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ESslashdefESslashbfeESleftbrackESleftbrackESslashbf eESrightbrackESrightbrack ESslashdefESslashbffESleftbrackESleftbrackESslashbf fESrightbrackESrightbrack ESslashdefESslashbfgESleftbrackESleftbrackESslashbf gESrightbrackESrightbrack ESslashdefESslashbfhESleftbrackESleftbrackESslashbf hESrightbrackESrightbrack ESslashdefESslashbfiESleftbrackESleftbrackESslashbf iESrightbrackESrightbrack ESslashdefESslashbfjESleftbrackESleftbrackESslashbf jESrightbrackESrightbrack ESslashdefESslashbfkESleftbrackESleftbrackESslashbf kESrightbrackESrightbrack ESslashdefESslashbflESleftbrackESleftbrackESslashbf lESrightbrackESrightbrack ESslashdefESslashbfmESleftbrackESleftbrackESslashbf mESrightbrackESrightbrack ESslashdefESslashbfnESleftbrackESleftbrackESslashbf nESrightbrackESrightbrack ESslashdefESslashbfoESleftbrackESleftbrackESslashbf oESrightbrackESrightbrack ESslashdefESslashbfpESleftbrackESleftbrackESslashbf pESrightbrackESrightbrack ESslashdefESslashbfqESleftbrackESleftbrackESslashbf qESrightbrackESrightbrack ESslashdefESslashbfrESleftbrackESleftbrackESslashbf rESrightbrackESrightbrack ESslashdefESslashbfsESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashdefESslashbftESleftbrackESleftbrackESslashbf tESrightbrackESrightbrack ESslashdefESslashbfuESleftbrackESleftbrackESslashbf uESrightbrackESrightbrack ESslashdefESslashbfvESleftbrackESleftbrackESslashbf vESrightbrackESrightbrack ESslashdefESslashbfwESleftbrackESleftbrackESslashbf wESrightbrackESrightbrack ESslashdefESslashbfxESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashdefESslashbfyESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashdefESslashbfzESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashdefESslashbfAESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashdefESslashbfBESleftbrackESleftbrackESslashbf BESrightbrackESrightbrack ESslashdefESslashbfCESleftbrackESleftbrackESslashbf CESrightbrackESrightbrack ESslashdefESslashbfDESleftbrackESleftbrackESslashbf DESrightbrackESrightbrack ESslashdefESslashbfEESleftbrackESleftbrackESslashbf EESrightbrackESrightbrack ESslashdefESslashbfFESleftbrackESleftbrackESslashbf FESrightbrackESrightbrack ESslashdefESslashbfGESleftbrackESleftbrackESslashbf GESrightbrackESrightbrack ESslashdefESslashbfHESleftbrackESleftbrackESslashbf HESrightbrackESrightbrack ESslashdefESslashbfIESleftbrackESleftbrackESslashbf IESrightbrackESrightbrack ESslashdefESslashbfJESleftbrackESleftbrackESslashbf JESrightbrackESrightbrack ESslashdefESslashbfKESleftbrackESleftbrackESslashbf KESrightbrackESrightbrack ESslashdefESslashbfLESleftbrackESleftbrackESslashbf LESrightbrackESrightbrack ESslashdefESslashbfMESleftbrackESleftbrackESslashbf MESrightbrackESrightbrack ESslashdefESslashbfNESleftbrackESleftbrackESslashbf NESrightbrackESrightbrack ESslashdefESslashbfOESleftbrackESleftbrackESslashbf OESrightbrackESrightbrack ESslashdefESslashbfPESleftbrackESleftbrackESslashbf PESrightbrackESrightbrack ESslashdefESslashbfQESleftbrackESleftbrackESslashbf QESrightbrackESrightbrack ESslashdefESslashbfRESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashdefESslashbfSESleftbrackESleftbrackESslashbf SESrightbrackESrightbrack ESslashdefESslashbfTESleftbrackESleftbrackESslashbf TESrightbrackESrightbrack ESslashdefESslashbfUESleftbrackESleftbrackESslashbf UESrightbrackESrightbrack ESslashdefESslashbfVESleftbrackESleftbrackESslashbf VESrightbrackESrightbrack ESslashdefESslashbfWESleftbrackESleftbrackESslashbf WESrightbrackESrightbrack ESslashdefESslashbfXESleftbrackESleftbrackESslashbf XESrightbrackESrightbrack ESslashdefESslashbfYESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashdefESslashbfZESleftbrackESleftbrackESslashbf ZESrightbrackESrightbrack ESslashdefESslashcalAESleftbrackESleftbrackESslashcal AESrightbrackESrightbrack ESslashdefESslashcalBESleftbrackESleftbrackESslashcal BESrightbrackESrightbrack ESslashdefESslashcalCESleftbrackESleftbrackESslashcal CESrightbrackESrightbrack ESslashdefESslashcalDESleftbrackESleftbrackESslashcal DESrightbrackESrightbrack ESslashdefESslashcalEESleftbrackESleftbrackESslashcal EESrightbrackESrightbrack ESslashdefESslashcalFESleftbrackESleftbrackESslashcal FESrightbrackESrightbrack ESslashdefESslashcalGESleftbrackESleftbrackESslashcal GESrightbrackESrightbrack ESslashdefESslashcalHESleftbrackESleftbrackESslashcal HESrightbrackESrightbrack ESslashdefESslashcalIESleftbrackESleftbrackESslashcal IESrightbrackESrightbrack ESslashdefESslashcalJESleftbrackESleftbrackESslashcal JESrightbrackESrightbrack ESslashdefESslashcalKESleftbrackESleftbrackESslashcal KESrightbrackESrightbrack ESslashdefESslashcalLESleftbrackESleftbrackESslashcal LESrightbrackESrightbrack ESslashdefESslashcalMESleftbrackESleftbrackESslashcal MESrightbrackESrightbrack ESslashdefESslashcalNESleftbrackESleftbrackESslashcal NESrightbrackESrightbrack ESslashdefESslashcalOESleftbrackESleftbrackESslashcal OESrightbrackESrightbrack ESslashdefESslashcalPESleftbrackESleftbrackESslashcal PESrightbrackESrightbrack ESslashdefESslashcalQESleftbrackESleftbrackESslashcal QESrightbrackESrightbrack ESslashdefESslashcalRESleftbrackESleftbrackESslashcal RESrightbrackESrightbrack ESslashdefESslashcalSESleftbrackESleftbrackESslashcal SESrightbrackESrightbrack ESslashdefESslashcalTESleftbrackESleftbrackESslashcal TESrightbrackESrightbrack ESslashdefESslashcalUESleftbrackESleftbrackESslashcal UESrightbrackESrightbrack ESslashdefESslashcalVESleftbrackESleftbrackESslashcal VESrightbrackESrightbrack ESslashdefESslashcalWESleftbrackESleftbrackESslashcal WESrightbrackESrightbrack ESslashdefESslashcalXESleftbrackESleftbrackESslashcal XESrightbrackESrightbrack ESslashdefESslashcalYESleftbrackESleftbrackESslashcal YESrightbrackESrightbrack ESslashdefESslashcalZESleftbrackESleftbrackESslashcal ZESrightbrackESrightbrack ESslashdefESslashhatxESleftbrackESslashhatESleftbrackxESrightbrackESrightbrack ESslashdefESslashhatXESleftbrackESslashhatESleftbrackXESrightbrackESrightbrack ESslashdefESslashhatyESleftbrackESslashhatESleftbrackyESrightbrackESrightbrack ESslashdefESslashhatYESleftbrackESslashhatESleftbrackYESrightbrackESrightbrack ESslashdefESslashhatWESleftbrackESslashhatESleftbrackWESrightbrackESrightbrack ESslashdefESslashtKESleftbrackESslashtildeESleftbrackKESrightbrackESrightbrack ESslashdefESslashtXESleftbrackESslashtildeESleftbrackXESrightbrackESrightbrack ESslashdefESslashtYESleftbrackESslashtildeESleftbrackYESrightbrackESrightbrack ESslashdefESslashhbfhESleftbrackESslashhatESleftbrackESslashbfhESrightbrackESrightbrack ESslashdefESslashbfalphaESleftbrackESslashmboxESleftbrackESslashboldmathESdollarESslashalphaESdollarESrightbrackESrightbrack ESslashdefESslashbfbetaESleftbrackESslashmboxESleftbrackESslashboldmathESdollarESslashbetaESdollarESrightbrackESrightbrack ESslashdefESslashtildefESleftbrackESslashtildeESleftbrackfESrightbrackESrightbrack ESslashdefESslashtildeKESleftbrackESslashtildeESleftbrackKESrightbrackESrightbrack ESslashdefESslashscIESleftbrackESslashmathfrakESleftbrackIESrightbrackESrightbrackESannopend) /EscapedPreamble 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/Subject (ESannotESslashnoindent ESdollarX, YESdollar, and ESdollarZESdollar are mutually independent, denoted ``ESdollarESslashperp (X, Y, Z)ESdollar": ESslash[ H(X,Y, Z) = H(X) + H(Y) + H(Z). 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eESrightbrackESrightbrack ESslashdefESslashbffESleftbrackESleftbrackESslashbf fESrightbrackESrightbrack ESslashdefESslashbfgESleftbrackESleftbrackESslashbf gESrightbrackESrightbrack ESslashdefESslashbfhESleftbrackESleftbrackESslashbf hESrightbrackESrightbrack ESslashdefESslashbfiESleftbrackESleftbrackESslashbf iESrightbrackESrightbrack ESslashdefESslashbfjESleftbrackESleftbrackESslashbf jESrightbrackESrightbrack ESslashdefESslashbfkESleftbrackESleftbrackESslashbf kESrightbrackESrightbrack ESslashdefESslashbflESleftbrackESleftbrackESslashbf lESrightbrackESrightbrack ESslashdefESslashbfmESleftbrackESleftbrackESslashbf mESrightbrackESrightbrack ESslashdefESslashbfnESleftbrackESleftbrackESslashbf nESrightbrackESrightbrack ESslashdefESslashbfoESleftbrackESleftbrackESslashbf oESrightbrackESrightbrack ESslashdefESslashbfpESleftbrackESleftbrackESslashbf pESrightbrackESrightbrack ESslashdefESslashbfqESleftbrackESleftbrackESslashbf qESrightbrackESrightbrack ESslashdefESslashbfrESleftbrackESleftbrackESslashbf rESrightbrackESrightbrack ESslashdefESslashbfsESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashdefESslashbftESleftbrackESleftbrackESslashbf tESrightbrackESrightbrack ESslashdefESslashbfuESleftbrackESleftbrackESslashbf uESrightbrackESrightbrack ESslashdefESslashbfvESleftbrackESleftbrackESslashbf vESrightbrackESrightbrack ESslashdefESslashbfwESleftbrackESleftbrackESslashbf wESrightbrackESrightbrack ESslashdefESslashbfxESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashdefESslashbfyESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashdefESslashbfzESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashdefESslashbfAESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashdefESslashbfBESleftbrackESleftbrackESslashbf BESrightbrackESrightbrack ESslashdefESslashbfCESleftbrackESleftbrackESslashbf CESrightbrackESrightbrack ESslashdefESslashbfDESleftbrackESleftbrackESslashbf DESrightbrackESrightbrack ESslashdefESslashbfEESleftbrackESleftbrackESslashbf EESrightbrackESrightbrack ESslashdefESslashbfFESleftbrackESleftbrackESslashbf FESrightbrackESrightbrack ESslashdefESslashbfGESleftbrackESleftbrackESslashbf GESrightbrackESrightbrack ESslashdefESslashbfHESleftbrackESleftbrackESslashbf HESrightbrackESrightbrack ESslashdefESslashbfIESleftbrackESleftbrackESslashbf IESrightbrackESrightbrack ESslashdefESslashbfJESleftbrackESleftbrackESslashbf JESrightbrackESrightbrack ESslashdefESslashbfKESleftbrackESleftbrackESslashbf KESrightbrackESrightbrack ESslashdefESslashbfLESleftbrackESleftbrackESslashbf LESrightbrackESrightbrack ESslashdefESslashbfMESleftbrackESleftbrackESslashbf MESrightbrackESrightbrack ESslashdefESslashbfNESleftbrackESleftbrackESslashbf NESrightbrackESrightbrack ESslashdefESslashbfOESleftbrackESleftbrackESslashbf OESrightbrackESrightbrack ESslashdefESslashbfPESleftbrackESleftbrackESslashbf PESrightbrackESrightbrack ESslashdefESslashbfQESleftbrackESleftbrackESslashbf QESrightbrackESrightbrack ESslashdefESslashbfRESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashdefESslashbfSESleftbrackESleftbrackESslashbf SESrightbrackESrightbrack ESslashdefESslashbfTESleftbrackESleftbrackESslashbf TESrightbrackESrightbrack ESslashdefESslashbfUESleftbrackESleftbrackESslashbf UESrightbrackESrightbrack ESslashdefESslashbfVESleftbrackESleftbrackESslashbf VESrightbrackESrightbrack ESslashdefESslashbfWESleftbrackESleftbrackESslashbf WESrightbrackESrightbrack ESslashdefESslashbfXESleftbrackESleftbrackESslashbf XESrightbrackESrightbrack ESslashdefESslashbfYESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashdefESslashbfZESleftbrackESleftbrackESslashbf ZESrightbrackESrightbrack ESslashdefESslashcalAESleftbrackESleftbrackESslashcal AESrightbrackESrightbrack ESslashdefESslashcalBESleftbrackESleftbrackESslashcal BESrightbrackESrightbrack ESslashdefESslashcalCESleftbrackESleftbrackESslashcal CESrightbrackESrightbrack ESslashdefESslashcalDESleftbrackESleftbrackESslashcal DESrightbrackESrightbrack ESslashdefESslashcalEESleftbrackESleftbrackESslashcal EESrightbrackESrightbrack ESslashdefESslashcalFESleftbrackESleftbrackESslashcal FESrightbrackESrightbrack ESslashdefESslashcalGESleftbrackESleftbrackESslashcal GESrightbrackESrightbrack ESslashdefESslashcalHESleftbrackESleftbrackESslashcal HESrightbrackESrightbrack ESslashdefESslashcalIESleftbrackESleftbrackESslashcal IESrightbrackESrightbrack ESslashdefESslashcalJESleftbrackESleftbrackESslashcal JESrightbrackESrightbrack ESslashdefESslashcalKESleftbrackESleftbrackESslashcal KESrightbrackESrightbrack ESslashdefESslashcalLESleftbrackESleftbrackESslashcal LESrightbrackESrightbrack ESslashdefESslashcalMESleftbrackESleftbrackESslashcal MESrightbrackESrightbrack ESslashdefESslashcalNESleftbrackESleftbrackESslashcal NESrightbrackESrightbrack ESslashdefESslashcalOESleftbrackESleftbrackESslashcal OESrightbrackESrightbrack ESslashdefESslashcalPESleftbrackESleftbrackESslashcal PESrightbrackESrightbrack ESslashdefESslashcalQESleftbrackESleftbrackESslashcal QESrightbrackESrightbrack ESslashdefESslashcalRESleftbrackESleftbrackESslashcal RESrightbrackESrightbrack ESslashdefESslashcalSESleftbrackESleftbrackESslashcal SESrightbrackESrightbrack ESslashdefESslashcalTESleftbrackESleftbrackESslashcal TESrightbrackESrightbrack ESslashdefESslashcalUESleftbrackESleftbrackESslashcal UESrightbrackESrightbrack ESslashdefESslashcalVESleftbrackESleftbrackESslashcal VESrightbrackESrightbrack ESslashdefESslashcalWESleftbrackESleftbrackESslashcal WESrightbrackESrightbrack ESslashdefESslashcalXESleftbrackESleftbrackESslashcal XESrightbrackESrightbrack ESslashdefESslashcalYESleftbrackESleftbrackESslashcal YESrightbrackESrightbrack ESslashdefESslashcalZESleftbrackESleftbrackESslashcal ZESrightbrackESrightbrack ESslashdefESslashhatxESleftbrackESslashhatESleftbrackxESrightbrackESrightbrack ESslashdefESslashhatXESleftbrackESslashhatESleftbrackXESrightbrackESrightbrack ESslashdefESslashhatyESleftbrackESslashhatESleftbrackyESrightbrackESrightbrack ESslashdefESslashhatYESleftbrackESslashhatESleftbrackYESrightbrackESrightbrack ESslashdefESslashhatWESleftbrackESslashhatESleftbrackWESrightbrackESrightbrack ESslashdefESslashtKESleftbrackESslashtildeESleftbrackKESrightbrackESrightbrack ESslashdefESslashtXESleftbrackESslashtildeESleftbrackXESrightbrackESrightbrack ESslashdefESslashtYESleftbrackESslashtildeESleftbrackYESrightbrackESrightbrack ESslashdefESslashhbfhESleftbrackESslashhatESleftbrackESslashbfhESrightbrackESrightbrack ESslashdefESslashbfalphaESleftbrackESslashmboxESleftbrackESslashboldmathESdollarESslashalphaESdollarESrightbrackESrightbrack ESslashdefESslashbfbetaESleftbrackESslashmboxESleftbrackESslashboldmathESdollarESslashbetaESdollarESrightbrackESrightbrack ESslashdefESslashtildefESleftbrackESslashtildeESleftbrackfESrightbrackESrightbrack ESslashdefESslashtildeKESleftbrackESslashtildeESleftbrackKESrightbrackESrightbrack ESslashdefESslashscIESleftbrackESslashmathfrakESleftbrackIESrightbrackESrightbrackESannopend) /EscapedPreamble 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zpg_XQKFAǿ=ȼ:ɹ8ʷ6˶5̵5͵6ζ7ϸ9к<Ѿ?DINU\dlvۀ܊ݖޢ)߯6DScs 2F[p(@Xr4Pm8Ww)KmLExifMM*i<-C      C  -<" }!1AQa"q2#BR$3br %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz w!1AQaq"2B #3Rbr $4%&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz ? mI?g=m>eIj"ݴN VOjYaʤJz/R4>7|,M+ xlOi3 6 lt4׍[ <)fs* Vb zTԤUKTj-|ЅU<]#WoiZa-=~ȸ'q\g޼']3fxVR)/%UxE ''xcjBMM$$ Ntv} EDžxO[?h&3E4 Qg$`{bڼOMoZ6K'۸uuF7!"Mo R0EǣJ>'R噖grMZZj]c2V5bֺ=oٮgMQ_{S- e֒Jay S$,c</)#IHcY?𦴏_$P%wAgiwZu45(4SH?zO#;זe}7S^jvWsV^-&VgX*oH2+Mi*5?HΑ?+£S ? |A6sx^ǦY&nyddE7++cܿMi*5?HΑ?+£S?Mi*5?H O U_;./td7VI2;)/\ Oq/27Y9'*] ]]PKqjEg=thumbs/mt0-16.jpgJFIF XICC_PROFILE HLinomntrRGB XYZ  1acspMSFTIEC sRGB-HP cprtP3desclwtptbkptrXYZgXYZ,bXYZ@dmndTpdmddvuedLview$lumimeas $tech0 rTRC< gTRC< bTRC< textCopyright (c) 1998 Hewlett-Packard CompanydescsRGB IEC61966-2.1sRGB IEC61966-2.1XYZ QXYZ XYZ o8XYZ bXYZ $descIEC http://www.iec.chIEC http://www.iec.chdesc.IEC 61966-2.1 Default RGB colour space - sRGB.IEC 61966-2.1 Default RGB colour space - sRGBdesc,Reference Viewing Condition in IEC61966-2.1,Reference Viewing Condition in IEC61966-2.1view_. \XYZ L VPWmeassig CRT curv #(-27;@EJOTY^chmrw| %+28>ELRY`gnu| &/8AKT]gqz !-8COZfr~ -;HUcq~ +:IXgw'7HYj{+=Oat 2FZn  % : O d y  ' = T j " 9 Q i  * C \ u & @ Z t .Id %A^z &Ca~1Om&Ed#Cc'Ij4Vx&IlAe@e Ek*Qw;c*R{Gp@j>i  A l !!H!u!!!"'"U"""# #8#f###$$M$|$$% %8%h%%%&'&W&&&''I'z''( (?(q(())8)k))**5*h**++6+i++,,9,n,,- -A-v--..L.../$/Z///050l0011J1112*2c223 3F3334+4e4455M555676r667$7`7788P8899B999:6:t::;-;k;;<' >`>>?!?a??@#@d@@A)AjAAB0BrBBC:C}CDDGDDEEUEEF"FgFFG5G{GHHKHHIIcIIJ7J}JK KSKKL*LrLMMJMMN%NnNOOIOOP'PqPQQPQQR1R|RSS_SSTBTTU(UuUVV\VVWDWWX/X}XYYiYZZVZZ[E[[\5\\]']x]^^l^__a_``W``aOaabIbbcCccd@dde=eef=ffg=ggh?hhiCiijHjjkOkklWlmm`mnnknooxop+ppq:qqrKrss]sttptu(uuv>vvwVwxxnxy*yyzFz{{c{|!||}A}~~b~#G k͂0WGrׇ;iΉ3dʋ0cʍ1fΏ6n֑?zM _ɖ4 uL$h՛BdҞ@iءG&vVǥ8nRĩ7u\ЭD-u`ֲK³8%yhYѹJº;.! 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zpg_XQKFAǿ=ȼ:ɹ8ʷ6˶5̵5͵6ζ7ϸ9к<Ѿ?DINU\dlvۀ܊ݖޢ)߯6DScs 2F[p(@Xr4Pm8Ww)KmLExifMM*i<-C      C  -<" }!1AQa"q2#BR$3br %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz w!1AQaq"2B #3Rbr $4%&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz ?ju]W|:WcIQl/$ d4lg5"|_[Cnksp?KEd!q~3ge] $*xp~eǭs^|AգԵt,*谨OGÿѰװA2ڱ%b<-wr9O nM{!me[\&cwFsq N 5𦴏_$V'SKÁ5s'O64"IGۏ31@Q@1 닯pgL2 6YNLl_KxkTo_\e]R;}v!AeTPEPEPPKqjE0Ythumbs/mt0-11.jpgJFIF XICC_PROFILE HLinomntrRGB XYZ  1acspMSFTIEC sRGB-HP cprtP3desclwtptbkptrXYZgXYZ,bXYZ@dmndTpdmddvuedLview$lumimeas $tech0 rTRC< gTRC< bTRC< textCopyright (c) 1998 Hewlett-Packard CompanydescsRGB IEC61966-2.1sRGB IEC61966-2.1XYZ QXYZ XYZ o8XYZ bXYZ $descIEC http://www.iec.chIEC http://www.iec.chdesc.IEC 61966-2.1 Default RGB colour space - sRGB.IEC 61966-2.1 Default RGB colour space - sRGBdesc,Reference Viewing Condition in IEC61966-2.1,Reference Viewing Condition in IEC61966-2.1view_. \XYZ L VPWmeassig CRT curv #(-27;@EJOTY^chmrw| %+28>ELRY`gnu| &/8AKT]gqz !