Content-Type: multipart/mixed; boundary="-------------0403181116567" This is a multi-part message in MIME format. ---------------0403181116567 Content-Type: text/plain; name="04-87.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-87.keywords" Topological methods, multidimensional dynamamical systems, local Brouwer degree. ---------------0403181116567 Content-Type: application/postscript; name="COV.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="COV.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software %%Title: COV.dvi %%CreationDate: Thu Mar 18 08:37:21 2004 %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR12 CMR7 CMR10 CMBX9 CMR9 CMMI10 MSBM10 CMSY10 %%+ CMR6 CMR8 CMMI7 CMBX10 CMTI10 CMSY7 CMMI5 CMSY5 CMR5 CMEX10 %%+ TimesNewRomanPSMT CMMI12 CMMI8 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf COV %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2004.03.18:0837 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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{A A 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/IEn 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 IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/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 A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 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 A 1 get A mul exch 0 get A 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/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 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/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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%%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 84 /T put dup 99 /c put dup 105 /i put dup 107 /k put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 164 /asteriskmath put dup 167 /plusminus put dup 50 /element put dup 106 /bar put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 174 /ff put dup 175 /fi put dup 182 /acute put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 54 /six put dup 55 /seven put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W 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 196 /dieresis put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 175 /fi put dup 177 /ffi put dup 40 /parenleft put dup 41 /parenright put dup 46 /period 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 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a 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 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 174 /alpha put dup 177 /delta put dup 178 /epsilon1 put dup 184 /lambda put dup 188 /pi put dup 190 /sigma put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater 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 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 82 /R put dup 83 /S put dup 85 /U put dup 86 /V put dup 88 /X put dup 90 /Z put dup 97 /a put dup 99 /c put dup 100 /d 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 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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05AA57BD77B0AB2506EBEF3A5AB3EB6E18C449F9133377E7D1E4FA6547001CA2 080D2E9119A8FFE980639FCBB3099943EA7BF33B8BB05EF83AB44078B6E6A8EF 7E7CE75F84E4C34CF1C0A002D074CC093D6405C4D49A54A3F3A4525A102E98CC B99FF6731977E40186DDF81CCD26D4FB10DEFADE281311922DCCD310D5CA0969 2B638B83346AB8E5DE35471E9D1B7B014E28E0DB5A83CD63055215DCBDE003F3 E6A2F698C9183CC6ACF52F02A2338974B58BD43A5374491159B4FFFEA3D02D57 E66E8756F1531087E7F7E5E5D4271B3731F07DFC2E5CE20DC49C308B400A4571 DFDCBA7F7ACF01D870BBBF316D9381A0AB0A59A0A752A20D3E5CFDFB53F00B64 B92EE9030A7799BE3E7DFC94DD10922D4A10D23E4AE8459B9EB7D46A4BFA7956 416B2F0E0DE7479388928329B43AC82F0C4D2DF415516E64039EB6C9431FCBEE 6F2FE2789C4640A11C424B3AFA1BFDEB52AE6E25F935438556FA7CFC14B482D6 240E6D3EDBD75A3E1D69BC2B11B20EB3D90F137D7DF6A8F5EA4312689A355052 25B706ED2E37DD5986E43A3C95C06A1837C45CE25E8CCA877F0849BFA22C65BA B88B4A6359AE947592CEDC2E47621831CE62CD4CB596E13C4A787B5F4BE21936 BE5AACA722A3CFC17D1B7DD6FF54B58A1732884E81C9F6860E96A87B8855C3EB 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%%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 162 /Delta put dup 166 /Pi put dup 167 /Sigma put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 182 /acute put dup 185 /macron put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright 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 61 /equal 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 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright 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 126 /tilde put dup 196 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 162 /Delta put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 182 /acute put dup 49 /one put dup 61 /equal put dup 71 /G put dup 77 /M put dup 80 /P put dup 90 /Z put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 105 /i put dup 107 /k put dup 108 /l put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 121 /y put dup 122 /z put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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30250 30167 2789 45 v 246 w Fq(B)31090 31256 y Fl(s)p Fv(\()p Fl(N)83 b Fv(\))33039 31056 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fq(:)6863 33730 y Fu(With)370 b(an)f(abuse)g(of)h (notation,)i(w)-31 b(e)370 b(will)h(denote)f(suc)-31 b(h)369 b(a)g(quadruple)h(b)-31 b(y)369 b Fq(N)121 b Fu(.)493 b(W)-92 b(e)368 b(denote)18047 35996 y Fq(N)18936 36162 y Fl(c)19695 35996 y Fu(=)p 20863 35107 2895 45 v 307 w Fq(B)21703 36196 y Fl(u)p Fv(\()p Fl(N)83 b Fv(\))23758 35996 y Fu(\(0)p Fq(;)184 b Fu(1\))248 b Fo(\243)p 27570 35107 2789 45 v 246 w Fq(B)28410 36196 y Fl(s)p Fv(\()p Fl(N)83 b Fv(\))30359 35996 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fq(;)16981 37810 y(N)17991 37353 y Fi(\241)17870 38084 y Fl(c)19045 37810 y Fu(=)308 b Fq(@)p 20863 36921 2895 45 v 61 w(B)21703 38010 y Fl(u)p Fv(\()p Fl(N)83 b Fv(\))23758 37810 y Fu(\(0)p Fq(;)184 b Fu(1\))248 b Fo(\243)p 27570 36921 2789 45 v 246 w Fq(B)28410 38010 y Fl(s)p Fv(\()p Fl(N)83 b Fv(\))30359 37810 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fq(;)16993 39624 y(N)18003 39167 y Fv(+)17882 39898 y Fl(c)19045 39624 y Fu(=)p 20214 38735 2895 45 v 308 w Fq(B)21054 39824 y Fl(u)p Fv(\()p Fl(N)83 b Fv(\))23108 39624 y Fu(\(0)p Fq(;)184 b Fu(1\))249 b Fo(\243)d Fq(@)p 27570 38735 2789 45 v 61 w(B)28410 39824 y Fl(s)p Fv(\()p Fl(N)83 b Fv(\))30359 39624 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fq(;)16391 41433 y(N)17401 40976 y Fi(\241)18456 41433 y Fu(=)308 b Fq(c)20104 40960 y Fi(\241)p Fv(1)20104 41759 y Fl(N)21292 41433 y Fu(\()p Fq(N)22732 40976 y Fi(\241)22611 41707 y Fl(c)23480 41433 y Fu(\))p Fq(;)1292 b(N)26519 40976 y Fv(+)27562 41433 y Fu(=)307 b Fq(c)29209 40960 y Fi(\241)p Fv(1)29209 41759 y Fl(N)30398 41433 y Fu(\()p Fq(N)31838 40976 y Fv(+)31717 41707 y Fl(c)32573 41433 y Fu(\))p Fq(:)8524 43699 y Fu(Hence)520 b(an)h Fq(h)p Fu(-set)g Fq(N)641 b Fu(is)521 b(a)g(pro)31 b(duct)521 b(of)g(t)-31 b(w)g(o)523 b(closed)e(balls)g(with)h(resp)31 b(ect)520 b(to)i(some)6863 45027 y(co)31 b(ordinate)442 b(system.)709 b(The)441 b(n)-31 b(um)g(b)31 b(ers)440 b Fq(u)p Fu(\()p Fq(N)121 b Fu(\))442 b(and)f Fq(s)p Fu(\()p Fq(N)121 b Fu(\))441 b(stand)h(for)f(the)g(dimensions)h(of)6863 46356 y(nominally)464 b(unstable)d(and)h(stable)f(directions,)485 b(resp)31 b(ectiv)-31 b(ely)-92 b(.)768 b(The)461 b(subscript)g Fq(c)f Fu(refers)6863 47684 y(to)382 b(the)e(new)h(co)31 b(ordinates)382 b(giv)-31 b(en)382 b(b)-31 b(y)380 b(homeomorphism)j Fq(c)31578 47850 y Fl(N)32418 47684 y Fu(.)527 b(Notice)382 b(that)f(if)h Fq(u)p Fu(\()p Fq(N)121 b Fu(\))327 b(=)f(0,)6863 49012 y(then)370 b Fq(N)10395 48611 y Fi(\241)11449 49012 y Fu(=)308 b Fo(;)369 b Fu(and)h(if)f Fq(s)p Fu(\()p Fq(N)121 b Fu(\))309 b(=)e(0,)370 b(then)g Fq(N)25335 48611 y Fv(+)26377 49012 y Fu(=)308 b Fo(;)p Fu(.)6863 51278 y Fk(De\257nition)426 b(2)554 b Fj(Assume)585 b Fq(N)59 b(;)184 b(M)708 b Fj(ar)-57 b(e)586 b Fq(h)p Fj(-sets,)632 b(such)585 b(that)g Fq(u)p Fu(\()p Fq(N)121 b Fu(\))652 b(=)f Fq(u)p Fu(\()p Fq(M)121 b Fu(\))651 b(=)g Fq(u)585 b Fj(and)6863 52606 y Fq(s)p Fu(\()p Fq(N)121 b Fu(\))309 b(=)e Fq(s)p Fu(\()p Fq(M)121 b Fu(\))308 b(=)f Fq(s)p Fj(.)493 b(L)-57 b(et)347 b Fq(f)426 b Fu(:)308 b Fq(N)429 b Fo(!)307 b Fp(R)23106 52117 y Fl(n)24054 52606 y Fj(b)-57 b(e)347 b(a)e(c)-57 b(ontinuous)347 b(map.)493 b(L)-57 b(et)346 b Fq(f)37067 52772 y Fl(c)37826 52606 y Fu(=)307 b Fq(c)39473 52772 y Fl(M)40592 52606 y Fo(\261)137 b Fq(f)257 b Fo(\261)137 b Fq(c)43250 52133 y Fi(\241)p Fv(1)43250 52932 y Fl(N)44746 52606 y Fu(:)6863 53935 y Fq(N)7752 54101 y Fl(c)8511 53935 y Fo(!)307 b Fp(R)10725 53446 y Fl(u)11550 53935 y Fo(\243)246 b Fp(R)13456 53446 y Fl(s)13929 53935 y Fj(.)510 b(We)396 b(say)g(that)23657 55589 y Fq(N)25607 54909 y Fl(f)24974 55589 y Fu(=)-184 b Fo(\))307 b Fq(M)6863 57480 y Fj(\()p Fq(N)517 b(f)119 b Fj(-c)-57 b(overs)396 b Fq(M)121 b Fj(\))396 b(i\256)g(the)g(fol)57 b(lowing)396 b(c)-57 b(onditions)397 b(ar)-57 b(e)396 b(satis\257e)-57 b(d)6863 59558 y Fk(1.)554 b Fj(Ther)-57 b(e)379 b(exists)e(a)i(c)-57 b(ontinuous)379 b(homotopy)f Fq(h)307 b Fu(:)h([0)p Fq(;)184 b Fu(1])211 b Fo(\243)d Fq(N)32043 59724 y Fl(c)32801 59558 y Fo(!)308 b Fp(R)35015 59069 y Fl(u)35802 59558 y Fo(\243)208 b Fp(R)37670 59069 y Fl(s)38143 59558 y Fj(,)383 b(such)378 b(that)g(the)9631 60886 y(fol)57 b(lowing)395 b(c)-57 b(onditions)398 b(hold)d(true)28286 63152 y Fq(h)28924 63318 y Fv(0)30527 63152 y Fu(=)1107 b Fq(f)33037 63318 y Fl(c)33488 63152 y Fq(;)20704 64813 y(h)p Fu(\([0)p Fq(;)184 b Fu(1])p Fq(;)g(N)25484 64355 y Fi(\241)25363 65086 y Fl(c)26235 64813 y Fu(\))247 b Fo(\\)f Fq(M)28970 64979 y Fl(c)30527 64813 y Fu(=)1107 b Fo(;)p Fq(;)20716 66473 y(h)p Fu(\([0)p Fq(;)184 b Fu(1])p Fq(;)g(N)25375 66639 y Fl(c)25830 66473 y Fu(\))247 b Fo(\\)f Fq(M)28686 66016 y Fv(+)28565 66747 y Fl(c)30527 66473 y Fu(=)1107 b Fo(;)p Fq(:)25681 70071 y Fu(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 6863 6974 a Fk(2.)554 b Fj(Ther)-57 b(e)396 b(exists)f(a)h(line)-57 b(ar)397 b(map)f Fq(A)308 b Fu(:)g Fp(R)23693 6485 y Fl(u)24579 6974 y Fo(!)g Fp(R)26793 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b(et)396 b Fq(N)517 b Fj(b)-57 b(e)397 b(an)g Fq(h)p Fj(-set.)509 b(We)396 b(de\257ne)g(the)g Fq(h)p Fj(-set)g Fq(N)34313 33964 y Fl(T)35406 34365 y Fj(as)g(fol)57 b(lows)8524 36579 y Fo(\262)554 b Fj(The)561 b(c)-57 b(omp)g(act)563 b(subset)e(of)h(the)g(quadruple)f Fq(N)29641 36178 y Fl(T)30900 36579 y Fj(is)h(the)g(c)-57 b(omp)g(act)562 b(subset)f(of)i(the)9631 37908 y(quadruple)395 b Fq(N)121 b Fj(,)397 b(also)e(denote)-57 b(d)396 b(by)h Fq(N)121 b Fj(,)8524 40122 y Fo(\262)554 b Fq(u)p Fu(\()p Fq(N)11705 39720 y Fl(T)12402 40122 y Fu(\))308 b(=)g Fq(s)p Fu(\()p Fq(N)121 b Fu(\))p Fj(,)397 b Fq(s)p Fu(\()p Fq(N)19393 39720 y Fl(T)20091 40122 y Fu(\))308 b(=)f Fq(u)p Fu(\()p Fq(N)121 b Fu(\))8524 42596 y Fo(\262)554 b Fj(The)396 b(home)-57 b(omorphism)396 b Fq(c)20591 42814 y Fl(N)21375 42593 y Fh(T)22350 42596 y Fu(:)308 b Fp(R)23764 42107 y Fl(n)24674 42596 y Fo(!)g Fp(R)26888 42107 y Fl(n)27798 42596 y Fu(=)g Fp(R)29766 42107 y Fl(u)p Fv(\()p Fl(N)31419 41773 y Fh(T)32032 42107 y Fv(\))32679 42596 y Fo(\243)246 b Fp(R)34585 42107 y Fl(s)p Fv(\()p Fl(N)36133 41773 y Fh(T)36746 42107 y Fv(\))37543 42596 y Fj(is)396 b(de\257ne)-57 b(d)397 b(by)22642 45031 y Fq(c)23121 45249 y Fl(N)23905 45028 y Fh(T)24573 45031 y Fu(\()p Fq(x)p Fu(\))308 b(=)f Fq(j)63 b Fu(\()p Fq(c)28970 45197 y Fl(N)29810 45031 y Fu(\()p Fq(x)p Fu(\)\))p Fq(;)9631 47596 y Fj(wher)-57 b(e)395 b Fq(j)371 b Fu(:)308 b Fp(R)14998 47107 y Fl(u)p Fv(\()p Fl(N)83 b Fv(\))17298 47596 y Fo(\243)246 b Fp(R)19205 47107 y Fl(s)p Fv(\()p Fl(N)83 b Fv(\))21462 47596 y Fo(!)307 b Fp(R)23676 47107 y Fl(s)p Fv(\()p Fl(N)83 b Fv(\))25871 47596 y Fo(\243)246 b Fp(R)27777 47107 y Fl(u)p Fv(\()p Fl(N)83 b Fv(\))30228 47596 y Fj(is)396 b(given)i(by)e Fq(j)63 b Fu(\()p Fq(p;)184 b(q)40 b Fu(\))310 b(=)d(\()p Fq(q)40 b(;)184 b(p)p Fu(\))p Fj(.)6863 50031 y Fu(Notice)517 b(that)g Fq(N)14039 49629 y Fl(T)71 b(;)p Fv(+)16182 50031 y Fu(=)551 b Fq(N)18604 49629 y Fi(\241)19866 50031 y Fu(and)516 b Fq(N)23175 49629 y 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b(de\257ne)-57 b(d)488 b(and)f(c)-57 b(ontinuous.)785 b(We)488 b(say)f(that)f Fq(N)30229 55955 y Fl(g)29567 56634 y Fo(\()-184 b Fu(=)473 b Fq(M)608 b Fj(\()p Fq(N)g(g)40 b Fj(-b)-57 b(ackc)g(overs)488 b Fq(M)121 b Fj(\))487 b(i\256)6863 58515 y Fq(M)8058 58114 y Fl(T)9209 57836 y(g)9669 57502 y Fg(\241)p Ff(1)9063 58515 y Fu(=)-184 b Fo(\))306 b Fq(N)12163 58114 y Fl(T)12861 58515 y Fj(.)8524 60951 y Fu(F)-92 b(ollo)-31 b(wing)267 b([1)q(],)286 b(let)266 b(us)e(p)31 b(oin)-31 b(t)266 b(out)g(that,)287 b(although)267 b(co)-31 b(v)g(ering)266 b(and)f(bac)-31 b(k)g(co)g(v)g(ering)268 b(o)31 b(ccur)6863 62279 y(often)430 b(sim)-31 b(ultaneously)-92 b(,)447 b(they)430 b(are)f(not)h(equiv)-61 b(alen)-31 b(t,)446 b(for)429 b(example)i(it)f(can)f(happ)31 b(en)429 b(that)6863 63607 y(the)370 b(map)g Fq(f)488 b Fu(is)369 b(not)h(de\257ned)f(on)g Fq(N)121 b Fu(.)25681 70071 y(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 6863 6974 a Fk(Theorem)426 b(4)554 b Fj(L)-57 b(et)371 b Fq(N)16233 7140 y Fl(i)16601 6974 y Fj(,)376 b Fq(i)308 b Fu(=)f(0)p Fq(;)184 b(:)g(:)g(:)189 b(;)184 b(k)406 b Fj(b)-57 b(e)371 b(h-sets)e(and)h Fq(N)30627 7140 y Fl(k)31480 6974 y Fu(=)307 b Fq(N)33537 7140 y Fv(0)34034 6974 y Fj(.)501 b(Assume)370 b(that)f(for)h(e)-57 b(ach)6863 8302 y Fq(i)308 b Fu(=)f(1)p Fq(;)184 b(:)g(:)g(:)189 b(;)184 b(k)432 b Fj(we)396 b(have)g(either)22781 9957 y Fq(N)23670 10123 y Fl(i)p Fi(\241)p Fv(1)25980 9277 y Fl(f)26412 9388 y Fh(i)25479 9957 y Fu(=)-184 b Fo(\))307 b Fq(N)28459 10123 y Fl(i)28828 9957 y Fq(;)14504 b Fu(\(3\))6863 11949 y Fj(or)16939 13389 y Fq(N)17828 13555 y Fl(i)18505 13389 y Fo(\275)307 b Fj(dom)185 b Fu(\()p Fq(f)22986 12915 y Fi(\241)p Fv(1)22867 13697 y Fl(i)24175 13389 y Fu(\))1132 b Fj(and)f Fq(N)29511 13555 y Fl(i)p Fi(\241)p Fv(1)31821 12709 y Fl(f)32253 12820 y Fh(i)31321 13389 y Fo(\()-184 b Fu(=)307 b Fq(N)34301 13555 y Fl(i)34670 13389 y Fq(:)8662 b Fu(\(4\))8524 15381 y Fj(Then)396 b(ther)-57 b(e)396 b(exists)f(a)h(p)-57 b(oint)397 b 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b Fq(i)308 b Fu(=)f(0)p Fq(;)184 b(:)g(:)g(:)k(;)c(k)282 b Fo(\241)246 b Fu(1)q Fq(;)18701 26008 y(f)19243 26174 y Fl(i)19918 26008 y Fu(=)308 b Fq(f)21629 26174 y Fl(i;c)22656 26008 y Fq(;)2399 b Fu(for)369 b Fq(i)308 b Fu(=)f(1)p Fq(;)184 b(:)g(:)g(:)189 b(;)184 b(k)36 b(;)21323 27668 y(N)22212 27834 y Fl(i)22889 27668 y Fu(=)307 b Fq(N)24946 27834 y Fl(c;i)25974 27668 y Fq(;)1292 b(N)28583 27195 y Fi(\247)28462 27977 y Fl(i)29637 27668 y Fu(=)308 b Fq(N)31816 27195 y Fi(\247)31695 27977 y Fl(i;c)32722 27668 y Fq(:)8524 30104 y Fu(In)428 b(order)h(to)h(simplify)g(the)g(exp)31 b(osition)431 b(w)-31 b(e)429 b(set)g Fq(N)30272 30270 y Fi(\241)p Fv(1)31869 30104 y Fu(=)407 b Fq(N)34026 30270 y Fl(k)19 b Fi(\241)p Fv(1)36133 30104 y Fu(and)429 b Fq(N)39234 30270 y Fl(k)40187 30104 y Fu(=)407 b Fq(N)42344 30270 y Fv(0)43270 30104 y Fu(and)6863 31432 y(accordingly)352 b Fq(f)13262 31598 y Fv(0)14066 31432 y Fu(=)308 b Fq(f)15777 31598 y Fl(k)16321 31432 y Fu(.)486 b(W)-92 b(e)350 b(also)h(de\257ne)f Fq(g)24910 31598 y Fl(i)25586 31432 y Fu(=)307 b Fq(f)27415 30958 y Fi(\241)p Fv(1)27296 31741 y Fl(i)28604 31432 y Fu(,)354 b(for)c(those)h Fq(i)f Fu(for)g(whic)-31 b(h)351 b(w)-31 b(e)351 b(ha)-31 b(v)g(e)351 b(the)6863 33180 y(bac)-31 b(k-co)g(v)g(ering)372 b(relation)f Fq(N)18736 33346 y Fl(i)p Fi(\241)p Fv(1)21046 32500 y Fl(f)21478 32611 y Fh(i)20545 33180 y Fo(\()-184 b Fu(=)307 b Fq(N)23525 33346 y Fl(i)23894 33180 y Fu(.)8524 34508 y(Notice)426 b(that)g(from)f(the)g(de\257nition)h(of)f(co)-31 b(v)g(ering)426 b(relation,)441 b(it)426 b(follo)-31 b(ws)427 b(immediately)6863 35837 y(that)503 b(there)f(exist)g Fq(u)528 b Fo(\270)g Fu(0,)536 b Fq(s)528 b Fo(\270)h Fu(0,)535 b(suc)-31 b(h)502 b(that)h Fq(u)p Fu(\()p Fq(N)30004 36003 y Fl(i)30373 35837 y Fu(\))529 b(=)f Fq(u)501 b Fu(and)h Fq(s)p Fu(\()p Fq(N)37979 36003 y Fl(i)38349 35837 y Fu(\))529 b(=)f Fq(s)p Fu(,)535 b(for)502 b(all)6863 37165 y Fq(i)308 b Fu(=)f(0)p Fq(;)184 b(:)g(:)g(:)189 b(;)184 b(k)282 b Fo(\241)246 b Fu(1.)8524 38493 y(The)313 b(idea)g(of)g(the)g(pro)31 b(of)313 b(is)g(to)g(\257nd)g(a)g(solution)h(of)f(the)g(equation)i Fq(x)133 b Fo(\241)g Fu(\()p Fq(f)38672 38659 y Fl(k)39350 38493 y Fo(\261)g Fq(f)40578 38659 y Fl(k)19 b Fi(\241)p Fv(1)42389 38493 y Fo(\261)133 b(\242)184 b(\242)g(\242)136 b(\261)6863 39822 y Fq(f)7405 39988 y Fv(1)7902 39822 y Fu(\)\()p Fq(x)p Fu(\))449 b(=)e(0)454 b(of)g(nonzero)f(lo)31 b(cal)455 b(Brou)-31 b(w)g(er)454 b(degree.)745 b(Eac)-31 b(h)454 b(mapping)h Fq(f)37485 39988 y Fl(i)38307 39822 y Fu(corresp)31 b(onding)6863 41150 y(to)509 b(a)f(direct)g(co)-31 b(v)g(ering)509 b(is)e(homotopic)k(to)d(some)g(linear)h(map,)543 b(and)508 b(eac)-31 b(h)508 b(mapping)i Fq(f)44685 41316 y Fl(i)6863 42478 y Fu(corresp)31 b(onding)513 b(to)g(a)g(bac)-31 b(k)g(co)g(v)g(ering)516 b(has)c(its)i(in)-31 b(v)g(erse)512 b Fq(g)31494 42644 y Fl(i)32375 42478 y Fu(homotopic)k(to)d(some)g (linear)6863 43807 y(map.)483 b(W)-92 b(e)338 b(will)j(pro)-31 b(v)g(e)339 b(that)h(an)f(appropriate)h(comp)31 b(osition)342 b(of)d(these)f(linear)i(maps)f(has)g(a)6863 45135 y(non-degenerate)290 b(\257xed)f(p)31 b(oin)-31 b(t,)307 b(and)289 b(use)f(the)h(homotop)-31 b(y)292 b(prop)31 b(ert)-31 b(y)289 b(of)g(the)g(lo)31 b(cal)291 b(Brou)-31 b(w)g(er)6863 46463 y(degree)369 b(to)h(conclude)g(that)g Fq(f)19154 46629 y Fl(k)19945 46463 y Fo(\261)246 b Fq(f)21286 46629 y Fl(k)19 b Fi(\241)p Fv(1)23210 46463 y Fo(\261)247 b(\242)184 b(\242)g(\242)248 b(\261)e Fq(f)26888 46629 y Fv(1)27754 46463 y Fu(has)369 b(a)g(\257xed)h(p)31 b(oin)-31 b(t.)8524 47792 y(As)285 b(a)i(to)31 b(ol)287 b(for)g(k)-31 b(eeping)287 b(trac)-31 b(k)287 b(of)g(the)f(o)31 b(ccurrences)285 b(of)i(co)-31 b(v)g(erings)287 b(and)f(bac)-31 b(k)g(co)g(v)g(erings,)6863 49446 y(w)g(e)491 b(de\257ne)g(the)g(map)h Fq(\261)736 b Fu(:)511 b Fo(f)p Fu(0)p Fq(;)184 b(:)g(:)g(:)189 b(;)184 b(k)35 b Fo(g)511 b(!)f(f)p Fu(0)p Fq(;)184 b Fu(1)p Fo(g)494 b Fu(b)-31 b(y)491 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\))512 b(=)e(1)491 b(if)g Fq(N)37215 49612 y Fl(i)p Fi(\241)p Fv(1)39728 48766 y Fl(f)40160 48877 y Fh(i)39228 49446 y Fu(=)-184 b Fo(\))509 b Fq(N)42410 49612 y Fl(i)43270 49446 y Fu(and)6863 51162 y Fq(\261)227 b Fu(\()p Fq(i)p Fu(\))487 b(=)g(0)477 b(if)g Fq(N)13699 51328 y Fl(i)p Fi(\241)p Fv(1)16188 50482 y Fl(f)16620 50593 y Fh(i)15688 51162 y Fo(\()-184 b Fu(=)486 b Fq(N)18847 51328 y Fl(i)19216 51162 y Fu(.)815 b(Let)477 b Fq(h)23067 51328 y Fl(i)23911 51162 y Fu(b)31 b(e)476 b(a)h(homotop)-31 b(y)480 b(map)d(from)g(the)g (de\257nition)i(of)6863 52878 y(co)-31 b(v)g(ering)518 b(relation)g(for)e Fq(N)18288 53044 y Fl(i)p Fi(\241)p Fv(1)20843 52198 y Fl(f)21275 52309 y Fh(i)20342 52878 y Fu(=)-184 b Fo(\))552 b Fq(N)23567 53044 y Fl(i)24452 52878 y Fu(or)516 b Fq(N)26844 53044 y Fl(i)p Fi(\241)p Fv(1)29399 52198 y Fl(f)29831 52309 y Fh(i)28898 52878 y Fo(\()-184 b Fu(=)552 b Fq(N)32123 53044 y Fl(i)32492 52878 y Fu(.)933 b(In)516 b(the)g(case)g(of)g(a)h(direct)6863 54207 y(co)-31 b(v)g(ering)371 b(\(i.e.)494 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(1\),)371 b(the)e(homotop)-31 b(y)372 b Fq(h)26346 54373 y Fl(i)27083 54207 y Fu(satis\257es)16883 56642 y Fq(h)17521 56808 y Fl(i)17890 56642 y Fu(\(0)p Fq(;)184 b(x)p Fu(\))1109 b(=)e Fq(f)24046 56808 y Fl(i)24414 56642 y Fu(\()p Fq(x)p Fu(\))p Fq(;)1293 b Fu(where)369 b Fq(x)307 b Fo(2)g Fp(R)33493 56153 y Fl(u)p Fv(+)p Fl(s)35169 56642 y Fq(;)8163 b Fu(\(7\))15072 58302 y Fq(h)15710 58468 y Fl(i)16079 58302 y Fu(\(1)p Fq(;)184 b Fu(\()p Fq(p;)g(q)40 b Fu(\)\))1111 b(=)c(\()p Fq(A)24764 58468 y Fl(i)25133 58302 y Fq(p;)184 b Fu(0\))p Fq(;)1294 b Fu(where)369 b Fq(p)307 b Fo(2)h Fp(R)34675 57813 y Fl(u)35623 58302 y Fu(and)369 b Fq(q)348 b Fo(2)307 b Fp(R)40462 57813 y Fl(s)40935 58302 y Fq(;)2397 b Fu(\(8\))10977 59963 y Fq(h)11615 60129 y Fl(i)11983 59963 y Fu(\([0)p Fq(;)184 b Fu(1])p Fq(;)g(N)16125 59489 y Fi(\241)16004 60271 y Fl(i)p Fi(\241)p Fv(1)17511 59963 y Fu(\))246 b Fo(\\)g Fq(N)20060 60129 y Fl(i)21536 59963 y Fu(=)1107 b Fo(;)p Fq(;)19275 b Fu(\(9\))10490 61760 y Fq(h)11128 61926 y Fl(i)11496 61760 y Fu(\([0)p Fq(;)184 b Fu(1])p Fq(;)g(N)15517 61926 y Fl(i)p Fi(\241)p Fv(1)17023 61760 y Fu(\))247 b Fo(\\)f Fq(N)19694 61286 y Fv(+)19573 62068 y Fl(i)21536 61760 y Fu(=)1107 b Fo(;)p Fq(:)18721 b Fu(\(10\))25681 70071 y(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 6863 6974 a Fu(In)369 b(the)g(case)g(of)h(a)g (bac)-31 b(k)g(co)g(v)g(ering)371 b(\(i.e.)494 b Fq(\261)227 b Fu(\()p Fq(i)p Fu(\))308 b(=)g(0\),)370 b(the)g(homotop)-31 b(y)372 b Fq(h)36346 7140 y Fl(i)37083 6974 y Fu(satis\257es)16895 9268 y Fq(h)17533 9434 y Fl(i)17901 9268 y Fu(\(0)p Fq(;)184 b(x)p Fu(\))1109 b(=)e Fq(g)24043 9434 y Fl(i)24412 9268 y Fu(\()p Fq(x)p Fu(\))p Fq(;)1292 b Fu(where)369 b Fq(x)308 b Fo(2)f Fp(R)33490 8779 y Fl(u)p Fv(+)p Fl(s)35167 9268 y Fq(;)7611 b Fu(\(11\))15084 10928 y Fq(h)15722 11094 y Fl(i)16090 10928 y Fu(\(1)p Fq(;)184 b Fu(\()p Fq(p;)g(q)40 b 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Fq(i)370 b Fu(=)g(0)p Fq(;)184 b(:)g(:)g(:)k(;)c(k)308 b Fo(\241)271 b Fu(1)406 b(suc)-31 b(h)6863 17871 y(that)18112 20165 y Fq(f)18654 20331 y Fl(i)19022 20165 y Fu(\()p Fq(x)20085 20331 y Fl(i)p Fi(\241)p Fv(1)21587 20165 y Fu(\))308 b(=)g Fq(x)24127 20331 y Fl(i)24495 20165 y Fq(;)1292 b Fu(if)369 b Fq(\261)227 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(1)q Fq(;)18310 21826 y(g)18838 21992 y Fl(i)19207 21826 y Fu(\()p Fq(x)20270 21992 y Fl(i)20639 21826 y Fu(\))h(=)f Fq(x)23178 21992 y Fl(i)p Fi(\241)p Fv(1)24679 21826 y Fq(;)1292 b Fu(if)370 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\))309 b(=)f(0)p Fq(:)11495 b Fu(\(15\))8524 24119 y(W)-92 b(e)413 b(will)k(treat)e(\(15\))h(as)f(a)g(m)-31 b(ultidimensional)418 b(system)d(of)g(equations)h(to)g(b)31 b(e)414 b(solv)-31 b(ed.)6863 25448 y(T)-92 b(o)370 b(this)f(end,)h (let)g(us)e(de\257ne)19307 27742 y(\246)307 b(=)h Fq(N)22502 27908 y Fv(0)23245 27742 y Fo(\243)246 b Fq(N)25241 27908 y Fv(1)25984 27742 y Fo(\243)g(\242)184 b(\242)g(\242)248 b(\243)e Fq(N)30624 27908 y Fl(k)19 b Fi(\241)p Fv(1)32302 27742 y Fq(:)6863 30035 y Fu(A)301 b(p)31 b(oin)-31 b(t)302 b Fq(x)307 b Fo(2)h Fu(\246)301 b(will)i(b)31 b(e)300 b(represen)-31 b(ted)300 b(b)-31 b(y)302 b Fq(x)307 b Fu(=)g(\()p Fq(x)27852 30201 y Fv(0)28349 30035 y Fq(;)184 b(x)29473 30201 y Fv(1)29970 30035 y Fq(;)g(:)g(:)g(:)k(;)c(x)33062 30201 y Fl(k)19 b Fi(\241)p Fv(1)34740 30035 y Fu(\).)471 b(W)-92 b(e)300 b(set)h Fq(x)40079 30201 y Fi(\241)p Fv(1)41574 30035 y Fu(=)308 b Fq(x)43376 30201 y Fl(k)19 b Fi(\241)p Fv(1)6863 31364 y Fu(and)370 b Fq(x)9649 31530 y Fl(k)10500 31364 y Fu(=)308 b Fq(x)12302 31530 y Fv(0)12798 31364 y Fu(.)8524 32692 y(W)-92 b(e)368 b(de\257ne)h(a)g(map)h Fq(F)461 b Fu(=)308 b(\()p Fq(F)20494 32858 y Fv(0)20991 32692 y Fq(;)184 b(F)22194 32858 y Fv(1)22691 32692 y Fq(;)g(:)g(:)g(:)k(;)c(F)25862 32858 y Fl(k)19 b Fi(\241)p Fv(1)27541 32692 y Fu(\))308 b(:)g(\246)f Fo(!)h Fp(R)32245 32203 y Fv(\()p Fl(u)p Fv(+)p Fl(s)p Fv(\))p Fl(k)35471 32692 y Fu(b)-31 b(y)13637 35982 y Fq(F)14349 36148 y Fl(i)14718 35982 y Fu(\()p Fq(x)15781 36148 y Fv(0)16277 35982 y Fq(;)184 b(x)17401 36148 y Fv(1)17899 35982 y Fq(;)g(:)g(:)g(:)k(;)c(x)20991 36148 y Fl(k)19 b Fi(\241)p Fv(1)22669 35982 y Fu(\))308 b(=)24575 34089 y Fe(\()25467 35227 y Fq(x)26100 35393 y Fl(i)26714 35227 y Fo(\241)246 b Fq(f)28363 35393 y Fl(i)28732 35227 y Fu(\()p Fq(x)29795 35393 y Fl(i)p Fi(\241)p Fv(1)31297 35227 y Fu(\))1107 b(if)370 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(1)q Fq(;)25467 36821 y(x)26100 36987 y Fl(i)p Fi(\241)p Fv(1)27847 36821 y Fo(\241)246 b Fq(g)29482 36987 y Fl(i)29851 36821 y Fu(\()p Fq(x)30914 36987 y Fl(i)31283 36821 y Fu(\))1121 b(if)370 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(0)q Fq(:)6863 39272 y Fu(With)i(this)g(notation,)324 b(solving)311 b(the)e(system)g(\(15\)) h(is)f(equiv)-61 b(alen)-31 b(t)310 b(to)g(solving)g(the)f(equation) 6863 40601 y Fq(F)154 b Fu(\()p Fq(x)p Fu(\))308 b(=)f(0)370 b(in)f(in)-31 b(t)186 b(\246.)8524 42058 y(W)-92 b(e)368 b(de\257ne)h(a)g(homotop)-31 b(y)372 b Fq(H)398 b Fu(=)307 b(\()p Fq(H)23552 42224 y Fv(0)24049 42058 y Fq(;)184 b(:)g(:)g(:)k(;)c(H)27428 42224 y Fl(k)19 b Fi(\241)p Fv(1)29107 42058 y Fu(\))308 b(:)g([0)p Fq(;)184 b Fu(1])249 b Fo(\243)d Fu(\246)308 b Fo(!)f Fp(R)37379 41569 y Fv(\()p Fl(u)p Fv(+)p Fl(s)p Fv(\))p Fl(k)40605 42058 y Fu(b)-31 b(y)12348 45348 y Fq(H)13268 45514 y Fl(i)13636 45348 y Fu(\()p Fq(\270;)184 b(x)15836 45514 y Fv(0)16334 45348 y Fq(;)g(x)17458 45514 y Fv(1)17955 45348 y Fq(;)g(:)g(:)g(:)k(;)c(x) 21047 45514 y Fl(k)19 b Fi(\241)p Fv(1)22725 45348 y Fu(\))308 b(=)24632 43456 y Fe(\()25523 44594 y Fq(x)26156 44760 y Fl(i)26771 44594 y Fo(\241)245 b Fq(h)28515 44760 y Fl(i)28884 44594 y Fu(\()p Fq(\270;)184 b(x)31084 44760 y Fl(i)p Fi(\241)p Fv(1)32586 44594 y Fu(\))1108 b(if)369 b Fq(\261)227 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(1)q Fq(;)25523 46188 y(x)26156 46354 y Fl(i)p Fi(\241)p Fv(1)27904 46188 y Fo(\241)246 b Fq(h)29649 46354 y Fl(i)30017 46188 y Fu(\()p Fq(\270;)184 b(x)32217 46354 y Fl(i)32586 46188 y Fu(\))1108 b(if)369 b Fq(\261)227 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(0)q Fq(:)6863 48639 y Fu(Notice)406 b(that)g Fq(H)90 b Fu(\(0)p Fq(;)184 b(x)p Fu(\))369 b(=)e Fq(F)154 b Fu(\()p Fq(x)p Fu(\).)599 b(The)405 b(assertion)g(of)g(the)g(theorem) g(is)g(a)g(consequence)g(of)6863 49967 y(the)413 b(follo)-31 b(wing)417 b(t)-31 b(w)g(o)414 b(lemmas,)426 b(whic)-31 b(h)414 b(will)g(b)31 b(e)413 b(pro)-31 b(v)g(ed)413 b(after)h(w)-31 b(e)413 b(complete)h(the)f(curren)-31 b(t)6863 51295 y(pro)31 b(of.)6863 53589 y Fk(Lemma)424 b(5)554 b Fj(F)-85 b(or)457 b(al)57 b(l)458 b Fq(\270)419 b Fo(2)h Fu([0)p Fq(;)184 b Fu(1])461 b Fj(the)d(lo)-57 b(c)g(al)459 b(Br)-57 b(ouwer)457 b(de)-57 b(gr)g(e)g(e)460 b Fu(deg)16 b(\()p Fq(H)36592 53755 y Fl(\270)37173 53589 y Fq(;)184 b Fu(in)-31 b(t)187 b(\246)p Fq(;)d Fu(0\))460 b Fj(is)e(wel)57 b(l)6863 54918 y(de\257ne)-57 b(d)397 b(and)f(do)-57 b(es)396 b(not)g(dep)-57 b(end)397 b(on)f Fq(\270)p Fj(.)509 b(Namely,)397 b(for)f(al)57 b(l)395 b Fq(\270)307 b Fo(2)h Fu([0)p Fq(;)184 b Fu(1])399 b Fj(we)d(have)17197 57211 y Fu(deg)16 b(\()p Fq(H)20223 57377 y Fl(\270)20804 57211 y Fq(;)184 b Fu(in)-31 b(t)186 b(\246)p Fq(;)e Fu(0\))310 b(=)d(deg)16 b(\()p Fq(H)29610 57377 y Fv(1)30107 57211 y Fq(;)184 b Fu(in)-31 b(t)187 b(\246)p Fq(;)d Fu(0\))p Fq(:)6863 59505 y Fk(Lemma)424 b(6)20445 60834 y Fu(deg)16 b(\()p Fq(H)23471 61000 y Fv(1)23968 60834 y Fq(;)184 b Fu(in)-31 b(t)187 b(\246)p Fq(;)d Fu(0\))310 b(=)d Fo(\247)p Fu(1)p Fq(:)8524 63127 y Fu(W)-92 b(e)322 b(con)-31 b(tin)g(ue)324 b(the)g(pro)31 b(of)323 b(of)h(Theorem)f(4.)478 b(Since)323 b Fq(F)461 b Fu(=)307 b Fq(H)32761 63293 y Fv(0)33258 63127 y Fu(,)333 b(from)323 b(the)h(ab)31 b(o)-31 b(v)g(e)324 b(lemmas)6863 64456 y(it)370 b(follo)-31 b(ws)372 b(immediately)g(that)11909 66750 y(deg)16 b(\()p Fq(F)31 b(;)184 b Fu(in)-31 b(t)187 b(\246)p Fq(;)d Fu(0\))309 b(=)f(deg)16 b(\()p Fq(H)23565 66916 y Fv(0)24062 66750 y Fq(;)184 b Fu(in)-31 b(t)186 b(\246)p Fq(;)e Fu(0\))310 b(=)e(deg)15 b(\()p Fq(H)32868 66916 y Fv(1)33366 66750 y Fq(;)184 b Fu(in)-31 b(t)186 b(\246)p Fq(;)e Fu(0\))310 b Fo(6)p Fu(=)d(0)p Fq(:)25681 70071 y Fu(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 6863 6974 a Fu(Hence)369 b(there)g(exists)h Fq(x)307 b Fo(2)g Fu(\246)370 b(suc)-31 b(h)368 b(that)j Fq(F)154 b Fu(\()p Fq(x)p Fu(\))307 b(=)h(0.)p 44721 6974 333 554 v 6863 8802 a Fk(Pro)35 b(of)362 b(of)f(Lemma)e(5:)465 b Fu(F)-92 b(rom)313 b(the)g(homotop)-31 b(y)316 b(prop)31 b(ert)-31 b(y)314 b(\(see)f(App)31 b(endix\))314 b(it)g(is)f(enough) 6863 10130 y(to)370 b(pro)-31 b(v)g(e)370 b(that)15778 11458 y Fq(H)16698 11624 y Fl(\270)17278 11458 y Fu(\()p Fq(x)p Fu(\))308 b Fo(6)p Fu(=)g(0)p Fq(;)1292 b Fu(for)370 b(all)g Fq(x)307 b Fo(2)h Fq(@)61 b Fu(\246)369 b(and)h Fq(\270)307 b Fo(2)g Fu([0)p Fq(;)184 b Fu(1].)6950 b(\(16\))8524 13204 y(In)408 b(order)g(to)h(pro)-31 b(v)g(e)409 b(\(16\),)421 b(let)409 b(us)f(\257x)h Fq(x)373 b Fu(=)g(\()p Fq(x)28003 13370 y Fv(0)28500 13204 y Fq(;)184 b(x)29624 13370 y Fv(1)30121 13204 y Fq(;)g(:)g(:)g(:)k(;)c(x)33213 13370 y Fl(k)19 b Fi(\241)p Fv(1)34891 13204 y Fu(\))374 b Fo(2)f Fq(@)61 b Fu(\246.)611 b(It)409 b(is)g(easy)f(to)6863 14532 y(see)384 b(that)i(the)f(exists)f Fq(i)334 b Fo(2)f(f)p Fu(0)p Fq(;)184 b Fu(1)p Fq(;)g(:)g(:)g(:)190 b(k)292 b Fo(\241)256 b Fu(1)p Fo(g)p Fu(,)389 b(suc)-31 b(h)385 b(that)g Fq(x)31327 14698 y Fl(i)32029 14532 y Fo(2)333 b Fq(@)61 b(N)34638 14698 y Fl(i)35340 14532 y Fu(=)333 b Fq(N)37544 14058 y Fv(+)37423 14840 y Fl(i)38535 14532 y Fo([)256 b Fq(N)40539 14058 y Fi(\241)40418 14840 y Fl(i)41287 14532 y Fu(.)538 b(Hence)6863 15860 y(at)370 b(least)g(one)g(of)f(the)h(follo)-31 b(wing)373 b(conditions)e(hold)e (true)22556 17883 y Fq(x)23189 18049 y Fl(i)23864 17883 y Fo(2)308 b Fq(N)25920 17410 y Fv(+)25799 18192 y Fl(i)26655 17883 y Fq(;)16123 b Fu(\(17\))22728 19544 y Fq(x)23361 19710 y Fl(i)24037 19544 y Fo(2)307 b Fq(N)26092 19070 y Fi(\241)25971 19852 y Fl(i)26839 19544 y Fq(:)15939 b Fu(\(18\))6863 21567 y(F)-92 b(or)333 b(eac)-31 b(h)335 b(of)f(the)g(t)-31 b(w)g(o)336 b(ab)31 b(o)-31 b(v)g(e)335 b(cases,)341 b(w)-31 b(e)335 b(ha)-31 b(v)g(e)335 b(to)g(consider)e (the)i(follo)-31 b(wing)337 b(four)d(p)31 b(ossibil-)6863 22895 y(ities)19500 25082 y Fq(N)20389 25248 y Fl(i)p Fi(\241)p Fv(1)23499 24402 y Fl(f)23931 24513 y Fh(i)22998 25082 y Fu(=)-184 b Fo(\))1107 b Fq(N)26778 25248 y Fl(i)27482 24350 y(f)27914 24461 y Fh(i)p Ff(+1)27454 25082 y Fu(=)-184 b Fo(\))307 b Fq(N)30434 25248 y Fl(i)p Fv(+1)31924 25082 y Fq(;)10854 b Fu(\(19\))19500 27182 y Fq(N)20389 27348 y Fl(i)p Fi(\241)p Fv(1)23499 26502 y Fl(f)23931 26613 y Fh(i)22998 27182 y Fu(=)-184 b Fo(\))1107 b Fq(N)26778 27348 y Fl(i)27482 26450 y(f)27914 26561 y Fh(i)p Ff(+1)27454 27182 y Fo(\()-184 b Fu(=)307 b Fq(N)30434 27348 y Fl(i)p Fv(+1)31924 27182 y Fq(;)10854 b Fu(\(20\))19500 29283 y Fq(N)20389 29449 y Fl(i)p Fi(\241)p Fv(1)23499 28603 y Fl(f)23931 28714 y Fh(i)22998 29283 y Fo(\()-184 b Fu(=)1107 b Fq(N)26778 29449 y Fl(i)27482 28551 y(f)27914 28662 y Fh(i)p Ff(+1)27454 29283 y Fu(=)-184 b Fo(\))307 b Fq(N)30434 29449 y Fl(i)p Fv(+1)31924 29283 y Fq(;)10854 b Fu(\(21\))19500 31383 y Fq(N)20389 31549 y Fl(i)p Fi(\241)p Fv(1)23499 30703 y Fl(f)23931 30814 y Fh(i)22998 31383 y Fo(\()-184 b Fu(=)1107 b Fq(N)26778 31549 y Fl(i)27482 30651 y(f)27914 30762 y Fh(i)p Ff(+1)27454 31383 y Fo(\()-184 b Fu(=)307 b Fq(N)30434 31549 y Fl(i)p Fv(+1)31924 31383 y Fq(:)10854 b Fu(\(22\))6863 33406 y(Assume)342 b(\257rst)g(that)h Fq(x)16200 33572 y Fl(i)16876 33406 y Fo(2)307 b Fq(N)18931 32933 y Fv(+)18810 33715 y Fl(i)19666 33406 y Fu(.)484 b(If)342 b(\(19\))i(or)e(\(20\))i(holds)f(true,)k(then)c(from)g(\(10\)) h(w)-31 b(e)342 b(obtain)22119 35429 y Fq(h)22757 35595 y Fl(i)23126 35429 y Fu(\()p Fq(t;)184 b(x)25080 35595 y Fl(i)p Fi(\241)p Fv(1)26582 35429 y Fu(\))308 b Fo(6)p Fu(=)f Fq(x)29121 35595 y Fl(i)29490 35429 y Fq(;)6863 37453 y Fu(for)361 b(ev)-31 b(ery)361 b Fq(t)307 b Fo(2)h Fu([0)p Fq(;)184 b Fu(1])363 b(and)e(ev)-31 b(ery)361 b Fq(x)21485 37619 y Fl(i)p Fi(\241)p Fv(1)23294 37453 y Fo(2)308 b Fq(N)25229 37619 y Fl(i)p Fi(\241)p Fv(1)26731 37453 y Fu(.)489 b(If)361 b(\(21\))i(or)d(\(22\))j(is)e(satis\257ed,)i (then)f(from)6863 38781 y(\(13\))371 b(it)f(results)f(that)22119 40109 y Fq(h)22757 40275 y Fl(i)23126 40109 y Fu(\()p Fq(t;)184 b(x)25080 40275 y Fl(i)25449 40109 y Fu(\))308 b Fo(6)p Fu(=)f Fq(x)27988 40275 y Fl(i)p Fi(\241)p Fv(1)29490 40109 y Fq(;)6863 41855 y Fu(for)531 b(ev)-31 b(ery)530 b Fq(t)576 b Fo(2)h Fu([0)p Fq(;)184 b Fu(1])533 b(and)e(ev)-31 b(ery)530 b Fq(x)22871 42021 y Fl(i)p Fi(\241)p Fv(1)24949 41855 y Fo(2)576 b Fq(N)27152 42021 y Fl(i)p Fi(\241)p Fv(1)28654 41855 y Fu(.)976 b(This)531 b(pro)-31 b(v)g(es)531 b(if)g Fq(x)38019 42021 y Fl(i)38963 41855 y Fo(2)577 b Fq(N)41288 41381 y Fv(+)41167 42163 y Fl(i)42023 41855 y Fu(,)571 b(then)6863 43183 y Fq(H)7783 43349 y Fl(t)8173 43183 y Fu(\()p Fq(x)p Fu(\))308 b Fo(6)p Fu(=)g(0)369 b(for)g(an)-31 b(y)370 b Fq(t)308 b Fo(2)f Fu([0)p Fq(;)184 b Fu(1].)8524 44511 y(Assume)280 b(no)-31 b(w)282 b(that)g Fq(x)17698 44677 y Fl(i)18373 44511 y Fo(2)308 b Fq(N)20429 44038 y Fi(\241)20308 44820 y Fl(i)21176 44511 y Fu(.)463 b(If)280 b(\(19\))j(or)d(\(21\))j(holds)e(true,)298 b(then)281 b(from)g(\(9\))h(it)g(follo)-31 b(ws)6863 45840 y(that)371 b(for)e(ev)-31 b(ery)369 b Fq(t)307 b Fo(2)h Fu([0)p Fq(;)184 b Fu(1])372 b(and)d(ev)-31 b(ery)369 b Fq(x)23925 46006 y Fl(i)p Fv(+1)25722 45840 y Fo(2)307 b Fq(N)27656 46006 y Fl(i)p Fv(+1)29515 45840 y Fu(w)-31 b(e)370 b(ha)-31 b(v)g(e)21565 47863 y Fq(h)22203 48029 y Fl(i)p Fv(+1)23692 47863 y Fu(\()p Fq(t;)184 b(x)25646 48029 y Fl(i)26016 47863 y Fu(\))307 b Fo(6)p Fu(=)h Fq(x)28555 48029 y Fl(i)p Fv(+1)30044 47863 y Fq(:)6863 49886 y Fu(If)369 b(\(20\))i(or)e(\(22\))i(is)e(satis\257ed,)h(then)g(from)f(\(14\))i(w) -31 b(e)370 b(obtain)21565 51909 y Fq(h)22203 52075 y Fl(i)p Fv(+1)23692 51909 y Fu(\()p Fq(t;)184 b(x)25646 52075 y Fl(i)p Fv(+1)27136 51909 y Fu(\))308 b Fo(6)p Fu(=)g Fq(x)29676 52075 y Fl(i)30044 51909 y Fq(;)6863 53932 y Fu(for)421 b(ev)-31 b(ery)421 b Fq(t)394 b Fo(2)f Fu([0)p Fq(;)184 b Fu(1])424 b(and)d(ev)-31 b(ery)421 b Fq(x)21958 54098 y Fl(i)p Fv(+1)23841 53932 y Fo(2)393 b Fq(N)25861 54098 y Fl(i)p Fv(+1)27351 53932 y Fu(.)648 b(This)421 b(pro)-31 b(v)g(es)421 b(that)h(if)g Fq(x)38509 54098 y Fl(i)39271 53932 y Fo(2)393 b Fq(N)41412 53459 y Fi(\241)41291 54241 y Fl(i)42160 53932 y Fu(,)434 b(then)6863 55261 y Fq(H)7783 55427 y Fl(t)8173 55261 y Fu(\()p Fq(x)p Fu(\))308 b Fo(6)p Fu(=)g(0)369 b(for)g(an)-31 b(y)370 b Fq(t)308 b Fo(2)f Fu([0)p Fq(;)184 b Fu(1].)p 44721 55261 V 6863 57088 a Fk(Pro)35 b(of)366 b(of)e(Lemma)g(6:)466 b Fu(Let)316 b(us)h(represen)-31 b(t)315 b Fq(x)26275 57254 y Fl(i)26960 57088 y Fu(as)i(a)g(pair)g Fq(x)31996 57254 y Fl(i)32671 57088 y Fu(=)308 b(\()p Fq(p)34827 57254 y Fl(i)35196 57088 y Fq(;)184 b(q)36181 57254 y Fl(i)36550 57088 y Fu(\),)328 b(where)317 b Fq(p)41321 57254 y Fl(i)42313 57088 y Fo(2)624 b Fp(R)44475 56600 y Fl(u)6863 58417 y Fu(and)577 b Fq(q)9717 58583 y Fl(i)10739 58417 y Fo(2)653 b Fp(R)12929 57928 y Fl(s)13402 58417 y Fu(.)1115 b(In)576 b(this)h(represen)-31 b(tation,)630 b(the)577 b(map)g Fq(H)32287 58583 y Fv(1)32784 58417 y Fu(\()p Fq(p)33771 58583 y Fv(0)34268 58417 y Fq(;)184 b(q)35253 58583 y Fv(0)35750 58417 y Fq(;)g(:)g(:)g(:)k(;)c(p)38766 58583 y Fl(k)19 b Fi(\241)p Fv(1)40445 58417 y Fq(;)184 b(q)41430 58583 y Fl(k)19 b Fi(\241)p Fv(1)43109 58417 y Fu(\))653 b(=)6863 59745 y(\()94 b(~)-647 b Fq(p)7850 59911 y Fv(0)8347 59745 y Fq(;)268 b Fu(~)-637 b Fq(q)9332 59911 y Fv(0)9830 59745 y Fq(;)184 b(:)g(:)g(:)k(;)279 b Fu(~)-648 b Fq(p)12846 59911 y Fl(k)19 b Fi(\241)p Fv(1)14524 59745 y Fq(;)268 b Fu(~)-637 b Fq(q)15509 59911 y Fl(k)19 b Fi(\241)p Fv(1)17188 59745 y Fu(\))370 b(has)f(the)g(follo)-31 b(wing)373 b(form)17521 61768 y(~)-647 b Fq(p)17984 61934 y Fl(i)18660 61768 y Fu(=)307 b Fq(p)20385 61934 y Fl(i)21000 61768 y Fo(\241)246 b Fq(A)22937 61934 y Fl(i)23306 61768 y Fq(p)23863 61934 y Fl(i)p Fi(\241)p Fv(1)25364 61768 y Fq(;)1292 b Fu(if)370 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(1,)10811 b(\(23\))22245 63429 y(~)-635 b Fq(q)22657 63595 y Fl(i)23333 63429 y Fu(=)308 b Fq(q)24996 63595 y Fl(i)25364 63429 y Fq(;)1292 b Fu(if)370 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(1,)10811 b(\(24\))20998 65089 y(~)-647 b Fq(p)21461 65255 y Fl(i)22137 65089 y Fu(=)308 b Fq(p)23863 65255 y Fl(i)p Fi(\241)p Fv(1)25364 65089 y Fq(;)1292 b Fu(if)370 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(0,)10811 b(\(25\))17698 66750 y(~)-636 b Fq(q)18109 66916 y Fl(i)18786 66750 y Fu(=)307 b Fq(q)20448 66916 y Fl(i)p Fi(\241)p Fv(1)22196 66750 y Fo(\241)246 b Fq(A)24133 66916 y Fl(i)24502 66750 y Fq(q)24996 66916 y Fl(i)25364 66750 y Fq(;)1292 b Fu(if)370 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(0.)10811 b(\(26\))25681 70071 y(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 6863 6974 a Fu(The)370 b(map)f Fq(H)12518 7140 y Fv(1)13384 6974 y Fu(is)g(linear.)494 b(F)-92 b(rom)369 b(\(100\))i(it)f(follo)-31 b(ws)372 b(that)e(to)g(pro)-31 b(v)g(e)370 b(that)6863 8302 y(deg)16 b(\()p Fq(H)9889 8468 y Fv(1)10386 8302 y Fq(;)184 b Fu(in)-31 b(t)187 b(\246)p Fq(;)d Fu(0\))310 b(=)d Fo(\247)p Fu(1)369 b(it)h(is)f(enough)h(to)g(sho)-31 b(w)370 b(that)h Fq(H)31398 8468 y Fv(1)32263 8302 y Fu(is)e(an)h(isomorphism.)8524 9631 y(Assume)453 b(that)i Fq(H)16113 9797 y Fv(1)16610 9631 y Fu(\()p Fq(p)17597 9797 y Fv(0)18094 9631 y Fq(;)184 b(q)19079 9797 y Fv(0)19577 9631 y Fq(;)g(:)g(:)g(:)k(;)c(p)22593 9797 y Fl(k)19 b Fi(\241)p Fv(1)24271 9631 y Fq(;)184 b(q)25256 9797 y Fl(k)19 b Fi(\241)p Fv(1)26935 9631 y Fu(\))449 b(=)f(0.)747 b(W)-92 b(e)453 b(ha)-31 b(v)g(e)455 b(to)g(sho)-31 b(w)454 b(that)h(for)f(all)6863 10959 y Fq(i)338 b Fu(=)g(0)p Fq(;)184 b(:)g(:)g(:)189 b(;)184 b(k)294 b Fo(\241)258 b Fu(1)388 b Fq(p)15281 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Fo(\241)152 b Fu(1.)p 6863 23004 333 554 v 8524 24837 a(It)519 b(is)g(imp)31 b(ortan)-31 b(t)522 b(to)e(remark)g(here)e(that)j(the)f(pro)31 b(of)519 b(of)h(Lemma)h(5)e(do)31 b(es)520 b(not)g(use)6863 26166 y(prop)31 b(ert)-31 b(y)369 b Fk(2)h Fu(from)g(the)f (de\257nition)i(of)f(co)-31 b(v)g(ering)370 b(relations.)8524 27494 y(In)429 b(the)g(view)i(of)f(Theorem)g(4,)446 b(it)430 b(mak)-31 b(es)430 b(practical)h(sense)e(to)h(mak)-31 b(e)431 b(no)f(distinction)6863 29149 y(b)31 b(et)-31 b(w)g(een)514 b(the)f(co)-31 b(v)g(ering)514 b(relations)h Fq(N)24645 28469 y Fl(f)24013 29149 y Fu(=)-184 b Fo(\))546 b Fq(M)633 b Fu(and)513 b Fq(N)32536 28469 y Fl(f)31903 29149 y Fo(\()-184 b Fu(=)546 b Fq(M)121 b Fu(.)923 b(F)-92 b(ollo)-31 b(wing)516 b([1)q(],)550 b(w)-31 b(e)6863 30477 y(in)g(tro)31 b(duce)6863 32373 y Fk(De\257nition)426 b(5)554 b Fj(L)-57 b(et)425 b Fq(N)546 b Fj(and)425 b Fq(M)545 b Fj(b)-57 b(e)426 b(h-sets.)595 b(We)425 b(say)g(that)f Fq(N)521 b Fu(generically)403 b Fq(f)119 b Fu(-co)-31 b(v)g(ers)401 b Fq(M)6863 34027 y Fj(\()396 b Fq(N)8601 34193 y Fv(1)10161 33347 y Fl(f)9405 34027 y Fo(\()-184 b(\))307 b Fq(N)12631 34193 y Fv(2)13128 34027 y Fj(\))396 b(if)h Fq(N)17002 33347 y Fl(f)16369 34027 y Fu(=)-184 b Fo(\))307 b Fq(M)517 b Fj(or)396 b Fq(N)23430 33347 y Fl(f)22797 34027 y Fo(\()-184 b Fu(=)307 b Fq(M)121 b Fj(.)8524 36311 y Fu(W)-92 b(e)368 b(emphasize)i(that)g(the)g (relation)h Fq(N)24981 36477 y Fv(1)26541 35631 y Fl(f)25785 36311 y Fo(\()-184 b(\))307 b Fq(N)29011 36477 y Fv(2)29877 36311 y Fu(is)369 b(not)h(symmetric)h(in)e(general.)6863 38348 y Fk(Collorary)425 b(7)554 b Fj(Assume)396 b(that)f(we)h(have)g (the)g(fol)57 b(lowing)395 b(chain)i(of)f(c)-57 b(overing)398 b(r)-57 b(elations)18304 40712 y Fq(N)19193 40878 y Fv(0)20580 40032 y Fl(f)21012 40143 y Ff(1)19997 40712 y Fo(\()-184 b(\))307 b Fq(N)23223 40878 y Fv(1)24611 40032 y Fl(f)25043 40143 y Ff(2)24028 40712 y Fo(\()-184 b(\))307 b Fq(N)27254 40878 y Fv(2)27935 40712 y Fq(:)184 b(:)g(:)30091 40032 y Fl(f)30523 40149 y Fh(k)29534 40712 y Fo(\()-184 b(\))307 b Fq(N)32760 40878 y Fl(k)33305 40712 y Fq(;)6863 42749 y Fj(Then)397 b(ther)-57 b(e)396 b(exists)f(a)h(p)-57 b(oint)396 b Fq(x)308 b Fo(2)f Fu(in)-31 b(t)186 b Fq(N)23682 42915 y Fv(0)24179 42749 y Fj(,)396 b(such)g(that)15236 44787 y Fq(f)15778 44953 y Fl(i)16392 44787 y Fo(\261)247 b Fq(f)17734 44953 y Fl(i)p Fi(\241)p Fv(1)19481 44787 y Fo(\261)g(\242)184 b(\242)g(\242)248 b(\261)f Fq(f)23160 44953 y Fv(1)23656 44787 y Fu(\()p Fq(x)p Fu(\))309 b Fo(2)e Fu(in)-31 b(t)186 b Fq(N)28899 44953 y Fl(i)29268 44787 y Fq(;)1316 b(i)308 b Fu(=)f(1)p Fq(;)184 b(:)g(:)g(:)189 b(;)184 b(k)35 b(:)6406 b Fu(\(27\))6863 46824 y Fj(Mor)-57 b(e)g(over)396 b(if)h Fq(N)13695 46990 y Fl(k)14548 46824 y Fu(=)307 b Fq(N)16605 46990 y Fv(0)17102 46824 y Fj(,)397 b(then)f Fq(x)g Fj(c)-57 b(an)396 b(b)-57 b(e)397 b(chosen)f(so)g(that) 19617 48862 y Fq(f)20159 49028 y Fl(k)20950 48862 y Fo(\261)246 b Fq(f)22291 49028 y Fl(k)19 b Fi(\241)p Fv(1)24215 48862 y Fo(\261)246 b(\242)184 b(\242)g(\242)249 b(\261)d Fq(f)27893 49028 y Fv(1)28390 48862 y Fu(\()p Fq(x)p Fu(\))308 b(=)f Fq(x:)10786 b Fu(\(28\))6863 50899 y Fk(Pro)35 b(of:)787 b Fu(The)516 b(statemen)-31 b(t)518 b(for)f(a)f(p)31 b(erio)g(dic)516 b(lo)31 b(op)517 b(\()p Fq(N)30003 51065 y Fv(0)31053 50899 y Fu(=)552 b Fq(N)33355 51065 y Fl(k)33900 50899 y Fu(\))516 b(follo)-31 b(ws)519 b(directly)e(from)6863 52228 y(Theorem)326 b(4.)479 b(The)325 b(nonp)31 b(erio)g(dic)326 b(case)f(will)i(b)31 b(e)325 b(reduced)g(to)h(the)f(p)31 b(erio)g(dic)326 b(one)f(b)-31 b(y)326 b(adding)6863 53556 y(a)370 b(new)f(co)-31 b(v)g(ering)371 b(relation)g(to)f(close)f (the)h(lo)31 b(op)370 b(as)f(follo)-31 b(ws.)8524 54884 y(Notice)458 b(that)g(from)f(the)g(de\257nition)i(of)e(co)-31 b(v)g(ering)458 b(relation)h(it)f(follo)-31 b(ws)459 b(immediately)6863 56213 y(that)371 b Fq(u)p Fu(\()p Fq(N)11215 56379 y Fl(i)11584 56213 y Fu(\))308 b(=)f Fq(u)369 b Fu(and)g Fq(s)p Fu(\()p Fq(N)18483 56379 y Fl(i)18853 56213 y Fu(\))308 b(=)f Fq(s)369 b Fu(for)g(some)h Fq(u;)184 b(s)369 b Fu(and)h(all)g Fq(i)308 b Fu(=)f(0)p Fq(;)184 b(:)g(:)g(:)189 b(;)184 b(k)282 b Fo(\241)246 b Fu(1.)8524 57541 y(W)-92 b(e)408 b(can)h(assume)g(that)h(for)f Fq(c)21055 57707 y Fl(N)21756 57818 y Fh(i)22535 57541 y Fu(=)374 b Fq(I)87 b(d)408 b Fu(for)h Fq(i)374 b Fu(=)g(0)p Fq(;)184 b Fu(1)p Fq(;)g(:)g(:)g(:)190 b(;)184 b(k)35 b Fu(.)612 b(It)409 b(easy)g(to)h(\257nd)e(a)h(map)6863 58869 y Fq(f)7405 59035 y Fl(k)19 b Fv(+1)9071 58869 y Fu(,)369 b(an)g(a\261ne)h(map)g Fq(A)17526 59035 y Fl(k)19 b Fv(+1)19499 58869 y Fu(:)308 b Fp(R)20914 58381 y Fl(u)21800 58869 y Fo(!)g Fp(R)24014 58381 y Fl(u)24593 58869 y Fu(,)369 b(and)h(a)f(homotop)-31 b(y)372 b Fq(h)34148 59035 y Fl(k)19 b Fv(+1)36182 58869 y Fu(suc)-31 b(h)369 b(that)23156 61335 y Fq(N)24045 61501 y Fl(k)24898 60596 y(f)25330 60713 y Fh(k)9 b Ff(+1)24937 61335 y Fu(=)-184 b Fo(\))346 b Fq(N)27956 61501 y Fv(0)28453 61335 y Fq(;)14325 b Fu(\(29\))6863 63373 y(where)369 b Fq(f)10606 63539 y Fl(k)19 b Fv(+1)12272 63373 y Fu(\()p Fq(p;)184 b(q)40 b Fu(\))309 b(=)e(\()p Fq(A)17451 63539 y Fl(k)19 b Fv(+1)19117 63373 y Fu(\()p Fq(p)p Fu(\))p Fq(;)184 b Fu(0\))372 b(and)e Fq(h)25171 63539 y Fl(k)19 b Fv(+1)26836 63373 y Fu(\()p Fq(t;)184 b(x)p Fu(\))309 b(=)e Fq(f)31239 63539 y Fl(k)19 b Fv(+1)32905 63373 y Fu(\()p Fq(x)p Fu(\).)8524 64701 y(No)-31 b(w)544 b(w)-31 b(e)543 b(ha)-31 b(v)g(e)544 b(a)f(closed)g(lo)31 b(op)544 b(of)f(co)-31 b(v)g(ering)544 b(relations)g(to)g(whic)-31 b(h)544 b(w)-31 b(e)543 b(can)g(apply)6863 66029 y(Theorem)370 b(4.)493 b(This)370 b(\257nishes)e(the)i(pro)31 b(of.)p 44721 66029 V 25681 70071 a(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 6863 6974 a Fr(3)1793 b(Multiple)599 b(wrapp)50 b(ed)599 b(co)-50 b(v)g(ering)599 b(relations)6863 9398 y Fu(The)428 b(goal)i(of)e(this)g(section)h(is)f(to)g(generalize)h (the)g(notion)g(of)f(co)-31 b(v)g(ering)430 b(relations)f(in)-31 b(tro-)6863 10727 y(duced)476 b(in)h(Section)g(2.)814 b(W)-92 b(e)475 b(will)j(c)-31 b(hange)477 b(condition)i Fk(2)d Fu(in)h(the)f(de\257nition)i(of)f(co)-31 b(v)g(ering)6863 12055 y(relations)311 b(in)g(order)e(to)h(allo)-31 b(w)313 b(for)d(more)g(general)g(maps)h(at)f(the)g(end)g(of)g(homotop)-31 b(y)313 b Fq(h)c Fu(\(this)6863 13383 y(means)370 b(that)g(w)-31 b(e)370 b(allo)-31 b(w)371 b(for)f(di\256eren)-31 b(t)369 b Fj(mo)-57 b(del)396 b(maps)p Fu(\).)6863 15819 y Fk(De\257nition)426 b(6)554 b Fj(Assume)427 b(that)f Fq(N)59 b(;)184 b(M)550 b Fj(ar)-57 b(e)427 b Fq(h)p Fj(-sets,)435 b(such)427 b(that)f Fq(u)p Fu(\()p Fq(N)121 b Fu(\))365 b(=)g Fq(u)p Fu(\()p Fq(M)121 b Fu(\))364 b(=)g Fq(u)427 b Fj(and)6863 17147 y Fq(s)p Fu(\()p Fq(N)121 b Fu(\))309 b(=)e Fq(s)p Fu(\()p Fq(M)121 b Fu(\))308 b(=)f Fq(s)p Fj(.)493 b(L)-57 b(et)347 b Fq(f)426 b Fu(:)308 b Fq(N)429 b Fo(!)307 b Fp(R)23106 16658 y Fl(n)24054 17147 y Fj(b)-57 b(e)347 b(a)e(c)-57 b(ontinuous)347 b(map.)493 b(L)-57 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Fo(!)321 b Fp(R)34855 25397 y Fl(u)35686 25886 y Fo(\243)251 b Fp(R)37597 25397 y Fl(s)38070 25886 y Fj(,)406 b(such)d(that)g(the)9631 27215 y(fol)57 b(lowing)395 b(c)-57 b(onditions)398 b(hold)d(true)28286 29650 y Fq(h)28924 29816 y Fv(0)30527 29650 y Fu(=)1107 b Fq(f)33037 29816 y Fl(c)33488 29650 y Fq(;)9290 b Fu(\(30\))20704 31310 y Fq(h)p Fu(\([0)p Fq(;)184 b Fu(1])p Fq(;)g(N)25484 30853 y Fi(\241)25363 31584 y Fl(c)26235 31310 y Fu(\))247 b Fo(\\)f Fq(M)28970 31476 y Fl(c)30527 31310 y Fu(=)1107 b Fo(;)p Fq(;)9730 b Fu(\(31\))20716 32971 y Fq(h)p Fu(\([0)p Fq(;)184 b Fu(1])p Fq(;)g(N)25375 33137 y Fl(c)25830 32971 y Fu(\))247 b Fo(\\)f Fq(M)28686 32514 y Fv(+)28565 33244 y Fl(c)30527 32971 y Fu(=)1107 b Fo(;)p Fq(:)9730 b Fu(\(32\))6863 35849 y Fk(2.)554 b Fj(Ther)-57 b(e)396 b(exists)f(a)h(map)g Fq(A)308 b Fu(:)g Fp(R)20567 35360 y Fl(u)21453 35849 y Fo(!)g Fp(R)23667 35360 y Fl(u)24246 35849 y Fj(,)397 b(such)e(that)14764 38284 y Fq(h)15402 38450 y Fv(1)15898 38284 y Fu(\()p Fq(p;)184 b(q)40 b Fu(\))1109 b(=)e(\()p Fq(A)p Fu(\()p Fq(p)p Fu(\))p Fq(;)184 b Fu(0\))p Fq(;)584 b Fj(for)396 b Fq(p)308 b Fo(2)p 30137 37395 1419 45 v 307 w Fq(B)30977 38450 y Fl(u)31556 38284 y Fu(\(0)p Fq(;)184 b Fu(1\))399 b Fj(and)d Fq(q)347 b Fo(2)p 38448 37395 1313 45 v 307 w Fq(B)39288 38450 y Fl(s)39761 38284 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fj(,)528 b Fu(\(33\))12123 39945 y Fq(A)p Fu(\()p Fq(@)61 b(B)14872 40111 y Fl(u)15452 39945 y Fu(\(0)p Fq(;)184 b Fu(1\)\))1110 b Fo(\275)d Fp(R)22216 39456 y Fl(u)23041 39945 y Fo(n)p 23840 38982 3879 45 v 246 w Fq(B)24680 40111 y Fl(u)25259 39945 y Fu(\(0)p Fq(;)184 b Fu(1\))s Fq(:)15059 b Fu(\(34\))9631 42380 y Fj(Mor)-57 b(e)g(over,)396 b(we)g(r)-57 b(e)g(quir)g(e)398 b(that)21648 44815 y Fu(deg)16 b(\()p Fq(A;)p 25075 43926 1419 45 v 184 w(B)25915 44981 y Fl(u)26495 44815 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fq(;)g Fu(0\))312 b(=)307 b Fq(w)30 b(;)8524 47693 y Fu(Note)478 b(that)i(in)e(the)g(case)g Fq(u)488 b Fu(=)h(0,)505 b(an)479 b(h-set)e(N)h(can)g(co)-31 b(v)g(er)479 b(an)f(h-set)g(M)f(only)i(with)6863 49022 y(degree)369 b Fq(w)338 b Fu(=)307 b(1.)8524 50350 y(The)297 b(previous)h(de\257nition)h(of)g(co)-31 b(v)g(ering)299 b(relation)g(\(De\257nition)g(1\))g(is)e(a)h(particular)h(case)6863 51678 y(of)457 b(the)h(presen)-31 b(t)456 b(one,)479 b(with)458 b(the)f(degree)g Fq(w)487 b Fu(equal)458 b(to)f(sgn)185 b(\(det)q(\()p Fq(A)p Fu(\)\))458 b(\(for)f Fq(u)d(>)f Fu(0\).)756 b(See)6863 53007 y(Figure)270 b(1)f(for)h(an)g(example)h (of)f(a)g(m)-31 b(ultiple)272 b(wrapp)31 b(ed)269 b(co)-31 b(v)g(ering)271 b(relation.)462 b(As)269 b(in)h(Section)g(2,)6863 54661 y(w)-31 b(e)370 b(will)h(call)f(the)g(map)g Fq(h)17618 54827 y Fv(1)18483 54661 y Fj(a)396 b(mo)-57 b(del)396 b(map)369 b Fu(for)g(the)h(relation)g Fq(N)34193 53981 y Fl(f)39 b(;w)33999 54661 y Fu(=)-184 b Fo(\))307 b Fq(M)121 b Fu(.)6863 57097 y Fk(Remark)424 b(8)554 b Fj(In)371 b(applic)-57 b(ations,)377 b(we)370 b(would)g(like)h(to)f(de) -57 b(cide)372 b(whether)e(two)g(h-sets)f(ar)-57 b(e)371 b(c)-57 b(or-)6863 58425 y(r)g(e)g(ctly)416 b(aligne)-57 b(d)416 b(b)-57 b(ase)g(d)416 b(essential)57 b(ly)414 b(on)h(the)g(information)h(on)f(their)g(b)-57 b(oundaries.)567 b(Condi-)6863 59753 y(tion)306 b(1)g(fr)-57 b(om)307 b(the)e(ab)-57 b(ove)307 b(de\257nition)g(is)f(state)-57 b(d)305 b(in)i(this)e(spirit.)480 b(In)306 b(c)-57 b(ondition)307 b(2,)324 b(we)306 b(c)-57 b(an)307 b(ex-)6863 61082 y(pr)-57 b(ess)319 b(the)g(lo)-57 b(c)g(al)320 b(Br)-57 b(ouwer)319 b(de)-57 b(gr)g(e)g(e)321 b(of)f Fq(A)f Fj(as)g(the)g(winding)i(numb) -57 b(er)321 b(of)e Fq(A)p Fu(\()p Fq(@)61 b(B)38718 61248 y Fl(u)39297 61082 y Fu(\(0)p Fq(;)184 b Fu(1\)\))323 b Fj(ab)-57 b(out)6863 62410 y(the)404 b(origin.)535 b(Mor)-57 b(e)404 b(pr)-57 b(e)g(cisely,)408 b(in)d(the)f(c)-57 b(ase)404 b Fq(u)322 b(>)g Fu(0)p Fj(,)407 b(sinc)-57 b(e)405 b(the)f(map)g Fq(A)322 b Fu(:)p 37998 61521 1443 45 v 323 w Fq(B)38838 62576 y Fl(n)39440 62410 y Fu(\(0)p Fq(;)184 b Fu(1\))325 b Fo(!)d Fp(R)44451 61921 y Fl(n)6863 63738 y Fj(satis\257es)21742 65067 y Fu(0)431 b Fq(=)-676 b Fo(2)307 b Fq(A)p Fu(\()p Fq(@)61 b(B)26397 65233 y Fl(u)26977 65067 y Fu(\(0)p Fq(;)184 b Fu(1\)\))p Fq(;)12914 b Fu(\(35\))25405 70071 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 12123 21867 a @beginspecial 5 @llx 251 @lly 657 @urx 628 @ury 1440 @rhi @setspecial %%BeginDocument: multiple.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: Adobe Illustrator(TM) 6.0 %%AI8_CreatorVersion: 9.0 %%For: (Marian Gidea) ( ) %%Title: (multiple.eps) %%CreationDate: 1/31/02 9:51 AM %%BoundingBox: 5 251 657 628 %%HiResBoundingBox: 5 251.5 657 628 %%DocumentProcessColors: Black %%DocumentFonts: TimesNewRomanPSMT %%DocumentNeededFonts: TimesNewRomanPSMT %%DocumentSuppliedResources: procset Adobe_level2_AI5 1.2 0 %%+ procset Adobe_typography_AI5 1.0 0 %%+ procset Adobe_blend_AI5 1.0 0 %%+ procset Adobe_ColorImage_AI6 1.1 0 %%+ procset Adobe_Illustrator_AI5 1.0 0 %%+ procset Adobe_pattern_AI5 1.0 0 %AI5_FileFormat 2.0 %AI3_ColorUsage: Color %AI3_IncludePlacedImages %%AI6_ColorSeparationSet: 1 1 (AI6 Default Color Separation Set) %%+ Options: 1 16 0 1 0 1 0 0 0 0 1 1 1 18 0 0 0 0 0 0 0 0 -1 -1 %%+ PPD: 1 21 0 0 60 45 2 2 1 0 0 1 0 0 0 0 0 0 0 0 0 0 () %AI3_TemplateBox: 306.5 395.5 306.5 395.5 %AI3_TileBox: -78 -77 510 691 %AI3_DocumentPreview: Header %AI5_ArtSize: 792 612 %AI5_RulerUnits: 2 %AI5_ArtFlags: 0 0 0 1 0 0 1 0 0 %AI5_TargetResolution: 800 %AI5_NumLayers: 1 %AI5_OpenToView: -181 739 1 1013 671 18 0 1 10 67 1 0 %AI5_OpenViewLayers: 7 %%PageOrigin:-78 -77 %%AI3_PaperRect:-12 781 600 -11 %%AI3_Margin:12 -13 -12 11 %AI7_GridSettings: 72 8 72 8 1 0 0.8 0.8 0.8 0.9 0.9 0.9 %AI9_Flatten: 1 %%EndComments %%BeginProlog %%BeginResource: procset Adobe_level2_AI5 1.2 0 %%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation) %%Version: 1.2 0 %%CreationDate: (04/10/93) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) userdict /Adobe_level2_AI5 26 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { (AI8_CMYK_CustomColor) 6 packedarray } bind def /findrgbcustomcolor { (AI8_RGB_CustomColor) 5 packedarray } bind def /setcustomcolor { exch aload pop dup (AI8_CMYK_CustomColor) eq { pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } { dup (AI8_RGB_CustomColor) eq { pop pop 3 { 1 exch sub 3 index mul 1 exch sub 3 1 roll } repeat 4 -1 roll pop setrgbcolor } { pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } ifelse } ifelse } def } if /setAIseparationgray { false setoverprint 0 setgray /setseparationgray where{ pop setseparationgray }{ /setcolorspace where{ pop [/Separation (All) /DeviceCMYK {dup dup dup}] setcolorspace 1 exch sub setcolor }{ setgray }ifelse }ifelse } def /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking %%EndResource %%BeginResource: procset Adobe_typography_AI5 1.0 1 %%Title: (Typography Operators) %%Version: 1.0 1 %%CreationDate:(03/26/93) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_typography_AI5 54 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def pop pop findfont _wv type /arraytype eq { _wv makeblendedfont } if dup length 2 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { /Tx { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { /Tx { dup currentpoint 4 2 roll gsave tr _psf grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave trj _pjsf grestore 3 1 roll moveto tr jsp } ddef } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { /Tx { dup currentpoint 4 2 roll currentpoint gsave newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll currentpoint gsave newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat 0 _rise translate _hs 1 scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { dup 1000 div /_fScl exch ddef % selectfont } def /Tl { pop 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { /_rise exch ddef currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { 100 div /_hs exch ddef iTm } def /TA { pop } def /Tq { pop } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { exch pop _fScl mul neg 0 rmoveto } def /TK { 2 npop } def /T* { _leading aload pop neg Td } def /T*- { _leading aload pop Td } def /T- { _ax neg 0 rmoveto _hyphen Tx } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ _fScl 1000 mul selectfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_blend_AI5 1.4 0 %%Title: (Adobe Illustrator (R) Version 5.0 Blend ProcSet) %%Version: 1.4 0 %%CreationDate: (11/19/93) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) userdict /defaultpacking currentpacking put true setpacking userdict /Adobe_blend_AI5 70 dict dup begin put /bd { bind def } bind def /xs { exch store } bd /nullProc { { } } def /initialize { pop pop Adobe_blend_AI5 begin Adobe_blend_AI5_vars begin /_contoneDevice where { pop } { /_contoneDevice false def } ifelse /_dpiThreshold where { pop } { /_dpiThreshold 600 def } ifelse /_screenFreqThreshold where { pop } { /_screenFreqThreshold 150 def } ifelse /tRectOK? deviceDPI _dpiThreshold le currentScreenFreq _screenFreqThreshold le and _contoneDevice not and def /invertXfer { [ { 1 exch sub } /exec load systemdict/currenttransfer get exec /exec load ] cvx systemdict/settransfer get exec } bd /spotDict 3 dict dup begin /nSpots 2 def /spot1 7 dict def /spot2 7 dict def end def composite? { /_setgray_ /setgray load def /_fill_ /fill load def /_image_ /image load def } { /_setgray_ systemdict/setgray get def /_fill_ systemdict/fill get def /_image_ systemdict/image get def } ifelse } bd /terminate { currentdict Adobe_blend_AI5_vars eq { end currentdict Adobe_blend_AI5 eq { end } if } if } bd /_compositeSpotDevice where { begin _compositeSpotDevice 0 ne {userdict /composite? true put} if end } { /_compositeSpotDevice 0 def } ifelse /nullString () def /d255 256 array def 0 1 255 { d255 exch dup 255 div put } bind for /d255- 256 array def 0 1 255 { d255- exch 1 d255 2 index get sub put } bind for /dUserSpace matrix defaultmatrix def currentdict /Adobe_blend_AI5_vars 89 dict dup begin put { /f /F /s /S /b /B } { null def } bind forall /byte 1 string def /sSave null def /setSSave { save /sSave exch store } bind def /Bm null def /doBlend null def /startC? false def /endC? false def /fCMYK? null def /startTint 0 def /endTint 0 def /bSMatrix matrix def /bUMatrix matrix def /dMatrix matrix def /inLine? true def /pTState? false def /bHi? false def /yHi 0 def /xHi 0 def /noImg /lv1Fix where { pop lv1Fix } { false } ifelse def /ccAry1 5 array def /ccTint 0 def /spotColor? false def /colorimage? true def [ /tint1Data /tint2Data /spotDict /bAxis /ubAxis /pChange /optimize? /nSamples /sInc /blendProc /_bn /xBCInc /yBCInc /bInc /bRender /cBName /cBType /nColors /color? /blend? /colorType /cData /cDataLen /bDataLen /rampPoint /midPoint /endPoint /blendLength /blackData /yeData /mgData /cyData /cnt1 /ndx /_fill /tmp counttomark { null def } bind repeat pop currentdict end currentdict end exch begin begin /unitSq { 0 0 moveto 0 1 lineto 1 1 lineto 1 0 lineto closepath } bd /gMark { counttomark 2 add -1 roll } bd /setCustomColor { dup /ccTint exch store 1 exch sub 6 1 roll ccAry1 astore exch setcustomcolor } bd /currentCustomColor { ccAry1 aload pop ccTint } bd /nsetcustomcolor where { pop } { /nsetcustomcolor { pop setcmykcolor } bd } ifelse /nsetcustomcolorend where { pop } { /nsetcustomcolorend { } bd } ifelse /setBSpace { newpath bUMatrix astore concat unitSq } bd /setCStop { dup 0 eq { pop spotColor? { dup 1 exch sub /ccTint exch def ccAry1 4 /Black put } if setgray } { 1 eq { setcmykcolor } { composite? not colorType 2 lt and { forceCMYK } { setCustomColor } ifelse } ifelse } ifelse } bd /makeByte { /tmp 0 store 255 mul cvi 8 string 8 { dup tmp 3 index put /tmp tmp 1 add store } repeat exch pop } bd /setImgSpace { cDataLen 1 8 2 index 0 0 1 0 0 dMatrix astore } bd /bwImage { setImgSpace cData /_image_ load { exec } stopped { $error /errorname get /undefinedresult ne { stop } { pop pop pop pop pop } ifelse } if } bd level2? { /bFill { _fill } def /bCImg { /cDataLen bDataLen store setImgSpace setSSave expandSpot cyData mgData yeData cData expandCMYK true 4 spotDict { ncolorimage } stopped { $error /errorname get /undefinedresult ne { stop } { 10 { pop } repeat } ifelse } if sSave restore } bd } if /expandOne { dup type /stringtype ne { cDataLen string exch dup 0 ne { 255 mul cvi 0 1 cDataLen 1 sub { 3 copy exch put pop } for } if pop } if } bd /expandSpot { spotColor? { spotDict begin spot1 begin tintImage type /nulltype ne { tintImage expandOne /tintImage exch def } if end spot2 begin tintImage type /nulltype ne { tintImage expandOne /tintImage exch def } if end end } if } bd /expandCMYK { 4 { expandOne 4 1 roll } repeat } bd /colorimage where dup { exch pop /ncolorimage where { pop } { /ncolorimage {pop colorimage} bd } ifelse } if not { /ncolorimage where { pop } { /colorimage? false store /ncolorimage { pop pop pop setSSave /blackData xs /yeData xs /mgData xs /cyData xs /cnt1 0 store [ byte dup 0 cyData dup type /stringtype eq { /cnt1 cvx /get cvx d255 /exch cvx /get cvx .3 /mul cvx } { .3 mul } ifelse mgData dup type /stringtype eq { /cnt1 cvx /get cvx d255 /exch cvx /get cvx .59 /mul cvx } { .59 mul } ifelse yeData dup type /stringtype eq { /cnt1 cvx /get cvx d255 /exch cvx /get cvx .11 /mul cvx } { .11 mul } ifelse blackData dup type /stringtype eq { /cnt1 cvx /get cvx d255 /exch cvx /get cvx } if /add cvx /add cvx /add cvx 1 /exch cvx /sub cvx /dup cvx 0 /lt cvx { pop 0 } /if cvx /dup cvx 1 /gt cvx { pop 1 } /if cvx 255 /mul cvx /cvi cvx 256 /mod cvx /dup cvx 0 /lt cvx { pop 0 } /if cvx /put cvx /cnt1 dup cvx 1 /add cvx /store cvx ] cvx bind _image_ sSave restore } bd } ifelse } if level2? not { /bCImg { /cDataLen bDataLen store setImgSpace setSSave expandSpot cyData mgData yeData cData colorimage? { expandCMYK } if true 4 spotDict { ncolorimage } stopped { $error /errorname get /undefinedresult ne { stop } { 10 { pop } repeat } ifelse } if sSave restore } bd /bwFill { setSSave /cDataLen 8 store /cData currentgray makeByte store bwImage sSave restore } bd /c1ImgFill { setSSave /cDataLen 8 store setImgSpace spotColor? { spotDict begin spot1 begin currentCustomColor makeByte /tintImage exch def /name exch def /spot_K exch def /spot_Y exch def /spot_M exch def /spot_C exch def end spot2 initSpotData end } if currentcmykcolor 4 { makeByte 4 1 roll } repeat true 4 spotDict { ncolorimage } stopped { $error /errorname get /undefinedresult ne { stop } { 10 { pop } repeat } ifelse } if sSave restore } bd /bFill noImg { { _fill } } { { color? { c1ImgFill } { bwFill } ifelse } } ifelse bd } if composite? { /bCFun { color? { cyData dup type /stringtype eq { /ndx cvx /get cvx d255 /exch cvx /get cvx } if mgData dup type /stringtype eq { /ndx cvx /get cvx d255 /exch cvx /get cvx } if yeData dup type /stringtype eq { /ndx cvx /get cvx d255 /exch cvx /get cvx } if cData dup type /stringtype eq { /ndx cvx /get cvx d255 /exch cvx /get cvx } if spotColor? { spotDict begin /spotDict cvx /begin cvx spot1 begin tintImage dup type /stringtype eq { /ndx cvx /get cvx d255- /exch cvx /get cvx } { dup type /nulltype ne { name type /nametype ne {1 exch sub} if } if } ifelse end /spot1 cvx /tintValue 3 -1 /roll cvx /put cvx spot2 begin tintImage dup type /stringtype eq { /ndx cvx /get cvx d255- /exch cvx /get cvx } { dup type /nulltype ne { name type /nametype ne {1 exch sub} if } if } ifelse end /spot2 cvx /tintValue 3 -1 /roll cvx /put cvx /end cvx end /spotDict cvx /nsetcustomcolor cvx } { /setcmykcolor cvx } ifelse } { cData /ndx cvx /get cvx d255 /exch cvx /get cvx /setgray cvx } ifelse } bd /Bc { newpath gsave setBSpace nColors 1 eq { pop pop setCStop } if bFill grestore } bd /linealBm { /nColors dup load 1 sub store newpath gsave setBSpace blend? { linImg } { bFill } ifelse grestore nColors 1 gt { getRData } if } bd /rdBm { /nColors dup load 1 sub store _fill gsave bUMatrix astore concat bHi? { xHi yHi bUMatrix idtransform /yHi exch store /xHi exch store rampPoint 1 lt { 1 rampPoint sub dup xHi mul exch yHi mul translate } if } if nColors { 0 0 rampPoint 0 360 arc _fill blend? bHi? or { rdBlend } if nColors 1 gt { getRData } if /nColors dup load 1 sub store } repeat /nColors 1 store grestore } bd /cGetRData { setCStop /blend? cData type /stringtype eq dup not color? and { pop cyData type /stringtype eq mgData type /stringtype eq yeData type /stringtype eq or or } if store } def /cGetRData } if /eCStop { mark 1 index 3 mul 3 add dup 8 gt { pop 8 } if 1 roll cleartomark } bd composite? not { /knockOut level2? { { 0 0 0 0 setcmykcolor _fill } } { /bFill noImg { { _fill } } { { _of true eq { currentgray 1 ne { bwFill } if } { bwFill } ifelse } } ifelse def /whiteByte 1 makeByte def noImg { { 0 0 0 0 setcmykcolor _fill } } { { cBType 0 eq { setSSave /cData whiteByte store /cDataLen 8 store bwImage sSave restore } { _fill } ifelse } } ifelse } ifelse bd /bCFun { cData dup type /stringtype ne { color? { 1 exch sub } if } { /ndx cvx /get cvx color? customColor? not and { d255- } { d255 } ifelse /exch cvx /get cvx } ifelse /_setgray_ cvx } bd /eCCBlend { dup 3 eq { pop mark 7 1 roll 6 copy ccThrough? dup /blend? xs { /startC? true store setCustomColor customColor? { /cData tint1Data store setCDataLen } if /endC? 3 index 3 eq { 4 index 1 ne } { false } ifelse store } if cleartomark stop } if 1 eq { pop pop pop } if pop /startC? false store 6 { 8 index } repeat ccThrough? dup /blend? xs { /endC? true store blend? not { stop } if customColor? { /cData tint1Data store setCDataLen } if } if } bd /handleOP { _of not { knockOut } if } bd /handleROP { _of not { 0 0 0 0 setcmykcolor _fill } { newpath } ifelse } bd /rdBm { /nColors dup load 1 sub store blend? { _fill } { handleROP } ifelse gsave bUMatrix astore concat bHi? { xHi yHi bUMatrix idtransform /yHi exch store /xHi exch store rampPoint 1 lt { 1 rampPoint sub dup xHi mul exch yHi mul translate } if } if nColors { 0 0 rampPoint 0 360 arc blend? { cData type /stringtype ne bHi? not and { cData color? { 1 exch sub } if _setgray_ _fill_ } { cData type /stringtype ne { /cDataLen 1 store /bDataLen 1 store } if rdBlend } ifelse } { handleROP pTState? { /bAxis rampPoint endPoint sub store xHi bAxis mul yHi bAxis mul translate } if } ifelse nColors 1 gt { getRData } if /nColors dup load 1 sub store } repeat /nColors 1 store grestore } bd /ccThrough? { gsave pop 0 setCustomColor currentcmykcolor grestore anyColor? } bd /forceCMYK { exch pop 1 exch sub 5 1 roll 4 { 4 index mul 4 1 roll } repeat 0 cCMYKData dup /cData ne { dup /yeData eq { pop 1 add } { /mgData eq { 2 } { 3 } ifelse add } ifelse 0 } if pop index 0 eq { pop pop pop pop 0 0 0 0 } if setcmykcolor pop /fCMYK? true store } bd /endCapSepBc { pop pop dup 0 eq { pop setgray } { 1 eq { setcmykcolor } { colorType 1 eq { forceCMYK } { fCMYK? { forceCMYK } { setCustomColor } ifelse } ifelse } ifelse } ifelse currentcmykcolor anyColor? blend? and { bFill } { handleOP } ifelse } bd } if /cCMYKData 0 def composite? dup not { pop customColor? } if not { /cCMYKData /cyData /mgData /yeData /cData black? not { yellow? { exch } { magenta? { 3 } { 4 } ifelse -1 roll } ifelse } if 4 1 roll pop pop pop store /Bc { gsave setBSpace nColors 1 gt { blend? currentcmykcolor anyColor? and { bFill } { handleOP } ifelse } { endCapSepBc } ifelse grestore newpath } bd /linealBm { /nColors dup load 1 sub store newpath gsave setBSpace blend? { cCMYKData load dup type /stringtype eq { dup length /cDataLen xs /cData xs gsave colorType 0 ne noImg not and { invertXfer } if linImg grestore } { pop bFill } ifelse } { handleOP } ifelse grestore nColors 1 gt { getRData } if } bd /cmykGetRData { /fCMYK? false store blend? { { cmykDataProcs colorType get exec } stopped pop blend? { /cData cCMYKData load store setCDataLen } if } if } def /cmykDataProcs [ { pop black? dup /blend? xs { setgray 0 } if pop } { cCMYKData load dup type /stringtype ne { 0 0 0 cyan? not { 4 magenta? { 1 } { yellow? { 2 } { 3 } ifelse } ifelse roll } if 4 copy add add add 0 eq { /blend? false store } if setcmykcolor /startC? true store /endC? true store eCStop stop } if pop dup 0 eq { pop setgray } { 1 eq { setcmykcolor } { forceCMYK } ifelse } ifelse } bind /eCCBlend load { cBType 1 eq { tint1Data tint2Data /tint1Data xs /tint2Data xs } if 0 eq { black? { setgray } { 0 0 0 4 -1 roll 1 exch sub setcmykcolor } ifelse black? { /blend? true store } if 6 { 8 index } repeat ccThrough? { /blend? true store } { black? { /cData tint1Data store setCDataLen } { /blend? false store } ifelse } ifelse } { mark 7 1 roll 6 copy ccThrough? { forceCMYK pop stop } if 9 index 0 eq { black? dup /blend? xs { pop 1 setgray /cData tint2Data store setCDataLen 0 } if pop } { /blend? 6 { 16 index } repeat ccThrough? store blend? { forceCMYK } if } ifelse cleartomark } ifelse } bind ] def /cmykGetRData } if composite? dup not { pop isCMYKSep? } if not { /endCapSepBc { /white? false store pop pop dup 0 eq { pop /white? 1 index 1 eq store setgray } { 1 eq { setcmykcolor } { setCustomColor } ifelse } ifelse % currentcmykcolor anyColor? endC? or blend? and { bFill } { handleOP } ifelse } bd /Bc { gsave setBSpace nColors 1 gt { blend? startC? and { bFill } { handleOP } ifelse } { endCapSepBc } ifelse grestore newpath } bd /linealBm { /nColors dup load 1 sub store newpath gsave setBSpace blend? { cData type /stringtype eq { linImg } { bFill } ifelse } { handleOP } ifelse grestore nColors 1 gt { getRData } if } bd /discardCMY { counttomark 4 add -3 roll pop pop pop } bd /testTopCC { 6 copy ccThrough? } bd /getCRamp { { ccDataProcs colorType 2 sub get exec } stopped pop blend? cDataLen 0 eq and { /cDataLen bDataLen store } if } bd /ccGetRData { /fCMYK? false store /startC? false store /endC? false store colorType 2 lt { /blend? false def } if blend? { getCRamp } { setCStop } ifelse blend? { /blend? cData 1 ne store blend? { cData dup type /stringtype ne { 1 exch sub /cData xs 0 } if pop } if } if } def /ccDataProcs [ /eCCBlend load { cBType 1 eq { tint1Data tint2Data /tint1Data xs /tint2Data xs } if 0 eq { /blend? false store pop } { mark 7 1 roll testTopCC { /blend? 1 index 1 ne store /startC? blend? store /endC? false store blend? not { cleartomark stop } if /cData tint1Data store setCDataLen setCustomColor pop stop } if cleartomark } ifelse 2 index 0 eq { /blend? false store } { mark 6 { 9 index } repeat testTopCC dup /blend? xs { /blend? 1 index 1 ne store /endC? blend? store /startC? false store blend? not { cleartomark stop } if /cData tint2Data store setCDataLen } if cleartomark } ifelse } bind ] def /ccGetRData } if load Adobe_blend_AI5_vars /getData 3 -1 roll put /setCDataLen { /cDataLen 0 cData dup type /stringtype eq { length exch } if pop store } bd /initSpotData { begin /name null def /tintImage null def /tintValue null def /spot_C null def /spot_M null def /spot_Y null def /spot_K null def end } bd /getRData { /colorType gMark store _compositeSpotDevice 0 ne { spotDict begin spot1 initSpotData spot2 initSpotData end /spotColor? colorType 2 eq colorType 3 eq or def } { /spotColor? false store } ifelse /blend? true store 0 0 0 0 setcmykcolor 100 div /rampPoint xs % (between 13 and 87%) 100 div /midPoint xs dup 0 eq { 2 } { dup 1 eq { 5 } { _compositeSpotDevice 0 ne { spotDict begin spot1 begin /name 3 index def /spot_K 4 index def /spot_Y 5 index def /spot_M 6 index def /spot_C 7 index def end end } if 7 } ifelse } ifelse /tmp exch def tmp index 100 div /endPoint xs _compositeSpotDevice 0 ne { tmp 2 add index 3 eq { /tmp tmp 4 add def tmp index dup spotDict begin spot1/name get ne { spot2 begin /name exch def /spot_K tmp 2 add index def /spot_Y tmp 3 add index def /spot_M tmp 4 add index def /spot_C tmp 5 add index def end } { pop } ifelse end } if } if /color? colorType 0 gt store colorType 3 eq { /tint2Data gMark store } if colorType 2 ge { /tint1Data gMark store } if _compositeSpotDevice 0 ne { spotDict begin colorType 2 ge { colorType 3 eq { spot2 begin /tintImage cBType 0 eq {tint2Data} {tint1Data} ifelse def name null eq {/name /Black def} if end } if spot1 begin /tintImage cBType 0 eq colorType 2 eq or {tint1Data} {tint2Data} ifelse def colorType 2 eq { name null eq { /name spot2/name get def spot2/name null put } if } { name null eq {/name /Black def} if } ifelse end } if end } if /cData gMark store setCDataLen colorType 0 gt { counttomark 4 add -3 roll /yeData xs /mgData xs /cyData xs } if blend? { /bDataLen cDataLen dup 0 eq color? and { [ cyData mgData yeData ] { dup type /stringtype eq { length exch pop exit } if pop } forall } if store bDataLen 0 eq { /bDataLen 1 store } if getData blend? { composite? cDataLen 0 eq and { /cDataLen bDataLen store } if } if } { setCStop } ifelse } bd /Bg { 0 0 0 0 setcmykcolor 6 { pop } repeat /blendLength xs pop pop pop /cBName xs /bRender xs bRender 2 ne { composite? not { _of setoverprint } if _eo {eoclip} {clip} ifelse _bn cBName 2 copy known { get mark exch aload pop /cBType xs /nColors xs mark exch aload pop 0 0 } if pop pop getRData cBType 0 eq { /linealBm } { bHi? { /pTState? nColors 2 gt store } if /doBlend /rdBlend load store /rdBm } ifelse } { inLine? not { mark mark } if /Bc dup { cleartomark mark } bd /nullProc } ifelse load /Bm xs } bd /linImg noImg { { newpath doRctBlend } } { { /doBlend color? composite? and { /bCImg } { /bwImage } ifelse load store 0 0 moveto tRectOK? composite? and { { mark 0 1 dtransform atan cvi 90 mod 0 eq 1 0 dtransform atan cvi 90 mod 0 eq } stopped { cleartomark } { and exch pop { newpath doRctBlend } { doBlend } ifelse } ifelse } { doBlend } ifelse } } ifelse bd /doRctBlend { gsave /sInc 1 store /nSamples bDataLen store /bInc 1 bDataLen div store /ubAxis 1 0 dtransform dUserSpace idtransform dup mul exch dup mul add sqrt store /pChange ubAxis 0 eq { 0 } { bDataLen ubAxis div } ifelse store pChange .5 gt noImg not and dup /optimize? xs { /nSamples ubAxis 2 div round cvi dup 1 le { pop 2 } if store /bInc 1 nSamples div store /sInc bDataLen 1 sub nSamples 1 sub div store } if 0 nSamples [ /dup cvx optimize? { /round cvx /cvi cvx } if /ndx /exch cvx /store cvx bCFun /rectfill where dup { exch pop _compositeSpotDevice 1 ne and } if { 0 0 bInc 1 /rectfill cvx } { 0 0 /moveto cvx bInc 0 /lineto cvx bInc 1 /lineto cvx 0 1 /lineto cvx /closepath cvx /_fill_ cvx } ifelse bInc 0 /translate cvx sInc /add cvx ] cvx bind repeat pop spotColor? {nsetcustomcolorend} if grestore } bd /rdPrep { /nSamples bDataLen dup 0 eq { pop 1 } if store /sInc -1 store /bAxis rampPoint endPoint sub store /bInc bAxis bDataLen div neg store /optimize? false store tRectOK? { /ubAxis bAxis 0 dtransform dUserSpace idtransform dup mul exch dup mul add sqrt 0 bAxis dtransform dUserSpace idtransform dup mul exch dup mul add sqrt 2 copy lt { exch } if pop store /pChange ubAxis 0 eq { 0 } { bDataLen ubAxis div } ifelse store pChange .5 gt noImg not and dup /optimize? xs { /nSamples ubAxis 2 div round cvi dup 1 le { pop 2 } if store /bInc bAxis nSamples div neg store /sInc bDataLen 1 sub nSamples 1 sub div neg store } if } if bHi? { /xBCInc xHi bAxis mul nSamples div store /yBCInc yHi bAxis mul nSamples div store } if } bd /rdBlend { newpath gsave rdPrep rampPoint bDataLen 1 sub nSamples [ /dup cvx optimize? { /round cvx /cvi cvx } if /ndx /exch cvx /store cvx bCFun 0 0 3 /index cvx 0 360 /arc cvx /_fill_ cvx /exch cvx bInc /add cvx /exch cvx sInc /add cvx bHi? { xBCInc yBCInc /translate cvx } if ] cvx bind repeat pop pop spotColor? {nsetcustomcolorend} if grestore pTState? { xHi bAxis mul yHi bAxis mul translate } if } bd /Bh { pop pop /pTState? false store 2 copy 0 ne exch 0 ne or dup /bHi? xs { /yHi xs /xHi xs 0 0 } if pop pop } bd /BD { inLine? not { ] nColors cBType ] _bn cBName 3 -1 roll put end } if } bd /Bn { 1 add dict dup nullString null put /_bn xs } bd /Bd { Adobe_blend_AI5_vars begin 3 -1 roll dup nullString eq dup { setSSave } if /inLine? exch def /cBName exch def /nColors exch def /cBType exch def } bd /Bb { sSave null eq { Adobe_blend_AI5_vars begin setSSave } if composite? { /_fill /fill load store } { /__fill /fill load store /_fill { _of true eq { currentgray 1 ne { __fill } if } { __fill } ifelse } def } ifelse /fill { } def } bd /BB { /cBType xs cleartomark cleartomark cBType dup bRender sSave dup type /savetype eq { restore 0 } if pop currentdict Adobe_blend_AI5_vars eq { end } if 2 ne exch 0 gt and { 2 eq { s } { S } ifelse } { pop newpath } ifelse } bd currentdict readonly pop end end defaultpacking setpacking %%EndResource %%BeginProcSet: Adobe_ColorImage_AI6 1.2 0 userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 50 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 40 dict def } if Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /_newproc null def /_proc1 null def /_proc2 null def /channelcount 0 def /sourcecount 0 def /sourcearray 4 array def /_ptispace null def /_ptiname null def /_pti0 0 def /_pti1 0 def /_ptiproc null def /_ptiscale 0 def /_pticomps 0 def /_ptibuf 0 string def /_gtigray 0 def /_cticmyk null def /_rtirgb null def /XIEnable true def /XIType 0 def /XIEncoding 0 def /XICompression 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIRowBytes 0 def /XIFile null def /XIBuffer1 null def /XIBuffer2 null def /XIDataProc null def /XIVersion 6 def /XIColorSpace /DeviceGray def /XIColorValues 0 def end currentdict /_colorimage known not { /colorimage where { /colorimage get /_colorimage exch def }{ /_colorimage null def } ifelse } if /_image systemdict /image get def /_currenttransfer systemdict /currenttransfer get def /FourEqual { 4 index ne { pop pop pop false }{ 4 index ne { pop pop false }{ 4 index ne { pop false }{ 4 index eq } ifelse } ifelse } ifelse } def /TestPlateIndex { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 FourEqual { /plateindex 0 def }{ 0 1 0 0 FourEqual { /plateindex 1 def }{ 0 0 1 0 FourEqual { /plateindex 2 def }{ 0 0 0 1 FourEqual { /plateindex 3 def }{ 0 0 0 0 FourEqual { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /concatprocs { /packedarray where { pop dup type /packedarraytype eq 2 index type /packedarraytype eq or }{ false } ifelse { /_proc2 exch cvlit def /_proc1 exch cvlit def _proc1 aload pop _proc2 aload pop _proc1 length _proc2 length add packedarray cvx }{ /_proc2 exch cvlit def /_proc1 exch cvlit def /_newproc _proc1 length _proc2 length add array def _newproc 0 _proc1 putinterval _newproc _proc1 length _proc2 putinterval _newproc cvx } ifelse } def /clrspaceissepn { type /arraytype eq } def /clrspacegetname { dup clrspaceissepn {dup length 2 sub get}{pop ()} ifelse } def /clrspacegetalt { aload pop pop pop colormake } def /clrspacegetcomps { dup /DeviceGray eq { pop 1 }{ dup /DeviceRGB eq { pop 3 }{ /DeviceCMYK eq { 4 }{ 1 } ifelse } ifelse } ifelse } def /clrspacemarksplate { dup /DeviceGray eq { pop plateindex 3 eq }{ dup /DeviceRGB eq { pop plateindex 5 ne }{ dup /DeviceCMYK eq { pop plateindex 5 ne }{ /findcmykcustomcolor where { pop dup length 2 sub get 0.1 0.1 0.1 0.1 5 -1 roll findcmykcustomcolor 1 setcustomcolor systemdict /currentgray get exec 1 ne }{ pop plateindex 5 ne } ifelse } ifelse } ifelse } ifelse } def /colormake { dup clrspacegetcomps exch 1 index 2 add 1 roll dup 1 eq {pop}{array astore} ifelse exch } def /colorexpand { dup clrspacegetname exch dup clrspaceissepn { clrspacegetalt exch 4 1 roll }{ 1 3 1 roll } ifelse } def /colortint { dup /DeviceGray eq { 3 1 roll 1 exch sub mul 1 exch sub exch }{ dup /DeviceRGB eq { 3 1 roll {1 exch sub 1 index mul 1 exch sub exch} forall pop 3 array astore exch }{ dup /DeviceCMYK eq { 3 1 roll {1 index mul exch} forall pop 4 array astore exch }{ 3 1 roll mul exch } ifelse } ifelse } ifelse } def /colortocmyk { dup /DeviceGray eq { pop 1 exch sub 0 0 0 4 -1 roll 4 array astore }{ dup /DeviceRGB eq { pop aload pop _rgbtocmyk 4 array astore }{ dup /DeviceCMYK eq { pop }{ clrspacegetalt colortint colortocmyk } ifelse } ifelse } ifelse } def /makeimagedict { 7 dict begin /ImageType 1 def /Decode exch def /DataSource exch def /ImageMatrix exch def /BitsPerComponent exch def /Height exch def /Width exch def currentdict end } def /stringinvert { 0 1 2 index length 1 sub { dup 2 index exch get 255 exch sub 2 index 3 1 roll put } for } def /stringknockout { 0 1 2 index length 1 sub { 255 2 index 3 1 roll put } for } def /stringapply { 0 1 4 index length 1 sub { dup 4 index exch get 3 index 3 1 roll 3 index exec } for pop exch pop } def /WalkRGBString { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval {} forall 5 index exec 3 index } for 5 {pop} repeat } def /WalkCMYKString { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval {} forall 6 index exec 3 index } for 5 { pop } repeat } def /StuffRGBIntoGrayString { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /StuffCMYKIntoGrayString { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /RGBToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 3 idiv string dup 3 1 roll /StuffRGBIntoGrayString load exch WalkRGBString end } def /CMYKToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 4 idiv string dup 3 1 roll /StuffCMYKIntoGrayString load exch WalkCMYKString end } def /SeparateCMYKImageProc { Adobe_ColorImage_AI6_Vars begin sourcecount 0 ne { sourcearray plateindex get exec }{ sourcearray 0 get exec dup length 4 idiv string 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop } ifelse end } def /ColorImageCompositeEmulator { pop true eq { Adobe_ColorImage_AI6_Vars /sourcecount get 5 add { pop } repeat }{ Adobe_ColorImage_AI6_Vars /channelcount get 1 ne { Adobe_ColorImage_AI6_Vars begin sourcearray 0 3 -1 roll put channelcount 3 eq {/RGBToGrayImageProc}{/CMYKToGrayImageProc} ifelse load end } if image } ifelse } def /colorimage { Adobe_ColorImage_AI6_Vars begin /channelcount 1 index def /sourcecount 2 index 1 eq { channelcount 1 sub } { 0 } ifelse def 4 sourcecount add index end dup 8 eq exch 1 eq or not { /_colorimage load null ne { _colorimage }{ Adobe_ColorImage_AI6_Vars /sourcecount get 7 add { pop } repeat } ifelse }{ dup 3 eq TestPlateIndex dup -1 eq exch 5 eq or or { /_colorimage load null eq { ColorImageCompositeEmulator }{ dup 1 eq { pop pop image }{ Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { gsave 0 _currenttransfer exec 1 _currenttransfer exec eq { 0 _currenttransfer exec 0.5 lt } { 0 _currenttransfer exec 1 _currenttransfer exec gt } ifelse { { pop 0 } } { { pop 1 } } ifelse systemdict /settransfer get exec } if _colorimage Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { grestore } if } ifelse } ifelse }{ dup 1 eq { pop pop image }{ pop pop Adobe_ColorImage_AI6_Vars begin sourcecount -1 0 { exch sourcearray 3 1 roll put } for /SeparateCMYKImageProc load end systemdict /image get exec } ifelse } ifelse } ifelse } def /proctintimage { /_ptispace exch store /_ptiname exch store /_pti1 exch store /_pti0 exch store /_ptiproc exch store /_pticomps _ptispace clrspacegetcomps store /_ptiscale _pti1 _pti0 sub store level2? { _ptiname length 0 gt version cvr 2012 ge and { [/Separation _ptiname _ptispace {_ptiproc}] setcolorspace [_pti0 _pti1] makeimagedict _image }{ [/Indexed _ptispace 255 {255 div _ptiscale mul _pti0 add _ptiproc}] setcolorspace [0 255] makeimagedict _image } ifelse }{ _pticomps 1 eq { { dup { 255 div _ptiscale mul _pti0 add _ptiproc 255 mul cvi put } stringapply } concatprocs _image }{ { dup length _pticomps mul dup _ptibuf length ne {/_ptibuf exch string store}{pop} ifelse _ptibuf { exch _pticomps mul exch 255 div _ptiscale mul _pti0 add _ptiproc _pticomps 2 add -2 roll _pticomps 1 sub -1 0 { 1 index add 2 index exch 5 -1 roll 255 mul cvi put } for pop pop } stringapply } concatprocs false _pticomps /_colorimage load null eq {7 {pop} repeat}{_colorimage} ifelse } ifelse } ifelse } def /graytintimage { /_gtigray 5 -1 roll store {1 _gtigray sub mul 1 exch sub} 4 1 roll /DeviceGray proctintimage } def /cmyktintimage { /_cticmyk 5 -1 roll store {_cticmyk {1 index mul exch} forall pop} 4 1 roll /DeviceCMYK proctintimage } def /rgbtintimage { /_rtirgb 5 -1 roll store {_rtirgb {1 exch sub 1 index mul 1 exch sub exch} forall pop} 4 1 roll /DeviceRGB proctintimage } def /tintimage { TestPlateIndex -1 eq { colorexpand 3 -1 roll 5 -1 roll {0}{0 exch} ifelse 4 2 roll dup /DeviceGray eq { pop graytintimage }{ dup /DeviceRGB eq { pop rgbtintimage }{ pop cmyktintimage } ifelse } ifelse }{ dup clrspacemarksplate { plateindex 5 lt { colortocmyk plateindex get 1 exch sub exch {1 0}{0 1} ifelse () graytintimage }{ pop exch {0}{0 exch} ifelse 0 3 1 roll () graytintimage } ifelse }{ pop pop pop {pop 1} 0 1 () /DeviceGray proctintimage } ifelse } ifelse } def /XINullImage { } def /XIImageMask { XIImageWidth XIImageHeight false [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load imagemask } def /XIImageTint { 0 setgray XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load XIType 3 eq XIColorValues XIColorSpace tintimage } def /XIImage { 0 setgray XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load XIChannelCount 1 eq {image}{false XIChannelCount colorimage} ifelse } def /XG { pop pop } def /XF { 13 {pop} repeat } def /Xh { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def /XIImageHeight exch def /XIImageWidth exch def /XIImageMatrix exch def 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale XIType 1 eq { /_lp /null ddef _fc /_lp /imagemask ddef } if /XIVersion 7 def end } def /XH { Adobe_ColorImage_AI6_Vars begin /XIVersion 6 def grestore end } def /XIEnable { Adobe_ColorImage_AI6_Vars /XIEnable 3 -1 roll put } def /XC { Adobe_ColorImage_AI6_Vars begin colormake /XIColorSpace exch def /XIColorValues exch def end } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def cvi dup 256 idiv /XICompression exch store 256 mod /XIEncoding exch store pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi }{ XIImageWidth XIChannelCount mul } ifelse /XIRowBytes exch def XIEncoding 0 ne XIVersion 6 le and { currentfile 128 string readline pop pop } if XICompression 0 eq { /XIBuffer1 XIRowBytes string def XIEncoding 0 eq { {currentfile XIBuffer1 readhexstring pop} }{ {currentfile XIBuffer1 readstring pop} } ifelse }{ /XIBuffer1 256 string def /XIBuffer2 XIRowBytes string def {currentfile XIBuffer1 readline pop (%) anchorsearch {pop} if} /ASCIIHexDecode filter /DCTDecode filter /XIFile exch def {XIFile XIBuffer2 readstring pop} } ifelse /XIDataProc exch def XIVersion 6 le { 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale XIType 1 eq { /_lp /null ddef _fc /_lp /imagemask ddef } if } if XIEnable { XIType 1 eq { XIImageMask }{ XIType 2 eq XIType 3 eq or { XIImageTint }{ XIImage } ifelse } ifelse }{ XINullImage } ifelse grestore end } def end %%EndProcSet %%BeginResource: procset Adobe_Illustrator_AI5 1.1 0 %%Title: (Adobe Illustrator (R) Version 5.0 Full Prolog) %%Version: 1.1 0 %%CreationDate: (3/7/1994) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 81 dict dup begin put /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_rise 0 def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_wv 0 def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 91 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /sw { dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add } def /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 ddef { 1 index eq { /_cnt _cnt 1 add ddef } if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /sp { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer readline not { stop } if endString eq { cleartomark stop } if } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer readline not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 4 npop 6 1 roll pop 4 1 roll pop pop pop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 3 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end end setpacking %%EndResource %%BeginResource: procset Adobe_pattern_AI5 1.1 0 %%Title: (Adobe Illustrator (R) Version 5.0 Pattern Operators) %%Version: 1.1 0 %%CreationDate: (03/26/93) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 95 dict put } if userdict /Adobe_Illustrator_AI5 get begin /@ { } def /& { } def /dp { dup null eq { pop _dp 0 ne { 0 1 _dp 1 sub _dl mod { _da exch get 3 get } for _dp 1 sub _dl mod 1 add packedarray _da 0 get aload pop 8 -1 roll 5 -1 roll pop 4 1 roll definepattern pop } if } { _dp 0 ne _dp _dl mod 0 eq and { null dp } if 7 packedarray _da exch _dp _dl mod exch put _dp _dl mod _da 0 get 4 get 2 packedarray /_dp _dp 1 add def } ifelse } def /E { _ed begin dup 0 get type /arraytype ne { 0 { dup 1 add index type /arraytype eq { 1 add } { exit } ifelse } loop array astore } if /_dd exch def /_ury exch def /_urx exch def /_lly exch def /_llx exch def /_n exch def /_y 0 def /_dl 4 def /_dp 0 def /_da _dl array def 0 1 _dd length 1 sub { /_d exch _dd exch get def 0 2 _d length 2 sub { /_x exch def /_c false def /_r _d _x 1 add get cvlit def _r _ ne { _urx _llx sub _ury _lly sub [ 1 0 0 1 0 0 ] [ /save cvx _llx neg _lly neg /translate cvx _c { nc /begin cvx } if _r dup type /stringtype eq { cvx } { { exec } /forall cvx } ifelse _c { /end cvx } if /restore cvx ] cvx /_fn 12 _n length add string def _y _fn cvs pop /_y _y 1 add def _fn 12 _n putinterval _fn _c false dp _d exch _x 1 add exch put } if } for } for null dp _n _dd /_pd end xput } def /fc { _fm dup concatmatrix pop } def /p { /_fm exch ddef 9 -2 roll _pm translate fc 7 -2 roll _pm scale fc 5 -1 roll _pm rotate fc 4 -2 roll exch 0 ne { dup _pm rotate fc 1 -1 _pm scale fc neg _pm rotate fc } { pop } ifelse dup _pm rotate fc exch dup sin exch cos div 1 0 0 1 0 6 2 roll _pm astore fc neg _pm rotate fc _pd exch get /_fdd exch ddef /_pf { save /_doClip 0 ddef 0 1 _fdd length 1 sub { /_fd exch _fdd exch get ddef _fd 0 2 _fd length 2 sub { gsave 2 copy get dup _ ne { cvx exec _fc } { pop } ifelse 2 copy 1 add get dup _ ne { aload pop findfont _fm patternfill } { pop fill } ifelse grestore pop } for pop } for restore newpath } ddef /_psf { save /_doClip 0 ddef 0 1 _fdd length 1 sub { /_fd exch _fdd exch get ddef _fd 0 2 _fd length 2 sub { gsave 2 copy get dup _ ne { cvx exec _fc } { pop } ifelse 2 copy 1 add get dup _ ne { aload pop findfont _fm 9 copy 6 npop patternashow } { pop 6 copy 3 npop hvashow } ifelse grestore pop } for pop } for restore sw rmoveto } ddef /_pjsf { save /_doClip 0 ddef 0 1 _fdd length 1 sub { /_fd exch _fdd exch get ddef _fd 0 2 _fd length 2 sub { gsave 2 copy get dup _ ne { cvx exec _fc } { pop } ifelse 2 copy 1 add get dup _ ne { aload pop findfont _fm 12 copy 6 npop patternawidthshow } { pop 9 copy 3 npop hvawidthshow } ifelse grestore pop } for pop } for restore swj rmoveto } ddef /_lp /none ddef } def /sc { _sm dup concatmatrix pop } def /P { /_sm exch ddef 9 -2 roll _pm translate sc 7 -2 roll _pm scale sc 5 -1 roll _pm rotate sc 4 -2 roll exch 0 ne { dup _pm rotate sc 1 -1 _pm scale sc neg _pm rotate sc } { pop } ifelse dup _pm rotate sc exch dup sin exch cos div 1 0 0 1 0 6 2 roll _pm astore sc neg _pm rotate sc _pd exch get /_sdd exch ddef /_ps { save /_doClip 0 ddef 0 1 _sdd length 1 sub { /_sd exch _sdd exch get ddef _sd 0 2 _sd length 2 sub { gsave 2 copy get dup _ ne { cvx exec _sc } { pop } ifelse 2 copy 1 add get dup _ ne { aload pop findfont _sm patternstroke } { pop stroke } ifelse grestore pop } for pop } for restore newpath } ddef /_pss { save /_doClip 0 ddef 0 1 _sdd length 1 sub { /_sd exch _sdd exch get ddef _sd 0 2 _sd length 2 sub { gsave 2 copy get dup _ ne { cvx exec _sc } { pop } ifelse 2 copy 1 add get dup _ ne { aload pop findfont _sm 10 copy 6 npop patternashowstroke } { pop 7 copy 3 npop ss } ifelse grestore pop } for pop } for restore pop sw rmoveto } ddef /_pjss { save /_doClip 0 ddef 0 1 _sdd length 1 sub { /_sd exch _sdd exch get ddef _sd 0 2 _sd length 2 sub { gsave 2 copy get dup _ ne { cvx exec _sc } { pop } ifelse 2 copy 1 add get dup _ ne { aload pop findfont _sm 13 copy 6 npop patternawidthshowstroke } { pop 10 copy 3 npop jss } ifelse grestore pop } for pop } for restore pop swj rmoveto } ddef /_lp /none ddef } def end userdict /Adobe_pattern_AI5 18 dict dup begin put /initialize { /definepattern where { pop pop pop } { begin begin Adobe_pattern_AI5 begin Adobe_pattern_AI5 { dup xcheck { bind } if pop pop } forall mark cachestatus 7 1 roll pop pop pop pop exch pop exch { { 10000 add dup 2 index gt { exit } if dup setcachelimit } loop } stopped cleartomark end end end Adobe_pattern_AI5 begin } ifelse } def /terminate { currentdict Adobe_pattern_AI5 eq { end } if } def errordict /nocurrentpoint { pop stop } put errordict /invalidaccess { pop stop } put /patternencoding 256 array def 0 1 255 { patternencoding exch ( ) 2 copy exch 0 exch put cvn put } for /definepattern { 17 dict begin /uniform exch def /cache exch def /key exch def /procarray exch def /mtx exch matrix invertmatrix def /height exch def /width exch def /ctm matrix currentmatrix def /ptm matrix def /str 32 string def /slice 9 dict def slice /s 1 put slice /q 256 procarray length div sqrt floor cvi put slice /b 0 put /FontBBox [ 0 0 0 0 ] def /FontMatrix mtx matrix copy def /Encoding patternencoding def /FontType 3 def /BuildChar { exch begin /setstrokeadjust where {pop true setstrokeadjust} if slice begin dup q dup mul mod s idiv /i exch def dup q dup mul mod s mod /j exch def q dup mul idiv procarray exch get /xl j width s div mul def /xg j 1 add width s div mul def /yl i height s div mul def /yg i 1 add height s div mul def uniform { 1 1 } { width 0 dtransform dup mul exch dup mul add sqrt dup 1 add exch div 0 height dtransform dup mul exch dup mul add sqrt dup 1 add exch div } ifelse width 0 cache { xl 4 index mul yl 4 index mul xg 6 index mul yg 6 index mul setcachedevice } { setcharwidth } ifelse gsave scale newpath xl yl moveto xg yl lineto xg yg lineto xl yg lineto closepath clip newpath end end exec grestore } def key currentdict definefont end } def /patterncachesize { gsave newpath 0 0 moveto width 0 lineto width height lineto 0 height lineto closepath patternmatrix setmatrix pathbbox exch ceiling 4 -1 roll floor sub 3 1 roll ceiling exch floor sub mul 1 add grestore } def /patterncachelimit { cachestatus 7 1 roll 6 npop 8 mul } def /patternpath { exch dup begin setfont ctm setmatrix concat slice exch /b exch slice /q get dup mul mul put FontMatrix concat uniform { width 0 dtransform round width div exch round width div exch 0 height dtransform round height div exch height div exch 0 0 transform round exch round exch ptm astore setmatrix } { ptm currentmatrix pop } ifelse { currentpoint } stopped not { 2 npop pathbbox true 4 index 3 index eq 4 index 3 index eq and { pop false { { 2 npop } { 3 npop true } { 7 npop true } { pop true } pathforall } stopped { 5 npop true } if } if { height div ceiling height mul 4 1 roll width div ceiling width mul 4 1 roll height div floor height mul 4 1 roll width div floor width mul 4 1 roll 2 index sub height div ceiling cvi exch 3 index sub width div ceiling cvi exch 4 2 roll moveto FontMatrix mtx invertmatrix dup dup 4 get exch 5 get rmoveto ptm ptm concatmatrix pop slice /s patterncachesize patterncachelimit div ceiling sqrt ceiling cvi dup slice /q get gt { pop slice /q get } if put 0 1 slice /s get dup mul 1 sub { slice /b get add gsave 0 1 str length 1 sub { str exch 2 index put } for pop dup { gsave ptm setmatrix 1 index str length idiv { str show } repeat 1 index str length mod str exch 0 exch getinterval show grestore 0 height rmoveto } repeat grestore } for 2 npop } { 4 npop } ifelse } if end } def /patternclip { _eo {eoclip} {clip} ifelse } def /patternstrokepath { strokepath } def /patternmatrix matrix def /patternfill { dup type /dicttype eq { Adobe_pattern_AI5 /patternmatrix get } if gsave patternclip Adobe_pattern_AI5 /patternpath get exec grestore newpath } def /patternstroke { dup type /dicttype eq { Adobe_pattern_AI5 /patternmatrix get } if gsave patternstrokepath true { { { newpath moveto } { lineto } { curveto } { closepath 3 copy Adobe_pattern_AI5 /patternfill get exec } pathforall 3 npop } stopped { 5 npop patternclip Adobe_pattern_AI5 /patternfill get exec } if } { patternclip Adobe_pattern_AI5 /patternfill get exec } ifelse grestore newpath } def /vpatternawidthshow { 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto gsave false charpath currentpoint 5 index 5 index 5 index Adobe_pattern_AI5 /patternfill get exec grestore _fontRotateAdjust sub moveto _hvwb eq { _hvcx _hvcy rmoveto } if _hvax _hvay rmoveto 90 rotate } { currentpoint _fontHeight sub _hvax sub 3 index _hvwb eq { _hvcx sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto gsave 2 index false charpath 6 index 6 index 6 index Adobe_pattern_AI5 /patternfill get exec grestore newpath moveto pop pop } ifelse } cforall 3 npop } def /hpatternawidthshow { { dup cstring exch gsave 3 index eq { 5 index 5 index rmoveto } if false charpath currentpoint 9 index 9 index 9 index Adobe_pattern_AI5 /patternfill get exec grestore newpath moveto 2 copy rmoveto } cforall 8 npop } def /patternashow { 0 0 0 6 3 roll patternawidthshow } def /patternawidthshow { 6 index type /dicttype eq { Adobe_pattern_AI5 /patternmatrix get 7 1 roll } if _lineorientation 0 eq { hpatternawidthshow } { vpatternawidthshow } ifelse } def /vpatternawidthshowstroke { 7 1 roll 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto gsave false charpath currentpoint 3 index setmatrix 6 index 6 index 6 index Adobe_pattern_AI5 /patternstroke get exec grestore _fontRotateAdjust sub moveto _hvwb eq { _hvcx _hvcy rmoveto } if _hvax _hvay rmoveto 90 rotate } { currentpoint _fontHeight sub _hvax sub 3 index _hvwb eq { _hvcx sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto gsave 2 index false charpath 4 index setmatrix 7 index 7 index 7 index Adobe_pattern_AI5 /patternstroke get exec grestore newpath moveto pop pop } ifelse } cforall 4 npop } def /hpatternawidthshowstroke { 7 1 roll { dup cstring exch gsave 3 index eq { 5 index 5 index rmoveto } if false charpath currentpoint 7 index setmatrix 10 index 10 index 10 index Adobe_pattern_AI5 /patternstroke get exec grestore newpath moveto 2 copy rmoveto } cforall 9 npop } def /patternashowstroke { 0 0 0 7 3 roll patternawidthshowstroke } def /patternawidthshowstroke { 7 index type /dicttype eq { patternmatrix /patternmatrix get 8 1 roll } if _lineorientation 0 eq { hpatternawidthshowstroke } { vpatternawidthshowstroke } ifelse } def end setpacking %%EndResource %%EndProlog %%BeginSetup %%IncludeFont: TimesNewRomanPSMT Adobe_level2_AI5 /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_blend_AI5 /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_pattern_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 130/quotesinglbase/florin/quotedblbase/ellipsis /dagger/daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE 145/quoteleft /quoteright/quotedblleft/quotedblright/bullet/endash/emdash/tilde/trademark /scaron/guilsinglright/oe/dotlessi 159/Ydieresis /space 164/currency 166/brokenbar 168/dieresis/copyright/ordfeminine 172/logicalnot/hyphen/registered/macron/ring /plusminus/twosuperior/threesuperior/acute/mu 183/periodcentered/cedilla /onesuperior/ordmasculine 188/onequarter/onehalf/threequarters 192/Agrave /Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla/Egrave/Eacute /Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis/Eth/Ntilde /Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply/Oslash/Ugrave /Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls/agrave/aacute /acircumflex/atilde/adieresis/aring/ae/ccedilla/egrave/eacute/ecircumflex /edieresis/igrave/iacute/icircumflex/idieresis/eth/ntilde/ograve/oacute /ocircumflex/otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex /udieresis/yacute/thorn/ydieresis TE %AI3_BeginEncoding: _TimesNewRomanPSMT TimesNewRomanPSMT 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1 0 k 0.9255 0 1 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 43.25 33 m 43.25 30.998 41.627 29.375 39.625 29.375 c 37.623 29.375 36 30.998 36 33 c 36 35.002 37.623 36.625 39.625 36.625 c 41.627 36.625 43.25 35.002 43.25 33 c f %AI6_EndPatternLayer ) & (0 O 0 R 0.0314 0.0196 0.9137 0 k 0.0314 0.0196 0.9137 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 13.25 43.75 m 13.25 40.9883 11.0117 38.75 8.25 38.75 c 5.4883 38.75 3.25 40.9883 3.25 43.75 c 3.25 46.5117 5.4883 48.75 8.25 48.75 c 11.0117 48.75 13.25 46.5117 13.25 43.75 c f %AI6_EndPatternLayer ) & (0 O 0 R 0.3922 0.9451 0 0 k 0.3922 0.9451 0 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 6.5 22.25 m 6.5 20.7314 5.2686 19.5 3.75 19.5 c 2.2314 19.5 1 20.7314 1 22.25 c 1 23.7686 2.2314 25 3.75 25 c 5.2686 25 6.5 23.7686 6.5 22.25 c f %AI6_EndPatternLayer ) & (0 O 0 R 0.0235 0.0118 0.6 0 k 0.0235 0.0118 0.6 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 34.4883 32.75 m 34.4883 31.9277 33.8223 31.2617 33 31.2617 c 32.1777 31.2617 31.5117 31.9277 31.5117 32.75 c 31.5117 33.5723 32.1777 34.2383 33 34.2383 c 33.8223 34.2383 34.4883 33.5723 34.4883 32.75 c f %AI6_EndPatternLayer ) & (0 O 0 R 0.0314 0.0196 0.9137 0 k 0.0314 0.0196 0.9137 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 49.25 43.75 m 49.25 40.9883 47.0117 38.75 44.25 38.75 c 41.4883 38.75 39.25 40.9883 39.25 43.75 c 39.25 46.5117 41.4883 48.75 44.25 48.75 c 47.0117 48.75 49.25 46.5117 49.25 43.75 c f %AI6_EndPatternLayer ) & (0 O 0 R 0.3922 0.9451 0 0 k 0.3922 0.9451 0 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 42.5 22.25 m 42.5 20.7314 41.2686 19.5 39.75 19.5 c 38.2314 19.5 37 20.7314 37 22.25 c 37 23.7686 38.2314 25 39.75 25 c 41.2686 25 42.5 23.7686 42.5 22.25 c f %AI6_EndPatternLayer ) & (0 O 0 R 0.0314 0.0196 0.9137 0 k 0.0314 0.0196 0.9137 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 49.25 7.752 m 49.25 4.9902 47.0117 2.752 44.25 2.752 c 41.4883 2.752 39.25 4.9902 39.25 7.752 c 39.25 10.5137 41.4883 12.752 44.25 12.752 c 47.0117 12.752 49.25 10.5137 49.25 7.752 c f %AI6_EndPatternLayer ) & (0 O 0 R 0.8353 0 0 0 k 0.8353 0 0 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 37.25 7.875 m 37.25 3.5264 33.7236 0 29.375 0 c 25.0264 0 21.5 3.5264 21.5 7.875 c 21.5 12.2236 25.0264 15.75 29.375 15.75 c 33.7236 15.75 37.25 12.2236 37.25 7.875 c f %AI6_EndPatternLayer ) & (0 O 0 R 0.0314 0.0196 0.9137 0 k 0.0314 0.0196 0.9137 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 13.25 7.75 m 13.25 4.9883 11.0117 2.75 8.25 2.75 c 5.4883 2.75 3.25 4.9883 3.25 7.75 c 3.25 10.5117 5.4883 12.75 8.25 12.75 c 11.0117 12.75 13.25 10.5117 13.25 7.75 c f %AI6_EndPatternLayer ) & (0 O 0 R 0.9255 0 1 0 k 0.9255 0 1 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 7.25 33 m 7.25 30.998 5.627 29.375 3.625 29.375 c 1.623 29.375 0 30.998 0 33 c 0 35.002 1.623 36.625 3.625 36.625 c 5.627 36.625 7.25 35.002 7.25 33 c f %AI6_EndPatternLayer ) & (0 O 0 R 0.0235 0.4078 0 0 k 0.0235 0.4078 0 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 39.6533 16.0381 m 39.6533 15.1064 38.8975 14.3506 37.9658 14.3506 c 37.0342 14.3506 36.2783 15.1064 36.2783 16.0381 c 36.2783 16.9697 37.0342 17.7256 37.9658 17.7256 c 38.8975 17.7256 39.6533 16.9697 39.6533 16.0381 c f %AI6_EndPatternLayer ) & ] E %AI3_EndPattern %AI3_BeginPattern: (Green Tablecloth) (Green Tablecloth) 0 0 24 24 [ %AI3_Tile (0 O 0 R 0.8 0.21 1 0 k 0.8 0.21 1 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 6 18 m 0 18 L 0 24 L 6 24 L 6 18 L f 18 18 m 12 18 L 12 24 L 18 24 L 18 18 L f %AI6_EndPatternLayer ) & (0 O 0 R 0.3798 0.0997 0.4747 0 k 0.3798 0.0997 0.4747 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 12 18 m 6 18 L 6 24 L 12 24 L 12 18 L f 24 18 m 18 18 L 18 24 L 24 24 L 24 18 L f %AI6_EndPatternLayer ) & (0 O 0 R 0 0 0 0 k 0 0 0 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 12 12 m 6 12 L 6 18 L 12 18 L 12 12 L f 24 12 m 18 12 L 18 18 L 24 18 L 24 12 L f %AI6_EndPatternLayer ) & (0 O 0 R 0.3798 0.0997 0.4747 0 k 0.3798 0.0997 0.4747 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 6 12 m 0 12 L 0 18 L 6 18 L 6 12 L f 18 12 m 12 12 L 12 18 L 18 18 L 18 12 L f %AI6_EndPatternLayer ) & (0 O 0 R 0.8 0.21 1 0 k 0.8 0.21 1 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 6 6 m 0 6 L 0 12 L 6 12 L 6 6 L f 18 6 m 12 6 L 12 12 L 18 12 L 18 6 L f %AI6_EndPatternLayer ) & (0 O 0 R 0.3798 0.0997 0.4747 0 k 0.3798 0.0997 0.4747 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 12 6 m 6 6 L 6 12 L 12 12 L 12 6 L f 24 6 m 18 6 L 18 12 L 24 12 L 24 6 L f %AI6_EndPatternLayer ) & (0 O 0 R 0 0 0 0 k 0 0 0 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 12 0 m 6 0 L 6 6 L 12 6 L 12 0 L f 24 0 m 18 0 L 18 6 L 24 6 L 24 0 L f %AI6_EndPatternLayer ) & (0 O 0 R 0.3798 0.0997 0.4747 0 k 0.3798 0.0997 0.4747 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 6 0 m 0 0 L 0 6 L 6 6 L 6 0 L f 18 0 m 12 0 L 12 6 L 18 6 L 18 0 L f %AI6_EndPatternLayer ) & ] E %AI3_EndPattern %AI3_BeginPattern: (Pyramids) (Pyramids) 1 1 19.5 19.5 [ %AI3_Tile (0 O 0 R 0.1882 0.1294 0.102 0.0118 k 0.1882 0.1294 0.102 0.0118 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 1 19.5 m 10.25 10.25 l 19.5 19.5 l 1 19.5 l f %AI6_EndPatternLayer ) & (0 O 0 R 0.6235 0.4863 0.4863 0.4784 k 0.6235 0.4863 0.4863 0.4784 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 19.5 1 m 10.25 10.25 l 1 1 l 19.5 1 l f %AI6_EndPatternLayer ) & (0 O 0 R 0.3529 0.2549 0.2235 0.0667 k 0.3529 0.2549 0.2235 0.0667 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 1 1 m 10.25 10.25 l 1 19.5 l 1 1 l f %AI6_EndPatternLayer ) & (0 O 0 R 0.4902 0.3647 0.3451 0.2196 k 0.4902 0.3647 0.3451 0.2196 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 1 w 4 M []0 d 0 XR 19.5 19.5 m 10.25 10.25 l 19.5 1 l 19.5 19.5 l f %AI6_EndPatternLayer ) & ] E %AI3_EndPattern %AI3_BeginPattern: (Red Stripe) (Red Stripe) 37 19 73 55 [ %AI3_Tile (0 O 0 R 0 0.9608 0.9137 0 k 0 0.9608 0.9137 0 K ) @ ( %AI6_BeginPatternLayer 0 J 0 j 18 w 4 M []0 d 0 XR 37 37 m 73 37 l S %AI6_EndPatternLayer ) & ] E %AI3_EndPattern %AI5_End_NonPrinting-- %AI5_BeginPalette 0 0 Pb 1 1 1 1 k Pc 0 0 0 0 k Pc 0 0 0 1 k Pc 0 0 0 0.8 k Pc 0 0 0 0.6 k Pc 0 0 0 0.4 k Pc 0 0 0 0.2 k Pc 0 1 1 0 k Pc 0 0.75 0.75 0 k Pc 0 0.5 0.5 0 k Pc 0 0.25 0.25 0 k Pc 0.8 0 1 0 k Pc 0.6 0 0.75 0 k Pc 0.4 0 0.5 0 k Pc 0.2 0 0.25 0 k Pc 1 0.5 0 0 k Pc 0.75 0.375 0 0 k Pc 0.5 0.25 0 0 k Pc 0.25 0.125 0 0 k Pc 0 0.5 1 0 k Pc 0 0.375 0.75 0 k Pc 0 0.25 0.5 0 k Pc 0 0.125 0.25 0 k Pc 0.5 0.9 0 0 k Pc 0.375 0.675 0 0 k Pc 0.25 0.45 0 0 k Pc 0.125 0.225 0 0 k Pc 0.8 0 0 0 k Pc 0.6 0 0 0 k Pc 0.4 0 0 0 k Pc 0.2 0 0 0 k Pc 0 0 1 0 k Pc 0 0 0.75 0 k Pc 0 0 0.5 0 k Pc 0 0 0.25 0 k Pc Bb 2 (White, Black) 0 0 0 1 1 0 0 1 0 0 Bg 0 BB Pc Bb 0 0 0 0 Bh 2 (Black, White Radial) 0 0 0 1 1 0 0 1 0 0 Bg 0 BB Pc Bb 2 (Steel Bar II) 0 0 0 1 1 0 0 1 0 0 Bg 0 BB Pc Bb 2 (Rainbow) 0 0 0 1 1 0 0 1 0 0 Bg 0 BB Pc Bb 0 0 0 0 Bh 2 (Emerald Eye) 0 0 0 1 1 0 0 1 0 0 Bg 0 BB Pc Bb 2 (Sunset) 0 0 0 1 1 0 0 1 0 0 Bg 0 BB Pc (Green Tablecloth) 0 0 1 1 0 0 0 0 0 [1 0 0 1 0 0] p Pc (Pyramids) 0 0 1 1 0 0 0 0 0 [1 0 0 1 0 0] p Pc (Azure Rings) 0 0 1 1 0 0 0 0 0 [1 0 0 1 0 0] p Pc (Red Stripe) 0 0 1 1 0 0 0 0 0 [1 0 0 1 0 0] p Pc (Clown Attack) 0 0 1 1 0 0 0 0 0 [1 0 0 1 0 0] p Pc (Camouflage) 0 0 1 1 0 0 0 0 0 [1 0 0 1 0 0] p Pc PB %AI5_EndPalette %%EndSetup %AI5_BeginLayer 1 1 1 1 0 0 0 79 128 255 Lb (Layer 1) Ln 0 A 0 O 0 g 0 R 0 0.9608 0.9137 0 K 0 J 0 j 18 w 4 M []0 d 0 XR q 418.2295 558.1943 m 395.1563 544.4839 345.5156 535 288 535 C 253.2061 535 221.2944 538.4707 196.4028 544.2485 C 180.1577 548.019 166.9023 552.772 157.7876 558.1841 C 151.563 561.8799 143.7676 567.7495 144.0049 575.8506 C 144.1875 581.917 149.1123 587.665 157.7876 592.8159 C 166.9023 598.228 180.1577 602.981 196.4028 606.7515 C 221.2944 612.5293 253.2061 616 288 616 C 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TC 100 100 200 TW 0 0 0 Ti 0 Ta 0 1 2 2 3 Th 0 Tq 0 0 Tl 0 Tc 0 Tw (M) Tx 1 0 Tk (\r) TX TO 0 To 1 0 0 1 5 416 0 Tp TP 0 Tr (f\(N\)) Tx 1 0 Tk (\r) TX TO LB %AI5_EndLayer-- %%PageTrailer gsave annotatepage grestore showpage %%Trailer Adobe_Illustrator_AI5 /terminate get exec Adobe_ColorImage_AI6 /terminate get exec Adobe_pattern_AI5 /terminate get exec Adobe_blend_AI5 /terminate get exec Adobe_typography_AI5 /terminate get exec Adobe_level2_AI5 /terminate get exec %%EOF %%EndDocument @endspecial 6863 24302 a Fu(Figure)395 b(1:)544 b(An)395 b(illustration)i(of)e(a)g(m)-31 b(ultiple)398 b(wrapp)31 b(ed)394 b(co)-31 b(v)g(ering)396 b(relation)h(in)e(the)g(plane.)6863 25631 y(The)315 b(arro)-31 b(ws)315 b(on)f(the)h(side)f(of)h Fq(f)119 b Fu(\()p Fq(N)i Fu(\))316 b(merely)f(suggest)g(that)g(a)g (cross)f(section)h(disk)f(in)h Fq(f)119 b Fu(\()p Fq(N)i Fu(\))6863 26959 y(wraps)369 b(around)h(t)-31 b(wice)370 b(the)g(corresp)31 b(onding)369 b(cross)f(section)i(disk)g(in)f Fq(M)6863 30495 y Fj(we)396 b(c)-57 b(an)397 b(de\257ne)g(a)e(map)h Fq(s)17679 30661 y Fl(A)18708 30495 y Fu(:)308 b Fq(S)20066 30093 y Fl(u)p Fi(\241)p Fv(1)22085 30495 y Fo(!)f Fq(S)24242 30093 y Fl(u)p Fi(\241)p Fv(1)26349 30495 y Fj(by)21851 33507 y Fq(s)22370 33673 y Fl(A)23091 33507 y Fu(\()p Fq(x)p Fu(\))i(=)26747 32758 y Fq(A)p Fu(\()p Fq(x)p Fu(\))p 26194 33252 3431 45 v 26194 34266 a Fo(k)p Fq(A)p Fu(\()p Fq(x)p Fu(\))p Fo(k)29757 33507 y Fq(:)13021 b Fu(\(36\))6863 36591 y Fj(The)546 b(de)-57 b(gr)g(e)g(e)547 b Fq(d)p Fu(\()p Fq(s)14246 36757 y Fl(A)14968 36591 y Fu(\))f Fj(of)g(a)f(mapping)h(of)g(a)f(spher)-57 b(e)546 b(is)f(de\257ne)-57 b(d)547 b(in)g(App)-57 b(endix)545 b(6.2.)959 b(By)6863 37919 y(L)-57 b(emma)297 b(23,)315 b(we)295 b(obtain)h Fu(deg)16 b(\()p Fq(A;)p 20701 37030 1419 45 v 184 w(B)21541 38085 y Fl(u)22121 37919 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fq(;)g Fu(0\))312 b(=)307 b Fq(d)p Fu(\()p Fq(s)29057 38085 y Fl(A)29779 37919 y Fu(\))p Fj(.)477 b(Thus,)315 b(the)295 b(de)-57 b(gr)g(e)g(e)297 b(of)f(a)f(c)-57 b(overing)6863 39635 y Fq(N)8375 38956 y Fl(f)39 b(;w)8181 39635 y Fu(=)-184 b Fo(\))307 b Fq(M)516 b Fj(c)-57 b(an)397 b(b)-57 b(e)397 b(c)-57 b(ompute)g(d)397 b(as)e Fq(w)338 b Fu(=)307 b Fq(d)p Fu(\()p Fq(s)25360 39801 y Fl(A)26082 39635 y Fu(\))p Fj(.)8524 42071 y Fu(W)-92 b(e)520 b(de\257ne)g(the)h(corresp)31 b(onding)521 b(notion)h(of)f(bac)-31 b(k)g(co)g(v)g(ering)524 b(for)c(this)h(new)g (t)-31 b(yp)31 b(e)522 b(of)6863 43399 y(co)-31 b(v)g(ering)371 b(relation.)6863 45613 y Fk(De\257nition)426 b(7)554 b Fj(Assume)585 b Fq(N)59 b(;)184 b(M)708 b Fj(ar)-57 b(e)586 b Fq(h)p Fj(-sets,)632 b(such)585 b(that)g Fq(u)p Fu(\()p Fq(N)121 b Fu(\))652 b(=)f Fq(u)p Fu(\()p Fq(M)121 b Fu(\))651 b(=)g Fq(u)585 b Fj(and)6863 46941 y Fq(s)p Fu(\()p Fq(N)121 b Fu(\))511 b(=)g Fq(s)p Fu(\()p Fq(M)121 b Fu(\))511 b(=)f Fq(s)p Fj(.)845 b(L)-57 b(et)509 b Fq(g)550 b Fu(:)511 b(\255)g Fo(\275)f Fp(R)24693 46452 y Fl(n)25806 46941 y Fo(!)h Fp(R)28223 46452 y Fl(n)28826 46941 y Fj(.)845 b(Assume)508 b(that)f Fq(g)37198 46540 y Fi(\241)p Fv(1)38896 46941 y Fu(:)k Fo(j)p Fq(M)121 b Fo(j)511 b(!)f Fp(R)44451 46452 y Fl(n)6863 48453 y Fj(is)431 b(a)f(wel)57 b(l)429 b(de\257ne)-57 b(d,)441 b(c)-57 b(ontinuous)431 b(map.)613 b(We)430 b(say)g(that)g Fq(N)32000 47773 y Fl(g)28 b(;w)31800 48453 y Fo(\()-184 b Fu(=)369 b Fq(M)551 b Fj(\()p Fq(N)h(g)40 b Fj(-b)-57 b(ackc)g(overs)431 b Fq(M)6863 50334 y Fj(with)396 b(de)-57 b(gr)g(e)g(e)397 b Fq(w)30 b Fj(\))396 b(i\256)h Fq(M)16844 49932 y Fl(T)17848 49654 y(g)18308 49320 y Fg(\241)p Ff(1)19339 49654 y Fl(;w)18164 50334 y Fu(=)-184 b Fo(\))623 b Fq(N)21581 49932 y Fl(T)22278 50334 y Fj(.)6863 52769 y Fk(Theorem)426 b(9)554 b Fj(L)-57 b(et)371 b Fq(N)16233 52935 y Fl(i)16601 52769 y Fj(,)376 b Fq(i)308 b Fu(=)f(0)p Fq(;)184 b(:)g(:)g(:)189 b(;)184 b(k)406 b Fj(b)-57 b(e)371 b(h-sets)e(and)h Fq(N)30627 52935 y Fl(k)31480 52769 y Fu(=)307 b Fq(N)33537 52935 y Fv(0)34034 52769 y Fj(.)501 b(Assume)370 b(that)f(for)h(e)-57 b(ach)6863 54097 y Fq(i)308 b Fu(=)f(1)p Fq(;)184 b(:)g(:)g(:)189 b(;)184 b(k)432 b Fj(we)396 b(have)g(either)22654 55752 y Fq(N)23543 55918 y Fl(i)p Fi(\241)p Fv(1)25353 55072 y Fl(f)25785 55183 y Fh(i)26135 55072 y Fl(;w)27038 55183 y Fh(i)25479 55752 y Fu(=)-184 b Fo(\))434 b Fq(N)28586 55918 y Fl(i)28955 55752 y Fq(;)13823 b Fu(\(37\))6863 57744 y Fj(or)16813 59184 y Fq(N)17702 59350 y Fl(i)18378 59184 y Fo(\275)308 b Fj(dom)184 b Fu(\()p Fq(f)22859 58710 y Fi(\241)p Fv(1)22740 59492 y Fl(i)24048 59184 y Fu(\))1132 b Fj(and)g Fq(N)29385 59350 y Fl(i)p Fi(\241)p Fv(1)31194 58504 y Fl(f)31626 58615 y Fh(i)31977 58504 y Fl(;w)32880 58615 y Fh(i)31321 59184 y Fo(\()-184 b Fu(=)433 b Fq(N)34427 59350 y Fl(i)34796 59184 y Fq(:)7982 b Fu(\(38\))8524 61176 y Fj(Then)396 b(ther)-57 b(e)396 b(exists)f(a)h(p)-57 b(oint)397 b Fq(x)307 b Fo(2)g Fu(in)-31 b(t)186 b Fq(N)25342 61342 y Fv(0)25839 61176 y Fj(,)397 b(such)e(that)14459 63612 y Fq(f)15001 63778 y Fl(i)15615 63612 y Fo(\261)247 b Fq(f)16957 63778 y Fl(i)p Fi(\241)p Fv(1)18704 63612 y Fo(\261)g(\242)184 b(\242)g(\242)248 b(\261)f Fq(f)22383 63778 y Fv(1)22879 63612 y Fu(\()p Fq(x)p Fu(\))1169 b Fo(2)g Fu(in)-31 b(t)186 b Fq(N)29844 63778 y Fl(i)30212 63612 y Fq(;)1317 b(i)308 b Fu(=)f(1)p Fq(;)184 b(:)g(:)g(:)189 b(;)184 b(k)35 b(;)5461 b Fu(\(39\))14107 65272 y Fq(f)14649 65438 y Fl(k)15439 65272 y Fo(\261)247 b Fq(f)16781 65438 y Fl(k)19 b Fi(\241)p Fv(1)18704 65272 y Fo(\261)247 b(\242)184 b(\242)g(\242)248 b(\261)f Fq(f)22383 65438 y Fv(1)22879 65272 y Fu(\()p Fq(x)p Fu(\))1108 b(=)f Fq(x:)14697 b Fu(\(40\))25405 70071 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 6863 6974 a Fk(Pro)35 b(of:)787 b Fu(The)516 b(pro)31 b(of)516 b(of)g(this)g(theorem)h(follo)-31 b(ws)518 b(the)e(same)g(pattern)h(as)e(the)h(pro)31 b(of)517 b(of)6863 8302 y(Theorem)469 b(4.)790 b(W)-92 b(e)467 b(de\257ne)h(\246,)493 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\),)495 b Fq(F)621 b Fu(and)469 b Fq(H)558 b Fu(as)468 b(it)g(w)-31 b(as)469 b(done)f(there.)790 b(Lemma)469 b(5)f(is)6863 9631 y(v)-61 b(alid,)406 b(with)399 b(the)f(same)g(pro)31 b(of,)406 b(b)31 b(ecause)397 b(condition)j Fk(2)e Fu(in)g(the)g (de\257nition)h(of)f(co)-31 b(v)g(ering)399 b(is)6863 10959 y(not)370 b(used)f(in)g(its)h(pro)31 b(of.)8524 12287 y(Instead)369 b(of)h(Lemma)g(6)f(w)-31 b(e)370 b(will)h(ha)-31 b(v)g(e)370 b(the)f(follo)-31 b(wing)6863 14423 y Fk(Lemma)424 b(10)16950 15751 y Fo(j)184 b Fu(deg)17 b(\()p Fq(H)20468 15917 y Fv(1)20965 15751 y Fq(;)184 b Fu(in)-31 b(t)187 b(\246)p Fq(;)d Fu(0\))p Fo(j)310 b Fu(=)e Fo(j)p Fq(w)28153 15917 y Fv(1)28896 15751 y Fo(\242)247 b Fq(w)30242 15917 y Fv(2)30985 15751 y Fo(\242)f Fq(:)184 b(:)g(:)j(w)33806 15917 y Fl(k)34351 15751 y Fo(j)p Fq(:)8524 17887 y Fu(W)-92 b(e)368 b(\257nish)h(the)h(pro)31 b(of)369 b(b)-31 b(y)369 b(the)h(same)f(argumen)-31 b(t)371 b(as)e(in)h(Theorem)g(4.)p 44721 17887 333 554 v 6863 21594 a Fd(3.1)1495 b(Pro)42 b(of)498 b(of)g(Lemma)h(10.)6863 23638 y Fu(Let)487 b(us)f(represen)-31 b(t)486 b Fq(x)16031 23804 y Fl(i)16886 23638 y Fu(as)g(a)h(pair)g Fq(x)22431 23804 y Fl(i)23303 23638 y Fu(=)503 b(\()p Fq(p)25654 23804 y Fl(i)26023 23638 y Fq(;)184 b(q)27008 23804 y Fl(i)27378 23638 y Fu(\),)516 b(where)487 b Fq(p)32507 23804 y Fl(i)33379 23638 y Fo(2)503 b Fp(R)35419 23149 y Fl(u)36485 23638 y Fu(and)487 b Fq(q)39249 23804 y Fl(i)40121 23638 y Fo(2)503 b Fp(R)42161 23149 y Fl(s)42634 23638 y Fu(.)845 b(As)6863 24966 y(usual)380 b(w)-31 b(e)380 b(set)f(\()p Fq(p)14135 25132 y Fi(\241)p Fv(1)15324 24966 y Fq(;)184 b(q)16309 25132 y Fi(\241)p Fv(1)17498 24966 y Fu(\))325 b(=)f(\()p Fq(p)20425 25132 y Fl(k)19 b Fi(\241)p Fv(1)22104 24966 y Fq(;)184 b(q)23089 25132 y Fl(k)19 b Fi(\241)p Fv(1)24767 24966 y Fu(\),)383 b(\()p Fq(p)26874 25132 y Fl(k)27419 24966 y Fq(;)184 b(q)28404 25132 y Fl(k)28950 24966 y Fu(\))325 b(=)f(\()p Fq(p)31877 25132 y Fv(0)32374 24966 y Fq(;)184 b(q)33359 25132 y Fv(0)33857 24966 y Fu(\))379 b(and)h Fq(A)37659 25132 y Fl(k)38528 24966 y Fu(=)324 b Fq(A)40543 25132 y Fv(0)41040 24966 y Fu(.)523 b(In)379 b(this)6863 26294 y(represen)-31 b(tation)399 b(the)e(map)i Fq(H)19473 26460 y Fv(1)19969 26294 y Fu(\()p Fq(p)20956 26460 y Fv(0)21453 26294 y Fq(;)184 b(q)22438 26460 y Fv(0)22936 26294 y Fq(;)g(:)g(:)g(:)k(;)c(p) 25952 26460 y Fl(k)19 b Fi(\241)p Fv(1)27631 26294 y Fq(;)184 b(q)28616 26460 y Fl(k)19 b Fi(\241)p Fv(1)30295 26294 y Fu(\))355 b(=)f(\()94 b(~)-647 b Fq(p)33282 26460 y Fv(0)33779 26294 y Fq(;)268 b Fu(~)-637 b Fq(q)34764 26460 y Fv(0)35262 26294 y Fq(;)184 b(:)g(:)g(:)k(;)279 b Fu(~)-648 b Fq(p)38278 26460 y Fl(k)19 b Fi(\241)p Fv(1)39957 26294 y Fq(;)267 b Fu(~)-636 b Fq(q)40942 26460 y Fl(k)19 b Fi(\241)p Fv(1)42620 26294 y Fu(\))398 b(has)6863 27623 y(the)370 b(follo)-31 b(wing)372 b(form)e(\(for)g Fq(\270)307 b Fu(=)h(0\))15292 29960 y(~)-647 b Fq(p)15755 30126 y Fl(i)16431 29960 y Fu(=)307 b(\(1)247 b Fo(\241)f Fq(\270)p Fu(\))p Fq(p)21569 30126 y Fl(i)22184 29960 y Fo(\241)g Fq(A)24121 30126 y Fl(i)24489 29960 y Fu(\()p Fq(p)25476 30126 y Fl(i)p Fi(\241)p Fv(1)26979 29960 y Fu(\))p Fq(;)1292 b Fu(if)370 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(1)q Fq(;)8765 b Fu(\(41\))24290 31621 y(~)-635 b Fq(q)24702 31787 y Fl(i)25378 31621 y Fu(=)307 b Fq(q)27040 31787 y Fl(i)27409 31621 y Fq(;)1292 b Fu(if)370 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(1)q Fq(;)8765 b Fu(\(42\))23043 33281 y(~)-647 b Fq(p)23506 33447 y Fl(i)24182 33281 y Fu(=)307 b Fq(p)25907 33447 y Fl(i)p Fi(\241)p Fv(1)27409 33281 y Fq(;)1292 b Fu(if)370 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(0)q Fq(;)8765 b Fu(\(43\))15653 34942 y(~)-635 b Fq(q)16065 35108 y Fl(i)16741 34942 y Fu(=)307 b(\(1)247 b Fo(\241)f Fq(\270)p Fu(\))p Fq(q)21816 35108 y Fl(i)p Fi(\241)p Fv(1)23564 34942 y Fo(\241)g Fq(A)25501 35108 y Fl(i)25870 34942 y Fu(\()p Fq(q)26794 35108 y Fl(i)27163 34942 y Fu(\))p Fq(;)1292 b Fu(if)370 b Fq(\261)227 b Fu(\()p Fq(i)p Fu(\))308 b(=)g(0)p Fq(:)8581 b Fu(\(44\))8524 37408 y(The)418 b(ab)31 b(o)-31 b(v)g(e)419 b(equations)h(de\257ne)d(a) i(homotop)-31 b(y)420 b Fq(C)468 b Fu(:)390 b([0)p Fq(;)184 b Fu(1])281 b Fo(\243)e Fu(\246)389 b Fo(!)f Fp(R)37573 36919 y Fv(\()p Fl(u)p Fv(+)p Fl(s)p Fv(\))p Fl(k)40430 37408 y Fu(.)639 b(W)-92 b(e)417 b(will)6863 38737 y(sho)-31 b(w)325 b(that)g(deg)16 b(\()p Fq(C)14811 38903 y Fl(\270)15392 38737 y Fq(;)184 b Fu(in)-31 b(t)187 b(\246)p Fq(;)d Fu(0\))326 b(is)e(indep)31 b(enden)-31 b(t)324 b(of)h Fq(\270)f Fu(and)g(then)g(w)-31 b(e)325 b(compute)g(the)f(degree)6863 40065 y(of)370 b Fq(C)8915 40231 y Fv(1)9412 40065 y Fu(.)6863 42403 y Fk(Lemma)424 b(11)555 b Fj(F)-85 b(or)395 b(any)i Fq(\270)307 b Fo(2)g Fu([0)p Fq(;)184 b Fu(1])17326 44740 y(deg)16 b(\()p Fq(C)20223 44906 y Fl(\270)20804 44740 y Fq(;)184 b Fu(in)-31 b(t)186 b(\246)p Fq(;)e Fu(0\))310 b(=)d(deg)16 b(\()p Fq(C)29481 44906 y Fv(1)29978 44740 y Fq(;)184 b Fu(in)-31 b(t)187 b(\246)p Fq(;)d Fu(0\))p Fq(:)6863 47078 y Fk(Pro)35 b(of:)779 b Fu(F)-92 b(rom)511 b(the)i(homotop)-31 b(y)514 b(prop)31 b(ert)-31 b(y)512 b(of)g(the)h(lo)31 b(cal)513 b(degree)f(\(see)f(App)31 b(endix\),)549 b(it)6863 48406 y(follo)-31 b(ws)372 b(that)e(it)g(is)f (enough)h(to)g(pro)-31 b(v)g(e)370 b(that)15842 50744 y Fq(C)16633 50910 y Fl(\270)17214 50744 y Fu(\()p Fq(x)p Fu(\))308 b Fo(6)p Fu(=)f(0)p Fq(;)1293 b Fu(for)369 b(all)i Fq(x)307 b Fo(2)g Fq(@)61 b Fu(\246)370 b(and)f Fq(\270)307 b Fo(2)h Fu([0)p Fq(;)184 b Fu(1].)7014 b(\(45\))6863 53082 y(Let)449 b(us)g(tak)-31 b(e)450 b Fq(x)440 b Fu(=)g(\()p Fq(p)16266 53248 y Fv(0)16763 53082 y Fq(;)184 b(q)17748 53248 y Fv(0)18246 53082 y Fq(;)g(p)19294 53248 y Fv(1)19792 53082 y Fq(;)g(q)20777 53248 y Fv(1)21274 53082 y Fq(;)g(:)g(:)g(:)k(;) c(p)24290 53248 y Fl(k)19 b Fi(\241)p Fv(1)25969 53082 y Fq(;)184 b(q)26954 53248 y Fl(k)19 b Fi(\241)p Fv(1)28633 53082 y Fu(\))441 b Fo(2)f Fq(@)61 b Fu(\246.)732 b(There)449 b(exists)g Fq(i)g Fu(suc)-31 b(h)449 b(that)6863 54410 y(one)370 b(the)f(follo)-31 b(wing)373 b(conditions)e(holds)e(true) 22805 56748 y Fq(p)23362 56914 y Fl(i)24038 56748 y Fo(2)308 b Fq(S)25827 56291 y Fl(u)26405 56748 y Fq(;)16373 b Fu(\(46\))23158 58408 y Fq(q)23652 58574 y Fl(i)24328 58408 y Fo(2)308 b Fq(S)26117 57951 y Fl(s)26590 58408 y Fq(:)16188 b Fu(\(47\))6863 60746 y(Assume)419 b(that)h Fq(p)14020 60912 y Fl(i)14779 60746 y Fo(2)390 b Fq(S)16650 60344 y Fl(u)17228 60746 y Fu(.)642 b(If)419 b Fq(\261)226 b Fu(\()p Fq(i)280 b Fu(+)f(1\))391 b(=)g(1,)432 b(then)513 b(~)-647 b Fq(p)29329 60912 y Fl(i)p Fv(+1)31209 60746 y Fo(6)p Fu(=)390 b(0,)432 b(b)31 b(ecause)418 b(from)i(condition)6863 62074 y(\(34\))371 b(it)f(follo)-31 b(ws)372 b(that)18162 64412 y Fo(k)p Fq(A)19545 64578 y Fl(i)p Fv(+1)21035 64412 y Fu(\()p Fq(p)22022 64578 y Fl(i)22391 64412 y Fu(\))p Fo(k)309 b Fq(>)e Fu(1)h Fo(\270)f(k)p Fu(\(1)248 b Fo(\241)e Fq(\270)p Fu(\))p Fq(p)31404 64578 y Fl(i)p Fv(+1)32893 64412 y Fo(k)p Fq(;)9332 b Fu(\(48\))6863 66750 y(for)369 b(an)-31 b(y)370 b Fq(p)11205 66916 y Fl(i)p Fv(+1)13002 66750 y Fo(2)p 14048 65787 3879 45 v 308 w Fq(B)14888 66916 y Fl(u)15466 66750 y Fu(\(0)p Fq(;)184 b Fu(1\))s(.)25405 70071 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 8524 6974 a Fu(If)369 b Fq(\261)226 b Fu(\()p Fq(i)247 b Fu(+)f(1\))308 b(=)g(0,)370 b(then)f(ob)-31 b(viously)465 b(~)-647 b Fq(p)24177 7140 y Fl(i)p Fv(+1)25974 6974 y Fu(=)307 b Fq(p)27699 7140 y Fl(i)28375 6974 y Fo(6)p Fu(=)h(0.)8524 8302 y(The)369 b(argumen)-31 b(t)371 b(for)e(the)g(case)g Fq(q)22190 8468 y Fl(i)22867 8302 y Fo(2)307 b Fq(S)24655 7900 y Fl(s)25497 8302 y Fu(is)369 b(similar.)p 44721 8302 333 554 v 8524 10154 a(No)-31 b(w)390 b(w)-31 b(e)390 b(turn)f(to)i(the)e(computation)k(of)d(the)g (degree)f(of)h Fq(C)33723 10320 y Fv(1)34219 10154 y Fu(.)554 b(Observ)-31 b(e)389 b(that)h Fq(C)42562 10320 y Fv(1)43448 10154 y Fu(has)6863 11482 y(the)370 b(follo)-31 b(wing)372 b(form)17707 13566 y(~)-647 b Fq(p)18170 13732 y Fl(i)18846 13566 y Fu(=)307 b Fo(\241)p Fq(A)21705 13732 y Fl(i)22074 13566 y Fu(\()p Fq(p)23061 13732 y Fl(i)p Fi(\241)p Fv(1)24563 13566 y Fu(\))p Fq(;)1293 b Fu(if)369 b Fq(\261)227 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(1)p Fq(;)11181 b Fu(\(49\))21875 15226 y(~)-636 b Fq(q)22286 15392 y Fl(i)22963 15226 y Fu(=)307 b Fq(q)24625 15392 y Fl(i)24994 15226 y Fq(;)1292 b Fu(if)369 b Fq(\261)227 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(1)p Fq(;)11181 b Fu(\(50\))20628 16887 y(~)-647 b Fq(p)21091 17053 y Fl(i)21767 16887 y Fu(=)307 b Fq(p)23492 17053 y Fl(i)p Fi(\241)p Fv(1)24994 16887 y Fq(;)1292 b Fu(if)369 b Fq(\261)227 b Fu(\()p Fq(i)p Fu(\))309 b(=)e(0)p Fq(;)11181 b Fu(\(51\))19139 18547 y(~)-636 b Fq(q)19550 18713 y Fl(i)20226 18547 y Fu(=)308 b Fo(\241)p Fq(A)23086 18713 y Fl(i)23455 18547 y Fu(\()p Fq(q)24379 18713 y Fl(i)24748 18547 y Fu(\))p Fq(;)1292 b Fu(if)370 b Fq(\261)226 b Fu(\()p Fq(i)p Fu(\))309 b(=)f(0)p Fq(:)10996 b Fu(\(52\))6863 20631 y(F)-92 b(rom)369 b(the)g(pro)31 b(duct)370 b(prop)31 b(ert)-31 b(y)369 b(of)h(the)f(degree)g(it)h(follo)-31 b(ws)371 b(that)32495 22714 y Fo(j)184 b Fu(deg)17 b(\()p Fq(C)35884 22880 y Fv(1)36381 22714 y Fq(;)184 b Fu(\246)p Fq(;)g Fu(0\))p Fo(j)311 b Fu(=)9048 23434 y Fe(\257)9048 24098 y(\257)9048 24762 y(\257)9417 24707 y Fu(\246)10247 25094 y Fl(i)p Fi(2)p Fq(\261)11842 24873 y Fg(\241)p Ff(1)12873 25094 y Fv(\(1\))14246 24707 y Fu(deg)16 b(\()p Fo(\241)p Fq(A)18043 24873 y Fl(i)18412 24707 y Fq(;)p 18903 23818 1419 45 v 184 w(B)19743 24873 y Fl(u)20322 24707 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fq(;)g Fu(0\))251 b Fo(\242)c Fu(\246)25888 25094 y Fl(i)p Fi(2)p Fq(\261)27483 24873 y Fg(\241)p Ff(1)28514 25094 y Fv(\(0\))29887 24707 y Fu(deg)16 b(\()p Fo(\241)p Fq(A)33684 24873 y Fl(i)34053 24707 y Fq(;)p 34544 23818 1313 45 v 184 w(B)35384 24873 y Fl(s)35858 24707 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fq(;)g Fu(0\))39789 23434 y Fe(\257)39789 24098 y(\257)39789 24762 y(\257)40347 24707 y Fq(:)6863 27193 y Fu(In)353 b(the)h(form)-31 b(ula)356 b(ab)31 b(o)-31 b(v)g(e)355 b(if)f Fq(\261)19008 26792 y Fi(\241)p Fv(1)20196 27193 y Fu(\()p Fq(i)p Fu(\))309 b(=)e Fo(;)p Fu(,)358 b(then)c(the)g (corresp)31 b(onding)354 b(pro)31 b(duct)354 b(is)f(set)h(to)h(b)31 b(e)6863 28522 y(equal)395 b(to)g(1.)567 b(In)394 b(the)g(situation)j (when)d Fq(u)349 b Fu(=)f(0)395 b(or)f Fq(s)348 b Fu(=)h(0)395 b(the)f(corresp)31 b(onding)394 b(pro)31 b(duct)394 b(is)6863 29850 y(also)370 b(set)f(equal)h(to)g(1.)8524 31179 y(F)-92 b(rom)368 b(\(104\))k(it)e(follo)-31 b(ws)371 b(that)17459 33262 y(deg)16 b(\()p Fo(\241)p Fq(A;)184 b(U)n(;)g Fu(0\))310 b(=)e(\()p Fo(\241)p Fu(1\))27728 32805 y Fl(u)28492 33262 y Fu(deg)16 b(\()p Fq(A;)184 b(U)n(;)g Fu(0\))p Fq(;)8631 b Fu(\(53\))6863 35346 y(whic)-31 b(h)370 b(completes)h(the)e (pro)31 b(of.)p 44721 35346 333 554 v 8524 37198 a(The)369 b(follo)-31 b(wing)373 b(corollary)e(is)e(an)g(immediate)j(consequence) d(of)h(Theorem)g(9.)6863 39130 y Fk(Collorary)425 b(12)555 b Fj(L)-57 b(et)478 b Fq(N)17234 39296 y Fl(i)17603 39130 y Fj(,)498 b Fq(i)455 b Fo(2)f Fp(Z)478 b Fj(b)-57 b(e)478 b(h-sets.)752 b(Assume)477 b(that)f(for)i(e)-57 b(ach)477 b Fq(i)455 b Fo(2)g Fp(Z)477 b Fj(we)g(have)6863 40458 y(either)22808 41898 y Fq(N)23697 42064 y Fl(i)p Fi(\241)p Fv(1)25507 41218 y Fl(f)25939 41329 y Fh(i)26289 41218 y Fl(;w)27192 41329 y Fh(i)25633 41898 y Fu(=)-184 b Fo(\))433 b Fq(N)28739 42064 y Fl(i)43085 41898 y Fu(\(54\))6863 43679 y Fj(or)16813 45118 y Fq(N)17702 45284 y Fl(i)18378 45118 y Fo(\275)308 b Fj(dom)184 b Fu(\()p Fq(f)22859 44645 y Fi(\241)p Fv(1)22740 45427 y Fl(i)24048 45118 y Fu(\))1132 b Fj(and)g Fq(N)29385 45284 y Fl(i)p Fi(\241)p Fv(1)31194 44439 y Fl(f)31626 44550 y Fh(i)31977 44439 y Fl(;w)32880 44550 y Fh(i)31321 45118 y Fo(\()-184 b Fu(=)433 b Fq(N)34427 45284 y Fl(i)34796 45118 y Fq(:)7982 b Fu(\(55\))8524 46900 y Fj(Then)396 b(ther)-57 b(e)396 b(exists)f(a)h(p)-57 b(oint)397 b Fq(x)307 b Fo(2)g Fu(in)-31 b(t)186 b Fq(N)25342 47066 y Fv(0)25839 46900 y Fj(,)397 b(such)e(that)15941 48983 y Fq(f)16483 49149 y Fl(i)17098 48983 y Fo(\261)246 b Fq(f)18439 49149 y Fl(i)p Fi(\241)p Fv(1)20187 48983 y Fo(\261)g(\242)184 b(\242)g(\242)249 b(\261)d Fq(f)23865 49149 y Fv(1)24362 48983 y Fu(\()p Fq(x)p Fu(\))1107 b Fo(2)g Fu(in)-31 b(t)186 b Fq(N)31203 49149 y Fl(i)31572 48983 y Fq(;)1316 b(i)308 b Fo(2)g Fp(Z)p Fq(:)7110 b Fu(\(56\))6863 51067 y Fj(Mor)-57 b(e)g(over,)348 b(if)337 b Fq(N)13926 51233 y Fl(i)p Fv(+)p Fl(k)15771 51067 y Fu(=)308 b Fq(N)17829 51233 y Fl(i)18533 51067 y Fj(for)336 b(some)f Fq(k)343 b(>)307 b Fu(0)336 b Fj(and)g(al)57 b(l)335 b Fq(i)p Fj(,)348 b(then)336 b(the)f(p)-57 b(oint)336 b Fq(x)g Fj(c)-57 b(an)336 b(b)-57 b(e)336 b(chosen)6863 52395 y(so)396 b(that)18510 54479 y Fq(f)19052 54645 y Fl(k)19843 54479 y Fo(\261)246 b Fq(f)21184 54645 y Fl(k)19 b Fi(\241)p Fv(1)23108 54479 y Fo(\261)246 b(\242)184 b(\242)g(\242)249 b(\261)d Fq(f)26786 54645 y Fv(1)27283 54479 y Fu(\()p Fq(x)p Fu(\))308 b(=)f Fq(x:)11893 b Fu(\(57\))6863 57517 y Fd(3.2)1495 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Fl(f)27162 66750 y Fu(=)-184 b Fo(\))307 b Fq(M)121 b Fu(.)25405 70071 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 6863 6974 a Fk(Theorem)426 b(13)554 b Fj(Assume)584 b Fq(N)59 b(;)184 b(M)706 b Fj(ar)-57 b(e)584 b(h-sets,)630 b(such)584 b(that)f Fq(u)p Fu(\()p Fq(N)121 b Fu(\))648 b(=)g Fq(u)p Fu(\()p Fq(M)121 b Fu(\))648 b(=)g Fq(u)583 b Fj(and)6863 8302 y Fq(s)p Fu(\()p Fq(N)121 b Fu(\))309 b(=)e Fq(s)p Fu(\()p Fq(M)121 b Fu(\))308 b(=)f Fq(s)p Fj(.)510 b(L)-57 b(et)397 b Fq(f)58 b(;)184 b(g)347 b Fu(:)308 b Fq(N)429 b Fo(!)307 b Fp(R)24171 7813 y Fl(n)25170 8302 y Fj(b)-57 b(e)396 b(c)-57 b(ontinuous.)511 b(Assume)396 b(that)23534 10691 y Fq(N)25045 10011 y Fl(f)39 b(;w)24851 10691 y Fu(=)-184 b Fo(\))307 b Fq(M)59 b(:)6863 12693 y Fj(Then)397 b(ther)-57 b(e)396 b(exists)f Fq(\262)308 b(>)f Fu(0)p Fj(,)397 b(such)f(that)f(if)i Fo(j)p Fq(f)25457 12859 y Fl(c)25908 12693 y Fu(\()p Fq(x)p Fu(\))247 b Fo(\241)f Fq(g)29283 12859 y Fl(c)29734 12693 y Fu(\()p Fq(x)p Fu(\))p 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b(basic)h(idea)g(is)e(to)i(construct)g(a)f(go)31 b(o)g(d)399 b(homotop)-31 b(y)400 b(connecting)g Fq(f)40737 21851 y Fl(c)41586 21685 y Fu(and)e Fq(g)44295 21851 y Fl(c)44746 21685 y Fu(.)6863 23014 y(Namely)-92 b(,)371 b(w)-31 b(e)370 b(construct)g(a)f(homotop)-31 b(y)23734 22722 y(~)23723 23014 y Fq(h)307 b Fu(:)h([0)p Fq(;)184 b Fu(1])249 b Fo(\243)d Fq(N)29739 23180 y Fl(c)30498 23014 y Fo(!)307 b Fp(R)32712 22525 y Fl(u)33536 23014 y Fo(\243)246 b Fp(R)35443 22525 y Fl(s)35916 23014 y Fu(,)369 b(suc)-31 b(h)369 b(that)20619 24723 y(~)20607 25015 y Fq(h)21245 25181 y Fv(0)22049 25015 y Fu(=)307 b Fq(g)23745 25181 y Fl(c)24196 25015 y Fq(;)26914 24723 y Fu(~)26902 25015 y Fq(h)27540 25181 y Fv(1)29144 25015 y Fu(=)1107 b Fq(f)31654 25181 y Fl(c)32104 25015 y Fq(;)10674 b Fu(\(58\))19331 26384 y(~)19320 26676 y Fq(h)p Fu(\([0)p Fq(;)184 b Fu(1])p Fq(;)g(N)24100 26219 y Fi(\241)23979 26949 y Fl(c)24851 26676 y Fu(\))247 b Fo(\\)f Fq(M)27586 26842 y Fl(c)29144 26676 y Fu(=)1107 b Fo(;)p Fq(;)11113 b Fu(\(59\))19344 28044 y(~)19332 28336 y Fq(h)p Fu(\([0)p Fq(;)184 b Fu(1])p Fq(;)g(N)23991 28502 y Fl(c)24447 28336 y Fu(\))246 b Fo(\\)g Fq(M)27302 27879 y Fv(+)27181 28610 y Fl(c)29144 28336 y Fu(=)1107 b Fo(;)p Fq(;)184 b(:)10622 b Fu(\(60\))6863 30337 y(W)-92 b(e)369 b(set)18627 31374 y(~)18615 31666 y Fq(h)p Fu(\()p Fq(t;)184 b(x)p Fu(\))309 b(=)f(\(1)247 b Fo(\241)e Fq(t)p Fu(\))p Fq(g)26809 31832 y Fl(c)27261 31666 y Fu(\()p Fq(x)p Fu(\))h(+)g Fq(tf)31049 31832 y Fl(c)31500 31666 y Fu(\()p Fq(x)p Fu(\))p Fq(:)9785 b Fu(\(61\))6863 33398 y(It)262 b(is)g(easy)f(to)i(see)e(that)i(for)f Fq(\262)g Fu(small)h(enough,)284 b(the)262 b(conditions)i(\(59\))f(and) f(\(60\))i(are)d(satis\257ed.)8524 35114 y(Let)500 b Fq(h)f Fu(b)31 b(e)500 b(a)g(homotop)-31 b(y)503 b(from)d(the)h (relation)g Fq(N)30470 34434 y Fl(f)39 b(;w)30276 35114 y Fu(=)-184 b Fo(\))525 b Fq(M)121 b Fu(.)885 b(It)500 b(is)f(easy)i(to)f(see)g(the)6863 36442 y(homotop)-31 b(y)372 b Fq(H)398 b Fu(:)308 b([0)p Fq(;)184 b Fu(1])248 b Fo(\243)e Fq(N)18417 36608 y Fl(c)19176 36442 y Fo(!)308 b Fp(R)21390 35953 y Fl(u)22215 36442 y Fo(\243)246 b Fp(R)24121 35953 y Fl(s)24963 36442 y Fu(giv)-31 b(en)371 b(b)-31 b(y)16654 39440 y Fq(H)90 b Fu(\()p Fq(t;)184 b(x)p Fu(\))309 b(=)21526 37547 y Fe(\()22429 38393 y Fu(~)22417 38685 y Fq(h)p Fu(\(2)p Fq(t;)184 b(x)p Fu(\))3015 b(for)370 b Fq(t)307 b Fo(2)g Fu([0)p Fq(;)33940 38249 y Fv(1)p 33940 38430 442 45 v 33940 39067 a(2)34514 38685 y Fu(])q Fq(;)22417 40279 y(h)p Fu(\(2)p Fq(t)247 b Fo(\241)f Fu(1)p Fq(;)184 b(x)p Fu(\))1108 b(for)370 b Fq(t)307 b Fo(2)g Fu([)32894 39843 y Fv(1)p 32895 40024 V 32895 40661 a(2)33469 40279 y Fq(;)184 b Fu(1])r Fq(:)43085 39440 y Fu(\(62\))6863 42732 y(satis\257es)369 b(all)i(the)e (conditions)i(from)f(Def.)492 b(6)370 b(for)f(a)h(co)-31 b(v)g(ering)370 b(relation)h Fq(N)38229 42052 y Fl(g)28 b(;w)38029 42732 y Fu(=)-184 b Fo(\))307 b Fq(M)p 44721 42732 333 554 v 8524 44550 a Fu(In)295 b(ab)31 b(o)-31 b(v)g(e)298 b(theorem)f(the)f(size)g(of)h(the)f(p)31 b(erturbation)297 b(i.e.)469 b(\()p Fo(j)p Fq(f)33593 44716 y Fl(c)34045 44550 y Fu(\()p Fq(x)p Fu(\))100 b Fo(\241)g Fq(g)37127 44716 y Fl(c)37579 44550 y Fu(\()p Fq(x)p Fu(\))p Fo(j)297 b Fu(for)f Fq(x)307 b Fo(2)g Fq(N)44171 44716 y Fl(c)44623 44550 y Fu(\))6863 45879 y(w)-31 b(as)488 b(giv)-31 b(en)488 b(in)g Fq(c)14006 46045 y Fl(N)15333 45879 y Fu(and)f Fq(c)18082 46045 y Fl(M)19552 45879 y Fu(co)31 b(ordinates.)848 b(No)-31 b(w)488 b(w)-31 b(e)488 b(state)g(the)f(result)g(in)-31 b(v)g(olving)491 b(the)6863 47207 y(di\256erence)369 b(b)31 b(et)-31 b(w)g(een)370 b Fq(f)488 b Fu(and)369 b Fq(g)409 b Fu(on)370 b Fq(N)121 b Fu(.)6863 49209 y Fk(Theorem)426 b(14)554 b Fj(Assume)584 b Fq(N)59 b(;)184 b(M)706 b Fj(ar)-57 b(e)584 b(h-sets,)630 b(such)584 b(that)f Fq(u)p Fu(\()p Fq(N)121 b Fu(\))648 b(=)g Fq(u)p Fu(\()p Fq(M)121 b Fu(\))648 b(=)g Fq(u)583 b Fj(and)6863 50537 y Fq(s)p Fu(\()p Fq(N)121 b Fu(\))309 b(=)e Fq(s)p Fu(\()p Fq(M)121 b Fu(\))308 b(=)f Fq(s)p Fj(.)510 b(L)-57 b(et)397 b Fq(f)58 b(;)184 b(g)347 b Fu(:)308 b Fq(N)429 b Fo(!)307 b Fp(R)24171 50048 y Fl(n)25170 50537 y Fj(b)-57 b(e)396 b(c)-57 b(ontinuous.)511 b(Assume)396 b(that)23534 52926 y Fq(N)25045 52246 y Fl(f)39 b(;w)24851 52926 y Fu(=)-184 b Fo(\))307 b Fq(M)59 b(;)6863 54927 y Fj(and)305 b(that)g(the)g(c)-57 b(o)g(or)g(dinate)306 b(map)f Fq(c)20877 55093 y Fl(M)22165 54927 y Fj(satis\257es)g(a)g(Lipschitz)g(c)-57 b(ondition.)481 b(Then)306 b(ther)-57 b(e)305 b(exists)6863 56256 y Fq(\262)j(>)f Fu(0)p Fj(,)397 b(such)f(that)f(if)i Fo(j)p Fq(f)119 b Fu(\()p Fq(x)p Fu(\))247 b Fo(\241)f Fq(g)40 b Fu(\()p Fq(x)p Fu(\))p Fo(j)309 b Fq(<)e(\262)396 b Fj(for)h(al)57 b(l)395 b Fq(x)307 b Fo(2)g Fq(N)121 b Fj(,)397 b(then)23534 58440 y Fq(N)25051 57760 y Fl(g)28 b(;w)24851 58440 y Fu(=)-184 b Fo(\))307 b Fq(M)59 b(:)6863 60442 y Fk(Pro)35 b(of:)445 b Fu(Let)274 b Fq(L)f Fu(b)31 b(e)272 b(a)i(Lipsc)-31 b(hitz)274 b(constan)-31 b(t)274 b(for)f 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a(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 6863 6974 a Fr(4)1793 b(Ho)-50 b(w)677 b(to)f(\257nd)h(a)f(homotop)-50 b(y)678 b(for)e(the)h(co)-50 b(v)g(ering)678 b(re-)9553 8966 y(lations)6863 11391 y Fu(The)499 b(goal)h(of)f(this)f(section)i(is)e(to)h(presen)-31 b(t)498 b(su\261cien)-31 b(t)499 b(conditions)h(whic)-31 b(h)500 b(ensure)d(that)6863 13045 y Fq(N)8813 12366 y Fl(f)8181 13045 y Fu(=)-184 b Fo(\))307 b Fq(M)121 b Fu(,)369 b(solely)h(based)f(on)h(the)f(kno)-31 b(wledge)371 b(of)f Fq(f)119 b Fu(\()p Fq(N)i Fu(\))370 b(and)g Fq(M)121 b Fu(.)6863 15481 y Fk(De\257nition)426 b(8)554 b Fj(L)-57 b(et)396 b Fq(N)517 b Fj(b)-57 b(e)397 b(a)f(h-set.)509 b(We)396 b(set)16686 17916 y Fq(S)64 b Fu(\()p Fq(N)121 b Fu(\))19299 17459 y Fi(\241)19299 18190 y Fl(c)20354 17916 y Fu(=)308 b Fo(f)p Fu(\()p Fq(p;)184 b(q)40 b Fu(\))309 b Fo(2)f Fp(R)26672 17427 y Fl(u)27497 17916 y Fo(\243)246 b Fp(R)29404 17427 y Fl(s)30123 17916 y Fo(j)g(k)p Fq(p)p Fo(k)309 b Fq(>)e Fu(1)p Fo(g)p 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Fo(\261)175 b Fq(c)40188 23642 y Fi(\241)p Fv(1)40188 24440 y Fl(N)41684 24115 y Fu(:)307 b Fq(N)43187 24281 y Fl(c)43946 24115 y Fo(!)6863 25443 y Fp(R)7663 24954 y Fl(u)8487 25443 y Fo(\243)246 b Fp(R)10394 24954 y Fl(s)10867 25443 y Fj(.)8524 26772 y(Assume)487 b(that)g(ther)-57 b(e)489 b(exists)e Fq(q)21571 26938 y Fv(0)22542 26772 y Fo(2)p 23754 25882 896 45 v 474 w Fq(B)24649 26938 y Fl(s)25123 26772 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fj(,)514 b(such)488 b(that)f(fol)57 b(lowing)488 b(c)-57 b(onditions)489 b(ar)-57 b(e)6863 28100 y(satis\257e)g(d)6863 30314 y Fk(1.)16605 32749 y Fq(f)17147 32915 y Fl(c)17598 32749 y Fu(\()p 18028 31860 V Fq(B)18924 32915 y Fl(u)19503 32749 y Fu(\(0)p Fq(;)184 b Fu(1\))249 b Fo(\243)d(f)p Fq(q)24363 32915 y Fv(0)24860 32749 y Fo(g)p Fu(\))1108 b Fo(\275)f Fu(in)-31 b(t)185 b(\()p Fq(S)64 b Fu(\()p Fq(M)121 b Fu(\))33653 32292 y Fi(\241)33653 33023 y Fl(c)34647 32749 y Fo([)246 b Fq(M)36705 32915 y Fl(c)37156 32749 y Fu(\))p Fq(;)5192 b Fu(\(64\))19478 34410 y Fq(f)20020 34576 y Fl(c)20471 34410 y Fu(\()p Fq(N)21911 33953 y Fi(\241)21790 34683 y Fl(c)22658 34410 y Fu(\))247 b Fo(\\)f Fq(M)25393 34576 y Fl(c)26951 34410 y Fu(=)1107 b Fo(;)p Fq(;)13306 b Fu(\(65\))19490 36070 y Fq(f)20032 36236 y Fl(c)20483 36070 y Fu(\()p Fq(N)21802 36236 y Fl(c)22254 36070 y Fu(\))246 b Fo(\\)g Fq(M)25109 35613 y Fv(+)24988 36344 y Fl(c)26951 36070 y Fu(=)1107 b Fo(;)p Fq(:)13306 b Fu(\(66\))6863 38948 y Fk(2.1)554 b Fj(Case)396 b Fq(u)307 b(>)h Fu(0)p Fj(.)510 b(We)396 b(de\257ne)h(a)e(map)h Fq(A)24731 39114 y Fl(q)25137 39225 y Ff(0)25932 38948 y Fu(:)308 b Fp(R)27347 38459 y Fl(u)28233 38948 y Fo(!)f Fp(R)30447 38459 y Fl(u)31421 38948 y Fj(by)21897 41384 y Fq(A)22727 41550 y Fl(q)23133 41661 y Ff(0)23621 41384 y Fu(\()p Fq(p)p Fu(\))h(=)g Fq(\274)27146 41550 y Fl(u)27724 41384 y Fu(\()p Fq(f)28696 41550 y Fl(c)29148 41384 y Fu(\()p Fq(p;)184 b(q)31120 41550 y Fv(0)31618 41384 y Fu(\)\))p Fq(;)10300 b Fu(\(67\))9631 43819 y Fj(wher)-57 b(e)472 b Fq(\274)13464 43985 y Fl(u)14488 43819 y Fu(:)446 b Fp(R)16041 43330 y Fl(u)16921 43819 y Fo(\243)301 b Fp(R)18883 43330 y Fl(s)19801 43819 y Fo(!)446 b Fp(R)22154 43330 y Fl(u)23205 43819 y Fj(is)472 b(the)g(pr)-57 b(oje)g(ction)474 b(onto)e Fp(R)34988 43330 y Fl(u)35567 43819 y Fj(,)492 b Fq(\274)37029 43985 y Fl(u)37608 43819 y Fu(\()p Fq(p;)184 b(q)40 b Fu(\))447 b(=)f Fq(p)p Fj(.)738 b(We)9631 45147 y(assume)395 b(that)21283 47583 y Fq(A)22113 47749 y Fl(q)22519 47860 y Ff(0)23006 47583 y Fu(\()p Fq(@)61 b(B)24925 47749 y Fl(u)25504 47583 y Fu(\(0)p Fq(;)184 b Fu(1\)\))1110 b Fo(\275)d Fp(R)32269 47094 y Fl(u)33094 47583 y Fo(n)p 33893 46620 3879 45 v 246 w Fq(B)34733 47749 y Fl(u)35311 47583 y Fu(\(0)p Fq(;)184 b Fu(1\))s Fq(;)5007 b Fu(\(68\))16420 49243 y(deg)16 b(\()p Fq(A)19356 49409 y Fl(q)19762 49520 y Ff(0)20250 49243 y Fq(;)p 20741 48354 1419 45 v 184 w(B)21581 49409 y Fl(u)22160 49243 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fq(;)g Fu(0\))313 b(=)307 b Fq(w)1137 b Fo(6)p Fu(=)1107 b(0)p Fq(:)10756 b Fu(\(69\))6863 52121 y Fk(2.2)554 b Fj(Case)396 b Fq(u)307 b Fu(=)h(0)p Fj(.)510 b(We)396 b(assume)f(that)23337 54556 y Fq(f)23879 54722 y Fl(c)24330 54556 y Fu(\()p Fq(N)25649 54722 y Fl(c)26101 54556 y Fu(\))308 b Fo(\275)f Fu(in)-31 b(t)186 b Fq(M)30588 54722 y Fl(c)31039 54556 y Fq(;)9631 56992 y Fj(and)396 b(set)f Fq(w)338 b Fu(=)307 b(1)p Fj(.)6863 59206 y(Then)23534 60645 y Fq(N)25045 59965 y Fl(f)39 b(;w)24851 60645 y Fu(=)-184 b Fo(\))307 b Fq(M)59 b(:)6863 63080 y Fk(Pro)35 b(of:)650 b Fu(W)-92 b(e)447 b(ha)-31 b(v)g(e)449 b(to)f(pro)-31 b(v)g(e)448 b(that)h(there)e(exists)h(a)g(homotop)-31 b(y)450 b Fq(h)438 b Fu(:)g([0)p Fq(;)184 b Fu(1])302 b Fo(\243)c Fq(N)41217 63246 y Fl(c)42106 63080 y Fo(!)438 b Fp(R)44451 62591 y Fl(n)6863 64409 y Fu(satisfying)371 b(conditions)g(from)f (De\257nition)g(6.)8524 65737 y(The)f(case)g Fq(u)307 b Fu(=)h(0)369 b(is)g(trivial)i(\()p Fq(w)339 b Fu(=)307 b(1\).)494 b(Th)-31 b(us)369 b(w)-31 b(e)370 b(fo)31 b(cus)369 b(on)g(the)h(case)f Fq(u)307 b(>)g Fu(0.)25405 70071 y(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 8524 6974 a Fu(F)-92 b(or)433 b(an)-31 b(y)435 b Fq(q)13225 7140 y Fv(1)14138 6974 y Fo(2)415 b Fp(R)16091 6485 y Fl(s)16998 6974 y Fu(w)-31 b(e)435 b(de\257ne)f(a)g(deformation)j(retraction)e(on)-31 b(to)436 b Fp(R)37722 6485 y Fl(u)38591 6974 y Fo(\243)289 b(f)p Fq(q)40788 7140 y Fv(1)41285 6974 y Fo(g)p Fu(,)451 b Fq(R)43437 7140 y Fl(q)43843 7251 y Ff(1)44746 6974 y Fu(:)6863 8302 y([0)p Fq(;)184 b Fu(1])249 b Fo(\243)d Fp(R)11229 7813 y Fl(u)12054 8302 y Fo(\243)g Fp(R)13961 7813 y Fl(s)14741 8302 y Fo(!)308 b Fp(R)16955 7813 y Fl(u)17780 8302 y Fo(\243)246 b Fp(R)19686 7813 y Fl(s)20159 8302 y Fu(,)370 b(b)-31 b(y)17986 10738 y Fq(R)18827 10904 y Fl(q)19233 11015 y Ff(1)19719 10738 y Fu(\()p Fq(\270;)184 b(p;)g(q)40 b Fu(\))310 b(=)e(\()p Fq(p;)184 b Fu(\(1)248 b Fo(\241)e Fq(\270)p Fu(\))p Fq(q)286 b Fu(+)246 b Fq(\270q)32696 10904 y Fv(1)33193 10738 y Fu(\))p Fq(:)9155 b 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y Fi(\241)22799 41502 y Fl(c)23884 41228 y Fu(and)339 b Fq(S)64 b Fu(\()p Fq(M)121 b Fu(\))28804 40826 y Fi(\241)28804 41502 y Fl(c)29734 41228 y Fo(\\)184 b Fq(M)31730 41394 y Fl(c)32488 41228 y Fu(=)308 b Fo(;)p Fu(.)482 b(Hence)338 b(condition)i(\(31\))6863 42556 y(is)369 b(satis\257ed.)8524 43885 y(It)354 b(remains)i(to)f(sho)-31 b(w)356 b(that)g(condition)h (\(32\))f(is)f(true)f(for)h Fq(h)p Fu(.)488 b(As)354 b(ab)31 b(o)-31 b(v)g(e,)360 b(consider)354 b(t)-31 b(w)g(o)6863 45213 y(cases)369 b Fq(\270)307 b Fo(\267)h Fu(1)p Fq(=)p Fu(2)370 b(and)f Fq(\270)308 b Fo(\270)f Fu(1)p Fq(=)p Fu(2.)494 b(F)-92 b(or)368 b Fq(\270)308 b Fo(\267)f Fu(1)p Fq(=)p Fu(2)370 b(from)g(\(66\))h(it)f(follo)-31 b(ws)372 b(that)17585 47649 y Fq(h)18223 47815 y Fl(\270)18804 47649 y Fu(\()p Fq(N)20123 47815 y Fl(c)20575 47649 y Fu(\))246 b Fo(\\)g Fq(M)23430 47191 y Fv(+)23309 47922 y Fl(c)24472 47649 y Fo(\275)307 b Fq(f)26182 47815 y Fl(c)26633 47649 y Fu(\()p Fq(N)27952 47815 y Fl(c)28404 47649 y 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b(are)e(close)h(to)6863 62592 y(pro)31 b(ducts)369 b(of)h(one)f(dimensional)j(maps.)493 b(Belo)-31 b(w)370 b(w)-31 b(e)370 b(will)h(discuss)d(the)i(case)f(of)g Fq(u)308 b Fu(=)f(1.)25405 70071 y(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 6863 6974 a Fd(4.1)1495 b(Case)499 b(of)f(one)h(nominally)g(expanding)g(direction,)h Fc(u)369 b Fw(=)g(1)6863 9017 y Fu(In)385 b(this)h(section)g(w)-31 b(e)385 b(discuss)g(the)g(case)g(of)h Fq(u)334 b Fu(=)g(1,)390 b(hence)385 b(w)-31 b(e)386 b(ha)-31 b(v)g(e)386 b(only)h(one)e (nominally)6863 10345 y(expanding)430 b(direction.)673 b(The)429 b(basic)g(idea)h(here)e(is)g(that)i(eac)-31 b(h)430 b(of)f(the)g(sets)f Fq(N)40194 9944 y Fi(\241)40941 10345 y Fu(,)444 b Fq(S)64 b Fu(\()p Fq(N)121 b Fu(\))44305 9944 y Fi(\241)6863 11674 y Fu(consists)372 b(of)h(t)-31 b(w)g(o)373 b(disjoin)-31 b(t)374 b(comp)31 b(onen)-31 b(ts,)374 b(allo)-31 b(wing)376 b(us)371 b(to)i(simplify)g(the)g (assumptions)g(of)6863 13002 y(Theorem)d(15.)6863 15138 y Fk(De\257nition)426 b(9)554 b Fj(L)-57 b(et)396 b Fq(N)517 b Fj(b)-57 b(e)397 b(an)g(h-set,)e(such)h(that)f Fq(u)p Fu(\()p Fq(N)121 b Fu(\))308 b(=)g(1)p Fj(.)510 b(We)396 b(set)20106 17273 y Fq(N)21116 16816 y Fl(l)t(e)20995 17547 y(c)22982 17273 y Fu(=)1107 b Fo(f\241)p Fu(1)p Fo(g)248 b(\243)p 28825 16384 896 45 v 246 w Fq(B)29720 17439 y Fl(s)30193 17273 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fq(;)19955 18934 y(N)20965 18477 y Fl(r)24 b(e)20844 19207 y(c)22982 18934 y Fu(=)1107 b Fo(f)p Fu(1)p Fo(g)248 b(\243)p 27964 18045 V 246 w Fq(B)28859 19100 y Fl(s)29332 18934 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fq(;)18811 20594 y(S)64 b Fu(\()p Fq(N)121 b Fu(\))21424 20137 y Fl(l)21424 20868 y(c)22982 20594 y Fu(=)1107 b(\()p Fo(\2411)p Fq(;)184 b Fo(\241)p Fu(1\))249 b Fo(\243)d Fp(R)31838 20105 y Fl(s)32311 20594 y Fq(;)18772 22255 y(S)64 b Fu(\()p Fq(N)121 b Fu(\))21385 21798 y Fl(r)21385 22528 y(c)22982 22255 y Fu(=)1107 b(\(1)p Fq(;)184 b Fo(1)p Fu(\))249 b Fo(\243)d Fp(R)30116 21766 y Fl(s)30589 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184 w(S)64 b Fu(\()p Fq(N)121 b Fu(\))18236 34933 y Fl(l)18578 35253 y Fq(;)p 19069 34290 3104 45 v 184 w(S)64 b Fu(\()p Fq(N)121 b Fu(\))21682 34933 y Fl(r)22174 35253 y Fu(\))370 b(represen)-31 b(ts)368 b(a)h(t-set,)h(as)f(it)h(w)-31 b(as)370 b(de\257ned)f(in)h([2].)6863 37227 y Fk(Theorem)426 b(16)554 b Fj(L)-57 b(et)398 b Fq(N)121 b Fj(,)398 b Fq(M)518 b Fj(b)-57 b(e)398 b(two)e(h-sets)h(in)h Fp(R)28071 36738 y Fl(n)28674 37227 y Fj(,)g(such)f(that)f Fq(u)p Fu(\()p Fq(N)121 b Fu(\))310 b(=)g Fq(u)p Fu(\()p Fq(M)121 b Fu(\))310 b(=)g(1)397 b Fj(and)6863 38555 y Fq(s)p Fu(\()p Fq(N)121 b Fu(\))309 b(=)e Fq(s)p Fu(\()p Fq(M)121 b Fu(\))308 b(=)f Fq(s)h Fu(=)f Fq(n)247 b Fo(\241)f Fu(1)p Fj(.)510 b(L)-57 b(et)396 b Fq(f)427 b Fu(:)308 b Fq(N)429 b Fo(!)307 b Fp(R)27220 38066 y Fl(n)28218 38555 y Fj(b)-57 b(e)397 b(c)-57 b(ontinuous.)8524 39883 y(Assume)491 b(that)f(ther)-57 b(e)492 b(exists)e Fq(q)21584 40049 y Fv(0)22561 39883 y Fo(2)p 23779 38994 896 45 v 480 w Fq(B)24675 40049 y 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Ff(0)33844 64600 y Fq(;)p 34335 63711 V 184 w(B)35175 64766 y Fl(u)35754 64600 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fq(;)g Fu(0\))364 b(=)358 b Fo(\241)p Fu(1)400 b(pro-)6863 65929 y(vided)370 b(that)g(\(79\))h(holds.)p 44721 65929 333 554 v 25405 70071 a(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 6863 6974 a Fr(5)1793 b(Some)598 b(examples,)i(mo)50 b(del)598 b(maps)6863 9398 y Fu(The)268 b(goal)h(of)f(this)g(section)g(is)g(to)g(discuss)f(what)i(are)e(all)i (p)31 b(ossible)267 b(mo)31 b(del)269 b(maps)f(and)g(to)g(giv)-31 b(e)6863 10727 y(non)g(trivial)372 b(examples)e(of)g(c)-31 b(haotic)371 b(b)31 b(eha)-31 b(vior)370 b(di\256eren)-31 b(t)370 b(from)f(top)31 b(ological)374 b(horsesho)31 b(es.)8524 12055 y(Let)444 b(us)h(remind)f(the)h(reader)f(that,)466 b(in)445 b(the)g(con)-31 b(text)446 b(of)f(De\257nition)h(6,)465 b(the)444 b(map)i Fq(h)44557 12221 y Fv(1)6863 13710 y Fu(is)454 b(called)g Fj(a)474 b(mo)-57 b(del)473 b(map)454 b Fu(for)f(the)h(relation)h Fq(N)28063 13030 y Fl(f)27430 13710 y Fu(=)-184 b Fo(\))447 b Fq(M)121 b Fu(,)475 b(and)454 b(that)g(the)g(degree)f Fq(w)484 b Fu(of)454 b(a)6863 15038 y(co)-31 b(v)g(ering)371 b(can)e(b)31 b(e)369 b(computed)h(as)f Fq(w)338 b Fu(=)308 b Fq(d)p Fu(\()p Fq(s)25025 15204 y Fl(A)25746 15038 y Fu(\),)370 b(where)f Fq(A)g Fu(is)g(the)h Fq(u)p Fu(-comp)31 b(onen)-31 b(t)370 b(of)g Fq(h)42832 15204 y Fv(1)43328 15038 y Fu(.)8524 16366 y(W)-92 b(e)382 b(ha)-31 b(v)g(e)385 b(to)f(consider)g(three)f(cases)g Fq(u)331 b Fu(=)h(0,)388 b Fq(u)331 b Fu(=)g(1)384 b(and)g Fq(u)331 b Fo(\270)g Fu(2.)537 b(The)383 b(case)h Fq(u)331 b Fu(=)g(0)384 b(is)6863 17695 y(trivial,)372 b(b)31 b(ecause)369 b Fq(h)15198 17861 y Fv(1)15694 17695 y Fu(\()p Fq(x)p Fu(\))308 b(=)g(0)369 b(b)-31 b(y)369 b(the)h(de\257nition.)8524 19023 y(F)-92 b(or)377 b Fq(u)321 b Fu(=)h(1,)381 b(from)d(Theorem)h(16)f(it)h(follo)-31 b(ws)380 b(that)f(w)-31 b(e)379 b(can)f(ha)-31 b(v)g(e)379 b(only)f Fq(w)353 b Fu(=)321 b Fo(\247)p Fu(1.)519 b(It)378 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Fu(has)e(sym)-31 b(b)31 b(olic)371 b(dynamics)f(on)g(t)-31 b(w)g(o)371 b(sym)-31 b(b)31 b(ols)369 b(\(see)h(also)g([22)q(])g(and)f(Section)h(5.2\).)6863 38681 y Fd(5.1)1495 b(Case)499 b Fc(u)369 b(>)g Fw(1)p Fd(.)6863 40724 y Fu(In)419 b(order)g(to)h(giv)-31 b(e)421 b(a)f(description)g(of)g(all)h(p)31 b(ossible)419 b(mo)31 b(del)421 b(maps)f(\(up)f(to)i(a)e(homotop)-31 b(y\),)6863 42052 y(w)g(e)370 b(\257rst)e(consider)i(maps)f(of)h(the)f(sphere)f Fq(S)25351 41651 y Fl(u)p Fi(\241)p Fv(1)27063 42052 y Fu(.)8524 43381 y(F)-92 b(or)310 b(simplicit)-31 b(y)-92 b(,)325 b(w)-31 b(e)312 b(will)g(represen)-31 b(t)310 b Fp(R)24793 42892 y Fl(u)25682 43381 y Fu(as)h Fp(C)130 b Fo(\243)f Fp(R)29701 42892 y Fl(u)p Fi(\241)p Fv(2)31413 43381 y Fu(,)323 b(where)311 b Fp(C)g Fu(is)f(a)i(set)e(of)i(complex) 6863 44709 y(n)-31 b(um)g(b)31 b(ers.)492 b(F)-92 b(or)369 b Fq(z)356 b Fo(2)307 b Fp(C)q Fu(,)369 b(b)-31 b(y)436 b(\271)-619 b Fq(z)417 b Fu(w)-31 b(e)370 b(denote)g(the)f(complex)i 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b(example)g(of)g(c)-42 b(haotic)777 b(b)42 b(eha)-42 b(vior)777 b(based)g(on)f(m)-42 b(ultiple)10267 8524 y(wrapp)42 b(ed)499 b(co)-42 b(v)g(erings)6863 10567 y Fu(In)323 b(this)h(section)h(w)-31 b(e)324 b(describ)31 b(e)323 b(a)h(class)g(of)g(c)-31 b(haotic)326 b(examples)f(based)e(on)h (the)g(mo)31 b(del)325 b(maps)6863 11895 y(of)370 b(higher)f(degree)g (in)-31 b(tro)31 b(duced)370 b(in)g(previous)f(subsection.)8524 13224 y(Let)358 b Fq(u)307 b Fu(=)g(2)359 b(and)f Fq(s)307 b Fo(\270)h Fu(0.)489 b(W)-92 b(e)357 b(will)j(represen)-31 b(t)357 b Fp(R)28544 12735 y Fl(u)p Fv(+)p Fl(s)30578 13224 y Fu(as)h Fp(C)224 b Fo(\243)g Fp(R)34833 12735 y Fl(s)35306 13224 y Fu(.)489 b(W)-92 b(e)357 b(de\257ne)h(an)g(h-set) 6863 14552 y Fq(N)7752 14718 y Fv(0)8618 14552 y Fu(b)-31 b(y)22038 16987 y Fq(N)22927 17153 y Fv(0)23731 16987 y Fu(=)p 24900 16098 896 45 v 308 w Fq(B)25795 17153 y Fv(2)26292 16987 y Fu(\(0)p Fq(;)184 b Fu(1\))248 b Fo(\243)p 30104 16098 V 246 w Fq(B)31000 17153 y Fl(s)31473 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Ff(1)15976 28500 y Fu(\()p Fq(z)49 b(;)184 b(q)40 b Fu(\))309 b(=)f(\()p Fq(z)295 b Fo(\241)246 b Fq(a;)184 b(q)40 b Fu(\))p Fq(;)2400 b(z)356 b Fo(2)307 b Fp(C)370 b Fu(and)f Fq(q)348 b Fo(2)307 b Fp(R)34921 28011 y Fl(s)6863 30935 y Fu(where)369 b Fq(a)308 b Fo(2)f Fp(C)q Fu(,)369 b Fo(j)p Fq(a)p Fo(j)309 b Fq(>)e Fu(2.)8524 32263 y(Notice)420 b(that)g Fq(N)15385 32429 y Fv(0)16160 32263 y Fo(\\)279 b Fq(N)18066 32429 y Fv(1)18953 32263 y Fu(=)390 b Fo(;)p Fu(.)641 b(F)-92 b(or)418 b(an)-31 b(y)420 b Fq(w)26674 32429 y Fv(1)27171 32263 y Fq(;)184 b(w)28454 32429 y Fv(2)29342 32263 y Fo(2)389 b Fp(Z)279 b Fo(n)h(f)p Fu(0)p Fo(g)420 b Fu(w)-31 b(e)419 b(de\257ne)f(a)h(map)g Fq(f)538 b Fu(on)6863 33592 y Fq(N)7752 33758 y Fv(0)8495 33592 y Fo([)246 b Fq(N)10368 33758 y Fv(1)11234 33592 y Fu(b)-31 b(y)15926 35855 y Fq(f)119 b Fu(\()p Fq(z)49 b(;)184 b(x)p Fu(\))308 b(=)20612 33962 y Fe(\()21503 35100 y Fq(\270)p Fu(\()p Fq(m)23551 35266 y Fl(w)24191 35377 y Ff(1)24679 35100 y Fu(\()p 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Fo(\))318 b Fq(N)22940 43819 y Fv(1)23437 43653 y Fq(;)2399 b(N)27032 43819 y Fv(1)27836 42973 y Fl(f)39 b(;w)29210 43084 y Ff(2)27848 43653 y Fu(=)-184 b Fo(\))318 b Fq(N)30839 43819 y Fv(0)31336 43653 y Fq(:)8524 46088 y Fu(Let)369 b Fq(g)347 b Fu(:)308 b Fp(R)12797 45600 y Fl(u)p Fv(+)p Fl(s)14780 46088 y Fo(!)g Fp(R)16994 45600 y Fl(u)p Fv(+)p Fl(s)19039 46088 y Fu(b)31 b(e)369 b(a)g(p)31 b(erturbation)370 b(of)g Fq(f)488 b Fu(giv)-31 b(en)371 b(b)-31 b(y)17238 48524 y Fq(g)40 b Fu(\()p Fq(z)49 b(;)184 b(p)p Fu(\))309 b(=)e Fq(f)119 b Fu(\()p Fq(z)49 b(;)184 b(q)40 b Fu(\))248 b(+)e(\()p Fq(\262)27099 48690 y Fl(z)27609 48524 y Fu(\()p Fq(z)49 b(;)184 b(q)40 b Fu(\))p Fq(;)184 b(\262)30998 48690 y Fl(q)31490 48524 y Fu(\()p Fq(z)49 b(;)184 b(q)40 b Fu(\)\))p Fq(;)8409 b Fu(\(87\))6863 50959 y(where)369 b Fq(\262)10513 51125 y Fl(z)11331 50959 y Fu(:)308 b Fp(C)246 b Fo(\243)g Fp(R)14898 50470 y Fl(s)15678 50959 y Fo(!)308 b Fp(C)369 b Fu(and)h Fq(\262)20863 51125 y Fl(q)21659 50959 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Ff(1)19745 63689 y Fu(=)-184 b Fo(\))313 b Fq(N)22731 63855 y Fv(0)23228 63689 y Fq(;)2448 b(N)26872 63855 y Fv(0)27676 63009 y Fl(g)28 b(;w)29039 63120 y Ff(1)27682 63689 y Fu(=)-184 b Fo(\))313 b Fq(N)30668 63855 y Fv(1)31165 63689 y Fq(;)18230 65471 y(N)19119 65637 y Fv(1)19924 64791 y Fl(g)28 b(;w)21287 64902 y Ff(2)19930 65471 y Fu(=)-184 b Fo(\))312 b Fq(N)22915 65637 y Fv(1)23412 65471 y Fq(;)2448 b(N)27056 65637 y Fv(1)27861 64791 y Fl(g)28 b(;w)29224 64902 y Ff(2)27867 65471 y Fu(=)-184 b Fo(\))312 b Fq(N)30852 65637 y Fv(0)31349 65471 y Fq(:)25405 70071 y Fu(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 6863 6974 a Fk(Pro)35 b(of:)558 b Fu(It)401 b(is)g(easy)g(to)h(see)f(that)h(for)g(all)g(co)-31 b(v)g(ering)403 b(relations,)411 b(the)402 b(common)g(homotop)-31 b(y)6863 8302 y Fq(h)p Fu(\()p Fq(t;)184 b Fu(\()p Fq(z)49 b(;)184 b(q)40 b Fu(\)\))310 b(=)e Fq(tf)119 b Fu(\()p Fq(z)49 b(;)184 b(q)40 b Fu(\))247 b(+)f(\(1)h Fo(\241)f Fq(t)p Fu(\))p Fq(g)40 b Fu(\()p Fq(z)49 b(;)184 b(q)40 b Fu(\))370 b(satis\257es)f(all)i(assumptions)f(of)g(Theorem)g(15.)p 44721 8302 333 554 v 8524 10223 a(Let)f Fq(N)428 b Fu(=)308 b Fq(N)13882 10389 y Fv(0)14625 10223 y Fo([)246 b Fq(N)16498 10389 y Fv(1)16995 10223 y Fu(.)492 b(W)-92 b(e)369 b(de\257ne)g(a)g (forw)-31 b(ard)370 b(in)-31 b(v)-61 b(arian)-31 b(t)371 b(set)e(of)h Fq(g)409 b Fu(b)-31 b(y)14363 12478 y(In)g(v)15931 11996 y Fv(+)16666 12478 y Fu(\()p Fq(N)59 b(;)184 b(g)40 b Fu(\))309 b(=)f Fo(f)p Fq(x)g Fo(2)f Fq(N)367 b Fo(j)247 b Fq(g)25928 12021 y Fl(k)26472 12478 y Fu(\()p Fq(x)p Fu(\))308 b Fo(2)f Fq(N)59 b(;)1293 b Fu(for)369 b Fq(k)343 b Fo(2)307 b Fp(N)370 b Fo(g)p Fq(:)8524 14734 y Fu(T)-92 b(o)331 b(pro)31 b(ceed)330 b(further)h(w)-31 b(e)331 b(will)h(recall)g(some)f(notions)h(from)g(sym)-31 b(b)31 b(olic)332 b(dynamics.)481 b(F)-92 b(or)6863 16224 y(an)-31 b(y)436 b Fq(k)454 b Fo(\270)417 b Fu(2,)453 b(w)-31 b(e)436 b(de\257ne)g(\247)18401 15751 y Fv(+)18401 16558 y Fl(k)19554 16224 y Fu(=)418 b 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Fq(\274)40 b Fu(\()p Fq(x)p Fu(\))385 b(=)g Fq(\256)442 b Fj(and)d Fq(x)f Fj(is)h(p)-57 b(erio)g(dic)440 b(p)-57 b(oint)439 b(of)g Fq(g)479 b Fj(with)438 b(the)g(same)6863 32630 y(princip)-57 b(al)397 b(p)-57 b(erio)g(d)397 b(as)e Fq(\256)t Fj(.)6863 34700 y Fk(Pro)35 b(of:)729 b Fu(F)-92 b(or)487 b(an)-31 b(y)488 b(\257nite)g(sequence)e(\()p Fq(\256)24140 34866 y Fv(0)24637 34700 y Fq(;)184 b(\256)25836 34866 y Fv(1)26334 34700 y Fq(;)g(:)g(:)g(:)k(;)c(\256)29501 34866 y Fl(l)t Fi(\241)p Fv(1)30975 34700 y Fu(\))505 b Fo(2)f(f)p Fu(0)p Fq(;)184 b Fu(1)p Fo(g)35855 34298 y Fl(l)36684 34700 y Fu(consider)487 b(a)h(closed)6863 36029 y(lo)31 b(op)339 b(of)g(co)-31 b(v)g(ering)339 b(relation)h(for)e Fq(g)378 b Fu(\(w)-31 b(e)339 b(drop)f(here)f(the)i (sym)-31 b(b)31 b(ol)339 b(of)g(the)f(function)h(and)g(the)6863 37357 y(degree\))16536 38685 y Fq(N)17425 38851 y Fl(\256)18000 38962 y Ff(0)18795 38685 y Fu(=)-184 b Fo(\))307 b Fq(N)21775 38851 y Fl(\256)22350 38962 y Ff(1)23146 38685 y Fu(=)-184 b Fo(\))307 b Fq(:)184 b(:)g(:)310 b Fu(=)-184 b Fo(\))307 b Fq(N)29816 38851 y Fl(l)t Fi(\241)p Fv(1)31596 38685 y Fu(=)-184 b Fo(\))307 b Fq(N)34576 38851 y Fv(0)35073 38685 y Fq(:)7705 b Fu(\(89\))6863 40570 y(F)-92 b(rom)433 b(Theorem)h(9)f(w)-31 b(e)434 b(obtain)g(the)g(existence)f(of)h(a)f(p) 31 b(oin)-31 b(t)435 b Fq(x)p Fu(,)449 b(suc)-31 b(h)432 b(that)j Fq(g)39755 40168 y Fl(i)40123 40570 y Fu(\()p Fq(x)p Fu(\))414 b Fo(2)g Fq(N)44071 40736 y Fl(\256)44646 40847 y Fh(i)6863 41898 y Fu(for)428 b Fq(i)407 b Fu(=)e(0)p Fq(;)184 b(:)g(:)g(:)189 b(;)184 b(l)308 b Fo(\241)286 b Fu(1)428 b(and)g Fq(g)19228 41497 y Fl(l)19568 41898 y Fu(\()p Fq(x)p Fu(\))406 b(=)g Fq(x)p Fu(.)669 b(This)429 b(sho)-31 b(ws)428 b(that)i(for)e(an)-31 b(y)429 b(p)31 b(erio)g(dic)429 b(sequence)6863 43227 y Fq(\256)345 b Fo(2)340 b Fu(\247)9793 43393 y Fv(2)10290 43227 y Fu(,)394 b Fq(\274)11662 42825 y Fi(\241)p Fv(1)12850 43227 y Fu(\()p Fq(\256)t Fu(\))c(con)-31 b(tains)391 b(a)e(p)31 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b(es,)369 b(but)g(w)-31 b(e)370 b(will)h(not)f(pursue)e (this)h(issue)g(here.)6863 55419 y Fr(6)1793 b(App)50 b(endix)6863 58065 y Fd(6.1)1495 b(Lo)42 b(cal)498 b(Brou)-42 b(w)g(er)501 b(degree)6863 60108 y Fu(In)341 b(this)h(section)h(w)-31 b(e)342 b(list)h(the)f(basic)g(prop)31 b(erties)341 b(of)h Fj(the)371 b(lo)-57 b(c)g(al)371 b(Br)-57 b(ouwer)370 b(de)-57 b(gr)g(e)g(e)344 b Fu(whic)-31 b(h)342 b(are)6863 61436 y(relev)-61 b(an)-31 b(t)370 b(for)f(us)g(in)g(this)h(pap)31 b(er.)492 b(The)370 b(pro)31 b(ofs)369 b(can)g(b)31 b(e)369 b(found)g(in)h([15)q(,)g(Ch.)493 b(I)31 b(I)g(I].)8524 62765 y(F)-92 b(or)455 b Fq(n)f Fu(=)f(0)j(w)-31 b(e)457 b(ha)-31 b(v)g(e)458 b Fp(R)19194 62276 y Fl(n)20250 62765 y Fu(=)453 b Fo(f)p Fu(0)p Fo(g)p Fu(.)755 b(W)-92 b(e)456 b(ha)-31 b(v)g(e)457 b(only)h(one)f(self)f(map)h(for)g(this)g (space,)6863 64093 y(namely)395 b Fq(f)119 b Fu(\(0\))349 b(=)f(0.)565 b(W)-92 b(e)393 b(formally)j(de\257ne)d Fj(the)418 b(lo)-57 b(c)g(al)418 b(Br)-57 b(ouwer)418 b(de)-57 b(gr)g(e)g(e)420 b(of)f Fq(f)537 b Fj(at)418 b Fu(0)g Fj(in)i(the)6863 65421 y(set)396 b Fo(f)p Fu(0)p Fo(g)370 b Fu(b)-31 b(y)21776 66750 y(deg)16 b(\()p Fq(f)58 b(;)184 b Fo(f)p Fu(0)p Fo(g)p Fq(;)g Fu(0\))312 b(=)c(1)25405 70071 y(20)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 8524 6974 a Fu(Assume)382 b Fq(n)330 b(>)g Fu(0.)533 b(Let)382 b Fq(D)361 b Fo(\275)330 b Fp(R)21481 6485 y Fl(n)22466 6974 y Fu(b)31 b(e)382 b(an)g(op)31 b(en)383 b(set)f(and)h Fq(f)449 b Fu(:)330 b Fq(S)394 b Fo(!)329 b Fp(R)37069 6485 y Fl(n)38054 6974 y Fu(b)31 b(e)382 b(con)-31 b(tin)g(uous,)6863 8302 y Fq(D)339 b Fo(\275)307 b Fq(S)433 b Fu(and)369 b Fq(c)308 b Fo(2)f Fp(R)15182 7813 y Fl(n)15784 8302 y Fu(.)493 b(Supp)31 b(ose)369 b(that)17795 10676 y(the)h(set)1107 b Fq(f)22829 10219 y Fi(\241)p Fv(1)24017 10676 y Fu(\()p Fq(c)p Fu(\))247 b Fo(\\)f Fq(D)1139 b Fu(is)369 b(compact.)8966 b(\(90\))6863 13051 y(Then)416 b Fj(the)439 b(lo)-57 b(c)g(al)440 b(Br)-57 b(ouwer)439 b(de)-57 b(gr)g(e)g(e)441 b(of)e Fq(f)558 b Fj(at)439 b Fq(c)g Fj(in)h(the)f(set)g Fq(D)447 b Fu(is)416 b(de\257ned.)633 b(W)-92 b(e)416 b(denote)g(it)6863 14379 y(b)-31 b(y)370 b(deg)15 b(\()p Fq(f)58 b(;)184 b(D)31 b(;)184 b(c)p Fu(\).)8524 15707 y(If)p 9631 14818 948 45 v 369 w Fq(D)338 b Fo(\275)308 b Fu(dom)q(\()p Fq(f)119 b Fu(\))370 b(and)p 18188 14818 V 369 w Fq(D)400 b Fu(is)369 b(compact,)j(then)d(\(90\))i(follo)-31 b(ws)372 b(from)d(the)h (condition)23329 18081 y Fq(c)431 b(=)-676 b Fo(2)307 b Fq(f)119 b Fu(\()p Fq(@)61 b(D)31 b Fu(\))p Fq(:)14500 b Fu(\(91\))8524 20456 y(Let)369 b(us)f(summarize)j(the)e(prop)31 b(erties)369 b(of)h(the)f(lo)31 b(cal)371 b(Brou)-31 b(w)g(er)369 b(degree)6863 22621 y Fk(Degree)427 b(is)d(an)g(in)-35 b(teger.)23369 23949 y Fu(deg)16 b(\()p Fq(f)58 b(;)184 b(D)31 b(;)184 b(c)p Fu(\))311 b Fo(2)c Fp(Z)p Fq(:)11771 b Fu(\(92\))6863 26335 y Fk(Solution)425 b(prop)35 b(ert)-35 b(y)-106 b(.)11431 28709 y Fu(If)1291 b(deg)16 b(\()p Fq(f)58 b(;)184 b(D)31 b(;)184 b(c)p Fu(\))310 b Fo(6)p Fu(=)e(0)p Fq(;)1292 b Fu(then)369 b(there)g(exists)h Fq(x)307 b Fo(2)g Fq(D)401 b Fu(with)370 b Fq(f)119 b Fu(\()p Fq(x)p Fu(\))308 b(=)g Fq(c:)1800 b Fu(\(93\))6863 31514 y Fk(Homotop)-35 b(y)426 b(prop)35 b(ert)-35 b(y)-106 b(.)555 b Fu(Let)369 b Fq(H)398 b Fu(:)308 b([0)p Fq(;)184 b Fu(1])248 b Fo(\243)e Fq(D)339 b Fo(!)308 b Fp(R)29695 31025 y Fl(n)30667 31514 y Fu(b)31 b(e)368 b(con)-31 b(tin)g(uous.)495 b(Supp)31 b(ose)368 b(that)20484 33058 y Fe([)19715 35484 y Fl(\270)p Fi(2)p Fv([0)p Fl(;)p Fv(1])22667 34110 y Fq(H)23677 33636 y Fi(\241)p Fv(1)23587 34444 y Fl(\270)24866 34110 y Fu(\()p Fq(c)p Fu(\))247 b Fo(\\)f Fq(D)1138 b Fu(is)369 b(compact)r Fq(:)8116 b Fu(\(94\))9631 37714 y(Then)16985 39042 y Fo(8)p Fq(\270)307 b Fo(2)h Fu([0)p Fq(;)184 b Fu(1])1294 b(deg)16 b(\()p Fq(H)26130 39208 y Fl(\270)26711 39042 y Fq(;)184 b(D)31 b(;)184 b(c)p Fu(\))310 b(=)e(deg)15 b(\()p Fq(H)34053 39208 y Fv(0)34551 39042 y Fq(;)184 b(D)31 b(;)184 b(c)p Fu(\))p Fq(:)5389 b Fu(\(95\))9631 41429 y(If)505 b([0)p Fq(;)184 b Fu(1])339 b Fo(\243)p 14622 40539 V 337 w Fq(D)566 b Fo(\275)534 b Fu(dom)q(\()p Fq(H)90 b Fu(\))506 b(and)p 24255 40539 V 506 w Fq(D)536 b Fu(is)505 b(compact,)542 b(then)505 b(\(94\))i(follo)-31 b(ws)508 b(from)e(the)9631 42757 y(follo)-31 b(wing)372 b(condition)23185 44085 y Fq(c)431 b(=)-676 b Fo(2)307 b Fq(H)90 b Fu(\([0)p Fq(;)184 b Fu(1])p Fq(;)g(@)61 b(D)31 b Fu(\))p Fq(:)11593 b Fu(\(96\))6863 46472 y Fk(Lo)35 b(cal)424 b(degree)j(is)d(a)g(lo)35 b(cally)424 b(constan)-35 b(t)424 b(function.)554 b Fu(Assume)246 b Fq(D)277 b Fu(is)246 b(b)31 b(ounded)246 b(and)h(op)31 b(en.)9631 47800 y(If)369 b Fq(p)f Fu(and)i Fq(q)409 b Fu(b)31 b(elong)370 b(to)g(the)f(same)h(comp)31 b(onen)-31 b(t)371 b(of)e Fp(R)32006 47311 y Fl(n)32854 47800 y Fo(n)247 b Fq(f)119 b Fu(\()p Fq(@)61 b(D)31 b Fu(\),)371 b(then)20837 50174 y(deg)16 b(\()p Fq(f)58 b(;)184 b(D)31 b(;)184 b(p)p Fu(\))310 b(=)e(deg)15 b(\()p Fq(f)58 b(;)184 b(D)31 b(;)184 b(q)40 b Fu(\))p Fq(:)9242 b Fu(\(97\))6863 52979 y Fk(Excision)425 b(prop)35 b(ert)-35 b(y)-106 b(.)554 b Fu(Supp)31 b(ose)369 b(that)i Fq(E)g Fo(\275)307 b Fq(D)31 b Fu(,)370 b Fq(E)433 b Fu(is)369 b(op)31 b(en)369 b(and)23326 55353 y Fq(f)23987 54896 y Fi(\241)p Fv(1)25176 55353 y Fu(\()p Fq(c)p Fu(\))247 b Fo(\\)f Fq(D)339 b Fo(\275)307 b Fq(E)64 b(:)11728 b Fu(\(98\))9631 57727 y(Then)20937 59056 y(deg)16 b(\()p Fq(f)58 b(;)184 b(E)64 b(;)184 b(c)p Fu(\))310 b(=)d(deg)16 b(\()p Fq(f)58 b(;)184 b(D)31 b(;)184 b(c)p Fu(\))p Fq(:)9341 b Fu(\(99\))6863 61442 y Fk(Lo)35 b(cal)424 b(degree)j(for)e(a\261ne)g(maps.)553 b Fu(Supp)31 b(ose)357 b(that)i Fq(f)119 b Fu(\()p Fq(x)p Fu(\))308 b(=)g Fq(A)p Fu(\()p Fq(x)223 b Fo(\241)h Fq(x)37653 61608 y Fv(0)38149 61442 y Fu(\))g(+)f Fq(c)p Fu(,)360 b(where)e Fq(A)9631 62770 y Fu(is)396 b(a)h(linear)h(map)f(and)g Fq(x)20127 62936 y Fv(0)20977 62770 y Fo(2)353 b Fp(R)22868 62282 y Fl(n)23470 62770 y Fu(.)575 b(If)397 b(the)g(equation)h 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b(map)g(\()p Fq(f)15391 8468 y Fv(1)15888 8302 y Fq(;)184 b(f)16921 8468 y Fv(2)17419 8302 y Fu(\))352 b(:)g Fp(R)19659 7813 y Fl(n)20206 7924 y Ff(1)20958 8302 y Fo(\243)263 b Fp(R)22882 7813 y Fl(n)23429 7924 y Ff(2)24268 8302 y Fo(!)352 b Fp(R)26526 7813 y Fl(n)27073 7924 y Ff(1)27824 8302 y Fo(\243)264 b Fp(R)29748 7813 y Fl(n)30295 7924 y Ff(2)31179 8302 y Fu(is)395 b(giv)-31 b(en)397 b(b)-31 b(y)396 b(\()p Fq(f)37771 8468 y Fv(1)38268 8302 y Fq(;)184 b(f)39301 8468 y Fv(2)39799 8302 y Fu(\)\()p Fq(x)41292 8468 y Fv(1)41789 8302 y Fq(;)g(x)42913 8468 y Fv(2)43410 8302 y Fu(\))352 b(=)9631 9631 y(\()p Fq(f)10603 9797 y Fv(1)11100 9631 y Fu(\()p Fq(x)12163 9797 y Fv(1)12659 9631 y Fu(\))p Fq(;)184 b(f)14122 9797 y Fv(2)14620 9631 y Fu(\()p Fq(x)15683 9797 y Fv(2)16180 9631 y Fu(\).)493 b(W)-92 b(e)368 b(ha)-31 b(v)g(e)10946 11981 y(deg)16 b(\(\()p Fq(f)14024 12147 y Fv(1)14521 11981 y Fq(;)184 b(f)15554 12147 y Fv(2)16052 11981 y Fu(\))p Fq(;)g(U)17729 12147 y Fv(1)18473 11981 y Fo(\243)246 b Fq(U)20336 12147 y Fv(2)20832 11981 y Fq(;)184 b Fu(\()p Fq(c)22232 12147 y Fv(1)22730 11981 y Fq(;)g(c)23700 12147 y Fv(2)24198 11981 y Fu(\)\))308 b(=)g(deg)16 b(\()p Fq(f)29183 12147 y Fv(1)29680 11981 y Fq(;)184 b(U)30927 12147 y Fv(1)31424 11981 y Fq(;)g(c)32394 12147 y Fv(1)32892 11981 y Fu(\))246 b Fo(\242)h Fu(deg)15 b(\()p Fq(f)36769 12147 y Fv(2)37266 11981 y Fq(;)184 b(U)38513 12147 y Fv(2)39011 11981 y Fq(;)g(c)39981 12147 y Fv(2)40478 11981 y Fu(\))p Fq(;)1317 b Fu(\(101\))9631 14331 y(whenev)-31 b(er)369 b(the)g(righ)-31 b(t)371 b(hand)e(side)g(is)g(de\257ned.)6863 16510 y Fk(Multiplication)424 b(prop)35 b(ert)-35 b(y)555 b Fu(Let)445 b Fq(D)464 b Fo(\275)434 b Fp(R)26133 16022 y Fl(n)27180 16510 y Fu(b)31 b(e)444 b(b)31 b(ounded)445 b(and)g(op)31 b(en.)719 b(Let)445 b Fq(f)552 b Fu(:)43070 16231 y(\271)42812 16510 y Fq(D)464 b Fo(\275)9631 17839 y Fp(R)10430 17350 y Fl(n)11340 17839 y Fo(!)308 b Fp(R)13554 17350 y Fl(n)14157 17839 y Fu(,)323 b Fq(g)348 b 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b(for)g(some)g Fq(q)33054 25411 y Fl(i)33986 25245 y Fo(2)564 b Fu(\242)36210 25411 y Fl(i)36579 25245 y Fu(.)954 b(F)-92 b(rom)522 b(equation)9631 26574 y(\(97\))328 b(it)g(follo)-31 b(ws)330 b(that)e(this)g (de\257nition)h(of)e(deg)16 b(\()p Fq(f)58 b(;)184 b(D)31 b(;)184 b Fu(\242)32709 26740 y Fl(i)33081 26574 y Fu(\))327 b(do)31 b(es)327 b(not)h(dep)31 b(end)326 b(on)i(the)9631 27902 y(c)-31 b(hoice)370 b(of)f Fq(q)14675 28068 y Fl(i)15044 27902 y Fu(.)6863 30082 y Fk(Addition)425 b(prop)35 b(ert)-35 b(y)-106 b(.)555 b Fu(If)373 b Fq(D)346 b Fu(=)21409 29252 y Fe(S)22332 30414 y Fl(i)p Fi(2)p Fl(I)23934 30082 y Fq(D)24850 30248 y Fl(i)25219 30082 y Fu(,)375 b(where)e(eac)-31 b(h)374 b Fq(D)32517 30248 y Fl(i)33259 30082 y Fu(is)f(op)31 b(en,)375 b(the)f(family)h Fo(f)p Fq(D)44130 30248 y Fl(i)44500 30082 y Fo(g)9631 31410 y Fu(is)369 b(disjoin)-31 b(t)371 b(and)e Fq(@)61 b(D)18403 31576 y Fl(i)19080 31410 y Fo(\275)307 b Fq(@)61 b(D)31 b Fu(,)370 b(then)g(for)f(ev)-31 b(ery)369 b Fq(c)431 b(=)-676 b Fo(2)307 b Fq(f)119 b Fu(\()p Fq(@)61 b(D)31 b Fu(\):)19843 33982 y(deg)16 b(\()p Fq(f)58 b(;)184 b(D)31 b(;)184 b(c)p Fu(\))310 b(=)26866 32930 y Fe(X)26984 35305 y Fl(i)p Fi(2)p Fl(I)28649 33982 y Fu(deg)16 b(\()p Fq(f)58 b(;)184 b(D)32762 34148 y Fl(i)33132 33982 y Fq(;)g(c)p Fu(\))p Fq(:)7693 b Fu(\(103\))8524 37862 y(F)-92 b(rom)368 b(Multiplication)373 b(prop)31 b(ert)-31 b(y)370 b(and)f(form)-31 b(ula)371 b(\(100\))g(w)-31 b(e)370 b(obtain)h(immediately)6863 40008 y Fk(Collorary)425 b(19)555 b Fj(L)-57 b(et)397 b Fq(D)338 b Fo(\275)308 b Fp(R)19487 39519 y Fl(n)20485 40008 y Fj(b)-57 b(e)397 b(op)-57 b(en)397 b(and)f(b)-57 b(ounde)g(d.)511 b(L)-57 b(et)397 b Fq(A)308 b Fu(:)g Fq(D)339 b Fo(!)307 b Fp(R)38397 39519 y Fl(n)38999 40008 y Fj(,)397 b(b)-57 b(e)397 b(c)-57 b(ontinu-)6863 41336 y(ous)396 b(and)g Fu(0)431 b Fq(=)-676 b Fo(2)307 b Fq(A)p Fu(\()p Fq(@)61 b(D)31 b Fu(\))p Fj(,)17447 43686 y Fu(deg)16 b(\()p Fo(\241)p Fq(A;)184 b(U)n(;)g Fu(0\))310 b(=)e(\()p Fo(\241)p Fu(1\))27716 43229 y Fl(n)28504 43686 y Fu(deg)16 b(\()p Fq(A;)184 b(U)n(;)g Fu(0\))p Fq(:)8066 b Fu(\(104\))8524 46036 y(As)328 b(the)i(consequence)f(of)h(Addition)h(and)e(Excision)i(prop)31 b(ert)-31 b(y)329 b(w)-31 b(e)330 b(obtain)h(the)e(follo)-31 b(w-)6863 47365 y(ing)6863 49510 y Fk(Collorary)425 b(20)555 b Fj(Supp)-57 b(ose)380 b(that)g Fq(D)411 b Fj(is)381 b(a)f(\257nite)h(union)h(of)e(op)-57 b(en)381 b(sets)f Fq(D)338 b Fu(=)38670 48680 y Fe(S)39592 48954 y Fl(n)39592 49842 y(i)p Fv(=1)41266 49510 y Fq(D)42182 49676 y Fl(i)42932 49510 y Fj(such)6863 50839 y(that)395 b(the)h(sets)f Fq(f)13804 50365 y Fi(\241)p Fv(1)13685 51216 y Fi(j)p Fl(D)14672 51327 y Fh(i)15078 50839 y Fu(\()p Fq(c)p Fu(\))i Fj(ar)-57 b(e)396 b(mutual)57 b(ly)396 b(disjoint)h(and)f Fq(c)430 b(=)-676 b Fo(2)308 b Fq(f)119 b Fu(\()p Fq(@)61 b(D)33905 51005 y Fl(i)34274 50839 y Fu(\))p Fj(.)511 b(Then)17792 54370 y Fu(deg)16 b(\()p Fq(f)58 b(;)184 b(D)31 b(;)184 b(c)p Fu(\))310 b(=)25340 52986 y Fl(n)24814 53318 y Fe(X)24897 55676 y Fl(i)p Fv(=1)26598 54370 y Fu(deg)16 b(\()p Fq(f)29246 54570 y Fi(j)p Fl(D)30233 54681 y Fh(i)30639 54370 y Fq(;)184 b(D)32046 54536 y Fl(i)32416 54370 y Fq(;)g(c)p Fu(\))p Fq(:)8409 b Fu(\(105\))8524 57688 y(Here)368 b(is)h(another)h(imp)31 b(ortan)-31 b(t)372 b(consequence)d(of)h(ab)31 b(o)-31 b(v)g(e)370 b(prop)31 b(erties)6863 59833 y Fk(Collorary)425 b(21)555 b Fj(Assume)485 b Fq(V)716 b Fo(\275)470 b Fp(R)22068 59345 y Fl(n)23156 59833 y Fj(is)486 b(b)-57 b(ounde)g(d)487 b(and)f(op)-57 b(en.)779 b(L)-57 b(et)487 b Fq(f)589 b Fu(:)p 38213 58944 892 45 v 471 w Fq(V)716 b Fo(!)470 b Fp(R)41951 59345 y Fl(n)43040 59833 y Fj(b)-57 b(e)486 b(a)6863 61162 y Fq(C)7733 60760 y Fv(1)8230 61162 y Fj(-map.)743 b(Assume)474 b(that)f Fq(c)448 b Fo(2)h Fp(R)21210 60673 y Fl(n)22115 61162 y Fo(n)303 b Fq(f)119 b Fu(\()p Fq(@)61 b(V)246 b Fu(\))475 b Fj(is)f(a)f(r)-57 b(e)g(gular)475 b(value)f(for)g Fq(f)119 b Fj(,)494 b(i.e.)744 b(for)474 b(e)-57 b(ach)6863 62490 y Fq(x)307 b Fo(2)h Fq(f)9510 62088 y Fi(\241)p Fv(1)10698 62490 y Fu(\()p Fq(c)p Fu(\))342 b Fj(the)f(Jac)-57 b(obian)341 b(matrix)g(of)g Fq(f)460 b Fj(at)340 b Fq(x)g Fj(denote)-57 b(d)341 b(by)h Fq(D)31 b(f)119 b Fu(\()p Fq(x)p Fu(\))341 b Fj(is)g(nonsingular,)353 b(then)16716 65062 y Fu(deg)15 b(\()p Fq(f)58 b(;)184 b(V)61 b(;)184 b(c)p Fu(\))311 b(=)24567 64010 y Fe(X)23498 66445 y Fl(x)p Fi(2)p Fl(f)25116 66224 y Fg(\241)p Ff(1)26147 66445 y Fv(\()p Fl(c)p Fv(\))27419 65062 y Fu(sgn)184 b(\(det)i Fq(D)31 b(f)119 b Fu(\()p Fq(x)p Fu(\)\))p Fq(:)25405 70071 y Fu(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 6863 6974 a Fd(6.2)1495 b(The)499 b(degree)f(of)g(maps)i Fc(S)24517 6492 y Fb(n)25512 6974 y Fa(!)369 b Fc(S)28086 6492 y Fb(n)6863 9017 y Fu(In)336 b(this)h(section)g(w)-31 b(e)336 b(recall)h(some)g(relev)-61 b(an)-31 b(t)337 b(facts)g(on)f(the)h(degree)f(of)h(maps)f Fq(S)39627 8615 y Fl(n)40537 9017 y Fo(!)307 b Fq(S)42694 8615 y Fl(n)43633 9017 y Fu(see)6863 10345 y(for)369 b(example)i([4)q(,)e(Ch.)493 b(7.5].)6863 12781 y Fk(De\257nition)426 b(10)554 b Fj(L)-57 b(et)440 b Fq(n)387 b Fo(\270)f Fu(1)p Fj(.)640 b(The)440 b(de)-57 b(gr)g(e)g(e)441 b(of)e(a)h(c)-57 b(ontinuous)440 b(map)f Fq(f)505 b Fu(:)387 b Fq(S)38990 12379 y Fl(n)39978 12781 y Fo(!)f Fq(S)42214 12379 y Fl(n)43256 12781 y Fj(is)439 b(a)6863 14109 y(unique)427 b(inte)-57 b(ger)427 b Fq(d)p Fu(\()p Fq(f)119 b Fu(\))427 b Fj(such)f(that)e Fq(f)22056 14275 y Fi(\244)22565 14109 y Fu(\()p Fq(u)p Fu(\))362 b(=)f Fq(d)p Fu(\()p Fq(f)119 b Fu(\))269 b Fo(\242)f Fq(u)p Fj(,)433 b(for)426 b(any)g(gener)-57 b(ator)426 b Fq(u)361 b Fo(2)g Fq(H)41905 14275 y Fl(n)42508 14109 y Fu(\()p Fq(S)43681 13707 y Fl(n)44283 14109 y Fu(\))p Fj(,)6863 15437 y(wher)-57 b(e)402 b Fq(H)10915 15603 y Fl(n)11518 15437 y Fu(\()p Fq(S)12691 15036 y Fl(n)13293 15437 y Fu(\))h Fj(is)f Fq(n)p Fj(-th)g(homolo)-57 b(gy)403 b(gr)-57 b(oup)402 b(of)g Fq(S)27657 15036 y Fl(n)28661 15437 y Fj(and)g Fq(f)31359 15603 y Fi(\244)32187 15437 y Fu(:)319 b Fq(H)33733 15603 y Fl(n)34335 15437 y Fu(\()p Fq(S)35508 15036 y Fl(n)36111 15437 y Fu(\))g Fo(!)f Fq(H)39205 15603 y Fl(n)39808 15437 y Fu(\()p Fq(S)40981 15036 y Fl(n)41584 15437 y Fu(\))402 b Fj(is)h(the)6863 16766 y(induc)-57 b(e)g(d)398 b(homomorphism.)8524 19201 y Fu(F)-92 b(or)368 b Fq(n)308 b Fu(=)f(0)370 b(w)-31 b(e)370 b(de\257ne)e(the)i(degree,)f Fq(d)p Fu(\()p Fq(f)119 b Fu(\),)371 b(as)e(follo)-31 b(ws,)372 b Fq(S)32880 18799 y Fv(0)33684 19201 y Fu(=)307 b Fo(f\241)p Fu(1)p Fq(;)184 b Fu(1)p Fo(g)p Fu(.)496 b(W)-92 b(e)368 b(set)15311 23363 y Fq(d)p Fu(\()p Fq(f)119 b Fu(\))308 b(=)18885 20762 y Fe(8)18885 21758 y(>)18885 22090 y(<)18885 24083 y(>)18885 24415 y(:)19869 21811 y Fu(1)p Fq(;)1969 b Fu(if)369 b Fq(f)119 b Fu(\(1\))309 b(=)f(1)369 b(and)h Fq(f)119 b Fu(\()p Fo(\241)p Fu(1\))309 b(=)e Fo(\241)p Fu(1)q Fq(;)19869 23405 y Fo(\241)p Fu(1)p Fq(;)1108 b Fu(if)369 b Fq(f)119 b Fu(\(1\))309 b(=)f Fo(\241)p Fu(1)369 b(and)h Fq(f)119 b Fu(\()p Fo(\241)p Fu(1\))309 b(=)e(1)q Fq(;)19869 24999 y Fu(0)p Fq(;)1969 b Fu(otherwise)q Fq(:)42532 23363 y Fu(\(106\))6863 27525 y Fk(Theorem)426 b(22)f(\(H.)f(Hopf)121 b(\))555 b Fj(L)-57 b(et)464 b Fq(n)432 b Fo(\270)e Fu(1)p Fj(.)715 b(Then)464 b Fq(f)58 b(;)184 b(g)471 b Fu(:)432 b Fq(S)32631 27124 y Fl(n)33663 27525 y Fo(!)f Fq(S)35944 27124 y Fl(n)37010 27525 y Fj(ar)-57 b(e)465 b(homotopic)e(if)6863 28854 y(and)396 b(only)h(if)g Fq(d)p Fu(\()p Fq(f)119 b Fu(\))308 b(=)g Fq(d)p Fu(\()p Fq(g)40 b Fu(\))p Fj(.)6863 31289 y Fk(Lemma)424 b(23)555 b Fj(L)-57 b(et)444 b Fq(u)394 b(>)h Fu(0)p Fj(,)457 b(Assume)444 b(that)f Fq(A)395 b Fu(:)p 27214 30400 1419 45 v 395 w Fq(B)28054 31455 y Fl(u)28632 31289 y Fu(\(0)p Fq(;)184 b Fu(1\))397 b Fo(!)e Fp(R)33787 30800 y Fl(u)34810 31289 y Fj(is)444 b(a)g(c)-57 b(ontinuous)445 b(map,)6863 32617 y(such)396 b(that)22004 33946 y Fu(0)430 b Fq(=)-676 b Fo(2)308 b Fq(A)p Fu(\()p Fq(@)61 b(B)56 b Fu(\(0)p Fq(;)184 b Fu(1\)\))p Fq(:)6863 35938 y Fj(L)-57 b(et)397 b(the)f(map)f Fq(s)13564 36104 y Fl(A)14593 35938 y Fu(:)308 b Fq(S)15951 35537 y Fl(u)p Fi(\241)p Fv(1)17970 35938 y Fo(!)f Fq(S)20127 35537 y Fl(u)p Fi(\241)p Fv(1)22235 35938 y Fj(b)-57 b(e)397 b(given)g(by)21851 38950 y Fq(s)22370 39116 y Fl(A)23091 38950 y Fu(\()p Fq(x)p Fu(\))309 b(=)26747 38201 y Fq(A)p Fu(\()p Fq(x)p Fu(\))p 26194 38696 3431 45 v 26194 39710 a Fo(k)p Fq(A)p Fu(\()p Fq(x)p Fu(\))p Fo(k)29757 38950 y Fq(:)6863 41973 y Fj(Then)19337 43301 y Fu(deg)15 b(\()p Fq(A;)p 22763 42412 1419 45 v 184 w(B)23603 43467 y Fl(u)24183 43301 y Fu(\(0)p Fq(;)184 b Fu(1\))p Fq(;)g Fu(0\))313 b(=)307 b Fq(d)p Fu(\()p Fq(s)31120 43467 y Fl(A)31842 43301 y Fu(\))p Fq(:)9953 b Fu(\(107\))6863 46962 y Fr(References)8862 49386 y Fu([1])555 b(G.)308 b(Arioli,)322 b Fj(Perio)-57 b(dic)341 b(orbits,)351 b(symb)-57 b(olic)341 b(dynamics)f(and)g(top) -57 b(olo)g(gic)g(al)340 b(entr)-57 b(opy)340 b(for)10584 50715 y(the)374 b(r)-57 b(estricte)g(d)375 b(3-b)-57 b(o)g(dy)376 b(pr)-57 b(oblem,)351 b Fu(Comm)-31 b(un.)348 b(Math.)f(Ph)-31 b(ys.)347 b(231)g(\(2002\))h(1,)k(1-24)8862 52929 y([2])555 b(G.)431 b(Arioli)i(and)f(P)-92 b(.)432 b(Zgliczy)31 b(\266)-584 b(nski,)450 b Fj(Symb)-57 b(olic)455 b(dynamics)f(for)g(the)f(H)-28 b(\266)-537 b(enon{Heiles)10584 54257 y(hamiltonian)299 b(on)g(the)g(critic)-57 b(al)299 b(ener)-57 b(gy)301 b(level)p Fu(,)569 b(J.)264 b(Di\256.)f(Eq,)286 b(171)265 b(\(2001\),)287 b(173{202.)8862 56471 y([3])555 b(L.)431 b(Blo)31 b(c)-31 b(k,)448 b(J.)431 b(Guc)-31 b(k)g(enheimer,)447 b(M.)431 b(Misiurewicz)h(and)f(L.-S.)h(Y)-92 b(oung,)447 b Fj(Perio)-57 b(dic)10584 57799 y(p)g(oints)524 b(of)f(one-dimensional)i(maps,)542 b Fu(Lecture)508 b(Notes)g(in)g (Mathematics,)545 b(819,)10584 59128 y(1980,)371 b(18{34.)8862 61342 y([4])555 b(J.)417 b(Dugundji)g(and)h(A.)f(Granas,)429 b Fj(Fixe)-57 b(d)440 b(Point)g(The)-57 b(ory,)881 b Fu(Monogra\257e)418 b(Matem-)10584 62670 y(at)-31 b(yczne)370 b(61,)h(PWN,)e(W)-92 b(arsza)-31 b(w)g(a,)371 b(1982.)8862 64884 y([5])555 b(R.)526 b(Easton,)567 b Fj(Isolating)540 b(Blo)-57 b(cks)541 b(and)g(Symb)-57 b(olic)542 b(Dynamics)p Fu(,)567 b(J.)526 b(Di\256.)g(Eq.,)567 b(17)10584 66212 y(\(1975\),)372 b(96{118.)25405 70071 y(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 8862 6974 a Fu([6])555 b(R.)278 b(Easton,)298 b Fj(Homo)-57 b(clinic)314 b(Phenomena)f(in)h (Hamiltonian)f(Systems)g(with)e(Sever)-57 b(al)10584 8302 y(De)g(gr)g(e)g(es)397 b(of)f(F)-85 b(r)-57 b(e)g(e)g(dom)p Fu(,)371 b(J.)e(Di\256.)g(Eq.,)i(29)f(\(1978\),)i(241{252.)8862 10516 y([7])555 b(Z.)531 b(Galias,)572 b Fj(Positive)544 b(top)-57 b(olo)g(gic)g(al)545 b(entr)-57 b(opy)545 b(of)f(Chua's)g (cir)-57 b(cuit:)807 b(a)544 b(c)-57 b(omputer)10584 11844 y(assiste)g(d)420 b(pr)-57 b(o)g(of.)398 b Fu(In)-31 b(t.)397 b(J.)g(Bifurcation)i(and)e(Chaos,)405 b(v)-31 b(ol.)399 b(7,)404 b(no.)397 b(2)h(\(1997\),)406 b(331{)10584 13173 y(349.)8862 15387 y([8])555 b(Z.)467 b(Galias)g(and)g(P)-92 b(.)466 b(Zgliczy)31 b(\266)-584 b(nski,)494 b Fj(Computer)485 b(assiste)-57 b(d)484 b(pr)-57 b(o)g(of)486 b(of)g(chaos)e(in)j(the) 10584 16715 y(L)-57 b(or)g(enz)397 b(system,)738 b Fu(Ph)-31 b(ysica)370 b(D,)f(115)i(\(1998\),)h(165{188.)8862 18929 y([9])555 b(J.)491 b(Kennedy)-92 b(,)522 b(S.)492 b(Ko)31 b(cak,)523 b(and)491 b(J.)h(Y)-92 b(ork)-31 b(e,)522 b Fj(A)509 b(Chaos)e(L)-57 b(emma)p Fu(,)523 b(Amer.)492 b(Math.)10584 20257 y(Mon)-31 b(thly)-92 b(,)371 b(V)-92 b(ol.)370 b(108,)h(No)f(5)f(\(2001\),)j(411-423.)8308 22471 y([10])556 b(J.)319 b(Kennedy)g(and)g(J.)g(Y)-92 b(ork)-31 b(e,)330 b Fj(T)-85 b(op)-57 b(olo)g(gic)g(al)351 b(horsesho)-57 b(es)p Fu(,)328 b(T)-92 b(rans.)319 b(AMS,)g(V)-92 b(ol.)320 b(353,)10584 23800 y(No)369 b(6)h(\(2001\),)i(2513{2530.)8308 26014 y([11])556 b(B.)483 b(Lani-W)-92 b(a)-31 b(yda)483 b(and)g(R.)f(Srzednic)-31 b(ki,)512 b Fj(A)500 b(gener)-57 b(alize)g(d)502 b(L)-57 b(efschetz)500 b(\257xe)-57 b(d)501 b(p)-57 b(oint)10584 27342 y(the)g(or)g(em)464 b(and)f(symb)-57 b(olic)465 b(dynamics)f(in)g(delay)f(e)-57 b(quations)p Fu(,)462 b(Ergo)31 b(d.)443 b(Th.)g(&)f(Dy-)10584 28670 y(nam.)370 b(Sys.,)g(22)g(\(2002\),)i(no.)e(4,)g(1215{1232)8308 30884 y([12])556 b(K.)583 b(Misc)-31 b(haik)g(o)g(w)585 b(and)e(M.)g(Mrozek,)636 b Fj(Isolating)593 b(Neighb)-57 b(orho)g(o)g(ds)592 b(and)h(Chaos,)10584 32213 y Fu(Japan)421 b(Journal)g(of)g(Industrial)g(and)f(Applied)i(Mathematics)g(12)f (\(1995\),)436 b(205{)10584 33541 y(236.)8308 35755 y([13])556 b(K.)322 b(Misc)-31 b(haik)g(o)g(w,)334 b(M.)321 b(Mrozek)h(and)f(P)-92 b(.)322 b(Zgliczy)31 b(\266)-584 b(nski)324 b(\(editors\),)333 b Fk(Conley)371 b(Index)10584 37083 y(Theory)p Fu(,)f(W)-92 b(arsa)-31 b(w)369 b(1999,)j(Banac)-31 b(h)370 b(Cen)-31 b(ter)370 b(Publications,)i(P)-31 b(oland.)8308 39297 y([14])556 b(M.)360 b(Misiurewicz)h(and)f(P)-92 b(.)361 b(Zgliczy)31 b(\266)-584 b(nski,)365 b Fj(T)-85 b(op)-57 b(olo)g(gic)g(al)389 b(entr)-57 b(opy)388 b(for)g(multidimen-)10584 40625 y(sional)332 b(p)-57 b(erturb)g(ations)332 b(of)h(one)f (dimensional)h(maps)p Fu(,)627 b(In)-31 b(t.)301 b(J.)f(of)g (Bifurcation)i(and)10584 41954 y(Chaos,)370 b(v)-31 b(ol.)371 b(11,)g(No.5)f(\(2001\),)j(1443{1446.)8308 44168 y([15])556 b(J.T.)280 b(Sc)-31 b(h)g(w)g(artz,)299 b Fj(Nonline)-57 b(ar)314 b(F)-85 b(unctional)313 b(A)-28 b(nalysis)p Fu(,)297 b(1969,)j(Gordon)279 b(and)g(Breac)-31 b(h)10584 45496 y(Science)369 b(Publishers)g(Inc.,)h(New)g(Y)-92 b(ork.)8308 47710 y([16])556 b(R.)363 b(Srzednic)-31 b(ki,)366 b Fj(A)390 b(gener)-57 b(alization)391 b(of)g(the)f(L)-57 b(efschetz)392 b(\257xe)-57 b(d)390 b(p)-57 b(oint)391 b(the)-57 b(or)g(em)391 b(and)10584 49038 y(dete)-57 b(ction)475 b(of)f(chaos,)h Fu(Pro)31 b(c.)455 b(Amer.)g(Math.)g(So)31 b(c.,)476 b(128,)i(No.)455 b(4)f(\(2000\),)479 b(1231{)10584 50367 y(1239.)8308 52581 y([17])556 b(K.)373 b(W\266)-553 b(ojcik)374 b(and)g(P)-92 b(.)373 b(Zgliczy)31 b(\266)-584 b(nski,)377 b Fj(Isolating)399 b(se)-57 b(gments,)402 b(\257xe)-57 b(d)400 b(p)-57 b(oint)400 b(index)g(and)10584 53909 y(symb)-57 b(olic)397 b(dynamics)p Fu(,)371 b(J.)e(Di\256.)g(Eq,) h(161)h(\(2000\),)h(245{288.)8308 56123 y([18])556 b(D.)499 b(Wilczak)i(and)f(P)-92 b(.)500 b(Zgliczy)31 b(\266)-584 b(nski,)535 b Fj(Heter)-57 b(o)g(clinic)518 b(Conne)-57 b(ctions)517 b(b)-57 b(etwe)g(en)517 b(Pe-)10584 57451 y(rio)-57 b(dic)553 b(Orbits)f(in)h(Planar)e(R)-57 b(estricte)g(d)554 b(Cir)-57 b(cular)552 b(Thr)-57 b(e)g(e)554 b(Bo)-57 b(dy)552 b(Pr)-57 b(oblem)553 b(-)f(A)10584 58780 y(Computer)395 b(Assiste)-57 b(d)396 b(Pr)-57 b(o)g(of)p Fu(,)370 b(Comm.)h(Math.)f (Ph)-31 b(ys.)370 b(234)h(\(2003\))h(1,)e(37-75)8308 60993 y([19])556 b(P)-92 b(.)273 b(Zgliczy)31 b(\266)-584 b(nski,)295 b Fj(Fixe)-57 b(d)308 b(p)-57 b(oint)308 b(index)f(for)h(iter)-57 b(ations,)326 b(top)-57 b(olo)g(gic)g(al)308 b(horsesho)-57 b(e)306 b(and)10584 62322 y(chaos)p Fu(,)342 b(T)-92 b(op)31 b(ological)339 b(Metho)31 b(ds)336 b(in)g(Nonlinear)h (Analysis,)344 b(V)-92 b(ol.)338 b(8,)343 b(No.)337 b(1)f(\(1996\),) 10584 63650 y(169{177.)25405 70071 y(24)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 8308 6974 a Fu([20])556 b(P)-92 b(.)558 b(Zgliczy)31 b(\266)-584 b(nski,)607 b Fj(Sharkovskii's)569 b(The)-57 b(or)g(em)570 b(for)g(multidimensional)g(p)-57 b(erturb)g(a-)10584 8302 y(tions)475 b(of)g(one{dimensional)i(maps)p Fu(,)953 b(Ergo)31 b(dic)456 b(Theory)g(and)f(Dynamical)i(Sys-)10584 9631 y(tems,)370 b(19)g(\(1999\),)i(1655{1684.)8308 11844 y([21])556 b(P)-92 b(.)428 b(Zgliczy)31 b(\266)-584 b(nski,)445 b Fj(On)450 b(p)-57 b(erio)g(dic)452 b(p)-57 b(oints)450 b(for)g(systems)g(of)g(we)-57 b(akly)450 b(c)-57 b(ouple)g(d)451 b(1-dim)10584 13173 y(maps)p Fu(,)369 b(Nonlinear)h(Analysis.)h(TMA,)f (V)-92 b(ol)370 b(46/7)h(\(2001\),)h(1039{1062.)8308 15387 y([22])556 b(P)-92 b(.)352 b(Zgliczy)31 b(\266)-584 b(nski,)358 b Fj(Computer)380 b(assiste)-57 b(d)380 b(pr)-57 b(o)g(of)380 b(of)g(chaos)g(in)h(the)f(H)-28 b(\266)-537 b(enon)379 b(map)h(and)10584 16715 y(in)397 b(the)e(R\304)-566 b(ossler)396 b(e)-57 b(quations)p Fu(,)739 b(Nonlinearit)-31 b(y,)372 b(V)-92 b(ol.)371 b(10,)f(No.)h(1)e(\(1997\))j(243{252.)8308 18929 y([23])556 b(P)-92 b(.)301 b(Zgliczy)31 b(\266)-584 b(nski,)318 b Fj(Multidimensional)334 b(p)-57 b(erturb)g(ations)334 b(of)g(one-dimensional)h(maps)10584 20257 y(and)410 b(stability)g(of)g (Sharkovskii)h(or)-57 b(dering)p Fu(,)777 b(In)-31 b(t.)385 b(J.)g(of)g(Bifurcation)h(and)f(Chaos,)10584 21586 y(V)-92 b(ol.)370 b(9,)g(No.)g(9)g(\(1999\),)i(1867{1876.)25405 70071 y(25)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0403181116567--