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cleartomark
%%EndFont 
%%BeginFont: MSAM10
%!PS-AdobeFont-1.1: MSAM10 2.1
%%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
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSAM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSAM10 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 /square put
readonly def
/FontBBox{8 -463 1331 1003}readonly def
/UniqueID 5031981 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI5
%!PS-AdobeFont-1.1: CMMI5 1.100
%%CreationDate: 1996 Aug 02 08:21:10
% 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 (CMMI5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI5 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
readonly def
/FontBBox{37 -250 1349 750}readonly def
/UniqueID 5087380 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633
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cleartomark
%%EndFont 
%%BeginFont: CMBX7
%!PS-AdobeFont-1.1: CMBX7 1.0
%%CreationDate: 1991 Aug 20 16:35:49
% 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 (CMBX7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX7 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 112 /p put
dup 120 /x put
readonly def
/FontBBox{-55 -250 1289 751}readonly def
/UniqueID 5000765 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171
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cleartomark
%%EndFont 
%%BeginFont: MSBM7
%!PS-AdobeFont-1.1: MSBM7 2.1
%%CreationDate: 1992 Oct 17 08:30:50
% 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 (MSBM7) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM7 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{0 -504 2615 1004}readonly def
/UniqueID 5032014 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 9 /bracerightbig put
dup 12 /vextendsingle put
dup 16 /parenleftBig put
dup 17 /parenrightBig put
dup 18 /parenleftbigg put
dup 19 /parenrightbigg put
dup 20 /bracketleftbigg put
dup 21 /bracketrightbigg put
dup 26 /braceleftbigg put
dup 27 /bracerightbigg put
dup 32 /parenleftBigg put
dup 33 /parenrightBigg put
dup 40 /braceleftBigg put
dup 41 /bracerightBigg put
dup 56 /bracelefttp put
dup 58 /braceleftbt put
dup 60 /braceleftmid put
dup 80 /summationtext put
dup 82 /integraltext put
dup 88 /summationdisplay put
dup 90 /integraldisplay put
dup 104 /bracketleftBig put
dup 105 /bracketrightBig put
dup 110 /braceleftBig put
dup 111 /bracerightBig put
dup 112 /radicalbig put
dup 113 /radicalBig put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueID 5000774 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR5
%!PS-AdobeFont-1.1: CMR5 1.00B
%%CreationDate: 1992 Feb 19 19:55:02
% 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 (CMR5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR5 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 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
readonly def
/FontBBox{-341 -250 1304 965}readonly def
/UniqueID 5000788 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI8
%!PS-AdobeFont-1.1: CMTI8 1.0
%%CreationDate: 1991 Aug 18 21:07:42
% 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 (CMTI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI8 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 44 /comma put
dup 45 /hyphen put
dup 46 /period 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 76 /L put
dup 77 /M put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T 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 121 /y put
readonly def
/FontBBox{-35 -250 1190 750}readonly def
/UniqueID 5000826 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 12 /fi put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 46 /period put
dup 70 /F put
dup 76 /L put
dup 84 /T put
dup 87 /W 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 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 121 /y put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueID 5000828 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53
% 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 (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 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 13 /gamma put
dup 21 /lambda put
dup 23 /nu put
dup 25 /pi put
dup 59 /comma put
dup 61 /slash put
dup 73 /I put
dup 99 /c put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 115 /s put
dup 116 /t put
dup 120 /x put
dup 122 /z put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueID 5087382 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 0 /minus put
dup 1 /periodcentered put
dup 33 /arrowright put
dup 48 /prime put
dup 49 /infinity put
dup 50 /element put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueID 5000817 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMIB10
%!PS-AdobeFont-1.1: CMMIB10 1.100
%%CreationDate: 1996 Jul 23 07:54:00
% 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 (CMMIB10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMIB10 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
readonly def
/FontBBox{-15 -250 1216 750}readonly def
/UniqueID 5087392 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 6 /plusminus put
dup 10 /circlemultiply put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 33 /arrowright put
dup 44 /arrowdblboth put
dup 49 /infinity put
dup 50 /element put
dup 54 /negationslash put
dup 55 /mapsto put
dup 56 /universal put
dup 66 /B put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
dup 107 /bardbl put
dup 112 /radical put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR7
%!PS-AdobeFont-1.1: CMR7 1.0
%%CreationDate: 1991 Aug 20 16:39:21
% 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 (CMR7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR7 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 33 /exclam put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus 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
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K 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
readonly def
/FontBBox{-27 -250 1122 750}readonly def
/UniqueID 5000790 def
currentdict end
currentfile eexec
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%%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 67 /C put
dup 78 /N put
dup 82 /R put
dup 126 /planckover2pi1 put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
/UniqueID 5031982 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 11 /alpha put
dup 12 /beta put
dup 13 /gamma put
dup 16 /zeta put
dup 18 /theta put
dup 21 /lambda put
dup 23 /nu put
dup 25 /pi put
dup 32 /psi put
dup 39 /phi1 put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 62 /greater put
dup 64 /partialdiff put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D 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 80 /P put
dup 81 /Q put
dup 84 /T put
dup 86 /V put
dup 87 /W put
dup 90 /Z 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 112 /p put
dup 113 /q put
dup 115 /s put
dup 116 /t put
dup 120 /x put
dup 122 /z put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueID 5087385 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 25 /germandbls 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 53 /five put
dup 54 /six put
dup 56 /eight put
dup 57 /nine put
dup 65 /A put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 77 /M put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U 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
dup 121 /y put
dup 127 /dieresis put
readonly def
/FontBBox{14 -250 1077 750}readonly def
/UniqueID 5000772 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 0 /Gamma put
dup 3 /Lambda put
dup 10 /Omega put
dup 11 /ff put
dup 12 /fi put
dup 14 /ffi put
dup 25 /germandbls put
dup 26 /ae 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 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 89 /Y 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
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueID 5000793 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
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end readonly def
/FontName /CMSY8 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
dup 20 /lessequal put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
/UniqueID 5000818 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 58 /period put
dup 90 /Z put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueID 5087383 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 11 /ff put
dup 12 /fi put
dup 25 /germandbls 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 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 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 87 /W put
dup 89 /Y put
dup 91 /bracketleft 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 -250 1070 750}readonly def
/UniqueID 5000791 def
currentdict end
currentfile eexec
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%%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 45 /hyphen put
dup 46 /period put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 58 /colon put
dup 65 /A put
dup 66 /B put
dup 69 /E put
dup 70 /F put
dup 72 /H put
dup 73 /I 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 82 /R put
dup 83 /S put
dup 84 /T put
dup 86 /V put
dup 89 /Y put
dup 97 /a put
dup 101 /e put
dup 102 /f put
dup 105 /i 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 120 /x put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
/UniqueID 5000768 def
currentdict end
currentfile eexec
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%%Page: 4 4
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%%Page: 5 5
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end
%%Page: 7 7
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y Fp(f)16704 52296 y Fj(\027)17255 52130 y Fr(\()p Fp(p)p
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51673 y Fn(2)21111 52130 y Fr(\()p Fp(p)p Fr(\))p Fp(h)23166
51673 y Fn(2)23664 52130 y Fr(\()p Fp(p)p Fr(\))25265
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63036 y Fk(00)21649 62596 y Fp(q)22143 62762 y Fj(l)t
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Fr(4)p Fp(p:)p eop end
%%Page: 10 10
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b(wing)345 b([4)q(].)d(According)h(to)6075 7651 y(\(32\))371
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10456 y Fr(\()p Fp(p)35137 9999 y Fk(0)35448 10456 y
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8359 14047 671 45 v 8359 15061 a(\031)9347 12795 y Ff(Z)10454
13069 y Fk(1)9962 15310 y Fn(0)11576 14302 y Fp(dp)g
Fm(j)13562 14010 y Fr(^)13324 14302 y Fp(f)13866 14468
y Fj(\027)14417 14302 y Fr(\()p Fp(p)p Fr(\))p Fm(j)16141
13845 y Fn(2)16946 14302 y Fp(A)17776 13845 y Fn(2)18273
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12795 y(Z)22120 13069 y Fk(1)21628 15310 y Fn(0)23531
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y Fk(0)28524 14302 y Fr(\))554 b(+)246 b Fp(h)31253 13845
y Fn(2)31750 14302 y Fr(\()p Fp(p)p Fr(\))33351 12795
y Ff(Z)34459 13069 y Fk(1)33967 15310 y Fn(0)35870 13553
y Fp(p)p 35714 14047 V 35714 15061 a(p)36271 14741 y
Fk(0)36715 14302 y Fp(dp)37848 13845 y Fk(0)38466 14302
y Fp(q)38960 14468 y Fj(l)t Fn(+2)p Fj(s)40838 14302
y Fr(\()41557 13553 y Fp(p)p 41401 14047 V 41401 15061
a(p)41958 14741 y Fk(0)42402 14302 y Fr(\))42832 12741
y Ff(\033)44155 14302 y Fp(:)6075 16960 y Fr(W)-92 b(e)486
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b(Then,)6075 18288 y(making)420 b(the)e(substitution)i
Fp(z)438 b Fr(:=)389 b Fp(p)21262 17887 y Fk(0)21573
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19617 y(and)369 b(\(40\))i(w)-31 b(e)370 b(\014nd)6075
22364 y(\(43\))5635 b Fp(I)14163 22530 y Fn(0)14966 22364
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y Fn(0)18960 21615 y Fp(p)p 18805 22110 868 45 v 18805
23123 a(p)19362 22804 y Fk(0)19805 22364 y Fp(dp)20938
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22364 y Fr(\()23109 21615 y Fp(p)p 22953 22110 V 22953
23123 a(p)23510 22804 y Fk(0)23954 22364 y Fr(\))554
b(+)245 b Fp(h)26682 21907 y Fn(2)27179 22364 y Fr(\()p
Fp(p)p Fr(\))28780 20857 y Ff(Z)29888 21131 y Fk(1)29396
23373 y Fn(0)31299 21615 y Fp(p)p 31143 22110 V 31143
23123 a(p)31700 22804 y Fk(0)32144 22364 y Fp(dp)33277
21907 y Fk(0)33895 22364 y Fp(q)34389 22530 y Fj(l)t
Fn(+1)35850 22364 y Fr(\()36568 21615 y Fp(p)p 36413
22110 V 36413 23123 a(p)36970 22804 y Fk(0)37413 22364
y Fr(\))9225 26255 y Fm(\024)10393 24748 y Ff(Z)11500
25022 y Fk(1)11008 27263 y Fn(0)12911 25506 y Fp(p)p
12756 26000 V 12756 27014 a(p)13313 26694 y Fk(0)13756
26255 y Fp(dp)14889 25798 y Fk(0)15507 26255 y Fp(q)16001
26421 y Fn(0)16498 26255 y Fr(\()17216 25506 y Fp(p)p
17061 26000 V 17061 27014 a(p)17618 26694 y Fk(0)18061
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26255 y Fr(\()p Fp(p)p Fr(\))22888 24748 y Ff(Z)23996
25022 y Fk(1)23504 27263 y Fn(0)25407 25506 y Fp(p)p
25251 26000 V 25251 27014 a(p)25808 26694 y Fk(0)26252
26255 y Fp(dp)27385 25798 y Fk(0)28003 26255 y Fp(q)28497
26421 y Fn(1)28994 26255 y Fr(\()29712 25506 y Fp(p)p
29557 26000 V 29557 27014 a(p)30114 26694 y Fk(0)30557
26255 y Fr(\))616 b(=)f Fp(p)307 b Fr(\()34506 25506
y Fp(\031)35177 25104 y Fn(2)p 34506 26000 1168 45 v
34813 27014 a Fr(2)36360 26255 y(+)246 b Fp(h)38105 25798
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%%Page: 11 11
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y Fn(1)p Fk(\000)19343 51683 y Fg(1)p 19343 51806 V 19343
52247 a(2)19908 51707 y Fr(\()p Fp(x)p Fr(\))g(can)f(b)31
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y(\(52\))946 b Fm(\000)p Fp(G)10718 54411 y Fk(0)10718
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V 11984 55509 a(2)12549 54868 y Fr(\()p Fp(x)p Fr(\))616
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58936 y(+)625 b Fp(\015)14526 57375 y Ff(\024)15111 58936
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30693 58681 1168 45 v 30723 59695 a Fr(16)31994 57375
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58681 671 45 v 34294 59695 a Fp(\031)35344 58936 y Fm(\000)36583
58187 y Fp(\031)37254 57785 y Fn(3)p 36583 58681 1168
45 v 36613 59695 a Fr(16)37884 57375 y Ff(\025\033)6075
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b(ositiv)-31 b(e)519 b(for)f(all)g Fp(x)554 b Fm(2)h
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61417 y Fn(2)34040 61737 y Fr(+)246 b(1)555 b Fm(\025)f
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y Fp(G)6945 63338 y Fn(1)p Fk(\000)8211 63041 y Fg(1)p
8211 63164 377 38 v 8211 63605 a(2)8776 63065 y Fr(\()p
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%%Page: 12 12
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y Fg(1)22301 7994 y Fr(,)h(all)g(con)-31 b(tributions)544
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(of)d(the)f(Lieb)6075 10866 y(and)407 b(Y)-92 b(au)407
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b(v)g(er,)407 b(for)h(the)f(Bro)-31 b(wn-Ra)g(v)g(enhall)410
b(op)31 b(erator,)408 b(an)6075 12194 y(impro)-31 b(v)g(ed)264
b(estimate)h(for)d(the)h Fp(l)330 b Fr(=)307 b(0)p Fp(;)493
b(s)308 b Fr(=)23823 11758 y Fn(1)p 23823 11939 442 45
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b(\()p Fp(f)14024 15057 y Fj(\027)14576 14891 y Fp(;)184
b Fr(\()p Fp(b)15972 15057 y Fn(0)p Fj(m)17501 14891
y Fr(+)246 b Fp(b)19083 14316 y Fn(\(1\))19083 15225
y Fj(l)t(sm)20626 14891 y Fr(\))184 b Fp(f)21782 15057
y Fj(\027)22334 14891 y Fr(\))615 b Fm(\025)24548 13384
y Ff(Z)25655 13658 y Fk(1)25163 15899 y Fn(0)26777 14891
y Fp(dp)308 b Fm(j)28764 14599 y Fr(^)28525 14891 y Fp(f)29067
15057 y Fj(\027)29618 14891 y Fr(\()p Fp(p)p Fr(\))p
Fm(j)31342 14434 y Fn(2)32147 14891 y Fp(e)p Fr(\()p
Fp(p)p Fr(\))248 b Fm(\001)e Fp(T)35527 15164 y Fn(0)36101
14866 y Fg(1)p 36101 14989 377 38 v 36101 15430 a(2)36666
14891 y Fr(\()p Fp(x)p Fr(\))p Fp(;)6656 18757 y(T)7303
19030 y Fn(0)7877 18733 y Fg(1)p 7877 18856 V 7877 19297
a(2)8442 18757 y Fr(\()p Fp(x)p Fr(\))308 b(:=)615 b(1)554
b Fm(\000)14373 18008 y Fp(\015)p 14373 18503 635 45
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y Fr(\()16646 17670 y Ff(p)p 17754 17670 3036 45 v 17754
18757 a Fp(x)18387 18437 y Fn(2)19129 18757 y Fr(+)246
b(1)h(+)f(1\))23567 18008 y(arctan)185 b Fp(x)p 23567
18503 3896 45 v 25198 19517 a(x)28148 18757 y Fr(+)29696
18008 y(\()30126 17086 y Fm(p)p 31048 17086 3036 45 v
922 x Fp(x)31681 17689 y Fn(2)32424 18008 y Fr(+)246
b(1)g Fm(\000)g Fr(1\)\()p Fp(x)h Fm(\000)f Fr(arctan)186
b Fp(x)p Fr(\))p 29696 18503 13468 45 v 31356 19517 a(\()p
Fp(x)32419 19197 y Fn(2)33162 19517 y Fr(+)246 b(1\))370
b(arctan)185 b Fp(x)246 b Fm(\000)g Fp(x)43480 16864
y Ff(\))44556 18757 y Fp(:)6075 21487 y Fr(v)-61 b(alid)370
b(for)f(all)i Fp(l)22 b(;)184 b(s)370 b Fr(according)g(to)g([4)q(].)
8594 22815 y(T)-92 b(ogether)426 b(with)g(Lemma)g(2)g(this)f(allo)-31
b(ws)427 b(for)e(the)g(follo)-31 b(wing)428 b(represen)-31
b(tation)426 b(of)g(\(32\))6075 24144 y(for)369 b Fp(s)307
b Fr(=)9897 23708 y Fn(1)p 9897 23889 442 45 v 9897 24525
a(2)10471 24144 y Fr(,)7015 26430 y(\()p Fp(f)7987 26596
y Fj(\027)8538 26430 y Fp(;)184 b Fr(\()p Fp(b)9934 26596
y Fn(0)p Fj(m)11464 26430 y Fr(+)245 b Fp(b)13045 25855
y Fn(\(1\))13045 26764 y Fj(l)t(sm)14835 26430 y Fr(+)h
Fp(b)16417 25855 y Fn(\(2\))16417 26764 y Fj(l)t(sm)17960
26430 y Fr(\))p Fp(f)18932 26596 y Fj(\027)19483 26430
y Fr(\))7322 28893 y Fm(\025)308 b Fr(\()p Fp(f)9463
29059 y Fj(\027)10014 28893 y Fp(;)184 b Fr(\()p Fp(b)11410
29059 y Fn(0)p Fj(m)12939 28893 y Fr(+)246 b Fp(b)14521
28318 y Fn(\(1\))14521 29227 y Fj(l)t(sm)16065 28893
y Fr(\))p Fp(f)17037 29059 y Fj(\027)17588 28893 y Fr(\))g
Fm(\000)19371 27386 y Ff(Z)20478 27660 y Fk(1)19986 29901
y Fn(0)21232 27386 y Ff(Z)22339 27660 y Fk(1)21847 29901
y Fn(0)23461 28893 y Fp(dpdp)25727 28436 y Fk(0)26038
28893 y Fm(j)26583 28601 y Fr(^)26345 28893 y Fp(f)26887
29059 y Fj(\027)27438 28893 y Fr(\()p Fp(p)p Fr(\))p
Fm(j)p Fp(b)29637 28318 y Fn(\(2\))29637 29227 y Fj(l)t(sm)31181
28893 y Fr(\()p Fp(p;)184 b(p)33216 28436 y Fk(0)33528
28893 y Fr(\))p Fm(j)34505 28601 y Fr(^)34265 28893 y
Fp(f)34807 29059 y Fj(\027)35359 28893 y Fr(\()p Fp(p)36346
28436 y Fk(0)36657 28893 y Fr(\))p Fm(j)7322 33306 y(\025)8798
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33306 y Fp(dp)308 b Fm(j)13014 33014 y Fr(^)12775 33306
y Fp(f)13317 33472 y Fj(\027)13868 33306 y Fr(\()p Fp(p)p
Fr(\))p Fm(j)15592 32849 y Fn(2)16398 33306 y Fp(e)p
Fr(\()p Fp(p)p Fr(\))247 b Fm(\001)f Fp(T)19777 33579
y Fn(0)20351 33281 y Fg(1)p 20351 33404 377 38 v 20351
33845 a(2)20916 33306 y Fr(\()p Fp(x)p Fr(\))554 b Fm(\000)24378
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y Fm(j)32082 33014 y Fr(^)31843 33306 y Fp(f)32385 33472
y Fj(\027)32936 33306 y Fr(\()p Fp(p)p Fr(\))p Fm(j)309
b Fp(b)35444 32731 y Fn(\(2\))35444 33756 y(0)36018 33458
y Fg(1)p 36018 33581 V 36018 34022 a(2)36528 33756 y
Fj(m)37369 33306 y Fr(\()p Fp(p;)184 b(p)39404 32849
y Fk(0)39716 33306 y Fr(\))308 b Fm(j)40999 33014 y Fr(^)40761
33306 y Fp(f)41303 33472 y Fj(\027)41854 33306 y Fr(\()p
Fp(p)42841 32849 y Fk(0)43152 33306 y Fr(\))p Fm(j)g
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g(is)g(estimated)i(b)-31 b(y)370 b(means)f(of)h(the)f(Lieb)h(and)f(Y)
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y Ff(Z)7220 37334 y Fk(1)6728 39576 y Fn(0)7973 37061
y Ff(Z)9080 37334 y Fk(1)8588 39576 y Fn(0)10203 38567
y Fp(dp)184 b(dp)12653 38110 y Fk(0)13148 38567 y Fm(j)13694
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38567 y Fr(\))308 b Fm(j)22612 38275 y Fr(^)22373 38567
y Fp(f)22915 38733 y Fj(\027)23466 38567 y Fr(\()p Fp(p)24453
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38567 y Fp(dp)308 b Fm(j)33980 38275 y Fr(^)33741 38567
y Fp(f)34283 38733 y Fj(\027)34834 38567 y Fr(\()p Fp(p)p
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42102 y Fp(N)121 b Fr(\()p Fp(p;)184 b(p)14507 41645
y Fk(0)14819 42102 y Fp(;)g(p)15867 41645 y Fk(00)16434
42102 y Fr(\))17048 40541 y Ff(\024)17633 42102 y Fp(q)18127
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40541 y Ff(\023)26389 42102 y Fp(h)27027 41645 y Fn(2)27523
42102 y Fr(\()p Fp(p)28510 41645 y Fk(00)29076 42102
y Fr(\))687 b(+)553 b Fp(q)32101 42268 y Fn(1)32783 40541
y Ff(\022)33730 41353 y Fp(p)34287 40951 y Fk(00)p 33730
41847 V 34013 42861 a Fp(p)34986 40541 y Ff(\023)35985
42102 y Fp(q)36479 42268 y Fn(1)37161 40541 y Ff(\022)38108
41353 y Fp(p)38665 40951 y Fk(00)p 38108 41847 V 38236
42861 a Fp(p)38793 42541 y Fk(0)39364 40541 y Ff(\023)40363
42102 y Fp(h)p Fr(\()p Fp(p)p Fr(\))p Fp(h)p Fr(\()p
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y(\(55\))3075 b Fm(\000)307 b Fp(q)12778 46078 y Fn(0)13459
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b Fp(p)32927 48142 y Fk(0)33615 48544 y Fr(and)f Fp(\020)404
b Fr(:=)322 b Fp(p)38710 48142 y Fk(00)39275 48544 y
Fp(=p)378 b Fr(for)g Fp(p)43023 48142 y Fk(00)43966 48544
y Fr(are)6075 49872 y(made.)433 b(Inserting)e(\(9\))i(for)e
Fp(A)18543 49470 y Fn(2)19040 49872 y Fr(\()p Fp(p)p
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Fp(p)411 b Fr(=)g Fp(mx)431 b Fr(as)h(b)31 b(efore,)431
b(\(54\))i(with)6075 51200 y(\(55\))371 b(is)e(cast)g(in)-31
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b Fm(j)26882 53500 y Fr(^)26643 53792 y Fp(f)27185 53958
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53335 y Fn(2)30265 53792 y Fp(e)p Fr(\()p Fp(p)p Fr(\))248
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60826 y Fm(\000)19210 60546 y Fr(~)19077 60826 y Fp(I)19564
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60826 y Fr(\()p Fp(x)p Fr(\))32436 59265 y Ff(\023)33436
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%%Page: 13 13
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7634 y(\020)25609 7232 y Fn(2)p 18894 8128 13363 45 v
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9142 a(z)39561 8103 y Fr(~)39241 8382 y Fp(N)428 b(q)41052
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12049 y(~)8856 12329 y Fp(I)9343 12495 y Fn(2)9840 12329
y Fr(\()p Fp(x)p Fr(\))308 b(:=)13424 10822 y Ff(Z)14531
11096 y Fk(1)14039 13338 y Fn(0)15654 12329 y Fp(\020)82
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11260 y Fn(2)21347 11580 y Fr(+)246 b(1)g(+)g(1)p 17802
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45 v 972 x Fp(x)20495 12948 y Fn(2)20991 13268 y Fp(\020)21557
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12495 y Fn(1)26345 12329 y Fr(\()p Fp(\020)82 b Fr(\))27955
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12948 y Fn(2)34923 13268 y Fr(+)246 b(1)g(+)g(1)39251
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16823 y Fr(+)246 b(1)23964 15884 y Fp(q)24458 16050 y
Fn(0)24955 15884 y Fr(\()p Fp(\020)82 b Fr(\))26565 14377
y Ff(Z)27673 14651 y Fk(1)27181 16893 y Fn(0)32444 15135
y Fp(dz)p 28928 15630 8172 45 v 28928 15851 a Ff(p)p
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15604 y(~)37540 15884 y Fp(N)429 b(q)39352 16050 y Fn(1)39849
15884 y Fr(\()p Fp(z)49 b Fr(\))13213 19076 y(~)13080
19356 y Fp(I)13567 19522 y Fn(4)14063 19356 y Fr(\()p
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23132 19323 4099 45 v 971 x Fp(x)23765 19975 y Fn(2)24261
20294 y Fp(\020)24827 19975 y Fn(2)25570 20294 y Fr(+)246
b(1)27671 19356 y Fp(q)28165 19522 y Fn(1)28661 19356
y Fr(\()p Fp(\020)82 b Fr(\))30271 17849 y Ff(Z)31380
18123 y Fk(1)30888 20365 y Fn(0)32635 18607 y Fp(dz)p
32635 19101 1140 45 v 32923 20115 a(z)34536 19076 y Fr(~)34215
19356 y Fp(N)429 b(q)36027 19522 y Fn(0)36523 19356 y
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22443 y(~)13411 22723 y Fp(N)429 b Fr(:=)22780 21974
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22690 5159 45 v 972 x Fp(x)18385 23342 y Fn(2)18882 23662
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23342 y Fn(2)21250 23662 y Fr(+)246 b(1)h(+)24264 22690
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23342 y Fn(2)26500 23662 y Fp(\020)27066 23342 y Fn(2)27808
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y(of)521 b(splitting)j(the)d(in)-31 b(tegrals)523 b(at)e(1)h(and)f
(making)i(a)e(v)-61 b(ariable)522 b(substitution)h Fp(z)609
b Fm(7!)561 b Fr(1)p Fp(=z)49 b(:)522 b Fr(It)6075 28038
y(is)534 b(found)h(n)-31 b(umerically)536 b(that)19572
27758 y(~)19321 28038 y Fp(G)20191 28311 y Fn(0)20765
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28038 y Fr(\()p Fp(x)p Fr(\))f(is)f(a)h(monotonically)j(decreasing)d
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b(its)e(in\014m)-31 b(um)370 b(at)g Fp(x)307 b Fm(!)h(1)p
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b(es)6075 32176 y(\(58\))3651 b(inf)11224 32895 y Fj(x)p
Fk(2)p Fe(R)12883 33006 y Fg(+)13941 31897 y Fr(~)13691
32176 y Fp(G)14561 32449 y Fn(0)15135 32152 y Fg(1)p
15135 32275 V 15135 32716 a(2)15700 32176 y Fr(\()p Fp(x)p
Fr(\))616 b(=)f(1)554 b Fm(\000)21631 31427 y Fp(\015)p
21631 31922 635 45 v 21672 32936 a Fr(2)22706 32176 y([)23146
31427 y Fp(\031)p 23146 31922 671 45 v 23205 32936 a
Fr(2)24196 32176 y(+)25494 31427 y(2)p 25436 31922 V
25436 32936 a Fp(\031)26239 32176 y Fr(])g Fm(\000)28942
31427 y Fp(\015)29576 31026 y Fn(2)p 28648 31922 1721
45 v 28648 32936 a Fr(8)p Fp(\031)29872 32616 y Fn(2)30809
32176 y Fr([)31249 31427 y Fp(\031)31920 31026 y Fn(4)p
31249 31922 1168 45 v 31556 32936 a Fr(4)32795 32176
y(+)246 b(4)h Fm(\000)f Fp(\031)36480 31719 y Fn(2)37222
32176 y Fm(\000)g Fp(\031)39000 31719 y Fn(2)39496 32176
y Fr(])184 b Fp(:)6075 34770 y Fr(The)364 b(limit)i Fp(x)307
b Fm(!)h(1)363 b Fr(of)i(Tix's)g([12)q(])f(appro)-31
b(ximation)368 b(\(53\))e(is)d(the)i(same)f(as)g(for)g(the)g(estimate)
6075 36098 y(\(45\))388 b(of)f(the)f(linear)i(term)e(in)h
Fp(\015)448 b Fr(in)-31 b(tro)31 b(duced)387 b(in)f(the)h(previous)g
(section.)g(The)g(critical)h(v)-61 b(alue)6075 37426
y(of)369 b Fp(\015)431 b Fr(is)369 b(obtained)i(from)6075
40204 y(\(59\))7715 b(1)554 b Fm(\000)18410 39455 y Fp(\015)p
18410 39949 635 45 v 18451 40963 a Fr(2)19362 38643 y
Ff(\022)20310 39455 y Fp(\031)p 20310 39949 671 45 v
20369 40963 a Fr(2)21360 40204 y(+)22658 39455 y(2)p
22599 39949 V 22599 40963 a Fp(\031)23403 38643 y Ff(\023)24771
40204 y Fm(\000)26318 39455 y Fp(\015)26952 39053 y Fn(2)p
26318 39949 1132 45 v 26607 40963 a Fr(8)27767 38643
y Ff(\022)28715 39455 y Fp(\031)p 28715 39949 671 45
v 28774 40963 a Fr(2)29764 40204 y Fm(\000)31063 39455
y Fr(2)p 31004 39949 V 31004 40963 a Fp(\031)31808 38643
y Ff(\023)32622 38872 y Fn(2)33734 40204 y Fr(=)615 b(0)6075
42817 y(and)363 b(is)f(giv)-31 b(en)364 b(b)-31 b(y)363
b Fp(\015)14315 42983 y Fj(c)14711 43094 y Fg(2)15506
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b(or)362 b Fp(\015)369 b(<)308 b(\015)25390 42983 y Fj(c)25786
43094 y Fg(2)26273 42817 y Fp(;)363 b Fr(the)g(l.h.s.)h(of)f(\(59\))h
(is)f(p)31 b(ositiv)-31 b(e.)365 b(F)-92 b(rom)362 b(\(56\))6075
44145 y(it)370 b(therefore)f(follo)-31 b(ws)371 b(that)g
Fp(b)18339 44418 y Fj(l)t(;)19019 44120 y Fg(1)p 19019
44243 377 38 v 19019 44685 a(2)19529 44418 y Fj(;m)20940
44145 y Fp(>)307 b Fr(0)677 b(for)370 b Fp(\015)f(<)307
b(\015)27716 44311 y Fj(c)28112 44422 y Fg(2)28599 44145
y Fp(:)8594 45730 y Fr(F)-92 b(or)369 b Fp(s)307 b Fr(=)h
Fm(\000)13570 45295 y Fn(1)p 13569 45476 442 45 v 13569
46112 a(2)14143 45730 y Fp(;)370 b Fr(w)-31 b(e)370 b(ha)-31
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Fp(f)20190 48165 y Fj(\027)20741 47999 y Fp(;)184 b Fr(\()p
Fp(b)22137 48165 y Fn(0)p Fj(m)23667 47999 y Fr(+)246
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47999 y Fr(+)g Fp(b)28620 47424 y Fn(\(2\))28620 48333
y Fj(l)t(sm)30163 47999 y Fr(\))184 b Fp(f)31319 48165
y Fj(\027)31871 47999 y Fr(\))6075 51252 y(\(60\))991
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51252 y Fp(;)184 b Fr(\()p Fp(b)13427 51418 y Fn(0)p
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51702 y(1)p Fj(;)p Fk(\000)18068 51404 y Fg(1)p 18068
51527 377 38 v 18068 51968 a(2)18577 51702 y Fj(;m)19681
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y Fr(^)31129 51252 y Fp(f)31671 51418 y Fj(\027)32223
51252 y Fr(\()p Fp(p)p Fr(\))p Fm(j)g Fp(b)34606 50677
y Fn(\(2\))34606 51702 y(1)p Fj(;)p Fk(\000)36137 51404
y Fg(1)p 36137 51527 V 36137 51968 a(2)36646 51702 y
Fj(;m)37750 51252 y Fr(\()p Fp(p;)g(p)39785 50795 y Fk(0)40097
51252 y Fr(\))p Fm(j)41073 50960 y Fr(^)40834 51252 y
Fp(f)41376 51418 y Fj(\027)41927 51252 y Fr(\()p Fp(p)42914
50795 y Fk(0)43225 51252 y Fr(\))p Fm(j)g Fp(:)6075 53822
y Fr(F)-92 b(or)372 b(the)i(linear)g(term,)g(again)i(a)d(b)31
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b(trast)431 b(to)g(\(42\),)i(the)e(factor)g Fp(h)p Fr(\()p
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55150 y Fr(\))g(is)f(k)-31 b(ept)432 b(in)e(the)h(k)-31
b(ernel)430 b(when)h(applying)h(the)6075 56478 y(Lieb)369
b(and)g(Y)-92 b(au)370 b(form)-31 b(ula)371 b(\(33\).)g(Then)6075
59053 y(\(61\))9416 57546 y Ff(Z)10523 57820 y Fk(1)10031
60061 y Fn(0)11277 57546 y Ff(Z)12384 57820 y Fk(1)11892
60061 y Fn(0)13506 59053 y Fp(dpdp)15772 58596 y Fk(0)16267
59053 y Fm(j)16813 58761 y Fr(^)16574 59053 y Fp(f)17116
59219 y Fj(\027)17667 59053 y Fr(\()p Fp(p)p Fr(\))p
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59503 y(1)p Fj(;)p Fk(\000)22934 59205 y Fg(1)p 22934
59328 V 22934 59770 a(2)23444 59503 y Fj(;m)24547 59053
y Fr(\()p Fp(p;)184 b(p)26582 58596 y Fk(0)26894 59053
y Fr(\))g Fm(j)28055 58761 y Fr(^)27815 59053 y Fp(f)28357
59219 y Fj(\027)28910 59053 y Fr(\()p Fp(p)29897 58596
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62553 y Fm(\001)12267 60992 y Ff(\022)13083 61047 y(Z)14190
61320 y Fk(1)13698 63562 y Fn(0)15600 61804 y Fp(p)p
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62719 y Fn(1)19248 60992 y Ff(\022)20351 61804 y Fp(p)p
20196 62299 V 20196 63313 a(p)20753 62993 y Fk(0)21197
60992 y Ff(\023)22565 62553 y Fr(+)553 b Fp(h)p Fr(\()p
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63562 y Fn(0)28737 61804 y Fp(p)p 28582 62299 V 28582
63313 a(p)29139 62993 y Fk(0)29582 62553 y Fp(dp)30715
62096 y Fk(0)31333 62553 y Fp(h)p Fr(\()p Fp(p)32958
62096 y Fk(0)33269 62553 y Fr(\))308 b Fp(q)34501 62719
y Fn(0)35182 60992 y Ff(\022)36285 61804 y Fp(p)p 36130
62299 V 36130 63313 a(p)36687 62993 y Fk(0)37130 60992
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%%Page: 14 14
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(second)g(one)g(the)6075 7330 y(substitution)371 b Fp(z)356
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(\014nds)6075 9949 y(\(62\))6724 b(\()p Fp(f)15737 10115
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13123 a Fp(x)41003 12804 y Fn(2)41745 13123 y Fr(+)g(1)h(+)e(1)p
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16045 y Ff(\022)11707 17326 y Fr(~)11439 17606 y Fp(J)12053
17772 y Fn(1)12549 17606 y Fr(\()p Fp(x)p Fr(\))431 b(+)18676
16857 y(1)p 16021 17351 V 16021 17572 a Fm(p)p 16943
17572 3036 45 v 923 x Fp(x)17576 18175 y Fn(2)18318 18495
y Fr(+)246 b(1)h(+)f(1)22593 17326 y(~)22325 17606 y
Fp(J)22939 17772 y Fn(2)23436 17606 y Fr(\()p Fp(x)p
Fr(\))h Fm(\000)26858 17326 y Fr(~)26590 17606 y Fp(J)27204
17772 y Fn(3)27700 17606 y Fr(\()p Fp(x)p Fr(\))431 b
Fm(\000)33827 16857 y Fr(1)p 31172 17351 5865 45 v 31172
17572 a Fm(p)p 32094 17572 3036 45 v 923 x Fp(x)32727
18175 y Fn(2)33469 18495 y Fr(+)246 b(1)h(+)f(1)37745
17326 y(~)37477 17606 y Fp(J)38091 17772 y Fn(4)38587
17606 y Fr(\()p Fp(x)p Fr(\))40080 16045 y Ff(\023)41080
17606 y Fp(;)6075 20220 y Fr(where)9693 19940 y(~)9425
20220 y Fp(J)10039 20386 y Fj(i)10408 20220 y Fr(\()p
Fp(x)p Fr(\))p Fp(;)1298 b(i)557 b Fr(=)g(1)p Fp(;)184
b(:::;)g Fr(4)1079 b(is)519 b(obtained)h(from)28812 19940
y(~)28679 20220 y Fp(I)29166 20386 y Fj(i)29535 20220
y Fr(\()p Fp(x)p Fr(\))f(b)-31 b(y)519 b(in)-31 b(terc)g(hanging)521
b Fp(q)40768 20386 y Fn(0)41783 20220 y Fr(with)f Fp(q)44948
20386 y Fn(1)6075 21549 y Fr(ev)-31 b(erywhere,)369 b(and)16732
23362 y(~)16464 23642 y Fp(J)17078 23808 y Fn(0)17574
23642 y Fr(\()p Fp(x)p Fr(\))308 b(:=)21159 22135 y Ff(Z)22266
22409 y Fk(1)21774 24651 y Fn(0)23388 23642 y Fp(dz)356
b(q)25329 23808 y Fn(0)25826 23642 y Fr(\()p Fp(z)49
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