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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 49 /one put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6CC3F1E9AE32F234EB60FE7D E34995B1ACFF52428EA20C8ED4FD73E3935CEBD40E0EAD70C0887A451E1B1AC8 47AEDE4191CCDB8B61345FD070FD30C4F375D8418DDD454729A251B3F61DAE7C 8882384282FDD6102AE8EEFEDE6447576AFA181F27A48216A9CAD730561469E4 78B286F22328F2AE84EF183DE4119C402771A249AAC1FA5435690A28D1B47486 1060C8000D3FE1BF45133CF847A24B4F8464A63CEA01EC84AA22FD005E74847E 01426B6890951A7DD1F50A5F3285E1F958F11FC7F00EE26FEE7C63998EA1328B C9841C57C80946D2C2FC81346249A664ECFB08A2CE075036CEA7359FCA1E90C0 F686C3BB27EEFA45D548F7BD074CE60E626A4F83C69FE93A5324133A78362F30 8E8DCC80DD0C49E137CDC9AC08BAE39282E26A7A4D8C159B95F227BDA2A281AF A9DAEBF31F504380B20812A211CF9FEB112EC29A3FB3BD3E81809FC6293487A7 455EB3B879D2B4BD46942BB1243896264722CB59146C3F65BD59B96A74B12BB2 9A1354AF174932210C6E19FE584B1B14C00E746089CBB17E68845D7B3EA05105 EEE461E3697FCF835CBE6D46C75523478E766832751CF6D96EC338BDAD57D53B 52F5340FAC9FE0456AD13101824234B262AC0CABA43B62EBDA39795BAE6CFE97 563A50AAE1F195888739F2676086A9811E5C9A4A7E0BF34F3E25568930ADF80F 0BDDAC3B634AD4BA6A59720EA4749236CF0F79ABA4716C340F98517F6F06D9AB 7ED8F46FC1868B5F3D3678DF71AA772CF1F7DD222C6BF19D8EF0CFB7A76FC6D1 0AD323C176134907AB375F20CFCD667AB094E2C7CB2179C4283329C9E435E7A4 1E042AD0BAA059B3F862236180B34D3FCED833472577BACD472A4DE3E3F6222F 7A252B780C86447859579C68E52691E144F836C1C62F19A12EFB710343D33262 1F7955FE5C37074CE5F9C7ABF1A241078519A4D7913A0AD861E0E357B50FB730 E757C0D26390E6028FAC61EB0E9414716AC8406A6E35DC70A7C1AA524804FC8E 985CC3604A2BE0A8235CC895B2B33CB7EE85FE4F2CD817BAC3D27ADD295D0A0E BC0E8D849952BCA7325DC261A785CD2305BC377AC61AC5E5B2CD3164CFF033CB 5436B8000673A4D763ED26273130702447C75A774C7799FB8C3E54A2E34D1710 CF7883A9B05285C7DF30F314455A4428A5369D92C0348D45BF4AEC5E16611D16 1E5EF015900F4DF63A58DC233BEE88417B204DBD110AACD1DE3D750F9C 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 76 /L put readonly def /FontBBox{-32 -250 1048 750}readonly 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 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 readonly def /FontBBox{-301 -250 1164 946}readonly 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 11 /ff put dup 12 /fi put dup 19 /acute put dup 44 /comma put dup 45 /hyphen put dup 46 /period 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 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 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 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 readonly def /FontBBox{-163 -250 1146 969}readonly 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 (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 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 19 /acute put dup 32 /suppress put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 53 /five put dup 54 /six put dup 55 /seven put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z 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 readonly def /FontBBox{14 -250 1077 750}readonly def currentdict end currentfile eexec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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 (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 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 65 /A put dup 67 /C put dup 71 /G put dup 72 /H put dup 77 /M put dup 78 /N put dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put readonly def /FontBBox{-33 -250 945 749}readonly def currentdict end currentfile eexec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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 0 /Gamma put dup 3 /Lambda put dup 6 /Sigma put dup 8 /Phi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 18 /grave put dup 19 /acute put dup 33 /exclam put dup 34 /quotedblright 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 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 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 95 /dotaccent 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 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 3645B82392D5CAE11A7CB49D7E2E82DCD485CBA04C77322EB2E6A79D73DC194E 59C120A2DABB9BF72E2CF256DD6EB54EECBA588101ABD933B57CE8A3A0D16B28 51D7494F73096DF53BDC66BBF896B587DF9643317D5F610CD9088F9849126F23 DDE030F7B277DD99055C8B119CAE9C99158AC4E150CDFC2C66ED92EBB4CC092A AA078CE16247A1335AD332DAA950D20395A7384C33FF72EAA31A5B89766E635F 45C4C068AD7EE867398F0381B07CB94D29FF097D59FF9961D195A948E3D87C31 821E9295A56D21875B41988F7A16A1587050C3C71B4E4355BB37F255D6B237CE 96F25467F70FA19E0F85785FF49068949CCC79F2F8AE57D5F79BB9C5CF5EED5D 9857B9967D9B96CDCF73D5D65FF75AFABB66734018BAE264597220C89FD17379 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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 (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 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 0 /minus put dup 48 /prime put dup 49 /infinity put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: LASY10 %!PS-AdobeFont-1.1: LASY10 1.001 %%CreationDate: 1992 Oct 23 20:19:17 %%RevisionDate: 2001 Jun 05 20:19:17 % Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def /FullName (LASY10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /LASY10 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 50 /a50 put readonly def /FontBBox{-19 -192 944 683}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % 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 (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 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 125 /circlering put readonly def /FontBBox{8 -463 1331 1003}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM7 %!PS-AdobeFont-1.1: EUFM7 2.1 %%CreationDate: 1992 Nov 20 17:36:25 % Euler fonts were designed by Hermann Zapf. % 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 (EUFM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM7 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 104 /h put dup 107 /k put dup 108 /l put dup 110 /n put readonly def /FontBBox{0 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % 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 (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 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 103 /g put dup 104 /h put dup 107 /k put dup 108 /l put dup 110 /n put dup 112 /p put dup 118 /v put dup 122 /z put readonly def /FontBBox{-26 -224 1055 741}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % 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 (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 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % 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 (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 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 61 /slash 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 115 /s put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec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0714F3FB9999614AA3B68652B033F45CC225F52843FA5668392E1D8C475E2D21 2EF48070E75BB2AD2E475D8589405500DEEF9B3C3FF899FCFFB4BE14827DD2B1 E306B07A0302E9E98BC4EE4E2681618E08C376121A514B00B8F0281821400C0B 66E6CAA6220DB66EA982C0C6C6BC766B81B90B2927345ED46DB587089C033D5E 0F56561259161532137FBD62F6695B0FD0A4397C6AAE1DE8EA0A067A18DA8B54 5E0E522BCAA8E0C7D238E37F6633E60B8B54F043BB16133CEC7175E5FD9CC0C6 BA3189EC794664FC917121C6E2521C43526E5BEED7859BFCF5CC72047A4C35D4 97232AEB6CBCEB6D9D783C23D336DFA0EF4CEC3FBAE947DA81E7EA98B95135FE 9F3FD40D3602E84A5404546E8F197B625169793DD134D28D686242C42E9B0D5B 354EFCDBFB805A8978AFC35E9A432D2D8241CE352E098B8AF76C167ED61298CF F3DC6EB9A7B474F603918399F05E7742C94B8033752D0A026A74CFE5D2421176 9D262B537C46B91CC88265FEDC88503325831D154064C36A7BADC07100081AD6 A200FB4544C29BB71997A2E0802FBC74B761B5C17597ABB237CD8C1F47F524BF C76540B9D00FBB2DB8C5B1985E5F9300C7AF086FEF3DFCDFD503B27258C98090 7DC0B577F0A8D91B068CB2369A0B30D3CDD27F7C69367B48CD77748DC89F1C53 346A42DCE20A3BD4A091E38AEF1190F36453B397BC37B9786DB26287D72C83BB A03B162AB9D53CC75BD41D2953E21B9B03A10B6F960D16DEDAE58E0312C532E9 3F0F66A4109035171DE4BDBBB764352180BAFB5268AD9179A06DB7200DC68D85 3B9E275E194D6867664F2C1E11D8734A658621C1C3CD09542A38C975C5C2981B 45011BC2FB1CC4A1155834B1583E42166D828803E213F6BC687433B140BCCE26 23BEBCDD3A1B538AE769CEA2E4CE13C75FF2910A794CD38BBC7B2A6A96EE0BDB 4AE2B71D80A86BBE8B8FC24FA8A85C5DC6604A9F2A5CE7891FAE796881413FD1 EE87FA66BAD0B6275CBB73161CAD8EA223CD4D2763D6CBA28D9F27679614FCA9 5C9CA8E72483A5365BEEFDC3CA078794F3D2AC3F2DD8D803AB8EE108C16809BD 1DE1381DC564602EF54F60202C89F305F61F79BE8EABB079CFB3171A1757720B 6EEEB80ACA27CAD2655D03B5BDF61D8D9073E4B7DAB84940180C7C692ABB4BAC 205D9055487145297D47CB982BB018E7A3923DC96673B494973D381420A6039F 6B96A368A5B3E0F6DCF9A3AFED4AE57120C130AEC84C84639AA948E0809EAF42 54337EAB826EF2592F6CD57AB3FB64F301A9EE8BCF0897E3B5A7B2F0023A9CA7 364FCCABB9D4DAED3A51ED3ECF21054762145DFC44BDBFCADF2474D136E90EFC 81ACA65DFB569BD568FF84F251CC2272CB8E08FB7437DCB10118C4BDAB07054A 025111B7F836CBD9BA0079D0F7EAD37FE5636C8ABD2F2A4B7953950896074BB8 9E1C7DE741F05A109B87C5FBE6541BAED1F9427B8042C53DA8B39CB9E20F30B6 5FB5454FDD974F7FFF83EDEFA2B890A58D292CF04AD29FC843C2D88652C9FE20 9D6944AD8030B33CFA2598618FB885E9707D5F62C47FFF0AC85D5FC96A32F2B0 E7F194205541DEAD2B024C3AF799E3955D64F1E8752CDD127910F77E1F18DA7E 2A95E07D442A60C987FE34AC9CD77D4B9CAFC504C8DB4EDCDE4664681627D64E 52E5220FDEE5500BE0C08F18F2E76E266A74A1E8BB90D04DE4CB786CDF3E848A DF5687CA6DA64918CB904E8F3C46A924B62F167CEF1A8457EEC7A812B72FDA43 9532C35BB80903295DD7F84B8C8278ADA4EE734DDF4A694A35A037124764F017 D0A6CE92FE593D8F672E338F0FB7B0548FFB9DABF2583D2DF669FD1948EAC19D 77D87E333DAD73DA1ED9AF0428621FF568F1F3852224177788DF5DB1DE00FAF8 2F4C2FA50DADB5D228FA636F0B6DBC9467A55730C498062F9C6B8FD372942BB3 2998941601DD0B38F6D58F3250039916CC2070551358061C8F9D4E7CB4480C0D 607242E7B5E7A6805D44DA096BDFA6A4B84A1D98670DED308CBD346CFCD6ACEA 9A5070EE019EFBF9B7B4ABEC2938556266CB055A7CA0FCCF2D696D4A4A787076 452D2307CF374BCDB62BFC53727E0DE8DF7BE78D53C80FD96FF7EE668F26117B 92A5F73E97F30EEE213BC7F5F3B257AA1A915D03C3FCA3BF47CB67B2D4E61E0C EA906F7C660F4083B9BFF7609B8487990F7FD0772F7A31B83D79C0B2E3EDBC10 864C4E8CD3020EED7695FEDD3BA0185F84B8D2560A4A0F546F4D85D9FF6239C8 95AEDC470847C7E66DDB4661F2404A4811905170411BA4C3A6D041A8B4C54FD3 DC693B9AB589F62131D3FA1881C6C096883E5615A18701B3AC3957D42C0126B0 91A266AB778CA41072ACEB4BA90B1533DA2858FC9E23773B2A9DECC3AEF7A90C 0B008BACF98C5BD6538D687F51839192B65301AFD015022B39896A334C8AA321 6C76C0DD500FC0C94A0E6ADCF5269F0B51E8EA09CA0C7A94F87326FB6B8572C2 49015B337F89C0E1C07CE985FE0D8CF1F150A79C6951E584807D28D9C5A6BCAB E34DFBDB118E045A273A8BABA158D387E3E35C997D1EEC61CBE41A49C95EB2DB C4E80EAF3D392E973BDA757CF6E7A34322E96DC9D75D9E3DB87B01D5D37BA9C8 00475928FDC857B6A82A700A4F19C7CB9937013F267D979F2BEB661D8FC95DBF 73A8BA4D394B9988 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 77 /circleplusdisplay put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 101 /tildewide put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 112 /radicalbig put dup 113 /radicalBig put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CAC6A7BEB5D02276E511FFAF2AE11910 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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 (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 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 0 /Gamma put dup 10 /Omega 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 52 /four put dup 53 /five put dup 54 /six put dup 58 /colon put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 100 /d put dup 101 /e put dup 105 /i put dup 109 /m put dup 115 /s put dup 116 /t put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 8 /circleplus put dup 14 /openbullet put dup 15 /bullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 65 /A put dup 66 /B put dup 69 /E put dup 71 /G put dup 72 /H put dup 76 /L put dup 77 /M put dup 78 /N put dup 84 /T put dup 85 /U put dup 91 /union put dup 92 /intersection put dup 94 /logicaland put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % 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 (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 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 11 /alpha put dup 14 /delta put dup 18 /theta put dup 19 /iota put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 29 /upsilon put dup 34 /epsilon put dup 37 /rho1 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 66 /B put dup 68 /D put dup 69 /E put dup 72 /H put dup 76 /L put dup 78 /N put dup 81 /Q put dup 83 /S put dup 84 /T put dup 85 /U put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 93 /sharp put dup 97 /a put dup 98 /b put dup 99 /c put dup 102 /f put dup 103 /g 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 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % 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 (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 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 11 /ff put dup 12 /fi put dup 19 /acute put dup 34 /quotedblright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 58 /colon put dup 65 /A 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 76 /L put dup 78 /N put dup 80 /P put dup 82 /R put dup 84 /T put dup 87 /W put dup 92 /quotedblleft 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 readonly def /FontBBox{-36 -251 1103 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 525003F3DBE5BF07B2E83E66B7F97DDD7CE0EEB75A78BD9227BF359D002B6ADB 8AC57A33FED4EF021A7085B1E2B933DE602F0FF71467ECD501744AE338AF29A0 26F7D368AC6F25CCB882DB7B7343566192BD687E1349225982823027D3B66703 3B0DB7A7E680A682B98023D39C7FAE81A5D5B867A0A66C8AA0DBC83B1596A84F 0436AC6A7900B767BDCCE0060A4811003C79FDCC71D73F7F2D0A6675E93AD21A 56B4CD8EF75EED3DE8C0A18BEBF7B9D1BE72504872D56EDB272F1E97FC726CB6 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % 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 (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 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 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 63 /perpendicular put dup 77 /M put dup 82 /R put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % 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 (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 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 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 37 /rho1 put dup 39 /phi1 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 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 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 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 readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 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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: 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 19 /acute put dup 34 /quotedblright put dup 40 /parenleft put dup 41 /parenright 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 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 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 88 /X put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright 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 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 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 39 /quoteright put dup 44 /comma 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 65 /A 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 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 88 /X put dup 90 /Z 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 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 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39139632 55387786 1000 600 600 (CELI.dvi) @start /Fa 138[44 44 3[44 1[44 44 3[44 1[44 2[44 2[44 32[44 17[44 46[{}11 83.022 /CMTT10 rf /Fb 206[33 49[{}1 58.1154 /CMR7 rf /Fc 179[57 76[{}1 83.022 /CMMI10 rf /Fd 198[48 48 48 48 48 48 48 48 48 48 48[{}10 83.022 /CMBX10 rf /Fe 133[34 40 39 55 38 45 28 34 35 38 42 42 47 68 21 38 1[25 42 38 25 38 42 38 38 42 9[83 62 62 59 47 61 64 56 1[62 74 52 2[32 62 64 54 56 63 59 58 62 18[25 30 25 24[42 6[47 51 11[{}51 83.022 /CMTI10 rf /Ff 133[42 51 1[69 51 51 49 38 50 1[46 53 51 62 43 53 35 25 51 53 44 46 52 49 48 51 6[56 68 1[92 68 68 65 51 66 1[62 70 68 82 57 70 47 34 68 71 59 62 69 65 64 68 9[46 46 46 2[46 46 2[27 31 27 11[27 12[46 19[{}58 83.022 /CMCSC10 rf /Fg 183[39 72[{}1 49.8132 /MSBM7 rf /Fh 214[51 51 40[{}2 143.462 /CMR17 rf /Fi 214[46 46 40[{}2 119.552 /CMR12 rf /Fj 206[61 18 47[48{}3 49.8132 /CMSY6 rf /Fk 173[48 3[48 63 4[52 52 3[48 1[48 65[{}7 66.4176 /MSBM7 rf /Fl 205[74 50[{}1 99.6264 /LASY10 rf /Fm 130[77 125[{}1 99.6264 /MSAM10 rf /Fn 145[36 1[19 27 2[36 104[{}4 66.4176 /EUFM7 rf /Fo 133[39 3[51 5[50 1[52 1[28 39 2[52 50 103[{}8 99.6264 /EUFM10 rf /Fp 204[30 30 30 30 48[{}4 49.8132 /CMR6 rf /Fq 140[29 1[28 32 1[38 54 20 33 25 22 43[32 61[{}10 49.8132 /CMMI6 rf /Fr 130[45 45 45 45 8[100 100 6[47 47 2[55 10[55 1[144 5[47 1[105 2[151 9[87 87 87 87 1[89 1[89 1[89 1[89 6[87 87 14[79 79 4[75 75 4[53 53 73 73 59 59 2[55 33 2[58 58 4[42 42 46 46{}42 99.6264 /CMEX10 rf /Fs 139[27 28 5[59 3[20 3[31 39 6[20 1[20 29[55 2[20 3[35 35 35 35 35 35 35 4[55 1[27 27 29[51 9[44{}22 66.4176 /CMR8 rf /Ft 141[83 1[83 1[50 2[50 28 2[50 50 7[66 1[66 66 5[62 54 5[82 120 69 3[84 59 1[53 2[65 80 10[0 2[66 66 100 15[100 6[77 4[77 77 2[77 1[50 50 5[77 4[50 77 28 77{}36 99.6264 /CMSY10 rf /Fu 144[77 24[72 3[72 3[72 94 4[77 77 3[72 1[72 65[{}9 99.6264 /MSBM10 rf /Fv 133[33 35 40 2[41 25 33 32 32 36 34 43 62 21 37 29 24 1[34 34 2[31 30 37 3[27 2[48 41 58 2[48 41 43 1[56 2[56 1[48 3[58 2[52 58 1[53 1[37 1[55 35 55 20 20 20[36 2[33 4[38 31 1[36 41 5[25 33 3[31 2[45 11[{}51 66.4176 /CMMI8 rf /Fw 133[40 47 45 65 45 52 32 40 41 45 50 50 55 80 25 45 30 30 50 45 30 45 50 45 45 50 4[50 4[97 2[70 1[71 1[66 1[72 1[61 2[38 72 1[64 66 74 70 1[72 6[30 8[50 2[30 35 30 9[50 14[50 6[55 60 11[{}49 99.6264 /CMTI12 rf /Fx 145[35 2[35 20 2[35 35 19[60 4[84 13[55 12[47 71 19 14[71 6[55 4[55 55 16[35 55 20 55{}19 66.4176 /CMSY8 rf /Fy 133[45 48 55 70 47 56 35 46 44 43 49 1[58 85 29 51 40 33 56 47 48 45 51 42 41 51 6[67 57 81 92 57 66 57 60 74 77 63 75 78 94 66 83 54 43 81 77 63 72 81 70 74 73 51 1[76 49 76 27 27 18[64 1[50 2[46 61 63 1[58 53 42 55 50 55 43 48 59 57 56 34 45 48 43 1[43 51 55 62 11[{}81 99.6264 /CMMI12 rf /Fz 165[47 1[58 1[58 58 55 43 57 60 52 60 58 70 48 2[28 58 60 1[52 59 55 1[58 7[38 38 38 38 38 38 38 38 38 38 3[21 44[{}31 74.7198 /CMR9 rf /FA 128[49 49 3[43 51 51 70 51 54 38 38 38 51 54 49 54 81 27 51 30 27 54 49 30 43 54 43 54 49 1[27 1[27 49 27 1[73 1[100 73 73 70 54 72 76 66 76 73 89 61 76 50 35 73 77 64 66 75 70 69 73 3[76 2[27 49 49 49 49 49 49 49 49 49 49 1[27 33 27 76 1[38 38 27 4[49 27 13[49 49 3[81 54 54 57 70 1[70 1[70 2[68 2[61{}88 99.6264 /CMR12 rf /FB 133[50 1[61 83 61 61 59 46 60 63 56 63 61 74 51 63 1[30 61 64 53 56 62 59 58 61 12[78 1[80 84 74 3[68 2[40 1[85 3[78 1[81 11[55 3[55 2[32 38 45[{}37 99.6264 /CMCSC10 rf /FC 132[42 37 44 44 60 44 46 32 33 33 1[46 42 46 69 23 44 1[23 46 42 25 37 46 37 46 42 3[23 42 23 1[62 62 1[62 62 60 46 61 1[57 65 62 76 52 1[43 30 62 65 54 57 63 60 59 62 6[23 42 42 42 42 42 42 42 42 42 42 42 23 28 23 2[32 32 5[42 14[42 19[{}69 83.022 /CMR10 rf /FD 133[50 59 59 1[59 62 44 44 46 59 62 56 62 93 31 59 1[31 62 56 34 51 62 50 62 54 6[68 1[85 2[86 78 62 84 84 77 84 88 106 67 2[42 88 1[70 74 86 81 1[85 7[56 56 56 56 56 56 56 56 56 56 1[31 1[31 4[31 26[62 65 11[{}58 99.6264 /CMBX12 rf end %%EndProlog 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Fy(;)17 b(Y)k FA(])p Fy(;)c FA(])p Fy(;)g(:)g(:)g(:)f(;)h FA(])97 b(and)h([)p Fy(X)r(;)17 b(Y)k FA(])2702 618 y Fs(0)2769 603 y FA(=)28 b Fy(Y)38 b(:)-2971 b FA(\(7\))24 900 y(Notice)33 b(that)g([)p Fy(X)r(;)17 b(Y)k FA(])801 915 y Fv(k)871 900 y FA(=)28 b(ad)1078 857 y Fv(k)1121 900 y FA(\()p Fy(Y)21 b FA(\).)124 1066 y(W)-8 b(e)48 b(can)f(express)j(the)e(group)g (op)s(eration)f(in)h(the)g(Lie)g(algebra)f(b)m(y)i(the)f(Bak)m (er-Campb)s(ell-)24 1183 y(Hausdor\013)33 b(form)m(ula,)g(that)f(w)m(e) i(presen)m(t)g(in)f(the)g(follo)m(wing)g(form)1282 1414 y Fy(X)d Fm(})23 b Fy(Y)49 b FA(=)1754 1290 y Fv(\035)1702 1319 y Fr(X)1713 1529 y Fv(j)t Fs(=1)1863 1414 y Fy(c)1905 1429 y Fv(n)1952 1414 y FA(\()p Fy(X)r(;)17 b(Y)k FA(\))p Fy(;)-2236 b FA(\(8\))24 1676 y(where)27 b Fy(c)341 1691 y Fs(1)380 1676 y FA(\()p Fy(X)r(;)17 b(Y)22 b FA(\))27 b(=)h Fy(X)16 b FA(+)8 b Fy(Y)20 b FA(,)27 b Fy(c)1147 1691 y Fs(2)1187 1676 y FA(\()p Fy(X)r(;)17 b(Y)k FA(\))28 b(=)f([)p Fy(X)r(;)17 b(Y)22 b FA(])p 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Fv(k)1094 2561 y Fp(1)1128 2552 y Fv(;:::)o(;k)1264 2561 y Fp(2)p Fq(p)1330 2552 y Fv(>)p Fs(0)1046 2626 y Fv(k)1083 2635 y Fp(1)1117 2626 y Fs(+)p Fx(\001\001\001)o Fv(k)1268 2635 y Fp(2)p Fq(p)1334 2626 y Fs(=)p Fv(n)1432 2435 y FA([)p Fy(c)1501 2450 y Fv(k)1538 2459 y Fp(1)1576 2435 y FA(\()p Fy(X)r(;)17 b(Y)k FA(\))p Fy(;)c FA([)p Ft(\001)g(\001)g(\001)32 b Fy(;)17 b FA([)p Fy(c)2191 2450 y Fv(k)2228 2459 y Fp(2)p Fq(p)2298 2435 y FA(\()p Fy(X)r(;)g(Y)k FA(\))p Fy(;)c(X)30 b FA(+)22 b Fy(Y)f FA(])p Fy(;)c FA(])p Fy(;)g(:)g(:)g(:)f(;)h FA(])p Fy(;)24 2779 y FA(see)34 b(Lemma)f(2.15.3)f(of)g([33)o(].)24 2946 y FD(Lemma)41 b(2.3.)h Fw(L)-5 b(et)37 b Fy(\027)g(>)30 b FA(0)36 b Fw(and)f(let)h Fy(n)31 b FA(=)f(2)p Fy(;)17 b(:)g(:)g(:)e(;)i(\023)p Fw(.)49 b(Then)36 b(ther)-5 b(e)36 b(exists)g(a)g(c)-5 b(onstant)36 b Fy(\013)3294 2961 y Fv(n)3341 2946 y FA(\()p Fy(\027)6 b FA(\))36 b Fw(only)24 3062 y(dep)-5 b(ending)33 b(on)i Fy(n)g Fw(and)f Fy(\027)42 b Fw(such)34 b(that)1157 3219 y Ft(k)p Fy(c)1249 3234 y Fv(n)1295 3219 y FA(\()p Fy(X)r(;)17 b(Y)22 b FA(\))p Ft(k)27 b(\024)h Fy(\013)1821 3234 y Fv(n)1868 3219 y FA(\()p Fy(\027)6 b FA(\))28 b Ft(k)p FA([)p Fy(X)r(;)17 b(Y)k FA(])p Ft(k)-2361 b FA(\(10\))24 3376 y Fw(whenever)34 b Ft(k)p Fy(X)8 b Ft(k)p Fy(;)17 b Ft(k)p Fy(Y)j Ft(k)27 b(\024)i Fy(\027)6 b Fw(.)124 3542 y FB(Pr)n(oof.)42 b FA(Our)28 b(statemen)m(t)h(is)g(trivial)f(for) f Fy(n)h FA(=)g(2,)g(b)s(eing)h Fy(c)2271 3557 y Fs(2)2310 3542 y FA(\()p Fy(X)r(;)17 b(Y)k FA(\))28 b(=)f([)p Fy(X)r(;)17 b(Y)22 b FA(])p Fy(=)p FA(2.)41 b(Assume)30 b(that)24 3658 y(it)k(is)g(true)h(for)e(ev)m(ery)j Fy(j)g FA(=)30 b(2)p Fy(;)17 b(:)g(:)g(:)f(;)h(n)p FA(,)34 b(with)g Fy(n)d Ft(\025)f FA(2.)48 b(W)-8 b(e)34 b(observ)m(e)i(that)e([)p Fy(c)2751 3673 y Fv(k)2788 3682 y Fp(2)p Fq(p)2858 3658 y FA(\()p Fy(X)r(;)17 b(Y)k FA(\))p Fy(;)c(X)31 b FA(+)23 b Fy(Y)e FA(])30 b Ft(6)p FA(=)g(0)24 3775 y(in)j(\(9\))f(implies)i Fy(k)677 3790 y Fs(2)p Fv(p)779 3775 y Fy(>)28 b 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Ft(\032)g Fo(p)33 b FA(for)f(ev)m(ery)i Fy(r)c(>)e FA(0.)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 169 246 a Fz(CONT)-6 b(A)n(CT)34 b(EQUA)-6 b(TIONS,)31 b(LIPSCHITZ)i(EXTENSIONS)f(AND)g(ISOPERIMETRIC)h (INEQUALITIES)106 b(9)24 446 y FD(Prop)s(osition)52 b(2.6.)c Fw(L)-5 b(et)47 b Fo(p)g Fw(and)f Fo(h)g Fw(b)-5 b(e)46 b(homo)-5 b(gene)g(ous)45 b(sub)-5 b(algebr)g(as)46 b(of)g Ft(G)53 b Fw(and)46 b(let)h Fy(P)60 b Fw(and)45 b Fy(H)24 562 y Fw(denote)e(their)h(c)-5 b(orr)g(esp)g(onding)42 b(homo)-5 b(gene)g(ous)42 b(sub)-5 b(gr)g(oups,)45 b(r)-5 b(esp)g(e)g(ctively.)71 b(Then)42 b(the)i(c)-5 b(ondition)24 678 y Fo(p)16 b Ft(\010)g Fo(h)29 b FA(=)e Ft(G)39 b Fw(is)32 b(e)-5 b(quivalent)32 b(to)g(r)-5 b(e)g(quir)g(e)32 b(that)h Fy(P)d Ft(\\)16 b Fy(H)36 b FA(=)27 b Ft(f)p Fy(e)p Ft(g)32 b Fw(and)g Fy(P)14 b(H)35 b FA(=)27 b Fu(G)p Fw(.)45 b(F)-7 b(urthermor)i(e,)31 b(if)i(one)24 794 y(of)i(these)f(c)-5 b(onditions)34 b(hold,)g(then)h(the)g(mapping) 803 967 y Fy(\036)28 b FA(:)f Fo(p)22 b Ft(\002)h Fo(h)28 b Ft(\000)-17 b(!)28 b Fu(G)p Fy(;)216 b(\036)p FA(\()p Fy(W)m(;)17 b(Y)k FA(\))28 b(=)f(exp)18 b Fy(W)58 b FA(exp)18 b Fy(Y)-2694 b FA(\(11\))24 1140 y Fw(is)35 b(a)f(di\013e)-5 b(omophism.)24 1323 y FD(Lemma)48 b(2.7.)e Fw(L)-5 b(et)42 b Fy(X)49 b Ft(2)41 b Fy(V)1119 1338 y Fs(1)1186 1323 y Ft(n)27 b(f)p FA(0)p Ft(g)p Fw(.)66 b(Then)41 b(ther)-5 b(e)42 b(exists)g(a)g(normal)f(homo)-5 b(gene)g(ous)40 b(sub)-5 b(gr)g(oup)24 1439 y Fy(N)38 b Ft(\032)28 b Fu(G)36 b Fw(such)e(that)i Fu(G)28 b FA(=)f Fy(N)33 b Fu(o)23 b Fy(H)1279 1454 y Fv(X)1346 1439 y Fw(.)124 1623 y FB(Pr)n(oof.)50 b FA(Let)35 b Fy(U)760 1638 y Fs(1)832 1623 y Ft(\032)d Fy(V)998 1638 y Fs(1)1072 1623 y FA(b)s(e)j(a)f(subspace)j(of)d(dimension)j(dim)17 b Fy(V)2513 1638 y Fs(1)2576 1623 y Ft(\000)24 b FA(1)35 b(that)f(do)s(es)i(not)e(con)m(tain)24 1739 y Fy(X)43 b FA(and)35 b(set)h Ft(N)47 b FA(=)32 b Fy(U)798 1754 y Fs(1)862 1739 y Ft(\010)24 b Fy(V)1020 1754 y Fs(2)1084 1739 y Ft(\010)g(\001)17 b(\001)g(\001)22 b(\010)j Fy(V)1484 1754 y Fv(\023)1513 1739 y FA(.)51 b(W)-8 b(e)36 b(notice)g(that)f Ft(N)50 b FA(is)36 b(a)f(homogeneous)h(ideal,)h(hence)24 1855 y Fy(N)53 b FA(=)43 b(exp)18 b Ft(N)56 b FA(is)42 b(a)f(normal)h(homogeneous)g(subgroup)g(of)f Fu(G)p FA(.)71 b(By)42 b(de\014nition)h(of)e Ft(N)15 b FA(,)43 b(w)m(e)g(ha)m(v)m(e)24 1971 y Ft(N)g(\\)29 b FA(span)q Ft(f)p Fy(X)8 b Ft(g)43 b FA(=)g Ft(f)p FA(0)p Ft(g)f FA(and)g(this)g(implies)i(that)e Fy(N)d Ft(\\)29 b Fy(H)2235 1986 y Fv(X)2346 1971 y FA(=)43 b Ft(f)p Fy(e)p Ft(g)f FA(and)g Fy(N)10 b(H)3020 1986 y Fv(X)3132 1971 y FA(=)43 b Fu(G)p FA(,)i(see)e(for)24 2087 y(instance)34 b(Prop)s(osition)f(2.6.)43 b(In)33 b(other)f(w)m(ords)i Fu(G)28 b FA(=)g Fy(N)k Fu(o)23 b Fy(H)2280 2102 y Fv(X)2347 2087 y FA(.)44 b Fl(2)24 2293 y FA(2.2.)49 b FD(Haar)e(measure)i(and)f(F)-9 b(ubini's)49 b(theorem.)h FA(W)-8 b(e)41 b(consider)i(a)e(graded)g(group)h Fu(G)f FA(with)24 2409 y(grading)k Ft(G)56 b FA(=)50 b Fy(V)687 2424 y Fs(1)757 2409 y Ft(\010)31 b(\001)17 b(\001)g(\001)29 b(\010)j Fy(V)1178 2424 y Fv(\023)1252 2409 y FA(and)46 b(w)m(e)h(\014x)f(a)f(left)h(in)m(v)-5 b(arian)m(t)46 b(Riemannian)g(metric)g Fy(g)j FA(on)c Fu(G)p FA(.)24 2526 y(Then)32 b(the)g(asso)s(ciated)g(v)m(olume)h (measure)g(v)m(ol)1745 2541 y Fv(g)1817 2526 y FA(is)f(clearly)g(left)f (in)m(v)-5 b(arian)m(t)32 b(and)g(de\014nes)h(the)e(Haar)24 2642 y(measure)d(of)f Fu(G)p FA(.)42 b(W)-8 b(e)27 b(will)h(denote)f(b) m(y)h Fy(\026)f FA(this)g(measure.)43 b(Let)27 b Fy(X)36 b Ft(2)28 b Fy(V)2552 2657 y Fs(1)2618 2642 y FA(and)f Fy(N)37 b FA(b)s(e)27 b(a)g(homogeneous)24 2758 y(normal)42 b(subgorup)i Fy(N)52 b FA(suc)m(h)44 b(that)f Fu(G)h FA(=)h Fy(N)39 b Fu(o)30 b Fy(H)1927 2773 y Fv(X)1994 2758 y FA(.)73 b(Then)43 b(w)m(e)h(ha)m(v)m(e)g(the)e(follo)m(wing)h(F) -8 b(ubini's)24 2874 y(theorem)33 b(with)h(resp)s(ect)f(to)g(this)g (factorization.)24 3058 y FD(Prop)s(osition)47 b(2.8.)f Fw(L)-5 b(et)43 b Fy(\026)f Fw(b)-5 b(e)43 b(the)f(Haar)h(me)-5 b(asur)g(e)42 b(of)g Fu(G)p Fw(.)69 b(Then)41 b(for)i(every)f(me)-5 b(asur)g(able)42 b(set)24 3174 y Fy(A)28 b Ft(\032)g Fu(G)p Fw(,)35 b(we)g(have)1174 3394 y Fy(\026)p FA(\()p Fy(A)p FA(\))27 b(=)1513 3258 y Fr(Z)1568 3484 y Fv(N)1652 3394 y Fy(\027)1700 3409 y Fv(X)1784 3394 y FA(\()p Fy(A)1895 3409 y Fv(n)1942 3394 y FA(\))33 b Fy(d\026)2123 3409 y Fv(N)2190 3394 y FA(\()p Fy(n)p FA(\))17 b Fy(;)-2344 b FA(\(12\))24 3623 y Fw(wher)-5 b(e)33 b Fy(A)371 3638 y Fv(n)446 3623 y FA(=)28 b Ft(f)p Fy(h)f Ft(2)h Fy(H)858 3638 y Fv(X)953 3623 y Ft(j)g Fy(nh)g Ft(2)g Fy(A)p Ft(g)p Fw(.)44 b(We)34 b(have)f(denote)-5 b(d)33 b(by)h Fy(\026)2381 3638 y Fv(N)2482 3623 y Fw(and)g Fy(\027)2719 3638 y Fv(X)2820 3623 y Fw(the)g(Haar)g(me)-5 b(asur)g(e)33 b(of)24 3739 y Fy(N)45 b Fw(and)35 b(of)f Fy(H)532 3754 y Fv(X)599 3739 y Fw(,)h(r)-5 b(esp)g(e)g(ctively,)124 3923 y FB(Pr)n(oof.)52 b FA(W)-8 b(e)36 b(\014x)h(an)e(orthonormal)g (basis)i(\()p Fy(X)1893 3938 y Fs(1)1932 3923 y Fy(;)17 b(:)g(:)g(:)f(;)h(X)2232 3938 y Fv(q)2270 3923 y FA(\))35 b(of)g Ft(G)42 b FA(with)36 b(resp)s(ect)h(to)e(the)h(metric)24 4039 y Fy(g)t FA(.)42 b(In)30 b(addition)g(w)m(e)h(assume)h(that)d (this)i(basis)g(is)f(adapted)g(to)g(the)g(grading)g(of)f Ft(M)p FA(,)i(suc)m(h)g(that)f Fy(X)3645 4054 y Fs(1)24 4155 y FA(is)j(prop)s(ortional)f(to)g Fy(X)8 b FA(.)43 b(By)34 b(Prop)s(osition)e(2.6,)h(the)g(mapping)519 4406 y Fy( )f FA(:)27 b Fu(R)22 b Ft(\002)h Fu(R)934 4365 y Fv(q)r Fx(\000)p Fs(1)1090 4406 y Ft(\000)-16 b(!)27 b Fu(M)p Fy(;)212 b FA(\()p Fy(\030)5 b(;)17 b(t)p FA(\))27 b Ft(\000)-16 b(!)27 b FA(exp)2195 4296 y Fr(\020)2326 4278 y Fv(q)2271 4312 y Fr(X)2281 4522 y Fv(j)t Fs(=2)2431 4406 y Fy(x)2486 4421 y Fv(j)2523 4406 y Fy(X)2604 4421 y Fv(j)2641 4296 y Fr(\021)2717 4406 y FA(exp)2882 4326 y Fr(\000)2928 4406 y Fy(x)2983 4421 y Fs(1)3023 4406 y Fy(X)3104 4421 y Fs(1)3143 4326 y Fr(\001)24 4678 y FA(is)34 b(a)e(di\013eomorphism.)47 b(Our)33 b(\014xed)h(basis)g(also)g 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Fv(X)3225 4979 y Fy(;)24 5170 y FA(see)39 b(for)f(instance)i (Prop)s(osition)e(2.3.47)g(of)g([24)o(].)61 b(Th)m(us,)41 b(the)e(fact)f(that)g Fy(X)2853 5185 y Fs(1)2930 5170 y Ft(2)g Fy(V)3091 5185 y Fs(1)3168 5170 y FA(implies)i(that)24 5286 y(the)35 b(mapping)g Fy(F)674 5250 y Fx(\000)p Fs(1)791 5286 y Ft(\016)24 b Fy( )38 b FA(has)d(jacobian)g(equal)g(to)f(one.)50 b(Com)m(bining)36 b(these)f(facts)g(with)g(classical)24 5403 y(F)-8 b(ubini's)33 b(theorem,)h(w)m(e)f(get)g(our)f(claim.)45 b Fl(2)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 24 246 a Fz(10)1315 b(V)-9 b(ALENTINO)25 b(MA)n(GNANI)24 446 y FA(2.3.)49 b FD(Di\013eren)m(tiabilit)m(y.)h FA(In)38 b(the)g(presen)m(t)i(subsection,)h(w)m(e)e(recall)g(the)f (notion)g(of)f(P)m(ansu)i(dif-)24 562 y(feren)m(tiabilit)m(y)-8 b(.)45 b Fu(G)33 b FA(and)g Fu(M)g FA(denote)g(t)m(w)m(o)g (strati\014ed)h(groups)e(and)h(\012)g(is)g(an)g(op)s(en)f(subset)j(of)d Fu(G)p FA(.)24 758 y FD(De\014nition)40 b(2.9)g FA(\(h-homomorphism\))p FD(.)j FA(A)35 b(group)f(homomorphism)h Fy(L)d FA(:)e Fu(G)i Ft(\000)-16 b(!)30 b Fu(M)k FA(suc)m(h)i(that)24 874 y Fy(L)p FA(\()p Fy(\016)175 838 y Fk(G)171 899 y Fv(r)231 874 y Fy(x)p FA(\))28 b(=)g Fy(\016)503 838 y Fk(M)499 899 y Fv(r)569 874 y Fy(L)p FA(\()p Fy(x)p FA(\))33 b(for)e(ev)m(ery)j Fy(x)28 b Ft(2)g Fu(G)k FA(and)g Fy(r)e(>)e FA(0)j(is)i(called)f Fw(homo)-5 b(gene)g(ous)33 b(homomorphism)p FA(,)d(in)24 990 y(short)j Fw(h-homomorphism)p FA(.)124 1186 y(Analogous)45 b(terminology)g(will)h(b)s(e)f(used)h(for) f(the)g(corresp)s(onding)h(Lie)g(algebra)e(homomor-)24 1302 y(phisms)34 b(of)e(graded)h(algebras)g(that)f(comm)m(ute)i(with)g (dilations.)24 1498 y FD(De\014nition)53 b(2.10)f FA(\(P-di\013eren)m (tiabilit)m(y\))p FD(.)e FA(Let)45 b Fy(d)g FA(and)g Fy(\032)g FA(b)s(e)h(homogeneous)g(distances)h(of)e Fu(G)24 1614 y FA(and)i Fu(M)p FA(,)52 b(resp)s(ectiv)m(ely)-8 b(.)91 b(W)-8 b(e)47 b(consider)i(the)f(mapping)g Fy(f)63 b FA(:)53 b(\012)h Ft(\000)-16 b(!)52 b Fu(M)p FA(.)88 b(W)-8 b(e)48 b(sa)m(y)g(that)f Fy(f)58 b FA(is)24 1731 y Fw(P-di\013er)-5 b(entiable)31 b FA(at)h Fy(x)d Ft(2)f FA(\012)33 b(if)f(there)h(exists)i(an)d(h-homomorphism)i Fy(L)28 b FA(:)f Fu(G)i Ft(\000)-16 b(!)27 b Fu(M)32 b FA(suc)m(h)i(that)973 1921 y Fy(\032)1023 1840 y Fr(\000)1069 1921 y Fy(f)11 b FA(\()p Fy(x)p FA(\))1259 1885 y Fx(\000)p Fs(1)1354 1921 y Fy(f)g FA(\()p Fy(xh)p FA(\))p Fy(;)17 b(L)p FA(\()p Fy(h)p FA(\))1842 1840 y Fr(\001)p 973 1972 915 4 v 1339 2063 a Fy(d)p FA(\()p Fy(h)p FA(\))1925 1995 y Ft(\000)-16 b(!)27 b FA(0)98 b(as)g Fy(h)27 b Ft(!)h Fy(e)17 b(:)24 2244 y FA(The)41 b(h-homomorphism)g Fy(L)g FA(satisfying)g(this)g(limit)g(is)g(unique)h(and)e(it)g(is)h (called)g Fw(P-di\013er)-5 b(ential)24 2360 y FA(of)48 b Fy(f)60 b FA(at)48 b Fy(x)p FA(.)93 b(W)-8 b(e)49 b(denote)h Fy(L)f FA(b)m(y)h Fy(D)s(f)11 b FA(\()p Fy(x)p FA(\),)52 b(when)e(w)m(e)g(read)f(the)h(P-di\013eren)m(tial)g(b)s(et)m(w)m(een)h (the)24 2476 y(corresp)s(onding)33 b(Lie)g(algebras,)g(w)m(e)h(will)f (denote)h(it)e(b)m(y)i Fy(d)-16 b(f)11 b FA(\()p Fy(x)p FA(\).)24 2672 y FD(Theorem)38 b(2.11.)k Fw(Every)35 b(Lipschitz)f(mapping)g Fy(f)k FA(:)28 b(\012)g Ft(\000)-16 b(!)27 b Fu(M)35 b Fw(is)g Fy(\026)p Fw(-a.e.)44 b(P-di\013er)-5 b(entiable.)124 2868 y FA(This)47 b(theorem)g(is)g(an)f(imp)s(ortan)m (t)g(result)h(due)g(to)f(P)m(ansu,)51 b([28)o(].)84 b(Here)47 b(ha)m(v)m(e)h(presen)m(ted)g(a)24 2984 y(sligh)m(tly)34 b(more)f(general)g(v)m(ersion)h(where)g Fu(M)e FA(is)h(graded,)g(but)g (it)g(migh)m(t)g(not)f(b)s(e)h(strati\014ed,)h([25)o(].)24 3180 y FD(De\014nition)40 b(2.12)f FA(\(Distributional)34 b(deriv)-5 b(ativ)m(es\))p FD(.)45 b FA(Let)34 b(\012)h(b)s(e)f(an)g (op)s(en)g(subset)i(of)d(a)h(strati\014ed)24 3296 y(group)h Fu(G)p FA(,)i(let)f Fy(X)44 b FA(b)s(e)36 b(a)f(left)h(in)m(v)-5 b(arian)m(t)36 b(v)m(ector)h(\014eld)f(of)f Fu(G)h FA(and)g(let)g Fy(E)42 b FA(b)s(e)36 b(a)f(\014nite)h(dimensional)24 3412 y(normed)47 b(space.)87 b(Then)48 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y Fy(X)2346 1144 y Fv(j)2382 1129 y Fy(f)2430 1144 y Fs(1)2470 1129 y Fy(;)24 1270 y FA(it)34 b(follo)m(ws)h(that)e(the)i(con)m(tact)f(equations)h(cannot)f(hold)g (at)g(ev)m(ery)i(p)s(oin)m(t.)47 b(As)35 b(a)e(consequence)k(of)24 1386 y(Theorem)29 b(1.1,)e(the)h(mapping)g Fy(f)38 b FA(is)27 b(not)g(Lipsc)m(hitz)i(on)f(ev)m(ery)h(op)s(en)e(subset)i(of)e Fu(H)3016 1350 y Fs(1)3055 1386 y FA(.)42 b(Clearly)-8 b(,)30 b(since)24 1503 y Fy(I)40 b FA(is)33 b(smo)s(oth,)g(all)g(comp)s (onen)m(ts)h(of)e(the)h(mapping)g Fy(I)40 b FA(are)33 b(con)m(tin)m(uously)i(P-di\013eren)m(tiable.)24 1687 y(5.2.)49 b FD(Con)m(tact)44 b(equations)i(and)g(Rumin)f(complex.)50 b FA(In)39 b(view)i(of)d(our)h(study)h(of)f(Lipsc)m(hitz)24 1803 y(mappings)g(in)f(the)h(three)f(dimensional)i(Heisen)m(b)s(erg)g (group)d Fu(H)2419 1767 y Fs(1)2459 1803 y FA(,)j(w)m(e)f(limit)f (ourselv)m(es)j(to)c(recall)24 1919 y(the)25 b(Rumin)h(complex)g(on)f Fu(H)1077 1883 y Fs(1)1116 1919 y FA(,)i(see)f([27)o(])f(for)g(the)g (case)g(of)g(general)g(con)m(tact)g(manifolds.)42 b(W)-8 b(e)25 b(denote)24 2036 y(b)m(y)33 b(\012)229 1999 y Fv(k)273 2036 y FA(\()p Fu(H)388 1999 y Fs(1)427 2036 y FA(\))g(the)g(mo)s(dule)g(of)f Fy(k)s FA(-forms)h(on)f Fu(H)1689 1999 y Fs(1)1761 2036 y FA(and)h(also)467 2199 y Fy(J)530 2158 y Fs(2)598 2199 y FA(=)701 2118 y Fr(\010)759 2199 y Fy(\013)c Ft(2)f FA(\012)1014 2158 y Fs(2)1054 2199 y FA(\()p Fy(M)10 b FA(\))28 b Ft(j)f Fy(\022)f Ft(^)c Fy(\013)28 b FA(=)g(0)1719 2118 y Fr(\011)1794 2199 y Fy(;)114 b(I)1986 2158 y Fs(1)2053 2199 y FA(=)2156 2118 y Fr(\010)2214 2199 y Fy(')17 b(\022)31 b Ft(j)c Fy(')h Ft(2)g Fy(C)2689 2158 y Fx(1)2763 2199 y FA(\()p Fu(H)2878 2158 y Fs(1)2918 2199 y FA(\))2956 2118 y Fr(\011)3048 2199 y Fy(;)24 2373 y FA(where)35 b Fy(\022)d FA(=)d Fy(dt)22 b FA(+)696 2292 y Fr(\000)742 2373 y Fy(x)797 2388 y Fs(2)836 2373 y Fy(dx)942 2337 y Fs(1)1005 2373 y Ft(\000)h Fy(x)1160 2388 y Fs(1)1200 2373 y Fy(dx)1306 2337 y Fs(2)1345 2292 y Fr(\001)1391 2373 y Fy(=)p FA(2)33 b(is)h(the)g(con)m(tact)f(form.)46 b(In)34 b(this)g(co)s(ordinates,)g (w)m(e)h(\014x)f(the)24 2489 y(left)f(in)m(v)-5 b(arian)m(t)33 b(v)m(ector)h(\014elds)768 2653 y Fy(X)849 2668 y Fs(1)916 2653 y FA(=)28 b Fy(@)1071 2668 y Fs(1)1133 2653 y Ft(\000)1242 2586 y Fy(x)1297 2601 y Fs(2)p 1242 2630 95 4 v 1265 2722 a FA(2)1347 2653 y Fy(@)1398 2668 y Fv(t)1428 2653 y Fy(;)114 b(X)1650 2668 y Fs(2)1717 2653 y FA(=)28 b Fy(@)1872 2668 y Fs(2)1934 2653 y FA(+)2042 2586 y Fy(x)2097 2601 y Fs(1)p 2042 2630 V 2065 2722 a FA(2)2147 2653 y Fy(@)2198 2668 y Fv(t)2326 2653 y FA(and)97 b Fy(X)2661 2668 y Fs(3)2728 2653 y FA(=)28 b Fy(@)2883 2668 y Fv(t)2913 2653 y Fy(:)24 2860 y FA(W)-8 b(e)33 b(also)g(set)g(\012)610 2824 y Fs(1)650 2860 y FA(\()p Fu(H)765 2824 y Fs(1)805 2860 y FA(\))p Fy(=I)943 2824 y Fs(1)1009 2860 y FA(=)1113 2780 y Fr(\010)1171 2860 y FA([)p Fy(\013)1260 2875 y Fs(1)1300 2860 y Fy(dx)1406 2875 y Fs(1)1467 2860 y FA(+)22 b Fy(\013)1627 2875 y Fs(2)1667 2860 y Fy(dx)1773 2875 y Fs(2)1834 2860 y FA(+)g Fy(\013)1994 2875 y Fs(3)2034 2860 y Fy(\022)s FA(])2109 2875 y Fx(R)2201 2860 y Ft(j)28 b Fy(\013)2319 2875 y Fv(j)2383 2860 y Ft(2)g Fy(C)2554 2824 y Fx(1)2629 2860 y FA(\()p Fu(H)2744 2824 y Fs(1)2783 2860 y FA(\))2821 2780 y Fr(\011)2896 2860 y Fy(:)33 b FA(Clearly)-8 b(,)921 3024 y([)p Fy(\013)1010 3039 y Fs(1)1049 3024 y Fy(dx)1155 3039 y Fs(1)1217 3024 y FA(+)22 b Fy(\013)1377 3039 y Fs(2)1416 3024 y Fy(dx)1522 3039 y Fs(2)1584 3024 y FA(+)g Fy(\013)1744 3039 y Fs(3)1784 3024 y Fy(\022)s FA(])1859 3039 y Fx(R)1951 3024 y FA(=)28 b([)p Fy(\013)2144 3039 y Fs(1)2183 3024 y Fy(dx)2289 3039 y Fs(1)2351 3024 y FA(+)22 b Fy(\013)2511 3039 y Fs(2)2550 3024 y Fy(dx)2656 3039 y Fs(2)2696 3024 y FA(])2723 3039 y Fx(R)24 3186 y FA(and)33 b(w)m(e)g(ha)m(v)m(e)h(the)f(follo)m (wing)24 3363 y FD(Theorem)38 b(5.1)g FA(\(Rumin,)33 b(1990\))p FD(.)40 b Fw(Ther)-5 b(e)35 b(exists)f Fy(D)d FA(:)c(\012)2154 3327 y Fs(1)2194 3363 y FA(\()p Fu(H)2309 3327 y Fs(1)2349 3363 y FA(\))p Fy(=I)2487 3327 y Fs(1)2554 3363 y Ft(\000)-16 b(!)27 b Fy(J)2805 3327 y Fs(2)2879 3363 y Fw(such)35 b(that)24 3554 y FA(\(37\))587 b(0)28 b Ft(\000)-16 b(!)27 b Fu(R)g Ft(\000)-16 b(!)28 b Fy(C)1415 3513 y Fx(1)1489 3554 y FA(\()p Fu(H)1604 3513 y Fs(1)1644 3554 y FA(\))g Ft(\000)-16 b(!)27 b FA(\012)1968 3513 y Fs(1)2008 3554 y FA(\()p Fu(H)2123 3513 y Fs(1)2163 3554 y FA(\))p Fy(=I)2301 3513 y Fs(1)2418 3497 y Fv(D)2367 3554 y Ft(\000)-16 b(!)28 b Fy(J)2619 3513 y Fs(2)2686 3554 y Ft(\000)-16 b(!)27 b FA(0)24 3716 y Fw(de\014nes)34 b(a)g(c)-5 b(omplex)34 b(whose)g(c)-5 b(ohomolo)g(gy)34 b(c)-5 b(oincides)33 b(with)i(the)g(De)f(R)n(ham)g(c)-5 b(ohomolo)g(gy,)33 b(wher)-5 b(e)76 3877 y Fy(D)s FA([)p Fy(\013)249 3892 y Fs(1)288 3877 y Fy(dx)394 3892 y Fs(1)456 3877 y FA(+)22 b Fy(\013)616 3892 y Fs(2)655 3877 y Fy(dx)761 3892 y Fs(2)801 3877 y FA(])828 3892 y Fx(R)920 3877 y FA(=)28 b Fy(d)1075 3797 y Fr(\000)1120 3877 y Fy(\013)1182 3892 y Fs(1)1221 3877 y Fy(dx)1327 3892 y Fs(1)1389 3877 y FA(+)22 b Fy(\013)1549 3892 y Fs(2)1589 3877 y Fy(dx)1695 3892 y Fs(2)1756 3877 y FA(+)g Fy(\013)1916 3892 y Fs(3)1956 3877 y Fy(\022)2004 3797 y Fr(\001)2077 3877 y Ft(2)28 b Fy(J)2234 3836 y Fs(2)2374 3877 y Fw(de\014ning)98 b Fy(\013)2869 3892 y Fs(3)2936 3877 y FA(=)28 b Fy(X)3121 3892 y Fs(1)3160 3877 y Fy(\013)3222 3892 y Fs(2)3284 3877 y Ft(\000)22 b Fy(X)3464 3892 y Fs(2)3504 3877 y Fy(\013)3566 3892 y Fs(1)3605 3877 y Fy(:)124 4055 y FA(W)-8 b(e)31 b(denote)h(b)m(y)h Fy(d)789 4070 y Fx(R)884 4055 y FA(the)f(di\013eren)m(tial)g(of)f(this)h(complex.)45 b(F)-8 b(or)30 b(more)i(information,)f(w)m(e)i(address)24 4171 y(the)g(reader)g(to)f([27].)43 b(In)33 b(the)g(next)h(prop)s (osition,)f(w)m(e)g(will)h(also)e(use)i(the)f(follo)m(wing)g(notation) 283 4394 y Fy(A)28 b FA(=)487 4254 y Fr(\022)602 4335 y Fy(a)653 4350 y Fs(11)811 4335 y Fy(a)862 4350 y Fs(12)602 4451 y Fy(a)653 4466 y Fs(21)811 4451 y Fy(a)862 4466 y Fs(22)979 4254 y Fr(\023)1069 4394 y Fy(;)114 b(b)28 b FA(=)1383 4254 y Fr(\022)1497 4335 y Fy(b)1538 4350 y Fs(1)1497 4451 y Fy(b)1538 4466 y Fs(2)1620 4254 y Fr(\023)1710 4394 y Fy(;)114 b(A)1924 4409 y Fv(j)1988 4394 y FA(=)2092 4254 y Fr(\022)2207 4335 y Fy(a)2258 4350 y Fs(1)p Fv(j)2207 4451 y Fy(a)2258 4466 y Fs(2)p Fv(j)2371 4254 y Fr(\023)2591 4394 y FA(and)130 b Fy(a)28 b FA(=)3061 4254 y Fr(\022)3176 4335 y Fy(a)3227 4350 y Fs(1)3176 4451 y Fy(a)3227 4466 y Fs(2)3308 4254 y Fr(\023)3398 4394 y Fy(:)24 4637 y FD(Prop)s(osition)35 b(5.2.)40 b Fw(L)-5 b(et)33 b Fy(f)38 b FA(=)28 b(\()p Fy(F)1298 4652 y Fs(1)1337 4637 y Fy(;)17 b(f)1429 4652 y Fs(3)1469 4637 y FA(\))27 b(:)h Fy(U)38 b Ft(\032)29 b Fu(H)1876 4601 y Fs(1)1943 4637 y Ft(\000)-16 b(!)27 b Fu(H)2208 4601 y Fs(1)2281 4637 y Fw(b)-5 b(e)32 b(a)h(smo)-5 b(oth)32 b(mapping,)g(wher)-5 b(e)32 b Fy(U)43 b Fw(is)24 4753 y(an)35 b(op)-5 b(en)35 b(neighb)-5 b(ourho)g(o)g(d)34 b(of)h(the)g(origin)g(and)g Fy(F)1842 4768 y Fs(1)1881 4753 y FA(\()p Fy(x)p FA(\))30 b(=)e Fy(Ax)23 b FA(+)g Fy(at)g FA(+)f Fy(b)36 b Fw(and)f Fy(f)2918 4768 y Fs(3)2986 4753 y FA(=)29 b Fy(f)3139 4768 y Fs(3)3178 4753 y FA(\()p Fy(x;)17 b(t)p FA(\))p Fw(.)47 b(Then)24 4870 y Fy(f)f Fw(is)34 b(lo)-5 b(c)g(al)5 b(ly)35 b(Lipschitz)f(if)h(and)f(only)h(if) f(for)h(some)f Fy(\034)39 b Ft(2)28 b Fu(R)35 b Fw(the)g(fol)5 b(lowing)33 b(c)-5 b(onditions)34 b(hold)470 5032 y FA(det)622 4951 y Fr(\000)667 5032 y Fy(A)740 5047 y Fs(1)808 5032 y Fy(a)859 4951 y Fr(\001)932 5032 y FA(=)28 b(det)1188 4951 y Fr(\000)1233 5032 y Fy(A)1306 5047 y Fs(2)1374 5032 y Fy(a)1425 4951 y Fr(\001)1498 5032 y FA(=)g(0)17 b Fy(;)-1671 b FA(\(38\))470 5278 y Fy(f)518 5293 y Fs(3)557 5278 y FA(\()p Fy(x;)17 b(t)p FA(\))28 b(=)f Fy(\034)34 b FA(+)1082 5204 y Fy(x)1137 5219 y Fs(2)1194 5204 y FA(det)1346 5124 y Fr(\000)1391 5204 y Fy(b)28 b(A)1533 5219 y Fs(2)1573 5124 y Fr(\001)1641 5204 y FA(+)22 b Fy(x)1794 5219 y Fs(1)1850 5204 y FA(det)2002 5124 y Fr(\000)2048 5204 y Fy(b)28 b(A)2190 5219 y Fs(1)2230 5124 y Fr(\001)p 1082 5255 1194 4 v 1654 5346 a FA(2)2308 5278 y(+)22 b Fy(t)2458 5108 y Fr( )2546 5204 y FA(det)2698 5124 y Fr(\000)2744 5204 y Fy(b)28 b(a)2864 5124 y Fr(\001)p 2546 5255 364 4 v 2704 5346 a FA(2)2942 5278 y(+)22 b(det)17 b Fy(A)3265 5108 y Fr(!)3377 5278 y Fy(:)-3380 b FA(\(39\))p eop end %%Page: 22 22 TeXDict begin 22 21 bop 24 246 a Fz(22)1315 b(V)-9 b(ALENTINO)25 b(MA)n(GNANI)124 446 y FB(Pr)n(oof.)43 b FA(Since)34 b Fy(f)43 b FA(is)34 b(Lipsc)m(hitz)g(and)f(smo)s(oth,)f(w)m(e)i(ha)m (v)m(e)971 535 y Fr(\032)1087 615 y Fy(X)1168 630 y Fs(1)1208 615 y Fy(f)1256 630 y Fs(3)1323 615 y FA(=)27 b Fy(\013)1488 630 y Fs(1)1556 615 y FA(=)1669 576 y Fs(1)p 1669 592 36 4 v 1669 650 a(2)1731 615 y FA(\()p Fy(f)1817 630 y Fs(1)1856 615 y Fy(X)1937 630 y Fs(1)1977 615 y Fy(f)2025 630 y Fs(2)2086 615 y Ft(\000)c Fy(f)2234 630 y Fs(2)2274 615 y Fy(X)2355 630 y Fs(1)2394 615 y Fy(f)2442 630 y Fs(1)2481 615 y FA(\))1087 732 y Fy(X)1168 747 y Fs(2)1208 732 y Fy(f)1256 747 y Fs(3)1323 732 y FA(=)k Fy(\013)1488 747 y Fs(2)1556 732 y FA(=)1669 693 y Fs(1)p 1669 709 V 1669 767 a(2)1731 732 y FA(\()p Fy(f)1817 747 y Fs(1)1856 732 y Fy(X)1937 747 y Fs(2)1977 732 y Fy(f)2025 747 y Fs(2)2086 732 y Ft(\000)c Fy(f)2234 747 y Fs(2)2274 732 y Fy(X)2355 747 y Fs(2)2394 732 y Fy(f)2442 747 y Fs(1)2481 732 y FA(\))24 675 y(\(40\))24 910 y(ev)m(erywhere)36 b(in)d Fy(U)10 b FA(.)44 b(Since)33 b Fy(d)1097 925 y Fx(R)1194 910 y FA(de\014nes)h(a)f(complex,)h(the)f(Rumin's)g(complex,) i(see)e([27],)g(then)980 1085 y Fy(d)1031 1100 y Fx(R)1111 1005 y Fr(\000\002)1199 1085 y Fy(\013)1261 1100 y Fs(1)1300 1085 y Fy(dx)1406 1100 y Fs(1)1468 1085 y FA(+)22 b Fy(\013)1628 1100 y Fs(2)1667 1085 y Fy(dx)1773 1100 y Fs(2)1813 1005 y Fr(\003)1854 1125 y Fx(R)1919 1005 y Fr(\001)1992 1085 y FA(=)28 b Fy(d)2147 1100 y Fx(R)2211 1005 y Fr(\000)2256 1085 y Fy(d)2307 1100 y Fx(R)2371 1085 y Fy(f)2419 1100 y Fs(3)2459 1005 y Fr(\001)2532 1085 y FA(=)g(0)17 b Fy(:)24 1259 y FA(T)-8 b(aking)33 b(in)m(to)g(accoun)m(t)g(that)264 1433 y Fy(d)315 1448 y Fx(R)380 1352 y Fr(\002)421 1433 y Fy(\013)483 1448 y Fs(1)523 1433 y Fy(dx)629 1448 y Fs(1)690 1433 y FA(+)22 b Fy(\013)850 1448 y Fs(2)890 1433 y Fy(dx)996 1448 y Fs(2)1035 1352 y Fr(\003)1077 1472 y Fx(R)1224 1433 y FA(=)83 b Fy(d)1434 1352 y Fr(\000)1479 1433 y Fy(\013)1541 1448 y Fs(1)1581 1433 y Fy(dx)1687 1448 y Fs(1)1749 1433 y FA(+)22 b Fy(\013)1909 1448 y Fs(2)1948 1433 y Fy(dx)2054 1448 y Fs(2)2116 1433 y FA(+)g Fy(\013)2276 1448 y Fs(3)2315 1433 y Fy(\022)2363 1352 y Fr(\001)1224 1594 y FA(=)1383 1513 y Fr(\000)1429 1594 y Fy(X)1510 1609 y Fs(1)1549 1594 y Fy(\013)1611 1609 y Fs(3)1673 1594 y Ft(\000)g Fy(X)1853 1609 y Fs(3)1893 1594 y Fy(\013)1955 1609 y Fs(1)1994 1594 y FA(\))17 b Fy(dx)2155 1609 y Fs(1)2217 1594 y Ft(^)22 b Fy(\022)j FA(+)2473 1513 y Fr(\000)2519 1594 y Fy(X)2600 1609 y Fs(2)2639 1594 y Fy(\013)2701 1609 y Fs(3)2763 1594 y Ft(\000)e Fy(X)2944 1609 y Fs(3)2983 1594 y Fy(\013)3045 1609 y Fs(2)3085 1594 y FA(\))17 b Fy(dx)3246 1609 y Fs(2)3307 1594 y Ft(^)22 b Fy(\022)24 1779 y FA(where)34 b Fy(\013)368 1794 y Fs(3)435 1779 y FA(=)28 b Fy(X)620 1794 y Fs(1)659 1779 y Fy(\013)721 1794 y Fs(2)783 1779 y Ft(\000)22 b Fy(X)963 1794 y Fs(2)1003 1779 y Fy(\013)1065 1794 y Fs(1)1137 1779 y FA(and)32 b Fy(\022)f FA(=)d Fy(dt)22 b FA(+)1712 1699 y Fr(\000)1757 1779 y Fy(x)1812 1794 y Fs(2)1852 1779 y Fy(dx)1958 1743 y Fs(1)2020 1779 y Ft(\000)h Fy(x)2175 1794 y Fs(1)2214 1779 y Fy(dx)2320 1743 y Fs(2)2360 1699 y Fr(\001)2406 1779 y Fy(=)p FA(2.)43 b(Then)33 b(the)g(system)1334 1884 y Fr(\032)1450 1965 y Fy(X)1531 1980 y Fs(1)1571 1965 y Fy(\013)1633 1980 y Fs(3)1694 1965 y Ft(\000)23 b Fy(@)1845 1980 y Fv(t)1875 1965 y Fy(\013)1937 1980 y Fs(1)2004 1965 y FA(=)28 b(0)1450 2081 y Fy(X)1531 2096 y Fs(2)1571 2081 y Fy(\013)1633 2096 y Fs(3)1694 2081 y Ft(\000)23 b Fy(@)1845 2096 y Fv(t)1875 2081 y Fy(\013)1937 2096 y Fs(2)2004 2081 y FA(=)28 b(0)24 2024 y(\(41\))24 2254 y(m)m(ust)34 b(hold.)43 b(One)33 b(can)g(c)m(hec)m(k)i(that)953 2430 y Fy(X)1034 2445 y Fs(1)1074 2430 y Fy(f)1122 2445 y Fs(2)1189 2430 y FA(=)27 b Fy(a)1343 2445 y Fs(21)1440 2430 y Ft(\000)1550 2362 y Fy(a)1601 2377 y Fs(2)1641 2362 y Fy(x)1696 2377 y Fs(2)p 1550 2407 186 4 v 1618 2498 a FA(2)2038 2430 y Fy(X)2119 2445 y Fs(1)2159 2430 y Fy(f)2207 2445 y Fs(1)2274 2430 y FA(=)g Fy(a)2428 2445 y Fs(11)2525 2430 y Ft(\000)2635 2362 y Fy(a)2686 2377 y Fs(1)2726 2362 y Fy(x)2781 2377 y Fs(2)p 2635 2407 V 2703 2498 a FA(2)953 2646 y Fy(X)1034 2661 y Fs(2)1074 2646 y Fy(f)1122 2661 y Fs(1)1189 2646 y FA(=)g Fy(a)1343 2661 y Fs(12)1440 2646 y FA(+)1548 2578 y Fy(a)1599 2593 y Fs(1)1639 2578 y Fy(x)1694 2593 y Fs(1)p 1548 2623 V 1617 2714 a FA(2)2037 2646 y Fy(X)2118 2661 y Fs(2)2157 2646 y Fy(f)2205 2661 y Fs(2)2272 2646 y FA(=)h Fy(a)2427 2661 y Fs(22)2524 2646 y FA(+)2632 2578 y Fy(a)2683 2593 y Fs(2)2722 2578 y Fy(x)2777 2593 y Fs(1)p 2632 2623 V 2700 2714 a FA(2)2844 2646 y Fy(:)24 2849 y FA(Th)m(us,)34 b(a)f(direct)g(calculation)g(sho)m (ws)h(that)411 3023 y(2)p Fy(\013)522 3038 y Fs(1)644 3023 y FA(=)83 b Fy(f)851 3038 y Fs(1)891 3023 y Fy(X)972 3038 y Fs(1)1011 3023 y Fy(f)1059 3038 y Fs(2)1121 3023 y Ft(\000)22 b Fy(f)1268 3038 y Fs(2)1308 3023 y Fy(X)1389 3038 y Fs(1)1428 3023 y Fy(f)1476 3038 y Fs(1)644 3254 y FA(=)803 3083 y Fr( )892 3180 y FA(det)1044 3099 y Fr(\000)1090 3180 y Fy(a)28 b(b)1210 3099 y Fr(\001)p 892 3231 364 4 v 1050 3322 a FA(2)1288 3254 y Ft(\000)23 b FA(det\()p Fy(A)p FA(\))1672 3083 y Fr(!)1768 3254 y Fy(x)1823 3269 y Fs(2)1885 3254 y FA(+)1993 3180 y(det)2145 3099 y Fr(\000)2190 3180 y Fy(a)28 b(A)2342 3195 y Fs(1)2382 3099 y Fr(\001)p 1993 3231 435 4 v 2186 3322 a FA(2)2437 3254 y Fy(x)2492 3269 y Fs(1)2532 3254 y Fy(x)2587 3269 y Fs(2)2649 3254 y FA(+)2757 3180 y(det)2909 3099 y Fr(\000)2955 3180 y Fy(a)g(A)3107 3195 y Fs(2)3146 3099 y Fr(\001)p 2757 3231 V 2950 3322 a FA(2)3202 3254 y Fy(x)3257 3213 y Fs(2)3257 3278 y(2)644 3500 y FA(+)83 b(det)955 3420 y Fr(\000)1001 3500 y Fy(a)28 b(A)1153 3515 y Fs(1)1192 3420 y Fr(\001)1238 3500 y Fy(t)22 b FA(+)g(det)1546 3420 y Fr(\000)1591 3500 y Fy(b)28 b(A)1733 3515 y Fs(1)1773 3420 y Fr(\001)24 3674 y FA(and)33 b(analogously)412 3847 y(2)p Fy(\013)523 3862 y Fs(2)645 3847 y FA(=)83 b Fy(f)852 3862 y Fs(1)892 3847 y Fy(X)973 3862 y Fs(2)1012 3847 y Fy(f)1060 3862 y Fs(2)1122 3847 y Ft(\000)22 b Fy(f)1269 3862 y Fs(2)1309 3847 y Fy(X)1390 3862 y Fs(2)1429 3847 y Fy(f)1477 3862 y Fs(1)645 4079 y FA(=)804 3908 y Fr( )893 4005 y FA(det)1045 3924 y Fr(\000)1091 4005 y Fy(b)28 b(a)1211 3924 y Fr(\001)p 893 4056 364 4 v 1050 4147 a FA(2)1289 4079 y(+)22 b(det\()p Fy(A)p FA(\))1671 3908 y Fr(!)1767 4079 y Fy(x)1822 4094 y Fs(1)1884 4079 y FA(+)1992 4005 y(det)2144 3924 y Fr(\000)2189 4005 y Fy(A)2262 4020 y Fs(2)2330 4005 y Fy(a)2381 3924 y Fr(\001)p 1992 4056 435 4 v 2185 4147 a FA(2)2437 4079 y Fy(x)2492 4094 y Fs(1)2531 4079 y Fy(x)2586 4094 y Fs(2)2648 4079 y FA(+)2756 4005 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2621 5345 364 4 v 2778 5436 a FA(2)3011 5368 y Fy(:)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 169 246 a Fz(CONT)-6 b(A)n(CT)34 b(EQUA)-6 b(TIONS,)31 b(LIPSCHITZ)i(EXTENSIONS)f(AND)g(ISOPERIMETRIC)h (INEQUALITIES)67 b(23)24 446 y FA(In)33 b(view)h(of)39 b(\(41\))o(,)33 b(w)m(e)h(get)24 611 y(\(42\))1044 b(det)1394 530 y Fr(\000)1439 611 y Fy(A)1512 626 y Fs(1)1580 611 y Fy(a)1631 530 y Fr(\001)1704 611 y FA(=)28 b(det)1960 530 y Fr(\000)2006 611 y Fy(A)2079 626 y Fs(2)2146 611 y Fy(a)2197 530 y Fr(\001)2270 611 y FA(=)g(0)17 b Fy(:)24 774 y FA(Notice)34 b(that)f(b)m(y)h(Rumin)f(complex,)i(this)f(is)g(a)f (necessary)i(condition)f(in)f(order)g(that)g(the)h(system)24 890 y(\(40\))43 b(admits)h(solutions.)78 b(In)44 b(order)g(to)f(solv)m (e)j(the)e(system)h(\(40\),)h(taking)e(in)m(to)g(accoun)m(t)g(\(42\),) 24 1006 y(direct)33 b(computations)h(yield)679 1081 y Fr(8)679 1170 y(<)679 1350 y(:)809 1195 y FA(2\()p Fy(f)944 1210 y Fs(3)1005 1195 y Ft(\016)22 b Fy(c)1119 1210 y Fv(p;X)1233 1219 y Fp(1)1271 1195 y FA(\))1309 1159 y Fx(0)1332 1195 y FA(\()p Fy(s)p FA(\))28 b(=)1585 1085 y Fr(\020)1655 1148 y Fs(det)q(\()p Fv(a)20 b(b)p Fs(\))p 1655 1172 241 4 v 1757 1230 a(2)1927 1195 y Ft(\000)i FA(det)c Fy(A)2252 1085 y Fr(\021)2328 1195 y Fy(p)2377 1210 y Fs(2)2438 1195 y FA(+)k(det)2688 1115 y Fr(\000)2734 1195 y Fy(b)28 b(A)2876 1210 y Fs(1)2916 1115 y Fr(\001)809 1375 y FA(2\()p Fy(f)944 1390 y Fs(3)1005 1375 y Ft(\016)22 b Fy(c)1119 1390 y Fv(p;X)1233 1399 y Fp(2)1271 1375 y FA(\))1309 1339 y Fx(0)1332 1375 y FA(\()p Fy(s)p FA(\))28 b(=)1585 1264 y Fr(\020)1655 1327 y Fs(det)q(\()p Fv(b)20 b(a)p Fs(\))p 1655 1352 V 1757 1409 a(2)1927 1375 y Ft(\000)i FA(det)c Fy(A)2252 1264 y Fr(\021)2328 1375 y Fy(p)2377 1390 y Fs(1)2438 1375 y FA(+)k(det)2688 1294 y Fr(\000)2734 1375 y Fy(b)28 b(A)2876 1390 y Fs(2)2916 1294 y Fr(\001)24 1569 y FA(where)34 b Fy(c)348 1584 y Fv(p;X)462 1594 y Fq(j)498 1569 y FA(\()p Fy(s)p FA(\))27 b(=)h Fy(p)17 b FA(exp\()p Fy(sX)1130 1584 y Fv(j)1167 1569 y FA(\).)43 b(Set)33 b Fy(p)28 b FA(=)f(\(0)p Fy(;)17 b(p)1803 1584 y Fs(2)1842 1569 y Fy(;)g(p)1935 1584 y Fs(3)1974 1569 y FA(\),)33 b(hence)150 1804 y(2)p Fy(f)247 1819 y Fs(3)303 1693 y Fr(\020)362 1804 y Fy(p)411 1819 y Fs(1)451 1804 y Fy(;)17 b(p)544 1819 y Fs(2)583 1804 y Fy(;)g(p)676 1819 y Fs(3)737 1804 y Ft(\000)846 1736 y Fy(p)895 1751 y Fs(1)935 1736 y Fy(p)984 1751 y Fs(2)p 846 1781 177 4 v 911 1872 a FA(2)1033 1693 y Fr(\021)1120 1804 y FA(=)28 b(2)p Fy(f)1321 1819 y Fs(3)1360 1804 y FA(\(0)p Fy(;)17 b(p)1540 1819 y Fs(2)1579 1804 y Fy(;)g(p)1672 1819 y Fs(3)1711 1804 y FA(\))22 b(+)g Fy(p)1918 1819 y Fs(1)1958 1804 y Fy(p)2007 1819 y Fs(2)2063 1663 y Fr(\022)2146 1736 y FA(det\()p Fy(a)28 b(b)p FA(\))p 2146 1781 332 4 v 2287 1872 a(2)2510 1804 y Ft(\000)22 b FA(det)17 b Fy(A)2834 1663 y Fr(\023)2930 1804 y FA(+)22 b Fy(p)3077 1819 y Fs(1)3133 1804 y FA(det)3285 1723 y Fr(\000)3331 1804 y Fy(b)28 b(A)3473 1819 y Fs(1)3513 1723 y Fr(\001)24 2038 y FA(and)33 b Fy(f)11 b FA(\(0)p Fy(;)17 b(p)453 2053 y Fs(2)491 2038 y Fy(;)g(p)584 2053 y Fs(3)623 2038 y FA(\))28 b(=)f Fy(f)840 2053 y Fs(3)880 2038 y FA(\(0)p Fy(;)17 b FA(0)p Fy(;)g(p)1153 2053 y Fs(3)1191 2038 y FA(\))22 b(+)g Fy(p)1398 2053 y Fs(2)1454 2038 y FA(det)1606 1958 y Fr(\000)1652 2038 y Fy(b)28 b(A)1794 2053 y Fs(2)1834 1958 y Fr(\001)1879 2038 y FA(.)44 b(It)33 b(follo)m(ws)g(that)89 2303 y Fy(f)137 2318 y Fs(3)177 2303 y FA(\()p Fy(x)p FA(\))28 b(=)f Fy(g)506 2192 y Fr(\020)566 2303 y Fy(t)22 b FA(+)731 2236 y Fy(x)786 2251 y Fs(1)826 2236 y Fy(x)881 2251 y Fs(2)p 731 2280 190 4 v 801 2371 a FA(2)931 2192 y Fr(\021)1012 2303 y FA(+)1120 2229 y Fy(x)1175 2244 y Fs(2)1232 2229 y FA(det)1384 2149 y Fr(\000)1430 2229 y Fy(b)28 b(A)1572 2244 y Fs(2)1611 2149 y Fr(\001)1679 2229 y FA(+)22 b Fy(x)1832 2244 y Fs(1)1889 2229 y FA(det)2041 2149 y Fr(\000)2086 2229 y Fy(b)28 b(A)2228 2244 y Fs(1)2268 2149 y Fr(\001)p 1120 2280 1194 4 v 1693 2371 a FA(2)2346 2303 y(+)2454 2236 y Fy(x)2509 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1714 2789 364 4 v 1872 2880 a FA(2)2110 2812 y(+)22 b(det)17 b Fy(A)2433 2642 y Fr(!)2535 2812 y FA(+)2643 2738 y(det)2795 2658 y Fr(\000)2840 2738 y Fy(b)28 b(A)2982 2753 y Fs(2)3022 2658 y Fr(\001)p 2643 2789 426 4 v 2831 2880 a FA(2)24 3061 y(and)33 b(our)f(form)m(ula) h(for)f Fy(f)942 3076 y Fs(3)1014 3061 y FA(giv)m(es)459 3310 y Fy(X)540 3325 y Fs(2)579 3310 y Fy(f)627 3325 y Fs(3)694 3310 y FA(=)c Fy(x)853 3325 y Fs(1)909 3310 y Fy(g)960 3269 y Fx(0)1000 3199 y Fr(\020)1059 3310 y Fy(t)22 b FA(+)1224 3243 y Fy(x)1279 3258 y Fs(1)1319 3243 y Fy(x)1374 3258 y Fs(2)p 1224 3287 190 4 v 1295 3378 a FA(2)1424 3199 y Fr(\021)1506 3310 y FA(+)1614 3236 y(det)1766 3155 y Fr(\000)1812 3236 y Fy(b)28 b(A)1954 3251 y Fs(2)1993 3155 y Fr(\001)p 1614 3287 426 4 v 1802 3378 a FA(2)2071 3310 y Ft(\000)2181 3243 y Fy(x)2236 3258 y Fs(1)p 2181 3287 95 4 v 2204 3378 a FA(2)2302 3140 y Fr( )2391 3236 y FA(det)2543 3155 y Fr(\000)2589 3236 y Fy(b)g(a)2709 3155 y Fr(\001)p 2391 3287 364 4 v 2548 3378 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y Fs(\(\012\))1370 5401 y FA(+)22 b(Lip)q(\()p Fy(f)11 b FA(\))1746 5320 y Fr(\001)1807 5401 y Fy(d)1858 5320 y Fr(\000)1904 5401 y Fy(c)1946 5416 y Fv(p;X)2068 5401 y FA(\()p Fy(t)p FA(\))p Fy(;)17 b(c)2265 5416 y Fv(p;X)2388 5401 y FA(\()p Fy(\034)11 b FA(\))2517 5320 y Fr(\001)2579 5401 y Fy(:)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 169 246 a Fz(CONT)-6 b(A)n(CT)34 b(EQUA)-6 b(TIONS,)31 b(LIPSCHITZ)i(EXTENSIONS)f(AND)g(ISOPERIMETRIC)h (INEQUALITIES)67 b(25)24 447 y FA(The)35 b(same)g(estimate)g(is)g (obtained)f(in)g(the)h(analogous)f(case)h Fy(\034)41 b Ft(2)31 b Fy(I)2534 462 y Fv(j)2605 447 y FA(and)j Fy(t)42 b(=)-61 b Ft(2)31 b Fy(c)3000 405 y Fx(\000)p Fs(1)3000 474 y Fv(p;X)3122 447 y FA(\(\012\).)48 b(If)34 b Fy(t)d Ft(2)f Fy(I)3647 462 y Fv(j)24 564 y FA(and)j Fy(\034)39 b Ft(2)28 b Fy(I)432 579 y Fv(k)507 564 y FA(with)33 b Fy(j)h Ft(6)p FA(=)28 b Fy(k)s FA(,)k(with)h(analogous)g (argumen)m(t)g(w)m(e)h(get)179 730 y Fy(\032)p Fi(\()296 704 y FA(~)274 730 y Fy(f)333 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y FA(\()p Fy(\034)11 b FA(\))51 b Ft(2)h Fy(E)6 b FA(,)49 b(where)f Fy(f)56 b FA(is)24 1026 y(Lipsc)m(hitz.)45 b(W)-8 b(e)33 b(ha)m(v)m(e)h(sho)m(wn)g(that)f(for)f(ev)m(ery)i Fy(r)m(;)17 b(r)1917 990 y Fx(0)1968 1026 y Ft(2)28 b Fy(c)2104 1041 y Fv(p;X)2226 1026 y FA(\()p Fu(R)p FA(\),)33 b(w)m(e)g(ha)m(v)m(e)626 1200 y Fy(\032)676 1119 y Fr(\000)744 1174 y FA(~)722 1200 y Fy(f)11 b FA(\()p Fy(r)s FA(\))p Fy(;)969 1174 y FA(~)948 1200 y Fy(f)f FA(\()p Fy(r)1091 1159 y Fx(0)1114 1200 y FA(\))1152 1119 y Fr(\001)1225 1200 y Ft(\024)1330 1119 y Fr(\000)1376 1200 y FA(2)p Fy(C)1502 1159 y Fx(0)1542 1200 y Ft(kr)1675 1215 y Fv(H)1742 1200 y Fy(G)1819 1215 y Fs(1)1858 1200 y Ft(k)1908 1216 y Fv(L)1956 1197 y Fj(1)2021 1216 y Fs(\(\012\))2153 1200 y FA(+)22 b(Lip\()p Fy(f)11 b FA(\))2528 1119 y Fr(\001)2590 1200 y Fy(d)p FA(\()p Fy(r)m(;)17 b(r)2811 1159 y Fx(0)2834 1200 y FA(\))g Fy(:)-2892 b FA(\(47\))24 1362 y(Finally)-8 b(,)31 b(w)m(e)h(adopt)e(the)h(same)g(argumen)m(t)h 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FA(with)g(\014rst)g (la)m(y)m(er)g Fo(v)24 2446 y FA(and)d(second)h(la)m(y)m(er)g Fo(z)p FA(.)42 b(W)-8 b(e)30 b(select)g(a)f(scalar)g(pro)s(duct)g(on)g Fo(n)h FA(suc)m(h)g(that)f Fo(v)f FA(and)i Fo(z)e FA(are)h (orthonormal.)24 2623 y FD(De\014nition)38 b(6.1.)k FA(Let)33 b Fy(a;)17 b(b)28 b FA(:)g([0)p Fy(;)17 b FA(1])27 b Ft(\000)-16 b(!)27 b Fo(v)32 b FA(b)s(e)h(Lipsc)m(hitz)i(lo)s(ops.)43 b(W)-8 b(e)33 b(sa)m(y)h(that)e(\000)c(:)f([0)p Fy(;)17 b FA(1])3378 2587 y Fs(2)3445 2623 y Ft(\000)-16 b(!)27 b Fo(v)24 2739 y FA(is)33 b(an)g Fw(isotr)-5 b(opic)34 b(homotopy)e FA(carrying)h Fy(a)g FA(to)f Fy(b)h FA(if)f(\()p Fy(\034)6 b(;)17 b(t)p FA(\))28 b Ft(\000)-16 b(!)27 b FA(\000\()p Fy(\034)6 b(;)17 b(t)p FA(\))32 b(is)h(Lipsc)m(hitz,)1235 2901 y([)p Fy(@)1313 2916 y Fv(\034)1356 2901 y FA(\000)p Fy(;)17 b(@)1512 2916 y Fv(t)1542 2901 y FA(\000])28 b(=)f(0)98 b(a.e.)43 b(in)33 b([0)p Fy(;)17 b FA(1])2407 2865 y Fs(2)2446 2901 y FA(,)24 3063 y(\000\()p Ft(\001)p Fy(;)g FA(0\))27 b(=)g Fy(a)p FA(,)33 b(\000\()p Ft(\001)p Fy(;)17 b FA(1\))27 b(=)g Fy(b)33 b FA(and)g(\000\(0)p Fy(;)17 b Ft(\001)p FA(\))27 b(=)g(\000\(1)p Fy(;)17 b Ft(\001)p FA(\).)24 3240 y FD(Remark)42 b(6.2.)h FA(F)-8 b(or)36 b(our)g(purp)s(oses,)i(the)e(p)s(oin)m(ts)h(\000\(0)p Fy(;)17 b Ft(\001)p FA(\))35 b(and)h(\000\(1)p Fy(;)17 b Ft(\001)p FA(\))35 b(need)i(not)f(coincide)h(with)24 3356 y(some)c(\014xed)h(p)s(oin)m(t.)124 3534 y(In)f(the)g(sequel,)h(w) m(e)g(will)f(use)h(the)f(follo)m(wing)f(linear)h(space)997 3759 y(Av)1121 3717 y Fx(1)1121 3784 y Fs(0)1224 3759 y FA(=)1327 3619 y Fr(\032)1402 3759 y Fy(\033)f Ft(2)c Fy(L)1649 3718 y Fx(1)1724 3679 y Fr(\000)1769 3759 y FA(]0)p Fy(;)17 b FA(1[)p Fy(;)g Fo(z)2048 3679 y Fr(\001)2121 3645 y(\014)2121 3704 y(\014)2121 3764 y(\014)2198 3624 y(Z)2298 3650 y Fs(1)2254 3849 y(0)2354 3759 y Fy(\033)32 b FA(=)27 b(0)2593 3619 y Fr(\033)2684 3759 y Fy(:)24 3983 y FA(In)37 b(the)f(next)h(de\014nition)h(the)e(same)h(sym)m(b)s (ol)h Ft(j)24 b(\001)g(j)36 b 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Fs(1)1055 2640 y Fy(;)17 b(X)1180 2655 y Fs(2)1219 2640 y FA(])28 b(=)h Fy(Z)7 b FA(.)44 b(Let)34 b Fy(\033)e Ft(2)d FA(Av)2007 2598 y Fx(1)2007 2665 y Fs(0)2114 2640 y FA(and)k Fy(\025)c(>)f FA(0,)33 b(where)h Fy(\033)f FA(=)28 b Fy(\033)3132 2655 y Fs(1)3172 2640 y Fy(Z)7 b FA(.)45 b(Then)34 b(w)m(e)24 2756 y(de\014ne)g(the)f(Lipsc)m (hitz)h(curv)m(e)458 3003 y Fy(a)28 b FA(=)g Fy(a)692 3018 y Fs(1)731 3003 y Fy(X)812 3018 y Fs(1)874 3003 y FA(+)22 b Fy(a)1023 3018 y Fs(2)1063 3003 y Fy(X)1144 3018 y Fs(2)1183 3003 y Fy(;)114 b FA(where)99 b Fy(a)1722 3018 y Fs(1)1789 3003 y Ft(\021)29 b Fy(\025)97 b FA(and)h Fy(a)2355 3018 y Fs(2)2394 3003 y FA(\()p Fy(t)p FA(\))28 b(=)2651 2936 y(1)p 2647 2981 57 4 v 2647 3072 a Fy(\025)2730 2868 y Fr(Z)2830 2894 y Fv(t)2785 3093 y Fs(0)2876 3003 y Fy(\033)2931 3018 y Fs(1)2971 3003 y FA(\()p Fy(s)p FA(\))17 b Fy(ds)g(:)24 3246 y FA(Clearly)34 b([)p Fy(a;)29 b FA(_)-39 b Fy(a)p FA(])28 b(=)f Fy(\033)37 b FA(and)32 b(the)h(corresp)s(onding)h(homotop)m(y)f(is)1197 3432 y(\000\()p Fy(\034)6 b(;)17 b(t)p FA(\))27 b(=)h Fy(\025X)1730 3447 y Fs(1)1791 3432 y FA(+)22 b(\(1)g Ft(\000)h Fy(t)p FA(\))p Fy(a)2222 3447 y Fs(2)2262 3432 y FA(\()p Fy(\034)11 b FA(\))p Fy(X)2472 3447 y Fs(2)24 3617 y FA(is)33 b(clearly)h (isotropic)f(and)g(carries)g Fy(a)g FA(to)g(the)g(p)s(oin)m(t)f Fy(\025X)2093 3632 y Fs(1)2161 3617 y Ft(2)c Fo(h)p FA(.)43 b(W)-8 b(e)34 b(ha)m(v)m(e)f Ft(j)p Fy(a)p FA(\(0\))p Ft(j)28 b FA(=)f Fy(\025)33 b FA(and)f(simple)24 3734 y(calculations)i(yield)1083 3932 y(Lip)q(\(\000\))27 b Ft(\024)1505 3865 y FA(2)17 b Ft(k)p Fy(\033)t Ft(k)1730 3880 y Fv(L)1778 3861 y Fj(1)p 1505 3909 342 4 v 1647 4001 a Fy(\025)1954 3932 y FA(and)98 b Ft(j)12 b FA(_)-39 b Fy(a)p Ft(j)27 b FA(=)2456 3865 y Ft(j)p Fy(\033)t Ft(j)p 2456 3909 115 4 v 2485 4001 a Fy(\025)2597 3932 y(:)24 4133 y FA(The)34 b(v)-5 b(alidit)m(y)33 b(of)39 b(\(49\))32 b(is)h(trivial,)g(since)h Ft(j)p FA(\000)1641 4148 y Fv(t)1693 4133 y Ft(^)22 b FA(\000)1842 4148 y Fv(\034)1885 4133 y Ft(j)28 b FA(=)f(0)32 b(at)h(an)m(y)g(di\013eren)m (tiabilit)m(y)i(p)s(oin)m(t)d(of)h(\000.)24 4324 y FD(Remark)50 b(6.7.)d FA(The)d(previous)h(example)f(sho)m(ws)h(that)e(Heisen)m(b)s (erg)i(groups)e Fu(H)3106 4287 y Fv(n)3197 4324 y FA(are)g(Allco)s(c)m (k)24 4440 y(groups,)49 b(since)f Fu(H)712 4404 y Fv(n)809 4440 y FA(=)i Fu(A)p Fo(l)1035 4404 y Fv(n)1035 4466 y Fn(h)1082 4440 y FA(,)f(where)e Fo(h)f FA(is)g(the)g(3-dimensional)h (Heisen)m(b)s(erg)g(algebra.)83 b(Let)46 b(us)24 4556 y(men)m(tion)i(that)g(for)f(Heisen)m(b)s(erg)i(groups)e(Lipsc)m(hitz)j (extensions)f(from)f(the)f(Euclidean)i(plane)24 4672 y(could)29 b(b)s(e)g(also)f(treated)h(b)m(y)g(a)g(di\013eren)m(t)g (metho)s(d,)h(according)f(to)f(F\177)-49 b(assler)29 b(Master's)h(thesis,)h(2007.)24 4863 y FD(Remark)44 b(6.8.)h FA(Arguing)38 b(as)g(in)h(the)f(previous)i(example,)h(one)d(can)h (\014nd)f(sev)m(eral)i(examples)g(of)24 4979 y(2-step)25 b(strati\014ed)h(algebras)f Fo(n)g FA(with)g(one)g(co)s(dimensional)h (horizon)m(tal)f(distribution)h(that)f(suitably)24 5096 y(yield)34 b(Allco)s(c)m(k)g(groups.)124 5286 y(On)26 b(the)g(other)h(hand,)g(it)f(is)h(easy)g(to)f(\014nd)h(Allco)s(c)m(k)g (groups)f(where)i(the)e(horizon)m(tal)h(distribution)24 5403 y(has)33 b(co)s(dimension)h(higher)f(than)g(one,)g(as)f(in)h(the)g (follo)m(wing)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 169 246 a Fz(CONT)-6 b(A)n(CT)34 b(EQUA)-6 b(TIONS,)31 b(LIPSCHITZ)i(EXTENSIONS)f(AND)g(ISOPERIMETRIC)h (INEQUALITIES)67 b(27)24 446 y FD(Example)29 b(6.9.)36 b FA(Let)25 b(us)g(consider)h(the)f(2-step)f(algebra)h Fo(k)2155 461 y Fv(s)2219 446 y FA(=)j Fo(v)6 b Ft(\010)g Fo(z)p FA(,)25 b(where)h Fo(v)i FA(=)f Ft(f)p Fy(X)3141 461 y Fs(1)3181 446 y Fy(;)17 b(:)g(:)g(:)e(;)i(X)3480 461 y Fv(s)p Fs(+1)3607 446 y Ft(g)p FA(,)24 562 y Fo(z)28 b FA(=)f(span)q Ft(f)p Fy(Z)507 577 y Fs(1)546 562 y Fy(;)17 b(:)g(:)g(:)f(;)h(Z)832 577 y Fv(s)868 562 y Ft(g)32 b FA(and)h(the)g(only)g(non)m(trivial)g(brac)m(k)m(et)h (relations)g(are)974 737 y([)p Fy(X)1082 752 y Fs(1)1122 737 y Fy(;)17 b(X)1247 752 y Fv(j)1283 737 y FA(])28 b(=)f Fy(Z)1508 752 y Fv(j)t Fx(\000)p Fs(1)1635 737 y Fy(;)114 b FA(and)98 b Fy(j)33 b FA(=)28 b(2)p Fy(;)17 b(:)g(:)g(:)e(;)i(s)22 b FA(+)g(1)17 b Fy(:)24 911 y FA(Let)33 b Fy(\033)e Ft(2)e FA(Av)504 869 y Fx(1)504 936 y Fs(0)611 911 y FA(and)k Fy(\025)28 b(>)f FA(0,)33 b(where)g Fy(\033)f FA(=)1570 837 y Fr(P)1675 863 y Fv(s)1675 940 y(k)r Fs(=1)1824 911 y Fy(\033)1879 926 y Fv(k)1922 911 y Fy(Z)1989 926 y Fv(k)2032 911 y FA(.)43 b(Then)34 b(w)m(e)g(de\014ne)f(the)g(Lipsc)m(hitz)i(curv)m(e)526 1184 y Fy(a)28 b FA(=)719 1060 y Fv(s)p Fs(+1)709 1090 y Fr(X)719 1300 y Fv(j)t Fs(=1)869 1184 y Fy(a)920 1199 y Fv(j)957 1184 y Fy(X)1038 1199 y Fv(j)1074 1184 y Fy(;)115 b FA(where)99 b Fy(a)1614 1199 y Fs(1)1681 1184 y Ft(\021)28 b Fy(\025)97 b FA(and)h Fy(a)2246 1199 y Fv(j)2283 1184 y FA(\()p Fy(t)p FA(\))28 b(=)2539 1117 y(1)p 2535 1161 57 4 v 2535 1253 a Fy(\025)2619 1049 y Fr(Z)2718 1075 y Fv(t)2674 1274 y Fs(0)2765 1184 y Fy(\033)2820 1199 y Fv(j)t Fx(\000)p Fs(1)2947 1184 y FA(\()p Fy(s)p FA(\))17 b Fy(ds)24 1468 y FA(for)33 b(ev)m(ery)i Fy(j)g FA(=)29 b(2)p Fy(;)17 b(:)g(:)g(:)f(;)h(s)22 b FA(+)g(1.)46 b(Clearly)35 b([)p Fy(a;)29 b FA(_)-39 b Fy(a)p FA(])29 b(=)g Fy(\033)t FA(,)34 b Ft(j)p Fy(a)p FA(\(0\))p Ft(j)28 b FA(=)h Fy(\025)k FA(and)45 b(_)-39 b Fy(a)30 b FA(=)f Fy(\025)2850 1432 y Fx(\000)p Fs(1)2960 1468 y Fy(\033)t FA(.)46 b(Moreo)m(v)m(er,)36 b(the)24 1584 y(mapping)1103 1807 y(\000\()p Fy(\034)6 b(;)17 b(t)p FA(\))27 b(=)h Fy(\025X)1636 1822 y Fs(1)1697 1807 y FA(+)22 b(\(1)g Ft(\000)h Fy(t)p FA(\))2104 1683 y Fv(s)p Fs(+1)2094 1713 y Fr(X)2104 1923 y Fv(j)t Fs(=2)2254 1807 y Fy(a)2305 1822 y Fv(j)2342 1807 y FA(\()p Fy(\034)11 b FA(\))17 b Fy(X)2569 1822 y Fv(j)24 2061 y FA(is)33 b(an)g(isotropic)g(homotop)m(y)g(carrying)g Fy(a)g FA(to)f Fy(\025X)1832 2076 y Fs(1)1899 2061 y Ft(2)c Fo(v)33 b FA(and)f(a)h(simple)h(computation)f(yields)1381 2305 y(Lip\(\000\))28 b Ft(\024)1803 2155 y(p)p 1886 2155 95 4 v 82 x FA(2)p Fy(s)p 1803 2282 178 4 v 1863 2373 a(\025)2007 2305 y Ft(k)p Fy(\033)t Ft(k)2166 2320 y Fv(L)2214 2301 y Fj(1)2300 2305 y Fy(:)24 2515 y FA(Finally)-8 b(,)33 b(to)f(pro)m(v)m(e)i(\(49\))e(w)m(e)i(observ)m(e)g(that)e(there) i(exists)g Fy(\014)2239 2530 y Fs(0)2306 2515 y Fy(>)28 b FA(0)k(suc)m(h)i(that)632 2648 y Fr(Z)688 2874 y Fs([0)p Fv(;)p Fs(1])818 2855 y Fp(2)872 2784 y Ft(j)p FA(\000)961 2799 y Fv(t)1013 2784 y Ft(^)22 b FA(\000)1162 2799 y Fv(\034)1205 2784 y Ft(j)17 b Fy(d\034)11 b(dt)27 b Ft(\024)i Fy(\014)1628 2799 y Fs(0)1700 2648 y Fr(Z)1756 2874 y Fs([0)p Fv(;)p Fs(1])1886 2855 y Fp(2)1940 2639 y Fr(\014)1940 2699 y(\014)1940 2759 y(\014)1940 2819 y(\014)2001 2659 y Fv(s)p Fs(+1)1990 2689 y Fr(X)2001 2899 y Fv(j)t Fs(=2)2151 2784 y Fy(a)2202 2799 y Fv(j)2238 2784 y Fy(X)2319 2799 y Fv(j)2356 2639 y Fr(\014)2356 2699 y(\014)2356 2759 y(\014)2356 2819 y(\014)2406 2639 y(\014)2406 2699 y(\014)2406 2759 y(\014)2406 2819 y(\014)2466 2659 y Fv(s)p Fs(+1)2455 2689 y Fr(X)2466 2899 y Fv(j)t Fs(=2)2628 2784 y FA(_)-39 b Fy(a)2667 2799 y Fv(j)2704 2784 y Fy(X)2785 2799 y Fv(j)2821 2639 y Fr(\014)2821 2699 y(\014)2821 2759 y(\014)2821 2819 y(\014)2871 2784 y Fy(d\034)11 b(dt)632 3136 y FA(=)28 b Fy(\014)791 3151 y Fs(0)863 3001 y Fr(Z)919 3226 y Fs([0)p Fv(;)p Fs(1])1049 3208 y Fp(2)1104 2992 y Fr(\014)1104 3052 y(\014)1104 3112 y(\014)1104 3171 y(\014)1153 3001 y(Z)1253 3027 y Fv(t)1209 3226 y Fs(0)1310 3012 y Fv(s)p Fs(+1)1299 3042 y Fr(X)1310 3252 y Fv(j)t Fs(=2)1472 3136 y FA(_)-39 b Fy(a)1511 3151 y Fv(j)1547 3136 y Fy(X)1628 3151 y Fv(j)1665 2992 y Fr(\014)1665 3052 y(\014)1665 3112 y(\014)1665 3171 y(\014)1715 2992 y(\014)1715 3052 y(\014)1715 3112 y(\014)1715 3171 y(\014)1775 3012 y Fv(s)p Fs(+1)1765 3042 y Fr(X)1775 3252 y Fv(j)t Fs(=2)1937 3136 y FA(_)g Fy(a)1976 3151 y Fv(j)2013 3136 y Fy(X)2094 3151 y Fv(j)2130 2992 y Fr(\014)2130 3052 y(\014)2130 3112 y(\014)2130 3171 y(\014)2180 3136 y Fy(d\034)11 b(dt)28 b Ft(\024)g Fy(\014)2558 3151 y Fs(0)2631 2996 y Fr(\022)2704 3001 y(Z)2804 3027 y Fs(1)2759 3226 y(0)2860 3136 y Ft(j)12 b FA(_)-39 b Fy(a)p Ft(j)17 b Fy(dt)3070 2996 y Fr(\023)3142 3011 y Fs(2)3215 3136 y Fy(:)24 3411 y FA(W)-8 b(e)33 b(ha)m(v)m(e)h(pro)m(v)m(ed)g(that)e Fo(k)984 3426 y Fv(s)1053 3411 y FA(is)h(surjectiv)m(e)i(on)e(isotrpic) g(lo)s(ops.)24 3595 y FD(Remark)57 b(6.10.)51 b FA(In)f(view)h(of)e (the)g(previous)i(example,)56 b(w)m(e)50 b(ha)m(v)m(e)h(obtained)f (other)g(Allco)s(c)m(k)24 3711 y(groups,)39 b(corresp)s(onding)g(to)e Fu(A)p Fo(l)1228 3675 y Fv(n)1228 3737 y Fn(k)1255 3745 y Fq(s)1292 3711 y FA(.)59 b(Notice)38 b(that)f(they)i(ha)m(v)m(e)g (horizon)m(tal)f(distribution)g(of)f(co)s(di-)24 3828 y(mension)c Fy(s)p FA(,)e(for)g(ev)m(ery)i(in)m(teger)g Fy(s)27 b Ft(\025)h FA(1.)43 b(Clearly)-8 b(,)33 b(a)e(2-step)g (algebra)h Fo(n)f FA(of)g Fy(s)p FA(-dimensional)i(second)24 3944 y(la)m(y)m(er,)h(ha)m(ving)f(a)f(subalgebra)i(isomorphic)f(to)g Fo(k)1823 3959 y Fv(s)1892 3944 y FA(is)g(surjectiv)m(e)i(on)d (isotropic)h(lo)s(ops.)24 4128 y FD(Example)27 b(6.11.)34 b FA(W)-8 b(e)23 b(de\014ne)h(the)e(\\m)m(ulti-Heisen)m(b)s(erg)j (algebra")d Ft(M)p Fo(h)2671 4092 y Fv(s)2707 4128 y FA(,)j(as)e(the)f(2-step)h(strati\014ed)24 4245 y(algebra,)50 b(where)e(\014rst)f(and)f(second)i(la)m(y)m(ers)g(are)f(spanned)h(b)m (y)f(the)g(bases)h(\()p Fy(X)3032 4260 y Fs(1)3071 4245 y Fy(;)17 b(:)g(:)g(:)f(;)h(X)3371 4260 y Fs(2)p Fv(s)3442 4245 y FA(\))47 b(and)24 4361 y(\()p Fy(Z)129 4376 y Fs(1)168 4361 y Fy(;)17 b(:)g(:)g(:)f(;)h(Z)454 4376 y Fv(s)490 4361 y FA(\),)33 b(resp)s(ectiv)m(ely)-8 b(,)35 b(and)e(the)g(only)g(non)m(trivial)h(brac)m(k)m(ets)g(are)1016 4535 y([)p Fy(X)1124 4550 y Fv(j)1160 4535 y Fy(;)17 b(X)1285 4550 y Fv(s)p Fs(+)p Fv(j)1409 4535 y FA(])28 b(=)f Fy(Z)1634 4550 y Fv(j)1768 4535 y FA(for)32 b(ev)m(ery)j Fy(j)e FA(=)28 b(1)p Fy(;)17 b(:)g(:)g(:)e(;)i(s:)24 4710 y FA(Let)33 b Fy(\033)e Ft(2)e FA(Av)504 4668 y Fx(1)504 4735 y Fs(0)579 4710 y FA(,)j(with)i Fy(\033)d FA(=)1051 4635 y Fr(P)1156 4662 y Fv(s)1156 4739 y(j)t Fs(=1)1299 4710 y Fy(\033)1354 4725 y Fv(j)1408 4710 y Fy(Z)1475 4725 y Fv(j)1544 4710 y FA(and)h(let)h Fy(\025)28 b(>)f FA(0.)44 b(W)-8 b(e)33 b(de\014ne)546 4982 y Fy(a)28 b FA(=)g Fy(\025)858 4857 y Fv(s)802 4887 y Fr(X)813 5097 y Fv(j)t Fs(=1)963 4982 y Fy(X)1044 4997 y Fv(j)1102 4982 y FA(+)1256 4857 y Fv(s)1200 4887 y Fr(X)1211 5097 y Fv(j)t Fs(=1)1361 4982 y Fy(a)1412 4997 y Fv(j)1448 4982 y Fy(X)1529 4997 y Fv(s)p Fs(+)p Fv(j)1653 4982 y Fy(;)147 b FA(where)132 b Fy(a)2258 4997 y Fv(j)2294 4982 y FA(\()p Fy(\034)11 b FA(\))28 b(=)2569 4915 y(1)p 2565 4959 57 4 v 2565 5050 a Fy(\025)2648 4846 y Fr(Z)2748 4873 y Fv(\034)2704 5072 y Fs(0)2808 4982 y Fy(\033)2863 4997 y Fv(j)2900 4982 y FA(\()p Fy(s)p FA(\))p Fy(ds)17 b(:)24 5264 y FA(Then)38 b(the)f(homotop)m(y)h(\000\()p Fy(\034)6 b(;)17 b(t)p FA(\))35 b(=)g Fy(\025)1399 5189 y Fr(P)1504 5215 y Fv(s)1504 5293 y(j)t Fs(=1)1647 5264 y Fy(X)1728 5279 y Fv(j)1789 5264 y FA(+)25 b(\(1)g Ft(\000)h Fy(t)p FA(\))2195 5189 y Fr(P)2300 5215 y Fv(s)2300 5293 y(j)t Fs(=1)2443 5264 y Fy(a)2494 5279 y Fv(j)2531 5264 y FA(\()p Fy(\034)11 b FA(\))p Fy(X)2741 5279 y Fv(s)p Fs(+)p Fv(j)2902 5264 y FA(is)37 b(clearly)h(isotropic)24 5394 y(and)30 b(carries)h Fy(a)f FA(to)f(the)i(p)s(oin)m(t)f Fy(\025)1207 5319 y Fr(P)1311 5345 y Fv(s)1311 5423 y(j)t Fs(=1)1455 5394 y Fy(X)1536 5409 y Fv(j)1572 5394 y FA(.)43 b(Clearly)-8 b(,)31 b Ft(j)p Fy(a)p FA(\(0\))p Ft(j)c FA(=)g Fy(\025)2433 5322 y Ft(p)p 2516 5322 46 4 v 72 x Fy(s)j FA(and)g Ft(j)12 b FA(_)-39 b Fy(a)p Ft(j)27 b FA(=)h Fy(\025)3074 5358 y Fx(\000)p Fs(1)3184 5394 y Ft(j)p Fy(\033)t Ft(j)p Fy(:)i FA(W)-8 b(e)30 b(also)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 24 246 a Fz(28)1315 b(V)-9 b(ALENTINO)25 b(MA)n(GNANI)24 446 y FA(ha)m(v)m(e)34 b(the)f(estimate)811 670 y(Lip)953 589 y Fr(\000)999 670 y FA(\000)1060 589 y Fr(\001)1133 670 y Ft(\024)28 b FA(ess)18 b(sup)1296 757 y Fs([0)p Fv(;)p Fs(1])1426 738 y Fp(2)1539 529 y Fr(\022)1684 545 y Fv(s)1629 575 y Fr(X)1639 785 y Fv(j)t Fs(=1)1789 670 y Fy(a)1840 629 y Fs(2)1840 695 y Fv(j)1902 670 y FA(+)34 b(_)-39 b Fy(a)2051 629 y Fs(2)2051 695 y Fv(j)2091 529 y Fr(\023)2164 552 y Fs(1)p Fv(=)p Fs(2)2302 670 y Ft(\024)2417 520 y(p)p 2500 520 95 4 v 83 x FA(2)p Fy(s)p 2417 647 178 4 v 2477 738 a(\025)2621 670 y Ft(k)p Fy(\033)t Ft(k)2780 685 y Fv(L)2828 666 y Fj(1)24 934 y FA(Finally)-8 b(,)33 b(arguing)f(exactly)i(as)f(in)g(Example)h(6.9,)e (w)m(e)i(get)77 1042 y Fr(Z)132 1268 y Fs([0)p Fv(;)p Fs(1])262 1249 y Fp(2)317 1178 y Ft(j)p FA(\000)406 1193 y Fv(t)457 1178 y Ft(^)23 b FA(\000)607 1193 y Fv(\034)650 1178 y Ft(j)17 b Fy(d\034)11 b(dt)27 b Ft(\024)h Fy(\014)1072 1193 y Fs(0)1145 1042 y Fr(Z)1200 1268 y Fs([0)p Fv(;)p Fs(1])1330 1249 y Fp(2)1385 1033 y Fr(\014)1385 1093 y(\014)1385 1153 y(\014)1385 1213 y(\014)1490 1053 y Fv(s)1435 1083 y Fr(X)1446 1293 y Fv(j)t Fs(=1)1595 1178 y Fy(a)1646 1193 y Fv(j)1683 1178 y Fy(X)1764 1193 y Fv(s)p Fs(+)p Fv(j)1888 1033 y Fr(\014)1888 1093 y(\014)1888 1153 y(\014)1888 1213 y(\014)1938 1033 y(\014)1938 1093 y(\014)1938 1153 y(\014)1938 1213 y(\014)2043 1053 y Fv(s)1988 1083 y Fr(X)1998 1293 y Fv(j)t Fs(=1)2160 1178 y FA(_)-39 b Fy(a)2199 1193 y Fv(j)2236 1178 y Fy(X)2317 1193 y Fv(j)2353 1033 y Fr(\014)2353 1093 y(\014)2353 1153 y(\014)2353 1213 y(\014)2403 1178 y Fy(d\034)11 b(dt)28 b Ft(\024)g Fy(\014)2781 1193 y Fs(0)2854 1037 y Fr(\022)2927 1042 y(Z)3027 1069 y Fs(1)2983 1268 y(0)3083 1178 y Ft(j)12 b FA(_)-39 b Fy(a)p Ft(j)17 b Fy(dt)3293 1037 y Fr(\023)3365 1052 y Fs(2)3438 1178 y Fy(:)24 1441 y FA(W)-8 b(e)47 b(ha)m(v)m(e)h(sho)m(wn)f(that)g Ft(M)p Fo(h)1152 1405 y Fv(s)1234 1441 y FA(is)g(surjectiv)m(e)i(on)e (isotropic)g(lo)s(ops,)j(then)d Fu(A)p Fo(l)2990 1405 y Fv(n)2990 1467 y Fx(M)p Fn(h)3110 1448 y Fq(s)3193 1441 y FA(are)g(Allco)s(c)m(k)24 1563 y(groups)33 b(for)f(ev)m(ery)i Fy(n;)17 b(s)28 b Ft(2)g Fu(N)22 b Ft(n)g(f)p FA(0)p Ft(g)p FA(.)24 1741 y FD(Example)45 b(6.12.)g FA(The)39 b Fw(c)-5 b(omplexi\014e)g(d)38 b(Heisenb)-5 b(er)g(g)39 b(algebr)-5 b(a)38 b FA(is)g(surjectiv)m(e)j(on)d(isotropic)h(lo)s (ops.)24 1858 y(Recall)45 b(that)f(this)i(algebra)e Fu(C)p Fo(h)k FA(=)g Fo(v)30 b Ft(\010)h Fo(z)44 b FA(is)h(an)g(H-t)m(yp)s(e)g (algebra,)j(with)d Fy(J)2944 1873 y Fv(Z)3048 1858 y FA(:)k Fo(v)f Ft(\000)-16 b(!)47 b Fo(v)e FA(and)24 1974 y Fy(J)87 1938 y Fs(2)78 1999 y Fv(Z)162 1974 y FA(=)28 b Ft(\000j)p Fy(Z)7 b Ft(j)473 1938 y Fs(2)512 1974 y FA(I)33 b(for)f(ev)m(ery)i Fy(Z)h Ft(2)28 b Fo(z)p FA(.)43 b(W)-8 b(e)33 b(\014x)g(an)g(orthonormal)f(basis)h(\()p Fy(Z)2635 1989 y Fs(1)2674 1974 y Fy(;)17 b(Z)2785 1989 y Fs(2)2824 1974 y FA(\))33 b(of)f Fo(z)g FA(and)h(de\014ne)h(the)24 2090 y(unit)40 b(v)m(ectors)g Fy(R)648 2105 y Fs(0)727 2090 y FA(=)f Fy(X)923 2105 y Fs(0)962 2090 y FA(,)i Fy(R)1104 2105 y Fs(1)1182 2090 y FA(=)e Fy(J)1351 2105 y Fv(Z)1399 2114 y Fp(1)1438 2090 y Fy(X)1519 2105 y Fs(0)1558 2090 y FA(,)i Fy(R)1700 2105 y Fs(2)1779 2090 y FA(=)d Fy(J)1947 2105 y Fv(Z)1995 2114 y Fp(2)2034 2090 y Fy(X)2115 2105 y Fs(0)2193 2090 y FA(and)i Fy(R)2464 2105 y Fs(3)2542 2090 y FA(=)f Fy(J)2711 2105 y Fv(Z)2759 2114 y Fp(1)2797 2090 y Fy(J)2851 2105 y Fv(Z)2899 2114 y Fp(2)2938 2090 y Fy(X)3019 2105 y Fs(0)3058 2090 y FA(,)i(that)e(form)g(an)24 2206 y(orthonormal)31 b(basis)h(of)f Fo(v)p FA(.)43 b(F)-8 b(or)30 b(more)i(information)f(on)g(the)g (complexi\014ed)j(Heisen)m(b)s(erg)f(algebra,)24 2323 y(see)41 b([30].)66 b(Let)41 b(us)g(\014x)g Fy(\025)f(>)h FA(0)f(and)g(c)m(ho)s(ose)h(a)f(curv)m(e)i Fy(\033)i FA(=)d Fy(\033)2341 2338 y Fs(1)2381 2323 y Fy(Z)2448 2338 y Fs(1)2514 2323 y FA(+)28 b Fy(\033)2673 2338 y Fs(2)2712 2323 y Fy(Z)2779 2338 y Fs(2)2859 2323 y Ft(2)41 b FA(Av)3091 2281 y Fx(1)3091 2347 y Fs(0)3166 2323 y FA(.)66 b(W)-8 b(e)41 b(de\014ne)24 2447 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2936 y Fs(1)1331 2921 y Fy(Z)1398 2936 y Fs(1)1461 2921 y FA(+)23 b Fy(\025)29 b FA(_)-39 b Fy(a)1685 2936 y Fs(2)1724 2921 y Fy(Z)1791 2936 y Fs(2)1862 2921 y FA(=)32 b Fy(\033)t FA(,)j Ft(j)p Fy(a)p FA(\(0\))p Ft(j)c FA(=)h Fy(\025)i FA(and)h Ft(j)12 b FA(_)-39 b Fy(a)p Ft(j)31 b FA(=)h Fy(\025)3048 2884 y Fx(\000)p Fs(1)3159 2921 y Ft(j)p Fy(\033)t Ft(j)p FA(.)50 b(W)-8 b(e)35 b(also)24 3038 y(notice)d(that)e(the)i(isotropic) f(homotop)m(y)h(\000\()p Fy(\034)6 b(;)17 b(t)p FA(\))27 b(=)h Fy(\025)17 b(R)2077 3053 y Fs(0)2135 3038 y FA(+)h(\(1)h Ft(\000)g Fy(t)p FA(\))2504 2958 y Fr(\000)2550 3038 y Fy(a)2601 3053 y Fs(1)2640 3038 y FA(\()p Fy(\034)11 b FA(\))p Fy(R)2843 3053 y Fs(1)2902 3038 y FA(+)19 b Fy(a)3048 3053 y Fs(2)3087 3038 y FA(\()p Fy(\034)11 b FA(\))p Fy(R)3290 3053 y Fs(2)3331 2958 y Fr(\001)3407 3038 y FA(carries)24 3161 y Fy(a)30 b FA(to)g Fy(\025R)353 3176 y Fs(0)420 3161 y Ft(2)e Fo(v)i FA(and)g(satis\014es)h (Lip\(\000\))d Ft(\024)g FA(2)17 b Ft(k)p Fy(\033)t 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b(b)m(y)f(the)g(orthonormal)f(basis)h(\()p Fy(Z)1674 3819 y Fs(1)1714 3804 y Fy(;)17 b(Z)1825 3819 y Fs(2)1863 3804 y Fy(;)g(Z)1974 3819 y Fs(3)2013 3804 y FA(\))46 b(and)2299 3723 y Fr(\000)2345 3804 y Fy(X)2426 3819 y Fs(0)2465 3804 y Fy(;)17 b(J)2563 3819 y Fv(Z)2611 3828 y Fp(1)2649 3804 y Fy(X)2730 3819 y Fs(0)2770 3804 y Fy(;)g(J)2868 3819 y Fv(Z)2916 3828 y Fp(2)2954 3804 y Fy(X)3035 3819 y Fs(0)3074 3804 y Fy(;)g(J)3172 3819 y Fv(Z)3220 3828 y Fp(3)3259 3804 y Fy(X)3340 3819 y Fs(0)3379 3723 y Fr(\001)3470 3804 y FA(is)46 b(an)24 3920 y(orthonormal)32 b(basis)i(of)e Fo(v)p FA(.)43 b Fy(X)1135 3935 y Fs(0)1207 3920 y FA(is)33 b(a)g(\014xed)g(unit)g(v)m (ector)h(of)e Fo(v)h FA(and)f(w)m(e)i(ha)m(v)m(e)718 4084 y Fy(J)772 4099 y Fv(Z)820 4108 y Fp(1)858 4084 y Fy(J)912 4099 y Fv(Z)960 4108 y Fp(2)1026 4084 y FA(=)28 b Fy(J)1184 4099 y Fv(Z)1232 4108 y Fp(3)1270 4084 y Fy(;)115 b(J)1466 4099 y Fv(Z)1514 4108 y Fp(1)1552 4084 y Fy(J)1606 4099 y Fv(Z)1654 4108 y Fp(3)1720 4084 y FA(=)28 b Ft(\000)p Fy(J)1955 4099 y Fv(Z)2003 4108 y Fp(2)2139 4084 y FA(and)98 b Fy(J)2448 4099 y Fv(Z)2496 4108 y Fp(2)2534 4084 y Fy(J)2588 4099 y Fv(Z)2636 4108 y Fp(3)2703 4084 y FA(=)27 b Fy(J)2860 4099 y Fv(Z)2908 4108 y Fp(1)2963 4084 y Fy(:)24 4249 y FA(W)-8 b(e)52 b(de\014ne)g(the)g(direct)g(pro)s(duct)g(algebra)f Fo(n)1790 4213 y Fs(3)1790 4274 y Fk(H)1906 4249 y FA(=)60 b Fo(v)35 b Ft(\010)g Fo(z)p FA(,)56 b(where)d Fo(v)60 b FA(=)f Fo(v)2960 4264 y Fs(1)3035 4249 y Ft(\010)35 b Fo(v)3198 4264 y Fs(2)3273 4249 y Ft(\010)g Fo(v)3436 4264 y Fs(3)3527 4249 y FA(and)24 4365 y(\()p Fy(R)136 4380 y Fv(l)q Fs(0)197 4365 y Fy(;)17 b(R)315 4380 y Fv(l)q Fs(1)377 4365 y Fy(;)g(R)495 4380 y Fv(l)q Fs(2)556 4365 y Fy(;)g(R)674 4380 y Fs(3)p Fv(l)735 4365 y FA(\))33 b(is)g(the)g(orthonormal)f (basis)h(of)g Fo(v)2032 4380 y Fv(l)2090 4365 y FA(for)f Fy(l)e FA(=)e(1)p Fy(;)17 b FA(2)p Fy(;)g FA(3.)42 b(W)-8 b(e)33 b(ha)m(v)m(e)h(de\014ned)684 4529 y Fy(R)758 4544 y Fv(l)q Fs(0)847 4529 y FA(=)28 b Fy(X)1032 4544 y Fv(l)1058 4529 y Fy(;)114 b(R)1273 4544 y Fv(l)q Fs(1)1362 4529 y FA(=)28 b Fy(J)1520 4544 y Fv(Z)1568 4553 y Fp(1)1606 4529 y Fy(X)1687 4544 y Fv(l)1713 4529 y Fy(;)114 b(R)1928 4544 y Fv(l)q Fs(2)2018 4529 y FA(=)27 b Fy(J)2175 4544 y Fv(Z)2223 4553 y Fp(2)2262 4529 y Fy(X)2343 4544 y Fv(l)2369 4529 y Fy(;)114 b(R)2584 4544 y Fv(l)q Fs(3)2673 4529 y FA(=)28 b Fy(J)2831 4544 y Fv(Z)2879 4553 y Fp(3)2917 4529 y Fy(X)2998 4544 y Fv(l)24 4693 y FA(where)34 b Fy(X)387 4708 y Fv(l)445 4693 y FA(is)f(a)g(unit)g(v)m(ector)g(of)f Fo(v)1285 4708 y Fv(l)1311 4693 y FA(.)44 b(F)-8 b(urthermoire,)33 b(when)m(v)m(er)i Fy(l)30 b Ft(6)p FA(=)d Fy(s)33 b FA(w)m(e)h(set)1025 4857 y([)p Fy(R)1126 4872 y Fv(l)q(i)1177 4857 y Fy(;)17 b(R)1295 4872 y Fv(sj)1364 4857 y FA(])28 b(=)f(0)98 b(for)32 b(ev)m(ery)i Fy(i;)17 b(j)34 b FA(=)27 b(1)p Fy(;)17 b FA(2)p Fy(;)g FA(3)p Fy(;)g FA(4.)24 5033 y(Let)33 b Fy(\025)27 b(>)h FA(0)k(and)h(let)g Fy(\033)e FA(=)989 4958 y Fr(P)1094 4984 y Fs(3)1094 5062 y Fv(j)t Fs(=1)1237 5033 y Fy(\033)1292 5048 y Fv(j)1329 5033 y Fy(Z)1396 5048 y Fv(j)1460 5033 y Ft(2)d FA(Av)1679 4991 y Fx(1)1679 5057 y Fs(0)1754 5033 y FA(.)43 b(W)-8 b(e)33 b(de\014ne)h(the)f(curv)m (e)637 5308 y Fy(a)28 b FA(=)g Fy(\025)877 5228 y Fr(\000)993 5184 y Fs(3)939 5214 y Fr(X)955 5426 y Fv(l)q Fs(=1)1099 5308 y Fy(R)1173 5323 y Fv(l)q Fs(0)1235 5228 y Fr(\001)1303 5308 y FA(+)1415 5241 y(1)p 1411 5285 V 1411 5377 a Fy(\025)1548 5184 y Fs(3)1494 5214 y Fr(X)1510 5426 y Fv(l)q Fs(=1)1655 5198 y Fr(\020)1731 5173 y(Z)1830 5199 y Fv(t)1786 5398 y Fs(0)1860 5308 y Fy(\033)1915 5323 y Fv(l)1941 5308 y FA(\()p Fy(s)p FA(\))17 b Fy(ds)2204 5198 y Fr(\021)2264 5308 y Fy(R)2338 5323 y Fv(l)q(l)2413 5308 y Ft(2)29 b FA(Lip)2650 5332 y Fs(0)2689 5228 y Fr(\000)2735 5308 y FA([0)p Fy(;)17 b FA(1])p Fy(;)g Fo(v)3026 5228 y Fr(\001)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 169 246 a Fz(CONT)-6 b(A)n(CT)34 b(EQUA)-6 b(TIONS,)31 b(LIPSCHITZ)i(EXTENSIONS)f(AND)g(ISOPERIMETRIC)h (INEQUALITIES)67 b(29)24 454 y FA(Then)41 b(one)g(can)f(easily)i(c)m (hec)m(k)g(that)e([)p Fy(a;)29 b FA(_)-39 b Fy(a)p FA(])41 b(=)1782 379 y Fr(P)1887 405 y Fs(3)1887 483 y Fv(l)q Fs(=1)2020 454 y Fy(\033)2075 469 y Fv(l)2118 454 y Fy(Z)2185 469 y Fv(l)2211 454 y FA(,)h Ft(j)p Fy(a)p FA(\(0\))p Ft(j)e FA(=)2668 371 y Ft(p)p 2751 371 49 4 v 83 x FA(3)17 b Fy(\025)40 b FA(and)g Ft(j)12 b FA(_)-39 b Fy(a)p Ft(j)40 b FA(=)h Fy(\025)3432 418 y Fx(\000)p Fs(1)3543 454 y Ft(j)p Fy(\033)t Ft(j)p FA(.)24 570 y(Finally)-8 b(,)33 b(the)g(isotropic)g(homotop)m(y)726 845 y(\000\()p Fy(\034)6 b(;)17 b(t)p FA(\))27 b(=)h Fy(\025)1178 765 y Fr(\000)1294 721 y Fs(3)1240 751 y Fr(X)1256 963 y Fv(l)q Fs(=1)1401 845 y Fy(R)1475 860 y Fv(l)q Fs(0)1536 765 y Fr(\001)1604 845 y FA(+)22 b(\(1)g Ft(\000)g Fy(t)p FA(\))1997 778 y(1)p 1993 822 57 4 v 1993 914 a Fy(\025)2131 721 y Fs(3)2077 751 y Fr(X)2093 963 y Fv(l)q Fs(=1)2237 735 y Fr(\020)2314 710 y(Z)2413 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)27 b FA(0)32 b(suc)m(h)i(that)24 818 y(\(60\))1019 b(Lip\(\000)1458 833 y Fs(5)1498 818 y FA(\))27 b Ft(\024)1691 793 y FA(~)1669 818 y Fy(C)1739 833 y Fs(1)1811 678 y Fr(\022)1884 818 y Fy(\025)22 b FA(+)2071 751 y Ft(k)p Fy(\033)t Ft(k)2230 766 y Fv(L)2278 747 y Fj(1)p 2071 795 276 4 v 2181 886 a Fy(\025)2357 678 y Fr(\023)2464 818 y Fy(:)24 1049 y FA(De\014ning)40 b Fy(\030)k FA(=)622 974 y Fr(P)727 1078 y Fv(i;j)824 1049 y Fy(\030)867 1064 y Fv(ij)927 1049 y Fy(X)1008 1064 y Fv(ij)1108 1049 y FA(and)39 b Fy(q)1347 1064 y Fs(0)1427 1049 y FA(=)1542 974 y Fr(P)1647 1078 y Fv(i;j)1744 1049 y Fy(\030)1787 1064 y Fv(ij)1847 1049 y Fy(e)1892 1064 y Fv(ij)1992 1049 y Ft(2)h Fu(R)2170 1013 y Fv(mn)2279 1049 y FA(,)h(one)f (immediately)i(c)m(hec)m(k)f(that)f(\000)3645 1064 y Fs(5)24 1168 y FA(mak)m(es)k Fy(q)h FA(m)m(ulti-isotropically)f (homotopic)e(the)g(a)g(p)s(oin)m(t)g Fy(q)2294 1183 y Fs(0)2377 1168 y Ft(2)i Fu(R)2559 1132 y Fv(mn)2668 1168 y FA(.)72 b(Th)m(us,)45 b(pasting)e(all)f(the)24 1285 y(previous)29 b(m)m(ulti-isotropic)f(homotopies,)h(w)m(e)g(get)e(the)h (follo)m(wing)f(mapping)h Fy(H)35 b FA(:)28 b([0)p Fy(;)17 b FA(1])3221 1248 y Fs(2)3287 1285 y Ft(\000)-16 b(!)28 b Fu(R)3548 1248 y Fv(mn)3657 1285 y FA(,)24 1401 y(de\014ned)34 b(as)923 1745 y Fy(H)8 b FA(\()p Fy(\034)e(;)17 b(t)p FA(\))27 b(=)1346 1422 y Fr(8)1346 1511 y(>)1346 1541 y(>)1346 1571 y(>)1346 1601 y(>)1346 1631 y(<)1346 1810 y(>)1346 1840 y(>)1346 1870 y(>)1346 1900 y(>)1346 1930 y(:)1476 1511 y FA(\000)1537 1526 y Fs(1)1576 1511 y FA(\()p Fy(\034)6 b(;)17 b FA(5)p Fy(t)p FA(\))351 b(0)27 b Ft(\024)h Fy(t)g(<)g FA(1)p Fy(=)p FA(5)1476 1628 y(\000)1537 1643 y Fs(2)1576 1628 y FA(\()p Fy(\034)6 b(;)17 b FA(5)p Fy(t)22 b Ft(\000)h FA(1\))82 b(1)p Fy(=)p FA(5)27 b Ft(\024)h Fy(t)g(<)g FA(2)p Fy(=)p FA(5)1476 1744 y(\000)1537 1759 y Fs(3)1576 1744 y FA(\()p Fy(\034)6 b(;)17 b FA(5)p Fy(t)22 b Ft(\000)h FA(2\))82 b(2)p Fy(=)p FA(5)27 b Ft(\024)h Fy(t)g(<)g FA(3)p Fy(=)p FA(5)1476 1860 y(\000)1537 1875 y Fs(4)1576 1860 y FA(\()p Fy(\034)6 b(;)17 b FA(5)p Fy(t)22 b Ft(\000)h FA(3\))82 b(3)p Fy(=)p FA(5)27 b Ft(\024)h Fy(t)g(<)g FA(4)p Fy(=)p FA(5)1476 1976 y(\000)1537 1991 y Fs(5)1576 1976 y FA(\()p Fy(\034)6 b(;)17 b FA(5)p Fy(t)22 b Ft(\000)h FA(4\))180 b(4)p Fy(=)p FA(5)27 b Ft(\024)h Fy(t)g(<)f FA(1)2758 1745 y Fy(:)24 2140 y FA(It)36 b(is)g(clearly)h(b)s(oth)f(con)m(tin)m(uous)h(and)f(a.e.)53 b(di\013eren)m(tiable)38 b(in)e([0)p Fy(;)17 b FA(1])2568 2103 y Fs(2)2607 2140 y FA(.)53 b(Moreo)m(v)m(er,)38 b(w)m(e)f(also)f(ha)m(v)m(e)24 2256 y Fy(H)113 2220 y Fx(\003)152 2256 y FA(\()p Fy(!)t FA(\))49 b(=)g(0)c(a.e.in)h([0)p Fy(;)17 b FA(1])1030 2220 y Fs(2)1069 2256 y FA(.)82 b(By)46 b(con)m(v)m(exit)m(y)j(of)c([0)p Fy(;)17 b FA(1])2111 2220 y Fs(2)2195 2256 y FA(and)46 b(triangle)f(inequalit)m(y)-8 b(,)51 b(one)46 b(easily)24 2372 y(notices)34 b(the)f(follo)m(wing)g (estimate)24 2630 y(\(61\))1130 b(Lip\()p Fy(H)8 b FA(\))27 b Ft(\024)h FA(5)1915 2506 y Fs(5)1860 2536 y Fr(X)1871 2745 y Fv(j)t Fs(=1)2021 2630 y FA(Lip\(\000)2262 2645 y Fv(j)2299 2630 y FA(\))17 b Fy(:)24 2899 y FA(Direct)33 b(computations)g(sho)m(w)h(that)1013 2980 y Fr(8)1013 3070 y(<)1013 3249 y(:)1143 3066 y FA(Lip\(\000)1384 3081 y Fs(1)1423 3066 y FA(\))28 b Ft(\024)g FA(6)p Fy(\031)20 b(L)1143 3182 y FA(Lip\(\000)1384 3197 y Fs(2)1423 3182 y FA(\))28 b Ft(\024)g FA(6)p Fy(\031)20 b(L)1143 3299 y FA(Lip\(\000)1384 3314 y Fs(3)1423 3299 y FA(\))28 b Ft(\024)g FA(6)p Fy(\031)t(L)22 b FA(+)1898 3259 y Fv(\031)p 1898 3276 43 4 v 1902 3333 a Fs(2)1968 3299 y FA(max)2149 3314 y Fs([0)p Fv(;)p Fs(1])2299 3299 y Ft(j)p Fy(\013)q Ft(j)2502 3184 y Fy(;)-2505 b FA(\(62\))24 3475 y(where)38 b(w)m(e)g(recall)g(that)f(w)m(e)h(ha)m(v)m(e)g (de\014ned)h Fy(L)c FA(=)h(Lip\()p Fy(c)p FA(\).)57 b(T)-8 b(aking)37 b(in)m(to)h(accoun)m(t)f(the)h(de\014nition)24 3591 y(of)31 b(Allco)s(c)m(k)h(group,)g(w)m(e)g(ha)m(v)m(e)g(to)f (estimate)i Ft(k)p Fy(\033)t Ft(k)1814 3606 y Fv(L)1862 3587 y Fj(1)1930 3591 y FA(.)43 b(Th)m(us,)33 b(taking)f(in)m(to)f (accoun)m(t)h(\(56\),)f(a)g(direct)24 3707 y(computation)i(yields)729 3933 y Ft(j)p Fy(!)822 3891 y Fv(k)863 3933 y FA(\()p Fy(\015)5 b FA(\()p Fy(\034)11 b FA(\))23 b Ft(^)37 b FA(_)-42 b Fy(\015)5 b FA(\()p Fy(\034)11 b FA(\)\))p Ft(j)28 b(\024)g FA(2)17 b Fy(n)g(m)1824 3792 y Fr(\022)1966 3933 y FA(max)1929 3992 y Fv(r)r Fs(=1)p Fv(;:::)o(;s)1897 4060 y Fs(1)p Fx(\024)p Fv(l)q(
)27 b FA(0)33 b(suc)m(h)h(that)24 5403 y(\(63\))1312 b(Lip)q(\(\000)1752 5418 y Fs(5)1791 5403 y FA(\))28 b Ft(\024)g Fy(C)2032 5418 y Fs(1)2088 5403 y Fy(L)17 b(:)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 169 246 a Fz(CONT)-6 b(A)n(CT)34 b(EQUA)-6 b(TIONS,)31 b(LIPSCHITZ)i(EXTENSIONS)f(AND)g(ISOPERIMETRIC)h (INEQUALITIES)67 b(35)24 446 y FA(In)33 b(addition,)g(b)m(y)i(\(57\))d (and)g(\(58\),)g(observing)i(that)e(max)18 b Ft(j)p Fy(a)p Ft(j)27 b FA(=)g(max)18 b Ft(j)p Fy(q)t Ft(j)p FA(,)31 b(w)m(e)j(also)f(obtain)1313 624 y(max)1339 690 y Fs([0)p Fv(;)p Fs(1])1511 624 y Ft(j)p Fy(q)t Ft(j)27 b(\024)h Fy(C)23 b FA(\(2)p Fy(\031)j FA(+)c Fy(C)2175 639 y Fs(0)2214 624 y FA(\))33 b Fy(L)17 b(;)24 856 y FA(where)46 b Fy(C)388 871 y Fs(0)477 856 y Fy(>)j FA(0)c(is)h(a)f(geometric)h(constan)m(t)g (dep)s(ending)g(on)f(the)h(group.)81 b(It)46 b(is)f(also)h(ob)m(vious) 24 972 y(that)34 b(Lip\()p Fy(q)t FA(\))d Ft(\024)g FA(Lip\(\000)882 987 y Fs(5)922 972 y FA(\))f Ft(\024)i Fy(C)1169 987 y Fs(1)1224 972 y Fy(L)p FA(.)50 b(Th)m(us,)36 b(joining)f(all)f(the)h (previous)g(estimates,)i(w)m(e)e(get)g(a)f(new)24 1088 y(constan)m(t)f Fy(C)487 1103 y Fs(2)554 1088 y Fy(>)28 b FA(0)k(suc)m(h)i(that)1505 1225 y(Lip\(\000)1746 1240 y Fs(4)1785 1225 y FA(\))28 b Ft(\024)g Fy(C)2026 1240 y Fs(2)2093 1225 y Fy(L)17 b(:)24 1383 y FA(By)26 b(last)h(inequalit)m (y)g(along)f(with)g(\(63\),)h(all)f(previous)h(estimates)h(for)d (Lip\(\000)2807 1398 y Fv(j)2844 1383 y FA(\))g(and)h(applying)i (\(61\),)24 1499 y(w)m(e)34 b(ha)m(v)m(e)g(found)e(a)g(geometric)i (constan)m(t)f Fy(\024)1641 1514 y Fs(1)1709 1499 y Fy(>)27 b FA(0,)33 b(only)g(dep)s(ending)g(on)g(the)g(group,)f(suc)m(h)i(that) 24 1677 y(\(64\))1320 b(Lip\()p Fy(H)8 b FA(\))27 b Ft(\024)h Fy(\024)2013 1692 y Fs(1)2080 1677 y Fy(L)17 b(:)24 1855 y FA(Then)45 b Fy(H)52 b FA(is)44 b(a)g(m)m(ulti-isotropic)h(homotop)m (y)g(suc)m(h)h(that)e Fy(H)8 b FA(\()p Ft(\001)p Fy(;)17 b FA(0\))45 b(=)j Fy(\013)g FA(:)f([0)p Fy(;)17 b FA(1])47 b Ft(\000)-16 b(!)46 b Fu(R)3373 1819 y Fv(mn)3527 1855 y FA(and)24 1971 y Fy(H)8 b FA(\()p Ft(\001)p Fy(;)17 b FA(1\))26 b Ft(\021)i Fy(q)484 1986 y Fs(0)524 1971 y FA(.)44 b(The)33 b(condition)g Fy(H)8 b FA(\(0)p Fy(;)17 b(t)p FA(\))27 b(=)g Fy(H)8 b FA(\(1)p Fy(;)17 b(t)p FA(\))32 b(for)g(ev)m(ery)j Fy(t)27 b Ft(2)i FA([0)p Fy(;)17 b FA(1])32 b(implies)i(that)844 2210 y Fy(')925 2130 y Fr(\000)970 2210 y Fy(\032)17 b(e)1082 2169 y Fv(i)p Fs(2)p Fv(\031)r(\022)1224 2130 y Fr(\001)1297 2210 y FA(=)1400 2070 y Fr(\032)1517 2152 y Fy(H)1606 2071 y Fr(\000)1651 2152 y Fy(\022)s(;)g FA(2\(1)k Ft(\000)i Fy(\032)p FA(\))2088 2071 y Fr(\001)2217 2152 y FA(if)k(1)p Fy(=)p FA(2)g Ft(\024)i Fy(\032)f Ft(\024)g FA(1)1517 2268 y Fy(q)1560 2283 y Fs(0)2217 2268 y FA(if)f(0)h Ft(\024)g Fy(\032)g Ft(\024)g FA(1)p Fy(=)p FA(2)24 2454 y(is)41 b(w)m(ell)h(de\014ned)g(on)f(the)g(closed)h(unit)f(disk)g Fy(D)k Ft(\032)c Fu(R)2037 2418 y Fs(2)2077 2454 y FA(.)68 b(F)-8 b(urthermore,)43 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Fy(')973 3039 y Fx(\003)1012 3075 y Fy(!)e FA(=)d(0)k(a.e.)44 b(in)33 b Fy(D)s FA(.)43 b(Our)32 b(claim)h(is)h(ac)m(hiev)m(ed.)45 b Fl(2)24 3262 y FD(Remark)c(7.4.)i FA(The)36 b(previous)g(theorem)g (extends)h(Theorem)f(2.3)f(of)f([1],)i(where)g(the)g(standard)24 3378 y(simplectic)31 b(space)e Fu(R)804 3342 y Fs(2)p Fv(n)915 3378 y FA(is)g(replaced)h(with)f(the)g Fu(R)1844 3342 y Fv(mn)1982 3378 y FA(equipp)s(ed)h(with)f(the)g(m)m (ulti-simplectic)i(form)24 3494 y Fy(!)g FA(=)220 3419 y Fr(P)325 3446 y Fv(s)325 3523 y(k)r Fs(=1)491 3494 y Fy(!)556 3458 y Fv(k)614 3494 y Fy(E)686 3509 y Fv(k)729 3494 y FA(.)124 3680 y(Next,)48 b(w)m(e)e(will)g(sho)m(w)f(ho)m(w)h (Theorem)g(7.3)e(leads)i(us)f(to)f(a)h(Lipsc)m(hitz)i(extension)f (theorem.)24 3797 y(W)-8 b(e)46 b(will)f(use)h(some)g(abstract)g(to)s (ols)f(in)g(metric)h(spaces,)k(follo)m(wing)45 b(the)h(w)m(ork)g(b)m(y) g(Lang)f(and)24 3913 y(Sc)m(hlic)m(henmaier,)36 b([21)o(].)24 4099 y FD(De\014nition)28 b(7.5.)35 b FA(W)-8 b(e)24 b(sa)m(y)g(that)g(a)f(metric)h(space)h Fy(Y)45 b FA(is)24 b Fw(Lipschitz)i Fy(m)p Fw(-c)-5 b(onne)g(cte)g(d)23 b FA(for)g(some)h Fy(m)k Ft(2)g Fu(N)24 4215 y FA(if)40 b(there)h(exists)i(a)d(constan)m(t)h Fy(c)1188 4230 y Fv(m)1296 4215 y Fy(>)g FA(0)f(suc)m(h)i(that)e(an)m(y)h(Lipsc)m(hitz)h (map)f(\000)g(:)g Fy(S)3026 4179 y Fv(m)3133 4215 y Ft(\000)-16 b(!)41 b Fy(Y)61 b FA(has)41 b(a)24 4332 y(Lipsc)m(hitz)34 b(extension)h(\010)28 b(:)g Fy(D)1109 4295 y Fv(m)p Fs(+1)1293 4332 y Ft(\000)-16 b(!)27 b Fy(Y)54 b FA(with)33 b(estimate)h (Lip\(\010\))28 b Ft(\024)g Fy(c)2668 4347 y Fv(m)2751 4332 y FA(Lip)q(\(\000\))17 b Fy(:)24 4518 y FD(De\014nition)37 b(7.6.)42 b FA(Let)32 b(\()p Fy(X)r(;)17 b(Y)k FA(\))32 b(b)s(e)h(a)e(couple)i(of)f(metric)h(spaces.)45 b(W)-8 b(e)32 b(sa)m(y)h(that)f(\()p Fy(X)r(;)17 b(Y)k FA(\))32 b(has)h(the)24 4634 y Fw(Lipschitz)40 b(extension)f(pr)-5 b(op)g(erty)38 b FA(if)h(there)g(exists)h Fy(C)k(>)38 b FA(0)g(suc)m(h)i(that)e(for)g(ev)m(ery)i(subset)g Fy(Z)k 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Fy(X)r(;)17 b(Y)k FA(\))24 5403 y Fw(has)34 b(the)h(Lipschitz)g (extension)e(pr)-5 b(op)g(erty.)p eop end %%Page: 36 36 TeXDict begin 36 35 bop 24 246 a Fz(36)1315 b(V)-9 b(ALENTINO)25 b(MA)n(GNANI)124 446 y FA(T)-8 b(aking)42 b(in)m(to)f(accoun)m(t)i (that)e(Nagata)f(dimension)k(of)c Fu(R)2263 410 y Fs(2)2344 446 y FA(is)i(clearly)h(t)m(w)m(o,)h(b)m(y)f(Theorem)f(7.7)24 562 y(it)g(follo)m(ws)h(that)g(b)s(oth)f(Lipsc)m(hitz)i (0-connectedness)h(and)d(Lipsc)m(hitz)j(1-connectedness)f(of)e Fu(A)p Fo(l)3637 526 y Fv(n)3637 587 y Fn(n)24 678 y FA(imply)31 b(Corollary)f(1.4.)42 b(The)31 b(former)e(prop)s(ert)m(y)i (is)f(a)f(consequence)k(of)c(the)h(fact)g(that)f Fu(A)p Fo(l)3311 642 y Fv(n)3311 703 y Fn(n)3358 678 y FA(,)h(as)g(an)m(y)24 794 y(strati\014ed)j(group,)g(is)g(connected)h(b)m(y)g(geo)s(desics.)45 b(The)33 b(latter)g(is)g(pro)m(v)m(ed)h(in)f(the)g(follo)m(wing)24 977 y FD(Theorem)38 b(7.8.)k Fu(A)p Fo(l)815 941 y Fv(n)897 977 y Fw(is)34 b(1-c)-5 b(onne)g(cte)g(d)34 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Fr(\000)1928 1681 y Fy(a)p FA(\()p Fy(t)p FA(\))22 b(+)g Fy(b)p FA(\()p Fy(t)p FA(\))2362 1600 y Fr(\001)2425 1681 y Fy(:)24 1853 y FA(W)-8 b(e)31 b(consider)g(our)f(\014xed)h(basis)g(\()p Fy(X)1327 1868 y Fv(ij)1388 1853 y FA(\))f(of)f Fy(V)1621 1868 y Fs(1)1661 1853 y FA(,)h(that)g(satis\014es)i(\(52\),)e(along)g (with)h(the)f(orthonormal)24 1969 y(basis)k(\()p Fy(Z)369 1984 y Fv(k)411 1969 y FA(\))e(of)g Fy(V)649 1984 y Fs(2)689 1969 y FA(.)43 b(W)-8 b(e)33 b(de\014ne)817 2220 y Fy(a)p FA(\()p Fy(t)p FA(\))28 b(=)1159 2126 y Fr(X)1125 2336 y Fs(1)p Fx(\024)p Fv(i)p Fx(\024)p Fv(n)1111 2401 y Fs(1)p Fx(\024)p Fv(j)t Fx(\024)p Fv(m)1367 2220 y Fy(\013)1429 2235 y Fv(ij)1490 2220 y FA(\()p Fy(t)p FA(\))17 b Fy(X)1699 2235 y Fv(ij)1857 2220 y FA(and)97 b Fy(b)p FA(\()p Fy(t)p FA(\))29 b(=)2451 2096 y Fv(s)2395 2126 y Fr(X)2403 2338 y Fv(k)r Fs(=1)2556 2220 y Fy(\014)2611 2235 y Fv(k)2653 2220 y FA(\()p Fy(t)p FA(\))17 b Fy(Z)2848 2235 y Fv(k)24 2561 y FA(where)37 b Fy(\013)c FA(=)f(\()p 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y Fs(1)1409 4080 y(0)1509 3990 y Ft(j)p Fy(H)1618 4005 y Fv(\022)1657 3909 y Fr(\000)1702 3990 y Fy(\022)s(;)17 b FA(2\(1)22 b Ft(\000)g Fy(\032)p FA(\))2139 3909 y Fr(\001)2207 3990 y Ft(^)h Fy(H)2377 4005 y Fv(\032)2417 3909 y Fr(\000)2463 3990 y Fy(\022)s(;)17 b FA(2\(1)k Ft(\000)i Fy(\032)p FA(\))2900 3909 y Fr(\001)2946 3990 y Ft(j)17 b Fy(d\022)i(d\032)e(:)24 4252 y FA(T)-8 b(aking)33 b(in)m(to)g(accoun)m(t)g(that)24 4495 y Fy(H)113 4414 y Fr(\000)158 4495 y Fy(\022)s(;)17 b FA(2\(1)p Ft(\000)p Fy(\032)p FA(\))551 4414 y Fr(\001)625 4495 y FA(=)27 b(\000)789 4510 y Fv(k)r Fs(+1)922 4414 y Fr(\000)968 4495 y Fy(\022)s(;)17 b FA(10\(1)p Ft(\000)p Fy(\032)p FA(\))p Ft(\000)p Fy(k)1541 4414 y Fr(\001)1684 4495 y FA(if)98 b(1)p Ft(\000)1975 4428 y Fy(k)25 b FA(+)d(1)p 1975 4472 V 2038 4563 a(10)2236 4495 y Fy(<)27 b(\032)h Ft(\024)g FA(1)p Ft(\000)2680 4428 y Fy(k)p 2658 4472 98 4 v 2658 4563 a FA(10)2864 4495 y(and)97 b Fy(k)31 b FA(=)d(0)p Fy(;)17 b FA(1)p Fy(;)g FA(2)p Fy(;)g FA(3)p Fy(;)g FA(4)g Fy(;)24 4719 y FA(a)32 b(simple)i(c)m(hange)g(of)e(v)-5 b(ariable)33 b(yields)380 4867 y Fr(Z)436 5093 y Fv(D)516 5003 y Ft(j)p Fy(')608 5018 y Fv(x)648 5027 y Fp(1)708 5003 y Ft(^)23 b Fy(')861 5018 y Fv(x)901 5027 y Fp(2)939 5003 y Ft(j)17 b Fy(dx)27 b FA(=)1275 4878 y Fs(4)1221 4908 y Fr(X)1229 5120 y Fv(k)r Fs(=0)1381 4867 y Fr(Z)1481 4894 y Fs(1)1437 5093 y(0)1537 4867 y Fr(Z)1637 4894 y Fs(1)1592 5093 y(0)1693 5003 y Ft(j)p FA(\(\000)1820 5018 y Fv(k)r Fs(+1)1952 5003 y FA(\))1990 5018 y Fv(\034)2033 4922 y Fr(\000)2079 5003 y Fy(\022)s(;)17 b(t)2206 4922 y Fr(\001)2274 5003 y Ft(^)22 b FA(\(\000)2461 5018 y Fv(k)r Fs(+1)2594 5003 y FA(\))2632 5018 y Fv(t)2662 4922 y Fr(\000)2707 5003 y Fy(\022)s(;)17 b(t)p FA(\))p Ft(j)g Fy(dt)g(d\022)i(:)-3138 b FA(\(73\))24 5286 y(No)m(w,)49 b(w)m(e)d(use)g(the)g(explicit)g(form)m(ulas)g(of)f(\000)1757 5301 y Fv(j)1838 5286 y FA(giv)m(en)i(in)e(the)g(pro)s(of)g(of)f (Theorem)j(7.3.)80 b(F)-8 b(rom)24 5403 y(de\014nition)42 b(of)f(\000)647 5418 y Fs(1)728 5403 y FA(it)g(is)h(ob)m(vious)g(that)g Ft(j)p FA(\(\000)1650 5418 y Fs(1)1688 5403 y FA(\))1726 5418 y Fv(\034)1798 5403 y Ft(^)28 b FA(\(\000)1991 5418 y Fs(1)2030 5403 y FA(\))2068 5418 y Fv(t)2098 5403 y Ft(j)42 b FA(=)h(0)e(a.e.)70 b(in)41 b(the)h(unit)g(square)g([0)p Fy(;)17 b FA(1])3618 5366 y Fs(2)3657 5403 y FA(,)p eop end %%Page: 42 42 TeXDict begin 42 41 bop 24 246 a Fz(42)1315 b(V)-9 b(ALENTINO)25 b(MA)n(GNANI)24 446 y FA(denoted)34 b(b)m(y)f Fy(Q)p FA(.)44 b(Simple)33 b(computations)h(yield)g(the)f(follo)m(wing)g (estimates)103 526 y Fr(Z)158 751 y Fv(Q)234 661 y Ft(j)p FA(\(\000)361 676 y Fs(2)400 661 y FA(\))438 676 y Fv(\034)503 661 y Ft(^)23 b FA(\(\000)691 676 y Fs(2)730 661 y FA(\))768 676 y Fv(t)798 661 y Ft(j)p Fy(d\034)11 b(dt)83 b FA(=)h(2)1325 526 y Fr(Z)1380 751 y Fv(Q)1456 661 y Ft(j)1509 635 y FA(_)1484 661 y Fy(\014)1539 676 y Fs(1)1578 661 y FA(\(2)p Fy(\034)34 b Ft(\000)22 b FA(1)g(+)g Fy(t)p FA(\))g Ft(^)2218 635 y FA(_)2193 661 y Fy(\014)2248 676 y Fs(2)2288 661 y FA(\(2)p Fy(\034)33 b Ft(\000)23 b FA(1\))p Ft(j)p Fy(d\034)11 b(dt)1099 954 y Ft(\024)83 b FA(2)1325 818 y Fr(Z)1424 844 y Fs(1)1380 1044 y(0)1480 813 y Fr(\022)1554 818 y(Z)1653 844 y Fs(1)1609 1044 y(0)1709 954 y Ft(j)1762 927 y FA(_)1737 954 y Fy(\014)1792 969 y Fs(1)1831 954 y FA(\(2)p Fy(\034)34 b Ft(\000)22 b FA(1)g(+)g Fy(t)p FA(\))p Ft(j)p Fy(dt)2449 813 y Fr(\023)2539 954 y Ft(j)2591 927 y FA(_)2567 954 y Fy(\014)2622 969 y Fs(2)2661 954 y FA(\(2)p Fy(\034)33 b Ft(\000)23 b FA(1\))p Ft(j)p Fy(d\034)1099 1254 y Ft(\024)83 b FA(2)1325 1119 y Fr(Z)1380 1344 y Fk(R)1449 1114 y Fr(\022)1522 1119 y(Z)1577 1344 y Fk(R)1646 1254 y Ft(j)10 b FA(~)-59 b Fy(\013)1737 1213 y Fx(0)1760 1254 y FA(\()p Fy(t)p FA(\))p Ft(j)p Fy(dt)1985 1114 y Fr(\023)2074 1254 y Ft(j)10 b FA(~)-59 b Fy(\013)2165 1213 y Fx(0)2188 1254 y FA(\(2)p Fy(\034)33 b Ft(\000)23 b FA(1\))p Ft(j)p Fy(d\034)38 b FA(=)28 b(2)2866 1114 y Fr(\022)2938 1119 y(Z)3038 1145 y Fs(1)2994 1344 y(0)3094 1254 y Ft(j)p Fy(\013)3185 1213 y Fx(0)3208 1254 y FA(\()p Fy(t)p FA(\))p Ft(j)p Fy(dt)3433 1114 y Fr(\023)3506 1129 y Fs(2)3578 1254 y Fy(:)24 1479 y FA(The)34 b(area)e(con)m(tributed)i(b)m(y)f(\000)1155 1494 y Fs(3)1227 1479 y FA(is)g(giv)m(en)h(b)m(y)32 1559 y Fr(Z)87 1785 y Fv(Q)164 1695 y Ft(j)p FA(\(\000)291 1710 y Fs(3)330 1695 y FA(\))368 1710 y Fv(\034)433 1695 y Ft(^)22 b FA(\(\000)620 1710 y Fs(3)660 1695 y FA(\))698 1710 y Fv(t)727 1695 y Ft(j)p Fy(d\034)11 b(dt)84 b FA(=)g Fy(\031)1264 1559 y Fr(Z)1319 1785 y Fv(Q)1396 1580 y Fr(\014)1396 1640 y(\014)1396 1700 y(\014)1429 1584 y(\020)1488 1695 y FA(cos)1636 1614 y Fr(\000)1681 1695 y Fy(\031)t(t=)p FA(2)1873 1614 y Fr(\001)1943 1669 y FA(_)1919 1695 y Fy(\014)1974 1710 y Fs(1)2013 1695 y FA(\(2)p Fy(\034)11 b FA(\))22 b(+)g(sin)2448 1614 y Fr(\000)2493 1695 y Fy(\031)t(t=)p FA(2)2685 1614 y Fr(\001)2764 1649 y FA(_)p 2731 1614 95 4 v 2731 1695 a Fy(\014)2786 1710 y Fs(1)2825 1695 y FA(\(2)p Fy(\034)11 b FA(\))23 b(+)3148 1669 y(_)3124 1695 y Fy(\014)3179 1710 y Fs(2)3218 1695 y FA(\(2)p Fy(\034)33 b Ft(\000)23 b FA(1\))3567 1584 y Fr(\021)1189 1950 y Ft(^)1272 1869 y Fr(\000)1317 1950 y Ft(\000)17 b FA(sin)1548 1869 y Fr(\000)1593 1950 y Fy(\031)t(t=)p FA(2)1785 1869 y Fr(\001)1831 1950 y Fy(\014)1886 1965 y Fs(1)1925 1950 y FA(\(2)p Fy(\034)11 b FA(\))22 b(+)g(cos)2371 1869 y Fr(\000)2416 1950 y Fy(\031)t(t=)p FA(2)2608 1869 y Fr(\001)p 2654 1868 V 81 x Fy(\014)2709 1965 y Fs(1)2748 1950 y FA(\(2)p Fy(\034)11 b FA(\))2926 1869 y Fr(\001)2989 1835 y(\014)2989 1895 y(\014)2989 1955 y(\014)3022 1950 y Fy(d\034)g(dt)1028 2187 y Ft(\024)84 b Fy(\031)1264 2052 y Fr(Z)1319 2277 y Fv(Q)1396 2073 y Fr(\014)1396 2133 y(\014)1396 2192 y(\014)1429 2077 y(\020)1488 2187 y FA(cos)1636 2107 y Fr(\000)1681 2187 y Fy(\031)t(t=)p FA(2)1873 2107 y Fr(\001)1943 2161 y FA(_)1919 2187 y Fy(\014)1974 2202 y Fs(1)2013 2187 y FA(\(2)p Fy(\034)11 b FA(\))22 b(+)g(sin)2448 2107 y Fr(\000)2493 2187 y Fy(\031)t(t=)p FA(2)2685 2107 y Fr(\001)2764 2141 y FA(_)p 2731 2106 V 2731 2187 a Fy(\014)2786 2202 y Fs(1)2825 2187 y FA(\(2)p Fy(\034)11 b FA(\))23 b(+)3148 2161 y(_)3124 2187 y Fy(\014)3179 2202 y Fs(2)3218 2187 y FA(\(2)p Fy(\034)33 b Ft(\000)23 b FA(1\))3567 2077 y Fr(\021)3626 2073 y(\014)3626 2133 y(\014)3626 2192 y(\014)1189 2332 y(\014)1189 2392 y(\014)1222 2336 y(\000)1267 2417 y Ft(\000)17 b FA(sin)1498 2336 y Fr(\000)1544 2417 y Fy(\031)t(t=)p FA(2)1736 2336 y Fr(\001)1781 2417 y Fy(\014)1836 2432 y Fs(1)1875 2417 y FA(\(2)p Fy(\034)11 b FA(\))23 b(+)f(cos)2321 2336 y Fr(\000)2367 2417 y Fy(\031)t(t=)p FA(2)2559 2336 y Fr(\001)p 2604 2336 V 2604 2417 a Fy(\014)2659 2432 y Fs(1)2698 2417 y FA(\(2)p Fy(\034)11 b FA(\))2876 2336 y Fr(\001)2922 2332 y(\014)2922 2392 y(\014)2972 2417 y Fy(d\034)g(dt)1028 2641 y Ft(\024)84 b FA(2)p Fy(\031)1313 2505 y Fr(Z)1412 2532 y Fs(1)1368 2731 y(0)1468 2560 y Fr(\000)1514 2641 y FA(2)p Ft(j)1615 2615 y FA(_)1591 2641 y Fy(\014)1646 2656 y Fs(1)1685 2641 y FA(\(2)p Fy(\034)11 b FA(\))p Ft(j)22 b FA(+)g Ft(j)2064 2615 y FA(_)2039 2641 y Fy(\014)2094 2656 y Fs(2)2133 2641 y FA(\(2)p Fy(\034)34 b Ft(\000)22 b FA(1\))p Ft(j)2510 2560 y Fr(\001)2572 2641 y Ft(j)p Fy(\014)2655 2656 y Fs(1)2694 2641 y FA(\(2)p Fy(\034)11 b FA(\))p Ft(j)17 b Fy(d\034)1028 2920 y Ft(\024)84 b Fy(\031)1264 2785 y Fr(Z)1364 2811 y Fs(1)1319 3010 y(0)1420 2920 y Ft(j)20 b FA(_)-47 b Fy(\013)p FA(\()p Fy(t)p FA(\))p Ft(j)p Fy(dt)1779 2785 y Fr(Z)1879 2811 y Fs(2)1834 3010 y(0)1935 2840 y Fr(\000)1980 2920 y FA(2)p Ft(j)10 b FA(~)-59 b Fy(\013)2120 2879 y Fx(0)2143 2920 y FA(\()p Fy(t)p FA(\))p Ft(j)22 b FA(+)g Ft(j)10 b FA(~)-59 b Fy(\013)2493 2879 y Fx(0)2515 2920 y FA(\()p Fy(t)23 b Ft(\000)f FA(1\))p Ft(j)2825 2840 y Fr(\001)2887 2920 y Fy(dt)1029 3216 y FA(=)84 b(3)p Fy(\031)1313 3076 y Fr(\022)1386 3081 y(Z)1486 3107 y Fs(1)1441 3306 y(0)1542 3216 y Ft(j)20 b FA(_)-47 b Fy(\013)p FA(\()p Fy(t)p FA(\))p Ft(j)p Fy(dt)1857 3076 y Fr(\023)1930 3091 y Fs(2)2003 3216 y Fy(:)24 3441 y FA(Concerning)34 b(the)f(isotropic)g(homotop)m(y)g (\000)1619 3456 y Fs(4)1659 3441 y FA(,)f(w)m(e)i(get)160 3521 y Fr(Z)215 3747 y Fv(Q)292 3657 y Ft(j)p FA(\(\000)419 3672 y Fs(4)458 3657 y FA(\))496 3672 y Fv(\034)561 3657 y Ft(^)22 b FA(\(\000)748 3672 y Fs(4)788 3657 y FA(\))826 3672 y Fv(t)855 3657 y Ft(j)p Fy(d\034)11 b(dt)84 b FA(=)1327 3589 y Fy(\031)p 1327 3634 59 4 v 1332 3725 a FA(2)1412 3521 y Fr(Z)1467 3747 y Fv(Q)1544 3542 y Fr(\014)1544 3602 y(\014)1544 3662 y(\014)1593 3576 y(\000)1639 3657 y FA(cos)1786 3576 y Fr(\000)1832 3657 y Fy(\031)t(t=)p FA(2)2024 3576 y Fr(\001)2069 3657 y FA(\()p 2107 3576 61 4 v Fy(\014)2168 3680 y Fs(1)2207 3657 y Fy(\014)2262 3672 y Fs(2)2302 3657 y FA(\))2340 3616 y Fx(0)2363 3657 y FA(\()p Fy(\034)11 b FA(\))23 b(+)f(sin)2749 3576 y Fr(\000)2795 3657 y Fy(\031)t(t=)p FA(2)2987 3576 y Fr(\001)3032 3657 y Fy(q)3079 3616 y Fx(0)3102 3657 y FA(\()p Fy(\034)11 b FA(\))3231 3576 y Fr(\001)1317 3912 y Ft(^)1400 3831 y Fr(\000)1445 3912 y Ft(\000)17 b FA(sin)1676 3831 y Fr(\000)1721 3912 y Fy(\031)t(t=)p FA(2)1913 3831 y Fr(\001)1959 3912 y FA(\()p Fy(\014)2052 3927 y Fs(1)2091 3912 y Fy(\014)2146 3927 y Fs(2)2186 3912 y FA(\)\()p Fy(\034)11 b FA(\))22 b(+)g(cos)2620 3831 y Fr(\000)2666 3912 y Fy(\031)t(t=)p FA(2)2858 3831 y Fr(\001)2903 3912 y Fy(q)t FA(\()p Fy(\034)11 b FA(\))3079 3831 y Fr(\001)3142 3797 y(\014)3142 3857 y(\014)3142 3917 y(\014)3175 3912 y Fy(d\034)g(dt)1156 4166 y Ft(\024)1327 4098 y Fy(\031)p 1327 4143 59 4 v 1332 4234 a FA(2)1412 4030 y Fr(Z)1512 4057 y Fs(1)1467 4256 y(0)1568 4085 y Fr(\000)1613 4166 y Ft(j)p FA(\()p 1679 4085 61 4 v Fy(\014)1740 4189 y Fs(1)1779 4166 y Fy(\014)1834 4181 y Fs(2)1874 4166 y FA(\))1912 4125 y Fx(0)1935 4166 y FA(\()p Fy(\034)g FA(\))p Ft(j)22 b FA(+)g Ft(j)p Fy(q)2287 4125 y Fx(0)2310 4166 y FA(\()p Fy(\034)11 b FA(\))p Ft(j)2467 4085 y Fr(\001)16 b(\000)2575 4166 y Ft(j)p FA(\()p 2641 4085 V Fy(\014)2701 4189 y Fs(1)2740 4166 y Fy(\014)2795 4181 y Fs(2)2835 4166 y FA(\)\()p Fy(\034)11 b FA(\))p Ft(j)22 b FA(+)g Ft(j)p Fy(q)t FA(\()p Fy(\034)11 b FA(\))p Ft(j)3382 4085 y Fr(\001)3444 4166 y Fy(d\034)24 4406 y FA(By)33 b(de\014nition)h(of)p 722 4325 V 32 w Fy(\014)782 4429 y Fs(1)822 4406 y Fy(\014)877 4421 y Fs(2)916 4406 y FA(,)f(one)g(easily)h(c)m(hec)m(ks)h(that)24 4642 y(\(74\))127 b(max)351 4709 y Fs([0)p Fv(;)p Fs(1])523 4642 y Ft(j)p FA(\()p 589 4561 V Fy(\014)649 4666 y Fs(1)689 4642 y Fy(\014)744 4657 y Fs(2)783 4642 y FA(\))p Ft(j)28 b(\024)g FA(2)1048 4507 y Fr(Z)1147 4533 y Fs(1)1102 4732 y(0)1203 4642 y Ft(j)20 b FA(_)-47 b Fy(\013)p FA(\()p Fy(t)p FA(\))p Ft(j)17 b Fy(dt)97 b FA(and)h Ft(j)p FA(\()p 1953 4561 V Fy(\014)2013 4666 y Fs(1)2052 4642 y Fy(\014)2107 4657 y Fs(2)2147 4642 y FA(\))2185 4601 y Fx(0)2208 4642 y FA(\()p Fy(t)p FA(\))p Ft(j)27 b(\024)i FA(2)2529 4561 y Fr(\000)2574 4642 y Ft(j)10 b FA(~)-59 b Fy(\013)2665 4601 y Fx(0)2688 4642 y FA(\(2)p Fy(t)p FA(\))p Ft(j)21 b FA(+)h Ft(j)10 b FA(~)-59 b Fy(\013)3086 4601 y Fx(0)3109 4642 y FA(\(2)p Fy(t)22 b Ft(\000)h FA(1\))p Ft(j)3468 4561 y Fr(\001)3529 4642 y Fy(:)24 4871 y FA(The)49 b(curv)m(e)h Fy(q)t FA(\()p Fy(t)p FA(\))e(is)h(giv)m(en)g(b)m(y)g(the)g(pro)s(of)e (of)h(Theorem)h(7.3,)j(where)e(it)e(has)h(b)s(een)f(obtained)24 4987 y(applying)38 b(De\014nition)f(6.3)g(with)h Fy(\025)d FA(=)h(length)1743 5010 y Fx(j\001j)1806 4987 y FA(\()p Fy(c)p FA(\).)57 b(It)37 b(follo)m(ws)h(that)f(there)h(exists)h Fy(C)j(>)36 b FA(0)g(suc)m(h)24 5110 y(that)c Ft(j)p Fy(q)t FA(\(0\))p Ft(j)27 b(\024)h Fy(C)c(\025)p FA(.)43 b(Th)m(us,)34 b(w)m(e)g(get)1238 5346 y Ft(j)p Fy(q)t FA(\()p Fy(t)p FA(\))p Ft(j)27 b(\024)i Fy(C)7 b(\025)22 b FA(+)1848 5279 y Fy(C)p 1848 5323 77 4 v 1858 5414 a(\025)1952 5210 y Fr(Z)2052 5237 y Fv(t)2007 5436 y Fs(0)2098 5346 y Ft(j)p Fy(\033)t FA(\()p Fy(t)p FA(\))p Ft(j)17 b Fy(dt)g(:)p eop end %%Page: 43 43 TeXDict begin 43 42 bop 169 246 a Fz(CONT)-6 b(A)n(CT)34 b(EQUA)-6 b(TIONS,)31 b(LIPSCHITZ)i(EXTENSIONS)f(AND)g(ISOPERIMETRIC)h (INEQUALITIES)67 b(43)24 446 y FA(Recall)33 b(that)974 623 y Fy(\033)t FA(\()p Fy(t)p FA(\))28 b(=)1331 498 y Fv(s)1275 528 y Fr(X)1283 740 y Fv(k)r Fs(=1)1436 623 y Fy(!)1501 581 y Fv(k)1559 542 y Fr(\000)1605 623 y FA(\()p 1643 541 61 4 v Fy(\014)1704 646 y Fs(1)1743 623 y Fy(\014)1798 638 y Fs(2)1838 623 y FA(\)\()p Fy(t)p FA(\))22 b Ft(^)g FA(\()p 2135 541 V Fy(\014)2196 646 y Fs(1)2235 623 y Fy(\014)2290 638 y Fs(2)2330 623 y FA(\))2368 581 y Fx(0)2391 623 y FA(\()p Fy(t)p FA(\))2502 542 y Fr(\001)2581 623 y Fy(Z)2648 638 y Fv(k)2707 623 y Fy(;)24 867 y FA(then)33 b(bilinearit)m(y)h(of)e Fy(!)888 830 y Fv(k)930 867 y FA('s)h(yields)i Fy(C)1373 882 y Fs(1)1440 867 y Fy(>)27 b FA(0)32 b(suc)m(h)i(that)1142 1030 y Ft(j)p Fy(\033)t FA(\()p Fy(t)p FA(\))p Ft(j)27 b(\024)h Fy(C)1570 1045 y Fs(1)1626 1030 y Ft(j)p FA(\()p 1692 948 V Fy(\014)1752 1053 y Fs(1)1791 1030 y Fy(\014)1846 1045 y Fs(2)1886 1030 y FA(\)\()p Fy(t)p FA(\))p Ft(j)17 b(j)p FA(\()p 2146 948 V Fy(\014)2205 1053 y Fs(1)2245 1030 y Fy(\014)2300 1045 y Fs(2)2339 1030 y FA(\))2377 988 y Fx(0)2401 1030 y FA(\()p Fy(t)p FA(\))p Ft(j)p Fy(;)24 1192 y FA(hence)34 b(in)f(view)h(of)39 b(\(74\))o(,)33 b(it)f(follo)m(ws)i(that)24 1356 y(\(75\))839 b Ft(j)p Fy(\033)t FA(\()p Fy(t)p FA(\))p Ft(j)27 b(\024)h FA(4)p Fy(C)1514 1371 y Fs(1)1570 1356 y Fy(\025)1644 1275 y Fr(\000)1689 1356 y Ft(j)10 b FA(~)-59 b Fy(\013)1780 1315 y Fx(0)1802 1356 y FA(\(2)p Fy(t)p FA(\))p Ft(j)22 b FA(+)g Ft(j)10 b FA(~)-59 b Fy(\013)2201 1315 y Fx(0)2224 1356 y FA(\(2)p Fy(t)22 b Ft(\000)h FA(1\))p Ft(j)2583 1275 y Fr(\001)2644 1356 y Fy(:)24 1519 y FA(As)33 b(a)f(result,)i(w)m (e)g(get)1275 1640 y(max)1265 1707 y Fv(t)p Fx(2)p Fs([0)p Fv(;)p Fs(1])1483 1640 y Ft(j)p Fy(q)t FA(\()p Fy(t)p FA(\))p Ft(j)27 b(\024)i Fy(C)23 b(\025)17 b FA(\(1)k(+)h(4)p Fy(C)2322 1655 y Fs(1)2361 1640 y FA(\))17 b Fy(;)24 1837 y FA(that)32 b(implies)188 1927 y Fr(Z)243 2153 y Fv(Q)319 2063 y Ft(j)p FA(\(\000)446 2078 y Fs(4)485 2063 y FA(\))523 2078 y Fv(\034)589 2063 y Ft(^)22 b FA(\(\000)776 2078 y Fs(4)815 2063 y FA(\))853 2078 y Fv(t)883 2063 y Ft(j)p Fy(d\034)11 b(dt)83 b Ft(\024)g Fy(\025)1439 1995 y(\031)p 1439 2040 59 4 v 1444 2131 a FA(2)1535 1982 y Fr(\000)1581 2063 y FA(2)22 b(+)g Fy(C)29 b FA(+)22 b(4)p Fy(C)2066 2078 y Fs(1)2105 2063 y Fy(C)2182 1982 y Fr(\001)2244 1927 y(Z)2344 1953 y Fs(1)2300 2153 y(0)2400 1982 y Fr(\000)2446 2063 y Ft(j)p FA(\()p 2512 1982 61 4 v Fy(\014)2572 2086 y Fs(1)2611 2063 y Fy(\014)2666 2078 y Fs(2)2706 2063 y FA(\))2744 2022 y Fx(0)2767 2063 y FA(\()p Fy(\034)11 b FA(\))p Ft(j)22 b FA(+)g Ft(j)p Fy(q)3119 2022 y Fx(0)3142 2063 y FA(\()p Fy(\034)11 b FA(\))p Ft(j)3299 1982 y Fr(\001)3372 2063 y Fy(d\034)28 b(:)24 2289 y FA(T)-8 b(aking)33 b(in)m(to)g(accoun) m(t)g(that)477 2380 y Fr(Z)577 2406 y Fs(1)533 2605 y(0)633 2515 y Ft(j)18 b FA(_)-45 b Fy(q)t FA(\()p Fy(t)p FA(\))p Ft(j)17 b Fy(dt)27 b Ft(\024)1092 2448 y Fy(C)p 1092 2492 77 4 v 1102 2584 a(\025)1195 2380 y Fr(Z)1295 2406 y Fs(1)1251 2605 y(0)1351 2515 y Ft(j)p Fy(\033)t FA(\()p Fy(t)p FA(\))p Ft(j)17 b Fy(dt)27 b Ft(\024)h FA(2)17 b Fy(C)23 b(C)2041 2530 y Fs(1)2096 2380 y Fr(Z)2196 2406 y Fs(1)2152 2605 y(0)2252 2515 y Ft(j)p FA(\()p 2318 2434 61 4 v Fy(\014)2378 2539 y Fs(1)2418 2515 y Fy(\014)2473 2530 y Fs(2)2512 2515 y FA(\))2550 2474 y Fx(0)2574 2515 y FA(\()p Fy(t)p FA(\))p Ft(j)k(\024)h FA(4)17 b Fy(C)23 b(C)3074 2530 y Fs(1)3130 2515 y Fy(\025)17 b(;)24 2729 y FA(w)m(e)34 b(are)e(then)h(lead)g(to)g(the)g(follo)m (wing)552 2803 y Fr(Z)607 3028 y Fv(Q)684 2938 y Ft(j)p FA(\(\000)811 2953 y Fs(4)850 2938 y FA(\))888 2953 y Fv(\034)953 2938 y Ft(^)22 b FA(\(\000)1140 2953 y Fs(4)1180 2938 y FA(\))1218 2953 y Fv(t)1247 2938 y Ft(j)p Fy(d\034)11 b(dt)83 b Ft(\024)h Fy(\031)1795 2858 y Fr(\000)1841 2938 y FA(2)22 b(+)g Fy(C)29 b FA(+)22 b(4)p Fy(C)2326 2953 y Fs(1)2365 2938 y Fy(C)2442 2858 y Fr(\001)16 b(\000)2550 2938 y FA(1)22 b(+)g(2)p Fy(C)7 b(C)2915 2953 y Fs(1)2954 2858 y Fr(\001)3016 2938 y Fy(\025)3073 2897 y Fs(2)3129 2938 y Fy(:)24 3165 y FA(Finally)-8 b(,)33 b(b)m(y)g(de\014nition)h(of) e(Allco)s(c)m(k)i(group)e(and)h(applying)g(De\014nition)g(6.3,)f(w)m(e) i(ha)m(v)m(e)552 3272 y Fr(Z)607 3497 y Fv(Q)683 3408 y Ft(j)p FA(\(\000)810 3423 y Fs(5)849 3408 y FA(\))887 3423 y Fv(\034)952 3408 y Ft(^)23 b FA(\(\000)1140 3423 y Fs(5)1179 3408 y FA(\))1217 3423 y Fv(t)1247 3408 y Ft(j)p Fy(d\034)11 b(dt)83 b Ft(\024)100 b Fy(C)1835 3267 y Fr(\022)1908 3272 y(Z)2008 3298 y Fs(1)1964 3497 y(0)2064 3408 y Ft(j)18 b FA(_)-45 b Fy(q)s FA(\()p Fy(t)p FA(\))p Ft(j)17 b Fy(dt)2380 3267 y Fr(\023)2453 3282 y Fs(2)2520 3408 y Ft(\024)28 b FA(16)17 b Fy(C)2817 3366 y Fs(3)2872 3408 y Fy(C)2949 3366 y Fs(2)2942 3432 y(1)3016 3408 y Fy(\025)3073 3366 y Fs(2)3129 3408 y Fy(:)24 3640 y FA(Joining)33 b(the)g(previous)h(estimates)g(with)f (\(73\),)f(estimate)i(\(72\))e(follo)m(ws.)44 b Fl(2)24 3817 y FD(Remark)g(8.7.)h FA(The)39 b(previous)h(theorem)f(could)g(b)s (e)g(seen)g(as)g(a)f(completion)h(of)f(Theorem)h(7.3,)24 3933 y(where)34 b(w)m(e)f(ha)m(v)m(e)h(\014xed)g(our)e(atten)m(tion)i (on)e(the)h(area)f(enclosed)j(b)m(y)e(the)g(extension)i Fy(')p FA(.)124 4111 y FB(Pr)n(oof)55 b(of)h(Theorem)f(1.5.)92 b FA(Up)49 b(to)f(left)h(translation,)k(w)m(e)d(can)f(assume)h(that)f (\000\(1)p Fy(;)17 b FA(0\))24 4227 y(coincides)34 b(with)g(the)f(unit) g(elemen)m(t.)45 b(Th)m(us,)34 b(w)m(e)g(de\014ne)995 4475 y(\000)27 b(=)h(exp)18 b Ft(\016)1420 4305 y Fr( )1548 4351 y Fv(n)1498 4381 y Fr(X)1513 4591 y Fv(i)p Fs(=1)1709 4351 y Fv(n)1658 4381 y Fr(X)1669 4591 y Fv(j)t Fs(=1)1819 4475 y Fy(c)1861 4490 y Fv(ij)1938 4475 y Fy(X)2019 4490 y Fv(ij)2102 4475 y FA(+)2255 4351 y Fv(s)2200 4381 y Fr(X)2207 4593 y Fv(k)r Fs(=1)2360 4475 y Fy(z)2405 4490 y Fv(k)2465 4475 y Fy(Z)2532 4490 y Fv(k)2574 4305 y Fr(!)2686 4475 y Fy(;)24 4747 y FA(where)41 b Fy(c)e FA(:)g Fy(S)526 4711 y Fs(1)605 4747 y Ft(\000)-17 b(!)39 b Fu(R)876 4711 y Fv(mn)986 4747 y FA(.)63 b(Arguing)40 b(as)f(in)h(the)g(b)s(eginning)g(of)e(the)i(pro)s(of)f(of)g(Theorem)h (7.8,)h(it)24 4863 y(follo)m(ws)34 b(that)557 4783 y Fr(R)604 4898 y Fv(c)655 4863 y Fy(\022)e FA(=)c(0)33 b(and)g Fy(c)p FA(\(1)p Fy(;)17 b FA(0\))28 b(=)h(0.)44 b(Then)35 b(w)m(e)f(apply)g(Theorem)g(8.6,)f(getting)g(a)g(Lipsc)m (hitz)24 4979 y(extension)h Fy(')28 b FA(:)g Fy(D)i Ft(\000)-16 b(!)27 b Fu(R)973 4943 y Fv(mn)1115 4979 y FA(suc)m(h)34 b(that)24 5189 y(\(76\))1058 5053 y Fr(Z)1113 5279 y Fv(D)1194 5189 y Ft(j)p Fy(@)1273 5204 y Fv(x)1313 5213 y Fp(1)1351 5189 y Fy(')22 b Ft(^)h Fy(@)1577 5204 y Fv(x)1617 5213 y Fp(2)1656 5189 y Fy(')p Ft(j)17 b Fy(dx)27 b Ft(\024)h Fy(K)35 b FA(length)2386 5213 y Fx(j\001j)2449 5189 y FA(\()p Fy(c)p FA(\))2567 5148 y Fs(2)2623 5189 y Fy(;)24 5403 y FA(Finally)-8 b(,)33 b(Prop)s(osition)g(8.2)f(and)h (Prop)s(osition)f(8.5)h(lead)g(us)g(to)f(the)h(conclusion.)45 b Fl(2)p eop end %%Page: 44 44 TeXDict begin 44 43 bop 24 246 a Fz(44)1315 b(V)-9 b(ALENTINO)25 b(MA)n(GNANI)24 446 y FD(Example)i(8.8.)34 b FA(Let)22 b(us)h(assume)h(the)e(setting)h(in)g(the)g(pro)s(of)e(of)h(Theorem)h (1.5,)h(where)g(the)f(Allco)s(c)m(k)24 562 y(group)29 b(is)h(the)g(7-dimensional)g(Heisen)m(b)s(erg)h(group)e Fu(H)2015 526 y Fs(2)2055 562 y FA(.)42 b(According)30 b(to)f(this)h(prop)s(osition,)g(w)m(e)g(set)24 678 y Fy(\036)d FA(=)h(\()p Fy(';)17 b( )t FA(\).)43 b(W)-8 b(e)33 b(equip)h Fu(H)1042 642 y Fs(2)1114 678 y FA(with)f(suitable)h (graded)f(co)s(ordinates)g(\()p Fy(x)2631 693 y Fs(1)2671 678 y Fy(;)17 b(:)g(:)g(:)e(;)i(x)2944 693 y Fs(5)2984 678 y FA(\))32 b(suc)m(h)i(that)649 1087 y Ft(r)p Fy(\036)p FA(\()p Fy(x)p FA(\))28 b(=)1053 767 y Fr(0)1053 942 y(B)1053 1002 y(B)1053 1062 y(B)1053 1122 y(B)1053 1185 y(@)1849 853 y Ft(r)p Fy(')1996 868 y Fs(1)1849 969 y Ft(r)p Fy(')1996 984 y Fs(2)1849 1085 y Ft(r)p Fy(')1996 1100 y Fs(3)1849 1201 y Ft(r)p Fy(')1996 1216 y Fs(4)1181 1317 y Fy(')1245 1332 y Fs(1)1285 1317 y Ft(r)p Fy(')1432 1332 y Fs(3)1493 1317 y Ft(\000)23 b Fy(')1657 1332 y Fs(3)1696 1317 y Ft(r)p Fy(')1843 1332 y Fs(1)1905 1317 y FA(+)f Fy(')2067 1332 y Fs(2)2106 1317 y Ft(r)p Fy(')2253 1332 y Fs(3)2314 1317 y Ft(\000)h Fy(')2478 1332 y Fs(3)2517 1317 y Ft(r)p Fy(')2664 1332 y Fs(2)2745 767 y Fr(1)2745 942 y(C)2745 1002 y(C)2745 1062 y(C)2745 1122 y(C)2745 1185 y(A)2866 1087 y Fy(;)-2869 b FA(\(77\))24 1495 y(where)38 b(the)g(con)m(tact)g(equations)g(imply)g(that)f Ft(r)p Fy( )k FA(is)d(equal)g(to)e(the)i(last)f(ro)m(w)h(of)43 b(\(77\).)57 b(Then)38 b(a)24 1611 y(simple)c(computation)f(yields)966 1789 y Ft(j)p Fy(@)1045 1804 y Fv(x)1085 1813 y Fp(1)1124 1789 y Fy(\036)22 b Ft(^)g Fy(@)1343 1804 y Fv(x)1383 1813 y Fp(2)1422 1789 y Fy(\036)p Ft(j)27 b(\024)1641 1699 y Fr(p)p 1740 1699 377 4 v 1740 1789 a FA(1)22 b(+)g(3)p Ft(j)p Fy(')p Ft(j)2078 1760 y Fs(2)2144 1789 y Ft(j)p Fy(@)2223 1804 y Fv(x)2263 1813 y Fp(1)2302 1789 y Fy(')g Ft(^)g Fy(@)2527 1804 y Fv(x)2567 1813 y Fp(2)2606 1789 y Fy(')p Ft(j)17 b Fy(:)24 1989 y FA(No)m(w,)30 b(if)d Fy(')h FA(is)h(the)f(extension)i(pro)m(vided)f(b)m(y)g(Theorem)h(8.6)d (and)h Fy(f)39 b FA(=)27 b(exp)2758 1878 y Fr(\020)2817 1914 y(P)2923 1940 y Fs(4)2923 2018 y Fv(j)t Fs(=1)3066 1989 y Fy(')3130 2004 y Fv(j)3183 1989 y Fy(X)3264 2004 y Fv(j)3322 1989 y FA(+)c Fy( )d(X)3585 2004 y Fs(5)3625 1878 y Fr(\021)24 2131 y FA(is)33 b(the)g(corresp)s(onding)h(Lipsc)m (hitz)g(mapping,)f(then)917 2325 y Ft(H)1002 2284 y Fs(2)1001 2349 y Fx(j\001j)1081 2325 y FA(\()p Fy(f)11 b FA(\()p Fy(D)s FA(\)\))27 b Ft(\024)h Fy(C)1601 2214 y Fr(\020)1661 2325 y FA(1)22 b(+)g(2)17 b(max)1956 2387 y Fv(D)2093 2325 y Ft(j)p Fy(')p Ft(j)2213 2214 y Fr(\021)2305 2325 y FA(length)2571 2348 y Fx(j\001j)2634 2325 y FA(\()p Fy(c)p FA(\))2752 2284 y Fs(2)24 2526 y FA(for)32 b(a)g(suitable)i (geometric)f(constan)m(t)h Fy(C)g(>)28 b FA(0.)1564 2754 y FB(References)66 2912 y FC([1])41 b Ff(D.Allcock)p FC(,)25 b Fe(A)n(n)k(isop)l(erimetric)j(ine)l(quality)e(for)h(the)f (Heisenb)l(er)l(g)g(gr)l(oups)p FC(,)e(GAF)-9 b(A,)28 b Fd(8)p FC(,)g(219-233,)c(\(1998\))66 3012 y([2])41 b Ff(J.Alonso)p FC(,)36 b Fe(In)n(\023)-40 b(egalit)n(\023)g(es)38 b(isop)n(\023)-40 b(erim)n(\023)g(etriques)39 b(et)e(quasi-isome)n (\023)-40 b(etries)p FC(,)40 b(C.R.Acad.Sci.P)n(aris,)c Fd(311)p FC(,)h(761-764,)195 3111 y(\(1990\))66 3211 y([3])k Ff(F.J.Almgren,)28 b(Jr.)p FC(,)f Fe(The)i(homotopy)i(gr)l (oups)d(of)i(the)e(inte)l(gr)l(al)h(cycle)h(gr)l(oups)p FC(,)d(T)-7 b(op)r(ology)g(,)25 b Fd(1)p FC(,)h(\(1962\),)g(257-)195 3310 y(299.)66 3410 y([4])41 b Ff(Z.M.Balogh,)31 b(R.Hofer-Isenegger,)j (J.T.Tyson)p FC(,)29 b Fe(Lifts)j(of)g(Lipschitz)g(maps)g(and)g (horizontal)g(fr)l(ac-)195 3510 y(tals)e(in)f(the)h(Heisenb)l(er)l(g)g (gr)l(oup)p FC(,)e(Ergo)r(dic)e(Theory)h(Dynam.)h(Systems,)f Fd(26)p FC(,)h(621-651,)c(\(2006\))66 3609 y([5])41 b Ff(M.R.Bridson)p FC(,)24 b Fe(The)j(ge)l(ometry)g(of)g(the)f(wor)l(d)h (pr)l(oblem)p FC(,)f(In)n(vitations)e(to)f(geometry)g(and)h(top)r (ology)-7 b(,)23 b(Oxford)195 3709 y(Univ)n(ersit)n(y)j(Press,)g (p.29-91,)g(\(2002\))66 3809 y([6])41 b Ff(J.Burillo,)36 b(J.T)-7 b(aba)n(ck)32 b Fe(Equivalenc)l(e)k(of)f(ge)l(ometric)g(and)g (c)l(ombinatorial)i(Dehn)d(functions)p FC(.)f(New)g(Y)-7 b(ork)195 3908 y(J.)27 b(Math.)h Fd(8)p FC(,)f(169-179,)e(\(2002\))66 4008 y([7])41 b Ff(L.Capogna,)33 b(M.Co)n(wling)p FC(,)d Fe(Conformality)k(and)e(Q-Harmonicity)g(in)g(Carnot)g(gr)l(oups)p FC(,)f(Duk)n(e)e(Math.)h(J.)195 4108 y Fd(135)p FC(,)d(n.3,)g(455-479,) e(\(2006\))66 4207 y([8])41 b Ff(D.B.A.Epstein,)35 b(J.W.Cannon,)g (D.F.Hol)-6 b(t,)35 b(S.V.F.Levy,)g(M.S.P)-7 b(a)h(terson,)36 b(W.P.Thurston)p FC(,)195 4307 y Fe(Wor)l(d)30 b(Pr)l(o)l(c)l(essing)g (in)g(Gr)l(oups)p FC(,)e(Jones)f(and)g(Bartlett,)h(Boston-London,)d (\(1992\))66 4406 y([9])41 b Ff(I.Cha)-7 b(vel)p FC(,)21 b Fe(R)n(iemannian)i(Ge)l(ometry.)i(A)e(Mo)l(dern)h(Intr)l(o)l (duction,)h(Se)l(c)l(ond)e(e)l(dition)p FC(,)h(Cam)n(bridge)19 b(Univ)n(ersit)n(y)195 4506 y(Press,)26 b(\(2006\).)24 4606 y([10])41 b Ff(J.Cheeger,)i(B.Kleiner)p FC(,)38 b Fe(Di\013er)l(entiating)g(maps)g(into)g Fc(L)2204 4576 y Fb(1)2278 4606 y Fe(and)g(the)g(ge)l(ometry)g(of)h(BV)f(functions)p FC(,)g(to)195 4705 y(app)r(ear)26 b(on)i(Annals)f(of)h(Mathematics)24 4805 y([11])41 b Ff(N.S.D)n(airbek)n(o)n(v)p FC(,)47 b Fe(Mappings)g(with)f(b)l(ounde)l(d)g(distortion)g(of)h(two-step)e (Carnot)h(gr)l(oups)p FC(,)j(Pro)r(c.)44 b(Anal.)195 4905 y(Geom.,)27 b(122-155,)e(Sob)r(olev)i(Institute)h(Press,)e(No)n(v) n(osibirsk,)g(\(2000\))24 5004 y([12])41 b Ff(L.C.Ev)-7 b(ans,)23 b(R.F.Gariepy)p FC(,)g Fe(Me)l(asur)l(e)h(the)l(ory)g(and)g (\014ne)g(pr)l(op)l(erties)h(of)f(functions)p FC(,)f(Studies)e(in)h (Adv)-5 b(anced)195 5104 y(Mathematics,)27 b(CR)n(C)g(Press,)g(Bo)r(ca) f(Raton,)i(\(1992\))24 5203 y([13])41 b Ff(H.Federer)p FC(,)27 b Fe(Ge)l(ometric)k(Me)l(asur)l(e)f(The)l(ory)p FC(,)f(Springer,)e(\(1969\).)24 5303 y([14])41 b Ff(G.B.)23 b(F)n(olland,)i(E.M.)f(Stein)p FC(,)f Fe(Har)l(dy)i(Sp)l(ac)l(es)f(on)h (Homo)l(gene)l(ous)f(gr)l(oups)p FC(,)f(Princeton)e(Univ)n(ersit)n(y)g (Press,)195 5403 y(\(1982\))p eop end %%Page: 45 45 TeXDict begin 45 44 bop 169 246 a Fz(CONT)-6 b(A)n(CT)34 b(EQUA)-6 b(TIONS,)31 b(LIPSCHITZ)i(EXTENSIONS)f(AND)g(ISOPERIMETRIC)h (INEQUALITIES)67 b(45)24 446 y FC([15])41 b Ff(N.Gar)n(of)-7 b(alo,)46 b(D.M.Nhieu)p FC(,)e Fe(Lipschitz)f(c)l(ontinuity,)j(glob)l (al)d(smo)l(oth)g(appr)l(oximation)h(and)e(extension)195 545 y(the)l(or)l(ems)f(for)h(Sob)l(olev)h(functions)e(in)g(Carnot-Car)l (ath)n(\023)-40 b(eo)l(dory)44 b(sp)l(ac)l(es)p FC(,)g(Jour.)c(Anal.)g (Math.,)j Fd(74)p FC(,)g(67-97)195 645 y(\(1998\))24 745 y([16])e Ff(M.Gr)n(omo)n(v)p FC(,)30 b Fe(Hyp)l(erb)l(olic)j(gr)l (oups)p FC(,)f(in)e(\\Essa)n(ys)e(in)i(Group)g(Theory")e (\(S.M.Gersten,)k(ed.\),)f(MSRI)f(Publi-)195 844 y(cations)c Fd(8)p FC(,)i(75-263,)d(Springer-V)-7 b(erlag,)25 b(\(1987\))24 944 y([17])41 b Ff(M.Gr)n(omo)n(v)p FC(,)e Fe(Asymptotic)g(invariants)g (for)h(in\014nite)e(gr)l(oups)p FC(,)i(in)e(\\Geometric)e(Group)h (Theory",)h(v)n(ol.)e(2)195 1043 y(\(G.A.Niblo,)k(M.A.Roller,)g (eds.\),)f(London)e(Mathematical)g(So)r(ciet)n(y)f(Lecture)h(Notes,)j Fd(182)p FC(,)e(Cam)n(bridge)195 1143 y(Univ)n(ersit)n(y)26 b(Press,)g(\(1993\))24 1243 y([18])41 b Ff(M.Gr)n(omo)n(v)p FC(,)29 b Fe(Carnot-Car)l(ath)n(\023)-40 b(eo)l(dory)34 b(sp)l(ac)l(es)e(se)l(en)g(fr)l(om)g(within)p FC(,)e(Subriemannian)g (Geometry)-7 b(,)29 b(Progress)195 1342 y(in)f(Mathematics,)f Fd(144)p FC(.)g(ed.)h(b)n(y)f(A.Bellaic)n(he)g(and)g(J.Risler,)g (Birkhauser)f(V)-7 b(erlag,)27 b(Basel,)f(1996.)24 1442 y([19])41 b Ff(P.Haj )-27 b(lasz,)24 b(P.K)n(oskela)p FC(,)d Fe(Sob)l(olev)k(met)f(Poinc)l(ar)l(e)p FC(,)g(Memoirs)d(of)h (the)g(American)f(Mathematical)g(So)r(ciet)n(y)-7 b(,)195 1542 y Fd(688)p FC(,)27 b(\(2000\).)24 1641 y([20])41 b Ff(A.K)n(or)464 1634 y(\023)461 1641 y(anyi,)g(H.M.Reimann)p FC(,)e Fe(F)-6 b(oundation)39 b(for)g(the)g(The)l(ory)h(of)f(Quasic)l (onformal)h(Mappings)g(on)f(the)195 1741 y(Heisenb)l(er)l(g)30 b(Gr)l(oup)p FC(,)e(Adv.)g(Math.,)g Fd(111)p FC(,)e(1-87,)g(\(1995\).) 24 1840 y([21])41 b Ff(U.Lang,)46 b(T.Schlichenmaier)p FC(,)d Fe(Nagata)f(dimension,)j(quasisymmetric)d(emb)l(e)l(ddings,)k (and)41 b(Lipschitz)195 1940 y(extensions)p FC(,)27 b(In)n(t.)h(Math.)g (Res.)f(Not.)h Fd(58)p FC(,)f(3625-3655,)d(\(2005\).)24 2040 y([22])41 b Ff(V.Ma)n(gnani)p FC(,)32 b Fe(Di\013er)l(entiability) j(and)f(A)n(r)l(e)l(a)f(formula)i(on)f(str)l(ati\014e)l(d)f(Lie)h(gr)l (oups)p FC(,)g(Houston)d(Jour.)g(Math.,)195 2139 y Fd(27)p FC(,)c(n.2,)g(297-323,)e(\(2001\))24 2239 y([23])41 b Ff(V.Ma)n(gnani)p FC(,)23 b Fe(Lipschitz)j(c)l(ontinuity,)g(A)n (leksandr)l(ov)g(the)l(or)l(em)f(and)g(char)l(acterizations)i(for)f (H-c)l(onvex)e(func-)195 2339 y(tions)p FC(,)j(Math.)h(Ann.,)h Fd(334)p FC(,)e(n.1,)g(199-233,)d(\(2006\))24 2438 y([24])41 b Ff(V.Ma)n(gnani)p FC(,)35 b Fe(Elements)h(of)h(Ge)l(ometric)g(Me)l (asur)l(e)g(The)l(ory)g(on)f(Sub-R)n(iemannian)g(gr)l(oups)p FC(,)h(PhD)d(theses)195 2538 y(series)26 b(of)i(Scuola)f(Normale)f(Sup) r(eriore,)h(\(2002\))24 2637 y([25])41 b Ff(V.Ma)n(gnani)p FC(,)27 b Fe(T)-6 b(owar)l(ds)31 b(Di\013er)l(ential)f(Calculus)g(in)g (str)l(ati\014e)l(d)g(gr)l(oups)p FC(,)e(arXiv:math/0701322v2,)22 b(\(2007\))24 2737 y([26])41 b Ff(A.Y.Olshanski)r(i,)29 b(M.V.Sapir)p FC(,)e Fe(Quadr)l(atic)k(isometric)g(functions)f(of)h (the)g(Heisenb)l(er)l(g)f(gr)l(oups.)h(A)f(c)l(om-)195 2837 y(binatorial)h(pr)l(o)l(of.)f FC(J.)d(Math.)h(Sci.,)g Fd(93)p FC(,)f(n.6,)g(921-927,)e(\(1999\))24 2936 y([27])41 b Ff(M.R)n(umin)p FC(,)36 b Fe(Un)g(c)l(omplexe)i(de)f(formes)h(di\013) n(\023)-40 b(er)l(entiel)t(les)38 b(sur)e(les)i(vari)n(\023)-40 b(et)n(\023)g(es)37 b(de)h(c)l(ontact,)f FC(C.)e(R.)h(Acad.)f(Sci.)195 3036 y(P)n(aris)25 b(S)n(\023)-39 b(er.)27 b(I)g(Math.)h Fd(310)p FC(,)f(n.6,)g(401-404,)e(\(1990\))24 3136 y([28])41 b Ff(P.P)-7 b(ansu)p FC(,)37 b Fe(M)n(\023)-40 b(etriques)39 b(de)f(Carnot-Car)l(ath)n(\023)-40 b(eo)l(dory)41 b(quasiisom)n(\023) -40 b(etries)40 b(des)f(esp)l(ac)l(es)g(sym)n(\023)-40 b(etriques)38 b(de)h(r)l(ang)195 3235 y(un)p FC(,)27 b(Ann.)h(Math.,)g Fd(129)p FC(,)f(1-60,)f(\(1989\))24 3335 y([29])41 b Ff(P.Petersen)p FC(,)27 b Fe(R)n(iemannian)j(Ge)l (ometry,)h(Se)l(c)l(ond)f(e)l(dition)p FC(,)e(Springer,)f(New)h(Y)-7 b(ork,)27 b(\(2006\))24 3434 y([30])41 b Ff(H.M.Reimann,)e(H.)g(M.,)h (F.Ricci)35 b Fe(The)i(c)l(omplexi\014e)l(d)h(Heisenb)l(er)l(g)f(gr)l (oup)p FC(,)g(Pro)r(ceedings)c(on)i(Analysis)195 3534 y(and)24 b(Geometry)g(\(Russian\))g(\(No)n(v)n(osibirsk)f(Ak)-5 b(ademgoro)r(dok,)23 b(1999\),)h(465{480,)d(Izdat.)k(Ross.)f(Ak)-5 b(ad.)24 b(Nauk)195 3634 y(Sib.)k(Otd.)f(Inst.)h(Mat.,)g(No)n(v)n (osibirsk,)d(\(2000\).)24 3733 y([31])41 b Ff(R.Young)p FC(,)49 b Fe(Sc)l(ale)l(d)e(r)l(elators)g(and)g(Dehn)g(functions)f(for) h(nilp)l(otent)f(gr)l(oups)p FC(,)51 b(arXiv:math/0601297v3,)195 3833 y(\(2006\).)24 3933 y([32])41 b Ff(R.Young)p FC(,)27 b Fe(Fil)t(ling)k(ine)l(qualities)g(for)g(nilp)l(otent)e(gr)l(oups)p FC(,)f(arXiv:math/0608174v4)23 b(\(2008\))24 4032 y([33])41 b Ff(V.S.V)-10 b(arad)n(arajan)p FC(,)43 b Fe(Lie)g(gr)l(oups,)j(Lie)d (algebr)l(as)h(and)e(their)h(r)l(epr)l(esentation)p FC(,)j(Springer-V) -7 b(erlag,)42 b(New)195 4132 y(Y)-7 b(ork,)27 b(\(1984\).)124 4319 y Ff(V)-10 b(alentino)25 b(Ma)n(gnani,)h(Dip)-6 b(ar)g(timento)27 b(di)e(Ma)-6 b(tema)g(tica,)28 b(Lar)n(go)d(Pontecor) -7 b(v)n(o)25 b(5,)h(I-56127,)f(Pisa)124 4418 y Fe(E-mail)30 b(addr)l(ess)7 b FC(:)38 b Fa(magnani@dm.unipi.)o(it)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF