65 (-,/F�9�3H;w® �9- � �81�3H6Z;56#zH356#C�X<756 2�aU�Ra��CM^%9�4#6 ��� � �8 � � ! �*� � � v'� � � � ! � � � � � � ! ��� v " � � 8 m

0�_ �)BDCu9p^�1'CEBgX<BDA'6�B]3qX<6#k'6i:29�3H;� �9- ���2Ó°��FI1'356|=r9#?);H6izH356PX<756Z�(RV¯ R�¥,�H��CM^%9�4#6 � � � �H� �)!p�V1�:uX<756 2�aU�Ra�CM^%9�4#6"��� � �8 � � ! �|BD3°9�X8\D69�CMXZXse21>e�9#?(C#t�1'356)e�9?�BDCpX<7H9�X �,� � �.� � � � ! �o9�35;�_�1�:@9�\D\V=fK5\gX<BgLOBD3H;5BD4#6C��1'_u\D6#3Hk�X<7� ��� � �c1'356Z7H9�C8 E� �6� � � " � � �(� �"!p�-1�: � � " � � �8 � �"!p�[tH9�3H1�X<7H6i:2e�9?YBDCuX<1f;56#zH356pBgX�9�C � ��� � �5� �)!Z� � ���5� �)!p� � � � 8 FePBgX<7 5 � *~1�:o351�XF%ePBgX<7y9�3�BD3qX<6#k'6#: �@Ó*��F�9�35;yePBgX<7°�E� > � � +� -�m � _�4#1'K(:<CE6'F%X<756#:<6@9�:<6@9c_�6#e�X<64[753HBD49�\JIK56#CMX<B]1�3HC�X<1)4}756#4[b�BD3>1�:<;56i:PX<1)CE751je�X<7%9�X�X<7H6#CE6pXse21c;56iz%35BgX<BD1'35C�4#1'BD354BD;56�m

�6#_�1�:<6pK5CEBD3Hk)CE49�\DBD3Hk)9�:<k'K5=c6#3qX<C�_�1�:��� � �5� �)!Z�dBgX�BDC�K5CE6#_�K5\�X<1f:<6=c9�:EbcX<7H9�XP9�3>6JIK5BgA'9�\]6#3qX�351�:<= BDC#� � � � � � Ó� � � � ��� � � �� � � � � FIBSm 6�m�_�1�:u® �(� �)� � 1'356�49�3)`�1'K535; � � � � � � � BD3fX<6i:<=cC�1'_H� � � � � � 9�35;c9�\]\,X<756�351�:<=WC� � �� � � � � _�1�:X<756u=fK5\gX<BgLOBD3H;5BD4#6C�1�_5\D635k�X<7{6il59�4iX<\g?86JIKH9�\qX<1 ~m � 356u=c9?{CMX[9�:EX�B]3{1'356u;HBD=W635CEBD1'3aF�3H1�X<B]4#BD35kPX<7H9�X2� ��;�� � � � � Ó � FeP756#:<6 ; BDCdX<756ono���,��QSUQ��V��_�K5354#X<BD1'3aF � � C � � �{��BDC26JIKH9�\%X<1Y�|3H69�:�®@9�35;YX<756i:<6_�1�:<6 � � C � � �{�[t(1'356o;56#;5KH4#6#CX<7H9�X@� ��;�� � � � � � � �(Ó � � � � F%_]:<1'=�eP75B]4[7w1'356{;56;5K54#6C�X<7H9�X�� � � � � � � ( � � ��;,� � v � � � � � � � � Ó � � � � � � v � � � � � � mP�IB]=WBD\]9�:<\g?qFBD_Z®�� � � �F21'356w:<6#^H\]9�4#6CWX<756yno���,��QSUQ��V�*_�K53H4iX<BD1'3�`I?�X<756y_�K5354#X<BD1'3 � ;56iz%356#;/`I? �¶� %<�E� �

� � /� � " � � � ;¶� %<�_�1�:@9�\D\ %}F�CE1>X<7H9�Xc� � ��� � � " � � � � � Ó ePBDX<7 � C � � �{�[F�9�35;�;H6i:<BgA(BD35k ��X<BD=c6#C89�35;�X[9�b(BD35kYX<756W4#1'3qAq1'\DK(X<B]1�3ePBgX<7 � 1�3H6fz%35;5CZX<7H9�X9� � � � � � � � � ( � � � �,� � v � � �� � � � � � Ó � � � � � � � v � � � � � � m)0�3�1�:<;56#:ZX<1�`�1'K535;5� � � � � � � F�1'356W;56351�X<6#C� ���O® � � º ��������� � ! �[FP� ���G�Ó � v >�� �)� � � º.��������� � ! ��9�35;6F��

�� F�CE1cX<7H9�Xo_�1�:p9�\D=W1'CMXo9�\D\TQ º ��������� Q ! 1'356@49�3K5CE6fX<7H6)1'356);5BD=c6#35CEB]1�3%9�\�:<6CEK5\gXZBD3�1�:<;56i:ZX<1>;56i:<BgAq6W9r`a1�KH35;�_�1�:ZX<756)351�:<= 1'_ � � �

�/v FwBD3~X<6i:<=WCZ1�_ Fy9�35; �� � �� �

"�� ���v FHt%X<756#3y1�3H6ZX[9�bq6#C�X<7H6Z^�1je26#: *~9�35;y1'3568B]3qX<6#k�:[9�X<6#CPBD3�Q º.��������� Q ! t%1'3568zH35BDCE7H6#CP`I?w9�3wBD3H;5K54iX<B]1�39�:<k'K5=W63qXP1'3 & m

2?�K5CEBD35k�9>CE49�\DBD35ky9�:<k'K5=W63qX{1'356c;56;5K54#6CZX<7%9�X � � � � � � � ( C5� � � � �¹v"�

� � � � � � �P� � � �� � � � � � � ePBgX<7 � �� ���� m¢�75BDC�B]=)^H\DB]6#C2X<7H9�XPBD_�® ��� � 1'35687H9�C � ���� �5� �"!p� � ���5� �)!Z��� � � " � � � � �¹v � � � � �5� �)!Z�[m

�%1�:{K5CEBD35k>X<756f:<6#BDX<6i:[9�X<BD1'3�X<756#1�:<6#= 1�_uGI9�4JIK56#CMLON�1'K5BDCZN�Q+R�T�Uc9�3H;�GI9'9�b~±d�%��� ����F 1'356W9�\DCE1w356#6;5CpX<1>4[7564<bX<7H9�X�� � � �H� �)!p� � � ��� � �5� �)!Z� � ���5� �)!p��� � � " � � � �� m8�H1�:8;51'BD35kYX<75BDC#Fa1'356@XE:<B]6#CoX<756fK5CEKH9�\V;H6#4#1'=W^�1'CEBgX<BD1'3 � �� � � � Óy� � > � � � � �[F'eP756#:<6 � � B]C�9pCM^�6#4B]9�\(CE=c1,1�X<7HBD35kpCE6jJqKH6#3546�tq_�1�:� �)� � �1'356�e�:<BgX<6#C* ���� � � � � �*�5

�� � �

��

ePBgX<7�� � �8Ó ��9�35; � �)� � �> �)9�35; � �*� � �%FICE1ZX<7%9�X � � � � � � � � � � � �

� � 1 � ( C � � " �@t,1'356�7H9�C � �GQa� >>� � � � � �}�GQa�H�I � 1 � � �+*H� � � �GQa�P> � �GQ�> *5��� O,*%F59�35;�BD_ � ��o1'356-�EK5CMX�K5CE6CuX<7H6o_�9�4iX2X<7H9�X�� � � > � ! � � � � ( C � * ��� � KML.NMO�� � ��� � � F,`HK5X�BD_� � �o1'356o=fK5CMX2`�6p=W1�:<6p49�:<6#_�K5\SF59�3H;Y`�6#CEBD;H6#CuX<756o41'35;5BgX<BD1'3�I�� 1 � v �+*5�4O,*���o1�3H6p9�\DCE1fBD=W^�1'CE6#CuX<756p4#1'35;5BgX<B]1�3HC

(,�

Page 72: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

I � 1 *�� v �+*5�4O,*$� ®~_�1�:c9�\]\u=fK5\gX<BgLOBD3H;5BD4#6C ��ePBgX<7*� ( � �*� ( �9>x��m � 356>K5CE6#C)X<756>¢��&�V¥5R�� 6il,^%9�35CEB]1�3¶ePBgX<7

BD3qX<6k�:[9�\�:<6#=r9�BD35;56#:�@��E����� @��O®I��Ó @ � �O®I��Ó ����� Ó � � � / � � � ���" v � � Ó I v�

� v " � � � / � � � � � � ���" v � � O&%P_�1�:oX<7H6f_�K53H4iX<BD1'3 @�;56#zH356;�`I?

@�� %<�H� � �GQ >�% *5�[FIK5CEB]35kZX<7H6P_+9�4iXdX<7H9�Xu_�1�:o�)((� �*� (��)>~�P1'356|7H9�C�I�� 1 � � �+*5� �� �GQa� *

� O&* �*®(F,9�35;r1'356|;56#;5KH4#6#CX<7H9�XW� � � > � � � � � � � ( C � �,F%9�35;y9�C�X<75BDC�;56#41'=)^�1'CEBgX<BD1'3>BDC2A'9�\DBD;>_�1�:P6iAq6#:E? ����®WBgX�^5:<1jAq6#C�X<756{9�CECE6#:EX<BD1'3am

ÚP6#^�69�X<6;�9�^5^H\DBD4j9�X<BD1'35C@1'_p�(RV¯ R�¥,�%� × C)BD={`�6;5;5BD3HkyX<7H6#1�:<6#=Û_�1�:E� v:� 8 � �"!p�{CE751je X<7H9�X)BD_ * � !� 1'356�7%9�C���� �5� �"!p� � � �� �)!p�[F�9�35;�9YCE49�\DBD35k�9�:<k'K5=W63qXZk�BDA'6C|X<7H6#3 � � � � � (�CE� � � � �¹v " �� � � � � � �P� � � E� � � � � � � ePBgX<7 � �!� � FP9�3H;�X<7HBDCc=W6j9�3HCWX<7H9�X � ���� �5� �)!Z� � ���5� �)!Z��� � �¹v � � �� �)!Z�@ePBDX<7 �#> � � !��� m*�H:<1�=�X<7HBDC#F2KHCEBD35k�X<7H6:<6#BDX<6i:[9�X<BD1'3�X<7561�:<6#=·1'356Y;56;5K546#CfX<7H9�Xf_�1�:W®��0- � !

�1'356Y7H9�C�� � � � � � ! � � � � � � � � 8'� � ! �89�35;���� � �8 � � ! � �

��� � � � � 8 � �)!p�dePBgX<7 v� � � � � v

�> �!m

�o351�X<756#:V^5:<1�`H\D6#=ØeP756#:<6�9�3)BD3qX<6#:E^�1'\]9�X<BD1'3{CM^%9�4#62ePBgX<7fCE64#1'35;f^%9�:[9�=c6iX<6#:��2BDCVK5CE6_�K5\(B]C�X<7H6�JIK56#CMX<BD1'3f1'_5XE:[9�46#C1'_ ; �@CM^�9�46#C#m��H1�:"- � vº _�K5354#X<BD1'35C8B]3 ; ��� � ! �p7H9Aq6r9>XE:[9�4#6W1'3 � ! " v F�eP7HBD4[7�`�6\D1'35k'CZX<1 ; � " � v � º � � � ! " v �[F9�35;°1'356W4j9�3�:<6BgX<6i:[9�X<6fX<75B]C89�:<k'K5=W63qXF�CE1wX<7H9�X{BD_ - � � º _�K5354#X<BD1'35C8BD3 ; ��� � ! �o7%9#Aq6r9�XE:[9�4#6W1'3 � ! " � F�eP75BD4}7`�6\D1'35k'CdX<1�; � " � � � º � � �)! " �'�[F(9�35;rX<756o4#1'3qX<BD3,K5BgXs?W1'_ _�K53H4iX<BD1'35CuBD3E; � � �"!p��_�1�:�- � ! º 9�^5^�69�:<CuX<756#3�9�C�9{3%9�X<K5:[9�\JIK56#CMX<B]1�3):<6#\]9�X<6#;fX<1pX[9�b,B]35koXE:[9�4#6C�1'3WCEK(`HCM^%9�46#C#m��o\gX<751'K5k'7)_�K5354#X<BD1'35C�BD3W;hv � º�� �)!Z��;518351�X-7H9A'6�XE:[9�4#6C�1�3 �)! " v�O9�4iX<KH9�\D\D?r_�K53H4iX<BD1'35C�BD3 C�� � �)!p�deP75BD4[7>A�9�35BDCE7>3569�: �)! " v|9�:<6Z;5635CE6ZBD3 ;hv � º�� �)!Z�E�[F(X<756ZCE\DBDk'7qX<\g?rCE=r9�\D\D6i:�CM^%9�46� ;°vj� �)!Z� � �uº�� �)!p��� v � º �¹v ;51c7H9A'68XE:[9�4#6C�1'3 �)! " vF%9�35;wX<756ZCM^�9�4681'_�XE:[9�46#CPBDC�6il59�4iX<\g? �dº�� �"! " vi�[mu¢�7H9�X�XE:[9�46#C6il(BDCMXd9�35;)`�6\D1'35kpX<1)�uº�� �"! " vi��_�1'\D\]1ePC�B]=W=W6;5B]9�X<6#\D?8_�:<1�= X<756�CMX[9�35;H9�:<;W6#CMX<BD=r9�X<6�� � � � � � � º ( C � � � � � v � ºº � � � ! � � � v � ºº F`HK(X�0 7H9�;@351�X�7H69�:<;f9�`a1�K5X�X<75BDC�:<6#=r9�:Ebp`a6#_�1�:<629�X[9�\Dbp`I?f�,75=@K56\I�Z¤��8R�T>BD3r�j��(�Ô,F�eP756#:<6d756u;HBDCE4#K5CECE6;@CE1'=W6 �M1'BD3IXe21�:Eb v ePBgX<7yN�9�:<Con �R����p��T������¶eP756i:<6@X<756i?y7H9�;y^H:<1Aq6;�CEK5:+�M64iX<BgA(BgXs?�`I?~9�3~9�:<k�KH=W6#3IXp1�_-�HKH354iX<BD1'3H9�\��o3H9�\g?(CEBDC#Fe21�:Eb(BD35kp6il,^H\DBD4BgX<\g?81'3@X<7H6uXE:[9�35CM^�1'CE6#;)1�^�6#:[9�X<1�:m � 3H624j9�3f41�3HCMXE:<K54iX-;5Bg:<6#4#X<\g?{9p\DBD_�X<BD35kP`,?f9�;H9�^5X<BD35kZ9�3W9�:<k�KH=W6#3IX}ºeP75BD4}7°BDC8:<6#\�9�X<6;�X<1>X<756cX<756#1�:<6=�1�_�Ùd=WB]\DBD1�Ðo��¤�¥,Q������-R�X<7H9�X@6iAq6i:E?�_�K5354#X<BD1'3°BD3 ��vj� �"! " vi�pBDC8X<756WXE:[9�46c1�_�9_�K5354iX<B]1�3>BD36� v'� vj� �"!p�[F%9�35;wBgX�BDC�KHCE6#_�K5\aX<1W;56#CE4#:<Bg`�6pX<7H9�X�:<6#CEKH\gX�z5:<CMXm

¢�756dBD;H69�BDC�CE751jeP3{1'38X<756-49�CE6d1'_ �8º�F�9�35;{k'BgAq6#3{9�_�K53H4iX<BD1'3#@ � ��v� �@��1'356-e�9�3qX<C�X<1P4#1'35CMXE:<K54#X � � � v'� vj� �{º�eP751'CE6@XE:[9�4#6@1'3yX<756#Q�9�l(BDCoB]C @ m � 356@KHCE6#CZ9cCMX[9�35;%9�:<;�9�^5^5:<1jl(BD=c9�X<BD1'3�9�:<k'K5=W63qXpK5CE6#;~BD3��oK5=c6i:<BD4j9�\��o3H9�\g?(CEBDC#F`%9�CE6#;w1'3w4#1'3qX<BD3,K51'K5C�^%BD6#46iePBDCE6Z9��c3568_�K5354#X<BD1'35C#m

�%1�:p9y�+^�1'CEBgX<BgAq6��u=c6#CE7yCEBD�6 (y1'356{41'35CEBD;56#:<C�X<7H68CM^%9�46 � � 1'_�_�K53H4iX<BD1'35CPB]3 ��v� �{�2eP75B]4[7y9�:<6@9��c356{1'3y6j9�4}7BD3qX<6#:EA�9�\a� � ( � � ��Óy��� (�� _�1�: � ��� FI9�35;)41'3qX<BD3IK51'K5Cd9�X�X<756�3H1I;H6#C � ( � � ��� tq`�6#4j9�K5CE6�X<756i:<6�6#l,B]CMX<C�® �,��� ��CEK54[7X<7H9�X �u� � K��O®I����Ó!� K��E����� � ( I v� � K��GQa��� OMQ ( �2� � K��O®q����Ó � K��E����� ��_�1�:29�\]\H9��c356P_�K5354#X<BD1'35Cd1'3y�O® � ���[Fq1'356P;56;5K546#C-X<7%9�Xu_�1�:K � � � 1'35687H9�C|­ ��( � � � � � K�� � (���� ((� � K � � v � I ��� K��GQa��� O Q+(/­ �"( � � � � � K�� � (����¹t(X<7H68BD=)^�1�:EX[9�3qXP1�`%CE6i:EA�9�X<BD1'3>BDCX<7H9�X � � � � v'� vj� �{�V9�35;c_�1�:*K � � � 1�3H6�7H9�C� � K � � � v � � � � � � K��E� �dÓ~��� (�� >�K�� ��(���� (/­ � � � � � K�� � (���� ( v� � � � K � � v m� _�K53H4iX<BD1'3!K � � � B]C@\DBD_�X<6#;°X<1~9�_�KH354iX<BD1'3hBD3,� v'� vj� �8ºj�Z`I?°X<756Y6il,^H\DBD4BgX@_�1�:<=@KH\]9 � �GQ � *5��� K��GQa� ( " � ! � � � F9�35;*1'356�4[7564<b(C�B]=W=W6;5B]9�X<6#\D?¶X<7%9�X�X<756�XE:[9�4#6~1'_ � BDCEK/9�35;*X<7H9�X!� � � � � v � ­ (#� � K � � v F �� �� � �� ! �� �� v � ­�� � K � � v 9�35;�� �� � ���� �� �� v � ­ (W� � K � � � v ( º� � � K � � v m � 354#6P1'356�b,3H1ePC-X<7H9�X-X<756PK535BD1'3W1'_�9�\D\ � � _�1�:( ��®ZBDC-;5635CE6P1�3H6�7H9�Cu9pe�9?fX<1Z\DB]_]X@ � �uº�� �{��`I?Ze�:<BgX<BD3Hk�BgX�9�C�9PCE6i:<BD6C � � ��v K � F�4[7H1I1'CEBD3Hk|® � � � �d9�35;{4[7H1I1'CEBD3Hk�K v CEK54}78X<7H9�X�� � @ >)K v � � v ( � � � @ � � v FX<75639K º CEK54}7>X<7H9�XW� �¹� @ > K v � > K º � � v ( � � � @ > K v � � v ( � º � � @H� � v F%9�35;>CE1c1'3aFHCE1)X<7%9�X � � ��v � � K � � � v � ( vv " � � � @ � � v m¢�756W;56#35CEBDXs?�BDCo^5:<1jAq6#;~`I?�9�^5^5:<1jl(BD=c9�X<BD3HkY_�K5354iX<B]1�3HCpBD3 C � � �{�[FaeP75B]4[7~B]CZ9Y;H6#35CE6)CEK5`HCM^%9�4#6W1'_ ��v� �@�[m{�%1�:� � C � � �{�d1'35684#1'35CMXE:<K54#X<C�X<7H6pB]3qX<6i:E^�1'\]9�X<6; � _�K5354iX<B]1�3�� � ��eP75BD4}7�B]C�X<756Z_�K53H4iX<BD1'3>1�_ � � CEK54[7>X<7H9�X�� � �u� �(��*��d� � (��@_�1�:Y9�\]\ � � � t|1'356w4[7564<b(Cr69�CEBD\g?hX<7H9�X � � � �d�GQa�8> �u�GQa���'( �P� (��@_�1�:Y9�\]\�Q�FueP756i:<6 ��B]C)X<756y=W1,;5K5\DKHC

vy¢�756#?>e26i:<6{e21�:Eb(BD35kr1'3~JIK56#CMX<BD1'35C|1'_�CE49�XEX<6#:<BD35kY9�3H;yX<756i?w3566#;56;�9cCM^�9�46{eP751'CE6@�-R����Q����¶XE:[9�35CE_�1�:<= 7%9�CXE:[9�4#6Cp1'3�CM^H7H6i:<6#C#F�ePBgX<7�XE:[9�4#6Co`�6\D1'35k'BD35kYX<1 �uº�t�X<756i?�BD3IXE:<1I;HK54#6;�X<7563~X<756)�-R����Q�����XE:[9�3HCE_�1�:<= 1�_dX<756)CM^%9�46;56CE4i:<Bg`�6;>756#:<6�mº 1 756#3w02e�9�Co9WCMX<K5;56#3IXFH0�7H9�;>XE:<BD6;>X<1W:<69�;wÙd=WB]\DBD1rÐo��¤V¥,Q���� �-R × CP9�:EX<BD4\D6�FH`HK(X|351�XPb(351jePBD3HkW=@KH4[7w0sX[9�\DB]9�3

9�XdX<756�X<BD=c6�0V7%9�;cXE:<1'K(`H\D6PK535;H6i:<CMX[9�35;5BD3Hk8eP7H9�Xu756�e�9�Cd;51'BD3HkZePBgX<7r9�\D\5X<756CE6P4#K5`�6#C-eP75BD4}7r9�^5^�69�:<6;cBD3c75BDC-^5:<1,1'_�m�~9�3q?>?q6j9�:<Cp9�_]X<6#:FH0�4#1'35CMXE:<K54#X<6#;y=@?>1jeP3y^5:<1,1�_sF%`HK(Xp9�\gX<751'K5k'7w0�7H9Aq6@351�XPXE:<B]6#;>X<1r4[7H6#4<byBD_V={?>K535;H6i:<CMX[9�35;5BD3Hk1'_a0sX[9�\DB]9�3rBDC2`�6iXEX<6#:2351je 9�35;�BD_a0d49�3YKH35;56i:<CMX[9�35;�7HBDCd^H:<1I1'_sF,0-X<75BD3(bcX<7H9�X2=@?r9�:<k'K5=W63qX�=@K5CMX2`a6oBD;H6#3qX<BD4j9�\HX<1f75BDC1�:<BDk'BD3H9�\aBD;56j9(m� no6#:<68e26f6#3541'K53qX<6i:pX<756@1�X<756i:p=c69�35BD3Hkr1'_VX<756{e21�:<;~BD3qX<6#:E^�1'\]9�X<BD1'3aF%k'1'BD35kr`%9�4<bwX<1YN���¤�����T�¤���F�9r4\]9�CECEBD49�\

X<1,1'\aBD3>�oK5=c6i:<BD4j9�\a�o3%9�\D?,CEB]C#m

(�­

Page 73: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

1'_oKH35BD_�1�:<=�41�3IX<BD3IK5BgXs?�1'_ �dF2CE1�X<7H9�XceP75639( � � ®(FuX<7H6>CE6jJIK56#3546 � � % �Ø4#1'3qAq6i:<k'6CcK535BD_�1�:<=W\g?¶X<1 �Ø9�35;�9�C- � * */. L�%i� � � �V� ��- � * */. L�%i� �V��Ó1! >�( � Ó ( #�1�3H6{9�\]CE1)7%9�CW� � � � % �6> �8� � v �·®(m

GI9'9�b�±d���%� ���y7%9�CPCE751jeP3>X<7%9�X�X<7H6i:<6Z;H1I6C�3H1�X|6il(BDCMX|9)\DBD356j9�:P4#1'3qX<BD3,K51'K5CP\DBD_�X<BD35kf_�:<1'= � v � �@�uBD3IX<1�� v'� v � �{��+027H9Aq68356iAq6#:P\D1,1�bq6;�_�1�:�X<756p^5:<1,1'_[�[m

( �

Page 74: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�@�,m�1°6#;53H6#CE;H9?��|^H:<BD\��j­IF�­�®'®'®(m�okq9�BD3hX<756YBD;H69yBDCfCE751jeP3�1'3hX<756�49�CE6Y1'_ �8º�Fd9�35;�k'BgAq6#3�9�_�KH354iX<BD1'3,@ � �uº�� �{�81'356re�9�3qX<CfX<1�4#1'35CMXE:<KH4iX

�6� � ;hvj� �8ºj� � �dº�� �8ºj� � v � º �¹v eP751'CE6uXE:[9�4#6u1'3@X<756 QW9�l,B]C�BDC3@ mV�H1�:-9@�+^�1'CEBgX<BgAq6��a=W6#CE7fCEB]�#6 (f1'356u41'35CEBD;56#:<C�X<7H6uCM^%9�46� � 1'_,_�K5354#X<BD1'35C�BD3"�uº�� �{��eP75BD4}7{9�:<6u9��c356d1'3{69�4[7{BD3qX<6#:EA�9�\H� � ( � � �%ÓW�j� (���_�1�: � ��� F�9�35;841'3qX<BD3IK51'K5C�9�X X<7H6-3H1I;H6#C� ( � � ��� tq`a6#49�K5CE6�X<756#:<6�6#l,B]CMX<C-® �,� � �~CEKH4[7WX<7H9�X �u� � K��O®I��� º%Ó!� K��E����� º�[v � º ( � I v� � K��GQa��� ºPOMQ � v � º ( �2� � K��O®I��� º%Ó� K��E�j��� º � v � º _�1�:�9�\D\,9��c356u_�KH354iX<BD1'35C�1'3Y�O® � ���[F�1'3562;56#;5KH4#6#C�X<7H9�XV_�1�:BK � � � 1'356u7H9�C � ­ � � ( � � � � � � K�� � (���� º � v � º (� � K � � º � � I���� K��GQa��� ºPOMQ � v � º ( � ­ � � ( � � � � � � K�� � (���� º � v � º tqX<7H6�BD=W^�1�:EX[9�3qX-1�`HCE6#:EA�9�X<BD1'3WBDC�X<7H9�X � � �,;°vj� �@��9�35;_�1�:�K � � � 1'35687H9�CW� � K � � � º � v� � � � � � � � K��E� �8Ó���� (��T> K�� � (���� º � v � º ( º� � � � � � � � K�� � (���� º � v � º ( � � � � K � � º m� _�KH354iX<BD1'3 K � � � BDCc\DB]_]X<6;�X<1¶9�_�K5354iX<B]1�3/BD3,;hvj� �8ºj�@`I?�X<756y6il,^H\DBD4BgXc_�1�:<=fK5\]9 � �GQ � *5�E�SK��GQa� ( " � ! � � � F9�35;�1'356�4[7564<b(C)BD=W=c6#;5B]9�X<6\g?~X<7H9�XfX<756YXE:[9�46Y1'_ � BDC�K°9�35;¶X<7H9�X6� � � � � º � � (#� � K � � º F �� �� � �� ! �� �� º � v� �

� � K � � º 9�35;�� �� � ���� �� �� º � � (#� � K � � � º ( �� � � � K � � º m � 356u7%9�C�X<7563 � � � � � � � � � � � ( � (#� � K � � º 9�35; � � � � � � / � � � � ( �� � � � K � � º _�1�:V® � (�� ��F9�35;~X<756#:<6#_�1�:<66� � � � � � � / � � � � � � � � � � � � / ��

� /((CE� � � � � v � º� / � � � � � � � � � v � º� � � � � � ( C�� � � K � � º mf¢�756#3�1'356)K5CE6CpX<756)_�9�4#XoX<7H9�XpX<7H6

K535B]1�3 1'_@9�\D\� � _�1�: ( �·®�BDC>;56#35CE6'FoCE1�X<7H9�Xw9�3q? @ � �uº�� �{�c4j9�3�`a6~e�:<BgXEX<6#3 9�Cw9¶CE6#:<BD6#C � � ��v K � F|ePBgX<7� � ��v � � K � � � º � ( C � � @ � � º m�o\gX<7H1�KHk�7h_�K5354#X<BD1'35C@B]3!;°v � º�� �{�89�:<6Y351�Xf41�3IX<BD3IK51'K5C#Fd9�C@X<7H6i?h9�:<6Y351�X)6iAq6#3h`�1'K535;56;aFV^%BD6#46iePBDCE6rCE=c1,1�X<7

_�K5354iX<B]1�3HC�eP7HBD4[7~9�:<6{;5BDCE41�3IX<BD3IK51'K5Cp9�Xp9W^�1'BD3qXo;51Y351�X|`�6\D1'35kWX<1�;hv � º�� �{�[mP�%1�:p6il59�=)^H\]6�F%\]6iX � � C�� � �{�2ePBgX<7�!�¸�{3569�:p®r9�35;�\D6iX � � ��;·eP756#:<6�;ÛBDCPX<7H6{no�%�,�VQSU#Q �V�~_�K5354#X<BD1'3at%X<756#3 � �T� � � Ó � ePBgX<7 � � C�� � �{��9�35;X<756#:<6#_�1�:<6f­ � � � � � � �I� ���uÓ � � � �q��CE1cX<7H9�X�� � � �,`�67H9A'6C|\DBDb'6 vº � � � 3569�:),/F�9�35;yX<756#:<6#_�1�:<6r�E�2Ó�� ���¹v � ºj��� � � � ��� º � �{�[FHCE1 � �� ; v � º � �{�[m

�oC�BgX�CE66#=WC�X<7H9�X�_�K5354#X<BD1'35C�BD3 ;hv � º�� �{�u49�35351�X�7%9#Aq68;5BDCE4#1'3qX<B]3IK5BgX<BD6C�9�X|9@^�1'BD3qXF51'35686il,^�64iX<C�CE1'=W6pb(BD3H;�1�_4#1'3qX<B]3IK5BgXs?qF%`HK5Xp1�_d9r;5Bg �6#:<6#3qXp3H9�X<K(:<6f9�C|X<756{A'9�\]K56f9�XZ9c^�1'BD3qXo;H1I6Co351�Xp=c9�bq6@CE6#35CE6'mp¢�756f_�1'\D\D1jePBD3HkcBD;H69�Co7H9Aq6`�66#3�BD3qXE:<1,;5K546#;~`I?~GI9�4jJqKH6#CMLON�1'K5B]CZN�QSR�T�UW9�35;�Ùd3(:<B]4#1>����¤V�%T���U'F�9�3H;�CE1'=W6f:<6\]9�X<6#;~e21�:Eb�7%9�CZ`a6#6#3�;51'356)`I?±dBD6i:E:<6@Ðo��Q+U#�5��� �2v#m� & ����$ §�0�_ � � ;hv � º'� �{�-X<7H6#3w1'356Z7H9�C

�� � � � " � � " � � �� ��� � �uº�� �{�[m

�2È<ÆjÆ5#[§ 2?�no����� � × CrBD356jJIKH9�\DBgXs?�1'356�7H9�C �� �� � � � � � " � � � � ���� �� �� º ( CE� � � � � � º _�1�: � � ;°vj� �{�[FP9�35;�CEBD=cBD\]9�:<\g?�1'356y7%9�C�� �� � � � " � � " � � � � ��

�� �� �� º (�CE� � � � � � º Fa9�35;yX<7H6i:<6#_�1�:<6 �� �� � � � � � " � � " � � ��

�� �� �� º ( ­ C�� � � � � � º _�1�:Z9�\]\ �<� ;°vj� �@�[m � 356f4#1'35CEBD;H6i:<CPX<7H6=c9�^5^%BD35k � �� � > �� eP75BD4[7>=c9�^HC�;°vj� �@�dBD3qX<1 �uºo_�1�:�X<756p=c69�CEK(:<6 � ��

�� � 9�35; �uº�� �{�dBD3IX<1 �uº|_�1�:�X<756Z=c69�CEK(:<6 O Q�F

9�35;yX<756#:<6#_�1�:<6{BDXo=c9�^HC ;hv � º�� �@� � � ;°vj� �@� � �uº�� �{��� v � º � º BD3qX<1cX<7H6@4#1�:E:<6CM^�1'35;5BD35kYBD3IX<6i:E^�1'\]9�X<BD1'3wCM^%9�4#6'F�eP7HBD4[7�BDC�uºp_�1�:�X<7568=W6j9�CEK5:<6 � ��

�� m

�IB]=WBD\]9�:<\g?qF�GI9�4JIK56CMLON�1'K5BDC>N�QSR�T�U�9�35; Ùd3(:<BD4#1�����¤V�%T���U�351�X<BD4#6;/X<7%9�X�eP7H6#3 4#1'35CEBD;H6i:<BD35k¶X<756�BD3qX<6#:E^�1'\]9�LX<BD1'3yCM^%9�4#6C � ;hv� ���P� � �uº����P� � � � º _�1�:p9c`�1'K535;56;�1�^�6#3�CE6iXoePBgX<7�9rN�Q A�U�²�³�Q���µr4#1'3qX<BD3IKH1�KHC|`�1'K535;H9�:E?'FaBgXo;51,6#C|k'BgAq6; v " �� ���P�2_�1�: � � vº �O9�3H;w1'356{7H9�C); �� ���P�*��; � ���P�2_�1�:o® ( -"( vº �[FH`HK(Xo_�1�: � � vº BDXPk'BgAq6#C|9W356#exCM^%9�4#6'FHeP75BD4}7X<756#?Y;H6#351�X<6#;�; v � º�'� ���P�[m� & ����$ §�0�_ � � ; v � º�'� �O® � ,��H� � ; v� �O® � ,�� � � º �O® � ,���� v � º%� º F5X<7563 �� � � � º �O® � ,��[m�2È<ÆjÆ5#[§u�oC � � ;hv� �O® � ,��2BD=W^H\DBD6C � � � �uº��O® � ,��2`I?wno��� � � × CPB]356JIKH9�\DBgXs?qFH1'356@7H9�C � ;°v� �O® � ,�� � �dº��O® � ,���� v � º%� º �

� �dº � ® � ,�t � �� � � � �dº��O® � ,�t:OMQa� � v � º%� º � �uº � ® � ,/t � �� � m� :<6#\�9�X<6;¶:<6CEK5\gXcBDC)X<7H9�XcBD_ � � ; v� ���P�8X<7H6>6il,X<6#3HCEBD1'3 �� 1'_ � `I?�®�1�K5X<CEBD;56 �!`�6\D1'35k'CfX<1 ; v � � ! �f9�35;

CEBD=cBD\]9�:<\g?�BD_ ��� ; �� ���P�W�S4#\D1'CEK(:<6Y1'_)C � ���P�pB]3!; � ���P�E�pX<7563 ���� ; � � �)!Z�Z_�1�:)® ( -+( �YBD_ - � vº FV`HK5Xf351�X_�1�:�-�� vº tdX<75BDCfB]C@69�CM?hCE6#63�9�C�� � ; v� º� ���P��� ;°v � º����P�89�35;¶X<756Y6il,X<6#3HCEBD1'3°`I?¶®�BDC@^HBD6#46iePBDCE6YCE=W1,1�X<7¶9�35;

vw±dBD6#:E:<6{Ðo��Q+U#�5��� �2FH�5:<6#3H4[7w=r9�X<756#=r9�X<B]4#B]9�3aF,L<�j�'��Õ5m-no6Ze21�:Ebq6;>BD3>�oBD46�m

(�Õ

Page 75: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

;5BDCE41'3qX<BD3IK51'K5CP9�35;YX<756i:<6_�1�:<6pBDC2351�X�BD39;°v � º�� �)!Z�[m��o4iX<KH9�\D\D?'F4; v � º�'� ���P�-4j9�3Y`�6p4[7%9�:[9�4#X<6i:<BD�6#;Y6#BgX<7H6i:�9�C2X<756pCM^%9�461'_�_�K53H4iX<BD1'35C2B]3�; v � º ���P�-CEK54[7YX<7H9�X �� � � � � � � º ���P�[FIeP7H6i:<6O%�GQa�-B]CuX<756p;5BDCMX[9�354#6o1'_ Q�X<1@X<756o`�1'K535;%9�:E?�� �oF51�:P9�CX<7568CM^%9�4#681'_�_�K5354#X<BD1'35C �6� ;°v � º����P�dCEK54}7>X<7H9�X ��6� ;hv � º�� �)!Z�[m

�Ø:<6#\]9�X<6;>;HBg�c4#K5\gXs?�BDCPX<7H9�X|9�3q?�^%9�:EX<B]9�\ ;H6i:<BgA'9�X<BDA'6 ���� �

=r9�^HC);hvj���P�uBD3IX<1$�dº����P�29�35;#�uº����P�uBD3IX<1�; " vj���P�[F9�35;>BgX�;51,6#C�=r9�^ ; � ���P�-B]3qX<1#; " � ���P�H� � ; �� ���P��� � _�1�:P®�(9-�(��Z9�3H; - � vº `HK(X�351�XP_�1�: - � vº F(`�6#4j9�K5CE6pBD3YX<75B]C49�CE68BgXP=c9�^HC�;hv � º����P�uBD3IX<1fX<756Z;HKH9�\ 1'_T; v � º�'� ���P�[m¢�75B]C8X<64[753HBD49�\d;HBg�c4#K5\gXs?°9�^5^�69�:<C@eP756#3¶1'356rCE1'\gAq6#C{`�1'K535;%9�:E?°A�9�\DK56c^5:<1�`H\D6=WC@\DBgbq6 >)? � � @¶BD3 � ePBgX<7X<756{`�1'K535;%9�:E? � ��=r9�;56@1�_VXse21Y;5BDC��E1�B]3qX|^HB]6#4#6C#F���� eP756#:<6f9c£ZQ���Q+²�³�¥,�%�¶4#1'35;HBgX<BD1'3yBDC|B]=)^�1'CE6#;�9�35;�� ! eP756#:<69��o����p��T�T/4#1'35;5BgX<B]1�3~BDCZBD=)^�1'CE6;at B]3�X<756)49�CE6feP756i:<6fX<7H6@^%9�:EX<C���� 9�3H;�� ! 7%9#Aq6W9Y41'=W=W1'3�`�1'K535;%9�:E?qFaX<7H6^5:<64#BDCE68^�9�BD:<CP1'_�;H9�X[9r9�\D\D1je26#;yBDCo9c\]BgXEX<\D6pX<6#4}7535BD4j9�\�X<1c4}7H9�:[9�4iX<6i:<B]�#6�m�0�_-9�\D\ X<756{`�1'K535;%9�:E?yBDC�� ! X<7H6#3yX<7568^5:<64#BDCE6CM^%9�4#6�BDC ; " v � º�� � �P�[F�X<756�;5K%9�\51'_�;°v � º�� � �P�d�+XE:[9�46#CV1'_%_�K5354#X<BD1'35CV1�_�;hvj���P�E�[F�`HK5XVX<756�BD=W^�1�:EX[9�3IX�^a1�B]3qX�BDC�X<7H9�X-1�3H649�353H1�X�:<6CMXE:<BD4iX-9�3)6\D6#=c6#3qX�1'_�; " v � º�� � �P��X<189|^�9�:EX� ! eP751'CE62`�1'K53H;H9�:E?@B]C�CE=W1,1�X<7f6351'K5k'7aF'`�6#4j9�KHCE62:<6#CMXE:<B]4iX<BD1'3BDCuX<7H6oXE:[9�35CM^�1'CE6#;�1'_ X<7H6o1�^�6#:[9�X<1�:�1'_�6il,X<6#35CEB]1�3r`I?�®(FH9�35;YX<75BDC26#l,X<6#35CEBD1'3�1�^a6i:[9�X<BD1'3 � ;H1I6C�351�X�9�4#X�1'39;°v � º��� ! �BD_�X<756c�S�|L<����LO;5BD=W635CEBD1'3H9�\ano� �U���R�� �?�>=W69�CEK(:<681'_����*BDC�^a1�CEBDX<BgAq6�m

¢�756CE6oX<6#4}7535BD4j9�\�;HBg�c4#K5\gX<B]6#Cu=c9?rCE66#=�JqKHBgX<6p9�49�;56#=cBD4�F,`HK5X�CE1'=W6p=W1,;56#\]BD�9�X<BD1'35CuBD3y¡u1'3qX<BD3IK5KH=���64[7H9�35B]4#C\D6j9�;yX<1rK5CEBD35kY1�^�6i:[9�X<BD1'35C|eP75BD4}7�9�:<6@351�Xo;56#zH356;�BD3~9�3�1�`IA(BD1'K5C|e�9?'Fa9�35;~BgXoBDC|BD=W^�1�:EX[9�3qXPX<1rK53H;56i:<CMX[9�35;~BD_�1�3H6CE751'K5\D;W:<6s�E6#4#X-CE1�=c6�\]9ePCd9�C�`�6BD35kZKH3(^H7q?(CEBD4j9�\H1�:dBD_�1'356�CE751'K5\D;WXE:E?@X<181Aq6#:<4#1'=W6�X<756�=r9�X<756#=r9�X<B]49�\(;5Bg�c4#KH\gXs?@X<7H9�XX<756#?r:<6i^5:<6CE6#3qXm � 356ZCEKH4[7w6il59�=)^H\]6pBDC�X<756ZCMX[9�X<BD4Z\]9e 1'_�_]:<BD4#X<BD1'3Y;5KH6pX<1Y¡�R��¥5R��8¯ º�F5eP75BD4[7wBD3qAq1'\gAq6#C�9)CEBDk'3>1'_�9351�:<=c9�\�_�1�:<46o9�XuX<756P`a1�KH35;H9�:E?r9�35;�9�3YBD356JIKH9�\DBDXs?)1�3rX<756oCMXE:<6#35k�X<7Y1'_ 98X[9�35k'6#3IX<B]9�\%_�1�:<4#6o9�XuX<756|`�1'K535;H9�:E?�t(B]_a1�3H6K5CE6C{N�BD356j9�:<B]�#6#;°Ùd\]9�CMX<BD4BgXs?'F�X<756r3H9�X<K(:[9�\dBD3H_�1�:<=c9�X<BD1'3�41'=WBD35ky_]:<1'= X<756czH35BDX<6#356CEC{1'_2X<756rCMX<1�:<6#;°6\]9�CMX<BD4W6356i:<k�?k'BgAq6#C�X<7H6uA�9�:<BD1'K5C�_�1�:<4#6CV9�CV351�:<=c9�\,XE:[9�46#CV1'_H_�K53H4iX<BD1'35CVBD3#;h�GO,��F%t �P�[F'BSm 6�m�6\D6#=c6#3qX<C�BD3#; " v � º � � �P�[t'K535_�1�:EX<K53H9�X<6#\D?'F1'356Z4j9�35351�X�;56#zH356ZX<75689�`%CE1�\]K(X<6pA�9�\DK56p1'_�9�3w9�:E`HBgXE:[9�:E?Y6#\D6=W63qX�BD3 ; " v � º�� � �P�[tH751je26iAq6#:F51'356Z49�3>;H6izH356ZeP7H9�XP9351'35356kq9�X<BDA'6�6#\D6=W63qX�BDCV`,?fCMX[9�X<BD35kpX<7H9�XdBgX�BDC-9Z351'35356k'9�X<BgAq6�=W69�CEK(:<6'FqCE1ZX<756PJIK56#CMX<BD1'3)B]CVX<1oz%35;c9pe�9#?fX<1Z6#lI^H:<6#CEC¡�R��¥(R��8¯ × CP\]9e eP75BD4}7w=c9�bq6#C�CE6#3HCE68_�:<1'= 9)=r9�X<756#=r9�X<B]49�\�^�1'BD3qXP1'_�A(BD6ie8F%9�\gX<751'K5k'7�X<756#:<689�:<68BD35;HBD49�X<BD1'35C�X<7H9�X¡�R��¥(R��8¯ × C�\�9#e�B]C�351�XP6il59�4iX<\g?reP7H9�X�:<69�\ =r9�X<6#:<B]9�\DC�_�1'\D\]1e��O9�CP;(?(3H9�=WBD4C2^H\]9?>9f:<1'\D6��[m

º°¡u7H9�:<\D6C��oK5k'K5CMX<BD3 �V��¡�R��¥5R��8¯�F5�H:<6#354}7>=r9�X<756#=r9�X<B]4#B]9�3aFa�)( ��Ò�L<� ��®qÒIm

(�Ô

Page 76: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�#Õ5m-��1'35;H9?��|^5:<B]\�� (IF�­�®'®'®(m� 356{7H9�CP^H:<1Aq6;wX<7H9�X|_�1�:p9�\D\�BD3qX<6k�6#:<C|® � � � �9�35;~9�\D\d�"( * (�, 1'356{7H9�C � ���� �5� �)!Z� � ���5� �)!p� � � �¹v �� � � �(� �"!p� � � ��� � �5� �)!p� � ���H� �)!p� � � � ePBgX<7 ��E�> � �� ��mc�%1�:�- � ®YeP75B]4[7�BDC8351�X@9�3�BD3qX<6k�6#:81�3H6W;56#zH356#C

X<756)�(RV¯ R�¥,�H��CM^%9�4#6W� � � �5� �)!Z�29�C � ��� � �5� �)!Z� � ���5� �)!p��� � � � ePBgX<7� � -{9�35;h�E��> �H� � -�F�9�3H;w1'356{;56#;5KH4#6#C_�:<1�=�X<756>:<6BgX<6i:[9�X<BD1'3�X<7561�:<6#=�1'_pGI9�4JIK56#CMLON�1'K5BDCrN�QSR�TVU�9�35;/GI9'9�b�±d�%��� ����X<7H9�Xr1'356�9�\DCE1°7H9�C � � � �5� �)!Z�9�� � � / � � � � ! � � � � � � � � � ! ��� � � � ePBgX<7y6jJIK5BgA�9�\D63qX|351�:<=cCoBD_ v�� º 9�:<6@351'353H6#kq9�X<BgAq6{B]3qX<6#k'6i:<CoCEK54[7�X<7H9�X -R�¸�E� >�I� v Ó � º _�1�:VCE1'=W6 � � �O® � �j�d�SBSm 6'm�6BgX<756i: v �9- � º 1�: v �9- � º �[F'9�3H;@X<75B]C�BDCVCMX<BD\]\qXE:<K5626iAq6#3fBD_� v�� º9�:<6f351'35356k'9�X<BgAq6{:<69�\DCp3IK5=@`�6i:<CoeP75BD4}7�9�:<6f351�Xp3H6#4#6CEC<9�:<BD\g?wBD3IX<6#k'6i:<C#F 9�\ge�9?(CoK535;56#:pX<756@41'35;5BgX<BD1'3 � � �O® � �j�[Fa1�_4#1'K(:<CE6'm

�IB]=WBD\]9�:<\g?qF�BD_%_�1�:- ��®p9�35;w��( 5 (-,�1'356�;56izH3H6#CVX<756 2�aU�Ra��CM^�9�46 � � � �8 � �)!Z��9�C � ��� � �5� �)!Z� � ���5� �)!p��� � � 8ePBgX<7 � -f9�3H;¶�E�)> �H� S� -�F�1'356f;56#;5KH4#6#C|X<7%9�Xp_�1�:p351'35356#kq9�X<BgAq68:<6j9�\]C - v�� - º CEK54[7�X<7H9�X -R�¸�E� > �I�6- v Ó���- ºePBgX<7 � � �O® � ���[F�1'356d7H9�C � � � �8 � �)!Z�H� � � � / � �H� �)!p� � � � � � �H� �)!p���� � 8 9�35; � � � �8 � �"!p�H� � � � / � �8 / � �)!p� � � � � � �8 � � �)!p���

� � 8ePBgX<7>6JIK5BgA'9�\D6#3qX�351�:<=cC#FH_�1�:|9�\D\��)( 5'v��65�º (-,�m0�_�-ZBDC�351�X|9�3>B]3qX<6#k'6i:FH1'356Z7%9�C� � � �5� �"!p�H��� � � �� � �)!p�[m�o\gX<7H1�KHk�7h1'356Y7H9�C�;hvj� �)!p��� � ;�º�� �)!Z� � �uº�� �"!p� � v � º%� º FVX<756r4j9�CE6r_�1�: *,� , BDC@;5Bg �6#:<6#3qX)9�35;hX<756rCM^%9�46

� v'� >� �)!p�feP75BD4}7/BDCcX<756yCM^%9�46y1'_pN�Q AaU�²�³�Q���µ°41'3qX<BD3IK51'K5CY_�K5354#X<BD1'35Cy�S1'_�X<6#3�;H6#351�X<6#; ��� *�� �)!p�E�[F�BDCr9�^H:<1�^a6i:CEK(`HCM^�9�46~1�_ � �xº � �� �"!p� � ���� �"!p��� v � º � F�eP75B]4[7/41'BD354#B];56#CWePBgX<7�9hCM^%9�4#6�BD3qXE:<1,;5K546#;/`I?��o3qX<1'35B����-¤����T��9�35;�;56351�X<6#; � v � �"!p�[F�eP75BD4[7�BDCcX<756�CM^%9�4#6~1�_Z_�K5354#X<BD1'35C � � � �� �)!Z�)_�1�:YeP75BD4[7/X<7H6i:<6y6il(BDCMX<C C CEKH4[7�X<7H9�X� � �GQ�>�(���Ó � �GQWÓ�(�� >h­ � �GQa���(�C � ( �,_�1�:o9�\D\ Q � ( � �)!{F�1�:o6JIK5BgA'9�\]6#3qX<\g?!� � � � � Ó � " � � >¶­ � � � (5CE� ( �,_�1�:p9�\D\( � �"!8m

� 356{;56;5K54#6Co9�\]CE1WX<7H9�X|BD_�- ����eP7H6i:<6 ��BDCo9)^�1'CEBgX<BgAq68BD3IX<6#k'6i:FHX<7563w_�1�:o6#A'6#:E?>=@K5\DX<BgLOBD35;56#l �hePBgX<7$� ��� � �X<756�;56i:<BDA�9�X<BD1'3 ���B]Cr\]BD3569�:r41�3IX<BD3IK51'K5C�_]:<1'= � � � �5� �"!p�)B]3qX<1 � � " � � � � �H� �)!p�c9�3H;*9�\]CE1h_]:<1'= � � � �8 � �"!p�)B]3qX<1� � "�� � � � �8 � �)!Z�[FZBD_Y� ( * �65 ( ,/m ¢�75BDCw_�1�\]\D1jePC>B]=W=W6;5B]9�X<6#\D?/_]:<1'= X<7H6�_+9�4iXwX<7H9�Xy_�1�:~9�3I? BD3qX<6k�6#: �

X<756�;56i:<BgA'9�X<BD1'35 �� BDC�\DBD3H69�:�4#1'3qX<BD3IKH1�KHC�_�:<1'= ���� �5� �"!p�cBD3IX<1,��� " � � � � �H� �)!p�[Fp9�3H; 9�_�X<6i:>4[751,1'CEBD35k v�� ºCEK54}7ØX<7%9�X � ( v � - � º 9�35; 41'=)^HK(X<6; � � �O® � ���>CEKH4[7 X<7H9�X - � �E��> �I� v Ó �H º F)1�3H6�9�^5L^H\DB]6#C�X<756@BD3qX<6i:E^a1�\�9�X<B]1�3>^5:<1�^�6#:EXs?'F�9�35; ���=c9�^HCo4#1'3qX<BD3IKH1�KHCE\g? � ��� / � �5� �"!p� � ��� � � �5� �"!p��� � � 8 � � � � �8 � �)!Z�2B]3qX<1

� � � / "�� � � � � � � ! � � � � � " � � � � � � � ! ��� � � 8 �9� �" � � � � �8 � � ! �[m

�%1�:�� 9�3�1�^�63�CEK(`HCE6#Xc1'_ �)!{Fu1'356>=c9?�;56iz%3566� � � �5���P�@_�1�:r9�\D\2^�1'CEBgX<BgAq6Y:<69�\ -�eP75BD4}7�9�:<6>351�XcBD3qX<6#k'6#:<CBD3h9�X@\D6j9�CMX{Xse21y;5Bg a6i:<6#3IX8e�9?(C#t�X<756cz5:<CMX@1'356Y9�C{X<756cCM^�9�46r1'_u:<6#CMXE:<B]4iX<BD1'35C{1'_2_�KH354iX<BD1'35C@_]:<1'= � � � �5� �"!p�[F�ePBgX<7X<756)JIK51�X<BD63qXp351�:<= � � � � � � �

� �

� J � � BD35_��

� � � � � � � � � �� �

� � 1 � F�eP75B]4[7�ePBD\D\�`�6f;56351�X<6#;�� � � �r_�1�:oX<7H6f;5BDCE4#KHCECEBD1'3ataX<7H6CE6#41'35;w1'356{9�C|9�3yBD3qX<6#:E^�1'\]9�X<BD1'3�CM^%9�4#6 � ��� / � �5���P� � ��� � � �5���P��� � � � ePBgX<7� v�� º 351'353H6#kq9�X<BgAq6ZBD3qX<6k'6i:<C�CEK54}7wX<7H9�X- �Ø�E��> �I� v Ó �H º ePBgX<7w® � � � ��F5eP75BD4[7>ePBD\D\�`a6Z;56351�X<6#;�� � � �)_�1�:�X<756Z;HBDCE4#K5CECEB]1�3°�S1�_�41'K(:<CE6�FH1'356849�3w9�\DCE1k'BgAq6pXse21W;56#zH35BgX<BD1'35C�_�1�:|;56iz%35BD35k)X<7568CM^%9�4#6#C�� � � �8 ���P�E�[m

�6#4j9�K5CE6{X<756{:<6CMXE:<BD4iX<BD1'3wX<1�� BDCo\]BD3569�:o4#1'3qX<B]3IK51'K5Co_�:<1�=S��� / � �5� �)!Z��BD3IX<1 ��� / � �5���P��9�35;�9�\DCE1r\DBD3H69�:o4#1'3(LX<BD3IKH1�KHCp_�:<1'= ��� � � �H� �)!Z�PBD3qX<1 ��� � � �5���P�[F BgXZBDCp4#1'3qX<B]3IK51'K5CZ_]:<1'= � � � �H� �)!p� � � ��� / � �5� �)!Z� � ��� � � �5� �)!Z� � � � �BD3qX<1 � � � � � � ��� / � �5���P� � ��� � � �5���P� � � � � FV9�35;h9�C{6#A'6#:E?�6#\D6=W6#3IX{1�_ � � � �wBDCZX<7H6W:<6#CMXE:<BD4#X<BD1'3�1'_�9�3°6\D6#=c6#3qX{_]:<1'=� � � �5� �"!p�[FH1'356Z;56;5K546#C�X<7H9�X � � � � � � � � �(m

0�_ZX<756y`�1'K535;%9�:E?/1'_"� BDCYCE=W1,1�X<7�6#3H1�KHk�7�CE1¶X<7H9�XrX<7H6i:<6�6il(BDCMX<C�9°41'3qX<BD3IK51'K5C�6il,X<6#3HCEBD1'3 ��eP75BD4}7�=c9�^HC��� / � �5���P�dBD3qX<1���� / � �H� �)!p�u9�35;6��� � � �5���P�-BD3IX<1 ��� � � �H� �)!p�[F(X<756#3�BDX�=c9�^HC�� � � ��� � ��� / � �5���P� � ��� � � �H���P��� � � �BD3qX<1�� � � � � � ! � � � � � / � � � � ! � � � � � � � � � ! ��� � � � Fa9�35;yX<7H6i:<6#_�1�:<6@6iAq6i:E?w6\D6#=c6#3qXo1'_ ��� � � BDC|X<756{:<6#CMXE:<BD4#X<BD1'3wX<1��1'_�9�3w6#\]6#=W63qX�1'_H� � � �H� �)!p�[FHBSm 6�m-1�3H6Z7H9�C � � � � � � � � �,m

¢�756y6#l,X<6#35CEBD1'3�^5:<1�^�6i:EXs?�7H9�CW`a6#6#3/CE751jeP3/_�1�:�� v'� �5���P�fBD_���BDCW`�1'K53H;56#;/ePBgX<7�9°N�Q A�U�²�³�Q���µ�41'3qX<BD3IK51'K5C`�1'K535;%9�:E?qF 9�35;�9�3�9�351'\D1'k'1'K5CoCEBDX<KH9�X<BD1'3�7H9�Co`�66#3�;56CE4i:<Bg`a6#;�eP75BD4}7ye21�:Eb(Cp_�1�:�� � � �5� �)!

��P9�35;�® ( �+(�~F 9�35;

BgX)6#lIX<635;5CfX<1�X<756�4j9�CE6�1'_�`�1'K53H;56#;�1�^a6#3�CE6iX<CfePBDX<7¶CE=W1,1�X<7¶`�1'K535;H9�:E?'m¶�o4iX<KH9�\D\D?¶�I���%Q�T�7H9�C)41�3HCMXE:<K54iX<6;�9�36il,X<635CEBD1'3YA'9�\DBD;>_�1�:|9�\D\T���� �5���P�uB]_ �*BDC�`�1'K535;56;>ePBgX<7>9WN�Q A�U�²�³�Q���µ)41'3qX<BD3IK51'K5C�`�1'K535;%9�:E?qm

(�Ò

Page 77: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�jÔ,m�1°6#;53H6#CE;H9?��|^H:<BD\��j�IF�­�®'®'®(m

�6#_�1�:<6@k'BgA(BD35k�9Y4[7H9�:[9�4iX<6#:<BD�9�X<BD1'3�1�_8� � � �5���P��BD3�X<756f4j9�CE6f1'_d9r`a1�KH35;56#;�1�^�6#3~CE6iXpePBgX<7~9YN�Q AaU�²�³�Q���µ�4#1'3(LX<BD3IKH1�KHC�`�1'K53H;H9�:E?'FHBgXPBDC�K5CE6_�K5\ X<1f`a6#k'BD3�ePBgX<7YX<7H6Z49�CE681'_��)!8m � 356ZCMX[9�:EX<C�ePBgX<7w9f^5:<6\DBD=WB]3H9�:E?c:<6#CEK5\gXm� & ����$ §�0�_d® � - � , 9�35;�� ( * ( , 1'356f7H9�C � � � � � � � v'� �5� �)!Z� � ���5� �)!Z��� v " � � ��� �!� ���5� �)!p�[FaX<756i:<66il(BDCMX<C C CEKH4[7>X<7%9�XW� � � > � � � � � � ( C � ( � � _�1�:o9�\D\ ( � �"!��qm�2È<ÆjÆ5#[§{0�_ � � � v'� �5� �)!Z�p1'356r7H9�C�� � � > � � � � � � ( � (B��� � KML.N4O%� � ��� � � 9�35;hBD_ � � ���5� �)!Z�o1'356r7%9�C�� � � > � � � � � � (­�� � � � � � F�9�35;/9�CcX<756w=c9�^5^HBD3Hk � �� � > � � � BDCc\DBD3H69�:Fu1'356wzH35;5CcX<7H9�Xc_�1�: � � �O® � ���{1'356y7H9�C � � � > � � � � � � (C � ( �¹v " � � � � � � � � / � �

� � 1 � � � �

� � 1�� � ��

� _�1�:|9�\D\ � � � � v'� �5� �)!Z� � ���5� �)!Z��� � � m¡u1'3qAq6i:<CE6\g?'Fo9�CECEK5=W6�X<7H9�X ��� ���H� �)!p�c9�35; � � � > � � � � � � ( C � (B� � _�1�:w9�\]\ ( � �)!{m � 3H6�;H6#4#1'=W^�1'CE6#C� � � � � � Ó°� � > � � � � �[FIeP756i:<6 � � BDC29ZCM^�64#B]9�\%CE=W1,1�X<75BD35k{CE6JIK56354#6'FIBSm 6�m � � �GQa�H� v

� 1 � v � � � �'FIePBgX<7 � v � C�� � �)!p�9�35;�I�� 1 � v �GQa�4O Q9�Ø��FI`%K(X21'356o9�;5;5C2X<756|7q?,^�1�X<756#CEBDCdX<7%9�X � v B]C29�3Y6iAq6#3r_�K5354#X<BD1'3am � 356|7H9�C � �GQa��>�� � � � � �}�GQa�H�I � 1 � � �+*H� � � �GQa� > � �GQ8> *5����O,*{9�35;@X<756#:<6#_�1�:<6R� � � > � � � � � � � ( I � 1 � � � �+*5����� � � > � ! � � � � O,* ( I � 1 C � � � �+*5����� * � � O&* �C�� � � m � 356Y7H9�C � � � � � � � � � (+� � � � � � v � � � � � � � C � � � � � � F�9�35;�_�1�:)9�3q?°;56#:<BgA�9�X<BgAq6 � � � ���� �

1'356Y7H9�C � �'� � � � � ���� � � � � � � � FP`HK(X>1'356~3566#;5C>9h`�6#XEX<6i:�6#CMX<B]=c9�X<6wX<7%9�3 � � � �'� � � � � ��� � � ( � �¹� � � � � ��� � v � � � � � � � � � � � � � � � FPeP75BD4}7�7%9�C351�XoKHCE6#;~9�\]\�X<756{BD35_�1�:<=c9�X<BD1'3w1'3 � t�4[7%9�3Hk�B]35k�*>BD3qX<16> *>BD3wX<7H6{B]3qX<6#k�:[9�\SF%1'356@4j9�3ye�:<BgX<6 � �'� � � � � �}�GQa��9�C|6BgX<756#:I � 1 � � � � � �}�+*H� � �GQ)> *H�MO,*{1�:8I � 1 � � � � � �}�:> *H� � �GQ2Ó�*H�MO,*{1�:u9�C�X<756�7H9�\D_%CEKH=�1�_%X<756CE6�Xse21pX<6i:<=cC#FI9�3H;fX<7HBDC�BDCVeP756#:<67H9A(BD35k>4[751,1'CE6#3�_�1�: � v 9�3�6iAq63�_�K53H4iX<BD1'3~B]CpKHCE6#_�K5\SF�9�C � � � � BDC89�3�1,;5;�_�K53H4iX<BD1'3�9�35;�X<756#:<6#_�1�:<6 �.�'� � � � � �}�GQa� �vº I � 1 � � � � � �}�+*5� � � �GQ�>�*5�3> � �GQYÓ�*5� � O,*%F�_�:<1�= eP75BD4[7�1'356W;56;5K546#C9� � � �'� � � � � ��� � � ( vº I � 1 �¹� � � � � �}�+*H����� � � ! � >� "�! � � � � O&* ( C I�� 1 � � �.� � � �}�+*5����� ( � � O,* � C � � � � " vmW�H1�: � � ��� v'� �5� �)!Z�o9�35;� v � ���5� �)!Z�[FaX<75B]CZCE751jePC8X<7H9�X�¶� % � � � ( C v �E�%Ó�� � " vi��Ó9C º % � � F,9�3H;)4}751,1'CEBD35k � � v� _�1�:�% ��Fq1'356�1�`5X[9�BD35C#�¶� % � � � ( C � %}v " � 9�35;W_�1�:d® ��% � �1'356f7H9�C��¶� % � � � ( %T� � � � � � �+`�6#4j9�K5CE6@BD3�X<756f49�CE6 � � ��� v 1�3H6)9�\De�9#?(Cp7H9�C|X<7H6f;56#41�=W^�1'CEBgX<BD1'3 � � ®pÓ � �[F 9�35;X<756#:<6#_�1�:<6 % " � �h� % � � � � � ��O® � ,��dePBgX<7 � �Ø��> -�m� ��%"-� j�D������ §P�H1�:Z®�1- � �@9�35;h� ( * � ,/F � � � � � � ! � ��� � � � � � � ! � � I � 1 I � 1

�� � � � " � � ! � � ���" ! � 1 � � �

O Q O,*�� ,�'F1�:P6JIK5BgA'9�\D6#3qX<\D? � � ���5� �"!p�29�3H; I � 1 �

� ��"� ��� �

�� ! � � � � � !� ! � 1 �-,�m�2È<ÆjÆ5#[§�N�6iX � � � � v:� � � � ! �W9�35; � v � � � � � ! �[m�0�_ ��� � � � � � � ! � � � � � � � v � � � � ePBgX<7 � � �#> -�FP1�3H67H9�Cw9�;564#1'=)^a1�CEBDX<BD1'3 � � � � Ó � v ePBgX<7 � � � � � � � v � � v 9�35;�� � KML.N4O%� � � ��� � � Ó %T� � � v � � � (Ö­ �¶� % � � �r9�35;% " � �¶� % � � � � ��� � ® � ,/t � �� �'mV�H1�:�* � �)!�1�3H6u7H9�C)� � � > � ! � � � � ( � � � � > � ! � � � � � Ó � � � v > � ! � v � � � ((� * ��� � KML.N4O�� � � ��� � � Ó­�� � � v � � � ( C � � * �'Ó v� � �¶� % � � �[tu1'356�4[751,1'CE6#C %�� v� ! ��9�35;�1'356�K5CE6#C6� � � > � ! � � � � ( C � * � � � v� ! � � � � m°¢�7HBDCfk'BgAq6CI � 1 �

� ��"� ��� �

�� ! � � � � � !� ! � 1 ((C I � 1 �� ! � ! � /�

� � � �� ! � � � � � !� ! � 1 F 9�35;�;56351�X<BD35kr`,? � !" v X<756�� & >*����LOno� VU��-R�� � ��=W6j9�CEK5:<6

1'_�X<7H6ZK535BDXPCM^H756i:<6'FHBgXPBDC ��C � !" v I � � � * � v " � � � v� ! � � � � � � � !� ! ����C � !

" v I � � % " � �¶� % � � � � � � �� �-,�m¡u1'3qAq6i:<CE6\g?'F{9�CECEK5=c6�X<7H9�X � � �uº�� �)!Z��9�35; I�� 1 I�� 1

�� � � � " � � ! � � ���"�! � 1 � � �

O Q O,* � ,/F8CE1�X<7H9�X�BD_f1�3H6°;56351�X<6#Cc���I� � � � � > � � � � � � _�1�: � � �"!{Fo1�3H6�7H9�C I � 1 � � � �� �'� 1 � � �

O � � ,�m¸N�6iX c�GLq�W`�6~X<756�9Aq6i:[9�k'6�1'_ Û1'3�X<7H6CM^H756#:<6<� * �B� L,F�CE1�X<7H9�X@`I?hn �R�¥��V�%�*B]356JIKH9�\DBgXs?�1'356Y7H9�C�I � ! �

� 7 � c�+*5��� � OM;+!" v ( I � ! �

� 7 � c�+*5��� � OM;+!" vF-9�35;

X<756#:<6#_�1�:<6�I � �� � 7 � � �

7 � �

� 77 � v� 1� / I�� 1 � � � �� �'� 1 � � �

O�� ( I�� 1 � � � �� �'� 1 � � �

O����-,/m¢�7563w9�C�_�1�:�X<756p^H:<6#4#6;5BD35kW\D6#=c=c9f1�3H6Z;564#1'=)^�1'CE6C � � � � � � Ó�� � > � � � � �u_�1�:|9�3>6iAq63>CM^�6#4B]9�\aCE=W1,1�X<7(LBD35k°CE6JIK56#3H4#6>ePBDX<79- � * */. L�%i� � v � � �Y�O® � ���[F�9�35;/1'356w1�`5X[9�BD3HC � � � >�� � � � � � � ( I � 1 � � � �+*5����� � � > � ! � � � � O&*��I�� 1 � � � �+*5��� c�+*5�4O&* ( �� 1 I � ! � $ � c�+*5�4O,* � �� 1 I � ! � $ � c�+*H�MO,*,� � � 1� /�

I �� c�GLq�MOML,t2X<756#3�K5CEBD35k�X<756>_�9�4iXWX<7H9�X� � � � BDC�1,;5; FH1'356{1�`5X[9�BD35CW� � � �'� � � � � ��� � � ( vº I � 1 � � � � � �+*5����� � � ! � > � "�! � � � � O,*�( vº I � 1 � � � � c�+*H� Ó c�:> *H����O,* (�� 1 / I � ! � $ � W�+*H�MO,* ( � � 1 /�

� I �� W�GLq�4O L,m � 356@;56;5K546#CPX<7%9�X��¶� % � � � (�C � �2Ó v��3I �� c�GLq�4O L �2Ó � �� I �� c�GLq�4OML,F

9�35;�X<756i:<6_�1�:<6@_�1�: % � �@1'356{X[9�bq6#C �W� v� 9�35;�1'356@7%9�C��¶� % � � � ( C*Ó C�% º I v� �� c�GLq�4OMLw�O9�3H;��¶� % � � � (�% � � � � � �

_�1�: %����j�[mW¢�756#3�ePBgX<7 � ��� >�-�F�1'356W7H9�C % " � �¶� % � � � � C�% " � Ó C�% � % I v � �� c�GLq�4OML,F�9�3H;�9�CpX<7H6);56#CEBD:<6#;�4#1'3(L;5BgX<B]1�3�% " � �¶� % � � � � ��� � ® � ,/t � �� �WBDC8BD3qA'9�:<B�9�3IXp`,?�X<756W4[7H9�35k'6c1�_�%pBD3qX<1 v� F�1'356W7H9�CZX<1>CE751je�X<7H9�XZX<756W_�K5354iX<B]1�3

( (

Page 78: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

� ;56izH3H6#;~1'3¶�O® � ���2`I? � � �'� � v�I �� �

� 7 �� � OMLW`�6\D1'35k'C|X<1 ��� � ® � ��t � �� �'Fa9�Cp9Y4#1'35CE6JIK56354#6f1'_VX<756f7q?,^�1�X<756#CEB]C|X<7H9�X

� � 7 �7 � � ��� � ® � ,�t � �� ��F�9�3H;�X<7HBDC�BDC�\]Bgbq6p;H6i:<BgA(BD35k)no��� � � × C�BD356JIKH9�\DBDXs?'m

� _�41�K5:<CE6�F5X<756Z^5:<1,1'_�9�\DCE1c9�^5^H\DB]6#C�X<1)X<756Z49�CE6'* ��,�F(`HK5XPBDC�41jA'6#:<6#;�`,?YX<756Z^5:<6iA(BD1'K5C�N�6#=W=r9(m

( �

Page 79: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�jÒ,m-�5:<BD;H9?��|^5:<BD\�­(��F%­�®'®'®,m� 356Y49�3h3H1e ;56#;HK54#6�9y4[7H9�:[9�4iX<6#:<BD�9�X<BD1'3°1'_ � � � �5���P�ZBD3°X<756r4j9�CE6r1'_P9w`�1'K535;56;h1�^�63hCE6#Xf1�_ �"!!ePBgX<7¶9

N�Q AaUj²a³�Q���µW4#1'3qX<BD3IKH1�KHC�`�1'K53H;H9�:E?'F5ePBgX<7w® ��- � �89�35;°��( * ( ,�m� ��%"-� j�D������ §�0�_��/BDC�9{`�1'K53H;56#;�1�^�63YCE6iX�1�_��)! ePBgX<7�9@N�Q A�U�²�³�Q���µf4#1'3qX<BD3IKH1�KHC2`�1'K535;%9�:E?qF,X<7563Y_�1�:�® ��- � �9�35;°��(+* �-,�F �6� � � � �5���P�uBD_�9�35;w1�3H\g?YBD_ � � ���5���P�29�35; I J I J �

� � � � " � � ! � � ���" ! � 1 � � �

OMQ O&* �-,/m�2È<ÆjÆ5#[§Y0�_ �(� � � � �5���P�[Fu;56izH3H6#;�9�C)X<756>:<6#CMXE:<BD4#X<BD1'3hX<1 ��1'_o9�_�K5354#X<BD1'3 � � � � � �5� �)!Z�[F-X<7H6#3�1'356w7H9�C � ����5� �)!Z�{9�35;�X<756i:<6_�1�:<6Y`I?h:<6CMXE:<BD4iX<B]1�3 ��� ���5���P�[Fd9�35; I J I J

�� � � � " � � ! � � ���"�! � 1 � � �

O Q O,* �SI J I J ��� � � " �

� ! � � ���"�! � 1 � � �

OMQ)O,* (I � 1 I � 1

��� � � " �

� ! � � ���" ! � 1 � � �

O Q O,*� ,/mN�6iX �6� ���5���P�dC<9�X<B]CE_]? I J I J

�� � � � " � � ! � � ���"�! � 1 � � �

OMQ)O,*� ,�m��H1�:�9@^%9�:EX<BgX<BD1'3r1'_�K535BDXs? � 7 � � � �%F(6j9�4}7 � 7 � C<9�X<BDCMzH6#C�9CEBD=cBD\]9�:�^5:<1�^�6#:EXs?'F'`�6#4j9�K5CE6#�¹� � 7 � �}�GQa� >�� � 7 � �}�+*5��� ((� � 7 �GQa����� � �GQa� > � �+*H����Ó � � 7 �GQa�M> � 7 �+*5����� � �+*5��� ( C � � �GQa� > � �+*5���+ÓC � Q�>�* ��� � �+*5��� F�K5CEBD35k>X<756W9�CECEK5=W^5X<BD1'3�X<7H9�X869�4}7 � 7 BDC89>N�Q A�U�²�³�Q���µ�41�3IX<BD3IK51'K5C8_�K5354#X<BD1'3am � 356W;56#;HK54#6CZX<7H9�X� � � 7 � �}�GQa��>8� � 7 � �}�+*5��� ��( C � � � �GQa��> � �+*5��� �IÓRC � � Q > * � � � � �+*5��� �(F�9�35;81'356-=@KHCMX�CE7H1e°X<7H9�XHI J I J �

�" ! �

�� � ! � � ��

�"�! � 1 � � �

OMQ O&* �,�t(X<75BDC2BDC�9@41�3HCE6JIK56#3H4#6p1�_ I J �

�"�! �

���"�! � 1 � � �

O Q#( � K53HBD_�1�:<=c\g?cBD3 * � �oF,eP75B]4[7Y7H1�\];5C2`�6#4j9�KHCE6pBD_���7H9�C�;5B]9�=W6#X<6i:O%F5X<756ZB]3qX<6#k�:[9�\aB]C�`�1'K535;56;�`,? I � �'� $ �

� �'��� �'� 1 � � �

O �#��C I �� ��� - � ,/F%9�C)� � �uÓ�-*�>�* � ��m-¦oCEB]35kc9f\]1I4j9�\a4[7H9�35k'6

1'_a`%9�CEBDC#F(1'356oBDC2\D6#;rX<1{41'35CEBD;56#:uX<756o4j9�CE6o1�_�� � ���I�GQ�� � Q ! � � Q ! � c�GQ�� �Z_�1�: ØN�Q AaU�²�³�Q���µ@41'3qX<BD3IK51'K5C#F59�35;�1�3H66il,X<635;5C � `,?@CM?(=c=W6iXE:E?qF�;H6izH35B]35k � � �GQ � � Q ! �*� � �GQ � � Q ! ��BD_�Q ! � c�GQ � ��9�35; � � �GQ � � Q ! �H� � �GQ � � ­ c�GQ � �M>#Q ! �BD_8Q ! � c�GQ����[mw0�_�_�1�:#Q,� �GQ�� � Q ! �p1'356Y;56izH3H6#C Q � �GQ�� � ­ c�GQ���� > Q ! �[FVCE1yX<7H9�X � Q���� Q�FVX<756#3°X<756YBD3qX<6#k�:[9�\I�� 1 I�� 1

� � � � � � " � � � ! � � ���"�! � 1 � � �

OMQ)O,*W4j9�3c`a6o4#K(X�BD3qX<1f_�1'K(:2^%9�:EX<C#F(1�3H6oBDC �W��I J�I�J��� � � � " � � ! � � ���"�! � 1 � � �

O Q O,*%F,eP75B]4[7YBDCuzH35BgX<6`I?°7I?I^a1�X<7H6#CEBDC#t�Xse21y^�9�:EX<Cf7H9Aq6cX<756r_�1�:<= I0J

�I�� 1 �

J��� � � � " � � ! � � ���" ! � 1 � � �

OMQ O&* ( � ���~`�6#4j9�K5CE6<� Q > * � ( � � Q >�* �_�1�:w9�\D\Q � � � � * � �)! � � � Fp9�35;*X<756�_�1'K(:EX<7*^%9�:EX>BDC I � 1 �

J�I � 1 �

J��� � � � " � � ! � � ���" ! � 1 � � �

OMQ O&* ( � � ��`�6#4j9�KHCE6� Q6> * � ( � � Q6>�* ��_�1�:�Q � * � � ! ��� � t2BD35;566#;hX<7H6Y=c9�^ Q5�»�GQ � � Q ! � �� � Q � � Q ! > c�GQ � ����BDCfN�Q AaU�²�³�Q���µ4#1'3qX<B]3IK51'K5CV_�:<1'= � � 1'3qX<1�� !� FI9�3H;fBgX<CVBD3IA'6#:<CE6�BDCVN�Q A�U�²�³�Q���µo4#1'3qX<B]3IK51'K5C-9�CVBgX�BDC ���Ø��� � � � ! ���� � � � � � ! Ó c��� � ����F9�35;@1'356uB]C�:<6;5K54#6;{X<1|CMX<K5;(?8X<756uC<9�=W6uBD3H6JIKH9�\DBgX<BD6C�_�1�: �)!

�F�BOm 6'm�BD3{X<756u4j9�CE6 �*®(F�eP7H6i:<6u1'356u7H9�C)� Q3> * � ((� Q3> * �

BD_TQ ! ��® ��* ! F%9�35;$� Q�> * � � � Q�>* �I_�1�:|9�\]\�Q � *%m� _�4#1'K(:<CE6'F5X<7H6p^5:<1,1'_V9�;H9�^5X<C�X<1)X<756849�CE6'* � ,/F5X<7H6Z1'35\g?�;5Bg a6i:<6#3H4#6Z41'=WBD35kW_�:<1'= X<7568_�9�4iX�X<7%9�XPX<7568351�:<=

BDC8351�X{6#lI^H:<6#CECE6#;�`I?°9�3�BD3qX<6k�:[9�\SmW0sX{BDC8B]=)^�1�:EX[9�3qXZX<1w351�X<BD4#6)X<7H9�X8ePBgX<751'K(X@9�3q?~:<6k�KH\]9�:<BgXs?�7I?I^a1�X<7H6#CEBDC81'3h9>CE6iX� � �)!{Fa9�3q?�N�Q A�U�²�³�Q���µr4#1'3qX<BD3IKH1�KHCo_�K5354#X<BD1'3y;56#zH356;�1'3�� 4j9�3y`�6@6il,X<6#3H;56#;yX<1 �)!{Fa9�35;�X<756@C<9�=W6@:<6#CEK5\gXpBDCXE:<K56f_�1�:Zn �R�¥��V����4#1'3qX<BD3IKH1�KHCp_�KH354iX<BD1'35C#mp�%1�:pCE751jePBD3HkrX<75BDC#Fa1�3H6)9�CECEKH=W6#CoX<7H9�XRC v ( � �GQa� ( C º _�1�:Z9�\D\TQ � �{F9�35;�X<7%9�XZX<756i:<6f6#l,B]CMX<C� � �O® � � #-9�3H; � x®YCEK54[7~X<7H9�XE� � �GQa�*> � �+*5����( � � Q > * � ��_�1�:Z9�\D\ Q � * � �{FaX<756#3�1�3H6;56#zH356#C8F��GQa�H��CEK5^ ! � � � �+*5��>�� � QW> * � � FH9�3H;YX<75BDC2k'BgAq6#C�9@n �R�¥��V�%��41'3qX<BD3IK51'K5C�_�KH354iX<BD1'3YeP75BD4[7�4#1'BD354BD;56CdePBgX<7� 1'3 �{FH9�35;>X<7563>1'356ZXE:<K5354j9�X<6C�BgX�`,? �@�GQa�*��=WBD3 �MC º � =r9�l �.C v�� F%�GQa� � �5m� 356�CE751'K5\];)351�X<BD4#62X<7H9�X � v'� >���P� 41'3qX[9�BD35CVX<756�CM^%9�4#6 ��� *����P��1'_%N�Q A�U�²�³�Q���µp4#1'3qX<BD3,K51'K5C�_�K5354#X<BD1'35C#F�`HK5X-=c9?`�6f;5Bg �6#:<6#3qXpBD_ �xBDCo351�XZ9c`�1'K53H;56#;�1�^a6#3�CE6#XoePBgX<7�9cN�Q AaU�²�³�Q���µ�4#1'3qX<BD3IKH1�KHC|`�1'K53H;H9�:E?'F�`a6#49�K5CE6 �!� � v'� >���P�BD=W^H\DBD6C�X<7H9�XdX<756#:<6|6il(BDCMX<C � CEK54[7cX<7%9�X�� � �GQa��> � �+*5��� ( �(O J �GQ � *5��eP756#:<6 O J �GQ � *5�VBDCdX<756|k'6#1,;56CEBD4P;5B]CMX[9�3H4#6|_]:<1'=Q~X<1�*%F�X<756@BD35zH=@K5= 1'_�X<756@\D635k�X<75C|1'_�^%9�X<75Co_�:<1�= Q~X<1�*�eP75B]4[7�CMX[9?yBD3 �oFa9�35;yX<756fk�61,;56#CEBD4@;5B]CMX[9�3H4#6@_]:<1'= QX<1"*Y4#1'K5\D;�`�6Z=fK54[7w\]9�:<k'6i:�X<7%9�3>X<756ZÙdK54\DBD;56j9�3>;5BDCMX[9�3546Z_�:<1'= QwX<1"*%m

�oCZ9re�9?yX<1�9�CE4#6#:EX[9�B]3yX<756)BD=)^�1�:EX[9�3546{1'_�X<7H6@:<6#k'K5\]9�:<BgXs?y1'_�X<756@`�1'K53H;H9�:E?�BD3y^5:<1jA(BD35k�CE1'=W6@^H:<1�^a6i:EX<BD6Co1�_�(RV¯ R�¥,�%�ØCM^%9�4#6C#Fu0f;56#CE4#:<Bg`�6w9�4#1'K53qX<6#:ELO6il59�=)^H\]6YeP75BD4[7�0)7H9�;�4#1'35CMXE:<K54#X<6#;�BD3�1�:<;56#:WX<1�9�35CMe26#:��+^%9�:EX<B]9�\D\g?%�89JIK56#CMX<B]1�3~X<7H9�X@�I6#:<k�6B�Ñ8R��-RBA � �%��T�Ra��vp7H9�;°9�CMbq6;�9Y_�6ie ?q6j9�:<C89�k'1�B]3�9YX[9�\Db�9�Xf¡u��¦@t�0|=c6#3qX<B]1�3H6#;~=@?y:<6#CEK5\DXX<1�={?�k'1I1,;~_�:<BD6#35;~Ùd;(e�9�:<;��V�����%T�����¥(º�F�eP751�7H9�;�CMX<K5;5BD6;~;51'=r9�B]35C|ePBDX<7�BD:E:<6#k'K5\]9�:o`�1'K535;%9�:<B]6#C#F�9�35;~7H6f\]9�X<6#:

v~�I6#:<k'6#B!&Z1'35CMX[9�3qX<B]351jA,BDX<4[7)Ñ8R��-RBA � ����T�Ra�2FqÚPK5CECEB]9�3r=c9�X<756=c9�X<BD4#B�9�3 t'7H6�e21�:Eb(CdBD3WX<756p�5RV¯ R�¥I�%�~0�3HCMX<BgX<K(X<6P1�_�~9�X<756=c9�X<BD4#C#F��,Bg`�6i:<B�9�3 2:[9�354[7>1'_�X<756ZÚ|K5CECEB]9�3w�o49�;56#=@?�1�_-�I4BD6#3546#C#F5�o1jAq1'CEBg`HBg:<CMb�mº N�K5;(ePBDkcÙ-;5e�9�:<;y�V�����%T����%¥�FaÐo6i:<=r9�3(LS`�1�:<3w=c9�X<756#=r9�X<BD4#B]9�3aF �j�'­�(ItH7HBDCP_+9�X<7H6i:FHÙd;5KH9�:<;y��������T����%¥�F�� � � ��L

�j��(�®(Fue�9�Cc9�4#\]9�CECEBD4j9�\�CE4[751'\]9�:WeP751�6=WBDk�:[9�X<6#;�X<1�Ùd35k'\]9�35;�BD3�X<756 ��®�Cr9�35;�e�9�C)X<756>z5:<CMXr_�1�:<6#B]k�3H6i:)X<1�1�`5X[9�BD3

(��

Page 80: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

=W63qX<BD1'356;�BDX{X<1 �{× �V��������¥I¥ � F�eP751ykq9Aq6Y9w=W1�:<6rk'6#356#:[9�\u4#1'35CMXE:<K54#X<BD1'3amw�o_]X<6#:{\D6j9�:<35BD35k�9�`�1'K(X �@× �V�������%¥,¥ × C4#1'35CMXE:<KH4iX<BD1'3aF5024#1'3qX[9�4iX<6;��I6#:<k�6B�Ñ8R��-RBA � ����T�Ra�heP751W=c6#3qX<B]1�3H6#;�X<7H9�XP756Z7H9�;y9�\DCE1W;51'356ZX<756849�CE6&* �/­,m� ��%"-� j�D������ §|�H1�: & �­,F�X<756i:<6@6#l,B]CMX<Co9��+`�1'K535;56;���4#1'353H6#4iX<6;y1�^�63�CE6iX)��CEK54[7�X<7H9�X�� v:� ����P��BDCo351�Xo;5635CE6BD36� v'� �H���P�d_�1�:@��(+* �-,�m�2È<ÆjÆ5#[§Z0o1'35\g?~k'BgAq6)X<756W^5:<1,1�_2_�1�: & ��­>9�35;�_�1�:8X<756W49�CE6Y­ �-*�� ,/tV_�1�:8X<7H6)4j9�CE6y��( * ( ­,F �{× ���������%¥,¥7H9�CrBD3qXE:<1,;5K546#;/9�=c1�:<6>X<6#4}7535BD4j9�\�4#1'35CMXE:<K54#X<BD1'3am � 356w;56#zH356C �+� � � � � � C � F2eP756i:<6�� BDC)X<7H6w1�^�6#3/CE6iX�I�GQ � *5� � Q ��® � * ��® � Q�º2Ó�*(º ��� �qF�� BDC|X<7H6f1�^a6#3~CE6#X �I�GQ � *5� � Q �x® � * ��® � Q�º�Ó�*(º ��� �qF 9�35;�_�1�: � ��{X<7H6^%9�CEC<9�k'6EC � BDC8;H6izH356;°BD3�^�1'\]9�:{4#1,1�:<;5BD3%9�X<6CZ`I? �I�GQ � *5� � ®+( � ( � � ­

" � > � � � L �¸­ " � �qmY¢�756cCE6jJIK56#3546 � �C<9�X<BDCMzH6C{® � � � �¸­ " � " v CE1wX<7%9�X8X<7H6)^%9�CEC<9�k'6#C@;51w3H1�X@1jAq6i:<\]9�^�`HK5X@9�\DCE1 � � � ®wCEK(�c4#BD63qX<\g?~:[9�^HBD;5\g?�CE1wX<7H9�X� � � � ­ � � � , _�1�:P6iAq6#:E?"* � !©� � ,��[m� 356W4[7564<b(CZBD=c=W6;5B]9�X<6#\g?>X<7H9�XZX<756)_�K5354#X<BD1'3 � �@;56#zH356#;�`I? � �W��®�BD3�{F � �#� ��BD3 � 9�3H; � �W� � BD3�9�\D\C � `�6\D1'35k'CZX<1 � v:� �5���P�o_�1�:r� ( * � ,�m>¢�75BDC{_�K53H4iX<BD1'3 � �W49�35351�X{`�6r9�^5^5:<1q9�4[756;°BD3$� v'� �5���P�o`I?�_�K5354iX<B]1�3HCBD3 � v:� >���P�[m)0�35;H6#6#; F�`�649�K5CE6��!B]C89�`�1'K535;56;°1�^a6#3�CE6#XZePBgX<7°9>N�Q A�U�²�³�Q���µ>41�3IX<BD3IK51'K5C8`�1'K535;H9�:E?qF�_�K5354iX<B]1�3HCBD3,� v:� �5� �Z�Z7H9Aq6Y9�3¶6#l,X<6#35CEBD1'3hBD3 � v'� �H� �8ºj�[FVeP75BD4[7¶BDC)9y4#1'3qX<BD3,K51'K5Cf_�KH354iX<BD1'3¶9�C * � & � ­,F-9�3H;hX<756i:<6_�1�:<6�5� � v'� �H���P�oBD=W^H\DBD6C �5� C � � �Z�[FV9�35;°X<756r\DBD3H69�:{_�1�:<= � � � � � �O®I�pBDC@4#1'3qX<BD3IKH1�KHC#F�eP756i:<6 � �O®q�ZBDC{4#1'=W^HK(X<6#;_�:<1�= X<756pCEBD;56Z1'_ �{md¢�756ZC<9�=W6Z9�:<k'K5=W63qXP9�^5^H\DBD6C�_�1�: �rF%9�35;YX<7H6p\DBD356j9�:�_�1�:<= �

� � � � �O®I�-B]C241'3qX<BD3IK51'K5C#F5eP756#:<6� �O®I�oBDC8351je!4#1'=)^HK5X<6#;�_�:<1'= X<756cCEBD;56c1'_��Ym � 356c7H9�C �

�� � �� > � � � � �j� � ��FV9�35; � �)49�35351�X{`�6c9�^5^H:<1'9�4}756#;`I?�_�K5354iX<B]1�3HCr_�:<1'= � v:� ����P�W`�6#4j9�KHCE6+�

�� � � � � � � � �W_�1�:>9�\D\ � � � v'� >���P�[F|9�35;*X<756i:<6_�1�:<6~9�3q?�_�K5354iX<B]1�3

F¶`�6#\]1�3Hk�B]35k�X<1�X<7H6y4#\D1'CEK(:<6y1'_�� v'� >���P�f=fK5CMXYC<9�X<B]CE_]? ���GF(� � � � �GF(�[m*0�3H;56#6;aF�BD_ � � � v'� >���P�f1'356y7%9�C

� � �GQa��> � �+*5����((� � KML.NMO�� � ��� � O J �GQ � *5��eP756#:<6 O J �GQ � *5��BDC-X<756oCE751�:EX<6#CMX2;5BDCMX[9�354#6|_�:<1'= QYX<1 *feP756#3Y1�3H6|CMX[9?(CuBD3 �oF9�35;>\D6iXEX<BD3Hk�Q>X<635;YX<1)®)_]:<1'=�X<756pCEBD;56Z1�_ �x9�35; *WX<6#35;�X<1)®f_�:<1�=�X<756pCEB];56p1'_ �YF51�3H6p7H9�C8O J �GQ � *5�#�·®@`�6#4j9�KHCE6X<756#:<689�:<689�:E`HBgXE:[9�:E?YCE7H1�:EXP^%9�X<75C�`I?�K5CEBD35kWX<756p^�9�CEC<9�k'6#CC � _�1�:P\�9�:<k'6 � m

�%1�:YX<756~k�6356i:[9�\o49�CE6¶� ( * � ,/F �@× �V�������%¥,¥ × C�CMX[9�:EX<CcePBDX<7/9�¡u��T���R�� � CE6#XYePBgX<7�^�1'CEBgX<BgAq6y=W6j9�CEK5:<6BD3¶9wCE6k�=c6#3qXfBD={`�6;5;56;°BD3¶9�3h1�^�63°CE6#X@1'_'�8º�F�9�35;h7H6r4#1'35CMXE:<K54#X<C{^%9�CEC<9�k'6C{X<7(:<1'K5k'7°X<756r41'=)^H\D6=W63qX{1'_uX<7H6¡u��T���R���CE6iX{BD3�CEK54}7�9Ye�9?�X<7%9�X8k'1'BD35k>_]:<1'= 1'356WCEBD;56fX<1�X<756W1�X<756i:Z`I?~KHCEBD35k�X<756f^%9�CEC<9�k'6#C{49�3~`�6);51'356)ePBgX<7O J �GQ � *H� (�C O%�GQ � *5��_�1�:Z9�\D\BQ � * � �otaX<756#3~756f;56iz%356#CZ9r_�K53H4iX<BD1'3yX[9�b(BD35kr;HBg �6i:<63qX|A'9�\DK56#Co1'3�`�1�X<7�CEBD;H6#Co1'_VX<7H6¡u��T���R��/CE6iXF eP75B]4[7�49�35351�XZ`a6W9�^5^5:<1q9�4[756;�`,?~_�KH354iX<BD1'35CZBD3 � v'� >���P��`a6#49�K5CE6)X<756#?yX[9�bq6)X<756)C<9�=W6)A'9�\]K56)1'3`�1�X<7yCEBD;56C|1'_VX<756)¡u��T���R��hCE6#XF�9�35;�_�1�:p_�KH354iX<BD1'35C|BD3 � v'� vj���P�2X<756{:<6#CMXE:<BD4#X<BD1'3�X<1�9�3I?wCEBD;56{1'_VX<756W¡u��T���R��¶CE6iXBDC�41�3IX<BD3IK51'K5C#m

¢�756W4#\]9�CECEBD4j9�\d¡u��T���R���CE6#XZ7H9�CZ=c69�CEK(:<6)�6i:<15F�9�35;�B]CpK5CE6#;�BD3~X<756)4#1'35CMXE:<KH4iX<BD1'3~1'_29Y351'35;H6#4i:<6j9�CEBD35k>_�K53H4iLX<BD1'3�@ 1'3 ! ® � � #SFpePBgX<7�@��O®I� � ® � @��E��� � ��Fo`%K(XwBDC>41'35CMX[9�3qXy1�3 ;HBDC��M1'BD3IX<C>BD3qX<6#:EA�9�\DC � � � � ��CEKH4[7 X<7H9�X� � � ( � K&%6( � � � �#����tdX<75BDCf4#1'35CMXE:<K54#X<BD1'3hBDCf1'_�X<6#3¶49�\D\D6;°X<756h«<;56#A(BD\ × C@CMX[9�Bg:<49�CE6¬(Fd9�CfBgXf7H9�CfBD35zH35BgX<6\g?�=r9�3q? %%9�X\D6#A'6\DCZ�SCMX<6#^HC}�29�35;wBgX�k'1,6#C�K5^w_�:<1'= ®fX<1w�pePBgX<751'K(X�7%9#A(BD3HkW9�3q?@�MKH=)^ m

¢�756o^%9�:EX<BD4K5\]9�:2_�K5354#X<BD1'3�@ 4#1'35CMXE:<KH4iX<6#;�C<9�X<BDCMzH6#C @ �E�3> Qa�H�Ø� > @ �GQa�-_�1�:�9�\D\ Q � ! ® � � #SFH9�3H; @ �.�.Qa�H�­ @ �GQa�-_�1�:P9�\D\a®�(,Q ( v� FH9�35;>BgX�BDC�n �R�¥��V�%��4#1'3qX<BD3,K51'K5C�1'_�1�:<;56#:)� � � � � �

� � �Hº t5BgX�BDC2X<756pKH35B]JIK56oz5l(6#;�^�1'BD3qX�1'_ X<7H6=c9�^5^%BD35k �*;56#zH356#;Y1'3Y_�K5354#X<BD1'35C �~eP75BD4}7�9�:<6o41'3qX<BD3IK51'K5C�1�3! ® � � #�ePBgX<7 �d�O®I�H�*®f9�3H; �d�E���H�Ø��F59�35; �8� ���H� �=W6j9�35C � �GQa��� vº �u�.�.Qa�p_�1�:)® ( Q ( v

� F � �+*H��� vº _�1�:#Q � � ® � v� �r9�3H; � ���I��� � > � �E� > �I�Z_�1�: º� ( � ( ��t1'356{K5CEKH9�\D\g?>CMX[9�:EX<C|_]:<1'= @ � �GQa�3��Qy_�1�:o9�\D\ Q~9�35;y1�3H68;56#zH356C@ � � �8� @ � " v �u_�1�: � x�89�35; @ � 4#1'3qAq6i:<k'6C�X<19@ K535B]_�1�:<=W\D?'m

9w^5:<6CMX<BDk'BD1'K5C@4[7H9�Bg:W9�X � l(_�1�:<;am�Ùd;(e�9�:<;¶CMX<KH;5BD6#;hB]3�¡29�3H9�;H9y;HK(:<BD35k 1°1�:<\D; 1h9�:f0E0}FV9�3H;hCM^�64#B]9�\DB]�#6#;°BD3¶�V\DK5BD;£o?(3H9�=WB]4#C#t8756�\]9�X<6i:�`�6#4j9�=W6�^H:<1�_�6#CECE1�:wBD3x¡29�={`5:<BD;Hk�6'FPeP756#:<6�75BDCYX<1,1�=c9�X<756#=r9�X<BD49�\o9�^5^H:<1'9�4}7 Bg:E:<BgX[9�X<6;�75BDC4#1'\D\]69�k'K56#CdeP751fe26i:<6oe21�:Eb(BD35k@B]3cX<7H6|XE:[9�;5BgX<B]1�3%9�\ 2:<BgX<BDCE7ce�9?Y1'_ ;51'BD35kf�|^5^%\DBD6#;Y�~9�X<756#=r9�X<B]4#C#FIeP756#:<6o1'356oe�:<BgX<6C6il,^%9�35CEB]1�3HCPePBgX<751'K(X|e21�3H;56i:<BD3HkWX<1,1r=@K54}7yBD_�X<756i?w4#1'3qAq6i:<k'6{1�:pBD_�X<756i?>:<69�\D\g?w9�^5^5:<1q9�4[7yX<756{CE1'\DK(X<BD1'3y1�_VX<7568:<6j9�\^5:<1�`H\D6=!X<7%9�X�1'356pB]C2B]3qX<6i:<6CMX<6#;YB]3am�Ùd;(e�9�:<; × C�9�^5^5:<1q9�4[7wBDC�=W1�:<6p9�b(BD3YX<1@X<756Z�5:<6#3H4[7�e�9#?�1�_�;51'BD35k)�~9�X<756#=r9�X<BD4#CePBgX<7>9�^5^H\]BD49�X<BD1'35C2X<1f:<6j9�\a^5:<1�`H\D6=WC8�+eP75BD4[7>1'356ZCE751'K5\D;>46i:EX[9�BD35\D?r4i:<BgX<BD4BD�#6o_�1�:�`�6#B]35kf1'_�X<6#3�X<1,1W9�`HCMXE:[9�4iXi�[F(eP756#:<61'356{BDC|CEK(^H^�1'CE6#;yX<1c^5:<1jAq6@9�\D\�X<756@9�CECE6i:EX<B]1�3HCP=r9�;56�mPÙ-;5e�9�:<;ye�9�CP`�1'K53H;wX<1r7H9Aq6{;HBg�c4#K5\gX<B]6#C�ePBgX<7y75BDC|4#1'\D\]69�k'K56_�1�:�756ZK5CE6;YX<1)4#1'35CMXE:<KH4iX�4#1'K53qX<6#:ELO6il59�=)^H\]6#C�CE751jePBD35kfX<7%9�X�X<756p=W6#X<751,;5C2X<7H9�X�e26i:<6pX[9�K5k'7qX�;5BD;�351�XP9�\ge�9?,C�e21�:Eb�mno6pe21�:Eb(C|9�X�X<756Z¦o3HBgAq6i:<CEBgXs?Y1'_� �9�X<7 F5Ùd35k'\]9�35;am� �o3qX<7H1�3I?yÐ@m �{× ���������%¥,¥�F50s:<BDCE7w=c9�X<756#=r9�X<BD4#B]9�3at(756Ze21�:Eb(CP9�XP�p9�X<BD1'3H9�\a¦o35BgAq6i:<CEBgXs?�1�_�0s:<6#\�9�3H;aF5��9#?(351,1�X<7am� Ðo61�:<kc�H6#:<;5BD3H9�35;>N�K5;(ePBDkW±d75BD\]Bg^5^~¡u��T���R��-F�Ðo6#:<=c9�3>=r9�X<7H6#=c9�X<BD4B]9�3aFa� ��ÕIÔ�L<�j�(���,m-no6Ze21�:Ebq6;>B]3�np9�\D\D6'm

��®

Page 81: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�)(Im-��1'35;H9?��|^5:<B]\�­�Õ5F�­�®'®'®(m¢�756�4[7%9�:[9�4#X<6i:<BD�j9�X<B]1�3HCc1'_�� � � �5���P�W9�35;�� � � �5� �)!Z�f^5:<1jA,B];56�9h4[7%9�:[9�4#X<6i:<BD�j9�X<B]1�3/1�_peP75B]4[7/_�K5354#X<BD1'35C � �

� � � � ���P�29�:<6ZCEK54}7>X<7H9�X�X<75686il,X<6#35CEB]1�3>1'_ � `,?>®)1'K(X<CEBD;56 �oFH;56#3H1�X<6; �� F5`�6#\D1'35k)X<1E� � � � � � ! �[m� & ����$ §VN�6#X ��`�6{9@`a1�KH35;56#;w1�^�63>CE6iXP1'_�� ! ePBDX<7>9)N�Q AaUj²a³�Q���µW4#1'3qX<BD3IKH1�KHC�`�1'K53H;H9�:E?'md¢�7563aF5_�1�:|® ��- � �9�35;�� ( * ( , 1'356)7H9�C ��!� � � � �5� �)!Z��BD_d9�35;~1'35\g?�BD_ � � � � � �H���P�P9�3H; O " � �!� ���5���P�[F�eP756#:<6#O%�GQa�P;56351�X<6#CX<7568;5BDCMX[9�354#68_]:<1'= QwX<1)X<756p`a1�KH35;H9�:E?�� �om�2È<ÆjÆ5#[§ � 356fCE751jePCoX<756@4j9�CE6 * � ,/F�X<756@49�CE6 * � , `a6#BD35kY69�CEBD6i:FaK5CEBD35krX<756f_�9�4iXoX<7H9�XoX<756@_�K5354#X<BD1'35C|KHCE6#;�9�:<6n �R�¥��V�%�h4#1'3qX<BD3IKH1�KHCP1'_�1�:<;56i: -�m �� � ���5� �"!p�uBDCP6JIK5BDA�9�\D6#3IX�X<1 � � ���5���P�29�35; I � 1 I � 1

� � � � � � " � � � ! � � ���" ! � 1 � � �

OMQ O&* � ,BDCr6jJIK5BgA�9�\D63qXWX<1 I0J�I0J �

� � � � " � � ! � � ���"�! � 1 � � �

O Q O,* � , 9�35;�I0J�I � 1 �J �

� � � � � ���" ! � 1 � � �

OMQ O&* � ,/F�BSm 6�m � � � � � �5���P�)9�35;� ��� ���5���P�ZeP756i:<6<� �d�GQa��� � � I � 1 �

J v��"�! � 1 � � �

O,*°_�1�:#Q � �om �6#49�K5CE6�� � ® � O��GQa����� �¸1'356Y7%9�C � �d�GQa��� � (I � �'� � � � � � v� �'� 1 � � �

O�� � � 1� � � � � �

F�CE1cX<7%9�X � �d�GQa��� (�C O��GQa� " � mo¢�75BDC|CE751jePCoX<7H9�XoBD_ � � � � � �H���P��9�35;6O " � � � ���H���P�X<7563 �� � � � � � � � ! �[m

0�3h1�:<;56#:{X<1w^H:<1Aq6cX<7H6c1�X<756#:@B]=)^H\DB]49�X<BD1'3aF�1'356rKHCE6#Cf9>^�9�:EX<BDX<BD1'3�1'_�K535BDXs?'F � 7 � � � �%F�9�35;h1'356r3H1�X<B]4#6#C{X<7H9�XF 7 � � 7 �$� � � � �5���P�|9�35;�BD_ ��$� � � � �5� �"!p�P1'356W7H9�C �F 7 � � 7 ��$� � � � �H� �)!p�[F�9�35;�9�C��>B]Coz%35BgX<6fBgXZB]Cp6#351'K5k'7~X<1CE751je�X<7H9�X8_�1�:869�4}7 �P1'356W7H9�C O " � F 7 � ���5���P�[m)¢�75B]Cp41�:E:<6#CM^�1'35;5CZX<1�^5:<1jA(BD35k�X<7%9�X9� �8� C O " � BD3~X<756W49�CE6W1�_� � eP7563 BDC8N�Q A�U�²�³�Q���µ>4#1'3qX<BD3,K51'K5C#t�`I?�K5CEBD35k>X<756W=r9�^H^HBD35k��GQ � � Q ! � �� � Q � � Q ! > c�GQ � ���)eP751'CE6WB]3qAq6i:<CE6WBDC� Q�� � Q ! �#�� �GQ�� � Q ! Ó c�GQ���� � eP75BD4[7w9�:<6p`�1�X<7>N�Q A�U�²�³�Q���µ)4#1'3qX<BD3,K51'K5C#FH1'356Z7H9�C�X<1W4#1'35CEBD;56#:�X<756Z4j9�CE6 �*®(FH9�35;BD3~X<7H9�XZ4j9�CE6)1'356W7H9�C�I !

1 � � v��" ! � 1 � � �

O,*6� � � Q ! �" �{BD_3Q ! �Ø®,F�`I?~9�3°9�:<k�KH=W6#3IXZ1'_u751'=W1'k'6#356BgXs?'F�9�3H; � BDC

X<756ZA�9�\DK56Z1'_�X<7H6ZBD3qX<6k�:[9�\ _�1�:)Q ! �Ø��m¢�756�;H6iA(BD\ × CVCMX[9�Bg:<4j9�CE6�BDC-9�3W6il59�=)^H\D6�1�_a9o_�K5354#X<BD1'3feP75BD4}7)BDC-351�Xd9�`HCE1'\DK(X<6#\D?@4#1'3qX<BD3,K51'K5C#FI9pX<6i:<= 6JIK5BgA'9�\D6#3qXVX<1

7H9A(BD35k{X<756o;56i:<BgA'9�X<BgAq6�BD3 ��vF(9�35;rX<756|;H6i:<BgA'9�X<BDA'6PBD3cX<756|CE635CE6o1'_a;HBDCMXE:<Bg`HK(X<B]1�3cB]C29�4#X<KH9�\D\g?c9{351'35356#kq9�X<BgAq6|=W6j9�CEK5:<6eP751'CE6YCEK(^5^�1�:EX)BDC{X<756w¡u��T���R��*CE6iXF-9�35;¶BgXfBDC@K5CE6#_�K5\uX<1yCE751je N�9�K5:<6#3qXr�(²�³�´8���5��µ × C@^5:<1,1'_�X<7H9�Xf351'35356k'9�X<BgAq6;5BDCMXE:<BD`HK(X<BD1'35CP9�:<68ÚP����R�T¶=W6j9�CEK(:<6#C#m� & ����$ §�0�_d9Y;5B]CMXE:<Bg`HK(X<BD1'35C � � � �O���P��BDCo3H1�3H356#kq9�X<BgAq6fBD3�X<756fCE6#3HCE6@X<7H9�X ��� � ��� �®Y_�1�:89�\]\#� � C � ���P�PCEK54[7X<7H9�X �! �®(F(X<7H6#3���BDCP9)351'35356k'9�X<BgAq68ÚP���-R�Th=c69�CEK(:<6�m�2È<ÆjÆ5#[§�N�6iX�� `�6�9o41'=)^%9�4iX-BD3 ��9�3H;)\D6iX � � C�� ���P��C<9�X<BDCE_]? � �®p6iAq6i:E?,eP756#:<6�9�3H; � � �21'3 �°m�¢�7563)_�1�:-6#A'6#:E?� � C � ���P�uePBgX<7�- � * */. L�%i� �V� � �hF%1�3H6{7H9�C�>�� � �8� � � ( � ( � � �8� � � F%9�35;wX<7H6i:<6#_�1�:<6 >�� � ��� � ��� � � �'( ��� � ���'(� � ��� � ��� � � �[F5BSm 6�m� ��� � ������( C ! � � ��� � FHePBgX<7�C ! � ��� � � �[m-¢�756#3w1'35686il,X<6#35;HC�X<75BDC�BD356jJIKH9�\DBgXs?cX<1)X<756Z4j9�CE6peP756#:<6� � C � ���P��ePBgX<7 - � * *�. L�%i� ����� �hF,CE751jePBD3Hk8X<7H9�X ��BDCu98Ú|���-R�T~=c69�CEK(:<6'FI`I?c9�^5^H\D?,B]35kZX<7H6�^H:<6#4#6;5BD35k{BD356jJIKH9�\DBgXs?X<1 � � � ��_�1�:Z9cCE=W1,1�X<75BD3HkrCE6JIK56354#6 � � �O9�35;�K5CEBD35k C ! _�1�:Z9c\�9�:<k'6#:|41'=)^%9�4iXpCE6iXi�[m � _�4#1'K(:<CE6f1�3H6f9�\DCE1c^5:<1jAq6#CBD3�X<7HBDC�e�9?rX<7%9�X ��� � ��� �®)_�1�:|9�\D\ � � C � ���P�uCEK54[7wX<7H9�X �! /®(m

0�3�1'356);5BD=c6#35CEB]1�3 F 1�3H6)C<9?,CZX<7%9�X{9�_�KH354iX<BD1'3!@�7%9�CZ`�1'K535;56;~A'9�:<B]9�X<BD1'3�B]_�X<7H6i:<6)6#l,B]CMX<CZ9Y41'35CMX[9�3qX�C CEK54[7X<7H9�Xo_�1�:Z9�\]\ & 9�3H;~9�\D\�BD3H4i:<69�CEBD3HkcCE6jJqKH6#3546#C Q v � Q º � ����� �5Q ! 1'356@7H9�C � !

" v7��v � @��GQ 7 �H>$@��GQ 7 ��v ��� (�Cfm � 3H6^5:<1jAq6#C�X<756#3)X<7H9�X-CEK54[7r9o_�K5354#X<BD1'3�@W7%9�Cd9p\]BD=WBgXV1'3)X<756�\D6_]Xd9�35;c9o\]BD=WBgX�1�3)X<7562:<BDk'7qXd9�X-6#A'6#:E?@^�1'BD3qXF�ePBgX<7c9�X-=c1'CMX

9)41'K53qX[9�`H\D683IK5={`a6i:P1'_�^�1'BD3qX<C�1'_�;5BDCE41�3IX<BD3IK5BgXs?qFH9�35;>X<7H9�X@ � K�> (>eP756i:<6 KY9�35;�(y9�:<68351'35;56#4#:<69�CEBD35k5F%9�35;y9�C1'35684[7H6#4<b(CP69�CEBD\g?Y`I?Y:<6k�KH\]9�:<BD�9�X<BD1'3YX<7H9�X�X<7568;56#:<BgA�9�X<BgAq6p1'_V9)351'35;H6#4i:<6j9�CEBD35kW_�K5354#X<BD1'3>BDCP9W351'35356#kq9�X<BgAq6ZÚ|���-R�T=W6j9�CEK(:<6�FH1'356pzH3H;5C�X<7%9�XPBD_H@w7%9�C�`a1�KH35;56#;>A'9�:<B�9�X<B]1�3YX<756#3 @�� BDCP9)ÚP���-R�Th=c69�CEK(:<6pePBgX<7�zH35BgX<6pX<1�X[9�\a=c9�CEC8�O9�35;X<756r41'3qAq6i:<CE6YBDC{XE:<K56�FV`�6#4j9�KHCE6Y9�3q?°ÚP���-R�T*=c69�CEK(:<6 �*49�3°`�6ce�:<BDXEX<6#3h9�C �

�> � " _�1�:f351'35356k'9�X<BgAq6rÚ|���-R�T

=W6j9�CEK(:<6#C �� � � " F%9�35;w6iAq6i:E?�351'353H6#kq9�X<BgAq6pÚ|���-R�T¶=W69�CEK(:<68BDC�X<7568;56i:<BgA'9�X<BgAq6oBD3>X<756ZCE635CE681'_�;5B]CMXE:<Bg`HK(X<BD1'35C�1'_�9351'35;564i:<6j9�CEB]35kW_�K53H4iX<BD1'3am

0�3r1�:<;56#:uX<1{;56#zH356|_�K5354#X<BD1'35Cd1'_�`�1'K53H;56#;cA'9�:<B]9�X<BD1'3WBD3r=W1�:<6PX<7H9�3r1'356|CM^%9�4#6o;5BD=W635CEBD1'3aF,1'356|356#6;5CdX<1@;56izH3H6X<7568CM^%9�4#6 � - 1�_�`�1'K535;H6#;wÚP����R�T¶=W6j9�CEK(:<6#C#m0�_��ÖBDCp9Y4#1'=)^�9�4#XF�X<7563 C)������X<756fCM^%9�4#6f1'_-4#1'3qX<BD3IKH1�KHCp_�KH354iX<BD1'35Co1'3 �Ö6JIK5Bg^H^�6#;yePBDX<7wX<756fCEK(^�3H1�:<= BDC9 ���T���²a³�CM^%9�4#6'F%eP7H1�CE6f;5KH9�\ � ������BDCPX<756@CM^%9�46{1'_�ÚP���-R�T�=W6j9�CEK5:<6#C|1'3��hF�6JIK5Bg^H^�6#;wePBgX<7wX<7H6{;HKH9�\�351�:<=� � � � � ��CEK(^ � � � � � � $ v � ��� � �����¹mV0�_ � � 9�:<6Z;5BDCMX<BD354#X2^�1'BD3qX<C�BD3 � X<756#3 �6� � � 4 � � ��% `a6#\D1'35k'C2X<1 � �����-B]_�9�35;>1�3H\g?BD_ � � � 4 � � �-, 9�35;w1�3H6Z7H9�CW� � � � � � � � � 4 � �¹m

�(�

Page 82: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

0�_ ��BDCp1�^a6#3aF�X<7H6#3 C � ���P�P9�35;~BgX<Co;5KH9�\ � ���P��1�_d9�\D\�X<756fÚP���-R�T�=W6j9�CEK5:<6#CpBD3+�Ø9�:<6@351�X ���T���²a³�CM^%9�4#6#C�S351�:u�V� ��a²�³�����CM^%9�4#6#Cu6#BgX<7H6i:i�[FI9�35;rBD_ 9ZCE6jJqKH6#3546 � � 1'_�^�1'BD3qX<C-X<6#35;HC-X<18X<756�`�1'K535;%9�:E? � ��X<756#3 �6� � � 4 � � ��%`�6\D1'35k'C�X<1 � ���P�u_�1�:|9�\D\a41I6#�c4#BD63qX<C 4 � �O9�C�6j9�4[7>41�=W^%9�4iX��»1'35\g?Y41'3qX[9�BD35CP9fz%35BgX<6Z3IK5=@`�6i:|1'_�X<756CE6p^�1'BD3IX<C}�[m

¢�756oCM^%9�4#6 � - ���P�[FIX<756oCM^%9�4#6|1'_a`a1�KH35;56#;YÚP���-R�T�=W6j9�CEK5:<6#C#F59�\DCE1@49�\D\D6;WX<756|CM^%9�46o1�_ =W6j9�CEK(:<6#CuePBgX<7czH35BgX<6�+X<1�X[9�\��2=c9�CECo1�:oX<756@CM^%9�4#6f1'_�=c69�CEK(:<6#C|ePBDX<7wzH35BgX<6{X<1�X[9�\�A�9�:<B]9�X<BD1'3aF%B]CPX<756fCM^%9�4#6@1'_�� � � ���P��CEK54[7�X<7H9�XoX<756i:<66il(BDCMX<C�C ePBgX<7(� ��� � �����&( C�� � ��� � _�1�:r9�\]\�� � C � ���P�[F29�35;�BgXW49�3�`�6wCE751jeP3�X<7H9�X � 4j9�3�`�6>6il,X<635;56#;�X<1°9�\D\� � C - ���P�[FqX<756|CM^%9�4#6P1'_�`a1�KH35;56#;r4#1'3qX<B]3IK51'K5Cu_�K5354iX<B]1�3HC-BD3 �oFIeP75BD4[7cB]Cd9 ���T���²�³~CM^%9�46�ePBDX<7)X<756PCEK5^r351�:<=wFICE1X<7H9�X � - ���P�VBDCdX<756P;5K%9�\�1'_ C - ���P��9�3H;cBDC29� ���T���²�³~CM^%9�46�mV0�_�� � 9�:<6P;5BDCMX<BD3H4iXd^�1'BD3qX<CdBD3 ��X<756#3 �6� � � 4 � � ��%`�6\D1'35k'C�X<1 � - ���P�dB]_�9�35;>1'35\D?YB]_ � � � 4 � � �-, 9�35;>1'35687%9�CW� � � � � � � � � 4 � � m

¢�756|6#lIX<635CEBD1'3cX<1{=W1�:<6�X<7H9�3r1'356|CM^%9�46|;5BD=c6#35CEB]1�3c7%9�Cd`a6#6#3rCMX<K5;5B]6#;c`I?WÙd3535BD1@£p�yÐoQSR���¤VQ�FI��� �V�%���%��9�35;1°635;56#\]\a�V¥,� �ZQ�T�¤2F(`HK5X�X<756Z6j9�:<\]BD6i:P3H9�=W6#C�1'_V¢PR�T���¥,¥,Q�v|9�3H;>N�9�={`�6#:EX<1r¡u��U����QSºZ9�:<6Z1'_]X<63>=W63qX<BD1'356;am� &������D������ §Z�H1�:{9�3�1�^�63~CE6iX � � �)!{F�9Y_�KH354iX<BD1'3 � `a6#\D1'35k'CpX<1 � �c���P�[FaX<756)CM^%9�4#6W1�_d_�K5354#X<BD1'35CZ1�_-`�1'K53H;56#;A�9�:<B]9�X<BD1'3�BD3#�*BD_ � � � v ���P�29�35; ���� � � ��� - ���P�u_�1�: "#�Ø� ���������:& m� ��%"-� j�D������ §"� �c� �)!p�ZBDC{4#1'3qX<B]3IK51'K5CE\g?�BD=@`�6#;H;56#;°BD3 ��v � � vj� �"!p�p_�1�: & �­,FV9�35;hBD3 � �� �{�Z_�1�: & � ��m�0�_� � BDCr9°CE=W1,1�X<75BD3Hk�CE6JIK56354#6�9�35; � � ���W� �)!Z�@X<7H6#3�� � � � BDCc`�1'K535;56;/BD3�� v:�¹vj� �)!p�[m � �c� �)!Z�9� � � ���v� �)!Z� � � � � � � > � � � v ( C � ( �q_�1�:|9�\]\ ( � �)!��qm�2È<ÆjÆ5#[§¶�H1�:9" � � ���������'& 9�35; � � C � � �)!Z�c1'356�7H9�C � � � �

�� � ���� � � ��� � > � � � � � � ������ � � � > � � � �� � � � �� � � � �

> � � � �����

�� � ���� �

�H� �������� � � �� � � ���[F(9�35;cX<756#:<6#_�1�:<6|9�C �� � � �~BDCd4#1'3qX<BD3IKH1�KHCdePBgX<7c4#1'=)^�9�4#XuCEK(^5^�1�:EX29�35;Y7H9�Cu98CEK(^�351�:<=

( C � � �8� � 1'356�;56#;HK54#6C�X<7%9�X �� � � � � �� � ���� � � ��� �� ( C � � ��� � 9�35;W9�C � � �

�� � ���� � � C ����P� BgX�=W69�35C�X<7H9�X �� �� � � � �

� � ���� �

�� �� v ( Cfm 2?°GI9'9�b°±d���%����� × C)BD=)^H:<1Aq6=W6#3IX@1'_p�(RV¯ R�¥,�H� × CfBD={`�6;5;5BD3HkwX<7561�:<6#=>F�� � � � CMX[9#?(C)BD3�9w`a1�KH35;56#;�CE6#X)1�_�X<7H6N�R����%T���µ�CM^%9�46���v � �¹vj� �)!Z�ZBD_ & �­h�S1�:)9y`�1'K53H;56#;�CE6iX)1'_ � >� �{�ZBD_ & ���j�[tu1�_P4#1'K(:<CE6'F�� � � � 4#1'3qAq6i:<k'6#CX<1 � BD3+��vj� �)!Z��CMXE:<1'35k5F 9�35;~`�6#4j9�KHCE6���v � �¹vj� �)!Z��BDCp9Y;HKH9�\2�O9�C|ePBD\]\�`�6@^5:<1jAq6#;~\�9�X<6#:i�[F�1'356f7H9�C � � ��v � � vj� �)!p�[m�H1�: �,� ���W� �)!Z�|1'356c7H9�C � � � > �,� ��v� �)!Z�[F�9�35; � � � � � � � > � � � � � � > � BD3 ��vj� �"!p�p9�C � � ®,F�`HK(X@9�CBgX{BDCf9�\]CE1 � � � � � � � � > � � � � � �p1'356r7H9�C � � � � � � � � � � > � � � � � ��� � v ( � ( ��� � KML.N4O�� � � � � ��� � v ( C � ( �¹F�eP75B]4[7°k'BgAq6C� � � � � > � � � v (�C � ( �,_�1�:p9�\D\ ( � �)!{m|¡u1'3qAq6i:<CE6\g?>BD_ � � ��v� �)!Z��9�35; � � � � � > � � � v (5C � ( �I_�1�:o9�\D\ ( � �)!{FHX<7563_�1�:"($� %�( �r1'356r7H9�C ��� � � �

"�� � ���

��� �BD3°X<756rCE6#35CE6Y1'_�;5BDCMXE:<Bg`HK5X<BD1'35C{9�C % � ®(F�`HK(X)9�C ��� � � �

"�� BDC{`�1'K535;56;hBD3

��v� �)!Z�29�35; ��vj� �)!p� � � C - � �)!p��� � � � - � �"!p�[F(X<7568\DBD=cBgX2`�6\D1'35k'C�X<1 � - � �)!Z�[m¢�756#:<6#_�1�:<6YX<756�_�9�4#XfX<7H9�X � � � v:� �5� �)!Z�8B]Cf6jJqKHBgA�9�\D63qX@X<1 ��� ���H� �)!p�@9�35;�� � � � � > � � � � ( C � (B��_�1�:c9�\D\

( � �)! B]CPXE:<K56@_�1�:)� � * ( , `HK(Xp351�Xp_�1�: * �¸��m|no1e26#Aq6i:|X<7H6{;HBg �6i:<6354#6{`�6iXse266#3 � v:�¹vj� �)!p�P9�35;�� �c� �)!p�BDC�351�XPCE66#3w_�1�:PCE1'=W68BD3qX<6#:E^�1'\]9�X<BD1'3�CM^%9�4#6#CP;56#zH356#;>ePBgX<7�X<756CE6pXse21WCM^�9�46#C#m� & ����$ §��%1�:>® � � �Ö��9�35;Ø� ( 5 ( , 1'356�7H9�C � � - � �)!Z� � � �� �)!Z��� � � 8 � � ��v� �)!Z� � � �� �)!Z��� � � 8 ���� � 8 � �"!p��_�1�: * � vv " � Fa9�35; � � �c� �)!p� � � - � �)!Z��� � � 8 � � ���W� �)!Z� � ��v� �"!p��� � � 8 � � � v:�¹vj� �)!Z� � � - � �)!Z��� � � 8 �� � v:�¹vj� �)!p� � ��v� �)!Z� � � � 8 �9�Yv

"� � v8 � �"!p�[m

�2È<ÆjÆ5#[§ � 356)K5CE6CpX<756)_+9�4iXpX<7%9�X"� v � � ! ��� � - � � ! �o9�35;�_�1�: � � � v � � ! ��X<7H6)351�:<= 1'_ � B]3 � v � � ! �|9�35;~X<7H6351�:<= 1'_ � BD3 � - � �)!Z��4#1'BD3H4#BD;56'm�N�6iX � � � � - � �)!Z�[F � v � � >� �)!Z�[F�9�35; � � ��vj� �)!p�[F%X<7H6#3 � � � � 9�35;X<756#:<6#_�1�:<6f� � � � v � � � 8 ��� � � � � v � � � 8 t54#1'3qAq6i:<CE6\g? N � � � � � � v � � � 8 =W69�35C N���I � � � %<� � �� ePBgX<7 � � %<� � � � � v 9,m 6�m% � �O® � ,��V9�35;"%

"� =c9�l � � � � � %<��� � � � � %T� � � � %<��� � � / � � � 8 � ® � ,�t � �� ��FI`HK(XdX<7563r9�C � � � v � � � v 9�35; � � � � %<��� � � � �� � � � %<��� � � � F-1�3H6r7%9�C � � %<� � � �� v 9(m 6�m�% � �O® � ,��89�35; %

"� =c9�l � � � � � %<��� � � � � %T� � � � %<��� � � / � � � 8 � ® � ,/t � �� �'F-9�35;X<756#:<6#_�1�:<6�N � � � � � v � � � 8 mr¢�7H6)C<9�=c6c9�:<k'K5=W63qX{k'BgAq6#C � � �c� �)!Z� � � - � �)!Z� � � � 8 � � � �c� �)!p� � ��v#� �"!p� � � � 8 9�35;� � v:�¹vj� �)!p� � � - � �)!Z� � � � 8 � � � v'� vj� �"!p� � ��vj� �)!Z� � � � 8 m� 356)1�`HCE6i:EAq6CpX<7H9�X � � �c� �)!p� � ��v� �)!Z��� � � � � � v'� vj� �)!Z� � ��vj� �)!p��� � � F�9�35;�X<756i:<6_�1�:<6@X<756CE6fXse21>CM^%9�4#6#C9�:<6o6JIKH9�\SF(9�35;�1'356|X<7H6#3YK5CE6#C2X<756|:<6#BgX<6#:[9�X<BD1'3WX<7561�:<6#=!1'_�GI9�4JIK56CMLON�1'K5BDC2N�QSR�TVUZ9�35;YGI9'9�br±d�%��� ����mI0�35;H6#6#; FIX<7H6

\DBD3H69�:d=c9�^ � �� � � � > �6� ��vj� �)!p�V7H9�C298351�:<= ( C � (B��BD3�� �c� �)!Z�V9�35;Y98351�:<= (/­Z1'3$��vj� �)!p��9�35;cX<756i:<6_�1�:<6vwN�6#1'35BD;H9P¢PR�T��%¥,¥,Q�F0sX[9�\DB�9�38=c9�X<756=c9�X<BD4#B�9�3 Fq� � ��Ô@=��j��ÕqÒ,mV¢�756d£p6#^%9�:EX<=W63qX�1'_(�~9�X<756#=r9�X<BD4#C 1'_,X<756d¦o35BgAq6i:<CEBgXs?

1'_�±dBDC<9)BDC�3%9�=c6#;�«<N�6#1'35BD;H9W¢PR�T��%¥,¥,Q�¬(mº N�9�=@`�6i:EX<1Y¡u��U����QsF(0sX[9�\]B]9�3(LS`�1�:<3�=r9�X<756#=r9�X<B]4#B]9�3aF��j�(�®�=%�j�'��®(m-no6oe21�:Ebq6#;y9�XP¦o3HBgAq6i:<CEBgXs?r1'_���BD4[7HBDkq9�3 F5�o353

�|:E`�1�:m

�'­

Page 83: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

98351�:<= ( C � ( � v " � 1�3 � ���W� �)!Z� � ��v� �"!p��� � � F(BSm 6'mV1'356|7H9�CR� � � � � > � � � v ( C � (B�¹v " � 1'3cX<75B]C-CM^�9�46�FIeP7HBD4[7rBDC-X<7H64[7%9�:[9�4#X<6i:<BD�j9�X<B]1�3�1'_�6#\D6=W63qX<C�1'_ � � v'� vj� �)!Z� � ��vj� �)!p��� � � m

���

Page 84: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

� �,m-��1'35;H9?��|^5:<B]\�­�Ò,F�­�®'®'®(m�I1'\gA(BD3Hkc351'3y\DB]3569�:|^%9�:EX<B]9�\�;5Bg �6#:<6#3qX<B]9�\�6JIKH9�X<BD1'35C|CE1'=W6#X<BD=W6C�:<6JIK5BD:<6#Co9c49�:<6_�K5\�K5CE6@1'_-9�;%9�^HX<6#;�_�K53H4iX<BD1'3H9�\

CM^%9�4#6C#mp¢�756@b(351jeP\D6#;5k'6@1'_VX<756f¢�7H6#1�:E?w1'_V0�3IX<6i:E^�1'\]9�X<BD1'3w7H6#\g^HCo_�1�:p4i:<6j9�X<B]35kr9r\]9�:<k'6{_+9�=WBD\g?>1'_-CEK54[7~CM^%9�46#C#Fa9�35;CE1'=W681�_�X<756= =r9?YB]35;56#6;�`�68K5CE6#_�K5\Sm

�~9�3q?�1'_VX<756@351'3�\]BD3569�:|^%9�:EX<B]9�\�;5Bg �6#:<6#3qX<B]9�\�6JIKH9�X<BD1'35CPeP7HBD4[7~9�:<6@CMX<K5;5B]6#;�7H9Aq6{X<7H6#Bg:p1�:<B]k�B]3yBDC8¡u1'3qX<BD3IK5KH=��64[7H9�35BD4CP1�:P±d7q?(CEBD4C#FH`HK(XPA'6#:E?�_�6iex=c9�X<756=c9�X<BD4#B�9�3HC2;H1cCM^�6#35;wX<756ZX<BD=c6oX<1WXE:E?YX<1cK535;56#:<CMX[9�3H;>eP7H9�XPX<756Z:<B]k�7IX6JIKH9�X<BD1'35C�9�35;{X<756d:<BDk'7qXVJIK56#CMX<B]1�3HC�CE751'K5\D;{`�6�F'9�35;@=c9�3q?Ze21�:Eb8_�1�:�?q6j9�:<C�1'3@;5BDCMX<1�:EX<6#;{6JIKH9�X<BD1'35C�ePBgX<7H1�K5X�b(351jePBD35kBgXt5X<756i:<6{9�:<6ZK535_�1�:EX<K53H9�X<6#\D?Y=c9�3q?YeP751Wb,3H1e�X<7568;56#_�6#4#X<C�1�_�X<756Z=c1I;H6#\DC�X<7H9�X�X<756i?�K5CE6Z`HK5X�^5:<6_�6#:�X<1W75BD;56ZX<756#=BD3�1�:<;H6i:2X<1@^5:<6#X<6#35;YX<7H9�XuX<7H6i?ce21�:Ebr1'3�CE1�=c6oK5CE6#_�K5\�:<69�\DBDCMX<BD4|^5:<1�`H\D6=>m�0sX�B]CuePBDCE6i:2X<1@`�6Z9e�9�:<6Z1�_ X<756p;56_�64iX<C21�_X<756P=c1,;56#\DC v 9�35;c=W63qX<BD1'3)X<7H6#=>F,9�35;c6#l,^H\]9�BD3)eP7H9�Xd1'356�XE:<BD6C�X<1@9�4}75BD6#A'6�`I?)e21�:Eb,B]35k81'3cCEBD=)^H\]BgzH6#;W1�:u;56#_�6#4#X<K51'K5C=W1,;56\DC#t�BD3H;56#6;�BD_u9rX<64[7535B]49�\V;5BD�W4K5\gXs?�7H9�CZ`�6#63�BD;56#3IX<BgzH6#;�1'3�9r:<6j9�\]BDCMX<BD4f=W1,;56\SFaBDXZBDCZ=@K54}7~`a6iXEX<6i:ZX<1YXE:E?yX<11jAq6i:<4#1'=c6ZX<7H9�XP^%9�:EX<BD4#KH\]9�:P;5Bg�c4#K5\DXs?Y`I?>e21�:Eb,B]35k)z5:<CMXo1'3y9WCEB]=)^H\DBDzH6#;>=c1,;56#\SFHeP75BD4[7y=c9?>7H9Aq68\]1�CMXoCE1�=c681'_�X<7H6:<69�\DB]CMX<BD4o_�69�X<K(:<6C�1'_�X<756p^H:<1�`%\D6#= X<7H9�XP1'356pe21'K5\];>\DBgbq6pX<1WCE1�\DA'6'm

¢�756uCM^%9�4#6�� �/7H9�C�`a6#6#38ePBD;56\g?ZK5CE6#;{B]38CEBgX<K%9�X<B]1�3HC eP756#:<6-CE1'\DK5X<BD1'35C�9�:<6d;HBDCE4#1'3qX<BD3,K51'K5C#F�`HK5X�X<756i:<6u9�:<6-:<6j9�CE1'35CX<1oX<75B]3(b8X<7%9�X�X<75BDCV_�K5354iX<B]1�3%9�\(CM^�9�46�BDCV351�X�:<69�\D\g?f9�;H9�^5X<6#;fX<1p=W1'CMX-351'35\DBD356j9�:�^�9�:EX<B�9�\(;5Bg �6#:<6#3qX<B�9�\(6JIKH9�X<BD1'35C�eP756#:<6;5BDCE41'3qX<BD3IK51'K5CPCE1'\DK(X<BD1'35CP9�:<68_�1'K535; m

� _54#1'K(:<CE6'F�X<756i:<629�:<6d^5:<1�`H\D6=WC�eP756i:<6dX<756�����CM^%9�4#6uBDC�9�CMbq6#;f_�1�:F�9�35;@CE1'=W6d^H:<1�`%\D6#=cC�BD3)Ðo6#1'=c6iXE:<BD4-�~69�CEK(:<6¢�7561�:E?Y9�:<6o1�_ X<75BDCuXs?I^a6�F(9�35;YX<756i:<6p9�:<6p9�^5^H\DB]49�X<BD1'35C-X<1fBD=c9�k'6P^5:<1,4#6CECEBD35k@_�1�:26#l59�=W^H\D6'FI`HK(X�1'356o7H9�C2BD=W^�1'CE6#;cX<1\D1,1�b�_�1�:o9c;51'=c9�BD3wePBgX<7>z%35BgX<68^�6i:<BD=c6iX<6#:{�SBSm 6�m2eP751'CE6@4[7H9�:[9�4iX<6#:<BDCMX<BD48_�K53H4iX<BD1'3>`a6#\D1'35k'CPX<1����@�[F�1�:|X<1r\D1,1�b�_�1�:p9CE6iX�ePBDX<7YzH3HBgX<6W� & >��j��LO;HBD=W635CEBD1'3H9�\ano� �U �-R�� �?��=c69�CEK(:<6'm

¢�756�=c9�BD3@4\]9�CECV1�_H^%9�:EX<B]9�\,;5Bg �6#:<6#3qX<B]9�\I6jJIKH9�X<BD1'35C�eP756i:<62;HBDCE4#1'3qX<BD3,K51'K5CVCE1�\]K(X<BD1'35C�9�^5^�69�:�_�1�:�BD3IXE:<BD35CEBD4d:<6j9�CE1'35CBDC-X<7H9�Xu1'_a7q?,^�6i:E`�1'\DB]4�4#1'35CE6i:EA'9�X<BD1'35Cd\]9ePC#tIX<75BDCd4#\�9�CECu4#1jAq6i:<CuBD=)^�1�:EX[9�3qXdCEBDX<KH9�X<BD1'35CdBD3�¡u1'3qX<BD3IK5KH=¸��6#4}7H9�35BD4C#F,9�35;X<1,18\]BgXEX<\D6�B]C-KH35;56i:<CMX<1,1,;r_]:<1'=!98=c9�X<756=c9�X<BD49�\(^a1�B]3qXu1�_�A(BD6ie8m� �6#49�K5CE6|1'_�X<7H6#Bg:d^5:[9�4iX<B]49�\HBD=)^�1�:EX[9�3546�Fq3,K5=W6#:<BD49�\CEBD=fK5\]9�X<BD1'35C21�_ X<756CE6o^5:<1�`H\D6=WC�9�:<6o^a6i:<_�1�:<=c6#;aF(_�1�:�6#l59�=W^H\D6oBD3Y1�:<;56#:2X<1f41'=)^HK(X<6oX<756 %%1e 1'_d�S4#1'=)^H:<6#CECEBg`H\]6��d9�BD:9�:<1'K535;�9�3�9�Bg:E^H\]9�356'm8¢�756��S�H:[9�3H4#1�L- 2:<BgX<BDCE7��o¡u1�3H4#1�:<;56)BDC|X<756f1'35\g?y4#1'=c=W6i:<4B]9�\�^H\]9�356@eP751 %HBD6Co1jAq6i:Z�~��²�³�ºr��F`HK(X|=W1'CMX|4#1'=W=c6i:<4B]9�\�^H\]9�356C'%H?�_+9�CMX|6#3H1�KHk�7>X<1W:<6JIK5Bg:<6Z4#1'=W^HK(X[9�X<BD1'35C�1'_�XE:[9�35CE1'35BD4�%H1jePC#t%BD35;H6#6#;>X<7568CM^�6#6;w1�_CE1'K535;Y;56#^�6#35;HCuK(^�1'3rX<756PX<6#=)^a6i:[9�X<K(:<6|9�35;cX<7H6�^H:<6#CECEK(:<6'F(9�35;rX<756|CE7H9�^�6ok'BgAq6#3cX<18X<756PePB]35k'Cd1'_�X<7H6�^%\]9�356|=c9�bq6#C=W1�:<6Y9�Bg:)k'1y`�6#\]1e!X<756rePBD3Hk~9�35;¶4i:<6j9�X<6#Cf9yCE\DBDk'7qX)CEK(:E^5:<6CECEBD1'3h`�6#\]1e!X<756rePBD3Hky`HK(XW9y75BDk'7¶;56i^5:<6CECEBD1'3�9�`�1jAq6X<756fePBD35k5F�9�3H;�9�XZX<756)4i:<KHBDCE6@Aq6\D1,4#BgXs?y1'_d\]9�:<k�6)4#1'=c=W6i:<4B]9�\H�E6iX<CoX<756)CM^�66#;�1'_dCE1'K535;�BD3~X<7H9�XZ;56i^H:<6#CECEBD1'3��+eP75BD4[7CEK54[b,C�X<756p^H\]9�356ZK(^Ie�9�:<;��2BDC2X<7563�B]35_�6#:<BD1�:�X<1fX<756pAq6#\D1,4#BDXs?r1�_�X<7H6o^%\]9�356�m-£p1'BD35kfX<756CE6Z3IK5=W6#:<BD49�\aCEBD=fK5\]9�X<BD1'35C27%9�C`�64#1'=W6{=fK54[7~\D6#CECo6il,^�6#3HCEBgAq68X<7%9�3�K5CEBD3HkrCE=r9�\D\�CE4j9�\]68^%\]9�356#CoBD3wePB]35;wX<KH35356#\]C#Fa9�35;yX<756@CE7H9�^�6f1�_VX<7H68^%\]9�356@49�3`�6)BD=)^H:<1Aq6;¶�S=c1'CMX<\g?y_�1�:Z;5B]=WBD35B]CE75BD35kcX<7H6f_�K56#\V41�3HCEK5=)^5X<B]1�3~1'_-X<756f^H\]9�356�Fa:[9�:<6#\g?�_�1�:Z;5BD=WB]35BDCE75BD3HkcX<7H6f351'BDCE6fBDX

v~�I1'=c6YA'6#:E?hk�1,1,;�9�35;¶A'6#:E?h751'356CMX)=r9�X<756#=r9�X<B]4#B]9�35C@=c9?¶CMX[9#?¶K53H9e�9�:<6�1'_PCE1'=c6Y^5:[9�4iX<BD4j9�\u\]BD=WBgX[9�X<BD1'35C@1�_X<756{6JIKH9�X<BD1'35C�X<7H9�X�X<7H6i?>7H9Aq68CMX<KH;5BD6#; F�9�35;y9�3y6#l(9�=W^H\D6pe�9�C|k'BgAq6#3>X<1c=W6Z`,?>Gq69�3yN��������uF5eP7H6#3y756{6il,^H\]9�BD3H6#;X<1y=W6rBD3/�j� ��ÕyX<756r1�:<BDk'BD3°1�_2X<756c^�1'\DBgX<B]49�\�;5Bg�c4#KH\gX<BD6#CZX<7%9�Xf756r7H9�;°6354#1'K53IX<6i:<6#; mw�oC@9�3h1��W46i:@B]3�X<756r�5:<6354[79�:<={?qFa756)7H9�;�`�66#3~X[9�bq63y^H:<BDCE1'356i:89�35;�7H6fCM^�6#3qX8=c1�CMXZ1'_$1°1�:<\D; 1h9�:o0E0|B]3~9wÐo6#:<=c9�3~49�=)^��+eP75BD\D6f9Y_+9�=W1'K5C=W6={`�6#:-1'_%X<756 �R����¯�����Q%k�:<1'K(^c7H9�;c;51,;5k'6#;)X<7H6�;(:[9�_]Xi�[Fq`HK(Xd756�41'3qX<BD3IK56;)X<1Z;51Z:<6CE69�:<4[7aF,9�35;c6iAq6#3c1�:<k'9�35B]�#6#;9~K535BDA'6#:<CEBgXs?��S1'_�eP75BD4}7¶7H6Ye�9�CW4}7H9�354#6\D\D1�:i�ZBD3HCEBD;56YX<756�4j9�=W^ mhno6YX<1'\D;¶=W6YX<7%9�XW756�7H9�;�CMX<1�^H^�6#;�e21�:Eb(BD3Hk~1'3�o�,�VQ���� L��I��R�����UY6jJIKH9�X<BD1'35C#F�_�1�:)_�6j9�:fX<7%9�X)75BDC@:<6CEK5\gX<Cf4#1'K5\];h`�6YK5CE6#;¶`I?hX<756wÐo6i:<=r9�3 tu_�1�K5:EXs?�?q69�:<C)9�_�X<6i:)75BDC;564#BDCEBD1'3w756{CMX<B]\D\ CE6#6=W6;wK53H9e�9�:<6{X<7H9�XPX<7568e21�:Eb,C|1'_V=c9�X<756=c9�X<BD4#B�9�3HC�1'3wX<75BDCP1jAq6i:<CEB]=)^H\DBDzH6#;w=W1,;56\ 7H9�C|7%9�:<;H\g?9�3q?Y^H:[9�4#X<BD49�\aK5CE6'm

�oBD41'\]9�C� PR����¯�����Q�BDC-X<756�^HCE6#K5;H1�3I?,=!1'_a98k�:<1'K(^r1'_�=c9�X<756=c9�X<BD4B]9�35C#Fq=W1'CMX<\g?f�H:<6#354}7atIX<751'CE6�eP7H1{4[751'CE6�X<7H63H9�=W684#6#:EX[9�B]35\g?cb(356iex9�`�1'K(Xo9)�5:<6354[7wk'6#356#:[9�\ 3H9�=W6; PR����¯�����Qsm

¡u7H9�:<\D6C�£Z6#35BDCZ�,9�K(X<6i:� PR����¯�����QsF5�H:<6#354}7yk'6#356#:[9�\OF�� �,�jÒ�L<� �'� (It�1�_dÐ|:<66ib>9�3546#CMXE:E?qFH7H68;564#\DB]356#;w9�3y1� a6i:|1�_X<756ZX<7(:<1'356Z1'_-Ð|:<6#6#46ZBD3°���'Ò'­,mº Ù-:<35CMXp����²�³uFa¡u�#6#4}7yCE4BD6#3IX<BDCMXF������ ��L<�j�,�jÒ,m�no68e21�:Ebq6#;�BD3>ÑpBD6353H9(mP¢�756{����²�³�3IKH={`�6#:|B]C�X<7H68:[9�X<BD1W1'_�X<7H6

Aq6#\D1,4BgXs?r1'_�X<756p^%\]9�356pX<1fX<7H6ZCM^�66#;w1'_�CE1'K535; m

��Õ

Page 85: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

=c9�bq6#C}�[FH`HK(XPX<756{=c9�X<756=c9�X<BD4j9�\�b(351jeP\D6#;5k'681'_�X<756#CE6{JIK56CMX<BD1'35C#F%=W1'CMX<\g?�;5K568X<1)X<7H6Ze21�:Eb�1'_dÐo�����V¯����VQ���T � 9�35;1'_�¡29�X<75\]6#6#3>��R����,´>�%��µ � FHBDC�CMX<BD\D\aBD3HCEK(�c4#BD63qXP_�1�:P41�:E:<1�`�1�:[9�X<BD35k)X<756ZBD3IX<K5BgX<BD1'3�1'_�X<7H6Z6#3Hk�B]356#6#:<C#m

�%1�:Y� �-*�� ,�F X<7H6� ���U�R �*CM^�9�46"� v � � � � � �{�� � � v'� �5� �@� � ���(� �@��� v � � � � � �,� ���5� �{� � � � � � � > � � � � (C � ( � v � � _�1�:|9�\D\ ( � � �{41�3IX[9�B]35C�;5BDCE41�3IX<BD3IK51'K5C{�+^HBD64#6iePB]CE6pCE=W1,1�X<7��d_�K5354#X<BD1'35C8�+eP75BD\]6p_�1�:{��� 5 �-,/FHX<756ZCM^%9�46� v � � � �8 � �{��;51,6C-3H1�Xu41�3IX[9�B]3)^HBD64#6#ePBDCE6�CE=W1,1�X<7W;5B]CE4#1'3qX<BD3IKH1�KHC-_�K5354#X<BD1'35C}�[m�¢�756P6il(4\DK5;56;W49�CE6 * �Ø��41�:E:<6#CM^�1'35;5CX<1"� �c� �{�[FH9�35;�* ��, 4#1�:E:<6#CM^a1�3H;5C�X<1 � >� �{�[F5CE1WeP7q?YX<7HBD3(brX<7H9�X�X<756Z4j9�CE6c���+* �-, BDC�`�6#XEX<6i: �

N�BD3569�:o^%9�:EX<B]9�\�;5BD �6i:<63qX<B]9�\�6JIKH9�X<BD1'35C|ePBgX<7y41'35CMX[9�3qXp41,6i�c4#BD63qX<C|4j9�3y`a6@CE1'\gAq6#;y`,?wKHCEBD35kY6\D6#=c6#3qX[9�:E?wCE1'\DK(LX<BD1'35C#F�9�35;{_�1�:�6#\]\DBg^5X<BD4�6jJIKH9�X<BD1'35C X<75BDC�\]69�;5C X<1|CEBD35k'K5\]9�:�B]3qX<6#k�:[9�\DC eP75BD4}784j9�3{`�6dCMX<K5;5BD6;Z`I?f¡u��¥��������R�T�L ����¤���T��X<7561�:<6=>F%eP75BD4[7y:<6JIK5BD:<6#C{� � * � ,/mP�H1�:o6#l59�=W^H\D6'F%CE1'\gA(BD35kE? � � @�_�1�:R@ � ���5� �)!Z��k'BgAq6#C �

��

��� � ��� � � ���5� �)!p�_�1�:�� � "#�� ���������'& FHBD_d���+* �-,/F5`HK(X�X<756p:<6#CEK5\DX�BDC�_�9�\DCE6p_�1�: * ��Z1�: * ��,�FH9�35;6@ � � v � � ! �d;51,6C�351�XPBD=W^H\g?X<7H9�X�X<7568;56i:<BgA'9�X<BgAq6#C ���

��� � `a6#\D1'35kfX<1"���W� �)!Z�[m¢�756#:<6c9�:<6c751je26iAq6#:{1�X<756#:{CM^%9�4#6#C{eP75BD4[7°49�3�`�6WK5CE6;°_�1�:{:<6i^H\]9�46#=W63qXF�K5CEB]35k>X<756cno����� �*CM^%9�46�=�vj� �)!p�BD35CMX<6j9�;~1'_ ��vj� �)!Z�[F 9�35;~X<756)CM^%9�4#6�� � � � �)!Z�PBD35CMX<6j9�;~1'_ � �� �)!Z�[F 9�35;�B]35;56#6;~CEBD35k'K5\]9�:ZBD3IX<6#k�:[9�\DCp9�4#X<Cp_]:<1'== v � � ! �|X<1wBgX<CE6\D_u9�35;°_�:<1'= � � � � � ! �oBD3IX<1>BDX<CE6#\D_stV9�4iX<KH9�\D\g?qFaX<7H6WBD3qX<6#:E^�1'\]9�X<BD1'3�CM^%9�4#6CZ`�6#Xse26#63 = v � � ! �o9�35;� � � � �)!Z�29�:<68X<756{C<9�=c6ZX<7H9�3wX<756{9�\g:<69�;(?>CMX<K5;5BD6;wBD3IX<6i:E^�1'\]9�X<BD1'3>CM^%9�4#6#CP`�6#Xse26#63 ��v� �)!Z��9�35;#��>� �)!Z�[FH`HK(XX<756CE6Z:<6#CEK5\gX<C|49�35351�XP`�68;56i:<BDA'6;wCE1c69�CEBD\g?qmdno1je26iAq6i:PX<756i:<68B]CP9�351�X<756#:|1�`HCMX[9�4#\D6'F5eP75BD4}7>CEKHk�k'6CMX<C�X<7%9�XPX<75684}751'BD4#6* � ­fBDC�X<75681�3H\g?Y:<BDk'7qX�1'356'm

¢�756~�SCE49�\]9�:i�|e�9Aq6r6jJqK%9�X<B]1�3°BD3¶9wk'6356i:[9�\d=c6#;5B]K5= 7H9�C@X<756c_�1�:<= ���GQa� � � �� � � > � !7 � � ��v ���� � � N 7 �'�GQa� ������ � � ��®,F

eP756#:<6dX<756u41,6i�c4#BD63qX<C � N 7 � � � � "#�Ø� ���������:& `�6#\]1�3HkPX<1 � >� �)!Z�[F�C<9�X<B]CE_]?8X<756uCM?(=W=W6#XE:E?84#1'35;5BgX<B]1�3�N � 7 �GQa�H� N 7 �'�GQa�9(m 6'm Q � �)! _�1�:f9�\D\�� � " � � ���������:& F�C<9�X<BDCE_�?�X<756c^�1'CEBgX<BgA(BgXs?�^5:<1�^�6#:EXs? ���GQa�# � " �¸®y9(m 6�m�Q � �)! 9�35;¶9�\DCE1C<9�X<BDCE_�?�X<756W6#\D\]Bg^5X<BD4BgXs?>^5:<1�^�6#:EXs?�X<7H9�X8_�1�:{CE1�=c6 � � ®>1'356W7H9�C � !7 � � ��v N 7 �'�GQa� � 7 � �9 ��� � � ºf_�1�:@9�\]\ � � �)!{F�9,m 6�mQ � �)!{m�¦o35;H6i:VX<756#CE624#1'35;HBgX<BD1'35C�X<756|¡u� V²a³ �>^H:<1�`%\D6#= BDC�e26\D\I^�1'CE6;fBD_H1�3H62BD=)^a1�CE6C � � � � � � � � � ;°vj� �)!Z��9�35;���� � �� � � � � � v � �dº�� �"!p�[F'9�3H;@1'356u7H9�C�41�3HCE6i:EA�9�X<BD1'3{1'_(X<1�X[9�\I6#356#:<k�?qF�CEK5=Ø1'_(X<756db(BD356#X<BD4-6356i:<k�? vº I � 1 ���GQa� �� ���� � �� º OMQ9�35;cX<756�^�1�X<6#3IX<B]9�\56356i:<k�? vº I � 1 � � !7 � � ��v N 7 �'�GQa� ������ � � ���� � � O Q�mV0sXde21'K5\D;cCE6#6=¸3H9�X<K(:[9�\HX<1{6il,^�6#4#X�X<7%9�XdePBgX<7cCE=c1,1�X<74#1,6#�W4BD6#3IX<C#FH\]Bgbq6Z_�1�:|X<7568CEB]=)^H\DBDzH6#;�e�9Aq6{6JIKH9�X<BD1'3wBD3y9W751'=c1�k'6356#1'K5CPB]CE1�XE:<1�^HB]48=c9�X<6i:<B]9�\ �

��� � � > 4 º ? � � ®(F�1�3H64#1'K5\];rX[9�bq6 � � � � v'� �H� �)!p�d9�35; � v �*®@_�1�:�6#l(9�=W^H\D6o9�35;Yz%35;YX<756oCE1'\DK(X<BD1'3 � � � � %<� � � v'� �H� �)!p�-_�1�:% ��®Y�S1�:�_�1�:

% ��®Y9�CoX<756@e�9Aq6f6JIKH9�X<BD1'3�BDCoBD3IA�9�:<B]9�3qXoX<7(:<1'K5k'7�X<BD=W6{:<6#A'6#:<C<9�\��[F�`HK(X1h9�\gX<6#:pN�Q�� � �p��T � 7H9�CpCE751jeP3~X<7H9�XoX<75B]C1'35\g?Y7%9�^H^�6#35C|_�1�: *9� ­,m

¡u1'K5\D;~X<756)CM^%9�46�� v � º � º � �)!p�P`�6)K5CE6;~X<7563~_�1�:Z3H1�3H\DBD356j9�:87q?,^�6i:E`�1'\DB]4@6JIKH9�X<BD1'35CpeP756i:<6)1'356W6il,^�6#4#X<CoCE1'=c6;5BDCE41'3qX<BD3IK5BDX<BD6#C2X<1W1,4#4#K5: �

0sX|B]CPK5CE6#_�K5\�X<1W4#1'=W^%9�:<6 ���W� �)!Z�uePBgX<7>X<756 ���U�R ��CM^%9�4#6 � v � � � � � �)!Z�[FHeP75BD4}7wBDC�X<7H68CM^%9�46{1'_ � � ���5� �)!p�CEK54}7>X<7H9�X#� � � � � > � � � � ( C � (B�¹v � �W_�1�:|9�\D\ ( � �"!{m� & ����$ § � 356�7H9�C�� �c� �)!p� � >� �)!p��� � v � �.� � � �)!p�[F|9�35;�=W1�:<6w^5:<64#BDCE6\g? � � �c� �)!p� � � �� �)!p� � v � � � � �� v � � � � � � ! �[m

0�_ � B]C�9)4}7H9�:[9�4iX<6i:<B]CMX<BD4p_�K5354iX<B]1�3 F(X<7563 �6� � v � �.� � � �)!p�uB]=)^H\DB]6#C �6� � �c� �)!Z�o�O9�3H;>41'3qAq6i:<CE6#\D?H�[m�2È<ÆjÆ5#[§�¢�7H6@\DBD3H69�:|=c9�^5^%BD35k � �� � � � > � BDCo4#1'3qX<BD3IKH1�KHCo_]:<1'= � �c� �)!p��BD3qX<1 ��vj� �)!p��ePBgX<7y351�:<= (�C � ( �¹F�9�35;_�:<1�= � � � ! �2BD3qX<1WBDX<CE6#\D_�ePBgX<7w351�:<= (�­,F%9�35;>X<756#:<6#_�1�:<68BgX�BDCP4#1'3qX<B]3IK51'K5C|_]:<1'= � ���W� � ! � � � � � ! ��� v � � � � B]3qX<1� ��vj� �"!p� � � �� �"!p� � v � � � � � ���5� �)!Z�dePBgX<7>351�:<= ( C � (B�¹v � �(m� ±�9�K5\HÚ{m,Ðo�����V¯�� ��Q���TdF'�o=W6#:<BD49�3)=r9�X<7H6#=c9�X<BD4B]9�3am�no6�e21�:Eb(Cd9�X-X<756o¡�R������T���0�35CMX<BgX<K(X<6�1'_���9�X<7H6#=c9�X<BD4j9�\

�I4BD6#3546#C#FH�o6ie �-1�:Eb�¦o35BgAq6i:<CEBDXs?'m� ¡29�X<75\D66#3y����T�¤V�>��R����,´>����µ(F�¡29�3H9�;5B]9�3(LS`�1�:<3>=c9�X<756#=r9�X<BD4#B]9�3aF��j�'­��,m2�I756oe21�:Eb(C�9�X�X<756@¡�R������T���0�35CMX<BgL

X<K(X<6�1�_��~9�X<756#=r9�X<B]49�\%�I4#B]6#3546#C#Fq�o6#e �-1�:Eb)¦o35BgAq6i:<CEBgXs?qm�no6i:d_�9�X<756#:FqGq1�7H3WN�B]k�7IX<1�3Y���VT�¤���F�0s:<BDCE7W=c9�X<756=c9�X<BD4B]9�3aF� �'� (jL<�j�'�'ÔIF%e�9�CP_]:<1'=��j��ÕIÒfX<1w�j��Õ��fX<7H68no69�;w1'_�X<756{�~9�X<756=c9�X<BD4#C�£p6i^�9�:EX<=c6#3qXo9�X�X<7H6f¡u����T���¤VQ��w0�35CMX<BDX<K(X<6Z1�X¢�6#4[7H351'\D1'k�?�BD3>±dBgXEX<CM`HK(:<k'7aFH\]9�X<6i:�X<1)`�6#41�=c6@¡u����T��a¤�Q��y����¥,¥(R�T�¦o35BgAq6i:<CEBgXs?qm

�o35;5:<6ie ¡u����T���¤VQ���F��I41�XEX<B]CE7(LS`�1�:<3�`HK5CEB]356#CECE=r9�3y9�35;>^H75B]\]9�3qX<7(:<1�^HBDCMXF�� � �'Ô@L|��(�j�,m�o35;5:<6ie 1 BD\D\DB]9�=�����¥,¥5R�TdFH�o=c6i:<BD4j9�3YzH3H9�354#B]6i:|9�35;>^H75BD\�9�3IX<7(:<1�^HBDCMXF � �'Ô�Ô�L<�j����(qm

� 1h9�\gX<6#:�N�Q�� ���p��TdF5�o=c6i:<BD4j9�3>=c9�X<756=c9�X<BD4B]9�3am�no6Ze21�:Eb,CP9�XP¦o35BDA'6#:<CEBgXs?Y1'_���BD35356CE1�X[9,FH��BD3H3569�^�1'\DBDC#m

�'Ô

Page 86: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

0�_ � � � v � � � � � �)!Z�c1'356�7H9�C6I�� 1 � � �GQ<>1(�� > � �GQa��� � O Q ( C5� ( �-_�1�:y9�\]\ ( � �)!8t{B]_8=c1�:<61jA'6#: � B]C>94[7%9�:[9�4#X<6i:<BDCMX<BD4d_�K5354#X<BD1'3aF�X<756#3 � �GQ > (�� > � �GQa�a4j9�3f1'35\g?ZX[9�b'6dX<7H6dA�9�\DK56C >8� � ® � �u9�35;@X<756i:<6_�1�:<6R� � �GQ3> (�� > � �GQa��� � �� � �GQ9>�(��H> � �GQa��� F�_�:<1'= eP75B]4[7�1'356f;56;5K546#CoX<7H9�X�� � � � � > � � � v ( C � (B�5_�1�:p6iAq6i:E?�( � �)!{Fa9�35;�9�Cp1�3H6)9�\]CE1r7%9�C� � � � � v � � � � � � �� FH1'3568;56#;HK54#6C�X<7H9�X �6� ���W� �)!Z�o�+X<756Zz5:<CMX�^%9�:EXPB]=)^H\g?(BD3Hk@X<756Z41�3IA'6#:<CE6��[m

�'Ò

Page 87: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

�j�,m-�5:<BD;H9?��|^5:<BD\�­ �,F%­�®'®'®,m'oKH9�CEBD\DBD356j9�:p7q?,^�6#:E`�1'\DBD4f6JIKH9�X<BD1'35Cp7H9Aq6f^5:<1�^�6#:EX<BD6#CZJIK5BgX<6f;5BD �6i:<63qXp_�:<1'= CE6=WBD\]BD3569�:p7q?,^�6i:E`�1'\DB]4f6JIKH9�X<BD1'35C#F

=W1'CMX<\g?*`�649�K5CE6°;HBDCE4#1'3qX<BD3,K5BgX<BD6C>=c9? 9�^5^�6j9�:�6iAq6#3 eP7H6#3�BD3HBgX<B]9�\Z;H9�X[9�9�:<6�Aq6i:E? CE=c1,1�X<7 m �H1�:~CE6#4#1'35;x1�:<;H6i:6JIKH9�X<BD1'35C�=W1,;56\D6#;�1'3�X<756�\DBD356j9�:re�9Aq6~6jJIKH9�X<BD1'3 �

��� � � > 4 º ? � �Ö®(F|CE6=WBD\]BD3569�:Y6jJIKH9�X<BD1'35C�9�:<6~1'_ZX<756�_�1�:<=

���� � ��> 4 º�? � � � � � � �� � � ���� � / ��������� ������ 1 �peP7H6i:<6pX<7H6p7HBDk'756i:P1�:<;56#:�^%9�:EXPBDC�\DB]3569�:�ePBDX<7�4#1'35CMX[9�3qX|4#1,6i�c4#B]6#3qX<C#F(eP75B]\D6_�1�:pJIKH9�CEBD\DB]3569�:|6jJqK%9�X<B]1�3HC�X<7H6{75BDk'756#:|1�:<;56#:|^%9�:EXoBDC|CMX<BD\D\�\DB]3569�:P`HK5X|ePBgX<7y41I6#�c4#BD63qX<C�eP75BD4}7y;H6i^�635;yK(^�1'3�\D1je26i:1�:<;56i:W;56#:<BgA�9�X<BgAq6#C#tuBD3¶X<756�49�CE6 & �»�Y1'356�7%9�CW_�1�:W6il59�=)^H\D6 �

��� � � >5@ � ������ � �

��� �� � ®(mh¢�7HBDCf6jJIKH9�X<BD1'3he�9�Cfz5:<CMX

CMX<K5;5B]6#;�`,?�±uR�QSU�U�R�T*BD3/� ��®�(IFV9�Cf9w=c1I;H6#\-_�1�:@9w41�=W^5:<6#CECEBD`H\D6Wkq9�C#t�X<756c4\]9�CECEBD49�\�:<6\]9�X<BD1'3�* � 4 �~\D69�;5C8X<1�9�3BD3541�:E:<6#4iXZA�9�\DK56)1'_dX<756)A'6\D1,4#BgXs?�1'_uCE1�KH35;��+eP75BD4}7�7H9�;�`�66#3�6#CMX<B]=c9�X<6#;~`I?~�o�%´>��R�T��[F�9�35;�N��TA�¥,��²��hCE66#=WCpX<17H9Aq6r^5:<1�^�1'CE6;hX<1~K5CE6YX<756r:<6\]9�X<BD1'3 * � 4 �

� t2±uR�QSU�U�R�T 7H9�;¶_�1'K535;�CE1'=W6YCM^�64#B]9�\uCE1'\DK(X<BD1'35C��+eP7HBD4[7�9�:<6Y49�\D\]6#;CEBD=W^H\D6�e�9Aq6#C23H1ep�[F,`HK(X�756|\D6_]XuX<756#=!BD3�9�3YBD=)^H\]BD4#BgXd_�1�:<=wF,CE1{X<7%9�X2BgXdX<1,1�bW_�1'K(:EXs?W?'6j9�:<Cd`�6_�1�:<6oCE1�=c6#1'356|^�1'BD3qX<6#;1'K(XuX<7H9�XdX<7H6i:<6Pe�9�C298^5:<1�`H\D6=>FIeP75BD4}7>�I��R����aUo6il,^H\]9�BD356;cBD3�����Õ��Z`I?WX<7H6P_�1�:<=c9�X<BD1'3c1'_a;HBDCE4#1'3qX<BD3,K5BgX<BD6C#m-�I��R�����U4#1'=W^HK(X<6#;�X<756c4#1�:E:<6#4#X2�EK5=)^°41'35;5BgX<BD1'35CZX<1wBD=)^�1'CE6)_�1�:{;5BDCE41�3IX<BD3IK51'K5C{CE1'\DK(X<BD1'35C#F�`I?�K5CEBD3Hk>X<756W4#1'35CE6#:EA�9�X<BD1'3�1�_=c9�CECP9�3H;�X<756p41�3HCE6i:EA�9�X<BD1'3�1'_�=W1'=c6#3qX<K5=wF59�3H;�X<756#CE6Z4#1'35;5BgX<B]1�3HCue26#:<6p:<6#;5BDCE41jA'6#:<6#;�`I?YÚPQ�� �p��T�ThBD3�� �'Ò�®(F(`HK(XBD35CMX<6j9�;h1�_�`�6#B]35ky49�\D\D6;�X<756w�I��R����aUr41�3H;5BgX<BD1'35C@1�:@X<756>�I��R����aU#LOÚPQ�� �Z��T�T 41�3H;5BgX<BD1'35C#F�X<7H6i?°9�:<6Y351je!b(351jeP39�C|X<7H6fÚP��T���Q�T���vMLOn�V¤-R�T�QSR���º{4#1'35;5BgX<B]1�3HC#mZ¢�756f;56#_�6#4#X<Co1'_�X<756#CE6��SBDCE6#3IXE:<1�^%BD4���=W1,;56#\]C|e26i:<6@3H1�XZ1�`IA(BD1'K5CoX<756#3aF9�C{¢�756i:<=c1,;(?(3H9�=WBD4Coe�9�CZ`%9�:<6\g?�B]3�BgX<C8BD35_+9�354i?�9�XZX<756)X<BD=c6�F�9�3H;�6iAq6#3¶�I��R����aUWe�9�CZe�:<1�3Hk�\D?~4#1'3qA(BD3546#;�\]9�X<6#:`I?°¢�³VR��{UjR�T �+eP751w\]9�X<6i:8`�6#4j9�=W6WN�1�:<; &Z��¥}�VQ�T-�p9�35;�`I?�ÚP����¥,��Q+¤V³ � F eP751>X<1'\D;�7HBD= X<7H9�X{75BDC8;5BDCE41'3qX<BD3IK51'K5CCE1'\DK(X<BD1'35C�e26#:<6Z351�X�^H7I?,CEB]49�\a`�649�K5CE6pX<7H6i?�;5BD;>3H1�XP41'35CE6i:EAq6Z6356i:<k�? � m

¢�756�9�^5^%9�:<BgX<B]1�3)1'_%;5BDCE41�3IX<BD3IK5BgX<B]6#C�BDC�=W1�:<6�69�CEBD\g?{CE66#3)1'3fzH:<CMX-1�:<;H6i:�6jJIKH9�X<BD1'35C#F'_�1�:�eP75BD4}7)9p4#\�9�CECEB]49�\,=c1I;H6#\ �BDC8X<7H6 ����¤V�%�VUr6JIKH9�X<BD1'3 � �

� � Ó � ������ ��®(m>¦oCEB]35k>X<756r=W6#X<751,;°1�_�4}7H9�:[9�4#X<6i:<BDCMX<B]4W4#K5:EA'6C#F�1'356rzH35;5C@69�CEBD\D?�X<7H64i:<BDX<BD49�\�X<B]=W6p1'_�6il(BDCMX<6#3H4#6Z1'_�9)CE=W1,1�X<7>CE1'\DK(X<B]1�3 m� & ����$ §�N�6#X � � `a6{9)CE=W1,1�X<7>`�1'K535;56;w_�K5354#X<BD1'3>1'3 �fmd0�_ � � BDCP351'35;564i:<6j9�CEB]35k5F5X<7H6i:<686il(BDCMX<CP9WK535B]JIK568CE=c1,1�X<7CE1'\DK(X<BD1'3w1'_"� �

� � Ó 4 � ������ � ®W_�1�: % ��®WC<9�X<BDCE_�?(BD35k � �GQ � ®I�3� � � �GQa��_�1�: Q � �fmu0�_VBD3H_ � � � � � �� � �GQa�3� >�� ��®(F%X<756i:<6BDC�9fK535B�JqKH6oCE=c1I1�X<7�CE1�\]K(X<BD1'3Y_�1�:�® ��% � � � � v� FH9�3H;YX<756i:<6pBDC2351)CE=W1,1�X<7YCE1'\DK5X<BD1'3Y1jAq6i:P9�3�BD3qX<6i:EA'9�\��O® � �o��ePBgX<7� � � � m�2È<ÆjÆ5#[§y�oCECEK5=c6wX<7H9�XcX<756i:<6y6#l(BDCMX<Cr9�CE=W1,1�X<7�CE1'\DK5X<BD1'3�_�1�:Y®�� % � �Zm �H1�:�* � � 1�3H6y;56izH3H6#CcX<756y4[7H9�:[9�4iLX<6i:<B]CMX<BD4�4#K(:EAq6�ePBDX<7�BD35BgX<B]9�\u^�1'BD3qX�*h`I? � � � � �� � � � ���H� %<� � %<�@9�35; �5�O®I����*%F�9�35;�1'356w;56;5K546#C)X<7%9�X �

� � � � � � � � � � �� � ����������5� %<� � %<� �

� � � �� � Ó � �

� � ���5� %<� � %<�#� ®(Fu9�35;�X<756i:<6_�1�:<6 � ���H� %<� � %<�W� � ���H�O®I� � ®I�#� � � �+*5�8_�1�:W® � % � �ZtuX<7HBDCfk'BgAq6C� � � � �� � � � � �+*H�[F%BSm 6�m��5� %<� � *{Ó % � � �+*5�[m�¢�7HBDCPCE751jePCPX<7%9�Xo1�3wX<7H68\DB]356RQ � *8Ó�% � � �+*5��_�1�:o® ��% � � X<756{CE=c1,1�X<7CE1'\DK(X<BD1'3�BDC|k'BgAq63w`I? � �GQ � %<� � � � �+*5�[mP0�_ � � B]C|351'35;564i:<6j9�CEB]35k5F�X<756@=r9�^5^HBD35k�* �� *{Ó�% � � �+*H��BDCp9ck'\D1�`%9�\�;5Bg �61�L=W1�:E^H7HBDCE=�_�:<1'= � 1'3qX<1 � _�1�:|9�3q? % ��®>�SBgXPBDC�B]354i:<6j9�CEB]35k,�[FH9�35;wX<756i:<6_�1�:<6pX<7H6i:<686il(BDCMX<CP9WK535B]JIK568k'\D1�`%9�\ CE=c1,1�X<7

v 1 BD\D\DB�9�=�Gq1'753w�~9�4jJqKH1�:<3wÚP��T���Q�T���F��,4#1�XEX<BDCE7>6#35k'BD3566i:F�� ��­�®�L<� ��(�­,m-no6Ze21�:Ebq6;>B]3�Ðo\]9�CEk'1e8mº~±dBD6#:E:<6pno63(:<B n�V¤-R�T�QSR���F5�H:<6#354}7>635k'BD3566i:F����'Ô(�iL<� � ��(Im� Gq1'753�1 BD\D\DB�9�= �I� � �� ��F�¢�75BD:<; �9�:<1'3wÚP����¥,��Q+¤V³uFHÙd35k'\DB]CE7�^%7q?(CEBD4#BDCMXF�����ÕI­�L<�j�,�j�,m2no68:<6#4#6BgAq6#;�X<7H68�ZRV¯���¥

^5:<BD�6ZBD3�±d7q?(CEBD4C�BD3��j��®�Õ(m-no6Z756\D;�X<756@¡u�,�V�%T��VQ+U³h±-:<1'_�6CECE1�:<CE75Bg^y9�Xp¡29�={`5:<B];5k'6�F � ��(���L<� � ��Õ(F�9�_�X<6i:P��� � ´>��¥,¥%m� 0sXYCE751jePCrX<7H9�X�9�:<1�KH35;Ø� � ��®/�+eP7563 �I��R����aUye�9�CY6;5BgX<BD35k°75B]Cce21�:Eb�1'_W����Õ��°9�35;/;HBD;/351�Xr:<6i^5:<1,;5KH4#6�75BDC

;56#:<BgA�9�X<BD1'3�1'_ZX<756c�EK5=W^*41�3H;5BgX<BD1'3*BD3�7HBDC�4#1'=)^%\D6iX<6ye21�:Eb(C#Fo9�35; 9�^�1'\D1'k'BD�#6;�_�1�:>75BDC¶«<=WBDCMX[9�bq6#¬,�[F|ÚP�&�V¥,�%QS¤�³uF�I��R����aU>9�35;�¢�³VR��{UjR�T¸�+eP751~`�6#4j9�=c6YN�1�:<; &Z��¥}�VQ�T�BD3*� �'�'­q�8;5BD;¶351�XWK535;56#:<CMX[9�35;¶X<7H9�X)=W6#4}7H9�35BD4j9�\u6356i:<k�?4#1'K5\];w`�6{XE:[9�3HCE_�1�:<=W6;yB]3qX<1r756j9�XmP0�_�1'356@7H9�Co\D69�:<356;�¢�756i:<=c1,;(?(3H9�=WBD4C#F%1'356@CE7H1�KH\D;y3H1�Xp;5BDCM^%9�:[9�k'6{X<756#CE6fk�:<6j9�XCE4#B]6#3qX<BDCMX<Cd1�_�X<756{�j� ��� ¡u6#3IX<K(:E?)_�1�:dX<756#Bg:u4#K(:<BD1'K5Cd=cBDCMX[9�b'6'FI9�35;r1'356PCE751'K5\D;c:<6#41'k�3HBD�#6�X<7H9�XdX<756i:<6|9�:<6�X<75BD35k'C-eP75BD4}7X[9�bq6oX<BD=W6|X<1)K535;H6i:<CMX[9�35;��O9�3H;�1'356p\]69�:<35C�351je�9�\DCE1fX<7H9�X�¢�756#:<=W1,;(?(3H9�=WBD4C2BDC�351�XP9�`�1'K(X�;(?(3H9�=WBD4C#m �[m��o4iX<K%9�\]\g?'F¢�756#:<=W1,;(?(3H9�=WBD4CZBDC{CMX<BD\D\d9>CEK5`,�E6#4#X8eP75B]4[7°BDC{351�X{CE1we26\D\-K535;H6i:<CMX<1,1,;aFV9�35;°=r9�X<7H6#=c9�X<BD4B]9�35CZCE751'K5\D;�^%9?�=W1�:<69�XEX<6#3qX<B]1�3�X<1wBgXt�BDk'351�:<BD35ky¢�756i:<=c1I;5?,3%9�=cBD4#C#F�9�35;�^HK(`H\]BDCE75BD35k>X<1I1w=fK54[7�1'3°BDCE63qXE:<1�^HBD4W6JIKH9�X<BD1'35C#F�_�1�:{6#l59�=W^H\D6'FX<6#3H;5CWX<1�=c9�bq6w6#35k'BD3H6#6i:<Cr9�35;�^H7q?(CEBD4BDCMX<CW`�6\DBD6iAq6�X<7H9�Xc=r9�X<7H6#=c9�X<BD4B]9�35C);51°351�Xcb,3H1e eP7H9�XcX<756i?�9�:<6>X[9�\gb(BD35k9�`�1'K(Xm��¢�756{_�K5354iX<B]1�3 � 7H9�CPX<756{;5BD=W635CEBD1'3 � �

" v 1'_�9WA'6\D1,4#BgXs?�tHCE1'=W68^H7q?(CEBD4#B]CMX<C�^5:<6#_�6i:|X<1We�:<BgX<6 ���� � Ó 4 � ������ ��®,FeP756#:<6 4 BDCP9W4[7H9�:[9�4iX<6#:<BDCMX<BD4oAq6\D1,4#BgXs?Y9�35; � 7H9�C�351W;5B]=W6#3HCEBD1'3am

��(

Page 88: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

CE1'\DK(X<BD1'3�_�1�:{9�\D\ % �Ø®(m@0�_dX<756i:<6)6#l(BDCMX * v � * º ePBgX<7 � � �+* v � � � � �+* º �[F X<756#3�X<756)4}7H9�:[9�4iX<6i:<B]CMX<BD4f\DBD356@ePBDX<7�BD35BgX<B]9�\^�1'BD3qX * v 4j9�X<4[756CPK(^�1'3>X<7H684}7H9�:[9�4#X<6i:<BDCMX<B]4Z\DBD3H6ZePBgX<7wBD35BgX<B]9�\a^�1'BD3qX * º 9�3H;�9WCE=W1,1�X<7wCE1'\DK(X<BD1'3w49�353H1�Xo6il(BDCMX|K(^wX<1X<756WX<BD=W6)1'_26#3541'K53qX<6i:{1'_uX<756)Xse21y4[7H9�:[9�4iX<6#:<BDCMX<BD4)\]BD356#C{9�C8BgX8e21'K5\D;�7H9Aq6WX<1�X[9�bq6)Xse21y;5Bg �6#:<6#3qX8A�9�\DK56C89�X8X<756#BD:BD3qX<6#:<CE6#4#X<BD1'3ata1�3H6f4j9�3~6j9�CEB]\g?w4[7H6#4<byX<7H9�Xp_�1�:Z9�3q? � � � � 1'356f49�3�zH35;~Xse21Y4}7H9�:[9�4#X<6i:<BDCMX<B]4{\]BD356#C|eP7HBD4[7~BD3qX<6#:<CE6#4#X`�6_�1�:<6��Zm � 356w49�3�4[7564<b�;5BD:<6#4iX<\D?°X<7H9�XWX<7H6>CE1'\DK5X<BD1'3�BDC)e26#\D\�;56#zH356;�_�1�:r® �0% � � `�649�K5CE6>X<756w=c9�^5^HB]35k* �� *oÓ)% � � �+*5�dB]C�9fk'\D1�`%9�\�;HBg �6#1'=c1�:E^%75BDCE= _�:<1'= �Ø1'3qX<1 � _�1�:|® ��% � � � FH9�35;>B]3�1�:<;H6i:�X<1WCE751je X<7H9�X�X<756#:<6pBDC351rCE1'\DK(X<BD1'3w1'3h�O® � �o�2ePBgX<7�� � � � F%1'356{;56#zH356C F�� � ���� 9�35;�1�3H684}756#4[b,C|X<7H9�X �

� � � � � � � � � � �� � � ! F%���H� %<� � %<� #�º�F%CE1rX<7H9�XF����5� %<� � %<�H� � � � ! �v � � � � � ! � F%9�35;>X<756i:<6_�1�:<6pX<7568CE=W1,1�X<7>CE1'\DK(X<BD1'3>C<9�X<B]CMzH6#CP\DBD=���� � CEK(^ � � � ���� � � Ó),/m

¢�756>9�3H9�\D1'k�1'_�X<756�BD=)^%\DBD4#BDX{_�1�:<=@KH\]9yK5CE6;h`,?°±2R�QSU�UjR�T e21'K5\D;h`a6cX<1~C<9?°X<7H9�XfX<756YCE1'\DK(X<BD1'3¶=@K5CMX)C<9�X<BDCE_�?� �GQ � %<�H� � � � Q�> % � �GQ � %<� � F�9�3H;>CE66#BD3HkfX<756Z\]BD=WBgX[9�X<BD1'3YBD3�X<B]=W6p1'3>X<75BDC�_�1�:<=@KH\]9)BDC�\D6#CEC�1�`IA(BD1'K5C#m�o_�X<6i:|X<756{4#:<BgX<BD4j9�\ X<BD=W6 � � 1'356{4j9�35351�Xo7H9Aq6@CE=W1,1�X<7yCE1'\DK5X<BD1'35C#F�9�35;wX<7H6{41�:E:<6#4iX<\D?�;56#zH356#;~CE1�\]K(X<BD1'3>ePBD\]\a`a6;5BDCE41'3qX<BD3IK51'K5C{9�X{CE1'=W6)^�1'BD3IX<C#F�9�35;�X<756i:<6_�1�:<6)X<756)^5:<1,;5K54#X81'_ � `I? ������ B]C8351�X8;56#zH356#; F�9�35;�BgX8BDC8B]=)^�1�:EX[9�3qXpX<1e�:<BgX<6�X<756>6jJqK%9�X<B]1�3�BD3�4#1'35CE6#:EA�9�X<BD1'3�_�1�:<=>F ���

� � Ó � � �� � º ����

� ®(F29�35;�X<1�41'35CEBD;56#:)X<7H6�e269�b�CE1'\DK(X<B]1�3HC#FdBOm 6'm¶X<7H6CE1'\DK(X<BD1'35C@BD3°X<7H6rCE6#35CE6Y1'_�;5BDCMXE:<BD`HK(X<BD1'35C#m�¢�756Z�EK5=W^¶4#1'35;5BgX<B]1�3HC8eP7HBD4[7��I��R�����UY7H9�;h41'=)^HK(X<6;h4#1�:E:<6#CM^a1�3H;°X<1C<9?(BD35k�X<7%9�X)B]_ � �

� � Ó ���� � � ®�9�3H; � � F°9�:<6�4#1'3qX<BD3,K51'K5Cf1'3�69�4}7¶CEB];56�1'_P9~4#K(:EAq6rePBDX<7¶6JIKH9�X<BD1'3 Q5� �u� %<�89�35;- � ���S� %<�2;56#3H1�X<6CPX<756ZAq6\D1,4#BgXs?�1'_�X<756c�+^a1�CECEBD`H\D6��2;5BDCE41'3qX<BD3IK5BDXs?>9�XPX<BD=c6 %}FHX<7563w1'356{7H9�C8" � *��GF(� � - " � *�� � �[FeP756#:<6 " � *�� �Z�8� � � �u� %<� �3� % � > � � �d� %<� " � % � � �

�> � " �S351�X<BD4#BD3HkWX<7H9�XZ4[7H9�35k'BD3Hk�Q�BD3qX<1 >�Q�4}7H9�35k'6#CF>B]3qX<1

>8F%FH4}7H9�35k'6#C�X<756ZCEBDk'3w1'_�" � *�� � �29�3H;>1'_ " � *��GF(�[F(`HK5XP4[7H9�35k'6#Co9�\]CE1fX<7H6ZCEBDk'3>1'_�-j�[m¦o35_�1�:EX<K53H9�X<6#\D?'F-X<756i:<6>9�:<6�X<1I1�=r9�3q?he269�b¶CE1'\DK(X<BD1'35C#t�_�1�:c6il59�=)^H\D6�_�1�: � � � ®�X<756#:<6�B]CW9~k'\D1�`%9�\2CE=c1,1�X<7

CE1'\DK(X<BD1'3ceP7HBD4[7rB]C � � ®(FI`HK(X2X<756i:<6p9�:<6oBD3(zH35BDX<6#\g?W=c9�3q?We269�bcCE1'\DK(X<BD1'35Cd`,?WX[9�b(BD35k�Q � 9�:E`HBgXE:[9�:E?'F4N ��®f9�:E`%BgXE:[9�:E?'F9�35;~;H6izH35B]35k � �GQ � %<�)� ®Y_�1�: Q�� Q � > % N�F � �GQ � %<�)� >P­.Ny_�1�: Q >)% N ��Q � F � �GQ � %<�)� ­.Ny_�1�:8® ��Q�� % N�F 9�35;� �GQ � %<�H�*®W_�1�: % N� Q�m0�3h1�:<;H6i:{X<1w:<6s�E6#4iX@9�\D\dK53(^H7q?(CEBD4j9�\�e26j9�b�CE1'\DK(X<BD1'35C#F�1'356r;564#BD;56CZX<7H6#3�X<1>bq6#6#^h1'35\g?~X<756r;5BDCE41'3qX<BD3IK5BDX<BD6#C8_�1�:eP75BD4}7 � " � � � t�X<756@=W6#X<751,;y1�_�4}7H9�:[9�4iX<6i:<B]CMX<BD4{4#K5:EA'6C|k'BgAq6#Cp9c\D1,49�\�N�Q A�U�²�³�Q���µr4#1'3qX<BD3IKH1�KHCoCE1�\]K(X<BD1'3weP756#3~1�3H6CMX[9�:EX<C2_�:<1�= 9@3H1�3H;56#4#:<69�CEBD35kf_�K5354#X<BD1'3Y9�35;Y`,?r4#1'3qX<BD3IKHBgXs?cBgX2k'BgAq6C2CEK54[7>9@N�Q AaU�²�³�Q���µf4#1'3qX<BD3IKH1�KHC2CE1'\DK5X<BD1'3rB]3cX<7H61�X<756i:�49�CE6 � " � � � F(BSm 6�mVeP7563Y1'356oCMX[9�:EX<C2_�:<1�=!X<7H6o;5BDCE4#1'3qX<B]3IK51'K5C2_�K5354#X<BD1'3Z�EK5=W^HBD35kf_]:<1'= � " K(^YX<1 � � �S49�\D\]6#;9Y:[9�:<6_�9�4#X<BD1'3ye�9Aq6��[m)¢�75BDCZCE6#\D64iX<BD1'3~4i:<BDX<6i:<BDK5=wFa49�\D\D6;~9�3*«<6#3IXE:<1�^,?,¬>4#1'35;HBgX<BD1'3 � F 6#l,X<6#35;5CpX<1YX<756)=W1�:<6)k'6#356#:[9�\6JIKH9�X<BD1'3 ���

� � Ó ��� � � �� � �!®�1'35\g?�BD_-X<756c351'35\DBD3H69�:<BgXs? @°B]C89>4#1'3qAq6il�_�K53H4iX<BD1'3at�X<756W4j9�CE6c1'_29�k'6356i:[9�\P�SCE=W1,1�X<7�� @7H9�Cp`a6#6#3�CE1'\gAq6;~`I? � \]k'9 � ¥,��Q�T�Q����jF�9�35; � ¥,��Q�T�Q�� × CZ4i:<BDX<6i:<BDK5= B]CoX<7%9�X81�3H6W9�44#6i^HX<Cp1�:Z:<6s�E6#4#X<C89Y;5B]CE4#1'3qX<BD3IKHBgXs?9�4#41�:<;HBD35k@X<1@X<756o^a1�CEBDX<BD1'3r1'_�X<756o4}751�:<;{�M1'BD3HBD35k � � " � @�� � " ���Z9�3H; � � � � @�� � � ���|ePBDX<7r:<6#CM^a6#4iX�X<1@X<756pk�:[9�^H7�1'_aX<7H6_�K5354iX<B]1�3�@ F59�44#6i^HX<BD35k{X<756o;5BDCE41'3qX<BD3IK5BDXs? � " � � � B]_ 9�35;�1�3H\g?WX<756o4[751�:<;�BDC29�`�1jAq6oX<756ok�:[9�^%7aF59�35;�9�4#46i^5X<BD3Hk8X<7H6;5BDCE41'3qX<BD3IK5BDXs? � " � � � BD_�9�35;w1'35\g?YX<75684[751�:<;wBDC�`�6\D1je X<7568k�:[9�^%7am¦o35;H6i: � ¥,�%Q�T�Q�� × Cf4#:<BgX<6i:<B]K5=»X<756i:<6�BDC)9~KH35B]JIK56r^%BD6#46iePBDCE6�CE=W1,1�X<7¶CE1'\DK(X<BD1'3aF�`HK5XWBD3¶1�:<;56#:)X<1~CMX<K5;5?°e269�bCE1'\DK(X<BD1'35CdePBgX<751'K(X27H9A,B]35k8X<1@9�CECEK5=c6|X<7H9�XdX<756#?c9�:<6�^%BD6#46iePBDCE6PCE=c1,1�X<7 F,Ù-`�6i:<7H9�:<;rnZRBA9� �P;56#:<BgAq6#;cX<756|6jJqKHBgA�9�\D63qX_�1�:<=fK5\]9�X<BD1'3 ��� � � �

� � Ó � �&� � ����

(x®>BD3�X<756WCE6#35CE6W1'_d;5B]CMXE:<Bg`HK(X<BD1'35CZ_�1�:{9�\D\V41�3IA'6#l¶�i«<63qXE:<1�^I?(¬I�|_�K5354#X<BD1'35C �-F eP756#:<6� BDCc9�4#1�:E:<6#CM^�1'35;HBD35k «<6#3IXE:<1�^,? %HK(l(¬5F2;56iz%356#;�`I? � � � ��� @��+m/±V6iX<6i:rN�� � 6#lIX<635;56;�X<756wBD;569~X<1�CM?,CMX<6=WCc1�_4#1'35CE6#:EA�9�X<BD1'3W\]9ePC#F � �

� � Ó ��� � � ����

�*®(FIePBgX<7 �f�GQ � %<� � � �(FI`HK(Xu351�Xu9�\D\H_�K53H4iX<BD1'35C �~1�3 � �{9�:<6P6#3IXE:<1�^%BD6#C#Fq`�649�K5CE6PBD_1'356{7H9�C � � � � � � X<756#3wX<756i:<6@9�:<6{4#1'=)^�9�X<BD`HBD\gXs?Y41'35;5BgX<BD1'35CPX<7H9�X��h=fK5CMXoC<9�X<BDCE_]?qF%BSm 6�m 4 � L � � � � � f�3��®,F9�35;>X<756pXE:<BDA,B�9�\a6#3qXE:<1�^HB]6#C �d� �{�H� � � 7 _�1�:�� �Ø� ��������� *)�EK5CMX|4#1�:E:<6#CM^�1'35;>X<1)X<756Zk'BgAq6#3>CM?(CMX<6=�1'_�6jJIKH9�X<BD1'35C#v � m

� 356�e�9?{X<1Z41�3HCMXE:<K54iXdCEK54[7c9�;5=cBDCECEBg`H\D62e26j9�bfCE1'\DK(X<B]1�3HCVBDCVX<1Z4#1'35CEB];56i:u9o:<6#k'K5\]9�:<BD�j9�X<B]1�3@`,?f9�:EX<Bgz%4#B]9�\,A(BDCE4#1'CEBgXs?� �

�� � Ó ��� � ���

���>� � � �

�� ����*®�_�1�: � ��®(m�¢�756-CE4j9�\]9�:�4j9�CE6�ePBDX<7�@�� � �*� �

�º �O9�\]CE1�4j9�\]\D6#;ZX<756$ %���¤V����U#LOnZRBA!�86JIKH9�X<BD1'3��e�9�C2z5:<CMX�CE1�\DA'6;r`I?YÙ-`�6i:<7%9�:<;�nZRBA9�YKHCEBD35kW9@351'35\DBD356j9�:�4[7H9�35k'6p1'_�_�K5354iX<B]1�3reP75B]4[7YXE:[9�35CE_�1�:<=cCuX<756p6JIKH9�X<BD1'3YB]3qX<1

� 2?Y9�3H9�\D1'k�?cePBgX<7Y4#1'35;5BgX<B]1�3HC2BD=)^a1�CE6;r`I?�¢�756#:<=W1,;(?(3H9�=WBD4Cu_�1�:�X<756pCM?(CMX<6#= 1�_�41�=W^5:<6#CECEBD`H\D6pk'9�C�;(?(3H9�=WB]4#C#m� � \Dkq9��|:<CE6#3 × 6iA(3H9 � ¥I��Q�T�Q���FjÚPK5CECEB]9�3@=c9�X<756=c9�X<BD4B]9�3aF`a1�:<3@BD3c�j�'­'Ô,m-�I756-e21�:Eb(C�9�X���1'CE4#1je/�qX[9�X<6-¦o3HBgAq6i:<CEBgXs?qm��Ù-`�6#:<7H9�:<;�nZRBA9�aF%Ðo6#:<=c9�3(LS`�1�:<3>=c9�X<756=c9�X<BD4#B�9�3 F��j��®q­�L<�j� � �,m-no6p6=WBDk�:[9�X<6#;rX<1)¦o3HBgX<6#;y�qX[9�X<6#C�BD3��j��Õq�,FH9�35;

e21�:Ebq6#;y9�X�0�35;5B]9�3H9)¦o35BgAq6#:<CEBgXs?'F& �\D1,1'=WBD3Hk�X<1'3aF,eP756#:<6p02=c6iXP75BD=�BD3��j� ��®(m��okq9�BD3aF(X<75BDC2BDC�1'35\g?c`,?Y9�3H9�\D1'k�?cePBgX<7�¢�7H6i:<=W1,;(?(3H9�=cBD4#C#F(9�35;�X<756CE6Y«<6#3qXE:<1�^HBD6CE¬)9�:<6o:[9�:<6#\g?c:<6\]9�X<6#;rePBDX<7cX<7H6

X<756#:<=W1,;(?(3H9�=WBD4p6#3IXE:<1�^,?'mv � 0sXpBDCo3H1�Xp4\D69�:pBD_VX<756{:<B]k�7IXp351�X<BD1'3y1'_-CE1�\]K(X<BD1'3�7H9�Co`�6#63�_�1'K53H;aF�`HK(XZ9�\D\�X<756{^H7I?,CEB]49�\�6#l59�=W^H\D6C|CE6#6= X<1c`a66#3H;51je26#;>ePBgX<7w9)CMXE:<BD4#X<\g?r41'3qAq6il¶«<6#3qXE:<1�^I?(¬5F5eP75BD4}7>CE1'=W6#X<BD=W6C2B]C�X<756pX<1�X[9�\a6356i:<k�?P2

� �

Page 89: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

X<756c\DBD356j9�:8756j9�X86JIKH9�X<BD1'3aF�9�35;�X<7HBDCpXE:[9�35CE_�1�:<=c9�X<BD1'3~e�9�C@9�\DCE1>_�1�KH35;�`I?�GqK5\DB]9�3¶¡�R�¥I��v<v#F�9�35;�B]C8351je�b,3H1eP3h9�CX<756)nZRBA!�5L�¡�R�¥I��XE:[9�3HCE_�1�:<=>m@¢�7H6fCE4j9�\�9�:849�CE6@ePBDX<7~9Yk'6356i:[9�\3@�e�9�CZCE1�\DA'6;�`I? � \Dkq9 � ¥,��Q�T�Q���F 9�3H;~X<756fCE4j9�\]9�:49�CE6ZePBgX<7>=W1�:<6pX<7H9�3w1'356ZCM^%9�46ZA�9�:<B]9�`H\D6 vsº F � �

� � Ó � !� ��v ��� � � � ���� ��*®)e�9�C�1�`5X[9�BD356;�`I?'&Z� �µ'³���Ra� v � m

¢�756rBD=W^�1�:EX[9�3IX{;5Bg �6#:<6#3546)`�6#Xse26#6#3°X<756cCE49�\]9�:@49�CE6Y9�3H;�X<756cAq6#4iX<1�:<B]9�\-49�CE6��+:<6#k'K5\]9�:<BD�6#;�`I?°9�:EX<Bgz%4#B]9�\VA(BDCML4#1'CEBgXs?%�[F�BDCpX<7%9�X8X<756i:<6c9�:<6cCEBD=)^%\D6 ����6#CMX<BD=r9�X<6CZ_�1�:ZX<7H6)CE4j9�\]9�:Z4j9�CE6�F�eP75BD4}7�9�:<6WK53(b(351jeP3�_�1�:ZX<756)Aq6#4#X<1�:{49�CE6�tX<756 ���Ø6#CMX<BD=r9�X<6C29�:<6oK5CE6#;YX<1{^5:<1jAq6o41'3qAq6i:<k'6#3546|`I?Y9{41'=)^%9�4iX<3H6#CEC�9�:<k'K5=c6#3qXF,`HK5X�9�3q?cK53HBD_�1�:<=�6#CMX<BD=r9�X<6|BD3�9 ���U�R ��CM^%9�4#6r�+� � � �8 � ��� � � �)!Z�dePBgX<7�- �/®)e21'K5\D;>`a68CEK(�c4#BD63qXmu¦o35_�1�:EX<K53%9�X<6\g?'F5X<75686CMX<BD=c9�X<6#C�_�1�:�X<7H68CE49�\]9�:P4j9�CE69�:<6p`%9�CE6;>1'3>X<7568=c9�l(BD=@K5= ^5:<BD354Bg^H\D6'F59�35;>X<756ZC<9�=c689�:<k'K5=W63qXP49�35351�X�`�686il,X<6#3H;56#;�X<1WCM?(CMX<6#=cC#m

¢�756#:<6@BDCo9�351�X<756#:|=c6iX<751,;y;5KH6{X<1rGI9�=c6#C8Ðo¥IQ ���)v � F%eP75BD4}7w^H:<1Aq6Co6il(BDCMX<6#3H4#6{_�1�:oCE1'=c6{CM?(CMX<6#=WCoBD_�X<7H68X<1�X[9�\A�9�:<B]9�X<BD1'3�BDC�CE=r9�\D\a6#351'K5k'7am

0 7H9A'6dBD3IXE:<1I;HK54#6;{9�351�X<756#:�9�^5^5:<1q9�4[7aF�`%9�CE6#;{1'38X<7562¡u1'=)^a6#35C<9�X<6#;)¡u1'=)^%9�4#X<356#CEC���6iX<751,;ZeP75B]4[780a7H9�;8^%9�:EX<\g?;56#A'6\D1�^�6#;~ePBDX<7��5:[9�3

4#1'BDC8� ����H� v ��FaeP75B]4[7�;51,6#C8351�X83H6#6#;�6#CMX<B]=c9�X<6#CZBD3 ��aUjR �*CM^%9�4#6C#F�`HK(X8eP75BD4}7~:<6jJIK5Bg:<6#C89CM^�64#B]9�\�KH35;56i:<CMX[9�35;HBD35kY1'_�751jexX<1rK5CE6f6#3qXE:<1�^HB]6#CPX<1rk'6#356#:[9�X<6@9Wb(BD35;�1'_V4#1'=W^%9�4iX<356CEC#t�ÚP1'3�£pQ�±d�%��T��uv � e�9�CPX<7H6z5:<CMXPX<1)zH35;y9fe�9#?�X<1c9�^H^H\g?Y=@?Y=c6iX<751,;�X<1WCM?,CMX<6=WC#m

� _�4#1'K(:<CE6'F5X<756Z^5:<64#6#;HBD35kW\DBDCMXP1�_V=W6iX<7H1I;HC�BDC�3H1�X|6il(7H9�K5CMX<BDA'6'FH9�35;>X<756#:<687H9�C�`a6#6#3w1�X<756#:�^%9�:EX<B]9�\D\g?YCEK54#46#CE_�K5\9�^5^5:<1q9�4[756C#m

¢�756 � �!6#CMX<B]=c9�X<68B]3wX<756@CE4j9�\�9�:p49�CE6@4j9�3y`�6@\DBD3(bq6;wX<1Y9$��vj� �{��41'3qXE:[9�4iX<BD1'3w^5:<1�^�6#:EXs?'F�eP75B]4[7ye�9�Co3H1�X<B]4#6#;`I? �9�:E`%9�:[9 &8�����5Q���µ(v��rBD3�1'356�;5BD=c6#35CEB]1�3�9�3H;¶`I? &Z� �µ�³��VR �xBD3�;5BD=W635CEBD1'3 & mh¢�75BDCf^5:<1�^�6#:EXs?hBDC)CMXE:<1'35k'\g?:<6#\�9�X<6;¶X<1�X<756w=c9�l(BD=@KH=·^5:<BD354Bg^H\D6'Fu9�35;�0f7H9Aq6w351�X<BD4#6;�ePBgX<7��~BD4[7H9�6\|¡u����T�����¥I¥ v � X<7H9�XWB]_|9�=c9�^ � _]:<1'=��v���P�)BD3qX<1hBgX<CE6\D_oC<9�X<BDCMzH6C I0J � � @a�4O Q � I J�@)OMQ*_�1�:�9�\D\ @ F�X<756#3 � BDCY9 ��vr41'3qXE:[9�4iX<BD1'3/BD_Z9�3H;�1�3H\g?�BD_ � BDC1�:<;56i:f^H:<6#CE6i:EA(BD3Hk(t29�Cf1�:<;56i:f^H:<6#CE6i:EA(BD3Hky^5:<1�^�6i:EX<B]6#C@;H1�3H1�X)1,44#K(:)_�1�:f:<69�\DB]CMX<BD4cCM?(CMX<6=WC#F�1'356�49�35351�X)6#lI^a6#4iX ��v4#1'3qXE:[9�4#X<BD1'3w^5:<1�^�6i:EX<BD6C|_�1�:oCM?(CMX<6#=cC#t���BD4}7H9�6#\-¡u����T�����¥,¥w9�3H;��o35;(:<6#eØ���BU ���uv {7H9Aq6@\]9�X<6#:o9�^5^%\DBD6#;wX<7H6{C<9�=c6BD;56j9)_�1�:P;5BDCE4#:<6iX<689�^5^5:<1jl(BD=c9�X<BD1'35C#m

¢�756rCEBD=W^H\D6#CMX{;HBDCE4i:<6#X<6r9�^H^5:<1jl,B]=c9�X<BD1'3�1'_$� �� � Ó ��� � � ���� � ®wBDC8X<756rN�� � LO����Q�� �V��QS²�³�UcCE4[7H6#=W6'F v

�� � � � �Vv7 >

vº � � �7 " v Ó � �7��v ���{Ó vº � � � @�� � �7 ��v �)> @�� � �7 " v ���$�Ö®(F�eP756#:<6 � �7 BDCY6il,^�64iX<6;/X<1¶9�^5^5:<1jl(BD=r9�X<6 �)�+�4? Q � � ?�%<�[tCMX[9�:EX<BD35kfePBgX<7>9f`�1'K535;56;�B]35BgX<B]9�\�;H9�X[9 � � F5C<9�X<BDCE_�?,B]35kW� ( � � �GQa� ( �¶9(m 6�m*Q � �)F51'356Z4[7H1I1'CE6C�_�1�:�6il59�=)^%\D6 � �7 �

v� �

I � 7 �Vv � � �7� � � � �+*5�4O,*w_�1�:89�\]\ �EF�9�35;�X<756f6#l,^H\DBD4BgX|CE4[7H6#=W6fk'6356i:[9�X<6#CoX<756f3IKH={`�6#:<C�� �7 _�1�: � ��®(F�`HK(XZ1'356f=fK5CMX

BD=W^�1'CE6uX<7H6o¡�R������T��aLO�V��Q����V��Q+²�³�U#LON��%´ �uº � 41�3H;5BgX<BD1'3>�+b,3H1eP3c9�C-X<756|¡u�VNy4#1'35;HBgX<BD1'3�� ��

� �CEK(^ ��� � ��� � � � @��O�GF(��� (��FdeP75B]4[7�BD=W^�1'CE6#CfX<7H9�XWX<756>3IK5=c6i:<BD4j9�\�Aq6#\D1,4#BDXs?°1'_|^5:<1�^%9�k'9�X<BD1'3 � �

�� BDCc9�Xc\D6j9�CMXW6JIKH9�\2X<1�X<756�:<6j9�\2Aq6#\D1,4BgXs?h1�_

^5:<1�^%9�kq9�X<BD1'3at(K53H;56i:2X<7HBDCu41'35;5BgX<BD1'3r1'356o7%9�C�� ( � �7 ( ��_�1�:�9�\D\���9�35;�9�\D\ � ��®(F59�35;�BgX2BDC26il59�4iX<\g?)X<7H6o41�3H;5BgX<BD1'3eP75BD4}7fBD=)^a1�CE6C�X<7H9�X.� � ��v7 BDC�9o351'35;56#4#:<69�CEBD35kp_�K5354iX<B]1�3f1'_ � �7 " v 9�35;f1'_ � �7 �Vv Fq9�35;@X<756 � v� � ��41�3IXE:[9�4#X<BD1'3{^5:<1�^�6i:EXs?_�1�\]\D1jePC#m � _a41�K5:<CE6�X<7HBDCu9�^5^5:<1q9�4[7Y4i:<69�X<6C-CE1'\DK5X<BD1'35CdCEK54}7cX<7H9�XR� � � � � � � � %<��> � � � � %<��� � v ((� � � � � � > � � � � v FIeP75BD4}7ck'BgAq6C9 ��� 6#CMX<B]=c9�X<6uB]_ � � � � �c� �{�[mV¢�7HBDCVCE4}756#=c62B]CV1'35\g?@1'_%1�:<;56#:P��9�35;fX<6#3H;5CVX<1pCE=W1,1�X<7)1�K5XVX<756�;5BDCE41'3qX<BD3IK5BDX<BD6#C�X<1,1=@KH4[7aF�`%K(Xo75B]k�7H6i:o1�:<;56i:pCE4[756=W6#Cp9�:<6@351�Xo1�:<;56#:|^5:<6CE6i:EA(BD35k5t�X<756i:<6@BDC|751je26#A'6#:Z9c4#\�9�CECo1'_�75BDk'756i:o1�:<;56#:oCE4[7H6#=W6C#F49�\D\]6#;>¢2Ñp£ CE4}756#=c6#C{�+X<1�X[9�\�A'9�:<B�9�X<B]1�3�;HBD=WBD3HBDCE75BD35k,�[F(_�1�:PeP75BD4}7�X<756ZX<1�X[9�\�A'9�:<B�9�X<B]1�3�B]C�351�XPBD354i:<6j9�CEBD35k5m

vEvyGqK5\]B]9�3>£fm�¡�R�¥I��FH�o=W6i:<B]49�3�=c9�X<756=c9�X<BD4#B�9�3 mVno6pe21�:Eb(C|9�XPÚP6#35CECE6\]9�6i:P±V1'\g?,X<6#4}7535BD4p0�35CMX<BgX<K5X<6�FH¢�:<1?qFH� �fmvsº�� CE49�\]9�:f6jJqK%9�X<B]1�3¶BD3 & A'9�:<B]9�`H\D6#C@BDCf351�X)9�k'1,1I;h^H7q?(CEBD4j9�\u=c1I;H6#\u_�1�: & � ��F�`�6#4j9�KHCE6YBgXfBD=W^H\DBD6C@9yA'6#:E?CMXE:<1'35kc9�35B]CE1�XE:<1�^I?�1'_�X<756ZCM^%9�46c�S;5K56pX<1c9f^%9�:EX<BD4K5\]9�:�;5BD:<6#4iX<B]1�3�1'_�^5:<1�^%9�kq9�X<B]1�3��[mv � �qX[9�35B]CE\]9A��oBDb'1'\]9�6#A,B]4[7 &8� Vµ'³��VR �uFdÚPK5CECEB�9�3/=r9�X<7H6#=c9�X<BD4B]9�3aFo�j� �'Ò�L<�j�'� (Im no6>e21�:Ebq6#;�9�Xr��1'CE4#1je �qX[9�X<6¦o35BgAq6#:<CEBgXs?'mv � GI9�=W6CoÐ@maÐo¥,Q ���fF��o=W6#:<BD49�3>=c9�X<756#=r9�X<BD4#B]9�3aF(`�1�:<3wBD3°�����Õ5muno6Ze21�:Eb(C|9�XZ�qX[9�X<6Z¦o35BDA'6#:<CEBgXs?�1'_��o6ie �d1�:Eb�F�qX<1'3q? 2:<1,1�b�mv � �H:[9�3

4#1'BDCp� ����%��Fa�5:<6354[7�=r9�X<7H6#=c9�X<BD4B]9�3aF�`�1�:<3~BD3��j��Õ�(Im{no6@e21�:Eb(C89�XZ¦o35BgAq6#:<CEBgXs?y1�_d±�9�:<BDCoÑo0@�S±dBD6i:E:<6f6#X�~9�:<BD6f¡ ���Q�� �[mv � ÚP1'3H9�\D;fGHm�£pQ�±d����T��dF��o=W6#:<BD49�3{=r9�X<756#=r9�X<B]4#B]9�3aFjL<�j� �'�,mVno6de21�:Ebq6;f9�X�¦o3HBgAq6i:<CEBgXs?81'_�¡29�\DBD_�1�:<35B]9�BD30 �6i:Ebq6\D6i?qmv � �9�:E`%9�:[9)N�6#6 '��Q�T�T &Z�����5Q���µ5F5�o=c6i:<BD4j9�3�=c9�X<756#=r9�X<BD4#B]9�3amu�I756pe21�:Eb(C|9�XP¦o3HBgAq6i:<CEBgXs?Y1'_�no1'K5CMX<1'3amv.����B]4[7H9�6#\�Ð@m ¡u����T�����¥,¥�F%�o=W6#:<BD49�3w=r9�X<756#=r9�X<B]4#B]9�3aF5`�1�:<3yB]3°�j��Õ'®(mPno68e21�:Eb,Cp9�X|¦o3HBgAq6i:<CEBgXs?>1�_u¡29�\DB]_�1�:<35B]9(F�,9�3qX[9� �9�:E`%9�:[9(mv.��o35;5:<6ie�GHm ���BU ���uFa�o=c6i:<BD4j9�3~=r9�X<7H6#=c9�X<BD4B]9�3am8no6)e21�:Eb(C89�XZX<756r¡�R������T���0�35CMX<BgX<K(X<6f_�1�:Z��9�X<7H6#=c9�X<BD4j9�\�I4BD6#3546#C#FH�o6ie �-1�:Eb�¦o35BgAq6i:<CEBDXs?'mº � np9�35CZN��%´ �2F�Ðo6i:<=r9�3(LS`�1�:<3�=c9�X<756=c9�X<BD4#B�9�3 FV�j��®�Õ�L<�� � �,mfno6f:<6#46#BgAq6#;�X<756#1 R�¥ ��^H:<BD�#6)B]3��j� ��Õ5mfno6)6#=WBDLk�:[9�X<6#;�X<1W¦o35BgX<6;y�IX[9�X<6C�BD3�X<756���®�CP9�35;>e21�:Ebq6#;y9�XP¦o35BgAq6i:<CEBDXs?r1'_-¡29�\]BD_�1�:<35B�9,F& �6i:Ebq6#\D6#?'m

�'�

Page 90: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

� `5X[9�BD35BD35k���� 6#CMX<BD=r9�X<6C@_�1�:)k'6#3H6i:[9�\2CM?(CMX<6#=cCf1�_|4#1'35CE6i:EA'9�X<BD1'3¶\]9ePC#F-1�:W=W1�:<6Yk'6356i:[9�\D\g?h1�`HX[9�B]35BD35k�CE1'=c66#CMX<B]=c9�X<6#Cu1�3Y_]:[9�4iX<B]1�3%9�\%;H6i:<BgA'9�X<BDA'6CdK5CEBD35k@0�3qX<6i:E^�1'\]9�X<BD1'3cCM^%9�46#C2BDCu4#6#:EX[9�B]35\g?c98;HBg�c4#K5\gX21�^�6#3YJIK56CMX<BD1'3aF(9�3H;rCE1'=c6356#e�BD;569�C�1�:|CE1�=c6Z356#e�_�KH354iX<BD1'3H9�\ CM^%9�4#6C�=r9#?Y`�68356#6;56#;w_�1�:�X<7H9�XPBD=W^�1�:EX[9�3IX�JIK56CMX<BD1'3am

��®

Page 91: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

­�®(m-��1'35;H9?��~9?���F�­�®'®�®(m¢�756{^�1'CECEBg`%BD\DBgXs?>1'_�;56#zH35BD35krX<756@CM^%9�4#6#Cf� � � � � v � � � � _�1�:Z®� � � �@9�35;¶� ( * ( , 9�CoCM^%9�4#6Co1�_VXE:[9�46#C|7%9�C

`�66#3�=W63qX<BD1'356;aF59�35;>BgX�BDC2X<BD=c6|X<1f6#l,^H\]9�BD3rX<7H6o^5:<1,1'_�m � 356Z351�X<BD46#CuX<7H9�X�BD_ % " � �¶� % � N5�dBDC2`�1'K535;56;YX<7563�1'356Z49�3;564#1'=)^�1'CE6�NE��N � � %<��Ó N v � %<�uePBgX<7$� � N � � %<��� � � Ó %T� � N v � %<��� � v ( ­ �h� % � NH�'(�C�% � 9�3H;wX<756i:<6_�1�:<6�N v � %<� �Ö®cBD3 � v 9�C%�� ,/F�9�3H;�`a6#49�K5CE6ZX<756pXE:[9�46#CP9�:<6pX[9�bq6#3w9�C %#�Û®,FHX<756i:<6ZePBD\D\�`a689)4}7H9�35k'681'_ %2BD3IX<1 v � m� ��%"-� j�D������ §dN�6#X�® � � � �Z9�35;��)(+*#(-,/m�0�_ F�� %<� � � � 9�3H;9F �O� %<� � � v ePBgX<7�% � � � F%� %<��� � � � ��� � ® � ,�t � �� �Z9�35;% � � � F �O� %<��� � v � ��� � ® � ,�t � �� � F�X<756#36F��O®I� � � � � � � v � � � � m8¡u1'3qAq6i:<CE6\g?'Fa6#A'6#:E?w6#\]6#=W63qXo1'_P� � � � � v � � � � 49�3�`�6{e�:<BDXEX<6#39�C�F��O®I�dePBgX<7 FrC<9�X<BDCE_]?(BD3HkfX<756p^H:<6#4#6;5BD35k)^5:<1�^�6i:EX<BD6C#m�2È<ÆjÆ5#[§°N�6#X9N � � � � � � v � � � � F|BSm 6�m ( " � � �¶� ( � � NH� � � �5� � �[m��%1�: � � � 1'356�4[751,1'CE6#C>9¶;564#1'=)^a1�CEBDX<BD1'3 N(�N � � � Ó�N v'� � ePBDX<7 N � � ��� � � � N v:� ��� � v CEKH4[7�X<7H9�X � � N � � � � � � Ó ( � � � N v'� � � � v ( ­ �¶� ( � � NH�[F29�35;�1'356w351�X<BD4#6C)X<7H9�X� � N � � � ��v >�N � � � � � v � � � N v:� � ��v >�N v:� � � � v (/­ ( " � � ��v � �¶� ( � ��v � N5�%Ó°­ ( " � �¶� ( � � NH� (�Õ ( " � �¶� ( � � NH�[m � 3H6o;56#zH356CuX<7H6_�K5354iX<B]1�3 � ePBgX<7�A'9�\]K56#C@BD3 � � `I? � � ( � ��� N � � � F�9�35;¶1'356r6il,X<635;5C � X<1w`�6�9��c356YBD3h69�4[7¶BD3qX<6#:EA�9�\P� ( � � ( � ��v}�[tX<75BDC)k'BgAq6#C6� � � � %<��� � � ( =c9�l � � � N � � � � � � � � � N � � � ��v � � � � (�­ �¶� ( � ��v � NH�$( ­ ( �¶� ( � � NH�81'3¶X<756�B]3qX<6i:EA'9�\p� ( � � ( � ��vi�[t2BD3X<756~49�CE6 * � ,/Fo1�3H6�;H6#;5K546#C�X<7H9�X�% " � � � � � %<��� � � ( ( " � � ­ ( �h� ( � � NH�W1'3�X<756~BD3IX<6i:EA�9�\f� ( � � ( � ��vi�r9�35;�X<756#:<6iL_�1�:<6�% " � � � � � %<��� � � � � � ® � ,/t � �� �'F�eP75BD\D6)BD_o� ( *�� , 1'356W7H9�C#I �

% � /�% % " � � � � � � %<��� � � � � �� ( ( " � � �H��­ (�� � �¶� ( � � N5� �

9�35;hX<756i:<6_�1�:<6�% " � � � � � %<��� � � � ��� � ® � ,�t � �� � m � 3hX<756rBD3qX<6#:EA�9�\|� ( � � ( � ��vi�p1'356Y7H9�C � ��� + ��

% � / " + � �

%�% � / " � % F�9�35;hX<756#:<6iL

_�1�:<6 � � � �O� %<��� � v ( ��� % ! � � % � + ��% � / " � % � �

�" v ( " º � �¶� ( � � NH�[tZB]3�X<756�49�CE6 * � ,�F|1'356�;H6#;5K546#C�X<7H9�X %Eº " � � � � �O� %<��� � v (( � º " � � � � ��v � �

�" v ( " º � �h� ( � � NH� � ���

� ��" v ( " � � �¶� ( � � N5�@1'3�X<756>B]3qX<6i:EA'9�\Z� ( � � ( � ��v �f9�35;�X<756i:<6_�1�:<6 % º " � � � � � � %<��� � v �

� � ® � ,�t � �� ��F�eP7HBD\D6cBD_p� (�*�� , 1'356r7H9�C I �% � /�% % � º " � � � � � � � � %<��� � � v � �� ( ( � º " � � � � �Vv � � � �� � " v � �

( " º � � �h� ( � � NH� � �C ( " � � � �¶� ( � � NH� � 9�35;>X<756#:<6#_�1�:<6 % º " � � � � � � %<��� � v � � � � ® � ,�t � �� �'m-£p6#zH35BD35k�F�� -��H� � � v� �8k'BgAq6#C�F�� -���� N�9�C -��·®9�35;w=W1�:<6#1jAq6i: % � � � F%� %<��� � � � ��� � ® � ,�t � �� � 9�35; % � � � F ��� %<��� � v � ��� � ® � ,�t � �� � m¡u1'3qAq6i:<CE6\g?'FP9�CECEKH=W6yX<7H9�X�% � � � F%� %<��� � � 9�35; % � � � F ��� %<��� � v `�6#\]1�3Hk�X<1 ��� � ® � ,/t � �� �'m ¢�756#3�_�1�:�% � � � ®°1�3H67H9�C � � F%� %<��>,F�� ����� � v ( I �

�� � F �O� -���� � v O - ( C %}v " � FuCE1�X<7H9�X F%� %<�{X<6#3H;5C)X<1°9~\]BD=WBgX)B]3 � � Ó � v 9�C�-�X<6#35;HC)X<1°®,m¦oCEBD35k°X<7H6#3�X<7H6y;564#1'=)^a1�CEBDX<BD1'3,F%�O®I� � F%� %<�2Ó � F��O®q�)>5F%� %<����FP1�3H6�7H9�C�% � � � F%� %<��� � � � ��� � ® � ,�t � �� ��F|9�35;�_]:<1'=� �

� � � � " � � � � � � /� ( v� I �� � � F �O� -���� � v O -f1'356f;56;5K54#6Co`I?�no����� � × CpBD356jJqK%9�\]BgXs?�X<7%9�X % � " v.� � F��O®I�H>!F%� %<��� � v � ��� � ® � ,�t � �� � tX<7563>1'35684[7H9�35k'6C %2BD3IX<1 v� FH1�:P1'356Z3H1�X<B]4#6#C�X<7%9�XPBgXPC<9?(C�X<7H9�X)F��O®I� � � � v�� � � � v " � � � F5eP75BD4}7>BDC8� � � � � v � � � � m¢�756)BD35BDX<B]9�\�;56izH3HBgX<BD1'3�1'_-XE:[9�46@CM^%9�46#Cp`I?�GI9�4JIK56CMLON�1'K5BDCpN�Q+R�T�U)9�35;�GI9'9�b�±d���%������K5CE6;�_�1'K(:Z^%9�:[9�=c6iX<6#:<C

9�35;Y4#1'35CEBD;H6i:<6#;Y_�K5354#X<BD1'35CuC<9�X<BDCE_]?(BD3Hk % � � � � ��� � �O® � ,�t � � �VePBDX<7"% � / � � � ��� / �O® � ,/t � v �[F(_�1�:2CEK5BgX[9�`H\D6|^%9�:[9�=c6iX<6#:<C� � � � v�� * � � * v t(X<756i?c7H9�;Y351�X<BD46#;cX<7H9�X2X<756o_+9�=cBD\g?W;56i^a6#35;56;Y1�3�9�X�=W1'CMXuX<75:<6#6P^�9�:[9�=c6iX<6i:<Cu`I?r4[7H9�35k'BD35k %-B]3qX<1 % � F`HK(X8X<7H6i?�7H9�;h9�\]CE1>B]3qXE:<1,;5K54#6;�X<756cBD=)^�1�:EX[9�3qXZ^%9�:[9�=W6#X<6i: � t�BDXZe�9�C8GI9'9�b�±d���%�����heP751w\]9�X<6i:v8_�1'K535;�X<7H9�XZX<7H6_+9�=cBD\g?�;56i^�635;56;�1'35\g?~K(^�1'3�Xse21>^%9�:[9�=c6iX<6#:<C#F�9�35;�;H6iAq6#\D1�^�6;~X<756WCEBD=)^%\D6i: � 9�3H; D�=W6#X<751,;5CpX<7H9�X87H9Aq6)`�6#63_�1�\]\D1je26#;>BD3�X<75BDC�41'K(:<CE6�m

1 BgX<7�X<756yC<9�=c6y9�:<k'K5=c6#3qX<CcKHCE6#;/BD3�X<756>^5:<64#6;5BD35k�^H:<1�^a1�CEBDX<BD1'3aFdX<756y4[7H9�:[9�4iX<6#:<BD�9�X<BD1'3�1�_oXE:[9�4#6wCM^%9�4#6CcBDCCEBD=cBD\]9�:�X<1cCMX<K5;5?,B]35k)X<756{_�1�\]\D1jePBD35kfA'9�:<B�9�3IXF5eP756i:<6{1'356{;56izH3H6#C � � � � � / � % � NH�*� BD3H_ + � + � � + / � � N � � � � �� Ó %T� � N v � � � /v F�9�35;1'356f;56#zH356#CW� � � � � v � � � � � � 9�CpX<756@CM^%9�46f1'_*N � � � Ó � v CEK54}7~X<7H9�X % " � � � � � � / � % � NH� � ��� � ® � ,�t � �� �'m8¢�756f\�9�4[by1�_751'=W1'k'6#3H6#BgXs?W=c9?c\D1,1�bWCMXE:[9�35k'6�F(9�35;YBD_�XE:[9�4#6|CM^%9�46#C27H9�;r3H1�Xu`a6#6#3Y;56izH3H6#;r`�6#_�1�:<6|BgXde21'K5\D;Y351�X26iAq6#3c`a6|1�`IA(BD1'K5CX<7H9�X{� � � � � v � � � � � � B]CP9@Aq64iX<1�:PCEK(`HCM^%9�46�F�9�35;>BgXPBDCP9�4iX<KH9�\D\g?�9)CM^�9�4689�\g:<6j9�;5?Y;H6izH356;amvwGI9�4JIK56CMLON�1'K5BDC�N�QSR�TVU × CuBD3IX<6i:<6#CMX<C27H9�;YCMePBgX<4[7H6#;rX<1@1�X<756#:�JIK56CMX<BD1'35Cu41'354#6#:<35BD35k{X<756oK5CE6p1'_ �HKH354iX<BD1'3H9�\��o3%9�\DL

?(CEBDCPBD3w\DB]3569�:|9�35;>X<7H6#3w3H1�3H\DBD356j9�:�^�9�:EX<B�9�\�;5Bg �6#:<6#3qX<B]9�\ 6JIKH9�X<BD1'35C#F%BD3w1�^5X<BD=cBD�9�X<BD1'3>9�35;w41'3qXE:<1'\a^H:<1�`%\D6#=cCP9�35;yBD3X<756Bg:u3IKH=W6i:<B]49�\�9�^5^5:<1jl(BD=c9�X<BD1'35C#mV�o_]X<6#:de�:<BgX<B]35k87HBDC-`�1,1�b(CdePBgX<7cÙd3(:<BD41{����¤V�%T��aU�F,75BDCuB]3qX<6i:<6CMXuBD3rBD3qX<6#:E^�1'\]9�X<BD1'3CM^%9�4#6CW`a6#49�=c6>=c9�:<k'BD3H9�\SFu`HK(Xc7H6wK5CE6;�X<7H6>B];569�CWeP7563�3564#6CEC<9�:E?�to9�_]X<6#:WzH35;HBD35kh9�351'35\DBD356j9�:c_�:[9�=c6ie21�:Eb�_�1�:BD3qX<6#:E^�1'\]9�X<BD35k>:<6#k'K5\�9�:<BDXs?�_�1�:@A�9�:<B]9�X<BD1'3H9�\-BD3H6JIKH9�\DBgX<BD6C#F�756rX<751'K5k'7qXf1'_�9yk'6#356#:[9�\DBD�9�X<BD1'3°9�35;¶756r^5:<1�`%9�`H\g?�_�1'K535;X<7H9�XfBgX@e�9�C)9yk'1I1,;°^5:<1�`H\]6#=»_�1�:)9yCMX<K5;H6#3qX)BD35CMX<69�;¶1'_�BD3qAq6CMX<BDkq9�X<B]35kwX<756�JIK56#CMX<BD1'3h75B]=WCE6#\]_�my£p6#A'6\D1�^5^HBD3HkwX<75B]CBD;56j9>=c9�;56WX<756Wz5:<CMX{^%9�:EX@1�_�={?~X<7H6#CEBDC#FV9�35;�X<756cCE64#1'35;�^�9�:EXf9�35CMe26i:<6;h9�351�X<756#:r�SCE\DB]k�7IX<\g?�9�49�;56=WBD4��oJIK56#CMX<B]1�3X<7H9�Xc756>7H9�;�X<751'K5k'7qXc1'_sF29�3H;�0f4[7%9�:[9�4#X<6i:<BD�6#;¶X<7H6�XE:[9�4#6#Cc1'_|_�K5354iX<B]1�3HC)C<9�X<BDCE_]?(BD3Hk � � � �uº � ® � �Zt';°vj���P� � 9�35;���� � � �uº � ® � �Zt �uº����P����m

�(�

Page 92: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

� & ����$ §r�H1�:r® � � � �>9�35; � ( * � � * v�� * ( ,/Fu1�3H6�7H9�C>� � � � � v � � � � � � �·� � � � � v � � � � F-ePBgX<7 � ;56izH3H6#;�`I?v " ��� v " �

�� �� /F%9�35; *9� � �E�8> � � * � Ó � * v � *�m

�2È<ÆjÆ5#[§�0�_o1'356y;56iz%356#C � 8 � % � NH��� � BD35_ + � + � � + / � � N � � � 8 � Ó�% 84� � N v � � 8 v � v � � 9�3H;�1'356y\D6iX<C 5 X<6#3H;�X<1 ,�F21'356w1�`5X[9�BD35C� � % � N5�H��B]35_ + � + � � + / =c9�l � � � N � � � � � %T� � N v � � �qF(9�35;�1'356o7%9�C � � % � NH� ( �h� % � NH� (�­ � � % � NH�-_�1�:�9�\D\ N � � � Ó�� v mÐo6#1'=c6iXE:<BD4j9�\]\g?'F�_�1�:VX<756PÐo��¤V¥,Q���� �-RwCE6iX-9�CECE1I4B]9�X<6#;@X<1 N�F'BSm 6'm �I�GQ � � Q v � � X<756#:<6u6#l(BDCMX<CV9o;56#41�=W^�1'CEBgX<BD1'3�N�� N � ÓWN vePBgX<7 � � N � � � � (,Q � � � � N v � � v (,Q v �qFHX<7568\DB]356 Q � ��%4Q v BD3qX<6#:<CE6#4#X<C�BgX|9�X Q � � � � % � NH� � Q v � ! � � � � + �� m��,BD=WBD\�9�:<\D?r1�3H6;56#zH356#C"� � - � NH� ��BD35_ + � + � � + / =c9�l � � � N � � � � �� � - � � N v � � � /v �qF�9�35;�1�3H6W7H9�C"� � - � N5�)( � � � � � / � - � NH�)( ­ � � - � NH�|_�1�:9�\D\ N � � � Ó � v F,9�35;WBD3)1�:<;56i:-X<1pzH35;)X<756�C<9�=c62^�1'BD3IX-1�_%X<756oÐo��¤�¥,Q������-R�CE6iX-1'356�4[7H1I1'CE6C�- � % � / � � % � NH� � � " � / FCE1>X<7%9�X � � % � NH� � � �0- ! � � � � + � � /�

� / FV9�35;�1'356W;H6#;5K546#C � � - � NH�� � � % � NH� � � t�X<7H6#3�1'356c351�X<BD4#6CpX<7H9�X %��� - �% � / � � % � NH� � � " � / BDC@9>k'1,1I;°4[7H9�35k'6c1'_uA�9�:<B]9�`H\]6�F `�649�K5CE6"% �� � � % � N5�oB]C8351'35;56#4#:<69�CEBD35k�9�35;% �� ! � � � � + �� BDC351'35BD3H4i:<69�CEBD3Hk(Fu9�35;�1'356>;56#;5KH4#6#C)X<7H9�X � �� � CW� %<� � �� ePBgX<7¶=WB]3 � * � � * v � ( CW� %<� ( =c9�l � * � � * v �w_�1�:c9�\D\�% � ®,F9�35;¶X<7H6i:<6#_�1�:<6�I � - " � � � � - � NH� � � �� � , BDC)6JIK5BgA'9�\D6#3qXfX<1<I � % " � � � / � � - � NH� �� � " � � � � � " � / � � �� � ,�m¶¢�75BDCk'BgAq6#C|X<756f4#1'35;5BDX<BD1'3 * � * * � > � *�� * � > * v �8�!�E� > � � * � *rÓ � * v *�Fa9�35; � * � � * v * � *E>��E� > � � * � *�F�CE1YX<7H9�X�E�8> � � *9� �E�8> � � * � *�9�35;w6#\]BD=WBD3%9�X<B]35k *y`�6iXse266#3�X<756CE6Z\]9�CMX�Xse21c_�1�:<=@KH\]9�C�k'BgAq6#C�X<7568;56#CEBD:<6#;>_�1�:<=@K5\]9)_�1�: � m�{%���.+.�$H��� §�0�_ �!� � v:� � � � ! �[F�X<756#3~BgX<C|XE:[9�4#6f1'36Q ! � ®rC<9�X<BDCMzH6C�� � �!� � v � �� � � � � ! " v ��� � v � �� � �� � � ! " v ���� � v:� �5� �)! " vi� � ���(� �"! " vi��� v � � � � m�2È<ÆjÆ5#[§�¦oCEBD3HkQ ! 9�C-X<756�A�9�:<B]9�`H\D6%��O9�35;WK5CEBD35k81�3H\g?@X<756�_�9�4#X�X<7H9�X �6� � v'� �5� �)!

��E�[Fq1'356�zH35;5C-X<7H9�X �6� � v'� �H� �)!p�

BD=W^H\DBD6C � � ��� � ® � ,�t � v'� �H� �)! " vi���W9�3H; � � � ��� � ® � ,/t ���5� �)! " vi����F 9�35;~X<75BDCo4#1�:E:<6CM^�1'35;5CoX<1 � � v�_�1�: � � �

� v'� �5� �)! " vi�29�35;�� v � ���H� �)! " vi�[m0�3¶;56iz%35BD35k�B]3qX<6i:E^�1'\]9�X<BD1'3hCM^%9�4#6#C#F-1'356Y7H9�Cf351�X@:<69�\D\D?�K5CE6#;hX<7H6r_�9�4#X@X<7H9�X � � 9�35; � v 9�:<6Y351�:<=W6;°Aq6#4#X<1�:CM^%9�4#6C#F59�3H;r1'356o49�3r6#lIX<635;cX<756PX<7561�:E?)X<1{X<756|49�CE6|1'_ 4#1'=W=fK(X[9�X<BgAq6{�S�|`�6#\]B]9�3���k�:<1'K(^HC#FH9�35;r=W1�:<61Aq6#:uX<756|351�:<=

49�3>`�6Z:<6#^H\]9�4#6;�`I?>9cJIKH9�CEBgLO351�:<=>FHC<9�X<BDCE_]?(BD3Hk ! N,#T �® � ! >8N,#�� ! N,# _�1�:|9�\D\ N�F ! N,#�� ®WBD_�9�35;w1'35\g?�BD_BN�� ®,F�9�3H;�X<7H64 LSXE:<B]9�35k'\D6WBD356JIKH9�\DBDXs?�! N)Ó 2 # ( 4 �3! N&#�Ó ! 2 #��o_�1�:@9�\D\HN � 2 �S1�3H6W49�\D\]CpX<75639! � #u9 4 LO351�:<=��[m � 3H6W351�X<BD46#CpX<7%9�X{B]_u1�3H6;56#zH356#C �W`I?���­ 4 � � � ­8X<7H6#3YX<756#:<6pBDC�9Y�iLO351�:<= � � � � �qCEK54[7>X<7H9�X�� � N � � ( ! N,# � (�­�� � N � �'_�1�:P9�\]\PN�F59�35;>CEK54[7w9f351�:<= BDC;56#zH356#;>`I? � � N � � ��BD35_ + � � %

�� / + � ! N 7 # � F5eP756i:<6ZX<7568BD3(zH=fK5=�BDC�X[9�bq6#3>1jAq6i:|9�\D\ � 9�35;y9�\]\a;564#1'=)^a1�CEBDX<BD1'35C�1�_BN�m

� 356Z4j9�3w;56izH3H6oX<7H6ZCM^%9�4#6c� � � � � v � � � � BD_ � � � � v 9�:<6{JIKH9�CEBgLO351�:<=W6;��|`�6\DB]9�3Yk�:<1'K(^HC#F�9�3H;�ePBgX<7w® � � � �89�CK5CEKH9�\SFH`HK5X|_�1�:o9W\�9�:<k'6#:P_+9�=WBD\g?�1�_ *�F�9�CP1'3568=r9?YX[9�bq6{®�� *+( ,/F%1�`5X[9�BD35BD3Hkc9cJIKH9�CEBgLO351�:<=W6;wCM^%9�4#6{BD3>X<75684j9�CE6® �+* � ��FH6#Aq6#3>BD_ � � 9�35;�� v 9�:<6Z3H1�:<=c6#;>CM^�9�46#C#m0�_B� � � � �iBDC�9@�iLO351�:<=Ø9�35;@_�1�: � ��®P1'356u;56iz%356#C ! � #�� � � � � � � F�X<7563 ! � #,BDC�9 4 LO3H1�:<=ØB]_(1�3H6d7H9�C2�GNaÓ 2 � � ( 4 �GN ��Ó 2 ���_�1�:�9�\D\ N � 2 �®(F59�35;Y1'356o4[7564<b(C269�CEBD\g?)X<7%9�X21'356o=r9?)X[9�bq6 4 �Ø�|BD_ � ( �o9�35; 4 � ­.�

" v2B]_ � ��m � 356o=r9#?WX<75634#1'35CEB];56i:�X<756ZA�9�:<B]9�3qX � � � � � / � % � NH�u_�1�:|® �+* � � * v (-,�F%9�C|9@^�9�:EX<B]4#K5\]9�:P49�CE6Z1'_�K5CEBD35krJqK%9�CEBDLO351�:<=WC#m

�'­

Page 93: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

­(��m�1°6#;53H6#CE;H9?���9#?2�,F%­�®'®'®�%1�:)4}7H9�:[9�4iX<6i:<B]�#BD35kwX<7H6r;HKH9�\u1'_{� � � � � v � � � � _�1�:W®�� � � �Y9�35;*�+( * � ,�F�1'356�356#6;5C)9y_�6ie!X<64[753HBD49�\

:<6#CEKH\gX<C#m�VBg:<CMX�1'_�9�\]\%X<7H6i:<6pBDC�9f356ie 7q?,^�1�X<756CEBDC#F,X<7H9�X � � � v BDC2;H6#35CE6pB]3�� � 9�35;>;56#35CE6ZBD3 � v t(X<7HBDC2BD=W^H\DBD6CdX<7H9�X ����9�35; ���v 9�:<6pCEK(`HCM^%9�4#6C�1�_d� � � � v � �SF,`�6#4j9�K5CE6oB]_�" � BDCdX<7H6oBD3��E6#4iX<B]1�3Y1�_�� � � v BD3qX<1 � � F,X<756#3 " � BDC�9{41'3qX<BD3IK51'K5C=c9�^5^%BD35kY_�:<1'= ���� BD3IX<1�� � � �� v � ��eP751'CE6@:[9�35k'6)BDCp;5635CE6�FaeP75B]\D6@X<756f7q?,^�1�X<756CEBDCoX<7H9�XpX<756@:[9�35k'6)1'_B" � B]Co;5635CE6BD=W^H\DBD6C�X<7H9�X�" � BDC|BD3��E6#4iX<BDA'6'm�¢�756#:<6#_�1�:<6 � �� �� �v 9�35; � �� Ó � �v 9�:<68e26\D\�;H6izH356;am � 3H6{;H6#351�X<6#C � � � � � � X<756@351�:<=1'3���9�35;>`I?!� � � � � � � X<756Z3H1�:<= 1'3 ���

�_�1�: ��� � � ­,m

� & ����$ §��oCECEKH=W6�X<7H9�X � � � v B]C�;56#35CE6PBD3 � � 9�35;W;56#35CE6PBD3 � v mV�%1�:�% ��®(FqX<756�;5KH9�\HCM^%9�4#6�1'_ � � � v 6JIK5BD^5^�6#;ePBgX<7�X<756r351�:<= Dd� % � N%t � � � � v �R��=c9�l � � � N � � � � %T� � N � � v �YBDC � �� Ó ���v 6JIK5Bg^5^a6#;�ePBgX<7�X<756r351�:<= � � v� �32 t � �� � ���v � �BD35_ - � �&- / �&- � � � 2 � � � � � Ó v� � � 2jv � � � v � m�2È<ÆjÆ5#[§a0�_4N � � � � v 9�35; 2 � � �� Ó � �v ePBgX<789�;56#41'=)^�1'CEBgX<BD1'3 2 � 2 � Ó 2�v�� 2 � � � �� � 2�v � � �v F�X<756381'356d7H9�C�� � 2 � N ��� (��� 2 � � N ���#Ó���� 2jv�� N ��� ((� � 2 � � � � � � � N � � � Ó v � � � 2�v � � � v %T� � N � � v ( � � � 2 � � � � � Ó v � � � 2jv � � � v ��=c9�l � � � N � � � � % � � N � � v �qF%9�35;�X<756#:<6#_�1�:<6p`I?=WB]35BD=WB]�#BD35kY9�=W1'35k>9�\D\�X<756);H6#4#1'=W^�1'CEBgX<BD1'35Co1'_ 2 1�3H6f;56;5K546#CpX<7H9�XE��� 20� N ��� ( � � v � � 2 t ���� � ���v �.Dd� % � N�t � � � � v �|_�1�:9�\D\ N � � � � v�� 2 � ���� Ó ���v � % ��®(m¡u1'3qAq6i:<CE6\g?'Fp\]6iX � `�6°9�\DBD356j9�:>41�3IX<BD3IK51'K5Cw_�1�:<= 1'3�� � � v 6JIK5Bg^H^�6#;*ePBgX<7*X<756�351�:<= Dd� % � N�t � � � � v �[F9�35;�\]6iX�� � � � �� � FuCE1�X<7H9�X � �P�GNH��� (0� =c9�l � � � N � � � � %T� � N � � v �>_�1�:W9�\D\ N � � � � v t21�3H6�K5CE6#C)no��³�T�L- ���T���²a³X<7561�:<6=·BD3�1�:<;H6i:)X<1yz%35;�9�\DBD3H69�:)4#1'3qX<BD3,K51'K5Cf_�1�:<= � �r1'3 � � � � v F-6jJqKHBg^5^�6;hePBgX<7hX<756�351�:<= � �¹�GN � � N v ��� � �=c9�l � � � N � � � � � %T� � N v � � v �'F%eP75BD4}7y6il,X<6#35;HC�X<756{\DBD356j9�:|_�1�:<= ���;56izH3H6#;y1'3wX<756{;5B]9�k'1�3%9�\�1'_2� � � �� v � �°� � � � v �2`I?��P�GN � NH�H� �P�GN5�[F,_�1�:ueP7HBD4[7r1'356|7H9�C�� ��P�GN � NH��� � � �P�GNH��� (�� =c9�l � � � N � � � � %T� � N � � v ���� � � �GN � NH��� �¹FICE1@X<7H9�XR� � ��� � (��9�35;�X<756#:<6#_�1�:<6fX<756i:<6W6#l,B]CMX<CZ9�3�6#l,X<6#35CEBD1'3 � �)1�3� � � � v C<9�X<BDCE_]?(BD3Hk!� � � �M� � ( �¸mW�o3q?~\DB]3569�:84#1'3qX<BD3,K51'K5C8_�1�:<=1'3 � � � � v 49�3/`�6�e�:<BgXEX<63�9�C��GN � � N v � �� � 2 � � N � �2Ó � 2jv�� N v �)ePBgX<7 2 � � � �� 9�35; 2jv � � �v F|9�3H;/X<756~351�:<=1'_�X<7H9�Xf\DB]3569�:@41�3IX<BD3IK51'K5Cf_�1�:<=»B]C@1�`IA(BD1'K5CE\g? (S� � 2 � � � � � Ó v� � � 2�v � � � v FV`HK(X)BgXfBDC)9�4iX<KH9�\D\g?�6JIKH9�\uX<1wX<7H9�XWJqK%9�3IX<BgXs?`�649�K5CE6W1'356c4j9�3�4}751,1'CE6�N � � � � ePBgX<75� � N � � � � � �c9�35; � 2 � � N � � �+� � 2 � � � � � 9�35;<N v � � v ePBgX<75� � N v � � v � v� 9�35;� 2�v�� N v �W� v� � � 2jv � � � v �O9�kq9�BD3¶`I?¶9�3�9�^5^H\DBD4j9�X<BD1'3h1'_Pno��³�T�L- ���T���²�³ X<7H6#1�:<6#=��[td1'356�;H6#;5K546#CfX<7H6#3¶X<7H9�X �P�GNH�W���P�GN � NH� � � 2 � � N ��Ó � 2�v�� N �8� � 2 � Ó 2�v�� N ��_�1�:Z9�\D\TN � � � � v Fa9�35; � � 2 � � � � � Ó v� � � 2�v � � � v (1�¸F�CE1YX<7H9�X �*BDCok'BgAq6#3`I?�X<756Z6\D6#=c6#3qX 2 � 2 � Ó 2jv � ���� Ó ���v F5eP7HBD4[7>C<9�X<BDCMz%6#C � � v� � 2 t ���� � ���v � ( ��m

0sX-:<6=c9�BD35C-X<18CE751je�X<7%9�X 2 BDC-;H6izH356;cBD3c9ZKH35B]JIK56�e�9#?qFIBOm 6'm�X<7H9�X � 2 � N �H� ®Z_�1�:u9�\D\�N � � � � v BD=W^H\DBD6C 2 �*®,tBD35;H6#6#; F�`�6#4j9�K5CE6���� 2 � � N ���4� � � 2�v�� N ����(�C � � N � � v _�1�:Z9�\D\TN � � � �� v 9�35;�� � � v BDCo;5635CE6fBD3 � v F 2 � 6#lIX<635;5CoBD39cK535B]JIK568e�9?�X<1r9�3�6#\D6=W6#3IXP1�_ � �v F%eP75BD4}7>X<7563y4#1'BD354BD;56C�ePBgX<7!> 2�v 1�3wX<7H68;5635CE6{CEK(`%CM^%9�4#6 � � �� v 1'_� v F�CE1X<7H9�XP1'356Z7%9�C 2 � � 2�v � ���� ���v 9�35; 2 � Ó 2�v �*®,m� & ����$ §r�oCECEK5=W6YX<7H9�X � � � v B]Cf;5635CE6>BD3 � � 9�35;�;H6#35CE6>BD3 � v mh�H1�:�- ��®(F�X<7H6�;5KH9�\�CM^%9�4#6>1�_ � � Ó � v6JIK5BD^5^�6#;wePBDX<7wX<756{351�:<= �¶� - � N�t � � � � v � � B]35_ + � � + / � + � � � N � � � � Ó9- � � N v � � v ��BDC ���� ����v 6JIK5Bg^5^a6#;wePBgX<7wX<756@351�:<=D � v� � 2 t ���� � ���v � ��=r9�l � � � 2 � � � � � v� � � 2 � � � v � m�2È<ÆjÆ5#[§�0�_�N � � � Ó � v 9�35; 2 � � �� ���v ePBDX<7 9�;56#41�=W^�1'CEBgX<BD1'35N � N � Ó�N v FpX<756#3 1'356°7H9�C$��� 20� N ��� (��� 20� N � ���Ó(��� 2 � N v ��� (�� � 2 � � � � � � N � � � � Ó v� � � 2 � � � v - � � N v � � v ( =r9�l � � � 2 � � � � � v� � � 2 � � � v � � � � N � � � Ó9- � � N � � v �[F 9�35;�X<756#:<6#_�1�:<6{`I?=WB]35BD=WB]�#BD35kr9�=W1'35k�9�\D\�X<756@;H6#4#1'=W^�1'CEBgX<BD1'35C|1'_HNw1'356f;56;5K54#6C|X<7H9�X�� � 2 � N ����(�D � v� � 2 t ���� � ���v ���¶� - � N�t � � � � v ��_�1�:9�\D\ N � � � Ó � v�� 2 � � �� ���v � - ��®(m¡u1'3qAq6i:<CE6\g?'F BD_ � BDCp9Y\]BD3569�:p41�3IX<BD3IK51'K5Cp_�1�:<= 1'3�� � Ó � v 6JIK5Bg^5^a6#;�ePBgX<7yX<756)351�:<= �¶� - � N�t � � � � v �[F�X<7563BgXPBDCP9)\DB]3569�:�41'3qX<BD3IK51'K5CP_�1�:<=�1'3�� � 9�3H;w9�\DCE1c9)\DBD3H69�:�41�3IX<BD3IK51'K5CP_�1�:<= 1�3�� v 9�35;>X<756#:<6#_�1�:<6)��BDC�k'BgAq6#3�`,?>9�36#\]6#=W63qX 2 � ���� ���v m�%1�:Y4#1'=W^HK(X<BD35k�X<756y351�:<=�1'_ ��F�_�1�:Y® � ���·��F8Q � * � ®�1'356y4[751,1'CE6#C�N � � � � ePBgX<7(� � N � � � � � Q�9�35;� 2 � N � � �E� > ��� Q�� � 2 � � � � 9�35;�N v � � v ePBgX<7 � � N v � � v �9*W9�35; � 2 � N v � �E�B> ��� *R� � 2 � � � v F,CE1{X<7H9�X N � N � Ó�N v C<9�X<BDCMz%6#C�¶� - � N�t � � � � v � (,QZÓ)-�*c9�35;<��� 2 � N ��� ��E�*> �'�}�GQW� � 2 � � � � Ó�*R� � 2 � � � v �[F(9�35;rX<756#:<6#_�1�:<6 � � �� �M ��E�*> �'� �

� � - � � � � � !�� � - � � � /� � � ! F9�35;w\D6iXEX<B]35k �{X<6#35;>X<1c®W9�3H;>6BgX<756#:�Qy1�:�*rX<6#35;�X<1c®(FH1'356ZzH35;5CW� � � � �M �=c9�l �P� � 2 � � � � � v� � � 2 � � � v �(m

¢�756|=c9�BD3):<6#CEK5\DXu9�`�1'K(Xu;5KH9�\DBgXs?)_�1�:uBD3qX<6#:E^�1'\]9�X<BD1'3)CM^%9�4#6C#FqeP75B]4[7c7H9�Cu9�\g:<69�;(?f`�66#3c=W6#3IX<BD1'356#;c_�1�:uN�R����%T���µCM^%9�4#6C#FHBDC�X<7568_�1'\D\]1ePB]35k5m

���

Page 94: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

� ��%"-� j�D������ §P�oCECEK5=W6{X<7H9�X � � �� v BDC|;5635CE6@B]3 � � 9�35;y;H6#35CE6fBD3 � v m|¢�7563aF�_�1�:p® � � � �@9�35;¶� ( 5 � ,/F1'356Z7%9�C � � � � � � v � � � 8 � � �Ø� ���� � ���v � � � 8 ePBDX<7�6jJqKHBgA�9�\D63qX�351�:<=WC#F5eP756#:<6 v8 Ó v8 �Ø��m

�2È<ÆjÆ5#[§ 1°6cCE7H9�\D\�^H:<1Aq6cX<7H9�X � � � � � � v � � � 8 � �� � ��� ���� � ���v � � � 8 � ! 9�35;/� ���� � ���v � � � 8 � �

� � � � � � � v � � � 8 � ! � � FV9�3H;�X<7H6:<6#CEKH\gX�ePBD\D\a_�1'\D\D1je�_]:<1'= X<756ZB];56#3qX<BDXs?~�+ePBgX<7�6JIK5BDA�9�\D6#3IX�3H1�:<=cC}�d`�6iXse266#3�X<7H6ZBD3qX<6#:E^�1'\]9�X<BD1'3�CM^%9�4#6C�;56#zH356;�`,?rX<7H6G'LO=W6#X<751,;w9�35;>X<756&ZLO=W6#X<751,;am

0�3r1�:<;56i:uX<18^H:<1Aq6PX<756Pz5:<CMX2BD3H4#\DK5CEB]1�3 FI1�3H6�X[9�bq6#C N � � � � � � � � v � � � 8 � �� � F,9�35;�9�C N �%;56#zH356C298\]BD3569�:d41'3qX<BD3IK51'K5C

_�1�:<=Û1'3 � � �� v Fu9�^5:<6#46#;5BD3Hk~\D6=W=c9~9�CECE6#:EX<CfX<7H9�XW_�1�:W6iAq6i:E? � ��®�X<756i:<6�6#l(BDCMX<C)9~CE6JIK56354#6 2 ��� � � �� vCEK54}7~X<7H9�X�Dd��­ � � 2 � t � � � � v � � �W9�35; �¶��­ " � � N � t � �� � � �v �*> �{=cBD3 �I� � ­ " � � ( � N � �32 � �[m@�H1�:@9YCE6JIK56#3H4#6�� � CEK54[7X<7H9�XZ­ " � � � �6� � 8 � � �[F�1'356@;56izH3H6#C N������ � � � � � � � 2 � F�9�35;�1�3H6{7H9�C N������ � � � � � � v � � � 8 � � ePBDX<7$� � N�������� � � � 8 � �(

� � ­ " � � � � � � 8 `�6#4j9�K5CE6o1'_ X<756p351�:<=c9�\DBD�j9�X<BD1'3c4}751,1�CE63�_�1�:2X<7H6oCE6JIK56#3H4#6 2 � t,X<756o^%9�:EX<BD4K5\]9�:24[751'BD46o_�1�:uX<7H6oCE6jJqKH6#35462 � BD=)^%\DBD6#C � � � � � � � �¶��­ " � � N �St ���� � ���v �B> �{=cBD3 �q� � ­ " � � � ( � N � � N����-��� ( � � N � � ��� � N�������� � � � 8 � �

( � � N � � ��� � ­ " � � � � � � 8�S351�X<BD46WX<7H9�XW� � =cBD3 �q� � ­" � � � � vi�[t�`I?h\D6iXEX<BD3Hk �YX<635;°X<1~®y1'356Y;56;5K54#6C{X<7H9�X � � � � � � �¶��­ " � � N �St ���� � � �v � (� � N � � ��� � ­ " � � � � � � 8 _�1�:)9�\D\dCEK54}7hCE6jJqKH6#3546#C#� � m� �6#4j9�K5CE6 � � � � � � � � ( � � � ­ " � � � � � � 8 _�1�:)9�\D\dCE6JIK56354#6C�� � BDC6JIK5BDA�9�\D6#3IX2X<1 � � ­ � � � � � � 8 (���FH1'3568;56#;HK54#6C�X<7H9�XW� � ­ � � �¶��­ " � � N �St ���� � ���v ��� � 8 ((� � N � � � FHBSm 6�mHN � � � ���� � ���v � � � 8 � ! m0�3�1�:<;56i:8X<1Y^5:<1jAq6)X<7H6)CE6#41�3H;�BD354\DK5CEBD1'3aF 1'356)X[9�bq6#C N � � � ���� � ���v � � � 8 � �

F�9�35;�1�3H6fe�:<BDX<6#C N �H� � � N �� ePBgX<7N �� � � �� � �v 9�35;r­ " � � Dd��­ � � N �� t ���� � � �v � � � 8 s� � �[t'X<7563)_�1�:*N � � � � � � v � � � 8 � ! 1'356�7H9�C ��� N � � N ��� ( � � ��� N �� � N ��� (� � Dd��­ � � N �� t ���� � � �v � �¶��­ " � � N%t � � � � v � (�� � � ­ � � �¶��­ " � � N�t � � � � v ��� � 8 (�� � � N � � � � 8 � ! m

1 756#3*0We�9�C>9¶CMX<K5;56#3IX�0r7H9�;*351�X<BD4#6;/X<7%9�X>_�1�:w® � � � �yX<7H6~CM^%9�4#6�� ���� � ���v � � �¹v BDC>9�4#X<KH9�\D\g?�9h;5KH9�\SF9�\gX<751'K5k'7h351�X{X<756Y;5KH9�\u1'_Z� � � � � v � � � t�087H9�;¶=W63qX<BD1'356#;°X<7H9�X{X<1�={?°9�;(A(BDCE1�:FVGI9�4JIK56CMLON�1'K5BDCfN�QSR�TVU'F�9�35;h7H67H9�;�X<1'\D;~=W6@X<7H9�XpGI9'9�b�±-����� ����7%9�;�9�\g:<69�;(?y=c9�;56@X<7%9�Xp1�`HCE6#:EA�9�X<BD1'3avm8¢�7H6fBD;569rBDCoX<1Y1�`HCE6#:EAq6@X<7H9�X � vpBDC|X<7H6;5KH9�\d1'_ 4 � FV9�35;°X<7H9�X{1�3H6c4j9�3°;56#zH356Y9>356#epºWBD3qX<6i:E^a1�\�9�X<B]1�3�CM^%9�4#6r=W1,;56\D6#;�1'3 4 � F�41'35CEBD;56#:<BD35k>X<7H9�X8X<7H6cK5CEKH9�\BD3qX<6#:E^�1'\]9�X<BD1'3~CM^%9�4#6WB]35;56il(6;~`I? � � *°B]C89�4iX<KH9�\D\D?�=W1,;56#\]6#;~1'3 � �5� � �[m)�%1�:{® � � ���W9�35;�Xse21 ���T���²�³/CM^%9�4#6#C� � � � v 4#1'3qX<BD3IKH1�KHCE\g?rBD={`a6#;5;56;YBD3IX<1)9f4#1'=W=c1'3cX<1�^�1'\D1'k'BD4j9�\�A'64iX<1�:�CM^%9�46�F51'356p;56#zH356C2X<756pCM^%9�4#6W� � � � � v � � � � � 9�CX<756)CM^%9�4#6)1'_*N � � � Ó � v CEKH4[7~X<7H9�X{­

"� � �¶��­ � � NH� � 4 � � � �[F 6JIK5Bg^5^a6#;�ePBgX<7�C<9�=W6)351�:<= 9�CW� � � � � v � � � �+X<75B]C

CM^%9�4#6oBDC-X<7H6P4#\D1'CEK(:<6|1'_ � � � v B]3y� � � � � v � � � �[m-¢�756P^5:<1,1'_a1�_�X<7H6�^H:<6iA(BD1'K5Cd±-:<1�^�1'CEBgX<BD1'3c6j9�CEB]\g?)6il,X<6#3H;5C-X<1{CE7H1eX<7H9�X�X<7568;5KH9�\ 1'_u� � � � � v � � � � � BDC8� � �� � � �v � � � v ePBgX<7w9�3w6jJqKHBgA�9�\D63qX�351�:<=>m�o351�X<756#:PK5CE6#_�K5\a:<6CEK5\gXP41�3H4#6i:<3HBD35k)BD3IX<6i:E^�1'\]9�X<BD1'3�CM^%9�4#6C�B]C�X<7568JIK56CMX<BD1'3�1'_�4#1'=)^%9�4#X<356#CEC#m

� ��%"-� j�D������ §V0�_��/B]Cd9Z\]BD3569�:-=r9�^H^HBD35k8_�:<1'=¸9Z351�:<=c6#;WCM^�9�46�xBD3IX<1 � � � v FICEK54}7WX<7H9�X���BDCd\DBD356j9�:-41'3qX<BD3IK51'K5C_�:<1�= B]3qX<1�� � 9�35;�4#1'=W^%9�4iXZ_�:<1'= BD3qX<1�� v F X<7H6#3�_�1�:{® � � �!�fX<756)=r9�^5^HBD35k �¸BDCZ4#1'=)^�9�4#XZ_�:<1�= �B]3qX<1� � � � � v � � �¹v �O9�35;>X<756i:<6_�1�:<6Z4#1'=)^�9�4#X�_�:<1'= �BD3qX<1w� � � � � v � � � � _�1�:{��( * (-,��[m0�_ ��B]C29{\DBD3H69�:u=r9�^H^HBD35kf_]:<1'= � � Ó�� v B]3qX<1f9{351�:<=W6;rCM^%9�4#6 � F,CEK54[7YX<7H9�X��B]Cu\DBD356j9�:24#1'3qX<BD3,K51'K5C2_�:<1�= � �BD3qX<1 � 9�35;�41�=W^%9�4iX�_�:<1'= � v BD3qX<1 � F,X<7563�_�1�:�® � � � �oX<756p=c9�^5^HBD3Hk �¸BDC�4#1'=)^�9�4#X2_�:<1'= � � � � � v � � � BD3qX<1 ��O9�35;�X<756#:<6#_�1�:<6Z4#1'=W^%9�4iX�_�:<1'= � � � � � v � � � � BD3IX<1 � _�1�:{�)(+*#(-,��[m�2È<ÆjÆ5#[§¶0�_�� � @ � � � � ( ��FoX<756#3 � @ � B]C�`�1'K535;56; BD3 � � 9�35; `�6\D1'35k'C�X<1�9�4#1'=W^%9�4iXwCEK(`HCE6iXy1'_ � v FpCE1�X<7H9�Xy9CEK(`HCE6jJIK56#3546 @ � 4#1'3qAq6#:<k�6CoBD3 � v 9�35;�X<756i:<6_�1�:<6{BDCp9>¡u� �²�³���CE6jJqKH6#3546�ta1'356@7H9�C�� � QH� � � �¹v (�C � � QH� � v

"�� � � QH� � � v _�1�:9�\D\8Q � � � � � � v � � �¹v F�9�35;/9�^5^%\g?(BD35k~X<75BDCcBD356jJqK%9�\]BgXs?°X<1 Q�� @ � >�@ � 1'356w;56;5K546#C)X<7%9�X�� @ � BDCc9h¡u� V²�³��

CE6JIK56354#68BD3�� � � � � v � � � v F%9�3H;�X<7H6i:<6#_�1�:<6 �xBDC�4#1'=)^�9�4#X�_�:<1'= �BD3qX<1w� � � � � v � � � v mN�6iX � � ( � � � � � ( ��CE1oX<7%9�X-_�1�:-69�4[7 � ��®|X<756#:<626#l(BDCMX<C�9o;H6#4#1'=W^�1'CEBgX<BD1'3�( � � ( �� Ó (�v� ePBgX<7 � � ( �� � � � Ó ��� � (�v� � � v (­ �¶� � � ( � � (»­ � � tpX<756#3 � � � ( �� � � � (»­�� � � � � � � � � � � � � � F|9�35;�� (�v� `a6#\D1'35k'CcX<1�9°41'=)^%9�4iX�CEK(`HCE6iX>1'_ � CE1¶X<7H9�X9�CEK(`HCE6JIK56354#6 � ( v� 4#1'3qAq6#:<k�6CWBD3 � F29�3H;�X<756#:<6#_�1�:<6�_�1�:)X<75BDC)CEK5`HCE6JIK56354#6w1'356>7H9�Cc\DBD=�CEK(^ �� � � � � � ( v� >� (�v� � � � (/Õ � � ��� � � � � � � � � � � t%KHCEBD35k�¡u��T���R�� × CP;5B]9�k'1'3H9�\�CEK(`HCE6JIK56354#6f9�:<k'K5=c6#3qXo1�3H6ZzH3H;5CPX<7H9�X�� ( � 41'3qX[9�BD35C|94#1'3qAq6#:<k�B]35kWCEK(`HCE6JIK56354#68B]3 � F%9�35;>X<756i:<6_�1�:<6 ��BDC�41�=W^%9�4iXP_]:<1'= � � � � � v � � � BD3qX<1YÐ@m

v>0sX�B]C2JIK5BgX<6o3%9�X<K5:[9�\�X<7H9�X�B]3cX<7H6|^5:<1,46#CEC21'_�;51'BD35k@:<6CE69�:<4[7�1�3H6|zH3H;5C2:<6CEK5\gX<CueP75B]4[7Y7H9Aq6Z9�\g:<69�;(?c`�6#63Y_�1�KH35;`�6_�1�:<6�F-9�35;¶Ùd3535BD1�£p��ÐoQSR���¤�Q�7H9�;h1'3546rC<9�BD;x«<4[75Bu46i:<4j9>XE:<1jA�9(F�4[7HBd:<BD4#6#:<49w:<BgXE:<1jA�9�¬h�+X<756cz5:<CMXf^%9�:EX@:<6=WBD35;HC1'_VX<7H6y«<CE6#6#by9�35;�?'1'K�ePBD\D\�zH35;5¬�_]:<1'= X<756@k�1'CM^�6\DC#F�`HK(XoX<756@^H\]9?y1'3yX<756{^H:<6iz5ly;51,6#Co3H1�Xoe21�:Ebye26#\D\�BD3~Ù-3Hk�\]BDCE7aF9�\gX<751'K5k'7°1'356c4#1'K5\D;�:<6#^H\]9�4#6cCE6#6ib°9�35;�zH35;�`,?�CE69�:<4[7¶9�3H;�;HBDCE4#1jAq6i:@B]3�1�:<;56i:{X<1wK5CE6c:<6#CE6j9�:<4[7h9�35;°:<6#;5B]CE4#1jAq6i:i�[m�I1'=W6#X<BD=W6C#FH1'356{=c9?�zH35;wX<7568:<6#CEKH\gX�`,?w9W;5BD �6i:<63qX|=W6#X<751,;y9�3H;wBgX|=c9?�`�68e21�:EX<7w^HK(`H\DB]CE75BD35kcBD_�X<756{356iex^5:<1,1�_BDCPCEBD=)^%\D6i:�X<7H9�3>X<756Z^5:<6#A(BD1'K5C�1'356'F%1�:oBD_�BDXP4#1'3qX[9�BD35CPB];569�C�eP75BD4[7w41�KH\D;>`�68K5CE6#_�K5\�_�1�:o1�X<7H6i:P^5:<1�`H\D6#=cC#tH1'_�4#1'K(:<CE6'F1'356ZCE7H1�KH\D;w=W6#3IX<BD1'3YX<756{9�K(X<751�:P1'_�X<756pzH:<CMX�^5:<1,1�_sFH6#A'63>BD_�756�¨�CE75687H9�C�3H1�X�^%K(`H\DBDCE7H6#;>BgXmº � `IA(BD1'K5CE\g?'FI1'356|49�3c;H6#CE4i:<BD`�6�=c1�:<6Pk'6#356#:[9�\%4#\]9�CECE6C-1'_aBD3qX<6i:E^a1�\�9�X<B]1�3WCM^%9�46#C#F(9�3H;cGI9'9�b)±d���%�����Y7%9�C29�4iX<KH9�\D\g?

;56#A'6\D1�^�6#;w9WJIK5BDX<6pk'6#356#:[9�\�_]:[9�=W6#e21�:EbY_�1�:P;51'BD3HkfX<7%9�Xm

��Õ

Page 95: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

����������� � �%�j&I��"��.�$5������� #"�$%�'&( N�KH48¢����5�(���dF 1 ����Thnp9�\D\�Ò'­,�j­,F�­�Ò ��LsÔ�( ��Õ5F5X[9�:EX[9�: �o9�35;5:<6ie8m 4#=@K m 6;5K

­'­,m-�5:<BD;H9?��~9?wÔ,F�­�®�®'®(m0�3����(�Õ,¨ (�Ô,Fa0|CM^�6#3IXpX<7H6@?'6j9�:89�Xp¦o35BDA'6#:<CEBgXs?y1'_�1*B]CE4#1'35CEBD3aF �~9�;5BDCE1'3aF�9�35;~0|1'_]X<63�;HBDCE4#K5CECE6;~ePBgX<7���B]4[7H9�6#\

¡u����T�����¥I¥�ta9�=W1'35kY1'K(:d�E1'BD3qXo:<6CEK5\gX<CoX<7H9�Xoe26@;5BD;~351�Xo^HK5`H\DBDCE7�X<756#:<6{e�9�CZ9YJIK56#CMX<BD1'3y:<6\]9�X<6#;yX<1YBD3IX<6i:E^�1'\]9�X<BD1'3am¢�756)=W1�X<BgA'9�X<B]1�3�_�1�:8\D1,1�b(BD3HkY9�XpX<7H6@^5:<1�`H\D6#= e�9�CZCE1'=W6fb(BD35;~1'_dk'6#356#:[9�\DBD�9�X<BD1'3yeP75BD4}7~7H9�;~`�66#3�^HK(`H\]BDCE756#; F _�1�:eP75BD4}7�BgXoe�9�Cp351�XZ4#\D6j9�:pBD_VX<756#:<6{e�9�Cp9�3q?�6il59�=)^H\]6{CE751jePBD35kYX<7H9�XpBgX|e�9�CpBD35;566#;~9rk'6#3IKHBD356�vpk'6356i:[9�\DBD�j9�X<BD1'3aF�9�35;9�CZ1'K(:Z:<6#CEKH\gXZ;5BD;�3H1�X841Aq6#:pX<7H6fC<9�=W6WCEBgX<KH9�X<BD1'35CoX<7H9�3~X<756)^HK(`H\DB]CE756#;~X<7561�:<6#=>F BDXp=cBDk'7qXpe26#\]\V7H9Aq6f`�66#3�=W1�:<6k'6#356#:[9�\aX<7%9�3wX<7H6Z^5:<6#A(BD1'K5CP1'356#C|BD3wCE1'=W6{49�CE6C#m2�o\gX<7H1�KHk�7we268CMX[9�:EX<6;>`I?>^5:<1jA(BD35kcCE1'=W6{1�`HCE6i:EA'9�X<B]1�3HCP_�1�:|\]BD3569�:=c9�^5^%BD35k'C#FH9�35;>X<7563>6il,X<635;56#;>X<756Z=c6iX<751,;�X<1W9W351'35\DBD3H69�:�CE6#XEX<BD35k5F(0u^5:<6#CE63qXPX<756p:<6CEK5\gX<C�BD3�:<6#Aq6i:<CE6Z1�:<;56#:m� ��%"-� j�D������ §�N�6#X&��`�6|9Z`�1'K535;56;y�SN���¯��aU�¤��� ��=W6j9�CEK5:[9�`%\D6�CEK5`HCE6iX21�_ �)! 9�35;c\D6#X �`�6|98351'35\DBD356j9�:d=c9�^5^HB]35k_�:<1�= � ����P�uBD3qX<1)BgX<CE6\D_�C<9�X<BDCE_�?,B]35k@X<7H6Z_�1�\]\D1jePBD35kf^5:<1�^�6i:EX<BD6C#§

B+� �BDC�N�Q AaU�²�³�Q���µW4#1'3qX<BD3,K51'K5CP_�:<1�= � ���P�dBD3qX<1WBgX<CE6\D_sFB]B�� ØBDC2=W1'351�X<1'356f�SBD3cX<756 �dºPCE635CE6��[F5BSm 6�m I J � c� � º �P> c� � v ���q� � º > � v �MOMQ6 �®{_�1�:�9�\D\ � v�� � º � ��>���P�[FX<7563r_�1�:26iAq6i:E?�* ��­,F�xB]CuN�Q A�U�²�³�Q���µ{4#1'3qX<BD3IKH1�KHCu_�:<1'= ���5���P��BD3qX<1{BDX<CE6#\D_��SBSm 6'm xB]CuN�Q A�U�²�³�Q���µ{4#1'3qX<BD3IKH1�KHCdePBgX<7

:<6#CM^a6#4iXPX<1)X<756"� � ;5BDCMX[9�354#6'F%9�35;w7H6#35468BgXP6il,X<6#3H;5C�B]3w9WK535B]JIK56Ze�9?w9�Co9)N�Q AaUj²a³�Q���µW41�3IX<BD3IK51'K5CP=r9�^5^HBD35kc_]:<1'=���5���P�dB]3qX<1WBgX<CE6#\D_}�[m�2È<ÆjÆ5#[§pN�6#X � `a6fX<756)N�Q AaU�²�³�Q���µ>4#1'35CMX[9�3qX8_�1�: ePBgX<7~:<6CM^�6#4#XpX<1�X<756 � ;5BDCMX[9�354#6'mf�%1�:@® � � � v� � 9�35;F � � >���P�aX<756#:<6uBDC�9|K53HB]JIK56 �6� � >���P�aCE1'\DK(X<B]1�3f1'_(X<7H626JIKH9�X<BD1'3 � Ó � c� � �H� FHFq9�CVBgXVB]C�X<7562K535B]JIK56uz5l(6#;@^�1'BD3IX1'_-X<756W=c9�^5^HBD3Hk � �� F > � c� � �[F eP75BD4}7�BDC89YCMXE:<BD4#XZ4#1'3qXE:[9�4#X<BD1'3at�=W1�:<6#1jAq6i:ZX<756)=r9�^5^HBD35k9F�� � B]CpN�Q AaU�²�³�Q���µ4#1'3qX<B]3IK51'K5C|ePBgX<7~9c4#1'35CMX[9�3qX�( vv " � � � Fa9�3H;�BD_�1'356@;56iz%356#CoX<756@=c9�^5^HB]35k � � `I? � � �GF5�8� F#> � � � c� � �2X<7563� � BDC�N�Q AaU�²�³�Q���µW4#1'3qX<B]3IK51'K5C�ePBgX<7>9W4#1'35CMX[9�3qX"( �

� �v " � � � m�%1�:WF v�� F º � � >���P�[FV1'356YCEK(`HXE:[9�4#X<C � v Ó � c� � v ��� F v _�:<1�= � º Ó � c� � º �#� F º 9�35;¶1'356Y=fK5\gX<Bg^%\DBD6#C8`I?c� � º � > c� � v �[F'k'BgA(BD3Hk �WI0JE� c� � º � >�c� � v ��� ºPOMQ#( �WI0JE� c� � º � >�c� � v ��� ºPOMQuÓEI0J � c� � º � >�c� � v ���'� � º > � v �MOMQ �I0J � c� � º � > c� � v � � �GF º >�F v �4OMQ�Fq9�35;fX<756#:<6#_�1�:<6�� � � � �GF º � > � � �GF v ��� � º ((� � F º >#F v � � º m�0�3f^%9�:EX<BD4#K5\�9�: � � 7H9�C-9pK535B]JIK566il,X<635CEBD1'3YX<1 �uº����P�|�+eP7HBD4[7>B]CP9f41'3qXE:[9�4iX<BD1'3��[m

np9A(BD35k�CE751jeP3~X<7H9�X � � BDCpN�Q AaUj²a³�Q���µ�4#1'3qX<BD3,K51'K5Cp1'3 � ����P��ePBgX<7�9Y41�3HCMX[9�3IX ( �� �

v " � � � 9�35;~N�Q AaU�²�³�Q���µ4#1'3qX<B]3IK51'K5C)1'3 �uº����P�ZePBgX<7�9�4#1'35CMX[9�3qX ( ��F-1'356>;56#;HK54#6C@`I?¶3H1�3H\DBD356j9�:)B]3qX<6i:E^�1'\]9�X<BD1'3hX<7H9�X � � BDCfN�Q AaU�²�³�Q���µ4#1'3qX<B]3IK51'K5C�1'3 ���5���P��ePBgX<7@9P4#1'35CMX[9�3qX ( � � � �v " � � � � � BD_ � B]C�;56izH3H6#;{`,? v

�� � Ó v " �º F'9�35;f9�C�1'356u7H9�C � � �

" º�

��®1'356P;56#;5KH4#6#C-X<7H9�X�X<756�41'35CMX[9�3qXd=c9?f`�6�=c9�;56�CE=r9�\]\,`I?fX[9�b(BD35k �oCE=c9�\D\SmV�oCECEK5=WB]35kpX<7563)X<7H9�X �o7H9�C�`�6#63W4[751,1'CE6#3CE=c9�\D\V6351'K5k'7~CE1�X<7H9�XpX<756)N�Q A�U�²�³�Q���µY41'35CMX[9�3qX81�_ � � BD3 ���5���P�PBDC�( � �¸��FaX<7563~_�1�:86iAq6i:E? � v�� � º � ��>���P�[F1'3568;56#zH356#C)F0� � � �uÓ � c� � ���u_�1�: "#�Ø� � ­,F%9�35;w1'3568;56#;5KH4#6#CPX<7H9�X ��� � c� � º �B>�c� � v ��� � � � � � � �GF º �B> � �GF v ��� � � (� � � F º > F v � � � ( ��� � � � º > � v � � � Ó ��� � c� � º �*> c� � v ��� � � �[F�9�35;�X<756#:<6#_�1�:<6 � � c� � º �3> c� � v ��� � � ( � � � � � º > � v � � �ePBgX<7 � � � !

� � v " ! � m0�3{X<756u49�CE6d1'_5\DBD356j9�:�=c9�^5^HB]35k'C#F�X<756uBD;569PBDC�X<7%9�X�1'356u=c9?8K5CE6uK53q`a1�KH35;56#;��S;56#3HCE6#\g?Z;H6izH356;��a4#\]1�CE6;{1�^�6i:[9�X<1�:<C

`I?>4#1'35CEBD;56#:<BD35k)X<756Bg:�:<6#CE1'\gAq6#3IX<C#FHBOm 6'm�X<756Z`�1'K535;56;w1�^�6#:[9�X<1�:<C{� � > � �(� " v�_�1�:|CE1'=W6 � � C)m-0�3>^%9�:EX<BD4K5\]9�:F5BD_�_�1�:9r:<69�\�noQ�¥(¯��%�H�¶CM^�9�46#; 9Y4\D1'CE6#;~K53q`�1'K53H;56#;�1�^�6#:[9�X<1�: � 7%9�C89Y;56#35CE6);51'=c9�BD3�9�35;�C<9�X<B]CMzH6#C � � � � � � �®Y_�1�:9�\D\ �6� y� �Z�[F(X<756#3>_�1�: ����®@X<756p:<6CE1�\DA'63qX�6il(BDCMX<C�9�35;>BDC�9f4#1'3qXE:[9�4iX<B]1�3 m-¢�7563�_�1�: � ��®@X<756o`a1�KH35;56#;w1�^a6i:[9�X<1�:�)>~� �2Ó � �Z� " vuBDC29�\DCE1@9{4#1'3qXE:[9�4iX<B]1�3 FqeP75BD\D6PB]_��*B]C-`�1'K535;H6#;rBgX<Cu351�:<=!BDC � � �'�[F,9�3H;c`I?cBD3qX<6#:E^�1'\]9�X<BD1'3WBgXdePBD\D\%7H9Aq69>CE=r9�\D\�351�:<=�BD3°9�3°BD3qX<6#:E^�1'\]9�X<BD1'3~CM^%9�4#6r9�35;°BgX8ePB]\D\�CE751je�X<7%9�X8X<7H6W1�^�6i:[9�X<1�:{BDC8`�1'K535;56;°BD3�X<7H9�X{CM^%9�4#6'mY¢�7H6

vw�y?�9�;5A,B]CE1�:{7H9�;�1'354#6W=c6#3qX<BD1'356;~X<7H9�XZeP7563~:<6j9�;5BD35kw9�X<1I1y9�`%CMXE:[9�4#X{9�:EX<BD4\D6f1'356WCE751'K5\];~z5:<CMX{\]1I1�b�9�XpX<7H66il59�=)^%\D6#C#m � 3�9r^H:<6#4#6;5BD35k�1,4#49�CEBD1'3aFa0|7H9�;�9�^5^H\DB]6#;�75B]CZ9�;(A(BD4#6)9�35;�CE751jeP3~X<7H9�X89Yk'6#356#:[9�\]BD�9�X<BD1'3�1�_-X<756fN�� � L��Q�¥5¤�������\D6#=c=c9Ye�9�C{351�X@9>k'6#3IKHBD356c1'356�FV9�35;�X<7H9�X@9�\]\VX<756c6#l59�=W^H\D6CZ1'_dX<7H6)^5:<1�^�1'CE6#;h356ie�X<7H6#1�:E?�4#1'K5\D;�`a6;56j9�\DX�ePBDX<7fBD3)9p4#\]9�CECEBD4j9�\,e�9?'Fq1'354#6�9o^%9�:EX<BD4#KH\]9�:V1�`HCE6i:EA'9�X<B]1�3f7%9�;)`�6#63f=r9�;H6�tI9p_]:<BD635;fBD35CEB]CMX<6#;fX<7H9�X�0�^HK(`H\]BDCE7@X<7H61�`HCE6i:EA'9�X<BD1'3aFI9�35;cBgX�`a6#49�=c6�={?)CE751�:EX<6#CMX29�:EX<B]4#\D6'm � _�4#1'K(:<CE6'FIBD3WCEK54}7cCEBgX<KH9�X<BD1'35C#FIBgX-B]CV`�6#XEX<6i:-X<1{9A'1'BD;c=W63qX<BD1'35BD35k3H9�=W6C#F 9�3H;~1'356fCE751'K5\D;�:<6=W6#=@`�6i:oX<7H9�Xp6#Aq6#3yX<756@`a6#CMXp=c9�X<756=c9�X<BD4B]9�35C|7H9A'6f=r9�;56f=WBDCMX[9�bq6#Cf�+0�e�9�CoX<1'\D;�X<7H9�X9)k�:<69�X�=r9�X<7H6#=c9�X<BD4B]9�3Y7H9�;>75BDC�6k�1c9f\]BgXEX<\D6o`5:<K5BDCE6;>9�_�X<6i:�^%K(`H\DBDCE7HBD35kW9f^�6#:<_�64iX<\g?rk'1,1,;�^5:<1,1'_�F(eP7563>756Z:<69�\DBD�6#;X<7H9�XP351W1�`,�E6#4#XPC<9�X<BDCMzH6#;y9�\]\�X<75687q?,^�1�X<756#CE6C�1'_�75BDC�X<7561�:<6#=>F5eP75BD4}7�X<756#:<6#_�1�:<6pe�9�C|9WJIK5BgX<6pKHCE6#\D6CEC�1'356��[m

�'Ô

Page 96: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

^5:<64#6#;HBD35kr±-:<1�^�1'CEBgX<BD1'3y7H9�Co_�1�\]\D1je26#;wX<756@C<9�=W6@CE4#63H9�:<BD1rBD3~9c351'35\]BD3569�:oCE6iXEX<B]35k5F%`HK5Xo1'356@4j9�3y;56;5K546@=W1�:<6{B]3�9\DBD3H69�:�CE6#XEX<BD35kf`I?�K5CEB]35k)X<7568CM^�6#4#XE:[9�\�:[9�;5BDKHC�1'_V9�3w1�^a6i:[9�X<1�:}º�m� & ����$ §�N�6#X ��`a6{9)\DBD3H69�:�=r9�^H^HBD35kc_]:<1'= � � Ó � v BD3qX<1cBgX<CE6#\D_�C<9�X<BDCE_�?(BD35k"� � � � � � � � � �dePBgX<7wCM^�6#4#XE:[9�\ :[9�;HBDK5C� � 9�35;�� � � � � v%� � v �2ePBgX<7wCM^a6#4iXE:[9�\�:[9�;5BDKHC � v �+eP756i:<6 � � 9�35; � v 9�:<6 ���T���²�³�CM^%9�4#6#C}�[m|¢�756#3�_�1�:p®� � �Ø�9�35;°��(+*#(-,�F5X<7568CM^�6#4#XE:[9�\a:[9�;5BDK5C ��� � � * �-1'_��x1'3�� � � � � v � � � � C<9�X<BDCMzH6C�X<756ZBD3H6JIKH9�\DBgXs? ��� � � *a� ( � v

"�� � � v m

�2È<ÆjÆ5#[§ � 3H6wK5CE6CWX<756w_+9�4iXWX<7%9�X ��� �Z�E� \DBD= � � � � � � � � v � � F�9�35;�X<7H6>_+9�4iXcX<7H9�X � � � � � � � � � � � � � / � � � �� � � � � � / � � � �

� (C � � � � � � v " �� � � � � � � � � � � � � � �� � � / � � / � ePBgX<79C�;56i^�635;5BD3Hk{K(^�1'3ceP75B]4[7r6jJqKHBgA�9�\D63qXd351�:<=!BDCuK5CE6;r_�1�:p� � � � � v � � � � �+`HK5Xu351�X1'3 � �[t(X[9�b(BD3HkfX<756p^a1e26#: v� 9�3H;>\]6iXEX<BD35k � X<6#3H;�X<1 , k'BgAq6#C2X<7H6p:<6#CEKH\gXm

0�_ @~BDC|751'\D1'=W1�:E^H7HBD4�F%X<756#3yX<756{CM^�64iXE:<K5= 1�_ @�� �Z��BDCPX<756@BD=c9�k'6Z`,? @~1�_VX<7H6{CM^�6#4#XE:<K5= 1'_��{F�9�35;�K5CEBD35kcX<7H6^5:<64#6#;HBD35k>\D6#=c=c9Y1'356W;56#;HK54#6CpX<7H9�X8B]_ � � B]CpX<756)CM^a6#4iXE:<KH= 1'_��B]3 � � F � v BDCZX<756WCM^�6#4#XE:<K5= 1'_��!BD3 � v F�9�35;� � � � BDC�X<7568CM^a6#4iXE:<KH=�1'_��xBD3�� � � � � v � � � � FHX<756#3w_�1�:|6iAq6#:E?Y7H1�\]1�=c1�:E^H75BD4Z_�K5354iX<B]1�3 @y1�3H687H9�CP=c9�l � �"! �

� �

� @����,��� (�S=c9�l � � ! � � @����,��� �[v

"� �S=c9�l � �"! / � @����,��� �

� my0Z1'3546Y9�CMbq6#;�¡uBg^5:<B]9�3°�-R�Q��VU � BD_�756rb(356ie�CE1�=c6YCEBgX<KH9�X<BD1'3�eP756i:<6cX<7H6CM^�64iXE:<K5= CMXE:<1�3Hk�\D?Y;56i^a6#35;5CPK(^�1'3>X<7568CM^%9�4#6ZKHCE6#;aFH`HK(X�02;HBD;>351�X|K535;56#:<CMX[9�3H;>7HBDCP9�35CMe26i:m

0{;51�351�Xfb(351je�9�3I?hk'1,1I;¶6il59�=)^%\D6r1'_P9�^5^%\DBD49�X<BD1'35Cf1'_�X<756#CE6Y:<6CEK5\gX<Cf1�`5X[9�BD356;°ePBgX<7¶��BD4}7H9�6#\P¡u����T�����¥,¥�FeP75BD4}7�BDC81'356W:<69�CE1'3�eP7I?~0p356#A'6#:8e�:<1�X<6)X<7H9�X8^5:<1,1'_d`a6#_�1�:<6'F�`HK(X80pzH35;°BD3qX<6#:<6#CMX<BD35k�X<756c_�9�4iX8X<7%9�X8ePBgX<7�:<6CM^�6#4#XX<1WBD3qX<6#:E^�1'\]9�X<BD1'3�9)=W1'351�X<1'35BD4BgXs?c^5:<1�^�6i:EXs?�BDCP9�\D=W1'CMX�\]Bgbq6Z9)N�Q A�U�²�³�Q���µ)41�3H;5BgX<BD1'3am��o4iX<K%9�\]\g?'F,X<7568\]9�CMX�BD356jJIKH9�\DBgXs?k'BgA(BD35kW9)\D1,4j9�\]BD�9�X<BD1'3Y1'_�X<7568CM^�6#4#XE:<K5=�CEK5k'k'6#CMX<C�X<7H9�XP1'35684#1'K5\D;>;H6iAq6#\D1�^w3H1�X<B]1�3HC�1�_�BD3qX<6#:E^�1'\]9�X<BD1'3Y1'_�CE6iX<C#m

�VBD3H9�\D\D?'F�0�e�9�3qX-X<1Z=c6#3qX<B]1�3c9o:<6#CEKH\gX�eP7HBD4[7)0V_�1'K535;r9p_�6ie�?q69�:<Cd9�k'1(FqeP75BD\D620�e�9�C�X<6j9�4[75BD3HkZ9Zk�:[9�;HKH9�X<6�41�K5:<CE61'3�=c9�X<756=c9�X<BD49�\u=c6iX<751,;5C)B]3�4#1'3qXE:<1'\SFd`�6#4j9�KHCE6�0@e�9�3IX<6#;�X<1�6il,^H\�9�B]3hX<7H6�_�1'\D\D1jePBD35k�:<6#CEKH\gXW1'_ �oA'6CW�����V��� � FeP75BD4}7>75687H9�;>KHCE6#;wBD3>41�3H356#4#X<BD1'3YePBgX<7y9f41'3qXE:<1'\a^5:<1�`H\D6= � m� & ����$ §ZN�6#X�O"��`�6c9>ÚP���-R�T�=c69�CEK(:<6c1�3 � 9�35;� � ®(mr��356#46#CEC<9�:E?�9�35;�CEK5�W4BD6#3IX{41�3H;5BgX<BD1'3�X<7H9�X8X<756i:<66il(BDCMX<Cr9�4#1'35CMX[9�3qX CW� �o�{CEKH4[7�X<7H9�X I ��� ��@�� �I��� º O"� ( C)� �o� I � � @��GQa��� º O Q�_�1�:Y9�\]\�_�K5354iX<B]1�35@ � � º � �@�{eP75BD4}7A�9�35B]CE7>1'K(X<CEBD;56c�O® � �o�dBDC�X<7%9�XPCEK5^ � � � �-�3! � � �{Ó�� #+� ( C�� �-,/m

�o_�X<6i:P\D1,1�b(BD3Hk)9�XP75BDC�^H:<1I1'_sF502_�1�KH35;>X<7H9�X�ePBgX<7�Aq6#:E?Y\]BgXEX<\D6o4[7%9�3Hk�681'35684#1'K5\D;>^5:<1jA'6ZX<7H6Z_�1�\]\D1jePBD35kfA�9�:<B]9�3qXm� & ����$ mVN�6iX)O ��`a689)ÚP����R�T¶=W6j9�CEK(:<6Z1'3 �fmu¢�756Z41�3H;5BgX<BD1'3>CEK(^

� � � �-�3! � � �8Ó*� #��&( C � � , BDC�6jJIK5BgA�9�\D63qX2X<1X<75686#l,B]CMX<6#3546Z1'_V9W4#1'35CMX[9�3qX C CEK54[7wX<7%9�X I � � ��@�� �I��� ºPO � (5C � I � � @��GQa��� º�O Q�� v � º � I � �E�uÓ Q�ºj��� @��GQa��� ºPOMQ�� v � º _�1�:9�\D\ _�K53H4iX<BD1'35C @yCEK54[7>X<7H9�XI��2�E�dÓ Q�ºj��� @��GQa��� ºPOMQ��-,�m

¦oCEBD3Hk/X<756h4j9�:[9�4#X<6i:<BDC<9�X<BD1'3x1'_)X<756hCM^�9�46*� � � � � v � v � º%� v 1'_cGI9�4jJIK56#CMLON�1�KHBDC~N�QSR�TVU�9�35;ØGI9'9�b�±d���%������F|BDX=W6j9�35CcX<7H9�XY1�3H6y4j9�3/:<6i^H\�9�46>X<756y:<BDk'7qXYCEBD;56y1'_pX<756yBD356jJIKH9�\DBgXs?¶`I?�X<7H6y351�:<= 1'_�@�B]3/9°41�:E:<6#CM^�1'35;5B]35k°BD3IX<6i:EL^�1'\]9�X<BD1'3*CM^%9�46�tf756i:<6� v BDC �uº�� �{�r_�1�:wX<756�N���¯��aU�¤���x=W6j9�CEK5:<6 OMQ�eP75BD\D6 � � BDC �uº~_�1�:wX<756�=c69�CEK(:<6��E�@ÓQ�º�4OMQ�Fd9�35;¶BgX)BDCf351�X);5Bg�c4#KH\gX@X<1�4}7H9�:[9�4iX<6i:<B]�#6~� � � � � v � v � º%� v 9�CfX<756YCM^%9�4#6�1'_P_�K53H4iX<BD1'35C @ � �uº�� �{�8CEK54[7�X<7H9�X��� v � I º � $ � � � $ º ��� / ­ ��� @��GQa��� º � v � º O Q � ,�mu�%1�:|^5:<1jA,B]35k)X<7H9�XPX<75684#1'35;HBgX<BD1'3>BDCP3H6#4#6CEC<9�:E?'F �oAq6#CP�����V���h41�3HCEBD;(L6i:<C{9�_�K5354iX<B]1�3 � � �uº�� �{�PeP751'CE6W�-R����Q�����XE:[9�35CE_�1�:<= ;51,6#C8351�XZA�9�35B]CE7�1�3��O® � ���@�SBD_ � � 4#1'3qAq6i:<k'6#CZX<1�� � X<7563� � � 41'3qAq6i:<k'6#C�X<1>���[FH9�35;w9�^5^H\DBD6C2X<756ZBD356JIKH9�\DBDXs?WX<1�@y;56izH3H6#;�`I? @��GQa�H� (jº 7 � � �d�GQa�dCE1)X<7H9�X�_�1�:�� � ! � � �ZÓ��$#º�0�_ � � � � � � �f��_�1�:29� ���T���²a³�CM^�9�46 �WFqX<756PCM^�64iXE:<K5=!1�_ �*BDC-X<756P351'356=)^5Xs?)4\D1'CE6#;cCE6iXu1'_ � � C�_�1�:deP75BD4}7

� >� �rBDC�351�X|BD3qAq6i:EX<Bg`%\D6�F59�35;>X<756ZCM^a6#4iXE:[9�\a:[9�;5BDK5C ��� �Z�uBDC�X<7568=c9�l(BD=@K5=�1�_)� � �q_�1�: �wBD3�X<7H6ZCM^�64iXE:<K5= 1�_��{m� ¡uBg^H:<B]9�3��-R�Q��VU'F ÚPK5=c9�35B�9�35LS`�1�:<3~=c9�X<756=c9�X<BD4B]9�3amZno6@e�9�CZ={?�4#1'\D\]69�k'K56@BD3 � :<C<9?~BD3����( �q¨ (��,m@no6@e21�:Eb(C

9�X�0�35;5B]9�3H9)¦o35BDA'6#:<CEBgXs?'F& �\D1,1'=WB]35k�X<1�3 m� �oAq6#C-�2m'���������-F'�5:<6354[7c=c9�X<756=c9�X<BD4#B�9�3 F�`�1�:<3)B]3>�j� �'�,mVno6�e�9�C-={?f41�\]\D69�k'K56�BD3 � :<C<9#?)_]:<1'= ���(�Ô|X<1W�j��(��Im

no6pe21�:Eb(C|9�XPÙd4#1'\D6Z�o1�:<=r9�\]6{�IK(^%"6#:<BD6#K5:<6�F�¡29�4}7H9�3aFH�H:[9�3H4#6�m� ¢�756wX<BgX<\D6>1'_o75B]Cr9�:EX<B]4#\D6>=W63qX<BD1'356#;�X<756w4#1'3qXE:<1'\P1'_o;56_�1�:<=r9�`%\D6wCMXE:<K54iX<K(:<6CcBD3�CM^%9�4#6�F�`HK(Xr1'35\g?�41'3qX[9�BD356#;

9w:<6CEK5\gXf1'_�4#1'3qXE:<1'\u_�1�:@X<756rCE4j9�\]9�:{e�9Aq6Y6JIKH9�X<BD1'3aF�9�\gX<7H1�KHk�7¶9�\DBgXEX<\D6cBD;569�\DB]CMX<BD4�FV9�C{X<7H6r4#1'3qXE:<1'\de�9�C)9�^5^H\DBD6;h9�X9w^�1'BD3IX@B]35CEBD;56cX<7H6r;51'=c9�BD3amy08k'K56CEC@X<7H9�X@GI9�4jJqKH6#CMLON�1'K5B]C@N�QSR�T�UY7H9�;hKH35;56i:<CMX<1,1,;hX<7H9�X{X<7H6r4#1'3qXE:<1'\u1'_'%H6il(Bg`%\D6CMXE:<K54#X<K(:<6#CuBD3rCM^%9�4#6oBDCu9�3rB]=)^�1�:EX[9�3qX2JIK56#CMX<BD1'3aFI`HK(Xu`a6#49�K5CE6|Ùd\]9�CMX<BD4BgXs?fePBgX<7c\]9�:<k'6|;5BDCM^H\]9�46#=W63qXuBDCu98X<1,1{;5BD�W4K5\gXCEK(`(�M64iXF�9�3H;w6iAq63>X<756{\DBD356j9�:<BD�#6;YAq6i:<CEB]1�3y1'_�Ùd\]9�CMX<BD4#BgXs?�BDC|9W4#1'=W^H\DBD4j9�X<6;�7I?I^a6i:E`�1'\DBD48CM?(CMX<6#=>F%756{7H9�;wCMX[9�:EX<6;>`I?4#1'35CEB];56i:<BD3Hk)JIK56CMX<BD1'35C2:<6#\]9�X<6#;YX<1)9)CE49�\]9�:�e�9#Aq6Z6JIKH9�X<BD1'3aF(`HK(XP756Z7H9�;Y^5:<1�`%9�`H\D?r_�1�:<k�1�XEX<63rX<1f^�1'BD3IX�1'K(X�751je�_�9�:X<756CE68JIK56#CMX<B]1�3HC�:<69�\D\g?�9�:<68_]:<1'=�4#1'3qXE:<1'\D\DB]35kf\�9�:<k'6Z;56_�1�:<=c9�`H\]6pCMXE:<KH4iX<K(:<6C#m

�'Ò

Page 97: Mathematical Sciences - Mellon College of Science at CMU ...Int.pdf · 9zce751 :exubd3qxe:

1'356{7H9�CW� ��@�� �I��� �6� =cBD3� � � � � v � � � �u� �H��� m�¢�7H9�XPX<756{4#1'35;5BDX<BD1'3>BDCPCEK(�c4BD6#3qXoBDCo9W4#1'35CE6JIK56354#6{1'_�X<7568_�1'\D\D1jePBD35k

\D6=W=c9(m� & ����$ §P¢�756#:<6@6il(BDCMX<Cp9Y4#1'35CMX[9�3qX�C�� ��CEK54}7�X<7H9�X �

� � � CEK(^ � � ��.� ���v � � ��@�� �q��� º (�C�� � � I � � @��GQa��� ºPO Q � v � º � I � �E��Ó

Q º ��� @��GQa��� º OMQ � v � º _�1�:|9�\D\ _�KH354iX<BD1'35C @ � � º � �{�uCEK54}7>X<7H9�X I �u�E�uÓ!Q º ��� @��GQa��� º OMQ �-,�m�2È<ÆjÆ5#[§2N�6#X N

�� � I � � � � ��v � � ��@�� �q��� º O � � v � º 9�35; 2 � � � I � �.� ���v � �� �

� � � � ���� �� º O � � v � º t�X<756@K5CEKH9�\�^H:<1I1'_�1�_�41'3qX<BD3IK5BDXs?>1�__�K5354iX<B]1�3HC21'_ X<756{�(RV¯ R�¥,�H�hCM^%9�46; v � �O® � �����pk'BgAq6#C2CEK(^ � � �

� � � ��v � � ��@�� �I��� º ( � º N��GN º�Ó 2 º� � v

� º FH9�35;�CEK5=W=cBD35kfBD3�h9�3H;hK5CEB]35k°¡u� V²a³ ��L��(²�³�´8����µ�k�BDA'6C �

� � � CEK5^ � � �� � � ��v � � ��@�� �q��� º ( ��º � I ��� ��@�� �I��� ºPO � � v � º � I �u� � ��@�� �I��� ºPÓ�� � � � � � ���� �� º O � � v � º F5eP75BD4}7>BDC�6#CECE63qX<B]9�\D\g?rX<7568;56#CEBg:<6;�:<6#CEKH\gXm

1 BgX<7�X<756):<6=c9�:Eb�4#1'354#6#:<35BD35kwBD3IX<6i:E^�1'\]9�X<BD1'3aF�9�35;°6il(4[7H9�35k'BD3Hk�X<7H6):<1'\D6#C81'_)@¶9�35; ��@ F�X<75BDC8\]6#=W=r9YC<9?(C@9\DBgXEX<\]6u=W1�:<62X<7H9�XVX<756�_�9�4iX�X<7H9�XVX<756�_�KH354iX<BD1'35C�_]:<1'= � ;°vj� �{� � �uº�� �{��� v � º �¹v 9�:<6�4#1'3qX<BD3,K51'K5C-9�35;)X<6#35;fX<18®p9�X&,�FI9�C�BDXk'BgAq6#C�CE1'=W6o^5:<6#4BDCE6oe�9?Y751je X<756p_�K5354#X<BD1'35C2X<6#35;YX<1)®W9�X ,/FH9�C�BDX�BD=)^%\DBD6#CdX<7%9�X � � � � CEK(^ � � � � � � ��v � � � �GQa��� º �-,/m0�3°X<756c49�CE6r1�_�_�K53H4iX<BD1'35C8ePBgX<7°CEK(^5^�1�:EX@B]3��O® � �o�[F�={?�A�9�:<B]9�3qX@k'BgAq6#C8X<756cC<9�=W6ck�:<1je�X<7°BD3/�PÓ �¸_�1'K53H;�`I?�oAq6#C����������-F,eP751)351�X<BD46#CuX<7H9�X2X<7H6pk�:<1e�X<7�B]C21�^5X<BD=c9�\�_�1�:�\]9�:<k�6oA�9�\DKH6#C21'_ �ZF(`�6#4j9�K5CE6pBD_B@wBDC2X<756o4}7H9�:[9�4iX<6i:<B]CMX<BD4_�K5354iX<B]1�3�1�_{�O® � �o�[F�X<7563 I ��� ��@�� �I��� ºPO � � 4 �Zm�0�3¶X<756�49�CE6�1'_ �"!�1'356�49�3�69�CEBD\g?°k'6#356#:[9�\DBD�#6rX<756Y^5:<1,1'_P9�35;4#1'35CEB];56i:ZÚ|���-R�T�=c69�CEK(:<6#C O"� x®Y_�1�:peP75B]4[7�X<756i:<6)6il(BDCMX<CZ9Y4#1'35CMX[9�3qX�C!CEKH4[7~X<7H9�X �-���{� ((C!_�1�:Z6iAq6i:E?�4#K(`a61'_-CEBD�6Y��F 9�35;~4#1'35CEB];56i:Z_�KH354iX<BD1'35CR@�CEKH4[7~X<7H9�X �

�� v � I º � $ � � � $ º ��� / ­ ��! � @��GQa��� º � v � º O Q�� ,�t _�1�:Z_�K53H4iX<BD1'35C|ePBgX<7

CEK(^5^a1�:EX�BD3>9{`a1�KH35;56#;�CE6#X �°F51'356p1�`5X[9�BD35C�9f4#1'35CMX[9�3qX�k�:<1jePBD35k)\DBgbq6 � �VÓ O&��N4 ( % (�L(������� ! F(`HK(X�BgX�BDC2351�X�4#\D6j9�:uX<1=W68BD_�X<7568;5B]9�=W6#X<6i:�1'_ �»BDC�X<75684#1�:E:<6#4#XPk'6#1'=W6#XE:<BD4ZJIKH9�3qX<BgXs?rX<1WK5CE68B]3�CEK54}7y9�3>BD3H6JIKH9�\DBgXs?qm

��(