%!FontType1-1.0: cmbx10 1.1/12-Nov-94 %%VMusage: 25027 25304 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmbx10) readonly def /FamilyName (cmbx10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 444 def /Ascender 694 def /Descender -194 def /CapHeight 686 def end readonly def /FontName /cmbx10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-60 -250 1164 750] readonly def /UniqueID 4701489 def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmr10 %!FontType1-1.0: cmr10 1.1/12-Nov-94 %%VMusage: 25429 25707 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmr10) readonly def /FamilyName (cmr10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmr10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-44 -250 1009 750] readonly def /UniqueID 4701481 def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmr7 %!FontType1-1.0: cmr7 1.1/12-Nov-94 %%VMusage: 25529 25807 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmr7) readonly def /FamilyName (cmr7) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmr7 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-31 -251 1122 750] readonly def /UniqueID 4701478 def /StrokeWidth 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmmi7 %!FontType1-1.0: cmmi7 1.1/12-Nov-94 %%VMusage: 32155 32728 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmmi7) readonly def /FamilyName (cmmi7) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmmi7 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 128 /kappa1 put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 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cleartomark %%EndFont %%BeginFont: cmti10 %!FontType1-1.0: cmti10 1.1/12-Nov-94 %%VMusage: 31328 31604 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmti10) readonly def /FamilyName (cmti10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmti10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-35 -250 1125 750] readonly def /UniqueID 4701494 def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmbx12 %!FontType1-1.0: cmbx12 1.1/12-Nov-94 %%VMusage: 24974 25247 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmbx12) readonly def /FamilyName (cmbx12) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 444 def /Ascender 694 def /Descender -194 def /CapHeight 686 def end readonly def /FontName /cmbx12 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-57 -250 1139 750] readonly def /UniqueID 4701490 def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmmi10 %!FontType1-1.0: cmmi10 1.1/12-Nov-94 %%VMusage: 31220 31787 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmmi10) readonly def /FamilyName (cmmi10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmmi10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmsy10 %!FontType1-1.0: cmsy10 1.1/12-Nov-94 %%VMusage: 26658 28948 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmsy10) readonly def /FamilyName (cmsy10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmsy10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 160 /space put dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-29 -272 1123 779] readonly def /UniqueID 4701534 def /StrokeWidth 0 def currentdict end currentfile eexec 9B9C0887DB83FB1ECD8335B0BB39CEF0AF64F656FC6E5C230CC9D3A7346AAC7A06BD9A40393C A15D3773A21E06B9B4254D3050E90726BBB09120935A8D448CDBC799D90205A21291254FA633 CC0C2ED88781EF21A5D45B72445C284FA44F8F39CF566976075A92E9E2947A6FC93D879C29EB 26809ECB4409CBC8666526CCED9265DE661A2F8B0A16BE45A9DA17EEAF89B0E023DE2B0373DC AB997D60B7D7DC1B4F7E650A2A9F13CA0543F3B080AE33D65F2802B8C5214D5B60D0D2956096 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmex10 %!FontType1-1.0: cmex10 1.1/12-Nov-94 %%VMusage: 20734 24250 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmex10) readonly def /FamilyName (cmex10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /Ascender 694 def /Descender -194 def /CapHeight 683 def /XHeight 431 def end readonly def /FontName /cmex10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 36 /floorleftBigg put dup 37 /floorrightBigg put dup 38 /ceilingleftBigg put dup 39 /ceilingrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 42 /angbracketleftBigg put dup 43 /angbracketrightBigg put dup 44 /slashBigg put dup 45 /backslashBigg put dup 46 /slashBig put dup 47 /backslashBig put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 54 /bracketleftex put dup 55 /bracketrightex put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 62 /braceex put dup 63 /arrowvertex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 70 /unionsqtext put dup 71 /unionsqdisplay put dup 72 /contintegraltext put dup 73 /contintegraldisplay put dup 74 /circledottext put dup 75 /circledotdisplay put dup 76 /circleplustext put dup 77 /circleplusdisplay put dup 78 /circlemultiplytext put dup 79 /circlemultiplydisplay put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 83 /uniontext put dup 84 /intersectiontext put dup 85 /unionmultitext put dup 86 /logicalandtext put dup 87 /logicalortext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 93 /unionmultidisplay put dup 94 /logicalanddisplay put dup 95 /logicalordisplay put dup 96 /coproducttext put dup 97 /coproductdisplay put dup 98 /hatwide put dup 99 /hatwider put dup 100 /hatwidest put dup 101 /tildewide put dup 102 /tildewider put dup 103 /tildewidest put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 106 /floorleftBig put dup 107 /floorrightBig put dup 108 /ceilingleftBig put dup 109 /ceilingrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 113 /radicalBig put dup 114 /radicalbigg put dup 115 /radicalBigg put dup 116 /radicalbt put dup 117 /radicalvertex put dup 118 /radicaltp put dup 119 /arrowvertexdbl put dup 120 /arrowtp put dup 121 /arrowbt put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put dup 126 /arrowdbltp put dup 160 /space put dup 161 /parenleftbig put dup 162 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmr8 %!FontType1-1.0: cmr8 1.1/12-Nov-94 %%VMusage: 25661 25938 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmr8) readonly def /FamilyName (cmr8) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmr8 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-40 -250 1070 750] readonly def /UniqueID 4701479 def /StrokeWidth 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Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmr5) readonly def /FamilyName (cmr5) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /Descender -194 def /CapHeight 683 def /XHeight 431 def /Ascender 694 def end readonly def /FontName /cmr5 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /less put dup 61 /equal put dup 62 /greater put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /arrowup put dup 175 /arrowdown put dup 176 /quotesingle put dup 177 /exclamdown put dup 178 /questiondown put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-14 -250 1304 750] readonly def 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cleartomark %%EndFont %%BeginFont: cmmi5 %!FontType1-1.0: cmmi5 1.1/12-Nov-94 %%VMusage: 32182 32751 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmmi5) readonly def /FamilyName (cmmi5) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmmi5 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 128 /kappa1 put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [0 -221 1349 750] readonly def /UniqueID 4701517 def /StrokeWidth 0 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmsy7 %!FontType1-1.0: cmsy7 1.1/12-Nov-94 %%VMusage: 27597 30113 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmsy7) readonly def /FamilyName (cmsy7) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmsy7 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 160 /space put dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-15 -270 1257 786] readonly def /UniqueID 4701531 def /StrokeWidth 0 def currentdict end currentfile eexec 9B9C0887DB83FB1ECD8335B0BB39CEF0AF64F656FC6E5C230CC9D3A7346AAC7A06BD9A40393C A15D3773A21E06B9B4254D3050E90726BBB09120935A8D448CDBC799D90205A21291254FA633 CC0C2ED88781EF21A5D45B72445C284FA44F8F39CF566976075A92E9E2947A6FC93D879C29EB 26809ECB4409CBC8666526CCED9265DE661A2F8B0A16BE45A9DA17EEAF89B0E023DE2B0373DC AB997D60B7D7DC1B4F7E650A2A9F13CA0543F3B080AE33D65F2802B8C7BE7B9F15E795D04A36 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5289 50 4 v 1367 5380 a(2)1446 5231 y Fe(\000)1510 5312 y Fl(\026)-69 b Fg(m)1578 5270 y Fk(2)1646 5312 y Ff(\000)22 b Fl(1)1795 5231 y Fe(\001)1868 5312 y Fl(=)1985 5244 y(3)p 1985 5289 V 1985 5380 a(2)2064 5201 y Fe(\020)2123 5312 y Fl(tanh)2323 5267 y Fk(2)2384 5201 y Fe(\020)2455 5244 y Fg(x)p 2455 5289 57 4 v 2459 5380 a Fl(2)2524 5201 y Fe(\021)2606 5312 y Ff(\000)g Fl(1)2755 5201 y Fe(\021)2865 5312 y Fg(:)752 b Fl(\(1)p Fg(:)p Fl(15\))0 5579 y Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)31 b(22:40)1127 b Fl(4)p eop %%Page: 5 5 5 4 bop 120 100 a Fl(The)33 b(op)s(erator)g Ff(A)g Fl(has)h(a)f(sp)s (ectral)g(gap:)0 332 y Fm(Lemma)j(1.2)31 b Fi(In)j(the)g(sp)-5 b(e)g(ctrum)34 b(of)f Ff(A)p Fi(,)g(0)g(is)g(an)h(isolate)-5 b(d)33 b(eigenvalue)g(of)g(multiplicity)i(one.)44 b(In)34 b(fact,)1408 470 y Fe(Z)1525 605 y Fg(v)t Fl(\()p Fg(x)p Fl(\))p Ff(A)p Fg(v)t Fl(\()p Fg(x)p Fl(\)d)p Fg(x)26 b Ff(\025)2234 538 y Fl(3)p 2234 582 50 4 v 2234 674 a(4)2296 605 y Ff(k)p Fg(v)t Ff(k)2448 564 y Fk(2)2448 630 y(2)0 890 y Fi(for)35 b(al)5 b(l)37 b Fg(v)i Fi(with)606 810 y Fe(R)689 890 y Fg(v)t Fl(\()p Fg(x)p Fl(\))18 b(\026)-68 b Fg(m)963 854 y Fa(0)990 890 y Fl(\()p Fg(x)p Fl(\)d)p Fg(x)28 b Fl(=)g(0)p Fi(.)0 1122 y Fm(Pro)s(of:)45 b Fl(W)-8 b(e)33 b(consider)h(the)f(op)s(erator)g Fg(H)40 b Fl(giv)m(en)33 b(b)m(y)g Fg(H)8 b(v)t Fl(\()p Fg(x)p Fl(\))27 b(=)h Ff(\000)p Fg(v)2546 1086 y Fa(00)2597 1122 y Fl(\()p Fg(x)p Fl(\))21 b(+)h Fg(V)g Fl(\()p Fg(x)p Fl(\).)43 b(W)-8 b(e)33 b(kno)m(w)g(that)51 b(\026)-68 b Fg(m)3872 1086 y Fa(0)0 1251 y Fl(is)34 b(an)h(eigen)m(v)m(ector,)g (and)g(that)f(the)h(corresp)s(onding)h(eigen)m(v)-6 b(alue)35 b(is)g Ff(\000)p Fl(1.)48 b(Let)34 b Ff(\000)p Fl(1)d(=)f Fg(e)3287 1266 y Fk(0)3332 1251 y Fg(;)17 b(e)3423 1266 y Fk(1)3467 1251 y Fg(;)g(e)3558 1266 y Fk(2)3603 1251 y Fg(;)g(:)g(:)g(:)32 b Fl(b)s(e)0 1381 y(the)j(negativ)m(e)f(eigen)m (v)-6 b(alues)36 b(of)e Fg(H)8 b Fl(,)35 b(rep)s(eated)g(according)g (to)f(their)g(m)m(ultiplicit)m(y)-8 b(.)47 b(Then)35 b(b)m(y)g(a)f(b)s(ound)0 1510 y(of)f(Lieb)h(and)g(Thirring)f([9],)f (one)i(has)1278 1695 y Fe(X)1332 1907 y Fj(j)1439 1790 y Ff(j)p Fg(e)1513 1805 y Fj(j)1555 1790 y Ff(j)1583 1748 y Fk(3)p Fj(=)p Fk(2)1735 1790 y Ff(\024)1877 1722 y Fl(3)p 1853 1767 100 4 v 1853 1858 a(16)1981 1654 y Fe(Z)2036 1880 y Fj(I)-19 b(R)2134 1790 y Ff(j)p Fg(V)22 b Fl(\()p Fg(x)p Fl(\))p Ff(j)2405 1748 y Fk(2)2449 1790 y Fl(d)p Fg(x)33 b(:)0 2118 y Fl(The)f(in)m(tegral)g(is)g(easily)f(ev) -6 b(aluated)32 b(and)h(equals)f(6.)43 b(Keeping)33 b(only)e(the)i (\014rst)f(t)m(w)m(o)g(terms)g(in)g(the)g(sum)0 2248 y(on)h(the)g(left)g(1)22 b(+)g Ff(j)p Fg(e)728 2263 y Fk(1)773 2248 y Ff(j)801 2212 y Fk(3)p Fj(=)p Fk(2)953 2248 y Ff(\024)28 b Fl(18)p Fg(=)p Fl(16)k(and)i(this)f(implies)g(that) f Ff(j)p Fg(e)2355 2263 y Fk(1)2400 2248 y Ff(j)27 b(\024)i Fl(1)p Fg(=)p Fl(4.)43 b(Th)m(us)34 b Fg(e)3080 2263 y Fk(1)3153 2248 y Ff(\025)28 b(\000)p Fl(1)p Fg(=)p Fl(4,)k(and)i(this)0 2377 y(completes)g(the)g(pro)s(of.)p 1075 2377 70 70 v 120 2711 a(As)h(indicated)h(in)g(Theorem)g(1.1,)g(w)m (e)g(shall)f(start)h(out)f(with)h Ff(k)p Fg(v)t Ff(k)2624 2726 y Fk(2)2703 2711 y Fl(small,)g(and)g(then,)h(b)s(ecause)g(of)0 2840 y(the)j(smo)s(othing)f(prop)s(erties)h(of)g(the)g(equation)f ([3],[5],)g(it)g(will)f(b)s(e)i(the)g(case)g(that)f(at)g(least)g(a)h (short)0 2970 y(time)d(later,)g Ff(k)p Fg(v)t Ff(k)648 2985 y Fk(2)729 2970 y Fl(is)f(still)g(small,)h(and)h(then)f Ff(k)p Fg(v)1853 2934 y Fa(0)1881 2970 y Ff(k)1931 2985 y Fk(2)2012 2970 y Fl(is)g(small)f(as)h(w)m(ell.)55 b(W)-8 b(e)37 b(shall)g(obtain)g(a)f(n)m(um)m(b)s(er)0 3099 y(of)31 b Fi(a{priori)f Fl(estimates)h(that)f(hold)h(when)h Ff(k)p Fg(v)t Ff(k)1769 3114 y Fk(2)1844 3099 y Fl(and)f Ff(k)p Fg(v)2137 3063 y Fa(0)2165 3099 y Ff(k)2215 3114 y Fk(2)2290 3099 y Fl(are)g(b)s(oth)g(small,)f(and)h(shall)g(use)h (them)f(in)0 3229 y(the)39 b(\014nal)g(section)g(of)g(the)g(pap)s(er)g (to)g(pro)m(v)m(e)g(that)f(this)h(condition)g(p)s(ersists)g (inde\014nitely)-8 b(.)60 b(The)40 b(\014rst)0 3358 y(estimate)31 b(that)f(w)m(e)i(obtain)e(under)j(these)f(conditions)f(sho)m(ws)h(that) e(the)i(excess)g(free)g(energy)f(of)50 b(\026)-69 b Fg(m)18 b Fl(+)g Fg(v)0 3488 y Fl(is)33 b(comparable)h(to)f Ff(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)31 b Fl(.)0 3720 y Fm(Lemma)43 b(1.3)37 b Fi(F)-8 b(or)40 b(al)5 b(l)40 b Fg(\017)34 b(>)h Fl(0)p Fi(,)40 b(ther)-5 b(e)38 b(ar)-5 b(e)39 b Fg(\016)n(;)17 b(\024)35 b(>)f Fl(0)39 b Fi(so)g(that)g(whenever)g Ff(k)p Fg(v)t Ff(k)2968 3735 y Fk(2)3046 3720 y Ff(\024)c Fg(\016)43 b Fi(and)c Ff(k)p Fg(v)3544 3684 y Fa(0)3572 3720 y Ff(k)3622 3735 y Fk(2)3700 3720 y Ff(\024)c Fg(\024)p Fi(,)0 3849 y(then)862 3953 y Fl(1)22 b Ff(\000)h Fg(\017)p 862 3998 213 4 v 943 4089 a Fl(2)1086 4021 y Ff(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)26 b(\024)i(F)10 b Fl(\()18 b(\026)-68 b Fg(m)22 b Fl(+)h Fg(v)t Fl(\))e Ff(\000)i(F)10 b Fl(\()18 b(\026)-68 b Fg(m)p Fl(\))27 b Ff(\024)2457 3953 y Fl(1)22 b(+)h Fg(\017)p 2457 3998 V 2538 4089 a Fl(2)2681 4021 y Ff(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)34 b Fg(:)0 4434 y Fm(Pro)s(of:)45 b Fl(One)35 b(easily)d(computes)j(that)769 4708 y Ff(F)10 b Fl(\()18 b(\026)-68 b Fg(m)22 b Fl(+)h Fg(v)t Fl(\))e Ff(\000)i(F)10 b Fl(\()18 b(\026)-68 b Fg(m)p Fl(\))27 b(=)1702 4640 y(1)p 1702 4685 50 4 v 1702 4776 a(2)1764 4708 y Ff(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)j Fl(+)2203 4640 y(1)p 2203 4685 V 2203 4776 a(4)2281 4572 y Fe(Z)2397 4627 y(\000)2443 4708 y Fl(2)f(\026)-69 b Fg(mv)2632 4666 y Fk(3)2699 4708 y Fl(+)23 b Fg(v)2851 4666 y Fk(4)2895 4627 y Fe(\001)2958 4708 y Fl(d)p Fg(x)33 b(:)0 4993 y Fl(Using)g(the)h(inequalit)m(y)e Ff(k)p Fg(v)t Ff(k)1066 4957 y Fk(2)1066 5018 y Fa(1)1177 4993 y Ff(\024)d Fl(2)p Ff(k)p Fg(v)t Ff(k)1485 5008 y Fk(2)1529 4993 y Ff(k)p Fg(v)1631 4957 y Fa(0)1658 4993 y Ff(k)1708 5008 y Fk(2)1753 4993 y Fl(,)k(one)g(obtains)979 5141 y Fe(\014)979 5201 y(\014)979 5260 y(\014)979 5320 y(\014)1012 5150 y(Z)1128 5145 y(\022)1202 5285 y Fl(2)18 b(\026)-68 b Fg(mv)1391 5244 y Fk(3)1458 5285 y Fl(+)1569 5218 y Fg(v)1621 5182 y Fk(4)p 1569 5262 97 4 v 1593 5354 a Fl(2)1678 5145 y Fe(\023)1768 5285 y Fl(d)p Fg(x)1880 5141 y Fe(\014)1880 5201 y(\014)1880 5260 y(\014)1880 5320 y(\014)1941 5285 y Ff(\024)2046 5175 y Fe(\020)2106 5285 y Fl(2)2156 5195 y Ff(p)p 2239 5195 156 4 v 90 x Fl(2)p Fg(\024\016)26 b Fl(+)d Fg(\024\016)2621 5175 y Fe(\021)2697 5285 y Ff(k)p Fg(v)t Ff(k)2849 5244 y Fk(2)2849 5310 y(2)2893 5285 y Fg(:)0 5579 y Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)31 b(22:40)1127 b Fl(5)p eop %%Page: 6 6 6 5 bop 0 115 a Fl(By)28 b(the)h(previous)g(lemma,)g(for)g Fg(\024)g Fl(and)g Fg(\016)k Fl(small)28 b(enough,)2180 4 y Fe(\020)2239 115 y Fl(2)2289 30 y Ff(p)p 2372 30 156 4 v 85 x Fl(2)p Fg(\024\016)e Fl(+)d Fg(\024\016)2754 4 y Fe(\021)2831 115 y Ff(k)p Fg(v)t Ff(k)2983 78 y Fk(2)2983 139 y(2)3054 115 y Ff(\024)29 b Fl(\()p Fg(\017=)p Fl(2\))p Ff(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)p Fl(,)26 b(and)0 289 y(this)33 b(completes)h(the)g(pro)s(of.)p 1225 289 70 70 v 120 538 a(The)f(\014rst)h(k)m(ey)f(result)g(is)h(a)f(lo)m(w)m (er)g(b)s(ound)h(on)g(the)f(dissipation)g(in)h(terms)f(of)h Ff(A)p Fl(:)0 788 y Fm(Theorem)k(1.4)33 b Fi(F)-8 b(or)36 b(any)g Fg(\017)28 b(>)g Fl(0)p Fi(,)625 1102 y Ff(I)7 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))28 b(=)g Ff(\000)1206 1034 y Fg(d)p 1187 1079 88 4 v 1187 1170 a(dt)1287 1021 y Fe(\002)1329 1102 y Ff(F)1411 1021 y Fe(\000)1456 1102 y Fg(m)p Fl(\()p Fg(t)p Fl(\)\))22 b Ff(\000)h(F)1900 1021 y Fe(\000)1963 1102 y Fl(\026)-68 b Fg(m)2032 1021 y Fe(\001\003)2147 1102 y Ff(\025)28 b Fl(\(1)22 b Ff(\000)g Fg(\017)p Fl(\))2558 966 y Fe(Z)2674 1102 y Fl([\()p Ff(A)p Fg(v)t Fl(\))2912 1060 y Fa(0)2939 1102 y Fl(\()p Fg(x)p Fl(\)])3101 1046 y Fk(2)3162 1102 y Fl(d)p Fg(x)371 b Fl(\(1)p Fg(:)p Fl(16\))0 1420 y Fi(whenever)41 b Ff(jj)p Fg(v)548 1384 y Fa(0)575 1420 y Ff(jj)631 1435 y Fk(2)714 1420 y Ff(\024)f Fg(\024)888 1435 y Fk(1)933 1420 y Fl(\()p Fg(\017)p Fl(\))i Fi(and)f Ff(jj)p Fg(v)t Ff(jj)1456 1435 y Fk(2)1539 1420 y Ff(\024)e Fg(\016)1699 1435 y Fk(1)1744 1420 y Fl(\()p Fg(\017)p Fl(\))j Fi(for)f(some)h(strictly)g (p)-5 b(ositive)40 b(c)-5 b(onstants)41 b Fg(\024)3537 1435 y Fk(1)3582 1420 y Fl(\()p Fg(\017)p Fl(\))h Fi(and)0 1550 y Fg(\016)44 1565 y Fk(1)89 1550 y Fl(\()p Fg(\017)p Fl(\))p Fi(.)j(Mor)-5 b(e)g(over,)35 b(ther)-5 b(e)35 b(exists)g(a)h(c)-5 b(onstant)35 b Fg(\015)e(>)28 b Fl(0)35 b Fi(so)g(that)1433 1733 y Fe(Z)1549 1787 y(\002)1590 1868 y Fl(\()p Ff(A)p Fg(v)t Fl(\))1799 1818 y Fa(0)1827 1787 y Fe(\003)1869 1804 y Fk(2)1930 1868 y Fl(d)p Fg(x)28 b Ff(\025)g Fg(\015)6 b Ff(jj)p Fg(v)2341 1827 y Fa(0)2367 1868 y Ff(jj)2423 1827 y Fk(2)2423 1893 y(2)3645 1868 y Fl(\(1)p Fg(:)p Fl(17\))0 2192 y Fi(whenever)434 2112 y Fe(R)517 2192 y Fg(v)t Fl(\()p Fg(x)p Fl(\))18 b(\026)-68 b Fg(m)791 2156 y Fa(0)819 2192 y Fl(\()p Fg(x)p Fl(\))p Fg(dx)27 b Fl(=)h(0)p Fi(.)120 2442 y Fl(This)33 b(theorem)h(is)f(pro)m (v)m(ed)h(in)g(section)f(2.)45 b(W)-8 b(e)33 b(use)h(\(1.16\))e(only)h (when)h Ff(I)7 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))28 b Fg(<<)3392 2361 y Fe(\002)3433 2442 y Ff(F)3515 2361 y Fe(\000)3561 2442 y Fg(m)p Fl(\()p Fg(t)p Fl(\)\))22 b Ff(\000)0 2571 y(F)82 2490 y Fe(\000)146 2571 y Fl(\026)-69 b Fg(m)215 2490 y Fe(\001)o(\003)302 2571 y Fl(;)40 b(i.e.,)e(when)h (the)g(dissipation)f(is)g(v)m(ery)g(small)f(compared)i(to)f(the)h (excess)g(free)g(energy)-8 b(.)58 b(The)0 2701 y(p)s(oin)m(t)29 b(is)g(that)f(in)h(this)g(case,)h Fg(v)j Fl(m)m(ust)c(b)s(e)h(v)m(ery)e (\\smo)s(oth)h(and)g(spread)h(out")f(and)g(so)g(b)s(oth)g Ff(h)p Fg(v)3520 2664 y Fa(0)3548 2701 y Fg(;)17 b(v)3645 2664 y Fa(0)3672 2701 y Ff(i)28 b Fl(and)0 2830 y Ff(h)p Fg(v)t(;)17 b(V)k(v)t Ff(i)33 b Fl(are)h(negligible)f(compared)i(to)e Ff(h)p Fg(v)t(;)17 b(v)t Ff(i)p Fl(.)44 b(That)33 b(is,)h Ff(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)26 b(\031)j(h)p Fg(v)t(;)17 b(v)t Ff(i)p Fl(,)32 b(and)i(so)g(the)g(op)s(erator)f Ff(A)0 2960 y Fl(di\013ers)41 b(negligibly)d(from)i(the)g(iden)m(tit)m (y)-8 b(.)64 b Fi(But)42 b(if)f(we)g(r)-5 b(eplac)g(e)42 b Ff(A)f Fi(by)h(the)f(identity)h(in)f(the)g(line)-5 b(arize)g(d)0 3089 y(evolution)40 b(e)-5 b(quation,)41 b(it)f(simply)h(b)-5 b(e)g(c)g(omes)39 b(the)h(he)-5 b(at)40 b(e)-5 b(quation.)59 b Fl(Therefore,)40 b(when)f(the)g (dissipation)0 3219 y(is)d(small)g(compared)h(to)e(the)i(excess)g(free) g(energy)-8 b(,)36 b(w)m(e)h(exp)s(ect)f(heat)h(equation)e(b)s(eha)m (vior,)i(whic)m(h)g(w)m(e)0 3348 y(can)d(calculate,)e(to)h(go)m(v)m (ern)h(this)f(dissipation)g(pro)s(cess.)120 3479 y(On)27 b(the)g(other)f(hand,)j(when)e(the)g(dissipation)f(is)h Fi(not)f Fl(small)g(compared)i(to)e(the)g(excess)i(free)f(energy)-8 b(,)0 3608 y(w)m(e)37 b(are)f(in)g(a)g(p)s(osition)g(to)g(b)s(ene\014t) h(from)g(the)f(plen)m(tiful)h(dissipativit)m(y)e(to)h(compute)h(b)s (ounds)g(on)g(the)0 3738 y(rate)c(at)g(whic)m(h)h(it)e(o)s(ccurs.)45 b(This)34 b(is)f(the)g Fi(dissip)-5 b(ation{dichotomy)31 b Fl(in)j(our)f(argumen)m(t.)120 3868 y(T)-8 b(o)31 b(mak)m(e)g (precise)i(use)g(of)f(the)g(dic)m(hotom)m(y)-8 b(,)32 b(in)m(tro)s(duce)g(a)g(small)f(parameter)h Fg(\017)3102 3883 y Fk(1)3179 3868 y Fl(to)f(b)s(e)h(\014xed)h(later,)0 3998 y(and)h(distinguish)f(b)s(et)m(w)m(een)i(the)f(times)f Fg(t)g Fl(for)h(whic)m(h)1235 4245 y Ff(I)7 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))28 b Ff(\024)g Fg(\017)1748 4260 y Fk(1)1793 4165 y Fe(\002)1834 4245 y Ff(F)10 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))22 b Ff(\000)g(F)10 b Fl(\()18 b(\026)-68 b Fg(m)p Fl(\))2563 4165 y Fe(\003)2637 4245 y Fg(:)980 b Fl(\(1)p Fg(:)p Fl(18\))0 4493 y(or)1235 4622 y Ff(I)7 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))28 b Ff(\025)g Fg(\017)1748 4637 y Fk(1)1793 4542 y Fe(\002)1834 4622 y Ff(F)10 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))22 b Ff(\000)g(F)10 b Fl(\()18 b(\026)-68 b Fg(m)p Fl(\))2563 4542 y Fe(\003)2637 4622 y Fg(:)980 b Fl(\(1)p Fg(:)p Fl(19\))0 4821 y(When)37 b(\(1.19\))d(is)i(true,)g(there)g(is)g(plen)m (t)m(y)g(of)g(dissipation,)g(and)g(the)g(excess)g(free)h(energy)f(is)g (deca)m(ying)0 4950 y(at)d(an)g(exp)s(onen)m(tial)g(rate,)g(and)h(it)e (will)h(b)s(e)g(relativ)m(ely)f(simple)h(to)g(exploit)f(this.)120 5081 y(W)-8 b(e)33 b(ha)m(v)m(e)h(already)e(explained)h(that)g (condition)g(\(1.18\))f(will)g(help)i(us)f(b)s(ecause)i(under)f(this)f (condi-)0 5210 y(tion,)f(w)m(e)h(will)e(b)s(e)i(able)g(to)f(sho)m(w)h (that)f Ff(k)p Fg(v)t Ff(k)1644 5225 y Fk(1)1721 5210 y Fl(dissipates)h(a)m(w)m(a)m(y)f Fi(as)j(though)d Fl(it)g(w)m(ere)h(a) g(solution)f(of)h(the)0 5340 y(heat)i(equation)g(with)861 5260 y Fe(R)908 5375 y Fj(I)-19 b(R)1005 5340 y Fg(v)t Fl(\()p Fg(t)p Fl(\)d)p Fg(x)31 b Fl(=)g(0)k(for)g(all)f Fg(t)p Fl(.)50 b(W)-8 b(e)35 b(will)f(return)i(to)f(this)g(shortly)-8 b(,)34 b(but)i(it)e(dep)s(ends)0 5579 y Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)d(22:40)1127 b Fl(6)p eop %%Page: 7 7 7 6 bop 0 100 a Fl(on)28 b(the)g(fact)g(that)f(when)i(\(1.18\))d (holds,)j Fg(v)i Fl(is)d(v)m(ery)f(\\smo)s(oth)h(and)g(spread)h(out".) 41 b(This)28 b(is)g(used)h(sev)m(eral)0 229 y(w)m(a)m(ys)k(in)h(the)f (pro)s(of.)44 b(Indeed,)35 b(com)m(bining)f(\(1.18\))d(with)i(the)h(k)m (ey)f(b)s(ound)h(\(1.16\),)1208 434 y Ff(k)1258 353 y Fe(\000)1303 434 y Ff(A)p Fg(v)1435 353 y Fe(\001)1480 374 y Fa(0)1508 434 y Ff(k)1558 393 y Fk(2)1558 458 y(2)1631 434 y Ff(\024)28 b Fg(\017)1776 449 y Fk(1)1821 353 y Fe(\002)1862 434 y Ff(F)10 b Fl(\()18 b(\026)-68 b Fg(m)22 b Fl(+)h Fg(v)t Fl(\))f Ff(\000)g(F)10 b Fl(\()18 b(\026)-68 b Fg(m)p Fl(\))2651 353 y Fe(\003)0 638 y Fl(under)35 b(appropriate)g(conditions)f(on)g Fg(v)t Fl(.)47 b(Then)35 b(since)g Ff(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)32 b Fl(is)i(comparable)h(with)e(the)i(excess)g(free)0 768 y(energy)j(of)57 b(\026)-69 b Fg(m)26 b Fl(+)f Fg(v)t Fl(,)39 b(when)g(\(1.18\))d(holds,)j(one)f(has)g Ff(k)2045 687 y Fe(\000)2091 768 y Ff(A)p Fg(v)2223 687 y Fe(\001)2268 708 y Fa(0)2296 768 y Ff(k)2346 731 y Fk(2)2346 792 y(2)2426 768 y Fg(<<)e Ff(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)p Fl(.)55 b(An)m(y)38 b(function)h Fg(v)i Fl(suc)m(h)0 897 y(that)33 b(this)g(is)g(the)h(case)g(is)f(so)g(smo)s(oth)g(and)h (spread)g(out)f(that)1077 1102 y Ff(kA)p Fg(v)1259 1017 y Fe(\015)1259 1077 y(\015)1313 1042 y Fk(2)1313 1142 y(2)1386 1102 y Ff(\031)28 b(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)26 b(\031)i Fl(2[)p Ff(F)10 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))20 b Ff(\000)j(F)10 b Fl(\()18 b(\026)-68 b Fg(m)p Fl(\)])32 b Fg(:)822 b Fl(\(1)p Fg(:)p Fl(20\))0 1306 y(The)31 b(precise)i(v)m(ersion)e(of)g(this)g(is)g(giv)m (en)f(in)h(Theorem)h(2.3,)e(and)i(it)e(is)h(the)g(k)m(ey)g(inequalit)m (y)f(b)s(ehind)i(the)0 1436 y Fi(dissip)-5 b(ation{dichotomy)35 b Fl(argumen)m(t.)58 b(It)37 b(enables)h(us)g(to)f(\\drop")h(extra)f(p) s(o)m(w)m(ers)h(of)g Ff(A)f Fl(when)h(\(1.18\))0 1565 y(holds.)120 1696 y(W)-8 b(e)35 b(shall)f(also)g(need)i(certain)f (momen)m(t)g(inequalities,)f(whic)m(h)i(sho)m(w)f(that)g Fg(v)t Fl(\()p Fg(t)p Fl(\))f(can't)g(spread)i(out)0 1825 y Fi(to)-5 b(o)33 b Fl(fast.)0 2055 y Fm(Theorem)e(1.5)e Fi(L)-5 b(et)31 b Fg(m)d Fl(=)47 b(\026)-69 b Fg(m)12 b Fl(+)g Fg(v)35 b Fi(b)-5 b(e)30 b(a)h(solution)g(of)f(\(1.5\))g(and)g (let)i Fg(C)38 b Fi(b)-5 b(e)30 b(a)h(p)-5 b(ositive)29 b(numb)-5 b(er.)44 b(De\014ne)0 2185 y Fg(\036)p Fl(\()p Fg(t)p Fl(\))35 b Fi(by)809 2385 y Fg(\036)p Fl(\()p Fg(t)p Fl(\))28 b(=)g(1)22 b(+)1286 2250 y Fe(Z)1342 2476 y Fj(I)-20 b(R)1439 2385 y Ff(j)p Fg(x)17 b Fl(\()o Ff(A)p Fg(v)t Fl(\))f Ff(j)1794 2344 y Fk(2)1838 2385 y Fl(d)p Fg(x)23 b Fl(+)f Fg(C)i Fl([)p Ff(F)10 b Fl(\()18 b(\026)-68 b Fg(m)22 b Fl(+)g Fg(v)t Fl(\))g Ff(\000)h(F)10 b Fl(\()18 b(\026)-68 b Fg(m)o Fl(\)])52 b Fg(:)554 b Fl(\(1)p Fg(:)p Fl(21\))0 2635 y Fi(Then)36 b(for)f(any)h Fg(\017)28 b(>)g Fl(0)p Fi(,)35 b(ther)-5 b(e)35 b(is)g(a)h(choic)-5 b(e)34 b(of)h Fg(C)g(<)28 b Ff(1)36 b Fi(and)f(an)h Fg(\017)2481 2650 y Fk(1)2554 2635 y Fg(>)28 b Fl(0)35 b Fi(so)h(that)f(one)h(has) 1170 2838 y Fl(d)p 1152 2883 92 4 v 1152 2974 a(d)p Fg(t)1255 2906 y(\036)p Fl(\()p Fg(t)p Fl(\))27 b Ff(\024)i Fl(4\(1)21 b(+)i Fg(\017)p Fl(\))17 b([)o Ff(F)10 b Fl(\()18 b(\026)-68 b Fg(m)22 b Fl(+)h Fg(v)t Fl(\))e Ff(\000)i(F)10 b Fl(\()18 b(\026)-68 b Fg(m)p Fl(\)])884 b(\(1)p Fg(:)p Fl(22\))0 3164 y Fi(whenever)38 b(\(1.18\))g(holds,)i(and)f Ff(jj)p Fg(v)t Ff(jj)1392 3179 y Fk(2)1469 3164 y Ff(\024)34 b Fg(\016)1624 3179 y Fk(1)1669 3164 y Fl(\()p Fg(\017)p Fl(\))p Fi(,)39 b Ff(jj)p Fg(v)1965 3128 y Fa(0)1992 3164 y Ff(jj)2048 3179 y Fk(2)2126 3164 y Ff(\024)c Fg(\024)2295 3179 y Fk(1)2340 3164 y Fl(\()p Fg(\017)p Fl(\))p Fi(,)k(and)g Ff(j)p Fg(a)p Fl(\()p Fg(t)p Fl(\))p Ff(j)32 b(\024)i Fl(1)39 b Fi(for)g(some)g(strictly)0 3293 y(p)-5 b(ositive)43 b(c)-5 b(onstants)43 b Fg(\024)871 3308 y Fk(1)916 3293 y Fl(\()p Fg(\017)p Fl(\))i Fi(and)f Fg(\016)1325 3308 y Fk(1)1370 3293 y Fl(\()p Fg(\017)p Fl(\))p Fi(.)71 b(R)-5 b(e)g(gar)g(d)5 b(less)43 b(of)h(whether)g(\(1.18\))g(hold)g(or) g(not,)i(ther)-5 b(e)44 b(is)g(a)0 3423 y(c)-5 b(onstant)35 b Fg(K)g(<)28 b Ff(1)1293 3536 y Fl(d)p 1276 3580 V 1276 3672 a(d)p Fg(t)1379 3603 y(\036)p Fl(\()p Fg(t)p Fl(\))f Ff(\024)h Fg(K)c Fl([)o Ff(F)10 b Fl(\()19 b(\026)-69 b Fg(m)22 b Fl(+)h Fg(v)t Fl(\))f Ff(\000)g(F)10 b Fl(\()18 b(\026)-68 b Fg(m)p Fl(\)])1008 b(\(1)p Fg(:)p Fl(23\))0 3836 y Fi(for)35 b(as)g(long)h(as)g Ff(jj)p Fg(v)736 3800 y Fa(0)763 3836 y Ff(jj)819 3851 y Fk(2)890 3836 y Ff(\024)29 b Fg(\024)1053 3851 y Fk(1)1098 3836 y Fl(\()p Fg(\017)p Fl(\))p Fi(,)35 b Ff(jj)p Fg(v)t Ff(jj)1446 3851 y Fk(2)1517 3836 y Ff(\024)28 b Fg(\016)1666 3851 y Fk(1)1711 3836 y Fl(\()p Fg(\017)p Fl(\))35 b Fi(and)h Ff(j)p Fg(a)p Fl(\()p Fg(t)p Fl(\))p Ff(j)26 b(\024)i Fl(1)p Fi(.)120 4067 y Fl(Theorem)44 b(1.5)f(is)h(pro)m(v)m(ed)g(in)g (section)g(3.)75 b(Theorems)45 b(1.4)e(and)h(1.5)f(are)h(the)g(main)g (ingredien)m(ts)0 4196 y(of)37 b(our)g(argumen)m(t)h(sp)s(eci\014c)g (to)f(the)g(Cahn{Hilliard)f(equation.)55 b(The)37 b(other)g(t)m(w)m(o)g (ingredien)m(ts)h(are)f(a)0 4326 y(constrained)h(form)f(of)g(the)g (uncertain)m(t)m(y)h(principle)g(inequalit)m(y)d(and)j(deca)m(y)f (estimate)g(for)g(a)g(system)0 4455 y(of)c(di\013eren)m(tial)h (inequalities)e(in)m(tro)s(duced)j(in)e([5].)120 4586 y(W)-8 b(e)35 b(will)f(no)m(w)i(explain)f(what)g(these)h(are,)g(and)g (ho)m(w)f(they)h(w)m(ork)e(together)i(to)f(pro)m(vide)g(the)h(pro)s(of) 0 4715 y(of)d(Theorem)h(1.1.)120 4846 y(The)e(constrained)h(form)f(of)g (the)g(uncertain)m(t)m(y)g(principle)h(inequalit)m(y)e([5])g(is)g(the)h (follo)m(wing:)43 b Fi(Under)0 4975 y(either)35 b(of)g(the)g(c)-5 b(onstr)g(aints)1072 4895 y Fe(R)1155 4975 y Fg( )t Fl(\()p Fg(x)p Fl(\)d)p Fg(x)27 b Fl(=)h(0)35 b Fi(or)h Fg( )t Fl(\(0\))26 b(=)j(0)p Fi(,)35 b(one)g(has)742 5145 y Fe(\022)816 5150 y(Z)932 5285 y Fg(x)989 5244 y Fk(2)1034 5285 y Ff(j)p Fg( )t Fl(\()p Fg(x)p Fl(\))p Ff(j)1294 5244 y Fk(2)1337 5285 y Fl(d)p Fg(x)1449 5145 y Fe(\023\022)1595 5150 y(Z)1712 5285 y Ff(j)p Fg( )1809 5244 y Fa(0)1836 5285 y Fl(\()p Fg(x)p Fl(\))p Ff(j)1999 5244 y Fk(2)2043 5285 y Fl(d)p Fg(x)2155 5145 y Fe(\023)2256 5285 y Ff(\025)2373 5218 y Fl(9)p 2373 5262 50 4 v 2373 5354 a(4)2435 5145 y Fe(\022)2508 5150 y(Z)2624 5285 y Ff(j)p Fg( )t Fl(\()p Fg(x)p Fl(\))p Ff(j)2884 5244 y Fk(2)2927 5285 y Fl(d)p Fg(x)3039 5145 y Fe(\023)3113 5165 y Fk(2)3645 5285 y Fl(\(1)p Fg(:)p Fl(24\))0 5579 y Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)c(22:40)1127 b Fl(7)p eop %%Page: 8 8 8 7 bop 120 230 a Fl(The)32 b(di\013erence)h(b)s(et)m(w)m(een)f (\(1.24\))e(and)i(the)g(usual)g(uncertain)m(t)m(y)g(principle)g(is)g(a) f(factor)g(of)h(9)f(in)h(the)0 360 y(constan)m(t,)j(and,)h(as)e(w)m(e)h (sho)m(wn)h(in)f([5],)f(this)g(is)h(crucial)g(for)g Fg(L)2358 323 y Fk(1)2437 360 y Fl(con)m(trol.)48 b(W)-8 b(e)35 b(wish)g(to)f(apply)h(this)f(to)0 489 y Fg( )k Fl(=)c Ff(A)p Fg(v)t Fl(.)55 b(It)37 b(is)g(clear)g(that)g Ff(A)p Fg(v)j Fl(will)c(ha)m(v)m(e)i(a)e(zero)i(somewhere,)h(a)e(tec)m(hnical) g(argumen)m(t)h(is)f(needed)0 619 y(to)44 b(con)m(trol)h(the)g(lo)s (cation.)77 b(T)-8 b(o)45 b(explain)f(ho)m(w)h(all)f(of)h(the)g(pieces) h(of)f(the)g(argumen)m(t)g(\014t)g(together,)0 748 y(assume)38 b(for)e(the)h(momen)m(t)g(that)f(the)h(initial)e(data)i(is)f(an)m (tisymmetric.)54 b(Then)37 b(the)g(solution)f(will)g(b)s(e)0 878 y(an)m(tisymmetric)d(for)g(all)g(time)g(and)g(so)1689 1007 y Ff(A)p Fg(v)t Fl(\(0)p Fg(;)17 b(t)p Fl(\))26 b(=)i(0)1434 b(\(1)p Fg(:)p Fl(25\))0 1203 y(for)34 b(all)g Fg(t)p Fl(.)47 b(The)34 b(tec)m(hnical)h(argumen)m(t)g(needed)h(to)e (remo)m(v)m(e)g(the)h(an)m(tisymmetry)f(assumption)g(will)g(b)s(e)0 1332 y(giv)m(en)f(in)g(section)h(2.)44 b(Ho)m(w)m(ev)m(er,)33 b(assuming)h(\(1.25\),)d(w)m(e)j(ha)m(v)m(e)g(from)f(\(1.16\))f(and)i (\(1.24\))e(that)1013 1589 y Fg(d)p 995 1633 88 4 v 995 1724 a(dt)1094 1575 y Fe(\002)1136 1656 y Ff(F)1218 1575 y Fe(\000)1263 1656 y Fg(m)p Fl(\()p 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b(problem)f(is)g(th)m(us)h(one)f(of)g(closure:)58 b(w)m(e)40 b(ha)m(v)m(e)g(to)0 2360 y(relate)33 b(the)h(quan)m(tit)m(y) e(on)h(the)h(righ)m(t)f(hand)h(side)g(to)f(the)g(excess)i(free)f (energy)-8 b(.)120 2491 y(No)m(w)41 b(w)m(e)h(are)g(ready)g(to)f(put)h (the)h(pieces)f(together.)70 b(When)43 b(\(1.20\))e(is)g(v)-6 b(alid,)43 b(in)m(terpreting)g(the)0 2620 y(appro)m(ximation)32 b(sign)i(appropriately)e(in)i(terms)f(of)h Fg(\017)p Fl(,)f(w)m(e)h(can)f(rewrite)h(\(1.26\))d(as)777 2877 y Fg(d)p 759 2921 88 4 v 759 3012 a(dt)859 2863 y Fe(\002)900 2944 y Ff(F)982 2863 y Fe(\000)1027 2944 y Fg(m)p Fl(\()p Fg(t)p Fl(\)\))22 b Ff(\000)h(F)1471 2863 y Fe(\000)1535 2944 y Fl(\026)-69 b Fg(m)1603 2863 y Fe(\001\003)1718 2944 y Ff(\024)28 b(\000)p Fl(9\(1)22 b Ff(\000)h Fg(\017)p Fl(\))2252 2877 y([)p Ff(F)10 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))21 b Ff(\000)h(F)10 b Fl(\()19 b(\026)-69 b Fg(m)p Fl(\)])3036 2840 y Fk(2)p 2252 2921 829 4 v 2500 3012 a Ff(k)p Fg(x)p Ff(A)p Fg(v)t Ff(k)2789 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5355 a(d)p Fg(t)2315 5287 y(\036)p Fl(\()p Fg(t)p Fl(\))28 b Ff(\024)2645 5261 y Fl(~)2621 5287 y Fg(B)5 b(f)11 b Fl(\()p Fg(t)p Fl(\))769 b(\(1)p Fg(:)p Fl(29\))0 5579 y Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)31 b(22:40)1127 b Fl(8)p eop %%Page: 9 9 9 8 bop 0 100 a Fl(with)36 b(the)h(di\013erence)g(b)s(et)m(w)m(een)1267 74 y(~)1241 100 y Fg(A=)p Fl(\()1431 74 y(~)1405 100 y Fg(A)23 b Fl(+)1628 74 y(~)1605 100 y Fg(B)t Fl(\))36 b(and)h(9)p Fg(=)p Fl(13)e(arbitrarily)g(small)h(for)g Fg(\017)h Fl(small)f(enough)h(for)0 229 y(all)32 b(times)i Fg(t)f Fl(suc)m(h)h(that)f(\(1.18\))f(holds,)i Ff(k)p Fg(v)t Fl(\()p Fg(t)p Fl(\))p Ff(k)1757 244 y Fk(2)1800 229 y Fl(,)f Ff(k)p Fg(v)1963 193 y Fa(0)1991 229 y Fl(\()p Fg(t)p Fl(\))p Ff(k)2155 244 y Fk(2)2232 229 y Fl(are)g(su\016cien)m (tly)h(small)f(and)g Ff(j)p Fg(a)p Fl(\()p Fg(t)p Fl(\))p Ff(j)26 b(\024)j Fl(1.)120 361 y(On)38 b(the)g(other)f(hand,)j(when)e (\(1.19\))e(holds,)j(there)f(is)f(plen)m(t)m(y)h(of)g(dissipation,)g 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y(our)c(system.)120 1269 y(Therefore)h(w)m(e)g (consider)g(the)g(system)f(of)h(di\013eren)m(tial)f(inequalities)1055 1532 y(d)p 1037 1577 92 4 v 1037 1668 a(d)p Fg(t)1141 1600 y(f)11 b Fl(\()p Fg(t)p Fl(\))26 b Ff(\024)j(\000)p Fg(A)1611 1532 y(f)11 b Fl(\()p Fg(t)p Fl(\))1785 1496 y Fk(2)p 1611 1577 218 4 v 1633 1668 a Fg(\036)p Fl(\()p Fg(t)p Fl(\))1940 1600 y(and)2230 1532 y(d)p 2212 1577 92 4 v 2212 1668 a(d)p Fg(t)2315 1600 y(\036)p Fl(\()p Fg(t)p Fl(\))28 b Ff(\024)g Fg(B)5 b(f)11 b Fl(\()p Fg(t)p Fl(\))769 b(\(1)p Fg(:)p Fl(30\))0 1930 y(with)33 b Fg(A)27 b Fl(=)h(9)34 b(and)f Fg(B)f Fl(=)d(4.)43 b(Theorem)34 b(5.1)f(of)g([4])g(sa)m(ys)g(that)g(for)g(an)m(y)h(solution)f(of)g (\(1.30\),)1008 2281 y Fg(f)11 b Fl(\()p Fg(t)p Fl(\))26 b Ff(\024)i Fg(f)11 b Fl(\(0\))1501 2240 y Fk(1)p Fa(\000)p Fj(q)1646 2281 y Fg(\036)p Fl(\(0\))1833 2240 y Fj(q)1877 2140 y Fe(\022)1962 2213 y Fg(\036)p Fl(\(0\))p 1962 2258 187 4 v 1962 2349 a Fg(f)g Fl(\(0\))2183 2281 y(+)23 b(\()p Fg(A)e Fl(+)i 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Fl(=)g(0)d(so)g(that)g(lim)2783 858 y Fj(t)p Fa(!1)2994 843 y Fg(a)p Fl(\()p Fg(t)p Fl(\))i(=)h Fg(a)d Fl(where)g Fg(a)g Fl(is)0 973 y(determined)39 b(through)f(\(1.10\).)56 b(Indeed,)40 b(the)e(map)g Fg(a)d Ff(7!)2228 892 y Fe(R)2275 1008 y Fj(I)-19 b(R)2373 973 y Fl(\()18 b(\026)-69 b Fg(m)2498 988 y Fj(a)2547 973 y Fl(\()p Fg(x)p Fl(\))22 b Ff(\000)g Fg(m)p Fl(\()p Fg(x;)17 b Fl(0\)\))f(d)p Fg(x)38 b Fl(is)f(linear,)h(and)0 1102 y(the)c(slop)s(e)f(is)g Ff(\000)618 1022 y Fe(R)665 1138 y Fj(I)-19 b(R)782 1102 y Fl(\026)-69 b Fg(m)850 1066 y Fa(0)850 1127 y Fj(a)899 1102 y Fl(\()p Fg(x)p Fl(\)d)p Fg(x)27 b Fl(=)h Ff(\000)p Fl(2,)33 b(as)h(one)f(sees)i (simply)e(b)m(y)g(di\013eren)m(tiating.)44 b(Th)m(us,)1010 1271 y Fe(\014)1010 1331 y(\014)1010 1391 y(\014)1010 1451 y(\014)1044 1280 y(Z)1099 1507 y Fj(I)-19 b(R)1197 1335 y Fe(\000)1261 1416 y Fl(\026)-69 b Fg(m)1329 1434 y Fj(a)p Fk(\()p Fj(t)p Fk(\))1470 1416 y Fl(\()p Fg(x)p Fl(\))22 b Ff(\000)h Fg(m)p Fl(\()p Fg(x;)17 b Fl(0\))2044 1335 y Fe(\001)2105 1416 y Fl(d)p Fg(x)2217 1271 y Fe(\014)2217 1331 y(\014)2217 1391 y(\014)2217 1451 y(\014)2278 1416 y Fl(=)28 b(2)p Ff(j)p Fg(a)p Fl(\()p Fg(t)p Fl(\))21 b Ff(\000)h Fg(a)p Ff(j)33 b Fg(:)0 1835 y Fh(2)45 b(F)-11 b(ree)45 b(Energy)g(Estimates)120 1966 y Fl(It)32 b(follo)m(ws)i(from)f (\(1.6\))f(and)i(the)f(de\014nition)h(of)g Ff(A)p Fl(,)e(one)i(has)846 2222 y(d)p 828 2267 92 4 v 828 2358 a(d)p Fg(t)931 2289 y Ff(F)10 b Fl(\()p Fg(m)p Fl(\))27 b(=)i Ff(\000)1405 2154 y Fe(Z)1460 2380 y Fj(I)-19 b(R)1558 2149 y Fe(\024)1651 2222 y Fl(d)p 1622 2267 113 4 v 1622 2358 a(d)p Fg(x)1763 2149 y Fe(\022)1837 2289 y Ff(A)p Fg(v)25 b Ff(\000)2102 2222 y Fl(1)p 2102 2267 50 4 v 2102 2358 a(2)2180 2209 y Fe(\000)2226 2289 y Fl(3)18 b(\026)-68 b Fg(mv)2415 2248 y Fk(2)2482 2289 y Fl(+)22 b Fg(v)2633 2248 y Fk(3)2678 2209 y Fe(\001)2724 2149 y(\023\025)2849 2170 y Fk(2)2911 2289 y Fl(d)p Fg(x)33 b(:)611 b Fl(\(2)p Fg(:)p Fl(1\))0 2587 y(F)-8 b(or)33 b(con)m(v)m(enience)i(of)e(notation,)g(de\014ne)774 2877 y Fg(U)39 b Fl(=)998 2809 y(1)p 998 2854 V 998 2945 a(2)1100 2809 y(d)p 1071 2854 113 4 v 1071 2945 a(d)p Fg(x)1212 2796 y Fe(\000)1258 2877 y Fl(3)19 b(\026)-69 b Fg(mv)1447 2836 y Fk(2)1514 2877 y Fl(+)22 b Fg(v)1665 2836 y Fk(3)1710 2796 y Fe(\001)1783 2877 y Fl(=)29 b Ff(\000)1978 2809 y Fl(3)p 1978 2854 50 4 v 1978 2945 a(2)2056 2796 y Fe(\000)2121 2877 y Fl(\026)-69 b Fg(m)2189 2836 y Fa(0)2217 2877 y Fg(v)2269 2836 y Fk(2)2336 2877 y Fl(+)23 b(2)18 b(\026)-68 b Fg(mv)t(v)2677 2836 y Fa(0)2727 2877 y Fl(+)22 b Fg(v)2878 2836 y Fk(2)2923 2877 y Fg(v)2975 2836 y Fa(0)3003 2796 y Fe(\001)3098 2877 y Fg(:)569 b Fl(\(2)p Fg(:)p Fl(2\))0 3160 y(No)m(w)33 b(for)g(an)m(y)h Fg(f)43 b Fl(and)34 b Fg(g)i Fl(in)e Fg(L)1123 3124 y Fk(2)1200 3160 y Fl(and)g(for)f(an)m(y)h(0)27 b Fg(<)h(\017)g(<)h Fl(1,)1253 3445 y Ff(k)p Fg(f)k Fl(+)22 b Fg(g)t Ff(k)1586 3404 y Fk(2)1586 3469 y(2)1657 3445 y Ff(\025)29 b Fl(\(1)21 b Ff(\000)i Fg(\017)p Fl(\))p Ff(k)p Fg(f)11 b Ff(k)2212 3404 y Fk(2)2212 3469 y(2)2278 3445 y Ff(\000)2390 3377 y Fl(1)p 2390 3422 V 2395 3513 a Fg(\017)2451 3445 y Ff(k)p Fg(g)t Ff(k)2603 3404 y Fk(2)2603 3469 y(2)3695 3445 y Fl(\(2)p Fg(:)p Fl(3\))0 3722 y(Com)m(bining)33 b(\(2.1\),\(2.2\))e(and)j(\(2.3\),)d(w)m(e)j(ha)m(v)m(e)256 4017 y Ff(\000)363 3949 y Fl(d)p 345 3994 92 4 v 345 4085 a(d)p Fg(t)449 4017 y Ff(F)10 b Fl(\()p Fg(m)p Fl(\))27 b(=)828 3881 y Fe(Z)883 4108 y Fj(I)-19 b(R)981 4017 y Ff(j)p Fl(\()p Ff(A)p Fg(v)t Fl(\))1219 3976 y Fa(0)1268 4017 y Fl(+)22 b Fg(U)11 b Ff(j)1474 3962 y Fk(2)1535 4017 y Fl(d)p Fg(x)28 b Ff(\025)g Fl(\(1)22 b Ff(\000)h Fg(\017)p Fl(\))2087 3881 y Fe(Z)2142 4108 y Fj(I)-19 b(R)2240 4017 y Ff(j)o Fl(\()p Ff(A)p Fg(v)t Fl(\))2477 3976 y Fa(0)2504 4017 y Ff(j)2532 3962 y Fk(2)2593 4017 y Fl(d)p Fg(x)22 b Ff(\000)2839 3949 y Fl(1)p 2839 3994 50 4 v 2844 4085 a Fg(\017)2917 3881 y Fe(Z)2973 4108 y Fj(I)-19 b(R)3071 4017 y Ff(j)o Fg(U)11 b Ff(j)3205 3967 y Fk(2)3266 4017 y Fl(d)p Fg(x)34 b(:)255 b Fl(\(2)p Fg(:)p Fl(4\))0 4316 y(The)41 b(follo)m(wing)e(lemma)h(is)h(closely)e (based)j(on)e(lemmas)g(and)h(argumen)m(ts)g(in)f(section)h(3)f(of)h ([4].)64 b(W)-8 b(e)0 4446 y(ha)m(v)m(e)29 b(stated)f(it)f(so)i(that)e (it)h(applied)h(to)e(a)h(general)g(class)h(of)f(p)s(oten)m(tials)g(b)s (ecause)i(the)e(pro)s(of,)h(although)0 4575 y(somewhat)k(in)m(v)m(olv)m (ed,)g(dep)s(ends)j(only)c(on)i(fairly)e(general)h(prop)s(erties)h(of) 53 b(\026)-69 b Fg(m)33 b Fl(and)h Ff(A)p Fl(.)0 4876 y Fm(Theorem)47 b(2.1)42 b Fi(L)-5 b(et)44 b Fg(v)i Ff(2)c Fg(L)1134 4840 y Fk(2)1178 4876 y Fl(\()p Fg(I)-18 b(R)q Fl(\))p Fi(,)44 b Fg(v)1486 4840 y Fa(0)1556 4876 y Ff(2)e Fg(L)1732 4840 y Fk(2)1776 4876 y Fl(\()p Fg(I)-18 b(R)q Fl(\))43 b Fi(and)2201 4796 y Fe(R)2284 4876 y Fg(v)t Fl(\()p Fg(x)p Fl(\))18 b(\026)-68 b Fg(m)2558 4840 y Fa(0)2586 4876 y Fl(\()p Fg(x)p Fl(\))p Fg(dx)41 b Fl(=)h(0)h Fi(then)h(ther)-5 b(e)43 b(exists)f(a)0 5005 y(p)-5 b(ositive)34 b(c)-5 b(onstant)35 b Fg(\015)6 b Fi(,)34 b(such)i(that)1433 5158 y Fe(Z)1549 5213 y(\002)1590 5294 y Fl(\()p Ff(A)p Fg(v)t Fl(\))1799 5244 y Fa(0)1827 5213 y Fe(\003)1869 5230 y Fk(2)1930 5294 y Fl(d)p Fg(x)28 b Ff(\025)g Fg(\015)6 b Ff(jj)p Fg(v)2341 5253 y Fa(0)2367 5294 y Ff(jj)2423 5253 y Fk(2)2423 5319 y(2)3695 5294 y Fl(\(2)p Fg(:)p Fl(5\))0 5579 y Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)31 b(22:40)1102 b Fl(10)p eop %%Page: 11 11 11 10 bop 0 100 a Fi(wher)-5 b(e)35 b Ff(A)h Fi(is)e(the)i(line)-5 b(ar)35 b(op)-5 b(er)g(ator)35 b(de\014ne)-5 b(d)35 b(in)h(\(1.14\).)0 318 y Fm(Pro)s(of:)57 b Fl(First)38 b(observ)m(e)i(that)e(\()p Ff(A)p Fg(v)t Fl(\))1405 268 y Fa(0)1470 318 y Fl(=)f Ff(A)p Fg(v)1716 282 y Fa(0)1770 318 y Fl(+)26 b Fg(V)1953 282 y Fa(0)1981 318 y Fg(v)43 b Fl(where)c Fg(V)62 b Fl(is)38 b(giv)m(en)h(in)g(\(1.15\).)59 b(Next,)39 b Fg(v)t Fl(\()p Fg(x)p Fl(\))e(=)0 448 y Fg(v)t Fl(\()p Fg(y)t Fl(\))10 b(+)280 367 y Fe(R)345 392 y Fj(x)326 483 y(y)412 448 y Fg(v)464 411 y Fa(0)492 448 y Fl(\()p Fg(z)t Fl(\)d)p Fg(z)t Fl(.)43 b(Multiply)26 b(b)s(oth)h(sides)h(b)m(y) 46 b(\026)-69 b Fg(m)1880 411 y Fa(0)1909 448 y Fl(\()p Fg(y)t Fl(\),)27 b(and)g(in)m(tegrate)g(in)g Fg(y)t Fl(.)41 b(Since)3180 367 y Fe(R)3263 448 y Fg(v)t Fl(\()p Fg(y)t Fl(\))18 b(\026)-68 b Fg(m)3533 411 y Fa(0)3561 448 y Fl(\()p Fg(y)t Fl(\))p Fg(dy)30 b Fl(=)0 617 y(0,)j(and)g(since)548 536 y Fe(R)650 617 y Fl(\026)-68 b Fg(m)719 580 y Fa(0)747 617 y Fl(\()p Fg(y)t Fl(\))p Fg(dy)30 b Fl(=)e(2,)1129 959 y Fg(v)t Fl(\()p Fg(x)p Fl(\))f(=)1460 892 y(1)p 1460 936 50 4 v 1460 1028 a(2)1539 824 y Fe(Z)1638 848 y Fa(1)1594 1050 y(\0001)1776 959 y Fl(\026)-69 b Fg(m)1844 918 y Fa(0)1873 959 y Fl(\()p Fg(y)t Fl(\))2021 819 y Fe(\022)2093 824 y(Z)2192 848 y Fj(x)2148 1050 y(y)2259 959 y Fg(v)2311 918 y Fa(0)2339 959 y Fl(\()p Fg(z)t Fl(\)d)p Fg(z)2572 819 y Fe(\023)2663 959 y Fl(d)p Fg(y)928 b Fl(\(2)p Fg(:)p Fl(6\))120 1303 y(Hence)1539 1441 y(\()p Ff(A)p Fg(v)t Fl(\))1749 1400 y Fa(0)1803 1441 y Fl(=)28 b Ff(A)p Fg(v)2040 1400 y Fa(0)2090 1441 y Fl(+)23 b Fg(K)7 b(v)2334 1400 y Fa(0)3695 1441 y Fl(\(2)p Fg(:)p Fl(7\))0 1649 y(where)969 1842 y Fg(K)g(\036)p Fl(\()p Fg(x)p Fl(\))27 b(=)h Fg(V)1467 1801 y Fa(0)1495 1842 y Fl(\()p Fg(x)p Fl(\))1642 1775 y(1)p 1642 1819 V 1642 1910 a(2)1720 1706 y Fe(Z)1819 1731 y Fa(1)1775 1933 y(\0001)1957 1842 y Fl(\026)-69 b Fg(m)2025 1801 y Fa(0)2054 1842 y Fl(\()p Fg(y)t Fl(\))2202 1701 y Fe(\022)2274 1706 y(Z)2373 1731 y Fj(x)2329 1933 y(y)2440 1842 y Fg(\036)p Fl(\()p Fg(z)t Fl(\)d)p Fg(z)2732 1701 y Fe(\023)2824 1842 y Fl(d)p Fg(y)120 2149 y Fl(The)32 b(op)s(erator)f Fg(K)39 b Fl(is)32 b(compact)g(on)g Fg(L)1548 2113 y Fk(2)1593 2149 y Fl(.)43 b(A)32 b(detailed)g(pro)s(of)g(in)g(a)f (closely)h(related)g(case)g(is)g(giv)m(en)g(in)0 2279 y([4].)43 b(No)m(w)33 b(consider)h(the)g(quadratic)f(form)g Ff(Q)p Fl(\()p Fg(\036)p Fl(\))h(giv)m(en)f(b)m(y)1454 2536 y Ff(Q)p Fl(\()p Fg(\036)p Fl(\))28 b(=)g Ff(k)17 b Fl(\()o Ff(A)22 b Fl(+)h Fg(K)7 b Fl(\))16 b Fg(\036)p Ff(k)2368 2495 y Fk(2)2368 2561 y(2)0 2794 y Fl(for)33 b Fg(\036)h Fl(in)f(the)h(domain)f(of)h Ff(A)p Fl(.)120 2927 y(W)-8 b(e)42 b(next)f(sho)m(w)i(that)e Ff(Q)p Fl(\()p Fg(\036)p Fl(\))h Fg(>)h Fl(0)e(for)h(all)f Fg(\036)h Fl(in)g(its)g(domain.)69 b(Supp)s(ose)44 b(on)e(the)g(con)m(trary)f (that)0 3057 y Ff(Q)p Fl(\()p Fg(\036)p Fl(\))28 b(=)g(0)33 b(for)h(some)f Fg(\036)h Fl(in)f(the)h(domain)f(of)g Ff(Q)p Fl(,)g(whic)m(h)i(is)e(the)g(op)s(erator)g(domain)g(of)h Ff(A)p Fl(.)43 b(De\014ne)1266 3390 y Fg(\021)t Fl(\()p Fg(x)p Fl(\))28 b(=)1587 3255 y Fe(Z)1686 3280 y Fj(x)1642 3481 y Fk(0)1753 3390 y Fg(\036)p Fl(\()p Fg(y)t Fl(\)d)p Fg(y)j Fl(=)d Ff(h)p Fl(1)2272 3408 y Fk([0)p Fj(;x)p Fk(])2430 3390 y Fg(;)17 b(\036)p Ff(i)33 b Fg(:)0 3718 y Fl(It)g(follo)m(ws)g(b)m(y)g(the)h(Sc)m(h)m(w)m(arz)h(inequalit)m(y)d (that)1217 3976 y Ff(j)p Fg(\021)t Fl(\()p Fg(x)p Fl(\))p Ff(j)27 b(\024)h(k)p Fg(\036)p Ff(k)1752 3991 y Fk(2)1797 3886 y Fe(p)p 1896 3886 113 4 v 1896 3976 a Ff(j)p Fg(x)p Ff(j)199 b Fl(for)33 b(all)99 b Fg(x)33 b(:)1012 b Fl(\(2)p Fg(:)p Fl(8\))0 4233 y(It)34 b(then)i(follo)m(ws)e(that)g Fg(K)7 b(\036)30 b Fl(=)h Fg(V)1252 4197 y Fa(0)1280 4233 y Fg(\021)c Ff(\000)1468 4194 y Fk(1)p 1468 4210 40 4 v 1468 4268 a(2)1520 4233 y Fg(V)1600 4197 y Fa(0)1628 4233 y Ff(h)19 b Fl(\026)-69 b Fg(m)1754 4197 y Fa(0)1782 4233 y Fg(;)17 b(\021)t Ff(i)p Fl(,)34 b(where)h(the)g(inner)g(pro)s (duct)h(on)e(the)h(righ)m(t)g(is)f(w)m(ell)0 4363 y(de\014ned)h(b)s (ecause)g(of)e(the)h(exp)s(onen)m(tial)f(deca)m(y)h(of)52 b(\026)-69 b Fg(m)2008 4327 y Fa(0)2070 4363 y Fl(and)33 b(\(2.8\).)43 b(Hence)659 4687 y(\()p Ff(A)22 b Fl(+)g Fg(K)7 b Fl(\))16 b Fg(\036)28 b Fl(=)h Ff(A)p Fg(\021)1372 4646 y Fa(0)1421 4687 y Fl(+)23 b Fg(V)1601 4646 y Fa(0)1629 4687 y Fg(\021)j Ff(\000)1816 4620 y Fl(1)p 1816 4664 50 4 v 1816 4756 a(2)1878 4687 y Fg(V)1958 4646 y Fa(0)1986 4687 y Ff(h)18 b Fl(\026)-68 b Fg(m)2112 4646 y Fa(0)2140 4687 y Fg(;)17 b(\021)t Ff(i)27 b Fl(=)h(\()p Ff(A)p Fg(\021)t Fl(\))2619 4637 y Fa(0)2669 4687 y Ff(\000)2781 4620 y Fl(1)p 2781 4664 V 2781 4756 a(2)2842 4687 y Fg(V)2923 4646 y Fa(0)2951 4687 y Ff(h)18 b Fl(\026)-68 b Fg(m)3077 4646 y Fa(0)3105 4687 y Fg(;)17 b(\021)t Ff(i)0 4993 y Fl(Since)34 b(the)g(righ)m(t)f(side)h(is)f(a)g(total)f(deriv)-6 b(ativ)m(e,)32 b(w)m(e)i(ha)m(v)m(e)1495 5312 y Ff(A)p Fg(\021)26 b Ff(\000)1761 5244 y Fl(1)p 1761 5289 V 1761 5380 a(2)1823 5312 y Fg(V)c Ff(h)d Fl(\026)-69 b Fg(m)2029 5270 y Fa(0)2058 5312 y Fg(;)17 b(\021)t Ff(i)26 b Fl(=)j Fg(C)1297 b Fl(\(2)p Fg(:)p Fl(9\))0 5579 y Fc(15)p Fb(=nov)r(ember)r (=)p Fc(2000;)31 b(22:40)1102 b Fl(11)p eop %%Page: 12 12 12 11 bop 0 100 a Fl(where)38 b Fg(C)45 b Fl(is)37 b(a)g(constan)m(t.) 57 b(T)-8 b(o)37 b(determine)h Fg(C)7 b Fl(,)38 b(m)m(ultiply)f(b)s (oth)g(sides)i(b)m(y)56 b(\026)-69 b Fg(m)2931 63 y Fa(0)2959 100 y Fl(,)38 b(and)g(in)m(tegrate.)56 b(Note)0 229 y(that)219 149 y Fe(R)321 229 y Fl(\026)-69 b Fg(m)389 193 y Fa(0)434 229 y Fl(\()p Ff(A)p Fg(\021)t Fl(\))16 b(d)p Fg(x)34 b Fl(=)g(0)j Fg(;)f Fl(b)s(ecause)j(\(2.8\))c(p)s(ermits)i(the)g(in)m (tegration)g(b)m(y)f(parts.)55 b(The)38 b(computaion)0 359 y(then)c(yields)f Fg(C)i Fl(=)28 b(\(1)p Fg(=)p Fl(2\))p Ff(h)18 b Fl(\026)-68 b Fg(m)1072 323 y Fa(0)1099 359 y Fg(;)17 b(\021)t Ff(i)p Fl(.)43 b(Putting)33 b(this)g(in)g(\(2.9\))f (yields)i Ff(A)17 b Fl(\()n Fg(\021)27 b Ff(\000)22 b Fl(\(1)p Fg(=)p Fl(2\))p Ff(h)c Fl(\026)-68 b Fg(m)3166 323 y Fa(0)3193 359 y Fg(;)17 b(\021)t Ff(i)p Fl(\))27 b(=)h(0.)120 489 y(No)m(w)40 b(an)m(y)h(solution)g Fg( )j Fl(of)d Ff(A)p Fg( )j Fl(=)d(0)g(either)g(deca)m(ys)h(exp)s(onen)m (tially)e(or)g(div)m(erges)i(exp)s(onen)m(tially)0 619 y(at)37 b(in\014nit)m(y)-8 b(,)38 b(since,)i(due)e(the)g(the)g(rapid)g (deca)m(y)g(of)57 b(\026)-69 b Fg(m)2067 583 y Fa(0)2096 619 y Fl(,)38 b(and)g(hence)i Fg(V)22 b Fl(,)38 b Fg(\036)2847 583 y Fa(00)2934 619 y Ff(\031)d Fg(\036)p Fl(.)58 b(The)38 b(only)f(option)0 748 y(consisten)m(t)c(with)g(\(2.8\))e(is)h(exp)s (onen)m(tial)g(deca)m(y)-8 b(.)44 b(Hence)34 b(w)m(e)f(m)m(ust)g(ha)m (v)m(e)f(that)g Fg(\021)25 b Ff(\000)c Fl(\(1)p Fg(=)p Fl(2\))p Ff(h)d Fl(\026)-68 b Fg(m)3522 712 y Fa(0)3549 748 y Fg(;)17 b(\021)t Ff(i)32 b Fl(is)g(in)0 878 y(the)f Fg(L)237 842 y Fk(2)312 878 y Fl(k)m(ernel)h(of)f Ff(A)p Fl(.)42 b(Ho)m(w)m(ev)m(er,)32 b(w)m(e)f(kno)m(w)g(from)g(Lemma)g(1.2)f (that)h(this)g(is)f(spanned)j(b)m(y)50 b(\026)-69 b Fg(m)3551 842 y Fa(0)3579 878 y Fl(.)44 b(So)31 b(w)m(e)0 1007 y(m)m(ust)j(ha)m(v)m(e)g Fg(\021)26 b Ff(\000)d Fl(\(1)p Fg(=)p Fl(2\))p Ff(h)17 b Fl(\026)-67 b Fg(m)1006 971 y Fa(0)1033 1007 y Fg(;)17 b(\021)t Ff(i)27 b Fl(=)i Fg(\013)18 b Fl(\026)-68 b Fg(m)1454 971 y Fa(0)1482 1007 y Fl(.)44 b(In)m(tegrating)34 b(b)s(oth)f(sides)h(against)52 b(\026)-69 b Fg(m)2975 971 y Fa(0)3037 1007 y Fl(yields)33 b Fg(\013)28 b Fl(=)h(0.)44 b(Hence)0 1137 y Fg(\021)37 b Fl(is)c(constan)m(t,)h(and)g(so)f Fg(\036)28 b Fl(=)g(0,)33 b(as)g(w)m(as)h(to)e(b)s(e)i(sho)m(wn.)120 1267 y(W)-8 b(e)33 b(will)f(no)m(w)i(sho)m(w)g(that)f(there)g(is)h(a)f Fg(\015)f(>)d Fl(0)k(so)g(that)1644 1499 y Ff(Q)p Fl(\()p Fg(\036)p Fl(\))28 b Ff(\025)g Fg(\015)6 b Ff(k)p Fg(\036)p Ff(k)2212 1458 y Fk(2)2212 1524 y(2)3645 1499 y Fl(\(2)p Fg(:)p Fl(10\))0 1731 y(for)37 b(all)g Fg(\036)p Fl(.)56 b(The)38 b(pro)s(of)f(is)h(similar)e(to)h(the)g(pro)s(of)h(of)f(W)-8 b(eyl's)37 b(lemma,)h(though)f(note)h(that)e Ff(A)25 b Fl(+)g Fg(K)44 b Fl(is)0 1861 y(not)33 b(self)h(adjoin)m(t.)120 1991 y(If)d(\(2.10\))f(w)m(ere)h(false,)h(there)g(w)m(ould)f(exist)g (an)g(in\014nite)g(orthonormal)g(sequence)i Ff(f)p Fg(\036)3316 2006 y Fj(n)3370 1991 y Ff(g)e Fl(in)g Fg(L)3633 1955 y Fk(2)3708 1991 y Fl(suc)m(h)0 2121 y(that)k(lim)356 2136 y Fj(n)p Fa(!1)586 2121 y Ff(Q)p Fl(\()p Fg(\036)765 2136 y Fj(n)820 2121 y Fl(\))c(=)g(0.)51 b(Since)36 b(the)g(sequence)h Ff(f)p Fg(\036)2087 2136 y Fj(n)2141 2121 y Ff(g)e Fl(is)h (orthonormal,)f(it)f(con)m(v)m(erges)j(w)m(eakly)d(to)0 2250 y(zero.)49 b(Next,)34 b(let)g Fg(c)715 2265 y Fj(n)799 2250 y Fl(=)d Ff(h)p Fg(\036)1005 2265 y Fj(n)1059 2250 y Fg(;)k Fl(\026)-68 b Fg(m)1191 2214 y Fa(0)1219 2250 y Ff(i)34 b Fl(and)i(note)e(that)h(lim)2066 2265 y Fj(n)p Fa(!1)2296 2250 y Fg(c)2339 2265 y Fj(n)2423 2250 y Fl(=)c(0.)48 b(If)35 b(the)g Fg(c)2974 2265 y Fj(n)3063 2250 y Fl(are)g(not)f(all)g (zero,)h(let)0 2380 y Fg(n)60 2395 y Fk(0)133 2380 y Fl(b)s(e)29 b(suc)m(h)h(that)e Ff(j)p Fg(c)766 2395 y Fj(n)815 2405 y Fc(0)859 2380 y Ff(j)f(\025)i(j)p Fg(c)1091 2395 y Fj(n)1145 2380 y Ff(j)f Fl(for)h(all)e Fg(n)p Fl(,)i(and)g(de\014ne)2084 2353 y(~)2071 2380 y Fg(\036)2130 2395 y Fj(n)2213 2380 y Fl(=)f Fg(\036)2377 2395 y Fj(n)2444 2380 y Ff(\000)13 b Fl(\()p Fg(c)2616 2395 y Fj(n)2670 2380 y Fg(=c)2763 2395 y Fj(n)2812 2405 y Fc(0)2857 2380 y Fl(\))p Fg(\036)2955 2395 y Fj(n)3004 2405 y Fc(0)3048 2380 y Fl(.)42 b(It)28 b(is)h(clear)f(that)h(the)13 2483 y(~)0 2509 y Fg(\036)59 2524 y Fj(n)145 2509 y Fl(are)j(all)f (orthogonal)g(to)50 b(\026)-69 b Fg(m)1151 2473 y Fa(0)1179 2509 y Fl(,)32 b(and)g(moreo)m(v)m(er)g(the)g(mo)s(di\014ed)h(sequence) g(still)e(con)m(v)m(erges)i(w)m(eakly)d(to)0 2639 y(zero,)i(and)g (still)f(satis\014es)h(lim)1125 2654 y Fj(n)p Fa(!1)1355 2639 y Ff(Q)p Fl(\()1488 2612 y(~)1475 2639 y Fg(\036)1534 2654 y Fj(n)1589 2639 y Fl(\))27 b(=)h(0)k(and)g(lim)2172 2654 y Fj(n)p Fa(!1)2402 2639 y Ff(k)2465 2612 y Fl(~)2452 2639 y Fg(\036)2511 2654 y Fj(n)2565 2639 y Ff(k)2615 2602 y Fk(2)2615 2663 y(2)2687 2639 y Fl(=)d(1.)43 b(\(If)31 b(all)g(of)h(the)g Fg(c)3512 2654 y Fj(n)3598 2639 y Fl(v)-6 b(anish,)0 2768 y(w)m(e)34 b(simply)e(tak)m(e)694 2742 y(~)681 2768 y Fg(\036)740 2783 y Fj(n)823 2768 y Fl(=)c Fg(\036)987 2783 y Fj(n)1075 2768 y Fl(for)33 b(all)f Fg(n)p Fl(.\))44 b(Moreo)m(v)m(er,)33 b(b)m(y)h(Lemma)f(1.2,) 1494 3067 y Ff(kA)1637 3041 y Fl(~)1624 3067 y Fg(\036)1683 3082 y Fj(n)1737 3067 y Ff(k)1787 3026 y Fk(2)1787 3091 y(2)1859 3067 y Ff(\025)2001 2999 y Fl(9)p 1976 3044 100 4 v 1976 3135 a(16)2088 3067 y Ff(k)2151 3041 y Fl(~)2138 3067 y Fg(\036)2197 3082 y Fj(n)2251 3067 y Ff(k)2301 3026 y Fk(2)2301 3091 y(2)2379 3067 y Fg(:)1238 b Fl(\(2)p Fg(:)p Fl(11\))120 3371 y(Since)34 b(the)f(sequence)i Ff(f)1027 3345 y Fl(~)1014 3371 y Fg(\036)1073 3386 y Fj(n)1128 3371 y Ff(g)e Fl(con)m(v)m(erges)h(w)m(eakly)e(to)h(zero,) 1679 3603 y(lim)1644 3663 y Fj(n)p Fa(!1)1868 3603 y Fg(K)1973 3577 y Fl(~)1960 3603 y Fg(\036)2019 3618 y Fj(n)2101 3603 y Fl(=)c(0)1388 b(\(2)p Fg(:)p Fl(12\))0 3896 y(strongly)37 b(in)h Fg(L)579 3860 y Fk(2)624 3896 y Fl(.)58 b(Also,)39 b(it)e(is)h(clear)g(that)g(the)g(op)s(erator)g (domain)g(of)g Ff(A)g Fl(is)g(the)g(form)g(domain)g(of)h Ff(Q)0 4026 y Fl(and)34 b(that)f Ff(kA)p Fg(\036)p Ff(k)649 3990 y Fk(2)649 4050 y(2)720 4026 y Ff(\024)c Fl(2)893 3945 y Fe(\000)938 4026 y Ff(Q)p Fl(\()p Fg(\036)p Fl(\))22 b(+)h Ff(k)p Fg(K)7 b(\036)p Ff(k)1529 3990 y Fk(2)1529 4050 y(2)1573 3945 y Fe(\001)1652 4026 y Fl(on)33 b(this)h(domain.)44 b(Th)m(us,)1178 4317 y Ff(kA)1321 4291 y Fl(~)1308 4317 y Fg(\036)1367 4332 y Fj(n)1421 4317 y Ff(k)1471 4276 y Fk(2)1471 4342 y(2)1543 4317 y Ff(\024)28 b Fl(2)1715 4207 y Fe(\020)1774 4317 y Ff(Q)p Fl(\()1907 4291 y(~)1894 4317 y Fg(\036)1953 4332 y Fj(n)2008 4317 y Fl(\))22 b(+)g Ff(k)p Fg(K)7 b Ff(k)2360 4276 y Fk(2)2405 4317 y Ff(k)2468 4291 y Fl(~)2455 4317 y Fg(\036)2514 4332 y Fj(n)2568 4317 y Ff(k)2618 4276 y Fk(2)2618 4342 y(2)2662 4207 y Fe(\021)3645 4317 y Fl(\(2)p Fg(:)p Fl(13\))0 4617 y(where)29 b Ff(k)p Fg(K)7 b Ff(k)28 b Fl(denote)h(the)f(op)s (erator)g(norm)h(of)f Fg(K)35 b Fl(on)28 b Fg(L)2067 4581 y Fk(2)2112 4617 y Fl(.)42 b(In)29 b(particular,)f(the)h Ff(kA)3095 4591 y Fl(~)3082 4617 y Fg(\036)3141 4632 y Fj(n)3195 4617 y Ff(k)3245 4632 y Fk(2)3318 4617 y Fl(are)f(uniformly)0 4747 y(b)s(ounded)35 b(b)m(y)e(a)g(\014nite)h (constan)m(t.)44 b(No)m(w,)883 4979 y Ff(Q)p Fl(\()1016 4952 y(~)1003 4979 y Fg(\036)1062 4994 y Fj(n)1117 4979 y Fl(\))27 b Ff(\024)h(kA)1431 4952 y Fl(~)1418 4979 y Fg(\036)1477 4994 y Fj(n)1531 4979 y Ff(k)1581 4937 y Fk(2)1581 5003 y(2)1648 4979 y Fl(+)23 b Ff(k)p Fg(K)1902 4952 y Fl(~)1890 4979 y Fg(\036)1949 4994 y Fj(n)2003 4979 y Ff(k)2053 4937 y Fk(2)2053 5003 y(2)2119 4979 y Fl(+)g(2)p Ff(kA)2412 4952 y Fl(~)2399 4979 y Fg(\036)2458 4994 y Fj(n)2512 4979 y Ff(k)2562 4994 y Fk(2)2606 4979 y Ff(k)p Fg(K)2761 4952 y Fl(~)2748 4979 y Fg(\036)2807 4994 y Fj(n)2861 4979 y Ff(k)2911 4994 y Fk(2)2989 4979 y Fg(:)628 b Fl(\(2)p Fg(:)p Fl(14\))0 5210 y(By)31 b(\(2.12\))g(and)i (\(2.13\),)e(the)i(last)e(t)m(w)m(o)i(terms)f(on)h(the)g(righ)m(t)f(in) g(\(2.14\))f(tend)i(to)f(zero)h(with)f Fg(n)p Fl(.)43 b(Hence)0 5340 y(for)c(an)m(y)f Fg(\017)f(>)g Fl(0,)i(w)m(e)g(obtain)f (that)g Ff(kA)1490 5314 y Fl(~)1477 5340 y Fg(\036)1536 5355 y Fj(n)1590 5340 y Ff(k)1640 5304 y Fk(2)1640 5365 y(2)1721 5340 y Ff(\024)e Fg(\017)p Ff(k)1937 5314 y Fl(~)1924 5340 y Fg(\036)1983 5355 y Fj(n)2038 5340 y Ff(k)2088 5304 y Fk(2)2088 5365 y(2)2171 5340 y Fl(for)j(all)e (su\016cien)m(tly)i(large)f Fg(n)p Fl(,)h(whic)m(h)h(w)m(ould)0 5579 y Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)31 b(22:40)1102 b Fl(12)p eop %%Page: 13 13 13 12 bop 0 100 a Fl(con)m(tradict)34 b(\(2.11\).)42 b(This)34 b(pro)m(v)m(es)g(\(2.10\).)42 b(No)m(w)33 b(b)m(y)h(\(2.7\),) d(when)k Ff(h)18 b Fl(\026)-68 b Fg(m)2680 63 y Fa(0)2708 100 y Fg(;)17 b(v)t Ff(i)26 b Fl(=)j(0,)j Ff(k)17 b Fl(\()o Ff(A)p Fg(v)t Fl(\))3362 50 y Fa(0)3406 100 y Ff(k)3456 63 y Fk(2)3456 124 y(2)3528 100 y Fl(=)29 b Ff(Q)p Fl(\()p Fg(v)3806 63 y Fa(0)3834 100 y Fl(\),)0 229 y(and)34 b(hence)g(w)m(e)g(ha)m(v)m(e)g(the)f(result.)p 1379 229 70 70 v 120 483 a(Com)m(bining)g(this)g(result)h(with)f(\(2.4\),)e(w)m (e)j(ha)m(v)m(e)582 806 y Ff(\000)690 738 y Fl(d)p 671 783 92 4 v 671 874 a(d)p Fg(t)775 806 y Ff(F)10 b Fl(\()p Fg(m)p Fl(\))27 b Ff(\025)h Fl(\(1)22 b Ff(\000)h Fl(2)p Fg(\017)p Fl(\))1511 670 y Fe(Z)1566 896 y Fj(I)-19 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2321 y Fk(2)2570 2382 y Fa(1)2681 2357 y Ff(\024)28 b Fl(2)p Ff(k)p Fg(v)t Ff(k)2988 2372 y Fk(2)3032 2357 y Ff(k)p Fg(v)3134 2321 y Fa(0)3162 2357 y Ff(k)3212 2372 y Fk(2)3290 2357 y Fl(.)p 3406 2357 70 70 v 0 2672 a Fm(Pro)s(of)41 b(of)f(Theorem)f(1.4:) 47 b Fl(No)m(w)34 b(c)m(ho)s(ose)h Fg(\024)g Fl(and)g Fg(\016)j Fl(so)d(that)f Fg(\017)2458 2687 y Fk(0)2533 2672 y Ff(\024)c Fg(\017)2680 2636 y Fk(2)2725 2672 y Fg(\015)6 b Fl(,)33 b(and)i(then)h(from)e(\(2.15\),w)m(e)0 2801 y(ha)m(v)m(e)g(the)f(inequalit)m(y)f(of)i(Theorem)g(2.1.)p 1641 2801 V 120 3116 a(W)-8 b(e)27 b(no)m(w)g(pro)m(v)m(e)h(a)f(b)s (ound)h(that)f(will)f(enable)i(us)f(to)g(apply)g(the)g(dissipation{dic) m(hotom)m(y)g(argumen)m(t)0 3245 y(describ)s(ed)35 b(in)e(the)h(in)m (tro)s(duction.)0 3438 y Fm(Theorem)f(2.3)c Fi(F)-8 b(or)32 b(al)5 b(l)32 b Fg(\017)c(>)h Fl(0)p Fi(,)i(ther)-5 b(e)32 b(is)f(an)g Fg(\017)1789 3453 y Fk(0)1862 3438 y Fg(>)d Fl(0)k Fi(such)f(that)h(for)f(or)h(al)5 b(l)32 b Fg(v)k Fi(ortho)-5 b(gonal)31 b(to)51 b Fl(\026)-69 b Fg(m)3659 3402 y Fa(0)3719 3438 y Fi(with)1231 3689 y Ff(I)7 b Fl(\()20 b(\026)-70 b Fg(m)23 b Fl(+)g Fg(v)t Fl(\))k(=)h Ff(k)p Fl(\()p Ff(A)p Fg(v)t Fl(\))2024 3648 y Fa(0)2051 3689 y Ff(k)2101 3648 y Fk(2)2101 3714 y(2)2173 3689 y Ff(\024)g Fg(\017)2318 3648 y Fk(2)2318 3714 y(0)2364 3689 y Ff(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)974 b Fl(\(2)p Fg(:)p Fl(17\))0 3940 y Fi(one)35 b(has)1068 4072 y Fl(\(1)21 b Ff(\000)i Fg(\017)p Fl(\))p Ff(kA)p Fg(v)t Ff(k)1589 4031 y Fk(2)1589 4096 y(2)1661 4072 y Ff(\024)28 b(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)26 b(\024)i Fl(\(1)21 b(+)i Fg(\017)p Fl(\))p Ff(kA)p Fg(v)t Ff(k)2725 4031 y Fk(2)2725 4096 y(2)2805 4072 y Fg(:)812 b Fl(\(2)p Fg(:)p Fl(18\))0 4465 y Fm(Pro)s(of:)45 b Fl(First,)33 b(b)m(y)g(Lemma)h(1.2,)e(inserting)i Ff(A)1798 4429 y Fk(1)p Fj(=)p Fk(2)1922 4465 y Fg(v)j Fl(in)d(place)f(of)h Fg(v)t Fl(,)1556 4778 y Ff(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)26 b(\024)2006 4711 y Fl(4)p 2006 4755 50 4 v 2006 4846 a(3)2068 4778 y Ff(kA)p Fg(v)t Ff(k)2300 4737 y Fk(2)2300 4803 y(2)3645 4778 y Fl(\(2)p Fg(:)p Fl(19\))0 5089 y(so)33 b(w)m(e)h(ha)m(v)m(e)g(that)f Ff(k)p Fl(\()p Ff(A)p Fg(v)t Fl(\))975 5053 y Fa(0)1001 5089 y Ff(k)1051 5053 y Fk(2)1051 5114 y(2)1123 5089 y Ff(\024)c Fl(\(4)p Fg(\017)1358 5053 y Fk(2)1358 5114 y(0)1402 5089 y Fg(=)p Fl(3\))p Ff(k)p Fg(v)t Ff(k)1693 5053 y Fk(2)1693 5114 y(2)1737 5089 y Fl(.)44 b(Then,)34 b(using)f(the)h(notation)e(of)i (Lemma)f(1.2,)445 5255 y Fe(\014)445 5315 y(\014)479 5340 y Ff(kA)p Fg(v)t Ff(k)711 5299 y Fk(2)711 5365 y(2)776 5340 y Ff(\000)23 b(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)1183 5255 y Fe(\014)1183 5315 y(\014)1242 5340 y Fl(=)28 b Ff(jh)p Fg(v)1466 5299 y Fa(00)1516 5340 y Fg(;)17 b Ff(A)p Fg(v)t Ff(i)k Fl(+)i Ff(h)p Fg(V)f(v)t(;)17 b Ff(A)p Fg(v)t Ff(ij)25 b(\024)j(jh)p Fg(v)2517 5299 y Fa(0)2545 5340 y Fg(;)17 b Fl(\()p Ff(A)p Fg(v)t Fl(\))2800 5299 y Fa(0)2826 5340 y Ff(ij)k Fl(+)i Ff(jh)p Fg(V)f(v)t(;)17 b Ff(A)p Fg(v)t Ff(ij)0 5579 y Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)31 b(22:40)1102 b Fl(13)p eop %%Page: 14 14 14 13 bop 120 100 a Fl(No)m(w)32 b Ff(jh)p Fg(V)22 b(v)t(;)17 b Ff(A)p Fg(v)t Ff(ij)26 b(\024)i(k)p Fg(v)t Ff(k)1072 115 y Fk(2)1116 100 y Ff(k)p Fg(V)22 b Ff(k)1296 115 y Fk(2)1341 100 y Ff(kA)p Fg(v)t Ff(k)1573 115 y Fa(1)1689 100 y Fl(and)34 b(b)m(y)f(\(2.17\))f(and)i(\(2.19\))1058 409 y Ff(kA)p Fg(v)t Ff(k)1290 367 y Fk(2)1290 433 y Fa(1)1401 409 y Ff(\024)29 b Fl(2)p Ff(kA)p Fg(v)t Ff(k)1789 424 y Fk(2)1832 409 y Ff(k)p Fl(\()p Ff(A)p Fg(v)t Fl(\))2092 367 y Fa(0)2119 409 y Ff(k)2169 424 y Fk(2)2241 409 y Ff(\024)2358 341 y Fl(8)p Fg(\017)2448 356 y Fk(0)p 2358 386 135 4 v 2401 477 a Fl(3)2505 409 y Ff(kA)p Fg(v)t Ff(k)2737 367 y Fk(2)2737 433 y(2)2814 409 y Fg(:)0 704 y Fl(Then,)j(b)m(y)e(Lemma)h(1.2)f(and)g(Sc)m(h)m(w)m(arz's)i (inequalit)m(y)-8 b(,)30 b Ff(k)p Fg(v)t Ff(k)2203 719 y Fk(2)2275 704 y Ff(\024)e Fl(\(4)p Fg(=)p Fl(3\))p Ff(kA)p Fg(v)t Ff(k)2840 719 y Fk(2)2882 704 y 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y(to)33 b(con)m(trol.)120 2264 y(T)-8 b(o)32 b(con)m(trol)h(the)h(term)f(in)m (v)m(olving)45 b(_)-41 b Fg(a)p Fl(\()p Fg(t)p Fl(\),)32 b(\014rst)i(note)f(that)1325 2460 y Fe(Z)1424 2595 y Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\))22 b Ff(\000)41 b Fl(\026)-68 b Fg(m)1873 2613 y Fj(a)p Fk(\()p Fj(t)p Fk(\))2014 2595 y Fl(\))18 b(\026)-68 b Fg(m)2140 2554 y Fa(0)2140 2622 y Fj(a)p Fk(\()p Fj(t)p Fk(\))2280 2595 y Fl(d)p Fg(x)28 b Fl(=)h(0)0 2937 y(whic)m(h)c(holds)h(for)e(all)g Fg(t)p Fl(.)41 b(Di\013eren)m(tiating)25 b(this)f(equation)h(in)f Fg(t)p Fl(,)i(one)f(obtains)37 b(_)-40 b Fg(a)p Fl(\()p Fg(t)p Fl(\))3025 2856 y Fe(\000)3070 2937 y Ff(k)18 b Fl(\026)-68 b Fg(m)3207 2900 y Fa(0)3207 2961 y Fj(a)3255 2937 y Ff(k)3305 2900 y Fk(2)3305 2961 y(2)3355 2937 y Ff(\000)5 b(h)p Fg(v)t(;)35 b Fl(\026)-68 b Fg(m)3660 2900 y Fa(00)3660 2961 y Fj(a)3710 2937 y Ff(i)3749 2856 y Fe(\001)3823 2937 y Fl(=)0 3066 y Ff(\000)94 2986 y Fe(R)161 3066 y Fl(\()p Fg(@)6 b(m=@)g(t)p Fl(\))17 b(\026)-67 b Fg(m)617 3030 y Fa(0)617 3091 y Fj(a)664 3066 y Fl(.)44 b(Th)m(us,)34 b(w)m(e)f(ha)m(v)m(e)873 3426 y Ff(j)12 b Fl(_)-40 b Fg(a)o Fl(\()p Fg(t)p Fl(\))p Ff(j)27 b(\024)h Fl(2)1277 3282 y Fe(\014)1277 3341 y(\014)1277 3401 y(\014)1277 3461 y(\014)1310 3290 y(Z)1410 3286 y(\022)1498 3359 y Fg(\016)t Ff(F)p 1495 3403 136 4 v 1495 3494 a Fg(\016)t(m)1643 3286 y Fe(\023)1716 3306 y Fa(0)1763 3426 y Fl(\026)-69 b Fg(m)1831 3385 y Fa(00)1831 3451 y Fj(a)1882 3426 y Fl(d)p Fg(x)1994 3282 y Fe(\014)1994 3341 y(\014)1994 3401 y(\014)1994 3461 y(\014)2089 3426 y Ff(\024)28 b Fl(2)2244 3336 y Fe(p)p 2343 3336 341 4 v 2343 3426 a Ff(I)7 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))p Ff(k)19 b Fl(\026)-69 b Fg(m)2820 3385 y Fa(00)2872 3426 y Ff(k)2922 3441 y Fk(2)2999 3426 y Fg(;)668 b Fl(\(3)p Fg(:)p Fl(6\))0 3776 y(as)26 b(long)g(as)g Ff(k)p Fg(v)t Ff(k)591 3791 y Fk(2)661 3776 y Fl(is)g(su\016cien)m(tly)g(small)f(that)1707 3696 y Fe(\000)1753 3776 y Ff(k)18 b Fl(\026)-68 b Fg(m)1890 3740 y Fa(0)1890 3801 y Fj(a)1938 3776 y Ff(k)1988 3740 y Fk(2)1988 3801 y(2)2040 3776 y Ff(\000)8 b(h)p Fg(v)t(;)35 b Fl(\026)-68 b Fg(m)2348 3740 y Fa(00)2348 3801 y Fj(a)2398 3776 y Ff(i)2437 3696 y Fe(\001)2510 3776 y Fg(>)29 b Fl(1)p Fg(=)p Fl(2.)41 b(Since)k(\026)-69 b Fg(m)3174 3740 y Fa(0)3229 3776 y Fl(has)26 b(exp)s(onen)m(tial)0 3906 y(deca)m(y)-8 b(,)38 b(this)g(giv)m(es)f(us)h(the)f(b)s(ounds)i(w) m(e)f(will)e(need)i(to)f(con)m(trol)g(the)h(e\013ects)g(of)g(the)f (terms)h(in)m(v)m(olving)12 4035 y(_)-40 b Fg(a)p Fl(\()p Fg(t)p Fl(\),)25 b(as)g(w)m(e)h(will)e(see)i(b)s(elo)m(w.)41 b(The)25 b(non-linear)h(terms)f(are)g(easily)g(handled)h(without)e(an)m (y)h(preparatory)0 4165 y(analysis.)120 4299 y(W)-8 b(e)41 b(no)m(w)h(turn)g(to)g(the)f(linear)h(part,)h(whic)m(h)g(will)d(pro)m (vide)i(all)f(of)g(the)h(most)g(imp)s(ortan)m(t)f(terms.)0 4429 y(Consider)34 b(the)g(gro)m(wth)f(of)g Fg( )t Fl(\()p Fg(t)p Fl(\))f(when)i Fg(v)j Fl(ev)m(olv)m(es)d(according)f(to)g(the)h (linearized)f(equation)1660 4694 y Fg(@)p 1642 4738 95 4 v 1642 4829 a(@)6 b(t)1748 4761 y(v)32 b Fl(=)c(\()p Ff(A)p Fg(v)t Fl(\))2142 4711 y Fa(00)2242 4761 y Fg(:)1425 b Fl(\(3)p Fg(:)p Fl(7\))0 5069 y(The)34 b(computations)g(that)f(follo) m(w)g(can)h(b)s(e)g(more)g(clearly)e(and)i(compactly)f(represen)m(ted)j (if)e(w)m(e)g(in)m(tro-)0 5199 y(duce)g(the)g(notation)1227 5340 y Fg(\030)e Fl(=)c Fg(x)17 b Fl(\()p Ff(A)p Fg(v)t Fl(\))1691 5290 y Fa(0)1935 5340 y Fl(and)199 b Fg(\021)32 b Fl(=)c Ff(A)p Fg(v)37 b(:)1022 b Fl(\(3)p Fg(:)p Fl(8\))0 5579 y Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)31 b(22:40)1102 b Fl(15)p eop %%Page: 16 16 16 15 bop 0 100 a Fm(Lemma)43 b(3.1)75 b Fi(L)-5 b(et)39 b Fg(v)t Fl(\()p Fg(x;)17 b(t)p Fl(\))38 b Fi(solve)h(\(3.7\),)g(and)g (let)h Fg( )t Fl(\()p Fg(t)p Fl(\))e Fi(b)-5 b(e)39 b(de\014ne)-5 b(d)39 b(in)g(terms)g(of)g Fg(v)k Fi(thr)-5 b(ough)40 b(\(3.3\).)0 229 y(Then)c(for)f(any)h Fg(\013)28 b(>)g Fl(0)p Fi(,)237 491 y Fl(d)p 219 536 92 4 v 219 627 a(d)p Fg(t)323 558 y( )t Fl(\()p Fg(t)p Fl(\))e(=)637 418 y Fe(\022)711 558 y Fl(12)c(+)944 491 y(1)g(+)h(4)p Ff(k)p Fg(V)f Ff(k)1346 506 y Fk(1)p 944 536 446 4 v 1110 627 a Fl(2)p Fg(\013)1402 418 y Fe(\023)1492 558 y Ff(h)p Fg(\021)1584 517 y Fa(0)1612 558 y Fg(;)17 b(\021)1710 517 y Fa(0)1737 558 y Ff(i)22 b Fl(+)1897 448 y Fe(\020)1957 558 y Fl(2)g(+)2141 491 y Fg(\013)p 2141 536 65 4 v 2148 627 a Fl(2)2217 558 y Ff(k)p Fl(\()p Fg(x)2363 517 y Fk(2)2407 558 y Fg(V)2487 517 y Fa(0)2515 558 y Fl(\))p Ff(k)2604 517 y Fk(2)2604 583 y Fa(1)2710 558 y Fl(+)h(2)p Fg(\013)p Ff(k)p Fg(V)f Ff(k)3104 573 y Fk(1)3148 448 y Fe(\021)3224 558 y Ff(h)p Fg(\021)t(;)17 b(\021)t Ff(i)242 b Fl(\(3)p Fg(:)p Fl(9\))33 888 y(where)34 b Fg(\021)e Fl(=)c Ff(A)p Fg(v)t Fl(.)0 1148 y Fm(Pro)s(of:)44 b Fl(Let)32 b Fg(V)53 b Fl(b)s(e)32 b(the)f(p)s(oten)m(tial)g(de\014ned)i (in)e(\(1.15\).)42 b(Then)32 b(one)f(easily)g(computes)h(the)g(comm)m (uta-)0 1277 y(tors)1048 1327 y Fe(\024)1141 1400 y Fg(@)p 1113 1445 116 4 v 1113 1536 a(@)6 b(x)1240 1468 y(;)17 b Ff(A)1365 1327 y Fe(\025)1444 1468 y Fl(=)28 b Fg(V)1629 1427 y Fa(0)1857 1468 y Fl(and)216 b([)p Fg(x;)17 b Ff(A)p Fl(])26 b(=)i(2)2692 1400 y Fg(@)p 2664 1445 V 2664 1536 a(@)6 b(x)2824 1468 y(:)793 b Fl(\(3)p Fg(:)p Fl(10\))0 1747 y(Clearly)-8 b(,)1210 1882 y(d)p 1192 1926 92 4 v 1192 2018 a(d)p Fg(t)1296 1949 y( )t Fl(\()p Fg(t)p Fl(\))26 b(=)i(2)1677 1814 y Fe(Z)1732 2040 y Fj(I)-19 b(R)1830 1949 y Fg(x)1887 1908 y Fk(2)1948 1949 y Fl(\()p Ff(A)p Fg(v)t Fl(\))16 b Ff(A)h Fl(\()o Ff(A)p Fg(v)t Fl(\))2479 1899 y Fa(00)2546 1949 y Fl(d)p Fg(x)34 b(:)0 2225 y Fl(No)m(w)28 b(one)h(comm)m(utes)g(deriv)-6 b(ativ)m(es)28 b(and)h(m)m(ultiples)g(of)f Fg(x)h Fl(past)f Ff(A)g Fl(and)h(in)m (tegrates)f(b)m(y)h(parts)f(to)g(obtain)0 2354 y(a)39 b(dissipativ)m(e)f(term)h(of)g(the)g(form)h Ff(\000)1458 2274 y Fe(R)1505 2390 y Fj(I)-19 b(R)1602 2354 y Fg(x)17 b Fl(\()p Ff(A)p Fg(v)t Fl(\))1885 2304 y Fa(0)1929 2354 y Ff(A)2026 2274 y Fe(\000)2071 2354 y Fg(x)g Fl(\()p Ff(A)p Fg(v)t Fl(\))2354 2304 y Fa(0)2382 2274 y Fe(\001)2444 2354 y Fl(d)p Fg(x)39 b Fl(in)m(to)g(whic)m(h)g(p)s(ositiv)m(e)f(terms) i(can)0 2484 y(b)s(e)34 b(absorb)s(ed.)120 2616 y(The)f(result,)h(in)f (the)g(notation)g(\(3.8\),)f(is)h(that)729 2876 y(d)p 711 2921 V 711 3012 a(d)p Fg(t)814 2943 y( )t Fl(\()p Fg(t)p Fl(\))27 b(=)h Ff(\000)p Fl(2)p Ff(h)p Fg(\030)5 b(;)17 b Ff(A)p Fg(\030)5 b Ff(i)19 b(\000)k Fl(4)p Ff(h)p Fg(\030)5 b(;)17 b(\021)1912 2902 y Fa(00)1961 2943 y Ff(i)22 b(\000)g Fl(4)p Ff(h)p Fg(x\021)t(;)17 b Ff(A)p Fg(\021)2498 2902 y Fa(0)2525 2943 y Ff(i)22 b(\000)g Fl(2)p Ff(h)p Fg(\021)2827 2902 y Fa(0)2855 2943 y Fg(;)17 b(x)2957 2902 y Fk(2)3001 2943 y Fg(V)3081 2902 y Fa(0)3109 2943 y Fg(\021)t Ff(i)0 3246 y Fl(The)34 b(last)f(three)g(terms)h (require)f(further)i(manipulation.)43 b(First:)764 3563 y Ff(h)p Fg(\030)5 b(;)17 b(\021)950 3522 y Fa(00)1000 3563 y Ff(i)27 b Fl(=)h Ff(\000h)p Fg(\030)1336 3522 y Fa(0)1363 3563 y Fg(;)17 b(\021)1461 3522 y Fa(0)1489 3563 y Ff(i)27 b Fl(=)h Ff(\000h)p Fg(x\021)1886 3522 y Fa(00)1937 3563 y Fg(;)17 b(\021)2035 3522 y Fa(0)2063 3563 y Ff(i)22 b(\000)g(h)p Fg(\021)2315 3522 y Fa(0)2343 3563 y Fg(;)17 b(\021)2441 3522 y Fa(0)2468 3563 y Ff(i)28 b Fl(=)g Ff(\000)2729 3496 y Fl(1)p 2729 3540 50 4 v 2729 3631 a(2)2791 3563 y Ff(h)p Fg(\021)2883 3522 y Fa(0)2911 3563 y Fg(;)17 b(\021)3009 3522 y Fa(0)3036 3563 y Ff(i)33 b Fg(:)0 3866 y Fl(This)h(term)f(is)g(con)m(trolled)h(b) m(y)f(the)h(deriv)-6 b(ativ)m(e)33 b(of)g(the)h(excess)g(free)h(energy) -8 b(.)44 b(Second,)34 b(one)g(has,)g(using)0 3995 y(\(3.10\))835 4131 y Ff(h)p Fg(x\021)t(;)17 b Ff(A)p Fg(\021)1162 4090 y Fa(0)1189 4131 y Ff(i)27 b Fl(=)h Ff(h)p Fg(\021)t(;)17 b Ff(A)p Fg(\030)5 b Ff(i)20 b Fl(+)j(2)p Ff(h)p Fg(\021)t(;)17 b(\021)2025 4090 y Fa(00)2074 4131 y Ff(i)28 b Fl(=)g Ff(h)p Fg(\021)t(;)17 b Ff(A)p Fg(\030)5 b Ff(i)20 b(\000)i Fl(2)p Ff(h)p Fg(\021)2812 4090 y Fa(0)2840 4131 y Fg(;)17 b(\021)2938 4090 y Fa(0)2965 4131 y Ff(i)33 b Fg(:)0 4336 y Fl(Finally)-8 b(,)31 b(for)j(an)m(y)f Fg(\013)28 b(>)g Fl(0,)220 4653 y Ff(h)p Fg(\021)312 4612 y Fa(0)340 4653 y Fg(;)17 b(x)442 4612 y Fk(2)486 4653 y Fg(V)566 4612 y Fa(0)594 4653 y Fg(\021)t Ff(i)28 b(\024)g(h)p Fg(\021)911 4612 y Fa(0)938 4653 y Fg(;)17 b(\021)1036 4612 y Fa(0)1064 4653 y Ff(i)1103 4612 y Fk(1)p Fj(=)p Fk(2)1228 4653 y Ff(h)p Fl(\()p Fg(x)1363 4612 y Fk(2)1407 4653 y Fg(V)1487 4612 y Fa(0)1515 4653 y Fl(\))p Fg(\021)t(;)g Fl(\()p Fg(x)1748 4612 y Fk(2)1791 4653 y Fg(V)1872 4612 y Fa(0)1900 4653 y Fl(\))p Fg(\021)t Ff(i)2031 4612 y Fk(1)p Fj(=)p Fk(2)2183 4653 y Ff(\024)2332 4585 y Fl(1)p 2300 4630 114 4 v 2300 4721 a(2)p Fg(\013)2426 4653 y Ff(h)p Fg(\021)2518 4612 y Fa(0)2546 4653 y Fg(;)g(\021)2644 4612 y Fa(0)2671 4653 y Ff(i)22 b Fl(+)2844 4585 y Fg(\013)p 2844 4630 65 4 v 2851 4721 a Fl(2)2920 4653 y Ff(k)p Fl(\()p Fg(x)3066 4612 y Fk(2)3110 4653 y Fg(V)3190 4612 y Fa(0)3218 4653 y Fl(\))p Ff(k)3307 4612 y Fk(2)3307 4677 y Fa(1)3391 4653 y Ff(h)p Fg(\021)t(;)17 b(\021)t Ff(i)32 b Fg(:)0 4956 y Fl(Putting)h(ev)m(erything)g(together,)g(one)g (obtains:)366 5218 y(d)p 348 5262 92 4 v 348 5354 a(d)p Fg(t)452 5285 y( )t Fl(\()p Fg(t)p Fl(\))26 b Ff(\024)j(\000)p Fl(2)p Ff(h)p Fg(\030)5 b(;)17 b Ff(A)p Fg(\030)5 b Ff(i)19 b(\000)j Fl(4)p Ff(h)p Fg(\030)5 b(;)17 b Ff(A)p Fg(\021)t Ff(i)j Fl(+)1787 5145 y Fe(\022)1861 5285 y Fl(10)h(+)2126 5218 y(1)p 2094 5262 114 4 v 2094 5354 a(2)p Fg(\013)2220 5145 y Fe(\023)2310 5285 y Ff(h)p Fg(\021)2402 5244 y Fa(0)2430 5285 y Fg(;)c(\021)2528 5244 y Fa(0)2555 5285 y Ff(i)22 b Fl(+)2727 5218 y Fg(\013)p 2727 5262 65 4 v 2734 5354 a Fl(2)2803 5285 y Ff(k)p Fl(\()p Fg(x)2949 5244 y Fk(2)2994 5285 y Fg(V)3074 5244 y Fa(0)3102 5285 y Fl(\))p Ff(k)3191 5244 y Fk(2)3191 5310 y Fa(1)3275 5285 y Ff(h)p Fg(\021)t(;)17 b(\021)t Ff(i)32 b Fg(:)0 5579 y Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)f(22:40)1102 b Fl(16)p eop %%Page: 17 17 17 16 bop 0 100 a Fl(No)m(w)33 b(one)h(uses)g(that)699 332 y Ff(\000)p Fl(2)p Ff(h)p Fg(\030)5 b(;)17 b Ff(A)p Fg(\030)5 b Ff(i)19 b(\000)k Fl(4)p Ff(h)p Fg(\030)5 b(;)17 b Ff(A)p Fg(\021)t Ff(i)25 b Fl(=)j Ff(\000)p Fl(2)p Ff(h)p Fl(\()p Fg(\030)e Fl(+)d Fg(\021)t Fl(\))p Fg(;)17 b Ff(A)p Fl(\()p Fg(\030)25 b Fl(+)d Fg(\021)t Fl(\))p Ff(i)g Fl(+)h(2)p Ff(h)p Fg(\021)t(;)17 b Ff(A)p Fg(\021)t Ff(i)31 b Fg(:)444 b Fl(\(3)p Fg(:)p Fl(11\))0 564 y(But)33 b Ff(h)p Fg(\021)t(;)17 b Ff(A)p Fg(\021)t Ff(i)26 b Fl(=)i Ff(h)p Fg(\021)730 528 y Fa(0)758 564 y Fg(;)17 b(\021)856 528 y Fa(0)883 564 y Ff(i)22 b Fl(+)h Ff(h)p Fg(\021)t(;)17 b(V)k(\021)t Ff(i)h Fl(+)g Ff(h)p Fg(\021)t(;)17 b(\021)t Ff(i)27 b(\024)h(h)p Fg(\021)1926 528 y Fa(0)1954 564 y Fg(;)17 b(\021)2052 528 y Fa(0)2079 564 y Ff(i)22 b Fl(+)h Ff(k)p Fg(V)f Ff(k)2420 579 y Fk(1)2464 564 y Ff(k)p Fg(\021)t Ff(k)2617 528 y Fk(2)2617 589 y Fa(1)2723 564 y Fl(+)h Ff(h)p Fg(\021)t(;)17 b(\021)t Ff(i)p Fl(,)32 b(and)1039 863 y Ff(k)p Fg(\021)t Ff(k)1192 822 y Fk(2)1192 888 y Fa(1)1304 863 y Ff(\024)c Fl(2)p Ff(k)p Fg(\021)1562 822 y Fa(0)1590 863 y Ff(k)1640 878 y Fk(2)1684 863 y Ff(k)p Fg(\021)t Ff(k)1837 878 y Fk(2)1909 863 y Ff(\024)2034 796 y Fl(1)p 2026 841 65 4 v 2026 932 a Fg(\013)2102 863 y Ff(h)p Fg(\021)2194 822 y Fa(0)2222 863 y Fg(;)17 b(\021)2320 822 y Fa(0)2347 863 y Ff(i)22 b Fl(+)h Fg(\013)p Ff(h)p Fg(\021)t(;)17 b(\021)t Ff(i)32 b Fg(:)0 1144 y Fl(Altogether)814 1347 y Ff(h)p Fg(\021)t(;)17 b Ff(A)p Fg(\021)t Ff(i)26 b(\024)1254 1207 y Fe(\022)1327 1347 y Fl(1)c(+)1511 1280 y Ff(k)p Fg(V)g Ff(k)1691 1295 y Fk(1)p 1511 1324 225 4 v 1591 1416 a Fg(\013)1748 1207 y Fe(\023)1837 1347 y Ff(h)p Fg(\021)1929 1306 y Fa(0)1957 1347 y Fg(;)17 b(\021)2055 1306 y Fa(0)2082 1347 y Ff(i)22 b Fl(+)h(\(1)f(+)g Fg(\013)p Ff(k)p Fg(V)g Ff(k)2697 1362 y Fk(1)2742 1347 y Fl(\))16 b Ff(h)p Fg(\021)t(;)h(\021)t Ff(i)32 b Fg(:)559 b Fl(\(3)p Fg(:)p Fl(12\))0 1619 y(Putting)33 b(\(3.12\))e(in)m(to)j(\(3.11\))d(giv)m(es)i(the)h(result.)p 1921 1619 70 70 v 0 1974 a Fm(Lemma)k(3.2)1384 2103 y Ff(h)p Fg(\021)t(;)17 b(\021)t Ff(i)26 b(\024)j(h)p Fg(\021)1837 2062 y Fa(0)1864 2103 y Fg(;)17 b(\021)1962 2062 y Fa(0)1990 2103 y Ff(i)22 b Fl(+)g Ff(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)31 b Fg(:)0 2295 y Fm(Pro)s(of:)45 b Fl(By)33 b(Sc)m(h)m(w)m(arz,) h(for)g(an)m(y)f Fg(\013)28 b(>)g Fl(0,)377 2588 y Ff(h)p Fg(\021)t(;)17 b(\021)t Ff(i)26 b Fl(=)i Ff(hA)856 2547 y Fk(1)p Fj(=)p Fk(2)981 2588 y Fg(\021)t(;)17 b Ff(A)1159 2547 y Fk(1)p Fj(=)p Fk(2)1283 2588 y Fg(v)t Ff(i)27 b(\024)i(h)p Fg(\021)t(;)17 b Ff(A)p Fg(\021)t Ff(i)1816 2547 y Fk(1)p Fj(=)p Fk(2)1939 2588 y Ff(h)p Fg(v)t(;)g Ff(A)p Fg(v)t Ff(i)2246 2547 y Fk(1)p Fj(=)p Fk(2)2397 2588 y Ff(\024)2514 2521 y Fg(\013)p 2514 2566 65 4 v 2521 2657 a Fl(2)2590 2588 y Ff(h)p Fg(\021)t(;)g Ff(A)p Fg(\021)t Ff(i)k Fl(+)3063 2521 y(1)p 3031 2566 114 4 v 3031 2657 a(2)p Fg(\013)3157 2588 y Ff(h)p Fg(v)t(;)c Ff(A)p Fg(v)t Ff(i)32 b Fg(;)0 2874 y Fl(and)38 b Ff(h)p Fg(\021)t(;)17 b Ff(A)p Fg(\021)t Ff(i)33 b(\024)i(h)p Fg(\021)744 2838 y Fa(0)771 2874 y Fg(;)17 b(\021)869 2838 y Fa(0)897 2874 y Ff(i)24 b Fl(+)i Ff(k)p Fg(V)47 b Fl(+)25 b(1)p Ff(k)1420 2889 y Fa(1)1504 2874 y Ff(h)p Fg(\021)t(;)17 b(\021)t Ff(i)p Fl(.)55 b(Since)38 b Ff(k)p Fg(V)47 b Fl(+)26 b(1)p Ff(k)2438 2889 y Fa(1)2556 2874 y Fl(=)35 b(1,)j(w)m(e)g(can)g(c)m(ho)s(ose)g Fg(\013)c Fl(=)h(1)i(and)0 3004 y(com)m(bine)d(the)g(ab)s(o)m(v)m(e)f(to)g (obtain)g(the)g(result.)p 1808 3004 70 70 v 0 3359 a Fm(Pro)s(of)h(of)f(Theorem)f(1.5)d Fl(First,)g(w)m(e)g(deal)g(with)f (the)h(inhomogenous)g(terms)g(in)m(v)m(olving)40 b(_)-41 b Fg(a)p Fl(\()p Fg(t)p Fl(\))28 b(on)h(the)0 3488 y(righ)m(t)k(in)g (\(3.4\),)f(as)h(they)g(con)m(tribute)h(to)1239 3656 y Fe(\014)1239 3715 y(\014)1239 3775 y(\014)1239 3835 y(\014)1272 3665 y(Z)1327 3891 y Fj(I)-19 b(R)1425 3800 y Fg(x)1482 3759 y Fk(2)1543 3719 y Fe(\000)1589 3800 y Ff(A)1669 3818 y Fj(a)p Fk(\()p Fj(t)p Fk(\))1809 3800 y Fg(v)1861 3719 y Fe(\001)1953 3733 y Fg(@)p 1935 3777 95 4 v 1935 3868 a(@)6 b(t)2058 3719 y Fe(\000)2104 3800 y Ff(A)2184 3818 y Fj(a)p Fk(\()p Fj(t)p Fk(\))2324 3800 y Fg(v)2376 3719 y Fe(\001)2438 3800 y Fl(d)p Fg(x)2550 3656 y Fe(\014)2550 3715 y(\014)2550 3775 y(\014)2550 3835 y(\014)2633 3800 y Fg(:)0 4112 y Fl(Using)33 b(the)h(b)s(ound)g (\(3.6\),)d(w)m(e)j(ha)m(v)m(e)g(that)602 4280 y Fe(\014)602 4339 y(\014)602 4399 y(\014)602 4459 y(\014)635 4424 y Fl(3)702 4289 y Fe(Z)757 4515 y Fj(I)-19 b(R)854 4343 y Fe(\000)900 4424 y Ff(A)980 4442 y Fj(a)p Fk(\()p Fj(t)p Fk(\))1120 4424 y Fg(v)1172 4343 y Fe(\001)1234 4424 y Fg(x)1291 4383 y Fk(2)1355 4424 y Fl(\026)-69 b Fg(m)1423 4383 y Fa(0)1423 4451 y Fj(a)p Fk(\()p Fj(t)p Fk(\))1564 4424 y Fl(dx)1672 4280 y Fe(\014)1672 4339 y(\014)1672 4399 y(\014)1672 4459 y(\014)1722 4424 y Ff(j)12 b Fl(_)-40 b Fg(a)o Fl(\()p Fg(t)p Fl(\))p Ff(j)27 b(\024)h Fl(3)p Ff(kA)2256 4442 y Fj(a)p Fk(\()p Fj(t)p Fk(\))2396 4424 y Fg(x)2453 4383 y Fk(2)2516 4424 y Fl(\026)-69 b Fg(m)2584 4383 y Fa(0)2584 4451 y Fj(a)p Fk(\()p Fj(t)p Fk(\))2725 4424 y Ff(k)2775 4439 y Fk(2)2820 4424 y Ff(k)p Fg(v)t Ff(k)2972 4439 y Fk(2)3016 4424 y Ff(j)12 b Fl(_)-40 b Fg(a)o Fl(\()p Fg(t)p Fl(\))p Ff(j)33 b Fg(:)0 4736 y Fl(No)m(w)g(applying)g(\(3.6\),)e(the)j(con)m(tribution)g(of)f(the)h (term)f(in)m(v)m(olving)45 b(_)-41 b Fg(a)p Fl(\()p Fg(t)p Fl(\))32 b(is)i(b)s(ounded)g(ab)s(o)m(v)m(e)g(b)m(y)1044 4968 y(6)p Ff(kA)1224 4986 y Fj(a)p Fk(\()p Fj(t)p Fk(\))1363 4968 y Fg(x)1420 4927 y Fk(2)1484 4968 y Fl(\026)-69 b Fg(m)1552 4986 y Fj(a)p Fk(\()p Fj(t)p Fk(\))1693 4968 y Ff(k)1743 4983 y Fk(2)1787 4968 y Ff(k)p Fg(v)t Ff(k)1939 4983 y Fk(2)1984 4968 y Ff(k)18 b Fl(\026)-68 b Fg(m)2121 4927 y Fa(00)2121 4996 y Fj(a)p Fk(\()p Fj(t)p Fk(\))2261 4968 y Ff(k)2311 4983 y Fk(2)2356 4878 y Fe(p)p 2456 4878 341 4 v 90 x Ff(I)7 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))33 b Fg(:)0 5201 y Fl(It)40 b(is)g(here)h(that)f(w)m(e)g(b)s (egin)h(using)f(the)h(h)m(yp)s(othesis)g(that)e Ff(j)p Fg(a)p Fl(\()p Fg(t)p Fl(\))p Ff(j)f(\024)i Fl(1.)64 b(The)41 b(exp)s(onen)m(tial)f(deca)m(y)h(of)19 5330 y(\026)-69 b Fg(m)87 5294 y Fa(0)87 5362 y Fj(a)p Fk(\()p Fj(t)p Fk(\))264 5330 y Fl(w)m(ould)36 b(not)g(giv)m(e)g(a)g(b)s(ound)h (on)f Ff(kA)1612 5348 y Fj(a)p Fk(\()p Fj(t)p Fk(\))1752 5330 y Fg(x)1809 5294 y Fk(2)1873 5330 y Fl(\026)-69 b Fg(m)1941 5294 y Fa(0)1941 5362 y Fj(a)p Fk(\()p Fj(t)p Fk(\))2082 5330 y Ff(k)2132 5345 y Fk(2)2212 5330 y Fl(that)36 b(is)g(uniform)h(in)f Fg(t)g Fl(if)g Ff(j)p Fg(a)p Fl(\()p Fg(t)p Fl(\))p Ff(j)f Fl(gets)h(large.)0 5579 y Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)31 b(22:40)1102 b Fl(17)p eop %%Page: 18 18 18 17 bop 0 100 a Fl(Since)34 b(this)f(is)g(precluded)h(b)m(y)f(the)h (h)m(yp)s(otheses,)g(for)f(an)m(y)g Fg(\013)27 b(>)h Fl(0,)33 b(there)g(is)g(a)g(univ)m(ersal)g(constan)m(t)h Fg(K)3844 115 y Fj(\013)0 229 y Fl(so)f(that)761 269 y Fe(\014)761 329 y(\014)761 389 y(\014)761 448 y(\014)794 413 y Fl(3)861 278 y Fe(Z)916 504 y Fj(I)-19 b(R)1014 413 y Fg(x)1071 372 y Fk(2)1132 333 y Fe(\000)1177 413 y Ff(A)1257 431 y Fj(a)p Fk(\()p Fj(t)p Fk(\))1398 413 y Fg(v)1450 333 y Fe(\001)1531 413 y Fl(\026)-69 b Fg(m)1599 372 y Fa(0)1599 441 y Fj(a)p Fk(\()p Fj(t)p Fk(\))1740 413 y Fl(dx)1848 269 y Fe(\014)1848 329 y(\014)1848 389 y(\014)1848 448 y(\014)1897 413 y Ff(j)12 b Fl(_)-40 b Fg(a)p Fl(\()p Fg(t)p Fl(\))p Ff(j)27 b(\024)2264 346 y Fg(K)2349 361 y Fj(\013)p 2264 391 142 4 v 2302 482 a Fg(\013)2417 413 y Ff(I)7 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))22 b(+)h Fg(\013)p Ff(k)p Fg(v)t Ff(k)3095 372 y Fk(2)3095 438 y(2)3645 413 y Fl(\(3)p Fg(:)p Fl(13\))0 685 y(Note)f(that)g(the)h(\014rst)g(term)g(on)f(the)h(righ)m(t)g(in)f (\(3.13\))g(can)g(b)s(e)h(absorb)s(ed)h(in)m(to)e(the)h(negativ)m(e)g (con)m(tribution)0 815 y(from)34 b(the)h(inclusion)g(of)f(the)h(m)m (ultiple)f Fg(C)41 b Fl(of)35 b(the)f(excess)h(free)h(energy)e(in)g Fg(\036)p Fl(,)h(at)e(least)h(if)h Fg(C)41 b Fl(is)34 b(c)m(hosen)0 944 y(appropriately)39 b(large.)64 b(Therefore,)42 b(since)f(w)m(e)f(can)g(tak)m(e)g Fg(\013)g Fl(arbitrarily)e(small,)i (and)h(can)f(b)s(ound)h(of)0 1074 y Ff(k)p Fg(v)t Ff(k)152 1038 y Fk(2)152 1099 y(2)229 1074 y Fl(in)34 b(terms)f(of)h(the)f (excess)h(free)h(energy)e(b)m(y)h(Lemma)f(1.3,)f(this)i(term)f(is)g (under)i(con)m(trol.)120 1205 y(One)29 b(ev)m(en)f(more)h(easily)e (handles)i(the)f(con)m(tributions)h(of)f(the)h(nonlinear)f(terms)g(in)g (\(3.5\))f(using)i(the)0 1334 y(b)s(ound)j Ff(k)p Fg(v)t Ff(k)457 1298 y Fk(2)457 1359 y Fa(1)568 1334 y Ff(\024)c Fl(2)p Ff(k)p Fg(v)t Ff(k)875 1349 y Fk(2)919 1334 y Ff(k)p Fg(v)1021 1298 y Fa(0)1049 1334 y Ff(k)1099 1349 y Fk(2)1143 1334 y Fl(.)44 b(W)-8 b(e)31 b(do)f(not)h(giv)m(e)f(the)h (details)g(here,)g(but)g(turn)g(to)g(the)g(application)f(of)0 1464 y(the)k(Lemmas)f(from)h(this)f(section)h(to)f(con)m(trol)g(the)g (con)m(tribution)h(from)f(the)h(linear)f(terms.)120 1594 y(T)-8 b(o)32 b(apply)h(Lemma)h(3.1,)e(c)m(ho)s(ose)i Fg(\013)f Fl(so)h(that)974 1761 y Fe(\020)1034 1872 y Fl(2)22 b(+)1217 1804 y Fg(\013)p 1217 1849 65 4 v 1224 1940 a Fl(2)1293 1872 y Ff(k)p Fl(\()p Fg(x)1439 1831 y Fk(2)1484 1872 y Fg(V)1564 1831 y Fa(0)1592 1872 y Fl(\))p Ff(k)1681 1831 y Fk(2)1681 1896 y Fa(1)1787 1872 y Fl(+)h(2)p Fg(\013)p Ff(k)p Fg(V)f Ff(k)2181 1887 y Fk(1)2225 1761 y Fe(\021)2312 1872 y Ff(\024)28 b Fl(2)2484 1761 y Fe(\020)2543 1872 y Fl(1)22 b(+)2732 1804 y Fg(\017)p 2727 1849 50 4 v 2727 1940 a Fl(4)2789 1761 y Fe(\021)2898 1872 y Fg(:)0 2159 y Fl(Then,)34 b(for)f(this)h(c)m(hoice)g(of)f Fg(\013)p Fl(,)g(and)h(using)g(the)f(notation)g(from)g(\(3.8\),)767 2404 y(d)p 749 2449 92 4 v 749 2540 a(d)p Fg(t)852 2472 y( )t Fl(\()p Fg(t)p Fl(\))27 b(=)1167 2331 y Fe(\022)1240 2472 y Fl(12)22 b(+)1474 2404 y(1)g(+)g(4)p Ff(k)p Fg(V)g Ff(k)1875 2419 y Fk(1)p 1474 2449 446 4 v 1640 2540 a Fl(2)p Fg(\013)1932 2331 y Fe(\023)2021 2472 y Ff(h)p Fg(\021)2113 2431 y Fa(0)2141 2472 y Fg(;)17 b(\021)2239 2431 y Fa(0)2266 2472 y Ff(i)22 b Fl(+)h(2)2494 2361 y Fe(\020)2553 2472 y Fl(1)f(+)2741 2404 y Fg(\017)p 2736 2449 50 4 v 2736 2540 a Fl(4)2798 2361 y Fe(\021)2874 2472 y Ff(h)p Fg(\021)t(;)17 b(\021)t Ff(i)32 b Fg(:)482 b Fl(\(3)p Fg(:)p Fl(14\))0 2779 y(Next,)32 b(b)m(y)i(Theorem)f(1.4,) 992 3012 y(d)p 974 3056 92 4 v 974 3147 a(d)p Fg(t)1077 3079 y(C)24 b Fl([)p Ff(F)10 b Fl(\()18 b(\026)-68 b Fg(m)21 b Fl(+)i Fg(v)t Fl(\))f Ff(\000)g(F)10 b Fl(\()18 b(\026)-68 b Fg(m)p Fl(\)])27 b Ff(\024)h(\000)p Fg(C)7 b Fl(\(1)23 b Ff(\000)f Fg(\017)p Fl(\))p Ff(h)p Fg(\021)2685 3038 y Fa(0)2713 3079 y Fg(;)17 b(\021)2811 3038 y Fa(0)2839 3079 y Ff(i)33 b Fg(:)0 3366 y Fl(Therefore,)h(if)g(w)m(e)f(c)m(ho)s (ose)h Fg(C)41 b Fl(so)33 b(that)g Fg(C)7 b Fl(\(1)22 b Ff(\000)h Fg(\017)p Fl(\))28 b Ff(\025)g Fl(\(12)21 b(+)i(\(1)f(+)g(4)p Ff(k)p Fg(V)g Ff(k)2684 3381 y Fk(1)2729 3366 y Fl(\))p Fg(=)p Fl(\(2)p Fg(\013)p Fl(\)\),)31 b(w)m(e)j(get)1401 3604 y(d)p 1383 3648 V 1383 3740 a(d)p Fg(t)1487 3671 y(\036)p Fl(\()p Fg(t)p Fl(\))27 b Ff(\024)h Fl(2)1859 3561 y Fe(\020)1918 3671 y Fl(1)22 b(+)2107 3604 y Fg(\017)p 2102 3648 50 4 v 2102 3740 a Fl(4)2164 3561 y Fe(\021)2240 3671 y Ff(h)p Fg(\021)t(;)17 b(\021)t Ff(i)32 b Fg(:)120 3965 y Fl(It)38 b(remains)h(to)g(b)s(ound)g Ff(k)p Fg(\021)t Ff(k)1203 3929 y Fk(2)1203 3990 y(2)1248 3965 y Fl(.)60 b(There)40 b(are)f(t)m(w)m(o)f(cases.)62 b(First)38 b(supp)s(ose)j(that)d(the)h(dissipation)g(is)0 4095 y(small)44 b(compared)i(to)e(the)h(excess)h(free)g(energy)f(so)g (that)f(\(1.18\))g(holds.)79 b(Then)46 b(b)m(y)f(Theorem)g(2.3,)0 4225 y Ff(k)p Fg(\021)t Ff(k)153 4188 y Fk(2)153 4249 y(2)234 4225 y Ff(\024)37 b Fl(\(1)25 b(+)h Fg(\017)p Fl(\))p Ff(h)p Fg(v)t(;)17 b Ff(A)p Fg(v)t Ff(i)p Fl(,)38 b(and)h(then)g(b)m(y)g(Lemma)f(1.3,)h Ff(k)p Fg(\021)t Ff(k)2300 4188 y Fk(2)2300 4249 y(2)2381 4225 y Ff(\024)e Fl(\(1)25 b(+)h Fg(\017)p Fl(\))2791 4188 y Fk(3)2836 4225 y Fl([)p Ff(F)10 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))24 b Ff(\000)j(F)10 b Fl(\()18 b(\026)-68 b Fg(m)o Fl(\)],)39 b(for)g Fg(\016)0 4354 y Fl(and)34 b Fg(\024)f Fl(su\016cien)m(tly)h(small.)43 b(Rede\014ning)35 b Fg(\017)p Fl(,)f(w)m(e)f(ha)m(v)m(e)h(pro)m(v)m(ed)g(\(1.22\))e(under)j(the)e(h)m (yp)s(othesis)h(\(1.18\).)120 4485 y(If)f(w)m(e)h(don't)f(assume)h (\(1.18\),)e(w)m(e)h(use)649 4718 y Ff(k)p Fg(\021)t Ff(k)802 4677 y Fk(2)802 4743 y(2)874 4718 y Fl(=)28 b Ff(h)p Fg(v)1070 4677 y Fa(0)1098 4718 y Fg(;)17 b(\021)1196 4677 y Fa(0)1223 4718 y Ff(i)22 b Fl(+)g Ff(h)p Fl(\()p Fg(V)44 b Fl(+)23 b(1\))p Fg(v)t(;)17 b(\021)t Ff(i)26 b(\024)i(k)p Fg(v)2174 4677 y Fa(0)2202 4718 y Ff(k)2252 4733 y Fk(2)2296 4628 y Fe(p)p 2396 4628 341 4 v 90 x Ff(I)7 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))22 b(+)h Ff(k)p Fg(v)t Ff(k)3010 4733 y Fk(2)3054 4718 y Ff(k)p Fg(\021)t Ff(k)3207 4733 y Fk(2)0 4951 y Fl(since)37 b Ff(k)p Fg(v)28 b Fl(+)d(1)p Ff(k)575 4966 y Fa(1)693 4951 y Fl(=)33 b(1.)54 b(This)37 b(leads)g(to)f Ff(k)p Fg(\021)t Ff(k)1697 4915 y Fk(2)1697 4976 y(2)1774 4951 y Ff(\024)e Fl(\(2)p Fg(=\015)6 b Fl(\))p Ff(I)h Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))23 b(+)h(4)p Ff(k)p Fg(v)t Ff(k)2787 4915 y Fk(2)2787 4976 y(2)2868 4951 y Fl(where)37 b Fg(\015)42 b Fl(is)36 b(the)h(constan)m(t)0 5081 y(in)k(Theorem)g (1.4.)65 b(Again,)42 b(the)f(term)f(in)m(v)m(olving)g Ff(I)7 b Fl(\()p Fg(m)p Fl(\()p Fg(t)p Fl(\)\))41 b(can)g(b)s(e)g (absorb)s(ed)g(b)m(y)g(an)g(appropriate)0 5210 y(c)m(hoice)34 b(of)g Fg(C)7 b Fl(.)45 b(The)35 b(remaining)e(term)h(is)f(easily)g (handled)i(b)m(y)e(Lemma)h(1.2)f(and)h(Lemma)g(1.3,)f(and)h(so)0 5340 y(\(1.23\))e(is)h(established.)p 981 5340 70 70 v 0 5579 a Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)e(22:40)1102 b Fl(18)p eop %%Page: 19 19 19 18 bop 0 100 a Fh(4)45 b(Pro)t(of)g(of)g(the)g(Main)g(Theorem)120 230 y Fl(W)-8 b(e)28 b(will)f(b)s(e)h(brief)h(in)f(the)g(presen)m (tation)h(of)f(this)h(pro)s(of)f(since)h(from)f(this)g(p)s(oin)m(t)g (on,)h(it)e(is)h(v)m(ery)g(close)0 360 y(to)33 b(the)g(one)h(w)m(e)g (ha)m(v)m(e)f(giv)m(en)g(for)h(the)g(LOP)f(equation)g(in)g(section)h(4) f(of)g([5].)120 490 y(Let)g Fg(m)p Fl(\()p Fg(t)p 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Fl(\))32 b(and)i Fg(\036)p Fl(\()p Fg(t)p Fl(\))f(b)s(e)h(giv)m(en)f(in)g(terms)h(of)f Fg(m)p Fl(\()p Fg(t)p Fl(\))g(as)g(in)h(the)f(in)m(tro)s(duction.)120 2047 y(W)-8 b(e)33 b(b)s(egin)h(b)m(y)f(assuming)h(that)f(at)f(time)h Fg(t)p Fl(,)g(\(1.18\))f(holds.)44 b(Then)35 b(b)m(y)e(Theorem)h(1.4,) 1382 2296 y(d)p 1364 2340 92 4 v 1364 2432 a(d)p Fg(t)1467 2363 y(f)11 b Fl(\()p Fg(t)p Fl(\))26 b Ff(\024)j(\000)p Fl(\(1)22 b Ff(\000)g Fg(\017)p Fl(\))p Ff(k)p Fl(\()p Ff(A)p Fg(v)t Fl(\))2399 2322 y Fa(0)2426 2363 y Ff(k)2476 2322 y Fk(2)2476 2388 y(2)2521 2363 y Fg(:)0 2667 y Fl(b)m(y)37 b(con)m(v)m(exit)m(y)f Ff(k)p Fl(\()p Ff(A)p Fg(v)t Fl(\))848 2631 y Fa(0)875 2667 y Ff(k)925 2631 y Fk(2)925 2692 y(2)1003 2667 y Ff(\025)e(k)p Fl(\()p Ff(A)p Fg(\032)23 b Ff(\003)i Fg(v)t Fl(\))1524 2631 y Fa(0)1551 2667 y Ff(k)1601 2631 y Fk(2)1601 2692 y(2)1683 2667 y Fl(where)37 b Fg(\032)c Fl(=)h(\(1)p Fg(=)p Fl(2\))18 b(\026)-68 b Fg(m)2485 2631 y Fa(0)2513 2667 y Fl(,)37 b(whic)m(h)h(is)e(a)h 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Fg(\016)42 b Fl(and)c Ff(k)p Fg(v)1327 2371 y Fa(0)1354 2407 y Fl(\()p Fg(t)p Fl(\))p Ff(k)1518 2422 y Fk(2)1597 2407 y Ff(\024)d Fg(\024)j Fl(hold)g(for)g(all)e Fg(t)p Fl(,)i(and)g(so)g Fg(f)11 b Fl(\()p Fg(t)p Fl(\))36 b(deca)m(ys)j(all)d(the)i(w)m(a)m(y)f(to)0 2536 y(zero)f(at)f(a)h(rate)f(close)h(to)g Fg(t)1032 2500 y Fa(\000)p Fk(9)p Fj(=)p Fk(13)1259 2536 y Fl(,)g(as)g(in)f (Theorem)i(1.1.)50 b(As)36 b(explained)g(at)f(the)h(end)h(of)f(section) g(1)g(of)0 2666 y(this)42 b(pap)s(er,)h(this)f(means)g(that)f Ff(kA)p Fg(v)t Fl(\()p Fg(t)p Fl(\))p Ff(k)1606 2681 y Fk(1)1691 2666 y Fl(deca)m(ys)h(to)f(zero)h(at)f(an)h(algebraic)f (rate,)i(and)f(that)f(this)0 2795 y(forces)34 b(lim)419 2810 y Fj(t)p Fa(!1)629 2795 y Fg(a)p Fl(\()p Fg(t)p Fl(\))27 b(=)h Fg(a)33 b Fl(where)h Fg(a)f Fl(is)g(giv)m(en)g(b)m(y)h (the)f(conserv)-6 b(ation)34 b(la)m(w.)1681 3025 y Fm(References)-35 3297 y Fl([1])49 b(A.)34 b(Asselah,)i Fi(Stability)i(of)f(a)h(wave)f (fr)-5 b(ont)38 b(for)f(a)g(nonlo)-5 b(c)g(al)38 b(c)-5 b(onservative)36 b(evolution.)51 b Fl(Pro)s(c.Ro)m(y)m(al)120 3427 y(So)s(c.)44 b(Edin)m(burgh)34 b Fm(128)39 b(A)33 b Fl(no.)44 b(2,)33 b(219{234)f(\(1998\).)-35 3628 y([2])49 b(J.)28 b(Bricmon)m(t,)h(A.)f(Kupiainen,)i(J.)f(T)-8 b(askinen,)29 b Fi(Stability)i(of)g(Cahn-Hil)5 b(liar)-5 b(d)31 b(F)-8 b(r)j(onts)p Fl(,)29 b(Comm.)43 b(Pure)120 3757 y(and)33 b(Appli.)44 b(Math.,)33 b Fm(52)h Fl(no.)44 b(7,)32 b(839{871)g(\(1999\).)-35 3959 y([3])49 b(L.)40 b(Ca\013arelli,)i(N.)e(E.)h(Muler,)i Fi(A)m(n)h Fg(L)1564 3923 y Fa(1)1691 3959 y Fi(b)-5 b(ound)42 b(for)h(solutions)f(of)h(the) f(Cahn-Hil)5 b(liar)-5 b(d)43 b(e)-5 b(quation)p Fl(,)120 4088 y(Arc)m(h.)44 b(Rational)33 b(Mec)m(h.)45 b(Anal.,)32 b Fm(133)p Fl(,)i(129{144)e(\(1995\).)-35 4290 y([4])49 b(E.)40 b(A.)g(Carlen,)i(M.)f(C.)f(Carv)-6 b(alho,)42 b(E.)e(Orlandi,)i Fi(A)m(lgebr)-5 b(aic)43 b(r)-5 b(ate)42 b(of)g(de)-5 b(c)g(ay)42 b(for)g(the)g(exc)-5 b(ess)42 b(fr)-5 b(e)g(e)120 4419 y(ener)g(gy)30 b(and)g(stability)g(of)g(fr)-5 b(onts)29 b(for)h(a)g(non{lo)-5 b(c)g(al)31 b(phase)e(kinetics)g(e)-5 b(quation)30 b(with)g(a)g(c)-5 b(onservation)120 4549 y(law)36 b(I)g Fl(,)d(Jour.)44 b(Stat.)g(Ph)m(ys,.)g Fm(95)33 b Fl(no.)44 b(5/6,)33 b(1069{1117)e(\(1999\).)-35 4750 y([5])49 b(E.)40 b(A.)g(Carlen,)i(M.)f(C.)f(Carv)-6 b(alho,)42 b(E.)e(Orlandi,)i Fi(A)m(lgebr)-5 b(aic)43 b(r)-5 b(ate)42 b(of)g(de)-5 b(c)g(ay)42 b(for)g(the)g(exc)-5 b(ess)42 b(fr)-5 b(e)g(e)120 4880 y(ener)g(gy)30 b(and)g(stability)g (of)g(fr)-5 b(onts)29 b(for)h(a)g(non{lo)-5 b(c)g(al)31 b(phase)e(kinetics)g(e)-5 b(quation)30 b(with)g(a)g(c)-5 b(onservation)120 5009 y(law)36 b(II)p Fl(,)d(Comm.)44 b(P)-8 b(.D.E.,)32 b Fm(25)h Fl(no.5/6,)f(847{886)g(\(2000\).)-35 5210 y([6])49 b(A.)36 b(De)h(Masi,)g(E.)g(Orlandi,)h(E.)e(Presutti,)h (L.)g(T)-8 b(riolo,)36 b Fi(Stability)j(of)f(the)h(interfac)-5 b(e)38 b(in)g(a)h(mo)-5 b(del)40 b(of)120 5340 y(phase)35 b(sep)-5 b(ar)g(ation)31 b Fl(Pro)s(c.)44 b(Ro)m(y)m(al)33 b(So)s(c.)44 b(Edin)m(burgh)35 b Fm(124A)e Fl(1013-1022)f(\(1994\).)0 5579 y Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)f(22:40)1102 b Fl(20)p eop %%Page: 21 21 21 20 bop -35 100 a Fl([7])49 b(G.)43 b(Giacomin,)k(J.)d(Leb)s(o)m (witz,)i Fi(Phase)f(se)-5 b(gr)g(e)g(gation)44 b(dynamics)h(in)h(p)-5 b(article)45 b(systems)g(with)h(long)120 229 y(r)-5 b(ange)35 b(inter)-5 b(actions)34 b(I:)h(macr)-5 b(osc)g(opic)35 b(limits)e Fl(J.)g(Stat.)44 b(Ph)m(ys.)g Fm(87)34 b Fl(no.1/2,)e(37{61) g(\(1997\).)-35 431 y([8])49 b(G.)32 b(Hardy)-8 b(,)31 b(J.)h(Littlew)m(o)s(o)s(d,)f(and)h(G.)g(P)m(oly)m(a)g Fi(Ine)-5 b(qualities)31 b Fl(Cam)m(bridge)i(Univ.)43 b(Press,)32 b(Cam)m(bridge)120 560 y(\(1932\))-35 761 y([9])49 b(E.H.)32 b(Lieb,)h(W.)g(Thirring,)g Fi(Ine)-5 b(qualities)35 b(of)g(the)h(moments)g(of)f(the)h(eigenvalues)f(of)g (the)120 891 y(Schr)-5 b(o\177)-51 b(odinger)27 b(Hamiltonian)h(and)g (their)g(r)-5 b(e)g(altion)28 b(to)g(Sob)-5 b(olev)28 b(ine)-5 b(qualities)p Fl(,)26 b(pp)g(269{303)e(in)h(Stud-)120 1020 y(ies)38 b(in)g(Mathematical)g(Ph)m(ysics,)i(Essa)m(ys)e(in)g (honor)h(of)g(V)-8 b(alen)m(tine)38 b(Bargmann,)g(edited)h(b)m(y)g (Lieb,)120 1150 y(Simon)33 b(and)h(Wigh)m(tman,)f(Princeton)g(Univ)m (ersit)m(y)g(Press,)h(Princeton,)f(\(1976\).)-85 1351 y([10])49 b(J.)32 b(L.)h(Leb)s(o)m(witz,)g(E.)f(Orlandi,)h(E.)f (Presutti,)g Fi(A)k(Particle)e(mo)-5 b(del)36 b(for)e(spino)-5 b(dal)35 b(de)-5 b(c)g(omp)g(osition)31 b Fl(J.)120 1481 y(Stat.)43 b(Ph)m(ys.)h Fm(63)p Fl(,)34 b(933-974)e(\(1991\).)-85 1682 y([11])49 b(H.)26 b(W)-8 b(eyl,)27 b Fi(Gr)q(\177)-52 b(upp)-5 b(enthe)g(orie)29 b(und)i(Quantenme)-5 b(chanik)27 b Fl(Wissensc)m(haftlic)m(h)i(V)-8 b(erlag,)27 b(Leipzig)g(\(1926\))0 5579 y Fc(15)p Fb(=nov)r(ember)r(=)p Fc(2000;)k(22:40)1102 b Fl(21)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF