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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 82 /R put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
/UniqueXX 5031982 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02

% 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 (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 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 43 /plus put
dup 48 /zero put
dup 49 /one put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueXX 5000789 def
currentdict end
currentfile eexec
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cleartomark

%%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 0 /minus put
dup 3 /asteriskmath put
readonly def
/FontBBox{-4 -948 1329 786}readonly def
/UniqueXX 5000816 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 12 /vextendsingle put
dup 16 /parenleftBig put
dup 17 /parenrightBig put
dup 32 /parenleftBigg put
dup 33 /parenrightBigg put
dup 48 /parenlefttp put
dup 49 /parenrighttp put
dup 63 /arrowvertex put
dup 64 /parenleftbt put
dup 65 /parenrightbt put
dup 66 /parenleftex put
dup 67 /parenrightex put
dup 91 /uniondisplay put
dup 101 /tildewide put
dup 120 /arrowtp put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueXX 5000774 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 11 /alpha put
dup 71 /G put
dup 78 /N put
dup 103 /g put
dup 105 /i put
dup 107 /k put
dup 108 /l put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueXX 5087381 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 8 /circleplus put
dup 10 /circlemultiply put
dup 14 /openbullet put
dup 18 /reflexsubset put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 24 /similar put
dup 26 /propersubset put
dup 32 /arrowleft put
dup 33 /arrowright put
dup 50 /element put
dup 54 /negationslash put
dup 55 /mapsto put
dup 56 /universal put
dup 68 /D put
dup 70 /F put
dup 84 /T put
dup 86 /V put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
dup 110 /backslash put
dup 112 /radical put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueXX 5000820 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 11 /ff put
dup 12 /fi put
dup 34 /quotedblright 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 54 /six put
dup 58 /colon put
dup 59 /semicolon put
dup 65 /A put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 72 /H put
dup 73 /I put
dup 76 /L put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 83 /S put
dup 84 /T put
dup 91 /bracketleft put
dup 92 /quotedblleft put
dup 93 /bracketright 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
readonly def
/FontBBox{-36 -251 1103 750}readonly def
/UniqueXX 5000829 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 17 /eta put
dup 22 /mu put
dup 23 /nu put
dup 24 /xi put
dup 58 /period put
dup 59 /comma put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 71 /G put
dup 74 /J put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 83 /S put
dup 84 /T put
dup 89 /Y put
dup 97 /a put
dup 98 /b 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 116 /t put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueXX 5087383 def
currentdict end
currentfile eexec
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cleartomark

%%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 48 /prime put
dup 49 /infinity put
dup 50 /element put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