-8COZfr~ -;HUcq~ +:IXgw'7HYj{+=Oat 2FZn  % : O d y  ' = T j " 9 Q i  * C \ u & @ Z t .Id %A^z &Ca~1Om&Ed#Cc'Ij4Vx&IlAe@e Ek*Qw;c*R{Gp@j>i  A l !!H!u!!!"'"U"""# #8#f###$$M$|$$% %8%h%%%&'&W&&&''I'z''( (?(q(())8)k))**5*h**++6+i++,,9,n,,- -A-v--..L.../$/Z///050l0011J1112*2c223 3F3334+4e4455M555676r667$7`7788P8899B999:6:t::;-;k;;<' >`>>?!?a??@#@d@@A)AjAAB0BrBBC:C}CDDGDDEEUEEF"FgFFG5G{GHHKHHIIcIIJ7J}JK KSKKL*LrLMMJMMN%NnNOOIOOP'PqPQQPQQR1R|RSS_SSTBTTU(UuUVV\VVWDWWX/X}XYYiYZZVZZ[E[[\5\\]']x]^^l^__a_``W``aOaabIbbcCccd@dde=eef=ffg=ggh?hhiCiijHjjkOkklWlmm`mnnknooxop+ppq:qqrKrss]sttptu(uuv>vvwVwxxnxy*yyzFz{{c{|!||}A}~~b~#G k͂0WGrׇ;iΉ3dʋ0cʍ1fΏ6n֑?zM _ɖ4 uL$h՛BdҞ@iءG&vVǥ8nRĩ7u\ЭD-u`ֲK³8%yhYѹJº;.! 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zpg_XQKFAǿ=ȼ:ɹ8ʷ6˶5̵5͵6ζ7ϸ9к<Ѿ?DINU\dlvۀ܊ݖޢ)߯6DScs 2F[p(@Xr4Pm8Ww)KmLExifMM*i<-C      C  -<" }!1AQa"q2#BR$3br %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz w!1AQaq"2B #3Rbr $4%&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz ?'tإ׋=?JAyp#9 pq}+s >"?){.^Ÿ7뫢9O^Ÿ7){.(V]3ŞbHI]UCHO(wGӝuto8όǦM弛7i*,8'w NU|Fu]VOu&ho.%F"E!vEE|) P]<+XaGRK.mC~6$+k|?sxWu;-jZ~!$Ȅ?X$ лO(?3;-k2fah1ڀ24 GXM4ZȩSX͚$D@k((((PKqjE-thumbs/st6-2.jpgJFIF XICC_PROFILE HLinomntrRGB XYZ  1acspMSFTIEC sRGB-HP cprtP3desclwtptbkptrXYZgXYZ,bXYZ@dmndTpdmddvuedLview$lumimeas $tech0 rTRC< gTRC< bTRC< textCopyright (c) 1998 Hewlett-Packard CompanydescsRGB IEC61966-2.1sRGB IEC61966-2.1XYZ QXYZ XYZ o8XYZ bXYZ $descIEC http://www.iec.chIEC http://www.iec.chdesc.IEC 61966-2.1 Default RGB colour space - sRGB.IEC 61966-2.1 Default RGB colour space - sRGBdesc,Reference Viewing Condition in IEC61966-2.1,Reference Viewing Condition in IEC61966-2.1view_. \XYZ L VPWmeassig CRT curv #(-27;@EJOTY^chmrw| %+28>ELRY`gnu| &/8AKT]gqz !-8COZfr~ -;HUcq~ +:IXgw'7HYj{+=Oat 2FZn  % : O d y  ' = T j " 9 Q i  * C \ u & @ Z t .Id %A^z &Ca~1Om&Ed#Cc'Ij4Vx&IlAe@e Ek*Qw;c*R{Gp@j>i  A l !!H!u!!!"'"U"""# #8#f###$$M$|$$% %8%h%%%&'&W&&&''I'z''( (?(q(())8)k))**5*h**++6+i++,,9,n,,- -A-v--..L.../$/Z///050l0011J1112*2c223 3F3334+4e4455M555676r667$7`7788P8899B999:6:t::;-;k;;<' >`>>?!?a??@#@d@@A)AjAAB0BrBBC:C}CDDGDDEEUEEF"FgFFG5G{GHHKHHIIcIIJ7J}JK KSKKL*LrLMMJMMN%NnNOOIOOP'PqPQQPQQR1R|RSS_SSTBTTU(UuUVV\VVWDWWX/X}XYYiYZZVZZ[E[[\5\\]']x]^^l^__a_``W``aOaabIbbcCccd@dde=eef=ffg=ggh?hhiCiijHjjkOkklWlmm`mnnknooxop+ppq:qqrKrss]sttptu(uuv>vvwVwxxnxy*yyzFz{{c{|!||}A}~~b~#G k͂0WGrׇ;iΉ3dʋ0cʍ1fΏ6n֑?zM _ɖ4 uL$h՛BdҞ@iءG&vVǥ8nRĩ7u\ЭD-u`ֲK³8%yhYѹJº;.! 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/Subject (ESannotESslashnoindent ESleftbrackESslashbf Theorem 12.5ESrightbrack ESslash Let ESdollarTESdollar and ESdollarQ_iESdollar be disjoint index sets and ESdollarESslashblue W_iESdollar be a subset of ESdollarQ_iESdollar for ESdollar1 ESslashleq i ESslashleq kESdollar, where ESdollark ESslashgeq 2ESdollar. 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ESslashdefESslashtKESleftbrackESslashtildeESleftbrackKESrightbrackESrightbrack ESslashdefESslashtXESleftbrackESslashtildeESleftbrackXESrightbrackESrightbrack ESslashdefESslashtYESleftbrackESslashtildeESleftbrackYESrightbrackESrightbrack ESslashdefESslashhbfhESleftbrackESslashhatESleftbrackESslashbfhESrightbrackESrightbrack ESslashdefESslashbfalphaESleftbrackESslashmboxESleftbrackESslashboldmathESdollarESslashalphaESdollarESrightbrackESrightbrack ESslashdefESslashbfbetaESleftbrackESslashmboxESleftbrackESslashboldmathESdollarESslashbetaESdollarESrightbrackESrightbrack ESslashdefESslashtildefESleftbrackESslashtildeESleftbrackfESrightbrackESrightbrack ESslashdefESslashtildeKESleftbrackESslashtildeESleftbrackKESrightbrackESrightbrack ESslashdefESslashscIESleftbrackESslashmathfrakESleftbrackIESrightbrackESrightbrackESannopend) /EscapedPreamble 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/Subject (ESannotESslashnoindent ESleftbrackESslashbf Theorem 12.5ESrightbrack ESslash Let ESdollarTESdollar and ESdollarQ_iESdollar be disjoint index sets and ESdollarESslashblue W_iESdollar be a subset of ESdollarQ_iESdollar for ESdollar1 ESslashleq i ESslashleq kESdollar, where ESdollark ESslashgeq 2ESdollar. Assume that there exist at least two ESdollariESdollar such that ESdollarW_i ESslashneq ESslashemptysetESdollar. If ESslash[ ESslashperp (X_ESleftbrackQ_iESrightbrack, 1 ESslashleq i ESslashleq k) ESslash, | ESslash, X_T , ESslash] then ESslash[ ESslashperp (X_ESleftbrackESleftbrackESslashblue W_iESrightbrackESrightbrack, 1 ESslashleq i ESslashleq k) ESslash, | ESslash, X_T, (X_ESleftbrackQ_i - ESleftbrackESleftbrackESslashblue W_iESrightbrackESrightbrackESrightbrack, 1 ESslashleq i ESslashleq k). 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ESslashdefESslashbfeESleftbrackESleftbrackESslashbf eESrightbrackESrightbrack ESslashdefESslashbffESleftbrackESleftbrackESslashbf fESrightbrackESrightbrack ESslashdefESslashbfgESleftbrackESleftbrackESslashbf gESrightbrackESrightbrack ESslashdefESslashbfhESleftbrackESleftbrackESslashbf hESrightbrackESrightbrack ESslashdefESslashbfiESleftbrackESleftbrackESslashbf iESrightbrackESrightbrack ESslashdefESslashbfjESleftbrackESleftbrackESslashbf jESrightbrackESrightbrack ESslashdefESslashbfkESleftbrackESleftbrackESslashbf kESrightbrackESrightbrack ESslashdefESslashbflESleftbrackESleftbrackESslashbf lESrightbrackESrightbrack ESslashdefESslashbfmESleftbrackESleftbrackESslashbf mESrightbrackESrightbrack ESslashdefESslashbfnESleftbrackESleftbrackESslashbf nESrightbrackESrightbrack ESslashdefESslashbfoESleftbrackESleftbrackESslashbf oESrightbrackESrightbrack ESslashdefESslashbfpESleftbrackESleftbrackESslashbf pESrightbrackESrightbrack ESslashdefESslashbfqESleftbrackESleftbrackESslashbf qESrightbrackESrightbrack ESslashdefESslashbfrESleftbrackESleftbrackESslashbf rESrightbrackESrightbrack ESslashdefESslashbfsESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashdefESslashbftESleftbrackESleftbrackESslashbf tESrightbrackESrightbrack ESslashdefESslashbfuESleftbrackESleftbrackESslashbf uESrightbrackESrightbrack ESslashdefESslashbfvESleftbrackESleftbrackESslashbf vESrightbrackESrightbrack ESslashdefESslashbfwESleftbrackESleftbrackESslashbf wESrightbrackESrightbrack ESslashdefESslashbfxESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashdefESslashbfyESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashdefESslashbfzESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashdefESslashbfAESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashdefESslashbfBESleftbrackESleftbrackESslashbf BESrightbrackESrightbrack ESslashdefESslashbfCESleftbrackESleftbrackESslashbf CESrightbrackESrightbrack ESslashdefESslashbfDESleftbrackESleftbrackESslashbf DESrightbrackESrightbrack ESslashdefESslashbfEESleftbrackESleftbrackESslashbf EESrightbrackESrightbrack ESslashdefESslashbfFESleftbrackESleftbrackESslashbf FESrightbrackESrightbrack ESslashdefESslashbfGESleftbrackESleftbrackESslashbf GESrightbrackESrightbrack ESslashdefESslashbfHESleftbrackESleftbrackESslashbf HESrightbrackESrightbrack ESslashdefESslashbfIESleftbrackESleftbrackESslashbf IESrightbrackESrightbrack ESslashdefESslashbfJESleftbrackESleftbrackESslashbf JESrightbrackESrightbrack ESslashdefESslashbfKESleftbrackESleftbrackESslashbf KESrightbrackESrightbrack ESslashdefESslashbfLESleftbrackESleftbrackESslashbf LESrightbrackESrightbrack ESslashdefESslashbfMESleftbrackESleftbrackESslashbf MESrightbrackESrightbrack ESslashdefESslashbfNESleftbrackESleftbrackESslashbf NESrightbrackESrightbrack ESslashdefESslashbfOESleftbrackESleftbrackESslashbf OESrightbrackESrightbrack ESslashdefESslashbfPESleftbrackESleftbrackESslashbf PESrightbrackESrightbrack ESslashdefESslashbfQESleftbrackESleftbrackESslashbf QESrightbrackESrightbrack ESslashdefESslashbfRESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashdefESslashbfSESleftbrackESleftbrackESslashbf SESrightbrackESrightbrack ESslashdefESslashbfTESleftbrackESleftbrackESslashbf TESrightbrackESrightbrack ESslashdefESslashbfUESleftbrackESleftbrackESslashbf UESrightbrackESrightbrack ESslashdefESslashbfVESleftbrackESleftbrackESslashbf VESrightbrackESrightbrack ESslashdefESslashbfWESleftbrackESleftbrackESslashbf WESrightbrackESrightbrack ESslashdefESslashbfXESleftbrackESleftbrackESslashbf XESrightbrackESrightbrack ESslashdefESslashbfYESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashdefESslashbfZESleftbrackESleftbrackESslashbf ZESrightbrackESrightbrack ESslashdefESslashcalAESleftbrackESleftbrackESslashcal AESrightbrackESrightbrack ESslashdefESslashcalBESleftbrackESleftbrackESslashcal BESrightbrackESrightbrack ESslashdefESslashcalCESleftbrackESleftbrackESslashcal CESrightbrackESrightbrack ESslashdefESslashcalDESleftbrackESleftbrackESslashcal DESrightbrackESrightbrack ESslashdefESslashcalEESleftbrackESleftbrackESslashcal EESrightbrackESrightbrack ESslashdefESslashcalFESleftbrackESleftbrackESslashcal FESrightbrackESrightbrack ESslashdefESslashcalGESleftbrackESleftbrackESslashcal GESrightbrackESrightbrack ESslashdefESslashcalHESleftbrackESleftbrackESslashcal HESrightbrackESrightbrack ESslashdefESslashcalIESleftbrackESleftbrackESslashcal IESrightbrackESrightbrack ESslashdefESslashcalJESleftbrackESleftbrackESslashcal JESrightbrackESrightbrack ESslashdefESslashcalKESleftbrackESleftbrackESslashcal KESrightbrackESrightbrack ESslashdefESslashcalLESleftbrackESleftbrackESslashcal LESrightbrackESrightbrack ESslashdefESslashcalMESleftbrackESleftbrackESslashcal MESrightbrackESrightbrack ESslashdefESslashcalNESleftbrackESleftbrackESslashcal NESrightbrackESrightbrack ESslashdefESslashcalOESleftbrackESleftbrackESslashcal OESrightbrackESrightbrack ESslashdefESslashcalPESleftbrackESleftbrackESslashcal PESrightbrackESrightbrack ESslashdefESslashcalQESleftbrackESleftbrackESslashcal QESrightbrackESrightbrack ESslashdefESslashcalRESleftbrackESleftbrackESslashcal RESrightbrackESrightbrack ESslashdefESslashcalSESleftbrackESleftbrackESslashcal SESrightbrackESrightbrack ESslashdefESslashcalTESleftbrackESleftbrackESslashcal TESrightbrackESrightbrack ESslashdefESslashcalUESleftbrackESleftbrackESslashcal UESrightbrackESrightbrack ESslashdefESslashcalVESleftbrackESleftbrackESslashcal VESrightbrackESrightbrack ESslashdefESslashcalWESleftbrackESleftbrackESslashcal WESrightbrackESrightbrack ESslashdefESslashcalXESleftbrackESleftbrackESslashcal XESrightbrackESrightbrack ESslashdefESslashcalYESleftbrackESleftbrackESslashcal YESrightbrackESrightbrack ESslashdefESslashcalZESleftbrackESleftbrackESslashcal ZESrightbrackESrightbrack ESslashdefESslashhatxESleftbrackESslashhatESleftbrackxESrightbrackESrightbrack ESslashdefESslashhatXESleftbrackESslashhatESleftbrackXESrightbrackESrightbrack ESslashdefESslashhatyESleftbrackESslashhatESleftbrackyESrightbrackESrightbrack ESslashdefESslashhatYESleftbrackESslashhatESleftbrackYESrightbrackESrightbrack ESslashdefESslashhatWESleftbrackESslashhatESleftbrackWESrightbrackESrightbrack