Content-Type: multipart/mixed; boundary="-------------0509011046930" This is a multi-part message in MIME format. ---------------0509011046930 Content-Type: text/plain; name="05-301.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-301.keywords" Kolmogorov Theorem, KAM Theory, Hamiltonian systems ---------------0509011046930 Content-Type: application/postscript; name="kolmogorov.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="kolmogorov.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: kolmo.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR12 CMR10 CMTT10 CMBX10 CMSY10 CMMI10 CMR7 %%+ CMMI7 CMSL10 MSBM10 CMMI8 CMSY8 CMR8 CMEX10 CMMI6 MSBM7 CMTI10 CMR6 %%+ CMSY6 MSAM10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips kolmo %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.09.01:1659 %%BeginProcSet: 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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 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1C87678CE98C24B934A76220 4DD9B2FF3A49786028E35DDE10AD2C926BD30AD47015FFE9469DE1F793D1C53A C8812CBCD402444EAEA7A50EC5FD93D6A04C2783B50EA48059E3E7407537CB8D 4C206846EF0764C05289733920E2399E58AD8F137C229F3CE3E34D2D1EAB2D53 20D44EFAC8EFA4D14A2EFE389D952527F98D0E49BD5BD2C8D58FF9CB9C78D974 75C2AB5467D73D2B5E277A3FDC35909938A9DF0EB91BD9159D3437BE22EE4544 3429AC8E2BFBE34AE54D3BA3AD04BDF3F4F43A2B43992DF88678681B3AB32CFD A23E2C98D1AF00AB206AC95B78BBE6316F7A0AB6BD3236C28C76288B3C25D1EB E9ABB3576C5EC15A71D26177F5883E9B48293D59015615E2EEAF2E9BA04151ED 5497B9A1C41CBA44BAFF13EA218F5EAC11952EE336AD1DBE6CE92F002EAA3B3D 3BE4C3792F3405763C4BD93EFC3B4FC34193439561841BA989DD8D9F9AEE7A7B 24AEB4654B35023C9720B8F31AA9452E29753FB7915CB29977E725611E37C0B7 784BCC26FACF8A7A0EB1E54290D27FFE52B2D87FAD080AD15EE1984C37E0EB30 122C3012D3A16B09C28903D138352AB5462674B6CFB63F1371768D094DDF288C 36FB9B58443F872D61F2CD8CED42FE0EFF3D7E9952A172BB1AFECB60BF79F2B6 04265FDE4F78BC9FD619AA733CD0412F1D9A7C13B271BF827DCBDC8ABAE24FF0 74D3C220621D7FF0EFE62D835A221D0A7C139E2E6681FC2BBA58FA3B80D416EC 3854C63BA040A4262B458340DAA18AA6AEA3BBAC61615CB85982B18664D3D3AF 340C65B969071CF2D0CABEB80E04623D0526F862ECA8280EEE236C535F70561A 854181132E677674AD5E14C6636F57541D3C821F0776D2CB9B8526D4B826791A 0B179B387D47B08982D36BD61C90BA2CDFE045E8BBAE1C5D4DBA9760AC2D9587 67FB0C22BBD0A905E37D82FD03BBFA69003B5FC6EFCA8CCA8516F461AA33D6F3 6F0B583AECD7A2374014C81033772FE29A7CC38DB81E8462044EF083ECA78849 AC11327BA7EB2FE5389E333AB1FEEDCDC90C78C4713BA7769975CD6290D03B5C 6635F5650B5C1898C7108E6A2A09814C43297E4051B12B263C433646D8555F45 C0B380F639A5A2B5E307C67E6B4208970FB2F32CB0D495CBC1A6D78B36524C44 4FDEA1286022EEC44210FC4ABCE3B8E8BD7DC9A57E5CF409205B78672AD1052E 23813AC24D 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % 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 (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 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 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 0 /.notdef put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 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 0 /.notdef put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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65AB48E09F5B75969FEAE53DB0D8DE69C8AFDF2F606F0A04118A10D0D2DE3D10 D973DB9C038DED5D1B36DEE2505CCBA233C205C77AD4DC6D244054ECFA2C9C7A B1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % 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 (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 68 /D put dup 84 /T put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 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 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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807C5EB16D536B9CA5B9F839F8A164D1764B06043E7FFE3197A3103B728333DC CEBCED9E461EDEFD83B4D63BE9F40C7D09B3D16E1EAE0BFE23D5031D90FC1D0E DDCD795A8B43DF420D9FC3D188EEA9CB9EE8828DF0DEF647252D8B35E7E80339 692B40C3A929D11297C95678B1C55E84A79398408558ECB4401BD037F7FD1BE0 C8F1DF49C5AA79F5715F6D9B26F76C9EA18B222EA31C709EB9A01C41E96B0034 278ACD8764647117CF800CFA3BE771B3D47DE1D69EBA57612C9608A342E29D43 1EE148ED81FAEEE22BD133A8049655F6EACCB782E8D05F4EE4D1CEE75B91DFF1 E02EB7EBFB8A3D502A962AB436422C14B4642130DFB8688EC8DC9EC0EB5046A9 97CC6FEAEEDB95B0C5CCCEEF450D207649231CA3C39263E5640230E0A1FB8705 702714677C6EBBE6C51EE52636D8EDB19B31A096C7C074CF9817400D8D88CCC3 D34373921A975C3E93FD10C2AD8E011BF46007F0DD04C720FEEFFDA404989E72 B941F3C7AC593A364F1C743585092A59745C88ED3927070D48F52A7F2045D821 A96832E5CA10B345B98AE7DA359C8400EE54C09AE31A37A41954CE913645CC56 80B6B84C9153766AF7D38ED0EE0419941511C1B63021C35C0983A497D9517428 77AA2E5BCEE84F82D83D4E76C2B65B31C567DC7FC341CF89B446D43A8281A672 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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 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 101 /tildewide put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % 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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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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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 /.notdef put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A 27D1663E0B62F461F6E40A5D6676D1D12B51E641C1D4E8E2771864FC104F8CBF 5B78EC1D88228725F1C453A678F58A7E1B7BD7CA700717D288EB8DA1F57C4F09 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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. 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All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 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 /.notdef put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec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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 0 /.notdef put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 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 1E042AD0BAA059B3F862236180B34D3FCED833472577BACD472A4B067A46F8EE 2AFACDE591ADF7304939394F221B5B9B316BC47DC7772711A35B466E1790D4F5 C02C57DE57A0498128C6041CB7E702F4D8500433633B8358C438237BBA39C647 F659FACF75574B1AAD3FE61FB97D6C5D0B19E4AC762321891092E73D192C4816 2F14A6FFAA1B22AABB65E8F611F1E9AF66DF68DA1B2B64B4CA8870261F8663FA 28953254FA64F0AD6EFBFAB15830AF856A65835C803B70963595060F91F4079B F8BF5396F04C070254AC5DCACB20C0B1FADD38825886322366927C97E3C5AA1B EA858D5974EBC14D17D4ADDCB3C7EA74A8A2B75293747CFCE0B484713D631F95 43BD5A359229F1276D2F652078ADA5000D1E060CE5CDF2E57401E487F3C10975 09AACD27CEE8581EBADC25D0FC9CEA5329026548341B42511E2D4018645C180E 6BC63116F4C9A7B4F32DB028E0DF7A5422FCF6EF5FD6DDAFA0343674E7FEF67B 06C9CA35222A785DFBB429856BF0B31A4A29FD22FA3AF52F53825082A03CECF8 249D53A256638240A95ABA01457CF5DFE1AD0BE65BE2DC8912D4FCC5DCE4A79F 4E315311FC1DEB1853167728AAB7D7771DF8CD6C837E44E0D983F63A478F2D9D DEA1FBFB5A468F55332DA71C38597CE57432014D1099995D9D1CC7912402AC98 1A32C4480F437D14870D8239C82AE81A20E12BBB1DD25B1C986888CACDDF3990 FA888F6B2697C09C6A6D74EE82260A52E0A3EDFC3C7F7C889FC6996A88466CA7 16CDFDF8CAD4BAEF6BA2065BBA2350E0DBCF958BB2 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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b(w)m(e)g(state)h (Kolmogoro)m(v)g(Theorem.)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 71 274 a Fn(J.)30 b(Villan)m(uev)-5 b(a)3217 b Fx(7)71 573 y Fo(Theorem)33 b(2)h(\(Kolmogoro)m(v)h (Theorem\).)40 b Fd(L)-5 b(et)32 b Fm(H)38 b Fd(b)-5 b(e)32 b(a)f(Hamiltonian)i(written)g(in)e(action-angle)h(vari-)71 686 y(ables,)39 b Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))34 b Fl(2)g Fk(T)721 653 y Fj(n)791 686 y Fl(\002)23 b Fk(R)951 653 y Fj(n)998 686 y Fd(,)38 b Fm(n)c Fl(\025)g Fx(2)p Fd(,)k Fx(2)p Fm(\031)s Fd(-p)-5 b(erio)g(dic)40 b(in)d Fm(\022)s Fd(,)h(r)-5 b(e)g(al)39 b(analytic)f(in)g(the)f(interior)i (of)e Fk(D)p Fx(\()p Fm(\032)3473 700 y Fh(0)3513 686 y Fm(;)15 b(r)3594 700 y Fh(0)3634 686 y Fx(\))37 b Fd(and)71 799 y(b)-5 b(ounde)g(d)32 b(up)f(to)g(the)g(b)-5 b(oundary,)33 b(for)e(c)-5 b(ertain)32 b Fm(\032)1709 813 y Fh(0)1774 799 y Fm(>)25 b Fx(0)31 b Fd(and)g Fx(0)26 b Fm(<)f(r)2328 813 y Fh(0)2393 799 y Fl(\024)g Fx(1)p Fd(.)41 b(We)30 b(supp)-5 b(ose)33 b(that)f Fm(H)37 b Fd(is)31 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b(Then,)29 b Fm(\022)f Fl(2)c Fk(T)2694 4021 y Fj(n)2766 4054 y Fl(7!)h Fx(\011)2953 4021 y Fi(\003)2993 4054 y Fx(\()p Fm(\022)s(;)15 b Fx(0\))26 b(=)f(\(\004)3412 4021 y Fi(\003)3451 4054 y Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(h)3659 4021 y Fi(\003)3699 4054 y Fx(\()p Fm(\022)s Fx(\)\))71 4167 y Fd(gives)23 b(the)g(p)-5 b(ar)g(ameterization)28 b(of)23 b(an)h(invariant)h(L)-5 b(agr)g(angian)25 b(torus)f(of)g Fm(H)7 b Fd(,)25 b(with)f Fm(!)s Fd(-quasi-p)-5 b(erio)g(dic)24 b(dynamics)71 4280 y(on)33 b Fm(\022)s Fd(.)41 b(Mor)-5 b(e)g(over,)34 b(we)e(have)h(the)g(fol)5 b(lowing)34 b(estimates)362 4526 y Fl(k)p Fx(\011)478 4488 y Fi(\003)538 4526 y Fl(\000)20 b Fx(Id)o Fl(k)757 4540 y Fj(\032)793 4520 y Fb(\003)830 4540 y Fj(;r)884 4520 y Fb(\003)1006 4526 y Fl(\024)83 b Fm(\016)1200 4540 y Fh(0)1240 4526 y Fm(;)932 b Fl(k)p Fm(h)2294 4488 y Fi(\003)2334 4526 y Fl(k)2379 4540 y Fj(\032)2415 4520 y Fb(\003)2481 4526 y Fl(\024)25 b Fx(3)20 b Fl(\002)g Fx(2)2778 4488 y Fh(8)2818 4526 y Fm(\013)9 b Fx(\026)-54 b Fm(\013)q(\033)3094 4464 y(")p 3000 4505 230 4 v 3000 4597 a(\015)5 b(\016)3095 4559 y Fj(\034)j Fh(+1)3092 4623 y(0)3240 4526 y Fm(;)422 4757 y Fl(k)p Fx(\004)528 4719 y Fi(\003)588 4757 y Fl(\000)20 b Fx(Id)762 4772 y Fj(\022)801 4757 y Fl(k)846 4771 y Fj(\032)882 4751 y Fb(\003)1006 4757 y Fl(\024)83 b Fx(9)21 b Fl(\002)f Fx(2)1362 4719 y Fh(7)1401 4757 y Fm(\013)1459 4719 y Fh(2)1508 4757 y Fx(\026)-54 b Fm(\013)q(\033)1613 4719 y Fh(2)1793 4695 y Fm(")p 1662 4736 304 4 v 1662 4828 a(\015)1714 4801 y Fh(2)1754 4828 y Fm(\016)1797 4789 y Fh(2)p Fj(\034)8 b Fh(+1)1794 4854 y(0)1976 4757 y Fm(;)202 b Fl(kB)2311 4719 y Fi(\003)2350 4757 y Fl(k)2395 4771 y Fj(\032)2431 4751 y Fb(\003)2497 4757 y Fl(\024)25 b Fx(9)c Fl(\002)e Fx(2)2794 4719 y Fh(9)2834 4757 y Fm(n\013)2947 4719 y Fh(2)2996 4757 y Fx(\026)-54 b Fm(\013\033)3100 4719 y Fh(2)3308 4695 y Fm(")p 3150 4736 359 4 v 3150 4844 a(\015)3202 4818 y Fh(2)3242 4844 y Fm(\016)3285 4796 y Fh(2\()p Fj(\034)8 b Fh(+1\))3282 4870 y(0)3519 4757 y Fm(;)648 4984 y Fl(k)p Fm(A)761 4946 y Fi(\003)801 4984 y Fl(k)846 4998 y Fj(\032)882 4979 y Fb(\003)1006 4984 y Fl(\024)83 b Fx(4)p Fm(\013;)202 b Fl(k)p Fx(\()1595 4961 y(\026)1570 4984 y Fm(A)1638 4946 y Fi(\003)1679 4984 y Fx(\))1714 4946 y Fi(\000)p Fh(1)1808 4984 y Fl(k)1853 4998 y Fj(\032)1889 4979 y Fb(\003)1955 4984 y Fl(\024)25 b Fx(4)9 b(\026)-54 b Fm(\013)q(;)201 b Fl(k)p Fx([)p Fm(H)2534 4946 y Fi(\003)2574 4984 y Fx(])2599 4998 y Fh(3)2639 4984 y Fl(k)2684 4998 y Fj(\032)2720 4979 y Fb(\003)2757 4998 y Fj(;R)2859 4984 y Fl(\024)25 b Fx(4)7 b Fg(e)-58 b Fm(\013)q(R)3129 4946 y Fh(3)3168 4984 y Fm(;)71 5216 y Fd(for)33 b(any)g Fx(0)26 b Fl(\024)f Fm(R)h Fl(\024)f Fm(r)792 5183 y Fi(\003)856 5216 y Fx(=)g Fm(r)993 5230 y Fh(0)1033 5216 y Fm(=)p Fx(2)c Fl(\000)f Fx(11)p Fm(\016)1365 5230 y Fh(0)1405 5216 y Fd(.)71 5440 y Fo(Remark)32 b(3.)39 b Fd(As)30 b(we)g(claime)-5 b(d,)31 b(c)-5 b(ondition)32 b(\(11\))f(is)f(ful\014l)5 b(le)-5 b(d)30 b(if)g Fm(\015)35 b Fd(is)29 b(b)-5 b(ounde)g(d)32 b(away)f(fr)-5 b(om)31 b(zer)-5 b(o)31 b(by)f Fl(O)s Fx(\()3669 5375 y Fl(p)p 3745 5375 43 4 v 65 x Fm(")p Fx(\))p Fd(.)71 5553 y(Mor)-5 b(e)g(over,)34 b(the)g(displac)-5 b(ement)35 b(in)e(action)h(variables)g(of)f(the)g(p)-5 b(erturb)g(e)g(d)35 b(torus)f(with)g(r)-5 b(esp)g(e)g(ct)35 b(to)f(the)f(unp)-5 b(er-)71 5666 y(turb)g(e)g(d)36 b(one)g(is)g(given) f(by)g Fm(h)1033 5633 y Fi(\003)1073 5666 y Fx(\()p Fm(\022)s Fx(\))p Fd(,)h(for)g(which)g(we)g(have)g(an)g(estimate)g(of)g Fl(O)s Fx(\()p Fm("=\015)5 b Fx(\))p Fd(.)52 b(We)35 b(p)-5 b(oint)37 b(out)f(that)h(the)71 5779 y(displac)-5 b(ement)35 b(in)d(the)h(angular)h(variables,)f(given)f(by)h Fx(\004)2021 5746 y Fi(\003)2080 5779 y Fl(\000)20 b Fx(Id)2254 5794 y Fj(\022)2293 5779 y Fd(,)33 b(c)-5 b(an)33 b(b)-5 b(e)32 b(bigger)g(if)h Fm(\015)k Fd(is)c(smal)5 b(l.)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 71 274 a Fx(8)2472 b Fn(Kolmogoro)m(v)32 b(Theorem)e(Revisited)71 573 y Fp(4)135 b(Canonical)46 b(transformations)71 776 y Fx(Let)26 b(us)e(start)i(b)m(y)g(describing) f(\(formally\))h(a)g(general)h(step)e(of)h(Kolmogoro)m(v's)h (\(iterativ)m(e\))i(metho)s(d.)38 b(F)-8 b(or)26 b(this)71 889 y(purp)s(ose)j(w)m(e)h(consider)h(a)f(Hamiltonian)i(of)f(the)f (form)889 1092 y Fm(H)7 b Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))26 b(=)e Fm(a)p Fx(\()p Fm(\022)s Fx(\))d(+)f Fl(h)p Fm(!)j Fx(+)d Fm(b)p Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(I)7 b Fl(i)21 b Fx(+)2177 1031 y(1)p 2177 1071 46 4 v 2177 1155 a(2)2232 1092 y Fl(h)p Fm(I)7 b(;)15 b(A)p Fx(\()p Fm(\022)s Fx(\))p Fm(I)7 b Fl(i)22 b Fx(+)e Fm(F)13 b Fx(\()p Fm(\022)s(;)i(I)7 b Fx(\))p Fm(;)657 b Fx(\(12\))71 1300 y(with)30 b Fm(A)346 1267 y Fi(>)431 1300 y Fx(=)25 b Fm(A)31 b Fx(and)e Fm(F)39 b Fx(=)25 b([)p Fm(H)7 b Fx(])1128 1314 y Fh(3)1168 1300 y Fx(.)41 b(W)-8 b(e)32 b(also)f(supp)s(ose)e(that)i Fm(!)j Fx(v)m(eri\014es)d (\(1\))g(and)f(that)h(det\()3213 1277 y(\026)3188 1300 y Fm(A)q Fx(\))26 b Fl(6)p 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Fm(H)3399 1831 y Fh(\(1\))3519 1864 y Fx(=)25 b Fm(H)j Fl(\016)21 b Fx(\010)71 1977 y(and)29 b(expand)g Fm(H)645 1944 y Fh(\(1\))768 1977 y Fx(as)h(in)g(\(12\))h(\(see)f(\(24\)\).)43 b(W)-8 b(e)30 b(w)m(an)m(t)g Fg(e)-50 b Fm(a)2045 1944 y Fh(\(1\))2169 1977 y Fx(and)29 b Fm(b)2384 1944 y Fh(\(1\))2508 1977 y Fx(to)h(b)s(ecome)g(of)g Fl(O)s Fx(\()p Fm(")3198 1944 y Fh(2)3238 1977 y Fx(\))f(after)i(this)e(\014rst)71 2090 y(transformation.)45 b(If)31 b(w)m(e)h(iterate)i(this)e(pro)s (cess,)f(the)h(size)h(of)f(these)g(\\error)g(terms")g(after)g Fm(s)f Fx(steps)h(turns)f(to)71 2203 y(b)s(e)d(\(\\roughly)h(sp)s (eaking"\))h(of)f Fl(O)s Fx(\()p Fm(")1311 2170 y Fh(2)1346 2146 y Ff(s)1384 2203 y Fx(\).)40 b(This)28 b(\\sup)s(er-con)m(v)m (ergence",)k(in)m(tro)s(duced)c(b)m(y)h(Siegel)h([14)q(],)f(is)g(used) 71 2316 y(to)i(o)m(v)m(ercome)i(the)d(small)h(divisors)f(of)h(the)f (problem)g(\(see)h(\(39\)\).)212 2429 y(W)-8 b(e)35 b(use)e(canonical)i 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Fh(0)1574 3782 y Fx(\)\))20 b Fl(\000)g Fm(D)1830 3797 y Fj(\022)1870 3782 y Fm(d)p Fx(\()p Fm(\022)1998 3744 y Fh(0)2037 3782 y Fx(\))2072 3708 y Fg(\001)2114 3728 y Fi(>)2188 3782 y Fm(ds)g Fl(\000)2389 3658 y Fg(Z)2480 3684 y Fj(t)2440 3864 y Fh(0)2510 3782 y Fx(\()p Fm(D)2620 3797 y Fj(\022)2659 3782 y Fm(d)p Fx(\()p Fm(\002)2813 3744 y Fj(s)2851 3782 y Fx(\()p Fm(\022)2932 3744 y Fh(0)2971 3782 y Fx(\)\)\))3076 3744 y Fi(>)3136 3782 y Fm(B)3210 3744 y Fj(s)3246 3782 y Fx(\()p Fm(\022)3327 3744 y Fh(0)3366 3782 y Fx(\))p Fm(ds;)173 b Fx(\(21\))416 4036 y Fm(g)462 3998 y Fj(t)493 4036 y Fx(\()p Fm(\022)574 3998 y Fh(0)613 4036 y Fx(\))83 b(=)g Fl(\000)971 3912 y Fg(Z)1061 3938 y Fj(t)1021 4118 y Fh(0)1091 4036 y Fx(\()p Fm(\030)25 b Fx(+)20 b Fl(r)1358 4051 y Fj(\022)1396 4036 y Fm(c)p Fx(\()p Fm(\002)1542 3998 y Fj(s)1580 4036 y Fx(\()p Fm(\022)1661 3998 y Fh(0)1700 4036 y Fx(\)\)\))p Fm(ds)h Fl(\000)2007 3912 y Fg(Z)2098 3938 y Fj(t)2057 4118 y Fh(0)2127 4036 y Fx(\()p Fm(D)2237 4051 y Fj(\022)2277 4036 y Fm(d)p Fx(\()p Fm(\002)2431 3998 y Fj(s)2468 4036 y Fx(\()p Fm(\022)2549 3998 y Fh(0)2588 4036 y Fx(\)\)\))2693 3998 y Fi(>)2753 4036 y Fm(g)2799 3998 y Fj(s)2837 4036 y Fx(\()p Fm(\022)2918 3998 y Fh(0)2957 4036 y Fx(\))p Fm(ds;)582 b Fx(\(22\))419 4289 y(^)-48 b Fm(g)462 4252 y Fj(t)493 4289 y Fx(\()p Fm(\022)574 4252 y Fh(0)613 4289 y Fx(\))83 b(=)g Fl(\000)971 4166 y Fg(Z)1061 4192 y Fj(t)1021 4372 y Fh(0)1091 4289 y Fx(\()p Fl(r)1202 4304 y Fj(\022)1241 4289 y Fm(c)p Fx(\()p Fm(\002)1387 4252 y Fj(s)1425 4289 y Fx(\()p Fm(\022)1506 4252 y Fh(0)1545 4289 y Fx(\)\))20 b Fl(\000)g(r)1802 4304 y Fj(\022)1841 4289 y Fm(c)p Fx(\()p Fm(\022)1961 4252 y Fh(0)2001 4289 y Fx(\)\))p Fm(ds)g Fl(\000)2272 4166 y Fg(Z)2363 4192 y Fj(t)2323 4372 y Fh(0)2393 4289 y Fx(\()p Fm(D)2503 4304 y Fj(\022)2542 4289 y Fm(d)p Fx(\()p Fm(\002)2696 4252 y Fj(s)2734 4289 y Fx(\()p Fm(\022)2815 4252 y Fh(0)2854 4289 y Fx(\)\)\))2959 4252 y Fi(>)3019 4289 y Fm(g)3065 4252 y Fj(s)3102 4289 y Fx(\()p Fm(\022)3183 4252 y Fh(0)3222 4289 y Fx(\))p Fm(ds:)317 b Fx(\(23\))71 4579 y(No)m(w,)34 b(w)m(e)f(sho)m(w)f(the)h(action)g(of)g(\010)1274 4546 y Fj(G)1365 4579 y Fx(on)g(the)f(Hamiltonian)i(\(12\).)49 b(As)32 b(\010)2613 4546 y Fj(G)2704 4579 y Fx(is)h(canonical)h(this)e (means)g(that,)71 4692 y(if)43 b(p)s(erform)g(the)g(c)m(hange)i(of)f (co)s(ordinates)g(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))48 b(=)e(\010)2055 4659 y Fj(G)2114 4692 y Fx(\()p Fm(\022)2195 4659 y Fi(0)2218 4692 y Fm(;)15 b(I)2305 4659 y Fi(0)2329 4692 y Fx(\))44 b(on)f(the)h(Hamiltonian)h(system)f(de\014ned)71 4805 y(b)m(y)34 b Fm(H)7 b Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\),)37 b(w)m(e)e(obtain)g(again)g(a)g(Hamiltonian)h(system,)g (with)e(Hamilton)i(function)e Fm(H)3219 4772 y Fh(\(1\))3314 4805 y Fx(\()p Fm(\022)3395 4772 y Fi(0)3418 4805 y Fm(;)15 b(I)3505 4772 y Fi(0)3528 4805 y Fx(\))33 b(=)f Fm(H)e Fl(\016)71 4918 y Fx(\010)137 4885 y Fj(G)196 4918 y Fx(\()p Fm(\022)277 4885 y Fi(0)300 4918 y Fm(;)15 b(I)387 4885 y Fi(0)410 4918 y Fx(\).)41 b(Let)31 b(us)f(compute)g (\(formally\))i(the)e(expression)g(of)h Fm(H)2370 4885 y Fh(\(1\))2464 4918 y Fx(.)40 b(F)-8 b(or)31 b(simplicit)m(y)h(w)m(e)f (skip)f(the)g(primes)71 5031 y(of)g(the)h(new)f(v)-5 b(ariables.)41 b(W)-8 b(e)32 b(obtain,)430 5264 y Fm(H)513 5226 y Fh(\(1\))608 5264 y Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))83 b(=)g Fm(a)p Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))19 b(+)h Fl(h)p Fm(!)k Fx(+)c Fm(b)p Fx(\()p Fm(\022)i Fx(+)e Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(;)15 b Fx(\(Id)2406 5278 y Fj(n)2473 5264 y Fx(+)20 b Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(I)27 b Fx(+)20 b Fm(g)s Fx(\()p Fm(\022)s Fx(\))p Fl(i)1048 5440 y Fx(+)1129 5379 y(1)p 1129 5419 46 4 v 1129 5502 a(2)1184 5440 y Fl(h)p Fx(\(Id)1338 5454 y Fj(n)1405 5440 y Fx(+)g Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(I)27 b Fx(+)20 b Fm(g)s Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(A)p Fx(\()p Fm(\022)24 b Fx(+)c Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))15 b(\(\(Id)2717 5454 y Fj(n)2784 5440 y Fx(+)20 b Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(I)27 b Fx(+)20 b Fm(g)s Fx(\()p Fm(\022)s Fx(\)\))q Fl(i)1048 5604 y Fx(+)p Fm(F)13 b Fx(\()p Fm(\022)22 b Fx(+)e Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b Fx(\(Id)1711 5618 y Fj(n)1778 5604 y Fx(+)20 b Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(I)27 b Fx(+)20 b Fm(g)s Fx(\()p Fm(\022)s Fx(\)\))894 5780 y(=)83 b Fm(a)1096 5742 y Fh(\(1\))1190 5780 y Fx(\()p Fm(\022)s Fx(\))20 b(+)g Fl(h)p Fm(!)k Fx(+)c Fm(b)1663 5742 y Fh(\(1\))1757 5780 y Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(I)7 b Fl(i)21 b Fx(+)2117 5718 y(1)p 2117 5759 V 2117 5842 a(2)2172 5780 y Fl(h)p Fm(I)7 b(;)15 b(A)2362 5742 y Fh(\(1\))2457 5780 y Fx(\()p Fm(\022)s Fx(\))p Fm(I)7 b Fl(i)21 b Fx(+)f Fm(F)2838 5742 y Fh(\(1\))2932 5780 y Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))p Fm(;)529 b Fx(\(24\))p eop end %%Page: 10 10 TeXDict begin 10 9 bop 71 274 a Fx(10)2427 b Fn(Kolmogoro)m(v)32 b(Theorem)e(Revisited)71 573 y Fx(with)114 804 y Fm(a)162 766 y Fh(\(1\))339 804 y Fx(=)83 b Fm(a)p Fx(\()p Fm(\022)22 b Fx(+)e Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))20 b(+)g Fl(h)p Fm(!)j Fx(+)d Fm(b)p Fx(\()p Fm(\022)j Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\))p Fl(i)21 b Fx(+)2051 742 y(1)p 2051 783 46 4 v 2051 866 a(2)2106 804 y Fl(h)p Fm(g)s Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(A)p Fx(\()p Fm(\022)25 b Fx(+)20 b Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(g)s Fx(\()p Fm(\022)s Fx(\))p Fl(i)20 b Fx(+)g Fm(F)13 b Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\)\))p Fm(;)123 987 y(b)162 949 y Fh(\(1\))339 987 y Fx(=)83 b Fm(B)5 b Fx(\()p Fm(\022)s Fx(\))683 949 y Fi(>)741 987 y Fm(!)23 b Fx(+)d(\(Id)1031 1001 y Fj(n)1098 987 y Fx(+)g Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))1414 949 y Fi(>)1488 987 y Fx(\()p Fm(b)p Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))20 b(+)g Fm(A)p Fx(\()p Fm(\022)i Fx(+)e Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(g)s Fx(\()p Fm(\022)s Fx(\))21 b(+)f Fl(r)2886 1001 y Fj(I)2925 987 y Fm(F)13 b Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\)\)\))h Fm(;)93 1125 y(A)161 1087 y Fh(\(1\))339 1125 y Fx(=)83 b(\(Id)611 1139 y Fj(n)679 1125 y Fx(+)19 b Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))994 1087 y Fi(>)1068 1051 y Fg(\000)1110 1125 y Fm(A)p Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))20 b(+)g Fm(D)1765 1087 y Fh(2)1762 1147 y Fj(I)1804 1125 y Fm(F)13 b Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\)\))2475 1051 y Fg(\001)2532 1125 y Fx(\(Id)2651 1139 y Fj(n)2718 1125 y Fx(+)20 b Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(;)91 1277 y(F)162 1239 y Fh(\(1\))339 1277 y Fx(=)83 b([)p Fm(F)13 b Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b Fx(\(Id)1110 1291 y Fj(n)1177 1277 y Fx(+)20 b Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(I)27 b Fx(+)20 b Fm(g)s Fx(\()p Fm(\022)s Fx(\)\)])1874 1304 y Fh(3)1929 1277 y Fm(:)71 1492 y Fx(No)m(w,)31 b(w)m(e)g(recall)h(that)f Fm(\030)t Fx(,)f Fm(c)h Fx(and)f Fm(d)g Fx(v)m(erify)h(\(15\).)42 b(Then,)30 b(w)m(e)h(ha)m(v)m(e)g(\(see)h(the)e(de\014nition)g(of)h Fm(L)3295 1506 y Fj(!)3345 1492 y Fx(\),)114 1708 y Fl(h)p Fm(!)s(;)15 b(g)s Fl(i)84 b Fx(=)f Fl(\000h)p Fm(!)s(;)15 b(\030)25 b Fx(+)20 b Fl(r)1006 1723 y Fj(\022)1044 1708 y Fm(c)p Fx(\()p Fm(\022)s Fx(\))p Fl(i)h Fx(+)f Fl(h)p Fm(!)s(;)f Fx(^)-49 b Fm(g)t Fx(\()p Fm(\022)s Fx(\))p Fl(i)26 b Fx(=)f Fl(\000h)p Fm(!)s(;)15 b(\030)t Fl(i)21 b(\000)d Fg(e)-49 b Fm(a)p Fx(\()p Fm(\022)s Fx(\))20 b(+)g Fl(h)p Fm(!)s(;)e Fx(^)-48 b Fm(g)t Fx(\()p Fm(\022)s Fx(\))p Fl(i)p Fm(;)114 1904 y Fl(h)p Fm(!)s(;)19 b Fx(^)-49 b Fm(g)t Fl(i)83 b Fx(=)g Fl(\000h)p Fm(!)s(;)774 1780 y Fg(Z)865 1807 y Fh(1)825 1987 y(0)905 1904 y Fx(\()p Fl(r)1016 1919 y Fj(\022)1055 1904 y Fm(c)p Fx(\()p Fm(\002)1201 1867 y Fj(s)1238 1904 y Fx(\()p Fm(\022)s Fx(\)\))21 b Fl(\000)f(r)1577 1919 y Fj(\022)1616 1904 y Fm(c)p Fx(\()p Fm(\022)s Fx(\)\))p Fm(ds)p Fl(i)h(\000)f(h)p Fm(!)s(;)2178 1780 y Fg(Z)2269 1807 y Fh(1)2229 1987 y(0)2308 1904 y Fx(\()p Fm(D)2418 1919 y Fj(\022)2458 1904 y Fm(d)p Fx(\()p Fm(\002)2612 1867 y Fj(s)2649 1904 y Fx(\()p Fm(\022)s Fx(\)\)\))2835 1867 y Fi(>)2895 1904 y Fm(g)2941 1867 y Fj(s)2978 1904 y Fx(\()p Fm(\022)s Fx(\))p Fm(ds)p Fl(i)470 b Fx(\(25\))414 2160 y(=)83 b Fl(\000)654 2036 y Fg(Z)745 2063 y Fh(1)705 2242 y(0)784 2160 y Fx(\()o Fg(e)-50 b Fm(a)q Fx(\()p Fm(\002)975 2122 y Fj(s)1012 2160 y Fx(\()p Fm(\022)s Fx(\)\))20 b Fl(\000)f Fg(e)-50 b Fm(a)p Fx(\()p Fm(\022)s Fx(\)\))p Fm(ds)21 b Fl(\000)1675 2036 y Fg(Z)1765 2063 y Fh(1)1725 2242 y(0)1820 2160 y Fx(\()p Fm(b)p Fx(\()p Fm(\002)2001 2122 y Fj(s)2038 2160 y Fx(\()p Fm(\022)s Fx(\)\))g Fl(\000)f Fm(A)p Fx(\()p Fm(\002)2476 2122 y Fj(s)2513 2160 y Fx(\()p Fm(\022)s Fx(\)\)\()p Fm(\030)25 b Fx(+)20 b Fl(r)2931 2175 y Fj(\022)2969 2160 y Fm(c)p Fx(\()p Fm(\002)3115 2122 y Fj(s)3153 2160 y Fx(\()p Fm(\022)s Fx(\)\)\)\))3375 2118 y Fi(>)3449 2160 y Fm(g)3495 2122 y Fj(s)3532 2160 y Fx(\()p Fm(\022)s Fx(\))p Fm(ds;)139 2362 y(B)213 2324 y Fi(>)272 2362 y Fm(!)85 b Fx(=)e Fl(\000)p Fm(D)714 2377 y Fj(\022)753 2362 y Fm(d)p Fx(\()p Fm(\022)s Fx(\))p Fm(!)23 b Fx(+)1109 2339 y(^)1087 2362 y Fm(B)5 b Fx(\()p Fm(\022)s Fx(\))1277 2324 y Fi(>)1336 2362 y Fm(!)28 b Fx(=)d Fl(\000)p Fm(b)p Fx(\()p Fm(\022)s Fx(\))20 b(+)g Fm(A)p Fx(\()p Fm(\022)s Fx(\)\()p Fm(\030)k Fx(+)c Fl(r)2304 2377 y Fj(\022)2343 2362 y Fm(c)p Fx(\()p Fm(\022)s Fx(\)\))h(+)2666 2339 y(^)2645 2362 y Fm(B)t Fx(\()p Fm(\022)s Fx(\))2834 2324 y Fi(>)2893 2362 y Fm(!)s(;)161 2535 y Fx(^)139 2558 y Fm(B)213 2520 y Fi(>)272 2558 y Fm(!)85 b Fx(=)e Fl(\000)654 2434 y Fg(Z)745 2461 y Fh(1)705 2640 y(0)800 2558 y Fx(\()p Fm(D)910 2573 y Fj(\022)949 2558 y Fm(d)p Fx(\()p Fm(\002)1103 2520 y Fj(s)1141 2558 y Fx(\()p Fm(\022)s Fx(\)\))20 b Fl(\000)g Fm(D)1478 2573 y Fj(\022)1517 2558 y Fm(d)p Fx(\()p Fm(\022)s Fx(\)\))c Fm(!)s(ds)k Fl(\000)1992 2434 y Fg(Z)2083 2461 y Fh(1)2042 2640 y(0)2137 2558 y Fm(B)2211 2520 y Fj(s)2248 2558 y Fx(\()p Fm(\022)s Fx(\))2364 2520 y Fi(>)2423 2558 y Fm(D)2498 2573 y Fj(\022)2537 2558 y Fm(d)p Fx(\()p Fm(\002)2691 2520 y Fj(s)2728 2558 y Fx(\()p Fm(\022)s Fx(\)\))p Fm(!)s(ds)414 2814 y Fx(=)83 b Fl(\000)654 2690 y Fg(Z)745 2716 y Fh(1)705 2896 y(0)784 2814 y Fx(\()p Fm(b)p Fx(\()p Fm(\002)965 2776 y Fj(s)1003 2814 y Fx(\()p Fm(\022)s Fx(\)\))20 b Fl(\000)g Fm(b)p Fx(\()p Fm(\022)s Fx(\)\))p Fm(ds)h Fx(+)1656 2690 y Fg(Z)1747 2716 y Fh(1)1707 2896 y(0)1802 2814 y Fx(\()p Fm(A)p Fx(\()p Fm(\002)2012 2776 y Fj(s)2050 2814 y Fx(\()p Fm(\022)s Fx(\)\)\()p Fm(\030)j Fx(+)c Fl(r)2467 2829 y Fj(\022)2506 2814 y Fm(c)p Fx(\()p Fm(\002)2652 2776 y Fj(s)2689 2814 y Fx(\()p Fm(\022)s Fx(\)\)\))h Fl(\000)f Fm(A)p Fx(\()p Fm(\022)s Fx(\)\()p Fm(\030)25 b Fx(+)20 b Fl(r)3438 2829 y Fj(\022)3476 2814 y Fm(c)p Fx(\()p Fm(\022)s Fx(\)\)\))c Fm(ds)568 3070 y Fl(\000)654 2946 y Fg(Z)745 2972 y Fh(1)705 3152 y(0)800 3070 y Fm(B)874 3032 y Fj(s)910 3070 y Fx(\()p Fm(\022)s Fx(\))1026 3032 y Fi(>)1100 3070 y Fx(\()p Fm(b)p Fx(\()p Fm(\002)1281 3032 y Fj(s)1318 3070 y Fx(\()p Fm(\022)s Fx(\)\))21 b Fl(\000)f Fm(A)p Fx(\()p Fm(\002)1756 3032 y Fj(s)1793 3070 y Fx(\()p Fm(\022)s Fx(\)\)\()p Fm(\030)25 b Fx(+)20 b Fl(r)2211 3085 y Fj(\022)2249 3070 y Fm(c)p Fx(\()p Fm(\002)2395 3032 y Fj(s)2433 3070 y Fx(\()p Fm(\022)s Fx(\)\)\)\))c Fm(ds:)904 b Fx(\(26\))71 3346 y(Using)30 b(these)h(expressions)f(w)m(e)h(can)g(rewrite)f Fm(H)1725 3313 y Fh(\(1\))1850 3346 y Fx(in)m(to)h(the)g(follo)m(wing)h (form:)236 3561 y Fm(a)284 3524 y Fh(\(1\))461 3561 y Fx(=)84 b(\026)-46 b Fm(a)20 b Fl(\000)g(h)p Fm(!)s(;)15 b(\030)t Fl(i)21 b Fx(+)f(\()o Fg(e)-50 b Fm(a)q Fx(\()p Fm(\022)22 b Fx(+)e Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))20 b Fl(\000)f Fg(e)-50 b Fm(a)p Fx(\()p Fm(\022)s Fx(\)\))21 b(+)e Fl(h)p Fm(!)s(;)g Fx(^)-49 b Fm(g)5 b Fx(\()p Fm(\022)s Fx(\))p Fl(i)20 b Fx(+)g Fl(h)p Fm(b)p Fx(\()p Fm(\022)j Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\))p Fl(i)615 3737 y Fx(+)696 3676 y(1)p 696 3717 V 696 3800 a(2)751 3737 y Fl(h)p Fm(g)s Fx(\()p Fm(\022)s Fx(\))p Fm(;)g(A)p Fx(\()p Fm(\022)24 b Fx(+)c Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(g)s Fx(\()p Fm(\022)s Fx(\))p Fl(i)21 b Fx(+)f Fm(F)13 b Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\)\))p Fm(;)1229 b Fx(\(27\))245 3920 y Fm(b)284 3883 y Fh(\(1\))461 3920 y Fx(=)637 3897 y(^)615 3920 y Fm(B)5 b Fx(\()p Fm(\022)s Fx(\))805 3883 y Fi(>)864 3920 y Fm(!)23 b Fx(+)d(\()p Fm(b)p Fx(\()p Fm(\022)j Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))19 b Fl(\000)h Fm(b)p Fx(\()p Fm(\022)s Fx(\)\))h(+)f Fm(B)5 b Fx(\()p Fm(\022)s Fx(\))2109 3883 y Fi(>)2167 3920 y Fm(b)p Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))20 b(+)f(\()p Fm(A)p Fx(\()p Fm(\022)24 b Fx(+)c Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))19 b Fl(\000)h Fm(A)p Fx(\()p Fm(\022)s Fx(\)\))p Fm(g)s Fx(\()p Fm(\022)s Fx(\))615 4058 y(+)p Fm(A)p Fx(\()p Fm(\022)s Fx(\))s(^)-48 b Fm(g)t Fx(\()p Fm(\022)s Fx(\))20 b(+)g Fm(B)5 b Fx(\()p Fm(\022)s Fx(\))1334 4021 y Fi(>)1392 4058 y Fm(A)p Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(g)s Fx(\()p Fm(\022)s Fx(\))20 b(+)g(\(Id)2250 4072 y Fj(n)2317 4058 y Fx(+)g Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))2633 4021 y Fi(>)2692 4058 y Fl(r)2768 4072 y Fj(I)2807 4058 y Fm(F)13 b Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\)\))p Fm(;)186 b Fx(\(28\))216 4196 y Fm(A)284 4158 y Fh(\(1\))461 4196 y Fx(=)83 b Fm(A)p Fx(\()p Fm(\022)s Fx(\))20 b(+)g(\()p Fm(A)p Fx(\()p Fm(\022)k Fx(+)c Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))19 b Fl(\000)h Fm(A)p Fx(\()p Fm(\022)s Fx(\)\))h(+)f Fm(B)5 b Fx(\()p Fm(\022)s Fx(\))2043 4158 y Fi(>)2101 4196 y Fm(A)p Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\)\(Id)2685 4210 y Fj(n)2752 4196 y Fx(+)20 b Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))615 4334 y(+)p Fm(A)p Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(B)5 b Fx(\()p Fm(\022)s Fx(\))19 b(+)h(\(Id)1570 4348 y Fj(n)1638 4334 y Fx(+)g Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))1954 4296 y Fi(>)2012 4334 y Fm(D)2090 4296 y Fh(2)2087 4356 y Fj(I)2130 4334 y Fm(F)13 b Fx(\()p Fm(\022)22 b Fx(+)e Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\)\)\(Id)2920 4348 y Fj(n)2987 4334 y Fx(+)20 b Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(;)361 b Fx(\(29\))213 4472 y Fm(F)284 4434 y Fh(\(1\))461 4472 y Fx(=)83 b Fm(F)13 b Fx(\()p Fm(\022)s(;)i(I)7 b Fx(\))21 b(+)f([)p Fm(F)13 b Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b Fx(\(Id)1618 4486 y Fj(n)1685 4472 y Fx(+)20 b Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(I)27 b Fx(+)20 b Fm(g)s Fx(\()p Fm(\022)s Fx(\)\))h Fl(\000)f Fm(F)13 b Fx(\()p Fm(\022)s(;)i(I)7 b Fx(\)])2767 4486 y Fh(3)2807 4472 y Fm(:)857 b Fx(\(30\))71 4780 y Fp(5)135 b(The)44 b(Iterativ)l(e)j(Lemma)71 4990 y Fx(In)32 b(this)h(section)h(w)m(e)g(state)g(and)f(pro)m(v)m(e)h(a)f (result)g(con)m(trolling)i(a)e(general)h(step)g(of)f(Kolmogoro)m(v's)i (iterativ)m(e)71 5103 y(pro)s(cess)30 b(\(Lemma)h(5)f(can)h(b)s(e)f (seen)g(as)h(a)g(quan)m(titativ)m(e)i(v)m(ersion)e(of)f(Section)h(4\).) 71 5305 y Fo(Lemma)46 b(5)g(\(Iterativ)m(e)e(Lemma\).)j Fd(L)-5 b(et)41 b Fm(H)49 b Fd(b)-5 b(e)41 b(a)h(Hamiltonian)h(written) f(in)f(action-angle)i(variables,)71 5418 y Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))26 b Fl(2)e Fk(T)446 5385 y Fj(n)509 5418 y Fl(\002)15 b Fk(R)661 5385 y Fj(n)708 5418 y Fd(,)30 b Fm(n)25 b Fl(\025)g Fx(2)p Fd(,)31 b Fx(2)p Fm(\031)s Fd(-p)-5 b(erio)g(dic)32 b(in)f Fm(\022)s Fd(,)f(r)-5 b(e)g(al)32 b(analytic)f(in)g(the)g(interior)g(of)g Fk(D)p Fx(\()p Fm(\032;)15 b(r)s Fx(\))31 b Fd(and)g(b)-5 b(ounde)g(d)32 b(up)71 5531 y(to)h(the)g(b)-5 b(oundary,)34 b(for)g(c)-5 b(ertain)33 b Fm(\032)25 b(>)g Fx(0)33 b Fd(and)h Fx(0)26 b Fm(<)f(r)i Fl(\024)e Fx(1)p Fd(.)42 b(We)33 b(exp)-5 b(and)34 b Fm(H)39 b Fd(as)889 5780 y Fm(H)7 b Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))26 b(=)e Fm(a)p Fx(\()p Fm(\022)s Fx(\))d(+)f Fl(h)p Fm(!)j Fx(+)d Fm(b)p Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(I)7 b Fl(i)21 b Fx(+)2177 5718 y(1)p 2177 5759 V 2177 5842 a(2)2232 5780 y Fl(h)p Fm(I)7 b(;)15 b(A)p Fx(\()p Fm(\022)s Fx(\))p Fm(I)7 b Fl(i)22 b Fx(+)e Fm(F)13 b Fx(\()p Fm(\022)s(;)i(I)7 b Fx(\))p Fm(;)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 71 274 a Fn(J.)30 b(Villan)m(uev)-5 b(a)3171 b Fx(11)71 573 y Fd(wher)-5 b(e)34 b Fm(F)k Fx(=)25 b([)p Fm(H)7 b Fx(])653 587 y Fh(3)693 573 y Fd(,)32 b(and)i(we)e(assume)i Fx(det\()1568 550 y(\026)1544 573 y Fm(A)q Fx(\))25 b Fl(6)p Fx(=)g(0)33 b Fd(and)h(the)f(fol)5 b(lowing)33 b(b)-5 b(ounds:)409 751 y Fl(k)o Fg(e)-50 b Fm(a)q Fl(k)548 765 y Fj(\032)614 751 y Fl(\024)29 b Fg(e)-55 b Fm(\026;)201 b Fl(k)p Fm(b)p Fl(k)1120 765 y Fj(\032)1186 751 y Fl(\024)32 b Fx(\026)-52 b Fm(\026;)201 b Fl(k)p Fm(A)p Fl(k)1721 765 y Fj(\032)1788 751 y Fl(\024)25 b Fm(m;)201 b Fl(j)p Fx(\()2275 728 y(\026)2250 751 y Fm(A)q Fx(\))2354 713 y Fi(\000)p Fh(1)2448 751 y Fl(j)26 b(\024)42 b Fx(\026)-62 b Fm(m;)201 b Fl(k)p Fm(F)13 b Fl(k)3062 765 y Fj(\032;R)3202 751 y Fl(\024)39 b Fg(e)-65 b Fm(m)o(R)3447 713 y Fh(3)3487 751 y Fm(;)177 b Fx(\(31\))71 928 y Fd(for)35 b(any)g Fx(0)30 b Fl(\024)e Fm(R)i Fl(\024)f Fm(r)s Fd(,)34 b(with)i Fm(m;)c Fx(\026)-62 b Fm(m;)30 b Fg(e)-66 b Fm(m)35 b Fd(gr)-5 b(e)g(ater)36 b(than)g(one.)48 b(Mor)-5 b(e)g(over,)36 b(we)f(supp)-5 b(ose)36 b(that)g Fm(!)c Fl(2)c Fk(R)3489 895 y Fj(n)3570 928 y Fd(veri\014es)71 1041 y(the)33 b(Diophantine)h(c)-5 b(ondition)34 b(\(1\),)f(for)h(c)-5 b(ertain)33 b Fx(0)26 b Fm(<)f(\015)30 b Fl(\024)25 b Fx(1)33 b Fd(and)h Fm(\034)h Fl(\025)25 b Fm(n)20 b Fl(\000)g Fx(1)p Fd(.)212 1154 y(Given)32 b(a)h(\014xe)-5 b(d)34 b Fx(0)26 b Fm(<)f(\016)k Fl(\024)c Fx(min)o Fl(f)p Fm(\032;)15 b(r)s Fl(g)p Fm(=)p Fx(6)p Fd(,)34 b(we)f(de\014ne)1652 1377 y Fm(\026)25 b Fx(=)33 b(\026)-53 b Fm(\026)20 b Fx(+)2096 1315 y Fg(e)-55 b Fm(\026)p 2004 1356 230 4 v 2004 1439 a(\015)5 b(\016)2099 1413 y Fj(\034)j Fh(+1)2244 1377 y Fm(;)1420 b Fx(\(32\))71 1593 y Fd(and)33 b(supp)-5 b(ose)1441 1720 y Fx(13)p Fm(n)1586 1682 y Fh(2)1626 1720 y Fm(m)1706 1682 y Fh(3)1762 1720 y Fx(\026)-62 b Fm(m)1825 1682 y Fh(2)1879 1720 y Fg(e)d Fm(m)o(\033)1999 1682 y Fh(2)2136 1658 y Fm(\026)p 2049 1699 V 2049 1782 a(\015)5 b(\016)2144 1756 y Fj(\034)j Fh(+2)2313 1720 y Fl(\024)25 b Fx(1)p Fm(;)1210 b Fx(\(33\))71 1917 y Fd(wher)-5 b(e)30 b Fm(\033)f Fx(=)c Fm(\033)s Fx(\()p Fm(n;)15 b(\034)5 b(;)15 b(!)s Fx(\))26 b Fl(\025)f Fx(1)k Fd(is)g(pr)-5 b(ovide)g(d)32 b(by)d(L)-5 b(emma)30 b(8.)41 b(Then,)30 b(ther)-5 b(e)30 b(exists)g(a)f(c)-5 b(anonic)g(al)32 b(tr)-5 b(ansformation)71 2030 y Fx(\010\()p Fm(\022)s(;)15 b(I)7 b Fx(\))25 b(=)g(\()p Fm(\002)s Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b Fl(I)7 b Fx(\()p 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b(that)i(we)71 2611 y(exp)-5 b(and)34 b(as)653 2757 y Fm(H)736 2720 y Fh(\(1\))830 2757 y Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))26 b(=)f Fm(a)1203 2720 y Fh(\(1\))1297 2757 y Fx(\()p Fm(\022)s Fx(\))20 b(+)g Fl(h)p Fm(!)k Fx(+)c Fm(b)1770 2720 y Fh(\(1\))1864 2757 y Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(I)7 b Fl(i)21 b Fx(+)2224 2696 y(1)p 2224 2736 46 4 v 2224 2820 a(2)2279 2757 y Fl(h)p Fm(I)7 b(;)15 b(A)2469 2720 y Fh(\(1\))2565 2757 y Fx(\()p Fm(\022)s Fx(\))p Fm(I)7 b Fl(i)20 b Fx(+)g Fm(F)2945 2720 y Fh(\(1\))3040 2757 y Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))p Fm(;)71 2951 y Fd(b)-5 b(eing)32 b Fm(F)374 2918 y Fh(\(1\))494 2951 y Fx(=)25 b([)p Fm(H)698 2918 y Fh(\(1\))792 2951 y Fx(])817 2965 y Fh(3)857 2951 y Fd(,)32 b(with)i(b)-5 b(ounds)34 b(in)e(the)h(domain)h Fk(D)p Fx(\()p Fm(\032)2150 2918 y Fh(\(1\))2245 2951 y Fm(;)15 b(r)2329 2918 y Fh(\(1\))2423 2951 y Fx(\))p Fd(,)33 b(wher)-5 b(e)33 b Fm(r)2819 2918 y Fh(\(1\))2939 2951 y Fx(=)25 b Fm(r)d Fl(\000)e Fx(5)p Fm(\016)s 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y Fj(\034)2433 5762 y Fm(;)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 71 274 a Fx(12)2427 b Fn(Kolmogoro)m(v)32 b(Theorem)e(Revisited)71 573 y Fx(and)g(b)m(y)g(applying)g(Cauc)m(h)m (y)h(estimates)h(\(6\),)1330 819 y Fl(kr)1451 834 y Fj(\022)1491 819 y Fm(c)p Fl(k)1575 834 y Fj(\032)p Fi(\000)p Fh(2)p Fj(\016)1765 819 y Fl(\024)1871 756 y(k)p Fm(c)p Fl(k)2000 771 y Fj(\032)p Fi(\000)p Fj(\016)p 1871 798 260 4 v 1978 882 a Fm(\016)2165 819 y Fl(\024)25 b Fm(\033)2418 758 y Fg(e)-55 b Fm(\026)p 2326 798 230 4 v 2326 882 a(\015)5 b(\016)2421 855 y Fj(\034)j Fh(+1)2566 819 y Fm(:)71 1055 y Fx(Cauc)m(h)m(y)34 b(estimates)h(on)e(the)g(deriv)-5 b(ativ)m(es)35 b(are)f(extensiv)m(ely)h(used)e(along)h(the)g(pap)s(er)e (without)h(an)m(y)h(explicit)71 1168 y(men)m(tion.)41 b(If)29 b(w)m(e)g(compute)h(the)f(a)m(v)m(erage)j(with)d(resp)s(ect)g (to)h Fm(\022)h Fx(of)e(the)h(second)f(equation)h(of)f(\(15\),)i(w)m(e) f(obtain)1527 1364 y Fm(\030)f Fx(=)c(\()1752 1341 y(\026)1727 1364 y Fm(A)q 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y(using)e(Lemma)g(8)h(again,)h(w)m(e)f(obtain)f(the)h(follo)m (wing)h(estimate)g(for)e Fm(d)p Fx(:)204 2846 y Fl(k)p Fm(d)p Fl(k)341 2861 y Fj(\032)p Fi(\000)p Fh(3)p Fj(\016)532 2846 y Fl(\024)680 2785 y Fm(\033)p 638 2825 140 4 v 638 2909 a(\015)5 b(\016)733 2882 y Fj(\034)802 2846 y Fx(\()q Fl(k)p Fm(b)p Fl(k)967 2860 y Fj(\032)1028 2846 y Fx(+)20 b Fl(k)p Fm(A)p Fl(k)1277 2860 y Fj(\032)1318 2846 y Fx(\()p Fl(j)p Fm(\030)t Fl(j)h Fx(+)f Fl(kr)1680 2861 y Fj(\022)1719 2846 y Fm(c)p Fl(k)1803 2861 y Fj(\032)p Fi(\000)p Fh(2)p Fj(\016)1968 2846 y Fx(\)\))26 b Fl(\024)2212 2785 y Fm(\033)p 2170 2825 V 2170 2909 a(\015)5 b(\016)2265 2882 y Fj(\034)2319 2846 y Fx(\()i(\026)-52 b Fm(\026)21 b Fx(+)f(2)p Fm(m)2646 2809 y Fh(2)2703 2846 y Fx(\026)-63 b Fm(m\033)s(\026)p Fx(\))26 b Fl(\024)f Fx(3)p Fm(m)3157 2809 y Fh(2)3214 2846 y Fx(\026)-63 b Fm(m\033)3331 2809 y Fh(2)3423 2785 y Fm(\026)p 3381 2825 V 3381 2909 a(\015)5 b(\016)3476 2882 y Fj(\034)3530 2846 y Fm(:)134 b Fx(\(41\))71 3087 y(From)30 b(\(41\))i(w)m(e)f(ha)m(v)m(e,)685 3333 y(max)p Fl(fk)p Fm(D)1019 3348 y Fj(\022)1059 3333 y Fm(d)p Fl(k)1151 3348 y Fj(\032)p Fi(\000)p Fh(4)p Fj(\016)1316 3333 y Fm(;)15 b Fl(k)p Fx(\()p Fm(D)1511 3348 y Fj(\022)1552 3333 y Fm(d)p Fx(\))1634 3296 y Fi(>)1694 3333 y Fl(k)1739 3348 y Fj(\032)p Fi(\000)p Fh(4)p Fj(\016)1903 3333 y Fl(g)26 b(\024)f Fm(n)2135 3270 y Fl(k)p Fm(d)p Fl(k)2272 3285 y Fj(\032)p Fi(\000)p Fh(3)p Fj(\016)p 2135 3312 303 4 v 2264 3396 a Fm(\016)2472 3333 y Fl(\024)g Fx(3)p Fm(nm)2748 3296 y Fh(2)2804 3333 y Fx(\026)-62 b Fm(m\033)2922 3296 y Fh(2)3059 3272 y Fm(\026)p 2972 3312 230 4 v 2972 3396 a(\015)5 b(\016)3067 3369 y Fj(\034)j Fh(+1)3211 3333 y Fm(:)71 3584 y Fx(No)m(w)25 b(w)m(e)f(con)m(trol)i(the)e(action) i(of)e(the)h(canonical)g(transformation)g(\010)2436 3551 y Fj(G)2495 3584 y Fx(.)38 b(F)-8 b(or)25 b(this)f(purp)s(ose,)g(w)m(e) h(consider)f(the)71 3697 y(system)k(of)f(ordinary)g(di\013eren)m(tial)i (equations)f(\(16\).)42 b(First)28 b(w)m(e)g(solv)m(e)g(the)g(equation) 3026 3673 y(_)3009 3697 y Fm(\022)f Fx(=)e Fm(d)p Fx(\()p Fm(\022)s Fx(\).)40 b(W)-8 b(e)29 b(observ)m(e)71 3810 y(that)f(from)f(\(41\))h(and)f(condition)h(\(33\))h(w)m(e)f(ha)m(v)m(e) g Fl(k)p Fm(d)p Fl(k)1898 3825 y Fj(\032)p Fi(\000)p Fh(3)p Fj(\016)2089 3810 y Fl(\024)d Fm(\016)s Fx(.)40 b(Then,)27 b(b)m(y)h(using)e(elemen)m(tal)k(argumen)m(ts)d(on)71 3923 y(the)i(solutions)h(of)f(an)g(ordinary)f(di\013eren)m(tial)i (equation)g(\(more)g(precisely)-8 b(,)30 b(the)f(solutions)h(are)f (de\014ned)f(un)m(til)71 4036 y(they)34 b(reac)m(h)h(the)f(b)s(oundary) e(of)i(the)g(domain\),)h(w)m(e)g(ha)m(v)m(e)g(that)f Fm(\002)2374 4003 y Fj(t)2404 4036 y Fx(\()p Fm(\022)s Fx(\))g(is)g(de\014ned)e(for)i(an)m(y)h Fm(\022)e Fl(2)e Fx(\001\()p Fm(\032)22 b Fl(\000)h Fx(5)p Fm(\016)s Fx(\))71 4148 y(and)30 b(0)25 b Fl(\024)g Fm(t)g Fl(\024)g Fx(1,)31 b(and)f(v)m(eri\014es)h(\(see)g(\(17\))h(and)e(\(19\)\),)478 4364 y(sup)446 4447 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])648 4364 y Fl(fk)p Fm(\002)810 4327 y Fj(t)860 4364 y Fl(\000)20 b Fx(Id)1034 4379 y Fj(\022)1074 4364 y Fl(k)1119 4379 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1283 4364 y Fl(g)26 b Fx(=)57 b(sup)1450 4447 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])1652 4364 y Fl(fk)p Fm(f)1797 4327 y Fj(t)1827 4364 y Fl(k)1872 4379 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2036 4364 y Fl(g)26 b(\024)57 b Fx(sup)2203 4447 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])2405 4364 y Fl(f)p Fm(t)p Fl(k)p Fm(d)p Fl(k)2620 4379 y Fj(\032)p Fi(\000)p Fh(3)p Fj(\016)2785 4364 y Fl(g)26 b(\024)f Fx(3)p Fm(m)3077 4327 y Fh(2)3134 4364 y Fx(\026)-62 b Fm(m\033)3252 4327 y Fh(2)3343 4303 y Fm(\026)p 3301 4343 140 4 v 3301 4426 a(\015)5 b(\016)3396 4400 y Fj(\034)3450 4364 y Fm(:)214 b Fx(\(42\))71 4633 y(W)-8 b(e)33 b(observ)m(e)g(that,)g(in)f (particular,)h(the)f(expression)g(\(42\))i(is)e(smaller)h(than)e Fm(\016)s Fx(.)47 b(After)32 b(w)m(e)h(ha)m(v)m(e)g(solv)m(ed)g(the)71 4746 y(equation)h(for)g Fm(\022)s 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Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2818 5760 y Fm(ds;)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 71 274 a Fn(J.)30 b(Villan)m(uev)-5 b(a)3171 b Fx(13)71 573 y(for)30 b(an)m(y)h(0)25 b Fl(\024)g Fm(t)g Fl(\024)g Fx(1.)42 b(If)29 b(w)m(e)i(apply)f(Gron)m(w)m(all's)i (inequalit)m(y)g(w)m(e)f(obtain)g(\(see)g(\(33\)\),)758 835 y(sup)725 917 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])927 835 y Fl(fk)p Fm(g)1063 797 y Fj(t)1094 835 y Fl(k)1139 850 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1304 835 y Fl(g)26 b(\024)f Fx(2)p Fm(m)17 b Fx(\026)-62 b Fm(m\033)s(\026)15 b Fx(exp)1955 706 y Fg(\032)2023 835 y Fx(3)p Fm(nm)2203 797 y Fh(2)2260 835 y Fx(\026)-63 b Fm(m\033)2377 797 y Fh(2)2514 773 y Fm(\026)p 2427 814 230 4 v 2427 897 a(\015)5 b(\016)2522 871 y Fj(\034)j Fh(+1)2666 706 y Fg(\033)2760 835 y Fl(\024)25 b Fx(4)p Fm(m)17 b Fx(\026)-62 b Fm(m\033)s(\026:)493 b Fx(\(43\))71 1124 y(W)-8 b(e)31 b(rep)s(eat)g(the)g(same)f(pro)s(cess)g(for)g Fm(B)1422 1091 y Fj(t)1482 1124 y Fx(\(see)h(\(20\)\).)43 b(Then,)29 b(for)i(an)m(y)f(0)c Fl(\024)f Fm(t)g Fl(\024)g Fx(1,)31 b(w)m(e)g(ha)m(v)m(e)657 1392 y Fl(k)p Fm(B)776 1355 y Fj(t)805 1392 y Fl(k)850 1407 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1098 1392 y Fl(\024)1251 1269 y Fg(Z)1342 1295 y Fj(t)1302 1475 y Fh(0)1387 1392 y Fl(k)p Fx(\()p Fm(D)1542 1407 y Fj(\022)1582 1392 y Fm(d)p Fx(\))1664 1355 y Fi(>)1724 1392 y Fl(k)1769 1407 y Fj(\032)p Fi(\000)p Fh(4)p Fj(\016)1934 1392 y Fx(\(1)21 b(+)f Fl(k)p Fm(B)2245 1355 y Fj(s)2281 1392 y Fl(k)2326 1407 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2491 1392 y Fx(\))p Fm(ds)1098 1646 y Fl(\024)82 b Fx(3)p Fm(nm)1431 1609 y Fh(2)1488 1646 y Fx(\026)-62 b Fm(m\033)1606 1609 y Fh(2)1742 1585 y Fm(\026)p 1655 1625 V 1655 1709 a(\015)5 b(\016)1750 1682 y Fj(\034)j Fh(+1)1915 1646 y Fx(+)2006 1523 y Fg(Z)2096 1549 y Fj(t)2056 1729 y Fh(0)2141 1646 y Fx(3)p Fm(nm)2321 1609 y Fh(2)2378 1646 y Fx(\026)-62 b Fm(m)o(\033)2495 1609 y Fh(2)2632 1585 y Fm(\026)p 2545 1625 V 2545 1709 a(\015)5 b(\016)2640 1682 y Fj(\034)j Fh(+1)2784 1646 y Fl(k)p Fm(B)2903 1609 y Fj(s)2940 1646 y Fl(k)2985 1661 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3150 1646 y Fm(ds:)71 1899 y Fx(Using)30 b(Gron)m(w)m(all)i(again)g(w)m(e)e(obtain) 417 2156 y(sup)384 2238 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])586 2156 y Fl(fk)p Fm(B)750 2118 y Fj(t)780 2156 y Fl(k)825 2171 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)990 2156 y Fl(g)25 b(\024)g Fx(3)p Fm(nm)1336 2118 y Fh(2)1393 2156 y Fx(\026)-62 b Fm(m)o(\033)1510 2118 y Fh(2)1647 2094 y Fm(\026)p 1560 2135 V 1560 2218 a(\015)5 b(\016)1655 2192 y Fj(\034)j Fh(+1)1814 2156 y Fx(exp)1968 2028 y Fg(\032)2037 2156 y Fx(3)p Fm(nm)2217 2118 y Fh(2)2273 2156 y Fx(\026)-62 b Fm(m\033)2391 2118 y Fh(2)2528 2094 y Fm(\026)p 2440 2135 V 2440 2218 a(\015)5 b(\016)2535 2192 y Fj(\034)j Fh(+1)2680 2028 y Fg(\033)2773 2156 y Fl(\024)25 b Fx(6)p Fm(nm)3049 2118 y Fh(2)3105 2156 y Fx(\026)-62 b Fm(m\033)3223 2118 y Fh(2)3360 2094 y Fm(\026)p 3273 2135 V 3273 2218 a(\015)5 b(\016)3368 2192 y Fj(\034)j Fh(+1)3512 2156 y Fm(:)71 2435 y Fx(Then,)29 b(w)m(e)i(ha)m(v)m(e)308 2646 y(sup)275 2729 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])477 2646 y Fl(fkI)624 2609 y Fj(t)674 2646 y Fl(\000)20 b Fx(Id)848 2660 y Fj(I)888 2646 y Fl(k)933 2661 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)o(;r)r Fi(\000)p Fj(\016)1236 2646 y Fl(g)84 b(\024)115 b Fx(sup)1518 2729 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])1720 2646 y Fl(fk)p Fm(B)1884 2609 y Fj(t)1914 2646 y Fl(k)1959 2661 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2124 2646 y Fl(g)p Fx(\()p Fm(r)23 b Fl(\000)d Fm(\016)s Fx(\))i(+)52 b(sup)2550 2729 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])2752 2646 y Fl(fk)p Fm(g)2888 2609 y Fj(t)2919 2646 y Fl(k)2964 2661 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3128 2646 y Fl(g)1365 2889 y(\024)82 b Fx(6)p Fm(nm)1698 2851 y Fh(2)1755 2889 y Fx(\026)-62 b Fm(m)o(\033)1872 2851 y Fh(2)1912 2889 y Fx(\()p Fm(r)23 b Fl(\000)d Fm(\016)s Fx(\))2278 2827 y Fm(\026)p 2190 2868 V 2190 2951 a(\015)5 b(\016)2285 2925 y Fj(\034)j Fh(+1)2451 2889 y Fx(+)19 b(4)p Fm(m)f Fx(\026)-63 b Fm(m)q(\033)s(\026)25 b Fl(\024)g Fx(8)p Fm(nm)3158 2851 y Fh(2)3214 2889 y Fx(\026)-62 b Fm(m\033)3332 2851 y Fh(2)3469 2827 y Fm(\026)p 3381 2868 V 3381 2951 a(\015)5 b(\016)3476 2925 y Fj(\034)j Fh(+1)3621 2889 y Fm(;)43 b Fx(\(44\))71 3154 y(whic)m(h)37 b(from)h(\(33\))h(is)f (smaller)h(than)e Fm(\016)s Fx(.)64 b(Bounds)37 b(\(42\))j(and)d (\(44\))i(on)f(the)g(comp)s(onen)m(ts)g(of)g(\010)3433 3121 y Fj(G)3433 3177 y(t)3530 3154 y Fx(giv)m(e)h(the)71 3267 y(inequalit)m(y)34 b Fl(k)p Fx(\010)22 b Fl(\000)f Fx(Id)o Fl(k)848 3293 y Fj(\032)884 3273 y Fc(\(1\))967 3293 y Fj(;r)1021 3273 y Fc(\(1\))1137 3267 y Fl(\024)28 b Fm(\016)37 b Fx(on)32 b(the)h(statemen)m(t,)i(and)d(justify)g(that)h (the)g(comp)s(ositions)g(in)m(v)m(olv)m(ed)i(in)71 3397 y(the)27 b(computation)h(of)f Fm(H)934 3364 y Fh(\(1\))1055 3397 y Fx(\(according)i(to)e(form)m(ulas)g(of)g(Section)h(4\))g(are)f (w)m(ell)h(de\014ned)e(in)h Fk(D)p Fx(\()p Fm(\032)13 b Fl(\000)g Fx(5)p Fm(\016)n(;)i(r)j Fl(\000)13 b Fm(\016)s Fx(\).)212 3512 y(T)-8 b(o)25 b(con)m(trol)g Fm(H)722 3479 y Fh(\(1\))841 3512 y Fx(w)m(e)g(also)g(ha)m(v)m(e)g(to)g (consider)i(^)-48 b Fm(g)28 b Fx(and)2064 3489 y(^)2043 3512 y Fm(B)t Fx(.)39 b(F)-8 b(or)28 b(^)-48 b Fm(g)28 b Fx(w)m(e)c(ha)m(v)m(e)i(\(see)f(\(23\))h(for)e Fm(t)h Fx(=)g(1)g(and)e(\(25\)\),)443 3724 y Fl(k)s Fx(^)-48 b Fm(g)t Fl(k)580 3739 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)828 3724 y Fl(\024)115 b Fx(sup)981 3807 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])1183 3724 y Fl(fkr)1349 3739 y Fj(\022)1389 3724 y Fm(c)p Fx(\()p Fm(\002)1535 3687 y Fj(t)1565 3724 y Fx(\()p Fm(\022)s Fx(\)\))21 b Fl(\000)e(r)1903 3739 y Fj(\022)1942 3724 y Fm(c)p Fx(\()p Fm(\022)s Fx(\))p Fl(k)2142 3739 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2307 3724 y Fl(g)i Fx(+)f Fl(k)p Fx(\()p Fm(D)2619 3739 y Fj(\022)2659 3724 y Fm(d)p Fx(\))2741 3687 y Fi(>)2800 3724 y Fl(k)2845 3739 y Fj(\032)p Fi(\000)p Fh(4)p Fj(\016)3058 3724 y Fx(sup)3025 3807 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])3227 3724 y Fl(fk)p Fm(g)3363 3687 y Fj(t)3394 3724 y Fl(k)3439 3739 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3604 3724 y Fl(g)828 4003 y(\024)82 b Fx(3)p Fm(nm)1161 3965 y Fh(2)1218 4003 y Fx(\026)-62 b Fm(m)o(\033)1335 3965 y Fh(3)1487 3941 y Fg(e)-56 b Fm(\026\026)p 1385 3982 304 4 v 1385 4065 a(\015)1437 4039 y Fh(2)1477 4065 y Fm(\016)1520 4039 y Fh(2)p Fj(\034)8 b Fh(+2)1719 4003 y Fx(+)20 b(12)p Fm(nm)2035 3965 y Fh(3)2092 4003 y Fx(\026)-62 b Fm(m)2155 3965 y Fh(2)2194 4003 y Fm(\033)2249 3965 y Fh(3)2366 3941 y Fm(\026)2421 3908 y Fh(2)p 2299 3982 230 4 v 2299 4065 a Fm(\015)5 b(\016)2394 4039 y Fj(\034)j Fh(+1)2563 4003 y Fl(\024)25 b Fx(15)p Fm(nm)2884 3965 y Fh(3)2941 4003 y Fx(\026)-62 b Fm(m)3004 3965 y Fh(2)3043 4003 y Fm(\033)3098 3965 y Fh(3)3215 3941 y Fm(\026)3270 3908 y Fh(2)p 3148 3982 V 3148 4065 a Fm(\015)5 b(\016)3243 4039 y Fj(\034)j Fh(+1)3387 4003 y Fm(;)277 b Fx(\(45\))272 4195 y Fl(kh)p Fm(!)s(;)19 b Fx(^)-49 b Fm(g)5 b Fl(ik)580 4210 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)828 4195 y Fl(\024)115 b Fx(sup)981 4277 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])1183 4195 y Fl(fk)o Fg(e)-50 b Fm(a)q Fx(\()p Fm(\002)1429 4157 y Fj(t)1459 4195 y Fx(\()p Fm(\022)s Fx(\)\))20 b Fl(\000)f Fg(e)-50 b Fm(a)p Fx(\()p Fm(\022)s Fx(\))p Fl(k)1930 4210 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2095 4195 y Fl(g)981 4415 y Fx(+)p Fm(n)p Fx(\()p Fl(k)p Fm(b)p Fl(k)1271 4429 y Fj(\032)1332 4415 y Fx(+)20 b Fl(k)p Fm(A)p Fl(k)1581 4429 y Fj(\032)1623 4415 y Fx(\()p Fl(j)p Fm(\030)t Fl(j)h Fx(+)f Fl(kr)1985 4430 y Fj(\022)2024 4415 y Fm(c)p Fl(k)2108 4430 y Fj(\032)p Fi(\000)p Fh(2)p Fj(\016)2273 4415 y Fx(\)\))48 b(sup)2358 4497 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])2560 4415 y Fl(fk)p Fm(g)2696 4377 y Fj(t)2728 4415 y Fl(k)2773 4430 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2937 4415 y Fl(g)828 4683 y(\024)82 b Fx(3)p Fm(nm)1161 4646 y Fh(2)1218 4683 y Fx(\026)-62 b Fm(m)o(\033)1335 4646 y Fh(2)1445 4622 y Fm(\026)5 b Fg(e)-56 b Fm(\026)p 1385 4662 V 1385 4746 a(\015)5 b(\016)1480 4719 y Fj(\034)j Fh(+1)1644 4683 y Fx(+)20 b(4)p Fm(nm)d Fx(\026)-62 b Fm(m\033)s(\026)p Fx(\()7 b(\026)-52 b Fm(\026)20 b Fx(+)g(2)p Fm(m)2431 4646 y Fh(2)2488 4683 y Fx(\026)-62 b Fm(m\033)s(\026)p Fx(\))25 b Fl(\024)g Fx(12)p Fm(nm)3042 4646 y Fh(3)3099 4683 y Fx(\026)-62 b Fm(m)3162 4646 y Fh(2)3201 4683 y Fm(\033)3256 4646 y Fh(2)3296 4683 y Fm(\026)3351 4646 y Fh(2)3390 4683 y Fm(;)274 b Fx(\(46\))71 4939 y(where)30 b(w)m(e)h(use)f(the)g (Mean)h(V)-8 b(alue)32 b(Theorem)e(\(8\))h(and)f(Cauc)m(h)m(y)h (estimates)h(to)f(b)s(ound)315 5199 y(sup)282 5281 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])484 5199 y Fl(fk)o Fg(e)-50 b Fm(a)q Fx(\()p Fm(\002)730 5161 y Fj(t)760 5199 y Fx(\()p Fm(\022)s Fx(\)\))20 b Fl(\000)f Fg(e)-50 b Fm(a)p Fx(\()p Fm(\022)s Fx(\))p Fl(k)1231 5214 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1396 5199 y Fl(g)26 b(\024)f(k)p Fm(D)1683 5214 y Fj(\022)1721 5199 y Fg(e)-50 b Fm(a)p Fl(k)1815 5214 y Fj(\032)p Fi(\000)p Fj(\016)1992 5199 y Fx(sup)1960 5281 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])2162 5199 y Fl(fk)p Fm(f)2307 5161 y Fj(t)2337 5199 y Fl(k)2382 5214 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2546 5199 y Fl(g)26 b(\024)f Fm(n)2778 5137 y Fl(k)o Fg(e)-50 b Fm(a)p Fl(k)2916 5151 y Fj(\032)p 2778 5178 180 4 v 2845 5261 a Fm(\016)3014 5199 y Fx(sup)2982 5281 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])3184 5199 y Fl(fk)p Fm(f)3329 5161 y Fj(t)3359 5199 y Fl(k)3404 5214 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3568 5199 y Fl(g)p Fm(;)71 5478 y Fx(and,)30 b(b)s(ounding)e(comp)s(onen)m(t)j(b)m(y)f(comp)s(onen)m(t,)744 5734 y(sup)711 5816 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])913 5734 y Fl(fkr)1079 5749 y Fj(\022)1119 5734 y Fm(c)p Fx(\()p Fm(\002)1265 5696 y Fj(t)1295 5734 y Fx(\()p Fm(\022)s Fx(\)\))21 b Fl(\000)f(r)1634 5749 y Fj(\022)1672 5734 y Fm(c)p Fx(\()p Fm(\022)s Fx(\))p Fl(k)1872 5749 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2037 5734 y Fl(g)26 b(\024)f Fm(n)2269 5671 y Fl(k)p Fm(c)p Fl(k)2398 5686 y Fj(\032)p Fi(\000)p Fj(\016)p 2269 5713 260 4 v 2356 5796 a Fm(\016)2399 5770 y Fh(2)2585 5734 y Fx(sup)2553 5816 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])2755 5734 y Fl(fk)p Fm(f)2900 5696 y Fj(t)2929 5734 y Fl(k)2974 5749 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3139 5734 y Fl(g)p Fm(:)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 71 274 a Fx(14)2427 b Fn(Kolmogoro)m(v)32 b(Theorem)e(Revisited)71 573 y Fx(Moreo)m(v)m(er,)36 b(w)m(e)e(also)g(use)e(that)i Fl(j)p Fx(\()p Fl(\001)p Fx(\))1300 540 y Fi(>)1360 573 y Fl(j)c(\024)g Fm(n)p Fl(j)p Fx(\()p Fl(\001)p Fx(\))p Fl(j)p Fx(,)35 b(b)s(eing)e(\()p Fl(\001)p Fx(\))h(a)f Fm(n)p Fx(-dimensional)g(v)m(ector)i(or)e(a)h (matrix)f(with)g Fm(n)71 686 y Fx(ro)m(ws.)41 b(F)-8 b(or)31 b(further)e(uses)h(w)m(e)h(observ)m(e)g(that)f(this)h(implies)f (the)h(b)s(ound)d Fl(jh)p Fx(\()p Fl(\001)p Fx(\))2711 700 y Fh(1)2752 686 y Fm(;)15 b Fx(\()p Fl(\001)p Fx(\))2887 700 y Fh(2)2928 686 y Fl(ij)26 b(\024)f Fm(n)p Fl(j)p Fx(\()p Fl(\001)p Fx(\))3285 700 y Fh(1)3325 686 y Fl(jj)p Fx(\()p Fl(\001)p Fx(\))3470 700 y Fh(2)3511 686 y Fl(j)p Fx(.)212 799 y(T)-8 b(o)28 b(b)s(ound)641 776 y(^)620 799 y Fm(B)694 766 y Fi(>)752 799 y Fm(!)i Fx(w)m(e)e(consider)f (\(26\))i(for)e Fm(t)e Fx(=)g(1.)40 b(Then,)27 b(w)m(e)h(add)e(and)h (subtract)g Fm(A)p Fx(\()p Fm(\022)s Fx(\)\()p Fm(\030)18 b Fx(+)c Fl(r)3442 814 y Fj(\022)3480 799 y Fm(c)p Fx(\()p Fm(\002)3626 766 y Fj(s)3664 799 y Fx(\()p Fm(\022)s Fx(\)\)\))71 912 y(inside)30 b(the)h(second)f(in)m(tegral,)i(th)m(us)e (obtaining:)552 1097 y Fl(k)619 1074 y Fx(^)597 1097 y Fm(B)671 1060 y Fi(>)730 1097 y Fm(!)s Fl(k)835 1112 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1082 1097 y Fl(\024)115 b Fx(sup)1236 1180 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])1438 1097 y Fl(fk)p Fm(b)p Fx(\()p Fm(\002)1674 1060 y Fj(t)1705 1097 y Fx(\()p Fm(\022)s Fx(\)\))20 b Fl(\000)g Fm(b)p Fx(\()p Fm(\022)s Fx(\))p Fl(k)2167 1112 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2331 1097 y Fl(g)1236 1317 y Fx(+)47 b(sup)1322 1400 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])1524 1317 y Fl(fk)p Fm(A)p Fx(\()p Fm(\002)1789 1280 y Fj(t)1820 1317 y Fx(\()p Fm(\022)s Fx(\)\))20 b Fl(\000)g Fm(A)p Fx(\()p Fm(\022)s Fx(\))p Fl(k)2311 1332 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2476 1317 y Fl(g)p Fx(\()p Fl(j)p Fm(\030)t Fl(j)h Fx(+)f Fl(kr)2883 1332 y Fj(\022)2922 1317 y Fm(c)p Fl(k)3006 1332 y Fj(\032)p Fi(\000)p Fh(2)p Fj(\016)3171 1317 y Fx(\))1236 1537 y(+)p Fl(k)p Fm(A)p Fl(k)1465 1551 y Fj(\032)1554 1537 y Fx(sup)1521 1620 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])1723 1537 y Fl(fkr)1889 1552 y Fj(\022)1929 1537 y Fm(c)p Fx(\()p Fm(\002)2075 1500 y Fj(t)2105 1537 y Fx(\()p Fm(\022)s Fx(\)\))g Fl(\000)g(r)2443 1552 y Fj(\022)2482 1537 y Fm(c)p Fx(\()p Fm(\022)s Fx(\))p Fl(k)2682 1552 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2847 1537 y Fl(g)1236 1762 y Fx(+)47 b(sup)1322 1845 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])1524 1762 y Fl(fk)p Fx(\()p Fm(B)1723 1725 y Fj(t)1753 1762 y Fx(\))1788 1725 y Fi(>)1847 1762 y Fl(k)1892 1777 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2057 1762 y Fl(g)15 b Fx(\()q Fl(k)p Fm(b)p Fl(k)2282 1776 y Fj(\032)2343 1762 y Fx(+)20 b Fl(k)p Fm(A)p Fl(k)2592 1776 y Fj(\032)2634 1762 y Fx(\()p Fl(j)p Fm(\030)t Fl(j)h Fx(+)f Fl(kr)2996 1777 y Fj(\022)3035 1762 y Fm(c)p Fl(k)3119 1777 y Fj(\032)p Fi(\000)p Fh(2)p Fj(\016)3284 1762 y Fx(\)\))1082 2036 y Fl(\024)83 b Fx(3)p Fm(m)1361 1998 y Fh(2)1418 2036 y Fx(\026)-62 b Fm(m)o(\033)1535 1998 y Fh(2)1627 1974 y Fm(\026)p 1585 2015 140 4 v 1585 2098 a(\015)5 b(\016)1680 2072 y Fj(\034)1749 1908 y Fg(\022)1816 2036 y Fm(n)1888 1974 y Fx(\026)-52 b Fm(\026)p 1881 2015 55 4 v 1886 2098 a(\016)1966 2036 y Fx(+)19 b(2)p Fm(n)2156 1998 y Fh(2)2196 2036 y Fm(m)2276 1998 y Fh(2)2332 2036 y Fx(\026)-62 b Fm(m\033)2460 1974 y(\026)p 2460 2015 V 2465 2098 a(\016)2545 2036 y Fx(+)20 b Fm(nm\033)2927 1974 y Fg(e)-55 b Fm(\026)p 2836 2015 230 4 v 2836 2098 a(\015)5 b(\016)2931 2072 y Fj(\034)j Fh(+2)3075 1908 y Fg(\023)1236 2256 y Fx(+6)p Fm(n)1407 2219 y Fh(2)1446 2256 y Fm(m)1526 2219 y Fh(2)1582 2256 y Fx(\026)-62 b Fm(m\033)1700 2219 y Fh(2)1837 2195 y Fm(\026)p 1750 2235 V 1750 2318 a(\015)5 b(\016)1845 2292 y Fj(\034)j Fh(+1)1989 2256 y Fx(\()f(\026)-52 b Fm(\026)20 b Fx(+)g(2)p Fm(m)2315 2219 y Fh(2)2372 2256 y Fx(\026)-62 b Fm(m\033)s(\026)p Fx(\))1082 2507 y Fl(\024)83 b Fx(26)p Fm(n)1381 2469 y Fh(2)1421 2507 y Fm(m)1501 2469 y Fh(4)1557 2507 y Fx(\026)-62 b Fm(m)1620 2469 y Fh(2)1659 2507 y Fm(\033)1714 2469 y Fh(3)1831 2445 y Fm(\026)1886 2412 y Fh(2)p 1764 2486 V 1764 2569 a Fm(\015)5 b(\016)1859 2543 y Fj(\034)j Fh(+1)2003 2507 y Fm(:)1661 b Fx(\(47\))71 2736 y(Here)31 b(w)m(e)g(use)f(\(b)s(ounding)f(again)i(comp)s(onen)m(t)g(b)m(y)f(comp) s(onen)m(t\),)841 2970 y(sup)808 3053 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])1010 2970 y Fl(fk)p Fm(b)p Fx(\()p Fm(\002)1246 2933 y Fj(t)1277 2970 y Fx(\()p Fm(\022)s Fx(\)\))20 b Fl(\000)g Fm(b)p Fx(\()p Fm(\022)s Fx(\))p Fl(k)1739 2985 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1904 2970 y Fl(g)83 b(\024)g Fm(n)2251 2908 y Fl(k)p Fm(b)p Fl(k)2380 2922 y Fj(\032)p 2251 2949 171 4 v 2314 3033 a Fm(\016)2478 2970 y Fx(sup)2446 3053 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])2648 2970 y Fl(fk)p Fm(f)2793 2933 y Fj(t)2823 2970 y Fl(k)2868 2985 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3032 2970 y Fl(g)p Fm(;)782 3242 y Fx(sup)750 3325 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])952 3242 y Fl(fk)p Fm(A)p Fx(\()p Fm(\002)1217 3204 y Fj(t)1248 3242 y Fx(\()p Fm(\022)s Fx(\)\))20 b Fl(\000)g Fm(A)p Fx(\()p Fm(\022)s Fx(\))p Fl(k)1739 3257 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1904 3242 y Fl(g)83 b(\024)g Fm(n)2241 3204 y Fh(2)2290 3180 y Fl(k)p Fm(A)p Fl(k)2448 3194 y Fj(\032)p 2290 3221 200 4 v 2368 3304 a Fm(\016)2547 3242 y Fx(sup)2514 3325 y Fj(t)p Fi(2)p Fh([0)p Fj(;)p Fh(1])2716 3242 y Fl(fk)p Fm(f)2861 3204 y Fj(t)2891 3242 y Fl(k)2936 3257 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3101 3242 y Fl(g)p Fm(:)71 3514 y Fx(A)m(t)27 b(this)g(time)g(w)m(e)h(ha)m (v)m(e)g(all)f(the)g(ingredien)m(ts)g(needed)g(to)g(b)s(ound)e(the)h (\(transformed\))h(Hamiltonian)h Fm(H)3660 3481 y Fh(\(1\))3780 3514 y Fx(=)71 3627 y Fm(H)f Fl(\016)21 b Fx(\010)306 3594 y Fj(G)395 3627 y Fx(\(see)31 b(\(24\)\))h(in)e(the)h(domain)f Fk(D)p Fx(\()p Fm(\032)20 b Fl(\000)g Fx(5)p Fm(\016)n(;)15 b(r)25 b Fl(\000)20 b Fx(5)p Fm(\016)s Fx(\).)42 b(F)-8 b(or)31 b Fm(a)2330 3594 y Fh(\(1\))2455 3627 y Fx(w)m(e)g(ha)m(v)m(e)g (\(see)h(\(27\)\),)314 3813 y Fl(k)p Fm(a)407 3775 y Fh(\(1\))522 3813 y Fl(\000)21 b Fx(\026)-46 b Fm(a)20 b Fx(+)g Fl(h)p Fm(!)s(;)15 b(\030)t Fl(ik)1031 3828 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1280 3813 y Fl(\024)82 b(k)o Fg(e)-50 b Fm(a)q Fx(\()p Fm(\002)s Fx(\()p Fm(\022)s Fx(\)\))21 b Fl(\000)d Fg(e)-50 b Fm(a)p Fx(\()p Fm(\022)s Fx(\))p Fl(k)2105 3828 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2290 3813 y Fx(+)20 b Fl(kh)p Fm(!)s(;)f Fx(^)-49 b Fm(g)5 b Fx(\()p Fm(\022)s Fx(\))p Fl(ik)2805 3828 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2990 3813 y Fx(+)20 b Fm(n)p Fl(k)p Fm(b)p Fl(k)3265 3827 y Fj(\032)3306 3813 y Fl(k)p Fm(g)s Fl(k)3442 3828 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1433 3975 y Fx(+)1514 3913 y Fm(n)p 1514 3954 55 4 v 1519 4037 a Fx(2)1579 3975 y Fl(k)p Fm(A)p Fl(k)1737 3989 y Fj(\032)1778 3975 y Fl(k)p Fm(g)s Fl(k)1914 3937 y Fh(2)1914 3998 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2100 3975 y Fx(+)g Fl(k)p Fm(F)13 b Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\)\))p Fl(k)2952 3990 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1280 4197 y Fl(\024)82 b Fx(3)p Fm(nm)1613 4160 y Fh(2)1670 4197 y Fx(\026)-62 b Fm(m)o(\033)1787 4160 y Fh(2)1897 4136 y Fm(\026)5 b Fg(e)-56 b Fm(\026)p 1837 4176 230 4 v 1837 4260 a(\015)5 b(\016)1932 4233 y Fj(\034)j Fh(+1)2096 4197 y Fx(+)20 b(12)p Fm(nm)2412 4160 y Fh(3)2469 4197 y Fx(\026)-62 b Fm(m)2532 4160 y Fh(2)2571 4197 y Fm(\033)2626 4160 y Fh(2)2666 4197 y Fm(\026)2721 4160 y Fh(2)2780 4197 y Fx(+)20 b(4)p Fm(nm)d Fx(\026)-62 b Fm(m\033)s(\026)7 b Fx(\026)-52 b Fm(\026)1433 4391 y Fx(+8)p Fm(nm)1684 4353 y Fh(3)1741 4391 y Fx(\026)-63 b Fm(m)1803 4353 y Fh(2)1843 4391 y Fm(\033)1898 4353 y Fh(2)1937 4391 y Fm(\026)1992 4353 y Fh(2)2052 4391 y Fx(+)20 b(64)p Fm(m)2313 4353 y Fh(3)2370 4391 y Fx(\026)-62 b Fm(m)2433 4353 y Fh(3)2487 4391 y Fg(e)c Fm(m\033)2607 4353 y Fh(3)2646 4391 y Fm(\026)2701 4353 y Fh(3)1280 4529 y Fl(\024)82 b Fx(25)p Fm(nm)1658 4491 y Fh(3)1715 4529 y Fx(\026)-62 b Fm(m)1778 4491 y Fh(2)1817 4529 y Fm(\033)1872 4491 y Fh(2)1912 4529 y Fm(\026)1967 4491 y Fh(2)2006 4529 y Fm(;)71 4714 y Fx(where)30 b(w)m(e)h(use)f(\(see)h(h)m(yp)s(othesis)f(\(31\))i(on)e Fm(F)13 b Fx(\))1246 4900 y Fl(k)p Fm(F)g Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\)\))p Fl(k)2007 4915 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2198 4900 y Fl(\024)39 b Fg(e)-65 b Fm(m)o Fl(k)p Fm(g)s Fl(k)2509 4863 y Fh(3)2509 4923 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3689 4900 y Fx(\(48\))71 5086 y(and)23 b(that)g(64)c(\026)-64 b Fm(m)16 b Fg(e)-66 b Fm(m\033)s(\026)25 b Fl(\024)g Fm(n)e Fx(\(see)h(\(33\))h(and)e (recall)h(that)g Fm(\016)30 b Fl(\024)24 b Fx(1)p Fm(=)p Fx(6\).)40 b(T)-8 b(o)24 b(obtain)g(\(48\))h(w)m(e)f(require)f Fl(k)p Fm(g)s Fl(k)3495 5101 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3686 5086 y Fl(\024)i Fm(r)s Fx(,)71 5199 y(whic)m(h)30 b(is)g(guaran)m(teed)i(b)m(y)e(\(33\))i(and)e(\(43\).)42 b(From)30 b(here,)h(w)m(e)g(deriv)m(e)g(the)f(estimate)i(\(see)g (\(7\)\))878 5385 y Fl(k)o Fg(e)-50 b Fm(a)971 5347 y Fh(\(1\))1065 5385 y Fl(k)1110 5400 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1300 5385 y Fl(\024)25 b Fx(2)p Fl(k)p Fm(a)1534 5347 y Fh(\(1\))1650 5385 y Fl(\000)c Fx(\026)-46 b Fm(a)20 b Fx(+)g Fl(h)p Fm(!)s(;)15 b(\030)t Fl(ik)2159 5400 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2350 5385 y Fl(\024)25 b Fx(50)p Fm(nm)2671 5347 y Fh(3)2727 5385 y Fx(\026)-62 b Fm(m)2790 5347 y Fh(2)2830 5385 y Fm(\033)2885 5347 y Fh(2)2924 5385 y Fm(\026)2979 5347 y Fh(2)3018 5385 y Fm(:)71 5571 y Fo(Remark)29 b(6.)38 b Fd(If)28 b(inste)-5 b(ad)29 b(of)f(the)g(b)-5 b(ound)29 b(\(46\))g(for)g Fl(kh)p Fm(!)s(;)19 b Fx(^)-49 b Fm(g)t Fl(ik)2125 5586 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2290 5571 y Fd(,)29 b(that)g(fol)5 b(lows)29 b(fr)-5 b(om)29 b(formula)h(\(25\),)g(we)e (use)71 5684 y(the)g(dir)-5 b(e)g(ct)30 b(\(but)e(worst\))h(estimate)g Fm(n)p Fl(j)p Fm(!)s Fl(jk)s Fx(^)-48 b Fm(g)t Fl(k)1586 5699 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1751 5684 y Fd(,)29 b(with)g(the)f(b)-5 b(ound)29 b(on)j Fx(^)-48 b Fm(g)32 b Fd(given)27 b(by)h(\(45\),)i(then)f(the)f(estimate)71 5796 y(of)33 b Fl(O)s Fx(\()p Fm(\026)343 5763 y Fh(2)382 5796 y Fx(\))g Fd(for)g Fl(k)o Fg(e)-50 b Fm(a)688 5763 y Fh(\(1\))783 5796 y Fl(k)828 5811 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1025 5796 y Fd(turns)33 b(to)h(b)-5 b(e)32 b(of)h Fl(O)s Fx(\()p Fm(\026)1756 5763 y Fh(2)1795 5796 y Fm(=)p Fx(\()p Fm(\015)5 b(\016)1970 5763 y Fj(\034)j Fh(+1)2106 5796 y Fx(\)\))p Fd(.)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 71 274 a Fn(J.)30 b(Villan)m(uev)-5 b(a)3171 b Fx(15)212 573 y(F)-8 b(or)31 b Fm(b)414 540 y Fh(\(1\))539 573 y Fx(w)m(e)g(ha)m(v)m(e)g(\(see)h(\(28\)\),)217 789 y Fl(k)p Fm(b)301 751 y Fh(\(1\))396 789 y Fl(k)441 804 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)689 789 y Fl(\024)83 b(k)910 766 y Fx(^)888 789 y Fm(B)962 751 y Fi(>)1021 789 y Fm(!)s Fl(k)1126 804 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1310 789 y Fx(+)20 b Fl(k)p Fm(b)p Fx(\()p Fm(\002)s Fx(\()p Fm(\022)s Fx(\)\))h Fl(\000)f Fm(b)p Fx(\()p Fm(\022)s Fx(\))p Fl(k)2055 804 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2240 789 y Fx(+)g Fl(k)p Fm(B)2450 751 y Fi(>)2509 789 y Fl(k)2554 804 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2718 789 y Fl(k)p Fm(b)p Fl(k)2847 803 y Fj(\032)843 927 y Fx(+)p Fl(k)p Fm(A)p Fx(\()p Fm(\002)s Fx(\()p Fm(\022)s Fx(\)\))h Fl(\000)e Fm(A)p Fx(\()p Fm(\022)s Fx(\))p Fl(k)1625 942 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1790 927 y Fl(k)p Fm(g)s Fl(k)1926 942 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2112 927 y Fx(+)h Fl(k)p Fm(A)p Fl(k)2361 941 y Fj(\032)2402 927 y Fl(k)s Fx(^)-48 b Fm(g)t Fl(k)2539 942 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2724 927 y Fx(+)20 b Fl(k)p Fm(B)2934 889 y Fi(>)2993 927 y Fl(k)3038 942 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3203 927 y Fl(k)p Fm(A)p Fl(k)3361 941 y Fj(\032)3402 927 y Fl(k)p Fm(g)s Fl(k)3538 942 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)843 1064 y Fx(+\(1)g(+)g Fl(k)p Fm(B)1224 1027 y Fi(>)1283 1064 y Fl(k)1328 1079 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1493 1064 y Fx(\))p Fl(kr)1649 1078 y Fj(I)1689 1064 y Fm(F)13 b Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\)\))p Fl(k)2405 1079 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)689 1263 y Fl(\024)83 b Fx(26)p Fm(n)988 1225 y Fh(2)1027 1263 y Fm(m)1107 1225 y Fh(4)1164 1263 y Fx(\026)-62 b Fm(m)1227 1225 y Fh(2)1266 1263 y Fm(\033)1321 1225 y Fh(3)1438 1201 y Fm(\026)1493 1168 y Fh(2)p 1371 1242 230 4 v 1371 1325 a Fm(\015)5 b(\016)1466 1299 y Fj(\034)j Fh(+1)1630 1263 y Fx(+)20 b(3)p Fm(nm)1901 1225 y Fh(2)1957 1263 y Fx(\026)-62 b Fm(m\033)2075 1225 y Fh(2)2184 1201 y Fm(\026)7 b Fx(\026)-52 b Fm(\026)p 2125 1242 V 2125 1325 a(\015)5 b(\016)2220 1299 y Fj(\034)j Fh(+1)2384 1263 y Fx(+)20 b(6)p Fm(n)2575 1225 y Fh(2)2614 1263 y Fm(m)2694 1225 y Fh(2)2751 1263 y Fx(\026)-62 b Fm(m)o(\033)2868 1225 y Fh(2)2978 1201 y Fm(\026)7 b Fx(\026)-52 b Fm(\026)p 2918 1242 V 2918 1325 a(\015)5 b(\016)3013 1299 y Fj(\034)j Fh(+1)843 1513 y Fx(+12)p Fm(n)1059 1476 y Fh(2)1098 1513 y Fm(m)1178 1476 y Fh(4)1235 1513 y Fx(\026)-63 b Fm(m)1297 1476 y Fh(2)1337 1513 y Fm(\033)1392 1476 y Fh(3)1509 1452 y Fm(\026)1564 1419 y Fh(2)p 1441 1492 V 1441 1575 a Fm(\015)5 b(\016)1536 1549 y Fj(\034)j Fh(+1)1701 1513 y Fx(+)20 b(15)p Fm(nm)2017 1476 y Fh(4)2074 1513 y Fx(\026)-63 b Fm(m)2136 1476 y Fh(2)2176 1513 y Fm(\033)2231 1476 y Fh(3)2348 1452 y Fm(\026)2403 1419 y Fh(2)p 2280 1492 V 2280 1575 a Fm(\015)5 b(\016)2375 1549 y Fj(\034)j Fh(+1)2540 1513 y Fx(+)20 b(24)p Fm(n)2776 1476 y Fh(2)2815 1513 y Fm(m)2895 1476 y Fh(4)2952 1513 y Fx(\026)-62 b Fm(m)3015 1476 y Fh(2)3054 1513 y Fm(\033)3109 1476 y Fh(3)3226 1452 y Fm(\026)3281 1419 y Fh(2)p 3159 1492 V 3159 1575 a Fm(\015)5 b(\016)3254 1549 y Fj(\034)j Fh(+1)843 1764 y Fx(+36)p Fm(m)1084 1726 y Fh(2)1141 1764 y Fx(\026)-63 b Fm(m)1203 1726 y Fh(2)1257 1764 y Fg(e)e Fm(m)o(\033)1377 1726 y Fh(2)1417 1764 y Fm(r)1470 1702 y(\026)1525 1669 y Fh(2)p 1471 1743 95 4 v 1496 1826 a Fm(\016)1590 1635 y Fg(\022)1657 1764 y Fx(1)21 b(+)e(6)p Fm(n)1913 1726 y Fh(2)1953 1764 y Fm(m)2033 1726 y Fh(2)2089 1764 y Fx(\026)-62 b Fm(m\033)2207 1726 y Fh(2)2344 1702 y Fm(\026)p 2257 1743 230 4 v 2257 1826 a(\015)5 b(\016)2352 1800 y Fj(\034)j Fh(+1)2496 1635 y Fg(\023)689 2021 y Fl(\024)83 b Fx(80)p Fm(n)988 1983 y Fh(2)1027 2021 y Fm(m)1107 1983 y Fh(4)1164 2021 y Fx(\026)-62 b Fm(m)1227 1983 y Fh(2)1281 2021 y Fg(e)c Fm(m\033)1401 1983 y Fh(3)1518 1959 y Fm(\026)1573 1926 y Fh(2)p 1450 2000 V 1450 2083 a Fm(\015)5 b(\016)1545 2057 y Fj(\034)j Fh(+1)1690 2021 y Fm(;)1974 b Fx(\(49\))71 2275 y(where)30 b(w)m(e)h(use)f(Lemma)g(9)h (to)g(b)s(ound)1111 2506 y Fl(kr)1232 2520 y Fj(I)1272 2506 y Fm(F)13 b Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\)\))p Fl(k)1988 2521 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2178 2506 y Fl(\024)2284 2445 y Fx(9)p 2284 2485 46 4 v 2284 2569 a(4)2354 2506 y Fg(e)-65 b Fm(m)2430 2445 y(r)p 2429 2485 44 4 v 2429 2569 a(\016)2483 2506 y Fl(k)p Fm(g)s Fl(k)2619 2469 y Fh(2)2619 2529 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2785 2506 y Fm(:)879 b Fx(\(50\))71 2748 y(T)-8 b(o)33 b(write)g(\(50\))h(w)m(e)f(recall)h(that)f(0)d Fm(<)e(\016)33 b Fl(\024)c Fm(r)s(=)p Fx(3)k(and)f(observ)m(e)h(that)g (from)f(\(33\))i(w)m(e)f(ha)m(v)m(e)h Fl(k)p Fm(g)s Fl(k)3329 2763 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3524 2748 y Fl(\024)29 b Fm(r)24 b Fl(\000)d Fm(\016)s Fx(.)71 2861 y(Moreo)m(v)m(er,)33 b(in)d(\(49\))i(w)m(e)e(also)i(use)e(6)p Fm(n)1367 2828 y Fh(2)1406 2861 y Fm(m)1486 2828 y Fh(2)1543 2861 y Fx(\026)-62 b Fm(m)o(\033)1660 2828 y Fh(2)1700 2861 y Fm(\026=)p Fx(\()p Fm(\015)5 b(\016)1930 2828 y Fj(\034)j Fh(+1)2065 2861 y Fx(\))25 b Fl(\024)g Fx(1)31 b(and)f(1)p Fm(=\016)g Fl(\024)25 b Fx(1)p Fm(=)p Fx(\(6)p Fm(\016)2943 2828 y Fj(\034)8 b Fh(+1)3079 2861 y Fx(\).)71 3064 y Fo(Remark)26 b(7.)35 b Fd(A)n(nother)25 b(te)-5 b(chnic)g(al)26 b(c)-5 b(omment)27 b(r)-5 b(efers)25 b(to)h(the)f(b)-5 b(ound)26 b(on)f Fl(k)2600 3041 y Fx(^)2578 3064 y Fm(B)2652 3031 y Fi(>)2711 3064 y Fm(!)s Fl(k)2816 3079 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3005 3064 y Fd(we)g(use)g(to)g(obtain)h(\(49\).)71 3186 y(Fr)-5 b(om)26 b(formula)g(\(26\))g(we)f(have)g(obtaine)-5 b(d)26 b(in)f(\(47\))g(an)h (estimate)f(for)g Fl(k)2487 3163 y Fx(^)2465 3186 y Fm(B)2539 3153 y Fi(>)2598 3186 y Fm(!)s Fl(k)2703 3201 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2892 3186 y Fd(of)g Fl(O)s Fx(\()p Fm(\026)3156 3153 y Fh(2)3196 3186 y Fm(=)p Fx(\()p Fm(\015)5 b(\016)3371 3153 y Fj(\034)j Fh(+1)3506 3186 y Fx(\)\))p Fd(.)39 b(How-)71 3308 y(ever,)28 b(we)g(observe)g(that)h(the)f(dir)-5 b(e)g(ct)29 b(b)-5 b(ound)29 b Fm(n)p Fl(j)p Fm(!)s Fl(jk)1800 3285 y Fx(^)1778 3308 y Fm(B)5 b Fl(k)1897 3323 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2062 3308 y Fd(,)28 b(by)g(taking)2521 3285 y Fx(^)2499 3308 y Fm(B)k Fd(fr)-5 b(om)29 b(\(21\),)h(pr)-5 b(o)g(duc)g(es)30 b(an)e(estimate)71 3421 y(of)33 b Fl(O)s Fx(\()p Fm(\026)343 3388 y Fh(2)382 3421 y Fm(=)p Fx(\()p Fm(\015)514 3388 y Fh(2)555 3421 y Fm(\016)598 3388 y Fh(2)p Fj(\034)8 b Fh(+2)767 3421 y Fx(\)\))p Fd(,)33 b(le)-5 b(ading)34 b(to)f(a)g(b)-5 b(ound)33 b(for)h Fl(k)p Fm(b)1885 3388 y Fh(\(1\))1979 3421 y Fl(k)2024 3436 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2222 3421 y Fd(of)e(the)i(same)f(or)-5 b(der)34 b(\(worst)g(than)g(\(49\)\).)212 3625 y Fx(No)m(w)d(it)g(is)f(the)h(turn)e(of)i Fm(A)1132 3592 y Fh(\(1\))1257 3625 y Fx(\(see)g(\(29\)\),)87 3840 y Fl(k)p Fm(A)200 3803 y Fh(\(1\))316 3840 y Fl(\000)19 b Fm(A)p Fl(k)519 3855 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)767 3840 y Fl(\024)83 b(k)p Fm(A)p Fx(\()p Fm(\002)s Fx(\()p Fm(\022)s Fx(\)\))21 b Fl(\000)f Fm(A)p Fx(\()p Fm(\022)s Fx(\))p Fl(k)1633 3855 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1818 3840 y Fx(+)g Fl(k)p Fm(A)p Fl(k)2067 3854 y Fj(\032)2108 3840 y Fx(\()p Fl(k)p Fm(B)5 b Fl(k)2307 3855 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2493 3840 y Fx(+)20 b Fl(k)p Fm(B)2703 3803 y Fi(>)2762 3840 y Fl(k)2807 3855 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2991 3840 y Fx(+)g Fl(k)p Fm(B)5 b Fl(k)3246 3855 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3411 3840 y Fl(k)p Fm(B)3530 3803 y Fi(>)3589 3840 y Fl(k)3634 3855 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)3799 3840 y Fx(\))921 3978 y(+\(1)21 b(+)f Fl(k)p Fm(B)5 b Fl(k)1348 3993 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1512 3978 y Fx(\)\(1)22 b(+)e Fl(k)p Fm(B)1859 3941 y Fi(>)1918 3978 y Fl(k)1963 3993 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2127 3978 y Fx(\))p Fl(k)p Fm(D)2285 3941 y Fh(2)2282 4001 y Fj(I)2325 3978 y Fm(F)13 b Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\)\))p Fl(k)3041 3993 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)767 4171 y Fl(\024)83 b Fx(3)p Fm(n)1021 4133 y Fh(2)1060 4171 y Fm(m)1140 4133 y Fh(3)1197 4171 y Fx(\026)-62 b Fm(m\033)1315 4133 y Fh(2)1451 4109 y Fm(\026)p 1364 4150 230 4 v 1364 4233 a(\015)5 b(\016)1459 4207 y Fj(\034)j Fh(+1)1624 4171 y Fx(+)19 b(6)p Fm(nm)1894 4133 y Fh(3)1951 4171 y Fx(\026)-62 b Fm(m\033)2069 4133 y Fh(2)2205 4109 y Fm(\026)p 2118 4150 V 2118 4233 a(\015)5 b(\016)2213 4207 y Fj(\034)j Fh(+1)2373 4043 y Fg(\022)2440 4171 y Fx(1)20 b(+)g Fm(n)g Fx(+)g(6)p Fm(n)2862 4133 y Fh(2)2901 4171 y Fm(m)2981 4133 y Fh(2)3038 4171 y Fx(\026)-63 b Fm(m\033)3155 4133 y Fh(2)3292 4109 y Fm(\026)p 3205 4150 V 3205 4233 a(\015)5 b(\016)3300 4207 y Fj(\034)j Fh(+1)3444 4043 y Fg(\023)921 4422 y Fx(+12)p Fm(nm)17 b Fx(\026)-62 b Fm(m)14 b Fg(e)-65 b Fm(mr)1421 4385 y Fh(2)1460 4422 y Fm(\033)1539 4361 y(\026)p 1525 4401 84 4 v 1525 4485 a(\016)1568 4458 y Fh(2)1633 4294 y Fg(\022)1700 4422 y Fx(1)21 b(+)f(6)p Fm(nm)2037 4385 y Fh(2)2093 4422 y Fx(\026)-62 b Fm(m\033)2211 4385 y Fh(2)2348 4361 y Fm(\026)p 2260 4401 230 4 v 2260 4485 a(\015)5 b(\016)2355 4458 y Fj(\034)j Fh(+1)2500 4294 y Fg(\023)15 b(\022)2649 4422 y Fx(1)20 b(+)g(6)p Fm(n)2905 4385 y Fh(2)2945 4422 y Fm(m)3025 4385 y Fh(2)3081 4422 y Fx(\026)-62 b Fm(m\033)3199 4385 y Fh(2)3336 4361 y Fm(\026)p 3248 4401 V 3248 4485 a(\015)5 b(\016)3343 4458 y Fj(\034)j Fh(+1)3488 4294 y Fg(\023)767 4643 y Fl(\024)83 b Fx(39)p Fm(n)1066 4605 y Fh(2)1106 4643 y Fm(m)1186 4605 y Fh(3)1242 4643 y Fx(\026)-62 b Fm(m)15 b Fg(e)-66 b Fm(m\033)1440 4605 y Fh(2)1577 4581 y Fm(\026)p 1489 4622 V 1489 4705 a(\015)5 b(\016)1584 4679 y Fj(\034)j Fh(+1)1729 4643 y Fm(;)71 4902 y Fx(where)30 b(w)m(e)h(use)f(Lemma)g(9) h(again,)h(com)m(bined)e(with)g(the)h(inequalit)m(y)h Fl(k)p Fm(g)s Fl(k)2621 4917 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2811 4902 y Fl(\024)25 b Fm(r)e Fl(\000)d Fx(2)p Fm(\016)s Fx(,)32 b(to)f(obtain)1073 5178 y Fl(k)p Fm(D)1196 5141 y Fh(2)1193 5201 y Fj(I)1236 5178 y Fm(F)13 b Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(g)s Fx(\()p Fm(\022)s Fx(\)\))p Fl(k)1952 5193 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2142 5178 y Fl(\024)25 b Fx(3)p Fm(n)15 b Fg(e)-66 b Fm(m)2428 5117 y(r)2472 5084 y Fh(2)p 2428 5157 84 4 v 2428 5241 a Fm(\016)2471 5214 y Fh(2)2521 5178 y Fl(k)p Fm(g)s Fl(k)2657 5193 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2823 5178 y Fm(:)71 5428 y Fx(Moreo)m(v)m(er,)33 b(w)m(e)d(also)i(p)s(oin)m(t)e(out)h(that) g(1)p Fm(=\016)1541 5395 y Fh(2)1607 5428 y Fl(\024)25 b Fx(1)p Fm(=\016)1836 5395 y Fj(\034)8 b Fh(+1)1971 5428 y Fx(.)212 5545 y(A)m(t)31 b(this)g(p)s(oin)m(t)f(w)m(e)h(can)f (con)m(trol)i(\()1424 5522 y(\026)1399 5545 y Fm(A)1467 5512 y Fh(\(1\))1562 5545 y Fx(\))1597 5512 y Fi(\000)p Fh(1)1692 5545 y Fx(.)40 b(W)-8 b(e)32 b(start)f(b)m(y)f(writing)927 5760 y(\026)903 5783 y Fm(A)971 5746 y Fh(\(1\))1090 5783 y Fx(=)1210 5760 y(\026)1186 5783 y Fm(A)21 b Fx(+)f(\()p 1401 5696 342 4 v Fm(A)1469 5757 y Fh(\(1\))1584 5783 y Fl(\000)g Fm(A)p Fx(\))25 b(=)1923 5760 y(\026)1899 5783 y Fm(A)1983 5682 y Fg(\020)2037 5783 y Fx(Id)2120 5797 y Fj(n)2187 5783 y Fx(+)20 b(\()2338 5760 y(\026)2313 5783 y Fm(A)q Fx(\))2417 5746 y Fi(\000)p Fh(1)2512 5783 y Fx(\()p 2547 5696 V Fm(A)2615 5757 y Fh(\(1\))2730 5783 y Fl(\000)f Fm(A)q Fx(\))2924 5682 y Fg(\021)2993 5783 y Fm(:)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 71 274 a Fx(16)2427 b Fn(Kolmogoro)m(v)32 b(Theorem)e(Revisited)71 573 y Fx(By)h(using)e(again)j(h)m(yp)s (othesis)e(\(33\),)i(w)m(e)f(ha)m(v)m(e)522 820 y Fl(j)p Fx(\()606 797 y(\026)582 820 y Fm(A)p Fx(\))685 782 y Fi(\000)p Fh(1)780 820 y Fx(\()p 815 732 342 4 v Fm(A)883 793 y Fh(\(1\))998 820 y Fl(\000)20 b Fm(A)p Fx(\))p Fl(j)26 b(\024)f(j)p Fx(\()1424 797 y(\026)1399 820 y Fm(A)q Fx(\))1503 782 y Fi(\000)p Fh(1)1597 820 y Fl(jk)p Fm(A)1735 782 y Fh(\(1\))1851 820 y Fl(\000)20 b Fm(A)p Fl(k)2055 835 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2245 820 y Fl(\024)25 b Fx(39)p Fm(n)2486 782 y Fh(2)2526 820 y Fm(m)2606 782 y Fh(3)2662 820 y Fx(\026)-62 b Fm(m)2725 782 y Fh(2)2779 820 y Fg(e)c Fm(m\033)2899 782 y Fh(2)3036 758 y Fm(\026)p 2949 799 230 4 v 2949 882 a(\015)5 b(\016)3044 856 y Fj(\034)j Fh(+1)3213 820 y Fl(\024)3319 758 y Fx(1)p 3319 799 46 4 v 3319 882 a(2)3374 820 y Fm(:)71 1072 y Fx(Hence,)31 b(w)m(e)g(obtain)g(\(using)f(Neumann's)g(series)h(to)g (compute)f(the)h(in)m(v)m(erse\))338 1344 y Fl(j)p Fx(\()423 1321 y(\026)398 1344 y Fm(A)466 1307 y Fh(\(1\))561 1344 y Fx(\))596 1307 y Fi(\000)p Fh(1)691 1344 y Fl(j)83 b(\024)1340 1283 y(j)p Fx(\()1424 1260 y(\026)1400 1283 y Fm(A)p Fx(\))1503 1250 y Fi(\000)p Fh(1)1598 1283 y Fl(j)p 963 1323 1038 4 v 963 1411 a Fx(1)21 b Fl(\000)e(j)p Fx(\()1204 1388 y(\026)1179 1411 y Fm(A)q Fx(\))1283 1384 y Fi(\000)p Fh(1)1378 1411 y Fl(jk)p Fm(A)1516 1384 y Fh(\(1\))1631 1411 y Fl(\000)h Fm(A)p Fl(k)1835 1426 y Fj(\032)p Fi(\000)p Fh(5)p 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Fi(\000)p Fh(5)p Fj(\016)2338 1572 y Fl(\024)43 b Fx(\026)-63 b Fm(m)20 b Fx(+)g(78)p Fm(n)2770 1535 y Fh(2)2810 1572 y Fm(m)2890 1535 y Fh(3)2947 1572 y Fx(\026)-62 b Fm(m)3010 1535 y Fh(3)3063 1572 y Fg(e)d Fm(m\033)3184 1535 y Fh(2)3320 1511 y Fm(\026)p 3233 1551 230 4 v 3233 1634 a(\015)5 b(\016)3328 1608 y Fj(\034)j Fh(+1)3472 1572 y Fm(:)192 b Fx(\(51\))71 1839 y(T)-8 b(o)31 b(\014nish)e(the)h(pro)s(of)g(of)g(Lemma)h(5)g(it)g (only)f(remains)g(to)h(consider)g Fm(F)2512 1806 y Fh(\(1\))2636 1839 y Fx(\(see)h(\(30\)\).)42 b(W)-8 b(e)32 b(de\014ne)890 2047 y Fl(F)9 b Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))26 b(=)f Fm(F)13 b Fx(\()p Fm(\022)23 b Fx(+)d Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b Fx(\(Id)1881 2061 y Fj(n)1948 2047 y Fx(+)20 b Fm(B)5 b Fx(\()p Fm(\022)s Fx(\)\))p Fm(I)27 b Fx(+)20 b Fm(g)s Fx(\()p Fm(\022)s Fx(\)\))h Fl(\000)f Fm(F)13 b Fx(\()p Fm(\022)s(;)i(I)7 b Fx(\))p Fm(;)71 2256 y Fx(and)30 b(w)m(e)g(ha)m(v)m(e)i Fm(F)662 2223 y Fh(\(1\))782 2256 y Fx(=)25 b Fm(F)33 b Fx(+)20 b([)p Fl(F)9 b Fx(])1184 2270 y Fh(3)1224 2256 y Fx(.)41 b(From)30 b(part)h(\(b\))f(of)h(Lemma)f(9)h(w)m(e)g(can)f (express)g([)p Fl(F)9 b Fx(])3130 2270 y Fh(3)3201 2256 y Fx(as)1145 2523 y([)p Fl(F)g Fx(])1269 2537 y Fh(3)1335 2523 y Fx(=)1431 2399 y Fg(Z)1522 2425 y Fh(1)1481 2605 y(0)1586 2461 y Fx(1)p 1586 2502 46 4 v 1586 2585 a(2)1641 2523 y(\(1)21 b Fl(\000)f Fm(s)p Fx(\))1911 2485 y Fh(2)1951 2523 y Fm(D)2029 2485 y Fh(3)2026 2545 y Fj(I)2068 2523 y Fl(F)9 b Fx(\()p Fm(\022)s(;)15 b(sI)7 b Fx(\)[)p Fm(I)g(;)15 b(I)7 b(;)15 b(I)7 b Fx(])p Fm(ds:)915 b Fx(\(52\))71 2777 y(T)-8 b(o)31 b(con)m(trol)h([)p Fl(F)9 b Fx(])635 2791 y Fh(3)705 2777 y Fx(\014rst)30 b(w)m(e)h(use)f(the)g(Mean)h(V)-8 b(alue)32 b(Theorem)e(\(8\))h(to)g(b)s(ound)e Fl(F)9 b Fx(.)41 b(W)-8 b(e)31 b(ha)m(v)m(e,)394 2986 y Fl(kF)9 b(k)558 3001 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)o(;r)r Fi(\000)p Fh(2)p Fj(\016)979 2986 y Fl(\024)83 b(k)p Fm(D)1253 3001 y Fj(\022)1293 2986 y Fm(F)13 b Fl(k)1409 3001 y Fj(\032)p Fi(\000)p Fj(\016)o(;r)1588 2986 y Fl(k)p Fm(f)d Fl(k)1733 3001 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1917 2986 y Fx(+)20 b Fl(k)p Fm(D)2128 3001 y Fj(\022)2168 2986 y Fm(F)13 b Fl(k)2284 3001 y Fj(\032;r)r Fi(\000)p Fj(\016)2467 2986 y Fl(k)p Fm(B)5 b Fx(\()p Fm(\022)s Fx(\))p Fm(I)27 b Fx(+)20 b Fm(g)s Fx(\()p Fm(\022)s Fx(\))p Fl(k)3067 3001 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)o(;r)r Fi(\000)p Fh(2)p Fj(\016)979 3177 y Fl(\024)83 b Fm(n)1198 3115 y Fl(k)p Fm(F)13 b Fl(k)1359 3129 y Fj(\032;r)p 1198 3156 256 4 v 1304 3239 a Fm(\016)1478 3177 y Fx(\()q Fl(k)p Fm(f)d Fl(k)1659 3192 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)1843 3177 y Fx(+)20 b Fl(k)p Fm(B)5 b Fl(k)2098 3192 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2263 3177 y Fx(\()p Fm(r)23 b Fl(\000)d Fx(2)p Fm(\016)s Fx(\))i(+)e Fl(k)p Fm(g)s Fl(k)2825 3192 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2990 3177 y Fx(\))979 3410 y Fl(\024)83 b Fm(n)15 b Fg(e)-66 b Fm(m)1278 3348 y(r)1322 3315 y Fh(3)p 1278 3389 83 4 v 1297 3472 a Fm(\016)1386 3282 y Fg(\022)1453 3410 y Fx(3)p Fm(m)1578 3372 y Fh(2)1634 3410 y Fx(\026)k Fm(m\033)1752 3372 y Fh(2)1844 3348 y Fm(\026)p 1802 3389 140 4 v 1802 3472 a(\015)5 b(\016)1897 3446 y Fj(\034)1971 3410 y Fx(+)20 b(6)p Fm(nm)2242 3372 y Fh(2)2298 3410 y Fx(\026)-62 b Fm(m\033)2416 3372 y Fh(2)2456 3410 y Fx(\()p Fm(r)23 b Fl(\000)d Fx(2)p Fm(\016)s Fx(\))2867 3348 y Fm(\026)p 2779 3389 230 4 v 2779 3472 a(\015)5 b(\016)2874 3446 y Fj(\034)j Fh(+1)3040 3410 y Fx(+)20 b(4)p Fm(m)d Fx(\026)-62 b Fm(m\033)s(\026)3446 3282 y Fg(\023)979 3630 y Fl(\024)83 b Fx(12)p Fm(n)1278 3593 y Fh(2)1318 3630 y Fm(m)1398 3593 y Fh(2)1455 3630 y Fx(\026)-63 b Fm(m)15 b Fg(e)-66 b Fm(m\033)1652 3593 y Fh(2)1789 3569 y Fm(\026)p 1702 3609 V 1702 3692 a(\015)5 b(\016)1797 3666 y Fj(\034)j Fh(+2)1941 3630 y Fm(:)71 3882 y Fx(T)-8 b(o)31 b(obtain)h(this)f(estimate)h(for)f Fl(F)40 b Fx(w)m(e)32 b(recall)g(that)g Fl(k)p Fm(f)10 b Fl(k)1997 3897 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)2187 3882 y Fl(\024)26 b Fm(\016)35 b Fx(and)30 b Fl(kI)e(\000)20 b Fx(Id)2833 3896 y Fj(I)2873 3882 y Fl(k)2918 3897 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)o(;r)r Fi(\000)p Fh(2)p Fj(\016)3282 3882 y Fl(\024)26 b Fm(\016)s Fx(.)44 b(Th)m(us,)30 b(w)m(e)71 3995 y(deriv)m(e)h(the)f(follo)m(wing)i(b)s(ound)d(for)h (the)g(third)g(order)g(deriv)-5 b(ativ)m(es)31 b(of)g Fl(F)9 b Fx(,)1113 4217 y Fl(k)p Fm(@)1211 4180 y Fh(3)1206 4240 y Fj(I)1237 4250 y Ff(j)1269 4240 y Fj(I)1300 4252 y Ff(k)1338 4240 y Fj(I)1369 4252 y Ff(l)1397 4217 y Fl(F)g(k)1516 4232 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)o(;r)r Fi(\000)p Fh(5)p Fj(\016)1880 4217 y Fl(\024)25 b Fx(12)p Fm(n)2121 4180 y Fh(2)2161 4217 y Fm(m)2241 4180 y Fh(2)2297 4217 y Fx(\026)-62 b Fm(m)14 b Fg(e)-65 b Fm(m\033)2495 4180 y Fh(2)2631 4156 y Fm(\026)p 2544 4196 V 2544 4280 a(\015)5 b(\016)2639 4253 y Fj(\034)j Fh(+5)2783 4217 y Fm(;)881 b Fx(\(53\))71 4464 y(and)30 b(hence,)g(for)h(an)m(y)f(0)c Fl(\024)f Fm(R)h Fl(\024)f Fm(r)e Fl(\000)d Fx(5)p Fm(\016)s Fx(,)31 b(w)m(e)g(obtain)214 4726 y Fl(k)p Fm(F)330 4689 y Fh(\(1\))445 4726 y Fl(\000)20 b Fm(F)13 b Fl(k)652 4741 y Fj(\032)p Fi(\000)p Fh(5)p Fj(\016)o(;r)r Fi(\000)p Fh(5)p Fj(\016)1016 4726 y Fl(\024)1111 4603 y Fg(Z)1202 4629 y Fh(1)1162 4809 y(0)1257 4726 y Fx(6)p Fm(n)1357 4689 y Fh(5)1396 4726 y Fm(m)1476 4689 y Fh(2)1533 4726 y Fx(\026)-62 b Fm(m)14 b Fg(e)-65 b Fm(m)o(\033)1730 4689 y Fh(2)1867 4665 y Fm(\026)p 1780 4705 V 1780 4789 a(\015)5 b(\016)1875 4762 y Fj(\034)j Fh(+5)2019 4726 y Fm(R)2089 4689 y Fh(3)2128 4726 y Fx(\(1)21 b Fl(\000)f Fm(s)p Fx(\))2398 4689 y Fh(2)2438 4726 y Fm(ds)25 b Fl(\024)g Fx(2)p Fm(n)2749 4689 y Fh(5)2788 4726 y Fm(m)2868 4689 y Fh(2)2925 4726 y Fx(\026)-62 b Fm(m)14 b Fg(e)-66 b Fm(m\033)3122 4689 y Fh(2)3259 4665 y Fm(\026)p 3172 4705 V 3172 4789 a(\015)5 b(\016)3267 4762 y Fj(\034)j Fh(+5)3411 4726 y Fm(R)3481 4689 y Fh(3)3520 4726 y Fm(;)144 b Fx(\(54\))71 4988 y(where)30 b(w)m(e)h(observ)m(e)g(that)g(b)s (ounding)d(the)j(m)m(ulti-linear)g(op)s(erator)g Fm(D)2484 4955 y Fh(3)2481 5015 y Fj(I)2523 4988 y Fl(F)40 b Fx(con)m(tributes)31 b(with)f(a)h(factor)g Fm(n)3699 4955 y Fh(3)3738 4988 y Fx(.)3780 5183 y Fa(\003)71 5478 y Fp(6)135 b(Pro)t(of)45 b(of)g(Theorem)g(2)71 5684 y Fx(This)20 b(section)h(is)g(dev)m(oted)g (to)g(pro)m(v)m(e)h(Kolmogoro)m(v)g(Theorem)e(itself.)39 b(This)19 b(result)i(follo)m(ws)g(from)f(an)h(inductiv)m(e)71 5796 y(application)32 b(of)e(Lemma)h(5.)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 71 274 a Fn(J.)30 b(Villan)m(uev)-5 b(a)3171 b Fx(17)212 573 y(F)-8 b(or)33 b(this)g(purp)s(ose,)f(the)h (\014rst)f(step)h(is)f(to)i(arrange)f(the)g(initial)h(Hamiltonian)g Fm(H)i Fx(=)29 b Fl(H)22 b Fx(+)g Fl(F)42 b Fx(of)33 b(\(10\))h(so)71 686 y(that)d(Lemma)f(5)h(can)g(b)s(e)e(applied)i(to)g (it.)41 b(Th)m(us,)30 b(w)m(e)g(set)h Fm(H)2128 653 y Fh(\(0\))2248 686 y Fx(=)25 b Fm(H)37 b Fx(and)30 b(expand)g Fm(H)3033 653 y Fh(\(0\))3157 686 y Fx(as)h(in)f(\(12\),)653 920 y Fm(H)736 883 y Fh(\(0\))830 920 y Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))26 b(=)f Fm(a)1203 883 y Fh(\(0\))1297 920 y Fx(\()p Fm(\022)s Fx(\))20 b(+)g Fl(h)p Fm(!)k Fx(+)c Fm(b)1770 883 y Fh(\(0\))1864 920 y Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b(I)7 b Fl(i)21 b Fx(+)2224 859 y(1)p 2224 899 46 4 v 2224 982 a(2)2279 920 y Fl(h)p Fm(I)7 b(;)15 b(A)2469 883 y Fh(\(0\))2565 920 y Fx(\()p Fm(\022)s Fx(\))p Fm(I)7 b Fl(i)20 b Fx(+)g Fm(F)2945 883 y Fh(\(0\))3040 920 y Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))p Fm(;)71 1137 y Fx(with)445 1250 y Fm(a)493 1213 y Fh(\(0\))612 1250 y Fx(=)25 b Fm(H)7 b Fx(\()p Fm(\022)s(;)15 b Fx(0\))p Fm(;)198 b(b)1254 1213 y Fh(\(0\))1374 1250 y Fx(=)25 b Fl(r)1546 1264 y Fj(I)1586 1250 y Fl(F)9 b Fx(\()p Fm(\022)s(;)15 b Fx(0\))p Fm(;)198 b(A)2152 1213 y Fh(\(0\))2272 1250 y Fx(=)25 b Fm(D)2446 1213 y Fh(2)2443 1273 y Fj(I)2485 1250 y Fm(H)7 b Fx(\()p Fm(\022)s(;)15 b Fx(0\))p Fm(;)198 b(F)3063 1213 y Fh(\(0\))3182 1250 y Fx(=)25 b([)p Fm(H)7 b Fx(])3411 1264 y Fh(3)3451 1250 y Fm(:)71 1413 y Fx(Moreo)m(v)m(er,) 31 b(w)m(e)d(observ)m(e)g(that)g Fg(e)-50 b Fm(a)1189 1380 y Fh(\(0\))1308 1413 y Fx(=)25 b Fl(F)9 b Fx(\()p Fm(\022)s(;)15 b Fx(0\))g Fl(\000)g(hF)9 b Fx(\()p Fm(\022)s(;)15 b Fx(0\))p Fl(i)2125 1428 y Fj(\022)2167 1413 y Fx(.)40 b(If)28 b(w)m(e)g(de\014ne)f Fm(\032)2759 1380 y Fh(\(0\))2879 1413 y Fx(=)d Fm(\032)3021 1427 y Fh(0)3089 1413 y Fx(and)j Fm(r)3307 1380 y Fh(\(0\))3426 1413 y Fx(=)e Fm(r)3563 1427 y Fh(0)3602 1413 y Fm(=)p Fx(2,)30 b(w)m(e)71 1526 y(obtain)h(the)f(follo)m(wing)i(b)s(ounds)c(\(w)m(e)j(use)g(\(6\))g (and)f(\(7\)\),)939 1722 y Fl(k)o Fg(e)-50 b Fm(a)1032 1684 y Fh(\(0\))1127 1722 y Fl(k)1172 1748 y Fj(\032)1208 1728 y Fc(\(0\))1378 1722 y Fl(\024)83 b Fx(2)p Fl(kF)9 b(k)1741 1736 y Fj(\032)1777 1745 y Fc(0)1814 1736 y Fj(;r)1866 1745 y Fc(0)1929 1722 y Fl(\024)25 b Fx(2)p Fm(";)948 1924 y Fl(k)p Fm(b)1032 1886 y Fh(\(0\))1127 1924 y Fl(k)1172 1950 y Fj(\032)1208 1930 y Fc(\(0\))1378 1924 y Fl(\024)1542 1862 y(kF)9 b(k)1706 1876 y Fj(\032)1742 1885 y Fc(0)1778 1876 y Fj(;r)1830 1885 y Fc(0)p 1542 1903 327 4 v 1665 1986 a Fm(r)1706 2000 y Fh(0)1904 1924 y Fl(\024)2029 1862 y Fm(")p 2009 1903 81 4 v 2009 1986 a(r)2050 2000 y Fh(0)2100 1924 y Fm(;)735 2134 y Fl(k)p Fm(A)848 2096 y Fh(\(0\))964 2134 y Fl(\000)20 b(Ak)1173 2159 y Fj(\032)1209 2140 y Fc(\(0\))1378 2134 y Fl(\024)1656 2072 y Fm(n)p 1542 2113 282 4 v 1542 2196 a Fx(\()p Fm(r)1618 2210 y Fh(0)1658 2196 y Fm(=)p Fx(2\))1783 2170 y Fh(2)1834 2134 y Fl(kF)9 b(k)1998 2148 y Fj(\032)2034 2157 y Fc(0)2070 2148 y Fj(;r)2122 2157 y Fc(0)2185 2134 y Fl(\024)25 b Fx(4)p Fm(n)2411 2072 y(")p 2391 2113 83 4 v 2391 2198 a(r)2435 2167 y Fh(2)2432 2224 y(0)2484 2134 y Fm(;)1180 b Fx(\(55\))919 2358 y Fl(k)p Fm(A)1032 2320 y Fh(\(0\))1127 2358 y Fl(k)1172 2383 y Fj(\032)1208 2364 y Fc(\(0\))1378 2358 y Fl(\024)83 b(kAk)1695 2372 y Fj(\032)1731 2381 y Fc(0)1791 2358 y Fx(+)20 b Fl(k)p Fm(A)1995 2320 y Fh(\(0\))2110 2358 y Fl(\000)g(Ak)2319 2383 y Fj(\032)2355 2364 y Fc(\(0\))2467 2358 y Fl(\024)25 b Fm(\013)c Fx(+)e(4)p Fm(n)2863 2296 y(")p 2842 2337 V 2842 2422 a(r)2886 2391 y Fh(2)2883 2448 y(0)2961 2358 y Fl(\024)25 b Fx(2)p Fm(\013;)71 2615 y Fx(where)33 b(w)m(e)g(use)g(that)h(4)p Fm(n"=r)1064 2582 y Fh(2)1061 2640 y(0)1134 2615 y Fl(\024)c Fx(1)h Fl(\024)e Fm(\013)34 b Fx(\(see)g(\(11\)\).)52 b(W)-8 b(e)34 b(also)g(observ)m(e)g(that)g(\(55\))h(implies)e(the)h (follo)m(wing)71 2728 y(b)s(ound)28 b(on)i(the)h(a)m(v)m(erage)i(of)e Fm(A)1132 2695 y Fh(\(0\))1256 2728 y Fx(\(compare)h(\(51\)\),)1074 2980 y Fl(j)p Fx(\()1158 2957 y(\026)1134 2980 y Fm(A)1202 2942 y Fh(\(0\))1297 2980 y Fx(\))1332 2942 y Fi(\000)p Fh(1)1426 2980 y Fl(j)26 b(\024)1941 2918 y(j)p Fx(\()2033 2895 y(\026)2001 2918 y Fl(A)p Fx(\))2109 2885 y Fi(\000)p Fh(1)2204 2918 y Fl(j)p 1583 2959 1005 4 v 1583 3046 a Fx(1)20 b Fl(\000)g(j)p Fx(\()1831 3023 y(\026)1799 3046 y Fl(A)p Fx(\))1907 3020 y Fi(\000)p Fh(1)2002 3046 y Fl(jk)p Fm(A)2140 3020 y Fh(\(0\))2256 3046 y Fl(\000)f(Ak)2464 3072 y Fj(\032)2500 3052 y Fc(\(0\))2623 2980 y Fl(\024)25 b Fx(2)9 b(\026)-54 b Fm(\013;)71 3252 y Fx(where)29 b(w)m(e)i(use)f(no)m(w)g(that)h(4)p Fm(n)9 b Fx(\026)-54 b Fm(\013"=r)1299 3219 y Fh(2)1296 3276 y(0)1364 3252 y Fl(\024)25 b Fx(1)p Fm(=)p Fx(2.)42 b(Moreo)m(v)m(er,)33 b(if)d(w)m(e)g(use)g(part)g(\()p Fm(b)p Fx(\))h(of)f(Lemma)h(9)f(w)m(e) h(obtain,)g(for)71 3365 y(an)m(y)g(0)25 b Fl(\024)g Fm(R)h Fl(\024)f Fm(r)644 3332 y Fh(\(0\))768 3365 y Fx(\(compare)32 b(\(52\),)g(\(53\))g(and)d(\(54\)\),)823 3619 y Fl(k)p Fx([)p Fl(F)9 b Fx(])992 3633 y Fh(3)1033 3619 y Fl(k)1078 3645 y Fj(\032)1114 3625 y Fc(\(0\))1197 3645 y Fj(;R)1299 3619 y Fl(\024)1395 3496 y Fg(Z)1486 3522 y Fh(1)1446 3702 y(0)1551 3558 y Fx(1)p 1551 3598 46 4 v 1551 3682 a(2)1606 3619 y Fm(n)1661 3582 y Fh(3)1710 3557 y Fl(kF)g(k)1874 3571 y Fj(\032)1910 3580 y Fc(0)1946 3571 y Fj(;r)1998 3580 y Fc(0)p 1710 3598 327 4 v 1733 3682 a Fx(\()p Fm(r)1809 3696 y Fh(0)1848 3682 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Fx(on)f(the)h(statemen)m(t)h(of)f (the)f(lemma)i(as)e(the)h(v)-5 b(alues)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 71 274 a Fx(18)2427 b Fn(Kolmogoro)m(v)32 b(Theorem)e(Revisited)71 573 y Fx(with)h(the)h(sup)s(er-script)e(\(0\)) j(in)m(tro)s(duced)e(b)s(efore.)43 b(When)32 b(applying)f(Lemma)h(5)g (to)g Fm(H)3089 540 y Fh(\()p Fj(j)t Fh(\))3180 573 y Fx(,)g(for)g Fm(j)g Fl(\025)27 b Fx(1,)33 b(these)71 686 y(quan)m(tities)f(are)e(replaced)h(b)m(y)f(di\013eren)m(t)h(ones,)g (no)m(w)f(lab)s(eled)h(with)f(the)h(sup)s(er-script)e(\()p Fm(j)5 b Fx(\),)32 b(where)464 923 y Fm(\016)507 885 y Fh(\()p Fj(j)t Fh(\))625 923 y Fx(=)732 861 y Fm(\016)772 875 y Fh(0)p 731 902 82 4 v 731 985 a Fx(2)776 959 y Fj(j)822 923 y Fm(;)198 b(\032)1092 885 y Fh(\()p Fj(j)t Fh(\))1208 923 y Fx(=)25 b Fm(\032)1351 885 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))1553 923 y Fl(\000)20 b Fx(5)p Fm(\016)1732 885 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))1915 923 y Fm(;)197 b(r)2181 885 y Fh(\()p Fj(j)t Fh(\))2297 923 y Fx(=)25 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1578 y Fm(m)700 1541 y Fh(\()p Fj(j)t Fh(\))816 1578 y Fl(\024)25 b Fx(2)p Fm(m)1037 1541 y Fh(\(0\))1132 1578 y Fm(;)214 b Fx(\026)-62 b Fm(m)1434 1541 y Fh(\()p Fj(j)t Fh(\))1550 1578 y Fl(\024)25 b Fx(2)18 b(\026)-63 b Fm(m)1771 1541 y Fh(\(0\))1866 1578 y Fm(;)212 b Fg(e)-66 b Fm(m)2168 1541 y Fh(\()p Fj(j)t Fh(\))2285 1578 y Fl(\024)25 b Fx(2)15 b Fg(e)-66 b Fm(m)2506 1541 y Fh(\(0\))2600 1578 y Fm(;)197 b(j)31 b Fx(=)25 b(1)p Fm(;)15 b(:)g(:)g(:)i(;)e(s:)388 b Fx(\(57\))71 1773 y(T)-8 b(o)24 b(start)h(the)f(induction)g(pro)s (cess,)h(w)m(e)g(remark)f(that)h(the)f(condition)h Fm(\016)2505 1740 y Fh(\()p Fj(j)t Fh(\))2622 1773 y Fl(\024)g Fx(min)p Fl(f)p Fm(\032)2962 1740 y Fh(\()p Fj(j)t Fh(\))3054 1773 y Fm(;)15 b(r)3138 1740 y Fh(\()p Fj(j)t Fh(\))3229 1773 y Fl(g)p Fm(=)p Fx(6)25 b(of)g(Lemma)f(5)71 1886 y(is)36 b(automatically)k(ful\014lled)c(from)g(the)h(h)m(yp)s(othesis)f Fm(\016)1958 1900 y Fh(0)2033 1886 y Fl(\024)f Fx(min)p Fl(f)p Fm(\032)2383 1900 y Fh(0)2423 1886 y Fm(=)p Fx(16)p Fm(;)15 b(r)2639 1900 y Fh(0)2680 1886 y Fm(=)p Fx(32)p Fl(g)p Fx(.)61 b(In)36 b(order)g(to)h(c)m(hec)m(k)h(this,)71 1998 y(w)m(e)31 b(observ)m(e)h(that)f(lim)854 2012 y Fj(j)t Fi(!)p Fh(+)p Fi(1)1101 1998 y Fm(\032)1148 1965 y Fh(\()p Fj(j)t Fh(\))1266 1998 y Fx(=)26 b Fm(\032)1410 2012 y Fh(0)1470 1998 y Fl(\000)20 b Fx(10)p Fm(\016)1691 2012 y Fh(0)1758 1998 y Fx(=)26 b Fm(\032)1902 1965 y Fi(\003)1962 1998 y Fx(+)20 b Fm(\016)2093 2012 y Fh(0)2164 1998 y Fx(and)30 b(lim)2468 2012 y Fj(j)t Fi(!)p Fh(+)p Fi(1)2715 1998 y Fm(r)2759 1965 y Fh(\()p Fj(j)t Fh(\))2876 1998 y Fx(=)c Fm(r)3014 2012 y Fh(0)3054 1998 y Fm(=)p Fx(2)21 b Fl(\000)g Fx(10)p Fm(\016)3387 2012 y Fh(0)3454 1998 y Fx(=)k Fm(r)3594 1965 y Fi(\003)3654 1998 y Fx(+)20 b Fm(\016)3785 2012 y Fh(0)3825 1998 y Fx(.)71 2111 y(F)-8 b(urthermore,)30 b(condition)h(\(33\))h(of)f(Lemma)f(5)h(written)g(for) f Fm(H)2259 2078 y Fh(\()p Fj(j)t Fh(\))2380 2111 y Fx(reads)g(as)1152 2368 y(13)p Fm(n)1297 2331 y Fh(2)1337 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Fj(s)p Fh(\))2152 5659 y Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))25 b(=)g(\(Id)2595 5673 y Fj(n)2662 5659 y Fx(+)20 b Fl(B)2816 5621 y Fh(\()p Fj(s)p Fh(\))2907 5659 y Fx(\()p Fm(\022)s Fx(\)\))p Fm(I)28 b Fx(+)20 b Fm(h)3269 5621 y Fh(\()p Fj(s)p Fh(\))3361 5659 y Fx(\()p Fm(\022)s Fx(\))p Fm(;)456 5796 y Fx(\011)527 5759 y Fi(\003)567 5796 y Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))83 b(=)g(\(\004)1103 5759 y Fi(\003)1142 5796 y Fx(\()p Fm(\022)s Fx(\))p Fm(;)15 b Fl(J)1377 5759 y Fi(\003)1416 5796 y Fx(\()p Fm(\022)s(;)g(I)7 b Fx(\)\))p Fm(;)363 b Fl(J)2120 5759 y Fi(\003)2160 5796 y Fx(\()p Fm(\022)s(;)15 b(I)7 b Fx(\))25 b(=)g(\(Id)2603 5810 y Fj(n)2670 5796 y Fx(+)20 b Fl(B)2824 5759 y Fi(\003)2863 5796 y Fx(\()p Fm(\022)s Fx(\)\))p Fm(I)28 b Fx(+)19 b Fm(h)3224 5759 y Fi(\003)3264 5796 y Fx(\()p Fm(\022)s Fx(\))p Fm(:)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 71 274 a Fx(20)2427 b Fn(Kolmogoro)m(v)32 b(Theorem)e(Revisited)71 573 y Fx(Th)m(us,)f(what)i(w)m(e)g(ha)m(v)m(e) g(to)g(pro)m(v)m(e)h(is)e(the)g(con)m(v)m(ergence)j(of)e(\004)2169 540 y Fh(\()p Fj(s)p Fh(\))2260 573 y Fx(,)g Fl(B)2379 540 y Fh(\()p Fj(s)p Fh(\))2500 573 y Fx(and)f Fm(h)2729 540 y Fh(\()p Fj(s)p Fh(\))2851 573 y Fx(to)h(\004)3023 540 y Fi(\003)3062 573 y Fx(,)g Fl(B)3181 540 y Fi(\003)3250 573 y Fx(and)f Fm(h)3479 540 y Fi(\003)3519 573 y Fx(,)g(resp)s(ec-)71 686 y(tiv)m(ely)-8 b(.)43 b(First,)31 b(w)m(e)f(obtain)h(the)g(form)m (ula,)1266 879 y(\004)1327 842 y Fh(\()p Fj(s)p Fh(\))1418 879 y Fx(\()p Fm(\022)s Fx(\))25 b(=)g Fm(\002)1727 842 y Fh(\(0\))1841 879 y Fl(\016)c Fm(\002)1979 842 y Fh(\(1\))2093 879 y Fl(\016)g(\001)15 b(\001)g(\001)21 b(\016)g Fm(\002)2423 842 y Fh(\()p Fj(s)p Fh(\))2515 879 y Fx(\()p Fm(\022)s Fx(\))p Fm(;)71 1073 y Fx(and)30 b(the)g(follo)m(wing)i(recurrences)e (for)g Fl(B)1466 1040 y Fh(\()p Fj(s)p Fh(\))1588 1073 y Fx(and)f Fm(h)1816 1040 y Fh(\()p Fj(s)p Fh(\))1908 1073 y Fx(:)740 1266 y Fl(B)803 1228 y Fh(\()p Fj(s)p Fh(\))895 1266 y Fx(\()p Fm(\022)s Fx(\))82 b(=)h Fl(B)1310 1228 y Fh(\()p Fj(s)p Fi(\000)p Fh(1\))1492 1266 y Fx(\()p Fm(\002)1599 1228 y Fh(\()p Fj(s)p Fh(\))1690 1266 y Fx(\()p Fm(\022)s Fx(\)\))21 b(+)f(\(Id)2071 1280 y Fj(n)2139 1266 y Fx(+)g Fl(B)2293 1228 y Fh(\()p Fj(s)p Fi(\000)p Fh(1\))2474 1266 y Fx(\()p Fm(\002)2581 1228 y Fh(\()p Fj(s)p Fh(\))2673 1266 y Fx(\()p Fm(\022)s Fx(\)\)\))p Fm(B)2933 1228 y Fh(\()p Fj(s)p Fh(\))3025 1266 y Fx(\()p Fm(\022)s Fx(\))p Fm(;)523 b Fx(\(63\))750 1404 y Fm(h)802 1366 y Fh(\()p Fj(s)p Fh(\))895 1404 y Fx(\()p Fm(\022)s Fx(\))82 b(=)h Fm(h)1299 1366 y Fh(\()p Fj(s)p Fi(\000)p Fh(1\))1481 1404 y Fx(\()p Fm(\002)1588 1366 y Fh(\()p Fj(s)p Fh(\))1680 1404 y Fx(\()p Fm(\022)s Fx(\)\))21 b(+)f(\(Id)2061 1418 y Fj(n)2129 1404 y Fx(+)f Fl(B)2282 1366 y Fh(\()p Fj(s)p Fi(\000)p Fh(1\))2464 1404 y Fx(\()p Fm(\002)2571 1366 y Fh(\()p Fj(s)p Fh(\))2663 1404 y Fx(\()p Fm(\022)s Fx(\)\)\))p Fm(g)2895 1366 y Fh(\()p Fj(s)p Fh(\))2988 1404 y Fx(\()p Fm(\022)s Fx(\))p Fm(;)560 b Fx(\(64\))71 1597 y(starting)31 b(up)e(with)h Fl(B)811 1564 y Fh(\(0\))930 1597 y Fx(=)25 b Fm(B)1100 1564 y Fh(\(0\))1225 1597 y Fx(and)k Fm(h)1453 1564 y Fh(\(0\))1573 1597 y Fx(=)c Fm(g)1715 1564 y Fh(\(0\))1810 1597 y Fx(.)212 1710 y(T)-8 b(o)31 b(pro)m(v)m(e)g(the)g(con)m(v)m(ergence)h(of)f(\004) 1415 1677 y Fh(\()p Fj(s)p Fh(\))1537 1710 y Fx(in)f(\001\()p Fm(\032)1801 1677 y Fi(\003)1840 1710 y Fx(\))h(w)m(e)g(write)f(it)h (as)842 1973 y(\004)903 1935 y Fh(\()p Fj(s)p Fh(\))995 1973 y Fx(\()p Fm(\022)s Fx(\))20 b Fl(\000)g Fm(\022)27 b Fx(=)1437 1859 y Fj(s)1388 1886 y Fg(X)1393 2082 y Fj(j)t Fh(=1)1535 1872 y Fg(\020)1589 1973 y Fx(\004)1650 1935 y Fh(\()p Fj(j)t Fh(\))1741 1973 y Fx(\()p Fm(\022)s Fx(\))20 b Fl(\000)g Fx(\004)2029 1935 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))2210 1973 y Fx(\()p Fm(\022)s Fx(\))2326 1872 y Fg(\021)2401 1973 y Fx(+)2491 1872 y Fg(\020)2546 1973 y Fm(\002)2618 1935 y Fh(\(0\))2712 1973 y Fx(\()p Fm(\022)s Fx(\))g Fl(\000)g Fm(\022)2985 1872 y Fg(\021)3054 1973 y Fm(;)610 b Fx(\(65\))71 2276 y(and)32 b(discuss)g(the)h (absolute)g(con)m(v)m(ergence)j(of)d(the)g(sum)2042 2207 y Fg(P)2138 2234 y Fh(+)p Fi(1)2138 2302 y Fj(j)t Fh(=1)2283 2276 y Fl(k)p Fx(\004)2389 2243 y Fh(\()p Fj(j)t Fh(\))2502 2276 y Fl(\000)22 b Fx(\004)2656 2243 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))2837 2276 y Fl(k)2882 2290 y Fj(\032)2918 2270 y Fb(\003)2959 2276 y Fx(.)48 b(F)-8 b(or)33 b(this)g(purp)s(ose,) f(w)m(e)71 2406 y(consider)e(the)h(follo)m(wing)h(expression)e(for)g (\004)1604 2373 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))1785 2406 y Fx(,)h(if)f Fm(j)h Fl(\025)25 b Fx(2,)222 2684 y(\004)283 2647 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))464 2684 y Fx(\()p Fm(\022)s Fx(\))20 b Fl(\000)g Fm(\022)28 b Fx(=)862 2566 y Fj(j)t Fi(\000)p Fh(2)857 2598 y Fg(X)867 2796 y Fj(l)q Fh(=0)1004 2583 y Fg(\020)1058 2684 y Fm(\002)1130 2647 y Fh(\()p Fj(l)q Fh(\))1231 2684 y Fl(\016)21 b(\001)15 b(\001)g(\001)21 b(\016)g Fm(\002)1561 2647 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))1742 2684 y Fx(\()p Fm(\022)s Fx(\))f Fl(\000)g Fm(\002)2041 2647 y Fh(\()p Fj(l)q Fh(+1\))2232 2684 y Fl(\016)h(\001)15 b(\001)g(\001)21 b(\016)g Fm(\002)2562 2647 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))2743 2684 y Fx(\()p Fm(\022)s Fx(\))2859 2583 y Fg(\021)2934 2684 y Fx(+)3025 2583 y Fg(\020)3079 2684 y Fm(\002)3151 2647 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))3332 2684 y Fx(\()p Fm(\022)s Fx(\))g Fl(\000)e Fm(\022)3605 2583 y Fg(\021)3674 2684 y Fm(:)71 2976 y Fx(From)32 b(the)g(iterativ)m(e)j(application)e(of)g(Lemma)f(5,)h(w)m(e)f(ha)m(v)m (e)i(that)e Fm(\002)2461 2943 y Fh(\()p Fj(l)q Fh(+1\))2654 2976 y Fl(\016)21 b(\001)15 b(\001)g(\001)23 b(\016)f Fm(\002)2987 2943 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))3168 2976 y Fx(\()p Fm(\022)s Fx(\))28 b Fl(2)g Fx(\001\()p Fm(\032)3559 2943 y Fh(\()p Fj(l)q Fh(+1\))3730 2976 y Fx(\))k(if)71 3089 y Fm(\022)27 b Fl(2)e Fx(\001\()p Fm(\032)385 3056 y Fh(\()p Fj(j)t Fh(\))477 3089 y Fx(\),)31 b(for)f(an)m(y)g Fm(l)e Fx(=)d(0)p Fm(;)15 b(:)g(:)g(:)i(;)e(j)26 b Fl(\000)20 b Fx(2.)41 b(Th)m(us,)29 b(w)m(e)i(obtain)g(\(see)g (\(38\)\),)405 3372 y Fl(k)p Fx(\004)511 3335 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))713 3372 y Fl(\000)20 b Fx(Id)887 3387 y Fj(\022)926 3372 y Fl(k)971 3398 y Fj(\032)1007 3378 y Fc(\()p Ff(j)s Fc(\))1175 3372 y Fl(\024)1333 3254 y Fj(j)t Fi(\000)p Fh(1)1329 3286 y Fg(X)1339 3483 y Fj(l)q Fh(=0)1475 3372 y Fl(k)p Fm(\002)1592 3335 y Fh(\()p Fj(l)q Fh(\))1694 3372 y Fl(\000)g Fx(Id)1868 3387 y Fj(\022)1907 3372 y Fl(k)1952 3398 y Fj(\032)1988 3378 y Fc(\()p Ff(l)p Fc(+1\))2167 3372 y Fl(\024)2268 3254 y Fj(j)t Fi(\000)p Fh(1)2263 3286 y Fg(X)2273 3483 y Fj(l)q Fh(=0)2410 3372 y Fx(3\()p Fm(m)2570 3335 y Fh(\()p Fj(l)q Fh(\))2651 3372 y Fx(\))2686 3335 y Fh(2)2743 3372 y Fx(\026)-62 b Fm(m)2806 3335 y Fh(\()p Fj(l)q Fh(\))2887 3372 y Fm(\033)2942 3335 y Fh(2)3069 3311 y Fm(\026)3124 3278 y Fh(\()p Fj(l)q Fh(\))p 2991 3351 291 4 v 2991 3438 a Fm(\015)5 b Fx(\()p Fm(\016)3121 3412 y Fh(\()p Fj(l)q Fh(\))3204 3438 y Fx(\))3239 3412 y Fj(\034)1175 3697 y Fl(\024)1333 3579 y Fj(j)t Fi(\000)p Fh(1)1329 3611 y Fg(X)1339 3808 y Fj(l)q Fh(=0)1475 3697 y Fx(3)21 b Fl(\002)f Fx(2)1677 3660 y Fh(6)p Fi(\000)p Fj(\034)1811 3697 y Fm(\013)1869 3660 y Fh(2)1918 3697 y Fx(\026)-54 b Fm(\013\033)2022 3660 y Fh(2)2062 3697 y Fm(\016)2102 3711 y Fh(0)2142 3697 y Fx(2)2187 3660 y Fi(\000)p Fh(\()p Fj(l)q Fh(+1\))2423 3636 y Fm(\037)2480 3603 y Fh(2)2515 3579 y Ff(l)p 2423 3676 121 4 v 2463 3760 a Fg(e)-59 b Fm(\037)2579 3697 y Fl(\024)25 b Fx(9)20 b Fl(\002)g Fx(2)2876 3660 y Fh(7)2916 3697 y Fm(\013)2974 3660 y Fh(2)3023 3697 y Fx(\026)-54 b Fm(\013)q(\033)3128 3660 y Fh(2)3308 3636 y Fm(")p 3177 3676 304 4 v 3177 3768 a(\015)3229 3742 y Fh(2)3269 3768 y Fm(\016)3312 3730 y Fh(2)p Fj(\034)8 b Fh(+1)3309 3794 y(0)3491 3697 y Fm(;)173 b Fx(\(66\))71 3997 y(where)36 b(w)m(e)h(con)m(trol)h(2)837 3964 y Fi(\000)p Fh(\()p Fj(l)q Fh(+1\))1063 3997 y Fm(\037)1120 3964 y Fh(2)1155 3941 y Ff(l)1220 3997 y Fx(b)m(y)f(the)g(geometric)h (progression)f(\()p Fm(\037=)p Fx(2\))2636 3964 y Fj(l)q Fh(+1)2754 3997 y Fx(,)h(of)f(ratio)g(smaller)g(than)g(1)p Fm(=)p Fx(2.)71 4110 y(W)-8 b(e)32 b(observ)m(e)g(that)g(if)f(w)m(e)g (assume)g Fd(a-priori)42 b Fx(con)m(v)m(ergence)34 b(of)d(\004)2298 4077 y Fh(\()p Fj(j)t Fh(\))2420 4110 y Fx(to)h(\004)2593 4077 y Fi(\003)2663 4110 y Fx(in)f(\001\()p Fm(\032)2928 4077 y Fi(\003)2967 4110 y Fx(\),)h(then)f(\(66\))i(also)f(holds)71 4223 y(for)f Fl(k)p Fx(\004)317 4190 y Fi(\003)377 4223 y Fl(\000)20 b Fx(Id)552 4238 y Fj(\022)591 4223 y Fl(k)636 4237 y Fj(\032)672 4218 y Fb(\003)712 4223 y Fx(.)43 b(In)31 b(particular,)g(as)h(\(66\))g(is)f(smaller)h(than)f Fm(\016)2305 4237 y Fh(0)2345 4223 y Fm(=n)g Fx(\(see)h(\(11\)\),)h (then)e Fl(k)p Fx(\004)3226 4190 y Fi(\003)3286 4223 y Fl(\000)21 b Fx(Id)3461 4238 y Fj(\022)3500 4223 y Fl(k)3545 4237 y Fj(\032)3581 4218 y Fb(\003)3648 4223 y Fl(\024)26 b Fm(\016)3785 4237 y Fh(0)3825 4223 y Fx(.)71 4336 y(Moreo)m(v)m(er,)33 b(if)d(w)m(e)h(tak)m(e)h Fm(\022)s(;)15 b(\022)1042 4303 y Fi(0)1089 4336 y Fl(2)25 b Fx(\001\()p Fm(\032)1333 4303 y Fh(\()p Fj(j)t Fh(\))1445 4336 y Fl(\000)20 b Fm(\016)1576 4350 y Fh(0)1615 4336 y Fx(\),)31 b(for)f Fm(j)h Fl(\025)25 b Fx(1,)31 b(and)f(use)g(\(6\))i(and)d (\(8\),)j(w)m(e)f(obtain)482 4529 y Fl(j)p Fx(\004)568 4492 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))749 4529 y Fx(\()p Fm(\022)830 4492 y Fi(0)853 4529 y Fx(\))21 b Fl(\000)e Fx(\004)1060 4492 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))1242 4529 y Fx(\()p Fm(\022)s Fx(\))p Fl(j)83 b(\024)g(j)p Fm(\022)1691 4492 y Fi(0)1733 4529 y Fl(\000)20 b Fm(\022)s Fl(j)g Fx(+)g Fl(j)p Fx(\(\004)2127 4492 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))2309 4529 y Fx(\()p Fm(\022)2390 4492 y Fi(0)2413 4529 y Fx(\))g Fl(\000)g Fm(\022)2605 4492 y Fi(0)2628 4529 y Fx(\))g Fl(\000)g Fx(\(\004)2870 4492 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))3052 4529 y Fx(\()p Fm(\022)s Fx(\))g Fl(\000)g Fm(\022)s Fx(\))p Fl(j)1466 4745 y(\024)1620 4589 y Fg( )1692 4745 y Fx(1)g(+)g Fm(n)1913 4671 y Fl(k)p Fx(\004)2019 4638 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))2220 4671 y Fl(\000)g Fx(Id)2395 4686 y Fj(\022)2434 4671 y Fl(k)2479 4696 y Fj(\032)2515 4677 y Fc(\()p Ff(j)s Fc(\))p 1913 4724 688 4 v 2216 4807 a Fm(\016)2256 4821 y Fh(0)2610 4589 y Fg(!)2697 4745 y Fl(j)p Fm(\022)2768 4707 y Fi(0)2811 4745 y Fl(\000)g Fm(\022)s Fl(j)25 b(\024)g Fx(2)p Fl(j)p Fm(\022)3210 4707 y Fi(0)3253 4745 y Fl(\000)20 b Fm(\022)s Fl(j)p Fm(:)249 b Fx(\(67\))71 5030 y(No)m(w,)33 b(w)m(e)e(pic)m(k)i(up)d Fm(\022)f Fl(2)e Fx(\001\()p Fm(\032)1083 4997 y Fh(\()p Fj(j)t Fh(+1\))1286 5030 y Fl(\000)20 b Fm(\016)1417 5044 y Fh(0)1457 5030 y Fx(\))32 b(and)f(set)h Fm(\022)1891 4997 y Fi(0)1940 5030 y Fx(=)27 b Fm(\002)2110 4997 y Fh(\()p Fj(j)t Fh(\))2201 5030 y Fx(\()p Fm(\022)s Fx(\))32 b(in)f(\(67\).)45 b(From)32 b(the)g(b)s(ound)d(\(37\))k(on)e(\010)3759 4997 y Fh(\()p Fj(j)t Fh(\))71 5143 y Fx(w)m(e)37 b(ha)m(v)m(e)h Fl(k)p Fm(\002)544 5110 y Fh(\()p Fj(j)t Fh(\))660 5143 y Fl(\000)24 b Fx(Id)838 5158 y Fj(\022)877 5143 y Fl(k)922 5168 y Fj(\032)958 5149 y Fc(\()p Ff(j)s Fc(+1\))1156 5143 y Fl(\024)35 b Fm(\016)1305 5110 y Fh(\()p Fj(j)t Fh(\))1434 5143 y Fx(and)h(hence)g Fm(\022)s(;)15 b(\022)2007 5110 y Fi(0)2065 5143 y Fl(2)35 b Fx(\001\()p Fm(\032)2319 5110 y Fh(\()p Fj(j)t Fh(\))2435 5143 y Fl(\000)24 b Fm(\016)2570 5157 y Fh(0)2610 5143 y Fx(\).)59 b(Then,)38 b(b)m(y)e(using)g(no)m(w)h(\(38\))h(to)71 5272 y(b)s(ound)28 b Fl(k)p Fm(\002)468 5239 y Fh(\()p Fj(j)t Fh(\))580 5272 y Fl(\000)20 b Fx(Id)754 5287 y Fj(\022)793 5272 y Fl(k)838 5298 y Fj(\032)874 5278 y Fc(\()p Ff(j)s Fc(+1\))1067 5272 y Fx(and)30 b(b)m(y)g(recalling)i(that)f Fm(\032)1978 5239 y Fh(\()p Fj(j)t Fh(\))2089 5272 y Fl(\000)20 b Fm(\016)2220 5286 y Fh(0)2285 5272 y Fl(\025)25 b Fm(\032)2428 5239 y Fi(\003)2468 5272 y Fx(,)30 b(w)m(e)h(obtain)333 5551 y Fl(k)p Fx(\004)439 5513 y Fh(\()p Fj(j)t Fh(\))550 5551 y Fl(\000)20 b Fx(\004)702 5513 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))884 5551 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Fm(h)3811 4393 y Fi(\003)71 4539 y Fx(in)d(\001\()p Fm(\032)335 4506 y Fi(\003)374 4539 y Fx(\),)h(then)f(\(70\))i(also)g(giv)m(es)f(a) g(b)s(ound)d(for)j Fl(k)p Fm(h)1862 4506 y Fi(\003)1902 4539 y Fl(k)1947 4553 y Fj(\032)1983 4534 y Fb(\003)2024 4539 y Fx(.)212 4652 y(By)g(using)f(\(64\))h(again)h(w)m(e)f(ha)m(v)m (e)334 4822 y Fm(h)386 4784 y Fh(\()p Fj(j)t Fh(\))478 4822 y Fx(\()p Fm(\022)s Fx(\))20 b Fl(\000)g Fm(h)757 4784 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))939 4822 y Fx(\()p Fm(\022)s Fx(\))25 b(=)g(\()p Fm(h)1263 4784 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))1445 4822 y Fx(\()p Fm(\002)1552 4784 y Fh(\()p Fj(j)t Fh(\))1644 4822 y Fx(\()p Fm(\022)s Fx(\)\))20 b Fl(\000)g Fm(h)1958 4784 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))2140 4822 y Fx(\()p Fm(\022)s Fx(\)\))g(+)g(\(Id)2521 4836 y Fj(n)2588 4822 y Fx(+)g Fl(B)2742 4784 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))2923 4822 y Fx(\()p Fm(\002)3030 4784 y Fh(\()p Fj(j)t Fh(\))3122 4822 y Fx(\()p Fm(\022)s Fx(\)\)\))p Fm(g)3354 4784 y Fh(\()p Fj(j)t 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Fm(n)p Fx(\()p Fm(m)1849 5452 y Fh(\()p Fj(j)t Fh(\))1940 5489 y Fx(\))1975 5452 y Fh(2)2032 5489 y Fx(\026)-62 b Fm(m)2095 5452 y Fh(\()p Fj(j)t Fh(\))2186 5489 y Fm(\033)2241 5452 y Fh(2)2408 5428 y Fm(\026)2463 5395 y Fh(\()p Fj(j)t Fh(\))p 2291 5468 382 4 v 2291 5555 a Fm(\015)5 b(\016)2383 5569 y Fh(0)2423 5555 y Fx(\()p Fm(\016)2501 5529 y Fh(\()p Fj(j)t Fh(\))2593 5555 y Fx(\))2628 5529 y Fj(\034)2702 5489 y Fx(+)20 b(8)p Fm(m)2918 5452 y Fh(\()p Fj(j)t Fh(\))3027 5489 y Fx(\026)-62 b Fm(m)3090 5452 y Fh(\()p Fj(j)t Fh(\))3181 5489 y Fm(\033)3236 5452 y Fh(2)3276 5489 y Fm(\026)3331 5452 y Fh(\()p Fj(j)t Fh(\))1480 5762 y Fl(\024)83 b Fx(2)1679 5725 y Fh(6)1719 5762 y Fm(\013)1777 5725 y Fh(2)1826 5762 y Fx(\026)-54 b Fm(\013\033)1930 5725 y Fh(2)1980 5701 y Fm(\037)2037 5668 y Fh(2)2072 5644 y Ff(j)p 1980 5741 129 4 v 2024 5824 a Fg(e)-59 b Fm(\037)2144 5762 y Fl(\024)25 b Fm(\037)2297 5725 y Fj(j)t Fh(+1)2423 5762 y Fm(:)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 71 274 a Fx(22)2427 b Fn(Kolmogoro)m(v)32 b(Theorem)e(Revisited)71 573 y Fx(Th)m(us,)j(the)h(con)m(v)m(ergence)h (of)f Fm(h)1154 540 y Fh(\()p Fj(s)p Fh(\))1279 573 y Fx(follo)m(ws)g(from)f(the)h(con)m(v)m(ergence)h(of)f(the)f(sum)2925 505 y Fg(P)3020 531 y Fh(+)p Fi(1)3020 600 y Fj(j)t Fh(=1)3165 573 y Fl(k)p Fm(h)3262 540 y Fh(\()p Fj(j)t Fh(\))3377 573 y Fl(\000)21 b Fm(h)3521 540 y Fh(\()p Fj(j)t Fi(\000)p Fh(1\))3703 573 y Fl(k)3748 587 y Fj(\032)3784 568 y Fb(\003)3825 573 y Fx(.)71 686 y(T)-8 b(o)31 b(\014nish)e(the)h(pro)s (of)g(it)h(only)f(remains)g(to)h(pro)m(v)m(e)h(that)f Fl(kJ)2149 653 y Fi(\003)2208 686 y Fl(\000)20 b Fx(Id)2382 700 y Fj(I)2422 686 y Fl(k)2467 700 y Fj(\032)2503 680 y Fb(\003)2540 700 y Fj(;r)2594 680 y Fb(\003)2659 686 y Fl(\024)25 b Fm(\016)2795 700 y Fh(0)2834 686 y Fx(,)31 b(whic)m(h)f(follo)m(ws)i(from)435 901 y Fl(kJ)559 863 y Fi(\003)618 901 y Fl(\000)20 b Fx(Id)793 915 y Fj(I)833 901 y Fl(k)878 915 y Fj(\032)914 895 y Fb(\003)950 915 y Fj(;r)1004 895 y Fb(\003)1127 901 y Fl(\024)82 b(kB)1388 863 y Fi(\003)1428 901 y Fl(k)1473 915 y Fj(\032)1509 895 y Fb(\003)1550 901 y Fm(r)1594 863 y Fi(\003)1653 901 y Fx(+)20 b Fl(k)p Fm(h)1841 863 y Fi(\003)1881 901 y Fl(k)1926 915 y Fj(\032)1962 895 y Fb(\003)1127 1065 y Fl(\024)82 b Fx(9)21 b Fl(\002)f Fx(2)1482 1027 y Fh(9)1522 1065 y Fm(n\013)1635 1027 y Fh(2)1683 1065 y Fx(\026)-54 b Fm(\013)q(\033)1788 1027 y Fh(2)1996 1003 y Fm(")p 1838 1044 359 4 v 1838 1153 a(\015)1890 1126 y Fh(2)1929 1153 y Fm(\016)1972 1105 y Fh(2\()p Fj(\034)8 b Fh(+1\))1969 1179 y(0)2222 964 y Fg(\020)2286 1003 y Fm(r)2327 1017 y Fh(0)p 2286 1044 81 4 v 2303 1127 a Fx(2)2396 1065 y Fl(\000)20 b Fx(11)p Fm(\016)2617 1079 y Fh(0)2658 964 y Fg(\021)2733 1065 y Fx(+)f(3)i Fl(\002)f Fx(2)3025 1027 y Fh(8)3065 1065 y Fm(\013)9 b Fx(\026)-54 b Fm(\013)q(\033)3341 1003 y(")p 3247 1044 230 4 v 3247 1136 a(\015)5 b(\016)3342 1098 y Fj(\034)j Fh(+1)3339 1162 y(0)1127 1312 y Fl(\024)82 b Fx(3)21 b Fl(\002)f Fx(2)1482 1275 y Fh(10)1557 1312 y Fm(n\013)1670 1275 y Fh(2)1719 1312 y Fx(\026)-54 b Fm(\013\033)1823 1275 y Fh(2)2031 1251 y Fm(")p 1873 1292 359 4 v 1873 1400 a(\015)1925 1374 y Fh(2)1964 1400 y Fm(\016)2007 1352 y Fh(2\()p Fj(\034)8 b Fh(+1\))2004 1426 y(0)2242 1312 y Fm(:)71 1675 y Fp(Ac)l(kno)l(wledgemen)l(ts)71 1884 y Fx(This)28 b(w)m(ork)h(has)f(b)s(een)g(partially)i(supp)s(orted) d(b)m(y)i(the)g(Spanish)e(MCyT/FEDER)i(gran)m(t)h(BFM2003-07521-)71 1997 y(C02-01)i(and)e(the)g(Catalan)i(CIRIT)d(gran)m(t)i(2001SGR-70.)71 2304 y Fp(References)116 2514 y Fx([1])47 b(Abraham,)i(R.H.)e(and)e (Marsden,)50 b(J.E.,)g Fd(F)-7 b(oundations)49 b(of)e(Me)-5 b(chanics)p Fx(,)51 b(second)46 b(edition,)k(revised)258 2626 y(and)40 b(enlarged,)j Fd(Benjamin/Cummings)h(Publishing)d(Co.)h (Inc.)g(A)-5 b(dvanc)g(e)g(d)43 b(Bo)-5 b(ok)43 b(Pr)-5 b(o)g(gr)g(am)p Fx(,)46 b(Read-)258 2739 y(ing,)31 b(Mass.,)g(1978.)116 2941 y([2])47 b(Arnol'd,)34 b(V.I.,)i(Pro)s(of)d(of)h(a)g(theorem)h(of) f(A.N.)g(Kolmogoro)m(v)i(on)e(the)g(in)m(v)-5 b(ariance)35 b(of)f(quasi-p)s(erio)s(dic)258 3054 y(motions)24 b(under)e(small)j(p)s (erturbations)e(of)h(the)g(Hamiltonian,)i Fd(R)n(ussian)h(Math.)h (Surveys)j Fo(18:5)25 b Fx(\(1963\),)258 3167 y(9{36.)116 3369 y([3])47 b(Barrar,)24 b(R.B.,)h(Con)m(v)m(ergence)f(of)e(the)h(v)m (on)f(Zeip)s(el)h(pro)s(cedure,)g Fd(Celestial)i(Me)-5 b(ch.)23 b Fo(2:4)f Fx(\(1970\),)27 b(494{504.)116 3570 y([4])47 b(Benettin,)34 b(G.,)g(Galgani,)h(L.,)f(Giorgilli,)h(A.)e(and) g(Strelcyn,)g(J.-M.,)h(A)f(Pro)s(of)f(of)h(Kolmogoro)m(v's)i(the-)258 3683 y(orem)e(on)h(in)m(v)-5 b(arian)m(t)34 b(tori)g(using)f(canonical) i(transformations)f(de\014ned)e(b)m(y)h(the)h(Lie)f(metho)s(d,)h Fd(Nuovo)258 3796 y(Cimento)f(B)40 b Fo(79:2)31 b Fx(\(1984\),)i (201{223.)116 3998 y([5])47 b(Celletti,)42 b(A.)d(and)f(Chierc)m(hia,)j (L.,)f(On)e(the)g(stabilit)m(y)i(of)f(realistic)h(three-b)s(o)s(dy)d (problems,)j Fd(Comm.)258 4111 y(Math.)33 b(Phys.)d Fo(186:2)i Fx(\(1997\),)h(413{449.)116 4312 y([6])47 b(de)42 b(la)h(Lla)m(v)m(e,)k (R.,)e(A)e(tutorial)g(on)f(KAM)h(theory)-8 b(,)46 b Fd(Smo)-5 b(oth)45 b(er)-5 b(go)g(dic)45 b(the)-5 b(ory)45 b(and)g(its)e(applic) -5 b(ations)258 4425 y(\(Se)g(attle,)32 b(W)-7 b(A,)30 b(1999\))p Fx(,)i Fd(Pr)-5 b(o)g(c.)31 b(Symp)-5 b(os.)33 b(Pur)-5 b(e)31 b(Math.)e Fx(v)m(ol)h Fo(69)p Fx(,)g(Amer.)f(Math.)g (So)s(c.,)g(Pro)m(vidence,)h(RI,)258 4538 y(2001,)i(175{292.)116 4740 y([7])47 b(de)33 b(la)h(Lla)m(v)m(e,)i(R.,)f(Gonz\023)-45 b(alez,)37 b(A.,)d(Jorba,)1774 4717 y(\022)1763 4740 y(A.)f(and)g(Villan)m(uev)-5 b(a,)36 b(J.,)f(KAM)e(theory)h(without)g (action{)258 4853 y(angle)d(v)-5 b(ariables,)31 b Fd(Nonline)-5 b(arity)40 b Fo(18:2)31 b Fx(\(2005\),)j(855{895.)116 5054 y([8])47 b(Hubbard,)24 b(J.)g(and)f(Ily)m(ashenk)m(o,)k(Y.,)f(A)e (pro)s(of)f(of)i(Kolmogoro)m(v's)h(theorem,)g Fd(Discr)-5 b(ete)26 b(and)i(Continious)258 5167 y(Dynamic)-5 b(al)34 b(Systems)k Fo(10:1)e(&)f(2)c Fx(\(2004\),)i(367{385.)116 5369 y([9])47 b(Jorba,)541 5346 y(\022)530 5369 y(A.)22 b(and)f(Villan)m(uev)-5 b(a,)25 b(J.,)e(On)e(the)h(normal)f(b)s(eha)m (viour)g(of)h(partially)g(elliptic)i(lo)m(w)m(er-dimensional)258 5482 y(tori)31 b(of)f(Hamiltonian)i(systems,)f Fd(Nonline)-5 b(arity)40 b Fo(10:4)31 b Fx(\(1997\),)i(783{822.)71 5684 y([10])47 b(Kolmogoro)m(v,)j(A.N.,)f(On)44 b(the)h(p)s(ersistence) f(of)h(conditionally)h(p)s(erio)s(dic)e(motions)h(under)e(a)i(small)258 5796 y(c)m(hange)31 b(of)g(the)f(Hamilton)i(function,)e Fd(Dokl.)j(A)n(kad.)f(Nauk)g(SSSR)j Fo(98:4)c Fx(\(1954\),)i(527{530.)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 71 274 a Fn(J.)30 b(Villan)m(uev)-5 b(a)3171 b Fx(23)71 573 y([11])47 b(Ne)-10 b(\025)-35 b(\020sh)m(tadt,)47 b(A.I.,)g(Estimates)d(in)f(the)g(Kolmogoro)m(v)i (theorem)e(on)g(conserv)-5 b(ation)44 b(of)f(conditionally)258 686 y(p)s(erio)s(dic)29 b(motions,)j Fd(J.)g(Appl.)h(Math.)g(Me)-5 b(ch.)30 b Fo(45:6)h Fx(\(1981\),)i(1016{1025.)71 872 y([12])47 b(P\177)-45 b(osc)m(hel,)30 b(J.,)f(In)m(tegrabilit)m(y)h(of) f(Hamiltonian)g(systems)g(on)f(Can)m(tor)h(sets,)g Fd(Comm.)j(Pur)-5 b(e)31 b(Appl.)g(Math.)258 984 y Fo(35:5)g Fx(\(1982\),)i(653{695.)71 1170 y([13])47 b(R)s(\177)-48 b(ussmann,)24 b(H.,)j(On)d(optimal)h (estimates)i(for)d(the)h(solutions)h(of)f(linear)g(partial)g (di\013eren)m(tial)i(equations)258 1283 y(of)32 b(\014rst)g(order)g (with)g(constan)m(t)i(co)s(e\016cien)m(ts)g(on)e(the)h(torus,)g Fd(Dynamic)-5 b(al)36 b(Systems,)f(The)-5 b(ory)36 b(and)g(Ap-)258 1396 y(plic)-5 b(ations)36 b(\(R)-5 b(enc)g(ontr)g(es,)37 b(Battel)5 b(le)34 b(R)-5 b(es.)35 b(Inst.,)g(Se)-5 b(attle,)35 b(Wash.,)g(1974\))p Fx(,)g(editor)d(J.)g(Moser,)i Fd(L)-5 b(e)g(ctur)g(e)258 1509 y(Notes)32 b(in)h(Physics)e Fx(v)m(ol)g Fo(38)p Fx(,)g(1975,)i(598{624.)71 1695 y([14])47 b(Siegel,)32 b(C.L.,)e(Iteration)i(of)e(analytic)i(functions,)e Fd(A)n(nn.)i(of)h (Math.)g(\(2\))p Fx(,)e Fo(43)g Fx(\(1942\),)i(607{612.)71 1881 y([15])47 b(T)-8 b(reshc)m(h)m(\177)-43 b(ev,)25 b(D.V.,)g(A)e(mec)m(hanism)g(for)f(the)g(destruction)h(of)f(resonance)h (tori)g(in)f(Hamiltonian)i(systems,)258 1994 y Fd(Math.)33 b(USSR-Sb.)d Fo(68:1)h Fx(\(1991\),)i(181{203.)71 2279 y Fp(A)134 b(Some)45 b(tec)l(hnical)h(results)71 2482 y Fx(In)30 b(this)g(section)h(w)m(e)g(presen)m(t)g(some)f(basic)h (results)f(used)g(throughout)g(the)g(pap)s(er.)71 2665 y Fo(Lemma)46 b(8)h(\(R)s(\177)-55 b(ussmann)47 b(estimates,)i([13)q (]\).)d Fd(L)-5 b(et)43 b Fm(g)s Fx(\()p Fm(\022)s Fx(\))p Fd(,)h(with)f Fm(\022)h Fl(2)e Fk(T)2755 2632 y Fj(n)2802 2665 y Fd(,)i Fm(n)d Fl(\025)h Fx(2)p Fd(,)i(b)-5 b(e)42 b(a)h Fx(2)p Fm(\031)s Fd(-p)-5 b(erio)g(dic)71 2778 y(function,)49 b(analytic)e(in)f(the)g(interior)g(of)g Fx(\001\()p Fm(\032)p Fx(\))g Fd(and)h(b)-5 b(ounde)g(d)47 b(up)f(to)h(the)f(b)-5 b(oundary,)50 b(and)d Fm(!)52 b Fl(2)d Fk(R)3712 2745 y Fj(n)3804 2778 y Fd(a)71 2891 y(Diophantine)36 b(ve)-5 b(ctor)36 b(verifying)f(\(1\),)i(for)e(c)-5 b(ertain)37 b Fm(\015)e(>)29 b Fx(0)36 b Fd(and)g Fm(\034)k Fl(\025)30 b Fm(n)21 b Fl(\000)h Fx(1)p Fd(.)50 b(If)38 b Fx(\026)-48 b Fm(g)34 b Fx(=)29 b(0)p Fd(,)36 b(ther)-5 b(e)36 b(is)g(a)f(unique)71 3004 y Fx(2)p Fm(\031)s Fd(-p)-5 b(erio)g(dic)30 b(function)f Fm(f)10 b Fx(\()p Fm(\022)s Fx(\))p Fd(,)29 b(with)g(the)g(normalization)2057 2980 y Fx(\026)2038 3004 y Fm(f)34 b Fx(=)25 b(0)p Fd(,)30 b(solving)f(the)g(e)-5 b(quation)30 b Fm(L)3182 3018 y Fj(!)3232 3004 y Fm(f)k Fx(=)25 b Fm(g)33 b Fd(\(se)-5 b(e)29 b(\(4\)\),)71 3117 y(that)34 b(is)e(analytic)i(in)f(the)g (interior)g(of)g Fx(\001\()p Fm(\032)p Fx(\))g Fd(and)h(satis\014es)f (the)g(b)-5 b(ound)1468 3364 y Fl(k)p Fm(f)10 b Fl(k)1613 3379 y Fj(\032)p Fi(\000)p Fj(\016)1768 3364 y Fl(\024)25 b Fm(\033)s Fx(\()p Fm(n;)15 b(\034)5 b(;)15 b(!)s Fx(\))2239 3302 y Fl(k)p Fm(g)s Fl(k)2375 3316 y Fj(\032)p 2239 3343 178 4 v 2259 3426 a Fm(\015)5 b(\016)2354 3400 y Fj(\034)2428 3364 y Fm(;)71 3601 y Fd(for)33 b(any)g Fx(0)26 b Fm(<)f(\016)k Fl(\024)c Fm(\032)p Fd(,)32 b(wher)-5 b(e)34 b Fm(\033)29 b Fl(\025)c Fx(1)33 b Fd(c)-5 b(an)33 b(b)-5 b(e)32 b(taken)h(as)913 3847 y Fm(\033)s Fx(\()p Fm(n;)15 b(\034)5 b(;)15 b(!)s Fx(\))27 b(=)1450 3786 y(3)p Fm(\031)p 1411 3826 179 4 v 1411 3910 a Fx(2)1456 3883 y Fj(\034)8 b Fh(+1)1600 3847 y Fx(6)1645 3810 y Fj(n=)p Fh(2)1762 3765 y Fg(p)p 1853 3765 274 4 v 82 x Fm(\034)i Fx(\000\(2)p Fm(\034)g Fx(\))2142 3719 y Fg(\022)2218 3786 y Fl(j)p Fm(!)2300 3800 y Fh(1)2340 3786 y Fl(j)20 b Fx(+)g Fl(\001)15 b(\001)g(\001)21 b Fx(+)f Fl(j)p Fm(!)2775 3800 y Fj(n)2822 3786 y Fl(j)p 2218 3826 629 4 v 2478 3910 a(j)p Fm(!)s Fl(j)2857 3719 y Fg(\023)2924 3742 y Fj(\034)2983 3847 y Fm(;)71 4096 y Fd(with)33 b Fx(\000\()p Fl(\001)p Fx(\))h Fd(the)f(Gamma)h (function.)71 4279 y Fo(Lemma)48 b(9.)g Fd(L)-5 b(et)43 b Fm(F)13 b Fx(\()p Fm(I)7 b Fx(\))44 b Fd(b)-5 b(e)43 b(an)g(analytic)h(function)f(in)g(the)h(interior)g(of)f Fk(B)p Fx(\()p Fm(r)s Fx(\))g Fd(and)h(b)-5 b(ounde)g(d)44 b(up)f(to)h(the)71 4392 y(b)-5 b(oundary.)132 4575 y(\(a\))46 b(If)33 b Fl(j)p Fm(F)13 b Fx(\()p Fm(I)7 b Fx(\))p Fl(j)26 b(\024)f Fm(A)p Fl(j)p Fm(I)7 b Fl(j)919 4542 y Fh(3)959 4575 y Fd(,)33 b Fl(8)p Fm(I)f Fl(2)25 b Fk(B)p Fx(\()p Fm(r)s Fx(\))p Fd(,)32 b(and)h Fx(0)26 b Fm(<)f(\016)k Fl(\024)c Fm(r)s(=)p Fx(3)p Fd(,)33 b(then)g(if)f Fm(j;)15 b(k)30 b Fl(2)25 b(f)p Fx(1)p Fm(;)15 b(:)g(:)g(:)i(;)e(n)p Fl(g)p Fd(,)1104 4752 y Fg(\014)1104 4807 y(\014)1134 4830 y Fm(@)1182 4844 y Fj(I)1213 4854 y Ff(j)1249 4830 y Fm(F)e Fx(\()p Fm(I)7 b Fx(\))1437 4752 y Fg(\014)1437 4807 y(\014)1494 4830 y Fl(\024)1600 4768 y Fx(9)p 1600 4809 46 4 v 1600 4892 a(4)1655 4830 y Fm(A)1733 4768 y(r)p 1733 4809 44 4 v 1733 4892 a(\016)1787 4830 y Fl(j)p Fm(I)g Fl(j)1884 4792 y Fh(2)1924 4830 y Fm(;)2150 4725 y Fg(\014)2150 4780 y(\014)2150 4834 y(\014)2181 4830 y Fm(@)2234 4792 y Fh(2)2229 4852 y Fj(I)2260 4862 y Ff(j)2292 4852 y Fj(I)2323 4864 y Ff(k)2365 4830 y Fm(F)13 b Fx(\()p Fm(I)7 b Fx(\))2553 4725 y Fg(\014)2553 4780 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Fx(\()p Fm(x)p Fx(\)[)p Fm(I)7 b(;)15 b(I)7 b(;)15 b(I)7 b Fx(])28 b(=)1260 5682 y Fg(P)1356 5777 y Fj(j;k)r(;l)q Fi(2f)p Fh(1)p Fj(;:::)m(;n)p Fi(g)1797 5750 y Fm(@)1850 5717 y Fh(3)1845 5777 y Fj(I)1876 5787 y Ff(j)1908 5777 y Fj(I)1939 5789 y Ff(k)1977 5777 y Fj(I)2008 5789 y Ff(l)2036 5750 y Fm(F)13 b Fx(\()p Fm(x)p Fx(\))p Fm(I)2269 5764 y Fj(j)2306 5750 y Fm(I)2346 5765 y Fj(k)2389 5750 y Fm(I)2429 5765 y Fj(l)2454 5750 y Fd(.)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 71 274 a Fx(24)2427 b Fn(Kolmogoro)m(v)32 b(Theorem)e(Revisited)71 573 y Fo(Pro)s(of)36 b(:)k Fx(W)-8 b(e)32 b(start)f(b)m(y)f(pro)m(ving)g(\(a\).)42 b(If)30 b Fl(j)p Fm(I)7 b Fl(j)26 b(\024)f Fx(2)p Fm(r)s(=)p Fx(3)31 b(w)m(e)g(ha)m(v)m(e,)h(b)m(y)e(using)g(Cauc)m(h)m(y)h (estimates)h(\(6\),)1194 832 y Fl(j)p Fm(@)1267 846 y Fj(I)1298 856 y Ff(j)1334 832 y Fm(F)13 b Fx(\()p Fm(I)7 b Fx(\))p Fl(j)27 b(\024)e Fm(A)1748 771 y Fx(\()p Fl(j)p Fm(I)7 b Fl(j)21 b Fx(+)f Fl(j)p Fm(I)7 b Fl(j)p Fm(=)p Fx(2\))2214 738 y Fh(3)p 1748 812 508 4 v 1907 895 a Fl(j)p Fm(I)g Fl(j)p Fm(=)p Fx(2)2290 832 y(=)2396 771 y(27)p 2396 812 91 4 v 2419 895 a(4)2497 832 y Fm(A)p Fl(j)p Fm(I)g Fl(j)2662 795 y Fh(2)2702 832 y Fm(:)71 1085 y Fx(If)30 b(2)p Fm(r)s(=)p Fx(3)c Fl(\024)f(j)p Fm(I)7 b Fl(j)26 b(\024)f Fm(r)d Fl(\000)e Fm(\016)35 b Fx(w)m(e)30 b(ha)m(v)m(e)559 1345 y Fl(j)p Fm(@)632 1359 y Fj(I)663 1369 y Ff(j)700 1345 y Fm(F)13 b Fx(\()p Fm(I)7 b Fx(\))p Fl(j)27 b(\024)e Fm(A)1198 1284 y(r)1242 1251 y Fh(3)p 1114 1324 253 4 v 1114 1407 a Fm(r)d Fl(\000)e(j)p Fm(I)7 b Fl(j)1401 1345 y Fx(=)25 b Fm(A)1764 1284 y(r)1808 1251 y Fh(3)p 1575 1324 460 4 v 1575 1407 a Fx(\()p Fm(r)e Fl(\000)d(j)p Fm(I)7 b Fl(j)p Fx(\))p Fl(j)p Fm(I)g Fl(j)1994 1381 y Fh(2)2045 1345 y Fl(j)p Fm(I)g Fl(j)2142 1308 y Fh(2)2207 1345 y Fl(\024)25 b Fm(A)2516 1284 y(r)2560 1251 y Fh(3)p 2381 1324 353 4 v 2381 1407 a Fm(\016)s Fx(\()p Fm(r)g Fl(\000)19 b Fm(\016)s Fx(\))2693 1381 y Fh(2)2744 1345 y Fl(j)p Fm(I)7 b Fl(j)2841 1308 y Fh(2)2906 1345 y Fl(\024)3012 1284 y Fx(9)p 3012 1324 46 4 v 3012 1407 a(4)3068 1345 y Fm(A)3146 1284 y(r)p 3146 1324 44 4 v 3146 1407 a(\016)3200 1345 y Fl(j)p Fm(I)g Fl(j)3297 1308 y Fh(2)3337 1345 y Fm(;)71 1606 y Fx(where)32 b(w)m(e)i(use)f (that)g(min)984 1624 y Fj(x)p Fi(2)p Fh([2)p Fj(r)r(=)p Fh(3)p Fj(;r)r Fi(\000)p Fj(\016)r Fh(])1395 1606 y Fl(f)p Fx(\()p Fm(r)26 b Fl(\000)21 b Fm(x)p Fx(\))p Fm(x)1773 1573 y Fh(2)1813 1606 y Fl(g)30 b Fx(=)f Fm(\016)s Fx(\()p Fm(r)d Fl(\000)c Fm(\016)s Fx(\))2304 1573 y Fh(2)2377 1606 y Fx(and)33 b(0)d Fm(<)f(\016)34 b Fl(\024)29 b Fm(r)s(=)p Fx(3.)49 b(Hence,)35 b(the)e(desired)71 1718 y(estimate)42 b(on)e Fm(@)629 1732 y Fj(I)660 1742 y Ff(j)696 1718 y Fm(F)54 b Fx(is)40 b(straigh)m(tforw)m(ard)h(from)e(b)s (oth)h(inequalities.)71 b(The)40 b(b)s(ound)e(on)i(the)h(second)f (order)71 1831 y(deriv)-5 b(ativ)m(es)32 b(is)e(obtained)h(from)f(the)g (t)m(w)m(o)i(next)f(inequalities.)42 b(If)30 b Fl(j)p Fm(I)7 b Fl(j)25 b(\024)g Fm(r)s(=)p Fx(3)31 b(w)m(e)g(ha)m(v)m(e)1212 2091 y Fl(j)p Fm(@)1290 2053 y Fh(2)1285 2113 y Fj(I)1316 2123 y Ff(j)1348 2113 y Fj(I)1379 2125 y Ff(k)1421 2091 y Fm(F)13 b Fx(\()p Fm(I)7 b Fx(\))p Fl(j)27 b(\024)d Fm(A)1834 2030 y Fx(\()p Fl(j)p Fm(I)7 b Fl(j)22 b Fx(+)e(2)p Fl(j)p Fm(I)7 b Fl(j)p Fx(\))2256 1997 y Fh(3)p 1834 2070 462 4 v 1997 2153 a Fl(j)p Fm(I)g Fl(j)2094 2127 y Fh(2)2332 2091 y Fx(=)25 b(27)p Fm(A)p Fl(j)p Fm(I)7 b Fl(j)p Fm(;)71 2344 y Fx(and)30 b(if)g Fm(r)s(=)p Fx(3)c Fl(\024)f(j)p Fm(I)7 b Fl(j)25 b(\024)g Fm(r)e Fl(\000)d Fx(2)p Fm(\016)35 b Fx(then)393 2604 y Fl(j)p Fm(@)471 2566 y Fh(2)466 2626 y Fj(I)497 2636 y Ff(j)530 2626 y Fj(I)561 2638 y Ff(k)603 2604 y Fm(F)13 b Fx(\()p Fm(I)7 b Fx(\))p Fl(j)26 b(\024)f Fm(A)1237 2542 y(r)1281 2509 y Fh(3)p 1016 2583 525 4 v 1016 2666 a Fx(\(\()p Fm(r)f Fl(\000)19 b(j)p Fm(I)7 b Fl(j)p Fx(\))p Fm(=)p Fx(2\))1498 2640 y Fh(2)1575 2604 y Fx(=)25 b(4)p Fm(A)1983 2542 y(r)2027 2509 y Fh(3)p 1794 2583 460 4 v 1794 2666 a Fx(\()p Fm(r)f Fl(\000)c(j)p Fm(I)7 b Fl(j)p Fx(\))2117 2640 y Fh(2)2157 2666 y Fl(j)p Fm(I)g Fl(j)2265 2604 y(j)p Fm(I)g Fl(j)26 b(\024)f Fm(A)2719 2542 y(r)2763 2509 y Fh(3)p 2562 2583 398 4 v 2562 2666 a Fm(\016)2605 2640 y Fh(2)2645 2666 y Fx(\()p Fm(r)e Fl(\000)d Fx(2)p Fm(\016)s Fx(\))2970 2604 y Fl(j)p Fm(I)7 b Fl(j)26 b(\024)e Fx(3)p Fm(A)3311 2542 y(r)3355 2509 y Fh(2)p 3311 2583 84 4 v 3311 2666 a Fm(\016)3354 2640 y Fh(2)3405 2604 y Fl(j)p Fm(I)7 b Fl(j)p Fm(:)71 2856 y Fx(P)m(art)31 b(\()p Fm(b)p Fx(\))g(of)f(the)h(statemen)m(t)h(is)f(obtained)f(from)g (the)h(follo)m(wing)h(in)m(tegral)g(expression:)357 3099 y([)p Fm(F)13 b Fx(\()p Fm(I)7 b Fx(\)])595 3113 y Fh(3)718 3099 y Fx(=)83 b Fm(F)13 b Fx(\()p Fm(I)7 b Fx(\))21 b Fl(\000)f Fm(F)13 b Fx(\(0\))22 b Fl(\000)e(hr)1582 3113 y Fj(I)1622 3099 y Fm(F)13 b Fx(\(0\))p Fm(;)i(I)7 b Fl(i)22 b(\000)2053 3037 y Fx(1)p 2053 3078 46 4 v 2053 3161 a(2)2108 3099 y Fl(h)p Fm(I)7 b(;)15 b(D)2308 3061 y Fh(2)2305 3121 y Fj(I)2348 3099 y Fm(F)e Fx(\(0\))p Fm(I)7 b Fl(i)718 3335 y Fx(=)872 3211 y Fg(Z)963 3237 y Fh(1)923 3417 y(0)1049 3273 y Fm(d)p 1028 3314 90 4 v 1028 3397 a(ds)1143 3207 y Fg(\022)1210 3335 y Fm(F)13 b Fx(\()p Fm(sI)7 b Fx(\))20 b(+)g(\(1)h Fl(\000)f Fm(s)p Fx(\))p Fl(hr)1933 3349 y Fj(I)1973 3335 y Fm(F)13 b Fx(\()p Fm(sI)7 b Fx(\))p Fm(;)15 b(I)7 b Fl(i)22 b Fx(+)2449 3273 y(1)p 2449 3314 46 4 v 2449 3397 a(2)2504 3335 y(\(1)f Fl(\000)f Fm(s)p Fx(\))2774 3297 y Fh(2)2813 3335 y Fl(h)p Fm(I)7 b(;)15 b(D)3013 3297 y Fh(2)3010 3357 y Fj(I)3053 3335 y Fm(F)e Fx(\()p Fm(sI)7 b Fx(\))p Fm(I)g Fl(i)3366 3207 y Fg(\023)3449 3335 y Fm(ds:)3780 3582 y Fa(\003)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0509011046930--