/UniqueXX 5000818 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 16 /zeta put
dup 17 /eta put
dup 18 /theta put
dup 22 /mu put
dup 23 /nu put
dup 24 /xi put
dup 25 /pi put
dup 27 /sigma put
dup 28 /tau put
dup 30 /phi put
dup 31 /chi put
dup 32 /psi put
dup 44 /arrowhookleft put
dup 58 /period put
dup 59 /comma put
dup 61 /slash put
dup 62 /greater put
dup 64 /partialdiff 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 78 /N put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
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dup 86 /V put
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dup 88 /X put
dup 89 /Y put
dup 90 /Z put
dup 97 /a put
dup 98 /b 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 108 /l put
dup 109 /m put
dup 110 /n put
dup 112 /p 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
/UniqueXX 5087386 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 1 /Delta put
dup 8 /Phi put
dup 11 /ff put
dup 12 /fi put
dup 13 /fl put
dup 14 /ffi put
dup 34 /quotedblright 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 63 /question 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 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 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 87 /W put
dup 88 /X put
dup 91 /bracketleft put
dup 92 /quotedblleft 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
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueXX 5000794 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMBX12
%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54

% 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 (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX12 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 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 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 73 /I put
dup 76 /L put
dup 79 /O 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 107 /k 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
dup 119 /w put
dup 120 /x put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueXX 5000769 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMSL10
%!PS-AdobeFont-1.1: CMSL10 1.0
%%CreationDate: 1991 Aug 20 16:40:20

% 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 (CMSL10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -9.46 def
/isFixedPitch false def
end readonly def
/FontName /CMSL10 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 /zero put
dup 49 /one put
dup 50 /two put
dup 54 /six put
dup 69 /E put
dup 77 /M put
dup 80 /P put
dup 97 /a put
dup 98 /b put
dup 100 /d put
dup 101 /e 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
dup 121 /y put
readonly def
/FontBBox{-62 -250 1123 750}readonly def
/UniqueXX 5000798 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMTT10
%!PS-AdobeFont-1.1: CMTT10 1.00B
%%CreationDate: 1992 Apr 26 10:42:42

% 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 (CMTT10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
end readonly def
/FontName /CMTT10 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 64 /at put
dup 97 /a put
dup 98 /b 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 107 /k 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
dup 120 /x put
dup 121 /y put
readonly def
/FontBBox{-4 -235 731 800}readonly def
/UniqueXX 5000832 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 14 /ffi put
dup 34 /quotedblright put
dup 38 /ampersand 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 47 /slash 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 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 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 92 /quotedblleft put
dup 93 /bracketright 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 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{-251 -250 1009 969}readonly def
/UniqueXX 5000793 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 50 /two put
dup 56 /eight put
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
/UniqueXX 5000768 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 11 /ff put
dup 12 /fi put
dup 19 /acute 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 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight 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 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 80 /P put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 88 /X 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 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
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueXX 5000828 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 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 61 /equal put
dup 77 /M put
dup 79 /O put
dup 97 /a put
dup 98 /b put
dup 101 /e put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 118 /v put
dup 120 /x put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueXX 5000791 def
currentdict end
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eop
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6000 825 m
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5700 900 m
gs 1 -1 sc (B) col0 sh gr
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5550 825 m
gs 1 -1 sc (\() col0 sh gr
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5325 900 m
gs 1 -1 sc (t) col0 sh gr
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5175 900 m
gs 1 -1 sc (\() col0 sh gr
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8775 825 m
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7425 675 m
gs 1 -1 sc (\() col0 sh gr
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7575 675 m
gs 1 -1 sc (N) col0 sh gr
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7875 675 m
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2315 y(sections)34 b(\(for)e(more)h(details)h(see,)f(e.g.,)g
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2501 y Fm(1)3083 2537 y Fo(\()p Fn(B)5 b Fo(\))37 b(con-)166
2658 y(sisting)g(of)e(all)g(functions)h(v)-5 b(anishing)37
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3813 y(for)e(the)h(section)h(\(\\the)f(\014eld"\))g Fn( )j
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4054 y(the)d(T)-8 b(a)m(ylor)30 b(series)i(of)d Fn( )k
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y Fl(k)1069 4279 y Fo(:)d Fn(C)1200 4238 y Fm(1)1275
4279 y Fo(\()p Fn(\030)5 b Fo(\))27 b Fj(\000)-16 b(!)27
b Fn(C)1691 4238 y Fm(1)1766 4279 y Fo(\()p Fn(J)1867
4238 y Fl(k)1910 4279 y Fo(\()p Fn(\030)5 b Fo(\)\))26
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y Fl(k)2053 4953 y Fo(\()p Fn(\030)5 b Fo(\))2177 4968
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y Fn(;)17 b(u)1798 5219 y Fl(a)1839 5255 y Fo(\))35 b(\()p
Fn(\026)30 b Fo(=)h(1)p Fn(;)17 b(:)g(:)g(:)f(;)h Fo(dim)f
Fn(B)5 b Fo(;)35 b Fn(a)c Fo(=)g(1)p Fn(;)17 b(:)g(:)g(:)e(;)i
Fo(dim)g Fn(\030)5 b Fo(\))166 5375 y(of)33 b Fn(\030)38
b Fo(generate)c(lo)s(cal)f(co)s(ordinates)h(\()p Fn(x)1593
5339 y Fl(\026)1640 5375 y Fn(;)17 b(u)1740 5339 y Fl(a)1781
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y Fj(\024)g Fn(:)17 b(:)g(:)29 b Fj(\024)g Fn(\026)3319
5390 y Fl(i)3376 5375 y Fj(\024)1769 5712 y Fo(8)p eop
%%Page: 9 9
9 8 bop 166 83 a Fo(dim)17 b Fn(B)5 b Fo(;)33 b Fn(i)28
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Fn(J)1214 47 y Fl(k)1257 83 y Fo(\()p Fn(\030)5 b Fo(\),)32
b(where)969 369 y Fn(u)1025 328 y Fl(a)1025 393 y(\026)1067
402 y Ff(1)1101 393 y Fl(:::)o(\026)1202 403 y Fi(i)1249
272 y Fh(\020)1299 369 y Fn(J)1362 328 y Fl(k)1405 369
y Fo(\()p Fn( )t Fo(\)\()p Fn(b)p Fo(\))1665 272 y Fh(\021)1742
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527 4 v 1883 437 a Fn(@)5 b(x)1994 408 y Fl(\026)2036
417 y Ff(1)2093 437 y Fn(:)17 b(:)g(:)f(@)5 b(x)2335
408 y Fl(\026)2377 418 y Fi(i)2436 369 y Fn( )2503 328
y Fl(a)2545 369 y Fo(\()p Fn(b)p Fo(\))33 b Fn(;)607
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y Fo(\()p Fn(\030)5 b Fo(\))27 b(:=)g Fn(\030)5 b Fo(.)32
b(The)i(dimension)g(of)e(the)h(v)m(ector)h(bundle)f Fn(J)2273
926 y Fl(k)2316 963 y Fo(\()p Fn(\030)5 b Fo(\))32 b(is)1098
1250 y(dim)18 b Fn(J)1341 1209 y Fl(k)1383 1250 y Fo(\()p
Fn(\030)5 b Fo(\))27 b(=)1638 1104 y Fh( )1704 1183 y
Fo(dim)17 b Fn(B)27 b Fo(+)22 b Fn(k)1893 1319 y(k)2136
1104 y Fh(!)2235 1250 y Fo(dim)17 b Fn(\030)37 b(:)166
1545 y Fo(Let)g Fn(\031)404 1509 y Fl(k)r(;l)523 1545
y Fo(:)d Fn(J)647 1509 y Fl(k)690 1545 y Fo(\()p Fn(\030)5
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Fo(\()p Fn(\030)5 b Fo(\),)36 b Fn(k)h(>)e(l)i Fj(\025)e
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1666 y(o\013)7 b(")32 b(all)g(deriv)-5 b(ativ)m(es)35
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Fo(,)33 b Fn(l)r Fo(\).)166 1888 y(Let)d Fn(\030)k Fo(and)c
Fn(\021)k Fo(b)s(e)c(v)m(ector)h(bundles)g(o)m(v)m(er)g
Fn(B)5 b Fo(.)30 b(A)g Fk(di\013er)-5 b(ential)32 b(op)-5
b(er)g(ator)40 b Fo(\(DO\))29 b(of)h(order)g Fn(k)166
2009 y Fo(from)i Fn(\030)37 b Fo(to)c Fn(\021)j Fo(is)d(a)f(mapping)
1084 2234 y Fn(D)e Fo(:)e Fn(C)1327 2193 y Fm(1)1402
2234 y Fo(\()p Fn(\030)5 b Fo(\))27 b Fj(\000)-17 b(!)28
b Fn(C)1818 2193 y Fm(1)1892 2234 y Fo(\()p Fn(\021)t
Fo(\))g(:)f Fn( )32 b Fj(7\000)-16 b(!)27 b Fn(D)s( )166
2460 y Fo(suc)m(h)g(that)e Fn(J)646 2423 y Fl(k)689 2460
y Fo(\()p Fn( )t Fo(\)\()p Fn(b)p Fo(\))j(=)f(0)f(implies)h
Fn(D)s( )t Fo(\()p Fn(b)p Fo(\))g(=)h(0.)d(If)g Fn(D)k
Fo(is)c(a)h(linear)f(mapping,)h(it)g(is)g(called)166
2580 y(a)c Fk(line)-5 b(ar)25 b(di\013er)-5 b(ential)25
b(op)-5 b(er)g(ator)11 b Fo(.)22 b(Let)h(Di\013)1725
2595 y Fl(k)1768 2580 y Fo(\()p Fn(\030)5 b(;)17 b(\021)t
Fo(\))k(and)h(LDi\013)2415 2595 y Fl(k)2457 2580 y Fo(\()p
Fn(\030)5 b(;)17 b(\021)t Fo(\))22 b(stand)h(resp)s(ectiv)m(ely)166
2700 y(for)32 b(the)h(v)m(ector)h(spaces)g(of)e(DOs)g(and)h(linear)g
(DOs)f(of)g(order)h Fn(k)j Fo(from)c Fn(\030)37 b Fo(to)32
b Fn(\021)t Fo(.)166 2923 y(There)38 b(exists)h(a)e(bijectiv)m(e)i
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2938 y Fl(k)2868 2923 y Fo(\()p Fn(\030)5 b(;)17 b(\021)t
Fo(\))36 b(and)h(the)166 3043 y(space)c(Hom\()p Fn(J)730
3007 y Fl(k)773 3043 y Fo(\()p Fn(\030)5 b Fo(\))p Fn(;)17
b(\021)t Fo(\))31 b(of)g(v)m(ector)i(bundle)g(morphisms)h(o)m(v)m(er)f
(the)f(iden)m(tit)m(y)i(in)e Fn(B)5 b Fo(.)32 b(This)166
3164 y(corresp)s(ondence)41 b(is)e(based)h(on)f(the)g(fact)g(that)f(ev)
m(ery)j(v)m(ector)f(bundle)g(morphism)g Fn(T)52 b Fo(:)166
3284 y Fn(\020)35 b Fj(!)27 b Fn(\021)36 b Fo(o)m(v)m(er)e(the)f(iden)m
(tit)m(y)h(in)f Fn(B)38 b Fo(generates)c(naturally)f(a)f(mapping)821
3510 y Fn(T)878 3525 y Fm(\003)945 3510 y Fo(:)c Fn(C)1077
3468 y Fm(1)1151 3510 y Fo(\()p Fn(\020)8 b Fo(\))27
b Fj(\000)-16 b(!)27 b Fn(C)1570 3468 y Fm(1)1645 3510
y Fo(\()p Fn(\021)t Fo(\))g(:)h Fn(\036)f Fj(7\000)-16
b(!)27 b Fn(T)2185 3525 y Fm(\003)2225 3510 y Fo(\()p
Fn(\036)p Fo(\))g(:=)h Fn(T)36 b Fj(\016)22 b Fn(\036)32
b(;)166 3735 y Fo(and)26 b(eac)m(h)g(linear)g(DO)e Fn(D)31
b Fj(2)d Fo(LDi\013)1432 3750 y Fl(k)1475 3735 y Fo(\()p
Fn(\030)5 b(;)17 b(\021)t Fo(\))24 b(corresp)s(onds)i(to)f(a)g(v)m
(ector)i(bundle)f(morphism)166 3855 y Fn(T)41 b Fj(2)29
b Fo(Hom\()p Fn(J)663 3819 y Fl(k)705 3855 y Fo(\()p
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Fn(D)e Fo(=)e Fn(T)1698 3870 y Fm(\003)1760 3855 y Fj(\016)22
b Fn(J)1895 3819 y Fl(k)1937 3855 y Fo(,)33 b(i.e.,)g(that)g(the)g
(diagram)p 1383 4351 4 189 v 1385 4351 a Fc(?)1292 4279
y Fl(D)2020 4189 y Fc(\020)1937 4217 y(\020)1854 4245
y(\020)1771 4272 y(\020)1688 4300 y(\020)1605 4328 y(\020)1536
4350 y(\020)-83 b(\))2003 4275 y Fl(T)2044 4283 y Fg(\003)p
1593 4069 265 4 v 1774 4067 a Fc(-)1681 4037 y Fl(J)1726
4018 y Fi(k)1245 4482 y Fn(C)1322 4446 y Fm(1)1397 4482
y Fo(\()p Fn(\021)t Fo(\))1247 4099 y Fn(C)1324 4063
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4063 y Fm(1)2064 4099 y Fo(\()p Fn(J)2165 4063 y Fl(k)2208
4099 y Fo(\()p Fn(\030)5 b Fo(\)\))166 4676 y(comm)m(utes.)25
b(The)e(morphism)h Fn(T)37 b Fo(is)23 b(called)g(the)h
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5139 y(P)m(aren)m(thetically)-8 b(,)37 b(in)f(the)f(language)g(of)f
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5259 y(\()p Fn(J)267 5223 y Fl(k)310 5259 y Fn(;)17 b(J)417
5223 y Fl(k)459 5259 y Fo(\()p Fn(\030)5 b Fo(\)\))37
b(is)i(a)f(univ)m(ersal)i(elemen)m(t)f(for)f(the)g(co)m(v)-5
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5259 y Fo(\))166 5380 y(from)42 b(the)h(category)g Fj(V)8
b Fo(\()p Fn(B)d Fo(\))42 b(of)g(v)m(ector)i(bundles)g(o)m(v)m(er)f
Fn(B)48 b Fo(to)42 b(the)h(category)f(of)g(linear)1769
5712 y(9)p eop
%%Page: 10 10
10 9 bop 166 83 a Fo(DOs)38 b(of)f(order)h Fn(k)j Fo(from)d
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428 y Fj(F)838 443 y Fv(Ob)936 428 y Fo(\()p Fn(\021)t
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443 y Fv(Mor)2124 428 y Fo(\()p Fn(T)14 b Fo(\))27 b(=)h
Fn(T)2459 443 y Fm(\003)2526 428 y Fo(=)g Fn(T)36 b Fj(\016)54
b Fn(:)166 754 y Fo(One)44 b(can)g(di\013eren)m(tiate)i(sim)m
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f(this)h(is)f(the)h(follo)m(wing.)g(Let)f Fn(D)h Fo(=)e
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958 y Fl(k)2898 994 y Fj(2)36 b Fo(Di\013)3165 1009 y
Fl(k)3208 994 y Fo(\()p Fn(\030)5 b(;)17 b(\021)t Fo(\).)166
1115 y(There)35 b(exists)g(a)e(unique)i(\014b)s(er)f(preserving)h
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Fn(T)14 b Fo(\))29 b(:)g Fn(J)2615 1079 y Fl(k)r Fv(+)p
Fl(l)2735 1115 y Fo(\()p Fn(\030)5 b Fo(\))28 b Fj(!)h
Fn(J)3079 1079 y Fl(l)3105 1115 y Fo(\()p Fn(\021)t Fo(\))k(suc)m(h)166
1235 y(that)f(the)h(diagram)1133 1396 y Fn(C)1210 1360
y Fm(1)1284 1396 y Fo(\()p Fn(J)1385 1360 y Fl(k)r Fv(+)p
Fl(l)1505 1396 y Fo(\()p Fn(\030)5 b Fo(\)\))1744 1333
y Fl(P)1799 1310 y Fi(l)1823 1333 y Fv(\()p Fl(T)10 b
Fv(\))1928 1341 y Fg(\003)1715 1396 y Fj(\000)-43 b(\000)-21
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1396 y Fo(\()p Fn(J)2296 1360 y Fl(l)2322 1396 y Fo(\()p
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y(?)2237 1481 y(x)2237 1531 y(?)2237 1581 y(?)2293 1585
y Fl(J)2338 1562 y Fi(l)1262 1766 y Fn(C)1339 1730 y
Fm(1)1414 1766 y Fo(\()p Fn(\030)5 b Fo(\))192 b Fj(\000)-30
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2623 y Fn(J)229 2587 y Fl(k)r Fv(+)p Fl(l)349 2623 y
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11 10 bop 166 83 a Fk(\(wher)-5 b(e)39 b Fn(x)p Fk(,)h
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%%Page: 13 13
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eop
%%Page: 14 14
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15 14 bop 166 83 a Fo(system)1481 205 y Fn(P)1558 164
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%%Page: 16 16
16 15 bop 166 83 a Fk(The)32 b(gener)-5 b(ators)31 b(of)h
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1050 y Fv(0)899 1035 y Fo(\(3\))o Fk(-invariant)f(harmonic)g(functions)
g(satisfy)h(the)f(same)g(e)-5 b(quation)166 1155 y(as)35
b(the)f Fn(O)527 1170 y Fv(0)566 1155 y Fo(\(1)p Fn(;)17
b Fo(2\))p Fk(-invariant)34 b(solutions)g(of)h(the)g(wave)f(e)-5
b(quation.)166 1627 y Fp(6)112 b(Dimensional)40 b(Reduction)d(of)h(a)g
(Group)f(Action)166 1978 y Fo(W)-8 b(e)29 b(need)h(some)f(details)g
(concerning)h(the)f(dimensional)h(reduction)g(of)e(a)g(group)g(action.)
166 2098 y(Let)g Fn(\030)k Fo(=)27 b(\()p Fn(E)6 b(;)17
b(\031)t(;)g(B)5 b Fo(\))27 b(b)s(e)h(a)f Fn(G)p Fo(-reducible)i(v)m
(ector)g(bundle,)g(the)f(Lie)g(group)f Fn(K)35 b Fo(act)27
b(on)h Fn(\030)k Fo(b)m(y)166 2219 y(v)m(ector)37 b(bundle)g(morphisms)
g(\()p Fn(F)s(;)17 b(f)11 b Fo(\),)35 b(and)g(the)h(actions)g(of)g
Fn(G)f Fo(and)h Fn(K)43 b Fo(comm)m(ute.)37 b(Due)166
2339 y(to)28 b(the)h(m)m(utual)g(comm)m(utativit)m(y)i(of)d(the)h
(actions)g(\()p Fn(T)8 b(;)17 b(t)p Fo(\))29 b(and)f(\()p
Fn(F)s(;)17 b(f)11 b Fo(\))28 b(of)g Fn(G)h Fo(and)f
Fn(K)36 b Fo(on)28 b Fn(\030)5 b Fo(,)166 2459 y(the)34
b(action)g(of)g Fn(k)e Fj(2)f Fn(K)40 b Fo(maps)35 b(the)f
Fn(G)p Fo(-in)m(v)-5 b(arian)m(t)34 b(sections)h(of)f
Fn(\030)k Fo(in)m(to)c Fn(G)p Fo(-in)m(v)-5 b(arian)m(t)34
b(ones)166 2580 y(\(i.e.,)g Fn(C)466 2544 y Fm(1)540
2580 y Fo(\()p Fn(\030)5 b Fo(\))664 2595 y Fl(G)756
2580 y Fo(is)33 b(a)g Fn(K)7 b Fo(-in)m(v)-5 b(arian)m(t)33
b(subset)i(of)d Fn(C)1956 2544 y Fm(1)2031 2580 y Fo(\()p
Fn(\030)5 b Fo(\)\).)32 b(This)i(fact)f(allo)m(ws)h(us)g(to)e(de\014ne)
166 2700 y(a)j(natural)h(action)g(\()p Fn(F)992 2715
y Fl(G)1051 2700 y Fn(;)17 b(f)1143 2715 y Fl(G)1202
2700 y Fo(\))35 b(of)g Fn(K)43 b Fo(on)36 b(the)g(reduced)h(bundle)g
Fn(\030)2557 2715 y Fl(G)2648 2700 y Fo(=)c(\(st)17 b
Fn(\030)2935 2724 y Fl(N)3003 2700 y Fn(;)g(\031)3102
2715 y Fl(G)3161 2700 y Fn(;)g(N)10 b Fo(\))35 b(as)166
2820 y(follo)m(ws.)43 b(Let)g Fn(k)k Fj(2)d Fn(K)7 b
Fo(,)43 b Fn(b)h Fj(2)h Fn(N)10 b Fo(,)43 b Fn(\033)48
b Fj(2)c Fo(\()p Fn(\030)1724 2835 y Fl(G)1783 2820 y
Fo(\))1821 2850 y Fl(b)1899 2820 y Fo(=)g(st)17 b Fn(\030)2160
2844 y Fl(b)2194 2820 y Fo(,)42 b(and)h(let)f Fn(g)48
b Fj(2)c Fn(G)e Fo(is)h(suc)m(h)h(that)166 2941 y Fn(t)201
2956 y Fl(g)264 2941 y Fj(\016)21 b Fn(f)383 2956 y Fl(k)426
2941 y Fo(\()p Fn(b)p Fo(\))28 b Fj(2)g Fn(N)10 b Fo(,)33
b(then)423 3291 y(\()p Fn(t)496 3306 y Fl(G)556 3291
y Fo(\))594 3306 y Fl(k)636 3291 y Fo(\()p Fn(b)p Fo(\))17
b(:=)g Fn(t)925 3306 y Fl(g)987 3291 y Fj(\016)22 b Fn(f)1107
3306 y Fl(k)1150 3291 y Fo(\()p Fn(b)p Fo(\))33 b Fn(;)361
3474 y Fo(\()p Fn(F)462 3489 y Fl(G)521 3474 y Fo(\))559
3503 y Fl(k)619 3474 y Fo(\()p Fn(\033)t Fo(\))16 b(:=)h
Fn(T)947 3489 y Fl(g)1009 3474 y Fj(\016)22 b Fn(F)1144
3489 y Fl(k)1187 3474 y Fo(\()p Fn(\033)t Fo(\))27 b
Fj(2)h Fo(\()p Fn(\030)1524 3489 y Fl(G)1583 3474 y Fo(\))1621
3503 y Fv(\()p Fl(t)1673 3514 y Fi(G)1726 3503 y Fv(\))1753
3515 y Fi(k)1791 3503 y Fv(\()p Fl(b)p Fv(\))1930 3474
y Fn(:)166 3715 y Fo(The)34 b(action)e Fn(k)710 3730
y Fl(G)802 3715 y Fo(of)g Fn(k)e Fj(2)f Fn(K)39 b Fo(on)33
b Fn(C)1424 3679 y Fm(1)1515 3715 y Fo(\()p Fn(\030)1596
3730 y Fl(G)1655 3715 y Fo(\))f(is)547 3956 y Fn(k)598
3971 y Fl(G)684 3956 y Fo(:)c Fn(C)816 3914 y Fm(1)907
3956 y Fo(\()p Fn(\030)988 3971 y Fl(G)1047 3956 y Fo(\))g
Fj(\000)-17 b(!)28 b Fn(C)1378 3914 y Fm(1)1469 3956
y Fo(\()p Fn(\030)1550 3971 y Fl(G)1609 3956 y Fo(\))f(:)h
Fn(S)34 b Fj(7\000)-17 b(!)28 b Fn(k)2062 3971 y Fl(G)2121
3956 y Fo(\()p Fn(S)6 b Fo(\))27 b(:=)h Fn(\022)2469
3914 y Fm(\000)p Fv(1)2585 3956 y Fj(\016)22 b Fn(k)j
Fj(\016)d Fn(\022)k Fj(\016)c Fn(S)38 b(:)166 4307 y
Fo(Induced)g(represen)m(tations)i(are)c(an)h(example)h(of)f
(dimensional)h(reduction)g(of)e(a)h(group)166 4427 y(action.)166
4658 y(Let)29 b Fn(\030)34 b Fo(and)29 b Fn(\021)k Fo(b)s(e)c
Fn(G)p Fo(-reducible)i(v)m(ector)f(bundles)h(o)m(v)m(er)f
Fn(B)k Fo(with)c(the)g(same)g(action)f Fn(t)g Fo(of)g
Fn(G)166 4778 y Fo(on)34 b Fn(B)5 b Fo(.)34 b(Let)h(the)f(Lie)g(group)g
Fn(K)42 b Fo(act)34 b(on)g Fn(\030)k Fo(and)c Fn(\021)k
Fo(b)m(y)d(v)m(ector)g(bundle)g(morphisms)h(with)166
4898 y(the)i(action)f Fn(f)49 b Fo(of)37 b Fn(K)44 b
Fo(on)37 b Fn(B)5 b Fo(.)38 b(Let)g(the)g(action)f(\()p
Fn(T)2019 4862 y Fl(\030)2057 4898 y Fn(;)17 b(t)p Fo(\))37
b(of)g Fn(G)g Fo(on)g Fn(\030)42 b Fo(comm)m(ute)d(with)f(the)166
5019 y(action)j(\()p Fn(F)582 4983 y Fl(\030)619 5019
y Fn(;)17 b(f)11 b Fo(\))40 b(of)g Fn(K)47 b Fo(on)40
b Fn(\030)5 b Fo(,)40 b(and)h(the)g(same)g(b)s(e)f(v)-5
b(alid)41 b(for)f(the)h(actions)g(\()p Fn(T)3098 4983
y Fl(\021)3139 5019 y Fn(;)17 b(t)p Fo(\))41 b(and)166
5139 y(\()p Fn(F)281 5103 y Fl(\021)323 5139 y Fn(;)17
b(f)11 b Fo(\))25 b(on)h Fn(\021)t Fo(.)g(It)g(is)h(easy)g(to)f(see)h
(that)f(if)g(a)g(DO)f Fn(D)31 b Fj(2)d Fo(Di\013)2296
5154 y Fl(k)2339 5139 y Fo(\()p Fn(\030)5 b(;)17 b(\021)t
Fo(\))25 b(is)h(sim)m(ultaneously)j Fn(K)7 b Fo(-)166
5259 y(and)34 b Fn(G)p Fo(-in)m(v)-5 b(arian)m(t,)34
b(then)h(the)f(reduced)i(DO)d Fn(D)1927 5274 y Fl(G)2016
5259 y Fj(2)e Fo(Di\013)2278 5274 y Fl(k)2337 5259 y
Fo(\()p Fn(\030)2418 5274 y Fl(G)2477 5259 y Fn(;)17
b(\021)2569 5274 y Fl(G)2628 5259 y Fo(\))34 b(is)g(in)m(v)-5
b(arian)m(t)35 b(under)166 5380 y(the)e(reduced)h(actions)f(\()p
Fn(F)1143 5333 y Fl(\030)1129 5404 y(G)1189 5380 y Fn(;)17
b(f)1281 5395 y Fl(G)1339 5380 y Fo(\))33 b(and)g(\()p
Fn(F)1715 5333 y Fl(\021)1701 5404 y(G)1760 5380 y Fn(;)17
b(f)1852 5395 y Fl(G)1911 5380 y Fo(\))32 b(of)g Fn(K)40
b Fo(on)32 b Fn(\030)2393 5395 y Fl(G)2485 5380 y Fo(and)g
Fn(\021)2722 5395 y Fl(G)2782 5380 y Fo(.)1745 5712 y(16)p
eop
%%Page: 17 17
17 16 bop 166 83 a Fp(Ac)m(kno)m(wledgmen)m(ts)166 423
y Fo(W)-8 b(e)41 b(w)m(ould)h(lik)m(e)g(to)e(express)k(our)c(gratitude)
h(to)g(Lub)s(omir)f(Chorbadjiev)i(for)f(the)g(in-)166
544 y(n)m(umerable)36 b(discussions)h(on)d(symmetry)i(and)e(reduction.)
i(W)-8 b(e)34 b(are)g(indebted)i(to)e(Prof.)166 664 y(Luis)29
b(J.)g(Bo)m(y)m(a)h(who)f(read)g(the)g(article)g(and)g(made)g(man)m(y)h
(v)-5 b(aluable)29 b(commen)m(ts.)i(W)-8 b(e)29 b(are)166
785 y(grateful)j(to)g(Prof.)g(Rafael)g(de)g(la)g(Lla)m(v)m(e)h(for)f
(his)h(help)g(and)f(encouragemen)m(t.)i(N.P)-8 b(.)33
b(w)m(as)166 905 y(partially)g(supp)s(orted)g(b)m(y)h(gran)m(ts)f(DMS)f
(9802156)f(and)i(T)-8 b(ARP-003658-071.)166 1287 y Fp(References)166
1620 y Fs([1])71 b(R.L.)52 b(Bry)m(an)m(t,)g(S.S.)f(Chern,)f(R.B.)i
(Gardner,)e(H.L.)i(Goldsc)m(hmidt,)g(P)-8 b(.A.)52 b(Gri\016ths,)332
1733 y Fu(Exterior)34 b(Di\013er)-5 b(ential)33 b(Systems)p
Fs(.)e(Springer,)f(New)g(Y)-8 b(ork,)31 b(1991.)166 1916
y([2])71 b(C.)21 b(Go)s(dbillon,)h Fu(G)n(\023)-44 b(eometri)n(\023)g
(e)25 b(di\013)n(\023)-44 b(er)-5 b(entiel)5 b(le)24
b(et)g(m)n(\023)-44 b(ec)-5 b(anique)24 b(analytique)p
Fs(.)e(Hermann,)f(P)m(aris,)332 2029 y(1969.)166 2213
y([3])71 b(I.S.)40 b(Krasil'shc)m(hik,)g(V.V.)h(Lyc)m(hagin,)g(A.M.)f
(Vinogrado)m(v,)i Fu(Ge)-5 b(ometry)43 b(of)e(Jet)g(Sp)-5
b(ac)g(es)332 2326 y(and)45 b(Nonline)-5 b(ar)46 b(Partial)f(Di\013er)
-5 b(ential)45 b(Equations)p Fs(.)e(Gordon)g(&)f(Breac)m(h,)j(New)e(Y)
-8 b(ork,)332 2439 y(1986.)166 2623 y([4])71 b(S.)30
b(MacLane,)i(G.)f(Birkho\013,)g Fu(A)n(lgebr)-5 b(a)p
Fs(.)31 b(Macmillan,)h(New)e(Y)-8 b(ork,)32 b(1967.)166
2806 y([5])71 b(P)-8 b(.A.)38 b(Nik)m(olo)m(v,)i(On)c(the)h
(dimensional)g(reduction)g(of)g(in)m(v)-5 b(arian)m(t)38
b(\014eld)f(and)f(di\013eren)m(tial)332 2919 y(op)s(erators,)31
b Fu(J.)h(Math.)h(Phys.)e Ft(28)g Fs(\(1987\))i(2354.)166
3103 y([6])71 b(P)-8 b(.A.)52 b(Nik)m(olo)m(v,)h(N.P)-8
b(.)52 b(P)m(etro)m(v,)g(A)f(lo)s(cal)h(approac)m(h)e(to)i(dimensional)
f(reduction:)f(I)s(I.)332 3216 y(Conformal)31 b(in)m(v)-5
b(ariance)31 b(in)f(Mink)m(o)m(wski)i(space,)f Fu(J.)h(Ge)-5
b(om.)34 b(Phys.)c Fs(\(submitted,)h(2001\).)166 3400
y([7])71 b(P)-8 b(.J.)53 b(Olv)m(er,)g Fu(Applic)-5 b(ations)55
b(of)e(Lie)f(Gr)-5 b(oups)55 b(to)e(Di\013er)-5 b(ential)54
b(Equations)p Fs(.)f(2nd)e(ed.,)332 3513 y(Springer,)30
b(New)g(Y)-8 b(ork,)32 b(1993.)166 3696 y([8])71 b(R.)32
b(P)m(alais,)i Fu(Seminar)h(on)f(the)g(A)n(tiyah-Singer)g(Index)g(The)
-5 b(or)g(em)p Fs(.)34 b(Princeton)e(Univ)m(ersit)m(y)332
3809 y(Press,)f(Princeton,)f(1965,)j(Ch.)d(4.)166 3993
y([9])71 b(J.F.)97 b(P)m(ommaret,)h Fu(Systems)c(of)f(Partial)i
(Di\013er)-5 b(ential)93 b(Equations)h(and)g(Lie)332
4106 y(Pseudo)-5 b(gr)g(oups)p Fs(.)33 b(Gordon)d(and)g(Breac)m(h,)i
(New)e(Y)-8 b(ork,)32 b(1978.)166 4290 y([10])26 b(J.F.)41
b(P)m(ommaret,)h Fu(Lie)g(Pseudo)-5 b(gr)g(oups)44 b(and)e(Me)-5
b(chanics)p Fs(.)41 b(Gordon)g(and)f(Breac)m(h,)h(New)332
4403 y(Y)-8 b(ork,)32 b(1988.)166 4586 y([11])26 b(J.F.)52
b(P)m(ommaret,)g Fu(Partial)h(Di\013er)-5 b(ential)52
b(Equations)g(and)h(Gr)-5 b(oup)53 b(The)-5 b(ory)p Fs(.)53
b(Klu)m(w)m(er,)332 4699 y(Dordrec)m(h)m(t,)32 b(1994.)166
4883 y([12])26 b(D.J.)48 b(Saunders,)d Fu(The)j(Ge)-5
b(ometry)49 b(of)f(Jet)f(Bund)5 b(les)p Fs(.)47 b(Cam)m(bridge)g(Univ)m
(ersit)m(y)h(Press,)332 4996 y(Cam)m(bridge,)31 b(1989.)166
5180 y([13])26 b(V.V.)43 b(Zharino)m(v,)f Fu(L)-5 b(e)g(ctur)g(e)44
b(Notes)f(on)h(Ge)-5 b(ometric)g(al)45 b(Asp)-5 b(e)g(cts)44
b(of)g(Partial)g(Di\013er)-5 b(ential)332 5293 y(Equations)p
Fs(.)31 b(W)-8 b(orld)31 b(Scien)m(ti\014c,)h(Riv)m(er)f(Edge,)g(NJ,)f
(1992.)1745 5712 y Fo(17)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF