This is a multi-part message in MIME format. ---------------0603130401878 Content-Type: text/plain; name="06-61.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-61.comments" Agrachev@sissa.it S.B.Kuksin@ma.hw.ac.uk Andrey.Sarychev@dmd.unifi.it Armen.Shirikyan@math.u-psud.fr ---------------0603130401878 Content-Type: text/plain; name="06-61.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-61.keywords" 2D Navier--Stokes system, analytic transformations, random perturbations ---------------0603130401878 Content-Type: application/postscript; name="aihpps2006.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="aihpps2006.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software %%Title: aihpps2006.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMR12 CMR8 CMBX9 CMR9 CMBX12 CMBX10 CMR10 CMR6 %%+ CMTT8 MSBM10 CMR7 CMSY10 CMMI10 CMMI7 CMSY7 CMMI5 CMEX10 CMSS10 %%+ CMTI10 CMMI8 CMMI6 CMR5 CMBXTI10 CMMIB10 CMMIB7 EUFM10 MSAM10 CMSY8 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o aihpps2006.ps aihpps2006.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2006.03.13:0943 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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All Rights Reserved) readonly def /FullName (CMMIB7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMMIB7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 17 /eta put readonly def /FontBBox{0 -250 1294 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0525392EECAC163E584A9104D99AD0BC 1B1844A0E222653FA481B8809B26A46F4C483A5D7E95816EA6582584156CFEDE B994ADCFF4645140E3617E4D7E1B0E4541CB9F562E55829B4DD880AABE2229E9 4A9FA259A734D29BBA91BA1E2055CBEA4339BCBFF98D32CEFF11F296225CAABA DCA10577A5D431B714726C1278D8101ABD1BD8D0BD0174FFF9148F8C61C241D9 2AD360A28616CB4A0670BAB68BB2A3981BBAA823C8858EF31989BDBABDF4098A E4EF75BB1764F1578F9CFAAF2948789888274ABDFD1043B1979B55446F38D4A9 BEE0FA78F366D6A169A173AA6011E82B445A8797E37C48107C750383AAB274E4 8EA55C83AFBD4D7CA454D8D0B21B556D7ACB02B73A82444FA1B1D6F5BD26D69D E9E4C809F92B4A969F8953AE78FFDBF365F24DE9C46852EF06999ACF1AF23DCA C20F84FEEB9BFC15D6796CC805729E436B6D4FFAE09F971F0B863021CD0AA3DD 648543DB7B2919D93A47C7BE06FA63199D2697B931F831AB560D301B759E72C7 0B9B20F9955B9D4EB96888A7F16B7B75E10C7F9B415D1443A152450E84669799 8BFC554C45650CDE076205E109A1E3D453664BA189E4E782FC320A7D64C140B5 1A9805779CF1E5E3A2472704B86C7063EA80DEB00782DF225A155AA968A78D8E 7A7D6349517378B46444669ACF48916FB453EA83799897FF177D6479B1C4D74E DDC0E907586009102FDEFD1CFAEE2798FBA979DF5C292454A6B5423F01C6923B 22A74651C32CBA3434D016D9659A5DEE6CB6D7AAEC53858197D177D3E2CE658C 0C09E5DA54E975FEF16EBB05EF82698407D3AE2C29A773FE41B535035FAF1882 964518468C3D0C7FA4618555F52670EF9E09C975A2AAEA4A3FF91290CD705070 6C97C740FD7F1FC881D1190D4346DFBF5591DD5CF6B37DF81109B347FC9751D5 4E02ACE998F1498CC1F441D3667571369CCBBE584F28D5903482AD0A907FE9D3 EBEF19D6284A5D738000A435B451557F3E22C445D7F68CC48AF26AC804526A84 B6EAB895E37D19D313ED0C24D579BAE9FAD8740A3EA351CAA601BA8BDA0E31DE 743A40EDB6BD78D4568344EAE32B6E8D2193932FBE899CF2272E7D0DB348D5B4 B38B945D8B2A9420802B29FB1613A051DB85694F166BFCA0F7EBBEFC2C3535F5 5D47C07151912D4CC1CAC7B4D26621D1A65A0CA83A5B8BF925F29E5E7CF8B302 AC408C181642C66DB5F08F8729D855869E5FDA094F265D13A11028768A6BC44E 1C5BA8C5C6DB4DE0949BF1A8A2D2705B6ACA8E29348CD0E1F4B6B4CEECA9B8E1 567D0EF61264676FA3B9DFBDF61444376987F1F264109E614EFDCF2B3290EBA4 A5EACFFB8EFA65D2A4CDE26F1BAFD42B1B4D0FA1ABFEFB8FAA5C8B64EDC6D63B 9FDF9D399DEE687576E26EC0A2B6B56E946E84B699F4D789486CE396296951CB BE5A6871970E58A2C460B9ACB91949EA279775B17885F220BFDD98FF274D4A18 5CB610DBEBC924C51AF1E2EE4E55412E7B3D41B97AF6995CC2E17710A552332B 58DADED20D90666C86022E897F38738B83026FE5CB85A10F3C9F6C5388F46D38 F5EB6C5AB08BF4CD44DB351F379470C7CEECA0D799F543E6F632C82933B54911 C17D22DB8B761D8070F7F3ABC559E337EDF5D1748819DDDA3546543FC0A356C5 5B2D0080D1DDCFADC6EA5FA3C2A3B911D5FA8E828BC0D612DA79E3D2C058B683 83 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMIB10 %!PS-AdobeFont-1.1: CMMIB10 1.100 %%CreationDate: 1996 Jul 23 07:54:00 % 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 (CMMIB10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMIB10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 17 /eta put readonly def /FontBBox{-15 -250 1216 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBXTI10 %!PS-AdobeFont-1.1: CMBXTI10 1.0 %%CreationDate: 1991 Aug 18 17:46:30 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBXTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMBXTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 72 /H put dup 84 /T put dup 88 /X put readonly def /FontBBox{-29 -250 1274 754}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put readonly def /FontBBox{-341 -250 1304 965}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 106 /j put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 23 /nu put dup 26 /rho put dup 62 /greater put dup 78 /N put dup 88 /X put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec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%%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put dup 55 /seven put dup 58 /colon 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 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec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73EBD0448B0E15602D912DCA6C 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSS10 %!PS-AdobeFont-1.1: CMSS10 1.0 %%CreationDate: 1991 Aug 20 17:33:34 % 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 (CMSS10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSS10 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 80 /P put readonly def /FontBBox{-61 -250 999 759}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 79 /circlemultiplydisplay put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 98 /hatwide put dup 101 /tildewide put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 72 /H put dup 84 /T put dup 88 /X put dup 99 /c put dup 106 /j put dup 107 /k put dup 109 /m put dup 112 /p put readonly def /FontBBox{37 -250 1349 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 63 /perpendicular put dup 66 /B put dup 75 /K put dup 107 /bardbl put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /delta put dup 17 /eta put dup 23 /nu put dup 34 /epsilon put dup 59 /comma put dup 61 /slash put dup 66 /B put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 82 /R put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 89 /Y put dup 98 /b put dup 99 /c put dup 102 /f put dup 103 /g put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 112 /p put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 65 /A put dup 67 /C put dup 68 /D put dup 70 /F put dup 72 /H put dup 73 /I put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 8 /Phi put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 17 /eta 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 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 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 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 96 /lscript put dup 97 /a put dup 98 /b 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 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 61 /equal put dup 91 /bracketleft put dup 97 /a put dup 99 /c put dup 108 /l put dup 111 /o put dup 114 /r put dup 118 /v put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec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1F2ADCA4EA90D15CBFA7F70712A1B93547540570C76E8AED1DC16EA2B418E666 BD0E9451655FD7E6DC82A7C6CBEBF4F63DF714043F9CBB83F036E377B5D1EC4E 3C289C5F86E5C59D9C9CC734B787950DED635F0A58595B99E60C1256B1D4FA59 BAD7DA3789871949575679F1C188A652FD538DF296442A22004599D74E320951 C5BB9F7FBB916C5BA63284F52F103F87C2EC3047374CD668E6C008A832F2A200 7A0C373A2A4F6B9F79AA8F5D1F717FEE3804D3F246AFCA1992CE3A6F1695D086 43944029EA478B5E7BBC976FA3EE8731F9D83702C245E6E70DAC763536022E80 30B78574B070627F5CD722877E3B1C13D48F67A8263AB5C6E421EF10EE6AA055 41E34E0C7CB4D57C1C47116DBA3B7CB02594596CD00AACFDCFE5921E3D92AB86 E2A5F09A4FE83F9442C48DE01009B0985295B4108B4659F2F5CAC6097DC960E9 5C425398FE1C87B6CE067520A4DF87E575B1C4E3CED4B5FD025689536C0797DD 9E1880377035CE71AD07C4F6E65328FFE80CDBC31194CD2BF233C85900A431D7 CF1AA3DBB1F818F97B7B1A5215B99B84B4C392F9AB2B8710DAB7E029A64D565F 934C5665F6D19002E05D7582503D51745398D952D41E822648427BAD4285D608 0E28C1180EBFB425037DE6AD36EB0AFC550AE1340AD5631FAAC4AD2A0EEE51B3 A5581B1BDDF2A67670648F80B9FF0D87E54E2212E529EA821B5B2BA2B7C3EB1A AEF4CEADDFB2A5D424F63AF6466123C111899AAF295C17B59318DC843181226D 6F000B9AF565302EF5B70653DE9096A6D44CB2CBDDF7FDF5C70557B278DDFEF1 E41D71CF0F6C256E1B91A0A9207689E5852A3ECDF000CB088B5B895C51D477BC B523879F7320D5079ACE1C3B3AABF6A5099CC6E09375A923D9EFCF870F8CF0BE AC7FA0FF2848162937474787AA9251ABD4508A0A3DB27C61F1FCACCC05FC166D 2AC56867A5AFD6899DB8FE57D301428E18813A6BF90BCDE2F06091B4B7D68B07 F5902EE67011847F970ADFE8BCEF943440A48466FBAA782D48DB38D467170C63 DCB038A6BD4BC7EA42DAC6A2BF618EA73C6DD6BE53835A1A6A7985FB4D142626 8A448AA0F71BE5E4E0361DC912A8404595D3C5E6C86CCB5FAE9DE64499FEE114 2881997A551E8711620F8A2A1FF070584D025EE38FC0A6BCDCA3FD9C0960C1AB E1F9FA0F37ED6AAB3F2F9EBC5B85A38CAEBE7C7CD8F5E3A62CF0106362736044 519EAFD47B7E42B0FBA27E92240D07B510BF152A18EAE82E6146EEC3193973A9 6056EB01AFCC7B1B5771B12B2BFF51DD5449AAB7EF028512AEE44665B2458CFB 75896B74001D78BA8FB97D47067A7EA214B52C32D15D51533C21CFE682AE3C00 519D58CE21649879F664757CACFA762ED1C3ECBC326C936B8D914DACF7CD2093 648A96DC2CE98D9725FCEA2BDEFE9D4DA76480951053401A13C8222FF8024F6A 35DCDA57AF3E0C0B9F3FB98BDE3096E0A95D95F7C0AAC5F178B6C43D955EC011 B6FE338F51FCA93B183A2EE28B548ED825B6B7691165560CFE2E19E8CB0ACD74 2ACD5D49F79B3358352FDAA9FE4FAFF960AF8E3769B80103BD855183D8BB8501 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 69 /E put dup 80 /P put dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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cleartomark %%EndFont %%BeginFont: CMTT8 %!PS-AdobeFont-1.1: CMTT8 1.0 %%CreationDate: 1991 Aug 20 16:46:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTT8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 64 /at put dup 65 /A put dup 66 /B put dup 75 /K put dup 83 /S put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-5 -232 545 699}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 5 /Pi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 16 /dotlessi put dup 19 /acute put dup 21 /breve put dup 34 /quotedblright put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 72 /H put dup 73 /I put dup 76 /L put dup 80 /P put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put 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cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 14 /ffi put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put dup 54 /six put dup 57 /nine put dup 67 /C put dup 68 /D put dup 72 /H put dup 76 /L put dup 78 /N put dup 81 /Q put dup 83 /S put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 121 /y put dup 123 /endash put readonly def /FontBBox{-39 -250 1036 750}readonly def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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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 12 /fi put dup 19 /acute put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 88 /X put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 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cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 46 /period put dup 65 /A put dup 75 /K put dup 83 /S put dup 97 /a put dup 99 /c put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 110 /n put dup 114 /r put dup 115 /s put dup 117 /u put dup 118 /v put dup 121 /y put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 39 /quoteright put dup 45 /hyphen put dup 68 /D put dup 69 /E put dup 79 /O put dup 80 /P put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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/CMMIB7 rf /Fe 238[50 17[{}1 83.022 /CMMIB10 rf /Ff 167[72 3[66 11[74 72[{}3 83.022 /CMBXTI10 rf /Fg 205[28 28 28 48[{}3 41.511 /CMR5 rf /Fh 149[25 106[{}1 49.8132 /CMMI6 rf /Fi 167[58 9[56 15[55 35[36 2[35 23[{}5 66.4176 /CMMI8 rf /Fj 132[42 34 40 39 55 38 45 28 34 35 38 42 42 47 68 21 38 25 25 42 38 25 38 42 38 38 42 6[51 3[62 62 59 47 61 1[56 1[62 74 52 2[32 62 64 54 56 63 59 58 62 6[25 2[42 1[42 1[42 42 42 2[25 30 25 31[47 51 11[{}57 83.022 /CMTI10 rf /Fk 175[53 80[{}1 83.022 /CMSS10 rf /Fl 154[46 2[46 7[46 106 120 5[39 1[88 125 59[61 61 8[48 48 5[35 38 38{}15 83.022 /CMEX10 rf /Fm 143[31 2[51 1[31 24 6[27 10[46 3[35 11[46 72[{}8 41.511 /CMMI5 rf /Fn 148[34 31[51 8[44 2[52 12[45 66 19 26[52 52 16[34 1[20 52{}12 58.1154 /CMSY7 rf /Fo 135[38 2[39 25 31 2[34 2[59 21 35 27 23 1[32 32 2[30 29 8[39 54 1[39 45 39 1[50 3[53 63 46 56 37 1[54 1[42 49 54 1[50 4[34 1[20 24[31 10[33 5[34 2[30 14[{}36 58.1154 /CMMI7 rf /Fp 132[56 50 59 59 81 59 62 44 44 46 59 62 56 62 93 31 59 1[31 62 56 34 51 62 50 62 54 12[78 62 84 1[77 1[88 7[70 1[86 2[85 12[56 56 56 56 2[31 37 32[62 12[{}41 99.6264 /CMBX12 rf /Fq 133[39 41 47 1[40 48 30 39 37 1[42 1[50 73 25 43 34 29 48 40 41 39 43 1[36 44 35 6[48 69 78 48 57 49 51 63 66 53 1[67 81 57 71 46 36 69 1[53 61 69 59 63 62 44 1[65 42 65 23 23 18[54 4[39 52 54 4[47 43 47 36 41 50 48 3[41 36 1[37 43 47 53 2[55 8[{}70 83.022 /CMMI10 rf /Fr 141[69 3[42 2[42 23 2[42 42 7[55 1[55 55 2[59 4[50 70 1[58 1[68 1[57 63 4[60 44 64 1[55 10[0 0 2[55 55 83 15[83 4[83 65 65 4[65 65 2[65 1[42 4[65 7[65 23 65{}37 83.022 /CMSY10 rf /Fs 137[35 3[26 2[33 2[19 8[30 1[33 5[19 29[51 8[33 33 33 33 33 4[51 1[26 26 39[41{}17 58.1154 /CMR7 rf /Ft 165[55 5[55 1[60 1[51 10[55 69[{}5 83.022 /MSBM10 rf /Fu 134[35 1[35 35 35 35 35 35 1[35 1[35 35 1[35 1[35 35 35 35 35 35 35 1[35 13[35 7[35 8[35 35 35 17[35 35 45[{}26 66.4176 /CMTT8 rf /Fv 203[30 30 30 30 49[{}4 49.8132 /CMR6 rf /Fw 129[42 2[42 37 44 44 60 44 46 32 33 33 44 46 42 46 69 23 44 25 23 46 42 25 37 46 37 46 42 1[23 42 23 42 23 51 62 1[85 62 62 60 46 61 65 57 65 62 76 52 65 43 30 62 65 54 57 63 60 59 62 3[65 1[23 23 42 42 42 42 42 42 42 42 42 42 1[23 28 23 65 1[32 32 23 65 3[42 12[42 1[42 2[23 1[69 1[46 48 60 4[62 3[69 52{}90 83.022 /CMR10 rf /Fx 134[50 50 2[53 37 38 39 1[53 48 53 80 27 2[27 53 48 29 44 53 42 53 46 12[66 3[65 3[57 2[36 75 3[73 69 1[72 7[48 1[48 48 48 48 48 48 48 48 1[27 4[37 37 27[53 12[{}41 83.022 /CMBX10 rf /Fy 134[71 71 2[75 52 53 55 1[75 67 75 112 37 2[37 1[67 41 61 75 60 75 65 14[100 8[50 105 3[103 97 1[102 12[67 67 67 67 67 48[{}30 119.552 /CMBX12 rf /Fz 132[38 1[41 1[55 41 43 30 30 30 41 43 38 43 64 21 41 1[21 43 38 23 34 43 34 43 38 9[79 2[55 43 1[60 2[58 1[48 3[58 3[59 55 9[38 2[38 38 1[38 38 38 38 1[21 26 21 29[64 1[43 12[{}45 74.7198 /CMR9 rf /FA 134[47 1[64 1[49 34 35 36 2[44 49 1[25 1[27 25 3[41 49 39 49 43 13[49 5[84 1[69 9[67 6[25 45[49 12[{}22 74.7198 /CMBX9 rf /FB 132[35 31 37 37 51 37 39 27 28 28 37 39 35 39 59 20 37 1[20 39 35 22 31 39 31 39 35 2[35 20 1[20 2[53 72 1[53 51 39 52 1[48 55 1[65 44 55 1[25 53 55 46 48 54 51 50 53 5[20 20 35 35 35 35 35 35 35 35 35 35 35 20 24 20 2[27 27 20[35 6[39 12[{}69 66.4176 /CMR8 rf /FC 134[51 2[51 54 1[38 38 3[54 2[51 1[27 54 49 1[43 1[43 1[49 13[54 7[76 9[73 18[27 1[27 44[{}18 99.6264 /CMR12 rf /FD 134[70 3[73 51 52 51 1[73 66 73 111 36 1[40 36 2[40 58 73 58 73 66 16[90 103 9[90 101 22[43 5[36 26[72 12[{}25 143.462 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 537 872 a FD(On)43 b(\014nite-dimensional)g(pro)7 b(jections)44 b(of)e(distributions)826 1054 y(for)g(solutions)g(of)h (randomly)f(forced)i(PDE's)747 1295 y FC(A.)33 b(Agrac)m(hev)1293 1259 y FB(1)1334 1295 y FC(,)g(S.)f(Kuksin)1823 1259 y FB(2)1864 1295 y FC(,)h(A.)g(Saryc)m(hev)2453 1259 y FB(3)2495 1295 y FC(,)f(A.)h(Shiriky)m(an)3105 1259 y FB(4)1778 1594 y FA(Abstract)837 1724 y Fz(The)e(pap)r(er)g(is)h(dev) 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Fl(Z)1727 1637 y Fo(X)1804 1562 y Fr(k)p Fq(x)p Fr(k)1935 1527 y Fs(2)1935 1582 y Fo(X)1998 1562 y Fq(\026)p Fw(\()p Fq(dx)p Fw(\))d Fq(<)f Fr(1)p Fq(:)788 b Fw(\(1.4\))722 1791 y(Moreo)n(v)n(er,)20 b(ev)n(ery)f(measure)h Fq(\026)1684 1803 y Fo(j)1739 1791 y Fw(in)h(\(1.1\))g(p)r(ossesses)e(a)h(con)n(tin) n(uous)f(densit)n(y)i Fq(\032)3162 1803 y Fo(j)3218 1791 y Fw(with)722 1891 y(resp)r(ect)28 b(to)f(the)h(Leb)r(esgue)f(measure)g (on)g Fq(X)2118 1903 y Fo(j)2153 1891 y Fw(.)639 2073 y(The)e(follo)n(wing)f(lemma)g(describ)r(es)g(the)h(random)f(v)-5 b(ariables)23 b(whose)h(distribution)h(sat-)515 2173 y(is\014es)i(prop)r(ert)n(y)f(\(P\);)i(its)g(pro)r(of)f(is)h(ob)n (vious.)515 2339 y Fx(Lemma)j(1.3.)40 b Fj(The)31 b(distribution)g(of)f (an)g Fq(X)7 b Fj(-value)l(d)29 b(r)l(andom)i(variable)h Fq(\030)h Fj(satis\014es)d(c)l(on-)515 2439 y(dition)g Fw(\(P\))g Fj(if)h(and)f(only)g(if)h Fq(\030)j Fj(has)c(the)g(form)1685 2688 y Fq(\030)d Fw(=)1862 2584 y Fn(1)1836 2609 y Fl(X)1838 2786 y Fo(j)s Fs(=1)1969 2688 y Fq(b)2005 2700 y Fo(j)2040 2688 y Fq(\030)2076 2700 y Fo(j)2111 2688 y Fq(g)2151 2700 y Fo(j)2186 2688 y Fq(;)999 b Fw(\(1.5\))515 2959 y Fj(wher)l(e)39 b Fr(f)p Fq(g)840 2971 y Fo(j)874 2959 y Fr(g)f Fj(is)h(an)g(orthonormal)h(b)l(asis)f(in)g Fq(X)7 b Fj(,)40 b Fr(f)p Fq(\030)2202 2971 y Fo(j)2237 2959 y Fr(g)e Fj(is)h(a)g(se)l(quenc)l(e)g(of)g(sc)l(alar)h(inde-)515 3058 y(p)l(endent)35 b(r)l(andom)h(variables)h(such)e(that)g Ft(E)14 b Fq(\030)1964 3028 y Fs(2)1960 3080 y Fo(j)2035 3058 y Fw(=)32 b(1)p Fj(,)k(and)g Fq(b)2438 3070 y Fo(j)2506 3058 y Fq(>)c Fw(0)j Fj(ar)l(e)g(some)h(c)l(onstants)515 3158 y(satisfying)31 b(the)f(c)l(ondition)1762 3216 y Fn(1)1735 3241 y Fl(X)1738 3418 y Fo(j)s Fs(=1)1869 3320 y Fq(b)1905 3285 y Fs(2)1905 3340 y Fo(j)1965 3320 y Fq(<)22 b Fr(1)p Fq(:)1050 b Fw(\(1.6\))515 3574 y Fj(Example)39 b Fw(1.4)p Fj(.)j Fw(Let)30 b Fq(\026)c Fr(2)h(P)7 b Fw(\()p Fq(X)g Fw(\))30 b(b)r(e)g(a)f(Gaussian)g(measure)g(with)h(a)f (mean)h(v)-5 b(alue)29 b Fq(a)e Fr(2)g Fq(X)515 3674 y Fw(and)22 b(a)g(co)n(v)-5 b(ariance)20 b(op)r(erator)g Fq(K)6 b Fw(.)35 b(W)-7 b(e)22 b(claim)g(that)h Fq(\026)f Fw(satis\014es)f(condition)h(\(P\))h(if)f(and)g(only)515 3773 y(if)k(it)h(is)e(non-degenerate,)g(i.e.,)h(all)g(eigen)n(v)-5 b(alues)25 b(of)h Fq(K)31 b Fw(are)25 b(p)r(ositiv)n(e.)36 b(Indeed,)26 b(F)-7 b(ernique's)515 3873 y(theorem)27 b(\(see)h([Bog98)n(]\))g(implies)g(that)h(condition)e(\(1.4\))h(is)g (satis\014ed)f(for)h(an)n(y)f(Gaussian)515 3973 y(measure.)45 b(F)-7 b(urthermore,)30 b(it)h(follo)n(ws)f(from)g(\(1.3\))g(that)h (the)g(pro)5 b(jection)30 b Fq(\026)2927 3985 y Fo(j)2992 3973 y Fw(of)h Fq(\026)g Fw(to)f Fq(X)3344 3985 y Fo(j)515 4072 y Fw(is)f(a)g(one-dimensional)f(Gaussian)g(measure)g(with)i(v)-5 b(ariance)28 b Fq(\025)2531 4084 y Fo(j)2566 4072 y Fw(.)42 b(Th)n(us,)29 b Fq(\026)2916 4084 y Fo(j)2981 4072 y Fw(p)r(ossesses)e(a)515 4172 y(con)n(tin)n(uous)f(densit)n(y)i(with)g (resp)r(ect)f(to)h(the)g(Leb)r(esgue)f(measure)f(if)i(and)g(only)f(if)h Fq(\025)3161 4184 y Fo(j)3220 4172 y Fq(>)22 b Fw(0.)639 4272 y(F)-7 b(or)39 b(the)h(sequel,)i(w)n(e)d(note)g(that)h(if)g Fq(\026)j Fr(2)g(P)7 b Fw(\()p Fq(X)g Fw(\))39 b(is)g(a)g (non-degenerate)f(Gaussian)515 4371 y(measure,)26 b(then)1378 4471 y Fq(\026)p Fw(\()p Fq(B)t Fw(\))d Fq(>)g Fw(0)83 b(for)27 b(an)n(y)f(ball)i Fq(B)f Fr(\032)c Fq(X)6 b Fw(;)692 b(\(1.7\))515 4620 y(see)27 b(Section)h(3.5)e(in)i([Bog98)n(]) g(for)f(a)g(pro)r(of.)1926 5255 y(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 515 523 a Fp(1.2)112 b(Supp)s(ort)38 b(of)g(decomp)s(osable)h(measures)515 676 y Fw(Let)29 b Fq(X)35 b Fw(b)r(e)29 b(a)f(separable)f(Hilb)r(ert)j(space)d(and)i (let)g Fq(\026)c Fr(2)g(P)7 b Fw(\()p Fq(X)g Fw(\))28 b(b)r(e)i(a)e(measure)f(p)r(ossessing)515 776 y(prop)r(ert)n(y)e (\(P\).)h(Denote)g(b)n(y)g Fr(f)p Fq(g)1502 788 y Fo(j)1536 776 y Fr(g)f Fw(the)i(orthonormal)d(basis)h(in)h Fq(X)33 b Fw(for)25 b(whic)n(h)h(represen)n(ta-)515 876 y(tion)j(\(1.1\))g (holds)g(and,)h(for)f(an)n(y)f(in)n(teger)h Fq(N)34 b Fr(\025)26 b Fw(1,)j(de\014ne)h Fq(X)2454 891 y Fs(\()p Fo(N)6 b Fs(\))2598 876 y Fw(as)28 b(the)i Fq(N)9 b Fw(-dimensional)515 975 y(space)27 b(spanned)g(b)n(y)g Fq(g)1215 987 y Fo(j)1250 975 y Fw(,)h(1)22 b Fr(\024)h Fq(j)28 b Fr(\024)22 b Fq(N)9 b Fw(.)37 b(Let)1330 1221 y Fq(\026)1380 1236 y Fs(\()p Fo(N)6 b Fs(\))1518 1221 y Fw(=)1639 1117 y Fo(N)1606 1142 y Fl(O)1611 1319 y Fo(j)s Fs(=1)1745 1221 y Fq(\026)1795 1233 y Fo(j)1830 1221 y Fq(;)97 b(\027)1991 1236 y Fs(\()p Fo(N)6 b Fs(\))2129 1221 y Fw(=)2296 1117 y Fn(1)2266 1142 y Fl(O)2217 1320 y Fo(j)s Fs(=)p Fo(N)g Fs(+1)2455 1221 y Fq(\026)2505 1233 y Fo(j)2540 1221 y Fq(:)645 b Fw(\(1.8\))515 1472 y(If)28 b Fq(a)d Fr(2)f Fq(X)815 1487 y Fs(\()p Fo(N)6 b Fs(\))958 1472 y Fw(and)28 b Fq(R)d(>)f Fw(0,)k(then)h(w)n(e)f(denote)h(b)n(y)f Fq(B)2151 1484 y Fo(N)2214 1472 y Fw(\()p Fq(a;)14 b(R)q Fw(\))28 b(the)h(closed)e(ball)h(in)h Fq(X)3169 1487 y Fs(\()p Fo(N)6 b Fs(\))3312 1472 y Fw(of)515 1571 y(radius)26 b Fq(R)j Fw(cen)n(tred)e(at)g Fq(a)p Fw(.)37 b(In)28 b(the)g(case)f Fq(a)c Fw(=)f(0,)28 b(w)n(e)f(write)g Fq(B)2421 1583 y Fo(N)2484 1571 y Fw(\()p Fq(R)q Fw(\).)515 1720 y Fx(Prop)s(osition)34 b(1.5.)41 b Fj(L)l(et)33 b Fq(\026)28 b Fr(2)g(P)7 b Fw(\()p Fq(X)g Fw(\))32 b Fj(b)l(e)h(a)g(me)l(asur)l(e)f(satisfying)j(c)l(ondition)e Fw(\(P\))p Fj(.)g(Then)515 1820 y(ther)l(e)d(is)g(a)g(p)l(oint)1628 1982 y Fq(A)24 b Fw(=)1828 1878 y Fn(1)1801 1903 y Fl(X)1804 2080 y Fo(j)s Fs(=1)1935 1982 y Fq(a)1979 1994 y Fo(j)2014 1982 y Fq(g)2054 1994 y Fo(j)2112 1982 y Fr(2)f Fq(X)949 b Fw(\(1.9\))515 2212 y Fj(such)30 b(that)858 2182 y Fs(1)1484 2312 y Fq(\032)1527 2324 y Fo(j)1562 2312 y Fw(\()p Fq(a)1638 2324 y Fo(j)1673 2312 y Fw(\))23 b Fq(>)g Fw(0)84 b Fj(for)31 b(al)t(l)f Fq(j)e Fr(\025)23 b Fw(1)p Fj(.)757 b Fw(\(1.10\))515 2448 y Fj(F)-6 b(urthermor)l(e,)31 b(if)g Fq(A)25 b Fr(2)g Fq(X)37 b Fj(is)31 b(a)f(p)l(oint)h(satisfying) 40 b Fw(\(1.10\))o Fj(,)31 b(then)f(for)i(any)f(inte)l(ger)f Fq(N)j Fr(\025)24 b Fw(1)515 2548 y Fj(ther)l(e)30 b(is)g Fq(r)848 2560 y Fo(N)934 2548 y Fq(>)23 b Fw(0)29 b Fj(such)h(that)1214 2782 y Fw(supp)13 b Fq(\026)24 b Fr(\033)e Fq(B)1622 2794 y Fo(N)1685 2782 y Fw(\()p Fq(A)1779 2794 y Fo(N)1843 2782 y Fq(;)14 b(r)1917 2794 y Fo(N)1980 2782 y Fw(\))p Fq(;)99 b(A)2196 2794 y Fo(N)2282 2782 y Fw(=)2401 2678 y Fo(N)2370 2703 y Fl(X)2372 2880 y Fo(j)s Fs(=1)2504 2782 y Fq(a)2548 2794 y Fo(j)2583 2782 y Fq(g)2623 2794 y Fo(j)2657 2782 y Fq(:)487 b Fw(\(1.11\))515 3037 y Fj(Pr)l(o)l(of.)43 b Fw(T)-7 b(o)31 b(pro)n(v)n(e)e(the)j(existence)f (of)g Fq(A)p Fw(,)h(denote)f(b)n(y)g Fq(A)2299 3007 y Fs(0)2366 3037 y Fw(=)2459 2975 y Fl(P)2547 3062 y Fo(j)2596 3037 y Fq(a)2640 3007 y Fs(0)2640 3059 y Fo(j)2677 3037 y Fq(g)2717 3049 y Fo(j)2783 3037 y Fw(an)n(y)f(p)r(oin)n(t)h(in)h(the) 515 3149 y(supp)r(ort)25 b(of)h Fq(\026)p Fw(.)36 b(Then)26 b Fq(a)1281 3119 y Fs(0)1281 3171 y Fo(j)1341 3149 y Fr(2)d Fw(supp)14 b Fq(\026)1654 3161 y Fo(j)1715 3149 y Fw(for)25 b(an)n(y)f Fq(j)k Fr(\025)23 b Fw(1.)36 b(Since)26 b Fq(\026)2510 3161 y Fo(j)2568 3149 y Fr(\034)d Fq(`)2709 3161 y Fo(X)2763 3169 y Fm(j)2798 3149 y Fw(,)j(there)f(is)h Fq(a)3183 3161 y Fo(j)3241 3149 y Fr(2)d Ft(R)515 3258 y Fw(suc)n(h)30 b(that)h Fr(j)p Fq(a)955 3270 y Fo(j)1010 3258 y Fr(\000)20 b Fq(a)1139 3228 y Fs(0)1139 3280 y Fo(j)1177 3258 y Fr(j)28 b(\024)f Fq(j)1359 3228 y Fn(\000)p Fs(1)1479 3258 y Fw(and)k(\(1.10\))f(holds.)45 b(De\014ning)31 b Fq(A)e Fr(2)f Fq(X)37 b Fw(b)n(y)31 b(relation)e(\(1.9\),)515 3358 y(w)n(e)e(obtain)g(the)h(required)f(result.)639 3478 y(When)i(pro)n(ving)d(the)i(second)f(part)g(of)h(the)g(prop)r (osition,)f(w)n(e)g(shall)h(assume,)f(without)515 3578 y(loss)21 b(of)i(generalit)n(y)-7 b(,)22 b(that)h Fq(A)g Fw(=)g(0.)35 b(Let)22 b(us)h(\014x)f(an)n(y)g(in)n(teger)g Fq(N)31 b Fr(\025)23 b Fw(1.)35 b(It)23 b(follo)n(ws)e(from)h(\(1.8\)) 515 3677 y(and)27 b(\(1.10\))g(that)1560 3777 y(supp)14 b Fq(\026)1795 3792 y Fs(\()p Fo(N)6 b Fs(\))1932 3777 y Fr(\033)23 b Fq(B)2083 3789 y Fo(N)2146 3777 y Fw(\()p Fq(r)2215 3789 y Fo(N)2279 3777 y Fw(\))p Fq(;)833 b Fw(\(1.12\))515 3914 y(where)35 b Fq(r)800 3926 y Fo(N)900 3914 y Fq(>)i Fw(0)e(is)h(su\016cien)n(tly)g(small.)61 b(W)-7 b(e)37 b(claim)e(that)h(\(1.11\))g(holds)f(for)g(the)i(same)515 4013 y(constan)n(t)32 b Fq(r)892 4025 y Fo(N)988 4013 y Fq(>)f Fw(0.)53 b(Indeed,)35 b(let)e Fq(x)g Fr(2)f Fq(B)1857 4025 y Fo(N)1920 4013 y Fw(\()p Fq(r)1989 4025 y Fo(N)2053 4013 y Fw(\))h(and)g Fq(")f(>)g Fw(0.)52 b(Consider)32 b(an)h Fq(X)7 b Fw(-v)-5 b(alued)515 4113 y(random)32 b(v)-5 b(ariable)32 b Fq(\030)37 b Fw(with)c(distribution)g Fq(\026)p Fw(.)53 b(By)33 b(Lemma)f(1.3,)i(it)f(has)f(the)i(form)e (\(1.5\).)515 4213 y(W)-7 b(e)28 b(wish)f(to)h(sho)n(w)e(that)1543 4312 y Ft(P)1594 4245 y Fl(\010)1642 4312 y Fr(k)p Fq(\030)c Fr(\000)c Fq(x)p Fr(k)1914 4324 y Fo(X)2001 4312 y Fr(\024)k Fq(")2127 4245 y Fl(\011)2198 4312 y Fq(>)h Fw(0)p Fq(:)816 b Fw(\(1.13\))515 4449 y(T)-7 b(o)27 b(this)h(end,)g(de\014ne)1245 4687 y Fq(')1299 4699 y Fo(N)1385 4687 y Fw(=)1503 4583 y Fo(N)1473 4608 y Fl(X)1475 4785 y Fo(j)s Fs(=1)1607 4687 y Fq(b)1643 4699 y Fo(j)1677 4687 y Fq(\030)1713 4699 y Fo(j)1749 4687 y Fq(g)1789 4699 y Fo(j)1823 4687 y Fq(;)97 b( )1997 4699 y Fo(N)2083 4687 y Fw(=)2250 4583 y Fn(1)2223 4608 y Fl(X)2171 4786 y Fo(j)s Fs(=)p Fo(N)6 b Fs(+1)2409 4687 y Fq(b)2445 4699 y Fo(j)2480 4687 y Fq(\030)2516 4699 y Fo(j)2551 4687 y Fq(g)2591 4699 y Fo(j)2626 4687 y Fq(:)p 515 4850 1146 4 v 607 4904 a Fv(1)642 4927 y FB(In)28 b(what)g(follo)n(ws,)f(w)n(e)h(iden)n (tify)f Fi(X)1596 4937 y Fh(j)1656 4927 y FB(with)h(the)g(real)f(line)h (and)g(regard)f Fi(\032)2625 4937 y Fh(j)2685 4927 y FB(as)h(a)f(function)h(of)g(a)f(real)515 5006 y(v)l(ariable.)1926 5255 y Fw(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 515 523 a Fw(It)28 b(is)f(clear)g(that)h (\(1.13\))e(will)i(b)r(e)g(established)g(if)g(w)n(e)f(pro)n(v)n(e)f (that)1005 699 y Ft(P)1056 632 y Fl(\010)1105 699 y Fr(k)p Fq(')1201 711 y Fo(N)1282 699 y Fr(\000)18 b Fq(x)p Fr(k)1454 711 y Fo(X)1540 699 y Fr(\024)23 b Fq("=)p Fw(2)1751 632 y Fl(\011)1821 699 y Fq(>)f Fw(0)p Fq(;)97 b Ft(P)2121 632 y Fl(\010)2169 699 y Fr(k)p Fq( )2265 711 y Fo(N)2328 699 y Fr(k)2370 711 y Fo(X)2455 699 y Fr(\024)23 b Fq("=)p Fw(2)2666 632 y Fl(\011)2736 699 y Fq(>)g Fw(0)p Fq(:)278 b Fw(\(1.14\))515 876 y(The)18 b(\014rst)g(inequalit)n(y)g(follo)n(ws)g (immediately)h(from)f(\(1.12\))o(.)34 b(T)-7 b(o)18 b(pro)n(v)n(e)f (the)i(second,)g(c)n(ho)r(ose)515 975 y(an)27 b(in)n(teger)g Fq(M)k(>)23 b(N)36 b Fw(and)28 b(write)1067 1228 y Fr(k)p Fq( )1163 1240 y Fo(N)1225 1228 y Fr(k)1267 1194 y Fs(2)1267 1249 y Fo(X)1353 1228 y Fw(=)1518 1125 y Fo(M)1493 1149 y Fl(X)1441 1328 y Fo(j)s Fs(=)p Fo(N)6 b Fs(+1)1679 1228 y Fq(b)1715 1194 y Fs(2)1715 1249 y Fo(j)1752 1228 y Fq(\030)1792 1194 y Fs(2)1788 1249 y Fo(j)1848 1228 y Fw(+)2016 1125 y Fn(1)1989 1149 y Fl(X)1931 1328 y Fo(j)s Fs(=)p Fo(M)g Fs(+1)2180 1228 y Fq(b)2216 1194 y Fs(2)2216 1249 y Fo(j)2253 1228 y Fq(\030)2293 1194 y Fs(2)2289 1249 y Fo(j)2353 1228 y Fw(=)23 b Fq(S)2492 1240 y Fo(M)2584 1228 y Fw(+)18 b Fq(R)2730 1240 y Fo(M)2804 1228 y Fq(:)340 b Fw(\(1.15\))515 1500 y(In)28 b(view)f(of)34 b(\(1.6\))27 b(and)h(the)g(relation)e Ft(E)14 b Fq(\030)1827 1470 y Fs(2)1823 1521 y Fo(j)1888 1500 y Fw(=)23 b(1,)k(w)n(e)g(ha)n(v) n(e)1287 1755 y Ft(E)14 b Fq(R)1419 1767 y Fo(M)1516 1755 y Fw(=)1689 1651 y Fn(1)1662 1676 y Fl(X)1604 1854 y Fo(j)s Fs(=)p Fo(M)6 b Fs(+1)1853 1755 y Fq(b)1889 1721 y Fs(2)1889 1775 y Fo(j)1949 1755 y Fr(!)23 b Fw(0)83 b(as)27 b Fq(M)k Fr(!)23 b(1)p Fq(:)515 2016 y Fw(By)k(Cheb)n(yshev's)g (inequalit)n(y)-7 b(,)27 b(it)h(follo)n(ws)f(that)1389 2193 y Ft(P)1440 2125 y Fl(\010)1488 2193 y Fq(R)1551 2205 y Fo(M)1648 2193 y Fr(\024)c Fq(\016)1776 2125 y Fl(\011)1847 2193 y Fr(!)g Fw(1)83 b(as)27 b Fq(M)k Fr(!)24 b(1)p Fq(;)662 b Fw(\(1.16\))515 2369 y(where)37 b Fq(\016)42 b(>)d Fw(0)e(is)g(an)g(arbitrary)f(constan)n(t.)65 b(F)-7 b(urthermore,)39 b(inequalities)e(\(1.10\))g(with)515 2469 y Fq(a)559 2481 y Fo(j)617 2469 y Fw(=)22 b(0)27 b(imply)h(that,)g(for)f(an)n(y)g(\014xed)h Fq(M)k(>)22 b(N)37 b Fw(and)27 b Fq(\016)f(>)d Fw(0,)k(w)n(e)g(ha)n(v)n(e)1648 2645 y Ft(P)1699 2578 y Fl(\010)1747 2645 y Fq(S)1798 2657 y Fo(M)1895 2645 y Fr(\024)22 b Fq(\016)2022 2578 y Fl(\011)2094 2645 y Fq(>)g Fw(0)p Fq(:)921 b Fw(\(1.17\))515 2821 y(Com)n(bining)40 b(\(1.15\))f({)h(\(1.17\))f(and)h(recalling)f (that)h Fq(R)2310 2833 y Fo(M)2424 2821 y Fw(and)g Fq(S)2649 2833 y Fo(M)2762 2821 y Fw(are)f(indep)r(enden)n(t,)515 2921 y(w)n(e)33 b(arriv)n(e)g(at)h(the)g(second)f(inequalit)n(y)h(in)g (\(1.14\).)56 b(This)34 b(completes)g(the)g(pro)r(of)g(of)g(the)515 3021 y(prop)r(osition.)p 3318 3021 4 57 v 3322 2968 50 4 v 3322 3021 V 3372 3021 4 57 v 515 3252 a Fp(1.3)112 b(A)37 b(zero-one)h(la)m(w)g(for)f(analytic)i(functions)515 3405 y Fw(Let)23 b Fq(X)30 b Fw(b)r(e)24 b(a)f(separable)f(Hilb)r(ert)i (space)f(and)g(let)h Fq(f)32 b Fw(:)23 b Fq(X)29 b Fr(!)23 b Ft(R)h Fw(b)r(e)g(a)f(con)n(tin)n(uous)f(function.)515 3505 y(Recall)27 b(that)h Fq(f)36 b Fw(is)28 b(said)f(to)g(b)r(e)h Fj(analytic)34 b Fw(if)28 b(for)f(an)n(y)g Fq(x)2215 3517 y Fs(0)2276 3505 y Fr(2)c Fq(X)34 b Fw(there)27 b(is)h Fq(\016)e(>)d Fw(0)k(suc)n(h)g(that)1006 3743 y Fq(f)9 b Fw(\()p Fq(x)p Fw(\))24 b(=)e Fq(f)9 b Fw(\()p Fq(x)1407 3755 y Fs(0)1445 3743 y Fw(\))19 b(+)1617 3639 y Fn(1)1590 3664 y Fl(X)1579 3840 y Fo(m)p Fs(=1)1736 3743 y Fq(L)1793 3755 y Fo(m)1855 3743 y Fw(\()p Fq(x)g Fr(\000)f Fq(x)2083 3755 y Fs(0)2121 3743 y Fw(\))83 b(for)27 b Fq(x)d Fr(2)f Fq(B)2575 3755 y Fo(X)2638 3743 y Fw(\()p Fq(x)2717 3755 y Fs(0)2755 3743 y Fq(;)14 b(\016)s Fw(\))q Fq(;)279 b Fw(\(1.18\))515 3996 y(where)28 b Fq(L)813 4008 y Fo(m)900 3996 y Fw(:)c Fq(X)31 b Fr(!)25 b Ft(R)j Fw(is)h(an)f 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b(of)e(the)h(results)515 2002 y Fw(Let)20 b(\()p Fq(H)r(;)14 b(d)p Fw(\))20 b(b)r(e)g(a)g (metric)f(space)g(and)h(let)g Fq(X)26 b Fw(and)19 b Fq(F)32 b Fw(b)r(e)20 b(\014nite-dimensional)g(v)n(ector)e(spaces.)515 2102 y(Consider)29 b(a)h(con)n(tin)n(uous)f(op)r(erator)f Fq(f)36 b Fw(:)28 b Fq(H)f Fr(\002)19 b Fq(X)34 b Fr(!)27 b Fq(F)12 b Fw(.)45 b(F)-7 b(or)30 b(an)n(y)f(probabilit)n(y)g(measure) 515 2201 y Fq(\026)23 b Fr(2)g(P)7 b Fw(\()p Fq(X)g Fw(\))27 b(and)h(an)n(y)f Fq(u)22 b Fr(2)i Fq(H)7 b Fw(,)27 b(denote)h(b)n(y)f Fq(f)1916 2213 y Fn(\003)1954 2201 y Fw(\()p Fq(u;)14 b(\026)p Fw(\))27 b(the)h(image)f(of)h Fq(\026)f Fw(under)h Fq(f)9 b Fw(\()p Fq(u;)14 b Fr(\001)p Fw(\).)515 2353 y Fx(Theorem)39 b(2.1.)44 b Fj(Supp)l(ose)37 b(that,)h(for)f(any)g Fq(u)d Fr(2)h Fq(H)7 b Fj(,)38 b(the)e(function)g Fq(f)9 b Fw(\()p Fq(u;)14 b Fr(\001)p Fw(\))36 b Fj(is)h(analytic)515 2452 y(and)c(the)h(interior)f(of)h(the)f(set)g Fq(f)9 b Fw(\()p Fq(u;)14 b(X)7 b Fw(\))32 b Fj(is)i(non-empty.)48 b(L)l(et)33 b Fq(\026)c Fr(2)g(P)7 b Fw(\()p Fq(X)g Fw(\))33 b Fj(b)l(e)g(a)g(me)l(asur)l(e)515 2552 y(p)l(ossessing)25 b(a)g(c)l(ontinuous)f(density)h Fq(\032)p Fw(\()p Fq(x)p Fw(\))h Fj(with)f(r)l(esp)l(e)l(ct)f(to)h(the)g(L)l(eb)l(esgue)f(me)l (asur)l(e)h(on)f Fq(X)7 b Fj(.)515 2652 y(Then)30 b(the)g(fol)t(lowing) i(assertions)f(hold.)580 2803 y Fx(\(i\))41 b Fj(F)-6 b(or)34 b(any)g Fq(u)1092 2815 y Fs(0)1159 2803 y Fr(2)d Fq(H)7 b Fj(,)35 b(the)f(me)l(asur)l(e)f Fq(f)1889 2815 y Fn(\003)1927 2803 y Fw(\()p Fq(u)2007 2815 y Fs(0)2044 2803 y Fq(;)14 b(\026)p Fw(\))34 b Fj(is)g(absolutely)h(c)l(ontinuous)e (with)h(r)l(e-)722 2902 y(sp)l(e)l(ct)c(to)g Fq(`)1059 2914 y Fo(F)1113 2902 y Fj(.)553 3061 y Fx(\(ii\))41 b Fj(The)d(function)f Fq(f)1274 3073 y Fn(\003)1312 3061 y Fw(\()p Fr(\001)p Fq(;)14 b(\026)p Fw(\))38 b Fj(fr)l(om)f Fq(H)44 b Fj(to)37 b(the)g(sp)l(ac)l(e)h Fr(P)7 b Fw(\()p Fq(F)12 b Fw(\))37 b Fj(endowe)l(d)h(with)g(the)f(total)722 3161 y(variation)32 b(norm)d(is)h(c)l(ontinuous.)639 3312 y Fw(Our)37 b(next)h(goal)e(is)h(to)g(study)h(the)g(case)e(in)i (whic)n(h)f Fq(X)44 b Fw(is)37 b(an)g(in\014nite-dimensional)515 3412 y(space.)57 b(More)34 b(precisely)-7 b(,)36 b(supp)r(ose)e(that)h Fq(X)41 b Fw(is)35 b(a)f(separable)g(Hilb)r(ert)h(space)f(and)g Fq(F)47 b Fw(is)515 3511 y(a)36 b(\014nite-dimensional)h(v)n(ector)e (space.)64 b(Recall)37 b(that)g(condition)f(\(P\))h(is)g(in)n(tro)r (duced)g(in)515 3611 y(Section)27 b(1.1.)515 3762 y Fx(Theorem)g(2.2.) 36 b Fj(L)l(et)26 b Fq(f)31 b Fw(:)23 b Fq(H)18 b Fr(\002)10 b Fq(X)29 b Fr(!)23 b Fq(F)38 b Fj(b)l(e)26 b(a)g(c)l(ontinuous)g (function)g(such)g(that)g Fq(f)9 b Fw(\()p Fq(u;)14 b Fr(\001)p Fw(\))26 b Fj(is)515 3862 y(analytic)31 b(for)h(any)e Fq(u)24 b Fr(2)h Fq(H)37 b Fj(and)31 b(the)f(derivative)j Fq(D)2131 3874 y Fo(x)2172 3862 y Fq(f)9 b Fw(\()p Fq(u;)14 b(x)p Fw(\))31 b Fj(is)g(c)l(ontinuous)e(with)i(r)l(esp)l(e)l(ct)515 3961 y(to)25 b Fw(\()p Fq(u;)14 b(x)p Fw(\))p Fj(.)38 b(Supp)l(ose)25 b(that,)i(for)f(any)g Fq(u)d Fr(2)g Fq(H)7 b Fj(,)27 b(ther)l(e)e(is)h(a)g(b)l(al)t(l)g Fq(B)2498 3973 y Fo(u)2567 3961 y Fj(in)f(a)h(\014nite-dimensional)515 4061 y(subsp)l(ac)l(e)i Fq(X)915 4073 y Fo(u)981 4061 y Fr(\032)23 b Fq(X)34 b Fj(such)28 b(that)g(the)h(interior)f(of)h(the) f(set)g Fq(f)9 b Fw(\()p Fq(u;)14 b(B)2557 4073 y Fo(u)2600 4061 y Fw(\))28 b Fj(is)g(non-empty.)38 b(Then)515 4161 y(for)e(any)g(me)l(asur)l(e)f Fq(\026)e Fr(2)h(P)7 b Fw(\()p Fq(X)g Fw(\))34 b Fj(satisfying)j(c)l(ondition)g Fw(\(P\))e Fj(statements)f Fw(\(i\))i Fj(and)g Fw(\(ii\))g Fj(of)515 4260 y(The)l(or)l(em)31 b(2.1)g(take)f(plac)l(e.)515 4490 y Fp(2.2)112 b(Pro)s(of)37 b(of)h(Theorem)g(2.1)515 4643 y Fw(Let)24 b(us)f(\014x)h(an)n(y)f(p)r(oin)n(t)h Fq(u)1290 4655 y Fs(0)1350 4643 y Fr(2)g Fq(H)30 b Fw(and)24 b(denote)g(b)n(y)f Fq(D)2130 4655 y Fo(x)2172 4643 y Fq(f)9 b Fw(\()p Fq(u)2302 4655 y Fs(0)2338 4643 y Fq(;)14 b(x)p Fw(\))25 b(the)f(deriv)-5 b(ativ)n(e)23 b(\(Jacobian\))515 4742 y(of)38 b(the)h(map)f Fq(f)9 b Fw(\()p Fq(u)1099 4754 y Fs(0)1135 4742 y Fq(;)14 b Fr(\001)p Fw(\))42 b(:)f Fq(X)47 b Fr(!)41 b Fq(F)50 b Fw(at)38 b(the)h(p)r(oin)n(t)f Fq(x)j Fr(2)h Fq(X)7 b Fw(.)68 b(W)-7 b(e)39 b(\014rst)f(sho)n(w)f (that)i(the)515 4842 y(matrix)27 b Fq(D)854 4854 y Fo(x)895 4842 y Fq(f)9 b Fw(\()p Fq(u)1025 4854 y Fs(0)1062 4842 y Fq(;)14 b(x)p Fw(\))28 b(has)f(a)h(minor)f Fq(m)p Fw(\()p Fq(x)p Fw(\))h(of)g(the)g(size)f(dim)14 b Fq(F)40 b Fw(suc)n(h)27 b(that)1267 5006 y Fq(m)p Fw(\()p Fq(x)p Fw(\))d Fr(6)p Fw(=)f(0)82 b(for)27 b Fq(\026)p Fw(-almost)g(ev)n(ery)f Fq(x)e Fr(2)f Fq(X)7 b(:)581 b Fw(\(2.1\))1926 5255 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 515 523 a Fw(T)-7 b(o)32 b(this)h(end,)i(recall) d(that)h(a)g(p)r(oin)n(t)g Fq(x)f Fr(2)g Fq(X)40 b Fw(is)32 b(said)h(to)g(b)r(e)g Fj(r)l(e)l(gular)42 b Fw(for)32 b Fq(f)9 b Fw(\()p Fq(u)3020 535 y Fs(0)3057 523 y Fq(;)14 b Fr(\001)p Fw(\))33 b(if)h(the)515 623 y(rank)j(of)i Fq(D)892 635 y Fo(x)933 623 y Fq(f)9 b Fw(\()p Fq(u)1063 635 y Fs(0)1100 623 y Fq(;)14 b(x)p Fw(\))39 b(is)f(maximal.)69 b(An)n(y)38 b(p)r(oin)n(t)h(that)g(is)f(not)g(regular)f(is)h(said)g(to) h(b)r(e)515 722 y Fj(singular)9 b Fw(.)42 b(In)30 b(view)f(of)g(Sard's) g(theorem)f(\(see)i([Ste83)o(]\),)g(the)g(image)e(under)i(the)f(smo)r (oth)515 822 y(function)37 b Fq(f)9 b Fw(\()p Fq(u)979 834 y Fs(0)1016 822 y Fq(;)14 b Fr(\001)p Fw(\))37 b(of)f(the)i(set)e (of)h(its)g(singular)e(p)r(oin)n(ts)i(has)f(zero)g(Leb)r(esgue)g (measure.)515 922 y(Since)j(the)h(in)n(terior)e(of)h Fq(f)9 b Fw(\()p Fq(u)1439 934 y Fs(0)1476 922 y Fq(;)14 b(X)7 b Fw(\))39 b(is)g(non-empt)n(y)-7 b(,)42 b(w)n(e)d(conclude)g (that)g Fq(f)9 b Fw(\()p Fq(u)3009 934 y Fs(0)3046 922 y Fq(;)14 b Fr(\001)p Fw(\))40 b(has)e(a)515 1021 y(regular)26 b(p)r(oin)n(t)h Fq(x)1060 1033 y Fs(0)1121 1021 y Fr(2)c Fq(X)7 b Fw(.)639 1121 y(Let)30 b Fq(m)p Fw(\()p Fq(x)p Fw(\))h(b)r(e)f(the)g(minor)g(of)f Fq(D)1673 1133 y Fo(x)1715 1121 y Fq(f)9 b Fw(\()p Fq(u)1845 1133 y Fs(0)1881 1121 y Fq(;)14 b(x)p Fw(\))31 b(that)f(is)g(non-zero)e(at)h Fq(x)2785 1133 y Fs(0)2823 1121 y Fw(.)43 b(Since)30 b Fq(m)p Fw(\()p Fq(x)p Fw(\))h(is)515 1220 y(analytic,)25 b(w)n(e)g(see)g(that)g Fq(m)p Fw(\()p Fq(x)p Fw(\))f Fr(6)p Fw(=)f(0)i(almost)f(ev)n(erywhere)g(with)i(resp)r(ect)f(to)g (the)h(Leb)r(esgue)515 1320 y(measure)j Fq(`)876 1332 y Fo(X)938 1320 y Fw(.)45 b(Since)30 b Fq(\026)g Fw(is)g(absolutely)g (con)n(tin)n(uous)f(with)h(resp)r(ect)g(to)g Fq(`)2824 1332 y Fo(X)2887 1320 y Fw(,)h(w)n(e)e(conclude)515 1420 y(that)f(\(2.1\))f(holds.)639 1519 y(F)-7 b(or)27 b(an)n(y)g Fq(")c(>)f Fw(0,)28 b(w)n(e)f(denote)1262 1702 y Fq(X)1338 1668 y Fo(")1396 1702 y Fw(=)22 b Fr(f)p Fq(x)h Fr(2)h Fq(X)29 b Fw(:)23 b Fr(j)p Fq(x)p Fr(j)h(\024)e Fq(")2061 1668 y Fn(\000)p Fs(1)2150 1702 y Fq(;)14 b Fr(j)p Fq(m)p Fw(\()p Fq(x)p Fw(\))p Fr(j)24 b(\025)f Fq(")p Fr(g)p Fq(:)515 1885 y Fw(Then)1330 1984 y Fq(\027)1371 1996 y Fo(")1430 1984 y Fw(:=)g Fq(\026)p Fw(\()p Fq(X)i Fr(n)18 b Fq(X)1853 1950 y Fo(")1888 1984 y Fw(\))23 b Fr(!)g Fw(0)83 b(as)f Fq(")23 b Fr(!)g Fw(0)p Fq(:)515 2134 y Fw(Let)36 b(us)h(tak)n(e)f(an)n(y)g Fq(x)i Fr(2)g Fq(X)1397 2104 y Fo(")1469 2134 y Fw(and)e(write)g(it)h(as)f Fq(x)j Fw(=)e(\()p Fq(x)2330 2146 y Fs(1)2368 2134 y Fq(;)14 b(x)2452 2146 y Fs(2)2490 2134 y Fw(\),)39 b(where)d Fq(x)2880 2146 y Fs(1)2954 2134 y Fw(denotes)h(the)515 2233 y(v)-5 b(ariables)21 b(en)n(tering)g(the)i(minor)f Fq(m)p Fw(\()p Fq(x)p Fw(\).)36 b(Accordingly)-7 b(,)22 b(the)h(space)f Fq(X)29 b Fw(can)22 b(b)r(e)g(represen)n(ted)515 2333 y(as)27 b(a)h(direct)h(pro)r(duct)f Fq(X)j Fw(=)24 b Fq(X)1494 2345 y Fs(1)1550 2333 y Fr(\002)18 b Fq(X)1702 2345 y Fs(2)1739 2333 y Fw(.)40 b(Applying)28 b(the)h(implicit)g (function)g(theorem,)f(w)n(e)515 2433 y(can)38 b(\014nd)h(op)r(en)g (balls)f Fq(V)1320 2445 y Fs(1)1400 2433 y Fr(\032)j Fq(X)1575 2445 y Fs(1)1651 2433 y Fw(and)d Fq(V)1871 2445 y Fs(2)1951 2433 y Fr(\032)j Fq(X)2126 2445 y Fs(2)2202 2433 y Fw(suc)n(h)d(that)h(for)f(an)n(y)g Fq(x)2944 2445 y Fs(2)3024 2433 y Fr(2)k Fq(V)3169 2445 y Fs(2)3245 2433 y Fw(and)515 2532 y Fq(u)24 b Fr(2)g Fq(B)733 2502 y Fo(")793 2532 y Fw(=)g Fq(B)945 2544 y Fo(X)1008 2532 y Fw(\()p Fq(u)1088 2544 y Fs(0)1125 2532 y Fq(;)14 b(r)1199 2544 y Fo(")1235 2532 y Fw(\))p Fq(;)29 b Fw(the)f(map)h Fq(f)9 b Fw(\()p Fq(u;)14 b Fr(\001)p Fq(;)g(x)1922 2544 y Fs(2)1959 2532 y Fw(\))28 b(is)h(a)e(di\013eomorphism)h(of)h(the)f (domain)g Fq(V)3341 2544 y Fs(1)515 2632 y Fw(on)n(to)k(its)i(image)e Fq(W)12 b Fw(\()p Fq(u;)i(x)1328 2644 y Fs(2)1366 2632 y Fw(\).)54 b(Here)33 b Fq(r)1714 2644 y Fo(")1783 2632 y Fq(>)f Fw(0)h(is)g(a)g(constan)n(t)g(that)g(go)r(es)g(to)g(zero)f (with)i Fq(")p Fw(.)515 2731 y(Accordingly)-7 b(,)24 b(w)n(e)f(can)h(write)g Fq(x)1513 2743 y Fs(1)1575 2731 y Fw(in)h(terms)e(of)i Fq(u)d Fr(2)i Fq(B)2204 2701 y Fo(")2239 2731 y Fw(,)h Fq(x)2334 2743 y Fs(2)2395 2731 y Fr(2)e Fq(V)2521 2743 y Fs(2)2583 2731 y Fw(and)h Fq(y)i Fw(=)d Fq(f)9 b Fw(\()p Fq(u;)14 b(x)3110 2743 y Fs(1)3147 2731 y Fq(;)g(x)3231 2743 y Fs(2)3268 2731 y Fw(\))24 b Fr(2)515 2831 y Fq(W)12 b Fw(\()p Fq(u;)i(x)769 2843 y Fs(2)806 2831 y Fw(\):)1659 2931 y Fq(x)1706 2943 y Fs(1)1767 2931 y Fw(=)23 b Fq(g)s Fw(\()p Fq(u;)14 b(y)s(;)g(x)2143 2943 y Fs(2)2179 2931 y Fw(\))p Fq(:)515 3080 y Fw(The)30 b(sets)g Fq(V)47 b Fw(=)27 b Fq(V)1088 3092 y Fs(1)1146 3080 y Fr(\002)20 b Fq(V)1279 3092 y Fs(2)1347 3080 y Fw(corresp)r(onding)28 b(to)j(v)-5 b(arious)29 b Fq(x)f Fr(2)g Fq(X)2509 3050 y Fo(")2574 3080 y Fw(form)i(an)g(op)r(en)h(co)n (v)n(er)d(of)515 3180 y(the)d(compact)f(set)h Fq(X)1187 3150 y Fo(")1222 3180 y Fw(.)36 b(Let)25 b(us)f(\014nd)h(a)g(\014nite)g (sub-co)n(v)n(er)e Fr(f)p Fq(V)2442 3150 y Fo(j)2477 3180 y Fr(g)p Fw(.)35 b(W)-7 b(e)26 b(denote)e(b)n(y)h Fr(f)p Fq(')3191 3192 y Fo(j)3225 3180 y Fw(\()p Fq(x)p Fw(\))p Fr(g)515 3279 y Fw(a)32 b(con)n(tin)n(uous)g(partition)g(of)g (unit)n(y)h(on)f Fq(X)1878 3249 y Fo(")1946 3279 y Fw(sub)r(ordinate)g (to)g Fr(f)p Fq(V)2621 3249 y Fo(j)2656 3279 y Fr(g)p Fw(.)51 b(That)33 b(is,)h Fq(')3152 3291 y Fo(j)3218 3279 y Fr(\025)d Fw(0,)515 3379 y(supp)14 b Fq(')754 3391 y Fo(j)812 3379 y Fr(\032)23 b Fq(V)967 3349 y Fo(j)1002 3379 y Fw(,)k(and)h(\()1246 3317 y Fl(P)1347 3379 y Fq(')1401 3391 y Fo(j)1437 3379 y Fw(\)\()p Fq(x)p Fw(\))c(=)f(1)k(for)g Fq(x)d Fr(2)f Fq(X)2113 3349 y Fo(")2148 3379 y Fw(.)37 b(Let)27 b Fq(\026)2406 3391 y Fo(j)2465 3379 y Fw(=)22 b Fq(')2606 3391 y Fo(j)2641 3379 y Fq(\026)p Fw(.)37 b(Then)630 3562 y Fq(\026)680 3574 y Fo(j)738 3562 y Fw(=)23 b Fq(')880 3574 y Fo(j)915 3562 y Fw(\()p Fq(x)p Fw(\))p Fq(\032)p Fw(\()p Fq(x)p Fw(\))14 b Fq(dx)1284 3574 y Fs(1)1338 3562 y Fq(dx)1428 3574 y Fs(2)1489 3562 y Fw(=)22 b Fq(')1630 3574 y Fo(j)1666 3562 y Fw(\()p Fq(x)1745 3574 y Fs(1)1783 3562 y Fq(;)14 b(x)1867 3574 y Fs(2)1904 3562 y Fw(\))p Fq(\032)p Fw(\()p Fq(x)2058 3574 y Fs(1)2096 3562 y Fq(;)g(x)2180 3574 y Fs(2)2218 3562 y Fw(\))p Fr(j)p Fq(m)p Fw(\()p Fq(u;)g(x)2510 3574 y Fs(1)2548 3562 y Fq(;)g(x)2632 3574 y Fs(2)2669 3562 y Fw(\))p Fr(j)2724 3527 y Fn(\000)p Fs(1)2827 3562 y Fq(dy)j(dx)3018 3574 y Fs(2)3070 3562 y Fq(;)115 b Fw(\(2.2\))515 3744 y(where)18 b Fq(m)p Fw(\()p Fq(u;)c(x)p Fw(\))k(is)h(the)f(minor)g (of)h Fq(D)1627 3756 y Fo(x)1668 3744 y Fq(f)9 b Fw(\()p Fq(u;)14 b(x)p Fw(\))19 b(corresp)r(onding)d(to)i Fq(x)2596 3756 y Fs(1)2634 3744 y Fw(,)i(and)f Fq(x)2877 3756 y Fs(1)2937 3744 y Fw(=)k Fq(g)s Fw(\()p Fq(u;)14 b(y)s(;)g(x)3313 3756 y Fs(2)3349 3744 y Fw(\))515 3844 y(on)27 b(the)h(righ)n(t-hand)e (side)i(of)34 b(\(2.2\).)j(Hence,)27 b Fq(f)1990 3856 y Fn(\003)2028 3844 y Fw(\()p Fq(u;)14 b(\026)2195 3856 y Fo(j)2230 3844 y Fw(\))23 b(=)g Fq(g)2413 3856 y Fo(j)2447 3844 y Fw(\()p Fq(u;)14 b(y)s Fw(\))g Fq(dy)s Fw(,)28 b(where)1075 4072 y Fq(g)1115 4084 y Fo(j)1149 4072 y Fw(\()p Fq(u;)14 b(y)s Fw(\))23 b(=)1452 3959 y Fl(Z)1563 4072 y Fw(~)-56 b Fq(')1603 4084 y Fo(j)1639 4072 y Fw(\()p Fq(y)s(;)14 b(x)1799 4084 y Fs(2)1836 4072 y Fw(\))8 b(~)-50 b Fq(\032)p Fw(\()p Fq(y)s(;)14 b(x)2071 4084 y Fs(2)2109 4072 y Fw(\))p Fr(j)f Fl(e)-59 b Fq(m)p Fw(\()p Fq(u;)14 b(y)s(;)g(x)2482 4084 y Fs(2)2519 4072 y Fw(\))p Fr(j)2574 4038 y Fn(\000)p Fs(1)2677 4072 y Fq(dx)2767 4084 y Fs(2)515 4297 y Fw(with)48 b(~)-55 b Fq(')765 4309 y Fo(j)800 4297 y Fw(\()p Fq(y)s(;)14 b(x)960 4309 y Fs(2)998 4297 y Fw(\))34 b(=)g Fq(')1217 4309 y Fo(j)1253 4297 y Fw(\()p Fq(g)s Fw(\()p Fq(u;)14 b(y)s(;)g(x)1573 4309 y Fs(2)1610 4297 y Fw(\))p Fq(;)g(x)1726 4309 y Fs(2)1763 4297 y Fw(\),)37 b(etc.)58 b(Let)34 b(us)h(denote)f Fq(\026)2635 4309 y Fo(")2705 4297 y Fw(=)g(\()2836 4235 y Fl(P)2938 4297 y Fq(')2992 4309 y Fo(j)3027 4297 y Fw(\))p Fq(\026)p Fw(.)58 b(Then)515 4397 y Fr(k)p Fq(\026)18 b Fr(\000)g Fq(\026)758 4409 y Fo(")793 4397 y Fr(k)835 4409 y Fs(v)l(ar)952 4397 y Fr(\024)23 b Fq(\027)1081 4409 y Fo(")1117 4397 y Fw(.)36 b(W)-7 b(e)28 b(ha)n(v)n(e)1532 4579 y Fq(f)1573 4591 y Fn(\003)1611 4579 y Fw(\()p Fq(u;)14 b(\026)1778 4591 y Fo(")1813 4579 y Fw(\))24 b(=)e Fq(g)1996 4591 y Fo(")2031 4579 y Fw(\()p Fq(u;)14 b(y)s Fw(\))g Fq(dy)j(;)846 b Fw(\(2.3\))515 4762 y(where)27 b Fq(g)795 4774 y Fo(")830 4762 y Fw(\()p Fq(u;)14 b(y)s Fw(\))23 b(=)1133 4700 y Fl(P)1235 4762 y Fq(g)1275 4774 y Fo(j)1309 4762 y Fw(\()p Fq(u;)14 b(y)s Fw(\))28 b(is)f(a)g(con)n(tin)n(uous)g (function)h(of)g Fq(u)22 b Fr(2)i Fq(B)2734 4732 y Fo(")2797 4762 y Fw(and)j Fq(y)s Fw(.)37 b(Clearly)-7 b(,)1097 4945 y Fr(k)p Fq(f)1180 4957 y Fn(\003)1218 4945 y Fw(\()p Fq(u;)14 b(\026)1385 4957 y Fo(")1420 4945 y Fw(\))19 b Fr(\000)f Fq(f)1595 4957 y Fn(\003)1633 4945 y Fw(\()p Fq(u;)c(\026)p Fw(\))p Fr(k)1874 4957 y Fs(v)l(ar)1991 4945 y Fr(\024)22 b Fq(\027)2119 4957 y Fo(")2238 4945 y Fw(for)27 b(an)n(y)g Fq(u)22 b Fr(2)i Fq(B)2738 4915 y Fo(")2773 4945 y Fq(:)412 b Fw(\(2.4\))1905 5255 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 639 523 a Fw(Relations)24 b(\(2.3\))g(and)g (\(2.4\))g(with)g Fq(u)f Fw(=)g Fq(u)1942 535 y Fs(0)2003 523 y Fw(and)h Fq(")e Fr(!)h Fw(0)h(imply)h(assertion)d(\(i\).)37 b(Indeed,)515 623 y(for)27 b(an)n(y)g(measurable)f(set)h Fq(Q)c Fr(\032)g Fq(F)39 b Fw(with)29 b(zero)d(Leb)r(esgue)h(measure,)g (w)n(e)g(ha)n(v)n(e)960 799 y Fq(f)1001 811 y Fn(\003)1039 799 y Fw(\()p Fq(u;)14 b(\026)p Fw(\)\()p Fq(Q)p Fw(\))23 b(=)f(\()p Fq(f)1551 811 y Fn(\003)1590 799 y Fw(\()p Fq(u;)14 b(\026)p Fw(\))k Fr(\000)g Fq(f)1931 811 y Fn(\003)1969 799 y Fw(\()p Fq(u;)c(\026)2136 811 y Fo(")2171 799 y Fw(\)\)\()p Fq(Q)p Fw(\))19 b(+)f Fq(f)2508 811 y Fn(\003)2546 799 y Fw(\()p Fq(u;)c(\026)2713 765 y Fo(")2748 799 y Fw(\)\()p Fq(Q)p Fw(\))p Fq(;)515 975 y Fw(so)27 b Fq(f)658 987 y Fn(\003)695 975 y Fw(\()p Fq(u;)14 b(\026)p Fw(\)\()p Fq(Q)p Fw(\))24 b Fr(\024)e Fq(\027)1176 987 y Fo(")1240 975 y Fw(for)27 b(eac)n(h)f Fq(")p Fw(.)37 b(Hence,)28 b Fq(f)1963 987 y Fn(\003)2001 975 y Fw(\()p Fq(u;)14 b(\026)p Fw(\)\()p Fq(Q)p Fw(\))23 b(=)g(0.)639 1075 y(T)-7 b(o)31 b(pro)n(v)n(e)f(\(ii\),)i(w)n(e)f(\014x)g(an)n(y)g Fq(\015)i(>)c Fw(0)h(and)h(c)n(ho)r(ose)f Fq(")f(>)g Fw(0)h(suc)n(h)h(that)g Fq(\027)2889 1087 y Fo(")2954 1075 y Fq(<)3057 1042 y Fs(1)p 3057 1056 34 4 v 3057 1103 a(3)3114 1075 y Fq(\015)5 b Fw(.)48 b(Due)515 1174 y(to)27 b(\(2.4\),)h(for)f Fq(u)22 b Fr(2)i Fq(B)1181 1144 y Fo(")1244 1174 y Fw(w)n(e)j(ha)n(v)n(e)752 1350 y Fr(k)p Fq(f)835 1362 y Fn(\003)872 1350 y Fw(\()p Fq(u;)14 b(\026)p Fw(\))k Fr(\000)g Fq(f)1213 1362 y Fn(\003)1251 1350 y Fw(\()p Fq(u)1331 1362 y Fs(0)1368 1350 y Fq(;)c(\026)p Fw(\))p Fr(k)1529 1362 y Fs(v)l(ar)1647 1350 y Fr(\024)22 b Fw(2)p Fq(\027)1817 1362 y Fo(")1871 1350 y Fw(+)c Fr(k)p Fq(f)2037 1362 y Fn(\003)2074 1350 y Fw(\()p Fq(u;)c(\026)2241 1362 y Fo(")2276 1350 y Fw(\))19 b Fr(\000)f Fq(f)2451 1362 y Fn(\003)2489 1350 y Fw(\()p Fq(u)2569 1362 y Fs(0)2606 1350 y Fq(;)c(\026)2693 1362 y Fo(")2728 1350 y Fw(\))p Fr(k)2802 1362 y Fs(v)l(ar)1647 1526 y Fr(\024)22 b Fw(2)p Fq(\027)1817 1538 y Fo(")1871 1526 y Fw(+)1964 1470 y(1)p 1964 1507 42 4 v 1964 1583 a(2)2043 1413 y Fl(Z)2140 1526 y Fr(j)p Fq(g)2203 1538 y Fo(")2238 1526 y Fw(\()p Fq(u;)14 b(y)s Fw(\))k Fr(\000)g Fq(g)2572 1538 y Fo(")2607 1526 y Fw(\()p Fq(u)2687 1538 y Fs(0)2724 1526 y Fq(;)c(y)s Fw(\))p Fr(j)g Fq(dy)26 b Fr(\024)d Fq(\015)5 b(;)515 1740 y Fw(if)25 b Fq(u)f Fw(is)g(su\016cien)n(tly)h(close)e(to)h Fq(u)1494 1752 y Fs(0)1531 1740 y Fw(.)36 b(Since)25 b Fq(u)1852 1752 y Fs(0)1913 1740 y Fw(is)f(an)h(arbitrary)d(p)r(oin)n (t,)j(what)g(has)f(b)r(een)h(said)515 1840 y(implies)j(\(ii\).)515 1970 y Fj(R)l(emark)42 b Fw(2.3)p Fj(.)h Fw(Analysing)31 b(the)h(pro)r(of)f(giv)n(en)g(ab)r(o)n(v)n(e,)g(one)g(easily)g(sees)g (that,)i(instead)e(of)515 2070 y(assuming)g(the)h(existence)f(of)h(in)n (terior)e(p)r(oin)n(ts)i(for)f(the)h(set)g Fq(f)9 b Fw(\()p Fq(u;)14 b(X)7 b Fw(\),)31 b(w)n(e)h(could)f(require)515 2169 y(that)g(the)h(function)f Fq(f)9 b Fw(\()p Fq(u;)14 b Fr(\001)p Fw(\))31 b(should)g(ha)n(v)n(e)f(at)h(least)f(one)h (regular)e(p)r(oin)n(t)j(for)e(an)n(y)h Fq(u)d Fr(2)h Fq(H)515 2269 y Fw(\(cf.)37 b(the)28 b(b)r(eginning)g(of)f(the)h(pro)r (of)6 b(\).)515 2500 y Fp(2.3)112 b(Pro)s(of)37 b(of)h(Theorem)g(2.2) 515 2653 y Fj(Step)24 b(1)p Fw(.)35 b(Let)22 b(us)g(\014x)f(an)n(y)g Fq(u)1348 2665 y Fs(0)1408 2653 y Fr(2)i Fq(H)29 b Fw(and)21 b(sho)n(w)g(that)h(there)f(is)h(\014nite-dimensional)f(subspace)515 2753 y Fq(X)584 2765 y Fs(1)647 2753 y Fr(\032)26 b Fq(X)36 b Fw(spanned)30 b(b)n(y)f(some)g(v)n(ectors)f(of)i(the)g(basis)f Fr(f)p Fq(g)2310 2765 y Fo(j)2344 2753 y Fr(g)g Fw(suc)n(h)g(that)h (dim)14 b Fq(X)3007 2765 y Fs(1)3071 2753 y Fw(=)26 b(dim)14 b Fq(F)515 2853 y Fw(and)39 b(\(2.1\))h(holds,)j(where)c Fq(m)p Fw(\()p Fq(x)p Fw(\))i(denotes)e(the)h(determinan)n(t)g(of)g (the)g(matrix)f(for)h(the)515 2952 y(restriction)29 b(of)g Fq(D)1080 2964 y Fo(x)1122 2952 y Fq(f)9 b Fw(\()p Fq(u)1252 2964 y Fs(0)1289 2952 y Fq(;)14 b(x)p Fw(\))30 b(to)g Fq(X)1608 2964 y Fs(1)1645 2952 y Fw(.)43 b(Indeed,)31 b(b)n(y)f(the)g(h)n(yp)r(othesis,)g(there)f(is)h(a)g(ball)f Fq(B)3303 2964 y Fo(u)3342 2972 y Fg(0)515 3052 y Fw(in)c(a)f (\014nite-dimensional)g(subspace)g Fq(X)1754 3064 y Fo(u)1793 3072 y Fg(0)1853 3052 y Fr(\032)e Fq(X)31 b Fw(suc)n(h)25 b(that)g(the)g(in)n(terior)e(of)i Fq(f)9 b Fw(\()p Fq(u)3054 3064 y Fs(0)3090 3052 y Fq(;)14 b(B)3190 3064 y Fo(u)3229 3072 y Fg(0)3266 3052 y Fw(\))25 b(is)515 3152 y(non-empt)n(y)-7 b(.)34 b(Since)23 b Fr(f)p Fq(g)1247 3164 y Fo(j)1281 3152 y Fr(g)f Fw(is)g(a)g(basis)g(in)h Fq(X)7 b Fw(,)23 b(for)f(an)n(y)f Fq(\016)26 b(>)d Fw(0)f(there)g(is)g(a)g (\014nite-dimensional)515 3251 y(subspace)27 b Fq(Y)41 b Fr(\032)23 b Fq(X)34 b Fw(spanned)27 b(b)n(y)h(some)f(v)n(ectors)f (of)h Fr(f)p Fq(g)2247 3263 y Fo(j)2281 3251 y Fr(g)g Fw(suc)n(h)h(that)1312 3427 y Fr(k)p Fk(P)1407 3439 y Fo(Y)1464 3427 y Fq(y)21 b Fr(\000)d Fq(y)s Fr(k)1695 3439 y Fo(X)1780 3427 y Fr(\024)23 b Fq(\016)86 b Fw(for)27 b(an)n(y)g Fq(y)f Fr(2)d Fq(B)2483 3439 y Fo(u)2522 3447 y Fg(0)2559 3427 y Fq(;)626 b Fw(\(2.5\))515 3604 y(where)21 b Fk(P)802 3616 y Fo(Y)883 3604 y Fw(:)i Fq(X)29 b Fr(!)23 b Fq(X)29 b Fw(is)22 b(the)g(orthogonal)e(pro)5 b(jection)21 b(in)h Fq(X)28 b Fw(on)n(to)21 b(the)i(subspace)e Fq(Y)d Fw(.)36 b(No)n(w)515 3703 y(note)24 b(that)h Fq(f)9 b Fw(\()p Fq(u)1003 3715 y Fs(0)1040 3703 y Fq(;)14 b Fr(\001)p Fw(\))25 b(is)g(con)n(tin)n(uous)f(and)g Fq(B)1872 3715 y Fo(u)1911 3723 y Fg(0)1973 3703 y Fw(is)g(compact.)36 b(Therefore)23 b(for)i(an)n(y)f Fq(")e(>)h Fw(0)h(w)n(e)515 3803 y(can)j(\014nd)h Fq(\016)e(>)d Fw(0)k(suc)n(h)g(that)947 3979 y Fr(k)p Fq(f)9 b Fw(\()p Fq(u)1119 3991 y Fs(0)1156 3979 y Fq(;)14 b(z)t Fw(\))k Fr(\000)g Fq(f)9 b Fw(\()p Fq(u)1499 3991 y Fs(0)1535 3979 y Fq(;)14 b(y)s Fw(\))p Fr(k)1690 3991 y Fo(F)1768 3979 y Fr(\024)22 b Fq(")83 b Fw(for)27 b Fq(y)f Fr(2)d Fq(B)2312 3991 y Fo(u)2351 3999 y Fg(0)2388 3979 y Fw(,)28 b Fr(k)p Fq(z)21 b Fr(\000)d Fq(y)s Fr(k)2710 3991 y Fo(X)2796 3979 y Fr(\024)k Fq(\016)s(:)262 b Fw(\(2.6\))515 4156 y(Com)n(bining)21 b(\(2.5\))g(and)g(\(2.6\))o(,)i (w)n(e)d(see)h(that)g(for)f(an)n(y)g Fq(")j(>)g Fw(0)d(there)h(is)g(a)f (\014nite-dimensional)515 4255 y(subspace)27 b Fq(Y)41 b Fr(\032)23 b Fq(X)34 b Fw(spanned)27 b(b)n(y)h(some)f(v)n(ectors)f (of)h Fr(f)p Fq(g)2247 4267 y Fo(j)2281 4255 y Fr(g)g Fw(suc)n(h)h(that)1159 4431 y Fr(k)p Fq(f)9 b Fw(\()p Fq(u)1331 4443 y Fs(0)1367 4431 y Fq(;)14 b Fk(P)1457 4443 y Fo(Y)1515 4431 y Fq(y)s Fw(\))k Fr(\000)g Fq(f)9 b Fw(\()p Fq(u)1822 4443 y Fs(0)1859 4431 y Fq(;)14 b(y)s Fw(\))p Fr(k)2014 4443 y Fo(F)2091 4431 y Fr(\024)23 b Fq(")83 b Fw(for)27 b Fq(y)e Fr(2)f Fq(B)2636 4443 y Fo(u)2675 4451 y Fg(0)2712 4431 y Fq(:)515 4608 y Fw(Cho)r(osing)g Fq(")f(>)g Fw(0)i(su\016cien)n(tly)g(small)g(and)h(applying)e(Prop)r (osition)g(4.1)h(of)g(the)h(App)r(endix)515 4707 y(\(see)j(Section)g (4\),)g(w)n(e)g(conclude)f(that)i(the)f(in)n(terior)f(of)h(the)g(set)g Fq(f)9 b Fw(\()p Fq(u)2689 4719 y Fs(0)2726 4707 y Fq(;)14 b(Y)k Fw(\))30 b(is)e(non-empt)n(y)-7 b(.)515 4807 y(Since)33 b(dim)14 b Fq(Y)50 b(<)31 b Fr(1)p Fw(,)j(Sard's)e(theorem)g(implies)h (that)g(the)g(function)g Fq(f)9 b Fw(\()p Fq(u)2886 4819 y Fs(0)2923 4807 y Fq(;)14 b Fr(\001)p Fw(\))32 b(:)f Fq(Y)51 b Fr(!)31 b Fq(F)515 4907 y Fw(has)25 b(at)g(least)g(one)h (regular)e(p)r(oin)n(t)h Fq(x)1644 4919 y Fs(0)1705 4907 y Fr(2)e Fq(Y)c Fw(.)36 b(Let)26 b(us)g(denote)f(b)n(y)h Fq(X)2609 4919 y Fs(1)2671 4907 y Fw(a)f(subspace)g(spanned)515 5006 y(b)n(y)f(some)h(v)n(ectors)e(of)i(the)h(basis)e Fr(f)p Fq(g)1628 5018 y Fo(j)1662 5006 y Fr(g)h Fw(suc)n(h)f(that)i (the)f(restriction)f(of)h Fq(D)2787 5018 y Fo(x)2828 5006 y Fq(f)9 b Fw(\()p Fq(u)2958 5018 y Fs(0)2995 5006 y Fq(;)14 b(x)3079 5018 y Fs(0)3117 5006 y Fw(\))25 b(to)g Fq(X)3342 5018 y Fs(1)1905 5255 y Fw(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 515 523 a Fw(is)25 b(an)g(isomorphism)e(from)i Fq(X)1454 535 y Fs(1)1516 523 y Fw(on)n(to)g Fq(F)12 b Fw(.)36 b(Then)25 b Fq(m)p Fw(\()p Fq(x)2191 535 y Fs(0)2229 523 y Fw(\))e Fr(6)p Fw(=)g(0.)36 b(Since)25 b Fq(m)p Fw(\()p Fq(x)p Fw(\))h(is)f(an)g(analytic)515 623 y(function,)j(the)g(required)f(result)g(follo)n(ws)f(from)i (Theorem)f(1.6.)639 744 y Fj(Step)i(2)p Fw(.)37 b(W)-7 b(e)27 b(no)n(w)f(rep)r(eat)g(the)h(argumen)n(t)f(used)g(in)h(the)g (pro)r(of)f(of)g(Theorem)g(2.1.)36 b(Let)515 844 y(us)c(represen)n(t)f Fq(X)39 b Fw(as)32 b(the)h(direct)f(pro)r(duct)g Fq(X)38 b Fw(=)30 b Fq(X)2179 856 y Fs(1)2238 844 y Fr(\002)21 b Fq(X)2393 856 y Fs(2)2430 844 y Fw(,)34 b(where)e Fq(X)2801 856 y Fs(1)2870 844 y Fw(is)h(constructed)515 944 y(in)39 b(Step)h(1)f(and)g Fq(X)1147 956 y Fs(2)1223 944 y Fw(denotes)g(the)h (orthogonal)d(complemen)n(t)i(of)g Fq(X)2767 956 y Fs(1)2843 944 y Fw(in)h Fq(X)7 b Fw(.)71 b(Denote)515 1043 y(b)n(y)23 b Fq(\025)674 1055 y Fs(1)735 1043 y Fw(and)g Fq(\025)940 1055 y Fs(2)1002 1043 y Fw(the)h(pro)5 b(jections)22 b(of)h Fq(\026)h Fw(to)f(the)h(subspaces)e Fq(X)2408 1055 y Fs(1)2469 1043 y Fw(and)h Fq(X)2695 1055 y Fs(2)2732 1043 y Fw(,)i(resp)r(ectiv)n(ely)-7 b(,)23 b(and)515 1143 y(b)n(y)29 b Fq(\032)p Fw(\()p Fq(x)754 1155 y Fs(1)792 1143 y Fw(\))i(the)f(densit)n(y)g(of)g Fq(\025)1434 1155 y Fs(1)1502 1143 y Fw(with)g(resp)r(ect)g(to)g Fq(`)2119 1155 y Fo(X)2173 1163 y Fg(1)2210 1143 y Fw(.)44 b(F)-7 b(or)29 b(an)n(y)h Fq(")c(>)h Fw(0,)j(let)h(us)f(c)n(ho)r(ose)e(a)515 1243 y(compact)f(set)g Fq(X)1052 1212 y Fo(")1110 1243 y Fr(\032)c Fq(X)34 b Fw(suc)n(h)27 b(that)826 1411 y Fr(k)p Fq(x)p Fr(k)957 1423 y Fo(X)1042 1411 y Fr(\024)c Fq(")1169 1381 y Fn(\000)p Fs(1)1258 1411 y Fw(,)k Fr(j)p Fq(m)p Fw(\()p Fq(x)p Fw(\))p Fr(j)e(\025)d Fq(")28 b Fw(for)f Fq(x)c Fr(2)h Fq(X)2069 1381 y Fo(")2103 1411 y Fq(;)180 b(\027)2347 1423 y Fo(")2406 1411 y Fw(:=)23 b Fq(\026)p Fw(\()p Fq(X)i Fr(n)18 b Fq(X)2829 1377 y Fo(")2863 1411 y Fw(\))24 b Fr(\024)e Fq(":)515 1580 y Fw(As)d(in)h(the)f(pro)r(of)g(of)g(Theorem)f(2.1,)i(w)n(e)f(can)g (\014nd)h(a)e(constan)n(t)h Fq(r)2479 1592 y Fo(")2538 1580 y Fq(>)k Fw(0)18 b(going)g(to)h(zero)g(with)g Fq(")515 1679 y Fw(and)30 b(a)g(\014nite)h(co)n(v)n(er)d Fr(f)p Fq(V)1292 1649 y Fo(j)1354 1679 y Fw(=)f Fq(V)1513 1639 y Fo(j)1494 1701 y Fs(1)1569 1679 y Fr(\002)19 b Fq(V)1720 1639 y Fo(j)1701 1701 y Fs(2)1755 1679 y Fr(g)30 b Fw(of)g(the)h (compact)f(set)g Fq(X)2613 1649 y Fo(")2679 1679 y Fw(suc)n(h)g(that)g Fq(V)3118 1639 y Fo(j)3099 1701 y Fs(1)3181 1679 y Fr(\032)d Fq(X)3342 1691 y Fs(1)515 1788 y Fw(and)35 b Fq(V)751 1748 y Fo(j)732 1810 y Fs(2)822 1788 y Fr(\032)g Fq(X)991 1800 y Fs(2)1063 1788 y Fw(are)g(balls,)h(and)f(for)g(an)n(y)g Fq(x)1952 1800 y Fs(2)2025 1788 y Fr(2)h Fq(V)2164 1800 y Fs(2)2237 1788 y Fw(and)g Fq(u)f Fr(2)h Fq(B)2648 1758 y Fo(")2720 1788 y Fw(=)f Fq(B)2883 1800 y Fo(X)2946 1788 y Fw(\()p Fq(u)3026 1800 y Fs(0)3063 1788 y Fq(;)14 b(r)3137 1800 y Fo(")3173 1788 y Fw(\))p Fq(;)36 b Fw(the)515 1888 y(map)30 b Fq(f)9 b Fw(\()p Fq(u;)14 b Fr(\001)p Fq(;)g(x)976 1900 y Fs(2)1013 1888 y Fw(\))31 b(is)g(a)f (di\013eomorphism)g(of)h(the)g(domain)f Fq(V)2412 1900 y Fs(1)2481 1888 y Fw(on)n(to)g(its)g(image)g Fq(W)12 b Fw(\()p Fq(u;)i(x)3286 1900 y Fs(2)3324 1888 y Fw(\).)515 1987 y(W)-7 b(e)27 b(denote)g(b)n(y)f Fq(x)1085 1999 y Fs(1)1146 1987 y Fw(=)d Fq(g)s Fw(\()p Fq(u;)14 b Fr(\001)p Fq(;)g(x)1501 1999 y Fs(2)1538 1987 y Fw(\))27 b(the)g(in)n(v)n(erse)e (function)j(of)f Fq(f)9 b Fw(\()p Fq(u;)14 b Fr(\001)p Fq(;)g(x)2706 1999 y Fs(2)2742 1987 y Fw(\).)37 b(Let)27 b Fr(f)p Fq(')3078 1999 y Fo(j)3113 1987 y Fw(\()p Fq(x)p Fw(\))p Fr(g)h Fw(b)r(e)515 2087 y(a)j(con)n(tin)n(uous)f(partition)h (of)h(unit)n(y)f(on)h Fq(X)1872 2057 y Fo(")1938 2087 y Fw(sub)r(ordinate)f(to)g Fr(f)p Fq(V)2611 2057 y Fo(j)2646 2087 y Fr(g)g Fw(and)g(let)h Fq(\026)3058 2099 y Fo(j)3122 2087 y Fw(=)e Fq(')3271 2099 y Fo(j)3306 2087 y Fq(\026)p Fw(.)515 2186 y(Then)e(w)n(e)f(ha)n(v)n(e)f(\(cf.)i(\(2.2\)\))645 2355 y Fq(\026)695 2367 y Fo(j)753 2355 y Fw(=)22 b Fq(')894 2367 y Fo(j)930 2355 y Fw(\()p Fq(x)p Fw(\))p Fq(\032)p Fw(\()p Fq(x)1163 2367 y Fs(1)1202 2355 y Fw(\))14 b Fq(dx)1338 2367 y Fs(1)1389 2355 y Fq(\025)p Fw(\()p Fq(dx)1559 2367 y Fs(2)1598 2355 y Fw(\))23 b(=)g Fq(')1795 2367 y Fo(j)1830 2355 y Fw(\()p Fq(x)1909 2367 y Fs(1)1947 2355 y Fq(;)14 b(x)2031 2367 y Fs(2)2069 2355 y Fw(\))p Fq(\032)p Fw(\()p Fq(x)2223 2367 y Fs(1)2261 2355 y Fw(\))p Fr(j)p Fq(m)p Fw(\()p Fq(u;)g(x)2553 2367 y Fs(1)2591 2355 y Fq(;)g(x)2675 2367 y Fs(2)2712 2355 y Fw(\))p Fr(j)2767 2321 y Fn(\000)p Fs(1)2871 2355 y Fq(dy)i(\025)p Fw(\()p Fq(dx)3141 2367 y Fs(2)3180 2355 y Fw(\))e Fq(;)515 2523 y Fw(where)33 b Fq(m)p Fw(\()p Fq(u;)14 b(x)p Fw(\))35 b(denotes)e(the)i(determinan)n(t)f(of)g(the)g(restriction)f(of)h Fq(D)2820 2535 y Fo(x)2862 2523 y Fq(f)9 b Fw(\()p Fq(u;)14 b(x)p Fw(\))34 b(to)g Fq(X)3319 2535 y Fs(1)3356 2523 y Fw(,)515 2623 y(and)24 b Fq(x)720 2635 y Fs(1)780 2623 y Fw(=)f Fq(g)s Fw(\()p Fq(u;)14 b(y)s(;)g(x)1156 2635 y Fs(2)1193 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4972 y Fs(2)2890 5006 y Fw(\))19 b Fr(\\)g Fq(H)r(;)1905 5255 y Fw(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 515 523 a Fw(endo)n(w)n(ed)40 b(with)h(natural)f(norms.)75 b(Here)41 b Fq(H)1978 493 y Fo(s)2013 523 y Fw(\()p Ft(T)2100 493 y Fs(2)2138 523 y Fq(;)14 b Ft(R)2235 493 y Fs(2)2272 523 y Fw(\))41 b(denotes)g(the)g(space)f(of)g(v)n(ector)515 623 y(functions)22 b(\()p Fq(u)947 635 y Fs(1)984 623 y Fq(;)14 b(u)1069 635 y Fs(2)1106 623 y Fw(\))22 b(whose)f(comp)r(onen)n(ts)g(b)r(elong)g (to)h(the)g(Sob)r(olev)f(space)g(of)h(order)e Fq(s)p Fw(.)35 b(Let)515 722 y(\005)26 b(:)f Fq(L)708 692 y Fs(2)745 722 y Fw(\()p Ft(T)832 692 y Fs(2)870 722 y Fq(;)14 b Ft(R)967 692 y Fs(2)1004 722 y Fw(\))26 b Fr(!)g Fq(H)36 b Fw(b)r(e)29 b(the)h(orthogonal)d(pro)5 b(jection)28 b(in)i Fq(L)2504 692 y Fs(2)2540 722 y Fw(\()p Ft(T)2627 692 y Fs(2)2665 722 y Fq(;)14 b Ft(R)2762 692 y Fs(2)2799 722 y Fw(\))30 b(on)n(to)e Fq(H)7 b Fw(.)42 b(After)515 822 y(applying)35 b(the)i(pro)5 b(jection)35 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2509 y Fq(;)14 b Fw(0)p Fq(;)g Fe(\021)1904 2530 y Fo(m)1967 2509 y Fw(\))23 b(=)g Fk(P)2163 2521 y Fo(F)2218 2509 y Fr(R)2288 2521 y Fs(1)2326 2509 y Fw(\()p Fq(g)s Fw(\))p Fq(:)711 b Fw(\(3.10\))515 2692 y(Com)n(bining)27 b(\(3.9\))f(and)g(\(3.10\),)g(w)n(e)g(conclude)h (that)f(there)h(is)f(a)g(\014nite-dimensional)g(sub-)515 2792 y(space)h Fq(Y)41 b Fr(\032)23 b Fq(Y)962 2804 y Fo(m)1053 2792 y Fw(suc)n(h)k(that)1512 2974 y Fq(f)1553 2986 y Fo(m)1615 2974 y Fw(\()p Fq(m)1720 2940 y Fn(\000)p Fs(1)1810 2974 y Fq(;)14 b Fw(0)p Fq(;)g(Y)k Fw(\))23 b Fr(\033)g Fq(B)2198 2986 y Fo(F)2253 2974 y Fw(\(1\))p Fq(:)515 3157 y Fw(Sard's)j(theorem)i(no)n(w)f(implies)g(that)h(the)g (deriv)-5 b(ativ)n(e)1343 3339 y(\()p Fq(D)1444 3351 y Fd(\021)1486 3367 y Fm(m)1546 3339 y Fq(f)1587 3351 y Fo(m)1649 3339 y Fw(\)\()p Fq(m)1786 3305 y Fn(\000)p Fs(1)1876 3339 y Fq(;)14 b Fw(0)p Fq(;)g Fe(\021)2044 3360 y Fo(m)2107 3339 y Fw(\))24 b(:)f Ff(X)2294 3351 y Fo(m)2380 3339 y Fr(!)g Fq(F)515 3522 y 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Fw(denotes)g(the)h(image)f(of)g(a)h(linear)e(op)r (erator)g Fq(A)p Fw(.)46 b(In)31 b(view)f(of)37 b(\(3.11\))30 b(and)515 623 y(the)e(con)n(tin)n(uit)n(y)f(of)g Fq(D)1213 635 y Fd(\021)1255 651 y Fm(m)1315 623 y Fq(f)1356 635 y Fo(m)1418 623 y Fw(,)h(w)n(e)f(can)g(\014nd)i Fq(r)c(>)e Fw(0)k(suc)n(h)g(that)1040 814 y(Im)p Fr(f)p Fw(\()p Fq(D)1282 826 y Fd(\021)1324 842 y Fm(m)1383 814 y Fq(f)1424 826 y Fo(m)1487 814 y Fw(\)\()p Fq(T)7 b(;)14 b(v)s(;)g Fe(\021)1777 779 y Fs(0)1777 834 y Fo(m)1840 814 y Fw(\))p Fr(g)23 b Fw(=)f Fq(F)95 b Fw(for)27 b(an)n(y)g Fq(v)f Fr(2)e Fq(B)2664 826 y Fo(H)2727 814 y Fw(\()p Fq(r)r Fw(\))q Fq(;)313 b Fw(\(3.13\))515 996 y(F)-7 b(urthermore,)19 b(the)g(dissipation)f(prop)r(ert)n(y)g(of)g(the)h(homogeneous)e(Na)n (vier{Stok)n(es)f(system)515 1096 y(implies)28 b(that)1267 1196 y Fq(u)p Fw(\()p Fq(l)r(T)12 b Fw(\))21 b Fr(2)j Fq(B)1630 1208 y Fo(H)1692 1196 y Fw(\()p Fq(r)r Fw(\))85 b(for)27 b(an)n(y)g Fq(u)2212 1208 y Fs(0)2272 1196 y Fr(2)c Fq(B)2413 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b(of)g(the)h(random)e(parameter)g(the)i(Cauc)n(h)n(y)e(problem)h(for)g (\(3.1\))o(,)j(\(3.18\))o(,)f(\(3.19\))515 3572 y(has)g(a)g(unique)h (solution)f(in)h(the)g(space)f Fr(X)12 b Fw(,)23 b(and)f(w)n(e)f (denote)h(b)n(y)f Fq(S)2547 3584 y Fo(t)2599 3572 y Fw(:)j Fq(H)29 b Fr(!)23 b Fq(H)29 b Fw(its)22 b(resolving)515 3672 y(\(random\))27 b(op)r(erator.)515 3838 y Fx(Theorem)33 b(3.7.)41 b Fj(Ther)l(e)32 b(is)g(an)f(inte)l(ger)1824 3808 y Fs(3)1892 3838 y Fq(N)k Fr(\025)25 b Fw(1)31 b Fj(not)g(dep)l(ending)i(on)e Fq(\027)g(>)26 b Fw(0)31 b Fj(such)g(that)515 3938 y(if)j Fq(h)29 b Fr(2)h Fq(L)818 3907 y Fs(2)818 3961 y(lo)r(c)905 3938 y Fw(\()p Ft(R)997 3950 y Fs(+)1052 3938 y Fq(;)14 b(H)7 b Fw(\))33 b Fj(is)h(a)g(given)f (function)h(and)f(the)h(image)g(of)g Fq(Q)f Fj(c)l(ontains)h Fq(H)3102 3950 y Fo(N)3165 3938 y Fj(,)g(then)515 4037 y(for)c(any)g Fq(u)854 4049 y Fs(0)914 4037 y Fr(2)24 b Fq(H)36 b Fj(and)30 b Fq(t)23 b(>)g Fw(0)29 b Fj(assertions)i Fw(\(i\))f Fj(and)g Fw(\(ii\))h Fj(of)f(The)l(or)l(em)h(3.5)g(hold,)h (and)1666 4220 y Fw(supp)14 b Fq(\025)1899 4232 y Fo(u)1943 4220 y Fw(\()p Fq(t)p Fw(\))24 b(=)f Fq(F)r(:)940 b Fw(\(3.20\))515 4402 y Fj(Pr)l(o)l(of.)43 b(Step)28 b(1)p Fw(.)37 b(Let)27 b(us)f(\014x)g Fq(T)34 b(>)23 b Fw(0)j(and)g(study)g(our)g(problem)g (on)g(the)g(in)n(terv)-5 b(al)26 b Fq(J)3124 4414 y Fo(T)3176 4402 y Fw(.)37 b(The)515 4502 y(solution)26 b(of)34 b(\(3.1\),)27 b(\(3.3\),)g(\(3.18\))o(,)g(\(3.19\))g(can)f(b)r(e)i(represen)n(ted)e (in)h(the)h(form)f Fq(u)22 b Fw(=)h Fq(v)d Fw(+)d Fq(\020)6 b Fw(,)515 4602 y(where)27 b Fq(v)f Fr(2)e(X)959 4614 y Fo(T)1039 4602 y Fw(is)j(a)g(solution)g(of)h(the)g(problem)569 4784 y(_)-36 b Fq(v)11 b Fw(+)c Fq(\027)e(Lv)10 b Fw(+)d Fq(B)t Fw(\()p Fq(v)s(;)14 b(v)s Fw(\))7 b(+)g Fq(B)t Fw(\()p Fq(v)s(;)14 b(\020)6 b Fw(\))h(+)g Fq(B)t Fw(\()p Fq(\020)f(;)14 b(v)s Fw(\))28 b(=)23 b Fq(h)p Fw(\()p Fq(t)p Fw(\))7 b Fr(\000)g Fq(\027)e(L\020)14 b Fr(\000)7 b Fq(B)t Fw(\()p Fq(\020)f(;)14 b(\020)6 b Fw(\))p Fq(;)98 b(v)s Fw(\(0\))24 b(=)e Fq(u)3065 4796 y Fs(0)3102 4784 y Fq(:)42 b Fw(\(3.21\))p 515 4858 1146 4 v 607 4912 a Fv(3)642 4935 y FB(A)23 b(more)h(precise)f(description)i(of)e Fi(N)31 b FB(can)24 b(b)r(e)g(found)g(in)f(Remark)h(3.2)1905 5255 y Fw(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 515 523 a Fw(Let)28 b Fr(F)j Fw(:)23 b Fq(H)j Fr(\002)18 b Fq(L)1036 493 y Fs(2)1072 523 y Fw(\()p Fq(J)1150 535 y Fo(T)1203 523 y Fq(;)c(U)9 b Fw(\))23 b Fr(!)h(X)1527 535 y Fo(T)1607 523 y Fw(b)r(e)k(the)g(op)r (erator)e(that)i(tak)n(es)f(\()p Fq(u)2671 535 y Fs(0)2708 523 y Fq(;)14 b(\020)6 b Fw(\))29 b(to)e(the)i(solution)515 623 y Fq(v)j Fr(2)e(X)731 635 y Fo(T)815 623 y Fw(of)37 b(\(3.21\))o(.)48 b(Using)31 b(standard)g(metho)r(ds)g(of)g(the)h (theory)f(of)g(2D)g(NS)h(equations,)515 722 y(one)f(can)h(sho)n(w)f (that)i Fr(F)40 b Fw(is)32 b(w)n(ell)g(de\014ned)g(and)g(analytic)f (\(cf.)i(Prop)r(osition)e(4.2\).)49 b(Th)n(us,)515 822 y(denoting)27 b(b)n(y)g Fr(F)1031 834 y Fo(t)1060 822 y Fw(\()p Fq(u)1140 834 y Fs(0)1177 822 y Fq(;)14 b(\020)6 b Fw(\))29 b(the)f(restriction)e(of)i Fr(F)8 b Fw(\()p Fq(u)2100 834 y Fs(0)2137 822 y Fq(;)14 b(\020)6 b Fw(\))28 b(to)f(the)h(time)h Fq(t)p Fw(,)e(w)n(e)h(can)f(write)1152 979 y Fk(P)1205 991 y Fo(F)1261 979 y Fq(u)p Fw(\()p Fq(t)p Fw(\))c(=)f Fk(P)1566 991 y Fo(F)1622 979 y Fq(\020)6 b Fw(\()p Fq(t)p Fw(\))19 b(+)f Fk(P)1913 991 y Fo(F)1968 979 y Fr(F)2028 991 y Fo(t)2057 979 y Fw(\()p Fq(u)2137 991 y Fs(0)2174 979 y Fq(;)c(\020)6 b Fw(\))24 b(=:)f Fq(f)2461 991 y Fo(t)2490 979 y Fw(\()p Fq(u)2570 991 y Fs(0)2607 979 y Fq(;)14 b(\020)6 b Fw(\))p Fq(:)515 1137 y Fw(It)43 b(follo)n(ws)g(that)g Fq(\025)1152 1149 y Fo(u)1191 1157 y Fg(0)1229 1137 y Fw(\()p Fq(t)p Fw(\))50 b(=)e Fr(D)r Fw(\()p Fk(P)1637 1149 y Fo(F)1693 1137 y Fq(u)p Fw(\()p Fq(t)p Fw(\)\))c(is)f(the)h(image)f(of)g Fq(\026)2585 1149 y Fo(T)2681 1137 y Fw(under)g(the)h(analytic)515 1236 y(map)27 b Fq(f)740 1248 y Fo(t)769 1236 y Fw(\()p Fq(u)849 1248 y Fs(0)886 1236 y Fq(;)14 b Fr(\001)p Fw(\),)28 b(where)f Fq(\026)1319 1248 y Fo(T)1399 1236 y Fw(is)g(the)h (distribution)g(of)f(the)h(restriction)f(of)g Fq(\020)34 b Fw(to)28 b Fq(J)3025 1248 y Fo(T)3077 1236 y Fw(.)37 b(Hence,)515 1336 y(assertions)32 b(\(i\))k(and)e(\(ii\))h(of)f (Theorem)g(3.5)g(will)g(b)r(e)h(established)f(if)h(w)n(e)f(sho)n(w)g (that)h(the)515 1435 y(h)n(yp)r(othesis)27 b(of)g(Theorem)g(2.2)g(are)g (satis\014ed.)639 1555 y Fj(Step)k(2)p Fw(.)40 b(Let)29 b(us)f(regard)f Fq(\026)1497 1567 y Fo(T)1578 1555 y Fw(as)h(a)g(measure)g(on)g Fq(L)2249 1525 y Fs(2)2286 1555 y Fw(\()p Fq(J)2364 1567 y Fo(T)2416 1555 y Fq(;)14 b(U)2510 1567 y Fs(0)2547 1555 y Fw(\),)30 b(where)e Fq(U)2930 1567 y Fs(0)2995 1555 y Fw(denote)h(the)515 1655 y(closure)g(in)h Fq(U)39 b Fw(of)31 b(the)f(image)g(of)g Fq(Q)p Fw(.)45 b(In)30 b(this)h(case,)f Fq(\026)2230 1667 y Fo(T)2313 1655 y Fw(is)g(a)g(non-degenerate)e(Gaussian)515 1754 y(measure)f(that)i(satis\014es)f(prop)r(ert)n(y)f(\(P\))i(\(see)f (Examples)g(1.2)f(and)i(1.4\).)39 b(W)-7 b(e)28 b(claim)h(that)515 1854 y(for)j(an)n(y)h Fq(u)858 1866 y Fs(0)927 1854 y Fr(2)g Fq(H)7 b Fw(,)34 b Fq(t)e Fr(2)h Fq(J)1344 1866 y Fo(T)1418 1854 y Fr(n)22 b(f)p Fw(0)p Fr(g)p Fw(,)33 b(and)g Fq(r)i(>)d Fw(0)h(there)g(is)g(a)g(ball)g Fq(B)2687 1866 y Fo(u)2726 1874 y Fg(0)2795 1854 y Fw(=)f Fq(B)2955 1866 y Fo(u)2994 1874 y Fg(0)3031 1854 y Fw(\()p Fq(t;)14 b(r)r Fw(\))34 b(in)f(a)515 1953 y(\014nite-dimensional)27 b(subspace)g(of)g Fq(L)1682 1923 y Fs(2)1719 1953 y Fw(\()p Fq(J)1797 1965 y Fo(T)1850 1953 y Fq(;)14 b(H)1956 1965 y Fo(N)2019 1953 y Fw(\))28 b(suc)n(h)f(that)1571 2111 y Fq(f)1612 2123 y Fo(t)1641 2111 y Fw(\()p Fq(u)1721 2123 y Fs(0)1758 2111 y Fq(;)14 b(B)1858 2123 y Fo(u)1897 2131 y Fg(0)1934 2111 y Fw(\))23 b Fr(\033)g Fq(B)2140 2123 y Fo(F)2195 2111 y Fw(\()p Fq(r)r Fw(\))p Fq(:)846 b Fw(\(3.22\))515 2268 y(Indeed,)28 b(for)f(an)n(y)g(deterministic)g (function)i Fq(\030)e Fr(2)c Fq(C)2122 2238 y Fs(1)2160 2268 y Fw(\()p Fq(J)2238 2280 y Fo(T)2290 2268 y Fq(;)14 b(H)2396 2280 y Fo(N)2459 2268 y Fw(\),)28 b(w)n(e)f(can)h(write)1315 2434 y Fq(u)p Fw(\()p Fq(t)p Fw(\))23 b(=)g Fq(\030)t Fw(\()p Fq(t)p Fw(\))c(+)f Fr(F)1864 2446 y Fo(t)1893 2434 y Fw(\()p Fq(u)1973 2446 y Fs(0)2010 2434 y Fq(;)c(\030)t Fw(\))23 b(=)g Fr(R)2300 2446 y Fo(t)2329 2434 y Fw(\()p Fq(u)2409 2446 y Fs(0)2446 2434 y Fq(;)2492 2413 y Fw(^)2483 2434 y Fq(\030)t Fw(\))p Fq(;)589 b Fw(\(3.23\))515 2592 y(where)38 b Fr(R)836 2604 y Fo(t)905 2592 y Fw(is)h(the)g(resolving)f (op)r(erator)f(for)h(the)i(NS)f(system)g(\(see)g(Section)g(4.3\))f(and) 523 2678 y(^)515 2700 y Fq(\030)27 b Fw(=)c Fq(@)710 2712 y Fo(t)739 2700 y Fq(\030)t Fw(.)36 b(It)25 b(follo)n(ws)e(from)h (Prop)r(osition)f(4.5)g(\(cf.)i(Step)g(2)f(in)g(the)h(pro)r(of)f(of)g (Theorem)g(3.1\))515 2810 y(that)f(if)h Fq(N)32 b Fr(\025)22 b Fw(1)h(is)g(su\016cien)n(tly)g(large,)f(then)i(there)f(is)g(a)g(ball) 2436 2789 y Fl(b)2418 2810 y Fq(B)2481 2822 y Fo(u)2520 2830 y Fg(0)2580 2810 y Fw(in)h(a)e(\014nite-dimensional)515 2910 y(subspace)27 b(of)g Fq(C)6 b Fw(\()p Fq(J)1099 2922 y Fo(T)1152 2910 y Fq(;)14 b(H)1258 2922 y Fo(N)1321 2910 y Fw(\))28 b(suc)n(h)f(that)1502 3078 y Fk(P)1555 3090 y Fo(F)1611 3078 y Fr(R)1681 3090 y Fo(t)1710 3078 y Fw(\()p Fq(u)1790 3090 y Fs(0)1827 3078 y Fq(;)1882 3057 y Fl(b)1864 3078 y Fq(B)1927 3090 y Fo(u)1966 3098 y Fg(0)2003 3078 y Fw(\))c Fr(\033)g Fq(B)2209 3090 y Fo(F)2264 3078 y Fw(\()p Fq(r)r Fw(\))p Fq(:)777 b Fw(\(3.24\))515 3245 y(It)25 b(is)f(clear)g(that)h(the)g(image)f(of)1541 3224 y Fl(b)1524 3245 y Fq(B)1587 3257 y Fo(u)1626 3265 y Fg(0)1687 3245 y Fw(under)h(the)g(linear)f(op)r(erator)2628 3223 y(^)2620 3245 y Fq(\030)j Fr(7!)2789 3179 y Fl(R)2844 3199 y Fn(\001)2828 3275 y Fs(0)2890 3223 y Fw(^)2882 3245 y Fq(\030)t Fw(\()p Fq(s)p Fw(\))14 b Fq(ds)25 b Fw(is)g(con-)515 3345 y(tained)j(in)g(a)g(\014nite-dimensional)f(ball)h Fq(B)1830 3357 y Fo(u)1869 3365 y Fg(0)1906 3345 y Fw(.)38 b(Com)n(bining)28 b(this)g(with)h(\(3.23\))e(and)h(\(3.24\))o(,)515 3445 y(w)n(e)36 b(arriv)n(e)f(at)i(\(3.22\))o(.)65 b(Hence,)39 b(Theorem)d(2.2)g(is)h(applicable,)h(and)f(w)n(e)f(obtain)h(asser-)515 3544 y(tions)27 b(\(i\))h(and)g(\(ii\).)639 3664 y Fj(Step)40 b(3)p Fw(.)71 b(Relation)38 b(\(3.20\))g(is)g(a)g(standard)g (consequence)g(of)g(con)n(trollabilit)n(y)-7 b(,)40 b(and)515 3763 y(w)n(e)35 b(con\014ne)g(ourselv)n(es)e(to)j(outlining)f(its)g (pro)r(of.)60 b(Let)35 b Fq(y)k Fr(2)d Fq(F)48 b Fw(b)r(e)36 b(an)f(arbitrary)e(p)r(oin)n(t)515 3863 y(and)27 b Fq(")c(>)g Fw(0.)36 b(W)-7 b(e)28 b(wish)g(to)f(sho)n(w)g(that)1567 4020 y Fq(\025)1615 4032 y Fo(u)1654 4040 y Fg(0)1691 4020 y Fw(\()p Fq(t)p Fw(\)\()p Fq(B)1880 4032 y Fo(F)1936 4020 y Fw(\()p Fq(y)s(;)14 b(")p Fw(\)\))23 b Fq(>)g Fw(0)p Fq(:)839 b Fw(\(3.25\))515 4178 y(In)29 b(view)g(of)36 b(\(3.22\))o(,)30 b(there)f(is)g Fq(\030)1516 4190 y Fs(0)1580 4178 y Fr(2)d Fq(L)1718 4148 y Fs(2)1755 4178 y Fw(\()p Fq(J)1833 4190 y Fo(T)1886 4178 y Fq(;)14 b(H)1992 4190 y Fo(N)2055 4178 y Fw(\))29 b(suc)n(h)g(that)h Fq(f)2528 4190 y Fo(t)2557 4178 y Fw(\()p Fq(u)2637 4190 y Fs(0)2674 4178 y Fq(;)14 b(\030)2747 4190 y Fs(0)2784 4178 y Fw(\))26 b(=)g Fq(y)s Fw(.)42 b(By)29 b(con)n(ti-)515 4277 y(n)n(uit)n(y)-7 b(,)27 b(there)h(is)f Fq(\016)f(>)d Fw(0)k(suc)n(h)g(that)h(if)1569 4435 y Fr(k)p Fq(\030)22 b Fr(\000)c Fq(\030)1788 4447 y Fs(0)1826 4435 y Fr(k)1868 4450 y Fo(L)1914 4433 y Fg(2)1945 4450 y Fs(\()p Fo(J)2008 4458 y Fm(T)2054 4450 y Fo(;U)6 b Fs(\))2178 4435 y Fr(\024)23 b Fq(\016)o(;)842 b Fw(\(3.26\))515 4592 y(then)45 b Fr(k)p Fq(f)804 4604 y Fo(t)833 4592 y Fw(\()p Fq(u)913 4604 y Fs(0)950 4592 y Fq(;)14 b(\030)t Fw(\))30 b Fr(\000)g Fq(f)1225 4604 y Fo(t)1254 4592 y Fw(\()p Fq(u)1334 4604 y Fs(0)1371 4592 y Fq(;)14 b(\030)1444 4604 y Fs(0)1481 4592 y Fw(\))p Fr(k)1555 4604 y Fo(F)1662 4592 y Fr(\024)52 b Fq(")p Fw(.)89 b(Since)45 b Fq(\026)2214 4604 y Fo(T)2311 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Fj(,)31 b(the)f(pr)l(oje)l(ction)i Fk(P)1619 2472 y Fo(F)1674 2460 y Fq(\026)f Fj(p)l(ossesses)g(an)f(almost)h(everywher)l(e)h(p)l (ositive)g(den-)515 2560 y(sity)e(with)g(r)l(esp)l(e)l(ct)g(to)f(the)h (L)l(eb)l(esgue)g(me)l(asur)l(e)f(on)h Fq(F)12 b Fj(.)515 2833 y Fy(4)134 b(App)t(endix)515 3031 y Fp(4.1)112 b(Small)39 b(p)s(erturbations)f(of)g(smo)s(oth)g(functions)515 3185 y Fw(Let)26 b Fq(E)32 b Fw(and)26 b Fq(F)38 b Fw(b)r(e)27 b(\014nite-dimensional)f(v)n(ector)f(spaces,)h(let)h Fq(D)e Fr(\032)d Fq(E)32 b Fw(b)r(e)27 b(an)f(op)r(en)g(subset,)515 3284 y(and)h(let)h Fq(f)k Fw(:)23 b Fq(D)i Fr(!)e Fq(F)40 b Fw(b)r(e)28 b(an)f(in\014nitely)h(di\013eren)n(tiable)g(function.)515 3445 y Fx(Prop)s(osition)40 b(4.1.)45 b Fj(Supp)l(ose)38 b(that)f Fq(f)9 b Fw(\()p Fq(D)r Fw(\))38 b Fj(has)g(a)g(non-empty)f (interior.)63 b(Then)38 b(ther)l(e)515 3544 y(is)26 b Fq(")c(>)h Fw(0)i Fj(such)h(that)f(if)i Fq(g)e Fw(:)f Fq(D)h Fr(!)e Fq(F)37 b Fj(is)26 b(a)g(c)l(ontinuous)f(function)g 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Fw(=)g(0)p Fq(;)515 5006 y Fw(where)k Fq(m)18 b Fr(^)h Fq(n)k Fw(=)f Fq(m)1153 5018 y Fs(1)1191 5006 y Fq(n)1241 5018 y Fs(2)1296 5006 y Fr(\000)c Fq(m)1452 5018 y Fs(2)1489 5006 y Fq(n)1539 5018 y Fs(1)1576 5006 y Fw(.)1905 5255 y(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 515 523 a Fx(De\014nition)41 b(4.6.)k Fw(A)37 b(symmetric)g(subset)f Fr(K)k(\032)d Ft(Z)2202 493 y Fs(2)2277 523 y Fw(con)n(taining)e(\(0)p Fq(;)14 b Fw(0\))37 b(is)f(said)g(to)h(b)r(e)515 623 y Fj(satur)l(ating)d Fw(if)28 b Fr([)1040 635 y Fo(i)p Fn(\025)p Fs(0)1153 623 y Fr(K)1217 593 y Fo(i)1269 623 y Fw(=)22 b Ft(Z)1411 593 y Fs(2)1449 623 y Fw(.)639 789 y(F)-7 b(or)39 b(an)n(y)f(subset)h Fr(K)44 b(\032)e Ft(Z)1503 759 y Fs(2)1540 789 y Fw(,)g(w)n(e)d(denote)g(b)n(y)g Fq(H)2214 801 y Fn(K)2308 789 y Fw(the)g(v)n(ector)f(space)h(spanned)f (b)n(y)515 888 y(the)e(family)g Fr(f)p Fq(e)1009 858 y Fs(1)1009 909 y(0)1045 888 y Fq(;)14 b(e)1121 858 y Fs(2)1121 909 y(0)1158 888 y Fq(;)g(e)1234 900 y Fo(j)1269 888 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b Fj(for)31 b(the)f(pr)l(oje)l(ction)h(to)f Fq(F)12 b Fj(.)515 1727 y Fy(References)515 1909 y Fw([AS05])103 b(A.)24 b(A.)g(Agrac)n(hev)d(and)i(A.)h(V.)g(Saryc)n(hev,)f Fj(Navier-Stokes)j(e)l(quations:)37 b(c)l(ontr)l(ol-)856 2009 y(lability)28 b(by)f(me)l(ans)g(of)g(low)g(mo)l(des)h(for)l(cing)p Fw(,)e(J.)d(Math.)i(Fluid)f(Mec)n(h.)g Fx(7)g Fw(\(2005\),)856 2109 y(no.)j(1,)h(108{152.)515 2275 y([AS06])p 856 2275 250 4 v 366 w(,)38 b Fj(Contr)l(ol)t(lability)h(of)f(2D)f(Euler)g(and)h (Navier{Stokes)g(e)l(quations)f(by)856 2374 y(de)l(gener)l(ate)30 b(for)l(cing)p Fw(,)f(Comm)n(un.)f(Math.)g(Ph)n(ys.)e(\(2006\),)h(to)g (app)r(ear.)515 2540 y([Bog98])68 b(V.)22 b(I.)g(Bogac)n(hev,)f Fj(Gaussian)26 b(Me)l(asur)l(es)p Fw(,)d(Mathematical)f(Surv)n(eys)e (and)i(Mono-)856 2640 y(graphs,)f(v)n(ol.)f(62,)i(American)e (Mathematical)h(So)r(ciet)n(y)-7 b(,)22 b(Pro)n(vidence,)f(RI,)g(1998.) 515 2806 y([BP98])95 b(V.)37 b(Bally)g(and)f(E.)h(P)n(ardoux,)g Fj(Mal)t(liavin)k(c)l(alculus)e(for)g(white)g(noise)g(driven)856 2906 y(p)l(ar)l(ab)l(olic)32 b(SPDEs)p Fw(,)27 b(P)n(oten)n(tial)g (Anal.)g Fx(9)h Fw(\(1998\),)e(no.)i(1,)f(27{64.)515 3072 y([CF88])97 b(P)-7 b(.)37 b(Constan)n(tin)g(and)h(C.)g(F)-7 b(oias,)39 b Fj(Navier-Stokes)h(Equations)p Fw(,)g(Univ)n(ersit)n(y)d (of)856 3171 y(Chicago)26 b(Press,)g(Chicago,)h(1988.)515 3337 y([DF98])94 b(R.)27 b(C.)g(Dalang)f(and)h(N.)g(E.)f(F)-7 b(rangos,)26 b Fj(The)k(sto)l(chastic)g(wave)f(e)l(quation)h(in)f(two) 856 3437 y(sp)l(atial)i(dimensions)p Fw(,)e(Ann.)f(Probab.)e Fx(26)h Fw(\(1998\),)g(no.)g(1,)g(187{212.)515 3603 y([Din79])79 b(S.)38 b(Dineen,)i Fj(Zer)l(o)f(one)g(laws)h(for)f(pr)l(ob)l(ability)i (me)l(asur)l(es)e(on)g(lo)l(c)l(al)t(ly)h(c)l(onvex)856 3703 y(sp)l(ac)l(es)p Fw(,)28 b(Math.)g(Ann.)g Fx(243)f Fw(\(1979\),)g(no.)g(2,)g(95{102.)515 3869 y([EH01])92 b(J.-P)-7 b(.)35 b(Ec)n(kmann)f(and)i(M.)g(Hairer,)g Fj(Uniqueness)h(of)h(the)f(invariant)h(me)l(asur)l(e)856 3968 y(for)30 b(a)f(sto)l(chastic)h(PDE)f(driven)g(by)h(de)l(gener)l (ate)f(noise)p Fw(,)f(Comm.)e(Math.)h(Ph)n(ys.)856 4068 y Fx(219)g Fw(\(2001\),)f(523{565.)515 4234 y([F)-7 b(el71])104 b(W.)25 b(F)-7 b(eller,)25 b Fj(A)n(n)i(Intr)l(o)l(duction)f(to)h(Pr)l (ob)l(ability)i(The)l(ory)g(and)e(Its)g(Applic)l(ations.)856 4334 y(Vol.)k(II)p Fw(,)c(John)g(Wiley)h(&)g(Sons,)f(New)h(Y)-7 b(ork,)26 b(1971.)515 4500 y([Fla94])92 b(F.)62 b(Flandoli,)71 b Fj(Dissip)l(ativity)63 b(and)f(invariant)g(me)l(asur)l(es)f(for)h (sto)l(chastic)856 4599 y(Navier{Stokes)31 b(e)l(quations)p Fw(,)d(NoDEA)g Fx(1)f Fw(\(1994\),)g(403{426.)515 4765 y([GS80])100 b(I.)43 b(I.)g(Gihman)g(and)f(A.)h(V.)g(Sk)n(oroho)r(d,)i Fj(The)g(The)l(ory)g(of)f(Sto)l(chastic)h(Pr)l(o-)856 4865 y(c)l(esses.)30 b(I)p Fw(,)e(Springer-V)-7 b(erlag,)25 b(Berlin,)i(1980.)1905 5255 y(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 515 523 a Fw([Has80])74 b(R.)38 b(Z.)f(Has)1226 493 y Fn(0)1249 523 y Fw(minski)-9 b(\025)-32 b(\020,)39 b Fj(Sto)l(chastic)h(Stability)f(of)h(Di\013er)l(ential)f (Equations)p Fw(,)i(Si-)856 623 y(jtho\013)28 b(&)g(No)r(ordho\013,)f (Alphen)h(aan)f(den)h(Rijn,)g(1980.)515 789 y([Hen81])66 b(D.)31 b(Henry)-7 b(,)31 b Fj(Ge)l(ometric)i(The)l(ory)h(of)f (Semiline)l(ar)g(Par)l(ab)l(olic)i(Equations)p Fw(,)d(Lec-)856 888 y(ture)c(Notes)f(in)h(Mathematics,)f(v)n(ol.)g(840,)f(Springer-V)-7 b(erlag,)25 b(Berlin,)j(1981.)515 1054 y([Kuk82])56 b(S.)46 b(B.)g(Kuksin,)k Fj(Di\013e)l(omorphisms)f(of)e(function)g(sp)l(ac)l (es)g(that)g(c)l(orr)l(esp)l(ond)856 1154 y(to)34 b(quasiline)l(ar)i(p) l(ar)l(ab)l(olic)h(e)l(quations)p Fw(,)d(Mat.)f(Sb.)g(\(N.S.\))h Fx(117\(159\))d Fw(\(1982\),)856 1254 y(no.)c(3,)h(359{378,)c(431.)515 1420 y([LNP00])40 b(J.)e(A.)h(Le\023)-42 b(on,)41 b(D.)f(Nualart,)g (and)f(R.)g(P)n(ettersson,)g Fj(The)i(sto)l(chastic)g(Bur)l(gers)856 1519 y(e)l(quation:)35 b(\014nite)22 b(moments)g(and)h(smo)l(othness)g (of)h(the)e(density)p Fw(,)g(In\014n.)e(Dimens.)856 1619 y(Anal.)28 b(Quan)n(tum)f(Probab.)f(Relat.)i(T)-7 b(op.)27 b Fx(3)g Fw(\(2000\),)g(no.)g(3,)g(363{385.)515 1785 y([MP06])78 b(J.)26 b(Mattingly)f(and)h(E.)f(P)n(ardoux,)f Fj(Mal)t(liavin)31 b(c)l(alculus)d(for)h(the)f(sto)l(chastic)h(2D)856 1885 y(Navier-Stokes)h(e)l(quation)p Fw(,)e(Comm.)g(Pure)f(Appl.)h (Math.)g(\(2006\),)e(to)i(app)r(ear.)515 2051 y([Nua95])61 b(D.)44 b(Nualart,)j Fj(The)f(Mal)t(liavin)h(Calculus)e(and)g(R)l (elate)l(d)g(T)-6 b(opics)p Fw(,)49 b(Springer-)856 2150 y(V)-7 b(erlag,)27 b(New)g(Y)-7 b(ork,)27 b(1995.)515 2316 y([Oco88])67 b(D.)34 b(Ocone,)g Fj(Sto)l(chastic)h(c)l(alculus)g (of)h(variations)h(for)f(sto)l(chastic)g(p)l(artial)g(dif-)856 2416 y(fer)l(ential)31 b(e)l(quations)p Fw(,)d(J.)f(F)-7 b(unct.)29 b(Anal.)e Fx(79)g Fw(\(1988\),)g(no.)g(2,)g(288{331.)515 2582 y([Ste83])96 b(S.)35 b(Stern)n(b)r(erg,)h Fj(L)l(e)l(ctur)l(es)f (on)h(Di\013er)l(ential)h(Ge)l(ometry)p Fw(,)h(Chelsea)c(Publishing)856 2682 y(Co.,)27 b(New)h(Y)-7 b(ork,)27 b(1983.)515 2848 y([T)-7 b(em79])52 b(R.)d(T)-7 b(emam,)54 b Fj(Navier{Stokes)c (Equations)p Fw(,)55 b(North-Holland,)e(Amsterdam,)856 2947 y(1979.)515 3113 y([VF88])95 b(M.)26 b(I.)g(Vishik)g(and)f(A.)h (V.)g(F)-7 b(ursik)n(o)n(v,)25 b Fj(Mathematic)l(al)30 b(Pr)l(oblems)f(in)f(Statistic)l(al)856 3213 y(Hydr)l(ome)l(chanics)p Fw(,)i(Klu)n(w)n(er,)c(Dordrec)n(h)n(t,)g(1988.)1905 5255 y(24)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0603130401878--