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%%CreationDate: 1993 Sep 17 09:05:00
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
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/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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0 1 255 {1 index exch /.notdef put} for
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dup 4 /squaresolid put
readonly def
/FontBBox{8 -463 1331 1003}readonly def
currentdict end
currentfile eexec
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%%EndFont 
%%BeginFont: CMSY6
%!PS-AdobeFont-1.1: CMSY6 1.0
%%CreationDate: 1991 Aug 15 07:21:34
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 48 /prime put
readonly def
/FontBBox{-4 -948 1329 786}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI12
%!PS-AdobeFont-1.1: CMTI12 1.0
%%CreationDate: 1991 Aug 18 21:06:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 12 /fi put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 79 /O put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 86 /V put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-36 -251 1103 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMCSC10
%!PS-AdobeFont-1.1: CMCSC10 1.0
%%CreationDate: 1991 Aug 18 17:46:49
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMCSC10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMCSC10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 46 /period put
dup 67 /C put
dup 72 /H put
dup 79 /O put
dup 80 /P put
dup 83 /S put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
readonly def
/FontBBox{14 -250 1077 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX8
%!PS-AdobeFont-1.1: CMBX8 1.0
%%CreationDate: 1991 Aug 20 16:36:07
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 99 /c put
readonly def
/FontBBox{-59 -250 1235 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSL12
%!PS-AdobeFont-1.1: CMSL12 1.0
%%CreationDate: 1991 Aug 20 16:40:41
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSL12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -9.46 def
/isFixedPitch false def
end readonly def
/FontName /CMSL12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 45 /hyphen put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 103 /g put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
readonly def
/FontBBox{-56 -251 1102 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 77 /M put
dup 82 /R put
dup 99 /c put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 110 /n put
readonly def
/FontBBox{11 -250 1241 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 69 /E put
dup 73 /I put
dup 80 /P put
dup 82 /R put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /parenleftbig put
dup 1 /parenrightbig put
dup 2 /bracketleftbig put
dup 3 /bracketrightbig put
dup 8 /braceleftbig put
dup 9 /bracerightbig put
dup 12 /vextendsingle put
dup 13 /vextenddouble put
dup 18 /parenleftbigg put
dup 19 /parenrightbigg put
dup 26 /braceleftbigg put
dup 27 /bracerightbigg put
dup 48 /parenlefttp put
dup 49 /parenrighttp put
dup 64 /parenleftbt put
dup 65 /parenrightbt put
dup 66 /parenleftex put
dup 67 /parenrightex put
dup 68 /angbracketleftBig put
dup 69 /angbracketrightBig put
dup 80 /summationtext put
dup 88 /summationdisplay put
dup 90 /integraldisplay put
dup 98 /hatwide put
dup 104 /bracketleftBig put
dup 105 /bracketrightBig put
dup 112 /radicalbig put
dup 114 /radicalbigg put
dup 116 /radicalbt put
dup 117 /radicalvertex put
dup 118 /radicaltp put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /alpha put
dup 22 /mu put
dup 23 /nu put
dup 58 /period put
dup 59 /comma put
dup 61 /slash put
dup 62 /greater put
dup 65 /A put
dup 67 /C put
dup 68 /D put
dup 77 /M put
dup 78 /N put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 88 /X put
dup 89 /Y put
dup 99 /c put
dup 100 /d put
dup 103 /g put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 109 /m put
dup 110 /n put
dup 115 /s put
dup 120 /x put
