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%   author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry",
%   version = "0.6",
%   date = "14 April 1995",
%   filename = "8r.enc",
%   email = "kb@cs.umb.edu",
%   address = "135 Center Hill Rd. // Plymouth, MA 02360",
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%   checksum = "xx",
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% 
% Idea is to have all the characters normally included in Type 1 fonts
% available for typesetting. This is effectively the characters in Adobe
% Standard Encoding + ISO Latin 1 + extra characters from Lucida.
% 
% Character code assignments were made as follows:
% 
% (1) the Windows ANSI characters are almost all in their Windows ANSI
% positions, because some Windows users cannot easily reencode the
% fonts, and it makes no difference on other systems. The only Windows
% ANSI characters not available are those that make no sense for
% typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
% (173). quotesingle and grave are moved just because it's such an
% irritation not having them in TeX positions.
% 
% (2) Remaining characters are assigned arbitrarily to the lower part
% of the range, avoiding 0, 10 and 13 in case we meet dumb software.
% 
% (3) Y&Y Lucida Bright includes some extra text characters; in the
% hopes that other PostScript fonts, perhaps created for public
% consumption, will include them, they are included starting at 0x12.
% 
% (4) Remaining positions left undefined are for use in (hopefully)
% upward-compatible revisions, if someday more characters are generally
% available.
% 
% (5) hyphen appears twice for compatibility with both ASCII and Windows.
% 
/TeXBase1Encoding [
% 0x00 (encoded characters from Adobe Standard not in Windows 3.1)
  /.notdef /dotaccent /fi /fl
  /fraction /hungarumlaut /Lslash /lslash
  /ogonek /ring /.notdef
  /breve /minus /.notdef 
% These are the only two remaining unencoded characters, so may as
% well include them.
  /Zcaron /zcaron 
% 0x10
 /caron /dotlessi 
% (unusual TeX characters available in, e.g., Lucida Bright)
 /dotlessj /ff /ffi /ffl 
 /.notdef /.notdef /.notdef /.notdef
 /.notdef /.notdef /.notdef /.notdef
 % very contentious; it's so painful not having quoteleft and quoteright
 % at 96 and 145 that we move the things normally found there down to here.
 /grave /quotesingle 
% 0x20 (ASCII begins)
 /space /exclam /quotedbl /numbersign
 /dollar /percent /ampersand /quoteright
 /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash
% 0x30
 /zero /one /two /three /four /five /six /seven
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% 0x40
 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O
% 0x50
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% 0x70
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% 0x80
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% 0x90
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% 0xA0
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 /macron
% 0xD0
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% 0xC0
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 /Igrave /Iacute /Icircumflex /Idieresis
% 0xD0
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 /Ocircumflex /Otilde /Odieresis /multiply
 /Oslash /Ugrave /Uacute /Ucircumflex
 /Udieresis /Yacute /Thorn /germandbls
% 0xE0
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 /adieresis /aring /ae /ccedilla
 /egrave /eacute /ecircumflex /edieresis
 /igrave /iacute /icircumflex /idieresis
% 0xF0
 /eth /ntilde /ograve /oacute
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%%EndFont
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%%CreationDate: 1993 May 30 16:26:28

% Copyright (c) 1992, 1993 The TeXplorators Corporation
% Hinting Copyright (c) 1992, 1993 Y&Y, Inc.

11 dict begin
/FontInfo 9 dict dup begin
/version (1.1) readonly def
/Notice (Copyright (C) 1992, 1993 The TeXplorators Corporation) readonly def
/FullName (MTSY) readonly def
/FamilyName (MathTime) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -100 def
/UnderlineThickness 50 def
end readonly def
/FontName /MTSY 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 /tie put
dup 66 /openbullet1 put
dup 67 /plus put
dup 68 /equal put
dup 69 /vector put
dup 70 /triangleright put
dup 71 /triangleleft put
dup 72 /equal1 put
dup 73 /semicolon put
dup 74 /grave put
dup 75 /acute put
dup 76 /caron put
dup 77 /breve put
dup 78 /macron put
dup 79 /circumflex put
dup 80 /dotaccent put
dup 81 /tilde put
dup 82 /dieresis put
dup 83 /overbar 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 /radical1 put
dup 124 /club put
dup 125 /diamond put
dup 126 /heart put
dup 127 /spade put
dup 160 /space put
dup 128 /arrowleft put
readonly def
/FontBBox{0 -954 1043 796}readonly def
/UniqueID 5018947 def
currentdict end
currentfile eexec

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cleartomark


%%EndFont
%%BeginFont: MTMI
%!PS-AdobeFont-1.1: MTMI 1.1
%%CreationDate: 1993 May 10 16:52:31

% Copyright (c) 1992, 1993 The TeXplorators Corporation
% Hinting Copyright (c) 1992, 1993 Y&Y, Inc.

11 dict begin
/FontInfo 9 dict dup begin
/version (1.1) readonly def
/Notice (Copyright (C) 1992, 1993 The TeXplorators Corporation) readonly def
/FullName (MTMI) def
/FamilyName (MathTime) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.036 def
/isFixedPitch false def
/UnderlinePosition -100 def
/UnderlineThickness 50 def
end readonly def
/FontName /MTMI 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 /Omega1 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 /parenleft put
dup 47 /parenright put
dup 48 /Gamma1 put
dup 49 /Delta1 put
dup 50 /Theta1 put
dup 51 /Lambda1 put
dup 52 /Xi1 put
dup 53 /Pi1 put
dup 54 /Sigma1 put
dup 55 /Upsilon1 put
dup 56 /Phi1 put
dup 57 /Psi1 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 /kappa1 put
dup 127 /Omega1 put
dup 160 /space put
dup 128 /psi put
readonly def
/FontBBox{0 -213 987 680}readonly def
/UniqueID 5018946 def
currentdict end
currentfile eexec

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cleartomark


%%EndFont
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37

% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 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: MTEX
%!PS-AdobeFont-1.1: MTEX 1.1
%%CreationDate: 1993 Jun 26 10:30:36

% Copyright (c) 1992, 1993 The TeXplorators Corporation
% Hinting Copyright (c) 1992, 1993 Y&Y, Inc.

11 dict begin
/FontInfo 9 dict dup begin
/version (1.1) readonly def
/Notice (Copyright (C) 1992, 1993 The TeXplorators Corporation) readonly def
/FullName (MTEX) readonly def
/FamilyName (MathTime) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -100 def
/UnderlineThickness 50 def
end readonly def
/FontName /MTEX 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 128 /parenleftBigg 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 159 /radical put
dup 160 /space put
readonly def
/FontBBox{-23 -2958 1456 766}readonly def
/UniqueID 5018948 def
currentdict end
currentfile eexec

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%%Page: 12 12
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%%Page: 15 15
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%%Page: 19 19
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%%Page: 27 27
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%%Page: 28 28
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Fv(2)2171 3583 y Fw(.)p Fu(R)2280 3542 y Ft(n)2321 3583
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3799 y Fw(/)3012 3763 y Fv(1)p Ft(=)p Fv(2)3143 3799
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FB(belonging)1 5099 y(to)j Fu(B)p Fw(.D)t(.P)15 b(/;)i(L)575
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(\(when)h(this)e(is)g(used)h(to)p eop
%%Page: 31 31
31 30 bop 1076 208 a Fm(SPECTRAL)22 b(PR)m(OPER)l(TIES)f(OF)j(WITTEN)f
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y Fv(2)1384 1263 y Fw(.x)6 b(;)17 b(D)t(/)22 b Fs(C)1780
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Fw(;)17 b(g)t(/)38 b FB(since)c(both)j Fw(K)1914 2466
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3095 y(8)3055 3069 y FB(is)c(self-adjoint:)48 b(for)35
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230 5 v 2705 2878 a Ft(C)t(.)p Fv(1)p FA(C)p Ft(!)q(/)2944
2842 y Fw(.)p FB(1)22 b Fs(\000)g Fw(t)3190 2801 y Fv(1)p
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y Fw(8)s(.x)6 b(/)22 b Fs(\000)g Fw(8)s(.t)9 b(x)d(/)27
b Fs(\025)1639 3382 y(j)p Fw(x)i Fs(\000)21 b Fw(t)9
b(x)d Fs(j)1966 3346 y Fv(1)p FA(C)p Ft(!)p 1639 3430
462 5 v 1676 3523 a Fw(C)g(.)p FB(1)21 b Fs(C)h Fw(!)r(/)2210
3452 y FB(for)100 b Fs(j)p Fw(x)6 b Fs(j)p Fw(;)34 b
Fs(j)p Fw(t)9 b(x)d Fs(j)28 b(\025)g Fw(C)s(:)674 b FB(\(7.39\))1
3664 y(Ne)o(xt)24 b(one)h(may)g(for)g(each)30 b Fw(k)i
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Fw(C)2662 3679 y Fv(1)2728 3664 y FB(such)c(that)f(when)29
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Fs(D)h Fw(.)p FB(1)22 b Fs(C)g(j)p Fw(x)6 b Fs(j)851
3790 y Fv(2)889 3827 y Fw(/)936 3740 y Fv(1)p 936 3757
34 5 v 936 3815 a(2)1013 3827 y FB(in)24 b(the)h(rest)g(of)g(this)e
(proof\))902 4028 y Fs(h)p Fw(x)28 b Fs(\000)22 b Fw(y)6
b Fs(i)1210 3987 y Ft(k)1247 4028 y Fw(e)1293 3987 y
FA(\000)p Ft(.8)r(.x)t(/)p FA(\000)p Ft(8)r(.y)t(//=)p
Fv(2)1825 4028 y Fs(\024)28 b Fw(C)1997 4043 y Fv(1)2133
4028 y FB(for)d(all)100 b Fw(y)33 b Fs(2)27 b FB(ch)q
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(x)g(hull)f(of)28 b Fw(A)t FB(.)i(Indeed,)24 b(if)k Fs(j)p
Fw(y)6 b Fs(j)28 b(\025)g Fw(C)33 b FB(the)24 b(inequality)e(in)h
(\(7.39\))g(yields)g(that)g(the)1 4385 y(left)i(hand)f(side,)h(when)k
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b Fs(i)27 b(2)h Fu(R)1537 4400 y FA(C)1597 4385 y FB(,)d(is)f
(estimated)g(by)1286 4587 y Fw(.r)1370 4545 y Ft(\014)1417
4587 y Fw(e)1463 4545 y FA(\000)p Ft(r)1550 4587 y Fw(/)1587
4545 y Ft(\015)1735 4587 y FB(for)h(some)99 b Fw(\014)t(;)34
b(\015)43 b(>)28 b FB(0)p Fs(I)1060 b FB(\(7.41\))1 4788
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b(can)i(let)i Fw(z)g Fs(D)1257 4748 y Ft(C)p 1245 4766
76 5 v 1245 4824 a FA(j)p Ft(x)t FA(j)1330 4788 y Fw(x)35
b FB(and)24 b(reduce)i(to)e(the)h(case)k Fs(j)p Fw(y)6
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4949 y Ft(k)1558 4990 y Fs(\024)28 b(h)p Fw(z)q Fs(i)1781
4949 y Ft(k)1817 4990 y Fs(h)p Fw(x)f Fs(\000)22 b Fw(z)q
Fs(i)2114 4949 y Ft(k)2178 4990 y Fs(\024)28 b Fw(.)p
FB(1)22 b Fs(C)g Fw(C)2565 4949 y Fv(2)2602 4990 y Fw(/)2639
4949 y Ft(k)2677 4990 y Fs(h)p Fw(x)28 b Fs(\000)21 b
Fw(z)q Fs(i)2974 4949 y Ft(k)3658 4990 y FB(\(7.42\))941
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Fs(j)28 b Fw(<)g(C)35 b FB(is)24 b(easy)-6 b(,)612 5638
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5638 y Fs(j)p Fw(x)6 b Fs(j)889 5597 y Ft(k)927 5638
y Fs(j)p Fw(f)18 b(.x)6 b(/)p Fs(j)28 b(\024)38 b FB(sup)1315
5723 y Ft(x)t FA(2)p Fp(R)1442 5704 y Fo(n)1490 5503
y Fq(Z)1588 5528 y Fv(1)1543 5727 y(0)1626 5638 y Fs(h)p
Fw(x)27 b Fs(\000)22 b Fw(t)9 b(x)d Fs(i)1970 5597 y
Ft(k)s FA(C)p Fv(1)2092 5638 y Fw(e)2138 5597 y FA(\000)p
Ft(.8)r(.x)t(/)p FA(\000)p Ft(8)r(.t)g(x)t(//=)p Fv(2)2683
5638 y Fw(d)h(t)1805 5802 y Fs(\002)22 b FB(sup)2043
5721 y Fq(\010)2107 5802 y Fs(h)p Fw(y)6 b Fs(i)2237
5760 y Ft(k)s FA(C)p Fv(1)2359 5802 y Fs(j)p Fw(v)t(.y)g(/)p
Fs(j)2626 5717 y Fq(\014)2626 5777 y(\014)2686 5802 y
Fw(y)33 b Fs(2)28 b Fu(R)2932 5760 y Ft(n)2989 5721 y
Fq(\011)3064 5802 y Fw(<)f Fs(1)p Fw(:)3658 5687 y FB(\(7.44\))p
eop
%%Page: 34 34
34 33 bop 1 208 a Fm(34)1628 b(JON)25 b(JOHNSEN)294 501
y FB(It)g(remains,)f(therefore,)h(to)g(sho)n(w)e(\(7.35\).)31
b(If)e Fw(v)i Fs(2)d Fw(Z)t(.1)2242 453 y Ft(.)p Fv(1)p
