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cleartomark
%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /alpha put
dup 13 /gamma put
dup 14 /delta put
dup 15 /epsilon1 put
dup 17 /eta put
dup 18 /theta put
dup 21 /lambda put
dup 25 /pi put
dup 26 /rho put
dup 28 /tau put
dup 30 /phi put
dup 32 /psi put
dup 33 /omega put
dup 39 /phi1 put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 62 /greater put
dup 64 /partialdiff put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 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 88 /X put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l 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 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR17
%!PS-AdobeFont-1.1: CMR17 1.0
%%CreationDate: 1991 Aug 20 16:38:24
% 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 (CMR17) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR17 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 45 /hyphen put
dup 65 /A put
dup 68 /D put
dup 78 /N put
dup 83 /S put
dup 84 /T put
dup 94 /circumflex put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 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 117 /u put
dup 121 /y put
readonly def
/FontBBox{-33 -250 945 749}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX6
%!PS-AdobeFont-1.1: CMBX6 1.0
%%CreationDate: 1991 Aug 20 16:35:30
% 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 (CMBX6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX6 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 104 /h put
readonly def
/FontBBox{-49 -250 1367 753}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBXTI10
%!PS-AdobeFont-1.1: CMBXTI10 1.0
%%CreationDate: 1991 Aug 18 17:46:30
% 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 (CMBXTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMBXTI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 45 /hyphen put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 65 /A put
dup 76 /L put
dup 80 /P put
dup 83 /S put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f 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 119 /w put
dup 121 /y put
readonly def
/FontBBox{-29 -250 1274 754}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX8
%!PS-AdobeFont-1.1: CMBX8 1.0
%%CreationDate: 1991 Aug 20 16:36:07
% 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 (CMBX8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 104 /h put
readonly def
/FontBBox{-59 -250 1235 750}readonly def
currentdict end
currentfile eexec
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1A1DD0D7D49A07BAE7A65E3C0E69D869D4E584E6E5FC98461807E726BA673D7B
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cleartomark
%%EndFont 
%%BeginFont: MSBM7
%!PS-AdobeFont-1.1: MSBM7 2.1
%%CreationDate: 1992 Oct 17 08:30:50
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM7) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 82 /R put
dup 90 /Z put
readonly def
/FontBBox{0 -504 2615 1004}readonly def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171
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D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758
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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 58 /period put
dup 110 /n put
dup 112 /p put
readonly def
/FontBBox{37 -250 1349 750}readonly def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633
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0B1F411C5B37D52202C7D02B1EE8B4CA06EB60F7A229D4126481D6A7F38C96E4
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0B8E2E2AFA552F7C3A6D8BC39542D5E4CC802CEBDEFB26720D9BEB85C6BD2D34
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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
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 3 /asteriskmath put
readonly def
/FontBBox{21 -944 1448 791}readonly def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964
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221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBAF552B11EFFB6A16C
F03FB920C15AE724EFDF0CCBF00A838D34440FF9FED532F44036AD22561184C5
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0AA3DF47FB1376A658F06026C2BD785EB1F9D9C858DC9A0316CF7DCE59E64F4E
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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
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 53 /five put
readonly def
/FontBBox{-341 -250 1304 965}readonly 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 14 /delta put
dup 59 /comma put
dup 65 /A put
dup 97 /a put
dup 98 /b put
dup 110 /n put
dup 116 /t put
readonly def
/FontBBox{0 -250 1171 750}readonly 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
readonly def
/FontBBox{-15 -951 1252 782}readonly 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 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 94 /circumflex put
readonly def
/FontBBox{-27 -250 1122 750}readonly 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 30 /phi put
dup 58 /period put
dup 59 /comma put
dup 65 /A put
dup 66 /B put
dup 68 /D put
dup 76 /L put
dup 78 /N 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 108 /l 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 120 /x put
readonly def
/FontBBox{-32 -250 1048 750}readonly 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 8 /Phi put
dup 40 /parenleft put
dup 41 /parenright put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 61 /equal put
dup 65 /A put
dup 68 /D put
dup 76 /L put
dup 80 /P put
dup 94 /circumflex put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 110 /n put
dup 111 /o put
dup 114 /r put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 120 /x put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY6
%!PS-AdobeFont-1.1: CMSY6 1.0
%%CreationDate: 1991 Aug 15 07:21:34
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 3 /asteriskmath put
dup 48 /prime put
readonly def
/FontBBox{-4 -948 1329 786}readonly 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 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 40 /braceleftBigg put
dup 56 /bracelefttp put
dup 58 /braceleftbt put
dup 60 /braceleftmid put
dup 62 /braceex put
dup 80 /summationtext put
dup 83 /uniontext put
dup 84 /intersectiontext put
dup 88 /summationdisplay put
dup 89 /productdisplay put
dup 90 /integraldisplay put
dup 91 /uniondisplay put
dup 92 /intersectiondisplay put
readonly def
/FontBBox{-24 -2960 1454 772}readonly 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 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSS8
%!PS-AdobeFont-1.1: CMSS8 1.0
%%CreationDate: 1991 Aug 20 17:34:12
% 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 (CMSS8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMSS8 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 116 /t put
readonly def
/FontBBox{-65 -250 1062 761}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX12
%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /ff put
dup 12 /fi put
dup 13 /fl put
dup 14 /ffi put
dup 41 /parenright put
dup 43 /plus put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 61 /equal put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 72 /H put
dup 73 /I put
dup 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 94 /circumflex put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 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{-53 -251 1139 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSL12
%!PS-AdobeFont-1.1: CMSL12 1.0
%%CreationDate: 1991 Aug 20 16:40:41
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSL12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -9.46 def
/isFixedPitch false def
end readonly def
/FontName /CMSL12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 45 /hyphen put
dup 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 55 /seven 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 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 87 /W put
dup 88 /X put
dup 89 /Y put
readonly def
/FontBBox{-56 -251 1102 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI12
%!PS-AdobeFont-1.1: CMTI12 1.0
%%CreationDate: 1991 Aug 18 21:06:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /ff put
dup 12 /fi put
dup 13 /fl put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 47 /slash put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 58 /colon put
dup 59 /semicolon put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 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 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{-36 -251 1103 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 17 /eta put
dup 18 /theta put
dup 26 /rho put
dup 28 /tau put
dup 32 /psi put
dup 59 /comma put
dup 69 /E put
dup 78 /N put
dup 84 /T put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 104 /h 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 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 120 /x put
readonly def
/FontBBox{11 -250 1241 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 1 /periodcentered put
dup 2 /multiply put
dup 3 /asteriskmath put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 24 /similar put
dup 33 /arrowright put
dup 48 /prime put
dup 49 /infinity put
dup 50 /element put
dup 68 /D put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
readonly def
/FontBBox{-30 -955 1185 779}readonly 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 42 /notsubseteql put
dup 82 /R put
dup 84 /T put
dup 90 /Z put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSS12
%!PS-AdobeFont-1.1: CMSS12 1.0
%%CreationDate: 1991 Aug 20 17:33:47
% 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 (CMSS12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMSS12 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 116 /t put
readonly def
/FontBBox{-62 -251 978 758}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 59 /semicolon put
dup 61 /equal put
dup 94 /circumflex put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /alpha put
dup 13 /gamma put
dup 14 /delta put
dup 15 /epsilon1 put
dup 17 /eta put
dup 18 /theta put
dup 21 /lambda put
dup 26 /rho put
dup 28 /tau put
dup 30 /phi put
dup 32 /psi put
dup 58 /period put
dup 59 /comma put
dup 61 /slash put
dup 65 /A put
dup 66 /B put
dup 68 /D put
dup 69 /E put
dup 72 /H put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 80 /P put
dup 82 /R put
dup 84 /T put
dup 88 /X put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 102 /f put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 120 /x put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 1 /Delta put
dup 8 /Phi put
dup 10 /Omega put
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 38 /ampersand 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 47 /slash put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 59 /semicolon put
dup 61 /equal put
dup 63 /question put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 87 /W put
dup 89 /Y put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 94 /circumflex put
dup 95 /dotaccent put
dup 96 /quoteleft put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 126 /tilde put
dup 127 /dieresis put
readonly def
/FontBBox{-34 -251 988 750}readonly def
currentdict end
currentfile eexec
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rf /Fd 187[92 68[{}1 119.552 /CMSY10 rf /Fe 151[38 104[{}1
49.8132 /CMBX6 rf /Ff 190[88 4[91 60[{}2 119.552 /CMMI12
rf /Fg 134[56 1[76 53 62 38 49 50 1[59 59 65 94 30 2[35
2[40 53 59 53 53 59 13[65 2[78 3[70 10[86 13[59 59 59
3[41 45[{}27 99.6264 /CMBXTI10 rf /Fh 151[45 52[41 41
41 41 48[{}5 66.4176 /CMBX8 rf /Fi 165[44 7[48 82[{}2
66.4176 /MSBM7 rf /Fj 143[31 1[37 51[19 58[{}3 41.511
/CMMI5 rf /Fk 252[31 2[45{}2 41.511 /CMSY5 rf /Fl 202[28
1[28 28 28 28 48[{}5 41.511 /CMR5 rf /Fm 139[25 5[41
11[29 36 31[50 5[20 44[30 14[{}7 58.1154 /CMMI7 rf /Fn
177[68 9[64 42[65 24[65{}4 83.022 /CMSY10 rf /Fo 252[34
2[52{}2 58.1154 /CMSY7 rf /Fp 161[33 42[33 33 33 33 4[51
1[26 26 40[{}8 58.1154 /CMR7 rf /Fq 135[47 2[48 30 39
37 1[42 40 2[25 2[29 48 40 1[39 43 36 36 44 18[67 1[57
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42 42 42 42 42 42 42 42 42 1[23 28 3[32 32 31[60 8[{}37
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59 3[33 2[58 58 4[42 42 46 46{}32 99.6264 /CMEX10 rf
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35 46 44 1[49 47 58 1[29 51 40 33 56 47 48 45 51 42 41
51 8[81 1[57 1[57 60 74 1[63 75 78 94 66 83 54 43 81
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49 3[43 51 51 70 51 54 38 38 38 51 54 49 54 81 27 51
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129 116 71 106 129 103 129 113 11[179 162 129 2[159 1[182
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(de\014ned)h(b)s(elo)m(w.)p Black Black eop end
%%Page: 6 24
TeXDict begin 6 23 bop Black 591 266 a FJ(6)206 b Fx(CHAPTER)34
b(1.)76 b(D)m(YNAMICAL)34 b(SYSTEMS)g(AND)f(ST)-8 b(ABILITY)34
b(THEOR)-8 b(Y)p Black Black Black Black 1601 434 1300
4 v 1601 1578 4 1145 v 1660 1553 a @beginspecial 167
@llx 399 @lly 371 @urx 588 @ury 1417 @rwi @setspecial
%%BeginDocument: eps/hdorff.eps
%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Adobe Illustrator(TM) for Windows, version 4.0
%%For: (Russel Stonier) (Central Queensland University)
%%Title: (hdorff.eps)
%%CreationDate: (5/19/99) (11:39 PM)
%%BoundingBox: 167 399 371 588
%%DocumentProcessColors: Black
%%DocumentFonts: TimesNewRomanPS-ItalicMT
%%+ TimesNewRomanPSMT
%%DocumentNeededFonts: TimesNewRomanPS-ItalicMT
%%+ TimesNewRomanPSMT
%%DocumentSuppliedResources: procset Adobe_packedarray 2.0 0
%%+ procset Adobe_cshow 1.1 0
%%+ procset Adobe_customcolor 1.0 0
%%+ procset Adobe_typography_AI3 1.0 1
%%+ procset Adobe_pattern_AI3 1.0 0
%%+ procset Adobe_Illustrator_AI3 1.0 1
%AI3_ColorUsage: Black&White
%AI3_IncludePlacedImages
%%CMYKCustomColor: 1 0 0.55 0 (Aqua)
%%+ 1 0.5 0 0 (Blue)
%%+ 0.5 0.4 0.3 0 (Blue Gray)
%%+ 0.8 0.05 0 0 (Blue Sky)
%%+ 0.5 0.85 1 0 (Brown)
%%+ 1 0.9 0.1 0 (Dark Blue)
%%+ 1 0.55 1 0 (Forest Green)
%%+ 0.05 0.2 0.95 0 (Gold)
%%+ 0.75 0.05 1 0 (Grass Green)
%%+ 0 0.45 1 0 (Orange)
%%+ 0.15 1 1 0 (Red)
%%+ 0.45 0.9 0 0 (Violet)
%AI3_TemplateBox: 306 396 306 396
%AI3_TileBox: 13 13 599 780
%AI3_DocumentPreview: Header
%%PageOrigin:13 13
%%AI3_PaperRect:-13 780 599 -12
%%AI3_Margin:13 -13 -13 12
%AI7_GridSettings: 72 8 72 8 1 0 0.8 0.8 0.8 0.9 0.9 0.9
%%EndComments
%%BeginProlog
%%BeginResource: procset Adobe_packedarray 2.0 0
%%Title: (Packed Array Operators)
%%Version: 2.0 0
%%CreationDate: (8/2/90) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
userdict /Adobe_packedarray 5 dict dup begin put
/initialize
{
/packedarray where
	{
	pop
	}
	{
	Adobe_packedarray begin
	Adobe_packedarray
		{
		dup xcheck
			{
			bind
			} if
		userdict 3 1 roll put
		} forall
 end
	} ifelse
} def
/terminate
{
} def
/packedarray
{
array astore readonly
} def
/setpacking
{
pop
} def
/currentpacking
{
false
} def
currentdict readonly pop end
%%EndResource
Adobe_packedarray /initialize get exec
%%BeginResource: procset Adobe_cshow 1.1 0
%%Title: (cshow Operator)
%%Version: 1.1 0
%%CreationDate: (1/23/89) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_cshow 3 dict dup begin put
/initialize
{
/cshow where
	{
	pop
	}
	{
	userdict /Adobe_cshow_vars 1 dict dup begin put
	/_cshow
		{} def
	Adobe_cshow begin
	Adobe_cshow
		{
		dup xcheck
			{
			bind
			} if
		userdict 3 1 roll put
		} forall
 end
 end
	} ifelse
} def
/terminate
{
} def
/cshow
{
exch
Adobe_cshow_vars
	exch /_cshow
	exch put
	{
	0 0 Adobe_cshow_vars /_cshow get exec
	} forall
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_customcolor 1.0 0
%%Title: (Custom Color Operators)
%%Version: 1.0 0
%%CreationDate: (5/9/88) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_customcolor 5 dict dup begin put
/initialize
{
/setcustomcolor where
	{
	pop
	}
	{
	Adobe_customcolor begin
	Adobe_customcolor
		{
		dup xcheck
			{
			bind
			} if
		pop pop
		} forall
 end
	Adobe_customcolor begin
	} ifelse
} def
/terminate
{
currentdict Adobe_customcolor eq
	{
 end
	} if
} def
/findcmykcustomcolor
{
5 packedarray
}  def
/setcustomcolor
{
exch
aload pop pop
4
	{
	4 index mul 4 1 roll
	} repeat
5 -1 roll pop
setcmykcolor
} def
/setoverprint
{
pop
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_typography_AI3 2.0 0
%%Title: (Typography Operators)
%%Version: 2.0 0
%%CreationDate:(5/31/90) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_typography_AI3 48 dict dup begin put
/initialize
{
/TZ
 where
	{
	pop
	}
	{
	Adobe_typography_AI3 begin
	Adobe_typography_AI3
		{
		dup xcheck
			{
			bind
			} if
		pop pop
		} forall
 end
	Adobe_typography_AI3 begin
	} ifelse
} def
/terminate
{
currentdict Adobe_typography_AI3 eq
	{
 end
	} if
} def
/modifyEncoding
{
	/_tempEncode exch ddef
	
	/_pntr 0 ddef
	
	{
		counttomark -1 roll
		dup type dup /marktype eq 		
		{
			pop pop exit
		}
		{
			/nametype eq
			{
				_tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll
				put
			}
			{
				/_pntr exch ddef					
			}
			ifelse
		}
		ifelse
	}
	loop	
	
	_tempEncode
}
def
/TE
{
	StandardEncoding 256 array copy modifyEncoding 
	/_nativeEncoding exch def
} def
%
/TZ		
{
	dup type /arraytype eq {/_wv exch def} {/_wv 0 def} ifelse 
	/_useNativeEncoding exch def
	pop pop
	
	findfont _wv type /arraytype eq {_wv makeblendedfont} if dup length 2 add dict
	
 begin
	
		mark exch
		{
			1 index /FID ne { def } if cleartomark mark
		}
		forall
		pop
		
		/FontName exch def
		
		counttomark 0 eq
		{
			1 _useNativeEncoding eq
			{
				/Encoding _nativeEncoding def
			}
			if
			cleartomark
		}
		{	
			/Encoding load 256 array copy 
			modifyEncoding /Encoding exch def
		}
		ifelse		
		FontName currentdict
 end
	
	definefont pop
}
def
/tr
{
_ax _ay 3 2 roll
} def
/trj
{
_cx _cy _sp _ax _ay 6 5 roll
} def
/a0
{
/Tx
	{
	dup 
	currentpoint 3 2 roll
	tr _psf
	newpath moveto
	tr _ctm _pss
	} ddef
/Tj
	{
	dup
	currentpoint 3 2 roll
	trj _pjsf
	newpath moveto
	trj _ctm _pjss
	} ddef
	
} def
/a1
{
/Tx
	{
	dup currentpoint 4 2 roll gsave
	dup currentpoint 3 2 roll
	tr _psf
	newpath moveto
	tr _ctm _pss
	grestore 3 1 roll moveto tr sp
	} ddef
/Tj
	{
	dup currentpoint 4 2 roll gsave
	dup currentpoint 3 2 roll
	trj _pjsf
	newpath moveto
	trj _ctm _pjss
	grestore 3 1 roll moveto tr sp
	} ddef
	
} def
/e0
{
/Tx
	{
	tr _psf
	} ddef
/Tj
	{
	trj _pjsf
	} ddef
} def
/e1
{
/Tx
	{
	dup currentpoint 4 2 roll gsave 
	tr _psf  
	grestore 3 1 roll moveto tr sp 
	} ddef
/Tj
	{
	dup currentpoint 4 2 roll gsave 
	trj _pjsf
	grestore 3 1 roll moveto tr sp 
	} ddef
} def
/i0
{
/Tx
	{
	tr sp
	} ddef
/Tj
	{
	trj jsp
	} ddef
} def
/i1
{
W N
} def
/o0
{
/Tx
	{
	tr sw rmoveto
	} ddef
/Tj
	{
	trj swj rmoveto
	} ddef
} def
/r0
{
/Tx
	{
	tr _ctm _pss
	} ddef
/Tj
	{
	trj _ctm _pjss
	} ddef
} def
/r1
{
/Tx
	{
	dup currentpoint 4 2 roll currentpoint gsave newpath moveto
	tr _ctm _pss 
	grestore 3 1 roll moveto tr sp 
	} ddef
/Tj
	{
	dup currentpoint 4 2 roll currentpoint gsave newpath moveto
	trj _ctm _pjss
	grestore 3 1 roll moveto tr sp 
	} ddef
} def
/To
{
	pop _ctm currentmatrix pop
} def
/TO
{
	iTe _ctm setmatrix newpath
} def
/Tp
{
	pop _tm astore pop _ctm setmatrix 
	_tDict begin /W {} def /h {} def
} def
/TP
{
 end
	iTm 0 0 moveto
} def
/Tr
{
	_render 3 le {currentpoint newpath moveto} if
	dup 8 eq {pop 0} {dup 9 eq {pop 1} if} ifelse
	dup /_render exch ddef
	_renderStart exch get load exec
} def
/iTm
{
_ctm setmatrix _tm concat 0 _rise translate _hs 1 scale
} def
/Tm
{
_tm astore pop iTm 0 0 moveto
} def
/Td
{
_mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto
} def
/iTe
{
	_render -1 eq {} {_renderEnd _render get dup null ne {load exec} {pop} ifelse} ifelse
	/_render -1 ddef
} def
/Ta
{
pop
} def
/Tf
{
dup 1000 div /_fScl exch ddef
exch findfont exch scalefont setfont
} def
/Tl
{
pop
0 exch _leading astore pop
} def
/Tt
{
pop
} def
/TW
{
3 npop
} def
/Tw
{
/_cx exch ddef
} def
/TC
{
3 npop
} def
/Tc
{
/_ax exch ddef
} def
/Ts
{
/_rise exch ddef
currentpoint
iTm
moveto
} def
/Ti
{
3 npop
} def
/Tz
{
100 div /_hs exch ddef
iTm
} def
/TA
{
pop
} def
/Tq
{
pop
} def
/Th
{
pop pop pop pop pop
} def
/TX {pop} def
%/Tx
%/Tj
/Tk
{
exch pop _fScl mul neg 0 rmoveto
} def
/TK
{
2 npop
} def
/T*
{
_leading aload pop neg Td
} def
/T*-
{
_leading aload pop Td
} def
/T-
{
_hyphen Tx
} def
/T+
{} def
/TR
{
_ctm currentmatrix pop 	
_tm astore pop 
iTm 0 0 moveto 
} def
/TS
{
0 eq {Tx} {Tj} ifelse
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_pattern_AI3 1.1 0
%%Title: (Adobe Illustrator (R) Version 3.0 Pattern Operators)
%%Version: 1.1 0
%%CreationDate: (7/21/89) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_pattern_AI3 16 dict dup begin put
/initialize
{
/definepattern where
	{
	pop
	}
	{
	Adobe_pattern_AI3 begin
	Adobe_pattern_AI3
		{
		dup xcheck
			{
			bind
			} if
		pop pop
		} forall
	mark
	cachestatus 7 1 roll pop pop pop pop exch pop exch
		{
		{
		10000 add
		dup 2 index gt
			{
			exit
			} if
		dup setcachelimit
		} loop
		} stopped
	cleartomark
	} ifelse
} def
/terminate
{
currentdict Adobe_pattern_AI3 eq
	{
 end
	} if
} def
errordict
/nocurrentpoint
{
pop
stop
} put
errordict
/invalidaccess
{
pop
stop
} put
/patternencoding
256 array def
0 1 255
{
patternencoding exch ( ) 2 copy exch 0 exch put cvn put
} for
/definepattern
{
17 dict begin
/uniform exch def
/cache exch def
/key exch def
/procarray exch def
/mtx exch matrix invertmatrix def
/height exch def
/width exch def
/ctm matrix currentmatrix def
/ptm matrix def
/str 32 string def
/slice 9 dict def
slice /s 1 put
slice /q 256 procarray length div sqrt floor cvi put
slice /b 0 put
/FontBBox [0 0 0 0] def
/FontMatrix mtx matrix copy def
/Encoding patternencoding def
/FontType 3 def
/BuildChar
	{
	exch
 begin
	/setstrokeadjust where {pop true setstrokeadjust} if
	slice begin
	dup q dup mul mod s idiv /i exch def
	dup q dup mul mod s mod /j exch def
	q dup mul idiv procarray exch get
	/xl j width s div mul def
	/xg j 1 add width s div mul def
	/yl i height s div mul def
	/yg i 1 add height s div mul def
	uniform
		{
		1 1
		}
		{
		width 0 dtransform
		dup mul exch dup mul add sqrt dup 1 add exch div
		0 height dtransform
		dup mul exch dup mul add sqrt dup 1 add exch div
		} ifelse
	width 0 cache
		{
		xl 4 index mul yl 4 index mul xg 6 index mul yg 6 index mul
		setcachedevice
		}
		{
		setcharwidth
		} ifelse
	gsave
	scale
	newpath
	xl yl moveto
	xg yl lineto
	xg yg lineto
	xl yg lineto
	closepath
	clip
	newpath
 end
 end
	exec
	grestore
	} def
key currentdict definefont
end
} def
/patterncachesize
{
gsave
newpath
0 0 moveto
width 0 lineto
width height lineto
0 height lineto
closepath
patternmatrix setmatrix
pathbbox
exch ceiling 4 -1 roll floor sub 3 1 roll
ceiling exch floor sub
mul 1 add
grestore
} def
/patterncachelimit
{
cachestatus 7 1 roll 6 npop 8 mul
} def
/patternpath
{
exch dup begin setfont
ctm setmatrix
concat
slice exch /b exch slice /q get dup mul mul put
FontMatrix concat
uniform
	{
	width 0 dtransform round width div exch round width div exch
	0 height dtransform round height div exch height div exch
	0 0 transform round exch round exch
	ptm astore setmatrix
	}
	{
	ptm currentmatrix pop
	} ifelse
{currentpoint} stopped not
	{
	2 npop
	pathbbox
	true
	4 index 3 index eq
	4 index 3 index eq
	and
		{
		pop false
			{
			{2 npop}
			{3 npop true}
			{7 npop true}
			{pop true}
			pathforall
			} stopped
			{
			5 npop true
			} if
		} if
		{
		height div ceiling height mul 4 1 roll
		width div ceiling width mul 4 1 roll
		height div floor height mul 4 1 roll
		width div floor width mul 4 1 roll
		2 index sub height div ceiling cvi exch
		3 index sub width div ceiling cvi exch
		4 2 roll moveto
		FontMatrix mtx invertmatrix
		dup dup 4 get exch 5 get rmoveto
		ptm ptm concatmatrix pop
		slice /s
		patterncachesize patterncachelimit div ceiling sqrt ceiling cvi
		dup slice /q get gt
			{
			pop slice /q get
			} if
		put
		0 1 slice /s get dup mul 1 sub
			{
			slice /b get add
			gsave
			0 1 str length 1 sub
				{
				str exch 2 index put
				} for
			pop
			dup
				{
				gsave
				ptm setmatrix
				1 index str length idiv {str show} repeat
				1 index str length mod str exch 0 exch getinterval show
				grestore
				0 height rmoveto
				} repeat
			grestore
			} for
		2 npop
		}
		{
		4 npop
		} ifelse
	} if
end
} def
/patternclip
{
clip
} def
/patternstrokepath
{
strokepath
} def
/patternmatrix
matrix def
/patternfill
{
dup type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get
	} if
gsave
patternclip
Adobe_pattern_AI3 /patternpath get exec
grestore
newpath
} def
/patternstroke
{
dup type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get
	} if
gsave
patternstrokepath
true
	{
		{
			{
			newpath 
			moveto
			}
			{
			lineto
			}
			{
			curveto
			}
			{
			closepath
			3 copy
			Adobe_pattern_AI3 /patternfill get exec
			} pathforall
		3 npop
		} stopped
			{
			5 npop
			patternclip
			Adobe_pattern_AI3 /patternfill get exec
			} if
	}
	{
	patternclip
	Adobe_pattern_AI3 /patternfill get exec
	} ifelse
grestore
newpath
} def
/patternashow
{
3 index type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get 4 1 roll
	} if
	{
	2 npop (0) exch
	2 copy 0 exch put pop
	gsave
	false charpath currentpoint
	6 index 6 index 6 index
	Adobe_pattern_AI3 /patternfill get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
5 npop
} def
/patternawidthshow
{
6 index type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get 7 1 roll
	} if
	{
	2 npop (0) exch
	2 copy 0 exch put 
	gsave
	_sp eq {5 index 5 index rmoveto} if
	false charpath currentpoint
	9 index 9 index 9 index
	Adobe_pattern_AI3 /patternfill get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
8 npop
} def
/patternashowstroke
{
4 index type /dicttype eq
	{
	patternmatrix /patternmatrix get 5 1 roll
	} if
4 1 roll
	{
	2 npop (0) exch
	2 copy 0 exch put pop
	gsave
	false charpath
	currentpoint
	4 index setmatrix
	7 index 7 index 7 index
	Adobe_pattern_AI3 /patternstroke get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
6 npop
} def
/patternawidthshowstroke
{
7 index type /dicttype eq
	{
	patternmatrix /patternmatrix get 8 1 roll
	} if
7 1 roll
	{
	2 npop (0) exch
	2 copy 0 exch put
	gsave
	_sp eq {5 index 5 index rmoveto} if
	false charpath currentpoint
	7 index setmatrix
	10 index 10 index 10 index
	Adobe_pattern_AI3 /patternstroke get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
9 npop
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_Illustrator_AI3 1.1 0
%%Title: (Adobe Illustrator (R) Version 3.0 Full Prolog)
%%Version: 1.1 0
%%CreationDate: (3/7/1994) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_Illustrator_AI3 71 dict dup begin put
/initialize
{
userdict /Adobe_Illustrator_AI3_vars 67 dict dup begin put
/_lp /none def
/_pf {} def
/_ps {} def
/_psf {} def
/_pss {} def
/_pjsf {} def
/_pjss {} def
/_pola 0 def
/_doClip 0 def
/cf	currentflat def
/_tm matrix def
/_renderStart [/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0] def 
/_renderEnd [null null null null /i1 /i1 /i1 /i1] def
/_render -1 def
/_rise 0 def
/_ax 0 def
/_ay 0 def
/_cx 0 def
/_cy 0 def
/_leading [0 0] def
/_ctm matrix def
/_mtx matrix def
/_sp 16#020 def
/_hyphen (-) def
/_fScl 0 def
/_cnt 0 def
/_hs 1 def
/_nativeEncoding 0 def
/_useNativeEncoding 0 def
/_tempEncode 0 def
/_pntr 0 def
/_tDict 2 dict def
/_wv 0 def
/Tx {} def
/Tj {} def
/CRender {} def
/_AI3_savepage {} def
/_gf null def
/_cf 4 array def
/_if null def
/_of false def
/_fc {} def
/_gs null def
/_cs 4 array def
/_is null def
/_os false def
/_sc {} def
/_pd 1 dict def
/_ed 15 dict def
/_pm matrix def
/_fm null def
/_fd null def
/_fdd null def
/_sm null def
/_sd null def
/_sdd null def
/_i null def
Adobe_Illustrator_AI3 begin
Adobe_Illustrator_AI3 dup /nc get begin
	{
	dup xcheck
		{
		bind
		} if
	pop pop
	} forall
end
end
end
Adobe_Illustrator_AI3 begin
Adobe_Illustrator_AI3_vars begin
newpath
} def
/terminate
{
end
end
} def
/_
null def
/ddef
{
Adobe_Illustrator_AI3_vars 3 1 roll put
} def
/xput
{
dup load dup length exch maxlength eq
	{
	dup dup load dup
	length 2 mul dict copy def
	} if
load begin def end
} def
/npop
{
	{
	pop
	} repeat
} def
/sw
{
dup length exch stringwidth
exch 5 -1 roll 3 index mul add
4 1 roll 3 1 roll mul add
} def
/swj
{
dup 4 1 roll
dup length exch stringwidth 
exch 5 -1 roll 3 index mul add
4 1 roll 3 1 roll mul add 
6 2 roll /_cnt 0 ddef
{1 index eq {/_cnt _cnt 1 add ddef} if} forall pop
exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop
} def
/ss
{
4 1 roll
	{
	2 npop 
	(0) exch 2 copy 0 exch put pop
	gsave
	false charpath currentpoint
	4 index setmatrix
	stroke
	grestore
	moveto
	2 copy rmoveto
	} exch cshow
3 npop
} def
/jss
{
4 1 roll
	{
	2 npop 
	(0) exch 2 copy 0 exch put 
	gsave
	_sp eq 
		{
		exch 6 index 6 index 6 index 5 -1 roll widthshow  
		currentpoint
		}
		{
		false charpath currentpoint
		4 index setmatrix stroke
		}ifelse
	grestore
	moveto
	2 copy rmoveto
	} exch cshow
6 npop
} def
/sp
{
	{
	2 npop (0) exch
	2 copy 0 exch put pop
	false charpath
	2 copy rmoveto
	} exch cshow
2 npop
} def
/jsp
{
	{
	2 npop 
	(0) exch 2 copy 0 exch put 
	_sp eq 
		{
		exch 5 index 5 index 5 index 5 -1 roll widthshow  
		}
		{
		false charpath
		}ifelse
	2 copy rmoveto
	} exch cshow
5 npop
} def
/pl
{
transform
0.25 sub round 0.25 add exch
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F
%%PageTrailer
gsave annotatepage grestore showpage
%%Trailer
Adobe_Illustrator_AI3 /terminate get exec
Adobe_pattern_AI3 /terminate get exec
Adobe_typography_AI3 /terminate get exec
Adobe_customcolor /terminate get exec
Adobe_cshow /terminate get exec
Adobe_packedarray /terminate get exec
%%EOF

%%EndDocument
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1775 a FJ(Figure)33 b(1.1:)43 b(Hausdor\013)33 b(Semi-Metric)p
Black Black Black 591 2182 a FI(De\014nition)k(1.2.2)f
FJ(\(Stable)d(Set\))p FI(.)p Black 41 w Fy(A)i(nonempty)e(c)-5
b(omp)g(act)33 b(subset)i FH(A)3251 2197 y FE(0)3318
2182 y FG(\032)28 b FH(E)6 b Fy(,)34 b(\()p FH(E)40 b
Fy(op)-5 b(en)591 2320 y(and)40 b FH(E)46 b FG(\032)40
b FC(R)1093 2284 y FF(d)1133 2320 y Fy(\))i(is)f FI(stable)h
Fy(under)f(the)g(semi-gr)-5 b(oup)40 b(mapping)g FG(f)p
FH(S)3119 2335 y FF(t)3188 2320 y FJ(:)g FH(t)g FG(2)g
FC(R)3508 2284 y FE(+)3567 2320 y FG(g)p Fy(,)j(if)e(for)591
2459 y(every)34 b FH(\017)28 b(>)g FJ(0)35 b Fy(ther)-5
b(e)34 b(exists)h(a)g FH(\016)t FJ(\()p FH(\017)p FJ(\))27
b FH(>)h FJ(0)35 b Fy(such)f(that)1478 2718 y FH(H)1567
2677 y FB(\003)1606 2718 y FJ(\()p FH(S)1704 2733 y FF(t)1733
2718 y FJ(\()p FG(N)1853 2733 y FF(\016)1891 2718 y FJ(\()p
FH(A)2002 2733 y FE(0)2041 2718 y FJ(\)\))p FH(;)17 b(A)2234
2733 y FE(0)2273 2718 y FJ(\))28 b FH(<)g(\017;)216 b
FG(8)p FH(t)28 b FG(\025)g FJ(0)p FH(:)687 b FJ(\(1.5\))p
Black 591 2977 a FI(De\014nition)50 b(1.2.3)g FJ(\(Asymptotically)c
(Stable)d(Set\))p FI(.)p Black 48 w Fy(A)i(nonempty)f(c)-5
b(omp)g(act)44 b(subset)591 3115 y FH(A)664 3130 y FE(0)751
3115 y FG(\032)49 b FH(E)6 b Fy(,)49 b(\()p FH(E)j Fy(op)-5
b(en)45 b(and)g FH(E)55 b FG(\032)48 b FC(R)1962 3079
y FF(d)2003 3115 y Fy(\))e(is)f FI(asymptotically)53
b(stable)47 b Fy(if)f(it)g(is)g FJ(stable)p Fy(,)591
3254 y(and)e(in)h(addition,)i(if)e(ther)-5 b(e)45 b(exists)g(a)g
FH(\016)51 b(>)c FJ(0)p Fy(,)h(so)c(that)i(for)f(e)-5
b(ach)45 b FH(\017)i(>)g FJ(0)p Fy(,)g(ther)-5 b(e)46
b(is)f(a)591 3392 y FH(T)c FJ(=)27 b FH(T)14 b FJ(\()p
FH(\017)p FJ(\))28 b FH(>)g FJ(0)34 b Fy(such)h(that)g(for)g(every)g
FH(x)2083 3407 y FE(0)2150 3392 y FG(2)28 b(N)2326 3407
y FF(\016)2364 3392 y FJ(\()p FH(A)2475 3407 y FE(0)2514
3392 y FJ(\))1507 3652 y FH(H)1596 3610 y FB(\003)1652
3652 y FJ(\()p FH(S)1750 3667 y FF(t)1780 3652 y FJ(\()p
FH(x)1873 3667 y FE(0)1912 3652 y FJ(\))p FH(;)17 b(A)2067
3667 y FE(0)2107 3652 y FJ(\))27 b FH(<)h(\017;)216 b
FG(8)p FH(t)28 b FG(\025)g FH(T)14 b FJ(\()p FH(\017)p
FJ(\))p FH(:)717 b FJ(\(1.6\))591 3982 y(If)35 b(an)m(y)h
FH(\017)p FJ(-neigh)m(b)s(ourho)s(o)s(d)f(of)f FH(A)1805
3997 y FE(0)1880 3982 y FJ(can)h(b)s(e)h(reac)m(hed)g(in)g(\014nite)g
(time)g(b)m(y)g(ev)m(ery)h(b)s(ounded)591 4120 y(subset)d(of)e
FC(R)1073 4084 y FF(d)1113 4120 y FJ(,)h(then)g FH(A)1468
4135 y FE(0)1540 4120 y FJ(is)g(said)g(to)f(b)s(e)h FI(globally)39
b(asymptotically)g(stable)p FJ(.)591 4329 y(A)32 b(nonempt)m(y)i(set)g
FH(B)j FJ(is)c(said)h(to)e(b)s(e)h FI(p)s(ositiv)m(ely)38
b(in)m(v)-6 b(arian)m(t)33 b FJ(if)1754 4589 y FH(S)1814
4604 y FF(t)1844 4589 y FJ(\()p FH(B)5 b FJ(\))28 b FG(\022)g
FH(B)5 b(;)212 b FG(8)p FH(t)28 b(>)g FJ(0)p FH(:)591
4848 y FJ(Note)42 b(that)g(stable)h(and)g(asymptotically)h(stable)f
(sets)h(are)e(p)s(ositiv)m(ely)i(in)m(v)-5 b(arian)m(t)43
b(\(see)591 4986 y([34)o(]\).)h(Additionally)-8 b(,)33
b(a)g(non-empt)m(y)g(set)h FH(B)j FJ(is)c(said)g(to)g(b)s(e)f
FI(S-in)m(v)-6 b(arian)m(t)34 b FJ(if)1755 5246 y FH(S)1815
5261 y FF(t)1845 5246 y FJ(\()p FH(B)5 b FJ(\))28 b(=)f
FH(B)5 b(;)212 b FG(8)p FH(t)28 b(>)g FJ(0)p FH(:)p Black
Black eop end
%%Page: 7 25
TeXDict begin 7 24 bop Black 591 266 a Fx(1.2.)75 b(A)m(UTONOMOUS)34
b(D)m(YNAMICAL)g(SYSTEMS)1168 b FJ(7)p Black Black Black
Black 1129 434 2245 4 v 1129 1783 4 1349 v 1188 1758
a @beginspecial 98 @llx 420 @lly 373 @urx 588 @ury 2551
@rwi @setspecial
%%BeginDocument: eps/stab.eps
%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Adobe Illustrator(TM) 7.0
%%For: (Russel Stonier) (Central Queensland University)
%%Title: (stab2.eps)
%%CreationDate: (5/21/99) (2:05 PM)
%%BoundingBox: 98 420 373 588
%%HiResBoundingBox: 98.6667 420.2947 372.2856 587.0803
%%DocumentProcessColors: Black
%%DocumentFonts: SymbolMT
%%+ TimesNewRomanPSMT
%%DocumentSuppliedFonts: SymbolMT
%%+ TimesNewRomanPSMT
%%DocumentSuppliedResources: procset Adobe_level2_AI5 1.2 0
%%+ procset Adobe_typography_AI5 1.0 1
%%+ procset Adobe_ColorImage_AI6 1.1 0
%%+ procset Adobe_Illustrator_AI5 1.2 0
%%+ procset Adobe_cshow 2.0 8
%AI5_FileFormat 3.0
%AI3_ColorUsage: Black&White
%AI3_IncludePlacedImages
%AI7_ImageSettings: 1
%%CMYKCustomColor: 1 0 0.55 0 (Aqua)
%%+ 1 0.5 0 0 (Blue)
%%+ 0.5 0.4 0.3 0 (Blue Gray)
%%+ 0.8 0.05 0 0 (Blue Sky)
%%+ 0.5 0.85 1 0 (Brown)
%%+ 1 0.9 0.1 0 (Dark Blue)
%%+ 1 0.55 1 0 (Forest Green)
%%+ 0.05 0.2 0.95 0 (Gold)
%%+ 0.75 0.05 1 0 (Grass Green)
%%+ 0 0.45 1 0 (Orange)
%%+ 0.15 1 1 0 (Red)
%%+ 0.45 0.9 0 0 (Violet)
%%AI6_ColorSeparationSet: 1 1 (AI6 Default Color Separation Set) 
%%+ Options: 1 16 0 1 0 1 1 1 0 1 1 1 1 18 0 0 0 0 0 0 0 0 -1 -1
%%+ PPD: 1 21 0 0 60 45 2 2 1 0 0 1 0 0 0 0 0 0 0 0 0 0 () 
%AI3_TemplateBox: 306 396 306 396
%AI3_TileBox: 13 13 599 780
%AI3_DocumentPreview: Header
%AI5_ArtSize: 612 792
%AI5_RulerUnits: 2
%AI5_ArtFlags: 1 0 0 1 0 0 1 1 0
%AI5_TargetResolution: 800
%AI5_NumLayers: 1
%AI5_OpenToView: -30 652 1.5 1008 693 18 0 1 8 67 1 0
%AI5_OpenViewLayers: 7
%%PageOrigin:13 13
%%AI3_PaperRect:-13 780 599 -12
%%AI3_Margin:13 -13 -13 12
%AI7_GridSettings: 72 8 72 8 1 0 0.8 0.8 0.8 0.9 0.9 0.9
%%EndComments
%%BeginProlog
%%BeginResource: procset Adobe_level2_AI5 1.2 0
%%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation)
%%Version: 1.2 0
%%CreationDate: (04/10/93) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
userdict /Adobe_level2_AI5 25 dict dup begin
	put
	/packedarray where not
	{
		userdict begin
		/packedarray
		{
			array astore readonly
		} bind def
		/setpacking /pop load def
		/currentpacking false def
	 end
		0
	} if
	pop
	userdict /defaultpacking currentpacking put true setpacking
	/initialize
	{
		Adobe_level2_AI5 begin
	} bind def
	/terminate
	{
		currentdict Adobe_level2_AI5 eq
		{
		 end
		} if
	} bind def
	mark
	/setcustomcolor where not
	{
		/findcmykcustomcolor
		{
			0
			6 packedarray
		} bind def
		/findrgbcustomcolor
		{
			1
			5 packedarray
		} bind def
		/setcustomcolor
		{
			exch 
			aload pop 
			0 eq
			{
				pop
				4
				{
					4 index mul
					4 1 roll
				} repeat
				5 -1 roll pop
				setcmykcolor
			}
			{
				pop
				3
				{
					1 exch sub
					3 index mul 
					1 exch sub
					3 1 roll
				} repeat
				4 -1 roll pop
				setrgbcolor
			} ifelse
		}
		def
	} if
	
	/gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def
	userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put
	userdict /level2?
	systemdict /languagelevel known dup
	{
		pop systemdict /languagelevel get 2 ge
	} if
	put
/level2ScreenFreq
{
 begin
		60
		HalftoneType 1 eq
		{
			pop Frequency
		} if
		HalftoneType 2 eq
		{
			pop GrayFrequency
		} if
		HalftoneType 5 eq
		{
			pop Default level2ScreenFreq
		} if
 end
} bind def
userdict /currentScreenFreq  
	level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put
level2? not
	{
		/setcmykcolor where not
		{
			/setcmykcolor
			{
				exch .11 mul add exch .59 mul add exch .3 mul add
				1 exch sub setgray
			} def
		} if
		/currentcmykcolor where not
		{
			/currentcmykcolor
			{
				0 0 0 1 currentgray sub
			} def
		} if
		/setoverprint where not
		{
			/setoverprint /pop load def
		} if
		/selectfont where not
		{
			/selectfont
			{
				exch findfont exch
				dup type /arraytype eq
				{
					makefont
				}
				{
					scalefont
				} ifelse
				setfont
			} bind def
		} if
		/cshow where not
		{
			/cshow
			{
				[
				0 0 5 -1 roll aload pop
				] cvx bind forall
			} bind def
		} if
	} if
	cleartomark
	/anyColor?
	{
		add add add 0 ne
	} bind def
	/testColor
	{
		gsave
		setcmykcolor currentcmykcolor
		grestore
	} bind def
	/testCMYKColorThrough
	{
		testColor anyColor?
	} bind def
	userdict /composite?
	level2?
	{
		gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore
		add add add 4 eq
	}
	{
		1 0 0 0 testCMYKColorThrough
		0 1 0 0 testCMYKColorThrough
		0 0 1 0 testCMYKColorThrough
		0 0 0 1 testCMYKColorThrough
		and and and
	} ifelse
	put
	composite? not
	{
		userdict begin
		gsave
		/cyan? 1 0 0 0 testCMYKColorThrough def
		/magenta? 0 1 0 0 testCMYKColorThrough def
		/yellow? 0 0 1 0 testCMYKColorThrough def
		/black? 0 0 0 1 testCMYKColorThrough def
		grestore
		/isCMYKSep? cyan? magenta? yellow? black? or or or def
		/customColor? isCMYKSep? not def
	 end
	} if
 end defaultpacking setpacking
%%EndResource
%%BeginResource: procset Adobe_typography_AI5 1.0 1
%%Title: (Typography Operators)
%%Version: 1.0 1
%%CreationDate:(6/10/1996) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_typography_AI5 68 dict dup begin
put
/initialize
{
 begin
 begin
	Adobe_typography_AI5 begin
	Adobe_typography_AI5
	{
		dup xcheck
		{
			bind
		} if
		pop pop
	} forall
 end
 end
 end
	Adobe_typography_AI5 begin
} def
/terminate
{
	currentdict Adobe_typography_AI5 eq
	{
	 end
	} if
} def
/modifyEncoding
{
	/_tempEncode exch ddef
	/_pntr 0 ddef
	{
		counttomark -1 roll
		dup type dup /marktype eq
		{
			pop pop exit
		}
		{
			/nametype eq
			{
				_tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll
				put
			}
			{
				/_pntr exch ddef
			} ifelse
		} ifelse
	} loop
	_tempEncode
} def
/havefont
{
	systemdict /languagelevel known
		{
		/Font resourcestatus dup
			{ exch pop exch pop }
		if
		}
		{
		systemdict /FontDirectory get 1 index known
			{ pop true }
			{
			systemdict /fileposition known
				{
				dup length 6 add exch
				Ss 6 250 getinterval
				cvs pop
				Ss exch 0 exch getinterval
				status
					{ pop pop pop pop true }
					{ false }
				ifelse
				}
				{
				pop false
				}
			ifelse
			}
		ifelse
		}
	ifelse
} def
/TE
{
	StandardEncoding 256 array copy modifyEncoding
	/_nativeEncoding exch def
} def
/subststring {
	exch 2 index exch search
	{
		exch pop
		exch dup () eq
		{
			pop exch concatstring
		}
		{
			3 -1 roll
			exch concatstring
			concatstring
		} ifelse
		exch pop true
	}
	{
		pop pop false
	} ifelse
} def
/concatstring {
	1 index length 1 index length
	1 index add
	string
	dup 0 5 index putinterval
	dup 2 index 4 index putinterval
	4 1 roll pop pop pop
} def
%
/TZ
{
	dup type /arraytype eq
	{
		/_wv exch def
	}
	{
		/_wv 0 def
	} ifelse
	/_useNativeEncoding exch def
	2 index havefont
	{
		3 index
		255 string
		cvs
		
		dup
		(_Symbol_)
		eq
		{
			pop
			2 index
			findfont
			
		}
		{
			dup length 1 sub
			1 exch
			getinterval
			
			cvn
			findfont
		} ifelse
	}
	{
		dup 1 eq
		{
			2 index 64 string cvs
			dup (-90pv-RKSJ-) (-83pv-RKSJ-) subststring
			{
				exch pop dup havefont
				{
					findfont false
				}
				{
					pop true
				} ifelse
			}
			{
				pop	dup
				(-90ms-RKSJ-) (-Ext-RKSJ-) subststring
				{
					exch pop dup havefont
					{
						findfont false
					}
					{
						pop true
					} ifelse
				}
				{
					pop pop true
				} ifelse
			} ifelse
			{
				/Ryumin-Light-83pv-RKSJ-H havefont
					{/Ryumin-Light-83pv-RKSJ-H}
					{/Courier}
					ifelse
					findfont
					[1 0 0.5 1 0 0] makefont
			} if
		}
		{
			/Courier findfont
		} ifelse
	} ifelse
	_wv type /arraytype eq
	{
		_wv makeblendedfont
	} if
	dup length 10 add dict
 begin
	mark exch
	{
		1 index /FID ne
		{
			def
		} if
		cleartomark mark
	} forall
	pop
	/FontScript exch def
	/FontDirection exch def
	/FontRequest exch def
	/FontName exch def
	counttomark 0 eq
	{
		1 _useNativeEncoding eq
		{
			/Encoding _nativeEncoding def
		} if
		cleartomark
	}
	{
		/Encoding load 256 array copy
		modifyEncoding /Encoding exch def
	} ifelse
	FontName currentdict
 end
	definefont pop
} def
/tr
{
	_ax _ay 3 2 roll
} def
/trj
{
	_cx _cy _sp _ax _ay 6 5 roll
} def
/a0
{
	/Tx
	{
		dup
		currentpoint 3 2 roll
		tr _psf
		newpath moveto
		tr _ctm _pss
	} ddef
	/Tj
	{
		dup
		currentpoint 3 2 roll
		trj _pjsf
		newpath moveto
		trj _ctm _pjss
	} ddef
} def
/a1
{
W B
} def
/e0
{
	/Tx
	{
		tr _psf
	} ddef
	/Tj
	{
		trj _pjsf
	} ddef
} def
/e1
{
W F 
} def
/i0
{
	/Tx
	{
		tr sp
	} ddef
	/Tj
	{
		trj jsp
	} ddef
} def
/i1
{
	W N
} def
/o0
{
	/Tx
	{
		tr sw rmoveto
	} ddef
	/Tj
	{
		trj swj rmoveto
	} ddef
} def
/r0
{
	/Tx
	{
		tr _ctm _pss
	} ddef
	/Tj
	{
		trj _ctm _pjss
	} ddef
} def
/r1
{
W S
} def
/To
{
	pop _ctm currentmatrix pop
} def
/TO
{
	iTe _ctm setmatrix newpath
} def
/Tp
{
	pop _tm astore pop _ctm setmatrix
	_tDict begin
	/W
	{
	} def
	/h
	{
	} def
} def
/TP
{
 end
	iTm 0 0 moveto
} def
/Tr
{
	_render 3 le
	{
		currentpoint newpath moveto
	} if
	dup 8 eq
	{
		pop 0
	}
	{
		dup 9 eq
		{
			pop 1
		} if
	} ifelse
	dup /_render exch ddef
	_renderStart exch get load exec
} def
/iTm
{
	_ctm setmatrix _tm concat
	_shift aload pop _lineorientation 1 eq { exch } if translate
	_scale aload pop _lineorientation 1 eq _yokoorientation 1 eq or { exch } if scale
} def
/Tm
{
	_tm astore pop iTm 0 0 moveto
} def
/Td
{
	_mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto
} def
/iTe
{
	_render -1 eq
	{
	}
	{
		_renderEnd _render get dup null ne
		{
			load exec
		}
		{
			pop
		} ifelse
	} ifelse
	/_render -1 ddef
} def
/Ta
{
	pop
} def
/Tf
{
	1 index type /nametype eq
	{
		dup 0.75 mul 1 index 0.25 mul neg
	} if
	/_fontDescent exch ddef
	/_fontAscent exch ddef
	/_fontSize exch ddef
	/_fontRotateAdjust _fontAscent _fontDescent add 2 div neg ddef
	/_fontHeight _fontSize ddef
	findfont _fontSize scalefont setfont
} def
/Tl
{
	pop neg 0 exch
	_leading astore pop
} def
/Tt
{
	pop
} def
/TW
{
	3 npop
} def
/Tw
{
	/_cx exch ddef
} def
/TC
{
	3 npop
} def
/Tc
{
	/_ax exch ddef
} def
/Ts
{
	0 exch
	_shift astore pop
	currentpoint
	iTm
	moveto
} def
/Ti
{
	3 npop
} def
/Tz
{
	count 1 eq { 100 } if
	100 div exch 100 div exch
	_scale astore pop
	iTm
} def
/TA
{
	pop
} def
/Tq
{
	pop
} def
/Tg
{
	pop
} def
/TG
{
	pop
} def
/Tv
{
	/_lineorientation exch ddef
} def
/TV
{
	/_charorientation exch ddef
} def
/Ty
{
	dup /_yokoorientation exch ddef 1 sub neg Tv
} def
/TY
{
	pop
} def
/T~
{
	Tx
} def
/Th
{
	pop pop pop pop pop
} def
/TX
{
	pop
} def
/Tk
{
	_fontSize mul 1000 div
	_lineorientation 0 eq { neg 0 } { 0 exch } ifelse
	rmoveto
	pop
} def
/TK
{
	2 npop
} def
/T*
{
	_leading aload pop
	_lineorientation 0 ne { exch } if
	Td
} def
/T*-
{
	_leading aload pop
	_lineorientation 0 ne { exch } if
	exch neg exch neg
	Td
} def
/T-
{
	_ax neg 0 rmoveto
	_lineorientation 1 eq _charorientation 0 eq and { 1 TV _hyphen Tx 0 TV } { _hyphen Tx } ifelse
} def
/T+
{
} def
/TR
{
	_ctm currentmatrix pop
	_tm astore pop
	iTm 0 0 moveto
} def
/TS
{
	currentfont 3 1 roll
	/_Symbol_ findfont _fontSize scalefont setfont
	
	0 eq
	{
		Tx
	}
	{
		Tj
	} ifelse
	setfont
} def
/Xb
{
	pop pop
} def
/Tb /Xb load def
/Xe
{
	pop pop pop pop
} def
/Te /Xe load def
/XB
{
} def
/TB /XB load def
currentdict readonly pop
end
setpacking
%
/X^
{
	currentfont 5 1 roll
	dup havefont
		{
		findfont _fontSize scalefont setfont
		}
		{
		pop
		exch
		} ifelse
	2 index 0 eq
	{
		Tx
	}
	{
		Tj
	} ifelse
	pop	pop
	setfont
} def
/T^	/X^	load def
%%EndResource
%%BeginProcSet: Adobe_ColorImage_AI6 1.1 0
userdict /Adobe_ColorImage_AI6 known not
{
	userdict /Adobe_ColorImage_AI6 24 dict put 
} if
userdict /Adobe_ColorImage_AI6 get begin
/initialize
{ 
	Adobe_ColorImage_AI6 begin
	Adobe_ColorImage_AI6
	{
		dup type /arraytype eq
		{
			dup xcheck
			{
				bind
			} if
		} if
		pop pop
	} forall
} def
/terminate { end } def
currentdict /Adobe_ColorImage_AI6_Vars known not
{
	/Adobe_ColorImage_AI6_Vars 15 dict def
} if
Adobe_ColorImage_AI6_Vars begin
	/channelcount 0 def
	/sourcecount 0 def
	/sourcearray 4 array def
	/plateindex -1 def
	/XIMask 0 def
	/XIBinary 0 def
	/XIChannelCount 0 def
	/XIBitsPerPixel 0 def
	/XIImageHeight 0 def
	/XIImageWidth 0 def
	/XIImageMatrix null def
	/XIBuffer null def
	/XIDataProc null def
	/XIVersion 6 def
end
/WalkRGBString null def
/WalkCMYKString null def
/StuffRGBIntoGrayString null def
/RGBToGrayImageProc null def
/StuffCMYKIntoGrayString null def
/CMYKToGrayImageProc null def
/ColorImageCompositeEmulator null def
/SeparateCMYKImageProc null def
/FourEqual null def
/TestPlateIndex null def
currentdict /_colorimage known not
{
	/colorimage where
	{
		/colorimage get /_colorimage exch def
	}
	{
		/_colorimage null def
	} ifelse
} if
/_currenttransfer systemdict /currenttransfer get def
/colorimage null def
/XI null def
/WalkRGBString
{
	0 3 index
	dup length 1 sub 0 3 3 -1 roll
	{
		3 getinterval { } forall
		5 index exec
		3 index
	} for
	
	 5 { pop } repeat
} def
/WalkCMYKString
{
	0 3 index
	dup length 1 sub 0 4 3 -1 roll
	{
		4 getinterval { } forall
		
		6 index exec
		
		3 index
		
	} for
	
	5 { pop } repeat
	
} def
/StuffRGBIntoGrayString
{
	.11 mul exch
	
	.59 mul add exch
	
	.3 mul add
	
	cvi 3 copy put
	
	pop 1 add
} def
/RGBToGrayImageProc
{	
	Adobe_ColorImage_AI6_Vars begin 
		sourcearray 0 get exec
		dup length 3 idiv string
		dup 3 1 roll 
		
		/StuffRGBIntoGrayString load exch
		WalkRGBString
 end
} def
/StuffCMYKIntoGrayString
{
	exch .11 mul add
	
	exch .59 mul add
	
	exch .3 mul add
	
	dup 255 gt { pop 255 } if
	
	255 exch sub cvi 3 copy put
	
	pop 1 add
} def
/CMYKToGrayImageProc
{	
	Adobe_ColorImage_AI6_Vars begin
		sourcearray 0 get exec
		dup length 4 idiv string
		dup 3 1 roll 
		
		/StuffCMYKIntoGrayString load exch
		WalkCMYKString
 end
} def
/ColorImageCompositeEmulator
{
	pop true eq
	{
		Adobe_ColorImage_AI6_Vars /sourcecount get 5 add { pop } repeat
	}
	{
		Adobe_ColorImage_AI6_Vars /channelcount get 1 ne
		{
			Adobe_ColorImage_AI6_Vars begin
				sourcearray 0 3 -1 roll put
			
				channelcount 3 eq 
				{ 
					/RGBToGrayImageProc 
				}
				{ 
					/CMYKToGrayImageProc
				} ifelse
				load
		 end
		} if
		image
	} ifelse
} def
/SeparateCMYKImageProc
{	
	Adobe_ColorImage_AI6_Vars begin
		sourcecount 0 ne
		{
			sourcearray plateindex get exec
		}
		{			
			sourcearray 0 get exec
			
			dup length 4 idiv string
			
			0 2 index
			
			plateindex 4 2 index length 1 sub
			{
				get 255 exch sub
				
				3 copy put pop 1 add
				
				2 index
			} for
			pop pop exch pop
		} ifelse
 end
} def
	
/FourEqual
{
	4 index ne
	{
		pop pop pop false
	}
	{
		4 index ne
		{
			pop pop false
		}
		{
			4 index ne
			{
				pop false
			}
			{
				4 index eq
			} ifelse
		} ifelse
	} ifelse
} def
/TestPlateIndex
{
	Adobe_ColorImage_AI6_Vars begin
		/plateindex -1 def
		/setcmykcolor where
		{
			pop
			gsave
			1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
			0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
			0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
			0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub
			grestore
			1 0 0 0 FourEqual 
			{ 
				/plateindex 0 def
			}
			{
				0 1 0 0 FourEqual
				{ 
					/plateindex 1 def
				}
				{
					0 0 1 0 FourEqual
					{
						/plateindex 2 def
					}
					{
						0 0 0 1 FourEqual
						{ 
							/plateindex 3 def
						}
						{
							0 0 0 0 FourEqual
							{
								/plateindex 5 def
							} if
						} ifelse
					} ifelse
				} ifelse
			} ifelse
			pop pop pop pop
		} if
		plateindex
 end
} def
/colorimage
{
	Adobe_ColorImage_AI6_Vars begin
		/channelcount 1 index def
		/sourcecount 2 index 1 eq { channelcount 1 sub } { 0 } ifelse def
		4 sourcecount add index dup 
		8 eq exch 1 eq or not
 end
	
	{
		/_colorimage load null ne
		{
			_colorimage
		}
		{
			Adobe_ColorImage_AI6_Vars /sourcecount get
			7 add { pop } repeat
		} ifelse
	}
	{
		dup 3 eq
		TestPlateIndex
		dup -1 eq exch 5 eq or or
		{
			/_colorimage load null eq
			{
				ColorImageCompositeEmulator
			}
			{
				dup 1 eq
				{
					pop pop image
				}
				{
					Adobe_ColorImage_AI6_Vars /plateindex get 5 eq
					{
						gsave
						
						0 _currenttransfer exec
						1 _currenttransfer exec
						eq
						{ 0 _currenttransfer exec 0.5 lt }
						{ 0 _currenttransfer exec 1 _currenttransfer exec gt } ifelse
						
						{ { pop 0 } } { { pop 1 } } ifelse
						systemdict /settransfer get exec
					} if
					
					_colorimage
					
					Adobe_ColorImage_AI6_Vars /plateindex get 5 eq
					{
						grestore
					} if
				} ifelse
			} ifelse
		}
		{
			dup 1 eq
			{
				pop pop
				image
			}
			{
				pop pop
				Adobe_ColorImage_AI6_Vars begin
					sourcecount -1 0
					{			
						exch sourcearray 3 1 roll put
					} for
					/SeparateCMYKImageProc load
			 end
				systemdict /image get exec
			} ifelse
		} ifelse
	} ifelse
} def
/XG
{
	pop pop
} def
/XF
{
	13 {pop} repeat
} def
/Xh
{
	Adobe_ColorImage_AI6_Vars begin
		gsave
		/XIMask exch 0 ne def
		/XIImageHeight exch def
		/XIImageWidth exch def
		/XIImageMatrix exch def
		0 0 moveto
		XIImageMatrix concat
		XIImageWidth XIImageHeight scale
		
		XIMask
		{
			/_lp /null ddef
			_fc
			/_lp /imagemask ddef
		}
		if
		/XIVersion 7 def
 end
} def
/XH
{
	Adobe_ColorImage_AI6_Vars begin
		/XIVersion 6 def
		grestore
 end
} def
/XI
{
	Adobe_ColorImage_AI6_Vars begin
		gsave
		/XIMask exch 0 ne def
		/XIBinary exch 0 ne def
		pop
		pop
		/XIChannelCount exch def
		/XIBitsPerPixel exch def
		/XIImageHeight exch def
		/XIImageWidth exch def
		pop pop pop pop
		/XIImageMatrix exch def
		XIBitsPerPixel 1 eq
		{
			XIImageWidth 8 div ceiling cvi
		}
		{
			XIImageWidth XIChannelCount mul
		} ifelse
		/XIBuffer exch string def
		XIBinary
		{
			/XIDataProc { currentfile XIBuffer readstring pop } def
			XIVersion 6 le
			{
				currentfile 128 string readline pop pop
			}
			if
		}
		{
			/XIDataProc { currentfile XIBuffer readhexstring pop } def
		} ifelse
		
		XIVersion 6 le
		{
			0 0 moveto
			XIImageMatrix concat
			XIImageWidth XIImageHeight scale
			XIMask
			{
				/_lp /null ddef
				_fc
				/_lp /imagemask ddef
			} if
		} if
		
		XIMask
		{
			XIImageWidth XIImageHeight
			false
			[ XIImageWidth 0 0 XIImageHeight neg 0 0 ]
			/XIDataProc load
			imagemask
		}
		{
			XIImageWidth XIImageHeight
			XIBitsPerPixel
			[ XIImageWidth 0 0 XIImageHeight neg 0 0 ]
			/XIDataProc load
			
			XIChannelCount 1 eq
			{
				gsave
				0 setgray
				image
				grestore
			}
			{
				false
				XIChannelCount
				colorimage
			} ifelse
		} ifelse
		grestore
 end
} def
end
%%EndProcSet
%%BeginResource: procset Adobe_Illustrator_AI5 1.2 0
%%Title: (Adobe Illustrator (R) Version 7.0 Full Prolog)
%%Version: 1.2 0
%%CreationDate: (3/7/1994) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_Illustrator_AI5_vars 107 dict dup begin
put
/_eo false def
/_lp /none def
/_pf
{
} def
/_ps
{
} def
/_psf
{
} def
/_pss
{
} def
/_pjsf
{
} def
/_pjss
{
} def
/_pola 0 def
/_doClip 0 def
/cf currentflat def
/_lineorientation 0 def
/_charorientation 0 def
/_yokoorientation 0 def
/_tm matrix def
/_renderStart
[
/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0
] def
/_renderEnd
[
null null null null /i1 /i1 /i1 /i1
] def
/_render -1 def
/_shift [0 0] def
/_ax 0 def
/_ay 0 def
/_cx 0 def
/_cy 0 def
/_leading
[
0 0
] def
/_ctm matrix def
/_mtx matrix def
/_sp 16#020 def
/_hyphen (-) def
/_fontSize 0 def
/_fontAscent 0 def
/_fontDescent 0 def
/_fontHeight 0 def
/_fontRotateAdjust 0 def
/Ss 256 string def
Ss 0 (fonts/) putinterval
/_cnt 0 def
/_scale [1 1] def
/_nativeEncoding 0 def
/_useNativeEncoding 0 def
/_tempEncode 0 def
/_pntr 0 def
/_tDict 2 dict def
/_hfname 100 string def
/_hffound false def
/Tx
{
} def
/Tj
{
} def
/CRender
{
} def
/_AI3_savepage
{
} def
/_gf null def
/_cf 4 array def
/_rgbf 3 array def
/_if null def
/_of false def
/_fc
{
} def
/_gs null def
/_cs 4 array def
/_rgbs 3 array def
/_is null def
/_os false def
/_sc
{
} def
/_pd 1 dict def
/_ed 15 dict def
/_pm matrix def
/_fm null def
/_fd null def
/_fdd null def
/_sm null def
/_sd null def
/_sdd null def
/_i null def
/_lobyte 0 def
/_hibyte 0 def
/_cproc null def
/_cscript 0 def
/_hvax 0 def
/_hvay 0 def
/_hvwb 0 def
/_hvcx 0 def
/_hvcy 0 def
/_bitfont null def
/_bitlobyte 0 def
/_bithibyte 0 def
/_bitkey null def
/_bitdata null def
/_bitindex 0 def
/discardSave null def
/buffer 256 string def
/beginString null def
/endString null def
/endStringLength null def
/layerCnt 1 def
/layerCount 1 def
/perCent (%) 0 get def
/perCentSeen? false def
/newBuff null def
/newBuffButFirst null def
/newBuffLast null def
/clipForward? false def
end
userdict /Adobe_Illustrator_AI5 known not {
	userdict /Adobe_Illustrator_AI5 95 dict put
} if
userdict /Adobe_Illustrator_AI5 get begin
/initialize
{
	Adobe_Illustrator_AI5 dup begin
	Adobe_Illustrator_AI5_vars begin
	discardDict
	{
		bind pop pop
	} forall
	dup /nc get begin
	{
		dup xcheck 1 index type /operatortype ne and
		{
			bind
		} if
		pop pop
	} forall
 end
	newpath
} def
/terminate
{
 end
 end
} def
/_
null def
/ddef
{
	Adobe_Illustrator_AI5_vars 3 1 roll put
} def
/xput
{
	dup load dup length exch maxlength eq
	{
		dup dup load dup
		length 2 mul dict copy def
	} if
	load begin
	def
 end
} def
/npop
{
	{
		pop
	} repeat
} def
/hswj
{
	dup stringwidth 3 2 roll
	{
		_hvwb eq { exch _hvcx add exch _hvcy add } if
		exch _hvax add exch _hvay add
	} cforall
} def
/vswj
{
	0 0 3 -1 roll
	{
		dup 255 le
		_charorientation 1 eq
		and
		{
			dup cstring stringwidth 5 2 roll
			_hvwb eq { exch _hvcy sub exch _hvcx sub } if
			exch _hvay sub exch _hvax sub
			4 -1 roll sub exch
			3 -1 roll sub exch
		}
		{
			_hvwb eq { exch _hvcy sub exch _hvcx sub } if
			exch _hvay sub exch _hvax sub
			_fontHeight sub
		} ifelse
	} cforall
} def
/swj
{
	6 1 roll
	/_hvay exch ddef
	/_hvax exch ddef
	/_hvwb exch ddef
	/_hvcy exch ddef
	/_hvcx exch ddef
	_lineorientation 0 eq { hswj } { vswj } ifelse
} def
/sw
{
	0 0 0 6 3 roll swj
} def
/vjss
{
	4 1 roll
	{
		dup cstring
		dup length 1 eq
		_charorientation 1 eq
		and
		{
			-90 rotate
			currentpoint
			_fontRotateAdjust add
			moveto
			gsave
			false charpath currentpoint
			5 index setmatrix stroke
			grestore
			_fontRotateAdjust sub
			moveto
			_sp eq
			{
				5 index 5 index rmoveto
			} if
			2 copy rmoveto
			90 rotate
		}
		{
			currentpoint
			_fontHeight sub
			5 index sub
			3 index _sp eq
			{
				9 index sub
			} if
	
			currentpoint
			exch 4 index stringwidth pop 2 div sub
			exch _fontAscent sub
			moveto
	
			gsave
			2 index false charpath
			6 index setmatrix stroke
			grestore
	
			moveto pop pop
		} ifelse
	} cforall
	6 npop
} def
/hjss
{
	4 1 roll
	{
		dup cstring
		gsave
		false charpath currentpoint
		5 index setmatrix stroke
		grestore
		moveto
		_sp eq
		{
			5 index 5 index rmoveto
		} if
		2 copy rmoveto
	} cforall
	6 npop
} def
/jss
{
	_lineorientation 0 eq { hjss } { vjss } ifelse
} def
/ss
{
	0 0 0 7 3 roll jss
} def
/vjsp
{
	4 1 roll
	{
		dup cstring
		dup length 1 eq
		_charorientation 1 eq
		and
		{
			-90 rotate
			currentpoint
			_fontRotateAdjust add
			moveto
			false charpath
            currentpoint
			_fontRotateAdjust sub
			moveto
			_sp eq
			{
				5 index 5 index rmoveto
			} if
			2 copy rmoveto
			90 rotate
		}
		{
			currentpoint
			_fontHeight sub
			5 index sub
			3 index _sp eq
			{
				9 index sub
			} if
	
			currentpoint
			exch 4 index stringwidth pop 2 div sub
			exch _fontAscent sub
			moveto
	
			2 index false charpath
	
			moveto pop pop
		} ifelse
	} cforall
	6 npop
} def
/hjsp
{
    4 1 roll
    {
        dup cstring
        false charpath
        _sp eq
        {
            5 index 5 index rmoveto
        } if
        2 copy rmoveto
    } cforall
    6 npop
} def
/jsp
{
	matrix currentmatrix
    _lineorientation 0 eq {hjsp} {vjsp} ifelse
} def
/sp
{
    matrix currentmatrix
    0 0 0 7 3 roll
    _lineorientation 0 eq {hjsp} {vjsp} ifelse
} def
/pl
{
	transform
	0.25 sub round 0.25 add exch
	0.25 sub round 0.25 add exch
	itransform
} def
/setstrokeadjust where
{
	pop true setstrokeadjust
	/c
	{
		curveto
	} def
	/C
	/c load def
	/v
	{
		currentpoint 6 2 roll curveto
	} def
	/V
	/v load def
	/y
	{
		2 copy curveto
	} def
	/Y
	/y load def
	/l
	{
		lineto
	} def
	/L
	/l load def
	/m
	{
		moveto
	} def
}
{
	/c
	{
		pl curveto
	} def
	/C
	/c load def
	/v
	{
		currentpoint 6 2 roll pl curveto
	} def
	/V
	/v load def
	/y
	{
		pl 2 copy curveto
	} def
	/Y
	/y load def
	/l
	{
		pl lineto
	} def
	/L
	/l load def
	/m
	{
		pl moveto
	} def
} ifelse
/d
{
	setdash
} def
/cf
{
} def
/i
{
	dup 0 eq
	{
		pop cf
	} if
	setflat
} def
/j
{
	setlinejoin
} def
/J
{
	setlinecap
} def
/M
{
	setmiterlimit
} def
/w
{
	setlinewidth
} def
/XR
{
	0 ne
	/_eo exch ddef
} def
/H
{
} def
/h
{
	closepath
} def
/N
{
	_pola 0 eq
	{
		_doClip 1 eq
		{
			_eo {eoclip} {clip} ifelse /_doClip 0 ddef
		} if
		newpath
	}
	{
		/CRender
		{
			N
		} ddef
	} ifelse
} def
/n
{
	N
} def
/F
{
	_pola 0 eq
	{
		_doClip 1 eq
		{
			gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc
			/_doClip 0 ddef
		}
		{
			_pf
		} ifelse
	}
	{
		/CRender
		{
			F
		} ddef
	} ifelse
} def
/f
{
	closepath
	F
} def
/S
{
	_pola 0 eq
	{
		_doClip 1 eq
		{
			gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc
			/_doClip 0 ddef
		}
		{
			_ps
		} ifelse
	}
	{
		/CRender
		{
			S
		} ddef
	} ifelse
} def
/s
{
	closepath
	S
} def
/B
{
	_pola 0 eq
	{
		_doClip 1 eq
		gsave F grestore
		{
			gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc
			/_doClip 0 ddef
		}
		{
			S
		} ifelse
	}
	{
		/CRender
		{
			B
		} ddef
	} ifelse
} def
/b
{
	closepath
	B
} def
/W
{
	/_doClip 1 ddef
} def
/*
{
	count 0 ne
	{
		dup type /stringtype eq
		{
			pop
		} if
	} if
	newpath
} def
/u
{
} def
/U
{
} def
/q
{
	_pola 0 eq
	{
		gsave
	} if
} def
/Q
{
	_pola 0 eq
	{
		grestore
	} if
} def
/*u
{
	_pola 1 add /_pola exch ddef
} def
/*U
{
	_pola 1 sub /_pola exch ddef
	_pola 0 eq
	{
		CRender
	} if
} def
/D
{
	pop
} def
/*w
{
} def
/*W
{
} def
/`
{
	/_i save ddef
	clipForward?
	{
		nulldevice
	} if
	6 1 roll 4 npop
	concat pop
	userdict begin
	/showpage
	{
	} def
	0 setgray
	0 setlinecap
	1 setlinewidth
	0 setlinejoin
	10 setmiterlimit
	[] 0 setdash
	/setstrokeadjust where {pop false setstrokeadjust} if
	newpath
	0 setgray
	false setoverprint
} def
/~
{
 end
	_i restore
} def
/O
{
	0 ne
	/_of exch ddef
	/_lp /none ddef
} def
/R
{
	0 ne
	/_os exch ddef
	/_lp /none ddef
} def
/g
{
	/_gf exch ddef
	/_fc
	{
		_lp /fill ne
		{
			_of setoverprint
			_gf setgray
			/_lp /fill ddef
		} if
	} ddef
	/_pf
	{
		_fc
		_eo {eofill} {fill} ifelse
	} ddef
	/_psf
	{
		_fc
		hvashow
	} ddef
	/_pjsf
	{
		_fc
		hvawidthshow
	} ddef
	/_lp /none ddef
} def
/G
{
	/_gs exch ddef
	/_sc
	{
		_lp /stroke ne
		{
			_os setoverprint
			_gs setgray
			/_lp /stroke ddef
		} if
	} ddef
	/_ps
	{
		_sc
		stroke
	} ddef
	/_pss
	{
		_sc
		ss
	} ddef
	/_pjss
	{
		_sc
		jss
	} ddef
	/_lp /none ddef
} def
/k
{
	_cf astore pop
	/_fc
	{
		_lp /fill ne
		{
			_of setoverprint
			_cf aload pop setcmykcolor
			/_lp /fill ddef
		} if
	} ddef
	/_pf
	{
		_fc
		_eo {eofill} {fill} ifelse
	} ddef
	/_psf
	{
		_fc
		hvashow
	} ddef
	/_pjsf
	{
		_fc
		hvawidthshow
	} ddef
	/_lp /none ddef
} def
/K
{
	_cs astore pop
	/_sc
	{
		_lp /stroke ne
		{
			_os setoverprint
			_cs aload pop setcmykcolor
			/_lp /stroke ddef
		} if
	} ddef
	/_ps
	{
		_sc
		stroke
	} ddef
	/_pss
	{
		_sc
		ss
	} ddef
	/_pjss
	{
		_sc
		jss
	} ddef
	/_lp /none ddef
} def
/Xa
{
	_rgbf astore pop
	/_fc
	{
		_lp /fill ne
		{
			_of setoverprint
			_rgbf aload pop setrgbcolor
			/_lp /fill ddef
		} if
	} ddef
	/_pf
	{
		_fc
		_eo {eofill} {fill} ifelse
	} ddef
	/_psf
	{
		_fc
		hvashow
	} ddef
	/_pjsf
	{
		_fc
		hvawidthshow
	} ddef
	/_lp /none ddef
} def
/XA
{
	_rgbs astore pop
	/_sc
	{
		_lp /stroke ne
		{
			_os setoverprint
			_rgbs aload pop setrgbcolor
			/_lp /stroke ddef
		} if
	} ddef
	/_ps
	{
		_sc
		stroke
	} ddef
	/_pss
	{
		_sc
		ss
	} ddef
	/_pjss
	{
		_sc
		jss
	} ddef
	/_lp /none ddef
} def
/_rgbtocmyk
{
3
	{
	1 exch sub 3 1 roll
	} repeat
3 copy 1 4 1 roll
3
	{
	3 index 2 copy gt
		{
		exch
		} if
	pop 4 1 roll
	} repeat
pop pop pop
4 1 roll
3
	{
	3 index sub
	3 1 roll
	} repeat
4 -1 roll
} def
/Xx
{
	exch
	/_gf exch ddef
	0 eq
	{
		findcmykcustomcolor
	}
	{
		/findrgbcustomcolor where not {
			4 1 roll _rgbtocmyk
			5 -1 roll
			findcmykcustomcolor
		}
		{
			pop
			findrgbcustomcolor
		} ifelse
	} ifelse
	/_if exch ddef
	/_fc
	{
		_lp /fill ne
		{
			_of setoverprint
			_if _gf 1 exch sub setcustomcolor
			/_lp /fill ddef
		} if
	} ddef
	/_pf
	{
		_fc
		_eo {eofill} {fill} ifelse
	} ddef
	/_psf
	{
		_fc
		hvashow
	} ddef
	/_pjsf
	{
		_fc
		hvawidthshow
	} ddef
	/_lp /none ddef
} def
/XX
{
	exch
	/_gs exch ddef
	0 eq
	{
		findcmykcustomcolor
	}
	{
		/findrgbcustomcolor where not {
			4 1 roll _rgbtocmyk
			5 -1 roll
			findcmykcustomcolor
		}
		{
			pop
			findrgbcustomcolor
		} ifelse
	} ifelse
	/_is exch ddef
	/_sc
	{
		_lp /stroke ne
		{
			_os setoverprint
			_is _gs 1 exch sub setcustomcolor
			/_lp /stroke ddef
		} if
	} ddef
	/_ps
	{
		_sc
		stroke
	} ddef
	/_pss
	{
		_sc
		ss
	} ddef
	/_pjss
	{
		_sc
		jss
	} ddef
	/_lp /none ddef
} def
/x
{
	/_gf exch ddef
	findcmykcustomcolor
	/_if exch ddef
	/_fc
	{
		_lp /fill ne
		{
			_of setoverprint
			_if _gf 1 exch sub setcustomcolor
			/_lp /fill ddef
		} if
	} ddef
	/_pf
	{
		_fc
		_eo {eofill} {fill} ifelse
	} ddef
	/_psf
	{
		_fc
		hvashow
	} ddef
	/_pjsf
	{
		_fc
		hvawidthshow
	} ddef
	/_lp /none ddef
} def
/X
{
	/_gs exch ddef
	findcmykcustomcolor
	/_is exch ddef
	/_sc
	{
		_lp /stroke ne
		{
			_os setoverprint
			_is _gs 1 exch sub setcustomcolor
			/_lp /stroke ddef
		} if
	} ddef
	/_ps
	{
		_sc
		stroke
	} ddef
	/_pss
	{
		_sc
		ss
	} ddef
	/_pjss
	{
		_sc
		jss
	} ddef
	/_lp /none ddef
} def
/A
{
	pop
} def
/annotatepage
{
userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse
} def
/XT {
	pop pop
} def
/discard
{
	save /discardSave exch store
	discardDict begin
	/endString exch store
	gt38?
	{
		2 add
	} if
	load
	stopped
	pop
 end
	discardSave restore
} bind def
userdict /discardDict 7 dict dup begin
put
/pre38Initialize
{
	/endStringLength endString length store
	/newBuff buffer 0 endStringLength getinterval store
	/newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store
	/newBuffLast newBuff endStringLength 1 sub 1 getinterval store
} def
/shiftBuffer
{
	newBuff 0 newBuffButFirst putinterval
	newBuffLast 0
	currentfile read not
	{
	stop
	} if
	put
} def
0
{
	pre38Initialize
	mark
	currentfile newBuff readstring exch pop
	{
		{
			newBuff endString eq
			{
				cleartomark stop
			} if
			shiftBuffer
		} loop
	}
	{
	stop
	} ifelse
} def
1
{
	pre38Initialize
	/beginString exch store
	mark
	currentfile newBuff readstring exch pop
	{
		{
			newBuff beginString eq
			{
				/layerCount dup load 1 add store
			}
			{
				newBuff endString eq
				{
					/layerCount dup load 1 sub store
					layerCount 0 eq
					{
						cleartomark stop
					} if
				} if
			} ifelse
			shiftBuffer
		} loop
	} if
} def
2
{
	mark
	{
		currentfile buffer readline not
		{
		stop
		} if
		endString eq
		{
			cleartomark stop
		} if
	} loop
} def
3
{
	/beginString exch store
	/layerCnt 1 store
	mark
	{
		currentfile buffer readline not
		{
		stop
		} if
		dup beginString eq
		{
			pop /layerCnt dup load 1 add store
		}
		{
			endString eq
			{
				layerCnt 1 eq
				{
					cleartomark stop
				}
				{
					/layerCnt dup load 1 sub store
				} ifelse
			} if
		} ifelse
	} loop
} def
end
userdict /clipRenderOff 15 dict dup begin
put
{
	/n /N /s /S /f /F /b /B
}
{
	{
		_doClip 1 eq
		{
			/_doClip 0 ddef _eo {eoclip} {clip} ifelse
		} if
		newpath
	} def
} forall
/Tr /pop load def
/Bb {} def
/BB /pop load def
/Bg {12 npop} def
/Bm {6 npop} def
/Bc /Bm load def
/Bh {4 npop} def
end
/Lb
{
	4 npop
	6 1 roll
	pop
	4 1 roll
	pop pop pop
	0 eq
	{
		0 eq
		{
			(%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard
		}
		{
			
			/clipForward? true def
			
			/Tx /pop load def
			/Tj /pop load def
			
			currentdict end clipRenderOff begin begin
		} ifelse
	}
	{
		0 eq
		{
			save /discardSave exch store
		} if
	} ifelse
} bind def
/LB
{
	discardSave dup null ne
	{
		restore
	}
	{
		pop
		clipForward?
		{
			currentdict
		 end
		 end
		 begin
					
			/clipForward? false ddef
		} if
	} ifelse
} bind def
/Pb
{
	pop pop
	0 (%AI5_EndPalette) discard
} bind def
/Np
{
	0 (%AI5_End_NonPrinting--) discard
} bind def
/Ln /pop load def
/Ap
/pop load def
/Ar
{
	72 exch div
	0 dtransform dup mul exch dup mul add sqrt
	dup 1 lt
	{
		pop 1
	} if
	setflat
} def
/Mb
{
	q
} def
/Md
{
} def
/MB
{
	Q
} def
/nc 4 dict def
nc begin
/setgray
{
	pop
} bind def
/setcmykcolor
{
	4 npop
} bind def
/setrgbcolor
{
	3 npop
} bind def
/setcustomcolor
{
	2 npop
} bind def
currentdict readonly pop
end
end
setpacking
%%EndResource
%%BeginResource: procset Adobe_cshow 2.0 8
%%Title: (Writing System Operators)
%%Version: 2.0 8
%%CreationDate: (1/23/89) ()
%%Copyright: ((C) 1992-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_cshow 14 dict dup begin put
/initialize
{
	Adobe_cshow begin
	Adobe_cshow
	{
		dup xcheck
		{
			bind
		} if
		pop pop
	} forall
 end
	Adobe_cshow begin
} def
/terminate
{
currentdict Adobe_cshow eq
	{
 end
	} if
} def
/cforall
{
	/_lobyte 0 ddef
	/_hibyte 0 ddef
	/_cproc exch ddef
	/_cscript currentfont /FontScript known { currentfont /FontScript get } { -1 } ifelse ddef
	{
		/_lobyte exch ddef
		_hibyte 0 eq
		_cscript 1 eq
		_lobyte 129 ge _lobyte 159 le and
		_lobyte 224 ge _lobyte 252 le and or and
		_cscript 2 eq
		_lobyte 161 ge _lobyte 254 le and and
		_cscript 3 eq
		_lobyte 161 ge _lobyte 254 le and and
    	_cscript 25 eq
		_lobyte 161 ge _lobyte 254 le and and
    	_cscript -1 eq
		or or or or and
		{
			/_hibyte _lobyte ddef
		}
		{
			_hibyte 256 mul _lobyte add
			_cproc
			/_hibyte 0 ddef
		} ifelse
	} forall
} def
/cstring
{
	dup 256 lt
	{
		(s) dup 0 4 3 roll put
	}
	{
		dup 256 idiv exch 256 mod
		(hl) dup dup 0 6 5 roll put 1 4 3 roll put
	} ifelse
} def
/clength
{
	0 exch
	{ 256 lt { 1 } { 2 } ifelse add } cforall
} def
/hawidthshow
{
	{
		dup cstring
		show
		_hvax _hvay rmoveto
		_hvwb eq { _hvcx _hvcy rmoveto } if
	} cforall
} def
/vawidthshow
{
	{
		dup 255 le
		_charorientation 1 eq
		and
		{
			-90 rotate
			0 _fontRotateAdjust rmoveto
			cstring
			_hvcx _hvcy _hvwb _hvax _hvay 6 -1 roll awidthshow
			0 _fontRotateAdjust neg rmoveto
			90 rotate
		}
		{
			currentpoint
			_fontHeight sub
			exch _hvay sub exch _hvax sub
			2 index _hvwb eq { exch _hvcy sub exch _hvcx sub } if
			3 2 roll
			cstring
			dup stringwidth pop 2 div neg _fontAscent neg rmoveto
			show
			moveto
		} ifelse
	} cforall
} def
/hvawidthshow
{
	6 1 roll
	/_hvay exch ddef
	/_hvax exch ddef
	/_hvwb exch ddef
	/_hvcy exch ddef
	/_hvcx exch ddef
	_lineorientation 0 eq { hawidthshow } { vawidthshow } ifelse
} def
/hvwidthshow
{
	0 0 3 -1 roll hvawidthshow
} def
/hvashow
{
	0 0 0 6 -3 roll hvawidthshow
} def
/hvshow
{
	0 0 0 0 0 6 -1 roll hvawidthshow
} def
currentdict readonly pop end
setpacking
%%EndResource
%%EndProlog
%%BeginSetup
Adobe_level2_AI5 /initialize get exec
Adobe_cshow /initialize get exec
Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec
Adobe_ColorImage_AI6 /initialize get exec
Adobe_Illustrator_AI5 /initialize get exec
%AI3_BeginRider
currentpacking true setpacking
%%BeginFont: TimesNewRomanPSMT
/TimesNewRomanPSMT
15 dict dup begin
/FontName /TimesNewRomanPSMT def
/FontType 3 def
/FontMatrix [ 0.000977 0 0 0.000977 0 0 ] def
/FontAscent 1024 def
/FontDescent -314 def
/FontScript 0 def
/FontBBox [ 0 -512 1024 1024 ] def
/Encoding 256 array dup 0 1 255 { /.notdef put dup } for pop def
/l /lineto load def
/m /moveto load def
/c /curveto load def
/BuildChar {
	Adobe_Illustrator_AI5_vars exch /_bitlobyte exch put
	Adobe_Illustrator_AI5_vars exch /_bitfont exch put
	Adobe_Illustrator_AI5_vars /_bithibyte 0 put

	_bitlobyte 16 4 string cvrs dup length (K) dup length
	dup 4 -1 roll add string Adobe_Illustrator_AI5_vars exch /_bitkey exch put
	exch _bitkey copy pop _bitkey exch 3 -1 roll putinterval
	_bitfont /CharMetrics get _bitkey cvn get 0 setcharwidth
	_bitfont /CharStrings get _bitkey cvn get exec
} bind def
25 dict dup begin
/KA {
} bind def
/KD {
} bind def
/K28 {
{
newpath
237 -126 m
216 -101 199 -74 187 -46 c
174 -18 164 19 155 65 c
144 123 138 188 138 260 c
138 326 145 389 158 449 c
171 510 191 559 218 599 c
244 638 278 669 320 692 c
320 712 l
237 675 170 614 119 529 c
68 444 42 349 42 246 c
42 175 55 106 81 41 c
107 -25 145 -82 193 -131 c
227 -165 269 -195 320 -220 c
320 -200 l
286 -177 258 -152 237 -126 c
closepath
} exec
fill
} bind def
/K29 {
{
newpath
105 618 m
126 593 143 566 156 538 c
168 511 179 474 187 428 c
198 369 204 304 204 233 c
204 167 197 104 184 43 c
171 -17 152 -67 125 -107 c
98 -146 64 -177 22 -200 c
22 -220 l
105 -183 172 -122 223 -37 c
274 48 300 143 300 247 c
300 318 287 386 261 452 c
235 517 197 575 149 623 c
116 657 73 687 22 712 c
22 692 l
56 669 84 644 105 618 c
closepath
} exec
fill
} bind def
/K30 {
{
newpath
92 103 m
135 26 189 -12 254 -12 c
286 -12 320 0 355 25 c
391 50 419 90 442 145 c
465 200 476 267 476 346 c
476 458 450 547 399 613 c
357 666 311 692 259 692 c
227 692 196 681 166 660 c
127 632 96 591 73 535 c
50 479 38 412 38 335 c
38 244 56 167 92 103 c
92 103 l
closepath
147 497 m
156 557 173 600 197 628 c
215 648 235 658 258 658 c
278 658 295 653 308 642 c
328 626 343 598 356 560 c
372 509 380 444 380 367 c
380 263 372 182 357 125 c
347 87 332 60 312 44 c
292 28 273 20 255 20 c
217 20 188 45 169 94 c
146 153 134 229 134 322 c
134 380 138 438 147 497 c
closepath
} exec
fill
} bind def
/K41 {
{
newpath
511 129 m
524 98 531 75 531 61 c
531 50 526 41 517 33 c
507 26 489 21 462 20 c
462 0 l
731 0 l
731 20 l
703 22 682 29 666 42 c
651 56 634 85 615 130 c
381 694 l
361 694 l
122 135 l
101 87 82 55 65 39 c
57 31 38 25 10 20 c
10 0 l
224 0 l
224 20 l
191 22 170 27 160 35 c
151 43 146 52 146 62 c
146 75 151 94 163 120 c
208 226 l
471 226 l
511 129 l
closepath
224 264 m
342 538 l
455 264 l
224 264 l
closepath
} exec
fill
} bind def
/K61 {
{
newpath
305 9 m
315 -2 328 -8 344 -8 c
379 -8 414 17 452 66 c
452 94 l
432 74 419 62 412 58 c
408 55 403 54 398 54 c
393 54 389 56 385 59 c
381 62 378 69 376 79 c
375 89 374 116 374 160 c
374 316 l
374 361 371 390 365 405 c
356 427 341 443 322 453 c
296 467 262 474 222 474 c
169 474 127 462 98 437 c
69 413 54 386 54 357 c
54 342 58 330 66 321 c
73 312 84 308 96 308 c
108 308 119 312 126 320 c
134 329 138 340 138 356 c
137 384 l
137 399 143 412 156 423 c
168 434 185 440 206 440 c
233 440 254 432 268 415 c
283 398 290 367 290 322 c
290 304 l
212 275 156 251 122 231 c
88 211 65 189 51 167 c
41 150 36 131 36 108 c
36 73 46 44 67 23 c
87 1 113 -10 145 -10 c
165 -10 184 -6 202 3 c
214 9 243 31 290 69 c
290 41 295 21 305 9 c
closepath
191 52 m
172 52 156 60 141 75 c
127 90 120 110 120 133 c
120 152 126 169 137 184 c
148 200 167 214 193 229 c
208 237 240 251 290 270 c
290 101 l
249 68 216 52 191 52 c
191 52 l
closepath
} exec
fill
} bind def
/K62 {
{
newpath
160 712 m
137 712 l
-2 656 l
5 640 l
19 645 31 648 40 648 c
48 648 55 646 60 641 c
66 637 70 629 72 616 c
75 604 76 571 76 518 c
76 46 l
105 26 133 11 160 1 c
188 -9 215 -14 242 -14 c
299 -14 351 7 397 50 c
451 100 478 166 478 248 c
478 318 461 373 426 413 c
391 454 350 474 303 474 c
252 474 204 443 160 381 c
160 712 l
closepath
210 389 m
228 398 245 402 263 402 c
296 402 325 386 351 355 c
377 323 390 278 390 220 c
390 156 377 107 351 74 c
325 41 297 24 267 24 c
249 24 230 28 212 37 c
194 45 177 58 160 75 c
160 349 l
180 369 197 382 210 389 c
210 389 l
closepath
} exec
fill
} bind def
/K63 {
{
newpath
408 180 m
391 139 373 111 355 96 c
329 75 299 64 267 64 c
221 64 185 85 158 127 c
131 169 118 218 118 273 c
118 327 132 370 159 403 c
179 428 206 440 239 440 c
260 440 276 435 288 425 c
299 414 306 399 308 380 c
310 360 314 346 320 339 c
329 326 344 320 364 320 c
378 320 390 324 398 332 c
406 340 410 350 410 363 c
410 389 395 414 366 438 c
337 462 300 474 255 474 c
195 474 143 451 100 406 c
57 361 36 302 36 230 c
36 155 55 96 94 52 c
133 8 179 -14 231 -14 c
275 -14 315 2 351 34 c
387 66 412 112 424 173 c
408 180 l
closepath
} exec
fill
} bind def
/K64 {
{
newpath
356 -14 m
379 -14 l
514 42 l
509 58 l
494 52 483 50 474 50 c
467 50 461 52 455 57 c
450 61 446 69 444 81 c
441 93 440 126 440 180 c
440 712 l
418 712 l
282 656 l
288 640 l
302 645 313 648 322 648 c
329 648 336 646 341 641 c
346 637 350 629 352 617 c
355 604 356 571 356 518 c
356 432 l
330 460 297 474 256 474 c
192 474 139 447 98 392 c
57 337 36 277 36 211 c
36 145 55 91 92 49 c
129 7 172 -14 222 -14 c
247 -14 270 -9 291 1 c
312 12 334 29 356 53 c
356 -14 l
closepath
270 42 m
234 42 202 60 173 96 c
144 131 130 182 130 249 c
130 316 145 366 175 401 c
197 427 223 440 252 440 c
267 440 283 435 299 426 c
315 416 328 402 338 384 c
348 365 354 344 356 321 c
356 88 l
328 57 299 42 270 42 c
270 42 l
closepath
} exec
fill
} bind def
/K65 {
{
newpath
424 286 m
424 343 407 389 374 423 c
341 457 299 474 248 474 c
187 474 137 452 96 408 c
56 364 36 303 36 224 c
36 151 56 94 95 51 c
134 8 181 -14 236 -14 c
287 -14 329 5 363 43 c
397 80 417 123 424 170 c
408 180 l
393 139 375 111 353 94 c
331 78 305 70 275 70 c
231 70 192 89 159 127 c
126 166 110 219 110 286 c
424 286 l
closepath
148 406 m
170 427 194 438 221 438 c
239 438 256 433 273 422 c
289 411 301 397 310 378 c
315 366 318 345 320 316 c
110 316 l
114 354 126 384 148 406 c
closepath
} exec
fill
} bind def
/K67 {
{
newpath
88 103 m
79 87 74 73 74 61 c
74 50 78 41 85 33 c
92 24 104 15 121 4 c
90 -29 73 -47 70 -50 c
53 -72 41 -89 35 -102 c
32 -110 30 -118 30 -126 c
30 -144 43 -162 68 -178 c
112 -207 166 -222 231 -222 c
315 -222 383 -198 435 -149 c
470 -115 488 -80 488 -42 c
488 -15 479 9 460 27 c
442 46 417 58 386 62 c
366 65 321 68 253 70 c
215 71 192 72 183 74 c
167 76 156 80 150 86 c
145 91 142 97 142 102 c
142 109 144 117 149 127 c
154 137 164 148 178 161 c
201 154 225 150 247 150 c
302 150 346 165 379 195 c
412 226 428 262 428 304 c
428 342 419 372 400 396 c
460 396 l
475 396 484 397 486 398 c
489 399 490 400 491 402 c
493 405 494 411 494 420 c
494 428 493 433 491 436 c
490 438 488 440 486 441 c
484 442 475 442 461 442 c
363 442 l
331 463 293 474 249 474 c
196 474 152 458 117 425 c
83 393 66 354 66 309 c
66 279 74 252 89 228 c
104 203 125 184 153 171 c
119 141 97 119 88 103 c
closepath
306 415 m
329 386 340 345 340 293 c
340 252 332 223 316 204 c
299 185 279 176 253 176 c
227 176 205 187 188 209 c
165 237 154 278 154 330 c
154 370 162 399 179 419 c
195 438 216 448 241 448 c
267 448 289 437 306 415 c
306 415 l
closepath
322 -12 m
373 -13 407 -18 425 -25 c
442 -32 450 -44 450 -61 c
450 -85 436 -108 407 -129 c
379 -149 335 -160 275 -160 c
212 -160 164 -150 131 -129 c
112 -117 102 -103 102 -87 c
102 -75 106 -62 114 -47 c
122 -33 133 -17 149 0 c
189 -6 247 -10 322 -12 c
322 -12 l
closepath
} exec
fill
} bind def
/K68 {
{
newpath
143 712 m
4 656 l
11 640 l
27 645 38 648 46 648 c
54 648 61 646 67 641 c
72 637 76 629 78 617 c
81 604 82 571 82 518 c
82 104 l
82 75 80 55 76 45 c
72 36 67 29 61 26 c
49 19 34 16 14 16 c
14 0 l
240 0 l
240 16 l
217 16 200 19 191 24 c
181 29 175 36 171 44 c
168 52 166 72 166 104 c
166 346 l
190 373 211 390 229 399 c
246 408 263 412 279 412 c
295 412 309 408 320 400 c
331 392 339 381 344 365 c
349 349 352 316 352 266 c
352 104 l
352 77 352 61 351 56 c
347 43 341 33 333 26 c
325 19 310 16 289 16 c
278 16 l
278 0 l
504 0 l
504 16 l
483 16 468 18 460 23 c
452 28 446 35 443 43 c
438 54 436 74 436 104 c
436 267 l
436 324 433 363 427 385 c
418 415 404 438 385 452 c
367 467 344 474 318 474 c
297 474 275 468 254 456 c
232 445 203 418 166 377 c
166 712 l
143 712 l
closepath
} exec
fill
} bind def
/K69 {
{
newpath
113 697 m
103 687 98 675 98 661 c
98 647 103 635 113 625 c
123 615 135 610 149 610 c
163 610 175 615 185 625 c
195 635 200 647 200 661 c
200 675 195 687 185 697 c
175 707 163 712 149 712 c
135 712 123 707 113 697 c
closepath
168 474 m
28 418 l
35 402 l
49 407 60 410 70 410 c
78 410 85 407 90 403 c
95 399 99 391 101 380 c
104 365 106 332 106 280 c
106 104 l
106 75 104 55 99 45 c
95 35 89 27 82 23 c
74 18 60 16 38 16 c
38 0 l
258 0 l
258 16 l
237 16 222 18 214 23 c
206 28 200 35 196 45 c
192 55 190 75 190 104 c
190 474 l
168 474 l
closepath
} exec
fill
} bind def
/K6C {
{
newpath
168 712 m
32 656 l
40 640 l
53 645 64 648 72 648 c
80 648 87 646 92 641 c
97 637 100 629 103 617 c
105 604 106 571 106 518 c
106 104 l
106 75 104 55 99 45 c
95 35 89 27 81 23 c
74 18 59 16 38 16 c
38 0 l
264 0 l
264 16 l
240 16 224 18 216 23 c
207 28 200 36 196 45 c
192 55 190 75 190 104 c
190 712 l
168 712 l
closepath
} exec
fill
} bind def
/K6D {
{
newpath
170 474 m
148 474 l
8 418 l
16 402 l
29 407 40 410 50 410 c
58 410 65 407 70 403 c
75 399 79 391 81 380 c
84 365 86 332 86 280 c
86 104 l
86 75 84 55 80 45 c
75 35 69 28 61 23 c
53 18 39 16 18 16 c
18 0 l
244 0 l
244 16 l
219 16 203 18 195 23 c
186 28 180 35 176 44 c
172 53 170 73 170 104 c
170 349 l
185 367 206 383 234 398 c
251 407 269 412 287 412 c
313 412 333 402 348 381 c
359 365 364 338 364 301 c
364 104 l
364 76 363 59 362 53 c
358 41 351 31 342 25 c
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Black Black eop end
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%%Page: 11 29
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%%EndData
showpage
%%Trailer
end
%%EOF

%%EndDocument
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%%EndData
showpage
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end
%%EOF

%%EndDocument
 @endspecial Black 1336 2418 a(Figure)32 b(2.5:)43 b(Complete)34
b(Asymptotic)g(Stabilit)m(y)p Black Black Black 630 2814
a Fy(Pr)-5 b(o)g(of:)p Black 49 w FJ(W)d(e)39 b(sho)m(w)h(b)m(y)h(con)m
(tradiction)f(that)2344 2788 y(^)2319 2814 y FH(A)f FJ(is)h(uniformly)g
(forw)m(ard)f(stable,)874 2952 y(and)33 b(hence)h(b)m(y)f(de\014nition)
h(completely)g(stable.)874 3135 y(Assume)1266 3110 y(^)1241
3135 y FH(A)j FJ(is)h(not)g(uniformly)g(forw)m(ards)h(stable,)g(but)f
(uniformly)h(pull-)874 3274 y(bac)m(k)f(stable.)56 b(Then)38
b(for)e(eac)m(h)h FH(\017)e(>)g FJ(0,)i(there)h(exists)g(a)e
FH(\016)j FJ(=)34 b FH(\016)t FJ(\()p FH(\017)p FJ(\))j(as)g(de-)874
3412 y(\014ned)25 b(for)f(uniform)g(pullbac)m(k)i(stabilit)m(y)-8
b(.)42 b(No)m(w,)26 b(since)2890 3387 y(^)2864 3412 y
FH(A)e FJ(is)h(not)f(uniformly)874 3551 y(forw)m(ards)37
b(stable,)i(there)f(exists)g(at)f(least)g(one)g(initial)g(state)h
(within)f(the)874 3689 y(neigh)m(b)s(ourho)s(o)s(d)c(system)1896
3664 y(^)1864 3689 y FG(N)1946 3718 y FF(\016)o(;)2014
3701 y FE(^)1996 3718 y FF(A)2085 3689 y FJ(that)f(will)h(escap)s(e)h
(an)e FH(\017)p FJ(-neigh)m(b)s(ourho)s(o)s(d)874 3827
y(of)1015 3802 y(^)989 3827 y FH(A)37 b FJ(at)f(some)h(later)f(time.)56
b(That)37 b(is,)h(there)f(exists)h(a)f FH(p)d FG(2)g
FH(P)14 b FJ(,)37 b FH(t)3322 3791 y FB(\003)3396 3827
y FG(2)e FC(R)3569 3791 y FE(+)874 3966 y FJ(and)e(at)f(least)h(one)g
(initial)g(state)g FH(x)2166 3981 y FE(0)2233 3966 y
FG(2)28 b(N)2409 3981 y FF(\016)2447 3966 y FJ(\()p FH(A)p
FJ(\()p FH(p)p FJ(\)\))k(suc)m(h)i(that)1651 4208 y(dist\(\010)1916
4223 y FE(\()p FF(t)1968 4204 y Fs(\003)2006 4223 y FF(;p)p
FE(\))2092 4208 y FJ(\()p FH(x)2185 4223 y FE(0)2225
4208 y FJ(\))p FH(;)17 b(A)p FJ(\()p FH(\022)2463 4223
y FF(t)2488 4204 y Fs(\003)2529 4208 y FH(p)p FJ(\)\))27
b(=)h FH(\017:)874 4450 y FJ(Ho)m(w)m(ev)m(er)i(b)m(y)f(uniform)f
(pullbac)m(k)h(stabilit)m(y)g(at)f FH(\022)2674 4465
y FF(t)2699 4446 y Fs(\003)2740 4450 y FH(p)p FJ(,)h(it)e(m)m(ust)i
(necessarily)874 4588 y(b)s(e)k(that)1651 4731 y(dist\(\010)1916
4746 y FE(\()p FF(t)1968 4727 y Fs(\003)2006 4746 y FF(;p)p
FE(\))2092 4731 y FJ(\()p FH(x)2185 4746 y FE(0)2225
4731 y FJ(\))p FH(;)17 b(A)p FJ(\()p FH(\022)2463 4746
y FF(t)2488 4727 y Fs(\003)2529 4731 y FH(p)p FJ(\)\))27
b FH(<)h(\017:)874 4928 y FJ(This)47 b(is)f(a)f(con)m(tradiction.)83
b(Hence)2314 4903 y(^)2288 4928 y FH(A)46 b FJ(m)m(ust)g(b)s(e)g
(uniformly)g(forw)m(ards)874 5066 y(stable,)33 b(and)g(th)m(us)h
(completely)g(stable.)p 3841 5391 4 66 v 3845 5329 59
4 v 3845 5391 V 3903 5391 4 66 v Black Black eop end
%%Page: 46 64
TeXDict begin 46 63 bop Black 591 266 a FJ(46)413 b Fx(CHAPTER)34
b(2.)76 b(NON-A)m(UTONOMOUS)34 b(D)m(YNAMICAL)g(SYSTEMS)p
Black Black 591 530 a FI(Lemma)45 b(2.1.2.)p Black 45
w Fy(If)1433 505 y FJ(^)1408 530 y FH(A)37 b FJ(=)h FG(f)p
FH(A)p FJ(\()p FH(p)p FJ(\);)17 b FH(p)36 b FG(2)i FH(P)14
b FG(g)40 b Fy(is)f(uniformly)h(forwar)-5 b(ds)40 b(stable)f(then)h(it)
h(is)591 669 y(c)-5 b(ompletely)34 b(stable.)p Black
630 960 a(Pr)-5 b(o)g(of:)p Black 49 w FJ(Similarly)d(,)51
b(w)m(e)d(will)g(sho)m(w)g(b)m(y)g(con)m(tradiction)g(that)3009
934 y(^)2984 960 y FH(A)f FJ(is)g(uniformly)874 1098
y(pullbac)m(k)34 b(stable,)g(and)e(hence)i(b)m(y)g(de\014nition)f
(completely)i(stable.)874 1275 y(Assume)1272 1249 y(^)1246
1275 y FH(A)44 b FJ(is)f(not)g(uniformly)i(pullbac)m(k)f(stable,)j(but)
d(uniformly)g(for-)874 1413 y(w)m(ards)g(stable.)74 b(Then)44
b(for)e(eac)m(h)i FH(\017)h(>)f FJ(0,)h(there)f(exists)g(a)f
FH(\016)48 b FJ(=)d FH(\016)t FJ(\()p FH(\017)p FJ(\))e(as)874
1551 y(de\014ned)34 b(for)e(uniform)h(forw)m(ard)g(stabilit)m(y)-8
b(.)874 1728 y(No)m(w,)52 b(since)1423 1703 y(^)1397
1728 y FH(A)c FJ(is)g(not)g(pullbac)m(k)i(stable,)i(there)d(exists)h
(at)d(least)i(one)874 1867 y(initial)36 b(sequence)i(within)e(the)g
(neigh)m(b)s(ourho)s(o)s(d)f(system)3071 1841 y(^)3039
1867 y FG(N)3121 1895 y FF(\016)o(;)3188 1878 y FE(^)3171
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%%EndData
showpage
%%Trailer
end
%%EOF

%%EndDocument
 @endspecial Black 1143 4975 a(Figure)h(2.9:)43 b(Complete)34
b(A)m(ttractor)f(in)f(T)-8 b(rac)m(king)34 b(Con)m(trol)p
Black Black 3841 5391 4 66 v 3845 5329 59 4 v 3845 5391
V 3903 5391 4 66 v Black Black eop end
%%Page: 60 78
TeXDict begin 60 77 bop Black 591 266 a FJ(60)413 b Fx(CHAPTER)34
b(2.)76 b(NON-A)m(UTONOMOUS)34 b(D)m(YNAMICAL)g(SYSTEMS)p
Black 591 530 a Fw(2.2.4)136 b(A)l(ttractors)45 b(and)g(Asymptotic)g
(Stabilit)l(y)591 806 y FJ(The)h(follo)m(wing)h(theorem)f(and)g(its)h
(lemma)g(pro)m(vide)g(an)f(equiv)-5 b(alen)m(t)47 b(coun)m(terpart)g
(in)591 945 y(non-autonomous)41 b(dynamical)h(systems)i(for)d(the)g
(relationship)i(b)s(et)m(w)m(een)g(semi-group)591 1083
y(attractors)28 b(and)g(asymptotically)i(stable)g(sets)f(in)g
(autonomous)g(systems)i(giv)m(en)e(in)g(The-)591 1222
y(orem)j(1.2.1.)p Black 591 1452 a FI(Theorem)38 b(2.2.1.)p
Black 42 w Fy(A)d(pul)5 b(lb)-5 b(ack)34 b(attr)-5 b(actor)36
b(is)e(pul)5 b(lb)-5 b(ack)34 b(e)-5 b(qui-asymptotic)g(al)5
b(ly)35 b(stable.)p Black 630 1784 a(Pr)-5 b(o)g(of:)p
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(automatically)f(satis\014es)h(the)874 1922 y(pullbac)m(k)27
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3895 y FJ(\()p FH(A)p FJ(\()p FH(\022)2027 3910 y FB(\000)p
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FH(t)28 b(>)g(T)8 b(:)874 4140 y FJ(No)m(w)45 b(for)f
FH(j)51 b FJ(large)45 b(enough)g(so)g(that)f(b)s(oth)h
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y Fs(\000)p FA(t)1756 4564 y(j)1792 4539 y FE(\()p FF(p)p
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1718 y FE(0)2656 1703 y FG(2)i FH(B)5 b FJ(\()p FH(p)p
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b FJ(0,)874 1842 y(the)33 b FH(x)1097 1857 y FF(j)1130
1838 y Fs(0)1189 1842 y FJ(satisfy)1244 2116 y FH(x)1299
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2112 y Fs(0)3080 2116 y FH(>)27 b(\034)6 b(:)874 2466
y FJ(As)33 b FH(x)1073 2481 y FE(0)1146 2466 y FJ(is)g(the)g(limit,)g
(w)m(e)g(ha)m(v)m(e)h(for)e(an)m(y)i FH(\034)39 b FG(\025)28
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FG(2)p 1913 2637 865 4 v 1917 2659 a Ft([)1913 2869 y
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3103 y FJ(and)33 b(hence)1624 3277 y FH(x)1679 3292 y
FE(0)1746 3277 y FG(2)1850 3183 y Ft(\\)1841 3393 y FF(\034)8
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3393 y FF(t)p FB(\025)p FF(\034)2122 3277 y FJ(\010)2192
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874 3732 y(dist)q(\()p FH(x)1125 3747 y FE(0)1165 3732
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FJ(w)m(as)h(arbitrary)-8 b(,)34 b(\(2.19\))f(holds)874
3870 y(true)g(for)f(all)g FH(p)c FG(2)g FH(P)14 b FJ(.)874
4080 y(F)-8 b(rom)36 b(\(2.19\),)h(for)f(an)m(y)i FH(\017)d(>)f
FJ(0,)k(and)e(eac)m(h)i FH(p)c FG(2)i FH(P)14 b FJ(,)37
b(there)g(exists)h(a)f FH(t)3478 4095 y FE(1)3552 4080
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FH(\017;)17 b(p)p FJ(\))32 b(suc)m(h)i(that)1532 4492
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4508 y FE(\()p FF(t)1820 4517 y Fu(1)1856 4508 y FF(;\022)1909
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4507 y FB(\000)p FF(t)2366 4516 y Fu(1)2406 4492 y FH(p)p
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4878 y(^)3014 4905 y FH(\016)k FJ(de\014nes)3437 4879
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%%Page: 67 85
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y Fs(\000)p FA(t)2301 1664 y Fu(1)2340 1640 y FF(p)p
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1763 y(describ)s(ed)39 b(in)f(the)f(de\014nition)i(of)d(the)i(pullbac)m
(k)h(absorbing)f(neigh)m(b)s(our-)874 1901 y(ho)s(o)s(d.)67
b(Hence)1504 1876 y(^)1478 1901 y FH(A)41 b FJ(satis\014es)i(the)e
(pullbac)m(k)h(prop)s(ert)m(y)g(for)e(eac)m(h)i FH(p)f
FG(2)h FH(P)14 b FJ(,)874 2040 y(that)32 b(is,)1391 2178
y(lim)1375 2238 y FF(t)p FB(!1)1558 2178 y FH(H)1647
2137 y FB(\003)1686 2178 y FJ(\(\010)1794 2193 y FE(\()p
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2174 y Fs(\003)2194 2209 y FA(p)2237 2178 y FJ(\()p FH(A)p
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b(e)30 b FJ(:)43 b(W)-8 b(e)33 b(are)f(required)h(to)f(pro)m(v)m(e)h
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2532 y(^)1616 2557 y FH(A)f FJ(is)g(equiv)-5 b(arian)m(t)38
b(as)f(de\014ned)h(in)f(\(2.13\).)55 b(Consider)874 2696
y(an)38 b(elemen)m(t)i(of)e(the)g(family)h FH(A)p FJ(\()p
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2696 y FH(p)p FJ(\))e FG(2)2505 2671 y FJ(^)2479 2696
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3060 y FH(A)p FJ(\()p FH(\022)1353 3075 y FB(\000)p FF(t)1433
3056 y Fs(\003)1474 3060 y FH(p)p FJ(\))g(=)1702 2965
y Ft(\\)1692 3176 y FF(\034)8 b FB(\025)p FE(0)p 1838
2943 1191 4 v 1842 2965 a Ft([)1838 3176 y FF(t)p FB(\025)p
FF(\034)1974 3060 y FJ(\010)2044 3075 y FE(\()p FF(t;\022)2149
3090 y Fs(\000)p Fu(\()p FA(t)p Fu(+)p FA(t)2316 3076
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3705 y FF(;\022)1203 3720 y Fs(\000)p FA(t)1275 3706
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4065 y Fs(\000)p FA(t)1747 4051 y Fs(\003)1787 4050 y
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a([)1949 4150 y FF(t)p FB(\025)p FF(\034)2085 4034 y
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Fu(\()p FA(t)p Fu(+)p FA(t)2427 4051 y Fs(\003)2463 4065
y Fu(\))2491 4050 y FF(p)p FE(\))2558 4034 y FJ(\()p
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4034 y FH(p)p FJ(\)\))3139 3864 y Ft(!)3235 4034 y FH(:)3379
3869 y FJ(\(2.22\))874 4336 y(The)37 b(inclusion)g(")p
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FG(\033)p FJ(",)g(let)f FH(x)g FJ(b)s(e)g(an)g(elemen)m(t)874
4474 y(of)c(the)h(righ)m(t)g(hand)g(side,)1171 4763 y
FH(x)28 b FG(2)1357 4669 y Ft(\\)1348 4879 y FF(\034)8
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y Fs(\000)p FA(t)1778 4780 y Fs(\003)1818 4779 y FF(p)p
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%%Page: 68 86
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2581 y FF(\034)8 b FB(\025)p FE(0)p 1824 2349 V 1828
2371 a Ft([)1824 2581 y FF(t)p FB(\025)p FF(\034)1960
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%%Trailer
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 @endspecial 591 4326 a(Figure)47 b(2.11:)73 b(Pullbac)m(k)49
b(A)m(ttraction)f(to)f(a)g(Non-Autonomously)h(P)m(erturb)s(ed)h(Limit)
591 4464 y(Cycle)p Black 591 4838 a(By)32 b(Theorem)h(2.3.1)e(w)m(e)h
(v)m(erify)h(the)f(existence)j(of)c(a)g(pullbac)m(k)i(attractor,)3423
4812 y(^)3398 4838 y FH(A)e FJ(con)m(tained)591 4976
y(within)36 b FH(B)5 b FJ(.)51 b(In)35 b(fact,)h(b)m(y)f(n)m
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(w)m(e)g(obtain)g(a)f(go)s(o)s(d)f(appro)m(ximation)j(for)2926
5089 y(^)2901 5114 y FH(A)p FJ(.)42 b(Using)28 b(this)g(approac)m(h,)
591 5253 y(initial)33 b(states)i(from)e(the)h(inner)g(and)g(outer)f(b)s
(oundaries)i(of)e FH(B)38 b FJ(w)m(ere)d(mapp)s(ed)g(\(using)f(a)591
5391 y(Runge-Kutta)22 b(metho)s(d\))h(from)g(a)g(pulled)h(bac)m(k)g
(initial)g(time)f(to)m(w)m(ards)h(a)f(\014xed)h(\014nal)g(time.)p
Black Black eop end
%%Page: 71 89
TeXDict begin 71 88 bop Black 591 266 a Fx(2.3.)75 b(ABSORBING)33
b(NEIGHBOURHOODS)1397 b FJ(71)p Black 591 530 a(This)36
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b(v)-5 b(alues)36 b(of)e(the)i(\014nal)f(time)h(b)s(et)m(w)m(een)i
FD(t)32 b FJ(=)g(40)591 669 y(and)43 b FD(t)j FJ(=)f(50,)h(and)d(the)h
(\014nal)f(appro)m(ximations)h(for)f FH(A)p FJ(\()p FD(t)p
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1222 y FJ(.)p 3841 1531 4 66 v 3845 1469 59 4 v 3845
1531 V 3903 1531 4 66 v 591 1741 a(A)d(con)m(v)m(erse)j(result)e(for)f
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(conditions)591 1879 y(on)45 b(the)g(neigh)m(b)s(ourho)s(o)s(d.)81
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591 2018 y(pullbac)m(k)40 b(attractor)e(for)f(whic)m(h)k(its)e(pullbac)
m(k)h(attracting)e(neigh)m(b)s(ourho)s(o)s(d)g(system)j(is)591
2156 y(kno)m(wn)k(to)f(b)s(e)h(indep)s(enden)m(t)h(of)e
FH(p)h FJ(\(that)f(is,)2373 2130 y(^)2369 2156 y FH(\016)51
b FJ(=)d FH(\016)t FJ(\),)f(then)e(it)g(can)f(b)s(e)h(sho)m(wn)h(that)
591 2295 y(there)37 b(exists)i(an)e(asso)s(ciated)h(pullbac)m(k)g
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(pro)m(v)m(ed)i(in)d([15],)j(and)e(reiterated)g(in)g(a)g(similar)591
2710 y(fashion)32 b(here)i(for)e(completeness.)p Black
591 2948 a FI(Theorem)42 b(2.3.2.)p Black 44 w Fy(L)-5
b(et)39 b FG(f)p FJ(\010)1667 2964 y FE(\()p FF(t;p)p
FE(\))1807 2948 y FJ(;)17 b FH(t)34 b FG(2)h FC(R)2093
2912 y FE(+)2152 2948 y FH(;)17 b(p)33 b FG(2)i FH(P)14
b FG(g)38 b Fy(b)-5 b(e)38 b(a)g(c)-5 b(o)g(cycle)38
b(of)g(c)-5 b(ontinuous)38 b(map-)591 3086 y(pings)d(on)h
FH(E)43 b Fy(with)36 b(a)g(pul)5 b(lb)-5 b(ack)36 b(attr)-5
b(actor)2205 3061 y FJ(^)2179 3086 y FH(A)31 b FJ(=)f
FG(f)p FH(A)p FJ(\()p FH(p)p FJ(\);)17 b FH(p)31 b FG(2)g
FH(P)14 b FG(g)35 b Fy(that)i(pul)5 b(lb)-5 b(ack)36
b(attr)-5 b(acts)591 3225 y(a)39 b FH(\016)t Fy(-neighb)-5
b(ourho)g(o)g(d)37 b(system)1753 3200 y FJ(^)1721 3225
y FG(N)1803 3253 y FF(\016)o(;)1870 3237 y FE(^)1853
3253 y FF(A)1948 3225 y Fy(for)i(some)g FH(\016)g(>)d
FJ(0)p Fy(.)58 b(Then)39 b(ther)-5 b(e)39 b(exists)g(a)g(Pul)5
b(lb)-5 b(ack)591 3363 y(A)n(bsorbing)33 b(Neighb)-5
b(ourho)g(o)g(d)1718 3338 y FJ(^)1694 3363 y FH(B)5 b
Fy(,)35 b(asso)-5 b(ciate)g(d)34 b(with)2531 3338 y FJ(^)2506
3363 y FH(A)p Fy(.)p Black 630 3706 a(Pr)-5 b(o)g(of:)p
Black 874 3844 a FJ(By)42 b(Theorem)h(2.2.1,)g(the)f(pullbac)m(k)i
(attractor)2751 3819 y(^)2725 3844 y FH(A)e FJ(is)g Fy(pul)5
b(lb)-5 b(ack)43 b(asymp-)874 3982 y(totic)-5 b(al)5
b(ly)40 b(stable)p FJ(,)f(and)f(hence)h(also)f Fy(pul)5
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s8Drts8<$"rVZ`PrPJSb~>
a8Z,>hZ!QUqSrR9_uB]:qPsSr`;]f;m#V,Gn;dPLjo5;\JS'IFk%4SngA_6Trr<&trr<&urr<&t
rrW9$rrE&urr<-#!!&YkJ,~>
a8Z,>hZ!QUqSrR9_uB]:qPsSr`;]f;m#V,Gn;dPLjo5;\JS'IFk%4SngA_6Trr<&trr<&urr<&t
rrW9$rrE&urr<-#!!&YkJ,~>
a8Gu<hYdESqSrR9_u0Q8qPsSr`;KZ9m#V,Gn;dPLjo#/ZJS'IFk%4SngAM*Rr;Zirr;Zisr;Zir
r<!&ur;ciqrr<-!!!(mS_Z,,~>
a8Z,>hZ!QUqSrR9_uB]:q5XJq`W#o<m>q5HmuIGKjo5;\JS'IFk%4Sng].9R!ri6#rr;uurr2ru
rVls"s8N)trrW9$!!&YkJ,~>
a8Z,>hZ!QUqSrR9_uB]:q5XJq`W#o<m>q5HmuIGKjo5;\JS'IFk%4Sng].9R!ri6#rr;uurr2ru
rVls"s8N)trrW9$!!&YkJ,~>
a8Gu<hYdESqSrR9_u0Q8q5XJq`Vfc:m>q5HmuIGKjo#/ZJS'IFk%4Sng\q-P!rW*!rr)isrqufs
rVZfurVZ`pr<!&u!!(mS_Z,,~>
a8Z,>hZ!QUqSrR9_uB]:q5XJq`W#o<m>q5Hn;dPLjSo2[JS'IFk%4Sng].9Rs8N'!rr2ruq>^Hp
s8N'!rVls"s8N(ks*t~>
a8Z,>hZ!QUqSrR9_uB]:q5XJq`W#o<m>q5Hn;dPLjSo2[JS'IFk%4Sng].9Rs8N'!rr2ruq>^Hp
s8N'!rVls"s8N(ks*t~>
a8Gu<hYdESqSrR9_u0Q8q5XJq`Vfc:m>q5Hn;dPLjS]&YJS'IFk%4Sng\q-Ps8;otrqufsq>L<n
s8;otrVZfurVZ`QrPJSb~>
huETSq>^9krVlithZ!QUqSrR9_uB]:q5XJq`W#o<m>q8ImuIGKjSo2[JS'IFk%4Sng]%6Rrr;uu
s8N'!qu?Wqrr2rurVls"s8N(ks*t~>
huETSq>^9krVlithZ!QUqSrR9_uB]:q5XJq`W#o<m>q8ImuIGKjSo2[JS'IFk%4Sng]%6Rrr;uu
s8N'!qu?Wqrr2rurVls"s8N(ks*t~>
hu3HQq>L-irVZ]rhYdESqSrR9_u0Q8q5XJq`Vfc:m>q8ImuIGKjS]&YJS'IFk%4Sng\h*Prr)is
s8;otqu-KorqufsrVZfurVZ`QrPJSb~>
i;`fWrr;uuqu?Zrp\t3nhZ!QUq8WI8`;]f;q5XJq`W#o<mZ7>In;dPLj8T)ZJS'IFk%4Snh#I3N
s8N'!r;Zcsr;Q`srVls"s8N(ks*t~>
i;`fWrr;uuqu?Zrp\t3nhZ!QUq8WI8`;]f;q5XJq`W#o<mZ7>In;dPLj8T)ZJS'IFk%4Snh#I3N
s8N'!r;Zcsr;Q`srVls"s8N(ks*t~>
i;NZUrr)isqu-Npp\b'lhYdESq8WI8`;KZ9q5XJq`Vfc:mZ7>In;dPLj8ArXJS'IFk%4Snh#7'L
s8;otr;HWqr;?TqrVZfurVZ`QrPJSb~>
i;`fWrr;rtr;Zcsp\t3nhZ!QUq8WI8`;]f;q5XJq`W#o<mZ7>In;dPLj8T)ZJS'IFk%4Snh#IBS
rVls"s8N)ts8N)rrr<&trrW9$rrA\kJ,~>
i;`fWrr;rtr;Zcsp\t3nhZ!QUq8WI8`;]f;q5XJq`W#o<mZ7>In;dPLj8T)ZJS'IFk%4Snh#IBS
rVls"s8N)ts8N)rrr<&trrW9$rrA\kJ,~>
i;NZUrr)frr;HWqp\b'lhYdESq8WI8`;KZ9q5XJq`Vfc:mZ7>In;dPLj8ArXJS'IFk%4Snh#76Q
rVZfurVZ`prVllpr;Zirr<!&ur;b^Q_Z,,~>
iW&oXr;Zcsr;ZWoqu6WrhZ!QUq8WI8`;]f;q5XJq`W#o<muRGJn;dPLir8uYJS'IFk%4Snh#@?S
r;Zcs!WN/urr<&qs8N*!rr<&urr<%ks*t~>
iW&oXr;Zcsr;ZWoqu6WrhZ!QUq8WI8`;]f;q5XJq`W#o<muRGJn;dPLir8uYJS'IFk%4Snh#@?S
r;Zcs!WN/urr<&qs8N*!rr<&urr<%ks*t~>
iVicVr;HWqr;HKmqu$KphYdESq8WI8`;KZ9q5XJq`Vfc:muRGJn;dPLir&iWJS'IFk%4Snh#.3Q
r;HWq!W<#sr;ZiorVlltr;Zisr;ZiSrPJSb~>
iW&oXr;Zcsr;Zcsrr;uur;ZQmjo5;\q8WI8`;]f;q5XJq`W#o<muRGJn;dPLir8uYJS'IFk%4Sn
h>dKTr;Zcs"TJH%s8Voqrr;oss8W*!TDsE~>
iW&oXr;Zcsr;Zcsrr;uur;ZQmjo5;\q8WI8`;]f;q5XJq`W#o<muRGJn;dPLir8uYJS'IFk%4Sn
h>dKTr;Zcs"TJH%s8Voqrr;oss8W*!TDsE~>
iVicVr;HWqr;HWqrr)isr;HEkjo#/Zq8WI8`;KZ9q5XJq`Vfc:muRGJn;dPLir&iWJS'IFk%4Sn
h>R?Rr;HWq"T8<#rVcQmrr)cqs8DrthYk=kJ,~>
iW&oXr;Zcsr;Q`srVuisrVlithZ!QUq8WI8`;]f;q5XJq`W#o<muRJKn;mSLir8uYJS'IFk%4Sn
df0:In,NCfTDsE~>
iW&oXr;Zcsr;Q`srVuisrVlithZ!QUq8WI8`;]f;q5XJq`W#o<muRJKn;mSLir8uYJS'IFk%4Sn
df0:In,NCfTDsE~>
iVicVr;HWqr;?TqrVc]qrVZ]rhYdESq8WI8`;KZ9q5XJq`Vfc:muRJKn;mSLir&iWJS'IFk%4Sn
des.Gn,<7dhYk=kJ,~>
iW&oXr;ZcspAb-mrVlithZ!QUq8WI8`;]f;q5XJq`W#o<n;mPKn;mSLir8uYJS'IFk%4Sndf0:I
n,E@fT)X<~>
iW&oXr;ZcspAb-mrVlithZ!QUq8WI8`;]f;q5XJq`W#o<n;mPKn;mSLir8uYJS'IFk%4Sndf0:I
n,E@fT)X<~>
iVicVr;HWqpAP!krVZ]rhYdESq8WI8`;KZ9q5XJq`Vfc:n;mPKn;mSLir&iWJS'IFk%4Sndes.G
n,34dh>P4jJ,~>
iW&oXr;ZcspAb-mrVlithZ!QUq8WI8`;]f;q5XJq`W#o<n;mPKnW3YLir8uYJS'IFk%4SndJj1H
nG`IgT)X<~>
iW&oXr;ZcspAb-mrVlithZ!QUq8WI8`;]f;q5XJq`W#o<n;mPKnW3YLir8uYJS'IFk%4SndJj1H
nG`IgT)X<~>
iVicVr;HWqpAP!krVZ]rhYdESq8WI8`;KZ9q5XJq`Vfc:n;mPKnW3YLir&iWJS'IFk%4SndJX%F
nGN=eh>P4jJ,~>
iW&lWrr;uuqu6Wrr;ZcsrVlithZ!QUq8WI8`;]f;po=Ap`r?#=nW3YLn;mSLiVrlXJS'IFk%4Sn
JcFL)J,~>
iW&lWrr;uuqu6Wrr;ZcsrVlithZ!QUq8WI8`;]f;po=Ap`r?#=nW3YLn;mSLiVrlXJS'IFk%4Sn
JcFL)J,~>
iVi`Urr)isqu$Kpr;HWqrVZ]rhYdESq8WI8`;KZ9po=Ap`r,l;nW3YLn;mSLiV``VJS'IFk%4Sn
Rf1,'J,~>
i;`fWrr;uuqu?Zrrr;uur;Q`shZ!QUq8WI8`;]f;po=Ap`r?#=nW3YLnW3YLiVrlXJS'IFk%4Sn
JcFL)J,~>
i;`fWrr;uuqu?Zrrr;uur;Q`shZ!QUq8WI8`;]f;po=Ap`r?#=nW3YLnW3YLiVrlXJS'IFk%4Sn
JcFL)J,~>
i;NZUrr)isqu-Nprr)isr;?TqhYdESq8WI8`;KZ9po=Ap`r,l;nW3YLnW3YLiV``VJS'IFk%4Sn
Rf1,'J,~>
huETSrr;uurr;lrqu6WrhZ!QUq8WI8`;]f;po=Ap`r?#=nW3YLnrN_LiVrlXJS'IFk%4SnJcFL)
J,~>
huETSrr;uurr;lrqu6WrhZ!QUq8WI8`;]f;po=Ap`r?#=nW3YLnrN_LiVrlXJS'IFk%4SnJcFL)
J,~>
hu3HQrr)isrr)`pqu$KphYdESq8WI8`;KZ9po=Ap`r,l;nW3YLnrN_LiV``VJS'IFk%4SnRf1,'
J,~>
a8Z,>hZ!QUq8WI8`;]f;po=Ap`r?#=nrNbMnrN\KiVrlXJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUq8WI8`;]f;po=Ap`r?#=nrNbMnrN\KiVrlXJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESq8WI8`;KZ9po=Ap`r,l;nrNbMnrN\KiV``VJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpr<@7`W#o<po=Ap`r?#=nrNbMo8ibKiVrlXJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpr<@7`W#o<po=Ap`r?#=nrNbMo8ibKiVrlXJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpr<@7`Vfc:po=Ap`r,l;nrNbMo8ibKiV``VJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpr<@7`W#o<po=Ap`r?#=nrNeNnrNYJiVrlXJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpr<@7`W#o<po=Ap`r?#=nrNeNnrNYJiVrlXJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpr<@7`Vfc:po=Ap`r,l;nrNeNnrNYJiV``VJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpr<@7`W#o<po=Ap`r?#=o8ikNo8i_JiVrlXJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpr<@7`W#o<po=Ap`r?#=o8ikNo8i_JiVrlXJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpr<@7`Vfc:po=Ap`r,l;o8ikNo8i_JiV``VJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpr<@7`W#o<po=Ap`r?#=o8ikNnW3SJiVrlXJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpr<@7`W#o<po=Ap`r?#=o8ikNnW3SJiVrlXJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpr<@7`Vfc:po=Ap`r,l;o8ikNnW3SJiV``VJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpr<@7`W#o<po=Ap`r?#=oT/tOn;d\Ps8N'!iVrlXJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpr<@7`W#o<po=Ap`r?#=oT/tOn;d\Ps8N'!iVrlXJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpr<@7`Vfc:po=Ap`r,l;oT/tOn;d\PrVZ]riV``VJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpr<@7`W#o<pT"8oa8Z,>oT/tOn;dPLh>[HTJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpr<@7`W#o<pT"8oa8Z,>oT/tOn;dPLh>[HTJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpr<@7`Vfc:pT"8oa8Gu<oT/tOn;dPLh>I<RJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpr<@7`W#o<pT"8oa8Z,>oT/tOn;dPLh>[HTJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpr<@7`W#o<pT"8oa8Z,>oT/tOn;dPLh>[HTJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpr<@7`Vfc:pT"8oa8Gu<oT/tOn;dPLh>I<RJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpr<@7`W#o<pT"8oa8Z,>ooK(Pn;dPLh#@?SJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpr<@7`W#o<pT"8oa8Z,>ooK(Pn;dPLh#@?SJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpr<@7`Vfc:pT"8oa8Gu<ooK(Pn;dPLh#.3QJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpr<@7`W#o<pT"8oa8Z,>ooK(Pn;dPLh#@?SJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpr<@7`W#o<pT"8oa8Z,>ooK(Pn;dPLh#@?SJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpr<@7`Vfc:pT"8oa8Gu<ooK(Pn;dPLh#.3QJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpW!76`r?#=pT"8oa8Z,>ooK(Pn;dPLh#@?SJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpW!76`r?#=pT"8oa8Z,>ooK(Pn;dPLh#@?SJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpW!76`r,l;pT"8oa8Gu<ooK(Pn;dPLh#.3QJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpW!76`r?#=pT"8oa8Z,>p5f1Qn;dPLg]%6RJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpW!76`r?#=pT"8oa8Z,>p5f1Qn;dPLg]%6RJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpW!76`r,l;pT"8oa8Gu<p5f1Qn;dPLg\h*PJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpW!76`r?#=pT"8oa8Z,>p5f1Qn;dPLg]%6RJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpW!76`r?#=pT"8oa8Z,>p5f1Qn;dPLg]%6RJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpW!76`r,l;pT"8oa8Gu<p5f1Qn;dPLg\h*PJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpW!76`r?#=pT"8oa8Z,>p5f1Qn;dPLg]%6RJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpW!76`r?#=pT"8oa8Z,>p5f1Qn;dPLg]%6RJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpW!76`r,l;pT"8oa8Gu<p5f1Qn;dPLg\h*PJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpW!76`r?#=pT"8oa8Z,>pQ,:Rn;dPLgA_-QJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpW!76`r?#=pT"8oa8Z,>pQ,:Rn;dPLgA_-QJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpW!76`r,l;pT"8oa8Gu<pQ,:Rn;dPLgAM!OJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpW!76`r?#=p8\/naSu5?pQ,:Rn;dPLgA_-QJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpW!76`r?#=p8\/naSu5?pQ,:Rn;dPLgA_-QJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpW!76`r,l;p8\/naSc)=pQ,:Rn;dPLgAM!OJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpW!76`r?#=p8\/naSu5?pQ,:Rn;dPLgA_-QJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpW!76`r?#=p8\/naSu5?pQ,:Rn;dPLgA_-QJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpW!76`r,l;p8\/naSc)=pQ,:Rn;dPLgAM!OJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpW!76`r?#=p8\/naSu5?plGCSmuIGKgA_-QJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpW!76`r?#=p8\/naSu5?plGCSmuIGKgA_-QJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpW!76`r,l;p8\/naSc)=plGCSmuIGKgAM!OJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUpW!76`r?#=p8\/naSu5?plGCSn;dPLg&D$PJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUpW!76`r?#=p8\/naSu5?plGCSn;dPLg&D$PJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESpW!76`r,l;p8\/naSc)=plGCSn;dPLg&1mNJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUp;[.5a8Z,>p8\/naSu5?plGCSn;dPLg&D$PJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUp;[.5a8Z,>p8\/naSu5?plGCSn;dPLg&D$PJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESp;[.5a8Gu<p8\/naSc)=plGCSn;dPLg&1mNJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUp;[.5a8Z,>p8\/naSu5?q2bLTmuIGKg&D$PJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUp;[.5a8Z,>p8\/naSu5?q2bLTmuIGKg&D$PJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESp;[.5a8Gu<p8\/naSc)=q2bLTmuIGKg&1mNJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUp;[.5a8Z,>p8\/naSu5?q2bLTn;dPLf`(pOJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUp;[.5a8Z,>p8\/naSu5?q2bLTn;dPLf`(pOJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESp;[.5a8Gu<p8\/naSc)=q2bLTn;dPLf_kdMJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUp;[.5a8Z,>orA&mao;>@q2bLTn;dPLf`(pOJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUp;[.5a8Z,>orA&mao;>@q2bLTn;dPLf`(pOJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESp;[.5a8Gu<orA&mao)2>q2bLTn;dPLf_kdMJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUp;[.5a8Z,>orA&mao;>@qN(RTn;dPLf`(pOJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUp;[.5a8Z,>orA&mao;>@qN(RTn;dPLf`(pOJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESp;[.5a8Gu<orA&mao)2>qN(RTn;dPLf_kdMJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUp;[.5a8Z,>orA&mao;>@qN(UUmuIGKf`(pOJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUp;[.5a8Z,>orA&mao;>@qN(UUmuIGKf`(pOJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESp;[.5a8Gu<orA&mao)2>qN(UUmuIGKf_kdMJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUp;[.5a8Z,>orA&mao;>@qN(UUn;dPLfDbgNJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUp;[.5a8Z,>orA&mao;>@qN(UUn;dPLfDbgNJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESp;[.5a8Gu<orA&mao)2>qN(UUn;dPLfDP[LJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUp;[.5a8Z,>orA&mao;>@qiC[Un;dPLfDbgNJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUp;[.5a8Z,>orA&mao;>@qiC[Un;dPLfDbgNJS'IFk%4SnJcFL)J,~>
a8Gu<hYdESp;[.5a8Gu<orA&mao)2>qiC[Un;dPLfDP[LJS'IFk%4SnRf1,'J,~>
a8Z,>hZ!QUp;[.5a8Z,>orA&mao;>@qiC^VmuIGKfDbgNJS'IFk%4SnJcFL)J,~>
a8Z,>hZ!QUp;[.5a8Z,>orA&mao;>@qiC^VmuIGKfDbgNJS'IFk%4SnJcFL)J,~>
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%%EndData
showpage
%%Trailer
end
%%EOF

%%EndDocument
 @endspecial Black 1354 3675 a(Figure)32 b(2.12:)1931
3649 y(^)1905 3675 y FH(A)h FJ(is)g(not)f(a)h(F)-8 b(orw)m(ard)32
b(A)m(ttractor)p Black Black 591 4016 a(Ho)m(w)m(ev)m(er,)39
b(as)c(seen)j(in)e(Figure)f(2.12,)h(it)g(is)g(clear)h(that)e
FH(B)41 b FJ(do)s(es)36 b(not)g(forw)m(ard)g(con)m(v)m(erge)591
4154 y(to)29 b(a)h(distinctly)i(de\014ned)g(forw)m(ard)e(attractor,)g
(nor)g(ev)m(en)i(to)d(the)i(pullbac)m(k)h(attractor)3837
4129 y(^)3811 4154 y FH(A)p FJ(.)591 4363 y(Analytically)i(this)f(is)h
(clear)f(up)s(on)g(in)m(v)m(estigation)h(of)f(the)g(solutions)h
(expressed)h(using)f(a)591 4502 y(co)s(cycle)g(represen)m(tation,)911
4754 y(\010)981 4770 y FE(\()p FF(t;t)1078 4779 y Fu(0)1114
4770 y FE(\))1146 4754 y FJ(\()p FH(x)1239 4769 y FE(0)1278
4754 y FJ(\))28 b(=)g(tanh\(\()p FH(t)22 b FJ(+)g FH(t)1909
4769 y FE(0)1949 4754 y FJ(\))p FH(=)p FJ(2\))f(+)2252
4687 y(1)h(+)g(10)p FH(e)2564 4651 y FB(\000)p FE(\()p
FF(t)p FE(+)p FF(t)2751 4660 y Fu(0)2787 4651 y FE(\))p
2252 4731 567 4 v 2320 4823 a FJ(1)g(+)g(10)p FH(e)2632
4794 y FB(\000)p FF(t)2712 4803 y Fu(0)2828 4754 y FJ(\()p
FH(x)2921 4769 y FE(0)2983 4754 y FG(\000)h FJ(tanh\()p
FH(t)3351 4769 y FE(0)3390 4754 y FH(=)p FJ(2\)\))p FH(:)591
4985 y FJ(Solutions)28 b(originating)f(at)g(an)m(y)h(\()p
FH(t)1863 5000 y FE(0)1903 4985 y FH(;)17 b(x)2002 5000
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%%Page: 100 118
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b FJ(\))34 b(and)f FH(t)d FG(\025)g FH(T)14 b FJ(\()p
FH(\017;)j(\015)5 b FJ(\).)46 b(This)35 b(giv)m(es)g(the)f(desired)i
(asymptotic)874 4310 y(limit)d(since)h FH(T)14 b FJ(\()p
FH(\015)5 b FJ(\))27 b FG(!)h(1)k FJ(as)h FH(\015)f FG(!)c
FJ(0)2216 4274 y FE(+)2274 4310 y FJ(.)p 3841 4627 4
66 v 3845 4565 59 4 v 3845 4627 V 3903 4627 4 66 v Black
Black eop end
%%Page: 101 119
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b(EXAMPLE)2453 b FJ(101)p Black 591 530 a Fz(4.2)160
b(Example)591 872 y Fw(4.2.1)136 b(In)l(tro)t(duction)591
1145 y FJ(The)34 b(pullbac)m(k)g(attractor)f(arising)g(from)g(p)s
(erturbation)h(of)e(a)h(con)m(tin)m(uous)i(autonomous)591
1284 y(system)i(with)f(a)g(kno)m(wn)g(semi-group)h(attractor)d(as)i(in)
g(Section)g(4.1)g(w)m(as)g(sho)m(wn)h(to)e(b)s(e)591
1422 y(upp)s(er)28 b(semi-con)m(tin)m(uous)i(in)d(its)i(con)m(v)m
(ergence)h(with)e(resp)s(ect)h(to)e(the)h(initial)g(semi-group)591
1561 y(attractor)j(it)i(is)g(deriv)m(ed)h(from,)f(that)f(is,)1461
1770 y(lim)1437 1837 y FF(\017)p FB(!)p FE(0)1572 1818
y Fu(+)1638 1770 y FH(H)1727 1729 y FB(\003)1766 1770
y FJ(\()p FH(A)1877 1729 y FF(\017)1910 1770 y FJ(\()p
FH(t)1983 1785 y FE(0)2023 1770 y FJ(\))p FH(;)17 b(A)2178
1785 y FE(0)2217 1770 y FJ(\))27 b(=)h(0)p FH(;)253 b
FG(8)p FH(t)2805 1785 y FE(0)2872 1770 y FG(2)28 b FC(R)p
FH(:)591 1996 y FJ(Lo)m(w)m(er)33 b(semi-con)m(tin)m(uit)m(y)-8
b(,)35 b(that)e(is)1763 2205 y(lim)1738 2273 y FF(\017)p
FB(!)p FE(0)1873 2254 y Fu(+)1940 2205 y FH(H)2029 2164
y FB(\003)2068 2205 y FJ(\()p FH(A)2179 2220 y FE(0)2219
2205 y FH(;)17 b(A)2336 2164 y FF(\017)2368 2205 y FJ(\()p
FH(t)2441 2220 y FE(0)2481 2205 y FJ(\)\))27 b(=)h(0)p
FH(;)591 2431 y Fy(do)-5 b(es)34 b(not)h(ne)-5 b(c)g(essarily)31
b FJ(hold,)i(as)g(the)g(follo)m(wing)g(coun)m(ter-example)h(sho)m(ws.)
591 2809 y Fw(4.2.2)136 b(De\014ning)45 b(the)g(P)l(erturb)t(ed)g
(System)591 3083 y FJ(Consider)34 b(the)f(2-dimensional)g(autonomous)g
(dynamical)h(system)1751 3292 y(_)-44 b FH(x)28 b FJ(=)g
FH(y)d FG(\000)e FH(x)p FJ(\()p FH(x)2242 3251 y FE(2)2304
3292 y FJ(+)f FH(y)2454 3251 y FE(2)2515 3292 y FG(\000)h
FJ(1\))2702 3251 y FE(2)2741 3292 y FH(;)943 b FJ(\(4.8\))1715
3501 y(_)-44 b FH(y)30 b FJ(=)e FG(\000)p FH(x)23 b FG(\000)g
FH(y)t FJ(\()p FH(x)2280 3460 y FE(2)2341 3501 y FJ(+)f
FH(y)2491 3460 y FE(2)2551 3501 y FG(\000)h FJ(1\))2738
3460 y FE(2)2777 3501 y FH(:)591 3681 y FJ(In)32 b(p)s(olar)g
(co-ordinates)h(this)h(system)g(is)f(expressed)j(b)m(y)d(the)g
(equations,)1916 3891 y(_)-43 b FH(r)31 b FJ(=)c FG(\000)p
FH(r)s FJ(\()p FH(r)2287 3850 y FE(2)2349 3891 y FG(\000)22
b FJ(1\))2535 3850 y FE(2)2574 3891 y FH(;)1110 b FJ(\(4.9\))2103
4074 y(_)2085 4100 y FH(\022)30 b FJ(=)e FG(\000)p FJ(1)p
FH(:)591 4280 y FJ(It)34 b(p)s(ossesses)j(a)d(semi-stable)i(limit)f
(cycle)g(at)f FH(r)g FJ(=)c(1,)35 b(with)g(tra)5 b(jectories)35
b(con)m(v)m(erging)h(to)591 4419 y(the)e(limit)g(cycle)h(from)f
(outside)g(the)g(unit)h(circle,)g(and)f(tra)5 b(jectories)34
b(con)m(v)m(erging)i(to)d(the)591 4557 y(origin)g(from)h(inside)h(the)f
(unit)h(circle.)48 b(In)34 b(this)h(case)g(the)f(global)g(semi-group)g
(attractor)591 4696 y(for)27 b(the)h(system)i(is)f(the)f(disc)h(at)f
(the)g(origin)g(with)g(unit)h(radius,)g(as)f(illustrated)h(in)f(Figure)
591 4834 y(4.1.)591 5043 y(W)-8 b(e)29 b(no)m(w)h(p)s(erturb)f(the)h
(original)e(system)j(of)e(equations)h(\(4.8\))f(with)h(a)e
(non-autonomous)591 5182 y(p)s(erturbation)k(so)h(that,)1454
5391 y(_)-44 b FH(x)28 b FJ(=)g FH(y)d FG(\000)d FH(x)p
FJ(\()p FH(x)1944 5350 y FE(2)2007 5391 y FJ(+)g FH(y)2157
5350 y FE(2)2218 5391 y FG(\000)g FJ(1\))2404 5350 y
FE(2)2465 5391 y FG(\000)h FH(\017x)p FG(j)17 b FJ(tanh\()p
FD(t)p FJ(\))p FG(j)p FH(;)597 b FJ(\(4.10\))p Black
Black eop end
%%Page: 102 120
TeXDict begin 102 119 bop Black 591 266 a FJ(102)237
b Fx(CHAPTER)34 b(4.)76 b(NON-A)m(UTONOMOUSL)-8 b(Y)33
b(PER)-8 b(TURBED)34 b(SYSTEMS)p Black Black Black Black
1129 2186 a @beginspecial 63 @llx 206 @lly 550 @urx 587
@ury 2692 @rwi @setspecial
%%BeginDocument: eps/nlcac.eps
%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Adobe Illustrator(TM) for Windows, version 4.0
%%For: (Russel Stonier) (Central Queensland University)
%%Title: (nlcac.eps)
%%CreationDate: (10/27/99) (4:17 PM)
%%BoundingBox: 63 206 550 587
%%DocumentProcessColors: Cyan Magenta Yellow Black
%%DocumentFonts: Helvetica
%%DocumentNeededFonts: Helvetica
%%DocumentSuppliedResources: procset Adobe_packedarray 2.0 0
%%+ procset Adobe_cmykcolor 1.1 0
%%+ procset Adobe_cshow 1.1 0
%%+ procset Adobe_customcolor 1.0 0
%%+ procset Adobe_typography_AI3 1.0 1
%%+ procset Adobe_pattern_AI3 1.0 0
%%+ procset Adobe_Illustrator_AI3 1.0 1
%AI3_ColorUsage: Color
%AI3_IncludePlacedImages
%%CMYKCustomColor: 1 0 0.55 0 (Aqua)
%%+ 1 0.5 0 0 (Blue)
%%+ 0.5 0.4 0.3 0 (Blue Gray)
%%+ 0.8 0.05 0 0 (Blue Sky)
%%+ 0.5 0.85 1 0 (Brown)
%%+ 1 0.9 0.1 0 (Dark Blue)
%%+ 1 0.55 1 0 (Forest Green)
%%+ 0.05 0.2 0.95 0 (Gold)
%%+ 0.75 0.05 1 0 (Grass Green)
%%+ 0 0.45 1 0 (Orange)
%%+ 0.15 1 1 0 (Red)
%%+ 0.45 0.9 0 0 (Violet)
%AI3_TemplateBox: 306 396 306 396
%AI3_TileBox: 13 13 599 780
%AI3_DocumentPreview: Header
%%PageOrigin:13 13
%%AI3_PaperRect:-13 780 599 -12
%%AI3_Margin:13 -13 -13 12
%AI7_GridSettings: 28.3465 8 28.3465 8 1 0 0.8 0.8 0.8 0.9 0.9 0.9
%%EndComments
%%BeginProlog
%%BeginResource: procset Adobe_packedarray 2.0 0
%%Title: (Packed Array Operators)
%%Version: 2.0 0
%%CreationDate: (8/2/90) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
userdict /Adobe_packedarray 5 dict dup begin put
/initialize
{
/packedarray where
	{
	pop
	}
	{
	Adobe_packedarray begin
	Adobe_packedarray
		{
		dup xcheck
			{
			bind
			} if
		userdict 3 1 roll put
		} forall
 end
	} ifelse
} def
/terminate
{
} def
/packedarray
{
array astore readonly
} def
/setpacking
{
pop
} def
/currentpacking
{
false
} def
currentdict readonly pop end
%%EndResource
Adobe_packedarray /initialize get exec
%%BeginResource: procset Adobe_cmykcolor 1.1 0
%%Title: (CMYK Color Operators)
%%Version: 1.1 0
%%CreationDate: (1/23/89) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_cmykcolor 4 dict dup begin put
/initialize
{
/setcmykcolor where
	{
	pop
	}
	{
	userdict /Adobe_cmykcolor_vars 2 dict dup begin put
	/_setrgbcolor
		/setrgbcolor load def
	/_currentrgbcolor
		/currentrgbcolor load def
	Adobe_cmykcolor begin
	Adobe_cmykcolor
		{
		dup xcheck
			{
			bind
			} if
		pop pop
		} forall
 end
 end
	Adobe_cmykcolor begin
	} ifelse
} def
/terminate
{
currentdict Adobe_cmykcolor eq
	{
 end
	} if
} def
/setcmykcolor
{
1 sub 4 1 roll
3
	{
	3 index add neg dup 0 lt
		{
		pop 0
		} if
	3 1 roll
	} repeat
Adobe_cmykcolor_vars /_setrgbcolor get exec
pop
} def 
/currentcmykcolor
{
Adobe_cmykcolor_vars /_currentrgbcolor get exec
3
	{
	1 sub neg 3 1 roll
	} repeat
0
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_cshow 1.1 0
%%Title: (cshow Operator)
%%Version: 1.1 0
%%CreationDate: (1/23/89) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_cshow 3 dict dup begin put
/initialize
{
/cshow where
	{
	pop
	}
	{
	userdict /Adobe_cshow_vars 1 dict dup begin put
	/_cshow
		{} def
	Adobe_cshow begin
	Adobe_cshow
		{
		dup xcheck
			{
			bind
			} if
		userdict 3 1 roll put
		} forall
 end
 end
	} ifelse
} def
/terminate
{
} def
/cshow
{
exch
Adobe_cshow_vars
	exch /_cshow
	exch put
	{
	0 0 Adobe_cshow_vars /_cshow get exec
	} forall
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_customcolor 1.0 0
%%Title: (Custom Color Operators)
%%Version: 1.0 0
%%CreationDate: (5/9/88) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_customcolor 5 dict dup begin put
/initialize
{
/setcustomcolor where
	{
	pop
	}
	{
	Adobe_customcolor begin
	Adobe_customcolor
		{
		dup xcheck
			{
			bind
			} if
		pop pop
		} forall
 end
	Adobe_customcolor begin
	} ifelse
} def
/terminate
{
currentdict Adobe_customcolor eq
	{
 end
	} if
} def
/findcmykcustomcolor
{
5 packedarray
}  def
/setcustomcolor
{
exch
aload pop pop
4
	{
	4 index mul 4 1 roll
	} repeat
5 -1 roll pop
setcmykcolor
} def
/setoverprint
{
pop
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_typography_AI3 2.0 0
%%Title: (Typography Operators)
%%Version: 2.0 0
%%CreationDate:(5/31/90) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_typography_AI3 48 dict dup begin put
/initialize
{
/TZ
 where
	{
	pop
	}
	{
	Adobe_typography_AI3 begin
	Adobe_typography_AI3
		{
		dup xcheck
			{
			bind
			} if
		pop pop
		} forall
 end
	Adobe_typography_AI3 begin
	} ifelse
} def
/terminate
{
currentdict Adobe_typography_AI3 eq
	{
 end
	} if
} def
/modifyEncoding
{
	/_tempEncode exch ddef
	
	/_pntr 0 ddef
	
	{
		counttomark -1 roll
		dup type dup /marktype eq 		
		{
			pop pop exit
		}
		{
			/nametype eq
			{
				_tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll
				put
			}
			{
				/_pntr exch ddef					
			}
			ifelse
		}
		ifelse
	}
	loop	
	
	_tempEncode
}
def
/TE
{
	StandardEncoding 256 array copy modifyEncoding 
	/_nativeEncoding exch def
} def
%
/TZ		
{
	dup type /arraytype eq {/_wv exch def} {/_wv 0 def} ifelse 
	/_useNativeEncoding exch def
	pop pop
	
	findfont _wv type /arraytype eq {_wv makeblendedfont} if dup length 2 add dict
	
 begin
	
		mark exch
		{
			1 index /FID ne { def } if cleartomark mark
		}
		forall
		pop
		
		/FontName exch def
		
		counttomark 0 eq
		{
			1 _useNativeEncoding eq
			{
				/Encoding _nativeEncoding def
			}
			if
			cleartomark
		}
		{	
			/Encoding load 256 array copy 
			modifyEncoding /Encoding exch def
		}
		ifelse		
		FontName currentdict
 end
	
	definefont pop
}
def
/tr
{
_ax _ay 3 2 roll
} def
/trj
{
_cx _cy _sp _ax _ay 6 5 roll
} def
/a0
{
/Tx
	{
	dup 
	currentpoint 3 2 roll
	tr _psf
	newpath moveto
	tr _ctm _pss
	} ddef
/Tj
	{
	dup
	currentpoint 3 2 roll
	trj _pjsf
	newpath moveto
	trj _ctm _pjss
	} ddef
	
} def
/a1
{
/Tx
	{
	dup currentpoint 4 2 roll gsave
	dup currentpoint 3 2 roll
	tr _psf
	newpath moveto
	tr _ctm _pss
	grestore 3 1 roll moveto tr sp
	} ddef
/Tj
	{
	dup currentpoint 4 2 roll gsave
	dup currentpoint 3 2 roll
	trj _pjsf
	newpath moveto
	trj _ctm _pjss
	grestore 3 1 roll moveto tr sp
	} ddef
	
} def
/e0
{
/Tx
	{
	tr _psf
	} ddef
/Tj
	{
	trj _pjsf
	} ddef
} def
/e1
{
/Tx
	{
	dup currentpoint 4 2 roll gsave 
	tr _psf  
	grestore 3 1 roll moveto tr sp 
	} ddef
/Tj
	{
	dup currentpoint 4 2 roll gsave 
	trj _pjsf
	grestore 3 1 roll moveto tr sp 
	} ddef
} def
/i0
{
/Tx
	{
	tr sp
	} ddef
/Tj
	{
	trj jsp
	} ddef
} def
/i1
{
W N
} def
/o0
{
/Tx
	{
	tr sw rmoveto
	} ddef
/Tj
	{
	trj swj rmoveto
	} ddef
} def
/r0
{
/Tx
	{
	tr _ctm _pss
	} ddef
/Tj
	{
	trj _ctm _pjss
	} ddef
} def
/r1
{
/Tx
	{
	dup currentpoint 4 2 roll currentpoint gsave newpath moveto
	tr _ctm _pss 
	grestore 3 1 roll moveto tr sp 
	} ddef
/Tj
	{
	dup currentpoint 4 2 roll currentpoint gsave newpath moveto
	trj _ctm _pjss
	grestore 3 1 roll moveto tr sp 
	} ddef
} def
/To
{
	pop _ctm currentmatrix pop
} def
/TO
{
	iTe _ctm setmatrix newpath
} def
/Tp
{
	pop _tm astore pop _ctm setmatrix 
	_tDict begin /W {} def /h {} def
} def
/TP
{
 end
	iTm 0 0 moveto
} def
/Tr
{
	_render 3 le {currentpoint newpath moveto} if
	dup 8 eq {pop 0} {dup 9 eq {pop 1} if} ifelse
	dup /_render exch ddef
	_renderStart exch get load exec
} def
/iTm
{
_ctm setmatrix _tm concat 0 _rise translate _hs 1 scale
} def
/Tm
{
_tm astore pop iTm 0 0 moveto
} def
/Td
{
_mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto
} def
/iTe
{
	_render -1 eq {} {_renderEnd _render get dup null ne {load exec} {pop} ifelse} ifelse
	/_render -1 ddef
} def
/Ta
{
pop
} def
/Tf
{
dup 1000 div /_fScl exch ddef
exch findfont exch scalefont setfont
} def
/Tl
{
pop
0 exch _leading astore pop
} def
/Tt
{
pop
} def
/TW
{
3 npop
} def
/Tw
{
/_cx exch ddef
} def
/TC
{
3 npop
} def
/Tc
{
/_ax exch ddef
} def
/Ts
{
/_rise exch ddef
currentpoint
iTm
moveto
} def
/Ti
{
3 npop
} def
/Tz
{
100 div /_hs exch ddef
iTm
} def
/TA
{
pop
} def
/Tq
{
pop
} def
/Th
{
pop pop pop pop pop
} def
/TX {pop} def
%/Tx
%/Tj
/Tk
{
exch pop _fScl mul neg 0 rmoveto
} def
/TK
{
2 npop
} def
/T*
{
_leading aload pop neg Td
} def
/T*-
{
_leading aload pop Td
} def
/T-
{
_hyphen Tx
} def
/T+
{} def
/TR
{
_ctm currentmatrix pop 	
_tm astore pop 
iTm 0 0 moveto 
} def
/TS
{
0 eq {Tx} {Tj} ifelse
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_pattern_AI3 1.1 0
%%Title: (Adobe Illustrator (R) Version 3.0 Pattern Operators)
%%Version: 1.1 0
%%CreationDate: (7/21/89) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_pattern_AI3 16 dict dup begin put
/initialize
{
/definepattern where
	{
	pop
	}
	{
	Adobe_pattern_AI3 begin
	Adobe_pattern_AI3
		{
		dup xcheck
			{
			bind
			} if
		pop pop
		} forall
	mark
	cachestatus 7 1 roll pop pop pop pop exch pop exch
		{
		{
		10000 add
		dup 2 index gt
			{
			exit
			} if
		dup setcachelimit
		} loop
		} stopped
	cleartomark
	} ifelse
} def
/terminate
{
currentdict Adobe_pattern_AI3 eq
	{
 end
	} if
} def
errordict
/nocurrentpoint
{
pop
stop
} put
errordict
/invalidaccess
{
pop
stop
} put
/patternencoding
256 array def
0 1 255
{
patternencoding exch ( ) 2 copy exch 0 exch put cvn put
} for
/definepattern
{
17 dict begin
/uniform exch def
/cache exch def
/key exch def
/procarray exch def
/mtx exch matrix invertmatrix def
/height exch def
/width exch def
/ctm matrix currentmatrix def
/ptm matrix def
/str 32 string def
/slice 9 dict def
slice /s 1 put
slice /q 256 procarray length div sqrt floor cvi put
slice /b 0 put
/FontBBox [0 0 0 0] def
/FontMatrix mtx matrix copy def
/Encoding patternencoding def
/FontType 3 def
/BuildChar
	{
	exch
 begin
	/setstrokeadjust where {pop true setstrokeadjust} if
	slice begin
	dup q dup mul mod s idiv /i exch def
	dup q dup mul mod s mod /j exch def
	q dup mul idiv procarray exch get
	/xl j width s div mul def
	/xg j 1 add width s div mul def
	/yl i height s div mul def
	/yg i 1 add height s div mul def
	uniform
		{
		1 1
		}
		{
		width 0 dtransform
		dup mul exch dup mul add sqrt dup 1 add exch div
		0 height dtransform
		dup mul exch dup mul add sqrt dup 1 add exch div
		} ifelse
	width 0 cache
		{
		xl 4 index mul yl 4 index mul xg 6 index mul yg 6 index mul
		setcachedevice
		}
		{
		setcharwidth
		} ifelse
	gsave
	scale
	newpath
	xl yl moveto
	xg yl lineto
	xg yg lineto
	xl yg lineto
	closepath
	clip
	newpath
 end
 end
	exec
	grestore
	} def
key currentdict definefont
end
} def
/patterncachesize
{
gsave
newpath
0 0 moveto
width 0 lineto
width height lineto
0 height lineto
closepath
patternmatrix setmatrix
pathbbox
exch ceiling 4 -1 roll floor sub 3 1 roll
ceiling exch floor sub
mul 1 add
grestore
} def
/patterncachelimit
{
cachestatus 7 1 roll 6 npop 8 mul
} def
/patternpath
{
exch dup begin setfont
ctm setmatrix
concat
slice exch /b exch slice /q get dup mul mul put
FontMatrix concat
uniform
	{
	width 0 dtransform round width div exch round width div exch
	0 height dtransform round height div exch height div exch
	0 0 transform round exch round exch
	ptm astore setmatrix
	}
	{
	ptm currentmatrix pop
	} ifelse
{currentpoint} stopped not
	{
	2 npop
	pathbbox
	true
	4 index 3 index eq
	4 index 3 index eq
	and
		{
		pop false
			{
			{2 npop}
			{3 npop true}
			{7 npop true}
			{pop true}
			pathforall
			} stopped
			{
			5 npop true
			} if
		} if
		{
		height div ceiling height mul 4 1 roll
		width div ceiling width mul 4 1 roll
		height div floor height mul 4 1 roll
		width div floor width mul 4 1 roll
		2 index sub height div ceiling cvi exch
		3 index sub width div ceiling cvi exch
		4 2 roll moveto
		FontMatrix mtx invertmatrix
		dup dup 4 get exch 5 get rmoveto
		ptm ptm concatmatrix pop
		slice /s
		patterncachesize patterncachelimit div ceiling sqrt ceiling cvi
		dup slice /q get gt
			{
			pop slice /q get
			} if
		put
		0 1 slice /s get dup mul 1 sub
			{
			slice /b get add
			gsave
			0 1 str length 1 sub
				{
				str exch 2 index put
				} for
			pop
			dup
				{
				gsave
				ptm setmatrix
				1 index str length idiv {str show} repeat
				1 index str length mod str exch 0 exch getinterval show
				grestore
				0 height rmoveto
				} repeat
			grestore
			} for
		2 npop
		}
		{
		4 npop
		} ifelse
	} if
end
} def
/patternclip
{
clip
} def
/patternstrokepath
{
strokepath
} def
/patternmatrix
matrix def
/patternfill
{
dup type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get
	} if
gsave
patternclip
Adobe_pattern_AI3 /patternpath get exec
grestore
newpath
} def
/patternstroke
{
dup type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get
	} if
gsave
patternstrokepath
true
	{
		{
			{
			newpath 
			moveto
			}
			{
			lineto
			}
			{
			curveto
			}
			{
			closepath
			3 copy
			Adobe_pattern_AI3 /patternfill get exec
			} pathforall
		3 npop
		} stopped
			{
			5 npop
			patternclip
			Adobe_pattern_AI3 /patternfill get exec
			} if
	}
	{
	patternclip
	Adobe_pattern_AI3 /patternfill get exec
	} ifelse
grestore
newpath
} def
/patternashow
{
3 index type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get 4 1 roll
	} if
	{
	2 npop (0) exch
	2 copy 0 exch put pop
	gsave
	false charpath currentpoint
	6 index 6 index 6 index
	Adobe_pattern_AI3 /patternfill get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
5 npop
} def
/patternawidthshow
{
6 index type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get 7 1 roll
	} if
	{
	2 npop (0) exch
	2 copy 0 exch put 
	gsave
	_sp eq {5 index 5 index rmoveto} if
	false charpath currentpoint
	9 index 9 index 9 index
	Adobe_pattern_AI3 /patternfill get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
8 npop
} def
/patternashowstroke
{
4 index type /dicttype eq
	{
	patternmatrix /patternmatrix get 5 1 roll
	} if
4 1 roll
	{
	2 npop (0) exch
	2 copy 0 exch put pop
	gsave
	false charpath
	currentpoint
	4 index setmatrix
	7 index 7 index 7 index
	Adobe_pattern_AI3 /patternstroke get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
6 npop
} def
/patternawidthshowstroke
{
7 index type /dicttype eq
	{
	patternmatrix /patternmatrix get 8 1 roll
	} if
7 1 roll
	{
	2 npop (0) exch
	2 copy 0 exch put
	gsave
	_sp eq {5 index 5 index rmoveto} if
	false charpath currentpoint
	7 index setmatrix
	10 index 10 index 10 index
	Adobe_pattern_AI3 /patternstroke get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
9 npop
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_Illustrator_AI3 1.1 0
%%Title: (Adobe Illustrator (R) Version 3.0 Full Prolog)
%%Version: 1.1 0
%%CreationDate: (3/7/1994) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_Illustrator_AI3 71 dict dup begin put
/initialize
{
userdict /Adobe_Illustrator_AI3_vars 67 dict dup begin put
/_lp /none def
/_pf {} def
/_ps {} def
/_psf {} def
/_pss {} def
/_pjsf {} def
/_pjss {} def
/_pola 0 def
/_doClip 0 def
/cf	currentflat def
/_tm matrix def
/_renderStart [/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0] def 
/_renderEnd [null null null null /i1 /i1 /i1 /i1] def
/_render -1 def
/_rise 0 def
/_ax 0 def
/_ay 0 def
/_cx 0 def
/_cy 0 def
/_leading [0 0] def
/_ctm matrix def
/_mtx matrix def
/_sp 16#020 def
/_hyphen (-) def
/_fScl 0 def
/_cnt 0 def
/_hs 1 def
/_nativeEncoding 0 def
/_useNativeEncoding 0 def
/_tempEncode 0 def
/_pntr 0 def
/_tDict 2 dict def
/_wv 0 def
/Tx {} def
/Tj {} def
/CRender {} def
/_AI3_savepage {} def
/_gf null def
/_cf 4 array def
/_if null def
/_of false def
/_fc {} def
/_gs null def
/_cs 4 array def
/_is null def
/_os false def
/_sc {} def
/_pd 1 dict def
/_ed 15 dict def
/_pm matrix def
/_fm null def
/_fd null def
/_fdd null def
/_sm null def
/_sd null def
/_sdd null def
/_i null def
Adobe_Illustrator_AI3 begin
Adobe_Illustrator_AI3 dup /nc get begin
	{
	dup xcheck
		{
		bind
		} if
	pop pop
	} forall
end
end
end
Adobe_Illustrator_AI3 begin
Adobe_Illustrator_AI3_vars begin
newpath
} def
/terminate
{
end
end
} def
/_
null def
/ddef
{
Adobe_Illustrator_AI3_vars 3 1 roll put
} def
/xput
{
dup load dup length exch maxlength eq
	{
	dup dup load dup
	length 2 mul dict copy def
	} if
load begin def end
} def
/npop
{
	{
	pop
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/sw
{
dup length exch stringwidth
exch 5 -1 roll 3 index mul add
4 1 roll 3 1 roll mul add
} def
/swj
{
dup 4 1 roll
dup length exch stringwidth 
exch 5 -1 roll 3 index mul add
4 1 roll 3 1 roll mul add 
6 2 roll /_cnt 0 ddef
{1 index eq {/_cnt _cnt 1 add ddef} if} forall pop
exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop
} def
/ss
{
4 1 roll
	{
	2 npop 
	(0) exch 2 copy 0 exch put pop
	gsave
	false charpath currentpoint
	4 index setmatrix
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	grestore
	moveto
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3 npop
} def
/jss
{
4 1 roll
	{
	2 npop 
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	gsave
	_sp eq 
		{
		exch 6 index 6 index 6 index 5 -1 roll widthshow  
		currentpoint
		}
		{
		false charpath currentpoint
		4 index setmatrix stroke
		}ifelse
	grestore
	moveto
	2 copy rmoveto
	} exch cshow
6 npop
} def
/sp
{
	{
	2 npop (0) exch
	2 copy 0 exch put pop
	false charpath
	2 copy rmoveto
	} exch cshow
2 npop
} def
/jsp
{
	{
	2 npop 
	(0) exch 2 copy 0 exch put 
	_sp eq 
		{
		exch 5 index 5 index 5 index 5 -1 roll widthshow  
		}
		{
		false charpath
		}ifelse
	2 copy rmoveto
	} exch cshow
5 npop
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/pl
{
transform
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0.25 sub round 0.25 add exch
itransform
} def
/setstrokeadjust where
	{
	pop true setstrokeadjust
	/c
	{
	curveto
	} def
	/C
	/c load def
	/v
	{
	currentpoint 6 2 roll curveto
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	/V
	/v load def
	/y
	{
	2 copy curveto
	} def
	/Y
	/y load def
	/l
	{
	lineto
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	/l load def
	/m
	{
	moveto
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	{
	/c
	{
	pl curveto
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	/C
	/c load def
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	{
	currentpoint 6 2 roll pl curveto
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	/v load def
	/y
	{
	pl 2 copy curveto
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	/Y
	/y load def
	/l
	{
	pl lineto
	} def
	/L
	/l load def
	/m
	{
	pl moveto
	} def
	} ifelse
/d
{
setdash
} def
/cf	{} def
/i
{
dup 0 eq
	{
	pop cf
	} if
setflat
} def
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{
setlinejoin
} def
/J
{
setlinecap
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/M
{
setmiterlimit
} def
/w
{
setlinewidth
} def
/H
{} def
/h
{
closepath
} def
/N
{
_pola 0 eq 
	{
	_doClip 1 eq {clip /_doClip 0 ddef} if 
	newpath
	} 
	{
	/CRender {N} ddef
	}ifelse
} def
/n
{N} def
/F
{
_pola 0 eq 
	{
	_doClip 1 eq 
		{
		gsave _pf grestore clip newpath /_lp /none ddef _fc 
		/_doClip 0 ddef
		}
		{
		_pf
		}ifelse
	} 
	{
	/CRender {F} ddef
	}ifelse
} def
/f
{
closepath
F
} def
/S
{
_pola 0 eq 
	{
	_doClip 1 eq 
		{
		gsave _ps grestore clip newpath /_lp /none ddef _sc 
		/_doClip 0 ddef
		}
		{
		_ps
		}ifelse
	} 
	{
	/CRender {S} ddef
	}ifelse
} def
/s
{
closepath
S
} def
/B
{
_pola 0 eq 
	{
	_doClip 1 eq
	gsave F grestore 
		{
		gsave S grestore clip newpath /_lp /none ddef _sc
		/_doClip 0 ddef
		} 
		{
		S
		}ifelse
	}
	{
	/CRender {B} ddef
	}ifelse
} def
/b
{
closepath
B
} def
/W
{
/_doClip 1 ddef
} def
/*
{
count 0 ne 
	{
	dup type (stringtype) eq {pop} if
	} if 
_pola 0 eq {newpath} if
} def
/u
{} def
/U
{} def
/q
{
_pola 0 eq {gsave} if
} def
/Q
{
_pola 0 eq {grestore} if
} def
/*u
{
_pola 1 add /_pola exch ddef
} def
/*U
{
_pola 1 sub /_pola exch ddef 
_pola 0 eq {CRender} if
} def
/D
{pop} def
/*w
{} def
/*W
{} def
/`
{
/_i save ddef
6 1 roll 4 npop
concat pop
userdict begin
/showpage {} def
0 setgray
0 setlinecap
1 setlinewidth
0 setlinejoin
10 setmiterlimit
[] 0 setdash
/setstrokeadjust where {pop false setstrokeadjust} if
newpath
0 setgray
false setoverprint
} def
/~
{
end
_i restore
} def
/@
{} def
/&
{} def
/O
{
0 ne
/_of exch ddef
/_lp /none ddef
} def
/R
{
0 ne
/_os exch ddef
/_lp /none ddef
} def
/g
{
/_gf exch ddef
/_fc
{
_lp /fill ne
	{
	_of setoverprint
	_gf setgray
	/_lp /fill ddef
	} if
} ddef
/_pf
{
_fc
fill
} ddef
/_psf
{
_fc
ashow
} ddef
/_pjsf
{
_fc
awidthshow
} ddef
/_lp /none ddef
} def
/G
{
/_gs exch ddef
/_sc
{
_lp /stroke ne
	{
	_os setoverprint
	_gs setgray
	/_lp /stroke ddef
	} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
/k
{
_cf astore pop
/_fc
{
_lp /fill ne
	{
	_of setoverprint
	_cf aload pop setcmykcolor
	/_lp /fill ddef
	} if
} ddef
/_pf
{
_fc
fill
} ddef
/_psf
{
_fc
ashow
} ddef
/_pjsf
{
_fc
awidthshow
} ddef
/_lp /none ddef
} def
/K
{
_cs astore pop
/_sc
{
_lp /stroke ne
	{
	_os setoverprint
	_cs aload pop setcmykcolor
	/_lp /stroke ddef
	} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
/x
{
/_gf exch ddef
findcmykcustomcolor
/_if exch ddef
/_fc
{
_lp /fill ne
	{
	_of setoverprint
	_if _gf 1 exch sub setcustomcolor
	/_lp /fill ddef
	} if
} ddef
/_pf
{
_fc
fill
} ddef
/_psf
{
_fc
ashow
} ddef
/_pjsf
{
_fc
awidthshow
} ddef
/_lp /none ddef
} def
/X
{
/_gs exch ddef
findcmykcustomcolor
/_is exch ddef
/_sc
{
_lp /stroke ne
	{
	_os setoverprint
	_is _gs 1 exch sub setcustomcolor
	/_lp /stroke ddef
	} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
/dp
{
dup null eq
{
pop
_dp 0 ne
	{
	0 1 _dp 1 sub _dl mod
		{
		_da exch get 3 get
		} for
	_dp 1 sub _dl mod 1 add packedarray
	_da 0 get aload pop 8 -1 roll 5 -1 roll pop 4 1 roll
	definepattern pop
	} if
}
{
_dp 0 ne _dp _dl mod 0 eq and
	{
	null dp
	} if
7 packedarray _da exch _dp _dl mod exch put
_dp _dl mod _da 0 get 4 get 2 packedarray
/_dp _dp 1 add def
} ifelse
} def
/E
{
_ed begin
dup 0 get type /arraytype ne
	{
	0
		{
		dup 1 add index type /arraytype eq
			{
			1 add
			}
			{
			exit
			} ifelse
		} loop
	array astore
	} if
/_dd exch def
/_ury exch def
/_urx exch def
/_lly exch def
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/_n exch def
/_y 0 def
/_dl 4 def
/_dp 0 def
/_da _dl array def
0 1 _dd length 1 sub
	{
	/_d exch _dd exch get def
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		{
		/_x exch def
		/_c _d _x get _ ne def
		/_r _d _x 1 add get cvlit def
		_r _ ne
			{
			_urx _llx sub _ury _lly sub [1 0 0 1 0 0] 
				[
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				_llx neg _lly neg /translate cvx
				_c
					{
					nc /begin cvx
					} if
				_r dup type /stringtype eq
					{
					cvx
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					{
					{exec} /forall cvx
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				_c
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			/_fn 12 _n length add string def
			_y _fn cvs pop
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			_fn 12 _n putinterval
			_fn _c false dp
			_d exch _x 1 add exch put
			} if
		} for
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null dp
_n _dd /_pd
end xput
} def
/fc
{
_fm dup concatmatrix pop
} def
/p
{
/_fm exch ddef
9 -2 roll _pm translate fc
7 -2 roll _pm scale fc
5 -1 roll _pm rotate fc
4 -2 roll exch 0 ne
	{
	dup _pm rotate fc
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	neg _pm rotate fc
	}
	{
	pop
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dup _pm rotate fc
exch dup sin exch cos div 1 0 0 1 0 6 2 roll
_pm astore fc
neg _pm rotate fc
_pd exch get /_fdd exch ddef
/_pf
{
save
/_doClip 0 ddef
0 1 _fdd length 1 sub
	{
	/_fd exch _fdd exch get ddef
	_fd
	0 2 _fd length 2 sub
		{
		gsave
		2 copy get dup _ ne
			{
			cvx exec _fc
			}
			{
			pop
			} ifelse
		2 copy 1 add get dup _ ne
			{
			aload pop findfont _fm
			patternfill
			}
			{
			pop
			fill
			} ifelse
		grestore
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	pop
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restore
newpath
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/_psf
{
save
/_doClip 0 ddef
0 1 _fdd length 1 sub
	{
	/_fd exch _fdd exch get ddef
	_fd
	0 2 _fd length 2 sub
		{
		gsave
		2 copy get dup _ ne
			{
			cvx exec _fc
			}
			{
			pop
			} ifelse
		2 copy 1 add get dup _ ne
			{
			aload pop findfont _fm
			9 copy 6 npop patternashow
			}
			{
			pop
			6 copy 3 npop ashow
			} ifelse
		grestore
		pop
		} for
	pop
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restore
%3 npop newpath
sw rmoveto
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{
save
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0 1 _fdd length 1 sub
	{
	/_fd exch _fdd exch get ddef
	_fd
	0 2 _fd length 2 sub
		{
		gsave
		2 copy get dup _ ne
			{
			cvx exec _fc
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			{
			pop
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		2 copy 1 add get dup _ ne
			{
			aload pop findfont _fm
			12 copy 6 npop patternawidthshow
			}
			{
			pop 9 copy 3 npop awidthshow
			} ifelse
		grestore
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		} for
	pop
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restore
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} ddef
/_lp /none ddef
} def
/sc
{
_sm dup concatmatrix pop
} def
/P
{
/_sm exch ddef
9 -2 roll _pm translate sc
7 -2 roll _pm scale sc
5 -1 roll _pm rotate sc
4 -2 roll exch 0 ne
	{
	dup _pm rotate sc
	1 -1 _pm scale sc
	neg _pm rotate sc
	}
	{
	pop
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dup _pm rotate sc
exch dup sin exch cos div 1 0 0 1 0 6 2 roll
_pm astore sc
neg _pm rotate sc
_pd exch get /_sdd exch ddef
/_ps
{
save
/_doClip 0 ddef
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	{
	/_sd exch _sdd exch get ddef
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		{
		gsave
		2 copy get dup _ ne
			{
			cvx exec _sc
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			{
			pop
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		2 copy 1 add get dup _ ne
			{
			aload pop findfont _sm
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			{
			pop stroke
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restore
newpath
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{
save
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	{
	/_sd exch _sdd exch get ddef
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	0 2 _sd length 2 sub
		{
		gsave
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			{
			cvx exec _sc
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			pop
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			{
			aload pop findfont _sm
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			{
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restore
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			{
			aload pop findfont _sm
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			{
			pop 10 copy 3 npop jss
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		grestore
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		} for
	pop
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restore
pop swj rmoveto
} ddef
/_lp /none ddef
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/A
{
pop
} def
/nc 3 dict def
nc begin
/setgray
{
pop
} bind def
/setcmykcolor
{
4 npop
} bind def
/setcustomcolor
{
2 npop
} bind def
currentdict readonly pop end
currentdict readonly pop end
setpacking
/annotatepage
{
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%%EndResource
%%EndProlog
%%BeginSetup
%%IncludeFont: Helvetica
Adobe_cmykcolor /initialize get exec
Adobe_cshow /initialize get exec
Adobe_customcolor /initialize get exec
Adobe_typography_AI3 /initialize get exec
Adobe_pattern_AI3 /initialize get exec
Adobe_Illustrator_AI3 /initialize get exec
[
39/quotesingle 96/grave 130/quotesinglbase/florin/quotedblbase/ellipsis
/dagger/daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE 145/quoteleft
/quoteright/quotedblleft/quotedblright/bullet/endash/emdash/tilde/trademark
/scaron/guilsinglright/oe/dotlessi 159/Ydieresis 164/currency 166/brokenbar
168/dieresis/copyright/ordfeminine 172/logicalnot 174/registered/macron/ring
/plusminus/twosuperior/threesuperior/acute/mu 183/periodcentered/cedilla
/onesuperior/ordmasculine 188/onequarter/onehalf/threequarters 192/Agrave
/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla/Egrave/Eacute
/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis/Eth/Ntilde
/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply/Oslash/Ugrave
/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls/agrave/aacute
/acircumflex/atilde/adieresis/aring/ae/ccedilla/egrave/eacute/ecircumflex
/edieresis/igrave/iacute/icircumflex/idieresis/eth/ntilde/ograve/oacute
/ocircumflex/otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex
/udieresis/yacute/thorn/ydieresis
TE
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50.18 67.112 L
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37.668 67.113 L
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1 1 L
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54.046 55.212 L
51.657 62.566 L
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50.18 67.112 L
47.791 74.467 L
45.402 67.113 L
37.668 67.113 L
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37.668 25.23 L
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50.18 25.229 L
f
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14.207 25.23 L
6.473 25.23 L
12.729 20.684 L
10.34 13.33 L
16.596 17.874 L
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26.719 25.229 L
18.985 25.229 L
f
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6.5948 1 L
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57.128 76.154 m
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63.384 84.766 L
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25.936 76.155 m
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26.5951 84.766 L
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41.535 55.213 m
47.791 59.757 L
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50.18 67.112 L
47.791 74.467 L
45.402 67.113 L
37.668 67.113 L
43.924 62.567 L
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45.402 25.23 L
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41.535 13.33 L
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51.657 20.683 L
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57.128 76.154 m
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%AI6_EndPatternLayer
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8.2 51.4 m
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8.2 65.8001 m
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8.2 29.8001 m
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313.4167 399.5 m
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313.1667 399.75 m
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313.1667 399.75 m
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313.1667 399.75 m
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313.4167 399.5 m
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313.1667 399.8333 m
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313.5833 399.9166 m
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313.75 399.5833 m
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313.4167 399.5 m
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313.25 399.8333 m
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313.5833 399.9166 m
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313.4167 399.5 m
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313.75 399.5833 m
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313.4167 399.5 m
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313.25 399.8333 m
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313.5833 399.9166 m
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313.75 399.5833 m
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313.4167 399.5 m
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313.5833 399.9166 m
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313.75 399.5833 m
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313.4167 399.5 m
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313.25 399.8333 m
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313.5833 399.9166 m
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313.75 399.5833 m
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313.4167 399.5 m
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313.25 399.8333 m
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313.5833 399.9166 m
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313.75 399.5833 m
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313.4167 399.5 m
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313.25 399.8333 m
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313.5833 399.9166 m
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313.75 399.5833 m
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313.4167 399.5 m
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313.25 399.8333 m
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174.75 450.75 m
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313.5833 399.9166 m
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313.75 399.5833 m
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313.4167 399.5 m
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313.25 399.8333 m
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175.4167 451.8333 m
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313.5833 399.9166 m
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313.75 399.5833 m
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313.4167 399.5 m
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313.25 399.8333 m
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176.0833 452.9166 m
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313.5833 399.9166 m
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313.75 399.5833 m
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313.4167 399.5 m
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313.25 399.8333 m
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382.4167 507.5 m
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244.5833 291.9167 m
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313.5833 399.9166 m
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313.75 399.5833 m
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313.4167 399.5 m
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313.25 399.8333 m
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383.75 506.9167 m
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177.5 455.0833 m
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313.6666 399.9166 m
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313.75 399.5833 m
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313.3333 399.5 m
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313.25 399.8333 m
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385.0833 506.4166 m
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313.6666 399.9166 m
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313.75 399.5833 m
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313.3333 399.5 m
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313.25 399.8333 m
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386.5 505.8333 m
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240.5 293.5833 m
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178.9167 457.25 m
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313.6666 399.9166 m
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313.75 399.5833 m
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313.3333 399.5 m
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313.25 399.8333 m
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387.8333 505.25 m
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179.6667 458.3333 m
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313.6666 399.9166 m
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313.75 399.5833 m
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313.3333 399.5 m
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313.25 399.8333 m
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389.1667 504.6667 m
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180.4167 459.3333 m
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313.6666 399.9166 m
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313.75 399.5833 m
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313.3333 399.5 m
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313.25 399.8333 m
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390.5 504 m
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445.8333 339 m
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236.5 295.4167 m
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181.1667 460.4166 m
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313.6666 399.9166 m
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313.75 399.5833 m
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313.3333 399.5 m
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313.25 399.8333 m
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391.75 503.4166 m
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445 338 m
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235.25 296 m
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182 461.4166 m
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313.6666 399.9166 m
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313.75 399.5833 m
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313.3333 399.5 m
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313.25 399.8333 m
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393.0833 502.8333 m
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233.9166 296.5833 m
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313.6666 399.9166 m
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313.75 399.5833 m
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313.3333 399.5 m
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313.25 399.8333 m
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443.4166 335.9167 m
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232.5833 297.25 m
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183.5833 463.5 m
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313.6666 399.9166 m
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313.75 399.5833 m
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313.3333 399.5 m
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313.25 399.8333 m
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395.6667 501.5 m
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442.5833 334.9167 m
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231.3333 297.9167 m
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184.4167 464.5 m
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313.6666 399.8333 m
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313.75 399.5833 m
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313.3333 399.5833 m
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313.25 399.8333 m
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397 500.8333 m
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441.75 333.9167 m
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230 298.5833 m
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185.25 465.5 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
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313.3333 399.8333 m
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398.25 500.1667 m
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440.9166 332.9167 m
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228.75 299.25 m
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186.0833 466.5 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
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313.3333 399.8333 m
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399.5 499.5 m
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440.0833 331.9167 m
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227.5 299.9167 m
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186.9167 467.5 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
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313.3333 399.8333 m
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400.75 498.8333 m
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439.25 330.9167 m
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226.25 300.5833 m
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187.75 468.5 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
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313.3333 399.8333 m
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402 498.1666 m
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438.3333 329.9167 m
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225 301.25 m
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188.6667 469.5 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
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313.3333 399.8333 m
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403.25 497.4166 m
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437.4166 328.9167 m
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223.75 302 m
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189.5833 470.5 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
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313.3333 399.8333 m
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404.5 496.7499 m
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436.5 328 m
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222.5 302.6667 m
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190.5 471.4167 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
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313.3333 399.8333 m
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405.75 495.9999 m
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435.5833 327 m
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221.25 303.4167 m
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191.4167 472.4166 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
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313.3333 399.8333 m
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434.6667 326.0833 m
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220.0833 304.1667 m
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192.3333 473.3333 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
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313.3333 399.8333 m
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408.1666 494.4999 m
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433.75 325.0833 m
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218.8333 304.9167 m
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193.25 474.3333 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
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313.3333 399.8333 m
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409.3333 493.7499 m
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432.75 324.1667 m
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217.6667 305.6667 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
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313.3333 399.8333 m
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410.5 492.9999 m
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431.8333 323.25 m
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216.5 306.4167 m
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195.1667 476.1666 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
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313.3333 399.8333 m
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430.8333 322.3333 m
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215.3333 307.1667 m
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196.1667 477.0833 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
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313.3333 399.8333 m
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412.8333 491.4166 m
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429.8333 321.4167 m
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214.1667 308 m
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197.1667 478 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
313.3333 399.5833 l
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313.3333 399.8333 m
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414 490.6666 m
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428.8333 320.5 m
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213 308.75 m
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198.1667 478.9167 m
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313.6666 399.8333 m
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313.6666 399.5833 m
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313.3333 399.5833 m
313.3333 399.5833 l
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313.3333 399.8333 m
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415.1667 489.8333 m
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427.8333 319.5833 m
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211.8333 309.5833 m
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199.1667 479.8333 m
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313.6666 399.8333 m
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313.6666 399.5833 m
313.6666 399.5833 l
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313.3333 399.5833 m
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q
0 J 1 w 4 M
90.3333 575.6666 m
90.3333 223.6667 l
536.7499 223.6667 l
536.7499 575.6666 l
90.3333 575.6666 l
h
W
n
0 R
0.0117 0.9687 0.918 0 K
1 J 3 w 10 M
261.3333 509.5833 m
260.5 509.3333 l
259.5833 509.0833 l
258.75 508.7499 l
257.8333 508.4999 l
257 508.2499 l
256.1667 507.9166 l
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253.5834 507.0832 l
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251.0001 506.1665 l
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249.3334 505.4998 l
248.5001 505.2498 l
247.6668 504.9164 l
246.7501 504.5831 l
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245.0834 503.9164 l
244.3334 503.5831 l
243.5001 503.1664 l
242.6668 502.8331 l
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241.0001 502.1664 l
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%%PageTrailer
gsave annotatepage grestore showpage
%%Trailer
Adobe_Illustrator_AI3 /terminate get exec
Adobe_pattern_AI3 /terminate get exec
Adobe_typography_AI3 /terminate get exec
Adobe_customcolor /terminate get exec
Adobe_cshow /terminate get exec
Adobe_cmykcolor /terminate get exec
Adobe_packedarray /terminate get exec
%%EOF

%%EndDocument
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b FH(t;)17 b(r)2657 5406 y FE(0)2696 5391 y FJ(\))p FH(:)p
Black Black eop end
%%Page: 104 122
TeXDict begin 104 121 bop Black 591 266 a FJ(104)237
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b(our)g(original)g(equation)h(for)f(the)591 669 y(deriv)-5
b(ativ)m(e)34 b(of)e(the)h(system)h(\(4.12\))e(w)m(e)i(ha)m(v)m(e)1106
855 y FH(d)p 1063 899 137 4 v 1063 991 a(ds)1160 962
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FH(\017r)2380 881 y FE(2)2419 922 y FG(j)17 b FJ(tanh\()p
FH(s)2743 881 y FB(\003)2804 922 y FJ(+)22 b FH(s)2948
937 y FE(1)3010 922 y FJ(+)g FH(t)3143 937 y FE(0)3205
922 y FG(\000)g FH(t)p FJ(\))p FG(j)p FH(;)1378 1108
y FG(\024)84 b(\000)p FJ(2)p FH(\017r)1751 1067 y FE(2)1791
1108 y FG(j)17 b FJ(tanh)o(\()p FH(s)2114 1067 y FB(\003)2176
1108 y FJ(+)22 b FH(s)2320 1123 y FE(1)2381 1108 y FJ(+)g
FH(t)2514 1123 y FE(0)2576 1108 y FG(\000)h FH(t)p FJ(\))p
FG(j)p FH(:)591 1330 y FJ(If)32 b(w)m(e)i(c)m(ho)s(ose)f
FH(t)g FJ(large)f(enough)h(so)g(that)f FH(s)2160 1294
y FB(\003)2228 1330 y FG(\024)2343 1291 y FE(1)p 2343
1307 36 4 v 2343 1364 a(2)2388 1330 y FH(t)22 b FG(\000)h
FH(s)2591 1345 y FE(1)2663 1330 y FJ(and)32 b FG(j)17
b FJ(tanh\()p FH(t)3165 1345 y FE(0)3227 1330 y FG(\000)3336
1291 y FE(1)p 3336 1307 V 3336 1364 a(2)3381 1330 y FH(t)p
FJ(\))p FG(j)28 b(\025)3625 1291 y FE(1)p 3625 1307 V
3625 1364 a(2)3703 1330 y FJ(then)1609 1526 y FH(d)p
1566 1570 137 4 v 1566 1661 a(ds)1663 1633 y FB(\003)1712
1593 y FH(r)1759 1552 y FE(2)1882 1593 y FG(\024)83 b(\000)p
FJ(2)p FH(\017r)2254 1552 y FE(2)2294 1593 y FG(j)17
b FJ(tanh)o(\()p FH(t)2606 1608 y FE(0)2668 1593 y FG(\000)2784
1526 y FH(t)p 2778 1570 49 4 v 2778 1661 a FJ(2)2836
1593 y(\))p FG(j)p FH(;)1882 1826 y FG(\024)83 b(\000)p
FJ(2)p FH(\017r)2254 1785 y FE(2)2304 1759 y FJ(1)p 2304
1803 V 2304 1895 a(2)2363 1826 y FH(;)1882 2012 y FJ(=)h
FG(\000)p FH(\017r)2205 1971 y FE(2)2245 2012 y FH(:)591
2234 y FJ(In)m(tegrating,)1283 2456 y FH(r)1330 2415
y FE(2)1370 2456 y FJ(\()p FH(s)1454 2415 y FB(\003)1515
2456 y FJ(+)22 b FH(s)1659 2471 y FE(1)1698 2456 y FH(;)17
b(t)1777 2471 y FE(0)1839 2456 y FG(\000)23 b FH(t;)17
b(r)2062 2471 y FE(0)2101 2456 y FJ(\))83 b FG(\024)g
FH(r)2429 2415 y FE(2)2426 2481 y(1)2485 2456 y FJ(exp)18
b FG(f\000)p FH(\017s)2863 2415 y FB(\003)2903 2456 y
FG(g)e FH(;)2222 2620 y FG(\024)83 b FJ(\(1)22 b(+)g
FH(\016)t FJ(\))17 b(exp)g FG(f\000)p FH(\017s)3068 2579
y FB(\003)3108 2620 y FG(g)g FH(;)2222 2783 y FG(\024)83
b FJ(1)22 b FG(\000)h FH(\016)n(;)591 3005 y FJ(for)32
b(all)g FH(s)921 2969 y FB(\003)988 3005 y FG(\025)c
FJ(ln)17 b(\(\(1)22 b(+)g FH(\016)t FJ(\))p FH(=)p FJ(\(1)f
FG(\000)i FH(\016)t FJ(\)\))17 b FH(=\017)p FJ(.)43 b(F)-8
b(or)32 b(ease)h(of)f(notation)h(de\014ne)g(the)g(v)-5
b(alue)591 3144 y FH(s)637 3159 y FE(2)704 3144 y FJ(=)27
b(ln)17 b(\(\(1)22 b(+)g FH(\016)t FJ(\))p FH(=)p FJ(\(1)f
FG(\000)i FH(\016)t FJ(\)\))16 b FH(=\017)p FJ(.)44 b(Th)m(us)34
b(w)m(e)g(no)m(w)f(ha)m(v)m(e)1199 3389 y FH(r)1246 3348
y FE(2)1285 3389 y FJ(\()p FH(s;)17 b(t)1448 3404 y FE(0)1509
3389 y FG(\000)23 b FH(t;)17 b(r)1732 3404 y FE(0)1771
3389 y FJ(\))28 b FG(\024)g FJ(1)22 b FG(\000)g FH(\016)395
b FG(8)p FH(s)2651 3404 y FE(1)2712 3389 y FJ(+)22 b
FH(s)2856 3404 y FE(2)2924 3389 y FG(\024)28 b FH(s)f
FG(\024)3224 3322 y FH(t)p 3217 3366 V 3217 3458 a FJ(2)3276
3389 y FH(;)591 3612 y FJ(and)1741 3834 y(1\))97 b FH(t)28
b FG(\025)g FJ(2\()p FH(s)2226 3849 y FE(1)2287 3834
y FJ(+)22 b FH(s)2431 3849 y FE(2)2470 3834 y FJ(\))p
FH(:)1741 4040 y FJ(2\))97 b FG(j)17 b FJ(tanh)o(\()p
FH(t)2237 4055 y FE(0)2299 4040 y FG(\000)2415 3973 y
FH(t)p 2408 4017 V 2408 4109 a FJ(2)2467 4040 y(\))p
FG(j)28 b(\025)2676 3973 y FJ(1)p 2676 4017 V 2676 4109
a(2)2734 4040 y FH(:)901 b FJ(\(4.13\))591 4619 y Fw(4.2.5)136
b(Beha)l(viour)45 b(F)-11 b(or)44 b(r)h Ff(<)g Fw(1)591
4892 y FJ(Consider)30 b(the)g(ev)m(olution)h(of)e(the)h(tra)5
b(jectory)30 b(within)g(the)g(unit)g(circle.)44 b(Let)29
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FH(s)3781 4907 y FE(1)3836 4892 y FJ(+)591 5031 y FH(s)637
5046 y FE(2)676 5031 y FH(;)17 b(t)755 5046 y FE(0)816
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5031 y FJ(\))33 b(with)g FH(s)1417 5046 y FE(1)1456 5031
y FH(;)17 b(s)1546 5046 y FE(2)1618 5031 y FJ(and)33
b FH(t)g FJ(de\014ned)h(as)e(ab)s(o)m(v)m(e.)45 b(Here)33
b FH(r)2923 4994 y FE(2)2920 5055 y(2)2990 5031 y FG(\024)28
b FJ(1)22 b FG(\000)h FH(\016)t FJ(.)43 b(Then)34 b(w)m(e)f(are)591
5169 y(in)m(terested)h(in)f(the)g(b)s(eha)m(viour)h(of)1727
5391 y FH(r)s FJ(\()p FH(s)1858 5350 y FB(\003)1919 5391
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FH(s)2268 5406 y FE(2)2308 5391 y FH(;)17 b(t)2387 5406
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%%Page: 105 123
TeXDict begin 105 122 bop Black 591 266 a Fx(4.2.)75
b(EXAMPLE)2453 b FJ(105)p Black 591 530 a(where)33 b
FH(s)28 b FJ(=)g FH(s)1096 494 y FB(\003)1157 530 y FJ(+)22
b FH(s)1301 545 y FE(1)1363 530 y FJ(+)g FH(s)1507 545
y FE(2)1546 530 y FJ(.)591 740 y(No)m(w)34 b FH(r)861
703 y FE(2)900 740 y FJ(\()p FH(s)984 703 y FB(\003)1047
740 y FJ(+)22 b FH(s)1191 755 y FE(1)1254 740 y FJ(+)h
FH(s)1399 755 y FE(2)1438 740 y FH(;)17 b(t)1517 755
y FE(0)1580 740 y FG(\000)23 b FH(t;)17 b(r)1803 755
y FE(0)1842 740 y FJ(\))30 b FG(\024)h FH(r)2065 703
y FE(2)2062 764 y(2)2134 740 y FG(\024)f FJ(1)23 b FG(\000)h
FH(\016)t FJ(.)47 b(Then)35 b(from)f(the)g(original)g(equation)591
878 y(\(4.12\))d(w)m(e)j(obtain)1003 1075 y FH(d)p 961
1119 137 4 v 961 1210 a(ds)1058 1182 y FB(\003)1107 1142
y FH(r)1154 1101 y FE(2)1276 1142 y FJ(=)84 b FG(\000)p
FJ(2)p FH(r)1609 1101 y FE(2)1649 1142 y FJ(\(1)21 b
FG(\000)i FH(r)1904 1101 y FE(2)1943 1142 y FJ(\))1981
1101 y FE(2)2043 1142 y FG(\000)f FJ(2)p FH(\017r)2277
1101 y FE(2)2317 1142 y FG(j)17 b FJ(tanh)o(\()p FH(s)2640
1101 y FB(\003)2701 1142 y FJ(+)22 b FH(s)2845 1157 y
FE(1)2907 1142 y FJ(+)g FH(s)3051 1157 y FE(2)3112 1142
y FJ(+)g FH(t)3245 1157 y FE(0)3307 1142 y FG(\000)h
FH(t)p FJ(\))p FG(j)p FH(;)1276 1327 y FG(\024)83 b(\000)p
FJ(2)p FH(r)1609 1286 y FE(2)1649 1327 y FH(\016)1696
1286 y FE(2)1735 1327 y FH(:)591 1560 y FJ(In)m(tegrating,)1187
1793 y FH(r)1234 1752 y FE(2)1273 1793 y FJ(\()p FH(s)1357
1752 y FB(\003)1419 1793 y FJ(+)22 b FH(s)1563 1808 y
FE(1)1624 1793 y FJ(+)g FH(s)1768 1808 y FE(2)1808 1793
y FH(;)17 b(t)1887 1808 y FE(0)1948 1793 y FG(\000)23
b FH(t;)17 b(r)2171 1808 y FE(0)2210 1793 y FJ(\))28
b FG(\024)g FH(r)2428 1752 y FE(2)2425 1817 y(2)2484
1793 y FJ(exp)2649 1712 y Ft(\010)2708 1793 y FG(\000)p
FJ(2)p FH(\016)2881 1752 y FE(2)2920 1793 y FH(s)2966
1752 y FB(\003)3006 1712 y Ft(\011)3080 1793 y FH(;)2276
1956 y FG(\024)g FJ(\(1)22 b FG(\000)h FH(\016)t FJ(\))17
b(exp)2857 1875 y Ft(\010)2915 1956 y FG(\000)p FJ(2)p
FH(\016)3088 1915 y FE(2)3127 1956 y FH(s)3173 1915 y
FB(\003)3213 1875 y Ft(\011)3288 1956 y FH(;)1227 2119
y(r)s FJ(\()p FH(s)1358 2078 y FB(\003)1419 2119 y FJ(+)22
b FH(s)1563 2134 y FE(1)1624 2119 y FJ(+)g FH(s)1768
2134 y FE(2)1808 2119 y FH(;)17 b(t)1887 2134 y FE(0)1948
2119 y FG(\000)23 b FH(t;)17 b(r)2171 2134 y FE(0)2210
2119 y FJ(\))28 b FG(\024)2381 2034 y(p)p 2464 2034 218
4 v 85 x FJ(1)22 b FG(\000)h FH(\016)d FJ(exp)2864 2039
y Ft(\010)2922 2119 y FG(\000)p FH(\016)3046 2078 y FE(2)3086
2119 y FH(s)3132 2078 y FB(\003)3171 2039 y Ft(\011)3246
2119 y FH(:)591 2501 y Fw(4.2.6)136 b(The)44 b(Pullbac)l(k)h(A)l
(ttractor)591 2774 y FJ(Allo)m(wing)24 b(the)h(pullbac)m(k)g(term)f(to)
g(run)g(to)g(completion,)i(that)e(is)g(b)m(y)h(letting)f
FH(s)3428 2738 y FB(\003)3472 2774 y FJ(+)t FH(s)3598
2789 y FE(1)3642 2774 y FJ(+)t FH(s)3768 2789 y FE(2)3836
2774 y FJ(=)591 2913 y FH(t)p FJ(,)32 b(and)h(for)f FH(t)h
FJ(satisfying)g(\(4.13\),)f(w)m(e)i(ha)m(v)m(e)1044 3145
y FH(r)s FJ(\()p FH(t;)17 b(t)1243 3160 y FE(0)1304 3145
y FG(\000)23 b FH(t;)17 b(r)1527 3160 y FE(0)1566 3145
y FJ(\))83 b FG(\024)1848 3060 y(p)p 1931 3060 V 85 x
FJ(1)22 b FG(\000)g FH(\016)f FJ(exp)2331 3065 y Ft(\010)2389
3145 y FH(\016)2436 3104 y FE(2)2475 3145 y FJ(\()p FH(s)2559
3160 y FE(1)2621 3145 y FJ(+)h FH(s)2765 3160 y FE(2)2804
3145 y FJ(\))2842 3065 y Ft(\011)2917 3145 y FJ(exp)3082
3065 y Ft(\010)3140 3145 y FG(\000)p FH(\016)3264 3104
y FE(2)3304 3145 y FH(t)3339 3065 y Ft(\011)3414 3145
y FH(;)1688 3309 y FJ(=)84 b FH(A)p FJ(\()p FH(r)2003
3324 y FE(0)2042 3309 y FH(;)17 b(\016)n(;)g(\017)p FJ(\))g(exp)2431
3228 y Ft(\010)2489 3309 y FG(\000)p FH(\016)2613 3268
y FE(2)2653 3309 y FH(t)2688 3228 y Ft(\011)2763 3309
y FH(;)591 3541 y FJ(where)30 b FH(A)f FJ(is)h(a)e(constan)m(t)i
(replacing)g(and)f(simplifying)i(the)e(expression)i(on)e(the)h
(previous)591 3680 y(line.)44 b(The)33 b(terms)h(that)e(constitute)i
(the)f(de\014nition)g(of)g FH(A)f FJ(are:)1382 3912 y
FH(A)p FJ(\()p FH(r)1537 3927 y FE(0)1576 3912 y FH(;)17
b(\016)n(;)g(\017)p FJ(\))83 b(=)2024 3827 y FG(p)p 2107
3827 V 85 x FJ(1)22 b FG(\000)h FH(\016)d FJ(exp)2507
3832 y Ft(\010)2565 3912 y FH(\016)2612 3871 y FE(2)2652
3912 y FJ(\()p FH(s)2736 3927 y FE(1)2797 3912 y FJ(+)i
FH(s)2941 3927 y FE(2)2980 3912 y FJ(\))3018 3832 y Ft(\011)3093
3912 y FH(:)1697 4127 y(s)1743 4142 y FE(1)1865 4127
y FJ(=)2078 4059 y(1)p 2034 4104 136 4 v 2034 4195 a(2)p
FH(\016)2130 4166 y FE(2)2196 4127 y FJ(ln)2294 3986
y Ft(\022)2442 4059 y FH(r)2489 4023 y FE(2)2486 4084
y(0)p 2377 4104 216 4 v 2377 4195 a FJ(1)g(+)g FH(\016)2603
3986 y Ft(\023)2693 4127 y FH(:)1697 4391 y(s)1743 4406
y FE(2)1865 4391 y FJ(=)2034 4324 y(1)p 2034 4368 49
4 v 2039 4460 a FH(\017)2110 4391 y FJ(ln\()2240 4324
y(1)g(+)g FH(\016)p 2239 4368 218 4 v 2239 4460 a FJ(1)g
FG(\000)g FH(\016)2466 4391 y FJ(\))p FH(:)591 4649 y
FJ(Therefore)33 b(the)g(tra)5 b(jectory)33 b(of)f(an)m(y)h(initial)f
(state)h FH(x)2525 4664 y FE(0)2593 4649 y FG(2)28 b
FC(R)2759 4613 y FE(2)2830 4649 y FJ(pullbac)m(k)34 b(con)m(v)m(erges)h
(to)d(the)591 4788 y(origin)g(for)g(an)m(y)h(time)g FH(t)1457
4803 y FE(0)1497 4788 y FJ(.)1184 5020 y(lim)1169 5080
y FF(t)p FB(!1)1352 5020 y FH(r)s FJ(\()p FH(t;)17 b(t)1551
5035 y FE(0)1612 5020 y FG(\000)23 b FH(t;)17 b(r)1835
5035 y FE(0)1874 5020 y FJ(\))28 b FG(\024)43 b FJ(lim)2045
5080 y FF(t)p FB(!1)2228 5020 y FH(A)p FJ(\()p FH(r)2383
5035 y FE(0)2423 5020 y FH(;)17 b(\016)n(;)g(\017)p FJ(\))g(exp)2811
4940 y Ft(\010)2869 5020 y FG(\000)p FH(\016)2993 4979
y FE(2)3033 5020 y FH(t)3068 4940 y Ft(\011)3154 5020
y FJ(=)28 b(0)p FH(:)591 5253 y FJ(Hence,)1760 5391 y
FH(A)1833 5350 y FF(\017)1866 5391 y FJ(\()p FH(t)1939
5406 y FE(0)1978 5391 y FJ(\))g(=)f(0)195 b FG(8)p FH(t)2481
5406 y FE(0)2549 5391 y FG(2)28 b FC(R)p FH(:)p Black
Black eop end
%%Page: 106 124
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b Fx(CHAPTER)34 b(4.)76 b(NON-A)m(UTONOMOUSL)-8 b(Y)33
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b(motion)f(of)g(initial)g(states)h(for)f(this)h(system)h(can)f(b)s(e)f
(seen)i(illustrated)f(b)s(elo)m(w)g(for)591 669 y(four)h(p)s(oin)m(ts)i
(originating)e(outside)i(the)f(unit)g(circle.)70 b(The)42
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(approac)m(hes)h(the)591 946 y(original)30 b(attractor,)h(it)g(is)g
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(erturbation,)591 1084 y(falling)44 b(in)m(to)h(a)g(region)f(spiraling)
i(in)m(to)f(the)g(origin)f(as)h(in)g(the)h(original)e(autonomous)591
1222 y(system.)83 b(The)46 b(p)s(erturbation)f(has)h(caused)g(a)f
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Black Black 1129 3371 a @beginspecial 63 @llx 206 @lly
550 @urx 587 @ury 2692 @rwi @setspecial
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%%Version: 1.1 0
%%CreationDate: (1/23/89) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_cmykcolor 4 dict dup begin put
/initialize
{
/setcmykcolor where
	{
	pop
	}
	{
	userdict /Adobe_cmykcolor_vars 2 dict dup begin put
	/_setrgbcolor
		/setrgbcolor load def
	/_currentrgbcolor
		/currentrgbcolor load def
	Adobe_cmykcolor begin
	Adobe_cmykcolor
		{
		dup xcheck
			{
			bind
			} if
		pop pop
		} forall
 end
 end
	Adobe_cmykcolor begin
	} ifelse
} def
/terminate
{
currentdict Adobe_cmykcolor eq
	{
 end
	} if
} def
/setcmykcolor
{
1 sub 4 1 roll
3
	{
	3 index add neg dup 0 lt
		{
		pop 0
		} if
	3 1 roll
	} repeat
Adobe_cmykcolor_vars /_setrgbcolor get exec
pop
} def 
/currentcmykcolor
{
Adobe_cmykcolor_vars /_currentrgbcolor get exec
3
	{
	1 sub neg 3 1 roll
	} repeat
0
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_cshow 1.1 0
%%Title: (cshow Operator)
%%Version: 1.1 0
%%CreationDate: (1/23/89) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_cshow 3 dict dup begin put
/initialize
{
/cshow where
	{
	pop
	}
	{
	userdict /Adobe_cshow_vars 1 dict dup begin put
	/_cshow
		{} def
	Adobe_cshow begin
	Adobe_cshow
		{
		dup xcheck
			{
			bind
			} if
		userdict 3 1 roll put
		} forall
 end
 end
	} ifelse
} def
/terminate
{
} def
/cshow
{
exch
Adobe_cshow_vars
	exch /_cshow
	exch put
	{
	0 0 Adobe_cshow_vars /_cshow get exec
	} forall
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_customcolor 1.0 0
%%Title: (Custom Color Operators)
%%Version: 1.0 0
%%CreationDate: (5/9/88) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_customcolor 5 dict dup begin put
/initialize
{
/setcustomcolor where
	{
	pop
	}
	{
	Adobe_customcolor begin
	Adobe_customcolor
		{
		dup xcheck
			{
			bind
			} if
		pop pop
		} forall
 end
	Adobe_customcolor begin
	} ifelse
} def
/terminate
{
currentdict Adobe_customcolor eq
	{
 end
	} if
} def
/findcmykcustomcolor
{
5 packedarray
}  def
/setcustomcolor
{
exch
aload pop pop
4
	{
	4 index mul 4 1 roll
	} repeat
5 -1 roll pop
setcmykcolor
} def
/setoverprint
{
pop
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_typography_AI3 2.0 0
%%Title: (Typography Operators)
%%Version: 2.0 0
%%CreationDate:(5/31/90) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_typography_AI3 48 dict dup begin put
/initialize
{
/TZ
 where
	{
	pop
	}
	{
	Adobe_typography_AI3 begin
	Adobe_typography_AI3
		{
		dup xcheck
			{
			bind
			} if
		pop pop
		} forall
 end
	Adobe_typography_AI3 begin
	} ifelse
} def
/terminate
{
currentdict Adobe_typography_AI3 eq
	{
 end
	} if
} def
/modifyEncoding
{
	/_tempEncode exch ddef
	
	/_pntr 0 ddef
	
	{
		counttomark -1 roll
		dup type dup /marktype eq 		
		{
			pop pop exit
		}
		{
			/nametype eq
			{
				_tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll
				put
			}
			{
				/_pntr exch ddef					
			}
			ifelse
		}
		ifelse
	}
	loop	
	
	_tempEncode
}
def
/TE
{
	StandardEncoding 256 array copy modifyEncoding 
	/_nativeEncoding exch def
} def
%
/TZ		
{
	dup type /arraytype eq {/_wv exch def} {/_wv 0 def} ifelse 
	/_useNativeEncoding exch def
	pop pop
	
	findfont _wv type /arraytype eq {_wv makeblendedfont} if dup length 2 add dict
	
 begin
	
		mark exch
		{
			1 index /FID ne { def } if cleartomark mark
		}
		forall
		pop
		
		/FontName exch def
		
		counttomark 0 eq
		{
			1 _useNativeEncoding eq
			{
				/Encoding _nativeEncoding def
			}
			if
			cleartomark
		}
		{	
			/Encoding load 256 array copy 
			modifyEncoding /Encoding exch def
		}
		ifelse		
		FontName currentdict
 end
	
	definefont pop
}
def
/tr
{
_ax _ay 3 2 roll
} def
/trj
{
_cx _cy _sp _ax _ay 6 5 roll
} def
/a0
{
/Tx
	{
	dup 
	currentpoint 3 2 roll
	tr _psf
	newpath moveto
	tr _ctm _pss
	} ddef
/Tj
	{
	dup
	currentpoint 3 2 roll
	trj _pjsf
	newpath moveto
	trj _ctm _pjss
	} ddef
	
} def
/a1
{
/Tx
	{
	dup currentpoint 4 2 roll gsave
	dup currentpoint 3 2 roll
	tr _psf
	newpath moveto
	tr _ctm _pss
	grestore 3 1 roll moveto tr sp
	} ddef
/Tj
	{
	dup currentpoint 4 2 roll gsave
	dup currentpoint 3 2 roll
	trj _pjsf
	newpath moveto
	trj _ctm _pjss
	grestore 3 1 roll moveto tr sp
	} ddef
	
} def
/e0
{
/Tx
	{
	tr _psf
	} ddef
/Tj
	{
	trj _pjsf
	} ddef
} def
/e1
{
/Tx
	{
	dup currentpoint 4 2 roll gsave 
	tr _psf  
	grestore 3 1 roll moveto tr sp 
	} ddef
/Tj
	{
	dup currentpoint 4 2 roll gsave 
	trj _pjsf
	grestore 3 1 roll moveto tr sp 
	} ddef
} def
/i0
{
/Tx
	{
	tr sp
	} ddef
/Tj
	{
	trj jsp
	} ddef
} def
/i1
{
W N
} def
/o0
{
/Tx
	{
	tr sw rmoveto
	} ddef
/Tj
	{
	trj swj rmoveto
	} ddef
} def
/r0
{
/Tx
	{
	tr _ctm _pss
	} ddef
/Tj
	{
	trj _ctm _pjss
	} ddef
} def
/r1
{
/Tx
	{
	dup currentpoint 4 2 roll currentpoint gsave newpath moveto
	tr _ctm _pss 
	grestore 3 1 roll moveto tr sp 
	} ddef
/Tj
	{
	dup currentpoint 4 2 roll currentpoint gsave newpath moveto
	trj _ctm _pjss
	grestore 3 1 roll moveto tr sp 
	} ddef
} def
/To
{
	pop _ctm currentmatrix pop
} def
/TO
{
	iTe _ctm setmatrix newpath
} def
/Tp
{
	pop _tm astore pop _ctm setmatrix 
	_tDict begin /W {} def /h {} def
} def
/TP
{
 end
	iTm 0 0 moveto
} def
/Tr
{
	_render 3 le {currentpoint newpath moveto} if
	dup 8 eq {pop 0} {dup 9 eq {pop 1} if} ifelse
	dup /_render exch ddef
	_renderStart exch get load exec
} def
/iTm
{
_ctm setmatrix _tm concat 0 _rise translate _hs 1 scale
} def
/Tm
{
_tm astore pop iTm 0 0 moveto
} def
/Td
{
_mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto
} def
/iTe
{
	_render -1 eq {} {_renderEnd _render get dup null ne {load exec} {pop} ifelse} ifelse
	/_render -1 ddef
} def
/Ta
{
pop
} def
/Tf
{
dup 1000 div /_fScl exch ddef
exch findfont exch scalefont setfont
} def
/Tl
{
pop
0 exch _leading astore pop
} def
/Tt
{
pop
} def
/TW
{
3 npop
} def
/Tw
{
/_cx exch ddef
} def
/TC
{
3 npop
} def
/Tc
{
/_ax exch ddef
} def
/Ts
{
/_rise exch ddef
currentpoint
iTm
moveto
} def
/Ti
{
3 npop
} def
/Tz
{
100 div /_hs exch ddef
iTm
} def
/TA
{
pop
} def
/Tq
{
pop
} def
/Th
{
pop pop pop pop pop
} def
/TX {pop} def
%/Tx
%/Tj
/Tk
{
exch pop _fScl mul neg 0 rmoveto
} def
/TK
{
2 npop
} def
/T*
{
_leading aload pop neg Td
} def
/T*-
{
_leading aload pop Td
} def
/T-
{
_hyphen Tx
} def
/T+
{} def
/TR
{
_ctm currentmatrix pop 	
_tm astore pop 
iTm 0 0 moveto 
} def
/TS
{
0 eq {Tx} {Tj} ifelse
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_pattern_AI3 1.1 0
%%Title: (Adobe Illustrator (R) Version 3.0 Pattern Operators)
%%Version: 1.1 0
%%CreationDate: (7/21/89) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_pattern_AI3 16 dict dup begin put
/initialize
{
/definepattern where
	{
	pop
	}
	{
	Adobe_pattern_AI3 begin
	Adobe_pattern_AI3
		{
		dup xcheck
			{
			bind
			} if
		pop pop
		} forall
	mark
	cachestatus 7 1 roll pop pop pop pop exch pop exch
		{
		{
		10000 add
		dup 2 index gt
			{
			exit
			} if
		dup setcachelimit
		} loop
		} stopped
	cleartomark
	} ifelse
} def
/terminate
{
currentdict Adobe_pattern_AI3 eq
	{
 end
	} if
} def
errordict
/nocurrentpoint
{
pop
stop
} put
errordict
/invalidaccess
{
pop
stop
} put
/patternencoding
256 array def
0 1 255
{
patternencoding exch ( ) 2 copy exch 0 exch put cvn put
} for
/definepattern
{
17 dict begin
/uniform exch def
/cache exch def
/key exch def
/procarray exch def
/mtx exch matrix invertmatrix def
/height exch def
/width exch def
/ctm matrix currentmatrix def
/ptm matrix def
/str 32 string def
/slice 9 dict def
slice /s 1 put
slice /q 256 procarray length div sqrt floor cvi put
slice /b 0 put
/FontBBox [0 0 0 0] def
/FontMatrix mtx matrix copy def
/Encoding patternencoding def
/FontType 3 def
/BuildChar
	{
	exch
 begin
	/setstrokeadjust where {pop true setstrokeadjust} if
	slice begin
	dup q dup mul mod s idiv /i exch def
	dup q dup mul mod s mod /j exch def
	q dup mul idiv procarray exch get
	/xl j width s div mul def
	/xg j 1 add width s div mul def
	/yl i height s div mul def
	/yg i 1 add height s div mul def
	uniform
		{
		1 1
		}
		{
		width 0 dtransform
		dup mul exch dup mul add sqrt dup 1 add exch div
		0 height dtransform
		dup mul exch dup mul add sqrt dup 1 add exch div
		} ifelse
	width 0 cache
		{
		xl 4 index mul yl 4 index mul xg 6 index mul yg 6 index mul
		setcachedevice
		}
		{
		setcharwidth
		} ifelse
	gsave
	scale
	newpath
	xl yl moveto
	xg yl lineto
	xg yg lineto
	xl yg lineto
	closepath
	clip
	newpath
 end
 end
	exec
	grestore
	} def
key currentdict definefont
end
} def
/patterncachesize
{
gsave
newpath
0 0 moveto
width 0 lineto
width height lineto
0 height lineto
closepath
patternmatrix setmatrix
pathbbox
exch ceiling 4 -1 roll floor sub 3 1 roll
ceiling exch floor sub
mul 1 add
grestore
} def
/patterncachelimit
{
cachestatus 7 1 roll 6 npop 8 mul
} def
/patternpath
{
exch dup begin setfont
ctm setmatrix
concat
slice exch /b exch slice /q get dup mul mul put
FontMatrix concat
uniform
	{
	width 0 dtransform round width div exch round width div exch
	0 height dtransform round height div exch height div exch
	0 0 transform round exch round exch
	ptm astore setmatrix
	}
	{
	ptm currentmatrix pop
	} ifelse
{currentpoint} stopped not
	{
	2 npop
	pathbbox
	true
	4 index 3 index eq
	4 index 3 index eq
	and
		{
		pop false
			{
			{2 npop}
			{3 npop true}
			{7 npop true}
			{pop true}
			pathforall
			} stopped
			{
			5 npop true
			} if
		} if
		{
		height div ceiling height mul 4 1 roll
		width div ceiling width mul 4 1 roll
		height div floor height mul 4 1 roll
		width div floor width mul 4 1 roll
		2 index sub height div ceiling cvi exch
		3 index sub width div ceiling cvi exch
		4 2 roll moveto
		FontMatrix mtx invertmatrix
		dup dup 4 get exch 5 get rmoveto
		ptm ptm concatmatrix pop
		slice /s
		patterncachesize patterncachelimit div ceiling sqrt ceiling cvi
		dup slice /q get gt
			{
			pop slice /q get
			} if
		put
		0 1 slice /s get dup mul 1 sub
			{
			slice /b get add
			gsave
			0 1 str length 1 sub
				{
				str exch 2 index put
				} for
			pop
			dup
				{
				gsave
				ptm setmatrix
				1 index str length idiv {str show} repeat
				1 index str length mod str exch 0 exch getinterval show
				grestore
				0 height rmoveto
				} repeat
			grestore
			} for
		2 npop
		}
		{
		4 npop
		} ifelse
	} if
end
} def
/patternclip
{
clip
} def
/patternstrokepath
{
strokepath
} def
/patternmatrix
matrix def
/patternfill
{
dup type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get
	} if
gsave
patternclip
Adobe_pattern_AI3 /patternpath get exec
grestore
newpath
} def
/patternstroke
{
dup type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get
	} if
gsave
patternstrokepath
true
	{
		{
			{
			newpath 
			moveto
			}
			{
			lineto
			}
			{
			curveto
			}
			{
			closepath
			3 copy
			Adobe_pattern_AI3 /patternfill get exec
			} pathforall
		3 npop
		} stopped
			{
			5 npop
			patternclip
			Adobe_pattern_AI3 /patternfill get exec
			} if
	}
	{
	patternclip
	Adobe_pattern_AI3 /patternfill get exec
	} ifelse
grestore
newpath
} def
/patternashow
{
3 index type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get 4 1 roll
	} if
	{
	2 npop (0) exch
	2 copy 0 exch put pop
	gsave
	false charpath currentpoint
	6 index 6 index 6 index
	Adobe_pattern_AI3 /patternfill get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
5 npop
} def
/patternawidthshow
{
6 index type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get 7 1 roll
	} if
	{
	2 npop (0) exch
	2 copy 0 exch put 
	gsave
	_sp eq {5 index 5 index rmoveto} if
	false charpath currentpoint
	9 index 9 index 9 index
	Adobe_pattern_AI3 /patternfill get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
8 npop
} def
/patternashowstroke
{
4 index type /dicttype eq
	{
	patternmatrix /patternmatrix get 5 1 roll
	} if
4 1 roll
	{
	2 npop (0) exch
	2 copy 0 exch put pop
	gsave
	false charpath
	currentpoint
	4 index setmatrix
	7 index 7 index 7 index
	Adobe_pattern_AI3 /patternstroke get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
6 npop
} def
/patternawidthshowstroke
{
7 index type /dicttype eq
	{
	patternmatrix /patternmatrix get 8 1 roll
	} if
7 1 roll
	{
	2 npop (0) exch
	2 copy 0 exch put
	gsave
	_sp eq {5 index 5 index rmoveto} if
	false charpath currentpoint
	7 index setmatrix
	10 index 10 index 10 index
	Adobe_pattern_AI3 /patternstroke get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
9 npop
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_Illustrator_AI3 1.1 0
%%Title: (Adobe Illustrator (R) Version 3.0 Full Prolog)
%%Version: 1.1 0
%%CreationDate: (3/7/1994) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_Illustrator_AI3 71 dict dup begin put
/initialize
{
userdict /Adobe_Illustrator_AI3_vars 67 dict dup begin put
/_lp /none def
/_pf {} def
/_ps {} def
/_psf {} def
/_pss {} def
/_pjsf {} def
/_pjss {} def
/_pola 0 def
/_doClip 0 def
/cf	currentflat def
/_tm matrix def
/_renderStart [/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0] def 
/_renderEnd [null null null null /i1 /i1 /i1 /i1] def
/_render -1 def
/_rise 0 def
/_ax 0 def
/_ay 0 def
/_cx 0 def
/_cy 0 def
/_leading [0 0] def
/_ctm matrix def
/_mtx matrix def
/_sp 16#020 def
/_hyphen (-) def
/_fScl 0 def
/_cnt 0 def
/_hs 1 def
/_nativeEncoding 0 def
/_useNativeEncoding 0 def
/_tempEncode 0 def
/_pntr 0 def
/_tDict 2 dict def
/_wv 0 def
/Tx {} def
/Tj {} def
/CRender {} def
/_AI3_savepage {} def
/_gf null def
/_cf 4 array def
/_if null def
/_of false def
/_fc {} def
/_gs null def
/_cs 4 array def
/_is null def
/_os false def
/_sc {} def
/_pd 1 dict def
/_ed 15 dict def
/_pm matrix def
/_fm null def
/_fd null def
/_fdd null def
/_sm null def
/_sd null def
/_sdd null def
/_i null def
Adobe_Illustrator_AI3 begin
Adobe_Illustrator_AI3 dup /nc get begin
	{
	dup xcheck
		{
		bind
		} if
	pop pop
	} forall
end
end
end
Adobe_Illustrator_AI3 begin
Adobe_Illustrator_AI3_vars begin
newpath
} def
/terminate
{
end
end
} def
/_
null def
/ddef
{
Adobe_Illustrator_AI3_vars 3 1 roll put
} def
/xput
{
dup load dup length exch maxlength eq
	{
	dup dup load dup
	length 2 mul dict copy def
	} if
load begin def end
} def
/npop
{
	{
	pop
	} repeat
} def
/sw
{
dup length exch stringwidth
exch 5 -1 roll 3 index mul add
4 1 roll 3 1 roll mul add
} def
/swj
{
dup 4 1 roll
dup length exch stringwidth 
exch 5 -1 roll 3 index mul add
4 1 roll 3 1 roll mul add 
6 2 roll /_cnt 0 ddef
{1 index eq {/_cnt _cnt 1 add ddef} if} forall pop
exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop
} def
/ss
{
4 1 roll
	{
	2 npop 
	(0) exch 2 copy 0 exch put pop
	gsave
	false charpath currentpoint
	4 index setmatrix
	stroke
	grestore
	moveto
	2 copy rmoveto
	} exch cshow
3 npop
} def
/jss
{
4 1 roll
	{
	2 npop 
	(0) exch 2 copy 0 exch put 
	gsave
	_sp eq 
		{
		exch 6 index 6 index 6 index 5 -1 roll widthshow  
		currentpoint
		}
		{
		false charpath currentpoint
		4 index setmatrix stroke
		}ifelse
	grestore
	moveto
	2 copy rmoveto
	} exch cshow
6 npop
} def
/sp
{
	{
	2 npop (0) exch
	2 copy 0 exch put pop
	false charpath
	2 copy rmoveto
	} exch cshow
2 npop
} def
/jsp
{
	{
	2 npop 
	(0) exch 2 copy 0 exch put 
	_sp eq 
		{
		exch 5 index 5 index 5 index 5 -1 roll widthshow  
		}
		{
		false charpath
		}ifelse
	2 copy rmoveto
	} exch cshow
5 npop
} def
/pl
{
transform
0.25 sub round 0.25 add exch
0.25 sub round 0.25 add exch
itransform
} def
/setstrokeadjust where
	{
	pop true setstrokeadjust
	/c
	{
	curveto
	} def
	/C
	/c load def
	/v
	{
	currentpoint 6 2 roll curveto
	} def
	/V
	/v load def
	/y
	{
	2 copy curveto
	} def
	/Y
	/y load def
	/l
	{
	lineto
	} def
	/L
	/l load def
	/m
	{
	moveto
	} def
	}
	{
	/c
	{
	pl curveto
	} def
	/C
	/c load def
	/v
	{
	currentpoint 6 2 roll pl curveto
	} def
	/V
	/v load def
	/y
	{
	pl 2 copy curveto
	} def
	/Y
	/y load def
	/l
	{
	pl lineto
	} def
	/L
	/l load def
	/m
	{
	pl moveto
	} def
	} ifelse
/d
{
setdash
} def
/cf	{} def
/i
{
dup 0 eq
	{
	pop cf
	} if
setflat
} def
/j
{
setlinejoin
} def
/J
{
setlinecap
} def
/M
{
setmiterlimit
} def
/w
{
setlinewidth
} def
/H
{} def
/h
{
closepath
} def
/N
{
_pola 0 eq 
	{
	_doClip 1 eq {clip /_doClip 0 ddef} if 
	newpath
	} 
	{
	/CRender {N} ddef
	}ifelse
} def
/n
{N} def
/F
{
_pola 0 eq 
	{
	_doClip 1 eq 
		{
		gsave _pf grestore clip newpath /_lp /none ddef _fc 
		/_doClip 0 ddef
		}
		{
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2164 1894 1112 4 v 2692 1985 a FH(h)3286 1917 y(;)1854
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b(N)2885 2673 y FF(\016)2922 2658 y FJ(\()p FH(A)p FJ(\()p
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b FG(\000)26 b FH(t)p FJ(\),)37 b(the)874 5253 y(suprem)m(um)f(ma)m(y)f
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b(al)35 b(indicated.)49 b(If)874 5391 y(the)e(suprem)m(um)i(is)e
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Black Black eop end
%%Page: 137 155
TeXDict begin 137 154 bop Black 591 266 a Fx(5.3.)75
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FH(;)17 b(A)p FJ(\()p FH(p)p FJ(\)\).)54 b(Also,)38 b(consider)874
946 y(some)h(^)-55 b FH(x)1173 886 y Fs(0)1196 909 y
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1561 y FF(p)p FE(\))3041 1546 y FJ(\()p FH(x)3134 1505
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y FJ(\))p FH(;)17 b(A)p FJ(\()p FH(p)p FJ(\)\))1619 1778
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2199 y FF(p)p FE(\))2261 2184 y FJ(\()p FH(x)2354 2119
y Fs(0)2376 2143 y FB(\003)2354 2208 y FF(\034)i FE(+)p
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FJ(,)k(and)p Black Black eop end
%%Page: 138 156
TeXDict begin 138 155 bop Black 591 266 a FJ(138)1492
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1689 y(\014)2779 1749 y(\014)2828 1744 y FJ(;)g(0)27
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y FJ(\()p FH(x)2616 1954 y FF(\034)i FE(+)p FF(t)2739
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740 y FJ(in)591 878 y(a)g(\014nite)h(n)m(um)m(b)s(er)h(of)e(steps.)p
Black 591 1088 a FI(De\014nition)38 b(6.1.1)f FJ(\(A1\))p
FI(.)p Black 42 w Fy(De\014ne)1639 1312 y FH(\015)5 b
FJ(\()p FH(h)p FJ(\))28 b(=)f(2)p FH(LC)2143 1327 y FF(r)2181
1312 y FH(h)2237 1270 y FF(r)r FE(+1)2365 1312 y FH(=)p
FJ(\(1)22 b FG(\000)h FH(e)2668 1270 y FB(\000)p FF(ch)2798
1312 y FJ(\))p FH(;)591 1536 y Fy(and)806 1511 y FJ(^)780
1536 y FH(A)853 1500 y FF(h)925 1536 y FJ(=)28 b FG(f)p
FH(A)1152 1500 y FF(h)1197 1536 y FJ(\()p FH(p)1284 1551
y FF(n)1331 1536 y FJ(\);)17 b FH(n)27 b FG(2)h FC(Z)p
FG(g)35 b Fy(by)1334 1760 y FH(A)1407 1719 y FF(h)1452
1760 y FJ(\()p FH(p)1539 1775 y FF(n)1586 1760 y FJ(\))27
b(=)h FG(f)p FH(x)p FJ(;)17 b FH(x)28 b FG(2)g FH(B)2160
1719 y FF(h)2205 1760 y FJ(\()p FH(p)2292 1775 y FF(n)2339
1760 y FJ(\))p FH(;)17 b(V)k FJ(\()p FH(p)2586 1775 y
FF(n)2633 1760 y FH(;)c(x)p FJ(\))28 b FG(\024)g FH(\015)5
b FJ(\()p FH(h)p FJ(\))p FG(g)p FH(;)543 b FJ(\(6.4\))591
1984 y Fy(for)34 b(any)h FH(h)28 b FG(2)g FJ([0)p FH(;)17
b(h)1287 1999 y FE(1)1326 1984 y FJ(\))p Fy(.)591 2264
y FJ(W)-8 b(e)23 b(pro)s(ceed)h(to)e(sho)m(w)i(that)1668
2239 y(^)1643 2264 y FH(A)1716 2228 y FF(h)1783 2264
y FJ(is)g(a)e(discrete)j(uniformly)f(asymptotically)g(stable)g(family)
591 2403 y(for)32 b(the)h(discretised)i(dynamical)e(system.)591
2612 y(Note)f FH(\015)5 b FJ(\()p FH(h)p FJ(\))28 b FG(\024)g
FH(a)p FJ(\()p FH(\016)1279 2627 y FE(0)1319 2612 y FJ(\),)k(and)h(th)m
(us)h FH(A)1894 2576 y FF(h)1939 2612 y FJ(\()p FH(p)2026
2627 y FF(n)2073 2612 y FJ(\))e(is)h(strictly)h(a)e(subset)i(of)f
FH(B)3146 2576 y FF(h)3190 2612 y FJ(\()p FH(p)3277 2627
y FF(n)3324 2612 y FJ(\))g(for)f(eac)m(h)h FH(n)p FJ(.)p
Black 591 2821 a FI(Lemma)39 b(6.1.3)e FJ(\(A2\))p FI(.)p
Black 42 w FH(A)1599 2785 y FF(h)1644 2821 y FJ(\()p
FH(p)1731 2836 y FF(n)1778 2821 y FJ(\))d Fy(is)h(b)-5
b(ounde)g(d)34 b(and)g(c)-5 b(omp)g(act)35 b(for)f(e)-5
b(ach)34 b FH(n)28 b FG(2)g FC(Z)p Fy(.)p Black 630 3123
a(Pr)-5 b(o)g(of:)p Black 49 w FH(A)p FJ(\()p FH(p)1096
3138 y FF(n)1143 3123 y FJ(\))39 b(is)i(b)s(ounded)f(for)f(eac)m(h)i
FH(n)f FG(2)h FC(Z)p FJ(.)65 b(Consequen)m(tly)43 b FH(A)3162
3087 y FF(h)3207 3123 y FJ(\()p FH(p)3294 3138 y FF(n)3341
3123 y FJ(\))d(m)m(ust)874 3261 y(also)27 b(b)s(e)h(b)s(ounded)g(since)
g FH(A)1891 3225 y FF(h)1936 3261 y FJ(\()p FH(p)2023
3276 y FF(n)2070 3261 y FJ(\))g FG(\032)g FH(B)2320 3225
y FF(h)2365 3261 y FJ(\()p FH(p)2452 3276 y FF(n)2499
3261 y FJ(\))f FG(\032)h(N)2751 3276 y FF(\016)2782 3285
y Fu(0)2821 3261 y FJ(\()p FH(A)p FJ(\()p FH(p)3019 3276
y FF(n)3066 3261 y FJ(\)\).)42 b(Compact-)874 3400 y(ness)34
b(follo)m(ws)f(from)g(the)g(con)m(tin)m(uit)m(y)h(of)e
FH(V)22 b FJ(.)p 3841 3701 V 3845 3638 59 4 v 3845 3701
V 3903 3701 4 66 v Black 591 3910 a FI(Lemma)39 b(6.1.4)e
FJ(\(A3\))p FI(.)p Black 1552 3885 a FJ(^)1526 3910 y
FH(A)1599 3874 y FF(h)1679 3910 y Fy(is)d(p)-5 b(ositively)35
b(invariant)f(under)h(the)f(discr)-5 b(etisation.)p Black
630 4212 a(Pr)g(o)g(of:)p Black 49 w FJ(Let)32 b FH(x)1165
4227 y FF(n)1240 4212 y FG(2)c FH(A)1407 4175 y FF(h)1452
4212 y FJ(\()p FH(p)1539 4227 y FF(n)1586 4212 y FJ(\).)44
b(It)32 b(then)i(follo)m(ws)f(that)1011 4436 y FH(V)22
b FJ(\()p FH(\022)1173 4451 y FF(h)1218 4436 y FH(p)1267
4451 y FF(n)1314 4436 y FH(;)17 b(x)1413 4451 y FF(n)1460
4436 y FJ(\))28 b FG(\024)g FH(V)21 b FJ(\()p FH(\022)1792
4451 y FF(h)1838 4436 y FH(p)1887 4451 y FF(n)1934 4436
y FH(;)c FJ(\010)2048 4451 y FE(\()p FF(h;p)2172 4459
y FA(n)2213 4451 y FE(\))2245 4436 y FJ(\()p FH(x)2338
4451 y FF(n)2385 4436 y FJ(\)\))22 b(+)g FH(L)p FG(j)p
FH(x)2730 4451 y FF(n)p FE(+1)2890 4436 y FG(\000)h FJ(\010)3060
4451 y FE(\()p FF(h;p)3184 4459 y FA(n)3226 4451 y FE(\))3258
4436 y FJ(\()p FH(x)3351 4451 y FF(n)3398 4436 y FJ(\))p
FG(j)p FH(;)1526 4599 y FG(\024)28 b FH(e)1676 4558 y
FB(\000)p FF(ch)1806 4599 y FH(V)22 b FJ(\()p FH(p)1972
4614 y FF(n)2019 4599 y FH(;)17 b(x)2118 4614 y FF(n)2165
4599 y FJ(\))22 b(+)g FH(LC)2459 4614 y FF(r)2497 4599
y FH(h)2553 4558 y FF(r)r FE(+1)2681 4599 y FH(;)1526
4805 y FG(\024)28 b FH(e)1676 4764 y FB(\000)p FF(ch)1806
4805 y FH(\015)5 b FJ(\()p FH(h)p FJ(\))23 b(+)2125 4738
y(1)p 2125 4782 49 4 v 2125 4874 a(2)2183 4805 y FH(\015)5
b FJ(\()p FH(h)p FJ(\)\(1)22 b FG(\000)h FH(e)2625 4764
y FB(\000)p FF(ch)2755 4805 y FJ(\))p FH(;)1526 5004
y FG(\024)28 b FH(\015)5 b FJ(\()p FH(h)p FJ(\))p FH(:)874
5391 y FJ(Hence)34 b FH(x)1219 5406 y FF(n)p FE(+1)1384
5391 y FG(2)28 b FH(A)1551 5355 y FF(h)1596 5391 y FJ(\()p
FH(p)1683 5406 y FF(n)p FE(+1)1820 5391 y FJ(\).)p Black
Black eop end
%%Page: 147 165
TeXDict begin 147 164 bop Black 591 266 a Fx(6.1.)75
b(UNIF)m(ORM)33 b(ASYMPTOTIC)i(ST)-8 b(ABILITY)1192 b
FJ(147)p Black 3841 530 4 66 v 3845 468 59 4 v 3845 530
V 3903 530 4 66 v Black 591 737 a FI(Lemma)39 b(6.1.5)e
FJ(\(A4\))p FI(.)p Black 1552 712 a FJ(^)1526 737 y FH(A)1599
701 y FF(h)1679 737 y Fy(uniformly)d(attr)-5 b(acts)2499
712 y FJ(^)2476 737 y FH(B)40 b Fy(in)34 b(a)h(\014nite)f(numb)-5
b(er)35 b(of)f(steps.)p Black 630 1034 a(Pr)-5 b(o)g(of:)p
Black 49 w FJ(Let)32 b FH(x)1165 1049 y FE(0)1233 1034
y FG(2)c FH(B)1406 998 y FF(h)1451 1034 y FJ(\()p FH(p)1538
1049 y FE(0)1577 1034 y FJ(\))p FG(n)p FH(A)1738 998
y FF(h)1783 1034 y FJ(\()p FH(p)1870 1049 y FE(0)1909
1034 y FJ(\).)44 b(Then)33 b FH(V)22 b FJ(\()p FH(p)2438
1049 y FE(0)2477 1034 y FH(;)17 b(x)2576 1049 y FE(0)2616
1034 y FJ(\))27 b FG(\025)i FH(\015)5 b FJ(\()p FH(h)p
FJ(\))32 b(and)h(w)m(e)g(ha)m(v)m(e)1303 1282 y FH(V)21
b FJ(\()p FH(p)1468 1297 y FE(1)1507 1282 y FH(;)c(x)1606
1297 y FE(1)1646 1282 y FJ(\))28 b FG(\024)g FH(e)1862
1241 y FB(\000)p FF(ch)1992 1282 y FH(V)22 b FJ(\()p
FH(p)2158 1297 y FE(0)2197 1282 y FH(;)17 b(x)2296 1297
y FE(0)2336 1282 y FJ(\))22 b(+)2504 1214 y(1)p 2504
1259 49 4 v 2504 1350 a(2)2562 1282 y FH(\015)5 b FJ(\()p
FH(h)p FJ(\)\(1)22 b FG(\000)h FH(e)3004 1241 y FB(\000)p
FF(ch)3134 1282 y FJ(\))p FH(;)1712 1515 y FG(\024)1827
1447 y FJ(1)p 1827 1492 V 1827 1583 a(2)1885 1515 y(\(1)f(+)g
FH(e)2137 1474 y FB(\000)p FF(ch)2268 1515 y FJ(\))p
FH(V)f FJ(\()p FH(p)2471 1530 y FE(0)2511 1515 y FH(;)c(x)2610
1530 y FE(0)2649 1515 y FJ(\))p FH(:)874 1934 y FJ(If)50
b(w)m(e)g(de\014ne)h FH(h)1504 1949 y FE(2)1601 1934
y FJ(=)57 b(min)p FG(f)p FH(h)2002 1949 y FE(1)2042 1934
y FH(;)17 b( )t FG(g)49 b FJ(where)i FH( )i FJ(satis\014es)f(\(1)33
b(+)h FH(e)3319 1898 y FB(\000)p FF(c )3457 1934 y FJ(\))57
b(=)874 2073 y(2)p FH(e)968 2036 y FB(\000)p FF(c )r(=)p
FE(4)1177 2073 y FJ(,)32 b(then)h(for)g(all)f FH(h)c
FG(2)g FJ([0)p FH(;)17 b(h)2097 2088 y FE(2)2136 2073
y FJ(\),)1813 2293 y(\(1)22 b(+)g FH(e)2065 2252 y FB(\000)p
FF(ch)2196 2293 y FJ(\))27 b FG(\024)i FJ(2)p FH(e)2461
2252 y FB(\000)p FF(ch=)p FE(4)2661 2293 y FH(:)874 2514
y FJ(Consequen)m(tly)-8 b(,)1667 2653 y FH(V)21 b FJ(\()p
FH(p)1832 2668 y FE(1)1871 2653 y FH(;)c(x)1970 2668
y FE(1)2010 2653 y FJ(\))28 b FG(\024)g FH(e)2226 2611
y FB(\000)p FF(ch=)p FE(4)2427 2653 y FH(V)21 b FJ(\()p
FH(p)2592 2668 y FE(0)2632 2653 y FH(;)c(x)2731 2668
y FE(0)2770 2653 y FJ(\))p FH(:)874 2838 y FJ(Supp)s(ose)34
b FH(x)1312 2853 y FF(j)1376 2838 y FG(2)28 b FH(B)5
b FJ(\()p FH(p)1636 2853 y FF(j)1673 2838 y FJ(\))p FG(n)p
FH(A)1834 2801 y FF(h)1879 2838 y FJ(\()p FH(p)1966 2853
y FF(j)2002 2838 y FJ(\))32 b(for)g FH(j)i FJ(=)28 b(1)p
FH(;)17 b(:)g(:)g(:)e(;)i(n)22 b FG(\000)h FJ(1.)43 b(Then)1638
3058 y FH(V)21 b FJ(\()p FH(p)1803 3073 y FF(n)1850 3058
y FH(;)c(x)1949 3073 y FF(n)1996 3058 y FJ(\))28 b FG(\024)g
FH(e)2212 3017 y FB(\000)p FF(nch=)p FE(4)2456 3058 y
FH(V)21 b FJ(\()p FH(p)2621 3073 y FE(0)2661 3058 y FH(;)c(x)2760
3073 y FE(0)2799 3058 y FJ(\))p FH(;)2062 3222 y FG(\024)28
b FH(e)2212 3181 y FB(\000)p FF(nch=)p FE(4)2456 3222
y FH(a)p FJ(\()p FH(\016)2588 3237 y FE(0)2628 3222 y
FJ(\))p FH(:)874 3606 y FJ(If)33 b(w)m(e)g(de\014ne)h
FH(N)k FJ(=)28 b FH(N)10 b FJ(\()p FH(h;)17 b(\016)1886
3621 y FE(0)1926 3606 y FJ(\))32 b(b)m(y)1686 3866 y
FH(N)10 b FJ(\()p FH(h;)17 b(\016)1955 3881 y FE(0)1995
3866 y FJ(\))27 b(=)2198 3799 y(4)p 2174 3843 99 4 v
2174 3934 a FH(ch)2298 3866 y FJ(ln)2396 3726 y Ft(\022)2480
3799 y FH(a)p FJ(\()p FH(\016)2612 3814 y FE(0)2651 3799
y FJ(\))p 2480 3843 210 4 v 2491 3934 a FH(\015)5 b FJ(\()p
FH(h)p FJ(\))2699 3726 y Ft(\023)2789 3866 y FH(;)874
4146 y FJ(then)29 b(for)e(all)i FH(n)e(>)h(N)39 b FJ(\(recalling)28
b(that)2330 4121 y(^)2304 4146 y FH(A)2377 4110 y FF(h)2450
4146 y FJ(is)h(p)s(ositiv)m(ely)h(in)m(v)-5 b(arian)m(t)29
b(under)874 4284 y(the)k(discretisation\),)1879 4423
y FH(V)21 b FJ(\()p FH(p)2044 4438 y FF(n)2091 4423 y
FH(;)c(x)2190 4438 y FF(n)2237 4423 y FJ(\))28 b FG(\024)g
FH(\015)5 b FJ(\()p FH(h)p FJ(\))p FH(:)874 4608 y FJ(Hence)42
b FH(x)1227 4623 y FF(n)1317 4608 y FG(2)g FH(A)1498
4571 y FF(h)1543 4608 y FJ(\()p FH(p)1630 4623 y FF(n)1677
4608 y FJ(\))f(for)f(all)h FH(n)h(>)g(N)10 b FJ(.)69
b(Note)41 b(that)f(the)i(attraction)e(is)874 4746 y(uniform.)k(That)33
b(is,)g FH(N)43 b FJ(is)33 b(indep)s(enden)m(t)i(of)d
FH(p)2579 4761 y FE(0)2651 4746 y FJ(and)g FH(x)2895
4761 y FE(0)2935 4746 y FJ(.)p 3841 5044 4 66 v 3845
4981 59 4 v 3845 5044 V 3903 5044 4 66 v 591 5253 a(T)-8
b(o)44 b(ful\014ll)h(the)g(requiremen)m(ts)j(of)c(uniform)h(asymptotic)
h(stabilit)m(y)-8 b(,)49 b(w)m(e)d(construct)f(an)591
5391 y(attracting)32 b FH(\016)t FJ(-neigh)m(b)s(ourho)s(d)g(of)1877
5366 y(^)1851 5391 y FH(A)1924 5355 y FF(h)1969 5391
y FJ(,)h(and)g(also)f(sho)m(w)i(that)2893 5366 y(^)2867
5391 y FH(A)2940 5355 y FF(h)3018 5391 y FJ(is)f(stable.)p
Black Black eop end
%%Page: 148 166
TeXDict begin 148 165 bop Black 591 266 a FJ(148)853
b Fx(CHAPTER)34 b(6.)76 b(NUMERICAL)34 b(APPR)m(O)m(XIMA)-8
b(TION)35 b(-)d(I)p Black Black 591 530 a FI(Lemma)39
b(6.1.6)e FJ(\(A5)c(-)f(A)m(ttracting)h FH(\016)t FJ(-Neigh)m(b)s
(ourho)s(o)s(d\))p FI(.)p Black 41 w Fy(De\014ne)1903
791 y FH(\016)f FJ(=)2091 724 y(1)p 2091 768 49 4 v 2091
860 a(2)2150 791 y FH(b)2191 750 y FB(\000)p FE(1)2286
791 y FJ(\()p FH(a)p FJ(\()p FH(\016)2456 806 y FE(0)2496
791 y FJ(\)\))p FH(:)591 1058 y Fy(Then)870 1033 y FJ(^)845
1058 y FH(A)918 1022 y FF(h)997 1058 y Fy(uniformly)j(attr)-5
b(acts)35 b(the)g(neighb)-5 b(ourho)g(o)g(d)34 b(system)h
FG(N)2989 1087 y FF(\016)o(;)3056 1070 y FE(^)3039 1087
y FF(A)3092 1068 y FA(h)3135 1058 y Fy(.)p Black 630
1372 a(Pr)-5 b(o)g(of:)p Black 49 w FJ(First)32 b(w)m(e)i(sho)m(w)f
FG(N)1640 1387 y FF(\016)1678 1372 y FJ(\()p FH(A)1789
1335 y FF(h)1834 1372 y FJ(\()p FH(p)1921 1387 y FF(n)1968
1372 y FJ(\)\))27 b FG(\022)i FH(B)5 b FJ(\()p FH(p)2343
1387 y FF(n)2390 1372 y FJ(\))32 b(for)g(eac)m(h)i FH(n)27
b FG(2)i FC(Z)p FJ(.)874 1551 y(F)-8 b(or)32 b(arbitrary)g
FH(n)p FJ(,)h(consider)h(an)m(y)f FH(x)28 b FG(2)h(N)2405
1566 y FF(\016)2442 1551 y FJ(\()p FH(A)2553 1515 y FF(h)2598
1551 y FJ(\()p FH(p)2685 1566 y FF(n)2732 1551 y FJ(\)\).)43
b(Then)1120 1786 y(dist)q(\()p FH(x;)17 b(A)p FJ(\()p
FH(p)1575 1801 y FF(n)1622 1786 y FJ(\)\))27 b FG(\024)h
FJ(dist)q(\()p FH(x;)17 b(A)2198 1744 y FF(h)2243 1786
y FJ(\()p FH(p)2330 1801 y FF(n)2377 1786 y FJ(\)\))22
b(+)g FH(H)2662 1744 y FB(\003)2701 1786 y FJ(\()p FH(A)2812
1744 y FF(h)2857 1786 y FJ(\()p FH(p)2944 1801 y FF(n)2991
1786 y FJ(\))p FH(;)17 b(A)p FJ(\()p FH(p)3233 1801 y
FF(n)3279 1786 y FJ(\)\))p FH(;)1725 1992 y FG(\024)1840
1924 y FJ(1)p 1840 1969 V 1840 2060 a(2)1899 1992 y FH(b)1940
1951 y FB(\000)p FE(1)2035 1992 y FJ(\()p FH(a)p FJ(\()p
FH(\016)2205 2007 y FE(0)2244 1992 y FJ(\)\))23 b(+)f
FH(a)2492 1951 y FB(\000)p FE(1)2586 1992 y FJ(\()p FH(\015)5
b FJ(\()p FH(h)p FJ(\)\))p FH(;)1725 2225 y FG(\024)1840
2158 y FJ(1)p 1840 2202 V 1840 2293 a(2)1899 2225 y FH(b)1940
2184 y FB(\000)p FE(1)2035 2225 y FJ(\()p FH(a)p FJ(\()p
FH(\016)2205 2240 y FE(0)2244 2225 y FJ(\)\))23 b(+)f
FH(a)2492 2184 y FB(\000)p FE(1)2586 2225 y FJ(\()2634
2158 y(1)p 2634 2202 V 2634 2293 a(2)2693 2225 y FH(a)p
FJ(\()p FH(\016)2825 2240 y FE(1)2864 2225 y FJ(\)\))p
FH(;)1725 2458 y FG(\024)1840 2391 y FJ(1)p 1840 2435
V 1840 2527 a(2)1899 2458 y FH(b)1940 2417 y FB(\000)p
FE(1)2035 2458 y FJ(\()p FH(a)p FJ(\()p FH(\016)2205
2473 y FE(0)2244 2458 y FJ(\)\))h(+)f FH(\016)2484 2473
y FE(1)2523 2458 y FH(;)1725 2657 y(<)28 b(b)1870 2615
y FB(\000)p FE(1)1964 2657 y FJ(\()p FH(a)p FJ(\()p FH(\016)2134
2672 y FE(0)2174 2657 y FJ(\)\))p FH(;)874 3055 y FJ(since)34
b FH(\016)1156 3070 y FE(1)1223 3055 y FJ(=)1337 3015
y FE(1)p 1337 3032 36 4 v 1337 3089 a(2)1382 3055 y FH(b)1423
3018 y FB(\000)p FE(1)1518 3055 y FJ(\()p FH(a)p FJ(\()p
FH(\016)1688 3070 y FE(0)1728 3055 y FJ(\)\))e(from)g(\(6.3\).)43
b(As)33 b(a)g(result)1478 3289 y FH(V)22 b FJ(\()p FH(p)1644
3304 y FF(n)1691 3289 y FH(;)17 b(x)p FJ(\))27 b FG(\024)i
FH(b)p FJ(\(dist)q(\()p FH(x;)17 b(A)p FJ(\()p FH(p)2495
3304 y FF(n)2542 3289 y FJ(\)\)\))27 b FH(<)h(a)p FJ(\()p
FH(\016)2919 3304 y FE(0)2959 3289 y FJ(\))p FH(:)874
3524 y FJ(Th)m(us)d FH(x)j FG(2)g FH(B)5 b FJ(\()p FH(p)1455
3539 y FF(n)1502 3524 y FJ(\).)40 b(Since)1878 3498 y(^)1852
3524 y FH(A)1925 3487 y FF(h)1993 3524 y FJ(uniformly)24
b(attracts)2806 3498 y(^)2782 3524 y FH(B)29 b FJ(and)23
b(since)h FG(N)3376 3552 y FF(\016)o(;)3444 3535 y FE(^)3426
3552 y FF(A)3479 3533 y FA(h)3550 3524 y FG(\022)897
3637 y FJ(^)874 3662 y FH(B)5 b FJ(,)1038 3637 y(^)1013
3662 y FH(A)1086 3626 y FF(h)1163 3662 y FJ(uniformly)34
b(attracts)e FG(N)2053 3691 y FF(\016)o(;)2121 3674 y
FE(^)2103 3691 y FF(A)2156 3672 y FA(h)2200 3662 y FJ(.)p
3841 3975 4 66 v 3845 3913 59 4 v 3845 3975 V 3903 3975
4 66 v Black 591 4194 a FI(Lemma)39 b(6.1.7)e FJ(\(A6)c(-)f(Stabilit)m
(y\))p FI(.)p Black 2012 4169 a FJ(^)1986 4194 y FH(A)2059
4158 y FF(h)2139 4194 y Fy(is)j(a)g(discr)-5 b(ete)34
b(uniformly)h(stable)f(family.)p Black 630 4507 a(Pr)-5
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%%Page: 161 179
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%%Page: 162 180
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%%Page: 163 181
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