dup 121 /y put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 61 /equal put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 1 /periodcentered put
dup 2 /multiply put
dup 6 /plusminus put
dup 8 /circleplus put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 26 /propersubset put
dup 33 /arrowright put
dup 35 /arrowdown put
dup 39 /similarequal put
dup 49 /infinity put
dup 50 /element put
dup 54 /negationslash put
dup 65 /A put
dup 68 /D put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 77 /M put
dup 80 /P put
dup 92 /intersection put
dup 100 /ceilingleft put
dup 101 /ceilingright put
dup 102 /braceleft put
dup 103 /braceright put
dup 104 /angbracketleft put
dup 105 /angbracketright put
dup 106 /bar put
dup 107 /bardbl put
dup 110 /backslash put
dup 112 /radical put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /alpha put
dup 12 /beta put
dup 13 /gamma put
dup 14 /delta put
dup 22 /mu put
dup 23 /nu put
dup 27 /sigma put
dup 31 /chi put
dup 33 /omega put
dup 34 /epsilon put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 62 /greater put
dup 65 /A put
dup 67 /C put
dup 68 /D put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 82 /R put
dup 83 /S put
dup 87 /W put
dup 88 /X put
dup 89 /Y put
dup 90 /Z put
dup 96 /lscript put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 109 /m put
dup 110 /n put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI10
%!PS-AdobeFont-1.1: CMTI10 1.00B
%%CreationDate: 1992 Feb 19 19:56:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 73 /I put
dup 84 /T put
dup 100 /d put
dup 101 /e put
dup 109 /m put
dup 110 /n put
dup 114 /r put
dup 115 /s put
dup 120 /x put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY7
%!PS-AdobeFont-1.1: CMSY7 1.0
%%CreationDate: 1991 Aug 15 07:21:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 3 /asteriskmath put
dup 121 /dagger put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 61 /slash put
dup 79 /O put
dup 88 /X put
dup 107 /k put
dup 110 /n put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /ff put
dup 12 /fi put
dup 13 /fl put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 65 /A put
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dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 74 /J put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
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dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
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dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
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dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 124 /emdash put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 65 /A put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 114 /r put
dup 115 /s put
dup 116 /t put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSS12
%!PS-AdobeFont-1.1: CMSS12 1.0
%%CreationDate: 1991 Aug 20 17:33:47
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSS12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMSS12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 45 /hyphen put
dup 46 /period put
dup 50 /two put
dup 64 /at put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
readonly def
/FontBBox{-62 -251 978 758}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 10 /Omega put
dup 11 /ff put
dup 12 /fi put
dup 13 /fl put
dup 14 /ffi put
dup 18 /grave put
dup 19 /acute put
dup 38 /ampersand put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 59 /semicolon put
dup 61 /equal put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 87 /W put
dup 89 /Y put
dup 90 /Z put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 94 /circumflex put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 123 /endash put
dup 126 /tilde put
dup 127 /dieresis put
readonly def
/FontBBox{-34 -251 988 750}readonly def
currentdict end
currentfile eexec
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%%EndFont 
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 3 /asteriskmath put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 24 /similar put
dup 33 /arrowright put
dup 48 /prime put
dup 49 /infinity put
dup 50 /element put
dup 65 /A put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 77 /M put
dup 98 /floorleft put
dup 99 /floorright put
dup 102 /braceleft put
dup 103 /braceright put
dup 107 /bardbl put
dup 121 /dagger put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
TeXDict begin 39139632 55387786 1000 600 600 (clusteringfinal.dvi)
@start /Fa 205[30 30 30 48[{}3 49.8132 /CMR6 rf /Fb 134[71
4[52 53 55 1[75 67 75 112 37 71 1[37 75 1[41 61 1[60
1[65 12[94 75 2[92 12[97 13[67 67 67 67 67 2[37 46[{}26
119.552 /CMBX12 rf /Fc 251[77 77 3[{}2 99.6264 /MSAM10
rf /Fd 207[18 48[{}1 49.8132 /CMSY6 rf /Fe 133[40 47
45 65 45 52 32 40 41 45 50 50 55 80 25 45 30 30 50 45
30 45 50 45 45 50 10[72 1[70 55 71 75 66 75 1[87 61 75
51 38 72 75 64 66 74 70 69 72 6[30 50 50 3[50 50 50 50
50 1[30 35 30 2[40 40 30 26[55 12[{}61 99.6264 /CMTI12
rf /Ff 137[61 61 59 46 60 1[56 63 61 74 51 2[30 2[53
56 62 59 1[61 50[32 46[{}17 99.6264 /CMCSC10 rf /Fg 156[36
99[{}1 66.4176 /CMBX8 rf /Fh 138[54 38 38 38 51 54 49
54 81 27 51 30 27 1[49 1[43 54 43 1[49 51[33 45[{}19
99.6264 /CMSL12 rf /Fi 145[38 2[33 25 22 5[28 16[46 4[58
77[{}7 49.8132 /CMMI6 rf /Fj 173[72 1[61 6[39 3[66 69[{}4
99.6264 /MSBM10 rf /Fk 137[105 105 105 1[100 1[100 6[47
47 5[55 7[55 1[144 7[105 10[61 61 87 87 87 87 14[87 87
20[75 75 6[73 73 4[55 33 2[58 58 4[42 42 46 46{}31 99.6264
/CMEX10 rf /Fl 134[59 1[81 1[62 44 44 46 1[62 56 62 93
31 59 1[31 62 56 34 51 62 50 1[54 12[78 1[84 1[77 3[67
1[58 6[81 1[85 12[56 56 56 56 2[31 37 3[44 44 40[{}35
99.6264 /CMBX12 rf /Fm 134[35 40 4[33 4[43 62 1[37 29
24 1[34 2[36 31 9[41 58 4[43 53 1[45 1[56 68 8[58 50
1[53 2[55 35 1[20 20 34[35 42 10[45 11[{}28 66.4176 /CMMI8
rf /Fn 194[55 8[35 35 35 35 35 4[55 1[27 27 40[{}9 66.4176
/CMR8 rf /Fo 143[83 1[50 2[50 28 39 39 50 50 44 44 7[66
11[69 2[120 4[84 59 72 1[77 2[80 10[0 3[66 100 9[77 3[50
1[100 6[77 4[77 77 11[77 1[77 3[77 28 77{}32 99.6264
/CMSY10 rf /Fp 133[45 48 55 1[47 56 35 46 4[58 85 1[51
40 33 1[47 48 45 51 42 41 51 41 5[67 57 81 92 3[60 74
2[75 78 94 66 7[81 70 1[73 2[76 49 76 27 27 23[46 61
1[61 3[55 3[48 59 7[43 51 55 62 11[{}48 99.6264 /CMMI12
rf /Fq 138[90 63 64 66 1[90 81 90 134 45 1[49 45 1[81
49 74 90 72 90 78 14[120 1[110 6[60 126 4[117 14[81 81
81 81 49[{}27 143.462 /CMBX12 rf /Fr 135[42 4[37 38 3[51
74 7[42 46 15[65 10[35 73[{}9 90.9091 /CMTI10 rf /Fs
134[30 117[34 3[{}2 58.1154 /CMSY7 rf /Ft 143[91 112[{}1
90.9091 /CMEX10 rf /Fu 145[55 2[47 18[75 8[69 17[45 61[{}5
90.9091 /CMMI10 rf /Fv 131[91 1[40 48 48 66 48 51 35
36 36 48 51 45 51 76 25 48 28 25 51 45 28 40 51 40 51
45 9[93 68 2[51 67 2[71 2[57 1[47 1[68 2[62 1[66 64 16[45
2[25 30 25 2[35 35 26[51 51 53 11[{}47 90.9091 /CMR10
rf /Fw 139[41 41 43 14[46 58 51 31[79 65[{}7 90.9091
/CMBX10 rf /Fx 137[45 50 35 37 33 1[50 49 50 77 23 2[23
50 49 30 43 1[43 50 47 32[65 13[49 3[27 33 45[{}22 99.6264
/CMSS12 rf /Fy 134[44 44 60 1[46 32 33 33 1[46 42 46
1[23 44 1[23 46 42 25 37 46 37 46 42 12[60 46 61 65 57
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48 b(Since)35 b Fo(H)f Fz(is)g(assumed)h(to)f(b)s(e)f(separable,)i(w)m
(e)g(ma)m(y)324 2354 y(iden)m(tify)25 b(it)e(with)h(the)g(space)h
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y Fz(\))2515 2369 y Fm(\013)p FA(\025)p Fn(1)2678 2354
y Fz(suc)m(h)d(that)3092 2279 y Fk(P)3197 2305 y FA(1)3197
2383 y Fm(\013)p Fn(=1)3353 2354 y Fp(x)3408 2318 y Fn(2)3408
2378 y Fm(\013)3486 2354 y Fp(<)324 2474 y Fo(1)p Fz(.)52
b(Eac)m(h)37 b(data)e(p)s(oin)m(t)h Fp(X)1313 2489 y
Fm(i)1377 2474 y Fz(\()p Fp(i)d Fz(=)g(1)p Fp(;)17 b(:)g(:)g(:)f(;)h(n)
p Fz(\))35 b(is)i(no)m(w)f(represen)m(ted)i(b)m(y)f(a)e(v)m(ector)i(in)
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2623 y Fm(i)2358 2608 y Fz(=)28 b(\()p Fp(X)2589 2557
y Fn(\(1\))2581 2634 y Fm(i)2683 2608 y Fp(;)17 b(X)2816
2557 y Fn(\(2\))2808 2634 y Fm(i)2910 2608 y Fp(;)g(:)g(:)g(:)o
Fz(\).)470 2728 y(Let)47 b Fp(s)f Fz(b)s(e)h(a)f(p)s(ositiv)m(e)h(in)m
(teger,)k(and)c(let)f(\()p Fp(N)2208 2743 y Fn(1)p Fm(\013)2293
2728 y Fz(\))2331 2743 y Fm(\013)p FA(\025)p Fn(1)2471
2728 y Fp(;)17 b(:)g(:)g(:)f(;)h Fz(\()p Fp(N)2806 2743
y Fm(s\013)2888 2728 y Fz(\))2926 2743 y Fm(\013)p FA(\025)p
Fn(1)3111 2728 y Fz(b)s(e)47 b Fp(s)f Fz(inde-)324 2849
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y(v)-5 b(ariance)33 b(1)p Fp(=s)p Fz(.)43 b(F)-8 b(or)32
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Fp(X)1299 3224 y Fm(D)1291 3290 y(i;j)1399 3265 y Fz(=)1545
3141 y Fm(D)1503 3171 y Fk(X)1507 3380 y Fm(\013)p Fn(=1)1663
3265 y Fp(N)1741 3280 y Fm(j)t(\013)1823 3265 y Fp(X)1912
3214 y Fn(\()p Fm(\013)p Fn(\))1904 3291 y Fm(i)2016
3265 y Fp(;)115 b(j)33 b Fz(=)28 b(1)p Fp(;)17 b(:)g(:)g(:)e(;)i(s;)324
3551 y Fz(or,)32 b(in)h(matrix)g(form,)953 3674 y Fk(0)953
3849 y(B)953 3913 y(@)1082 3761 y Fp(X)1171 3724 y Fm(D)1163
3785 y(i;)p Fn(1)1151 3858 y Fz(.)1151 3891 y(.)1151
3924 y(.)1083 4044 y Fp(X)1172 4008 y Fm(D)1164 4069
y(i;s)1288 3674 y Fk(1)1288 3849 y(C)1288 3913 y(A)1402
3904 y Fz(=)1506 3674 y Fk(0)1506 3849 y(B)1506 3913
y(@)1635 3762 y Fp(N)1713 3777 y Fn(11)1871 3762 y Fp(:)17
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3891 y(.)1698 3924 y(.)1914 3858 y(.)1914 3891 y(.)1914
3924 y(.)2143 3858 y(.)2143 3891 y(.)2143 3924 y(.)1636
4044 y Fp(N)1714 4059 y Fm(s)p Fn(1)1871 4044 y Fp(:)17
b(:)g(:)83 b(N)2147 4059 y Fm(sD)2287 3674 y Fk(1)2287
3849 y(C)2287 3913 y(A)2391 3674 y(0)2391 3849 y(B)2391
3913 y(@)2532 3762 y Fp(X)2621 3711 y Fn(\(1\))2613 3788
y Fm(i)2610 3858 y Fz(.)2610 3891 y(.)2610 3924 y(.)2519
4061 y Fp(X)2608 4010 y Fn(\()p Fm(D)r Fn(\))2600 4086
y Fm(i)2769 3674 y Fk(1)2769 3849 y(C)2769 3913 y(A)2905
3904 y Fp(:)470 4270 y Fz(Conditioned)50 b(on)e Fp(X)1271
4285 y Fn(1)1311 4270 y Fp(;)17 b(:)g(:)g(:)e(;)i(X)1610
4285 y Fm(n)1657 4270 y Fz(,)53 b(for)48 b(\014xed)h
Fp(i)g Fz(and)g Fp(j)6 b Fz(,)52 b(the)d(sequence)i(\()p
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(enden)m(t)i(cen)m(tered)g(random)f(v)-5 b(ariables,)50
b(and)d(therefore)g(it)f(is)h(a)324 4511 y(martingale.)42
b(Moreo)m(v)m(er,)30 b(denoting)e(b)m(y)g Fj(E)1891 4526
y Fm(N)1986 4511 y Fz(exp)s(ectation)g(tak)m(en)h(with)f(resp)s(ect)g
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y Fp(;)17 b(:)g(:)g(:)f(;)h(X)2536 4646 y Fm(n)2582 4631
y Fz(\),)1182 4927 y Fj(E)1248 4942 y Fm(N)1316 4847
y Fk(\000)1361 4927 y Fp(X)1450 4886 y Fm(D)1442 4952
y(i;j)1522 4847 y Fk(\001)1568 4869 y Fn(2)1635 4927
y Fz(=)1781 4803 y Fm(D)1739 4833 y Fk(X)1743 5042 y
Fm(\013)p Fn(=1)1909 4773 y Fk(\000)1955 4854 y Fp(X)2044
4803 y Fn(\()p Fm(\013)p Fn(\))2036 4879 y Fm(i)2148
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4904 249 4 v 2487 4996 a Fp(s)2676 4927 y(:)1918 5251
y Fz(9)p eop end
%%Page: 10 10
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Fp(X)1303 512 y Fm(D)1295 573 y(i;j)1375 548 y Fz(\))1413
563 y Fm(D)r FA(\025)p Fn(1)1601 548 y Fz(is)e(a)f(martingale)g(b)s
(ounded)h(in)g Fp(L)2855 563 y Fn(2)2895 548 y Fz(.)48
b(Consequen)m(tly)-8 b(,)324 668 y(it)31 b(con)m(v)m(erges)j(almost)d
(surely)i(and)f(in)f Fp(L)1820 683 y Fn(2)1892 668 y
Fz(to)g(some)h(random)f(v)-5 b(ariable)32 b Fp(X)3055
683 y Fm(i;j)3166 668 y Fz(\(see,)h(e.g.,)324 789 y(Williams)d([45)o
(]\).)43 b(Moreo)m(v)m(er,)31 b(there)f(exists)g(a)f(random)g(v)-5
b(ariable)29 b Fp(Z)2813 804 y Fm(i;j)2922 789 y Fz(in)g
Fp(L)3098 804 y Fn(2)3166 789 y Fz(suc)m(h)i(that)324
909 y Fo(j)p Fp(X)441 873 y Fm(D)433 934 y(i;j)513 909
y Fo(j)c(\024)h Fp(Z)740 924 y Fm(i;j)820 909 y Fz(.)43
b(It)33 b(follo)m(ws)g(b)m(y)h(dominated)f(con)m(v)m(ergence)i(that)
1200 1201 y(lim)1167 1263 y Fm(D)r FA(!1)1384 1201 y
Fj(E)1450 1216 y Fm(N)1518 1120 y Fk(\000)1564 1201 y
Fp(X)1653 1160 y Fm(D)1645 1226 y(i;j)1725 1120 y Fk(\001)1770
1143 y Fn(2)1838 1201 y Fo(!)27 b Fj(E)2031 1216 y Fm(N)2099
1201 y Fp(X)2188 1160 y Fn(2)2180 1226 y Fm(i;j)2287
1201 y Fz(=)2401 1134 y Fo(k)p Fp(X)2532 1149 y Fm(i)2560
1134 y Fo(k)2610 1097 y Fn(2)p 2401 1178 249 4 v 2502
1269 a Fp(s)2692 1201 y(:)470 1469 y Fz(The)33 b(v)m(ector)p
961 1389 89 4 v 32 w Fp(X)1050 1484 y Fm(i)1106 1469
y Fz(=)27 b(\()p Fp(X)1328 1484 y Fm(i;)p Fn(1)1411 1469
y Fp(;)17 b(:)g(:)g(:)f(;)h(X)1711 1484 y Fm(i;s)1791
1469 y Fz(\))28 b Fo(2)g Fj(R)2023 1433 y Fm(s)2091 1469
y Fz(ma)m(y)k(b)s(e)g(regarded)g(as)f(a)h Fh(random)f(pro-)324
1589 y(jection)j Fz(of)f Fp(X)835 1604 y Fm(i)897 1589
y Fz(to)g(an)h Fp(s)p Fz(-dimensional)g(subspace.)49
b(Clearly)-8 b(,)35 b(for)e(\014xed)i Fp(X)3046 1604
y Fm(i)3074 1589 y Fz(,)f(eac)m(h)h(com-)324 1710 y(p)s(onen)m(t)c
Fp(X)728 1725 y Fm(i;j)839 1710 y Fz(is)h(a)e(normal)h(random)g(v)-5
b(ariable)31 b(with)h(mean)g(0)e(and)h(v)-5 b(ariance)32
b Fo(k)p Fp(X)3323 1725 y Fm(i)3351 1710 y Fo(k)3401
1673 y Fn(2)3440 1710 y Fp(=s)p Fz(.)324 1830 y(Therefore,)j
Fj(E)860 1845 y Fm(N)927 1830 y Fo(k)p 977 1750 V Fp(X)1066
1845 y Fm(i)1094 1830 y Fo(k)1144 1794 y Fn(2)1213 1830
y Fz(=)29 b Fo(k)p Fp(X)1449 1845 y Fm(i)1477 1830 y
Fo(k)1527 1794 y Fn(2)1599 1830 y Fz(and)34 b(\(with)g
Fp(X)2132 1845 y Fm(i)2194 1830 y Fz(\014xed\))g Fp(s)p
Fo(k)p 2564 1750 V Fp(X)2653 1845 y Fm(i)2681 1830 y
Fo(k)2731 1794 y Fn(2)2770 1830 y Fp(=)p Fo(k)p Fp(X)2950
1845 y Fm(i)2978 1830 y Fo(k)3028 1794 y Fn(2)3101 1830
y Fz(has)f Fp(\037)3336 1794 y Fn(2)3409 1830 y Fz(dis-)324
1950 y(tribution)g(with)g Fp(s)f Fz(degrees)i(of)e(freedom.)45
b(Similarly)-8 b(,)33 b(for)f(an)m(y)h Fp(i)28 b Fo(6)p
Fz(=)g Fp(i)2863 1914 y FA(0)2914 1950 y Fo(2)g(f)p Fz(1)p
Fp(;)17 b(:)g(:)g(:)f(;)h(n)p Fo(g)p Fz(,)1301 2170 y
Fj(E)1367 2185 y Fm(N)1435 2170 y Fo(k)p 1485 2090 V
Fp(X)1573 2185 y Fm(i)1624 2170 y Fo(\000)p 1723 2090
V 22 w Fp(X)1812 2185 y Fm(i)1836 2166 y Fd(0)1863 2170
y Fo(k)1913 2129 y Fn(2)1979 2170 y Fz(=)28 b Fo(k)p
Fp(X)2214 2185 y Fm(i)2264 2170 y Fo(\000)23 b Fp(X)2445
2185 y Fm(i)2469 2166 y Fd(0)2495 2170 y Fo(k)2545 2129
y Fn(2)324 2390 y Fz(and)29 b Fp(s)p Fo(k)p 606 2310
V Fp(X)695 2405 y Fm(i)738 2390 y Fo(\000)p 830 2310
V 15 w Fp(X)920 2405 y Fm(i)944 2386 y Fd(0)971 2390
y Fo(k)1021 2354 y Fn(2)1060 2390 y Fp(=)p Fo(k)p Fp(X)1240
2405 y Fm(i)1283 2390 y Fo(\000)15 b Fp(X)1456 2405 y
Fm(i)1480 2386 y Fd(0)1508 2390 y Fo(k)1558 2354 y Fn(2)1626
2390 y Fz(has)30 b Fp(\037)1858 2354 y Fn(2)1927 2390
y Fz(distribution)g(with)g Fp(s)f Fz(degrees)i(of)e(freedom.)324
2511 y(No)m(w)34 b(b)m(y)h(a)f(simple)h(Cherno\013)g(b)s(ound)f
(\(Cherno\013)h([10]\))f(for)f(the)i Fp(\037)2817 2475
y Fn(2)2890 2511 y Fz(distribution,)h(w)m(e)324 2631
y(ha)m(v)m(e)726 2795 y Fj(P)787 2810 y Fm(N)854 2654
y Fk(\032)939 2727 y Fo(k)p 989 2647 V Fp(X)1077 2742
y Fm(i)1128 2727 y Fo(\000)p 1227 2647 V 22 w Fp(X)1316
2742 y Fm(i)1340 2723 y Fd(0)1367 2727 y Fo(k)1417 2691
y Fn(2)p 939 2772 518 4 v 947 2863 a Fo(k)p Fp(X)1078
2878 y Fm(i)1128 2863 y Fo(\000)22 b Fp(X)1308 2878 y
Fm(i)1332 2859 y Fd(0)1359 2863 y Fo(k)1409 2834 y Fn(2)1488
2795 y Fo(\000)h Fz(1)k Fp(>)h(")1814 2654 y Fk(\033)1916
2795 y Fo(\024)g Fz(exp)2186 2684 y Fk(h)2245 2727 y
Fp(s)p 2243 2772 49 4 v 2243 2863 a Fz(2)2302 2714 y
Fk(\000)2370 2795 y Fo(\000)23 b Fp(")e Fz(+)h(ln)q(\(1)g(+)g
Fp(")p Fz(\))3008 2714 y Fk(\001)3053 2684 y(i)3132 2795
y Fp(;)324 3021 y Fz(and,)33 b(similarly)-8 b(,)747 3304
y Fj(P)808 3319 y Fm(N)875 3163 y Fk(\032)960 3236 y
Fo(k)p 1010 3156 89 4 v Fp(X)1099 3251 y Fm(i)1149 3236
y Fo(\000)p 1249 3156 V 23 w Fp(X)1337 3251 y Fm(i)1361
3232 y Fd(0)1388 3236 y Fo(k)1438 3200 y Fn(2)p 960 3281
518 4 v 968 3372 a Fo(k)p Fp(X)1099 3387 y Fm(i)1149
3372 y Fo(\000)23 b Fp(X)1330 3387 y Fm(i)1354 3368 y
Fd(0)1380 3372 y Fo(k)1430 3343 y Fn(2)1509 3304 y Fo(\000)g
Fz(1)k Fp(<)h Fo(\000)p Fp(")1912 3163 y Fk(\033)2014
3304 y Fo(\024)h Fz(exp)2285 3193 y Fk(h)2344 3236 y
Fp(s)p 2342 3281 49 4 v 2342 3372 a Fz(2)2401 3223 y
Fk(\000)2447 3304 y Fp(")21 b Fz(+)h(ln)q(\(1)g Fo(\000)g
Fp(")p Fz(\))2986 3223 y Fk(\001)3031 3193 y(i)3111 3304
y Fp(:)324 3575 y Fz(By)33 b(the)g(union)g(b)s(ound)g(w)m(e)g(obtain)g
(the)g(follo)m(wing:)324 3804 y Fl(Theorem)38 b(3.1)f
(\(Johnson-Lindenstrauss)k(lemma\))50 b Fe(L)-5 b(et)25
b Fo(H)h Fe(b)-5 b(e)25 b(a)f(sep)-5 b(ar)g(able)24 b(Hilb)-5
b(ert)324 3924 y(sp)g(ac)g(e.)44 b(F)-7 b(or)34 b(any)g
Fp(";)17 b(\016)31 b Fo(2)d Fz(\(0)p Fp(;)17 b Fz(1\))34
b Fe(and)h(any)f(p)-5 b(ositive)35 b(inte)-5 b(ger)34
b Fp(n)p Fe(,)h(let)g Fp(s)g Fe(b)-5 b(e)34 b(a)h(p)-5
b(ositive)34 b(in-)324 4044 y(te)-5 b(ger)35 b(such)f(that)1315
4182 y Fp(s)27 b Fo(\025)i Fz(4\()p Fp(")1627 4141 y
Fn(2)1665 4182 y Fp(=)p Fz(2)22 b Fo(\000)h Fp(")1931
4141 y Fn(3)1970 4182 y Fp(=)p Fz(3\))2106 4141 y FA(\000)p
Fn(1)2216 4182 y Fz(log)2405 4115 y Fp(n)p 2369 4159
130 4 v 2369 4179 a Fo(p)p 2452 4179 47 4 v 85 x Fp(\016)2544
4182 y(:)324 4407 y Fe(De\014ne)42 b(a)g(line)-5 b(ar)43
b(map)f Fp(f)53 b Fz(:)43 b Fo(H)h(!)e Fj(R)1751 4370
y Fm(s)1831 4407 y Fe(as)h(a)g(r)-5 b(andom)42 b(pr)-5
b(oje)g(ction)42 b(describ)-5 b(e)g(d)42 b(ab)-5 b(ove.)324
4527 y(Then,)35 b(for)h(any)g(set)g Fo(D)i Fe(of)e Fp(n)g
Fe(p)-5 b(oints)35 b(in)h Fo(H)q Fe(,)g(with)g(pr)-5
b(ob)g(ability)35 b(at)i(le)-5 b(ast)35 b Fz(1)23 b Fo(\000)g
Fp(\016)t Fe(,)36 b(for)g(al)5 b(l)324 4647 y Fz(\()p
Fp(u;)17 b(v)t Fz(\))26 b Fo(2)i Fp(D)755 4611 y Fn(2)795
4647 y Fe(,)802 4867 y Fz(\(1)22 b Fo(\000)h Fp(")p Fz(\))p
Fo(k)p Fp(u)e Fo(\000)i Fp(v)t Fo(k)1423 4826 y Fn(2)1489
4867 y Fo(\024)28 b(k)p Fp(f)11 b Fz(\()p Fp(u)p Fz(\))21
b Fo(\000)i Fp(f)11 b Fz(\()p Fp(v)t Fz(\))p Fo(k)2192
4826 y Fn(2)2258 4867 y Fo(\024)29 b Fz(\(1)21 b(+)h
Fp(")p Fz(\))p Fo(k)p Fp(u)f Fo(\000)i Fp(v)t Fo(k)2982
4826 y Fn(2)3056 4867 y Fp(:)1894 5251 y Fz(10)p eop
end
%%Page: 11 11
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(particularly)h(w)m(ell)h(suited)f(for)f(the)g(purp)s(oses)h(of)f
Fp(k)s Fz(-)324 789 y(means)33 b(clustering.)470 909
y(Let)24 b Fp(s)k Fz(=)g Fo(d)p Fz(4\()p Fp(")991 873
y Fn(2)1030 909 y Fp(=)p Fz(2)5 b Fo(\000)g Fp(")1261
873 y Fn(3)1300 909 y Fp(=)p Fz(3\))1436 873 y FA(\000)p
Fn(1)1546 909 y Fz(log)q(\()p Fp(n=)1817 824 y Fo(p)p
1900 824 47 4 v 85 x Fp(\016)t Fz(\))p Fo(e)24 b Fz(and)h(let)p
2367 829 89 4 v 24 w Fp(X)2456 924 y Fn(1)2495 909 y
Fp(;)17 b(:)g(:)g(:)f(;)p 2714 829 V 17 w(X)2803 924
y Fm(n)2877 909 y Fo(2)28 b Fj(R)3043 873 y Fm(s)3104
909 y Fz(b)s(e)d(the)f(ran-)324 1029 y(domly)e(pro)5
b(jected)24 b(data)d(p)s(oin)m(ts)i Fp(X)1604 1044 y
Fn(1)1643 1029 y Fp(;)17 b(:)g(:)g(:)f(;)h(X)1943 1044
y Fm(n)1990 1029 y Fz(.)40 b(W)-8 b(e)22 b(prop)s(ose)g(the)g(follo)m
(wing)h(algorithm)p 3123 1065 412 4 v(:)470 1270 y(Determine)j(a)e
(collection)p 1437 1214 50 4 v 26 w Fl(c)1487 1285 y
Fm(n)1561 1270 y Fz(=)k(\()p 1703 1215 42 4 v Fp(c)1745
1285 y Fm(n)p Fn(1)1827 1270 y Fp(;)17 b(:)g(:)g(:)f(;)p
2046 1215 V 17 w(c)2087 1285 y Fm(nk)2173 1270 y Fz(\))27
b Fo(2)i Fz(\()p Fj(R)2443 1234 y Fm(s)2479 1270 y Fz(\))2517
1234 y Fm(k)2584 1270 y Fz(of)c(cluster)h(cen)m(ters)g(whic)m(h)324
1391 y(minimizes)44 b(the)f(empirical)h(clustering)g(risk)f(in)g
Fj(R)2248 1354 y Fm(s)2327 1391 y Fz(based)g(on)f(the)h(pro)5
b(jected)44 b(data)p 324 1431 89 4 v 324 1511 a Fp(X)413
1526 y Fn(1)452 1511 y Fp(;)17 b(:)g(:)g(:)f(;)p 671
1431 V 17 w(X)759 1526 y Fm(n)806 1511 y Fz(,)26 b(and)d(let)p
1171 1431 66 4 v 24 w Fp(S)1237 1526 y Fm(n)p Fn(1)1319
1511 y Fp(;)17 b(:)g(:)g(:)f(;)p 1538 1431 V 17 w(S)1603
1526 y Fm(nk)1716 1511 y Fo(\032)29 b Fj(R)1894 1475
y Fm(s)1954 1511 y Fz(b)s(e)23 b(the)h(asso)s(ciated)g(V)-8
b(oronoi)23 b(cells.)42 b(De\014ne)324 1631 y(the)33
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%%Page: 16 16
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%%Page: 18 18
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Fm(c)1991 2042 y Fe(b)-5 b(e)34 b(the)h(r)-5 b(e)g(al-value)g(d)34
b(map)g(de\014ne)-5 b(d)34 b(by)1217 2262 y Fp(`)1258
2277 y Fm(c)1292 2262 y Fz(\()p Fp(x)p Fz(\))28 b(=)g
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Fo(k)p Fp(c)p Fo(k)2162 2220 y Fn(2)2201 2262 y Fp(;)116
b(x)28 b Fo(2)h(H)36 b Fp(:)324 2482 y Fe(Then)e(the)h(fol)5
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2685 y Fz(\()p Fp(i)p Fz(\))709 2981 y Fj(E)h Fz(sup)792
3074 y Fg(c)p FA(2F)932 3051 y Fi(k)925 3097 y(R)993
2900 y Fk(\000)p 1039 2901 106 4 v 81 x Fp(W)13 b Fz(\()p
Fl(c)p Fp(;)k(\026)1335 2996 y Fm(n)1381 2981 y Fz(\))22
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Fp(;)j(\026)p Fz(\))1876 2900 y Fk(\001)1948 2981 y Fo(\024)28
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3051 y Fi(k)2318 3097 y(R)2400 2913 y Fz(1)p 2395 2958
59 4 v 2395 3049 a Fp(n)2531 2856 y Fm(n)2480 2886 y
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2996 y Fm(i)2724 2900 y Fk(\002)2831 2981 y Fz(min)2782
3044 y Fm(j)t Fn(=1)p Fm(;:::)n(;k)3058 2981 y Fp(`)3099
2996 y Fm(c)3130 3006 y Fi(j)3166 2981 y Fz(\()p Fp(X)3285
2996 y Fm(i)3313 2981 y Fz(\))3351 2900 y Fk(\003)3393
2981 y Fp(:)376 3332 y Fz(\()p Fp(ii)p Fz(\))631 3627
y Fj(E)h Fz(sup)714 3721 y Fg(c)p FA(2F)854 3698 y Fi(k)847
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3627 y Fz(min)1312 3690 y Fm(j)t Fn(=1)p Fm(;:::)n(;k)1587
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3547 y Fk(\003)1950 3627 y Fo(\024)28 b Fz(2)p Fp(k)2158
3517 y Fk(h)2205 3627 y Fj(E)k Fz(sup)2288 3709 y Fm(c)p
FA(2F)2416 3720 y Fi(R)2497 3560 y Fz(1)p 2493 3605 V
2493 3696 a Fp(n)2628 3503 y Fm(n)2577 3533 y Fk(X)2592
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3627 y Fo(h)p Fp(X)2942 3642 y Fm(i)2969 3627 y Fp(;)17
b(c)p Fo(i)22 b Fz(+)3262 3560 y Fp(R)3337 3524 y Fn(2)p
3224 3605 191 4 v 3224 3696 a Fz(2)3273 3624 y Fo(p)p
3356 3624 59 4 v 72 x Fp(n)3424 3517 y Fk(i)3471 3627
y Fp(:)343 3978 y Fz(\()p Fp(iii)p Fz(\))1287 4181 y
Fj(E)33 b Fz(sup)1370 4262 y Fm(c)p FA(2F)1498 4273 y
Fi(R)1580 4114 y Fz(1)p 1575 4158 V 1575 4249 a Fp(n)1711
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%%Page: 19 19
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668 y(the)33 b Fp(\033)547 683 y Fm(i)575 668 y Fz('s.)44
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y Fz(\()p Fp(X)2519 1170 y FA(0)2511 1236 y Fm(i)2542
1211 y Fz(\))2580 1130 y Fk(\003)532 1534 y Fo(\024)28
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a Fp(n)1066 1409 y Fm(n)1015 1439 y Fk(X)1030 1649 y
Fm(i)p Fn(=1)1176 1534 y Fp(\033)1231 1549 y Fm(i)1260
1453 y Fk(\002)1366 1534 y Fz(min)1318 1597 y Fm(j)t
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1559 y Fi(j)1701 1534 y Fz(\()p Fp(X)1820 1549 y Fm(i)1849
1534 y Fz(\))1887 1453 y Fk(\003)1950 1534 y Fz(+)22
b Fj(E)36 b Fz(sup)2131 1627 y Fg(c)p FA(2F)2271 1605
y Fi(k)2264 1651 y(R)2346 1467 y Fz(1)p 2342 1511 V 2342
1602 a Fp(n)2477 1409 y Fm(n)2426 1439 y Fk(X)2441 1649
y Fm(i)p Fn(=1)2570 1534 y Fz(\()p Fo(\000)p Fp(\033)2740
1549 y Fm(i)2769 1534 y Fz(\))2807 1453 y Fk(\002)2914
1534 y Fz(min)2865 1597 y Fm(j)t Fn(=1)p Fm(;:::)n(;k)3141
1534 y Fp(`)3182 1549 y Fm(c)3213 1559 y Fi(j)3249 1534
y Fz(\()p Fp(X)3376 1493 y FA(0)3368 1559 y Fm(i)3399
1534 y Fz(\))3437 1453 y Fk(\003)532 1857 y Fz(=)28 b(2)p
Fj(E)35 b Fz(sup)768 1950 y Fg(c)p FA(2F)908 1927 y Fi(k)901
1973 y(R)982 1789 y Fz(1)p 978 1834 V 978 1925 a Fp(n)1113
1732 y Fm(n)1063 1762 y Fk(X)1077 1972 y Fm(i)p Fn(=1)1223
1857 y Fp(\033)1278 1872 y Fm(i)1307 1776 y Fk(\002)1413
1857 y Fz(min)1365 1920 y Fm(j)t Fn(=1)p Fm(;:::)n(;k)1640
1857 y Fp(`)1681 1872 y Fm(c)1712 1882 y Fi(j)1749 1857
y Fz(\()p Fp(X)1868 1872 y Fm(i)1896 1857 y Fz(\))1934
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Fz(\))98 b(T)-8 b(o)38 b(pro)m(v)m(e)i(statemen)m(t)g(\()p
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Fp(R)1298 3229 y Fm(n)1345 3214 y Fz(\()p Fo(G)1442 3229
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3229 y Fn(1)2900 3214 y Fz(+)23 b Fp(g)3046 3229 y Fn(2)3116
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y Fp(;)17 b(g)3388 3229 y Fn(2)3427 3214 y Fz(\))30 b
Fo(2)568 3335 y(G)627 3350 y Fn(1)688 3335 y Fo(\002)23
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3829 y Fj(E)f Fz(sup)573 3910 y Fm(c)p FA(2F)701 3921
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%%Page: 20 20
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(con)m(v)m(ergence)k(in)d(the)524 3861 y(source)39 b(co)s(ding)e
(theorem,)j(in)e(empirical)g(quan)m(tizer)i(design,)f(and)f(in)g(univ)m
(ersal)524 3981 y(lossy)28 b(source)f(co)s(ding,)h Fe(IEEE)g(T)-7
b(r)i(ansactions)28 b(on)g(Information)g(The)-5 b(ory)p
Fz(,)27 b(V)-8 b(ol.)26 b(40,)524 4101 y(pp.)33 b(1728{1740.)324
4305 y([32])48 b(Linial,)53 b(N.,)f(London,)h(E.)48 b(and)h(Rabino)m
(vic)m(h,)54 b(Y.)48 b(\(1995\).)g(The)h(geometry)g(of)524
4425 y(graphs)27 b(and)g(some)h(of)e(its)i(algorithmic)f(applications,)
i Fe(Combinatoric)-5 b(a)p Fz(,)26 b(V)-8 b(ol.)27 b(15,)524
4545 y(pp.)33 b(215{245.)324 4749 y([33])48 b(Mardia,)28
b(K.V.,)f(Ken)m(t,)h(J.T.)e(and)g(Bibb)m(y)-8 b(,)28
b(J.M.)e(\(1979\).)f Fe(Multivariate)k(A)n(nalysis)p
Fz(,)524 4869 y(Academic)34 b(Press,)h(London.)1894 5251
y(26)p eop end
%%Page: 27 27
TeXDict begin 27 26 bop 324 548 a Fz([34])48 b(McDiarmid,)38
b(C.)f(\(1989\).)f(On)g(the)h(metho)s(d)f(of)g(b)s(ounded)h
(di\013erences,)j(in)d Fe(Sur-)524 668 y(veys)f(in)g(Combinatorics)e
(1989)p Fz(,)g(pp.)g(148{188,)e(Cam)m(bridge)j(Univ)m(ersit)m(y)i
(Press,)524 789 y(Cam)m(bridge.)324 992 y([35])48 b(P)m(ollard,)26
b(D.)c(\(1981\).)g(Strong)g(consistency)k(of)c Fp(k)s
Fz(-means)h(clustering,)j Fe(The)g(A)n(nnals)524 1112
y(of)35 b(Statistics)p Fz(,)e(V)-8 b(ol.)32 b(9,)g(pp.)h(135{140.)324
1316 y([36])48 b(P)m(ollard,)33 b(D.)f(\(1982\).)f(A)h(cen)m(tral)g
(limit)h(theorem)g(for)e Fp(k)s Fz(-means)i(clustering,)g
Fe(The)524 1436 y(A)n(nnals)h(of)h(Pr)-5 b(ob)g(ability)p
Fz(,)32 b(V)-8 b(ol.)33 b(10,)f(pp.)h(919{926.)324 1640
y([37])48 b(P)m(ollard,)53 b(D.)47 b(\(1982\).)g(Quan)m(tization)h(and)
g(the)h(metho)s(d)f(of)f Fp(k)s Fz(-means,)53 b Fe(IEEE)524
1760 y(T)-7 b(r)i(ansactions)33 b(on)i(Information)e(The)-5
b(ory)p Fz(,)33 b(V)-8 b(ol.)32 b(28,)g(pp.)h(199{205.)324
1963 y([38])48 b(Rac)m(hev,)e(S.T.)e(and)e(R)s(\177)-51
b(usc)m(hendorf,)46 b(L.)c(\(1998\).)g Fe(Mass)h(T)-7
b(r)i(ansp)g(ortation)43 b(Pr)-5 b(ob-)524 2084 y(lems,)34
b(V)-7 b(olume)35 b(I:)f(The)-5 b(ory)p Fz(,)32 b(Springer-V)-8
b(erlag,)33 b(New)g(Y)-8 b(ork.)324 2287 y([39])48 b(Rac)m(hev,)e(S.T.)
e(and)e(R)s(\177)-51 b(usc)m(hendorf,)46 b(L.)c(\(1998\).)g
Fe(Mass)h(T)-7 b(r)i(ansp)g(ortation)43 b(Pr)-5 b(ob-)524
2408 y(lems,)34 b(V)-7 b(olume)35 b(II:)f(Applic)-5 b(ations)p
Fz(,)31 b(Springer-V)-8 b(erlag,)33 b(New)h(Y)-8 b(ork.)324
2611 y([40])48 b(Ramsa)m(y)-8 b(,)44 b(J.O.)d(and)g(Silv)m(erman,)j
(B.W.)d(\(1997\).)f Fe(F)-7 b(unctional)41 b(Data)h(A)n(nalysis)p
Fz(,)524 2731 y(Springer,)34 b(New)f(Y)-8 b(ork.)324
2935 y([41])48 b(Sc)m(h\177)-49 b(olk)m(opf,)34 b(B.)e(and)g(Smola,)g
(A.J.)h(\(2002\).)e Fe(L)-5 b(e)g(arning)33 b(with)h(Kernels)p
Fz(,)e(The)h(MIT)524 3055 y(Press,)i(Cam)m(bridge.)324
3259 y([42])48 b(Sha)m(w)m(e-T)-8 b(a)m(ylor,)30 b(J.)d(and)g
(Cristianini,)i(N.)e(\(2004\).)f Fe(Kernel)j(Metho)-5
b(ds)30 b(for)f(Pattern)524 3379 y(A)n(nalysis)p Fz(,)j(Cam)m(bridge)i
(Univ)m(ersit)m(y)h(Press,)g(Cam)m(bridge.)324 3582 y([43])48
b(V)-8 b(apnik,)40 b(V.N.)e(\(1998\).)e Fe(Statistic)-5
b(al)40 b(L)-5 b(e)g(arning)38 b(The)-5 b(ory)p Fz(,)39
b(John)f(Wiley)h(&)e(Sons,)524 3703 y(New)d(Y)-8 b(ork.)324
3906 y([44])48 b(V)-8 b(empala,)49 b(S.)c(and)g(W)-8
b(ang,)48 b(G.)c(\(2002\).)g(A)h(sp)s(ectral)g(algorithm)g(for)f
(learning)524 4027 y(mixtures)35 b(of)d(distributions,)j(in)e
Fe(Pr)-5 b(o)g(c)g(e)g(e)g(dings)34 b(of)h(the)g(43r)-5
b(d)35 b(IEEE)g(F)-7 b(oundations)524 4147 y(of)35 b(Computer)f(Scienc)
-5 b(e)34 b(\(F)n(OCS)f('02\))p Fz(.)324 4350 y([45])48
b(Williams,)27 b(D.)d(\(1991\).)g Fe(Pr)-5 b(ob)g(ability)27
b(with)g(Martingales)p Fz(,)e(Cam)m(bridge)h(Univ)m(ersit)m(y)524
4471 y(Press,)35 b(Cam)m(bridge.)1894 5251 y(27)p eop
end
%%Trailer

userdict /end-hook known{end-hook}if
%%EOF