Content-Type: multipart/mixed; boundary="-------------0410110436795" This is a multi-part message in MIME format. ---------------0410110436795 Content-Type: text/plain; name="04-321.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-321.keywords" Hill's equation, Quasi-periodic motion, Lindstedt series, Renormalization group, Reducibility, Cantor set, Divergent series ---------------0410110436795 Content-Type: application/postscript; name="bcg.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bcg.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: hill_draft_15.dvi %%Pages: 40 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMSY8 CMSY6 CMR9 CMBX9 CMMI9 CMR6 CMSY9 CMR10 %%+ CMMI10 CMR7 CMSY10 CMMI7 CMEX10 CMSY7 CMMIB10 CMR5 CMBX10 CMTI10 %%+ CMMIB7 CMMI5 LASY10 CMBX7 CMSY5 CMMIB5 CMBSY10 TeX-cmex7 CMBX5 %%+ TeX-cmex9 CMMI6 rsfs10 rsfs7 CMR8 CMMI8 TeX-cmex8 EUFM10 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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. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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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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(001.001) readonly def /Copyright (Conversion of metafont curves by Metafog (c) 1995 Richard Kinch) readonly def /Notice (Copyright (c) Taco Hoekwater, 1998. All rights reserved.) readonly def /FullName (rsfs7) readonly def /FamilyName (rsfs7) readonly def /ItalicAngle -12 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Roman) readonly def end readonly def /FontName /rsfs7 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 84 /T put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox {16 -302 1349 728} readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: rsfs10 %!PS-AdobeFont-1.0: rsfs10 001.000 %%CreationDate: Sat Mar 21 18:47:14 1998 %%VMusage: 120000 150000 11 dict begin /FontInfo 14 dict dup begin /version (001.001) readonly def /Copyright (Conversion from mf curves by Metafog (c) 1995 Richard Kinch) readonly def /Notice (Copyright (c) Taco Hoekwater, 1998. All rights reserved.) readonly def /FullName (rsfs10) readonly def /FamilyName (rsfs10) readonly def /ItalicAngle -12 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Roman) readonly def end readonly def /FontName /rsfs10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 84 /T put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox {-2 -300 1240 728} readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMIB7 %!PS-AdobeFont-1.1: CMMIB7 001.100 %%CreationDate: 1996 Jul 27 07:35:50 % Computer Modern fonts were designed by Donald E. Knuth. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (001.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMIB7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMMIB7 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 23 /nu put dup 33 /omega put readonly def /FontBBox{0 -250 1294 750}readonly def /UniqueID 5087389 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 D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBSY10 %!PS-AdobeFont-1.1: CMBSY10 1.00 %%CreationDate: 1992 Jul 23 21:21:18 % 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 (CMBSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMBSY10 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 -940 1332 825}readonly def /UniqueID 5000762 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF17EFB0FFE6C69FEBA8389DCC1923D30683D A8CD93F7195D5A07BA2F18CB3FD5FFEDA4D83BF758062134D84AC0100187A6CD 1F80F5DC15B47D73F69655445AD218A8AD78C16EF96F385C9E2D46F8A330C7B5 A859EB0610C78FC5CE39715A1C5458D30498C0A339504A74C7E8F84B3DEC1516 B3ABAA0A06DEDCD5F9FEAA5AC4AE8D5A5BA5EC0B64784454F58049E13467D705 8F13A22BDED5F93EDDCAB7A1886A5168D25B120F8BBCC23546BC7398D4E3EC17 138921404C390EB84C3CC243C0FF3DEC9EBFFF3DEA73365F1E4BC2F3AB911B2F 780946F4F6F49935A54EF955D9894FEB37239C896CF98240162F6A6E9677EA24 06BEE1F04463C033047F7F972C560213C7A02BFEE5AE5AE5BF72377CED942A6D 8059E59CF03CD6782BD34BC02AA4FD1BA25A5CBE32569D7FED28EFB4C0F5F7C8 6DADC1A047CB514E19B36A84D4DB390FFE5B841C390666FE27C712E23E22FC84 A8670626E8B72700B9EE9F06F2121264C1CF69FEEC3E20897D0D9057032830FE A18A4BA2AD5CE10EE4FED4BB9E2A9C06965779827D7CBA93926793A7161454E3 C5AC6A3AAEB75EC64556142508DE6E37B71058F8B97C1A9B4CEBF74FBD2D6D84 F5DAA2B04AD30B313070B33789935E83DB470FAB8EC65165679F247964BD0C20 78291B6E13C29E8B86429C1B90C396729D6BDE4CCF24BE000390D798DA73BBEC AC5C9B1AC19B2C660CF1CDEC05289F6CAEF0E43465E3627DE26670BAA825429B 4B8FE57928267D5EBE38C5BF93F90304EB89DE120F81362FB5A3D374AB25B33C D03A8E9E176E41C964625E58A65EA958EF2B089933C06B71E29249A96D5A2395 DE687A0C60B837B5657B90F8642A27B037E4FFFA82343351B7C36571EE7ECCC2 4F4FCD1A82249D1B505767ED8116330D97A131E0B60E2E29F2E118BA6E5EED6E 9951CEADF3E9F130C8EFAC9E2AE3AB99A386E4022EECC24AB7B18591598E592D 3267C8A9186796FB53866EBDEE6C832C7074F02CB550FFB1BA69F4BA5CBF4DEC 928DAEA497B74267619AB939A9155A986B7DE16932E97220A561A933F3C8CEC7 988C8CD499564AA7C4FD0AE2D95873A9A20545946D41EB84619ADE7E0775132E C5C337E2EF18C66DBB6276ED114BFB0CB35DC206D865F018E38370CF8901D039 BB67BEC69B518B7F274CD895580FB32FB10FBCBD0EAEAC2068CCE355AB7EB83F 52D884EC0F7766B7C3834E65E7DB5AA779E4937DAA7EBD6AA2FD283211651187 092838E5CC4B7AE6BF563FA63B553E7099C88FD25A02325D 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMIB10 %!PS-AdobeFont-1.1: CMMIB10 1.100 %%CreationDate: 1996 Jul 23 07:54:00 % 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 (CMMIB10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMIB10 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{-15 -250 1216 750}readonly def /UniqueID 5087392 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D004B836D34E88C25F6CE738846C8E2E59A2BCF 4ACF80A26D78872E9343A0537BC3BD7715F32ACD958D5AAED865BFE129278935 063A31C2634DE2F9077E0AAAAEB224466B779096D8E3FF0A12AD5157F6603DED 1A82F3511359143311179080C510740B401C930C96270FD1AB3ECBCFEF5DE53F E846BAAE95828D5790922640EF8AB9D7CEBE7669FEA02B72F86872D3D8754A18 A1629C40A7C00C956F140BC63362478279C36EE353638CD3E249897207A94504 4400668C8E702058EBF7284C9BDF830A3FC79C7EE900CC4C3664F9767A237275 CEE3671644A75F1E696DA906B4C66870DBE87F5B4A176920C078ADBE24F55C09 3D18CDE21B5FBC1C6A8AB18E05EDBEF9D1C1C18B3E6377BA2A688579D4F708F9 A5CF4F56C5E39E2726106E9713E638775E606464CD674E5DC25CE9A696A65806 C8E9D206B421E246F18013ACC6C7B2985BA93B1B7D7745CCB25B09957F50128C B523A55ACA6A7A2A0193A536E590291ED9D577B527CAD0372E05BFCA1829FED1 662D06144A5FFA628C587A4FA05B179F1A7E3B23B47765FDC054271A0DBF9C2B B4F4771F80D1F7AAD9024868C30DAD5CF728DB2A71D86D53B0E674996E8C01F7 EF97B225A28872F8AD4752A466E078C2B020EB832F237CB9B5631EB2D2EDDB00 709D3864CA3A6C3EF18085EAEABC011E9F35C9BE4B5D0B608361F329B5784DAC 5557A602E9E3C204909D84DB988F0BAB914E87CD685C7DA55C5E0B9F0176184F FC39B570873BBF346A0D1DE3942DA05434949A65CE64D8BAB0A091C40F7FF47A 4FC57CB4420221C7B3EB8B891044B5FB0227009F0F6028D3F28545E63BC996D4 ADD6E35FA301B8E06658043FBC01B8546ABA729DEFAE61CEB0ED3122C08E5553 C43864EECACC5C8E810B84724EEBF0179D35C3A94E3FD99632240AE85BC7CAD1 44F2ABA942F36F70AB0E497FC10636AF0196D5FE0A9394CD21AB46A3633DE57F 8E104D18F2078654CB91BCADAB47511515C9A8AC30DCA5116A6B25E652F244A2 421974CF2E0A0A3D370EFB146DF759A5FBCCB79F0024F69C666246ADB7BE380B 86CEE31582B5D04DD4BDD079BA3BA786F16227C44A9ADDE27CABA1E633869A9B 176D0C49619E3BC3CA1B5D99E660808DE9D6D967F7B73160EE80F0579EE233F4 E2060C5D7053282080DC3BA7E0ABC53EAFE9E4E6DB21DB5E9E02EFB692741D83 21391952005FEE7477BF88EFF241D06897698D0A7CEFA6F71DE5390A66442039 B0EB0D22E419EB625AB1EA99122596774A35B39F8DE847D0E3E476008C750A77 02838C814FB1CB0D82F42D25BFF551AD25506FAD3F6B35CC6704F231CDB05362 143C8648A539809797A135F585348A4698CA26D4E146D1B96D5E9BCF6CD3C0DD 9079D5ECF80195C1AAD1889F940292B387870CB16326E218F6839BD4A1446158 779E7A5257000E24AE5F98F3F686E871BAE0D302A5F499C60310BD79B11383DC 97D12D559DC6403653CBA52A59FBCC609DC1E4AE5CA9DDE2B97B93F2BA110E5C A44D71049A687B233A1EA56BD1EBAC828DD1D6255BC4EAC194B6BE7705D37A67 5D2602DF3EF450753B113ED13C26859C4D6433D37CEF3B20CF93EFC75C271FE6 7CDBDF5B0811B13596B578F4F1A1B04D1ACB004E34BADFB28665718E46E38924 46BEFBC3B47499E081761B5173F67DB8E4C5A4D4681FECE52F6923748DE61717 D0AAE3A68B343ED9A807E9F902A70D7FAACF0BFE3D7B4FF2E8599DC247284FD4 C34362BFFA005D74D5793C0567CE6841C215A42C1BAF41DE1B86AB8390B1C8E1 AAEBCAC9990811E2665B1ECB52EEC21EE956321AFD9832FE664A385D4C46EC88 8432F6C0C075CDA0CCA44131153618BD535771571370D638F353F0F810D67BB3 B89D231806D45280C00AA0DB2E815BAC421CD60CCA5C210712CBBFFD8B47DE35 0FD6E498AAC49F0BC4569D73F40447133FC8C630F6F47DE272983D636A8A986E BBAE5717CDA6F9FD4BC07F372CB2F5A766FFC72DC2CE964870B0ABDCE13DFE1F C5B4082623F37230B6D625730E61764BC29160374DEB2DAA7268C029AAC0EAD0 208F320A4709E2973313D3D2C8708CDA272871FD820CC1AE03C179891A2E8BF4 2082C3E59D098F79DDE2CBCA5F8C69E997CD0360392886637E9F027D9420557E D3F5CACA5381DEA6132F00AAB2B64773 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMIB5 %!PS-AdobeFont-1.1: CMMIB5 001.100 %%CreationDate: 1996 Jul 31 22:13:01 % Computer Modern fonts were designed by Donald E. Knuth. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (001.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMIB5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMMIB5 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 23 /nu put readonly def /FontBBox{0 -250 1516 750}readonly def /UniqueID 5087387 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0525392EECAC163E584A9104D99AD0BC 1B1844A0E222653FA481B8809B26A46F4C483A5D7E95816EA6582584156CFEDE B994ADCFF4645140E3617E4D7E1B0E4541CB9F562E55829B4DD880AABE2229E9 4A9FA259A734D29BBA91BA1E2055CBEA4339BCBFF98D32CEFF11F296225CAABA DCA10577A5D431B714726C1278D8101ABD1BD8D0BD0174FFF9148F8C61C241D9 2AD360A28616CB4A0670BAB68BB2ACDAE11479ED1EF664C6F672D3B0CB0D6710 B4784FC8C2DB518F6F61D12D8BFB01655FD31F538D8F278EFB700ED09C7FB174 63EA75F41A2F6CC9FE77ECE7F3F42102F565C46540904D337E2E87DEDA1EF669 6AE3DAE88BF72C076705BB2FF626CD8310961B7D50544FC1FC00DFC71A7201E4 B6FAD251DC743BDA37378D95FCC4CE2A0E45A1FF4009EC05937A2107B09268B7 C94BFC8DBDF56F29DD97E19CA1DCB0E4EFAFA8C9A740C29621FF88CFBCE29124 1E24A1E3D8CCF5B38FCB3A1F925A081252713A2F032BA95856F987E18BC6AEED C0A91EF66D690DAD364C6195F3C70D178D9C834A4A5DE468B707F12B72B51287 F34B7A15CEEE2992AB6F6979E4A54934A967860309E700EF783E0499B832E5AC 3534AE03379F41C7EEDF08487C4C24AE19D8FAA264341C0EB4C802E935D125B8 2BF0821B3F4EEF148B691B68A2EB16BE901CF95C0DDBF89BBA3AF038D886584C 8DD53E721D2375A21DFA2C33970BEEF9AC42AE4050991BFC82AF1DCE6BAC76DE E6743B298308943CB284707C97C698C60DC4C908822D1AF1E2F7F42BECB4E7C8 34373C2BE22329D430251A1B500821FE295A8F23221D652FCB8B7AEA42BE90C2 591F1D67644E037AF99AC68827269A25C4210BB71E903398C7CB24023E4BD0CC 59ABE549C12598A1085E46169D76D68DB12E037D079F57C5583FACA5144C4A20 9185B327D311DE08B2059BB3860A8F34F65C6A64E285737ECA1F6D429EADDA86 31D40B7AAD19B33BE82ABA58144202FB70C52565FC966003DC094F76FCBDFC30 955B9E08814822B57BDAAA4DBAC45D3BDE12DB9D4CE871E5CC7EB30C5231D89B 49C42B03447886C6C626B5E6F33C0CBDFA4AAB15C1E9721689F6EFF710094D3A 2900340367475C40ED58FFD577468E9ACF91028D38BF103149B2444A7324D957 7124A7729B4B18DC5EE4714E1166B417DDC298D6073DDC6CD84B131FD16E1480 D6526F758B756C493C70A4DC0B4586ACB5B7E92E42658BEC6CD865CED760E16F 372B74EB54BA6909CFD9895F0FBF494F2265CFA15FED3D9DF838706E392F1C2B 728F0B4A6DF13D20728AA5C99B80D934E9E9147C9833524F97F1E97CDFC77710 3F8626CD9A098001132EF78D26B563A1C3AA28B20CCDAEC9637FC3D507C07DA0 DA2B510220BB582DB3FFA7B7C7809D6222FE2C0AD97342B166FD502C31E47BE9 FB9A96BB28A122BBDEB65B18B04D3135142B772C8A378245F552F4054BB8CE99 77413171CEFA6422528B8B39BC4FC4147C21BCD145037E467B0F248C26264AE8 A393BE9B109C88DDD553308244EC9C8A4F0126E1F7204EFF0CCA3C7B0CFB1D12 431B71B7C0841D4075C7A58008787CDF3310EED20E50876750E4F9BDA386F02A 6B816DA3FC77D63E5506D6BEBD8A 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX7 %!PS-AdobeFont-1.1: CMBX7 1.0 %%CreationDate: 1991 Aug 20 16:35:49 % 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 (CMBX7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX7 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{-55 -250 1289 751}readonly def /UniqueID 5000765 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: LASY10 %!PS-AdobeFont-1.1: LASY10 1.001 %%CreationDate: 1992 Oct 23 20:19:17 %%RevisionDate: 2001 Jun 05 20:19:17 % Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def /FullName (LASY10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /LASY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 50 /a50 put readonly def /FontBBox{-19 -192 944 683}readonly def /UniqueID 5011949 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BB201D5DB4627484A80A431B6AFDBBBF23D4157D4AFE17E6B1C853DD417 25F84CD55402AB88AB7EEFDEDBF2C2C731BD25567C53B474CCF739188A930039 098A197F9C4BE7594D79442B2C8A67447DE44698321145D7689B91EF235EA80E B600AA8E238064F154284096C4C2554EFE8DDF13AFF8D3CE30E0999375C0FEE6 F992DEA5FC3897E2CC8B7A90238E61E41622DE80F438DD994C73275CC52249D9 F6686F87F394FB7BB668138B210BEC9E46415A1B58C990B81E7D7DD301143517 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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 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 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. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D251491EBF65A98C9FE2B1CF8D725A70281949 8F4AFFE638BBA6B12386C7F32BA350D62EA218D5B24EE612C2C20F43CD3BFD0D F02B185B692D7B27BEC7290EEFDCF92F95DDEB507068DE0B0B0351E3ECB8E443 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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 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig 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 28 /angbracketleftbigg put dup 29 /angbracketrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 83 /uniontext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 101 /tildewide put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CAC6A7BEB5D02276E511FFAF2AE11910 DE076F24311D94D07CACC323F360887F1EA11BDDA7927FF3325986FDB0ABDFC8 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % 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. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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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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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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% 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 (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 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{-30 -958 1146 777}readonly def /UniqueID 5000819 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 D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 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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. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 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 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D6F5CA5FED4F9AC9ADE41E04F9FC52E758C9F45A92BED935 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 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{-58 -250 1195 750}readonly def /UniqueID 5000767 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0364CD5660F74BEE96790DE35AFA90CCF712 B1805DA88AE375A04D99598EADFC625BDC1F9C315B6CF28C9BD427F32C745C99 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 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{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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y(c)l(oncludes)f(that)779 4931 y FG(~)772 4953 y FF(\022)26 b FD(2)927 4932 y FA(e)915 4953 y FD(T)981 4909 y FE(\()p FB(p)p FE(\))960 4978 y FB(k)q(;)p Fu(\027)1072 4953 y Fv(.)38 b(Ther)l(efor)l(e,)1549 4932 y FA(e)1536 4953 y FD(T)1581 4965 y FB(k)q Fz(\000)p FB(p;)p Fu(\027)1793 4953 y FD(\022)1893 4932 y FA(e)1881 4953 y FD(T)1947 4909 y FE(\()p FB(p)p FE(\))1926 4978 y FB(k)q(;)p Fu(\027)2037 4953 y Fv(.)231 5180 y Fw(Lemma)32 b(3.5)40 b Fv(L)l(et)50 b FF(k)64 b FD(\025)c FG(1)p Fv(,)c(then)1458 5113 y FA(\012)1497 5180 y FG([)p FF(Qu)1634 5150 y FE(2)1670 5180 y FG(])1693 5150 y FE(\()p FB(k)q Fz(\000)p FE(1\))1871 5113 y FA(\013)1971 5180 y FG(=)2096 5118 y FA(P)2184 5138 y FB(k)q Fz(\000)p FE(1)2184 5205 y FB(p)p FE(=0)2324 5180 y FF(\013)2377 5150 y FE(\()p FB(p)p FE(\))2467 5180 y FF(G)2532 5192 y FB(k)q Fz(\000)p FB(p)2660 5180 y Fv(,)g(wher)l(e,)h(for)51 b(al)t(l)h FF(j)66 b FD(\025)60 b FG(1)p Fv(,)c FF(G)3733 5192 y FB(j)3829 5180 y FG(:=)231 5217 y FA(P)319 5308 y FB(\022)r Fz(2)407 5293 y Fl(e)398 5308 y Fz(T)436 5316 y Fs(j;)p Fk(0)531 5280 y FG(V)-7 b(al\()p FF(\022)r FG(\))p Fv(.)3073 b Fr(2)2032 5532 y FG(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 231 169 a Fv(Pr)l(o)l(of.)40 b FG(F)-7 b(rom)27 b(the)h(de\014nition,)1119 279 y FA(D)1169 372 y FG([)p FF(Qu)1306 337 y FE(2)1343 372 y FG(])1366 337 y FE(\()p FB(k)q Fz(\000)p FE(1\))1544 279 y FA(E)1618 372 y FG(=)1825 293 y FA(X)1705 472 y FB(k)1740 480 y Fx(1)1773 472 y FE(+)p FB(k)1859 480 y Fx(2)1892 472 y FE(=)p FB(k)q Fz(\000)p FE(1)2225 293 y FA(X)2078 469 y Fu(\027)2120 477 y Fx(0)2152 469 y FE(+)p Fu(\027)2245 477 y Fx(1)2278 469 y FE(+)p Fu(\027)2370 477 y Fx(2)2403 469 y FE(=)p Fq(0)2505 372 y FF(Q)2571 384 y Fu(\027)2613 392 y Fx(0)2649 372 y FF(u)2697 337 y FE(\()p FB(k)2758 345 y Fx(1)2791 337 y FE(\))2697 392 y Fu(\027)2739 400 y Fx(1)2821 372 y FF(u)2869 337 y FE(\()p FB(k)2930 345 y Fx(2)2962 337 y FE(\))2869 392 y Fu(\027)2910 400 y Fx(2)3006 372 y FF(;)231 642 y FG(so)d(that)h(b)n(y)f(using)g(the)h (de\014nition)f(of)h(tree)f(v)-5 b(alue)25 b(\(2.20\))g(and)g(the)h (notations)e(of)i(De\014nition)g(2.6)e(one)h(immediately)231 741 y(realizes)i(that)g(one)h(can)f(write)1541 749 y FA(D)1592 841 y FG([)p FF(Qu)1729 807 y FE(2)1765 841 y FG(])1788 807 y FE(\()p FB(k)q Fz(\000)p FE(1\))1966 749 y FA(E)2040 841 y FG(=)2169 762 y FA(X)2127 941 y FB(\022)r Fz(2T)2244 950 y Fs(k)q(;)p Fk(0)2344 841 y FG(V)-7 b(al\()p FF(\022)r FG(\))14 b FF(:)231 1111 y FG(No)n(w)27 b(let)h Fy(\027)594 1123 y FE(1)654 1111 y FG(=)23 b(\()p FF(m)p 774 1124 73 4 v 21 x FE(1)884 1111 y FF(;)14 b(n)971 1068 y FE(\(1\))971 1133 y(1)1060 1111 y FF(;)g(n)1147 1068 y FE(\(1\))1147 1133 y(2)1236 1111 y FG(\))23 b FD(2)h FC(Z)1432 1081 y FB(d)1498 1111 y FG(and)k Fy(\027)1713 1123 y FE(2)1774 1111 y FG(=)22 b(\()p FF(m)p 1893 1124 V 1967 1132 a FE(2)2004 1111 y FF(;)14 b(n)2091 1068 y FE(\(2\))2091 1133 y(1)2180 1111 y FF(;)g(n)2267 1068 y FE(\(2\))2267 1133 y(2)2356 1111 y FG(\))23 b FD(2)g FC(Z)2551 1081 y FB(d)2590 1111 y FG(.)37 b(F)-7 b(rom)27 b(Lemma)h(3.2,)1301 1311 y FF(n)1351 1268 y FE(\(1\))1351 1333 y(2)1523 1311 y FD(2)84 b(f\000)p FG(2)p FF(k)1854 1323 y FE(1)1890 1311 y FF(;)14 b FD(\000)p FG(2\()p FF(k)2109 1323 y FE(1)2164 1311 y FD(\000)k FG(1\))p FF(;)c(:)g(:)g(:)f(;)h FD(\000)p FG(2)p FF(;)g FG(0)p FF(;)g FG(2)p FD(g)1301 1453 y FF(n)1351 1410 y FE(\(2\))1351 1475 y(2)1523 1453 y FD(2)84 b(f\000)p FG(2)p FF(k)1854 1465 y FE(2)1890 1453 y FF(;)14 b FD(\000)p FG(2\()p FF(k)2109 1465 y FE(2)2164 1453 y FD(\000)k FG(1\))p FF(;)c(:)g(:)g(:)f(;)h FD(\000)p FG(2)p FF(;)g FG(0)p FF(;)g FG(2)p FD(g)g FF(:)231 1655 y FG(T)-7 b(o)26 b(b)r(e)g(more)f(precise,)h(let)g FF(\022)1121 1667 y FB(j)1179 1655 y FD(2)e(T)1303 1667 y FB(k)1338 1675 y Fs(j)1369 1667 y FB(;)p Fu(\027)1431 1675 y Fs(j)1466 1655 y FG(,)i FF(j)i FG(=)23 b(1)p FF(;)14 b FG(2,)25 b(b)r(e)i(a)e(tree)h(con)n(tributing)f(to)h FF(u)2797 1610 y FE(\()p FB(k)2858 1618 y Fs(j)2889 1610 y FE(\))2797 1666 y Fu(\027)2838 1674 y Fs(j)2919 1655 y FG(,)g(then)h FF(n)3206 1612 y FE(\(1\))3206 1677 y(2)3318 1655 y FG(=)22 b(2\()p FF(b)3515 1667 y FE(1)3567 1655 y FD(\000)15 b FF(v)3687 1667 y FE(1)3724 1655 y FG(\))27 b(and)231 1779 y FF(n)281 1736 y FE(\(2\))281 1801 y(2)398 1779 y FG(=)h(2\()p FF(b)601 1791 y FE(2)658 1779 y FD(\000)20 b FF(v)783 1791 y FE(2)820 1779 y FG(\),)32 b(where)e FF(b)1186 1791 y FB(j)1251 1779 y FG(and)h FF(v)1456 1791 y FB(j)1522 1779 y FG(are)e(the)i(n)n(um)n(b)r(er)g(of)f(blac)n(k) g(bullets)h(and)f(the)h(n)n(um)n(b)r(er)f(of)h(v)n(ertices)e(in)i FF(\022)3858 1791 y FB(j)3893 1779 y FG(,)231 1879 y(resp)r(ectiv)n (ely)-7 b(.)37 b(F)-7 b(rom)27 b FF(\022)967 1891 y FE(1)1033 1879 y FG(and)g FF(\022)1233 1891 y FE(2)1298 1879 y FG(w)n(e)h(w)n(ould)f(lik)n(e)h(to)f(construct)h(a)f(tree)h FF(\022)d FD(2)f(T)2707 1891 y FB(k)q(;)p Fq(0)2806 1879 y FG(,)k FF(k)e FG(=)d FF(k)3057 1891 y FE(1)3113 1879 y FG(+)18 b FF(k)3239 1891 y FE(2)3295 1879 y FG(+)g(1,)28 b(con)n(tributing)231 1978 y(to)337 1911 y FA(\012)376 1978 y FG([)p FF(Qu)513 1948 y FE(2)550 1978 y FG(])573 1948 y FE(\()p FB(k)q Fz(\000)p FE(1\))750 1911 y FA(\013)790 1978 y FG(.)49 b(First)31 b(one)h(m)n(ust)f(note)h(that)g(the)g(ro)r (ot)f(lines)h(of)f FF(\022)2467 1990 y FE(1)2536 1978 y FG(and)h FF(\022)2741 1990 y FE(2)2810 1978 y FG(en)n(ter)f(a)g(v)n (ertex)g(in)h FF(\022)i FG(with)e(mo)r(de)231 2102 y Fy(\027)285 2114 y FE(0)359 2102 y FG(=)k(\(0)p 492 2115 42 4 v FF(;)14 b(n)621 2059 y FE(\(0\))621 2124 y(1)709 2102 y FF(;)g FD(\000)p FG(2\).)61 b(As)36 b(the)g(line)g(whic)n(h)f (exits)h(this)g(v)n(ertex)f(\(ro)r(ot)g(line\))h(carries)e(zero)h (momen)n(tum,)j(one)d(has)231 2219 y(the)e(constrain)n(t)e FD(\000)p FG(2)21 b(+)g FF(n)1038 2176 y FE(\(1\))1038 2242 y(2)1148 2219 y FG(+)g FF(n)1284 2176 y FE(\(2\))1284 2242 y(2)1404 2219 y FG(=)30 b(0.)51 b(Th)n(us,)33 b(\()p FF(b)1922 2231 y FE(1)1980 2219 y FG(+)21 b FF(b)2102 2231 y FE(2)2139 2219 y FG(\))h FD(\000)f FG(\()p FF(v)2351 2231 y FE(1)2411 2219 y FG(+)g FF(v)2537 2231 y FE(2)2596 2219 y FG(+)g(1\))30 b(=)h(0.)50 b(This)32 b(last)g(relation)f(implies) 231 2319 y(that)e FD(j)p FF(E)496 2331 y FB(B)554 2319 y FG(\()p FF(\022)r FG(\))p FD(j)c FG(=)g FD(j)p FF(V)19 b FG(\()p FF(\022)r FG(\))p FD(j)p FG(,)30 b(so)e(\(see)g(Remark)g (3.1\))g(the)h(tree)g FF(\022)i FG(con)n(tributing)d(to)2777 2252 y FA(\012)2817 2319 y FG([)p FF(Qu)2954 2289 y FE(2)2990 2319 y FG(])3013 2289 y FE(\()p FB(k)q Fz(\000)p FE(1\))3191 2252 y FA(\013)3259 2319 y FG(m)n(ust)h(ha)n(v)n(e)e(exactly)231 2419 y(one)36 b(white)h(bullet)g(\(with)g(some)e(order)g(lab)r(el)h FF(p)p FG(\).)63 b(Of)37 b(course)e FD(j)p FF(V)18 b FG(\()p FF(\022)r FG(\))p FD(j)26 b FG(+)d FF(p)38 b FG(=)f FF(k)i FG(and)d(1)h FD(\024)g(j)p FF(V)19 b FG(\()p FF(\022)r FG(\))p FD(j)38 b(\024)f FF(k)s FG(,)h(hence)231 2518 y(0)23 b FD(\024)f FF(p)h FD(\024)g FF(k)e FD(\000)d FG(1.)37 b(Therefore,)26 b(one)h(can)h(write)805 2686 y FA(D)856 2778 y FG([)p FF(Qu)993 2744 y FE(2)1030 2778 y FG(])1053 2744 y FE(\()p FB(k)q Fz(\000)p FE(1\))1230 2686 y FA(E)1376 2778 y FG(=)1535 2674 y FB(k)q Fz(\000)p FE(1)1536 2699 y FA(X)1536 2875 y FB(p)p FE(=0)1716 2699 y FA(X)1670 2899 y FB(\022)r Fz(2T)1801 2869 y Fx(\()p Fs(p)p Fx(\))1787 2919 y Fs(k)q(;)p Fk(0)1895 2778 y FG(V)-7 b(al\()p FF(\022)r FG(\))47 b(=)2278 2674 y FB(k)q Fz(\000)p FE(1)2278 2699 y FA(X)2279 2875 y FB(p)p FE(=0)2459 2699 y FA(X)2413 2899 y FB(\022)r Fz(2T)2544 2869 y Fx(\()p Fs(p)p Fx(\))2530 2919 y Fs(k)q(;)p Fk(0)2638 2778 y FF(\013)2691 2744 y FE(\()p FB(p)p FE(\))2795 2778 y FG(V)-7 b(al\()p FF(\022)21 b FD(n)d FF(E)3128 2790 y FB(W)3204 2778 y FG(\()p FF(\022)r FG(\)\))1376 3121 y(=)1535 3018 y FB(k)q Fz(\000)p FE(1)1536 3042 y FA(X)1536 3218 y FB(p)p FE(=0)1670 3121 y FF(\013)1723 3087 y FE(\()p FB(p)p FE(\))1873 3042 y FA(X)1828 3242 y FB(\022)r Fz(2)1916 3227 y Fl(e)1907 3242 y Fz(T)1959 3212 y Fx(\()p Fs(p)p Fx(\))1945 3262 y Fs(k)q(;)p Fk(0)2053 3121 y FG(V)-7 b(al)o(\()p FF(\022)r FG(\))47 b(=)2435 3018 y FB(k)q Fz(\000)p FE(1)2436 3042 y FA(X)2437 3218 y FB(p)p FE(=0)2571 3121 y FF(\013)2624 3087 y FE(\()p FB(p)p FE(\))2807 3042 y FA(X)2728 3237 y FB(\022)r Fz(2)2816 3222 y Fl(e)2807 3237 y Fz(T)2845 3246 y Fs(k)q Fp(\000)p Fs(p;)p Fk(0)3020 3121 y FG(V)-7 b(al\()p FF(\022)r FG(\))1364 3464 y(=:)1535 3361 y FB(k)q Fz(\000)p FE(1)1536 3386 y FA(X)1536 3561 y FB(p)p FE(=0)1670 3464 y FF(\013)1723 3430 y FE(\()p FB(p)p FE(\))1814 3464 y FF(G)1879 3476 y FB(k)q Fz(\000)p FB(p)2020 3464 y FF(;)231 3730 y FG(where)33 b(Remark)f(3.4)g(w)n(as)g (used.)53 b(Note)33 b(that,)h(b)n(y)f(construction,)g FF(G)2443 3742 y FB(j)2479 3730 y FG(,)h FF(j)j FD(\025)31 b FG(1,)j(is)f(expressed)f(b)n(y)h(a)f(sum)h(of)g(trees)231 3830 y(with)28 b(no)g(white)g(bullets)g(suc)n(h)f(that)h(they)g(ha)n(v) n(e)e(exactly)h FF(j)33 b FG(blac)n(k)26 b(bullets)i(and)g FF(j)k FG(v)n(ertices.)p 3869 3830 48 48 v 231 4030 a Fw(De\014nition)f(3.6)41 b Fv(L)l(et)24 b FF(k)h FD(\025)e FG(1)h Fv(and)1369 4009 y FA(e)1357 4030 y FD(T)1402 4042 y FB(k)q(;)p Fu(\027)1529 4030 y Fv(as)g(in)h(De\014nition)f(3.3.) 38 b(We)25 b(split)2617 4009 y FA(e)2604 4030 y FD(T)2649 4042 y FB(k)q(;)p Fu(\027)2776 4030 y Fv(into)g(two)f(disjoint)i(sets)e (as)h(fol)t(lows:)244 4122 y FA(e)231 4143 y FD(T)276 4155 y FB(k)q(;)p Fu(\027)402 4143 y FG(=:)525 4122 y FA(e)513 4143 y FD(T)579 4113 y FB(c)558 4167 y(k)q(;)p Fu(\027)674 4081 y FA(S)770 4122 y(e)757 4143 y FD(T)824 4113 y FB(nc)802 4167 y(k)q(;)p Fu(\027)905 4143 y Fv(,)30 b(wher)l(e)356 4352 y FD(\017)451 4331 y FA(e)439 4352 y FD(T)505 4321 y FB(c)484 4375 y(k)q(;)p Fu(\027)587 4352 y Fv(:)38 b(set)30 b(of)g(tr)l(e)l(es)f(in)1186 4331 y FA(e)1173 4352 y FD(T)1218 4364 y FB(k)q(;)p Fu(\027)1351 4352 y Fv(such)g(that)h(the)g(amputate)l(d)g(line)g(is)g(c)l(onne)l (cte)l(d)p 2498 4365 346 4 v 30 w(to)g(the)g(r)l(o)l(ot)f(line;)356 4542 y FD(\017)451 4521 y FA(e)439 4542 y FD(T)505 4512 y FB(nc)484 4566 y(k)q(;)p Fu(\027)587 4542 y Fv(:)38 b(set)30 b(of)g(tr)l(e)l(es)f(in)1186 4521 y FA(e)1173 4542 y FD(T)1218 4554 y FB(k)q(;)p Fu(\027)1351 4542 y Fv(such)g(that)h(the)g(amputate)l(d)g(line)g(is)g(not)g(c)l(onne)l (cte)l(d)p 2498 4555 493 4 v 29 w(to)g(the)g(r)l(o)l(ot)g(line.)231 4750 y(We)e(c)l(al)t(l)i(a)e(tr)l(e)l(e)g(in)873 4729 y FA(e)861 4750 y FD(T)927 4720 y FB(c)906 4774 y(k)q(;)p Fu(\027)1037 4750 y Fv(as)g(a)g FF(c)p FG(-class)f Fv(tr)l(e)l(e)h(and) g(a)h(tr)l(e)l(e)e(in)2130 4729 y FA(e)2118 4750 y FD(T)2184 4720 y FB(nc)2163 4774 y(k)q(;)p Fu(\027)2293 4750 y Fv(as)i(a)f FF(nc)p FG(-class)f Fv(tr)l(e)l(e.)37 b(Note)28 b(any)h FF(nc)p Fv(-class)f(tr)l(e)l(e)g(has)231 4850 y(or)l(der)j FF(k)26 b FD(\025)c FG(2)p Fv(.)356 5050 y FG(An)n(y)29 b(tree)h(in)815 5030 y FA(e)802 5050 y FD(T)869 5020 y FB(nc)847 5074 y(k)q(;)p Fu(\027)980 5050 y FG(can)f(b)r(e)h(transformed)e(to)i(b)r(e)g(dra)n(wn)e(in)i(its) g(\\canonical)d(form")i(as)g(depicted)h(in)g(Figure)f(1.)231 5175 y(Indeed,)c(let)f FF(\022)h FD(2)792 5154 y FA(e)780 5175 y FD(T)846 5144 y FB(nc)825 5198 y(k)q(;)p Fu(\027)927 5175 y FG(,)g FF(k)h FD(\025)c FG(2,)i Fy(\027)29 b FD(2)23 b FC(Z)1437 5144 y FB(d)1476 5175 y FG(,)i(b)r(e)f(arbitrary)-7 b(.)33 b(Let)24 b FF(v)2198 5144 y Fz(0)2195 5195 y FE(1)2256 5175 y FG(b)r(e)g(the)g(v)n(ertex)f(connected)g(to)h(the)f(amputated)h (line)231 5274 y(of)31 b FF(\022)r FG(.)47 b(De\014ne)32 b FF(v)744 5244 y Fz(0)741 5295 y FE(2)809 5274 y FG(as)f(the)g(v)n (ertex)f(suc)n(h)g(that)i(one)e(of)h(its)g(en)n(tering)f(lines)h(is)g (exactly)f(the)i(line)f(exiting)f FF(v)3582 5244 y Fz(0)3579 5295 y FE(1)3617 5274 y FG(.)47 b(De\014ne)2032 5532 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 231 169 a FF(v)274 139 y Fz(0)271 190 y FB(j)337 169 y FG(inductiv)n(ely)29 b(as)h(the)g(v)n(ertex)f(suc) n(h)h(that)g(one)f(of)h(its)g(en)n(tering)f(lines)h(is)g(exactly)f(the) i(line)f(exiting)g FF(v)3524 139 y Fz(0)3521 190 y FB(j)s Fz(\000)p FE(1)3641 169 y FG(.)44 b(If,)31 b(for)231 268 y(some)c FF(j)h FD(\025)22 b FG(2,)27 b FF(v)723 238 y Fz(0)720 290 y FB(j)782 268 y FG(is)g(the)h(v)n(ertex)d (connected)i(to)g(the)g(ro)r(ot)g(line,)g(then)h(w)n(e)e(set)h FF(n)c FG(:=)g FF(j)5 b FG(.)36 b(No)n(w)27 b(relab)r(el)f(the)i FF(n)e FG(v)n(ertices)231 368 y(de\014ned)33 b(ab)r(o)n(v)n(e)e(as)g (follo)n(ws:)45 b FF(v)1222 380 y FB(j)1288 368 y FG(=)30 b FF(v)1426 338 y Fz(0)1423 390 y FB(n)p Fz(\000)p FB(j)s FE(+1)1635 368 y FG(,)k(1)c FD(\024)g FF(j)36 b FD(\024)30 b FF(n)p FG(.)50 b(The)32 b(v)n(ertices)g FF(v)2668 380 y FB(j)2735 368 y FG(will)g(b)r(e)h(called)f Fv(c)l(anonic)l(al)j (vertic)l(es)p FG(.)231 468 y(Set)28 b FF(\022)413 480 y FB(j)448 468 y FG(,)f(1)c FD(\024)g FF(j)28 b FD(\024)22 b FF(n)c FD(\000)f FG(1,)27 b(as)g(the)g(subtree)g(whose)g(ro)r(ot)f (line)i(is)f(the)h(one)f(en)n(tering)f FF(v)2894 480 y FB(j)2957 468 y FG(and)h(not)g(exiting)g FF(v)3579 480 y FB(j)s FE(+1)3698 468 y FG(;)h FF(\022)3788 480 y FB(n)3861 468 y FG(is)231 567 y(de\014ned)j(as)f(the)h(subtree)f (whose)g(ro)r(ot)g(line)g(en)n(ters)g FF(v)1939 579 y FB(n)1984 567 y FG(,)i(not)e(b)r(eing)h(the)g(amputated)f(line.)46 b(The)31 b(subtrees)f FF(\022)3722 579 y FB(j)3787 567 y FG(will)231 667 y(b)r(e)c(called)g Fv(c)l(anonic)l(al)j(subtr)l(e)l (es)p FG(.)36 b(No)n(w)25 b(dra)n(w)g(the)h(tree)f(in)h(suc)n(h)g(a)f (w)n(a)n(y)f(that)j(the)f(ro)r(ot)f(line)h(of)f(eac)n(h)g FF(\022)3457 679 y FB(j)3492 667 y FG(,)i(1)22 b FD(\024)h FF(j)28 b FD(\024)22 b FF(n)p FG(,)231 780 y(is)28 b(the)h(upp)r(er)g (line)f(en)n(tering)g(the)g(v)n(ertex)g FF(v)1613 792 y FB(j)1648 780 y FG(:)38 b(in)29 b(this)f(w)n(a)n(y)f FF(\022)g FD(2)2296 759 y FA(e)2283 780 y FD(T)2350 750 y FB(nc)2328 803 y(k)q(;)p Fu(\027)2459 780 y FG(is)i(as)e(represen)n (ted)g(in)i(Figure)f(1.)38 b(F)-7 b(rom)28 b(no)n(w)231 904 y(on,)g(an)n(y)e(tree)i(in)802 883 y FA(e)790 904 y FD(T)856 874 y FB(nc)835 927 y(k)q(;)p Fu(\027)965 904 y FG(is)f(though)n(t)h(of)f(as)g(b)r(eing)h(dra)n(wn)f(in)g(its)h (\\canonical)e(form".)429 2260 y @beginspecial 0 @llx 0 @lly 190 @urx 155 @ury 1805 @rwi @setspecial %%BeginDocument: eps/canonical_form.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: canonical_form.eps %%Creator: fig2dev Version 3.2 Patchlevel 3d %%CreationDate: Wed Jan 14 16:02:26 2004 %%For: dacortez@fma (Daniel Augusto Cortez) %%BoundingBox: 0 0 190 155 %%Magnification: 1.0000 %%EndComments /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 155 moveto 0 0 lineto 190 0 lineto 190 155 lineto closepath clip newpath -442.1 398.5 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index oldshow % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manual % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P1 exch def /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 10 setmiterlimit 0 slj 0 slc 0.06299 0.06299 sc % % Fig objects follow % /Symbol ff 225.00 scf sf 7360 4923 m gs 1 -1 sc (n) col0 sh gr 7.500 slw % Ellipse n 8447 4091 127 127 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 554.67 264.27] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 8997 4634 127 127 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 591.33 300.47] PATmp PATsp ef gr PATusp gs col0 s gr [60] 0 sd % Rotated Ellipse gs 8242 4288 tr -225.001 rot n 0 0 500 293 0 360 DrawEllipse 225.001 rot gs col0 s gr gr [] 0 sd [60] 0 sd % Rotated Ellipse gs 8792 4846 tr -225.001 rot n 0 0 500 293 0 360 DrawEllipse 225.001 rot gs col0 s gr gr [] 0 sd % Ellipse n 9595 5219 127 127 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 631.20 339.47] PATmp PATsp ef gr PATusp gs col0 s gr [60] 0 sd % Rotated Ellipse gs 9393 5427 tr -225.001 rot n 0 0 500 293 0 360 DrawEllipse 225.001 rot gs col0 s gr gr [] 0 sd % Polyline gs clippath 8051 4920 m 8009 4962 l 8115 5069 l 8052 4963 l 8158 5026 l cp eoclip n 8041 4952 m 8266 5177 l gs col0 s gr gr % arrowhead n 8158 5026 m 8052 4963 l 8115 5069 l 8158 5026 l cp gs 0.00 setgray ef gr col0 s % Polyline n 8041 4952 m 7816 4727 l gs col0 s gr % Polyline n 8266 5177 m 8356 5267 l gs col0 s gr % Polyline n 8552 5464 m 8327 5239 l gs col0 s gr % Polyline gs clippath 8558 5028 m 8600 5070 l 8707 4964 l 8601 5028 l 8664 4921 l cp eoclip n 8590 5039 m 8815 4814 l gs col0 s gr gr % arrowhead n 8664 4921 m 8601 5028 l 8707 4964 l 8664 4921 l cp gs 0.00 setgray ef gr col0 s % Polyline n 8815 4814 m 8905 4724 l gs col0 s gr % Polyline n 8590 5039 m 8365 5264 l gs col0 s gr % Polyline gs clippath 8009 4488 m 8051 4530 l 8158 4424 l 8052 4488 l 8115 4381 l cp eoclip n 8041 4499 m 8266 4274 l gs col0 s gr gr % arrowhead n 8115 4381 m 8052 4488 l 8158 4424 l 8115 4381 l cp gs 0.00 setgray ef gr col0 s % Polyline n 8266 4274 m 8356 4184 l gs col0 s gr % Polyline n 8041 4499 m 7816 4724 l gs col0 s gr % Polyline gs clippath 7308 4695 m 7308 4755 l 7459 4755 l 7339 4725 l 7459 4695 l cp eoclip n 7830 4725 m 7323 4725 l gs col0 s gr gr % arrowhead n 7459 4695 m 7339 4725 l 7459 4755 l 7459 4695 l cp gs 0.00 setgray ef gr col0 s % Polyline n 7339 4725 m 7031 4725 l gs col0 s gr % Polyline gs clippath 9156 5613 m 9198 5655 l 9305 5549 l 9199 5613 l 9262 5506 l cp eoclip n 9188 5624 m 9413 5399 l gs col0 s gr gr % arrowhead n 9262 5506 m 9199 5613 l 9305 5549 l 9262 5506 l cp gs 0.00 setgray ef gr col0 s % Polyline n 9413 5399 m 9503 5309 l gs col0 s gr % Polyline n 9188 5624 m 8963 5849 l gs col0 s gr % Polyline n 9021 5910 m 8796 5685 l gs col0 s gr % Polyline n 9136 6025 m 8911 5800 l gs col0 s gr % Polyline gs clippath 9128 5975 m 9086 6017 l 9192 6124 l 9129 6018 l 9235 6081 l cp eoclip n 9118 6007 m 9343 6232 l gs col0 s gr gr % arrowhead n 9235 6081 m 9129 6018 l 9192 6124 l 9235 6081 l cp gs 0.00 setgray ef gr col0 s % Polyline n 9335 6224 m 9425 6314 l gs col0 s gr /Times-Roman ff 240.00 scf sf 8597 5516 m gs 1 -1 sc 315.0 rot (...) col0 sh gr /NewCenturySchlbk-Italic ff 225.00 scf sf 7716 4921 m gs 1 -1 sc (v) col0 sh gr /NewCenturySchlbk-Italic ff 120.00 scf sf 7799 4970 m gs 1 -1 sc (1) col0 sh gr /NewCenturySchlbk-Italic ff 225.00 scf sf 8227 5404 m gs 1 -1 sc (v) col0 sh gr /NewCenturySchlbk-Italic ff 120.00 scf sf 8302 5453 m gs 1 -1 sc (2) col0 sh gr /NewCenturySchlbk-Italic ff 225.00 scf sf 8801 5977 m gs 1 -1 sc (v) col0 sh gr /NewCenturySchlbk-Italic ff 120.00 scf sf 8883 6027 m gs 1 -1 sc (n) col0 sh gr /NewCenturySchlbk-Bold ff 210.00 scf sf 9300 6129 m gs 1 -1 sc (0) col0 sh gr /Symbol ff 225.00 scf sf 8697 4044 m gs 1 -1 sc (q) col0 sh gr /NewCenturySchlbk-Italic ff 120.00 scf sf 8814 4074 m gs 1 -1 sc (1) col0 sh gr /Symbol ff 225.00 scf sf 9240 4603 m gs 1 -1 sc (q) col0 sh gr /NewCenturySchlbk-Italic ff 120.00 scf sf 9369 4630 m gs 1 -1 sc (2) col0 sh gr /Symbol ff 225.00 scf sf 9829 5175 m gs 1 -1 sc (q) col0 sh gr /NewCenturySchlbk-Italic ff 120.00 scf sf 9958 5197 m gs 1 -1 sc (n) col0 sh gr $F2psEnd rs end %%EndDocument @endspecial 2137 1306 a FN(Figure)31 b(1:)46 b(Canonical)32 b(form)g(of)f(a)h(tree)f(in)3466 1287 y Fj(e)3454 1306 y FJ(T)3516 1275 y Fi(nc)3496 1322 y(k)q(;)p Fh(\027)3592 1306 y FN(.)50 b(The)2137 1398 y(dashed)18 b(bullet)f(represen)n(ts)h (a)h(general)g(subtree)e(con)n(tain-)2137 1489 y(ing)25 b(only)g(blac)n(k)g(bullets.)35 b(Eac)n(h)25 b(canonical)i(subtree)d FL(\022)3741 1497 y Fi(j)3774 1489 y FN(,)2137 1580 y(1)34 b FJ(\024)f FL(j)38 b FJ(\024)33 b FL(n)p FN(,)i(is)f(of)g(order)f FL(k)2994 1589 y Fi(\022)3023 1600 y Fs(j)3091 1580 y FN(and)g(con)n(tains)g(exactly)2137 1672 y FL(k)2177 1681 y Fi(\022)2206 1692 y Fs(j)2265 1672 y FN(v)n(ertices)23 b(and)g FL(k)2727 1681 y Fi(\022)2756 1692 y Fs(j)2803 1672 y FN(+)12 b(1)24 b(blac)n(k)f(bullets.)34 b(T)-6 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2476 y FE(\))3302 2545 y Fu(\027)3343 2553 y Fs(j)3489 2535 y Fv(if)h FF(k)3616 2547 y FB(\022)3648 2555 y Fs(j)3716 2535 y Fv(is)f(the)231 2634 y(or)l(der)28 b(of)g FF(\022)581 2646 y FB(j)644 2634 y Fv(and)f(if)h Fy(\027)934 2646 y FB(j)996 2634 y Fv(is)g(the)f(momentum)f(\015owing)i(thr)l(ough)g (its)f(r)l(o)l(ot)g(line.)39 b(Of)27 b(c)l(ourse)g Fy(\027)3128 2646 y FB(j)3186 2634 y FD(6)p FG(=)c Fw(0)j Fv(sinc)l(e)i(this)g (would)231 2734 y(give)j(a)f(c)l(ontribution)g(to)g FF(\013)1092 2704 y FE(\()p FB(k)1153 2712 y Fs(j)1184 2704 y FE(\))1244 2734 y Fv(and)g(white)h(bul)t(lets)f(ar)l(e)g(disc)l(ar)l(de)l(d)h (along)g(the)f(c)l(onstruction.)38 b(Mor)l(e)30 b(gener)l(al)t(ly,)i (as)231 2834 y(e)l(ach)i(line)e(in)h FF(L)p FG(\()p FF(\022)r FG(\))p Fv(,)h(which)g(is)e(neither)h(the)g(r)l(o)l(ot)f(line)h(nor)f (the)g(line)h(exiting)g(fr)l(om)f(the)h(amputate)l(d)g(white)g(bul)t (let,)231 2933 y(c)l(an)e(b)l(e)g(se)l(en)g(as)g(the)g(r)l(o)l(ot)g (line)h(of)g(a)f(subtr)l(e)l(e,)g(then)g(it)g(must)f(have)i(a)f (momentum)f(di\013er)l(ent)h(fr)l(om)h(zer)l(o,)g(as)f(ther)l(e)231 3033 y(ar)l(e)f(no)g(other)h(white)f(bul)t(lets.)231 3220 y Fw(Remark)i(3.8)41 b Fv(Note)26 b(that)g(ther)l(e)h(ar)l(e)g FG(2)1504 3190 y FB(n)1575 3220 y Fv(ine)l(quivalent)g(tr)l(e)l(es)f (in)2332 3199 y FA(e)2319 3220 y FD(T)2386 3190 y FB(nc)2364 3244 y(k)q(;)p Fu(\027)2493 3220 y Fv(admitting)h(the)g(same)g(c)l (anonic)l(al)g(form)h(with)231 3320 y FF(n)i Fv(c)l(anonic)l(al)h (subtr)l(e)l(es.)356 3507 y FG(One)c(no)n(w)g(writes)g(the)h(v)-5 b(alue)28 b(of)f(a)g(canonical)g(subtree)g FF(\022)2165 3519 y FB(j)2228 3507 y FG(as)1175 3755 y(V)-7 b(al\()p FF(\022)1366 3767 y FB(j)1402 3755 y FG(\))23 b(=)1613 3695 y FF(b)1649 3652 y FE(\()p FB(j)s FE(\))1649 3706 y Fu(\027)1691 3714 y Fs(j)p 1555 3735 240 4 v 1555 3811 a FF(i)p Fy(!)d FD(\001)f Fy(\027)1759 3823 y FB(j)1827 3755 y FD(\021)k FF(B)1982 3720 y FE(\()p FB(j)s FE(\))1978 3775 y Fu(\027)2020 3783 y Fs(j)2083 3755 y FF(;)97 b Fy(\027)2256 3767 y FB(j)2314 3755 y FD(6)p FG(=)23 b Fw(0)14 b FF(;)97 b FG(1)22 b FD(\024)h FF(j)28 b FD(\024)22 b FF(n)14 b(:)231 3971 y FG(Therefore,)27 b FF(\022)670 3983 y FB(j)732 3971 y FG(giv)n(es)g(a)g(con)n(tribution)g(to)g(the)h (function)1592 4201 y FF(B)1655 4213 y FB(j)1690 4201 y FG(\()p FF(t)p FG(\))c(=)1895 4088 y FA(Z)1978 4108 y FB(t)1941 4276 y FE(0)2021 4201 y FG(d)p FF(t)2097 4166 y Fz(0)2134 4201 y FF(b)2170 4213 y FB(j)2205 4201 y FG(\()p FF(t)2267 4166 y Fz(0)2291 4201 y FG(\))18 b(+)g FF(C)2483 4213 y FB(j)2533 4201 y FF(;)1190 b FG(\(3.1\))231 4412 y(where)29 b(the)g(in)n(tegration)f(constan)n(t)h FF(C)1437 4424 y FB(j)1501 4412 y FG(is)g(c)n(hosen)f(in)i(suc)n(h)e(a) h(w)n(a)n(y)f(that)h(summed)h(to)f(the)g(constan)n(t)g(term)g(arising) 231 4512 y(from)f(the)g(de\014nite)g(in)n(tegral)e(giv)n(es)g(the)i (zero)f(F)-7 b(ourier)27 b(mo)r(de)g(of)h FF(B)2359 4524 y FB(j)2394 4512 y FG(:)1622 4756 y FF(C)1681 4768 y FB(j)1735 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b(the)i(momen)n(tum)f(\015o)n(wing)231 1640 y(through)c(the)h(ro)r(ot)f(line)h(is)f(zero,)g(whic)n(h)g(means)g (that)1716 1766 y FB(n)1677 1791 y FA(X)1679 1968 y FB(j)s FE(=1)1810 1870 y Fy(\027)1864 1882 y FB(j)1917 1870 y FG(+)2040 1766 y FB(n)2000 1791 y FA(X)2003 1968 y FB(j)s FE(=1)2134 1870 y Fy(\027)2188 1882 y FE(0)p FB(;j)2299 1870 y FG(=)22 b Fw(0)14 b FF(:)231 2122 y FG(Therefore,)27 b(b)n(y)g(an)g(explicit)h(computation,)745 2291 y FA(X)760 2469 y Fz(N)814 2477 y Fk(0)878 2370 y FG(V)-7 b(al\()p FF(\022)r FG(\))84 b(=)1373 2266 y FB(n)1340 2291 y FA(Y)1335 2467 y FB(r)r FE(=1)1465 2291 y FA(X)1487 2465 y Fu(\027)1529 2473 y Fs(r)1600 2291 y FA(X)1599 2465 y Fu(\027)1641 2473 y Fx(0)p Fs(;r)2300 2239 y FA(Q)2379 2259 y FB(n)2379 2326 y(j)s FE(=1)2512 2301 y FF(Q)2578 2313 y Fu(\027)2619 2322 y Fx(0)p Fs(;j)2716 2301 y FF(b)2752 2258 y FE(\()p FB(j)s FE(\))2752 2312 y Fu(\027)2793 2320 y Fs(j)p 1745 2351 1648 4 v 1745 2378 a FA(Q)1824 2399 y FB(n)1824 2466 y(j)s FE(=1)1943 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4231 8134 l 4294 8240 l 4337 8197 l cp gs 0.00 setgray ef gr col0 s % Polyline n 4445 8348 m 4535 8438 l gs col0 s gr % Polyline n 3370 7275 m 3460 7365 l gs col0 s gr % Polyline gs clippath 3155 7018 m 3113 7060 l 3219 7167 l 3156 7061 l 3262 7124 l cp eoclip n 3145 7050 m 3370 7275 l gs col0 s gr gr % arrowhead n 3262 7124 m 3156 7061 l 3219 7167 l 3262 7124 l cp gs 0.00 setgray ef gr col0 s % Polyline n 3145 7050 m 2920 6825 l gs col0 s gr % Polyline gs clippath 3653 7121 m 3695 7163 l 3802 7057 l 3696 7121 l 3759 7014 l cp eoclip n 3685 7132 m 3910 6907 l gs col0 s gr gr % arrowhead n 3759 7014 m 3696 7121 l 3802 7057 l 3759 7014 l cp gs 0.00 setgray ef gr col0 s % Polyline n 3685 7132 m 3460 7357 l gs col0 s gr % Polyline n 3910 6907 m 4000 6817 l gs col0 s gr % Polyline gs clippath 4186 7664 m 4228 7706 l 4335 7600 l 4229 7664 l 4292 7557 l cp eoclip n 4218 7675 m 4443 7450 l gs col0 s gr gr % arrowhead n 4292 7557 m 4229 7664 l 4335 7600 l 4292 7557 l cp gs 0.00 setgray ef gr col0 s % Polyline n 4218 7675 m 3993 7900 l gs col0 s gr % Polyline n 4443 7450 m 4533 7360 l gs col0 s gr % Polyline n 3910 7815 m 4000 7905 l gs col0 s gr % Polyline gs clippath 3695 7558 m 3653 7600 l 3759 7707 l 3696 7601 l 3802 7664 l cp eoclip n 3685 7590 m 3910 7815 l gs col0 s gr gr % arrowhead n 3802 7664 m 3696 7601 l 3759 7707 l 3802 7664 l cp gs 0.00 setgray ef gr col0 s % Polyline n 3685 7590 m 3460 7365 l gs col0 s gr % Polyline n 4220 8123 m 3995 7898 l gs col0 s gr % Polyline gs clippath 4727 7124 m 4769 7166 l 4876 7060 l 4770 7124 l 4833 7017 l cp eoclip n 4759 7135 m 4984 6910 l gs col0 s gr gr % arrowhead n 4833 7017 m 4770 7124 l 4876 7060 l 4833 7017 l cp gs 0.00 setgray ef gr col0 s % Polyline n 4759 7135 m 4534 7360 l gs col0 s gr % Polyline n 4984 6910 m 5074 6820 l gs col0 s gr % Polyline gs clippath 5262 7651 m 5304 7693 l 5411 7587 l 5305 7651 l 5368 7544 l cp eoclip n 5294 7662 m 5519 7437 l gs col0 s gr gr % arrowhead n 5368 7544 m 5305 7651 l 5411 7587 l 5368 7544 l cp gs 0.00 setgray ef gr col0 s % Polyline n 5294 7662 m 5069 7887 l gs col0 s gr % Polyline n 5519 7437 m 5609 7347 l gs col0 s gr % Polyline gs clippath 5817 8206 m 5859 8248 l 5966 8142 l 5860 8206 l 5923 8099 l cp eoclip n 5849 8217 m 6074 7992 l gs col0 s gr gr % arrowhead n 5923 8099 m 5860 8206 l 5966 8142 l 5923 8099 l cp gs 0.00 setgray ef gr col0 s % Polyline n 5849 8217 m 5624 8442 l gs col0 s gr % Polyline n 6074 7992 m 6164 7902 l gs col0 s gr % Polyline n 4981 8889 m 5071 8979 l gs col0 s gr % Polyline gs clippath 4767 8632 m 4725 8674 l 4831 8781 l 4768 8675 l 4873 8738 l cp eoclip n 4757 8664 m 4981 8889 l gs col0 s gr gr % arrowhead n 4873 8738 m 4768 8675 l 4831 8781 l 4873 8738 l cp gs 0.00 setgray ef gr col0 s % Polyline n 4757 8664 m 4532 8439 l gs col0 s gr % Polyline gs clippath 4870 9214 m 4828 9172 l 4721 9278 l 4828 9215 l 4764 9321 l cp eoclip n 4839 9204 m 4614 9429 l gs col0 s gr gr % arrowhead n 4764 9321 m 4828 9215 l 4721 9278 l 4764 9321 l cp gs 0.00 setgray ef gr col0 s % Polyline n 4839 9204 m 5063 8980 l gs col0 s gr % Polyline n 4614 9429 m 4523 9520 l gs col0 s gr % Polyline n 5522 9429 m 5611 9519 l gs col0 s gr % Polyline gs clippath 5306 9172 m 5264 9214 l 5370 9320 l 5307 9215 l 5413 9278 l cp eoclip n 5296 9204 m 5522 9429 l gs col0 s gr gr % arrowhead n 5413 9278 m 5307 9215 l 5370 9320 l 5413 9278 l cp gs 0.00 setgray ef gr col0 s % Polyline n 5296 9204 m 5071 8979 l gs col0 s gr % Polyline gs clippath 5845 9711 m 5803 9753 l 5909 9860 l 5846 9754 l 5952 9817 l cp eoclip n 5835 9743 m 6060 9968 l gs col0 s gr gr % arrowhead n 5952 9817 m 5846 9754 l 5909 9860 l 5952 9817 l cp gs 0.00 setgray ef gr col0 s % Polyline n 5835 9743 m 5610 9518 l gs col0 s gr % Polyline n 6060 9968 m 6150 10058 l gs col0 s gr % Polyline gs clippath 5412 9747 m 5370 9705 l 5263 9811 l 5370 9748 l 5306 9854 l cp eoclip n 5381 9737 m 5156 9962 l gs col0 s gr gr % arrowhead n 5306 9854 m 5370 9748 l 5263 9811 l 5306 9854 l cp gs 0.00 setgray ef gr col0 s % Polyline n 5381 9737 m 5606 9512 l gs col0 s gr % Polyline n 5156 9962 m 5066 10052 l gs col0 s gr % Polyline gs clippath 4339 8679 m 4297 8637 l 4190 8743 l 4297 8680 l 4233 8786 l cp eoclip n 4308 8669 m 4083 8894 l gs col0 s gr gr % arrowhead n 4233 8786 m 4297 8680 l 4190 8743 l 4233 8786 l cp gs 0.00 setgray ef gr col0 s % Polyline n 4308 8669 m 4533 8444 l gs col0 s gr % Polyline n 4083 8894 m 3993 8984 l gs col0 s gr % Polyline n 3353 9441 m 3443 9531 l gs col0 s gr % Polyline n 2453 9452 m 2363 9541 l gs col0 s gr % Polyline n 2812 9986 m 2902 10076 l gs col0 s gr % Polyline gs clippath 2598 9729 m 2556 9771 l 2662 9878 l 2599 9772 l 2704 9835 l cp eoclip n 2588 9761 m 2812 9986 l gs col0 s gr gr % arrowhead n 2704 9835 m 2599 9772 l 2662 9878 l 2704 9835 l cp gs 0.00 setgray ef gr col0 s % Polyline n 2588 9761 m 2363 9536 l gs col0 s gr % Polyline gs clippath 2701 10311 m 2659 10269 l 2552 10375 l 2659 10312 l 2595 10418 l cp eoclip n 2670 10301 m 2445 10526 l gs col0 s gr gr % arrowhead n 2595 10418 m 2659 10312 l 2552 10375 l 2595 10418 l cp gs 0.00 setgray ef gr col0 s % Polyline n 2670 10301 m 2894 10077 l gs col0 s gr % Polyline n 2445 10526 m 2354 10617 l gs col0 s gr % Polyline n 3353 10526 m 3442 10616 l gs col0 s gr % Polyline gs clippath 3137 10269 m 3095 10311 l 3201 10417 l 3138 10312 l 3244 10375 l cp eoclip n 3127 10301 m 3353 10526 l gs col0 s gr gr % arrowhead n 3244 10375 m 3138 10312 l 3201 10417 l 3244 10375 l cp gs 0.00 setgray ef gr col0 s % Polyline n 3127 10301 m 2902 10076 l gs col0 s gr % Polyline gs clippath 3676 10808 m 3634 10850 l 3740 10957 l 3677 10851 l 3783 10914 l cp eoclip n 3666 10840 m 3891 11065 l gs col0 s gr gr % arrowhead n 3783 10914 m 3677 10851 l 3740 10957 l 3783 10914 l cp gs 0.00 setgray ef gr col0 s % Polyline n 3666 10840 m 3441 10615 l gs col0 s gr % Polyline n 3891 11065 m 3981 11155 l gs col0 s gr % Polyline gs clippath 3243 10844 m 3201 10802 l 3094 10908 l 3201 10845 l 3137 10951 l cp eoclip n 3212 10834 m 2987 11059 l gs col0 s gr gr % arrowhead n 3137 10951 m 3201 10845 l 3094 10908 l 3137 10951 l cp gs 0.00 setgray ef gr col0 s % Polyline n 3212 10834 m 3437 10609 l gs col0 s gr % Polyline n 2987 11059 m 2897 11149 l gs col0 s gr % Polyline gs clippath 2170 9776 m 2128 9734 l 2021 9840 l 2128 9777 l 2064 9883 l cp eoclip n 2139 9766 m 1914 9991 l gs col0 s gr gr % arrowhead n 2064 9883 m 2128 9777 l 2021 9840 l 2064 9883 l cp gs 0.00 setgray ef gr col0 s % Polyline n 2139 9766 m 2364 9541 l gs col0 s gr % Polyline n 1914 9991 m 1824 10081 l gs col0 s gr % Polyline 2 slj [60] 0 sd n 2487 8319 m 2486 8319 l 2482 8320 l 2476 8321 l 2467 8322 l 2454 8324 l 2438 8327 l 2418 8330 l 2396 8334 l 2370 8339 l 2343 8344 l 2313 8350 l 2283 8357 l 2251 8364 l 2218 8372 l 2183 8381 l 2147 8392 l 2110 8403 l 2071 8416 l 2030 8430 l 1987 8447 l 1942 8465 l 1895 8486 l 1848 8508 l 1802 8531 l 1758 8554 l 1719 8575 l 1684 8594 l 1653 8610 l 1627 8623 l 1605 8634 l 1587 8642 l 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sc (2) col0 sh gr /NewCenturySchlbk-Italic ff 210.00 scf sf 4549 6917 m gs 1 -1 sc (T) col0 sh gr /NewCenturySchlbk-Italic ff 210.00 scf sf 5333 7634 m gs 1 -1 sc (T) col0 sh gr /NewCenturySchlbk-Italic ff 150.00 scf sf 5449 7695 m gs 1 -1 sc (3) col0 sh gr /NewCenturySchlbk-Italic ff 210.00 scf sf 5169 8703 m gs 1 -1 sc (T) col0 sh gr /NewCenturySchlbk-Italic ff 150.00 scf sf 5273 8782 m gs 1 -1 sc (4) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 2125 2102 a(Figure)23 b(4:)33 b(Examples)23 b(of)g(self-energy)g(graphs)g(in)f(a)h(tree)2125 2194 y(of)j(order)f FL(k)e FN(=)e(7.)35 b(Note)25 b(that,)g(in)g(accordance) h(with)f(the)2125 2285 y(de\014nition,)i(there)f(is)h(only)f(one)h(en)n (tering)f(line)h(and)f(one)2125 2376 y(exiting)c(line)h(\(carrying)f (the)g(same)g(momen)n(tum\))g(in)g(the)2125 2468 y(self-energy)33 b(graphs)f FL(T)2811 2476 y FK(1)2845 2468 y FN(,)i FL(T)2945 2476 y FK(2)2979 2468 y FN(,)f FL(T)3078 2476 y FK(3)3144 2468 y FN(and)f FL(T)3345 2476 y FK(4)3379 2468 y FN(.)53 b(It)31 b(is)h(clear)2125 2559 y(that)e(the)g(scales)j(on)d(the)g (lines)h(of)h FL(T)3219 2567 y FK(1)3253 2559 y FN(,)g FL(T)3351 2567 y FK(2)3386 2559 y FN(,)g FL(T)3484 2567 y FK(3)3518 2559 y FN(,)g FL(T)3616 2567 y FK(4)3681 2559 y FN(are)2125 2650 y(strictly)25 b(less)i(then)d(the)h(scales)h (on)f(their)g(external)h(lines)2125 2742 y(\(after)g(all,)h (self-energy)g(graphs)f(are)g(clusters\).)231 3161 y FG(is)h(also)e(on)h(scale)g FF(n)841 3173 y FB(`)869 3157 y Fp(0)919 3161 y FD(\035)d FG(1.)36 b(This)26 b(pro)r(cess)g (could)g(rep)r(eat)g(itself)h(sev)n(eral)e(times,)i(resulting)f(at)h (the)f(end)h(in)g(a)f(bunc)n(h)231 3261 y(of)i(lines)f FF(`)550 3273 y FE(1)587 3261 y FF(;)14 b(:)g(:)g(:)g(;)g(`)807 3273 y FB(N)897 3261 y FG(on)27 b(scales)g FF(n)1294 3273 y FB(`)1322 3281 y Fs(i)1375 3261 y FD(\035)c FG(1,)k(i.e.)37 b(w)n(e)27 b(end)h(up)g(with)g(an)g(accum)n(ulation)e(of)i(small)f (divisors)3518 3231 y FE(3)3554 3261 y FG(.)356 3378 y(Actually)-7 b(,)27 b(from)f(a)h(more)f(precise)g(p)r(oin)n(t)h(of)g (view,)f(the)h(whole)g(problem)f(with)h(the)h(self-energy)d(graphs)g (is)i(that)231 3478 y(w)n(e)i(are)g(not)h(able)f(to)g(giv)n(e)g(a)g (satisfactory)f(b)r(ound)i(on)f(the)h(n)n(um)n(b)r(er)f(of)h (self-energy)e(lines)i(in)f(a)h(giv)n(en)e(tree)h FF(\022)r FG(.)44 b(On)231 3578 y(the)32 b(other)e(hand,)i(it)g(is)f(easy)f(to)h (sho)n(w)f(that)i(\(see,)g(e.g.,)f(Ref.)h([12)o(]\))g(if)f(w)n(e)g (denote)g(b)n(y)g FF(N)3106 3548 y FE(norm)3097 3598 y FB(n)3260 3578 y FG(\()p FF(\022)r FG(\))h(the)g(n)n(um)n(b)r(er)e (of)231 3677 y(normal)d(lines)g(in)h(a)f(tree)h FF(\022)r FG(,)g(then)g(there)f(exists)g(a)h(p)r(ositiv)n(e)f(constan)n(t)g FF(c)g FG(suc)n(h)h(that)1510 3853 y FF(N)1586 3818 y FE(norm)1577 3873 y FB(n)1740 3853 y FG(\()p FF(\022)r FG(\))c FD(\024)e FF(c)14 b FG(2)2048 3818 y Fz(\000)p FB(n=\034)2279 3774 y FA(X)2230 3956 y FB(v)r Fz(2)p FB(B)s FE(\()p FB(\022)r FE(\))2462 3853 y FD(j)p Fy(\027)2539 3865 y FB(v)2578 3853 y 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Fz(R)3435 1731 y FB(k)q(;n)3532 1739 y Fx(0)3598 1708 y FG(eac)n(h)i(line)231 1807 y FF(`)23 b FD(2)g FF(L)p FG(\()p FF(T)12 b FG(\))25 b(has)g(scale)f FF(n)966 1819 y FB(`)1021 1807 y FG(=)f FF(n)1159 1819 y FE(0)1196 1807 y FG(\).)36 b(Nev)n(ertheless)25 b(the)g(corresp)r(onding)f (propagators)e(do)j(not)h(dep)r(end)g(on)f(the)g(scales.)231 1907 y(Hence)1573 1928 y FA(X)1520 2115 y FB(T)9 b Fz(2S)1658 2095 y Fp(R)1654 2135 y Fs(k)q(;n)p 1760 1940 58 4 v 1760 2007 a FD(V)1817 2019 y FB(T)1870 2007 y FG(\(0;)14 b FF(")p FG(\))22 b(=)2203 1928 y FA(X)2162 2123 y FB(\022)r Fz(2)2250 2108 y Fl(e)2241 2123 y Fz(T)2279 2132 y Fs(k)q(;)p Fk(0)2378 2007 y FG(V)-7 b(al\()p FF(\022)r FG(\))p FF(;)231 2260 y FG(so)27 b(that)h(the)g(assertion)e(follo)n(ws)h(from)g(the)h (de\014nition)g(of)g FF(G)2144 2272 y FB(k)2212 2260 y FG(\(see)g(Lemma)f(3.5\).)p 3869 2260 48 48 v 231 2440 a Fw(Lemma)32 b(6.2)40 b Fv(F)-6 b(or)30 b(any)g(self-ener)l(gy)h FF(T)41 b Fv(one)30 b(has)1280 2615 y FG(1)18 b 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b FF(e)2700 3877 y Fz(\000)p FB(B)2802 3885 y Fx(2)2834 3877 y FE(2)2867 3852 y Fs(n)2904 3864 y Fx(0)2936 3852 y Fs(=\034)2995 3864 y Fx(1)3036 3911 y FF(;)1075 3999 y FA(\014)1075 4049 y(\014)1075 4099 y(\014)1103 4095 y FF(@)1147 4107 y FB(")1196 4002 y FA(\020)1246 4095 y FD(M)1346 4051 y FE([)p FB(n)1406 4059 y Fx(0)1438 4051 y FE(])1346 4118 y FB(j)1373 4126 y Fx(0)1461 4095 y FG(\(0;)g FF(")p FG(\))k FD(\000)p 1744 4028 V 18 w(M)1843 4042 y FE([)p FB(n)1903 4050 y Fx(0)1936 4042 y FE(])1843 4115 y FB(j)1870 4123 y Fx(0)1959 4095 y FG(\(0;)c FF(")p FG(\))2141 4002 y FA(\021)2190 3999 y(\014)2190 4049 y(\014)2190 4099 y(\014)2301 4095 y FD(\024)82 b FF(B)2511 4107 y FE(1)2548 4095 y FF(e)2587 4060 y Fz(\000)p FB(B)2689 4068 y Fx(2)2722 4060 y FE(2)2755 4035 y Fs(n)2792 4047 y Fx(0)2824 4035 y Fs(=\034)2883 4047 y Fx(1)2923 4095 y FF(;)231 4278 y Fv(for)31 b(suitable)f(c)l (onstants)f FF(B)1093 4290 y FE(1)1160 4278 y Fv(and)h FF(B)1384 4290 y FE(2)1421 4278 y Fv(,)h(dep)l(ending)g(on)f FF(j)2016 4290 y FE(0)2083 4278 y Fv(but)f(indep)l(endent)h(of)h FF(n)2831 4290 y FE(0)2868 4278 y Fv(.)961 b Fr(2)231 4459 y Fv(Pr)l(o)l(of.)39 b FG(One)25 b(can)g(write)g FF(M)1110 4428 y FE([)p FB(n)1170 4436 y Fx(0)1202 4428 y FE(])1225 4459 y FG(\(0;)14 b FF(")p FG(\))25 b(as)f(in)i(\(5.11\))e (with)i FF(x)d FG(=)g(0,)i(where)g FD(V)2587 4471 y FB(T)2639 4459 y FG(\(0;)14 b FF(")p FG(\))25 b(is)g(giv)n(en)f(b)n(y)h(\(5.9\))g (with)h FF(n)3687 4471 y FB(`)3742 4459 y FG(=)c FF(n)3879 4471 y FE(0)231 4558 y FG(and)1457 4690 y FF(g)1500 4647 y FE([)p FB(n)1560 4656 y Fs(`)1588 4647 y FE(])1497 4715 y FB(`)1634 4690 y FG(=)1732 4634 y FF( )1786 4646 y FB(n)1827 4654 y Fx(0)1864 4634 y FG(\()p FD(j)p FF(x)1966 4604 y FE(0)1966 4658 y FB(`)2004 4634 y FD(j)p FG(\))p 1732 4671 328 4 v 1839 4747 a FF(ix)1915 4719 y FE(0)1915 4772 y FB(`)2092 4690 y FG(=)2190 4634 y(1)c FD(\000)g FF(\037)2385 4646 y FB(n)2426 4654 y Fx(0)2462 4634 y FG(\()p FD(j)p FF(x)2564 4604 y FE(0)2564 4658 y FB(`)2602 4634 y FD(j)p FG(\))p 2190 4671 468 4 v 2367 4747 a FF(ix)2443 4719 y FE(0)2443 4772 y FB(`)2668 4690 y FF(:)231 4915 y FG(F)-7 b(urthermore)27 b FD(M)813 4872 y FE([)p FB(n)873 4880 y Fx(0)905 4872 y FE(])813 4938 y FB(j)840 4946 y Fx(0)928 4915 y FG(\()p FF(x)p FG(;)14 b FF(")p FG(\))28 b(and)p 1304 4848 100 4 v 27 w FD(M)1404 4862 y FE([)p FB(n)1464 4870 y Fx(0)1496 4862 y FE(])1404 4935 y FB(j)1431 4943 y Fx(0)1519 4915 y FG(\(0;)14 b FF(")p FG(\))28 b(are)e(p)r(olynomials)h(of)h(degree)e FF(j)2716 4927 y FE(0)2781 4915 y FG(in)i FF(")p FG(,)f(hence)h(one)f(has)972 5175 y FD(M)1072 5132 y FE([)p FB(n)1132 5140 y Fx(0)1164 5132 y FE(])1072 5198 y FB(j)1099 5206 y Fx(0)1187 5175 y FG(\(0;)14 b FF(")p FG(\))k FD(\000)p 1470 5109 V 18 w(M)1570 5122 y FE([)p FB(n)1630 5130 y Fx(0)1662 5122 y FE(])1570 5196 y FB(j)1597 5204 y Fx(0)1685 5175 y FG(\(0;)c FF(")p FG(\))23 b(=)2008 5068 y FB(j)2035 5076 y Fx(0)1978 5096 y FA(X)1978 5275 y FB(k)q FE(=1)2181 5096 y FA(X)2112 5283 y FB(T)9 b Fz(2S)2250 5263 y Fp(R)2246 5303 y Fs(k)q(;n)2334 5315 y Fx(0)2384 5108 y FA(\000)2422 5175 y FD(V)2473 5187 y FB(T)2525 5175 y FG(\(0;)14 b FF(")p FG(\))19 b FD(\000)p 2809 5109 58 4 v 18 w(V)2866 5187 y FB(T)2919 5175 y FG(\(0;)14 b FF(")p FG(\))3101 5108 y FA(\001)3152 5175 y FF(;)2032 5532 y FG(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 231 169 a FG(whic)n(h)28 b(is)f(trivially)g (di\013eren)n(tiable)h(with)g(resp)r(ect)f(to)g FF(")p FG(.)37 b(By)27 b(applying)g(Lemma)h(6.2,)f(w)n(e)g(obtain)814 277 y FA(\014)814 327 y(\014)814 377 y(\014)842 280 y(\020)891 373 y FD(M)991 329 y FE([)p FB(n)1051 337 y Fx(0)1083 329 y FE(])991 396 y FB(j)1018 404 y Fx(0)1106 373 y FG(\(0;)14 b FF(")p FG(\))k FD(\000)p 1389 306 100 4 v 18 w(M)1489 320 y FE([)p FB(n)1549 328 y Fx(0)1581 320 y FE(])1489 393 y FB(j)1516 401 y Fx(0)1604 373 y FG(\(0;)c FF(")p FG(\))1786 280 y FA(\021)1835 277 y(\014)1835 327 y(\014)1835 377 y(\014)3746 373 y FG(\(6.5\))1146 630 y FD(\024)1264 522 y FB(j)1291 530 y Fx(0)1234 551 y FA(X)1234 730 y FB(k)q FE(=1)1368 630 y FD(j)p FF(")p FD(j)1453 588 y FB(k)1577 551 y FA(X)1508 738 y FB(T)9 b Fz(2S)1646 718 y Fp(R)1642 758 y Fs(k)q(;n)1730 770 y Fx(0)1840 551 y FA(X)1780 733 y FB(`)1808 716 y Fp(0)1830 733 y Fz(2)p FB(L)p FE(\()p FB(T)g FE(\))2034 630 y FF(\037)2086 642 y FB(n)2127 650 y Fx(0)2163 630 y FG(\()p FD(j)p FF(x)2265 596 y FE(0)2265 650 y FB(`)2293 634 y Fp(0)2321 630 y FD(j)p FG(\))2390 463 y FA(0)2390 613 y(@)2519 551 y(Y)2463 733 y FB(`)p Fz(2)p FB(L)p FE(\()p FB(T)g FE(\))2749 574 y FG(1)p 2704 611 131 4 v 2704 687 a FD(j)p FF(x)2774 658 y FE(0)2774 712 y FB(`)2812 687 y FD(j)2845 463 y FA(1)2845 613 y(A)2932 463 y(0)2932 613 y(@)3068 551 y(Y)3004 733 y FB(v)r Fz(2)p FB(B)s FE(\()p FB(T)g FE(\))3251 630 y FD(j)p FF(F)3327 642 y FB(v)3367 630 y FD(j)3390 463 y FA(1)3390 613 y(A)3477 630 y FF(;)231 966 y FG(and)27 b(for)f FF(@)562 978 y FB(")598 966 y FG(\()p FD(M)730 923 y FE([)p FB(n)790 931 y Fx(0)822 923 y FE(])730 990 y FB(j)757 998 y Fx(0)845 966 y FG(\(0;)14 b FF(")p FG(\))j FD(\000)p 1125 900 100 4 v 16 w(M)1225 914 y FE([)p FB(n)1285 922 y Fx(0)1317 914 y FE(])1225 987 y FB(j)1252 995 y Fx(0)1340 966 y FG(\(0;)d FF(")p FG(\)\))27 b(the)g(same)f(b)r(ound)i(can)e(b)r(e)h(obtained)g(with)g FF(k)17 b FD(j)p FF(")p FD(j)3123 925 y FB(k)q Fz(\000)p FE(1)3275 966 y FG(instead)27 b(of)g FD(j)p FF(")p FD(j)3740 936 y FB(k)3781 966 y FG(.)36 b(In)231 1077 y(\(6.5\))25 b(the)g(factor)e FF(\037)853 1089 y FB(n)894 1097 y Fx(0)931 1077 y FG(\()p FD(j)p FF(x)1033 1047 y FE(0)1033 1100 y FB(`)1061 1084 y Fp(0)1088 1077 y FD(j)p FG(\))i(requires)e FD(j)p FF(x)1547 1047 y FE(0)1547 1100 y FB(`)1575 1084 y Fp(0)1602 1077 y FD(j)g(\024)g FF(C)1795 1089 y FE(1)1833 1077 y FG(2)1875 1047 y Fz(\000)p FB(n)1968 1055 y Fx(0)2004 1077 y FG(,)i(so)f(that)h(b)n(y)f(the)h(Diophan)n(tine)g(condition)f (\(2.14\))g(one)g(has)231 1176 y FD(j)p Fy(\027)308 1146 y FE(0)308 1200 y FB(`)336 1183 y Fp(0)362 1176 y FD(j)f(\025)g FG(2)538 1146 y FB(n)579 1154 y Fx(0)611 1146 y FB(=\034)687 1176 y FG(,)k(hence)h(in)535 1295 y FA(0)535 1444 y(@)671 1383 y(Y)608 1564 y FB(v)r Fz(2)p FB(B)s FE(\()p FB(T)9 b FE(\))855 1461 y FD(j)p FF(F)931 1473 y FB(v)971 1461 y FD(j)994 1295 y FA(1)994 1444 y(A)1089 1461 y FD(\024)23 b(Q)1245 1427 y Fz(j)p FB(B)s FE(\()p FB(T)9 b FE(\))p Fz(j)1455 1295 y FA(0)1455 1444 y(@)1591 1383 y(Y)1528 1564 y FB(v)r Fz(2)p FB(B)s FE(\()p FB(T)g FE(\))1775 1461 y FF(e)1814 1427 y Fz(\000)p FB(\024)p Fz(j)p Fu(\027)1966 1435 y Fs(v)2001 1427 y Fz(j)p FB(=)p FE(2)2092 1295 y FA(1)2092 1444 y(A)2179 1295 y(0)2179 1444 y(@)2315 1383 y(Y)2252 1564 y FB(v)r Fz(2)p FB(B)s FE(\()p FB(T)g FE(\))2498 1461 y FF(e)2537 1427 y Fz(\000)p FB(\024)p Fz(j)p Fu(\027)2690 1435 y Fs(v)2725 1427 y Fz(j)p FB(=)p FE(4)2816 1295 y FA(1)2816 1444 y(A)2903 1295 y(0)2903 1444 y(@)3039 1383 y(Y)2975 1564 y FB(v)r Fz(2)p FB(B)s FE(\()p FB(T)g FE(\))3222 1461 y FF(e)3261 1427 y Fz(\000)p FB(\024)p Fz(j)p Fu(\027)3413 1435 y Fs(v)3449 1427 y Fz(j)p FB(=)p FE(4)3540 1295 y FA(1)3540 1444 y(A)231 1739 y FG(one)27 b(can)h(b)r(ound)g(the)g(third)f(pro)r(duct)h(b)n(y) 990 1850 y FA(0)990 1999 y(@)1126 1938 y(Y)1063 2120 y FB(v)r Fz(2)p FB(B)s FE(\()p FB(T)9 b FE(\))1310 2017 y FF(e)1349 1982 y Fz(\000)p FB(\024)p Fz(j)p Fu(\027)1501 1990 y Fs(v)1536 1982 y Fz(j)p FB(=)p FE(4)1627 1850 y FA(1)1627 1999 y(A)1723 2017 y FD(\024)23 b FF(e)1850 1982 y Fz(\000)p FB(\024)p Fz(j)p Fu(\027)2002 1957 y Fx(0)2002 2004 y Fs(`)2027 1992 y Fp(0)2053 1982 y Fz(j)p FB(=)p FE(4)2167 2017 y FD(\024)g FF(e)2294 1982 y Fz(\000)p FB(\024)p FE(2)2418 1957 y Fs(n)2455 1969 y Fx(0)2486 1957 y Fs(=\034)2554 1982 y FB(=)p FE(4)2649 2017 y FF(<)f(e)2775 1982 y Fz(\000)p FB(\024)p FE(2)2899 1957 y Fs(n)2936 1969 y Fx(0)2968 1957 y Fs(=\034)3027 1969 y Fx(1)3063 1982 y FB(=)p FE(4)3134 2017 y FF(;)231 2294 y FG(if)30 b FF(\034)345 2306 y FE(1)408 2294 y FF(>)25 b(\034)534 2306 y FE(0)571 2294 y FG(,)30 b(while)f(using)g(the)g(second)g(pro)r (duct)g(to)g(p)r(erform)f(the)i(sum)f(o)n(v)n(er)e(the)j(mo)r(de)f(lab) r(els)g(and)g(the)g(\014rst)g(one)231 2394 y(to)f(\014nd,)g(b)n(y)f (reasoning)f(as)h(for)g(\(4.3\))g(to)h(\(4.5\),)638 2509 y FA(0)638 2659 y(@)766 2597 y(Y)710 2779 y FB(`)p Fz(2)p FB(L)p FE(\()p FB(T)9 b FE(\))997 2620 y FG(1)p 952 2657 131 4 v 952 2733 a FD(j)p FF(x)1022 2705 y FE(0)1022 2758 y FB(`)1060 2733 y FD(j)1093 2509 y FA(1)1093 2659 y(A)1179 2509 y(0)1179 2659 y(@)1316 2597 y(Y)1252 2779 y FB(v)r Fz(2)p FB(B)s FE(\()p FB(T)g FE(\))1499 2676 y FF(e)1538 2642 y Fz(\000)p FB(\024)p Fz(j)p Fu(\027)1690 2650 y Fs(v)1726 2642 y Fz(j)p FB(=)p FE(2)1817 2509 y FA(1)1817 2659 y(A)1912 2676 y FD(\024)2000 2559 y FA(\022)2085 2620 y FF(\034)g FG(!)p 2071 2657 97 4 v 2071 2733 a FF(C)2130 2745 y FE(0)2191 2559 y FA(\022)2263 2620 y FG(2)p FD(j)p FF(L)p FG(\()p FF(T)j FG(\))p FD(j)p 2263 2657 269 4 v 2373 2733 a FF(\024)2541 2559 y FA(\023)2602 2576 y FB(\034)2644 2559 y FA(\023)2705 2572 y Fz(j)p FB(L)p FE(\()p FB(T)d FE(\))p Fz(j)2917 2676 y FD(\024)p 3005 2610 52 4 v 23 w FG(\000)3057 2688 y FE(1)p 3094 2610 V 3094 2676 a FG(\000)3146 2623 y FB(k)3181 2631 y Fs(T)3146 2697 y FE(2)3245 2676 y FG(\()p FF(k)3320 2688 y FB(T)3373 2676 y FG(!\))3428 2635 y FB(\014)3487 2676 y FF(;)231 2966 y FG(with)31 b FF(k)466 2978 y FB(T)547 2966 y FG(=)d FF(k)33 b FG(for)d FF(T)39 b FD(2)29 b(S)1074 2936 y Fz(R)1068 2990 y FB(k)q(;n)1165 2998 y Fx(0)1202 2966 y FG(,)i(so)f(that,)i(b)n(y)e(collecting)g(together)g(the)h(b)r (ounds)g(and)f(inserting)g(them)h(in)n(to)g(\(6.5\),)231 3101 y(w)n(e)c(pro)n(v)n(e)f(the)i(assertion.)35 b(In)28 b(particular)e FF(B)1649 3113 y FE(1)1713 3101 y FG(is)i(prop)r (ortional)d(to)p 2375 3034 V 27 w(\000)2427 3048 y FB(j)2454 3056 y Fx(0)2427 3121 y FE(2)2491 3101 y FG(\()p FF(j)2557 3113 y FE(0)2595 3101 y FG(!\))2650 3070 y FB(\014)2695 3101 y FG(,)j(while)f FF(B)3025 3113 y FE(2)3090 3101 y FG(is)g(indep)r(enden)n(t)h(of)g FF(j)3772 3113 y FE(0)3809 3101 y FG(.)p 3869 3101 48 48 v 231 3301 a Fw(Lemma)k(6.4)40 b Fv(L)l(et)25 b FF(G)937 3313 y FB(j)972 3301 y Fv(,)h FF(j)i FD(\025)23 b FG(1)p Fv(,)j(b)l(e)f(as)g(the)g(pr)l(evious)h(se)l (ctions.)37 b(Assume)25 b(that)f(ther)l(e)i(is)f FF(j)3053 3313 y FE(0)3113 3301 y FD(2)f FC(N)h Fv(such)g(that)g FF(G)3700 3313 y FB(j)3727 3321 y Fx(0)3787 3301 y FD(6)p FG(=)e(0)231 3401 y Fv(and)31 b FF(G)458 3413 y FB(j)516 3401 y FG(=)22 b(0)30 b Fv(for)g(al)t(l)h FG(1)23 b FD(\024)f FF(j)28 b(<)23 b(j)1262 3413 y FE(0)1299 3401 y Fv(.)39 b(Ther)l(e)31 b(exists)e(two)h(c)l(onstants)f FF(c)2385 3413 y FE(1)2452 3401 y Fv(and)h FF(c)2649 3413 y FE(2)2686 3401 y Fv(,)g(dep)l(ending)h(on)f FF(j)3280 3413 y FE(0)3318 3401 y Fv(,)g(such)g(that)f(for)1547 3636 y FF(n)1597 3648 y FE(0)1657 3636 y FD(\025)23 b FF(\034)1781 3648 y FE(1)1832 3636 y FG(log)1940 3656 y FE(2)1991 3519 y FA(\022)2052 3636 y FF(c)2088 3648 y FE(1)2143 3636 y FG(+)18 b FF(c)2262 3648 y FE(2)2313 3636 y FG(log)2466 3580 y(1)p 2444 3617 85 4 v 2444 3693 a FD(j)p FF(")p FD(j)2539 3519 y FA(\023)3746 3636 y FG(\(6.6\))231 3864 y Fv(one)30 b(has)1477 3888 y FA(\014)1477 3938 y(\014)1477 3987 y(\014)1505 3983 y FF(@)1549 3995 y FB(")1584 3983 y FD(M)1684 3940 y FE([)p FB(n)1744 3948 y Fx(0)1776 3940 y FE(])1684 4006 y FB(j)1711 4014 y Fx(0)1799 3983 y FG(\(0;)14 b FF(")p FG(\))1981 3888 y FA(\014)1981 3938 y(\014)1981 3987 y(\014)2032 3983 y FD(\025)2129 3927 y FF(j)2163 3939 y FE(0)p 2129 3964 72 4 v 2144 4040 a FG(2)2224 3983 y FD(j)p FF(")p FD(j)2309 3941 y FB(j)2336 3949 y Fx(0)2369 3941 y Fz(\000)p FE(1)2472 3983 y FD(j)p FF(G)2560 3995 y FB(j)2587 4003 y Fx(0)2624 3983 y FD(j)p FF(:)1076 b FG(\(6.7\))231 4156 y Fv(pr)l(ovide)l(d)32 b FF(")d Fv(is)i(smal)t(l)f(enough.)39 b(If)31 b FF(j)1375 4168 y FE(0)1435 4156 y FG(=)23 b(1)29 b Fv(one)h(c)l(an)g(take)g FF(c)2116 4168 y FE(2)2176 4156 y FG(=)23 b(0)p Fv(.)1523 b Fr(2)231 4356 y Fv(Pr)l(o)l(of.)40 b FG(One)27 b(can)h(write)p 1477 4469 100 4 v 1477 4535 a FD(M)1577 4483 y FE([)p FB(n)1637 4491 y Fx(0)1669 4483 y FE(])1577 4556 y FB(j)1604 4564 y Fx(0)1692 4535 y FG(\(0;)14 b FF(")p FG(\))23 b(=)2015 4428 y FB(j)2042 4436 y Fx(0)1985 4456 y FA(X)1984 4635 y FB(k)q FE(=1)2119 4535 y FF(")2158 4501 y FB(k)2199 4535 y FF(G)2264 4547 y FB(k)2328 4535 y FG(=)f FF(")2454 4501 y FB(j)2481 4509 y Fx(0)2518 4535 y FF(G)2583 4547 y FB(j)2610 4555 y Fx(0)2647 4535 y FF(;)1076 b FG(\(6.8\))231 4759 y(b)n(y)28 b(Lemma)f(6.1,)g(so)g(that)1575 4880 y FF(@)1619 4892 y FB(")p 1654 4813 V 1654 4880 a FD(M)1754 4827 y FE([)p FB(n)1814 4835 y Fx(0)1846 4827 y FE(])1754 4901 y FB(j)1781 4909 y Fx(0)1869 4880 y FG(\(0;)14 b FF(")p FG(\))23 b(=)f FF(j)2195 4892 y FE(0)2233 4880 y FF(")2272 4846 y FB(j)2299 4854 y Fx(0)2331 4846 y Fz(\000)p FE(1)2421 4880 y FF(G)2486 4892 y FB(j)2513 4900 y Fx(0)2550 4880 y FF(:)1173 b FG(\(6.9\))231 5030 y(By)28 b(Lemma)f(6.3,)g(w)n(e)g(can)g(b)r(ound)816 5151 y FA(\014)816 5201 y(\014)816 5251 y(\014)844 5247 y FF(@)888 5259 y FB(")938 5155 y FA(\020)987 5247 y FD(M)1087 5204 y FE([)p FB(n)1147 5212 y Fx(0)1179 5204 y FE(])1087 5270 y FB(j)1114 5278 y Fx(0)1202 5247 y FG(\(0;)14 b FF(")p FG(\))k FD(\000)p 1485 5180 V 18 w(M)1585 5194 y FE([)p FB(n)1645 5202 y Fx(0)1677 5194 y FE(])1585 5267 y FB(j)1612 5275 y Fx(0)1700 5247 y FG(\(0;)c FF(")p FG(\))1882 5155 y FA(\021)1931 5151 y(\014)1931 5201 y(\014)1931 5251 y(\014)1982 5247 y FD(\024)23 b FF(\014)2117 5259 y FE(1)2154 5247 y FF(B)2217 5259 y FE(1)2254 5247 y FF(e)2293 5212 y Fz(\000)p FB(\014)2383 5220 y Fx(2)2415 5212 y FB(B)2465 5220 y Fx(2)2497 5212 y FE(2)2530 5187 y Fs(n)2567 5199 y Fx(0)2599 5187 y Fs(=\034)2658 5199 y Fx(1)2722 5247 y FD(\024)2820 5191 y FG(1)p 2820 5228 42 4 v 2820 5304 a(2)2871 5247 y FD(j)p FF(")p FD(j)2956 5212 y FB(j)2983 5220 y Fx(0)3016 5212 y Fz(\000)p FE(1)3119 5247 y FD(j)p FF(G)3207 5259 y FB(j)3234 5267 y Fx(0)3271 5247 y FD(j)14 b FF(;)373 b FG(\(6.10\))2032 5532 y(26)p eop end 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y(\014)1402 3381 y FD(M)1502 3347 y FE([)p FB(n)p FE(])1585 3381 y FG(\(0;)14 b FF(")p FG(\))k FD(\000)p 1868 3314 V 18 w(M)1967 3328 y FE([)p FB(n)2027 3336 y Fx(0)2060 3328 y FE(])1967 3402 y FB(j)1994 3410 y Fx(0)2083 3381 y FG(\(0;)c FF(")p FG(\))2265 3286 y FA(\014)2265 3335 y(\014)2265 3385 y(\014)2315 3381 y FF(<)23 b FG(2)p FF(G)2510 3393 y FB(j)2537 3401 y Fx(0)2587 3381 y FD(j)p FF(")p FD(j)2672 3339 y FB(j)2699 3347 y Fx(0)2750 3381 y FF(;)231 3595 y FG(if)28 b FF(n)357 3607 y FE(0)422 3595 y FG(is)g(\014xed)f(as)g(said)g(in)h(Remark)f (6.6,)g(so)g(that)h(2)p FF(\014)1917 3607 y FE(1)1953 3595 y FF(B)2016 3607 y FE(1)2067 3595 y FD(j)p FF(")14 b FD(j)p FF(e)2205 3565 y Fz(\000)p FB(\014)2295 3573 y Fx(2)2327 3565 y FB(B)2377 3573 y Fx(2)2409 3565 y FE(2)2442 3540 y Fs(n)2479 3552 y Fx(0)2511 3540 y Fs(=\034)2570 3552 y Fx(1)2634 3595 y FD(\024)22 b(j)p FF(")p FD(j)2806 3565 y FB(j)2833 3573 y Fx(0)2870 3595 y FD(j)p FF(G)2958 3607 y FB(j)2985 3615 y Fx(0)3023 3595 y FD(j)p 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FE(\()p FB(t)p FE(\))1442 1330 y FG(~)1428 1351 y FF(U)9 b FG(\()p FF(!)p 1526 1364 55 4 v 1581 1371 a FE(1)1618 1351 y FF(t;)14 b(!)1737 1363 y FE(0)1774 1351 y FF(t)p FG(\).)41 b(As)29 b(a)g(consequence)f FF(\036)p FG(\()p FF(t)p FG(\))i(is)f(a)f(quasi-p)r (erio)r(dic)g(function)i(with)231 1450 y FF(d)j FG(fundamen)n(tal)g (frequencies)f FF(!)p 1222 1463 V 1277 1471 a FE(1)1314 1450 y FF(;)14 b(!)1403 1462 y FE(0)1440 1450 y FF(;)g FG(\012)1537 1462 y FB(")1572 1450 y FG(,)34 b(and)f(the)g(dep)r (endence)g(on)g(the)g(last)f(frequency)g(is)h(only)f(through)g(the)231 1550 y(factor)e FF(e)511 1520 y FB(i)p FE(\012)581 1528 y Fs(")614 1520 y FB(t)643 1550 y FG(,)i(exactly)e(as)g(in)i(the)f(unp) r(erturb)r(ed)g(case)f(\(2.3\).)47 b(As)31 b(an)n(ticipated)f(in)h (Remark)f(2.2)g(the)h(same)g(result)231 1649 y(can)c(b)r(e)h(obtained)g (b)n(y)f(starting)g(from)g(the)h(unp)r(erturb)r(ed)g(solution)f(giv)n (en)g(b)n(y)g(the)h(second)f(function)h(in)g(\(2.4\),)g(and)231 1749 y(an)g(analogous)d(result)i(is)h(found,)g(so)f(that)g(w)n(e)h(can) f(conclude)g(that)h(the)g(system)f(is)h(reducible)f(for)g FF(")c FD(2)h(E)7 b FG(.)356 1867 y(So)28 b(the)h(solution)p 931 1821 48 4 v 28 w FF(u)p FG(\()p FF(t)p FG(\))g(describ)r(es)f(the)i (motion)e(on)g(a)h FF(d)p FG(-dimensional)f(maximal)g(torus)g(whic)n(h) g(is)h(the)g(con)n(tin)n(u-)231 1966 y(ation)24 b(in)g FF(")g FG(of)g(an)f(unp)r(erturb)r(ed)i FF(d)p FG(-dimensional)e (torus.)35 b(The)24 b(rotation)f(v)n(ector)g(of)h(the)g(latter)g(is)f Fy(!)j FG(=)d(\()p FF(!)p 3509 1979 55 4 v 3564 1987 a FE(1)3601 1966 y FF(;)14 b(!)3690 1978 y FE(0)3727 1966 y FF(;)g FG(\012)3824 1978 y FE(0)3861 1966 y FG(\),)231 2066 y(while,)28 b(as)g(an)f(e\013ect)i(of)e(the)i(p)r(erturbation,)e (only)g(the)i(last)e(comp)r(onen)n(t)h(of)g(the)g(rotation)f(v)n(ector) f(is)i(c)n(hanged)f(in)n(to)231 2165 y(a)g(new)g(frequency)f(\012)905 2177 y FB(")964 2165 y FG(=)d(\012)1112 2177 y FE(0)1166 2165 y FG(+)17 b FD(h)p FF(g)s FD(i)p 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Fv(,)g FF(v)26 b FD(2)e FF(B)t FG(\()p FF(T)12 b FG(\))p Fv(,)30 b(satisfy)h(the)f(r)l(elation)2289 1976 y FA(P)2376 2063 y FB(v)r Fz(2)p FB(B)s FE(\()p FB(T)9 b FE(\))2627 2038 y FD(j)p Fy(\027)2704 2050 y FB(v)2744 2038 y FD(j)23 b FF(<)g FG(2)2920 2008 y FE(\()p FB(n)2987 2016 y Fx(ext)3068 2008 y Fz(\000)p FE(4\))p FB(=\034)3255 2038 y Fv(,)30 b(wher)l(e)h FF(n)3595 2050 y FE(ext)3719 2038 y Fv(is)f(the)439 2138 y(minimum)f(b)l(etwe)l(en)h (the)g(sc)l(ales)g(of)h(the)f(external)f(lines)h(of)h FF(T)12 b Fv(.)231 2307 y(We)30 b(c)l(al)t(l)h FG(self-energy)26 b(line)j Fv(any)h(line)h FF(`)1460 2277 y FE(out)1460 2330 y FB(T)1589 2307 y Fv(which)g(exits)e(fr)l(om)h(a)g(self-ener)l (gy)h(gr)l(aph)g FF(T)12 b Fv(.)37 b(We)30 b(c)l(al)t(l)h FG(normal)26 b(line)k Fv(any)231 2406 y(line)h(which)g(is)f(not)f(a)i (self-ener)l(gy)f(line.)356 2589 y FG(The)i(self-energy)g(v)-5 b(alue)32 b(is)g(then)h(de\014ned)g(as)f(b)r(efore)g(\(see)h 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FH(References)273 1077 y FG([1])41 b(J.)21 b(C.)g(A.)h(Barata,)f Fv(On)i(F)-6 b(ormal)25 b(Quasi-Perio)l(dic)h(Solutions)e(of)h(the)f(Schr\177)-42 b(odinger)25 b(Equation)g(for)g(a)f(Two-L)l(evel)402 1177 y(System)29 b(with)g(a)h(Hamiltonian)g(Dep)l(ending)g(Quasi-Perio) l(dic)l(al)t(ly)i(on)e(Time)p FG(,)e(Rev.)f(Math.)g(Ph)n(ys.)f Fw(12)h FG(\(2000\),)402 1276 y(no.)g(1,)h(25-64.)273 1443 y([2])41 b(M.)27 b(V.)g(Bartuccelli,)g(G.)g(Gen)n(tile,)g Fv(Lindste)l(dt)i(series)h(for)g(p)l(erturb)l(ations)f(of)h(iso)l(chr)l (onous)g(systems:)38 b(a)29 b(r)l(eview)402 1542 y(of)i(the)e(gener)l (al)i(the)l(ory)p FG(,)d(Rev.)g(Math.)g(Ph)n(ys.)e Fw(14)h FG(\(2002\),)g(no.)g(2,)g(121-171.)273 1708 y([3])41 b(H.)c(Bohr,)h Fv(Ueb)l(er)g(fastp)l(erio)l(dische)j(eb)l(ene)d(Bewe)l (gungen)p FG(,)h(Commen)n(t.)e(Math.)g(Helv.)g Fw(4)f FG(\(1932\),)i(51-64.)c(H.)402 1808 y(Bohr,)d Fv(Kleiner)l(e)j (Beitr\177)-42 b(age)34 b(zur)f(The)l(orie)h(der)g(F)-6 b(astp)l(erio)l(dischen)35 b(F)-6 b(unktionen)p FG(,)32 b(Dansk)n(e)e(Vid.)i(Selsk.)f(Mat.-)402 1907 y(Fys.)c(Medd.)h Fw(10)g FG(\(1930\),)e(no.)h(10,)g(5-11.)273 2073 y([4])41 b(H.)30 b(Bro)r(er,)f(J.)g(Puig,)g(C.)h(Sim\023)-42 b(o,)30 b Fv(R)l(esonanc)l(e)h(tongues)h(and)g(instability)g(p)l(o)l(ckets)g (in)g(the)g(quasi-p)l(erio)l(dic)i(Hil)t(l-)402 2173 y(Schr\177)-42 b(odinger)31 b(e)l(quation)p FG(,)d(Comm.)g(Math.)g(Ph)n (ys.)e Fw(241)h FG(\(2003\),)g(no.)g(2-3,)f(467-503.)273 2339 y([5])41 b(H.)24 b(Bro)r(er,)f(C.)h(Sim\023)-42 b(o,)24 b Fv(Hil)t(l's)j(e)l(quation)g(with)f(quasi-p)l(erio)l(dic)j (for)l(cing:)38 b(r)l(esonanc)l(e)26 b(tongues,)h(instability)g(p)l(o)l (ck-)402 2439 y(ets)i(and)i(glob)l(al)g(phenomena)p FG(,)e(Bol.)e(So)r (c.)h(Bras.)e(Mat.)h Fw(29)h FG(\(1998\),)e(no.)h(2,)h(253-293.)273 2605 y([6])41 b(Ch.-Q.)34 b(Cheng,)h Fv(Birkho\013-Kolmo)l(gor)l(ov-A)n (rnold-Moser)40 b(tori)c(in)g(c)l(onvex)g(Hamiltonian)h(systems)p FG(,)e(Comm.)402 2704 y(Math.)28 b(Ph)n(ys.)e Fw(177)h FG(\(1996\),)g(no.)g(3,)g(529-559.)273 2870 y([7])41 b(Ch.-Q.)27 b(Cheng,)h Fv(L)l(ower-dimensional)k(invariant)f(tori)f(in) h(the)f(r)l(e)l(gions)g(of)h(instability)g(for)g(ne)l(arly)g(inte)l(gr) l(able)402 2970 y(Hamiltonian)g(systems)p FG(,)c(Comm.)h(Math.)f(Ph)n (ys.)g Fw(203)g FG(\(1999\),)f(no.)i(2,)f(385-419.)273 3136 y([8])41 b(L.)27 b(Chierc)n(hia,)e Fv(Quasi-p)l(erio)l(dic)31 b(Schr\177)-42 b(odinger)30 b(op)l(er)l(ators)g(in)f(one)g(dimension,)i (absolutely)e(c)l(ontinuous)f(sp)l(e)l(c-)402 3236 y(tr)l(a,)45 b(Blo)l(ch)f(waves)f(and)f(inte)l(gr)l(able)h(Hamiltonian)g(systems)p FG(,)h(Quaderni)c(del)i(Consiglio)d(Nazionale)h(delle)402 3335 y(Ricerc)n(he,)27 b(Firenze,)g(1986.)273 3501 y([9])41 b(L.)20 b(Chierc)n(hia,)g Fv(A)n(bsolutely)j(c)l(ontinuous)f(sp)l(e)l (ctr)l(a)h(of)g(quasi-p)l(erio)l(dic)j(Schr\177)-42 b(odinger)24 b(op)l(er)l(ators)p FG(,)f(J.)c(Math.)h(Ph)n(ys.)402 3601 y Fw(28)27 b FG(\(1987\),)f(no.)i(12,)f(2891-2898.)231 3767 y([10])41 b(E.)25 b(I.)g(Dinaburg,)h(Ja.)e(G.)i(Sina)-9 b(\025)-32 b(\020,)25 b Fv(The)j(one-dimensional)i(Schr\177)-42 b(odinger)29 b(e)l(quation)f(with)g(quasip)l(erio)l(dic)j(p)l(oten-)402 3867 y(tial)p FG(,)d(F)-7 b(unk)n(cional.)27 b(Anal.)h(i)g(Prilo)n (\024)-39 b(zen.)25 b Fw(9)i FG(\(1975\),)g(no.)g(4,)g(8-21.)231 4033 y([11])41 b(L.)48 b(H.)g(Eliasson,)k Fv(Flo)l(quet)c(solutions)h (for)g(the)g FG(1)p Fv(-dimensional)g(quasi-p)l(erio)l(dic)i(Schr\177) -42 b(odinger)50 b(e)l(quation)p FG(,)402 4132 y(Comm.)28 b(Math.)f(Ph)n(ys.)g Fw(146)g FG(\(1992\),)f(no.)h(3,)h(447-482.)231 4298 y([12])41 b(G.)25 b(Galla)n(v)n(otti,)g(G.)g(Gen)n(tile,)h Fv(Hyp)l(erb)l(olic)j(low-dimensional)h(invariant)e(tori)g(and)g (summations)f(of)i(diver)l(gent)402 4398 y(series)p FG(,)f(Comm.)g (Math.)g(Ph)n(ys.)e Fw(227)h FG(\(2002\),)g(no.)g(3,)g(421-460.)231 4564 y([13])41 b(G.)19 b(Gen)n(tile,)j Fv(Quasi-p)l(erio)l(dic)h (solutions)g(for)f(two-level)h(systems)p FG(,)e(Comm.)e(Math.)g(Ph)n (ys.)g Fw(242)f FG(\(2002\),)i(no.)f(1-2,)402 4664 y(221-250.)231 4830 y([14])41 b(G.)28 b(Gen)n(tile,)g(G.)g(Galla)n(v)n(otti,)e Fv(De)l(gener)l(ate)k(el)t(liptic)h(tori)p FG(,)d(Preprin)n(t,)f(2004.) 231 4996 y([15])41 b(G.)32 b(Gen)n(tile,)g(V.)g(Mastropietro,)e Fv(Metho)l(ds)35 b(for)f(the)f(analysis)i(of)f(the)f(Lindste)l(dt)g (series)h(for)g(KAM)f(tori)h(and)402 5095 y(r)l(enormalizability)40 b(in)e(classic)l(al)h(me)l(chanics.)h(A)d(r)l(eview)i(with)f(some)g (applic)l(ations)p FG(,)j(Rev.)36 b(Math.)g(Ph)n(ys.)g Fw(8)402 5195 y FG(\(1996\),)26 b(no.)i(3,)f(393-444.)2032 5532 y(39)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 231 169 a FG([16])41 b(M.)23 b(Herman,)g Fv(Some)i(op)l(en)g(pr)l(oblems)h(in)f(dynamic)l(al)i (systems)p FG(,)c(Pro)r(ceedings)e(of)h(the)h(In)n(ternational)e (Congress)402 268 y(of)27 b(Mathematicians,)h(V)-7 b(ol.)27 b(I)r(I)h(\(Berlin,)g(1998\),)e(Do)r(c.)i(Math.)f(1998,)f(Extra)g(V)-7 b(ol.)28 b(I)r(I,)g(797-808.)231 434 y([17])41 b(D.B.)30 b(Hin)n(ton,)h(J.K.)e(Sha)n(w,)h Fv(On)h(the)h(absolutely)h(c)l (ontinuous)e(sp)l(e)l(ctrum)h(of)g(the)g(p)l(erturb)l(e)l(d)g(Hil)t (l's)h(e)l(quation)p FG(,)402 534 y(Pro)r(c.)26 b(London)h(Math.)h(So)r (c.)g(\(3\))f(50)g(\(1985\),)f(no.)i(1,)f(175{192.)231 700 y([18])41 b(R.)26 b(Johnson,)f(J.)h(Moser,)f Fv(The)k(r)l(otation)g (numb)l(er)e(for)i(almost)g(p)l(erio)l(dic)h(p)l(otentials)p FG(,)d(Comm)n(un.)f(Math.)g(Ph)n(ys.)402 800 y Fw(84)h FG(\(1982\),)f(no.)i(3,)f(403-438.)231 966 y([19])41 b(Y.)28 b(Katznelson,)e Fv(A)n(n)j(Intr)l(o)l(duction)h(to)f(harmonic)j (analysis)p FG(,)d(Do)n(v)n(er,)d(New)i(Y)-7 b(ork,)27 b(1978.)231 1132 y([20])41 b(R.)35 b(Krik)n(orian,)f Fv(R)n(\023)-40 b(eductibilit)n(\023)g(e)37 b(pr)l(esque)f(p)l(artout)g (des)h(\015ots)e(\014br)n(\023)-40 b(es)37 b(quasi-p)n(\023)-40 b(erio)l(diques)38 b(\022)-42 b(a)37 b(valeurs)f(dans)h(des)402 1231 y(gr)l(oup)l(es)30 b(c)l(omp)l(acts)p FG(,)e(Ann.)g(Sci.)1443 1210 y(\023)1435 1231 y(Ecole)f(Norm.)g(Sup.)h(\(4\))g Fw(32)f FG(\(1999\),)f(no.)i(2,)f(187-240.)231 1397 y([21])41 b(R.)25 b(Krik)n(orian,)e Fv(Glob)l(al)28 b(density)g(of)g(r)l(e)l (ducible)g(quasi-p)l(erio)l(dic)h(c)l(o)l(cycles)f(on)f FF(T)2883 1367 y FE(1)2932 1397 y FD(\002)13 b FG(SU\(2\),)25 b(Ann.)h(of)e(Math.)h(\(2\))402 1497 y Fw(154)i FG(\(2001\),)f(no.)i (2,)f(269-326.)231 1663 y([22])41 b(W.)28 b(Magn)n(us,)f(S.)g(Winkler,) h Fv(Hil)t(l's)j(Equation)p FG(,)d(Do)n(v)n(er,)f(New)g(Y)-7 b(ork,)27 b(1979.)231 1829 y([23])41 b(J.)28 b(Moser,)f(J.)i(P\177)-42 b(osc)n(hel,)27 b Fv(A)n(n)i(extension)h(of)h(a)g(r)l(esult)f(by)h (Dinabur)l(g)f(and)h(Sina)-8 b(\025)-34 b(\020)31 b(on)f(quasip)l(erio) l(dic)k(p)l(otentials)p FG(,)402 1929 y(Commen)n(t.)28 b(Math.)f(Helv.)h Fw(59)f FG(\(1984\),)g(no.)g(1,)g(39-85.)231 2095 y([24])41 b(F.S.)26 b(Rofe-Bek)n(eto)n(v,)f Fv(A)j(\014niteness)f (test)h(for)h(the)f(numb)l(er)g(of)h(discr)l(ete)g(levels)g(which)g(c)l (an)g(b)l(e)f(intr)l(o)l(duc)l(e)l(d)g(into)402 2194 y(the)d(gaps)g(of)h(the)e(c)l(ontinuous)g(sp)l(e)l(ctrum)g(by)h(p)l (erturb)l(ations)f(of)i(a)f(p)l(erio)l(dic)h(p)l(otential)d FG(\(Russian\),)g(Dokl.)f(Ak)-5 b(ad.)402 2294 y(Nauk)27 b(SSSR)h Fw(156)f FG(\(1964\),)g(515-518.)231 2460 y([25])41 b(H.)28 b(R)r(\177)-44 b(ussmann,)27 b Fv(On)i(the)h(one-dimensional)i (Schr\177)-42 b(odinger)31 b(e)l(quation)f(with)h(a)f(quasip)l(erio)l (dic)i(p)l(otential)p FG(,)d(Non-)402 2560 y(linear)j(dynamics)h(\(In)n (ternat.)g(Conf.,)i(New)e(Y)-7 b(ork,)34 b(1979\),)g(Ann.)f(New)h(Y)-7 b(ork)32 b(Acad.)h(Sci.,)i(357,)f(New)f(Y)-7 b(ork)402 2659 y(Acad.)27 b(Sci.,)h(New)g(Y)-7 b(ork,)27 b(1980.)231 2825 y([26])41 b(E.)26 b(Sorets,)f(T.)h(Sp)r(encer,)g Fv(Positive)k(Lyapunov)f(exp)l(onents)f(for)h(Schr\177)-42 b(odinger)30 b(op)l(er)l(ators)f(with)g(quasi-p)l(erio)l(dic)402 2925 y(p)l(otentials)p FG(,)f(Comm)n(un.)g(Math.)g(Ph)n(ys.)e Fw(142)h FG(\(1991\),)g(no.)g(3,)g(543-566.)231 3091 y([27])41 b(V.A.)608 3070 y(\024)603 3091 y(Zeludev,)31 b Fv(The)i(eigenvalues)h(of)f(a)g(p)l(erturb)l(e)l(d)f(Schr\177)-42 b(odinger)34 b(op)l(er)l(ator)g(with)f(p)l(erio)l(dic)h(p)l(otential)d FG(\(Rus-)402 3191 y(sian\),)e(Problems)f(of)g(Mathematical)h(Ph)n (ysics,)f(no.)g(2,)h(Sp)r(ectral)g(Theory)-7 b(,)28 b(Di\013raction)h (Problems,)f(pp.)h(108-)402 3290 y(123,)d(Izdat.)i(Leningrad.)e(Univ.,) i(Leningrad,)f(1967.)231 3456 y([28])41 b(V.A.)612 3435 y(\024)607 3456 y(Zeludev,)36 b Fv(The)i(p)l(erturb)l(ation)e(of)h(the) f(sp)l(e)l(ctrum)g(of)h(the)f(one-dimensional)i(selfadjoint)h(Schr\177) -42 b(odinger)402 3556 y(op)l(er)l(ator)40 b(with)h(p)l(erio)l(dic)g(p) l(otential)e FG(\(Russian\),)i(Problems)c(of)i(Mathematical)e(Ph)n (ysics,)j(no.)e(4,)j(Sp)r(ectral)402 3656 y(Theory)-7 b(.)27 b(W)-7 b(a)n(v)n(e)26 b(Pro)r(cesses,)g(pp.)i(61-82,)d(Izdat.)j (Leningrad.)e(Univ.,)i(Leningrad,)f(1970.)2032 5532 y(40)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0410110436795--