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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 48 /zero put
readonly def
/FontBBox{-341 -250 1304 965}readonly def
/UniqueXX 5000788 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMMI5
%!PS-AdobeFont-1.1: CMMI5 1.100
%%CreationDate: 1996 Aug 02 08:21:10

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 59 /comma put
dup 97 /a put
dup 106 /j put
dup 107 /k put
readonly def
/FontBBox{37 -250 1349 750}readonly def
/UniqueXX 5087380 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMMI9
%!PS-AdobeFont-1.1: CMMI9 1.100
%%CreationDate: 1996 Jul 23 07:53:55

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI9 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 18 /theta put
dup 59 /comma put
dup 77 /M put
dup 80 /P put
dup 115 /s put
readonly def
/FontBBox{-29 -250 1075 750}readonly def
/UniqueXX 5087384 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMBX9
%!PS-AdobeFont-1.1: CMBX9 1.0
%%CreationDate: 1991 Aug 20 16:36:25

% 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 (CMBX9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX9 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 46 /period put
dup 48 /zero put
dup 49 /one put
dup 51 /three put
dup 52 /four put
dup 54 /six put
dup 56 /eight put
dup 57 /nine put
dup 70 /F put
dup 97 /a put
dup 98 /b put
dup 103 /g put
dup 105 /i put
readonly def
/FontBBox{-58 -250 1195 750}readonly def
/UniqueXX 5000767 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 91 /uniondisplay put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueXX 5000774 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 78 /N put
dup 82 /R put
dup 90 /Z put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
/UniqueXX 5031982 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 14 /openbullet put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 24 /similar put
dup 26 /propersubset put
dup 33 /arrowright put
dup 39 /similarequal put
dup 49 /infinity put
dup 50 /element put
dup 54 /negationslash put
dup 56 /universal put
dup 59 /emptyset put
dup 76 /L put
dup 81 /Q put
dup 92 /intersection put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
dup 116 /unionsq put
dup 117 /intersectionsq put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueXX 5000820 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMSY7
%!PS-AdobeFont-1.1: CMSY7 1.0
%%CreationDate: 1991 Aug 15 07:21:52

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 3 /asteriskmath put
dup 48 /prime put
dup 50 /element put
dup 54 /negationslash put
dup 81 /Q put
dup 102 /braceleft put
dup 103 /braceright put
dup 110 /backslash put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueXX 5000817 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMTI10
%!PS-AdobeFont-1.1: CMTI10 1.00B
%%CreationDate: 1992 Feb 19 19:56:16

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /ff put
dup 12 /fi put
dup 14 /ffi put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 55 /seven put
dup 58 /colon put
dup 67 /C put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 73 /I put
dup 76 /L put
dup 77 /M put
dup 80 /P put
dup 83 /S put
dup 84 /T put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueXX 5000828 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 13 /gamma put
dup 18 /theta put
dup 27 /sigma put
dup 28 /tau put
dup 59 /comma put
dup 67 /C put
dup 71 /G put
dup 83 /S put
dup 97 /a put
dup 105 /i put
dup 107 /k put
dup 110 /n 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
/UniqueXX 5087382 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMR7
%!PS-AdobeFont-1.1: CMR7 1.0
%%CreationDate: 1991 Aug 20 16:39:21

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 40 /parenleft put
dup 41 /parenright put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 61 /equal put
readonly def
/FontBBox{-27 -250 1122 750}readonly def
/UniqueXX 5000790 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 1 /Delta put
dup 5 /Pi put
dup 6 /Sigma put
dup 8 /Phi put
dup 11 /alpha put
dup 13 /gamma put
dup 15 /epsilon1 put
dup 17 /eta put
dup 18 /theta put
dup 21 /lambda put
dup 22 /mu put
dup 25 /pi put
dup 26 /rho put
dup 27 /sigma put
dup 28 /tau 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 66 /B put
dup 67 /C put
dup 68 /D put
dup 71 /G put
dup 75 /K put
dup 77 /M put
dup 78 /N put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 97 /a put
dup 105 /i put
dup 107 /k put
dup 110 /n put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueXX 5087385 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMR9
%!PS-AdobeFont-1.1: CMR9 1.0
%%CreationDate: 1991 Aug 20 16:39:59

% 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 (CMR9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR9 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 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 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 87 /W put
dup 88 /X put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 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
readonly def
/FontBBox{-39 -250 1036 750}readonly def
/UniqueXX 5000792 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 69 /E put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 110 /n put
dup 111 /o put
dup 115 /s put
dup 116 /t put
dup 120 /x put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueXX 5000769 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: 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 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 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 70 /F put
dup 72 /H put
dup 73 /I put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
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dup 91 /bracketleft put
dup 93 /bracketright put
dup 97 /a put
dup 98 /b put
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dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 123 /endash put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueXX 5000793 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 12 /fi put
dup 44 /comma put
dup 46 /period 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 65 /A put
dup 68 /D put
dup 71 /G put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 78 /N put
dup 79 /O put
dup 82 /R put
dup 84 /T put
dup 86 /V put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 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 122 /z put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
/UniqueXX 5000768 def
currentdict end
currentfile eexec
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4481 y(class)f(of)h(problems)f(along)g(these)g(lines)h(w)n(ould)g(b)r
(e)g(to)g(\014nd)g(curv)n(es)f Fm(\015)2214 4493 y Fi(\003)2279
4481 y Fh(2)e(Q)2429 4493 y Fk(k)q(;a)2555 4481 y Fp(suc)n(h)0
4581 y(that)997 4709 y Fm(\001)p Fp([)p Fm(\015)1132
4721 y Fi(\003)1171 4709 y Fp(])c(=)78 b(sup)1304 4779
y Fk(\015)15 b Fi(2Q)1453 4788 y Fc(k)q(;a)1555 4709
y Fm(\001)p Fp([)p Fm(\015)5 b Fp(])p eop
%%Page: 11 11
11 10 bop 0 17 a Fn(Existence)26 b(of)g(ideal)h(knots)1840
b(11)0 248 y Fp(where)65 419 y Fh(Q)133 431 y Fk(k)q(;a)253
419 y Fp(=)23 b Fh(f)p Fm(\015)k Fh(2)c Fm(C)596 385
y Fl(1)634 419 y Fp(\()p Fm(S;)14 b Fg(R)814 385 y Fl(3)851
419 y Fp(\))23 b Fh(j)h Fm(\015)5 b Fp(\(0\))22 b(=)h(0)p
Fm(;)36 b(\015)28 b Fh(')23 b Fm(k)s(;)37 b(G)p Fp([)p
Fm(\015)5 b Fp(])23 b Fh(\024)f Fm(a;)37 b Fh(j)p Fm(\015)2027
385 y Fi(0)2050 419 y Fp(\()p Fm(s)p Fp(\))p Fh(j)24
b Fp(=)e(1)p Fm(;)37 b(s)23 b Fh(2)g Fm(S)5 b Fh(g)p
Fm(:)0 589 y Fp(Here)27 b Fm(k)j Fp(is)e(a)f(simple)h(curv)n(e)e(of)h
(prescrib)r(ed)g(knot)g(t)n(yp)r(e,)h Fm(G)p Fp([)p Fm(\015)5
b Fp(])28 b(is)f(a)g(giv)n(en)g(functional)0 689 y(\(for)32
b(example,)f(radius)g(of)h(gyration\))f(and)g Fm(a)h
Fp(is)g(a)f(constan)n(t.)g(When)i Fm(G)p Fp([)p Fm(\015)5
b Fp(])32 b(is)f(lo)n(w)n(er)0 789 y(semicon)n(tin)n(uous)22
b(in)h(the)g Fm(C)868 758 y Fl(1)928 789 y Fp(top)r(ology)f(on)g
Fh(Q)p Fp(,)h(w)n(e)f(\014nd)h(that)g(existence)g(of)f(maximiz-)0
888 y(ers)31 b(follo)n(ws)f(b)n(y)i(the)g(same)f(argumen)n(ts)f(used)i
(in)g(the)f(pro)r(of)h(of)f(Theorem)g(1.)g(Th)n(us,)0
988 y(for)25 b(closed)h(curv)n(es,)e(w)n(e)i(obtain)g(existence)f(for)h
(v)-5 b(arious)24 b(optimal)i(pac)n(king)f(problems)0
1087 y(as)i(considered)g(in)g([9].)0 1320 y Fa(A)l(cknow)t(le)l
(dgements)83 b Fn(W)-6 b(e)19 b(wish)i(to)e(thank)g(J.)h(Can)n(tarella)
i(for)e(bringing)g(the)f(reference)h([11])0 1411 y(to)29
b(our)f(atten)n(tion,)h(J.)g(Sulliv)l(an)f(for)h(his)g(v)l(arious)f
(commen)n(ts,)h(and)f(the)g(National)i(Science)0 1503
y(F)-6 b(oundation)25 b(for)i(its)f(supp)r(ort.)0 1789
y Fq(References)38 1973 y Fn(1.)43 b(G.)27 b(Buc)n(k)g(and)g(J.)g
(Simon,)h Fa(Ener)l(gy)i(and)f(length)h(of)e(knots)p
Fn(,)h(in)e(Lectures)g(at)h(Knots)f(96,)140 2065 y(ed.)e(S.)h(Suzuki,)f
(W)-6 b(orld)25 b(Scien)n(ti\014c,)h(Singap)r(ore)h(\(1997\).)38
2156 y(2.)43 b(R.A.)21 b(Litherland,)g(J.)h(Simon,)g(O.)f(Durumeric)g
(and)g(E.)h(Ra)n(wdon,)f Fa(Thickness)k(of)e(knots)p
Fn(,)140 2247 y(T)-6 b(op)r(ology)27 b(and)e(its)h(Applications)g
Fe(91)g Fn(\(1999\))h(233{244.)38 2339 y(3.)43 b(A.)24
b(Stasiak,)i(V.)e(Katritc)n(h)h(and)f(L.H.)h(Kau\013man,)g(Editors,)h
Fa(Ide)l(al)g(Knots)p Fn(,)g(W)-6 b(orld)25 b(Sci-)140
2430 y(en)n(ti\014c)g(Publishing,)h(Singap)r(ore)h(\(1998\).)38
2521 y(4.)43 b(V.)38 b(Katritc)n(h,)g(J.)i(Bednar,)f(D.)f(Mic)n(houd,)h
(R.G.)f(Sc)n(harein,)h(J.)g(Dub)r(o)r(c)n(het)f(and)g(A.)140
2613 y(Stasiak,)26 b Fa(Ge)l(ometry)j(and)f(physics)h(of)e(knots)p
Fn(,)h(Nature)d Fe(384)h Fn(\(1996\))h(142{145.)38 2704
y(5.)43 b(V.)18 b(Katritc)n(h,)h(W.K.)g(Olson,)h(P)-6
b(.)19 b(Pieranski,)h(J.)g(Dub)r(o)r(c)n(het)e(and)g(A.)h(Stasiak,)h
Fa(Pr)l(op)l(erties)140 2795 y(of)27 b(ide)l(al)g(c)l(omp)l(osite)i
(knots)p Fn(,)e(Nature)f Fe(388)g Fn(\(1997\))h(148{151.)38
2887 y(6.)43 b(A.)e(Stasiak,)i(V.)e(Katritc)n(h,)h(J.)g(Bednar,)h(D.)e
(Mic)n(houd)h(and)f(J.)i(Dub)r(o)r(c)n(het,)e Fa(Ele)l(c-)140
2978 y(tr)l(ophor)l(etic)30 b(mobility)d(of)g(DNA)g(knots)p
Fn(,)h(Nature)d Fe(384)h Fn(\(1996\))h(122.)38 3069 y(7.)43
b(O.)26 b(Gonzalez)h(and)f(J.H.)h(Maddo)r(c)n(ks,)g Fa(Glob)l(al)h
(Curvatur)l(e,)i(Thickness,)f(and)f(the)h(Ide)l(al)140
3161 y(Shap)l(e)f(of)g(Knots)p Fn(,)f(The)f(Pro)r(ceedings)i(of)e(the)g
(National)h(Academ)n(y)e(of)h(Sciences,)h(USA)140 3252
y Fe(96)f Fn(\(1999\))h(4769{4773.)38 3343 y(8.)43 b(O.)24
b(Gonzalez,)j(J.H.)e(Maddo)r(c)n(ks,)h(F.)f(Sc)n(h)n(uric)n(h)n(t)e
(and)i(H.)f(v)n(on)g(der)h(Mosel,)i Fa(Glob)l(al)g(cur-)140
3435 y(vatur)l(e)c(and)e(self-c)l(ontact)j(of)d(nonline)l(arly)h
(elastic)f(curves)i(and)f(r)l(o)l(ds)p Fn(,)f(Calculus)f(of)f(V)-6
b(ari-)140 3526 y(ations)26 b Fe(14)g Fn(\(2002\))h(29{68)h(\(DOI)c
(10.1007/s00526010)q(00)q(89,)32 b(April)26 b(2001\).)38
3617 y(9.)43 b(A.)29 b(Maritan,)i(C.)f(Mic)n(heletti,)h(A.)f(T)-6
b(ro)n(v)l(ato)30 b(and)f(J.R.)h(Bana)n(v)l(ar,)h Fa(Optimal)f(shap)l
(es)j(of)140 3708 y(c)l(omp)l(act)c(strings)p Fn(,)e(Nature)e
Fe(406)i Fn(\(2000\))g(287{290.)0 3800 y(10.)43 b(A.)32
b(Stasiak)g(and)g(J.H.)h(Maddo)r(c)n(ks,)g Fa(Best)h(p)l(acking)h(in)e
(pr)l(oteins)i(and)f(DNA)p Fn(,)e(Nature)140 3891 y Fe(406)26
b Fn(\(2000\))h(251{253.)0 3982 y(11.)43 b(J.)31 b(Can)n(tarella,)i(R.)
e(Kusner)f(and)g(J.M.)i(Sulliv)l(an,)f Fa(On)h(the)h(minimum)e(r)l(op)l
(elength)j(of)140 4074 y(knots)29 b(and)f(links)p Fn(,)d(In)n(v)n(en)n
(tiones)g(mathematicae,)i(to)f(app)r(ear.)0 4165 y(12.)43
b(M.)27 b(Spiv)l(ak,)f Fa(A)j(c)l(ompr)l(ehensive)i(intr)l(o)l(duction)
f(to)g(di\013er)l(ential)f(ge)l(ometry,)h(V)-6 b(olume)29
b(I)p Fn(,)140 4256 y(Second)c(Edition,)h(Publish)g(or)g(P)n(erish,)h
(Houston)e(\(1979\).)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF