Content-Type: multipart/mixed; boundary="-------------0003040929798" This is a multi-part message in MIME format. ---------------0003040929798 Content-Type: text/plain; name="00-99.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-99.keywords" attractors, evolution equations, entropy ---------------0003040929798 Content-Type: application/postscript; name="attractor.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="attractor.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: paper3.dvi %%Pages: 18 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMMI10 Times-Bold Times-Italic Times-Roman CMMI8 CMR8 %%+ CMSY6 CMCSC10 CMMI7 CMR10 CMSY10 CMSY7 CMMI12 CMEX10 CMR7 CMMI5 %%+ CMSY5 CMR5 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips -o paper3.ps paper3 %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 2000.03.01:1424 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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{dup dup 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 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 dup 1 get dup mul exch 0 get dup 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 /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /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 %%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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota 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 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec 80347982AB3942D930E069A70D0D48311D725E830D1C76FBA12E12486E989C9874C2B527F092 5722787027F44470D484262C360CDFDDDF3657533A57BB16F73048BFBBFCB73A650484015441 FDC837ADD94AC8FBD2022E3EC8F115D4B4BB7B7F15388F22CC6198EFE768BD9FCEB3446EE4A8 DC27D6CD152485384EF5F59381FFDA43F2D20C8FB08AA27AB2015B774DB10DACFDCD33E60F17 8C461553146AB427BDD7DA12534BA078AD3D780414930DA4F8D58ABEFD45DB119B10EB409DD8 97923C6E705479464A4B310B58348C4B42393988FEF4925CF984423AAF65FEA9F0E64629953B CF50B919D968D99BD185F83112B2759CC411764E9BDE677F57C5EE5AC555448D5F81A16259DE 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Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry", % version = "0.6", % date = "22 June 1996", % filename = "8r.enc", % email = "kb@@mail.tug.org", % address = "135 Center Hill Rd. // Plymouth, MA 02360", % codetable = "ISO/ASCII", % checksum = "119 662 4424", % docstring = "Encoding for TrueType or Type 1 fonts to be used with TeX." % @} % % Idea is to have all the characters normally included in Type 1 fonts % available for typesetting. This is effectively the characters in Adobe % Standard Encoding + ISO Latin 1 + extra characters from Lucida. % % Character code assignments were made as follows: % % (1) the Windows ANSI characters are almost all in their Windows ANSI % positions, because some Windows users cannot easily reencode the % fonts, and it makes no difference on other systems. The only Windows % ANSI characters not available are those that make no sense for % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen % (173). quotesingle and grave are moved just because it's such an % irritation not having them in TeX positions. % % (2) Remaining characters are assigned arbitrarily to the lower part % of the range, avoiding 0, 10 and 13 in case we meet dumb software. % % (3) Y&Y Lucida Bright includes some extra text characters; in the % hopes that other PostScript fonts, perhaps created for public % consumption, will include them, they are included starting at 0x12. % % (4) Remaining positions left undefined are for use in (hopefully) % upward-compatible revisions, if someday more characters are generally % available. % % (5) hyphen appears twice for compatibility with both ASCII and Windows. % /TeXBase1Encoding [ % 0x00 (encoded characters from Adobe Standard not in Windows 3.1) /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef % These are the only two remaining unencoded characters, so may as % well include them. /Zcaron /zcaron % 0x10 /caron /dotlessi % (unusual TeX characters available in, e.g., Lucida Bright) /dotlessj /ff /ffi /ffl /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef % very contentious; it's so painful not having quoteleft and quoteright % at 96 and 145 that we move the things normally found there down to here. /grave /quotesingle % 0x20 (ASCII begins) /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash % 0x30 /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question % 0x40 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O % 0x50 /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore % 0x60 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o % 0x70 /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % rubout; ASCII ends % 0x80 /.notdef /.notdef /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /.notdef /.notdef /.notdef % 0x90 /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /.notdef /.notdef /Ydieresis % 0xA0 /.notdef % nobreakspace /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen % Y&Y (also at 45); Windows' softhyphen /registered /macron % 0xD0 /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown % 0xC0 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis % 0xD0 /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls % 0xE0 /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis % 0xF0 /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def %%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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota 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 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk 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 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright 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 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 127 /spade put dup 128 /arrowleft put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /arrowup put dup 175 /arrowdown put dup 176 /quotesingle put dup 177 /exclamdown put dup 178 /questiondown put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /arrowup put dup 12 /arrowdown put dup 13 /quotesingle put dup 14 /exclamdown put dup 15 /questiondown put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk 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 60 /less put dup 61 /equal put dup 62 /greater put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec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1ACAD9ED23FDF218A06EE1F634F3F0076CC5555870CFEF5E36AAC3DBC7C669B96FC01FDB921D 1FCA9FC7BE5DD2ACE448CC833AB3BD48A8A74B1068C1E5374C149622093B774A7C9FFE05180000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota 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 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk 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 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec 8053514D28EC28DA1630165FAB262882D3FCA78881823C5537FE6C3DDA8EE5B897E17CB027F5 C73FDBB56B0A7C25FC3512B55FE8F3ACFBFFCC7F4A382D8299CC8FD37D3CEA49DABDCA92847A F0560B404EF71134B0F3D99934FC9D0B4E602011B9CFB856C23F958F3C5A2FBE0EF8587D1F57 74879C324E51FCB22888B74F241550D7401EB990D4F3A7AF635198422283CAC1B6CD446DDBCB D915DB9BFF88844E784C6BF7389803D9450B0C21756A017306457C7E62C1D269F306BD3402E2 66DEFC3B5E7D8A8D2F5BF0FE6DDD40D07391DF4FAD4A6018DCE29A2B8F692B29F2023A7C0E66 DE8ED85C14F1F8492167357F51A7E84CC5D92E0FEE4D81CF7FBC8DE52D2E7BB57142033993F9 C08C315ABADE8DBC4A732E84E142D3BEE51557910E12CD8AA37C459A5E6B7F5269F59078ABA3 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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 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 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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 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright 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 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 127 /spade put dup 128 /arrowleft put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota 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 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec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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 161 /parenleftbig put dup 162 /parenrightbig put dup 163 /bracketleftbig put dup 164 /bracketrightbig put dup 165 /floorleftbig put dup 166 /floorrightbig put dup 167 /ceilingleftbig put dup 168 /ceilingrightbig put dup 169 /braceleftbig put dup 170 /bracerightbig put dup 173 /angbracketleftbig put dup 174 /angbracketrightbig put dup 175 /vextendsingle put dup 176 /vextenddouble put dup 177 /slashbig put dup 178 /backslashbig put dup 179 /parenleftBig put dup 180 /parenrightBig put dup 181 /parenleftbigg put dup 182 /parenrightbigg put dup 183 /bracketleftbigg put dup 184 /bracketrightbigg put dup 185 /floorleftbigg put dup 186 /floorrightbigg put dup 187 /ceilingleftbigg put dup 188 /ceilingrightbigg put dup 189 /braceleftbigg put dup 190 /bracerightbigg put dup 191 /angbracketleftbigg put dup 192 /angbracketrightbigg put dup 193 /slashbigg put dup 194 /backslashbigg put dup 195 /parenleftBigg put dup 196 /arrowdblbt put dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 4 /floorleftbig put dup 5 /floorrightbig put dup 6 /ceilingleftbig put dup 7 /ceilingrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 14 /slashbig put dup 15 /backslashbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 22 /floorleftbigg put dup 23 /floorrightbigg put dup 24 /ceilingleftbigg put dup 25 /ceilingrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 28 /angbracketleftbigg put dup 29 /angbracketrightbigg put dup 30 /slashbigg put dup 31 /backslashbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 36 /floorleftBigg put dup 37 /floorrightBigg put dup 38 /ceilingleftBigg put dup 39 /ceilingrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 42 /angbracketleftBigg put dup 43 /angbracketrightBigg put dup 44 /slashBigg put dup 45 /backslashBigg put dup 46 /slashBig put dup 47 /backslashBig put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 54 /bracketleftex put dup 55 /bracketrightex put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 62 /braceex put dup 63 /arrowvertex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 70 /unionsqtext put dup 71 /unionsqdisplay put dup 72 /contintegraltext put dup 73 /contintegraldisplay put dup 74 /circledottext put dup 75 /circledotdisplay put dup 76 /circleplustext put dup 77 /circleplusdisplay put dup 78 /circlemultiplytext put dup 79 /circlemultiplydisplay put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 83 /uniontext put dup 84 /intersectiontext put dup 85 /unionmultitext put dup 86 /logicalandtext put dup 87 /logicalortext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 93 /unionmultidisplay put dup 94 /logicalanddisplay put dup 95 /logicalordisplay put dup 96 /coproducttext put dup 97 /coproductdisplay put dup 98 /hatwide put dup 99 /hatwider put dup 100 /hatwidest put dup 101 /tildewide put dup 102 /tildewider put dup 103 /tildewidest put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 106 /floorleftBig put dup 107 /floorrightBig put dup 108 /ceilingleftBig put dup 109 /ceilingrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 113 /radicalBig put dup 114 /radicalbigg put dup 115 /radicalBigg put dup 116 /radicalbt put dup 117 /radicalvertex put dup 118 /radicaltp put dup 119 /arrowvertexdbl put dup 120 /arrowtp put dup 121 /arrowbt put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put dup 126 /arrowdbltp put dup 127 /arrowdblbt put dup 128 /parenleftBigg put dup 160 /space put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec 80347982AB3942D930E069A70D0D48311D7190FA2D133A583138F76695558E7AE9348D37CAC6 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk 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 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec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B5FBF780CF8754FA7422C3F57E62CCEEAD9AA3470000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota 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 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright 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 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 127 /spade put dup 128 /arrowleft put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 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%%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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /arrowup put dup 175 /arrowdown put dup 176 /quotesingle put dup 177 /exclamdown put dup 178 /questiondown put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /arrowup put dup 12 /arrowdown put dup 13 /quotesingle put dup 14 /exclamdown put dup 15 /questiondown put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk 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 60 /less put dup 61 /equal put dup 62 /greater put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 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Fn(1)1092 5323 y(\0001)1239 5232 y Fw(')1304 5259 y Fr(y)1352 5232 y Fq(\()p Fw(x)p Fq(\))17 b Fs(d)p Fw(x)43 b Fq(=)1765 5097 y Fk(Z)1865 5139 y Fn(1)1820 5323 y(\0001)1967 5232 y Fw(\030)2016 5189 y Fj(\()p Fr(n)p Fj(\))2011 5259 y Fr(y)2132 5232 y Fq(\()p Fw(x)p Fq(\))17 b Fs(d)p Fw(x)43 b Fq(=)h Fs(1)25 b Fw(;)p eop %%Page: 4 4 4 3 bop 94 -45 a Fw(")p Fv({Entr)n(opy)38 b(of)h(Driven)e(Equa)-7 b(tions)2105 b Fu(4)94 194 y Fs(where)26 b(the)e(last)h(equation)f (de\002nes)h Fw(Q)1427 151 y Fj(\()p Fr(n)p Fj(\))1568 194 y Fs(for)g Fw(n)j(>)g Fs(1.)316 314 y(W)-8 b(e)43 b(consider)f(the)g(e)n(v)n(olution)f(de\002ned)i(by)f(Eq.\(2.1\))g(in)g (the)h(Banach)g(space)g(H)3257 271 y Fm(1)3257 341 y Fr(')3357 314 y Fs(which)f(is)g(the)94 433 y(completion)24 b(of)h Fo(C)734 390 y Fn(1)728 460 y Fm(b)843 433 y Fs(\(bounded)f (smooth)g(functions\))g(in)g(the)h(norm)1120 718 y Fo(k)p Fw(f)11 b Fo(k)1280 675 y Fm(2)1280 745 y(H)1330 725 y Fh(1)1330 772 y Fg(')1429 718 y Fq(=)49 b Fs(sup)1551 803 y 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Fj(\()p Fr(n)p Fj(\))2697 4881 y Fr(y)2818 4870 y Fq(\()p Fw(f)32 b Fo(\000)23 b Fw(g)t Fq(\))p Fo(k)3179 4897 y Fm(2)3242 4870 y Fw(:)94 5172 y Fi(Moreo)o(v)o(er)l(,)h Fq(\006)h Fi(is)f Fw(#)777 5129 y Fr(t)812 5172 y Fi(\226in)l(v)n (ariant:)1717 5291 y Fw(#)1776 5248 y Fr(t)1811 5291 y Fq(\(\006\))44 b(=)h(\006)25 b Fw(:)p eop %%Page: 5 5 5 4 bop 94 -45 a Fw(")p Fv({Entr)n(opy)38 b(of)h(Driven)e(Equa)-7 b(tions)2105 b Fu(5)316 194 y Fs(W)-8 b(e)29 b(denote)f(by)g Fq(\010)969 151 y Fr(t)969 221 y(f)1049 194 y Fs(the)h(semi\255\003o)n (w)e(in)h(H)1802 151 y Fm(1)1802 221 y Fr(')1888 194 y Fs(generated)h(by)f(Eq.\(2.1\))g(where)h Fw(F)14 b Fq(\()p Fw(x;)j(t)p Fq(\))32 b(=)h Fw(#)3554 151 y Fr(t)3589 113 y Fk(\000)3634 194 y Fw(f)3694 113 y Fk(\001)3740 194 y Fq(\()p Fw(x)p Fq(\))94 314 y Fs(with)d Fw(f)45 b Fo(2)36 b Fq(\006)p Fs(.)52 b(Using)29 b(terminology)f(from)i(random) g(dynamical)f(systems)g(\(see)i([A]\),)f(we)h(introduce)e(the)94 433 y(sk)o(e)n(w)c(product)1280 554 y 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Fs(See)h(Section)e(6.)p eop %%Page: 6 6 6 5 bop 94 -45 a Fw(")p Fv({Entr)n(opy)38 b(of)h(Driven)e(Equa)-7 b(tions)2105 b Fu(6)94 194 y(Remark.)37 b Fs(W)-8 b(e)37 b(sk)o(etch)f(here)h(a)g(relation)e(between)i(this)e(result)h(and)g (the)g(theory)g(of)h(\002nite)f(dimensional)94 314 y(random)25 b(dynamical)f(systems:)34 b(the)25 b(quantity)1357 516 y Fw(H)1440 543 y Fr(")1482 516 y Fq(\()p Fp(A)17 b Fo(j)g Fq(\006\))43 b(=)h Fw(H)2018 543 y Fr(")2061 516 y Fq(\()p Fp(A)p Fq(\))21 b Fo(\000)i Fw(H)2420 543 y Fr(")2462 516 y Fq(\(\006\))94 718 y Fs(is)i(usually)f(called)h(the)g (conditional)f(entrop)o(y)g(\(see,)i FE(e)o(.g)o(.,)e Fs([KI]\).)i(It)f(is)g(interpreted)g(as)g(the)g(entrop)o(y)f(created)94 838 y(by)g(the)g(\003o)n(w)g(be)o(yond)f(that)h(created)h(by)f(the)g (forcing)g(alone.)36 b(Authors)23 b(often)h(consider)g(systems)f(dri)n (v)o(en)f(by)94 958 y(a)28 b(noise)e(which)g(is)h(a)g(random)f(process) h(with)f(probability)g(distrib)n(ution)e(gi)n(v)o(en)h(by)i(a)g (measure)g Fu(P)g 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Fs(In)23 b(this)f(section)g(we)g (restrict)h(Eq.\(2.1\))f(to)g(the)g(case)h Fw(\032)28 b Fq(=)g Fw(\016)j Fo(\034)d Fs(1)22 b(and)h Fp(N)q Fq(\()p Fw(x)p Fq(\))j(=)i Fo(\000)p Fs(1)18 b Fq(+)f Fp(N)3186 1986 y Fm(1)3225 1959 y Fq(\()p Fw(x)p Fq(\))p Fs(.)35 b(Namely)22 b(we)94 2079 y(consider)j(solutions)e Fw(u)899 2106 y Fr(t)934 2079 y Fq(\()p Fw(x)p Fq(\))k(=)h Fw(u)p Fq(\()p Fw(x;)17 b(t)p Fq(\))24 b Fs(of)1076 2281 y Fw(@)1129 2308 y Fr(t)1164 2281 y Fw(u)44 b Fq(=)h(\()p Fs(1)22 b Fq(+)g Fs(i)p Fw(\013)p Fq(\))p Fw(@)1787 2238 y Fm(2)1781 2308 y Fr(x)1831 2281 y Fw(u)g Fo(\000)h Fw(u)f Fq(+)h Fp(N)2267 2308 y Fm(1)2305 2281 y Fq(\()p Fo(j)p Fw(u)p Fo(j)2457 2238 y Fm(2)2496 2281 y Fq(\))p Fw(u)f Fq(+)h Fw(\016)t(F)39 b(:)776 b Fq(\()p Fs(3)p Fw(:)p Fs(1)p Fq(\))94 2483 y Fs(W)-8 b(e)26 b(strengthen)e(Assumption)e(2.1)j(and)f (Assumption)f(2.3)h(with)g(the)h(additional)94 2675 y Fu(Assumption)h(3.1.)35 b Fi(W)-8 b(e)25 b(ha)n(v)o(e)g Fs(3)p Fw(\013)1293 2632 y Fm(2)1360 2675 y 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Fw(F)14 b Fq(\()p Fw(x;)j(\034)32 b Fq(+)22 b Fw(t)p Fq(\))g Fo(\000)h Fw(F)14 b Fq(\()p Fw(x;)j(t)2442 5244 y Fr(k)2512 5218 y Fq(+)22 b Fw(t)p Fq(\))p Fo(j)44 b(\024)h Fw(C)2958 5175 y Fn(\000)p Fm(1)3060 5218 y Fw(")26 b(:)p eop %%Page: 7 7 7 6 bop 94 -45 a Fw(")p Fv({Entr)n(opy)38 b(of)h(Driven)e(Equa)-7 b(tions)2105 b Fu(7)316 194 y Fs(Item)27 b(1\))h(in)f(Assumption)e(3.1) i(is)g(the)g(de\002nition)g(of)h(almost)e(periodic)h(functions,)g(item) g(2\))g(requires)94 314 y(that)f(the)f(amplitude)g(of)g(the)h (oscillations)e(of)h Fw(F)40 b Fs(are)27 b(of)e(order)h(1)g(and)g(item) f(3\))g(asserts)h(that)f(frequencies)h(of)94 433 y(oscillation)e(at)g (dif)n(ferent)h(locations)e(on)i Fu(R)g Fs(are)g(in)g(some)f(sense)h (independent)f(of)g(each)i(other)-5 b(.)316 553 y(W)d(e)41 b(ne)o(xt)f(pro)o(vide)f(tw)o(o)i(e)o(xamples)e(of)i(functions)e Fw(F)55 b Fs(that,)44 b(although)c(not)g(written)g(in)g(the)g(form)94 672 y(required)25 b(by)g(Assumption)d(2.3,)j(do)f(satisfy)g(Assumption) f(3.1.)35 b(The)25 b(\002rst)g(one)g(is)f(inspired)g(by)h([C2]:)949 947 y Fw(F)1013 974 y Fm(1)1053 947 y Fq(\()p Fw(x;)17 b(t)p Fq(\))43 b(=)1442 852 y Fk(Y)1434 1066 y Fr(k)r Fn(2)p Ff(Z)1595 947 y Fs(sign)o Fq(\()p Fo(\000)p Fw(k)s Fq(\))17 b Fs(tanh)2159 866 y Fk(\000)2204 947 y Fw(x)22 b Fo(\000)h Fw(z)2429 974 y Fr(k)2478 947 y Fq(\()p Fw(t)p Fq(\))f Fo(\000)h Fs(4)p Fw(k)s(x)2876 904 y Fn(\003)2921 866 y Fk(\001)2992 947 y Fw(;)94 1305 y Fs(where)j Fw(z)409 1332 y Fr(k)458 1305 y Fq(\()p Fw(t)p Fq(\))h(=)h Fw(x)761 1262 y Fn(\003)824 1305 y Fs(cos)p Fq(\()p Fw(!)1058 1332 y Fr(k)1106 1305 y Fw(t)22 b Fq(+)h Fw(')1329 1332 y Fr(k)1378 1305 y Fq(\))i Fs(with)f Fo(f)p Fw(!)1756 1332 y Fr(k)1804 1305 y Fo(g)1854 1332 y Fr(k)r Fn(2)p Ff(Z)2028 1305 y Fs(independent)g(o)o(v)o(er)g(the)h(rationals.)35 b(The)24 b(phases)h Fw(')3825 1332 y Fr(k)94 1424 y Fs(are)h(arbitrary) f(real)g(numbers.)35 b(Remark)26 b(that)e Fw(F)1766 1451 y Fm(1)1831 1424 y Fs(is)g(a)h(well\255de\002ned)g(bounded)f(analytic)g (function.)316 1544 y(The)j(second)g(e)o(xample)g(is)g(v)o(ery)g (similar:)39 b(let)27 b Fw(f)1984 1571 y Fm(0)2024 1544 y Fq(\()p Fw(x)p Fq(\))f Fs(be)i(a)g Fo(C)2437 1501 y Fn(1)2549 1544 y Fs(function)e(supported)g(by)i([)p Fo(\000)p Fw(x)3623 1501 y Fn(\003)3669 1544 y Fw(;)17 b(x)3771 1501 y Fn(\003)3816 1544 y Fs(],)94 1664 y Fo(j)p Fw(f)171 1690 y Fm(0)211 1664 y Fq(\()p Fw(x)p Fq(\))p Fo(j)27 b(\025)h Fs(3)p Fw(=)p Fs(4)c(for)h Fo(j)p Fw(x)p Fo(j)i(\024)h Fw(x)1123 1621 y Fn(\003)1169 1664 y Fs(.)36 b(W)-8 b(e)25 b(de\002ne)1011 2004 y Fw(F)1075 2031 y Fm(2)1115 2004 y Fq(\()p Fw(x;)17 b(t)p Fq(\))43 b(=)1579 1879 y Fn(1)1547 1909 y Fk(X)1496 2124 y Fr(k)r Fj(=)p Fn(\0001)1759 2004 y Fs(cos)p Fq(\()p Fw(!)1993 2031 y Fr(k)2042 2004 y Fw(t)22 b Fq(+)h Fw(')2265 2031 y Fr(k)2314 2004 y Fq(\))p Fw(f)2402 2031 y Fm(0)2441 2004 y Fq(\()p Fw(x)f Fq(+)h Fs(4)p Fw(x)2766 1961 y Fn(\003)2811 2004 y Fw(k)s Fq(\))i Fw(;)94 2357 y 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b(It)28 b(follo)n(ws)e(from)h(Eq.\(5.1\))g(and)h(Assumption)d(3.1)j (that)f Fo(j)p Fw(\020)3489 2896 y Fr(t)3523 2869 y Fq(\()p Fw(x)p Fq(\))p Fo(j)k(\024)h Fw(\016)94 2988 y Fs(for)26 b(all)e Fq(\()p Fw(x;)17 b(t)p Fq(\))p Fs(.)35 b(Moreo)o(v)o(er)l(,)23 b(the)i(follo)n(wing)e(holds:)94 3180 y Fu(Lemma)e(5.1.)34 b Fi(There)20 b(is)g(a)g Fw(C)35 b(>)28 b Fs(0)20 b Fi(such)f(that)h (if)g Fw(\016)k Fi(is)19 b(suf)n(\002ciently)g(small,)h Fw(\013)28 b(<)g Fs(1)p Fw(=)2941 3097 y Fo(p)p 3023 3097 50 4 v 3023 3180 a Fs(3)p Fi(,)21 b(and)f Fo(k)p Fw(u)3390 3207 y Fm(0)3441 3180 y Fo(\000)11 b Fw(\020)3573 3207 y Fm(0)3613 3180 y Fo(k)3663 3207 y Fm(H)3713 3187 y Fh(1)3713 3234 y Fg(')3797 3180 y Fo(\024)94 3343 y Fw(C)c(\016)220 3300 y Fm(3)261 3343 y Fi(,)25 b(then)f Fo(k)p Fw(u)614 3370 y Fr(t)671 3343 y Fo(\000)f Fw(\020)815 3370 y Fr(t)849 3343 y Fo(k)899 3370 y Fm(H)949 3350 y Fh(1)949 3397 y Fg(')1033 3343 y Fo(\024)28 b Fw(C)7 b(\016)1264 3300 y Fm(3)1329 3343 y Fi(for)25 b(all)g 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Fq(\()p Fw(x)2659 4473 y Fm(2)2698 4447 y Fw(;)17 b(t)p Fq(\))22 b Fs(if)g Fo(j)p Fw(x)3008 4473 y Fm(1)3064 4447 y Fo(\000)17 b Fw(x)3215 4473 y Fm(2)3255 4447 y Fo(j)27 b(\025)h Fs(2)p Fw(A)17 b Fs(log)f Fw(")3747 4404 y Fn(\000)p Fm(1)3849 4447 y Fs(.)94 4566 y(The)25 b(follo)n(wing)e(proof)i(is)f(a)h(rigorous)f(implementation)f(of)i (this)e(idea.)316 4686 y(In)e(the)f(sequel)h(we)g(assume)f(\(suf)n (\002ciently)g(lar)n(ge\))i Fw(T)35 b Fs(and)21 b Fw(L)p Fs(,)g(and)g(a)g(\(suf)n(\002ciently)f(small\))g Fw(")h Fs(ha)n(v)o(e)g(been)94 4805 y(\(arbitrarily\))k(\002x)o(ed.)36 b(W)-8 b(e)25 b(\002rst)g(sho)n(w)f(that)h(for)g(all)f Fw(x)1908 4832 y Fm(0)1976 4805 y Fo(2)k Fs([)p Fo(\000)p Fw(L;)17 b(L)p Fs(],)25 b Fw(t)j Fo(7!)f Fw(u)p Fq(\()p Fw(x)2788 4832 y Fm(0)2828 4805 y Fw(;)17 b(t)p Fq(\))24 b Fs(is)g(almost)g(periodic.)35 b(Let)94 4925 y Fw(I)h Fq(=)28 b Fs([)p Fw(x)369 4952 y Fm(0)423 4925 y Fo(\000)14 b Fw(A)j Fs(log)f Fw(")796 4882 y Fn(\000)p Fm(1)899 4925 y Fw(;)h(x)1001 4952 y Fm(0)1054 4925 y Fq(+)d Fw(A)j Fs(log)f Fw(")1427 4882 y Fn(\000)p Fm(1)1529 4925 y Fs(])22 b(and)f(let)g Fw(\034)39 b Fq(=)28 b Fw(T)14 b Fq(\()p Fw(";)j(I)8 b Fq(\))20 b Fs(be)h(an)h Fw(")p Fs(\226period)f(for)g Fw(F)36 b Fs(\(see)21 b(Assumption)94 5044 y(3.1\).)36 b(De\002ne)104 5291 y Fw(v)152 5318 y Fr(t)188 5291 y Fq(\()p Fw(x)p Fq(\))43 b(=)i Fw(u)p Fq(\()p Fw(x)641 5318 y Fm(0)703 5291 y Fq(+)22 b Fw(x;)17 b(t)22 b Fq(+)g Fw(\034)11 b Fq(\))22 b Fo(\000)h Fw(u)p Fq(\()p Fw(x)1430 5318 y Fm(0)1491 5291 y Fq(+)g Fw(x;)17 b(t)p Fq(\))24 b Fw(;)116 b(G)p Fq(\()p Fw(x;)17 b(t)p Fq(\))43 b(=)i Fw(\016)t(F)14 b Fq(\()p Fw(x)2617 5318 y Fm(0)2679 5291 y Fq(+)22 b Fw(x;)17 b(t)k Fq(+)i Fw(\034)11 b Fq(\))22 b Fo(\000)g Fw(\016)t(F)14 b Fq(\()p Fw(x)3474 5318 y Fm(0)3536 5291 y Fq(+)23 b Fw(x;)17 b(t)p Fq(\))24 b Fw(;)p eop %%Page: 11 11 11 10 bop 94 -45 a Fw(")p Fv({Entr)n(opy)38 b(of)h(Driven)e(Equa)-7 b(tions)2055 b Fu(11)94 194 y Fs(and)25 b(assume)f Fw(t)k(>)g(T)42 b(>)28 b(B)21 b Fs(log)16 b Fw(")1242 151 y Fn(\000)p Fm(1)1345 194 y Fs(.)36 b(By)25 b(Eq.\(3.1\),)f(we)h(ha)n(v)o(e)g (\(using)f(Eq.\(2.2\))g(and)h(Lemma)f(5.1\),)283 446 y Fw(@)336 473 y Fr(t)387 310 y Fk(Z)503 446 y Fw(')568 473 y Fr(y)616 446 y Fo(j)p Fw(v)692 473 y Fr(t)728 446 y Fo(j)756 403 y Fm(2)839 446 y Fo(\024)45 b(\000)p Fq(\()p Fs(2)22 b Fq(+)h Fs(2)p Fw(\016)1347 403 y Fm(2)1387 446 y Fq(\))1443 310 y Fk(Z)1558 446 y Fw(')1623 473 y Fr(y)1671 446 y Fo(j)p Fw(v)1747 473 y Fr(t)1782 446 y Fo(j)1810 403 y Fm(2)1872 446 y Fq(+)g Fw(\016)2036 310 y Fk(Z)2152 446 y Fw(')2217 473 y Fr(y)2266 446 y Fo(j)p Fw(G)2372 473 y Fr(t)2407 446 y Fo(j)2435 403 y Fm(2)2518 446 y Fo(\024)45 b(\000)2734 310 y Fk(Z)2850 446 y Fw(')2915 473 y Fr(y)2964 446 y Fo(j)p Fw(v)3040 473 y Fr(t)3075 446 y Fo(j)3103 403 y Fm(2)3164 446 y Fq(+)23 b Fw(C)7 b(")3388 403 y Fm(2)3453 446 y Fw(;)188 b Fq(\()p Fs(5)p Fw(:)p Fs(4)p Fq(\))94 693 y Fs(thus)572 848 y Fo(j)p Fw(v)648 875 y Fr(t)683 848 y Fq(\()p Fs(0)p Fq(\))p Fo(j)44 b Fq(=)1004 703 y Fk(\014)1004 763 y(\014)1004 823 y(\014)1004 883 y(\014)1038 848 y Fw(e)1084 805 y Fn(\000)p Fr(t)1181 767 y Fk(\000)1227 848 y Fw(g)1275 875 y Fr(t)1332 848 y Fw(?)22 b(v)1452 875 y Fm(0)1492 767 y Fk(\001)1537 848 y Fq(\()p Fs(0)p Fq(\))g(+)1787 712 y Fk(Z)1886 755 y Fr(t)1842 939 y Fm(0)1938 848 y Fw(e)1984 805 y Fn(\000)p Fr(s)2089 767 y Fk(\000)2135 848 y Fw(g)2183 875 y Fr(s)2247 848 y Fw(?)g Fo(j)2361 821 y Fq(~)2347 848 y Fp(N)2425 875 y Fr(t)p Fn(\000)p Fr(s)2559 848 y Fw(v)2607 875 y Fr(t)p Fn(\000)p Fr(s)2764 848 y Fq(+)h Fw(G)2942 875 y Fr(t)p Fn(\000)p Fr(s)3077 767 y Fk(\001)3123 848 y Fq(\()p Fs(0)p Fq(\))17 b Fs(d)p Fw(s)3365 703 y Fk(\014)3365 763 y(\014)3365 823 y(\014)3365 883 y(\014)883 1118 y Fo(\024)44 b Fw(C)7 b(e)1128 1075 y Fn(\000)p Fr(t)1249 1118 y Fq(+)22 b Fw(C)7 b(\016)1474 1075 y Fm(2)1531 983 y Fk(Z)1631 1025 y Fr(t)1586 1209 y Fm(0)1682 1118 y Fw(e)1728 1075 y Fn(\000)p Fr(s)1834 1037 y Fk(\000)1879 1118 y Fo(j)p Fw(g)1955 1145 y Fr(s)1997 1118 y Fo(j)22 b Fw(?)g Fo(j)p Fw(v)2195 1145 y Fr(t)p Fn(\000)p Fr(s)2330 1118 y Fo(j)2358 1037 y Fk(\001)2403 1118 y Fq(\()p Fs(0)p Fq(\))17 b Fs(d)p Fw(s)k Fq(+)h Fw(C)7 b(")883 1385 y Fo(\024)44 b Fw(C)7 b(")23 b Fq(+)g Fw(C)7 b(\016)1377 1342 y Fm(2)1434 1250 y Fk(Z)1533 1292 y Fr(t)1489 1476 y Fm(0)1585 1385 y Fw(e)1631 1342 y Fn(\000)p Fr(s)1736 1385 y Fo(k)p Fw(g)1834 1412 y Fr(s)1876 1385 y Fw(=)1926 1323 y Fo(p)p 2009 1323 106 4 v 62 x Fw(')2074 1412 y Fm(0)2114 1385 y Fo(k)2164 1412 y Fm(2)2204 1305 y Fk(\000)2249 1385 y Fw(e)2295 1340 y Fn(\000)2370 1301 y Fm(1)p 2370 1317 35 4 v 2370 1375 a(2)2417 1340 y Fj(\()p Fr(t)p Fn(\000)p Fr(s)p Fj(\))2636 1385 y Fq(+)23 b Fw(")2782 1342 y Fm(2)2822 1305 y Fk(\001)883 1601 y Fo(\024)44 b Fw(C)7 b(")26 b(;)3669 1195 y Fq(\()p Fs(5)p Fw(:)p Fs(5)p Fq(\))94 1770 y Fs(where)h(we)g(ha)n(v)o(e)f(assumed)f(that)h Fw(\016)31 b Fs(w)o(as)26 b(small)f(enough)h(and)g Fw(A;)17 b(B)30 b Fs(where)d(lar)n(ge)g(enough.)39 b(Hence)27 b(an)o(y)f Fw(")p Fs(\226)94 1889 y(period)f(for)f Fw(F)39 b Fs(on)24 b Fw(I)32 b Fs(is)25 b(an)f Fw(")p Fs(\226period)h(for)f Fw(t)k Fo(7!)f Fw(u)p Fq(\()p Fw(x)1887 1916 y Fm(0)1927 1889 y Fw(;)17 b(t)p Fq(\))p Fs(.)35 b(Moreo)o(v)o(er)l(,)23 b Fw(u)p Fq(\()p Fw(x)2695 1916 y Fm(0)2735 1889 y Fw(;)17 b(t)p Fq(\))23 b Fs(oscillates)h(with)g(amplitude)94 2009 y Fp(O)p Fq(\()p Fw(\016)251 1966 y Fm(2)292 2009 y Fq(\))p Fs(,)g(for)h Fw(\020)7 b Fq(\()p Fw(x)668 2036 y Fm(0)708 2009 y Fw(;)17 b(T)d Fq(\))24 b Fs(does)g(\(and)h(Lemma)f (5.1)h(holds\):)35 b(let)24 b Fw(x)2270 2036 y Fm(0)2310 2009 y Fs(,)h Fw(t)2396 1966 y Fn(\003)2469 2009 y Fw(>)j(T)39 b Fs(and)25 b Fw(\034)36 b Fs(be)25 b(such)f(that)1050 2205 y Fk(\014)1050 2264 y(\014)1083 2154 y(Z)1182 2197 y Fr(t)1212 2154 y Fc(\003)1138 2380 y Fm(0)1276 2289 y Fw(e)1322 2247 y Fn(\000)p Fr(s)1427 2209 y Fk(\000)1473 2289 y Fw(F)1537 2316 y Fr(t)1567 2296 y Fc(\003)1609 2316 y Fj(+)p Fr(\034)8 b Fn(\000)p Fr(s)1819 2289 y Fq(\()p Fw(x)p Fq(\))22 b Fo(\000)h Fw(F)2140 2316 y Fr(t)2170 2296 y Fc(\003)2211 2316 y Fj(+)p Fr(s)2315 2289 y Fq(\()p Fw(x)p Fq(\))2450 2209 y Fk(\001)2512 2289 y Fs(d)p Fw(s)2609 2205 y Fk(\014)2609 2264 y(\014)2686 2289 y Fo(\025)2819 2250 y Fm(1)p 2819 2266 V 2819 2324 a(2)2891 2289 y Fw(;)94 2561 y Fs(for)j(all)e Fw(x)h Fs(with)f Fo(j)p Fw(x)e Fo(\000)g Fw(x)907 2588 y Fm(0)947 2561 y Fo(j)27 b(\024)i Fw(x)1165 2519 y Fn(\003)1210 2561 y Fs(.)36 b(Assume)24 b(that)h Fo(j)p Fw(\020)1864 2588 y Fr(t)1894 2568 y Fc(\003)1935 2588 y Fj(+)p Fr(\034)2045 2561 y Fq(\()p Fw(x)p Fq(\))d Fo(\000)h Fw(\020)2346 2588 y Fr(t)2376 2568 y Fc(\003)2422 2561 y Fq(\()p Fw(x)p Fq(\))p Fo(j)k Fw(<)2729 2522 y Fm(1)p 2729 2539 V 2729 2596 a(4)2776 2561 y Fw(\016)i Fs(for)c(all)f(such)h Fw(x)p Fs(.)36 b(Then)383 2850 y Fo(j)p Fw(\020)455 2877 y Fr(t)485 2857 y Fc(\003)526 2877 y Fj(+)p Fr(\034)637 2850 y Fq(\()p Fw(x)p Fq(\))22 b Fo(\000)h Fw(\020)938 2877 y Fr(t)968 2857 y Fc(\003)1014 2850 y Fq(\()p Fw(x)p Fq(\))p Fo(j)43 b Fq(=)1342 2705 y Fk(\014)1342 2765 y(\014)1342 2825 y(\014)1342 2885 y(\014)1375 2714 y(Z)1475 2757 y Fr(t)1505 2714 y Fc(\003)1547 2757 y Fj(+)p Fr(\034)1431 2940 y Fm(0)1674 2850 y Fw(e)1720 2807 y Fn(\000)p Fr(s)1825 2769 y Fk(\000)1871 2850 y Fq(\()p Fs(1)22 b Fq(+)g Fs(i)p Fw(\013)p Fq(\))p Fw(@)2271 2807 y Fm(2)2265 2877 y Fr(x)2315 2850 y Fw(\020)2359 2877 y Fr(t)2389 2857 y Fc(\003)2430 2877 y Fj(+)p Fr(\034)8 b Fn(\000)p Fr(s)2641 2850 y Fq(\()p Fw(x)p Fq(\))22 b(+)g Fw(\016)t(F)3009 2877 y Fr(t)3039 2857 y Fc(\003)3081 2877 y Fj(+)p Fr(\034)8 b Fn(\000)p Fr(s)3292 2850 y Fq(\()p Fw(x)p Fq(\))3427 2769 y Fk(\001)3489 2850 y Fs(d)p Fw(s)1403 3143 y Fo(\000)1502 3007 y Fk(Z)1602 3050 y Fr(t)1632 3007 y Fc(\003)1558 3234 y Fm(0)1695 3143 y Fw(e)1741 3100 y Fn(\000)p Fr(s)1847 3062 y Fk(\000)1892 3143 y Fq(\()p Fs(1)22 b Fq(+)h Fs(i)p Fw(\013)p Fq(\))p Fw(@)2293 3100 y Fm(2)2287 3170 y Fr(x)2336 3143 y Fw(\020)2380 3170 y Fr(t)2410 3150 y Fc(\003)2452 3170 y Fn(\000)p Fr(s)2556 3143 y Fq(\()p Fw(x)p Fq(\))f(+)h Fw(\016)t(F)2925 3170 y Fr(t)2955 3150 y Fc(\003)2996 3170 y Fn(\000)p Fr(s)3101 3143 y Fq(\()p Fw(x)p Fq(\))3236 3062 y Fk(\001)3298 3143 y Fs(d)p Fw(s)3395 2998 y Fk(\014)3395 3058 y(\014)3395 3118 y(\014)3395 3178 y(\014)1220 3452 y Fo(\025)45 b(\000)p Fw(C)7 b(e)1543 3409 y Fn(\000)p Fr(t)1635 3366 y Fc(\003)1705 3452 y Fq(+)23 b Fw(\016)1870 3277 y Fk(\014)1870 3337 y(\014)1870 3397 y(\014)1870 3457 y(\014)1870 3517 y(\014)1903 3316 y(Z)2002 3359 y Fr(t)2032 3316 y Fc(\003)1958 3543 y Fm(0)2096 3452 y Fw(e)2142 3409 y Fn(\000)p Fr(s)2247 3371 y Fk(\000)2293 3452 y Fw(F)2357 3479 y Fr(t)2387 3459 y Fc(\003)2428 3479 y Fj(+)p Fr(\034)8 b Fn(\000)p Fr(s)2639 3452 y Fq(\()p Fw(x)p Fq(\))22 b Fo(\000)g Fw(F)2959 3479 y Fr(t)2989 3459 y Fc(\003)3031 3479 y Fj(+)p Fr(s)3135 3452 y Fq(\()p Fw(x)p Fq(\))3270 3371 y Fk(\001)3332 3452 y Fs(d)p Fw(s)3429 3277 y Fk(\014)3429 3337 y(\014)3429 3397 y(\014)3429 3457 y(\014)3429 3517 y(\014)1403 3775 y Fo(\000)g Fw(C)1597 3639 y Fk(Z)1697 3682 y Fr(t)1727 3639 y Fc(\003)1653 3866 y Fm(0)1790 3775 y Fw(e)1836 3732 y Fn(\000)p Fr(s)1958 3690 y Fk(\014)1958 3750 y(\014)1991 3775 y Fw(@)2050 3732 y Fm(2)2044 3802 y Fr(x)2094 3694 y Fk(\000)2140 3775 y Fw(\020)2184 3802 y Fr(t)2214 3782 y Fc(\003)2255 3802 y Fj(+)p Fr(\034)8 b Fn(\000)p Fr(s)2466 3775 y Fq(\()p Fw(x)p Fq(\))22 b Fo(\000)g Fw(\020)2766 3802 y Fr(t)2796 3782 y Fc(\003)2838 3802 y Fn(\000)p Fr(s)2943 3775 y Fq(\()p Fw(x)p Fq(\))3078 3694 y Fk(\001)3123 3690 y(\014)3123 3750 y(\014)1220 4068 y Fo(\025)1354 4029 y Fm(1)p 1354 4045 V 1354 4103 a(2)1401 4068 y Fw(\016)k Fo(\000)d Fw(")f Fo(\000)h Fw(C)1834 3933 y Fk(Z)1933 3975 y Fr(t)1963 3933 y Fc(\003)1889 4159 y Fm(0)2027 4068 y Fw(e)2073 4025 y Fn(\000)p Fr(s)2195 3984 y Fk(\014)2195 4043 y(\014)2228 4068 y Fw(@)2287 4025 y Fm(2)2281 4095 y Fr(x)2331 3987 y Fk(\000)2376 4068 y Fw(\020)2420 4095 y Fr(t)2450 4075 y Fc(\003)2492 4095 y Fj(+)p Fr(\034)8 b Fn(\000)p Fr(s)2703 4068 y Fq(\()p Fw(x)p Fq(\))21 b Fo(\000)i Fw(\020)3003 4095 y Fr(t)3033 4075 y Fc(\003)3074 4095 y Fn(\000)p Fr(s)3179 4068 y Fq(\()p Fw(x)p Fq(\))3314 3987 y Fk(\001)3359 3984 y(\014)3359 4043 y(\014)3434 4068 y Fw(:)94 4344 y Fs(This)15 b(implies)g(that)g Fo(j)p Fw(@)849 4301 y Fm(2)843 4371 y Fr(x)891 4344 y Fq(\()p Fw(\020)974 4371 y Fr(t)1004 4351 y Fc(\003)1045 4371 y Fj(+)p Fr(\034)8 b Fn(\000)p Fr(s)1256 4344 y Fq(\()p Fw(x)p Fq(\))q Fo(\000)q Fw(\020)1514 4371 y Fr(t)1544 4351 y Fc(\003)1585 4371 y Fn(\000)p Fr(s)1690 4344 y Fq(\()p Fw(x)p Fq(\)\))p Fo(j)15 b Fs(has)g(a)g(maximum)g(of)g(order)g Fw(\016)k Fs(some)n(where)c(in)g Fo(j)p Fw(x)q Fo(\000)q Fw(x)3702 4371 y Fm(0)3741 4344 y Fo(j)28 b(\024)94 4479 y Fw(x)151 4436 y Fn(\003)222 4479 y Fs(and)d Fw(s)i(<)h(t)606 4436 y Fn(\003)677 4479 y Fs(hence)d Fo(j)p Fw(\020)1006 4506 y Fr(t)1036 4486 y Fc(\003)1077 4506 y Fj(+)p Fr(\034)8 b Fn(\000)p Fr(s)1288 4479 y Fq(\()p Fw(x)p Fq(\))22 b Fo(\000)g Fw(\020)1588 4506 y Fr(t)1618 4486 y Fc(\003)1660 4506 y Fn(\000)p Fr(s)1765 4479 y Fq(\()p Fw(x)p Fq(\))p Fo(j)k(\025)i Fw(C)7 b(\016)2185 4436 y Fm(2)2251 4479 y Fs(for)25 b(some)f Fw(x)p Fs(,)h Fw(s)f Fs(nearby)-6 b(.)316 4614 y(W)e(e)15 b(ha)n(v)o(e)g(thus)g(sho)n(wn)g(that)g(for)g (all)g Fo(f)p Fw(x)1635 4641 y Fn(\000)p Fr(M)1782 4614 y Fw(;)i(:)g(:)g(:)d(;)j(x)2061 4641 y Fr(M)2149 4614 y Fo(g)27 b(\032)h Fs([)p Fo(\000)p Fw(L;)17 b(L)p Fs(])e(with)g Fw(M)37 b Fq(=)28 b Fs([)p Fw(L=)p Fq(\()p Fs(2)p Fw(A)17 b Fs(log)e Fw(")3623 4571 y Fn(\000)p Fm(1)3725 4614 y Fq(\))p Fs(])p Fo(\000)94 4749 y Fs(1)g(and)g Fo(j)p Fw(x)403 4776 y Fr(k)452 4749 y Fo(\000)p Fw(x)586 4776 y Fr(j)628 4749 y Fo(j)27 b(\025)i Fw(\016)833 4776 y Fr(k)r(j)919 4749 y Fw(A)17 b Fs(log)e Fw(")1200 4706 y Fn(\000)p Fm(1)1303 4749 y Fs(,)i(there)e(is)g(a)g(bounded)g (function)g Fw(u)2464 4706 y Fn(\003)2522 4749 y Fs(such)g(that)g(for)g (all)g Fo(f)p Fw(a)3234 4776 y Fn(\000)p Fr(M)3383 4749 y Fw(;)i(:)g(:)g(:)d(;)j(a)3658 4776 y Fr(M)3745 4749 y Fo(g)28 b(2)94 4889 y Fs([)p Fo(\000)p Fw(C)7 b(\016)330 4846 y Fm(2)371 4889 y Fw(;)17 b(C)7 b(\016)542 4846 y Fm(2)582 4889 y Fs(])33 b(there)g(are)g Fw(t)1070 4916 y Fn(\000)p Fr(M)1221 4889 y Fw(;)17 b(:)g(:)g(:)d(;)j(t)1479 4916 y Fr(M)1599 4889 y Fs(for)33 b(which)g Fw(u)p Fq(\()p Fw(x)2178 4916 y Fr(k)2226 4889 y Fw(;)17 b(t)2307 4916 y Fr(k)2356 4889 y Fq(\))38 b(=)h Fw(u)2606 4846 y Fn(\003)2652 4889 y Fq(\()p Fw(x)2748 4916 y Fr(k)2797 4889 y Fq(\))26 b(+)h Fw(a)3019 4916 y Fr(k)3068 4889 y Fs(.)59 b(Moreo)o(v)o(er)l(,)34 b(by)e(As\255)94 5008 y(sumption)23 b(3.1)i(there)g(is)f(a)h Fw(\034)36 b Fs(such)25 b(that)555 5212 y(sup)609 5294 y Fr(t)710 5212 y Fs(max)774 5278 y Fr(k)1055 5212 y Fs(sup)898 5304 y Fn(j)p Fr(x)p Fn(j\024)p Fr(A)13 b Fm(log)h Fr(")1267 5284 y Fc(\000)p Fh(1)1367 5212 y Fo(j)p Fw(F)g Fq(\()p Fw(x)1569 5239 y Fr(k)1639 5212 y Fq(+)23 b Fw(x;)17 b(\034)32 b Fq(+)23 b Fw(t)p Fq(\))f Fo(\000)g Fw(F)14 b Fq(\()p Fw(x)2387 5239 y Fr(k)2458 5212 y Fq(+)23 b Fw(x;)17 b(t)2696 5239 y Fr(k)2766 5212 y Fq(+)23 b Fw(t)p Fq(\))p Fo(j)43 b(\024)i Fw(C)3212 5169 y Fn(\000)p Fm(1)3315 5212 y Fw(")25 b(:)p eop %%Page: 12 12 12 11 bop 94 -45 a Fw(")p Fv({Entr)n(opy)38 b(of)h(Driven)e(Equa)-7 b(tions)2055 b Fu(12)94 194 y Fs(By)19 b(repeating)e(the)h (calculations)f(made)h(abo)o(v)o(e)f(\(Eq.\(5.4\))g(and)h(Eq.\(5.5\)\)) g(we)g(obtain)f(that)h(there)g(is)f(a)h Fw(t)3623 151 y Fn(\003)3697 194 y Fw(>)28 b(T)94 314 y Fs(such)d(that)1231 433 y(max)1093 499 y Fr(k)r Fj(=)p Fn(\000)p Fr(M)s(;:::)o(;M)1558 433 y Fo(j)p Fw(u)p Fq(\()p Fw(x)1739 460 y Fr(k)1787 433 y Fw(;)17 b(t)1868 390 y Fn(\003)1913 433 y Fq(\))22 b Fo(\000)g Fw(a)2126 460 y Fr(k)2197 433 y Fo(\000)h Fw(u)2354 390 y Fn(\003)2400 433 y Fq(\()p Fw(x)2496 460 y Fr(k)2544 433 y Fq(\))p Fo(j)44 b(\024)h Fw(")25 b(:)94 655 y Fs(This)g(pro)o(v)o(es)e(that)758 574 y Fk(\000)804 580 y(S)887 685 y Fr(t>T)1059 655 y Fq(\010)1131 612 y Fr(t)1131 682 y(f)1182 655 y Fq(\()p Fs(0)p Fq(\))1310 574 y Fk(\001)1355 695 y Fr(L)1440 655 y Fs(contains)h(an)h Fw(")p Fs(\226distinguishable)d(set)j Fp(Y)g Fs(with)1531 913 y(card)q Fq(\()p Fp(Y)p Fq(\))44 b Fo(\025)2006 772 y Fk(\022)2091 846 y Fs(2)p Fw(\016)2189 803 y Fm(2)p 2091 890 138 4 v 2137 981 a Fw(")2241 772 y Fk(\023)2314 799 y Fm(2)p Fr(M)94 1138 y Fs(\()p Fp(Y)29 b Fq(=)f Fo(f)p Fw(x)427 1165 y Fm(1)466 1138 y Fw(;)17 b(:)g(:)g(:)e(;)i(x)746 1165 y Fr(n)799 1138 y Fo(g)28 b(\032)g Fp(X)23 b Fs(is)g(called)g(an)g Fw(")p Fs(\226distinguishable)e(set)h(in)h Fp(X)g Fs(if)g(dist)p Fq(\()p Fw(x)2920 1165 y Fr(j)2961 1138 y Fw(;)17 b(x)3063 1165 y Fr(k)3111 1138 y Fq(\))28 b Fw(>)g(")c Fs(for)f(all)g Fw(j)33 b Fo(6)p Fq(=)28 b Fw(k)s Fs(,)94 1257 y(see)e([KT]\).)f(Hence) 801 1517 y Fw(H)884 1544 y Fr(")926 1436 y Fk(\000)991 1423 y([)972 1636 y Fr(t>T)1138 1517 y Fq(\010)1210 1474 y Fr(t)1210 1544 y(f)1262 1517 y Fq(\()p Fs(0)p Fq(\))1390 1436 y Fk(\001)1479 1517 y Fo(\025)1613 1450 y Fs(log)o Fq(\(\()p Fs(2)p Fw(\016)1916 1407 y Fm(2)1955 1450 y Fw(")2001 1407 y Fn(\000)p Fm(1)2104 1450 y Fq(\))2143 1407 y Fr(L=)p Fj(\()p Fr(A)13 b Fm(log)h Fr(")2484 1364 y Fc(\000)p Fh(1)2567 1407 y Fj(\))p Fn(\000)p Fm(2)2700 1450 y Fq(\))p 1613 1494 1127 4 v 2117 1585 a Fs(2)p Fw(L)2795 1517 y Fo(\025)2991 1450 y Fs(1)p 2929 1494 175 4 v 2929 1585 a(16)p Fw(A)3140 1517 y(:)94 1787 y Fs(W)l(ith)25 b(the)f(proof)h(of)g(Lemma)f(5.1)h(belo)n(w)-6 b(,)23 b(the)i(proof)f(of)h(Theorem)g(3.2)f(is)h(complete.)p 3794 1711 80 4 v 3794 1783 4 72 v 3870 1783 V 3794 1787 80 4 v 94 1949 a Fu(Pr)n(oof)f(of)g(Lemma)h(5.1.)e Fs(Let)g Fw(\015)33 b Fq(=)1331 1865 y Fo(p)p 1414 1865 276 4 v 84 x Fs(1)22 b Fq(+)h Fw(\013)1650 1921 y Fm(2)1690 1949 y Fs(,)h Fw(y)31 b Fo(2)d Fu(R)p Fs(,)c Fw(A)j(>)i Fs(0,)23 b Fw(\021)32 b(>)c Fs(0.)36 b(De\002ne)24 b Fw(v)2974 1976 y Fr(t)3037 1949 y Fq(=)k Fw(u)3199 1976 y Fr(t)3254 1949 y Fo(\000)20 b Fw(\020)3395 1976 y Fr(t)3430 1949 y Fs(.)35 b(Using)23 b(the)94 2069 y(cutof)n(f)i(function)f Fw(')777 2096 y Fr(y)850 2069 y Fs(de\002ned)h(by)f(Eq.\(2.2\),)h(we)g (ha)n(v)o(e)158 2261 y Fm(1)p 158 2278 35 4 v 158 2335 a(2)204 2300 y Fw(@)257 2327 y Fr(t)309 2165 y Fk(Z)425 2300 y Fw(')490 2327 y Fr(y)538 2220 y Fk(\000)584 2300 y Fw(A)p Fo(j)p Fw(v)735 2327 y Fr(t)770 2300 y Fo(j)798 2258 y Fm(2)859 2300 y Fq(+)e Fo(j)p Fw(v)1039 2258 y Fn(0)1035 2327 y Fr(t)1070 2300 y Fo(j)1098 2258 y Fm(2)1138 2220 y Fk(\001)173 2562 y Fq(=)45 b Fo(\000)389 2426 y Fk(Z)505 2562 y Fw(')570 2589 y Fr(y)619 2451 y Fk(\020)678 2562 y Fo(j)p Fw(v)758 2519 y Fn(00)754 2589 y Fr(t)808 2562 y Fo(j)836 2519 y Fm(2)898 2562 y Fq(+)23 b Fw(A)p Fo(j)p Fw(v)1149 2589 y Fr(t)1184 2562 y Fo(j)1212 2519 y Fm(2)1273 2562 y Fq(+)g(\()p Fw(A)e Fo(\000)i Fs(1)p Fq(\))p Fo(j)p Fw(v)1777 2519 y Fn(0)1773 2589 y Fr(t)1808 2562 y Fo(j)1836 2519 y Fm(2)1875 2451 y Fk(\021)1957 2562 y Fo(\000)f Fs(Re)2200 2426 y Fk(Z)2317 2562 y Fw(')2382 2519 y Fn(0)2382 2589 y Fr(y)2430 2562 y Fq(\()p Fs(1)f Fq(+)i Fs(i)p Fw(\013)p Fq(\))2771 2451 y Fk(\020)2830 2562 y Fw(A)p 2905 2507 52 4 v(v)2957 2589 y Fr(t)2992 2562 y Fw(v)3044 2519 y Fn(0)3040 2589 y Fr(t)3097 2562 y Fq(+)p 3197 2507 V 23 w Fw(v)3249 2519 y Fn(0)3249 2589 y Fr(t)3284 2562 y Fw(v)3336 2519 y Fn(00)3332 2589 y Fr(t)3387 2451 y Fk(\021)295 2823 y Fq(+)g Fs(Re)539 2687 y Fk(Z)655 2823 y Fw(')720 2850 y Fr(y)768 2712 y Fk(\020)827 2823 y Fw(A)p 902 2768 V(v)954 2850 y Fr(t)989 2823 y Fp(N)1067 2850 y Fm(1)1106 2823 y Fq(\()p Fo(j)p Fw(u)1230 2850 y Fr(t)1265 2823 y Fo(j)1293 2780 y Fm(2)1332 2823 y Fq(\))p Fw(u)1428 2850 y Fr(t)1485 2823 y Fq(+)p 1585 2768 V 23 w Fw(v)1637 2780 y Fn(0)1637 2850 y Fr(t)1672 2823 y Fw(@)1725 2850 y Fr(x)1775 2742 y Fk(\000)1820 2823 y Fp(N)1898 2850 y Fm(1)1937 2823 y Fq(\()p Fo(j)p Fw(u)2061 2850 y Fr(t)2096 2823 y Fo(j)2124 2780 y Fm(2)2163 2823 y Fq(\))p Fw(u)2259 2850 y Fr(t)2294 2742 y Fk(\001)2340 2712 y(\021)173 3106 y Fo(\024)45 b(\000)372 2970 y Fk(Z)489 3106 y Fw(')554 3133 y Fr(y)619 2965 y Fk(\022)675 2995 y(\020)735 3106 y Fo(j)p Fw(v)815 3063 y Fn(00)811 3133 y Fr(t)865 3106 y Fo(j)893 3063 y Fm(2)955 3106 y Fo(\000)23 b Fw(\015)6 b Fo(j)p Fw(v)1193 3063 y Fn(00)1189 3133 y Fr(t)1242 3106 y Fo(jj)p Fw(v)1350 3063 y Fn(0)1346 3133 y Fr(t)1381 3106 y Fo(j)22 b Fq(+)1542 3067 y Fm(1)p 1542 3083 35 4 v 1542 3140 a(4)1589 3106 y Fw(\015)1647 3063 y Fm(2)1686 3106 y Fo(j)p Fw(v)1766 3063 y Fn(0)1762 3133 y Fr(t)1797 3106 y Fo(j)1825 3063 y Fm(2)1865 2995 y Fk(\021)1930 3106 y Fq(+)2013 2995 y Fk(\020)2072 3106 y Fq(\()p Fw(A)g Fo(\000)g Fs(1)g Fo(\000)2491 3038 y Fw(\015)2549 2996 y Fm(2)p 2491 3083 97 4 v 2514 3174 a Fs(4)2600 3106 y Fq(\))p Fo(j)p Fw(v)2719 3063 y Fn(0)2715 3133 y Fr(t)2750 3106 y Fo(j)2778 3063 y Fm(2)2839 3106 y Fo(\000)h Fw(A\015)6 b Fo(j)p Fw(v)3152 3063 y Fn(0)3148 3133 y Fr(t)3182 3106 y Fo(jj)p Fw(v)3286 3133 y Fr(t)3320 3106 y Fo(j)22 b Fq(+)h Fw(A)p Fo(j)p Fw(v)3621 3133 y Fr(t)3656 3106 y Fo(j)3684 3063 y Fm(2)3723 2995 y Fk(\021)3766 2965 y(\023)295 3376 y Fq(+)g Fs(Re)539 3240 y Fk(Z)655 3376 y Fw(')720 3403 y Fr(y)768 3265 y Fk(\020)827 3376 y Fw(A)p 902 3321 52 4 v(v)954 3403 y Fr(t)989 3376 y Fp(N)1067 3403 y Fm(1)1106 3376 y Fq(\()p Fo(j)p Fw(u)1230 3403 y Fr(t)1265 3376 y Fo(j)1293 3333 y Fm(2)1332 3376 y Fq(\))p Fw(u)1428 3403 y Fr(t)1485 3376 y Fq(+)p 1585 3321 V 23 w Fw(v)1637 3333 y Fn(0)1637 3403 y Fr(t)1672 3376 y Fw(@)1725 3403 y Fr(x)1775 3295 y Fk(\000)1820 3376 y Fp(N)1898 3403 y Fm(1)1937 3376 y Fq(\()p Fo(j)p Fw(u)2061 3403 y Fr(t)2096 3376 y Fo(j)2124 3333 y Fm(2)2163 3376 y Fq(\))p Fw(u)2259 3403 y Fr(t)2294 3295 y Fk(\001)2340 3265 y(\021)173 3637 y Fo(\024)45 b(\000)p Fw(\021)442 3502 y Fk(Z)559 3637 y Fw(')624 3664 y Fr(y)672 3556 y Fk(\000)717 3637 y Fw(A)p Fo(j)p Fw(v)868 3664 y Fr(t)903 3637 y Fo(j)931 3594 y Fm(2)993 3637 y Fq(+)22 b Fo(j)p Fw(v)1172 3594 y Fn(0)1168 3664 y Fr(t)1203 3637 y Fo(j)1231 3594 y Fm(2)1271 3556 y Fk(\001)1339 3637 y Fo(\000)1438 3502 y Fk(Z)1555 3637 y Fw(')1620 3664 y Fr(y)1668 3527 y Fk(n)1734 3637 y Fq(\()p Fw(A)g Fo(\000)g Fs(1)g Fo(\000)2153 3598 y Fm(3)p 2153 3614 35 4 v 2153 3671 a(4)2200 3637 y Fw(\015)2258 3594 y Fm(2)2319 3637 y Fo(\000)g Fw(\021)t Fq(\))p Fo(j)p Fw(v)2590 3594 y Fn(0)2586 3664 y Fr(t)2621 3637 y Fo(j)2649 3594 y Fm(2)2711 3637 y Fq(+)g Fw(A)p Fq(\()p Fs(1)g Fo(\000)g Fw(\021)27 b Fo(\000)3282 3598 y Fm(1)p 3282 3614 V 3282 3671 a(2)3329 3637 y Fw(A)p Fq(\))p Fo(j)p Fw(v)3519 3664 y Fr(t)3554 3637 y Fo(j)3582 3594 y Fm(2)3621 3527 y Fk(o)295 3898 y Fq(+)c Fs(Re)539 3763 y Fk(Z)655 3898 y Fw(')720 3925 y Fr(y)768 3788 y Fk(\020)827 3898 y Fw(A)p 902 3844 52 4 v(v)954 3925 y Fr(t)989 3898 y Fp(N)1067 3925 y Fm(1)1106 3898 y Fq(\()p Fo(j)p Fw(u)1230 3925 y Fr(t)1265 3898 y Fo(j)1293 3856 y Fm(2)1332 3898 y Fq(\))p Fw(u)1428 3925 y Fr(t)1485 3898 y Fq(+)p 1585 3844 V 23 w Fw(v)1637 3856 y Fn(0)1637 3925 y Fr(t)1672 3898 y Fw(@)1725 3925 y Fr(x)1775 3818 y Fk(\000)1820 3898 y Fp(N)1898 3925 y Fm(1)1937 3898 y Fq(\()p Fo(j)p Fw(u)2061 3925 y Fr(t)2096 3898 y Fo(j)2124 3856 y Fm(2)2163 3898 y Fq(\))p Fw(u)2259 3925 y Fr(t)2294 3818 y Fk(\001)2340 3788 y(\021)2424 3898 y Fw(:)94 4129 y Fs(If)29 b(we)g(choose)f Fw(A)33 b Fq(=)g(\()p Fs(11)23 b Fq(+)i Fs(3)p Fw(\013)1235 4087 y Fm(2)1275 4129 y Fq(\))p Fw(=)p Fs(6)j(and)g Fw(\021)37 b Fq(=)c(\()p Fs(1)23 b Fo(\000)i Fs(3)p Fw(\013)2138 4087 y Fm(2)2178 4129 y Fq(\))p Fw(=)p Fs(12)j(\()p Fw(\021)36 b(>)e Fs(0)28 b(by)g(Assumption)e(3.1\),)j(then)f(the)94 4249 y(term)d(in)f(curly)h(brack)o(ets)g(v)n(anishes.)34 b(W)-8 b(e)26 b(obtain)241 4462 y Fw(@)294 4489 y Fr(t)346 4326 y Fk(Z)462 4462 y Fw(')527 4489 y Fr(y)575 4381 y Fk(\000)621 4462 y Fw(A)p Fo(j)p Fw(v)772 4489 y Fr(t)806 4462 y Fo(j)834 4419 y Fm(2)896 4462 y Fq(+)d Fo(j)p Fw(v)1076 4419 y Fn(0)1072 4489 y Fr(t)1107 4462 y Fo(j)1135 4419 y Fm(2)1174 4381 y Fk(\001)1264 4462 y Fo(\024)45 b(\000)p Fs(2)p Fw(\021)1583 4326 y Fk(Z)1699 4462 y Fw(')1764 4489 y Fr(y)1812 4381 y Fk(\000)1858 4462 y Fw(A)p Fo(j)p Fw(v)2009 4489 y Fr(t)2044 4462 y Fo(j)2072 4419 y Fm(2)2133 4462 y Fq(+)23 b Fo(j)p Fw(v)2313 4419 y Fn(0)2309 4489 y Fr(t)2344 4462 y Fo(j)2372 4419 y Fm(2)2411 4381 y Fk(\001)2479 4462 y Fq(+)g Fw(\021)2648 4326 y Fk(Z)2765 4462 y Fw(')2830 4489 y Fr(y)2878 4381 y Fk(\000)2923 4462 y Fw(A)p Fo(j)p Fw(v)3074 4489 y Fr(t)3109 4462 y Fo(j)3137 4419 y Fm(2)3199 4462 y Fq(+)f Fo(j)p Fw(v)3378 4419 y Fn(0)3374 4489 y Fr(t)3410 4462 y Fo(j)3438 4419 y Fm(2)3477 4381 y Fk(\001)1386 4723 y Fq(+)1485 4587 y Fk(Z)1602 4723 y Fw(')1667 4750 y Fr(y)1715 4723 y Fw(\021)1768 4680 y Fn(\000)p Fm(1)1870 4642 y Fk(\000)1916 4723 y Fw(AQ)2070 4680 y Fm(2)2070 4750 y(3)2109 4723 y Fo(j)p Fw(u)2194 4750 y Fr(t)2229 4723 y Fo(j)2257 4680 y Fm(6)2318 4723 y Fq(+)h Fs(2)p Fw(Q)2547 4680 y Fm(2)2547 4750 y(3)2586 4723 y Fo(j)p Fw(u)2671 4750 y Fr(t)2706 4723 y Fo(j)2734 4680 y Fm(4)2774 4723 y Fo(j)p Fw(u)2859 4680 y Fn(0)2859 4750 y Fr(t)2894 4723 y Fo(j)2922 4680 y Fm(2)2961 4642 y Fk(\001)1264 4984 y Fo(\024)1386 4903 y Fk(\000)1432 4984 y Fs(4)p Fw(\021)1535 4941 y Fn(\000)p Fm(1)1637 4984 y Fo(k)p Fw(u)1744 5011 y Fr(t)1778 4984 y Fo(k)1828 4941 y Fm(4)1828 5011 y Fn(1)1913 4984 y Fw(Q)1992 4941 y Fm(2)1992 5011 y(3)2053 4984 y Fo(\000)g Fw(\021)2206 4903 y Fk(\001)2268 4849 y(Z)2385 4984 y Fw(')2450 5011 y Fr(y)2498 4903 y Fk(\000)2543 4984 y Fw(A)p Fo(j)p Fw(v)2694 5011 y Fr(t)2729 4984 y Fo(j)2757 4941 y Fm(2)2819 4984 y Fq(+)f Fo(j)p Fw(v)2998 4941 y Fn(0)2994 5011 y Fr(t)3029 4984 y Fo(j)3057 4941 y Fm(2)3097 4903 y Fk(\001)1386 5245 y Fq(+)g Fs(4)p Fw(\021)1588 5203 y Fn(\000)p Fm(1)1690 5245 y Fo(k)p Fw(u)1797 5272 y Fr(t)1832 5245 y Fo(k)1882 5203 y Fm(4)1882 5272 y Fn(1)1967 5245 y Fw(Q)2046 5203 y Fm(2)2046 5272 y(3)2102 5110 y Fk(Z)2218 5245 y Fw(')2283 5272 y Fr(y)2331 5165 y Fk(\000)2377 5245 y Fw(A)p Fo(j)p Fw(\020)2524 5272 y Fr(t)2558 5245 y Fo(j)2586 5203 y Fm(2)2647 5245 y Fq(+)h Fo(j)p Fw(\020)2826 5203 y Fn(0)2819 5272 y Fr(t)2854 5245 y Fo(j)2882 5203 y Fm(2)2921 5165 y Fk(\001)2992 5245 y Fw(:)3669 4854 y Fq(\()p Fs(5)p Fw(:)p Fs(6)p Fq(\))p eop %%Page: 13 13 13 12 bop 94 -45 a Fw(")p Fv({Entr)n(opy)38 b(of)h(Driven)e(Equa)-7 b(tions)2055 b Fu(13)94 194 y Fs(Let)31 b Fp(P)36 b Fq(=)473 113 y Fk(\010)531 194 y Fw(f)i Fo(2)29 b Fs(L)760 151 y Fn(1)872 194 y Fs(:)e Fo(k)p Fw(f)11 b Fo(k)1087 151 y Fm(4)1087 221 y Fn(1)1198 194 y Fo(\024)1316 155 y Fm(1)p 1316 171 35 4 v 1316 228 a(8)1362 194 y Fw(\021)1415 151 y Fm(2)1455 194 y Fw(Q)1534 151 y Fn(\000)p Fm(2)1534 222 y(3)1636 113 y Fk(\011)1694 194 y Fs(.)54 b(If)31 b Fo(k)p Fw(v)1968 221 y Fr(t)2003 194 y Fo(k)2053 221 y Fm(H)2103 201 y Fh(1)2103 248 y Fg(')2195 194 y Fq(=)36 b Fo(k)p Fw(u)2415 221 y Fr(t)2476 194 y Fo(\000)26 b Fw(\020)2623 221 y Fr(t)2657 194 y Fo(k)2707 221 y Fm(H)2757 201 y Fh(1)2757 248 y Fg(')2849 194 y Fo(\024)36 b Fw(\016)t Fs(,)d(then)d(by)h(Eq.\(2.6\))f(we)94 335 y(ha)n(v)o(e)391 569 y Fo(k)p Fw(u)498 595 y Fr(t)533 569 y Fo(k)583 526 y Fm(2)583 595 y Fn(1)712 569 y Fo(\024)45 b Fs(2)p Fo(k)p Fw(u)991 595 y Fr(t)1048 569 y Fo(\000)22 b Fw(\020)1191 595 y Fr(t)1226 569 y Fo(k)1276 526 y Fm(2)1276 595 y Fn(1)1382 569 y Fq(+)h Fs(2)p Fw(\032)1584 526 y Fm(2)1623 569 y Fw(\016)1671 526 y Fm(2)1755 569 y Fo(\024)1889 501 y Fw(\031)p 1889 546 61 4 v 1894 637 a Fs(2)1961 569 y Fq(\()p Fs(1)f Fq(+)2172 479 y Fo(p)p 2255 479 50 4 v 90 x Fs(5)o Fq(\))p Fo(k)p Fw(u)2450 595 y Fr(t)2507 569 y Fo(\000)h Fw(\020)2651 595 y Fr(t)2685 569 y Fo(k)2735 526 y Fm(2)2735 595 y(H)2785 576 y Fh(1)2785 622 y Fg(')2863 569 y Fq(+)g Fs(2)p Fw(\032)3065 526 y Fm(2)3104 569 y Fw(\016)3152 526 y Fm(2)3236 569 y Fo(\024)45 b Fw(C)7 b(\016)3484 526 y Fm(2)3549 569 y Fw(;)94 836 y Fs(hence)29 b(there)f(is)f(a)h Fw(K)39 b Fq(=)33 b Fw(K)7 b Fq(\()p Fw(Q)1192 863 y Fm(2)1231 836 y Fw(;)17 b(Q)1355 863 y Fm(3)1394 836 y Fw(;)g(\013;)g(\032)p Fq(\))25 b Fs(such)j(that)f Fw(u)2109 863 y Fr(t)2176 836 y Fo(2)33 b Fp(P)28 b Fs(for)g(an)o(y)f Fw(\016)36 b(<)c(K)7 b Fs(.)45 b(In)28 b(that)f(case,)j(Eq.\(5.6\))94 956 y(implies)261 1237 y Fw(@)314 1264 y Fr(t)365 1101 y Fk(Z)481 1237 y Fw(')546 1264 y Fr(y)595 1156 y Fk(\000)640 1237 y Fw(A)p Fo(j)p Fw(v)791 1264 y Fr(t)826 1237 y Fo(j)854 1194 y Fm(2)916 1237 y Fq(+)22 b Fo(j)p Fw(v)1095 1194 y Fn(0)1091 1264 y Fr(t)1126 1237 y Fo(j)1154 1194 y Fm(2)1194 1156 y Fk(\001)1284 1237 y Fo(\024)44 b(\000)1494 1198 y Fm(1)p 1495 1214 35 4 v 1495 1271 a(2)1542 1237 y Fw(\021)1611 1101 y Fk(Z)1728 1237 y Fw(')1793 1264 y Fr(y)1841 1156 y Fk(\000)1886 1237 y Fw(A)p Fo(j)p Fw(v)2037 1264 y Fr(t)2072 1237 y Fo(j)2100 1194 y Fm(2)2162 1237 y Fq(+)22 b Fo(j)p Fw(v)2341 1194 y Fn(0)2337 1264 y Fr(t)2372 1237 y Fo(j)2400 1194 y Fm(2)2440 1156 y Fk(\001)2508 1237 y Fq(+)g Fw(C)7 b(\016)2733 1194 y Fm(4)2790 1101 y Fk(Z)2906 1237 y Fw(')2971 1264 y Fr(y)3020 1156 y Fk(\000)3065 1237 y Fw(A)p Fo(j)p Fw(\020)3212 1264 y Fr(t)3246 1237 y Fo(j)3274 1194 y Fm(2)3336 1237 y Fq(+)23 b Fo(j)p Fw(\020)3515 1194 y Fn(0)3508 1264 y Fr(t)3542 1237 y Fo(j)3570 1194 y Fm(2)3610 1156 y Fk(\001)3680 1237 y Fw(;)94 1537 y Fs(namely)i(the)f(ball)h Fo(f)p Fw(f)38 b Fs(:)27 b Fo(k)p Fw(f)32 b Fo(\000)23 b Fw(\020)1201 1564 y Fr(t)1236 1537 y Fo(k)1286 1564 y Fm(H)1336 1544 y Fh(1)1336 1591 y Fg(')1419 1537 y Fo(\024)28 b Fw(C)7 b(\016)1650 1494 y Fm(3)1691 1537 y Fo(g)24 b Fs(is)h(absorbing)f(in)g Fp(P)p Fs(.)p 3794 1461 80 4 v 3794 1533 4 72 v 3870 1533 V 3794 1537 80 4 v 94 1927 a Ft(6.)30 b(Pr)n(oof)f(of)h(Theor)n(em)f(2.5)94 2117 y Fs(Let)c Fw(N)11 b Fq(\()p Fw(";)17 b Fq(\()p Fp(A)p Fq(\))628 2144 y Fr(L)686 2117 y Fq(\))25 b Fs(be)g(the)g (number)f(of)h(balls)f(of)h(radius)f(at)h(most)f Fw(")h Fs(needed)g(to)g(co)o(v)o(er)f Fq(\()p Fp(A)p Fq(\))3237 2144 y Fr(L)3296 2117 y Fs(.)36 b(W)-8 b(e)25 b(ha)n(v)o(e)94 2308 y Fu(Pr)n(oposition)i(6.1.)41 b Fi(There)27 b(are)g Fw(A;)17 b(C)37 b(>)30 b Fs(0)d Fi(such)f(that)g(for)h(all)f Fw(L)k(>)h Fs(0)26 b Fi(and)h(suf)n(\002ciently)e(small)h Fw(")31 b(>)f Fs(0)p Fi(,)d(one)94 2428 y(has)1057 2528 y Fw(N)11 b Fq(\()p Fw(";)17 b Fq(\()p Fw(S)1384 2485 y Fm(1)1423 2528 y Fq(\()p Fp(A)p Fq(\)\))1617 2555 y Fr(L)1675 2528 y Fq(\))p 1057 2573 658 4 v 1063 2664 a Fw(N)11 b Fq(\()p Fw(";)17 b Fq(\()p Fw(#)1382 2635 y Fm(1)1421 2664 y Fq(\(\006\)\))1610 2691 y Fr(L)1669 2664 y Fq(\))1770 2596 y Fo(\024)1904 2522 y Fw(N)11 b Fq(\()p Fs(2)p Fw(";)17 b Fq(\()p Fp(A)p Fq(\))2330 2549 y Fr(L)p Fj(+)p Fr(C)2511 2522 y Fq(\))p 1904 2573 647 4 v 1945 2664 a Fw(N)11 b Fq(\()p Fs(2)p Fw(";)17 b Fq(\006)2288 2691 y Fr(L)p Fj(+)p Fr(C)2470 2664 y Fq(\))2562 2596 y Fw(A)2637 2553 y Fr(L)p Fj(+)p Fm(1)p Fr(=")2896 2596 y Fw(:)745 b Fq(\()p Fs(6)p Fw(:)p Fs(1)p Fq(\))316 2905 y Fs(W)-8 b(e)33 b(continue)f(the)g(proof)h(of)g (Theorem)f(2.5.)60 b(W)-8 b(e)33 b(kno)n(w)e(from)i(Proposition)e(2.2)h (that)h(there)g(is)f(an)94 3025 y Fw(R)40 b Fo(\024)g Fw(\032)26 b Fq(+)h Fw(R)585 3051 y Fm(0)658 3025 y Fs(such)32 b(that)h Fw(N)11 b Fq(\()p Fw(R)q(;)17 b Fq(\()p Fp(A)p Fq(\))1463 3051 y Fr(L)1520 3025 y Fq(\))39 b(=)g Fs(1)33 b(and)g Fw(N)11 b Fq(\()p Fw(R)q(;)17 b Fq(\006)2298 3051 y Fr(L)2356 3025 y Fq(\))39 b(=)h Fs(1.)60 b(Let)33 b Fw(T)47 b Fs(be)33 b(the)g(smallest)f(inte)o(ger)94 3144 y(lar)n(ger)26 b(than)e Fq(\()p Fs(log)16 b(2)p Fq(\))818 3101 y Fn(\000)p Fm(1)936 3144 y Fs(log)p Fq(\()p Fw(R)q(=")p Fq(\))p Fs(.)35 b(By)25 b(iterating)f Fw(T)39 b Fs(times)24 b(the)g(bound)g(\(6.1\),)h(we)g(obtain)1032 3384 y Fw(N)11 b Fq(\()p Fw(";)17 b Fq(\()p Fw(S)1359 3341 y Fr(T)1421 3384 y Fq(\()p Fp(A)p Fq(\)\))1615 3411 y Fr(L)1674 3384 y Fq(\))p 1032 3428 681 4 v 1038 3520 a Fw(N)11 b Fq(\()p Fw(";)17 b Fq(\()p Fw(#)1357 3491 y Fr(T)1419 3520 y Fq(\(\006\)\))1608 3546 y Fr(L)1667 3520 y Fq(\))1769 3451 y Fo(\024)44 b Fw(A)1965 3393 y Fr(L)2033 3326 y Fk(P)2139 3431 y Fg(n)p Fb(=)p Fh(1)p Fg(;:::)o(;T)2428 3393 y Fj(\()p Fm(1)p Fj(+)p Fr(nC)s(=)p Fj(\()p Fr("L)p Fj(\)\))2920 3451 y Fw(:)94 3761 y Fs(Since)26 b Fw(N)11 b Fq(\()p Fw(";)17 b Fq(\()p Fw(S)668 3719 y Fr(T)729 3761 y Fq(\()p Fp(A)p Fq(\)\))923 3788 y Fr(L)982 3761 y Fq(\))28 b(=)g Fw(N)11 b Fq(\()p Fw(";)17 b Fq(\()p Fp(A)p Fq(\))1530 3788 y Fr(L)1588 3761 y Fq(\))25 b Fs(and)f Fw(N)11 b Fq(\()p Fw(";)17 b Fq(\()p Fw(#)2139 3719 y Fr(T)2201 3761 y Fq(\(\006\)\))2390 3788 y Fr(L)2449 3761 y Fq(\))27 b(=)i Fw(N)11 b Fq(\()p Fw(";)17 b Fq(\006)2914 3788 y Fr(L)2973 3761 y Fq(\))p Fs(,)24 b(we)h(\002nd)841 4055 y(lim)801 4119 y Fr(L)p Fn(!1)1043 3987 y Fs(log)16 b Fw(N)11 b Fq(\()p Fw(";)17 b Fq(\()p Fp(A)p Fq(\))1563 4014 y Fr(L)1621 3987 y Fq(\))p 1043 4032 617 4 v 1293 4123 a Fs(2)p Fw(L)1716 4055 y Fo(\024)1850 4016 y Fm(1)p 1850 4032 35 4 v 1850 4089 a(2)1897 4055 y Fw(T)31 b Fs(log)16 b Fw(A)21 b Fq(+)63 b Fs(lim)2326 4119 y Fr(L)p Fn(!1)2568 3987 y Fs(log)16 b Fw(N)11 b Fq(\()p Fw(";)17 b Fq(\006)3005 4014 y Fr(L)3064 3987 y Fq(\))p 2568 4032 535 4 v 2777 4123 a Fs(2)p Fw(L)3140 4055 y(:)94 4320 y Fs(By)26 b(De\002nition)e(2.4,)g(that)g(is)1207 4554 y Fw(H)1290 4581 y Fr(")1332 4554 y Fq(\()p Fp(A)p Fq(\))44 b Fo(\024)g Fw(C)7 b Fq(\()p Fw(R)q(;)17 b(A)p Fq(\))g Fs(log)d Fw(")2210 4512 y Fn(\000)p Fm(1)2335 4554 y Fq(+)22 b Fw(H)2517 4581 y Fr(")2560 4554 y Fq(\(\006\))i Fw(:)316 4788 y Fs(W)-8 b(e)28 b(ne)o(xt)f(pro)o(v)o(e)g(the)h(lo)n (wer)f(bound.)44 b(Fix)28 b Fw(L;)17 b(")28 b Fs(and)g(let)f Fo(f)p Fw(f)2391 4815 y Fm(1)2431 4788 y Fw(;)17 b(:)g(:)g(:)d(;)j(f) 2702 4815 y Fr(N)2777 4788 y Fo(g)27 b Fs(be)h(an)h Fw(")p Fs(\226net)f(for)g Fq(\006)3561 4815 y Fr(L)3648 4788 y Fs(\()p FE(i.e)o(.)g Fs(a)94 4908 y(set)g Fp(X)j Fo(\032)h Fq(\006)516 4935 y Fr(L)603 4908 y Fs(such)27 b(that)f(dist)o Fq(\()p Fp(X)p Fw(;)17 b Fq(\006)1360 4935 y Fr(L)1420 4908 y Fq(\))30 b Fo(\024)i Fw(")p Fs(,)c(see)g([KT]\))g(with)e Fw(N)42 b Fq(=)32 b Fw(N)11 b Fq(\()p Fw(";)17 b Fq(\006)2840 4935 y Fr(L)2899 4908 y Fq(\))31 b Fo(\031)g Fs(e)o(xp)o Fq(\()p Fs(2)p Fw(LH)3459 4935 y Fr(")3501 4908 y Fq(\(\006\)\))p Fs(.)43 b(By)94 5027 y(the)25 b(de\002nition)f(Eq.\(2.8\))g(of)h(the)g (attractor)l(,)g(one)f(has)922 5261 y Fp(A)44 b Fo(3)h Fw(x)1211 5288 y Fr(j)1297 5261 y Fo(\021)73 b Fs(lim)1419 5321 y Fr(t)p Fn(!1)1608 5181 y Fk(\000)1654 5261 y Fq(\010)1726 5219 y Fr(t)1726 5288 y(#)1774 5268 y Fc(\000)p Fg(t)1861 5288 y Fr(f)1900 5306 y Fg(j)1942 5261 y Fq(\()p Fs(0)p Fq(\))p Fw(;)17 b(f)2164 5288 y Fr(j)2204 5181 y Fk(\001)2275 5261 y Fw(;)116 b(j)50 b Fq(=)44 b Fs(1)p Fw(;)17 b(:)g(:)g(:)d(;)j(N) 35 b(:)p eop %%Page: 14 14 14 13 bop 94 -45 a Fw(")p Fv({Entr)n(opy)38 b(of)h(Driven)e(Equa)-7 b(tions)2055 b Fu(14)94 194 y Fs(By)26 b(our)e(choice)h(of)g(the)g (product)f(topology)f(in)i(De\002nition)f(2.4,)g Fo(f)p Fw(x)2454 221 y Fm(1)2494 194 y Fw(;)17 b(:)g(:)g(:)d(;)j(x)2773 221 y Fr(N)2848 194 y Fo(g)24 b Fs(is)h(an)g Fw(")p Fs(\226net)f(for)i Fp(A)p Fs(.)p 3794 118 80 4 v 3794 190 4 72 v 3870 190 V 3794 194 80 4 v 94 364 a Fu(Pr)n(oof)j(of)g(Pr)n(oposition)g(6.1.)f Fs(Let)h Fw(u)g Fs(and)f Fw(v)33 b Fs(be)c(tw)o(o)f(orbits)g(of)h (Eq.\(2.1\))f(with)g(initial)f(conditions)g Fw(u)3661 391 y Fm(0)3730 364 y Fs(and)94 484 y Fw(v)142 511 y Fm(0)210 484 y Fs(such)g(that)g Fq(\()p Fw(u)693 511 y Fm(0)733 484 y Fw(;)17 b(f)827 511 y Fm(1)866 484 y Fq(\))27 b Fs(and)h Fq(\()p Fw(v)1191 511 y Fm(0)1231 484 y Fw(;)17 b(f)1325 511 y Fm(2)1364 484 y Fq(\))27 b Fs(belong)g(to)g Fp(A)g Fs(for)h(some)f Fw(f)2369 511 y Fm(1)2408 484 y Fw(;)17 b(f)2502 511 y Fm(2)2569 484 y Fs(in)27 b Fq(\006)p Fs(.)44 b(The)27 b(dif)n(ference)h Fw(w)35 b Fq(=)c Fw(u)24 b Fo(\000)g Fw(v)94 603 y Fs(satis\002es)h (the)g(equation)1008 802 y Fw(@)1061 828 y Fr(t)1096 802 y Fw(w)47 b Fq(=)e(\()p Fs(1)21 b Fq(+)i Fs(i)p Fw(\013)p Fq(\))p Fw(@)1736 759 y Fm(2)1730 828 y Fr(x)1780 802 y Fw(w)i Fq(+)d Fw(G)2053 759 y Fj(\()p Fm(1)p Fj(\))2156 802 y Fw(w)j Fq(+)e Fw(G)2430 759 y Fj(\()p Fm(2)p Fj(\))p 2532 747 75 4 v 2532 802 a Fw(w)i Fq(+)e Fw(G)2806 759 y Fj(\()p Fm(3)p Fj(\))2933 802 y Fw(;)708 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1391 y Fj(\()p Fm(1)p Fj(\))1319 1467 y Fr(t)1421 1442 y Fq(\()p Fw(x)p Fq(\))p Fo(j)p Fw(;)g Fo(j)p Fw(G)1735 1391 y Fj(\()p Fm(2)p Fj(\))1735 1467 y Fr(t)1836 1442 y Fq(\()p Fw(x)p Fq(\))p Fo(j)1999 1362 y Fk(\001)2088 1442 y Fo(\024)45 b Fs(1)22 b Fq(+)g Fs(2)2431 1349 y Fk(p)p 2531 1349 274 4 v 93 x Fs(1)g Fq(+)g Fw(\014)2763 1414 y Fm(2)2804 1442 y Fw(R)2881 1400 y Fm(2)2880 1469 y(0)2945 1442 y Fw(:)696 b Fq(\()p Fs(6)p Fw(:)p Fs(9)p Fq(\))316 1617 y Fs(Let)25 b Fo(j)p Fw(f)11 b Fo(j)590 1644 y Fr(d)663 1617 y Fo(\021)28 b Fs(sup)906 1644 y Fn(j)p Fr(x)p Fn(j\024)p Fr(d)1124 1617 y Fo(j)p Fw(f)11 b Fq(\()p Fw(x)p Fq(\))p Fo(j)p Fs(.)34 b(Assume)24 b Fw(f)1829 1644 y Fm(1)1869 1617 y Fw(;)17 b(f)1963 1644 y Fm(2)2029 1617 y Fo(2)28 b Fq(\006)d Fs(are)h(such)e(that)872 1868 y(sup)838 1959 y Fn(j)p Fr(y)s Fn(j\024)p Fr(L)1062 1868 y Fo(k)1112 1734 y Fk(q)p 1211 1734 165 4 v 1211 1868 a Fw(\030)1260 1816 y Fj(\()p Fr(n)p Fj(\))1255 1880 y Fr(y)1376 1868 y Fq(\()p Fw(f)1464 1895 y Fm(1)1525 1868 y Fo(\000)f Fw(f)1674 1895 y Fm(2)1713 1868 y Fq(\))p Fo(k)1802 1895 y Fm(2)1886 1868 y Fq(=)79 b Fs(sup)2008 1959 y Fn(j)p Fr(y)s Fn(j\024)p Fr(L)2232 1868 y Fo(k)2282 1734 y Fk(q)p 2381 1734 V 2381 1868 a Fw(\030)2430 1816 y Fj(\()p Fr(n)p Fj(\))2425 1880 y Fr(y)2546 1868 y Fw(G)2624 1816 y Fj(\()p Fm(3)p Fj(\))2624 1897 y Fm(0)2726 1868 y Fo(k)2776 1895 y Fm(2)2860 1868 y Fo(\024)45 b Fs(2)p Fw(")25 b(:)94 2118 y Fs(Then)g(by)g(Assumption)d(2.3,)1414 2292 y(sup)1380 2382 y Fn(j)p Fr(y)s Fn(j\024)p Fr(L)1604 2292 y Fo(k)1654 2157 y Fk(q)p 1753 2157 V 1753 2292 a Fw(\030)1802 2240 y Fj(\()p Fr(n)p Fj(\))1797 2303 y Fr(y)1918 2292 y Fw(G)1996 2240 y Fj(\()p Fm(3)p Fj(\))1996 2316 y Fr(t)2098 2292 y Fo(k)2148 2319 y Fm(2)2232 2292 y Fo(\024)45 b Fs(2)p Fw(e)2450 2249 y Fd({)s Fr(t)2542 2292 y Fw(")1031 b Fq(\()p Fs(6)p Fw(:)p Fs(10)p Fq(\))94 2556 y Fs(if)26 b Fw(t)h(<)i Fs(2.)36 b(Assume)25 b Fo(j)p Fw(w)907 2582 y Fm(0)946 2556 y Fo(j)974 2582 y Fr(L)p Fj(+)p Fr(C)5 b Fj(\()p Fr(")p Fj(\))1285 2556 y Fo(\024)28 b Fs(2)p Fw(")p Fs(.)37 b(If)26 b Fw(C)7 b Fq(\()p Fw(")p Fq(\))28 b Fw(>)h Fq(\(\()p Fw(C)2125 2582 y Fr(n)2201 2556 y Fq(+)22 b Fw(Q)2379 2513 y Fj(\()p Fr(n)p Fj(\))2496 2556 y Fq(\))p Fw(=")p Fq(\))2670 2513 y Fm(1)p Fr(=n)2799 2556 y Fs(,)k(Lemma)e(6.3)h(and)g(Eq.\(6.10\))94 2675 y(imply)1237 2849 y(sup)1203 2940 y Fn(j)p Fr(y)s Fn(j\024)p Fr(L)1427 2849 y Fo(k)1477 2715 y Fk(q)p 1576 2715 V 1576 2849 a Fw(\030)1625 2797 y Fj(\()p Fr(n)p Fj(\))1620 2861 y Fr(y)1741 2849 y Fw(w)1812 2876 y Fr(t)1847 2849 y Fo(k)1897 2876 y Fm(2)1981 2849 y Fo(\024)45 b Fs(2)p Fw("e)2245 2806 y Fr(M)7 b(t)2364 2769 y Fk(\000)2410 2849 y Fs(1)22 b Fq(+)g Fw(e)2627 2806 y Fd({)s Fr(t)2720 2769 y Fk(\001)94 3122 y Fs(for)33 b(all)f Fw(t)38 b(<)g Fs(2.)58 b(W)-8 b(e)32 b(ne)o(xt)g(use)g(that)g(Lemma)f(6.2)h(implies)f Fo(j)p Fw(K)2348 3070 y Fj(\()p Fm(1)p Fj(\))2341 3151 y Fm(1)2475 3122 y Fw(?)26 b(w)2622 3149 y Fm(0)2663 3122 y Fo(j)2691 3149 y Fr(L)2788 3122 y Fo(\024)38 b Fw(C)7 b(")33 b Fs(and)f Fo(j)p Fw(K)3356 3070 y Fj(\()p Fm(2)p Fj(\))3349 3151 y Fm(1)3484 3122 y Fw(?)26 b(w)3631 3149 y Fm(0)3671 3122 y Fo(j)3699 3149 y Fr(L)3797 3122 y Fo(\024)94 3280 y Fs(2)p Fw(C)215 3307 y Fr(n)270 3280 y Fw(e)316 3237 y Fn(\000)p Fj(\()p Fr(p)450 3194 y Fc(\003)492 3237 y Fj(\))523 3194 y Fh(2)553 3237 y Fr(=)p Fm(2)634 3280 y Fw(")p Fs(.)36 b(Inserting)23 b(those)g(bounds)g(into)g(\(6.6\)) h(and)f(\(6.7\))h(gi)n(v)o(es)e(\(by)i(Cauchy\226Schw)o(arz)h (inequal\255)94 3399 y(ity\))204 3628 y Fo(j)p Fw(w)306 3576 y Fj(\()p Fm(1)p Fj(\))303 3657 y Fm(1)408 3628 y Fo(j)436 3655 y Fr(L)539 3628 y Fo(\024)45 b(j)p Fw(K)781 3576 y Fj(\()p Fm(1)p Fj(\))774 3657 y Fm(1)905 3628 y Fw(?)22 b(w)1048 3655 y Fm(0)1088 3628 y Fo(j)1116 3655 y Fr(L)1197 3628 y Fq(+)h Fw(C)58 b Fs(sup)1392 3718 y Fn(j)p Fr(y)s Fn(j\024)p Fr(L)1616 3492 y Fk(Z)1715 3535 y Fm(1)1671 3719 y(0)1772 3628 y Fo(k)p Fw(K)1914 3576 y Fj(\()p Fm(1)p Fj(\))1907 3657 y Fm(1)p Fn(\000)p Fr(s)2046 3628 y Fw(=)2096 3496 y Fk(q)p 2195 3496 V 2195 3628 a Fw(\030)2244 3576 y Fj(\()p Fr(n)p Fj(\))2239 3657 y Fm(0)2360 3628 y Fo(k)2410 3655 y Fm(2)2450 3547 y Fk(\000)2495 3628 y Fo(k)2545 3494 y Fk(q)p 2645 3494 V 134 x Fw(\030)2694 3576 y Fj(\()p Fr(n)p Fj(\))2689 3640 y Fr(y)2809 3628 y Fw(w)2880 3655 y Fr(s)2923 3628 y Fo(k)2973 3655 y Fm(2)3035 3628 y Fq(+)23 b Fo(k)3185 3494 y Fk(q)p 3284 3494 V 3284 3628 a Fw(\030)3333 3576 y Fj(\()p Fr(n)p Fj(\))3328 3640 y Fr(y)3449 3628 y Fw(G)3527 3585 y Fj(\()p Fm(3)p Fj(\))3527 3655 y Fr(s)3629 3628 y Fo(k)3679 3655 y Fm(2)3719 3547 y Fk(\001)539 3925 y Fo(\024)45 b Fw(C)7 b(")23 b Fq(+)g Fs(2)p Fw(C)7 b(")1099 3790 y Fk(Z)1198 3833 y Fm(1)1154 4016 y(0)1355 3858 y Fw(C)1426 3885 y Fr(n)p 1267 3903 302 4 v 1267 3922 a Fo(p)p 1350 3922 219 4 v 80 x Fs(1)22 b Fo(\000)g Fw(s)1580 3925 y(e)1626 3883 y Fj(\()p Fr(M)7 b Fj(+)p Fd({)s Fj(\))p Fr(s)1950 3925 y Fs(d)p Fw(s)539 4143 y Fo(\024)45 b Fw(A")25 b(;)3619 4262 y Fq(\()p Fs(6)p Fw(:)p Fs(11)p Fq(\))94 4382 y Fs(where)h Fw(A)e Fs(depends)h(on)f Fw(n)h Fs(b)n(ut)f(not)g(on)h Fw(p)1500 4339 y Fn(\003)1546 4382 y Fs(,)g(and)204 4610 y Fo(j)p Fw(w)306 4559 y Fj(\()p Fm(2)p Fj(\))303 4639 y Fm(1)408 4610 y Fo(j)436 4637 y Fr(L)539 4610 y Fo(\024)45 b(j)p Fw(K)781 4559 y Fj(\()p Fm(2)p Fj(\))774 4639 y Fm(1)905 4610 y Fw(?)22 b(w)1048 4637 y Fm(0)1088 4610 y Fo(j)1116 4637 y Fr(L)1197 4610 y Fq(+)h Fw(C)58 b Fs(sup)1392 4701 y Fn(j)p Fr(y)s Fn(j\024)p Fr(L)1616 4475 y Fk(Z)1715 4518 y Fm(1)1671 4701 y(0)1772 4610 y Fo(k)p Fw(K)1914 4559 y Fj(\()p Fm(2)p Fj(\))1907 4639 y Fm(1)p Fn(\000)p Fr(s)2046 4610 y Fw(=)2096 4478 y Fk(q)p 2195 4478 165 4 v 2195 4610 a Fw(\030)2244 4559 y Fj(\()p Fr(n)p Fj(\))2239 4639 y Fm(0)2360 4610 y Fo(k)2410 4637 y Fm(2)2450 4530 y Fk(\000)2495 4610 y Fo(k)2545 4476 y Fk(q)p 2645 4476 V 134 x Fw(\030)2694 4559 y Fj(\()p Fr(n)p Fj(\))2689 4622 y Fr(y)2809 4610 y Fw(w)2880 4637 y Fr(s)2923 4610 y Fo(k)2973 4637 y Fm(2)3035 4610 y Fq(+)23 b Fo(k)3185 4476 y Fk(q)p 3284 4476 V 3284 4610 a Fw(\030)3333 4559 y Fj(\()p Fr(n)p Fj(\))3328 4622 y Fr(y)3449 4610 y Fw(G)3527 4567 y Fj(\()p Fm(3)p Fj(\))3527 4637 y Fr(s)3629 4610 y Fo(k)3679 4637 y Fm(2)3719 4530 y Fk(\001)539 4955 y Fo(\024)45 b Fw(e)707 4912 y Fn(\000)p Fj(\()p Fr(p)841 4869 y Fc(\003)884 4912 y Fj(\))915 4869 y Fh(2)945 4912 y Fr(=)p Fm(2)1025 4955 y Fw(")23 b Fq(+)f Fs(2)p Fw(C)7 b(")1384 4819 y Fk(Z)1484 4862 y Fm(1)1440 5045 y(0)1553 4887 y Fw(C)1624 4914 y Fr(n)1678 4887 y Fw(e)1724 4844 y Fn(\000)p Fj(\()p Fr(p)1858 4802 y Fc(\003)1901 4844 y Fj(\))1932 4802 y Fh(2)1962 4844 y Fj(\()p Fm(1)p Fn(\000)p Fr(s)p Fj(\))p Fr(=)p Fm(2)p 1553 4932 687 4 v 1745 4952 a Fo(p)p 1828 4952 219 4 v 79 x Fs(1)22 b Fo(\000)h Fw(s)2251 4955 y(e)2297 4912 y Fj(\()p Fr(M)7 b Fj(+)p Fd({)s Fj(\))p Fr(s)2621 4955 y Fs(d)p Fw(s)539 5172 y Fo(\024)45 b Fw(B)5 b Fq(\()p Fw(p)831 5129 y Fn(\003)876 5172 y Fq(\))p Fw(")25 b(;)3619 5291 y Fq(\()p Fs(6)p Fw(:)p Fs(12)p Fq(\))p eop %%Page: 16 16 16 15 bop 94 -45 a Fw(")p Fv({Entr)n(opy)38 b(of)h(Driven)e(Equa)-7 b(tions)2055 b Fu(16)94 194 y Fs(where)26 b Fw(B)5 b Fq(\()p Fw(p)533 151 y Fn(\003)578 194 y Fq(\))27 b Fo(!)h Fs(0)c(as)h Fw(p)1004 151 y Fn(\003)1078 194 y Fo(!)i(1)p Fs(.)36 b(W)-8 b(e)25 b(choose)g Fw(p)1873 151 y Fn(\003)1943 194 y Fs(so)g(lar)n(ge)g(that)f Fw(B)5 b Fq(\()p Fw(p)2623 151 y Fn(\003)2669 194 y Fq(\))27 b Fw(<)2852 155 y Fm(1)p 2852 171 35 4 v 2852 228 a(2)2899 194 y Fs(.)316 314 y(Assume)f(ne)o(xt)h(that)g(we)h(ha)n(v)o(e)f(a)h(co)o(v)o(ering)e(of)h Fq(\()p Fp(A)p Fq(\))2105 341 y Fr(L)p Fj(+)p Fr(C)5 b Fj(\()p Fr(")p Fj(\))2415 314 y Fs(by)27 b(open)g(sets)g(of)h(radius) f(at)g(most)g(2)p Fw(")p Fs(.)44 b(If)94 472 y Fq(\()p Fw(u;)17 b(f)284 499 y Fm(1)323 472 y Fq(\))32 b Fs(and)h Fq(\()p Fw(v)t(;)17 b(f)756 499 y Fm(2)794 472 y Fq(\))33 b Fs(belong)e(to)h(one)h(of)f(those)g(sets,)i(then)e Fw(w)2298 499 y Fm(0)2376 472 y Fq(=)39 b Fw(u)26 b Fo(\000)h Fw(v)36 b Fs(and)d Fw(G)3018 420 y Fj(\()p Fm(3)p Fj(\))3018 501 y Fm(0)3158 472 y Fq(=)39 b Fw(f)3323 499 y Fm(1)3389 472 y Fo(\000)27 b Fw(f)3542 499 y Fm(2)3614 472 y Fs(satisfy)94 592 y(the)37 b(hypotheses)f(stated)g(abo)o(v)o(e)g(that)h(led)f(to)h (inequalities)e(\(6.11\))i(and)g(\(6.12\).)72 b(W)-8 b(e)37 b(ne)o(xt)f(use)h(a)h(result)94 711 y(of)31 b(Cartwright)e (\(see)i([KT],)f(Eq.\(191\)\):)46 b(for)30 b(all)g Fw(f)40 b Fs(in)30 b(the)g(Bernstein)g(class)g Fw(B)2910 738 y Fr(R;)p Fm(2)p Fr(p)3070 718 y Fc(\003)3146 711 y Fs(\(see)g (Eq.\(6.8\)\),)h(the)94 831 y(follo)n(wing)23 b(identity)h(holds:)814 1101 y Fw(f)11 b Fq(\()p Fw(x)p Fq(\))44 b(=)1186 1033 y Fs(sin)p Fq(\()p Fs(8)p Fw(p)1442 990 y Fn(\003)1487 1033 y Fw(x)p Fq(\))p 1186 1078 397 4 v 1228 1169 a Fs(32)p Fq(\()p Fw(p)1417 1140 y Fn(\003)1462 1169 y Fq(\))1501 1140 y Fm(2)1698 976 y Fn(1)1665 1006 y Fk(X)1611 1215 y Fr(n)p Fj(=)p Fn(\0001)1863 1101 y Fq(\()p Fo(\000)p Fs(1)p Fq(\))2068 1058 y Fr(n)2123 1101 y Fw(f)11 b Fq(\()p Fw(x)2279 1128 y Fr(n)2332 1101 y Fq(\))2383 1033 y Fs(sin)o Fq(\()p Fs(4)p Fw(p)2638 990 y Fn(\003)2683 1033 y Fq(\()p Fw(x)22 b Fo(\000)h Fw(x)2958 1060 y Fr(n)3012 1033 y Fq(\)\))p 2383 1078 707 4 v 2532 1169 a(\()p Fw(x)f Fo(\000)h Fw(x)2807 1196 y Fr(n)2861 1169 y Fq(\))2900 1140 y Fm(2)3126 1101 y Fw(;)465 b Fq(\()p Fs(6)p Fw(:)p Fs(13)p Fq(\))94 1387 y Fs(where)26 b Fw(x)420 1413 y Fr(n)502 1387 y Fq(=)628 1347 y Fr(n\031)p 619 1364 118 4 v 619 1421 a Fm(8)p Fr(p)695 1401 y Fc(\003)748 1387 y Fs(.)36 b(Let)25 b Fw(f)5 b(;)17 b(g)27 b Fs(be)e(in)f Fw(B)1438 1413 y Fr(R;)p Fm(2)p Fr(p)1598 1393 y Fc(\003)1645 1387 y Fs(.)36 b(A)24 b(simple)g(application)g(of)h(Eq.\(6.13\))f(sho)n(ws)f (that)750 1604 y Fo(j)p Fw(f)33 b Fo(\000)22 b Fw(g)t Fo(j)1039 1631 y Fr(L)1142 1604 y Fo(\024)45 b Fw(C)404 b Fs(sup)1359 1695 y Fn(j)p Fr(n)p Fn(j\024)p Fm([8)p Fr(p)1617 1675 y Fc(\003)1658 1695 y Fr(L=\031)s Fm(])p Fj(+)p Fm(8)p Fr(C)5 b(p)2026 1675 y Fc(\003)2067 1695 y Fr(=)p Fj(\()p Fr("\031)s Fj(\))2273 1604 y Fo(j)p Fw(f)11 b Fq(\()p Fw(x)2457 1631 y Fr(n)2511 1604 y Fq(\))21 b Fo(\000)i Fw(g)t Fq(\()p Fw(x)2819 1631 y Fr(n)2872 1604 y Fq(\))p Fo(j)f Fq(+)3072 1565 y Fm(1)p 3072 1581 35 4 v 3072 1639 a(4)3119 1604 y Fw(")j(:)94 1874 y Fs(Hence,)37 b(among)c(all)g(the)g(functions)g(in)g Fw(B)1595 1901 y Fr(R;)p Fm(2)p Fr(p)1755 1881 y Fc(\003)1835 1874 y Fs(that)g(are)h(bounded)f(by)g Fw(A")h Fs(in)f([)p Fo(\000)p Fw(L;)17 b(L)p Fs(])33 b(\(by)h(Eq.\(6.11\),)94 2033 y Fw(w)168 1982 y Fj(\()p Fm(1)p Fj(\))165 2062 y Fm(1)299 2033 y Fs(is)29 b(such)f(a)h(function\),)g(at)g(most)f Fq(\()p Fs(4)p Fw(A)p Fq(\))1626 1990 y Fr(C)5 b(p)1729 1948 y Fc(\003)1770 1990 y Fr(L)1829 2033 y Fq(\()p Fs(4)p Fw(R)q(=")p Fq(\))2130 1990 y Fr(C)g(p)2233 1948 y Fc(\003)2274 1990 y Fr(=")2386 2033 y Fs(of)29 b(them)f(dif)n(fer)h(pairwise)f(by)h (more)f(than)94 2153 y Fw("=)p Fs(2)h(in)e([)p Fo(\000)p Fw(L;)17 b(L)p Fs(].)46 b(By)28 b(taking)f(a)i(ball)e(of)h(radius)g Fw(")g 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Fo(p)p 1165 3434 36 4 v 81 x Fw(t)1213 3356 y Fk(\001)1303 3437 y Fq(=)1424 3351 y Fo(p)p 1507 3351 V 86 x Fw(ty)1596 3394 y Fr(n)1650 3437 y Fw(K)1742 3385 y Fj(\()p Fm(1)p Fj(\))1735 3461 y Fr(t)1844 3437 y Fq(\()p Fw(y)t Fq(\))1303 3705 y(=)1467 3638 y Fs(1)p 1436 3682 111 4 v 1436 3773 a(2)p Fw(\031)1575 3569 y Fk(Z)1675 3612 y Fn(1)1630 3796 y(\0001)1777 3705 y Fw(y)1830 3662 y Fr(n)1884 3705 y Fw(e)1930 3662 y Fm(i)p Fr(k)r(y)2042 3705 y Fw(e)2088 3662 y Fn(\000)p Fj(\()p Fm(1)p Fj(+)p Fm(i)p Fr(\013)p Fj(\))p Fr(k)2423 3619 y Fh(2)2459 3705 y Fw(\037)p Fq(\()p Fo(j)p Fw(k)s(=)2693 3619 y Fo(p)p 2775 3619 36 4 v 2775 3705 a Fw(tp)2861 3662 y Fn(\003)2907 3705 y Fo(j)p Fq(\))17 b Fs(d)p Fw(k)27 b(;)1303 3979 y Fq(=)1467 3912 y Fs(1)p 1436 3957 111 4 v 1436 4048 a(2)p Fw(\031)1575 3844 y Fk(Z)1675 3887 y Fn(1)1630 4070 y(\0001)1777 3979 y Fw(e)1823 3937 y Fm(i)p Fr(k)r(y)1935 3899 y Fk(\000)1981 3979 y Fo(\000)p Fs(i)p Fw(@)2139 4006 y Fr(k)2188 3899 y Fk(\001)2233 3937 y Fr(n)2304 3869 y Fk(\020)2364 3979 y Fw(e)2410 3937 y Fn(\000)p Fj(\()p Fm(1)p Fj(+)p Fm(i)p Fr(\013)p Fj(\))p Fr(k)2745 3894 y Fh(2)2781 3979 y Fw(\037)p Fq(\()p Fo(j)p Fw(k)s(=)3015 3894 y Fo(p)p 3097 3894 36 4 v 3097 3979 a Fw(tp)3183 3937 y Fn(\003)3229 3979 y Fo(j)p Fq(\))3296 3869 y Fk(\021)3388 3979 y Fs(d)p Fw(k)h(;)94 4252 y Fs(which)h(is)f(bounded)g(because)h Fw(\037)g Fs(is)f Fo(C)1422 4209 y Fn(1)1535 4252 y Fs(and)h(e)o(xp)o Fq(\()p Fo(\000)p Fw(k)2021 4209 y Fm(2)2061 4252 y Fq(\))g Fs(decays)g(f)o(aster)g(than)f(an)o(y)g(po)n(wer)g(of)h Fw(k)s Fs(.)47 b(The)29 b(an\255)94 4372 y(alyticity)38 b(follo)n(ws)f(from)i(the)f(P)o(ale)o(y\226W)l(iener)h(Theorem)f(using) g(that)g Fw(\037)h Fs(has)g(compact)g(support,)i(hence)94 4507 y(for)46 b(comple)o(x)e Fw(z)t Fs(,)51 b Fo(j)17 b Fs(e)o(xp)o Fq(\()p Fs(i)p Fw(pz)t Fq(\))p Fw(\037)p Fq(\()p Fo(j)p Fw(p=p)1441 4465 y Fn(\003)1487 4507 y Fo(j)p Fq(\))p Fo(j)55 b(\024)i Fs(e)o(xp)o Fq(\()p Fs(2)p Fw(p)2052 4465 y Fn(\003)2098 4507 y Fo(j)p Fs(Im)16 b Fw(z)t Fo(j)p Fq(\))p Fs(.)97 b(The)46 b(decay)g(of)f Fw(K)3191 4456 y Fj(\()p Fm(2)p Fj(\))3184 4532 y Fr(t)3338 4507 y Fs(follo)n(ws)f(from)94 4627 y(the)c(f)o(act)f(that)g(1)30 b Fo(\000)h Fw(\037)p Fq(\()p Fo(j)p Fw(p=p)1099 4584 y Fn(\003)1145 4627 y Fo(j)p Fq(\))38 b Fs(has)i(support)e(on)h Fo(f)p Fw(p)48 b(>)g(p)2223 4584 y Fn(\003)2269 4627 y Fo(g)f Fq(=)i Fo(f)p Fw(k)h(>)2770 4546 y Fo(p)p 2853 4546 V 81 x Fw(tp)2939 4584 y Fn(\003)2985 4627 y Fo(g)39 b Fs(where)h(e)o(xp)o Fq(\()p Fo(\000)p Fw(k)3670 4584 y Fm(2)3710 4627 y Fq(\))48 b Fo(\024)94 4746 y Fs(e)o(xp)p Fq(\()p Fo(\000)p Fq(\()p Fw(p)442 4704 y Fn(\003)488 4746 y Fq(\))527 4704 y Fm(2)566 4746 y Fw(t=)p Fs(2)p Fq(\))17 b Fs(e)o(xp)o Fq(\()p Fo(\000)p Fw(k)1071 4704 y Fm(2)1111 4746 y Fw(=)p Fs(2)p Fq(\))24 b Fs(and)h(we)g(still)e(get)i (a)g(con)l(v)o(er)n(gent)f(inte)o(gral.)316 4866 y(W)-8 b(e)30 b(ne)o(xt)f(pro)o(v)o(e)g(Lemma)g(6.3.)51 b(W)-8 b(e)30 b(compute)f(a)h(dif)n(ferential)f(inequality)g(for)h(the)g (weighted)f(norm)94 4986 y(\(see)d(Section)f(7\),)f(using)g(the)h (bound)f Fo(k)p Fq(\()p Fw(@)1540 5012 y Fr(x)1589 4986 y Fw(\030)1638 4943 y Fj(\()p Fr(n)p Fj(\))1633 5012 y Fr(y)1754 4986 y Fq(\))p Fw(=\030)1892 4943 y Fj(\()p Fr(n)p Fj(\))1887 5012 y Fr(y)2007 4986 y Fo(k)2057 5012 y Fn(1)2169 4986 y Fq(=)k Fw(n)p Fs(:)205 5245 y Fw(@)258 5272 y Fr(t)310 5110 y Fk(Z)426 5245 y Fw(\030)475 5203 y Fj(\()p Fr(n)p Fj(\))470 5272 y Fr(y)591 5245 y Fo(j)p Fw(w)690 5272 y Fr(t)725 5245 y Fo(j)753 5203 y Fm(2)836 5245 y Fo(\024)958 5165 y Fk(\000)1004 5245 y Fo(k)p Fw(G)1132 5194 y Fj(\()p Fm(1)p Fj(\))1132 5270 y Fr(t)1234 5245 y Fo(k)1284 5272 y Fn(1)1390 5245 y Fq(+)23 b Fo(k)p Fw(G)1618 5194 y Fj(\()p Fm(2)p Fj(\))1618 5270 y Fr(t)1720 5245 y Fo(k)1770 5272 y Fn(1)1877 5245 y Fq(+)1988 5206 y Fm(1)p 1988 5222 35 4 v 1988 5280 a(4)2035 5245 y Fw(n)2095 5203 y Fm(2)2135 5245 y Fq(\()p Fs(1)e Fq(+)i Fw(\013)2409 5203 y Fm(2)2449 5245 y Fq(\))2488 5165 y Fk(\001)2550 5110 y(Z)2666 5245 y Fw(\030)2715 5203 y Fj(\()p Fr(n)p Fj(\))2710 5272 y Fr(y)2831 5245 y Fo(j)p Fw(w)2930 5272 y Fr(t)2965 5245 y Fo(j)2993 5203 y Fm(2)3055 5245 y Fq(+)3154 5110 y Fk(Z)3271 5245 y Fw(\030)3320 5203 y Fj(\()p Fr(n)p Fj(\))3315 5272 y Fr(y)3435 5245 y Fo(j)p Fw(G)3541 5194 y Fj(\()p Fm(3)p Fj(\))3541 5270 y Fr(t)3643 5245 y Fo(j)3671 5203 y Fm(2)3736 5245 y Fw(;)p eop %%Page: 17 17 17 16 bop 94 -45 a Fw(")p Fv({Entr)n(opy)38 b(of)h(Driven)e(Equa)-7 b(tions)2055 b Fu(17)94 194 y Fs(hence)26 b(using)e(Eq.\(6.9\),)594 445 y Fo(k)644 311 y Fk(q)p 743 311 165 4 v 743 445 a Fw(\030)792 394 y Fj(\()p Fr(n)p Fj(\))787 457 y Fr(y)908 445 y Fw(w)979 472 y Fr(t)1014 445 y Fo(k)1064 472 y Fm(2)1148 445 y Fo(\024)45 b Fw(e)1316 403 y Fr(M)7 b(t)1435 445 y Fo(k)1485 311 y Fk(q)p 1584 311 V 1584 445 a Fw(\030)1633 394 y Fj(\()p Fr(n)p Fj(\))1628 457 y Fr(y)1749 445 y Fw(w)1820 472 y Fm(0)1860 445 y Fo(k)1910 472 y Fm(2)1972 445 y Fq(+)2072 310 y Fk(Z)2171 353 y Fr(t)2127 536 y Fm(0)2206 365 y Fk(\000)2252 445 y Fw(e)2298 403 y Fr(M)g Fj(\()p Fr(t)p Fn(\000)p Fr(s)p Fj(\))2579 445 y Fo(k)2629 311 y Fk(q)p 2729 311 V 134 x Fw(\030)2778 394 y Fj(\()p Fr(n)p Fj(\))2773 457 y Fr(y)2893 445 y Fw(G)2971 403 y Fj(\()p Fm(3)p Fj(\))2971 472 y Fr(s)3074 445 y Fo(k)3124 472 y Fm(2)3163 365 y Fk(\001)3226 445 y Fs(d)p Fw(s)24 b(:)p 3794 628 80 4 v 3794 700 4 72 v 3870 700 V 3794 704 80 4 v 94 1034 a Ft(7.)30 b(Pr)n(oof)f(of)h(Pr)n(oposition)g(2.2)94 1224 y Fs(W)-8 b(e)41 b(gi)n(v)o(e)e(here)i(a)f(proof)h(of)f (Proposition)f(2.2)h(assuming)e(e)o(xistence,)44 b(uniqueness,)f(and)d (re)o(gularity)f(of)94 1343 y(solutions)25 b(o)o(v)o(er)h(compact)h (interv)n(als)f(of)h(time,)f(which)h(are)g(standard)g(results)f(for)h (smooth)e(enough)i Fw(F)41 b Fs(\(see)94 1463 y([P]\).)26 b(The)f(proof)g(is)f(essentially)g(identical)g(to)g(the)h(one)f(gi)n(v) o(en)g(by)g(Mielk)o(e)g([M].)316 1582 y(W)-8 b(e)18 b(deri)n(v)o(e)f Fw(y)t Fs(\226independent)f(dif)n(ferential)h(inequalities)f(for)2403 1502 y Fk(R)2486 1582 y Fw(')2551 1609 y Fr(y)2599 1582 y Fo(j)p Fw(u)2684 1609 y Fr(t)2719 1582 y Fo(j)2747 1539 y Fm(2)2804 1582 y Fs(and)2965 1502 y Fk(R)3049 1582 y Fw(')3114 1609 y Fr(y)3162 1582 y Fo(j)p Fw(u)3247 1539 y Fn(0)3247 1609 y Fr(t)3281 1582 y Fo(j)3309 1539 y Fm(2)3349 1582 y Fs(,)j(we)f(inte)o(grate)94 1702 y(them)26 b(\(using)f(Gronw)o(all\255lik)o(e)g(estimates\).)39 b(W)-8 b(e)27 b(\002rst)f(compute)g(the)f(time)h(deri)n(v)n(ati)n(v)o (e)e(of)i(the)g(\002rst)g(of)g(these)94 1821 y(tw)o(o)f(inte)o(grals:) 590 2066 y Fw(@)643 2093 y Fr(t)695 1930 y Fk(Z)811 2066 y Fw(')876 2093 y Fr(y)924 2066 y Fo(j)p Fw(u)1009 2093 y Fr(t)1044 2066 y Fo(j)1072 2023 y Fm(2)1155 2066 y Fq(=)45 b Fo(\000)p Fs(2)1421 1930 y Fk(Z)1537 2066 y Fw(')1602 2093 y Fr(y)1650 2066 y Fo(j)p Fw(u)1735 2023 y Fn(0)1735 2093 y Fr(t)1770 2066 y Fo(j)1798 2023 y Fm(2)1860 2066 y Fo(\000)22 b Fs(2Re)2153 1930 y Fk(Z)2269 2066 y Fw(')2334 2023 y Fn(0)2334 2093 y Fr(y)2382 2066 y Fq(\()p Fs(1)g Fq(+)h Fs(i)p Fw(\013)p Fq(\))p 2724 2011 58 4 v Fw(u)2780 2093 y Fr(t)2815 2066 y Fw(u)2872 2023 y Fn(0)2872 2093 y Fr(t)1294 2327 y Fq(+)f Fs(2)1460 2191 y Fk(Z)1576 2327 y Fw(')1641 2354 y Fr(y)1689 2327 y Fo(j)p Fw(u)1774 2354 y Fr(t)1809 2327 y Fo(j)1837 2284 y Fm(2)1876 2327 y Fs(Re)c Fp(N)q Fq(\()p Fo(j)p Fw(u)2207 2354 y Fr(t)2241 2327 y Fo(j)2269 2284 y Fm(2)2308 2327 y Fq(\))k(+)h Fs(2)2536 2191 y Fk(Z)2651 2327 y Fw(')2716 2354 y Fr(y)2764 2327 y Fs(Re)18 b Fq(\()p 2931 2272 V Fw(u)2988 2354 y Fr(t)3023 2327 y Fw(F)3087 2354 y Fr(t)3122 2327 y Fq(\))1155 2588 y Fo(\024)1277 2453 y Fk(Z)1393 2588 y Fw(')1458 2615 y Fr(y)1507 2588 y Fo(j)p Fw(u)1592 2615 y Fr(t)1626 2588 y Fo(j)1654 2545 y Fm(2)1710 2508 y Fk(\000)1768 2549 y Fm(3)p 1768 2565 35 4 v 1768 2623 a(2)1837 2588 y Fq(+)1949 2549 y Fm(1)p 1949 2565 V 1949 2623 a(2)1995 2588 y Fw(\013)2059 2545 y Fm(2)2121 2588 y Fq(+)23 b Fs(2)p Fw(Q)2350 2615 y Fm(1)2412 2588 y Fo(\000)f Fs(2)p Fo(j)p Fw(u)2646 2615 y Fr(t)2681 2588 y Fo(j)2709 2545 y Fm(2)p Fr(q)2787 2508 y Fk(\001)2855 2588 y Fq(+)2955 2453 y Fk(Z)3071 2588 y Fw(')3136 2615 y Fr(y)3184 2588 y Fo(j)p Fw(F)3276 2615 y Fr(t)3311 2588 y Fo(j)3339 2545 y Fm(2)1155 2850 y Fo(\024)45 b Fw(A)22 b Fo(\000)1474 2714 y Fk(Z)1590 2850 y Fw(')1655 2876 y Fr(y)1703 2850 y Fo(j)p Fw(u)1788 2876 y Fr(t)1823 2850 y Fo(j)1851 2807 y Fm(2)1915 2850 y Fw(;)94 3112 y Fs(where)k Fw(A)h Fq(=)h Fw(A)p Fq(\()p Fw(\032;)17 b(\013)845 3069 y Fm(2)884 3112 y Fw(;)g(Q)1008 3139 y Fm(1)1046 3112 y Fw(;)g(q)t Fq(\))p Fs(.)35 b(By)26 b(simple)d(inte)o(gration)g(we)i(get)867 3235 y Fk(Z)983 3371 y Fw(')1048 3398 y Fr(y)1096 3371 y Fo(j)p Fw(u)1181 3398 y Fr(t)1216 3371 y Fo(j)1244 3328 y Fm(2)1327 3371 y Fo(\024)45 b Fs(max)1637 3230 y Fk(\032)1712 3371 y Fw(A)25 b(;)1881 3260 y Fk(\020)1940 3235 y(Z)2057 3371 y Fw(')2122 3398 y Fr(y)2170 3371 y Fo(j)p Fw(u)2255 3398 y Fm(0)2294 3371 y Fo(j)2322 3328 y Fm(2)2384 3371 y Fo(\000)e Fw(A)2559 3260 y Fk(\021)2618 3371 y Fw(e)2664 3328 y Fn(\000)p Fr(t)2784 3371 y Fq(+)f Fw(A)2958 3230 y Fk(\033)3074 3371 y Fw(:)567 b Fq(\()p Fs(7)p Fw(:)p Fs(1)p Fq(\))316 3620 y Fs(W)-8 b(e)25 b(ne)o(xt)f(estimate)g(the)h (time)f(deri)n(v)n(ati)n(v)o(e)e(of)j(the)g(second)f(inte)o(gral)g (described)h(abo)o(v)o(e:)297 3864 y Fw(@)350 3891 y Fr(t)402 3728 y Fk(Z)518 3864 y Fw(')583 3891 y Fr(y)631 3864 y Fo(j)p Fw(u)716 3821 y Fn(0)716 3891 y Fr(t)751 3864 y Fo(j)779 3821 y Fm(2)863 3864 y Fq(=)44 b Fo(\000)p Fs(2)1128 3728 y Fk(Z)1244 3864 y Fw(')1309 3891 y Fr(y)1358 3864 y Fo(j)p Fw(u)1443 3821 y Fn(00)1443 3891 y Fr(t)1493 3864 y Fo(j)1521 3821 y Fm(2)1583 3864 y Fo(\000)22 b Fs(2Re)1876 3728 y Fk(Z)1992 3864 y Fw(')2057 3821 y Fn(0)2057 3891 y Fr(y)2105 3864 y Fq(\()p Fs(1)g Fq(+)h Fs(i)p Fw(\013)p Fq(\))p 2447 3809 58 4 v Fw(u)2503 3821 y Fn(0)2503 3891 y Fr(t)2538 3864 y Fw(u)2595 3821 y Fn(00)2595 3891 y Fr(t)2669 3864 y Fq(+)f Fs(2)2835 3728 y Fk(Z)2951 3864 y Fw(')3016 3891 y Fr(y)3064 3864 y Fs(Re)17 b Fq(\()p 3230 3809 V Fw(u)3287 3821 y Fn(0)3287 3891 y Fr(t)3322 3864 y Fw(F)3400 3821 y Fn(0)3386 3891 y Fr(t)3428 3864 y Fq(\))1001 4125 y(+)23 b Fs(2Re)1294 3990 y Fk(Z)1411 4125 y Fw(')1476 4152 y Fr(y)1524 4015 y Fk(\020)1583 4125 y Fo(j)p Fw(u)1668 4082 y Fn(0)1668 4152 y Fr(t)1703 4125 y Fo(j)1731 4082 y Fm(2)1770 4125 y Fp(N)q Fq(\()p Fo(j)p Fw(u)1973 4152 y Fr(t)2007 4125 y Fo(j)2035 4082 y Fm(2)2075 4125 y Fq(\))f(+)p 2235 4070 V 22 w Fw(u)2292 4082 y Fn(0)2292 4152 y Fr(t)2328 4125 y Fw(u)p Fp(N)2480 4082 y Fn(0)2508 4125 y Fq(\()p Fo(j)p Fw(u)2632 4152 y Fr(t)2666 4125 y Fo(j)2694 4082 y Fm(2)2734 4125 y Fq(\))p Fw(@)2826 4152 y Fr(x)2875 4125 y Fo(j)p Fw(u)2960 4152 y Fr(t)2995 4125 y Fo(j)3023 4082 y Fm(2)3063 4015 y Fk(\021)863 4386 y Fo(\024)44 b(\000)1078 4251 y Fk(Z)1195 4386 y Fw(')1260 4413 y Fr(y)1308 4386 y Fo(j)p Fw(u)1393 4344 y Fn(00)1393 4413 y Fr(t)1443 4386 y Fo(j)1471 4344 y Fm(2)1533 4386 y Fq(+)23 b(\()p Fs(2)e Fq(+)i Fw(\013)1907 4344 y Fm(2)1969 4386 y Fq(+)g Fs(2)p Fw(Q)2198 4413 y Fm(1)2237 4386 y Fq(\))2293 4251 y Fk(Z)2409 4386 y Fw(')2474 4413 y Fr(y)2522 4386 y Fo(j)p Fw(u)2607 4344 y Fn(0)2607 4413 y Fr(t)2642 4386 y Fo(j)2670 4344 y Fm(2)2731 4386 y Fq(+)2831 4251 y Fk(Z)2947 4386 y Fw(')3012 4413 y Fr(y)3060 4386 y Fo(j)p Fw(F)3152 4413 y Fr(t)3187 4386 y Fo(j)3215 4344 y Fm(2)1001 4648 y Fq(+)g Fs(2)p Fw(Q)1230 4675 y Fm(2)1286 4512 y Fk(Z)1402 4648 y Fw(')1467 4675 y Fr(y)1515 4648 y Fo(j)p Fw(u)1600 4605 y Fn(0)1600 4675 y Fr(t)1635 4648 y Fo(j)1663 4605 y Fm(2)1702 4648 y Fo(j)p Fw(u)1787 4675 y Fr(t)1822 4648 y Fo(j)1850 4605 y Fm(4)1914 4648 y Fw(:)3669 4256 y Fq(\()p Fs(7)p Fw(:)p Fs(2)p Fq(\))94 4888 y Fs(The)37 b(last)f(term)g(can)h(be)f(bounded)g (by)g(some)g(v)n(ariant)g(of)g(the)g(Gagliardo\226Nirenber)n(g)g (inequalities,)i(see)94 5007 y([BGO,M],)25 b(we)h(also)e(use)h(that)f (for)h(all)g Fw(\025)i(>)i Fs(0)24 b(one)h(has)885 5245 y Fo(\000)979 5110 y Fk(Z)1095 5245 y Fw(')1160 5272 y Fr(y)1208 5245 y Fo(j)p Fw(f)1296 5203 y Fn(00)1346 5245 y Fo(j)1374 5203 y Fm(2)1458 5245 y Fo(\024)45 b(\000)p Fw(\025)1732 5110 y Fk(Z)1848 5245 y Fw(')1913 5272 y Fr(y)1961 5245 y Fo(j)p Fw(f)2049 5203 y Fn(0)2076 5245 y Fo(j)2104 5203 y Fm(2)2166 5245 y Fq(+)22 b Fw(\025)p Fq(\()p Fs(1)g Fq(+)h Fw(\025)p Fq(\))2648 5110 y Fk(Z)2764 5245 y Fw(')2829 5272 y Fr(y)2877 5245 y Fo(j)p Fw(f)11 b Fo(j)2993 5203 y Fm(2)3056 5245 y Fw(:)p eop %%Page: 18 18 18 17 bop 94 -45 a FB(")p Fv({Entr)n(opy)38 b(of)g(Driven)f(Equa)-7 b(tions)2063 b Fu(18)94 194 y Fs(It)25 b(follo)n(ws)f(that)g (Eq.\(7.2\))g(can)h(be)g(estimated)f(by)417 476 y Fw(@)470 503 y Fr(t)522 340 y Fk(Z)638 476 y Fw(')703 503 y Fr(y)751 476 y Fo(j)p Fw(u)836 433 y Fn(0)836 503 y Fr(t)871 476 y Fo(j)899 433 y Fm(2)982 476 y Fo(\024)45 b(\000)1193 437 y Fm(3)p 1194 453 35 4 v 1194 510 a(4)1257 340 y Fk(Z)1373 476 y Fw(')1438 503 y Fr(y)1486 476 y Fo(j)p Fw(u)1571 433 y Fn(00)1571 503 y Fr(t)1622 476 y Fo(j)1650 433 y Fm(2)1712 476 y Fq(+)22 b Fw(C)1882 503 y Fm(1)1944 476 y Fq(+)h Fw(C)2115 503 y Fm(2)2172 340 y Fk(Z)2288 476 y Fw(')2353 503 y Fr(y)2401 476 y Fo(j)p Fw(u)2486 433 y Fn(0)2486 503 y Fr(t)2521 476 y Fo(j)2549 433 y Fm(2)2610 476 y Fq(+)g Fw(C)2781 503 y Fm(3)2821 365 y Fk(\020)2881 476 y Fs(sup)2928 557 y Fr(y)3036 340 y Fk(Z)3152 476 y Fw(')3217 503 y Fr(y)3265 476 y Fo(j)p Fw(u)3350 503 y Fr(t)3385 476 y Fo(j)3413 433 y Fm(2)3452 365 y Fk(\021)3512 404 y Fm(5)982 756 y Fo(\024)45 b(\000)1198 620 y Fk(Z)1314 756 y Fw(')1379 783 y Fr(y)1428 756 y Fo(j)p Fw(u)1513 713 y Fn(0)1513 783 y Fr(t)1547 756 y Fo(j)1575 713 y Fm(2)1637 756 y Fq(+)23 b Fw(A)e Fq(+)i Fw(B)2014 645 y Fk(\020)2073 756 y Fs(sup)2121 837 y Fr(y)2228 620 y Fk(Z)2344 756 y Fw(')2409 783 y Fr(y)2457 756 y Fo(j)p Fw(u)2542 783 y Fr(t)2577 756 y Fo(j)2605 713 y Fm(2)2645 645 y Fk(\021)2704 684 y Fm(5)2769 756 y Fw(;)94 1053 y Fs(where)29 b Fw(C)437 1080 y Fm(1)477 1053 y Fs(,)g Fw(C)602 1080 y Fm(2)642 1053 y Fs(,)h Fw(C)768 1080 y Fm(3)808 1053 y Fs(,)f Fw(A)p Fs(,)g(and)f Fw(B)33 b Fs(are)c(constants)e(independent)g(of)i Fw(u)p Fs(,)g Fw(t)p Fs(,)g(and)f Fw(y)t Fs(.)46 b(The)28 b(claim)g(is)f(pro)o (v)o(ed)g(by)94 1173 y(inserting)d(Eq.\(7.1\))g(and)h(by)g(inte)o (gration.)94 1528 y Ft(Refer)n(ences)94 1777 y FD([A])154 b(Arnold,)19 b(L.:)28 b Fa(Random)19 b(Dynamical)g(Systems)p FD(,)i(Springer)m(,)e(Berlin)h(Heidelber)o(g)g(\(1998\).)94 1867 y([B2])119 b(Bogensch)681 1866 y(\250)674 1867 y(utz,)37 b(T)-6 b(.:)54 b(Entrop)o(y)-5 b(,)34 b(pressure,)i(and)c(a)i(v)n (ariational)f(principle)g(for)f(random)g(dynamical)i(systems.)67 b(Random)360 1956 y(Comput.)20 b(Dynamics)g FC(1)p FD(,)g(99\226116)f (\(1992\).)94 2046 y([BCD])48 b(Bartucelli,)25 b(M.,)f(Constantin,)h(P) -9 b(.,)24 b(Doering,)f(C.R.,)i(Gibbon,)e(J.D.,)g(and)h(Gisself)2662 2045 y(\250)2658 2046 y(alt,)g(M.:)36 b(On)22 b(the)i(possibility)f(of) g(soft)g(and)360 2136 y(hard)d(turb)n(ulence)g(in)g(the)h(comple)o(x)f (Ginzb)n(ur)o(g\226Landau)f(equation.)29 b(Physica)20 b(D)g FC(44)p FD(,)f(421\226444)g(\(1990\).)94 2225 y([BGO])43 b(Bartucelli,)29 b(M.V)-10 b(.,)27 b(Gibbon,)g(J.D.)e(and)i(Oli)n(v)o (er)m(,)g(M.:)41 b(Length)26 b(scales)i(in)e(solutions)g(of)g(the)h (comple)o(x)f(Ginzb)n(ur)o(g\226Landau)360 2315 y(equation.)j(Physica) 21 b(D)e FC(89)p FD(,)h(267\226286)f(\(1996\).)94 2405 y([BV])101 b(Babin,)26 b(A.V)-10 b(.)24 b(and)h(V)-5 b(ishik,)25 b(M.I.:)38 b(Attractors)24 b(of)h(partial)g(dif)n (ferential)g(e)n(v)n(olution)g(equations)g(in)g(an)g(unbounded)f (domain.)360 2494 y(Proc.)c(R.)g(Soc.)g(Edinb)n(ur)o(gh)f(A)g FC(116)p FD(,)h(221\226243)f(\(1990\).)94 2584 y([C1])119 b(Collet,)21 b(P)-9 b(.:)29 b(Thermodynamic)20 b(limit)g(of)g(the)g (Ginzb)n(ur)o(g\226Landau)f(equations.)29 b(Nonlinearity)20 b FC(7)p FD(,)g(1175\2261190)e(\(1994\).)94 2674 y([C2])119 b(Collet,)24 b(P)-9 b(.:)33 b(Sampling)23 b(almost\255periodic)f (functions)g(with)g(random)g(probes)f(of)h(\002nite)h(density.)35 b(Proc.)23 b(R.)f(Soc.)g(London)f(A)360 2763 y FC(452)p FD(,)e(2263\2262277)g(\(1996\).)94 2853 y([CE1])70 b(Collet,)29 b(P)-9 b(.)27 b(and)g(Eckmann,)i(J.\255P)-9 b(.:)42 b(Extensi)n(v)o(e) 27 b(Properties)g(of)f(the)h(Comple)o(x)g(Ginzb)n(ur)o(g\226Landau)e (Equation.)49 b(Commun.)360 2943 y(Math.)20 b(Phys.)g FC(200)p FD(,)f(699\226722)g(\(1999\).)94 3032 y([CE2])70 b(Collet,)25 b(P)-9 b(.)24 b(and)g(Eckmann,)h(J.\255P)-9 b(.:)36 b(The)24 b(de\002nition)g(and)f(measurement)i(of)e(the)h (topological)g(entrop)o(y)g(per)f(unit)h(v)n(olume)g(in)360 3122 y(parabolic)d(PDEs.)28 b(Nonlinearity)20 b FC(12)p FD(,)f(451\226473)g(\(1999\).)94 3212 y([CE3])70 b(Collet,)23 b(P)-9 b(.)22 b(and)g(Eckmann,)g(J.\255P)-9 b(.:)33 b(T)-6 b(opological)21 b(entrop)o(y)h(and)f FB(")p FD(\226entrop)o(y)h(for)e (damped)i(hyperbolic)g(equations.)34 b(Preprint.)360 3301 y(\(1999\).)94 3391 y([F])168 b(Frisch,)20 b(U.:)29 b Fa(T)l(urb)n(ulence)o(,)21 b(The)f(le)m(gacy)h(of)f(A.N.)f(K)m(olmo)o (gor)l(o)o(v)p FD(,)h(Cambridge)h(Uni)n(v)-5 b(.)19 b(Press,)h (Cambridge)g(\(1995\).)94 3481 y([GV])96 b(Ginibre,)25 b(J.)f(and)g(V)-9 b(elo,)25 b(G.:)36 b(The)24 b(Cauchy)h(problem)e(in)h (local)h(spaces)g(for)e(the)i(comple)o(x)f(Ginzb)n(ur)o(g\226Landau)f (equation)i(I.)360 3570 y(Compactness)c(methods.)29 b(Physica)20 b(D)g FC(95)p FD(,)f(191\226228)g(\(1996\).)94 3660 y([K1])114 b(K)o(uksin,)19 b(S.B.:)29 b(On)20 b(turb)n(ulence)g(in)g(nonlinear)g (Schr)1845 3659 y(\250)1838 3660 y(odinger)g(equations.)29 b(GAF)-6 b(A)19 b FC(7)p FD(,)h(783\226822)e(\(1997\).)94 3750 y([K2])114 b(K)o(uksin,)24 b(S.B.:)38 b(Spectral)26 b(properties)d(of)h(solutions)g(for)f(nonlinear)i(pdes)f(in)g(the)h (turb)n(ulent)f(re)o(gime.)42 b(GAF)-6 b(A)23 b FC(9)p FD(,)i(141\226184)360 3839 y(\(1999\).)94 3929 y([KI])127 b(Kifer)m(,)19 b(Y)-10 b(.:)28 b Fa(Random)19 b(perturbations)g(of)h (dynamical)g(systems)p FD(,)h(Birkh)2326 3928 y(\250)2322 3929 y(auser)e(Boston,)h(Boston,)g(MA)f(\(1988\).)94 4019 y([KT])105 b(K)m(olmogoro)o(v)-5 b(,)21 b(A.N.)h(and)g(T)m (ikhomiro)o(v)-5 b(,)22 b(V)-10 b(.M.:)33 b FB(")p FD(\226entrop)o(y)22 b(and)h FB(")p FD(\226capacity)h(of)e(sets)h(in)f(functional)h(spaces.) 38 b(In)21 b Fa(Selected)360 4108 y(W)-7 b(orks)20 b(of)g(A.N.)g(K)m (olmo)o(gor)l(o)o(v)-6 b(,)20 b(V)-9 b(ol)20 b(III)p FD(,)e(\(Shiryaye)n(v)-5 b(,)20 b(A.N.,)e(ed.\).)29 b(Kluwer)m(,)18 b(Dordrecht)i(\(1993\).)94 4198 y([LD])105 b(Luce)20 b(B.P)-9 b(.)20 b(and)f(Doering)g(C.R.:)29 b(Scaling)21 b(of)d(turb)n(ulent)i(spik)o(e)g(amplitudes)g(in)g(the)g(comple)o(x)g (Ginzb)n(ur)o(g\255Landau)e(equation.)360 4288 y(Phys.)i(Lett.)g(A)f FC(178)p FD(,)h(92\22698)f(\(1993\).)94 4377 y([LO])105 b(Le)n(v)o(ermore,)20 b(C.D.)f(and)i(Oli)n(v)o(er)m(,)e(M.:)29 b(The)20 b(comple)o(x)h(Ginzb)n(ur)o(g\226Landau)e(equation)h(as)h(a)f (model)h(problem.)29 b(In)19 b Fa(Dynamical)360 4467 y(systems)27 b(and)e(pr)l(obabilistic)i(methods)f(in)g(partial)g(dif)o (fer)m(ential)h(equations)p FD(,)g(\(Deift,)g(P)-9 b(.,)28 b(Le)n(v)o(ermore,)f(C.D.,)g(and)f(W)-6 b(ayne,)360 4557 y(C.E.)20 b(ed.\).)28 b(AMS,)20 b(Pro)o(vidence,)g(RI)g(\(1996\).)94 4646 y([M])141 b(Mielk)o(e,)30 b(A.:)43 b(Bounds)27 b(for)g(the)g (solutions)h(of)e(the)i(comple)o(x)g(Ginzb)n(ur)o(g\226Landau)e (equation)i(in)f(terms)h(of)f(the)g(dispersion)360 4736 y(parameters.)i(Physica)21 b(D)e FC(117)p FD(,)h(106\226116)f (\(1998\).)94 4826 y([MS])97 b(Mielk)o(e,)32 b(A.)c(and)h(Schneider)m (,)i(G.:)46 b(Attractors)29 b(for)f(modulation)g(equations)i(on)e (unbounded)g(domains\227e)o(xistence)i(and)360 4915 y(comparison.)f (Nonlinearity)19 b FC(8)p FD(,)h(743\226768)f(\(1995\).)94 5005 y([P])168 b(P)o(azy)-5 b(,)41 b(A.:)61 b Fa(Semigr)l(oups)35 b(of)i(Linear)f(Oper)o(ator)o(s)g(and)g(Applications)g(to)g(P)-6 b(artial)36 b(Dif)o(fer)m(ential)g(Equations)p FD(,)k(Springer)m(,)360 5095 y(Ne)n(w\255Y)-9 b(ork)18 b(\(1983\).)94 5184 y([VC])101 b(V)-5 b(ishik,)28 b(M.I.)d(and)i(Chep)o(yzho)o(v)-5 b(,)27 b(V)-10 b(.V)g(.:)40 b(K)m(olmogoro)o(v)25 b FB(")p FD(\226entrop)o(y)h(estimates)i(for)d(the)i(uniform)e(attractors)h(of)g (non\255auto\255)360 5274 y(nomous)19 b(reaction\226dif)n(fusion)h (systems.)29 b(Sbornik:)f(Mathematics)22 b FC(189)p FD(,)d(235\226263)g (\(1998\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0003040929798--