Content-Type: multipart/mixed; boundary="-------------9909290559870" This is a multi-part message in MIME format. ---------------9909290559870 Content-Type: text/plain; name="99-362.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-362.keywords" Radial Schr\"odinger equation, Langer modification, exact WKB analysis (method), phase shift, scattering matrix, quantization condition ---------------9909290559870 Content-Type: application/postscript; name="fura2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fura2.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.76 (OzTeX) %%Title: final.dvi %%Pages: 22 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMBX12 CMR10 CMMI10 CMSY10 CMR7 CMBX10 %%+ CMSY7 CMMI7 CMEX10 Symbol Times-Roman CMTI10 MSBM10 CMR5 CMMI5 %%+ CMSL10 LASY10 CMSY5 %%EndComments %DVIPSCommandLine: dvips -T210mm,297mm -PCMPS -ofinal.ps final.dvi %DVIPSParameters: dpi=300 %DVIPSSource: TeX output 1999.09.29:1113 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]{ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]} if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{dup length product length le{dup length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false} ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot} imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M}B /d{ -3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w} B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @psencodingfile{ % author = "S. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSL10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-62 -250 1123 750}readonly def /UniqueID 5000798 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmmi5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmr5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /arrowup put dup 175 /arrowdown put dup 176 /quotesingle put dup 177 /exclamdown put dup 178 /questiondown put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /arrowup put dup 12 /arrowdown put dup 13 /quotesingle put dup 14 /exclamdown put dup 15 /questiondown put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /less put dup 61 /equal put dup 62 /greater put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /lessornotequal put dup 162 /greaterornotequal put dup 163 /notlessequal put dup 164 /notgreaterequal put dup 165 /notless put dup 166 /notgreater put dup 167 /notprecedes put dup 168 /notfollows put dup 169 /lessornotdbleql put dup 170 /greaterornotdbleql put dup 173 /notlessorslnteql put dup 174 /notgreaterorslnteql put dup 175 /lessnotequal put dup 176 /greaternotequal put dup 177 /notprecedesoreql put dup 178 /notfollowsoreql put dup 179 /precedeornoteqvlnt put dup 180 /followornoteqvlnt put dup 181 /lessornotsimilar put dup 182 /greaterornotsimilar put dup 183 /notlessdblequal put dup 184 /notgreaterdblequal put dup 185 /precedenotslnteql put dup 186 /follownotslnteql put dup 187 /precedenotdbleqv put dup 188 /follownotdbleqv put dup 189 /lessnotdblequal put dup 190 /greaternotdblequal put dup 191 /notsimilar put dup 192 /notapproxequal put dup 193 /upslope put dup 194 /downslope put dup 195 /notsubsetoreql put dup 196 /epsiloninv put dup 0 /lessornotequal put dup 1 /greaterornotequal put dup 2 /notlessequal put dup 3 /notgreaterequal put dup 4 /notless put dup 5 /notgreater put dup 6 /notprecedes put dup 7 /notfollows put dup 8 /lessornotdbleql put dup 9 /greaterornotdbleql put dup 10 /notlessorslnteql put dup 11 /notgreaterorslnteql put dup 12 /lessnotequal put dup 13 /greaternotequal put dup 14 /notprecedesoreql put dup 15 /notfollowsoreql put dup 16 /precedeornoteqvlnt put dup 17 /followornoteqvlnt put dup 18 /lessornotsimilar put dup 19 /greaterornotsimilar put dup 20 /notlessdblequal put dup 21 /notgreaterdblequal put dup 22 /precedenotslnteql put dup 23 /follownotslnteql put dup 24 /precedenotdbleqv put dup 25 /follownotdbleqv put dup 26 /lessnotdblequal put dup 27 /greaternotdblequal put dup 28 /notsimilar put dup 29 /notapproxequal put dup 30 /upslope put dup 31 /downslope put dup 32 /notsubsetoreql put dup 33 /notsupersetoreql put dup 34 /notsubsetordbleql put dup 35 /notsupersetordbleql put dup 36 /subsetornotdbleql put dup 37 /supersetornotdbleql put dup 38 /subsetornoteql put dup 39 /supersetornoteql put dup 40 /subsetnoteql put dup 41 /supersetnoteql put dup 42 /notsubseteql put dup 43 /notsuperseteql put dup 44 /notparallel put dup 45 /notbar put dup 46 /notshortbar put dup 47 /notshortparallel put dup 48 /notturnstile put dup 49 /notforces put dup 50 /notsatisfies put dup 51 /notforcesextra put dup 52 /nottriangeqlright put dup 53 /nottriangeqlleft put dup 54 /nottriangleleft put dup 55 /nottriangleright put dup 56 /notarrowleft put dup 57 /notarrowright put dup 58 /notdblarrowleft put dup 59 /notdblarrowright put dup 60 /notdblarrowboth put dup 61 /notarrowboth put dup 62 /dividemultiply put dup 63 /emptyset put dup 64 /notexistential 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 /hatwide put dup 92 /hatwider put dup 93 /tildewide put dup 94 /tildewider put dup 96 /Finv put dup 97 /Gmir put dup 102 /Omegainv put dup 103 /eth put dup 104 /equalorsimilar put dup 105 /beth put dup 106 /gimel put dup 107 /daleth put dup 108 /lessdot put dup 109 /greaterdot put dup 110 /multicloseleft put dup 111 /multicloseright put dup 112 /barshort put dup 113 /parallelshort put dup 114 /integerdivide put dup 115 /similar put dup 116 /approxequal put dup 117 /approxorequal put dup 118 /followsorequal put dup 119 /precedesorequal put dup 120 /archleftdown put dup 121 /archrightdown put dup 122 /Digamma put dup 123 /kappa put dup 124 /k put dup 125 /planckover2pi put dup 126 /planckover2pi1 put dup 127 /epsiloninv put dup 128 /notsubsetoreql put dup 160 /space put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: cmti10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /sterling put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmex10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /parenleftbig put dup 162 /parenrightbig put dup 163 /bracketleftbig put dup 164 /bracketrightbig put dup 165 /floorleftbig put dup 166 /floorrightbig put dup 167 /ceilingleftbig put dup 168 /ceilingrightbig put dup 169 /braceleftbig put dup 170 /bracerightbig put dup 173 /angbracketleftbig put dup 174 /angbracketrightbig put dup 175 /vextendsingle put dup 176 /vextenddouble put dup 177 /slashbig put dup 178 /backslashbig put dup 179 /parenleftBig put dup 180 /parenrightBig put dup 181 /parenleftbigg put dup 182 /parenrightbigg put dup 183 /bracketleftbigg put dup 184 /bracketrightbigg put dup 185 /floorleftbigg put dup 186 /floorrightbigg put dup 187 /ceilingleftbigg put dup 188 /ceilingrightbigg put dup 189 /braceleftbigg put dup 190 /bracerightbigg put dup 191 /angbracketleftbigg put dup 192 /angbracketrightbigg put dup 193 /slashbigg put dup 194 /backslashbigg put dup 195 /parenleftBigg put dup 196 /arrowdblbt put dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 4 /floorleftbig put dup 5 /floorrightbig put dup 6 /ceilingleftbig put dup 7 /ceilingrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 14 /slashbig put dup 15 /backslashbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 22 /floorleftbigg put dup 23 /floorrightbigg put dup 24 /ceilingleftbigg put dup 25 /ceilingrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 28 /angbracketleftbigg put dup 29 /angbracketrightbigg put dup 30 /slashbigg put dup 31 /backslashbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 36 /floorleftBigg put dup 37 /floorrightBigg put dup 38 /ceilingleftBigg put dup 39 /ceilingrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 42 /angbracketleftBigg put dup 43 /angbracketrightBigg put dup 44 /slashBigg put dup 45 /backslashBigg put dup 46 /slashBig put dup 47 /backslashBig put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 54 /bracketleftex put dup 55 /bracketrightex put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 62 /braceex put dup 63 /arrowvertex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 70 /unionsqtext put dup 71 /unionsqdisplay put dup 72 /contintegraltext put dup 73 /contintegraldisplay put dup 74 /circledottext put dup 75 /circledotdisplay put dup 76 /circleplustext put dup 77 /circleplusdisplay put dup 78 /circlemultiplytext put dup 79 /circlemultiplydisplay put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 83 /uniontext put dup 84 /intersectiontext put dup 85 /unionmultitext put dup 86 /logicalandtext put dup 87 /logicalortext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 93 /unionmultidisplay put dup 94 /logicalanddisplay put dup 95 /logicalordisplay put dup 96 /coproducttext put dup 97 /coproductdisplay put dup 98 /hatwide put dup 99 /hatwider put dup 100 /hatwidest put dup 101 /tildewide put dup 102 /tildewider put dup 103 /tildewidest put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 106 /floorleftBig put dup 107 /floorrightBig put dup 108 /ceilingleftBig put dup 109 /ceilingrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 113 /radicalBig put dup 114 /radicalbigg put dup 115 /radicalBigg put dup 116 /radicalbt put dup 117 /radicalvertex put dup 118 /radicaltp put dup 119 /arrowvertexdbl put dup 120 /arrowtp put dup 121 /arrowbt put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put dup 126 /arrowdbltp put dup 127 /arrowdblbt put dup 128 /parenleftBigg put dup 160 /space put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec 80347982AB3942D930E069A70D0D48311D7190FA2D133A583138F76695558E7A E9348D37CAC6651806D08527C1BB4A062A4835AC37784CC39AD8841404E438B4 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmmi7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmsy7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 127 /spade put dup 128 /arrowleft put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 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cleartomark %%EndFont %%BeginFont: cmbx10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmr7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u 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cleartomark %%EndFont %%BeginFont: cmsy10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 127 /spade put dup 128 /arrowleft put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 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%!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmr10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmbx12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec 9B9C1569015F2C1D2BF560F4C0D52257BACDD6500ABDA5ED9835F6A016CFC8F0 0B6C052ED76A87856B50F4D80DFAEB508C97F8281F3F88B17E4D3B90C0F65EC3 79791AACDC162A66CBBC5BE2F53AAD8DE72DD113B55A022FBFEE658CB95F5BB3 2BA0357B5E050FDDF264A07470BEF1C52119B6FBD5C77EBED964AC5A2BBEC9D8 B3E48AE5BB003A63D545774B922B9D5FF6B0066ECE43645A131879B032137D6D 823385FE55F3402D557FD3B4486BE79011D1F5BFAE5C1F476EE6F05EB1D2CAEB 269958B194521197B312FCCED4867F3C8FBD030BD715D8FFDA1DCD454B174E7A 1A97B59FE770E67702519D9D9B23D61AC08424D555242A8CA08C49AEF300945D 99B999A79CE74804AE6BFDE623F4463371442F6523A5F6CE19C839A708C02513 2E22C696C8CCADE45680E5197189D0F98E7F0D5F955E353970B392CF530A68CC 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: cmr17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueID 5000795 def currentdict end currentfile eexec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0 currenttransfer exec .5 ge def /level2 /languagelevel where {pop languagelevel 1 ne}{false}ifelse def /colorexists level2 { statusdict/processcolors known {statusdict/processcolors get exec}{1}ifelse 4 eq def } {systemdict/setcmykcolor known def} ifelse /tc1 [0 0 0 1] def /tc2 [0 0 0 1] def /fc [0 0 0 1] def /sc [0 0 0 1] def /concatprocs{ /packedarray where {pop dup type /packedarraytype eq 2 index type /packedarraytype eq or}{false}ifelse { /proc2 exch cvlit def/proc1 exch cvlit def proc1 aload pop proc2 aload pop proc1 length proc2 length add packedarray cvx } { /proc2 exch cvlit def/proc1 exch cvlit def /newproc proc1 length proc2 length add array def newproc 0 proc1 putinterval newproc proc1 length proc2 putinterval newproc cvx }ifelse }bdf /storerect{/top xdf/right xdf/bottom xdf/left xdf}bdf /rectpath{newpath left bottom m left top L right top L right bottom L closepath}bdf /i{dup 0 eq {pop df dup} {dup} ifelse /clipflatness xdf setflat }bdf version cvr 38.0 le {/setrgbcolor{ currenttransfer exec 3 1 roll currenttransfer exec 3 1 roll currenttransfer exec 3 1 roll setrgbcolor}bdf}if /gettint{0 get}bdf /puttint{0 exch put}bdf /vms {/vmsv save def} bdf /vmr {vmsv restore} bdf /vmrs{vmsv restore /vmsv save def}bdf /eomode{ {/filler /eofill load def /clipper /eoclip load def} {/filler /fill load def /clipper /clip load def} ifelse }bdf /CD{/NF exch def{exch dup/FID ne 1 index/UniqueID ne and{exch NF 3 1 roll put} {pop pop}ifelse}forall NF}bdf /MN{1 index length/Len exch def dup length Len add string dup Len 4 -1 roll putinterval dup 0 4 -1 roll putinterval}bdf /RC{4 -1 roll /ourvec xdf 256 string cvs(|______)anchorsearch {1 index MN cvn/NewN exch def cvn findfont dup maxlength dict CD dup/FontName NewN put dup /Encoding ourvec put NewN exch definefont pop}{pop}ifelse}bdf /RF{dup FontDirectory exch known{pop 3 -1 roll pop}{RC}ifelse}bdf /FF{dup 256 string cvs(|______)exch MN cvn dup FontDirectory exch known {exch pop findfont 3 -1 roll pop}{pop dup findfont dup maxlength dict CD dup dup /Encoding exch /Encoding get 256 array copy 7 -1 roll {3 -1 roll dup 4 -2 roll put}forall put definefont}ifelse}bdf userdict begin /BDFontDict 20 dict def end BDFontDict begin /bu{}def /bn{}def /setTxMode{av 70 ge{pop}if pop}def /gm{m}def /show{pop}def /gr{pop}def /fnt{pop pop pop}def /fs{pop}def /fz{pop}def /lin{pop pop}def end /MacVec 256 array def MacVec 0 /Helvetica findfont /Encoding get 0 128 getinterval putinterval MacVec 127 /DEL put MacVec 16#27 /quotesingle put MacVec 16#60 /grave put /NUL/SOH/STX/ETX/EOT/ENQ/ACK/BEL/BS/HT/LF/VT/FF/CR/SO/SI /DLE/DC1/DC2/DC3/DC4/NAK/SYN/ETB/CAN/EM/SUB/ESC/FS/GS/RS/US MacVec 0 32 getinterval astore pop /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis /dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/notequal/AE/Oslash /infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation /product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash /questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft /guillemotright/ellipsis/nbspace/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge /ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright/fi/fl /daggerdbl/periodcentered/quotesinglbase/quotedblbase /perthousand/Acircumflex/Ecircumflex/Aacute /Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave/Oacute/Ocircumflex /apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde /macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron MacVec 128 128 getinterval astore pop /fps{ currentflat exch dup 0 le{pop 1}if { dup setflat 3 index stopped {1.3 mul dup 3 index gt{pop setflat pop pop stop}if} {exit} ifelse }loop pop setflat pop pop }bdf /fp{100 currentflat fps}bdf /clipper{clip}bdf /W{/clipper load 100 clipflatness fps}bdf end%. FreeHandDict end%. FHIODict %%EndProcSet %%EndProlog %%BeginSetup FHIODict begin FreeHandDict begin 18.418 298.7222 513.6289 639.8179 storerect rectpath clip newpath /onlyk{false}ndf /ccmyk{dup 5 -1 roll sub 0 max exch}ndf /setcmykcolor{1 exch sub ccmyk ccmyk ccmyk pop setrgbcolor}ndf /setcmykcoloroverprint{4{dup -1 eq{pop 0}if 4 1 roll}repeat setcmykcolor}ndf /findcmykcustomcolor{5 /packedarray where{pop packedarray}{array astore readonly}ifelse}ndf /setcustomcolor{exch aload pop pop 4{4 index mul 4 1 roll}repeat setcmykcolor pop}ndf /setseparationgray{1 exch sub dup dup dup setcmykcolor}ndf /setoverprint{pop}ndf /currentoverprint false ndf /colorimage{pop pop [5 -1 roll/exec cvx 6 -1 roll/exec cvx 7 -1 roll/exec cvx 8 -1 roll/exec cvx /cmykbufs2gray cvx]cvx image} %. version 47.1 on Linotronic of Postscript defines colorimage incorrectly (rgb model only) version cvr 47.1 le isLino and{userdict begin bdf end}{ndf}ifelse /customcolorimage{ colorexists { aload pop pop 4 array astore setimagecmyk currentcolortransfer {ik mul ik sub 1 add}concatprocs 4 1 roll {iy mul iy sub 1 add}concatprocs 4 1 roll {im mul im sub 1 add}concatprocs 4 1 roll {ic mul ic sub 1 add}concatprocs 4 1 roll setcolortransfer /magentabuf 0 string def /yellowbuf 0 string def /blackbuf 0 string def {invbuf dup length magentabuf length ne {dup length dup dup /magentabuf exch string def /yellowbuf exch string def /blackbuf exch string def}if dup magentabuf copy yellowbuf copy blackbuf copy pop}concatprocs {magentabuf}{yellowbuf}{blackbuf}true 4 colorimage } { pop image } ifelse }ndf /separationimage{image}ndf /newcmykcustomcolor{6 /packedarray where{pop packedarray}{array astore readonly}ifelse}ndf /inkoverprint false ndf /setinkoverprint{pop}ndf /overprintprocess{pop}ndf /setspotcolor {spots exch get 0 5 getinterval exch setcustomcolor}ndf /currentcolortransfer{currenttransfer dup dup dup}ndf /setcolortransfer{systemdict begin settransfer end pop pop pop}ndf /getcmyk { dup length 4 eq {aload pop} {aload pop spots exch get 0 4 getinterval aload pop 4 {4 index mul 4 1 roll}repeat 5 -1 roll pop} ifelse }bdf /setimagecmyk{ getcmyk/ik xdf /iy xdf /im xdf /ic xdf }ndf /autospread{pop}ndf /fhsetspreadsize{pop}ndf /fhsetspreadallow{pop}ndf /strokeopf false def /fillopf false def /R{0 ne /strokeopf xdf}bdf /O{0 ne /fillopf xdf}bdf /filler{fill}bdf /F{fc fhsetcolor fillopf setoverprint false autospread gsave /filler load fp grestore false setoverprint}bdf /f{closepath F}bdf /S{sc fhsetcolor strokeopf setoverprint true autospread {stroke}fp false setoverprint}bdf /s{closepath S}bdf /B{fc fhsetcolor fillopf setoverprint gsave /filler load fp grestore sc fhsetcolor strokeopf setoverprint true autospread {stroke}fp false setoverprint}bdf /b{closepath B}bdf /setcolorscreen where{pop}{/setcolorscreen{setscreen pop pop pop pop pop pop pop pop pop}bdf}ifelse /fhsetcolor{dup length 4 eq {aload overprintprocess setcmykcolor} {aload 1 get spots exch get 5 get setinkoverprint setspotcolor} ifelse }ndf /settextcolor{ dup fhsetcolor textblackopf { dup length 4 eq {onlyk{3 get 1.0 eq{true setinkoverprint}if}{pop}ifelse} {pop} ifelse } {pop}ifelse }ndf /ka{/fc xdf}bdf /Ka{/sc xdf}bdf /xa{/fc xdf} bdf /Xa{/sc xdf} bdf /bc2[0 0]def /bc4[0 0 0 0]def /absmax{2 copy abs exch abs gt{exch}if pop}bdf /calcstep { colorexists not and{calcgraysteps}{maxsteps}ifelse tc1 length 4 eq { 0 1 3 {tc1 1 index get tc2 3 -1 roll get sub }for absmax absmax absmax } { bc2 tc1 1 get 1 exch put tc1 gettint tc2 gettint sub abs } ifelse mul abs round dup 0 eq{pop 1}if dup /numsteps xdf 1 sub dup 0 eq{pop 1}if /numsteps1 xdf }bdf /cblend{ tc1 length 4 eq { 0 1 3 {bc4 exch tc1 1 index get tc2 2 index get 1 index sub frac mul add put }for bc4 } { bc2 tc1 gettint tc2 gettint 1 index sub frac mul add puttint bc2 } ifelse fhsetcolor }bdf /logtaper{/frac frac 9 mul 1 add log def}bdf FHIODict begin /origmtx matrix currentmatrix def /iminv false def /invbuf{0 1 2 index length 1 sub{dup 2 index exch get 255 exch sub 2 index 3 1 roll put}for}bdf /cmykbufs2gray{ dup length 0 1 3 -1 roll 1 sub {4 index 1 index get 4 index 2 index get 4 index 3 index get 4 index 4 index get 255 exch sub ccmyk ccmyk ccmyk pop 5 mul exch 45 mul add exch 14 mul add -6 bitshift 2 index 3 1 roll put}for 4 1 roll pop pop pop }bdf end /textopf false def /textblackopf true def /curtextmtx{}def /otw .25 def /msf{dup/curtextmtx xdf makefont setfont}bdf /makesetfont/msf load def /curtextheight{.707104 .707104 curtextmtx dtransform dup mul exch dup mul add sqrt}bdf /ta{1 index {tempstr 0 2 index put tempstr 2 index gsave exec grestore tempstr stringwidth rmoveto 5 index eq{6 index 6 index rmoveto}if 3 index 3 index rmoveto }forall 7{pop}repeat}bdf /sts{settextcolor textopf setoverprint/ts{awidthshow}def exec false setoverprint}bdf /stol{/xt currentlinewidth def setlinewidth settextcolor textopf setoverprint newpath /ts{{false charpath stroke}ta}def exec false setoverprint xt setlinewidth}bdf /s8 8 string def /pattnt 1 def /patcol[]def /pa{1 0 matrix defaultmatrix dtransform exch atan}bdf /por{1 0 dtransform 0 eq exch 0 ne and}bdf /patcolor{/pattnt xdf dup length array copy/patcol xdf colorexists {/pattnt 1 pattnt sub def patcol length 4 eq {0 1 3{patcol exch 2 copy get .25 lt{0}{pattnt}ifelse put}for} {patcol dup gettint .25 lt{0}{pattnt}ifelse puttint}ifelse patcol fhsetcolor} {patcol fhsetcolor currentgray 1 ne{pattnt systemdict /setgray get exec}if}ifelse}bdf /pat{3 1 roll s8 copy pop 9.375 pa por not{-90. add}if {1 add 4 mul cvi s8 exch get exch 1 add 4 mul cvi 7 sub bitshift 1 and inverted?{1 exch sub}if}setscreen inverted? systemdict begin{}settransfer end {.9921875 exch sub}if patcolor }bdf /pF{gsave fc pat /filler load fp grestore}bdf /pf{closepath pF}bdf /pS{gsave sc pat {stroke}fp grestore newpath}bdf /ps{closepath pS}bdf /currentpacking where{pop false setpacking}if /spots[1 0 0 0 (Process Cyan) false newcmykcustomcolor 0 1 0 0 (Process Magenta) false newcmykcustomcolor 0 0 1 0 (Process 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Fd(=)p Fe(2)736 1609 y Fi(dx=h)810 1635 y Fm(f)846 1583 y Fj(1)832 1595 y Fh(X)831 1683 y Fi(n)p Fl(=0)894 1635 y Fo(\()p Fn(J)h Fm(\016)c Fn(I)994 1641 y Fl(+)1022 1635 y Fo(\))1038 1618 y Fi(n)1061 1635 y Fo([1])f Fm(\000)1170 1583 y Fj(1)1156 1595 y Fh(X)1155 1683 y Fi(n)p Fl(=0)1225 1635 y Fn(I)1243 1641 y Fl(+)1280 1635 y Fm(\016)h Fo(\()p Fn(J)k Fm(\016)c Fn(I)1410 1641 y Fl(+)1438 1635 y Fo(\))1454 1618 y Fi(n)1477 1635 y Fo([1])p Fm(g)p Fn(:)262 1764 y Fo(The)k(essen)o(tial)g(step)h(is)f(to)g(obtain)f(suitable)h (estimates)g(for)g(the)g(op)q(erators)h Fn(I)1485 1770 y Fl(+)1513 1764 y Fo([)p Fn(f)t Fo(])262 1814 y(and)f Fn(J)t Fo([)p Fn(f)t Fo(]:)535 1885 y Fn(I)553 1891 y Fl(+)580 1885 y Fo([)p Fn(f)t Fo(])f(=)684 1829 y Fh(Z)707 1923 y Fl(\000)728 1927 y Fd(z)753 1885 y Fn(e)772 1868 y Fl(2\()p Fi(z)818 1872 y Fe(1)834 1868 y Fj(\000)p Fi(z)q Fl(\))p Fi(=h)929 1885 y Fn(H)s Fo(\()p Fn(z)1002 1891 y Fl(1)1021 1885 y Fn(;)7 b(h)p Fo(\))p Fn(f)t Fo(\()p Fn(z)1139 1891 y Fl(1)1158 1885 y Fn(;)g(h)p Fo(\))p Fn(dz)1258 1891 y Fl(1)1276 1885 y Fn(;)535 1990 y(J)t Fo([)p Fn(f)t Fo(])k(=)665 1933 y Fh(Z)688 2028 y Fl(\000)709 2032 y Fd(z)735 1990 y Fn(H)s Fo(\()p Fn(z)808 1996 y Fl(1)826 1990 y Fn(;)c(h)p Fo(\))p Fn(f)t Fo(\()p Fn(z)944 1996 y Fl(1)964 1990 y Fn(;)g(h)p Fo(\))p Fn(dz)1064 1996 y Fl(1)1082 1990 y Fn(:)1487 1941 y Fo(\(22\))262 2098 y(The)17 b(function)g Fn(H)s Fo(\()p Fn(z)r(;)7 b(h)p Fo(\))17 b(is)g(de\014ned)h(b)o(y)f Fn(H)s Fo(\()p Fn(z)r(;)7 b(h)p Fo(\))17 b(=)g Fm(\000)p Fn(P)1184 2083 y Fj(0)1178 2108 y Fi(y)1198 2098 y Fo(\()p Fn(y)q(;)7 b(h)p Fo(\))p Fn(=)p Fo(\(4)p Fn(P)f Fo(\()p Fn(y)q(;)h(h)p Fo(\))1481 2083 y Fl(3)p Fi(=)p Fl(2)1533 2098 y Fo(\),)262 2148 y(and)17 b(the)h(in)o(tegration)f(path)h(\000)763 2154 y Fi(z)799 2148 y Fo(in)f(\(22\))h(is)f(tak)o(en)h(to)f(b)q(e)h (the)h(image)d(b)o(y)i Fn(y)h Fm(7!)262 2197 y Fn(z)r Fo(\()p Fn(y)q(;)7 b(h)p Fo(\))17 b(of)g Fn(\015)468 2203 y Fi(y)505 2197 y Fo(=)g Fm(f)p Fn(y)595 2203 y Fl(1)614 2197 y Fo(;)7 b Fm(\0001)16 b Fn(<)h Fo(Re)7 b Fn(y)848 2203 y Fl(1)884 2197 y Fn(<)17 b Fo(Re)7 b Fn(y)q(;)g Fo(Im)g Fn(y)1106 2203 y Fl(1)1142 2197 y Fo(=)17 b(Im)6 b Fn(y)q Fm(g)p Fo(.)28 b(W)m(e)17 b(c)o(ho)q(ose)h Fn(y)262 2247 y Fo(with)d(0)h Fn(<)f Fm(j)7 b Fo(Im)f Fn(y)q Fm(j)16 b Fn(<)g(\031)h Fo(in)f(case)h(\(I\))f(and)g(with)g(0)f Fn(<)h Fm(j)7 b Fo(Im)f Fn(y)q Fm(j)16 b Fn(<)f Fo(2)p Fn(\031)i Fo(in)f(case)h(\(I)q(I\))262 2297 y(\(see)f(Figure)f(3\),)f (so)h(that)g(the)g(path)g Fn(\015)875 2303 y Fi(y)910 2297 y Fo(a)o(v)o(oids)f(the)h(turning)g(p)q(oin)o(t\(s\))g(of)f Fn(P)20 b Fo(for)890 2469 y(10)p eop %%Page: 11 11 11 10 bop 262 307 a Fo(all)12 b Fn(h)p Fo('s.)18 b(In)c(the)g(Langer)g (plane,)g(w)o(e)g(ha)o(v)o(e)454 412 y Fn(I)472 418 y Fl(+)500 412 y Fo([)p Fn(f)t Fo(])d(=)h Fm(\000)640 384 y Fo(1)p 640 402 21 2 v 640 440 a(4)673 355 y Fh(Z)696 449 y Fi(\015)713 453 y Fd(y)720 412 y Fn(e)739 383 y Fj(\000)p Fl(2)787 353 y Fh(R)815 363 y Fd(y)807 401 y(y)822 407 y Fe(1)846 383 y Fi(P)t Fl(\()p Fi(y)901 387 y Fe(2)917 383 y Fi(;h)p Fl(\))959 371 y Fe(1)p Fd(=)p Fe(2)1005 383 y Fi(dy)1039 387 y Fe(2)1056 383 y Fi(=h)1099 380 y Fn(P)1132 364 y Fj(0)1126 390 y Fi(y)1146 380 y Fo(\()p Fn(y)1182 386 y Fl(1)1201 380 y Fn(;)7 b(h)p Fo(\))p 1099 402 161 2 v 1106 440 a Fn(P)f Fo(\()p Fn(y)1175 446 y Fl(1)1194 440 y Fn(;)h(h)p Fo(\))1264 412 y Fn(f)12 b(dy)1338 418 y Fl(1)1357 412 y Fn(;)640 554 y(J)t Fo([)p Fn(f)t Fo(])g(=)g Fm(\000)808 526 y Fo(1)p 808 544 21 2 v 808 583 a(4)840 498 y Fh(Z)864 592 y Fi(\015)881 596 y Fd(y)913 522 y Fn(P)946 507 y Fj(0)940 532 y Fi(y)960 522 y Fo(\()p Fn(y)996 528 y Fl(1)1015 522 y Fn(;)7 b(h)p Fo(\))p 913 544 161 2 v 920 583 a Fn(P)f Fo(\()p Fn(y)989 589 y Fl(1)1008 583 y Fn(;)h(h)p Fo(\))1078 554 y Fn(f)12 b(dy)1152 560 y Fl(1)1170 554 y Fn(:)262 653 y Fo(Notice)22 b(here)i(that)e Fn(H)s Fo(\()p Fn(z)670 659 y Fl(1)689 653 y Fn(;)7 b(h)p Fo(\))p Fn(dz)789 659 y Fl(1)833 653 y Fo(=)26 b Fm(f\000)p Fn(P)977 638 y Fj(0)971 663 y Fi(y)991 653 y Fo(\()p Fn(y)1027 659 y Fl(1)1046 653 y Fn(;)7 b(h)p Fo(\))p Fn(=)p Fo(4)p Fn(P)f Fo(\()p Fn(y)1216 659 y Fl(1)1233 653 y Fn(;)h(h)p Fo(\))p Fm(g)p Fn(dy)1355 659 y Fl(1)1395 653 y Fo(and)23 b(that)262 703 y Fm(\000)p Fn(P)327 688 y Fj(0)321 713 y Fi(y)340 703 y Fo(\()p Fn(y)376 709 y Fl(1)396 703 y Fn(;)7 b(h)p Fo(\))p Fn(=)p Fo(4)p Fn(P)f Fo(\()p Fn(y)566 709 y Fl(1)583 703 y Fn(;)h(h)p Fo(\))15 b(is)g(exp)q(onen)o(tially)f(decreasing)j(as)e(Re)7 b Fn(y)1288 709 y Fl(1)1322 703 y Fo(tends)17 b(to)e Fm(\0001)262 752 y Fo(if)f Fn(h)g(>)h Fo(0.)22 b(Hence)17 b(the)f(in)o(tegrals)f(in)g(\(22\))g(are)h(absolutely)f(con)o(v)o (ergen)o(t)h(when)g Fn(f)262 802 y Fo(is)d(b)q(ounded)i(for)e(example.) 324 852 y(W)m(e)g(in)o(tro)q(duce)i(new)f(real)g(v)n(ariables)f Fn(r)i Fo(and)f Fn(\022)h Fo(b)o(y)490 971 y Fn(r)d Fo(=)565 900 y Fh(\()619 928 y Fm(j)p Fn(e)650 913 y Fi(y)670 928 y Fm(j)p Fn(=h;)619 1013 y Fm(j)p Fn(e)650 998 y Fi(y)q(=)p Fl(2)704 1013 y Fm(j)p Fn(=h;)804 971 y(\022)h Fo(=)880 900 y Fh(\()934 929 y Fo(Im)7 b Fn(y)43 b Fo(in)13 b(case)i(\(I\))q Fn(;)934 1012 y Fo(Im)7 b Fn(y)q(=)p Fo(2)41 b(in)14 b(case)h(\(I)q(I\))p Fn(:)1487 971 y Fo(\(23\))262 1090 y(Since)g Fn(\022)i Fo(is)e(constan)o(t)h(on)f(the)h (in)o(tegral)f(con)o(tour,)g(w)o(e)h(can)f(rewrite)i(the)e(in)o(tegral) 262 1140 y(op)q(erators)g Fn(I)464 1146 y Fl(+)507 1140 y Fo(and)g Fn(J)k Fo(in)14 b(terms)i(of)e Fn(r)i Fo(and)f Fn(h)p Fo(.)21 b(Noticing)14 b(that)h Fn(dy)1320 1146 y Fi(j)1351 1140 y Fo(=)e Fn(dr)1437 1146 y Fi(j)1454 1140 y Fn(=r)1494 1146 y Fi(j)1526 1140 y Fo(in)262 1190 y(case)h(\(I\))g(\(resp.)20 b Fn(dy)573 1196 y Fi(j)602 1190 y Fo(=)12 b(2)p Fn(dr)708 1196 y Fi(j)724 1190 y Fn(=r)764 1196 y Fi(j)795 1190 y Fo(in)h(case)i(\(I)q(I\)\),)f(w)o(e)g (ha)o(v)o(e)491 1293 y Fn(I)509 1299 y Fl(+)536 1293 y Fo([)p Fn(f)t Fo(]\()p Fn(r)o(;)7 b(h)p Fo(\))k(=)732 1237 y Fh(Z)773 1247 y Fi(r)755 1331 y Fl(0)799 1293 y Fn(e)818 1276 y Fj(\000)p Fi(\020)r Fl(\()p Fi(r)o(;r)913 1280 y Fe(1)930 1276 y Fi(;h)p Fl(\))974 1293 y Fn(K)s Fo(\()p Fn(r)1047 1299 y Fl(1)1066 1293 y Fn(;)c(h)p Fo(\))p Fn(f)t Fo(\()p Fn(r)1184 1299 y Fl(1)1203 1293 y Fn(;)g(h)p Fo(\))p Fn(dr)1303 1299 y Fl(1)1320 1293 y Fn(;)155 b Fo(\(24\))588 1423 y Fn(J)t Fo([)p Fn(f)t Fo(]\()p Fn(r)o(;)7 b(h)p Fo(\))j(=)810 1366 y Fh(Z)852 1377 y Fi(r)833 1461 y Fl(0)877 1423 y Fn(K)s Fo(\()p Fn(r)950 1429 y Fl(1)969 1423 y Fn(;)d(h)p Fo(\))p Fn(f)t Fo(\()p Fn(r)1087 1429 y Fl(1)1106 1423 y Fn(;)g(h)p Fo(\))p Fn(dr)1206 1429 y Fl(1)1223 1423 y Fn(;)252 b Fo(\(25\))262 1511 y(where)469 1579 y Fn(\020)s Fo(\()p Fn(r)o(;)7 b(r)562 1585 y Fl(1)580 1579 y Fn(;)g(h)p Fo(\))k(=)h Fn(c)712 1585 y Fl(1)737 1522 y Fh(Z)779 1532 y Fi(r)760 1616 y(r)775 1620 y Fe(1)804 1579 y Fn(P)837 1561 y Fl(1)p Fi(=)p Fl(2)895 1550 y Fn(dr)936 1556 y Fl(2)p 894 1569 62 2 v 894 1607 a Fn(r)913 1613 y Fl(2)931 1607 y Fn(h)960 1579 y(;)48 b(K)s Fo(\()p Fn(r)1093 1585 y Fl(1)1111 1579 y Fn(;)7 b(h)p Fo(\))12 b(=)f Fn(c)1243 1585 y Fl(2)1278 1546 y Fn(P)1311 1531 y Fj(0)1305 1557 y Fi(y)p 1267 1569 70 2 v 1267 1607 a Fn(r)1286 1613 y Fl(1)1304 1607 y Fn(P)1342 1579 y(;)133 b Fo(\(26\))262 1673 y(with)17 b Fn(c)378 1679 y Fl(1)415 1673 y Fo(=)h(2)f(\(resp.)31 b(4\))18 b(and)g Fn(c)793 1679 y Fl(2)829 1673 y Fo(=)h Fm(\000)p Fo(1)p Fn(=)p Fo(4)e(\(resp.)31 b Fm(\000)p Fo(1)p Fn(=)p Fo(2\).)f(In)17 b(terms)i(of)e(the)262 1723 y(v)n(ariables)c(\()p Fn(r)o(;)7 b(h)p Fo(\),)12 b(the)j(functions)f Fn(P)19 b Fo(and)14 b Fn(P)962 1708 y Fj(0)956 1733 y Fi(y)989 1723 y Fo(are)g(written)h(as)f(follo)o(ws:) 502 1809 y Fn(P)6 b Fo(\()p Fn(r)570 1815 y Fl(1)588 1809 y Fn(;)h(h)p Fo(\))k(=)h Fn(h)726 1792 y Fl(2)744 1809 y Fm(f)p Fn(r)785 1792 y Fl(2)784 1820 y(1)803 1809 y Fn(e)822 1792 y Fl(2)p Fi(i\022)870 1809 y Fn(V)894 1815 y Fl(0)913 1809 y Fo(\()p Fn(hr)972 1815 y Fl(1)990 1809 y Fn(e)1009 1792 y Fi(i\022)1040 1809 y Fo(\))d(+)h(\()p Fn(`)g Fo(+)f(1)p Fn(=)p Fo(2\))1270 1792 y Fl(2)1288 1809 y Fm(g)p Fn(;)166 b Fo(\(27\))482 1888 y Fn(P)515 1871 y Fj(0)509 1898 y Fi(y)529 1888 y Fo(\()p Fn(r)564 1894 y Fl(1)582 1888 y Fn(;)7 b(h)p Fo(\))k(=)h Fn(h)720 1871 y Fl(2)738 1888 y Fn(r)758 1871 y Fl(2)757 1898 y(1)777 1888 y Fn(e)796 1871 y Fl(2)p Fi(i\022)843 1888 y Fm(f)p Fo(2)p Fn(V)909 1894 y Fl(0)928 1888 y Fo(\()p Fn(hr)987 1894 y Fl(1)1005 1888 y Fn(e)1024 1871 y Fi(i\022)1055 1888 y Fo(\))d(+)h Fn(V)1146 1894 y Fl(1)1165 1888 y Fo(\()p Fn(hr)1224 1894 y Fl(1)1242 1888 y Fn(e)1261 1871 y Fi(i\022)1292 1888 y Fo(\))p Fm(g)p Fn(;)146 b Fo(\(28\))262 1955 y(where)378 2048 y Fn(V)402 2054 y Fl(0)420 2048 y Fo(\()p Fn(\030)r Fo(\))12 b(=)528 1990 y Fh(\032)580 2023 y Fn(V)e Fo(\()p Fn(\030)r Fo(\))p Fn(;)580 2073 y(\030)600 2058 y Fl(2)619 2073 y Fn(V)f Fo(\()p Fn(\030)688 2058 y Fl(2)707 2073 y Fo(\))p Fn(;)809 2048 y(V)833 2054 y Fl(1)852 2048 y Fo(\()p Fn(\030)r Fo(\))j(=)959 1990 y Fh(\032)1011 2023 y Fn(\030)r(V)1065 2008 y Fj(0)1055 2033 y Fi(x)1077 2023 y Fo(\()p Fn(\030)r Fo(\))41 b(in)14 b(case)h(\(I\))p Fn(;)1011 2073 y(\030)1031 2058 y Fl(4)1050 2073 y Fn(V)1083 2058 y Fj(0)1074 2083 y Fi(x)1095 2073 y Fo(\()p Fn(\030)1131 2058 y Fl(2)1150 2073 y Fo(\))42 b(in)13 b(case)i(\(I)q(I\))p Fn(:)262 2150 y Fo(Remark)e(that)379 2243 y(lim)375 2270 y Fi(\030)q Fj(!)p Fl(0)447 2243 y Fn(V)471 2249 y Fl(0)490 2243 y Fo(\()p Fn(\030)r Fo(\))f(=)598 2185 y Fh(\032)650 2218 y Fn(q)669 2224 y Fl(0)687 2218 y Fn(;)650 2268 y(q)669 2274 y Fj(\000)p Fl(1)713 2268 y Fn(;)803 2243 y Fo(lim)799 2270 y Fi(\030)q Fj(!)p Fl(0)871 2243 y Fn(V)895 2249 y Fl(1)914 2243 y Fo(\()p Fn(\030)r Fo(\))g(=)1022 2185 y Fh(\032)1074 2218 y Fo(0)41 b(in)13 b(case)i(\(I\))q Fn(;)1074 2268 y Fm(\000)p Fn(q)1125 2274 y Fj(\000)p Fl(1)1211 2268 y Fo(in)e(case)i(\(I)q(I\))p Fn(:)262 2344 y Fo(W)m(e)e(ha)o(v)o(e)h(the)g(follo)o(wing)d(lemmas:)890 2469 y(11)p eop %%Page: 12 12 12 11 bop 262 307 a Fk(Lemma)15 b(8)20 b Fg(Supp)n(ose)13 b(that)f Fn(\017)f(<)h Fm(j)p Fn(\022)q Fm(j)f Fn(<)h(\031)t Fm(\000)r Fn(\017)f Fg(for)g(a)h(given)f(p)n(ositive)h Fn(\017)p Fg(.)17 b(Then)12 b(ther)n(e)262 357 y(exist)g(p)n(ositive)h Fn(\016)i Fg(and)f Fn(c)f Fg(such)g(that)g(for)g Fn(e)906 342 y Fl(Re)6 b Fi(y)984 357 y Fo(\(=)12 b Fm(j)p Fn(x)p Fm(j)p Fo(\))f Fn(<)h(\016)r Fg(,)h(for)f(any)i Fn(y)1371 363 y Fl(1)1401 357 y Fm(2)e Fn(\015)1462 363 y Fi(y)1495 357 y Fg(and)262 407 y(for)i(any)h(smal)r(l)g Fn(h)p Fg(,)f(we)h(have)g(in)g(b)n(oth)g(c)n(ases)657 496 y(R)n(e)g Fn(P)6 b Fo(\()p Fn(r)787 502 y Fl(1)805 496 y Fn(;)h(h)p Fo(\))864 479 y Fl(1)p Fi(=)p Fl(2)927 496 y Fm(\025)12 b Fn(ch)p Fo(\(1)d(+)g Fn(r)1119 502 y Fl(1)1138 496 y Fo(\))p Fn(:)262 585 y Fc(Pro)q(of:)41 b Fo(Notice)16 b(that)f(the)h(function)f Fn(V)899 591 y Fl(0)918 585 y Fo(\()p Fn(r)953 591 y Fl(1)972 585 y Fn(;)7 b(h)p Fo(\))14 b(sta)o(ys)i(close)g(to)f Fn(q)1323 591 y Fl(0)1355 585 y Fn(<)f Fo(0)h(in)g(case)262 635 y(\(I\))f(and)f(to)h Fn(q)473 641 y Fj(\000)p Fl(1)529 635 y Fn(<)e Fo(0)h(in)h(case)h(\(I)q (I\))f(when)g Fm(j)p Fn(x)p Fm(j)f Fo(is)h(small.)j(This)c(means)h (that)g(there)262 684 y(exists)g Fn(\016)g(>)e Fo(0)h(suc)o(h)i(that)f (if)f Fm(j)p Fn(x)p Fm(j)d Fn(<)i(\016)r Fo(,)h(w)o(e)h(ha)o(v)o(e)g (for)g(all)e Fn(h)g(>)g Fo(0,)673 773 y Fm(j)7 b Fo(arg)g Fn(e)776 756 y Fl(2)p Fi(i\022)823 773 y Fn(V)847 779 y Fl(0)866 773 y Fo(\()p Fn(r)901 779 y Fl(1)920 773 y Fn(;)g(h)p Fo(\))p Fm(j)j Fn(<)i(\031)e Fm(\000)g Fn(\017:)262 862 y Fo(Since)16 b(\()p Fn(`)11 b Fo(+)g(1)p Fn(=)p Fo(2\))538 847 y Fl(2)572 862 y Fo(is)16 b(a)g(p)q(ositiv)o(e)g (constan)o(t,)h(it)f(is)g(easy)g(to)g(see)h(from)f(\(27\))g(that)262 912 y(Re)6 b Fn(P)350 897 y Fl(1)p Fi(=)p Fl(2)402 912 y Fn(=h)14 b Fo(is)f(b)q(ounded)i(b)q(elo)o(w)e(b)o(y)h(a)g(p)q(ositiv) o(e)f(constan)o(t.)324 962 y(T)m(ogether)k(with)g(the)g(fact)g(that)g (Re)7 b Fn(P)940 947 y Fl(1)p Fi(=)p Fl(2)991 962 y Fn(=h)17 b Fo(b)q(eha)o(v)o(es)h(lik)o(e)e Fm(j)p Fn(\030)1320 968 y Fl(1)1338 962 y Fm(j)g Fo(=)h Fn(r)1434 968 y Fl(1)1469 962 y Fo(as)g Fn(r)1542 968 y Fl(1)262 1012 y Fo(tends)d(to)g(+)p Fm(1)p Fo(,)f(w)o(e)h(get)h(the)f(Lemma.)659 b Fb(2)262 1125 y Fk(Lemma)15 b(9)20 b Fg(Ther)n(e)e(exists)g(a)h(c)n(onstant)g Fn(C)954 1131 y Fl(1)990 1125 y Fn(>)f Fo(0)g Fg(such)g(that)h(for)e (any)i Fn(y)1437 1131 y Fl(1)1474 1125 y Fm(2)e Fn(\015)1540 1131 y Fi(y)262 1175 y Fg(and)e(for)f(any)i(smal)r(l)e Fn(h)p Fg(,)h(we)f(have)i(in)f(b)n(oth)g(c)n(ases,)722 1264 y Fm(j)p Fn(P)767 1246 y Fj(0)761 1274 y Fi(y)780 1264 y Fo(\()p Fn(r)815 1270 y Fl(1)834 1264 y Fn(;)7 b(h)p Fo(\))p Fm(j)j(\024)i Fn(C)989 1270 y Fl(1)1007 1264 y Fn(h)1031 1246 y Fl(2)1050 1264 y Fn(r)1070 1246 y Fl(2)1069 1274 y(1)1089 1264 y Fn(:)262 1441 y Fc(Pro)q(of:)48 b Fo(F)m(rom)16 b(\(28\),)h(w)o(e)g(see)h(that)f Fm(j)p Fn(P)909 1426 y Fj(0)903 1452 y Fi(y)923 1441 y Fo(\()p Fn(r)958 1447 y Fl(1)976 1441 y Fn(;)7 b(h)p Fo(\))p Fm(j)p Fn(=h)1092 1426 y Fl(2)1109 1441 y Fn(r)1129 1426 y Fl(2)1128 1452 y(1)1165 1441 y Fo(sta)o(ys)17 b(close)g(to)g(2)p Fm(j)p Fn(q)1480 1447 y Fl(0)1498 1441 y Fm(j)f Fo(in)262 1491 y(case)e(\(I\))g(and)g(to)g Fm(j)p Fn(q)573 1497 y Fj(\000)p Fl(1)617 1491 y Fm(j)f Fo(and)h(is)f(b)q(ounded.)599 b Fb(2)262 1604 y Fk(Prop)q(osition)14 b(10)20 b Fg(Supp)n(ose)g(that)f Fn(\017)f(<)h Fm(j)p Fn(\022)q Fm(j)f Fn(<)h(\031)13 b Fm(\000)g Fn(\017)18 b Fg(for)h(a)g(given)g(p)n(ositive)f Fn(\017)p Fg(.)262 1654 y(Ther)n(e)d(exist)g(p)n(ositive)h(c)n (onstants)g Fn(\016)r Fg(,)g Fn(C)i Fg(and)f Fn(c)e Fg(such)i(that)e (if)g Fn(e)1266 1639 y Fl(Re)6 b Fi(y)1345 1654 y Fo(\(=)14 b Fm(j)p Fn(x)p Fm(j)p Fo(\))e Fn(<)h(\016)r Fg(,)262 1704 y(then)i(in)g(b)n(oth)g(c)n(ases)g(\(I\))649 1710 y Fj(\000)692 1704 y Fg(and)g(\(II\))838 1710 y Fj(\000)881 1704 y Fg(we)g(have)574 1820 y Fm(j)p Fo(\()p Fn(J)e Fm(\016)c Fn(I)686 1826 y Fl(+)714 1820 y Fo(\))730 1803 y Fi(n)753 1820 y Fo([1]\()p Fn(r)o(;)e(h)p Fo(\))p Fm(j)i(\024)965 1792 y Fn(C)998 1777 y Fl(2)p Fi(n)p 961 1810 80 2 v 961 1848 a Fo(2)982 1836 y Fi(n)1004 1848 y Fn(n)p Fo(!)1046 1820 y(\()1111 1792 y Fn(r)p 1067 1810 110 2 v 1067 1848 a Fo(1)g(+)g Fn(cr)1181 1820 y Fo(\))1197 1803 y Fl(4)p Fi(n)1236 1820 y Fn(;)239 b Fo(\(29\))490 1941 y Fm(j)p Fn(I)520 1947 y Fl(+)556 1941 y Fm(\016)9 b Fo(\()p Fn(J)14 b Fm(\016)9 b Fn(I)687 1947 y Fl(+)714 1941 y Fo(\))730 1924 y Fi(n)753 1941 y Fo([1]\()p Fn(r)o(;)e(h)p Fo(\))p Fm(j)i(\024)965 1913 y Fn(C)998 1898 y Fl(2)p Fi(n)p Fl(+1)p 961 1932 123 2 v 961 1970 a Fo(2)982 1958 y Fi(n)p Fj(\000)p Fl(1)1047 1970 y Fn(n)p Fo(!)1154 1913 y Fn(r)1174 1898 y Fl(4)p Fi(n)p Fl(+2)p 1093 1932 223 2 v 1093 1970 a Fo(\(1)g(+)h Fn(cr)q Fo(\))1235 1958 y Fl(4)p Fi(n)p Fl(+3)1321 1941 y Fn(:)154 b Fo(\(30\))262 2054 y Fc(Pro)q(of:)60 b Fo(W)m(e)20 b(pro)o(v)o(e)g(the)g(estimate)h(\(29\))e(b)o(y)h (induction.)36 b(It)20 b(is)g(ob)o(vious)f(for)262 2104 y Fn(n)11 b Fo(=)h(0,)f(and)h(w)o(e)g(supp)q(ose)h(it)e(holds)h(for)f Fn(n)p Fo(.)17 b(F)m(or)12 b(brevit)o(y)g(w)o(e)g(write)g(\()p Fn(J)d Fm(\016)c Fn(I)1407 2110 y Fl(+)1434 2104 y Fo(\))1450 2089 y Fi(n)1473 2104 y Fo([1])10 b(=)262 2154 y Fn(p)283 2160 y Fi(n)305 2154 y Fo(\()p Fn(y)q(;)d(h)p Fo(\).)18 b(It)12 b(follo)o(ws)f(\014rst)j(from)d(Lemma)h(8)g(and)g(Lemma)g(9)g (\(noticing)g(also)g(that)262 2204 y Fm(j)p Fn(P)6 b Fm(j)k(\025)h Fo(\(Re)c Fn(P)477 2189 y Fl(1)p Fi(=)p Fl(2)529 2204 y Fo(\))545 2189 y Fl(2)564 2204 y Fo(\),)13 b(that)h(there)h(exists)g Fn(C)f(>)e Fo(0)i(suc)o(h)g(that)523 2322 y Fm(j)p Fn(I)553 2328 y Fl(+)581 2322 y Fo([)p Fn(f)t Fo(])p Fm(j)d(\024)g Fn(C)s(e)747 2305 y Fj(\000)p Fi(cr)807 2322 y Fn(r)827 2305 y Fj(\000)p Fi(c)876 2266 y Fh(Z)918 2276 y Fi(r)899 2360 y Fl(0)943 2322 y Fn(e)962 2305 y Fi(cr)992 2309 y Fe(1)1056 2294 y Fn(r)1076 2276 y Fi(c)p Fl(+1)1075 2305 y(1)p 1016 2313 160 2 v 1016 2351 a Fo(\(1)e(+)g Fn(r)1122 2357 y Fl(1)1141 2351 y Fo(\))1157 2339 y Fl(2)1181 2322 y Fm(j)p Fn(f)t Fm(j)p Fn(dr)1270 2328 y Fl(1)1287 2322 y Fn(;)890 2469 y Fo(12)p eop %%Page: 13 13 13 12 bop 262 307 a Fo(and)637 370 y Fm(j)p Fn(J)t Fo([)p Fn(f)t Fo(])p Fm(j)11 b(\024)h Fn(C)830 314 y Fh(Z)872 324 y Fi(r)853 408 y Fl(0)963 342 y Fn(r)982 348 y Fl(1)p 902 361 160 2 v 902 399 a Fo(\(1)d(+)h Fn(r)1009 405 y Fl(1)1027 399 y Fo(\))1043 387 y Fl(2)1067 370 y Fm(j)p Fn(f)t Fm(j)p Fn(dr)1156 376 y Fl(1)1173 370 y Fn(;)262 467 y Fo(where)16 b Fn(c)f Fo(is)g(the)g(same)g(as)h(in)e(Lemma)h(8)f (and)h(ma)o(y)f(b)q(e)i(assumed)g(to)f(b)q(e)g(smaller)262 517 y(than)e(1.)18 b(So)c(w)o(e)g(ha)o(v)o(e)483 627 y Fm(j)p Fn(I)513 633 y Fl(+)541 627 y Fo([)p Fn(p)574 633 y Fi(n)596 627 y Fo(])p Fm(j)c(\024)679 599 y Fn(C)712 584 y Fl(2)p Fi(n)p Fl(+1)p 679 617 114 2 v 696 655 a Fo(2)717 643 y Fi(n)740 655 y Fn(n)p Fo(!)798 627 y Fn(e)817 610 y Fj(\000)p Fi(cr)883 570 y Fh(Z)925 581 y Fi(r)906 665 y Fl(0)950 627 y Fn(e)969 610 y Fi(cr)999 614 y Fe(1)1093 599 y Fn(r)1113 581 y Fl(4)p Fi(n)p Fl(+1)1112 610 y(1)p 1023 617 241 2 v 1023 655 a Fo(\(1)f(+)h Fn(cr)1148 661 y Fl(1)1166 655 y Fo(\))1182 643 y Fl(4)p Fi(n)p Fl(+2)1268 627 y Fn(dr)1309 633 y Fl(1)1327 627 y Fn(:)262 773 y Fo(If)j(w)o(e)h(put)g Fn(g)q Fo(\()p Fn(r)q Fo(\))e(=)g Fn(e)588 758 y Fi(cr)687 745 y Fn(r)707 730 y Fl(4)p Fi(n)p Fl(+2)p 627 764 223 2 v 627 802 a Fo(\(1)d(+)g Fn(cr)q Fo(\))768 790 y Fl(4)p Fi(n)p Fl(+3)854 773 y Fo(,)14 b(w)o(e)g(get)566 920 y Fn(g)587 902 y Fj(0)599 920 y Fo(\()p Fn(r)q Fo(\))e(=)f Fn(e)725 902 y Fi(cr)825 892 y Fn(r)845 876 y Fl(4)p Fi(n)p Fl(+1)p 764 910 V 764 948 a Fo(\(1)e(+)h Fn(cr)q Fo(\))906 936 y Fl(4)p Fi(n)p Fl(+2)997 892 y Fn(c)1015 876 y Fl(2)1033 892 y Fn(r)1053 876 y Fl(2)1081 892 y Fo(+)g(4)p Fn(n)e Fo(+)i(2)p 997 910 243 2 v 1038 948 a(\(1)f(+)h Fn(cr)q Fo(\))1180 936 y Fl(2)1245 920 y Fn(;)262 1033 y Fo(so)j(that)683 1099 y Fn(g)704 1082 y Fj(0)716 1099 y Fo(\()p Fn(r)q Fo(\))f Fm(\025)828 1071 y Fo(1)p 828 1089 21 2 v 828 1127 a(2)854 1099 y Fn(e)873 1082 y Fi(cr)972 1071 y Fn(r)992 1056 y Fl(4)p Fi(n)p Fl(+1)p 912 1089 223 2 v 912 1127 a Fo(\(1)d(+)g Fn(cr)q Fo(\))1053 1115 y Fl(4)p Fi(n)p Fl(+2)262 1196 y Fo(and)627 1262 y Fm(j)p Fn(I)657 1268 y Fl(+)685 1262 y Fo([)p Fn(p)718 1268 y Fi(n)740 1262 y Fo(])p Fm(j)h(\024)827 1234 y Fn(C)860 1219 y Fl(2)p Fi(n)p Fl(+1)p 823 1252 123 2 v 823 1290 a Fo(2)844 1278 y Fi(n)p Fj(\000)p Fl(1)909 1290 y Fn(n)p Fo(!)1016 1234 y Fn(r)1036 1219 y Fl(4)p Fi(n)p Fl(+2)p 955 1252 223 2 v 955 1290 a Fo(\(1)g(+)f Fn(cr)q Fo(\))1097 1278 y Fl(4)p Fi(n)p Fl(+3)1183 1262 y Fn(:)292 b Fo(\(31\))262 1358 y(Then)14 b(w)o(e)g(obtain)262 1469 y Fm(j)p Fn(J)7 b Fm(\016)s Fn(I)346 1475 y Fl(+)372 1469 y Fo([)p Fn(p)405 1475 y Fi(n)427 1469 y Fo(])p Fm(j)j(\024)514 1441 y Fn(C)547 1426 y Fl(2)p Fi(n)p Fl(+2)p 510 1459 123 2 v 510 1497 a Fo(2)531 1485 y Fi(n)p Fj(\000)p Fl(1)596 1497 y Fn(n)p Fo(!)637 1412 y Fh(Z)679 1422 y Fi(r)660 1507 y Fl(0)765 1441 y Fn(r)785 1423 y Fl(4)p Fi(n)p Fl(+3)784 1452 y(1)p 695 1459 241 2 v 695 1497 a Fo(\(1)g(+)f Fn(cr)820 1503 y Fl(1)838 1497 y Fo(\))854 1485 y Fl(4)p Fi(n)p Fl(+5)941 1469 y Fn(dr)982 1475 y Fl(1)1011 1469 y Fo(=)1116 1441 y Fn(C)1149 1426 y Fl(2)p Fi(n)p Fl(+2)p 1060 1459 226 2 v 1060 1497 a Fo(2)1081 1485 y Fi(n)p Fl(+1)1145 1497 y Fo(\()p Fn(n)h Fo(+)f(1\)!)1290 1469 y(\()1356 1441 y Fn(r)p 1311 1459 110 2 v 1311 1497 a Fo(1)h(+)f Fn(cr)1426 1469 y Fo(\))1442 1452 y Fl(4\()p Fi(n)p Fl(+1\))1549 1469 y Fn(:)262 1584 y Fo(and)j(this)i(completes)f (the)h(pro)q(of)f(of)f(\(29\).)18 b(The)13 b(estimate)h(\(30\))f(follo) o(ws)e(immedi-)262 1634 y(ately)i(from)g(\(29\))h(and)g(\(31\).)813 b Fb(2)324 1709 y Fo(Theorem)15 b(1)e(follo)o(ws)g(easily)h(from)f (these)j(estimates:)j(w)o(e)c(ha)o(v)o(e)f(b)o(y)g(\(29\))g(and)262 1759 y(\(30\))f(that)282 1850 y(\()p Fn(J)d Fm(\016)c Fn(I)376 1856 y Fl(+)404 1850 y Fo(\))420 1833 y Fi(n)442 1850 y Fo([1])11 b(=)h Fn(O)q Fo(\()p Fn(r)611 1833 y Fl(4)p Fi(n)650 1850 y Fo(\))p Fn(;)48 b(I)744 1856 y Fl(+)777 1850 y Fm(\016)6 b Fo(\()p Fn(J)k Fm(\016)c Fn(I)898 1856 y Fl(+)925 1850 y Fo(\))941 1833 y Fi(n)964 1850 y Fo([1])k(=)i Fn(O)q Fo(\()p Fn(r)1132 1833 y Fl(4)p Fi(n)p Fl(+2)1213 1850 y Fo(\))42 b(as)f Fn(r)13 b Fm(!)e Fo(0)p Fn(:)20 b Fo(\(32\))262 1941 y(Hence)344 2024 y Fn(x)368 2009 y Fl(1)p Fi(=)p Fl(2)420 2024 y Fn(w)451 2009 y Fl(0)469 2024 y Fo(\()p Fn(y)q(;)7 b(h)p Fo(\))42 b(=)12 b Fn(e)670 2009 y Fj(\000)p Fi(\031)q(i=)p Fl(4)764 2024 y Fn(e)783 2009 y Fi(y)q(=)p Fl(2)837 2024 y Fn(P)870 2009 y Fj(\000)p Fl(1)p Fi(=)p Fl(4)955 2024 y Fo(exp\()1034 1990 y Fh(R)1062 2001 y Fi(y)1089 2024 y Fn(P)1122 2009 y Fl(1)p Fi(=)p Fl(2)1173 2024 y Fn(dt=h)p Fo(\)\(1)d(+)g Fn(O)q Fo(\()p Fn(r)1427 2009 y Fl(2)1446 2024 y Fo(\)\))607 2110 y(=)j Fn(e)670 2095 y Fj(\000)p Fi(\031)q(i=)p Fl(4)774 2100 y Fo(~)764 2110 y Fn(Q)797 2095 y Fj(\000)p Fl(1)p Fi(=)p Fl(4)882 2110 y Fo(exp)q(\()962 2077 y Fh(R)989 2087 y Fi(x)1027 2100 y Fo(~)1017 2110 y Fn(Q)1050 2095 y Fl(1)p Fi(=)p Fl(2)1102 2110 y Fn(dt=h)p Fo(\)\(1)d(+)g Fn(O)q Fo(\()p Fn(r)1356 2095 y Fl(2)1375 2110 y Fo(\)\))p Fn(;)262 2200 y Fo(as)h Fn(r)h Fo(tends)g(to)f(0)f(and)h(in)g (particular)g(as)g Fn(x)g Fo(tends)h(to)f(0)f(while)h Fn(h)h(>)h Fo(0)e(is)g(\014xed.)17 b(Since)271 2239 y(~)262 2250 y Fn(Q)c Fm(\030)i Fn(h)379 2234 y Fl(2)397 2250 y Fo(\()p Fn(`)c Fo(+)g(1)p Fn(=)p Fo(2\))563 2234 y Fl(2)581 2250 y Fn(=x)626 2234 y Fl(2)644 2250 y Fo(,)k(the)h(righ)o(t) f(hand)g(side)h(b)q(eha)o(v)o(es)g(lik)o(e)f Fn(x)1294 2234 y Fi(`)p Fl(+1)1352 2250 y Fo(.)22 b(Therefore)262 2299 y Fn(x)286 2284 y Fj(\000)p Fl(1)p Fi(=)p Fl(2)363 2299 y Fn(w)394 2284 y Fl(0)427 2299 y Fo(is)13 b(prop)q(ortional)g(to) h Fn(u)782 2284 y Fl(0)800 2299 y Fo(,)g(and)g(this)f(giv)o(es)h (Theorem)h(1.)890 2469 y(13)p eop %%Page: 14 14 14 13 bop 324 307 a Fo(Next)18 b(w)o(e)g(pro)o(v)o(e)g(\(4\))f(of)g (Theorem)i(2.)29 b(W)m(e)17 b(need)i(t)o(w)o(o)e(other)h(exact)g(WKB) 262 357 y(solutions:)e(w)o(e)c(c)o(ho)q(ose)g(t)o(w)o(o)g(base)g(p)q (oin)o(ts)f Fn(y)947 363 y Fj(\006)987 357 y Fo(indep)q(enden)o(t)i(of) f Fn(h)f Fo(with)g Fn(\022)j Fo(=)d Fn(\022)1488 363 y Fj(\006)1528 357 y Fo(=)262 407 y Fm(\006)p Fn(\031)q(=)p Fo(2)i(as)h(in)g(Figure)g(3.)j(W)m(e)d(set)585 493 y Fn(w)616 478 y Fl(+)643 493 y Fo(\()p Fn(y)q(;)7 b(h)p Fo(\))12 b(=)g Fn(e)814 478 y Fl(+)p Fi(\031)q(i=)p Fl(4)908 493 y Fn(w)938 499 y Fj(\000)965 493 y Fo(\()p Fn(y)q(;)7 b(h)p Fo(;)g Fn(\021)q Fo(\()p Fn(h)p Fo(\))p Fn(;)g(y)1181 499 y Fl(+)1209 493 y Fo(\))p Fn(;)585 545 y(w)616 530 y Fj(\000)644 545 y Fo(\()p Fn(y)q(;)g(h)p Fo(\))12 b(=)f Fn(e)814 530 y Fl(+)p Fi(\031)q(i=)p Fl(4)908 545 y Fn(w)938 551 y Fj(\000)966 545 y Fo(\()p Fn(y)q(;)c(h)p Fo(;)g Fn(\021)q Fo(\()p Fn(h)p Fo(\))p Fn(;)g(y)1182 551 y Fj(\000)1210 545 y Fo(\))p Fn(:)262 628 y Fo(The)15 b(p)q(oin)o(t)f(is) g(that)h(w)o(e)g(kno)o(w)f(w)o(ell)g(the)h(asymptotic)f(form)g(of)g (these)i(t)o(w)o(o)e(exact)262 678 y(WKB)c(solutions)f(on)h(the)g(real) g(in)o(terv)n(al)f Fn(I)s Fo(.)17 b(So)10 b(it)g(su\016ces)h(to)e (express)j Fn(w)1386 663 y Fl(0)1414 678 y Fo(in)e(terms)262 728 y(of)h Fn(w)338 713 y Fl(+)378 728 y Fo(and)h Fn(w)488 713 y Fj(\000)516 728 y Fo(,)g(and)g(Theorem)h(2)f(\(4\))h(follo)o(ws)e (from)g(the)i(follo)o(wing)d(connection)262 778 y(form)o(ula:)262 863 y Fk(Prop)q(osition)k(11)20 b Fg(Ther)n(e)11 b(exist)g(two)g (functions)h Fn(A)1079 869 y Fl(+)1107 863 y Fo(\()p Fn(h)p Fo(\))g Fg(and)g Fn(A)1283 869 y Fj(\000)1311 863 y Fo(\()p Fn(h)p Fo(\))g Fg(and)g(a)g(p)n(os-)262 913 y(itive)i(numb)n(er)h Fn(A)g Fg(satisfying)417 998 y Fn(A)448 1004 y Fl(+)476 998 y Fo(\()p Fn(h)p Fo(\))d(=)f Fm(\000)p Fn(A)f Fo(+)g Fn(o)p Fo(\(1\))p Fn(;)49 b(A)867 1004 y Fj(\000)895 998 y Fo(\()p Fn(h)p Fo(\))11 b(=)h Fn(A)e Fo(+)f Fn(o)p Fo(\(1\))43 b Fg(as)f Fn(h)12 b Fm(!)f Fo(0)p Fn(;)262 1083 y Fg(such)k(that)509 1168 y Fn(w)540 1151 y Fl(0)559 1168 y Fo(\()p Fn(y)q(;)7 b(h)p Fo(\))12 b(=)f Fn(A)741 1174 y Fl(+)769 1168 y Fo(\()p Fn(h)p Fo(\))p Fn(w)856 1151 y Fl(+)884 1168 y Fo(\()p Fn(y)q(;)c(h)p Fo(\))i(+)h Fn(A)1062 1174 y Fj(\000)1090 1168 y Fo(\()p Fn(h)p Fo(\))p Fn(w)1177 1151 y Fj(\000)1205 1168 y Fo(\()p Fn(y)q(;)d(h)p Fo(\))p Fn(:)262 1253 y Fc(Pro)q(of:)36 b Fo(As)14 b(in)g(Section)g(2)f(\(see)j (\(14\))d(and)h(\(15\)\),)f(w)o(e)h(ha)o(v)o(e)262 1364 y Fn(A)293 1370 y Fl(+)320 1364 y Fo(\()p Fn(h)p Fo(\))e(=)g Fm(\000)469 1336 y Fn(W)514 1321 y Fl(+)508 1346 y(ev)o(en)575 1336 y Fo(\()p Fn(z)r Fo(\()p Fn(y)648 1342 y Fj(\000)677 1336 y Fo(\);)7 b Fn(z)r Fo(\()p Fm(\0001)p Fo(\))p Fn(;)g(h)p Fo(\))p 469 1354 428 2 v 482 1395 a Fn(W)527 1377 y Fl(+)521 1400 y(ev)o(en)588 1395 y Fo(\()p Fn(z)r Fo(\()p Fn(y)661 1401 y Fj(\000)690 1395 y Fo(\);)g Fn(z)r Fo(\()s(^)-24 b Fn(y)782 1401 y Fl(+)809 1395 y Fo(\))p Fn(;)7 b(h)p Fo(\))902 1364 y Fn(;)27 b(A)972 1370 y Fj(\000)1000 1364 y Fo(\()p Fn(h)p Fo(\))12 b(=)1117 1336 y Fn(W)1162 1321 y Fl(+)1156 1346 y(ev)o(en)1222 1336 y Fo(\()p Fn(z)r Fo(\()p Fn(y)1295 1342 y Fl(+)1324 1336 y Fo(\);)7 b Fn(z)r Fo(\()p Fm(\0001)p Fo(\))p Fn(;)g(h)p Fo(\))p 1117 1354 V 1130 1395 a Fn(W)1175 1377 y Fl(+)1169 1400 y(ev)o(en)1235 1395 y Fo(\()p Fn(z)r Fo(\()p Fn(y)1308 1401 y Fj(\000)1337 1395 y Fo(\);)g Fn(z)r Fo(\()s(^)-24 b Fn(y)1429 1401 y Fl(+)1457 1395 y Fo(\))p Fn(;)7 b(h)p Fo(\))1549 1364 y Fn(;)262 1476 y Fo(where)19 b(^)-24 b Fn(y)403 1482 y Fl(+)447 1476 y Fo(is,)16 b(as)g(in)f(\(16\),)g(the)i (cop)o(y)e(of)h(the)g(p)q(oin)o(t)f Fn(y)1132 1482 y Fl(+)1176 1476 y Fo(on)g(the)i(other)f(sheet)h(of)262 1526 y(the)h(Riemann)e(surface.)30 b(W)m(e)17 b(ha)o(v)o(e)g(exactly)h (as)f(in)g(Section)h(2)f(in)g(spite)h(of)f(the)262 1576 y(dep)q(endence)f(of)d Fn(P)20 b Fo(on)13 b Fn(h)h Fo(that)596 1661 y Fn(W)641 1644 y Fl(+)635 1671 y(ev)o(en)702 1661 y Fo(\()p Fn(z)r Fo(\()p Fn(y)775 1667 y Fj(\000)804 1661 y Fo(\);)7 b Fn(z)r Fo(\()s(^)-24 b Fn(y)896 1667 y Fl(+)924 1661 y Fo(\))p Fn(;)7 b(h)p Fo(\))k(=)h(1)d(+)g Fn(O)q Fo(\()p Fn(h)p Fo(\))p Fn(:)324 1746 y Fo(On)19 b(the)g(other)h(hand,)f(w)o(e)h(need)f(to)g(study)h(the)f(b)q(eha)o (viour)g(as)g Fn(h)g Fm(!)h Fo(0)e(of)262 1796 y Fn(W)307 1781 y Fl(+)301 1806 y(ev)o(en)367 1796 y Fo(\()p Fn(z)r Fo(\()p Fn(y)q Fo(\);)7 b Fn(z)r Fo(\()p Fm(\0001)p Fo(\))p Fn(;)g(h)p Fo(\))15 b(for)f(\014xed)h Fn(y)f Fo(=)e Fn(y)940 1802 y Fj(\006)969 1796 y Fo(.)19 b(As)14 b(ab)q(o)o(v)o(e,)g(w)o(e)h (will)d(consider)k(this)262 1845 y(function)d(of)g Fn(y)j Fo(and)e Fn(h)g Fo(as)f(a)h(function)g(of)f Fn(r)i Fo(and)e Fn(h)p Fo(;)494 1961 y Fn(W)539 1944 y Fl(+)533 1972 y(ev)o(en)600 1961 y Fo(\()p Fn(z)r Fo(\()p Fn(y)q Fo(\);)7 b Fn(z)r Fo(\()p Fm(\0001)p Fo(\))p Fn(;)g(h)p Fo(\))12 b(=)966 1910 y Fj(1)952 1922 y Fh(X)951 2010 y Fi(n)p Fl(=0)1013 1961 y Fo(\()p Fn(J)i Fm(\016)9 b Fn(I)1114 1967 y Fl(+)1142 1961 y Fo(\))1158 1944 y Fi(n)1180 1961 y Fo([1]\()p Fn(r)o(;)e(h)p Fo(\))p Fn(:)157 b Fo(\(33\))262 2087 y(W)m(e)9 b(put)g(for)h(short)g Fn(F)c Fo(\()p Fn(r)o(;)h(h)p Fo(\))j(=)742 2056 y Fh(P)786 2066 y Fj(1)786 2099 y Fi(n)p Fl(=0)850 2087 y Fo(\()p Fn(J)5 b Fm(\016)q Fn(I)934 2093 y Fl(+)961 2087 y Fo(\))977 2072 y Fi(n)1000 2087 y Fo([1]\()p Fn(r)o(;)i(h)p Fo(\).)14 b(Applying)9 b(the)h(op)q(erator) 262 2137 y Fn(J)j Fm(\016)c Fn(I)346 2143 y Fl(+)387 2137 y Fo(to)14 b(b)q(oth)g(sides,)g(w)o(e)g(get)455 2221 y Fn(F)6 b Fo(\()p Fn(r)o(;)h(h)p Fo(\))41 b(=)11 b(1)e(+)h(\()p Fn(J)j Fm(\016)c Fn(I)837 2227 y Fl(+)865 2221 y Fo(\)[)p Fn(F)d Fo(]\()p Fn(r)o(;)h(h)p Fo(\))622 2325 y(=)k(1)e(+)737 2269 y Fh(Z)785 2325 y Fn(\037)811 2332 y Fl([0)p Fi(;r)q Fl(])875 2325 y Fo(\()p Fn(r)910 2331 y Fl(1)928 2325 y Fo(\))p Fn(K)s Fo(\()p Fn(r)1017 2331 y Fl(1)1036 2325 y Fn(;)e(h)p Fo(\))p Fn(I)1113 2331 y Fl(+)1140 2325 y Fo([)p Fn(F)f Fo(]\()p Fn(r)1232 2331 y Fl(1)1249 2325 y Fn(;)h(h)p Fo(\))p Fn(dr)1349 2331 y Fl(1)890 2469 y Fo(14)p eop %%Page: 15 15 15 14 bop 262 307 a Fo(where)15 b Fn(\037)408 314 y Fl([0)p Fi(;r)q Fl(])486 307 y Fo(is)g(the)g(c)o(haracteristic)h(function)e(of) g(the)h(in)o(terv)n(al)e([0)p Fn(;)7 b(r)q Fo(],)13 b(and)h Fn(K)k Fo(is)262 357 y(the)c(k)o(ernel)g(de\014ned)h(in)f(\(26\).)324 407 y(No)o(w)f(w)o(e)g(sho)o(w)h(the)f(existence)i(of)e(a)g(limit)f (for)h Fn(F)6 b Fo(\()p Fn(r)o(;)h(h)p Fo(\))j(=)i Fn(F)6 b Fo(\()p Fn(r)q Fo(\()p Fn(y)q(;)h(h)p Fo(\))p Fn(;)g(h)p Fo(\))12 b(as)i Fn(h)262 457 y Fo(tends)i(to)e(0.)22 b(Recall)14 b(that)h Fn(r)h Fo(is)f(here)h(a)f(function)f(of)h Fn(h)g Fo(de\014ned)h(b)o(y)f(\(23\))f(with)h(a)262 506 y(\014xed)f Fn(y)q Fo(.)324 556 y(First,)20 b(it)f(follo)o(ws)f(from)h (the)h(lemmas)e(8)h(and)g(9)g(that,)i(for)e(some)g(p)q(ositiv)o(e)262 606 y(constan)o(t)c Fn(M)5 b Fo(,)14 b Fm(j)p Fn(K)s Fo(\()p Fn(r)586 612 y Fl(1)604 606 y Fn(;)7 b(h)p Fo(\))p Fm(j)12 b(\024)g Fn(M)5 b(r)795 612 y Fl(1)814 606 y Fn(=)p Fo(\(1)k(+)h Fn(r)942 612 y Fl(1)960 606 y Fo(\))976 591 y Fl(2)995 606 y Fo(.)20 b(W)m(e)14 b(also)g(ha)o(v)o(e)g Fm(j)p Fn(I)1309 612 y Fl(+)1337 606 y Fo([)p Fn(F)6 b Fo(]\()p Fn(r)1429 612 y Fl(1)1446 606 y Fn(;)h(h)p Fo(\))p Fm(j)k(\024)262 656 y Fn(M)307 641 y Fj(0)318 656 y Fn(=)p Fo(\(1)h(+)h Fn(cr)470 662 y Fl(1)488 656 y Fo(\))19 b(b)o(y)g(\(30\),)h(so)f(that)g(the)g(function)g(under)g (the)h(in)o(tegral)e(sign)h(is)262 706 y(dominated)f(b)o(y)g(an)h(in)o (tegrable)g(function)f(indep)q(enden)o(t)j(of)d Fn(h)p Fo(.)33 b(Moreo)o(v)o(er)19 b(for)262 756 y(eac)o(h)i(\014xed)g Fn(r)488 762 y Fl(1)506 756 y Fo(,)h(it)f(is)g(not)f(di\016cult)g(to)h (see)h(that)f Fn(\037)1132 763 y Fl([0)p Fi(;r)1183 767 y Fa(\006)1208 763 y Fl(])1219 756 y Fo(\()p Fn(r)1254 762 y Fl(1)1273 756 y Fo(\),)h Fn(K)s Fo(\()p Fn(r)1396 762 y Fl(1)1415 756 y Fn(;)7 b(h)p Fo(\))20 b(and)262 805 y Fn(I)280 811 y Fl(+)307 805 y Fo([)p Fn(F)6 b Fo(]\()p Fn(r)399 811 y Fl(1)416 805 y Fn(;)h(h)p Fo(\))20 b(ha)o(v)o(e)g(a)g (limit)e(as)i Fn(h)i Fm(!)g Fo(0,)f(and)f(the)g(existence)i(of)e(a)g (limit)e(for)262 855 y Fn(F)6 b Fo(\()p Fn(r)o(;)h(h)p Fo(\))13 b(as)j Fn(h)f Fo(tends)h(to)f(0)g(follo)o(ws)e(from)i(Leb)q (esgue's)h(dominated)f(con)o(v)o(ergence)262 905 y(theorem.)j(In)c 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Fl(0)309 1305 y Fo(\()p Fn(r)o(;)7 b(r)381 1311 y Fl(1)399 1305 y Fo(\))12 b(=)f Fn(\020)s Fo(\()p Fn(r)o(;)c(r)563 1311 y Fl(1)581 1305 y Fn(;)g Fo(0\))k(=)692 1220 y Fh(8)692 1258 y(<)692 1332 y(:)750 1261 y Fo(2)778 1228 y Fh(R)805 1238 y Fi(r)797 1276 y(r)812 1280 y Fe(1)830 1261 y Fm(f\000)p Fn(q)902 1267 y Fl(0)921 1261 y Fn(r)941 1246 y Fl(2)940 1272 y(2)968 1261 y Fo(+)f(\()p Fn(`)g Fo(+)f(1)p Fn(=)p Fo(2\))1173 1246 y Fl(2)1191 1261 y Fm(g)1212 1246 y Fl(1)p Fi(=)p Fl(2)1264 1261 y Fn(dr)1305 1267 y Fl(2)1323 1261 y Fn(=r)1363 1267 y Fl(2)1395 1261 y Fo(\(I\))1442 1274 y Fj(\000)1470 1261 y Fn(;)750 1347 y Fo(4)778 1313 y Fh(R)805 1323 y Fi(r)797 1361 y(r)812 1365 y Fe(1)830 1347 y Fm(f\000)p Fn(q)902 1353 y Fj(\000)p Fl(1)947 1347 y Fn(r)967 1331 y Fl(2)966 1357 y(2)994 1347 y Fo(+)h(\()p Fn(`)g Fo(+)f(1)p Fn(=)p Fo(2\))1199 1331 y Fl(2)1217 1347 y Fm(g)1238 1331 y Fl(1)p Fi(=)p Fl(2)1290 1347 y Fn(dr)1331 1353 y Fl(2)1349 1347 y Fn(=r)1389 1353 y Fl(2)1421 1347 y Fo(\(I)q(I\))1484 1359 y Fj(\000)1512 1347 y Fn(;)262 1444 y Fo(and)403 1600 y Fn(K)438 1606 y Fl(0)457 1600 y Fo(\()p Fn(r)492 1606 y Fl(1)511 1600 y Fo(\))i(=)h Fn(K)s Fo(\()p Fn(r)655 1606 y Fl(1)674 1600 y Fn(;)7 b Fo(0\))k(=)785 1490 y Fh(8)785 1527 y(>)785 1540 y(>)785 1552 y(<)785 1627 y(>)785 1639 y(>)785 1652 y(:)847 1518 y Fo(1)p 847 1536 21 2 v 847 1574 a(2)1011 1518 y Fn(q)1030 1524 y Fl(0)1048 1518 y Fn(r)1067 1524 y Fl(1)p 878 1536 340 2 v 878 1574 a Fm(\000)p Fn(q)929 1580 y Fl(0)947 1574 y Fn(r)967 1560 y Fl(2)966 1585 y(1)995 1574 y Fo(+)f(\()p Fn(`)f Fo(+)h(1)p Fn(=)p Fo(2\))1200 1562 y Fl(2)1264 1546 y Fo(\(I\))1312 1558 y Fj(\000)1340 1546 y Fn(;)847 1621 y Fo(1)p 847 1639 21 2 v 847 1677 a(2)1011 1621 y Fn(q)1030 1627 y Fj(\000)p Fl(1)1074 1621 y Fn(r)1093 1627 y Fl(1)p 878 1639 366 2 v 878 1677 a Fm(\000)p Fn(q)929 1683 y Fj(\000)p Fl(1)973 1677 y Fn(r)993 1663 y Fl(2)992 1688 y(1)1021 1677 y Fo(+)g(\()p Fn(`)f Fo(+)h(1)p Fn(=)p Fo(2\))1226 1665 y Fl(2)1290 1649 y Fo(\(I)q(I\))1354 1661 y Fj(\000)1382 1649 y Fn(:)262 1764 y Fo(W)m(e)j(then)i(see)g(that)f Fn(A)d Fo(=)h Fn(A)702 1770 y Fl(+)741 1764 y Fo(=)g Fn(A)816 1770 y Fj(\000)856 1764 y Fn(>)g Fo(1.)17 b(This)d(completes)h(the)f(pro)q(of.)115 b Fb(2)262 1926 y Fp(4)66 b(Asymptotics)22 b(of)g(the)g(actions)262 2017 y Fo(W)m(e)15 b(pro)o(v)o(e)g(no)o(w)h(Corollary)e(4.)23 b(W)m(e)15 b(w)o(an)o(t)g(to)g(compute)h(the)h(T)m(a)o(ylor)d (expansion)262 2067 y(with)f(resp)q(ect)j(to)e Fn(h)g Fo(of)f(the)i Fn(h)p Fo(-dep)q(enden)o(t)g(action)e(in)o(tegral)h(in)f (\(4\))720 2181 y(\010\()p Fn(h)p Fo(\))f(=)862 2125 y Fh(Z)903 2135 y Fi(x)922 2139 y Fe(1)888 2219 y Fl(~)-20 b Fi(!)947 2127 y Fh(q)p 989 2127 66 2 v 54 x Fm(\000)1031 2171 y Fo(~)1021 2181 y Fn(Qdt;)262 2294 y Fo(with)16 b Fn(x)383 2300 y Fl(1)419 2294 y Fm(2)h Fn(I)s Fo(.)28 b(As)18 b(one)f(w)o(ould)g(exp)q(ect,)i(the)f(\014rst)g(term)g(is)f (easily)f(seen)j(to)e(b)q(e)262 2343 y(the)e Fn(h)p Fo(-indep)q(enden)o 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0 currenttransfer exec .5 ge def /level2 /languagelevel where {pop languagelevel 1 ne}{false}ifelse def /colorexists level2 { statusdict/processcolors known {statusdict/processcolors get exec}{1}ifelse 4 eq def } {systemdict/setcmykcolor known def} ifelse /tc1 [0 0 0 1] def /tc2 [0 0 0 1] def /fc [0 0 0 1] def /sc [0 0 0 1] def /concatprocs{ /packedarray where {pop dup type /packedarraytype eq 2 index type /packedarraytype eq or}{false}ifelse { /proc2 exch cvlit def/proc1 exch cvlit def proc1 aload pop proc2 aload pop proc1 length proc2 length add packedarray cvx } { /proc2 exch cvlit def/proc1 exch cvlit def /newproc proc1 length proc2 length add array def newproc 0 proc1 putinterval newproc proc1 length proc2 putinterval newproc cvx }ifelse }bdf /storerect{/top xdf/right xdf/bottom xdf/left xdf}bdf /rectpath{newpath left bottom m left top L right top L right bottom L closepath}bdf /i{dup 0 eq {pop df dup} {dup} ifelse /clipflatness xdf setflat }bdf version cvr 38.0 le {/setrgbcolor{ currenttransfer exec 3 1 roll currenttransfer exec 3 1 roll currenttransfer exec 3 1 roll setrgbcolor}bdf}if /gettint{0 get}bdf /puttint{0 exch put}bdf /vms {/vmsv save def} bdf /vmr {vmsv restore} bdf /vmrs{vmsv restore /vmsv save def}bdf /eomode{ {/filler /eofill load def /clipper /eoclip load def} {/filler /fill load def /clipper /clip load def} ifelse }bdf /CD{/NF exch def{exch dup/FID ne 1 index/UniqueID ne and{exch NF 3 1 roll put} {pop pop}ifelse}forall NF}bdf /MN{1 index length/Len exch def dup length Len add string dup Len 4 -1 roll putinterval dup 0 4 -1 roll putinterval}bdf /RC{4 -1 roll /ourvec xdf 256 string cvs(|______)anchorsearch {1 index MN cvn/NewN exch def cvn findfont dup maxlength dict CD dup/FontName NewN put dup /Encoding ourvec put NewN exch definefont pop}{pop}ifelse}bdf /RF{dup FontDirectory exch known{pop 3 -1 roll pop}{RC}ifelse}bdf /FF{dup 256 string cvs(|______)exch MN cvn dup FontDirectory exch known {exch pop findfont 3 -1 roll pop}{pop dup findfont dup maxlength dict CD dup dup /Encoding exch /Encoding get 256 array copy 7 -1 roll {3 -1 roll dup 4 -2 roll put}forall put definefont}ifelse}bdf userdict begin /BDFontDict 20 dict def end BDFontDict begin /bu{}def /bn{}def /setTxMode{av 70 ge{pop}if pop}def /gm{m}def /show{pop}def /gr{pop}def /fnt{pop pop pop}def /fs{pop}def /fz{pop}def /lin{pop pop}def end /MacVec 256 array def MacVec 0 /Helvetica findfont /Encoding get 0 128 getinterval putinterval MacVec 127 /DEL put MacVec 16#27 /quotesingle put MacVec 16#60 /grave put /NUL/SOH/STX/ETX/EOT/ENQ/ACK/BEL/BS/HT/LF/VT/FF/CR/SO/SI /DLE/DC1/DC2/DC3/DC4/NAK/SYN/ETB/CAN/EM/SUB/ESC/FS/GS/RS/US MacVec 0 32 getinterval astore pop /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis /dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/notequal/AE/Oslash /infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation /product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash /questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft /guillemotright/ellipsis/nbspace/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge /ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright/fi/fl /daggerdbl/periodcentered/quotesinglbase/quotedblbase /perthousand/Acircumflex/Ecircumflex/Aacute /Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave/Oacute/Ocircumflex /apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde /macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron MacVec 128 128 getinterval astore pop /fps{ currentflat exch dup 0 le{pop 1}if { dup setflat 3 index stopped {1.3 mul dup 3 index gt{pop setflat pop pop stop}if} {exit} ifelse }loop pop setflat pop pop }bdf /fp{100 currentflat fps}bdf /clipper{clip}bdf /W{/clipper load 100 clipflatness fps}bdf end%. FreeHandDict end%. FHIODict %%EndProcSet %%EndProlog %%BeginSetup FHIODict begin FreeHandDict begin 58.0872 575.2227 535.7064 813.7564 storerect rectpath clip newpath /onlyk{false}ndf /ccmyk{dup 5 -1 roll sub 0 max exch}ndf /setcmykcolor{1 exch sub ccmyk ccmyk ccmyk pop setrgbcolor}ndf /setcmykcoloroverprint{4{dup -1 eq{pop 0}if 4 1 roll}repeat setcmykcolor}ndf /findcmykcustomcolor{5 /packedarray where{pop packedarray}{array astore readonly}ifelse}ndf /setcustomcolor{exch aload pop pop 4{4 index mul 4 1 roll}repeat setcmykcolor pop}ndf /setseparationgray{1 exch sub dup dup dup setcmykcolor}ndf /setoverprint{pop}ndf /currentoverprint false ndf /colorimage{pop pop [5 -1 roll/exec cvx 6 -1 roll/exec cvx 7 -1 roll/exec cvx 8 -1 roll/exec cvx /cmykbufs2gray cvx]cvx image} %. version 47.1 on Linotronic of Postscript defines colorimage incorrectly (rgb model only) version cvr 47.1 le isLino and{userdict begin bdf end}{ndf}ifelse /customcolorimage{ colorexists { aload pop pop 4 array astore setimagecmyk currentcolortransfer {ik mul ik sub 1 add}concatprocs 4 1 roll {iy mul iy sub 1 add}concatprocs 4 1 roll {im mul im sub 1 add}concatprocs 4 1 roll {ic mul ic sub 1 add}concatprocs 4 1 roll setcolortransfer /magentabuf 0 string def /yellowbuf 0 string def /blackbuf 0 string def {invbuf dup length magentabuf length ne {dup length dup dup /magentabuf exch string def /yellowbuf exch string def /blackbuf exch string def}if dup magentabuf copy yellowbuf copy blackbuf copy pop}concatprocs {magentabuf}{yellowbuf}{blackbuf}true 4 colorimage } { pop image } ifelse }ndf /separationimage{image}ndf /newcmykcustomcolor{6 /packedarray where{pop packedarray}{array astore readonly}ifelse}ndf /inkoverprint false ndf /setinkoverprint{pop}ndf /overprintprocess{pop}ndf /setspotcolor {spots exch get 0 5 getinterval exch setcustomcolor}ndf /currentcolortransfer{currenttransfer dup dup dup}ndf /setcolortransfer{systemdict begin settransfer end pop pop pop}ndf /getcmyk { dup length 4 eq {aload pop} {aload pop spots exch get 0 4 getinterval aload pop 4 {4 index mul 4 1 roll}repeat 5 -1 roll pop} ifelse }bdf /setimagecmyk{ getcmyk/ik xdf /iy xdf /im xdf /ic xdf }ndf /autospread{pop}ndf /fhsetspreadsize{pop}ndf /fhsetspreadallow{pop}ndf /strokeopf false def /fillopf false def /R{0 ne /strokeopf xdf}bdf /O{0 ne /fillopf xdf}bdf /filler{fill}bdf /F{fc fhsetcolor fillopf setoverprint false autospread gsave /filler load fp grestore false setoverprint}bdf /f{closepath F}bdf /S{sc fhsetcolor strokeopf setoverprint true autospread {stroke}fp false setoverprint}bdf /s{closepath S}bdf /B{fc fhsetcolor fillopf setoverprint gsave /filler load fp grestore sc fhsetcolor strokeopf setoverprint true autospread {stroke}fp false setoverprint}bdf /b{closepath B}bdf /setcolorscreen where{pop}{/setcolorscreen{setscreen pop pop pop pop pop pop pop pop pop}bdf}ifelse /fhsetcolor{dup length 4 eq {aload overprintprocess setcmykcolor} {aload 1 get spots exch get 5 get setinkoverprint setspotcolor} ifelse }ndf /settextcolor{ dup fhsetcolor textblackopf { dup length 4 eq {onlyk{3 get 1.0 eq{true setinkoverprint}if}{pop}ifelse} {pop} ifelse } {pop}ifelse }ndf /ka{/fc xdf}bdf /Ka{/sc xdf}bdf /xa{/fc xdf} bdf /Xa{/sc xdf} bdf /bc2[0 0]def /bc4[0 0 0 0]def /absmax{2 copy abs exch abs gt{exch}if pop}bdf /calcstep { colorexists not and{calcgraysteps}{maxsteps}ifelse tc1 length 4 eq { 0 1 3 {tc1 1 index get tc2 3 -1 roll get sub }for absmax absmax absmax } { bc2 tc1 1 get 1 exch put tc1 gettint tc2 gettint sub abs } ifelse mul abs round dup 0 eq{pop 1}if dup /numsteps xdf 1 sub dup 0 eq{pop 1}if /numsteps1 xdf }bdf /cblend{ tc1 length 4 eq { 0 1 3 {bc4 exch tc1 1 index get tc2 2 index get 1 index sub frac mul add put }for bc4 } { bc2 tc1 gettint tc2 gettint 1 index sub frac mul add puttint bc2 } ifelse fhsetcolor }bdf /logtaper{/frac frac 9 mul 1 add log def}bdf FHIODict begin /origmtx matrix currentmatrix def /iminv false def /invbuf{0 1 2 index length 1 sub{dup 2 index exch get 255 exch sub 2 index 3 1 roll put}for}bdf /cmykbufs2gray{ dup length 0 1 3 -1 roll 1 sub {4 index 1 index get 4 index 2 index get 4 index 3 index get 4 index 4 index get 255 exch sub ccmyk ccmyk ccmyk pop 5 mul exch 45 mul add exch 14 mul add -6 bitshift 2 index 3 1 roll put}for 4 1 roll pop pop pop }bdf end /textopf false def /textblackopf true def /curtextmtx{}def /otw .25 def /msf{dup/curtextmtx xdf makefont setfont}bdf /makesetfont/msf load def /curtextheight{.707104 .707104 curtextmtx dtransform dup mul exch dup mul add sqrt}bdf /ta{1 index {tempstr 0 2 index put tempstr 2 index gsave exec grestore tempstr stringwidth rmoveto 5 index eq{6 index 6 index rmoveto}if 3 index 3 index rmoveto }forall 7{pop}repeat}bdf /sts{settextcolor textopf setoverprint/ts{awidthshow}def exec false setoverprint}bdf /stol{/xt currentlinewidth def setlinewidth settextcolor textopf setoverprint newpath /ts{{false charpath stroke}ta}def exec false setoverprint xt setlinewidth}bdf /currentpacking where{pop false setpacking}if /spots[1 0 0 0 (Process Cyan) false newcmykcustomcolor 0 1 0 0 (Process Magenta) false newcmykcustomcolor 0 0 1 0 (Process Yellow) false newcmykcustomcolor 0 0 0 1 (Process Black) false newcmykcustomcolor 0 0 0 0 (Blanc) false newcmykcustomcolor ]def %%EndSetup %%IncludeFont: Symbol 0 dict dup begin end /f0 /Symbol FF def [] 0 d 3.863708 M 1 w 0 j 0 J 0 O 0 R 0 i false eomode [0 0 0 1] Ka [0 0 0 1] ka vms u true fhsetspreadallow 58.5872 680.1297 m 198 680 L S 253.54 795.1083 m 254 636 L S 226.7936 680.4797 m 366.2064 680.35 L S 422.54 795.1083 m 423 634 L S 395.7936 680.4797 m 535.2064 680.35 L S vmrs %%IncludeFont: Times-Roman MacVec 256 array copy /f1 /|______Times-Roman dup RF findfont def { f1 [24 0 0 24 0 0] makesetfont 105 589 m 0 0 32 0 0 (A) ts } [0 0 0 1] sts vmrs %%IncludeFont: Times-Roman MacVec 256 array copy /f1 /|______Times-Roman dup RF findfont def { f1 [24 0 0 24 0 0] makesetfont 286 590 m 0 0 32 0 0 (B) ts } [0 0 0 1] sts vmrs %%IncludeFont: Times-Roman 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%%Trailer end % FreeHandDict end % FHIODict %%EndDocument @endspecial 486 2157 a Fg(Figur)n(e)h(5:)k(The)c(e\013e)n(ctive)g(and) g(physic)n(al)g(p)n(otentials.)324 2245 y Fo(Our)e(aim)e(is)i(to)f (calculate)h(the)g(scattering)h(matrix)d(near)i(the)h(critical)e (energy)262 2295 y(lev)o(el)j Fn(E)i Fo(=)e Fn(V)477 2301 y Fi(m)525 2295 y Fo(and)h(to)g(get)g(quan)o(tization)f(rules)h (for)g(the)g(resonances)i(close)f(to)262 2344 y Fn(V)286 2350 y Fi(m)317 2344 y Fo(.)g(Let)12 b(us)g(\014rst)g(recall)f(what)g (the)h(scattering)g(matrix)e(and)i(resonances)h(for)e(this)890 2469 y(18)p eop %%Page: 19 19 19 18 bop 262 307 a Fo(op)q(erator)15 b(are.)22 b(Under)15 b(our)g(short-range)h(assumption)f(on)f Fn(V)c Fo(,)k(the)i(solutions)e (of)262 357 y(the)g(radial)f(Sc)o(hr\177)-21 b(odinger)14 b(equation)g(b)q(eha)o(v)o(e)g(as)g Fn(x)d Fm(!)g Fo(+)p Fm(1)j Fo(lik)o(e)570 453 y Fn(s)589 459 y Fl(+)617 453 y Fo(\()p Fn(E)r(;)7 b(h)p Fo(\))p Fn(e)744 436 y Fi(i)756 411 y Fj(p)p 783 411 26 2 v 25 x 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Fn(;)515 998 y(D)549 1004 y Fl(3)568 998 y Fo(\()p Fn(r)o(;)g(h)p Fo(\))k(=)h Fn(D)q Fo(\()p Fn(r)q(h)p Fo(\))p Fn(;)262 1080 y Fo(where)j(w)o(e)f(de\014ne)h(for)e(a)h(real)g (in)o(terv)n(al)f Fn(I)s Fo(,)574 1154 y(^)564 1164 y Fn(D)q Fo(\()p Fn(R;)7 b(I)s Fo(\))12 b(=)g Fm(f)p Fn(E)h Fm(2)873 1154 y Fo(^)863 1164 y Fn(D)r Fo(\()p Fn(R)p Fo(\);)7 b(arg\006)1070 1170 y Fl(1)1088 1164 y Fo(\()p Fn(E)r Fo(\))12 b Fm(2)g Fn(I)s Fm(g)p Fn(:)262 1248 y Fo(Another)i(function)e Fn(N)5 b Fo(\()p Fn(z)r Fo(\))14 b(pla)o(ys)f(an)g(imp)q(ortan)o(t)f(role)h(when)h(the)g(energy)g Fn(E)h Fo(is)e(in)262 1298 y(the)h(domain)f Fn(D)515 1304 y Fl(3)534 1298 y Fo(.)18 b(It)c(is)f(de\014ned)i(on)f(the)g(univ) o(ersal)g(co)o(v)o(ering)g(of)f(C)p Fm(nf)p Fo(0)p Fm(g)g Fo(b)o(y)649 1413 y Fn(N)5 b Fo(\()p Fn(z)r Fo(\))12 b(=)857 1350 y Fm(p)p 892 1350 46 2 v 35 x Fo(2)p Fn(\031)p 801 1403 193 2 v 801 1441 a Fo(\000\(1)p Fn(=)p Fo(2)d(+)g Fn(z)r Fo(\))998 1413 y Fn(e)1017 1396 y Fi(z)e Fl(log\()p Fi(z)q(=e)p Fl(\))1161 1413 y Fn(;)314 b Fo(\(37\))262 1519 y(where)15 b(log)6 b Fn(z)16 b Fo(is)e(real)g(when)g(arg)p Fn(z)g Fo(=)e(0.)18 b(With)13 b(these)i(notations)f(w)o(e)g(ha)o(v)o(e) 262 1603 y Fk(Theorem)g(13)21 b Fc(Scattering)14 b(matrix)262 1653 y Fg(L)n(et)f Fn(R)g Fg(b)n(e)h(a)f(su\016ciently)g(smal)r(l)g(p)n (ositive)h(numb)n(er.)k(Then)c(ther)n(e)f(exist)g(functions)262 1703 y Fn(\017)279 1709 y Fi(j)310 1703 y Fo(=)h Fn(\017)373 1709 y Fi(j)390 1703 y Fo(\()p Fn(E)r(;)7 b(h)p Fo(\))16 b(\(1)d Fm(\024)h Fn(j)i Fm(\024)e Fo(12\))i Fg(analytic)g(with)g(r)n (esp)n(e)n(ct)f(to)h Fn(E)i Fg(with)d(p)n(ar)n(ameter)h Fn(h)262 1753 y Fg(satisfying)607 1802 y Fn(\017)624 1808 y Fi(j)641 1802 y Fo(\()p Fn(E)r(;)7 b(h)p Fo(\))k(=)h(1)d(+)g Fn(o)p Fo(\(1\))43 b(\(1)11 b Fm(\024)h Fn(j)i Fm(\024)e Fo(12\))262 1873 y Fg(such)j(that)g(the)g(fol)r(lowing)f(formulas)g (hold:)324 1923 y(\(i\))g(Ther)n(e)g(exists)h Fn(r)637 1929 y Fl(0)667 1923 y Fn(>)c Fo(0)k Fg(such)g(that)g(uniformly)f(in)h (any)g(c)n(omp)n(act)g(subset)g(of)262 1972 y Fn(D)296 1978 y Fl(1)315 1972 y Fo(\()p Fn(R;)7 b(r)401 1978 y Fl(0)418 1972 y Fn(;)g(h)p Fo(\))15 b Fg(we)f(have)583 2091 y Fk(S)610 2097 y Fi(`)626 2091 y Fo(\()p Fn(E)r(;)7 b(h)p Fo(\))k(=)h Fn(ie)822 2074 y Fj(\000)p Fl(2)p Fi(i)p Fl(\006)901 2078 y Fd(r)917 2074 y Fi(=h)961 2063 y Fn(e)980 2048 y Fl(2)p Fi(i)1013 2041 y Fl(~)1009 2048 y(\006)1033 2052 y Fe(01)1062 2048 y Fi(=h)1101 2063 y Fn(\017)1118 2069 y Fl(1)1145 2063 y Fo(+)e Fn(\017)1204 2069 y Fl(2)p 961 2082 262 2 v 961 2126 a Fn(e)980 2114 y Fl(2)p Fi(i)1013 2107 y Fl(~)1009 2114 y(\006)1033 2118 y Fe(01)1062 2114 y Fi(=h)1101 2126 y Fn(\017)1118 2132 y Fl(3)1145 2126 y Fo(+)g Fn(\017)1204 2132 y Fl(4)1227 2091 y Fn(;)248 b Fo(\(38\))262 2202 y Fg(and)15 b(uniformly)g(in)f(any)i(c)n(omp)n (act)f(subset)g(of)g Fn(D)1028 2208 y Fl(2)1047 2202 y Fo(\()p Fn(R;)7 b(r)1133 2208 y Fl(0)1151 2202 y Fn(;)g(h)p Fo(\))14 b Fg(we)h(have)513 2320 y Fk(S)540 2326 y Fi(`)556 2320 y Fo(\()p 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Fo(\()p Fn(e)874 390 y Fj(\000)p Fi(i\031)q(=)p Fl(2)969 405 y Fo(\006)999 411 y Fl(1)1017 405 y Fn(=\031)q(h)p Fo(\))p Fn(e)1122 390 y Fl(2)p Fi(i)1155 383 y Fl(~)1151 390 y(\006)1175 394 y Fe(01)1205 390 y Fi(=h)1243 405 y Fn(\017)1260 411 y Fl(9)1288 405 y Fo(+)10 b Fn(\017)1347 411 y Fl(10)p 801 424 581 2 v 806 468 a Fn(e)825 456 y Fl(2)p Fi(i)858 449 y Fl(~)854 456 y(\006)878 460 y Fe(01)908 456 y Fi(=h)946 468 y Fn(\017)963 474 y Fl(11)1008 468 y Fo(+)f Fn(N)c Fo(\()p Fn(e)1122 456 y Fi(i\031)q(=)p Fl(2)1190 468 y Fo(\006)1220 474 y Fl(1)1239 468 y Fn(=\031)q(h)p Fo(\))p Fn(\017)1342 474 y Fl(12)1387 433 y Fn(:)88 b Fo(\(40\))324 616 y(Recalling)13 b(that)i(resonances)i(are)e(the)h(p)q(oles)f(of)f (the)i(scattering)g(matrix,)d(w)o(e)262 666 y(also)i(get)i(a)f(quan)o (tization)f(condition)h(for)g(the)g(resonances)j(in)c(eac)o(h)i(domain) e(of)262 716 y(the)f(energy)h(plane:)262 807 y Fk(Theorem)f(14)21 b Fc(Quan)o(tization)13 b(conditions)262 857 y Fg(Supp)n(ose)j(that)e Fn(E)j Fg(is)e(a)g(r)n(esonanc)n(e)g(in)g Fn(D)q Fo(\()p Fn(R)p Fo(\))p Fg(.)311 940 y(1.)20 b(If)13 b Fn(E)h Fm(2)d Fn(D)525 946 y Fl(1)544 940 y Fo(\()p Fn(R;)c(r)630 946 y Fl(0)648 940 y Fn(;)g(h)p Fo(\))f Fm([)g Fn(D)781 946 y Fl(2)798 940 y Fo(\()p Fn(R;)h(r)884 946 y Fl(0)902 940 y Fn(;)g(h)p Fo(\))p Fg(,)12 b(then)i(ther)n(e)f(is)g(an)g Fn(n)f Fm(2)f Ff(N)i Fg(such)h(that)599 1051 y Fo(\006)629 1057 y Fl(01)664 1051 y Fo(\()p Fn(E)r Fo(\))e(=)g Fm(f)p Fn(n)d Fo(+)887 1022 y(1)p 887 1041 21 2 v 887 1079 a(2)922 1051 y(+)g Fn(\025)p Fo(\()p Fn(`)h Fo(+)1076 1022 y(1)p 1076 1041 V 1076 1079 a(2)1102 1051 y(\))p Fm(g)p Fn(\031)q(h)f Fo(+)h Fn(o)p Fo(\()p Fn(h)p Fo(\))p Fn(:)311 1172 y Fg(2.)20 b(If)15 b Fn(E)e Fm(2)f Fn(D)527 1178 y Fl(3)546 1172 y Fo(\()p Fn(r)o(;)7 b(h)p Fo(\))p Fg(,)13 b(then)j(ther)n(e)e(is) h(an)g Fn(n)d Fm(2)f Ff(N)k Fg(such)g(that)412 1288 y Fo(\006)442 1294 y Fl(01)477 1288 y Fo(\()p Fn(E)r Fo(\))10 b(+)f Fn(i)612 1260 y(h)p 612 1278 24 2 v 614 1316 a Fo(2)648 1288 y(log)e Fn(N)e Fo(\()p Fn(i)782 1260 y Fo(\006)812 1266 y Fl(1)831 1260 y Fo(\()p Fn(E)r Fo(\))p 782 1278 114 2 v 815 1316 a Fn(\031)q(h)901 1288 y Fo(\))12 b(=)g Fm(f)p Fn(n)c Fo(+)1074 1260 y(1)p 1074 1278 21 2 v 1074 1316 a(2)1109 1288 y(+)h Fn(\025)p Fo(\()p Fn(`)h Fo(+)1264 1260 y(1)p 1264 1278 V 1264 1316 a(2)1289 1288 y(\))p Fm(g)p Fn(\031)q(h)g Fo(+)f Fn(o)p Fo(\()p Fn(h)p Fo(\))p Fn(:)262 1483 y Fp(References)262 1566 y Fo([Be-Mo])20 b(M.V.Berry)m(,)15 b(K.E.Moun)o(t:)20 b Fc(Semiclassical)15 b(appro)o(ximations)e(in)i(w)o(a)o(v)o(e)347 1607 y(mec)o(hanics)p Fo(,)f(Rep.)f(Prog.)h(Ph)o(ys.,)f Fk(35)p Fo(,)g(315{397)f(\(1972\).) 262 1682 y([Br])20 b(D.M.Brink:)g Fc(Semi-classical)15 b(metho)q(ds)h(in)f(n)o(ucleus-n)o(ucleus)i(scattering)f Fo(,)347 1723 y(Cam)o(bridge)d(Monographs)h(on)f(Mathematical)g(Ph)o (ysics)i(\(1985\).)262 1798 y([Ec])20 b(J.Ecalle:)31 b Fc(Les)21 b(fonctions)f(r)o(\023)-20 b(esurgen)o(tes)p Fo(,)24 b(Publications)19 b(Math)o(\023)-20 b(ematiques)347 1840 y(d'Orsa)o(y)m(,)13 b Fk(81-05)g Fo(\(1981\).)262 1914 y([F)m(r-F)m(r])19 b(N.F)m(r\177)-21 b(oman,)13 b(P)m(.O.F)m(r\177)-21 b(oman:)16 b Fc(Phase-In)o(tegral)f(Metho)q(d)g (Allo)o(wing)d(Near-)347 1956 y(lying)h(T)m(ransition)h(P)o(oin)o(ts)p Fo(,)g(Springer)h(T)m(racts)g(in)g(Natural)f(Philosoph)o(y)m(,)f Fk(40)347 1997 y Fo(\(1996\).)262 2072 y([F)m(u-Ra)f(1])20 b(S.F)m(ujii)o(\023)-20 b(e,)14 b(T.Ramond:)21 b Fc(Matrice)c(de)f (scattering)h(et)g(r)o(\023)-20 b(esonances)18 b(as-)347 2114 y(so)q(ci)o(\023)-20 b(es)13 b(\022)-21 b(a)12 b(une)g(orbite)g(h) o(\023)-20 b(et)o(\023)g(ero)q(cline)p Fo(,)13 b(Annales)g(de)f (l'I.H.P)m(.)e(section)i(Ph)o(ysique)347 2155 y(Th)o(\023)-20 b(eorique,)14 b Fk(69-1)p Fo(,)f(31-82)f(\(1998\).)262 2230 y([F)m(u-Ra)g(2])20 b(S.F)m(ujii)o(\023)-20 b(e,)17 b(T.Ramond:)25 b Fc(Semiclassical)18 b(resonances)i(for)e(the)h(radial) 347 2271 y(Sc)o(hr\177)-21 b(odinger)14 b(equation)g(at)f(\014xed)i (angular)e(momen)o(tum)p Fo(,)f(in)i(preparation.)890 2469 y(21)p eop %%Page: 22 22 22 21 bop 262 307 a Fo([Ge-Gr])19 b(C.G)o(\023)-20 b(erard,)17 b(A.Grigis:)22 b Fc(Precise)c(estimates)g(of)e(tunneling)g(and)h (eigen-)347 349 y(v)n(alues)g(near)h(a)g(p)q(oten)o(tial)f(barrier)p Fo(,)h(J.)g(of)f(Di\013.)f(Equations)i Fk(42)p Fo(,)f(149{177)347 390 y(\(1988\).)262 465 y([La)c(1])20 b(R.E.Langer:)28 b Fc(On)20 b(the)g(asymptotic)g(solutions)f(of)g(ordinary)g (di\013eren-)347 506 y(tial)d(equations,)h(with)g(reference)i(to)e(the) h(Stok)o(es')f(phenomenon)g(ab)q(out)g(a)347 548 y(singular)c(p)q(oin)o (t)p Fo(,)g(T)m(rans.)g(Amer.)h(Math.)f(So)q(c.)h Fk(37)p Fo(,)f(397{416)f(\(1935\).)262 623 y([La)h(2])20 b(R.E.Langer:)c Fc(On)d(the)g(connection)h(form)o(ulas)d(and)i(the)g(solutions)f(of)g (the)347 664 y(w)o(a)o(v)o(e)h(equation)p Fo(,)g(Ph)o(ysical)h(Review)g Fk(51)p Fo(,)f(669{676)f(\(1937\).)262 739 y([Ra])19 b(T.Ramond:)d Fc(Semiclassical)d(study)h(of)f(quan)o(tum)g(scattering)i (on)e(the)i(line)p Fo(,)347 780 y(Comm.)d(Math.)i(Ph)o(ys.)f Fk(177)p Fo(,)g(221{254)f(\(1996\).)262 855 y([Ol])19 b(F.W.J.Olv)o(er:)i Fc(Asymptotics)16 b(and)f(sp)q(ecial)h(functions)p Fo(,)g(Academic)f(Press,)347 897 y(\(1974\).)262 971 y([V)m(o])k(A.V)m(oros:)c Fc(The)c(Return)g(of)f(the)h(Quartic)g (Oscillator.)e(The)i(Complex)e(WKB)347 1013 y(Metho)q(d)p Fo(,)17 b(Ann.)f(Inst.)g(H.)g(P)o(oincar)o(\023)-20 b(e)17 b(Ph)o(ys.)f(Th)o(\023)-20 b(eor.,)17 b Fk(39)h(\(3\))d Fo(\(1983\),)h(pp.)347 1054 y(211-338.)890 2469 y(22)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9909290559870--