%!PS-Adobe-2.0 %%Creator: dvips 5.76 Copyright 1997 Radical Eye Software (www.radicaleye.com) %%Title: v2.dvi %%CreationDate: Mon Oct 29 16:16:18 2001 %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CaslonOpenfaceBT-Regular cmr10 cmcsc10 cmbx10 cmmi10 %%+ cmsy10 cmmi9 cmr9 GoudyHandtooledBT-Regular cmsy9 cmex10 cmti10 %%+ cmsl10 cmmi7 cmtt10 cmsy7 cmr7 %%EndComments %DVIPSCommandLine: dvips -h /data/home/luc/notes/CMFONTS -o bessel.ps v2 %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2001.10.29:1616 %%BeginProcSet: /data/home/luc/notes/CMFONTS %!PS-AdobeFont-1.1: cmr10 1.0-prerelease %%VMusage: 36895 36984 12 dict begin /FontInfo 11 dict dup begin /UnderlineThickness 49.95 def /Weight(Medium) def /ItalicAngle 0 def /isFixedPitch false def /Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmr10) def /version(1.0-prerelease) def /UnderlinePosition -125.0 def /FamilyName(cmr10) def end readonly def /UniqueID 4100015 def /PaintType 0 def /FontBBox{-42 -250 1009 749} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmr10 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 /polishlcross put dup 33 /exclam put dup 34 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/UnderlineThickness 49.95 def /Weight(Medium) def /ItalicAngle 0 def /isFixedPitch false def /Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmmi10) def /version(1.0-prerelease) def /UnderlinePosition -125.0 def /FamilyName(cmmi10) def end readonly def /UniqueID 4100070 def /PaintType 0 def /FontBBox{-34 -250 1047 749} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmmi10 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 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/ItalicAngle 0 def /isFixedPitch false def /Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 6 def /FullName(cmsy10) def /version(1.0-prerelease) def /UnderlinePosition -480.0 def /FamilyName(cmsy10) def end readonly def /UniqueID 4100063 def /PaintType 0 def /FontBBox{-28 -960 1122 778} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmsy10 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 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 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Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmti10) def /version(1.0-prerelease) def /UnderlinePosition -125.0 def /FamilyName(cmti10) def end readonly def /UniqueID 4100040 def /PaintType 0 def /FontBBox{-35 -250 1124 749} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmti10 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 /polishlcross put dup 33 /exclam put dup 34 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/isFixedPitch false def /Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. 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Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. 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Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmsl9) def /version(1.0-prerelease) def /UnderlinePosition -125.5 def /FamilyName(cmsl9) def end readonly def /UniqueID 4100025 def /PaintType 0 def /FontBBox{-62 -251 1150 750} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmsl9 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 /polishlcross put dup 33 /exclam put dup 34 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %!PS-AdobeFont-1.1: cmsl8 1.0-prerelease %%VMusage: 39079 39168 12 dict begin /FontInfo 11 dict dup begin /UnderlineThickness 50.0 def /Weight(Medium) def /ItalicAngle 0 def /isFixedPitch false def /Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmsl8) def /version(1.0-prerelease) def /UnderlinePosition -125.5 def /FamilyName(cmsl8) def end readonly def /UniqueID 4100024 def /PaintType 0 def /FontBBox{-62 -251 1183 749} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmsl8 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 /polishlcross put dup 33 /exclam put dup 34 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Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmr9) def /version(1.0-prerelease) def /UnderlinePosition -125.5 def /FamilyName(cmr9) def end readonly def /UniqueID 4100021 def /PaintType 0 def /FontBBox{-41 -251 1037 750} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmr9 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 /polishlcross put dup 33 /exclam put dup 34 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false def /Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmmi9) def /version(1.0-prerelease) def /UnderlinePosition -125.0 def /FamilyName(cmmi9) def end readonly def /UniqueID 4100076 def /PaintType 0 def /FontBBox{-32 -250 1077 750} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmmi9 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %!PS-AdobeFont-1.1: cmsy9 1.0-prerelease %%VMusage: 38032 40152 12 dict begin /FontInfo 11 dict dup begin /UnderlineThickness 86.85 def /Weight(Medium) def /ItalicAngle 0 def /isFixedPitch false def /Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 5 def /FullName(cmsy9) def /version(1.0-prerelease) def /UnderlinePosition -479.0 def /FamilyName(cmsy9) def end readonly def /UniqueID 4100068 def /PaintType 0 def /FontBBox{-29 -958 1152 779} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmsy9 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 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 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/UnderlineThickness 50.05 def /Weight(Medium) def /ItalicAngle 0 def /isFixedPitch false def /Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmti9) def /version(1.0-prerelease) def /UnderlinePosition -125.5 def /FamilyName(cmti9) def end readonly def /UniqueID 4100044 def /PaintType 0 def /FontBBox{-35 -251 1149 750} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmti9 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 /polishlcross put dup 33 /exclam put dup 34 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/Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmbx9) def /version(1.0-prerelease) def /UnderlinePosition -125.0 def /FamilyName(cmbx9) def end readonly def /UniqueID 4100009 def /PaintType 0 def /FontBBox{-61 -250 1195 750} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmbx9 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 /polishlcross put dup 33 /exclam put dup 34 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Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmr7) def /version(1.0-prerelease) def /UnderlinePosition -125.5 def /FamilyName(cmr7) def end readonly def /UniqueID 4100019 def /PaintType 0 def /FontBBox{-34 -251 1122 749} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmr7 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 /polishlcross put dup 33 /exclam put dup 34 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def /ItalicAngle 0 def /isFixedPitch false def /Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmmi7) def /version(1.0-prerelease) def /UnderlinePosition -125.0 def /FamilyName(cmmi7) def end readonly def /UniqueID 4100074 def /PaintType 0 def /FontBBox{-3 -250 1172 749} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmmi7 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 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Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 5 def /FullName(cmsy7) def /version(1.0-prerelease) def /UnderlinePosition -476.0 def /FamilyName(cmsy7) def end readonly def /UniqueID 4100066 def /PaintType 0 def /FontBBox{-15 -952 1260 787} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmsy7 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 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 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Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmti7) def /version(1.0-prerelease) def /UnderlinePosition -125.5 def /FamilyName(cmti7) def end readonly def /UniqueID 4100042 def /PaintType 0 def /FontBBox{-26 -251 1270 749} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmti7 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 /polishlcross put dup 33 /exclam put dup 34 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Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmbx7) def /version(1.0-prerelease) def /UnderlinePosition -125.5 def /FamilyName(cmbx7) def end readonly def /UniqueID 4100007 def /PaintType 0 def /FontBBox{-60 -251 1290 750} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmbx7 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 /polishlcross put dup 33 /exclam put dup 34 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %!PS-AdobeFont-1.1: cmex10 1.0-prerelease %%VMusage: 29724 32867 12 dict begin /FontInfo 11 dict dup begin /UnderlineThickness 186.6 def /Weight(Medium) def /ItalicAngle 0 def /isFixedPitch false def /Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 0 def /FullName(cmex10) def /version(1.0-prerelease) def /UnderlinePosition -1480.0 def /FamilyName(cmex10) def end readonly def /UniqueID 4100061 def /PaintType 0 def /FontBBox{-25 -2960 1455 772} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmex10 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 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 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Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmcsc10) def /version(1.0-prerelease) def /UnderlinePosition -125.0 def /FamilyName(cmcsc10) def end readonly def /UniqueID 4100047 def /PaintType 0 def /FontBBox{12 -250 1078 749} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmcsc10 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 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Bitstream Inc. % U.S. GOVERNMENT RESTRICTED RIGHTS % This software typeface product is provided with RESTRICTED RIGHTS. Use, % duplication or disclosure by the Government is subject to restrictions % as set forth in the license agreement and in FAR 52.227-19 (c) (2) (May, 1987), % when applicable, or the applicable provisions of the DOD FAR supplement % 252.227-7013 subdivision (a) (15) (April, 1988) or subdivision (a) (17) % (April, 1988). Contractor/manufacturer is Bitstream Inc., % 215 First Street, Cambridge, MA 02142. % Bitstream is a registered trademark of Bitstream Inc. 11 dict begin /FontInfo 9 dict dup begin /version (003.001) readonly def /Notice (Copyright 1990-1992 as an unpublished work by Bitstream Inc. All rights reserved. Confidential.) readonly def /FullName (Goudy Handtooled) readonly def /FamilyName (Goudy Handtooled) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -102 def /UnderlineThickness 71 def end readonly def /FontName /GoudyHandtooledBT-Regular def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding StandardEncoding def /FontBBox {-167 -236 1019 963} readonly def /UniqueID 15531975 def currentdict end currentfile eexec D9D66F632A4812FBAE89F8A58F28761E72B1FFC5625083034B3E1ADCDDE02FA4 B44AE8F3A7B85787C8C90047C8CC19AF3F685AB532A42DE6DDE2B6C900F290EA 0E843501E6741A47C20B736B9394214370E11946173895F8DE508780F72F0EBA B617E6A7468FB2D84028D377CDC3C4E337E3066313976E514621C47725A95FE2 55D6B29C4B5E91721651BF1D56CDCD84D4FA9F072BE27E14FBB961CCD91008A1 CDEFB20D6B8674ACB2DC8A1C918BC55D8CBCA5C32F80A63C54119B24DAB62F09 9A91E000DC8726FCD5AB1CEC8160DC0ADB17E317C54D62AA50BCDB4D031CF07A 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %!PS-AdobeFont-1.0: CaslonOpenfaceBT-Regular 003.001 %%CreationDate: Tue Jun 23 03:26:06 1992 %%VMusage: 46525 50754 % Bitstream Type 1 Font Program % Copyright 1990-1992 as an unpublished work by Bitstream Inc., Cambridge, MA. % All rights reserved. % Confidential and proprietary to Bitstream Inc. % U.S. GOVERNMENT RESTRICTED RIGHTS % This software typeface product is provided with RESTRICTED RIGHTS. Use, % duplication or disclosure by the Government is subject to restrictions % as set forth in the license agreement and in FAR 52.227-19 (c) (2) (May, 1987), % when applicable, or the applicable provisions of the DOD FAR supplement % 252.227-7013 subdivision (a) (15) (April, 1988) or subdivision (a) (17) % (April, 1988). Contractor/manufacturer is Bitstream Inc., % 215 First Street, Cambridge, MA 02142. % Bitstream is a registered trademark of Bitstream Inc. 11 dict begin /FontInfo 9 dict dup begin /version (003.001) readonly def /Notice (Copyright 1990-1992 as an unpublished work by Bitstream Inc. All rights reserved. Confidential.) readonly def /FullName (Caslon Openface) readonly def /FamilyName (Caslon Openface) readonly def /Weight (Normal) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -102 def /UnderlineThickness 108 def end readonly def /FontName /CaslonOpenfaceBT-Regular def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding StandardEncoding def /FontBBox {-167 -236 1135 963} readonly def /UniqueID 15531576 def currentdict end currentfile eexec D9D66F632A4812FBAE89F8A58F28761E72B1FFC5625083034B3E1ADCDDE02FA4 B44AE8F3A7B85787C8C90047C8CC19AF3F685AB532A42DE6DDE2B6C900F290EA 0E843501E6741A47C20B736B9394214370E11946173895F8DE508780F72F0EBA B617E6A7468FB2D84028D377CDC3C4E337E3066313976E514621C47725A95FE2 55D6B29C4B5E91721651BF1D56CDCD84D4FA9F072BE27E14FBB961CCD91008A1 CDEFB20D6B8674ACB2DC8A1C918BC55D8CBCA5C32F80A63C54119B24DAB62F09 9A91E000DC8726FCD5AB1CEC8160DC0ADB17E317C54D62AA5C0F9DCE91FC1D04 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Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmtt10) def /version(1.0-prerelease) def /UnderlinePosition -117.0 def /FamilyName(cmtt10) def end readonly def /UniqueID 4100057 def /PaintType 0 def /FontBBox{-6 -234 539 697} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmtt10 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 /visiblespace 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Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmtt9) def /version(1.0-prerelease) def /UnderlinePosition -117.0 def /FamilyName(cmtt9) def end readonly def /UniqueID 4100060 def /PaintType 0 def /FontBBox{-10 -234 543 699} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmtt9 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for 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 /visiblespace put dup 33 /exclam put dup 34 /quotedbl 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 /backslash put dup 93 /bracketright put dup 94 /asciicircum put dup 95 /underscore 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 /braceleft put dup 124 /bar put dup 125 /braceright put dup 126 /asciitilde put dup 127 /dieresis put readonly def currentdict end currentfile eexec D9D66F633B846A989B9974B0179FC6CC445BC7C8A959A39A32E9DCE7FAEF17EE 3BEC9F50CF7269C04E6A63459A9211B2F19CE3B85116D00C0080DED86FE95FED 7261DA657C3D97CA8E203F5D840114FCDF36CC39021D69046B9667441E78D684 7B79AFD6950C60389C705C3DAFBD5557BE942A2E4CD1B3DC46AB91FE47C5D1B9 C399C08EF2B2AB6AAA3D5C12212D1FA283482F3937BA47AD143A82B13FC33E6E 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2770 y Fm(p)953 2782 y Fk(n)1021 2770 y Fp(=)1537 2713 y(\()p Fm(a=)p Fp(2\))1729 2683 y Fj(2)p Fk(n)p Fj(+)p Fk(\027)1899 2638 y Fl(p)p 1969 2638 133 4 v 1969 2713 a Fm(\031)s(a)2063 2689 y Fj(3)p 1119 2751 1401 4 v 1119 2767 a Fl(p)p 1188 2767 42 4 v 69 x Fp(2\()p Fm(a)14 b Fp(cosh)o(\()p Fm(a)p Fp(\))19 b Fl(\000)f Fp(sinh\()p Fm(a)p Fp(\)\))c Fm(n)p Fp(!\000\()p Fm(n)19 b Fp(+)f Fm(\027)24 b Fp(+)18 b(1\))2557 2770 y Fm(;)42 b(n)23 b Fl(\025)f Fp(0)27 b Fm(:)0 3320 y Fo(Lemma)34 b(5.)46 b Fe(Let)31 b Fm(p)625 3332 y Fk(n)700 3320 y Fe(describ)r(e)g(a)f (unimo)r(dal)g(distribution)h(on)g(the)g(in)n(tegers)e(with)i(mo)r(de)g (at)f Fm(m)p Fe(,)i(and)e(let)h Fm(\033)3485 3290 y Fj(2)3554 3320 y Fe(b)r(e)g(the)0 3436 y(second)c(momen)n(t)h(cen)n(tered)f(at)g (the)h(mo)r(de,)1410 3374 y Fg(P)1497 3460 y Fk(n)1543 3436 y Fp(\()p Fm(n)18 b Fl(\000)g Fm(m)p Fp(\))1831 3406 y Fj(2)1870 3436 y Fm(p)1912 3448 y Fk(n)1957 3436 y Fe(.)37 b(Then,)28 b(if)g Fm(p)2375 3448 y Fk(m)2462 3436 y Fl(\024)23 b Fp(1)p Fm(=)p Fp(3)p Fe(,)j(w)n(e)h(ha)n(v)n(e)1611 3602 y Fm(\033)1661 3568 y Fj(2)1700 3602 y Fm(p)1742 3568 y Fj(2)1742 3623 y Fk(m)1829 3602 y Fl(\025)22 b Fp(1)p Fm(=)p Fp(648)k Fm(:)0 4100 y Fo(Pr)n(oof.)45 b Fp(Assume)27 b(without)h(loss)f(of)g(generalit)n(y)e(that)j Fm(m)23 b Fp(=)g(0.)36 b(De\014ne)28 b Fm(p)23 b Fp(=)2456 4038 y Fg(P)2544 4125 y Fk(n>)p Fj(0)2690 4100 y Fm(p)2732 4112 y Fk(n)2777 4100 y Fp(,)28 b Fm(q)e Fp(=)2978 4038 y Fg(P)3066 4125 y Fk(n<)p Fj(0)3212 4100 y Fm(p)3254 4112 y Fk(n)3300 4100 y Fp(,)h Fm(r)f Fp(=)d Fm(p)3543 4114 y Fj(0)3581 4100 y Fp(.)36 b(It)28 b(is)0 4216 y(clear)f(that)927 4333 y Fm(\033)977 4298 y Fj(2)1039 4333 y Fp(=)1131 4254 y Fg(X)1127 4435 y Fk(n>)p Fj(0)1269 4333 y Fm(n)1319 4298 y Fj(2)1357 4333 y Fm(p)1399 4345 y Fk(n)1462 4333 y Fp(+)1550 4254 y Fg(X)1545 4435 y Fk(n<)p Fj(0)1688 4333 y Fm(n)1738 4298 y Fj(2)1776 4333 y Fm(p)1818 4345 y Fk(n)1886 4333 y Fl(\025)c Fm(r)2074 4254 y Fg(X)2027 4435 y Fk(s)p Fh(\025)p Fk(n>)p Fj(0)2254 4333 y Fm(n)2304 4298 y Fj(2)2361 4333 y Fp(+)18 b Fm(r)2567 4254 y Fg(X)2497 4435 y Fh(\000)p Fk(t)p Fh(\024)p Fk(n<)p Fj(0)2771 4333 y Fm(n)2821 4298 y Fj(2)0 4568 y Fp(where)27 b Fm(s)c Fp(=)g Fl(b)p Fm(p=r)r Fl(c)k Fp(and)h Fm(t)23 b Fp(=)f Fl(b)p Fm(q)s(=r)r Fl(c)p Fp(.)37 b(Th)n(us,)28 b(b)r(ecause)1712 4506 y Fg(P)1800 4529 y Fk(s)1800 4593 y(n)p Fj(=1)1946 4568 y Fm(n)1996 4538 y Fj(2)2057 4568 y Fp(=)23 b Fm(s)2184 4538 y Fj(3)2222 4568 y Fm(=)p Fp(3)17 b(+)h Fm(s)2445 4538 y Fj(2)2484 4568 y Fm(=)p Fp(2)f(+)h Fm(s=)p Fp(6,)27 b(w)n(e)g(ha)n(v)n(e)667 4734 y Fm(\033)717 4700 y Fj(2)779 4734 y Fl(\025)22 b Fm(r)r Fp(\()p Fm(s)976 4700 y Fj(3)1015 4734 y Fm(=)p Fp(3)c(+)g Fm(s)1239 4700 y Fj(2)1277 4734 y Fm(=)p Fp(2)g(+)g Fm(s=)p Fp(6\))f(+)h Fm(r)r Fp(\()p Fm(t)1818 4700 y Fj(3)1858 4734 y Fm(=)p Fp(3)f(+)h Fm(t)2072 4700 y Fj(2)2110 4734 y Fm(=)p Fp(2)g(+)g Fm(t=)p Fp(6\))k Fl(\025)h Fm(r)r Fp(\()p Fm(s)2661 4700 y Fj(3)2700 4734 y Fm(=)p Fp(3)18 b(+)g Fm(t)2915 4700 y Fj(3)2953 4734 y Fm(=)p Fp(3\))27 b Fm(:)1858 5280 y(3)p eop %%Page: 4 4 4 3 bop 0 83 a Fp(If)28 b Fm(p)23 b Fl(\025)f Fm(r)r Fp(,)29 b(then)f Fm(s)23 b Fl(\025)g Fm(p=)p Fp(2)p Fm(r)r Fp(.)36 b(If)28 b Fm(q)e Fl(\025)d Fm(r)r Fp(,)28 b(then)h Fm(t)23 b Fl(\025)f Fm(q)s(=)p Fp(2)p Fm(r)r Fp(.)37 b(Th)n(us,)1219 291 y Fm(\033)1269 257 y Fj(2)1331 291 y Fl(\025)22 b Fi(1)1463 312 y Fj([)p Fk(p)p Fh(\025)p Fk(r)r Fj(])1639 174 y Fg(\022)1751 235 y Fm(p)1793 205 y Fj(3)p 1710 272 162 4 v 1710 349 a Fp(24)p Fm(r)1833 325 y Fj(2)1881 174 y Fg(\023)1961 291 y Fp(+)c Fi(1)2089 312 y Fj([)p Fk(q)r Fh(\025)p Fk(r)r Fj(])2264 174 y Fg(\022)2376 235 y Fm(q)2416 205 y Fj(3)p 2335 272 V 2335 349 a Fp(24)p Fm(r)2458 325 y Fj(2)2506 174 y Fg(\023)1331 532 y Fl(\025)k Fi(1)1463 553 y Fj([max\()p Fk(p;q)r Fj(\))p Fh(\025)p Fk(r)r Fj(])1867 476 y Fp(max\()p Fm(p;)14 b(q)s Fp(\))2205 446 y Fj(3)p 1867 513 377 4 v 1975 590 a Fp(24)p Fm(r)2098 566 y Fj(2)1331 752 y Fl(\025)22 b Fi(1)1463 773 y Fj([\(1)p Fh(\000)p Fk(r)r Fj(\))p Fk(=)p Fj(2)p Fh(\025)p Fk(r)r Fj(])1842 696 y Fp(\(1)d Fl(\000)f Fm(r)r Fp(\))2089 666 y Fj(3)p 1842 733 286 4 v 1884 810 a Fp(192)p Fm(r)2049 786 y Fj(2)1331 953 y Fl(\025)k Fi(1)1463 974 y Fj([)p Fk(r)r Fh(\024)p Fj(1)p Fk(=)p Fj(3])1784 896 y Fp(1)p 1704 934 203 4 v 1704 1010 a(648)p Fm(r)1869 986 y Fj(2)1944 953 y Fm(:)p 1995 903 65 4 v 1995 961 4 59 v 2056 961 V 1995 964 65 4 v 0 1501 a Fo(Lemma)31 b(6.)46 b Fe(F)-7 b(or)27 b(all)g(discrete)g (log-conca)n(v)n(e)d(distributions)k(with)g(mo)r(de)g(at)f Fm(m)p Fe(,)h(w)n(e)f(ha)n(v)n(e,)g(for)g(all)g Fm(n)p Fe(,)1192 1667 y Fm(p)1234 1679 y Fk(n)1302 1667 y Fl(\024)c Fm(p)1432 1679 y Fk(m)1510 1667 y Fp(min)14 b(\(1)p Fm(;)g Fp(exp)f(\(1)18 b Fl(\000)g Fm(p)2130 1679 y Fk(m)2194 1667 y Fl(j)p Fm(n)h Fl(\000)f Fm(m)p Fl(j)p Fp(\)\))42 b Fm(:)0 1833 y Fe(F)-7 b(urthermore,)1225 1949 y Fm(p)1267 1961 y Fk(n)1335 1949 y Fl(\024)23 b Fm(p)1465 1961 y Fk(m)1543 1949 y Fp(min)14 b(\(1)p Fm(;)g Fp(exp)f(\(1)18 b Fl(\000)g Fm(q)s Fl(j)p Fm(n)h Fl(\000)f Fm(m)p Fl(j)p Fp(\)\))42 b Fm(;)0 2107 y Fe(where)27 b Fm(q)f Fp(=)d(min)543 2015 y Fg(\020)686 2074 y Fj(1)p 603 2088 201 4 v 603 2151 a Fk(\033)644 2096 y Fh(p)p 701 2096 103 3 v 55 x Fj(648)813 2107 y Fm(;)860 2074 y Fj(1)p 860 2088 35 4 v 860 2135 a(3)904 2015 y Fg(\021)954 2107 y Fe(.)0 2663 y Fo(Pr)n(oof.)45 b Fp(Devro)n(y)n(e)25 b(\(1987\))g(deriv)n(ed)g (a)h(general)f(inequalit)n(y)g(for)h(all)g(discrete)g(log-conca)n(v)n (e)d(distributions)j(with)h(mo)r(de)0 2779 y(at)g Fm(m)p Fp(:)1077 2895 y Fm(p)1119 2907 y Fk(n)1187 2895 y Fl(\024)c Fm(p)1317 2907 y Fk(m)1395 2895 y Fp(min)14 b(\(1)p Fm(;)g Fp(exp)f(\()q(1)18 b Fl(\000)g Fm(p)2016 2907 y Fk(m)2080 2895 y Fl(j)p Fm(n)g Fl(\000)g Fm(m)p Fl(j)p Fp(\)\))42 b Fm(;)g Fp(all)27 b Fm(n:)0 3045 y Fp(Assume)32 b Fm(p)354 3057 y Fk(m)447 3045 y Fl(\024)e Fp(1)p Fm(=)p Fp(3.)47 b(Then)32 b(b)n(y)f(Lemma)h(5,)g Fm(p)1518 3057 y Fk(m)1612 3045 y Fl(\025)d Fp(1)p Fm(=\033)1840 2976 y Fl(p)p 1909 2976 125 4 v 69 x Fp(648)n(,)k(whic)n(h)e(sho)n(ws)g(that)h Fm(p)2799 3057 y Fk(m)2893 3045 y Fl(\025)d Fm(q)s Fp(.)49 b(This)31 b(concludes)g(the)0 3161 y(pro)r(of.)p 249 3111 65 4 v 249 3169 4 59 v 311 3169 V 249 3172 65 4 v 0 3659 a Fo(Lemma)g(7.)46 b Fe(F)-7 b(or)27 b(all)g(log-conca)n(v)n (e)d(densit)n(y)k(functions,)g Fm(p)1842 3671 y Fk(m)1906 3659 y Fm(\033)e Fl(\024)2067 3594 y(p)p 2136 3594 265 4 v 65 x Fp(28)18 b(+)g(4)p Fm(e)o Fe(.)1858 5280 y Fm(4)p eop %%Page: 5 5 5 4 bop 0 83 a Fo(Pr)n(oof.)45 b Fp(Note)28 b(that,)g(b)n(y)f(Lemma)g (6,)416 249 y Fm(\033)466 215 y Fj(2)528 249 y Fl(\024)615 170 y Fg(X)656 355 y Fk(k)749 249 y Fm(k)795 215 y Fj(2)833 249 y Fm(p)875 261 y Fk(m)953 249 y Fp(min)1105 157 y Fg(\020)1155 249 y Fp(1)p Fm(;)14 b(e)1273 215 y Fj(1)p Fh(\000)p Fk(p)1394 222 y Fd(m)1456 215 y Fh(j)p Fk(k)r Fh(j)1536 157 y Fg(\021)528 492 y Fp(=)22 b(2)673 413 y Fg(X)671 597 y Fk(k)r(>)p Fj(0)809 492 y Fm(k)855 457 y Fj(2)893 492 y Fm(p)935 504 y Fk(m)1013 492 y Fp(min)1165 400 y Fg(\020)1215 492 y Fp(1)p Fm(;)14 b(e)1333 457 y Fj(1)p Fh(\000)p Fk(p)1454 464 y Fd(m)1517 457 y Fk(k)1558 400 y Fg(\021)528 729 y Fl(\024)22 b Fp(2)782 650 y Fg(X)671 839 y Fj(3)p Fk(=p)773 846 y Fd(m)836 839 y Fk(>k)r(>)p Fj(0)1028 729 y Fm(k)1074 695 y Fj(2)1112 729 y Fm(p)1154 741 y Fk(m)1236 729 y Fp(+)c(2)1443 650 y Fg(X)1375 839 y Fk(k)r Fh(\025)p Fj(3)p Fk(=p)1568 846 y Fd(m)1645 729 y Fm(k)1691 695 y Fj(2)1729 729 y Fm(ep)1810 741 y Fk(m)1874 729 y Fm(e)1913 695 y Fh(\000)p Fk(p)2000 702 y Fd(m)2063 695 y Fk(k)528 992 y Fl(\024)k Fp(2)p Fm(p)699 1004 y Fk(m)777 900 y Fg(\020)826 992 y Fp(\(1)p Fm(=)p Fp(3\)\(3)p Fm(=p)1174 1004 y Fk(m)1237 992 y Fp(\))1269 958 y Fj(3)1326 992 y Fp(+)c(\(1)p Fm(=)p Fp(2\)\(3)p Fm(=p)1757 1004 y Fk(m)1819 992 y Fp(\))1851 958 y Fj(2)1908 992 y Fp(+)g(\(1)p Fm(=)p Fp(6\)\(3)p Fm(=p)2339 1004 y Fk(m)2402 992 y Fp(\))2434 900 y Fg(\021)2502 992 y Fp(+)g(2)2709 913 y Fg(X)2641 1102 y Fk(k)r Fh(\025)p Fj(3)p Fk(=p)2834 1109 y Fd(m)2910 992 y Fm(k)2956 958 y Fj(2)2995 992 y Fm(ep)3076 1004 y Fk(m)3139 992 y Fm(e)3178 958 y Fh(\000)p Fk(p)3265 965 y Fd(m)3328 958 y Fk(k)528 1273 y Fp(=)k(18)p Fm(=p)783 1238 y Fj(2)783 1293 y Fk(m)864 1273 y Fp(+)c(9)p Fm(=p)1073 1285 y Fk(m)1155 1273 y Fp(+)g(1)g(+)g(2)1437 1160 y Fg(Z)1482 1348 y Fk(x)p Fh(\025)p Fj(3)p Fk(=p)1676 1355 y Fd(m)1740 1348 y Fh(\000)p Fj(1)1845 1273 y Fm(x)1892 1238 y Fj(2)1931 1273 y Fm(ep)2012 1285 y Fk(m)2075 1273 y Fm(e)2114 1238 y Fh(\000)p Fk(p)2201 1245 y Fd(m)2264 1238 y Fk(x)2321 1273 y Fm(dx)528 1527 y Fl(\024)637 1471 y Fp(28)p 625 1508 107 4 v 625 1584 a Fm(p)667 1560 y Fj(2)667 1605 y Fk(m)760 1527 y Fp(+)866 1471 y(4)p Fm(e)p 853 1508 V 853 1584 a(p)895 1560 y Fj(2)895 1605 y Fk(m)983 1414 y Fg(Z)1029 1602 y Fk(t)p Fh(\025)p Fj(0)1168 1471 y Fm(t)1198 1441 y Fj(2)1237 1471 y Fm(e)1276 1441 y Fh(\000)p Fk(t)p 1168 1508 189 4 v 1242 1584 a Fp(2)1381 1527 y Fm(dt)28 b(:)0 1743 y Fp(Th)n(us,)f Fm(p)275 1713 y Fj(2)275 1764 y Fk(m)339 1743 y Fm(\033)389 1713 y Fj(2)451 1743 y Fl(\024)c Fp(28)17 b(+)h(4)p Fm(e)p Fp(.)p 863 1693 65 4 v 863 1751 4 59 v 925 1751 V 863 1754 65 4 v 0 2241 a Fo(Lemma)31 b(8.)46 b Fe(F)-7 b(or)27 b(the)h(Bessel)e Fp(\()p Fm(\027)q(;)14 b(a)p Fp(\))28 b Fe(distribution,)g(if)g(w)n(e)f(set)h Fm(A)23 b Fp(=)2160 2169 y Fl(p)p 2229 2169 269 4 v 72 x Fm(a)2273 2217 y Fj(2)2330 2241 y Fp(+)18 b Fm(\027)2459 2217 y Fj(2)2497 2241 y Fe(,)28 b Fm(B)f Fp(=)2726 2168 y Fg(p)p 2809 2168 476 4 v 73 x Fm(a)2853 2217 y Fj(2)2909 2241 y Fp(+)18 b(\()p Fm(\027)24 b Fp(+)18 b(1\))3246 2217 y Fj(2)3285 2241 y Fe(,)27 b(then)908 2471 y Fm(\033)958 2437 y Fj(2)1020 2471 y Fl(\024)c Fm(Q)1196 2424 y Fj(def)1211 2471 y Fp(=)1441 2415 y Fm(a)1485 2385 y Fj(2)p 1324 2452 317 4 v 1324 2528 a Fp(2\()p Fm(\027)g Fp(+)18 b Fm(A)p Fp(\))1668 2471 y(+)1751 2354 y Fg(\022)1812 2471 y Fp(1)g(+)2074 2415 y Fm(a)2118 2385 y Fj(2)2156 2415 y Fp(\(1)h(+)f Fm(B)k Fl(\000)c Fm(A)p Fp(\))p 1965 2452 739 4 v 1965 2528 a(2\()p Fm(\027)24 b Fp(+)18 b Fm(A)p Fp(\)\()p Fm(\027)24 b Fp(+)18 b(1)g(+)g Fm(B)t Fp(\))2714 2354 y Fg(\023)2775 2365 y Fj(2)2855 2471 y Fm(:)0 2998 y Fo(Pr)n(oof.)45 b Fp(Note)28 b(that)584 3164 y Fm(\033)634 3129 y Fj(2)696 3164 y Fp(=)23 b Fm(\037)836 3129 y Fj(2)892 3164 y Fp(+)18 b(\()p Fm(m)h Fl(\000)f Fm(\026)p Fp(\))1264 3129 y Fj(2)696 3382 y Fp(=)23 b Fm(\026)18 b Fp(+)945 3326 y(1)p 945 3363 42 4 v 945 3439 a(4)996 3382 y Fm(a)1040 3348 y Fj(2)1079 3382 y Fm(R)1142 3394 y Fk(\027)1184 3382 y Fp(\()p Fm(a)p Fp(\)\()p Fm(R)1387 3396 y Fk(\027)t Fj(+1)1517 3382 y Fp(\()p Fm(a)p Fp(\))h Fl(\000)f Fm(R)1790 3394 y Fk(\027)1832 3382 y Fp(\()p Fm(a)p Fp(\)\))h(+)2074 3240 y Fg( )2140 3382 y Fl(b)2187 3254 y(p)p 2256 3254 269 4 v 72 x Fm(a)2300 3302 y Fj(2)2357 3326 y Fp(+)f Fm(\027)2486 3302 y Fj(2)2543 3326 y Fl(\000)g Fm(\027)p 2187 3363 486 4 v 2409 3439 a Fp(2)2682 3382 y Fl(c)g(\000)2830 3326 y Fm(aR)2937 3338 y Fk(\027)2979 3326 y Fp(\()p Fm(a)p Fp(\))p 2830 3363 258 4 v 2938 3439 a(2)3098 3240 y Fg(!)3163 3261 y Fj(2)696 3684 y Fl(\024)23 b Fm(\026)18 b Fp(+)935 3542 y Fg( )1011 3556 y Fl(p)p 1080 3556 269 4 v 72 x Fm(a)1124 3604 y Fj(2)1181 3628 y Fp(+)g Fm(\027)1310 3604 y Fj(2)1367 3628 y Fl(\000)g Fm(\027)p 1011 3665 486 4 v 1233 3741 a Fp(2)1524 3684 y(+)g Fm(\022)j Fl(\000)2198 3628 y Fm(a)2242 3598 y Fj(2)p 1760 3665 958 4 v 1760 3755 a Fp(2\()p Fm(\027)j Fp(+)18 b Fm(\020)24 b Fp(+)2126 3682 y Fg(p)p 2209 3682 477 4 v 73 x Fm(a)2253 3731 y Fj(2)2309 3755 y Fp(+)18 b(\()p Fm(\027)24 b Fp(+)18 b Fm(\020)6 b Fp(\))2646 3731 y Fj(2)2685 3755 y Fp(\))2727 3542 y Fg(!)2793 3562 y Fj(2)696 3876 y(def)711 3923 y Fp(=)37 b Fm(I)26 b Fp(+)18 b Fm(I)7 b(I)34 b(;)0 4089 y Fp(b)n(y)27 b(Lemmas)g(1)h(and)f(2,)g(where)g Fm(\020)j Fl(2)23 b(f)p Fp(0)p Fm(;)14 b Fp(1)p Fl(g)p Fp(,)26 b Fm(\022)f Fl(2)f Fp([0)p Fm(;)14 b Fp(1].)36 b(Consider)26 b(eac)n(h)h(term)h(separately)-7 b(.)35 b(Clearly)-7 b(,)1179 4297 y Fm(I)30 b Fp(=)23 b Fm(\026)g Fp(=)g 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Fj(2)3002 4741 y Fm(:)1858 5280 y(5)p eop %%Page: 6 6 6 5 bop 0 83 a Fp(F)-7 b(or)30 b Fm(\020)k Fp(=)27 b(0,)j(w)n(e)g(ha)n (v)n(e)f Fm(I)7 b(I)35 b Fl(\024)27 b Fm(\022)975 53 y Fj(2)1041 83 y Fl(\024)g Fp(1.)44 b(F)-7 b(or)30 b Fm(\020)k Fp(=)27 b(1,)j(b)n(y)g(Lemma)g(1,)h(the)g(last)e(term)i(in)f (the)h(brac)n(k)n(ets)d(is)i(smaller)g(than)0 199 y(the)e(middle)g (term,)g(so)f(that)h(w)n(e)f(ma)n(y)g(b)r(ound)h(as)f(follo)n(ws:)869 475 y Fm(I)7 b(I)30 b Fl(\024)1065 308 y Fg(0)1065 457 y(@)1138 475 y Fp(1)18 b(+)1376 388 y Fm(a)1420 358 y Fj(2)1472 296 y Fg(\020)1521 388 y Fp(1)g(+)1664 315 y Fg(p)p 1747 315 476 4 v 73 x Fm(a)1791 364 y Fj(2)1848 388 y Fp(+)g(\()p Fm(\027)24 b Fp(+)18 b(1\))2185 364 y Fj(2)2242 388 y Fl(\000)2325 317 y(p)p 2394 317 269 4 v 71 x Fm(a)2438 364 y Fj(2)2494 388 y Fp(+)g Fm(\027)2623 364 y Fj(2)2662 296 y Fg(\021)p 1291 456 1506 4 v 1291 546 a Fp(2\()p Fm(\027)23 b Fp(+)1512 474 y Fl(p)p 1582 474 269 4 v 1582 546 a Fm(a)1626 522 y Fj(2)1682 546 y Fp(+)18 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Fp(,)h(v)n(ol.)f(52,)f(pp.)i(438{447,)d(2000.)1838 5280 y Fm(11)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF