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/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
] def

%%EndProcSet
%%BeginProcSet: 8r.enc
% File  8r.enc as of 2002-03-12 for PSNFSS 9
%
% This is the encoding vector for Type1 and TrueType fonts to be used
% with TeX.  This file is part of the PSNFSS bundle, version 9
% 
% Authors: S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, W. Schmidt
%
% Idea is to have all the characters normally included in Type 1 fonts
% available for typesetting. This is effectively the characters in Adobe
% Standard Encoding + ISO Latin 1 + extra characters from Lucida + Euro.
% 
% Character code assignments were made as follows:
% 
% (1) the Windows ANSI characters are almost all in their Windows ANSI
% positions, because some Windows users cannot easily reencode the
% fonts, and it makes no difference on other systems. The only Windows
% ANSI characters not available are those that make no sense for
% typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
% (173). quotesingle and grave are moved just because it's such an
% irritation not having them in TeX positions.
% 
% (2) Remaining characters are assigned arbitrarily to the lower part
% of the range, avoiding 0, 10 and 13 in case we meet dumb software.
% 
% (3) Y&Y Lucida Bright includes some extra text characters; in the
% hopes that other PostScript fonts, perhaps created for public
% consumption, will include them, they are included starting at 0x12.
% 
% (4) Remaining positions left undefined are for use in (hopefully)
% upward-compatible revisions, if someday more characters are generally
% available.
% 
% (5) hyphen appears twice for compatibility with both ASCII and Windows.
%
% (6) /Euro is assigned to 128, as in Windows ANSI
% 
/TeXBase1Encoding [
% 0x00 (encoded characters from Adobe Standard not in Windows 3.1)
  /.notdef /dotaccent /fi /fl
  /fraction /hungarumlaut /Lslash /lslash
  /ogonek /ring /.notdef
  /breve /minus /.notdef 
% These are the only two remaining unencoded characters, so may as
% well include them.
  /Zcaron /zcaron 
% 0x10
 /caron /dotlessi 
% (unusual TeX characters available in, e.g., Lucida Bright)
 /dotlessj /ff /ffi /ffl 
 /.notdef /.notdef /.notdef /.notdef
 /.notdef /.notdef /.notdef /.notdef
 % very contentious; it's so painful not having quoteleft and quoteright
 % at 96 and 145 that we move the things normally found there down to here.
 /grave /quotesingle 
% 0x20 (ASCII begins)
 /space /exclam /quotedbl /numbersign
 /dollar /percent /ampersand /quoteright
 /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash
% 0x30
 /zero /one /two /three /four /five /six /seven
 /eight /nine /colon /semicolon /less /equal /greater /question
% 0x40
 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O
% 0x50
 /P /Q /R /S /T /U /V /W
 /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore
% 0x60
 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o
% 0x70
 /p /q /r /s /t /u /v /w
 /x /y /z /braceleft /bar /braceright /asciitilde
 /.notdef % rubout; ASCII ends
% 0x80
 /Euro /.notdef /quotesinglbase /florin
 /quotedblbase /ellipsis /dagger /daggerdbl
 /circumflex /perthousand /Scaron /guilsinglleft
 /OE /.notdef /.notdef /.notdef
% 0x90
 /.notdef /.notdef /.notdef /quotedblleft
 /quotedblright /bullet /endash /emdash
 /tilde /trademark /scaron /guilsinglright
 /oe /.notdef /.notdef /Ydieresis
% 0xA0
 /.notdef % nobreakspace
 /exclamdown /cent /sterling
 /currency /yen /brokenbar /section
 /dieresis /copyright /ordfeminine /guillemotleft
 /logicalnot
 /hyphen % Y&Y (also at 45); Windows' softhyphen
 /registered
 /macron
% 0xD0
 /degree /plusminus /twosuperior /threesuperior
 /acute /mu /paragraph /periodcentered
 /cedilla /onesuperior /ordmasculine /guillemotright
 /onequarter /onehalf /threequarters /questiondown
% 0xC0
 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla
 /Egrave /Eacute /Ecircumflex /Edieresis
 /Igrave /Iacute /Icircumflex /Idieresis
% 0xD0
 /Eth /Ntilde /Ograve /Oacute
 /Ocircumflex /Otilde /Odieresis /multiply
 /Oslash /Ugrave /Uacute /Ucircumflex
 /Udieresis /Yacute /Thorn /germandbls
% 0xE0
 /agrave /aacute /acircumflex /atilde
 /adieresis /aring /ae /ccedilla
 /egrave /eacute /ecircumflex /edieresis
 /igrave /iacute /icircumflex /idieresis
% 0xF0
 /eth /ntilde /ograve /oacute
 /ocircumflex /otilde /odieresis /divide
 /oslash /ugrave /uacute /ucircumflex
 /udieresis /yacute /thorn /ydieresis
] def

%%EndProcSet
%%BeginProcSet: 09fbbfac.enc
% Thomas Esser, Dec 2002. public domain
%
% Encoding for:
%     cmsltt10 cmtt10 cmtt12 cmtt8 cmtt9
/TeX09fbbfacEncoding [
/Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi
/Omega /arrowup /arrowdown /quotesingle /exclamdown /questiondown
/dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla
/germandbls /ae /oe /oslash /AE /OE /Oslash /visiblespace /exclam
/quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft
/parenright /asterisk /plus /comma /hyphen /period /slash /zero /one
/two /three /four /five /six /seven /eight /nine /colon /semicolon /less
/equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N
/O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright
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/m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright
/asciitilde /dieresis /visiblespace /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
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%%EndProcSet
%%BeginProcSet: texps.pro
%!
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%%EndProcSet
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%!
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%%EndProcSet
%%BeginFont: CMTT8
%!PS-AdobeFont-1.1: CMTT8 1.0
%%CreationDate: 1991 Aug 20 16:46:05
% 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 (CMTT8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
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/FontName /CMTT8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
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/UniqueID 5000830 def
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cleartomark
%%EndFont 
%%BeginFont: MSBM7
%!PS-AdobeFont-1.1: MSBM7 2.1
%%CreationDate: 1992 Oct 17 08:30:50
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM7) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 67 /C put
readonly def
/FontBBox{0 -504 2615 1004}readonly def
/UniqueID 5032014 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 0 /.notdef put
readonly def
/FontBBox{-59 -250 1235 750}readonly def
/UniqueID 5000766 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY5
%!PS-AdobeFont-1.1: CMSY5 1.0
%%CreationDate: 1991 Aug 15 07:21:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{21 -944 1448 791}readonly def
/UniqueID 5000815 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: LASY10
%!PS-AdobeFont-1.1: LASY10 1.001
%%CreationDate: 1992 Oct 23 20:19:17
%%RevisionDate: 2001 Jun 05 20:19:17
% Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.001) readonly def
/Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def
/FullName (LASY10) readonly def
/FamilyName (LaTeX) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /LASY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 50 /a50 put
readonly def
/FontBBox{-19 -192 944 683}readonly def
/UniqueID 5011949 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: 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 161 /parenleftbig put
dup 162 /parenrightbig put
dup 175 /vextendsingle put
dup 176 /vextenddouble put
dup 179 /parenleftBig put
dup 180 /parenrightBig put
dup 181 /parenleftbigg put
dup 182 /parenrightbigg put
dup 195 /parenleftBigg put
dup 33 /parenrightBigg put
dup 40 /braceleftBigg put
dup 48 /parenlefttp put
dup 49 /parenrighttp put
dup 64 /parenleftbt put
dup 65 /parenrightbt put
dup 80 /summationtext put
dup 81 /producttext put
dup 88 /summationdisplay put
dup 89 /productdisplay put
dup 90 /integraldisplay put
dup 101 /tildewide put
dup 104 /bracketleftBig put
dup 105 /bracketrightBig put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueID 5000774 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR5
%!PS-AdobeFont-1.1: CMR5 1.00B
%%CreationDate: 1992 Feb 19 19:55:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-341 -250 1304 965}readonly def
/UniqueID 5000788 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI5
%!PS-AdobeFont-1.1: CMMI5 1.100
%%CreationDate: 1996 Aug 02 08:21:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{37 -250 1349 750}readonly def
/UniqueID 5087380 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI10
%!PS-AdobeFont-1.1: CMTI10 1.00B
%%CreationDate: 1992 Feb 19 19:56:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueID 5000828 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 /.notdef put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueID 5000817 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI8
%!PS-AdobeFont-1.1: CMTI8 1.0
%%CreationDate: 1991 Aug 18 21:07:42
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-35 -250 1190 750}readonly def
/UniqueID 5000826 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 0 /.notdef put
readonly def
/FontBBox{-27 -250 1122 750}readonly def
/UniqueID 5000790 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 67 /C put
dup 82 /R put
dup 90 /Z put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
/UniqueID 5031982 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 0 /.notdef put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueID 5087382 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueID 5087385 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMCSC10
%!PS-AdobeFont-1.1: CMCSC10 1.0
%%CreationDate: 1991 Aug 18 17:46:49
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMCSC10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMCSC10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{14 -250 1077 750}readonly def
/UniqueID 5000772 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueID 5000793 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 0 /.notdef put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueID 5087381 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 0 /.notdef put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueID 5087383 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueID 5000791 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
/UniqueID 5000768 def
currentdict end
currentfile eexec
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7330 y(obtains)416 b(a)f(hierarc)-31 b(h)g(y)415 b(of)h(equations)g
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b(ely)-92 b(.)630 b(The)415 b(pro)31 b(of)415 b(is)f(v)-31
b(ery)6075 8658 y(explicit)371 b(and)e(is)g(the)h(basis)f(of)h(a)f
(practical)i(algorithm)h(whic)-31 b(h)370 b(w)-31 b(e)370
b(implemen)-31 b(t)371 b(in)f(Section)g(5)8289 10616
y(The)494 b(high)g(order)g(part)g(of)g(the)g(parameterization)j
Fp(K)31935 10215 y Fo(>)33176 10616 y Fr(will)f(b)31
b(e)493 b(found)i(as)e(the)i(\257xed)6075 11945 y(p)31
b(oin)-31 b(t)394 b(of)f(the)g(con)-31 b(traction)395
b(op)31 b(erator)393 b Fk(N)510 b Fr(=)346 b Fk(S)25907
11543 y Fm(0)26666 11945 y Fk(\261)262 b(F)502 b Fr(\(see)392
b(\(4.53\))j(b)31 b(elo)-31 b(w\).)565 b(It)393 b(is)f(w)-31
b(ell)394 b(kno)-31 b(wn)6075 13273 y(that)306 b(if)g
Fp(K)10379 12871 y Fo(>)10300 13547 y Fj(\244)11431 13273
y Fr(v)-31 b(eri\257es)306 b Fk(k)p Fp(K)16639 12871
y Fo(>)16560 13547 y Fj(\244)17505 13273 y Fk(\241)118
b(N)163 b Fr(\()p Fp(K)21004 12871 y Fo(>)20925 13547
y Fj(\244)21752 13273 y Fr(\))p Fk(k)308 b(\267)g Fp(")d
Fr(then)h(the)f(distance)h(of)g Fp(K)35796 12871 y Fo(>)35717
13547 y Fj(\244)36848 13273 y Fr(to)g(the)g(true)f(solution)6075
14634 y Fp(K)7094 14233 y Fo(>)8157 14634 y Fr(is)316
b Fk(k)p Fp(K)10789 14233 y Fo(>)11677 14634 y Fk(\241)140
b Fp(K)13697 14233 y Fo(>)13618 14908 y Fj(\244)14444
14634 y Fk(k)308 b(\267)16964 14199 y Fo(`")p 16607 14380
1504 45 v 16607 15016 a Fm(1)p Fj(\241)p Fo(`)18243 14634
y Fr(,)327 b(where)316 b Fp(`)308 b Fr(=)f(Lip)185 b(\()p
Fk(N)163 b Fr(\).)476 b(This)316 b(is)h(\\a)f(p)31 b(osteriori")318
b(estimate)g(whic)-31 b(h)6075 16152 y(pro)g(vides)369
b(an)h(explicit)h(b)31 b(ound)369 b(of)h(the)f(error)f(in)i(the)f
(computed)h(appro)-31 b(ximation,)374 b Fp(K)42200 15750
y Fj(\267)42947 16152 y Fr(.)17777 19382 y(3.)555 b Fq(Pol)-74
b(ynomial)417 b(Appr)-25 b(o)g(xima)-74 b(tion)8289 22004
y Fr(W)-92 b(e)499 b(shall)j(denote)f(b)-31 b(y)500 b
Fp(E)19230 22170 y Fm(1)20227 22004 y Fr(the)h(one-dimensional)i
(subspace)d(generated)h(b)-31 b(y)500 b(the)h(\257rst)6075
23333 y(v)-61 b(ariable)390 b(and)g(b)-31 b(y)389 b Fp(E)14796
23499 y Fm(2)15682 23333 y Fr(the)h Fp(n)p Fr(-dimensional)i(space)d
(generated)h(b)-31 b(y)389 b(the)h(last)g Fp(n)g Fr(v)-61
b(ariables.)553 b(W)-92 b(e)6075 24661 y(shall)362 b(denote)f
Fp(\274)12845 24827 y Fo(i)13214 24661 y Fr(,)i Fp(i)308
b Fr(=)f(1)p Fp(;)184 b Fr(2,)365 b(the)c(corresp)31
b(onding)362 b(pro)61 b(jectors)361 b(on)-31 b(to)363
b Fp(E)35516 24827 y Fm(1)36374 24661 y Fr(and)e Fp(E)39335
24827 y Fm(2)40192 24661 y Fr(resp)31 b(ectiv)-31 b(ely)-92
b(,)6075 25989 y(and)385 b Fp(\274)8874 26155 y Fm(2)p
Fo(;l)10253 25989 y Fr(=)393 b(~)-611 b Fp(\274)12080
26155 y Fo(l)12419 25989 y Fp(\274)13050 26155 y Fm(2)13882
25989 y Fr(:)335 b Fn(R)15323 25588 y Fm(1+)p Fo(n)17381
25989 y Fk(!)g Fn(R)p Fr(,)390 b(1)336 b Fk(\267)e Fp(l)357
b Fk(\267)335 b Fp(n)p Fr(,)390 b(where)444 b(~)-612
b Fp(\274)29495 26155 y Fo(l)30220 25989 y Fr(is)386
b(the)f(pro)61 b(jector)387 b(from)f Fp(E)41511 26155
y Fm(2)42393 25989 y Fr(on)-31 b(to)387 b(its)6075 27318
y Fp(l)k Fr(co)31 b(ordinate.)494 b(Also)370 b Fp(E)15986
27484 y Fm(2)p Fo(;l)17337 27318 y Fr(=)308 b Fp(\274)19137
27484 y Fm(2)p Fo(;l)20181 27318 y Fp(E)20998 27484 y
Fm(2)21494 27318 y Fr(.)8289 29394 y(W)-92 b(e)368 b(write)i
Fp(F)461 b Fr(=)308 b(\()p Fp(F)16665 28992 y Fm(1)17161
29394 y Fp(;)184 b(F)18518 28992 y Fm(2)19015 29394 y
Fr(\))308 b(=)g(Id)430 b(+)23474 28564 y Ff(P)24642 28838
y Fo(r)24642 29726 y(j)45 b Fm(=)p Fo(L)26580 29394 y
Fp(F)27292 29560 y Fo(j)28004 29394 y Fr(+)29360 29114
y(~)29111 29394 y Fp(F)29823 29560 y Fo(r)24 b Fm(+1)31803
29394 y Fr(in)370 b(the)f(form)15127 32400 y Fp(F)15993
31943 y Fm(1)16489 32400 y Fr(\()p Fp(x;)184 b(y)40 b
Fr(\))1108 b(=)f Fp(x)245 b Fr(+)24837 31016 y Fo(r)24255
31349 y Ff(X)24117 33723 y Fo(j)45 b Fm(=)p Fo(N)26177
32400 y Fp(F)27043 31943 y Fm(1)26889 32674 y Fo(j)27539
32400 y Fr(\()p Fp(x;)184 b(y)40 b Fr(\))247 b(+)f Fp(o)p
Fr(\()p Fk(j)p Fr(\()p Fp(x;)184 b(y)40 b Fr(\))p Fk(j)35608
31943 y Fo(r)36100 32400 y Fr(\))p Fp(;)6962 b Fr(\(3.1\))15127
36255 y Fp(F)15993 35798 y Fm(2)16489 36255 y Fr(\()p
Fp(x;)184 b(y)40 b Fr(\))1108 b(=)f Fp(y)285 b Fr(+)24859
34871 y Fo(r)24277 35203 y Ff(X)24067 37578 y Fo(j)45
b Fm(=)p Fo(M)26270 36255 y Fp(F)27136 35798 y Fm(2)26982
36529 y Fo(j)27632 36255 y Fr(\()p Fp(x;)184 b(y)40 b
Fr(\))247 b(+)f Fp(o)p Fr(\()p Fk(j)p Fr(\()p Fp(x;)184
b(y)40 b Fr(\))p Fk(j)35701 35798 y Fo(r)36193 36255
y Fr(\))p Fp(;)6869 b Fr(\(3.2\))6075 39436 y(where)333
b Fp(F)10106 39034 y Fm(1)9952 39724 y Fo(j)10602 39436
y Fr(,)341 b Fp(F)12116 39034 y Fm(2)11962 39724 y Fo(j)12946
39436 y Fr(are)333 b(homogeneous)j(p)31 b(olynomials)336
b(of)e(degree)f Fp(j)397 b Fr(taking)335 b(v)-61 b(alues)334
b(in)g Fp(E)41814 39602 y Fm(1)42644 39436 y Fr(and)g
Fp(E)45578 39602 y Fm(2)6075 41004 y Fr(resp)31 b(ectiv)-31
b(ely)370 b(and)14511 40724 y(~)14263 41004 y Fp(F)14975
41170 y Fo(r)24 b Fm(+1)16586 41004 y Fr(\()p Fp(x;)184
b(y)40 b Fr(\))309 b(=)e Fp(o)p Fr(\()p Fk(j)p Fr(\()p
Fp(x;)184 b(y)40 b Fr(\))p Fk(j)24778 40602 y Fo(r)25270
41004 y Fr(\).)8289 43092 y(Moreo)-31 b(v)g(er)430 b(w)-31
b(e)430 b(will)i(write)f Fp(F)20815 42690 y Fo(i)20661
43380 y(j)21592 43092 y Fr(=)409 b Fp(\274)23493 43258
y Fo(i)23861 43092 y Fp(F)24573 43258 y Fo(j)25469 43092
y Fr(and)431 b Fp(F)28549 42560 y Fm(2)p Fo(;l)28395
43400 y(j)30001 43092 y Fr(=)409 b Fp(\274)31902 43258
y Fm(2)p Fo(;l)32946 43092 y Fp(F)33658 43258 y Fo(j)34124
43092 y Fr(,)445 b(etc.)676 b(W)-92 b(e)429 b(also)i(in)-31
b(tro)31 b(duce)6075 44430 y(the)369 b(notation)10613
46343 y Fp(F)11479 45886 y Fm(1)11325 46617 y Fo(k)11975
46343 y Fr(\()p Fp(x;)184 b(y)40 b Fr(\))309 b(=)17005
45292 y Ff(X)16019 47718 y Fo(j)45 b Fm(+)p Fj(j)p Fo(m)p
Fj(j)p Fm(=)p Fo(k)19775 46343 y Fp(a)20360 46509 y Fo(j)m(;m)21827
46343 y Fp(x)22460 45886 y Fo(j)22926 46343 y Fp(y)23509
45886 y Fo(m)24349 46343 y Fp(;)2399 b(F)27921 45811
y Fm(2)p Fo(;l)27767 46677 y(k)28965 46343 y Fr(\()p
Fp(x;)184 b(y)40 b Fr(\))308 b(=)33995 45292 y Ff(X)33009
47718 y Fo(j)45 b Fm(+)p Fj(j)p Fo(m)p Fj(j)p Fm(=)p
Fo(k)36764 46343 y Fp(b)37239 45886 y Fo(l)37239 46617
y(j)m(;m)38706 46343 y Fp(x)39339 45886 y Fo(j)39805
46343 y Fp(y)40388 45886 y Fo(m)41229 46343 y Fp(;)6075
49648 y Fr(where)246 b Fp(y)9736 49247 y Fo(m)10883 49648
y Fr(=)308 b Fp(y)12635 49156 y Fo(m)13421 49267 y Fg(1)12595
49944 y Fm(1)14092 49648 y Fk(\242)184 b(\242)g(\242)j
Fp(y)16151 49247 y Fo(m)16937 49358 y Fh(n)16111 49922
y Fo(n)17534 49648 y Fr(,)271 b Fp(a)18697 49814 y Fo(N)35
b(;)p Fm(0)20501 49648 y Fr(=)307 b Fp(a)22254 49814
y Fo(N)35 b(;)p Fm(0)p Fo(;:::)q(;)p Fm(0)25506 49648
y Fr(,)271 b(the)247 b(v)-31 b(ector)247 b Fp(b)31544
49814 y Fo(M)35 b(;)p Fm(0)33490 49648 y Fr(=)308 b(\()p
Fp(b)35564 49247 y Fm(1)35564 49953 y Fo(M)35 b(;)p Fm(0)p
Fo(;:::;)p Fm(0)38959 49648 y Fp(;)184 b(:)g(:)g(:)k(;)c(b)41893
49247 y Fo(n)41893 49953 y(M)35 b(;)p Fm(0)p Fo(;:::)q(;)p
Fm(0)45289 49648 y Fr(\))45719 49247 y Fm(T)6075 50983
y Fr(and)369 b(the)h(matrix)14139 54005 y Fp(B)14979
54171 y Fo(M)83 b Fj(\241)p Fm(1)p Fo(;)p Fm(1)18106
54005 y Fr(=)19275 51780 y Ff(0)19275 53773 y(@)20797
52609 y Fp(b)21272 52207 y Fm(1)21272 52913 y Fo(M)g
Fj(\241)p Fm(1)p Fo(;)p Fm(1)p Fo(;)p Fm(0)p Fo(;:::;)p
Fm(0)27658 52609 y Fp(:)184 b(:)g(:)1294 b(b)30716 52207
y Fm(1)30716 52913 y Fo(M)83 b Fj(\241)p Fm(1)p Fo(;)p
Fm(0)p Fo(;)p Fm(0)p Fo(;:::)q(;)p Fm(1)22936 53994 y
Fp(:)184 b(:)g(:)20797 55322 y(b)21272 54921 y Fo(n)21272
55627 y(M)83 b Fj(\241)p Fm(1)p Fo(;)p Fm(1)p Fo(;)p
Fm(0)p Fo(;:::;)p Fm(0)27658 55322 y Fp(:)184 b(:)g(:)1294
b(b)30716 54921 y Fo(n)30716 55627 y(M)83 b Fj(\241)p
Fm(1)p Fo(;)p Fm(0)p Fo(;)p Fm(0)p Fo(;:::)q(;)p Fm(1)36550
51780 y Ff(1)36550 53773 y(A)37703 54005 y Fp(:)6075
57303 y Fr(Condition)388 b(\(2.4\))g(on)f Fp(F)539 b
Fr(implies)387 b(that)g Fp(a)23498 57469 y Fo(N)35 b(;)p
Fm(0)25330 57303 y Fp(<)335 b Fr(0,)391 b Fp(b)28252
57469 y Fo(M)35 b(;)p Fm(0)30226 57303 y Fr(=)336 b(0,)390
b(and)d(Condition)h(\(2.5\))g(implies)6075 58632 y(that)370
b(the)g(eigen)-31 b(v)-61 b(alues)370 b(of)g Fp(B)18206
58798 y Fo(M)83 b Fj(\241)p Fm(1)p Fo(;)p Fm(1)21394
58632 y Fr(ha)-31 b(v)g(e)370 b(p)31 b(ositiv)-31 b(e)371
b(real)f(part.)6075 60693 y Fw(Lemma)580 b(3.1.)635 b
Fi(L)-57 b(et)521 b Fp(F)687 b Fr(=)534 b(\()p Fp(F)19485
60291 y Fm(1)19982 60693 y Fp(;)184 b(F)21339 60291 y
Fm(2)21836 60693 y Fr(\))534 b(:)g Fp(U)654 b Fk(\275)534
b Fn(R)27245 60291 y Fm(1+)p Fo(n)29502 60693 y Fk(!)f
Fn(R)31941 60291 y Fm(1+)p Fo(n)34186 60693 y Fi(b)-57
b(e)522 b(a)e Fp(C)37625 60291 y Fo(r)38637 60693 y Fi(map,)552
b(such)521 b(that)6075 62021 y Fp(F)154 b Fr(\(0)p Fp(;)184
b Fr(0\))518 b(=)d(0)p Fi(,)540 b Fp(D)31 b(F)154 b Fr(\(0)p
Fp(;)184 b Fr(0\))518 b(=)e(Id)695 b Fi(and)510 b(satis\257es)g
(\(2.2\){\(2.5\).)853 b(Given)512 b Fr(2)k Fk(\267)f
Fp(m)g Fk(\267)h Fp(r)31 b Fi(,)540 b(ther)-57 b(e)6075
63494 y(exist)514 b(p)-57 b(olynomials)515 b Fp(K)601
b Fr(=)522 b Fp(K)18780 63660 y Fm(1)19609 63494 y Fr(+)20801
62664 y Ff(P)21970 62938 y Fo(m)p Fj(\241)p Fo(L)p Fm(+1)21970
63827 y Fo(j)45 b Fm(=2)25415 63494 y Fp(K)26355 63660
y Fo(j)27336 63494 y Fi(and)515 b Fp(R)530 b Fr(=)522
b Fp(R)33200 63660 y Fm(1)34028 63494 y Fr(+)35220 62664
y Ff(P)36389 62938 y Fo(m)36389 63827 y(j)45 b Fm(=)p
Fo(N)38504 63494 y Fp(R)39345 63660 y Fo(j)39810 63494
y Fi(,)545 b(wher)-57 b(e)514 b Fp(L)523 b Fr(=)p eop
end
%%Page: 6 6
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b(.)297 b(MAR)-71 b(T)38297 3149 y(\266)38358 3345 y(IN)6075
6001 y Fr(min)q(\()p Fp(N)59 b(;)184 b(M)121 b Fr(\))379
b Fi(and)f Fp(K)14865 6167 y Fo(j)15709 6001 y Fi(and)f
Fp(R)18681 6167 y Fo(j)19524 6001 y Fi(ar)-57 b(e)378
b(homo)-57 b(gene)g(ous)378 b(p)-57 b(olynomials)378
b(of)g(de)-57 b(gr)g(e)g(e)379 b Fp(j)63 b Fi(,)382 b(with)377
b Fp(K)43149 6167 y Fm(1)43646 6001 y Fr(\()p Fp(t)p
Fr(\))308 b(=)6075 7330 y(\()p Fp(t;)184 b Fr(0\))p Fi(,)398
b Fp(R)9957 7496 y Fm(1)10453 7330 y Fr(\()p Fp(t)p Fr(\))309
b(=)e Fp(t)p Fi(,)396 b(such)g(that)19005 9244 y Fp(F)j
Fk(\261)247 b Fp(K)79 b Fr(\()p Fp(t)p Fr(\))247 b Fk(\241)f
Fp(K)325 b Fk(\261)246 b Fp(R)9 b Fr(\()p Fp(t)p Fr(\))308
b(=)f Fp(o)p Fr(\()p Fp(t)31566 8787 y Fo(m)32407 9244
y Fr(\))p Fp(:)6075 11159 y Fi(Mor)-57 b(e)g(over,)396
b(we)g(c)-57 b(an)397 b(cho)-57 b(ose)396 b Fp(R)404
b Fi(and)396 b Fp(K)475 b Fi(of)396 b(the)g(form)13662
14008 y Fp(R)316 b Fr(=)15987 12115 y Ff(\()16879 13253
y Fp(R)17720 13419 y Fm(1)18462 13253 y Fr(+)246 b Fp(R)20410
13419 y Fo(N)26988 13253 y Fr(if)1218 b Fp(N)429 b Fk(\267)307
b Fp(m)h(<)f Fr(2)p Fp(N)368 b Fk(\241)246 b Fr(1)16879
14847 y Fp(R)17720 15013 y Fm(1)18462 14847 y Fr(+)g
Fp(R)20410 15013 y Fo(N)21495 14847 y Fr(+)g Fp(R)23443
15013 y Fm(2)p Fo(N)83 b Fj(\241)p Fm(1)26988 14847 y
Fr(if)1218 b Fp(m)308 b Fk(\270)f Fr(2)p Fp(N)368 b Fk(\241)246
b Fr(1)13492 17772 y Fp(K)387 b Fr(=)15987 15879 y Ff(\()16879
16182 y(P)18047 16456 y Fo(m)p Fj(\241)p Fo(N)83 b Fm(+1)18047
17345 y Fo(j)45 b Fm(=1)21670 17013 y Fp(K)22610 17179
y Fo(j)32735 17013 y Fr(if)1218 b Fp(N)429 b Fk(\267)307
b Fp(M)16879 18611 y Fr(\()17309 17781 y Ff(P)18478 18055
y Fo(m)p Fj(\241)p Fo(N)83 b Fm(+1)18478 18943 y Fo(j)45
b Fm(=1)22100 18611 y Fp(K)23119 18209 y Fm(1)23040 18899
y Fo(j)23616 18611 y Fp(;)24723 17781 y Ff(P)25891 18055
y Fo(m)p Fj(\241)p Fo(M)83 b Fm(+1)25891 18943 y Fo(j)45
b Fm(=1)29657 18611 y Fp(K)30676 18209 y Fm(2)30597 18899
y Fo(j)31173 18611 y Fr(\))1132 b(if)1218 b Fp(M)428
b(<)308 b(N)59 b(:)6075 20830 y Fw(Remark)496 b(3.2.)591
b Fi(If)454 b Fp(N)533 b(<)412 b(M)574 b Fi(we)453 b(do)g(not)h(ne)-57
b(e)g(d)455 b(to)e(use)g(Condition)i(\(2.5\))e(to)g(determine)h
Fp(K)6075 22159 y Fi(and)396 b Fp(R)9 b Fi(.)6075 24954
y(Pr)-57 b(o)g(of.)554 b Fr(Since)325 b(w)-31 b(e)325
b(w)-31 b(an)g(t)327 b(that)f Fp(K)79 b Fr(\(0\))309
b(=)f(0)325 b(and)g Fp(R)9 b Fr(\(0\))308 b(=)f(0)325
b(w)-31 b(e)325 b(lo)31 b(ok)327 b(for)d Fp(K)404 b Fr(and)325
b Fp(R)333 b Fr(in)325 b(the)g(form)16496 27756 y Fp(K)79
b Fr(\()p Fp(t)p Fr(\))309 b(=)20659 26372 y Fo(m)20252
26705 y Ff(X)20286 29062 y Fo(j)45 b Fm(=1)22036 27756
y Fp(K)22976 27922 y Fo(j)23442 27756 y Fr(\()p Fp(t)p
Fr(\))p Fp(;)2400 b(R)9 b Fr(\()p Fp(t)p Fr(\))307 b(=)31401
26372 y Fo(m)30995 26705 y Ff(X)31028 29062 y Fo(j)45
b Fm(=1)32778 27756 y Fp(R)33619 27922 y Fo(j)34085 27756
y Fr(\()p Fp(t)p Fr(\))p Fp(:)8147 b Fr(\(3.3\))6075
30789 y(W)-92 b(e)368 b(in)-31 b(tro)31 b(duce)370 b(the)f(notation)
11347 33314 y Fp(K)12366 32857 y Fm(1)12863 33314 y Fr(\()p
Fp(t)p Fr(\))308 b(=)16006 31931 y Fo(m)15599 32263 y
Ff(X)15682 34620 y Fo(i)p Fm(=1)17383 33314 y Fp(c)17862
32857 y Fm(1)17862 33588 y Fo(i)18358 33314 y Fp(t)18758
32857 y Fo(i)19127 33314 y Fp(;)2399 b(K)22852 32857
y Fm(2)p Fo(;l)23896 33314 y Fr(\()p Fp(t)p Fr(\))308
b(=)27039 31931 y Fo(m)26632 32263 y Ff(X)26715 34620
y Fo(i)p Fm(=2)28416 33314 y Fp(c)28895 32782 y Fm(2)p
Fo(;l)28895 33623 y(i)29939 33314 y Fp(t)30339 32857
y Fo(i)30707 33314 y Fp(;)2399 b(R)9 b Fr(\()p Fp(t)p
Fr(\))307 b(=)37405 31931 y Fo(m)36999 32263 y Ff(X)37081
34620 y Fo(i)p Fm(=1)38782 33314 y Fp(d)39358 33480 y
Fo(i)39727 33314 y Fp(t)40127 32857 y Fo(i)40495 33314
y Fp(:)6075 36258 y Fr(W)-92 b(e)328 b(substitute)i(\(3.3\))h(in)-31
b(to)331 b Fp(F)319 b Fk(\261)166 b Fp(K)245 b Fk(\241)166
b Fp(K)245 b Fk(\261)166 b Fp(R)315 b Fr(=)308 b(0)329
b(to)h(determine)g Fp(K)34343 36424 y Fo(j)35138 36258
y Fr(and)f Fp(R)38091 36424 y Fo(j)38886 36258 y Fr(order)f(b)-31
b(y)330 b(order.)6075 37587 y(A)-31 b(t)370 b(\257rst)e(order)h(w)-31
b(e)370 b(obtain)20980 39036 y Fp(K)21920 39202 y Fm(1)22663
39036 y Fk(\241)246 b Fp(K)24710 39202 y Fm(1)25452 39036
y Fk(\261)h Fp(R)27093 39202 y Fm(1)27896 39036 y Fr(=)308
b Fp(o)p Fr(\()p Fp(t)p Fr(\))p Fp(:)6075 40718 y Fr(Here)368
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b(p)31 b(ossible)369 b(c)-31 b(hoices,)371 b(ho)-31 b(w)g(ev)g(er)370
b(w)-31 b(e)370 b(tak)-31 b(e)14900 42633 y Fp(K)15840
42799 y Fm(1)16337 42633 y Fr(\()p Fp(t)p Fr(\))308 b(=)g(\()p
Fp(K)20523 42176 y Fm(1)20444 42907 y(1)21020 42633 y
Fr(\()p Fp(t)p Fr(\))p Fp(;)184 b(K)23790 42176 y Fm(2)23711
42907 y(1)24289 42633 y Fr(\()p Fp(t)p Fr(\)\))309 b(=)e(\()p
Fp(t;)184 b Fr(0\))p Fp(;)2401 b(R)33309 42799 y Fm(1)33805
42633 y Fr(\()p Fp(t)p Fr(\))308 b(=)g Fp(t:)6075 44548
y Fw(Case)474 b Fp(N)501 b Fk(\267)379 b Fp(M)121 b Fw(.)620
b Fr(If)412 b(2)380 b Fk(\267)f Fp(j)442 b(<)379 b(N)533
b Fr(\(of)414 b(course)d(this)i(case)f(is)g(v)-31 b(oid)413
b(if)g Fp(N)500 b Fr(=)380 b(2\))413 b(from)f Fp(F)429
b Fk(\261)275 b Fp(K)354 b Fk(\241)6075 45876 y Fp(K)325
b Fk(\261)246 b Fp(R)316 b Fr(=)307 b(0)370 b(w)-31 b(e)370
b(can)f(write)10903 47791 y Fp(K)11843 47957 y Fm(1)12585
47791 y Fr(+)246 b Fp(K)14632 47957 y Fm(2)15375 47791
y Fr(+)g Fk(\242)184 b(\242)g(\242)248 b Fr(+)e Fp(K)20066
47957 y Fo(j)20779 47791 y Fk(\241)g Fp(K)22826 47957
y Fm(1)23569 47791 y Fk(\261)g Fp(R)254 b Fk(\241)246
b Fp(K)27510 47957 y Fm(2)28253 47791 y Fk(\261)g Fp(R)254
b Fk(\241)246 b(\242)184 b(\242)g(\242)249 b(\241)d Fp(K)34839
47957 y Fo(j)35551 47791 y Fk(\261)h Fp(R)315 b Fr(=)308
b Fp(o)p Fr(\()p Fp(t)40043 47334 y Fo(j)40509 47791
y Fr(\))p Fp(:)6075 49705 y Fr(Comparing)399 b(the)f(terms)f(of)h
(order)f(t)-31 b(w)g(o)399 b(w)-31 b(e)398 b(get)g Fp(K)27676
49871 y Fm(2)28438 49705 y Fk(\241)264 b Fp(K)30503 49871
y Fm(1)31265 49705 y Fk(\261)h Fp(R)32924 49871 y Fm(2)33685
49705 y Fk(\241)g Fp(K)35751 49871 y Fm(2)36512 49705
y Fk(\261)h Fp(R)38172 49871 y Fm(1)39022 49705 y Fr(=)355
b(0)397 b(and)h(hence)6075 51034 y Fp(R)6916 51200 y
Fm(2)7719 51034 y Fr(=)308 b(0)319 b(and)g Fp(K)12802
51200 y Fm(2)13617 51034 y Fr(is)g(free.)475 b(Assuming)320
b(inductiv)-31 b(ely)321 b(that)f Fp(R)31081 51200 y
Fo(p)31900 51034 y Fr(=)307 b(0)319 b(for)g(2)308 b Fk(\267)g
Fp(p)f(<)g(l)330 b Fk(\267)307 b Fp(j)382 b Fr(w)-31
b(e)320 b(ha)-31 b(v)g(e)11101 53042 y Fp(K)12041 53208
y Fm(1)12784 53042 y Fr(+)246 b Fk(\242)184 b(\242)g(\242)248
b Fr(+)e Fp(K)17475 53208 y Fo(l)18061 53042 y Fk(\241)g
Fp(K)20108 53208 y Fm(1)20851 53042 y Fk(\261)g Fr(\()p
Fp(R)22921 53208 y Fm(1)23664 53042 y Fr(+)g Fp(R)25612
53208 y Fo(l)25951 53042 y Fr(\))g Fk(\241)g(\242)184
b(\242)g(\242)249 b(\241)d Fp(K)31319 53208 y Fo(l)31905
53042 y Fk(\261)g Fr(\()p Fp(R)33975 53208 y Fm(1)34718
53042 y Fr(+)g Fp(R)36666 53208 y Fo(l)37005 53042 y
Fr(\))308 b(=)f Fp(o)p Fr(\()p Fp(t)40278 52585 y Fo(l)40618
53042 y Fr(\))6075 54957 y(and)363 b(comparing)j(the)d(terms)h(of)g
(order)e Fp(l)386 b Fr(w)-31 b(e)364 b(get)g(the)g(condition)h
Fp(K)34627 55123 y Fo(l)35202 54957 y Fk(\241)234 b Fp(K)37237
55123 y Fm(1)37968 54957 y Fk(\261)h Fp(R)39597 55123
y Fo(l)40171 54957 y Fk(\241)g Fp(K)42207 55123 y Fo(l)42781
54957 y Fk(\261)g Fp(R)44410 55123 y Fm(1)45214 54957
y Fr(=)6075 56285 y(0.)824 b(Pro)61 b(jecting)482 b(on)-31
b(to)481 b(the)f(\257rst)f(comp)31 b(onen)-31 b(t)481
b(w)-31 b(e)481 b(obtain)g Fp(R)32186 56451 y Fo(l)33017
56285 y Fr(=)491 b(0.)825 b(Moreo)-31 b(v)g(er)480 b
Fp(K)41969 56451 y Fo(l)42788 56285 y Fr(is)f(free.)6075
57614 y(Therefore)369 b Fp(R)11934 57780 y Fm(2)12738
57614 y Fr(=)307 b Fk(\242)184 b(\242)g(\242)310 b Fr(=)d
Fp(R)17514 57780 y Fo(N)83 b Fj(\241)p Fm(1)19794 57614
y Fr(=)307 b(0.)8289 59572 y(F)-92 b(or)368 b Fp(j)j
Fk(\270)307 b Fp(N)490 b Fr(w)-31 b(e)370 b(write)11856
61108 y Fo(j)11262 61492 y Ff(X)11345 63850 y Fo(i)p
Fm(=1)13046 62544 y Fp(K)13986 62710 y Fo(i)14600 62544
y Fr(+)16390 61108 y Fo(j)15796 61492 y Ff(X)15707 63867
y Fo(i)p Fm(=)p Fo(N)17669 62544 y Fp(F)18381 62710 y
Fo(i)18995 62544 y Fk(\261)247 b Fp(K)325 b Fk(\241)22761
61108 y Fo(j)22167 61492 y Ff(X)22249 63850 y Fo(i)p
Fm(=1)23950 62544 y Fp(K)24890 62710 y Fo(i)25505 62544
y Fk(\261)247 b Fr(\()p Fp(R)27576 62710 y Fm(1)28318
62544 y Fr(+)f Fp(R)30266 62710 y Fo(N)31351 62544 y
Fr(+)g Fk(\242)184 b(\242)g(\242)249 b Fr(+)d Fp(R)35944
62710 y Fo(j)36409 62544 y Fr(\))308 b(=)g Fp(o)p Fr(\()p
Fp(t)39683 62087 y Fo(j)40149 62544 y Fr(\))p Fp(:)p
eop end
%%Page: 7 7
TeXDict begin 7 6 bop 16168 3345 a Fm(ONE-DIMENSIONAL)389
b(INV)-95 b(ARIANT)389 b(MANIF)-24 b(OLDS)9653 b(7)6075
6001 y Fr(When)368 b Fp(j)j Fr(=)308 b Fp(N)121 b Fr(,)369
b(comparing)i(terms)e(of)h(order)e Fp(N)490 b Fr(w)-31
b(e)370 b(ha)-31 b(v)g(e)15808 8157 y Fp(K)16748 8323
y Fo(N)17834 8157 y Fr(+)246 b Fp(F)19653 8323 y Fo(N)20738
8157 y Fk(\261)g Fp(K)22477 8323 y Fm(1)23220 8157 y
Fk(\241)g Fp(K)25267 8323 y Fm(1)26010 8157 y Fk(\261)g
Fp(R)27650 8323 y Fo(N)28735 8157 y Fk(\241)g Fp(K)30782
8323 y Fo(N)31868 8157 y Fk(\261)h Fp(R)33509 8323 y
Fm(1)34312 8157 y Fr(=)308 b(0)p Fp(:)7458 b Fr(\(3.4\))6075
10312 y(Since)397 b Fp(R)9834 10478 y Fm(1)10330 10312
y Fr(\()p Fp(t)p Fr(\))354 b(=)f Fp(t)396 b Fr(w)-31
b(e)397 b(get)h(that)g Fp(K)20850 10478 y Fo(N)22086
10312 y Fr(is)f(free)f(and)h Fp(F)28425 9910 y Fm(1)28271
10617 y Fo(N)29375 10312 y Fk(\261)264 b Fp(K)31132 10478
y Fm(1)31983 10312 y Fr(=)353 b Fp(R)34038 10478 y Fo(N)34877
10312 y Fr(.)575 b(Th)-31 b(us)397 b Fp(d)39167 10478
y Fo(N)40360 10312 y Fr(=)353 b Fp(a)42159 10478 y Fo(N)35
b(;)p Fm(0)43655 10312 y Fr(.)575 b(W)-92 b(e)6075 11640
y(note)394 b(that)g(the)f(pro)61 b(jection)395 b(on)-31
b(to)395 b Fp(E)21528 11806 y Fm(2)22417 11640 y Fr(of)f(the)f
(left-hand)i(side)e(of)480 b(\(3.4\))395 b(v)-61 b(anishes.)564
b(Indeed,)399 b(if)6075 12968 y Fp(M)442 b(>)321 b(N)121
b Fr(,)380 b(b)-31 b(y)377 b(the)h(form)g(of)g Fp(F)154
b Fr(,)380 b Fp(F)20243 12567 y Fm(2)20089 13273 y Fo(N)21249
12968 y Fr(=)321 b(0.)518 b(If)377 b Fp(M)442 b Fr(=)322
b Fp(N)498 b Fr(w)-31 b(e)378 b(ha)-31 b(v)g(e)378 b
Fp(F)34074 12567 y Fm(2)33920 13273 y Fo(N)35011 12968
y Fk(\261)252 b Fp(K)36756 13134 y Fm(1)37253 12968 y
Fr(\()p Fp(t)p Fr(\))322 b(=)f Fp(F)40883 12567 y Fm(2)40729
13273 y Fo(M)41712 12968 y Fr(\()p Fp(t;)184 b Fr(0\))323
b(=)e(0)6075 14297 y(b)-31 b(y)369 b(Condition)j(\(2.5\).)8289
16255 y(When)c Fp(j)j(>)307 b(N)121 b Fr(,)370 b(the)f(terms)h(of)f
(order)g Fp(j)432 b Fr(can)369 b(b)31 b(e)369 b(obtained)i(from)12173
18410 y Fp(K)13113 18576 y Fm(1)13856 18410 y Fr(+)246
b Fk(\242)184 b(\242)g(\242)248 b Fr(+)e Fp(K)18547 18576
y Fo(j)19260 18410 y Fr(+)g Fp(F)21079 18576 y Fo(N)22164
18410 y Fk(\261)h Fr(\()p Fp(K)24334 18576 y Fm(1)25077
18410 y Fr(+)f Fk(\242)184 b(\242)g(\242)248 b Fr(+)e
Fp(K)29768 18576 y Fo(j)45 b Fj(\241)p Fo(N)83 b Fm(+1)32832
18410 y Fr(\))246 b(+)g Fk(\242)184 b(\242)g(\242)249
b Fr(+)d Fp(F)37972 18576 y Fo(j)38684 18410 y Fk(\261)g
Fp(K)40423 18576 y Fm(1)13280 20139 y Fr(=)1107 b Fp(K)16188
20305 y Fm(1)16931 20139 y Fk(\261)246 b Fr(\()p Fp(R)19001
20305 y Fm(1)19744 20139 y Fr(+)g Fp(R)21692 20305 y
Fo(N)22777 20139 y Fr(+)g Fk(\242)184 b(\242)g(\242)248
b Fr(+)e Fp(R)27369 20305 y Fo(j)27835 20139 y Fr(\))h(+)f
Fk(\242)184 b(\242)g(\242)248 b Fr(+)e Fp(K)33203 20305
y Fo(j)33916 20139 y Fk(\261)g Fp(R)35556 20305 y Fm(1)36298
20139 y Fr(+)g Fp(o)p Fr(\()p Fp(t)38772 19682 y Fo(j)39238
20139 y Fr(\))p Fp(:)3824 b Fr(\(3.5\))8289 22924 y(Actually)354
b(from)g(\(3.5\))h(w)-31 b(e)353 b(will)h(obtain)h Fp(K)26045
23090 y Fo(j)45 b Fj(\241)p Fo(N)83 b Fm(+1)29461 22924
y Fr(and)352 b Fp(R)32437 23090 y Fo(j)33256 22924 y
Fr(assuming)h(w)-31 b(e)354 b(already)f(kno)-31 b(w)6075
24253 y Fp(K)7015 24419 y Fo(p)7908 24253 y Fr(for)382
b Fp(p)328 b(<)h(j)317 b Fk(\241)254 b Fp(N)376 b Fr(+)254
b(1)382 b(and)g Fp(R)19899 24419 y Fo(p)20792 24253 y
Fr(for)g Fp(p)328 b(<)g(j)63 b Fr(.)531 b(Indeed,)384
b(pro)61 b(jecting)400 b(\(3.5\))383 b(on)-31 b(to)384
b Fp(E)41117 24419 y Fm(2)p Fo(;l)42542 24253 y Fr(b)-31
b(y)382 b Fp(\274)44723 24419 y Fm(2)p Fo(;l)45767 24253
y Fr(,)6075 25581 y(and)369 b(equating)i(the)f(terms)f(of)h(order)e
Fp(j)432 b Fr(w)-31 b(e)370 b(get)9337 27804 y Fp(c)9816
27272 y Fm(2)p Fo(;l)9816 28112 y(j)11106 27804 y Fr(+)246
b Fp(b)12688 27347 y Fo(l)12688 28078 y(N)83 b Fj(\241)p
Fm(1)p Fo(;)p Fm(1)p Fo(;)p Fm(0)p Fo(;:::)q(;)p Fm(0)17825
27804 y Fp(c)18304 27272 y Fm(2)p Fo(;)p Fm(1)18304 28129
y Fo(j)45 b Fj(\241)p Fo(N)83 b Fm(+1)21613 27804 y Fr(+)246
b Fp(b)23195 27347 y Fo(l)23195 28078 y(N)83 b Fj(\241)p
Fm(1)p Fo(;)p Fm(0)p Fo(;)p Fm(1)p Fo(;:::)q(;)p Fm(0)28332
27804 y Fp(c)28811 27272 y Fm(2)p Fo(;)p Fm(2)28811 28129
y Fo(j)45 b Fj(\241)p Fo(N)83 b Fm(+1)32121 27804 y Fr(+)245
b Fk(\242)184 b(\242)g(\242)249 b Fr(+)d Fp(b)36347 27347
y Fo(l)36347 28078 y(N)83 b Fj(\241)p Fm(1)p Fo(;)p Fm(0)p
Fo(;)p Fm(0)p Fo(;:::;)p Fm(1)41484 27804 y Fp(c)41963
27272 y Fm(2)p Fo(;n)41963 28129 y(j)45 b Fj(\241)p Fo(N)83
b Fm(+1)11551 29793 y Fr(=)308 b(\()p Fp(j)h Fk(\241)246
b Fp(N)367 b Fr(+)246 b(1\))p Fp(c)18847 29261 y Fm(2)p
Fo(;l)18847 30118 y(j)45 b Fj(\241)p Fo(N)83 b Fm(+1)21912
29793 y Fp(d)22488 29959 y Fo(N)23573 29793 y Fr(+)246
b Fp(c)25159 29261 y Fm(2)p Fo(;l)25159 30101 y(j)26449
29793 y Fr(+)g(\241)28248 29261 y Fm(2)p Fo(;l)28248
30101 y(j)29292 29793 y Fp(;)6075 32238 y Fr(where)397
b(\241)9996 31836 y Fm(2)9996 32526 y Fo(j)10848 32238
y Fr(=)355 b(\(\241)13186 31706 y Fm(2)p Fo(;)p Fm(1)13186
32546 y Fo(j)14387 32238 y Fp(;)184 b(:)g(:)g(:)k(;)c
Fr(\241)17538 31706 y Fm(2)p Fo(;n)17538 32546 y(j)18845
32238 y Fr(\))398 b(dep)31 b(ends)397 b(on)h Fp(F)31
b(;)184 b(K)27722 32404 y Fm(2)28219 32238 y Fp(;)g(:)g(:)g(:)k(;)c(K)
31618 32404 y Fo(j)45 b Fj(\241)p Fo(N)33562 32238 y
Fp(;)184 b(R)34894 32404 y Fo(N)35734 32238 y Fp(;)g(:)g(:)g(:)k(;)c(R)
39034 32404 y Fo(j)45 b Fj(\241)p Fm(1)40634 32238 y
Fr(.)578 b(Hence,)405 b(in)6075 33566 y(matrix)370 b(notation)15487
35721 y(\()p Fp(B)16757 35887 y Fo(N)83 b Fj(\241)p Fm(1)p
Fo(;)p Fm(1)19680 35721 y Fk(\241)246 b Fr(\()p Fp(j)310
b Fk(\241)246 b Fp(N)367 b Fr(+)246 b(1\))p Fp(a)27021
35887 y Fo(N)35 b(;)p Fm(0)28518 35721 y Fr(Id)184 b(\))p
Fp(c)30626 35264 y Fm(2)30626 35995 y Fo(j)45 b Fj(\241)p
Fo(N)83 b Fm(+1)33998 35721 y Fr(=)307 b(\241)35858 35264
y Fm(2)35858 35995 y Fo(j)36355 35721 y Fp(:)7137 b Fr(\(3.6\))8289
38543 y(If)343 b Fp(N)429 b Fr(=)307 b Fp(M)464 b Fr(the)344
b(matrix)i Fp(B)19691 38709 y Fo(N)83 b Fj(\241)p Fm(1)p
Fo(;)p Fm(1)22562 38543 y Fk(\241)195 b Fr(\()p Fp(j)259
b Fk(\241)195 b Fp(N)316 b Fr(+)195 b(1\))p Fp(a)29648
38709 y Fo(N)35 b(;)p Fm(0)31145 38543 y Fr(Id)528 b(is)344
b(in)-31 b(v)g(ertible)345 b(b)31 b(ecause,)349 b(b)-31
b(y)344 b(the)6075 39871 y(prop)31 b(ert)-31 b(y)303
b(on)g(the)g(eigen)-31 b(v)-61 b(alues)304 b(of)f Fp(B)21539
40037 y Fo(M)83 b Fj(\241)p Fm(1)p Fo(;)p Fm(1)24661
39871 y Fr(and)303 b(the)g(fact)h(that)g Fp(a)33621 40037
y Fo(N)35 b(;)p Fm(0)35425 39871 y Fp(<)307 b Fr(0,)317
b(all)304 b(its)f(eigen)-31 b(v)-61 b(alues)6075 41200
y(ha)-31 b(v)g(e)407 b(p)31 b(ositiv)-31 b(e)408 b(real)e(part,)417
b(and)406 b(hence)g(they)h(are)f(non-zero.)604 b(If)406
b Fp(N)490 b(<)369 b(M)527 b Fr(then)406 b Fp(B)41245
41366 y Fo(N)83 b Fj(\241)p Fm(1)p Fo(;)p Fm(1)44291
41200 y Fr(=)369 b(0)6075 42528 y(and)449 b(\()p Fp(j)364
b Fk(\241)299 b Fp(N)421 b Fr(+)299 b(1\))p Fp(a)14755
42694 y Fo(N)35 b(;)p Fm(0)16253 42528 y Fr(Id)633 b(is)450
b(in)-31 b(v)g(ertible.)735 b(Hence)450 b(in)f(b)31 b(oth)450
b(cases)f(w)-31 b(e)450 b(can)g(solv)-31 b(e)451 b(\(3.6\))g(and)6075
43857 y(obtain)371 b Fp(K)10538 43455 y Fm(2)10459 44161
y Fo(j)45 b Fj(\241)p Fo(N)83 b Fm(+1)13522 43857 y Fr(.)8289
45954 y(Pro)61 b(jecting)387 b(\(3.5\))371 b(on)-31 b(to)371
b Fp(E)19720 45552 y Fm(1)20585 45954 y Fr(w)-31 b(e)370
b(ha)-31 b(v)g(e)9439 48199 y Fp(c)9918 47742 y Fm(1)9918
48473 y Fo(j)10661 48199 y Fr(+)246 b Fp(N)121 b(a)13363
48365 y Fo(N)35 b(;)p Fm(0)p Fo(;)p Fm(0)p Fo(;:::;)p
Fm(0)17319 48199 y Fp(c)17798 47742 y Fm(1)17798 48473
y Fo(j)45 b Fj(\241)p Fo(N)83 b Fm(+1)21107 48199 y Fr(+)246
b Fp(a)22799 48365 y Fo(N)83 b Fj(\241)p Fm(1)p Fo(;)p
Fm(1)p Fo(;)p Fm(0)p Fo(;:::;)p Fm(0)27936 48199 y Fp(c)28415
47667 y Fm(2)p Fo(;)p Fm(1)28415 48525 y Fo(j)45 b Fj(\241)p
Fo(N)83 b Fm(+1)31724 48199 y Fr(+)246 b Fp(a)33416 48365
y Fo(N)83 b Fj(\241)p Fm(1)p Fo(;)p Fm(0)p Fo(;)p Fm(1)p
Fo(;:::)q(;)p Fm(0)38553 48199 y Fp(c)39032 47667 y Fm(2)p
Fo(;)p Fm(2)39032 48525 y Fo(j)45 b Fj(\241)p Fo(N)83
b Fm(+1)42342 48199 y Fr(+)245 b Fp(:)184 b(:)g(:)11653
50074 y Fr(=)307 b Fp(d)13397 50240 y Fo(j)14110 50074
y Fr(+)246 b(\()p Fp(j)310 b Fk(\241)245 b Fp(N)368 b
Fr(+)246 b(1\))p Fp(c)21345 49617 y Fm(1)21345 50348
y Fo(j)45 b Fj(\241)p Fo(N)83 b Fm(+1)24409 50074 y Fp(d)24985
50240 y Fo(N)26071 50074 y Fr(+)246 b Fp(c)27657 49617
y Fm(1)27657 50348 y Fo(j)28399 50074 y Fr(+)g(\241)30198
49617 y Fm(1)30198 50348 y Fo(j)30695 50074 y Fp(;)12797
b Fr(\(3.7\))6075 52337 y(where)369 b(\241)9968 51936
y Fm(1)9968 52625 y Fo(j)10833 52337 y Fr(dep)31 b(ends)368
b(on)i Fp(F)31 b(;)184 b(K)18825 52503 y Fm(2)19322 52337
y Fp(;)g(:)g(:)g(:)k(;)c(K)22721 52503 y Fo(j)45 b Fj(\241)p
Fo(N)24665 52337 y Fp(;)184 b(R)25997 52503 y Fo(N)26837
52337 y Fp(;)g(:)g(:)g(:)k(;)c(R)30137 52503 y Fo(j)45
b Fj(\241)p Fm(1)31737 52337 y Fr(.)8289 54347 y(Condition)371
b(\(3.7\))h(b)31 b(ecomes)14777 56518 y(\(2)p Fp(N)368
b Fk(\241)246 b Fp(j)309 b Fk(\241)246 b Fr(1\))p Fp(a)21564
56684 y Fo(N)35 b(;)p Fm(0)23061 56518 y Fp(c)23540 56061
y Fm(1)23540 56791 y Fo(j)45 b Fj(\241)p Fo(N)83 b Fm(+1)26850
56518 y Fk(\241)246 b Fp(d)28533 56684 y Fo(j)29307 56518
y Fr(=)676 b(kno)-31 b(wn)371 b(terms)p Fp(:)6427 b Fr(\(3.8\))6075
58780 y(Therefore,)407 b(if)399 b Fp(j)421 b Fk(6)p Fr(=)357
b(2)p Fp(N)388 b Fk(\241)265 b Fr(1)400 b(w)-31 b(e)399
b(can)h(tak)-31 b(e)400 b Fp(d)25209 58946 y Fo(j)26033
58780 y Fr(=)357 b(0)399 b(and)h(determine)f Fp(c)36063
58379 y Fm(1)36063 59085 y Fo(j)45 b Fj(\241)p Fo(N)83
b Fm(+1)39526 58780 y Fr(from)399 b(\(3.8\))s(.)582 b(If)6075
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%%Page: 8 8
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%%Page: 9 9
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Fp(;)p eop end
%%Page: 10 10
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%%Page: 11 11
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44414 y Fo(r)31693 44816 y Fr(\).)673 b(T)-92 b(aking)431
b Fp(K)37872 44414 y Fj(\267)38620 44816 y Fr(\()p Fp(t)p
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%%Page: 12 12
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18284 y Fp(\274)12007 18450 y Fm(2)12504 18284 y Fp(K)13523
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b Fp(t)18026 17172 y Fo(N)19050 16123 y Ff(Z)20157 16397
y Fm(1)19665 18638 y(0)20838 17630 y Fp(D)31 b(K)22804
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b Fr(=)e Fk(\241)p Fp(T)31958 17796 y Fo(r)32580 17630
y Fk(\241)131 b(N)163 b Fr(\()p Fp(K)36092 17172 y Fo(>)36841
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17172 y Fm(2)p Fo(r)24 b Fj(\241)p Fo(N)42508 17630 y
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(determine)f Fp(\264)473 b Fr(suc)-31 b(h)434 b(that)h
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%%Page: 15 15
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%%Page: 18 18
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%%Page: 19 19
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b(that)h Fk(S)26757 40324 y Fo(j)27224 40726 y Fr(\()p
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%%Page: 20 20
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%%Page: 21 21
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%%Page: 22 22
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%%Page: 23 23
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%%Page: 24 24
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61044 y Fo(k)19890 60878 y Fk(\261)h Fp(K)21806 60476
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%%Page: 25 25
TeXDict begin 25 24 bop 16168 3345 a Fm(ONE-DIMENSIONAL)389
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9450 y Fk(\261)g Fp(K)35788 8993 y Fj(\267)36536 9450
y Fk(k)37089 9616 y Fo(r)-24 b(;k)38284 8553 y Ff(\242)38792
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y Fo(k)43743 11491 y Fk(k)44296 11657 y Fo(r)-24 b(;k)45492
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12989 y Fk(\261)282 b Fp(K)10183 12587 y Fj(\267)10930
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12587 y Fj(\267)26077 12989 y Fr(\()p Fp(t)p Fr(\))282
b(+)f Fp(K)29780 12587 y Fo(>)30528 12989 y Fr(\()p Fp(t)p
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23179 y(W)-92 b(e)368 b(write)17712 24507 y Fk(F)110
b Fr(\()p Fp(K)20067 24050 y Fo(>)20814 24507 y Fr(\)\()p
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b Fr(\()p Fp(t;)184 b(K)31998 24050 y Fo(>)32746 24507
y Fr(\()p Fp(t)p Fr(\)\))6075 26068 y(with)7621 27861
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Fp(C)26154 29253 y Fo(r)26645 29655 y Fr(.)8289 31613
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b(e)378 b(collect)i(some)f(prop)31 b(erties)378 b(of)g
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32941 y(after.)6075 34745 y Fw(Lemma)513 b(4.12.)601
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b Fk(2)g(X)18604 37873 y Fo(k)18442 38563 y(j)19148 38275
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y Fi(.)7957 41534 y Fr(\(3\))10057 41098 y Fo(@)48 b(H)p
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b Fr(\(Id)185 b Fp(;)f(K)15680 41133 y Fo(>)16428 41534
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b Fr(\()p Fp(K)32677 46356 y Fj(\267)33424 46813 y Fr(\()p
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48936 y Fo(>)17673 49337 y Fk(2)307 b(X)19670 48936 y
Fm(1)19508 49625 y Fo(j)20167 49337 y Fr(,)328 b(0)308
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b Fk(j)p Fr(\()p Fp(x;)184 b(y)40 b Fr(\))p Fk(j)41210
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b Fp(Q)22738 50993 y Fo(k)23447 50827 y Fk(\261)164 b
Fr(\()p Fp(K)25613 50425 y Fj(\267)26525 50827 y Fr(+)g
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b Fp(h)14556 52475 y Fm(2)15650 52309 y Fr(=)h Fp(K)18128
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y Fo(>)8392 53863 y Fr(\()p Fp(t)p Fr(\)\))10082 52966
y Ff(\241)10590 53863 y Fp(K)11609 53461 y Fo(>)12357
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b(X)26049 53461 y Fo(k)25887 54151 y(j)26593 53863 y
Fr(.)8289 55893 y(T)-92 b(o)328 b(establish)h(\(2\),)338
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b(substituting)i Fp(z)356 b Fr(=)308 b Fp(K)17448 56820
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y Fr(\()p Fp(t)p Fr(\))246 b(+)g Fp(K)25197 59134 y Fo(>)25945
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y Fj(\267)30350 59591 y Fr(\()p Fp(t)p Fr(\))17247 62526
y(+)18354 61019 y Ff(Z)19461 61293 y Fm(1)18969 63534
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b Fr(\()p Fp(K)27033 62069 y Fj(\267)27780 62526 y Fr(\()p
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62526 y Fr(\()p Fp(t)p Fr(\)\))p Fp(K)35420 62069 y Fo(>)36168
62526 y Fr(\()p Fp(t)p Fr(\))p Fp(D)g(K)39394 62069 y
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b(:)p eop end
%%Page: 26 26
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b Fp(K)25467 7057 y Fo(>)26215 7459 y Fr(\))p Fp(D)31
b(K)28611 7057 y Fj(\267)29801 7459 y Fk(2)442 b(X)31933
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Fo(\264)36146 7767 y(j)38477 7459 y Fr(and)451 b(using)f(b\))g(of)6075
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Fp(K)23574 8560 y Fj(\267)24585 8962 y Fr(+)264 b Fp(K)26729
8560 y Fo(>)27871 8962 y Fr(and)396 b Fp(h)30688 9128
y Fm(1)31536 8962 y Fr(=)351 b Fp(K)33767 8560 y Fo(>)34514
8962 y Fr(,)402 b Fp(h)35861 9128 y Fm(2)36709 8962 y
Fr(=)351 b Fp(D)31 b(K)39887 8560 y Fj(\267)40635 8962
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9956 y Fm(3)7519 10358 y Fp(P)154 b Fr(\()p Fp(K)9833
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9956 y Fo(>)14250 10358 y Fr(\))p Fp(K)15699 9956 y Fo(>)16447
10358 y Fp(D)g(K)18413 9956 y Fj(\267)19468 10358 y Fk(2)308
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y(j)31175 10358 y Fr(.)8289 12388 y(Finally)-92 b(,)530
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y Fp(z)356 b Fr(=)307 b Fp(K)9133 13314 y Fo(>)9881 13716
y Fr(\()p Fp(t)p Fr(\))369 b(w)-31 b(e)370 b(can)f(write)7343
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a(@)g(z)9135 16348 y Fr(\()p Fp(t;)184 b(K)11475 15891
y Fj(\267)12223 16348 y Fr(\()p Fp(t)p Fr(\)\))309 b(=)f
Fp(D)31 b(Q)17213 16514 y Fo(k)17758 16348 y Fr(\()p
Fp(K)19207 15891 y Fj(\267)19955 16348 y Fr(\()p Fp(t)p
Fr(\))246 b(+)g Fp(K)23587 15891 y Fo(>)24335 16348 y
Fr(\()p Fp(t)p Fr(\)\))h(+)27379 14841 y Ff(Z)28486 15115
y Fm(1)27994 17356 y(0)29167 16348 y Fp(D)30114 15891
y Fm(2)30611 16348 y Fp(P)154 b Fr(\()p Fp(K)32925 15891
y Fj(\267)33672 16348 y Fr(\()p Fp(t)p Fr(\))247 b(+)f
Fp(\265)31 b(K)37856 15891 y Fo(>)38603 16348 y Fr(\()p
Fp(t)p Fr(\)\))p Fp(K)41312 15891 y Fo(>)42060 16348
y Fr(\()p Fp(t)p Fr(\))184 b Fp(d\265)31 b(:)6075 18809
y Fr(Applying)371 b(prop)31 b(ert)-31 b(y)369 b(b\))g(of)h(Prop)31
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b(4.13.)518 b Fi(Assume)359 b Fp(F)512 b Fi(is)359 b
Fp(C)21017 20467 y Fo(r)21867 20868 y Fi(and)g Fp(k)200
b Fk(\241)165 b Fr(2)p Fp(N)287 b Fr(+)165 b Fp(L)308
b(>)g Fr(0)p Fi(.)498 b(L)-57 b(et)359 b Fk(F)469 b Fi(b)-57
b(e)360 b(the)e(op)-57 b(er)g(ator)359 b(de\257ne)-57
b(d)6075 22197 y(in)509 b Fr(\(4.52\))s Fi(.)591 b(L)-57
b(et)424 b Fp(\275)13814 22363 y Fo(r)14661 22197 y Fr(=)357
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Fk(k)p Fp(T)20418 22363 y Fo(k)20964 22197 y Fk(k)21517
22363 y Fo(r)-24 b(;k)22978 22197 y Fr(+)266 b Fk(k)p
Fp(Q)25533 22363 y Fo(k)26344 22197 y Fk(\261)g Fp(K)28182
21795 y Fj(\267)28930 22197 y Fk(k)29483 22363 y Fo(r)-24
b(;k)30678 22197 y Fr(\))423 b Fi(and)h Fp(\275)357 b
Fk(\270)f Fp(\275)36427 22363 y Fo(r)36918 22197 y Fi(.)591
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23123 y Fj(\244)7291 23525 y Fk(2)307 b Fr(\(0)p Fp(;)184
b Fr(1\))399 b Fi(such)d(that,)g(for)g(any)g Fr(0)308
b Fp(<)g(t)22632 23691 y Fm(0)23435 23525 y Fp(<)g(t)25004
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b(d)397 b(on)g Fk(B)20757 25337 y Fo(k)19 b Fj(\241)p
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y Fi(and)396 b Fk(F)110 b Fr(\()p Fk(B)28541 25337 y
Fo(k)19 b Fj(\241)p Fo(N)83 b Fm(+1)28507 26119 y Fo(j)m(;\275)31682
25810 y Fr(\))308 b Fk(\275)g(X)34541 25408 y Fo(k)34379
26098 y(j)35481 25810 y Fi(for)396 b Fr(0)308 b Fk(\267)f
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Fr(\(2\))554 b Fk(F)418 b Fr(:)307 b Fk(B)12513 26918
y Fo(k)19 b Fj(\241)p Fo(N)83 b Fm(+1)12479 27687 y Fo(r)24
b Fj(\241)p Fm(1)p Fo(;\275)15962 27392 y Fk(!)308 b(X)18329
26990 y Fo(k)18167 27666 y(r)24 b Fj(\241)p Fm(1)20186
27392 y Fi(is)396 b(Lipschitz,)g(with)g(Lipschitz)g(c)-57
b(onstant)396 b(b)-57 b(ounde)g(d)397 b(by)21321 29472
y Fr(Lip)185 b Fk(F)417 b(\267)308 b Fp(C)79 b(t)26772
28998 y Fo(k)19 b Fj(\241)p Fm(2)p Fo(N)83 b Fm(+)p Fo(L)26772
29767 y Fm(0)30521 29472 y Fp(:)12418 b Fr(\(4.55\))6075
31491 y Fw(Remark)364 b(4.14.)507 b Fr(Ev)-31 b(en)318
b(though)g(in)f(this)g(lemma)i(the)e(condition)i(on)f
Fp(k)351 b Fr(is)317 b(to)h(b)31 b(e)317 b(greater)g(than)6075
32819 y(2)p Fp(N)366 b Fk(\241)244 b Fp(L)p Fr(,)369
b(in)g(other)g(previous)f(results,)h(namely)h(Prop)31
b(osition)370 b(4.10,)h(w)-31 b(e)369 b(ha)-31 b(v)g(e)369
b(to)g(require)f(the)6075 34148 y(stronger)h(condition)i
Fp(k)343 b(>)307 b Fr(2)p Fp(N)368 b Fk(\241)246 b Fr(1.)6075
36837 y Fi(Pr)-57 b(o)g(of.)554 b Fr(W)-92 b(e)326 b(use)h(the)g
(expression)g Fk(F)110 b Fr(\()p Fp(K)22790 36435 y Fo(>)23538
36837 y Fr(\)\()p Fp(t)p Fr(\))308 b(=)g Fk(\241)p Fp(T)28213
37003 y Fo(k)28757 36837 y Fr(\()p Fp(t)p Fr(\))162 b
Fk(\241)g Fp(H)90 b Fr(\()p Fp(t;)184 b(K)34552 36435
y Fo(>)35301 36837 y Fr(\()p Fp(t)p Fr(\)\))328 b(with)h
Fp(H)417 b Fr(de\257ned)326 b(b)-31 b(y)6075 38165 y(\(4.54\))s(.)8289
40194 y(Let)436 b Fp(K)11358 39793 y Fo(>)12526 40194
y Fk(2)420 b(B)14445 39721 y Fo(k)19 b Fj(\241)p Fo(N)83
b Fm(+1)14411 40503 y Fo(j)m(;\275)17586 40194 y Fr(.)695
b(By)437 b(Lemma)h(4.12)g(the)f(fact)h(that)g Fk(F)110
b Fr(\()p Fk(B)35604 39721 y Fo(k)19 b Fj(\241)p Fo(N)83
b Fm(+1)35570 40503 y Fo(j)m(;\275)38745 40194 y Fr(\))420
b Fk(\275)g(X)41828 39793 y Fo(k)41666 40483 y(j)42809
40194 y Fr(follo)-31 b(ws)6075 41533 y(from)15540 43373
y Fk(kF)110 b Fr(\()p Fp(K)18448 42916 y Fo(>)19196 43373
y Fr(\))p Fk(k)20179 43539 y Fo(j)m(;k)21659 43373 y
Fk(\267)307 b(k)p Fp(T)24027 43539 y Fo(k)24572 43373
y Fk(k)25125 43539 y Fo(j)m(;k)26543 43373 y Fr(+)246
b Fk(k)p Fp(H)336 b Fk(\261)247 b Fr(\(Id)184 b Fp(;)g(K)33398
42916 y Fo(>)34147 43373 y Fr(\))p Fk(k)35130 43539 y
Fo(j)m(;k)36302 43373 y Fp(:)8289 46091 y Fr(T)-92 b(o)257
b(establish)h(the)f(second)g(statemen)-31 b(t)259 b(w)-31
b(e)258 b(tak)-31 b(e)258 b Fp(K)29364 45651 y Fo(>)29285
46386 y Fm(1)30111 46091 y Fp(;)184 b(K)31621 45651 y
Fo(>)31542 46386 y Fm(2)32677 46091 y Fk(2)308 b(B)34484
45618 y Fo(k)19 b Fj(\241)p Fo(N)83 b Fm(+1)34450 46386
y Fo(r)24 b Fj(\241)p Fm(1)p Fo(;\275)37882 46091 y Fr(and)257
b(0)308 b Fk(\267)f Fp(l)330 b Fk(\267)307 b Fp(r)52
b Fk(\241)22 b Fr(1.)6075 47478 y(Then,)370 b(using)f(\(1\))i(of)e
(Lemma)i(4.12,)8392 49372 y Fk(j)p Fp(D)9646 48915 y
Fo(l)9986 49372 y Fr(\()p Fk(F)110 b Fr(\()p Fp(K)12771
48915 y Fo(>)12692 49650 y Fm(1)13519 49372 y Fr(\))247
b Fk(\241)f(F)110 b Fr(\()p Fp(K)17658 48915 y Fo(>)17579
49650 y Fm(2)18405 49372 y Fr(\)\)\()p Fp(t)p Fr(\))p
Fk(j)310 b Fr(=)d Fk(j)p Fp(D)23564 48915 y Fo(l)23905
49372 y Fr([)p Fp(H)90 b Fr(\()p Fp(t;)184 b(K)27562
48915 y Fo(>)27483 49650 y Fm(1)28311 49372 y Fr(\()p
Fp(t)p Fr(\)\))247 b Fk(\241)f Fp(H)90 b Fr(\()p Fp(t;)184
b(K)34705 48915 y Fo(>)34626 49650 y Fm(2)35454 49372
y Fr(\()p Fp(t)p Fr(\)\)])p Fk(j)14257 51264 y(\267)914
b Fr(sup)15426 52245 y Fo(\265)24 b Fj(2)p Fm(\(0)p Fo(;)p
Fm(1\))18490 51264 y Fk(j)p Fp(D)19744 50807 y Fo(l)t
Fm(+1)21205 51264 y Fp(H)90 b Fr(\()p Fp(t;)184 b(\265)31
b(K)25106 50807 y Fo(>)25027 51542 y Fm(1)25854 51264
y Fr(\()p Fp(t)p Fr(\))247 b(+)f(\(1)h Fk(\241)f Fp(\265)31
b Fr(\))p Fp(K)32805 50807 y Fo(>)32726 51542 y Fm(2)33552
51264 y Fr(\()p Fp(t)p Fr(\)\))p Fk(j)184 b(j)p Fp(K)37059
50807 y Fo(>)36980 51542 y Fm(1)37809 51264 y Fr(\()p
Fp(t)p Fr(\))247 b Fk(\241)f Fp(K)41442 50807 y Fo(>)41363
51542 y Fm(2)42189 51264 y Fr(\()p Fp(t)p Fr(\))p Fk(j)14257
54030 y(\267)308 b Fp(C)79 b(t)16696 53573 y Fo(k)19
b Fj(\241)p Fm(\()p Fo(l)t Fm(+1\))p Fo(\264)20513 54030
y Fp(t)20913 53573 y Fo(k)g Fj(\241)p Fo(N)83 b Fm(+1)24055
54030 y Fk(k)p Fp(K)25627 53573 y Fo(>)25548 54308 y
Fm(1)26621 54030 y Fk(\241)246 b Fp(K)28747 53573 y Fo(>)28668
54308 y Fm(2)29494 54030 y Fk(k)30047 54196 y Fo(r)24
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54030 y Fp(:)6075 55871 y Fr(Hence,)369 b(since)g Fp(\264)347
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Fo(;k)22227 57711 y Fr(=)1312 b(sup)23395 58649 y Fm(0)p
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57711 y Fk(k)p Fp(D)28757 57254 y Fo(l)29097 57711 y
Fr(\()p Fk(F)110 b Fr(\()p Fp(K)31882 57254 y Fo(>)31803
57989 y Fm(1)32630 57711 y Fr(\))247 b Fk(\241)f(F)110
b Fr(\()p Fp(K)36769 57254 y Fo(>)36690 57989 y Fm(2)37516
57711 y Fr(\)\))p Fk(k)38929 57877 y Fm(0)p Fo(;k)19
b Fj(\241)p Fo(l)t(\264)15765 60341 y Fk(\267)1312 b
Fr(sup)16933 61279 y Fm(0)p Fj(\267)p Fo(l)t Fj(\267)p
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61322 y Fm(\))24073 60341 y Fp(C)79 b(t)25343 59884 y
Fj(\241)p Fo(\264)26574 60341 y Fp(t)26974 59884 y Fo(k)19
b Fj(\241)p Fo(N)83 b Fm(+1)30115 60341 y Fk(k)p Fp(K)31687
59884 y Fo(>)31608 60619 y Fm(1)32681 60341 y Fk(\241)246
b Fp(K)34807 59884 y Fo(>)34728 60619 y Fm(2)35555 60341
y Fk(k)36108 60507 y Fo(r)24 b Fj(\241)p Fm(1)p Fo(;k)19
b Fj(\241)p Fo(N)83 b Fm(+1)15765 63081 y Fk(\267)307
b Fp(C)79 b(t)18203 62607 y Fo(k)19 b Fj(\241)p Fm(2)p
Fo(N)83 b Fm(+)p Fo(L)18203 63376 y Fm(0)21952 63081
y Fk(k)p Fp(K)23524 62624 y Fo(>)23445 63359 y Fm(1)24518
63081 y Fk(\241)246 b Fp(K)26644 62624 y Fo(>)26565 63359
y Fm(2)27391 63081 y Fk(k)27944 63247 y Fo(r)24 b Fj(\241)p
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y Fp(:)p eop end
%%Page: 27 27
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b(INV)-95 b(ARIANT)389 b(MANIF)-24 b(OLDS)9212 b(27)45248
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b(op)g(osition)396 b(4.11.)554 b Fr(T)-92 b(aking)411
b Fp(\275)f Fr(as)g(in)g(Lemma)h(4.13)h(and)e(using)g(Prop)31
b(osition)412 b(4.10)6075 10229 y(w)-31 b(e)369 b(ha)-31
b(v)g(e)19001 11751 y Fk(S)19754 11294 y Fm(0)20496 11751
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Fo(k)19 b Fj(\241)p Fo(N)83 b Fm(+1)23851 12046 y Fo(r)24
b Fj(\241)p Fm(1)p Fo(;\275)27334 11751 y Fk(!)307 b(X)29700
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y Fo(r)24 b Fj(\241)p Fm(1)32841 11751 y Fp(:)6075 13745
y Fr(Moreo)-31 b(v)g(er,)390 b(for)c(an)-31 b(y)387 b
Fp(K)16105 13344 y Fo(>)17187 13745 y Fk(2)336 b(B)19022
13272 y Fo(k)19 b Fj(\241)p Fo(N)83 b Fm(+1)18988 14040
y Fo(r)24 b Fj(\241)p Fm(1)p Fo(;\275)22163 13745 y Fr(,)390
b(adding)d(and)f(subtracting)h Fk(F)110 b Fr(\(0\))336
b(=)f Fk(\241)p Fp(T)39901 13911 y Fo(k)40703 13745 y
Fk(\241)257 b Fp(Q)42696 13911 y Fo(k)43498 13745 y Fk(\261)g
Fp(K)45327 13344 y Fj(\267)6075 15074 y Fr(w)-31 b(e)369
b(can)h(write)7551 17061 y Fk(kS)8857 16604 y Fm(0)9600
17061 y Fk(\261)246 b(F)110 b Fr(\()p Fp(K)12754 16604
y Fo(>)13502 17061 y Fr(\))p Fk(k)14485 17227 y Fo(r)24
b Fj(\241)p Fm(1)p Fo(;k)19 b Fj(\241)p Fo(N)83 b Fm(+1)19766
17061 y Fk(\267)307 b(kS)22240 16604 y Fm(0)22738 17061
y Fk(k)23291 16164 y Ff(\241)23799 17061 y Fk(kF)110
b Fr(\(0\))p Fk(k)27224 17227 y Fo(r)24 b Fj(\241)p Fm(1)p
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16604 y Fo(>)34610 17061 y Fr(\))247 b Fk(\241)e(F)110
b Fr(\(0\))p Fk(k)39265 17227 y Fo(r)24 b Fj(\241)p Fm(1)p
Fo(;k)41642 16164 y Ff(\242)11613 18969 y Fk(\267)307
b(kS)14087 18512 y Fm(0)14584 18969 y Fk(k)15137 18072
y Ff(\241)15645 18969 y Fk(k)p Fp(T)16845 19135 y Fo(k)17390
18969 y Fk(k)17943 19135 y Fo(r)-24 b(;k)19384 18969
y Fr(+)246 b Fk(k)p Fp(Q)21919 19135 y Fo(k)22711 18969
y Fk(\261)g Fp(K)24529 18512 y Fj(\267)25277 18969 y
Fk(k)25830 19135 y Fo(r)-24 b(;k)27025 18072 y Ff(\242)27778
18969 y Fr(+)246 b Fp(C)79 b(t)30155 18495 y Fo(k)19
b Fj(\241)p Fm(2)p Fo(N)83 b Fm(+)p Fo(L)30155 19264
y Fm(0)33904 18969 y Fk(kS)35210 18512 y Fm(0)35707 18969
y Fk(k)184 b(k)p Fp(K)38016 18512 y Fo(>)38765 18969
y Fk(k)39318 19135 y Fo(r)24 b Fj(\241)p Fm(1)p Fo(;k)19
b Fj(\241)p Fo(N)83 b Fm(+1)44291 18969 y Fp(:)6075 21093
y Fr(Then,)370 b(w)-31 b(e)369 b(can)h(c)-31 b(ho)31
b(ose)369 b Fp(t)16843 21259 y Fm(0)17709 21093 y Fr(suc)-31
b(h)368 b(that)j Fk(kS)23910 20691 y Fm(0)24653 21093
y Fk(\261)247 b(F)110 b Fr(\()p Fp(K)27808 20691 y Fo(>)28555
21093 y Fr(\))p Fk(k)29538 21259 y Fo(r)24 b Fj(\241)p
Fm(1)p Fo(;k)19 b Fj(\241)p Fo(N)83 b Fm(+1)34819 21093
y Fp(<)308 b(\275)p Fr(,)369 b(and)15747 23191 y(Lip)185
b(\()p Fk(S)18729 22734 y Fm(0)19472 23191 y Fk(\261)247
b(F)110 b Fr(\))21608 23553 y Fj(jB)22474 23194 y Fh(k)9
b Fd(\241)p Fh(N)60 b Fg(+1)22455 23806 y Fh(r)19 b Fd(\241)p
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/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def
/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/Pnt { stroke [] 0 setdash
   gsave 1 setlinecap M 0 0 V stroke grestore } def
/Dia { stroke [] 0 setdash 2 copy vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V closepath stroke
  Pnt } def
/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V
  currentpoint stroke M
  hpt neg vpt neg R hpt2 0 V stroke
  } def
/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M
  0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke
  Pnt } def
/Crs { stroke [] 0 setdash exch hpt sub exch vpt add M
  hpt2 vpt2 neg V currentpoint stroke M
  hpt2 neg 0 R hpt2 vpt2 V stroke } def
/TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke
  Pnt  } def
/Star { 2 copy Pls Crs } def
/BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M
  0 vpt2 neg V  hpt2 0 V  0 vpt2 V
  hpt2 neg 0 V  closepath fill } def
/TriUF { stroke [] 0 setdash vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath fill } def
/TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V closepath stroke
  Pnt  } def
/TriDF { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V closepath fill} def
/DiaF { stroke [] 0 setdash vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V closepath fill } def
/Pent { stroke [] 0 setdash 2 copy gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore Pnt } def
/PentF { stroke [] 0 setdash gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath fill grestore } def
/Circle { stroke [] 0 setdash 2 copy
  hpt 0 360 arc stroke Pnt } def
/CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def
/C0 { BL [] 0 setdash 2 copy moveto vpt 90 450  arc } bind def
/C1 { BL [] 0 setdash 2 copy        moveto
       2 copy  vpt 0 90 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C2 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 90 180 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C3 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 0 180 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C4 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 180 270 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C5 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 0 90 arc
       2 copy moveto
       2 copy  vpt 180 270 arc closepath fill
               vpt 0 360 arc } bind def
/C6 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 90 270 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C7 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 0 270 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C8 { BL [] 0 setdash 2 copy moveto
      2 copy vpt 270 360 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C9 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 270 450 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
       2 copy moveto
       2 copy vpt 90 180 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C11 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 0 180 arc closepath fill
       2 copy moveto
       2 copy  vpt 270 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C12 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 180 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C13 { BL [] 0 setdash  2 copy moveto
       2 copy  vpt 0 90 arc closepath fill
       2 copy moveto
       2 copy  vpt 180 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C14 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 90 360 arc closepath fill
               vpt 0 360 arc } bind def
/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/Rec   { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto
       neg 0 rlineto closepath } bind def
/Square { dup Rec } bind def
/Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def
/S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def
/S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def
/S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def
/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill
       2 copy vpt Square fill
       Bsquare } bind def
/S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def
/S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def
/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill
       Bsquare } bind def
/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill
       Bsquare } bind def
/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
       2 copy vpt Square fill Bsquare } bind def
/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
       2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def
/D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def
/D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def
/D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def
/D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def
/D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def
/D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def
/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def
/D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def
/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def
/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def
/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def
/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def
/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def
/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def
/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def
/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def
/DiaE { stroke [] 0 setdash vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V closepath stroke } def
/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M
  0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke } def
/TriUE { stroke [] 0 setdash vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke } def
/TriDE { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V closepath stroke } def
/PentE { stroke [] 0 setdash gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore } def
/CircE { stroke [] 0 setdash 
  hpt 0 360 arc stroke } def
/Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def
/DiaW { stroke [] 0 setdash vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V Opaque stroke } def
/BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M
  0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V Opaque stroke } def
/TriUW { stroke [] 0 setdash vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V Opaque stroke } def
/TriDW { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V Opaque stroke } def
/PentW { stroke [] 0 setdash gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  Opaque stroke grestore } def
/CircW { stroke [] 0 setdash 
  hpt 0 360 arc Opaque stroke } def
/BoxFill { gsave Rec 1 setgray fill grestore } def
/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont
dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall
currentdict end definefont
end
%%EndProlog
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742 280 M
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742 2576 M
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11 -1952 R
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11 -1929 R
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10 -1929 R
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10 -1906 R
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11 -1883 R
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11 -1860 R
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10 -1630 R
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11 -1630 R
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11 -1607 R
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11 -1584 R
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10 -1584 R
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10 -1561 R
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10 -1538 R
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11 -1538 R
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0 1515 V
11 -1515 R
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11 -1492 R
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10 -1469 R
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11 -1469 R
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11 -1446 R
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0 1423 V
11 -1423 R
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10 -1423 R
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10 -1401 R
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10 -1378 R
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11 -1355 R
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10 -1355 R
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11 -1355 R
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11 -1332 R
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11 -1309 R
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11 -1286 R
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11 -1125 R
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11 -987 R
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11 -918 R
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10 -872 R
0 872 V
11 -872 R
0 872 V
10 -849 R
0 849 V
11 -849 R
0 849 V
10 -849 R
0 849 V
currentpoint stroke M
11 -827 R
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10 -827 R
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11 -827 R
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10 -804 R
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11 -804 R
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10 -804 R
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11 -781 R
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10 -781 R
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11 -758 R
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10 -758 R
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11 -758 R
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11 -735 R
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10 -735 R
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11 -735 R
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10 -735 R
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10 -712 R
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11 -712 R
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10 -712 R
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11 -689 R
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10 -689 R
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11 -689 R
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10 -689 R
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11 -689 R
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11 -666 R
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11 -643 R
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11 -643 R
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11 -643 R
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11 -620 R
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10 -620 R
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11 -620 R
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11 -597 R
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10 -597 R
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11 -597 R
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10 -597 R
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11 -597 R
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10 -597 R
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11 -597 R
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10 -574 R
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11 -574 R
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10 -574 R
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11 -574 R
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10 -574 R
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11 -574 R
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11 -574 R
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10 -574 R
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11 -574 R
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10 -574 R
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11 -574 R
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10 -574 R
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11 -574 R
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10 -574 R
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11 -551 R
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10 -551 R
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11 -551 R
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10 -551 R
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11 -551 R
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10 -551 R
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11 -551 R
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10 -551 R
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11 -551 R
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10 -551 R
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10 -574 R
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11 -574 R
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11 -574 R
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10 -574 R
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11 -574 R
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10 -574 R
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11 -574 R
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10 -574 R
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11 -574 R
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10 -574 R
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11 -574 R
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10 -574 R
0 574 V
11 -597 R
0 597 V
10 -597 R
0 597 V
11 -597 R
0 597 V
10 -597 R
0 597 V
11 -597 R
0 597 V
10 -597 R
0 597 V
11 -597 R
0 597 V
10 -620 R
0 620 V
11 -620 R
0 620 V
10 -620 R
0 620 V
11 -620 R
0 620 V
11 -620 R
0 620 V
10 -643 R
0 643 V
11 -643 R
0 643 V
10 -643 R
0 643 V
11 -643 R
0 643 V
10 -643 R
0 643 V
11 -666 R
0 666 V
10 -666 R
0 666 V
11 -666 R
0 666 V
10 -689 R
0 689 V
11 -689 R
0 689 V
10 -689 R
0 689 V
11 -689 R
0 689 V
10 -712 R
0 712 V
11 -712 R
0 712 V
10 -712 R
0 712 V
11 -735 R
0 735 V
10 -735 R
0 735 V
11 -735 R
0 735 V
10 -758 R
0 758 V
11 -758 R
0 758 V
11 -758 R
0 758 V
10 -781 R
0 781 V
11 -781 R
0 781 V
10 -804 R
0 804 V
11 -804 R
0 804 V
10 -804 R
0 804 V
11 -827 R
0 827 V
10 -827 R
0 827 V
11 -849 R
0 849 V
10 -849 R
0 849 V
11 -872 R
0 872 V
10 -872 R
0 872 V
11 -895 R
0 895 V
10 -895 R
0 895 V
11 -918 R
0 918 V
10 -918 R
0 918 V
11 -941 R
0 941 V
10 -941 R
0 941 V
11 -964 R
0 964 V
10 -964 R
0 964 V
11 -987 R
0 987 V
11 -987 R
0 987 V
10 -1010 R
0 1010 V
11 -1033 R
0 1033 V
10 -1033 R
0 1033 V
11 -1056 R
0 1056 V
10 -1056 R
0 1056 V
11 -1079 R
0 1079 V
10 -1102 R
0 1102 V
11 -1102 R
0 1102 V
10 -1125 R
0 1125 V
11 -1148 R
0 1148 V
10 -1148 R
0 1148 V
11 -1171 R
0 1171 V
10 -1194 R
0 1194 V
11 -1194 R
0 1194 V
10 -1217 R
0 1217 V
11 -1240 R
0 1240 V
10 -1263 R
0 1263 V
11 -1263 R
0 1263 V
10 -1286 R
0 1286 V
11 -1309 R
0 1309 V
11 -1332 R
0 1332 V
10 -1355 R
0 1355 V
11 -1355 R
0 1355 V
10 -1378 R
0 1378 V
11 -1401 R
0 1401 V
10 -1423 R
0 1423 V
11 -1446 R
0 1446 V
10 -1469 R
0 1469 V
11 -1469 R
0 1469 V
10 -1492 R
0 1492 V
11 -1515 R
0 1515 V
10 -1538 R
0 1538 V
11 -1561 R
0 1561 V
10 -1584 R
0 1584 V
11 -1607 R
0 1607 V
10 -1630 R
0 1630 V
11 -1653 R
0 1653 V
10 -1676 R
0 1676 V
11 -1699 R
0 1699 V
10 -1722 R
0 1722 V
11 -1745 R
0 1745 V
11 -1768 R
0 1768 V
10 -1791 R
0 1791 V
11 -1814 R
0 1814 V
10 -1837 R
0 1837 V
11 -1860 R
0 1860 V
10 -1883 R
0 1883 V
11 -1906 R
0 1906 V
10 -1929 R
0 1929 V
11 -1952 R
0 1952 V
10 -1997 R
0 1997 V
11 -2020 R
0 2020 V
10 -2043 R
0 2043 V
11 -2066 R
0 2066 V
10 -2089 R
0 2089 V
11 -2112 R
0 2112 V
10 -2158 R
0 2158 V
11 -2181 R
0 2181 V
10 -2204 R
0 2204 V
11 -2227 R
0 2227 V
currentpoint stroke M
10 -2273 R
0 2273 V
11 -2296 R
0 2296 V
11 -2319 R
0 2319 V
10 -2342 R
0 2342 V
11 -2388 R
0 2388 V
10 -2411 R
0 2411 V
11 -2434 R
0 2434 V
10 -2480 R
0 2480 V
11 -2503 R
0 2503 V
10 -2526 R
0 2526 V
11 -2571 R
0 2571 V
10 -2594 R
0 2594 V
11 -2640 R
0 2640 V
10 -2663 R
0 2663 V
11 -2686 R
0 2686 V
10 -2732 R
0 2732 V
11 -2755 R
0 2755 V
10 -2801 R
0 2801 V
11 -2824 R
0 2824 V
10 -2870 R
0 2870 V
11 -2893 R
0 2893 V
10 -2939 R
0 2939 V
11 -2962 R
0 2962 V
11 -3008 R
0 3008 V
10 -3054 R
0 3054 V
11 -3077 R
0 3077 V
10 -3123 R
0 3123 V
11 -3145 R
0 3145 V
10 -3191 R
0 3191 V
11 -3237 R
0 3237 V
10 -3260 R
0 3260 V
11 -3306 R
0 3306 V
10 -3352 R
0 3352 V
11 -3375 R
0 3375 V
10 -3421 R
0 3421 V
11 -3467 R
0 3467 V
10 -3513 R
0 3513 V
1.000 UL
LT1
742 280 M
0 2273 V
753 280 M
0 2296 V
763 280 M
0 2296 V
774 280 M
0 2296 V
784 280 M
0 2296 V
795 280 M
0 2296 V
805 280 M
0 2296 V
816 280 M
0 2296 V
826 280 M
0 2296 V
837 280 M
0 2296 V
847 280 M
0 2296 V
858 280 M
0 2296 V
868 280 M
0 2296 V
879 280 M
0 2296 V
889 280 M
0 2296 V
900 280 M
0 2296 V
910 280 M
0 2296 V
921 280 M
0 2296 V
931 280 M
0 2296 V
942 280 M
0 2319 V
952 280 M
0 2319 V
963 280 M
0 2319 V
974 280 M
0 2319 V
984 280 M
0 2319 V
995 280 M
0 2319 V
1005 280 M
0 2319 V
1016 280 M
0 2319 V
1026 280 M
0 2342 V
1037 280 M
0 2342 V
1047 280 M
0 2342 V
1058 280 M
0 2342 V
1068 280 M
0 2342 V
1079 280 M
0 2342 V
1089 280 M
0 2365 V
1100 280 M
0 2365 V
1110 280 M
0 2365 V
1121 280 M
0 2365 V
1131 280 M
0 2365 V
1142 280 M
0 2388 V
1152 280 M
0 2388 V
1163 280 M
0 2388 V
1173 280 M
0 2388 V
1184 280 M
0 2388 V
1195 280 M
0 2411 V
1205 280 M
0 2411 V
1216 280 M
0 2411 V
1226 280 M
0 2411 V
1237 280 M
0 2411 V
1247 280 M
0 2434 V
1258 280 M
0 2434 V
1268 280 M
0 2434 V
1279 280 M
0 2434 V
1289 280 M
0 2457 V
1300 280 M
0 2457 V
1310 280 M
0 2457 V
1321 280 M
0 2480 V
1331 280 M
0 2480 V
1342 280 M
0 2480 V
1352 280 M
0 2480 V
1363 280 M
0 2503 V
1373 280 M
0 2503 V
1384 280 M
0 2503 V
1394 280 M
0 2503 V
1405 280 M
0 2526 V
1416 280 M
0 2526 V
1426 280 M
0 2526 V
1437 280 M
0 2549 V
1447 280 M
0 2549 V
1458 280 M
0 2549 V
1468 280 M
0 2572 V
1479 280 M
0 2572 V
1489 280 M
0 2572 V
1500 280 M
0 2594 V
1510 280 M
0 2594 V
1521 280 M
0 2594 V
1531 280 M
0 2617 V
1542 280 M
0 2617 V
1552 280 M
0 2617 V
1563 280 M
0 2640 V
1573 280 M
0 2640 V
1584 280 M
0 2640 V
1594 280 M
0 2663 V
1605 280 M
0 2663 V
1615 280 M
0 2663 V
1626 280 M
0 2686 V
1637 280 M
0 2686 V
1647 280 M
0 2686 V
1658 280 M
0 2709 V
1668 280 M
0 2709 V
1679 280 M
0 2709 V
1689 280 M
0 2732 V
1700 280 M
0 2732 V
1710 280 M
0 2755 V
1721 280 M
0 2755 V
1731 280 M
0 2755 V
1742 280 M
0 2778 V
1752 280 M
0 2778 V
1763 280 M
0 2778 V
1773 280 M
0 2801 V
1784 280 M
0 2801 V
1794 280 M
0 2824 V
1805 280 M
0 2824 V
1815 280 M
0 2824 V
1826 280 M
0 2847 V
1836 280 M
0 2847 V
1847 280 M
0 2870 V
1858 280 M
0 2870 V
1868 280 M
0 2870 V
1879 280 M
0 2893 V
1889 280 M
0 2893 V
1900 280 M
0 2916 V
1910 280 M
0 2916 V
1921 280 M
0 2916 V
1931 280 M
0 2939 V
1942 280 M
0 2939 V
1952 280 M
0 2962 V
1963 280 M
0 2962 V
1973 280 M
0 2962 V
1984 280 M
0 2985 V
1994 280 M
0 2985 V
2005 280 M
0 3008 V
2015 280 M
0 3008 V
2026 280 M
0 3008 V
2036 280 M
0 3031 V
2047 280 M
0 3031 V
2057 280 M
0 3054 V
2068 280 M
0 3054 V
2079 280 M
0 3077 V
2089 280 M
0 3077 V
2100 280 M
0 3077 V
2110 280 M
0 3100 V
2121 280 M
0 3100 V
2131 280 M
0 3123 V
2142 280 M
0 3123 V
2152 280 M
0 3123 V
2163 280 M
0 3146 V
2173 280 M
0 3146 V
2184 280 M
0 3168 V
2194 280 M
0 3168 V
2205 280 M
0 3191 V
2215 280 M
0 3191 V
2226 280 M
0 3191 V
2236 280 M
0 3214 V
2247 280 M
0 3214 V
2257 280 M
0 3237 V
2268 280 M
0 3237 V
2278 280 M
0 3237 V
2289 280 M
0 3260 V
2300 280 M
0 3260 V
2310 280 M
0 3283 V
2321 280 M
0 3283 V
2331 280 M
0 3283 V
2342 280 M
0 3306 V
2352 280 M
0 3306 V
2363 280 M
0 3329 V
2373 280 M
0 3329 V
2384 280 M
0 3352 V
2394 280 M
0 3352 V
2405 280 M
0 3352 V
2415 280 M
0 3375 V
2426 280 M
0 3375 V
2436 280 M
0 3398 V
2447 280 M
0 3398 V
2457 280 M
0 3398 V
2468 280 M
0 3421 V
2478 280 M
0 3421 V
2489 280 M
0 3421 V
2499 280 M
0 3444 V
2510 280 M
0 3444 V
2521 280 M
0 3467 V
2531 280 M
0 3467 V
2542 280 M
0 3467 V
2552 280 M
0 3490 V
2563 280 M
0 3490 V
2573 280 M
0 3513 V
2584 280 M
0 3513 V
2594 280 M
0 3513 V
2605 280 M
0 3536 V
2615 280 M
0 3536 V
2626 280 M
0 3536 V
2636 280 M
0 3559 V
2647 280 M
0 3559 V
2657 280 M
0 3582 V
2668 280 M
0 3582 V
2678 280 M
0 3582 V
2689 280 M
0 3605 V
2699 280 M
0 3605 V
2710 280 M
0 3605 V
2720 280 M
0 3628 V
2731 280 M
0 3628 V
2742 280 M
0 3628 V
2752 280 M
0 3651 V
2763 280 M
0 3651 V
2773 280 M
0 3651 V
2784 280 M
0 3674 V
2794 280 M
0 3674 V
2805 280 M
0 3674 V
2815 280 M
0 3697 V
2826 280 M
0 3697 V
2836 280 M
0 3697 V
currentpoint stroke M
2847 280 M
0 3720 V
2857 280 M
0 3720 V
2868 280 M
0 3720 V
2878 280 M
0 3742 V
2889 280 M
0 3742 V
2899 280 M
0 3742 V
2910 280 M
0 3765 V
2920 280 M
0 3765 V
2931 280 M
0 3765 V
2941 280 M
0 3765 V
2952 280 M
0 3788 V
2962 280 M
0 3788 V
2973 280 M
0 3788 V
2984 280 M
0 3811 V
2994 280 M
0 3811 V
3005 280 M
0 3811 V
3015 280 M
0 3811 V
3026 280 M
0 3834 V
3036 280 M
0 3834 V
3047 280 M
0 3834 V
3057 280 M
0 3834 V
3068 280 M
0 3857 V
3078 280 M
0 3857 V
3089 280 M
0 3857 V
3099 280 M
0 3857 V
3110 280 M
0 3857 V
3120 280 M
0 3880 V
3131 280 M
0 3880 V
3141 280 M
0 3880 V
3152 280 M
0 3880 V
3162 280 M
0 3903 V
3173 280 M
0 3903 V
3183 280 M
0 3903 V
3194 280 M
0 3903 V
3205 280 M
0 3903 V
3215 280 M
0 3926 V
3226 280 M
0 3926 V
3236 280 M
0 3926 V
3247 280 M
0 3926 V
3257 280 M
0 3926 V
3268 280 M
0 3926 V
3278 280 M
0 3949 V
3289 280 M
0 3949 V
3299 280 M
0 3949 V
3310 280 M
0 3949 V
3320 280 M
0 3949 V
3331 280 M
0 3949 V
3341 280 M
0 3949 V
3352 280 M
0 3949 V
3362 280 M
0 3972 V
3373 280 M
0 3972 V
3383 280 M
0 3972 V
3394 280 M
0 3972 V
3404 280 M
0 3972 V
3415 280 M
0 3972 V
3426 280 M
0 3972 V
3436 280 M
0 3972 V
3447 280 M
0 3972 V
3457 280 M
0 3972 V
3468 280 M
0 3972 V
3478 280 M
0 3972 V
3489 280 M
0 3972 V
3499 280 M
0 3972 V
3510 280 M
0 3995 V
3520 280 M
0 3995 V
3531 280 M
0 3995 V
3541 280 M
0 3995 V
3552 280 M
0 3995 V
3562 280 M
0 3995 V
3573 280 M
0 3995 V
3583 280 M
0 3995 V
3594 280 M
0 3995 V
3604 280 M
0 3995 V
3615 280 M
0 3972 V
3625 280 M
0 3972 V
3636 280 M
0 3972 V
3647 280 M
0 3972 V
3657 280 M
0 3972 V
3668 280 M
0 3972 V
3678 280 M
0 3972 V
3689 280 M
0 3972 V
3699 280 M
0 3972 V
3710 280 M
0 3972 V
3720 280 M
0 3972 V
3731 280 M
0 3972 V
3741 280 M
0 3972 V
3752 280 M
0 3949 V
3762 280 M
0 3949 V
3773 280 M
0 3949 V
3783 280 M
0 3949 V
3794 280 M
0 3949 V
3804 280 M
0 3949 V
3815 280 M
0 3949 V
3825 280 M
0 3926 V
3836 280 M
0 3926 V
3846 280 M
0 3926 V
3857 280 M
0 3926 V
3868 280 M
0 3926 V
3878 280 M
0 3903 V
3889 280 M
0 3903 V
3899 280 M
0 3903 V
3910 280 M
0 3903 V
3920 280 M
0 3903 V
3931 280 M
0 3880 V
3941 280 M
0 3880 V
3952 280 M
0 3880 V
3962 280 M
0 3857 V
3973 280 M
0 3857 V
3983 280 M
0 3857 V
3994 280 M
0 3857 V
4004 280 M
0 3834 V
4015 280 M
0 3834 V
4025 280 M
0 3834 V
4036 280 M
0 3811 V
4046 280 M
0 3811 V
4057 280 M
0 3811 V
4067 280 M
0 3788 V
4078 280 M
0 3788 V
4089 280 M
0 3788 V
4099 280 M
0 3765 V
4110 280 M
0 3765 V
4120 280 M
0 3742 V
4131 280 M
0 3742 V
4141 280 M
0 3742 V
4152 280 M
0 3720 V
4162 280 M
0 3720 V
4173 280 M
0 3697 V
4183 280 M
0 3697 V
4194 280 M
0 3674 V
4204 280 M
0 3674 V
4215 280 M
0 3651 V
4225 280 M
0 3651 V
4236 280 M
0 3628 V
4246 280 M
0 3628 V
4257 280 M
0 3605 V
4267 280 M
0 3605 V
4278 280 M
0 3582 V
4288 280 M
0 3582 V
4299 280 M
0 3559 V
4310 280 M
0 3559 V
4320 280 M
0 3536 V
4331 280 M
0 3513 V
4341 280 M
0 3513 V
4352 280 M
0 3490 V
4362 280 M
0 3490 V
4373 280 M
0 3467 V
4383 280 M
0 3444 V
4394 280 M
0 3444 V
4404 280 M
0 3421 V
4415 280 M
0 3398 V
4425 280 M
0 3398 V
4436 280 M
0 3375 V
4446 280 M
0 3352 V
4457 280 M
0 3352 V
4467 280 M
0 3329 V
4478 280 M
0 3306 V
4488 280 M
0 3283 V
4499 280 M
0 3283 V
4509 280 M
0 3260 V
4520 280 M
0 3237 V
4531 280 M
0 3214 V
4541 280 M
0 3191 V
4552 280 M
0 3191 V
4562 280 M
0 3168 V
4573 280 M
0 3146 V
4583 280 M
0 3123 V
4594 280 M
0 3100 V
4604 280 M
0 3077 V
4615 280 M
0 3077 V
4625 280 M
0 3054 V
4636 280 M
0 3031 V
4646 280 M
0 3008 V
4657 280 M
0 2985 V
4667 280 M
0 2962 V
4678 280 M
0 2939 V
4688 280 M
0 2916 V
4699 280 M
0 2893 V
4709 280 M
0 2870 V
4720 280 M
0 2847 V
4730 280 M
0 2824 V
4741 280 M
0 2801 V
4752 280 M
0 2778 V
4762 280 M
0 2755 V
4773 280 M
0 2732 V
4783 280 M
0 2709 V
4794 280 M
0 2686 V
4804 280 M
0 2663 V
4815 280 M
0 2640 V
4825 280 M
0 2617 V
4836 280 M
0 2594 V
4846 280 M
0 2549 V
4857 280 M
0 2526 V
4867 280 M
0 2503 V
4878 280 M
0 2480 V
4888 280 M
0 2457 V
4899 280 M
0 2434 V
4909 280 M
0 2388 V
4920 280 M
0 2365 V
4930 280 M
0 2342 V
4941 280 M
0 2319 V
currentpoint stroke M
4951 280 M
0 2273 V
4962 280 M
0 2250 V
4973 280 M
0 2227 V
4983 280 M
0 2204 V
4994 280 M
0 2158 V
5004 280 M
0 2135 V
5015 280 M
0 2112 V
5025 280 M
0 2066 V
5036 280 M
0 2043 V
5046 280 M
0 2020 V
5057 280 M
0 1975 V
5067 280 M
0 1952 V
5078 280 M
0 1906 V
5088 280 M
0 1883 V
5099 280 M
0 1860 V
5109 280 M
0 1814 V
5120 280 M
0 1791 V
5130 280 M
0 1745 V
5141 280 M
0 1722 V
5151 280 M
0 1676 V
5162 280 M
0 1653 V
5172 280 M
0 1607 V
5183 280 M
0 1584 V
5194 280 M
0 1538 V
5204 280 M
0 1492 V
5215 280 M
0 1469 V
5225 280 M
0 1424 V
5236 280 M
0 1401 V
5246 280 M
0 1355 V
5257 280 M
0 1309 V
5267 280 M
0 1286 V
5278 280 M
0 1240 V
5288 280 M
0 1194 V
5299 280 M
0 1171 V
5309 280 M
0 1125 V
5320 280 M
0 1079 V
5330 280 M
0 1033 V
1.000 UL
LT2
742 2576 Pnt
747 2576 Pnt
751 2576 Pnt
756 2576 Pnt
760 2576 Pnt
764 2576 Pnt
769 2576 Pnt
773 2577 Pnt
778 2577 Pnt
782 2577 Pnt
787 2577 Pnt
791 2578 Pnt
796 2578 Pnt
800 2578 Pnt
805 2578 Pnt
809 2579 Pnt
814 2579 Pnt
818 2580 Pnt
823 2580 Pnt
827 2581 Pnt
832 2581 Pnt
836 2582 Pnt
841 2582 Pnt
845 2583 Pnt
850 2583 Pnt
854 2584 Pnt
859 2584 Pnt
863 2585 Pnt
868 2586 Pnt
872 2586 Pnt
877 2587 Pnt
881 2588 Pnt
885 2589 Pnt
890 2589 Pnt
894 2590 Pnt
899 2591 Pnt
903 2592 Pnt
908 2593 Pnt
912 2594 Pnt
917 2595 Pnt
921 2596 Pnt
926 2597 Pnt
930 2597 Pnt
935 2598 Pnt
939 2600 Pnt
943 2601 Pnt
948 2602 Pnt
952 2603 Pnt
957 2604 Pnt
961 2605 Pnt
966 2606 Pnt
970 2607 Pnt
975 2609 Pnt
979 2610 Pnt
983 2611 Pnt
988 2612 Pnt
992 2613 Pnt
997 2615 Pnt
1001 2616 Pnt
1006 2617 Pnt
1010 2619 Pnt
1014 2620 Pnt
1019 2622 Pnt
1023 2623 Pnt
1028 2624 Pnt
1032 2626 Pnt
1037 2627 Pnt
1041 2629 Pnt
1045 2630 Pnt
1050 2632 Pnt
1054 2633 Pnt
1059 2635 Pnt
1063 2637 Pnt
1068 2638 Pnt
1072 2640 Pnt
1076 2642 Pnt
1081 2643 Pnt
1085 2645 Pnt
1090 2647 Pnt
1094 2648 Pnt
1099 2650 Pnt
1103 2652 Pnt
1107 2654 Pnt
1112 2655 Pnt
1116 2657 Pnt
1121 2659 Pnt
1125 2661 Pnt
1129 2663 Pnt
1134 2665 Pnt
1138 2667 Pnt
1143 2668 Pnt
1147 2670 Pnt
1151 2672 Pnt
1156 2674 Pnt
1160 2676 Pnt
1165 2678 Pnt
1169 2680 Pnt
1173 2682 Pnt
1178 2684 Pnt
1182 2687 Pnt
1187 2689 Pnt
1191 2691 Pnt
1195 2693 Pnt
1200 2695 Pnt
1204 2697 Pnt
1209 2699 Pnt
1213 2701 Pnt
1217 2704 Pnt
1222 2706 Pnt
1226 2708 Pnt
1230 2710 Pnt
1235 2713 Pnt
1239 2715 Pnt
1244 2717 Pnt
1248 2720 Pnt
1252 2722 Pnt
1257 2724 Pnt
1261 2727 Pnt
1265 2729 Pnt
1270 2731 Pnt
1274 2734 Pnt
1279 2736 Pnt
1283 2739 Pnt
1287 2741 Pnt
1292 2744 Pnt
1296 2746 Pnt
1300 2748 Pnt
1305 2751 Pnt
1309 2753 Pnt
1314 2756 Pnt
1318 2759 Pnt
1322 2761 Pnt
1327 2764 Pnt
1331 2766 Pnt
1335 2769 Pnt
1340 2772 Pnt
1344 2774 Pnt
1348 2777 Pnt
1353 2779 Pnt
1357 2782 Pnt
1361 2785 Pnt
1366 2787 Pnt
1370 2790 Pnt
1375 2793 Pnt
1379 2796 Pnt
1383 2798 Pnt
1388 2801 Pnt
1392 2804 Pnt
1396 2807 Pnt
1401 2809 Pnt
1405 2812 Pnt
1409 2815 Pnt
1414 2818 Pnt
1418 2821 Pnt
1422 2824 Pnt
1427 2827 Pnt
1431 2829 Pnt
1435 2832 Pnt
1440 2835 Pnt
1444 2838 Pnt
1448 2841 Pnt
1453 2844 Pnt
1457 2847 Pnt
1461 2850 Pnt
1466 2853 Pnt
1470 2856 Pnt
1474 2859 Pnt
1479 2862 Pnt
1483 2865 Pnt
1487 2868 Pnt
1492 2871 Pnt
1496 2874 Pnt
1500 2877 Pnt
1505 2880 Pnt
1509 2883 Pnt
1513 2886 Pnt
1518 2889 Pnt
1522 2893 Pnt
1526 2896 Pnt
1531 2899 Pnt
1535 2902 Pnt
1539 2905 Pnt
1544 2908 Pnt
1548 2911 Pnt
1552 2915 Pnt
1557 2918 Pnt
1561 2921 Pnt
1565 2924 Pnt
1569 2927 Pnt
1574 2931 Pnt
1578 2934 Pnt
1582 2937 Pnt
1587 2940 Pnt
1591 2944 Pnt
1595 2947 Pnt
1600 2950 Pnt
1604 2954 Pnt
1608 2957 Pnt
1613 2960 Pnt
1617 2964 Pnt
1621 2967 Pnt
1626 2970 Pnt
1630 2974 Pnt
1634 2977 Pnt
1638 2980 Pnt
1643 2984 Pnt
1647 2987 Pnt
1651 2990 Pnt
1656 2994 Pnt
1660 2997 Pnt
1664 3001 Pnt
1669 3004 Pnt
1673 3007 Pnt
1677 3011 Pnt
1681 3014 Pnt
1686 3018 Pnt
1690 3021 Pnt
1694 3025 Pnt
1699 3028 Pnt
1703 3032 Pnt
1707 3035 Pnt
1711 3039 Pnt
1716 3042 Pnt
1720 3046 Pnt
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%%EndDocument
 @endspecial 39615 30092 a
 currentpoint grestore moveto
 39615 30092 a 369 w Fr(Figure)369
b(2)8289 32703 y(This)413 b(is)f(consequence)h(of)g(the)g(fact)h(that)
27105 32249 y Fo(@)27626 31915 y Fh(M)28452 32249 y Fo(F)29013
32360 y Fg(2)p 26364 32448 3823 45 v 26364 33092 a Fo(@)48
b(x)27387 32870 y Fh(M)60 b Fd(\241)p Fg(1)29189 33092
y Fo(@)48 b(y)30699 32703 y Fr(=)380 b(1)g Fp(>)g Fr(0)413
b(whic)-31 b(h)414 b(pro)-31 b(vides)413 b(a)g(w)-31
b(eak)6075 34122 y(expansion)370 b(in)f(the)h Fp(y)408
b Fr(direction)371 b(and)e(is)g(the)h(reason)f(of)g(the)h(uniqueness)f
(of)g(the)h(manifold.)8289 36080 y(>F)-92 b(rom)344 b(no)-31
b(w)346 b(on)f(w)-31 b(e)345 b(restrict)g(ourselv)-31
b(es)345 b(to)g(the)g(in)-31 b(terv)-61 b(al)346 b(\(0)p
Fp(;)184 b Fr(1\),)353 b(since)345 b(the)g(stable)h(man-)6075
37409 y(ifold)370 b(is)f(the)h(giv)-31 b(en)370 b(b)-31
b(y)370 b Fp(K)79 b Fr(\(\(0)p Fp(;)184 b Fr(1\)\).)8289
39367 y(In)339 b(order)h(to)h(c)-31 b(hec)g(k)341 b(that)g
Fp(K)20177 38965 y Fj(\267)21265 39367 y Fr(is)f(a)g(go)31
b(o)g(d)341 b(appro)-31 b(ximation)344 b(of)d(an)f(in)-31
b(v)-61 b(arian)-31 b(t)342 b(manifold)h(of)6075 40695
y Fp(F)563 b Fr(in)411 b([0)p Fp(;)184 b(t)10434 40861
y Fm(0)10932 40695 y Fr(],)421 b(w)-31 b(e)411 b(ha)-31
b(v)g(e)411 b(computed)g(the)g(distance)g(b)31 b(et)-31
b(w)g(een)411 b(the)f(curv)-31 b(es)410 b Fp(F)38237
40293 y Fo(m)39077 40695 y Fr(\()p Fp(K)40526 40293 y
Fj(\267)41274 40695 y Fr(\))h(and)f Fp(K)45327 40293
y Fj(\267)6075 42024 y Fr(for)369 b Fp(m)307 b Fr(=)h(1)p
Fp(;)184 b Fr(5)p Fp(;)g Fr(10)p Fp(;)g Fr(15)p Fp(;)g
Fr(20)377 b(b)-31 b(y)369 b(the)g(form)-31 b(ula)11797
44440 y(max)11151 45154 y Fo(i)p Fm(=0)p Fo(;)p Fj(\242\242\242)151
b Fo(;I)14688 44440 y Fr(dist\()p Fp(F)17773 43983 y
Fo(m)18614 44440 y Fr(\()p Fp(K)20063 43983 y Fj(\267)20811
44440 y Fr(\()p Fp(t)21641 44606 y Fo(i)22010 44440 y
Fr(\)\))p Fp(;)184 b(K)24380 43983 y Fj(\267)25130 44440
y Fr(\))615 b(with)h Fp(I)394 b Fr(=)308 b(1000)p Fp(;)802
b Fr(and)615 b Fp(t)37111 44606 y Fo(i)37787 44440 y
Fr(=)307 b Fp(t)39355 44606 y Fm(0)40081 43691 y Fp(i)p
39985 44185 574 45 v 39985 45199 a(I)40691 44440 y(:)2801
b Fr(\(5.3\))6075 46906 y(The)369 b(results)g(are)g(displa)-31
b(y)g(ed)371 b(in)e(the)h(next)f(table.)20414 49100 y
Fp(m)p 21985 49499 45 1332 v 1174 w Fr(dist\()p Fp(F)25645
48698 y Fo(m)26487 49100 y Fr(\()p Fp(K)27936 48698 y
Fj(\267)28684 49100 y Fr(\))p Fp(;)184 b(K)30624 48698
y Fj(\267)31373 49100 y Fr(\))p 19793 49543 12564 45
v 20346 50473 a(20)p 21985 50871 45 1329 v 1871 w(5)p
Fp(:)p Fr(801640)250 b Fk(\242)c Fr(10)29410 50071 y
Fj(\241)p Fm(12)p 19793 50916 12564 45 v 20346 51845
a Fr(15)p 21985 52244 45 1329 v 1871 w(1)p Fp(:)p Fr(389271)k
Fk(\242)c Fr(10)29410 51444 y Fj(\241)p Fm(13)p 19793
52288 12564 45 v 20346 53218 a Fr(10)p 21985 53617 45
1329 v 1871 w(3)p Fp(:)p Fr(369653)k Fk(\242)c Fr(10)29410
52816 y Fj(\241)p Fm(15)p 19793 53661 12564 45 v 20623
54591 a Fr(5)p 21985 54989 45 1329 v 2147 w(7)p Fp(:)p
Fr(892456)k Fk(\242)c Fr(10)29410 54189 y Fj(\241)p Fm(17)p
19793 55033 12564 45 v 20623 55963 a Fr(1)p 21985 56362
45 1329 v 2147 w(1)p Fp(:)p Fr(944933)k Fk(\242)c Fr(10)29410
55562 y Fj(\241)p Fm(18)p 19793 56406 12564 45 v 8289
58221 a Fr(Finally)360 b(w)-31 b(e)360 b(ha)-31 b(v)g(e)359
b(compared)h(the)f(parameterization)j(metho)31 b(d)360
b(with)g(the)f(graph)h(trans-)6075 59549 y(form)553 b(metho)31
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b(A)-26 b(ustin,)335 b(TX)g(78712-1082,)329 b(U.S.A,)8289
19556 y Fl(E-mail)340 b(addr)-49 b(ess)p Fv(,)313 b(R.)h(de)g(la)g(Lla)
-26 b(v)g(e:)419 b Fa(llave@math.utexas.edu)8289 22068
y Fu(Dep)-59 b(ar)g(t)g(ament)408 b(de)f(Ma)-59 b(tem)19124
21994 y(\265)19097 22068 y(atica)408 b(Aplicad)-20 b(a)407
b(IV,)f(Universit)-59 b(a)g(t)408 b(Polit)36283 21994
y(\265)36281 22068 y(ecnica)f(de)g(Ca)-59 b(t)g(aluny)g(a,)6075
23175 y(Ed-C3,)333 b(Jordi)h(Gir)-20 b(ona,)335 b(1-3,)e(08034)f(Bar)
-20 b(celona,)335 b(Sp)-59 b(ain,)8289 24912 y Fl(E-mail)340
b(addr)-49 b(ess)p Fv(,)313 b(P)-78 b(.)313 b(Mart)-105
b(\266)-366 b(\263n:)420 b Fa(martin@ma4.upc.edu)p eop
end
%%Trailer

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