Content-Type: multipart/mixed; boundary="-------------0001231634141" This is a multi-part message in MIME format. ---------------0001231634141 Content-Type: text/plain; name="00-36.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-36.keywords" Dirac operator, magnetic field, Aharonov-Casher theorem ---------------0001231634141 Content-Type: application/postscript; name="zerofin.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="zerofin.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: zerofin.dvi %%Pages: 51 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: cmr17 msbm10 cmr12 cmsy10 cmbx10 cmr10 cmr8 cmbx12 %%+ cmti12 cmsy7 cmex10 msbm8 cmr6 cmmi12 cmmi8 cmsy8 cmmi6 cmsy6 cmmi10 %%+ cmtt12 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -o zerofin.ps zerofin.dvi %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 2000.01.23:1725 %%BeginProcSet: tex.pro /TeXDict 250 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 product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{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 %%BeginFont: cmr17 %!FontType1-1.0: cmr17 1.1/12-Nov-94 %%VMusage: 27721 27997 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmr17) readonly def /FamilyName (cmr17) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 695 def /Descender -195 def /CapHeight 683 def end readonly def /FontName /cmr17 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-35 -249 944 749] readonly def /UniqueID 4701483 def 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Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.2/19-Jan-95) readonly def /Notice (Copyright \(C\) 1994, 1995, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-C.) readonly def /FullName (msbm10) readonly def /FamilyName (msbm10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /CapHeight 683 def /Ascender 694 def /Descender -194 def end readonly def /FontName /msbm10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space 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 160 /hardspace put 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 dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-56 -420 2332 920] readonly def /UniqueID 4702309 def /StrokeWidth 0 def currentdict end currentfile eexec 9B9C0887DB83FB1ECD8335B0BB39CEF0AF64F656FC6E5C230CC9D3A7346AAC7A06BD9A40393C A15D3773A21E06B9B4254D3050E90726BBB09120935A8D448CDBC799D90205A21291254FA633 CC0C2ED88781EF21A5D45B72445C284FA44F8F39CF566976075A92E9E2947A6FC93D879C29EB 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cleartomark %%EndFont %%BeginFont: cmr12 %!FontType1-1.0: cmr12 1.1/12-Nov-94 %%VMusage: 25445 25725 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmr12) readonly def /FamilyName (cmr12) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmr12 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-38 -251 988 750] readonly def /UniqueID 4701482 def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmsy10 %!FontType1-1.0: cmsy10 1.1/12-Nov-94 %%VMusage: 26658 28948 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmsy10) readonly def /FamilyName (cmsy10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmsy10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 160 /space put dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-29 -272 1123 779] readonly def /UniqueID 4701534 def /StrokeWidth 0 def currentdict end currentfile eexec 9B9C0887DB83FB1ECD8335B0BB39CEF0AF64F656FC6E5C230CC9D3A7346AAC7A06BD9A40393C A15D3773A21E06B9B4254D3050E90726BBB09120935A8D448CDBC799D90205A21291254FA633 CC0C2ED88781EF21A5D45B72445C284FA44F8F39CF566976075A92E9E2947A6FC93D879C29EB 26809ECB4409CBC8666526CCED9265DE661A2F8B0A16BE45A9DA17EEAF89B0E023DE2B0373DC AB997D60B7D7DC1B4F7E650A2A9F13CA0543F3B080AE33D65F2802B8C5214D5B60D0D2956096 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmbx10 %!FontType1-1.0: cmbx10 1.1/12-Nov-94 %%VMusage: 25027 25304 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmbx10) readonly def /FamilyName (cmbx10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 444 def /Ascender 694 def /Descender -194 def /CapHeight 686 def end readonly def /FontName /cmbx10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-60 -250 1164 750] readonly def /UniqueID 4701489 def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmr10 %!FontType1-1.0: cmr10 1.1/12-Nov-94 %%VMusage: 25429 25707 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmr10) readonly def /FamilyName (cmr10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmr10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-44 -250 1009 750] readonly def /UniqueID 4701481 def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmr8 %!FontType1-1.0: cmr8 1.1/12-Nov-94 %%VMusage: 25661 25938 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmr8) readonly def /FamilyName (cmr8) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmr8 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-40 -250 1070 750] readonly def /UniqueID 4701479 def /StrokeWidth 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Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmbx12) readonly def /FamilyName (cmbx12) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 444 def /Ascender 694 def /Descender -194 def /CapHeight 686 def end readonly def /FontName /cmbx12 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-57 -250 1139 750] readonly def /UniqueID 4701490 def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmti12 %!FontType1-1.0: cmti12 1.1/12-Nov-94 %%VMusage: 30959 31233 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmti12) readonly def /FamilyName (cmti12) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmti12 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-36 -251 1104 750] readonly def /UniqueID 4701495 def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmsy7 %!FontType1-1.0: cmsy7 1.1/12-Nov-94 %%VMusage: 27597 30113 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmsy7) readonly def /FamilyName (cmsy7) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmsy7 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 160 /space put dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-15 -270 1257 786] readonly def /UniqueID 4701531 def /StrokeWidth 0 def currentdict end currentfile eexec 9B9C0887DB83FB1ECD8335B0BB39CEF0AF64F656FC6E5C230CC9D3A7346AAC7A06BD9A40393C A15D3773A21E06B9B4254D3050E90726BBB09120935A8D448CDBC799D90205A21291254FA633 CC0C2ED88781EF21A5D45B72445C284FA44F8F39CF566976075A92E9E2947A6FC93D879C29EB 26809ECB4409CBC8666526CCED9265DE661A2F8B0A16BE45A9DA17EEAF89B0E023DE2B0373DC AB997D60B7D7DC1B4F7E650A2A9F13CA0543F3B080AE33D65F2802B8C7BE7B9F15E795D04A36 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmex10 %!FontType1-1.0: cmex10 1.1/12-Nov-94 %%VMusage: 20734 24250 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmex10) readonly def /FamilyName (cmex10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /Ascender 694 def /Descender -194 def /CapHeight 683 def /XHeight 431 def end readonly def /FontName /cmex10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 36 /floorleftBigg put dup 37 /floorrightBigg put dup 38 /ceilingleftBigg put dup 39 /ceilingrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 42 /angbracketleftBigg put dup 43 /angbracketrightBigg put dup 44 /slashBigg put dup 45 /backslashBigg put dup 46 /slashBig put dup 47 /backslashBig put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 54 /bracketleftex put dup 55 /bracketrightex put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 62 /braceex put dup 63 /arrowvertex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 70 /unionsqtext put dup 71 /unionsqdisplay put dup 72 /contintegraltext put dup 73 /contintegraldisplay put dup 74 /circledottext put dup 75 /circledotdisplay put dup 76 /circleplustext put dup 77 /circleplusdisplay put dup 78 /circlemultiplytext put dup 79 /circlemultiplydisplay put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 83 /uniontext put dup 84 /intersectiontext put dup 85 /unionmultitext put dup 86 /logicalandtext put dup 87 /logicalortext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 93 /unionmultidisplay put dup 94 /logicalanddisplay put dup 95 /logicalordisplay put dup 96 /coproducttext put dup 97 /coproductdisplay put dup 98 /hatwide put dup 99 /hatwider put dup 100 /hatwidest put dup 101 /tildewide put dup 102 /tildewider put dup 103 /tildewidest put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 106 /floorleftBig put dup 107 /floorrightBig put dup 108 /ceilingleftBig put dup 109 /ceilingrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 113 /radicalBig put dup 114 /radicalbigg put dup 115 /radicalBigg put dup 116 /radicalbt put dup 117 /radicalvertex put dup 118 /radicaltp put dup 119 /arrowvertexdbl put dup 120 /arrowtp put dup 121 /arrowbt put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put dup 126 /arrowdbltp put dup 160 /space put dup 161 /parenleftbig put dup 162 /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 dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-24 -2959 1454 750] readonly def /UniqueID 4701543 def /StrokeWidth 0 def currentdict end currentfile eexec 9B9C0887DB83FB1ECD8335B0BB39CEF0AF64F656FC6E5C230CC9D3A7346AAC7A06BD9A40393C A15D3773A21E06B9B4254D3050E90726BBB09120935A8D448CDBC799D90205A21291254FA633 CC0C2ED88781EF21A5D45B72445C284FA44F8F39CF566976075A92E9E2947A6FC93D879C29EB 26809ECB4409CBC8666526CCED9265DE661A2F8B0A16BE45A9DA17EEAF89B0E023DE2B0373DC AB997D60B7D7DC1B4F7E650A2A9F13CA0543F3B080AE33D65F2802B8C5214D5B60D0D2956096 89997B955E43A4D99ACFF7BF2FBF2CC4D0F447C1D54FB3D42B7E474427B6DD55A05308D91C3D 62C926F6E6AE84EA2B0934AEBC9F41694A4EC2254C5863450A4027A97B85652572E1D5DE078B 35903D1BAA896C24F2A3A773197487169B2FDD71110B5E3F8E74E62DBFCCEED1B0289C5B5E9D 1A138738D8455BA3F58C60544D7F35B83CB5EAE52BC73D66522AE7E3A2C68AB9AF55C7B14D79 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: msbm8 %!FontType1-1.0: msbm8 1.2/19-Jan-95 %%VMusage: 32482 33913 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.2/19-Jan-95) readonly def /Notice (Copyright \(C\) 1994, 1995, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-C.) readonly def /FullName (msbm8) readonly def /FamilyName (msbm8) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /Descender -194 def /Ascender 694 def /CapHeight 683 def /XHeight 431 def end readonly def /FontName /msbm8 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space 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 160 /hardspace put 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 dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-59 -474 2478 974] readonly def /UniqueID 4702307 def /StrokeWidth 0 def currentdict end currentfile eexec 9B9C0887DB83FB1ECD8335B0BB39CEF0AF64F656FC6E5C230CC9D3A7346AAC7A06BD9A40393C A15D3773A21E06B9B4254D3050E90726BBB09120935A8D448CDBC799D90205A21291254FA633 CC0C2ED88781EF21A5D45B72445C284FA44F8F39CF566976075A92E9E2947A6FC93D879C29EB 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cleartomark %%EndFont %%BeginFont: cmr6 %!FontType1-1.0: cmr6 1.1/12-Nov-94 %%VMusage: 25711 25989 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmr6) readonly def /FamilyName (cmr6) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmr6 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /polishlcross put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-24 -250 1193 750] readonly def /UniqueID 4701477 def /StrokeWidth 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2450CA024A9FA6EE57F4D7726598755E2BB79F4847B2E4FCAB4CD2C4B0DB4D2BB37FB5E6749C D602B2CC38EC959D72CED7C46FFA53FBF39C93F36DD5324A982017BB5454B4423BA3723B4685 D2A8A1D813F7C61736736EB7B96DAE195723CFFECC72C97500666639734483CFBB279CD1A216 8A513DC21BB32D2564BF98CE6C1FA294F70E202447BC45B578B797381F2E64BDE3A50000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmmi12 %!FontType1-1.0: cmmi12 1.1/12-Nov-94 %%VMusage: 31226 31798 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmmi12) readonly def /FamilyName (cmmi12) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmmi12 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 128 /kappa1 put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-33 -218 1026 750] readonly def /UniqueID 4701523 def /StrokeWidth 0 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmmi8 %!FontType1-1.0: cmmi8 1.1/12-Nov-94 %%VMusage: 32271 32844 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmmi8) readonly def /FamilyName (cmmi8) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmmi8 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 128 /kappa1 put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-27 -218 1110 750] readonly def /UniqueID 4701520 def /StrokeWidth 0 def currentdict end currentfile eexec 9B9C0887DB83FB1ECD8335B0BB39CEF0AF64F656FC6E5C230CC9D3A7346AAC7A06BD9A40393C A15D3773A21E06B9B4254D3050E90726BBB09120935A8D448CDBC799D90205A21291254FA633 CC0C2ED88781EF21A5D45B72445C284FA44F8F39CF566976075A92E9E2947A6FC93D879C29EB 26809ECB4409CBC8666526CCED9265DE661A2F8B0A16BE45A9DA17EEAF89B0E023DE2B0373DC AB997D60B7D7DC1B4F7E650A2A9F13CA0543F3B080AE33D65F2802B8C6F032ED6F1F9EC1620E 5D5EAC22D6B7A2C527D76FC035BF25A0299FD60A72AE7E0C045FEB341646AF23C463ED2A5DC1 919C6FF451154C679ACFF08A89CE1DF2E249199FA1BDA096E03FDF154361C76BDFA752EDBDDD 62DB3AB84D264814C5B276341F7271CE9A705401E402E7F19B0B7B41E9E3406776BC4A9F4979 979660367C62DF3C0EFA8E8452C767E501CC39413F7CAF0897FDEE7EA241F331303744E4E050 B9D6C012D2B489895CCEB25415A5C2FE967F55DBA9A8700F2AD92A1AB0A6E9C4819F10E12F31 A096DE2FEFFCCAAC9CF6B6129A01EEA2CB8D4502D2539490E4C8F10E3A785A36EF693E8ADBB8 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmsy8 %!FontType1-1.0: cmsy8 1.1/12-Nov-94 %%VMusage: 27115 29406 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmsy8) readonly def /FamilyName (cmsy8) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /CapHeight 683 def /XHeight 431 def /Ascender 694 def /Descender -194 def end readonly def /FontName /cmsy8 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 160 /space put dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-30 -271 1192 782] readonly def /UniqueID 4701532 def /StrokeWidth 0 def currentdict end currentfile eexec 9B9C0887DB83FB1ECD8335B0BB39CEF0AF64F656FC6E5C230CC9D3A7346AAC7A06BD9A40393C A15D3773A21E06B9B4254D3050E90726BBB09120935A8D448CDBC799D90205A21291254FA633 CC0C2ED88781EF21A5D45B72445C284FA44F8F39CF566976075A92E9E2947A6FC93D879C29EB 26809ECB4409CBC8666526CCED9265DE661A2F8B0A16BE45A9DA17EEAF89B0E023DE2B0373DC AB997D60B7D7DC1B4F7E650A2A9F13CA0543F3B080AE33D65F2802B8C6F032ED6F1F9EC1620E 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmmi6 %!FontType1-1.0: cmmi6 1.1/12-Nov-94 %%VMusage: 31418 31988 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmmi6) readonly def /FamilyName (cmmi6) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmmi6 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 128 /kappa1 put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 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177C7E0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmsy6 %!FontType1-1.0: cmsy6 1.1/12-Nov-94 %%VMusage: 27478 29768 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmsy6) readonly def /FamilyName (cmsy6) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /Descender -194 def /CapHeight 683 def /XHeight 431 def /Ascender 694 def end readonly def /FontName /cmsy6 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 160 /space put dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-4 -271 1335 790] readonly def /UniqueID 4701530 def /StrokeWidth 0 def currentdict end currentfile eexec 9B9C0887DB83FB1ECD8335B0BB39CEF0AF64F656FC6E5C230CC9D3A7346AAC7A06BD9A40393C A15D3773A21E06B9B4254D3050E90726BBB09120935A8D448CDBC799D90205A21291254FA633 CC0C2ED88781EF21A5D45B72445C284FA44F8F39CF566976075A92E9E2947A6FC93D879C29EB 26809ECB4409CBC8666526CCED9265DE661A2F8B0A16BE45A9DA17EEAF89B0E023DE2B0373DC AB997D60B7D7DC1B4F7E650A2A9F13CA0543F3B080AE33D65F2802B8C6F032ED6F1F9EC1620E 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cleartomark %%EndFont %%BeginFont: cmmi10 %!FontType1-1.0: cmmi10 1.1/12-Nov-94 %%VMusage: 31220 31787 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmmi10) readonly def /FamilyName (cmmi10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 694 def /Descender -194 def /CapHeight 683 def end readonly def /FontName /cmmi10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 128 /kappa1 put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-34 -219 1048 750] readonly def /UniqueID 4701522 def /StrokeWidth 0 def currentdict end currentfile eexec 9B9C0887DB83FB1ECD8335B0BB39CEF0AF64F656FC6E5C230CC9D3A7346AAC7A06BD9A40393C A15D3773A21E06B9B4254D3050E90726BBB09120935A8D448CDBC799D90205A21291254FA633 CC0C2ED88781EF21A5D45B72445C284FA44F8F39CF566976075A92E9E2947A6FC93D879C29EB 26809ECB4409CBC8666526CCED9265DE661A2F8B0A16BE45A9DA17EEAF89B0E023DE2B0373DC AB997D60B7D7DC1B4F7E650A2A9F13CA0543F3B080AE33D65F2802B8C5214D5B60D0D2956096 89997B955E43A4D99ACFF7BF2FBF2CC4D0F447C1D54FB3D42B7E474427B6DD55A05308D91C3D 62C926F6E6AE84EA2B0934AEBC9F41694A4EC2254C5863450A4027A97B85652572E1D5DE078B 35903D1BA14975A9E9F46055D92A37D4BAF7EACDE7D30966108AB2A62A087A6B19AFF720617E 3E1535BB6773B757456D9426F1C8DCF2E62F77177D2EC038AB246F094F752CECC5681A5EED71 75102D01900546C622D45E50E53F598D88860C45F4AB927CF5C377C6A488CE40284AF020ED8B 72B394E1D20CDA8E1D8A54D96C6806D2401E2DFF956BEF41FB60329FD117AA133C9100E22C6C 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmtt12 %!FontType1-1.0: cmtt12 1.1/12-Nov-94 %%VMusage: 27047 27319 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmtt12) readonly def /FamilyName (cmtt12) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /Ascender 611 def /Descender -222 def /CapHeight 611 def end readonly def /FontName /cmtt12 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /space 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 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /arrowup put dup 175 /arrowdown put dup 176 /quotesingle put dup 177 /exclamdown put dup 178 /questiondown put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /visiblespace put dup 196 /dieresis put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-3 -234 526 697] readonly def 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b(ase)34 b(when)g Fl(M)46 b Fu(is)35 b(3-dimensional)d(also)i(de\014ne) g Fl(e)2381 4369 y Fx(3)2456 4354 y Fu(such)h(that)1269 4622 y Fl(\013)q Fv(\()p Fl(e)1415 4637 y Fx(3)1454 4622 y Fv(\))28 b(=)f(\()p Fl(\030)1704 4637 y Fx(+)1763 4622 y Fl(;)17 b(\033)t Fv(\()p Fl(\013)q Fv(\))p Fl(\030)2048 4637 y Fx(+)2106 4622 y Fv(\))88 4890 y Fu(for)34 b(al)5 b(l)35 b Fl(\013)q Fu(.)44 b(Then)34 b Fl(e)819 4905 y Fx(1)859 4890 y Fl(;)17 b(e)948 4905 y Fx(2)987 4890 y Fl(;)g(e)1076 4905 y Fx(3)1150 4890 y Fu(is)35 b(a)g(p)-5 b(ositively)34 b(oriente)-5 b(d)34 b(orthonormal)g(b)-5 b(asis.)1682 5251 y Fv(6)p eop %%Page: 7 7 7 6 bop 88 548 a Fu(Pr)-5 b(o)g(of.)48 b Fv(W)-8 b(e)24 b(treat)g(the)h(3-dimensional)f(case,)j(the)d(2-dimensional)h(case)g (is)g(similar.)41 b(W)-8 b(e)88 716 y(pro)m(v)m(e)33 b(instead)g(that)g(there)g(is)f(an)h(orthonormal)f(basis)h(of)f(forms)g Fl(e)2601 680 y Fx(1)2641 716 y Fl(;)17 b(e)2730 680 y Fx(2)2769 716 y Fl(;)g(e)2858 680 y Fx(3)2930 716 y Fv(for)32 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1235 y Fx(3)2584 1220 y Fv(that)f(the)h(matrices)88 1388 y(of)f Fl(\033)t Fv(\()p Fl(e)343 1352 y Fx(1)382 1388 y Fv(\),)i Fl(\033)t Fv(\()p Fl(e)625 1352 y Fx(2)664 1388 y Fv(\),)g Fl(\033)t Fv(\()p Fl(e)907 1352 y Fx(3)946 1388 y Fv(\),)f(in)g(the)h(basis)f Fl(\030)1617 1403 y Fi(\006)1711 1388 y Fv(are)f(the)i(standard)f(P)m (auli)g(matrices)h(\(note)88 1556 y(that)c(\()p Fl(\030)380 1571 y Fx(+)439 1556 y Fl(;)17 b(\033)t Fv(\()p Fl(\013)q Fv(\))p Fl(\030)724 1571 y Fx(+)781 1556 y Fv(\))28 b(=)g Fq(\000)p Fv(\()p Fl(\030)1109 1571 y Fi(\000)1168 1556 y Fl(;)17 b(\033)t Fv(\()p Fl(\013)q Fv(\))p Fl(\030)1453 1571 y Fi(\000)1511 1556 y Fv(\))32 b(since)i Fl(\033)t Fv(\()p Fl(\013)q Fv(\))e(is)h(traceless\))895 1649 y Fo(\022)985 1711 y Fv(0)97 b(1)985 1868 y(1)g(0)1196 1649 y Fo(\023)1286 1790 y Fl(;)1427 1649 y Fo(\022)1517 1711 y Fv(0)h Fq(\000)p Fl(i)1525 1868 y(i)137 b Fv(0)1791 1649 y Fo(\023)1881 1790 y Fl(;)2022 1649 y Fo(\022)2112 1711 y Fv(1)f(0)2112 1868 y(0)98 b Fq(\000)p Fv(1)2401 1649 y Fo(\023)2491 1790 y Fl(:)88 2018 y Fv(The)39 b(orthonormalit)m (y)f(and)g(p)s(ositivit)m(y)i(follo)m(ws)e(from)g(simple)i(matrix)e (calculations.)p 3255 2186 4 66 v 3259 2123 59 4 v 3259 2186 V 3317 2186 4 66 v 88 2392 a Fw(8.)50 b(DEFINITION)35 b(\()p Fl(S)6 b(pin)1204 2356 y Fk(c)1276 2392 y Fw(Connection\).)49 b Fv(A)26 b(connection)g Fq(r)f Fv(on)g(a)g(spinor)h(bun-)88 2560 y(dle)k(\011)g(is)h(said)f(to)g(b)s(e)g(a)g Fl(S)6 b(pin)1174 2524 y Fk(c)1239 2560 y Fv(connection)32 b(if)e(for)f(all)h (tangen)m(t)h(v)m(ectors)g Fl(X)36 b Fq(2)28 b Fl(T)3040 2575 y Fi(\003)3079 2560 y Fl(M)41 b Fv(w)m(e)88 2728 y(ha)m(v)m(e)136 2934 y(\(i.\))j Fl(X)8 b Fv(\()p Fl(\030)d(;)17 b(\021)t Fv(\))26 b(=)h(\()p Fq(r)869 2949 y Fk(X)937 2934 y Fl(\030)5 b(;)17 b(\021)t Fv(\))k(+)h(\()p Fl(\030)5 b(;)17 b Fq(r)1451 2949 y Fk(X)1517 2934 y Fl(\021)t Fv(\))32 b(for)g(all)h(sections)h Fl(\030)5 b(;)17 b(\021)35 b Fv(in)e(\011.)136 3170 y(\(ii.\))55 b([)p Fq(r)458 3185 y Fk(X)525 3170 y Fl(;)17 b(\033)t Fv(\()p Fl(\013)q Fv(\)])33 b(=)h Fl(\033)t Fv(\()p Fq(r)1117 3185 y Fk(X)1185 3170 y Fl(\013)q Fv(\))h(for)h(all)g(one-forms)g Fl(\013)h Fv(on)f Fl(M)10 b Fv(.)56 b(Here)36 b Fq(r)2805 3185 y Fk(X)2873 3170 y Fl(\013)h Fv(refers)g(to)331 3338 y(the)c(Levi-Civita)h(connection)g(acting)f(on)f(one-forms.)88 3545 y Fw(9.)50 b(PR)m(OPOSITION)35 b(\(Lo)s(cal)j(expression)h(for)e Fq(r)2203 3560 y Fk(X)2271 3545 y Fw(\).)48 b Fu(L)-5 b(et)51 b Fl(\030)2621 3560 y Fi(\006)2730 3545 y Fu(b)-5 b(e)50 b(a)g(lo)-5 b(c)g(al)50 b(or-)88 3713 y(thonormal)29 b(b)-5 b(asis)30 b(of)g(spinor)f(\014elds)h(on)g(a)g(3-dimensional)e (manifold)h(and)h(let)g Fl(e)3029 3728 y Fx(1)3069 3713 y Fl(;)17 b(e)3158 3728 y Fx(2)3197 3713 y Fl(;)g(e)3286 3728 y Fx(3)88 3881 y Fu(b)-5 b(e)34 b(the)h(orthonormal)f(b)-5 b(asis)34 b(de\014ne)-5 b(d)34 b(in)g(Pr)-5 b(op.)35 b(7.)45 b(Then)34 b(for)g(al)5 b(l)35 b(ve)-5 b(ctors)34 b Fl(X)88 3960 y Fo(0)88 4135 y(B)88 4199 y(@)191 4033 y Fv(\()p Fl(\030)272 4048 y Fx(+)331 4033 y Fl(;)17 b Fq(r)458 4048 y Fk(X)525 4033 y Fl(\030)568 4048 y Fx(+)627 4033 y Fv(\))99 b(\()p Fl(\030)845 4048 y Fx(+)904 4033 y Fl(;)17 b Fq(r)1031 4048 y Fk(X)1098 4033 y Fl(\030)1141 4048 y Fi(\000)1200 4033 y Fv(\))191 4346 y(\()p Fl(\030)272 4361 y Fi(\000)331 4346 y Fl(;)g Fq(r)458 4361 y Fk(X)525 4346 y Fl(\030)568 4361 y Fx(+)627 4346 y Fv(\))99 b(\()p Fl(\030)845 4361 y Fi(\000)904 4346 y Fl(;)17 b Fq(r)1031 4361 y Fk(X)1098 4346 y Fl(\030)1141 4361 y Fi(\000)1200 4346 y Fv(\))1254 3960 y Fo(1)1254 4135 y(C)1254 4199 y(A)171 4868 y Fv(=)82 b Fl(i)372 4828 y Fx(1)p 372 4845 36 4 v 372 4902 a(2)435 4637 y Fo(0)435 4813 y(B)435 4877 y(@)873 4711 y Fv(\()p Fl(e)956 4726 y Fx(1)996 4711 y Fl(;)17 b Fq(r)1123 4726 y Fk(X)1190 4711 y Fl(e)1235 4726 y Fx(2)1275 4711 y Fv(\))434 b Fq(\000)p Fv(\()p Fl(e)1907 4726 y Fx(3)1948 4711 y Fl(;)17 b Fq(r)2075 4726 y Fk(X)2142 4711 y Fl(e)2187 4726 y Fx(2)2226 4711 y Fv(\))22 b Fq(\000)h Fl(i)p Fv(\()p Fl(e)2502 4726 y Fx(3)2542 4711 y Fl(;)17 b Fq(r)2669 4726 y Fk(X)2736 4711 y Fl(e)2781 4726 y Fx(1)2821 4711 y Fv(\))538 5024 y Fq(\000)p Fv(\()p Fl(e)698 5039 y Fx(3)738 5024 y Fl(;)g Fq(r)865 5039 y Fk(X)933 5024 y Fl(e)978 5039 y Fx(2)1017 5024 y Fv(\))22 b(+)g Fl(i)p Fv(\()p Fl(e)1291 5039 y Fx(3)1331 5024 y Fl(;)17 b Fq(r)1458 5039 y Fk(X)1525 5024 y Fl(e)1570 5039 y Fx(1)1610 5024 y Fv(\))397 b Fq(\000)p Fv(\()p Fl(e)2205 5039 y Fx(1)2245 5024 y Fl(;)17 b Fq(r)2372 5039 y Fk(X)2439 5024 y Fl(e)2484 5039 y Fx(2)2523 5024 y Fv(\))2875 4637 y Fo(1)2875 4813 y(C)2875 4877 y(A)2984 4868 y Fq(\000)23 b Fl(i\013)q Fv(\()p Fl(X)8 b Fv(\))p Fl(I)g(;)1682 5251 y Fv(7)p eop %%Page: 8 8 8 7 bop 88 548 a Fu(wher)-5 b(e)34 b Fl(\013)h Fu(is)g(the)g(r)-5 b(e)g(al)34 b(lo)-5 b(c)g(al)35 b(one-form)e(given)h(by)892 772 y Fl(\013)q Fv(\()p Fl(X)8 b Fv(\))27 b(=)g Fl(i)1293 732 y Fx(1)p 1293 749 36 4 v 1293 806 a(2)1339 772 y Fv(\()p Fl(\030)1420 787 y Fx(+)1478 772 y Fl(;)17 b Fq(r)1605 787 y Fk(X)1673 772 y Fl(\030)1716 787 y Fx(+)1774 772 y Fv(\))22 b(+)g Fl(i)1975 732 y Fx(1)p 1975 749 V 1975 806 a(2)2021 772 y Fv(\()p Fl(\030)2102 787 y Fi(\000)2160 772 y Fl(;)17 b Fq(r)2287 787 y Fk(X)2354 772 y Fl(\030)2397 787 y Fi(\000)2456 772 y Fv(\))p Fl(:)234 995 y Fu(The)27 b(same)g(formulas)g(ar)-5 b(e)28 b(true)h(in)e(the)h (2-dimensional)e(c)-5 b(ase)27 b(if)g(we)h(simply)f(r)-5 b(eplac)g(e)88 1163 y Fl(e)133 1178 y Fx(3)207 1163 y Fu(by)35 b Fv(0)g Fu(everywher)-5 b(e.)88 1377 y(Pr)g(o)g(of.)48 b Fv(Let)41 b Fl(e)621 1341 y Fx(1)661 1377 y Fl(;)17 b(e)750 1341 y Fx(2)789 1377 y Fl(;)g(e)878 1341 y Fx(3)959 1377 y Fv(b)s(e)42 b(the)f(dual)h(basis)g(to)f Fl(e)1924 1392 y Fx(1)1964 1377 y Fl(;)17 b(e)2053 1392 y Fx(2)2092 1377 y Fl(;)g(e)2181 1392 y Fx(3)2221 1377 y Fv(.)69 b(Using)42 b(that)f Fl(X)8 b Fv(\()p Fl(!)t Fv(\()p Fl(e)3095 1392 y Fk(j)3131 1377 y Fv(\)\))43 b(=)88 1545 y Fq(r)171 1560 y Fk(X)238 1545 y Fl(!)t Fv(\()p Fl(e)386 1560 y Fk(j)422 1545 y Fv(\))22 b(+)g Fl(!)t Fv(\()p Fq(r)766 1560 y Fk(X)833 1545 y Fl(e)878 1560 y Fk(j)914 1545 y Fv(\))33 b(for)f(an)m(y)h(one-form)f Fl(!)k Fv(w)m(e)e(\014nd,)f (with)g Fl(!)e Fv(=)d Fl(e)2653 1509 y Fk(j)2689 1545 y Fv(,)33 b(that)822 1769 y(\()p Fl(e)905 1784 y Fk(j)942 1769 y Fl(;)17 b Fq(r)1069 1784 y Fk(X)1136 1769 y Fl(e)1181 1784 y Fx(1)1220 1769 y Fv(\))23 b(+)f Fl(i)p Fv(\()p Fl(e)1495 1784 y Fk(j)1531 1769 y Fl(;)17 b Fq(r)1658 1784 y Fk(X)1726 1769 y Fl(e)1771 1784 y Fx(2)1810 1769 y Fv(\))28 b(=)f Fl(e)2024 1727 y Fk(j)2061 1769 y Fv(\()p Fq(r)2182 1784 y Fk(X)2249 1769 y Fl(e)2294 1784 y Fx(1)2334 1769 y Fv(\))22 b(+)g Fl(ie)2570 1727 y Fk(j)2607 1769 y Fv(\()p Fq(r)2728 1784 y Fk(X)2796 1769 y Fl(e)2841 1784 y Fx(2)2880 1769 y Fv(\))905 1961 y(=)83 b Fl(X)8 b Fv(\()p Fl(\030)1234 1976 y Fi(\000)1292 1961 y Fl(;)17 b(\033)t Fv(\()p Fl(e)1478 1920 y Fk(j)1515 1961 y Fv(\))p Fl(\030)1596 1976 y Fx(+)1654 1961 y Fv(\))22 b Fq(\000)h Fv(\()p Fl(\030)1895 1976 y Fi(\000)1954 1961 y Fl(;)17 b(\033)t Fv(\()p Fq(r)2178 1976 y Fk(X)2245 1961 y Fl(e)2290 1920 y Fk(j)2326 1961 y Fv(\))p Fl(\030)2407 1976 y Fx(+)2466 1961 y Fv(\))905 2154 y(=)83 b(\()p Fq(r)1185 2169 y Fk(X)1252 2154 y Fl(\030)1295 2169 y Fi(\000)1354 2154 y Fl(;)17 b(\033)t Fv(\()p Fl(e)1540 2113 y Fk(j)1576 2154 y Fv(\))p Fl(\030)1657 2169 y Fx(+)1716 2154 y Fv(\))22 b(+)g(\()p Fl(\030)1955 2169 y Fi(\000)2014 2154 y Fl(;)17 b(\033)t Fv(\()p Fl(e)2200 2113 y Fk(j)2236 2154 y Fv(\))p Fq(r)2357 2169 y Fk(X)2424 2154 y Fl(\030)2467 2169 y Fx(+)2526 2154 y Fv(\))p Fl(:)88 2378 y Fv(Since)33 b Fl(\033)t Fv(\()p Fl(e)484 2342 y Fx(3)524 2378 y Fv(\))p Fl(\030)605 2393 y Fi(\006)691 2378 y Fv(=)28 b Fq(\006)p Fl(\030)915 2393 y Fi(\006)1006 2378 y Fv(\(see)34 b(the)f(pro)s(of)f (of)g(Prop.)h(7\))f(w)m(e)i(\014nd)225 2602 y(\()p Fl(e)308 2617 y Fx(3)348 2602 y Fl(;)17 b Fq(r)475 2617 y Fk(X)542 2602 y Fl(e)587 2617 y Fx(1)627 2602 y Fv(\))22 b(+)g Fl(i)p Fv(\()p Fl(e)901 2617 y Fx(3)941 2602 y Fl(;)17 b Fq(r)1068 2617 y Fk(X)1135 2602 y Fl(e)1180 2617 y Fx(2)1219 2602 y Fv(\))28 b(=)g(\()p Fq(r)1510 2617 y Fk(X)1577 2602 y Fl(\030)1620 2617 y Fi(\000)1679 2602 y Fl(;)17 b(\030)1766 2617 y Fx(+)1824 2602 y Fv(\))22 b Fq(\000)h Fv(\()p Fl(\030)2065 2617 y Fi(\000)2123 2602 y Fl(;)17 b Fq(r)2250 2617 y Fk(X)2317 2602 y Fl(\030)2360 2617 y Fx(+)2419 2602 y Fv(\))28 b(=)f Fq(\000)p Fv(2\()p Fl(\030)2795 2617 y Fi(\000)2854 2602 y Fl(;)17 b Fq(r)2981 2617 y Fk(X)3048 2602 y Fl(\030)3091 2617 y Fx(+)3150 2602 y Fv(\))88 2826 y(where)43 b(w)m(e)h(ha)m(v)m(e)g(also)e(used)i (that)e(0)j(=)f Fl(X)8 b Fv(\()p Fl(\030)1811 2841 y Fi(\000)1870 2826 y Fl(;)17 b(\030)1957 2841 y Fx(+)2015 2826 y Fv(\))45 b(=)f(\()p Fq(r)2339 2841 y Fk(X)2406 2826 y Fl(\030)2449 2841 y Fi(\000)2508 2826 y Fl(;)17 b(\030)2595 2841 y Fx(+)2653 2826 y Fv(\))29 b(+)g(\()p Fl(\030)2906 2841 y Fi(\000)2965 2826 y Fl(;)17 b Fq(r)3092 2841 y Fk(X)3159 2826 y Fl(\030)3202 2841 y Fx(+)3260 2826 y Fv(\).)88 2994 y(This)33 b(also)g(giv)m(es)478 3217 y(\()p Fl(\030)559 3232 y Fx(+)617 3217 y Fl(;)17 b Fq(r)744 3232 y Fk(X)811 3217 y Fl(\030)854 3232 y Fi(\000)913 3217 y Fv(\))28 b(=)f Fq(\000)p 1159 3131 474 4 v Fv(\()p Fl(\030)1240 3232 y Fi(\000)1299 3217 y Fl(;)17 b Fq(r)1426 3232 y Fk(X)1493 3217 y Fl(\030)1536 3232 y Fx(+)1595 3217 y Fv(\))28 b(=)1774 3178 y Fx(1)p 1774 3194 36 4 v 1774 3252 a(2)1819 3217 y Fv(\()p Fl(e)1902 3232 y Fx(3)1942 3217 y Fl(;)17 b Fq(r)2069 3232 y Fk(X)2136 3217 y Fl(e)2181 3232 y Fx(1)2221 3217 y Fv(\))22 b Fq(\000)g Fl(i)2423 3178 y Fx(1)p 2423 3194 V 2423 3252 a(2)2469 3217 y Fv(\()p Fl(e)2552 3232 y Fx(3)2591 3217 y Fl(;)17 b Fq(r)2718 3232 y Fk(X)2786 3217 y Fl(e)2831 3232 y Fx(2)2870 3217 y Fv(\))p Fl(:)88 3441 y Fv(Using)33 b Fl(\033)t Fv(\()p Fl(e)504 3405 y Fx(1)543 3441 y Fv(\))p Fl(\030)624 3456 y Fi(\006)711 3441 y Fv(=)27 b Fl(\030)857 3456 y Fi(\007)948 3441 y Fv(w)m(e)34 b(obtain)234 3665 y(\()p Fl(\030)315 3680 y Fx(+)373 3665 y Fl(;)17 b Fq(r)500 3680 y Fk(X)567 3665 y Fl(\030)610 3680 y 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Fv(\))32 b(=)h(\()p Fl(\021)3090 4499 y Fx(2)3129 4484 y Fl(;)17 b(\021)3221 4499 y Fx(1)3260 4484 y Fv(\).)p 3255 4652 4 66 v 3259 4589 59 4 v 3259 4652 V 3317 4652 4 66 v 1658 5251 a(11)p eop %%Page: 12 12 12 11 bop 88 548 a Fr(3)161 b(The)35 b(Dirac)h(and)f(Laplace)h(op)t (erators)f(on)g(Spinors)88 815 y Fv(Giv)m(en)28 b(a)f Fl(S)6 b(pin)647 778 y Fk(c)710 815 y Fv(connection)29 b(on)e(a)h(Spinor)g(bundle)h(\011)e(w)m(e)i(ma)m(y)f(de\014ne)h(the)f (\014rst)g(order)88 983 y(Dirac)k(op)s(erator)g(and)g(the)h(second)h (order)f(Laplace)g(op)s(erator)f(on)g(spinor)i(sections.)88 1234 y Fw(16.)50 b(DEFINITION)35 b(\(The)j(Dirac)f(op)s(erator\).)49 b Fv(The)23 b(Dirac)e(op)s(erator)g Fq(D)30 b Fv(:)e(\000\(\011\))g Fq(!)88 1402 y Fv(\000\(\011\))k(is)h(giv)m(en)h(b)m(y)1260 1570 y Fq(D)s Fl(\030)e Fv(:=)c Fq(\000)p Fl(i)1673 1475 y Fo(X)1729 1685 y Fk(j)1834 1570 y Fl(\033)t Fv(\()p Fl(e)1976 1528 y Fk(j)2012 1570 y Fv(\))p Fq(r)2133 1585 y Fk(e)2166 1595 y Fh(j)2203 1570 y Fl(\030)88 1850 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y Fh(j)2641 4368 y Fl(X)r(;)17 b(e)2813 4383 y Fk(j)2850 4368 y Fv(\))p Fl(:)43 b Fv(W)-8 b(e)43 b(then)88 4536 y(obtain)469 4803 y Fq(D)549 4762 y Fi(\003)616 4803 y Fv(=)27 b Fq(\000)p Fl(i)846 4709 y Fo(X)903 4919 y Fk(j)1007 4803 y Fl(\033)t Fv(\()p Fl(e)1149 4762 y Fk(j)1186 4803 y Fv(\))p Fq(r)1307 4818 y Fk(e)1340 4828 y Fh(j)1399 4803 y Fq(\000)22 b Fl(i)1548 4709 y Fo(X)1604 4919 y Fk(j)1709 4803 y Fl(\033)t Fv(\()p Fq(r)1889 4818 y Fk(e)1922 4828 y Fh(j)1958 4803 y Fl(e)2003 4762 y Fk(j)2040 4803 y Fv(\))g Fq(\000)g Fl(i)2249 4709 y Fo(X)2305 4919 y Fk(j)2410 4803 y Fv(div)q(\()p Fl(e)2626 4818 y Fk(j)2662 4803 y Fv(\))p Fl(\033)t Fv(\()p Fl(e)2842 4762 y Fk(j)2879 4803 y Fv(\))p Fl(:)1658 5251 y Fv(12)p eop %%Page: 13 13 13 12 bop 88 548 a Fv(Note)32 b(that)98 705 y Fo(X)154 915 y Fk(j)259 800 y Fq(r)342 815 y Fk(e)375 825 y Fh(j)411 800 y Fl(e)456 759 y Fk(j)521 800 y Fv(=)624 705 y Fo(X)680 915 y Fk(j)785 705 y Fo(X)837 917 y Fk(k)929 800 y Fv(\()p Fq(r)1050 815 y 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4339 y Fq(f)p Fl(k)25 b Fv(+)d Fl(c)g Fq(\000)1709 4300 y Fx(1)p 1709 4316 36 4 v 1709 4374 a(2)1755 4339 y Fq(g)p Fl(;)162 b Fu(if)34 b Fl(m)28 b(>)g(k)1312 4496 y Fq(;)p Fl(;)605 b Fu(if)34 b Fl(m)28 b Fv(=)g Fl(k)1312 4652 y Fq(f\000)p Fl(k)d Fq(\000)e Fl(c)f Fq(\000)1789 4613 y Fx(1)p 1789 4629 V 1789 4687 a(2)1834 4652 y Fq(g)p Fl(;)83 b Fu(if)34 b Fl(m)28 b(<)g(k)j(:)234 4834 y Fu(\(ii.\))60 b(The)39 b(multiplicity)h(of)g(an)g(eigenvalue)f (of)g Fq(D)2123 4849 y Fk(M)2242 4834 y Fu(is)h(e)-5 b(qual)40 b(to)g(the)g(numb)-5 b(er)40 b(of)88 5002 y(ways)f(it)h(c)-5 b(an)39 b(b)-5 b(e)40 b(written)g(as)f Fq(\006)1286 4917 y Fo(p)p 1386 4917 549 4 v 85 x Fl(\025)1443 4973 y Fx(2)1505 5002 y Fv(+)22 b(\()p Fl(k)j Fv(+)d Fl(c)p Fv(\))1895 4973 y Fx(2)1960 5002 y Fq(\000)2073 4963 y Fx(1)p 2073 4979 36 4 v 2073 5036 a(2)2158 5002 y Fu(with)40 b Fl(k)g Fq(2)d Fp(Z)j Fu(and)f Fl(\025)e Fq(2)g Fv(\006)3136 5017 y Fx(+)3195 5002 y Fv(\()p Fl(k)s Fv(\))1658 5251 y(32)p eop %%Page: 33 33 33 32 bop 88 548 a Fu(c)-5 b(ounte)g(d)44 b(with)h(multiplicity)f(or)h (as)f(an)h(element)f(in)g Fl(S)2175 563 y Fk(k)2262 548 y Fu(c)-5 b(ounte)g(d)45 b(with)f(multiplicity)88 716 y Fq(j)p Fl(m)22 b Fq(\000)g Fl(k)s Fq(j)p Fu(.)234 884 y(\(iii.\))50 b(The)37 b(eigensp)-5 b(ac)g(e)35 b(of)i Fq(D)1355 899 y Fk(M)1470 884 y Fu(with)g(eigenvalue)f(in)h Fl(S)2339 899 y Fk(k)2418 884 y Fu(c)-5 b(ontains)37 b(spinors)f(with)88 1052 y(de\014nite)e(spin)g(value)h(sgn)o Fv(\()p Fl(m)22 b Fq(\000)h Fl(k)s Fv(\))p Fu(,)35 b(\(i.e,)f(the)h (eigenvalue)f(of)g Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fu(\).)88 1271 y(36.)44 b(REMARK.)50 b Fv(F)-8 b(rom)50 b(the)i(assumption)g Fl(\014)1834 1286 y Fk(M)1972 1271 y Fv(=)59 b Fl(h)35 b Fq(\003)18 b Fl(\027)58 b Fv(it)51 b(follo)m(ws)g(that)g Fl(\014)3111 1286 y Fk(M)3250 1271 y Fv(=)88 1439 y Fl(\036)146 1403 y Fi(\003)185 1439 y Fv(\()p Fl(g)31 b(v)t(ol)428 1454 y Fk(N)495 1439 y Fv(\))h(for)g(some)h(function)h Fl(g)c Fq(2)f Fl(C)1590 1403 y Fi(1)1664 1439 y Fv(\()p Fl(N)10 b Fv(\))33 b(and)g(that)1092 1603 y(1)p 1005 1647 223 4 v 1005 1738 a(\(2)p Fl(\031)t Fv(\))1189 1710 y Fx(2)1254 1535 y Fo(Z)1310 1760 y Fk(M)1405 1670 y Fl(\014)1460 1685 y Fk(M)1562 1670 y Fq(^)22 b Fl(\027)34 b Fv(=)1875 1603 y(1)p 1846 1647 108 4 v 1846 1738 a(2)p Fl(\031)1980 1535 y Fo(Z)2035 1760 y Fk(N)2119 1670 y Fl(g)d(v)t(ol)2324 1685 y Fk(N)2391 1670 y Fl(:)734 b Fv(\(21\))88 1901 y(T)-8 b(o)33 b(see)i(this,)f(w)m(e)h(compute)f(0) 29 b(=)g Fl(d\014)1439 1916 y Fk(M)1547 1901 y Fv(=)g Fl(dh)23 b Fq(^)g(\003)p Fl(\027)35 b Fv(=)30 b Fl(n)p Fv(\()p Fl(h)p Fv(\))f Fl(v)t(ol)2456 1916 y Fk(M)2568 1901 y Fv(using)34 b(that)f Fl(d)23 b Fq(\003)6 b Fl(\027)36 b Fv(=)88 2069 y Fl(d)139 1958 y Fo(\020)198 2069 y Fl(\036)256 2033 y Fi(\003)295 2069 y Fv(\()p Fl(v)t(ol)460 2084 y Fk(N)527 2069 y Fv(\))565 1958 y Fo(\021)676 2069 y Fv(=)52 b(0)47 b(from)f(\(16\).)86 b(Hence)48 b Fl(h)f Fv(is)g(constan)m(t)h(along)e(the)i(Hopf)e(\014b)s(ers,)88 2237 y Fl(h)39 b Fv(=)g Fl(g)30 b Fq(\016)d Fl(\036)p Fv(,)41 b(and)e(therefore)i Fl(\014)1246 2252 y Fk(M)1364 2237 y Fv(is)f(a)f(pullbac)m(k)i(of)e(a)g(\(not)g(necessarily)j (exact\))f(t)m(w)m(o)88 2405 y(form)c(on)h Fl(N)10 b Fv(.)61 b(The)39 b(in)m(tegral)f(relation)g(\(21\))g(follo)m(ws)g(from) g(using)h(F)-8 b(ubini's)39 b(theorem)88 2573 y(in)e(lo)s(cal)g(patc)m (hes)h(and)f(the)h(fact)f(that)f(the)i(length)f(of)g(the)g(\014b)s(ers) h(is)g(2)p Fl(\031)t Fv(.)56 b(From)37 b(this)88 2741 y(relation)32 b(\(19\))g(is)h(straigh)m(t)g(forw)m(ard.)44 b(\(Recall)33 b(that)2049 2661 y Fo(R)2132 2741 y Fl(v)t(ol)2259 2756 y Fk(N)2354 2741 y Fv(=)27 b Fl(\031)t Fv(\).)234 2935 y(The)c(eigen)m(v)-5 b(alues)23 b(in)f Fl(S)1080 2950 y Fk(k)1145 2935 y Fv(corresp)s(ond)g(to)g(the)g(zero)g(eigen)m(v) -5 b(alues)24 b(of)d(the)h(2-dimensional)88 3103 y(op)s(erator)33 b Fq(D)559 3118 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))735 3103 y Fv(.)48 b(F)-8 b(or)34 b(these)h(w)m(e)g(ha)m(v)m(e)h(used)f (the)g(follo)m(wing)f(theorem)h(of)f(Aharono)m(v)88 3271 y(and)41 b(Casher)h([A)m(C)q(])f(to)f(sa)m(y)i(exactly)h(what)e(the)h (m)m(ultiplicit)m(y)h(is)f(and)f(exactly)h(what)88 3439 y(the)d(spin)h(is)g(of)f(the)h(eigenfunctions.)65 b(F)-8 b(or)39 b(completeness)j(w)m(e)e(giv)m(e)g(a)f(pro)s(of)g(of)g(this)88 3607 y(theorem)33 b(in)g(the)g(App)s(endix.)88 3826 y Fw(37.)50 b(THEOREM)36 b(\(Aharono)m(v)h(Casher)h(theorem)g(on)g Fp(S)2491 3790 y Fx(2)2531 3826 y Fw(\).)48 b Fu(L)-5 b(et)49 b Fq(r)g Fu(b)-5 b(e)49 b(a)g(c)-5 b(o-)88 3994 y(variant)41 b(derivative)g(on)h(the)g(spinor)f(bund)5 b(le)41 b Fv(\011)1903 4009 y Fk(n)1992 3994 y Fu(on)g Fp(S)2193 3958 y Fx(2)2275 3994 y Fu(with)h(Chern)f(numb)-5 b(er)42 b Fl(n)f Fv(=)119 4123 y Fx(1)p 98 4139 79 4 v 98 4196 a(2)p Fk(\031)202 4082 y Fo(R)285 4162 y Fl(\014)6 b Fu(,)50 b(wher)-5 b(e)46 b Fl(\014)52 b Fu(is)47 b(the)f(c)-5 b(orr)g(esp)g(onding)46 b(2-form.)79 b(L)-5 b(et)47 b Fq(D)j Fu(b)-5 b(e)46 b(the)h(c)-5 b(orr)g(esp)g(onding)88 4330 y(Dir)g(ac)33 b(op)-5 b(er)g(ator.)45 b(Then)34 b(the)g(dimension)f(of)i(the)f(sp)-5 b(ac)g(e)34 b(of)h(harmonic)e (spinors)h(ker)f Fq(D)88 4498 y Fu(is)27 b Fq(j)p Fl(n)p Fq(j)p Fu(.)42 b(Mor)-5 b(e)g(over,)28 b(ker)33 b Fq(D)e(\032)d(f)p Fl(\021)j Fq(j)c Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fl(\021)32 b Fv(=)27 b Fl(\021)t Fq(g)g Fu(if)g Fl(n)h(>)g Fv(0)f Fu(and)g(ker)33 b Fq(D)d(\032)f(f)p Fl(\021)i Fq(j)c Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fl(\021)32 b Fv(=)88 4666 y Fq(\000)p Fl(\021)t Fq(g)j Fu(if)f Fl(n)29 b(<)e Fv(0)p Fu(,)35 b(wher)-5 b(e)35 b(as)f(b)-5 b(efor)g(e)35 b(we)f(have)g(written)i Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))28 b(=)g Fq(\000)p Fl(i\033)t Fv(\()p Fl(f)2672 4630 y Fx(1)2712 4666 y Fv(\))p Fl(\033)t Fv(\()p Fl(f)2906 4630 y Fx(2)2945 4666 y Fv(\))35 b Fu(for)f(any)88 4834 y(p)-5 b(ositively)35 b(oriente)-5 b(d)35 b(\(lo)-5 b(c)g(al\))34 b(orthonormal)h(b)-5 b(asis)35 b Fl(f)2042 4798 y Fx(1)2081 4834 y Fl(;)17 b(f)2184 4798 y Fx(2)2259 4834 y Fu(of)35 b(one-forms.)46 b Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))36 b Fu(is)f(in-)88 5002 y(dep)-5 b(endent)33 b(of)i(the)g(choic)-5 b(e)34 b(of)g(one-form)g(b)-5 b(asis.)1658 5251 y Fv(33)p eop %%Page: 34 34 34 33 bop 88 548 a Fu(38.)44 b(REMARK.)50 b Fv(With)22 b(the)g(notation)g(of)f(Theorem)j(35)d(w)m(e)i(see)g(immediately)h (that)e(for)88 716 y(the)j(v)m(ery)h(sp)s(ecial)g(case)f(of)f(the)i (eigen)m(v)-5 b(alue)26 b(in)f Fl(S)1850 731 y Fx(0)1914 716 y Fv(w)m(e)g(can)g(sa)m(y)h(that)f(the)g(m)m(ultiplicit)m(y)i(is)88 884 y(precisely)i Fq(j)p Fl(m)p Fq(j)p Fv(.)41 b(If)27 b Fq(j)p Fl(c)p Fq(j)g Fl(<)g Fv(1)p Fl(=)p Fv(2)f(w)m(e)i(cannot)f(sa) m(y)h(this)g(for)e(the)h(eigen)m(v)-5 b(alues)29 b(in)e Fl(S)2955 899 y Fk(k)3024 884 y Fv(for)g Fl(k)k Fq(6)p Fv(=)88 1052 y(0)23 b(b)s(ecause)i(it)e(is)i(p)s(ossible)g(that)e (these)i(could)f(also)g(b)s(e)f(written)i(as)f Fq(\006)2573 967 y Fo(p)p 2673 967 572 4 v 85 x Fl(\025)2730 1023 y Fx(2)2791 1052 y Fv(+)e(\()p Fl(k)2981 1023 y Fi(0)3027 1052 y Fv(+)g Fl(c)p Fv(\))3205 1023 y Fx(2)3248 1052 y Fq(\000)98 1181 y Fx(1)p 98 1197 36 4 v 98 1254 a(2)176 1220 y Fv(for)34 b(some)g Fl(k)626 1184 y Fi(0)679 1220 y Fq(6)p Fv(=)c Fl(k)37 b Fv(and)c Fl(\025)d Fq(6)p Fv(=)f(0.)47 b(The)35 b(case)g(when)f Fl(c)c Fv(=)2231 1181 y Fx(1)p 2231 1197 V 2231 1254 a(2)2309 1220 y Fv(is)35 b(of)e(most)h(in)m (terest)h(to)f(us)88 1388 y(and)e(w)m(e)i(form)m(ulate)f(it)f(as)h(a)f (theorem.)88 1618 y Fw(39.)50 b(THEOREM)36 b(\(Dimension)i(of)g(the)f (space)h(of)g(harmonic)h(spinors)f(on)g Fp(S)3339 1582 y Fx(3)3378 1618 y Fw(\).)88 1786 y Fu(With)32 b(the)g(notation)g(of)g (The)-5 b(or)g(em)31 b(35)h(we)g(c)-5 b(onsider)31 b(the)h(c)-5 b(ase)31 b(when)h Fl(c)27 b Fv(=)h(1)p Fl(=)p Fv(2)p Fu(.)43 b(In)31 b(this)88 1954 y(c)-5 b(ase)39 b(the)i(eigenvalue)e(0)h (has)g(multiplicity)g Fl(m)h Fu(if)f Fl(m)e(>)g Fv(0)i Fu(and)g Fq(\000)p Fl(m)27 b Fq(\000)f Fv(1)40 b Fu(if)g Fl(m)f(<)e Fq(\000)p Fv(1)88 2122 y Fu(otherwise)e Fv(0)h Fu(is)g(not)g(an)g(eigenvalue.)48 b(The)36 b(eigenvalue)f Fq(\000)p Fv(1)h Fu(has)g(multiplicity)h Fq(\000)p Fl(m)g Fu(if)88 2290 y Fl(m)28 b(<)f Fv(0)35 b Fu(and)f Fl(m)23 b Fv(+)f(1)34 b Fu(if)h Fl(m)28 b(>)f Fq(\000)p Fv(1)p Fu(.)88 2520 y(Pr)-5 b(o)g(of.)48 b Fv(This)28 b(is)f(a)f(simple)i (consequence)h(of)d(Theorem)i(35.)41 b(The)27 b(eigen)m(v)-5 b(alue)28 b(in)f Fl(S)3118 2535 y Fx(0)3184 2520 y Fv(has)88 2688 y(m)m(ultiplicit)m(y)36 b(precisely)g Fq(j)p Fl(m)p Fq(j)d Fv(and)h(the)g(eigen)m(v)-5 b(alue)36 b(in)e Fl(S)2189 2703 y Fi(\000)p Fx(1)2317 2688 y Fv(has)g(eigen)m(v)-5 b(alue)35 b(precisely)88 2856 y Fq(j)p Fl(m)22 b Fv(+)g(1)p Fq(j)p Fv(.)43 b(The)33 b(theorem)g(no)m(w)h(follo)m(ws)f(from)f (considering)i(the)f(di\013eren)m(t)h(cases.)p 3255 2856 4 66 v 3259 2794 59 4 v 3259 2856 V 3317 2856 4 66 v 88 3092 a Fu(40.)44 b(REMARK.)50 b Fv(Th)m(us)29 b(for)d(an)m(y)j (natural)e(n)m(um)m(b)s(er)i(w)m(e)f(can)g(construct)h(a)e(Dirac)g(op)s (er-)88 3260 y(ator)g(on)i Fp(S)476 3224 y Fx(3)544 3260 y Fv(suc)m(h)g(that)f(the)h(dimension)h(of)e(the)h(k)m(ernel)h(is)f (that)f(giv)m(en)h(n)m(um)m(b)s(er.)44 b(Using)88 3428 y(the)d(conformal)f(in)m(v)-5 b(ariance)42 b(of)e(the)h(dimension)h(of) f(the)g(k)m(ernel)h(one)f(ma)m(y)g(arriv)m(e)h(at)88 3596 y(a)f(construction)h(on)f Fp(R)961 3560 y Fx(3)1042 3596 y Fv(whic)m(h)i(is)f(summarized)h(b)s(elo)m(w.)71 b(One)42 b(has)f(to)g(discuss)j(the)88 3764 y(b)s(eha)m(vior)33 b(at)f(in\014nit)m(y)-8 b(,)34 b(but)f(this)g(is)g(not)g(v)m(ery)h (di\016cult.)88 3994 y Fw(41.)50 b(THEOREM)36 b(\(Zero)h(mo)s(des)i(on) e Fp(R)1782 3958 y Fx(3)1822 3994 y Fw(\).)48 b Fu(L)-5 b(et)37 b Fl(\034)43 b Fv(:)32 b Fp(R)2331 3958 y Fx(3)2402 3994 y Fq(!)f Fl(M)j Fq(n)24 b(f)p Fl(p)p Fq(g)36 b Fu(b)-5 b(e)37 b(the)g(in-)88 4162 y(verse)j(of)h(the)g(ster)-5 b(e)g(o)g(gr)g(aphic)40 b(pr)-5 b(oje)g(ction)40 b(onto)h Fp(R)1975 4126 y Fx(3)2055 4162 y Fu(fr)-5 b(om)41 b(the)g(spher)-5 b(e)40 b Fl(M)50 b Fv(=)39 b(\()p Fp(S)3118 4126 y Fx(3)3158 4162 y Fl(;)17 b(g)3249 4177 y Fx(3)3287 4162 y Fv(\))88 4330 y Fu(with)33 b(a)f(p)-5 b(oint)33 b(r)-5 b(emove)g(d.)43 b(L)-5 b(et)34 b Fl(\014)f Fv(=)28 b Fl(\034)1456 4294 y Fi(\003)1495 4330 y Fv(\()p Fl(h)19 b Fq(\003)r Fl(\027)6 b Fv(\))27 b(=)h(\()p Fl(h)18 b Fq(\016)g Fl(\034)11 b Fv(\))p Fl(\034)2207 4294 y Fi(\003)2248 4330 y Fv(\()p Fq(\003)p Fl(\027)6 b Fv(\))33 b Fu(b)-5 b(e)33 b(the)g(pul)5 b(lb)-5 b(ack)32 b(of)h(an)88 4498 y(arbitr)-5 b(ary)39 b(close)-5 b(d)38 b(two)i(form)e Fl(h)26 b Fq(\003)9 b Fl(\027)45 b Fu(on)39 b Fp(S)1646 4462 y Fx(3)1686 4498 y Fu(.)57 b([In)39 b(other)g(wor)-5 b(ds,)39 b Fl(\014)j Fv(=)2708 4472 y Fo(e)2710 4498 y Fl(h)34 b(\034)2853 4462 y Fi(\003)2893 4498 y Fv(\()p Fq(\003)p Fl(\027)6 b Fv(\))39 b Fu(is)g(an)88 4666 y(arbitr)-5 b(ary)28 b(close)-5 b(d)26 b(two)i(form)f(on)g Fp(R)1371 4630 y Fx(3)1438 4666 y Fu(which)g(is)g(p)-5 b(ar)g(al)5 b(lel)27 b(with)g Fl(\034)2392 4630 y Fi(\003)2432 4666 y Fv(\()p Fq(\003)p Fl(\027)6 b Fv(\))28 b Fu(and)f(the)h(function)85 4808 y Fo(e)88 4834 y Fl(h)e Fq(\016)g Fl(\034)299 4798 y Fi(\000)p Fx(1)434 4834 y Fu(extends)40 b(to)h(a)f(r)-5 b(e)g(gular)40 b(function)h Fl(h)f Fu(on)g Fp(S)2014 4798 y Fx(3)2054 4834 y Fu(.])61 b(Then)40 b Fl(h)e Fv(=)g Fl(g)30 b Fq(\016)c Fl(\036)40 b Fu(with)g(some)88 5002 y(function)34 b Fl(g)d Fq(2)d Fl(C)720 4966 y Fi(1)795 5002 y Fv(\()p Fp(S)888 4966 y Fx(2)927 5002 y Fv(\))35 b Fu(and)f Fv(\()p Fl(\034)1280 4966 y Fi(\000)p Fx(1)1375 5002 y Fv(\))1413 4966 y Fi(\003)1452 5002 y Fl(\014)41 b Fu(extends)34 b(to)h(a)f(two)h(form)f Fl(\014)2567 5017 y Fk(M)2674 5002 y Fv(=)27 b(\()p Fl(g)e Fq(\016)d Fl(\036)p Fv(\))g Fq(\003)5 b Fl(\027)41 b Fu(on)1658 5251 y Fv(34)p eop %%Page: 35 35 35 34 bop 88 548 a Fl(M)10 b Fu(.)52 b(L)-5 b(et)38 b Fq(D)522 565 y Fn(R)570 546 y Fm(3)646 548 y Fu(b)-5 b(e)36 b(the)i(Dir)-5 b(ac)36 b(op)-5 b(er)g(ator)37 b(on)g Fp(R)1803 512 y Fx(3)1880 548 y Fu(with)g(magnetic)f(two)h(form) g Fl(\014)43 b Fu(and)36 b(let)88 716 y Fq(D)165 731 y Fk(M)278 716 y Fu(b)-5 b(e)35 b(the)g(Dir)-5 b(ac)34 b(op)-5 b(er)g(ator)34 b(on)h Fl(M)46 b Fu(with)34 b(magnetic)g(two)h (form)f Fl(\014)2591 731 y Fk(M)2670 716 y Fu(.)45 b(Then)1113 971 y(dim)34 b(Ker)28 b Fq(D)i Fv(=)e Fu(dim)34 b(Ker)27 b Fq(D)2166 986 y Fk(M)2273 971 y Fl(;)88 1227 y Fu(and)34 b(dim)g(Ker)28 b Fq(D)737 1242 y Fk(M)850 1227 y Fu(is)35 b(given)f(in)h(The)-5 b(or)g(em)33 b(39.)88 1467 y(Pr)-5 b(o)g(of.)48 b Fv(From)38 b(Remark)h(36,)h(w)m(e)g(easily)g(see)f(that) g(an)m(y)g(t)m(w)m(o)g(form)g(of)f(the)h(form)f Fl(\014)44 b Fv(=)88 1635 y Fl(\034)141 1599 y Fi(\003)180 1635 y Fv(\()p Fl(h)37 b Fq(\003)19 b Fl(\027)6 b Fv(\))31 b(is)h(closed)g(if)f(and)h(only)f(if)g Fl(h)d Fv(=)g Fl(g)22 b Fq(\016)e Fl(\036)30 b Fv(with)i(some)g(function)g Fl(g)i Fv(on)d Fp(S)2993 1599 y Fx(2)3033 1635 y Fv(.)43 b(Then)88 1803 y Fl(\014)143 1818 y Fk(M)254 1803 y Fv(de\014ned)34 b(as)f(\()p Fl(g)25 b Fq(\016)d Fl(\036)p Fv(\))g Fq(\003)6 b Fl(\027)38 b Fv(on)33 b Fl(M)43 b Fv(coincides)35 b(with)e(\()p Fl(\034)2151 1767 y Fi(\000)p Fx(1)2246 1803 y Fv(\))2284 1767 y Fi(\003)2323 1803 y Fl(\014)38 b Fv(on)33 b Fl(M)f Fq(n)22 b(f)p Fl(p)p Fq(g)p Fv(.)234 1971 y(Next,)28 b(w)m(e)e(recall)h(that)e(the)h(magnetic)g(t)m(w)m(o)h(form)e (determines)j(the)e(Dirac)f(op)s(erator)88 2139 y(on)f Fp(S)270 2103 y Fx(3)333 2139 y Fv(and)g Fp(R)586 2103 y Fx(3)650 2139 y Fv(up)g(to)g(bundle)h(isomorphism)h(and)e(gauge)g (transformation,)h(hence)h Fq(D)3239 2156 y Fn(R)3287 2137 y Fm(3)88 2307 y Fv(and)34 b Fq(D)356 2322 y Fk(M)469 2307 y Fv(in)h(the)g(theorem)g(are)f(w)m(ell)i(de\014ned.)51 b(They)35 b(can)g(b)s(e)g(considered)h(acting)e(on)88 2475 y(the)f(trivial)g(bundle)g(\011)941 2492 y Fn(R)989 2473 y Fm(3)1056 2475 y Fv(=)27 b Fp(R)1231 2439 y Fx(3)1293 2475 y Fq(\002)22 b Fp(C)1464 2439 y Fx(2)1536 2475 y Fv(and)33 b(\011)1802 2490 y Fk(M)1908 2475 y Fv(=)28 b Fl(M)33 b Fq(\002)22 b Fp(C)2310 2439 y Fx(2)2350 2475 y Fv(,)32 b(resp)s(ectiv)m(ely)-8 b(.)234 2643 y(The)43 b(stereographic)h(pro)5 b(jection)43 b(isometrically)i(iden)m(ti\014es) f(\()p Fl(M)c Fq(n)29 b(f)p Fl(p)p Fq(g)p Fl(;)17 b(g)3017 2658 y Fx(3)3055 2643 y Fv(\))43 b(with)88 2811 y(\()p Fp(R)198 2775 y Fx(3)237 2811 y Fl(;)17 b Fv(\012)351 2775 y Fx(2)390 2811 y Fl(ds)487 2775 y Fx(2)527 2811 y Fv(\),)53 b(where)e(\012\()p Fl(x)p Fv(\))56 b(:=)g(\(1)34 b(+)f Fl(x)1645 2775 y Fx(2)1685 2811 y Fv(\))1723 2775 y Fi(\000)p Fx(1)1817 2811 y Fv(.)94 b(F)-8 b(or)48 b(an)m(y)i (normalized)h(spinor)e Fl(\030)61 b Fq(2)88 2979 y Fv(Ker)53 b Fq(D)375 2994 y Fk(M)454 2979 y Fv(,)e Fq(D)609 2994 y Fk(M)688 2979 y Fl(\030)58 b Fv(=)53 b(0,)e(w)m(e)e(ha)m(v)m(e)g Fq(D)1521 2996 y Fn(R)1569 2977 y Fm(3)1607 2979 y Fv(\(\012)p Fl(\030)5 b Fv(\))53 b(=)g(0)48 b(b)m(y)g(Theorem)h(23.)88 b(By)48 b(elliptic)88 3147 y(regularit)m(y)33 b(of)f Fq(D)720 3162 y Fk(M)799 3147 y Fv(,)h Fl(\030)j Fv(is)e(smo)s(oth,)e (in)h(particular)g Fq(h)p Fl(\030)5 b(;)17 b(\030)5 b Fq(i)30 b Fv(is)k(b)s(ounded.)44 b(Hence)645 3267 y Fo(Z)700 3493 y Fn(R)748 3474 y Fm(3)787 3403 y Fq(h)p Fv(\012)p Fl(\030)5 b(;)17 b Fv(\012)p Fl(\030)5 b Fq(i)p Fv(\()p Fl(v)t(ol)1310 3419 y Fn(R)1358 3400 y Fm(3)1395 3403 y Fv(\))27 b Fq(\024)i Fv(max)1619 3465 y Fk(M)1747 3403 y Fq(h)p Fl(\030)5 b(;)17 b(\030)5 b Fq(i)1982 3267 y Fo(Z)2035 3493 y Fn(R)2083 3474 y Fm(3)2138 3403 y Fv(\012)2208 3362 y Fx(2)2248 3403 y Fv(\()p Fl(v)t(ol)2413 3419 y Fn(R)2461 3400 y Fm(3)2500 3403 y Fv(\))27 b Fl(<)h Fq(1)88 3658 y Fv(hence)34 b(\012)p Fl(\030)e Fq(2)c Fv(Ker)g Fq(D)860 3675 y Fn(R)908 3656 y Fm(3)947 3658 y Fv(.)234 3826 y(Con)m(v)m(ersely)-8 b(,)36 b(if)d Fl( )g Fq(2)c Fv(Ker)g Fq(D)1291 3843 y Fn(R)1339 3824 y Fm(3)1377 3826 y Fv(,)34 b(then)f Fq(D)1737 3841 y Fk(M)1816 3826 y Fl(\030)g Fv(=)c(0)k(a)m(w)m(a)m(y)h(from)f Fl(p)g Fv(with)h Fl(\030)f Fv(:=)c(\012)3137 3790 y Fi(\000)p Fx(1)3231 3826 y Fl( )t Fv(.)88 3994 y(It)k(follo)m(ws)h(from)f Fl( )f Fq(2)d Fl(L)1002 3958 y Fx(2)1042 3994 y Fv(\(\011)1156 4011 y Fn(R)1204 3992 y Fm(3)1243 3994 y Fv(\))k(that)g Fl(\030)g Fq(2)c Fl(L)1763 3958 y Fx(2)1802 3994 y Fv(\(\011)1916 4009 y Fk(M)1995 3994 y Fv(\),)34 b(since)g(\012)29 b Fq(\024)g Fv(1.)45 b(These)35 b(t)m(w)m(o)f(facts)88 4162 y(easily)27 b(imply)h(that)f Fl(\030)k Fv(extends)d(to)e Fl(p)h Fv(and)g Fq(D)1697 4177 y Fk(M)1775 4162 y Fl(\030)33 b Fv(=)27 b(0)f(ev)m(erywhere.)45 b(In)27 b(fact,)h(consider)g(a)88 4330 y(sequence)35 b(of)d(b)s(ounded)i(cuto\013)f(functions)h Fl(\037)1758 4345 y Fk(n)1833 4330 y Fq(2)28 b Fl(C)2004 4294 y Fi(1)2079 4330 y Fv(\()p Fl(M)10 b Fv(\),)33 b Fl(\037)2380 4345 y Fk(n)2427 4330 y Fv(\()p Fl(p)p Fv(\))28 b(=)g(0,)33 b Fl(\037)2854 4345 y Fk(n)2929 4330 y Fq(!)27 b Fv(1)33 b(suc)m(h)88 4498 y(that)g Fq(k)p Fl(d\037)462 4513 y Fk(n)509 4498 y Fq(k)559 4515 y Fk(L)607 4496 y Fm(2)641 4515 y Fx(\(\003)717 4496 y Fm(1)752 4515 y Fx(\()p Fk(M)7 b Fx(\)\))944 4498 y Fq(!)30 b Fv(0.)47 b(Suc)m(h)36 b(sequence)g(exists)g(in)e(three)h(dimensions.)50 b(Clearly)88 4666 y Fq(D)165 4681 y Fk(M)243 4666 y Fv(\()p Fl(\037)342 4681 y Fk(n)389 4666 y Fl(\030)5 b Fv(\))27 b Fq(!)h(D)707 4681 y Fk(M)786 4666 y Fl(\030)j Fv(and)c Fq(D)1121 4681 y Fk(M)1200 4666 y Fv(\()p Fl(\037)1299 4681 y Fk(n)1346 4666 y Fl(\030)5 b Fv(\))27 b(=)h Fq(\000)p Fl(i\033)t Fv(\()p Fl(d\037)1882 4681 y Fk(n)1929 4666 y Fv(\))p Fl(\030)k Fq(!)27 b Fv(0)g(as)h Fl(n)f Fq(!)h Fv(0,)g(hence)g Fq(D)3019 4681 y Fk(M)3098 4666 y Fl(\030)k Fv(=)c(0)88 4834 y(on)j Fl(M)43 b Fv(in)32 b(the)h(sense)h(of)d (distributions.)45 b(It)32 b(then)h(follo)m(ws)f(from)g(elliptic)h (regularit)m(y)g(of)88 5002 y Fq(D)165 5017 y Fk(M)276 5002 y Fv(and)g Fl(\030)f Fq(2)c Fl(L)701 4966 y Fx(2)741 5002 y Fv(\(\011)855 5017 y Fk(M)933 5002 y Fv(\))33 b(that)f Fl(\030)37 b Fv(is)c(smo)s(oth)g(on)f Fl(M)44 b Fv(and)32 b Fl(\030)g Fq(2)d Fv(Ker)e Fq(D)2631 5017 y Fk(M)p 3255 5002 4 66 v 3259 4940 59 4 v 3259 5002 V 3317 5002 4 66 v 1658 5251 a Fv(35)p eop %%Page: 36 36 36 35 bop 88 548 a Fu(42.)44 b(REMARK.)50 b Fv(Finally)39 b(one)h(ma)m(y)g(note)f(that)g(the)h(\014rst)g(zero)f(mo)s(de)h (constructed)88 716 y(b)m(y)c(Loss)f(and)g(Y)-8 b(au)35 b(in)h([L)-8 b(Y])35 b(is)h(the)f(stereographic)h(pro)5 b(jection)36 b(of)f(the)g(one)h(one)f(gets)88 884 y(according)43 b(to)f(Theorem)j(39)d(with)h Fl(m)j Fv(=)f(1)e(and)g Fl(c)i Fv(=)g(1)p Fl(=)p Fv(2.)73 b(This)45 b(corresp)s(onds)f(to)88 1052 y(c)m(ho)s(osing)33 b Fl(g)e Fv(=)c(3)33 b(in)f(Theorem)i(35.)234 1261 y(The)f(pro)s(of)f(of)g(Theorem)i(35)e(is)h(divided)h(in)m(to)f (subsections.)88 1595 y Fe(8.3)135 b(Rotationally)48 b(symmetric)d(eigen)l(basis)88 1827 y Fv(By)33 b(the)h(prop)s(erties)g (of)f Fl(\036)g Fv(and)h(our)f(assumption)i(on)e Fl(\014)2145 1842 y Fk(M)2257 1827 y Fv(the)h(rotation)f(along)g(the)h Fp(S)3286 1791 y Fx(1)88 1995 y Fv(\014b)s(ers)e(is)g(a)f(symmetry)j (of)d(the)h(data,)f(hence)i(the)f(generator)f(of)g(this)i(rotation)d (should)88 2163 y(comm)m(ute)k(with)f Fq(D)807 2178 y Fk(M)886 2163 y Fv(.)88 2414 y Fw(43.)50 b(PR)m(OPOSITION.)d Fu(L)-5 b(et)35 b Fl(n)g Fu(b)-5 b(e)35 b(the)g(ve)-5 b(ctor)34 b(dual)h(to)g Fl(\027)42 b Fu(and)34 b(let)h(us)g(de\014ne) 1249 2682 y Fl(Q)28 b Fv(:=)f Fq(\000)p Fl(i)p Fq(r)1677 2641 y Fk(M)1677 2707 y(n)1779 2682 y Fq(\000)1889 2615 y Fv(1)p 1889 2659 49 4 v 1889 2750 a(2)1948 2682 y Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fl(;)988 b Fv(\(22\))88 2950 y Fu(which)34 b(is)g(symmetric)h(sinc)-5 b(e)34 b(div)e Fl(n)c Fv(=)g(0)p Fu(.)44 b(Then)1451 3217 y Fv([)p Fq(D)1555 3232 y Fk(M)1634 3217 y Fl(;)17 b(Q)p Fv(])28 b(=)f(0)1190 b(\(23\))88 3485 y Fu(\(when)33 b(acting)i(on)f(smo)-5 b(oth)35 b(se)-5 b(ctions\),)34 b(and)g(we)g(also)h(have)f(that)1178 3753 y Fq(fD)1305 3768 y Fk(M)1384 3753 y Fl(;)17 b(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fq(g)27 b Fv(=)h(2)p Fl(Q)22 b Fq(\000)h Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))917 b(\(24\))88 4020 y Fu(\(on)34 b(the)h(domain)e(of)i Fq(D)969 4035 y Fk(M)1048 4020 y Fu(\).)88 4271 y(Pr)-5 b(o)g(of.)48 b Fv(W)-8 b(e)39 b(use)h(the)f(basis)g Fl(u)1217 4286 y Fx(1)1256 4271 y Fl(;)17 b(u)1356 4286 y Fx(2)1395 4271 y Fl(;)g(n)38 b Fv(constructed)j(in)d(Section)i(7,)g(and)f(let)g Fl(u)3050 4235 y Fx(1)3089 4271 y Fl(;)17 b(u)3189 4235 y Fx(2)3227 4271 y Fl(;)g(\027)88 4439 y Fv(b)s(e)29 b(the)g(dual)g(basis.)43 b(Sometimes)31 b(w)m(e)e(use)h(the)g(notation) e Fl(u)2266 4403 y Fx(3)2333 4439 y Fv(=)f Fl(\027)6 b Fv(,)30 b Fl(u)2603 4454 y Fx(3)2670 4439 y Fv(=)d Fl(n)i Fv(for)g(brevit)m(y)-8 b(.)88 4607 y(W)g(e)33 b(also)f(drop)h(the)g(sup)s(erscript)h Fl(M)43 b Fv(from)33 b Fq(r)1800 4571 y Fk(M)1879 4607 y Fv(.)43 b(Then)88 4912 y Fq(f)o(D)214 4927 y Fk(M)293 4912 y Fl(;)17 b(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fq(g)83 b Fv(=)g(\()p Fq(\000)p Fl(i)p Fv(\))1076 4787 y Fx(3)1021 4817 y Fo(X)1032 5027 y Fk(j)t Fx(=1)1182 4831 y Fo(\010)1240 4912 y Fl(\033)t Fv(\()p Fl(u)1393 4871 y Fk(j)1429 4912 y Fv(\))p Fq(r)1550 4927 y Fk(u)1591 4937 y Fh(j)1627 4912 y Fl(;)45 b(\033)t Fv(\()p Fl(u)1852 4871 y Fx(3)1891 4912 y Fv(\))1929 4831 y Fo(\011)1658 5251 y Fv(36)p eop %%Page: 37 37 37 36 bop 659 626 a Fv(=)83 b(\()p Fq(\000)p Fl(i)p Fv(\))1076 501 y Fx(3)1021 531 y Fo(X)1032 741 y Fk(j)t Fx(=1)1182 545 y Fo(\010)1240 626 y Fl(\033)t Fv(\()p Fl(u)1393 584 y Fk(j)1429 626 y Fv(\))p Fl(;)17 b(\033)t Fv(\()p Fl(u)1664 584 y Fx(3)1702 626 y Fv(\))1740 545 y Fo(\011)1815 626 y Fq(r)1898 641 y Fk(u)1939 651 y Fh(j)1998 626 y Fv(+)22 b(\()p Fq(\000)p Fl(i)p Fv(\))2353 501 y Fx(3)2299 531 y Fo(X)2310 741 y Fk(j)t Fx(=1)2460 626 y Fl(\033)t Fv(\()p Fl(u)2613 584 y Fk(j)2649 626 y Fv(\))2704 545 y Fo(\002)2745 626 y Fq(r)2828 641 y Fk(u)2869 651 y Fh(j)2905 626 y Fl(;)17 b(\033)t Fv(\()p Fl(u)3102 584 y Fx(3)3141 626 y Fv(\))3179 545 y Fo(\003)659 974 y Fv(=)83 b Fq(\000)p Fv(2)p Fl(i)p Fq(r)1060 989 y Fk(n)1130 974 y Fq(\000)22 b Fl(i)1334 850 y Fx(3)1279 880 y Fo(X)1290 1090 y Fk(j)t Fx(=1)1440 974 y Fl(\033)t Fv(\()p Fl(u)1593 933 y Fk(j)1629 974 y Fv(\))p Fl(\033)t Fv(\()p Fq(r)1847 989 y Fk(u)1888 999 y Fh(j)1924 974 y Fl(u)1980 933 y Fx(3)2019 974 y Fv(\))28 b(=)f Fq(\000)p Fv(2)p Fl(i)p Fq(r)2430 989 y Fk(n)2500 974 y Fq(\000)c Fv(2)p Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))27 b(=)h(2)p Fl(Q)22 b Fq(\000)h Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fl(:)88 1287 y Fv(In)27 b(the)g(last)g(step)g(w)m(e)h(used)g Fq(\000)p Fl(i\033)t Fv(\()p Fl(u)1363 1250 y Fx(1)1402 1287 y Fv(\))p Fl(\033)t Fv(\()p Fl(u)1593 1250 y Fx(2)1632 1287 y Fv(\))g(=)f Fl(\033)t Fv(\()p Fl(u)1954 1250 y Fx(3)1993 1287 y Fv(\))g(and)g(v)-5 b(arious)27 b Fq(r)2653 1302 y Fk(u)2694 1312 y Fh(i)2724 1287 y Fl(u)2780 1250 y Fk(j)2816 1287 y Fv('s)g(computed)88 1455 y(in)32 b(\(17\).)234 1623 y(T)-8 b(o)33 b(pro)m(v)m(e)g(23,)g(w)m(e)g(\014rst)g(compute)171 1913 y([)p Fq(D)275 1928 y Fk(M)354 1913 y Fl(;)17 b(\033)t Fv(\()p Fl(\027)6 b Fv(\)])83 b(=)g(\()p Fq(\000)p Fl(i)p Fv(\))1114 1789 y Fx(3)1059 1819 y Fo(X)1070 2029 y Fk(j)t Fx(=1)1220 1833 y Fo(\002)1261 1913 y Fl(\033)t Fv(\()p Fl(u)1414 1872 y Fk(j)1450 1913 y Fv(\))p Fq(r)1571 1928 y Fk(u)1612 1938 y Fh(j)1649 1913 y Fl(;)17 b(\033)t Fv(\()p Fl(u)1846 1872 y Fx(3)1885 1913 y Fv(\))1923 1833 y Fo(\003)697 2262 y Fv(=)83 b(\()p Fq(\000)p Fl(i)p Fv(\))1114 2137 y Fx(3)1059 2167 y Fo(X)1070 2377 y Fk(j)t Fx(=1)1220 2181 y Fo(\002)1261 2262 y Fl(\033)t Fv(\()p Fl(u)1414 2221 y Fk(j)1450 2262 y Fv(\))p Fl(;)17 b(\033)t Fv(\()p Fl(u)1685 2221 y Fx(3)1724 2262 y Fv(\))1762 2181 y Fo(\003)1820 2262 y Fq(r)1903 2277 y Fk(u)1944 2287 y Fh(j)2003 2262 y Fv(+)22 b(\()p Fq(\000)p Fl(i)p Fv(\))2358 2137 y Fx(3)2304 2167 y Fo(X)2315 2377 y Fk(j)t Fx(=1)2465 2262 y Fl(\033)t Fv(\()p Fl(u)2618 2221 y Fk(j)2654 2262 y Fv(\))2709 2181 y Fo(\002)2750 2262 y Fq(r)2833 2277 y Fk(u)2874 2287 y Fh(j)2910 2262 y Fl(;)17 b(\033)t Fv(\()p Fl(u)3107 2221 y Fx(3)3146 2262 y Fv(\))3184 2181 y Fo(\003)697 2517 y Fv(=)83 b(2)p Fl(\033)t Fv(\()p Fl(u)1058 2476 y Fx(1)1097 2517 y Fv(\))p Fq(r)1218 2532 y Fk(u)1259 2541 y Fm(2)1320 2517 y Fq(\000)22 b Fv(2)p Fl(\033)t Fv(\()p Fl(u)1621 2476 y Fx(2)1660 2517 y Fv(\))p Fq(r)1781 2532 y Fk(u)1822 2541 y Fm(1)1883 2517 y Fq(\000)g Fv(2)p Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fl(:)88 2771 y Fv(Next,)33 b(w)m(e)h(compute)321 3062 y([)p Fq(D)425 3077 y Fk(M)504 3062 y Fl(;)17 b Fq(r)631 3077 y Fk(n)678 3062 y Fv(])83 b(=)g(\()p Fq(\000)p Fl(i)p Fv(\))1204 2937 y Fx(2)1150 2967 y Fo(X)1161 3177 y Fk(j)t Fx(=1)1311 3062 y Fl(\033)t Fv(\()p Fl(u)1464 3021 y Fk(j)1500 3062 y Fv(\))1555 2981 y Fo(\002)1596 3062 y Fq(r)1679 3077 y Fk(u)1720 3087 y Fh(j)1756 3062 y Fl(;)17 b Fq(r)1883 3077 y Fk(n)1930 2981 y Fo(\003)1993 3062 y Fv(+)22 b Fl(i)2196 2937 y Fx(2)2141 2967 y Fo(X)2152 3177 y Fk(j)t Fx(=1)2302 2981 y Fo(\002)2343 3062 y Fq(r)2426 3077 y Fk(n)2473 3062 y Fl(;)17 b(\033)t Fv(\()p Fl(u)2670 3021 y Fk(j)2706 3062 y Fv(\))2744 2981 y Fo(\003)2802 3062 y Fq(r)2885 3077 y Fk(u)2926 3087 y Fh(j)788 3411 y Fv(=)83 b(\()p Fq(\000)p Fl(i)p 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Fl(R)1342 4234 y Fx(\011)1431 4219 y Fv(in)e(terms)h(of)e(the)g(Riemannian)i (curv)-5 b(ature)31 b Fl(R)g Fv(and)88 4387 y(the)d(magnetic)h(t)m(w)m (o)f(form)g Fl(\014)1124 4402 y Fk(M)1203 4387 y Fv(.)42 b(Then)29 b Fl(R)g Fv(can)g(b)s(e)f(computed)h(from)f Fq(r)2682 4402 y Fk(u)2723 4412 y Fh(i)2753 4387 y Fl(u)2809 4402 y Fk(j)2845 4387 y Fv('s)h(\(17\),)f(and)88 4555 y(the)33 b(result)g(is)376 4809 y Fl(R)q Fv(\()p Fl(u)545 4824 y Fx(1)584 4809 y Fl(;)17 b(u)684 4824 y Fx(3)723 4809 y Fv(\))p Fl(u)817 4824 y Fx(1)884 4809 y Fv(=)27 b Fq(\000)p Fl(u)1120 4824 y Fx(3)1402 4809 y Fl(R)q Fv(\()p Fl(u)1571 4824 y Fx(1)1610 4809 y Fl(;)17 b(u)1710 4824 y Fx(3)1749 4809 y Fv(\))p Fl(u)1843 4824 y Fx(2)1909 4809 y Fv(=)28 b(0)242 b Fl(R)q Fv(\()p Fl(u)2473 4824 y Fx(1)2512 4809 y Fl(;)17 b(u)2612 4824 y Fx(3)2650 4809 y Fv(\))p Fl(u)2744 4824 y Fx(3)2811 4809 y Fv(=)28 b Fl(u)2971 4824 y Fx(1)500 5002 y Fl(R)q Fv(\()p Fl(u)669 5017 y Fx(2)708 5002 y Fl(;)17 b(u)808 5017 y Fx(3)847 5002 y Fv(\))p Fl(u)941 5017 y Fx(1)1007 5002 y Fv(=)28 b(0)180 b Fl(R)q Fv(\()p Fl(u)1509 5017 y Fx(2)1548 5002 y Fl(;)17 b(u)1648 5017 y Fx(3)1687 5002 y Fv(\))p Fl(u)1781 5017 y Fx(2)1847 5002 y Fv(=)28 b Fq(\000)p Fl(u)2084 5017 y Fx(3)2304 5002 y Fl(R)q Fv(\()p Fl(u)2473 5017 y Fx(2)2512 5002 y Fl(;)17 b(u)2612 5017 y Fx(3)2650 5002 y Fv(\))p Fl(u)2744 5017 y Fx(3)2811 5002 y Fv(=)28 b Fl(u)2971 5017 y Fx(2)3010 5002 y Fl(:)1658 5251 y Fv(37)p eop %%Page: 38 38 38 37 bop 88 548 a Fv(By)34 b(the)g(assumption)h(on)e(the)i(magnetic)f (\014eld)g(in)g(Theorem)h(35,)f Fl(\014)2606 563 y Fk(M)2685 548 y Fv(\()p Fl(u)2779 563 y Fk(j)2815 548 y Fl(;)17 b(u)2915 563 y Fx(3)2954 548 y Fv(\))29 b(=)h(0)j(for)88 716 y Fl(j)g Fv(=)28 b(1)p Fl(;)17 b Fv(2.)42 b(W)-8 b(e)33 b(obtain)662 941 y Fl(R)736 956 y Fx(\011)795 941 y Fv(\()p Fl(u)889 956 y Fx(1)928 941 y Fl(;)17 b(u)1028 956 y Fx(3)1067 941 y Fv(\))27 b(=)h Fq(\000)1329 901 y Fk(i)p 1323 918 36 4 v 1323 975 a Fx(2)1369 941 y Fl(\033)t Fv(\()p Fl(u)1522 900 y Fx(2)1561 941 y Fv(\))f Fl(;)212 b(R)1939 956 y Fx(\011)1998 941 y Fv(\()p Fl(u)2092 956 y Fx(2)2131 941 y Fl(;)17 b(u)2231 956 y Fx(3)2270 941 y Fv(\))27 b(=)2455 901 y Fk(i)p 2449 918 V 2449 975 a Fx(2)2494 941 y Fl(\033)t Fv(\()p Fl(u)2647 900 y Fx(1)2686 941 y Fv(\))p Fl(;)88 1165 y Fv(hence)34 b(using)f([)p Fl(u)697 1180 y Fx(1)736 1165 y Fl(;)17 b(u)836 1180 y Fx(3)875 1165 y Fv(])27 b(=)h Fq(\000)p Fv(2)p Fl(u)1215 1180 y Fx(2)1254 1165 y Fv(,)33 b([)p Fl(u)1397 1180 y Fx(2)1436 1165 y Fl(;)17 b(u)1536 1180 y Fx(3)1574 1165 y Fv(])28 b(=)g(2)p Fl(u)1838 1180 y Fx(1)1909 1165 y Fv(w)m(e)34 b(conclude)g(that)387 1433 y(\()p Fq(\000)p Fl(i)p Fv(\))645 1308 y Fx(2)590 1338 y Fo(X)601 1548 y Fk(j)t Fx(=1)751 1433 y Fl(\033)t Fv(\()p Fl(u)904 1391 y Fk(j)940 1433 y Fv(\))995 1352 y Fo(\002)1036 1433 y Fq(r)1119 1448 y Fk(u)1160 1458 y Fh(j)1197 1433 y Fl(;)17 b Fq(r)1324 1448 y Fk(n)1370 1352 y Fo(\003)1439 1433 y Fv(=)28 b(2)p Fl(i\033)t Fv(\()p Fl(u)1778 1391 y Fx(1)1817 1433 y Fv(\))p Fq(r)1938 1448 y Fk(u)1979 1457 y Fm(2)2040 1433 y Fq(\000)22 b Fv(2)p Fl(i\033)t Fv(\()p Fl(u)2374 1391 y Fx(2)2413 1433 y Fv(\))p Fq(r)2534 1448 y Fk(u)2575 1457 y Fm(1)2636 1433 y Fq(\000)h Fl(i\033)t Fv(\()p Fl(u)2922 1391 y Fx(3)2961 1433 y Fv(\))p Fl(:)88 1705 y Fv(Therefore)756 1873 y([)p Fq(D)860 1888 y Fk(M)939 1873 y Fl(;)17 b Fq(r)1066 1888 y Fk(n)1113 1873 y Fv(])28 b(=)f Fl(i\033)t Fv(\()p Fl(u)1457 1832 y Fx(1)1497 1873 y Fv(\))p Fq(r)1618 1888 y Fk(u)1659 1897 y Fm(2)1719 1873 y Fq(\000)c Fl(i\033)t Fv(\()p Fl(u)2005 1832 y Fx(2)2044 1873 y Fv(\))p Fq(r)2165 1888 y Fk(u)2206 1897 y Fm(1)2267 1873 y Fq(\000)f Fl(i\033)t Fv(\()p Fl(u)2552 1832 y Fx(3)2592 1873 y Fv(\))p Fl(:)88 2077 y Fv(and)32 b(com)m(bining)i(this)f(with)g([)q Fq(D)1263 2092 y Fk(M)1341 2077 y Fl(;)17 b(\033)t Fv(\()p Fl(\027)6 b Fv(\)])33 b(computed)h(ab)s(o)m(v)m(e,)f(w)m(e)h(arriv)m(e)f(at)g(\(23\).)p 3255 2077 4 66 v 3259 2014 59 4 v 3259 2077 V 3317 2077 4 66 v 234 2307 a(Since)25 b Fq(D)557 2322 y Fk(M)659 2307 y Fv(has)f(a)g(pure)g(p)s(oin)m(t)g(sp)s(ectrum,)j(\(23\))c (implies)i(that)f(it)g(has)g(an)f(eigen)m(basis)88 2475 y(consisting)48 b(of)f(eigenspinors)i(of)e Fl(Q)p Fv(.)87 b(One)48 b(exp)s(ects)h(that)e(these)h(eigenspinors)h(are)88 2643 y(actually)f(pull-ups)g(of)g(some)g(spinors)h(in)f(an)f (appropriate)h(spinor)g(bundle)h(on)f Fl(N)10 b Fv(.)88 2811 y(This)33 b(is)g(correct)h(after)e(a)g(gauge)h(transformation)f (whic)m(h)i(w)m(e)g(describ)s(e)g(no)m(w.)234 2979 y(F)-8 b(or)26 b(an)m(y)h Fl(k)k Fq(2)d Fp(Z)f Fv(let)g(us)g(\014x)h(a)e (spinor)i(connection)f Fq(r)2164 2943 y Fk(N)2259 2979 y Fv(=)h Fq(r)2446 2943 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))2649 2979 y Fv(with)f(a)g(magnetic)88 3147 y(form)i Fl(\014)370 3162 y Fk(N)437 3147 y Fv(\()p Fl(k)s Fv(\))g(on)g(the)h(spinor)f (bundle)i(\011)1575 3162 y Fk(m)p Fi(\000)p Fk(k)1764 3147 y Fv(with)f(Chern)g(n)m(um)m(b)s(er)h Fl(m)15 b Fq(\000)g Fl(k)33 b Fv(on)c Fl(N)10 b Fv(.)43 b(W)-8 b(e)88 3315 y(iden)m(tify)33 b(\011)518 3330 y Fk(M)628 3315 y Fv(with)g(the)f(lift)g(of)f(\011)1357 3330 y Fk(m)p Fi(\000)p Fk(k)1517 3315 y Fv(.)43 b(Let)32 b Fq(r)1844 3279 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))2064 3315 y Fv(b)s(e)g(the)h (lift)e(of)h Fq(r)2711 3279 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))2919 3315 y Fv(according)88 3483 y(to)g(Prop)s(osition)h(27:)570 3708 y Fq(r)653 3657 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))653 3735 y Fk(X)869 3708 y Fv(:=)27 b Fl(\036)1057 3667 y Fi(\003)1096 3708 y Fv(\()p Fq(r)1217 3667 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))1394 3708 y Fv(\))1432 3723 y Fk(X)1521 3708 y Fq(\000)1631 3640 y Fv(1)p 1631 3685 49 4 v 1631 3776 a(2)1690 3708 y Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fl(\033)t Fv(\()p Fq(r)2059 3723 y Fk(X)2126 3708 y Fl(\027)g Fv(\))23 b(+)2357 3640 y Fl(i)p 2349 3685 V 2349 3776 a Fv(2)2408 3708 y Fl(\027)6 b Fv(\()p Fl(X)i Fv(\))p Fl(\033)t Fv(\()p Fl(\027)e Fv(\))p Fl(;)309 b Fv(\(25\))88 3932 y(where)39 b(w)m(e)g(also)f(used)h(\(16\).)59 b(Let)38 b Fl(\014)1448 3948 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1674 3932 y Fv(:=)e Fl(\036)1871 3896 y Fi(\003)1911 3932 y Fv(\()p Fl(\014)2004 3947 y Fk(N)2071 3932 y Fv(\()p Fl(k)s Fv(\)\))i(b)s(e)g(the)g(magnetic)h(form)f(of)88 4100 y Fq(r)171 4050 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))171 4127 y Fk(X)359 4100 y Fv(.)43 b(Finally)33 b(w)m(e)h(de\014ne)1109 4300 y Fo(e)1095 4325 y Fq(r)1178 4284 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1394 4325 y Fv(:=)28 b Fq(r)1608 4284 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1818 4325 y Fv(+)22 b Fl(i)p Fv(\()p Fl(c)g Fv(+)g Fl(k)s Fv(\))p Fl(\027)q(:)835 b Fv(\(26\))88 4550 y(The)33 b(magnetic)g(t)m(w)m(o)h(form)e(of)1245 4525 y Fo(e)1231 4550 y Fq(r)1314 4514 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1535 4550 y Fv(is)97 4775 y Fl(\014)152 4790 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))362 4775 y Fq(\000)23 b Fv(\()p Fl(c)f Fv(+)g Fl(k)s Fv(\))p Fl(d\027)89 b Fv(=)83 b Fl(\014)1156 4790 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1366 4775 y Fv(+)22 b(2\()p Fl(c)g Fv(+)g Fl(k)s Fv(\)\()p Fq(\003)p Fl(\027)6 b Fv(\))28 b(=)g Fl(\036)2175 4733 y Fi(\003)2214 4664 y Fo(\020)2273 4775 y Fl(\014)2328 4790 y Fk(N)2396 4775 y Fv(\()p Fl(k)s Fv(\))22 b(+)g(2\()p Fl(c)g Fv(+)g Fl(k)s Fv(\)\()p Fl(v)t(ol)3152 4790 y Fk(N)3218 4775 y Fv(\))3256 4664 y Fo(\021)942 4987 y Fv(=)83 b Fl(\036)1159 4946 y Fi(\003)1198 4877 y Fo(\020)1258 4987 y Fl(g)t Fv(\()p Fl(v)t(ol)1474 5002 y Fk(N)1540 4987 y Fv(\))1578 4877 y Fo(\021)1665 4987 y Fv(=)28 b Fl(\014)1824 5002 y Fk(M)1658 5251 y Fv(38)p eop %%Page: 39 39 39 38 bop 88 548 a Fv(b)m(y)31 b(\(18\).)43 b(It)31 b(is)g(no)m(w)h (clear)f(that)1321 523 y Fo(e)1308 548 y Fq(r)1391 512 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1610 548 y Fv(and)f Fq(r)1880 512 y Fk(M)1990 548 y Fv(are)h(gauge)g(equiv)-5 b(alen)m(t)33 b(since)f(their)88 716 y(magnetic)d(\014elds)h(are)e(the) h(same.)43 b(Therefore)30 b(there)f(exists)h(a)e(function)h Fl(f)2801 731 y Fk(k)2872 716 y Fq(2)f Fl(C)3043 680 y Fi(1)3118 716 y Fv(\()p Fl(M)10 b Fv(\),)88 884 y(dep)s(ending)33 b(on)g Fl(k)s Fv(,)g(suc)m(h)h(that)1252 1126 y Fo(e)1238 1152 y Fq(r)1321 1110 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1537 1152 y Fv(=)27 b Fl(e)1685 1110 y Fk(if)1743 1122 y Fh(k)1786 1152 y Fq(r)1869 1110 y Fk(M)1948 1152 y Fl(e)1993 1110 y Fi(\000)p Fk(if)2106 1122 y Fh(k)2148 1152 y Fl(:)88 1419 y Fv(Hence)36 b(the)h(sp)s(ectrum)g(of)e Fq(D)1172 1434 y Fk(M)1286 1419 y Fv(and)1499 1394 y Fo(e)1479 1419 y Fq(D)1556 1435 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1777 1419 y Fv(=)d Fq(\000)p Fl(i\033)t Fv(\()2106 1394 y Fo(e)2092 1419 y Fq(r)2175 1383 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))2364 1419 y Fv(\))k(are)f(the)h(same)h(for)e(an)m(y)88 1587 y Fl(k)30 b Fq(2)e Fp(Z)p Fv(,)33 b(and)g(w)m(e)h(will)f(w)m(ork)g (with)h(the)f(latter)f(op)s(erators.)234 1755 y(Let)1348 1898 y Fo(e)1328 1923 y Fl(Q)1405 1882 y Fx(\()p Fk(k)r Fx(\))1531 1923 y Fv(:=)c Fl(e)1707 1882 y Fk(if)1765 1894 y Fh(k)1807 1923 y Fl(Qe)1929 1882 y Fi(\000)p Fk(if)2042 1894 y Fh(k)88 2145 y Fv(then)33 b(b)m(y)g(unitary)g(transformation)g (w)m(e)g(get)g(from)f(\(23\))g(and)h(\(24\))f(that)1314 2302 y Fo(h)1381 2388 y(e)1361 2413 y Fq(D)1438 2428 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1626 2413 y Fl(;)1716 2388 y Fo(e)1697 2413 y Fl(Q)1774 2372 y Fx(\()p Fk(k)r Fx(\))1872 2302 y Fo(i)1947 2413 y Fv(=)27 b(0)1053 b(\(27\))1044 2589 y Fo(n)1131 2675 y(e)1111 2700 y Fq(D)1188 2716 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1376 2700 y Fl(;)44 b(\033)t Fv(\()p Fl(\027)6 b Fv(\))1636 2589 y Fo(o)1730 2700 y Fv(=)28 b(2)1902 2675 y Fo(e)1883 2700 y Fl(Q)1960 2659 y Fx(\()p Fk(k)r Fx(\))2080 2700 y Fq(\000)22 b Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))784 b(\(28\))88 2922 y(on)44 b(smo)s(oth)h(sections.)82 b(Since)1319 2897 y Fo(e)1298 2922 y Fq(D)1375 2938 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1608 2922 y Fv(has)45 b(compact)h(resolv)m(en)m(t,)k(eac)m (h)45 b(eigenspace)88 3090 y(is)d(\014nite)g(dimensional)h(and)f(b)m(y) g(elliptic)h(regularit)m(y)f(consists)i(of)d(smo)s(oth)h(sections.)88 3258 y(It)37 b(then)h(follo)m(ws)g(from)f(\(27\))g(and)h(\(28\))e(that) i(there)g(exists)h(an)e(eigen)m(basis)j(of)3081 3233 y Fo(e)3060 3258 y Fq(D)3137 3273 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))88 3426 y Fv(consisting)33 b(of)g(eigenspinors)h(of)1328 3401 y Fo(e)1309 3426 y Fl(Q)1386 3390 y Fx(\()p Fk(k)r Fx(\))1484 3426 y Fv(.)88 3677 y Fw(44.)50 b(PR)m(OPOSITION.)d Fu(The)35 b(sp)-5 b(e)g(ctrum)36 b(of)f Fl(Q)h Fu(b)-5 b(elongs)35 b(to)h(the)f(set)h Fp(Z)23 b Fv(+)g Fl(c)p Fu(.)47 b(Mor)-5 b(e-)88 3845 y(over,)34 b(for)h(any)f(inte)-5 b(ger)35 b Fl(k)s Fu(,)g(if)1352 4087 y Fo(e)1333 4113 y Fl(Q)1410 4072 y Fx(\()p Fk(k)r Fx(\))1508 4113 y Fl(\037)27 b Fv(=)h(\()p Fl(k)d Fv(+)d Fl(c)p Fv(\))p Fl(\037;)1072 b Fv(\(29\))88 4380 y Fu(then)43 b(ther)-5 b(e)44 b(exists)f(a)h(se)-5 b(ction)43 b Fl(\030)48 b Fu(of)c(the)g(spinor)f(bund)5 b(le)43 b Fv(\011)2360 4395 y Fk(m)p Fi(\000)p Fk(k)2564 4380 y Fu(on)g Fl(N)55 b Fu(with)43 b(Chern)88 4548 y(numb)-5 b(er)34 b Fl(m)22 b Fq(\000)h Fl(k)38 b Fu(such)d(that)g Fl(\037)28 b Fv(=)f Fl(\030)g Fq(\016)22 b Fl(\036)p Fu(.)88 4799 y(Pr)-5 b(o)g(of.)48 b Fv(Since)40 b Fl(Q)f Fv(and)985 4774 y Fo(e)966 4799 y Fl(Q)1043 4763 y Fx(\()p Fk(k)r Fx(\))1179 4799 y Fv(are)g(unitarily)g(equiv)-5 b(alen)m(t)41 b(for)d(an)m(y)h Fl(k)s Fv(,)i(it)d(is)i(enough)f(to)88 4967 y(compute)d(the)g(sp)s(ectrum)h(of)1219 4942 y Fo(e)1200 4967 y Fl(Q)1277 4931 y Fx(\()p Fk(k)r Fx(\))1375 4967 y Fv(.)52 b(Let)1650 4942 y Fo(e)1631 4967 y Fl(Q)1708 4931 y Fx(\()p Fk(k)r Fx(\))1806 4967 y Fl(\037)33 b Fv(=)f Fl(E)6 b(\037)p Fv(.)53 b(First)35 b(w)m(e)i(compute)3031 4942 y Fo(e)3012 4967 y Fl(Q)3089 4931 y Fx(\()p Fk(k)r Fx(\))3222 4967 y Fv(on)1658 5251 y(39)p eop %%Page: 40 40 40 39 bop 88 549 a Fv(an)m(y)35 b(pull-up)h(spinor)f Fl(\021)28 b Fq(\016)23 b Fl(\036)35 b Fv(with)g Fl(\021)h Fq(2)c Fv(\000\(\011)1731 564 y Fk(m)p Fi(\000)p Fk(k)1891 549 y Fv(\).)50 b(Notice)36 b(that)2547 524 y Fo(e)2528 549 y Fl(Q)2605 513 y Fx(\()p Fk(k)r Fx(\))2735 549 y Fv(=)31 b Fq(\000)p Fl(i)2966 524 y Fo(e)2952 549 y Fq(r)3035 498 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))3035 559 y Fk(n)3248 549 y Fq(\000)98 678 y Fx(1)p 98 694 36 4 v 98 751 a(2)143 717 y Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\),)33 b(hence)371 959 y Fo(e)352 985 y Fl(Q)429 944 y Fx(\()p Fk(k)r Fx(\))527 985 y Fv(\()p Fl(\021)26 b Fq(\016)c Fl(\036)p Fv(\))83 b(=)f Fq(\000)p Fl(i)p Fq(r)1241 944 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1241 1009 y Fk(n)1430 985 y Fv(\()p Fl(\021)26 b Fq(\016)c Fl(\036)p Fv(\))g(+)g(\()p Fl(k)j Fv(+)d Fl(c)p Fv(\)\()p Fl(\021)k Fq(\016)c Fl(\036)p Fv(\))g Fq(\000)2533 917 y Fv(1)p 2533 962 49 4 v 2533 1053 a(2)2592 985 y Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\)\()p Fl(\021)26 b Fq(\016)c Fl(\036)p Fv(\))890 1178 y(=)82 b(\()p Fl(k)26 b Fv(+)c Fl(c)p Fv(\)\()p Fl(\021)j Fq(\016)d Fl(\036)p Fv(\))1532 b(\(30\))88 1445 y(using)33 b(\(25\),)f(\(26\))g(and)g(that) h Fl(\036)1241 1409 y Fi(\003)1280 1445 y Fv(\()p Fq(r)1401 1409 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))1577 1445 y Fv(\))1615 1460 y Fk(n)1662 1445 y Fv(\()p Fl(\021)26 b Fq(\016)c Fl(\036)p Fv(\))27 b(=)h(0.)234 1613 y(Next)38 b(w)m(e)f(c)m(ho)s(ose)h (a)f(p)s(oin)m(t)f Fl(p)f Fq(2)g Fp(S)1522 1577 y Fx(3)1599 1613 y Fv(where)j Fl(\037)f Fv(do)s(es)g(not)g(v)-5 b(anish.)57 b(W)-8 b(e)37 b(c)m(ho)s(ose)g(an)88 1781 y(orthonormal)d(basis)j Fq(f)p Fl(\030)989 1745 y Fx(+)1047 1781 y Fl(;)17 b(\030)1139 1745 y Fi(\000)1197 1781 y Fq(g)35 b Fv(in)h(\011)1475 1796 y Fk(m)p Fi(\000)p Fk(k)1670 1781 y Fv(in)f(a)g(neigh)m(b)s(orho)s (o)s(d)g Fl(V)57 b Fv(around)36 b(the)f(p)s(oin)m(t)88 1949 y Fl(\036)p Fv(\()p Fl(p)p Fv(\))g(and)g(w)m(e)i(pull)f(it)f(up.) 53 b(This)37 b(giv)m(es)g(an)e(orthonormal)g(basis)i Fq(f)p Fl(\030)2639 1913 y Fx(+)2721 1949 y Fq(\016)24 b Fl(\036;)17 b(\030)2945 1913 y Fi(\000)3027 1949 y Fq(\016)24 b Fl(\036)p Fq(g)35 b Fv(in)88 2117 y Fl(\036)146 2081 y Fi(\000)p Fx(1)240 2117 y Fv(\()p Fl(V)21 b Fv(\),)32 b(whic)m(h)h(is)f(a)g(tubular)g(neigh)m(b)s(orho)s(o)s(d)f(of)h(the)g (circle)h(\014b)s(er)f Fl(C)38 b Fv(going)31 b(through)88 2285 y Fl(p)p Fv(.)44 b(In)33 b(this)g(neigh)m(b)s(orho)s(o)s(d)g(w)m (e)g(can)g(write)h(the)f(eigenspinor)h Fl(\037)f Fv(as)g Fl(\037)28 b Fv(=)g Fl(r)2834 2300 y Fx(+)2893 2285 y Fv(\()p Fl(\030)2979 2249 y Fx(+)3060 2285 y Fq(\016)22 b Fl(\036)p Fv(\))g(+)88 2453 y Fl(r)132 2468 y Fi(\000)191 2453 y Fv(\()p Fl(\030)277 2417 y Fi(\000)357 2453 y Fq(\016)g Fl(\036)p Fv(\))32 b(with)h(some)h(functions)f Fl(r)1488 2468 y Fi(\006)1575 2453 y Fq(2)28 b Fl(C)1746 2417 y Fi(1)1821 2453 y Fv(\()p Fl(M)10 b Fv(\).)44 b(Then)601 2696 y Fo(e)582 2721 y Fl(Q)659 2680 y Fx(\()p Fk(k)r Fx(\))757 2721 y Fl(\037)28 b Fv(=)f Fq(\000)p Fl(i)p Fv(\()p Fl(nr)1199 2736 y Fx(+)1259 2721 y Fv(\)\()p Fl(\030)1383 2680 y Fx(+)1464 2721 y Fq(\016)22 b Fl(\036)p Fv(\))f Fq(\000)i Fl(i)p Fv(\()p Fl(nr)1926 2736 y Fi(\000)1985 2721 y Fv(\)\()p Fl(\030)2109 2680 y Fi(\000)2190 2721 y Fq(\016)f Fl(\036)p Fv(\))g(+)g(\()p Fl(k)j Fv(+)d Fl(c)p Fv(\))p Fl(\037)88 2988 y Fv(and)40 b(b)m(y)447 2963 y Fo(e)428 2988 y Fl(Q)505 2952 y Fx(\()p Fk(k)r Fx(\))602 2988 y Fl(\037)h Fv(=)f Fl(E)6 b(\037)40 b Fv(and)g(linear)h(indep)s(endence)h(of)e Fl(\030)2248 3003 y Fi(\006)2334 2988 y Fq(\016)27 b Fl(\036)39 b Fv(w)m(e)i(get)f(that)g Fl(nr)3150 3003 y Fi(\006)3250 2988 y Fv(=)88 3156 y Fl(i)p Fv(\()p Fl(E)28 b Fq(\000)23 b Fv(\()p Fl(k)i Fv(+)d Fl(c)p Fv(\)\))p Fl(r)733 3171 y Fi(\006)792 3156 y Fv(.)43 b(Hence)34 b Fl(r)1196 3171 y Fi(\006)1287 3156 y Fv(m)m(ust)g(b)s(e)f(of)f(the)h(form)1046 3424 y Fl(r)1090 3439 y Fi(\006)1177 3424 y Fv(=)27 b Fl(r)1327 3373 y Fx(\(0\))1324 3446 y Fi(\006)1438 3424 y Fv(exp)17 b([)p Fl(i\022)k Fv(\()o Fl(E)29 b Fq(\000)22 b Fv(\()p Fl(k)j Fv(+)d Fl(c)p Fv(\)\)])17 b Fl(;)88 3692 y Fv(where)25 b Fl(\022)j Fv(is)d(the)g(arclength)g(parameter)g (along)f Fl(C)31 b Fv(in)25 b(the)g(direction)g(of)f Fl(n)h Fv(and)g Fl(n)p Fv(\()p Fl(r)3090 3641 y Fx(\(0\))3087 3713 y Fi(\006)3184 3692 y Fv(\))j(=)88 3860 y(0.)40 b(Since)26 b(the)g(total)e(length)i(of)e Fl(C)32 b Fv(is)26 b(2)p Fl(\031)i Fv(and)e(at)e(least)i(one)f(of)g Fl(r)2390 3875 y Fx(+)2449 3860 y Fv(,)h Fl(r)2546 3875 y Fi(\000)2630 3860 y Fv(is)g(not)e(iden)m(tically)88 4028 y(zero,)33 b(w)m(e)g(see)h(that)e Fl(E)c Fq(\000)23 b Fv(\()p Fl(k)i Fv(+)d Fl(c)p Fv(\))28 b Fq(2)g Fp(Z)p Fv(,)33 b(hence)h Fl(E)g Fq(2)28 b Fp(Z)22 b Fv(+)g Fl(c)p Fv(.)234 4196 y(No)m(w)35 b(the)g(second)h(statemen)m(t)g(in)f(Prop)s(osition)g(44)f (is)i(straigh)m(t)f(forw)m(ard.)50 b(If)34 b Fl(E)k Fv(=)88 4364 y Fl(k)33 b Fv(+)d Fl(c)44 b Fv(is)h(an)g(eigen)m(v)-5 b(alue)46 b(in)e(\(29\),)j(then)e Fl(nr)1809 4379 y Fx(+)1916 4364 y Fv(=)j Fl(nr)2142 4379 y Fi(\000)2249 4364 y Fv(=)g(0,)f(i.e.)80 b Fl(r)2744 4379 y Fi(\006)2847 4364 y Fv(are)45 b(pull-up)88 4532 y(functions,)h Fl(r)592 4547 y Fi(\006)696 4532 y Fv(=)g Fl(r)865 4496 y Fi(\003)862 4556 y(\006)950 4532 y Fq(\016)29 b Fl(\036)42 b Fv(with)i(some)g Fl(r)1664 4496 y Fi(\003)1661 4556 y(\006)1765 4532 y Fq(2)i Fl(C)1954 4496 y Fi(1)2028 4532 y Fv(\()p Fl(N)10 b Fv(\),)46 b(hence)e Fl(\037)i Fv(=)f(\()p Fl(r)2859 4496 y Fi(\003)2856 4556 y Fx(+)2915 4532 y Fl(\030)2958 4547 y Fx(+)3017 4532 y Fv(\))29 b Fq(\016)g Fl(\036)g Fv(+)88 4700 y(\()p Fl(r)173 4664 y Fi(\003)170 4724 y(\000)229 4700 y Fl(\030)272 4715 y Fi(\000)330 4700 y Fv(\))22 b Fq(\016)g Fl(\036)p Fv(,)33 b(and)f(it)h(is)g(the)g(pull-up)g(of)f Fl(\030)g Fv(=)c Fl(r)1806 4664 y Fi(\003)1803 4724 y Fx(+)1862 4700 y Fl(\030)1905 4715 y Fx(+)1985 4700 y Fv(+)22 b Fl(r)2130 4664 y Fi(\003)2127 4724 y(\000)2186 4700 y Fl(\030)2229 4715 y Fi(\000)2288 4700 y Fv(.)p 3255 4700 4 66 v 3259 4637 59 4 v 3259 4700 V 3317 4700 4 66 v 234 4942 a(W)-8 b(e)33 b(summarize)h(our)f(result)1658 5251 y(40)p eop %%Page: 41 41 41 40 bop 88 548 a Fw(45.)50 b(THEOREM.)d Fu(L)-5 b(et)44 b Fq(D)1201 563 y Fk(M)1280 548 y Fl( )j Fv(=)c Fl(e )k Fu(and)c Fl(Q )k Fv(=)c Fl(\026 )t Fu(.)70 b(Then)42 b Fl(\026)h Fv(=)g Fl(k)31 b Fv(+)d Fl(c)44 b Fu(with)88 716 y(some)34 b Fl(k)c Fq(2)e Fp(Z)p Fu(.)46 b(Fix)34 b(this)h Fl(k)s Fu(,)f(let)1286 690 y Fo(e)1269 716 y Fl( )e Fv(:=)c Fl(e)1540 680 y Fk(if)1598 692 y Fh(k)1640 716 y Fl( )t Fu(,)35 b(then)g(by)g(unitarity)882 958 y Fo(e)862 984 y Fq(D)939 999 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1144 957 y Fo(e)1127 984 y Fl( )d Fv(=)27 b Fl(e)1387 957 y Fo(e)1370 984 y Fl( )238 b Fu(and)2044 958 y Fo(e)2025 984 y Fl(Q)2102 942 y Fx(\()p Fk(k)r Fx(\))2217 957 y Fo(e)2200 984 y Fl( )32 b Fv(=)27 b Fl(\026)2474 957 y Fo(e)2457 984 y Fl( )t(:)601 b Fv(\(31\))88 1251 y Fu(Then)361 1225 y Fo(e)345 1251 y Fl( )37 b Fv(=)c Fl(\030)28 b Fq(\016)c Fl(\036)p Fu(,)39 b(with)e(some)g(se)-5 b(ction)37 b Fl(\030)43 b Fu(of)37 b(the)h(spinor)f(bund)5 b(le)37 b Fv(\011)2682 1266 y Fk(m)p Fi(\000)p Fk(k)2880 1251 y Fu(on)g Fl(N)49 b Fu(with)88 1419 y(Chern)34 b(numb)-5 b(er)34 b Fl(m)23 b Fq(\000)f Fl(k)s Fu(.)45 b(Mor)-5 b(e)g(over,)35 b(for)f(any)h(se)-5 b(ction)34 b Fl(\037)h Fu(of)g Fv(\011)2480 1434 y Fk(m)p Fi(\000)p Fk(k)2675 1419 y Fu(we)f(have)469 1662 y Fo(e)449 1687 y Fq(D)526 1702 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))714 1687 y Fv(\()p Fl(\037)22 b Fq(\016)g Fl(\036)p Fv(\))27 b(=)1134 1606 y Fo(\000)1180 1687 y Fq(D)1257 1702 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))1433 1687 y Fl(\037)1494 1606 y Fo(\001)1562 1687 y Fq(\016)22 b Fl(\036)f Fq(\000)1823 1619 y Fv(1)p 1823 1664 49 4 v 1823 1755 a(2)1882 1687 y Fl(\037)h Fq(\016)g Fl(\036)g Fv(+)g(\()p Fl(k)j Fv(+)d Fl(c)p Fv(\))p Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fl(\037)23 b Fq(\016)f Fl(\036)27 b(:)188 b Fv(\(32\))88 1954 y Fu(Pr)-5 b(o)g(of.)48 b Fv(All)39 b(statemen)m(ts)i(ha)m(v)m(e)g(b)s (een)f(pro)m(v)m(en)g(in)g(Prop)s(osition)f(44)g(except)h(the)g (\(32\),)88 2122 y(whic)m(h)34 b(is)f(a)f(straigh)m(t)h(forw)m(ard)g (calculation.)p 3255 2122 4 66 v 3259 2060 59 4 v 3259 2122 V 3317 2122 4 66 v 88 2456 a Fe(8.4)135 b(Pro)t(of)45 b(of)g(Theorem)g(35)88 2688 y Fu(Pr)-5 b(o)g(of.)48 b Fv(Let)22 b Fl(e)g Fv(b)s(e)g(an)f(eigen)m(v)-5 b(alue)24 b(of)d Fq(D)1503 2703 y Fk(M)1603 2688 y Fv(and)h(consider)h(the)f (corresp)s(onding)h(eigenspace,)88 2856 y(whic)m(h)30 b(is)g(\014nite)f(dimensional.)44 b(In)30 b(this)f(subspace)i(w)m(e)f (\014nd)g(a)f(sim)m(ultaneous)i(eigen)m(ba-)88 3024 y(sis)c(of)f Fl(Q)p Fv(,)i(hence)g(w)m(e)g(consider)f(spinors)h Fl( )i Fv(with)e Fq(D)1944 3039 y Fk(M)2022 3024 y Fl( )k Fv(=)c Fl(e )i Fv(and)d Fl(Q )32 b Fv(=)27 b Fl(\026 )k Fv(for)26 b(some)88 3192 y Fl(\026)p Fv(.)50 b(Then)36 b Fl(\026)31 b Fv(=)h Fl(k)27 b Fv(+)c Fl(c)35 b Fv(with)h(some)g Fl(k)e Fq(2)e Fp(Z)k Fv(and)f(\014x)g(this)h Fl(k)s Fv(.)51 b(F)-8 b(ollo)m(wing)35 b(Theorem)h(45,)88 3360 y(let)245 3334 y Fo(e)228 3360 y Fl( )c Fv(:=)c Fl(e)499 3324 y Fk(if)557 3336 y Fh(k)599 3360 y Fl( )37 b Fv(and)905 3334 y Fo(e)888 3360 y Fl( )32 b Fv(=)c Fl(\030)e Fq(\016)c Fl(\036)32 b Fv(with)i(some)f Fl(\030)f Fq(2)c Fv(\011)2030 3375 y Fk(m)p Fi(\000)p Fk(k)2190 3360 y Fv(.)43 b(From)33 b(\(32\))f(w)m(e)h(ha)m(v)m(e)553 3518 y Fo(\020)633 3603 y(e)613 3628 y Fq(D)690 3644 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))900 3628 y Fv(+)1008 3561 y(1)p 1008 3605 49 4 v 1008 3696 a(2)1067 3518 y Fo(\021)1126 3628 y Fv(\()p Fl(\030)27 b Fq(\016)21 b Fl(\036)p Fv(\))28 b(=)1532 3518 y Fo(h)1596 3547 y(\000)1642 3628 y Fq(D)1719 3644 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))1917 3628 y Fv(+)22 b(\()p Fl(k)j Fv(+)d Fl(c)p Fv(\))p Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))2496 3547 y Fo(\001)2559 3628 y Fl(\030)2607 3518 y Fo(i)2675 3628 y Fq(\016)22 b Fl(\036)28 b(:)292 b Fv(\(33\))88 3896 y(Using)33 b(\(32\))f(once)h(more)g(for)f Fl(\037)c Fv(=)1375 3815 y Fo(\000)1421 3896 y Fq(D)1498 3911 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))1696 3896 y Fv(+)22 b(\()p Fl(k)j Fv(+)d Fl(c)p Fv(\))p Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))2275 3815 y Fo(\001)2338 3896 y Fl(\030)37 b Fv(w)m(e)c(obtain)514 4053 y Fo(\020)594 4138 y(e)573 4163 y Fq(D)650 4179 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))860 4163 y Fv(+)968 4096 y(1)p 968 4140 V 968 4232 a(2)1027 4053 y Fo(\021)1087 4075 y Fx(2)1126 4163 y Fv(\()p Fl(\030)27 b Fq(\016)21 b Fl(\036)p Fv(\))28 b(=)1532 4053 y Fo(h)1596 4083 y(\000)1642 4163 y Fq(D)1719 4179 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))1917 4163 y Fv(+)22 b(\()p Fl(k)j Fv(+)d Fl(c)p Fv(\))p Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))2496 4083 y Fo(\001)2542 4105 y Fx(2)2598 4163 y Fl(\030)2646 4053 y Fo(i)2715 4163 y Fq(\016)22 b Fl(\036)27 b(:)88 4431 y Fv(Notice)37 b(that)g Fq(fD)741 4446 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))917 4431 y Fl(;)17 b(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fq(g)35 b Fv(=)f(0,)k(whic)m(h)h(easily)f(follo)m (ws)f(from)g(\(24\))f(and)h(\(30\).)56 b(In)88 4599 y(particular,)32 b(the)h(nonzero)h(sp)s(ectrum)g(of)e Fq(D)1710 4614 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))1918 4599 y Fv(is)h(symmetric)i(and)997 4867 y Fq(D)1077 4825 y Fx(2)1074 4891 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))1250 4867 y Fl(\030)d Fv(=)1429 4756 y Fo(h\020)1535 4867 y Fl(e)23 b Fv(+)1711 4799 y(1)p 1711 4844 V 1711 4935 a(2)1769 4756 y Fo(\021)1829 4778 y Fx(2)1890 4867 y Fq(\000)g Fv(\()p Fl(k)i Fv(+)d Fl(c)p Fv(\))2282 4825 y Fx(2)2322 4756 y Fo(i)2369 4867 y Fl(\030)1658 5251 y Fv(41)p eop %%Page: 42 42 42 41 bop 88 548 a Fv(i.e.)44 b Fl(\030)37 b Fv(is)c(an)f(eigenspinor)i (of)e Fq(D)1270 512 y Fx(2)1267 578 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))1476 548 y Fv(and)h(w)m(e)g(de\014ne)1113 813 y Fl(\025)27 b Fv(:=)1328 656 y Fo(r)p 1427 656 846 4 v 1427 702 a(\020)1487 813 y Fl(e)22 b Fv(+)1662 745 y(1)p 1662 790 49 4 v 1662 881 a(2)1721 702 y Fo(\021)1781 724 y Fx(2)1842 813 y Fq(\000)h Fv(\()p Fl(k)i Fv(+)d Fl(c)p Fv(\))2234 784 y Fx(2)2273 813 y Fl(:)88 1026 y Fv(If)33 b Fl(\025)c(>)g Fv(0,)34 b(then)g(clearly)h Fl(\025)29 b Fq(2)h Fv(\006)1275 1041 y Fx(+)1334 1026 y Fv(\()p Fl(k)s Fv(\))k(\(recall)g(that)f(\006)2079 1041 y Fx(+)2139 1026 y Fv(\()p Fl(k)s Fv(\))g(is)h(the)g(p)s(ositiv)m (e)h(sp)s(ectrum)88 1194 y(of)i Fq(D)281 1210 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))457 1194 y Fv(\).)60 b(The)39 b(m)m(ultiplicit)m (y)h(of)d Fl(e)h Fv(in)h(the)f(subspace)i Fq(f)p Fl( )77 b Fv(:)d Fl(Q )41 b Fv(=)c(\()p Fl(k)29 b Fv(+)c Fl(c)p Fv(\))p Fl( )t Fq(g)38 b Fv(is)88 1362 y(b)s(ounded)33 b(b)m(y)g(the)g(m)m(ultiplicit)m(y)i(of)d Fl(\025)h Fv(in)g(the)g(set)g (\006)2017 1377 y Fx(+)2076 1362 y Fv(\()p Fl(k)s Fv(\).)234 1530 y(If)28 b Fl(\025)f Fv(=)h(0,)h(then)f(b)m(y)h(Theorem)g(37)f(the) g(eigen)m(v)-5 b(alue)29 b(0)f(b)s(elongs)h(to)e(the)i(sp)s(ectrum)g (of)88 1698 y Fq(D)165 1714 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))375 1698 y Fv(if)k(and)h(only)h(if)f Fl(m)c Fq(6)p Fv(=)g Fl(k)s Fv(.)47 b(In)35 b(this)f(case)h(the)f(m)m(ultiplicit)m(y) j(of)c(the)i(0-eigen)m(v)-5 b(alue)88 1866 y(is)35 b Fq(j)p Fl(m)24 b Fq(\000)h Fl(k)s Fq(j)p Fv(,)35 b(and)g(the)h (eigenspinor)h(is)e(con)m(tained)i(in)e(the)h(subspace)h Fq(f)p Fl( )68 b Fv(:)c Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fl( )37 b Fv(=)88 2034 y([sgn\()p Fl(m)24 b Fq(\000)g Fl(k)s Fv(\)])p Fl( )t Fq(g)p Fv(.)51 b(Hence,)37 b(b)m(y)f(\(33\))f(w) m(e)h(obtain)f Fl(e)d Fv(=)f([sgn)q(\()p Fl(m)24 b Fq(\000)g Fl(k)s Fv(\)]\()p Fl(k)j Fv(+)d Fl(c)p Fv(\))g Fq(\000)3088 1995 y Fx(1)p 3088 2011 36 4 v 3088 2068 a(2)3168 2034 y Fv(and)88 2202 y(the)36 b(m)m(ultiplicit)m(y)j(of)d(this)h(eigen)m(v) -5 b(alue)38 b(of)1700 2177 y Fo(e)1679 2202 y Fq(D)1756 2218 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1981 2202 y Fv(is)e(at)g(most)h Fq(j)p Fl(m)25 b Fq(\000)g Fl(k)s Fq(j)p Fv(.)54 b(This)38 b(sho)m(ws)88 2370 y(that)32 b(Sp)s(ec)c Fq(D)601 2385 y Fk(M)713 2370 y Fv(is)33 b(included)h(in)f(the)g(union)g(giv)m(en)h (in)e(\(20\))g(with)i(m)m(ultiplicit)m(y)-8 b(.)234 2538 y(F)g(or)21 b(the)h(con)m(v)m(erse)i(statemen)m(t,)i(for)21 b(an)m(y)h(\014xed)h Fl(k)31 b Fq(2)d Fp(Z)22 b Fv(w)m(e)h(start)f (with)g(an)g(eigenspace)88 2706 y(of)32 b Fq(D)276 2722 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))484 2706 y Fv(with)i(eigen)m(v)-5 b(alue)34 b Fl(\025)p Fv(.)234 2874 y(If)25 b Fl(\025)j Fv(=)f(0,)g(then)f(the)f(same)i(space)f(is)g(also)f(an)g(eigenspace)j (of)c Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))26 b(with)g(eigen)m(v)-5 b(alue)88 3042 y(sgn\()p Fl(m)23 b Fq(\000)f Fl(k)s Fv(\))33 b(b)m(y)h(Theorem)g(37.)44 b(Hence)34 b(b)m(y)g(\(33\))e(the)i(lift)e (of)h(this)g(eigenspace)i(to)e(\011)3247 3057 y Fk(M)88 3210 y Fv(is)g(an)f(eigenspace)j(of)930 3185 y Fo(e)910 3210 y Fq(D)987 3226 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))1207 3210 y Fv(with)e(eigen)m(v)-5 b(alue)34 b Fl(e)28 b Fv(=)g([sgn\()p Fl(m)23 b Fq(\000)f Fl(k)s Fv(\)]\()p Fl(k)k Fv(+)c Fl(c)p Fv(\))g Fq(\000)3028 3171 y Fx(1)p 3028 3187 V 3028 3244 a(2)3074 3210 y Fv(.)234 3378 y(If)32 b Fl(\025)c(>)g Fv(0,)k(then)h(for)f(an)m(y)h(elemen)m(t)i Fl(\030)i Fv(of)32 b(this)h(eigenspace)i(w)m(e)e(form)903 3614 y Fl(\037)964 3573 y Fi(\006)1051 3614 y Fv(:=)28 b Fl(\030)e Fv(+)1359 3544 y Fq(\000)p Fl(\025)d Fq(\006)1615 3459 y Fo(p)p 1715 3459 549 4 v 85 x Fl(\025)1772 3516 y Fx(2)1833 3544 y Fv(+)f(\()p Fl(k)k Fv(+)c Fl(c)p Fv(\))2224 3516 y Fx(2)p 1359 3591 904 4 v 1703 3682 a Fl(k)j Fv(+)d Fl(c)2273 3614 y(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fl(\030)88 3827 y Fv(if)31 b Fl(k)24 b Fv(+)d Fl(c)27 b Fq(6)p Fv(=)h(0)k(and)g Fl(\037)852 3791 y Fx(+)938 3827 y Fv(:=)c Fl(\030)5 b Fv(,)32 b Fl(\037)1237 3791 y Fi(\000)1323 3827 y Fv(:=)c Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))p Fl(\030)37 b Fv(if)31 b Fl(k)24 b Fv(+)d Fl(c)28 b Fv(=)f(0.)43 b(The)33 b(sets)g Fq(f)p Fl(\037)2776 3786 y Fi(\000)2776 3853 y Fk(j)2835 3827 y Fq(g)f Fv(and)g Fq(f)p Fl(\037)3217 3786 y Fx(+)3217 3853 y Fk(j)3276 3827 y Fq(g)88 3995 y Fv(are)j(b)s(oth)f(linearly)i (indep)s(enden)m(t)h(as)f Fl(\030)5 b Fv('s)35 b(run)g(through)g(a)f (linearly)i(indep)s(enden)m(t)i(set)88 4163 y Fq(f)p Fl(\030)181 4178 y Fk(j)217 4163 y Fq(g)27 b(\032)h Fv(Ker)33 b(\()p Fq(D)704 4179 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))902 4163 y Fq(\000)23 b Fl(\025)p Fv(\).)43 b(F)-8 b(or,)32 b(if)1458 4089 y Fo(P)1563 4192 y Fk(j)1617 4163 y Fl(c)1659 4178 y Fk(j)1695 4163 y Fl(\037)1756 4122 y Fx(+)1756 4189 y Fk(j)1843 4163 y Fv(=)27 b(0)33 b(with)g(some)g(constan)m(ts)h Fl(c)2968 4178 y Fk(j)3037 4163 y Fv(then)809 4377 y(0)28 b(=)989 4266 y Fo(\020)1049 4377 y Fv(1)22 b(+)g Fl(\025)1275 4336 y Fi(\000)p Fx(1)1369 4377 y Fq(D)1446 4392 y Fk(N)s(;)p Fx(\()p Fk(k)r Fx(\))1622 4266 y Fo(\021)1698 4282 y(X)1754 4492 y Fk(j)1859 4377 y Fl(c)1901 4392 y Fk(j)1937 4377 y Fl(\037)1998 4336 y Fx(+)1998 4402 y Fk(j)2085 4377 y Fv(=)27 b(2)2254 4282 y Fo(X)2310 4492 y Fk(j)2414 4377 y Fl(c)2456 4392 y Fk(j)2493 4377 y Fl(\030)2541 4336 y Fk(j)2577 4377 y Fl(:)234 4666 y Fv(It)32 b(is)h(easy)h(to)e(c)m (hec)m(k)j(from)d(\(33\))f(that)i Fl(\037)1742 4630 y Fi(\006)1823 4666 y Fq(\016)21 b Fl(\036)32 b Fv(is)h(an)g(eigenspinor) h(of)2859 4641 y Fo(e)2838 4666 y Fq(D)2915 4682 y Fk(M)s(;)p Fx(\()p Fk(k)r Fx(\))3136 4666 y Fv(with)88 4834 y(eigen)m(v)-5 b(alue)36 b Fl(e)d Fv(=)f Fq(\006)819 4749 y Fo(p)p 919 4749 549 4 v 85 x Fl(\025)976 4805 y Fx(2)1037 4834 y Fv(+)22 b(\()p Fl(k)k Fv(+)c Fl(c)p Fv(\))1428 4805 y Fx(2)1491 4834 y Fq(\000)1602 4795 y Fx(1)p 1602 4811 36 4 v 1602 4868 a(2)1647 4834 y Fv(.)52 b(This)36 b(completes)h(the)f (pro)s(of)e(of)h(Theorem)88 5002 y(35.)1658 5251 y(42)p eop %%Page: 43 43 43 42 bop 3255 548 4 66 v 3259 485 59 4 v 3259 548 V 3317 548 4 66 v 88 935 a Fr(A)161 b(App)t(endix:)87 b(Spinor)61 b(bundles)f(on)i Ff(S)2632 883 y Fv(2)2747 935 y Fr(\(and)g Ff(S)3210 883 y Fv(3)3263 935 y Fr(\))371 1172 y(and)53 b(the)g(Aharono)l(v-Casher)e(Theorem)88 1439 y Fv(W)-8 b(e)28 b(shall)g(no)m(w)h(construct)g(spinor)f(bundles)h(\011)f(on)g Fp(S)2028 1403 y Fx(2)2068 1439 y Fv(.)41 b(First)28 b(w)m(e)h(c)m(ho)s(ose)g(co)s(ordinates.)88 1607 y(Let)38 b Fp(S)323 1571 y Fx(2)323 1632 y(+)419 1607 y Fv(=)f Fp(S)587 1571 y Fx(2)653 1607 y Fq(n)26 b(f)p Fl(S)6 b Fq(g)37 b Fv(and)h Fp(S)1182 1571 y Fx(2)1182 1632 y Fi(\000)1279 1607 y Fv(=)f Fp(S)1447 1571 y Fx(2)1512 1607 y Fq(n)26 b(f)p Fl(N)10 b Fq(g)p Fv(,)40 b(where)f Fl(N)49 b Fv(and)38 b Fl(S)44 b Fv(are)38 b(the)g(north)g(and)88 1775 y(south)33 b(p)s(oles)g(resp)s(ectiv)m(ely)-8 b(.)234 1943 y(Consider)34 b(the)f(stereographic)h(pro)5 b(jections)33 b Fl(z)1952 1958 y Fi(\006)2039 1943 y Fv(:)28 b Fp(S)2149 1907 y Fx(2)2149 1968 y Fi(\006)2236 1943 y Fq(!)g Fp(C)k Fv(de\014ned)i(b)m(y)523 2453 y Fl(!)d Fv(=)719 2099 y Fo(8)719 2189 y(>)719 2218 y(>)719 2248 y(>)719 2278 y(>)719 2308 y(>)719 2338 y(<)719 2517 y(>)719 2547 y(>)719 2577 y(>)719 2607 y(>)719 2637 y(>)719 2667 y(:)849 2083 y( )1006 2186 y Fq(\000)p Fv(4)p 1132 2100 245 4 v Fl(z)1177 2201 y Fi(\000)1237 2186 y Fv(\()p Fl(!)t Fv(\))p 938 2231 509 4 v 938 2322 a(4)22 b(+)g Fq(j)p Fl(z)1180 2337 y Fi(\000)1239 2322 y Fv(\()p Fl(!)t Fv(\))p Fq(j)1408 2293 y Fx(2)1456 2254 y Fl(;)49 b Fq(\000)1619 2186 y Fv(4)23 b Fq(\000)f(j)p Fl(z)1863 2201 y Fi(\000)1922 2186 y Fv(\()p Fl(!)t Fv(\))p Fq(j)2091 2150 y Fx(2)p 1619 2231 510 4 v 1620 2322 a Fv(4)g(+)g Fq(j)p Fl(z)1862 2337 y Fi(\000)1921 2322 y Fv(\()p Fl(!)t Fv(\))p Fq(j)2090 2293 y Fx(2)2139 2083 y Fo(!)2235 2254 y Fl(;)83 b Fv(for)32 b Fl(!)f Fq(2)d Fp(S)2735 2218 y Fx(2)2735 2278 y Fi(\000)849 2541 y Fo(\022)1040 2614 y Fv(4)p Fl(z)1134 2629 y Fx(+)1193 2614 y Fv(\()p Fl(!)t Fv(\))p 932 2658 509 4 v 932 2750 a(4)22 b(+)g Fq(j)p Fl(z)1174 2765 y Fx(+)1233 2750 y Fv(\()p Fl(!)t Fv(\))p Fq(j)1402 2721 y Fx(2)1450 2681 y Fl(;)1537 2614 y Fv(4)f Fq(\000)i(j)p Fl(z)1780 2629 y Fx(+)1839 2614 y Fv(\()p Fl(!)t Fv(\))p Fq(j)2008 2578 y Fx(2)p 1537 2658 510 4 v 1538 2750 a Fv(4)e(+)h Fq(j)p Fl(z)1779 2765 y Fx(+)1838 2750 y Fv(\()p Fl(!)t Fv(\))p Fq(j)2007 2721 y Fx(2)2056 2541 y Fo(\023)2146 2681 y Fl(;)172 b Fv(for)32 b Fl(!)f Fq(2)d Fp(S)2735 2645 y Fx(2)2735 2706 y(+)2863 2453 y Fl(;)88 2960 y Fv(where)38 b(w)m(e)h(ha)m(v)m(e)g(iden)m(ti\014ed)g Fp(S)1238 2923 y Fx(2)1314 2960 y Fq(\032)e Fp(C)26 b Fq(\002)g Fp(R)p Fv(.)58 b(Note)38 b(that)f(for)g Fl(!)j Fq(2)d Fp(S)2656 2923 y Fx(2)2656 2984 y Fi(\000)2740 2960 y Fq(\\)26 b Fp(S)2887 2923 y Fx(2)2887 2984 y(+)2984 2960 y Fv(w)m(e)39 b(ha)m(v)m(e)88 3128 y Fl(z)133 3143 y Fi(\000)192 3128 y Fv(\()p Fl(!)t Fv(\))45 b(=)h Fq(\000)p Fv(4)p Fl(z)671 3143 y Fx(+)730 3128 y Fv(\()p Fl(!)t Fv(\))871 3079 y Fi(\000)p Fx(1)965 3128 y Fv(.)75 b(With)44 b(the)g(ab)s(o)m(v)m(e)g (c)m(hoice)h(the)e(maps)h Fl(z)2586 3143 y Fi(\006)2689 3128 y Fv(are)f(orien)m(tation)88 3296 y(preserving)e(when)g(w)m(e)g(c) m(ho)s(ose)g(the)f(standard)g(orien)m(tations)h(of)e Fp(S)2598 3259 y Fx(2)2678 3296 y Fv(and)g Fp(C)h Fv(\(strictly)88 3464 y(sp)s(eaking)32 b Fl(z)533 3479 y Fi(\000)624 3464 y Fv(is)h(a)e(stereographic)i(pro)5 b(jection)33 b(follo)m(w)m(ed)f(b)m (y)h(a)f(re\015ection\).)44 b(If)32 b(w)m(e)g(use)88 3632 y(the)h(metric)1195 3800 y Fl(ds)1292 3758 y Fx(2)1358 3800 y Fv(=)28 b(\(1)22 b(+)1679 3760 y Fx(1)p 1679 3777 36 4 v 1679 3834 a(4)1724 3800 y Fq(j)p Fl(z)t Fq(j)1829 3758 y Fx(2)1869 3800 y Fv(\))1907 3758 y Fi(\000)p Fx(2)2001 3800 y Fl(dz)f(d)p 2169 3745 50 4 v(z)88 4020 y Fv(on)32 b Fp(C)h Fv(b)s(oth)f(maps)h Fl(z)858 4035 y Fi(\006)950 4020 y Fv(are)g(also)f(isometries.)88 4353 y Fe(A.1)134 b(Spinor)45 b(bundles)f(on)h Fd(S)1605 4310 y Fs(2)1696 4353 y Fe(and)g Fd(S)2022 4310 y Fs(3)88 4586 y Fv(Corresp)s(onding)e (to)f(eac)m(h)h Fl(n)h Fq(2)g Fp(Z)f Fv(w)m(e)g(de\014ne)h(a)e(spinor)g (bundle)i(\011)2674 4601 y Fk(n)2763 4586 y Fv(on)e Fp(S)2963 4550 y Fx(2)3045 4586 y Fv(b)m(y)h(the)88 4754 y(follo)m(wing)32 b(prop)s(erties:)233 5002 y Fq(\017)48 b Fv(There)34 b(are)f(op)s(en)g(subsets)h(\011)1425 4966 y Fi(\006)1425 5027 y Fk(n)1512 5002 y Fq(\032)28 b Fv(\011)1693 5017 y Fk(n)1772 5002 y Fv(suc)m(h)34 b(that)f(\011)2280 5017 y Fk(n)2354 5002 y Fv(=)28 b(\011)2534 4966 y Fx(+)2534 5027 y Fk(n)2615 5002 y Fq([)22 b Fv(\011)2779 4966 y Fi(\000)2779 5027 y Fk(n)1658 5251 y Fv(43)p eop %%Page: 44 44 44 43 bop 233 549 a Fq(\017)48 b Fv(There)34 b(are)f(di\013eomorphisms) h Fl(\036)1558 564 y Fi(\006)1645 549 y Fv(:)28 b(\011)1776 498 y Fx(\()p Fi(\006)p Fx(\))1776 559 y Fk(n)1917 549 y Fq(!)f Fp(S)2099 513 y Fx(2)2099 574 y Fi(\006)2181 549 y Fq(\002)c Fp(C)2353 513 y Fx(2)2392 549 y Fv(.)233 800 y Fq(\017)48 b Fv(If)33 b Fl(\021)e Fq(2)d Fv(\011)678 815 y Fk(n)758 800 y Fv(and)k Fl(\036)1005 815 y Fi(\006)1064 800 y Fv(\()p Fl(\021)t Fv(\))27 b(=)h(\()p Fl(!)1422 815 y Fi(\006)1481 800 y Fl(;)17 b(u)1581 815 y Fi(\006)1639 800 y Fv(\))32 b(then)803 1068 y Fl(!)864 1083 y Fx(+)950 1068 y Fv(=)c Fl(!)1115 1083 y Fi(\000)1271 1068 y Fv(and)98 b Fl(u)1582 1083 y Fi(\000)1668 1068 y Fv(=)28 b Fq(U)1834 1083 y Fk(n)1881 1068 y Fv(\()p Fl(z)1964 1083 y Fx(+)2023 1068 y Fv(\()p Fl(!)2122 1083 y Fx(+)2181 1068 y Fv(\)\))p Fq(W)8 b Fv(\()p Fl(z)2446 1083 y Fx(+)2506 1068 y Fv(\()p Fl(!)2605 1083 y Fx(+)2664 1068 y Fv(\)\))p Fl(u)2796 1083 y Fx(+)3152 1068 y Fv(\(34\))331 1335 y(where)514 1625 y Fq(U)576 1640 y Fk(n)624 1625 y Fv(\()p Fl(z)t Fv(\))28 b(=)881 1484 y Fo(\022)964 1557 y Fq(j)p Fl(z)t Fq(j)p 964 1602 106 4 v 991 1693 a Fl(z)1079 1484 y Fo(\023)1152 1506 y Fk(n)1313 1625 y Fv(and)98 b Fq(W)8 b Fv(\()p Fl(z)t Fv(\))29 b(=)1931 1454 y Fo( )2052 1553 y Fl(z)t Fq(j)p Fl(z)t Fq(j)2206 1517 y Fi(\000)p Fx(1)2484 1553 y Fv(0)2152 1709 y(0)p 2384 1654 50 4 v 183 w Fl(z)5 b Fq(j)p Fl(z)t Fq(j)2539 1673 y Fi(\000)p Fx(1)2675 1454 y Fo(!)2781 1625 y Fq(2)28 b Fl(S)6 b(U)k Fv(\(2\))233 2000 y Fq(\017)48 b Fv(If)31 b Fl(\013)e Fv(=)e Fl(a)p Fv(\()p Fl(z)t Fv(\))p Fl(d)p 848 1945 V(z)d Fv(+)p 1012 1914 177 4 v 19 w Fl(a)p Fv(\()p Fl(z)t Fv(\))q Fl(dz)35 b Fv(is)c(a)g(real)g(one-form)f(on)h Fp(C)g Fv(then)g(the)h(Cli\013ord) f(m)m(ulti-)331 2168 y(plication)i Fl(\033)k Fv(on)c(\011)1036 2183 y Fk(n)1115 2168 y Fv(is)g(de\014ned)h(b)m(y)331 2477 y Fl(\036)389 2492 y Fi(\006)465 2396 y Fo(\000)510 2477 y Fl(\033)586 2396 y Fo(\000)632 2477 y Fl(z)681 2436 y Fi(\003)677 2502 y(\006)736 2477 y Fv(\()p Fl(\013)q Fv(\))875 2396 y Fo(\001)937 2477 y Fl(\021)989 2396 y Fo(\001)1062 2477 y Fv(=)1166 2307 y Fo( )1244 2477 y Fl(!)t(;)1385 2396 y Fo(\000)1431 2477 y Fv(1)22 b(+)1609 2438 y Fx(1)p 1609 2454 36 4 v 1609 2511 a(4)1655 2477 y Fq(j)p Fl(z)1728 2492 y Fi(\006)1787 2477 y Fv(\()p Fl(!)t Fv(\))p Fq(j)1956 2436 y Fx(2)1994 2396 y Fo(\001)2056 2307 y( )2338 2405 y Fv(0)p 2631 2319 372 4 v 244 w Fl(a)p Fv(\()p Fl(z)2765 2420 y Fi(\006)2825 2405 y Fv(\()p Fl(!)t Fv(\)\))2177 2562 y Fl(a)p Fv(\()p Fl(z)2311 2577 y Fi(\006)2370 2562 y Fv(\()p Fl(!)t Fv(\)\))244 b(0)3044 2307 y Fo(!)3140 2477 y Fl(u)3196 2492 y Fi(\006)3255 2307 y Fo(!)3350 2477 y Fl(;)331 2791 y Fv(when)28 b Fl(\036)638 2806 y Fi(\006)697 2791 y Fv(\()p Fl(\021)t Fv(\))f(=)h(\()p Fl(!)t(;)17 b(u)1159 2806 y Fi(\006)1216 2791 y Fv(\).)42 b(Here)28 b Fl(z)1597 2755 y Fi(\003)1593 2816 y(\006)1652 2791 y Fv(\()p Fl(\013)q Fv(\))e(are)h(the)g(pull-bac) m(ks)i(of)d(the)h(one-form)g Fl(\013)331 2959 y Fv(to)32 b Fp(S)505 2923 y Fx(2)505 2984 y Fi(\006)564 2959 y Fv(.)44 b(Note)32 b(that)g(it)g(is)g(the)g(Cli\013ord)h(m)m (ultiplication)g(relativ)m(e)g(to)f(the)g(metric)331 3127 y Fl(ds)428 3091 y Fx(2)500 3127 y Fv(whic)m(h)i(is)f(b)s(eing)g (used)h(on)e Fp(C)1570 3091 y Fx(2)1610 3127 y Fv(.)234 3378 y(It)c(is)i(fairly)e(easy)i(to)e(c)m(hec)m(k)j(that)d(this)h (really)g(de\014nes)i(a)d(spinor)h(bundle)h(\011)2997 3393 y Fk(n)3072 3378 y Fv(on)f Fp(S)3259 3342 y Fx(2)3298 3378 y Fv(.)88 3546 y(In)38 b(particular,)i(w)m(e)f(notice)f(that)g(if) 48 b(~)-59 b Fl(\013)39 b Fv(is)f(a)g(one-form)g(on)f Fp(S)2353 3510 y Fx(2)2431 3546 y Fv(then)49 b(~)-59 b Fl(\013)2721 3512 y Fo(\014)2721 3572 y(\014)2721 3632 y(\014)2754 3627 y Fn(S)2791 3604 y Fm(2)2791 3647 y Fg(\006)2884 3546 y Fv(=)37 b Fl(z)3046 3510 y Fi(\003)3042 3571 y(\006)3101 3546 y Fv(\()p Fl(\013)3201 3561 y Fi(\006)3260 3546 y Fv(\),)88 3794 y(where)h Fl(\013)436 3809 y Fi(\006)531 3794 y Fv(=)e Fl(a)694 3809 y Fi(\006)753 3794 y Fv(\()p Fl(z)t Fv(\))p 878 3713 101 4 v Fl(dz)31 b Fv(+)p 1106 3708 236 4 v 25 w Fl(a)1157 3809 y Fi(\006)1217 3794 y Fv(\()p Fl(z)t Fv(\))p Fl(dz)42 b Fv(satis\014es)d Fl(a)1895 3809 y Fx(+)1954 3794 y Fv(\()p Fl(z)t Fv(\))e(=)f(4)p Fl(a)2328 3809 y Fi(\000)2387 3794 y Fv(\()p Fq(\000)p Fv(4)p Fl(z)2600 3758 y Fi(\000)p Fx(1)2695 3794 y Fv(\))p 2733 3739 50 4 v Fl(z)2783 3758 y Fi(\000)p Fx(2)2877 3794 y Fv(.)58 b(Th)m(us)39 b(w)m(e)88 3962 y(see)25 b(that)e(the)i(Cli\013ord)f(m)m(ultiplication)h(transforms)g(consisten) m(tly)h(b)s(et)m(w)m(een)g(\011)3007 3926 y Fx(\()p Fi(\006)p Fx(\))3121 3962 y Fv(,)g(i.e.,)206 4271 y(\(1)c(+)423 4232 y Fx(1)p 423 4248 36 4 v 423 4305 a(4)468 4271 y Fq(j)p Fl(z)541 4286 y Fi(\000)600 4271 y Fq(j)628 4230 y Fx(2)667 4271 y Fv(\))722 4101 y Fo( )963 4199 y Fv(0)p 1216 4112 291 4 v 204 w Fl(a)1267 4214 y Fi(\000)1326 4199 y Fv(\()p Fl(z)1409 4214 y Fi(\000)1468 4199 y Fv(\))842 4356 y Fl(a)893 4371 y Fi(\000)952 4356 y Fv(\()p Fl(z)1035 4371 y Fi(\000)1095 4356 y Fv(\))204 b(0)1548 4101 y Fo(!)998 4678 y Fv(=)83 b(\(1)22 b(+)1374 4639 y Fx(1)p 1374 4656 36 4 v 1374 4713 a(4)1419 4678 y Fq(j)p Fl(z)1492 4693 y Fx(+)1551 4678 y Fq(j)1579 4637 y Fx(2)1618 4678 y Fv(\))p Fq(W)8 b Fv(\()p Fl(z)1845 4693 y Fx(+)1905 4678 y Fv(\))1960 4508 y Fo( )2201 4607 y Fv(0)p 2453 4520 291 4 v 203 w Fl(a)2504 4622 y Fx(+)2563 4607 y Fv(\()p Fl(z)2646 4622 y Fx(+)2706 4607 y Fv(\))2080 4763 y Fl(a)2131 4778 y Fx(+)2190 4763 y Fv(\()p Fl(z)2273 4778 y Fx(+)2332 4763 y Fv(\))204 b(0)2785 4508 y Fo(!)2881 4678 y Fq(W)8 b Fv(\()p Fl(z)3070 4693 y Fx(+)3129 4678 y Fv(\))3167 4637 y Fi(\003)88 4993 y Fv(where)33 b Fl(z)414 5008 y Fi(\006)501 4993 y Fv(=)28 b Fl(z)650 5008 y Fi(\006)709 4993 y Fv(\()p Fl(!)t Fv(\).)1658 5251 y(44)p eop %%Page: 45 45 45 44 bop 88 548 a Fw(46.)50 b(PR)m(OPOSITION.)d Fu(If)29 b Fv(\011)g Fu(is)h(a)f(spinor)g(bund)5 b(le)29 b(over)g Fp(S)2405 512 y Fx(2)2475 548 y Fu(then)g Fv(\011)g Fu(is)h(di\013e)-5 b(omor-)88 716 y(phic)34 b(to)h Fv(\011)490 731 y Fk(n)572 716 y Fu(for)f(some)g Fl(n)28 b Fq(2)g Fp(Z)p Fu(.)88 957 y(Pr)-5 b(o)g(of.)48 b Fv(W)-8 b(e)37 b(just)g(sk)m(etc)m(h)i(this) f(standard)f(argumen)m(t.)56 b(Since)38 b(an)m(y)g(v)m(ector)g(bundle)f (on)88 1125 y Fp(C)k Fv(or)h(on)g Fp(S)530 1089 y Fx(2)530 1150 y Fi(\006)631 1125 y Fv(is)g(trivial)g(w)m(e)h(easily)g(see)g (that)f(an)m(y)g(spinor)h(bundle)g(on)f Fp(S)2881 1089 y Fx(2)2962 1125 y Fv(is)h(of)e(the)88 1293 y(form)46 b(describ)s(ed)j(ab)s(o)m(v)m(e)f(with)g Fq(U)1368 1308 y Fk(n)1415 1293 y Fv(\()p Fl(z)1498 1308 y Fx(+)1557 1293 y Fv(\()p Fl(!)t Fv(\)\))e(replaced)i(b)m(y)g(a)f(general)g (function)h Fq(U)62 b Fv(:)88 1461 y Fp(S)143 1425 y Fx(2)209 1461 y Fq(n)26 b(f)p Fl(N)5 b(;)17 b(S)6 b Fq(g)38 b(!)g Fl(U)10 b Fv(\(1\).)63 b(Let)39 b Fq(\000)p Fl(n)h Fv(b)s(e)f(the)g(degree)h(of)e(the)i(map)f Fq(U)49 b Fv(\(i.e.,)41 b(the)e(degree)88 1629 y(when)45 b(w)m(e)g(restrict)g(to) e(e.g.)i(the)f(equatorial)g(circle\).)79 b(Then)45 b Fp(S)2512 1593 y Fx(2)2582 1629 y Fq(n)30 b(f)p Fl(N)5 b(;)17 b(S)6 b Fq(g)46 b(3)h Fl(!)k Fq(7!)88 1797 y(U)10 b Fv(\()p Fl(!)t Fv(\))p Fq(U)373 1761 y Fi(\003)363 1822 y Fk(n)412 1797 y Fv(\()p Fl(z)495 1812 y Fx(+)554 1797 y Fv(\()p Fl(!)t Fv(\)\))45 b Fq(2)h Fl(U)10 b Fv(\(1\))43 b(is)h(a)e(map)i(of)e(degree)i(0.)75 b(W)-8 b(e)44 b(ma)m(y)f (therefore)h(\014nd)g(t)m(w)m(o)88 1965 y(functions)35 b Fq(U)572 1980 y Fi(\006)661 1965 y Fv(:)30 b Fp(S)773 1929 y Fx(2)836 1965 y Fq(n)23 b(f)p Fl(N)5 b(;)17 b(S)6 b Fq(g)30 b(!)f Fl(U)10 b Fv(\(1\))35 b(with)f Fq(U)1882 1980 y Fx(+)1976 1965 y Fv(equal)h(1)e(near)i Fl(N)44 b Fv(and)34 b(lik)m(ewise)j(for)88 2133 y Fl(S)h Fv(suc)m(h)c(that)1024 2301 y Fq(U)10 b Fv(\()p Fl(!)t Fv(\))p Fq(U)1309 2260 y Fi(\003)1299 2326 y Fk(n)1348 2301 y Fv(\()p Fl(z)1431 2316 y Fx(+)1490 2301 y Fv(\()p Fl(!)t Fv(\)\))27 b(=)h Fq(U)1862 2316 y Fx(+)1921 2301 y Fv(\()p Fl(!)t Fv(\))p Fq(U)2124 2316 y Fi(\000)2182 2301 y Fv(\()p Fl(!)t Fv(\))2323 2260 y Fi(\003)2362 2301 y Fl(:)88 2518 y Fv(If)48 b(w)m(e)h(no)m(w)g (use)g Fq(U)825 2533 y Fi(\006)932 2518 y Fv(to)f(c)m(hange)h(the)g(co) s(ordinates)f(on)h(the)f(\014b)s(ers)h(of)f(\011)g(o)m(v)m(er)h Fp(S)3266 2482 y Fx(2)3266 2543 y Fi(\006)88 2686 y Fv(resp)s(ectiv)m (ely)e(w)m(e)f(see)g(that)f(the)g(transformation)g(matrix)g Fq(U)55 b Fv(will)45 b(b)s(e)g(replaced)h(b)m(y)88 2854 y Fq(U)150 2869 y Fk(n)219 2854 y Fq(\016)22 b Fl(z)336 2869 y Fx(+)395 2854 y Fv(.)p 3255 2854 4 66 v 3259 2792 59 4 v 3259 2854 V 3317 2854 4 66 v 88 3094 a Fu(47.)44 b(REMARK.)50 b Fv(A)36 b(similar)h(argumen)m(t)g(immediately)h(pro)m(v) m(es)g(that)e(on)h Fp(S)2931 3058 y Fx(3)3007 3094 y Fv(there)g(is)88 3262 y(only)42 b(one)f Fl(S)6 b(pin)704 3226 y Fk(c)781 3262 y Fv(bundle.)71 b(This)43 b(bundle)f(is)g(in)g (fact)f(trivial,)j(since)f(w)m(e)g(can)f(\014nd)g(a)88 3430 y(global)29 b(orthonormal)g(frame)g(\()p Fl(e)1280 3445 y Fx(1)1320 3430 y Fl(;)17 b(e)1409 3445 y Fx(2)1448 3430 y Fl(;)g(e)1537 3445 y Fx(3)1577 3430 y Fv(\))27 b(:=)h(\()p Fl(u)1867 3445 y Fx(1)1906 3430 y Fl(;)17 b(u)2006 3445 y Fx(2)2044 3430 y Fl(;)g(n)p Fv(\))30 b(on)f Fp(S)2401 3394 y Fx(3)2470 3430 y Fv(\(see)i(Section)f(7\).)42 b(The)88 3598 y(Cli\013ord)34 b(m)m(ultiplication)i(on)e Fp(S)1264 3562 y Fx(3)1327 3598 y Fq(\002)23 b Fp(C)1499 3562 y Fx(2)1573 3598 y Fv(is)34 b(de\014ned)i(b)m(y)f Fl(\033)t Fv(\()p Fl(e)2289 3562 y Fk(j)2326 3598 y Fv(\))30 b(:=)g Fl(\033)2582 3613 y Fk(j)2619 3598 y Fv(,)35 b(where)g Fl(\033)3019 3613 y Fk(j)3090 3598 y Fv(is)g(the)88 3766 y Fl(j)6 b Fv(-th)32 b(P)m(auli)h(matrix.)88 4098 y Fe(A.2)134 b Fc(S)7 b(pin)676 4054 y Fb(c)761 4098 y Fe(connections)45 b(on)g FC(\011)1766 4116 y Fb(n)88 4330 y Fv(W)-8 b(e)37 b(\014rst)h(describ)s(e)h Fl(S)6 b(pin)1054 4294 y Fk(c)1126 4330 y Fv(connections)39 b(on)f Fp(C)f Fv(with)h(resp)s(ect)h(to)e(the) g(metric)i Fl(ds)3175 4294 y Fx(2)3250 4330 y Fv(=)88 4498 y(\(1+)261 4459 y Fx(1)p 261 4475 36 4 v 261 4532 a(4)306 4498 y Fq(j)p Fl(z)t Fq(j)411 4462 y Fx(2)450 4498 y Fv(\))488 4462 y Fi(\000)p Fx(2)582 4498 y Fl(dz)t(d)p 733 4443 50 4 v(z)5 b Fv(.)40 b(On)22 b Fp(C)g Fv(w)m(e)g(consider)i (the)e(trivial)g(spinor)g(bundle,)k(i.e.,)e(\011)k(=)f Fp(C)p Fq(\002)p Fp(C)3258 4462 y Fx(2)3298 4498 y Fv(.)234 4666 y(Since)e(the)f(metric)h Fl(ds)1031 4630 y Fx(2)1094 4666 y Fv(is)g(conformally)f(equiv)-5 b(alen)m(t)26 b(to)e(the)g (standard)h(metric)g Fl(dz)t(d)p 3276 4611 V(z)88 4834 y Fv(with)34 b(the)g(conformal)g(factor)f(\012\()p Fl(z)t Fv(\))e(=)f(\(1)22 b(+)1764 4795 y Fx(1)p 1764 4811 36 4 v 1764 4868 a(4)1809 4834 y Fq(j)p Fl(z)t Fq(j)1914 4798 y Fx(2)1954 4834 y Fv(\))1992 4798 y Fi(\000)p Fx(1)2120 4834 y Fv(w)m(e)35 b(ma)m(y)f(use)h(the)f(results)i(from)88 5002 y(Section)d(4.)43 b(The)33 b(spinor)g(sections)h(with)e(resp)s (ect)i(to)e(the)g(standard)h(metric)g(on)f Fp(C)g Fv(are)1658 5251 y(45)p eop %%Page: 46 46 46 45 bop 88 548 a Fv(giv)m(en)44 b(in)g(terms)g(of)g(a)f(\(real\))g (one-form)g Fl(\013)k Fv(=)g Fl(a)p Fv(\()p Fl(z)t Fv(\))p Fl(d)p 2130 493 50 4 v(z)34 b Fv(+)p 2315 461 177 4 v 30 w Fl(a)p Fv(\()p Fl(z)t Fv(\))q Fl(dz)48 b Fv(as)43 b(follo)m(ws.)77 b(The)88 716 y(co)m(v)-5 b(arian)m(t)33 b(deriv)-5 b(ativ)m(e)34 b(along)e(a)g(\(real\))h(v)m(ector)g(\014eld)h Fl(X)h Fv(=)28 b Fl(\030)5 b Fv(\()p Fl(z)t Fv(\))p Fl(@)2510 731 y Fk(z)2572 716 y Fv(+)p 2670 629 174 4 v 22 w Fl(\030)g Fv(\()p Fl(z)t Fv(\))p Fl(@)p 2894 693 40 3 v 15 x Fk(z)2967 716 y Fv(is)918 967 y Fl(\030)g Fv(\()p Fl(z)t Fv(\))1091 857 y Fo(\020)1150 967 y Fl(@)1201 982 y Fk(z)1264 967 y Fq(\000)22 b Fl(i)p 1396 881 177 4 v(a)p Fv(\()p Fl(z)t Fv(\))1573 857 y Fo(\021)1655 967 y Fv(+)p 1753 881 174 4 v 22 w Fl(\030)5 b Fv(\()p Fl(z)t Fv(\))1926 857 y Fo(\020)1986 967 y Fl(@)p 2037 945 40 3 v 15 x Fk(z)2099 967 y Fq(\000)23 b Fl(ia)p Fv(\()p Fl(z)t Fv(\))2409 857 y Fo(\021)2468 967 y Fl(:)657 b Fv(\(35\))88 1218 y(W)-8 b(e)30 b(ma)m(y)g(calculate)h(the)f(co)m(v)-5 b(arian)m(t)30 b(deriv)-5 b(ativ)m(e)31 b(of)f(spinors)h(in)f(the)g (conformal)f(metric)88 1386 y Fl(ds)185 1350 y Fx(2)256 1386 y Fv(from)j(Prop)h(22)f(using)650 1679 y Fl(\033)t Fv(\()p Fl(X)836 1638 y Fi(\003)875 1679 y Fv(\))c(=)1055 1640 y Fx(1)p 1055 1656 36 4 v 1055 1713 a(2)1100 1679 y Fl(\033)1175 1568 y Fo(\020)p 1235 1592 174 4 v 1235 1679 a Fl(\030)5 b Fv(\()p Fl(z)t Fv(\))p Fl(dz)27 b Fv(+)22 b Fl(\030)5 b Fv(\()p Fl(z)t Fv(\))p Fl(d)p 1853 1624 50 4 v(z)1902 1568 y Fo(\021)1989 1679 y Fv(=)2093 1508 y Fo( )2275 1607 y Fv(0)p 2469 1520 174 4 v 145 w Fl(\030)g Fv(\()p Fl(z)t Fv(\))2213 1763 y Fl(\030)g Fv(\()p Fl(z)t Fv(\))145 b(0)2684 1508 y Fo(!)88 1971 y Fv(and)935 2161 y Fl(\033)t Fv(\()p Fl(d)p Fv(\012\))28 b(=)g Fq(\000)1410 2122 y Fx(1)p 1410 2138 36 4 v 1410 2195 a(2)1455 2161 y Fv(\(1)22 b(+)1672 2122 y Fx(1)p 1672 2138 V 1672 2195 a(4)1717 2161 y Fq(j)p Fl(z)t Fq(j)1822 2120 y Fx(2)1862 2161 y Fv(\))1900 2120 y Fi(\000)p Fx(2)2011 1991 y Fo( )2132 2089 y Fv(0)p 2264 2034 50 4 v 83 w Fl(z)2131 2246 y(z)88 b Fv(0)2355 1991 y Fo(!)2451 2161 y Fl(:)88 2427 y Fv(According)32 b(to)f(Prop)h(22)f(the)h(co)m(v)-5 b(arian)m(t)32 b(deriv)-5 b(ativ)m(e)34 b(of)d(spinors)i(in)f(the)g (metric)g Fl(ds)3189 2391 y Fx(2)3260 2427 y Fv(is)186 2756 y Fq(r)269 2715 y Fk(\013)269 2780 y(X)364 2756 y Fv(=)c Fl(\030)5 b Fv(\()p Fl(z)t Fv(\))641 2645 y Fo(\020)700 2756 y Fl(@)751 2771 y Fk(z)813 2756 y Fq(\000)23 b Fl(i)p 946 2669 177 4 v(a)p Fv(\()p Fl(z)t Fv(\))1123 2645 y Fo(\021)1205 2756 y Fv(+)p 1303 2669 174 4 v 22 w Fl(\030)5 b Fv(\()p Fl(z)t Fv(\))1476 2645 y Fo(\020)1535 2756 y Fl(@)p 1586 2733 40 3 v 15 x Fk(z)1649 2756 y Fq(\000)22 b Fl(ia)p Fv(\()p Fl(z)t Fv(\))1959 2645 y Fo(\021)2040 2756 y Fq(\000)2150 2652 y Fl(i)p Fv(Im)2316 2542 y Fo(\020)2376 2652 y Fl(z)p 2425 2566 174 4 v 4 w(\030)5 b Fv(\()p Fl(z)t Fv(\))2598 2542 y Fo(\021)p 2150 2733 509 4 v 2209 2824 a Fv(\(4)22 b(+)g Fq(j)p Fl(z)t Fq(j)2521 2795 y Fx(2)2561 2824 y Fv(\))2685 2585 y Fo( )2805 2684 y Fv(1)121 b(0)2805 2840 y(0)83 b Fq(\000)p Fv(1)3105 2585 y Fo(!)3200 2756 y Fl(:)234 3048 y Fv(No)m(w)40 b(assume)i(that)e(w)m(e)h(ha)m(v)m(e)g(a)f(co)m(v)-5 b(arian)m(t)40 b(deriv)-5 b(ativ)m(e)42 b Fq(r)e Fv(on)g(\011)2723 3063 y Fk(n)2770 3048 y Fv(.)66 b(Let)40 b Fl(\021)j Fv(b)s(e)e(a)88 3216 y(section)33 b(in)g(\011)603 3231 y Fk(n)682 3216 y Fv(and)900 3191 y(~)872 3216 y Fl(X)40 b Fv(a)33 b(v)m(ector\014eld)h(on)f Fp(S)1737 3180 y Fx(2)1776 3216 y Fv(.)44 b(F)-8 b(or)32 b Fl(!)f Fq(2)d Fp(S)2263 3180 y Fx(2)2263 3241 y Fi(\006)2355 3216 y Fv(write)337 3467 y Fl(\036)395 3482 y Fi(\006)454 3467 y Fv(\()p Fl(\021)t Fv(\()p Fl(!)t Fv(\)\))f(=)g(\()p Fl(!)t(;)17 b(u)1056 3482 y Fi(\006)1114 3467 y Fv(\()p Fl(z)1197 3482 y Fi(\006)1256 3467 y Fv(\()p Fl(!)t Fv(\)\)\))97 b(and)g(\()p Fl(z)1907 3482 y Fi(\006)1967 3467 y Fv(\))2005 3482 y Fi(\003)2072 3442 y Fv(~)2044 3467 y Fl(X)8 b Fv(\()p Fl(!)t Fv(\))27 b(=)g(\()p Fl(!)t(;)17 b(X)2632 3482 y Fi(\006)2690 3467 y Fv(\()p Fl(z)2773 3482 y Fi(\006)2832 3467 y Fv(\()p Fl(!)t Fv(\)\)\))p Fl(:)88 3719 y Fv(Then)33 b(there)h(exists)g(one-forms)f Fl(\013)1371 3734 y Fi(\006)1462 3719 y Fv(on)g Fp(C)f Fv(suc)m(h)i(that)914 3970 y Fl(\036)972 3985 y Fi(\006)1047 3970 y Fv(\()p Fq(r)1188 3980 y Fx(~)1168 3997 y Fk(X)1235 3970 y Fl(\021)t Fv(\()p Fl(!)t Fv(\)\))27 b(=)1596 3859 y Fo(\020)1656 3970 y Fl(!)t(;)17 b Fq(r)1848 3920 y Fk(\013)1893 3929 y Fg(\006)1848 3997 y Fk(X)1906 4006 y Fg(\006)1961 3970 y Fl(u)2017 3985 y Fi(\006)2076 3970 y Fv(\()p Fl(z)2159 3985 y Fi(\006)2218 3970 y Fv(\()p Fl(!)t Fv(\)\))2397 3859 y Fo(\021)2472 3970 y Fl(:)653 b Fv(\(36\))234 4221 y(Note)33 b(that)f(b)m(y)h(\(34\))f(w)m(e)i(m)m (ust)g(ha)m(v)m(e)531 4485 y Fq(r)614 4434 y Fk(\013)659 4443 y Fg(\000)614 4512 y Fk(X)672 4521 y Fg(\000)728 4485 y Fl(u)784 4500 y Fi(\000)843 4485 y Fv(\()p Fl(z)926 4500 y Fi(\000)985 4485 y Fv(\()p Fl(!)t Fv(\)\))27 b(=)1294 4344 y Fo(\022)1378 4417 y Fq(j)p Fl(z)1451 4432 y Fx(+)1510 4417 y Fv(\()p Fl(!)t Fv(\))p Fq(j)p 1378 4462 300 4 v 1405 4553 a Fl(z)1450 4568 y Fx(+)1510 4553 y 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5002 y Fx(+)2655 4877 y Fo(\021)2731 4987 y Fv(\()p Fq(\000)p Fv(4)p Fl(z)2944 4946 y Fi(\000)p Fx(1)3039 4987 y Fv(\))75 b(\(37\))1658 5251 y(46)p eop %%Page: 47 47 47 46 bop 88 548 a Fv(With)43 b Fl(X)431 563 y Fi(\006)537 548 y Fv(=)j(2Re)16 b([)q Fl(\030)910 563 y Fi(\006)968 548 y Fv(\()p Fl(z)t Fv(\))p Fl(@)1144 563 y Fk(z)1185 548 y Fv(])44 b(w)m(e)g(ha)m(v)m(e)h(the)f(relation)g Fl(\030)2237 563 y Fx(+)2295 548 y Fv(\()p Fl(z)t Fv(\))j(=)2599 509 y Fx(1)p 2599 525 36 4 v 2599 582 a(4)2645 548 y Fl(z)2694 512 y Fx(2)2734 548 y Fl(\030)2777 563 y Fi(\000)2836 548 y Fv(\()p Fq(\000)p Fv(4)p Fl(z)3049 512 y Fi(\000)p Fx(1)3144 548 y Fv(\))p Fl(:)c Fv(A)88 716 y(straigh)m(tforw)m(ard)33 b(calculation)h(then)f(sho)m(ws)i(that)d(if)h Fl(\013)2134 731 y Fi(\006)2221 716 y Fv(=)c(2Re)16 b([)p Fl(a)2584 731 y Fi(\006)2643 716 y Fv(\()p Fl(z)t Fv(\))p Fl(d)p 2819 661 50 4 v(z)5 b Fv(])33 b(then)h(\(37\))88 884 y(implies)g(that)1005 1052 y Fl(a)1056 1067 y Fi(\000)1115 1052 y Fv(\()p Fl(z)t Fv(\))28 b(=)g(4)p 1421 997 V Fl(z)1470 1011 y Fi(\000)p Fx(2)1565 1052 y Fl(a)1616 1067 y Fx(+)1675 1052 y Fv(\()p Fq(\000)p Fv(4)p Fl(z)1888 1011 y Fi(\000)p Fx(1)1983 1052 y Fv(\))22 b(+)g Fl(i)2184 1013 y Fk(n)p 2184 1029 43 4 v 2188 1086 a Fx(2)p 2237 997 50 4 v 2237 1052 a Fl(z)2287 1011 y Fi(\000)p Fx(1)2381 1052 y Fl(:)744 b Fv(\(38\))88 1274 y(Con)m(v)m(ersely)-8 b(,)41 b(for)36 b(an)m(y)i(c)m(hoice)g(of)f(functions)h Fl(a)1834 1289 y Fi(\006)1930 1274 y Fv(on)f Fp(C)f Fv(satisfying)j(\(38\))d(the)h (relation)88 1442 y(\(36\))32 b(will)h(de\014ne)h(a)e Fl(S)6 b(pin)1047 1406 y Fk(c)1114 1442 y Fv(connection)34 b(on)f Fp(S)1793 1406 y Fx(2)1832 1442 y Fv(.)234 1610 y(It)27 b(is)h(easy)g(to)e(see)i(that)f Fl(\014)1162 1650 y Fq(j)p Fn(S)1227 1627 y Fm(2)1227 1671 y Fg(\006)1311 1610 y Fv(=)g Fl(z)1463 1574 y Fi(\003)1459 1635 y(\006)1519 1610 y Fv(\()p Fl(d\013)1670 1625 y Fi(\006)1728 1610 y Fv(\).)42 b(Using)28 b(Stok)m(es)g(la)m(w)g(w)m(e)g(then)f(ha)m(v)m (e)i(that)502 1787 y Fo(Z)558 2013 y Fn(S)595 1994 y Fm(2)650 1923 y Fl(\014)k Fv(=)842 1787 y Fo(Z)897 2013 y Fk(C)973 1923 y Fl(z)1022 1882 y Fi(\003)1018 1947 y(\000)1077 1923 y Fv(\()p Fl(\013)1177 1938 y Fi(\000)1236 1923 y Fv(\))22 b(+)g Fl(z)1443 1882 y Fi(\003)1439 1947 y Fx(+)1499 1923 y Fv(\()p Fl(\013)1599 1938 y Fx(+)1658 1923 y Fv(\))27 b(=)1837 1855 y Fl(in)p 1837 1900 92 4 v 1858 1991 a Fv(2)1955 1787 y Fo(Z)2010 2013 y Fi(j)p Fk(z)s Fi(j)p Fx(=2)2206 1855 y Fl(d)p 2257 1800 50 4 v(z)p 2206 1900 101 4 v 2231 1936 50 4 v 2231 1991 a(z)2339 1923 y Fq(\000)2448 1855 y Fl(dz)p 2448 1900 101 4 v 2474 1991 a(z)2586 1923 y Fv(=)h(2)p Fl(\031)t(n)f(;)242 b Fv(\(39\))88 2206 y(where)29 b Fl(C)34 b Fv(is)29 b(the)f (equatiorial)h(curv)m(e)g(orien)m(ted)g(appropriately)-8 b(,)30 b(corresp)s(onding)e(to)g(the)88 2374 y(circle)33 b Fq(j)p Fl(z)t Fq(j)28 b Fv(=)g(2)k(b)s(eing)h(orien)m(ted)g(coun)m (terclo)s(c)m(kwise.)47 b(Note)33 b(that)796 2506 y Fo(Z)851 2732 y Fi(j)p Fk(z)s Fi(j)p Fx(=2)1037 2642 y Fv(4)p 1086 2587 50 4 v Fl(z)1136 2601 y Fi(\000)p Fx(2)1230 2642 y Fl(a)1281 2657 y Fx(+)1340 2642 y Fv(\()p Fq(\000)p Fv(4)p Fl(z)1553 2601 y Fi(\000)p Fx(1)1649 2642 y Fv(\))p Fl(d)p 1738 2587 V(z)f Fv(=)27 b Fq(\000)2012 2506 y Fo(Z)2068 2732 y Fi(j)p Fk(z)s Fi(j)p Fx(=2)2253 2642 y Fl(a)2304 2657 y Fx(+)2364 2642 y Fv(\()p Fl(z)t Fv(\))p Fl(d)p 2540 2587 V(z)5 b(:)88 2937 y Fw(48.)50 b(PR)m(OPOSITION.)d Fu(F)-7 b(or)27 b(any)g(close)-5 b(d)27 b(2-form)g Fl(\014)33 b Fu(on)27 b Fp(S)2377 2901 y Fx(2)2445 2937 y Fu(with)2649 2856 y Fo(R)2732 2937 y Fl(\014)33 b Fv(=)28 b(2)p Fl(\031)t(n)f Fu(ther)-5 b(e)88 3105 y(is)34 b(a)h Fl(S)6 b(pin)483 3069 y Fk(c)553 3105 y Fu(c)-5 b(onne)g(ction)33 b(on)i Fv(\011)1254 3120 y Fk(n)1336 3105 y Fu(such)f(that)i Fl(\014)k Fu(is)35 b(the)f(magnetic)g(2-form.)88 3356 y(Pr)-5 b(o)g(of.)48 b Fv(W)-8 b(e)32 b(m)m(ust)h(sho)m(w)f(that)g (there)g(are)g(one-forms)g Fl(\013)2171 3371 y Fi(\006)2261 3356 y Fv(on)g Fp(C)f Fv(satisfying)i(\(38\))e(suc)m(h)88 3524 y(that)i Fl(\014)355 3564 y Fq(j)p Fn(S)420 3541 y Fm(2)420 3585 y Fg(\006)505 3524 y Fv(=)c Fl(z)659 3488 y Fi(\003)655 3548 y(\006)715 3524 y Fv(\()p Fl(d\013)866 3539 y Fi(\006)924 3524 y Fv(\).)47 b(W)-8 b(e)34 b(construct)g(them)h (explicitly)-8 b(.)48 b(As)34 b(b)s(efore)g(w)m(e)g(will)h(write)88 3692 y Fl(\013)150 3707 y Fi(\006)236 3692 y Fv(=)28 b(2Re[)p Fl(a)582 3707 y Fi(\006)641 3692 y Fv(\()p Fl(z)t Fv(\))p Fl(d)p 817 3637 V(z)5 b Fv(])33 b(with)g Fl(a)1200 3707 y Fi(\006)1292 3692 y Fv(de\014ned)h(b)s(elo)m(w.)234 3860 y(Let)423 3833 y(~)409 3860 y Fl(\014)464 3875 y Fi(\006)550 3860 y Fv(=)28 b(\()p Fl(z)741 3818 y Fi(\000)p Fx(1)737 3881 y Fi(\006)836 3860 y Fv(\))874 3824 y Fi(\003)913 3860 y Fv(\()p Fl(\014)6 b Fv(\).)43 b(W)-8 b(e)33 b(de\014ne)960 4127 y Fl(h)1016 4142 y Fx(+)1075 4127 y Fv(\()p Fl(z)t Fv(\))c(:=)e Fl(\031)1418 4086 y Fi(\000)p Fx(1)1529 3992 y Fo(Z)1584 4217 y Fn(C)1653 4127 y Fv(log)18 b Fq(j)p Fl(z)26 b Fq(\000)d Fl(z)2044 4086 y Fi(0)2068 4127 y Fq(j)2096 4086 y Fx(2)2149 4101 y Fv(~)2135 4127 y Fl(\014)2190 4142 y Fx(+)2249 4127 y Fv(\()p Fl(z)2336 4086 y Fi(0)2360 4127 y Fv(\))28 b Fl(;)699 b Fv(\(40\))1309 4458 y Fl(a)1360 4473 y Fx(+)1419 4458 y Fv(\()p Fl(z)t Fv(\))29 b(:=)1721 4390 y Fl(i)p 1713 4435 49 4 v 1713 4526 a Fv(4)1772 4458 y Fl(@)p 1823 4435 40 3 v 15 x Fk(z)1863 4458 y Fl(h)1919 4473 y Fx(+)1978 4458 y Fv(\()p Fl(z)t Fv(\))88 4680 y(and)1005 4848 y Fl(a)1056 4863 y Fi(\000)1115 4848 y Fv(\()p Fl(z)t Fv(\))f(:=)g(4)p 1448 4793 50 4 v Fl(z)1497 4807 y Fi(\000)p Fx(2)1592 4848 y Fl(a)1643 4863 y Fx(+)1702 4848 y Fv(\()p Fq(\000)p Fv(4)p Fl(z)1915 4807 y Fi(\000)p Fx(1)2010 4848 y Fv(\))22 b(+)g Fl(i)2211 4809 y Fk(n)p 2211 4825 43 4 v 2215 4882 a Fx(2)p 2264 4793 50 4 v 2264 4848 a Fl(z)2314 4807 y Fi(\000)p Fx(1)1658 5251 y Fv(47)p eop %%Page: 48 48 48 47 bop 88 548 a Fv(according)27 b(to)h(\(38\).)41 b(A)27 b(simple)i(calculation)f(sho)m(ws)i(that)d Fl(a)2298 563 y Fi(\000)2357 548 y Fv(\()p Fl(z)t Fv(\))h(is)g(smo)s(oth)g(on)f Fp(C)p Fv(,)i(i.e.)88 716 y(the)41 b(singularities)h(apparen)m(tly)g (presen)m(t)g(in)f(\(38\))f(exactly)i(cancel)g(eac)m(h)f(other.)68 b(F)-8 b(or)88 884 y(this)32 b(argumen)m(t)h(w)m(e)h(use)f(that)1236 804 y Fo(R)1283 919 y Fn(C)1352 884 y Fl(\014)1407 899 y Fx(+)1494 884 y Fv(=)27 b(2)p Fl(\031)t(n)32 b Fv(and)h(that)f Fl(\014)2251 899 y Fx(+)2342 884 y Fv(is)h(the)f(pushforw)m(ard)i(of)e (a)88 1052 y(\(smo)s(oth\))g(2)g(form)h(on)f Fp(S)1010 1016 y Fx(2)1050 1052 y Fv(.)234 1220 y(From)43 b(the)i(de\014nition)g (of)e Fl(h)1312 1235 y Fx(+)1415 1220 y Fv(w)m(e)i(see)g(that)f(\()p Fl(@)p 2051 1197 40 3 v 15 x Fk(z)2091 1220 y Fl(@)2142 1235 y Fk(z)2182 1220 y Fl(h)p Fv(\()p Fl(z)t Fv(\)\))q Fl(dz)34 b Fq(^)d Fl(d)p 2680 1165 50 4 v(z)51 b Fv(=)c Fq(\000)p Fv(2)p Fl(i)3072 1194 y Fv(~)3058 1220 y Fl(\014)3113 1235 y Fx(+)3173 1220 y Fv(\()p Fl(z)t Fv(\),)88 1388 y(whic)m(h)40 b(implies)g Fl(d\013)823 1403 y Fx(+)920 1388 y Fv(=)1048 1362 y(~)1034 1388 y Fl(\014)1089 1403 y Fx(+)1148 1388 y Fv(.)62 b(Finally)39 b Fl(d\013)1688 1403 y Fi(\000)1785 1388 y Fv(=)1913 1362 y(~)1899 1388 y Fl(\014)1954 1403 y Fi(\000)2051 1388 y Fv(follo)m(ws)h(from)e(the)h (relation)g(\(38\))88 1556 y(and)32 b(that)h Fl(z)538 1520 y Fi(\003)534 1581 y(\000)593 1556 y Fv(\()p Fl(d\013)744 1571 y Fi(\000)803 1556 y Fv(\))27 b(=)h Fl(z)1021 1520 y Fi(\003)1017 1581 y Fx(+)1076 1556 y Fv(\()p Fl(d\013)1227 1571 y Fx(+)1286 1556 y Fv(\))k(on)h Fp(S)1547 1520 y Fx(2)1547 1581 y Fi(\000)1628 1556 y Fq(\\)23 b Fp(S)1772 1520 y Fx(2)1772 1581 y(+)1831 1556 y Fv(.)p 3255 1724 4 66 v 3259 1661 59 4 v 3259 1724 V 3317 1724 4 66 v 88 2058 a Fe(A.3)134 b(The)45 b(Dirac)g(op)t(erator)h(on)f FC(\011)1893 2076 y Fb(n)88 2290 y Fv(Let)34 b Fq(D)i Fv(b)s(e)e(the)h(Dirac)e(op)s(erator)h(corresp)s(onding)h(to)e(the)i Fl(S)6 b(pin)2465 2254 y Fk(c)2534 2290 y Fv(connection)35 b Fq(r)f Fv(on)g(a)88 2458 y(spinor)g(bundle)g(\011)778 2473 y Fk(n)858 2458 y Fv(on)g Fp(S)1050 2422 y Fx(2)1123 2458 y Fv(with)g(Chern)g(n)m(um)m(b)s(er)h Fl(n)p Fv(.)46 b(W)-8 b(e)34 b(ma)m(y)g(then)g(for)f(an)m(y)h(spinor)88 2626 y(\014eld)f Fl(\021)j Fv(in)d(\011)573 2641 y Fk(n)652 2626 y Fv(write)1031 2894 y Fl(\036)1089 2909 y Fi(\006)1148 2894 y Fv(\()1210 2868 y(~)1186 2894 y Fq(D)s Fl(\021)t Fv(\()p Fl(!)t Fv(\)\))26 b(=)i(\()p Fl(!)t(;)17 b Fq(D)1851 2909 y Fi(\006)1909 2894 y Fl(u)1965 2909 y Fi(\006)2023 2894 y Fv(\()p Fl(z)2106 2909 y Fi(\006)2165 2894 y Fv(\()p Fl(!)t Fv(\)\)\))88 3161 y(for)k Fl(!)31 b Fq(2)d Fp(S)467 3125 y Fx(2)467 3186 y Fi(\006)526 3161 y Fv(.)40 b(Here)23 b Fq(D)890 3176 y Fi(\006)971 3161 y Fv(are)f(the)g(Dirac)f(op)s (erators)h(on)g Fp(C)g Fv(corresp)s(onding)h(to)e(the)h(metric)88 3329 y Fl(ds)185 3293 y Fx(2)224 3329 y Fv(.)51 b(According)35 b(to)g(Theorem)h(23)f(w)m(e)h(can)f(express)j Fq(D)2183 3344 y Fi(\006)2277 3329 y Fv(in)d(terms)h(of)e(the)i(standard)88 3497 y(Dirac)23 b(op)s(erators)h(on)g Fp(C)g 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b(\(34\))f(the)g(spinors)88 716 y Fl(v)135 731 y Fi(\006)226 716 y Fv(satisfy)e(the)f(transformation)f(prop)s(ert)m(y)278 977 y(\012\()p Fl(z)t Fv(\))473 935 y Fi(\000)p Fx(1)p Fk(=)p Fx(2)639 977 y Fl(v)686 992 y Fi(\000)745 977 y Fv(\()p Fl(z)t Fv(\))c(=)g(\012\()p Fq(\000)p Fv(4)p Fl(z)1285 935 y Fi(\000)p Fx(1)1381 977 y Fv(\))1419 935 y Fi(\000)p Fx(1)p Fk(=)p Fx(2)1583 977 y Fq(U)10 b Fv(\()p Fq(\000)p Fv(4)p Fl(z)1868 935 y Fi(\000)p Fx(1)1964 977 y Fv(\))p Fq(W)e Fv(\()p Fq(\000)p Fv(4)p Fl(z)2321 935 y Fi(\000)p Fx(1)2417 977 y Fv(\))p Fl(v)2502 992 y Fx(+)2561 977 y Fv(\()p Fq(\000)p Fv(4)p Fl(z)2774 935 y Fi(\000)p Fx(1)2869 977 y Fv(\))28 b Fl(:)190 b Fv(\(42\))234 1237 y(Clearly)35 b(the)f(map)f(from)h Fl(\021)j Fv(to)c(\()p Fl(v)1482 1252 y Fi(\000)1541 1237 y Fl(;)17 b(v)1632 1252 y Fx(+)1691 1237 y Fv(\))34 b(with)g(the)g(ab)s(o)m(v)m(e)g(transformation)g(prop-)88 1405 y(ert)m(y)f(is)g(a)f(linear)h(isomorphism.)234 1573 y(It)43 b(turns)g(out)g(to)f(b)s(e)h(fairly)g(simple)i(to)d(c)m (haracterize)j(the)e(elemen)m(ts)i Fl(v)2964 1588 y Fi(\006)3066 1573 y Fv(in)e(the)88 1741 y(k)m(ernel)48 b(of)e(the)h(standard)g (Dirac)f(op)s(erators.)85 b(Since)48 b Fl(\033)2230 1756 y Fx(3)2316 1741 y Fv(an)m(ticomm)m(utes)h(with)e(the)88 1909 y(standard)37 b(Dirac)f(op)s(erator,)i(w)m(e)g(can)f(consider)h (the)f(cases)i Fl(\033)2385 1924 y Fx(3)2424 1909 y Fl(v)2471 1924 y Fi(\006)2566 1909 y Fv(=)34 b Fl(v)2723 1924 y Fi(\006)2819 1909 y Fv(and)j Fl(\033)3068 1924 y Fx(3)3108 1909 y Fl(v)3155 1924 y Fi(\006)3250 1909 y Fv(=)88 2077 y Fq(\000)p Fl(v)212 2092 y Fi(\006)304 2077 y Fv(separately)-8 b(.)44 b(Assume)35 b(for)d(de\014niteness)j(that)e Fl(\033)2096 2092 y Fx(3)2136 2077 y Fl(v)2183 2092 y Fi(\006)2269 2077 y Fv(=)28 b Fl(v)2420 2092 y Fi(\006)2479 2077 y Fv(,)33 b(i.e.)1268 2338 y([)p Fl(@)1350 2353 y Fx(\026)-39 b Fk(z)1408 2338 y Fq(\000)23 b Fl(ia)1592 2353 y Fi(\006)1651 2338 y Fv(\()p Fl(z)t Fv(\)])p Fl(v)1850 2353 y Fi(\006)1938 2338 y Fv(=)28 b(0)f Fl(:)1007 b Fv(\(43\))234 2598 y(W)-8 b(e)33 b(write)g Fl(v)698 2613 y Fi(\006)790 2598 y Fv(in)g(the)g(form) 1237 2859 y Fl(v)1284 2874 y Fi(\006)1343 2859 y Fv(\()p Fl(z)t Fv(\))c(=)e Fl(f)1648 2874 y Fi(\006)1707 2859 y Fv(\()p Fl(z)t Fv(\))p Fl(e)1877 2814 y Fi(\000)1942 2775 y Fx(1)p 1942 2791 36 4 v 1942 2848 a(4)1988 2814 y Fk(h)2029 2823 y Fg(\006)2081 2814 y Fx(\()p Fk(z)s Fx(\))88 3120 y Fv(where)33 b Fl(h)425 3135 y Fx(+)517 3120 y Fv(is)g(de\014ned)h(in)f(\(40\))f(and)983 3380 y Fl(h)1039 3395 y Fi(\000)1099 3380 y Fv(\()p Fl(z)t Fv(\))c(:=)g Fl(h)1439 3395 y Fx(+)1498 3380 y Fv(\()p Fq(\000)p Fv(4)p Fl(z)1711 3339 y Fi(\000)p Fx(1)1806 3380 y Fv(\))22 b(+)g(2)p Fl(n)17 b Fv(log)g Fq(j)p Fl(z)t Fq(j)2335 3339 y Fx(2)2403 3380 y Fl(:)88 3641 y Fv(One)36 b(easily)h(computes)h(that)e Fl(@)1277 3656 y Fx(\026)-39 b Fk(z)1313 3641 y Fl(h)1369 3656 y Fi(\006)1429 3641 y Fv(\()p Fl(z)t Fv(\))34 b(=)g Fl(ia)1782 3656 y Fi(\006)1841 3641 y Fv(\()p Fl(z)t Fv(\),)k(and)e(using)h(\(43\))e(w)m(e)j(obtain)e (that)88 3809 y Fl(@)143 3824 y Fx(\026)-39 b Fk(z)179 3809 y Fl(f)227 3824 y Fi(\006)286 3809 y Fv(\()p Fl(z)t Fv(\))28 b(=)g(0,)f(i.e.)41 b(these)27 b(are)e(analytic)h(functions.)43 b(Finally)25 b(\(42\))g(giv)m(es)i(the)f(follo)m(wing)88 3977 y(relation)32 b(b)s(et)m(w)m(een)j Fl(f)870 3992 y Fi(\000)962 3977 y Fv(and)d Fl(f)1199 3992 y Fx(+)1076 4237 y Fl(f)1124 4252 y Fi(\000)1183 4237 y Fv(\()p Fl(z)t Fv(\))c(=)g(2\()p Fq(\000)p Fl(z)t Fv(\))1691 4196 y Fk(n)p Fi(\000)p Fx(1)1829 4237 y Fl(f)1877 4252 y Fx(+)1936 4237 y Fv(\()p Fq(\000)p Fv(4)p Fl(z)2149 4196 y Fi(\000)p Fx(1)2244 4237 y Fv(\))g Fl(:)88 4498 y Fv(Hence)47 b Fl(f)439 4513 y Fi(\000)544 4498 y Fv(is)f(an)g(analytic)h(function,)j (b)s(ounded)c(b)m(y)h(a)f(constan)m(t)h(times)g Fq(j)p Fl(z)t Fq(j)3056 4462 y Fk(n)p Fi(\000)p Fx(1)3239 4498 y Fv(at)88 4666 y(in\014nit)m(y)-8 b(.)41 b(Then)25 b Fl(n)j Fq(\025)g Fv(1)23 b(and)g Fl(f)1187 4681 y Fi(\000)1269 4666 y Fv(is)h(a)f(p)s(olynomial)g(of)g(degree)h(at)f(most)h Fl(n)s Fq(\000)s Fv(1.)41 b(A)23 b(basis)h(in)88 4834 y(the)h(k)m(ernel)h(of)e(the)h(Dirac)f(op)s(erator)g(is)i(obtained)f(b) m(y)g(c)m(ho)s(osing)g Fl(f)2478 4849 y Fi(\000)2538 4834 y Fv(\()p Fl(z)t Fv(\))j(=)f(1)p Fl(;)17 b(z)t(;)g(:)g(:)g(:)f(z) 3160 4798 y Fk(n)p Fi(\000)p Fx(1)3298 4834 y Fv(,)88 5002 y(and)32 b(the)h(dimension)h(is)g Fl(n)p Fv(.)1658 5251 y(49)p eop %%Page: 50 50 50 49 bop 234 548 a Fv(Similar)48 b(argumen)m(t)f(sho)m(ws)i(that)e(if) g Fl(\033)1715 563 y Fx(3)1755 548 y Fl(v)1802 563 y Fi(\006)1914 548 y Fv(=)52 b Fq(\000)p Fl(v)2166 563 y Fi(\006)2226 548 y Fv(,)f(then)d Fl(n)k Fq(\024)h(\000)p Fv(1,)f(and)47 b(the)88 716 y(dimension)29 b(of)e(the)h(space)h(of)e (suc)m(h)i(zero)f(mo)s(des)h(is)f Fq(j)p Fl(n)p Fq(j)p Fv(.)41 b(In)28 b(particular)g(all)g(zero)g(mo)s(des)88 884 y(ha)m(v)m(e)46 b(de\014nite)g(spin)g(and)f(only)h(one)f(of)g(the)g (t)m(w)m(o)h(eigenspaces)i(of)c Fl(\033)2731 899 y Fx(3)2820 884 y Fv(=)49 b Fl(\033)t Fv(\()p Fl(\027)6 b Fv(\))45 b(can)88 1052 y(accomo)s(date)32 b(zero)h(mo)s(des,)g(dep)s(ending)h (on)e(the)g(sign)h(of)f Fl(n)p Fv(.)44 b(Recalling)33 b(that)f Fl(n)g Fv(is)h(the)88 1220 y(total)f(\015ux)h(divided)h(b)m(y) g(2)p Fl(\031)t Fv(,)e(w)m(e)i(ha)m(v)m(e)f(completed)i(the)e(pro)s(of) e(of)h(Theorem)i(37.)p 3255 1220 4 66 v 3259 1157 59 4 v 3259 1220 V 3317 1220 4 66 v 88 1608 a Fr(References)88 1874 y Fv([AMN1])52 b(Adam,)31 b(Christoph)g(&)f(Muratori,)h(Bruno)g(&) f(Nash,)h(Charles,)h Fu(Zer)-5 b(o)32 b(mo)-5 b(des)478 2042 y(of)32 b(the)g(Dir)-5 b(ac)32 b(op)-5 b(er)g(ator)32 b(in)h(thr)-5 b(e)g(e)32 b(dimensions.)c Fv(Ph)m(ys.)k(Rev.)f Fw(D60)p Fv(,)f(\(1999\),)478 2210 y(125001.)88 2461 y([AMN2])52 b(Adam,)31 b(Christoph)h(&)e(Muratori,)h(Bruno)g(&)f(Nash,) h(Charles,)i Fu(De)-5 b(gener)g(acy)478 2629 y(of)37 b(zer)-5 b(o)37 b(mo)-5 b(des)37 b(of)g(the)h(Dir)-5 b(ac)37 b(op)-5 b(er)g(ator)37 b(in)h(thr)-5 b(e)g(e)37 b(dimensions.)d Fv(Preprin)m(t.)478 2797 y Fa (http://xxx.lanl.gov/abs/he)q(p-th)q(/991)q(013)q(9)p Fv(,)k(\(1999\))88 3048 y([A)m(C])196 b(Aharono)m(v,)64 b(Y)-8 b(akir)58 b(&)f(Casher,)65 b(Aharon,)f Fu(Gr)-5 b(ound)58 b(state)g(of)g(spin-1/2)478 3216 y(char)-5 b(ge)g(d)47 b(p)-5 b(article)47 b(in)h(a)g(two-dimensional)e(magnetic)h (\014eld.)f Fv(Ph)m(ys.)j(Rev.)478 3384 y(A\(3\))32 b Fw(19)p Fv(,)h(\(1979\),)e(2461{2462.)88 3635 y([B])267 b(B\177)-49 b(ar,)32 b(Christian,)i Fu(Metrics)h(with)g(harmonic)f (spinors.)d Fv(Geom.)i(F)-8 b(unc.)33 b(Anal.)478 3803 y Fw(6)p Fv(,)f(\(1996\),)g(899-942.)88 4054 y([E])270 b(Elton,)33 b(Daniel)f(M.,)h Fu(A)n(n)i(example)f(of)g(a)h(zer)-5 b(o)34 b(mo)-5 b(de.)32 b Fv(Preprin)m(t)i(\(1998\).)88 4305 y([ES])216 b(Erd})-49 b(os,)37 b(L\023)-49 b(aszl\023)g(o)36 b(&)f(Solo)m(v)m(ej,)j(Jan)d(Philip,)j Fu(On)f(the)g(kernel)g(of)h Fl(S)6 b(pin)3019 4269 y Fk(c)3091 4305 y Fu(Dir)-5 b(ac)478 4473 y(op)g(er)g(ators)36 b(on)h Fp(S)1100 4437 y Fx(3)1177 4473 y Fu(and)g Fp(R)1441 4437 y Fx(3)1480 4473 y Fu(.)e Fv(T)-8 b(o)35 b(app)s(ear)g(in)g(the)h(Pro)s(ceedings)g(of)f(the)g (1999)478 4641 y(GeorgiaT)-8 b(ec)m(h-UAB)43 b(In)m(ternational)h (Conference)h(on)e(Di\013eren)m(tial)g(Equa-)478 4809 y(tions)33 b(and)f(Mathematical)i(Ph)m(ysics.)1658 5251 y(50)p eop %%Page: 51 51 51 50 bop 88 548 a Fv([FLL])150 b(F)-8 b(r\177)-49 b(ohlic)m(h,)58 b(J)s(\177)-51 b(urg)52 b(&)h(Lieb,)59 b(Elliott)53 b(H.)g(&)g(Loss,)58 b(Mic)m(hael,)i Fu(Stability)54 b(of)478 716 y(Coulomb)45 b(systems)g(with)h(magnetic)e(\014elds.)h(I.)g(The)g(one-ele)-5 b(ctr)g(on)45 b(atom.)478 884 y Fv(Comm)m(un.)34 b(Math.)f(Ph)m(ys.)h Fw(104)f Fv(\(1986\),)e(no.)i(2,)f(251{270.)88 1135 y([GLP])132 b(Gilk)m(ey)-8 b(,)36 b(P)m(eter)h(B.)e(&)f(Leah)m(y)-8 b(,)37 b(John)e(V.)g(&)g(P)m(ark,)h(Jeongh)m(y)m(eong,)h Fu(Spinors,)478 1303 y(sp)-5 b(e)g(ctr)g(al)31 b(ge)-5 b(ometry,)32 b(and)f(R)n(iemannian)f(submersions.)d Fv(Lecture)k(Notes) e(Se-)478 1471 y(ries,)49 b(40,)f(Seoul)e(National)e(Univ)m(ersit)m(y) -8 b(,)51 b(Researc)m(h)c(Institute)g(of)d(Mathe-)478 1639 y(matics,)34 b(Global)f(Analysis)h(Researc)m(h)h(Cen)m(ter,)g (Seoul,)f(1998.)e(Can)i(also)f(b)s(e)478 1807 y(found)f(online)i(at)e Fa(http://www.emc.dk/EMIS/mon)q(ogr)q(aphs)q(/GLP)q(/)88 2058 y Fv([H])263 b(Hitc)m(hin,)47 b(Nigel,)f Fu(Harmonic)e(spinors.)e Fv(Adv)-5 b(ances)45 b(in)f(Math.)f Fw(14)g Fv(\(1974\),)478 2226 y(1{55.)88 2477 y([LL])214 b(Lieb,)31 b(Elliott)g(H.)g(&)g(Loss,)h (Mic)m(hael,)g Fu(Stability)i(of)e(Coulomb)h(systems)g(with)478 2645 y(magnetic)43 b(\014elds.)g(II.)h(The)f(many-ele)-5 b(ctr)g(on)44 b(atom)g(and)f(the)i(one-ele)-5 b(ctr)g(on)478 2813 y(mole)g(cule.)31 b Fv(Comm)m(un.)j(Math.)f(Ph)m(ys.)i Fw(104)d Fv(\(1986\),)g(no.)g(2,)h(271{282.)88 3064 y([L)-8 b(Y])210 b(Loss,)25 b(Mic)m(hael)e(&)f(Y)-8 b(au,)24 b(Horng-Tzer,)h Fu(Stability)g(of)g(Coulomb)g(systems)g(with)478 3232 y(magnetic)31 b(\014elds.)g(III.)g(Zer)-5 b(o)32 b(ener)-5 b(gy)32 b(b)-5 b(ound)32 b(states)g(of)g(the)h(Pauli)f(op)-5 b(er)g(ator.)478 3400 y Fv(Comm)m(un.)34 b(Math.)f(Ph)m(ys.)h Fw(104)f Fv(\(1986\),)e(no.)i(2,)f(283{290.)88 3676 y Fu(A)-5 b(ddr)g(ess)34 b(of)h(the)g(authors:)88 3844 y(L\023)-50 b(aszl\023)g(o)38 b(Er)-5 b(d})-50 b(os)p Fv(:)53 b(Sc)m(ho)s(ol)37 b(of)g(Mathematics,)k(Georgia)36 b(Institute)j(of)d(T)-8 b(ec)m(hnology)g(,)41 b(A)m(t-)88 4012 y(lan)m(ta)32 b(GA)g(30332,)g(USA.)88 4180 y Fu(E-mail)p Fv(:)42 b(lerdos@math.gatec)m(h.edu)88 4448 y Fu(Jan)34 b(Philip)g(Solovej)p Fv(:)42 b(Departmen)m(t)33 b(of)f(Mathematics,)i (Univ)m(ersit)m(y)h(of)c(Cop)s(enhagen,)88 4616 y(Univ)m(ersitetspark)m (en)36 b(5,)c(DK-2100)f(Cop)s(enhagen,)j(Denmark.)88 4784 y Fu(E-mail)p Fv(:)42 b(solo)m(v)m(ej@math.ku.dk)1658 5251 y(51)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0001231634141--