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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
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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 20 /kappa put
dup 59 /comma put
dup 61 /slash put
dup 65 /A put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 78 /N put
dup 80 /P put
dup 87 /W put
dup 96 /lscript put
dup 99 /c put
dup 110 /n put
readonly def
/FontBBox{37 -250 1349 750}readonly def
/UniqueID 5087380 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 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
dup 68 /D put
dup 70 /F put
dup 102 /f put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 115 /s put
dup 116 /t put
dup 117 /u put
readonly def
/FontBBox{-341 -250 1304 965}readonly def
/UniqueID 5000788 def
currentdict end
currentfile eexec
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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 67 /C put
dup 78 /N put
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: LASY10
%!PS-AdobeFont-1.1: LASY10 1.001
%%CreationDate: 1992 Oct 23 20:19:17
%%RevisionDate: 2001 Jun 05 20:19:17
% Copyright (C) 1997, 2001 American Mathematical Society. All Rights 
Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.001) readonly def
/Notice (Copyright (C) 1997, 2001 American Mathematical Society. All 
Rights Reserved) readonly def
/FullName (LASY10) readonly def
/FamilyName (LaTeX) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /LASY10 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 /a50 put
readonly def
/FontBBox{-19 -192 944 683}readonly def
/UniqueID 5011949 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 1 /Delta put
dup 3 /Lambda put
dup 5 /Pi put
dup 9 /Psi put
dup 10 /Omega put
dup 11 /ff put
dup 12 /fi put
dup 14 /ffi put
dup 19 /acute 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 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 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 126 /tilde put
dup 127 /dieresis put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueID 5000793 def
currentdict end
currentfile eexec
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%%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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%%EndFont
%%BeginFont: MSBM10
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%%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 
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/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
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end readonly def
/FontName /MSBM10 def
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/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
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dup 67 /C put
dup 82 /R put
dup 90 /Z 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: 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 45 /hyphen put
dup 46 /period put
dup 56 /eight put
dup 58 /colon put
dup 65 /A put
dup 67 /C put
dup 68 /D put
dup 70 /F put
dup 71 /G put
dup 73 /I put
dup 76 /L put
dup 77 /M 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 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{-163 -250 1146 969}readonly def
/UniqueID 5000828 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 3 /asteriskmath put
dup 6 /plusminus put
dup 8 /circleplus put
dup 10 /circlemultiply put
dup 14 /openbullet put
dup 15 /bullet put
dup 17 /equivalence put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 26 /propersubset put
dup 33 /arrowright put
dup 39 /similarequal put
dup 49 /infinity put
dup 50 /element put
dup 54 /negationslash put
dup 56 /universal put
dup 69 /E put
dup 70 /F put
dup 72 /H put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 83 /S put
dup 85 /U put
dup 86 /V put
dup 90 /Z put
dup 91 /union put
dup 92 /intersection put
dup 102 /braceleft put
dup 103 /braceright put
dup 104 /angbracketleft put
dup 105 /angbracketright put
dup 106 /bar put
dup 110 /backslash put
dup 114 /nabla put
dup 120 /section put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 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 2 /multiply put
dup 3 /asteriskmath put
dup 6 /plusminus put
dup 15 /bullet put
dup 21 /greaterequal put
dup 33 /arrowright put
dup 48 /prime put
dup 49 /infinity put
dup 50 /element put
dup 80 /P put
dup 86 /V put
dup 106 /bar put
dup 110 /backslash put
dup 114 /nabla put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueID 5000817 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 1 /Delta put
dup 9 /Psi put
dup 10 /Omega put
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
dup 87 /W put
dup 91 /bracketleft put
dup 105 /i put
dup 109 /m put
dup 110 /n put
readonly def
/FontBBox{-27 -250 1122 750}readonly def
/UniqueID 5000790 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 17 /eta put
dup 20 /kappa put
dup 21 /lambda put
dup 27 /sigma put
dup 30 /phi put
dup 31 /chi put
dup 32 /psi put
dup 59 /comma put
dup 61 /slash put
dup 65 /A put
dup 69 /E put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 76 /L put
dup 78 /N put
dup 80 /P put
dup 83 /S put
dup 86 /V put
dup 87 /W put
dup 96 /lscript put
dup 99 /c put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 115 /s put
dup 119 /w 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: 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 14 /delta put
dup 15 /epsilon1 put
dup 17 /eta put
dup 20 /kappa put
dup 21 /lambda put
dup 22 /mu put
dup 24 /xi put
dup 25 /pi put
dup 26 /rho put
dup 27 /sigma put
dup 30 /phi put
dup 31 /chi put
dup 32 /psi put
dup 33 /omega put
dup 34 /epsilon 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 65 /A 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 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 90 /Z put
dup 97 /a 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 106 /j put
dup 107 /k put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 117 /u put
dup 119 /w put
dup 120 /x put
dup 121 /y 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: 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 58 /colon put
dup 80 /P put
dup 83 /S put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 105 /i put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 114 /r put
dup 121 /y put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueID 5000791 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 12 /fi put
dup 40 /parenleft put
dup 41 /parenright 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 73 /I put
dup 75 /K 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 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 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 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{-301 -250 1164 946}readonly def
/UniqueID 5000768 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 12 /vextendsingle put
dup 13 /vextenddouble put
dup 16 /parenleftBig put
dup 17 /parenrightBig put
dup 18 /parenleftbigg put
dup 19 /parenrightbigg put
dup 26 /braceleftbigg put
dup 40 /braceleftBigg put
dup 76 /circleplustext put
dup 77 /circleplusdisplay put
dup 80 /summationtext put
dup 82 /integraltext put
dup 83 /uniontext put
dup 88 /summationdisplay put
dup 90 /integraldisplay put
dup 101 /tildewide put
dup 102 /tildewider put
dup 104 /bracketleftBig put
dup 105 /bracketrightBig put
dup 112 /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: CMSS10
%!PS-AdobeFont-1.1: CMSS10 1.0
%%CreationDate: 1991 Aug 20 17:33:34
% 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 (CMSS10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMSS10 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 47 /slash put
dup 58 /colon put
dup 64 /at 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 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
readonly def
/FontBBox{-61 -250 999 759}readonly def
/UniqueID 5000803 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 11 /ff put
dup 12 /fi put
dup 19 /acute put
dup 34 /quotedblright 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 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 76 /L put
dup 77 /M put
dup 78 /N 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 92 /quotedblleft 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
readonly def
/FontBBox{-62 -250 1123 750}readonly def
/UniqueID 5000798 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 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 50 /two put
dup 52 /four put
dup 57 /nine put
dup 65 /A put
dup 66 /B put
dup 69 /E put
dup 70 /F put
dup 74 /J put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 82 /R 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 101 /e put
dup 105 /i put
dup 110 /n put
dup 114 /r put
dup 117 /u put
dup 121 /y put
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueID 5000794 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 68 /D put
dup 69 /E put
dup 70 /F put
dup 72 /H 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 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 119 /w put
readonly def
/FontBBox{-56 -251 1102 750}readonly def
/UniqueID 5000799 def
currentdict end
currentfile eexec
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Fo(\(0)p Fp(;)p Fo(+)p Fn(1)p Fo(\))945 3882 y Fv(\()p
Fq(H)1053 3852 y Fp(c)1087 3882 y Fv(\))44 b Fl(if)31
b(and)f(only)g(if,)h(for)g(al)t(l)g Fq( )26 b Fm(2)d
Fq(H)2187 3830 y Ff(1)p 2187 3839 V 2187 3872 a(2)2229
3882 y Fv(\()p Fk(R)2321 3852 y Fo(3)2359 3882 y Fq(;)14
b Fk(C)2456 3852 y Fo(4)2493 3882 y Fv(\))p Fl(,)594
4017 y Ft(\015)594 4067 y(\015)594 4116 y(\015)640 4020
y(\020)689 4112 y Fm(\000)p Fq(c)790 4078 y Fo(2)827
4112 y Fv(\001)19 b(+)f Fq(c)1034 4078 y Fo(4)1071 4020
y Ft(\021)1131 4015 y Ff(1)p 1131 4024 V 1131 4057 a(4)1173
4020 y Ft(\020)1223 4112 y Fq(P)30 b Fm(\000)18 b Fv(\003)1447
4078 y Fp(c)1447 4133 y Fo(+)1502 4020 y Ft(\021)1551
4112 y Fq( )1608 4017 y Ft(\015)1608 4067 y(\015)1608
4116 y(\015)1654 4170 y Fp(L)1700 4154 y Ff(2)1732 4170
y Fo(\()p Fg(R)1800 4154 y Ff(3)1833 4170 y Fp(;)p Fg(C)1895
4154 y Ff(4)1927 4170 y Fo(\))1980 4112 y Fm(\024)23
b Fq(")2136 4017 y Ft(\015)2136 4067 y(\015)2136 4116
y(\015)2182 4020 y(\020)2232 4112 y Fm(\000)p Fq(c)2333
4078 y Fo(2)2369 4112 y Fv(\001)c(+)f Fq(c)2576 4078
y Fo(4)2613 4020 y Ft(\021)2673 4015 y Ff(1)p 2673 4024
V 2673 4057 a(4)2715 4112 y Fq( )2772 4017 y Ft(\015)2772
4067 y(\015)2772 4116 y(\015)2818 4170 y Fp(L)2864 4154
y Ff(2)2896 4170 y Fo(\()p Fg(R)2964 4154 y Ff(3)2997
4170 y Fp(;)p Fg(C)3059 4154 y Ff(4)3091 4170 y Fo(\))3150
4112 y Fq(:)446 4323 y Fv(In)28 b([14)o(])f(the)h(follo)n(wing)f
(result)g(is)h(pro)n(v)n(ed)e(:)321 4489 y Fs(Theorem)32
b(3)f(\([14)o(]\))45 b Fl(T)-6 b(ake)32 b Fq(V)44 b(;)39
b(N)g Fl(\014xe)l(d.)j(F)-6 b(or)31 b Fq(c)f Fl(lar)l(ge)i(and)f
Fq(\017)2386 4501 y Fo(0)2423 4489 y Fq(;)45 b(\024)30
b Fl(smal)t(l)i(enough,)g(for)g(al)t(l)45 b Fq(P)321
4588 y(")360 4600 y Fo(0)397 4588 y Fl(-close)31 b(to)43
b Fv(\003)802 4558 y Fp(c)802 4609 y Fo(+)857 4588 y
Fl(,)1138 4688 y Fq(c)p Fv(\()p Fq(P)12 b Fv(\))23 b(:=)233
b(inf)1437 4750 y Fp(W)1508 4733 y Ff(+)1555 4750 y Fn(2)p
Fp(G)1652 4758 y Fe(N)1705 4750 y Fo(\()p Fp(P)9 b(H)1840
4733 y Ff(1)p Fe(=)p Ff(2)1932 4750 y Fo(\))2111 4688
y Fv(sup)1982 4764 y Fe(W)e Fd(2)p Fe(G)2125 4778 y(N)2179
4764 y Ff(\()p Fe(H)2250 4749 y Ff(1)p Fe(=)p Ff(2)2342
4764 y(\))2027 4818 y Fe(P)h Ff(\()p Fe(W)g Ff(\)=)p
Fe(W)2272 4831 y Ff(+)2388 4688 y Fm(E)2439 4654 y Fp(c)2432
4708 y(\024)2475 4688 y Fv(\()p Fq(W)k Fv(\))321 4953
y Fl(is)33 b(indep)l(endent)g(of)h Fq(P)44 b Fl(and)33
b(we)g(denote)g(it)g(by)47 b Fq(E)1909 4923 y Fp(c)1904
4973 y(\024)1947 4953 y Fl(.)g(Mor)l(e)l(over,)35 b Fq(E)2479
4923 y Fp(c)2474 4973 y(\024)2550 4953 y Fl(is)e(achieve)l(d)i(by)e(a)g
(solution)321 5053 y Fq(W)399 5065 y Fp(\024)466 5053
y Fv(=)p Fl(sp)l(an)p Fm(f)p Fq( )788 5065 y Fo(1)824
5053 y Fq(;)14 b(:)g(:)g(:)g( )1026 5065 y Fp(N)1089
5053 y Fm(g)29 b Fl(of)i(the)f(Dir)l(ac-F)-6 b(o)l(ck)30
b(e)l(quations:)1277 5155 y Ft(\032)1380 5219 y Fq(H)1456
5189 y Fp(c)1449 5242 y(\024;W)1570 5250 y Fe(\024)1613
5219 y Fq( )1667 5231 y Fp(i)1718 5219 y Fv(=)22 b Fq(\017)1839
5189 y Fp(c)1839 5241 y(i)1887 5219 y Fq( )1944 5189
y Fp(c)1941 5241 y(i)1978 5219 y Fq(;)43 b Fv(0)23 b
Fq(<)f(\017)2230 5189 y Fp(c)2230 5241 y(i)2287 5219
y Fq(<)h Fv(1)p Fq(;)1380 5323 y Fv(Gram)1589 5338 y
Fp(L)1635 5321 y Ff(2)1671 5323 y Fv(\011)f(=)h(1)1888
5335 y Fg(N)3264 5272 y Fv(\(DF\))321 5487 y Fl(with)792
5614 y Fq(H)868 5579 y Fp(c)861 5634 y(\024;W)1009 5614
y Fq(')h Fv(:=)e(\()p Fq(H)1305 5579 y Fp(c)1358 5614
y Fv(+)c Fq(V)37 b Fv(+)18 b Fq(\024)c(\032)1714 5626
y Fo(\011)1787 5614 y Fm(\003)1883 5558 y Fv(1)p 1857
5595 94 4 v 1857 5671 a Fm(j)p Fq(x)p Fm(j)1961 5614
y Fv(\))p Fq(')19 b Fm(\000)f Fq(\024)2211 5501 y Ft(Z)2256
5689 y Fg(R)2298 5673 y Ff(3)2359 5558 y Fq(R)2422 5570
y Fo(\011)2477 5558 y Fv(\()p Fq(x;)c(y)s Fv(\))p Fq(')p
Fv(\()p Fq(y)s Fv(\))p 2359 5595 474 4 v 2476 5671 a
Fm(j)p Fq(x)19 b Fm(\000)f Fq(y)s Fm(j)2842 5614 y Fq(dy)26
b(:)276 b Fv(\(MF\))1861 5940 y Fi(4)p eop end
%%Page: 5 5
TeXDict begin 5 4 bop 321 343 a Fs(Remark.)63 b Fv(It)37
b(is)f(easy)g(to)g(v)n(erify)g(that)50 b Fq(")1716 355
y Fo(0)1791 343 y Fq(>)37 b Fv(0)50 b(giv)n(en,)38 b(for)50
b Fq(c)36 b Fv(large)f(and)h Fq(\024)51 b Fv(small)36
b(enough,)321 443 y Fq(\037)373 458 y Fo(\(0)p Fp(;)p
Fn(1)p Fo(\))548 443 y Fv(\()p Fq(H)656 413 y Fp(c)649
466 y(\024;W)770 474 y Fe(\024)813 443 y Fv(\))28 b(is)42
b Fq(")1010 455 y Fo(0)1046 443 y Fv(-close)27 b(to)41
b(\003)1446 413 y Fo(+)1446 463 y Fp(c)1501 443 y Fv(.)321
582 y Fs(Corollary)32 b(4)g(\([14)o(]\))45 b Fl(T)-6
b(ake)27 b Fq(V)5 b(;)28 b(N)36 b Fl(\014xe)l(d.)h(Cho)l(ose)28
b Fq(c)f Fl(lar)l(ge)g(and)g Fq(\024)g Fl(smal)t(l)h(enough.)38
b(If)27 b(we)g(de\014ne)321 682 y(the)j(pr)l(oje)l(ctor)1487
782 y Fq(P)1552 746 y Fo(+)1540 806 y Fp(\024;W)1698
782 y Fv(=)22 b Fq(\037)1837 797 y Fo(\(0)p Fp(;)p Fn(1)p
Fo(\))2012 782 y Fv(\()p Fq(H)2120 747 y Fp(c)2113 802
y(\024;W)2248 782 y Fv(\))321 917 y Fl(with)31 b Fq(H)578
886 y Fp(c)571 939 y(\024;W)734 917 y Fl(given)g(by)f(formula)h
(\(MF\),)f(then)975 1075 y Fq(E)1041 1041 y Fp(c)1036
1095 y(\024)1102 1075 y Fv(=)148 b(min)1200 1135 y Fe(W)7
b Fd(2)p Fe(G)1343 1143 y(n)1384 1135 y Ff(\()p Fe(H)1455
1120 y Ff(1)p Fe(=)p Ff(2)1547 1135 y(\))1226 1202 y
Fe(P)1269 1181 y Ff(+)1262 1225 y Fe(\024;W)1380 1202
y(W)g Ff(=)p Fe(W)1593 1075 y Fm(E)1644 1041 y Fp(c)1637
1095 y(\024)1680 1075 y Fv(\()p Fq(W)12 b Fv(\))24 b(=)223
b(min)2024 1135 y Fe(W)7 b Fd(2)p Fe(G)2167 1149 y(N)2221
1135 y Ff(\()p Fe(H)2292 1120 y Ff(1)p Fe(=)p Ff(2)2384
1135 y(\))1955 1188 y Fe(W)37 b Ff(solution)30 b(of)k(\(DF\))2499
1075 y Fm(E)2550 1041 y Fp(c)2543 1095 y(\024)2585 1075
y Fv(\()p Fq(W)12 b Fv(\))31 b Fq(:)547 b Fv(\(2\))446
1406 y(Another)30 b(v)-5 b(ariational)29 b(problem)h(w)n(as)f(in)n(tro)
r(duced)h(in)h(the)f(w)n(orks)f(of)i(Bac)n(h)e Fl(et)j(al.)46
b Fv(and)31 b(Bar-)321 1505 y(baroux)c Fl(et)i(al.)38
b Fv(\([10)o(,)28 b(11)o(,)g(12)o(]\))g(:)37 b(de\014ne)1025
1684 y Ft(e)1008 1705 y Fm(P)1066 1717 y Fp(\024)1132
1705 y Fv(=)23 b Fm(f)p Fq(P)1327 1669 y Fo(+)1315 1748
y Fp(\024;)1377 1733 y Fc(f)1374 1748 y Fp(W)1472 1705
y Fv(=)f Fq(\037)1611 1720 y Fo([0)p Fp(;)p Fn(1)p Fo(\))1779
1705 y Fv(\()p Fq(H)1887 1670 y Fp(c)1880 1742 y(\024;)1942
1727 y Fc(f)1939 1742 y Fp(W)2015 1705 y Fv(\))h(;)2133
1684 y Ft(f)2130 1705 y Fq(W)35 b Fm(2)23 b Fq(G)2386
1717 y Fp(N)2450 1705 y Fv(\()p Fq(H)2558 1670 y Fo(1)p
Fp(=)p Fo(2)2662 1705 y Fv(\))p Fm(g)p Fq(;)581 b Fv(\(3\))321
1860 y(and)728 2118 y Fq(S)784 2084 y Fp(N)779 2155 y(\024;)841
2140 y Fc(f)838 2155 y Fp(W)996 2118 y Fv(:=)83 b Fm(f)p
Fq(\015)27 b Fm(2)c Fq(S)1408 2130 y Fo(1)1446 2118 y
Fv(\()p Fq(L)1535 2084 y Fo(2)1572 2118 y Fv(\))14 b
Fq(;)41 b(\015)28 b Fv(=)23 b Fq(\015)1889 2084 y Fn(\003)1940
2118 y Fq(;)42 b(H)2081 2084 y Fp(c)2074 2155 y(\024;)2136
2140 y Fc(f)2133 2155 y Fp(W)2208 2118 y Fq(\015)27 b
Fm(2)d Fq(S)2408 2130 y Fo(1)2459 2118 y Fq(;)1222 2271
y(P)1287 2236 y Fo(+)1275 2314 y Fp(\024;)1337 2299 y
Fc(f)1334 2314 y Fp(W)1410 2271 y Fq(\015)5 b(P)1523
2236 y Fn(\000)1511 2314 y Fp(\024;)1573 2299 y Fc(f)1570
2314 y Fp(W)1668 2271 y Fv(=)22 b(0)14 b Fq(;)41 b Fm(\000)p
Fq(P)2005 2236 y Fn(\000)1993 2314 y Fp(\024;)2055 2299
y Fc(f)2052 2314 y Fp(W)2150 2271 y Fm(\024)23 b Fq(\015)28
b Fm(\024)22 b Fq(P)2461 2236 y Fo(+)2449 2314 y Fp(\024;)2511
2299 y Fc(f)2508 2314 y Fp(W)2597 2271 y Fq(;)42 b Fv(tr)14
b Fq(\015)27 b Fv(=)c Fq(N)9 b Fm(g)p Fq(;)321 2464 y
Fv(with)36 b(the)g(notation)f Fq(P)1074 2429 y Fn(\000)1062
2507 y Fp(\024;)1124 2492 y Fc(f)1121 2507 y Fp(W)1233
2464 y Fv(:=)g(1)-28 b(I)24 b Fm(\000)f Fv(P)1569 2429
y Fo(+)1569 2507 y Fp(\024;)1629 2492 y Fc(f)1628 2507
y Fo(W)1699 2464 y Fv(,)38 b(and)d Fq(S)1980 2476 y Fo(1)2053
2464 y Fv(b)r(eing)g(the)h(Banac)n(h)e(space)h(of)g(trace-class)321
2597 y(op)r(erators)26 b(on)h Fq(L)861 2567 y Fo(2)898
2597 y Fv(\()p Fk(R)990 2567 y Fo(3)1027 2597 y Fq(;)14
b Fk(C)1124 2567 y Fo(4)1162 2597 y Fv(\).)37 b(F)-7
b(or)27 b(all)g Fq(\015)h Fm(2)23 b Fq(S)1723 2567 y
Fp(N)1718 2638 y(\024;)1780 2623 y Fc(f)1777 2638 y Fp(W)1852
2597 y Fv(,)28 b(let)588 2840 y Fm(F)656 2806 y Fp(c)648
2861 y(\024)691 2840 y Fv(\()p Fq(\015)5 b Fv(\))23 b(=)g(tr)13
b(\()q(\()p Fq(H)1133 2806 y Fp(c)1185 2840 y Fv(+)18
b Fq(V)h Fv(\))p Fq(\015)5 b Fv(\))18 b(+)1558 2784 y
Fq(\024)p 1558 2821 48 4 v 1561 2897 a Fv(2)1630 2727
y Ft(Z)1737 2784 y Fq(\032)1780 2796 y Fp(\015)1822 2784
y Fv(\()p Fq(x)p Fv(\))p Fq(\032)1976 2796 y Fp(\015)2020
2784 y Fv(\()p Fq(y)s Fv(\))p 1737 2821 392 4 v 1813
2897 a Fm(j)p Fq(x)h Fm(\000)f Fq(y)s Fm(j)2138 2840
y Fv(d)p Fq(x)c Fv(d)p Fq(y)22 b Fm(\000)2447 2784 y
Fq(\024)p 2447 2821 48 4 v 2450 2897 a Fv(2)2519 2727
y Ft(Z)2625 2784 y Fm(j)p Fq(\015)5 b Fv(\()p Fq(x;)14
b(y)s Fv(\))p Fm(j)2911 2754 y Fo(2)p 2625 2821 324 4
v 2668 2897 a Fm(j)p Fq(x)19 b Fm(\000)f Fq(y)s Fm(j)2959
2840 y Fv(d)p Fq(x)c Fv(d)p Fq(y)s(:)321 3064 y Fv(Here,)26
b Fq(\032)582 3076 y Fp(\015)624 3064 y Fv(\()p Fq(x)p
Fv(\))e(:=)870 3002 y Ft(P)958 3022 y Fo(4)958 3089 y
Fp(s)p Fo(=1)1091 3064 y Fq(\015)1134 3076 y Fp(s;s)1221
3064 y Fv(\()p Fq(x;)k(x)p Fv(\))c(=)1542 3002 y Ft(P)1629
3089 y Fp(n)1688 3064 y Fq(w)1747 3076 y Fp(n)1793 3064
y Fm(j)p Fq( )1870 3076 y Fp(n)1915 3064 y Fv(\()p Fq(x)p
Fv(\))p Fm(j)2049 3034 y Fo(2)2088 3064 y Fv(,)i(with)g
Fq(w)2383 3076 y Fp(n)2454 3064 y Fv(the)g(eigen)n(v)-5
b(alues)24 b(of)h Fq(\015)30 b Fv(and)c Fq( )3401 3076
y Fp(n)321 3164 y Fv(the)k(eigenspinors)d(of)i Fq(\015)5
b Fv(,)29 b(and)g Fq(\015)5 b Fv(\()p Fq(x;)14 b(y)s
Fv(\))26 b(=)1649 3102 y Ft(P)1737 3189 y Fp(n)1796 3164
y Fq(w)1855 3176 y Fp(n)1901 3164 y Fq( )1955 3176 y
Fp(n)2000 3164 y Fv(\()p Fq(x)p Fv(\))20 b Fm(\012)p
2215 3096 100 4 v 19 w Fq( )2269 3176 y Fp(n)2315 3164
y Fv(\()p Fq(y)s Fv(\),)29 b(i.e.,)h Fq(\015)5 b Fv(\()p
Fq(x;)14 b(y)s Fv(\))30 b(is)f(the)g(k)n(ernel)f(of)321
3263 y Fq(\015)5 b Fv(.)446 3403 y(In)25 b([12])g(it)h(has)f(b)r(een)h
(pro)n(v)n(ed)e(that)i(for)f(ev)n(ery)38 b Fq(P)1990
3368 y Fo(+)1978 3446 y Fp(\024;)2040 3431 y Fc(f)2037
3446 y Fp(W)2135 3403 y Fm(2)2230 3382 y Ft(e)2214 3403
y Fm(P)2272 3415 y Fp(\024)2328 3403 y Fv(,)27 b(the)e(in\014m)n(um)i
(of)39 b Fm(F)3016 3373 y Fp(c)3008 3424 y(\024)3090
3403 y Fv(on)25 b(the)h(set)321 3536 y Fq(S)377 3505
y Fp(N)372 3577 y(\024;)434 3562 y Fc(f)431 3577 y Fp(W)539
3536 y Fv(is)18 b(actually)g(equal)g(to)g(the)h(in\014m)n(um)g
(de\014ned)g(in)f(the)h(smaller)f(class)f(of)i(Slater)f(determinan)n
(ts.)321 3654 y(More)27 b(precisely)-7 b(,)27 b(with)h(the)g(ab)r(o)n
(v)n(e)e(notations,)321 3814 y Fs(Theorem)32 b(5)f(\([12)o(]\))45
b Fl(F)-6 b(or)30 b Fq(\024)g Fl(smal)t(l)h(enough)f(and)g(for)h(al)t
(l)f Fq(P)2287 3779 y Fo(+)2275 3857 y Fp(\024;)2337
3842 y Fc(f)2334 3857 y Fp(W)2433 3814 y Fm(2)2528 3793
y Ft(e)2511 3814 y Fm(P)2569 3826 y Fp(\024)2612 3814
y Fl(,)g(one)g(has)1244 3998 y Fv(inf)1174 4060 y Fp(\015)t
Fn(2)p Fp(S)1302 4040 y Fe(N)1299 4093 y(\024;)1359 4083
y Fb(f)1352 4093 y Fe(W)1430 3998 y Fm(F)1498 3963 y
Fp(c)1490 4018 y(\024)1532 3998 y Fv(\()p Fq(\015)5 b
Fv(\))23 b(=)264 b(inf)1755 4066 y Fp(W)9 b Fn(2)p Fp(G)1923
4074 y Fe(N)1976 4066 y Fo(\()p Fp(P)2053 4039 y Ff(+)2044
4099 y Fe(\024;)2104 4089 y Fb(f)2097 4099 y Fe(W)2162
4066 y Fp(H)2220 4049 y Ff(1)p Fe(=)p Ff(2)2312 4066
y Fo(\))2352 3998 y Fm(E)2403 3963 y Fp(c)2396 4018 y(\024)2439
3998 y Fv(\()p Fq(W)j Fv(\))747 b(\(4\))321 4240 y Fl(Mor)l(e)l(over,)
31 b(the)e(in\014mum)f(is)h(achieve)l(d)i(by)e(a)g(solution)h(of)f(the)
g(pr)l(oje)l(cte)l(d)h(Dir)l(ac-F)-6 b(o)l(ck)29 b(e)l(quations,)321
4340 y(namely)1532 4516 y Fq(\015)1575 4528 y Fp(min)1726
4516 y Fv(=)1844 4412 y Fp(N)1813 4437 y Ft(X)1820 4614
y Fp(i)p Fo(=1)1933 4516 y Fm(h)p Fq( )2019 4528 y Fp(i)2061
4516 y Fq(;)14 b(:)p Fm(i)p Fq( )2207 4528 y Fp(i)321
4737 y Fl(with)31 b Fq(P)567 4702 y Fo(+)555 4780 y Fp(\024;)617
4765 y Fc(f)614 4780 y Fp(W)689 4737 y Fq( )743 4749
y Fp(i)794 4737 y Fv(=)22 b Fq( )935 4749 y Fp(i)993
4737 y Fl(\()p Fq(i)g Fv(=)h(1)p Fq(;)14 b(:)g(:)g(:)f(N)c
Fv(\))p Fl(,)30 b(and)h(for)f Fq(W)1890 4749 y Fo(min)2028
4737 y Fv(:=)23 b Fq(span)p Fv(\()p Fq( )2400 4749 y
Fo(1)2437 4737 y Fq(;)14 b(:)g(:)g(:)f(;)h( )2675 4749
y Fp(N)2738 4737 y Fv(\))30 b Fl(,)1075 4862 y Ft(\()1183
4942 y Fq(P)1248 4906 y Fo(+)1236 4985 y Fp(\024;)1298
4970 y Fc(f)1295 4985 y Fp(W)1371 4942 y Fq(H)1447 4912
y Fp(c)1440 4965 y(\024;W)1561 4973 y Ff(min)1663 4942
y Fq(P)1728 4906 y Fo(+)1716 4985 y Fp(\024;)1778 4970
y Fc(f)1775 4985 y Fp(W)1850 4942 y Fq( )1904 4954 y
Fp(i)1955 4942 y Fv(=)22 b Fq(\017)2076 4954 y Fp(i)2104
4942 y Fq( )2158 4954 y Fp(i)2185 4942 y Fq(;)44 b Fv(0)23
b Fq(<)f(\017)2438 4954 y Fp(i)2489 4942 y Fq(<)g Fv(1)p
Fq(;)1183 5066 y Fv(Gram)1392 5081 y Fp(L)1438 5064 y
Ff(2)1474 5066 y Fv(\011)g(=)h(1)1691 5078 y Fg(N)3340
5004 y Fv(\(5\))446 5272 y(Let)28 b(us)f(no)n(w)g(de\014ne)h(the)g
(follo)n(wing)e(sup-inf:)1162 5431 y Fq(e)1201 5397 y
Fp(c)1201 5451 y(\024)1266 5431 y Fv(:=)89 b(sup)1377
5517 y Fp(P)1428 5491 y Ff(+)1419 5550 y Fe(\024;)1479
5540 y Fb(f)1472 5550 y Fe(W)1537 5517 y Fn(2)1594 5502
y Fc(e)1582 5517 y Fn(P)1972 5431 y Fv(inf)1731 5499
y Fp(W)9 b Fn(2)p Fp(G)1899 5507 y Fe(N)1952 5499 y Fo(\()p
Fp(P)2029 5472 y Ff(+)2020 5532 y Fe(\024;)2080 5522
y Fb(f)2073 5532 y Fe(W)2138 5499 y Fp(H)2196 5483 y
Ff(1)p Fe(=)p Ff(2)2288 5499 y Fo(\))2328 5431 y Fm(E)2379
5397 y Fp(c)2372 5451 y(\024)2414 5431 y Fv(\()p Fq(W)j
Fv(\))i Fq(:)735 b Fv(\(6\))446 5691 y(Then)28 b(Theorem)e(2)i(has)f
(the)h(follo)n(wing)e(consequence:)1861 5940 y Fi(5)p
eop end
%%Page: 6 6
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b Fl(If)30 b Fq(\024)g Fl(is)g(smal)t(l)h(enough,)1377
526 y Fq(e)1416 491 y Fp(c)1416 546 y(\024)1482 526 y
Fv(=)105 b(sup)1569 612 y Fp(P)1620 585 y Ff(+)1611 645
y Fe(\024;)1671 635 y Fb(f)1664 645 y Fe(W)1730 612 y
Fn(2)1787 597 y Fc(e)1775 612 y Fn(P)1822 620 y Fe(\024)1990
526 y Fv(inf)1920 588 y Fp(\015)t Fn(2)p Fp(S)2048 568
y Fe(N)2045 621 y(\024;)2105 611 y Fb(f)2098 621 y Fe(W)2175
526 y Fm(F)2243 491 y Fp(c)2235 546 y(\024)2278 526 y
Fv(\()p Fq(\015)5 b Fv(\))446 811 y(F)-7 b(rom)37 b(the)g(ab)r(o)n(v)n
(e)f(de\014nitions,)k(Theorem)d(3,)i(Corollary)c(4)i(and)g(the)h
(remark)d(made)i(after)321 910 y(Theorem)27 b(3,)g(w)n(e)h(clearly)e
(see)h(that)h(for)f(all)g Fq(\024)h Fv(small)f(and)h
Fq(c)f Fv(large,)1724 1093 y Fq(E)1790 1059 y Fp(c)1785
1113 y(\024)1851 1093 y Fm(\025)c Fq(e)1978 1059 y Fp(c)1978
1113 y(\024)2020 1093 y Fq(:)1297 b Fv(\(7\))321 1276
y(One)28 b(can)f(hop)r(e)g(more:)321 1425 y Fs(Conjecture)i(M:)c
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1445 y(\024)1755 1425 y Fl(and)f Fq(e)1966 1395 y Fp(c)1966
1445 y(\024)2050 1425 y Fl(c)l(oincide,)30 b(and)e(ther)l(e)f(is)h(a)g
(solution)f Fq(W)3317 1395 y Fp(c)3305 1445 y(\024)3378
1425 y Fl(of)321 1525 y(the)j(DF)g(e)l(quations)g(such)g(that)1162
1707 y Fm(E)1213 1673 y Fp(c)1206 1728 y(\024)1249 1707
y Fv(\()p Fq(W)1371 1673 y Fp(c)1359 1728 y(\024)1405
1707 y Fv(\))23 b(=)g Fq(e)1587 1673 y Fp(c)1587 1728
y(\024)1652 1707 y Fv(=)271 b(inf)1740 1776 y Fp(V)14
b Fn(2)p Fp(G)1890 1784 y Fe(N)1943 1776 y Fo(\()p Fp(P)2020
1749 y Ff(+)2011 1797 y Fe(\024;W)2125 1785 y(c)2117
1814 y(\024)2161 1776 y Fp(H)2219 1759 y Ff(1)p Fe(=)p
Ff(2)2311 1776 y Fo(\))2350 1707 y Fm(E)2401 1673 y Fp(c)2394
1728 y(\024)2437 1707 y Fv(\()p Fq(V)19 b Fv(\))14 b
Fq(:)321 1997 y Fl(In)25 b(other)g(wor)l(ds,)i(the)f(max-min)e(level)i
Fq(e)1589 1966 y Fp(c)1589 2017 y(\024)1657 1997 y Fl(is)f(attaine)l(d)
g(by)h(a)f(p)l(air)h Fv(\()p Fq(W)n(;)14 b(P)2601 1961
y Fo(+)2589 2039 y Fp(\024;)2651 2024 y Fc(f)2648 2039
y Fp(W)2724 1997 y Fv(\))25 b Fl(such)g(that)3133 1976
y Ft(f)3130 1997 y Fq(W)35 b Fv(=)23 b Fq(W)12 b Fl(.)446
2246 y Fv(This)34 b(pap)r(er)f(is)h(dev)n(oted)f(to)h(discussing)f
(this)h(conjecture,)h(whic)n(h,)h(if)e(it)g(w)n(ere)f(true,)j(w)n(ould)
321 2345 y(allo)n(w)27 b(us)h(to)f(in)n(terpret)h(the)g(Dirac-F)-7
b(o)r(c)n(k)26 b(mo)r(del)i(as)f(a)h(v)-5 b(ariational)26
b(appro)n(ximation)g(of)i(QED.)f(In)321 2495 y(order)22
b(to)i(study)g(the)f(di\013eren)n(t)h(cases)e(that)i(can)f(app)r(ear)g
(when)h(studying)f(the)h(problems)37 b Fq(E)3237 2465
y Fp(c)3232 2515 y(\024)3312 2495 y Fv(and)321 2594 y
Fq(e)360 2564 y Fp(c)360 2615 y(\024)444 2594 y Fv(for)k
Fq(\024)h Fv(small,)27 b(w)n(e)g(b)r(egin)h(b)n(y)f(discussing)g(the)h
(case)41 b Fq(\024)23 b Fv(=)f(0.)321 2818 y Fs(Prop)s(osition)30
b(7)46 b Fl(Conje)l(ctur)l(e)30 b(M)g(is)g(true)f(in)h(the)g(c)l(ase)g
Fq(\024)23 b Fv(=)f(0)p Fl(.)321 2993 y Fs(Pro)s(of)p
Fv(.)70 b(The)38 b(case)g Fq(\024)j Fv(=)f(0)e(is)h(ob)n(vious.)68
b(Indeed,)41 b(all)d(pro)5 b(jectors)37 b Fq(P)2647 2957
y Fo(+)2635 3036 y(0)p Fp(;)2691 3021 y Fc(f)2688 3036
y Fp(W)2802 2993 y Fv(coincide)h(with)h(the)321 3118
y(pro)5 b(jector)41 b Fq(\037)746 3133 y Fo([0)p Fp(;)p
Fn(1)p Fo(\))913 3118 y Fv(\()p Fq(H)1021 3088 y Fp(c)1083
3118 y Fv(+)28 b Fq(V)19 b Fv(\).)80 b(The)42 b(lev)n(el)g
Fq(E)1833 3088 y Fp(c)1828 3138 y Fo(0)1867 3118 y Fv(,)j(seen)d(as)f
(the)i(minim)n(um)g(of)e(Corollary)f(2,)45 b(is)321 3217
y(ac)n(hiev)n(ed)37 b(b)n(y)i(an)n(y)e Fq(N)9 b Fv(-dimensional)38
b(space)f Fq(W)1842 3229 y Fp(min)2009 3217 y Fv(spanned)h(b)n(y)g
Fq(N)47 b Fv(orthogonal)37 b(eigen)n(v)n(ectors)321 3317
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b Fv(whose)30 b(eigen)n(v)-5 b(alues)30 b(are)g(the)h
Fq(N)40 b Fv(\014rst)31 b(p)r(ositiv)n(e)f(eigen)n(v)-5
b(alues)30 b(of)h Fq(H)2901 3287 y Fp(c)2955 3317 y Fv(+)21
b Fq(V)d Fv(,)32 b(coun)n(ted)321 3417 y(with)i(m)n(ultiplicit)n(y)-7
b(.)53 b(Then)33 b Fq(E)1290 3386 y Fp(c)1285 3437 y
Fo(0)1357 3417 y Fv(is)g(the)g(sum)g(of)g(these)g Fq(N)42
b Fv(\014rst)33 b(p)r(ositiv)n(e)f(eigen)n(v)-5 b(alues.)52
b(Clearly)-7 b(,)321 3516 y(\()p Fq(W)431 3528 y Fp(min)560
3516 y Fq(;)14 b(\037)649 3531 y Fo([0)p Fp(;)p Fn(1)p
Fo(\))816 3516 y Fv(\()p Fq(H)924 3486 y Fp(c)977 3516
y Fv(+)k Fq(V)h Fv(\)\))28 b(realizes)e Fq(e)1549 3486
y Fp(c)1549 3537 y Fo(0)1586 3516 y Fv(.)1747 b Fh(2)321
3757 y Fs(3)32 b(-)f(P)m(erturbation)i(from)f(the)g(linear)f(closed)g
(shells)f(case)p Fv(.)446 3904 y(By)d(\\closed)g(shells",)f(w)n(e)i
(mean)1463 4164 y Fq(N)k Fv(=)1692 4060 y Fp(I)1649 4085
y Ft(X)1656 4262 y Fp(i)p Fo(=1)1783 4164 y Fq(N)1859
4129 y Fp(c)1850 4184 y(i)1893 4164 y Fq(;)97 b(I)30
b Fm(2)23 b Fk(Z)2212 4129 y Fo(+)2281 4164 y Fq(:)1036
b Fv(\(8\))321 4423 y(W)-7 b(e)24 b(recall)f(that)h Fq(N)930
4393 y Fp(c)921 4445 y(i)987 4423 y Fv(is)g(the)g(dimension)g(of)f(the)
i(eigenspace)d Fm(M)2327 4393 y Fp(c)2327 4445 y(i)2384
4423 y Fv(=)g(Ker)o(\()p Fq(H)2713 4393 y Fp(c)2758 4423
y Fv(+)11 b Fq(V)29 b Fm(\000)11 b Fq(\025)3035 4393
y Fp(c)3035 4445 y(i)3082 4423 y Fv(1)-33 b(1)o(\))24
b(de\014ned)321 4523 y(in)c Fm(x)p Fv(2.)33 b(Under)20
b(assumption)f(\(8\),)i(for)e Fq(\024)j Fv(=)h(0,)e(there)e(is)g(a)g
(unique)g(solution,)i Fq(W)2755 4493 y Fp(c)2743 4544
y Fo(0)2789 4523 y Fv(,)g(to)e(the)h(v)-5 b(ariational)321
4623 y(problems)27 b(de\014ning)h Fq(E)1057 4593 y Fp(c)1052
4643 y Fo(0)1118 4623 y Fv(and)g Fq(e)1319 4593 y Fp(c)1319
4643 y Fo(0)1356 4623 y Fv(,)1618 4803 y Fq(W)1708 4769
y Fp(c)1696 4824 y Fo(0)1765 4803 y Fv(=)1899 4700 y
Fp(I)1853 4725 y Ft(M)1862 4901 y Fp(i)p Fo(=1)1992 4803
y Fm(M)2092 4769 y Fp(c)2092 4824 y(i)2126 4803 y Fq(:)321
5025 y Fv(The)37 b("shells")e(of)i(energy)e Fq(\025)1241
4995 y Fp(c)1241 5047 y(i)1313 5025 y Fq(;)52 b Fv(1)37
b Fm(\024)h Fq(i)f Fm(\024)h Fq(I)45 b(;)36 b Fv(are)g(\\closed":)53
b(eac)n(h)36 b(one)g(is)g(o)r(ccupied)h(b)n(y)f(the)321
5125 y(maximal)31 b(n)n(um)n(b)r(er)h(of)f(electrons)g(allo)n(w)n(ed)f
(b)n(y)h(the)h(P)n(auli)f(exclusion)g(principle.)48 b(The)32
b(subspace)321 5225 y Fq(W)411 5195 y Fp(c)399 5245 y
Fo(0)473 5225 y Fv(is)27 b(in)n(v)-5 b(arian)n(t)27 b(under)g(the)h
(represen)n(tation)e Fm(\017)h Fv(of)h Fq(S)5 b(U)k Fv(\(2\).)446
5324 y(W)-7 b(e)28 b(are)e(in)n(terested)h(in)h(solutions)f
Fq(W)1642 5294 y Fp(c)1630 5345 y(\024)1703 5324 y Fv(of)g(the)h
(Dirac-F)-7 b(o)r(c)n(k)26 b(equations)h(lying)g(in)g(a)h(neigh)n(b)r
(or-)321 5424 y(ho)r(o)r(d)43 b(\012)25 b Fm(\032)g Fq(G)784
5436 y Fp(N)847 5424 y Fv(\()p Fq(H)955 5394 y Fo(1)p
Fp(=)p Fo(2)1060 5424 y Fv(\))k(of)g Fq(W)1307 5394 y
Fp(c)1295 5445 y Fo(0)1341 5424 y Fv(,)g(for)g Fq(\024)g
Fv(small.)40 b(Using)29 b(the)g(implicit)h(function)g(theorem,)f(w)n(e)
f(are)321 5524 y(going)33 b(to)g(sho)n(w)g(that)h(for)f(eac)n(h)g
Fq(\024)h Fv(small,)g Fq(W)1799 5493 y Fp(c)1787 5544
y(\024)1867 5524 y Fv(exists,)h(is)e(unique,)j(and)d(is)h(a)f(smo)r
(oth)h(function)321 5623 y(of)h Fq(\024)p Fv(.)60 b(F)-7
b(or)34 b(this)i(purp)r(ose,)g(w)n(e)f(need)g(a)g(lo)r(cal)f(c)n(hart)g
(near)h Fq(W)2340 5593 y Fp(c)2328 5644 y Fo(0)2373 5623
y Fv(.)60 b(T)-7 b(ak)n(e)34 b(an)h(orthonormal)e(basis)1861
5940 y Fi(6)p eop end
%%Page: 7 7
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y Fo(1)445 343 y Fq(;)14 b Fm(\001)g(\001)g(\001)27 b
Fq(;)14 b( )697 355 y Fp(N)760 343 y Fv(\))28 b(of)f
Fq(W)1004 313 y Fp(c)992 364 y Fo(0)1038 343 y Fv(,)g(whose)g(elemen)n
(ts)g(are)g(eigen)n(v)n(ectors)e(of)i Fq(H)2445 313 y
Fp(c)2496 343 y Fv(+)18 b Fq(V)h Fv(,)27 b(the)h(asso)r(ciated)e
(eigen-)321 443 y(v)-5 b(alues)31 b(b)r(eing)f Fq(\026)848
455 y Fo(1)914 443 y Fm(\024)e(\001)14 b(\001)g(\001)28
b(\024)f Fq(\026)1274 455 y Fp(N)1368 443 y Fv(\()p Fw(i.e.)46
b Fq(\025)1600 413 y Fp(c)1600 463 y Fo(1)1638 443 y
Fq(;)14 b(:)g(:)g(:)g(;)g(\025)1871 413 y Fp(c)1871 466
y(I)1940 443 y Fv(coun)n(ted)30 b(with)h(m)n(ultiplicit)n(y\).)47
b(Let)31 b Fm(Z)38 b Fv(b)r(e)31 b(the)321 542 y(orthogonal)20
b(space)i(of)g Fq(W)1127 512 y Fp(c)1115 563 y Fo(0)1183
542 y Fv(for)g(the)g Fq(L)1499 512 y Fo(2)1558 542 y
Fv(scalar)f(pro)r(duct,)i(in)g Fq(H)2287 512 y Fo(1)p
Fp(=)p Fo(2)2391 542 y Fv(\()p Fk(R)2483 512 y Fo(3)2520
542 y Fq(;)14 b Fk(C)2617 512 y Fo(4)2655 542 y Fv(\).)35
b(Then)22 b Fm(Z)30 b Fv(is)22 b(a)g(Hilb)r(ert)321 642
y(space)27 b(for)g(the)h Fq(H)889 612 y Fo(1)p Fp(=)p
Fo(2)1021 642 y Fv(scalar)e(pro)r(duct.)37 b(The)27 b(map)874
825 y Fq(C)43 b Fv(:)37 b Fq(\037)23 b Fv(=)g(\()p Fq(\037)1283
837 y Fo(1)1320 825 y Fq(;)14 b Fm(\001)g(\001)g(\001)27
b Fq(;)14 b(\037)1570 837 y Fp(N)1633 825 y Fv(\))24
b Fm(!)f Fq(span)p Fv(\()p Fq( )2056 837 y Fo(1)2111
825 y Fv(+)18 b Fq(\037)2246 837 y Fo(1)2283 825 y Fq(;)c
Fm(\001)g(\001)g(\001)28 b Fq(;)14 b( )2536 837 y Fp(N)2617
825 y Fv(+)k Fq(\037)2752 837 y Fp(N)2815 825 y Fv(\))23
b Fq(;)321 1007 y Fv(de\014ned)30 b(on)f(a)g(small)g(neigh)n(b)r(orho)r
(o)r(d)f Fm(O)k Fv(of)d(0)g(in)g Fm(Z)1969 977 y Fp(N)2032
1007 y Fv(,)h(is)f(the)h(desired)f(lo)r(cal)f(c)n(hart.)42
b(Denote)29 b Fq(G)3400 1019 y Fp(\037)321 1107 y Fv(the)f
Fq(N)f Fm(\002)18 b Fq(N)37 b Fv(matrix)27 b(of)h(scalar)d(pro)r(ducts)
j(\()p Fq(\037)1774 1119 y Fp(l)1799 1107 y Fq(;)14 b(\037)1888
1119 y Fp(`)1920 1107 y Fv(\))1952 1122 y Fp(L)1998 1105
y Ff(2)2035 1107 y Fv(.)37 b(Then)1175 1290 y Fm(E)1226
1255 y Fp(c)1219 1310 y(\024)1280 1290 y Fm(\016)18 b
Fq(C)6 b Fv(\()p Fq(\037)p Fv(\))24 b(=)e Fm(E)1683 1255
y Fp(c)1676 1310 y(\024)1719 1222 y Ft(\000)1757 1290
y Fv(\()p Fq(I)k Fv(+)18 b Fq(G)1999 1302 y Fp(\037)2045
1290 y Fv(\))2077 1255 y Fn(\000)p Fo(1)p Fp(=)p Fo(2)2233
1290 y Fv(\()p Fq( )k Fv(+)c Fq(\037)p Fv(\))2508 1222
y Ft(\001)2570 1290 y Fq(:)321 1472 y Fv(The)39 b(di\013eren)n(tial)e
(of)i(this)f(functional)g(de\014nes)h(a)e(smo)r(oth)h(map)g
Fq(F)2521 1484 y Fp(\024)2619 1472 y Fv(:)55 b Fm(O)43
b(\032)d(Z)2978 1442 y Fp(N)3082 1472 y Fm(!)g Fv(\()p
Fm(Z)3304 1442 y Fn(0)3328 1472 y Fv(\))3360 1442 y Fp(N)3423
1472 y Fv(,)321 1572 y(where)30 b Fm(Z)631 1542 y Fn(0)681
1572 y Fm(\032)d Fq(H)849 1542 y Fn(\000)p Fo(1)p Fp(=)p
Fo(2)1036 1572 y Fv(is)j(the)g(top)r(ological)f(dual)h(of)g
Fm(Z)37 b Fv(for)30 b(the)h Fq(H)2425 1542 y Fo(1)p Fp(=)p
Fo(2)2559 1572 y Fv(top)r(ology)-7 b(,)30 b(iden)n(ti\014ed)h(with)321
1671 y(the)j(orthogonal)c(space)j(of)g Fq(W)1308 1641
y Fp(c)1296 1692 y Fo(0)1374 1671 y Fv(for)g(the)g(dualit)n(y)g(pro)r
(duct)g(in)g Fq(H)2437 1641 y Fn(\000)p Fo(1)p Fp(=)p
Fo(2)2615 1671 y Fm(\002)22 b Fq(H)2778 1641 y Fo(1)p
Fp(=)p Fo(2)2882 1671 y Fv(.)53 b(Note)33 b(that)h Fq(F)3403
1683 y Fp(\024)321 1771 y Fv(dep)r(ends)24 b(smo)r(othly)f(on)g(the)h
(parameter)e Fq(\024)p Fv(.)35 b(A)24 b(subspace)f Fq(C)6
b Fv(\()p Fq(\037)p Fv(\))24 b(is)f(solution)g(of)h(\(DF\))g(if)g(and)g
(only)321 1871 y(if)31 b Fq(F)453 1883 y Fp(\024)497
1871 y Fv(\()p Fq(\037)p Fv(\))d(=)f(0.)45 b(T)-7 b(o)31
b(apply)f(the)h(implicit)g(function)g(theorem,)g(w)n(e)f(just)h(ha)n(v)
n(e)e(to)h(c)n(hec)n(k)g(that)h(the)321 1970 y(op)r(erator)e
Fq(L)e Fv(:=)g Fq(D)927 1982 y Fp(\037)972 1970 y Fq(F)1025
1982 y Fo(0)1063 1970 y Fv(\(0\))k(is)f(an)g(isomorphism)f(from)h
Fm(Z)2158 1940 y Fp(N)2251 1970 y Fv(to)g(its)h(dual)f(\()p
Fm(Z)2760 1940 y Fn(0)2783 1970 y Fv(\))2815 1940 y Fp(N)2879
1970 y Fv(.)45 b(This)30 b(op)r(erator)321 2070 y(is)e(simply)f(the)h
(Hessian)f(of)h(the)g(DF)g(energy)f(expressed)f(in)i(our)f(lo)r(cal)g
(co)r(ordinates:)974 2253 y Fq(L\037)c Fv(=)1194 2185
y Ft(\000)1232 2253 y Fv(\()p Fq(H)1333 2265 y Fp(c)1385
2253 y Fv(+)18 b Fq(V)38 b Fm(\000)18 b Fq(\026)1687
2265 y Fo(1)1724 2253 y Fv(\))p Fq(\037)1808 2265 y Fo(1)1845
2253 y Fq(;)c Fm(\001)g(\001)g(\001)28 b Fq(;)14 b Fv(\()p
Fq(H)2145 2265 y Fp(c)2197 2253 y Fv(+)k Fq(V)38 b Fm(\000)18
b Fq(\026)2499 2265 y Fp(N)2562 2253 y Fv(\))p Fq(\037)2646
2265 y Fp(N)2709 2185 y Ft(\001)2770 2253 y Fq(:)547
b Fv(\(9\))321 2435 y(Under)21 b(assumption)g(\(8\),)h(the)f(scalars)e
Fq(\026)1598 2447 y Fp(k)1639 2435 y Fq(;)28 b(k)e Fv(=)c(1)p
Fq(;)14 b(:)g(:)g(:)f(N)c Fv(,)23 b(are)c(not)i(eigen)n(v)-5
b(alues)20 b(of)h(the)g(restriction)321 2535 y(of)h Fq(H)486
2505 y Fp(c)527 2535 y Fv(+)7 b Fq(V)41 b Fv(to)22 b(the)g
Fq(L)978 2505 y Fo(2)1015 2535 y Fv(-orthogonal)d(subspace)j(of)g
Fq(W)1973 2505 y Fp(c)1961 2555 y Fo(0)2006 2535 y Fv(.)35
b(This)22 b(implies)h(that)f Fq(L)g Fv(is)f(an)h(isomorphism.)321
2634 y(As)36 b(a)e(consequence,)i(there)f(exists)g(a)g(neigh)n(b)r
(orho)r(o)r(d)f(of)h Fq(W)2284 2604 y Fp(c)2272 2655
y Fo(0)2341 2634 y Fm(\002)23 b(f)p Fv(0)p Fm(g)34 b
Fv(in)h Fq(G)2758 2646 y Fp(N)2822 2634 y Fv(\()p Fq(H)2930
2604 y Fo(1)p Fp(=)p Fo(2)3034 2634 y Fv(\))24 b Fm(\002)f
Fk(R)p Fv(,)37 b(\012)23 b Fm(\002)321 2734 y Fv(\()p
Fm(\000)p Fq(\024)466 2746 y Fo(0)503 2734 y Fq(;)14
b(\024)588 2746 y Fo(0)625 2734 y Fv(\))52 b(and)38 b(a)f(smo)r(oth)g
(function)h Fq(h)1646 2704 y Fp(c)1720 2734 y Fv(:)i(\()p
Fm(\000)p Fq(\024)1928 2746 y Fo(0)1965 2734 y Fq(;)14
b(\024)2050 2746 y Fo(0)2087 2734 y Fv(\))40 b Fm(!)g
Fv(\012)d(suc)n(h)h(that)g(for)f Fq(\024)j Fm(2)g Fv(\()p
Fm(\000)p Fq(\024)3232 2746 y Fo(0)3269 2734 y Fq(;)14
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%%Page: 8 8
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5312 y(\024)3069 5271 y Fv(\()p Fq(W)3191 5240 y Fp(c)3179
5291 y(\024)3225 5271 y Fv(\))23 b(=)g(0)14 b Fq(:)321
5382 y Fv(Since)32 b Fq(V)50 b Fv(is)32 b(radial,)f Fq(W)1081
5352 y Fp(c)1069 5403 y(\024)1147 5382 y Fv(b)r(elongs)g(to)g(an)g
Fq(S)5 b(U)k Fv(\(2\))31 b(orbit)g(of)h(minimizers.)48
b(W)-7 b(e)32 b(are)f(in)n(terested)g(in)321 5482 y(cases)d(where)g
(this)i(orbit)e(is)h(not)g(reduced)f(to)h(a)g(p)r(oin)n(t.)41
b(Then)29 b(the)g(mean-\014eld)g(op)r(erator)e Fq(H)3288
5452 y Fp(c)3281 5505 y(\024;W)3411 5488 y Fe(c)3402
5521 y(\024)321 5592 y Fv(should)22 b(not)f(comm)n(ute)h(with)g(the)g
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b(and)e(one)h(exp)r(ects)g(the)g(follo)n(wing)e(prop)r(ert)n(y)321
5691 y(to)28 b(hold:)1861 5940 y Fi(9)p eop end
%%Page: 10 10
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Fq(c)f Fl(lar)l(ge)i(enough,)f(if)h Fq(\024)e Fl(is)h(smal)t(l)g(then)f
(for)i(any)f(minimizer)g Fq(W)2927 313 y Fp(c)2915 364
y(\024)2991 343 y Fl(of)h Fm(E)3141 313 y Fp(c)3134 364
y(\024)3207 343 y Fl(on)f Fq(S)3378 355 y Fp(\024)3421
343 y Fl(,)321 443 y(ther)l(e)f(is)g(a)g(matrix)g Fq(A)23
b Fm(2)h Fq(S)5 b(U)k Fv(\(2\))29 b Fl(such)h(that)1515
625 y Fq(P)1580 590 y Fn(\000)1568 650 y Fp(\024;W)1698
633 y Fe(c)1689 666 y(\024)1734 625 y Fv(\()p Fq(A)19
b Fm(\017)f Fq(W)1997 591 y Fp(c)1985 646 y(\024)2031
625 y Fv(\))23 b Fm(6)p Fv(=)g(0)14 b Fq(:)1045 b Fv(\(17\))446
908 y(Let)28 b(us)f(explain)g(wh)n(y)h Fs(\(P\))g Fv(con)n(tradicts)e
(Conjecture)h(M:)321 1082 y Fs(Prop)s(osition)j(12)45
b Fl(If)37 b Fs(\(P\))f Fl(is)h(satis\014e)l(d,)h(then)e(for)h
Fq(c)f Fl(lar)l(ge)g(enough)h(and)f Fq(\024)g Fl(smal)t(l,)j(given)e
(any)321 1182 y(solution)30 b Fq(W)725 1151 y Fp(c)713
1202 y(\024)789 1182 y Fl(of)g(the)g(nonline)l(ar)g(Dir)l(ac-F)-6
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1151 y Fp(c)2736 1202 y(\024)2782 1182 y Fv(\))24 b(=)e
Fq(E)2986 1194 y Fp(\024)3029 1182 y Fl(,)31 b(we)f(have)1243
1364 y Fq(E)1309 1330 y Fp(c)1304 1385 y(\024)1370 1364
y Fv(=)23 b Fm(E)1509 1330 y Fp(c)1502 1385 y(\024)1545
1364 y Fv(\()p Fq(W)1667 1330 y Fp(c)1655 1385 y(\024)1701
1364 y Fv(\))g Fq(>)173 b Fv(inf)1854 1424 y Fe(W)7 b
Fd(2)p Fe(G)1997 1438 y(N)2051 1424 y Ff(\()p Fe(H)2122
1409 y Ff(1)p Fe(=)p Ff(2)2214 1424 y(\))1887 1495 y
Fe(P)1930 1474 y Fd(\000)1923 1519 y Fe(\024;W)2037 1507
y(c)2029 1536 y(\024)2072 1495 y(W)g Ff(=0)2260 1364
y Fm(E)2311 1330 y Fp(c)2304 1385 y(\024)2347 1364 y
Fv(\()p Fq(W)12 b Fv(\))p Fq(:)774 b Fv(\(18\))321 1757
y Fs(Pro)s(of)p Fv(.)77 b(Since)42 b Fq(\024)e Fv(is)h(small,)j(for)d
(an)n(y)f(matrix)h Fq(A)k Fm(2)h Fq(S)5 b(U)k Fv(\(2\))41
b(the)g(map)g Fq(P)2817 1721 y Fo(+)2805 1781 y Fp(\024;A)p
Fn(\017)p Fp(W)3019 1765 y Fe(c)3010 1798 y(\024)3095
1757 y Fv(induces)h(a)321 1880 y(di\013eomorphism)27
b(b)r(et)n(w)n(een)h(the)g(submanifolds)f Fq(G)1935 1892
y Fp(N)1999 1880 y Fv(\()p Fq(P)2096 1844 y Fo(+)2084
1904 y Fp(\024;W)2214 1888 y Fe(c)2205 1921 y(\024)2249
1880 y Fq(H)2325 1850 y Fo(1)p Fp(=)p Fo(2)2429 1880
y Fv(\))h(and)g Fq(G)2716 1892 y Fp(N)2779 1880 y Fv(\()p
Fq(P)2876 1844 y Fo(+)2864 1904 y Fp(\024;)12 b(A)p Fn(\017)p
Fp(W)3090 1888 y Fe(c)3081 1921 y(\024)3125 1880 y Fq(H)3201
1850 y Fo(1)p Fp(=)p Fo(2)3305 1880 y Fv(\))i(.)446 1991
y(No)n(w,)37 b(w)n(e)e(\014x)h Fq(A)h Fm(2)g Fq(S)5 b(U)k
Fv(\(2\))35 b(suc)n(h)h(that)g(\(17\))f(holds.)61 b(Then)36
b(there)f(exists)h(a)f(unique)h(p)r(oin)n(t)321 2091
y Fq(W)411 2061 y Fo(+)489 2091 y Fm(2)24 b Fq(G)633
2103 y Fp(N)696 2091 y Fv(\()p Fq(H)804 2061 y Fo(1)p
Fp(=)p Fo(2)908 2091 y Fv(\))k(suc)n(h)g(that)1149 2274
y Fq(P)1214 2238 y Fn(\000)1202 2298 y Fp(\024;W)1332
2282 y Fe(c)1323 2315 y(\024)1367 2274 y Fq(W)1457 2239
y Fo(+)1535 2274 y Fv(=)23 b(0)p Fq(;)97 b(P)1850 2238
y Fo(+)1838 2298 y Fp(\024;)11 b(A)p Fn(\017)p Fp(W)2063
2282 y Fe(c)2054 2315 y(\024)2098 2274 y Fq(W)2188 2239
y Fo(+)2266 2274 y Fv(=)23 b Fq(A)c Fm(\017)f Fq(W)2585
2239 y Fp(c)2573 2294 y(\024)3298 2274 y Fv(\(19\))321
2468 y(By)28 b(\(17\),)f(w)n(e)g(ha)n(v)n(e)1612 2568
y Fq(W)1702 2534 y Fo(+)1780 2568 y Fm(6)p Fv(=)c Fq(A)c
Fm(\017)e Fq(W)2098 2534 y Fp(c)2086 2588 y(\024)2132
2568 y Fq(:)321 2717 y Fv(On)28 b(the)g(other)f(hand,)g(in)h([14)o(])g
(it)g(w)n(as)f(pro)n(v)n(ed)f(that)1219 2900 y Fm(E)1270
2866 y Fp(c)1263 2921 y(\024)1306 2900 y Fv(\()p Fq(A)19
b Fm(\017)f Fq(W)1569 2866 y Fp(c)1557 2921 y(\024)1603
2900 y Fv(\))23 b(=)234 b(sup)1828 2976 y Fe(W)7 b Fd(2)p
Fe(G)1971 2990 y(N)2025 2976 y Ff(\()p Fe(H)2096 2961
y Ff(1)p Fe(=)p Ff(2)2188 2976 y(\))1756 3047 y Fe(P)1799
3026 y Ff(+)1792 3071 y Fe(\024;A)p Fd(\017)p Fe(W)1979
3059 y(c)1972 3088 y(\024)2015 3047 y(W)g Ff(=)p Fe(A)p
Fd(\017)p Fe(W)2252 3035 y(c)2245 3064 y(\024)2307 2900
y Fm(E)2358 2866 y Fp(c)2351 2921 y(\024)2394 2900 y
Fv(\()p Fq(W)12 b Fv(\))750 b(\(20\))321 3245 y(and)28
b Fq(A)18 b Fm(\017)g Fq(W)713 3215 y Fp(c)701 3266 y(\024)775
3245 y Fv(is)27 b(the)h(unique)g(solution)f(of)h(this)g(maximization)f
(problem.)36 b(Therefore,)1449 3428 y Fm(E)1500 3394
y Fp(c)1493 3448 y(\024)1536 3428 y Fv(\()p Fq(A)19 b
Fm(\017)f Fq(W)1799 3394 y Fp(c)1787 3448 y(\024)1833
3428 y Fv(\))23 b Fq(>)g Fm(E)2027 3394 y Fp(c)2020 3448
y(\024)2062 3428 y Fv(\()p Fq(W)2184 3394 y Fo(+)2240
3428 y Fv(\))g Fq(:)321 3610 y Fv(But)1222 3710 y Fm(E)1273
3676 y Fp(c)1266 3731 y(\024)1309 3710 y Fv(\()p Fq(W)1431
3676 y Fo(+)1486 3710 y Fv(\))h Fm(\025)279 b Fv(inf)1629
3779 y Fp(W)9 b Fn(2)p Fp(G)1797 3787 y Fe(N)1850 3779
y Fo(\()p Fp(P)1927 3752 y Ff(+)1918 3800 y Fe(\024;W)2032
3788 y(c)2024 3817 y(\024)2068 3779 y Fp(H)2126 3762
y Ff(1)p Fe(=)p Ff(2)2218 3779 y Fo(\))2258 3710 y Fm(E)2309
3676 y Fp(c)2302 3731 y(\024)2344 3710 y Fv(\()p Fq(W)j
Fv(\))24 b Fq(;)321 3945 y Fv(hence,)k(b)n(y)f(in)n(v)-5
b(ariance)27 b(of)g Fm(E)1228 3915 y Fp(c)1221 3966 y(\024)1292
3945 y Fv(under)g(the)h(action)f(of)h Fq(S)5 b(U)k Fv(\(2\),)1055
4128 y Fq(E)1121 4094 y Fp(c)1116 4148 y(\024)1182 4128
y Fv(=)23 b Fm(E)1321 4094 y Fp(c)1314 4148 y(\024)1357
4128 y Fv(\()p Fq(A)c Fm(\017)f Fq(W)1620 4094 y Fp(c)1608
4148 y(\024)1654 4128 y Fv(\))23 b Fq(>)279 b Fv(inf)1797
4196 y Fp(W)9 b Fn(2)p Fp(G)1965 4204 y Fe(N)2018 4196
y Fo(\()p Fp(P)2095 4169 y Ff(+)2086 4218 y Fe(\024;W)2200
4206 y(c)2192 4234 y(\024)2235 4196 y Fp(H)2293 4180
y Ff(1)p Fe(=)p Ff(2)2385 4196 y Fo(\))2425 4128 y Fm(E)2476
4094 y Fp(c)2469 4148 y(\024)2512 4128 y Fv(\()p Fq(W)j
Fv(\))23 b Fq(;)321 4392 y Fv(and)28 b(the)g(Prop)r(osition)e(is)h(pro)
n(v)n(ed.)1961 b Fh(2)446 4541 y Fv(Since)30 b(there)h(are)e(no)h
(solutions)g(of)g(\(DF\))i(under)e(lev)n(el)g Fq(E)2296
4511 y Fp(c)2291 4562 y(\024)2334 4541 y Fv(,)i(and)e
Fq(e)2592 4511 y Fp(c)2592 4562 y(\024)2662 4541 y Fm(\024)e
Fq(E)2821 4511 y Fp(c)2816 4562 y(\024)2859 4541 y Fv(,)j(Prop)r
(osition)e(12)321 4641 y(has)e(the)h(follo)n(wing)f(consequence:)321
4815 y Fs(Corollary)32 b(13)45 b Fl(If)40 b Fs(\(P\))f
Fl(is)h(satis\014e)l(d,)i(then)d(for)h Fq(c)f Fl(lar)l(ge)h(enough)g
(and)f Fq(\024)g Fl(smal)t(l,)k(ther)l(e)c(is)h(no)321
4915 y(solution)33 b Fq(W)716 4927 y Fn(\003)788 4915
y Fl(of)g(the)g(nonline)l(ar)g(Dir)l(ac-F)-6 b(o)l(ck)34
b(e)l(quations)f(with)g(p)l(ositive)h(L)l(agr)l(ange)g(multipliers,)321
5014 y(such)c(that)g(the)g(c)l(ouple)1668 5114 y Fv(\()p
Fq(W)1778 5126 y Fn(\003)1817 5114 y Fq(;)14 b(P)1919
5078 y Fo(+)1907 5138 y Fp(\024;W)2028 5146 y Fd(\003)2067
5114 y Fv(\))321 5263 y Fl(r)l(e)l(alizes)31 b(the)f(max-min)f
Fq(e)1136 5233 y Fp(c)1136 5284 y(\024)1179 5263 y Fl(.)38
b(So)30 b(Conje)l(ctur)l(e)g Fq(M)38 b Fl(is)31 b(wr)l(ong)e(when)i
Fs(\(P\))f Fl(holds.)446 5587 y Fv(W)-7 b(e)28 b(no)n(w)f(exhibit)h(a)f
(case)g(where)g Fs(\(P\))h Fv(holds.)1838 5940 y Fi(10)p
eop end
%%Page: 11 11
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b(14)45 b Fl(Assume)29 b(that)44 b Fq(N)32 b Fv(=)1701
300 y Fp(I)1658 325 y Ft(X)1665 501 y Fp(i)p Fo(=1)1792
404 y Fq(N)1868 369 y Fp(c)1859 424 y(i)1920 404 y Fv(+)18
b(1)p Fq(;)37 b(I)30 b Fm(\025)22 b Fv(1)p Fq(:)30 b
Fl(Then)g Fs(\(P\))g Fl(is)g(satis\014e)l(d.)321 686
y Fs(Pro)s(of)i(of)g(Prop)s(osition)e(14)p Fv(.)321 830
y Fs(Step)35 b(0)p Fv(.)44 b(Fix)h Fq(c)f Fv(large)28
b(enough)i(and)g(tak)n(e)g(a)f(sequence)h(of)g(p)r(ositiv)n(e)g
(parameters)e(\()p Fq(\024)3082 842 y Fp(`)3114 830 y
Fv(\))3146 842 y Fp(`)p Fn(\025)p Fo(0)3294 830 y Fv(con-)321
930 y(v)n(erging)d(to)h(0.)36 b(Let)26 b(\()p Fq(W)1081
900 y Fp(c)1069 953 y(`)1115 930 y Fv(\))1147 942 y Fp(`)p
Fn(\025)p Fo(0)1291 930 y Fv(b)r(e)g(a)g(sequence)g(in)h
Fq(G)1974 942 y Fp(N)2037 930 y Fv(\()p Fq(H)2145 900
y Fo(1)p Fp(=)p Fo(2)2249 930 y Fv(\),)g(with)g Fq(W)2609
900 y Fp(c)2597 953 y(`)2669 930 y Fv(a)f(minimizer)g(of)g
Fm(E)3267 900 y Fp(c)3260 950 y(\024)3299 959 y Fe(`)3358
930 y Fv(on)321 1029 y Fq(S)372 1041 y Fp(\024)411 1050
y Fe(`)444 1029 y Fv(.)36 b(Let)g Fq( )717 999 y Fp(c)714
1053 y(`)774 1029 y Fm(2)24 b Fq(W)943 999 y Fp(c)931
1053 y(`)1000 1029 y Fv(b)r(e)f(an)g(eigen)n(v)n(ector)d(of)j(the)h
(mean-\014eld)e(Hamiltonian)h Fq(H)2819 999 y Fp(c)2812
1052 y(\024)2851 1061 y Fe(`)2880 1052 y Fp(;W)2971 1032
y Fe(c)2962 1072 y(`)3006 1029 y Fv(,)h(normalised)321
1148 y(in)f Fq(L)470 1118 y Fo(2)530 1148 y Fv(and)f(corresp)r(onding)f
(to)i(the)g(highest)f(o)r(ccupied)h(lev)n(el.)35 b(Extracting)21
b(a)h(subsequence)g(if)h(nec-)321 1248 y(essary)-7 b(,)24
b(w)n(e)h(ma)n(y)g(assume)g(that)39 b Fq( )1414 1218
y Fp(c)1411 1271 y(`)1471 1248 y Fm(!)23 b Fq( )1634
1218 y Fp(c)1691 1248 y Fm(2)g(M)1869 1218 y Fp(c)1869
1271 y(I)5 b Fo(+1)2014 1248 y Fv(=)23 b(Ker)o(\()p Fq(H)2344
1218 y Fp(c)2392 1248 y Fv(+)13 b Fq(V)33 b Fm(\000)13
b Fq(\025)2677 1218 y Fp(c)2677 1271 y(I)5 b Fo(+1)2800
1248 y Fv(\).)36 b(Moreo)n(v)n(er,)23 b(from)321 1347
y(Theorem)k(10)g(w)n(e)g(ha)n(v)n(e)1343 1508 y Fq(W)1433
1473 y Fp(c)1421 1528 y(`)1490 1508 y Fm(!)c Fq(W)1686
1473 y Fp(c)1674 1528 y Fo(0)1743 1508 y Fv(=)1876 1404
y Fp(I)1830 1429 y Ft(M)1839 1605 y Fp(i)p Fo(=1)1969
1508 y Fm(M)2069 1473 y Fp(c)2069 1528 y(i)2121 1508
y Fm(\010)18 b Fk(C)23 b Fq( )2344 1473 y Fp(c)2401 1508
y Fq(:)321 1735 y Fs(Step)38 b(1)p Fv(.)54 b(Fix)47 b
Fq(c)32 b Fm(\025)g Fv(1)14 b(.)53 b(Since)33 b Fq(P)1423
1699 y Fn(\000)1411 1759 y Fp(\024)1450 1768 y Fe(`)1479
1759 y Fp(;W)1570 1739 y Fe(c)1561 1779 y(`)1606 1735
y Fq( )1663 1705 y Fp(c)1660 1759 y(`)1729 1735 y Fv(=)f(0,)i(w)n(e)f
(can)f(write,)j(b)n(y)e(a)f(classical)g(result)h(due)g(to)321
1845 y(Kato,)404 2066 y Fq(P)469 2031 y Fn(\000)457 2091
y Fp(\024)496 2100 y Fe(`)525 2091 y Fp(;A)p Fn(\017)p
Fp(W)700 2071 y Fe(c)691 2111 y(`)736 2066 y Fq( )793
2032 y Fp(c)790 2087 y(`)850 2066 y Fv(=)972 2010 y(1)p
947 2047 92 4 v 947 2123 a(2)p Fq(\031)1063 1953 y Ft(Z)1146
1974 y Fo(+)p Fn(1)1109 2142 y(\0001)1253 1974 y Ft(h)1293
2066 y Fv(\()p Fq(H)1401 2032 y Fp(c)1394 2087 y(\024)1433
2096 y Fe(`)1462 2087 y Fp(;W)1553 2067 y Fe(c)1544 2107
y(`)1606 2066 y Fm(\000)18 b Fq(i\021)s Fv(\))1794 2032
y Fn(\000)p Fo(1)1874 2066 y Fm(\000)5 b Fv(\()p Fq(H)2052
2032 y Fp(c)2045 2087 y(\024)2084 2096 y Fe(`)2113 2087
y Fp(;A)p Fn(\017)p Fp(W)2288 2067 y Fe(c)2279 2107 y(`)2342
2066 y Fm(\000)18 b Fq(i\021)s Fv(\))2530 2032 y Fn(\000)p
Fo(1)2619 1974 y Ft(i)2672 2066 y Fq( )2729 2032 y Fp(c)2726
2087 y(`)2777 2066 y Fq(d\021)437 b Fv(\(21\))627 2355
y(=)749 2299 y(1)p 724 2336 V 724 2412 a(2)p Fq(\031)826
2242 y Ft(Z)909 2262 y Fo(+)p Fn(1)872 2430 y(\0001)1017
2355 y Fv(\()p Fq(H)1125 2320 y Fp(c)1118 2375 y(\024)1157
2384 y Fe(`)1186 2375 y Fp(;W)1277 2355 y Fe(c)1268 2395
y(`)1330 2355 y Fm(\000)19 b Fq(i\021)s Fv(\))1519 2320
y Fn(\000)p Fo(1)1608 2355 y Fv(\()p Fq(H)1716 2320 y
Fp(c)1709 2375 y(\024)1748 2384 y Fe(`)1777 2375 y Fp(;A)p
Fn(\017)p Fp(W)1952 2355 y Fe(c)1943 2395 y(`)1992 2355
y Fm(\000)5 b Fq(H)2138 2320 y Fp(c)2131 2375 y(\024)2170
2384 y Fe(`)2198 2375 y Fp(;W)2289 2355 y Fe(c)2280 2395
y(`)2324 2355 y Fv(\)\()p Fq(H)2464 2320 y Fp(c)2457
2375 y(\024)2496 2384 y Fe(`)2526 2375 y Fp(;A)p Fn(\017)p
Fp(W)2701 2355 y Fe(c)2692 2395 y(`)2754 2355 y Fm(\000)18
b Fq(i\021)s Fv(\))2942 2320 y Fn(\000)p Fo(1)3032 2355
y Fq( )3089 2320 y Fp(c)3086 2375 y(`)3136 2355 y Fq(d\021)605
2615 y Fv(=)709 2559 y Fq(\024)757 2571 y Fp(`)p 703
2596 V 703 2672 a Fv(2)p Fq(\031)818 2502 y Ft(Z)901
2523 y Fo(+)p Fn(1)865 2691 y(\0001)1009 2615 y Fv(\()p
Fq(H)1117 2581 y Fp(c)1169 2615 y Fv(+)g Fq(V)38 b Fm(\000)18
b Fq(i\021)s Fv(\))1526 2581 y Fn(\000)p Fo(1)1615 2615
y Fv(\(\012)1707 2627 y Fp(A)p Fn(\017)p Fp(W)1862 2608
y Fe(c)1853 2646 y Ff(0)1916 2615 y Fm(\000)g Fv(\012)2059
2627 y Fp(W)2130 2608 y Fe(c)2121 2646 y Ff(0)2166 2615
y Fv(\)\()p Fq(H)2306 2581 y Fp(c)2358 2615 y Fv(+)g
Fq(V)38 b Fm(\000)18 b Fq(i\021)s Fv(\))2715 2581 y Fn(\000)p
Fo(1)2804 2615 y Fq( )2861 2581 y Fp(c)2918 2615 y Fq(d\021)k
Fv(+)c Fq(o)p Fv(\()p Fq(\024)3227 2627 y Fp(`)3259 2615
y Fv(\))c Fq(;)321 2805 y Fv(where)27 b(b)n(y)41 b(\012)750
2817 y Fp(W)867 2805 y Fv(w)n(e)28 b(denote)f(the)h(nonlinear)f(part)g
(of)41 b Fq(H)2129 2775 y Fp(c)2122 2828 y(\024;W)2256
2805 y Fv(:)1427 2974 y Fq(H)1503 2940 y Fp(c)1496 2995
y(\024;W)1653 2974 y Fv(=)23 b Fq(H)1817 2940 y Fp(c)1869
2974 y Fv(+)18 b Fq(V)37 b Fv(+)18 b Fq(\024)c Fv(\012)2242
2986 y Fp(W)2331 2974 y Fq(:)446 3165 y Fv(But)28 b(note)f(that)h
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3190 y(i)p Fo(=1)1775 3165 y Fm(M)1875 3135 y Fp(c)1875
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3333 y(\012)1377 3345 y Fp(A)p Fn(\017)p Fp(W)1532 3326
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3828 y Fv(\(\012)1609 3840 y Fp(A)p Fn(\017)p Fp( )1739
3824 y Fe(c)1793 3828 y Fm(\000)g Fv(\012)1936 3840 y
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Fq(i\021)s Fv(\))2567 3794 y Fn(\000)p Fo(1)2656 3828
y Fq( )2713 3794 y Fp(c)2770 3828 y Fq(d\021)26 b Fm(6)p
Fv(=)d(0)14 b Fq(:)251 b Fv(\(22\))321 4045 y(Since)930
4167 y(\()p Fq(H)1038 4132 y Fp(c)1090 4167 y Fv(+)18
b Fq(V)38 b Fm(\000)18 b Fq(i\021)s Fv(\))1447 4132 y
Fn(\000)p Fo(1)1536 4167 y Fq( )1593 4132 y Fp(c)1650
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4148 345 4 v 1748 4224 a Fq(\025)1796 4195 y Fp(c)1796
4248 y(I)5 b Fo(+1)1937 4224 y Fm(\000)18 b Fq(i\021)2185
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Fv(:=)1390 4583 y Ft(Z)1473 4604 y Fo(+)p Fn(1)1436 4772
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4662 y Fn(\000)p Fo(1)2339 4640 y Fq(d\021)p 2210 4677
V 2210 4753 a(\025)2258 4725 y Fp(c)2258 4778 y(I)5 b
Fo(+1)2399 4753 y Fm(\000)18 b Fq(i\021)2579 4696 y(:)321
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5129 y(1)p 717 5166 74 4 v 717 5242 a Fq(c)753 5218 y
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y(\0001)1032 5093 y Ft(\020)1107 5129 y Fv(1)p 1091 5166
V 1091 5242 a Fq(c)1127 5218 y Fo(2)1175 5186 y Fv(\()p
Fq(H)1283 5151 y Fp(c)1335 5186 y Fv(+)18 b Fq(V)h Fv(\))f
Fm(\000)g Fq(i)1672 5129 y(\021)p 1657 5166 V 1657 5242
a(c)1693 5218 y Fo(2)1740 5093 y Ft(\021)1790 5111 y
Fn(\000)p Fo(1)1943 5129 y Fq(d)p Fv(\()p Fq(\021)s(=c)2140
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1899 5250 144 4 v 1940 5298 a Fp(c)1970 5281 y Ff(2)2071
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y Fo(2)2492 5068 y Ft(\022)2553 5186 y Fq(L)2610 5198
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y Fo(2)2953 5093 y Ft(\021)3003 5068 y(\023)3092 5186
y Fq(;)183 b Fv(\(23\))321 5433 y(where)27 b Fq(L)618
5445 y Fp(c)652 5433 y Fv(,)g(in)h(the)g(F)-7 b(ourier)27
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(the)h(matrix)920 5634 y(^)911 5655 y Fq(L)968 5667 y
Fp(c)1001 5655 y Fv(\()p Fq(p)p Fv(\))23 b(=)1218 5542
y Ft(Z)1301 5562 y Fo(+)p Fn(1)1264 5730 y(\0001)1422
5655 y Fv(\()p Fm(\000)p Fq(iu)18 b Fv(+)g Fq(\014)23
b Fv(+)18 b(\()p Fj(\013)g Fm(\001)h Fq(p)p Fv(\))p Fq(=c)p
Fv(\))2189 5620 y Fn(\000)p Fo(1)2278 5655 y Fv(\()p
Fm(\000)p Fq(iu)e Fv(+)h(1\))2626 5620 y Fn(\000)p Fo(1)2729
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eop end
%%Page: 12 12
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V 2427 1034 a Fm(\000)p Fq(iu)e Fm(\000)h Fq(!)2724 1010
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1699 y(2)p 1194 1736 51 4 v 1194 1812 a Fq(\031)1264
1734 y Fv(^)1255 1755 y Fq(L)1312 1767 y Fp(c)1345 1755
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1699 y(\()p Fj(\013)h Fm(\001)f Fq(p)p Fv(\))p 1868 1736
230 4 v 1964 1812 a Fq(c)2126 1755 y Fv(+)g Fq(O)2274
1663 y Ft(\020)2348 1699 y Fm(j)p Fq(p)p Fm(j)2436 1669
y Fo(2)p 2348 1736 126 4 v 2374 1812 a Fq(c)2410 1788
y Fo(2)2497 1663 y Ft(\021)2560 1755 y Fq(:)321 2087
y Fs(Step)32 b(3)p Fv(.)37 b(It)28 b(is)f(w)n(ell)h(kno)n(wn)e(\(see)i
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y Fv(can)28 b(b)r(e)g(written)f(as)1358 2320 y Fq( )1415
2286 y Fp(c)1472 2320 y Fv(=)1560 2203 y Ft(\022)1748
2259 y Fq(\036)1631 2342 y Fn(\000)p Fp(i)p Fo(\()p Fp(\033)r
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a Fo(2)p Fp(c)1924 2203 y Ft(\023)2003 2320 y Fv(+)18
b Fq(O)2151 2228 y Ft(\020)2241 2264 y Fv(1)p 2226 2301
74 4 v 2226 2377 a Fq(c)2262 2353 y Fo(2)2322 2228 y
Ft(\021)2386 2320 y Fq(;)321 2574 y(\036)24 b Fm(2)f
Fq(L)529 2544 y Fo(2)566 2574 y Fv(\()p Fk(R)658 2544
y Fo(3)695 2574 y Fq(;)14 b Fk(C)792 2544 y Fo(2)830
2574 y Fv(\))23 b(b)r(eing)f(an)h(eigenstate)f(of)h(\()1729
2542 y Fn(\000)p Fo(\001)p 1729 2556 107 4 v 1775 2603
a(2)1854 2574 y Fv(+)9 b Fq(V)18 b Fv(\),)24 b(with)f(eigen)n(v)-5
b(alue)36 b Fq(\026)23 b Fv(=)g(lim)2939 2586 y Fp(c)p
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y Fp(c)3250 2597 y(I)5 b Fo(+1)3381 2574 y Fm(\000)321
2674 y Fq(c)357 2644 y Fo(2)394 2674 y Fv(\))p Fq(:)39
b Fv(Since)f(w)n(e)g(ha)n(v)n(e)f(assumed)h(that)g Fq(V)57
b Fv(is)39 b(smo)r(oth,)h(this)f(asymptotic)e(result)h(holds)g(for)g
(the)321 2774 y(top)r(ology)27 b(of)g(the)h(Sc)n(h)n(w)n(artz)e(space)h
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b(So,)1110 2953 y(2)p Fq(c)1188 2923 y Fo(2)p 1110 2990
115 4 v 1142 3066 a Fq(\031)1234 3009 y Fm(L)1291 2975
y Fp(c)1325 3009 y Fv(\(\012)1417 3021 y Fp(A)p Fn(\017)p
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3009 y Fv(\))23 b(=)1844 2953 y Fq(i)p 1840 2990 36 4
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3066 y Fq(f)9 b Fv(\()p Fq(A;)14 b(\036)p Fv(\))2210
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3237 y Fv(where)753 3420 y Fq(f)9 b Fv(\()p Fq(A;)14
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y Fm(j)p Fq(A)19 b Fm(\017)f Fq(\036)p Fm(j)1435 3386
y Fo(2)1491 3420 y Fm(\003)1561 3364 y Fq(x)h Fm(\001)f
Fq(\033)p 1561 3401 158 4 v 1575 3477 a Fm(j)p Fq(x)p
Fm(j)1668 3453 y Fo(3)1729 3328 y Ft(\021)1778 3420 y
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Fm(\001)h Fq(\033)p 2580 3401 V 2594 3477 a Fm(j)p Fq(x)p
Fm(j)2687 3453 y Fo(3)2747 3328 y Ft(\021)2797 3420 y
Fv(\()p Fq(A)g Fm(\017)e Fq(\036)p Fv(\))24 b Fq(:)201
b Fv(\(25\))321 3647 y(What)28 b(remains)f(to)g(pro)n(v)n(e)f(is)i(:)
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2618 3777 55 4 v 2629 3825 a(2)2703 3796 y Fv(+)20 b
Fq(V)f Fv(,)32 b(there)f(exists)f(an)321 3896 y Fq(A)24
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%%Page: 13 13
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(Mol.)698 2921 y(Ph)n(ys.)j Fs(20)59 b Fv(\(1987\),)27
b(p.)g(639-649.)565 3049 y([4])45 b(M.J.)23 b(Esteban,)g(E.)f(S)n(\023)
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b(o)r(c)n(k)22 b(equations)g(for)g(atoms)g(and)698 3148
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y Fv(\(2000\),)26 b(p.)i(1123-1157.)565 3503 y([6])45
b(M.H.)21 b(Mittleman.)g(Theory)f(of)h(relativistic)f(e\013ects)h(on)f
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b(Ph)n(ys.)g(Rev.)g(A)h Fs(24\(3\))59 b Fv(\(1981\),)27
b(p.)g(1167-1175.)565 3730 y([7])45 b(J.)c(Suc)n(her.)f(F)-7
b(oundations)41 b(of)g(the)g(relativistic)g(theory)f(of)h(man)n
(y-particle)f(atoms.)698 3830 y(Ph)n(ys.)26 b(Rev.)i(A)g
Fs(22)j(\(2\))60 b Fv(\(1980\),)26 b(p.)i(348-362.)565
3958 y([8])45 b(J.)20 b(Suc)n(her.)f(Relativistic)h(man)n(y-electron)e
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4057 y(p.)28 b(271-281.)565 4185 y([9])45 b(P)-7 b(.)29
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4384 y(cem)n(b)r(er)27 b(1989\).)523 4512 y([10])45 b(V.)32
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(Stabilit)n(y)f(of)h(matter)f(for)698 4611 y(the)j(Hartree-F)-7
b(o)r(c)n(k)33 b(functional)i(of)g(the)h(relativistic)e(electron-p)r
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Fv(\(1998\),)e(p.)i(353-364.)523 4839 y([11])45 b(V.)31
b(Bac)n(h,)g(J.M.)g(Barbaroux,)f(B.)h(Hel\013er,)h(H.)g(Sieden)n(top.)f
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b(electron-p)r(ositron)g(\014eld.)h(Comm.)g(Math.)h(Ph)n(ys.)e
Fs(201)p Fv(\(2\))h(\(1999\),)i(p.)698 5038 y(445-460.)523
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b(B.)h(Hel\013er,)g(H.)g(Sieden)n(top.)f(On)h(the)g(Hartree-F)-7
b(o)r(c)n(k)698 5265 y(equations)26 b(of)i(the)g(electron-p)r(ositron)e
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b(Chaix,)h(D.)g(Iracane,)e(P)-7 b(.L.)31 b(Lions.)e(F)-7
b(rom)31 b(quan)n(tum)f(electro)r(dynamics)f(to)i(mean-)698
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b(\(De-)698 5691 y(cem)n(b)r(er)27 b(1989\).)1838 5940
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