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%%Title: del.dvi
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%%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
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dup 12 /fi put
dup 45 /hyphen put
dup 46 /period put
dup 58 /colon 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
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dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
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dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
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readonly def
/FontBBox{-35 -250 1190 750}readonly def
/UniqueID 5000826 def
currentdict end
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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 19 /acute put
dup 39 /quoteright put
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 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 86 /V put
dup 87 /W put
dup 89 /Y put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e 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 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w 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: CMSL8
%!PS-AdobeFont-1.1: CMSL8 1.0
%%CreationDate: 1991 Aug 20 16:41:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights  
Reserved) readonly def
/FullName (CMSL8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -9.46 def
/isFixedPitch false def
end readonly def
/FontName /CMSL8 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 48 /zero put
dup 50 /two put
dup 54 /six put
dup 65 /A put
dup 77 /M 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 104 /h put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 121 /y put
readonly def
/FontBBox{-57 -250 1183 750}readonly def
/UniqueID 5000796 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: TeX-cmex7
%!PS-AdobeFont-1.0: TeX-cmex7 001.001
% Filtered by type1fix.pl 0.05
%%EndComments
13 dict dup begin
/FontInfo 16 dict dup begin
/Copyright (see\040copyright\040of\040original\040TeX\040font) def
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/Notice (converted\040after\040April\0402001) def
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/UnderlineThickness 50 def
/Weight (Regular) def
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/version (001.001) def
end readonly def
/FontName /TeX-cmex7 def
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/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 {-14 -2954 1627 771} readonly def
/UniqueID 4314415 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633
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cleartomark
%%EndFont
%%BeginFont: MSBM5
%!PS-AdobeFont-1.1: MSBM5 2.1
%%CreationDate: 1992 Oct 17 08:36:07
% 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 (MSBM5) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM5 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 82 /R put
readonly def
/FontBBox{0 -574 3027 1074}readonly def
/UniqueID 5032013 def
currentdict end
currentfile eexec
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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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3BE4C3792F3405763C4BD93EFC3B4FC34193439561841BA989DD8D9F9AEE7A7B
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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 5 /Pi put
dup 11 /alpha put
dup 14 /delta put
dup 18 /theta put
dup 22 /mu put
dup 25 /pi put
dup 30 /phi put
dup 31 /chi put
dup 32 /psi put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 66 /B put
dup 67 /C put
dup 76 /L put
dup 88 /X put
dup 89 /Y put
dup 90 /Z put
dup 96 /lscript put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 102 /f put
dup 103 /g put
dup 110 /n put
dup 121 /y put
readonly def
/FontBBox{37 -250 1349 750}readonly def
/UniqueID 5087380 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMSY5
%!PS-AdobeFont-1.1: CMSY5 1.0
%%CreationDate: 1991 Aug 15 07:21:16
% 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 (CMSY5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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dup 48 /prime put
dup 49 /infinity put
dup 50 /element put
dup 54 /negationslash put
dup 76 /L put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
dup 107 /bardbl put
dup 110 /backslash put
readonly def
/FontBBox{21 -944 1448 791}readonly def
/UniqueID 5000815 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 44 /comma put
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 70 /F put
dup 73 /I put
dup 76 /L put
dup 77 /M put
dup 80 /P put
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dup 84 /T put
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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 103 /g put
dup 104 /h put
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dup 115 /s put
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dup 119 /w put
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dup 121 /y put
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readonly def
/FontBBox{-163 -250 1146 969}readonly def
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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
dup 126 /planckover2pi1 put
readonly def
/FontBBox{0 -504 2615 1004}readonly def
/UniqueID 5032014 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 2 /multiply put
dup 3 /asteriskmath put
dup 6 /plusminus put
dup 13 /circlecopyrt put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 33 /arrowright put
dup 37 /arrownortheast put
dup 48 /prime put
dup 49 /infinity put
dup 50 /element put
dup 54 /negationslash put
dup 65 /A put
dup 72 /H put
dup 76 /L put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
dup 107 /bardbl put
dup 110 /backslash put
dup 112 /radical put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueID 5000817 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 97 /a put
dup 110 /n put
dup 112 /p put
dup 115 /s put
readonly def
/FontBBox{-341 -250 1304 965}readonly def
/UniqueID 5000788 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights  
Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 67 /C put
dup 73 /I put
dup 78 /N put
dup 82 /R put
dup 86 /V 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: 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 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 49 /one put
dup 58 /colon put
dup 66 /B put
dup 70 /F put
dup 76 /L put
dup 83 /S put
dup 84 /T put
dup 85 /U 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 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-62 -250 1123 750}readonly def
/UniqueID 5000798 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights  
Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /parenleftbig put
dup 1 /parenrightbig put
dup 2 /bracketleftbig put
dup 3 /bracketrightbig put
dup 8 /braceleftbig put
dup 9 /bracerightbig put
dup 12 /vextendsingle put
dup 13 /vextenddouble put
dup 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 34 /bracketleftBigg put
dup 35 /bracketrightBigg put
dup 48 /parenlefttp put
dup 49 /parenrighttp put
dup 50 /bracketlefttp put
dup 51 /bracketrighttp put
dup 52 /bracketleftbt put
dup 53 /bracketrightbt put
dup 56 /bracelefttp put
dup 58 /braceleftbt put
dup 60 /braceleftmid put
dup 64 /parenleftbt put
dup 65 /parenrightbt 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
dup 114 /radicalbigg put
dup 115 /radicalBigg put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueID 5000774 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights  
Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 1 /periodcentered put
dup 2 /multiply put
dup 6 /plusminus put
dup 8 /circleplus put
dup 17 /equivalence put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 24 /similar put
dup 25 /approxequal put
dup 26 /propersubset put
dup 33 /arrowright put
dup 38 /arrowsoutheast put
dup 40 /arrowdblleft put
dup 41 /arrowdblright put
dup 49 /infinity put
dup 50 /element put
dup 51 /owner put
dup 54 /negationslash put
dup 55 /mapsto put
dup 56 /universal put
dup 65 /A put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 72 /H put
dup 75 /K put
dup 91 /union put
dup 92 /intersection put
dup 94 /logicaland put
dup 102 /braceleft put
dup 103 /braceright put
dup 104 /angbracketleft put
dup 105 /angbracketright put
dup 106 /bar put
dup 107 /bardbl put
dup 112 /radical put
dup 114 /nabla put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 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 5 /Pi put
dup 14 /delta put
dup 18 /theta put
dup 21 /lambda put
dup 22 /mu put
dup 23 /nu put
dup 25 /pi 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 62 /greater put
dup 64 /partialdiff put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 76 /L put
dup 82 /R put
dup 83 /S put
dup 88 /X put
dup 89 /Y put
dup 96 /lscript put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 101 /e put
dup 102 /f put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 120 /x put
dup 121 /y 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 3 /Lambda put
dup 5 /Pi put
dup 9 /Psi put
dup 11 /alpha put
dup 12 /beta put
dup 14 /delta put
dup 18 /theta put
dup 20 /kappa put
dup 21 /lambda put
dup 22 /mu put
dup 23 /nu put
dup 24 /xi put
dup 25 /pi put
dup 27 /sigma put
dup 30 /phi put
dup 31 /chi put
dup 32 /psi put
dup 34 /epsilon put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 62 /greater put
dup 64 /partialdiff put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 69 /E put
dup 71 /G put
dup 72 /H put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 88 /X put
dup 89 /Y put
dup 90 /Z put
dup 96 /lscript put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k 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 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: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights  
Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /ff put
dup 12 /fi put
dup 13 /fl put
dup 14 /ffi put
dup 22 /macron put
dup 33 /exclam put
dup 34 /quotedblright put
dup 35 /numbersign 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 87 /W put
dup 91 /bracketleft put
dup 92 /quotedblleft put
dup 93 /bracketright put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 123 /endash put
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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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 12 /fi put
dup 19 /acute put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 47 /slash put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 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 78 /N 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 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 122 /z put
dup 123 /endash put
dup 127 /dieresis put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueID 5000791 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights  
Reserved) readonly def
/FullName (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 49 /one put
dup 50 /two put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueID 5000789 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171
9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F
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62D41C2D23843E0A2A73F72D4B5286E3CB7B8695B2434479776A59CB3BC6D352
677CCABF4133E20F12D91CFD8978C09FB81DF801707194AB8FABA9A6599C20FA
B36DB0A166490D20BB0BD53581A8715B099754E6FA0F7D2892F5573380DDFF1E
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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 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 44 /comma 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 59 /semicolon put
dup 61 /equal put
dup 65 /A put
dup 76 /L put
dup 78 /N put
dup 80 /P put
dup 81 /Q put
dup 86 /V put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 99 /c put
dup 108 /l put
dup 111 /o put
readonly def
/FontBBox{-27 -250 1122 750}readonly def
/UniqueID 5000790 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 44 /comma 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 71 /G put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 80 /P put
dup 82 /R 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 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 119 /w put
dup 120 /x 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: CMBX12
%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights  
Reserved) readonly def
/FullName (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 12 /fi put
dup 82 /R put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 118 /v put
dup 121 /y put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueID 5000769 def
currentdict end
currentfile eexec
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y Fr(x)f Fp(\024)846 2965 y Fo(Z)892 3154 y Fj(R)934
3137 y Fl(3)995 3022 y Fp(j)1029 3000 y Ft(~)1018 3022
y Fr(\036)1067 3034 y Fq(n)1112 3022 y Fp(j)1135 2992
y Fw(2)p 995 3059 V 1037 3135 a Fp(j)p Fr(x)p Fp(j)1197
3078 y Fr(d)1240 3044 y Fw(3)1277 3078 y Fr(x)19 b Ft(+)1447
3022 y(1)p 1436 3059 64 4 v 1436 3135 a Fr(R)1523 2965
y Fo(Z)1570 3154 y Fj(R)1612 3137 y Fl(3)1662 3078 y
Fp(j)p Fr(\036)1734 3090 y Fq(n)1780 3078 y Fp(j)1803
3044 y Fw(2)1868 2961 y Fo(\022)1929 3078 y Ft(1)f Fp(\000)g
Fr(\037)2124 3044 y Fw(2)2175 2961 y Fo(\022)2246 3022
y Fp(j)p Fr(x)p Fp(j)p 2246 3059 94 4 v 2261 3135 a Fr(R)2349
2961 y Fo(\023\023)2499 3078 y Fr(d)2542 3044 y Fw(3)2580
3078 y Fr(x)757 3324 y Fp(\024)846 3211 y Fo(Z)892 3399
y Fj(R)934 3383 y Fl(3)995 3268 y Fp(j)1029 3246 y Ft(~)1018
3268 y Fr(\036)1067 3280 y Fq(n)1112 3268 y Fp(j)1135
3237 y Fw(2)p 995 3305 179 4 v 1037 3381 a Fp(j)p Fr(x)p
Fp(j)1197 3324 y Fr(d)1240 3289 y Fw(3)1277 3324 y Fr(x)h
Ft(+)1447 3268 y(1)p 1436 3305 64 4 v 1436 3381 a Fr(R)0
3575 y Ft(with)195 3553 y(~)184 3575 y Fr(\036)233 3587
y Fq(n)302 3575 y Ft(=)j Fr(\037)p Ft(\()p Fp(j)p Fr(R)560
3545 y Fk(\000)p Fw(1)658 3575 y Fp(\001)8 b(j)p Ft(\))14
b Fr(\036)807 3587 y Fq(n)853 3575 y Ft(,)23 b(it)g(is)f(therefore)g
(su\016cien)n(t)h(to)f(pro)n(v)n(e)f(that)2233 3508 y
Fo(R)2272 3605 y Fj(R)2314 3588 y Fl(3)2365 3575 y Fp(j)p
Fr(x)p Fp(j)2458 3545 y Fk(\000)p Fw(1)2561 3575 y Fp(j)2595
3553 y Ft(~)2584 3575 y Fr(\036)2633 3587 y Fq(n)2679
3575 y Fp(j)2702 3545 y Fw(2)2753 3575 y Fr(d)2796 3545
y Fw(3)2833 3575 y Fr(x)0 3671 y Ft(is)27 b(uniformly)h(b)r(ounded,)g
(for)f(some)g Fr(R)d(>)e Ft(0,)28 b(ev)n(en)n(tually)e(small.)i(Using)f
(the)h(estimate)1071 3907 y Fr(a)1115 3873 y Fw(2)1175
3907 y Fp(\025)1272 3851 y Ft(\()p Fr(a)19 b Ft(+)f Fr(b)p
Ft(\))1518 3821 y Fw(2)p 1272 3888 283 4 v 1323 3964
a Ft(1)g(+)g Fr(")1584 3907 y Fp(\000)1677 3851 y Fr(b)1713
3821 y Fw(2)p 1677 3888 73 4 v 1694 3964 a Fr(")1787
3907 y(;)0 4125 y Ft(w)n(e)27 b(get)h(the)g(follo)n(wing)e(lo)n(w)n(er)
g(b)r(ound)239 4248 y Fo(Z)285 4437 y Fj(R)327 4420 y
Fl(3)438 4305 y Fp(j)p Fr(P)514 4317 y Fq(B)571 4305
y Fr(\036)620 4317 y Fq(n)666 4305 y Fp(j)689 4275 y
Fw(2)p 387 4342 390 4 v 387 4430 a Ft(1)18 b(+)534 4408
y(~)530 4430 y Fr(\025)h Ft(+)692 4396 y Fq(\027)725
4404 y Fg(n)p 690 4411 78 4 v 690 4458 a Fk(j)p Fq(x)p
Fk(j)801 4361 y Fr(d)844 4327 y Fw(3)881 4361 y Fr(x)26
b Fp(\025)1043 4248 y Fo(Z)1089 4437 y Fj(R)1131 4420
y Fl(3)1192 4305 y Fp(j)p Fr(P)1268 4317 y Fq(B)1325
4305 y Fr(\036)1374 4317 y Fq(n)1420 4305 y Fp(j)1443
4275 y Fw(2)1494 4305 y Fr(\037)1546 4275 y Fw(2)p 1192
4342 392 4 v 1193 4430 a Ft(1)17 b(+)1339 4408 y(~)1335
4430 y Fr(\025)i Ft(+)1497 4396 y Fq(\027)1530 4404 y
Fg(n)p 1495 4411 78 4 v 1495 4458 a Fk(j)p Fq(x)p Fk(j)1607
4361 y Fr(d)1650 4327 y Fw(3)1687 4361 y Fr(x)954 4639
y Fp(\025)1043 4526 y Fo(Z)1089 4715 y Fj(R)1131 4699
y Fl(3)1404 4583 y Fp(j)p Fr(P)1480 4595 y Fq(B)1548
4561 y Ft(~)1537 4583 y Fr(\036)1586 4595 y Fq(n)1632
4583 y Fp(j)1655 4553 y Fw(2)p 1192 4620 713 4 v 1192
4708 a Ft(\(1)f(+)g Fr(")p Ft(\))1438 4640 y Fo(\000)1476
4708 y Ft(1)g(+)1622 4686 y(~)1619 4708 y Fr(\025)h Ft(+)1780
4674 y Fq(\027)1813 4682 y Fg(n)p 1779 4689 78 4 v 1779
4736 a Fk(j)p Fq(x)p Fk(j)1866 4640 y Fo(\001)1928 4639
y Fr(d)1971 4605 y Fw(3)2008 4639 y Fr(x)g Fp(\000)2167
4583 y Fr(C)p 2167 4620 66 4 v 2180 4696 a(")2256 4639
y Fp(k)2309 4618 y Ft(~)2298 4639 y Fr(\036)2347 4651
y Fq(n)2392 4639 y Fp(k)2434 4605 y Fw(2)2434 4662 y
Fq(L)2480 4645 y Fl(2)2512 4662 y Fw(\()p Fj(R)2580 4645
y Fl(3)2612 4662 y Fw(\))0 4915 y Ft(where)29 b Fr(C)36
b Ft(is)29 b(a)g(constan)n(t)g(whic)n(h)g(dep)r(ends)h(on)f
Fp(k)p Fr(\037)1603 4885 y Fk(0)1626 4915 y Fp(k)1668
4885 y Fw(2)1668 4942 y Fq(L)1714 4926 y Fh(1)1773 4942
y Fw(\()p Fj(R)1841 4926 y Fl(+)1888 4942 y Fw(\))1918
4915 y Ft(,)h Fr(B)j Ft(and)d Fr(R)q Ft(.)f(Next,)h(with)g(the)0
5023 y(same)d(t)n(yp)r(e)h(of)f(argumen)n(ts,)g(w)n(e)g(can)g(write)229
5269 y Fp(j)p Fr(P)305 5281 y Fq(B)373 5247 y Ft(~)362
5269 y Fr(\036)411 5281 y Fq(n)457 5269 y Fp(j)480 5235
y Fw(2)540 5269 y Fp(\025)638 5213 y(j)p Fr(\033)22 b
Fp(\001)c(r)851 5191 y Ft(~)840 5213 y Fr(\036)889 5225
y Fq(n)935 5213 y Fp(j)958 5183 y Fw(2)p 638 5250 358
4 v 726 5326 a Ft(1)g(+)g Fr(")1024 5269 y Fp(\000)1117
5213 y Ft(1)p 1117 5250 42 4 v 1118 5326 a Fr(")1182
5173 y Fo(\014)1182 5223 y(\014)1182 5273 y(\014)1224
5269 y Fr(B)g Fp(j)p Fr(x)p Fp(j)1423 5247 y Ft(~)1412
5269 y Fr(\036)1461 5281 y Fq(n)1521 5173 y Fo(\014)1521
5223 y(\014)1521 5273 y(\014)1548 5194 y Fw(2)1608 5269
y Fp(\025)1706 5213 y(j)p Fr(\033)k Fp(\001)d(r)1920
5191 y Ft(~)1909 5213 y Fr(\036)1958 5225 y Fq(n)2003
5213 y Fp(j)2026 5183 y Fw(2)p 1706 5250 358 4 v 1794
5326 a Ft(1)f(+)g Fr(")2092 5269 y Fp(\000)2185 5213
y Fr(B)2252 5183 y Fw(2)2303 5213 y Fr(R)2367 5183 y
Fw(2)p 2185 5250 220 4 v 2275 5326 a Fr(")2428 5269 y
Fp(j)2462 5247 y Ft(~)2451 5269 y Fr(\036)2500 5281 y
Fq(n)2546 5269 y Fp(j)2569 5235 y Fw(2)2629 5269 y Fr(:)p
eop end
%%Page: 8 8
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(these)h(estimates,)f(this)h(giv)n(es)315 365 y Fo(Z)361
554 y Fj(R)403 537 y Fl(3)659 422 y Fp(j)p Fr(\033)22
b Fp(\001)d(r)873 400 y Ft(~)862 422 y Fr(\036)911 434
y Fq(n)956 422 y Fp(j)979 392 y Fw(2)p 463 459 750 4
v 463 546 a Ft(\(1)f(+)g Fr(")p Ft(\))709 522 y Fw(2)747
479 y Fo(\000)785 546 y Ft(1)g(+)931 525 y(~)928 546
y Fr(\025)h Ft(+)1089 512 y Fq(\027)1122 520 y Fg(n)p
1088 527 78 4 v 1088 575 a Fk(j)p Fq(x)p Fk(j)1175 479
y Fo(\001)1236 478 y Fr(d)1279 444 y Fw(3)1317 478 y
Fr(x)k Fp(\024)g Fr(C)6 b Ft(\()p Fr(";)14 b(R)q(;)g(\037)p
Ft(\))k(+)1934 365 y Fo(Z)1981 554 y Fj(R)2023 537 y
Fl(3)2087 422 y Fr(\027)2128 434 y Fq(n)p 2083 459 94
4 v 2083 535 a Fp(j)p Fr(x)p Fp(j)2200 478 y(j)2234 456
y Ft(~)2223 478 y Fr(\036)2272 490 y Fq(n)2318 478 y
Fp(j)2341 444 y Fw(2)2392 478 y Fr(d)2435 444 y Fw(3)2473
478 y Fr(x)23 b(:)0 747 y Ft(Because)339 725 y(~)328
747 y Fr(\036)377 759 y Fq(n)461 747 y Ft(has)37 b(a)g(compact)h(supp)r
(ort)g(in)g(the)g(ball)g(of)g(radius)f(2)14 b Fr(R)q
Ft(,)37 b(if)i Fr(\016)k Ft(=)d Fr(\016)s Ft(\()p Fr(R)q
Ft(\))g Fr(>)0 854 y Ft(\(1)18 b(+)179 832 y(~)175 854
y Fr(\025)q Ft(\))p Fr(R)q(=\027)403 866 y Fq(n)475 854
y Ft(at)28 b(least)f(for)g Fr(n)g Ft(large)f(enough,)h(then)893
1017 y(1)p 719 1054 390 4 v 719 1141 a(1)18 b(+)865 1119
y(~)862 1141 y Fr(\025)h Ft(+)1023 1107 y Fq(\027)1056
1115 y Fg(n)p 1022 1122 78 4 v 1022 1170 a Fk(j)p Fq(x)p
Fk(j)1142 1073 y Fp(\025)1366 1017 y(j)p Fr(x)p Fp(j)p
1239 1054 348 4 v 1239 1130 a Fr(\027)1280 1142 y Fq(n)1339
1130 y Ft(\(1)g(+)f Fr(\016)s Ft(\))1680 1073 y Fp(8)23
b Fr(x)h Fp(2)f Fr(B)t Ft(\(0)p Fr(;)14 b(R)q Ft(\))0
1322 y(so)27 b(that)439 1459 y(1)p 137 1496 645 4 v 137
1572 a(\(1)19 b(+)f Fr(")p Ft(\))384 1548 y Fw(2)435
1572 y Fr(\027)476 1584 y Fq(n)535 1572 y Ft(\(1)g(+)g
Fr(\016)s Ft(\))806 1402 y Fo(Z)852 1591 y Fj(R)894 1574
y Fl(3)945 1515 y Fp(j)p Fr(x)p Fp(j)c(j)p Fr(\033)22
b Fp(\001)d(r)1266 1494 y Ft(~)1255 1515 y Fr(\036)1304
1527 y Fq(n)1349 1515 y Fp(j)1372 1481 y Fw(2)1424 1515
y Fr(d)1467 1481 y Fw(3)1504 1515 y Fr(x)24 b Fp(\024)e
Fr(C)6 b Ft(\()p Fr(";)14 b(R)q(;)g(\037)p Ft(\))19 b(+)2122
1402 y Fo(Z)2168 1591 y Fj(R)2210 1574 y Fl(3)2274 1459
y Fr(\027)2315 1471 y Fq(n)p 2270 1496 94 4 v 2270 1572
a Fp(j)p Fr(x)p Fp(j)2387 1515 y(j)2421 1494 y Ft(~)2410
1515 y Fr(\036)2459 1527 y Fq(n)2505 1515 y Fp(j)2528
1481 y Fw(2)2579 1515 y Fr(d)2622 1481 y Fw(3)2660 1515
y Fr(x)k(:)0 1736 y Ft(On)k(the)h(other)f(hand,)h(according)e(to)h([5,)
8 b(4],)694 1837 y Fo(Z)740 2025 y Fj(R)782 2009 y Fl(3)833
1950 y Fp(j)p Fr(x)p Fp(j)14 b(j)p Fr(\033)22 b Fp(\001)d(r)1154
1928 y Ft(~)1143 1950 y Fr(\036)1192 1962 y Fq(n)1237
1950 y Fp(j)1260 1915 y Fw(2)1312 1950 y Fr(d)1355 1915
y Fw(3)1392 1950 y Fr(x)24 b Fp(\025)1550 1837 y Fo(Z)1596
2025 y Fj(R)1638 2009 y Fl(3)1725 1893 y Ft(1)p 1699
1931 V 1699 2007 a Fp(j)p Fr(x)p Fp(j)1816 1950 y(j)1850
1928 y Ft(~)1839 1950 y Fr(\036)1888 1962 y Fq(n)1934
1950 y Fp(j)1957 1915 y Fw(2)2008 1950 y Fr(d)2051 1915
y Fw(3)2088 1950 y Fr(x)k(:)0 2185 y Ft(This)c(pro)n(vides)e(a)h
(uniform)h(upp)r(er)g(b)r(ound)g(on)1480 2118 y Fo(R)1520
2214 y Fj(R)1562 2198 y Fl(3)1612 2185 y Fp(j)p Fr(x)p
Fp(j)1705 2155 y Fk(\000)p Fw(1)1808 2185 y Fp(j)1842
2163 y Ft(~)1831 2185 y Fr(\036)1880 2197 y Fq(n)1926
2185 y Fp(j)1949 2155 y Fw(2)2000 2185 y Fr(d)2043 2155
y Fw(3)2081 2185 y Fr(x)g Ft(if)g Fr(")g Ft(and)f Fr(\016)k
Ft(are)c(c)n(hosen)0 2280 y(small)k(enough)g(in)h(order)e(that)1321
2430 y(1)p 1070 2467 545 4 v 1070 2543 a(\(1)18 b(+)g
Fr(")p Ft(\))1316 2519 y Fw(2)1367 2543 y Ft(\(1)g(+)g
Fr(\016)s Ft(\))1647 2486 y Fr(>)23 b(\027)1781 2452
y Fw(2)1776 2507 y Fq(n)0 2707 y Ft(for)30 b Fr(n)g Ft(large.)g(This)g
(can)g(alw)n(a)n(ys)f(b)r(e)i(done)f(since)g Fr(\027)1620
2719 y Fq(n)1696 2707 y Ft(con)n(v)n(erges)e(to)j Fr(\027)i
Fp(2)28 b Ft(\(0)p Fr(;)14 b Ft(1\))30 b(and)h Fr(\016)s
Ft(\()p Fr(R)q Ft(\))0 2793 y(can)26 b(b)r(e)g(tak)n(en)g(as)f(small)h
(as)g(desired)f(for)h Fr(R)e(>)e Ft(0)k(su\016cien)n(tly)g(small.)g
(This)g(concludes)g(the)0 2880 y(pro)r(of)h(of)h(\(8\).)106
3017 y(Summarizing,)f(w)n(e)h(obtain)f(that)818 3234
y Fr(J)886 3117 y Fo(\024)930 3234 y Fr(\036)979 3246
y Fq(n)1025 3234 y Fr(;)1071 3178 y Ft(1)p 1071 3215
42 4 v 1071 3291 a(2)1123 3234 y(\()1158 3212 y(~)1155
3234 y Fr(\025)19 b Ft(+)f Fr(\025)1353 3246 y Fw(1)1391
3234 y Ft(\()p Fr(\027)q(;)c(B)t Ft(\)\))p Fr(;)g(\027)q(;)g(B)1816
3117 y Fo(\025)1883 3234 y Fp(\024)22 b Ft(0)28 b Fr(;)0
3470 y Ft(for)35 b Fr(n)g Ft(large)f(enough:)g(hence)i
Fr(\025)p Ft([)p Fr(\036)1106 3482 y Fq(n)1152 3470 y
Fr(;)14 b(\027)q(;)g(B)t Ft(])35 b Fp(\024)1504 3438
y Fw(1)p 1504 3452 34 4 v 1504 3499 a(2)1547 3470 y Ft(\()1582
3448 y(~)1579 3470 y Fr(\025)19 b Ft(+)f Fr(\025)1777
3482 y Fw(1)1815 3470 y Ft(\()p Fr(\027)q(;)c(B)t Ft(\)\))36
b Fr(<)f(\025)2241 3482 y Fw(1)2279 3470 y Ft(\()p Fr(\027)q(;)14
b(B)t Ft(\),)36 b(a)f(con)n(tra-)0 3565 y(diction.)2544
b Ff(\003)106 3702 y Ft(Consider)27 b(no)n(w)g(the)h(e\013ect)g(of)g(a)
f(scaling)f(on)i Fr(J)8 b Ft(.)0 3859 y Fx(Lemma)32 b(1.)24
b Fi(L)l(et)30 b Fr(B)g Fp(\025)25 b Ft(0)p Fi(,)31 b
Fr(\025)25 b Fp(\025)g(\000)p Ft(1)p Fi(,)31 b Fr(\022)c(>)e
Ft(0)31 b Fi(and)g Fr(\036)1629 3871 y Fq(\022)1667 3859
y Ft(\()p Fr(x)p Ft(\))26 b(:=)f Fr(\022)1958 3829 y
Fw(3)p Fq(=)p Fw(2)2063 3859 y Fr(\036)p Ft(\()p Fr(\022)17
b(x)p Ft(\))32 b Fi(for)f(any)h Fr(x)26 b Fp(2)f Fm(R)2818
3829 y Fw(3)2855 3859 y Fi(.)0 3946 y(Then)559 4033 y
Fp(r)628 4045 y Fq(A)678 4064 y Fg(\022)707 4052 y Fl(2)740
4064 y Fg(B)793 4033 y Fr(\036)842 4045 y Fq(\022)880
4033 y Ft(\()p Fr(x)p Ft(\))f(=)f Fr(\022)1144 3998 y
Fw(5)p Fq(=)p Fw(2)1263 4033 y Ft([)p Fp(r)p Fr(\036)p
Ft(\()p Fr(\022)17 b(x)p Ft(\))i Fp(\000)f Fr(i)c(A)1778
4045 y Fq(B)1835 4033 y Ft(\()p Fr(\022)i(x)p Ft(\))e
Fr(\036)p Ft(\()p Fr(\022)k(x)p Ft(\)])44 b Fr(;)0 4183
y Fi(and)30 b(for)h(any)f Fr(a)23 b Fp(2)g Fm(R)p Fi(,)30
b Fr(\027)f Fp(2)23 b Ft(\(0)p Fr(;)14 b Ft(1\))p Fi(,)62
4429 y Fr(J)8 b Ft([)p Fr(\036)188 4441 y Fq(\022)226
4429 y Fr(;)14 b(\025;)g(\022)389 4395 y Fq(a)430 4429
y Fr(\027)q(;)g(\022)550 4395 y Fw(2)587 4429 y Fr(B)t
Ft(])23 b(=)788 4316 y Fo(Z)834 4505 y Fj(R)876 4488
y Fl(3)926 4287 y Fo( )992 4429 y Fr(\022)1033 4395 y
Fw(2)1204 4373 y Fp(j)p Fr(P)1280 4385 y Fq(B)1338 4373
y Fr(\036)p Fp(j)1410 4343 y Fw(2)p 1081 4410 491 4 v
1081 4497 a Ft(1)18 b(+)g Fr(\025)g Ft(+)1383 4464 y
Fq(\022)1417 4447 y Fg(a)p Fl(+1)1524 4464 y Fq(\027)p
1383 4478 178 4 v 1434 4525 a Fk(j)p Fq(x)p Fk(j)1600
4429 y Ft(+)g(\(1)g Fp(\000)g Fr(\025)p Ft(\))p Fp(j)p
Fr(\036)p Fp(j)2033 4395 y Fw(2)2090 4429 y Fp(\000)2183
4373 y Fr(\022)2224 4343 y Fq(a)p Fw(+1)2348 4373 y Fr(\027)p
2183 4410 212 4 v 2242 4486 a Fp(j)p Fr(x)p Fp(j)2418
4429 y(j)p Fr(\036)p Fp(j)2513 4395 y Fw(2)2551 4287
y Fo(!)2645 4429 y Fr(d)2688 4395 y Fw(3)2725 4429 y
Fr(x)24 b(:)2775 4610 y Ft(\(9\))0 4760 y(Using)31 b(this)h(scaling,)f
(w)n(e)g(pro)n(v)n(e)f(some)g(prop)r(erties)h(enjo)n(y)n(ed)g(b)n(y)g
(the)h(function)g Fr(\025)2610 4772 y Fw(1)2648 4760
y Ft(\()p Fr(\027)q(;)14 b(B)t Ft(\).)0 4857 y(T)-7 b(ak)n(e)27
b Fr(\025)c Ft(=)g Fr(\025)p Ft([)p Fr(\036)480 4869
y Fq(\022)518 4857 y Fr(;)14 b(\027)q(;)g(\022)675 4827
y Fw(2)712 4857 y Fr(B)t Ft(])24 b Fr(>)e Fp(\000)p Ft(1,)27
b Fr(\022)e(>)e Ft(1)k(and)h Fr(a)23 b Ft(=)f(0)27 b(in)h(\(9\).)g
(With)h(1)17 b(+)h Fr(\025)24 b Ft(=)f Fr(\022)r Ft(\(1)18
b(+)g Fr(\026)p Ft(\),)57 5030 y(0)25 b(=)f Fr(J)8 b
Ft([)p Fr(\036)339 5042 y Fq(\022)377 5030 y Fr(;)14
b(\025;)g(\027)q(;)g(\022)619 4995 y Fw(2)657 5030 y
Fr(B)t Ft(])124 5229 y(=)24 b Fr(\022)268 5116 y Fo(Z)314
5305 y Fj(R)356 5288 y Fl(3)407 5087 y Fo( )557 5173
y Fp(j)p Fr(P)633 5185 y Fq(B)690 5173 y Fr(\036)p Fp(j)762
5143 y Fw(2)p 482 5210 392 4 v 482 5286 a Ft(1)18 b(+)g
Fr(\026)h Ft(+)807 5253 y Fq(\027)p 787 5267 78 4 v 787
5315 a Fk(j)p Fq(x)p Fk(j)902 5229 y Ft(+)f(\(1)h Fp(\000)f
Fr(\026)p Ft(\))p Fp(j)p Fr(\036)p Fp(j)1338 5195 y Fw(2)1394
5229 y Fp(\000)1511 5173 y Fr(\027)p 1487 5210 94 4 v
1487 5286 a Fp(j)p Fr(x)p Fp(j)1605 5229 y(j)p Fr(\036)p
Fp(j)1700 5195 y Fw(2)1737 5087 y Fo(!)1817 5229 y Fr(d)1860
5195 y Fw(3)1897 5229 y Fr(x)h Ft(+)f(2\(1)g Fp(\000)g
Fr(\022)r Ft(\))2350 5116 y Fo(Z)2397 5305 y Fj(R)2439
5288 y Fl(3)2489 5229 y Fp(j)p Fr(\036)p Fp(j)2584 5195
y Fw(2)2645 5229 y Fr(d)2688 5195 y Fw(3)2725 5229 y
Fr(x)29 b(:)p eop end
%%Page: 9 9
TeXDict begin 9 8 bop 0 17 a Fu(Relativistic)24 b(h)n(ydrogenic)h
(atoms)f(in)f(strong)h(magnetic)h(\014elds)1152 b(9)0
248 y Ft(Assuming)27 b(that)h Fp(k)p Fr(\036)p Fp(k)694
263 y Fq(L)740 247 y Fl(2)772 263 y Fw(\()p Fj(R)840
247 y Fl(3)872 263 y Fw(\))925 248 y Ft(=)23 b(1,)k(w)n(e)g(get)963
465 y Fr(J)8 b Ft([)p Fr(\036;)14 b(\026;)g(\027)q(;)g(B)t
Ft(])23 b(=)g(2)1558 409 y Fr(\022)e Fp(\000)d Ft(1)p
1558 446 185 4 v 1629 522 a Fr(\022)1775 465 y(>)23 b
Ft(0)0 654 y(and)k(th)n(us,)931 792 y Fr(\025)p Ft([)p
Fr(\036;)14 b(\027)q(;)g(B)t Ft(])24 b Fr(>)f(\026)g
Ft(=)1539 736 y Fr(\025)p 1539 773 49 4 v 1543 849 a(\022)1616
792 y Fp(\000)1709 736 y Fr(\022)e Fp(\000)d Ft(1)p 1709
773 185 4 v 1781 849 a Fr(\022)1926 792 y(:)0 972 y Ft(On)27
b(the)h(other)f(hand,)759 939 y Fq(@)p 739 953 80 4 v
739 1001 a(@)t(\026)828 972 y Fr(J)8 b Ft([)p Fr(\036;)14
b(\026;)g(\027)q(;)g(B)t Ft(])24 b Fp(\024)e(\000)p Ft(1,)27
b(so)g(that)671 1199 y Fr(\025)p Ft([)p Fr(\036;)14 b(\027)q(;)g(B)t
Ft(])24 b Fp(\024)e Fr(\026)d Ft(+)f Fr(J)8 b Ft([)p
Fr(\036;)14 b(\026;)g(\027)q(;)g(B)t Ft(])23 b(=)1800
1143 y Fr(\025)p 1800 1180 49 4 v 1804 1256 a(\022)1877
1199 y Ft(+)1969 1143 y Fr(\022)e Fp(\000)d Ft(1)p 1969
1180 185 4 v 2041 1256 a Fr(\022)2187 1199 y(:)0 1388
y Ft(Summarizing,)27 b(w)n(e)g(ha)n(v)n(e)g(the)h(estimate)212
1545 y Fr(\025)p Ft([)p Fr(\036)332 1557 y Fq(\022)371
1545 y Fr(;)14 b(\027)q(;)g(\022)528 1515 y Fw(2)565
1545 y Fr(B)t Ft(])p 212 1582 443 4 v 413 1658 a Fr(\022)683
1601 y Fp(\000)776 1545 y Fr(\022)21 b Fp(\000)d Ft(1)p
776 1582 185 4 v 848 1658 a Fr(\022)994 1601 y Fp(\024)k
Fr(\025)p Ft([)p Fr(\036;)14 b(\027)q(;)g(B)t Ft(])24
b Fp(\024)1529 1545 y Fr(\025)p Ft([)p Fr(\036)1649 1557
y Fq(\022)1687 1545 y Fr(;)14 b(\027)q(;)g(\022)1844
1515 y Fw(2)1881 1545 y Fr(B)t Ft(])p 1529 1582 443 4
v 1729 1658 a Fr(\022)2000 1601 y Ft(+)2093 1545 y Fr(\022)20
b Fp(\000)e Ft(1)p 2093 1582 185 4 v 2164 1658 a Fr(\022)2370
1601 y Fp(8)23 b Fr(\022)i(>)e Ft(1)g Fr(:)0 1799 y Ft(The)28
b(ab)r(o)n(v)n(e)e(estimate,)h(whic)n(h)h(holds)f(pro)n(vided)g
Fr(\025)p Ft([)p Fr(\036)1678 1811 y Fq(\022)1716 1799
y Fr(;)14 b(\027)q(;)g(\022)1873 1769 y Fw(2)1910 1799
y Fr(B)t Ft(])24 b Fr(>)e Fp(\000)p Ft(1)27 b(is)g(equiv)-5
b(alen)n(t)28 b(to)147 1961 y Fr(\025)p Ft([)p Fr(\036;)14
b(\027)q(;)g(\022)424 1931 y Fw(2)462 1961 y Fr(B)t Ft(])p
147 1998 406 4 v 329 2074 a Fr(\022)580 2017 y Fp(\000)673
1961 y Fr(\022)21 b Fp(\000)d Ft(1)p 673 1998 185 4 v
745 2074 a Fr(\022)890 2017 y Fp(\024)23 b Fr(\025)p
Ft([)p Fr(\036)1098 2032 y Fw(1)p Fq(=\022)1204 2017
y Fr(;)14 b(\027)q(;)g(B)t Ft(])23 b Fp(\024)1530 1961
y Fr(\025)p Ft([)p Fr(\036;)14 b(\027)q(;)g(\022)1807
1931 y Fw(2)1845 1961 y Fr(B)t Ft(])p 1530 1998 406 4
v 1712 2074 a Fr(\022)1964 2017 y Ft(+)2057 1961 y Fr(\022)20
b Fp(\000)e Ft(1)p 2057 1998 185 4 v 2128 2074 a Fr(\022)2334
2017 y Fp(8)23 b Fr(\022)i(>)e Ft(1)136 b(\(10\))0 2216
y(under)28 b(the)g(condition)f Fr(\025)p Ft([)p Fr(\036;)14
b(\027)q(;)g(\022)1020 2186 y Fw(2)1058 2216 y Fr(B)t
Ft(])24 b Fr(>)f Fp(\000)p Ft(1.)k(As)h(a)f(consequence,)g(w)n(e)g(ha)n
(v)n(e)g(the)h(follo)n(wing)0 2302 y(result.)0 2426 y
Fx(Prop)s(osition)i(3.)24 b Fi(F)-6 b(or)33 b(al)t(l)g
Fr(\027)f Fp(2)c Ft(\(0)p Fr(;)14 b Ft(1\))p Fi(,)31
b(the)i(function)f Fr(B)f Fp(7!)c Fr(\025)2013 2438 y
Fw(1)2051 2426 y Ft(\()p Fr(\027)q(;)14 b(B)t Ft(\))32
b Fi(is)g(c)l(ontinuous)f(as)0 2517 y(long)f(as)g(it)g(takes)g(its)g
(values)g(in)g Ft(\()p Fp(\000)p Ft(1)p Fr(;)14 b Ft(+)p
Fp(1)p Ft(\))p Fi(.)29 b(Mor)l(e)l(over)465 2679 y Fr(\025)513
2691 y Fw(1)551 2679 y Ft(\()p Fr(\027)q(;)14 b(\022)703
2649 y Fw(2)740 2679 y Fr(B)t Ft(\))p 465 2716 375 4
v 632 2792 a Fr(\022)868 2735 y Fp(\000)961 2679 y Fr(\022)21
b Fp(\000)d Ft(1)p 961 2716 185 4 v 1033 2792 a Fr(\022)1178
2735 y Fp(\024)23 b Fr(\025)1314 2747 y Fw(1)1352 2735
y Ft(\()p Fr(\027)q(;)14 b(B)t Ft(\))23 b Fp(\024)1683
2679 y Fr(\025)1731 2691 y Fw(1)1768 2679 y Ft(\()p Fr(\027)q(;)14
b(\022)1920 2649 y Fw(2)1958 2679 y Fr(B)t Ft(\))p 1683
2716 375 4 v 1849 2792 a Fr(\022)2086 2735 y Ft(+)2179
2679 y Fr(\022)20 b Fp(\000)e Ft(1)p 2179 2716 185 4
v 2250 2792 a Fr(\022)2403 2735 y(;)307 b Ft(\(11\))0
2961 y Fi(if)26 b Fr(\025)124 2973 y Fw(1)162 2961 y
Ft(\()p Fr(\027)q(;)14 b(B)t Ft(\))23 b Fp(2)h Ft(\()p
Fp(\000)p Ft(1)p Fr(;)14 b Ft(+)p Fp(1)p Ft(\))25 b Fi(and)h
Fr(\022)f Fp(2)1154 2869 y Fo(\020)1204 2961 y Ft(1)p
Fr(;)1433 2928 y Fw(2)p 1292 2942 315 4 v 1292 2990 a(1)p
Fk(\000)p Fq(\025)1416 2998 y Fl(1)1449 2990 y Fw(\()p
Fq(\027;B)s Fw(\))1617 2869 y Fo(\021)1667 2961 y Fi(.)h(As)f(a)h(c)l
(onse)l(quenc)l(e,)g Fr(B)h Fp(7!)c Fr(\025)2633 2973
y Fw(1)2671 2961 y Ft(\()p Fr(\027)q(;)14 b(B)t Ft(\))0
3086 y Fi(is)42 b(Lipschitz)h(c)l(ontinuous)e(for)h(any)g
Fr(\027)50 b Fp(2)45 b Ft(\(0)p Fr(;)14 b Ft(1\))41 b
Fi(and)h Fr(B)49 b(>)44 b Ft(0)d Fi(such)g(that)h Fr(\025)2533
3098 y Fw(1)2571 3086 y Ft(\()p Fr(\027)q(;)14 b(B)t
Ft(\))45 b Fp(2)0 3177 y Ft(\()p Fp(\000)p Ft(1)p Fr(;)14
b Ft(+)p Fp(1)p Ft(\))p Fi(:)991 3264 y Fr(\025)1039
3276 y Fw(1)1095 3264 y Fp(\000)k Ft(1)p 991 3301 229
4 v 1051 3377 a(2)p Fr(B)1253 3320 y Fp(\024)1350 3264
y Fr(@)5 b(\025)1447 3276 y Fw(1)p 1350 3301 135 4 v
1359 3377 a Fr(@)g(B)1518 3320 y Fp(\024)1615 3264 y
Fr(\025)1663 3276 y Fw(1)1719 3264 y Ft(+)18 b(1)p 1615
3301 229 4 v 1675 3377 a(2)p Fr(B)1877 3320 y(:)0 3521
y Fi(Pr)l(o)l(of.)43 b Ft(Cho)r(ose)27 b Fr(a)c Fp(2)g
Ft(\()p Fp(\000)p Ft(1)p Fr(;)14 b(\025)918 3533 y Fw(1)955
3521 y Ft(\()p Fr(\027)q(;)g(B)t Ft(\)\))28 b(and)g(tak)n(e)f(an)n(y)g
Fr(\036)c Fp(2)g Fr(C)1939 3490 y Fk(1)1933 3541 y Fw(0)2010
3521 y Ft(\()p Fm(R)2102 3490 y Fw(3)2140 3521 y Fr(;)14
b Fm(C)2237 3490 y Fw(2)2274 3521 y Ft(\).)28 b(Since)1074
3673 y Fr(@)p 1049 3710 98 4 v 1049 3786 a(@)5 b(\025)1156
3729 y(J)j Ft([)p Fr(\036;)14 b(\025;)g(\027)q(;)g(B)t
Ft(])24 b Fp(\024)f(\000)p Ft(1)j Fr(;)0 3922 y Ft(an)h(in)n(tegration)
g(on)g(the)h(in)n(terv)-5 b(al)27 b([)p Fr(a;)14 b(\025)p
Ft([)p Fr(\036;)g(\027)q(;)g(B)t Ft(]])28 b(sho)n(ws)f(that)406
4135 y Fp(\000)p Fr(J)8 b Ft([)p Fr(\036;)14 b(a;)g(\027)q(;)g(B)t
Ft(])23 b(=)994 4042 y Fo(h)1033 4135 y Fr(J)8 b Ft([)p
Fr(\036;)14 b(\025;)g(\027)q(;)g(B)t Ft(])1450 4042 y
Fo(i)1490 4060 y Fq(\025)p Fw(=)p Fq(\025)p Fw([)p Fq(\036;\027;B)s
Fw(])1490 4193 y Fq(\025)p Fw(=)p Fq(a)1851 4135 y Fp(\024)22
b(\000)p Fr(\025)p Ft([)p Fr(\036;)14 b(\027)q(;)g(B)t
Ft(])19 b(+)f Fr(a)0 4329 y Ft(where)h(the)h(\014rst)f(equalit)n(y)g
(holds)h(b)n(y)f(de\014nition)h(of)f Fr(\025)p Ft([)p
Fr(\036;)14 b(\027)q(;)g(B)t Ft(],)21 b Fn(i.e.)e Fr(J)8
b Ft([)p Fr(\036;)14 b(\025)p Ft([)p Fr(\036;)g(\027)q(;)g(B)t
Ft(])p Fr(;)g(\027)q(;)g(B)t Ft(])24 b(=)0 4414 y(0.)j(As)h(a)f
(consequence,)843 4567 y Fr(J)8 b Ft([)p Fr(\036;)14
b(a;)g(\027)q(;)g(B)t Ft(])23 b Fp(\025)g Fr(\025)p Ft([)p
Fr(\036;)14 b(\027)q(;)g(B)t Ft(])19 b Fp(\000)f Fr(a)23
b(>)g Ft(0)f Fr(:)0 4729 y Ft(The)28 b(function)g Fr(\022)d
Fp(7!)e Fr(J)8 b Ft([)p Fr(\036;)14 b(a;)g(\027)q(;)g(\022)1030
4698 y Fw(2)1068 4729 y Fr(B)t Ft(])27 b(is)h(con)n(tin)n(uous,)f(so)g
(only)g(t)n(w)n(o)g(cases)f(are)h(p)r(ossible:)0 4814
y Fn(First)g(case:)p 0 4827 375 4 v 923 4911 a Fr(J)8
b Ft([)p Fr(\036;)14 b(a;)g(\027)q(;)g(\022)1287 4877
y Fw(2)1324 4911 y Fr(B)t Ft(])23 b Fp(\025)g Ft(0)82
b Fp(8)24 b Fr(\022)h(>)e Ft(1)f Fr(;)0 5051 y Fn(Second)27
b(case:)p 0 5064 454 4 v 28 w Ft(there)h(exists)f(a)g(constan)n(t)1334
5029 y(\026)1327 5051 y Fr(\022)e Ft(=)1486 5029 y(\026)1479
5051 y Fr(\022)r Ft(\()p Fr(a;)14 b(\036)p Ft(\))24 b
Fr(>)f Ft(1)k(suc)n(h)g(that)42 5193 y(\(i\))h Fr(J)8
b Ft([)p Fr(\036;)14 b(a;)g(\027)q(;)g(\022)521 5163
y Fw(2)558 5193 y Fr(B)t Ft(])23 b Fr(>)g Ft(0)k(for)g(an)n(y)g
Fr(\022)e Fp(2)f Ft(\(1)p Fr(;)1372 5171 y Ft(\026)1366
5193 y Fr(\022)r Ft(\),)42 5291 y(\(ii\))k Fr(J)8 b Ft([)p
Fr(\036;)14 b(a;)g(\027)q(;)509 5269 y Ft(\026)503 5291
y Fr(\022)544 5261 y Fw(2)581 5291 y Fr(B)t Ft(])23 b(=)g(0)k(or,)g
(equiv)-5 b(alen)n(tly)e(,)27 b Fr(\025)p Ft([)p Fr(\036;)14
b(\027)q(;)1697 5269 y Ft(\026)1689 5291 y Fr(\022)1730
5261 y Fw(2)1768 5291 y Fr(B)t Ft(])24 b(=)e Fr(a)p Ft(.)p
eop end
%%Page: 10 10
TeXDict begin 10 9 bop 0 17 a Fu(10)1325 b(Jean)25 b(Dolb)r(eault,)f
(Maria)e(J.)i(Esteban,)g(Mic)n(hael)g(Loss)0 248 y Ft(In)i(the)g
(second)g(case,)f(b)n(y)g(\(i\),)i(w)n(e)f(kno)n(w)f(that)h
Fr(\025)p Ft([)p Fr(\036;)14 b(\027)q(;)g(\022)1748 218
y Fw(2)1786 248 y Fr(B)t Ft(])23 b Fr(>)g(a)g(>)f Fp(\000)p
Ft(1)j(for)h(an)n(y)f Fr(\022)g Fp(2)f Ft(\(1)p Fr(;)2814
226 y Ft(\026)2808 248 y Fr(\022)r Ft(\))0 340 y(and)j(so)g(\(10\))h
(applies:)542 515 y Fr(\022)17 b(\025)646 527 y Fw(1)683
515 y Ft(\()p Fr(\027)q(;)d(B)t Ft(\))24 b Fp(\024)e
Fr(\022)16 b(\025)p Ft([)p Fr(\036)1179 530 y Fw(1)p
Fq(=\022)1285 515 y Fr(;)e(\027)q(;)g(B)t Ft(])23 b Fp(\024)f
Fr(\025)p Ft([)p Fr(\036;)14 b(\027)q(;)g(\022)1878 481
y Fw(2)1916 515 y Fr(B)t Ft(])19 b(+)f Fr(\022)j Fp(\000)d
Ft(1)k Fr(:)28 697 y Ft(In)27 b(the)h(limit)h(case)d
Fr(\022)g Ft(=)807 675 y(\026)800 697 y Fr(\022)s Ft(,)h(w)n(e)h(get)
771 845 y(\026)764 867 y Fr(\022)16 b(\025)867 879 y
Fw(1)905 867 y Ft(\()p Fr(\027)q(;)e(B)t Ft(\))k(+)g(1)g
Fp(\000)1366 845 y Ft(\026)1359 867 y Fr(\022)26 b Fp(\024)c
Fr(\025)p Ft([)p Fr(\036;)14 b(\027)q(;)1754 845 y Ft(\026)1747
867 y Fr(\022)1788 833 y Fw(2)1826 867 y Fr(B)t Ft(])23
b(=)g Fr(a)g(;)0 1029 y Ft(whic)n(h)28 b(giv)n(es)e(the)i(estimate)958
1216 y(\026)951 1237 y Fr(\022)d Fp(\025)1239 1181 y
Ft(1)18 b Fp(\000)g Fr(a)p 1113 1218 439 4 v 1113 1294
a Ft(1)g Fp(\000)g Fr(\025)1304 1306 y Fw(1)1342 1294
y Ft(\()p Fr(\027)q(;)c(B)t Ft(\))1585 1237 y(=:)23 b
Fr(\022)1737 1203 y Fk(\003)1775 1237 y Ft(\()p Fr(a)p
Ft(\))h Fr(:)0 1456 y Ft(Th)n(us)j(the)h(inequalit)n(y)566
1627 y Fr(\022)16 b(\025)669 1639 y Fw(1)706 1627 y Ft(\()p
Fr(\027)q(;)e(B)t Ft(\))24 b Fp(\024)e Fr(\022)16 b(\025)p
Ft([)p Fr(\036)1202 1642 y Fw(1)p Fq(=\022)1308 1627
y Fr(;)e(\027)q(;)g(B)t Ft(])23 b Fp(\024)f Fr(\025)p
Ft([)p Fr(\036;)14 b(\027)q(;)g(\022)1901 1592 y Fw(2)1939
1627 y Fr(B)t Ft(])19 b(+)f Fr(\022)j Fp(\000)d Ft(1)0
1807 y(holds)27 b(for)g(an)n(y)g Fr(\022)f Fp(2)d Ft([0)p
Fr(;)14 b(\022)787 1776 y Fk(\003)825 1807 y Ft(\()p
Fr(a)p Ft(\)])28 b(and)g(for)f(an)n(y)g Fr(\036)c Fp(2)h
Fr(C)1646 1776 y Fk(1)1640 1827 y Fw(0)1716 1807 y Ft(\()p
Fm(R)1808 1776 y Fw(3)1846 1807 y Fr(;)14 b Fm(C)1943
1776 y Fw(2)1980 1807 y Ft(\),)28 b(whic)n(h)g(pro)n(v)n(es)d(the)j
(r.h.s.)0 1898 y(inequalit)n(y)f(in)h(\(11\))f(b)n(y)h(letting)f
Fr(a)c Fp(!)g(\000)p Ft(1:)974 2111 y(lim)938 2163 y
Fq(a)p Fk(!\000)p Fw(1)1139 2111 y Fr(\022)1180 2077
y Fk(\003)1219 2111 y Ft(\()p Fr(a)p Ft(\))g(=)1646 2055
y(2)p 1448 2092 V 1448 2168 a(1)18 b Fp(\000)g Fr(\025)1639
2180 y Fw(1)1676 2168 y Ft(\()p Fr(\027)q(;)c(B)t Ft(\))1920
2111 y Fr(:)0 2330 y Ft(The)28 b(l.h.s.)f(inequalit)n(y)h(is)f
(obtained)g(in)h(the)g(same)f(manner.)893 b Ff(\003)106
2488 y Ft(With)38 b(the)e(appropriate)f(test)h(functions)h(one)f(can)g
(pro)n(v)n(e)f(that)h Fr(\025)2266 2500 y Fw(1)2304 2488
y Ft(\()p Fr(\027)q(;)14 b(B)t Ft(\))37 b(is)f(alw)n(a)n(ys)0
2579 y(b)r(elo)n(w)19 b Fp(\000)p Ft(1)g(for)h Fr(B)j
Ft(large.)c(W)-7 b(e)20 b(recall)f(that)h Fr(\025)1347
2591 y Fw(1)1385 2579 y Ft(\()p Fr(\027)q(;)14 b(B)t
Ft(\))23 b(=)g Fp(\000)p Ft(1)18 b(means)i(that)g(if)g
Fr(J)8 b Ft([)p Fr(\036;)14 b Fp(\000)p Ft(1)p Fr(;)g(\027)q(;)g(B)t
Ft(])22 b Fp(\024)0 2670 y Ft(0)27 b(for)g(an)n(y)g Fr(\036)d
Fp(2)f(A)p Ft(\()p Fr(\027)q(;)14 b(B)t Ft(\).)28 b(Let)g(us)g(giv)n(e)
e(some)h(details.)0 2822 y Fx(Prop)s(osition)j(4.)24
b Fi(L)l(et)i Fr(\027)i Fp(2)23 b Ft(\(0)p Fr(;)14 b
Ft(1\))p Fi(.)26 b(Then)h(for)f Fr(B)k Fi(lar)l(ge)d(enough,)f
Fr(\025)2120 2834 y Fw(1)2158 2822 y Ft(\()p Fr(\027)q(;)14
b(B)t Ft(\))23 b Fp(\024)g Ft(0)i Fi(and)i(ther)l(e)0
2914 y(exists)i Fr(B)294 2884 y Fk(\003)356 2914 y Fr(>)22
b Ft(0)30 b Fi(such)f(that)h Fr(\025)921 2926 y Fw(1)959
2914 y Ft(\()p Fr(\027)q(;)14 b(B)t Ft(\))23 b(=)g Fp(\000)p
Ft(1)29 b Fi(for)h(any)g Fr(B)d Fp(\025)c Fr(B)1952 2884
y Fk(\003)1990 2914 y Fi(.)0 3078 y(Pr)l(o)l(of.)43 b
Ft(Let)28 b(us)f(consider)g Fr(B)g(>)c Ft(0)k(and)g(the)h(trial)g
(function)1245 3298 y Fr( )e Ft(=)1413 3181 y Fo(\022)1474
3242 y Fr(\036)1478 3355 y Ft(0)1524 3181 y Fo(\023)1612
3298 y Fr(;)0 3517 y Ft(where)795 3668 y Fr(\036)d Ft(=)955
3536 y Fo(r)p 1038 3536 126 4 v 1067 3612 a Fr(B)p 1048
3649 106 4 v 1048 3725 a Ft(2)14 b Fr(\031)1177 3668
y(e)1216 3634 y Fk(\000)1278 3611 y Fg(B)p 1278 3620
45 3 v 1286 3654 a Fl(4)1333 3634 y Fw(\()p Fk(j)p Fq(x)1417
3642 y Fl(1)1448 3634 y Fk(j)1468 3609 y Fl(2)1501 3634
y Fw(+)p Fk(j)p Fq(x)1610 3642 y Fl(2)1641 3634 y Fk(j)1661
3609 y Fl(2)1693 3634 y Fw(\))1737 3551 y Fo(\022)1799
3612 y Fr(f)9 b Ft(\()p Fr(x)1928 3624 y Fw(3)1965 3612
y Ft(\))1877 3725 y(0)1997 3551 y Fo(\023)0 3867 y Ft(and)40
b Fr(f)54 b Fp(2)45 b Fr(C)434 3837 y Fk(1)428 3888 y
Fw(0)505 3867 y Ft(\()p Fm(R)p Fr(;)14 b Fm(R)p Ft(\))41
b(is)f(suc)n(h)h(that)f Fr(f)54 b Fp(\021)44 b Ft(1)c(for)h
Fp(j)p Fr(x)p Fp(j)k(\024)f Fr(\016)s Ft(,)d Fr(\016)j
Ft(small)c(but)h(\014xed,)g(and)0 3958 y Fp(k)p Fr(f)9
b Fp(k)134 3990 y Fg(L)174 3978 y Fl(2)205 3990 y(\()p
Fe(R)p Fl(\))311 3958 y Ft(=)23 b(1.)k(Note)h(that)f
Fr(\036)d Fp(2)f Fr(K)6 b(er)r Ft(\()p Fr(P)1262 3970
y Fq(B)1338 3958 y Ft(+)18 b Fr(i\033)1497 3970 y Fw(3)1549
3958 y Fr(@)1593 3970 y Fq(x)1631 3978 y Fl(3)1667 3958
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4282 y Fw(2)1510 4312 y Ft(\()p Fm(R)1602 4282 y Fw(3)1640
4312 y Ft(\).)g(With)g Fr(r)e Ft(=)d Fp(j)p Fr(x)p Fp(j)p
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4473 y Fr(\027)p 1731 4510 V 1734 4586 a(r)1801 4529
y Fp(j)p Fr(\036)p Fp(j)1896 4495 y Fw(2)1934 4437 y
Fo(\021)1997 4529 y Fr(d)2040 4495 y Fw(3)2077 4529 y
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y Fw(2)1635 4989 y Fp(\000)1772 4933 y Fr(\027)h Fp(j)p
Fr(f)9 b Ft(\()p Fr(x)1984 4945 y Fw(3)2021 4933 y Ft(\))p
Fp(j)2076 4903 y Fw(2)p 1728 4970 V 1728 4987 a Fo(p)p
1811 4987 346 4 v 71 x Fr(s)1850 5034 y Fw(2)1906 5058
y Ft(+)18 b Fp(j)p Fr(x)2059 5070 y Fw(3)2097 5058 y
Fp(j)2120 5034 y Fw(2)2167 4847 y Fo(!)2247 4989 y Fr(s)c(e)2339
4955 y Fk(\000)p Fq(B)f(s)2485 4930 y Fl(2)2518 4955
y Fq(=)p Fw(2)2603 4989 y Fr(ds)h(dx)2789 5001 y Fw(3)297
5250 y Ft(=)386 5137 y Fo(Z)c(Z)488 5326 y Fj(R)p Fk(\002)p
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5133 387 4 v 829 5194 a Fq(B)882 5169 y Fh(\000)p Fl(1)959
5194 y Fq(s)990 5169 y Fl(2)1023 5194 y Fw(+)p Fk(j)p
Fq(x)1132 5202 y Fl(3)1164 5194 y Fk(j)1184 5169 y Fl(2)p
759 5231 457 4 v 964 5307 a Fr(\027)1239 5250 y Fp(j)p
Fr(f)1312 5216 y Fk(0)1335 5250 y Ft(\()p Fr(x)1414 5262
y Fw(3)1452 5250 y Ft(\))p Fp(j)1507 5216 y Fw(2)1563
5250 y Fp(\000)1713 5194 y Fr(\027)19 b Fp(j)p Fr(f)9
b Ft(\()p Fr(x)1925 5206 y Fw(3)1963 5194 y Ft(\))p Fp(j)2018
5164 y Fw(2)p 1656 5231 V 1656 5248 a Fp(p)p 1726 5248
387 4 v 1726 5309 a Fq(B)1779 5284 y Fh(\000)p Fl(1)1856
5309 y Fq(s)1887 5284 y Fl(2)1920 5309 y Fw(+)p Fk(j)p
Fq(x)2029 5317 y Fl(3)2060 5309 y Fk(j)2080 5284 y Fl(2)2122
5133 y Fo(\023)2197 5250 y Fr(s)14 b(e)2289 5216 y Fk(\000)p
Fq(s)2372 5191 y Fl(2)2404 5216 y Fq(=)p Fw(2)2490 5250
y Fr(ds)g(dx)2676 5262 y Fw(3)2741 5250 y Fr(:)p eop
end
%%Page: 11 11
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1196 a Fr(\027)1297 1139 y Ft(+)18 b Fr(C)1439 1151 y
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2366 y Fw(1)1271 2354 y Ft(\()p Fr(\027)q(;)14 b(B)t
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Fr(\027)q(;)14 b(\022)1738 2695 y Fw(2)1776 2726 y Fr(B)t
Ft(\))29 b Fp(\024)f Ft(0.)i(This)h(is)g(made)f(p)r(ossible)0
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%%Page: 12 12
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%%Page: 13 13
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%%Page: 14 14
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Fp(j)1154 677 y Fw(2)1210 711 y Ft(+)k Fp(j)p Fr(q)s
Fp(j)1379 677 y Fw(2)1430 711 y Fp(j)p Fr(\036)p Fp(j)1525
677 y Fw(2)1581 711 y Fp(\000)g Fr(@)1708 723 y Fq(r)1745
711 y Fp(h)p Fr(\036;)c(\033)23 b Fp(\001)18 b Fr(q)f(\036)p
Fp(i)i Ft(+)f Fp(h)p Fr(\036;)c(\033)23 b Fp(\001)18
b Ft(\()p Fr(@)2516 723 y Fq(r)2553 711 y Fr(q)s Ft(\))c
Fr(\036)p Fp(i)618 911 y Ft(+)715 855 y(1)p 715 892 42
4 v 716 968 a Fr(r)781 819 y Fo(h)838 911 y Fp(\000)k
Fr(@)965 923 y Fq(r)1002 911 y Fp(h)p Fr(\036;)c(\033)23
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819 y Fo(i)2744 911 y Fr(:)0 1108 y Ft(As)28 b(a)f(last)g(preliminary)g
(remark,)f(w)n(e)h(notice)h(that)f Fr(r)17 b(@)1746 1120
y Fq(r)1783 1108 y Fr(q)26 b Ft(=)d Fr(q)s Ft(.)106 1242
y(Since)36 b(the)g(v)n(ector)e(p)r(oten)n(tial)h(gro)n(ws)f(linearly)-7
b(,)34 b(w)n(e)i(lo)r(calize)e(the)i(problem)f(near)g(the)0
1324 y(origin.)27 b(T)-7 b(o)27 b(this)h(end)f(consider)g(the)h
(function)1010 1593 y Fr(t)p Ft(\()p Fr(r)r Ft(\))d(=)1255
1423 y Fo(8)1255 1498 y(<)1255 1647 y(:)1342 1498 y Ft(1)128
b(if)23 b Fr(r)j Fp(\024)d Fr(R)28 b(;)1342 1685 y(R)q(=r)f
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Fr(\027)i Fp(\024)22 b Ft(1)28 b(w)n(e)f(get)g(the)h(lo)n(w)n(er)e(b)r
(ound)18 2089 y Fr(G)83 2101 y Fw(1)120 2089 y Ft([)p
Fr(\036)p Ft(])g Fp(\025)330 1976 y Fo(Z)377 2165 y Fj(R)419
2148 y Fl(3)479 2033 y Fr(t)p Ft(\()p Fr(r)r Ft(\))14
b Fr(r)p 479 2070 188 4 v 550 2146 a(\027)691 2089 y
Fp(j)p Fr(P)767 2101 y Fw(1)804 2089 y Fr(\036)p Fp(j)876
2055 y Fw(2)928 2089 y Fr(d)971 2055 y Fw(3)1008 2089
y Fr(x)19 b Fp(\000)1157 1976 y Fo(Z)1203 2165 y Fj(R)1245
2148 y Fl(3)1306 2033 y Fr(\027)p 1306 2070 47 4 v 1309
2146 a(r)1376 2089 y Fp(j)p Fr(\036)p Fp(j)1471 2055
y Fw(2)1522 2089 y Fr(d)1565 2055 y Fw(3)1603 2089 y
Fr(x)241 2319 y Fp(\025)24 b Fr(\027)391 2202 y Fo(\022)452
2206 y(Z)498 2394 y Fj(R)540 2378 y Fl(3)590 2319 y Fr(t)p
Ft(\()p Fr(r)r Ft(\))14 b Fr(r)k Fp(j)p Fr(P)868 2331
y Fw(1)906 2319 y Fr(\036)p Fp(j)978 2284 y Fw(2)1029
2319 y Fr(d)1072 2284 y Fw(3)1110 2319 y Fr(x)h Fp(\000)1259
2206 y Fo(Z)1305 2394 y Fj(R)1347 2378 y Fl(3)1407 2262
y Ft(1)p 1407 2299 42 4 v 1408 2376 a Fr(r)1472 2319
y Fp(j)p Fr(\036)p Fp(j)1567 2284 y Fw(2)1619 2319 y
Fr(d)1662 2284 y Fw(3)1699 2319 y Fr(x)1746 2202 y Fo(\023)1831
2319 y Ft(=)k Fr(\027)1979 2202 y Fo(\022)2040 2319 y
Fr(K)6 b Ft([)p Fr(\036)p Ft(])18 b Fp(\000)2313 2206
y Fo(Z)2360 2394 y Fj(R)2402 2378 y Fl(3)2462 2262 y
Ft(1)p 2462 2299 V 2463 2376 a Fr(r)2527 2319 y Fp(j)p
Fr(\036)p Fp(j)2622 2284 y Fw(2)2674 2319 y Fr(d)2717
2284 y Fw(3)2754 2319 y Fr(x)2801 2202 y Fo(\023)0 2548
y Ft(where)27 b(the)h(kinetic)g(part)f(is)g(de\014ned)h(b)n(y)g
Fr(K)6 b Ft([)p Fr(\036)p Ft(])23 b(:=)1624 2481 y Fo(R)1663
2577 y Fj(R)1705 2561 y Fl(3)1755 2548 y Fr(t)p Ft(\()p
Fr(r)r Ft(\))14 b Fr(r)k Fp(j)p Fr(P)2033 2560 y Fw(1)2070
2548 y Fr(\036)p Fp(j)2142 2518 y Fw(2)2194 2548 y Fr(d)2237
2518 y Fw(3)2275 2548 y Fr(x)28 b Ft(and)f(satis\014es)224
2723 y Fr(K)6 b Ft([)p Fr(\036)p Ft(])224 2897 y(=)311
2784 y Fo(Z)357 2973 y Fj(R)399 2956 y Fl(3)450 2897
y Fr(t)p Ft(\()p Fr(r)r Ft(\))14 b Fr(r)651 2780 y Fo(\024)695
2897 y Fp(j)p Fr(@)762 2909 y Fq(r)799 2897 y Fr(\036)p
Fp(j)871 2863 y Fw(2)927 2897 y Ft(+)1038 2841 y(1)p
1020 2878 78 4 v 1020 2954 a Fr(r)1059 2930 y Fw(2)1121
2897 y Fp(j)p Fr(\033)8 b Fp(\001)d Fr(L)14 b(\036)p
Fp(j)1370 2863 y Fw(2)1426 2897 y Ft(+)k Fp(j)p Fr(q)s
Fp(j)1595 2863 y Fw(2)1646 2897 y Fp(j)p Fr(\036)p Fp(j)1741
2863 y Fw(2)1779 2780 y Fo(\025)1836 2897 y Fr(d)1879
2863 y Fw(3)1917 2897 y Fr(x)390 3122 y Ft(+)469 3009
y Fo(Z)514 3198 y Fj(R)556 3181 y Fl(3)607 3122 y Fr(t)p
Ft(\()p Fr(r)r Ft(\))740 3030 y Fo(h)798 3122 y Fp(\000)g
Fr(r)f(@)979 3134 y Fq(r)1016 3122 y Fp(h)p Fr(\036;)d(\033)8
b Fp(\001)d Fr(q)17 b(\036)p Fp(i)i Ft(+)f Fr(r)f Fp(h)p
Fr(\036;)d(\033)8 b Fp(\001)d Ft(\()p Fr(@)1785 3134
y Fq(r)1822 3122 y Fr(q)s Ft(\))14 b Fr(\036)p Fp(i)20
b(\000)e Fr(@)2136 3134 y Fq(r)2173 3122 y Fp(h)p Fr(\036;)c(\033)8
b Fp(\001)d Fr(L)14 b(\036)p Fp(i)2526 3030 y Fo(i)2579
3122 y Fr(d)2622 3088 y Fw(3)2660 3122 y Fr(x)390 3344
y Ft(+)469 3231 y Fo(Z)514 3420 y Fj(R)556 3403 y Fl(3)607
3344 y Fr(t)p Ft(\()p Fr(r)r Ft(\))740 3252 y Fo(h)780
3344 y Fp(h)p Fr(\033)8 b Fp(\001)d Fr(L)14 b(\036;)g(\033)8
b Fp(\001)d Fr(q)16 b(\036)p Fp(i)k Ft(+)e Fp(h)p Fr(\033)8
b Fp(\001)d Fr(q)16 b(\036;)e(\033)8 b Fp(\001)d Fr(L)14
b(\036)p Fp(i)1861 3252 y Fo(i)1915 3344 y Fr(d)1958
3310 y Fw(3)1995 3344 y Fr(x)28 b(:)0 3560 y Ft(An)g(in)n(tegration)e
(b)n(y)i(parts)e(in)i(the)g Fr(r)j Ft(v)-5 b(ariable)26
b(yields)104 3721 y Fr(K)6 b Ft([)p Fr(\036)p Ft(])104
3896 y(=)191 3783 y Fo(Z)238 3971 y Fj(R)280 3955 y Fl(3)330
3896 y Fr(t)p Ft(\()p Fr(r)r Ft(\))14 b Fr(r)546 3779
y Fo(\024)589 3896 y Fp(j)p Fr(@)656 3908 y Fq(r)693
3896 y Fr(\036)p Fp(j)765 3861 y Fw(2)821 3896 y Ft(+)932
3840 y(1)p 914 3877 V 914 3953 a Fr(r)953 3929 y Fw(2)1015
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Fp(j)1264 3861 y Fw(2)1320 3896 y Ft(+)k Fp(j)p Fr(q)s
Fp(j)1489 3861 y Fw(2)1540 3896 y Fp(j)p Fr(\036)p Fp(j)1635
3861 y Fw(2)1673 3779 y Fo(\025)1730 3896 y Fr(d)1773
3861 y Fw(3)1811 3896 y Fr(x)127 4128 y Ft(+)206 4015
y Fo(Z)251 4204 y Fj(R)293 4187 y Fl(3)344 4128 y Fr(t)p
Ft(\()p Fr(r)r Ft(\))491 4011 y Fo(\024)536 4128 y Ft(4)c
Fp(h)p Fr(\036;)g(\033)8 b Fp(\001)d Fr(q)16 b(\036)p
Fp(i)k Ft(+)1040 4072 y(2)p 1040 4109 42 4 v 1041 4185
a Fr(r)1105 4128 y Fp(h)p Fr(\036;)14 b(\033)8 b Fp(\001)d
Fr(L)14 b(\036)p Fp(i)19 b Ft(+)f Fp(h)p Fr(\033)8 b
Fp(\001)d Fr(L)14 b(\036;)g(\033)8 b Fp(\001)d Fr(q)16
b(\036)p Fp(i)j Ft(+)f Fp(h)p Fr(\033)8 b Fp(\001)d Fr(q)17
b(\036;)d(\033)8 b Fp(\001)d Fr(L)14 b(\036)p Fp(i)2641
4011 y Fo(\025)2699 4128 y Fr(d)2742 4094 y Fw(3)2779
4128 y Fr(x)127 4353 y Ft(+)206 4240 y Fo(Z)251 4429
y Fj(R)293 4412 y Fl(3)344 4353 y Fr(t)374 4319 y Fk(0)397
4353 y Ft(\()p Fr(r)r Ft(\))500 4261 y Fo(h)554 4353
y Fr(r)j Fp(h)p Fr(\036;)d(\033)8 b Fp(\001)d Fr(q)17
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Fr(L)14 b(\036)p Fp(i)1413 4261 y Fo(i)1467 4353 y Fr(d)1510
4319 y Fw(3)1547 4353 y Fr(x)28 b(;)0 4569 y Ft(where)g(w)n(e)g(ha)n(v)
n(e)f(also)h(used)g(that)h Fr(r)17 b(@)1193 4581 y Fq(r)1230
4569 y Fr(q)27 b Ft(=)e Fr(q)s Ft(.)j(Consider)g(no)n(w)g(the)h(region)
e(of)h(in)n(tegration)0 4656 y(where)40 b Fr(r)47 b Fp(\024)c
Fr(R)e Ft(and)f(denote)h(the)f(corresp)r(onding)e(expression)h(b)n(y)h
Fr(K)2317 4668 y Fw(1)2354 4656 y Ft([)p Fr(\036)p Ft(].)h(There)f(the)
0 4747 y(deriv)-5 b(ativ)n(e)27 b(of)g Fr(t)p Ft(\()p
Fr(r)r Ft(\))i(v)-5 b(anishes)27 b(and)h(hence)f(collecting)g(terms)h
(w)n(e)f(\014nd)188 4996 y Fr(K)259 5008 y Fw(1)296 4996
y Ft([)p Fr(\036)p Ft(])d(=)502 4883 y Fo(Z)549 5072
y Fq(r)r Fk(\024)p Fq(R)701 4996 y Fr(t)14 b(r)799 4854
y Fo(")847 4996 y Fp(j)p Fr(@)914 5008 y Fq(r)951 4996
y Fr(\036)p Fp(j)1023 4962 y Fw(2)1079 4996 y Ft(+)1162
4876 y Fo(\014)1162 4926 y(\014)1162 4976 y(\014)1162
5025 y(\014)1190 4879 y(\024)1244 4940 y Ft(1)p 1244
4977 V 1245 5053 a Fr(r)1309 4996 y Ft(\()p Fr(\033)22
b Fp(\001)d Fr(L)f Ft(+)g(1\))g(+)g Fr(\033)k Fp(\001)c
Fr(q)1935 4879 y Fo(\025)1993 4996 y Fr(\036)2042 4876
y Fo(\014)2042 4926 y(\014)2042 4976 y(\014)2042 5025
y(\014)2070 4897 y Fw(2)2126 4996 y Fp(\000)2237 4940
y Ft(1)p 2219 4977 78 4 v 2219 5053 a Fr(r)2258 5029
y Fw(2)2320 4996 y Fp(j)p Fr(\036)p Fp(j)2415 4962 y
Fw(2)2453 4854 y Fo(#)2515 4996 y Fr(d)2558 4962 y Fw(3)2595
4996 y Fr(x)1767 5246 y Ft(+)c(2)1902 5133 y Fo(Z)1947
5322 y Fq(r)r Fk(\024)p Fq(R)2086 5246 y Fp(h)p Fr(\036;)g(\033)22
b Fp(\001)d Fr(q)d(\036)p Fp(i)e Fr(d)2506 5212 y Fw(3)2545
5246 y Fr(x)28 b(:)p eop end
%%Page: 15 15
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(decoupled)i(the)g(deriv)-5 b(ativ)n(es)31 b(with)i(resp)r(ect)g(to)f
Fr(r)j Ft(from)e(the)0 330 y(magnetic)g(\014eld)h(or,)f(to)h(b)r(e)g
(precise,)f(from)g Fr(q)s Ft(.)h(The)g(problem)f(is)g(that)h(the)g
(angular)e(mo-)0 413 y(men)n(tum)c(is)g(still)g(coupled)f(to)g(the)h
(magnetic)g(\014eld.)g(Ob)n(viously)363 514 y Fo(\014)363
564 y(\014)363 614 y(\014)363 663 y(\014)404 542 y(h)453
578 y Ft(1)p 453 615 42 4 v 454 691 a Fr(r)519 634 y
Ft(\()p Fr(\033)22 b Fp(\001)c Fr(L)g Ft(+)g(1\))g(+)g
Fr(\033)k Fp(\001)d Fr(q)1145 542 y Fo(i)1198 634 y Fr(\036)1247
514 y Fo(\014)1247 564 y(\014)1247 614 y(\014)1247 663
y(\014)1312 634 y Fp(\025)1400 514 y Fo(\014)1400 564
y(\014)1400 614 y(\014)1400 663 y(\014)1441 539 y(\014)1441
589 y(\014)1441 639 y(\014)1479 578 y Ft(1)p 1479 615
V 1480 691 a Fr(r)1544 634 y Ft(\()p Fr(\033)j Fp(\001)d
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y(\014)1996 639 y(\014)2042 634 y Fp(\000)k(j)c Fr(\033)21
b Fp(\001)e Fr(q)e(\036)d Fp(j)2426 514 y Fo(\014)2426
564 y(\014)2426 614 y(\014)2426 663 y(\014)2495 634 y
Fr(:)0 856 y Ft(F)-7 b(urther,)27 b(since)h(the)g(eigen)n(v)-5
b(alues)26 b(of)i Fr(\033)22 b Fp(\001)c Fr(L)g Ft(+)g(1)27
b(are)g(giv)n(en)g(b)n(y)g Fp(\006)p Ft(1,)g Fp(\006)p
Ft(2)14 b Fr(:)g(:)g(:)e Ft(,)742 1020 y Fp(k)h Ft(\()p
Fr(\033)23 b Fp(\001)18 b Fr(L)g Ft(+)g(1\))c Fr(\036)g
Fp(k)1290 1045 y Fq(L)1336 1028 y Fl(2)1368 1045 y Fw(\()p
Fq(S)1438 1028 y Fl(2)1471 1045 y Fw(\))1547 1020 y Fp(\025)106
b(k)o Fr(\036)14 b Fp(k)1864 1045 y Fq(L)1910 1028 y
Fl(2)1942 1045 y Fw(\()p Fq(S)2012 1028 y Fl(2)2044 1045
y Fw(\))2116 1020 y Fr(;)0 1203 y Ft(and)27 b(w)n(e)h(ha)n(v)n(e)e
(that)0 1288 y Fo(\015)0 1338 y(\015)0 1388 y(\015)0
1437 y(\015)46 1316 y(h)95 1352 y Ft(1)p 95 1389 V 96
1465 a Fr(r)161 1408 y Ft(\()p Fr(\033)8 b Fp(\001)d
Fr(L)18 b Ft(+)g(1\))g(+)g Fr(\033)8 b Fp(\001)d Fr(q)745
1316 y Fo(i)798 1408 y Fr(\036)861 1288 y Fo(\015)861
1338 y(\015)861 1388 y(\015)861 1437 y(\015)908 1491
y Fq(L)954 1475 y Fl(2)986 1491 y Fw(\()p Fq(S)1056 1475
y Fl(2)1088 1491 y Fw(\))1067 1408 y Fp(\025)1164 1352
y Ft(1)p 1164 1389 V 1165 1465 a Fr(r)1229 1408 y Fp(k)p
Fr(\036)p Fp(k)1362 1423 y Fq(L)1408 1407 y Fl(2)1440
1423 y Fw(\()p Fq(S)1510 1407 y Fl(2)1542 1423 y Fw(\))1572
1408 y Fp(\000)1637 1313 y Fo(\015)1637 1363 y(\015)1637
1412 y(\015)1697 1408 y Fp(j)p Fr(q)s Fp(j)14 b Fr(\036)1860
1313 y Fo(\015)1860 1363 y(\015)1860 1412 y(\015)1907
1466 y Fq(L)1953 1450 y Fl(2)1984 1466 y Fw(\()p Fq(S)2054
1450 y Fl(2)2087 1466 y Fw(\))2115 1408 y Fp(\025)2203
1316 y Fo(h)2252 1352 y Ft(1)p 2252 1389 V 2253 1465
a Fr(r)2317 1408 y Fp(\000)2393 1352 y Fr(r)p 2392 1389
V 2392 1465 a Ft(2)2443 1316 y Fo(i)2496 1408 y Fp(k)p
Fr(\036)p Fp(k)2629 1423 y Fq(L)2675 1407 y Fl(2)2707
1423 y Fw(\()p Fq(S)2777 1407 y Fl(2)2809 1423 y Fw(\))2858
1408 y Fr(;)0 1667 y Ft(since)34 b Fp(j)p Fr(q)s Ft(\()p
Fr(r)r Ft(\))p Fp(j)i(\024)d Fr(r)r(=)p Ft(2.)h(F)-7
b(or)34 b Fr(r)j Fp(\024)d Fr(R)g Fp(\024)1238 1598 y(p)p
1307 1598 V 69 x Ft(2,)g(the)h(factor)e([)1833 1634 y
Fw(1)p 1833 1648 34 4 v 1833 1695 a Fq(r)1913 1667 y
Fp(\000)2011 1634 y Fq(r)p 2010 1648 V 2010 1695 a Fw(2)2053
1667 y Ft(])i(is)f(nonnegativ)n(e.)f(Since)0 1788 y([)33
1755 y Fw(1)p 33 1769 V 33 1817 a Fq(r)108 1788 y Fp(\000)202
1755 y Fq(r)p 201 1769 V 201 1817 a Fw(2)244 1788 y Ft(])267
1758 y Fw(2)323 1788 y Fp(\000)432 1755 y Fw(1)p 416
1769 66 4 v 416 1817 a Fq(r)449 1800 y Fl(2)514 1788
y Ft(=)612 1755 y Fq(r)645 1730 y Fl(2)p 612 1769 V 628
1817 a Fw(4)705 1788 y Fp(\000)18 b Ft(1)27 b(and)h(since)f(2)14
b Fp(j)p Fr(q)s Ft(\()p Fr(r)r Ft(\))p Fp(j)24 b(\024)f
Fr(r)r Ft(,)28 b(w)n(e)g(obtain)f(the)h(lo)n(w)n(er)e(b)r(ound)620
2026 y Fr(K)691 2038 y Fw(1)728 2026 y Ft([)p Fr(\036)p
Ft(])e Fp(\025)934 1913 y Fo(Z)980 2101 y Fq(r)r Fk(\024)p
Fq(R)1133 2026 y Fr(r)1187 1909 y Fo(\024)1244 2026 y
Fp(j)p Fr(@)1311 2038 y Fq(r)1348 2026 y Fr(\036)p Fp(j)1420
1991 y Fw(2)1476 2026 y Ft(+)1559 1934 y Fo(h)1609 1970
y Fr(r)1648 1939 y Fw(2)p 1609 2007 78 4 v 1626 2083
a Ft(4)1714 2026 y Fp(\000)18 b Ft(2)1839 1934 y Fo(i)1878
2026 y Fp(j)p Fr(\036)p Fp(j)1973 1991 y Fw(2)2024 1909
y Fo(\025)2082 2026 y Fr(d)2125 1991 y Fw(3)2163 2026
y Fr(x)28 b(:)0 2282 y Ft(The)g(function)506 2249 y Fq(r)539
2224 y Fl(3)p 506 2263 66 4 v 522 2311 a Fw(4)599 2282
y Fp(\000)18 b Ft(2)p Fr(r)30 b Ft(is)e(a)f(decreasing)f(function)i(on)
f(the)h(in)n(terv)-5 b(al)2229 2215 y Fo(\002)2278 2282
y Ft(0)p Fr(;)2357 2213 y Fp(p)p 2425 2213 42 4 v 2425
2282 a Ft(2)2481 2215 y Fo(\003)2515 2282 y Ft(.)28 b(Hence)188
2520 y Fr(K)259 2532 y Fw(1)296 2520 y Ft([)p Fr(\036)p
Ft(])c Fp(\025)503 2407 y Fo(Z)549 2596 y Fq(r)r Fk(\024)p
Fq(R)701 2520 y Fr(r)17 b Fp(j)p Fr(@)822 2532 y Fq(r)859
2520 y Fr(\036)p Fp(j)931 2486 y Fw(2)983 2520 y Fr(d)1026
2486 y Fw(3)1063 2520 y Fr(x)i Ft(+)1212 2428 y Fo(h)1275
2464 y Fr(R)1339 2434 y Fw(3)p 1275 2501 101 4 v 1305
2577 a Ft(4)1404 2520 y Fp(\000)f Ft(2)p Fr(R)1606 2428
y Fo(i)1659 2407 y(Z)1705 2596 y Fq(r)r Fk(\024)p Fq(R)1858
2520 y Fp(j)p Fr(\036)p Fp(j)1953 2486 y Fw(2)2005 2520
y Fr(d)2048 2486 y Fw(3)2085 2520 y Fr(x)29 b(;)96 b
Ft(if)29 b Fr(R)23 b(<)2531 2447 y Fp(p)p 2600 2447 42
4 v 73 x Ft(2)k Fr(:)106 2807 y Ft(Next,)j(w)n(e)f(lo)r(ok)f(at)h(the)h
(con)n(tribution)e(to)i Fr(K)6 b Ft([)p Fr(\036)p Ft(])29
b(of)g(the)h(region)e(where)g Fr(r)h Fp(\025)c Fr(R)q
Ft(,)k(whic)n(h)0 2899 y(w)n(e)e(denote)h(b)n(y)f Fr(K)576
2911 y Fw(2)613 2899 y Ft([)p Fr(\036)p Ft(],)99 3067
y Fr(K)170 3079 y Fw(2)206 3067 y Ft([)p Fr(\036)p Ft(])99
3241 y(=)22 b Fr(t)230 3128 y Fo(Z)276 3317 y Fq(r)r
Fk(\025)p Fq(R)379 3241 y Fr(t)p Ft(\()p Fr(r)r Ft(\))14
b Fr(r)595 3124 y Fo(\024)638 3241 y Fp(j)p Fr(@)705
3253 y Fq(r)742 3241 y Fr(\036)p Fp(j)814 3207 y Fw(2)871
3241 y Ft(+)981 3185 y(1)p 964 3222 78 4 v 964 3298 a
Fr(r)1003 3274 y Fw(2)1064 3241 y Fp(j)p Fr(\033)8 b
Fp(\001)d Fr(L)14 b(\036)p Fp(j)1313 3207 y Fw(2)1369
3241 y Ft(+)k Fp(j)p Fr(q)s Fp(j)1538 3207 y Fw(2)1589
3241 y Fp(j)p Fr(\036)p Fp(j)1684 3207 y Fw(2)1722 3124
y Fo(\025)1779 3241 y Fr(d)1822 3207 y Fw(3)1860 3241
y Fr(x)122 3481 y Ft(+)201 3368 y Fo(Z)246 3557 y Fq(r)r
Fk(\025)p Fq(R)349 3481 y Fr(t)p Ft(\()p Fr(r)r Ft(\))496
3364 y Fo(\024)541 3481 y Ft(4)c Fp(h)p Fr(\036;)g(\033)8
b Fp(\001)d Fr(q)17 b(\036)p Fp(i)i Ft(+)1045 3425 y(2)p
1045 3462 42 4 v 1046 3538 a Fr(r)1110 3481 y Fp(h)p
Fr(\036;)14 b(\033)8 b Fp(\001)d Fr(L)14 b(\036)p Fp(i)19
b Ft(+)f Fp(h)p Fr(\033)8 b Fp(\001)d Fr(L)14 b(\036;)g(\033)8
b Fp(\001)d Fr(q)16 b(\036)p Fp(i)k Ft(+)e Fp(h)p Fr(\033)8
b Fp(\001)d Fr(q)16 b(\036;)e(\033)8 b Fp(\001)d Fr(L)14
b(\036)p Fp(i)2646 3364 y Fo(\025)2704 3481 y Fr(d)2747
3447 y Fw(3)2785 3481 y Fr(x)122 3719 y Fp(\000)201 3606
y Fo(Z)246 3795 y Fq(r)r Fk(\025)p Fq(R)409 3663 y Fr(t)p
Ft(\()p Fr(r)r Ft(\))p 409 3700 135 4 v 456 3776 a Fr(r)567
3627 y Fo(h)620 3719 y Fr(r)j Fp(h)p Fr(\036;)d(\033)8
b Fp(\001)d Fr(q)17 b(\036)p Fp(i)i Ft(+)f Fp(h)p Fr(\036;)c(\033)8
b Fp(\001)d Fr(L)14 b(\036)p Fp(i)1479 3627 y Fo(i)1546
3719 y Fr(;)0 3954 y Ft(using)27 b(the)h(fact)g(that)g
Fr(t)734 3923 y Fk(0)780 3954 y Ft(=)23 b Fp(\000)p Fr(t=r)r
Ft(.)k(Collecting)g(the)h(terms,)g(w)n(e)f(get)110 4122
y Fr(K)181 4134 y Fw(2)217 4122 y Ft([)p Fr(\036)p Ft(])110
4296 y(=)197 4183 y Fo(Z)243 4372 y Fq(r)r Fk(\025)p
Fq(R)297 4296 y Fr(t)p Ft(\()p Fr(r)r Ft(\))14 b Fr(r)498
4179 y Fo(\024)542 4296 y Fp(j)p Fr(@)609 4308 y Fq(r)646
4296 y Fr(\036)p Fp(j)718 4262 y Fw(2)774 4296 y Ft(+)885
4240 y(1)p 867 4277 78 4 v 867 4353 a Fr(r)906 4329 y
Fw(2)968 4296 y Fp(j)p Fr(\033)8 b Fp(\001)d Fr(L)14
b(\036)p Fp(j)1217 4262 y Fw(2)1272 4296 y Ft(+)k Fp(j)p
Fr(q)s Fp(j)1441 4262 y Fw(2)1493 4296 y Fp(j)p Fr(\036)p
Fp(j)1588 4262 y Fw(2)1625 4179 y Fo(\025)1683 4296 y
Fr(d)1726 4262 y Fw(3)1764 4296 y Fr(x)110 4536 y Ft(+)189
4423 y Fo(Z)234 4612 y Fq(r)r Fk(\025)p Fq(R)287 4536
y Fr(t)p Ft(\()p Fr(r)r Ft(\))434 4419 y Fo(\024)479
4536 y Ft(3)c Fp(h)p Fr(\036;)g(\033)8 b Fp(\001)d Fr(q)17
b(\036)p Fp(i)i Ft(+)983 4480 y(1)p 983 4517 42 4 v 984
4593 a Fr(r)1048 4536 y Fp(h)p Fr(\036;)14 b(\033)8 b
Fp(\001)d Fr(L)14 b(\036)p Fp(i)19 b Ft(+)f Fp(h)p Fr(\033)8
b Fp(\001)d Fr(L)14 b(\036;)g(\033)8 b Fp(\001)d Fr(q)17
b(\036)p Fp(i)i Ft(+)f Fp(h)p Fr(\033)8 b Fp(\001)d Fr(q)17
b(\036;)d(\033)8 b Fp(\001)d Fr(L)14 b(\036)p Fp(i)2585
4419 y Fo(\025)2642 4536 y Fr(d)2685 4502 y Fw(3)2723
4536 y Fr(x)28 b(:)0 4766 y Ft(This)g(can)f(b)r(e)h(rewritten)f(as)87
4996 y Fr(K)158 5008 y Fw(2)195 4996 y Ft([)p Fr(\036)p
Ft(])50 b Fp(\025)428 4883 y Fo(Z)474 5072 y Fq(r)r Fk(\025)p
Fq(R)627 4996 y Fr(t)p Ft(\()p Fr(r)r Ft(\))14 b Fr(r)828
4854 y Fo(")877 4996 y Fp(j)p Fr(@)944 5008 y Fq(r)981
4996 y Fr(\036)p Fp(j)1053 4962 y Fw(2)1109 4996 y Ft(+)1192
4876 y Fo(\014)1192 4926 y(\014)1192 4976 y(\014)1192
5025 y(\014)1234 4904 y(h)1297 4940 y Ft(1)p 1297 4977
V 1298 5053 a Fr(r)1362 4904 y Fo(h)1401 4996 y Fr(\033)22
b Fp(\001)c Fr(L)g Ft(+)1679 4940 y(1)p 1679 4977 V 1679
5053 a(2)1731 4904 y Fo(i)1789 4996 y Ft(+)g Fr(\033)j
Fp(\001)e Fr(q)2036 4904 y Fo(i)2075 4996 y Fr(\036)2138
4876 y Fo(\014)2138 4926 y(\014)2138 4976 y(\014)2138
5025 y(\014)2166 4897 y Fw(2)2222 4996 y Fp(\000)2360
4940 y Ft(1)p 2315 4977 133 4 v 2315 5053 a(4)14 b Fr(r)2410
5029 y Fw(2)2471 4996 y Fp(j)p Fr(\036)p Fp(j)2566 4962
y Fw(2)2604 4854 y Fo(#)2666 4996 y Fr(d)2709 4962 y
Fw(3)2746 4996 y Fr(x)506 5246 y Ft(+)23 b(2)650 5133
y Fo(Z)695 5322 y Fq(r)r Fk(\025)p Fq(R)848 5246 y Fr(t)p
Ft(\()p Fr(r)r Ft(\))14 b Fp(h)p Fr(\036;)g(\033)24 b
Fp(\001)18 b Fr(q)f(\036)p Fp(i)d Fr(d)1417 5212 y Fw(3)1455
5246 y Fr(x)28 b(:)p eop end
%%Page: 16 16
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(get)88 490 y Fr(K)159 502 y Fw(2)196 490 y Ft([)p Fr(\036)p
Ft(])d Fp(\025)402 377 y Fo(Z)449 565 y Fq(r)r Fk(\025)p
Fq(R)601 490 y Fr(t)p Ft(\()p Fr(r)r Ft(\))14 b Fr(r)k
Fp(j)p Fr(@)870 502 y Fq(r)907 490 y Fr(\036)p Fp(j)979
456 y Fw(2)1031 490 y Fr(d)1074 456 y Fw(3)1111 490 y
Fr(x)h Fp(\000)1302 434 y Ft(1)p 1270 471 106 4 v 1270
547 a(4)p Fr(R)1399 377 y Fo(Z)1445 565 y Fq(r)r Fk(\025)p
Fq(R)1598 490 y Fp(j)p Fr(\036)p Fp(j)1693 456 y Fw(2)1745
490 y Fr(d)1788 456 y Fw(3)1825 490 y Fr(x)g Fp(\000)f
Ft(2)p Fr(R)2093 377 y Fo(Z)2139 565 y Fq(r)r Fk(\025)p
Fq(R)2302 434 y Ft(1)p 2302 471 42 4 v 2303 547 a Fr(r)2367
490 y Fp(j)p Fr(q)s Fp(j)c(j)p Fr(\036)p Fp(j)2562 456
y Fw(2)2614 490 y Fr(d)2657 456 y Fw(3)2694 490 y Fr(x)28
b(;)0 751 y Ft(and,)f(using)h Fp(j)p Fr(q)s Fp(j)p Fr(=r)d
Fp(\024)e Ft(1)p Fr(=)p Ft(2,)401 997 y Fr(K)472 1009
y Fw(2)509 997 y Ft([)p Fr(\036)p Ft(])h Fp(\025)715
884 y Fo(Z)761 1073 y Fq(r)r Fk(\025)p Fq(R)914 997 y
Fr(t)p Ft(\()p Fr(r)r Ft(\))14 b Fr(r)k Fp(j)p Fr(@)1183
1009 y Fq(r)1220 997 y Fr(\036)p Fp(j)1292 963 y Fw(2)1343
997 y Fr(d)1386 963 y Fw(3)1424 997 y Fr(x)h Fp(\000)1573
905 y Fo(h)1612 997 y Fr(R)g Ft(+)1819 941 y(1)p 1787
978 106 4 v 1787 1054 a(4)p Fr(R)1902 905 y Fo(i)1955
884 y(Z)2001 1073 y Fq(r)r Fk(\025)p Fq(R)2154 997 y
Fp(j)p Fr(\036)p Fp(j)2249 963 y Fw(2)2301 997 y Fr(d)2344
963 y Fw(3)2381 997 y Fr(x)29 b(:)106 1324 y Ft(Th)n(us)f(w)n(e)f(can)g
(estimate)h Fr(K)6 b Ft([)p Fr(\036)p Ft(])23 b(=)f Fr(K)1277
1336 y Fw(1)1314 1324 y Ft([)p Fr(\036)p Ft(])d(+)f Fr(K)1582
1336 y Fw(2)1619 1324 y Ft([)p Fr(\036)p Ft(])28 b(as)f(follo)n(ws:)0
1581 y Fr(K)6 b Ft([)p Fr(\036)p Ft(])23 b Fp(\025)283
1468 y Fo(Z)329 1657 y Fj(R)371 1640 y Fl(3)421 1581
y Fr(t)p Ft(\()p Fr(r)r Ft(\))14 b Fr(r)k Fp(j)p Fr(@)690
1593 y Fq(r)727 1581 y Fr(\036)p Fp(j)799 1547 y Fw(2)850
1581 y Fr(d)893 1547 y Fw(3)931 1581 y Fr(x)7 b Ft(+)1057
1489 y Fo(h)1105 1525 y Fr(R)1169 1495 y Fw(3)p 1105
1562 101 4 v 1135 1638 a Ft(4)1223 1581 y Fp(\000)g Ft(2)p
Fr(R)1401 1489 y Fo(i)1452 1468 y(Z)1498 1657 y Fq(r)r
Fk(\024)p Fq(R)1651 1581 y Fp(j)p Fr(\036)p Fp(j)1746
1547 y Fw(2)1798 1581 y Fr(d)1841 1547 y Fw(3)1878 1581
y Fr(x)g Fp(\000)2004 1489 y Fo(h)2043 1581 y Fr(R)g
Ft(+)2226 1525 y(1)p 2194 1562 106 4 v 2194 1638 a(4)p
Fr(R)2309 1489 y Fo(i)2363 1468 y(Z)2409 1657 y Fq(r)r
Fk(\025)p Fq(R)2561 1581 y Fp(j)p Fr(\036)p Fp(j)2656
1547 y Fw(2)2708 1581 y Fr(d)2751 1547 y Fw(3)2789 1581
y Fr(x)22 b(:)0 1838 y Ft(Observ)n(e)k(that)i(generally)225
2103 y(0)d Fp(\024)381 1990 y Fo(Z)427 2179 y Fj(R)469
2162 y Fl(3)470 2103 y Fr(t)p Ft(\()p Fr(r)r Ft(\))14
b Fr(r)672 1983 y Fo(\014)672 2033 y(\014)672 2083 y(\014)672
2133 y(\014)699 2103 y Fr(@)743 2115 y Fq(r)780 2103
y Fr(\036)19 b Ft(+)941 2047 y(1)p 941 2084 42 4 v 942
2160 a Fr(r)1006 2103 y(\036)1055 1983 y Fo(\014)1055
2033 y(\014)1055 2083 y(\014)1055 2133 y(\014)1083 2004
y Fw(2)1148 2103 y Fr(d)1191 2069 y Fw(3)1229 2103 y
Fr(x)547 2329 y Ft(=)635 2216 y Fo(Z)681 2404 y Fj(R)723
2388 y Fl(3)723 2329 y Fr(t)p Ft(\()p Fr(r)r Ft(\))14
b Fr(r)k Fp(j)p Fr(@)992 2341 y Fq(r)1029 2329 y Fr(\036)p
Fp(j)1101 2294 y Fw(2)1153 2329 y Fr(d)1196 2294 y Fw(3)1233
2329 y Fr(x)h Fp(\000)1382 2216 y Fo(Z)1428 2404 y Fj(R)1470
2388 y Fl(3)1471 2329 y Fr(t)p Ft(\()p Fr(r)r Ft(\))1629
2272 y(1)p 1629 2309 V 1630 2385 a Fr(r)1694 2329 y Fp(j)p
Fr(\036)p Fp(j)1789 2294 y Fw(2)1841 2329 y Fr(d)1884
2294 y Fw(3)1921 2329 y Fr(x)g Fp(\000)2070 2216 y Fo(Z)2116
2404 y Fj(R)2158 2388 y Fl(3)2159 2329 y Fr(t)2189 2294
y Fk(0)2212 2329 y Ft(\()p Fr(r)r Ft(\))14 b Fp(j)p Fr(\036)p
Fp(j)2424 2294 y Fw(2)2477 2329 y Fr(d)2520 2294 y Fw(3)2557
2329 y Fr(x)29 b(:)0 2579 y Ft(Since)39 b Fr(t)258 2549
y Fk(0)281 2579 y Ft(\()p Fr(r)r Ft(\))k Fp(\021)e Ft(0)d(and)g
Fr(t)p Ft(\()p Fr(r)r Ft(\))43 b Fp(\021)e Ft(1)d(on)h([0)p
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2579 y Ft(\()p Fr(r)r Ft(\))k(=)f Fp(\000)p Fr(t)p Ft(\()p
Fr(r)r Ft(\))p Fr(=r)h Ft(on)d(\()p Fr(R)q(;)14 b Fp(1)p
Ft(\),)39 b(the)0 2666 y(follo)n(wing)27 b(estimate)g(holds)5
2800 y Fo(Z)52 2988 y Fj(R)94 2972 y Fl(3)130 2913 y
Fr(t)p Ft(\()p Fr(r)r Ft(\))14 b Fr(r)k Fp(j)p Fr(@)399
2925 y Fq(r)436 2913 y Fr(\036)p Fp(j)508 2878 y Fw(2)559
2913 y Fr(d)602 2878 y Fw(3)640 2913 y Fr(x)h Fp(\000)789
2800 y Fo(Z)835 2988 y Fj(R)877 2972 y Fl(3)937 2856
y Ft(1)p 937 2893 V 938 2970 a Fr(r)1002 2913 y Fp(j)p
Fr(\036)p Fp(j)1097 2878 y Fw(2)1149 2913 y Fr(d)1192
2878 y Fw(3)1230 2913 y Fr(x)25 b Fp(\025)f(\000)1470
2800 y Fo(Z)1516 2988 y Fj(R)1558 2972 y Fl(3)1604 2856
y Ft(1)18 b Fp(\000)g Fr(t)p Ft(\()p Fr(r)r Ft(\))p 1604
2893 278 4 v 1723 2970 a Fr(r)1906 2913 y Fp(j)p Fr(\036)p
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2913 y Fr(x)h Ft(+)2282 2800 y Fo(Z)2328 2988 y Fj(R)2370
2972 y Fl(3)2420 2913 y Fr(t)2450 2878 y Fk(0)2473 2913
y Ft(\()p Fr(r)r Ft(\))14 b Fp(j)p Fr(\036)p Fp(j)2685
2878 y Fw(2)2738 2913 y Fr(d)2781 2878 y Fw(3)2819 2913
y Fr(x)1414 3134 y Ft(=)23 b Fp(\000)1581 3021 y Fo(Z)1627
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3115 42 4 v 1790 3191 a Fr(r)1855 3134 y Fp(j)p Fr(\036)p
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3134 y Fr(x)g Fp(\025)g(\000)2326 3078 y Ft(1)p 2315
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y Fq(r)r Fk(\025)p Fq(R)2601 3134 y Fp(j)p Fr(\036)p
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y Fr(x)k Fp(\025)926 3535 y Fo(h)975 3571 y Fr(R)1039
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3627 y Fp(\000)18 b Ft(2)p Fr(R)1293 3535 y Fo(i)1346
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3608 106 4 v 2135 3684 a(4)p Fr(R)2250 3535 y Fo(i)2303
3514 y(Z)2349 3703 y Fq(r)r Fk(\025)p Fq(R)2502 3627
y Fp(j)p Fr(\036)p Fp(j)2597 3593 y Fw(2)2649 3627 y
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y Ft(Optimizing)g(on)f Fr(R)d Fp(2)f Ft(\(0)p Fr(;)825
3832 y Fp(p)p 894 3832 42 4 v 69 x Ft(2)o(],)28 b Fn(i.e.)g
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Fq(;)270 4177 y Fk(p)p 325 4177 34 3 v 48 x Fw(2\))397
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Fp(\000)2046 4102 y Ft(5)p 2014 4139 V 2014 4215 a(4)p
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4168 y Fk(p)p 2344 4168 34 3 v 48 x Fw(5)p Fq(=)p Fw(2)2471
4158 y Ft(=)k Fp(\000)2624 4085 y(p)p 2692 4085 42 4
v 2692 4158 a Ft(5)28 b Fr(;)0 4412 y Ft(w)n(e)f(get)402
4590 y Fr(G)467 4602 y Fw(1)505 4590 y Ft([)p Fr(\036)p
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y Fr(K)6 b Ft([)p Fr(\036)p Ft(])19 b Fp(\000)1106 4477
y Fo(Z)1152 4665 y Fj(R)1194 4649 y Fl(3)1254 4534 y
Ft(1)p 1254 4571 V 1255 4647 a Fr(r)1319 4590 y Fp(j)p
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4555 y Fw(3)1547 4590 y Fr(x)1594 4473 y Fo(\023)1678
4590 y Fp(\025)k(\000)p Fr(\027)1890 4517 y Fp(p)p 1960
4517 V 1960 4590 a Ft(5)2015 4477 y Fo(Z)2061 4665 y
Fj(R)2103 4649 y Fl(3)2153 4590 y Fp(j)p Fr(\036)p Fp(j)2248
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y Fr(x)28 b(:)0 4823 y Ft(Hence)g(the)g(condition)f Fr(G)819
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4835 y Fw(1)1336 4823 y Ft([)p Fr(\036)p Ft(])23 b Fp(\025)g(\000)p
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4823 y Ft(en)n(tails)27 b(that)1191 5066 y Fr(B)t Ft(\()p
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5047 139 4 v 1490 5123 a(5)14 b Fr(\027)1592 5099 y Fw(2)1666
5066 y Fr(:)2816 5291 y Ff(\003)p eop end
%%Page: 17 17
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%%Page: 18 18
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b Fp(\000)825 5190 y(p)p 894 5190 V 64 x Fr(\027)954
5187 y Fo(\001)1020 5254 y Ft(\()p Fr(\005)1121 5266
y Fk(L)1185 5254 y Fr(Z)q(;)d(P)25 b(\005)c(Z)6 b Ft(\))18
b(+)1644 5137 y Fo(\022)1705 5254 y Ft(1)g Fp(\000)1895
5198 y Ft(1)p 1858 5235 116 4 v 1858 5251 a Fp(p)p 1928
5251 47 4 v 1928 5311 a Fr(\027)1984 5137 y Fo(\023)2073
5254 y Ft(\()p Fr(\005)2181 5220 y Fq(c)2228 5254 y Fr(Z)q(;)c(P)26
b(\005)2478 5220 y Fq(c)2525 5254 y Fr(Z)6 b Ft(\))42
b Fr(:)p eop end
%%Page: 19 19
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(atoms)f(in)f(strong)h(magnetic)h(\014elds)1116 b(19)0
248 y Ft(This)28 b(simply)f(follo)n(ws)g(from)g(the)h(iden)n(tities)256
465 y(\()p Fr(a)19 b Ft(+)f Fr(b)p Ft(\))502 431 y Fw(2)562
465 y Fp(\024)23 b Fr(a)694 431 y Fw(2)750 465 y Ft(+)18
b Fr(b)869 431 y Fw(2)924 465 y Ft(+)g(2)c Fp(j)p Fr(a)g(b)p
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348 y Fo(\024)1492 398 y(\000)1530 465 y Ft(1)18 b(+)1673
401 y Fp(p)p 1742 401 47 4 v 64 x Fr(\027)1803 398 y
Fo(\001)1868 465 y Fr(a)1912 431 y Fw(2)1968 465 y Ft(+)2051
348 y Fo(\022)2112 465 y Ft(1)g(+)2302 409 y(1)p 2265
446 116 4 v 2265 463 a Fp(p)p 2334 463 47 4 v 59 x Fr(\027)2390
348 y Fo(\023)2479 465 y Fr(b)2515 431 y Fw(2)2552 348
y Fo(\025)2637 465 y Fr(;)256 698 y Ft(\()p Fr(a)h Ft(+)f
Fr(b)p Ft(\))502 663 y Fw(2)562 698 y Fp(\025)23 b Fr(a)694
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y(\000)1533 698 y Ft(1)c Fp(\000)1676 634 y(p)p 1745
634 V 64 x Fr(\027)1805 630 y Fo(\001)1871 698 y Fr(a)1915
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y Ft(1)g Fp(\000)2305 641 y Ft(1)p 2268 678 116 4 v 2268
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h(the)i(next)g(prop)r(osition.)0 1066 y Fx(Prop)s(osition)i(8.)24
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1036 y Fk(1)1112 1087 y Fw(0)1189 1066 y Ft(\()p Fm(R)1281
1036 y Fw(3)1318 1066 y Fr(;)14 b Fm(C)1415 1036 y Fw(4)1452
1066 y Ft(])p Fi(,)108 1230 y Fp(E)152 1249 y Fq(\027)t
Fw(+)p Fq(\027)277 1232 y Fl(3)p Fg(=)p Fl(2)373 1230
y Ft([)p Fr(\005)20 b(Z)6 b Ft(])18 b(+)g Fp(E)716 1249
y Fq(\027)t Fw(+)804 1207 y Fk(p)p 859 1207 38 3 v 42
x Fq(\027)914 1230 y Ft([)p Fr(\005)1013 1195 y Fq(c)1060
1230 y Fr(Z)6 b Ft(])53 b Fp(\024)f(E)1360 1242 y Fq(\027)1401
1230 y Ft([)p Fr(Z)6 b Ft(])53 b Fp(\024)f(E)1724 1249
y Fq(\027)t Fk(\000)p Fq(\027)1850 1232 y Fl(3)p Fg(=)p
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Fp(E)2289 1249 y Fq(\027)t Fk(\000)2378 1207 y(p)p 2433
1207 V 42 x Fq(\027)2488 1230 y Ft([)p Fr(\005)2587 1195
y Fq(c)2634 1230 y Fr(Z)6 b Ft(])30 b Fr(:)106 1400 y
Ft(The)c(follo)n(wing)e(result)i(will)g(allo)n(w)e(us)h(to)h(get)f(rid)
h(of)f(the)h(higher)f(Landau)g(lev)n(els)g(when)0 1487
y(lo)r(oking)33 b(for)h(the)h(ground)e(state)i(energy)-7
b(,)33 b Fn(i.e.)i Ft(of)f(the)h(term)f Fp(E)2009 1506
y Fq(\027)t Fw(+)2097 1464 y Fk(p)p 2151 1464 V 2151
1506 a Fq(\027)2207 1487 y Ft([)p Fr(\005)2306 1457 y
Fq(c)2353 1487 y Fr(Z)6 b Ft(])34 b(in)h(Prop)r(osi-)0
1590 y(tion)28 b(8.)f(Consider)k(\026)-46 b Fr(\027)28
b Fp(2)23 b Ft(\(0)p Fr(;)14 b Ft(1\))27 b(suc)n(h)h(that)1034
1770 y(2)14 b(\()5 b(\026)-47 b Fr(\027)24 b Ft(+)1270
1700 y Fp(p)p 1339 1700 47 4 v 1344 1770 a Ft(\026)-47
b Fr(\027)19 b Ft(\))k(=)g(2)18 b Fp(\000)1685 1697 y(p)p
1754 1697 42 4 v 73 x Ft(2)27 b Fr(;)0 1933 y Fn(i.e.)32
b Ft(\026)-46 b Fr(\027)28 b Fp(\031)23 b Ft(0)p Fr(:)p
Ft(056,)j(and)h(for)g(an)n(y)g Fr(\027)h Fp(2)c Ft(\(0)p
Fr(;)18 b Ft(\026)-46 b Fr(\027)5 b Ft(\),)28 b(de\014ne)117
2114 y Fr(d)p Ft(\()p Fr(\016)s Ft(\))c(:=)f(\(1)18 b
Fp(\000)g Ft(2)p Fr(\016)s Ft(\))688 2041 y Fp(p)p 757
2041 V 73 x Ft(2)g Fp(\000)g Ft(2)p Fr(\016)30 b(;)97
b(d)1172 2126 y Fk(\006)1228 2114 y Ft(\()p Fr(\027)5
b Ft(\))24 b(:=)f Fr(d)p Ft(\()p Fr(\016)1585 2126 y
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b Ft(\))24 b(:=)2450 2050 y Fp(p)p 2519 2050 47 4 v 64
x Fr(\027)g Fp(\006)18 b Fr(\027)33 b(:)0 2272 y Ft(W)-7
b(e)28 b(ha)n(v)n(e)846 2361 y Fr(d)p Ft(\()p Fr(\016)s
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b(<)22 b Ft(1)c Fp(\000)1804 2288 y(p)p 1873 2288 42
4 v 73 x Ft(2)p Fr(=)p Ft(2)c Fr(;)1025 2501 y(d)1068
2513 y Fk(\006)1124 2501 y Ft(\()p Fr(\027)5 b Ft(\))24
b Fr(>)f Ft(0)82 b(if)i Fr(\027)28 b(<)g Ft(\026)-47
b Fr(\027)33 b(:)0 2666 y Fx(Prop)s(osition)d(9.)24 b
Fi(L)l(et)34 b Fr(B)i(>)31 b Ft(0)j Fi(and)h Fr(\016)g
Fp(2)d Ft(\(1)21 b Fp(\000)1541 2597 y(p)p 1610 2597
V 69 x Ft(2)p Fr(=)p Ft(2\))p Fi(.)34 b(F)-6 b(or)34
b(any)2168 2645 y Ft(~)2150 2666 y Fr(Z)j Ft(=)2341 2598
y Fo(\000)2379 2629 y Fq(X)2392 2694 y Fw(0)2438 2598
y Fo(\001)2476 2666 y Fi(,)2553 2645 y Ft(\026)2536 2666
y Fr(Z)g Ft(=)2726 2598 y Fo(\000)2774 2629 y Fw(0)2764
2694 y Fq(Y)2817 2598 y Fo(\001)2855 2666 y Fi(,)0 2771
y Fr(X)r(;)14 b(Y)41 b Fp(2)24 b Fr(L)333 2740 y Fw(2)369
2771 y Ft(\()p Fm(R)461 2740 y Fw(3)499 2771 y Fr(;)14
b Fm(C)596 2740 y Fw(2)633 2771 y Ft(\))779 2953 y Fp(E)823
2965 y Fq(\016)873 2953 y Ft([)p Fr(\005)972 2918 y Fq(c)1037
2932 y Ft(~)1020 2953 y Fr(Z)6 b Ft(])23 b Fp(\025)f
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2878 68 4 v 1447 2953 a Fr(B)32 b Fp(k)14 b Fr(\005)1674
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2918 y Fw(2)1861 2992 y Fg(L)1901 2980 y Fl(2)1933 2992
y(\()p Fe(R)1985 2980 y Fl(3)2018 2992 y(\))2078 2953
y Fr(:)725 3165 y Fp(E)769 3177 y Fk(\000)p Fq(\016)871
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a Fr(B)32 b Fp(k)14 b Fr(\005)1737 3130 y Fq(c)1730 3185
y Fk(L)1793 3165 y Fr(Y)32 b Fp(k)1915 3130 y Fw(2)1915
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Fq(B)1503 3524 y Ft(\))1535 3490 y Fw(2)1587 3524 y Fm(I)h
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(diamagnetic)f(inequalit)n(y)g(\(see)h([1]\),)683 3788
y Fo(Z)729 3977 y Fj(R)771 3960 y Fl(3)822 3806 y Fo(\014)822
3856 y(\014)822 3906 y(\014)863 3901 y Ft(\()p Fp(r)19
b(\000)f Fr(iA)1157 3913 y Fq(B)1214 3901 y Ft(\))c Fr( )1331
3806 y Fo(\014)1331 3856 y(\014)1331 3906 y(\014)1359
3826 y Fw(2)1410 3901 y Fr(dx)24 b Fp(\025)1611 3788
y Fo(Z)1657 3977 y Fj(R)1699 3960 y Fl(3)1750 3806 y
Fo(\014)1750 3856 y(\014)1750 3906 y(\014)1791 3901 y
Fp(rj)p Fr( )s Fp(j)1977 3806 y Fo(\014)1977 3856 y(\014)1977
3906 y(\014)2005 3826 y Fw(2)2056 3901 y Fr(dx)29 b(;)0
4115 y Ft(Hardy's)e(inequalit)n(y)-7 b(,)848 4222 y Fo(Z)894
4411 y Fj(R)936 4394 y Fl(3)986 4240 y Fo(\014)986 4289
y(\014)986 4339 y(\014)1028 4335 y Fp(rj)p Fr( )s Fp(j)1214
4240 y Fo(\014)1214 4289 y(\014)1214 4339 y(\014)1242
4260 y Fw(2)1293 4335 y Fr(dx)24 b Fp(\025)1504 4279
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Fp(j)1821 4249 y Fw(2)p 1718 4316 141 4 v 1723 4392 a
Fp(j)p Fr(x)p Fp(j)1816 4368 y Fw(2)1891 4335 y Fr(dx)29
b(;)0 4551 y Ft(and)e(the)h(nonnegativit)n(y)f(of)g Fr(\033)22
b Fp(\001)d Fx(B)f Ft(+)g Fr(B)g Fm(I)p Ft(,)28 b(w)n(e)f(get)224
4767 y Fr(P)289 4733 y Fw(2)277 4788 y Fq(B)353 4767
y Ft(+)18 b(\(1)g(+)g Fr(B)t Ft(\))c Fm(I)24 b Ft(=)f(\()p
Fp(r)c(\000)f Fr(iA)1162 4779 y Fq(B)1219 4767 y Ft(\))1251
4733 y Fw(2)1302 4767 y Fm(I)h Ft(+)f Fr(\033)k Fp(\001)c
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4650 y Fo(\022)2172 4711 y Ft(1)p 2172 4748 42 4 v 2172
4824 a(4)2292 4711 y(1)p 2247 4748 131 4 v 2247 4824
a Fp(j)p Fr(x)p Fp(j)2340 4800 y Fw(2)2406 4767 y Ft(+)18
b(1)2531 4650 y Fo(\023)2634 4767 y Fr(:)28 4984 y Ft(Since)27
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(v)n(e)488 5158 y Fp(p)p 557 5158 68 4 v 74 x Fr(B)c
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Ft(+)f Fr(P)1008 5203 y Fw(2)996 5256 y Fq(B)1076 5232
y Fp(\025)1164 5132 y Fo(q)p 1247 5132 533 4 v 100 x
Fr(P)1312 5203 y Fw(2)1300 5256 y Fq(B)1376 5232 y Ft(+)g(\(1)g(+)g
Fr(B)t Ft(\))c Fm(I)24 b Fp(\025)1890 5087 y Fo(s)p 1973
5087 369 4 v 1983 5176 a Ft(1)p 1983 5213 42 4 v 1983
5289 a(4)2103 5176 y(1)p 2058 5213 131 4 v 2058 5289
a Fp(j)p Fr(x)p Fp(j)2151 5265 y Fw(2)2218 5232 y Ft(+)18
b(1)27 b Fr(:)p eop end
%%Page: 20 20
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(Maria)e(J.)i(Esteban,)g(Mic)n(hael)g(Loss)0 248 y Ft(No)n(w,)j(for)g
(an)n(y)g Fr(\016)f(>)d Ft(0,)252 354 y Fo(p)p 335 354
245 4 v 71 x Fm(I)18 b Ft(+)g Fr(P)533 397 y Fw(2)521
450 y Fq(B)597 425 y Fp(\000)713 393 y Fq(\016)p 690
407 78 4 v 690 454 a Fk(j)p Fq(x)p Fk(j)761 425 y Ft(=)k(\(1)d
Fp(\000)f Ft(2)p Fr(\016)s Ft(\))1152 354 y Fo(p)p 1234
354 245 4 v 1234 425 a Fm(I)h Ft(+)f Fr(P)1433 397 y
Fw(2)1421 450 y Fq(B)1497 425 y Ft(+)g(2)p Fr(\016)1675
354 y Fo(p)p 1758 354 V 71 x Fm(I)h Ft(+)f Fr(P)1957
397 y Fw(2)1945 450 y Fq(B)2021 425 y Fp(\000)2136 393
y Fq(\016)p 2114 407 78 4 v 2114 454 a Fk(j)p Fq(x)p
Fk(j)761 564 y Fp(\025)k Ft(\(1)d Fp(\000)f Ft(2)p Fr(\016)s
Ft(\))1152 493 y Fo(p)p 1234 493 245 4 v 1234 564 a Fm(I)h
Ft(+)f Fr(P)1433 535 y Fw(2)1421 588 y Fq(B)1497 564
y Ft(+)g(2)p Fr(\016)1675 475 y Fo(q)p 1758 475 340 4
v 1768 531 a Fw(1)p 1768 545 34 4 v 1768 592 a(4)1873
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Fk(j)761 699 y Fp(\025)g Ft(\(1)d Fp(\000)f Ft(2)p Fr(\016)s
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y Fp(\000)g Ft(2)p Fr(\016)1675 629 y Fp(p)p 1745 629
68 4 v 1745 699 a Fr(B)31 b(:)0 888 y Ft(On)g(the)h(range)e(of)h
Fr(\005)693 858 y Fq(c)686 911 y Fk(L)768 888 y Ft(the)g(op)r(erator)
1253 817 y Fo(p)p 1336 817 245 4 v 71 x Fm(I)19 b Ft(+)f
Fr(P)1535 859 y Fw(2)1523 912 y Fq(B)1594 888 y Fp(\025)1682
817 y Fo(p)p 1765 817 248 4 v 71 x Fm(I)g Ft(+)g Fr(L)1955
859 y Fw(2)1955 912 y Fq(B)2043 888 y Ft(is)32 b(b)r(ounded)f(from)g(b)
r(elo)n(w)0 995 y(b)n(y)115 928 y Fp(p)p 184 928 252
4 v 67 x Ft(1)18 b(+)g(2)p Fr(B)t Ft(.)28 b(Hence)610
1108 y Fo(q)p 693 1108 245 4 v 100 x Fm(I)19 b Ft(+)f
Fr(P)892 1180 y Fw(2)880 1233 y Fq(B)956 1208 y Fp(\000)1076
1152 y Fr(\016)p 1049 1189 94 4 v 1049 1265 a Fp(j)p
Fr(x)p Fp(j)1175 1208 y(\025)23 b Ft(\(1)18 b Fp(\000)g
Ft(2)p Fr(\016)s Ft(\))1566 1137 y Fp(p)p 1635 1137 252
4 v 71 x Ft(1)g(+)g(2)p Fr(B)k Fp(\000)c Ft(2)p Fr(\016)2084
1134 y Fp(p)p 2153 1134 68 4 v 74 x Fr(B)32 b(;)0 1427
y Ft(whic)n(h)c(is)f(equiv)-5 b(alen)n(t)27 b(to)608
1533 y Fo(q)p 691 1533 245 4 v 100 x Fm(I)18 b Ft(+)g
Fr(P)889 1604 y Fw(2)877 1657 y Fq(B)953 1633 y Fp(\000)g
Fr(\016)f(P)35 b Fp(\025)23 b Ft(\(1)18 b Fp(\000)g Ft(2)p
Fr(\016)s Ft(\))1569 1562 y Fp(p)p 1638 1562 252 4 v
71 x Ft(1)g(+)g(2)p Fr(B)k Fp(\000)c Ft(2)p Fr(\016)2086
1559 y Fp(p)p 2155 1559 68 4 v 74 x Fr(B)32 b(:)0 1819
y Ft(Since)758 1912 y(\(1)18 b Fp(\000)g Ft(2)p Fr(\016)s
Ft(\))1061 1842 y Fp(p)p 1130 1842 252 4 v 70 x Ft(1)g(+)g(2)p
Fr(B)k Fp(\000)c Ft(2)p Fr(\016)1578 1838 y Fp(p)p 1647
1838 68 4 v 74 x Fr(B)28 b Fp(\030)22 b Fr(d)p Ft(\()p
Fr(\016)s Ft(\))1986 1838 y Fp(p)p 2056 1838 V 2056 1912
a Fr(B)0 2049 y Ft(as)27 b Fr(B)g Fp(!)c(1)p Ft(,)28
b(and)812 2159 y(inf)794 2212 y Fq(B)s(>)p Fw(0)932 2159
y Ft([\(1)18 b Fp(\000)g Ft(2)p Fr(\016)s Ft(\))1258
2088 y Fp(p)p 1327 2088 252 4 v 71 x Ft(1)g(+)g(2)p Fr(B)k
Fp(\000)c Ft(2)p Fr(\016)1775 2084 y Fp(p)p 1845 2084
68 4 v 1845 2159 a Fr(B)t Ft(])23 b Fr(>)g Ft(0)0 2345
y(if)32 b Fr(d)p Ft(\()p Fr(\016)s Ft(\))e Fr(>)e Ft(0,)j(the)h(righ)n
(t)f(hand)g(side)g(is)g(p)r(ositiv)n(e)g(for)g(an)n(y)f(\014eld)i
(strength)e(pro)n(vided)h(that)0 2450 y(0)23 b Fp(\024)f
Fr(\016)k(<)d Ft(1)18 b Fp(\000)446 2381 y(p)p 515 2381
42 4 v 69 x Ft(2)p Fr(=)p Ft(2.)2152 b Ff(\003)0 2731
y Fi(3.3.)29 b(The)23 b(r)l(estricte)l(d)f(pr)l(oblem.)47
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(energy)f Fr(\025)2633 2743 y Fw(1)2671 2731 y Ft(\()p
Fr(\027)q(;)c(B)t Ft(\))0 2818 y(is)24 b(comparable)e(with)j(the)f(one)
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(in)h(the)h(lo)n(w)n(er)d(comp)r(onen)n(ts)h(of)h(the)g(w)n(a)n(v)n(e)f
(function,)h(pro)n(vided)0 2982 y(that)d(the)g(Coulom)n(b)f(p)r(oten)n
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(to)h(Theorem)0 3069 y(1)i(\(also)g(see)g(Theorem)g(3.1)f(in)i([5])f
(in)h(case)f Fr(B)h Ft(=)c(0\),)j(for)g(all)h(0)25 b
Fr(<)h(B)31 b(<)25 b(B)t Ft(\()p Fr(\027)5 b Ft(\),)31
b Fr(\027)h Fp(2)26 b Ft(\(0)p Fr(;)14 b Ft(1\),)0 3160
y Fr(\025)48 3172 y Fw(1)86 3160 y Ft(\()p Fr(\027)q(;)g(B)t
Ft(\))28 b(is)f(c)n(haracterized)f(as)554 3326 y Fr(\025)602
3338 y Fw(1)640 3326 y Ft(\()p Fr(\027)q(;)14 b(B)t Ft(\))23
b(=)173 b(inf)971 3379 y Fg(X)s Fh(2)p Fg(C)1103 3365
y Fh(1)1100 3400 y Fl(0)1164 3379 y(\()p Fe(R)1216 3367
y Fl(3)1249 3379 y Fg(;)p Fe(C)1298 3367 y Fl(2)1330
3379 y(\))1102 3437 y Fg(X)s Fh(6)p Fl(=0)1708 3326 y
Ft(sup)1582 3395 y Fg(Y)11 b Fh(2)p Fg(C)1710 3381 y
Fh(1)1707 3416 y Fl(0)1771 3395 y(\()p Fe(R)1823 3383
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3486 y Ft(\))2082 3326 y Fp(E)2126 3338 y Fq(\027)2167
3326 y Ft([)p Fr(Z)6 b Ft(])28 b Fr(:)406 b Ft(\(15\))0
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195 3866 y Fr(\025)243 3831 y Fk(L)243 3886 y Fw(1)293
3866 y Ft(\()p Fr(\027)q(;)14 b(B)t Ft(\))24 b(:=)330
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3939 y Fl(0)998 3919 y(\()p Fe(R)1050 3907 y Fl(3)1083
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3989 y Fg(\005)696 3975 y(c)693 4011 y Fh(L)737 3989
y Fg(X)s Fl(=0)11 b Fg(;)17 b Fl(0)p Fg(<)p Fh(k)p Fg(X)s
Fh(k)1088 3975 y Fl(2)1088 4021 y Fg(L)1128 4009 y Fl(2)1160
4021 y(\()p Fe(R)1212 4009 y Fl(3)1245 4021 y(\))1271
3989 y Fg(<)p Fl(1)1884 3866 y Ft(sup)1635 3948 y Fg(Y)12
b Fh(2)p Fg(C)1764 3934 y Fh(1)1761 3968 y Fl(0)1825
3948 y(\()p Fe(R)1877 3936 y Fl(3)1909 3948 y Fg(;)p
Fe(C)1958 3936 y Fl(2)1991 3948 y(\))f Fg(;)17 b(Z)s
Fl(=)2145 3958 y Ft(\()2176 3927 y Fg(X)2178 3970 y(Y)2226
3958 y Ft(\))1461 4028 y Fg(\005)1510 4014 y(c)1507 4050
y Fh(L)1552 4028 y Fg(Y)11 b Fl(=0)f Fg(;)34 b Fh(k)p
Fg(Y)12 b Fh(k)1839 4053 y Fg(L)1879 4041 y Fl(2)1910
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4061 y Fg(L)2289 4049 y Fl(2)2320 4061 y(\()p Fe(R)2372
4049 y Fl(3)2405 4061 y(\))2442 3866 y Fp(E)2486 3878
y Fq(\027)2527 3866 y Ft([)p Fr(Z)6 b Ft(])27 b Fr(:)0
4221 y Ft(W)-7 b(e)27 b(sho)n(w)f(b)r(elo)n(w)g(that)h(this)g
(restricted)f(problem)h(is)f(actually)g(a)h(one-dimensional)e(prob-)0
4315 y(lem.)j(F)-7 b(or)27 b(this)h(purp)r(ose,)f(let)h(us)f(de\014ne)h
(the)g(function)g Fr(a)1804 4285 y Fq(B)1804 4335 y Fw(0)1884
4315 y Ft(:)c Fm(R)18 b Fp(\002)g Fm(R)2152 4285 y Fw(+)2230
4315 y Fp(!)23 b Fm(R)28 b Ft(giv)n(en)f(b)n(y)864 4557
y Fr(a)908 4523 y Fq(B)908 4577 y Fw(0)965 4557 y Ft(\()p
Fr(z)t Ft(\))c(:=)g Fr(B)1301 4444 y Fo(Z)1384 4464 y
Fw(+)p Fk(1)1347 4632 y Fw(0)1529 4501 y Fr(s)14 b(e)1621
4471 y Fk(\000)p Fq(B)s(s)1757 4445 y Fl(2)1789 4471
y Fq(=)p Fw(2)p 1529 4538 332 4 v 1531 4554 a Fp(p)p
1600 4554 258 4 v 70 x Fr(s)1639 4600 y Fw(2)1695 4624
y Ft(+)k Fr(z)1821 4600 y Fw(2)1884 4557 y Fr(ds)28 b(:)0
4775 y Ft(and)f(implicitly)i(de\014ne)f Fr(\026)823 4787
y Fk(L)900 4775 y Ft(as)f(the)h(unique)g(solution)f(of)147
5112 y Fr(\026)197 5124 y Fk(L)247 5112 y Ft([)p Fr(f)t(;)14
b(\027)q(;)g(B)t Ft(])23 b(=)642 4885 y Fo(Z)688 5074
y Fj(R)731 4881 y Fo(\022)1106 4942 y Fp(j)p Fr(f)1179
4912 y Fk(0)1202 4942 y Ft(\()p Fr(z)t Ft(\))p Fp(j)1332
4912 y Fw(2)p 794 4979 887 4 v 794 5057 a Ft(1)18 b(+)g
Fr(\026)987 5069 y Fk(L)1037 5057 y Ft([)p Fr(f)t(;)c(\027)q(;)g(B)t
Ft(])k(+)g Fr(\027)h(a)1516 5029 y Fq(B)1516 5079 y Fw(0)1573
5057 y Ft(\()p Fr(z)t Ft(\))1709 4998 y(+)f(\(1)g Fp(\000)g
Fr(\027)h(a)2071 4964 y Fq(B)2071 5019 y Fw(0)2128 4998
y Ft(\()p Fr(z)t Ft(\)\))14 b Fp(j)p Fr(f)9 b Ft(\()p
Fr(z)t Ft(\))p Fp(j)2484 4964 y Fw(2)2513 4881 y Fo(\023)2588
4998 y Fr(dz)p 642 5093 2032 4 v 1435 5103 a Fo(Z)1481
5291 y Fj(R)1541 5216 y Fp(j)p Fr(f)g Ft(\()p Fr(z)t
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5112 y(:)p eop end
%%Page: 21 21
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Ft(\(0)p Fr(;)14 b Ft(1\))p Fi(,)725 418 y Fr(\025)773
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1121 y Fh(L)1164 1105 y Fq(\037)p Fw(=0)12 b Fq(;)18
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y Fp(\000)p Ft(1)k Fp(\000)2345 975 y Fq(\027)p 2345
989 V 2347 1036 a(r)2436 749 y Fo(1)2436 898 y(A)2509
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y Ft(or)113 1599 y Fr(\025)161 1564 y Fk(L)161 1619 y
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Fj(R)1214 1669 y Fl(3)1247 1694 y Fq(;)p Fj(C)1309 1669
y Fl(2)1341 1694 y Fw(\))922 1787 y Fq(\005)976 1795
y Fh(L)1020 1787 y Fq(\037)p Fk(6)p Fw(=0)12 b Fq(;)19
b( )r Fw(=)1293 1793 y Ft(\()1325 1760 y Fg(\036)1325
1807 y(\037)1361 1793 y Ft(\))931 1881 y Fk(k)p Fq(\005)e( )r
Fk(k)1116 1856 y Fl(2)1116 1906 y Fg(L)1156 1894 y Fl(2)1188
1906 y(\()p Fe(R)1240 1894 y Fl(3)1272 1906 y(\))1299
1881 y Fw(=1)1419 1432 y Fo(0)1419 1581 y(@)1492 1432
y(0)1492 1581 y(@)1640 1502 y Ft(1)h Fp(\000)1793 1470
y Fq(\027)p 1793 1484 V 1795 1531 a(r)1941 1502 y Fp(\000)c
Fr(i)g(\033)2110 1514 y Fw(3)2160 1502 y Fr(@)2204 1514
y Fq(x)2242 1522 y Fl(3)1565 1690 y Fp(\000)g Fr(i)g(\033)1734
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y Fl(3)1984 1690 y Fp(\000)p Ft(1)k Fp(\000)2202 1657
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1432 y Fo(1)2695 1581 y(A)0 2079 y Ft(with)k(the)g(notation)f
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2101 y(3)1547 2079 y Ft(.)24 b(F)-7 b(or)23 b(an)n(y)g(giv)n(en)g
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2102 y Fk(L)2656 2079 y Fr(\036)g Ft(=)e(0,)0 2169 y(the)28
b(suprem)n(um)f(in)h Fr(\037)g Ft(is)f(ac)n(hiev)n(ed)g(b)n(y)g(the)h
(function)403 2373 y Fr(\037)455 2385 y Fk(L)505 2373
y Ft([)p Fr(\036)p Ft(])c(=)711 2280 y Fo(\020)761 2373
y Fr(V)809 2385 y Fk(L)878 2373 y Ft(+)961 2305 y Fo(\000)999
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2298 y Fk(\000)p Fw(1)1829 2373 y Fr(\005)1898 2385 y
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y Fw(3)2213 2373 y Fr(@)2257 2385 y Fq(x)2295 2393 y
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y Fg(\036)821 3010 y(\037)857 2998 y Ft(\))872 2671 y
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y Ft(1)h Fp(\000)1246 2709 y Fq(\027)p 1246 2723 V 1248
2770 a(r)1394 2742 y Fp(\000)c Fr(i)g(\033)1563 2754
y Fw(3)1613 2742 y Fr(@)1657 2754 y Fq(x)1695 2762 y
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y Fp(\000)p Ft(1)k Fp(\000)1655 2897 y Fq(\027)p 1655
2911 V 1657 2958 a(r)1745 2671 y Fo(1)1745 2821 y(A)1973
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2910 y Fw(\()p Fj(R)2203 2893 y Fl(3)2236 2910 y Fw(\))2276
2838 y Fr(;)2468 2782 y(\005)h( )p 2323 2819 V 2323 2895
a Fp(k)p Fr(\005)f( )s Fp(k)2553 2910 y Fq(L)2599 2893
y Fl(2)2630 2910 y Fw(\()p Fj(R)2698 2893 y Fl(3)2731
2910 y Fw(\))2771 2671 y Fo(1)2771 2821 y(A)2857 2838
y Fr(:)0 3170 y Ft(Since)28 b Fr(\033)264 3182 y Fw(3)315
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3389 y Fr(\025)176 3355 y Fk(L)176 3410 y Fw(1)227 3389
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y Fg(\005)507 3489 y(c)504 3526 y Fh(L)549 3503 y Fg(\036)p
Fl(=0)10 b Fg(;)17 b Fh(k)p Fg(\036)p Fh(k)798 3528 y
Fg(L)838 3516 y Fl(2)869 3528 y(\()p Fe(R)921 3516 y
Fl(3)954 3528 y(\))980 3503 y(=1)1075 3276 y Fo(Z)1121
3465 y Fj(R)1163 3448 y Fl(3)1213 3272 y Fo(\024)1502
3333 y Fp(j)p Fr(@)1569 3345 y Fq(x)1607 3353 y Fl(3)1643
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y Ft(1)f Fp(\000)2301 3333 y Fr(\027)p 2301 3370 47 4
v 2304 3446 a(r)2357 3297 y Fo(\021)2407 3389 y Fp(j)p
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3389 y Fr(dx)28 b(:)0 3684 y Ft(No)n(w,)i(with)i(the)f(notations)f(of)h
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y(ha)n(v)n(e)191 3721 y Fo(R)231 3817 y Fj(R)273 3801
y Fl(2)323 3788 y Fr(\036)372 3800 y Fq(`)418 3788 y
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y Fw(1)844 3788 y Fr(dx)934 3800 y Fw(2)995 3788 y Ft(=)22
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b Fr(x)2376 3759 y Fw(2)2376 3810 y(3)2414 3788 y Ft(,)27
b(and)0 3988 y Fr(\025)48 3953 y Fk(L)48 4008 y Fw(1)99
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y Fh(k)p Fg(\036)p Fh(k)750 4057 y Fg(L)790 4045 y Fl(2)821
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y(=1)581 4108 y Fg(\036)p Fh(2)p Fg(C)699 4094 y Fh(1)696
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Fg(\036)1002 4123 y(`)1031 4108 y Fl(\)\))1050 3896 y
Fo(\020)1100 3988 y Ft(\()p Fr(V)1180 4000 y Fk(L)1232
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y Fq(L)463 4488 y Fl(2)495 4504 y Fw(\()p Fj(R)563 4488
y Fl(2)595 4504 y Fq(;)p Fj(C)657 4488 y Fl(2)689 4504
y Fw(\))742 4471 y Ft(=)830 4392 y Fo(X)834 4571 y Fq(`)p
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%%Page: 22 22
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Fr(:)p eop end
%%Page: 23 23
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eop end
%%Page: 24 24
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137 4 v 1570 1971 a Fp(p)p 1639 1971 68 4 v 71 x Fr(B)1716
1857 y Fo(\023)1800 1974 y Ft(=)23 b Fr(B)1955 1940 y
Fw(1)p Fq(=)p Fw(4)2073 1974 y Fr(g)s Ft(\()p Fr(y)s
Ft(\))28 b Fr(;)0 2195 y Ft(one)f(tranforms)g(the)h(ab)r(o)n(v)n(e)e
(minimization)i(problem)f(in)n(to)248 2416 y Fr(\025)296
2428 y Fk(L)347 2416 y Ft(\()p Fr(\016)o(;)14 b(B)t Ft(\))19
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x Fr(B)207 b Ft(inf)985 2462 y Fg(g)q Fh(2)p Fg(C)1098
2448 y Fh(1)1095 2482 y Fl(0)1160 2462 y(\()p Fe(R)p
Fg(;)p Fe(C)p Fl(\))p Fh(nf)p Fl(0)p Fh(g)979 2496 y
Fd(R)1002 2547 y Fe(R)1046 2530 y Fg(g)q Fl(\()p Fg(y)q
Fl(\))1151 2518 y(2)1196 2530 y Fg(d\026)p Fl(\()p Fg(y)q
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2416 y Fp(\000)18 b Fr(\016)f Fp(j)p Fr(g)s Ft(\()p Fr(y)s
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2416 y Fr(dy)30 b(:)0 2696 y Ft(Hence,)813 2807 y Fr(\025)861
2819 y Fk(L)911 2807 y Ft(\()p Fr(\016)o(;)14 b(B)t Ft(\))23
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Fr(\016)s Ft(,)g(let)h Fr(\024)h Ft(=)g Fr(\024)p Ft(\()p
Fr(\016)s Ft(\))g(:=)g Fr(\016)1257 2890 y Fo(\000)1295
2957 y Ft(1)24 b Fp(\000)h Fr(\025)1499 2969 y Fk(L)1549
2957 y Ft(\()p Fr(\016)o(;)14 b Ft(1\))1728 2890 y Fo(\001)1803
2957 y Ft(and)37 b Fr(\026)p Ft(\()p Fr(y)s Ft(\))i(:=)g(1)p
Fr(=a)2426 2927 y Fw(1)2426 2978 y(0)2462 2957 y Ft(\()p
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b(alue)31 b Fr(E)1932 3069 y Fw(1)1999 3057 y Ft(=)e
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3174 y Fq(y)169 3154 y Ft(+)18 b Fr(\024)p Ft(\()p Fr(\016)s
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3626 71 4 v 1064 3691 a Fr(\031)p 1064 3728 51 4 v 1068
3805 a Ft(2)1207 3748 y Fp(8)g Fr(z)k Fp(2)c Fm(R)28
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y Ft(\()p Fr(z)t Ft(\))23 b Fp(\030)1960 3691 y Ft(1)p
1936 3728 89 4 v 1936 3805 a Fp(j)p Fr(z)t Fp(j)2058
3748 y Ft(as)f Fp(j)p Fr(z)t Fp(j)h(!)g(1)k Fr(:)0 3983
y Ft(There)32 b(exists)g(therefore)f(a)h(constan)n(t)g
Fr(c)f(>)f Ft(0)i(suc)n(h)g(that)h Fr(\026)p Ft(\()p
Fr(y)s Ft(\))e Fp(\024)f Fr(c)14 b(e)2232 3953 y Fk(j)p
Fq(y)r Fk(j)2344 3983 y Ft(for)31 b(an)n(y)h Fr(y)i Fp(2)d
Fm(R)p Ft(.)0 4075 y(T)-7 b(o)36 b(get)g(an)g(upp)r(er)h(estimate)f(of)
g Fr(E)1154 4087 y Fw(1)1192 4075 y Ft(,)g(w)n(e)g(ma)n(y)g(no)n(w)g
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4075 y Ft(\()p Fr(y)s Ft(\))i(:=)0 4178 y(cos)o(\()p
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Ft(1)p Fr(;)14 b Ft(1\))22 b Fp(3)h Fr(y)30 b Ft(and)e(the)g(rescaled)e
(functions)i Fr(g)1948 4190 y Fq(\033)1992 4178 y Ft(\()p
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Fq(=)p Fw(2)2441 4178 y Fr(g)2481 4190 y Fw(1)2518 4110
y Fo(\000)2574 4178 y Fp(\001)d Fr(=\033)s Ft(\).)220
4421 y Fr(E)281 4433 y Fw(1)318 4421 y Ft(\()p Fr(\016)s
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544 4402 144 4 v 544 4478 a Ft(4)14 b Fr(\033)650 4454
y Fw(2)715 4421 y Ft(+)k Fr(\024)860 4308 y Fo(Z)943
4328 y Fw(1)906 4497 y Fk(\000)p Fw(1)1009 4421 y Fp(j)p
Fr(g)1072 4433 y Fw(1)1109 4421 y Fp(j)1132 4387 y Fw(2)1183
4421 y Fr(\026)p Ft(\()p Fr(\033)g(y)s Ft(\))c Fr(dy)27
b Fp(\024)1658 4365 y Fr(\031)1708 4335 y Fw(2)p 1631
4402 V 1631 4478 a Ft(4)14 b Fr(\033)1737 4454 y Fw(2)1802
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y Fw(1)2043 4497 y Fk(\000)p Fw(1)2146 4421 y Fr(e)2185
4387 y Fq(\033)r Fk(j)p Fq(y)r Fk(j)2304 4421 y Fp(j)p
Fr(g)2367 4433 y Fw(1)2404 4421 y Fp(j)2427 4387 y Fw(2)2478
4421 y Fr(dy)1787 4660 y Fp(\024)1912 4604 y Fr(\031)1962
4574 y Fw(2)p 1884 4641 V 1884 4717 a Ft(4)g Fr(\033)1990
4693 y Fw(2)2056 4660 y Ft(+)k(2)c Fr(\024)g(c)g Ft(\()o
Fr(e)2377 4626 y Fq(\033)2440 4660 y Fp(\000)k Ft(1\))41
b Fr(:)0 4864 y Ft(Optimizing)28 b(in)f Fr(\033)32 b
Ft(in)27 b(the)h(ab)r(o)n(v)n(e)f(expression,)f(w)n(e)h(c)n(ho)r(ose)f
Fr(\033)h Ft(=)22 b Fr(\033)s Ft(\()p Fr(\016)s Ft(\))29
b(satisfying)1140 5041 y Fr(\031)1190 5007 y Fw(2)1251
5041 y Ft(=)22 b(4)14 b Fr(\024)g(c)g(\033)1556 5007
y Fw(3)1607 5041 y Fr(e)1646 5007 y Fq(\033)1718 5041
y Fr(;)0 5199 y Ft(whic)n(h)28 b(implies)718 5291 y Fr(\033)s
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5291 y Fp(!)g(1)83 b Ft(as)f Fr(\016)26 b Fp(!)d Ft(0)28
b Fr(:)p eop end
%%Page: 25 25
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144 4 v 1317 816 a Ft(4)14 b Fr(\033)1423 788 y Fw(2)1420
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y Ft(\()p Fr(\016)s Ft(\),)h(w)n(e)f(consider)f(the)h(function)0
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(b)r(elo)n(w)f(the)g(function)h Fr(\024)14 b(\026)p Ft(\()p
Fr(y)s Ft(\).)33 b(Then)g(observ)n(e)f(that)0 1303 y(the)25
b(\014rst)g(eigen)n(v)-5 b(alue)24 b(of)h(the)g(op)r(erator)e
Fp(\000)p Fr(@)1381 1273 y Fw(2)1376 1324 y Fq(y)1430
1303 y Ft(+)13 b Fr(\026)1558 1315 y Fq(\016)1594 1303
y Ft(\()p Fr(y)s Ft(\),)26 b(that)f(w)n(e)f(denote)h
Fr(E)2378 1273 y Fq(\016)2373 1324 y Fw(1)2415 1303 y
Ft(,)g(satis\014es)f(the)0 1399 y(equation)733 1488 y
Fo(q)p 816 1488 103 4 v 103 x Fr(E)882 1562 y Fq(\016)877
1613 y Fw(1)932 1591 y Fr(\033)979 1603 y Fq(\016)1039
1591 y Ft(=)f(arctan)1371 1449 y Fo( )1437 1440 y(s)p
1520 1440 485 4 v 1530 1534 a Fr(\024)14 b(\026)p Ft(\()p
Fr(\033)1721 1546 y Fq(\016)1758 1534 y Ft(\))19 b Fp(\000)f
Fr(E)1958 1506 y Fq(\016)1953 1557 y Fw(1)p 1530 1571
465 4 v 1711 1650 a Fr(E)1777 1622 y Fq(\016)1772 1673
y Fw(1)2018 1449 y Fo(!)2125 1591 y Fr(;)0 1823 y Ft(and)27
b(as)g Fr(\016)k Ft(go)r(es)c(to)g(0)657 1793 y Fw(+)740
1823 y Ft(this)g(implies)887 2052 y Fr(E)948 2064 y Fw(1)986
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1996 y Fr(\031)1502 1966 y Fw(2)p 1424 2033 144 4 v 1424
2109 a Ft(4)14 b Fr(\033)1530 2081 y Fw(2)1527 2134 y
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y Ft(Summarizing,)27 b(what)h(w)n(e)f(ha)n(v)n(e)f(obtained)i(is)837
2511 y Fr(E)898 2523 y Fw(1)936 2511 y Ft(\()p Fr(\016)s
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2492 457 4 v 1161 2589 a Ft(4)1217 2522 y Fo(\000)1268
2589 y Ft(log)15 b Fr(\024)p Ft(\()p Fr(\016)s Ft(\))1542
2522 y Fo(\001)1580 2539 y Fw(2)1641 2443 y Fo(\000)1679
2511 y Ft(1)j(+)g Fr(o)p Ft(\(1\))1968 2443 y Fo(\001)2020
2511 y Fr(:)0 2781 y Ft(So,)27 b(imp)r(osing)h Fr(E)553
2793 y Fw(1)590 2781 y Ft(\()p Fr(\016)s Ft(\))c(=)f
Fr(\016)846 2751 y Fw(2)883 2781 y Ft(,)28 b(w)n(e)f(get)g
Fr(\024)p Ft(\()p Fr(\016)s Ft(\))d(=)e Fr(e)1496 2751
y Fk(\000)1568 2729 y Fg(\031)p 1558 2738 57 3 v 1558
2771 a Fl(2)p Fg(\016)1624 2751 y Fw(\(1+)p Fq(o)p Fw(\(1\)\))1882
2781 y Ft(.)28 b(Since)g(b)n(y)g(\(17\),)704 3010 y Fp(\000)p
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3010 y Fr(\016)o(;)14 b(B)1193 3022 y Fk(L)1243 3010
y Ft(\()p Fr(\016)s Ft(\))1347 2943 y Fo(\001)1409 3010
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v 75 x Fr(B)1785 3022 y Fk(L)1835 3010 y Ft(\()p Fr(\016)s
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3207 y Ft(w)n(e)27 b(get)982 3320 y Fr(B)1045 3332 y
Fk(L)1095 3320 y Ft(\()p Fr(\016)s Ft(\))38 b(=)e(4)14
b Fr(\016)1434 3286 y Fw(2)1484 3320 y Fr(e)1533 3263
y Fg(\031)p 1533 3272 36 3 v 1537 3306 a(\016)1590 3286
y Fw(\(1+)p Fq(o)p Fw(\(1\)\))1876 3320 y Fr(;)0 3465
y Ft(whic)n(h,)28 b(together)e(with)i(\(16\),)g(concludes)f(the)h(pro)r
(of.)1110 b Ff(\003)0 3741 y Fx(App)s(endix)408 3711
y Fw(1)445 3741 y Fx(:)31 b(selfadjoin)m(tness)g(of)g
Fr(H)1312 3753 y Fq(B)1388 3741 y Fp(\000)18 b Fr(\027)h
Fp(j)g(\001)f(j)1637 3711 y Fk(\000)p Fw(1)0 3905 y Ft(With)28
b(the)g(notation)g Fr(P)34 b Ft(=)23 b Fp(\000)p Fr(i)14
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b Fr(\013)c Fp(\001)f Fr(P)30 b Ft(+)18 b Fr(m)c(\014)23
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%%Page: 26 26
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(quation)p Fu(,)e(Ph)n(ys.)g(Rev.)f(Letters,)h(85)h(\(2000\),)g(pp.)e
(4020{4023.)35 3867 y(8.)42 b Fb(R.)d(C.)h(Duncan)p Fu(,)d
Fa(Physics)i(in)g(ultr)l(a-str)l(ong)h(magnetic)f(\014elds)p
Fu(.)75 b(Review)38 b(talk)h(giv)n(en)f(at)h(Fifth)132
3941 y(Hun)n(tsville)23 b(Gamma-Ra)n(y)h(Burst)g(Symp)r(osium.)35
4017 y(9.)42 b Fb(I.)20 b(Fushiki,)g(E.)h(H.)f(Gudmundsson,)g(C.)h(J.)e
(Pethick,)i(and)f(J.)g(Yngv)-6 b(ason)p Fu(,)17 b Fa(Matter)j(in)h(a)g
(magnetic)132 4091 y(\014eld)26 b(in)f(the)h(Thomas-Fermi)h(and)f(r)l
(elate)l(d)h(the)l(ories)p Fu(,)d(Ann.)f(Ph)n(ysics,)g(216)i(\(1992\),)
g(pp.)e(29{72.)0 4167 y(10.)42 b Fb(M.)22 b(Griesemer,)g(R.)h(T.)g
(Lewis,)g(and)g(H.)f(Siedentop)p Fu(,)f Fa(A)i(minimax)g(principle)h
(for)f(eigenvalues)g(in)132 4242 y(sp)l(e)l(ctr)l(al)g(gaps:)g(Dir)l
(ac)f(op)l(er)l(ators)i(with)e(Coulomb)i(p)l(otentials)p
Fu(,)c(Do)r(c.)g(Math.,)f(4)h(\(1999\),)h(pp.)f(275{283)132
4316 y(\(electronic\).)0 4392 y(11.)42 b Fb(M.)29 b(Griesemer)f(and)h
(H.)h(Siedentop)p Fu(,)d Fa(A)i(minimax)h(principle)g(for)g(the)f
(eigenvalues)g(in)g(sp)l(e)l(ctr)l(al)132 4466 y(gaps)p
Fu(,)24 b(J.)f(London)i(Math.)e(So)r(c.)h(\(2\),)h(60)f(\(1999\),)h
(pp.)e(490{500.)0 4542 y(12.)42 b Fb(B.)21 b(Janco)o(vici)p
Fu(,)d Fa(R)l(adiative)j(c)l(orr)l(e)l(ction)i(to)e(the)g(gr)l
(ound-state)h(ener)l(gy)f(of)h(an)g(ele)l(ctr)l(on)f(in)h(an)f(intense)
132 4616 y(magnetic)k(\014eld)p Fu(,)f(Ph)n(ys.)f(Rev.,)g(187)i
(\(1969\),)g(pp.)e(2275{2276.)0 4692 y(13.)42 b Fb(M.)32
b(Kla)o(us)g(and)g(R.)h(W)821 4686 y(\177)819 4692 y(ust)p
Fu(,)f Fa(Char)l(acterization)i(and)f(uniqueness)g(of)f(distinguishe)l
(d)i(selfadjoint)132 4766 y(extensions)25 b(of)h(Dir)l(ac)g(op)l(er)l
(ators)p Fu(,)f(Comm.)e(Math.)h(Ph)n(ys.,)f(64)h(\(1978/79\),)i(pp.)d
(171{176.)0 4842 y(14.)42 b Fb(C.)33 b(K)n(ouveliotou,)h(R.)f(C.)h
(Duncan,)e(and)h(C.)h(Thompson)p Fu(,)d Fa(Magnetars)p
Fu(,)h(Scien)n(ti\014c)h(American)132 4916 y(Magazine,)24
b(\(2003\),)h(pp.)f(35{41.)0 4992 y(15.)42 b Fb(E.)24
b(H.)h(Lieb,)g(J.)f(P.)h(Solo)o(vej,)f(and)g(J.)g(Yngv)-6
b(ason)p Fu(,)22 b Fa(Asymptotics)j(of)g(he)l(avy)g(atoms)h(in)f(high)g
(mag-)132 5067 y(netic)20 b(\014elds.)i(I.)g(L)l(owest)g(L)l(andau)h(b)
l(and)g(r)l(e)l(gions)p Fu(,)c(Comm.)g(Pure)g(Appl.)g(Math.,)g(47)h
(\(1994\),)h(pp.)e(513{)132 5141 y(591.)0 5217 y(16.)p
132 5204 191 4 v 233 w(,)33 b Fa(Asymptotics)i(of)h(he)l(avy)f(atoms)h
(in)f(high)h(magnetic)f(\014elds.)g(II.)h(Semiclassic)l(al)g(r)l(e)l
(gions)p Fu(,)132 5291 y(Comm.)23 b(Math.)g(Ph)n(ys.,)g(161)i
(\(1994\),)g(pp.)e(77{124.)p eop end
%%Page: 28 28
TeXDict begin 28 27 bop 0 17 a Fu(28)1325 b(Jean)25 b(Dolb)r(eault,)f
(Maria)e(J.)i(Esteban,)g(Mic)n(hael)g(Loss)0 248 y(17.)42
b Fb(G.)23 b(Nenciu)p Fu(,)f Fa(Self-adjointness)j(and)g(invarianc)l(e)
g(of)g(the)f(essential)h(sp)l(e)l(ctrum)g(for)g(Dir)l(ac)g(op)l(er)l
(ators)132 323 y(de\014ne)l(d)h(as)g(quadr)l(atic)h(forms)p
Fu(,)d(Comm.)f(Math.)h(Ph)n(ys.,)f(48)h(\(1976\),)h(pp.)e(235{247.)0
398 y(18.)p 132 385 191 4 v 233 w(,)i Fa(Existenc)l(e)i(of)g(the)g(sp)l
(ontane)l(ous)j(p)l(air)f(cr)l(e)l(ation)f(in)f(the)h(external)f
(\014eld)h(appr)l(oximation)i(of)132 472 y(Q.E.D.)p Fu(,)22
b(Comm)n(un.)i(Math.)f(Ph)n(ys.,)g(109)i(\(1987\),)g(pp.)e(303{312.)0
547 y(19.)42 b Fb(R.)27 b(F.)g(O'Connell)p Fu(,)e Fa(E\013e)l(ct)i(of)h
(the)f(anomalous)i(magnetic)e(moment)h(of)f(the)g(ele)l(ctr)l(on)h(on)g
(sp)l(onta-)132 622 y(ne)l(ous)22 b(p)l(air)h(pr)l(o)l(duction)h(in)d
(a)i(str)l(ong)f(magnetic)f(\014eld)p Fu(,)f(Ph)n(ys.)f(Rev.)g(Lett.,)h
(21)g(\(1968\),)h(pp.)e(397{398.)0 697 y(20.)42 b Fb(P.)27
b(Pickl)p Fu(,)e Fa(Existenc)l(e)i(of)g(Sp)l(ontane)l(ous)i(Pair)f(Cr)l
(e)l(ation)p Fu(,)e(PhD)f(thesis,)g(Mathematisc)n(hes)h(Institut)132
771 y(der)d(Ludwig-Maximilians-Univ)n(ersit\177)-35 b(at)22
b(M)r(\177)-37 b(unc)n(hen,)25 b(2005.)0 846 y(21.)42
b Fb(B.)35 b(Thaller)p Fu(,)f Fa(The)h(Dir)l(ac)g(e)l(quation)p
Fu(,)f(T)-6 b(exts)35 b(and)f(Monographs)h(in)e(Ph)n(ysics,)h
(Springer-V)-6 b(erlag,)132 921 y(Berlin,)22 b(1992.)p
eop end
%%Trailer

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