ESslashdefESslashtKESleftbrackESslashtildeESleftbrackKESrightbrackESrightbrack ESslashdefESslashtXESleftbrackESslashtildeESleftbrackXESrightbrackESrightbrack ESslashdefESslashtYESleftbrackESslashtildeESleftbrackYESrightbrackESrightbrack ESslashdefESslashhbfhESleftbrackESslashhatESleftbrackESslashbfhESrightbrackESrightbrack ESslashdefESslashbfalphaESleftbrackESslashmboxESleftbrackESslashboldmathESdollarESslashalphaESdollarESrightbrackESrightbrack ESslashdefESslashbfbetaESleftbrackESslashmboxESleftbrackESslashboldmathESdollarESslashbetaESdollarESrightbrackESrightbrack ESslashdefESslashtildefESleftbrackESslashtildeESleftbrackfESrightbrackESrightbrack ESslashdefESslashtildeKESleftbrackESslashtildeESleftbrackKESrightbrackESrightbrack ESslashdefESslashscIESleftbrackESslashmathfrakESleftbrackIESrightbrackESrightbrackESannopend) /EscapedPreamble 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/Subject (ESannotESslashnoindent ESleftbrackESslashbf Definition 12.11ESrightbrack ESslash A conditional mutual independency on ESdollarX_1, X_2, ESslashcdots, X_nESdollar is full if all ESdollarX_1, X_2, ESslashcdots, X_nESdollar are involved. 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/Subject (ESannotESslashnoindent ESleftbrackESslashbf Definition 12.11ESrightbrack ESslash A conditional mutual independency on ESdollarX_1, X_2, ESslashcdots, X_nESdollar is full if all ESdollarX_1, X_2, ESslashcdots, X_nESdollar are involved. Such a conditional mutual independency is called a full conditional mutual independency (FCMI). ESslashbigskip ESslashnoindent ESleftbrackESslashbf Example 12.11ESrightbrack ESslash For ESdollarn=5ESdollar, ESslash[ ESslashmboxESleftbrack ESdollarX_1ESdollar, ESdollarX_2,X_4ESdollar, and ESdollarX_5ESdollar are mutually independent conditioning on ESdollarX_3ESdollar ESrightbrack ESslash] is an FCMI. However, ESslash[ ESslashmboxESleftbrack ESdollarX_1ESdollar, ESdollarX_2ESdollar, and ESdollarX_5ESdollar are mutually independent conditioning on ESdollarX_3ESdollar ESrightbrack ESslash] is not an FCMI because ESdollarX_4ESdollar is not involved. ESannotend) /EscapedSubject (ESannoes%5Cnoindent%0A%7B%5Cbf%20Definition%2012.11%7D%20%5C%20A%20conditional%20mutual%20independency%20on%20$X_1,%20X_2,%20%5Ccdots,%20X_n$%0Ais%20full%20if%20all%20$X_1,%20X_2,%20%5Ccdots,%20X_n$%20are%20involved.%0ASuch%20a%20conditional%20mutual%20independency%20is%20called%20a%20full%0Aconditional%20mutual%20independency%20(FCMI).%0A%0A%5Cbigskip%0A%5Cnoindent%0A%7B%5Cbf%20Example%2012.11%7D%20%5C%0AFor%20$n=5$,%0A%5C%5B%0A%5Cmbox%7B%0A$X_1$,%20$X_2,X_4$,%20and%20$X_5$%20are%20mutually%20independent%20conditioning%20on%20$X_3$%0A%7D%0A%5C%5D%0Ais%20an%20FCMI.%20%20However,%0A%5C%5B%0A%5Cmbox%7B%0A$X_1$,%20$X_2$,%20and%20$X_5$%20are%20mutually%20independent%20conditioning%20on%20$X_3$%0A%7D%0A%5C%5D%0Ais%20not%20an%20FCMI%20because%20$X_4$%20is%20not%20involved.%0A%0AESannoesend) /Type (EEtype2EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag26.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj startxref 60833 %%EOFPKpjE_S|latexit-drag-27.pdf%PDF-1.5 % 5 0 obj << /Length 29 /Filter /FlateDecode >> stream x+2T0B˥kJ, endstream endobj 2 0 obj << /Type /XObject /Subtype /Form /BBox [0 0 853 497] /FormType 1 /Matrix [1 0 0 1 0 0] /Resources 8 0 R /Length 38 /Filter /FlateDecode >> stream x+2T0BC]CS 626VH5Tp l endstream endobj 1 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (/var/folders/ns/vq4jp5593zl9f3kk14264msw0000gn/T/LaTeXiT-2.6.1/latexit-1-2.pdf) /PTEX.PageNumber 1 /PTEX.InfoDict 9 0 R /BBox [0 0 1243 1732] /Resources << /XObject << /Fm1 10 0 R >>/ProcSet [ /PDF ] >> /Length 57 /Filter /FlateDecode >> stream x3T0BCC=KC#c#= \B.#=3$ BCr \) endstream endobj 10 0 obj << /Type /XObject /Subtype /Form /BBox [ 0 0 328.999 192] /FormType 1 /Matrix [ 1 0 0 1 0 0] /Resources 11 0 R /Length 43 /Filter /FlateDecode >> stream x+2T0BC]CK]sc=KKK\.}\C|@. endstream endobj 12 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./latexit-1.pdf) /PTEX.PageNumber 1 /PTEX.InfoDict 13 0 R /BBox [ 0 0 612 792] /Resources << /Font << /F14 14 0 R /F8 15 0 R /F10 16 0 R /F11 17 0 R /F7 18 0 R /F13 19 0 R /F1 20 0 R /F9 21 0 R >> /ProcSet [/PDF/Text] >> /Length 1033 /Filter /FlateDecode >> stream xڽWMo6WF 9O=4Er6@-ػm/V( z!fgߌ{s&WR1ϽL*ǭf倚-n+!e &7}xFٻd!ޢ KY%*ZU?y3J evDqd/90ǒ݇d\ !4[;Ysպ%I A}3WɅ@ZKx+~٬wz;q 5[o쟴d筠,i04 S 1jɧW.]IÄ1Eo.G¸~BaxTßX;JJ B{pK $H:2QT81MLMSzpЍ?-8Q^hqZfHh C*0<4B endstream endobj 54 0 obj << /Length1 1583 /Length2 7324 /Length3 0 /Length 8374 /Filter /FlateDecode >> stream xڍTk6t75HKww7Ð 14tJw Ҡ4H+H(HRR|sYk׮{}{f !xxAb9 =>qXXP8!  9  p@  @~HoC@ P 8,rp7?'G2ߏv0OTT;@4lWdF ;A~#&ڸz=$9>NG. *i S/ @/c Dx dv:@ X/nx W 'og0fs9읠:/ 2z‘66NP[et6 prCxz:AfCx:p]`pXߒWv^n@DE s Q!!A>;%sV52 ANN7%>>ĝ88p!{8^hk?C{f. !"m?(2(Hj!W#V m]`^ݐ9vȻi/T&.Y˿'YÝ;/dh|x6G1VVcL1se^Rdj/TFNp~5\@Q_=):}S; gƬ}K֡#6C7cK5~Urm*p2f tƙ02"M =̲U oQ.d;㩮?~7П)pRW+2_`$e,9OUV\2=5uZxt@GW\'Zs'VW@<)1u̲,v(Yj};ixyŅ:ǖx'a[r-%"_^ iCF>JW><}-.di螼A"aޫIG@~,KnSGrM6Uģ$Y#BٟV$O H}X13n]@m֛.m_s#a%ARhuK`γX[ήp.mY4k~sQ.+ &`[XU470X1Ds(Ѿg*xߌ9u,Q㵒ygzZ?4TQ䇷/@c-~2O-iC50MF&*6IPlp~˿ne[%\9=wthӋ*wy~m@vRvݖ8z5m= ʠxGnN^xWOPVˎNm? 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/Subject (ESannotESslashnoindent ESslashbeginESleftbrackitemizeESrightbrack ESslashitem Recall ESdollarESslashcalN_n = ESslashESleftbrack 1, 2, ESslashcdots, n ESslashESrightbrackESdollar. ESslashitem Consider a nonempty atom ESslash[ A = ESslashtildeESleftbrackYESrightbrack_1 ESslashcap ESslashtildeESleftbrackYESrightbrack_2 ESslashcap ESslashcdots ESslashcap ESslashtildeESleftbrackYESrightbrack_n ESslash] of ESdollarESleftbrackESslashcal FESrightbrack_nESdollar, where ESdollarESslashtildeESleftbrackYESrightbrack_i = ESslashtildeESleftbrackXESrightbrack_iESdollar or ESdollarESslashtildeESleftbrackXESrightbrack_i^cESdollar, and there exists at least one ESdollariESdollar such that ESdollarESslashtildeESleftbrackYESrightbrack_i = ESslashtildeESleftbrackXESrightbrack_iESdollar. ESslashitem Let ESslash[ U_A = ESslashESleftbrack i ESslashin ESslashcalN_n : ESslashtildeESleftbrackYESrightbrack_i = ESslashtildeESleftbrackXESrightbrack_i^c ESslashESrightbrack. ESslash] Note that ESdollarAESdollar and ESdollarU_AESdollar uniquely determine each other. ESslashitem By Theorem 3.19, ESslash[ ESslashmu^*(A) = ESslashsum_ESleftbrackJ ESslashsubseteq U_A^cESrightbrack (-1)^ESleftbrack|J|+1ESrightbrack H(J|U_A) . 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ESslashdefESslashtYESleftbrackESslashtildeESleftbrackYESrightbrackESrightbrack ESslashdefESslashhbfhESleftbrackESslashhatESleftbrackESslashbfhESrightbrackESrightbrack ESslashdefESslashbfalphaESleftbrackESslashmboxESleftbrackESslashboldmathESdollarESslashalphaESdollarESrightbrackESrightbrack ESslashdefESslashbfbetaESleftbrackESslashmboxESleftbrackESslashboldmathESdollarESslashbetaESdollarESrightbrackESrightbrack ESslashdefESslashtildefESleftbrackESslashtildeESleftbrackfESrightbrackESrightbrack ESslashdefESslashtildeKESleftbrackESslashtildeESleftbrackKESrightbrackESrightbrack ESslashdefESslashscIESleftbrackESslashmathfrakESleftbrackIESrightbrackESrightbrackESannopend) /EscapedPreamble 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/Subject (ESannotESslashnoindent ESleftbrackESslashbf Theorem 3.19ESrightbrack (Variation of the Inclusion-Exclusion Formula) ESslashba ESslashmu ESslashleft( ESslashbigcap_ESleftbrackk=1ESrightbrack^m A_k - B ESslashright) & = & ESslashsum_ESleftbrack1 ESslashle i ESslashle mESrightbrack ESslashmu(A_i - B) - ESslashsum_ESleftbrack1 ESslashle i < j ESslashle mESrightbrack ESslashmu(A_i ESslashcup A_j - B) + ESslashcdots ESslashESslash & & + (-1)^ESleftbrackm+1ESrightbrack ESslashmu(A_1 ESslashcup A_2 ESslashcup ESslashcdots ESslashcup A_m - B). 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/Subject (ESannotESslashnoindent ESslashbeginESleftbrackitemizeESrightbrack ESslashitem Let ESdollarESslashESleftbrack T, Q_i, 1 ESslashleq i ESslashleq k ESslashESrightbrackESdollar be a partition of ESdollarESslashcalN_nESdollar. ESslashitem The tuple ESdollar(T, Q_i, 1 ESslashleq i ESslashleq k)ESdollar defines the following FCMI on ESdollarX_1, X_2, ESslashcdots, X_nESdollar: ESslash[ K : X_ESleftbrackQ_1ESrightbrack, X_ESleftbrackQ_2ESrightbrack, ESslashcdots, X_ESleftbrackQ_kESrightbrack ESslash ESslashmboxESleftbrackare mutually independent conditioning on ESdollarX_TESdollar.ESrightbrack ESslash] ESslashitem Denote ESdollarKESdollar by ESdollar(T, Q_i, 1 ESslashleq i ESslashleq k)ESdollar. 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vESrightbrackESrightbrack ESslashdefESslashbfwESleftbrackESleftbrackESslashbf wESrightbrackESrightbrack ESslashdefESslashbfxESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashdefESslashbfyESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashdefESslashbfzESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashdefESslashbfAESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashdefESslashbfBESleftbrackESleftbrackESslashbf BESrightbrackESrightbrack ESslashdefESslashbfCESleftbrackESleftbrackESslashbf CESrightbrackESrightbrack ESslashdefESslashbfDESleftbrackESleftbrackESslashbf DESrightbrackESrightbrack ESslashdefESslashbfEESleftbrackESleftbrackESslashbf EESrightbrackESrightbrack ESslashdefESslashbfFESleftbrackESleftbrackESslashbf FESrightbrackESrightbrack ESslashdefESslashbfGESleftbrackESleftbrackESslashbf GESrightbrackESrightbrack ESslashdefESslashbfHESleftbrackESleftbrackESslashbf HESrightbrackESrightbrack ESslashdefESslashbfIESleftbrackESleftbrackESslashbf IESrightbrackESrightbrack ESslashdefESslashbfJESleftbrackESleftbrackESslashbf JESrightbrackESrightbrack ESslashdefESslashbfKESleftbrackESleftbrackESslashbf KESrightbrackESrightbrack ESslashdefESslashbfLESleftbrackESleftbrackESslashbf LESrightbrackESrightbrack ESslashdefESslashbfMESleftbrackESleftbrackESslashbf MESrightbrackESrightbrack ESslashdefESslashbfNESleftbrackESleftbrackESslashbf NESrightbrackESrightbrack ESslashdefESslashbfOESleftbrackESleftbrackESslashbf OESrightbrackESrightbrack ESslashdefESslashbfPESleftbrackESleftbrackESslashbf PESrightbrackESrightbrack ESslashdefESslashbfQESleftbrackESleftbrackESslashbf QESrightbrackESrightbrack ESslashdefESslashbfRESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashdefESslashbfSESleftbrackESleftbrackESslashbf SESrightbrackESrightbrack ESslashdefESslashbfTESleftbrackESleftbrackESslashbf TESrightbrackESrightbrack ESslashdefESslashbfUESleftbrackESleftbrackESslashbf UESrightbrackESrightbrack ESslashdefESslashbfVESleftbrackESleftbrackESslashbf VESrightbrackESrightbrack ESslashdefESslashbfWESleftbrackESleftbrackESslashbf WESrightbrackESrightbrack ESslashdefESslashbfXESleftbrackESleftbrackESslashbf XESrightbrackESrightbrack ESslashdefESslashbfYESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashdefESslashbfZESleftbrackESleftbrackESslashbf ZESrightbrackESrightbrack ESslashdefESslashcalAESleftbrackESleftbrackESslashcal AESrightbrackESrightbrack ESslashdefESslashcalBESleftbrackESleftbrackESslashcal BESrightbrackESrightbrack ESslashdefESslashcalCESleftbrackESleftbrackESslashcal CESrightbrackESrightbrack ESslashdefESslashcalDESleftbrackESleftbrackESslashcal DESrightbrackESrightbrack ESslashdefESslashcalEESleftbrackESleftbrackESslashcal EESrightbrackESrightbrack ESslashdefESslashcalFESleftbrackESleftbrackESslashcal FESrightbrackESrightbrack ESslashdefESslashcalGESleftbrackESleftbrackESslashcal GESrightbrackESrightbrack ESslashdefESslashcalHESleftbrackESleftbrackESslashcal HESrightbrackESrightbrack ESslashdefESslashcalIESleftbrackESleftbrackESslashcal IESrightbrackESrightbrack ESslashdefESslashcalJESleftbrackESleftbrackESslashcal JESrightbrackESrightbrack ESslashdefESslashcalKESleftbrackESleftbrackESslashcal KESrightbrackESrightbrack ESslashdefESslashcalLESleftbrackESleftbrackESslashcal LESrightbrackESrightbrack ESslashdefESslashcalMESleftbrackESleftbrackESslashcal MESrightbrackESrightbrack ESslashdefESslashcalNESleftbrackESleftbrackESslashcal NESrightbrackESrightbrack ESslashdefESslashcalOESleftbrackESleftbrackESslashcal OESrightbrackESrightbrack ESslashdefESslashcalPESleftbrackESleftbrackESslashcal PESrightbrackESrightbrack ESslashdefESslashcalQESleftbrackESleftbrackESslashcal QESrightbrackESrightbrack ESslashdefESslashcalRESleftbrackESleftbrackESslashcal RESrightbrackESrightbrack ESslashdefESslashcalSESleftbrackESleftbrackESslashcal SESrightbrackESrightbrack ESslashdefESslashcalTESleftbrackESleftbrackESslashcal TESrightbrackESrightbrack ESslashdefESslashcalUESleftbrackESleftbrackESslashcal UESrightbrackESrightbrack ESslashdefESslashcalVESleftbrackESleftbrackESslashcal VESrightbrackESrightbrack ESslashdefESslashcalWESleftbrackESleftbrackESslashcal WESrightbrackESrightbrack ESslashdefESslashcalXESleftbrackESleftbrackESslashcal XESrightbrackESrightbrack ESslashdefESslashcalYESleftbrackESleftbrackESslashcal YESrightbrackESrightbrack ESslashdefESslashcalZESleftbrackESleftbrackESslashcal ZESrightbrackESrightbrack ESslashdefESslashhatxESleftbrackESslashhatESleftbrackxESrightbrackESrightbrack ESslashdefESslashhatXESleftbrackESslashhatESleftbrackXESrightbrackESrightbrack ESslashdefESslashhatyESleftbrackESslashhatESleftbrackyESrightbrackESrightbrack ESslashdefESslashhatYESleftbrackESslashhatESleftbrackYESrightbrackESrightbrack ESslashdefESslashhatWESleftbrackESslashhatESleftbrackWESrightbrackESrightbrack ESslashdefESslashtKESleftbrackESslashtildeESleftbrackKESrightbrackESrightbrack ESslashdefESslashtXESleftbrackESslashtildeESleftbrackXESrightbrackESrightbrack ESslashdefESslashtYESleftbrackESslashtildeESleftbrackYESrightbrackESrightbrack ESslashdefESslashhbfhESleftbrackESslashhatESleftbrackESslashbfhESrightbrackESrightbrack ESslashdefESslashbfalphaESleftbrackESslashmboxESleftbrackESslashboldmathESdollarESslashalphaESdollarESrightbrackESrightbrack ESslashdefESslashbfbetaESleftbrackESslashmboxESleftbrackESslashboldmathESdollarESslashbetaESdollarESrightbrackESrightbrack ESslashdefESslashtildefESleftbrackESslashtildeESleftbrackfESrightbrackESrightbrack ESslashdefESslashtildeKESleftbrackESslashtildeESleftbrackKESrightbrackESrightbrack ESslashdefESslashscIESleftbrackESslashmathfrakESleftbrackIESrightbrackESrightbrackESannopend) /EscapedPreamble 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/Subject (ESannotESslashnoindent ESleftbrackESslashbf Definition 12.13ESrightbrack ESslash Let ESdollarK= (T, Q_i, 1 ESslashleq i ESslashleq k)ESdollar be an FCMI on ESdollarX_1, X_2, ESslashcdots,ESdollar ESdollarX_nESdollar. The image of ESdollarKESdollar, denoted by ESdollarESleftbrackESslashit ImESrightbrack (K)ESdollar, is the set of all atoms of ESdollarESslashcalF_nESdollar such that ESslash[ Y_j = ESslashleftESslashESleftbrack ESslashbeginESleftbrackarrayESrightbrackESleftbrackllESrightbrack ESslashtildeESleftbrackXESrightbrack_j & ESslashmboxESleftbrackif ESdollarj ESslashin W_iESdollarESrightbrack ESslashESslash ESslashtildeESleftbrackXESrightbrack_j^c & ESslashmboxESleftbrackif ESdollarT ESslashcup ESslashbigcup_ESleftbracki=1ESrightbrack^k (Q_i - W_i)ESdollar.ESrightbrack ESslashendESleftbrackarrayESrightbrack ESslashright. ESslash] where ESdollarW_i ESslashsubseteq Q_iESdollar, ESdollar1 ESslashleq i ESslashleq kESdollar, and there exist at least two ESdollariESdollar such that ESdollarW_i ESslashneq ESslashemptysetESdollar. 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/Subject (ESannotESslashnoindent ESleftbrackESslashbf Definition 12.13ESrightbrack ESslash Let ESdollarK= (T, Q_i, 1 ESslashleq i ESslashleq k)ESdollar be an FCMI on ESdollarX_1, X_2, ESslashcdots,ESdollar ESdollarX_nESdollar. The image of ESdollarKESdollar, denoted by ESdollarESleftbrackESslashit ImESrightbrack (K)ESdollar, is the set of all atoms of ESdollarESslashcalF_nESdollar such that ESslash[ Y_j = ESslashleftESslashESleftbrack ESslashbeginESleftbrackarrayESrightbrackESleftbrackllESrightbrack ESslashtildeESleftbrackXESrightbrack_j & ESslashmboxESleftbrackif ESdollarj ESslashin W_iESdollarESrightbrack ESslashESslash ESslashtildeESleftbrackXESrightbrack_j^c & ESslashmboxESleftbrackif ESdollarT ESslashcup ESslashbigcup_ESleftbracki=1ESrightbrack^k (Q_i - W_i)ESdollar.ESrightbrack ESslashendESleftbrackarrayESrightbrack ESslashright. ESslash] where ESdollarW_i ESslashsubseteq Q_iESdollar, ESdollar1 ESslashleq i ESslashleq kESdollar, and there exist at least two ESdollariESdollar such that ESdollarW_i ESslashneq ESslashemptysetESdollar. 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ESslashdefESslashscIESleftbrackESslashmathfrakESleftbrackIESrightbrackESrightbrackESannopend) /EscapedPreamble 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ESslashdefESslashscIESleftbrackESslashmathfrakESleftbrackIESrightbrackESrightbrackESannopend) /EscapedPreamble 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