Ft(/)2242 527 y(8)2329 501 y Fw(/)t FB(,)d(then)k Fw(v)i
Fs(D)2842 480 y(Q)2806 501 y Fw(K)2884 516 y Fv(2)2922
501 y Fw(.x)6 b(;)17 b(D)t(/v)8 b FB(,)24 b(hence)1169
708 y Fw(v)31 b Fs(D)1390 687 y(Q)1354 708 y Fw(K)1432
723 y Fv(2)1470 708 y Fw(.x)6 b(;)17 b(D)t(/)1734 667
y Ft(N)1794 708 y Fw(v)103 b FB(for)25 b(e)n(v)o(ery)99
b Fw(N)38 b Fs(2)27 b Fu(N)p Fs(I)943 b FB(\(7.45\))1
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Fs(Q)836 914 y Fw(K)914 929 y Fv(2)952 914 y Fw(.x)6
b(;)17 b(D)t(/)1216 878 y Ft(N)1317 914 y FB(by)36 b(Proposition)g(7.1)
g(has)h(entries)g(in)j(OPS)q Fw(.m)3000 878 y FA(\000)p
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b(/)p Fs(j)289 1033 y Ft(N)350 1069 y Fw(D)434 1033 y
Ft(\014)480 1069 y Fw(v)529 1084 y Ft(j)569 1069 y Fw(.x)g(/)29
b FB(is)24 b(in)29 b Fw(L)995 1033 y Fv(2)1032 1069 y
Fw(.)p Fu(R)1141 1033 y Ft(n)1181 1069 y Fw(/)g FB(for)d(all)e
(multiindices)j Fw(\014)36 b FB(and)25 b(all)j Fw(j)40
b FB(and)29 b Fw(N)14 b FB(.)31 b(Moreo)o(v)o(er)l(,)737
1275 y Fs(h)p Fw(x)6 b Fs(i)867 1234 y Ft(N)955 1275
y Fs(\024)28 b Fw(C)1133 1234 y FA(\000)p Fv(1)1221 1275
y Fs(j)p Fw(8)1328 1234 y FA(0)1351 1275 y Fw(.x)6 b(/)p
Fs(j)1509 1234 y Ft(N)g(=!)1673 1275 y Fs(\024)28 b Fw(C)1851
1234 y FA(\000)p Fv(1)1940 1275 y Fs(h)p Fw(8)2056 1234
y FA(0)2078 1275 y Fw(.x)6 b(/)p Fs(i)2245 1234 y Ft(N)2302
1210 y Fk(0)2350 1275 y Fs(\024)28 b Fw(C)2528 1234 y
FA(0)2550 1275 y Fw(.)p FB(1)22 b Fs(C)g(j)p Fw(8)2866
1234 y FA(0)2889 1275 y Fw(.x)6 b(/)p Fs(j)3047 1234
y Ft(N)3104 1210 y Fk(0)3125 1275 y Fw(/)496 b FB(\(7.46\))1
1482 y(when)31 b Fw(N)333 1445 y FA(0)387 1482 y Fs(\025)g
Fw(N)9 b(=!)d FB(;)28 b(cf.)f(\(IV)1022 1497 y Ft(!)1076
1482 y FB(\).)37 b(It)27 b(is)g(thus)f(sho)n(wn)g(that)k
Fw(v)35 b FB(has)27 b(coef)n(\002cients)g(in)j Fe(S)7
b Fw(.)p Fu(R)3023 1445 y Ft(n)3065 1482 y Fw(/)t FB(,)27
b(and)g(altogether)g(this)1 1636 y(sho)n(ws)d(the)g(theorem.)p
3827 1636 4 68 v 3831 1572 60 4 v 3831 1636 V 3890 1636
4 68 v 1 1871 a(7.3.)49 b Fx(Example.)i FB(Consider)24
b(the)h(potential,)e(as)i(done)g(by)f(Helf)n(fer)h([Hel98,)g(Hel97a])g
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b Fs(D)1316 2070 y Fv(1)p 1314 2087 37 5 v 1314 2145
a Ft(h)1430 1994 y(n)1378 2015 y Fq(X)1388 2218 y Ft(j)7
b FA(D)p Fv(1)1519 2110 y Fw(.)1581 2070 y Ft(\025)p
1566 2087 67 5 v 1566 2145 a Fv(12)1642 2110 y Fw(x)1698
2069 y Fv(4)1692 2137 y Ft(j)1758 2110 y Fs(C)1868 2070
y Ft(\027)p 1868 2087 36 5 v 1869 2145 a Fv(2)1913 2110
y Fw(x)1969 2069 y Fv(2)1963 2137 y Ft(j)2006 2110 y
Fw(/)22 b Fs(C)2177 2070 y Fv(1)p 2175 2087 37 5 v 2175
2145 a Ft(h)2232 2070 y Fd(I)p 2232 2087 44 5 v 2237
2145 a Fv(2)2354 1994 y Ft(n)2302 2015 y Fq(X)2312 2218
y Ft(j)7 b FA(D)p Fv(1)2460 2110 y Fs(j)p Fw(x)2538 2125
y Ft(j)2600 2110 y Fs(\000)22 b Fw(x)2750 2125 y Ft(j)7
b FA(C)p Fv(1)2875 2110 y Fs(j)2903 2069 y Fv(2)3658
2110 y FB(\(7.47\))1 2370 y(whereby)29 b Fw(x)423 2385
y Ft(n)p FA(C)p Fv(1)576 2370 y Fs(D)e Fw(x)731 2385
y Fv(1)798 2370 y FB(as)e(a)g(con)l(v)o(ention.)k(Here)g
Fw(h)g(>)e FB(0)i(and)g Fe(I)35 b Fw(>)27 b FB(0)i(while)1726
2577 y Fw(\025)f(>)f FB(0)h Fw(>)f(\027)s(:)1485 b FB(\(7.48\))1
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y Fq(R)490 2938 y Fw(e)536 2902 y FA(\000)p Ft(8)661
2938 y Fw(d)7 b(x)37 b Fs(D)30 b FB(1)h(may)26 b(be)h(ful\002lled)f(by)
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(,)26 b(\(III\))h(and)1 3092 y(\(V\))e(clearly)g(hold.)30
b(Concerning)25 b(\(IV)1325 3107 y Ft(!)1379 3092 y FB(\),)g(Euler')-5
b(s)24 b(formula)g(gi)n(v)o(es)635 3299 y Fw(x)k Fs(\001)22
b Fw(8)842 3258 y FA(0)864 3299 y Fw(.x)6 b(/)28 b Fs(\025)1136
3258 y Fv(1)p 1135 3276 37 5 v 1135 3334 a Ft(h)1198
3204 y Fq(X)1339 3299 y Fw(.)1386 3258 y Ft(\025)p 1386
3276 V 1388 3334 a Fv(3)1432 3299 y Fw(x)1488 3258 y
Fv(4)1482 3325 y Ft(j)1548 3299 y Fs(C)22 b Fw(\027)6
b(x)1757 3258 y Fv(2)1751 3325 y Ft(j)1794 3299 y Fw(/)28
b Fs(D)1974 3254 y FA(j)p Ft(x)t FA(j)2049 3230 y Fl(4)p
1974 3276 105 5 v 2008 3334 a Ft(h)2088 3299 y Fw(.)2135
3258 y Ft(\025)p 2135 3276 37 5 v 2137 3334 a Fv(3)2182
3299 y Fw(.)2230 3258 y Fv(1)p 2229 3276 V 2229 3334
a Ft(n)2275 3299 y Fw(/)2312 3258 y Fv(2)2372 3299 y
Fs(\000)21 b(j)p Fw(\027)6 b Fs(jj)p Fw(x)g Fs(j)2692
3258 y FA(\000)p Fv(2)2783 3299 y Fw(/)28 b Fs(\025)g
Fw(C)3026 3258 y FA(\000)p Fv(1)3115 3299 y Fs(j)p Fw(x)6
b Fs(j)3227 3258 y Fv(4)3658 3299 y FB(\(7.49\))1 3505
y(when)34 b Fs(j)p Fw(x)6 b Fs(j)38 b(\025)g Fw(C)i FB(for)30
b(some)g(suf)n(\002ciently)f(lar)n(ge)34 b Fw(C)43 b
Fs(D)38 b Fw(C)6 b(.h;)17 b(n;)g(\025)s(;)g(\027)6 b(/)t
FB(;)33 b(here)d(it)g(is)g(used)f(that)34 b Fs(j)p Fw(x)3386
3520 y Ft(j)3426 3505 y Fs(j)3454 3469 y Fv(2)3530 3505
y Fs(\025)j(j)p Fw(x)6 b Fs(j)3754 3469 y Fv(2)3792 3505
y Fw(=)f(n)1 3660 y FB(for)32 b(some)j Fw(j)50 b Fs(2)40
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FB(\))h(holds)e(for)i(all)j Fw(!)42 b Fs(2)17 b FB(]0)p
Fw(;)g FB(3])34 b(for)e(the)f(abo)o(v)o(e)k Fw(8)s(.x)6
b(/)t FB(.)50 b(\(By)1 3814 y(comparison,)24 b(the)g(assumptions)f(in)h
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b Fw(8)2652 3778 y FA(00)2652 3841 y Ft(j)7 b(k)2755
3814 y FB(are)25 b(unbounded)f(on)29 b Fu(R)3570 3778
y Ft(n)3614 3814 y FB(.\))294 3969 y(Because)c(of)f(this,)g(the)g
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3969 y FB(and)g Fw(A)2474 3984 y Fv(1)2540 3969 y FB(ha)n(v)o(e)c(the)g
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b Fw(A)2444 4138 y Fv(1)2509 4123 y Fw(>)e FB(0)t(.)294
4278 y(The)g(lo)n(wer)f(bound)k Fw(m.A)1195 4293 y Fv(1)1232
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Fw(h)h FB(and)f Fe(I)12 b FB(,)27 b(be)g(estimated)f(in)g(v)n(arious)g
(w)o(ays,)1 4433 y(see)f(for)g(e)o(xample)f([Hel98,)h(Hel97a].)294
4700 y(8.)51 b(F)t Fy(I)t(N)q(A)t(L)33 b FB(R)t Fy(E)t(M)t(A)t(R)t(K)
294 4913 y FB(The)28 b(essential)e(self-adjointness)g(of)32
b Fw(A)1688 4928 y Fv(0)1757 4913 y FB(and)g Fw(A)2006
4928 y Fv(1)2075 4913 y FB(\(or)g Fw(A)2296 4928 y Ft(k)2338
4913 y FB(\))c(holds)e(in)i(a)g(greater)g(generality)f(than)g(that)1
5068 y(established)19 b(in)h(Section)g(7.)30 b(F)o(or)20
b(scalar)g(Schr)8 b(\250)-41 b(odinger)21 b(operators)f(this)f(is)h
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b(Kato)1 5222 y([Kat73])38 b(and)g(S.)h(Agmon)e([Agm78)o(],)k(b)n(ut)d
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b Fw(1)1335 5329 y Ft(.k)s(/)1335 5402 y(8)1426 5377
y FB(.)57 b(In)33 b(f)o(act,)j(Simader')-5 b(s)33 b(ar)n(gument)g(for)
38 b Fs(\000)p Fw(1)16 b Fs(C)p Fw(V)55 b FB(specialised)33
b(to)g(the)1 5531 y(case)j Fw(V)59 b Fs(2)40 b Fw(C)506
5495 y Fv(0)543 5531 y Fw(.)p Fu(R)652 5495 y Ft(n)692
5531 y Fw(;)17 b Fu(R)p Fw(/)36 b FB(appears)c(in)f(a)h(recent)g
(lecture)g(note)f([Hel99,)i(Thm.)e(9.4.1],)h(and)g(in)f(this)g(form)g
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b FB(with)27 b Fw(V)45 b Fs(2)28 b Fw(C)2269 5650 y Fv(0)2306
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Fw(/)t FB(,)24 b(when)f(this)f(operator)i(is)e(positi)n(v)o(e)1
5841 y(on)29 b Fw(C)205 5804 y FA(1)199 5868 y Fv(0)277
5841 y Fw(.)p Fu(R)386 5804 y Ft(n)426 5841 y Fw(;)17
b Fu(R)548 5804 y Ft(n)588 5841 y Fw(/)t FB(,)25 b(hence)g(to)k
Fw(1)1123 5792 y Ft(.)p Fv(1)p Ft(/)1123 5866 y(8)1239
5841 y FB(and)g Fw(A)1485 5856 y Fv(1)1526 5841 y FB(.)p
eop
%%Page: 35 35
35 34 bop 1076 208 a Fm(SPECTRAL)22 b(PR)m(OPER)l(TIES)f(OF)j(WITTEN)f
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%%Page: 36 36
36 35 bop 1 208 a Fm(36)1628 b(JON)25 b(JOHNSEN)1 482
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3237 y Fu(C)2672 3196 y Ft(n)2712 3237 y Fw(/;)904 b
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y Fw(.)p Fu(R)2264 3711 y Ft(n)2305 3748 y Fw(;)17 b
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%%Trailer
end
userdict /end-hook known{end-hook}if
SafetyEnclosure restore
%%EOF