%!PS-Adobe-2.0 %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software %%Title: qpa5.dvi %%Pages: 38 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMBX12 CMR12 CMR10 CMBX10 MSBM10 CMSY10 CMR8 CMMI12 %%+ EURM10 CMMI8 CMR6 CMSL12 CMR7 CMMI10 CMMI7 CMR5 CMSL10 CMEX10 CMTI12 %%+ MSBM7 CMMI6 CMSY8 CMSY7 CMSY6 CMMI5 CMTI10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips qpa5 %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2007.08.15:1003 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %%EndProcSet %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 50 /two put dup 51 /three put dup 58 /colon put dup 65 /A put dup 67 /C put dup 69 /E put dup 72 /H put dup 73 /I put dup 77 /M put dup 79 /O put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 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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 105 /i put readonly def /FontBBox{37 -250 1349 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: 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 84 /T put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 21 /greaterequal put dup 33 /arrowright put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EURM10 %!PS-AdobeFont-1.1: EURM10 2.1 %%CreationDate: 1992 Nov 20 17:36:34 % Euler fonts were designed by Hermann Zapf. % 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 (EURM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EURM10 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 10 /Omega put dup 29 /upsilon put readonly def /FontBBox{-32 -206 1060 722}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 14 /delta put dup 18 /theta put dup 23 /nu put dup 24 /xi put dup 27 /sigma put dup 34 /epsilon put dup 59 /comma put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 80 /summationtext put dup 82 /integraltext put dup 83 /uniontext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 98 /hatwide put dup 101 /tildewide put dup 102 /tildewider put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 114 /radicalbigg put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 34 /quotedblright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 46 /period put dup 49 /one put dup 51 /three put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 68 /D put dup 70 /F put dup 73 /I put dup 76 /L put dup 80 /P put dup 84 /T put dup 85 /U put dup 87 /W put dup 92 /quotedblleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 123 /endash put readonly def /FontBBox{-36 -251 1103 750}readonly def currentdict end currentfile eexec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All Rights Reserved) readonly def /FullName (CMSL10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 70 /F put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-62 -250 1123 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 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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 38 /subsetornoteql put dup 67 /C put dup 78 /N put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 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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 46 /period put dup 49 /one put dup 50 /two put readonly def /FontBBox{-341 -250 1304 965}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 25 /pi put dup 28 /tau put dup 33 /omega put dup 34 /epsilon put dup 59 /comma put dup 62 /greater put dup 87 /W put dup 100 /d put dup 104 /h put dup 105 /i put dup 107 /k put dup 109 /m put dup 110 /n put dup 116 /t put dup 118 /v put dup 121 /y put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 23 /nu put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 67 /C put dup 72 /H put dup 75 /K put dup 77 /M put dup 78 /N put dup 79 /O put dup 86 /V put dup 87 /W put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 107 /k put dup 110 /n put dup 112 /p put dup 113 /q put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 101 /e put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSL12 %!PS-AdobeFont-1.1: CMSL12 1.0 %%CreationDate: 1991 Aug 20 16:40:41 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSL12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 66 /B put dup 68 /D put dup 69 /E put dup 80 /P put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 123 /endash put readonly def /FontBBox{-56 -251 1102 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 4 /divide put dup 15 /bullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 39 /similarequal put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 75 /K put dup 76 /L put dup 84 /T put dup 86 /V put dup 87 /W put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 120 /section put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 24 /xi put dup 28 /tau put dup 33 /omega put dup 34 /epsilon put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 67 /C put dup 74 /J put dup 75 /K put dup 97 /a put dup 100 /d put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put dup 125 /weierstrass put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec 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cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 77 /M put dup 78 /N put dup 79 /O put dup 81 /Q put dup 86 /V put dup 87 /W put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 125 /weierstrass put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec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16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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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 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 101 /e put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 22 /macron put dup 34 /quotedblright put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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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 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 75 /K put dup 77 /M put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 18 /grave put dup 19 /acute put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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f(attractors)g(in)h(dissipativ)n(e)261 2893 y(mec)n(hanical)27 b(mo)r(dels,)i(pro)n(vided)e(external)h(parameters)f(are)h(tuned)h (with)g(the)g(frequency)f(of)h(the)g(mo-)261 2992 y(tion.)38 b(An)g(application)f(to)h(the)g(dissipativ)n(e)g(\\spin{orbit)e(mo)r (del")i(of)g(celestial)f(mec)n(hanics)g(\(whic)n(h)261 3092 y(actually)27 b(motiv)-5 b(ated)27 b(the)h(analysis)f(in)g(this)h (pap)r(er\))g(is)f(presen)n(ted.)17 3316 y FK(Keyw)m(ords:)38 b FN(Quasi{p)r(erio)r(dic)f(attractors,)g(small)h(divisors,)f (celestial)h(mec)n(hanics,)g(spin{orbit)g(problem,)g(nearly)17 3416 y(Hamiltonian)27 b(systems,)g(dissipativ)n(e)g(systems,)g(KAM)h (theory)-7 b(,)27 b(Nash{Moser)e(theorem.)17 3561 y FK(MSC2000)30 b(n)m(um)m(b)s(ers:)d FN(34C27,)f(34C30,)g(37C70,)f(37J40,)h(70F10,)g (70F15,)g(70F40,)g(70K43.)17 3993 y FJ(Con)l(ten)l(ts)17 4212 y FI(1)90 b(In)m(tro)s(duction)37 b(and)i(results)2384 b(2)163 4333 y FH(1.1)99 b(Dissipativ)m(e)35 b(nearly{in)m(tegrable)f (\015o)m(ws)g(on)e FG(R)22 b 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b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)93 b(35)3684 1247 y FI(38)17 1578 y FJ(1)161 b(In)l(tro)t(duction)51 b(and)i(results)17 1819 y FH(In)28 b(ph)m(ysical)h(examples,)h (Hamiltonian)e(dynamics,)i(t)m(ypically)-8 b(,)29 b(arises)g(through)f (dissipativ)m(e)i(systems)17 1940 y(with)35 b(v)m(ery)h(small)g (dissipation.)g(It)e(is)h(therefore)h(natural)e(to)h(ask)g(whic)m(h)h (part)f(of)f(the)h(Hamiltonian)17 2060 y(dynamics)41 b(smo)s(othly)f(p)s(ersists)h(when)f(the)g(dissipation)g(is)g(turned)g (on.)g(In)f(particular,)h(if)f(the)h(ref-)17 2181 y(erence)35 b(Hamiltonian)e(system)j(is)e(nearly{in)m(tegrable,)h(a)e(basic)h (question)h(is)f(to)f(discuss)j(the)d(fate)h(of)17 2301 y(quasi{p)s(erio)s(dic)k(tra)5 b(jectories,)38 b(whic)m(h,)g(as)f(KAM)g (theory)h(sho)m(ws)g(\(see,)g(e.g.,)f([1]\),)g(are)g(common)g(in)17 2421 y(the)c(purely)g(Hamiltonian)g(nearly{in)m(tegrable)h(regime.)17 2564 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FN(is)g(a)g(set)g(of)g(full)h(Leb)r(esgue)17 5173 y(measure.)1882 5645 y FH(3)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 17 548 a FH(The)39 b Fn(leitmotiv)f FH(is)h(that)f(KAM)g(quasi{p)s(erio)s(dic)i(solutions,)f(for)f FA(\021)j FF(6)p FH(=)c(0,)h(smo)s(othly)h(bifurcate)g(in)m(to)17 668 y(quasi{p)s(erio)s(dic)27 b(attractors)1071 632 y FE(5)1136 668 y FH(pro)m(vided)h(the)f(\\external)g(frequency")h Fz(\035)e FH(is)h(tuned)g(with)g(the)g(\\in)m(ternal)17 789 y(frequency")34 b FA(!)i FH(in)d(\(1.2\).)f(This)i(is)f(the)g(con)m (ten)m(t)h(of)e(the)h(\014rst)g(result:)17 1017 y FI(Theorem)38 b(1)49 b Fn(Fix)35 b FH(0)c FA(<)f(\024)h(<)g FH(1)f FF(\024)i FA(\034)47 b Fn(and)36 b FA(\021)1667 1032 y FE(0)1738 1017 y FA(>)30 b FH(0)p Fn(.)36 b(Then,)g(ther)-5 b(e)36 b(exists)h FH(0)30 b FA(<)h(")2990 1032 y FE(0)3059 1017 y FA(<)g FH(1)36 b Fn(such)h(that)f(for)17 1137 y(any)e FA(")28 b FF(2)g FH([0)p FA(;)17 b(")537 1152 y FE(0)576 1137 y FH(])p Fn(,)34 b(any)h FA(\021)d FF(2)c FA(I)1071 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b(in)h(whic)n(h)17 5103 y(the)h(\()p Ft(x;)14 b(t)p FN(\))28 b(motion)g(linearizes.)1882 5645 y FH(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 17 548 a FH(is)36 b(a)f Fv(quasi{p)s(erio)s(dic)i (attractor)d FH(for)h(the)h(dynamics)h(asso)s(ciated)g(to)e(\(1.1\))g (with)h Fz(\035)g FH(as)f(in)h(\(1.4\))f(and)17 668 y(that)c(the)g (dynamics)i(on)f FF(T)1012 683 y Fy(";\021)1106 668 y FH(\()p FA(!)t FH(\))e(is)i(analytically)g(conjugated)g(to)f(the)g (linear)h(\015o)m(w)g FA(\022)f FF(!)c FA(\022)c FH(+)c(\()p FA(!)t(t;)e(t)p FH(\);)17 789 y(compare,)33 b(also,)g(p)s(oin)m(t)f (\(i\))h(of)f(Remark)h(3)f(b)s(elo)m(w.)17 934 y(\(iv\))42 b(The)g(result)h(is)f(p)s(erturbativ)m(e)h(in)e FA(")h FH(but)f(it)h(is)g Fv(uniform)g FH(in)g FA(\021)t FH(.)f(It)h(is)g (particularly)g(noticeable)17 1054 y(the)34 b(smo)s(oth)f(dep)s (endence)j(of)d FA(u)1223 1069 y Fy(")1293 1054 y FH(on)g FA(\021)38 b FH(as)33 b FA(\021)g FF(!)c FH(0,)34 b(whic)m(h)h(sho)m (ws)g(that)e(the)h(in)m(v)-5 b(arian)m(t)34 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g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.) f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.) g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)-206 b(.)2392 1106 y(.)2391 1103 y(.)2390 1100 y(.)2390 1097 y(.)2389 1093 y(.)2388 1090 y(.)2387 1087 y(.)2386 1084 y(.)2385 1081 y(.)2384 1077 y(.)2384 1074 y(.)2383 1071 y(.)2382 1068 y(.)2381 1064 y(.)2380 1061 y(.)2379 1058 y(.)2379 1055 y(.)2378 1051 y(.)2377 1048 y(.)2376 1045 y(.)2375 1042 y(.)2374 1039 y(.)2373 1035 y(.)2373 1032 y(.)2372 1029 y(.)2371 1026 y(.)2370 1022 y(.)2369 1019 y(.)2368 1016 y(.)2367 1013 y(.)2367 1009 y(.)2366 1006 y(.)2365 1003 y(.)2364 1000 y(.)2363 996 y(.)2362 993 y(.)2362 990 y(.)2361 987 y(.)2360 984 y(.)2359 980 y(.)2358 977 y(.)2357 974 y(.)2356 971 y(.)2356 967 y(.)2355 964 y(.)2354 961 y(.)2353 958 y(.)2352 954 y(.)2351 951 y(.)2351 948 y(.)2350 945 y(.)2349 941 y(.)2348 938 y(.)2347 935 y(.)2346 932 y(.)2345 929 y(.)2345 925 y(.)2344 922 y(.)2343 919 y(.)2342 916 y(.)2341 912 y(.)2340 909 y(.)2340 906 y(.)2339 903 y(.)2338 899 y(.)2337 896 y(.)2336 893 y(.)2335 890 y(.)2334 887 y(.)2334 883 y(.)2333 880 y(.)2332 877 y(.)2331 874 y(.)2330 870 y(.)2329 867 y(.)2329 864 y(.)2328 861 y(.)2327 857 y(.)2326 854 y(.)2325 851 y(.)2324 848 y(.)2323 844 y(.)2323 841 y(.)2322 838 y(.)2321 835 y(.)2320 832 y(.)2319 828 y(.)2318 825 y(.)2318 822 y(.)2317 819 y(.)2316 815 y(.)2315 812 y(.)2314 809 y(.)2313 806 y(.)2312 802 y(.)2312 799 y(.)2311 796 y(.)2310 793 y(.)2309 789 y(.)2308 786 y(.)2307 783 y(.)2307 780 y(.)2306 777 y(.)2305 773 y(.)2304 770 y(.)2303 767 y(.)2302 764 y(.)2301 760 y(.)2301 757 y(.)2300 754 y(.)2299 751 y(.)2298 747 y(.)2297 744 y(.)2296 741 y(.)2296 738 y(.)2295 734 y(.)2294 731 y(.)2293 728 y(.)2292 725 y(.)2291 722 y(.)2290 718 y(.)2290 715 y(.)2289 712 y(.)2288 709 y(.)2151 1110 y(.)2152 1106 y(.)2154 1103 y(.)2155 1100 y(.)2156 1097 y(.)2157 1094 y(.)2158 1091 y(.)2159 1088 y(.)2160 1084 y(.)2161 1081 y(.)2162 1078 y(.)2163 1075 y(.)2164 1072 y(.)2165 1069 y(.)2166 1065 y(.)2168 1062 y(.)2169 1059 y(.)2170 1056 y(.)2171 1053 y(.)2172 1050 y(.)2173 1047 y(.)2174 1043 y(.)2175 1040 y(.)2176 1037 y(.)2177 1034 y(.)2178 1031 y(.)2179 1028 y(.)2180 1024 y(.)2181 1021 y(.)2183 1018 y(.)2184 1015 y(.)2185 1012 y(.)2186 1009 y(.)2187 1005 y(.)2188 1002 y(.)2189 999 y(.)2190 996 y(.)2191 993 y(.)2192 990 y(.)2193 987 y(.)2194 983 y(.)2195 980 y(.)2197 977 y(.)2198 974 y(.)2199 971 y(.)2200 968 y(.)2201 964 y(.)2202 961 y(.)2203 958 y(.)2204 955 y(.)2205 952 y(.)2206 949 y(.)2207 945 y(.)2208 942 y(.)2209 939 y(.)2211 936 y(.)2212 933 y(.)2213 930 y(.)2214 927 y(.)2215 923 y(.)2216 920 y(.)2217 917 y(.)2218 914 y(.)2219 911 y(.)2220 908 y(.)2221 904 y(.)2222 901 y(.)2223 898 y(.)2224 895 y(.)2226 892 y(.)2227 889 y(.)2228 885 y(.)2229 882 y(.)2230 879 y(.)2231 876 y(.)2232 873 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2034 4567 a(8)2117 4499 y(e)2160 4458 y FE(4)2222 4499 y FH(+)2354 4431 y(5)p 2330 4476 V 2330 4567 a(16)2437 4499 y(e)2480 4458 y FE(6)2520 4388 y Fm(\021)2749 4431 y FH(1)p 2589 4476 369 4 v 2589 4567 a(\(1)g FF(\000)h FH(e)2841 4538 y FE(2)2881 4567 y FH(\))2919 4538 y FE(6)3001 4499 y FH(;)1179 4761 y Fz(\035)1237 4776 y FE(e)1300 4761 y FH(:=)1445 4693 y FA(N)1523 4708 y FE(e)p 1441 4738 123 4 v 1441 4829 a Fz(\012)1528 4844 y FE(e)1601 4761 y FH(=)28 b(1)22 b(+)g(6e)1966 4720 y FE(2)2027 4761 y FH(+)2135 4693 y(3)p 2135 4738 49 4 v 2135 4829 a(8)2194 4761 y(e)2237 4720 y FE(4)2299 4761 y FH(+)g FA(O)s FH(\(e)2556 4720 y FE(6)2595 4761 y FH(\))32 b FA(:)855 b FH(\(1.10\))170 5063 y Fx(\017)49 b FH(0)27 b FA(<)h(A)f(<)h(B)33 b(<)27 b(C)40 b FH(are)32 b(the)h(principal)h (momen)m(ts)g(of)e(inertia)h(of)f(the)h(satellite;)1882 5645 y(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 170 548 a Fx(\017)49 b FA(\032)311 563 y FE(e)346 548 y FH(\()p FA(t)p FH(\))24 b(and)g(f)692 563 y FE(e)727 548 y FH(\()p FA(t)p FH(\))g(are,)g(resp)s(ectiv)m(ely) -8 b(,)26 b(the)f(\(normalized\))f(orbital)g(radius)g(and)g(the)g(true) g(anomaly)261 668 y(of)40 b(the)i(Keplerian)f(motion,)g(whic)m(h)i(\(b) s(ecause)f(of)f(the)g(assumed)i(normalizations\))f(are)f(2)p FA(\031)t FH({)261 789 y(p)s(erio)s(dic)35 b(function)h(of)f(time)h FA(t)p FH(.)g(The)g(explicit)h(expression)h(for)c FA(\032)2683 804 y FE(e)2755 789 y FH(and)h(f)2977 804 y FE(e)3048 789 y FH(ma)m(y)h(b)s(e)f(describ)s(ed)261 909 y(as)d(follo)m(ws.)i (Let)e(u)c(=)g(u)1142 924 y FE(e)1178 909 y FH(\()p FA(t)p FH(\))k(b)s(e)h(the)g(2)p FA(\031)t FH({p)s(erio)s(dic)f(function)h (obtained)h(b)m(y)f(in)m(v)m(erting)1112 1118 y FA(t)28 b FH(=)f(u)c FF(\000)f FH(e)17 b(sin)h(u)32 b FA(;)212 b FH(\(\\Kepler)2344 1077 y Fj(0)2368 1118 y FH(s)33 b(equation)2805 1066 y Fj(00)2847 1118 y FH(\))g(;)602 b(\(1.11\))261 1326 y(then)1179 1535 y FA(\032)1229 1550 y FE(e)1265 1535 y FH(\()p FA(t)p FH(\))83 b(=)g(1)22 b FF(\000)g FH(e)17 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h(frequencies)i FA(!)c FF(2)d(D)3105 1817 y Fy(\015)t(;\034)3240 1802 y FH(is)33 b(smo)s(oth,)f(as)17 1922 y(explained)37 b(brie\015y)f(in)g(Remark)g(7)f(b)s(elo)m(w.)h(The)g(r^)-49 b(ole)36 b(of)e(the)i(bifurcation)g(parameter)g(is)f(pla)m(y)m(ed)i(b)m (y)17 2042 y FA(\015)j FH(and)35 b(in)g(the)g(application)g(to)g(the)g (spin{orbit)h(problem)g(b)m(y)f(the)h(eccen)m(tricit)m(y)i(e)d(of)f (the)i(reference)17 2163 y(Keplerian)d(orbit.)17 2496 y FJ(2)161 b(Pro)t(ofs)17 2744 y FD(2.1)135 b(Preliminaries)17 2928 y FH(Here,)26 b(w)m(e)h(discuss)i(some)d(more)h(prop)s(erties)g (of)e(the)h(small)h(divisor)g(op)s(erators)f FA(D)2983 2943 y Fy(\021)3051 2928 y FH(and)g(\001)3315 2943 y Fy(\021)3383 2928 y FH(and)g(pro)m(v)m(e)17 3049 y(the)33 b(compatibilit)m(y)h(condition)f(\(1.16\).)17 3277 y FI(Lemma)39 b(1)49 b FH(\(i\))34 b Fn(If)h FA(u)27 b FF(2)h FA(C)1013 3241 y FE(2)1052 3277 y FH(\()p FG(T)1156 3241 y FE(2)1196 3277 y FH(\))p Fn(,)35 b(then)1490 3497 y FF(h)p FA(D)1610 3512 y Fy(\021)1651 3497 y FA(u)p FF(i)27 b FH(=)h FA(\021)t FF(h)p FA(u)p FF(i)33 b FA(;)1424 b FH(\(2.28\))1490 3642 y FF(h)p FA(u)1585 3669 y Fk(\022)1607 3684 y Fw(1)1649 3642 y FH(\001)1730 3657 y Fy(\021)1772 3642 y FA(u)p FF(i)27 b FH(=)h FA(\021)t(!)t FF(h)p FH(\()p FA(u)2248 3669 y Fk(\022)2270 3684 y Fw(1)2311 3642 y FH(\))2349 3601 y FE(2)2389 3642 y FF(i)34 b FA(:)1058 b FH(\(2.29\))17 3983 y(\(ii\))35 b Fn(If)f FA(v)t(;)17 b(V)49 b FF(2)28 b FA(C)656 3947 y FE(2)695 3983 y FH(\()p FG(T)799 3947 y FE(2)839 3983 y FH(\))35 b Fn(and)f FA(V)22 b FH(\()p FA(\022)s FH(\))27 b FF(6)p FH(=)h(0)34 b FF(8)i FA(\022)30 b FF(2)f FG(T)1845 3947 y FE(2)1919 3983 y Fn(then)1169 4230 y FA(V)22 b FH(\001)1329 4245 y Fy(\021)1371 4230 y FA(v)k FF(\000)c FA(v)t FH(\001)1675 4245 y Fy(\021)1717 4230 y FA(V)50 b FH(=)27 b FA(D)2008 4245 y Fy(\021)2050 4120 y Fm(\020)2110 4230 y FA(V)2188 4189 y FE(2)2228 4230 y FA(D)2309 4245 y FE(0)2348 4230 y FH(\()2410 4163 y FA(v)p 2396 4207 79 4 v 2396 4298 a(V)2484 4230 y FH(\))2522 4120 y Fm(\021)2617 4230 y FA(:)903 b FH(\(2.30\))17 4506 y(\(iii\))30 b Fn(L)-5 b(et)31 b FA(!)g FF(2)d(D)631 4521 y Fy(\024;\034)734 4506 y Fn(;)i(let)h FA(\030)h(>)27 b(\030)1153 4470 y Fj(0)1204 4506 y FF(\025)h FH(0)p Fn(;)i(let)g FA(\021)i FF(2)c FG(R)i Fn(and)g(let)g FA(p)g Fn(and)g FA(s)g Fn(b)-5 b(e)30 b(non)g(ne)-5 b(gative)29 b(inte)-5 b(gers.)30 b(Then,)17 4636 y(for)k(any)h FA(u)27 b FF(2)h(H)621 4588 y Fy(\030)620 4660 y FE(0)660 4636 y Fn(,)933 4872 y FA(D)1017 4831 y Fj(\000)p FE(1)1014 4897 y Fy(\021)1138 4872 y FH(:)g FA(u)44 b FF(2)h(H)1489 4825 y Fy(\030)1488 4897 y FE(0)1625 4872 y FF(\000)-17 b(!)1965 4778 y Fm(X)1882 4995 y Fy(n)p Fj(2)p Fl(Z)2016 4976 y Fw(2)2051 4995 y Fj(nf)p FE(0)p Fj(g)2209 4872 y FA(\025)2266 4831 y Fj(\000)p FE(1)2266 4897 y Fy(\021)r(;n)2370 4872 y FA(u)2426 4887 y Fy(n)2489 4872 y Ff(e)2542 4831 y Fy(in)p Fj(\001)p Fy(\022)2712 4872 y FF(2)45 b(H)2908 4825 y Fy(\030)2942 4802 y Fe(0)2907 4897 y FE(0)3003 4872 y FA(;)933 5136 y FF(k)p FA(D)1067 5095 y Fj(\000)p Fy(s)1064 5161 y(\021)1174 5136 y FA(@)1230 5095 y Fy(p)1225 5173 y Fk(\022)1247 5188 y Fw(1)1291 5136 y FA(u)p FF(k)1397 5151 y Fy(\030)1431 5132 y Fe(0)1484 5136 y FF(\024)28 b FA(\033)1644 5151 y Fy(p;s)1737 5136 y FH(\()p FA(\030)e FF(\000)d FA(\030)1992 5095 y Fj(0)2015 5136 y FH(\))17 b FF(k)p FA(u)p FF(k)2226 5151 y Fy(\030)2297 5136 y FA(;)1223 b FH(\(2.31\))1858 5645 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 17 548 a Fn(wher)-5 b(e)34 b FA(\025)349 563 y Fy(\021)r(;n)481 548 y FH(=)27 b FA(i)p FH(\()p FA(!)t(n)778 563 y FE(1)840 548 y FH(+)22 b FA(n)996 563 y FE(2)1035 548 y FH(\))g(+)h FA(\021)38 b Fn(\(c)-5 b(omp)g(ar)g(e)34 b FH(\(1.19\))p Fn(\))g(and)484 772 y FA(\033)539 787 y Fy(p;s)631 772 y FH(\()p FA(\016)t FH(\))27 b(:=)h FA(\033)967 787 y Fy(p;s)1059 772 y FH(\()p FA(\016)t FH(;)17 b FA(!)t(;)g(\021)t FH(\))26 b(:=)109 b(sup)1544 859 y Fy(n)p Fj(2)p Fl(Z)1678 840 y Fw(2)1713 859 y Fj(nf)p FE(0)p Fj(g)1870 661 y Fm(\020)1930 772 y FF(j)p FA(i)p FH(\()p FA(!)t(n)2152 787 y FE(1)2213 772 y FH(+)22 b FA(n)2369 787 y FE(2)2409 772 y FH(\))g(+)g FA(\021)t FF(j)2647 730 y Fj(\000)p Fy(s)2754 772 y FF(j)p FA(n)2840 787 y FE(1)2880 772 y FF(j)2908 730 y Fy(p)2980 772 y Ff(e)3033 730 y Fj(\000)p Fy(\016)r Fj(j)p Fy(n)p Fj(j)3208 661 y Fm(\021)3302 772 y FA(:)17 1035 y Fn(F)-7 b(urthermor)i(e,)1208 1195 y FA(\033)1263 1210 y Fy(p;s)1355 1195 y FH(\()p FA(\016)t FH(\))27 b FF(\024)1757 1127 y FH(1)p 1621 1172 323 4 v 1621 1263 a FA(\024)1677 1234 y Fy(s)1730 1263 y FA(\016)1777 1234 y Fy(s\034)8 b FE(+)p Fy(p)1970 1084 y Fm(\020)2039 1127 y FA(s\034)34 b FH(+)22 b FA(p)p 2039 1172 269 4 v 2155 1263 a Ff(e)2318 1084 y Fm(\021)2377 1106 y Fy(s\034)8 b FE(+)p Fy(p)2578 1195 y FA(:)942 b FH(\(2.32\))17 1396 y Fn(Final)5 b(ly,)34 b(if)g FA(p)28 b(>)g FH(0)p Fn(,)34 b FH(\(2.31\))g Fn(holds)g(for)h (any)g FA(u)27 b FF(2)h(H)1908 1360 y Fy(\030)1946 1396 y Fn(.)17 1538 y FH(\(iv\))37 b Fn(L)-5 b(et)36 b FA(B)g FH(:=)31 b FG(T)687 1501 y FE(2)750 1538 y FF(\002)24 b FA(I)894 1553 y FE(0)933 1538 y Fn(.)37 b(L)-5 b(et)37 b FH(\()p FA(\022)s(;)17 b(\021)t FH(\))30 b FF(2)h FA(B)36 b FF(!)30 b FA(u)p FH(\()p FA(\022)s FH(;)17 b FA(\021)t FH(\))35 b Fn(b)-5 b(elong)36 b(to)h FA(C)2560 1501 y Fj(1)2634 1538 y FH(\()p FA(B)5 b FH(\))37 b Fn(and)f FA(\022)d FF(!)e FA(u)p FH(\()p FA(\022)s FH(;)17 b FA(\021)t FH(\))35 b Fn(b)-5 b(elong)17 1658 y(to)35 b FF(H)219 1622 y Fy(\030)291 1658 y Fn(for)g(some)f FA(\030)e(>)c FH(0)35 b Fn(and)f(any)h FA(\021)c FF(2)d FA(I)1551 1673 y FE(0)1591 1658 y Fn(.)34 b(Assume)h(that)1432 1851 y FA(u)1488 1866 y FE(0)1527 1851 y FH(\(0\))27 b(:=)h FF(h)p FA(u)p FH(\()p FF(\001)p FH(;)17 b(0\))p FF(i)26 b FH(=)h(0)35 b FA(;)17 2045 y Fn(and)26 b(let)i FA(!)j FF(2)d(D)591 2060 y Fy(\024;\034)694 2045 y Fn(.)f(Then,)g FA(D)1112 2008 y Fj(\000)p FE(1)1109 2069 y Fy(\021)1206 2045 y FA(u)g FF(2)h FA(C)1460 2008 y Fj(1)1534 2045 y FH(\()p FA(B)5 b FH(\))p Fn(;)28 b FA(\022)i FF(!)e FA(D)2034 2008 y Fj(\000)p FE(1)2031 2069 y Fy(\021)2128 2045 y FA(u)p FH(\()p FA(\022)s FH(;)17 b FA(\021)t FH(\))26 b Fn(b)-5 b(elongs)26 b(to)i FF(H)2951 2008 y Fy(\030)2985 1985 y Fe(0)3038 2045 y Fn(for)f(any)g FH(0)h FF(\024)g FA(\030)3595 2008 y Fj(0)3645 2045 y FA(<)g(\030)17 2165 y Fn(and)34 b(any)h FA(\021)c FF(2)d FA(I)609 2180 y FE(0)649 2165 y Fn(.)34 b(F)-7 b(urthermor)i(e,)737 2376 y FA(D)821 2335 y Fj(\000)p FE(1)818 2400 y Fy(\021)915 2376 y FA(u)27 b FH(=)1111 2308 y FA(u)1167 2323 y FE(0)p 1111 2353 95 4 v 1133 2444 a FA(\021)1238 2376 y FH(+)22 b FA(D)1420 2335 y Fj(\000)p FE(1)1417 2400 y Fy(\021)1514 2376 y FH(\()p FA(u)g FF(\000)h FA(u)1786 2391 y FE(0)1825 2376 y FH(\))34 b FA(;)737 2581 y(D)821 2539 y Fj(\000)p FE(1)818 2605 y Fy(\021)915 2581 y FA(u)p FH(\()p FA(\022)s FH(;)17 b(0\))26 b(=)i FF(\000)p FA(@)1446 2596 y Fy(\021)1489 2581 y FA(u)1545 2596 y FE(0)1584 2581 y FH(\(0\))22 b(+)g FA(D)1913 2539 y Fj(\000)p FE(1)1910 2605 y(0)2007 2581 y FA(u)p FH(\()p FA(\022)s FH(;)17 b(0\))34 b FA(;)1206 b FH(\(2.33\))737 2793 y FF(k)p FA(D)871 2752 y Fj(\000)p FE(1)868 2818 y Fy(\021)964 2793 y FA(u)p FH(\()p FF(\001)p FH(;)17 b FA(\021)t FH(\))p FF(k)1270 2808 y Fy(\030)1304 2790 y Fe(0)1357 2793 y FF(\024)1462 2679 y Fm(\014)1462 2739 y(\014)1462 2798 y(\014)1505 2726 y FA(u)1561 2741 y FE(0)1600 2726 y FH(\()p FA(\021)t FH(\))p 1505 2771 223 4 v 1590 2862 a FA(\021)1737 2679 y Fm(\014)1737 2739 y(\014)1737 2798 y(\014)1793 2793 y FH(+)22 b FA(\033)1946 2808 y FE(0)p Fy(;)p FE(1)2040 2793 y FH(\()p FA(\030)27 b FF(\000)c FA(\030)2296 2752 y Fj(0)2318 2793 y FH(\))17 b FF(k)p FA(u)p FH(\()p FF(\001)p FA(;)g FF(\001)p FH(;)g FA(\021)t FH(\))j FF(\000)i FA(u)2926 2808 y FE(0)2965 2793 y FH(\()p FA(\021)t FH(\))p FF(k)3143 2823 y Fy(\030)3215 2793 y FA(:)17 3084 y FI(Pro)s(of)37 b FH(Equalit)m(y)d(\(2.28\))e(is)h (ob)m(vious.)17 3225 y(The)g(op)s(erator)f FA(D)694 3189 y FE(2)691 3250 y(0)733 3225 y FA(@)784 3252 y Fk(\022)806 3267 y Fw(1)882 3225 y FH(is)i(sk)m(ew{symmetric,)i(hence,)e(b)m(y)g (in)m(tegration)f(b)m(y)g(parts,)1113 3352 y Fm(D)1174 3463 y FA(u)1230 3489 y Fk(\022)1252 3504 y Fw(1)1295 3463 y FH(\001)1376 3478 y Fy(\021)1418 3463 y FA(u)1474 3352 y Fm(E)1618 3463 y FH(=)83 b FF(h)p FA(u)1872 3489 y Fk(\022)1894 3504 y Fw(1)1936 3463 y FA(D)2020 3421 y FE(2)2017 3487 y(0)2059 3463 y FA(u)p FF(i)22 b FH(+)g FA(\021)t FF(h)p FA(u)2421 3489 y Fk(\022)2443 3504 y Fw(1)2485 3463 y FA(D)2566 3478 y FE(0)2605 3463 y FA(u)p FF(i)1618 3644 y FH(=)83 b FF(h)p FA(D)1900 3603 y FE(2)1897 3669 y(0)1938 3644 y FA(u)1994 3671 y Fk(\022)2016 3686 y Fw(1)2059 3644 y FA(u)p FF(i)22 b FH(+)g FA(\021)t FF(h)p FA(u)2421 3671 y Fk(\022)2443 3686 y Fw(1)2485 3644 y FA(D)2566 3659 y FE(0)2605 3644 y FA(u)p FF(i)1618 3790 y FH(=)83 b FA(\021)t FF(h)p FA(u)1924 3816 y Fk(\022)1946 3831 y Fw(1)1988 3790 y FA(D)2069 3805 y FE(0)2108 3790 y FA(u)p FF(i)1618 3949 y FH(=)g FA(\021)t(!)t FF(h)p FH(\()p FA(u)2027 3975 y Fk(\022)2049 3990 y Fw(1)2090 3949 y FH(\))2128 3907 y FE(2)2167 3949 y FF(i)33 b FA(;)17 4142 y FH(whic)m(h)h(is)f(\(2.29\).)17 4262 y(\(ii\))f(Relation)h (\(2.30\))f(follo)m(ws)h(from)f(the)h(de\014nitions)h(of)e FA(D)2235 4277 y Fy(\021)2310 4262 y FH(and)h(\001)2581 4277 y Fy(\021)2623 4262 y FH(:)695 4455 y FA(V)21 b FH(\001)854 4470 y Fy(\021)896 4455 y FA(v)26 b FF(\000)d FA(v)t FH(\001)1201 4470 y Fy(\021)1243 4455 y FA(V)104 b FH(=)83 b FA(V)22 b(D)1723 4470 y Fy(\021)1765 4455 y FA(D)1846 4470 y FE(0)1885 4455 y FA(v)k FF(\000)d FA(v)t(D)2190 4470 y Fy(\021)2231 4455 y FA(D)2312 4470 y FE(0)2352 4455 y FA(V)1404 4601 y FH(=)83 b FA(V)22 b(D)1726 4560 y FE(2)1723 4625 y(0)1765 4601 y FA(v)k FF(\000)c FA(v)t(D)2072 4560 y FE(2)2069 4625 y(0)2111 4601 y FA(V)44 b FH(+)22 b FA(\021)t FH(\()p FA(V)f(D)2559 4616 y FE(0)2599 4601 y FA(v)26 b FF(\000)c FA(v)t(D)2903 4616 y FE(0)2942 4601 y FA(V)g FH(\))32 b(;)17 4794 y(on)g(the)h(other) g(hand)g(one)g(has)702 4987 y FA(D)783 5002 y Fy(\021)825 4987 y FH(\()p FA(V)942 4946 y FE(2)981 4987 y FA(D)1062 5002 y FE(0)1101 4987 y FH(\()1163 4920 y FA(v)p 1149 4964 79 4 v 1149 5056 a(V)1238 4987 y FH(\)\))83 b(=)g FA(D)1637 5002 y Fy(\021)1679 4987 y FH(\()p FA(V)21 b(D)1876 5002 y FE(0)1915 4987 y FA(v)26 b FF(\000)d FA(v)t(D)2220 5002 y FE(0)2259 4987 y FA(V)f FH(\))1397 5173 y(=)83 b FA(V)21 b(D)1718 5132 y FE(2)1715 5197 y(0)1757 5173 y FA(v)26 b FF(\000)d FA(v)t(D)2065 5132 y FE(2)2062 5197 y(0)2104 5173 y FA(V)43 b FH(+)22 b FA(\021)t FH(\()p FA(V)g(D)2552 5188 y FE(0)2591 5173 y FA(v)k FF(\000)d FA(v)t(D)2896 5188 y FE(0)2935 5173 y FA(V)e FH(\))33 b FA(;)1858 5645 y FH(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 17 548 a FH(from)32 b(whic)m(h)i(\(2.30\))e (follo)m(ws.)17 693 y(\(iii\))g(\(2.31\))g(follo)m(ws)h(immediately)i (from)d(the)h(de\014nitions)h(of)e FA(D)2429 657 y Fj(\000)p FE(1)2426 718 y Fy(\021)2556 693 y FH(and)h(of)f FA(\033)2912 708 y Fy(p;s)3004 693 y FH(:)388 951 y FF(k)p FA(@)494 910 y Fy(p)489 988 y Fk(\022)511 1003 y Fw(1)555 951 y FA(D)639 910 y Fj(\000)p Fy(s)636 976 y(\021)730 951 y FA(u)p FF(k)836 966 y Fy(\030)870 947 y Fe(0)980 951 y FH(=)1222 857 y Fm(X)1139 1074 y Fy(n)p Fj(2)p Fl(Z)1273 1055 y Fw(2)1308 1074 y Fj(nf)p FE(0)p Fj(g)1466 841 y Fm(\020)1525 951 y FF(j)p FA(n)1611 966 y FE(1)1651 951 y FF(j)1679 910 y Fy(p)1735 951 y FF(j)p FA(i)p FH(\()p FA(!)t(n)1957 966 y FE(1)2018 951 y FH(+)22 b FA(n)2174 966 y FE(2)2213 951 y FH(\))g(+)g FA(\021)t FF(j)2451 910 y Fj(\000)p Fy(s)2576 951 y Ff(e)2629 910 y Fj(\000j)p Fy(n)p Fj(j)p FE(\()p Fy(\030)s Fj(\000)p Fy(\030)2917 887 y Fe(0)2937 910 y FE(\))2969 841 y Fm(\021)3061 951 y FF(j)p FA(u)3145 966 y Fy(n)3191 951 y FF(j)33 b Ff(e)3305 910 y Fj(j)p Fy(n)p Fj(j)p Fy(\030)979 1212 y FF(\024)83 b FA(\033)1194 1227 y Fy(p;s)1287 1212 y FH(\()p FA(\030)26 b FF(\000)d FA(\030)1542 1171 y Fj(0)1564 1212 y FH(\))33 b FF(k)p FA(u)p FF(k)1791 1227 y Fy(\030)1860 1212 y FA(:)17 1430 y FH(Since)1235 1550 y FF(j)p FA(i)p FH(\()p FA(!)t(n)1457 1565 y FE(1)1518 1550 y FH(+)22 b FA(n)1674 1565 y FE(2)1713 1550 y FH(\))g(+)g FA(\021)t FF(j)27 b(\025)i(j)p FA(!)t(n)2235 1565 y FE(1)2295 1550 y FH(+)22 b FA(n)2451 1565 y FE(2)2491 1550 y FF(j)32 b FA(;)17 1724 y FH(\(2.32\))g(follo)m(ws)h(at)f(once)h(from)g(the)g(Diophan)m (tine)g(estimate)g(\(1.3\))f(and)h(from)f(the)h(ev)-5 b(aluation)1412 1972 y(sup)1421 2051 y Fy(x>)p FE(0)1576 1862 y Fm(\020)1635 1972 y FA(x)1690 1931 y Fy(a)1749 1972 y Ff(e)1802 1931 y Fj(\000)p Fy(x)1900 1862 y Fm(\021)1988 1972 y FH(=)2091 1862 y Fm(\020)2170 1905 y FA(a)p 2161 1949 70 4 v 2178 2041 a Ff(e)2240 1862 y Fm(\021)2299 1884 y Fy(a)2374 1972 y FA(;)17 2241 y FH(v)g(alid)32 b(for)g(an)m(y)i FA(a)28 b FF(\025)g FH(0.)17 2386 y(\(iv\):)33 b(The)g(F)-8 b(ourier)33 b(co)s(e\016cien)m(ts)1207 2653 y FA(u)1263 2668 y Fy(n)1309 2653 y FH(\()p FA(\021)t FH(\))28 b(=)1665 2586 y(1)p 1578 2630 223 4 v 1578 2721 a(\(2)p FA(\031)t FH(\))1762 2692 y FE(2)1811 2517 y Fm(Z)1866 2743 y Fl(T)1910 2724 y Fw(2)1949 2653 y FA(u)p FH(\()p FA(\022)s FH(;)17 b FA(\021)t FH(\))33 b Ff(e)2311 2612 y Fj(\000)p Fy(in)p Fj(\001)p Fy(\022)2507 2653 y FA(d\022)17 2927 y FH(are)f FA(C)256 2891 y Fj(1)331 2927 y FH(\()p FA(I)412 2942 y FE(0)451 2927 y FH(\))h(and,)g(b)m(y)g (assumption,)857 3163 y FF(k)o FA(u)p FH(\()p FF(\001)p FH(;)17 b FA(\021)t FH(\))p FF(k)1211 3192 y Fy(\030)1276 3163 y FH(:=)1419 3068 y Fm(X)1407 3285 y Fy(n)p Fj(2)p Fl(Z)1541 3267 y Fw(2)1592 3163 y FF(j)p FA(u)1676 3178 y Fy(n)1722 3163 y FH(\()p FA(\021)t FH(\))p FF(j)33 b Ff(e)1964 3122 y Fj(j)p Fy(n)p Fj(j)p Fy(\030)2111 3163 y FA(<)28 b FF(1)k FA(;)211 b FF(8)p FA(\021)32 b FF(2)c FA(I)2857 3178 y FE(0)2929 3163 y FA(:)17 3485 y FH(Therefore,)34 b(b)m(y)g(the)f(assumption)h(on)f FA(!)t FH(,)g(it)g(follo)m(ws)g(immediately)i(that)e FA(D)2837 3449 y Fj(\000)p FE(1)2834 3510 y Fy(\021)2931 3485 y FA(u)f FH(b)s(elongs)i(to)f FA(C)3567 3449 y Fj(1)3641 3485 y FH(\()p FA(B)5 b FH(\))17 3618 y(and)36 b(that)f FA(D)508 3582 y Fj(\000)p FE(1)505 3643 y Fy(\021)602 3618 y FA(u)p FH(\()p FF(\001)p FA(;)17 b FF(\001)p FH(\))34 b(b)s(elongs)i(to)g FF(H)1429 3582 y Fy(\030)1463 3558 y Fe(0)1525 3618 y FH(for)f(an)m(y)h FA(\021)h FF(6)p FH(=)c(0)i(and)h FA(\030)2383 3582 y Fj(0)2439 3618 y FA(<)d(\030)5 b FH(;)35 b(the)h(ev)-5 b(aluations)37 b(in)f(the)g(\014rst)17 3738 y(t)m(w)m(o)44 b(lines)g(of)f(\(2.33\))f (sho)m(w)j(that)e(the)g(same)h(is)g(true)g Fv(for)f(an)m(y)g FA(\021)50 b FF(2)c FA(I)2700 3753 y FE(0)2783 3738 y FH(\(and)d(an)m(y)h FA(\030)3264 3702 y Fj(0)3333 3738 y FA(<)h(\030)5 b FH(\).)43 b(Last)17 3859 y(estimate)33 b(in)g(\(2.33\))f(follo)m(ws)h(at)g(once)g(from)f(p)s(oin)m(t)h (\(iii\).)p 2285 3859 41 83 v 17 4184 a FI(Corollary)38 b(1)48 b Fn(L)-5 b(et)36 b FF(F)860 4199 y Fy(\021)936 4184 y Fn(b)-5 b(e)35 b(as)f(in)h FH(\(1.17\))p Fn(.)f(If)g FA(u)27 b FF(2)h FA(C)1970 4148 y FE(2)2010 4184 y FH(\()p FG(T)2114 4148 y FE(2)2153 4184 y FH(\))35 b Fn(and)g FA(\021)c FF(2)d FG(R)35 b Fn(then)1027 4322 y Fm(D)1088 4433 y FH(\(1)22 b(+)g FA(u)1351 4459 y Fk(\022)1373 4474 y Fw(1)1416 4433 y FH(\))p FF(F)1526 4448 y Fy(\021)1567 4433 y FH(\()p FA(u)p FH(;)17 b FA(\015)5 b FH(\))1799 4322 y Fm(E)1942 4433 y FH(=)83 b FA(\021)t(!)20 b FF(h)p FH(\()p FA(u)2367 4459 y Fk(\022)2389 4474 y Fw(1)2431 4433 y FH(\))2469 4392 y FE(2)2509 4433 y FF(i)i FH(+)g FA(\015)39 b(:)762 b FH(\(2.34\))17 4678 y Fn(In)34 b(p)-5 b(articular,)35 b(if)f FF(F)786 4693 y Fy(\021)828 4678 y FH(\()p FA(u)p FH(;)17 b FA(\015)5 b FH(\))27 b(=)g(0)p Fn(,)35 b(then)f FH(\(1.16\))h Fn(holds.)17 4954 y FI(Pro)s(of)i FH(First)32 b(observ)m(e)j(that)d(b)m(y)i(\(2.28\))1367 5173 y FF(h)p FH(\001)1487 5188 y Fy(\021)1529 5173 y FA(u)p FF(i)27 b FH(=)g FF(h)p FA(D)1874 5188 y FE(0)1913 5173 y FH(\()p FA(D)2032 5188 y Fy(\021)2074 5173 y FA(u)p FH(\))p FF(i)g FH(=)h(0)k FA(:)1858 5645 y FH(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 17 548 a FH(Observ)m(e)34 b(also)f(that)831 750 y FF(h)p FH(\(1)22 b(+)g FA(u)1133 777 y Fk(\022)1155 792 y Fw(1)1197 750 y FH(\))p FA(g)1282 765 y Fy(x)1326 750 y FH(\()p FA(\022)1409 765 y FE(1)1471 750 y FH(+)g FA(u;)17 b(\022)1714 765 y FE(2)1753 750 y FH(\))p FF(i)27 b FH(=)h FF(h)p FA(@)2051 777 y Fk(\022)2073 792 y Fw(1)2138 750 y FF(\001)22 b FA(g)t FH(\()p FA(\022)2322 765 y FE(1)2384 750 y FH(+)g FA(u;)17 b(\022)2627 765 y FE(2)2666 750 y FH(\))p FF(i)27 b FH(=)h(0)k FA(:)17 953 y FH(By)h(these)h (observ)-5 b(ations)33 b(and)g(\(2.29\),)245 1072 y Fm(D)306 1183 y FH(\(1)22 b(+)g FA(u)569 1209 y Fk(\022)591 1224 y Fw(1)633 1183 y FH(\))p FF(F)743 1198 y Fy(\021)785 1183 y FH(\()p FA(u)p FH(;)17 b FA(\015)5 b FH(\))1017 1072 y Fm(E)1160 1183 y FH(=)1319 1072 y Fm(D)1380 1183 y FH(\(1)22 b(+)g FA(u)1643 1209 y Fk(\022)1665 1224 y Fw(1)1707 1183 y FH(\)\001)1826 1198 y Fy(\021)1869 1183 y FA(u)f FH(+)h(\(1)g(+)g FA(u)2307 1209 y Fk(\022)2329 1224 y Fw(1)2372 1183 y FH(\))p FA(g)2457 1198 y Fy(x)2500 1183 y FH(\()p FA(\022)2583 1198 y FE(1)2645 1183 y FH(+)g FA(u;)17 b(\022)2888 1198 y FE(2)2927 1183 y FH(\))22 b(+)g(\(1)g(+)g FA(u)3348 1209 y Fk(\022)3370 1224 y Fw(1)3413 1183 y FH(\))p FA(\015)3507 1072 y Fm(E)1160 1395 y FH(=)1319 1285 y Fm(D)1380 1395 y FA(u)1436 1422 y Fk(\022)1458 1437 y Fw(1)1500 1395 y FH(\001)1581 1410 y Fy(\021)1624 1395 y FA(u)f FH(+)h FA(@)1850 1422 y Fk(\022)1872 1437 y Fw(1)1938 1395 y FF(\001)g FA(g)t FH(\()p FA(\022)2122 1410 y FE(1)2183 1395 y FH(+)g FA(u;)17 b(\022)2426 1410 y FE(2)2465 1395 y FH(\))2503 1285 y Fm(E)2586 1395 y FH(+)22 b FA(\015)1160 1608 y FH(=)1319 1497 y Fm(D)1380 1608 y FA(u)1436 1634 y Fk(\022)1458 1649 y Fw(1)1500 1608 y FH(\001)1581 1623 y Fy(\021)1624 1608 y FA(u)1680 1497 y Fm(E)1762 1608 y FH(+)g FA(\015)1160 1790 y FH(=)83 b FA(\021)t(!)20 b FF(h)p FH(\()p FA(u)1585 1816 y Fk(\022)1607 1831 y Fw(1)1649 1790 y FH(\))1687 1748 y FE(2)1726 1790 y FF(i)i FH(+)g FA(\015)38 b(;)17 2004 y FH(pro)m(ving)33 b(the)g(claims.)p 981 2004 41 83 v 17 2380 a FD(2.2)135 b(Newton)46 b(sc)l(heme)17 2565 y FH(W)-8 b(e)28 b(describ)s(e,)i(no)m(w,)f(the)g(Newton)g (\(KAM\))f(sc)m(heme,)j(on)d(whic)m(h)i(the)f(pro)s(of)e(of)h(Theorem)h (3)f(is)h(based.)17 2707 y(W)-8 b(e)33 b(start)f(with)h(a)g(simple)h (lemma)f(on)g(the)g(di\013eren)m(tial)g FA(d)p FF(F)42 b FH(of)32 b(the)h(op)s(erator)1126 2910 y FF(F)1198 2925 y Fy(\021)1240 2910 y FH(\()p FA(v)t FH(;)17 b FA(\014)6 b FH(\))26 b(=)i(\001)1683 2925 y Fy(\021)1725 2910 y FA(v)e FH(+)c FA(g)1943 2925 y Fy(x)1986 2910 y FH(\()p FA(\022)2069 2925 y FE(1)2131 2910 y FH(+)g FA(v)t(;)17 b(\022)2369 2925 y FE(2)2409 2910 y FH(\))22 b(+)g FA(\014)38 b(;)17 3112 y FH(whic)m(h)c(is)f(giv)m(en)h(b)m(y)751 3076 y FE(16)1284 3233 y FA(d)p FF(F)1407 3248 y Fy(\021)r(;v)1532 3233 y FH(=)28 b(\001)1717 3248 y Fy(\021)1781 3233 y FH(+)22 b FA(g)1926 3248 y Fy(xx)2009 3233 y FH(\()p FA(\022)2092 3248 y FE(1)2154 3233 y FH(+)g FA(v)t(;)17 b(\022)2392 3248 y FE(2)2432 3233 y FH(\))32 b FA(:)1018 b FH(\(2.35\))17 3441 y FI(Lemma)39 b(2)49 b Fn(L)-5 b(et)35 b FA(v)c FF(2)d FA(C)936 3405 y FE(2)976 3441 y FH(\()p FG(T)1080 3405 y FE(2)1120 3441 y FH(\))34 b Fn(and)h FA(\014)e FF(2)28 b FG(R)p Fn(.)35 b(Assume)g(that)1204 3643 y FA(V)50 b FH(:=)27 b(1)22 b(+)g FA(v)1657 3670 y Fk(\022)1679 3685 y Fw(1)1750 3643 y FF(6)p FH(=)28 b(0)35 b FA(;)215 b FF(8)36 b FA(\022)31 b FF(2)d FG(T)2507 3602 y FE(2)2581 3643 y FA(;)939 b FH(\(2.36\))17 3846 y Fn(and)34 b(de\014ne)627 4048 y FA(e)28 b FH(:=)g FA(e)p FH(\()p FA(\022)s FH(;)17 b FA(\021)t FH(\))27 b(:=)h FF(F)1326 4063 y Fy(\021)1367 4048 y FH(\()p FA(v)t FH(;)17 b FA(\014)6 b FH(\))34 b FA(;)627 4193 y(W)42 b FH(:=)27 b FA(V)970 4152 y FE(2)1044 4193 y FA(;)2476 b FH(\(2.37\))627 4366 y FA(A)700 4381 y Fy(\021)r(;v)826 4366 y FH(:)28 b FA(w)i FF(2)e FA(C)1152 4325 y FE(2)1192 4366 y FH(\()p FG(T)1296 4325 y FE(2)1336 4366 y FH(\))f FF(!)g FA(A)1601 4381 y Fy(\021)r(;v)1700 4366 y FA(w)j FH(:=)e FA(V)2009 4325 y Fj(\000)p FE(1)2103 4366 y FA(D)2184 4381 y Fy(\021)2226 4255 y Fm(\020)2286 4366 y FA(W)14 b(D)2473 4381 y FE(0)2529 4366 y FH(\()p FA(V)2645 4325 y Fj(\000)p FE(1)2739 4366 y FA(w)s FH(\))2850 4255 y Fm(\021)2937 4366 y FF(2)28 b FA(C)7 b FH(\()p FG(T)3212 4325 y FE(2)3252 4366 y FH(\))35 b FA(:)195 b FH(\(2.38\))17 4600 y Fn(Then,)34 b(for)g(every)h FA(w)30 b FF(2)e FA(C)985 4564 y FE(2)1025 4600 y FH(\()p FG(T)1129 4564 y FE(2)1168 4600 y FH(\))35 b Fn(and)g(any)f FA(\021)e FF(2)c FG(R)p Fn(,)1257 4803 y FA(d)p FF(F)1380 4818 y Fy(\021)r(;v)1478 4803 y FH(\()p FA(w)s FH(\))f(=)g FA(A)1830 4818 y Fy(\021)r(;v)1929 4803 y FA(w)d FH(+)e FA(V)2200 4761 y Fj(\000)p FE(1)2294 4803 y FA(e)2339 4829 y Fk(\022)2361 4844 y Fw(1)2421 4803 y FA(w)38 b(:)991 b FH(\(2.39\))p 17 4907 1512 4 v 96 5017 a Fu(16)166 5047 y FN(By)29 b(de\014nition)h Ft(d)p Fx(F)772 5059 y Fs(\021)r(;v)868 5047 y FN(\()p Ft(w)r FN(\))d(:=)37 b(lim)1134 5098 y Fs(\034)7 b Fh(!)p Fu(0)1295 4991 y Fx(F)1355 5003 y Fs(\021)1395 4991 y FN(\()p Ft(v)22 b FN(+)c Ft(\034)9 b(w)r FN(;)14 b Ft(\014)t FN(\))20 b Fx(\000)e(F)1961 5003 y Fs(\021)2001 4991 y FN(\()p Ft(v)s FN(;)c Ft(\014)t FN(\))p 1295 5028 903 4 v 1723 5104 a Ft(\034)2207 5047 y FN(;)30 b(notice)g(that)g(since)f Ft(\014)34 b FN(app)r(ears)28 b(as)h(an)h(additiv)n(e)17 5173 y(constan)n(t)d(in)g(the)h(expression)f(for)g Fx(F)1179 5185 y Fs(\021)1219 5173 y FN(,)g(the)h(di\013eren)n(tial)g(of)f Fx(F)1982 5185 y Fs(\021)2050 5173 y FN(is)h(indep)r(enden)n(t)g(of)g Ft(\014)t FN(.)1858 5645 y FH(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 17 548 a FH(Notice)33 b(that)f(from)h(\(2.30\)) e(it)i(follo)m(ws)g(that)1291 750 y FA(A)1364 765 y Fy(\021)r(;v)1462 750 y FA(w)d FH(=)e(\001)1747 765 y Fy(\021)1789 750 y FA(w)c FF(\000)f FA(V)2062 709 y Fj(\000)p FE(1)2156 750 y FH(\001)2237 765 y Fy(\021)2279 750 y FA(V)54 b(w)35 b(:)1025 b FH(\(2.40\))17 997 y FI(Pro)s(of)37 b FH(\(of)32 b(Lemma)h(2\))f(F)-8 b(rom)32 b(the)h(de\014nition)h(of)e FA(e)p FH(\()p FA(\022)s FH(\))h(it)f(follo)m(ws)i(that)1260 1198 y FA(e)1305 1225 y Fk(\022)1327 1240 y Fw(1)1398 1198 y FH(=)28 b(\001)1583 1213 y Fy(\021)1625 1198 y FA(V)44 b FH(+)22 b FA(g)1871 1213 y Fy(xx)1954 1198 y FH(\()p FA(\022)2037 1213 y FE(1)2099 1198 y FH(+)g FA(v)t(;)17 b(\022)2337 1213 y FE(2)2377 1198 y FH(\))p FA(V)54 b(:)994 b FH(\(2.41\))17 1400 y(Then,)34 b(b)m(y)f(\(2.35\),)f (\(2.40\))g(and)g(\(2.41\))g(one)h(sees)h(that)741 1602 y FA(d)p FF(F)864 1617 y Fy(\021)r(;v)962 1602 y FH(\()p FA(w)s FH(\))82 b(=)h(\001)1433 1617 y Fy(\021)1475 1602 y FA(w)25 b FH(+)d FA(g)1715 1617 y Fy(xx)1798 1602 y FH(\()p FA(\022)1881 1617 y FE(1)1943 1602 y FH(+)g FA(v)t(;)17 b(\022)2181 1617 y FE(2)2221 1602 y FH(\))p FA(w)1193 1778 y FH(=)83 b FA(A)1425 1793 y Fy(\021)r(;v)1524 1778 y FA(w)24 b FH(+)e FA(V)1795 1737 y Fj(\000)p FE(1)1889 1667 y Fm(\020)1949 1778 y FH(\001)2030 1793 y Fy(\021)2072 1778 y FA(V)43 b FH(+)22 b FA(g)2317 1793 y Fy(xx)2401 1778 y FH(\()p FA(\022)2484 1793 y FE(1)2546 1778 y FH(+)g FA(v)t(;)17 b(\022)2784 1793 y FE(2)2823 1778 y FH(\))p FA(V)2940 1667 y Fm(\021)2999 1778 y FA(w)1193 1960 y FH(=)83 b FA(A)1425 1975 y Fy(\021)r(;v)1524 1960 y FA(w)24 b FH(+)e FA(V)1795 1919 y Fj(\000)p FE(1)1889 1960 y FA(e)1934 1987 y Fk(\022)1956 2002 y Fw(1)2032 1960 y FA(w)35 b(:)p 2282 1960 41 83 v 17 2184 a FH(The)40 b(idea)g(of)e(a)h (Newton)h(sc)m(heme)i(is)e(to)f(start)g(with)h(an)f Fv(appro)m(ximate)h (solution)g FH(of)f(the)g(equation)17 2305 y FF(F)89 2320 y Fy(\021)130 2305 y FH(\()p FA(u)p FH(;)17 b FA(\015)5 b FH(\))27 b(=)h(0,)k(namely)-8 b(,)34 b(a)e(function)h FA(v)e FH(:)d FG(T)1621 2268 y FE(2)1689 2305 y FF(!)f FG(R)32 b Fv(and)h FH(a)f(n)m(um)m(b)s(er)i FA(\014)39 b FH(suc)m(h)34 b(that)985 2507 y FA(e)83 b FH(:=)g FF(F)1371 2522 y Fy(\021)1413 2507 y FH(\()p FA(v)t FH(;)17 b FA(\014)6 b FH(\))26 b(:=)i(\001)1883 2522 y Fy(\021)1925 2507 y FA(v)e FH(+)c FA(g)2143 2522 y Fy(x)2187 2507 y FH(\()p FA(\022)2270 2522 y FE(1)2332 2507 y FH(+)g FA(v)t(;)17 b(\022)2570 2522 y FE(2)2609 2507 y FH(\))22 b(+)g FA(\014)17 2708 y FH(is)33 b(small)g(and,)g(then,)g(to)f(\014nd)h(a)g (\\quadratically)g(b)s(etter)g(appro)m(ximation")1148 2910 y FA(v)1199 2869 y Fj(0)1249 2910 y FH(=)28 b FA(v)e FH(+)c FA(w)230 b FH(and)e FA(\014)2270 2869 y Fj(0)2320 2910 y FH(=)28 b FA(\014)g FH(+)2619 2884 y(^)2605 2910 y FA(\014)17 3112 y FH(satisfying)497 3314 y FA(w)i FH(=)e FA(O)776 3329 y FE(1)842 3314 y FH(=)960 3288 y(^)946 3314 y FA(\014)201 b FH(and)227 b FF(F)1658 3329 y Fy(\021)1700 3314 y FH(\()p FA(v)1789 3273 y Fj(0)1812 3314 y FH(;)17 b FA(\014)1917 3273 y Fj(0)1940 3314 y FH(\))27 b(=)h FA(O)2184 3329 y FE(2)2320 3314 y FH(where)99 b FA(O)2742 3329 y Fy(k)2812 3314 y FH(=)28 b FA(O)s FH(\()p FF(k)p FA(e)p FF(k)3177 3273 y Fy(k)3218 3314 y FH(\))33 b FA(;)17 3528 y FH(T)-8 b(o)32 b(\014nd)h FA(w)i FH(and)664 3501 y(^)650 3528 y FA(\014)6 b FH(,)32 b(w)m(e)i(de\014ne)67 3729 y FA(Q)144 3744 y FE(1)211 3729 y FH(:=)28 b FA(g)389 3744 y Fy(x)432 3729 y FH(\()p FA(\022)515 3744 y FE(1)575 3729 y FH(+)20 b FA(v)25 b FH(+)20 b FA(w)s(;)d(\022)1001 3744 y FE(2)1040 3729 y FH(\))j FF(\000)h FA(g)1243 3744 y Fy(x)1286 3729 y FH(\()p FA(\022)1369 3744 y FE(1)1430 3729 y FH(+)f FA(v)t(;)d(\022)1666 3744 y FE(2)1705 3729 y FH(\))j FF(\000)h FA(g)1908 3744 y Fy(xx)1992 3729 y FH(\()p FA(\022)2075 3744 y FE(1)2135 3729 y FH(+)f FA(v)t(;)d(\022)2371 3744 y FE(2)2410 3729 y FH(\))p FA(w)34 b(;)212 b(Q)2868 3744 y FE(2)2935 3729 y FH(:=)28 b FA(V)3145 3688 y Fj(\000)p FE(1)3239 3729 y FA(e)3284 3756 y Fk(\022)3306 3771 y Fw(1)3366 3729 y FA(w)34 b FH(;)50 b(\(2.42\))17 3931 y(notice)33 b(that)f(the)h FA(Q)760 3946 y Fy(i)789 3931 y FH('s)g(are)f(quadratic)i(in)e FA(w)j FH(and)e FA(e)p FH(.)g(Then,)h(b)m(y)f(Lemma)g(2,)g(one)g(has) 951 4133 y FF(F)1023 4148 y Fy(\021)1064 4133 y FH(\()p FA(v)1153 4092 y Fj(0)1176 4133 y FH(;)17 b FA(\014)1281 4092 y Fj(0)1304 4133 y FH(\))83 b(:=)g FF(F)1683 4148 y Fy(\021)1725 4133 y FH(\()p FA(v)25 b FH(+)d FA(w)s FH(;)17 b FA(\014)27 b FH(+)2244 4107 y(^)2230 4133 y FA(\014)6 b FH(\))1439 4290 y(=)96 b FF(F)1683 4305 y Fy(\021)1725 4290 y FH(\()p FA(v)t FH(;)17 b FA(\014)6 b FH(\))21 b(+)2090 4264 y(^)2076 4290 y FA(\014)27 b FH(+)22 b FA(d)p FF(F)2379 4305 y Fy(\021)r(;v)2477 4290 y FH(\()p FA(w)s FH(\))f(+)h FA(Q)2822 4305 y FE(1)1439 4446 y FH(=)96 b FA(e)22 b FH(+)1790 4420 y(^)1776 4446 y FA(\014)28 b FH(+)22 b FA(A)2030 4461 y Fy(\021)r(;v)2128 4446 y FA(w)j FH(+)d FA(Q)2398 4461 y FE(1)2460 4446 y FH(+)g FA(Q)2635 4461 y FE(2)2707 4446 y FA(:)813 b FH(\(2.43\))17 4648 y(Next)33 b(result)h(sho)m(ws)g(ho)m(w)f(to)f(solv) m(e)i(the)f(equation)1521 4850 y FA(e)23 b FH(+)1701 4824 y(^)1687 4850 y FA(\014)28 b FH(+)22 b FA(A)1941 4865 y Fy(\021)r(;v)2039 4850 y FA(w)30 b FH(=)e(0)17 5052 y(under)g(suitable)h(conditions)f(up)s(on)g(the)g(function)g FA(v)t FH(;)f(in)h(suc)m(h)h(equation)f FA(v)j FH(and)d(\(hence\))h FA(e)e FH(are)h(giv)m(en,)17 5173 y(while)33 b FA(w)i FH(and)580 5146 y(^)566 5173 y FA(\014)j FH(are)33 b(unkno)m(wns.)1858 5645 y(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 17 548 a FI(Prop)s(osition)37 b(1)49 b Fn(L)-5 b(et)40 b FA(g)t Fn(,)e FA(!)43 b Fn(and)c FA(I)1360 563 y FE(0)1439 548 y Fn(b)-5 b(e)39 b(as)g(in)g(The)-5 b(or)g(em)39 b(3;)g(let)h FA(B)h FH(:=)36 b FG(T)2807 512 y FE(2)2872 548 y FF(\002)26 b FA(I)3018 563 y FE(0)3058 548 y Fn(.)39 b(L)-5 b(et)40 b FA(v)g FF(2)d FA(C)3567 512 y Fj(1)3641 548 y FH(\()p FA(B)5 b FH(\))17 668 y Fn(and)34 b(let)h FA(\022)c FF(!)c FA(v)t FH(\()p FA(\022)s FH(;)17 b FA(b)p FH(\))34 b Fn(b)-5 b(elong)34 b(to)h FF(H)1336 621 y Fy(\030)1335 693 y FE(0)1409 668 y Fn(for)g(some)f FA(\030)e(>)c FH(0)34 b Fn(and)g(any)h FA(\021)c FF(2)d FA(I)2668 683 y FE(0)2708 668 y Fn(;)34 b(let)h FA(\014)e FF(2)c FA(C)3169 632 y Fj(1)3243 668 y FH(\()p FA(I)3324 683 y FE(0)3364 668 y FH(\))p Fn(.)34 b(Final)5 b(ly,)17 789 y(let)38 b FA(V)60 b Fn(and)38 b FA(W)52 b Fn(b)-5 b(e)39 b(de\014ne)-5 b(d,)37 b(r)-5 b(esp)g(e)g(ctively,)38 b(as)g(in)g FH(\(2.36\))g Fn(and)g FH(\(2.37\))g Fn(and)f(assume)h (that)h FA(V)22 b FH(\()p FA(\022)s FH(\))34 b FF(6)p FH(=)g(0)17 909 y Fn(for)g(any)h FA(\022)c FF(2)d FG(T)595 873 y FE(2)635 909 y Fn(,)35 b FF(h)p FA(W)845 873 y Fj(\000)p FE(1)938 909 y FF(i)28 b(6)p FH(=)f(0)35 b Fn(and)f(that)1381 1129 y FA(\030)26 b FH(+)c FF(k)p FA(v)1645 1156 y Fk(\022)1667 1171 y Fw(1)1710 1129 y FF(k)1760 1144 y Fy(\030)1826 1129 y FA(<)1939 1103 y FH(\026)1929 1129 y FA(\030)40 b(;)1508 b FH(\(2.44\))1381 1420 y FF(j)o FA(\021)21 b FF(h)p FA(W)1622 1379 y Fj(\000)p FE(1)1715 1420 y FA(D)1799 1379 y Fj(\000)p FE(1)1796 1444 y Fy(\021)1893 1420 y FA(v)1940 1446 y Fk(\022)1962 1461 y Fw(1)2006 1420 y FF(ij)27 b FA(<)g FF(jh)p FA(W)2376 1379 y Fj(\000)p FE(1)2470 1420 y FF(ij)34 b FA(:)949 b FH(\(2.45\))17 1640 y Fn(De\014ne:)587 1860 y FA(E)6 b FH(\()p FA(\022)s FH(\))83 b(:=)g FA(E)6 b FH(\()p FA(\022)s FH(;)17 b FA(\021)t FH(\))27 b(:=)g FA(V)22 b(e)28 b FH(:=)g FA(V)21 b FF(F)1946 1875 y Fy(\021)1988 1860 y FH(\()p FA(v)t FH(;)c FA(\014)6 b FH(\))33 b FA(;)p 711 1925 79 4 v 711 2005 a(E)89 b FH(:=)p 1058 1925 V 83 w FA(E)6 b FH(\()p FA(\021)t FH(\))27 b(:=)h FF(h)p FA(E)6 b FF(i)34 b FA(;)606 2135 y Fm(e)587 2160 y FA(E)6 b FH(\()p FA(\022)s FH(\))83 b(:=)1077 2135 y Fm(e)1058 2160 y FA(E)6 b FH(\()p FA(\022)s FH(;)17 b FA(\021)t FH(\))27 b(:=)g FA(E)i FF(\000)p 1713 2080 V 22 w FA(E)41 b(;)737 2396 y(a)84 b FH(:=)f FA(a)p FH(\()p FA(\021)t FH(\))27 b(:=)1405 2315 y FF(h)p FA(W)1550 2279 y Fj(\000)p FE(1)1643 2315 y FA(D)1727 2279 y Fj(\000)p FE(1)1724 2340 y Fy(\021)1841 2290 y Fm(e)1821 2315 y FA(E)7 b FF(i)21 b(\000)p 2060 2235 V 23 w FA(E)6 b FF(h)p FA(W)2283 2279 y Fj(\000)p FE(1)2377 2315 y FA(D)2461 2279 y Fj(\000)p FE(1)2458 2340 y Fy(\021)2555 2315 y FA(v)2602 2342 y Fk(\022)2624 2357 y Fw(1)2667 2315 y FF(i)p 1405 2373 1302 4 v 1547 2464 a(h)p FA(W)1692 2435 y Fj(\000)p FE(1)1785 2464 y FF(i)22 b FH(+)g FA(\021)t FF(h)p FA(W)2141 2435 y Fj(\000)p FE(1)2235 2464 y FA(D)2319 2435 y Fj(\000)p FE(1)2316 2489 y Fy(\021)2413 2464 y FA(v)2460 2491 y Fk(\022)2482 2506 y Fw(1)2525 2464 y FF(i)2751 2396 y FA(;)742 2686 y FH(^)728 2713 y FA(\014)89 b FH(:=)1072 2686 y(^)1058 2713 y FA(\014)5 b FH(\()p FA(\021)t FH(\))28 b(:=)f FF(\000)p FH(\()p 1519 2633 79 4 v FA(E)i FH(+)22 b FA(\021)t FH(\))17 b FA(a)27 b FH(=)g FF(\000)2093 2636 y(h)p FA(W)2238 2600 y Fj(\000)p FE(1)2333 2636 y FF(i)p 2371 2556 V -1 w FA(E)i FH(+)22 b FA(\021)e FF(h)p FA(W)2783 2600 y Fj(\000)p FE(1)2877 2636 y FA(D)2961 2600 y Fj(\000)p FE(1)2958 2661 y Fy(\021)3074 2611 y Fm(e)3055 2636 y FA(E)6 b FF(i)p 2093 2690 1078 4 v 2124 2781 a(h)p FA(W)2269 2752 y Fj(\000)p FE(1)2363 2781 y FF(i)22 b FH(+)g FA(\021)t FF(h)p FA(W)2719 2752 y Fj(\000)p FE(1)2812 2781 y FA(D)2896 2752 y Fj(\000)p FE(1)2893 2806 y Fy(\021)2990 2781 y FA(v)3037 2808 y Fk(\022)3059 2823 y Fw(1)3103 2781 y FF(i)3216 2713 y FA(;)553 2970 y(E)625 2985 y FE(1)665 2970 y FH(\()p FA(\022)s FH(\))83 b(:=)g FA(E)1130 2985 y FE(1)1169 2970 y FH(\()p FA(\022)s FH(;)17 b FA(b)p FH(\))28 b(:=)g FA(D)1621 2929 y Fj(\000)p FE(1)1618 2995 y Fy(\021)1734 2945 y Fm(e)1715 2970 y FA(E)g FF(\000)23 b FA(a)1983 2860 y Fm(\020)2042 2970 y FH(1)f(+)g FA(\021)t(D)2347 2929 y Fj(\000)p FE(1)2344 2995 y Fy(\021)2441 2970 y FA(v)2488 2997 y Fk(\022)2510 3012 y Fw(1)2553 2860 y Fm(\021)2635 2970 y FF(\000)p 2734 2890 79 4 v 22 w FA(E)h(D)2913 2929 y Fj(\000)p FE(1)2910 2995 y Fy(\021)3007 2970 y FA(v)3054 2997 y Fk(\022)3076 3012 y Fw(1)3154 2970 y FA(:)366 b FH(\(2.46\))17 3212 y Fn(Then:)206 3333 y(\(i\))41 b(al)5 b(l)41 b(functions)g(in)h FH(\(2.46\))e Fn(ar)-5 b(e)42 b FA(C)1600 3296 y Fj(1)1674 3333 y FH(\()p FA(B)5 b FH(\))42 b Fn(\(or)f FA(C)2120 3296 y Fj(1)2195 3333 y FH(\()p FA(I)2276 3348 y FE(0)2316 3333 y FH(\))g Fn(if)g(do)h(not)f (dep)-5 b(end)41 b(on)g FA(\022)k Fn(explicitly\))17 3453 y(and,)34 b(for)h(any)f FA(\030)626 3417 y Fj(0)677 3453 y FA(<)27 b(\030)5 b Fn(,)34 b(they)i(b)-5 b(elong)34 b(to)h FF(H)1598 3417 y Fy(\030)1632 3393 y Fe(0)1658 3453 y Fn(,)g(for)f(al)5 b(l)35 b FA(\021)c FF(2)d FA(I)2234 3468 y FE(0)2274 3453 y Fn(;)206 3573 y(\(ii\))34 b(the)h(fol)5 b(lowing)33 b(e)-5 b(qualities)35 b(hold:)1599 3793 y FA(D)1680 3808 y Fy(\021)1722 3793 y FA(E)1794 3808 y FE(1)1861 3793 y FH(=)27 b FA(E)i FH(+)2177 3767 y(^)2163 3793 y FA(\014)5 b(V)57 b(;)1183 b FH(\(2.47\))1599 3939 y FF(h)p FA(W)1744 3898 y Fj(\000)p FE(1)1837 3939 y FA(E)1909 3954 y FE(1)1949 3939 y FF(i)27 b FH(=)h(0)35 b FA(:)1317 b FH(\(2.48\))206 4159 y Fn(\(iii\))34 b(If)g(we)h (de\014ne)858 4379 y FH(^)-69 b FA(w)r FH(\()p FA(\022)s FH(;)17 b FA(\021)t FH(\))27 b(:=)h FF(\000)p FA(V)22 b(D)1528 4338 y Fj(\000)p FE(1)1525 4403 y(0)1622 4379 y FH(\()p FA(W)1766 4338 y Fj(\000)p FE(1)1860 4379 y FA(E)1932 4394 y FE(1)1971 4379 y FH(\))35 b FA(;)216 b(w)30 b FH(:=)48 b(^)-69 b FA(w)24 b FF(\000)f FA(V)f FF(h)d FH(^)-68 b FA(w)r FF(i)606 b FH(\(2.49\))17 4599 y Fn(then)31 b(these)h(functions)g(ar)-5 b(e)31 b FA(C)1134 4563 y Fj(1)1209 4599 y FH(\()p FA(B)5 b FH(\))32 b Fn(\(or)f FA(C)1635 4563 y Fj(1)1710 4599 y FH(\()p FA(I)1791 4614 y FE(0)1831 4599 y FH(\))g Fn(if)h(do)g(not)g(dep)-5 b(end)30 b(on)i FA(\022)j Fn(explicitly\))c(and,)h(for)f(any)17 4719 y FA(\030)65 4683 y Fj(0)115 4719 y FA(<)d(\030)5 b Fn(,)34 b(they)h(b)-5 b(elong)34 b(to)h FF(H)1036 4683 y Fy(\030)1070 4659 y Fe(0)1096 4719 y Fn(,)g(for)g(al)5 b(l)34 b FA(\021)e FF(2)c FA(I)1673 4734 y FE(0)1712 4719 y Fn(.)35 b(F)-7 b(urthermor)i(e,)34 b(the)h(fol)5 b(lowing)33 b(identities)h(hold:)1574 4939 y FF(h)p FA(w)s FF(i)26 b FH(=)i(0)34 b FA(;)1582 b FH(\(2.50\))1574 5084 y FA(e)22 b FH(+)1753 5058 y(^)1739 5084 y FA(\014)28 b FH(+)22 b FA(A)1993 5099 y Fy(\021)r(;v)2091 5084 y FA(w)30 b FH(=)e(0)34 b FA(:)1142 b FH(\(2.51\))1858 5645 y(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 17 548 a FH(F)-8 b(rom)25 b(this)i(statemen)m (t)869 512 y FE(17)945 548 y FH(,)f(the)h(de\014nitions)g(in)f (\(2.42\))g(and)g(from)f(\(2.43\),)h(there)h(follo)m(ws)f(immediately) 17 668 y(the)33 b(follo)m(wing)17 875 y FI(Corollary)38 b(2)48 b Fn(Under)31 b(the)g(assumptions)f(of)g(Pr)-5 b(op)g(osition)30 b(1,)h(if)2459 849 y FH(^)2445 875 y FA(\014)37 b Fn(and)30 b FA(w)j Fn(ar)-5 b(e)30 b(de\014ne)-5 b(d)30 b(as)g(in)h FH(\(2.46\))17 996 y Fn(and)j FH(\(2.49\))p Fn(,)g(then)1161 1116 y FA(e)1206 1075 y Fj(0)1257 1116 y FH(:=)28 b FF(F)1460 1131 y Fy(\021)1501 1116 y FH(\()p FA(v)e FH(+)c FA(w)s FH(;)17 b FA(\014)27 b FH(+)2021 1090 y(^)2007 1116 y FA(\014)5 b FH(\))28 b(=)g FA(Q)2314 1131 y FE(1)2375 1116 y FH(+)22 b FA(Q)2550 1131 y FE(2)2625 1116 y FA(:)17 1367 y FI(Pro)s(of)37 b FH(\(of)32 b(Prop)s(osition)h (1\))206 1510 y Fn(\(i\))p FH(:)39 b(The)h(regularit)m(y)g(prop)s (erties)g(of)f(the)g(functions)i(de\014ned)f(in)g(\(2.46\))e(follo)m(w) i(from)f(the)g(as-)17 1630 y(sumption)26 b(\(2.45\))f(and)h(p)s(oin)m (t)f(\(iv\))h(of)f(Lemma)h(1.)g(Notice)g(that,)f(in)h(view)h(of)e(p)s (oin)m(t)g(\(vi\))h(of)f(Remark)h(4,)17 1751 y(assumption)36 b(\(2.44\))f(implies)i(that)e FA(\022)h FF(!)c FA(g)1628 1766 y Fy(x)1671 1751 y FH(\()p FA(\022)1754 1766 y FE(1)1818 1751 y FH(+)24 b FA(v)t FH(\()p FA(\022)s FH(\))p FA(;)17 b(\022)2182 1766 y FE(2)2221 1751 y FH(\))36 b(b)s(elongs)g(to)f FF(H)2855 1715 y Fy(\030)2928 1751 y FH(\(and,)g(hence,)i(so)f(do)s(es) 17 1871 y FA(e)p FH(\()p FA(\022)145 1886 y FE(1)185 1871 y FH(\)\).)206 2014 y Fn(\(ii\))p FH(:)c(F)-8 b(rom)32 b(the)h(de\014nitions)h(of)e FA(E)1484 2029 y FE(1)1556 2014 y FH(and)1760 1987 y(^)1746 2014 y FA(\014)38 b FH(\(and)33 b(of)f(the)h(op)s(erator)f FA(D)2820 2029 y Fy(\021)2861 2014 y FH(\),)h(there)g(follo)m(ws)1132 2228 y FA(D)1213 2243 y Fy(\021)1255 2228 y FA(E)1327 2243 y FE(1)1449 2228 y FH(=)1628 2203 y Fm(e)1608 2228 y FA(E)28 b FF(\000)23 b FA(D)1889 2243 y Fy(\021)1948 2228 y FA(a)f FF(\000)h FA(\021)d(a)d(v)2304 2255 y Fk(\022)2326 2270 y Fw(1)2391 2228 y FF(\000)p 2491 2148 79 4 v 23 w FA(E)6 b(v)2616 2255 y Fk(\022)2638 2270 y Fw(1)1449 2391 y FH(=)83 b FA(E)28 b FF(\000)23 b FH(\()p 1846 2311 V FA(E)28 b FH(+)22 b FA(\021)t(a)p FH(\))p FA(V)1449 2536 y FH(=)83 b FA(E)28 b FH(+)1820 2510 y(^)1806 2536 y FA(\014)6 b(V)54 b(;)17 2737 y FH(pro)m(ving)33 b(\(2.47\).)f(Again,) g(from)h(the)g(de\014nitions)h(of)e FA(E)2043 2752 y FE(1)2115 2737 y FH(and)h FA(a)p FH(,)g(there)g(follo)m(ws)221 2966 y FF(h)p FA(W)366 2925 y Fj(\000)p FE(1)460 2966 y FA(E)532 2981 y FE(1)572 2966 y FF(i)27 b FH(=)h FF(h)p FA(W)887 2925 y Fj(\000)p FE(1)980 2966 y FA(D)1064 2925 y Fj(\000)p FE(1)1061 2991 y Fy(\021)1178 2941 y Fm(e)1158 2966 y FA(E)6 b FF(i)22 b(\000)h FA(a)1448 2855 y Fm(\020)1508 2966 y FF(h)p FA(W)1653 2925 y Fj(\000)p FE(1)1746 2966 y FF(i)f FH(+)g FA(\021)t FF(h)p FA(W)2102 2925 y Fj(\000)p FE(1)2196 2966 y FA(D)2280 2925 y Fj(\000)p FE(1)2277 2991 y Fy(\021)2374 2966 y FA(v)2421 2993 y Fk(\022)2443 3008 y Fw(1)2486 2966 y FF(i)2525 2855 y Fm(\021)2606 2966 y FF(\000)p 2706 2886 V 23 w FA(E)6 b FF(h)p FA(W)2929 2925 y Fj(\000)p FE(1)3023 2966 y FA(D)3107 2925 y Fj(\000)p FE(1)3104 2991 y Fy(\021)3201 2966 y FA(v)3248 2993 y Fk(\022)3270 3008 y Fw(1)3313 2966 y FF(i)28 b FH(=)f(0)32 b FA(;)17 3195 y FH(pro)m(ving)h(\(2.48\).)206 3337 y Fn(\(iii\))p FH(:)f(The)i(regularit)m(y)f(claim)h(is)f(handled)h(as)f (ab)s(o)m(v)m(e.)h(Equation)g(\(2.50\))e(follo)m(ws)h(at)g(once)g(from) 17 3458 y(the)26 b(de\014nition)h(of)e FA(w)j FH(\(noting)e(that)f FF(h)p FA(V)c FF(i)28 b FH(=)f(1\).)f(T)-8 b(o)26 b(c)m(hec)m(k)i (\(2.51\),)d(\014rst,)h(observ)m(e)i(that)d(the)h(de\014nition)17 3578 y(of)32 b FA(w)j FH(implies)f(that)966 3801 y FA(D)1047 3816 y FE(0)1086 3801 y FH(\()p FA(V)1202 3760 y Fj(\000)p FE(1)1297 3801 y FA(w)s FH(\))82 b(=)h FA(D)1730 3816 y FE(0)1769 3691 y Fm(\020)1851 3801 y FF(\000)23 b FA(D)2035 3760 y Fj(\000)p FE(1)2032 3826 y(0)2129 3801 y FH(\()p FA(W)2273 3760 y Fj(\000)p FE(1)2367 3801 y FA(E)2439 3816 y FE(1)2478 3801 y FH(\))f FF(\000)h(h)d FH(^)-69 b FA(w)r FF(i)2788 3691 y Fm(\021)1490 3983 y FH(=)83 b FF(\000)p FA(W)1832 3942 y Fj(\000)p FE(1)1926 3983 y FA(E)1998 3998 y FE(1)2071 3983 y FA(:)17 4184 y FH(Whence,)34 b(recalling)f(\(2.38\))f(and)h(\(2.47\))e(w)m(e)j(\014nd)734 4413 y FA(e)22 b FH(+)913 4387 y(^)899 4413 y FA(\014)28 b FH(+)22 b FA(A)1153 4428 y Fy(\021)r(;v)1251 4413 y FA(w)86 b FH(=)d FA(V)1644 4372 y Fj(\000)p FE(1)1738 4302 y Fm(h)1785 4413 y FA(V)22 b(e)g FH(+)2043 4387 y(^)2029 4413 y FA(\014)6 b(V)44 b FH(+)22 b FA(V)f(A)2440 4428 y Fy(\021)r(;v)2539 4413 y FA(w)2612 4302 y Fm(i)1407 4625 y FH(=)83 b FA(V)1644 4584 y Fj(\000)p FE(1)1738 4515 y Fm(h)1785 4625 y FA(E)29 b FH(+)1998 4599 y(^)1984 4625 y FA(\014)5 b(V)44 b FH(+)22 b FA(D)2324 4640 y Fy(\021)2366 4515 y Fm(\020)2425 4625 y FA(W)14 b(D)2612 4640 y FE(0)2652 4625 y FH(\()p FA(V)2768 4584 y Fj(\000)p FE(1)2862 4625 y FA(w)s FH(\))2973 4515 y Fm(\021i)1407 4838 y FH(=)83 b FA(V)1644 4797 y Fj(\000)p FE(1)1738 4727 y Fm(h)1785 4838 y FA(E)29 b FH(+)1998 4812 y(^)1984 4838 y FA(\014)5 b(V)44 b FF(\000)23 b FA(D)2326 4853 y Fy(\021)2368 4838 y FA(E)2440 4853 y FE(1)2479 4727 y Fm(i)1407 5019 y FH(=)83 b(0)32 b FA(:)p 1987 5019 41 83 v 17 5081 1512 4 v 96 5143 a Fu(17)166 5173 y FN(Compare,)27 b(esp)r(ecially)-7 b(,)27 b(\(2.51\).)1858 5645 y FH(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 17 548 a FI(Remark)38 b(6)48 b FH(\(i\))38 b(Notice)g(that)f FA(a;)1323 522 y FH(^)1309 548 y FA(\014)42 b FH(=)36 b FA(O)s FH(\()p FF(k)p FA(e)p FF(k)p FH(\))g(so)i(that,)f(also,)h FA(E)2521 563 y FE(1)2561 548 y FA(;)17 b(w)38 b FH(=)e FA(O)s FH(\()p FF(k)p FA(e)p FF(k)p FH(\).)g(Th)m(us,)k FA(Q)3544 563 y Fy(i)3572 548 y FA(;)17 b(e)3661 512 y Fj(0)3720 548 y FH(=)17 668 y FA(O)s FH(\()p FF(k)p FA(e)p FF(k)278 632 y FE(2)316 668 y FH(\).)33 b(Explicit)h(estimates)g(will)f(b)s(e)g(pro)m(vided)h (in)f(the)g(next)g(paragraph.)17 811 y(\(ii\))f(F)-8 b(rom)32 b(\(2.34\))g(it)h(follo)m(ws)g(that)p 844 930 79 4 v 844 1010 a FA(E)h FH(:=)27 b FF(h)p FA(V)22 b(e)p FF(i)27 b FH(=)h FF(h)p FH(\(1)21 b(+)i FA(v)1706 1037 y Fk(\022)1728 1052 y Fw(1)1771 1010 y FH(\))p FF(F)1881 1025 y Fy(\021)1922 1010 y FH(\()p FA(v)t FH(;)17 b FA(\014)6 b FH(\))p FF(i)27 b FH(=)g FA(\014)h FH(+)22 b FA(\021)e(!)t FF(h)p FH(\()p FA(v)2761 1037 y Fk(\022)2783 1052 y Fw(1)2826 1010 y FH(\))2864 969 y FE(2)2903 1010 y FF(i)p FA(:)578 b FH(\(2.52\))17 1250 y(\(iii\))32 b(In)h(the)g(conserv)-5 b(ativ)m(e)35 b(case)e(\()p FA(\021)f FH(=)27 b(0\))32 b(w)m(e)i(ha)m(v)m(e)g(that)p 2173 1170 V 32 w FA(E)g FH(=)28 b FA(\014)38 b FH(and)1341 1450 y FA(E)33 b FH(=)28 b FA(V)21 b FF(F)1700 1465 y FE(0)1739 1450 y FH(\()p FA(v)t FH(;)c FA(\014)6 b FH(\))27 b(=)g FA(E)2173 1465 y FE(0)2235 1450 y FH(+)22 b FA(\014)6 b(V)17 1649 y FH(with)33 b(\(compare)g(\(2.52\)\))1217 1770 y FA(E)1289 1785 y FE(0)1356 1770 y FH(:=)28 b FA(V)22 b FF(F)1638 1785 y FE(0)1677 1770 y FH(\()p FA(v)t FH(;)17 b(0\))31 b FA(;)212 b FF(h)p FA(E)2278 1785 y FE(0)2317 1770 y FF(i)28 b FH(=)f(0)33 b FA(:)17 1935 y FH(Th)m(us,)h(from)e(the)h (de\014nitions)h(giv)m(en)g(in)f(\(2.46\))f(and)g(\(2.49\),)g(there)h (follo)m(ws)h(that)1468 2120 y Fm(e)1449 2145 y FA(E)f FH(=)28 b FA(E)g FF(\000)23 b FA(\014)33 b FH(=)27 b FA(E)2121 2160 y FE(0)2183 2145 y FH(+)22 b FA(\014)6 b(v)2389 2172 y Fk(\022)2411 2187 y Fw(1)2487 2145 y FA(;)1449 2371 y(a)27 b FH(=)1641 2304 y FF(h)p FA(W)1786 2268 y Fj(\000)p FE(1)1880 2304 y FA(D)1964 2263 y Fj(\000)p FE(1)1961 2328 y(0)2058 2304 y FA(E)2130 2319 y FE(0)2169 2304 y FF(i)p 1641 2348 568 4 v 1786 2440 a(h)p FA(W)1931 2411 y Fj(\000)p FE(1)2024 2440 y FF(i)2251 2371 y FA(;)1463 2567 y FH(^)1449 2593 y FA(\014)33 b FH(=)27 b FF(\000)p FA(\014)39 b(;)1449 2739 y(E)1521 2754 y FE(1)1588 2739 y FH(=)27 b FA(D)1775 2697 y Fj(\000)p FE(1)1772 2763 y(0)1869 2739 y FA(E)1941 2754 y FE(0)2003 2739 y FF(\000)c FA(a)33 b(;)17 2938 y FH(and)f FA(w)j FH(as)e(in)g(\(2.49\).)f(Th)m (us,)i FA(w)h Fv(is)e(indep)s(enden)m(t)i(of)d FA(\014)38 b Fv(and)32 b(so)h(is)g(the)g(new)h(appro)m(ximate)f(solution)1515 3138 y FA(e)1560 3097 y Fj(0)1612 3138 y FH(=)27 b FF(F)1787 3153 y FE(0)1826 3138 y FH(\()p FA(v)f FH(+)c FA(w)s FH(;)17 b(0\))31 b FA(;)17 3338 y FH(\(recall)f(that)f FA(\014)581 3302 y Fj(0)632 3338 y FH(=)e FA(\014)22 b FH(+)918 3312 y(^)904 3338 y FA(\014)33 b FH(=)28 b(0)h(in)h(this)h (case\).)f(This)h(sho)m(ws)g(that)f(in)g(the)g(conserv)-5 b(ativ)m(e)31 b(case)g Fv(one)f(can)17 3458 y(alw)m(a)m(ys)k(tak)m(e)f FA(\014)g FH(=)28 b(0.)17 3744 y FD(2.3)135 b(KAM)44 b(Estimates)17 3929 y FH(Here)39 b(w)m(e)h(collect)g(the)g(main)f (estimates)h(for)f(the)g(KAM)g(algorithm)g(describ)s(ed)i(in)e(Prop)s (osition)g(1.)17 4049 y(W)-8 b(e)33 b(start)f(with)h(the)g(follo)m (wing)17 4254 y FI(De\014nition)38 b(1)49 b Fn(L)-5 b(et)823 4228 y FH(\026)813 4254 y FA(\030)32 b(>)27 b(\030)33 b(>)27 b FH(0)p Fn(.)35 b(Then,)162 4460 y FF(\017)49 b(V)322 4475 y Fy(\030)404 4460 y Fn(denotes)43 b(the)h(set)h(of)e (functions)h FA(v)49 b FF(2)c FA(C)1937 4424 y Fj(1)2012 4460 y FH(\()p FG(T)2116 4424 y FE(2)2185 4460 y FF(\002)29 b FA(I)2334 4475 y FE(0)2374 4460 y FH(\))44 b Fn(such)g(that,)g(for)g (al)5 b(l)44 b FA(\021)49 b FF(2)c FA(I)3490 4475 y FE(0)3529 4460 y Fn(,)g FA(\022)j FF(!)261 4589 y FA(v)t FH(\()p FA(\022)s FH(;)17 b FA(\021)t FH(\))26 b FF(2)j(H)738 4542 y Fy(\030)737 4614 y FE(0)776 4589 y Fn(.)162 4802 y FF(\017)49 b(V)337 4738 y FE(\026)330 4755 y Fy(\030)322 4830 y(\030)403 4802 y Fn(denotes)35 b(the)h(subset)g(of)f FA(v)e FF(2)c(V)1559 4817 y Fy(\030)1633 4802 y Fn(verifying,)35 b(for)g(al)5 b(l)35 b FA(\021)e FF(2)d FA(I)2582 4817 y FE(0)2621 4802 y Fn(,)36 b FA(V)21 b FH(\()p FA(\022)s FH(\))29 b(:=)g(1)23 b(+)g FA(v)3268 4829 y Fk(\022)3290 4844 y Fw(1)3333 4802 y FH(\()p FA(\022)s FH(\))29 b FF(6)p FH(=)g(0)35 b Fn(for)261 4931 y(al)5 b(l)34 b FA(\022)d FF(2)d FG(T)636 4895 y FE(2)676 4931 y Fn(,)35 b FF(h)p FA(W)886 4895 y Fj(\000)p FE(1)979 4931 y FF(i)28 b(6)p FH(=)f(0)35 b Fn(wher)-5 b(e)34 b FA(W)41 b FH(:=)28 b FA(V)1851 4895 y FE(2)1925 4931 y Fn(and)982 5131 y FF(k)p FA(v)1079 5158 y Fk(\022)1101 5173 y Fw(1)1144 5131 y FF(k)1194 5146 y Fy(\030)1259 5131 y FA(<)1373 5105 y FH(\026)1363 5131 y FA(\030)e FF(\000)d FA(\030)39 b(;)216 b FF(j)p FA(\021)20 b FF(h)p FA(W)2098 5090 y Fj(\000)p FE(1)2192 5131 y FA(D)2276 5090 y Fj(\000)p FE(1)2273 5156 y Fy(\021)2370 5131 y FA(v)2417 5158 y Fk(\022)2439 5173 y Fw(1)2482 5131 y FF(ij)28 b FA(<)f FF(jh)p FA(W)2853 5090 y Fj(\000)p FE(1)2946 5131 y FF(ij)35 b FA(:)1858 5645 y FH(20)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 162 548 a FF(\017)49 b(W)359 563 y Fy(\030)432 548 y Fn(denotes)34 b(the)h(set)g FF(V)1158 563 y Fy(\030)1218 548 y FF(\002)22 b FA(C)1394 512 y Fj(1)1469 548 y FH(\()p FA(I)1550 563 y FE(0)1589 548 y FH(\))p Fn(.)162 770 y FF(\017)49 b(W)374 705 y FE(\026)367 723 y Fy(\030)359 798 y(\030)440 770 y Fn(denotes)34 b(the)h(set)g FF(V)1181 705 y FE(\026)1174 723 y Fy(\030)1166 798 y(\030)1234 770 y FF(\002)23 b FA(C)1411 734 y Fj(1)1485 770 y FH(\()p FA(I)1566 785 y FE(0)1606 770 y FH(\))p Fn(.)17 1030 y(If)34 b FH(\()p FA(v)t(;)17 b(\014)6 b FH(\))27 b FF(2)h(W)585 965 y FE(\026)578 983 y Fy(\030)570 1057 y(\030)616 1030 y Fn(,)35 b(we)f(de\014ne)1122 1274 y FF(K)q FH(\()p FA(v)t(;)17 b(\014)6 b FH(\))27 b(:=)h(\()p FA(v)1678 1233 y Fj(0)1701 1274 y FA(;)17 b(\014)1806 1233 y Fj(0)1828 1274 y FH(\))28 b(:=)g(\()p FA(v)d FH(+)d FA(w)s(;)17 b(\014)27 b FH(+)2544 1248 y(^)2530 1274 y FA(\014)6 b FH(\))35 b FA(;)17 1494 y Fn(wher)-5 b(e)34 b FA(w)j Fn(and)603 1468 y FH(^)588 1494 y FA(\014)k Fn(ar)-5 b(e)35 b(de\014ne)-5 b(d)33 b(as)i(in)f(Pr)-5 b(op)g(osition)35 b(1.)17 1722 y FH(Th)m(us,)f(b)m(y)f(Prop)s(osition)g (1,)987 1952 y FF(K)c FH(:)e FF(W)1259 1888 y FE(\026)1252 1905 y Fy(\030)1244 1980 y(\030)1319 1952 y FF(!)g(W)1544 1967 y Fy(\030)1578 1948 y Fe(0)1637 1952 y FA(;)407 b FF(8)2169 1926 y FH(\026)2159 1952 y FA(\030)32 b(>)27 b(\030)32 b(>)c(\030)2564 1911 y Fj(0)2614 1952 y FA(>)g FH(0)k FA(:)17 2212 y FI(Lemma)39 b(3)49 b Fn(\(i\))34 b(L)-5 b(et)35 b FH(\()p FA(v)t(;)17 b(\014)6 b FH(\))27 b FF(2)h(W)1297 2147 y FE(\026)1290 2165 y Fy(\030)1282 2240 y(\030)1329 2212 y Fn(;)35 b(let)g FH(\()p FA(w)s(;)1699 2186 y FH(^)1686 2212 y FA(\014)t FH(\))28 b(:=)g FF(K)q FH(\()p FA(v)t(;)17 b(\014)6 b FH(\))21 b FF(\000)i FH(\()p FA(v)t(;)17 b(\014)6 b FH(\))34 b Fn(\(c)-5 b(omp)g(ar)g(e)34 b(De\014nition)f(1\))i(and)17 2332 y(de\014ne)715 2519 y FA(V)49 b FH(:=)28 b(1)22 b(+)g FA(v)1168 2546 y Fk(\022)1190 2561 y Fw(1)1268 2519 y FA(;)216 b(W)42 b FH(:=)27 b FA(V)1854 2478 y FE(2)1928 2519 y FA(;)216 b(\032)28 b FH(:=)g FA(\021)2458 2439 y FF(h)p FA(W)2603 2403 y Fj(\000)p FE(1)2697 2439 y FA(D)2781 2403 y Fj(\000)p FE(1)2778 2464 y Fy(\021)2875 2439 y FA(v)2922 2466 y Fk(\022)2944 2481 y Fw(1)2987 2439 y FF(i)p 2458 2497 568 4 v 2603 2588 a(h)p FA(W)2748 2559 y Fj(\000)p FE(1)2842 2588 y FF(i)3071 2519 y FA(:)449 b FH(\(2.53\))17 2744 y Fn(Assume)35 b(that)g(ther)-5 b(e)35 b(exist)f FA(\027)6 b Fn(,)36 b FA(\013)f Fn(such)g(that,)g(for)g(any)f FA(\021)e FF(2)c FA(I)2276 2759 y FE(0)2315 2744 y Fn(,)1559 2964 y FF(k)p FA(v)1656 2991 y Fk(\022)1678 3006 y Fw(1)1721 2964 y FF(k)1771 2979 y Fy(\030)1836 2964 y FA(<)g(\027)34 b(<)2135 2938 y FH(\026)2125 2964 y FA(\030)27 b FF(\000)22 b FA(\030)40 b(;)1143 b FH(\(2.54\))1559 3109 y FF(j)p FA(\032)p FF(j)27 b FA(<)g(\013)i(<)e FH(1)35 b FA(:)1447 b FH(\(2.55\))17 3329 y Fn(Then,)43 b(ther)-5 b(e)44 b(exists)f FA(s)h FH(:=)h FA(s)p FH(\()p FA(\034)11 b FH(\))44 b FA(>)h FH(1)e Fn(and)1662 3293 y FE(18)1780 3329 y FA(c)i FH(:=)f FA(c)p FH(\()p FA(\024;)17 b(\034)6 b(;)17 b(M)5 b(;)17 b(\027)q(;)g(\013)q(;)g(\021)2677 3344 y FE(0)2715 3329 y FH(\))45 b FA(>)f FH(1)g Fn(such)f(that,)i(for) e(any)17 3450 y FH(0)27 b FA(<)h(\016)j(<)d(\030)39 b Fn(and)34 b(any)h FA(\021)d FF(2)c FA(I)1050 3465 y FE(0)1089 3450 y Fn(,)35 b(the)g(fol)5 b(lowing)33 b(estimates)i(hold:)1018 3670 y FF(j)1060 3644 y FH(^)1046 3670 y FA(\014)5 b FF(j)17 b FA(;)33 b FF(k)p FA(w)1331 3697 y Fk(\022)1353 3712 y Fw(1)1395 3670 y FF(k)1445 3685 y Fy(\030)s Fj(\000)p Fy(\016)1588 3670 y FA(;)g FF(k)p FA(D)1782 3629 y Fj(\000)p FE(1)1779 3695 y Fy(\021)1876 3670 y FA(w)1946 3697 y Fk(\022)1968 3712 y Fw(1)2011 3670 y FF(k)2061 3685 y Fy(\030)s Fj(\000)p Fy(\016)2232 3670 y FF(\024)28 b FA(c)17 b(\016)2443 3629 y Fj(\000)p Fy(s)2551 3670 y FF(k)p FA(e)p FF(k)2696 3685 y Fy(\030)2768 3670 y FA(:)752 b FH(\(2.56\))17 3890 y Fn(\(ii\))36 b(L)-5 b(et)37 b FA(W)469 3854 y Fj(0)529 3890 y Fn(and)f FA(\032)770 3854 y Fj(0)831 3890 y Fn(b)-5 b(e)37 b(de\014ne)-5 b(d)35 b(as,)i(r)-5 b(esp)g(e)g(ctively,)36 b FA(W)51 b Fn(and)36 b FA(\032)h Fn(with)g FA(v)j Fn(r)-5 b(eplac)g(e)g(d)36 b(by)h FA(v)3255 3854 y Fj(0)3310 3890 y FH(:=)31 b(\()p FA(v)c FH(+)d FA(w)s FH(\))p Fn(.)17 4010 y(If)1416 4131 y FF(k)p FA(w)1536 4157 y Fk(\022)1558 4172 y Fw(1)1600 4131 y FF(k)1650 4146 y Fy(\030)s Fj(\000)p Fy(\016)1804 4131 y FF(\024)k FA(\027)h FF(\000)23 b(k)p FA(v)2183 4157 y Fk(\022)2205 4172 y Fw(1)2248 4131 y FF(k)2298 4146 y Fy(\030)2370 4131 y FA(;)1150 b FH(\(2.57\))17 4305 y Fn(then)1320 4425 y FF(k)p FA(W)1476 4384 y Fj(0)1521 4425 y FF(\000)22 b FA(W)14 b FF(k)1776 4440 y Fy(\030)s Fj(\000)p Fy(\016)1930 4425 y FF(\024)28 b FA(c)17 b(\016)2141 4384 y Fj(\000)p Fy(s)2249 4425 y FF(k)p FA(e)p FF(k)2394 4440 y Fy(\030)2466 4425 y FH(;)1054 b(\(2.58\))17 4600 y FA(e)62 4564 y Fj(0)113 4600 y FH(:=)28 b FF(F)316 4615 y Fy(\021)357 4600 y FH(\()p FA(v)446 4564 y Fj(0)469 4600 y FH(;)17 b FA(\014)574 4564 y Fj(0)597 4600 y FH(\))27 b(:=)h FF(F)865 4615 y Fy(\021)906 4600 y FH(\()p FF(K)q FH(\()p FA(v)t(;)17 b(\014)6 b FH(\)\))34 b Fn(b)-5 b(elongs)34 b(to)h FF(H)1861 4564 y Fy(\030)s Fj(\000)p Fy(\016)1988 4600 y Fn(,)f(for)h(any)g FA(\021)c FF(2)d FA(I)2611 4615 y FE(0)2651 4600 y Fn(,)35 b(and)f(satis\014es)1463 4820 y FF(k)p FA(e)1558 4779 y Fj(0)1581 4820 y FF(k)1631 4835 y Fy(\030)s Fj(\000)p Fy(\016)1785 4820 y FF(\024)28 b FA(c)17 b(\016)1996 4779 y Fj(\000)p Fy(s)2104 4820 y FF(k)p FA(e)p FF(k)2249 4779 y FE(2)2249 4844 y Fy(\030)2323 4820 y FA(:)1197 b FH(\(2.59\))p 17 4925 1512 4 v 96 4987 a Fu(18)166 5017 y FN(Recall)27 b(the)h(de\014nition)g(of)g Ft(M)36 b FN(in)28 b(\(1.20\).)1858 5645 y FH(21)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 17 548 a Fn(Final)5 b(ly,)34 b(if)1339 722 y FF(j)p FA(W)1473 681 y Fj(0)1518 722 y FF(\000)22 b FA(W)14 b FF(j)1751 739 y Fl(T)1795 720 y Fw(2)1861 722 y FF(\024)1976 655 y FH(1)p 1976 699 49 4 v 1976 790 a(4)2062 655 y(\(1)22 b FF(\000)g FA(\027)6 b FH(\))2362 618 y FE(4)p 2062 699 341 4 v 2063 790 a FH(\(1)21 b(+)h FA(\027)6 b FH(\))2361 762 y FE(2)2447 722 y FA(;)1073 b FH(\(2.60\))17 942 y Fn(then)957 1096 y FH(inf)976 1164 y Fl(T)1020 1145 y Fw(2)1091 1096 y FA(W)1197 1055 y Fj(0)1248 1096 y FA(>)1362 1029 y FH(\(1)21 b FF(\000)i FA(\027)6 b FH(\))1662 993 y FE(2)p 1362 1073 V 1507 1165 a FH(2)1747 1096 y FA(;)216 b FF(h)p FH(\()p FA(W)2173 1055 y Fj(0)2195 1096 y FH(\))2233 1055 y Fj(\000)p FE(1)2328 1096 y FF(i)27 b FA(>)2507 1029 y FH(\(1)22 b(+)g FA(\027)6 b FH(\))2806 993 y FE(2)p 2507 1073 339 4 v 2652 1165 a FH(2)3547 1096 y(\(2.61\))17 1300 y Fn(and)994 1421 y FH(sup)1028 1508 y Fl(T)1072 1489 y Fw(2)1157 1421 y FF(j)p FH(\()p FA(W)1329 1379 y Fj(0)1352 1421 y FH(\))1390 1379 y Fj(\000)p FE(1)1506 1421 y FF(\000)23 b FA(W)1712 1379 y Fj(\000)p FE(1)1806 1421 y FF(j)17 b FA(;)32 b FF(j)p FA(\032)1988 1379 y Fj(0)2034 1421 y FF(\000)22 b FA(\032)p FF(j)45 b(\024)28 b FA(c)17 b(\016)2467 1379 y Fj(\000)p Fy(s)2575 1421 y FF(k)p FA(e)p FF(k)2720 1436 y Fy(\030)2792 1421 y FA(:)728 b FH(\(2.62\))17 1741 y FI(Pro)s(of)36 b FH(The)c(pro)s(of)f (of)g(this)h(Lemma)h(is)f(\(KAM\))g(routine)g(and)f(it)h(is)g(based)g (on)g(a)f(systematic)j(use)f(of)17 1862 y(p)s(oin)m(t)f(\(iii\))h(of)f (Lemma)h(1)g(and)f(of)g(p)s(oin)m(t)h(\(vi\))g(of)f(Remark)h(4.)17 2005 y(W)-8 b(e)33 b(b)s(egin)g(b)m(y)g(observing)h(that)e(\(2.54\))g (implies)i(that)716 2334 y FA(\026)775 2349 y FE(0)842 2334 y FH(:=)28 b(\(1)22 b(+)g FA(\027)6 b FH(\))1272 2293 y Fj(\000)p FE(2)1394 2334 y FA(<)28 b FF(j)p FA(W)1632 2293 y Fj(\000)p FE(1)1725 2334 y FH(\()p FA(\022)s FH(\))p FF(j)g FA(<)f(\026)2067 2349 y FE(1)2134 2334 y FH(:=)h(\(1)22 b FF(\000)g FA(\027)6 b FH(\))2565 2293 y Fj(\000)p FE(2)2692 2334 y FA(;)115 b FF(8)33 b FA(\022)d FF(2)f FG(S)3147 2293 y FE(2)3147 2359 y Fy(\030)3219 2334 y FA(;)716 2489 y(\026)775 2504 y FE(0)842 2489 y FA(<)f(W)1052 2448 y Fj(\000)p FE(1)1146 2489 y FH(\()p FA(\022)s FH(\))f FA(<)h(\026)1460 2504 y FE(1)1532 2489 y FA(;)114 b FF(8)33 b FA(\022)e FF(2)d FG(T)1997 2448 y FE(2)2069 2489 y FA(:)1451 b FH(\(2.63\))17 2697 y(Next,)33 b(from)g(the)g (de\014nitions)h(in)e(\(2.46\))g(and)h(\(2.54\))f(it)g(follo)m(ws)i (immediately)g(that)1021 2915 y FF(j)p 1049 2835 79 4 v FA(E)5 b FF(j)17 b FA(;)33 b FF(k)1300 2890 y Fm(e)1281 2915 y FA(E)6 b FF(k)1409 2930 y Fy(\030)1463 2915 y FA(;)33 b FF(k)p FA(E)6 b FF(k)1701 2930 y Fy(\030)1766 2915 y FF(\024)29 b FH(\(1)21 b(+)h FA(\027)6 b FH(\))17 b FF(k)p FA(e)p FF(k)2332 2930 y Fy(\030)2398 2915 y FA(<)27 b FH(2)p FF(k)p FA(e)p FF(k)2695 2930 y Fy(\030)2765 2915 y FA(:)17 3123 y FH(F)-8 b(rom)32 b(\(2.31\))g(it)g(follo)m(ws)888 3332 y(sup)922 3419 y Fl(T)966 3400 y Fw(2)1052 3332 y FF(j)p FA(D)1164 3290 y Fj(\000)p FE(1)1161 3356 y Fy(\021)1257 3332 y FA(v)1304 3358 y Fk(\022)1326 3373 y Fw(1)1370 3332 y FF(j)27 b(\024)h(k)p FA(D)1664 3290 y Fj(\000)p FE(1)1661 3356 y Fy(\021)1758 3332 y FA(v)1805 3358 y Fk(\022)1827 3373 y Fw(1)1870 3332 y FF(k)1920 3347 y FE(0)1987 3332 y FF(\024)h FA(\033)2148 3347 y FE(0)p Fy(;)p FE(1)2242 3332 y FH(\()p FA(\030)5 b FH(\))p FF(k)p FA(v)2463 3358 y Fk(\022)2485 3373 y Fw(1)2528 3332 y FF(k)2578 3347 y Fy(\030)2643 3332 y FA(<)28 b(\033)2802 3347 y FE(0)p Fy(;)p FE(1)2896 3332 y FH(\()p FA(\030)5 b FH(\))17 b FA(\027)888 3559 y FH(sup)922 3646 y Fl(T)966 3627 y Fw(2)1052 3559 y FF(j)p FA(D)1164 3518 y Fj(\000)p FE(1)1161 3584 y Fy(\021)1277 3534 y Fm(e)1257 3559 y FA(E)6 b FF(j)28 b(\024)g(k)p FA(D)1630 3518 y Fj(\000)p FE(1)1627 3584 y Fy(\021)1743 3534 y Fm(e)1724 3559 y FA(E)6 b FF(k)1852 3574 y FE(0)1919 3559 y FF(\024)28 b FA(\033)2079 3574 y FE(0)p Fy(;)p FE(1)2174 3559 y FH(\()p FA(\030)5 b FH(\))p FF(k)p FA(e)p FF(k)2443 3574 y Fy(\030)2512 3559 y FA(:)17 3827 y FH(F)-8 b(rom)32 b(\(2.63\))g(one)g(has)1243 3991 y(inf)1262 4059 y Fl(T)1306 4040 y Fw(2)1377 3991 y FA(W)42 b(>)1649 3924 y FH(1)p 1624 3969 99 4 v 1624 4060 a FA(\026)1683 4075 y FE(1)1765 3991 y FA(;)212 b FF(h)p FA(W)2149 3950 y Fj(\000)p FE(1)2242 3991 y FF(i)28 b FA(>)f(\026)2471 4006 y FE(0)2543 3991 y FA(;)17 4206 y FH(so)32 b(that)1003 4414 y FF(h)p FA(W)1148 4373 y Fj(\000)p FE(1)1242 4414 y FF(i)22 b FH(+)g FA(\021)t FF(h)p FA(W)1598 4373 y Fj(\000)p FE(1)1691 4414 y FA(D)1775 4373 y Fj(\000)p FE(1)1772 4438 y Fy(\021)1869 4414 y FA(v)1916 4440 y Fk(\022)1938 4455 y Fw(1)1982 4414 y FF(i)27 b(\025)h(h)p FA(W)2298 4373 y Fj(\000)p FE(1)2392 4414 y FF(i)p FH(\(1)21 b FF(\000)i(j)p FA(\032)p FF(j)p FH(\))2286 4573 y FA(>)k FF(h)p FA(W)2534 4531 y Fj(\000)p FE(1)2628 4573 y FF(i)p FH(\(1)22 b FF(\000)g FA(\013)q FH(\))2286 4731 y FA(>)27 b(\026)2448 4746 y FE(0)2504 4731 y FH(\(1)22 b FF(\000)h FA(\013)q FH(\))32 b FA(:)17 4965 y FH(W)-8 b(e)33 b(are,)f(no)m(w,)i(ready)f(to)f(estimate)i FF(j)1425 4938 y FH(^)1411 4965 y FA(\014)5 b FF(j)p FH(:)32 b(in)h(view)h(of)e(\(2.32\))g(and)h(the)g(ab)s(o)m(v)m(e)g (estimates,)h(w)m(e)g(get)1274 5173 y FF(j)1316 5146 y FH(^)1302 5173 y FA(\014)5 b FF(j)28 b(\024)g FA(c)1565 5188 y FE(1)1621 5173 y FA(\030)1669 5132 y Fj(\000)p Fy(\034)1783 5173 y FF(k)p FA(e)p FF(k)1928 5188 y Fy(\030)1993 5173 y FA(<)g(c)2139 5188 y FE(1)2195 5173 y FA(\016)2242 5132 y Fj(\000)p Fy(\034)2356 5173 y FF(k)p FA(e)p FF(k)2501 5188 y Fy(\030)1858 5645 y FH(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 17 548 a FH(with)463 703 y FA(c)505 718 y FE(1)572 703 y FH(:=)805 635 y(1)p 713 680 234 4 v 713 771 a(1)22 b FF(\000)g FA(\013)982 635 y(\026)1041 650 y FE(1)p 982 680 99 4 v 982 771 a FA(\026)1041 786 y FE(0)1107 592 y Fm(\020)1167 703 y FH(2)g(+)1345 635 y FA(\021)1393 650 y FE(0)p 1345 680 88 4 v 1361 771 a FA(\024)1460 703 y FH(\()p FA(\034)11 b(=)17 b Ff(e)p FH(\))1708 661 y Fy(\034)1750 592 y Fm(\021)1837 703 y FH(=)2043 635 y(1)p 1951 680 234 4 v 1951 771 a(1)22 b FF(\000)h FA(\013)2222 635 y FH(\(1)e(+)h FA(\027)6 b FH(\))2520 599 y FE(2)p 2221 680 341 4 v 2221 771 a FH(\(1)22 b FF(\000)g FA(\027)6 b FH(\))2521 742 y FE(2)2588 592 y Fm(\020)2647 703 y FH(2)22 b(+)2826 635 y FA(\021)2874 650 y FE(0)p 2826 680 88 4 v 2842 771 a FA(\024)2940 703 y FH(\()p FA(\034)11 b(=)17 b Ff(e)p FH(\))3188 661 y Fy(\034)3231 592 y Fm(\021)3323 703 y FA(:)17 927 y FH(Analogously)-8 b(,)33 b(w)m(e)h(\014nd)641 1147 y FF(j)p FA(a)p FF(j)82 b(\024)i FA(c)1033 1162 y FE(2)1089 1147 y FA(\030)1137 1106 y Fj(\000)p Fy(\034)1234 1147 y FF(k)p FA(e)p FF(k)1379 1162 y Fy(\030)1444 1147 y FA(<)28 b(c)1590 1162 y FE(2)1646 1147 y FA(\016)1693 1106 y Fj(\000)p Fy(\034)1791 1147 y FF(k)p FA(e)p FF(k)1936 1162 y Fy(\030)2006 1147 y FA(;)393 1438 y FF(k)p FA(E)515 1453 y FE(1)554 1438 y FF(k)604 1465 y Fy(\030)s Fj(\000)703 1438 y Fk(\016)p 703 1450 31 3 v 703 1491 a Fw(3)830 1438 y FF(\024)84 b(k)p FA(D)1125 1397 y Fj(\000)p FE(1)1122 1463 y Fy(\021)1238 1413 y Fm(e)1218 1438 y FA(E)7 b FF(k)1347 1465 y Fy(\030)s Fj(\000)1446 1438 y Fk(\016)p 1445 1450 V 1445 1491 a Fw(3)1512 1438 y FH(+)22 b FF(j)p FA(a)p FF(j)p FH(\()o(1)g(+)g FF(j)p FA(\021)t FF(jk)p FA(D)2165 1397 y Fj(\000)p FE(1)2162 1463 y Fy(\021)2258 1438 y FA(v)2305 1465 y Fk(\022)2327 1480 y Fw(1)2370 1438 y FF(k)2420 1465 y Fy(\030)s Fj(\000)2519 1438 y Fk(\016)p 2518 1450 V 2518 1491 a Fw(3)2563 1438 y FH(\))g(+)g FF(j)p 2749 1358 79 4 v FA(E)6 b FF(j)32 b(k)p FA(D)3021 1397 y Fj(\000)p FE(1)3018 1463 y Fy(\021)3115 1438 y FA(v)3162 1465 y Fk(\022)3184 1480 y Fw(1)3227 1438 y FF(k)3277 1465 y Fy(\030)s Fj(\000)3376 1438 y Fk(\016)p 3376 1450 31 3 v 3376 1491 a Fw(3)830 1622 y FF(\024)84 b FA(\033)1046 1637 y FE(0)p Fy(;)p FE(1)1140 1622 y FH(\()p FA(\016)t(=)p FH(3\))p FF(k)1430 1597 y Fm(e)1411 1622 y FA(E)6 b FF(k)1539 1637 y Fy(\030)1598 1622 y FH(+)22 b FF(j)p FA(a)p FF(j)p FH(\(1)g(+)g FF(j)p FA(\021)t FF(j)p FA(\033)2173 1637 y FE(0)p Fy(;)p FE(1)2267 1622 y FH(\()p FA(\016)t(=)p FH(3\))p FA(\027)6 b FH(\))22 b(+)g FF(j)p 2728 1542 79 4 v FA(E)5 b FF(j)p FA(\033)2888 1637 y FE(0)p Fy(;)p FE(1)2983 1622 y FH(\()p FA(\016)t(=)p FH(3\))p FA(\027)830 1767 y FF(\024)84 b FA(c)1033 1782 y FE(3)1089 1767 y FA(\016)1136 1726 y Fj(\000)p FE(2)p Fy(\034)1269 1767 y FF(k)p FA(e)p FF(k)1414 1782 y Fy(\030)1517 1767 y FA(;)2003 b FH(\(2.64\))17 1987 y(for)22 b(suitable)i(constan)m (ts)g FA(c)974 2002 y Fy(i)1030 1987 y FH(=)j FA(c)1175 2002 y Fy(i)1203 1987 y FH(\()p FA(\024;)17 b(\034)6 b(;)17 b(\021)1481 2002 y FE(0)1520 1987 y FA(;)g(\027)q(;)g(\013)q FH(\))22 b(\(whic)m(h,)i(b)s(eing)f(clear)g(the)h(ideas,)f(w)m(e)h(do)f (not)f(compute)17 2119 y(an)m(y)34 b(more\);)f(notice)h(that)f FF(k)p FA(E)1133 2134 y FE(1)1173 2119 y FF(k)g FH(has)g(b)s(een)h (estimated)h(on)e(an)g(in)m(termediate)i(space)f FF(H)3292 2082 y Fy(\030)s Fj(\000)3391 2055 y Fk(\016)p 3391 2067 31 3 v 3391 2108 a Fw(3)3469 2119 y FH(so)f(as)h(to)17 2239 y(b)s(e)e(able)h(to)g(con)m(trol)g(further)f(applications)i(of)e (the)h(op)s(erator)f FA(D)2429 2203 y Fj(\000)p FE(1)2426 2264 y Fy(\021)2555 2239 y FH(or)h FA(@)2726 2266 y Fk(\022)2748 2281 y Fw(1)2791 2239 y FH(.)17 2384 y(No)m(w,)j(recalling)g(the)f (de\014nition)h(of)f FA(w)j FH(and)55 b(^)-69 b FA(w)38 b FH(in)d(\(2.49\),)g(b)m(y)h(p)s(oin)m(t)g(\(iii\))f(of)g(Lemma)h(1,)f (b)m(y)h(\(2.64\),)17 2505 y(one)d(gets)884 2725 y FF(k)20 b FH(^)-69 b FA(w)r FF(k)1056 2750 y Fy(\030)s Fj(\000)1155 2723 y Fw(2)p 1155 2735 V 1155 2776 a(3)1195 2750 y Fy(\016)1316 2725 y FF(\024)84 b FH(\(1)21 b(+)h FF(k)p FA(v)1780 2751 y Fk(\022)1802 2766 y Fw(1)1846 2725 y FF(k)1896 2740 y Fy(\030)1933 2725 y FH(\))17 b FF(k)p FA(D)2122 2684 y Fj(\000)p FE(1)2119 2749 y(0)2216 2725 y FH(\()p FA(W)2360 2684 y Fj(\000)p FE(1)2453 2725 y FA(E)2525 2740 y FE(1)2565 2725 y FH(\))p FF(k)2653 2750 y Fy(\030)s Fj(\000)2752 2723 y Fw(2)p 2751 2735 V 2751 2776 a(3)2792 2750 y Fy(\016)1316 2893 y FF(\024)84 b FH(\(1)21 b(+)h FA(\027)6 b FH(\))p FA(\033)1830 2908 y FE(0)p Fy(;)p FE(1)1926 2893 y FH(\()p FA(\016)t(=)p FH(3\))17 b(\(1)j FF(\000)j FA(\027)6 b FH(\))2463 2852 y Fj(\000)p FE(2)2574 2893 y FF(k)p FA(E)2696 2908 y FE(1)2736 2893 y FF(k)2786 2920 y Fy(\030)s Fj(\000)2885 2893 y Fk(\016)p 2884 2905 V 2884 2946 a Fw(3)1316 3064 y FF(\024)84 b FA(c)1519 3079 y FE(4)1575 3064 y FA(\016)1622 3022 y Fj(\000)p FE(3)p Fy(\034)1771 3064 y FF(k)p FA(e)p FF(k)1916 3079 y Fy(\030)1987 3064 y FA(:)17 3284 y FH(Th)m(us,)34 b(since)g FF(jh)19 b FH(^)-68 b FA(w)r FF(i)o(j)28 b(\024)g(k)20 b FH(^)-69 b FA(w)r FF(k)1040 3308 y Fy(\030)s Fj(\000)1139 3281 y Fw(2)p 1139 3293 V 1139 3335 a(3)1179 3308 y Fy(\016)1217 3284 y FH(,)32 b(w)m(e)i(\014nd)1414 3526 y FF(k)p FA(w)s FF(k)1587 3551 y Fy(\030)s Fj(\000)1686 3524 y Fw(2)p 1684 3536 V 1684 3577 a(3)1725 3551 y Fy(\016)1790 3526 y FF(\024)28 b FA(c)1937 3541 y FE(5)1977 3526 y FA(\016)2024 3485 y Fj(\000)p FE(3)p Fy(\034)2157 3526 y FF(k)p FA(e)p FF(k)2302 3541 y Fy(\030)2372 3526 y FA(;)1148 b FH(\(2.65\))17 3746 y(and)847 3966 y FF(k)p FA(w)967 3993 y Fk(\022)989 4008 y Fw(1)1032 3966 y FF(k)1082 3981 y Fy(\030)s Fj(\000)p Fy(\016)1236 3966 y FF(\024)28 b FA(\033)1396 3981 y FE(1)p Fy(;)p FE(0)1491 3966 y FH(\()p FA(\016)t(=)p FH(3\))17 b FF(k)p FA(w)s FF(k)1902 3991 y Fy(\030)s Fj(\000)2001 3964 y Fw(2)p 1998 3976 V 1998 4017 a(3)2038 3991 y Fy(\016)2104 3966 y FF(\024)28 b FA(c)2251 3981 y FE(6)2307 3966 y FA(\016)2354 3925 y Fj(\000)p FE(\(3)p Fy(\034)8 b FE(+1\))2649 3966 y FF(k)p FA(e)p FF(k)2794 3981 y Fy(\030)2864 3966 y FA(;)847 4142 y FF(k)p FA(D)981 4100 y Fj(\000)p FE(1)978 4166 y Fy(\021)1075 4142 y FA(w)1145 4168 y Fk(\022)1167 4183 y Fw(1)1210 4142 y FF(k)1260 4157 y Fy(\030)s Fj(\000)p Fy(\016)1414 4142 y FF(\024)28 b FA(\033)1574 4157 y FE(0)p Fy(;)p FE(1)1669 4142 y FH(\()p FA(\016)t(=)p FH(3\))17 b FF(k)p FA(w)2027 4168 y Fk(\022)2049 4183 y Fw(1)2090 4142 y FF(k)2140 4167 y Fy(\030)s Fj(\000)2239 4139 y Fw(2)p 2239 4151 V 2239 4193 a(3)2279 4167 y Fy(\016)2345 4142 y FF(\024)28 b FA(c)2492 4157 y FE(7)2548 4142 y FA(\016)2595 4100 y Fj(\000)p FE(\(4)p Fy(\034)8 b FE(+1\))2890 4142 y FF(k)p FA(e)p FF(k)3035 4157 y Fy(\030)3105 4142 y FA(;)415 b FH(\(2.66\))17 4375 y(whic)m(h)34 b(pro)m(v)m(es)g(part)e(\(i\))h(of) f(the)h(Lemma.)17 4520 y(No)m(w,)g(b)m(y)g(\(2.57\))f(and)h(\(2.66\),) 722 4740 y FF(k)p FA(W)878 4699 y Fj(0)923 4740 y FF(\000)23 b FA(W)14 b FF(k)1179 4755 y Fy(\030)s Fj(\000)p Fy(\016)1389 4740 y FH(=)83 b FF(k)p FH(2)p FA(w)1717 4767 y Fk(\022)1739 4782 y Fw(1)1782 4740 y FA(V)43 b FH(+)22 b FA(w)2053 4699 y FE(2)2050 4777 y Fk(\022)2072 4792 y Fw(1)2115 4740 y FF(k)2165 4755 y Fy(\030)s Fj(\000)p Fy(\016)1388 4909 y FF(\024)83 b FH(2)p FA(\027)6 b FF(k)p FA(w)1771 4936 y Fk(\022)1793 4951 y Fw(1)1836 4909 y FF(k)1886 4924 y Fy(\030)s Fj(\000)p Fy(\016)2035 4909 y FH(+)22 b FF(k)p FA(w)2253 4936 y Fk(\022)2275 4951 y Fw(1)2317 4909 y FF(k)2367 4868 y FE(2)2367 4934 y Fy(\030)s Fj(\000)p Fy(\016)2521 4909 y FF(\024)28 b FH(3)p FA(\027)6 b FF(k)p FA(w)2849 4936 y Fk(\022)2871 4951 y Fw(1)2914 4909 y FF(k)2964 4924 y Fy(\030)s Fj(\000)p Fy(\016)1388 5075 y FF(\024)83 b FA(c)1590 5090 y FE(8)1630 5075 y FA(\016)1677 5034 y Fj(\000)p FE(\(3)p Fy(\034)8 b FE(+1\))1972 5075 y FF(k)p FA(e)p FF(k)2117 5090 y Fy(\030)2187 5075 y FA(:)1333 b FH(\(2.67\))1858 5645 y(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 17 548 a FH(Inequalit)m(y)37 b(\(2.57\))e(guaran)m(tees,)i(also,)f(that)g FA(e)1767 512 y Fj(0)1823 548 y FF(2)e(H)2008 512 y Fy(\030)s Fj(\000)p Fy(\016)2134 548 y FH(;)i(compare)g(p)s(oin)m(t)g(\(vi\),)g(Remark)g (4.)g(Then,)17 668 y(b)m(y)d(Corollary)g(2)f(and)h(\(2.65\),)f(one)h (gets)g(\(for)f(a)g(suitable)i FA(c)2217 683 y FE(9)2284 668 y FA(>)27 b FH(1\))324 885 y FF(k)p FA(e)419 844 y Fj(0)442 885 y FF(k)492 900 y Fy(\030)s Fj(\000)p Fy(\016)701 885 y FF(\024)84 b(k)p FA(Q)989 900 y FE(1)1028 885 y FF(k)1078 900 y Fy(\030)s Fj(\000)p Fy(\016)1227 885 y FH(+)22 b FF(k)p FA(Q)1452 900 y FE(2)1491 885 y FF(k)1541 900 y Fy(\030)s Fj(\000)p Fy(\016)1695 885 y FF(\024)28 b FA(M)g FF(k)p FA(w)s FF(k)2095 844 y FE(2)2095 910 y Fy(\030)s Fj(\000)p Fy(\016)2242 885 y FH(+)22 b(\(1)g FF(\000)h FA(\027)6 b FH(\))2641 844 y Fj(\000)p FE(1)2735 885 y FA(\033)2790 900 y FE(1)p Fy(;)p FE(0)2885 885 y FH(\()p FA(\016)t FH(\))p FF(k)p FA(e)p FF(k)3153 900 y Fy(\030)3191 885 y FF(k)p FA(w)s FF(k)3364 900 y Fy(\030)s Fj(\000)p Fy(\016)701 1040 y FF(\024)84 b FA(c)904 1055 y FE(9)960 1040 y FA(\016)1007 999 y Fj(\000)p FE(6)p Fy(\034)1157 1040 y FF(k)p FA(e)p FF(k)1302 999 y FE(2)1302 1065 y Fy(\030)1374 1040 y FA(:)17 1282 y FH(Let)194 1257 y Fm(f)191 1282 y FA(W)42 b FH(:=)27 b FA(W)561 1246 y Fj(0)606 1282 y FF(\000)c FA(W)14 b FH(.)32 b(F)-8 b(rom)32 b(\(2.60\))g(it)h(follo)m(ws)g(that)1117 1559 y(sup)1151 1646 y Fl(T)1195 1627 y Fw(2)1281 1559 y FF(j)1312 1533 y Fm(f)1309 1559 y FA(W)13 b FF(j)27 b(\024)1584 1491 y FH(1)p 1584 1536 49 4 v 1584 1627 a(4)1653 1491 y(\(1)22 b FF(\000)h FA(\027)6 b FH(\))1954 1455 y FE(4)p 1653 1536 341 4 v 1654 1627 a FH(\(1)22 b(+)g FA(\027)6 b FH(\))1953 1598 y FE(2)2031 1559 y FH(=)2145 1491 y(1)p 2145 1536 49 4 v 2145 1627 a(4)2230 1491 y FA(\026)2289 1506 y FE(0)p 2230 1536 99 4 v 2230 1627 a FA(\026)2289 1593 y FE(2)2289 1651 y(1)2366 1559 y FA(<)2528 1491 y FH(1)p 2479 1536 147 4 v 2479 1627 a(2)p FA(\026)2587 1642 y FE(1)2669 1559 y FA(;)851 b FH(\(2.68\))17 1839 y(so)32 b(that,)h(on)f FG(T)576 1803 y FE(2)649 1839 y FH(one)h(has)g(\(recall)g(\(2.63\)\))412 2048 y(1)p 412 2093 49 4 v 412 2184 a(2)498 2116 y FF(\024)28 b FH(1)22 b(+)g FA(W)878 2074 y Fj(\000)p FE(1)975 2090 y Fm(f)972 2116 y FA(W)41 b FF(\024)1221 2048 y FH(3)p 1221 2093 V 1221 2184 a(2)1312 2116 y FA(;)212 b(W)1657 2074 y Fj(0)1707 2116 y FH(=)28 b FA(W)36 b FH(+)2040 2090 y Fm(f)2037 2116 y FA(W)41 b(>)2308 2048 y FH(1)p 2283 2093 99 4 v 2283 2184 a FA(\026)2342 2199 y FE(1)2414 2116 y FF(\000)2572 2048 y FH(1)p 2523 2093 147 4 v 2523 2184 a(2)p FA(\026)2631 2199 y FE(1)2708 2116 y FH(=)2821 2048 y(\(1)22 b FF(\000)h FA(\027)6 b FH(\))3122 2012 y FE(2)p 2821 2093 341 4 v 2967 2184 a FH(2)3199 2116 y FA(>)28 b FH(0)k FA(:)17 2387 y FH(Let)g(no)m(w)i FA(Z)g FH(:=)28 b(\()p FA(W)771 2351 y Fj(0)794 2387 y FH(\))832 2351 y Fj(\000)p FE(1)948 2387 y FF(\000)23 b FA(W)1154 2351 y Fj(\000)p FE(1)1248 2387 y FH(.)32 b(Then,)i(b)m(y)g(\(2.67\))e (and)g(\(2.68\))g(one)h(\014nds)g(\(on)g FG(T)3130 2351 y FE(2)3170 2387 y FH(\))601 2684 y FF(j)p FA(Z)7 b FF(j)27 b FH(=)1169 2592 y Fm(f)1166 2617 y FA(W)p 871 2661 696 4 v 871 2773 a(W)977 2745 y FE(2)1016 2773 y FH(\(1)22 b(+)g FA(W)1329 2745 y Fj(\000)p FE(1)1426 2748 y Fm(f)1423 2773 y FA(W)14 b FH(\))1604 2684 y FF(\024)28 b FH(2)p FA(\026)1817 2643 y FE(2)1817 2709 y(1)1873 2684 y FF(j)1904 2659 y Fm(f)1901 2684 y FA(W)13 b FF(j)27 b(\024)i FA(c)2209 2699 y FE(10)2300 2684 y FH(min)q FF(f)p FA(\016)2560 2643 y Fj(\000)p FE(\(3)p Fy(\034)8 b FE(+1\))2854 2684 y FF(k)p FA(e)p FF(k)2999 2699 y Fy(\030)3053 2684 y FA(;)34 b FH(1)p FF(g)334 b FH(\(2.69\))17 2978 y(\(for)38 b(a)g(suitable)i FA(c)709 2993 y FE(10)821 2978 y FA(>)e FH(1\),)g(pro)m(ving)h(also)g(the)g(\014rst)g(inequalit)m(y)i(in)d (\(2.62\).)g(F)-8 b(urthermore,)39 b(b)m(y)h(the)17 3098 y(\014rst)33 b(inequalit)m(y)h(in)f(\(2.68\),)245 3360 y FF(h)p FH(\()p FA(W)428 3319 y Fj(0)451 3360 y FH(\))489 3319 y Fj(\000)p FE(1)583 3360 y FF(i)28 b FH(=)f FF(h)p FA(W)898 3319 y Fj(\000)p FE(1)992 3360 y FF(i)22 b FH(+)g FF(h)p FA(Z)7 b FF(i)27 b(\025)h(h)p FA(W)1580 3319 y Fj(\000)p FE(1)1674 3360 y FF(i)22 b(\000)g(jh)p FA(Z)7 b FF(i)o(j)28 b(\025)g FA(\026)2233 3375 y FE(0)2294 3360 y FF(\000)23 b FH(2)p FA(\026)2502 3319 y FE(2)2502 3384 y(1)2557 3360 y FF(j)2588 3335 y Fm(f)2585 3360 y FA(W)14 b FF(j)27 b(\025)2861 3292 y FA(\026)2920 3307 y FE(0)p 2861 3337 99 4 v 2886 3428 a FH(2)2997 3360 y(=)3280 3292 y(1)p 3110 3337 388 4 v 3110 3428 a(2\(1)22 b(+)g FA(\027)6 b FH(\))3458 3399 y FE(2)3540 3360 y FA(:)17 3633 y FH(Finally)-8 b(,)29 b(using)i(\(2.69\),)d(\(2.66\))h (and)h(\(2.57\))e(\(in)i(order)f(to)g(estimate)i FF(k)p FA(w)2690 3659 y Fk(\022)2712 3674 y Fw(1)2755 3633 y FF(k)e FH(in)h(terms)g(of)f FA(\027)34 b(<)28 b FH(1\),)h(one)17 3753 y(obtains)219 4032 y FF(j)p FA(\032)297 3991 y Fj(0)342 4032 y FF(\000)23 b FA(\032)p FF(j)28 b(\024)653 3858 y Fm(\014)653 3918 y(\014)653 3977 y(\014)653 4037 y(\014)653 4097 y(\014)686 4032 y FA(\021)748 3952 y FH(\()p FA(W)892 3916 y Fj(\000)p FE(1)1008 3952 y FH(+)22 b FA(Z)7 b FH(\)\()p FA(D)1340 3916 y Fj(\000)p FE(1)1337 3976 y Fy(\021)1433 3952 y FA(v)1480 3978 y Fk(\022)1502 3993 y Fw(1)1568 3952 y FH(+)22 b FA(D)1750 3916 y Fj(\000)p FE(1)1747 3976 y Fy(\021)1844 3952 y FA(w)1914 3978 y Fk(\022)1936 3993 y Fw(1)1979 3952 y FH(\))p 748 4009 1270 4 v 1108 4101 a FF(h)p FA(W)1253 4072 y Fj(\000)p FE(1)1346 4101 y FF(i)g FH(+)g FF(h)p FA(Z)7 b FF(i)2049 4032 y(\000)23 b FA(\021)2211 3952 y(W)2317 3916 y Fj(\000)p FE(1)2410 3952 y FA(D)2494 3916 y Fj(\000)p FE(1)2491 3976 y Fy(\021)2588 3952 y FA(v)2635 3978 y Fk(\022)2657 3993 y Fw(1)p 2210 4009 491 4 v 2317 4101 a FF(h)p FA(W)2462 4072 y Fj(\000)p FE(1)2555 4101 y FF(i)2711 3858 y Fm(\014)2711 3918 y(\014)2711 3977 y(\014)2711 4037 y(\014)2711 4097 y(\014)2771 4032 y FF(\024)29 b FA(c)2919 4047 y FE(11)3010 4032 y FA(\016)3057 3991 y Fj(\000)p FE(\(4)p Fy(\034)8 b FE(+1\))3352 4032 y FF(k)p FA(e)p FF(k)3497 4047 y Fy(\030)3567 4032 y FA(;)17 4336 y FH(pro)m(ving)33 b(also)g(the)g (second)g(inequalit)m(y)i(in)e(\(2.62\).)17 4481 y(The)g(theses)i(of)d (the)h(Lemma)g(follo)m(w,)g(no)m(w,)g(b)m(y)h(taking)e FA(c)c FH(:=)g(max)2490 4496 y Fy(i)2535 4481 y FA(c)2577 4496 y Fy(i)2638 4481 y FH(and)k FA(s)c FH(=)f(6)p FA(\034)11 b FH(.)p 3284 4481 41 83 v 17 4868 a FD(2.4)135 b(Con)l(v)l(ergence)46 b(of)f(the)h(Nash{Moser)f(algorithm)17 5052 y FH(Here)35 b(w)m(e)h(complete)h(the)e(quan)m(titativ)m(e)i(description)f(of)f(the) g(KAM)g(pro)s(cedure)h(giving)f(a)g(su\016cien)m(t)17 5173 y(condition)e(in)g(order)f(for)g(the)h(algorithm)g(to)f(con)m(v)m (erge.)1858 5645 y(24)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 17 548 a FH(F)-8 b(or)31 b(an)m(y)j FA(i)28 b FF(\025)g FH(0)k(and)h(for)f(0)27 b FA(<)h(\030)1185 563 y Fj(\003)1252 548 y FA(<)f(\030)32 b(<)1544 522 y FH(\026)1534 548 y FA(\030)t FH(,)h(w)m(e)g(let)527 812 y FA(\030)570 827 y Fy(i)625 812 y FH(:=)28 b FA(\030)799 827 y Fj(\003)860 812 y FH(+)968 744 y FA(\030)f FF(\000)22 b FA(\030)1180 759 y Fj(\003)p 968 789 252 4 v 1055 880 a FH(2)1104 851 y Fy(i)1262 812 y FA(;)212 b(\016)1544 827 y Fy(i)p FE(+1)1690 812 y FH(:=)28 b FA(\030)1864 827 y Fy(i)1913 812 y FF(\000)23 b FA(\030)2056 827 y Fy(i)p FE(+1)2202 812 y FH(=)2315 744 y FA(\030)k FF(\000)22 b FA(\030)2527 759 y Fj(\003)p 2315 789 V 2357 880 a FH(2)2406 851 y Fy(i)p FE(+1)2609 812 y FA(;)212 b FH(\()p FF(8)33 b FA(i)28 b FF(\025)g FH(0\))k(;)17 1067 y(as)g(ab)s(o)m(v)m(e) i(\(compare)f(\(2.63\)\))f(w)m(e)h(let)g FA(\026)1504 1082 y FE(0)1571 1067 y FH(:=)28 b(\(1)22 b(+)g FA(\027)6 b FH(\))2001 1031 y Fj(\000)p FE(2)2128 1067 y FH(and)33 b FA(\026)2377 1082 y FE(1)2443 1067 y FH(:=)28 b(\(1)22 b FF(\000)g FA(\027)6 b FH(\))2874 1031 y Fj(\000)p FE(2)2969 1067 y FH(;)33 b(w)m(e)g(let)885 1308 y(0)28 b FA(<)f(\026)1124 1323 y FE(0)1191 1308 y FA(<)35 b FH(\026)-56 b FA(\026)1354 1323 y FE(0)1421 1308 y FH(:=)27 b(inf)1571 1375 y Fl(T)1615 1357 y Fw(2)1686 1308 y FA(W)1792 1267 y Fj(\000)p FE(1)1913 1308 y FF(\024)36 b FH(\026)-56 b FA(\026)2078 1323 y FE(1)2144 1308 y FH(:=)28 b(sup)2309 1395 y Fl(T)2353 1376 y Fw(2)2439 1308 y FA(W)2545 1267 y Fj(\000)p FE(1)2666 1308 y FA(<)g(\026)2829 1323 y FE(1)2900 1308 y FH(;)620 b(\(2.70\))17 1542 y(\014nally)-8 b(,)33 b(w)m(e)h(let)f(\(recall)g (the)g(de\014nition)g(of)f FA(\032)h FH(in)g(\(1.21\)\))576 1815 y FA(\026)27 b FH(:=)44 b(min)988 1674 y Fm(\032)1080 1815 y FA(\027)28 b FF(\000)23 b(k)p FA(v)1353 1842 y Fk(\022)1375 1857 y Fw(1)1418 1815 y FF(k)1468 1830 y Fy(\030)1522 1815 y FA(;)1617 1747 y(\026)1676 1762 y FE(0)p 1593 1792 147 4 v 1593 1883 a FH(2)p FA(\026)1701 1849 y FE(2)1701 1908 y(1)1766 1815 y FA(;)33 b(\026)1885 1830 y FE(1)1947 1815 y FF(\000)d FH(\026)-57 b FA(\026)2105 1830 y FE(1)2161 1815 y FA(;)41 b FH(\026)-57 b FA(\026)2280 1830 y FE(0)2342 1815 y FF(\000)22 b FA(\026)2500 1830 y FE(0)2556 1815 y FA(;)33 b(\013)23 b FF(\000)g(j)p FA(\032)p FF(j)2924 1674 y Fm(\033)3025 1815 y FA(>)28 b FH(0)k FA(:)310 b FH(\(2.71\))17 2100 y FI(Prop)s(osition)37 b(2)49 b Fn(Under)39 b(the)g(same)g(assumptions)f(and)g(notations)h(of) g(p)-5 b(art)39 b(\(i\))g(of)g(L)-5 b(emma)38 b(3,)h(let)17 2221 y FH(0)f FA(<)g(\030)261 2236 y Fj(\003)338 2221 y FA(<)g(\030)45 b Fn(and)40 b(let)h FA(\030)921 2236 y Fy(i)949 2221 y Fn(,)f FA(\026)1078 2236 y Fy(i)1106 2221 y Fn(,)48 b FH(\026)-57 b FA(\026)1235 2236 y Fy(i)1304 2221 y Fn(and)40 b FA(\026)g Fn(b)-5 b(e)40 b(as)h(ab)-5 b(ove;)39 b(let)i FA(C)47 b Fn(and)40 b FA(m)h Fn(b)-5 b(e)41 b(p)-5 b(ositive)39 b(numb)-5 b(ers)40 b(such)17 2341 y(that)1107 2462 y FA(C)35 b FF(\025)28 b FA(c)17 b FH(4)1425 2421 y Fy(s)1478 2462 y FH(\()p FA(\030)26 b FF(\000)d FA(\030)1728 2477 y Fj(\003)1767 2462 y FH(\))1805 2421 y Fj(\000)p Fy(s)1931 2462 y FA(;)216 b(m)28 b FF(\024)h FH(2)2442 2421 y Fy(s)p Fj(\000)p FE(1)2585 2462 y FA(\026)35 b(;)841 b FH(\(2.72\))17 2636 y Fn(and)34 b(assume)g(that)1543 2756 y FA(C)23 b FF(k)p FA(e)p FF(k)1781 2771 y Fy(\030)1847 2756 y FF(\024)44 b Ff(e)2021 2715 y Fj(\000)2125 2688 y Fw(1)p 2086 2700 109 3 v 2097 2741 a Fc(e)10 b Fk(m)2243 2756 y FA(:)1277 b FH(\(2.73\))17 2949 y Fn(Then,)36 b FH(\()p FA(v)388 2964 y Fy(i)416 2949 y FA(;)17 b(\014)515 2964 y Fy(i)543 2949 y FH(\))31 b(:=)g FF(K)823 2913 y Fy(i)852 2949 y FH(\()p FA(v)t(;)17 b(\014)6 b FH(\))30 b(=)h(\()p FA(v)1306 2964 y Fy(i)p Fj(\000)p FE(1)1448 2949 y FH(+)23 b FA(w)1617 2964 y Fy(i)1645 2949 y FA(;)17 b(\014)1744 2964 y Fy(i)p Fj(\000)p FE(1)1886 2949 y FH(+)1999 2923 y(^)1985 2949 y FA(\014)2040 2964 y Fy(i)2069 2949 y FH(\))31 b FF(2)g(W)2348 2885 y FE(\026)2341 2902 y Fy(\030)2333 2977 y(\030)2364 2987 y Fk(i)2432 2949 y Fn(for)37 b(al)5 b(l)36 b FA(i)c FF(\025)f FH(1)p Fn(;)37 b(the)f(se)-5 b(quenc)g(es)36 b FF(f)p FA(v)3718 2964 y Fy(i)3746 2949 y FF(g)17 3070 y Fn(and)e FF(f)p FA(\014)311 3085 y Fy(i)339 3070 y FF(g)h Fn(c)-5 b(onver)g(ge)34 b(uniformly)g(on,)g(r)-5 b(esp)g(e)g(ctively,)35 b FG(S)2027 3034 y FE(2)2027 3096 y Fy(\030)2058 3104 y Fe(\003)2120 3070 y FF(\002)23 b FA(I)2263 3085 y FE(0)2337 3070 y Fn(and)35 b FA(I)2570 3085 y FE(0)2609 3070 y Fn(,)g(de\014ning)e(a)i (limit)710 3380 y FH(\()p FA(u;)17 b(\015)5 b FH(\))27 b(:=)47 b(lim)1100 3440 y Fy(j)t Fj(!1)1273 3380 y FH(\()p FA(v)1358 3395 y Fy(j)1395 3380 y FA(;)17 b(\014)1494 3395 y Fy(j)1530 3380 y FH(\))28 b(=)1699 3269 y Fm(\020)1759 3380 y FA(v)e FH(+)1967 3255 y Fj(1)1930 3285 y Fm(X)1945 3495 y Fy(i)p FE(=1)2090 3380 y FA(w)2160 3395 y Fy(i)2205 3380 y FA(;)33 b(\014)28 b FH(+)2483 3255 y Fj(1)2446 3285 y Fm(X)2461 3495 y Fy(i)p FE(=1)2620 3353 y FH(^)2606 3380 y FA(\014)2661 3395 y Fy(i)2690 3269 y Fm(\021)2777 3380 y FF(2)g(W)2984 3315 y FE(\026)2977 3332 y Fy(\030)2969 3407 y(\030)3000 3415 y Fe(\003)3075 3380 y FA(;)17 3687 y Fn(which)d(is)i(a)f(solution)h(of)f FH(\(1.18\))p Fn(,)g(i.e.,)g FF(F)1491 3702 y Fy(\021)1532 3687 y FH(\()p FA(u)p FH(;)17 b FA(\015)5 b FH(\))27 b(=)h(0)e Fn(for)h(al)5 b(l)26 b FA(\021)31 b FF(2)d FA(I)2465 3702 y FE(0)2505 3687 y Fn(.)e(F)-7 b(urthermor)i(e,)26 b(if)g FA(W)3324 3702 y Fy(i)3379 3687 y Fn(and)g FA(\032)3610 3702 y Fy(i)3665 3687 y Fn(ar)-5 b(e)17 3827 y(de\014ne)g(d)27 b(as,)g(r)-5 b(esp)g(e)g(ctively,)28 b FA(W)41 b Fn(and)28 b FA(\032)g Fn(in)g FH(\(1.21\))f Fn(with)h FA(v)k Fn(r)-5 b(eplac)g(e)g(d)27 b(by)h FA(v)2617 3842 y Fy(i)2645 3827 y Fn(,)g(then)g FA(W)3005 3842 y Fj(\003)3072 3827 y FH(:=)g(lim)17 b FA(W)3447 3842 y Fy(i)3503 3827 y FF(2)28 b(W)3710 3762 y FE(\026)3703 3779 y Fy(\030)3695 3854 y(\030)3726 3862 y Fe(\003)3766 3827 y Fn(,)17 3947 y FA(\032)67 3962 y Fj(\003)134 3947 y FH(:=)g(lim)17 b FA(\032)467 3962 y Fy(i)523 3947 y FF(2)28 b FA(C)694 3911 y Fj(1)769 3947 y FH(\()p FA(I)850 3962 y FE(0)889 3947 y FH(\))p Fn(,)35 b(and)f(\(for)h(al)5 b(l)34 b FA(\022)d FF(2)d FG(T)1752 3911 y FE(2)1827 3947 y Fn(and)34 b(any)h FA(\021)c FF(2)d FA(I)2419 3962 y FE(0)2459 3947 y Fn(\))789 4167 y FF(k)p FA(u)895 4194 y Fk(\022)917 4209 y Fw(1)960 4167 y FF(k)1010 4182 y Fy(\030)1041 4190 y Fe(\003)1109 4167 y FF(\024)g FA(\027)41 b(;)216 b FF(j)p FA(\032)1624 4182 y Fj(\003)1663 4167 y FF(j)28 b(\024)g FA(\013)36 b(;)1598 b FH(\(2.74\))789 4312 y FF(j)p FA(\015)27 b FF(\000)c FA(\014)6 b FF(j)17 b FA(;)32 b FF(k)p FA(u)1266 4339 y Fk(\022)1288 4354 y Fw(1)1353 4312 y FF(\000)22 b FA(v)1499 4339 y Fk(\022)1521 4354 y Fw(1)1565 4312 y FF(k)1615 4327 y Fy(\030)1646 4335 y Fe(\003)1702 4312 y FA(;)34 b FF(j)p FA(\032)22 b FF(\000)g FA(\032)2012 4327 y Fj(\003)2052 4312 y FF(j)17 b FA(;)33 b FF(k)p FA(W)i FF(\000)23 b FA(W)2526 4327 y Fj(\003)2565 4312 y FF(k)2615 4327 y Fy(\030)2646 4335 y Fe(\003)2714 4312 y FF(\024)28 b FA(C)2889 4327 y Fj(\003)2945 4312 y FF(k)p FA(e)p FF(k)3090 4327 y Fy(\030)3162 4312 y FA(;)358 b FH(\(2.75\))17 4532 y Fn(wher)-5 b(e)1351 4676 y FA(C)1421 4691 y Fj(\003)1488 4676 y FH(:=)1633 4609 y FA(C)p 1629 4653 86 4 v 1629 4744 a FH(2)1678 4716 y Fy(s)1759 4565 y Fm(\020)1819 4676 y FH(1)22 b FF(\000)39 b Ff(e)2059 4635 y Fj(\000)2162 4608 y Fw(1)p 2124 4620 109 3 v 2134 4661 a Fc(e)10 b Fk(m)2246 4565 y Fm(\021)2305 4588 y Fj(\000)p FE(1)2435 4676 y FA(:)1858 5645 y FH(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 17 548 a FI(Pro)s(of)37 b FH(W)-8 b(e)33 b(claim)g(that)f(\(1.24\))g(implies)i(that,)f(for)f(an)m(y)h FA(i)28 b FF(\025)g FH(1,)573 784 y(\()p FA(v)658 799 y Fy(i)686 784 y FA(;)17 b(\014)785 799 y Fy(i)813 784 y FH(\))28 b(:=)f FF(K)1086 743 y Fy(i)1115 784 y FH(\()p FA(v)t(;)17 b(\014)6 b FH(\))26 b FF(2)i(W)1580 719 y FE(\026)1573 736 y Fy(\030)1565 811 y(\030)1596 821 y Fk(i)3547 784 y FH(\(2.76\))573 953 y(max)p FF(fj)846 927 y FH(^)832 953 y FA(\014)887 968 y Fy(i)915 953 y FF(j)p FA(;)17 b FF(k)p FA(@)1088 980 y Fk(\022)1110 995 y Fw(1)1153 953 y FA(w)1223 968 y Fy(i)1251 953 y FF(k)1301 968 y Fy(\030)1332 978 y Fk(i)1362 953 y FA(;)g FF(j)p FA(\032)1484 968 y Fy(i)1534 953 y FF(\000)23 b FA(\032)1684 968 y Fy(i)p Fj(\000)p FE(1)1802 953 y FF(j)p FA(;)17 b FH(sup)1908 1040 y Fl(T)1952 1021 y Fw(2)2037 953 y FF(j)p FA(W)2157 968 y Fy(i)2207 953 y FF(\000)23 b FA(W)2399 968 y Fy(i)p Fj(\000)p FE(1)2517 953 y FF(j)p FA(;)17 b FH(sup)2623 1040 y Fl(T)2667 1021 y Fw(2)2752 953 y FF(j)p FA(W)2886 912 y Fj(\000)p FE(1)2872 979 y Fy(i)3002 953 y FF(\000)23 b FA(W)3208 912 y Fj(\000)p FE(1)3194 979 y Fy(i)p Fj(\000)p FE(1)3312 953 y FF(j)17 b(g)1028 1243 y(\024)1143 1176 y FH(\()p FA(C)7 b FF(k)p FA(e)p FF(k)1403 1191 y Fy(\030)1441 1176 y FH(\))1479 1140 y FE(2)1514 1116 y Fk(i)p Fe(\000)p Fw(1)p 1143 1220 480 4 v 1340 1311 a FH(2)1389 1283 y Fy(s)3547 1243 y FH(\(2.77\))573 1505 y FA(\022)31 b FF(!)c FA(e)821 1520 y Fy(i)849 1505 y FH(\()p FA(\022)s FH(\))h(:=)g FF(F)1204 1520 y Fy(\021)1245 1505 y FH(\()p FA(v)1330 1520 y Fy(i)1358 1505 y FH(;)17 b FA(\014)1457 1520 y Fy(i)1485 1505 y FH(\))28 b FF(2)g(H)1730 1464 y Fy(\030)1761 1474 y Fk(i)1889 1505 y FH(and)98 b FF(k)p FA(e)2239 1520 y Fy(i)2267 1505 y FF(k)2317 1520 y Fy(\030)2348 1530 y Fk(i)2406 1505 y FF(\024)2521 1437 y FH(\()p FA(C)7 b FF(k)p FA(e)p FF(k)2781 1452 y Fy(\030)2819 1437 y FH(\))2857 1401 y FE(2)2892 1378 y Fk(i)p 2521 1482 402 4 v 2614 1573 a FA(C)24 b FH(2)2757 1544 y Fy(s)11 b(i)3547 1505 y FH(\(2.78\))17 1756 y(where)27 b FA(W)384 1771 y FE(0)451 1756 y FH(:=)h FA(W)14 b FH(,)26 b FA(\032)791 1771 y FE(0)858 1756 y FH(:=)i FA(\032)p FH(.)e(W)-8 b(e)27 b(pro)m(v)m(e)g(the)g(claim)g(b)m(y)g(induction.)g(First)f(of)g (all,)g(observ)m(e)i(that)e(\(1.24\))17 1876 y(implies)34 b(immediately)g(that)1082 1840 y FE(19)1218 2096 y FH(2)1267 2055 y Fy(k)1309 2096 y FH(\()p FA(C)7 b FF(k)p FA(e)p FF(k)1569 2111 y Fy(\030)1607 2096 y FH(\))1645 2055 y FE(2)1680 2031 y Fk(k)1750 2096 y FF(\024)28 b FA(m)33 b(;)212 b FF(8)33 b FA(k)d FF(\025)f FH(0)j FA(:)952 b FH(\(2.79\))17 2316 y(No)m(w,)31 b(let)g(us)g(c)m(hec)m(k)h(\(2.76\)) p FF(\004)p FH(\(2.78\))e(for)g FA(i)e FH(=)f(1.)j(By)h(Lemma)g(3,)g (part)f(\(i\),)g(with)h FA(\016)h FH(:=)27 b FA(\016)3251 2331 y FE(1)3291 2316 y FH(,)k FA(\030)22 b FF(\000)c FA(\016)32 b FH(=)27 b FA(\030)3730 2331 y FE(1)3769 2316 y FH(,)17 2436 y(\()p FA(v)106 2400 y Fj(0)129 2436 y FA(;)17 b(\014)234 2400 y Fj(0)256 2436 y FH(\))36 b(=)g(\()p FA(v)527 2451 y FE(1)567 2436 y FA(;)17 b(\014)666 2451 y FE(1)705 2436 y FH(\))36 b(=)g FF(K)q FH(\()p FA(v)t(;)17 b(\014)6 b FH(\))36 b(=)g(\()p FA(v)29 b FH(+)c FA(w)1633 2451 y FE(1)1673 2436 y FA(;)17 b(\014)30 b FH(+)1918 2410 y(^)1904 2436 y FA(\014)1959 2451 y FE(1)1998 2436 y FH(\),)38 b(b)m(y)g(de\014nition)h(of)e FA(C)7 b FH(,)38 b FA(\016)2981 2451 y FE(1)3020 2436 y FH(,)g FA(m)g FH(and)f FA(\026)h FH(and)f(b)m(y)17 2557 y(\(2.79\))32 b(\(with)h FA(k)d FH(=)e(0\),)k(w)m(e)i(ha)m(v)m(e) 1007 2832 y FF(j)1049 2805 y FH(^)1035 2832 y FA(\014)1090 2847 y FE(1)1129 2832 y FF(j)17 b FA(;)32 b FF(k)p FA(@)1334 2858 y Fk(\022)1356 2873 y Fw(1)1400 2832 y FA(w)1470 2847 y FE(1)1509 2832 y FF(k)1559 2847 y Fy(\030)1590 2856 y Fw(1)1656 2832 y FF(\024)c FA(c\016)1850 2791 y Fj(\000)p Fy(s)1846 2856 y FE(1)1942 2832 y FF(k)p FA(e)p FF(k)2087 2847 y Fy(\030)2152 2832 y FH(=)2266 2764 y FA(C)7 b FF(k)p FA(e)p FF(k)2488 2779 y Fy(\030)p 2266 2809 260 4 v 2353 2900 a FH(2)2402 2871 y Fy(s)2563 2832 y FF(\024)2678 2764 y FA(\026)p 2678 2809 59 4 v 2683 2900 a FH(2)2779 2832 y FA(:)741 b FH(\(2.80\))17 3077 y(In)33 b(particular)1390 3260 y FF(k)p FA(@)1491 3287 y Fk(\022)1513 3302 y Fw(1)1556 3260 y FA(w)1626 3275 y FE(1)1665 3260 y FF(k)1715 3275 y Fy(\030)1746 3284 y Fw(1)1813 3260 y FF(\024)1928 3180 y FA(\027)28 b FF(\000)23 b(k)p FA(v)2201 3206 y Fk(\022)2223 3221 y Fw(1)2266 3180 y FF(k)2316 3195 y Fy(\030)p 1928 3237 427 4 v 2116 3328 a FH(2)2396 3260 y FA(;)17 3465 y FH(whic)m(h)43 b(allo)m(ws)h(to)d(apply)i(part)f(\(ii\))g(of)g(Lemma)h(3,)f(with)h FA(W)2340 3429 y Fj(0)2407 3465 y FH(=)h FA(W)2619 3480 y FE(1)2658 3465 y FH(,)e FA(e)2772 3429 y Fj(0)2840 3465 y FH(=)i FA(e)3005 3480 y FE(1)3044 3465 y FH(,)f FA(\032)3164 3429 y Fj(0)3231 3465 y FH(=)h FA(\032)3401 3480 y FE(1)3441 3465 y FH(,)e(and)h(to)17 3586 y(obtain,)32 b(as)h(ab)s(o)m(v)m(e,)564 3859 y FF(k)p FA(W)706 3874 y FE(1)767 3859 y FF(\000)23 b FA(W)14 b FF(k)1023 3874 y Fy(\030)1054 3883 y Fw(1)1120 3859 y FF(\024)1235 3792 y FA(C)7 b FF(k)p FA(e)p FF(k)1457 3807 y Fy(\030)p 1235 3837 260 4 v 1322 3928 a FH(2)1371 3899 y Fy(s)1532 3859 y FF(\024)1647 3792 y FA(\026)p 1647 3837 59 4 v 1652 3928 a FH(2)1748 3859 y FA(;)212 b FF(k)p FA(e)2082 3874 y FE(1)2122 3859 y FF(k)2172 3874 y Fy(\030)2203 3883 y Fw(1)2269 3859 y FF(\024)2388 3792 y FA(C)p 2384 3837 86 4 v 2384 3928 a FH(2)2433 3899 y Fy(s)2479 3859 y FF(k)p FA(e)p FF(k)2624 3818 y FE(2)2624 3884 y Fy(\030)2691 3859 y FH(=)2805 3792 y(\()p FA(C)7 b FF(k)p FA(e)p FF(k)3065 3807 y Fy(\030)3102 3792 y FH(\))3140 3756 y FE(2)p 2805 3837 375 4 v 2903 3928 a FA(C)23 b FH(2)3045 3899 y Fy(s)3222 3859 y FA(;)298 b FH(\(2.81\))17 4115 y(and,)32 b(since)i FA(\026)28 b FF(\024)g FA(\026)723 4130 y FE(0)762 4115 y FA(=)p FH(\(2)p FA(\026)957 4079 y FE(2)957 4139 y(1)996 4115 y FH(\),)k(b)m(y)i(\(2.62\))e(\(recall)h(the)g(iden)m(tit)m(y)h (in)f(\(2.68\)\),)f(w)m(e)h(ha)m(v)m(e)h(also)1178 4379 y(inf)1198 4447 y Fl(T)1242 4428 y Fw(2)1313 4379 y FA(W)1405 4394 y FE(1)1472 4379 y FA(>)1634 4312 y FH(1)p 1585 4356 147 4 v 1585 4447 a(2)p FA(\026)1693 4462 y FE(1)1775 4379 y FA(;)211 b FF(h)p FA(W)2158 4338 y Fj(\000)p FE(1)2144 4404 y(1)2252 4379 y FF(i)28 b FA(>)2432 4312 y(\026)2491 4327 y FE(0)p 2432 4356 99 4 v 2457 4447 a FH(2)1178 4642 y(sup)1212 4729 y Fl(T)1256 4710 y Fw(2)1341 4642 y FF(j)p FA(W)1475 4601 y Fj(\000)p FE(1)1461 4667 y(1)1591 4642 y FF(\000)23 b FA(W)1797 4601 y Fj(\000)p FE(1)1891 4642 y FF(j)17 b FA(;)32 b FF(j)p FA(\032)2073 4657 y FE(1)2135 4642 y FF(\000)23 b FA(\032)p FF(j)44 b(\024)2472 4575 y FA(C)7 b FF(k)p FA(e)p FF(k)2694 4590 y Fy(\030)p 2472 4619 260 4 v 2559 4710 a FH(2)2608 4682 y Fy(s)2774 4642 y FA(;)p 17 4807 1512 4 v 96 4868 a Fu(19)166 4898 y FN(If)34 b Ft(x)f FN(and)g Ft(y)j FN(are)c(p)r(ositiv)n(e)g(n)n(um)n (b)r(ers)h(suc)n(h)f(that)i Ft(x)e Fx(\024)g FN(exp\()p Fx(\000)p FN(1)p Ft(=)p FN(\(e)p Ft(y)s FN(\)\),)g(then)i Ft(tx)2742 4868 y Fs(t)2803 4898 y Fx(\024)e Ft(y)k FN(for)c(an)n(y)h Ft(t)f(>)f FN(0.)i(In)g(fact,)17 4998 y(let)28 b Ft(\025)c FN(:=)f(log)15 b Ft(x)489 4968 y Fh(\000)p Fu(1)606 4998 y FN(and)28 b(observ)n(e)e(that)j(the)f(h)n(yp)r(othesis)f(is)h(equiv) -5 b(alen)n(t)28 b(to)2411 4965 y Fu(1)p 2388 4979 81 4 v 2388 5027 a(e)11 b Fs(\025)2502 4998 y Fx(\024)23 b Ft(y)s FN(.)28 b(Then,)g Ft(tx)3002 4968 y Fs(t)3055 4998 y FN(=)3157 4965 y Fu(1)p 3154 4979 40 4 v 3154 5027 a Fs(\025)3217 4998 y FN(\()p Ft(\025t)p FN(\))14 b(exp\()p Fx(\000)p Ft(\025t)p FN(\))25 b Fx(\024)45 5065 y Fu(1)p 27 5079 69 4 v 27 5126 a(e)p Fs(\025)129 5098 y Fx(\024)d Ft(y)s FN(.)1858 5645 y FH(26)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 17 548 a FH(whic)m(h,)40 b(together)f(with)g (\(2.80\))f(and)h(\(2.81\),)f(pro)m(v)m(e)h(\(2.77\))f(and)h(\(2.78\))f (for)g FA(i)g FH(=)g(1.)g(Finally)-8 b(,)39 b(from)17 668 y(the)33 b(ab)s(o)m(v)m(e)g(estimates)h(and)f(de\014nitions,)h (there)f(follo)m(ws)575 948 y FF(k)p FA(@)676 975 y Fk(\022)698 990 y Fw(1)741 948 y FA(v)788 963 y FE(1)828 948 y FF(k)878 963 y Fy(\030)909 972 y Fw(1)975 948 y FF(\024)28 b(k)p FA(v)1177 975 y Fk(\022)1199 990 y Fw(1)1243 948 y FF(k)1293 963 y Fy(\030)1352 948 y FH(+)22 b FF(k)p FA(@)1551 975 y Fk(\022)1573 990 y Fw(1)1617 948 y FA(w)1687 963 y FE(1)1726 948 y FF(k)1776 963 y Fy(\030)1807 972 y Fw(1)1873 948 y FF(\024)28 b(k)p FA(v)2075 975 y Fk(\022)2097 990 y Fw(1)2141 948 y FF(k)2191 963 y Fy(\030)2250 948 y FH(+)2358 868 y FA(\027)h FF(\000)23 b(k)p FA(v)2632 894 y Fk(\022)2654 909 y Fw(1)2697 868 y FF(k)2747 883 y Fy(\030)p 2358 925 427 4 v 2547 1016 a FH(2)2822 948 y FA(<)28 b(\027)34 b(<)3121 922 y FH(\026)3111 948 y FA(\030)27 b FF(\000)22 b FA(\030)37 b(;)575 1192 y FF(j)p FA(\032)653 1207 y FE(1)692 1192 y FF(j)28 b(\024)g(j)p FA(\032)p FF(j)22 b FH(+)g FF(j)p FA(\032)1157 1207 y FE(1)1218 1192 y FF(\000)h FA(\032)p FF(j)28 b(\024)g(j)p FA(\032)p FF(j)21 b FH(+)1764 1124 y FA(\026)p 1764 1169 59 4 v 1769 1260 a FH(2)1861 1192 y FF(\024)28 b(j)p FA(\032)p FF(j)22 b FH(+)2202 1124 y FA(\013)h FF(\000)f(j)p FA(\032)p FF(j)p 2202 1169 291 4 v 2322 1260 a FH(2)2529 1192 y FA(<)28 b(\013)g(<)g FH(1)k FA(;)17 1468 y FH(sho)m(wing)h(that) g(\()p FA(v)685 1483 y FE(1)724 1468 y FA(;)17 b(\014)823 1483 y FE(1)863 1468 y FH(\))27 b FF(2)h(W)1135 1403 y FE(\026)1128 1421 y Fy(\030)1120 1496 y(\030)1151 1505 y Fw(1)1190 1468 y FH(,)33 b(pro)m(ving)g(also)g(\(2.76\))f(for)g FA(i)c FH(=)f(1.)17 1613 y(Let,)k(no)m(w,)i FA(j)g FF(\025)28 b FH(1)k(and)f(assume)i(that)e(\(2.76\))p FF(\004)p FH(\(2.78\))g(hold) g(true)h(for)f(1)c FF(\024)i FA(i)e FF(\024)i FA(j)6 b FH(;)31 b(w)m(e)i(w)m(an)m(t)f(to)f(pro)m(v)m(e)17 1733 y(\(2.76\))p FF(\004)p FH(\(2.78\))h(for)g FA(i)c FH(=)g FA(j)h FH(+)22 b(1.)33 b(Let)g(1)28 b FF(\024)g FA(i)h FF(\024)f FA(j)6 b FH(;)33 b(then,)h(b)m(y)g(de\014nition)f(of)g FA(v)2824 1748 y Fy(i)2852 1733 y FH(,)g(b)m(y)h(\(2.77\))e(and)h (\(2.79\),)17 1854 y(there)g(follo)m(ws)1087 2137 y FF(k)p FA(@)1188 2164 y Fk(\022)1210 2179 y Fw(1)1254 2137 y FA(v)1301 2152 y Fy(i)1329 2137 y FF(k)1379 2152 y Fy(\030)1410 2162 y Fk(i)1524 2137 y FH(=)84 b FF(k)p FA(v)1781 2164 y Fk(\022)1803 2179 y Fw(1)1868 2137 y FH(+)2026 2013 y Fy(i)1966 2043 y Fm(X)1974 2255 y Fy(k)r FE(=1)2127 2137 y FA(@)2178 2164 y Fk(\022)2200 2179 y Fw(1)2244 2137 y FA(w)2314 2152 y Fy(k)2356 2137 y FF(k)2406 2152 y Fy(\030)2437 2162 y Fk(i)1523 2476 y FF(\024)g(k)p FA(v)1781 2503 y Fk(\022)1803 2518 y Fw(1)1846 2476 y FF(k)1896 2491 y Fy(\030)1956 2476 y FH(+)2114 2352 y Fy(i)2054 2382 y Fm(X)2061 2594 y Fy(k)r FE(=1)2214 2476 y FF(k)p FA(@)2315 2503 y Fk(\022)2337 2518 y Fw(1)2381 2476 y FA(w)2451 2491 y Fy(k)2493 2476 y FF(k)2543 2491 y Fy(\030)2574 2503 y Fk(k)1523 2816 y FF(\024)g(k)p FA(v)1781 2842 y Fk(\022)1803 2857 y Fw(1)1846 2816 y FF(k)1896 2831 y Fy(\030)1956 2816 y FH(+)2114 2691 y Fy(i)2054 2721 y Fm(X)2061 2933 y Fy(k)r FE(=1)2224 2748 y FH(\()p FA(C)7 b FF(k)p FA(e)p FF(k)2484 2763 y Fy(\030)2522 2748 y FH(\))2560 2712 y FE(2)2595 2689 y Fk(k)q Fe(\000)p Fw(1)p 2224 2793 492 4 v 2427 2884 a FH(2)2476 2855 y Fy(s)1523 3155 y FF(\024)84 b(k)p FA(v)1781 3181 y Fk(\022)1803 3196 y Fw(1)1846 3155 y FF(k)1896 3170 y Fy(\030)1956 3155 y FH(+)2109 3087 y FA(m)p 2064 3132 176 4 v 2064 3223 a FH(2)2113 3194 y Fy(s)p Fj(\000)p FE(1)2326 3030 y Fy(i)2266 3060 y Fm(X)2274 3272 y Fy(k)r FE(=1)2458 3087 y FH(1)p 2437 3132 92 4 v 2437 3223 a(2)2486 3194 y Fy(k)1524 3494 y FH(=)g FF(k)p FA(v)1781 3520 y Fk(\022)1803 3535 y Fw(1)1846 3494 y FF(k)1896 3509 y Fy(\030)1956 3494 y FH(+)22 b FA(\026)2189 3369 y Fy(i)2129 3399 y Fm(X)2137 3611 y Fy(k)r FE(=1)2321 3426 y FH(1)p 2300 3471 V 2300 3562 a(2)2349 3533 y Fy(k)1524 3729 y FA(<)84 b FF(k)p FA(v)1781 3756 y Fk(\022)1803 3771 y Fw(1)1846 3729 y FF(k)1896 3744 y Fy(\030)1956 3729 y FH(+)22 b FA(\026)27 b(<)h(\027)39 b(:)1189 b FH(\(2.82\))17 3946 y(Analogously)-8 b(,)33 b(for)f(an)m(y)h(1)28 b FF(\024)g FA(i)g FF(\024)g FA(j)6 b FH(,)1298 4249 y FF(j)p FA(\032)1376 4264 y Fy(i)1404 4249 y FF(j)84 b FH(=)f FF(j)p FA(\032)22 b FH(+)1933 4125 y Fy(i)1874 4155 y Fm(X)1881 4367 y Fy(k)r FE(=1)2034 4249 y FA(\032)2084 4264 y Fy(k)2149 4249 y FF(\000)h FA(\032)2299 4264 y Fy(k)r Fj(\000)p FE(1)2432 4249 y FF(j)1515 4588 y(\024)83 b(j)p FA(\032)p FF(j)22 b FH(+)1961 4464 y Fy(i)1901 4494 y Fm(X)1909 4706 y Fy(k)r FE(=1)2062 4588 y FF(j)p FA(\032)2140 4603 y Fy(k)2204 4588 y FF(\000)h FA(\032)2354 4603 y Fy(k)r Fj(\000)p FE(1)2487 4588 y FF(j)1516 4824 y FA(<)83 b FF(j)p FA(\032)p FF(j)22 b FH(+)g FA(\026)28 b(<)f(\013)33 b(:)1334 b FH(\(2.83\))17 5041 y(Th)m(us,)34 b(w)m(e)g(can)f(apply)g (Lemma)h(3,)e(part)h(\(i\))f(\(with)h FA(\016)f FH(:=)c FA(\016)2195 5056 y Fy(j)t FE(+1)2322 5041 y FH(,)33 b FA(\030)f FH(=)c FA(\030)2604 5056 y Fy(j)2640 5041 y FH(,)33 b FA(v)t FH(,)f FA(\014)6 b FH(,)33 b FA(\032)g FH(replaced)g(b)m(y)h FA(v)3581 5056 y Fy(j)3618 5041 y FH(,)f FA(\014)3733 5056 y Fy(j)3769 5041 y FH(,)17 5173 y FA(\032)67 5188 y Fy(j)104 5173 y FH(,)j(\()p FA(v)256 5137 y Fj(0)279 5173 y FA(;)17 b(\014)384 5137 y Fj(0)406 5173 y FH(\))34 b(=)g(\()p FA(v)673 5188 y Fy(j)t FE(+1)800 5173 y FA(;)17 b(\014)899 5188 y Fy(j)t FE(+1)1026 5173 y FH(\))34 b(=)g FF(K)q FH(\()p FA(v)1370 5188 y Fy(j)1407 5173 y FA(;)17 b(\014)1506 5188 y Fy(j)1542 5173 y FH(\))34 b(=)g(\()p FA(v)1809 5188 y Fy(j)1870 5173 y FH(+)25 b FA(w)2041 5188 y Fy(j)t FE(+1)2167 5173 y FA(;)17 b(\014)2266 5188 y Fy(j)2327 5173 y FH(+)2442 5146 y(^)2428 5173 y FA(\014)2483 5188 y Fy(j)t FE(+1)2610 5173 y FH(\)\),)36 b(whic)m(h,)i(in)e(view)i(of)e(\(2.56\),)1858 5645 y(27)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 17 548 a FH(the)33 b(iden)m(tit)m(y)1579 711 y FA(c)17 b(\016)1685 670 y Fj(\000)p Fy(s)1681 737 y(i)p FE(+1)1827 711 y FH(=)1945 644 y FA(C)p 1941 688 86 4 v 1941 779 a FH(2)1990 751 y Fy(s)2053 711 y FH(2)2102 670 y Fy(s)11 b(i)2207 711 y FA(;)1313 b FH(\(2.84\))17 915 y(and)32 b(\(2.78\),)g(yields)1090 1132 y FF(j)1132 1106 y FH(^)1118 1132 y FA(\014)1173 1147 y Fy(j)t FE(+1)1300 1132 y FF(j)17 b FA(;)32 b FF(k)p FA(@)1505 1159 y Fk(\022)1527 1174 y Fw(1)1571 1132 y FA(w)1641 1147 y Fy(j)t FE(+1)1767 1132 y FF(k)1817 1147 y Fy(\030)1848 1157 y Fk(j)s Fw(+1)2045 1132 y FF(\024)83 b FA(c)17 b(\016)2311 1091 y Fj(\000)p Fy(s)2307 1158 y(j)t FE(+1)2466 1132 y FF(k)p FA(e)2561 1147 y Fy(j)2598 1132 y FF(k)2648 1147 y Fy(\030)2679 1157 y Fk(j)2046 1343 y FH(=)2220 1275 y FA(C)p 2215 1320 V 2215 1411 a FH(2)2264 1382 y Fy(s)2327 1343 y FH(2)2376 1301 y Fy(s)12 b(j)2474 1343 y FF(k)p FA(e)2569 1358 y Fy(j)2605 1343 y FF(k)2655 1358 y Fy(\030)2686 1368 y Fk(j)2045 1604 y FF(\024)2215 1537 y FH(\()p FA(C)7 b FF(k)p FA(e)p FF(k)2475 1552 y Fy(\030)2513 1537 y FH(\))2551 1501 y FE(2)2586 1477 y Fk(j)p 2215 1581 408 4 v 2376 1672 a FH(2)2425 1644 y Fy(s)2665 1604 y FA(;)855 b FH(\(2.85\))17 1852 y(whic)m(h)30 b(pro)m(v)m(es)h(the)e(\014rst)h(t) m(w)m(o)g(b)s(ounds)f(in)h(\(2.77\))e(with)i FA(i)e FH(=)f FA(j)21 b FH(+)15 b(1.)29 b(T)-8 b(o)29 b(apply)g(part)g(\(ii\))g(of)g (Lemma)g(3,)17 1973 y(w)m(e)k(ha)m(v)m(e)h(to)e(c)m(hec)m(k)j(\(2.57\)) d(with)h FA(w)i FH(and)e FA(v)j FH(replaced,)e(resp)s(ectiv)m(ely)-8 b(,)36 b(b)m(y)d FA(w)2817 1988 y Fy(j)t FE(+1)2976 1973 y FH(and)g FA(v)3213 1988 y Fy(j)3249 1973 y FH(.)g(Since)821 2284 y FF(k)p FA(@)922 2311 y Fk(\022)944 2326 y Fw(1)988 2284 y FA(v)1035 2299 y Fy(j)1072 2284 y FF(k)1122 2299 y Fy(\030)1153 2309 y Fk(j)1211 2284 y FH(+)22 b FF(k)p FA(@)1410 2311 y Fk(\022)1432 2326 y Fw(1)1476 2284 y FA(w)1546 2299 y Fy(j)t FE(+1)1672 2284 y FF(k)1722 2299 y Fy(\030)1753 2309 y Fk(j)s Fw(+1)1894 2284 y FF(\024)29 b(k)p FA(v)2097 2311 y Fk(\022)2119 2326 y Fw(1)2162 2284 y FF(k)2212 2299 y Fy(\030)2271 2284 y FH(+)2384 2159 y Fy(i)p FE(+1)2369 2189 y Fm(X)2377 2401 y Fy(k)r FE(=1)2530 2284 y FF(k)p FA(@)2631 2311 y Fk(\022)2653 2326 y Fw(1)2696 2284 y FA(w)2766 2299 y Fy(k)2809 2284 y FF(k)2859 2299 y Fy(\030)2890 2311 y Fk(k)2964 2284 y FA(;)17 2591 y FH(\(2.85\))c(sho)m(ws)j(that)e(the)g(inequalities)i (in)f(\(2.82\))e(hold)h(also)g(for)g FA(i)i FH(=)f FA(j)14 b FH(+)9 b(1)27 b(so)f(that)g(\(2.57\))f(is)i(satis\014ed.)17 2712 y(Th)m(us,)42 b(b)m(y)f(\(2.58\))e(\(with)i FA(W)1105 2675 y Fj(0)1168 2712 y FH(and)f FA(W)54 b FH(corresp)s(onding,)41 b(resp)s(ectiv)m(ely)-8 b(,)43 b(to)d FA(W)2949 2727 y Fy(j)t FE(+1)3116 2712 y FH(and)g FA(W)3405 2727 y Fy(j)3442 2712 y FH(\),)g(\(2.84\))17 2832 y(and)32 b(b)m(y)i(\(2.78\)) e(with)h FA(i)28 b FH(=)f FA(j)6 b FH(,)33 b(w)m(e)h(see)f(that)627 3122 y FF(k)p FA(W)769 3137 y Fy(j)t FE(+1)918 3122 y FF(\000)22 b FA(W)1109 3137 y Fy(j)1146 3122 y FF(k)1196 3137 y Fy(\030)1227 3147 y Fk(j)s Fw(+1)1368 3122 y FF(\024)28 b FA(c)17 b(\016)1579 3081 y Fj(\000)p Fy(s)1575 3147 y(i)p FE(+1)1693 3122 y FF(k)p FA(e)1788 3137 y Fy(j)1825 3122 y FF(k)1875 3137 y Fy(\030)1906 3147 y Fk(j)1970 3122 y FH(=)2088 3055 y FA(C)p 2084 3099 86 4 v 2084 3190 a FH(2)2133 3161 y Fy(s)2196 3122 y FH(2)2245 3081 y Fy(s)11 b(i)2317 3122 y FF(k)p FA(e)2412 3137 y Fy(j)2449 3122 y FF(k)2499 3137 y Fy(\030)2530 3147 y Fk(j)2594 3122 y FF(\024)2709 3055 y FH(\()p FA(C)c FF(k)p FA(e)p FF(k)2969 3070 y Fy(\030)3006 3055 y FH(\))3044 3018 y FE(2)3079 2995 y Fk(j)p 2709 3099 408 4 v 2870 3190 a FH(2)2919 3161 y Fy(s)3159 3122 y FA(;)361 b FH(\(2.86\))17 3370 y(sho)m(wing)30 b(that)g(\(2.77\))f(for)g FF(k)p FA(W)1161 3385 y Fy(i)1205 3370 y FF(\000)16 b FA(W)1390 3385 y Fy(i)p Fj(\000)p FE(1)1509 3370 y FF(k)29 b FH(holds)h(also)g (with)g FA(i)e FH(=)f FA(j)22 b FH(+)16 b(1.)30 b(No)m(w,)g(b)m(y)g (\(2.59\),)f(\(2.84\))g(and)17 3490 y(\(2.78\))j(with)h FA(i)28 b FH(=)f FA(j)6 b FH(,)33 b(w)m(e)g(\014nd)898 3780 y FF(k)p FA(e)993 3795 y Fy(j)t FE(+1)1120 3780 y FF(k)1170 3795 y Fy(\030)1201 3805 y Fk(j)s Fw(+1)1342 3780 y FF(\024)1461 3713 y FA(C)p 1457 3757 86 4 v 1457 3849 a FH(2)1506 3820 y Fy(s)1569 3780 y FH(2)1618 3739 y Fy(s)11 b(j)1725 3713 y FH(\()p FA(C)c FF(k)p FA(e)p FF(k)1985 3728 y Fy(\030)2023 3713 y FH(\))2061 3677 y FE(2)2096 3653 y Fk(j)s Fw(+1)p 1725 3757 485 4 v 1819 3849 a FA(C)1896 3820 y FE(2)1952 3849 y FH(2)2001 3820 y FE(2)p Fy(s)k(j)2247 3780 y FH(=)2361 3713 y(\()p FA(C)c FF(k)p FA(e)p FF(k)2621 3728 y Fy(\030)2658 3713 y FH(\))2696 3677 y FE(2)2731 3653 y Fk(j)s Fw(+1)p 2361 3757 V 2425 3849 a FA(C)23 b FH(2)2567 3820 y Fy(s)p FE(\()p Fy(j)t FE(+1\))2888 3780 y FA(;)632 b FH(\(2.87\))17 4028 y(i.e.,)30 b(\(2.78\))f(with)h FA(i)e FH(=)g FA(j)22 b FH(+)16 b(1.)30 b(Next,)g(from)g(\(2.86\),)f(\(2.79\),)g(the)h(fact)g(that)f FA(j)34 b FF(\025)28 b FH(1)h(and)h(the)g(de\014nition)17 4149 y(of)i(the)h FA(\026)p FH('s,)g(there)g(follo)m(ws)1161 4433 y FF(k)p FA(W)1303 4448 y Fy(j)t FE(+1)1452 4433 y FF(\000)22 b FA(W)1643 4448 y Fy(j)1680 4433 y FF(k)1730 4448 y Fy(\030)1761 4458 y Fk(j)s Fw(+1)1957 4433 y FF(\024)2128 4366 y FH(\()p FA(C)7 b FF(k)p FA(e)p FF(k)2388 4381 y Fy(\030)2425 4366 y FH(\))2463 4330 y FE(2)2498 4306 y Fk(j)p 2128 4410 408 4 v 2289 4502 a FH(2)2338 4473 y Fy(s)1957 4645 y FF(\024)2179 4578 y FA(m)p 2128 4622 188 4 v 2128 4713 a FH(2)2177 4685 y Fy(s)2230 4713 y FH(2)2279 4685 y Fy(j)2353 4645 y FH(=)2525 4578 y FA(\026)p 2467 4622 176 4 v 2467 4713 a FH(2)2516 4685 y Fy(j)t FE(+1)1957 4857 y FF(\024)2128 4790 y FA(\026)p 2128 4834 59 4 v 2133 4925 a FH(4)2224 4857 y FF(\024)2364 4790 y FA(\026)2423 4805 y FE(0)p 2339 4834 147 4 v 2339 4925 a FH(4)p FA(\026)2447 4891 y FE(2)2447 4950 y(1)1958 5129 y FH(=)2128 5062 y(1)p 2128 5106 49 4 v 2128 5198 a(4)2213 5062 y(\(1)22 b FF(\000)h FA(\027)6 b FH(\))2514 5026 y FE(4)p 2213 5106 341 4 v 2214 5198 a FH(\(1)22 b(+)g FA(\027)6 b FH(\))2513 5169 y FE(2)2596 5129 y FA(;)1858 5645 y FH(28)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 17 548 a FH(sho)m(wing)33 b(that)g(\(2.60\))f (holds)h(in)g(the)g(presen)m(t)h(case.)g(Therefore,)f(b)m(y)h (\(2.61\),)955 827 y(inf)975 895 y Fl(T)1019 876 y Fw(2)1090 827 y FA(W)1182 842 y Fy(j)t FE(+1)1336 827 y FA(>)1450 760 y FH(\(1)22 b FF(\000)g FA(\027)6 b FH(\))1750 724 y FE(2)p 1450 804 341 4 v 1596 895 a FH(2)1833 827 y FA(;)212 b FF(h)p FA(W)2217 786 y Fj(\000)p FE(1)2203 853 y Fy(j)t FE(+1)2329 827 y FF(i)27 b FA(>)2509 760 y FH(\(1)22 b(+)g FA(\027)6 b FH(\))2808 724 y FE(2)p 2509 804 339 4 v 2654 895 a FH(2)17 1082 y(and)637 1250 y(sup)671 1337 y Fl(T)715 1318 y Fw(2)800 1250 y FF(j)p FA(W)934 1209 y Fj(\000)p FE(1)920 1276 y Fy(j)t FE(+1)1069 1250 y FF(\000)23 b FA(W)1275 1209 y Fj(\000)p FE(1)1261 1276 y Fy(j)1369 1250 y FF(j)32 b FA(;)49 b FF(j)p FA(\032)1583 1265 y Fy(j)t FE(+1)1732 1250 y FF(\000)22 b FA(\032)1881 1265 y Fy(j)1918 1250 y FF(j)28 b(\024)g FA(c)17 b(\016)2185 1209 y Fj(\000)p Fy(s)2181 1276 y(j)t FE(+1)2307 1250 y FF(k)p FA(e)2402 1265 y Fy(j)2439 1250 y FF(k)2489 1265 y Fy(\030)2520 1275 y Fk(j)2584 1250 y FF(\024)2699 1183 y FH(\()p FA(C)7 b FF(k)p FA(e)p FF(k)2959 1198 y Fy(\030)2997 1183 y FH(\))3035 1147 y FE(2)3070 1123 y Fk(j)p 2699 1227 408 4 v 2860 1318 a FH(2)2909 1290 y Fy(s)3149 1250 y FA(;)371 b FH(\(2.88\))17 1484 y(sho)m(wing)33 b(that)f(\(2.77\))f(holds)i(for)e FA(i)d FH(=)g FA(j)e FH(+)21 b(1.)32 b(Finally)-8 b(,)32 b(b)m(y)h(\(2.88\),)f(one)g(sees)h (that)f(\(2.83\))f(holds)i(also)17 1604 y(for)f FA(i)c FH(=)f FA(j)h FH(+)22 b(1,)33 b(implying)g(that)1574 1725 y FF(j)p FA(\032)1652 1740 y Fy(j)t FE(+1)1778 1725 y FF(j)28 b FA(<)f(\013)h(<)g FH(1)k FA(;)17 1918 y FH(whic)m(h)i(sho)m (ws)g(that)e(\()p FA(v)872 1933 y Fy(j)t FE(+1)999 1918 y FA(;)17 b(\014)1098 1933 y Fy(j)t FE(+1)1225 1918 y FH(\))27 b FF(2)h(W)1497 1853 y FE(\026)1490 1871 y Fy(\030)1482 1946 y(\030)1513 1956 y Fk(j)s Fw(+1)1627 1918 y FH(.)33 b(The)h(claim)f(has)g(b)s(een)g(completely)h(pro)m(v)m(en.)17 2101 y(The)e(\(fast\))g(deca)m(y)g(of)g(\()p FA(C)7 b FF(k)p FA(e)p FF(k)p FH(\))1157 2065 y FE(2)1192 2042 y Fk(j)1260 2101 y FH(implies)33 b(that)e(\()p FA(v)1885 2116 y Fy(j)1922 2101 y FA(;)17 b(\014)2021 2116 y Fy(j)2057 2101 y FH(\))31 b(con)m(v)m(erge)j(uniformly)e(to)f(\()p FA(u;)17 b(\015)5 b FH(\))27 b FF(2)h(W)3551 2037 y FE(\026)3544 2054 y Fy(\030)3536 2129 y(\030)3567 2137 y Fe(\003)3639 2101 y FH(and)17 2222 y(that,)k(for)g(an)m(y)h FA(\021)f FF(2)c FA(I)805 2237 y FE(0)844 2222 y FH(,)1052 2442 y FF(F)1124 2457 y Fy(\021)1166 2442 y FH(\()p FA(u)p FH(;)17 b FA(\015)5 b FH(\))26 b(=)47 b(lim)1528 2502 y Fy(j)t Fj(!1)1718 2442 y FF(F)1790 2457 y Fy(\021)1832 2442 y FH(\()p FA(v)1917 2457 y Fy(j)1953 2442 y FH(;)17 b FA(\014)2052 2457 y Fy(j)2089 2442 y FH(\))27 b(=)47 b(lim)2258 2502 y Fy(j)t Fj(!1)2448 2442 y FA(e)2493 2457 y Fy(i)2549 2442 y FH(=)27 b(0)33 b FA(;)17 2708 y FH(sho)m(wing)g(that)g(\()p FA(u;)17 b(\015)5 b FH(\))31 b(is)j(a)e(solution)h(of)f(\(1.18\))g(for)g(an)m(y)h FA(\021)e FF(2)e FA(I)2355 2723 y FE(0)2394 2708 y FH(.)k(F)-8 b(urthermore,)33 b(since)1308 2928 y FF(k)p FA(@)1409 2955 y Fk(\022)1431 2970 y Fw(1)1474 2928 y FA(v)1521 2943 y Fy(i)1550 2928 y FF(k)1600 2943 y Fy(\030)1631 2953 y Fk(i)1689 2928 y FA(<)27 b(\027)39 b(;)212 b FF(j)p FA(\032)2196 2943 y Fy(i)2224 2928 y FF(j)27 b FA(<)h(\013)33 b(;)17 3148 y FH(w)m(e)46 b(see,)h(b)m(y)f(taking)g(limits,)g(that)f (\(1.25\))g(holds.)h(As)g(for)f(the)h(b)s(ounds)g(in)g(\(1.26\),)e(w)m (e)j(ha)m(v)m(e,)g(for)17 3268 y(example,)34 b(that)1188 3500 y FF(k)p FA(u)1294 3527 y Fk(\022)1316 3542 y Fw(1)1381 3500 y FF(\000)23 b FA(v)1528 3527 y Fk(\022)1550 3542 y Fw(1)1593 3500 y FF(k)1643 3515 y Fy(\030)1674 3523 y Fe(\003)1797 3500 y FF(\024)1957 3406 y Fm(X)1972 3616 y Fy(i)p Fj(\025)p FE(1)2118 3500 y FF(k)p FA(@)2219 3527 y Fk(\022)2241 3542 y Fw(1)2284 3500 y FA(w)2354 3515 y Fy(i)2382 3500 y FF(k)2432 3515 y Fy(\030)2463 3525 y Fk(i)1797 3802 y FF(\024)1986 3735 y FH(1)p 1967 3779 86 4 v 1967 3870 a(2)2016 3841 y Fy(s)2080 3707 y Fm(X)2094 3918 y Fy(i)p Fj(\025)p FE(1)2223 3802 y FH(\()p FA(C)7 b FF(k)p FA(e)p FF(k)2483 3817 y Fy(\030)2521 3802 y FH(\))2559 3761 y FE(2)2594 3737 y Fk(i)1797 4104 y FF(\024)1986 4036 y FH(1)p 1967 4081 V 1967 4172 a(2)2016 4143 y Fy(s)2096 4009 y Fm(X)2111 4220 y Fy(i)p Fj(\025)p FE(1)2240 4104 y FH(\()p FA(C)g FF(k)p FA(e)p FF(k)2500 4119 y Fy(\030)2538 4104 y FH(\))2576 4063 y Fy(i)1797 4349 y FF(\024)83 b FA(C)2027 4364 y Fj(\003)2083 4349 y FF(k)p FA(e)p FF(k)2228 4364 y Fy(\030)2298 4349 y FA(:)17 4569 y FH(whic)m(h)26 b(implies)g(the)f(second)g(inequalit)m(y) i(in)e(\(1.26\);)e(the)i(other)g(inequalities)i(are)d(obtained)h(in)g (exactly)17 4689 y(the)33 b(same)g(w)m(a)m(y)-8 b(.)p 764 4689 41 83 v 1858 5645 a(29)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 17 548 a FD(2.5)135 b(Lo)t(cal)45 b(uniqueness)17 756 y FH(Next,)36 b(w)m(e)h(pro)m(v)m(e)f(a)g(lo)s(cal) f(uniqueness)j(result)f(for)e(the)h(solutions)g(of)f(\(1.18\).)g(Suc)m (h)i(result)f(is)g(based)17 876 y(up)s(on)c(the)h(follo)m(wing)g (simple)h(observ)-5 b(ation)33 b(on)g(analytic)g(functions.)17 1089 y FI(Lemma)39 b(4)49 b Fn(L)-5 b(et)35 b FA(w)30 b FF(2)e(H)966 1053 y Fy(\030)997 1061 y Fe(\003)1072 1089 y Fn(and)34 b(assume)h(that)g(ther)-5 b(e)35 b(exist)f FA(c;)17 b(s)28 b(>)f FH(0)35 b Fn(such)g(that)1019 1296 y FF(k)p FA(w)s FF(k)1192 1311 y Fy(\030)s Fj(\000)p Fy(\016)1345 1296 y FF(\024)28 b FA(c)17 b FF(k)p FA(w)s FF(k)1682 1254 y FE(2)1682 1320 y Fy(\030)1755 1296 y FA(\016)1802 1254 y Fj(\000)p Fy(s)1928 1296 y FA(;)216 b FF(8)35 b FH(0)28 b FA(<)f(\016)32 b(<)c(\030)k FF(\024)c FA(\030)2843 1311 y Fj(\003)2917 1296 y FA(;)603 b FH(\(2.89\))1019 1586 y(\()o FA(c)17 b FH(4)1164 1545 y Fy(s)1217 1586 y FA(\030)1265 1545 y Fj(\000)p Fy(s)1260 1611 y Fj(\003)1356 1586 y FH(\))g FF(k)p FA(w)s FF(k)1584 1601 y Fy(\030)1615 1609 y Fe(\003)1682 1586 y FF(\024)28 b FH(1)35 b FA(:)1649 b FH(\(2.90\))17 1793 y Fn(Then)34 b FA(w)c FF(\021)e FH(0)p Fn(.)17 2052 y FI(Pro)s(of)37 b FH(Let,)c(for)f(all)g FA(i)c FF(\025)g FH(0,)33 b FA(\030)1130 2067 y Fy(i)1185 2052 y FH(:=)28 b FA(\030)1359 2067 y Fj(\003)1398 2052 y FA(=)p FH(2)1496 2015 y Fy(i)1556 2052 y FH(and)33 b FA(\016)1789 2067 y Fy(i)1845 2052 y FH(:=)27 b FA(\030)2018 2067 y Fj(\003)2058 2052 y FA(=)p FH(2)2156 2015 y Fy(i)p FE(+1)2273 2052 y FH(.)33 b(Then,)h(b)m(y)f(\(2.89\),)f(one)h(has)713 2258 y FF(k)p FA(w)s FF(k)886 2273 y Fy(\030)917 2283 y Fk(i)p Fw(+1)1052 2258 y FF(\024)28 b FA(B)1231 2273 y FE(0)1287 2258 y FA(B)1366 2217 y Fy(i)1361 2283 y FE(1)1433 2258 y FF(k)p FA(w)s FF(k)1606 2217 y FE(2)1606 2283 y Fy(\030)1637 2293 y Fk(i)1699 2258 y FA(;)212 b(B)2012 2273 y FE(0)2079 2258 y FH(:=)28 b FA(c)k FH(2)2333 2217 y Fy(s)2370 2258 y FA(\030)2418 2217 y Fj(\000)p Fy(s)2413 2283 y Fj(\003)2541 2258 y FA(;)115 b(B)2757 2273 y FE(1)2824 2258 y FH(:=)27 b(2)3003 2217 y Fy(s)3073 2258 y FA(:)17 2465 y FH(Iterating)32 b(this)i(relation)936 2428 y FE(20)1043 2465 y FH(one)f(gets,)g(for)f(all)h FA(i)28 b FF(\025)g FH(0,)856 2827 y FF(k)p FA(w)s FF(k)1029 2842 y FE(0)1095 2827 y FF(\024)g(k)p FA(w)s FF(k)1373 2842 y Fy(\030)1404 2852 y Fk(i)1461 2827 y FF(\024)1576 2613 y Fm(\020)1636 2723 y FA(C)7 b FF(k)p FA(w)s FF(k)1886 2738 y Fy(\030)1917 2746 y Fe(\003)1956 2613 y Fm(\021)2015 2635 y FE(2)2050 2611 y Fk(i)p 1576 2804 505 4 v 1713 2895 a FA(C)40 b FH(2)1872 2866 y Fy(s)11 b(i)2124 2827 y FA(;)212 b(C)34 b FH(:=)28 b FA(c)17 b FH(4)2706 2786 y Fy(s)2759 2827 y FA(\030)2807 2786 y Fj(\000)p Fy(s)2802 2851 y Fj(\003)2930 2827 y FA(;)17 3072 y FH(whic)m(h,)34 b(b)m(y)f(\(2.90\),)f(implies)i(that)f FF(k)p FA(w)s FF(k)1483 3087 y FE(0)1549 3072 y FH(:=)1680 2998 y Fm(P)1801 3072 y FF(j)p FA(w)1899 3087 y Fy(n)1946 3072 y FF(j)27 b FH(=)h(0)k(so)h(that)f FA(w)e FF(\021)e FH(0.)p 2949 3072 41 83 v 17 3393 a FI(Prop)s(osition)37 b(3)49 b Fn(L)-5 b(et)53 b FH(0)59 b FA(<)h(\030)1200 3408 y Fj(\003)1299 3393 y FA(<)1444 3367 y FH(\026)1434 3393 y FA(\030)d Fn(and)52 b(let)g FH(\()p FA(u;)17 b(\015)5 b FH(\))59 b FF(2)h(W)2425 3328 y FE(\026)2418 3346 y Fy(\030)2410 3421 y(\030)2441 3429 y Fe(\003)2534 3393 y Fn(b)-5 b(e)52 b(a)g(solution)g(of)f FH(\(1.18\))h Fn(\(i.e.,)17 3513 y FF(F)89 3528 y Fy(\021)130 3513 y FH(\()p FA(u)p FH(;)17 b FA(\015)5 b FH(\))27 b(=)h(0)p Fn(\),)34 b(satisfying)g(also)1065 3720 y FF(k)p FA(u)1171 3746 y Fk(\022)1193 3761 y Fw(1)1235 3720 y FF(k)1285 3735 y Fy(\030)1316 3743 y Fe(\003)1384 3720 y FF(\024)28 b FA(\027)34 b FF(\024)1686 3693 y FH(\026)1676 3720 y FA(\030)27 b FF(\000)22 b FA(\030)1888 3735 y Fj(\003)1962 3720 y FA(;)216 b FF(j)p FA(\032)p FF(j)28 b(\024)g FA(\013)g(<)g FH(1)34 b FA(;)17 3953 y Fn(for)39 b(some)f FA(\027)q(;)17 b(\013)36 b FF(2)h FH(\(0)p FA(;)17 b FH(1\))p Fn(.)38 b(Then,)g(ther)-5 b(e)39 b(exist)k FH(^)-53 b FA(s)36 b FH(:=)k(^)-53 b FA(s)p FH(\()p FA(\034)11 b FH(\))36 b FA(>)g FH(1)j Fn(and)2620 3928 y Fm(b)2601 3953 y FA(C)k FH(=)2845 3928 y Fm(b)2826 3953 y FA(C)7 b FH(\()p FA(\024;)17 b(\034)6 b(;)17 b(\021)3181 3968 y FE(0)3220 3953 y FA(;)g(M)5 b(;)17 b(\027)q(;)g(\013)q FH(\))35 b FA(>)g FH(1)17 4089 y Fn(such)f(that,)i(if)e FH(\()p FA(u)656 4053 y Fj(0)679 4089 y FA(;)17 b(\015)779 4053 y Fj(0)802 4089 y FH(\))28 b FF(2)g(W)1075 4024 y FE(\026)1068 4042 y Fy(\030)1060 4117 y(\030)1091 4125 y Fe(\003)1166 4089 y Fn(is)35 b(also)f(a)h(solution)f(of)h FH(\(1.18\))f Fn(\(i.e.)g FF(F)2603 4104 y Fy(\021)2645 4089 y FH(\()p FA(u)2739 4053 y Fj(0)2761 4089 y FH(;)17 b FA(\015)2861 4053 y Fj(0)2884 4089 y FH(\))28 b(=)g(0)p Fn(\))34 b(satisfying)1107 4321 y FF(k)p FA(u)1213 4280 y Fj(0)1213 4358 y Fk(\022)1235 4373 y Fw(1)1277 4321 y FF(k)1327 4336 y Fy(\030)1358 4344 y Fe(\003)1426 4321 y FF(\024)28 b FA(\027)42 b(;)1882 4296 y Fm(b)1863 4321 y FA(C)24 b(\030)2005 4280 y Fj(\000)s FE(^)-38 b Fy(s)2000 4346 y Fj(\003)2131 4321 y FF(k)p FA(u)2237 4280 y Fj(0)2282 4321 y FF(\000)22 b FA(u)p FF(k)2487 4336 y Fy(\030)2552 4321 y FF(\024)28 b FH(1)841 b(\(2.91\))17 4540 y Fn(then)34 b FA(u)28 b FF(\021)g FA(u)478 4504 y Fj(0)535 4540 y Fn(and)35 b FA(\015)d FH(=)c FA(\015)968 4504 y Fj(0)991 4540 y Fn(.)p 17 4624 1512 4 v 96 4685 a Fu(20)166 4715 y FN(If)g Ft(y)290 4727 y Fs(i)341 4715 y Ft(>)22 b FN(0)27 b(and)h Fx(f)p Ft(x)748 4727 y Fs(i)775 4715 y Fx(g)817 4727 y Fs(i)p Fh(\025)p Fu(0)957 4715 y FN(is)g(a)f(sequence)g(of)h(p)r(ositiv)n(e)f (n)n(um)n(b)r(ers)g(satisfying)1626 4888 y Ft(x)1673 4900 y Fs(i)p Fu(+1)1808 4888 y Fx(\024)c Ft(y)1937 4900 y Fu(0)1974 4888 y Ft(y)2018 4854 y Fs(i)2015 4909 y Fu(1)2052 4888 y Ft(x)2099 4854 y Fu(2)2099 4909 y Fs(i)2164 4888 y Ft(;)17 5073 y FN(then)31 b(one)f(also)g(has)g Ft(x)732 5085 y Fs(i)788 5073 y Fx(\024)e FN(\()p Ft(y)954 5085 y Fu(0)991 5073 y Ft(y)1032 5085 y Fu(1)1069 5073 y Ft(x)1116 5085 y Fu(0)1154 5073 y FN(\))1186 5043 y Fu(2)1219 5018 y Fb(i)1250 5073 y Ft(=)p FN(\()p Ft(y)1365 5085 y Fu(0)1401 5073 y Ft(y)1445 5038 y Fs(i)p Fu(+1)1442 5095 y(1)1557 5073 y FN(\),)j(as)f(it)h(follo)n(ws)f(m)n(ultiplying)g (b)r(oth)h(sides)g(of)f(the)h(ab)r(o)n(v)n(e)f(inequalit)n(y)17 5173 y(b)n(y)d Ft(y)173 5185 y Fu(0)210 5173 y Ft(y)254 5137 y Fs(i)p Fu(+2)251 5195 y(1)393 5173 y FN(so)g(as)g(to)g(obtain)h Ft(z)996 5185 y Fs(i)p Fu(+1)1130 5173 y Fx(\024)23 b Ft(z)1261 5143 y Fu(2)1257 5194 y Fs(i)1325 5173 y FN(with)28 b Ft(z)1553 5185 y Fs(i)1603 5173 y FN(:=)23 b Ft(y)1755 5185 y Fu(0)1792 5173 y Ft(y)1836 5137 y Fs(i)p Fu(+1)1833 5195 y(1)1947 5173 y Ft(x)1994 5185 y Fs(i)2022 5173 y FN(.)1858 5645 y FH(30)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 17 548 a FI(Pro)s(of)37 b FH(Let)1164 668 y FA(w)30 b FH(:=)e FA(u)1451 627 y Fj(0)1496 668 y FF(\000)23 b FA(u)32 b(;)410 b FH(^)-52 b FA(\015)32 b FH(:=)c FA(\015)2388 627 y Fj(0)2433 668 y FF(\000)23 b FA(\015)38 b(;)17 837 y FH(and)26 b(observ)m(e)h(that,)f(since)h FA(e)h FH(:=)g FF(F)1280 852 y Fy(\021)1321 837 y FH(\()p FA(u)p FH(;)17 b FA(\015)5 b FH(\))27 b(=)h(0,)d(then,)i(b)m(y)f (\(2.35\))f(and)h(\(2.39\),)f(one)h(has)h(the)f(follo)m(wing)17 957 y(iden)m(tities:)250 1163 y FA(A)323 1178 y Fy(\021)r(;u)426 1163 y FA(w)k FH(:=)d FA(V)735 1122 y Fj(\000)p FE(1)829 1163 y FA(D)910 1178 y Fy(\021)952 1053 y Fm(\020)1012 1163 y FA(W)14 b(D)1199 1178 y FE(0)1254 1163 y FH(\()p FA(V)1371 1122 y Fj(\000)p FE(1)1465 1163 y FA(w)s FH(\))1576 1053 y Fm(\021)1663 1163 y FH(=)27 b FA(d)p FF(F)1889 1178 y Fy(\021)r(;u)1991 1163 y FH(\()p FA(w)s FH(\))g(=)g(\001)2351 1178 y Fy(\021)2394 1163 y FA(w)d FH(+)e FA(g)2633 1178 y Fy(xx)2717 1163 y FH(\()p FA(\022)2800 1178 y FE(1)2861 1163 y FH(+)g FA(u;)17 b(\022)3104 1178 y FE(2)3144 1163 y FH(\))p FA(w)34 b(;)234 b FH(\(2.92\))17 1401 y(where)34 b FA(V)49 b FH(:=)27 b(1)22 b(+)g FA(u)760 1428 y Fk(\022)782 1443 y Fw(1)858 1401 y FH(and)32 b FA(W)42 b FH(:=)27 b FA(V)1390 1365 y FE(2)1429 1401 y FH(.)33 b(Th)m(us,)h(\(since)g (also)f FF(F)2308 1416 y Fy(\021)2349 1401 y FH(\()p FA(u)2443 1365 y Fj(0)2466 1401 y FH(;)17 b FA(\015)2566 1365 y Fj(0)2589 1401 y FH(\))28 b(=)f(0\))380 1607 y(0)96 b(=)h FF(F)770 1622 y Fy(\021)811 1607 y FH(\()p FA(u)905 1566 y Fj(0)928 1607 y FH(;)17 b FA(\015)1028 1566 y Fj(0)1051 1607 y FH(\))28 b(=)f FF(F)1292 1622 y Fy(\021)1334 1607 y FH(\()p FA(u)p FH(;)17 b FA(\015)5 b FH(\))21 b(+)h(\001)1766 1622 y Fy(\021)1808 1607 y FA(w)j FH(+)d FA(g)2048 1622 y Fy(x)2091 1607 y FH(\()p FA(\022)2174 1622 y FE(1)2236 1607 y FH(+)g FA(u)2390 1566 y Fj(0)2413 1607 y FA(;)17 b(\022)2502 1622 y FE(2)2542 1607 y FH(\))22 b FF(\000)g FA(g)2748 1622 y Fy(x)2792 1607 y FH(\()p FA(\022)2875 1622 y FE(1)2937 1607 y FH(+)g FA(u;)17 b(\022)3180 1622 y FE(2)3219 1607 y FH(\))22 b(+)k(^)-53 b FA(\015)525 1752 y FH(=)97 b(\001)779 1767 y Fy(\021)821 1752 y FA(w)24 b FH(+)e FA(g)1060 1767 y Fy(xx)1144 1752 y FH(\()p FA(\022)1227 1767 y FE(1)1289 1752 y FH(+)g FA(u;)17 b(\022)1532 1767 y FE(2)1571 1752 y FH(\))g FA(w)24 b FH(+)i(^)-53 b FA(\015)698 1928 y FH(+)774 1818 y Fm(\020)833 1928 y FA(g)880 1943 y Fy(x)924 1928 y FH(\()p FA(\022)1007 1943 y FE(1)1069 1928 y FH(+)22 b FA(u)f FH(+)h FA(w)s(;)17 b(\022)1504 1943 y FE(2)1543 1928 y FH(\))22 b FF(\000)h FA(g)1750 1943 y Fy(x)1793 1928 y FH(\()p FA(\022)1876 1943 y FE(1)1938 1928 y FH(+)f FA(u;)17 b(\022)2181 1943 y FE(2)2221 1928 y FH(\))22 b FF(\000)g FA(g)2427 1943 y Fy(xx)2511 1928 y FH(\()p FA(\022)2594 1943 y FE(1)2656 1928 y FH(+)g FA(u;)17 b(\022)2899 1943 y FE(2)2938 1928 y FH(\))p FA(w)3049 1818 y Fm(\021)512 2101 y FH(=:)83 b FA(A)771 2116 y Fy(\021)r(;u)873 2101 y FA(w)25 b FH(+)g(^)-52 b FA(\015)27 b FH(+)22 b FA(Q)1319 2116 y FE(1)1359 2101 y FH(\()p FA(w)s FH(\))31 b FA(:)1981 b FH(\(2.93\))17 2307 y(W)-8 b(e,)33 b(no)m(w,)g(claim)g(that,)g(if)f(w)m(e)i(de\014ne)936 2513 y FA(E)g FH(:=)28 b FA(V)21 b(Q)1328 2528 y FE(1)1401 2513 y FA(;)p 1639 2433 79 4 v 211 w(E)34 b FH(:=)28 b FF(h)p FA(E)6 b FF(i)32 b FA(;)2322 2488 y Fm(e)2303 2513 y FA(E)h FH(:=)28 b FA(E)g FF(\000)p 2739 2433 V 23 w FA(E)38 b(;)671 b FH(\(2.94\))17 2719 y(then)1199 2901 y(^)-52 b FA(\015)32 b FH(=)c FF(\000)1470 2825 y(h)p FA(W)1615 2789 y Fj(\000)p FE(1)1709 2825 y FF(i)p 1748 2745 V FA(E)g FH(+)22 b FA(\021)e FF(h)p FA(W)2159 2789 y Fj(\000)p FE(1)2253 2825 y FA(D)2337 2789 y Fj(\000)p FE(1)2334 2849 y Fy(\021)2450 2800 y Fm(e)2431 2825 y FA(E)6 b FF(i)p 1470 2878 1078 4 v 1496 2969 a(h)p FA(W)1641 2941 y Fj(\000)p FE(1)1735 2969 y FF(i)22 b FH(+)g FA(\021)t FF(h)p FA(W)2091 2941 y Fj(\000)p FE(1)2184 2969 y FA(D)2268 2941 y Fj(\000)p FE(1)2265 2994 y Fy(\021)2362 2969 y FA(u)2418 2996 y Fk(\022)2440 3011 y Fw(1)2483 2969 y FF(i)2590 2901 y FA(:)930 b FH(\(2.95\))17 3142 y(In)33 b(fact,)f(from)g(\(2.92\))g(and)h(\(2.93\),)f(there)h(follo)m(ws)957 3376 y FA(D)1038 3391 y Fy(\021)1080 3265 y Fm(\020)1140 3376 y FA(W)14 b(D)1327 3391 y FE(0)1366 3376 y FH(\()p FA(V)1482 3335 y Fj(\000)p FE(1)1577 3376 y FA(w)s FH(\))1688 3265 y Fm(\021)1769 3376 y FH(+)25 b(^)-52 b FA(\015)27 b FH(+)22 b FA(u)2099 3402 y Fk(\022)2121 3417 y Fw(1)2167 3376 y FH(^)-52 b FA(\015)27 b FH(+)2359 3350 y Fm(e)2340 3376 y FA(E)h FH(+)p 2538 3296 79 4 v 22 w FA(E)34 b FH(=)27 b(0)32 b FA(:)692 b FH(\(2.96\))17 3622 y(If)39 b FA(\021)44 b FH(=)c(0,)g(then)g(taking)g(a)m(v)m(erage)h(in)f (\(2.96\))f(yields)45 b(^)-52 b FA(\015)45 b FH(=)40 b FF(\000)p 2320 3542 V FA(E)6 b FH(,)40 b(whic)m(h)i(is)e(\(2.95\))f (when)i FA(\021)j FH(=)c(0.)f(If)17 3743 y FA(\021)31 b FF(6)p FH(=)d(0,)k(then,)h(observing)h(that)e FA(D)1289 3707 y Fj(\000)p FE(1)1286 3767 y Fy(\021)1383 3743 y FH(1)c(=)1575 3704 y FE(1)p 1573 3720 38 4 v 1573 3777 a Fy(\021)1621 3743 y FH(,)33 b(from)f(\(2.96\))g(one)h(obtains)533 4040 y FA(D)614 4055 y FE(0)654 4040 y FH(\()p FA(V)770 3998 y Fj(\000)p FE(1)864 4040 y FA(w)s FH(\))22 b(+)1105 3972 y FA(W)1211 3936 y Fj(\000)p FE(1)1309 3972 y FH(^)-53 b FA(\015)p 1105 4017 256 4 v 1207 4108 a(\021)1393 4040 y FH(+)26 b(^)-53 b FA(\015)5 b(W)1653 3998 y Fj(\000)p FE(1)1747 4040 y FA(D)1831 3998 y Fj(\000)p FE(1)1828 4064 y Fy(\021)1925 4040 y FA(u)1981 4066 y Fk(\022)2003 4081 y Fw(1)2068 4040 y FH(+)22 b FA(W)2272 3998 y Fj(\000)p FE(1)2366 4040 y FA(D)2450 3998 y Fj(\000)p FE(1)2447 4064 y Fy(\021)2563 4014 y Fm(e)2544 4040 y FA(E)28 b FH(+)2752 3972 y FA(W)2858 3936 y Fj(\000)p FE(1)p 2952 3892 79 4 v 2952 3972 a FA(E)p 2752 4017 279 4 v 2865 4108 a(\021)3068 4040 y FH(=)f(0)33 b(;)17 4291 y(m)m(ultiplying)45 b(b)m(y)g FA(\021)j FH(and)c(taking)g(a)m(v)m(erage)h(in)f(the)g (latter)g(relation,)g(yields,)i(up)s(on)e(solving)h(for)i(^)-53 b FA(\015)5 b FH(,)17 4411 y(\(2.95\).)17 4554 y(Th)m(us,)34 b FA(w)h FH(satis\014es)f(the)f(equation)1111 4760 y FA(A)1184 4775 y Fy(\021)r(;u)1287 4760 y FA(w)d FH(=)e FA(Q)k(;)407 b(Q)28 b FH(:=)g FF(\000)t FH(^)-53 b FA(\015)28 b FF(\000)22 b FA(Q)2602 4775 y FE(1)2674 4760 y FA(;)846 b FH(\(2.97\))17 4967 y(with)33 b FA(Q)g FH(\\quadratic")f(in)h FA(w)s FH(.)f(In)h(fact,)g(observing)h(that)1206 5173 y FF(k)p FA(Q)1333 5188 y FE(1)1373 5173 y FF(k)1423 5188 y Fy(\030)1488 5173 y FF(\024)28 b FA(M)f FF(k)p FA(w)s FF(k)1887 5132 y FE(2)1887 5197 y Fy(\030)1958 5173 y FA(;)212 b FF(8)33 b FA(\030)f FF(\024)c FA(\030)2508 5188 y Fj(\003)2580 5173 y FA(;)1858 5645 y FH(31)p eop end %%Page: 32 32 TeXDict begin 32 31 bop 17 548 a FH(\()p FA(M)43 b FH(b)s(eing)33 b(as)g(in)f(\(1.20\)\),)g(b)m(y)i(the)f(relations)g(in)g(\(2.97\),)f (\(2.95\))f(and)i(\(2.94\),)f(one)h(\014nds)g(that)1347 737 y FF(j)t FH(^)-53 b FA(\015)5 b FF(j)17 b FA(;)33 b FF(k)p FA(Q)p FF(k)1713 752 y Fy(\030)1778 737 y FF(\024)28 b FA(c)1925 752 y FE(11)2016 737 y FA(\030)2064 696 y Fj(\000)p Fy(\034)2194 737 y FF(k)p FA(w)s FF(k)2367 696 y FE(2)2367 762 y Fy(\030)2438 737 y FA(;)1082 b FH(\(2.98\))17 926 y(for)32 b(a)g(suitable)i FA(c)653 941 y FE(11)755 926 y FH(=)28 b FA(c)901 941 y FE(11)976 926 y FH(\()p FA(\024;)17 b(\034)6 b(;)17 b(\021)1254 941 y FE(0)1293 926 y FA(;)g(M)5 b(;)17 b(\027)q(;)g(\013)q FH(\))27 b FA(>)g FH(1)33 b(and)g(for)f(an)m(y)h FA(\030)f FF(\024)c FA(\030)2632 941 y Fj(\003)2671 926 y FH(.)33 b(Next,)h(from)e(the)h(de\014nition)17 1046 y(of)f FA(A)201 1061 y Fy(\021)r(;u)303 1046 y FH(,)h(one)g(\014nds)g(the)g(iden)m(tit) m(y)1285 1236 y FA(D)1366 1251 y FE(0)1406 1236 y FH(\()p FA(V)1522 1194 y Fj(\000)p FE(1)1617 1236 y FA(w)s FH(\))27 b(=)g FA(W)1964 1194 y Fj(\000)p FE(1)2058 1236 y FA(D)2142 1194 y Fj(\000)p FE(1)2139 1260 y Fy(\021)2236 1236 y FH(\()p FA(V)22 b(Q)p FH(\))32 b FA(;)1020 b FH(\(2.99\))17 1443 y(and,)32 b(since)i FF(h)p FA(w)s FF(i)27 b FH(=)h(0)k(\(as)h(it)f (follo)m(ws)h(from)g FA(u;)17 b(u)1797 1407 y Fj(0)1846 1443 y FF(2)29 b(H)2026 1396 y Fy(\030)2057 1404 y Fe(\003)2025 1468 y FE(0)2097 1443 y FH(\),)j(one)h(sees)h(that)f(\(2.99\))f(is)h (equiv)-5 b(alen)m(t)34 b(to)820 1660 y FA(w)c FH(=)47 b(^)-68 b FA(w)24 b FF(\000)f FA(V)e FF(h)f FH(^)-69 b FA(w)r FF(i)33 b FA(;)231 b FH(^)-69 b FA(w)31 b FH(:=)c FA(V)22 b(D)2111 1619 y Fj(\000)p FE(1)2108 1685 y(0)2205 1549 y Fm(\020)2264 1660 y FA(W)2370 1619 y Fj(\000)p FE(1)2464 1660 y FA(D)2548 1619 y Fj(\000)p FE(1)2545 1685 y Fy(\021)2642 1660 y FH(\()p FA(V)g(Q)p FH(\))2874 1549 y Fm(\021)2966 1660 y FA(:)17 1871 y FH(By)33 b(Lemma)g(1,)f(one)h (gets)g(the)g(estimate)905 2060 y FF(k)p FA(w)s FF(k)1078 2075 y Fy(\030)s Fj(\000)p Fy(\016)1231 2060 y FF(\024)28 b FA(c)1378 2075 y FE(12)1453 2060 y FA(\016)1500 2019 y Fj(\000)p FE(2)p Fy(\034)1666 2060 y FF(k)p FA(Q)p FF(k)1843 2075 y Fy(\030)1913 2060 y FA(;)212 b FH(\(0)27 b FA(<)h(\016)j(<)d(\030)k FF(\024)c FA(\030)2771 2075 y Fj(\003)2810 2060 y FH(\))33 b FA(;)17 2249 y FH(whic)m(h,)h(b)m(y)f (\(2.98\),)f(implies)906 2438 y FF(k)p FA(w)s FF(k)1079 2453 y Fy(\030)s Fj(\000)p Fy(\016)1232 2438 y FF(\024)d FA(c)1380 2453 y FE(13)1471 2438 y FA(\016)1518 2397 y Fj(\000)p FE(3)p Fy(\034)1651 2438 y FF(k)p FA(w)s FF(k)1824 2397 y FE(2)1824 2463 y Fy(\030)1895 2438 y FA(;)49 b(;)212 b FH(\(0)28 b FA(<)f(\016)32 b(<)27 b(\030)32 b FF(\024)d FA(\030)2830 2453 y Fj(\003)2869 2438 y FH(\))17 2650 y(sho)m(wing)35 b(that)f FA(w)j FH(satis\014es)e(the)g(estimate)g (\(2.89\).)f(Letting)2306 2625 y Fm(b)2287 2650 y FA(C)k FH(:=)30 b FA(c)2570 2665 y FE(13)2661 2650 y FH(4)2710 2614 y FE(3)p Fy(\034)2823 2650 y FH(and)38 b(^)-53 b FA(s)30 b FH(:=)h(3)p FA(\034)11 b FH(,)34 b(the)h(thesis)17 2770 y(follo)m(ws)e(from)f(Lemma)h(4.)p 1142 2770 41 83 v 17 2993 a(The)26 b(ab)s(o)m(v)m(e)h(analysis)g(sho)m(ws)h(the)e (smo)s(oth)g(\()p FA(C)1728 2957 y Fj(1)1802 2993 y FH(\))g(dep)s (endence)i(up)s(on)e(the)g(\\dissipation")h(parameter)17 3113 y FA(\021)t FH(.)i(W)-8 b(e)29 b(close)h(this)g(section)g(with)g (a)f(brief)g(remark)h(on)f(ho)m(w)h(solutions)g(dep)s(end)g(up)s(on)f (other)g(ev)m(en)m(tual)17 3234 y(\\external")k(parameters.)17 3427 y FI(Remark)38 b(7)48 b FH(\(i\))33 b(If)g(the)h(function)f FA(g)j FH(in)e(\(1.18\))e(dep)s(ends)j(also)e(in)g(a)g(real{analytic)g (w)m(a)m(y)h(on)f(one)h(\(or)17 3547 y(more\))39 b(external)g (parameters)h FA(})e FF(2)g FA(J)47 b FF(\032)38 b FG(C)1690 3511 y Fy(m)1757 3547 y FH(,)h(then)g(so)g(do)f(KAM)h(solutions)g FA(u)p FH(\()p FA(\022)s FH(;)17 b FA(\021)t(;)g(})p FH(\))38 b Fv(pro)m(vided)17 3667 y(the)e(smallness)i(condition)f FH(\(1.24\))e Fv(holds)i(uniformly)g(in)f FA(})p FH(,)h(i.e.,)f(pro)m (vided)i(suc)m(h)f(conditions)g(holds)17 3788 y(with)c(the)g FF(k)22 b(\001)g(k)579 3803 y Fy(\030)649 3788 y FH(norm)32 b(rede\014ned)j(as)e(the)g(norm)1222 4007 y FF(k)p FA(e)p FF(k)1367 4022 y Fy(\030)1433 4007 y FH(:=)1563 3913 y Fm(X)1568 4124 y Fy(n)p Fj(2)p Fl(Z)1724 3897 y Fm(\020)1800 4007 y FH(sup)1805 4089 y Fy(})p Fj(2)p Fy(J)1980 4007 y FF(j)p FA(e)2053 4022 y Fy(n)2100 4007 y FH(\()p FA(})p FH(\))p FF(j)2266 3897 y Fm(\021)2358 4007 y Ff(e)2411 3966 y Fj(j)p Fy(n)p Fj(j)p Fy(\030)2563 4007 y FA(:)17 4285 y FH(This)43 b(claim)g(follo)m(ws)g(from)f(the)g(uniform)h(con)m (v)m(ergence)i(of)c(the)i(KAM)g(sc)m(heme)h(and)e(W)-8 b(eierstrass)17 4405 y(theorem)38 b(on)f(analytic)h(limits)g(of)e (holomorphic)i(function;)g(for)f(more)g(details)h(compare,)g(e.g,)g (with)17 4525 y([5].)17 4666 y(\(ii\))25 b(The)h(dep)s(endence)i(up)s (on)d(the)h(frequency)h FA(!)t FH(,)d(as)i(w)m(ell)g(kno)m(wn,)h(is)f (more)f(delicate)i(since)f(it)f(in)m(v)m(olv)m(es)17 4787 y(the)38 b(small)g(divisors)i FA(\025)869 4802 y Fy(\021)r(;n)973 4787 y FH(:)e(it)g(is,)g(ho)m(w)m(ev)m(er,)i(standard) e(to)g(c)m(hec)m(k)i(that)e(this)g(dep)s(endence)i(is)f FA(C)3603 4751 y Fj(1)3715 4787 y FH(in)17 4907 y(the)e(sense)i(of)d (Whitney)j(on)e(a)f(b)s(ounded)i(set)g(of)e(Diophan)m(tine)i(n)m(um)m (b)s(ers,)h(sa)m(y)-8 b(,)38 b FF(D)3118 4922 y Fy(\024;\034)3246 4907 y FF(\\)26 b FH([1)f(+)g FA(r)m(;)17 b FH(1)p FA(=r)s FH(])17 5027 y(for)33 b(an)m(y)i(pre\014xed)h(0)30 b FA(<)h(r)i(<)d FH(0;)k(for)g(more)g(details)h(on)f(Whitney)i(smo)s (othness)g(and)e(pro)s(ofs)g(w)m(e)h(refer)17 5148 y(the)e(reader)g(to) f([7],)h([11)o(])g(and)g([2)o(].)1858 5645 y(32)p eop end %%Page: 33 33 TeXDict begin 33 32 bop 17 548 a FD(2.6)135 b(Pro)t(of)45 b(of)g(Theorem)g(3)17 733 y FH(W)-8 b(e)33 b(start)h(b)m(y)g(observing) g(that)f(the)h(h)m(yp)s(otheses)h(\(H1\))p FF(\004)p FH(\(H3\))f(of)e(Theorem)j(3)e(imply)h(that)f(\()p FA(v)t(;)17 b(\014)6 b FH(\))28 b FF(2)17 869 y(W)130 804 y FE(\026)123 822 y Fy(\030)115 896 y(\030)194 869 y FH(\(recall)33 b(De\014nition)g(1\))f(and)g(that)h(\(2.54\))f(and)g(\(2.55\))g(hold.) 17 1013 y(Next,)h(b)s(ecause)h(of)e(\(H2\))h(and)f(\(H3\),)h(w)m(e)g (ha)m(v)m(e)h(that,)f(on)f FG(T)2240 977 y FE(2)2280 1013 y FH(,)h(one)f(has)1411 1205 y(1)p 1235 1249 400 4 v 1235 1340 a(\(1)22 b FF(\000)h FA(\033)t(\027)6 b FH(\))1595 1312 y FE(2)1672 1272 y FF(\025)28 b FA(W)1883 1231 y Fj(\000)p FE(1)2005 1272 y FF(\025)2295 1205 y FH(1)p 2120 1249 398 4 v 2120 1340 a(\(1)22 b(+)g FA(\033)t(\027)6 b FH(\))2478 1312 y FE(2)2560 1272 y FA(;)17 1542 y FH(so)32 b(that)h(\(recall)g(the)g(de\014nitions)h(of)e FA(\026)1456 1557 y FE(0)1495 1542 y FH(,)h FA(\026)1614 1557 y FE(1)1653 1542 y FH(,)40 b(\026)-56 b FA(\026)1772 1557 y FE(0)1843 1542 y FH(and)41 b(\026)-57 b FA(\026)2092 1557 y FE(1)2163 1542 y FH(in)33 b(\(2.63\))f(and)h(\(2.70\)\))f(w)m(e)h(\014nd)1076 1816 y(\026)-56 b FA(\026)1128 1831 y FE(0)1189 1816 y FF(\000)23 b FA(\026)1348 1831 y FE(0)1414 1816 y FF(\025)1529 1748 y FH(\(1)f(+)g FA(\027)6 b FH(\))1828 1712 y FE(2)1890 1748 y FF(\000)23 b FH(\(1)f(+)g FA(\033)t(\027)6 b FH(\))2348 1712 y FE(2)p 1529 1793 859 4 v 1590 1884 a FH(\(1)22 b(+)g FA(\027)6 b FH(\))1889 1855 y FE(2)1929 1884 y FH(\(1)22 b(+)g FA(\033)t(\027)6 b FH(\))2287 1855 y FE(2)2425 1816 y FH(=:)36 b(^)-57 b FA(\026)2615 1831 y FE(0)2682 1816 y FA(>)27 b FH(0)33 b FA(;)1069 2088 y(\026)1128 2103 y FE(1)1189 2088 y FF(\000)d FH(\026)-56 b FA(\026)1348 2103 y FE(1)1414 2088 y FF(\025)1529 2021 y FH(\(1)22 b FF(\000)h FA(\033)t(\027)6 b FH(\))1889 1985 y FE(2)1951 2021 y FF(\000)23 b FH(\(1)e FF(\000)i FA(\027)6 b FH(\))2351 1985 y FE(2)p 1529 2066 862 4 v 1590 2157 a FH(\(1)22 b FF(\000)h FA(\027)6 b FH(\))1891 2128 y FE(2)1931 2157 y FH(\(1)22 b FF(\000)g FA(\033)t(\027)6 b FH(\))2290 2128 y FE(2)2429 2088 y FH(=:)35 b(^)-57 b FA(\026)2618 2103 y FE(1)2685 2088 y FA(>)28 b FH(0)k FA(:)17 2359 y FH(Th)m(us,)i(the)f(n)m(um)m(b)s(er)h FA(\026)e FH(de\014ned)i(in)f(\(2.71\))f(is)h(b)s(ounded)g(b)s(elo)m(w) h(b)m(y)305 2632 y FA(\026)28 b FF(\025)g FA(\026)556 2647 y Fj(\003)623 2632 y FH(=)f FA(\026)785 2647 y Fj(\003)824 2632 y FH(\()p FA(\027)q(;)17 b(\013)q(;)g(\033)t FH(\))27 b(:=)h(min)1496 2522 y Fm(n)1562 2632 y FH(\(1)22 b FF(\000)h FA(\033)t FH(\))p FA(\027)g(;)2033 2565 y FH(\(1)e FF(\000)i FA(\027)6 b FH(\))2333 2529 y FE(2)p 2009 2609 388 4 v 2009 2700 a FH(2\(1)22 b(+)g FA(\027)6 b FH(\))2357 2672 y FE(2)2423 2632 y FA(;)41 b FH(^)-57 b FA(\026)2542 2647 y FE(0)2598 2632 y FA(;)41 b FH(^)-57 b FA(\026)2717 2647 y FE(1)2773 2632 y FA(;)33 b FH(\(1)22 b FF(\000)h FA(\033)t FH(\))p FA(\013)3202 2522 y Fm(o)3296 2632 y FA(>)k FH(0)33 b FA(:)17 2897 y FH(Therefore,)g(taking)1503 3017 y FA(m)28 b FH(=)g FA(m)1805 3032 y Fj(\003)1872 3017 y FH(:=)g(2)2052 2976 y Fy(s)p Fj(\000)p FE(1)2179 3017 y FA(\026)2238 3032 y Fj(\003)17 3189 y FH(w)m(e)33 b(see)h(that)e(condition)i(\(2.73\))d(ma)m(y)j(b)s(e)e(rewritten)i(as) 1641 3403 y FA(k)1692 3418 y FE(1)1748 3403 y FF(k)p FA(e)p FF(k)1893 3418 y Fy(\030)1958 3403 y FF(\024)28 b FH(1)33 b FA(;)17 3618 y FH(with)862 3738 y FA(k)913 3753 y FE(1)980 3738 y FH(=)28 b FA(k)1135 3753 y FE(1)1174 3738 y FH(\()p FA(\030)5 b(;)17 b(\030)1347 3753 y Fj(\003)1385 3738 y FA(;)g(\024;)g(\034)6 b(;)17 b(\021)1669 3753 y FE(0)1708 3738 y FA(;)g(M)5 b(;)17 b(\027)q(;)g(\013)q(;)g(\033)t FH(\))27 b(:=)g FA(C)56 b Ff(e)2692 3662 y Fw(1)p 2538 3674 339 3 v 2549 3715 a Fc(e)10 b Fk(m)2646 3723 y Fe(\003)2682 3715 y Fw(\()p Fk(\027;\013;\033)r Fw(\))2923 3738 y FA(;)17 3910 y FH(while)33 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begin 34 33 bop 17 548 a FH(and)32 b(assume)i(that)1617 668 y FA(k)1668 683 y FE(4)1739 668 y FF(k)p FA(e)p FF(k)1884 683 y Fy(\030)1915 691 y Fe(\003)1983 668 y FF(\024)28 b FH(1)k FA(:)1302 b FH(\(2.101\))17 843 y(Then,)34 b(if)e FA(u)444 807 y Fj(0)499 843 y FH(and)h FA(\015)745 807 y Fj(0)801 843 y FH(solv)m(e)h(\(1.18\),)e(i.e.,)h FF(F)1605 858 y Fy(\021)1646 843 y FH(\()p FA(u)1740 807 y Fj(0)1763 843 y FH(;)17 b FA(\015)1863 807 y Fj(0)1886 843 y FH(\))28 b(=)f(0)33 b(for)f(eac)m(h)h FA(\021)f FF(2)c FA(I)2722 858 y FE(0)2761 843 y FH(,)33 b(and)f(if)1513 1063 y FA(k)1564 1078 y FE(3)1636 1063 y FF(k)p FA(u)1742 1022 y Fj(0)1787 1063 y FF(\000)23 b FA(v)t FF(k)1988 1078 y Fy(\030)2019 1086 y Fe(\003)2086 1063 y FF(\024)28 b FH(1)k FA(;)1199 b FH(\(2.102\))17 1283 y(then,)33 b(b)m(y)g(\(2.102\),)f(\(2.75\),)g(the)h(de\014nition)h(of)e FA(k)1832 1298 y FE(2)1903 1283 y FH(in)h(\(2.100\),)f(\(2.101\))f(,)i (w)m(e)h(see)f(that)243 1552 y FF(k)p FA(u)349 1511 y Fj(0)393 1552 y FF(\000)23 b FA(v)t FF(k)594 1567 y Fy(\030)625 1575 y Fe(\003)692 1552 y FF(\024)28 b(k)p FA(u)903 1511 y Fj(0)948 1552 y FF(\000)23 b FA(u)p FF(k)1154 1567 y Fy(\030)1185 1575 y Fe(\003)1246 1552 y FH(+)f FF(k)p FA(u)g FF(\000)h FA(v)t FF(k)1673 1567 y Fy(\030)1704 1575 y Fe(\003)1771 1552 y FF(\024)1907 1485 y FH(1)p 1886 1529 91 4 v 1886 1620 a FA(k)1937 1635 y FE(3)2008 1552 y FH(+)f FA(k)2157 1567 y FE(2)2213 1552 y FF(k)p FA(e)p FF(k)2358 1567 y Fy(\030)2389 1575 y Fe(\003)2457 1552 y FF(\024)2593 1485 y FH(1)p 2572 1529 V 2572 1620 a FA(k)2623 1635 y FE(3)2694 1552 y FH(+)2802 1485 y FA(k)2853 1500 y FE(2)p 2802 1529 V 2802 1620 a FA(k)2853 1635 y FE(4)2930 1552 y FH(=)3064 1485 y(2)p 3043 1529 V 3043 1620 a FA(k)3094 1635 y FE(3)3171 1552 y FH(=)3368 1485 y(1)p 3285 1529 217 4 v 3304 1613 a Fm(b)3285 1638 y FA(C)7 b(\030)3410 1597 y Fj(\000)s FE(^)-38 b Fy(s)3405 1651 y Fj(\003)3543 1552 y FA(;)17 1840 y FH(sho)m(wing)33 b(that)g(\(2.91\))f(is)h(satis\014ed)h(so)f(that,)f(b)m(y)h(Prop)s (osition)g(3,)g FA(u)2538 1804 y Fj(0)2588 1840 y FH(=)28 b FA(u)k FH(and)g FA(\015)3025 1804 y Fj(0)3076 1840 y FH(=)c FA(\015)5 b FH(.)17 1986 y(Th)m(us)36 b(\(since)g FA(k)596 2001 y FE(4)670 1986 y FH(is)f(greater)g(than)f FA(k)1383 2001 y FE(2)1457 1986 y FH(and)h FA(k)1700 2001 y FE(3)1739 1986 y FH(\),)g(w)m(e)g(see)h(that)f(all)f(claims)i (in)f(Theorem)h(3)e(follo)m(w)h(b)m(y)17 2106 y(taking)d FA(k)f FH(:=)d(max)p FF(f)p FA(k)812 2121 y FE(1)868 2106 y FA(;)33 b(k)979 2121 y FE(4)1018 2106 y FF(g)p FH(.)p 1246 2106 41 83 v 17 2494 a FD(2.7)135 b(Pro)t(of)45 b(of)g(Theorem)g(1)17 2704 y FH(W)-8 b(e)33 b(no)m(w)g(sho)m(w)h(ho)m (w)f(Theorem)h(1)e(can)h(b)s(e)g(obtained)g(as)f(a)h(corollary)f(of)h (Theorem)g(3.)17 2849 y(Since)40 b FA(f)50 b FH(in)40 b(\(1.6\))e(is)i(assumed)h(to)e(b)s(e)h(real{analytic,)g(there)g (exists)h(a)2746 2823 y(\026)2736 2849 y FA(\030)i(>)c FH(0)g(suc)m(h)i(that)e FA(f)50 b FF(2)39 b(H)3765 2796 y FE(\026)3758 2813 y Fy(\030)17 2970 y FH(\(p)s(oin)m(t)32 b(\(ii\))h(of)f(Remark)h(4\).)g(Assuming,)h(as)f(w)m(e)g(shall)g (henceforth)h(do,)e(that)1657 3190 y FF(j)p FA(")p FF(j)27 b(\024)h FA(")1937 3205 y FE(0)2004 3190 y FA(<)f FH(1)17 3410 y(w)m(e)33 b(can)g(tak)m(e)h(the)f(constan)m(t)g FA(M)43 b FH(in)33 b(Theorem)h(3)e(to)g(b)s(e)1599 3630 y FA(M)38 b FH(:=)28 b FF(k)p FA(@)1968 3589 y FE(3)1963 3654 y Fy(x)2008 3630 y FA(f)11 b FF(k)2124 3634 y FE(\026)2117 3651 y Fy(\030)2187 3630 y FA(:)17 3850 y FH(In)33 b(this)g(section,)h FF(k)22 b(\001)f(k)853 3865 y Fy(\030)923 3850 y FH(denotes)34 b(the)f(norm)g(\(compare)g(\(i\),)g(Remark)g(7\))607 4100 y FF(k)p FA(h)p FF(k)763 4115 y Fy(\030)828 4100 y FH(:=)958 4005 y Fm(X)963 4217 y Fy(n)p Fj(2)p Fl(Z)1119 3990 y Fm(\020)1195 4100 y FH(sup)1206 4182 y Fy(")p Fj(2)p Fy(J)1359 4100 y FF(j)p FA(h)1443 4115 y Fy(n)1489 4100 y FH(\()p FA(")p FH(\))p FF(j)1639 3990 y Fm(\021)1715 4100 y Ff(e)1768 4059 y Fj(j)p Fy(n)p 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Fk(\022)1662 2965 y Fw(1)1705 2924 y FF(k)1755 2939 y Fy(\030)1786 2947 y Fe(\003)1853 2924 y FF(\024)c FA(")2005 2939 y FE(0)2060 2924 y FA(k)s(M)44 b(;)17 3085 y FH(whic)m(h,)37 b(together)e(with)i(analyticit)m(y)f(in)g FA(")p FH(,)f(implies)i(that)f FA(u)c FH(=)g FA(O)s FH(\()p FA(")p FH(\),)i(i.e.,)j FA(u)p FF(j)2929 3100 y Fy(")p FE(=0)3087 3085 y FH(=)c(0.)i(Finally)-8 b(,)36 b(the)17 3206 y(relation)42 b(b)s(et)m(w)m(een)i FA(\015)j FH(and)42 b FA(!)j FH(in)d(\(1.6\))g(and)g(Eq.)g(\(1.16\))g(imply)h(\(1.4\),)e (completing)i(the)g(pro)s(of)e(of)17 3326 y(Theorem)34 b(1.)p 655 3326 41 83 v 17 3694 a FD(2.8)135 b(Pro)t(of)45 b(of)g(Theorem)g(2)17 3899 y FH(Also)33 b(the)g(pro)s(of)e(of)h (Theorem)i(2)e(is)h(based)h(up)s(on)e(Theorem)i(3)e(along)g(the)h (lines)h(of)e FF(x)h FH(2.7,)f(but,)h(\014rst,)17 4020 y(w)m(e)26 b(ha)m(v)m(e)g(to)f(in)m(v)m(estigate)i(the)f(analytical)f (prop)s(erties)h(of)f(the)h(spin{orbit)f(p)s(oten)m(tial)h(de\014ned)g (in)g(\(1.8\).)17 4140 y(F)-8 b(or)31 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y(\(1.13\)\))1148 1719 y FA(\021)31 b FH(=)d FA(K)7 b Fz(\012)1508 1734 y FE(e)1576 1719 y FA(;)212 b(\015)33 b FH(=)27 b FA(K)7 b Fz(\012)2179 1734 y FE(e)2215 1719 y FA(!)26 b FF(\000)c FA(K)7 b(N)2569 1734 y FE(e)2638 1719 y FA(:)833 b FH(\(2.112\))17 1893 y(By)33 b(de\014nition)g(of)f Fz(\035)772 1908 y FE(e)841 1893 y FH(and)k(~)-52 b FA(\015)5 b FH(,)32 b(w)m(e)i(can)f(rewrite)g (\(2.112\))f(as)1020 2113 y FA(\021)g FH(=)27 b FA(K)7 b Fz(\012)1380 2128 y FE(e)1449 2113 y FA(;)212 b(!)t(")1799 2072 y FE(2)1857 2113 y FH(~)-52 b FA(\015)5 b FH(\()p FA(\021)t(;)17 b(";)g FH(e)p FA(;)g(!)t FH(\))25 b(=)j Fz(\035)2511 2128 y FE(e)2569 2113 y FF(\000)23 b FA(!)35 b(:)706 b FH(\(2.113\))17 2333 y(Letting)1236 2453 y(^)-53 b FA(\015)5 b FH(\()p FA(K)r(;)17 b(";)g FH(e)p FA(;)g(!)t FH(\))26 b(:=)31 b(~)-52 b FA(\015)5 b FH(\()p FA(K)i Fz(\012)2163 2468 y FE(e)2199 2453 y FA(;)17 b(";)g FH(e)p FA(;)g(!)t FH(\))30 b FA(;)17 2628 y FH(the)j(second)h(relation)e(in)h 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